content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Quasi-Symétries pour les processus ponctuels pfaffiens
6 au 14 avril 2017
For a given domain E in Euclidean space, we denote Conf(E) the set of locally finite configurations on E and denote Diffc(E) the group of compactly supported C1-diffeomorphisms of E. The tautological
action of Diffc(E) on E induces a natural action of Diffc(E) on Conf(E). We say a point process P on E (i.e., a Borel probability measure P on Conf(E)) is Diffc(E)-quasi-symmetric, if the Diffc(E)
-action preserves the measure class of P. In the setting of determinantal point processes, the quasi-symmetries have been obtained by Bufetov for determinantal point processes on the real line R
induced by integrable kernels, including Dyson-sine process, Bessel processes, Airy process etc and have also been obtained by Bufetov-Qiu for determinantal point processes on the complex plane C or
the unit disk D ⊂ C associated with a class of Hilbert spaces of holomorphic functions. Note that in all these cases, the associated Radon-Nikodym derivatives can be expressed as certain regularized
multiplicative functionals. Pfaffian point processes arise naturally in many similar situations where determinantal point processes arise and they share certain similarity as determinantal point
processes, for instance, both of them possess the repulsive nature and their Laplace transforms are given by Fredholm determinants or Fredholm Pfaffians. A natural problem that we would like to
investigate is whether certain important Pfaffian point processes, for instance, the classical sine1 and sine4 point processes on the real line R, also possess the quasi-symmetries?
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement N°647113) | {"url":"https://fconferences.cirm-math.fr/1941.html","timestamp":"2024-11-09T23:35:28Z","content_type":"text/html","content_length":"73399","record_id":"<urn:uuid:e449854b-4f6d-4366-a57b-46e9a0b1b827>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00304.warc.gz"} |
AMC 8 Special Seminar B
Need Help?
Need help finding the right class? Have a question about how classes work?
Special AMC 8 Problem Seminar B
This course is a special two-day, 5-hour seminar to prepare for the AMC 8, which is the premier fall math contest for middle school students. The AMC 8 also gives students early problem-solving days
experience that is valuable towards the high-school level AMC 10 and AMC 12 contests, which are the first stage in determining the United States team for the International Math Olympiad. In this
course, students learn problem solving strategies and test-taking tactics over two lessons — during each lesson, class will meet over a 3-hour period, with a half-hour break in the middle. The
course also includes a practice AMC 8 test. This course covers entirely different problems than the Special AMC 8 Problem Seminar A.
2 days
Sat & Sun 4:00 - 7:00 $145
Jan 18 - Jan 19 PM ET – Ryo Kudo $145
Who Should Take?
This class is appropriate for students in grade 8 or below attempting to make the Honor Roll on the AMC 8.
• If a student already consistently scores above 18 on the AMC 8, this class is probably not necessary.
• If a student is unlikely to score more than 8 on the AMC 8, that student should consider our Prealgebra curriculum.
The material for this class is distinct from the
Special AMC 8 Problem Seminar A
1 Word Problems and Number Theory
2 Counting and Geometry | {"url":"https://artofproblemsolving.com/school/course/maa-amc8-special-b","timestamp":"2024-11-06T17:44:47Z","content_type":"text/html","content_length":"182381","record_id":"<urn:uuid:1ea53134-0dcd-4079-8ffb-ae103242ecfe>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00752.warc.gz"} |
High Pass Filter Calculator
A high pass filter prevents frequencies below its cut-off frequency from passing and lets through signals above it. In this article, you will learn how to calculate the various passive high-pass
filters. In addition, you will have access to an online high pass filter calculator.
General information about the high pass filter
A high pass filter circuit designates a circuit in electrical engineering with the purpose of attenuating or blocking low frequencies. High frequencies, however, should be as unhindered as possible.
The term high pass filter is also common. The high pass is passive if no amplifying element is used. Otherwise, it is considered active.
A high pass is used where low frequencies are undesirable and therefore should be filtered out. Examples include the construction of tweeters or the high-frequency signal transmission via power
lines. The low frequencies in these cases would make the signal almost unusable for further processing and must be eliminated.
Electricians distinguish between a high pass 1st order and 2nd order. High order high passes are achieved by switching lower orders in series. We explain how the high pass works and how a high pass
can be calculated. In addition, we provide a high pass calculator for the sake of simplicity.
Passive first order high pass filter
The simple high pass of the 1st order is built up with a capacitor and a resistor connected in series. Aspects of a high pass filter schematic follow. The capacitor has the abbreviation $C$ and the
resistor $R$, which is why the abbreviation $RC$ high pass is often used. A CR high pass is also often called, but designates the same circuit. The output voltage $V_{out}$ must here be tapped
parallel to the resistor, otherwise we would have a low-pass filter.
When a high frequency is applied to the input, an imperceptibly small voltage drops across the capacitor. The output voltage $V_{out}$ is thus almost identical to the input voltage $V_{in}$. However,
if a low frequency is present, part of the voltage across the capacitor will drop. As a result, the output voltage drops parallel to the resistor with a time delay.
RC high pass – how it works
In a single, erratic change in the input voltage $V_{in}$, there is a short voltage spike of the output voltage $V_{out}$. This is because the capacitor lets the changed voltage pass for a short
time. Its capacitive reactance $X_C$ takes a short time to build up.
However, if the input voltage has a frequency, $X_C$ depends on the level of that frequency. As the frequency increases, the voltage drop across the capacitor decreases. Consequently, the output
voltage increases. At a low frequency, $X_C$ increases and more voltage drops across the capacitor. The output voltage $V_{out}$ decreases.
Formula – RC high pass filter calculation
The basic formula for calculating an RC high pass is:
$$ \frac{V_{out}}{V_{in}} = \frac{R}{Z} $$
The following applies to the impedance Z:
$$ Z = \sqrt{R^2 + X_C^2} $$
The RC high pass filter transfer function is calculated according to:
$$ \frac{V_{out}}{V_{in}} = \frac{1}{\sqrt{1 + \frac{1}{(2 \pi f R C)^2}}} $$
$R$ stands for the ohmic resistance. $f$ is the frequency and $C$ is the capacitance of the capacitor.
Calculate cutoff frequency of high pass
The capacitive reactance $X_C$ decreases as the frequency increases, while the ohmic resistance $R$ remains constant. The cutoff frequency $f_c$ is the frequency at which the resistances are equal.
Consequently, at a frequency above $f_c$, $R > X_C$ and at a lower frequency $X_C > R$.
With this formula, the cutoff frequency can be calculated with an RC high pass:
$$ f_c = \frac{1}{2 \pi R C} $$
RC high pass calculator
The online calculator helps you to dimension the components for the desired cutoff frequency.
RC High Pass Filter Calculator
Our online calculators are provided "as is" without any warranty of any kind.
Alternative: RL high pass
The RL high pass is also a 1st order filter. Instead of the capacitor, however, an inductor is used and the output voltage tapped parallel to this. The mode of operation is exactly the opposite: the
inductive reactance $X_L$ increases along with the frequency.
The formula for the calculation is:
$$ \frac{V_{out}}{V_{in}} = \frac{1}{\sqrt{1 + (2 \pi f L)^2}} $$
The cutoff frequency for a RL high pass results from:
$$ f_c = \frac{R}{2 \pi L} $$
RL high pass calculator
The online calculator helps you to dimension the components for the desired cutoff frequency.
RL High Pass Filter Calculator
Our online calculators are provided "as is" without any warranty of any kind.
Passive second order high pass filter
The structure is identical to the high-pass filter 1st order, except that the ohmic resistance is replaced by an inductance. Consequently, in the 2nd order high pass filter, a coil is connected in
series with a capacitor. The term LC high pass is therefore common. The output voltage $V_{out}$ is tapped here over the inductive load.
A 2nd order high pass filters the low frequencies twice as effectively as a 1st order high pass. The edge is twice as steep. The difference comes from the coil, which, unlike the capacitor, reacts
quickly to high frequencies.
LC high pass operation
The function of the capacitor remains unchanged. At a low-frequency input voltage, it forms a high capacitive reactance $X_C$. A sudden change therefore causes a momentary voltage spike at the
output, because the capacitor’s reaction is delayed.
When applying a sinusoidal voltage, however, the coil fulfills its purpose. The capacitor forms a resistor at low frequencies and allows high frequencies through. The coil, on the other hand, reacts
immediately to an increase in frequency and forms an inductive reactance $X_L$. In contrast to the capacitor, their resistance increases together with the frequency. This ensures a faster and
stronger response to frequency increases.
Formula – calculate high pass 2nd order
The formulas for calculating an LC high pass are:
$$ L = \frac{Z}{2 \pi f} $$
$$ C = \frac{1}{2 \pi f Z} $$
$$ f = \frac{1}{2 \pi \sqrt{LC}} $$
$$ Z = \sqrt{\frac{L}{C}} $$
The associated high pass transfer function is:
$$ \frac{V_{out}}{V_{in}} = \frac{X_L}{X_L + X_C} $$
$L$ stands for the inductance of the coil, $Z$ for the impedance and $C$ for the capacitance of the capacitor.
Calculate cutoff frequency of 2nd order high pass
As described above, capacitive and inductive reactances always change in opposite directions. At the cutoff frequency, the resistors are identical. It is therefore: $X_L = X_C$. At a higher
frequency, therefore, $X_C > X_L$ and at a lower frequency $X_C < X_L$.
The formula for calculating the cutoff frequency is:
$$ f_c = \frac{1}{2 \pi \sqrt{LC}} $$
LC high pass filter calculator
The online calculator helps you to dimension the components for the desired cutoff frequency.
LC High Pass Filter Calculator
Our online calculators are provided "as is" without any warranty of any kind. | {"url":"https://electronicbase.net/high-pass-filter-calculator/","timestamp":"2024-11-12T05:02:55Z","content_type":"text/html","content_length":"119438","record_id":"<urn:uuid:8f04cf8f-5467-48aa-8ab3-0275a6e94a68>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00028.warc.gz"} |
On transport twistor spaces (Joint with IP seminar)
For 2-dimensional Riemannian manifolds there is a rich
interplay between the geodesic transport equation on the unit tangent
bundle and Fourier analysis in the vertical fibres. This interplay has
shaped the understanding of many geometric inverse problems and
rigidity questions since the late 1970s. The transport twistor space
is a (degenerate) complex 2-dimensional manifold Z which encodes this
interplay and sheds new light on numerous aspects of the transport
equation by translating them into a complex geometric language. The
focus of the talk will lie on these novel twistor correspondences, as
well as some new results regarding the algebra of holomorphic
functions on Z and its moduli space of holomorphic vector bundles.
This is based on joint work with Thibault Lefeuvre and Gabriel | {"url":"https://math.washington.edu/events/2023-03-29/transport-twistor-spaces-joint-ip-seminar","timestamp":"2024-11-05T23:34:08Z","content_type":"text/html","content_length":"50741","record_id":"<urn:uuid:053a024c-9520-4fe7-849e-cf9932bf64cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00400.warc.gz"} |
Just how long does a formal proof take to finish?
Formal methods are exhaustive in their nature. That’s what makes them special. That’s why I like using them over constrained random simulation based testing. If there’s ever a way a property within
your design can be made to fail, formal methods can find it.
That’s the good news.
The bad news is that because formal methods are exhaustive they can take exponential time to complete. The bigger and more complex your design is, the longer the solver will take to prove a property.
Eventually, there comes a complexity where the property becomes essentially impossible to prove.
In other words, the answer to “how long does the formal solver take to return an answer?” can be anywhere from trivial to infinite depending upon the problem.
That’s not helpful. Perhaps some statistics might be more useful.
Looking at some statistics
I’ve now been doing formal verification for almost two years, ever since my first humbling experience. Over the course of that time, I’ve kept the output directories created by SymbiYosys for nearly
900 of the proofs that I’ve completed. This includes both halves of any induction proofs, as well as quite a few cover proofs. With a bit of work, these proof durations can be organized into an
approximate cumulated distribution function, such as Fig. 1 shows.
Fig 1.
In this chart, the X axis is the number of seconds a given proof took to complete, whereas the Y axis is the percentage of all of the proofs that took less than that X amount of time. By plotting
this on a semilog scale in X, you can understand some of the realities of formal verification. For example,
• 82% of all of the proofs I’ve done have taken less than one minute
• 87% of all of the proofs I’ve done have taken less than two minutes
• 93% of all of the proofs I’ve done have taken less than five minutes
• 95% of all of the proofs I’ve done have taken less than ten minutes
Every now and again, I’ll post about how long a given proof takes. For example, I’ve had proofs require a couple of hours to return. A classic example would be some of the proofs associated with
verifying my open source generated FFT cores. Such proofs are the exception rather than the norm, however, and typically when I write about such extreme times its because I wasn’t expecting the proof
to take that long to accomplish.
The reality is that I don’t normally notice how long a proof takes. Why not? Because formal verification, in my experience, has typically been faster than simulation. It’s typically faster than
running a design through synthesis or place-and-route. This follows from the fact that 95% of all of these proofs were accomplished in less than 10 minutes, whereas it often takes longer than 10
minutes with Vivado to synthesize a design.
How do I keep my proofs that short?
This is a really good question, and there’s typically several parts to the answer.
In general, the amount of time a proof requires is a function of the number of items that need to be checked, and the number of steps they need to be checked in. Of these two, I usually have the most
control over the number of steps required by the proof. SymbiYosys calls this the “depth” of the proof.
How shall this depth be set?
1. For many simple peripheral cores, the depth can be set initially to however long it takes to perform the operation the core is required to perform.
This can often be determined by running a cover() check, and seeing how long it takes the core to complete an operation and to return to idle.
This doesn’t work for all cores, however, but it is a fairly good start. It does apply nicely to most SPI cores, as well as those that are similar such as my MDIO controller, since they all have
fixed transaction lengths. It can also apply to CPUs, where the depth is determined by the time it takes for a single instruction to go from when it is issued all the way to when it is retired.
2. For most of my proofs, I start with a depth set to default, 20 steps. If I struggle inexplicably at that depth, I may set it to longer as a result of a basic knee-jerk reaction.
The fact is, when you first start out with a formal proof, the solver can typically find assertion failures very quickly. It’s only as you slowly remove these initial failures that the proof
starts to take the solver more and more time to return an answer.
3. If the solver takes too long at a depth of 20, I’ll often shorten the depth.
This was the case with my AXI crossbar. AXI is such a complicated protocol, I couldn’t let the depth get too long at all. In the end, I fixed this depth to four time-steps. It was the shortest
depth I could find where all of the various constraints could be evaluated properly in the time interval.
One of the nice features of Yosys’ SMT solver is that it reports back periodic status messages showing how long each step has taken. This helps you know where the “limit” is. For example, if the
first five steps take less than 6 seconds each, but the six step has taken over an hour and it hasn’t yet completed, you may need to drop the depth to five and just work with it there.
4. The trick to setting the depth is induction.
If the inductive step ever passes, even if I don’t have all of the properties I want in place yet, I’ll set the depth to whatever it took to pass induction. This keeps the proof as short as it
will ever be.
For example, the ZipCPU can be formally verified in between 10 and 14 steps depending upon the configuration. Given that each step is longer than the step before hand, it makes sense to keep the
solver from doing too much. Those configurations that can be solved in 10 steps I set to be solved in 10 steps. Those that cannot, get set to however many steps they need. While this won’t speed
up the inductive step at all, it often shortens the associated basecase.
5. How do you know your depth is too shallow?
I’ve had several proofs that have required depths of much longer than ten or twenty steps. Examples include my serial port receiver (an asynchronous proof) at 110 clocks, my hyperram controller
at 40 clocks, and several of the slower configurations of my universal flash controller ranging from 26 steps all the way up to 610. Cover proofs tend to be worse than assertion based proofs,
with my serial port receiver requiring 720 steps, and the MDIO controller for my ethernet implementations requiring 258 steps.
The easy way to know that a proof isn’t too shallow is to work with induction until it passes as we just discussed above. In the case of cover, covering intermediate states will help to reveal
just how long the trace needs to be.
Knowing if an induction proof is too shallow requires understanding your core, and the trace produced during induction.
As I teach in my formal methods course, there are three kinds of assertion failures during induction: 1) those that fail at the last time step, 2) those that fail the time step before that, and
3) those whose failure can be tracked to earlier in the trace. Typically, in the third case, an assertion is sufficient to bring the design back in line. If the data necessary to make the
assertion isn’t part of the trace, such as if it’s dependent upon something that happened earlier, then you either need to add a register to capture the dependency or you need to increase the
depth of the trace.
The reason that my serial port receive proof is so long is that I had a criteria that the clock in the serial port transmitter would never be off by more than half a baud interval at the end of
the transmission. Measuring how far that would be at every time step required a multiplication function–something that doesn’t work well with formal methods. As a result, I was forced to only
checking this value at the end of every baud interval, and using power-of-two properties. This fixed the induction length to at least one baud interval in length.
6. Some problems are just too hard
Two classic examples are multiplies and encryption. Of the two, formally verifying designs with multipliers within them is an area of active research. I wouldn’t be surprised to see some
breakthroughs in the near future. Formally verifying designs with encryption within them should be and should remain a hard problem, otherwise the encryption isn’t worth its salt.
I like to get around this problem by replacing the internal multiplier or encryption result with a solver chosen value. This can work for DSP problems, making it possible to still apply formal
methods to DSP algorithms although the result is often not quite as satisfying.
The Beginner Mistake
The big mistake I’ve seen beginners make is to take a large and complex core, often one with several component files having no formal properties, and then try to formally verify that a single
property holds for all time.
This is a recipe for both frustration and failure.
A classic example would be a user who finds a CPU core on opencores, knows nothing about it, but still wants to know if an assertion about it will pass.
Instead, start your formal verification work at the bottom level of a design with what I often call “leaf modules”–modules that have no submodules beneath them. Formal verification, and particularly
verification using induction, is not a black box exercise. Passing an induction test requires an intimate knowledge of the design in question, and several assertions within the design. Building those
assertions from the bottom up makes it easier to get a property to pass later at the top level.
I should mention that there are several solvers that do not require this intimate internal knowledge, such as the abc pdr solver or either of the aiger solvers aiger avy or aiger suprove, and so I’ve
seen beginners attempt to use these solvers for this purpose as well. Sadly, these solvers are not well suited for such large designs, and they tend not to provide any progress feedback along the
way. The result tends to be user complaints that the solver hangs or crashes, when in reality the problem was that the user was expecting too much from the tool.
This is also one of those reasons why formal verification works so well at the design stage, rather than as a separate verification stage done by a new team of engineers. It is the designer who knows
how to constrain the values within his own design–not the verification engineer.
Despite its reputation for computational complexity, hardware formal verification tends to be very fast in practice today. It’s often faster than both simulation and synthesis, allowing a designer to
iterate on his design faster than he would with either of these other approaches.
If you’ve never tried formal verification, then let me invite you to work through my beginning verilog tutorial. Once you get past the second lesson, every design will involve formally verifying that
it works before ever trying to implement the design on actual hardware. Indeed, the background you will need for more complicated projects is to be gained by working on simpler projects–as it is in
many other fields.
But the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works
that are therein shall be burned up. (2Pet 3:10) | {"url":"http://zipcpu.com/formal/2019/08/03/proof-duration.html","timestamp":"2024-11-04T05:14:50Z","content_type":"text/html","content_length":"21993","record_id":"<urn:uuid:e4cdbced-2b28-4807-83b6-a05f717a2882>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00130.warc.gz"} |
Graphing Sinusoidal Functions Worksheet - Graphworksheets.com
Graphing Trigonometric Functions Worksheet – If you’re looking for graphing functions worksheets, you’ve come to the right place. There are several different types of graphing functions to choose
from. For example, Conaway Math has Valentine’s Day-themed graphing functions worksheets for you to use. This is a great way for your child to learn about these … Read more | {"url":"https://www.graphworksheets.com/tag/graphing-sinusoidal-functions-worksheet/","timestamp":"2024-11-11T04:18:54Z","content_type":"text/html","content_length":"47916","record_id":"<urn:uuid:a3504808-1780-4a43-912d-58b92a40e03f>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00801.warc.gz"} |
Targeting an Embedded Processor
The sections that follow describe issues that often arise when targeting a fixed-point design for use on an embedded processor, such as some general assumptions about integer sizes and operations
available on embedded processors. These assumptions lead to design issues and design rules that might be useful for your specific fixed-point design.
Size Assumptions
Embedded processors are typically characterized by a particular bit size. For example, the terms “8-bit micro,” “32-bit micro,” or “16-bit DSP” are common. It is generally safe to assume that the
processor is predominantly geared to processing integers of the specified bit size. Integers of the specified bit size are referred to as the base data type. Additionally, the processor typically
provides some support for integers that are twice as wide as the base data type. Integers consisting of double bits are referred to as the accumulator data type. For example a 16-bit micro has a
16-bit base data type and a 32-bit accumulator data type.
Although other data types may be supported by the embedded processor, this section describes only the base and accumulator data types.
Operation Assumptions
The embedded processor operations discussed in this section are limited to the needs of a basic simulation diagram. Basic simulations use multiplication, addition, subtraction, and delays.
Fixed-point models also need shifts to do scaling conversions. For all these operations, the embedded processor should have native instructions that allow the base data type as inputs. For
accumulator-type inputs, the processor typically supports addition, subtraction, and delay (storage/retrieval from memory), but not multiplication.
Multiplication is typically not supported for accumulator-type inputs because of complexity and size issues. A difficulty with multiplication is that the output needs to be twice as big as the inputs
for full precision. For example, multiplying two 16-bit numbers requires a 32-bit output for full precision. The need to handle the outputs from a multiplication operation is one of the reasons
embedded processors include accumulator-type support. However, if multiplication of accumulator-type inputs is also supported, then there is a need to support a data type that is twice as big as the
accumulator type. To restrict this additional complexity, multiplication is typically not supported for inputs of the accumulator type.
Design Rules
The important design rules that you should be aware of when modeling dynamic systems with fixed-point math follow.
Design Rule 1: Only Multiply Base Data Types
It is best to multiply only inputs of the base data type. Embedded processors typically provide an instruction for the multiplication of base-type inputs, but not for the multiplication of
accumulator-type inputs. If necessary, you can combine several instructions to handle multiplication of accumulator-type inputs. However, this can lead to large, slow embedded code.
You can insert blocks to convert inputs from the accumulator type to the base type prior to Product or Gain blocks, if necessary.
Design Rule 2: Delays Should Use the Base Data Type
There are two general reasons why a Unit Delay should use only base-type numbers:
• The Unit Delay essentially stores a variable's value to RAM and, one time step later, retrieves that value from RAM. Because the value must be in memory from one time step to the next, the RAM
must be exclusively dedicated to the variable and can't be shared or used for another purpose. Using accumulator-type numbers instead of the base data type doubles the RAM requirements, which can
significantly increase the cost of the embedded system.
• The Unit Delay typically feeds into a Gain block. The multiplication design rule requires that the input (the unit delay signal) use the base data type.
Design Rule 3: Temporary Variables Can Use the Accumulator Data Type
Except for unit delay signals, most signals are not needed from one time step to the next. This means that the signal values can be temporarily stored in shared and reused memory. This shared and
reused memory can be RAM or it can simply be registers in the CPU. In either case, storing the value as an accumulator data type is not much more costly than storing it as a base data type.
Design Rule 4: Summation Can Use the Accumulator Data Type
Addition and subtraction can use the accumulator data type if there is justification. The typical justification is reducing the buildup of errors due to roundoff or overflow.
For example, a common filter operation is a weighted sum of several variables. Multiplying a variable by a weight naturally produces a product of the accumulator type. Before summing, each product
can be converted back to the base data type. This approach introduces round-off error into each part of the sum.
Alternatively, the products can be summed using the accumulator data type, and the final sum can be converted to the base data type. Round-off error is introduced in just one point and the precision
is generally better. The cost of doing an addition or subtraction using accumulator-type numbers is slightly more expensive, but if there is justification, it is usually worth the cost. | {"url":"https://kr.mathworks.com/help/fixedpoint/ug/targeting-an-embedded-processor.html","timestamp":"2024-11-07T18:25:02Z","content_type":"text/html","content_length":"73476","record_id":"<urn:uuid:f8245748-3cdd-4d3a-8479-4ac0c6698fc7>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00785.warc.gz"} |
Who was Hermann Grassman? - MathMania
Hermann Grassmann was a German mathematician who is best known for his work in the area of vector spaces. He also made important contributions to the study of geometry and invariant theory.
Grassmann’s work laid the foundation for much of the modern theory of vector spaces. Hermann Grassmann was born in Stettin, Germany on April 15, 1809.
He studied at the University of Berlin and the University of Königsberg, and received his doctorate from the University of Halle in 1831. After spending several years as a Privatdozent at the
University of Halle, Grassmann was appointed to a chair at the Royal Prussian Academy of Sciences in Berlin in 1845. He remained at the Academy until his retirement in 1881. Grassmann died on
September 26, 1877, in Berlin.
Grassmann was required to write an essay on the theory of tides as part of one of his many examinations. He took the basic idea from Laplace’s Traité de mécanique céleste and Lagrange’s Méchanique
analytique, but he expounded his ideas using the vector techniques he had been considering since 1832
In 1844, Grassmann released his masterpiece (A1) and is widely recognized as the “extensive magnitudes theory.” The work began with rather general definitions of a metaphysical nature, since A1
proposed a new framework for all mathematics.
Following his father’s concept, A1 also defined the exterior product, also known as “combinatorial product” (in German: kombinatorisches Produkt äußeres Produkt “outer product”), which is the
fundamental operation of an algebra now known as exterior algebra. In 1878, William Kingdon Clifford added the exterior algebra to quaternions created by William Rowan Hamilton by changing
Grassmann’s rule epep = 0 to epep = 1. (The rule i2 = j2 = k2 = -1 applies to quaternions.)
It was too far ahead of its time to be appreciated, and the BA1 (Arithmetics I) was a groundbreaking text. The ministry requested Ernst Kummer for a report when Grassmann submitted it for a
professorship in 1847. It was clear that there were good ideas in it, but Kommer stated that the explanation was inadequate and suggested against hiring Grassmann to a university position. In an
effort to encourage others to take his theory seriously, Grassmann wrote several papers on algebraic curves and surfaces throughout the next ten years. | {"url":"https://playmathmania.com/who-was-hermann-grassman/","timestamp":"2024-11-05T03:55:58Z","content_type":"text/html","content_length":"214565","record_id":"<urn:uuid:f0bdd1c9-b641-4023-b5b3-1352116164f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00378.warc.gz"} |
Peter Mann Winkler is a research mathematician, author of more than 125 research papers in mathematics^[1] and patent holder in a broad range of applications, ranging from cryptography to marine
navigation.^[2] His research areas include discrete mathematics, theory of computation and probability theory. He is currently a professor of mathematics and computer science at Dartmouth College.^[3
Peter Winkler studied mathematics at Harvard University and later received his PhD in 1975 from Yale University under the supervision of Angus McIntyre.^[4] He has also served as an assistant
professor at Stanford, full professor and chair at Emory and as a mathematics research director at Bell Labs and Lucent Technologies.^[2] He was visiting professor at the Technische Universität
He has published three books on mathematical puzzles: Mathematical Puzzles: A connoisseur's collection (A K Peters, 2004, ISBN 978-1-56881-201-4, translated to German and Russian), Mathematical
Mind-Benders (A K Peters, 2007, ISBN 978-1-56881-336-3), and Mathematical Puzzles (A K Peters, 2021, ISBN 978-0-36720-693-2). And he is widely considered to be a pre eminent scholar in this domain.
He was the Visiting Distinguished Chair for Public Dissemination of Mathematics at the National Museum of Mathematics (MoMath), gave topical talks at the Gathering 4 Gardner conferences, and wrote
novel papers related to some of these puzzles.
Winkler's book Bridge at the Enigma Club^[6] was a runner up for the 2011 Master Point Press Book Of The Year award.^[7]
Also in 2011, Winkler received the David P. Robbins Prize of the Mathematical Association of America as coauthor of one of two papers^[8] in the American Mathematical Monthly.
According to a story included in Chapter One of "The Man Who Loved Only Numbers / The Story of Paul Erdös and the Search for Mathematical Truth",^[9] Paul Erdős attended the bar mitzvah celebration
for Peter Winkler's twins, and Winkler's mother-in-law tried to throw Erdős out. [Quote:]
"Erdös came to my twins' bar mitzvah, notebook in hand," said Peter Winkler, a colleague of Graham's at AT&T. "He also brought gifts for my children--he loved kids--and behaved himself very well.
But my mother-in-law tried to throw him out. She thought he was some guy who wandered in off the street, in a rumpled suit, carrying a pad under his arm. It is entirely possible that he proved a
theorem or two during the ceremony."^[9]
External links | {"url":"https://www.knowpia.com/knowpedia/Peter_Winkler","timestamp":"2024-11-08T22:11:57Z","content_type":"text/html","content_length":"80592","record_id":"<urn:uuid:62d47f9c-9a27-44cd-b0f1-0a51ffb614a3>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00082.warc.gz"} |
Using Runge's Theorem to Determine Zariski Density of Integral Points in Two and Three Dimensions
The German mathematician Carl Runge (1856-1927) came up with a theorem that said that any Diophantine equation in two variables satisfying a certain set of conditions has only finitely many integral
solutions. This thesis will provide a detailed proof of this theorem and some examples in which we can apply it. This proof makes use of two theorems from abstract algebra: The Symmetric Function
Theorem and Newton-Puiseux's Theorem. The statement and proof of these theorems will also be given. This thesis will then introduce the Zariski Topology in all dimensions and show the strong
connection between the notion of Zariski density in two dimensions and the property of having finitely or infinitely many integral solutions to a given Diophantine equation in two variables. The
concept of Zariski density makes it possible to formulate generalizations to Runge's Theorem in higher variables. After this introduction there will be an attempt of the writer to generalize Runge's
Theorem such that it can be applied to Diophantine equations of three variables. | {"url":"https://studenttheses.uu.nl/handle/20.500.12932/31987","timestamp":"2024-11-03T06:31:56Z","content_type":"text/html","content_length":"14695","record_id":"<urn:uuid:65116118-ee2b-4e37-8c46-b2a16ceb7ee3>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00638.warc.gz"} |
Oblate Spheroid Mass
The Mass or Weight of an Oblate Spheroid calculator computes the volume of an oblate spheroid based on the semi-major(b) and semi- minor (c) axis with the assumption that the spheroid is generated
via rotation around the minor axis (see diagram).
INSTRUCTIONS: Choose your length units for a and b (e.g. feet, meters, light-years), and enter the following:
• (b) - semi-major axis, the distance from the oblate spheroid's center along the longest axis of the spheroid
• (c) - semi-minor axis, the distance from the oblate spheroid's center along the shortest axis of the spheroid
• (mD) - the mean density of the substance comprising the oblate spheroid.
Oblate Spheroid Mass / Weight: The mass (M) is returned in kilograms. However, this can be automatically converted to other mass and weight units (e.g. tons, pounds) via the pull-down menu.
The Math / Science
The oblate spheroid is an ellipsoid that can be formed by rotating an ellipse about its minor axis. The rotational axis thus formed will appear to be the oblate spheroid's polar axis. The oblate
spheroid is fully described then by its semi-major and semi-minor axes.
One important shape in nature that is close to (though not exactly) an oblate spheroid is the Earth which has a semi-minor axis (c) which is the polar radius of 6,356 kilometers, and a semi-major
axis (b) which is the equatorial radius of 6,378 kilometers. Consideration: what force would make the equatorial radius larger than the polar radius?
• computes the volume of an ellipsoid based on the length of the three semi-axes (a, b, c)
• Ellipsoid - Surface Area computes the surface area of an ellipsoid based on the length of the three semi-axes (a, b, c)
• Ellipsoid - Mass or Weight computes the mass or weight of an ellipsoid based on the length of the three semi-axes (a, b, c) and the mean density.
• Ellipsoid Cap - Volume computes the volume of a section of an ellipsoid.
• Oblate Spheroid - Volume computes the volume of an Oblate Spheroid based on the length of the two semi-axes (b, c)
• Oblate Spheroid- Surface Area computes the surface area of an Oblate Spheroid based on the length of the two semi-axes (b, c)
• Oblate Spheroid- Mass or Weight computes the mass or weight of an Oblate Spheroid based on the length of the two semi-axes (b, c) and the mean density.
• Sphere - Volume computes the volume of a sphere based on the length of the radius (a)
• Sphere - Surface Area computes the surface area of a sphere based on the length of the radius (a)
• Sphere - Mass or Weight computes the mass or weight of a sphere based on the length of the radius (a) and the mean density.
• Circular - Volume: Computes the volume of a column with a circular top and bottom and vertical sides.
• Circular - Mass: Computes the mass/weight of circular volume based on its dimensions and mean density.
• Elliptical Volume: Computes the volume of a column with an elliptical top and bottom and vertical sides.
• Elliptical - Mass: Computes the mass/weight of an elliptical volume based on its dimensions and mean density.
• Ellipse Vertical Chord from Edge (VE): Computes the length of the vertical chord of an ellipse based on distance from the edge.
• Ellipse Vertical Chord from Center (VC): Computes the length of the vertical chord of an ellipse based on distance from the center.
• Ellipse Horizontal Chord from Edge (HE): Computes the length of the horizontal chord of an ellipse based on distance from the edge.
• Ellipse Horizontal Chord from Center (HC): Computes the length of the vertical chord of an ellipse based on distance from the center.
• Common Mean Density: Provides a lookup function to find the mean density of hundreds of materials (woods, metals, liquids, chemicals, food items, soils, and more)
Metals are materials characterized by its physical and chemical properties, primarily its ability to conduct electricity and heat, its luster or shine when polished, its malleability (ability to be
hammered or pressed into shapes), and its ductility (ability to be drawn into wires). Metals typically have a crystalline structure and are found naturally in solid form (with the exception of
mercury, which is a liquid at room temperature).
Metals Densities
• Density of Aluminum - 2,700 kg/m^3
• Density of Brass - 8,530 kg/m^3
• Density of Bronze - 8,150 kg/m^3
• Density of Chromium - 7190 kg/m^3
• Density of Cobalt - 8746 kg/m^3
• Density of Copper - 8,920 kg/m^3
• Density of Gallium - 5907 kg/m^3
• Density of Gold - 19,300 kg/m^3
• Density of Iron - 7,847 kg/m^3
• Density of Lead - 11,340 kg/m^3
• Density of Nickle - 8908 kg/m^3
• Density of Palladium - 12,023 kg/m^3
• Density of Platinum - 21,450 kg/m^3
• Density of Steel - 7,850 kg/m^3
• Density of Silver - 10,490 kg/m^3
• Density of Titanium - 4,500 kg/m^3
• Density of Tungsten - 19,600 kg/m^3
• Density of Uranium - 19,050 kg/m^3
• Density of Zinc - 7,135 kg/m^3
• Density of Zirconium - 6,570 kg/m³
Metals make up a large portion of the periodic table of elements, with examples including iron, copper, gold, silver, aluminum, and titanium, among many others. Metals are essential in various
industries such as construction, manufacturing, electronics, transportation, and energy production due to their unique properties and versatility.
Metals are generally dense materials. Density is a measure of how much mass is contained in a given volume. Metals tend to have high densities because their atoms are closely packed together in a
crystalline structure. This close packing of atoms contributes to their characteristic properties such as strength, malleability, and conductivity.
However, it's important to note that the density of metals can vary widely depending on factors such as their elemental composition, crystal structure, and any impurities present. For example, some
metals like lead and platinum are denser than others like aluminum or magnesium.
The Weight of Metal Calculator contains functions and data to compute the weight (mass) of metal objects based on their size, shape and the density of the metal. The Weight of Metal functions are:
• Cylinder Weight: Computes the weight (mass) of a cylinder based on the radius, length (height) and density of metal.
• Sphere Mass: Computes the mass (weight) of a sphere based on the radius and density of metal.
• Hemisphere Mass: Computes the mass (weight) of a hemisphere based on the radius and density of metal.
• Weight of Metal Bars: Computes the mass (weight) of a number of metal flats or metal bars based on the dimensions and density of metal.
• Weight of Metal Rods: Computes the mass (weight) of a number of metal rods based on the dimensions and density of metal.
For the mean densities of other substances click HERE.
Related Calculators
The following table contains links to calculators that compute the volume of other shapes:
Other Volume Calculators
Various Shapes Polygon Columns
Cube Triangular Prism Triangular
Box Paraboloid Quadrilateral
Cone Polygon based Pyramid Pentagon
Cone Frustum Pyramid Frustum Hexagon
Cylinder Sphere Heptagon
Slanted Cylinder Sphere Cap Octagon
Ellipsoid Oblate Spheroid Nonagon
Torus Capsule Decagon | {"url":"https://www.vcalc.com/wiki/vCalc/Oblate+Spheroid+-+Mass","timestamp":"2024-11-03T23:28:53Z","content_type":"text/html","content_length":"70306","record_id":"<urn:uuid:08d8eb8e-5b1a-4329-87e2-08e96cccc789>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00156.warc.gz"} |
Interagency Modeling and Analysis Group
This model describes the kinetics of an enzymatic reaction where an inhibitor can bind to the enzyme in a non-competitive manner.
This model describes the enzymatic conversion of a single substrate, S, to a single product, P, with an inhibitor, I, which can also bind to the enzyme, E, preventing it from forming the product. The
difference between competitive and non-competitive inhibition is that the enzyme-inhibitor complex, EI can still bind to the substrate in the non-competitive case. The resulting
enzyme-inhibitor-substrate complex, EIS, can then dissociate into ES and I. The ES complex can then yield the product through a reaction release step. The entire binding-inhibition-reaction-release
sequence may be represented symbolically as:
k1 --> k2 -->
S + I + E <-----------> ES + I <-----------> P + E
<-- k_1 <-- k_2
^ ^
| | ^ ^ | |
k3 | | | k_3 k_3 | | | k3
v | | | | v
v v
k1 -->
EI + S <---------------> EIS
<-- k_1
where k[1] is the forward binding rate of S to E to EI, k[-1] is the backwards reaction rate of ES dissociating to E and S and EIS to EI and S, k[2] is the forward reaction rate of ES forming E and
P, k[-2] is the reverse reaction rate of E and P producing ES, k[3] is the forward reaction rate of E and ES binding to I and k[-3] is the reaction rate of EI dissociating to form E and I and EIS to
ES and I.
This reaction is governed by a system of five ODEs which describe the concentrations of the substrate, inhibitor, enzyme complex ES, inhibition complex EI and product. The sixth equation to close the
system specifies the total amount of enzyme present, E[tot] which must be conserved. All of the substrate and no complex or product are present at time t=0. The system of equations are:
The backward reaction rates in this model are determined from the equilibrium dissociation rates of S binding to E, I binding to E and P binding to E. The expressions for the equilibrium dissociation
rates are given by:
where K[s] is the equilibrium dissociation rate of S binding to E and EI, K[i] of I binding to E and ES and K[p] of P binding to E. The reaction velocity is the current velocity of the reaction in
forming the product, v, divided by the maximal reaction velocity, V[max]. If we assume Michaelis-Menten kinetics the reaction is rate limited by the dissociation of ES into E and P. Therefore we
Download JSim model project file
Model Feedback
We welcome comments and feedback for this model. Please use the button below to send comments:
Segel IH.: "Enzyme Kinetics", John Wiley and Sons, New York, 1975
Chapter 3, Pages 125-136.
Key terms
Transport Physiology
Enzymatic Reaction
Non-competitive Inhibition
Michaelis-Menten Kinetics
Please cite https://www.imagwiki.nibib.nih.gov/physiome in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH
/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale
Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243
Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07. | {"url":"https://www.imagwiki.nibib.nih.gov/physiome/jsim/models/webmodel/NSR/noncompetitiveinhibition","timestamp":"2024-11-11T13:41:37Z","content_type":"text/html","content_length":"60725","record_id":"<urn:uuid:979a2e9c-a9ca-4265-9817-f653baad78b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00878.warc.gz"} |
Lesson Plan Ideas for Mathematics Classes
I. How to write the equation of a line:
A) Define slope
1) Do several examples. Math1 students need more examples. Math 3 and 5 need fewer examples.
2) Draw several lines on the board without a coordinate system. Ask the question “What’s my slope?” Ask “Which line has the greatest slope?”
3) Extend the concept in math 3 and 5 to examples where one of the points is variable and the slope is given. Have students solve. Draw lots of pictures. Make it real.
4) Give examples of lines that are perpendicular or parallel. Talk about a family of lines that share a common slope.
5) Talk about horizontal and vertical lines. Give lots of examples. Show students how they can recognize that a line is horizontal or vertical by quickly noting that the x coordinates or
y-coordinates match.
6) Create a quiz that is open ended. You might have students write a paragraph about slope. Ask for at least 5 facts or details. Ask them to include several graphical examples.
B) Draw a line on an x-y plane. Label the y-intercept (0, b) and a general point (x, y). Calculate the slope of the line. Simplify the slope equation. Your final result will be the slope-intercept
equation of the line: y = mx + b. Follow the derivation of the form with many specific examples. Again the higher level classes need fewer examples than the lower level classes.
C) Draw a line on an x-y plane. Label two points (x, y) and ( x sub 1 ,y sub1 ). Calculate the slope. Simplify. You now have the point slope equation of the line: y – y sub1 = m (x –xsub1). Again do
many specific examples.
D) Given 2 points on a line create a flow chart that describes the steps necessary to write the equation of a line.
1) Do you know “b”? If so, calculate m and you are done.
2) Do the x coordinates match? If so the line is vertical. The equation is x = whatever the matching x-coordinate happens to be.
3) Do the y-coordinates match? If so the line is horizontal. The equation is y = whatever the matching y coordinates happens to be.
4) Finally, if none of the above applies, find m. Then write the equation of the line in point slope form and simplify. Alternatively find “b” using a specific point and slope and write answer in
slope intercept or function form. Do lots of examples. Include several real world examples that would be of interest to the students. Depreciation after you have purchased a new car or piece of
machinery; costs of renting a car where a fixed cost and a price per mile are given; salaries based upon commission would all be examples.
II. Extension of Ax+By = C into several other types of problems.
A) Students should be able to recognize the slope of a line from the equation. If the equation is in y = mx+b, the slope of the line is the coefficient of x. If the line is in format Ax+By =C the
slope is –A/B. Derive this for the class. Then do lots of practice with the game “What’s my slope?”
B) If you have the problem of writing the equation of a line parallel to a given line and through a specific point there are many methods of solution. A method that students enjoy uses the fact that
the answer line and the original line have exactly the same A and B. (Remember A and B determine the slope of the line, and the lines are parallel.) Replace the x and y in the original equation with
the specific point that your answer line intersects. Find the new C. Write your answer using specific values for A, B, and C and the general x and y. Example: Find the equation of the line that
passes through (3, 5) that is parallel to 6x + 7y = 12. To find C: 6(3) + 7(5) = C; 18 + 35 = 53 = C. The answer is 6x + 7y = 53.
C) If your answer line is perpendicular to the original line you can interchange A and B and switch one sign. Remind students that perpendicular lines have negative reciprocal slopes. Find the new C.
Write your answer. For example: Find the equation of a line that is perpendicular to 4x +5y = 14 that passes through (6, 9). The new line will be of the form 5x-4y=C. Find C by 5(6) -4(9) = 30 – 36 =
-6 = C. The answer is 5x – 4y = -6.
D) In Math 3 and 5 you can talk about distinguishing lines from equations of higher degree. Talk about depreciation. Is it usually linear? Talk about linear regression and other curve fitting
III. Functions
A) To describe domain one of my favorite examples is to use “My Closet”. Picture entering your closet. It is full of beautiful outfits. One red dress however is too small and can’t be used. Your
domain of wearing apparel is everything in the closet but the red dress. Relate this to a rational function where the domain is the set of real numbers except whatever value makes the denominator
Extend the closet example to a closet that is shared by a husband and wife. Suppose that the clothes of the wife are on the right side of the closet (call it the non-negative side) and the husband’s
clothes are on the left side (call it the negative side). If you have a radical function with an even root your domain is the non-negative or wife’s side of the closet. Talk about domain as the set
of inputs that can safely be used in the function process.
Another example of domain is “The Birthday Party”. Your 5 year-old son has prepared a list of boys that he wants to take to the zoo to celebrate his birthday. Mom peruses the list; there is a problem
with Johnny! Poor Johnny is not in Mom’s domain because he is
one holy terror Johnny is not an acceptable input for this particular function (ha ha). Have fun!
B) Describe domain using the example of a child beginning his or her life in his domicile or domain. As the child matures he or she travels out on the range.
C) When I talk about function notation I always emphasize that f(x) notation is really
f ( ). You are in need of an input to put in the open parentheses. The x is simply a place holder for whatever your input happens to be. This seems to help students deal with problems of the f(x+h)
category. You can extend this to evaluating polynomials. If you are to evaluate 3x+5 rewrite the problem as 3( ) +5. You can then replace the spot in the open parentheses with a specific value of x.
This also helps students when their inputs are negative. | {"url":"https://mymathtutors.com/","timestamp":"2024-11-12T00:22:49Z","content_type":"text/html","content_length":"22832","record_id":"<urn:uuid:249b6b3f-12fb-49c8-8193-bd542751e9f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00795.warc.gz"} |
Sudo Null - Latest IT News
KNN classifier
stands for
k Nearest Neighbor
k Nearest Neighbors
- this is one of the simplest classification algorithms, also sometimes used in regression tasks. Due to its simplicity, it is a good example from which you can begin your acquaintance with the field
of Machine Learning. This article describes an example of writing the code for such a classifier in python, as well as visualizing the results.
Classification task
in machine learning, this is the task of assigning an object to one of the predefined classes based on its formalized features. Each of the objects in this problem is represented as a vector in
N-dimensional space, each dimension in which is a description of one of the features of the object. Suppose we need to classify monitors: measurements in our parameter space will be the diagonal in
inches, aspect ratio, maximum resolution, HDMI interface, cost, etc. The case of classifying texts is somewhat more complicated, they usually use a term-document matrix (
description on machinelearning .ru
To train the classifier, you must have a set of objects for which classes are predefined. This set is called
training sample
, its marking is done manually, with the involvement of specialists in the study area. For example, in the task of
Detecting Insults in Social Commentary
for pre-assembled tests of comments, a person puts down the opinion whether this comment is an insult to one of the participants in the discussion, the task itself is an example of binary
classification. In the classification problem, there can be more than two classes (multiclass), each of the objects can belong to more than one class (intersecting).
To classify each of the objects of the test sample, the following operations must be performed sequentially:
• Calculate the distance to each of the objects in the training set
• Select k objects of the training set, the distance to which is minimum
• The class of a classified object is the class most often found among k nearest neighbors
The examples below are implemented in python. For their correct execution, in addition to python, you must have
matplotlib installed
. The library initialization code is as follows:
import random
import math
import pylab as pl
import numpy as np
from matplotlib.colors import ListedColormap
Initial data
Consider the work of the classifier by example. To begin with, we need to generate data on which experiments will be made:
#Train data generator
def generateData (numberOfClassEl, numberOfClasses):
data = []
for classNum in range(numberOfClasses):
#Choose random center of 2-dimensional gaussian
centerX, centerY = random.random()*5.0, random.random()*5.0
#Choose numberOfClassEl random nodes with RMS=0.5
for rowNum in range(numberOfClassEl):
data.append([ [random.gauss(centerX,0.5), random.gauss(centerY,0.5)], classNum])
return data
For simplicity, I chose a two-dimensional space in which the location of the mathematical expectation of a two-dimensional Gaussian with a standard deviation of 0.5 is randomly selected from 0 to 5
along each axis. The value 0.5 is chosen so that the objects turn out to be fairly well separable (the
rule of three sigma
To look at the resulting selection, you need to run the following code:
def showData (nClasses, nItemsInClass):
trainData = generateData (nItemsInClass, nClasses)
classColormap = ListedColormap(['#FF0000', '#00FF00', '#FFFFFF'])
pl.scatter([trainData[i][0][0] for i in range(len(trainData))],
[trainData[i][0][1] for i in range(len(trainData))],
c=[trainData[i][1] for i in range(len(trainData))],
showData (3, 40)
Here is an example of the image resulting from the execution of this code:
Getting training and test samples
So, we have a set of objects, for each of which a class is defined. Now we need to break this set into two parts: training selection and test selection. To do this, use the following code:
#Separate N data elements in two parts:
# test data with N*testPercent elements
# train_data with N*(1.0 - testPercent) elements
def splitTrainTest (data, testPercent):
trainData = []
testData = []
for row in data:
if random.random() < testPercent:
return trainData, testData
Classifier Implementation
Now, having a training sample, we can implement the classification algorithm itself:
#Main classification procedure
def classifyKNN (trainData, testData, k, numberOfClasses):
#Euclidean distance between 2-dimensional point
def dist (a, b):
return math.sqrt((a[0] - b[0])**2 + (a[1] - b[1])**2)
testLabels = []
for testPoint in testData:
#Claculate distances between test point and all of the train points
testDist = [ [dist(testPoint, trainData[i][0]), trainData[i][1]] for i in range(len(trainData))]
#How many points of each class among nearest K
stat = [0 for i in range(numberOfClasses)]
for d in sorted(testDist)[0:k]:
stat[d[1]] += 1
#Assign a class with the most number of occurences among K nearest neighbours
testLabels.append( sorted(zip(stat, range(numberOfClasses)), reverse=True)[0][1] )
return testLabels
To determine the distance between objects, you can use not only the Euclidean distance: the Manhattan distance, cosine measure, Pearson's correlation criterion, etc. are also used.
Execution examples
Now you can evaluate how well our classifier works. To do this, we will generate the data, we will divide it into a training and test sample, we will classify the objects of the test sample and
compare the real value of the class with the result of the classification:
#Calculate classification accuracy
def calculateAccuracy (nClasses, nItemsInClass, k, testPercent):
data = generateData (nItemsInClass, nClasses)
trainData, testDataWithLabels = splitTrainTest (data, testPercent)
testData = [testDataWithLabels[i][0] for i in range(len(testDataWithLabels))]
testDataLabels = classifyKNN (trainData, testData, k, nClasses)
print "Accuracy: ", sum([int(testDataLabels[i]==testDataWithLabels[i][1]) for i in range(len(testDataWithLabels))]) / float(len(testDataWithLabels))
To evaluate the quality of the classifier, various algorithms and various measures are used; more details can be found here:
Now the most interesting thing remains: to show the classifier’s work graphically. In the pictures below, I used 3 classes, each with 40 elements, the value of k for the algorithm was taken to be
The following code was used to display these images:
#Visualize classification regions
def showDataOnMesh (nClasses, nItemsInClass, k):
#Generate a mesh of nodes that covers all train cases
def generateTestMesh (trainData):
x_min = min( [trainData[i][0][0] for i in range(len(trainData))] ) - 1.0
x_max = max( [trainData[i][0][0] for i in range(len(trainData))] ) + 1.0
y_min = min( [trainData[i][0][1] for i in range(len(trainData))] ) - 1.0
y_max = max( [trainData[i][0][1] for i in range(len(trainData))] ) + 1.0
h = 0.05
testX, testY = np.meshgrid(np.arange(x_min, x_max, h),
np.arange(y_min, y_max, h))
return [testX, testY]
trainData = generateData (nItemsInClass, nClasses)
testMesh = generateTestMesh (trainData)
testMeshLabels = classifyKNN (trainData, zip(testMesh[0].ravel(), testMesh[1].ravel()), k, nClasses)
classColormap = ListedColormap(['#FF0000', '#00FF00', '#FFFFFF'])
testColormap = ListedColormap(['#FFAAAA', '#AAFFAA', '#AAAAAA'])
pl.scatter([trainData[i][0][0] for i in range(len(trainData))],
[trainData[i][0][1] for i in range(len(trainData))],
c=[trainData[i][1] for i in range(len(trainData))],
is one of the simplest classification algorithms, so it often turns out to be ineffective in real problems. In addition to the accuracy of the classification, the problem of this classifier is the
speed of classification: if there are N objects in the training set, M objects in the test set, and the space dimension K, then the number of operations for classifying the test set can be estimated
as O (K * M * N). Nevertheless, the kNN algorithm is a good example to get started with Machine Learning.
List of references
The method of nearest neighbors on Machinelearning.ru
2. The
vector model on Machinelearning.ru
Book on Information Retrieval
Description of the method of nearest neighbors in the framework of scikit-learn
Book “Programming collective intelligence” | {"url":"https://sudonull.com/post/139014-KNN-classifier","timestamp":"2024-11-07T06:07:24Z","content_type":"text/html","content_length":"17037","record_id":"<urn:uuid:26cd6c98-19ad-4b3b-b74e-dde46b15ab72>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00653.warc.gz"} |
Applying Inlet Boundary Conditions | Ansys Courses
If I give mass flow rate, temperatures, pressure data from experiment for flow entering a pipe, I noticed the mass flow rate inlet boundary condition only applies the temperature/pressure if the flow
is supersonic. What would be the best method to apply these types of inlet boundary conditions to a compressible gas type problem?
For compressible flows, you can apply either a pressure inlet or mass flow inlet BC. Here's what happens in each case:
Pressure Inlet:
The total pressure and total temperature are known, along with the flow direction, turbulence variables and other scalars. For ** subsonic flow ** the static pressure at the inlet boundary is
obtained (predicted) from the CFD solution. You can then determine the Mach number, velocity magnitude, static density and temperature, and mass flow rate. So the mass flow rate is a result of the
computation since total pressure is fixed.
Mass Flow Inlet:
The mass flow and total temperature are known, along with the flow direction, turbulence variables and other scalars. For ** subsonic flow **, again the static pressure is obtained from the CFD
solution. Using the static pressure, you can derive an equation for Mach number knowing mass flow rate and total temperature, and thus obtain total pressure. Therefore, in this case the total
pressure is derived from the
CFD solution.
Refer to this document for more details. | {"url":"https://innovationspace.ansys.com/courses/courses/topics-in-compressible-flow-general/lessons/boundary-conditions-for-compressible-flow-simulations/","timestamp":"2024-11-07T07:48:59Z","content_type":"text/html","content_length":"177244","record_id":"<urn:uuid:8ed6ea4e-3a8b-4d28-968b-b635523496d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00430.warc.gz"} |
Dennis Chen’s Website
Hey there! I’m Dennis, and I’m studying computer science and mathematics at Carnegie Mellon University.
My interests change pretty frequently, but some things I like to do as of now:
• Logic puzzles (mostly sudoku)
□ Puzzle construction
□ Sudoku competitions
• Teaching
□ Fall 2024: I will be a CS251 TA
□ I taught contest math through my first three years of high school and the fall semester of my freshman year at college
• Math
□ Abstract algebra
□ Set theory/formal logic/model theory
• Computers
□ Rust
□ Functional programming
□ Linux
□ Typesetting (LaTeX/Typst)
□ System administration
□ Networking protocols (e.g. HTTP)
□ HTML/CSS and web design
• Writing
I also did design work before. The first draft of the MAT logo was by me. William changed it to a purple color scheme, which is currently on the final logo. I also have amateurish book, handout, and
CV designs, the last two of which have been used semi-often by other people.
• Variant Sudoku puzzles follow typical sudoku rules, but typically with a twist. Sometimes you may have to “color” cells that you know are the same but don’t know the value of yet, and sometimes
you will even have to construct the regions of the puzzle yourself. I have some beginner recommendations.
• Hanabi is a cooperative card game where you see everyone’s cards except your own. You give clues (color, number) to your teammates so they learn the identity of their cards, and they give you
clues so you learn your cards. The goal is to play stacks of cards like in solitaire — the default is 5 stacks of 5 different colors, but this can change.
• K-Pop
□ Seventeen
□ ATEEZ
□ P1Harmony
□ aespa
• Rock climbing (I am not good) | {"url":"https://dennisc.net/","timestamp":"2024-11-14T05:32:04Z","content_type":"text/html","content_length":"3475","record_id":"<urn:uuid:afb10492-16ec-4cf6-b353-ee7a8f3b767e>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00536.warc.gz"} |
How to create a vector in Python using NumPy - Learn Digital Marketing
NumPy is a powerful Python library for numerical computing that provides methods for creating and manipulating arrays. One common use case for NumPy is creating vectors, which are one-dimensional
arrays of data. In this article, we’ll explore how to create a vector in Python using NumPy.
Before we dive in, make sure you have NumPy installed in your Python environment. You can install NumPy using pip, a package manager for Python. Simply run the following command in your terminal:
pip install numpy
Once you have NumPy installed, you’re ready to create vectors!
Creating a vector with NumPy
To create a vector in NumPy, we’ll use the `numpy.array` method, which creates an array object from a list or tuple of data. For example, to create a vector of integers, we can pass a list of
integers to the `numpy.array` method:
import numpy as np
# create a vector of integers
int_vector = np.array([1, 2, 3, 4, 5])
[1 2 3 4 5]
Similarly, we can create a vector of floats or complex numbers by passing a list or tuple of floating-point numbers or complex numbers, respectively:
# create a vector of floats
float_vector = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
# create a vector of complex numbers
complex_vector = np.array([(1+2j), (3+4j), (5+6j)])
[1. 2. 3. 4. 5.]
[1.+2.j 3.+4.j 5.+6.j]
Vector operations
Once we have created a vector, we can perform a range of operations on it using NumPy. Here are some examples of common operations:
Accessing elements
We can access individual elements of a vector using square brackets and the index of the element we want. For example, to access the third element of a vector, we can use the following code:
import numpy as np
vector = np.array([1, 2, 3, 4, 5])
# access the third element
Vector arithmetic
We can perform arithmetic operations on vectors, such as addition, subtraction, multiplication, and division. These operations are performed element-wise, meaning that each element in the output
vector is the result of the corresponding elements of the input vectors.
import numpy as np
a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
# vector addition
c = a + b
# vector subtraction
d = a - b
# vector multiplication
e = a * b
# vector division
f = a / b
[5 7 9]
[-3 -3 -3]
[ 4 10 18]
[0.25 0.4 0.5 ]
Vector dot product
We can compute the dot product of two vectors using the `numpy.dot` method. The dot product of two vectors is the sum of the products of their corresponding elements. For example, to compute the dot
product of two vectors, we can use the following code:
import numpy as np
a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
# dot product
c = np.dot(a, b)
How do I create a vector with a specific length?
To create a vector of a specific length, we can use the `numpy.zeros` method, which creates an array of zeros with the specified shape. We can then modify the elements of the array as needed to
create the desired vector. For example, to create a vector of length 5 filled with zeros, we can use the following code:
import numpy as np
# create a vector of zeros with length 5
vector = np.zeros(5)
[0. 0. 0. 0. 0.]
How do I create a vector with regularly-spaced elements?
To create a vector with regularly-spaced elements, we can use the `numpy.arange` method, which returns an array of evenly spaced values within a given interval. We can specify the start, stop, and
step values for the range of values. For example, to create a vector with values from 0 to 9 with a step of 2, we can use the following code:
import numpy as np
# create a vector with regularly-spaced elements
vector = np.arange(0, 10, 2)
[0 2 4 6 8]
How do I create a vector with random elements?
To create a vector with random elements, we can use the `numpy.random` module, which contains functions for generating random numbers and arrays. We can use the `numpy.random.rand` method to generate
an array of random numbers between 0 and 1 with the specified shape. We can then scale and shift the values as needed to create a vector with the desired range of values. For example, to create a
vector of length 5 with random values between 0 and 10, we can use the following code:
import numpy as np
# create a vector with random elements
vector = 10 * np.random.rand(5)
[0.74922579 1.67779193 6.75908569 3.76741797 8.22730004]
NumPy provides a simple and powerful way to create and manipulate vectors in Python. With the methods and operations we’ve covered in this article, you can easily create and work with vectors for a
wide range of applications. | {"url":"https://thrivemyway.com/how-to-create-a-vector-in-python-using-numpy/","timestamp":"2024-11-02T05:58:39Z","content_type":"text/html","content_length":"303054","record_id":"<urn:uuid:0a435223-f8d0-468d-81af-0a994f4b42a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00553.warc.gz"} |
How Many Radians are in One Revolution? Explained - The Techy Life
How Many Radians are in One Revolution? Explained
One of the fundamental concepts in trigonometry is the measurement of angles. We often encounter angles in our daily lives, from the tilt of a sunflower towards the sunlight to the rotation of the
hands on a clock. While degrees serve as the conventional unit for measuring angles, there is another unit of measurement that offers a more elegant and versatile approach – radians. In this article,
we will delve into the topic of radians and explore the intriguing question of how many radians are present in one revolution.
To truly understand the concept of radians, it is essential to have a solid foundation in trigonometry. Trigonometry, a branch of mathematics that studies the relationships between angles and sides
of triangles, is crucial for various fields, including physics, engineering, and astronomy. The measurement of angles is a vital component of trigonometry, and radians provide a unique perspective to
approach this fundamental aspect. In this article, we will unravel the concepts surrounding radians, their applications, and the reasons why they are considered a more precise measurement system.
Additionally, we will address the perplexing question of how many radians exist in one revolution. Through a comprehensive exploration of this topic, we hope to enhance your understanding of angles
and provide you with new insights into the fascinating world of trigonometry.
**Definition of Radians and Revolutions**
In order to understand the relationship between radians and revolutions, it is essential to have a clear understanding of what these terms mean and how they are used to measure angles and rotations.
**Definition of Radians**
Radians are the unit of measurement commonly used to quantify angles. Unlike degrees, radians are based on the concept of measuring angles in terms of the radius of a circle. Specifically, one radian
is defined as the angle subtended at the center of a circle by an arc that is equal in length to the radius of that circle.
**Definition of Revolutions**
On the other hand, revolutions are used to measure complete rotations around a circle. A revolution occurs when an object or point moves in a circular motion and returns to its starting position. It
is equivalent to a full 360-degree rotation or 2π radians.
**Understanding the Relationship**
The relationship between radians and revolutions can be described by a conversion factor that allows for the interchange between the two units of measurement. This conversion factor is necessary as
radians and revolutions represent different aspects of circular motion.
**Derivation of the Conversion Formula**
The conversion formula between radians and revolutions is derived from the relationship between the circumference of a circle and the angle measured in radians. Since one revolution corresponds to a
full circumference of a circle, which is the same as 2π radians, the conversion factor between the two units is:
Revolutions = Radians / 2π
**Explanation of the Formula’s Significance**
The importance of π (pi) in the conversion formula is evident in its role as the constant that relates the circumference of a circle to its radius. This allows for the conversion of an angle measured
in radians to its equivalent representation in terms of revolutions.
**Detailed Calculation Steps**
To convert from radians to revolutions, the formula is applied by dividing the given angle in radians by 2π. The resulting value represents the equivalent angle in terms of revolutions. Similarly, to
convert from revolutions to radians, the formula is applied by multiplying the given number of revolutions by 2π.
Understanding the relationship between radians and revolutions is crucial for various applications in mathematics, physics, engineering, and navigation. By practicing and applying the conversion
formula in different contexts, individuals can develop a better grasp of these units and their significance. In the following sections, we will explore examples of converting between radians and
revolutions, real-life scenarios where these units are used, common misconceptions to watch out for, and the importance of radians in mathematics. It is through this comprehensive exploration that a
deeper understanding of radians and revolutions can be achieved.
Understanding the Relationship
A. Conversion factor between radians and revolutions
To comprehend the relationship between radians and revolutions, it is essential to understand the conversion factor that allows us to interchange between the two units of measuring angles. This
conversion factor, which is derived from the properties of a circle, plays a crucial role in many mathematical and scientific applications.
1. Derivation of the conversion formula
The conversion formula between radians and revolutions can be derived by considering the relationship between the circumference of a circle and the angle it subtends. A circle with a radius of one
unit has a circumference of 2π units. As one revolution corresponds to a complete rotation around the circle, which covers the entire circumference, it can be concluded that one revolution is equal
to 2π radians.
Mathematically, the conversion formula can be expressed as:
1 revolution = 2π radians
2. Explanation of why the formula is necessary
The conversion formula is necessary because radians and revolutions are different units used to measure angles. While revolutions provide a measure of the number of complete rotations, radians
represent a ratio between the length of an arc and the radius of a circle. As radians are commonly used in mathematics and scientific calculations, it is crucial to understand their relationship with
revolutions for accurate angle measurement.
Converting between radians and revolutions enables us to express angles in a way that is consistent with different mathematical and scientific contexts. It allows for seamless integration of
trigonometric functions, calculus, and other advanced mathematical concepts that rely on radians as the preferred unit of angle measurement.
In summary, the conversion factor between radians and revolutions allows for the interchangeability of these units and facilitates accurate angle measurement in both mathematical and scientific
applications. Understanding this relationship is fundamental for any individual working with angles and is crucial for the successful application of trigonometry, calculus, and other branches of
advanced mathematics.
IExplaining the Conversion Formula
A. Relationship between radians and the circumference of a circle
In order to understand the conversion formula between radians and revolutions, it is important to establish the relationship between radians and the circumference of a circle. The circumference of a
circle is defined as the distance around the outer edge of the circle. It can be calculated by multiplying the diameter of the circle by π (pi), which is approximately 3.14159.
A circle with a radius of 1 unit has a circumference of 2π units. This relationship is the key to understanding the conversion between radians and revolutions.
B. Importance of π (pi) in the conversion formula
Pi (π) is a mathematical constant that represents the ratio of a circle’s circumference to its diameter. It is an irrational number, meaning it cannot be expressed as a fraction and has an infinite
number of decimal places. In the context of converting between radians and revolutions, π plays a crucial role as it relates the circumference to the angle measured in radians.
C. Detailed calculation steps
The conversion formula between radians and revolutions is derived by considering the relationship between the circumference of a circle and the angle measured in radians.
To convert from radians to revolutions, the formula is as follows:
Number of revolutions = Angle in radians / (2π)
To convert from revolutions to radians, the formula is:
Angle in radians = Number of revolutions * (2π)
By using these formulas, it is possible to convert between radians and revolutions accurately.
It is important to note that when using these formulas, the angle measured in radians must be a positive value. Negative angles can be handled by considering them as rotations in the opposite
Overall, understanding the conversion formula between radians and revolutions allows for accurate and consistent conversion of angles between the two units of measurement. This knowledge is crucial
for various applications in mathematics, physics, engineering, and navigation systems, where the use of radians and revolutions is common. By practicing and applying the conversion formula in
different contexts, individuals can develop a solid understanding of this relationship and enhance their mathematical skills.
Examples of Conversion from Radians to Revolutions
A. Step-by-step explanation of example calculations
In this section, we will provide step-by-step explanations on how to convert angles given in radians to revolutions using the conversion formula derived earlier. These examples will allow readers to
apply the formula and gain a better understanding of its practical use.
To begin, let’s consider an angle of 3π/2 radians. To convert this angle to revolutions, we can use the following steps:
1. Recall the derived conversion formula: revolutions = radians / (2π).
2. Substitute the given angle into the formula: revolutions = (3π/2) / (2π).
3. Simplify by canceling out the common factor of π: revolutions = (3/2) / 2.
4. Divide 3 by 2: revolutions = 1.5 / 2.
5. Perform the division: revolutions = 0.75.
Therefore, an angle of 3π/2 radians is equivalent to 0.75 revolutions.
2. Variation in measurements with different circle sizes
It’s important to note that the conversion from radians to revolutions remains consistent regardless of the circle’s size. The conversion formula accounts for the relationship between the angle and
the complete rotation around the circle.
For instance, let’s consider a smaller circle with a circumference of 10 units. We want to convert an angle of π/2 radians to revolutions on this smaller circle.
1. Apply the conversion formula: revolutions = radians / (2π).
2. Substitute the angle into the formula: revolutions = (π/2) / (2π).
3. Simplify by canceling out the common factor of π: revolutions = 1/2.
4. Since the circle has a circumference of 10 units, we can calculate the converted value: revolutions = (1/2) * 10 = 5.
Therefore, an angle of π/2 radians on a smaller circle with a circumference of 10 units is equivalent to 5 revolutions.
These examples illustrate the step-by-step process of converting angles given in radians to revolutions, showcasing how the conversion formula remains consistent regardless of the circle’s size. By
practicing these calculations, readers will become more proficient in applying the conversion formula and better understanding the relationship between radians and revolutions.
Examples of Conversion from Revolutions to Radians
A. Step-by-step explanation of example calculations
In this section, we will delve into the conversion process from revolutions to radians. While the previous section focused on converting from radians to revolutions, this section will provide
step-by-step explanations for converting from revolutions to radians using the conversion formula.
To convert from revolutions to radians, we use the conversion factor derived in Section IThis conversion factor states that one revolution is equal to 2π radians.
Let’s consider an example calculation:
Example: Convert 3 revolutions to radians
Step 1: Start with the given number of revolutions, in this case, 3 revolutions.
Step 2: Multiply the number of revolutions by the conversion factor. In this case, we will multiply 3 revolutions by 2π radians/1 revolution.
Calculation: 3 revolutions * 2π radians/1 revolution = 6π radians
Therefore, 3 revolutions is equal to 6π radians.
B. Demonstration of the formula’s consistency
To further reinforce the consistency of the conversion formula, let’s consider another example:
Example: Convert 5 revolutions to radians
Step 1: Start with the given number of revolutions, in this case, 5 revolutions.
Step 2: Multiply the number of revolutions by the conversion factor. In this case, we will multiply 5 revolutions by 2π radians/1 revolution.
Calculation: 5 revolutions * 2π radians/1 revolution = 10π radians
Therefore, 5 revolutions is equal to 10π radians.
By following the same process, it is evident that the conversion formula consistently converts a given number of revolutions into radians based on the conversion factor of 2π radians per revolution.
It is important to note that this formula remains consistent regardless of the size of the circle. Whether we are considering a small circle or a large circle, the conversion from revolutions to
radians remains the same.
Understanding the conversion from revolutions to radians is crucial for various applications, such as navigation systems, physics, and engineering. By practicing and applying the conversion formula,
individuals can confidently work with angles expressed in both revolutions and radians.
VPractical Applications
A. Real-life scenarios where radians and revolutions are used
Radians and revolutions are fundamental units of measurement that have practical applications in various fields. Understanding their relationship is essential for accurate calculations and precise
measurements. Here are some real-life scenarios where radians and revolutions are commonly used:
1. Navigation systems and map calculations
In navigation systems, such as GPS devices, the use of radians and revolutions is crucial for determining accurate positions and calculating distances. The coordinates of a location on a map are
often specified in terms of latitude and longitude, which are measured in radians. By converting these values, navigation systems can provide precise directions and ensure accurate positioning.
Furthermore, map calculations also rely on radians and revolutions. For example, calculating the distance between two points on a map involves converting the angular displacement (in radians) into
the corresponding arc length on the Earth’s surface, using the formula derived from the relationship between radians and the circumference of a circle.
2. Physics and engineering measurements
Radians and revolutions are extensively used in physics and engineering to measure and analyze rotational motion. In physics, concepts such as angular velocity, angular acceleration, and moment of
inertia are expressed in radians, allowing for precise calculations and predictions related to rotational movements.
For engineering applications, radians are important in fields such as robotics, mechanical engineering, and design. The precise control of robotic arms, the analysis of gear ratios, and the design of
complex mechanical systems all require a thorough understanding of radians and revolutions.
The accurate measurement of rotational angles is critical in these applications, as slight errors can lead to significant deviations and miscalculations. By using radians and revolutions, engineers
and physicists can ensure the reliability and effectiveness of their designs and experiments.
In conclusion, radians and revolutions have practical applications in diverse fields, including navigation systems, map calculations, physics, and engineering. Understanding the relationship between
these units of measurement enables precise calculations, accurate measurements, and reliable predictions. As such, it is essential to grasp the conversion formula and apply it effectively in various
real-life scenarios.
Common Misconceptions
1. Incorrect calculation methods to watch out for
When converting between radians and revolutions, it is important to be aware of common mistakes that can lead to inaccurate results. One common misconception is confusing the conversion factors for
radians and degrees. Radians and degrees measure angles differently, so it is essential to use the correct conversion formula.
Another common error is miscalculating the number of revolutions in a given angle measurement. Since one revolution is equal to 2π radians, it is important to divide the given angle in radians by 2π
to obtain the correct number of revolutions. Failing to do so can lead to incorrect conversions and inaccurate results.
2. Providing tips to avoid errors
To avoid errors in converting between radians and revolutions, there are several tips that can be followed.
Firstly, always double-check the conversion formula being used. Ensure that the correct conversion factor between radians and revolutions is being applied. Remember that 1 revolution is equal to 2π
Secondly, it is helpful to use a calculator with a trigonometric function that can handle both radians and degrees. This will minimize the risk of manually calculating incorrect values.
Additionally, when performing calculations, it is recommended to keep the values as precise as possible until the final answer is obtained. Rounding off intermediate values can introduce errors into
the result.
Lastly, practice and familiarity with the conversion formula will improve accuracy. By working on different examples and exercises, you can become more proficient in converting between radians and
By being aware of common misconceptions and following these tips, one can avoid errors and accurately convert between radians and revolutions. As with any mathematical concept, practice and
understanding are key to mastering the conversion process.
Importance of Radians in Mathematics
A. Relation to trigonometric functions
Radians play a crucial role in mathematics, particularly in relation to trigonometric functions. Trigonometry is the branch of mathematics that deals with the relationships between angles and the
sides of a triangle. It is widely used in fields such as physics, engineering, and navigation.
Trigonometric functions, such as sine, cosine, and tangent, are defined using radians as the unit of measurement for angles. This is because radians provide a more natural and intuitive way to
express angles in the context of trigonometry. The sine function, for example, represents the ratio of the length of the side opposite an angle to the length of the hypotenuse of a right triangle.
When angles are expressed in radians, the formulas and calculations involved in trigonometry become simpler and more elegant.
B. Significance in calculus and advanced mathematics
In addition to its role in trigonometry, radians are also of great significance in calculus and advanced mathematics. Calculus is a branch of mathematics that deals with the study of change and
motion. Key concepts in calculus, such as derivatives and integrals, are defined using radians.
The use of radians in calculus allows for more precise and accurate calculations. When angles are measured in radians, the formulas for calculating the derivative and integral of trigonometric
functions simplify significantly. This simplification allows for more efficient and powerful techniques to solve a wide range of problems in physics, engineering, and other fields.
Furthermore, radians are used extensively in complex analysis, a branch of mathematics that deals with functions of complex numbers. Complex analysis relies heavily on trigonometric functions, and
thus radians, to study complex numbers, functions, and their properties.
In conclusion, understanding the relationship between radians and revolutions is of utmost importance in mathematics. Radians are closely related to trigonometric functions and play a fundamental
role in calculus and advanced mathematical concepts. Their use allows for more elegant and efficient solutions to problems in various fields, making them essential for students and professionals
alike to grasp and apply in their mathematical endeavors.
Recap of the importance of understanding the relationship between radians and revolutions
In conclusion, understanding the relationship between radians and revolutions is crucial in various fields that involve angle measurements. Radians provide a standardized unit for measuring angles,
allowing for accurate calculations and comparisons. Revolutions, on the other hand, represent a complete rotation around a circle and are essential in understanding rotational motion.
By knowing the conversion factors between radians and revolutions, individuals can easily convert between the two units and apply them in real-life scenarios. Whether it is in navigation systems, map
calculations, physics, engineering, or even advanced mathematics, the ability to convert between radians and revolutions is essential.
Encouragement to practice and apply the conversion formula in various contexts
To fully grasp the concept of radians and revolutions, it is important to practice and apply the conversion formula in various contexts. By working through example problems and calculations,
individuals can gain confidence and proficiency in converting between radians and revolutions.
Furthermore, it is crucial to understand the limitations and sources of error when converting between these units. This includes being aware of incorrect calculation methods and tips to avoid errors.
By addressing common misconceptions, individuals can ensure accuracy in their measurements and calculations.
Lastly, the importance of radians should not be underestimated in mathematics. Radians are closely related to trigonometric functions and play a significant role in calculus and advanced mathematics.
Mastering the concept of radians and their relationship with revolutions opens the door to a deeper understanding of mathematical concepts and their practical applications.
In summary, radians and revolutions are interconnected units that are vital in various fields. From navigation and physics to mathematics and engineering, the ability to convert between radians and
revolutions is essential. By practicing and understanding the conversion formula, individuals can confidently apply these units in real-world scenarios and further enhance their understanding of
Leave a Comment | {"url":"https://thetechy.life/how-many-radians-in-one-revolution/","timestamp":"2024-11-10T11:07:44Z","content_type":"text/html","content_length":"97384","record_id":"<urn:uuid:3d0b7056-91cb-4780-81cd-425c655eaf07>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00320.warc.gz"} |
Not a Function
In the previous post we talked about what is a function. In this post let’s talk about things that look like a function, but actually are not.
By definition #
Not all equations are functions. y = x + 1 is a function, but y² + x² = 1 is not, because “function is a many-to-one (or sometimes one-to-one) relation” (in this case there are 1 or 2 values of y
corresponds to one x).
Not all graphs (set of points in Cartesian coordinates) are functions. This graph represents a function:
But this one is not:
Not all tables (set of tuples (x, y)) are functions. This one is represents a function:
But this one is not:
All functions are relations, but not all relations are functions.
If we will draw a table of all possible relations between sets A and B, only two of those are functions (marked with f)
corresponds to a 0-1 element in set B 0-M 1 1-M
0-1 element in set A
1 f *
1-M f *
* Multivalued functions (or multiple-valued functions) are relations that map single points in the domain to possibly multiple points in the range (in the theory of complex functions).
More about domain #
Function from A to B is an object f such that every a in A is uniquely associated with an object f(a) in B. The set A of values at which a function is defined is called its domain.
So here is a possibly confusing bit - function requires every element of input set (domain) to correspond to some element in the output set (codomain).
What about y = 1/x, there is no output for 0 ( at least not one version which all agrees about). Explanation here is the following: 0 is not part of the domain of the given function, or you can say
that function 1/x is not defined for zero.
Consequence: if y₁=x²/x and y₂=x than y₁≠y₂ because y₁ is defined for all real numbers except 0, but y₂ is defined for all reals (ℝ).
Total function #
In programming they have related terminology:
Total functions — that is, a function
□ which is defined for all possible inputs
□ and is guaranteed to terminate.
🤔 It means that the domain of the function is not the same thing as the type of “all possible inputs”. Or maybe programming we need a slightly different definition of a function.
🤔 There are two conditions here (1) defined for all inputs and (2) function terminates. It seems to me that the second condition is redundant here because if function never terminates, we never have
an answer thus the result of the operation is not defined. For example, this what happens when you try to divide by 0 in a mechanical calculator.
Image credit: popularmechanics.com.
Non-functions in programming #
No input #
Should we consider “functions” which doesn’t have input to be a function?
🤔 Is it even appropriate to call it a function? Maybe a better name would be coroutine or procedure?
If they produce more than one output than one output than no:
Math.random(); // 0.8240352303263008
Math.random(); // 0.1830674266691794
Date.now(); // 1562502871898
Date.now(); // 1562502872905
🤔 What if they produce one output, for example, a function which returns singleton? Probably not (to explain in more details we need to talk about effects, which is a subject for an upcoming post).
More than one output for the same input #
Not a function:
let counter = 0;
const inc = x => (counter = counter + x);
inc(1); // 1
inc(1); // 2
🤔 Interesting that we consider one output as “one output over the time” (e.g. consequent calls). What about more than one output at once?
const y = x => {
if (x > 1 || x < -1)
throw new Error("Function undefined for x > 1 or x < -1");
const result = Math.sqrt(1 - x * x);
return [-result, result];
First, we need to define what is the same output means - how we can compare two values in programming.
Comparison #
When two values are equal in programming? We can consider two options:
• nominal comparison (identity) - objects are equal only when they are identical e.g. they have some unique nominal identifier which is in case of computers can be memory reference (or pointer).
• structural comparison (equality) - objects are equal if all of it’s “members” are equal, in the most simplified case we can compare memory bit by bit.
Side note: for primitive values, like integers, which values are so small that they are directly placed on stack instead of heap nominal comparison and structural comparison is the same thing.
For the given example:
y(0.5) === y(0.5); // false
y doesn’t produce nominally “same” results.
y(0.5)[0] === y(0.5)[0]; // true
y(0.5)[1] === y(0.5)[1]; // true
but it produces a structurally “same” result. We can choose any type of comparison and depend on this y will be or will not be a (mathematical) function.
As well we can make y to return nominally identical results:
const memoize = f => {
const cache = new Map();
return x => {
if (!cache.has(x)) {
cache.set(x, f(x));
return cache.get(x);
const y1 = memoize(y);
as you can see y1 returns nominally identical results for the same input
y1(0.5) === y1(0.5); // true
the trade-off here is that we need more memory to store outputs. Most likely it will allocate a bigger slice of memory for new Map() upfront, so we need to pay price (memory) even if we don’t call
On the other hand, structural comparison requires more CPU cycles - in the worst case we need to compare memory bit by bit.
Side note: in garbage-collected languages we can use less memory for nominal comparison, because we can track if output object is in use or not, and if it is not in use we can remove it from cache
(similar to how WeakMap works, except for values instead of keys).
There is no universal answer to this question, structural comparison will fail for recursive (cyclic graph) data structure, for example:
the nominal comparison will not work if we would want to compare values from two different functions
JSON.parse("[-0.8660254037844386,0.8660254037844386]") === y1(0.5); // false
🤔 How to compare functions (if we talk about functions as values)? If we would want to compare them structurally, shall we compare bytecode that they produce? What if bytecode for two functions was
produced by different compilers? What if it is the same function, but implementations are different, for example:
const fact1 = n => {
let res = 1;
for (let i = 1; i <= n; i++) {
res = res * i;
return res;
const fact2 = n => (n < 2 ? 1 : n * fact(n - 1));
🤔 How to implement nominal comparison for deserialized objects? Should we store all deserialized strings to always return the same reference?
On practice programming languages (machines) can use a combination of those two approaches, for example:
• compare references first, and fallback to structural comparison if the first check is falsy
• or compare structurally primitives (integers, strings, etc.) and compare nominally other variables (arrays, objects, etc.)
• etc.
So it’s up to you, developer, to decide which comparison use.
Are lists and structures valid function results? #
Function y declared above represents the same relation between x and y as y² + x² = 1. But earlier we concluded that y² + x² = 1 is equation and not a function. 🤔 Does this mean that y is not a
Well, I would say that it is still a function (y has a single output - list). This is one of the examples of how the idea of mathematical function (platonic idea), doesn’t always directly translates
to computations (which are in more closer relation to physics).
(a modified version of https://xkcd.com/435/)
In math, they do not talk about computational complexity (“big O notation”), as long as for the same input it produces the same output mathematicians would consider those to be the same functions,
for example, bubble sort and merge sort. From a computer science point of view they have different time and space complexity.
Where is the (platonic) idea of function is very useful in math in computer science it may need some adjustment or reinterpretation.
Read more: Function, procedure, method, operator..., Category vs Design pattern | {"url":"https://stereobooster.com/posts/not-a-function/","timestamp":"2024-11-08T13:46:59Z","content_type":"text/html","content_length":"54172","record_id":"<urn:uuid:f54b6b07-d208-4f7d-9a7d-e4e769619c17>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00033.warc.gz"} |
Electron Tomography
In the context of biology and medicine, electron tomography is a means of estimating the internal structure of an object from measurements of the intensity of a high-voltage electron beam impinging
upon it. The object is assumed to possess a degree of opacity with respect to individual electrons that results in an attenuation of the beam via effects such as scattering and absorption. The device
that generates the electron beam, positions and orients the object in the beam, and measures the beam's intensity is called a transmission electron microscope (TEM), with the label “transmission”
referring to the passage of the electron beam through the object being investigated. A measurement of the intensity of the electron beam usually takes the form of a one- or two-dimensional grey-scale
image referred to by various names, with “tilt”, “micrograph”, “electron micrograph”, “transmission electron micrograph”, “TEM micrograph”, and “projection” being among the most common. In the
present document, we will refer to all such measurements using the umbrella term projection.
Light microscopy versus electron microscopy
Theoretically speaking, the resolution limit of an imaging system depends directly on the range of energy wavelengths it is capable of detecting. Since the wavelength of visible light is between 390
to 750 nanometres (10^-9m), standard light microscopes are incapable of resolving features smaller than 1/10th of a micrometre (10^-7m). Because of the inverse relationship between the relativistic
momentum of an electron and its wavelength, electrons that are accelerated to an appreciable fraction (e.g., 70%) of the speed of light possess wavelengths on the order of picometres (10^-12m). In
terms of the electromagnetic spectrum, this is the wavelength of gamma rays. Modern TEMs can therefore achieve magnifications 1,000X that of standard light microscopes, easily resolving features in
the micro- to nanometre range (10^-6m--10^-9m), with the highest achieved resolutions (ca. 2005) in the 1/100ths of nanometres. Biological structures falling within this resolution range include
mammalian cell nuclei (~6μm in diameter), human red blood cells (~6--8μm in diameter), Caulobacter crescentus bacteria (4--6μm long), mitochrondria (0.5--10μm), Polio virus capsids (30nm in
diameter), and typical cell membranes (6--10nm thick).
Projection acquisition
When producing projections intended to be used for tomographic reconstruction (a term defined below), the object being imaged is usually required to be a thin section of tissue between 150--500nm
thick. The tissue is often stained to increase its electron opacity, most commonly with compounds containing heavy metals such as osmium, lead, or uranium (e.g., osmium tetroxide and uranyl acetate).
The high voltages used to accelerate the electrons in the TEM column require that the column be evacuated to prevent electrical arcing and undesirable interactions between the electrons and
atmospheric gas molecules. To withstand the vacuum of the TEM column, biological material must also be fixated prior to imaging. Both osmium tetroxide and uranyl acetate, in addition to their
staining properties, behave as biological fixatives; plastic embedding provides alternative means of fixation. At some point during sample preparation, the object is marked with small particles, such
as ~100nm diameter colloidal gold, which act as fiducial markers (or simply fiducials). If this marking occurs after fixation and any subsequent shaving down of the section block, only the exterior
surfaces of the block will be marked. Following the sample preparation, the object is placed on a specimen stage, which is the part of the TEM that positions and orients the object in the electron
beam. Projections of the object are taken at a series of orientations, typically 1- or 2-degree incremental rotations about a single axis (“single tilt”) or two orthogonal axes (“double tilt”) -- in
either case, the current rotation axis is often referred to as the tilt axis. Note that these two projection geometries are the most common only because they are the most mechanically convenient.
Figure 1: This diagram serves as an illustration of the initial projection geometry inside an electron microscope. In this and subsequent diagrams, the electron source is located at the origin, but
here the beam is turned off. The object being imaged is the opaque grey block in the center of the diagram; note the black spheres embedded in the surface of the block -- these represent fiducial
markers. There are also fiducial markers embedded on the far side of the object. The line parallel to the y-axis passing through the middle of the block is the tilt axis. The white square to the
right of the object is the projection screen: the plane onto which the object is projected.
Figure 2: In this diagram the electron beam has been turned on; the dashed line running along the negative z-axis to the middle of the projection is the optical axis. In this example, the material
comprising most of the object is assumed to be perfectly electron-transparent; on the other hand, the fiducial markers and any structures contained within the object are assumed to be perfectly
electron-opaque. As discussed further below, neither assumption holds in practice. Judging by the projection, there appear to be two or three structures inside the object. Note the projections of the
fiducial markers embedded on the far side of the object.
Figure 3: In this diagram, the opaque material of the object has been rendered transparent to give the viewer a better idea of what is going on. There are actually four structures inside the object:
a rectangular solid, a small sphere, an elliptical solid, and a bowl-shaped structure. Note that the projection of the rectangular solid at this orientation more-or-less completely shadows the small
sphere. A mathematical description of this orientation would be something like ω = (y-axis, 0.0).
Figure 4: In this animated diagram, the object is rotated about the y-axis in 15-degree increments between 0.0 and 60.0 degrees.
Figure 5: In this animated diagram, the specimen stage holding the object has been rotated 90.0 degrees about the z-axis and the object is rotated about the y-axis in 15-degree increments between 0.0
and 60.0 degrees. Note that having a rotation axis parallel to the x-axis would generate the same series of projections provided the projection screen were rotated (or were assumed to be rotated) by
-90.0 degrees about the z-axis as well.
A careful reader may have noticed a number of additional peculiarities in the preceding diagrams; for further discussion, click here. (A link to a page containing further discussion of the figures.)
In the animated diagrams above, the rotation angles about the tilt axis range in magnitude from 0.0 to 60.0 degrees. This matches what is usually found in practice: assuming that the object at
orientation ω = (tilt-axis ≠ z-axis, 0.0) has a thickness along the z-axis that is much less than its extents along the x- and y-axes, as the object's rotation about the tilt axis approaches +/-90.0
degrees, the thickness of the object along the z-axis becomes very large. For most biological samples oriented at high degree, the object's thickness along the z-axis becomes too great for the
electron beam to effectively penetrate.
Tomographic reconstruction
A single projection of an object gives some indication of its internal structure, but because of shadowing, particularly in the case of perfectly electron-opaque structures, much of the detail is
obscured. Furthermore -- especially in the case of perfectly electron-opaque structures -- a single projection leaves spatial arrangement along the z-axis completely ambiguous. Multiple projections
of an object remove some of this spatial ambiguity, but the shadowing problem remains. (It would be instructive at this point to consider why the geometries of only three of the four structures in
the object imaged in the diagrams above can be completely determined from their projections.) What is desired is the calculation of a 3D scalar function, a tomogram, representing the density of the
object with respect to the electron beam. The word “tomogram” is a modern combination of two words from Attic Greek: “tómos”, a cut or section, and “grámma”, something written or drawn. Indeed, once
calculated, a tomogram can be viewed as a stack of 2D slices, but there exist other techniques for viewing a tomogram as a whole volume. The technique of reconstructing an object's tomogram from its
projections is called tomographic reconstruction or, simply, tomography.
At this point is necessary to touch briefly upon the mathematical theory underpinning tomographic reconstruction. Tomographic reconstruction is what physicists and mathematicians refer to as an
inverse problem. An inverse problem is one that attempts to determine the values of a set of model parameters from a set of observations. Assume that the synthetic TEM in the diagrams above is
perfectly stable, and that its physical geometry is completely known. Furthermore, assume that we have a sensible model for the physics of electrons, the relationship between the attenuation of an
electron beam passing through matter and that matter's density, and the electron detector comprising the projection screen. The projections and their corresponding object orientations thus stand as a
set of observations, and the model parameters we seek are therefore associations between the points in the 3D space occupied by the object and scalar densities. Mathematical models of image formation
in a TEM are usually couched in the language of line integrals: the intensity associated with a particular point on a given projection is the integral along a line passing through that point and the
electron source. Tomographic reconstruction can be understood as the inverting of these line integrals. In theory, completely reconstructing an object's tomogram with perfect fidelity requires that
the object be of finite extent, that all line integrals not passing through the object equal zero, and that all line integrals passing through the object be computed. If a set of line integrals are
missing, they are assumed to equal some value (usually zero), which will degrade the accuracy of the tomogram. As mentioned in the previous section, in the case of typical biological specimens there
is an entire range of orientations at which projections cannot be taken of the object. These missing projections and their associated orientations are known as the missing wedge and are a source of
error in tomographic reconstructions.
Given a set of projections of an object, there are three common approaches to reconstructing its tomogram, only one of which concerns us at present: filtered back-projection. The filtered
back-projection algorithm is comprised of three steps. The first step, alignment, calculates a geometrical association -- a set of parameters referred to as a projection map -- between the various
orientations of the object and the detector at a fixed reference orientation (alternatively, the geometrical associations can be understood to be between projections oriented with respect to the
object at a fixed reference orientation). This is usually achieved via approximations that depend on tracked 2D projections of a set of fiducial markers, although, theoretically, anything that can be
tracked with tolerable accuracy throughout the set of projections can be used. The second step, filtration, is too mathematically abstract to be discussed much here, though it can accurately be
construed as a sharpening of the projections. It is worth noting, however, that while in theory filtration depends directly on alignment, often in practice this dependence is assumed to take a very
simplified form, which amounts to a sharpening along the horizontal pixel rows in the projection. The third step, reconstruction, is typically implemented as follows: for each 3D point x in the
tomogram, for each filtered projection indexed by orientation ω, find the 2D point x[p] in the current projection that x projects to according to projection map indexed by ω and add the filtered
intensity at x[p] to the intensity of the tomogram at x. The mapping from x to x[p] is called forward-projection, while the carrying of the filtered intensity at x[p] back to x is called
back-projection. Given some means of calculating the projection trajectory that passes through point x[p] in the filtered projection indexed by ω (alternatively, given a simplified projection
geometry), forward-projection is not a theoretical necessity, and the filtered intensity at x[p] can be back-projected at constant value along its trajectory through the tomogram. In practice,
however, particularly when the projection trajectories are curvilinear, this leads to sampling errors in the tomogram and foward-projection must be used. | {"url":"https://confluence.crbs.ucsd.edu/display/ncmir/Electron+Tomography","timestamp":"2024-11-02T07:38:31Z","content_type":"text/html","content_length":"70789","record_id":"<urn:uuid:0f41ef87-1e09-4c40-be81-a2b24712b5ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00818.warc.gz"} |
Anal cerci is present in cockroach a. in male cockroach
b. in f... | Filo
Question asked by Filo student
Anal cerci is present in cockroach a. in male cockroach b. in female c. in both d. none of these
Not the question you're searching for?
+ Ask your question
Sure, I can convert the solutions into step-by-step solutions using LaTeX. Here is an example: Problem: Simplify the expression . Solution: In this solution, I used the following steps: 1. Simplify
the fraction by canceling out common factors in the numerator and denominator. 2. Simplify the fraction further by dividing both the numerator and denominator by the greatest common factor of 3, 6,
and 3x^2y^2. 3. Simplify the expression by using the properties of exponents to simplify the variables. 4. Write the final answer in simplified form. By breaking down the solution into these steps,
students can more easily understand how to simplify similar expressions in the future.
Found 3 tutors discussing this question
Discuss this question LIVE
10 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions on Structural Organisation in Plants and Animals
View more
Students who ask this question also asked
View more
Stuck on the question or explanation?
Connect with our Biology tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text Anal cerci is present in cockroach a. in male cockroach b. in female c. in both d. none of these
Updated On Feb 12, 2024
Topic Structural Organisation in Plants and Animals
Subject Biology
Class Class 11
Answer Type Text solution:1 | {"url":"https://askfilo.com/user-question-answers-biology/anal-cerci-is-present-in-cockroach-a-in-male-cockroach-b-in-36383931323530","timestamp":"2024-11-06T21:15:52Z","content_type":"text/html","content_length":"133622","record_id":"<urn:uuid:9c172412-0774-4027-9277-8c7a817257e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00684.warc.gz"} |
Convert Kilometer/hour to Foot/second
Please provide values below to convert kilometer/hour [km/h] to foot/second [ft/s], or vice versa.
Definition: The unit kilometers per hour (symbol: km/h) is a unit of speed expressing the number of kilometers traveled in one hour.
History/origin: The unit of kilometers per hour is based on the meter, which was formally defined in 1799. According to the Oxford English Dictionary, the term kilometer first came into use in 1810.
It was not until later in the mid-late 19^th century however, that the use of kilometers per hour became more widespread; the myriametre (10,000 meters) per hour was preferred for expressing speed.
Current use: Km/h is currently the most commonly used unit of speed around the world and is typically used for car speeds and road signs. It is also common for both miles per hour as well as
kilometers per hour to be displayed on car speedometers. There are many abbreviations for the unit kilometers per hour (kph, kmph, k.p.h, KMph., etc.), but "km/h" is the SI unit symbol.
Feet per second
Definition: A foot per second (symbol: ft/s) is a unit of speed and velocity that expresses the time taken in seconds to travel a specific distance in feet. It is equal to 0.3048 meters per second,
the International System of Units (SI) derived unit of speed and velocity. It is also equal to 0.592484 knots and 0.681818 miles per hour.
History/origin: The foot per second is a measurement based in systems like the imperial and United States customary systems of units, where the foot is the preferred unit of length.
Current use: The foot per second is not widely used. The meter per second is the preferred measurement in scientific contexts, and either miles per hour or kilometers per hour are more common in
everyday use for describing road speeds. The foot per second is also a relatively small unit of measurement making it difficult for use on a larger scale.
Kilometer/hour to Foot/second Conversion Table
Kilometer/hour [km/h] Foot/second [ft/s]
0.01 km/h 0.0091134442 ft/s
0.1 km/h 0.0911344415 ft/s
1 km/h 0.9113444153 ft/s
2 km/h 1.8226888306 ft/s
3 km/h 2.7340332458 ft/s
5 km/h 4.5567220764 ft/s
10 km/h 9.1134441528 ft/s
20 km/h 18.2268883056 ft/s
50 km/h 45.5672207641 ft/s
100 km/h 91.1344415281 ft/s
1000 km/h 911.3444152814 ft/s
How to Convert Kilometer/hour to Foot/second
1 km/h = 0.9113444153 ft/s
1 ft/s = 1.09728 km/h
Example: convert 15 km/h to ft/s:
15 km/h = 15 × 0.9113444153 ft/s = 13.6701662292 ft/s
Popular Speed Unit Conversions
Convert Kilometer/hour to Other Speed Units | {"url":"https://www.tbarnyc.com/speed/kilometer-hour-to-foot-second.html","timestamp":"2024-11-08T18:25:56Z","content_type":"text/html","content_length":"12046","record_id":"<urn:uuid:cc87474c-b4ab-4f98-a69b-182b3cd95d6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00294.warc.gz"} |
SATURDAY, July 31, 2010 <br>Timothy L. Meaker
Theme: None
I'm going to go out on a limb here and predict that y'all had trouble with this one. It was hard, right?? I sort of picked and poked my way through most of it and then only had the northeast corner
left but it wasn't budging. I set it down for a while — played a little
sporcle, read some blogs
— and when I went back to it everything fell right into place. That's so cool how that happens.
The biggest problem I had in the northeast is that I assumed
5A: Census bureau, essentially
was some kind of COUNTER (I had the TER in place). Makes sense, right? That only left three letters up front though, and MAN COUNTER seemed a little … off. Also, with the ORS in place I wanted
FLEXORS for a while instead of TENSORS (
7D: Stretching muscles
). I was flailing around in the dark is what I'm saying.
Overall, I'd say there's nothing super sparkly about this grid, except maybe SCHOOLMARM, WIND TUNNEL, and MOJO (
53A: Old-time educator / 18A: Aerodynamics research tool / 58A: Mystical amulet
), and it includes an awful lot of three-letter "words," but nothing jumped out at me as blatantly horrible and when it was all said and done I felt like I'd had an actual workout. And that's a good
Several people tripped me up today. Most of whom I'd never heard of.
• 15A: Artist Bonheur (ROSA). Ringing vague, vague bells.
• 17A: Harpsichordist Kipnis (IGOR). Or maybe those aren't bells, maybe it's a harpsichord. (Seriously? Harpsichordist?)
• 44A: "Samson Agonistes" dramatist (MILTON). Obviously, I've heard of Milton, but the work title didn't do anything for me.
• 49A: Actress Van Devere (TRISH). No bells (or harpsichords) none.
• 51A: Beaumont, Texas, university (LAMAR). Again, back in the cobwebs somewhere.
• 44D: "Animal magnetism" coiner (MESMER). Never heard of him, but now that I've read a little about him that seems awfully weird.
• 22A: Servers with wheels (TEA WAGONS). Of course, I wanted this to be CARHOPS.
• 41A: Hands and feet (MEASURES). Great clue. Reminded me of The Beekeeper's Apprentice (great book!), which I just finished. The characters in that book often talk about weight in terms of
• 46A: White Sands and others (TEST SITES). I first entered MONUMENTS, which maybe doesn't make any sense to a lot of you — I used to live very near White Sands and was surprised to learn that it
is, in fact, a "monument." Obviously not the kind of monument that word generally evokes for me. Or maybe you all knew that already.
• 59A: Where to find waiters (TRAIN DEPOT). Another great clue.
• 2D: Subject of Joshua Kendall's "The Man Who Made Lists" (ROGET). I haven't heard of this book (is it a book? ... yep) but with a couple crosses in place, the answer became clear.
• 11D: Judgment for insufficient evidence (NONSUIT). I thought this was going to be something in Latin.
• 40D: Elvis sighting, e.g. (FACTOID). I recently read something about how the word FACTOID doesn't mean what people usually think it means. That is, it means (basically) "unverified fact" and not
"little fact." (I like what this site has to say about the confusion.)
• 48D: Man of letters? (SAJAK). Did anyone else try SUPER here? Whenever I see a question-mark clue with the word "letters" in it, I assume the answer is going to be something about renting
• 53D: Houston in NYC, et al. (STS.). Another great clue. And one of my favorite streets in New York. Not that I've ever spent any time there, but I love how it's pronounced.
Follow PuzzleGirl
on Twitter.]
Everything Else — 1A: Not clear-cut (GRAY); 16A: Sight from Sydney Harbour (OPERA HOUSE); 19A: Cares for (TENDS); 21A: Beginnings (SEEDS); 25A: Co. whose largest hub is at O'Hare (UAL); 28A: Shooting
sound (REPORT); 29A: Items in a nautical table (TIDES); 31A: Pub employees (BARMAIDS); 34A: Show-off (HOT DOG); 35A: Land in un lac (ILE); 36A: Lo-__ graphics (RES); 37A: Vigor (PEP); 38A: Suffix
with string (-ENT); 39A: Took off (DOFFED); 43A: Wind threat (SHEAR); 45A: Indirect route (ARC); 60A: Replacement for those left out (ET AL.); 61A: Weathers the struggle (SOLDIERS ON); 62A: Rink fake
(DEKE); 1D: Abrasive bits (GRIT); 3D: In unison (AS ONE); 4D: Spar part (YARDARM); 5D: Stock page name (DOW); 6D: Bee: Pref. (API-); 8D: Hot-blooded (ARDENT); 9D: "Gremlins" actress (CATES); 10D:
Former Israeli prime minister Olmert (EHUD); 12D: Napa vessel (TUN); 13D: Capt.'s heading (ESE); 14D: Family mem. (REL.); 20D: Oath taker (SWEARER); 23D: On foot, in France (À PIED); 24D: Jupiter and
Mars (GODS); 26D: Scary snake (ADDER); 27D: Freetown currency (LEONE); 29D: Nursery purchase (TOPSOIL); 30D: York and Snorkel: Abbr. (SGTS.); 31D: Orders (BIDS); 32D: Welcoming word (ALOHA); 33D:
Direct (REFER); 34D: Qualifying races (HEATS); 37D: Bombard (PELT); 41D: Accidents (MISHAPS); 42D: Not tractable (UNTAMED); 46D: Symbol of equivalence, in math (TILDE); 47D: Fake feelings (EMOTE);
50D: Noodle __: old product name (RONI); 52D: Part (ROLE); 54D: Zagreb's land, to the IOC (CRO); 55D: Holbrook of "Evening Shade" (HAL); 56D: Eeyore pal (ROO); 57D: K2, for one: Abbr. (MTN.).
21 comments:
OMG, it's noon and I'm the first to post???
I guess I wasn't the only one who struggled with this one. Actually it was painful! I had to declare a "DNF", which for me is hard to do because I usually like to SOLDIER ON. That SW corner was a
Since I did not finish 100% correct, I feel like I'm unworthy to critique, so therefore I won't say anything bad.
Thank God for Puzzlegirl and her terrific writeup or I'd consider this morning a total waste of my time. Thank you @PG, you salvaged my day.
Grid's actually not that interesting, and puzzle tries to make up for it with nutso cluing on a lot of proper nouns. Cheap.
But it was definitely tougher than normal for the LAT, so that's a plus.
Easily 20 years ago, one of the news weeklys, Time if I recall, had a piece on a phenomenon among undergraduates to boil a topic down to one concept and concentrate on that. I don't recall the
article well enough to say if the article was derogatory, implying that the writer took the word in its rigorous sense as in "not really a fact", or as a diminutive, meaning that it was a
distillation of the course material. Anyway, I'm glad to be set straight on this (I think).
Geez, I knew it was Saturday from the number of Proper Names I didn't know.
The same ones listed by PuzzleGirl.
I'm starting to think I'm getting dumber or maybe just need stronger coffee on Saturday.
First fills were that SCHOOL MARM and TRAIN DEPOT.
Then up top I got the OPERA HOURSE & WIND TUNNEL.
GRAY area gave me YARDARM.
AS ONE, yielded ROSA & IGOR.
Then there were the write-overs or wouldn't-fits.
Like dismissal for NONSUIT.
Carhops for TEA WAGONS.
Some great clues, like Hands and feet for MEASURES.
Some anwsers a bit obscure:
"Gremlins" actress, CATES. The only actor I remember from this 1984 movie was Hoyt Axton and the cute little criters.
"Animal magnestism" coiner, MESMER. Who? I'll look him up. Oh, that German Physician who died in 1815.
All-in-all, a total slog.
c'mon, MESMER is way more famous than those other folks. his name was even made into a verb.
i thought this wasn't any harder than a usual saturday, but i was pretty unimpressed with the fill. as PG astutely noted, there's a lot of 3-letter answers in the grid, of which few are words:
ARC, PEP, ROO, and TUN. the others: DOW, API-, ESE, REL, UAL, ILE, RES, ENT, STS, CRO-, HAL, MTN. the longer stuff was better, but a little blah. i liked SOLDIERS ON and FACTOID.
Pretty tough LAT puzzle, I think, not sure since I haven't had time to do a lot of puzzles the last couple of months. Other things take up so much of my day! I'm sure I'll pay for it at the
Lollapuzzoola... Especially the last two days the NYT and LAT puzzles have had an unusually high number of words or names I had never heard of.
Phoebe Cates is a nice actress, I think married to Kevin Klein. I read about Mr. Mesmer just recently, and his name the root of "mesmerize", but of course I forgot.....
My feelings about this puzzle? Well, yuck pretty much sums it up. It was a big DNF for me. There just didn't seem to be any wit or sparkle going on except for PG's excellent write up. MESMER did
not mesmerize me, nor did any of those other names - CATES, ROSA, IGOR, LAMAR. ROGET, MILTON and HAL I'm ok with, but the clues? Lots of crummy abbreviations didn't help either. Oh well - it's
off to the gym for me.
DNF for me. I agree with Rex: nutso clueing.
My first fill was VAT, which of course turned out to be TUN. But while this one looked daunting at first, it was fairly easy after I tackled the many three letter fills. UAL and ILE started
things rolling. I found the SE corner hardest - but not really.
I got it all except for the southeast corner- "Deke"? "Mojo"? Yike.
This was hard for me, but pleasantly so, despite a DNF due to one stinking "U" (NONSUIT x UAL). I'd guessed the latter had to be _AL, but that still left most U.S. carriers as possibilities --
especially as I too was expecting Latin for the down clue.
NE came easily -- OPERA HOUSE and WIND TUNNEL were my first fills.
I didn't get those NW names either, but Googled ROGET, which (a) makes perfect sense, and (b) opened up the sector -- _G_R as a first name was pretty obvious, even if I didn't get the reference.
The mini-theme of WIND TUNNEL and SHEAR had me looking for an aeronautical answer to "took off". The other, military mini-theme (SGTS, SOLDIERS ON, REPORT) was merely amusing.
MESMER was a gimme; that, plus STS, CRO, and HAL opened up the SW.
I'd heard of "Samson Agonistes", but not (or I forgot) who wrote it, but crosses gave me MILTON anyway.
Noodle RONI? Never heard of it. Guess it's a Rice-a-Roni spinoff? Yup. Horrid, klunky name though.
Last to fall (except for that "U"), was the SE. I'd never heard of LAMAR Univ., but it was at least somewhat plausible from crosses. I Googled it too, to check my guess.
I should have got MOJO from the MO__ ... but I didn't; it had to wait for the "J". (L.A. Times puzzle -> MOJO -> Mr. Mojo risin' -> "L.A. Woman" ... nice!)
Here's another +1 for "Hands and feet" -> MEASURES. Confused me up big-time, but it's good anyway.
I forgot to thank you for the FACTOID link.
I was thinking along the lines of the CNN trivia triffles.
Therefore 40d, Elvis sighting, had me thinking along the line of "urban legends."
Just because Franz MESMER became the source of the word mesmerize doesn't mean I would automatically associate him as the coiner of "Animal magnetism."
I think this was a bit obscure.
Same with the MILTON work sited.
The former wife of George C. Scott, TRISH.
The Harpsichordist, IGOR. All crosses and I said to myself "Looks good to me" and "Who cares?" ...
"WTF is a harpsicord? Something like a piano?"
Do all Captains head ESE (East SouthEast)?
Are all Indirect routes an ARC?
Finally, aren't COBRA's scarier than ADDER's ...
What does an adder look like?
Inquiring minds want to know.
Had to resort to Google, yig. It was hard--got up early to get in an 18 miler, so that's my excuse. I only did the puzzle after the run.
Oh yeah. I forgot TRISH. Never heard of her. K2=MTN? DEKE?
@Tinbeni I concur. Your points are precisely many of the answers that frustrated me.
However, my problems were with some of the smaller fill words. Although I got the 8-10 letter answers using a new ability I suddenly have of seeing the whole thing, I couldn't get some words that
don't yield to this, perhaps due to "nutso" clues.
I don't agree that ENT is a true suffix for STRING, as it changes the root meaning, whereas IER could be.
The longest words I didn't get were TOPSOIL and NONSUIT. I had babyOIL and NOcaUse. Never called a show off a HOTDOG.
As for names, I didn't know the harpsichordist and was amazed how many KIPNIS people there are.
Did not know LAMAR U. Didn't know
TRISH Cates, but liked the movie which to me was an allegory on babies vs. adolescents.
Lived in NYC for a while and knew HOUSe TON St. is so pronounced. Warning: when in Midtown don't look up, or you're a rube from Jardooville. For a time, it seemed, you'd better not wear anything
but black for same reason.
Naturally, Didn't know DEKE (sports). Hate, questions like IOC abbreviations.
Too me, MESMER and MILTON are mainstream, but I've been a SCHOOLMARM.
K2 is the name of the 2nd highest mountain. Has a real name, Godwin-Austin. I think we used to hear a lot more about it, before it was established that Everest was tallest.
Hmm, after reading the comments, I had to go back to make sure I got the answers entered correctly. I didn't have too much trouble with this puzzle, for some reason.
I started with DEKE (easy one for me, big hockey fan) and worked my way upwards. I stuttered on the NW corner but got GRIT quickly and the answers opened up.
Fun puzzle.
It was established that Everest was tallest in 1852. How old are you?
OK - now that I actually have time to kvetch, I want to say that at 58 I've never heard of K2. Furthermore, I don't get mountain climbing. I live a stone's away from a MTN with a 10,000 ft.
summit. I can see it from my bedroom window. People are constantly falling off, getting lost, scorching, freezing to death or being smothered by avalanches. What's the draw? Now as far as DEKE
goes - the only DEKE I've ever heard of is the frat DKE known for hunky football players and rowdy parties. Maybe TMI?
Stone's throw
CCL:I'm with you. Or if you want to put yourself in harms way like that, don't marry and have kids. I have read too many sad stories about that. In addition, the people in charge of trying to
save these fools also put their lifes on the line!
I'm feeling pretty good about finishing with no errors and no Google. I didn't find this one very difficult. I sat for half the time on one letter TENSORS/REPORT where I had TENDON and I knew
DEEDS wasn't correct. Oh well that's what happenes after doing a xword puzzle at 1 AM.
I didn't know quite a bit but the crosses helped me out everytime. An odd shape for the puzzle, this one felt similar in the cluing to a Newsday stumper. | {"url":"https://latcrossword.blogspot.com/2010/07/saturday-july-31-2010-timothy-l-meaker.html?showComment=1280603951629","timestamp":"2024-11-11T07:29:38Z","content_type":"application/xhtml+xml","content_length":"118561","record_id":"<urn:uuid:8107eb1a-df28-4fd3-873f-90891eba9e23>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00207.warc.gz"} |
Times Tables Multiplication Matching Card Game Flashcards - Kidpid
Times Tables Multiplication Matching Card Game Flashcards
Multiplication is one of the four foundational operations in mathematics. Students are taught multiplication from a very young age since it is essential for them to excel in their academics. It can
be understood as the repeated addition. A multiplicand is a quantity that is to be multiplied by another number. The other number with which a multiplicand is to be multiplied with is called the
multiplier. The result of the multiplication is known as the product. Hence, multiplication can be defined as one of the four elementary mathematical operations of arithmetic that is performed to
find the product of a multiplicand and a multiplier.
Multiplication is made easier by learning the multiplication tables. These are a list of multiples of a number and these tables are often taught to students before they start with multiplication
exercises. It can be extremely helpful while doing basic calculations hence it is efficient to teach these to students at a young age as it is when they are young that they retain most of what they
are taught easily.
Multiplication Time Table Flashcards
This week we bring to you these multiplication table matching flashcards to test your child’s knowledge and memory. These can also be used to help them memorize the multiplication tables of numbers 1
to 12. These flashcards are different from those multiplication table sheets that you find online. These are curated with special attention and focus to ensure that children pay attention to the task
at hand which is simple in nature. You’ll need to download and cut the flashcards into individual pieces. Students would then take turns matching the correct answers using the flashcards. You can
also ask them to write these in their notebooks or to paste the flashcards in the correct order in their scrapbooks.
Knowing multiplication table by heart helps one solve mathematical problems easily in no time. Since it is important for kids to learn these, it is necessary that they are taught in a proper manner.
Often, kids are told to memorize the tables without being told what it means. Subsequently, many children find it difficult to recall the tables in times when they are asked. Students should be
educated properly and must be told how 2 times 5 makes ten and so on. Explaining such a thing takes no time but it goes a long way in helping students.
What are you waiting for then? Download the flashcards and get started now!
You must be logged in to post a comment.
Algebra Worksheets For Grade 5
Algebra introduces students to mathematical concepts using variables, constants, and equations to solve problems. In Grade 5, students explore foundational algebraic ideas such as writing…
Single-digit Addition Math Worksheets & Free Printables
We have bought single-digit addition math foundational worksheets that will help your kid in building and brushing their counting skills. In this busy world, it…
Alphabet Match Game for Kids
The Alphabet Match Game is a fun activity designed to help kids learn letters by matching them with pictures. It boosts letter recognition and vocabulary…
Single Digit Addition Workbook
Mathematics is considered one of the toughest subjects as it requires a lot of practice. Check out these amazing and creative picture books that will…
GK Quiz for Class 3
Q: A polygon with 5 sides is called a _________. (A) Nonagon (B) Octagon (C) Hexagon (D) Pentagon Q: What is the young one… | {"url":"https://www.kidpid.com/times-tables-multiplication-matching-card-game-flashcards/","timestamp":"2024-11-04T15:00:13Z","content_type":"text/html","content_length":"167316","record_id":"<urn:uuid:9885fb72-88e5-40d0-a44e-86617fb253fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00184.warc.gz"} |
on a straight line
12.2 Angles on a straight line
Let us look at angles formed on one side of a straight line.
In this diagram, line segment \(AB\) meets line segment \(DC\). The angle at the vertex, \(C\), where they meet, is now split into two angles: \(\hat{C_1}\) and \(\hat{C_2}\).
\(\hat{C_1}\) is the name for the angle at vertex \(C\) labelled "1" (or \(A\hat{C}D\)).
The sum of the angles formed on a straight line
The sum of the angles that are formed on a straight line is always \(180^{\circ}\).
We can shorten this property as: \(\angle\)s on a straight line.
• Two angles that add up to \(180^{\circ}\) are called supplementary angles.
• Angles that share a vertex and a common side are said to be adjacent angles. \(\hat{C_1}\) and \(\hat{C_2}\) are supplementary angles.
• Hence, \(\hat{C_1}\) and \(\hat{C_2}\) are called adjacent supplementary angles.
supplementary angles
two angles that add up to \(180^{\circ}\)
adjacent angles
angles that share a vertex and a common side
You can have more than one line meeting at the same point on a straight line. Here are a few examples of angles on a straight line.
The sum of the angles on perpendicular lines
When two lines are perpendicular, the adjacent supplementary angles are both equal to \(90^{\circ}\).
In the diagram, \(\hat{M_1}=\hat{M_2} =90^{\circ}\).
A right angle is shown by forming a square at one of the right angles, like this: ⦜.
Finding unknown angles on straight lines
Worked example 12.1: Calculating unknown angles on a straight line
Calculate the size of \(x\).
In the diagram, we have two angles that are on the same side of the straight line. The first angle is \(100^{\circ}\) and the second angle is unknown (\(x\)). We need to calculate the size of \(x\).
We know that the two angles have a sum of \(180^{\circ}\), so we can say that:
\(100^{\circ}+x=180^{\circ}\) (\(\angle\)s on a straight line)
Now we can solve this equation.
\[ x&=180^{\circ}-100^{\circ} \\ x&=80^{\circ} \]
Worked example 12.2: Calculating unknown angles on a straight line
Calculate the size of \(x\).
Notice that there are three angles on the same side of the straight line. We have \(x\), an angle of \(29^{\circ}\) and an angle of \(90^{\circ}\). (Remember that the ⦜ symbol on the diagram
indicates a \(90^{\circ}\) angle.) These three angles have a sum of \(180^{\circ}\), so we can say that:
\(x+29^{\circ}+90^{\circ}=180^{\circ}\) (\(\angle\)s on a straight line)
Now we can solve this equation.
\[ x+29^{\circ}+90^{\circ}&=180^{\circ} \\ x+119^{\circ}&=180^{\circ} \\ x&=180^{\circ}-119^{\circ} \\ x&=61^{\circ} \]
There is a simpler way to solve for \(x\). It is given that we have a perpendicular line. Adjacent angles on a perpendicular line are both equal to \(90^{\circ}\). So, we have a different equation to
\[ x+29^{\circ}&=90^{\circ} \\ x&=90^{\circ}-29^{\circ} \\ x&=61^{\circ} \]
Worked example 12.3: Calculating unknown angles on a straight line
Calculate the size of \(y\).
Notice that we have three angles on the same side of the straight line. We have \(2y\), an angle of \(48^{\circ}\) and an angle of \(52^{\circ}\). These three angles have a sum of \(180^{\circ}\), so
we can say that:
\(2y+48^{\circ}+52^{\circ}=180^{\circ}\) (\(\angle\)s on a straight line)
Now we can solve this equation.
\[ 2y+48^{\circ}+52^{\circ}&=180^{\circ} \\ 2y+100^{\circ}&=180^{\circ} \\ 2y&=180^{\circ}-100^{\circ} \\ 2y&=80^{\circ} \\ y&=40^{\circ} \]
Exercise 12.1
Calculate the size of \(a\).
\[ a+63^{\circ}&=180^{\circ} &(\angle\text{s on a straight line}) \\ x&=180^{\circ}-63^{\circ} \\ x&=117^{\circ} \]
Calculate the size of:
1. \(x\)
2. \(\hat{ECB}\)
1. \[ x+3x+2x&=180^{\circ} &(\angle\text{s on a straight line}) \\ 6x&=180^{\circ} \\ x&=30^{\circ} \]
2. \[ \hat{ECB}&=2x \\ &=2(30^{\circ})\\ &=60^{\circ} \]
Calculate the size of:
1. \(x\)
2. \(\hat{GEH}\)
1. \[ (x+30^{\circ})+(x+40^{\circ}) +(2x+10^{\circ}) &=180^{\circ} &(\angle\text{s on a straight line}) \\ 4x+80^{\circ}&=180^{\circ} \\ 4x&=100^{\circ} \\ x&=25^{\circ} \]
2. \[ \hat{GEH}&=x+40^{\circ} \\ &=25^{\circ}+40^{\circ} \\ &=65^{\circ} \]
Hint: Remember that the matching curved lines “))” indicate that the angles are equal.
Calculate the size of:
1. \(k\)
2. \(\hat{TYP}\)
1. \[ (2k)+(k+65^{\circ}) +(2k) &=180^{\circ} &&(\angle\text{s on a straight line}) \\ 5k+65^{\circ}&=180^{\circ} \\ 5k &=115^{\circ} \\ &=23^{\circ} \]
2. \[ \hat{TYP}&=k+65^{\circ} \\ &=23^{\circ}+65^{\circ} \\ &=88^{\circ} \] | {"url":"https://www.siyavula.com/read/za/mathematics/grade-8/geometry-of-straight-lines/12-geometry-of-straight-lines-02","timestamp":"2024-11-08T13:48:52Z","content_type":"text/html","content_length":"103854","record_id":"<urn:uuid:00117b0a-2ca7-44f8-adb8-c6f8a7d7ad6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00848.warc.gz"} |
Model and Solve Statements
how to specify the variables to be maximized in the GAMS model and solve it over a defined sets?
For example, I would like to code the CVAR equation as the following:
1- Does GAMS coding requires to include the highlighted variables in the model/solver command to maximize the highlighted variables over the specified sets?
2- Or no need to do that since that would be implicitly specified by the objective and model constraints?
My code is:
ObjEq9… obj=e= Zeta - (1/(1-alpha))*sum(w,PiW(w)*Uw(w));
MODEL Aggregator /ALL/;
SOLVE Aggregator USING NLP MAXIMIZING obj;
If the non-used variables and sets are in your constraints, you are on the right track. | {"url":"https://forum.gams.com/t/model-and-solve-statements/2377","timestamp":"2024-11-14T05:11:02Z","content_type":"text/html","content_length":"16114","record_id":"<urn:uuid:058f97bf-eae0-421a-9fd7-1448935aee2c>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00846.warc.gz"} |
Does the Power Your Car Audio Amplifier Produces Really Matter?
Did you know that a difference of 20 watts of power between one car audio amplifier and another might be completely inaudible? That same 20-watt difference might mean having to keep your windows
rolled up on the highway to hear your music. Let’s look at the physics of reproducing sound with moving-coil loudspeakers and why choosing an amplifier with a few watts more than another may be
significant or irrelevant.
Car Audio Speakers and Amplifier Power
If you look at a typical higher-end 6.5-inch coaxial car audio speaker, you’ll find that it has an efficiency rating of 86 dB when driven with 1 watt of power and measured at a distance of 1 meter.
If the reference is 2.83 V, then that’s 2 watts into a 4-ohm driver, and the efficiency number will be 3 dB higher at 89 dB. If it’s a 2-ohm driver with a 2.83V spec, then that’s 4 watts, and they
will add another 3 dB. Aren’t specification games fun?
The first takeaway from this is that it takes a doubling of your amplifier’s power to increase a speaker’s output by 3 dB. At the same time, halving the power reduces the output by the same 3 dB. If
you only need 80 dB of output, then our 86 dB efficient speaker will only need 0.25 watt of power to reach that level.
Power required for specific output levels relative to an 86 dB SPL 1W/1M speaker.
Scenario 1 – Deck Power and Door Speakers
Suppose you have a modest audio system that is made up of a typical aftermarket radio and a pair of equally typical door speakers. Most radios can produce about 20 watts of output per channel, and
we’ll use our 89 dB example speakers (though now we have two of them, so the pair will produce 92 dB when each is powered with 1 watt). With 20 watts of power on tap, the system should produce just a
smidge over 104 dB of output. Of course, this assumes that the speakers increase the output linearly for every doubling of power. At 20 watts, especially if they are reproducing bass frequencies, you
are likely near their upper limit.
What if we switch the radio to a high-power unit like the Sony XAV-AX7000 that can produce 45 watts of power per channel? Now, assuming the speakers can handle the extra power, the system will
produce about 107.5 dB of output. That extra 25 watts increases how loudly our music will play by 3.5 decibels. It doesn’t sound like a lot, but it would be audible.
Sony’s High-Power head units use an amplifier to produce an honest 45 watts of power per channel.
Scenario 2 – Small Subwoofer Amplifier Versus Large
In our second example, let’s look at the higher power levels involved with driving a subwoofer. Say we have a Rockford Fosgate amplifier capable of producing 1,000 watts of power. Considering the
transfer function of the typical vehicle interior, a pair of 10-inch subwoofers in a vented enclosure might have an efficiency of about 101 dB at 40 Hz when each is driven with 1 watt. When we
increase the power to the subs to 500 watts each, the system should produce 128 decibels of output. That’s pretty darned loud! Keeping in mind that we need to double or half the power level to
produce a change of 3 dB, what happens when we pick an amplifier that can produce 1,050 watts or that same increase of 25 watts to each subwoofer? Well, the system will play 0.21 decibel louder.
While you might be able to measure that with a Term-Lab SPL system, it’s unlikely you can hear that small of an increase.
The ARC Audio X2 1100.1 subwoofer amplifier is conservatively rated to produce 1,100 watts of power in a 1-ohm load.
The T1000-1bdCP monoblock amp from Rockford Fosgate can deliver 1,000 watts of power into a 1- or 2-ohm load.
For those looking for a subwoofer amplifier that can deliver 1,000 watts of power into a 1- or 2-ohm load, check out the Hertz ML Power 1.
The M ONE X from Helix has a power rating of 1,030 watts into a 1-ohm load.
The Voce Uno from Audison is a Class-AB amplifier that can deliver up to 1,700 watts of power into a 1-ohm load.
Looking at Power Specifications Using Decibel-Watts
While most of us are used to seeing the decibel unit used in the context of measuring volume levels, it can be applied to a variety of electrical applications. If you take a close look at the power
measurements in any of the BestCarAudio.com Test Drive Reviews, you’ll see we list watts and a number called dBW, or decibel-watts. The unit dBW refers to decibels referenced to one watt of power. As
such, if the amp produced 1 watt of power, it would be rated at 0 dBW, or no increase or decrease relative to 1 watt. If it made 8 watts, then it would be rated at 9 dBW. One hundred watts is 20 dBW,
and 1,000 watts is 30 dBW.
Suppose your speakers or subwoofers can handle the power in terms of thermal capacity and cone excursion capability. In that case, you can add the dBW number to the 1-watt efficiency number of the
speaker to estimate how loudly it will play. Of course, no midrange speaker is going to able to deal with 200 watts of power, and a subwoofer isn’t going to increase its output linearly when driven
with 10,000 watts.
Backtracking for a moment, to provide some clarity, when we use an RTA or SPL meter to measure a sound source’s volume level, we measure dB SPL. Similar to the way that the decibel watt (dBW)
references 1 watt of power, the dB SPL unit references 20 micropascals or, 0 dB. A sound level of 20 micropascals is considered the lowest hearing threshold of a young, healthy ear. Of course, if we
can measure this pressure (20 micropascals), then 0 dB isn’t absolute silence or a vacuum. It’s just a reference level. In the same way, 0.25 watt is -6 dBW; it’s possible to have negative SPL
numbers or a room that’s quieter than 0 dB. Microsoft built an anechoic test chamber that has a noise level of -20.35 dB SPL, or 20.35 decibels quieter than 0 dB. It’s said that the sound that air
particles create from bumping into one another in a still room is -23 dB SPL.
The anechoic chamber in Microsoft’s Building B87 is the quietest room on earth, with a background noise level of -20.23 dB SPL.
When Watts Matter and When They Don’t
Let’s take all this information and put it into use. If you’re shopping for a new radio, then one that produces 45 watts is going to provide an audible improvement over one that produces only 20
watts. If you are shopping for an amplifier for your speakers, a 75-watt and a 100-watt amp will only increase output by just over 1 dB. If one subwoofer amplifier produces 25 watts more than one
that makes 750 watts, the difference is only 0.14 dB. The best way to think of amplifiers is by ranking them as small, medium and large. A small stand-alone speaker amplifier would be 50 watts per
channel. A large one would be 100-125 watts. A small sub amp would be 250-300 watts, a medium would be 600-800 watts, and a large would be 1,200-2,500 watts. Worrying about whether the amp makes
1,150 or 1,175 watts (a difference of 0.09 dBW) is a waste of time.
When it’s time to upgrade your car audio system with a new high-power radio or an amplifier, drop by your local specialty mobile enhancement retailer and ask to audition several options on the same
set of speakers or subwoofers and the same source unit. This is the best way to determine which solution offers the least distortion and most accurate sound. Just a reminder, don’t get hung up on a
few watts – that’s truly the least of your worries.
This article is written and produced by the team at www.BestCarAudio.com. Reproduction or use of any kind is prohibited without the express written permission of 1sixty8 media.
Leave a Reply Cancel reply | {"url":"https://www.shopstreetsmart.com/does-the-power-your-car-audio-amplifier-produces-really-matter/","timestamp":"2024-11-11T08:37:38Z","content_type":"text/html","content_length":"95233","record_id":"<urn:uuid:4e19bf95-bfbe-47cb-9dce-e763a82dbfb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00277.warc.gz"} |
The Nutils Book
Nutils is a Free and Open Source Python programming library for Finite Element Method computations, developed by Evalf and distributed under the permissive MIT license. Key features are a readable,
math centric syntax, an object oriented design, strict separation of topology and geometry, and high level function manipulations with support for automatic differentiation.
Nutils provides the tools required to construct a typical simulation workflow in just a few lines of Python code, while at the same time leaving full flexibility to build novel workflows or interact
with third party tools. With native support for Isogeometric Analysis (IGA), the Finite Cell method (FCM), multi-physics, mixed methods, and hierarchical refinement, Nutils forms an excellent
platform for numerical science. Efficient under-the-hood vectorization and built-in parallellisation provide for an effortless transition from academic research projects to full scale, real world
Since Nutils is a library for the development of numerical simulations, this book assumes that the reader is familiar with differential calculus, Galerkin methods, and the Finite Element Method. If
this is not the case, chances are that Nutils is not the tool they are looking for.
First time users who are eager to get their feet wet will want to begin with the getting started guide and build a functioning Poisson solver in three easy steps, no questions asked. Following this,
beginners are strongly advised to follow the hands on tutorial to gain an an in-depth understanding of Nutils concepts and get familiar with the syntax.
Novices and advanced users alike may find interest in the installation guide, which ranges from basic installation instructions to tips and tricks for optimizing the installation, instructions for
running a Docker style container, and suggestions for computing remotely.
The release history provides an overview of changes between releases. This is the place to monitor for long term users who want to keep up to date with the latest and greatest new features. The
release pages also provide links to the relevant API reference where all Nutils functions are documented.
Anybody looking to build their own Nutils simulations are encouraged to browse through the example projects. Most simulations will have components in common with existing scripts, so a mix-and-match
approach is a good way to start building your own. In case questions do remain, the support page lists ways of getting in touch with developers.
Finally, the science section provides an overview of publications that use Nutils in their research. Reproducing results from these articles is a great starting point for follow-up research, as well
as good scientific practice in its own right. Help others do the same by citing Nutils in your own publications!
The following is a quick start guide to running your first Nutils simulation in three simple steps. Afterward, be sure to read the installation guide for extra installation instructions, study the
tutorial to familiarize yourself with Nutils' concepts and syntax, and explore the examples for inspiration.
With Python version 3.7 or newer installed, Nutils and Matplotlib can be installed via the Python Package Index using the pip package installer. In a terminal window:
python -m pip install --user nutils matplotlib
Note that Nutils depends on Numpy, Treelog and Stringly, which means that these modules are pulled in automatically if they were not installed prior. Though most Nutils applications will require
Matplotlib for visualization, it is not a dependency for Nutils itself, and is therefore installed explicitly.
Open a text editor and create a file poisson.py with the following contents:
from nutils import mesh, function, solver, export, cli
def main(nelems: int = 10, etype: str = 'square'):
domain, x = mesh.unitsquare(nelems, etype)
u = function.dotarg('udofs', domain.basis('std', degree=1))
g = u.grad(x)
J = function.J(x)
cons = solver.optimize('udofs',
domain.boundary.integral(u**2 * J, degree=2), droptol=1e-12)
udofs = solver.optimize('udofs',
domain.integral((g @ g / 2 - u) * J, degree=1), constrain=cons)
bezier = domain.sample('bezier', 3)
x, u = bezier.eval([x, u], udofs=udofs)
export.triplot('u.png', x, u, tri=bezier.tri, hull=bezier.hull)
Note that while we could make the script even shorter by avoiding the main function and cli.run, the above structure is preferred as it automatically sets up a logging environment, activates a matrix
backend and handles command line parsing.
Back in the terminal, the simulation can now be started by running:
python poisson.py
This should produce the following output:
nutils v7.0
optimize > constrained 40/121 dofs
optimize > optimum value 0.00e+00
optimize > solve > solving 81 dof system to machine precision using arnoldi solver
optimize > solve > solver returned with residual 6e-17
optimize > optimum value -1.75e-02
log written to file:///home/myusername/public_html/poisson.py/log.html
If the terminal is reasonably modern (Windows users may want to install the new Windows Terminal) then the messages are coloured for extra clarity. The last line of the log shows the location of the
simultaneously generated html file that holds the same log, as well as a link to the generated image.
To run the same simulation on a mesh that is finer and made up or triangles instead of squares, arguments can be provided on the command line:
python poisson.py nelems=20 etype=triangle
Nutils requires a working installation of Python 3.7 or higher. Many different installers exist and there are no known issues with any of them. When in doubt about which to use, a safe option is to
go with the official installer. From there on Nutils can be installed following the steps below.
Depending on your system the Python executable may be installed as either python or python3, or both, not to mention alternative implementations such as pypy or pyston. In the following instructions,
python is to be replaced with the relevant executable name.
Nutils is installed via Python's Pip package installer, which most Python distributions install by default. In the following instructions we add the flag --user for a local installation that does not
require system privileges, which is recommended but not required.
The following command installs the stable version of Nutils from the package archive, along with its dependencies Numpy, Treelog and Stringly:
python -m pip install --user nutils
To install the most recent development version we use Github's ability to generate zip balls:
python -m pip install --user --force-reinstall \
Alternatively, if the Git version control system is installed, we can use pip's ability to interact with it directly to install the same version as follows:
python -m pip install --user --force-reinstall \
This notation has the advantage that even a specific commit (rather than a branch) can be installed directly by specifying it after the @.
Finally, if we do desire a checkout of Nutils' source code, for instance to make changes to it, then we can instruct pip to install directly from the location on disk:
git clone https://github.com/evalf/nutils.git
cd nutils
python -m pip install --user .
In this scenario it is possible to add the --editable flag to install Nutils by reference, rather than by making a copy, which is useful in situations of active development. Note, however, that pip
requires manual intervention to revert back to a subsequent installation by copy.
Nutils currently supports three matrix backends: Numpy, Scipy and MKL. Since Numpy is a primary dependency this backend is always available. Unfortunately it is also the least performant of the three
because of its inability to exploit sparsity. It is therefore strongly recommended to install one of the other two backends via the instructions below.
By default, Nutils automatically activates the best available matrix backend: MKL, Scipy or Numpy, in that order. A consequence of this is that a faulty installation may easily go unnoticed as Nutils
will silently fall back on a lesser backend. As such, to make sure that the installation was successful it is recommended to force the backend at least once by setting the NUTILS_MATRIX environment
variable. In Linux:
NUTILS_MATRIX=MKL python myscript.py
The Scipy matrix backend becomes available when Scipy is installed, either using the platform's package manager or via pip:
python -m pip install --user scipy
In addition to a sparse direct solver, the Scipy backend provides many iterative solvers such as CG, CGS and GMRES, as well as preconditioners. The direct solver can optionally be made more
performant by additionally installing the scikit-umfpack module.
Intel's oneAPI Math Kernel Library provides the Pardiso sparse direct solver, which is easily the most powerful direct solver that is currently supported. It is installed via the official
instructions, or, if applicable, by any of the steps below.
On a Debian based Linux system (such as Ubuntu) the libraries can be directly installed via the package manager:
sudo apt install libmkl-rt
For Fedora or Centos Linux, Intel maintains its own repository that can be added with the following steps:
sudo dnf config-manager --add-repo https://yum.repos.intel.com/mkl/setup/intel-mkl.repo
sudo rpm --import https://yum.repos.intel.com/intel-gpg-keys/GPG-PUB-KEY-INTEL-SW-PRODUCTS-2019.PUB
sudo dnf install intel-mkl
sudo tee /etc/ld.so.conf.d/mkl.conf << EOF > /dev/null
sudo ldconfig -v
Here we list some modules that are not direct requirements, but that can be used in conjunction with Nutils to make life a little bit better.
BottomBar is a context manager for Python that prints a status line at the bottom of a terminal window. When it is installed, cli.run automatically activates it to display the location of the html
log (rather than only logging it at the beginning and end of the simulation) as well as runtime and memory usage information.
python -m pip install bottombar
While Nutils is not (yet) the fastest tool in its class, with some effort it is possible to achieve sufficient performance to allow simulations of over a million degrees of freedom. The matrix
backend is the most important thing to get right, but there are a few other factors that are worth considering.
On multi-core architectures, the most straightforward acceleration path available is to use parallel assembly, activated using the NUTILS_NPROCS environment variable. Both Linux and OS X both are
supported. Unfortunately, the feature is currently disabled on Windows as it does not support the fork system call that is used by the current implementation.
On Windows, the easiest way to enjoy parallel speedup is to make use of the new Windows Subsystem for Linux (WSL2), which is complete Linux environment running on top of Windows. To install it simply
select one of the many Linux distributions from the Windows store, such as Ubuntu 20.04 LTS or Debian GNU/Linux.
Many Numpy installations default to using the openBLAS library to provide its linear algebra routines, which supports multi-threading using the openMP parallelization standard. While this is useful
in general, it is in fact detrimental in case Nutils is using parallel assembly, in which case the numerical operations are best performed sequentially. This can be achieved by setting the
OMP_NUM_THREADS environment variable.
In Linux this can be done permanently by adding the following line to the shell's configuration file. In Linux this is typically ~/.bashrc:
export OMP_NUM_THREADS=1
The downside to this approach is that multithreading is disabled for all applications that use openBLAS, not just Nutils. Alternatively in Linux the setting can be specified one-off in the form of a
OMP_NUM_THREADS=1 NUTILS_NPROCS=8 python myscript.py
The most commonly used Python interpreter is without doubt the CPython reference implementation, but it is not the only option. Before taking an application in production it may be worth testing if
other implementations have useful performance benefits.
One interpreter of note is Pyston, which brings just-in-time compilation enhancements that in a typical application can yield a 20% speed improvement. After Pyston is installed, Nutils and
dependencies can be installed as before simply replacing python by pyston3. As packages will be installed from source some development libraries may need to be installed, but what is missing can
usually be inferred from the error messages.
As an alternative to installing Nutils, it is possible to download a preinstalled system image with all the above considerations taken care of. Nutils provides OCI compatible containers for all
releases, as well as the current developement version, which can be run using tools such as Docker or Podman. The images are hosted in Github's container repository.
The container images include all the official examples. To run one, add the name of the example and any additional arguments to the command line. For example, you can run example laplace using the
latest version of Nutils with:
docker run --rm -it ghcr.io/evalf/nutils:latest laplace
HTML log files are generated in the /log directory of the container. If you want to store the log files in /path/to/log on the host, add -v /path/to/log:/log to the command line before the name of
the image. Extending the previous example:
docker run --rm -it -v /path/to/log:/log ghcr.io/evalf/nutils:latest laplace
To run a Python script in this container, bind mount the directory containing the script, including all files necessary to run the script, to /app in the container and add the relative path to the
script and any arguments to the command line. For example, you can run /path/to/myscript.py with Docker using:
docker run --rm -it -v /path/to:/app:ro ghcr.io/evalf/nutils:latest myscript.py
Computations beyond a certain size are usually moved to a remote computing facility, typically accessed using tools such as Secure Shell or Mosh, combined with a terminal multiplexer such as GNU
Screen or Tmux. In this scenario it is useful to install a webserver for remote viewing of the html logs.
The standard ~/public_html output directory is configured with the scenario in mind, as the Apache webserver uses this as the default user directory. As this is disabled by default, the module needs
to be enabled by editing the relevant configuration file or, in Debian Linux, by using the a2enmod utility:
sudo a2enmod userdir
Similar behaviour can be achieved with the Nginx by configuring a location pattern in the appropriate server block:
location ~ ^/~(.+?)(/.*)?$ {
alias /home/$1/public_html$2;
Finally, the terminal output can be made to show the http address rather than the local uri by adding the following line to the ~/.nutilsrc configuration file:
outrooturi = 'https://mydomain.tld/~myusername/'
In this tutorial we will explore Nutils' main building blocks by solving a simple 1D Laplace problem. The tutorial assumes knowledge of the Python programming language, as well as familiarity with
the third party modules Numpy and Matplotlib. It also assumes knowledge of advanced calculus, weak formulations, and the Finite Element Method, and makes heavy use of Einstein notation.
The computation that we will work towards amounts to about 20 lines of Nutils code, including visualization. The entire script is presented below, in copy-pasteable form suitable for interactive
exploration using for example ipython. In the sections that follow we will go over these lines ones by one and explain the relevant concepts involved.
from nutils import function, mesh, solver
from nutils.expression_v2 import Namespace
import numpy
from matplotlib import pyplot as plt
topo, geom = mesh.rectilinear([numpy.linspace(0, 1, 5)])
ns = Namespace()
ns.x = geom
ns.define_for('x', gradient='∇', normal='n', jacobians=('dV', 'dS'))
ns.basis = topo.basis('spline', degree=1)
ns.u = function.dotarg('lhs', ns.basis)
sqr = topo.boundary['left'].integral('u^2 dS' @ ns, degree=2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# optimize > constrained 1/5 dofs
# optimize > optimum value 0.00e+00
res = topo.integral('∇_i(basis_n) ∇_i(u) dV' @ ns, degree=0)
res -= topo.boundary['right'].integral('basis_n dS' @ ns, degree=0)
lhs = solver.solve_linear('lhs', residual=res, constrain=cons)
# solve > solving 4 dof system to machine precision using arnoldi solver
# solve > solver returned with residual 9e-16±1e-15
bezier = topo.sample('bezier', 32)
nanjoin = lambda array, tri: numpy.insert(array.take(tri.flat, 0).astype(float),
slice(tri.shape[1], tri.size, tri.shape[1]), numpy.nan, axis=0)
sampled_x = nanjoin(bezier.eval('x_0' @ ns), bezier.tri)
def plot_line(func, **arguments):
plt.plot(sampled_x, nanjoin(bezier.eval(func, **arguments), bezier.tri))
plt.xticks(numpy.linspace(0, 1, 5))
plot_line(ns.u, lhs=lhs)
You are encouraged to execute this code at least once before reading on, as the code snippets that follow may assume certain products to be present in the namespace. In particular the plot_line
function is used heavily in the ensuing sections.
We will introduce fundamental Nutils concepts based on the 1D homogeneous Laplace problem,
\[ u''(x) = 0 \]
with boundary conditions \( u(0) = 0 \) and \( u'(1) = 1 \). Even though the solution is trivially found to be \( u(x) = x \), the example serves to introduce many key concepts in the Nutils
paradigm, concepts that can then be applied to solve a wide class of physics problems.
A key step to solving a problem using the Finite Element Method is to cast it into weak form.
Let \( Ω \) be the unit line \( [0,1] \) with boundaries \( Γ_\text{left} \) and \( Γ_\text{right} \), and let \( H_0(Ω) \) be a suitable function space such that any \( u ∈ H_0(Ω) \) satisfies \( u
= 0 \) in \( Γ_\text{left} \). The Laplace problem is solved uniquely by the element \( u ∈ H_0(Ω) \) for which \( R(v, u) = 0 \) for all test functions \( v ∈ H_0(Ω) \), with \( R \) the bilinear
\[ R(v, u) := ∫_Ω \frac{∂v}{∂x_i} \frac{∂u}{∂x_i} \ dV - ∫_{Γ_\text{right}} v \ dS. \]
The final step before turning to code is to make the problem discrete.
To restrict ourselves to a finite dimensional subspace we adopt a set of Finite Element basis functions \( φ_n ∈ H_0(Ω) \). In this space, the Finite Element solution is established by solving the
linear system of equations \( R_n(\hat{u}) = 0 \), with residual vector \( R_n(\hat{u}) := R(φ_n, \hat{u}) \), and discrete solution
\[ \hat{u}(x) = φ_n(x) \hat{u}_n. \]
Note that discretization inevitably implies approximation, i.e. \( u ≠ \hat{u} \) in general. In this case, however, we choose \( {φ_n} \) to be the space of piecewise linears, which contains the
exact solution. We therefore expect our Finite Element solution to be exact.
Rather than having a single concept of what is typically referred to as the 'mesh', Nutils maintains a strict separation of topology and geometry. The nutils.topology.Topology represents a collection
of elements and inter-element connectivity, along with recipes for creating bases. It has no (public) notion of position. The geometry takes the nutils.topology.Topology and positions it in space.
This separation makes it possible to define multiple geometries belonging to a single nutils.topology.Topology, a feature that is useful for example in certain Lagrangian formulations.
While not having mesh objects, Nutils does have a nutils.mesh module, which hosts functions that return tuples of topology and geometry. Nutils provides two builtin mesh generators:
nutils.mesh.rectilinear, a generator for structured topologies (i.e. tensor products of one or more one-dimensional topologies), and nutils.mesh.unitsquare, a unit square mesh generator with square
or triangular elements or a mixture of both. The latter is mostly useful for testing. In addition to generators, Nutils also provides the nutils.mesh.gmsh importer for gmsh-generated meshes.
The structured mesh generator takes as its first argument a list of element vertices per dimension. A one-dimensional topology with four elements of equal size between 0 and 1 is generated by
mesh.rectilinear([[0, 0.25, 0.5, 0.75, 1.0]])
# (StructuredTopology<4>, Array<1>)
Alternatively we could have used numpy.linspace to generate a sequence of equidistant vertices, and unpack the resulting tuple:
topo, geom = mesh.rectilinear([numpy.linspace(0, 1, 5)])
We will use this topology and geometry throughout the remainder of this tutorial.
Note that the argument is a list of length one: this outer sequence lists the dimensions, the inner the vertices per dimension. To generate a two-dimensional topology, simply add a second list of
vertices to the outer list. For example, an equidistant topology with four by eight elements with a unit square geometry is generated by
mesh.rectilinear([numpy.linspace(0, 1, 5), numpy.linspace(0, 1, 9)])
# (StructuredTopology<4x8>, Array<2>)
Any topology defines a boundary via the nutils.topology.Topology.boundary attribute. Optionally, a topology can offer subtopologies via the getitem operator. The rectilinear mesh generator
automatically defines 'left' and 'right' boundary groups for the first dimension, making the left boundary accessible as:
# StructuredTopology<>
Optionally, a topology can be made periodic in one or more dimensions by passing a list of dimension indices to be periodic via the keyword argument periodic. For example, to make the second
dimension of the above two-dimensional mesh periodic, add periodic=[1]:
mesh.rectilinear([numpy.linspace(0, 1, 5), numpy.linspace(0, 1, 9)], periodic=[1])
# (StructuredTopology<4x8p>, Array<2>)
Note that in this case the boundary topology, though still available, is empty.
In Nutils, a basis is a vector-valued function object that evaluates, in any given point \( ξ \) on the topology, to the full array of basis function values \( φ_0(ξ), φ_1(ξ), \dots, φ_{n-1}(ξ) \).
It must be pointed out that Nutils will in practice operate only on the basis functions that are locally non-zero, a key optimization in Finite Element computations. But as a concept, it helps to
think of a basis as evaluating always to the full array.
Several nutils.topology.Topology objects support creating bases via the Topology.basis() method. A nutils.topology.StructuredTopology, as generated by nutils.mesh.rectilinear, can create a spline
basis with arbitrary degree and arbitrary continuity. The following generates a degree one spline basis on our previously created unit line topology topo:
basis = topo.basis('spline', degree=1)
The five basis functions are
We will use this basis throughout the following sections.
Change the degree argument to 2 for a quadratic spline basis:
plot_line(topo.basis('spline', degree=2))
By default the continuity of the spline functions at element edges is the degree minus one. To change this, pass the desired continuity via keyword argument continuity. For example, a quadratic
spline basis with \( C^0 \) continuity is generated with
plot_line(topo.basis('spline', degree=2, continuity=0))
\( C^0 \) continuous spline bases can also be generated by the 'std' basis:
plot_line(topo.basis('std', degree=2))
The 'std' basis is supported by topologies with square and/or triangular elements without hanging nodes.
Discontinuous basis functions are generated using the 'discont' type, e.g.
plot_line(topo.basis('discont', degree=2))
A function in Nutils is a mapping from a topology onto an n-dimensional array, and comes in the form of a functions: nutils.function.Array object. It is not to be confused with Python's own function
objects, which operate on the space of general Python objects. Two examples of Nutils functions have already made the scene: the geometry geom, as returned by nutils.mesh.rectilinear, and the bases
generated by Topology.basis(). Though seemingly different, these two constructs are members of the same class and in fact fully interoperable.
The nutils.function.Array functions behave very much like numpy.ndarray objects: the functions have a nutils.function.Array.shape, nutils.function.Array.ndim and a nutils.function.Array.dtype:
# (1,)
# (5,)
# 1
# <class 'float'>
The functions support numpy-style indexing. For example, to get the first element of the geometry geom you can write geom[0] and to select the first two basis functions you can write
The usual unary and binary operators are available:
Several trigonometric functions are defined in the nutils.function module. An example with a sine function:
The dot product is available via nutils.function.dot. To contract the basis with an arbitrary coefficient vector:
plot_line(function.dot(basis, [1,2,0,5,4]))
Recalling the definition of the discrete solution, the above is precisely the way to evaluate the resulting function. What remains now is to establish the coefficients for which this function solves
the Laplace problem.
A discrete model is often written in terms of an unknown, or a vector of unknowns. In Nutils this translates to a function argument, nutils.function.Argument. Usually an argument is used in an inner
product with a basis. For this purpose there exists the nutils.function.dotarg function. For example, the discrete solution can be written as
ns.u = function.dotarg('lhs', ns.basis)
with the argument identified by 'lhs' the vector of unknowns \( \hat{u}_n )).
Nutils functions behave entirely like Numpy arrays, and can be manipulated as such, using a combination of operators, object methods, and methods found in the nutils.function module. Though powerful,
the resulting code is often lengthy, littered with colons and brackets, and hard to read. Namespaces provide an alternative, cleaner syntax for a prominent subset of array manipulations.
A nutils.expression_v2.Namespace is a collection of nutils.function.Array functions. An empty nutils.expression_v2.Namespace is created as follows:
ns = Namespace()
New entries are added to a nutils.expression_v2.Namespace by assigning an nutils.function.Array to an attribute. For example, to assign the geometry geom to ns.x, simply type
ns.x = geom
You can now use ns.x where you would use geom. Usually you want to add the gradient, normal and jacobian of this geometry to the namespace as well. This can be done using
nutils.expression_v2.Namespace.define_for naming the geometry (as present in the namespace) and names for the gradient, normal, and the jacobian as keyword arguments:
ns.define_for('x', gradient='∇', normal='n', jacobians=('dV', 'dS'))
Note that any keyword argument is optional.
To assign a linear basis to ns.basis, type
ns.basis = topo.basis('spline', degree=1)
and to assign the discrete solution as the inner product of this basis with argument 'lhs', type
ns.u = function.dotarg('lhs', ns.basis)
You can also assign numbers and numpy.ndarray objects:
ns.a = 1
ns.b = 2
ns.c = numpy.array([1,2])
ns.A = numpy.array([[1,2],[3,4]])
In addition to inserting ready objects, a namespace's real power lies in its ability to be assigned string expressions. These expressions may reference any nutils.function.Array function present in
the nutils.expression_v2.Namespace, and must explicitly name all array dimensions, with the object of both aiding readibility and facilitating high order tensor manipulations. A short explanation of
the syntax follows; see nutils.expression_v2 for the complete documentation.
A term is written by joining variables with spaces, optionally preceeded by a single number, e.g. 2 a b. A fraction is written as two terms joined by /, e.g. 2 a / 3 b, which is equivalent to (2 a) /
(3 b). An addition or subtraction is written as two terms joined by + or -, respectively, e.g. 1 + a b - 2 b. Exponentation is written by two variables or numbers joined by ^, e.g. a^2. Several
trigonometric functions are available, e.g. 0.5 sin(a).
Assigning an expression to the namespace is then done as follows.
ns.e = '2 a / 3 b'
ns.e = (2*ns.a) / (3*ns.b) # equivalent w/o expression
The resulting ns.e is an ordinary nutils.function.Array. Note that the variables used in the expression should exist in the namespace, not just as a local variable:
localvar = 1
ns.f = '2 localvar'
# Traceback (most recent call last):
# ...
# nutils.expression_v2.ExpressionSyntaxError: No such variable: `localvar`.
# 2 localvar
# ^^^^^^^^
When using arrays in an expression all axes of the arrays should be labelled with an index, e.g. 2 c_i and c_i A_jk. Repeated indices are summed, e.g. A_ii is the trace of d and A_ij c_j is the
matrix-vector product of d and c. You can also insert a number, e.g. c_0 is the first element of c. All terms in an expression should have the same set of indices after summation, e.g. it is an error
to write c_i + 1.
When assigning an expression with remaining indices to the namespace, the indices should be listed explicitly at the left hand side:
ns.f_i = '2 c_i'
ns.f = 2*ns.c # equivalent w/o expression
The order of the indices matter: the resulting nutils.function.Array will have its axes ordered by the listed indices. The following three statements are equivalent:
ns.g_ijk = 'c_i A_jk'
ns.g_kji = 'c_k A_ji'
ns.g = ns.c[:,numpy.newaxis,numpy.newaxis]*ns.A[numpy.newaxis,:,:] # equivalent w/o expression
Function ∇, introduced to the namespace with ~nutils.expression_v2.Namespace.define_for using geometry ns.x, returns the gradient of a variable with respect ns.x, e.g. the gradient of the basis is
∇_i(basis_n). This works with expressions as well, e.g. ∇_i(2 basis_n + basis_n^2) is the gradient of 2 basis_n + basis_n^2.
Sometimes it is useful to evaluate an expression to an nutils.function.Array without inserting the result in the namespace. This can be done using the <expression> @ <namespace> notation. An example
with a scalar expression:
'2 a / 3 b' @ ns
# Array<>
(2*ns.a) / (3*ns.b) # equivalent w/o `... @ ns`
# Array<>
An example with a vector expression:
'2 c_i' @ ns
# Array<2>
2*ns.c # equivalent w/o `... @ ns`
# Array<2>
If an expression has more than one remaining index, the axes of the evaluated array are ordered alphabetically:
'c_i A_jk' @ ns
# Array<2,2,2>
ns.c[:,numpy.newaxis,numpy.newaxis]*ns.A[numpy.newaxis,:,:] # equivalent w/o `... @ ns`
# Array<2,2,2>
A central operation in any Finite Element application is to integrate a function over a physical domain. In Nutils, integration starts with the topology, in particular the integral() method.
The integral method takes a nutils.function.Array function as first argument and the degree as keyword argument. The function should contain the Jacobian of the geometry against which the function
should be integrated, using either nutils.function.J or dV in a namespace expression (assuming the jacobian has been added to the namespace using ns.define_for(..., jacobians=('dV', 'dS'))). For
example, the following integrates 1 against geometry x:
I = topo.integral('1 dV' @ ns, degree=0)
# Array<>
The resulting nutils.function.Array object is a representation of the integral, as yet unevaluated. To compute the actual numbers, call the Array.eval() method:
# 1.0±1e-15
Be careful with including the Jacobian in your integrands. The following two integrals are different:
topo.integral('(1 + 1) dV' @ ns, degree=0).eval()
# 2.0±1e-15
topo.integral('1 + 1 dV' @ ns, degree=0).eval()
# 5.0±1e-15
Like any other nutils.function.Array, the integrals can be added or subtracted:
J = topo.integral('x_0 dV' @ ns, degree=1)
# 1.5±1e-15
Recall that a topology boundary is also a nutils.topology.Topology object, and hence it supports integration. For example, to integrate the geometry x over the entire boundary, write
topo.boundary.integral('x_0 dS' @ ns, degree=1).eval()
# 1.0±1e-15
To limit the integral to the right boundary, write
topo.boundary['right'].integral('x_0 dS' @ ns, degree=1).eval()
# 1.0±1e-15
Note that this boundary is simply a point and the integral a point evaluation.
Integrating and evaluating a 1D nutils.function.Array results in a 1D numpy.ndarray:
>>> topo.integral('basis_i dV' @ ns, degree=1).eval()
array([0.125, 0.25 , 0.25 , 0.25 , 0.125])±1e-15
Since the integrals of 2D nutils.function.Array functions are usually sparse, the Array.eval() <nutils.function.Array.eval> method does not return a dense numpy.ndarray, but a Nutils sparse matrix
object: a subclass of nutils.matrix.Matrix. Nutils interfaces several linear solvers (more on this in Section solvers below) but if you want to use a custom solver you can export the matrix to a
dense, compressed sparse row or coordinate representation via the Matrix.export() method. An example:
M = topo.integral('∇_i(basis_m) ∇_i(basis_n) dV' @ ns, degree=1).eval()
# array([[ 4., -4., 0., 0., 0.],
# [-4., 8., -4., 0., 0.],
# [ 0., -4., 8., -4., 0.],
# [ 0., 0., -4., 8., -4.],
# [ 0., 0., 0., -4., 4.]])±1e-15
M.export('csr') # (data, column indices, row pointers) # doctest: +NORMALIZE_WHITESPACE
# (array([ 4., -4., -4., 8., -4., -4., 8., -4., -4., 8., -4., -4., 4.])±1e-15,
# array([0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4])±1e-15,
# array([ 0, 2, 5, 8, 11, 13])±1e-15)
M.export('coo') # (data, (row indices, column indices)) # doctest: +NORMALIZE_WHITESPACE
# (array([ 4., -4., -4., 8., -4., -4., 8., -4., -4., 8., -4., -4., 4.])±1e-15,
# (array([0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4])±1e-15,
# array([0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4])±1e-15))
Using topologies, bases and integrals, we now have the tools in place to start performing some actual functional-analytical operations. We start with what is perhaps the simplest of its kind, the
least squares projection, demonstrating the different implementations now available to us and working our way up from there.
Taking the geometry component \( x_0 \) as an example, to project it onto the basis \( {φ_n} \) means finding the coefficients \( \hat{u}_n \) such that
\[ \left(\int_Ω φ_n φ_m \ dV\right) \hat u_m = \int_Ω φ_n x_0 \ dV \]
for all \( φ_n \), or \( A_{nm} \hat{u}_m = f_n \). This is implemented as follows:
A = topo.integral('basis_m basis_n dV' @ ns, degree=2).eval()
f = topo.integral('basis_n x_0 dV' @ ns, degree=2).eval()
# solve > solving 5 dof system to machine precision using arnoldi solver
# solve > solver returned with residual 3e-17±1e-15
# array([0. , 0.25, 0.5 , 0.75, 1. ])±1e-15
Alternatively, we can write this in the slightly more general form
\[ R_n := \int_Ω φ_n (u - x_0) \ dV = 0. \]
res = topo.integral('basis_n (u - x_0) dV' @ ns, degree=2)
Taking the derivative of \( R_n \) to \( \hat{u}m \) gives the above matrix \( A{nm} \), and substituting for \( \hat{u} \) the zero vector yields \( -f_n \). Nutils can compute those derivatives for
you, using the method Array.derivative() to compute the derivative with respect to an nutils.function.Argument, returning a new nutils.function.Array.
A = res.derivative('lhs').eval()
f = -res.eval(lhs=numpy.zeros(5))
# solve > solving 5 dof system to machine precision using arnoldi solver
# solve > solver returned with residual 3e-17±1e-15
# array([0. , 0.25, 0.5 , 0.75, 1. ])±1e-15
The above three lines are so common that they are combined in the function nutils.solver.solve_linear:
solver.solve_linear('lhs', res)
# solve > solving 5 dof system to machine precision using arnoldi solver
# solve > solver returned with residual 3e-17±1e-15
# array([0. , 0.25, 0.5 , 0.75, 1. ])±1e-15
We can take this formulation one step further. Minimizing
\[ S := \int_Ω (u - x_0)^2 \ dV \]
for \( \hat{u} \) is equivalent to the above two variants. The derivative of \( S \) to \( \hat{u}_n \) gives \( 2 R_n \):
sqr = topo.integral('(u - x_0)^2 dV' @ ns, degree=2)
solver.solve_linear('lhs', sqr.derivative('lhs'))
# solve > solving 5 dof system to machine precision using arnoldi solver
# solve > solver returned with residual 6e-17±1e-15
# array([0. , 0.25, 0.5 , 0.75, 1. ])±1e-15
The optimization problem can also be solved by the nutils.solver.optimize function, which has the added benefit that \( S \) may be nonlinear in \( \hat{u} \) --- a property not used here.
solver.optimize('lhs', sqr)
# optimize > solve > solving 5 dof system to machine precision using arnoldi solver
# optimize > solve > solver returned with residual 0e+00±1e-15
# optimize > optimum value 0.00e+00±1e-15
# array([0. , 0.25, 0.5 , 0.75, 1. ])±1e-15
Nutils also supports solving a partial optimization problem. In the Laplace problem stated above, the Dirichlet boundary condition at \( Γ_\text{left} \) minimizes the following functional:
sqr = topo.boundary['left'].integral('(u - 0)^2 dS' @ ns, degree=2)
By passing the droptol argument, nutils.solver.optimize returns an array with nan ('not a number') for every entry for which the optimization problem is invariant, or to be precise, where the
variation is below droptol:
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# optimize > constrained 1/5 dofs
# optimize > optimum value 0.00e+00
# array([ 0., nan, nan, nan, nan])±1e-15
Consider again the Laplace problem stated above. The residual is implemented as
res = topo.integral('∇_i(basis_n) ∇_i(u) dV' @ ns, degree=0)
res -= topo.boundary['right'].integral('basis_n dS' @ ns, degree=0)
Since this problem is linear in argument lhs, we can use the nutils.solver.solve_linear method to solve this problem. The constraints cons are passed via the keyword argument constrain:
lhs = solver.solve_linear('lhs', res, constrain=cons)
# solve > solving 4 dof system to machine precision using arnoldi solver
# solve > solver returned with residual 9e-16±1e-15
# array([0. , 0.25, 0.5 , 0.75, 1. ])±1e-15
For nonlinear residuals you can use nutils.solver.newton.
Having obtained the coefficient vector that solves the Laplace problem, we are now interested in visualizing the function it represents. Nutils does not provide its own post processing functionality,
leaving that up to the preference of the user. It does, however, facilitate it, by allowing nutils.function.Array functions to be evaluated in samples. Bundling function values and a notion of
connectivity, these form a bridge between Nutils' world of functions and the discrete realms of matplotlib, VTK, etc.
The Topology.sample(method, ...) method generates a collection of points on the nutils.topology.Topology, according to method. The 'bezier' method generates equidistant points per element, including
the element vertices. The number of points per element per dimension is controlled by the second argument of Topology.sample(). An example:
bezier = topo.sample('bezier', 2)
The resulting nutils.sample.Sample object can be used to evaluate nutils.function.Array functions via the Sample.eval(func) method. To evaluate the geometry ns.x write
x = bezier.eval('x_0' @ ns)
# array([0. , 0.25, 0.25, 0.5 , 0.5 , 0.75, 0.75, 1. ])±1e-15
The first axis of the returned numpy.ndarray represents the collection of points. To reorder this into a sequence of lines in 1D, a triangulation in 2D or in general a sequence of simplices, use the
Sample.tri attribute:
x.take(bezier.tri, 0)
# array([[0. , 0.25],
# [0.25, 0.5 ],
# [0.5 , 0.75],
# [0.75, 1. ]])±1e-15
Now, the first axis represents the simplices and the second axis the vertices of the simplices.
If an nutils.function.Array function has arguments, those arguments must be specified by keyword arguments to Sample.eval(). For example, to evaluate ns.u with argument lhs replaced by solution
vector lhs, obtained using nutils.solver.solve_linear above, write
u = bezier.eval('u' @ ns, lhs=lhs)
# array([0. , 0.25, 0.25, 0.5 , 0.5 , 0.75, 0.75, 1. ])±1e-15
We can now plot the sampled geometry x and solution u using matplotlib_, plotting each line in Sample.tri with a different color:
>>> plt.plot(x.take(bezier.tri.T, 0), u.take(bezier.tri.T, 0))
Recall that we have imported matplotlib.pyplot as plt above. The plt.plot() function takes an array of x-values and and array of y-values, both with the first axis representing vertices and the
second representing separate lines, hence the transpose of bezier.tri.
The plt.plot() function also supports plotting lines with discontinuities, which are represented by nan values. We can use this to plot the solution as a single, but possibly discontinuous line. The
function numpy.insert can be used to prepare a suitable array. An example:
nanjoin = lambda array, tri: numpy.insert(array.take(tri.flat, 0).astype(float),
slice(tri.shape[1], tri.size, tri.shape[1]), numpy.nan, axis=0)
nanjoin(x, bezier.tri)
# array([0. , 0.25, nan, 0.25, 0.5 , nan, 0.5 , 0.75, nan, 0.75, 1. ])±1e-15
plt.plot(nanjoin(x, bezier.tri), nanjoin(u, bezier.tri))
Note the difference in colors between the last two plots.
All of the above was written for a one-dimensional example. We now extend the Laplace problem to two dimensions and highlight the changes to the corresponding Nutils implementation. Let \( Ω \) be a
unit square with boundary \( Γ \), on which the following boundary conditions apply:
\[ \begin{cases} u = 0 & Γ_\text{left} \\ \frac{∂u}{∂x_i} n_i = 0 & Γ_\text{bottom} \\ \frac{∂u}{∂x_i} n_i = \cos(1) \cosh(x_1) & Γ_\text{right} \\ u = \cosh(1) \sin(x_0) & Γ_\text{top} \end{cases}
The 2D homogeneous Laplace solution is the field \( u \) for which \( R(v, u) = 0 \) for all v, where
\[ R(v, u) := \int_Ω \frac{∂v}{∂x_i} \frac{∂u}{∂x_i} \ dV - \int_{Γ_\text{right}} v \cos(1) \cosh(x_1) \ dS. \]
Adopting a Finite Element basis \( {φ_n} \) we obtain the discrete solution \( \hat{u}(x) = φ_n(x) \hat{u}_n \) and the system of equations \( R(φ_n, \hat{u}) = 0 \).
Following the same steps as in the 1D case, a unit square mesh with 10x10 elements is formed using nutils.mesh.rectilinear:
nelems = 10
topo, geom = mesh.rectilinear([
numpy.linspace(0, 1, nelems+1), numpy.linspace(0, 1, nelems+1)])
Recall that nutils.mesh.rectilinear takes a list of element vertices per dimension. Alternatively you can create a unit square mesh using nutils.mesh.unitsquare, specifying the number of elements per
dimension and the element type:
topo, geom = mesh.unitsquare(nelems, 'square')
The above two statements generate exactly the same topology and geometry. Try replacing 'square' with 'triangle' or 'mixed' to generate a unit square mesh with triangular elements or a mixture of
square and triangular elements, respectively.
We start with a clean namespace, assign the geometry to ns.x, create a linear basis and define the solution ns.u as the contraction of the basis with argument lhs.
ns = Namespace()
ns.x = geom
ns.define_for('x', gradient='∇', normal='n', jacobians=('dV', 'dS'))
ns.basis = topo.basis('std', degree=1)
ns.u = function.dotarg('lhs', ns.basis)
Note that the above statements are identical to those of the one-dimensional example.
The residual is implemented as
res = topo.integral('∇_i(basis_n) ∇_i(u) dV' @ ns, degree=2)
res -= topo.boundary['right'].integral('basis_n cos(1) cosh(x_1) dS' @ ns, degree=2)
The Dirichlet boundary conditions are rewritten as a least squares problem and solved for lhs, yielding the constraints vector cons:
sqr = topo.boundary['left'].integral('u^2 dS' @ ns, degree=2)
sqr += topo.boundary['top'].integral('(u - cosh(1) sin(x_0))^2 dS' @ ns, degree=2)
cons = solver.optimize('lhs', sqr, droptol=1e-15)
# optimize > solve > solving 21 dof system to machine precision using arnoldi solver
# optimize > solve > solver returned with residual 3e-17±2e-15
# optimize > constrained 21/121 dofs
# optimize > optimum value 4.32e-10±1e-9
To solve the problem res=0 for lhs subject to lhs=cons excluding the nan values, we can use nutils.solver.solve_linear:
lhs = solver.solve_linear('lhs', res, constrain=cons)
# solve > solving 100 dof system to machine precision using arnoldi solver
# solve > solver returned with residual 2e-15±2e-15
Finally, we plot the solution. We create a nutils.sample.Sample object from topo and evaluate the geometry and the solution:
bezier = topo.sample('bezier', 9)
x, u = bezier.eval(['x_i', 'u'] @ ns, lhs=lhs)
We use plt.tripcolor to plot the sampled x and u:
plt.tripcolor(x[:,0], x[:,1], bezier.tri, u, shading='gouraud', rasterized=True)
This two-dimensional example is also available as the script examples/laplace.py.
Nutils is developed on Github and released at cadence of roughly one year, with the actual time of release depending on the level of maturity of newly introduced features. Major releases introduce
new features and may deprecate features that have been superceded. Minor releases contain only bugfixes and are always safe to upgrade to.
Every major release follows the following procedure:
• The development branch is branched off to release/x, where x is the major version number
• The release is assigned a code name, in alphabetical order, derived from a type of noodles dish
• The release commit is tagged as vx.0, with minor updates following as vx.1, vx.2 etc
• The package is uploaded to PyPi for easy installation using pip
• The changelog in the development branch is emptied and added to this release history
Since Nutils is under active development and releases are fairly infrequent, users may choose to work with the development version directly, taking for granted that their code may require continuous
updating as features develop — keep an eye on the changelog in the project root! The development version is continuously updated here:
Nutils 8.0 was released on July 28th, 2023.
These are the main additions and changes since Nutils 7 Hiyamugi.
The SI module provides a framework for checking the dimensional consistency of a formulation at runtime, as well as tools to convert between different units for post processing. Extensive
documentation can be found in the SI module. The framework can be seen in action in the cahnhilliard example, with more examples to be converted in future.
Support for Numpy operations on Nutils arrays has been extended to include:
• numpy.choose
• numpy.cross
• numpy.diagonal
• numpy.einsum
• numpy.interp (array support limited to x argument)
• numpy.linalg.det
• numpy.linalg.eig
• numpy.linalg.eigh
• numpy.linalg.inv
• numpy.linalg.norm
• numpy.searchsorted (array support limited to v argument)
The nutils.solver methods have been generalized to accept scalar valued functionals, from which residual vectors are derived through differentiation. To this end, a trial/test function pair can be
specified as a solve target separated by a colon, as in the following example:
ns.add_field(('u', 'v'), topo.basis('std', degree=1))
res = topo.integral('∇_i(u) ∇_i(v) dV' @ ns, degree=2)
args = solver.newton('u:v', res).solve(1e-10)
Multiple fields can either comma-joined or provided as a tuple. Note that the colon automatically triggers a new-style dictionary return value, even in absence of a trialing comma as in the above
The namespace from the nutils.expression_v2 module newly provides the nutils.expression_v2.Namespace.add_field method, as a convenient shorthand for creating fields with the same name as their
arguments. That is:
ns.add_field(('u', 'v'), topo.basis('std', degree=1), shape=(2,))
is equivalent to
basis = topo.basis('std', degree=1)
ns.u = function.dotarg('u', basis, shape=(2,))
ns.v = function.dotarg('v', basis, shape=(2,))
Multiple solver targets can now be specified as a comma-separated string, as a shorthand for the string tuple that will remain a valid argument. This means the following two invocations are
args = solver.newton(('u', 'p'), (ures, pres)).solve(1e-10)
args = solver.newton('u,p', (ures, pres)).solve(1e-10)
To distinguish single-length tuples from the single argument legacy notation, the former requires a trailing comma. I.e., the following are NOT equivalent:
args = solver.newton('u,', (ures,)).solve(1e-10)
u = solver.newton('u', ures).solve(1e-10)
The sample methods asfunction and basis have a new interpolation argument that take the string values "none" (default) or "nearest". The latter activates a new mode that allows evaluation of sampled
data on other samples than the original by selecting the point that is closest to the target.
Similar to derivative, the new function linearize takes the derivative of an array to one or more arguments, but with the derivative directions represented by arguments rather than array axes. This
is particularly useful in situations where weak forms are made up of symmetric, energy like components, combined with terms that require dedicated test fields.
The linesearch argument of solver.newton can now receive the None value to indicate that line search is to be disabled. Additionally, the legacy arguments searchrange and rebound have been
deprecated, and should be replaced by linesearch=solver.NormBased(minscale=searchrange[0], acceptscale=searchrange[1], maxscale=rebound).
The info struct returned by solve_withinfo newly contains the amount of iterations as the niter attribute:
res, info = solver.newton('u:v', res).solve_withinfo(1e-10, maxiter=10)
assert info.niter <= 10
The trim routine (which is used for the Finite Cell Method) is rewritten for speed and to produce more efficient quadrature schemes. The changes relate to the subdivision at the deepest refinement
level. While this step used to introduce auxiliary vertices at every dimension (lines, faces, volumes), the new implementation limits the introduction of vertices to the line segments only, resulting
in a subdivision that consists of fewer simplices and consequently fewer quadrature points.
Solver methods newton, minimize and pseudotime have their function signature slightly changed:
1. The tol argument (used to define the stop criterion) has been made mandatory. As the default value used to be 0 - an unreacheable value in practice - the argument was effectively mandatory
already, which this change formalizes.
2. The maxiter argument was off by 1, leading maxiter=n to accept n+1 iterations. This mistake is now fixed, which may break applications that relied on the former erroneous behaviour.
The locate method has a skip_missing argument that instructs the method to silently drop points that can not be located on the topology. This setting was partially ignored by trimmed topologies which
could lead to a LocateError despite the flag being set. This issue is now fixed.
Support for the Nutils configuration file (which used to be located in either ~/.nutilsrc or ~/.config/nutils/config) has been removed. Instead, the following environment variables can be set to
override the default Nutils settings:
• NUTILS_PDB = yes|no
• NUTILS_GRACEFULEXIT = yes|no
• NUTILS_OUTROOTDIR = path/to/html/logs
• NUTILS_OUTROOTURI = uri/to/html/logs
• NUTILS_SCRIPTNAME = myapp
• NUTILS_OUTDIR = path/to/this/html/log
• NUTILS_OUTURI = uti/to/this/html/log
• NUTILS_RICHOUTPUT = yes|no
• NUTILS_VERBOSE = 1|2|3|4
• NUTILS_MATRIX = numpy|scipy|mkl|auto
• NUTILS_NPROCS = 1|2|...
• NUTILS_CACHE = yes|no
• NUTILS_CACHEDIR = path/to/cache
The nutils.function methods that have direct equivalents in the numpy module (function.sum, function.sqrt, function.sin, etc) have been deprecated in favour of using Numpy's methods (numpy.sum,
numpy.sqrt, numpy.sin, etc) and will be removed in the next release. Ultimately, only methods that relate to the variable nature of function arrays and therefore have no Numpy equivalent, such as
function.grad and function.normal, will remain in the function module.
Be aware that some functions were not 100% equivalent to their Numpy counterpart. For instance, function.max is the equivalent to numpy.maximum, as the deprecation message helpfully points out. More
problematically, function.dot behaves very differently from both numpy.dot and numpy.matmul. Porting the code over to equivalent instructions will therefore require some attention.
The nutils.function.Array.dot method is incompatible with Numpy's equivalent method for arrays of ndim != 1, or when axes are specified (which Numpy does not allow). Aiming for 100% compatibility,
the next release cycle will remove the axis argument and temporarily forbid arguments of ndim != 1. The release cycle thereafter will re-enable arguments with ndim != 1, with logic equal to Numpy's
method. In the meantime, the advice is to rely on numpy.dot, numpy.matmul or the @ operator instead.
The nutils.function.Array.sum method by default operates on the last axis. This is different from Numpy's behavour, which by default sums all axes. Aiming for 100% compatibility, the next release
cycle will make the axis argument mandatory for any array of ndim > 1. The release cycle thereafter will reintroduce the default value to match Numpy's. To prepare for this, relying on the current
default now triggers a deprecation warning.
Nutils 7.0 was released on January 1st, 2022.
Nutils 7.1 was released on September 3rd, 2022.
Nutils 7.2 was released on November 4th, 2022.
Nutils 7.3 was released on June 20th, 2023.
These are the main additions and changes since Nutils 6 Garak-Guksu.
The nutils.expression module has been renamed to nutils.expression_v1, the nutils.function.Namespace class to nutils.expression_v1.Namespace and the nutils.expression_v2 module has been added,
featuring a new nutils.expression_v2.Namespace. The version 2 of the namespace v2 has an expression language that differs slightly from version 1, most notably in the way derivatives are written. The
old namespace remains available for the time being. All examples are updated to the new namespace. You are encouraged to use the new namespace for newly written code.
In the past using functions on products of nutils.topology.Topology instances required using function.bifurcate. This has been replaced by the concept of 'spaces'. Every topology is defined in a
space, identified by a name (str). Functions defined on some topology are considered constant on other topologies (defined on other spaces).
If you want to multiply two topologies, you have to make sure that the topologies have different spaces, e.g. via the space parameter of nutils.mesh.rectilinear. Example:
from nutils import mesh, function
Xtopo, x = mesh.rectilinear([4], space='X')
Ytopo, y = mesh.rectilinear([2], space='Y')
topo = Xtopo * Ytopo
geom = function.concatenate([x, y])
Resulting from to the function/evaluable split introduced in #574, variable length axes such as relating to integration points or sparsity can stay confined to the evaluable layer. In order to
benefit from this situation and improve compatibility with Numpy's arrays, nutils.function.Array objects are henceforth limited to constant shapes. Additionally:
• The sparsity construct nutils.function.inflate has been removed;
• The nutils.function.Elemwise function requires all element arrays to be of the same shape, and its remaining use has been deprecated in favor of nutils.function.get;
• Aligning with Numpy's API, nutils.function.concatenate no longer automatically broadcasts its arguments, but instead demands that all dimensions except for the concatenation axis match exactly.
The nutils.topology.Topology.locate method now allows tol to be left unspecified if eps is specified instead, which is repurposed as stop criterion for distances in element coordinates. Conversely,
if only tol is specified, a corresponding minimal eps value is set automatically to match points near element edges. The ischeme and scale arguments are deprecated and replaced by maxdist, which can
be left unspecified in general. The optional weights argument results in a sample that is suitable for integration.
The unit type has been moved into its own nutils.unit module, with the old location types.unit now holding a forward method. The forward emits a deprecation warning prompting to change
nutils.types.unit.create (or its shorthand nutils.types.unit) to nutils.unit.create.
Libraries that are installed in odd locations will no longer be automatically located by Nutils (see b8b7a6d5 for reasons). Instead the user will need to set the appropriate environment variable,
prior to starting Python. In Windows this is the PATH variable, in Linux and OS X LD_LIBRARY_PATH.
Crucially, this affects the MKL libraries when they are user-installed via pip. By default Nutils selects the best available matrix backend that it finds available, which could result in it silently
falling back on Scipy or Numpy. To confirm that the path variable is set correctly run your application with matrix=mkl to force an error if MKL cannot be loaded.
The function module has been split into a high-level, numpy-like function module and a lower-level evaluable module. The evaluable module is agnostic to the so-called points axis. Scripts that don't
use custom implementations of function.Array should work without modification.
Custom implementations of the old function.Array should now derive from evaluable.Array. Furthermore, an accompanying implementation of function.Array should be added with a prepare_eval method that
returns the former.
The following example implementation of an addition
class Add(function.Array):
def __init__(self, a, b):
super().__init__(args=[a, b], shape=a.shape, dtype=a.dtype)
def evalf(self, a, b):
return a+b
should be converted to
class Add(function.Array):
def __init__(self, a: function.Array, b: function.Array) -> None:
self.a = a
self.b = b
super().__init__(shape=a.shape, dtype=a.dtype)
def prepare_eval(self, **kwargs) -> evaluable.Array:
a = self.a.prepare_eval(**kwargs)
b = self.b.prepare_eval(**kwargs)
return Add_evaluable(a, b)
class Add_evaluable(evaluable.Array):
def __init__(self, a, b):
super().__init__(args=[a, b], shape=a.shape, dtype=a.dtype)
def evalf(self, a, b):
return a+b
In problems involving multiple fields, where formerly it was required to nutils.function.chain the bases in order to construct and solve a block system, an alternative possibility is now to keep the
residuals and targets separate and reference the several parts at the solving phase:
# old, still valid approach
ns.ubasis, ns.pbasis = function.chain([ubasis, pbasis])
ns.u_i = 'ubasis_ni ?dofs_n'
ns.p = 'pbasis_n ?dofs_n'
# new, alternative approach
ns.ubasis = ubasis
ns.pbasis = pbasis
ns.u_i = 'ubasis_ni ?u_n'
ns.p = 'pbasis_n ?p_n'
# common: problem definition
ns.σ_ij = '(u_i,j + u_j,i) / Re - p δ_ij'
ures = topo.integral('ubasis_ni,j σ_ij d:x d:x' @ ns, degree=4)
pres = topo.integral('pbasis_n u_,kk d:x' @ ns, degree=4)
# old approach: solving a single residual to a single target
dofs = solver.newton('dofs', ures + pres).solve(1e-10)
# new approach: solving multiple residuals to multiple targets
state = solver.newton(['u', 'p'], [ures, pres]).solve(1e-10)
In the new, multi-target approach, the return value is no longer an array but a dictionary that maps a target to its solution. If additional arguments were specified to newton (or any of the other
solvers) then these are copied into the return dictionary so as to form a complete state, which can directly be used as an arguments to subsequent evaluations.
If an argument is specified for a solve target then its value is used as an initial guess (newton, minimize) or initial condition (thetamethod). This replaces the lhs0 argument which is not supported
for multiple targets.
To explicitly refer to the history state in nutils.solver.thetamethod and its derivatives impliciteuler and cranknicolson, instead of specifiying the target through the target0 parameter, the new
argument historysuffix specifies only the suffix to be added to the main target. Hence, the following three invocations are equivalent:
# deprecated
solver.impliciteuler('target', residual, inertia, target0='target0')
# new syntax
solver.impliciteuler('target', residual, inertia, historysuffix='0')
# equal, since '0' is the default suffix
solver.impliciteuler('target', residual, inertia)
When nutils.solver.newton, nutils.solver.minimize or nutils.solver.pseudotime are used as iterators, the generated vectors are now modified in place. Therefore, if iterates are stored for analysis,
be sure to use the .copy method.
The function function.elemwise has been deprecated. Use function.Elemwise instead:
function.elemwise(topo.transforms, values) # deprecated
function.Elemwise(values, topo.f_index) # new
The transforms attribute of bases has been removed due to internal restructurings. The transforms attribute of the topology on which the basis was created can be used as a replacement:
reftopo = topo.refined
refbasis = reftopo.basis(...)
supp = refbasis.get_support(...)
#topo = topo.refined_by(refbasis.transforms[supp]) # no longer valid
topo = topo.refined_by(reftopo.transforms[supp]) # still valid
Nutils 6.0 was released on April 29th, 2020.
Nutils 6.1 was released on July 17th, 2020.
Nutils 6.2 was released on October 7th, 2020.
Nutils 6.3 was released on November 18th, 2021.
These are the main additions and changes since Nutils 5 Farfalle.
The new nutils.sparse module introduces a data type and a suite of manipulation methods for arbitrary dimensional sparse data. The existing integrate and integral methods now create data of this type
under the hood, and then convert it to a scalar, Numpy array or nutils.matrix.Matrix upon return. To prevent this conversion and receive the sparse objects instead use the new
nutils.sample.Sample.integrate_sparse or nutils.sample.eval_integrals_sparse.
The nutils.mesh.gmsh method now depends on the external meshio module to parse .msh files:
python3 -m pip install --user --upgrade meshio
When creating a vector basis using topo.basis(..).vector(nd), the order of the degrees of freedom changed from grouping by vector components to grouping by scalar basis functions:
[b0, 0] [b0, 0]
[b1, 0] [ 0, b0]
[.., ..] old [b1, 0]
[bn, 0] ------> [ 0, b1]
[ 0, b0] new [.., ..]
[.., ..] [bn, 0]
[ 0, bn] [ 0, bn]
This should not affect applications unless the solution vector is manipulated directly, such as might happen in unit tests. If required for legacy purposes the old vector can be retrieved using old =
new.reshape(-1,nd).T.ravel(). Note that the change does not extend to nutils.function.vectorize.
For nutils.cli.run to draw a status bar, it now requires the external bottombar module to be installed:
python3 -m pip install --user bottombar
This replaces stickybar, which is no longer used. In addition to the log uri and runtime the status bar will now show the current memory usage, if that information is available. On Windows this
requires psutil to be installed; on Linux and OSX it should work by default.
The nutils.mesh.gmsh method now supports input in the 'msh4' file format, in addition to the 'msh2' format which remains supported for backward compatibility. Internally, nutils.mesh.parsegmsh now
takes file contents instead of a file name.
The new boolean command line option gracefulexit determines what happens when an exception reaches nutils.cli.run. If true (default) then the exception is handled as before and a system exit is
initiated with an exit code of 2. If false then the exception is reraised as-is. This is useful in particular when combined with an external debugging tool.
The way exceptions are handled by nutils.cli.run is changed from logging the entire exception and traceback as a single error message, to logging the exceptions as errors and tracebacks as debug
messages. Additionally, the order of exceptions and traceback is fully reversed, such that the most relevant message is the first thing shown and context follows.
The nutils.solver.newton method now sets the relative tolerance of the linear system to 1e-3 unless otherwise specified via linrtol. This is mainly useful for iterative solvers which can save
computational effort by having their stopping criterion follow the current Newton residual, but it may also help with direct solvers to warn of ill conditioning issues. Iterations furthermore use
nutils.matrix.Matrix.solve_leniently, thus proceeding after warning that tolerances have not been met in the hope that Newton convergence might be attained regardless.
The methods nutils.solver.newton, nutils.solver.minimize, nutils.solver.pseudotime, nutils.solver.solve_linear and nutils.solver.optimize now receive linear solver arguments as keyword arguments
rather than via the solveargs dictionary, which is deprecated. To avoid name clashes with the remaining arguments, argument names must be prefixed by lin:
solver.solve_linear('lhs', res,
solveargs=dict(solver='gmres')) # deprecated syntax
solver.solve_linear('lhs', res,
linsolver='gmres') # new syntax
Direct solvers enter an iterative refinement loop in case the first pass did not meet the configured tolerance. In machine precision mode (atol=0, rtol=0) this refinement continues until the residual
The absolute and/or relative tolerance for solutions of a linear system can now be specified in nutils.matrix.Matrix.solve via the atol resp. rtol arguments, regardless of backend and solver. If the
backend returns a solution that violates both tolerances then an exception is raised of type nutils.matrix.ToleranceNotReached, from which the solution can still be obtained via the .best attribute.
Alternatively the new method nutils.matrix.Matrix.solve_leniently always returns a solution while logging a warning if tolerances are not met. In case both tolerances are left at their default value
or zero then solvers are instructed to produce a solution to machine precision, with subsequent checks disabled.
Nutils now depends on stringly (version 1.0b1) for parsing of command line arguments. The new implementation of nutils.cli.run is fully backwards compatible, but the preferred method of annotating
function arguments is now as demonstrated in all of the examples.
For new Nutils installations Stringly will be installed automatically as a dependency. For existing setups it can be installed manually as follows:
python3 -m pip install --user --upgrade stringly
The nutils.function.Namespace has two new arguments: length_<indices> and fallback_length. The former can be used to assign fixed lengths to specific indices in expressions, say index i should have
length 2, which is used for verification and resolving undefined lengths. The latter is used to resolve remaining undefined lengths:
ns = nutils.function.Namespace(length_i=2, fallback_length=3)
ns.eval_ij('δ_ij') # using length_i
# Array<2,2>
ns.eval_jk('δ_jk') # using fallback_length
# Array<3,3>
Nutils now depends on treelog version 1.0b5, which brings improved iterators along with other enhancements. For transitional convenience the backwards incompatible changes have been backported in the
nutils.log wrapper, which now emits a warning in case the deprecated methods are used. This wrapper is scheduled for deletion prior to the release of version 6.0. To update treelog to the most recent
version use:
python -m pip install -U treelog
The new nutils.types.unit allows for the creation of a unit system for easy specification of physical quantities. Used in conjunction with nutils.cli.run this facilitates specifying units from the
command line, as well as providing a warning mechanism against incompatible units:
U = types.unit.create(m=1, s=1, g=1e-3, N='kg*m/s2', Pa='N/m2')
def main(length=U('2m'), F=U('5kN')):
topo, geom = mesh.rectilinear([numpy.linspace(0,length,10)])
python myscript.py length=25cm # OK
python myscript.py F=10Pa # error!
Samples now provide a nutils.sample.Sample.basis: an array that for any point in the sample evaluates to the unit vector corresponding to its index. This new underpinning of
nutils.sample.Sample.asfunction opens the way for sampled arguments, as demonstrated in the last example below:
H1 = mysample.asfunction(mydata) # mysample.eval(H1) == mydata
H2 = mysample.basis().dot(mydata) # mysample.eval(H2) == mydata
ns.Hbasis = mysample.basis()
H3 = 'Hbasis_n ?d_n' @ ns # mysample.eval(H3, d=mydata) == mydata
Gmsh element support has been extended to include cubic and quartic meshes in 2D and quadratic meshes in 3D, and parsing the msh file is now a cacheable operation. Additionally, tetrahedra now define
bezier points at any order.
The Nutils repository has moved to https://github.com/evalf/nutils.git. For the time being the old address is maintained by Github as an alias, but in the long term you are advised to update your
remote as follows:
git remote set-url origin https://github.com/evalf/nutils.git
Nutils 5.0 was released on April 3rd, 2020.
Nutils 5.1 was released on September 3rd, 2019.
Nutils 5.2 was released on June 11th, 2019.
These are the main additions and changes since Nutils 4 Eliche.
The Matrix.matvec method has been deprecated in favour of the new __matmul__ (@) operator, which supports multiplication arrays of any dimension. The nutils.matrix.Matrix.solve method has been
extended to support multiple right hand sides:
matrix.matvec(lhs) # deprecated
matrix @ lhs # new syntax
matrix @ numpy.stack([lhs1, lhs2, lhs3], axis=1)
matrix.solve(numpy.stack([rhs1, rhs2, rhs3], axis=1)
Matrices produced by the MKL backend now support the nutils.matrix.Matrix.solve argument solver='fmgres' to use Intel MKL's fgmres method.
The nutils.solver.thetamethod class, as well as its special cases impliciteuler and cranknicolson, now have a timetarget argument to specify that the formulation contains a time variable:
res = topo.integral('...?t... d:x' @ ns, degree=2)
solver.impliciteuler('dofs', res, ..., timetarget='t')
In nutils.topology.Topology.trim, in case the levelset cannot be evaluated on the to-be-trimmed topology itself, the correct topology can now be specified via the new leveltopo argument.
nutils.testing.TestCase now facilitates comparison against base64 encoded, compressed, and packed data via the new method nutils.testing.TestCase.assertAlmostEqual64. This replaces
numeric.assert_allclose64 which is now deprecated and scheduled for removal in Nutils 6.
A special case nutils.topology.Topology.locate method for structured topologies checks of the geometry is an affine transformation of the natural configuration, in which case the trivial inversion is
used instead of expensive Newton iterations:
topo, geom = mesh.rectilinear([2, 3])
smp = topo.locate(geom/2-1, [[-.1,.2]])
# locate detected linear geometry: x = [-1. -1.] + [0.5 0.5] xi ~+2.2e-16
The introduction of sequence abstractions nutils.elementseq and nutils.transformseq, together with and a lazy implementation of nutils.function.Basis basis functions, help to prevent the unnecessary
generation of data. In hierarchically refined topologies, in particular, this results in large speedups and a much reduced memory footprint.
The nutils.log module is deprecated and will be replaced by the externally maintained treelog <https://github.com/evalf/treelog>_, which is now an installation dependency.
The nutils.parallel module is largely rewritten. The old methods pariter and parmap are replaced by the nutils.parallel.fork context, combined with the shared nutils.parallel.range iterator:
indices = parallel.range(10)
with parallel.fork(nprocs=2) as procid:
for index in indices:
print('procid={}, index={}'.format(procid, index))
Nutils 4.0 was released on June 11th, 2019.
Nutils 4.1 was released on August 28th, 2018.
These are the main additions and changes since Nutils 3 Dragon Beard.
In addition to the knotmultiplicities argument to define the continuity of basis function on structured topologies, the nutils.topology.Topology.basis method now supports the continuity argument to
define the global continuity of basis functions. With negative numbers counting backwards from the degree, the default value of -1 corresponds to a knot multiplicity of 1.
Functions of type nutils.function.Evaluable can receive arguments in addition to element and points by depending on instances of nutils.function.Argument and having their values specified via
f = geom.dot(function.Argument('myarg', shape=geom.shape))
f = 'x_i ?myarg_i' @ ns # equivalent operation in namespace
topo.sample('uniform', 1).eval(f, myarg=numpy.ones(geom.shape))
Namespace expression syntax now includes the d: Jacobian operator, allowing one to write 'd:x' @ ns instead of function.J(ns.x). Since including the Jacobian in the integrand is preferred over
specifying it separately, the geometry argument of nutils.topology.Topology.integrate is deprecated:
topo.integrate(ns.f, geometry=ns.x) # deprecated
topo.integrate(ns.f * function.J(ns.x)) # was and remains valid
topo.integrate('f d:x' @ ns) # new namespace syntax
Hierarchically refined topologies now support basis truncation, which reduces the supports of individual basis functions while maintaining the spanned space. To select between truncated and
non-truncated the basis type must be prefixed with 'th-' or 'h-', respectively. A non-prefixed basis type falls back on the default implementation that fails on all types but discont:
htopo.basis('spline', degree=2) # no longer valid
htopo.basis('h-spline', degree=2) # new syntax for original basis
htopo.basis('th-spline', degree=2) # new syntax for truncated basis
htopo.basis('discont', degree=2) # still valid
The nutils.cache module provides a memoizing function decorator nutils.cache.function which reads return values from cache in case a set of function arguments has been seen before. It is similar in
function to Python's functools.lru_cache, except that the cache is maintained on disk and nutils.types.nutils_hash is used to compare arguments, which means that arguments need not be Python
hashable. The mechanism is activated via nutils.cache.enable:
def f(x):
return x * 2
with cache.enable():
If nutils.cli.run is used then the cache can also be enabled via the new --cache command line argument. With many internal Nutils functions already decorated, including all methods in the
nutils.solver module, transparent caching is available out of the box with no further action required.
The new nutils.types module unifies and extends components relating to object types. The following preexisting objects have been moved to the new location:
• util.enforcetypes → types.apply_annotations
• util.frozendict → types.frozendict
• numeric.const → types.frozenarray
The new MKL backend generates matrices that are powered by Intel's Math Kernel Library, which notably includes the reputable Pardiso solver. This requires libmkl to be installed, which is
conveniently available through pip:
pip install mkl
When nutils.cli.run is used the new matrix type is selected automatically if it is available, or manually using --matrix=MKL.
For problems that adhere to an energy structure, the new solver method nutils.solver.minimize provides an alternative mechanism that exploits this structure to robustly find the energy minimum:
res = sqr.derivative('dofs')
solver.newton('dofs', res, ...)
solver.minimize('dofs', sqr, ...) # equivalent
Two new methods, nutils.numeric.pack and its inverse nutils.numeric.unpack, provide lossy compression to floating point data. Primarily useful for regression tests, the convenience method
numeric.assert_allclose64 combines data packing with zlib compression and base64 encoding for inclusion in Python codes.
Nutils 3.0 was released on August 22nd, 2018.
Nutils 3.1 was released on February 5th, 2018.
These are the main additions and changes since Nutils 2 Chuka Men.
The nutils.function.Namespace object represents a container of nutils.function.Array instances:
ns = function.Namespace()
ns.x = geom
ns.basis = domain.basis('std', degree=1).vector(2)
In addition to bundling arrays, arrays can be manipulated using index notation via string expressions using the nutils.expression syntax:
ns.sol_i = 'basis_ni ?dofs_n'
f = ns.eval_i('sol_i,j n_j')
Analogous to nutils.topology.Topology.integrate, which integrates a function and returns the result as a (sparse) array, the new method nutils.topology.Topology.integral with identical arguments
results in an nutils.sample.Integral object for postponed evaluation:
x = domain.integrate(f, geometry=geom, degree=2) # direct
integ = domain.integral(f, geometry=geom, degree=2) # indirect
x = integ.eval()
Integral objects support linear transformations, derivatives and substitutions. Their main use is in combination with routines from the nutils.solver module.
Transformation chains (sequences of transform items) are stored as standard tuples. Former class methods are replaced by module methods:
elem.transform.promote(ndims) # no longer valid
transform.promote(elem.transform, ndims) # new syntax
In addition, every edge_transform and child_transform of Reference objects is changed from (typically unit-length) TransformChain to nutils.transform.TransformItem.
Command line parsers nutils.cli.run or nutils.cli.choose dropped support for space separated arguments (--arg value), requiring argument and value to be joined by an equals sign instead:
python script.py --arg=value
Boolean arguments are specified by omitting the value and prepending 'no' to the argument name for negation:
python script.py --pdb --norichoutput
For convenience, leading dashes have been made optional:
python script.py arg=value pdb norichoutput
Intersections between topologies can be made using the & operator. In case the operands have different refinement patterns, the resulting topology will consist of the common refinements of the
intersection = topoA & topoB
interface = topo['fluid'].boundary & ~topo['solid'].boundary
The nutils.topology.Topology.indicator method is moved from subtopology to parent topology, i.e. the topology you want to evaluate the indicator on, and now takes the subtopology is an argument:
ind = domain.boundary['top'].indicator() # no longer valid ind = domain.boundary.indicator(domain.boundary['top']) # new syntax ind = domain.boundary.indicator('top') # equivalent
The nutils.function.Evaluable.eval method accepts a flexible number of keyword arguments, which are accessible to evalf by depending on the EVALARGS token. Standard keywords are _transforms for
transformation chains, _points for integration points, and _cache for the cache object:
f.eval(elem, 'gauss2') # no longer valid
ip, iw = elem.getischeme('gauss2')
tr = elem.transform, elem.opposite
f.eval(_transforms=tr, _points=ip) # new syntax
The numeric.const array represents an immutable, hashable array:
A = numeric.const([[1,2],[3,4]])
d = {A: 1}
Existing arrays can be wrapped into a const object by adding copy=False. The writeable flag of the original array is set to False to prevent subsequent modification:
A = numpy.array([1,2,3])
Aconst = numeric.const(A, copy=False)
A[1] = 4
# ValueError: assignment destination is read-only
The util.enforcetypes decorator applies conversion methods to annotated arguments:
def f(a:float, b:tuple)
print(type(a), type(b))
f(1, [2])
# <class 'float'> <class 'tuple'>
The decorator is by default active to constructors of cache.Immutable derived objects, such as function.Evaluable.
Evaluable objects have a default edit implementation that re-instantiates the object with the operand applied to all constructor arguments. In situations where the default implementation is not
sufficient it can be overridden by implementing the edit method (note: without the underscore):
class B(function.Evaluable):
def __init__(self, d):
assert isinstance(d, dict)
self.d = d
def edit(self, op):
return B({key: op(value) for key, value in self.d.items()})
The nutils.function.derivative axes argument has been removed; derivative(func, var) now takes the derivative of func to all the axes in var:
der = function.derivative(func, var,
axes=numpy.arange(var.ndim)) # no longer valid
der = function.derivative(func, var) # new syntax
The nutils.util.run function is deprecated and replaced by two new functions, nutils.cli.choose and nutils.cli.run. The new functions are very similar to the original, but have a few notable
• cli.choose requires the name of the function to be executed (typically 'main'), followed by any optional arguments
• cli.run does not require the name of the function to be executed, but only a single one can be specified
• argument conversions follow the type of the argument's default value, instead of the result of eval
• the --tbexplore option for post-mortem debugging is replaced by --pdb, replacing Nutils' own traceback explorer by Python's builtin debugger
• on-line debugging is provided via the ctrl+c signal handler
• function annotations can be used to describe arguments in both help messages and logging output (see examples)
The nutils.solver module provides infrastructure to facilitate formulating and solving complicated nonlinear problems in a structured and largely automated fashion.
Topologies have been made fully immutable, which means that the old setitem operation is no longer supported. Instead, to add a subtopology to the domain, its boundary, its interfaces, or points, any
of the methods withsubdomain, withboundary, withinterfaces, and withpoints, respectively, will return a copy of the topology with the desired groups added:
topo.boundary['wall'] = topo.boundary['left,top'] # no longer valid
newtopo = topo.withboundary(wall=topo.boundary['left,top']) # new syntax
newtopo = topo.withboundary(wall='left,top') # equivalent shorthand
Any topology can be revolved using the new nutils.topology.Topology.revolved method, which interprets the first geometry dimension as a radius and replaces it by two new dimensions, shifting the
remaining axes backward. In addition to the modified topology and geometry, simplifying function is returned as the third return value which replaces all occurrences of the revolution angle by zero.
This should only be used after all gradients have been computed:
rdomain, rgeom, simplify = domain.revolved(geom)
basis = rdomain.basis('spline', degree=2)
M = function.outer(basis.grad(rgeom)).sum(-1)
rdomain.integrate(M, geometry=rgeom, ischeme='gauss2', edit=simplify)
The gmsh importer was unintentionally misnamed as gmesh; this has been fixed. With that the old name is deprecated and will be removed in future. In addition, support for the non-physical mesh format
and externally supplied boundary labels has been removed (see the unit test tests/mesh.py for examples of valid .geo format). Support is added for periodicity and interface groups.
Nutils 2.0 was released on February 18th, 2016.
These are the main additions and changes since Nutils 1 Bakmi.
The jump operator has been changed according to the following definition: jump(f) = opposite(f) - f. In words, it represents the value of the argument from the side that the normal is pointing
toward, minus the value from the side that the normal is pointing away from. Compared to the old definition this means the sign is flipped.
The Topology base class no longer takes a list of elements in its constructor. Instead, the __iter__ method should be implemented by the derived class, as well as __len__ for the number of elements,
and getelem(index) to access individual elements. The 'elements' attribute is deprecated.
The nutils.topology.StructuredTopology object no longer accepts an array with elements. Instead, an 'axes' argument is provided with information that allows it to generate elements in the fly. The
'structure' attribute is deprecated. A newly added shape tuple is now a documented attribute.
Two global properties have been renamed as follows:
• dumpdir → outdir
• outdir → outrootdir
The outrootdir defaults to ~/public_html and can be redefined from the command line or in the .nutilsrc configuration file. The outdir defaults to the current directory and is redefined by util.run,
nesting the name/date/time subdirectory sequence under outrootdir.
The behaviour of nutils.function.sum is inconsistent with that of the Numpy counterparts. In case no axes argument is specified, Numpy sums over all axes, whereas Nutils sums over the last axis. To
undo this mistake and transition to Numpy's behaviour, calling sum without an axes argument is deprecated and will be forbidden in Nutils 3.0. In Nutils 4.0 it will be reintroduced with the corrected
The nutils.function.outer method allows arguments of different dimension by left-padding the smallest prior to multiplication. There is no clear reason for this generality and it hinders error
checking. Therefore in future in function.outer(a, b), a.ndim must equal b.ndim. In a brief transition period non-equality emits a warning.
Relevant only for custom nutils.function.Evaluable objects, the evalf method changes from constructor argument to instance/class method:
class MyEval( function.Evaluable):
def __init__(self, ...):
function.Evaluable(args=[...], shape=...)
def evalf( self, ...):
Moreover, the args argument may only contain Evaluable objects. Static information is to be passed through self.
At this point Nutils is pure Python. It is no longer necessary to run make to compile extension modules. The numeric.py module remains unchanged.
Touching elements of periodic domains are no longer part of the boundary topology. It is still available as boundary of an appropriate non-periodic subtopology:
domain.boundary['left'] # no longer valid
domain[:,:1].boundary['left'] # still valid
The new nutils.transform module provides objects and operations relating to affine coordinate transformations.
The new command line switch --tbexplore activates the traceback explorer on program failure. To change the default behavior add tbexplore=True to your .nutilsrc file.
The new command line switch --richoutput activates color and unicode output. To change the default behavior add richoutput=True to your .nutilsrc file.
Nutils 1.0 was released on August 4th, 2014.
Nutils 0.0 was released on October 28th, 2013.
The fastest way to build a new Nutils simulation is to borrow bits and pieces from existing scripts. Aiming to facilitate this practice, the following website provides an overview of concise examples
demonstrating different areas of physics and varying computational techniques:
⧉ https://examples.nutils.org
The examples are taken both from the Nutils repository and from user contributed repositories, and are tested regularly to confirm validity against different versions of Nutils.
Users are encouraged to contribute (concise versions of) their simulations to this collection of examples. In doing so, they help other users get up to speed, they help the developers by adding to a
large body of realistic codes to test Nutils against, and, in doing so, they may even help themselves by preventing future Nutils version from accidentally breaking their code.
Examples should resemble the official examples from the Nutils repository. In particular, they:
• use cli.run to call main function;
• have reasonable default parameters corresponding to a simulation that is relevant but not overly expensive;
• do not make use of undocumented functions (typically prefixed with an underscore);
• use the most recent version of the namespace, if applicable;
• generate one or more images that visualize the solution of the simulation;
• use treelog to communicate output (info or user for text, infofile or userfile for data);
• conform to the PEP 8 coding style;
• are concise enough to fit a single file.
Examples are submitted by means of a pull request to the examples repository, which should add a yaml file to the examples/user directory. The file should define the following entries:
• name — Title of the simulation.
• authors — List of author names.
• description — Markdown formatted description of the simulation.
• repository — URL of the Git repository that contains the script.
• commit — Commit hash.
• script — Path of the script.
• images — List of images that are selected as preview.
• tags — List of relevant tags.
Once merged, the script becomes part of the automated testing suite which runs it at regular intervals against the latest Nutils version. The code itself remains hosted on the external git
repository. In case new features merit updates to the script, the developers may reach out with concrete suggestions to keep the examples relevant.
Nutils has been used in scientific publications since its earliest releases, such as this 2015 analysis of a trabecular bone fragment by Verhoosel et al, combining several of Nutils' strengths
including the Finite Cell Method, hierarchical refinement, and isogeometric analysis. One of its images was later selected to feature as cover art for the Encyclopedia of Computational Mechanics.
Nutils has since been used in a wide range of applications, pushing the boundaries of computational techniques, studying physical phenomena, and testing new models. The publication overview lists an
up to date selection of Nutils powered research. If you are using Nutils in your own research, please consider citing Nutils in your publications.
Below is an overview of mostly peer reviewed articles that use Nutils for numerical experiments. In case your Nutils powered research is not listed here, please send the DOI to info@nutils.org or
submit a pull request with the new entry. Articles that cite Nutils will be picked up automatically.
• Error estimation and adaptive moment hierarchies for goal-oriented approximations of the Boltzmann equation by M.R.A. Abdelmalik and E.H. van Brummelen, Computer Methods in Applied Mechanics and
Engineering, October 2017.
• An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies by M. Baumann, R. Astudillo, Y. Qiu, E.Y.M. Ang, M.B. van Gijzen and R.E.
Plessix, Springer Computational Geosciences, June 2017.
• Mixed Isogeometric Finite Cell Methods for the Stokes problem by T. Hoang, C.V. Verhoosel, F. Auricchio, E.H. van Brummelen and A. Reali, Computer Methods in Applied Mechanics and Engineering,
April 2017.
• Elasto-capillarity Simulations based on the Navier-Stokes-Cahn-Hilliard Equations by E.H. Van Brummelen, M. Shokrpour-Roudbari and G.J. Van Zwieten, Advances in Computational Fluid-Structure
Interaction and Flow Simulation, October 2016.
• A fracture-controlled path-following technique for phase-field modeling of brittle fracture by N. Singh, C.V. Verhoosel, R. de Borst and E.H. van Brummelen, Finite Elements in Analysis and Design
, June 2016.
• Condition number analysis and preconditioning of the finite cell method by F. de Prenter, C.V. Verhoosel, G.J. van Zwieten and E.H. van Brummelen, Computer Methods in Applied Mechanics and
Engineering, January 2016.
• Stabilized second-order convex splitting schemes for Cahn–Hilliard models with application to diffuse-interface tumor-growth models by X. Wu, G.J. van Zwieten and K.G. van der Zee, Special Issue
of Numerical Methods and Applications of Multi-Physics in Biomechanical Modeling, September 2013.
• Adaptive Time-Stepping for Cahn-Hilliard-type Equations with Application to Diffuse-Interface Tumor-growth Models (pdf) by X. Wu, G.J. Van Zwieten, K.G. van der Zee and G. Simsek, ADMOS 2013,
March 2013.
• Shape-Newton Method for Isogeometric Discretizations of Free-Boundary Problems by K.G. van der Zee, G.J. van Zwieten, C.V. Verhoosel and E.H. van Brummelen, MARINE 2011, IV International
Conference on Computational Methods in Marine Engineering, February 2013.
To acknowledge Nutils in publications, authors are encouraged to cite the specific version of Nutils that was used to generate their results. To this end, Nutils releases are assigned a Digital
Object Identifier (DOI) by Zenodo which can be used for citations. For instance, a bibliography entry for Nutils 7.0 could look like this:
For LaTeX documents, the corresponding bibtex entry would be:
title = {Nutils 7.0},
author = {van Zwieten, J.S.B. and van Zwieten, G.J. and Hoitinga, W.},
publisher = {Zenodo},
year = {2022},
doi = {10.5281/zenodo.6006701},
Note that Zenodo can additionally host and assign a DOI to code that is specific to the publication, which is a great way to share digital artifacts and ensure reproducibility of the research.
For questions that are not answered by the tutorial, the API reference for the relevant release, or the examples, there are a few avenues for getting additional support.
Questions that lend themselves to be formulated in a concise and general way can be made into a Q&A topic, where both developers and advanced users can weigh in their answers, and where they may be
of benefit to others encountering the same issue. Be sure to check first if your issue was not discussed already!
If you believe that you have spotted a bug, the best thing to do is to file an issue. Issues should contain a description of the problem, the expected behaviour, and steps to reproduce, including the
version of Nutils that the issue relates to. If you believe that the bug was recently introduced you can help the developers by identifying the first failing commit, for instance using bisection.
Finally, for general discussions, questions, suggestions, or just to say hello, everybody is welcome to join the nutils-users support channel at #nutils-users:matrix.org. Note that you will need to
create an account at any Matrix server in order to join this channel. | {"url":"https://nutils.org/print.html","timestamp":"2024-11-04T13:31:19Z","content_type":"text/html","content_length":"163933","record_id":"<urn:uuid:261b45e3-60d9-4454-abf2-2910f543a1e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00830.warc.gz"} |
Question #dfd4e | Socratic
Question #dfd4e
1 Answer
Here's my take on this one.
I'm not really sure about what exactly you want to know, so I'll try and cover as many options as possible.
So, let's assume that you want to
• dilute a 0.1 mL sample by adding 0.9 mL of water
This one is pretty straightforward. When you dilute a 0.1-mL sample by adding 0.9 mL of water, you're essentially performing a $1 : 10$ dilution.
The volume of the final solution will be
${V}_{\text{final" = 0.1 + 0.9 = "1 mL}}$
The dilution factor, which is simply the ratio between the final volume of the solution and the initial volume of the sample, will be
$\text{DF" = V_"final"/V_"initial}$
#"DF" = (1color(red)(cancel(color(black)("mL"))))/(0.1color(red)(cancel(color(black)("mL")))) = 10#
• dilute a 0.1-mL sample by a specific dilution factor
If you want to dilute a 0.1-mL sample by a specific dilution factor, you can determine how much water you need to add by using the formula for the dilution factor to calculate the final volume of the
Let's say that you want to dilute this sample by a dilution factor of $50$. You would have
$\text{DF" = V_"final"/V_"initial" implies V_"final" = "DF" * V_"initial}$
${V}_{\text{final" = 50 * "0.1 mL" = "5 mL}}$
This means that you need to add
${V}_{\text{water" = V_"final" - V_"initial" = 5 - 0.1 = "4.9 mL water}}$
to dilute a 0.1-mL sample by a dilution factor of 50.
• dilute a 0.1-mL sample to a specific final volume
This one is easier to calculate, because you know what the volume of the final solution must be. The difference between this volume and the initial volume of the sample will represent the volume of
water you need to add.
${V}_{\text{final" = V_"water" + V_"initial}}$
Let's say that you want to dilute the sample to a final volume of 0.9 mL. This would mean that you must add
${V}_{\text{water" = 0.9 - 0.1 = "0.8 mL water}}$
In this case, the dilution factor would be
#"DF" = (0.9color(red)(cancel(color(black)("mL"))))/(0.1color(red)(cancel(color(black)("mL")))) = 9#
The formulas are true regardless of what the volume of the initial sample is, so you can use these as a guideline to help you with your dilution calculations.
Impact of this question
2810 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/55d5778a11ef6b0644adfd4e","timestamp":"2024-11-06T08:22:53Z","content_type":"text/html","content_length":"39539","record_id":"<urn:uuid:3796513a-8b60-436d-ac1b-4d8212d29e69>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00373.warc.gz"} |
Cubic yds to tons
Disclaimer: Whilst every effort has been made in building our calculator tools, we are not to be held liable for any damages or monetary losses arising out of or in connection with their use. Full
Calculate the amount of gravel or aggregate needed in tons and cubic yards by entering the dimensions below. Joe is the creator of Inch Calculator and has over 20 years of experience in engineering
and construction. He holds several degrees and certifications. Full bio. Sarabeth is an expert in the home and garden industry and was formerly a certified kitchen and bathroom designer. She is a
subject matter expert in home improvement and is often quoted in notable publications.
Cubic yds to tons
Before purchasing gravel, mulch, dirt or rock you will need to determine how much you need. Mulch and dirt are sold by the yard. To determine the area you wish to cover, you will need a tape measure.
A pen or pencil and a sheet of paper are also helpful. With the help of another person, if possible, stretch your measuring tape from one end of the area you wish to cover to the other. If your
measuring tape is too short to measure the entire length of the area at once, clearly mark its stopping point with a stick or other thin object and then measure the rest. Add the total measurements
to find the length. Write this number down on your sheet of paper. Once you have done so, write that number down, as well. One to 2 inches is common. Once you have made this decision, considering any
surrounding vegetation that might require rainwater, note the depth on your paper. Then, multiply your length by the width by the depth of gravel. This number is the cubic area.
Most suppliers sell gravel by the ton, but some may sell it by the yard, particularly if you only need a small amount.
Using this cubic yards to tons converter, you can estimate the mass in tons of any substance of known density from its volume in cubic yards. Not only that, with our tool, you can convert the mass in
metric tons to US tons, long tons, pounds, and kilograms and vice versa. We also explain how to convert the mass in metric tons to long tons and US tons. Similarly, our Gallons to pounds converter
will help you figure out how many pounds to a gallon of a given fluid. A cubic yard cu yd is the volume of a cube with the length, width, and height of one yard yd. For more on volume conversion,
head to our volume converter.
Cubic yds to tons
Disclaimer: Whilst every effort has been made in building our calculator tools, we are not to be held liable for any damages or monetary losses arising out of or in connection with their use. Full
disclaimer. To roughly convert cubic yards to tons you can multiply your cubic yard figure by 1. This conversion will give an approximation for many sand and gravel products. To get a more accurate
conversion, you'll need to involve a density figure in your calculation. This is because the cubic yard is a unit of volume and the ton or tonne is a unit of weight. To convert between the two,
you'll need to know the density of the material.
Kanal d evim şahane başvuru formu doldur
To roughly convert cubic yards to tons you can multiply your cubic yard figure by 1. Order Now. While this varies by supplier, many sell large volumes by weight in tons. To give a crude example, the
weight of a cubic yard of feathers will be much lighter than a cubic yard of sand. I have supplied the following material density approximations for converting cubic yards to US tons and metric tons.
Add the total measurements to find the length. For a super-accurate conversion, give your topsoil supplier a call to find out the exact density of their product. Check out our stone and mulch
calculators for estimating other landscape materials. Write this number down on your sheet of paper. The average pickup truck can carry 1 cubic yard of gravel, while a dump truck can carry 13 to 25
tons of gravel. Keep in mind that not all gravel needs to be compacted, and some varieties compact more than others, so this much additional material will not always be required for every project.
How do I convert cubic yards to tons? Order Now. Here are some instructions for how to enable JavaScript in your browser. This number is the cubic area. In addition, if you are backfilling an area
with gravel, there may be voids that you have not accounted for, which will increase the amount of gravel needed to complete the job. In addition, there are 2, pounds to a ton. Multiply the length,
width, and height together to find the volume of the space. With the help of another person, if possible, stretch your measuring tape from one end of the area you wish to cover to the other. The
figures below use estimates of densities from Engineering Toolbox and SI Metric and should be used as a rough guide only. Popup calculator Copy a link to this page Print this page Email a link to
this page Scroll up to form What does this mean? If your measuring tape is too short to measure the entire length of the area at once, clearly mark its stopping point with a stick or other thin
object and then measure the rest. This conversion will give an approximation for many sand and gravel products. See the chart below to see how many tons are in a cubic yard of gravel.
2 thoughts on “Cubic yds to tons”
1. What interesting question
2. I consider, that you are mistaken. I can defend the position. Write to me in PM. | {"url":"https://gorgoroth.com.pl/cubic-yds-to-tons.php","timestamp":"2024-11-07T16:07:20Z","content_type":"text/html","content_length":"26963","record_id":"<urn:uuid:03833cb9-1caf-4153-8fc5-f34a42939c0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00509.warc.gz"} |
Subtracting 36203 - math word problem (36203)
Subtracting 36203
The petals of each always have something in common. Can you figure out what number will be in the middle of the flower so that when subtracting and adding, the numbers come from the flowers?
The flowers are numbered 50, 30, 20, 40, and 10.
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Grade of the word problem:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/36203","timestamp":"2024-11-06T08:22:17Z","content_type":"text/html","content_length":"62131","record_id":"<urn:uuid:aca83d14-74ed-4e64-999e-e57b68e08722>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00427.warc.gz"} |
Intensive Mathematics
General Course Information and Notes
Version Description
For each year in which a student scores at Level 1 on FCAT 2.0 Mathematics, the student must receive remediation by completing an intensive mathematics course the following year or having the
remediation integrated into the student's required mathematics course. This course should be tailored to meet the needs of the individual student. Appropriate benchmarks from the following set of
standards should be identified to develop an appropriate curriculum.
General Information
Course Number: 1200400
Course Path:
Abbreviated Title: INTENS MATH
Number of Credits: Multiple Credit (more than 1 credit)
Course Type: Elective Course
Course Level: 2
Course Status: Course Approved
Grade Level(s): 9,10,11,12
Educator Certifications
One of these educator certification options is required to teach this course.
Student Resources
Vetted resources students can use to learn the concepts and skills in this course.
Original Student Tutorials
Educational Games
Solving Inequalities: Inequalities and Graphs of Inequalities:
In this challenge game, you will be solving inequalities and working with graphs of inequalities. Use the "Teach Me" button to review content before the challenge. During the challenge you get one
free solve and two hints! After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple
times as needed! Good luck!
Type: Educational Game
Timed Algebra Quiz:
In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students
to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the
topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Educational Game
Algebra Four:
In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to
practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including
background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.
Type: Educational Game
Educational Software / Tools
Two Way Frequency Excel Spreadsheet:
This Excel spreadsheet allows the educator to input data into a two way frequency table and have the resulting relative frequency charts calculated automatically on the second sheet. This resource
will assist the educator in checking student calculations on student-generated data quickly and easily.
Steps to add data: All data is input on the first spreadsheet; all tables are calculated on the second spreadsheet
1. Modify column and row headings to match your data.
2. Input joint frequency data.
3. Click the second tab at the bottom of the window to see the automatic calculations.
Type: Educational Software / Tool
Transformations Using Technology:
This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to
result in a specified final image.
Type: Educational Software / Tool
Lesson Plan
Do Credit Cards Make You Gain Weight? What is Correlation, and How to Distinguish It from Causation:
This lesson introduces the students to the concepts of correlation and causation, and the difference between the two. The main learning objective is to encourage students to think critically about
various possible explanations for a correlation, and to evaluate their plausibility, rather than passively taking presented information on faith. To give students the right tools for such analysis,
the lesson covers most common reasons behind a correlation, and different possible types of causation.
Type: Lesson Plan
Perspectives Video: Experts
Jumping Robots and Quadratics:
<p>Jump to it and learn more about how quadratic equations are used in robot navigation problem solving!</p>
Type: Perspectives Video: Expert
Problem Solving with Project Constraints:
<p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help
him solve problems!</p>
Type: Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Base 16 Notation in Computing:
Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Professional/Enthusiast
Unit Conversions:
<p>Get fired up as you learn more about ceramic glaze recipes and mathematical units.</p>
Type: Perspectives Video: Professional/Enthusiast
Making Candy: Illuminating Exponential Growth:
<p>No need to sugar coat it: making candy involves math and muscles. Learn how light refraction and exponential growth help make candy colors just right!</p>
Type: Perspectives Video: Professional/Enthusiast
Making Candy: Uniform Scaling:
<p>Don't be a shrinking violet. Learn how uniform scaling is important for candy production.</p>
Type: Perspectives Video: Professional/Enthusiast
Using Geometry and Computers to make Art with CNC Machining:
<p>See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.</p>
Type: Perspectives Video: Professional/Enthusiast
Estimating Oil Seep Production by Bubble Volume:
<p>You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!</p>
Type: Perspectives Video: Professional/Enthusiast
The Pythagorean Theorem: Geometry’s Most Elegant Theorem:
This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra
and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties
of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3)
Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and
real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more
time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the
activities include scissors, tape, string and markers.
Type: Presentation/Slideshow
Problem-Solving Tasks
Student Center Activity
What is a Function?:
This video will demonstrate how to determine what is and is not a function.
Type: Video/Audio/Animation
Relations and Functions:
This video demonstrates how to determine if a relation is a function and how to identify the domain.
Type: Video/Audio/Animation
Real-Valued Functions of a Real Variable:
Although the domain and codomain of functions can consist of any type of objects, the most common functions encountered in Algebra are real-valued functions of a real variable, whose domain and
codomain are the set of real numbers, R.
Type: Video/Audio/Animation
Roots and Unit Fraction Exponents:
Exponents are not only integers. They can also be fractions. Using the rules of exponents, we can see why a number raised to the power " one over n" is equivalent to the nth root of that number.
Type: Video/Audio/Animation
Rational Exponents:
Exponents are not only integers and unit fractions. An exponent can be any rational number expressed as the quotient of two integers.
Type: Video/Audio/Animation
Simplifying Radical Expressions:
Radical expressions can often be simplified by moving factors which are perfect roots out from under the radical sign.
Type: Video/Audio/Animation
Solving Mixture Problems with Linear Equations:
Mixture problems can involve mixtures of things other than liquids. This video shows how Algebra can be used to solve problems involving mixtures of different types of items.
Type: Video/Audio/Animation
Using Systems of Equations Versus One Equation:
When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?
Type: Video/Audio/Animation
Systems of Linear Equations in Two Variables:
The points of intersection of two graphs represent common solutions to both equations. Finding these intersection points is an important tool in analyzing physical and mathematical systems.
Type: Video/Audio/Animation
Why the Elimination Method Works:
This chapter presents a new look at the logic behind adding equations- the essential technique used when solving systems of equations by elimination.
Type: Video/Audio/Animation
Point-Slope Form:
The point-slope form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the slope and the coordinates of a single point which lies on the line.
Type: Video/Audio/Animation
Two Point Form:
The two point form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the coordinates of two points which lie on the line.
Type: Video/Audio/Animation
Linear Equations in the Real World:
Linear equations can be used to solve many types of real-word problems. In this episode, the water depth of a pool is shown to be a linear function of time and an equation is developed to model its
behavior. Unfortunately, ace Algebra student A. V. Geekman ends up in hot water anyway.
Type: Video/Audio/Animation
Solving Literal Equations:
Literal equations are formulas for calculating the value of one unknown quantity from one or more known quantities. Variables in the formula are replaced by the actual or 'literal' values
corresponding to a specific instance of the relationship.
Type: Video/Audio/Animation
Parallel Lines 2:
This video shows how to determine which lines are parallel from a set of three different equations.
Type: Video/Audio/Animation
Parallel Lines:
This video illustrates how to determine if the graphs of a given set of equations are parallel.
Type: Video/Audio/Animation
Perpendicular Lines 2:
This video describes how to determine the equation of a line that is perpendicular to another line. All that is given initially the equation of a line and an ordered pair from the other line.
Type: Video/Audio/Animation
Basic Linear Function:
This video demonstrates writing a function that represents a real-life scenario.
Type: Video/Audio/Animation
Quadratic Functions 2:
This video gives a more in-depth look at graphing quadratic functions than previously offered in Quadratic Functions 1.
Type: Video/Audio/Animation
MIT BLOSSOMS - Fabulous Fractals and Difference Equations :
This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school
algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework
assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a
blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and
practice calculating trajectories of difference equations.
Type: Video/Audio/Animation
Graphing Lines 1:
Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"
Type: Video/Audio/Animation
Fitting a Line to Data:
Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.
Type: Video/Audio/Animation
This Khan Academy video tutorial introduces averages and algebra problems involving averages.
Type: Video/Audio/Animation
Virtual Manipulatives
Circumscribe a Circle About a Triangle:
In this GeoGebraTube interactive worksheet, you can watch the step by step process of circumscribing a circle about a triangle. Â Using paper and pencil along with this resource will reinforce the
Type: Virtual Manipulative
Inscribe a Regular Hexagon in a Circle:
This geogebratube interactive worksheet shows the step by step process for inscribing a regular hexagon in a circle. There are other geogebratube interactive worksheets for the square and the
equilateral triangle.
Type: Virtual Manipulative
3-D Conic Section Explorer:
Using this resource, students can manipulate the measurements of a 3-D hourglass figure (double-napped cone) and its intersecting plane to see how the graph of a conic section changes. Â Students
will see the impact of changing the height and slant of the cone and the m and b values of the plane on the shape of the graph. Students can also rotate and re-size the cone and graph to view from
different angles.Â
Type: Virtual Manipulative
Combining Transformations:
In this manipulative activity, you can first get an idea of what each of the rigid transformations look like, and then get to experiment with combinations of transformations in order to map a
pre-image to its image.
Type: Virtual Manipulative
Solving Quadratics By Taking The Square Root:
This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square
root' by Khan Academy.
Type: Virtual Manipulative
Slope Slider:
In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in
slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes
supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Cross Section Flyer - Shodor:
With this online Java applet, students use slider bars to move a cross section of a cone, cylinder, prism, or pyramid. This activity allows students to explore conic sections and the 3-dimensional
shapes from which they are derived. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and
exploration questions for use with the java applet.
Type: Virtual Manipulative
Linear Function Machine:
In this activity, students plug values into the independent variable to see what the output is for that function. Then based on that information, they have to determine the coefficient (slope) and
constant(y-intercept) for the linear function. This activity allows students to explore linear functions and what input values are useful in determining the linear function rule. This activity
includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.
Type: Virtual Manipulative
Graphing Lines:
Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.
Type: Virtual Manipulative
Box Plot:
In this activity, students use preset data or enter in their own data to be represented in a box plot. This activity allows students to explore single as well as side-by-side box plots of different
data. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the
Java applet.
Type: Virtual Manipulative
Data Flyer:
Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider
bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes
supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Function Matching:
This is a graphing tool/activity for students to deepen their understanding of polynomial functions and their corresponding graphs. This tool is to be used in conjunction with a full lesson on
graphing polynomial functions; it can be used either before an in depth lesson to prompt students to make inferences and connections between the coefficients in polynomial functions and their
corresponding graphs, or as a practice tool after a lesson in graphing the polynomial functions.
Type: Virtual Manipulative
Normal Distribution Interactive Activity:
With this online tool, students adjust the standard deviation and sample size of a normal distribution to see how it will affect a histogram of that distribution. This activity allows students to
explore the effect of changing the sample size in an experiment and the effect of changing the standard deviation of a normal distribution. Tabs at the top of the page provide access to supplemental
materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Function Flyer:
In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of
functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the
topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Advanced Data Grapher:
This is an online graphing utility that can be used to create box plots, bubble graphs, scatterplots, histograms, and stem-and-leaf plots.
Type: Virtual Manipulative
Number Cruncher:
In this activity, students enter inputs into a function machine. Then, by examining the outputs, they must determine what function the machine is performing. This activity allows students to explore
functions and what inputs are most useful for determining the function rule. This activity includes supplemental materials, including background information about the topics covered, a description of
how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Curve Fitting:
With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic.
Best fit or adjustable fit can be displayed.
Type: Virtual Manipulative
Equation Grapher:
This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes
in the graph(s).
Type: Virtual Manipulative
Line of Best Fit:
This manipulative allows the user to enter multiple coordinates on a grid, estimate a line of best fit, and then determine the equation for a line of best fit.
Type: Virtual Manipulative
A Plethora of Polyhedra:
This program allows users to explore spatial geometry in a dynamic and interactive way. The tool allows users to rotate, zoom out, zoom in, and translate a plethora of polyhedra. The program is able
to compute topological and geometrical duals of each polyhedron. Geometrical operations include unfolding, plane sections, truncation, and stellation.
Type: Virtual Manipulative
Histogram Tool:
This virtual manipulative histogram tool can aid in analyzing the distribution of a dataset. It has 6 preset datasets and a function to add your own data for analysis.
Type: Virtual Manipulative
Multi Bar Graph:
This activity allows the user to graph data sets in multiple bar graphs. The color, thickness, and scale of the graph are adjustable which may produce graphs that are misleading. Users may input
their own data, or use or alter pre-made data sets. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the
application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
In this activity, students can create and view a histogram using existing data sets or original data entered. Students can adjust the interval size using a slider bar, and they can also adjust the
other scales on the graph. This activity allows students to explore histograms as a way to represent data as well as the concepts of mean, standard deviation, and scale. This activity includes
supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this course. | {"url":"https://www.cpalms.org/PreviewCourse/Preview/10333?isShowCurrent=false","timestamp":"2024-11-03T07:24:17Z","content_type":"text/html","content_length":"463915","record_id":"<urn:uuid:b49e8156-81a4-44bb-b014-5b73e844f126>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00614.warc.gz"} |
The Stacks project
Lemma 37.8.2. If $f : X \to S$ is a formally étale morphism, then given any solid commutative diagram
\[ \xymatrix{ X \ar[d]_ f & T \ar[d]^ i \ar[l] \\ S & T' \ar[l] \ar@{-->}[lu] } \]
where $T \subset T'$ is a first order thickening of schemes over $S$ there exists exactly one dotted arrow making the diagram commute. In other words, in Definition 37.8.1 the condition that $T$ be
affine may be dropped.
Comments (0)
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder,
this is tag 04FD. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 04FD, in case you are confused. | {"url":"https://stacks.math.columbia.edu/tag/04FD","timestamp":"2024-11-08T08:16:00Z","content_type":"text/html","content_length":"14557","record_id":"<urn:uuid:a15be7d4-42ea-4fee-9f85-8e3650620932>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00033.warc.gz"} |
ree - Kruskal with Disjoint Set Union
Minimum spanning tree - Kruskal with Disjoint Set Union¶
For an explanation of the MST problem and the Kruskal algorithm, first see the main article on Kruskal's algorithm.
In this article we will consider the data structure "Disjoint Set Union" for implementing Kruskal's algorithm, which will allow the algorithm to achieve the time complexity of $O(M \log N)$.
Just as in the simple version of the Kruskal algorithm, we sort all the edges of the graph in non-decreasing order of weights. Then put each vertex in its own tree (i.e. its set) via calls to the
make_set function - it will take a total of $O(N)$. We iterate through all the edges (in sorted order) and for each edge determine whether the ends belong to different trees (with two find_set calls
in $O(1)$ each). Finally, we need to perform the union of the two trees (sets), for which the DSU union_sets function will be called - also in $O(1)$. So we get the total time complexity of $O(M \log
N + N + M)$ = $O(M \log N)$.
Here is an implementation of Kruskal's algorithm with Union by Rank.
vector<int> parent, rank;
void make_set(int v) {
parent[v] = v;
rank[v] = 0;
int find_set(int v) {
if (v == parent[v])
return v;
return parent[v] = find_set(parent[v]);
void union_sets(int a, int b) {
a = find_set(a);
b = find_set(b);
if (a != b) {
if (rank[a] < rank[b])
swap(a, b);
parent[b] = a;
if (rank[a] == rank[b])
struct Edge {
int u, v, weight;
bool operator<(Edge const& other) {
return weight < other.weight;
int n;
vector<Edge> edges;
int cost = 0;
vector<Edge> result;
for (int i = 0; i < n; i++)
sort(edges.begin(), edges.end());
for (Edge e : edges) {
if (find_set(e.u) != find_set(e.v)) {
cost += e.weight;
union_sets(e.u, e.v);
Notice: since the MST will contain exactly $N-1$ edges, we can stop the for loop once we found that many.
Practice Problems¶
See main article on Kruskal's algorithm for the list of practice problems on this topic. | {"url":"https://gh.cp-algorithms.com/main/graph/mst_kruskal_with_dsu.html","timestamp":"2024-11-05T20:08:30Z","content_type":"text/html","content_length":"132502","record_id":"<urn:uuid:000cdbd9-0814-4e12-bc5e-1232dbb9425b>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00803.warc.gz"} |
Hydrogen Atom: Radial Solution
Mind Network - Samuel Solomon
Hydrogen Atom
Radial Solution; Any Orbital
In the last page, we have solved the angular solution to the Schrödinger equation for any spherically symmetric potential for any 'L' quantum number. This included the Hydrogen atom's angular
solution (the spherical harmonics). In order to finish off our quantum mechanics solution of the Hydrogen atom, we also need to solve the radial component. As a reminder, the two Schrödinger
equations for spherically symmetric potentials were as follows:
We did not need to define a potential to solve for the angular solution, but we will need to define a potential for the radial component. A one electron atom, like the Hydrogen atom, consists of only
a nucleus and a sole electron. The electron is held (bounded) by the nucleus by an electrostatic interaction (a radial force). Our quantum picture of this system is diagrammed below:
Where k = Coulomb constant; Z = nuclear charge (Hydrogen: Z = +1); q = magnitude of charge (for one electron or proton)
In the diagram, we only show the electron moving. This is because the nuclear mass is much bigger than the electron's mass, and hence, for the same force, the electron accelerates and moves a lot
more. It moves so much, that the nuclear movements are rather negligible in comparison (we don't really see the nucleus move that much). We call this the Born-Oppenheimer approximation (and sometimes
the Franck-Condon limit depending on your field). Regardless, the Schrödinger equation does not differentiate on this basis, but I thought I would point this out.
What is important to solve is the Schrödinger equation and for that all we need is the potential energy. From non-relativistic electromagnetism, we know the force and potential energy exerted on two
charged particles (shown in the above diagram on the right), and hence also know the potential energy.
For a quick check, we can see that the electrostatic forces are radial forces, which produces a spherically symmetric potential (which we assumed in equation 2). Therefore, we are now ready to solve
the radial Schrödinger equation for the Hydrogen atom. It will be a mathematically challenging proof, so we will make 3 simplifying substitutions to our radial wave equation first:
We can now plug these into the radial equation (equation 2):
In order to make sure everyone is following, let us review some key steps below:
2: The radial Schrödinger equation for any 'L' quantum number
2 to 8: Multiply both sides by R(r) and divide by 'r.' We also plugged in our Hydrogen atom potential for V(r)
8 to 9: Replace the red 'r R(r)' with 'U(r)' and the purple 'dr^2' with 'dP^2' (also moved the U(r) terms onto the RHS)
9 to 10: Divide out the purple term
10 to 11: Simplify the middle term on the right and repalce the purple term with 1/P^2 (equation 6)
11 to 12: Replace the blue middle term on the RHS with 'P_0/P' (equation 7)
Now this differential equation in line 12 is less of a headache to solve (don't have to worry about tons of constants). In order to solve it, like any other differential equations, we need an ansatz
(a trial function). In order to have a good anstaz, lets first get a general sense of the solution by looking at its extreme values (limiting cases): we know that as r (and by extension P) goes to
infinity, the wave function R(r) (and by extension U(r)) must go to zero. We can evaluate this limit below:
In order to make sure everyone is following, let us review some key steps below:
12: Our equation we are trying to solve
12 to 13: We take the limit of the equation as r (or P) goes to infinity (1/P is now essentially zero)
13 to 14: We solve the differential equation
14 to 15: As P goes to infinity, U(r) must go to zero, which cannot happen with the 'e^P' term; hence, B must be zero
We do a quick mathematical check on the right to make sure we solved the differential equation correctly.
We now have a general idea of how the solution acts as 'r' or 'P' goes to +/- infinity. As 'P' goes to +/- infinity, the exponential term in equation 15 dominates the value of U(r). However, as 'P'
becomes smaller and smaller, the co-factor 'A' is no longer an insignificant value. While it may have acted like a constant in the limit, it could possibly vary with 'P.' We can account for that in
our new general ansatz below:
Just like for the quantum harmonic oscillator and the one-electron atom case, we could keep going to solve for this function f(P). However, one will find this extremely difficult ... our ansatz isn't
good enough just yet (this is just my foresight into the problem; with a lot of pain it probably can still be solved). To add more information to our ansatz (our guess function), let us also evaluate
U(r) as 'r' or 'P' goes to zero. While we do not know this specific value, we do know that it has to be a finite number (we can never diverge to infinity or the area under the curve won't be finite /
normalizable). We can evaluate this boundary below:
In order to make sure everyone is following, let us review some key steps below:
12: Our equation we are trying to solve
12 to 20: We take the limit of the equation as r (or P) goes to zero
20 to 21: We solve the differential equation
21 to 22: As P goes to zero, U(r) must be finite, which cannot happen with the 'P^-L' term (~1/0); hence, D must be zero
We do a quick mathematical check on the right to make sure we solved the differential equation correctly.
We now have a general idea of how the solution acts as 'r' or 'P' goes to +/- infinity AND as 'r' or 'P' goes to zero. Using this combined understanding of how the function is supposed to look like,
we can make an even better educated guess (ansatz) for what U(r) should be. As 'P' goes to +/- infinity, the exponential term in equation 15 dominates the value of U(r) and as 'P' goes to zero the P^
(L+1) term dominates. We can account for that in our new general ansatz below:
Where h(P) is some function of 'P' we don't know yet. We can now plug back in our ansatz to equation 12 to solve for the explicit form of h(P)
In order to make sure everyone is following, let us review some key steps below:
28: Plug our ansatz into equation 12
28 to 30: Working with the middle term, take the first and second derivative of the ansatz (simplified it as well)
30 to 31: Reshow the RHS of equation 28
31: Combine the LHS (equation 30) and the RHS (equation 31). We cancel out the 'P^L * e^-P' terms
32 to 33: Simplify the expression
We now have a second order differential equation for the h(P) function. In order to solve a differential equation (just like before) we need another ansatz. This time, we will expand out h(P) into a
polynomial series (see power series for reference). For those unfamiliar with this trick, it is the same principle of Taylor series. We will show this below:
In order to make sure everyone is following, let us review some key steps below:
34: The definition of a power series
35 and 36: We apply the derivatives on the power series. We also reformat the indeces (see below)
Note the switch in indeces for equations 35 and 26. We do this because when J = 0 we are adding zero in our sum (which is trivial). We therefore skip over this index in the summation (as it is just
adding in zero). In order to do this, we add 1 to ever J term.
Let us now plug this ansatz into equation 33 to solve for the 'a_J' terms in the polynomial expansion of h(P):
In order to make sure everyone is following, let us review some key steps below:
33: Our differential equation we need to solve for h(P)
37: Plug in our power series expansion of h(P). Note: I split the h'(P) terms. My goal is to eventually pull out 'P^J'
37 to 38: Group the terms by 'a_J+1' and 'a_J.' I additionally pull out a 'P^J' term.
38 to 39: 'P' (a function of 'r') spans from +/- infinity. Hence, its coefficient must sum to zero at all 'P' values / indeces
39 to 40: Solve for 'a_j+1' in terms of 'a_j'
Some mathematicians might immediately see the problem with equation 40. However, it is not immediately obvious, so lets make it clear. Let us look at the limiting case as J goes to infinity (the
terms near the end of the summation):
In order to make sure everyone is following, let us review some key steps below:
41: Finding 'a_J+1' for large values of 'J' (as J goes to infinity)
41 to 42: Constants in the numerator and order(1) terms in the denominator are insignificant in the limit
42 to 43: Simplify the expression
43 to 44: We find The ratio of 'a_j+1' to ''a_j' to be 2/J
We can compare this answer to the expression on the right:
45: We can Taylor expand exp(P^2) into a format that matches the power series starting equation (for 'b' instead of 'a')
45 to 46: Now when we take the ratio, in the limit as J goes to infinity, we again find the same ratio 2/J
For very large 'J' values, our expression in equation 40 acts like exp(P^2). This is NOT good because, going back to the beginning, we MUST have the wave function normalize. If h(P) has an exp(P^2)
term then:
exp(P^2-P) does NOT normalize and cannot be how our equation acts as P (or r) goes to +/- infinity.
The only way to mathematically get around this fact is if we NEVER sum up these diverging terms at all. We need our summation to converge (as our wave function should never diverge to infinity);
hence, we need 'a_j+1' to go to zero at one point. Once 'a_j+1' = 0 at one point, then the rest of the terms in the infinite summation also equals zero and the finite summation will converge (not go
to infinity and we will not see these diverging terms)
In order to make sure everyone is following, let us review some key steps below:
48: Equation 40 (relationship between 'a' coefficients in the power series). For some J_max, 'a_j+1' must be zero
48 to 49: In order for this to be true, the numerator must be zero for some J_max (end of summation) value
49 to 50: Define a new constant 'n' to be J+L+1. 'J' and 'L' are integers so 'n' is also an integer
50 to 51: P_0 was a constant we defined in the beginning of the problem. We now plug back in its original value
51 to 52: Solve for the energy 'E'
NOTE: we initially stated that J (or n) started from zero. Now we know that n can actually never be zero, and the index really starts at n = 1 (we didn't know this initially, so I included the index
in equations 26 - 28). We can't have 1/0.Note that the 'n' values only go up to some 'n_max.'
And that is the energy of the Hydrogen atom. There are two very important notes about this energy:
1. Bohr got this EXACT SAME result earlier by just considering the electron as a standing wave going in a circle
2. The angular momentum 'L' quantum number does NOT effect the energy. Only the 'n' value
2. The energy of the Hydrogen atom is QUANTIZED (cannot take on certain energy values; constrained by 'n')
This energy quantization mathematically arose because of the necessity for the series to converge (for the wave function to be normalized). Because not all wave functions normalize, not all energies
are possible (remember, this is similar to the particle in a box situation / energy quantization).
A quick note about the Bohr model: it is wrong. Electrons do not just go in a 2D circle around the nucleus. It is crazy (lucky) that Bohr actually got the right answer. This is why the Bohr model was
so famous. Before people could really comprehend this mathematically, somehow Bohr was telling people the right answers for their experimental results. It had some things to do with luck, but Bohr
did use quantization of angular momentum (which back then, like now, was hard for people to think about). So Bohr's model did get some things right (and was very novel to think of). Going back to the
Bohr model, Bohr defined a very useful constant: the Bohr radius. We can use this constant below:
It is important to note that we can have further splitting of the energy due to spin magnetism, the Zeeman effect, hyperfine splitting, and relativistic effects, but for slow moving Hydrogen atoms
not in a magnetic field, this is the energy.
The last thing to do is to finish solving the wave function of the Hydrogen atom. To begin, let us plug back in our initial definitions from the top of the page:
And that is the brute force way of solving the Hydrogen atom radial equation for some 'n' index. We know that 'n' index's energy states as 'n' is related proportional to energy (so we call n = 1 the
first excited state, n = 2 the second excited state, and so on). To be clear, we start the index not at n = 0, but at n = 1 (but you can add +/- 1 to the constant as long as you are internally
consistent). For us, since energy is 1/n^2, n can never be zero.
Quantum numbers:
'n': We call 'n' a quantum number as it directly tells us information about an observable (the energy) of the atom
'J': We call 'J' a quantum number as mathematically (although not discussed) it is the number of radial nodes present
'L': We have already introduced 'L,' which tells us about the magnitude of the observable angular momentum.
'm': We have already found the magnetic quantum number 'm' (the z-projection of the angular momentum)
With all of our variables defined (and related to an experimental quantity), we can now solve the radial solution to the Hydrogen atom.
We have now solved the wave function psi of the Hydrogen atom (We found the radial component in this page and the angular component in the last page). Note, for the radial component, inputting an
energy will constrain the 'n' quantum number and inputting a specific orbital will constrain the 'L' quantum number. Altogether, knowing 'n' and 'L' constrained the quantum number 'J.' Once this is
know, everything else in the radial equation is either a constant or the variable 'r.'
The last steps when actually solving the Hydrogen atom would be to put the angular and radial components together (like in equation 58) and normalize the function over all 3D space. | {"url":"http://www.mindnetwork.us/hydrogen-atom-radial-solution.html","timestamp":"2024-11-02T23:34:09Z","content_type":"text/html","content_length":"116710","record_id":"<urn:uuid:d6bc8de6-d7d7-4e9a-94b4-42f3c22a9479>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00632.warc.gz"} |
People with Mathematician profession (First 49 people) - Page 0 - xwhos.com
Alan Greenspan
Former Chair of the Federal Reserve of the United States
James Harris Simons
American mathematician
Thomas Carlyle
John Redwood
Member of Parliament of the United Kingdom
Éamon de Valera
Former President of Ireland
Philip J. Hanlon
President of Dartmouth College
Wendy Hall
British computer scientist
Albert Einstein
Theoretical physicist
Isaac Newton
English mathematician
Galileo Galilei
Ferdinand Verbiest
Blaise Pascal
French mathematician
Evangelista Torricelli
Italian physicist
Greek philosopher
Brian Greene
American theoretical physicist
Robert Andrews Millikan
American physicist
Nicolaus Copernicus
Erwin Schrödinger
Austrian-Irish physicist
Paul Dirac
Theoretical physicist
Kenneth Arrow
American economist
Nicanor Parra
Chilean poet
Andrew Wiles
Pierre de Fermat
French mathematician
Marcus du Sautoy
British mathematician
Yutaka Taniyama
Japanese mathematician
Grigori Perelman
Russian mathematician
Steven Orszag
American mathematician
Greek philosopher
Ahmed Chalabi
Iraqi Politician
William of Ockham
English philosopher
Roger Penrose
British mathematician
Lady Byron
Thomas Harriot
English astronomer
John Forbes Nash Jr.
American mathematician
Ludwig Boltzmann
Austrian physicist
Wolfgang Schwarz
Austrian former figure skater
Guillermo Martínez
Cuban javelin thrower
Fred Sommers
American philosopher
Dugald Stewart
Scottish philosopher
Hero of Alexandria
Greek mathematician
Matteo Ricci
Italian football player
Joachim Bouvet
Charles Babbage
Christiaan Huygens
Dutch mathematician
Georg Ohm
German physicist
Benjamin Banneker
American naturalist
Vincenzo Viviani
Italian mathematician
Daniel Bernoulli
Swiss mathematician | {"url":"https://www.xwhos.com/job/mathematician.html","timestamp":"2024-11-05T23:19:34Z","content_type":"text/html","content_length":"53181","record_id":"<urn:uuid:bb1b4bf8-c792-485e-aa63-1a05c969cd02>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00880.warc.gz"} |
A particle originally at a rest at the highest point of a smoot-Turito
The pole of the straight line 9x + y – 28 = 0 with respect to the circle
Pole and polar are one of the important parts of the circle, many questions can be based on these parts. It should be remembered that while writing the equation of polar for a pole, the coefficients
of square terms in the equation of the circle is one. If the equation of the circle does not have coefficients as one, then make it one before solving the question. | {"url":"https://www.turito.com/ask-a-doubt/physics-a-particle-originally-at-a-rest-at-the-highest-point-of-a-smooth-circle-in-a-vertical-plane-is-gently-pushe-q56dbcf","timestamp":"2024-11-14T07:03:35Z","content_type":"application/xhtml+xml","content_length":"931120","record_id":"<urn:uuid:ff2fdd35-b9c9-4720-a3b4-7b8e539957de>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00198.warc.gz"} |
Aria Halavati
I am a 5th year PhD student in pure mathematics at Courant Institute of mathematical sciences and I obtained my Bachelors in Applied mathematics at Sharif University of Technology in 2019.
My research interests are in the areas of Calculus of Variations, Geometric Measure Theory and Partial Differential Equations.
PhD advisors: Guido De Philippis and Fang-Hua Lin
Undergraduate advisor: Morteza Fotouhi
Email: aria.halavati [at] cims.nyu.edu
Office: 710 Warren Weaver Hall
Photography by Jennifer Ah - Rollei - Jan 2024 - SF. | {"url":"https://cims.nyu.edu/~ah5160/","timestamp":"2024-11-08T09:22:02Z","content_type":"text/html","content_length":"4203","record_id":"<urn:uuid:2b736e02-72b1-463d-a145-0e4a4353ea84>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00675.warc.gz"} |
Academy of Chinese Culture and Health Sciences Linear Algebra Worksheet - Custom Scholars
Academy of Chinese Culture and Health Sciences Linear Algebra Worksheet
https://intervisualtechnology.us/pxudpothrb/167/12830/Unit1 Projectapply.pdf
+ 2 1
Fit to page
I Page view A’ Read aloud
passes through (5,-2), perpendicular to the graph of x + 2y = 8
Solve each system of equations. Check each answer algebraically.
11. -6x + 3y = 33
-4x + y = 16
12. 2y = 5x – 1
x + y = -1
13. x + y + z = -1
2x + 4y + z = 1
x + 2y – 3z = -3
Graph each system of inequalities. Name the coordinates of the
vertices of the feasible region. Find the maximum and minimum
values of the given function.
14. 52y2-3
4x + y s5
-2x + y s5
f(x, y) = 4x – 3y
15. x>-10
12y? -6
3x + 4y = -8
2y 2 X – 10
f(x, y) = 2x + y
Unit1 Projectapply.pdf X + V
@ https://intervisualtechnology.us/pxudpothrb/167/12830/Unit1 Projectapply.pdf
Fit to page
Page view AV Read aloud
Graph each relation and find the domain and range. Then
determine whether the relation is a function.
{(-4,-8), (-2, 2), (0, 5), (2, 3), (4, -9)}
y = 3x – 3
Find the functional value.
f(0) if f(x) = x – 3×2
Graph each equation or inequality.
y = x – 2
f(x) = [[3x]] + 3
y < 3|x - 21 6. 2 द Find the slope of the line that passes through the two points. 7. (5, -3), (6,2) YTTER (4,5) (-2.5) 8. Write an equation in slope-intercept form for the line that satisfies each
set of conditions. 9. passes through (-6, 15), parallel to the graph of 3x + 2y = 1 A Vlogy.us/pxudpothrb/167/17837/Algebra_2_Rubric.pdf of 20 - tn Fit to page 6 Page view A Read aloud In Add note
Algebra 2 All Classes Student Name: CATEGORY Mathematical Concepts ما Completion Outstanding Excellent Average Poor Explanation shows Explanation shows Explanation shows Explanation shows complete
substantial some understanding very limited understanding of the understanding of the of the mathematical understanding of the mathematical mathematical concepts needed to underlying concepts
concepts used to concepts used to solve the problem(s). needed to solve the All problems are Problems are Problems may be Problems do not show completed correctly competed correctly; completed
correctly; work and/or are work is clearly shown; work is clearly shown; work is not always incorrect answers are clearly shown; very checked/confirmed checking/confirmatio little All graphs are All
graphs are shown; Some graphs are Not all graphs are correctly shown may have errors on shown; may have shown; may have graphs errors on graphs errors on graphs very little Diagrams and Sketches
Mathematical Errors 90-100% of the steps Almost all (85-89%) of Most (75-84%) of the More than 75% of the and solutions have no the steps and steps and solutions steps and solutions mathematical
errors. solutions have no have no mathematical have mathematical mathematical errors. errors. errors. Neatness and Organization The work is presented The work is presented the work is presented the
work appears in a neat, clear, in a neat and in an organized sloppy and organized fashion that organized fashion that fashion but may be unorganized. It is hard is easy to read. is usually easy to
read. hard to read at times. to know what information goes © Copy BI K hp | {"url":"https://customscholars.com/academy-of-chinese-culture-and-health-sciences-linear-algebra-worksheet/","timestamp":"2024-11-14T07:42:08Z","content_type":"text/html","content_length":"54566","record_id":"<urn:uuid:3fbedc73-961b-4934-a5f8-1ef92e7584cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00893.warc.gz"} |
Newton's Laws, Part 5 - David The Maths TutorNewton’s Laws, Part 5
Newton’s Laws, Part 5
Please read the prior posts in this series if you have not been following along. I ended last post with two equations, each relating to a different phase of our model rockets flight: powered and
coasting. Let’s look at the powered phase (phase 1).
For this phase, Newton’s second law (F = ma) reduced down to:
5.08 = 0.4a
From this equation, a first class rocket scientist can use calculus to find the velocity and the distance from the launch pad at any time in seconds after launch. These equations are:
v(t) = 12.7t, x(t) = 6.35t²
where v(t) is the velocity t seconds after launch and x(t) is the distance from the launch pad t seconds after launch. Now I’ve introduced what is called functional notation. Instead of saying the
velocity or distance after 3 seconds, I can just say v(3) or x(3). Maths is full of shorthand notations.
These two equations assumes that we start the clock at 0 seconds and that velocity, acceleration, and distance are all 0 at 0 seconds. Now remember, these equations are only valid for the first 3.3
seconds of flight (see previous post) because the engine stops burning at 3.3 seconds.
So how fast is our rocket going at 3.3 seconds? Well, just replace t with 3.3 in the velocity equation and calculate it:
v(3.3) = 12.7×3.3 = 41.91 m/s
To give you a perspective of how fast this is, this is equivalent to almost 151 km/h. How high is the rocket at engine burnout? Let’s replace t with 3.3 in the distance equation and calculate it:
x(3.3) = 6.35×(3.3)² = 69.15 m
Is this the highest the rocket goes, a measly 69 meters? Well remember, at burnout, the rocket is going up very fast. It will take gravity a while to turn that around. Enter the phase 2 equations.
From my last post, Newton’s second law for phase 2 is:
a = -9.8
Again, using calculus, our friend, the first class rocket scientist generates the two equations (applicable only to phase 2):
v(t) = -9.8t + 74.25, x(t) = -4.9t² + 74.25t -122.514
These equations are a bit more complex because they have to take into account that at 3.3 seconds, the velocity is 41.91 m/s and the rocket is 69.15 m high.
So how high does our rocket go? At the peak of its travels, the velocity goes from positive (going up) to negative (going down). That is, it passes through 0. So in order to find the highest that our
rocket goes, we need to find when the velocity equals 0. So we use our velocity equation and set it equal to o, then solve for the time that makes that happen:
v(t) = -9.8t + 74.25 = 0
-9.8t = -74.25
t = -74.25/-9.8 = 7.58 seconds
So now we use the distance equation and replace t with 7.58:
x(7.58) = -4.9×(7.58)² + 74.25×7.58 -122.514 = 158.76 m
So now remember that we are not using a parachute. So the next two questions to ask is when does it hit the ground and how fast is it going when it does.
When the rocket hits the ground, its distance is 0. So now we use the distance equation, set it equal to 0 and find the value of t to make that happen:
x(t) = -4.9t² + 74.25t -122.514 = 0
Now you can solve this using the quadratic formula which I have covered in a previous post. Using this formula, you get two answers: 1.884s and 13.269s. The first answer is not greater than 3.3.
These phase 2 equations are only valid for t greater than 3.3 seconds. So we can reject that answer and choose 13.269 seconds. So the total flight time is a bit over 13 seconds.
Now how fast does it hit the ground? Put the time 13.269 into the velocity equation to get:
v(13.269) = -9.8×13.269 + 74.25 = -55.79 m/s
The velocity is negative because it is going down. So the rocket is going its fastest when it hits the ground, not when the engine burns out. 55.79 m/s is equivalent to 200.84 km/h. What are the odds
that we can launch this rocket again?
In my next post, let’s do the same problem but use a bigger rocket (since our model rocket is now one with the earth). | {"url":"https://davidthemathstutor.com.au/2019/06/18/newtons-laws-part-5/","timestamp":"2024-11-03T12:46:12Z","content_type":"text/html","content_length":"45985","record_id":"<urn:uuid:bccc5086-9f83-4e2f-8c3d-d1821ae4db29>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00697.warc.gz"} |
class sklearn.decomposition.NMF(n_components=None, *, init=None, solver='cd', beta_loss='frobenius', tol=0.0001, max_iter=200, random_state=None, alpha=0.0, l1_ratio=0.0, verbose=0, shuffle=False)
Non-Negative Matrix Factorization (NMF)
Find two non-negative matrices (W, H) whose product approximates the non- negative matrix X. This factorization can be used for example for dimensionality reduction, source separation or topic
The objective function is:
0.5 * ||X - WH||_Fro^2
+ alpha * l1_ratio * ||vec(W)||_1
+ alpha * l1_ratio * ||vec(H)||_1
+ 0.5 * alpha * (1 - l1_ratio) * ||W||_Fro^2
+ 0.5 * alpha * (1 - l1_ratio) * ||H||_Fro^2
||A||_Fro^2 = \sum_{i,j} A_{ij}^2 (Frobenius norm)
||vec(A)||_1 = \sum_{i,j} abs(A_{ij}) (Elementwise L1 norm)
For multiplicative-update (‘mu’) solver, the Frobenius norm (0.5 * ||X - WH||_Fro^2) can be changed into another beta-divergence loss, by changing the beta_loss parameter.
The objective function is minimized with an alternating minimization of W and H.
Read more in the User Guide.
n_componentsint or None
Number of components, if n_components is not set all features are kept.
initNone | ‘random’ | ‘nndsvd’ | ‘nndsvda’ | ‘nndsvdar’ | ‘custom’
Method used to initialize the procedure. Default: None. Valid options:
None: ‘nndsvd’ if n_components <= min(n_samples, n_features),
otherwise random.
‘random’: non-negative random matrices, scaled with:
sqrt(X.mean() / n_components)
‘nndsvd’: Nonnegative Double Singular Value Decomposition (NNDSVD)
initialization (better for sparseness)
‘nndsvda’: NNDSVD with zeros filled with the average of X
(better when sparsity is not desired)
‘nndsvdar’: NNDSVD with zeros filled with small random values
(generally faster, less accurate alternative to NNDSVDa for when sparsity is not desired)
○ ‘custom’: use custom matrices W and H
solver‘cd’ | ‘mu’
Numerical solver to use: ‘cd’ is a Coordinate Descent solver. ‘mu’ is a Multiplicative Update solver.
New in version 0.17: Coordinate Descent solver.
New in version 0.19: Multiplicative Update solver.
beta_lossfloat or string, default ‘frobenius’
String must be in {‘frobenius’, ‘kullback-leibler’, ‘itakura-saito’}. Beta divergence to be minimized, measuring the distance between X and the dot product WH. Note that values different
from ‘frobenius’ (or 2) and ‘kullback-leibler’ (or 1) lead to significantly slower fits. Note that for beta_loss <= 0 (or ‘itakura-saito’), the input matrix X cannot contain zeros. Used
only in ‘mu’ solver.
tolfloat, default: 1e-4
Tolerance of the stopping condition.
max_iterinteger, default: 200
Maximum number of iterations before timing out.
random_stateint, RandomState instance, default=None
Used for initialisation (when init == ‘nndsvdar’ or ‘random’), and in Coordinate Descent. Pass an int for reproducible results across multiple function calls. See Glossary.
alphadouble, default: 0.
Constant that multiplies the regularization terms. Set it to zero to have no regularization.
New in version 0.17: alpha used in the Coordinate Descent solver.
l1_ratiodouble, default: 0.
The regularization mixing parameter, with 0 <= l1_ratio <= 1. For l1_ratio = 0 the penalty is an elementwise L2 penalty (aka Frobenius Norm). For l1_ratio = 1 it is an elementwise L1
penalty. For 0 < l1_ratio < 1, the penalty is a combination of L1 and L2.
New in version 0.17: Regularization parameter l1_ratio used in the Coordinate Descent solver.
verbosebool, default=False
Whether to be verbose.
shuffleboolean, default: False
If true, randomize the order of coordinates in the CD solver.
New in version 0.17: shuffle parameter used in the Coordinate Descent solver.
components_array, [n_components, n_features]
Factorization matrix, sometimes called ‘dictionary’.
The number of components. It is same as the n_components parameter if it was given. Otherwise, it will be same as the number of features.
Frobenius norm of the matrix difference, or beta-divergence, between the training data X and the reconstructed data WH from the fitted model.
Actual number of iterations.
Cichocki, Andrzej, and P. H. A. N. Anh-Huy. “Fast local algorithms for large scale nonnegative matrix and tensor factorizations.” IEICE transactions on fundamentals of electronics, communications
and computer sciences 92.3: 708-721, 2009.
Fevotte, C., & Idier, J. (2011). Algorithms for nonnegative matrix factorization with the beta-divergence. Neural Computation, 23(9).
>>> import numpy as np
>>> X = np.array([[1, 1], [2, 1], [3, 1.2], [4, 1], [5, 0.8], [6, 1]])
>>> from sklearn.decomposition import NMF
>>> model = NMF(n_components=2, init='random', random_state=0)
>>> W = model.fit_transform(X)
>>> H = model.components_
fit(X[, y]) Learn a NMF model for the data X.
fit_transform(X[, y, W, H]) Learn a NMF model for the data X and returns the transformed data.
get_params([deep]) Get parameters for this estimator.
inverse_transform(W) Transform data back to its original space.
set_params(**params) Set the parameters of this estimator.
transform(X) Transform the data X according to the fitted NMF model
__init__(n_components=None, *, init=None, solver='cd', beta_loss='frobenius', tol=0.0001, max_iter=200, random_state=None, alpha=0.0, l1_ratio=0.0, verbose=0, shuffle=False)[source]¶
Initialize self. See help(type(self)) for accurate signature.
fit(X, y=None, **params)[source]¶
Learn a NMF model for the data X.
X{array-like, sparse matrix}, shape (n_samples, n_features)
Data matrix to be decomposed
fit_transform(X, y=None, W=None, H=None)[source]¶
Learn a NMF model for the data X and returns the transformed data.
This is more efficient than calling fit followed by transform.
X{array-like, sparse matrix}, shape (n_samples, n_features)
Data matrix to be decomposed
Warray-like, shape (n_samples, n_components)
If init=’custom’, it is used as initial guess for the solution.
Harray-like, shape (n_components, n_features)
If init=’custom’, it is used as initial guess for the solution.
Warray, shape (n_samples, n_components)
Transformed data.
Get parameters for this estimator.
deepbool, default=True
If True, will return the parameters for this estimator and contained subobjects that are estimators.
paramsmapping of string to any
Parameter names mapped to their values.
Transform data back to its original space.
W{array-like, sparse matrix}, shape (n_samples, n_components)
Transformed data matrix
X{array-like, sparse matrix}, shape (n_samples, n_features)
Data matrix of original shape
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
Estimator parameters.
Estimator instance.
Transform the data X according to the fitted NMF model
X{array-like, sparse matrix}, shape (n_samples, n_features)
Data matrix to be transformed by the model
Warray, shape (n_samples, n_components)
Transformed data
Examples using sklearn.decomposition.NMF¶ | {"url":"https://scikit-learn.org/0.23/modules/generated/sklearn.decomposition.NMF.html","timestamp":"2024-11-06T02:53:03Z","content_type":"text/html","content_length":"42597","record_id":"<urn:uuid:6ff8c31a-de33-4da9-a86a-431c2224b79c>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00076.warc.gz"} |
Best Online Math Tutor | Mgadvantagetutoring
top of page
Pre-algebra, Algebra I, Algebra II, Trigonometry, Pre-calculus, Differential Calculus, Integral Calculus, Multivariable Calculus, Differential Equations, Linear Algebra, Modern Algebra
Why Online Math Tutoring?
MG Advantage Tutoring helps students understand each mathematical theory using modern curriculum. We give each child a chance to explain themselves regarding every math problem, we tutor everyone at
their own pace. Math is important as it a basic in life. We teach each student how to calculate each Math problem.
Why is Math a Significant Subject
Math is applicable everywhere in life and especially for student who want to pursue careers in accounting, engineering and many math related careers. Math is also applicable in technical course and
subjects such as physics, and chemistry
Best Math Concept to Remember
You can master any math concept by using skills, Our tutors help you understand any math concept and problem at all levels of your study. The concepts include but not limited to:-
• Wholes and Parts
• Counting
• Comparable Thinking
• Sharing ideas relating to Math
Our Tutoring Pricing Plans
Our expert tutors have degrees in the subject they tutor, along with superb test scores and a demonstrated commitment to excellence. They are able to tutor high-achieving students, as well as
students in tricky or unique situations.
Our master tutors meet all the requirements of expert tutors, along with years of experience tutoring students in all manner of situations and with varying goals. They demonstrate leadership
capabilities in directing both academic programs and training other tutors.
Tutoring with the Founder himself, Michael Grantham. Perfect test scores, straight A’s, over a decade of experience tutoring all ages, and writing curricula both for MGAT tutors and MGAT students
uniquely qualify Michael to provide the best tutoring for your student available anywhere.
bottom of page | {"url":"https://www.mgadvantagetutoring.com/how-to-get-as-in-math","timestamp":"2024-11-02T14:20:21Z","content_type":"text/html","content_length":"908108","record_id":"<urn:uuid:7174a7a9-a80e-47e2-a542-7825468f2f4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00443.warc.gz"} |
The degree of the differential equation [1+(dxdy)2]23=dx2d2y... | Filo
Question asked by Filo student
The degree of the differential equation is
a. 4
c. 2
d. Not defined
Not the question you're searching for?
+ Ask your question
Video solutions (1)
Learn from their 1-to-1 discussion with Filo tutors.
2 mins
Uploaded on: 1/10/2024
Was this solution helpful?
Found 5 tutors discussing this question
Discuss this question LIVE for FREE
5 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Practice more questions on Differential Equations
View more
Students who ask this question also asked
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text The degree of the differential equation is
Updated On Jan 10, 2024
Topic Differential Equations
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 88
Avg. Video Duration 2 min | {"url":"https://askfilo.com/user-question-answers-mathematics/the-degree-of-the-differential-equation-is-36363933353639","timestamp":"2024-11-13T01:55:44Z","content_type":"text/html","content_length":"236915","record_id":"<urn:uuid:74ec775f-1586-4d7f-902a-157255c8612b>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00011.warc.gz"} |
Mastering Essential Math Functions for Python Calculations - Adventures in Machine Learning
Math Functions and Number Methods
Mathematics is one of the core subjects taught in schools all around the world. As we grow, we begin to learn about arithmetic, geometry, algebra, and many other fundamental concepts.
Among these concepts, we come across various math functions and number methods used to perform tasks like rounding numbers, calculating the exponents, and determining the absolute value of numbers.
In this article, we will dive deeper into some of the most commonly used math functions and number methods.
1. round()
The round() function
The round() function is used to round off a given number to the nearest integer. This function takes one or two arguments.
The first argument is the number to be rounded, and the second argument is the number of decimal places to round to. If the second argument is not provided, the number is rounded off to the nearest
Here’s an example:
round(3.14) # Output: 3
In the above example, we passed the value 3.14 to the round() function, which returned 3, the nearest integer. For instance, if we pass a value of 7.5 to the round() function, it can return either 7
or 8, depending on the rounding ties to even strategy.
Rounding ties to even strategy
The rounding ties to even strategy is a way of resolving ties during rounding. When a number is exactly halfway between two possible rounding-up values such as integers, it is rounded up or down to
the nearest even number.
Here’s an example:
round(2.5) # Output: 2
round(3.5) # Output: 4
In the above example, the value of 2.5 is being rounded. Since 2 is already an even number, it remains unchanged.
The value of 3.5 is being rounded, and since it lies halfway between 3 or 4, it gets rounded up to the nearest even number, which is 4.
Rounding to a given number of decimal places
As mentioned earlier, the second argument to the round() function specifies the number of decimal places to round off to. Here’s an example:
round(3.14159, 2) # Output: 3.14
The above code will round 3.14159 to 2 decimal places, resulting in the value 3.14.
2. abs()
The abs() function
The abs() function is used to determine the absolute value of a number. The absolute value of a number represents the distance of the number from zero on the number line.
The returned value is always positive. Here’s an example:
abs(-5) # Output: 5
abs(5) # Output: 5
In the above example, we passed -5 and 5 to the abs() function, which returned 5 in both cases.
3. pow()
The pow() function
The pow() function is used to raise a number to the power of another number. The pow() function takes two arguments, the base and the exponent.
Here’s an example:
pow(2, 3) # Output: 8
The above code will raise 2 to the power of 3, resulting in the value 8. Another commonly used operator for exponentiation is the ** operator.
Here’s an example:
2 ** 3 # Output: 8
The above code is an equivalent statement to pow(2,3). It will raise 2 to the power of 3, resulting in the value 8.
.is_integer() method
The .is_integer() method is a built-in method used with floating-point numbers. It returns True if the given number is an integer, and False if otherwise.
For example:
(5.0).is_integer() # Output: True
(5.5).is_integer() # Output: False
In the above example, we used the .is_integer() method to check whether 5.0 and 5.5 are integers. The first value returns True, while the second value returns False.
4. Finding the Absolute Value With abs()
Mathematical calculations can be performed with integers and decimal numbers.
However, we will sometimes need to work with negative numbers, which can produce unexpected results when performing certain calculations. The absolute value of a number negates any negative sign,
converting the value to a positive number, removing any ambiguity during mathematical calculations.
Calculation of absolute value
To calculate the absolute value of a number, we can use the built-in abs() function. The abs() function takes a single argument representing the number for which we want to calculate the absolute
Let’s take a look at an example:
x = -5
abs(x) # Output: 5
In this example, we assign the value -5 to a variable, and we then pass it to the abs() function. The output will be 5, which is the absolute value of -5.
The abs() function works with float numbers as well. For example:
y = -3.14159
abs(y) # Output: 3.14159
In this example, we assign the float value -3.14159 to a variable and pass it to the abs() function.
The output will be 3.14159, which is the absolute value of -3.14159. The abs() function can be used in combination with other mathematical operations to produce desired results.
For example:
x = -5
y = 3
z = abs(x) * y # Output: 15
In this example, we multiply the absolute value of -5 with 3 to obtain the value 15.
5. Raising a Number to a Power With pow()
The pow() function
The pow() function is a built-in function in Python used to raise a number to a given power.
The function takes in two arguments, the base number, and the exponent. It raises the base number to the power of the exponent provided as the second argument.
Using ** operator
In Python, the double asterisk (**) operator can be used to raise a number to a power, and it works in the same way as the pow() function. Here’s an example:
x = 2
y = x ** 3
print(y) # Output: 8
In this example, we assign the value 2 to a variable and then use the ** operator to raise it to the power of 3.
The output will be 8.
Difference between ** and pow()
Although the ** operator and pow() function produce the same result, there is a difference between the two.
The pow() function has an optional third argument, which specifies the modulus of the computation. The modulus determines what value to return when the base number is negative.
For example:
x = -2
y = 3
result = pow(x, y, 5)
print(result) # Output: 3
In this example, we use the pow() function to raise -2 to the power of 3 and take the modulus of the result with 5. The output will be 3.
On the other hand, using ** operator does not allow us to define the modulus. Here’s an example:
x = -2
y = 3
result = x ** y % 5
print(result) # Output: 3
In this example, we use the ** operator to raise -2 to the power of 3 and take the modulus of the result with 5.
The output will be 3, just like the pow() example.
6. Checking if a Float Is Integral
When working with numbers, we may need to determine whether a float value is integral or fractional. In Python, there are several ways to perform this task, such as using the round() function or the
int() function.
The easiest method is to use the .is_integer() method, which is a built-in method of float objects.
Number methods
Number methods are functions that can be used to manipulate numbers. Python provides many built-in number methods that can be used to validate user input or perform mathematical calculations.
Some of these methods are available for all number types in Python, while others are specific to certain types.
.is_integer() method
The .is_integer() method is a built-in method of float objects in Python.
This method returns True if the float value is integral, and False if it has a fractional part. Here’s an example:
x = 5.0
y = 5.5
print(x.is_integer()) # Output: True
print(y.is_integer()) # Output: False
In this example, we create two float variables, x and y.
We then use the .is_integer() method to check whether each value is integral. The output for x will be True because it has no decimal places, while the output for y will be False because it has a
fractional part.
We can use the .is_integer() method to validate user input. For example, suppose that we need to prompt the user to enter a float value that must be integral.
We can use the .is_integer() method to check whether the input is valid. Here’s an example:
value = input("Enter a float value: ")
if float(value).is_integer():
print("Valid input")
print("Invalid input")
In this example, we prompt the user to enter a float value, which we then convert to a float using the float() function.
We then use the .is_integer() method to see whether the input is valid. If the input is integral, we print “Valid input”, and if not, we print “Invalid input”.
The .is_integer() method is useful since it can help us perform validations while accepting float values as input. We can use it to build more complex applications, such as data analysis or
scientific computation, where precision is essential.
In this article, we have explored the .is_integer() method of float objects in Python, which allows us to check whether a float has an integral value. We have seen how the method can be used to
validate user input and how it can be incorporated into more complex applications.
Python provides many built-in number methods to assist with various mathematical operations, and the .is_integer() method is just one of them. Knowing these methods can help us perform calculations
more efficiently and produce more accurate results.
In this article, we delved into several important math functions and number methods that are used when working with mathematical calculations in Python. We explored the round(), abs(), pow()
functions, and the .is_integer() method for float objects.
The round() function is used to round off a given number while the abs() function calculates the absolute value of a number. The pow() function raises a number to a given power, and the ** operator
is an equivalent method.
The .is_integer() method returns true when a float is of integer value. We also learned about the differences between ** and pow() functions.
Understanding and using these functions can lead to more accurate mathematical calculations in Python, making them an essential tool for data analysis and scientific computations. | {"url":"https://www.adventuresinmachinelearning.com/mastering-essential-math-functions-for-python-calculations/","timestamp":"2024-11-11T21:16:47Z","content_type":"text/html","content_length":"74056","record_id":"<urn:uuid:b73ba8d5-1006-4dc6-baa4-e182cacdc2f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00465.warc.gz"} |
INSITU - Insitu Testing
The INSITU program interprets statical and dynamical geotechnical in situ tests.
In order to interpret the SPT tests, the program utilises the main correlations that are normally used to determine the geotechnical parameters of a soil.
Based on this simple information, the program determines the relative density, the friction angle, the effective vertical and total stress, the confined modulus, Young's modulus, the upper limits of
the settlement/admissible load ratio, the point resistance, the dynamic shear modulus and the cyclical stress of the terrain traversed, when possible using different calculation methods to permit
comparison of the results.
When interpreting the CPT test results, the program divides, as normal practise, granular terrains with a prevalence of sand from cohesive terrains made up mostly of clay.
For granular terrains, it is able to determine (from the data that has been entered) the friction angle, the drained compressibility and the relative density, while for cohesive terrains it evaluates
the shear resistance in undrained conditions, compressibility, sensitivity and the degree of over-consolidation.
For the interpretation of this test the data required are, yet again, extremely simple: the classification of the terrain that has to be used, the measured test values and any correction factors that
prove necessary to limit perturbations in the interpretation of the data.
With the DP test interpretation, which is mainly used in granular terrains and to localise resistant strata, the program supplies all the information previously mentioned for the SPT test.
The INSITU program can be adapted to any measurement instrument. | {"url":"https://www.geoandsoft.com/english/insitu_testing.htm","timestamp":"2024-11-15T00:51:24Z","content_type":"text/html","content_length":"19304","record_id":"<urn:uuid:8b2a2151-e29a-45e2-91c7-cabb11339a7c>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00184.warc.gz"} |
NYT: Game Probabilities - Week 7
Weekly game probabilities are available now at the nytimes.com Fifth Down. This week I also look at how the Chargers can be so dominant statistically yet only have two wins to show for it. It's
something more in depth than my usual lead-ins for the game probabilities.
Here's an excerpt along with a graph I did not include in the original post:
"Successful plays are not enough. Consecutive successes are required to win.
...Two equal teams could each have 12 first downs in a game. One team could have three drives of four consecutive first downs, each leading to a touchdown, and the rest of its drives could be
three-and-outs. The other team could have 12 drives consisting of one first down followed by a punt. Both teams could have equal yards, first downs and efficiency stats, and yet one team could win,
21-0. It’s easy to imagine a game in which one team has many more first downs and yards, but still loses. Could something like this bunching effect be cursing the Chargers?
It’s a given that N.F.L. offenses tend to score in proportion to their yards gained. It’s actually an extremely tight correlation, and the best–fit estimate of a team’s points per game is to take
just under 10 percent of its yards per game and subtract 10. For the Chargers, who lead the N.F.L. with 433 yards gained per game, we’d expect the offense to score about 32 points per game, but
they’ve actually scored only 26."
34 Responses to “NYT: Game Probabilities - Week 7”
TBD says:
Continuing what I said yesterday:
if the model gives the Patriots a 15% chance to win this weekend, and the multi million dollar betting market says the Patriots are +125 (44%), the model is busted. It needs to be
massaged with some qualitative input.
Giles says:
That's an interesting analysis given that all of San Diego's losses have been 8 points or less.
Bobby says:
Vegas is also adjusted for what the public believes...the stats don't consider the public.
TBD says:
I'll say it another way: the line isn't really reflective of "Vegas" it's much more reflective of the collective knowledge (ie $$) of every NFL bettor out there. it's a nearly efficient
market, as can be seen by the # of people who can consistently beat closing lines (very few). a discrepancy between the wisdom of the entire public and this model is very hard to believe.
TBD says:
I meant to say, "a discrepancy this large..."
Brian Burke says:
mwh-Put a gun to my head and I would not say SD is an 85% favorite over the Pats. However, consider their respective passing games: SD offense 7.9 net YPA (#1); NE defense 6.9 net YPA (#
Throw in that the game is at SD, and this suggests the betting markets are saying NE is the slightly better team. Bettors are basically banking that turnover rates and special teams
break-downs will continue on pace, and that's a bad bet.
So while I'd agree NE's chances are better than 15%, it's almost certainly a lot less than 45%.
tgt says:
Your point only makes sense if the betting market is rational. Economists often assume markets are rational, but social economists have shown in recent years that these assumptions are
incorrect. A stat that performs better than Vegas would be but one more piece of evidence in the trend.
DA Baracus says:
"Bettors are basically banking that turnover rates and special teams break-downs will continue on pace, and that's a bad bet."
Turnover rates, yes. But bad special teams play? Absolutely liable to continue. It's not just a matter of luck, it's a combination of bad coaching and bad players.
Dan Schlauch says:
I think a lot of readers are struggling over your use of the word "luck". Maybe another word like "unpredictable" is better. The idea is exactly the same, but people tend to view any
human-influenced event as inherently skill based. I can see people scoffing at the idea that a fumble was "luck", but recognizing that it was unpredictable.
I took the liberty of evaluating your week 6 picks by mean squared error and comparing to several other simple systems. The results are not very good, but I realize this was not an
ordinary week. I haven't checked but, I suspect the week 5 picks would have come out on top.
Vegas Picks (100% for favorite): .1818
Vegas moneyline (converted to %): .2128
Home picks (at 57%): .2485
Zero info picks (all 50%): .25
AdvancedNFLStats: .2813
TBD says:
"A stat that performs better than Vegas would be but one more piece of evidence in the trend"
it would be. however, all kinds of results over the years have shown that beating the NFL point spread, over any long sample size, is an extremely difficult or impossible thing to do,
-110 juice considered. which means the point spreads are pretty damn efficient, if perhaps not 100% perfectly so. which means that any discrepancy of 15% vs 44%, to me, points to
systematic error. now, Brian has said that he thinks 15% is probably too low as well. but I think the lowest it could possibly be is 30-35% or so.
Anonymous says:
Accuscores predictions are good. Just what vegas don't want, a lop sided bet. So,is the NFL riged. I hope not,but if it is people can be predictable too. I don't know if vegas shows the
balance of the bets on the weekly games before the games actually start, but I would like to see that before I layed my money down. Would an accurate prediction system force almighty
Vegas to cheat to win. I thank so. You?
Anonymous says:
Does your calculation of yards factor in special teams? We know the Chargers have been brutal on special teams this year, and in fact, your remark about adding and subtracting those 8 pts
/game is virtually identical to considering the impact of special teams (if you didn't in your analysis). Special teams has surrendered 30 pts in their 3 losses. In fact, if you simply
consider the average impact of each special teams play (such as, an average punt return, kick, etc.) then San Diego is 5-0. A lot of hypotheticals, but special teams has been massive for
San Diego this season.
Anonymous says:
I think the discrepancy between this model and a relatively efficient market raises very interesting questions. It's not impossible that as of today this model catches things that the
wisdom of the crowds does not. If so, the market would react very quickly and would incoporate the considerations found in the model into the odds. For now, the "market" seems to be
discounting this model for one reason or another.
On a related note - any objective model that closely mirrored vegas odds would be an impressive analytical feat.
tgt says:
Basically, your position is: if it hasn't been done before, it won't be done now. Also, vegas builds alot of wiggle room in, so you have to beat it badly to win consistently, which
somehow supports the vegas line being more accurate than this stat.
I don't follow your "logic."
@Aaron Gordon,
Brian ignores special teams as being impossible to predict confidently. It's a known flaw in his work.
The vegas market is not efficient. The casinos are, but the bettor are not. Also, without confidence in the various different advanced stats' confidence levels, it actually is more
rational and efficient for the bettors to not follow the new models.
Brian Burke says:
"Brian ignores special teams as being impossible to predict confidently. It's a known flaw in his work."
Speak for yourself. I challenge anyone to measure ST performance meaningfully and be able to show that previous ST outcomes predict future ST outcomes in a season. I would say that the
flaw lies in the fallacy that week-to-week ST performance is predictable.
Sure, by the end of the season some teams will have more missed FGs, and some will have a few kick returns, and some will have a blocked punt or two. But this would also be true if ST
outcomes were overwhelmingly random.
If I take 32 pennies, and flip them each 16 times, some pennies will appear to be far more capable than others at landing on heads. That does not mean that you can predict how each penny
will land on the next flip.
tgt says:
Your ratings do ignore special teams, you do think they are impossible to predict confidently, and special teams do have an impact on the game. I see a flaw in your ratings. Your
contention that flawlessness is impossible does not impact the existence of said flaw.
More interesting than the knee jerk defensiveness is the reasoning behind your belief.
The largest ST plays (touchdowns and blocks) cause a huge impact to the game without being predictable each week. Sure. Why that means you should ignore all special teams plays is
First off, some features of ST are predictable each week: Kickoff distance(touchbacks) and punting distance. Similarly, median kickoff return distance and punt return distance are
correlated from week to week.
The real difficulty I see is that you insist on rate stats. While offensive and defensive rate stats are easy to create, pretty well correlated with winning, and relatively consistent, ST
rate stats are pretty much junk. The main reason rate stats work for offense in defense is the size of the sample. 1 50 yard pass a week does not throw off the sample, and shows a
reasonable liklihood for something to occur each game, but the same number of 50 yard returns would be one every 4 games. Moreover, as special teams performance diverges from the mean,
the rates stats lose predictive value.
I'm not sure what the best solution is. Maybe you could try tying EPA of special teams plays in with the rate stats for offense and defense? Maybe use median values instead of averages
for special teams plays? Maybe vary the strength of each special teams factor based on how often that rating is expected in a given game (e.g., if two teams have awesome offenses and no
defenses, punting and punt returns would be set low while kickoffs and kick returns would be set high)? Maybe include more incertainty around teams who have more big special teams plays
(pos and neg), and less around teams with consistent special teams?
Like you, I don't think STs rate stats are worth even the 32 pennies you referenced, but that doesn't mean that there aren't other stats that could fill that hole.
Brian Burke says:
tgt-Allow me to rephrase your comment from "It's a known flaw in his work." to:
"I think it might be a flaw."
Let's keep separate things that are 'known' and accepted from the things that are personal theories.
Anonymous says:
Brian doesn't ignore special teams, as his post about my favorite punter, Zoltan Mesko, proved. He also understands how important it can be, as the WPA analysis of his punt suggested.
That is why I was so puzzled to see it not even mentioned in this post about the Chargers' WP this weekend against those very Patriots. It is just such an obvious consideration. One team
has consistently given up points in ST play through five games, while the other has consistently been the beneficiary of ST points. Even if he gave it lip service to say "...but ST is
completely unpredictable like fumbles."
Buzz says:
I think one of the biggest things to keep in consideration for the chargers this week against the pats is the health of gates, floyd,and naane (with vjax already out) Can their passing
efficiency remain so high with so many receivers out? That information is being taken into consideration for the vegas odds and not in the model. That isn't a flaw in the model but
instead something that is a given and can help explain a certain percentage of the variance between the two.
That said most people in my confidence level pool picked the Pats and I picked the chargers (although with a lower confidence than the model would suggest). Will be interesting to see if
this great team on paper can show up despite some injuries.
Unknown says:
The Vegas books have a bias to account for the proximity to the CA bettors, and especially the So. Cal market. So, online its Chargers -1, in my local book its Chargers -3.
Chuck Winkler says:
Like everyone, I was curious about how this model compares to Vegas because this is the best way to gauge success rate. Since Brian has predictions archived back to 2007, there is a
pretty good base to work with for anyone willing to dedicate the time. So I did.
Based on this post - http://www.advancednflstats.com/2010/10/how-accurate-is-prediction-model.html - the prediction model is too confident with large spreads as the highest accuracy
occurs within the 25-75% range. To give his model a fair chance, I disregarded all games that had a >75.0% favorite.
For the next step I converted all Vegas game lines to Win percentage to make for an easy comparison. Since low percentage differences do little to separate the two methods, a 10% and 20%
difference for Weeks 4-10 and 8% and 15% difference for Weeks 11-17 was used (the game win percent lines between Vegas and Brian's model become more equal as the season progresses so this
is why the range needed falls later in the year). This narrows down the number of games used to an average of 2-3 per week.
When Brian's predictions were different from Vegas, this how the prediction model did week-by-week:
04: 5-7
05: 7-2
06: 0-8
07: 4-0
08: 1-3
09: 4-2
10: 1-5
11: 5-0
12: 6-3
13: 5-0
14: 5-2
15: 5-5
16: 6-5
17: 2-0
Total: 56-42 (57.1%)
04: 4-1
05: 4-4
06: 3-1
07: 2-1
08: 1-2
09: 2-2
10: 0-4
11: 0-2
12: 2-3
13: 1-3
14: 1-4
15: 0-0
16: 4-1
17: 1-2
Total: 25-30 (45.5%)
Combined Total: 81-72 (52.9%)
Since a +9 Win record over a 153 game period falls within a common statistical variability for something that is 50/50, it can be said that the Game Prediction Model has thus far proved
no skill over Vegas.
To see if a small percentage advantage is legitimate, we would look for a higher win percentage in the "bigger difference" games. The fact that the biggest differential games (20%+/15%+)
came out with a -5 overall record in 55 attempts does more to show that this model falls close to the 50/50 range.
In compiling this data, I did notice that there were two periods that performed better than the rest - Most of the 2007 Season, and in the later weeks of last Season, 2009. It would be
interesting to know if the model has undergone any changes since the beginning. Also, is a 7% win percentage factored into the home team (it actually should be 8%, but the former is what
Vegas uses)? That variable could provide a significant difference in how this compares.
This has quickly become my favorite site to use for NFL information so please don't take this post to mean anything negative. My main motivation for doing the research and posting these
statistics is in hopes that Brian can re-evaluate his model and improve it or create something better. But as it stands now, Vegas is the better alternative for weekly game predictions.
tgt says:
Which of these premises do you not agree with:
* special teams affect who wins each game,
* special teams are not factored into the game probablilities, or
* a win probability system is flawed if it ignores factors which contribute toward winning?
You've shown support for the first two and the last one is just the definition of a flawed system.
I stand by my objective reasoning. The only thing possibly subjective is my definition of a flawed system, and I doubt many people would disagree with it. If that isn't a flaw, what would
As an aside, flawed does not mean bad. All the nfl, nba, and mlb ratings I've seen are flawed. All attempts to model the stock market and our economy are flawed. Flaws are. We just need
to handle them as best we can.
Instead of attempting to sweep an obvious flaw under the rug, you should explain the impact of that flaw. It looks to me like the following occurs:
* At the specials teams margins, your game probabilities are unreliable.
* At the special teams median, your game probabilities slightly underestimate confidence.
Much like Chuck above me, I'm just trying to be helpful.
JJB says:
I'm guessing the gist of the original post is to say that the Chargers' statistical performance indicates that they are a better team than their record. I don't think that's a surprise to
anyone. Their poor turnover and special teams numbers will regress to the mean in the remainder of the season and their record will recover. The only disappointment is that they've played
the softest part of their schedule (Chiefs, Jags, Cardinals, Seahawks, Raiders, and Rams) and are only 2 - 4 when they "should" be 6 - 0. But when you go 14-2 and fire your coach, you
gotta expect a few bad breaks.
Jeff Clarke says:
Thanks for doing that. I thought it would be cool if somebody did something like that.
I'm not sure you quite answered the question I would have asked. Do you still have the spreadsheet available? Could you answer some follow up questions?
The biggest question is would Brian's system have beaten Vegas over the last 5 years?
To beat Vegas long term, you don't need to win all the games or even a majority of the games, you just need to successfully identify games in which your teams have a greater chance of
winning than Vegas says they do. If you identify a group of games where you think a team has a 60% chance of winning and Vegas says 30%, you'll still win a lot of money if they only win
45%. If you say 60% and Vegas says 30% and they win 25%, well you've got a problem.
I guess the big question is what were the average odds via Vegas for those games and what were the average odds via Brian's model for those games.
Frankly, I don't think the Chargers have a 85% chance of winning, but I do think they have more than a 55% chance of winning. If I was to bet on the game, that would make it worthwhile to
bet on the Chargers even if I thought they only had a 60% shot.
Brian Burke says:
Aaron-Reread the article. You must have missed the paragraph detailing the impact of SD's ST.
tgt-You have to be kidding. Airlines don't provide airbags behind each seat. It would improve safety in crashes, right? So is that a " known flaw" in airliner design? Of course not. It is
a net negative effect due to the expense and complexity, and its benefit is negligible. Airline engineers know all about airbags and their potential benefit and choose to worry about more
important things. Not a flaw.
All the data is available. Produce an open model that will accurately predict ST performance measured in a way that correlates with team wins in weeks 9-17 this season, and I'll send you
a free ANS sweatshirt.
Chuck-Thanks. I'll take a +9 game system any day. To be fair, keep in mind this model is completely ignorant of playoff teams resting starters or top QBs or other players who are injured.
Anonymous says:
Since this system has been more accurate than Vegas, it's more likely that the true flaw is in trying to take special teams into account.
Andy says:
i think maybe using the word flaw brings up the wrong connotation (at least in my mind). "Flaw" makes it sound like Brian should run out and change his model to include special teams. But
if you are just trying to say that this is one aspect that is not modeled, and could result in a "delta" between what is predicted and what actually happens, then you are right.
Probability is all about what is known and unknown. Prediction models have to treat what is unknown (or unknowable) as random. I would say that even though one could predict small pockets
of special teams performance reliably (kickoff distance), special teams when taken as a whole are random (unpredictable). And randomness just is; that's why predictions are in terms of
probabilities. So I wouldn't call it a "flaw"...at least not a flaw in the model.
Andy says:
Also, I understand that the conversation is about over/underestimating game probabilities, but I would like to see a straight up wins comparison against Vegas...I don't remember where i
read it (somewhere on this site) but I thought this model had beaten the Vegas picks for a few years in a row?
Ian Simcox says:
Brian, if I could add my own two pence. This is what you're talking about when you mention the difference between descriptive or explanatory stats and predictive ones.
Yes, past special teams play correlates with past wins. It is a big factor in explaining why a team won a certain game. But past special teams play does not correlate with future wins, at
least not in any meaningful way yet found.
So if Brian was in the wild-card week explaining why your team missed out on the playoffs, he'd likely include special teams in his calculations. But sat here going into week 7, San
Diego's past special teams play doesn't provide any clue (mathematically speaking) as to how they'll do against the Patriots.
Brian Burke says:
Sampo says:
Wow. This blog sure is getting a lot of interest lately!
It's been a great ride.
tgt says:
I don't see the word flaw with as many connotations as other people, so I didn't see it as a huge negative. It does appear to have struck a nerve though, so feel free to replace "flaw"
with any of the following "design decision," "feature," "aspect," or "horrible, atrocious idea of horribleness" wherever you see fit. :)
We do seem to agree that SOME special teams are likely measurable and predictable. I think the incremental improvement might be worthwhile, but I don't have the patience or ability to
factor it in myself. Of course, Brian is of no compulsion to act on my wishes.
As above, pretend I didn't use the word flaw, instead used something with less negative connotations.
Continuing on my nitpicks though, your airplane example is pretty horrible. Modelling a system is not analogous to building an object. True, in both situations, you make tradeoffs between
improvements and costs, but that's roughly where the similarities end. That isn't even very similar as the basic definitions of improvements and costs are not even the same between the
types of situations.
Dan Schlauch says:
@Chuck Winkler
Where do you see the archived predictions? I would be very interested in doing something a little more in depth in evaluating the predictions.
Dan Schlauch says:
@Chuck Winkler
Nevermind. It looks like you went through all the old posts and compiled it yourself. I'm pretty sure I don't have the patience for that, but if you or Brian or anyone else wants to send
me what you put together, preferably with moneylines, I think I could add a lot to the discussion. | {"url":"https://www.advancedfootballanalytics.com/2010/10/nyt-game-probabilities-week-7.html","timestamp":"2024-11-03T09:24:42Z","content_type":"application/xhtml+xml","content_length":"154450","record_id":"<urn:uuid:60861e40-c03f-4ff5-8968-437a1ef23604>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00720.warc.gz"} |
Adventures in Chance-ylvania - Chalkdust
Adventures in Chance-ylvania
High stakes gambling with Paula Rowińska
“I’m really sorry, the vampirograph indicated that you are a vampire.” Imagine that you (or your mother/brother/girlfriend/pet scorpion) received such a message. You’re probably terrified, worrying
about the future, thinking about the upcoming treatment. Wait a moment! Before you start panicking, consult… a mathematician.
Medical tests aren’t perfect. Testing positive for an illness doesn’t necessarily mean that you’re sick; for many reasons, tests can detect things that aren’t really there. On the other hand, a
negative test result doesn’t exclude the disease without any doubt. The question is: can we quantify the level of uncertainty linked to a particular test? Or in this case: if you test positive on a
vampirograph, what’s the probability that you’re really a vampire?
A similar question was tackled in the 18th century by a Presbyterian minister, Reverend Thomas Bayes. Bayes’ theorem became a basis for statistical inference, even though conclusions drawn from it
are sometimes counter intuitive. His result gives us an explicit formula to update our prior belief about the probability of some event of interest based on additional evidence:
$$\mathbb{P}(\text{event}|\text{evidence})=\mathbb{P}(\text{event})\frac{\mathbb{P}(\text{evidence}|\text{event})}{\mathbb{P}(\text{evidence})}.$$Let’s get some intuition about this equation. I
assume you’re familiar with the notion of the probability measure $\mathbb{P}$; don’t worry about a rigorous definition, a common interpretation—ie how likely the event is—will suffice. The
mysterious symbols $\mathbb{P}\left(\text{something}|\text{something else}\right)$ denote a conditional probability—how likely $\text{something}$ is given that $\text{something else}$ happened.
Now we’re ready to take a look at Bayes’ theorem again. In the beginning we have some vague idea about the probability of the event, a so-called
probability $\mathbb{P}(\text{event})$. For example, let’s say we’re interested in the probability that this cute guy we just met is a nice person. We might base our estimation on our previous
experience with similar people or the fact that he’s a mathematician (mathematicians are usually nice people, aren’t we?), but our knowledge is pretty limited. However, later we gather some
additional observations (he smiles a lot, he helps us solve a ridiculously difficult equation etc) and we keep updating our prior probability. In the end we’re left with a
probability $\mathbb{P}(\text{event}|\text{evidence})$ of him being a nice person given the evidence we have.
Hold on, how did we get from vampires to estimating if a cute guy is also nice? (No, I’m not a Twilight fan.) Bayes’ theorem has many applications! Before we approach our vampire problem, we need to
make a few assumptions—all numbers come from my imagination [citation needed]. The scenario is as follows:
• Of the vampirography participants, approximately 2% are in fact vampires.
• When someone is a vampire, they have 0.85 chance of being detected, ie getting a positive result from a vampirograph (so there is a 0.15 chance they remain undetected).
• When someone is an actual human being, they have 0.1 chance of being falsely “detected” (so the remaining 90% of vampirographs give legitimate negative results).
In other [S:words:S] numbers:
vampires (0.02) humans (0.98)
positive result 0.85 0.1
negative result 0.15 0.9
Now assume that you tested positive on a vampirograph. What are the chances that you’re a genuine vampire? Time to ask Bayes for help.
We’re interested in $\mathbb{P}(\text{vampire}|\text{positive result})$—the probability that you’re a vampire if you tested positive. So what do the numbers tell us?
• The probability that you’re a vampire based only on the fact that you’re getting a vampirography: $\mathbb{P}(\text{vampire})=0.02$.
• The probability that you’re a human based only on the fact that you’re getting a vampirography: $\mathbb{P}(\text{human})=0.98$.
• The probability that you test positive if you’re a vampire: $\mathbb{P}(\text{positive result}|\text{vampire})=0.85$.
• The probability that you test positive if you’re a human: $\mathbb{P}(\text{positive result}|\text{human})=0.1$.
We also need the probability that you test positive regardless of what you are (you’re either a vampire or a human, we assume no other possibilities). This is a bit more tricky, but let’s see what we
can squeeze out of our data. We’ll need the law of total probability, which might be interpreted as a weighted average of probabilities, where we average over all possible cases. In our example we
have only two possibilities—someone is either a vampire or a human. A vampirograph gives a positive result, in each of these cases: rightly when we deal with an actual vampire and falsely when the
participant is human. Therefore we can split our calculation of the probability of the positive result into these two separate cases.
\mathbb{P}(\text{positive result})&=\mathbb{P}(\text{positive result}|\text{vampire})\mathbb{P}(\text{vampire})\\&\quad+\mathbb{P}(\text{positive result}|\text{human})\mathbb{P}(\text{human})
\\&=0.85\times 0.02 + 0.1\times 0.98\\&=0.115,
\end{align*}where we have used the law of total probability. Now we’re ready to plug everything into Bayes’ formula:
\mathbb{P}(\text{vampire}|\text{positive result})&=\mathbb{P}(\text{vampire})\frac{\mathbb{P}(\text{positive result}|\text{vampire})}{\mathbb{P}(\text{positive result})}\\&=0.02 \cdot \frac{0.85}
\end{align*}Yes, even though vampirography seems to be pretty good at detecting vampires, if you test positive the chance that you’re actually a vampire is only 14.8%! No need to panic yet, I guess.
Why is this number so small though? This is always the case with very rare conditions, when the prior probability has a big influence on the posterior. Before the test, the chance that a randomly
chosen participant prefers human blood to ketchup is very small, only 2%, because this is the proportion of vampires in the population. Getting a positive result significantly increases this value,
but we started from a low level, so the final probability remains quite low. Luckily most dangerous diseases, such as different types of cancer, tuberculosis or AIDS, are relatively rare, which means
that conclusions of our study would be similar if you replaced being a vampire with a real illness. This means that a worrying test result was most likely a mistake, not a real problem, and that you
should follow up with a doctor and possibly repeat the test.
Conclusion? Take care of yourself and get tested regularly (this article isn’t sponsored by the NHS in case you’re wondering). However, if you test positive, don’t panic and consult with a doctor… or
a clergyman. Preferably Thomas Bayes.
One thought on “Adventures in Chance-ylvania” | {"url":"https://chalkdustmagazine.com/features/adventures-chance-ylvania/","timestamp":"2024-11-02T01:34:37Z","content_type":"text/html","content_length":"93513","record_id":"<urn:uuid:71192724-312c-41bb-a7ef-6357b3a80abc>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00869.warc.gz"} |
What is RTK? | RTK F9P Positioning Solutions
First, let’s understand what RTK really stands for! Real Time Kinematics is a GNSS technology that allows to partially remove signal errors due to propagation in the atmosphere. These errors are
Antenna’s phase center variation
Ionosphere propagation is the most important effect. The ionization of the propagating medium causes reflections and refractions of the electromagnetic waves. The propagation time measurement time
performed by the receiver is therefore false.
Another important effect that cannot be modeled is multipath. It corresponds to wave reflection on obstacles near the receiver (trees, buildings...) that retard or duplicate signals. It can be
strongly attenuated with a good hardware.
RTK requires consequently two GNSS receivers, a “base” station, generally motionless and whose position is perfectly known, and a “rover” mobile receiver. The base sends correction data to the rover
(raw data) so that the rover can compute the double-difference RTK algorithm. This means that pseudoranges and carrier phases from the base will be “substracted” (it is a bit more complicated though)
from those from the rover. RTKLIB is used to perform those calculations, that can lead to a centimeter-level accuracy.
To get deeper into details, RTK uses carrier measurements in order to get centimetric. GPS signal wavelength is about 20 centimeters, so if you are able to measure the phase of this signal, that you
know satellite’s phase center and that you calculate the integer number of periods between the satellite and you, this is where you get to centimetric precision. We call ambiguity the previous
integer number of periods. If the algorithm finds an integer solution, then the solution is called a “fix“. This is when when maximum accuracy is reached. Otherwise, if the solution is a non-integer
(decimal values), the solution is called “float“. This solution is less precise. We will refer as “single” for the positions neither “fix” nor “float” (happens if no correction data is received, or
if signals are not good enough). Something really important about RTK is that if your antennas do not provide good enough signals, the computed position will easily be erroneous! | {"url":"https://drotek.gitbook.io/rtk-f9p-positioning-solutions/what-is-rtk","timestamp":"2024-11-09T19:21:05Z","content_type":"text/html","content_length":"192259","record_id":"<urn:uuid:b981b07e-7542-4351-b542-0d12ef0e3468>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00549.warc.gz"} |
Problem of multiple generality
Jump to navigation Jump to search
The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:
Some cat is feared by every mouse
then it follows logically that:
All mice are afraid of at least one cat
The syntax of traditional logic (TL) permits exactly four sentence types: "All As are Bs", "No As are Bs", "Some As are Bs" and "Some As are not Bs". Each type is a quantified sentence containing
exactly one quantifier. Since the sentences above each contain two quantifiers ('some' and 'every' in the first sentence and 'all' and 'at least one' in the second sentence), they cannot be
adequately represented in TL. The best TL can do is to incorporate the second quantifier from each sentence into the second term, thus rendering the artificial-sounding terms 'feared-by-every-mouse'
and 'afraid-of-at-least-one-cat'. This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms. Hence the sentence "Some cat is feared by
every mouse" is allotted the same logical form as the sentence "Some cat is hungry". And so the logical form in TL is:
Some As are Bs
All Cs are Ds
which is clearly invalid.
The first logical calculus capable of dealing with such inferences was Gottlob Frege's Begriffsschrift (1879), the ancestor of modern predicate logic, which dealt with quantifiers by means of
variable bindings. Modestly, Frege did not argue that his logic was more expressive than extant logical calculi, but commentators on Frege's logic regard this as one of his key achievements.
Using modern predicate calculus, we quickly discover that the statement is ambiguous.
Some cat is feared by every mouse
could mean (Some cat is feared) by every mouse (paraphrasable as Every mouse fears some cat), i.e.
For every mouse m, there exists a cat c, such that c is feared by m,
${\displaystyle \forall m.\,(\,{\text{Mouse}}(m)\rightarrow \exists c.\,({\text{Cat}}(c)\land {\text{Fears}}(m,c))\,)}$
in which case the conclusion is trivial.
But it could also mean Some cat is (feared by every mouse) (paraphrasable as There's a cat feared by all mice), i.e.
There exists one cat c, such that for every mouse m, c is feared by m.
${\displaystyle \exists c.\,(\,{\text{Cat}}(c)\land \forall m.\,({\text{Mouse}}(m)\rightarrow {\text{Fears}}(m,c))\,)}$
This example illustrates the importance of specifying the scope of such quantifiers as for all and there exists.
Further reading[edit]
• Patrick Suppes, Introduction to Logic, D. Van Nostrand, 1957, ISBN 978-0-442-08072-3.
• A. G. Hamilton, Logic for Mathematicians, Cambridge University Press, 1978, ISBN 0-521-29291-3.
• Paul Halmos and Steven Givant, Logic as Algebra, MAA, 1998, ISBN 0-88385-327-2.
This logic-related article is a stub. You can help Wikipedia by expanding it. | {"url":"https://static.hlt.bme.hu/semantics/external/pages/logikai_form%C3%A1t%C3%B3l/en.wikipedia.org/wiki/Problem_of_multiple_generality.html","timestamp":"2024-11-14T11:42:32Z","content_type":"text/html","content_length":"57204","record_id":"<urn:uuid:a2ea5796-c678-4aea-8525-966e92a1e217>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00470.warc.gz"} |
Algebra: Number Sentences & Fact Families - FLASH-PC
About This Product
Algebra: Number Sentences & Fact Families - FLASH-PC
FLASH-PC is a comprehensive teaching resource specifically designed to simplify the teaching and learning of number sentences and fact families in Algebra. This invaluable asset is ideal for grade 1
and 2 teachers or homeschoolers requiring curriculum-based math materials. It's not only user-friendly but also caters to varied learning styles making it quite versatile in its application.
What's Included:
• An array of tools including pre-assessment aids, lesson plans, real-world word problems, recurrent drill activities.
• A SMART Response assessment tool that harmoniously integrates technology into classroom instruction while enabling efficient monitoring of student progress.
• Dual language support - English and Spanish voice over/text options ensure accessibility across diverse linguistic backgrounds.
• A teacher guide within this package that renders seamless navigation through various components thus enhancing pedagogical efficiency.
This product can be used for whole group lessons revolving around interactive whiteboards showcasing real-world word problems or small group work using printable games coupled with timed drills.
Furthermore, homework assignments extracted from varied resources available cater to concept reinforcement at home thus facilitating continual learning outside classroom boundaries.
In Summary:The 'Algebra: Number Sentences & Fact Families - FLASH-PC' constitutes an all-rounded approach for educators seeking efficient ways to constructively engage students whilst manoeuvring
algebraic concepts fluently.
What's Included
1 zip file with PC software
Resource Tags
number sentences fact families algebra math lesson plan board games | {"url":"https://teachsimple.com/product/algebra-number-sentences-and-fact-families-flash-pc","timestamp":"2024-11-09T21:01:52Z","content_type":"text/html","content_length":"641724","record_id":"<urn:uuid:45e86cc0-1ad7-4c9c-b629-984854b66dd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00163.warc.gz"} |
What is 80 Fahrenheit to Kelvin? - ConvertTemperatureintoCelsius.info
80 Fahrenheit is equal to 299.15 Kelvin.
To understand this temperature conversion, it’s essential to grasp the concept of temperature scales. Fahrenheit and Kelvin are two different units of temperature measurement. Fahrenheit is commonly
used in the United States, while Kelvin is the SI unit for temperature.
The Fahrenheit scale is based on 32 degrees as the freezing point of water and 212 degrees as the boiling point of water at standard atmospheric pressure. In contrast, the Kelvin scale starts at
absolute zero, the theoretical coldest possible temperature, and is commonly used in scientific calculations.
To convert Fahrenheit to Kelvin, you can use the following formula:
T(K) = (T(°F) + 459.67) x (5/9)
Where T(K) is the temperature in Kelvin and T(°F) is the temperature in Fahrenheit.
Now, let’s apply this formula to the given temperature of 80 Fahrenheit:
T(°F) = 80
T(K) = (80 + 459.67) x (5/9)
T(K) ≈ 299.15
This means that 80 Fahrenheit is equivalent to approximately 299.15 Kelvin. This conversion is crucial in scientific research, especially in fields such as physics, chemistry, and engineering.
In real-world scenarios, understanding this conversion can be valuable. For example, in meteorology, scientists often need to convert temperatures between different units to analyze weather patterns
and predict climate changes. Additionally, in industries like manufacturing and healthcare, precise temperature conversions are essential for maintaining equipment and preserving sensitive materials.
In conclusion, understanding the conversion of temperatures from Fahrenheit to Kelvin is essential for anyone working in scientific or technical fields. This knowledge allows for accurate
measurements and calculations, ultimately contributing to advancements in various industries and scientific research. | {"url":"https://converttemperatureintocelsius.info/what-is-80-fahrenheit-in-kelvin/","timestamp":"2024-11-05T22:02:48Z","content_type":"text/html","content_length":"71926","record_id":"<urn:uuid:a861fb63-c486-4121-aa9a-64819029cb35>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00358.warc.gz"} |
The TameFlow Bibliography | Tameflow
ACHOUIANTZ-2013 Chris Achouiantz (2013): The Kanban Kick-start Field Guide v1.1 .
ACKERMAN-2010 Ackerman, L. and Gonzalez C. (2010): Patterns-Based Engineering: Successfully Delivering Solutions via Patterns. Addison-Wesley.
ADDISON-2002 Addison, T. and Vallab, S. (2002). Controlling software project risks: An empirical study of methods used by experienced project managers. SAICSIT 02: Proceedings of the 2002 annual
research conference of the South African institute of computer scientists and information technologists on Enablement through technology - South African Institute for Computer Scientists and
Information Technologists, 2002.
ALBERS-2009 Albers, A. et al. (2009): Design Patterns in Microtechnology. Proceedings of the International Conference on Engineering Design ICED‘09, Stanford.
ALDERFER-2011 Alderfer, C. P. (2011): The Practice of Organizational Diagnosis, Theory and Methods. Oxford University Press 2011.
ALEXANDER-1964 Alexander, C. (1964): Notes on the Synthesis of Form. Oxford University Press (15th printing, 1999).
ALEXANDER-1965 Alexander, C. (1965): A City is Not a Tree. Architectural Forum vol. 122 April No. 1, pp 58-61 and No. 2 pp 58-62.
ALEXANDER-1977 Alexander, C. (1977): A Pattern Language: Towns, Buildings, Construction
ALEXANDER-1979 Alexander, C. (1979): The Timeless Way of Building
ALEXANDER-1985 Alexander, C. et al. (1985): The Production of Houses. Oxford University Press.
ALEXANDER-1999 Alexander, C. (1999): The Origins of Pattern Theory, The Future of the Theory, and the Generation of a Living World. IEEE Software, September/October 1999. Transcript of Keynote speech
at OOPSLA ‘96.
ANDERSON-1993 Anderson, B. (1993): April. Workshop Report: Towards an Architecture Handbook. OOPSLA Messenger. 4 (2): 109–114.
ANDERSON-1994 Anderson, B., Coad, P., and Mayfield, M. 1994. Addendum to the Proceedings of OOPSLA ‘93. Workshop Report: Patterns: Building Blocks for Object Oriented Architectures. OOPS Messenger 5
(2): 107–109.
ANDERSON-2003 Anderson, D. J. (2003): Agile Management for Software Engineering: Applying the Theory of Constraints for Business Results
ANDERSON-2008 Anderson, D. J. (2008): Why We Lost Focus on Development Practices (Blog post).
ANDERSON-2010 Anderson, D. J. (2010): Kanban: Successful Evolutionary Change for Your Technology Business
ANDERSON-2012 Anderson, D. J. (2012): Lessons in Agile Management: On the Road to Kanban
ARGYRIS-1952 Argyris, C. (1952): The Impact of Budgets on People. School of Business and Public Administration, Cornell University.
ARGYRIS-1977 Argyris, C. (1977): Double Loop Learning in Organizations, Harvard Business Review, September 1977.
ARGYRIS-1978 Argyris, C. and Schon D. A. (1978): Organizational Learning: A Theory of Action Perspective (Addison-Wesley Series on Organization Development.) . Addison-Wesley.
ARGYRIS-1991 Argyris, C. (1991): Teaching Smart People How to Learn, Harvard Business Review, May-June 1991, pp. 99-109.
ARGYRIS-1999 Argyris, C. (1999): On Organizational Learning. Second Edition. Wiley-Blackwell.
AUSTIN-2003 Austin, R. et al. (2003): Artful Making: What Managers Need to Know About How Artists Work. Financial Times.
BABATUNDE-1994 Babatunde, A. O. and Harmon, R. (1994): Process Dynamics, Modeling, and Control
BAETJER-1998 Baetjer, H. (1998). Software as Capital, An Economic Perspective on Software Engineering. IEEE Computer Society Press.
BARTRAM-2006 Bartram, P (2006): Forecasting the End of Budgets. Director, Aug2006, Vol. 60 Issue 1, p30.
BECK-1987 Beck, K. and Cunningham, W. (1987): Using Pattern Languages for Object-Oriented Programs. Proceedings of OOPSLA ‘87, Orlando.
BECK-1991 Beck, K (1991): Think Like an Object. Unix Review, September 1991.
BECK-1994 Beck, K. and Johnson, R. (1994): Patterns Generate Architectures. University of Illinois.
BECK-2001 Beck, K. et al. (2001): Manifesto for Agile Software Development. Snowbird, Utah, 2001.
BECKER-2010 Becker, S. et al (2010): The Evolution of a Management Accounting Idea: The Case of Beyond Budgeting. Institute of Management Accounting and Control (IMC) WHU – Otto Beisheim School of
Management, Vallendar, Germany.
BEEDLE-2000 Beedle, M et al. (2000): SCRUM: An extension pattern language for hyperproductive software development.
BENEDICT-1934 Benedict, R. (1934): Patterns of culture. Houghton Mifflin Company.
BENKLER-2006 Benkler, Y. (2006): The Wealth of Networks: How Social Production Transforms Markets and Freedom. Yale University Press.
BENKLER-2011 Benkler, Y. (2011): The Penguin and the Leviathan: How Cooperation Triumphs over Self-Interest. Crown Business.
BENNETT-2007 Bennett, M. et al. (2007): An Architectural Pattern for Goal-Based Control. Proceedings of the IEEE Aerospace Conference, Big Sky.
BERGIN-2000 Bergin, J. (2000): Fourteen Pedagogical Patterns. Proceedings of the 5th European Conference on Pattern Languages of Programs EuroPLoP, Irsee.
BOEHM-1981 Boehm, B. W. (1981): Software Engineering Economics
BOEHM-1991 Boehm, B. (1991). Software risk management: Principles and practices.. IEEE Software, V8N1PP32-41.
BORCHERS-2000 Borchers, J. (2000): A Pattern Approach to Interaction Design. In the Proceedings of the ACM DIS 2000 International Conference on Designing Interactive Systems.
BORCHERS-2001 Borchers, J. (2001): A Pattern Approach to Interaction Design. Wiley
BORODITSKY-2011 Boroditsky, L. (2011): How Language Shapes Thought. The Languages we Speak Affect our Perceptions of the World. Scientific American, February 2011.
BRABHAM-2008 Brabham, D. C. (2008): Crowdsourcing as a Model for Problem Solving. Convergence: The International Journal of Research into New Media Technologies.
BRAGG-2007 Bragg S. M.: Throughput Accounting: A Guide to Constraint Management
BUSCHMANN-1996 Buschmann, F. et al. (1996): Pattern-Oriented Software Architecture, Volume 1 - A System of Patterns. John Wiley & Sons.
BUSCHMANN-2007 Buschmann, F. et al. (2007): Pattern-Oriented Software Architecture, Volume 4 - A Pattern Language for Distributed Computing. John Wiley & Sons.
CAIN-1996a Cain, B. C. and Coplien, J. O. (1996): A Role-Based Empirical Process Modeling Environment. AT&T Bell Laboratories. Proceedings of the Second International Conference on the Software
Process, IEEE Computer Press, pp. 125-133.
CAIN-1996b Cain, B. C. et al. (1996): Social patterns in productive software development organizations. Annals of Software Engineering, 1996, Volume 2, Issue 1, pp. 259-286. Springer.
CAMILLUS-2008 Camillus, J. C. (2008): Strategy as a Wicked Problem, Harvard Business Review, May 2008.
CASPARI-2004 Caspari, J. A. and Caspari P. (2004): Management Dynamics, Merging Constraints Accounting to Drive Improvement. John Wiley.
CASTELLS-2010 Castells, M (2010): The Rise of the Network Society: The Information Age: Economy, Society and Culture, Volume I. Wiley-Blackwell, 2nd Edition.
CHARETTE-1989 Charette, R. N. (1989): Software Engineering Risk Analysis and Management (Mcgraw Hill Software Engineering Series)
CHARLTON-2011 Charlton, I (2011): Theory of Constraints in Software Development.
CLOUTIER-2006 Cloutier, R. J. (2006): Applicability of Patterns to Architecting Complex Systems. Doctoral Dissertation, Stevens Institute of Technology, Hoboken, NJ. USA.
COAD-1992 Coad, P. (1992): Object-Oriented Patterns. Communications of the ACM 35 (9): 152–159.
COAD-1993 Coad, P. and Mayfield, M. (1993). Addendum to the Proceedings of OOPSLA ‘92. Workshop Report: Patterns. “OOPS Messenger” 4 (2): 93–95.
COCKBURN-2001 Cockburn, A. (2001): Agile Software Development. Addison-Wesley Professional. COCKBURN-2005 Cockburn, A. et al. (2005): The declaration of interdependence for modern management or DOI.
COHN-2005 Cohn, M (2005): Agile Estimating and Planning
CONKLIN-2005 Conklin, J (2005): Dialogue Mapping: Building Shared Understanding of Wicked Problems. Wiley.
CONSTANTINE-1995 Constantine, L. (1995). Constantine on Peopleware. Yourdon Press.
COPLIEN-1994 Coplien, J. O. (1994): Borland Software Craftsmanship: A New Look at Process, Quality and Productivity. Proceedings of the 5th Annual Borland International Conference, Orlando, Florida,
5 June 1994.
COPLIEN-1994b Coplien, J. O. (1994): A Development Process Generative Pattern Language. AT&T Bell Laboratories. Proceedings of PLoP/94. Also republished in COPLIEN-1995
COPLIEN-1995 Coplien, J. O. and Schmidt, D. (editors) (1995): Pattern Languages of Program Design
COPLIEN-1996 Coplien, J. O. and Harrison, N.B. (1996): Patterns of Productive Software Organizations. Bell Labs Technical Journal, Summer 1996. Lucent Technologies Inc.
COPLIEN-1996b Coplien, J. O. (1996): The Human Side of Patterns. AT&T Bell Laboratories / C++ Report 8(1)
COPLIEN-1997 Coplien, J. O. (1997) “Idioms and Patterns as Architectural Literature.* IEEE Software, January 1997..
COPLIEN-2004 Coplien, J. O. and Harrison, N. B. (2004): Organizational Patterns of Agile Software Development
COPLIEN-2004b Coplien, J. O. (2004): The Culture of Patterns. ComSIS Journal Vol. 1, No. 2.
COPLIEN-2007 Coplien, J. O. (2007): Organizational Patterns: A Key for Agile Software Development. INCOSE, May 28, 2007.
COPLIEN-2008 Coplien, J. O. (2008): Scrum Patterns Summary: The Patterns without which Scrum is unlikely to work.
CORBETT-1998 Corbett, T. (1998): Throughput Accounting
COSTAGLIOLA-2006 Costagliola, G. et al (2006): Effort estimation modeling techniques: a case study for web applications. Proceedings of the 6th international conference on Web engineering, Palo Alto,
CA, USA
COX-2010 Cox, J. and Schleier, J. (2010): Theory of Constraints Handbook
DAUM-2003 Daum, J. (2003): Interview with Lennart Francke: Managing without budgets at Svenska Handelsbanken. The new New Economy Analyst Report, Feb 24, 2003.
DEARDEN-2002 Dearden, A. et al (2002): Using Pattern Languages in Participatory Design. Sheffield University.
DEGRACE-1990 DeGrace, P. and Hulet Stahl, L. (1990): Wicked Problems, Righteous Solutions: A Catalog of Modern Software Engineering Paradigms. Prentice Hall.
DEKKERS-2001 Dekkers, C. and Gunter, I. (2000): Using “Backfiring” to Accurately Size Software: more Wishful Thinking Than Science?. IT Metrics Strategies, November 2000 Vol VI No. 11, Cutter
DEMARCO-1999 DeMarco, T. and Lister, T. (1999). Peopleware, Productive Projects and Teams. Dorset House Publishing.
DEMARCO-2002 DeMarco, T. (2002). Slack, Getting Past Burnout, Busywork, and the Myth of Total Efficiency. Broadway Books.
DEMARCO-2003 DeMarco, T. and Lister, T. (2003): Waltzing With Bears: Managing Risk on Software Projects
DEMING-1982 Deming, W. E. (1982): Out of the Crisis
DEMING-1993 Deming, W. E. (1993): The New Economics for Industry, Government, Education
DENNING-2008 Denning, P. J., Gunderson, C., and Hayes-Roth, R. (2008): The Profession of IT: Evolutionary System Development. CACM, December 2008, V51 N12 PP29-31.
DENNE-2004a Denne, M. and Cleland-Huang, J. (2004): Software by Numbers: Low-Risk, High-Return Development
DENNE-2004b Denne, M. and Cleland-Huang, J. (2004): The Incremental Funding Method: Data-Driven Software Development. IEEE Software, v21n3:39–47.
DERBY-2006 Derby, E. and Larsen, D. (2006): Agile Retrospectives: Making Good Teams Great
DETTMER-2007 Dettmer, H. W. (2007): The Logical Thinking Process: A Systems Approach to Complex Problem Solving
DIAMOND-2009 Diamond, M. A. and Allcorn, S. (2009): Private Selves in Public Organizations, The Psychodynamics of Organizational Diagnosis and Change. Palgrave Macmillan.
DIBONA-1999 DiBona, C. and Ockman, S., editors (1999). Open Sources, Voices from the Open Source Revolution. O’Reilly Media.
DOVEY-1990 Dovey, K. (1990): The Pattern Language and Its Enemies. Design Studies 11 (1): 3-9.
DUBAKOV-2011 Dubakov, M (September 27, 2011): The Future of Agile Software Development.
DUBNER-2006 Dubner, S. J. and Levitt, S. D. (May 7, 2006): A Star is Made. The New York Times.
ECKSTEIN-2001 Eckstein, J. and Voelter, M. (2001): Learning to Teach, Learning to Learn. Patterns and Pedagogy, A Winning Team. Net Object Days 2001.
ERICKSON-2000 Erickson, T. (2000): Pattern Languages as Languages. CHI 2000 Workshop.
FALLAH-2010 Fallah, M., et. al. (2010). Critical Chain Project Scheduling: Utilizing Uncertainty for Buffer Sizing. International Journal of Research and Review in Applied Sciences, June 2010, pp.
FEDURKO-2012 Fedurko, J (2012). What is the Current Reality Tree - Two Practical Approaches to Building a CRT. 2nd International TOCPA Conference, 19-20 May 2012, Moscow.
FISHER-1992 Fisher, L. M. (1992): The Borland Barbarian’s New Weapon. The New York Times.
FOGEL-2005 Fogel, K. (2005). Producing Open Source Software, How to Run a Successful Free Software Project. O’Reilly Media.
FUTRELL-2002 Futrell, R. T. et al. (2002): Quality Software Project Management
GABRIEL-1996 Gabriel, R. (1996): Gabriel, R. P. (1996): Patterns of Software: Tales from the Software Community
GALBRAITH-2001 Galbraith, J. et al (2001): Designing Dynamic Organizations: A Hands-on Guide for Leaders at All Levels. AMACOM
GAMMA-1994 Gamma, E. et al. (1994): Design Patterns: Elements of Reusable Object-Oriented Software. Addison-Wesley Professional.
GEEKIE-2006 Geekie, A. (2006): Buffer Sizing for the Critical Chain Project Management Method. Master’s thesis, Department of Engineering and Technology Management, Faculty of Engineering, University
of Pretoria.
GILB-1998 Gilb, T. (1988). Principles of Software Engineering Management. Addison-Wesley.
GOLDIN-2006 Goldin, D et al., editors (2006): Interactive Computation, The New Paradigm. Springer.
GOLDMAN-2005 Goldman, R. and Gabriel, R. P. (2005). Innovation Happens Elsewhere, Open Source as Business Strategy. Morgan Kaufmann Publishers.
GOLDRATT-1990a Goldratt, E. (1990): The Haystack Syndrome: Sifting Information Out of the Data Ocean
GOLDRATT-1990b Goldratt, E. (1990): What is this Thing Called Theory of Constraints
GOLDRATT-1994 Goldratt, E. (1994): It’s Not Luck
GOLDRATT-1992 Goldratt, E. (1992): The Goal: A Process of Ongoing Improvement
GOLDRATT-1997 Goldratt, E. (1997): Critical Chain
GOOLD-2002 Goold M. and Campbell A. (2002): Designing Effective Organizations: How to Create Structured Networks. Wiley
GOVINDARAJ-2011 Govindaraj, S. (2011): Using Class of Service to Manage Product Risk.. Silver Stripe Software.
GRAHAM-2006 Graham, P. (June, 2006): The Power of the Marginal.
GRIFFITHS-2007 Griffiths, M. (2007): Developments in Agile Project Management. PMI Global Conference Proceedings, Atlanta, Georgia.
HAMMARBERG-2013 Hammarberg, M and Sunden, J. (2013): Kanban in Action. Manning Publications.
HANMER-2004 Hanmer R. and Kocan K. (2004): Documenting Architectures with Patterns. Bell Labs Technical Journal 9(1): 143-163, Wiley Periodicals.
HARRISON-1996 Harrison, N. B. and Coplien, J. O. (1996): Patterns of Productive Software Organizations. Bell Labs Technical Journal.
HARRISON-1999 Harrison, N. et al. (editors) (1999): Pattern Languages of Program Design 4. Volume 4 of the Software Patterns Series. Addison-Wesley.
HARRISON-2000 Harrison, N. et al. (2000): Pattern Languages of Program Design 4
HAY-1995 Hay, D. C. (1995): Data Model Patterns, Conventions of Thought. Dorset House Publishing.
HEIN-2011 Hein, A. M. (2011): Project Icarus: Stakeholder Scenarios for an Interstellar Exploration Program. Technische Universität München, Institute of Astronautics.
HEIN-2012 Hein, A. M. (2012): Adopting Patterns for Space Mission and Space Systems Architecting. 5th International Workshop on Systems & Concurrent Engineering for Space Applications.
HIBBS-2009 Hibbs, C. et al. (2009): The Art of Lean Software Development
HOPE-2003 Hope, J and Fraser, R. (2003): Who Needs Budgets?, Harvard Business Review, February 2003.
HOWE-2009 Howe, J. (2009): Crowdsourcing: Why the Power of the Crowd Is Driving the Future of Business. Crown Business.
HUMPHREY-1995 Humphrey, W. S. (1995): A Discipline for Software Engineering
IKONEN-2011 IIkonen, M. et al. (2011): On the Impact of Kanban on Software Project Work: An Empirical Case Study Investigation. Department of Computer Science, University of Helsinki, Finland.
ISHIKAWA-1990 Ishikawa, K. (1990): Introduction to Quality Control
JACOB-2009 Jacob, D et al. (2009): Velocity: Combining Lean, Six Sigma and the Theory of Constraints to Achieve Breakthrough Performance - A Business Novel. Free Press.
JOHNSON-2004 Johnson E. (pseudonym) (2004): Function Points: Numerology for Software Devlopers in Hacknot: Essays on Software Development.
JONES-1995 Jones, C. (1995): Backfiring: Converting Lines-of-Code to Function Points. IEEE Software, November 1995 (vol. 28 no. 11).
JONES-1999 Jones, D. et al. (1999): Patterns: Using Proven Experience to Develop Online Learning. Interactive Multimedia, Queensland University.
JONES-1996 Jones, C. (1996): Applied Software Measurement: Assuring Productivity and Quality
JONES-2000 Jones, C. (2007): Software Assessments, Benchmarks, and Best Practices
JONES-2007 Jones, C. (2007): Estimating Software Costs: Bringing Realism to Estimating
KATZ-1978 Katz, D. and Kahn, R. L. (1978): The Social Psychology of Organizations. Wiley.
KEIDEL-1995 Keidel, R. W. (1995): Seeing Organizational Patterns. A New Theory and Language of Organizational Design.
KERTH-2001 Kerth, N. L. (2001): Project Retrospectives: A Handbook for Team Reviews
KIRCHER-2004 Kircher, M. et al. (2004): Pattern-Oriented Software Architecture, Volume 3 - Patterns for Resource Management. John Wiley & Sons.
KNIBERG-2010 Kniberg, H. and Skarin, M. (2010): Kanban and Scrum - making the most of both
KITCHENHAM-1997 Kitchenham, B. (1997): Counterpoint: The Problem with Function Points. IEEE Software, March 1997, pp 29-31.
KOENIG-1995 Koenig, A. (1995): Patterns and Antipatterns. Journal of Object-Oriented Programming 8(1): 46-48. Later republished in (Rising 1998).
KROEBER-1909 Kroeber, A. L (1909): Classificatory Systems of Relationships. The Journal of the Royal Anthropological Institute of Great Britain and Ireland, vol. XXXIX, pp. 77-84.
KROEBER-1938 Kroeber, A. L. (1938): Basic and Secondary Patterns of Social Organization. The Journal of the Royal Anthropological Institute of Great Britain and Ireland, vol. LXVIII, July-December,
pp. 299-309.
KROEBER-1944 Kroeber, A. L. (1944): Configurations of Culture Growth. Cambridge University Press.
KROEBER-1948 Kroeber, A. L. (1948): Anthropology: Culture, Patterns and Process. Harcourt, Brace and World.
KROEBER-1952 Kroeber, A. L. (1952): The Nature of Culture. University of Chicago Press. Organizational Design
LADAS-2008 Ladas, C. (2008): Scrumban - Essays on Kanban Systems for Lean Software Development
LAIRD-2006 Laird L. M. and Brennan M. C. (2006): Software Measurement and Estimation: A Practical Approach (Quantitative Software Engineering Series)
LANDY-2010 Landy, F. J. and Conte, J. M. (2010): Work in the 21st Century. An Introduction to Industrial and Organizational Psychology. Third Edition. Wiley-Blackwell.
LAVAZZA-2008 Lavazza L. A., et al. (2008): Model-based Functional Size Measurement. Proceedings of the Second ACM-IEEE international symposium on Empirical software engineering and measurement,
Kaiserslautern, Germany.
LEACH-2004 Leach, L. P. (2004): Critical Chain Project Management, Second Edition
LEVESQUE-2008 Levesque G. et al. (2008): Estimating Software Size with UML Models. Proceedings of the 2008 C3S2E conference, Montreal, Quebec, Canada.
LITTLE-1961 Little, J. D. C. (1961): A proof for the queuing formula: L = λW. Operations Research, 9(3) 383–387.
LITTLE-2008 Little, J. D. C and Graves, S. C. (2008). Little’s Law, pp 81-100, in Chhajed, D. and Lowe, T. J. (eds.) Building Intuition: Insights From Basic Operations Management Models and
Principles doi: 10.1007/978-0-387 -73699-0, (c) Springer Science + Business Media.
LITTLE-2011 Little, J. D. C. (2011): Little’s Law Viewed on its 50th Anniversary. Operations Research, 59(3) 536-549.
LOWE-2006 Lowe, J. D. (2006): A Design Pattern Language for Space Stations and Long-Term Residence Human Spacecraft. American Institute of Aeronautics and Astronautics.
MANNS-2004 Manns, M. L. and Rising, L. (2004): Fearless Change: Patterns for Introducing New Ideas. Addison-Wesley.
MARTIN-1998 Martin, R. C. et al. (editors) (1998): Pattern Languages of Program Design 3. Volume 3 of the Software Patterns Series. Addison-Wesley,
MARTIN-2012 Martin R (2012): Why I Decided to Rethink Hiring Smart People, Harvard Business Review Blog, October 2012.
MEDIRATTA-2007 Mediratta, B. (2007): The Google Way: Give Engineers Room. The New York Times, October 21, 2007.
MESZAROS-1997 Meszaros, G. and Doble, J.(1997) “A Pattern Language for Pattern Writing,” in “Pattern Languages of Program Design 3”, Martin R. et al. (ed.): Addison-Wesley Longman.
MCCARTHY-2012 McCarthy, J. and McCarthy M. (2012): Elements of the Core.
MCGRATH-1995 McGrath, R. G. and MacMilan (1995): Discovery-Driven Planning, Harvard Business Review, July 1995.
MCGRATH-2010 McGrath, R. G. (2010): Business Models: A Discovery Driven Approach. Long Range Planing, Elsevier.
MILLS-1972 Mills, Harlan D. (1972): Mathematical Foundations of Structured Programming. IBM Corporation Technical Report No. FSC 72-6012, IBM Federal Systems Division, Gaithersburg, Maryland.
MINTZBERG-1992 Mintzberg H. (1992): Structure in Five: Designing Effective Organizations. Prentice Hall.
MORENO-1931 Moreno, J. L. (1931): Group Method and Group Psychotherapy. Beacon House
MORENO-1934 Moreno, J. L. (1934): Who shall survive?: foundations of sociometry, group psychotherapy, and sociodrma. Washington, D. C.: Nervous and Mental Disease Publishing Co., 1934. Who Shall
Survive? Foundations of Sociometry, Group Psychotherapy and Sociodrama (1953 reprint)
MORENO-1951 Moreno, J. L. (1951): Sociometry, Experimental Method and the Science of Society. Beacon House.
MORENO-1953 Moreno, J. L. (1953/1977): Who shall survive? Foundations of Sociometry, Group Psychotherapy and Sociodrama. Beacon House.
MOROWSKI-2008 Morowski, P. (2008): The Borland Agile Journey - An Executive Perspective on Enterprise Transformation. Agile Journal, August 2008.
MULLINS-2006 Mullins, L. J. (2006): Essentials of Organisational Behaviour. Prentice Hall
MURRAY-2000 Murray, A. R. (2000): Discourse Structure of Software Explanation: Snapshot Theory, Cognitive Patterns and Grounded Theory Methods. Doctoral Thesis, University of Ottawa.
NERUR-2007 Nerur, S. and Balijepally, V. (2007): Theoretical Reflections on Agile Development Methodologies The traditional goal of optimization and control is making way for learning and
optimization. CACM, March 2007, V50 N30 PP79-83.
NOBEL-2010 Nobel, J. and Johnson, R.. (editors) (2010): Transactions on Pattern Languages of Programming I. Springer.
NOBEL-2011 Nobel, J. et al. (editors) (2011): Transactions on Pattern Languages of Programming II. Springer.
NOLAN-1990 Nolan, T. W. and Provost, L. P. (1990): Understanding Variation, Quality Progress.
NOREEN-1995 Noreen, E. W. et al.: Theory of Constraints and Its Implications for Management Accounting
OESTERGREN-2008 Oestergren, K and Stensaker, I (2008): Management control without budgets: A field study of “beyond budgeting” in practice. NHH Norwegian School of Economics and Business
OHNO-1988 Ohno, T. (1988): Toyota Production System: Beyond Large-Scale Production
PATTON-2009 Patton, J. (2000): Kanban Development Oversimplified (Online article).
PERZEL-1999 Perzel, K. and Kane, D. (1999): Usability Patterns for Applications on the World Wide Web. PLoP 1999 Conference.
POPPENDIECK-2003 Poppendieck, M. and Poppendieck, T. (2003): Lean Software Development: An Agile Toolkit
POPPENDIECK-2007 Poppendieck, M. and Poppendieck, T. (2007): Implementing Lean Software Development: From Concept to Cash
REIFER-2000 Reifer, D. (2000): Web Development: Estimating Quick-to-Market Software. IEEE Software, November/December 2000, pp 57-64.
REINERTSEN-2009 Reinertsen, D. J. (2009): The Principles of Product Development Flow: Second Generation Lean Product Development
REIS-2011 Reis, E. (2011): The Lean Startup: How Today’s Entrepreneurs Use Continuous Innovation to Create Radically Successful Businesses. Crown Business.
RICKETTS-2007 Ricketts, J. A. (2007): Reaching The Goal: How Managers Improve a Services Business Using Goldratt’s Theory of Constraints
RISING-1990 Rising, L. (1999): Patterns: A Way to Reuse Expertise. IEEE Communications Magazine, April 1999.
RISING-1998 Rising, L. (editor) (1998): The Patterns Handbook: Techniques, Strategies, and Applications. Cambridge University Press.
RITTEL-1973 Rittle, H. W. and Webber M. M. (1973): Dilemmas in a General Theory of Planning. Policy Sciences n. 4, pp 155-169, Elsevier Scientific Publishing Company.
ROLL-HANSEN-2009 Roll-Hansen, N. (2009): Why the distinction between basic (theoretical) and applied (practical) research is important in the politics of science. Center for the Philosophy of Natural
and Social Science, Contingency and Dissent in Science Project, Technical Report 04/09, London School of Economics.
SALINGAROS-2000 Salingaros, N. A. (2000): The Structure of Pattern Languages. Architectural Research Quarterly, vol. 4, pp. 149-161. Cambridge University Press.
SALUSTRI-2005 Salustri, F. A. (2005): Using Pattern Languages in Design Engineering. Proceedings of the International Conference on Engineering Design ICED'05, Melbourne, 2005.
SCHEINKOPF-1999 Scheinkopf, L. (1990): Thinking for a Change: Putting the TOC Thinking Processes to Use (The CRC Press Series on Constraints Management)
SCHRAGENHEIM-1999 Schragenheim, E. (1999): Management Dilemmas: The Theory of Constraints Approach to Problem Identification and Solutions (The CRC Press Series on Constraints Management)
SCHMIDT-2007 Schmidt, D. et al. (2007): Pattern-Oriented Software Architecture, Volume 2 - Patterns for Concurrent and Networked Objects. John Wiley & Sons.
SCHWABER-2001 Schwaber K. (2001): Agile Software Development with Scrum
SCHWABER-2011 Schwaber, K (2011-04-07): Scrum Fails? Ken Schwaber’s Blog: Telling It Like It Is.
SCHULER-2008 Schuler, D. (2008): Liberating Voices, A Pattern Language for Communication Revolution. MIT Press
SENGE-2006 Senge, P. (2006): The Fifth Discipline: The Art & Practice of the Learning Organization. Doubleday.
SHEWHART-1986 Shewart, W. A. (1986): Statistical Method from the Viewpoint of Quality Control (Dover Books on Mathematics)
SHUSTEK-2008 Shustek, L. (2008): Donald Knuth: A life’s work interrupted, Communications of the ACM, vol. 51, issue 8: ACM, New York, NY, USA, pp. 31-35, 08/2008.
SMITE-2010 Smite, D. et al. (2010): Agility Across Time and Space: Implementing Agile Methods in Global Software Projects Steve Tendon contributed Chapter 4: “Tailoring Agility: Promiscuous Pair
Story Authoring and Value Calculation.”
SMITH-1999 Smith, D: The Measurement Nightmare: How the Theory of Constraints Can Resolve Conflicting Strategies, Policies, and Measures (APICS Constraints Management)
SMITH-2003 Smith, F. J. (2003): Organizational Surveys, The Diagnosis and Betterment of Organizations Through Their Members. Lawrence Erlbaum Associates. SMITH-2012 Smith, J. McC. (2012): Elemental
Design Patterns. Addison-Wesley.
SPOLSKY-2007 Spolsky J. (2007): Evidence Based Scheduling.
SONG-2008 Song, J-M. (2008): Extending Performance-Based Design Methods by Applying Structural Engineering Design Patterns. Dissertation, University of California, Berkeley.
STANFORD-2007 Stanford, N. (2007): Guide to Organisations Design: Creating high-performing and adaptable enterprises. Bloomberg Press.
STOKES-1997 Stokes, D. E. (1997): Pasteur’s Quadrant, Basic Science and Technological Innovation. Brookings Institution Press.
SULLIVAN-2012 Sullivan, T. T. et al. (2012): The TOCICO Dictionary, Second Edition, 2012.
SUROWIECKI-2005 Surowiecki, J. (2005): The Wisdom of Crowds. Anchor Books. i
SUTHERLAND-2001 Sutherland, J. (2001): Inventing and Reinventing SCRUM in Five Companies. PatientKeeper, Inc.
SUTHERLAND-2003 Sutherland, J. (2003): Scrum: Another way to think about scaling a project.
SUTHERLAND-2005 Sutherland, J. et al. (2005): Future of Scrum: Parallel Pipelining of Sprints in Complex Projects . Agile 2005, July 24-29, 2005, Mariott Denver City Center.
SUTHERLAND-2006 Sutherland, J. et al. (2006): Adaptive Engineering of Large Software Projects with Distributed/Outsourced Teams . 6th International Conference on Complex Systems (ICCS), June 25-30,
2006; Boston, MA.
SUTHERLAND-2007 Sutherland, J. et al. (2007): Distributed Scrum: Agile Project Management with Outsourced Development Teams . 40th Annual Hawaii International Conference on System Sciences
SUTHERLAND-2007b Sutherland, J. (2007): Origins of Scrum . Blog post.
SUTHERLAND-2008 Sutherland, J. et al. (2008): Fully Distributed Scrum: The Secret Sauce for Hyperproductive Offshored Development Teams . 6th International Conference on Complex Systems (ICCS), June
25-30, 2006; Boston, MA.
SUTHERLAND-2008a Sutherland J. (2008): Pretty Good Scrum: Secret Sauce for Distributed Teams .
SUTHERLAND-2009 Sutherland, J. (2009): Shock Therapy Self Orgnization in Scrum .
SUTHERLAND-2009b Sutherland, J. (2009): Agile Architecture: Red Pill or Blue Pill .
SUTHERLAND-2010 Sutherland J. (2010): Agile Contracts: Money for Nothing and Your Change for Free .
SUTHERLAND-2010a Sutherland, J. (2010): The Roots of Scrum, How the Japanese Lean Experience Changed Global Software Developmment . ACCU conference 2010. A video of an earlier similar presentation is
here: The Roots of Scrum - InfoQ Presentation .
SUTHERLAND-2011 Sutherland, J. and Schwaber, K. (2011): The Scrum Papers: Nut, Bolt, and Origins of an Agile Framework . Draft, January 29, 2011, Scrum, Inc.
SUTHERLAND-2012 Sutherland, J. and Schwaber, K. (2012): The Scrum Papers: Nut, Bolt, and Origins of an Agile Framework . Version 1.1, April 2, 2012, Scrum, Inc.
SUTHERLAND-2013 Sutherland, J. et al. (2013): Teams that Finish Early Accelerate Faster: A Pattern Language for High Performing Scrum Teams .
SWIERINGA-1992 Swieringa, J and Wierdsma, A. (1992): Becoming a Learning Organization: Beyond the Learning Curve. Longman Group.
TAKEUCHI-1986 Takeuhci, I. and Nonaka, I. (1986): New New Product Development Game. Harvard Business Review Article.
TENDON-2002 Tendon, S. (2002): Mobile Marketing Patterns. Research with Prof. Douglas Lamont, Northwestern University, Chicago.
THOMAS-2005 Thomas, J. (2005): Patterns to Promote Individual and Collective Creativity. IBM Research.
TIDWELL Tidwell, J. (1998): Interaction Design Patterns. PloP 1998.
TIDWELL Tidwell, J. (2010): Designing Interfaces, Patterns for Effective Interaction Design. O’Reilly.
TUCKMAN-1965 Tuckman, B. W. (1965): Developmental Sequences in Small Groups. Psychological Bulletin, Vol 63(6), Jun 1965, 384-399.
VANWELIE-2000 van Welie, M. and Troettenberg H. (2000): Interaction Patterns in User Interfaces. PLoP 2000.
VLISSIDES Vlissides, J. M. et al (editors) (1996): Pattern Languages of Program Design 2. Volume 2 of the Software Patterns Series. Addison-Wesley
WEBER-1992 Weber, J. (1992): Kahn the Barbarian. Los Angeles Times.
WEGNER-1997 Wegner, P. (1997): Why interaction is more powerful than algorithms .
WEGNER-1999 Wegner P. and Goldin D (1999): Interaction, Computability, and Church’s Thesis. Draft, Brown University.
WEGNER-2006 Wegner P. and Goldin D (2006): * Principles of Problem Solving.* CACM, July 2006, V49 N7 PP27-29.
WEINBERG-1998 Weinberg, G. M. (1998). The Psychology of Computer Programming, Silver Anniversary Edition. Dorset House Publishing.
WILKINSON Wilkinson, N. M. (1998): Using CRC Cards. SIGS.
WOEPPEL-2005 Woeppel, M. (2005): Projects in Less Time: A Synopsis of Critical Chain
WOMACK-2003 Womack, J. P. and Jones, D. T. (2003): Lean Thinking: Banish Waste and Create Wealth in Your Corporation, Revised and Updated
WONG-2008 Wong, J. M. (2008): Extending performance-based design methods by applying structural engineering design patterns. University of California, Berkeley. | {"url":"https://tameflow.com/bibliography/","timestamp":"2024-11-11T07:31:22Z","content_type":"text/html","content_length":"80710","record_id":"<urn:uuid:52bf8a0b-9f61-4127-ac89-6f7fbf82e96d>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00544.warc.gz"} |
Fraction Bars
Fraction Bars Printable
Fraction Bars Printable - Web our online fraction flash cards use visual aids, circles or fraction bars, to introduce students the concept of fractions. Use them for matching, identifying, solving
and exploring fractions. Web what is a fraction bar? Web teach starter has created a printable activity to help your students understand how unit fractions are pieced together to create a whole. Use
them for adding, subtracting, multiplying, dividing, and creating worksheets. Web either you can purchase one from amazon or make one of your own. Web use these fractions worksheets to produce
rectangular fractions bars and pie wedge fractions to be used as visuals in your teaching lesson plans. These colorful fraction bars will enrich your teaching of whole and part relationships.
Students can cut out the bars and use them as. Web fraction strips 1 whole 1 2 1 2 1 3 1 3 1 3 1 4 1 4 1 4 1 4 1 5 1 5 1 5 1 5 1 5 1 6 1 6 1 6 1 6 1 6 1 6 1 8 1 8 1 8 1 8 1.
Free Printable Fraction Bars/Strips Chart (Up To 20) Number Dyslexia
Web what is a fraction bar? Web use these fractions worksheets to produce rectangular fractions bars and pie wedge fractions to be used as visuals in your teaching lesson plans. Use them for adding,
subtracting, multiplying, dividing, and creating worksheets. Web fraction strips 1/2 1/5 1/6 1/4 1/6 1/3 1/5 1/4 1/2 1/5 1/6 1/3 1/4 1/5. Students can cut.
Printable Fraction Bars Etsy
Web use these fractions worksheets to produce rectangular fractions bars and pie wedge fractions to be used as visuals in your teaching lesson plans. Web what is a fraction bar? Web in four seasons
with the thunder, pokuševski appeared in 150 games (65 starts) and averaged 7.5 points, 4.7 rebounds and 2.0 assists in 20.6 minutes per game. Visit classplayground.com.
Blank Fraction Bars Printable Printable Blank World
This packet is full of fun printable worksheets that will help your students learn fractions skills! Visit classplayground.com for more resources and printables about fraction strips. As per the
request, numberdyslexia.com is giving a free printable version of the fraction. Web download and print color or black and white fraction strips to teach kids about fractions. Web try our interactive.
Fraction Strip Templates for Kids School Math Printables
These flashcards of fractions are generated. Use the interactive fraction strips tool to help students. Web in four seasons with the thunder, pokuševski appeared in 150 games (65 starts) and averaged
7.5 points, 4.7 rebounds and 2.0 assists in 20.6 minutes per game. Web printable fraction bars by teach simple math > fractions | not grade specific, grade 3, 4,.
Free Fraction Strips Template Nyla's Crafty Teaching
Web our online fraction flash cards use visual aids, circles or fraction bars, to introduce students the concept of fractions. Web browse fraction bar printable resources on teachers pay teachers, a
marketplace trusted by millions of teachers for original educational resources. The fraction bars (often referred to as cuisenaire rods) have been made to work hand in hand with my.
Free Printable Fraction Bars/Strips Chart (Up To 20) Number Dyslexia
A variety of options are available on polypad for manipulating. A fraction bar is a rectangular manipulative that represents a whole or parts of a whole. These flashcards of fractions are generated.
Web fraction strips 1/2 1/5 1/6 1/4 1/6 1/3 1/5 1/4 1/2 1/5 1/6 1/3 1/4 1/5. Web either you can purchase one from amazon or make one.
Fraction Strips Printable File Folder Fun
Web fraction strips 1/2 1/5 1/6 1/4 1/6 1/3 1/5 1/4 1/2 1/5 1/6 1/3 1/4 1/5. Web try our interactive fraction circles. Fraction bars should be cut into their. As per the request, numberdyslexia.com
is giving a free printable version of the fraction. Visit classplayground.com for more resources and printables about fraction strips.
Fraction Strips Printable Activity for Students — Mashup Math
Web what is a fraction bar? A variety of worksheets with 55 pages. Printable fraction strips color blank. Web try our interactive fraction circles. Web fraction strips 1/2 1/5 1/6 1/4 1/6 1/3 1/5 1/4
1/2 1/5 1/6 1/3 1/4 1/5.
Fraction Strips for Fractions, Percents and Decimals Nyla's Crafty
Web either you can purchase one from amazon or make one of your own. Web download and print fraction strips in different colors and formats to help kids learn and visualize fractions. Visit
classplayground.com for more resources and printables about fraction strips. A fraction bar is a rectangular manipulative that represents a whole or parts of a whole. Use the.
Fraction Strip Templates for Kids School Math Printables
Web browse fraction bar printable resources on teachers pay teachers, a marketplace trusted by millions of teachers for original educational resources. The fraction bars (often referred to as
cuisenaire rods) have been made to work hand in hand with my fraction strips. Web printable fraction bars by teach simple math > fractions | not grade specific, grade 3, 4, 5,.
Printable Fraction Strips Color Blank.
Use them for matching, identifying, solving and exploring fractions. Web download and print fraction strips in different colors and formats to help kids learn and visualize fractions. Web use these
fractions worksheets to produce rectangular fractions bars and pie wedge fractions to be used as visuals in your teaching lesson plans. A variety of worksheets with 55 pages.
Web Fraction Strips 1 Whole 1 2 1 2 1 3 1 3 1 3 1 4 1 4 1 4 1 4 1 5 1 5 1 5 1 5 1 5 1 6 1 6 1 6 1 6 1 6 1 6 1 8 1 8 1 8 1 8 1.
These flashcards of fractions are generated. Web printable fraction bars by teach simple math > fractions | not grade specific, grade 3, 4, 5, 6 | activities printable fraction bars attributes
subject fractions grades not. Menu about us gift cards. Web teach starter has created a printable activity to help your students understand how unit fractions are pieced together to create a whole.
Web Try Our Interactive Fraction Circles.
Web new york attorney general letitia james brought the civil suit in 2022, seeking a penalty that grew to $370 million and asking the judge to bar trump from doing. Web our online fraction flash
cards use visual aids, circles or fraction bars, to introduce students the concept of fractions. Visit classplayground.com for more resources and printables about fraction strips. Web fraction strips
1/2 1/5 1/6 1/4 1/6 1/3 1/5 1/4 1/2 1/5 1/6 1/3 1/4 1/5.
Fraction Bars Should Be Cut Into Their.
Web in four seasons with the thunder, pokuševski appeared in 150 games (65 starts) and averaged 7.5 points, 4.7 rebounds and 2.0 assists in 20.6 minutes per game. As per the request,
numberdyslexia.com is giving a free printable version of the fraction. These colorful fraction bars will enrich your teaching of whole and part relationships. Web what is a fraction bar?
Related Post: | {"url":"https://elastic.almalnews.com/en/fraction-bars-printable.html","timestamp":"2024-11-11T23:39:59Z","content_type":"text/html","content_length":"26898","record_id":"<urn:uuid:5aeb0dbc-4b2b-4212-9af7-3de72301542d>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00469.warc.gz"} |
Jane Greer, Author at Lakes Region Repeater Association
The rules and regulations in this part are designed to provide an amateur radio service having a fundamental purpose as expressed in the following principles:
(a) Recognition and enhancement of the value of the amateur service to the public as a voluntary noncommercial communication service, particularly with respect to providing emergency communications.
(b) Continuation and extension of the amateur’s proven ability to contribute to the advancement of the radio art.
(c) Encouragement and improvement of the amateur service through rules which provide for advancing skills in both the communication and technical phases of the art.
(d) Expansion of the existing reservoir within the amateur radio service of trained operators, technicians, and electronics experts.
(e) Continuation and extension of the amateur’s unique ability to enhance international goodwill.
Scientific Details of The Linen Frequency Study
In 2003, a study was done by a Jewish doctor, Heidi Yellen, on the frequencies of fabric. According to this study, the human body has a signature frequency of 100, and organic cotton is the same –
100. The study showed that if the number is lower than 100, it puts a strain on the body. A diseased, nearly dead person has a frequency of about 15, and that is where polyester, rayon, and silk
register. Nonorganic cotton registers a signature frequency of about 70. However, if the fabric has a higher frequency, it gives energy to the body. This is where linen comes in as a super-fabric.
Its frequency is 5,000. Wool is also 5,000, but when mixed together with linen, the frequencies cancel each other out and fall to zero. Even wearing a wool sweater on top of a linen outfit in a study
collapsed the electrical field. The reason for this could be that the energy field of wool flows from left to right, while that of linen flows in the opposite direction, from right to left.
In an email dated 2/10/12, Dr. Yellen explained the process of this study:
“Frequency was determined by a technician named Ivanne Farr who used a digital instrument designed by a retired Texas A&M professor called the Ag-Environ machine. We had a public demonstration with
an audience at internationally known artist Bob Summers home.
“Bob Graham, the inventor, told us that his machine was created to analyze the signature frequencies of agricultural commodities to aid the farmer in determining the right time of harvest growth. The
gentleman identified signature frequencies that identified illness also and had turned to helping people get well. Bob Graham stated that it was a ‘signature frequency of that plant’s species
identity.’ The mHz is different, we were suggested that it would be the same as Rose essential oil.
“There could be better devices so we have been looking around for more options. There’s a device that a brilliant American agriculture scientist developed that does measure the frequency of Linen. We
have not yet acquired one but hope to soon!
“Dr. Philip Callahan, a noted physician and researcher, was able to prove the existence of this energy using plant leaves attached to an oscilloscope. About six months ago, he visited me in
California and showed me a new development. He had discovered that flax cloth, as suggested in the Books of Moses [the Torah or Pentatuch], acts as an antenna for the energy. He found that when the
pure flax cloth was put over a wound or local pain, it greatly accelerated the healing process. He was also using the flax seed cloth as a sophisticated antenna for his oscilloscope. This is the
instrument that he uses to determine energy of flax.”
Pgs. 19-20 of Whole Health, by Mark Mincolla Ph.D.
Bennett Hill Tower 12/18/2023
Bennett Hill Tower Ossipee, NH 12/18/2023
Rev 2 12/7/2023
Stephen Connell KC1SVE
There are matching tools available to calculate and design an impedance matching circuit in moments. One example is provided by Analog devices Inc.’s online design center tool RF Impedance Matching
Link: https://www.analog.com/en/design-center/interactive-design-tools/rf-impedance-matching-calculator.html
Another online tool that is helpful is the Telestrian Interactive Smith Chart. It doesn’t do the design for you, but it shows you the effects of adding matching components. You can plug in the load
impedance, set the series matching components to a short circuit and the parallel components to an open and it shows you where your impedance lies on the Smith chart. You can then add your matching
components to view how the impedance point moves on the chart.
Link: http://cgi.www.telestrian.co.uk/cgi-bin/www.telestrian.co.uk/smiths.pl
I was curious though, what’s the magic behind these tools? Sometimes it helps to know, for when things go awry. This section describes the procedure that I worked out should I ever have to “do it
My approach here is to try to limit this discussion to what MUST be done and WHY I am doing it. I’m trying to keep it simple with enough information to recall what I was thinking at the time, so I
can do it again when I want to.
A basic understanding of resistance, reactance, (resistors, inductors and capacitors) and the Smith chart is required. For learning about the Smith chart, I highly recommend Basics of the Smith
Chart, Alan Wolke’s (W2AEW) online presentation. This is the first time I’ve actually used a Smith Chart for a practical application so, I’m certainly not an expert. It might be helpful though to
follow me here to see how I finally got it to work for me.
I have a Radio Transceiver designed to operate with a 50-ohm antenna (load) impedance. I want to match my antenna (load) impedance to the Radio (source) impedance. Why? The match is needed so that
the Transmit signal does not reflect back from the load to the source AND when I receive a signal, I don’t want the signal to reflect back up the antenna. Reflections are bad because they return the
RF signal to the radio (or antenna) reducing the power for the antenna to radiate (or the radio to receive). In the transmit case it could also damage the radio if the reflected power exceeds the
radio’s ability to dissipate the energy. If we match the source and load impedances, we won’t have reflections, so all of the power will be radiated (or received). When matched(equal) their ratio
will be 1:1 of course. This is our goal; it is referred to as a Voltage Standing Wave Ratio (VSWR). We want this to be 1.
Objective – > VSWR = 1.0
Antenna Impedance
The antenna has impedance to signal flow. The impedance is due to its resistance, inductance and capacitance. These are the three components that we need to know about to describe what is impeding
the signal flow. The resistance (R) is kept very low by using conductors that are large enough to avoid dissipating heat energy. Resistance is NOT dependent on frequency. The inductance and
capacitance also impede the signal flow. This impedance IS frequency dependent, it’s called reactance, designated by X. Reactance depends on the geometry of the antenna, where you are connected to
it, and how it sits in the electric and magnetic fields that it generates and that exist around it (from things like the earth and other conductors). The geometries of the conductors effectively
create inductors and capacitors, just like the components we can buy. Because their impedance is frequency dependent, as we transmit or try to receive signals of different frequencies this impedance
changes as well. So, if we want the antenna to work well (limited signal loss due to reflections) we need to design matching circuits for the frequency ranges we plan on using. Because of this
frequency dependence, we need to measure and track this impedance separately from resistance. Since frequency is the common factor for both inductor and capacitor impedance, we can combine them into
a single reactance term (X). The resulting equation to describe impedance is:
Impedance equation Z = R + jX Where R (resistance) is NOT frequency dependent and X (reactance) IS frequency dependent.
Impedance (Z) and Admittance (Y = 1/Z)
So, we have impedance that can be described by physical passive components (resistors, inductors and capacitors). What is Admittance and why do we need it? Admittance is described by the equation G
+jB, where G is the conductance (1/R – frequency independent) and B is the susceptance (1/X- frequency dependent). Do we have passive components that can conduct? And, what is susceptance? It turns
out that when we add any of the three passive components in parallel to the load, instead of in series, the circuit will conduct more current and it is more susceptible to passing the frequency
dependent current (AC). We need to know about admittance because it gives us a way to describe what happens when we add components in parallel to the load, and even better a way to directly plot
these changes on a Smith chart. In short, it depends on how the component is placed in the circuit. If it’s added in series, it will increase the impedance. If it’s added in parallel it will decrease
the impedance by the inverse of its value. Therefore, admittance is defined by Y = 1/Z. Admittance will make it possible to graphically move on the Smith chart when we’re adding parallel components,
that’s why we need it.
The Smith Chart
The basic Smith chart is a plot of impedances with constant resistance (R) circles and constant reactance (X) arcs, both plotted from 0 to infinity. It’s a chart that we can plot our measured
impedance onto. The normalized resistance value of 1 (reactance = 0) is at the center of the chart. By normalized, it means we divide all of our impedances by the system (or source) impedance. So, in
our case the center normalized value represents a perfect 50-ohm resistance with 0-ohms reactance. Once normalized any measured impedance can be plotted on the chart. The center point (1) is going to
be the target location that we want to match the load impedance to. This is where the source and load impedances are equal to each other; the VSWR is 1.
To move our load impedance to the center point we will be adding only inductors and capacitors, no resistors. We don’t want to dissipate power as heat by adding resistors, the inductors and
capacitors will store and return the power to the circuit. As we add inductors and capacitors, we will be changing the reactance, the resistance does not change; it remains constant. This means we
will be moving along the circles only (constant resistance circles). How can we get to the center if we are stuck on circles?
This is why we need to know about admittance. If we draw admittance lines (constant conductance circles and constant susceptance arcs) we see that we get another set of lines, except that 0 and
infinity are swapped (everything is inverted). Combined charts are available, that already have both impedance and admittance lines plotted. It turns out that the plots are identical, except that
they are rotated 180 degrees from each other. This makes perfect sense because the right side of the impedance chart is 0-ohms and the left is infinite ohms, and we know that Y = 1/Z. So, if you flip
one around 180 degrees, you’ve got the other.
We need the admittance chart to add parallel components. This will allow us to add inductors and capacitors in parallel. In this case, we will be changing the susceptance (1/X reactance) as we move
along constant conductance circles (1/R in this case). If you don’t have a combined Smith chart no worries, it can be done with just the impedance lines plotted, which is what I will describe here.
Because they are the same plots, just rotated by 180 degrees, we can just replot impedance points to a rotated 180-degree location and then view the impedance chart as though it’s an admittance
chart. You can then go back and forth using one chart for parallel components and series components simply by doing this. It’s not as difficult as it sounds. The procedure below will show you how.
Circuit Requirements
• Design a circuit using inductors and capacitors to achieve the impedance match.
• To cover all potential load impedances will require parallel and series components that will be arranged in one of two possible topologies.
Z2series topo Z2parallel topo
• Determine if Z1 and Z2 are inductors or capacitors.
• Determine the values of the components.
1. Determine the Load impedance (Zload) at the Frequency (f) that you wish to match. I like using an inexpensive vector network analyzer, such as the nanoVNA for this.
Example: Impedance: ZL = 37 + j16, Frequency: f = 21.25MHz
• Normalize Zload: We are working in a 50-ohm system, so we divide the values by 50 to normalize.
Example: Zload = 37/50 + j16/50 = .74 + j.32
• Use the impedance Smith chart. Our goal is to plot our impedance as it is and then to add our matching components to move that impedance to the middle of the chart (1). As noted previously, It’s
where the VSWR = 1 (50-ohm match). We are only adding inductors and capacitors, no resistors. This means that we will only be changing reactance, so resistance will stay constant. Therefore, our
movement is confined to the constant resistance circle that we are on.
How can we get to the center (1) if we are stuck on a constant resistance circle? We can’t, so we add the admittance chart. We’ll get another set of circles and arcs that intersect the impedance
circles and arcs. These circles will be constant conductance circles. You can use a combined Smith chart that shows both if you have one. For this procedure, I’ll proceed with the simple impedance
only smith chart.
• Continuing with the Impedance only chart, we need to add an important constant conductance = 1 circle. Let’s add that circle (see below) and lets also highlight the constant resistance = 1
We need these two circles for two reasons: 1. They will define which of the two topographies we will use (Z2series or Z2 parallel). 2. They will identify all points on the chart where the resistance
and conductance are equal to our target of 1. (50 ohms). Since we can only move along circles, and we only want to add 2 components, when we add our first component, we want to move our impedance to
a point on one of these circles. Then, we will only need to add one more component to move along one of these circles to the get to the center of the chart.
Plot the normalized Zload (.74_j.32) on the chart.
The .74 resistance is read along the center horizontal. The .32 reactance is read from the perimeter, where the arcs terminate.
• Eventually, we want to move this impedance to the center point of the chart, but the first component we add must get us to a point on one of these two circles. We know that adding inductors and
capacitors moves us along constant resistance or constant conductance circles. Looking at our plotted impedance, we see that we can move along a constant resistance circle to get to the constant
conductance circle = 1. Let’s do that.
We moved along a constant resistance circle from .74+j.32 to .74+j.45, therefore, we have added .13 reactance in series with the load. This means that we must use the Z2series topology. Notice that
for any impedance we start with that lies inside the constant conductance circle we MUST use the Z2series topology because the only circles that intersect the conductance =1 circle are constant
resistance circles. The opposite holds true for the constant resistance circle. If we are inside the constant resistance circle, we MUST use the Z2parallel topology. What if we are outside these two
circles? In that case, you get a choice, either topology can be made to work.
What else did we already determine? We added .13 reactance. Anytime we ADD reactance we are adding an inductor. So, the series componentis an inductor. Notice that we could have moved in a negative j
direction and reached the bottom edge of the circle. In this case we would be subtracting reactance. Anytime we SUBTRACT reactance we are adding a capacitor. It’s your choice. Back to our inductor,
we also know the value of the inductor by the amount of reactance (.13) that we added. We’ll figure that out at later.
• We are now on the admittance circle =1, and it goes right thru the center point (1). This is good, we can move along this circle with one more component to get to the center. When we move along a
constant conductance circle, we are adding a component in parallel.
• If you have a combined Admittance/Impedance Smith chart: You can now move toward the center along this circle, noting that you will move -j.63. Moving in a negative j direction is subtracting
susceptance. Thismeans that this component is a capacitor. And, since this is a susceptance value (B=1/X) we need to invert it to find the reactance X= 1/.63 = 1.59.
If you only have an impedance Smith chart: Notice that if we move along the constant conductance = 1 circle there aren’t any arcs that go to the perimeter where we can read the change in reactance.
However, since we know that the admittance chart is the same as the impedance chart except that it is rotated 180 degrees, we can replot our impedance to a location 180 degrees around the chart. We
will then use the impedance Smith chart (with labeled arcs) as though it is an admittance Smith chart. We will have to remember though that we effectively rotated our chart 180 degrees. Up is now
down and down is up. So, moving in the negative j direction is really moving positive and vice versa. We will have to remember this since the + or – sign of our direction movement determines if it’s
a capacitor or an inductor that we adding.
To do this: Replot the impedance 180 degrees around the chart: Draw a circle, centered at the center of the chart with the radius set to the impedance. Next draw a line from the plotted impedance
thru the center to the opposite side of the same circle.
From this point we can now use the constant resistance=1 circle to get to the center point. We now have j values at the perimeter that we can use. We see that we are going to have to move +j.6 to get
to the center. But remember, the chart has been spun 180 degrees. Therefore, we are really moving -j.63 not +j.63. So, we are adding a capacitor (- movement) not an inductor (+ movement). Again, we
are working in the admittance domain, just using the impedance chart so we can read values from the perimeter. Therefore, the component is to be added in parallel and, we must invert the susceptance
value (B) to find the reactance (X=1/B) of the component we are adding. Therefore X = 1/.63 =1.59.
• Knowing the impedance, we can calculate the values for the Inductor and Capacitor.
ZL = .13*50 = 6.5, 6.5 = 2*
ZC = 1.59*50 = 79.5, 79.5 = 1/(2*
• The components we select must also be capable of handling the currents and voltages that will be present. You must meet the maximum voltage spec for the capacitor and the maximum current spec for
the inductor. If you don’t the components could be damaged, or even cause a fire hazard. For this example, our Transmitter can deliver 100 watts of power. Power is calculated using RMS values, so
the RMS voltages and currents can be calculated by these formulas; P=I^2*R and P=I*V, if we know the resistance. The Transmitter is designed to operate in a 50-ohm system. Using the equations, we
calculate; Vrms=70.7 volts, Irms=1.414 amps. The component ratings are in peak voltages, so we need to convert the rms values to peak values. Divide by .707 to get the peak values. The peak
voltages and currents are then V=100 volts, and I = 2 amps. We could see 2X’s these values due to reflections when we are not matched. That gets us up to V=200V and I=4A. For capacitors the rule
of thumb is to use a capacitor with a voltage rating of 2X the voltage, so a minimum capacitor rating of 400V would suffice. It’s easy to find inexpensive capacitors rated to 500V and into the
thousands. I like to wind my own inductors, so I would use magnet wire that can handle at least the 4A. #22 wire is rated to 7A for chassis wiring. This is a non-bundled spec. Depending on the
inductance our windings could become fairly dense, so take care for large values of inductance. I would expect #22 to be fine for most inductors being designed for this application. Anyway, I
have been using #20 wire rated to 11A chassis and #18 wire rated to 16A chassis without any issues.
Annual Business Meeting
October 21, the LRRA held their Annual Meeting at 4:30 PM in the old Lions Club Function Hall in Moultonborough on Old Route 109.
Elections where held for President, Vice President, Secretary, and Treasurer. Because all four incumbents have expressed the willingness to continue for another term, a nominating committee was not
convened. An election still is required, however.
To see the official documents please see this page.
As required by our constitution, the 2021 Annual Meeting and Biennial Election of Officers was held on Saturday Oct 23rd at the Town Hall in Center Ossipee, NH. Over twenty members and guests
attended and enjoyed a wonderful dinner catered by our very own Jason/KB1RFS (who was also celebrating a birthday that day) and his wife Nikki.
The current slate of officers ran unopposed in the election and were re-elected via a unanimous voice vote.
Continue reading 2021 Annual Meeting and Election of Officers
Lakes Region Repeater Association held a business meeting at Hart’s Turkey Farm in Meredith, New Hampshire. We had 26 people in attendance and the food was excellent. Our secretary Sarah Silk read
the Minutes of 11-13-18. With a motion from Rick Zach, K1RJZ, and a second from Sandy Percy, W1SND, the Minutes were accepted and approved as written.
The Treasurer’s Report was not presented, the Treasurer Jane Greer, W2REX was unable to make it due to bad weather.
Continue reading 2019 January Meeting Harts Turkey Farm
Our first Ham Radio Flea Market was a great social event and place to find good deals on equipment and parts. We even had a drawing for a door prize.
Our goal was to have two a year, one in the spring and the other in the fall. Most ham radio clubs sponsor at least one flea market a year.
The LRRA Flea and Ve was very well attended and a great place to connect an entire community. Our ham radio flea market ended up being a multi-purpose event.
People were buying and selling equipment and parts. Many new hams get their first radio at a flea market. Quite a few old and newer radios changed hands at good prices. Equally important is finding
parts and test equipment. We hams usually stock up on hard to find parts for future projects and repairs.
There was a lot of information on local club activities. It was a great time to find out what groups were active in our neighborhood and what they were doing. Lots of memberships got sold at our flea
Entry fees were five dollars which covered our costs. Vendors usually donate prizes and we had a door prize. The admission ticket entered you on our drawing plus we offered a 50-50. These events were
fund raisers for our club. Often, you can win a new radio, sometimes an expensive one.
At 1pm we held a Ve session for testing. We were happy to announce our testers passed their tests.
In the earlier days of amateur radio, most people built most of their own equipment. New parts were expensive, so hams started holding flea markets to share the wealth in their community. Sometimes
they are also combined with auctions. In addition, many larger clubs hold a “hamfest” or “swap meet” which is the same idea. Our first LRRA Flea and Ve was a success!
Ham radio field day 2019
Lakes Region Repeater Association’s Field Day was held the fourth full weekend of June 2019. Every year is the opportunity for thousands of amateur radio enthusiasts throughout the U.S and Canada to
set up temporary communications stations and make contact with Hams.
Our licensed hams operators spent the weekend practicing community outreach, emergency preparedness, and technical skills. LRRA was basically in radio heaven.
A contest is held each year with individuals, clubs and teams trying to make contact with as many stations as possible over 24 hours. Field Day 2019 took place with over 35,000 people taking part.
Our Field Day began at 18:00 UTC Saturday and ran through 20:59 UTC Sunday. We packed our camping equipment, threw up some temporary antennas, and spent the next 24 hours spinning the dials on our
radios, because this not-to-be-missed event was rich in history, tradition and technology.
If you are curious about what exactly is Field Day, it is an annual event conducted by the American Radio Relay League. Amateur radio operators across North America setup their equipment in fields,
parking lots, and parks.
We set up our ham radio field day 2019 at Constitution Park in Ossipee, New Hampshire using off grid electricity and in make-work conditions. Operators then make contact with similar groups across
North America.
ARRL Field Day stresses emergency preparedness. During our exercise, we took “Field” Day literally; we erected radio masts and towers, each bearing several antennas, in a parking lot at Constitution
We used generators to provide power to ham radio transceivers. We worked through logistical problems like transpiration, food, shelter and other accommodations for our group for up to 24 hours.
Field Day is rarely a single-man operation. In fact, Field Day is frequently used to highlight to the public, the virtues and utility of ham radio in an emergency situation.
LRRA has demonstrated in the past a wide range of technologies, including single sideband voice, Morse code, and a number of digital modes including APRS, packet radio, as well as satellite | {"url":"https://www.w1bst.org/author/jgreer16/","timestamp":"2024-11-04T16:52:02Z","content_type":"text/html","content_length":"194596","record_id":"<urn:uuid:b4ea5581-cb6e-4c42-a854-56f97386ef51>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00404.warc.gz"} |
Grey Matters: Blog
In the past 2 posts, we've looked at the precision of the purely mathematical approach to age guessing, and the skillful approach of estimating age by appearances.
In this post, you'll learn an approach that seems to be a math trick, yet seems impossible to explain in that way.
You starting by asking a spectator to put any 5-, 6-, or 7-digit number in their calculator. You then tell them to multiply that number by 9. The last steps are to add their age to that number, and
then show you the resulting number.
You examine the number for a few seconds, and instantly announce their age!
Why this is deceptive
Let's say you perform this, and the resulting number on the calculator is 1,248,695, and you announce that the person's age is 26, which is confirmed by them.
Mathematically, all they have is the formula 9x + y = 1,248,695. With two variables, that equation (known to mathematicians as a Diophantine equation) has an infinite number of solutions. How is it
possible that you could narrow down the possibilities so quickly, and in your head?
How it works
When you're shown the total, you first add the digits of the answer up in your head. Next, using the age-guessing skills you learned in the previous post, ask yourself if the person could be that
If the person seems older than that, add 9 to the number you got and ask if that seems to be a more reasonable age. If that doesn't seem right, move up or down in 9-year increments, and keep doing
that until you find an age that seems right.
In our example, you'd see the answer 1,248,695, so you add 1+2+4+8+6+9+5=35. Ask yourself if the person could reasonably be 35. Let's say they look younger than that, so you subtract 9. 35 - 9 = 26,
so you consider 26, which we'll say seems more reasonable, so you guess that number out loud.
Why it works
You start with a long random number, and then multiply it by 9. What happens when you multiply any number by 9? Square One TV's Nine, Nine, Nine song explains:
When you sum up the digits, the result is known as a digit sum. The digit sum of 99 is 18 because 9+9=18. In the video above, notice they keep repeating the process of taking the digit sum until they
get a 1-digit number. If you do this, the 1-digit number you get is called the digital root. The digital root of 99 is 9 because 9+9=18, and 1+8=9. The point of the above video, of course, is that
any number multiplied by 9 will have a digital root of 9.
What happens when you add a number to a multiple of 9? Let's take 5 as an example. 9+5=14, and the digital root of 14 is 5 (1+4=5). 18+5=23, and the digital root of 23 is 5 (2+3=5), and so on. Let's
18+14, which is a multiple of 9 plus a number with the digital root of 5. 18+14=32, and 32's digital root is 5! Also notice that the answers remain spaced by multiples of 9: 5, 14, 23, 32, and so on.
In short, any time you add a number to a multiple of 9, the answer will always have the same digital root as the number you added, and you'll always be a multiple of 9 away from another number with
the same digital root.
Applying this to the trick, when you multiply by 9 and add the age, the digit sum (1+2+4+8+6+9+5=35 in our above example) will not necessarily be their age, but will have the same digital root as
their age, and be some multiple of 9 away from the correct age (even if that multiple is 0).
Try this out for yourself. Get a calculator, enter any 5-, 6-, or 7-digit number, multiply that by 9, then add your age. Take the result, and enter it into the widget below, then click Submit. A
window will pop up showing all the possible ages (listed as the variable a) between 0 and 100 you could be, based on the number you entered.
Sneakier ways of getting to a multiple of 9
If someone is familiar with the effects of multiplying by 9, they might suspect what you're doing. There are other less obvious ways of getting to a multiple of 9:
• From a sidebar in in Karl J. Smith's Nature of Mathematics (available at Amazon.com): Mix up the serial number on a dollar bill. You now have two numbers, the original serial number and the
mixed-up one. Subtract the smaller from the larger. Assuming you didn't create two identical numbers, the result will have a digital root of 9, because you're subtracting 2 numbers with identical
digital roots (More about this principle here).
• Also from the same sidebar in in Karl J. Smith's Nature of Mathematics: Using a calculator keyboard or push-button phone, choose any 3-digit column, row, or diagonal, and arrange these digits in
any order. Multiply this number by another [3-digit] row, column, or diagonal. As it happens, most numeric keypads are arranged in such a way that any row, column, or diagonal of the numbers 1-9 will
make a multiple of 3. Multiplying two multiples of 3 together will always result in a multiple of 9.
• You could also adapt Scam School's first Pi Day Magic Trick (YouTube link). Have them multiply 1-digit numbers together as shown in the video, until you get to a number somewhere between 1 million
and 1 billion. Instead of having them remove a digit as in the original routine, however, have them add their age instead. As you see in the video, though, it is possible to get a number like
8,100,000,000. Adding their age to that would be obvious (assuming the guy in the video is 22, he'd get 8,100,000,022). To prevent this, tell them to avoid pressing the 5 and 0 keys, as this will
just result in a lot of zeros at the end (or just one in the case of multiplying by 0).
These aren't the only secret ways to get to a multiple of 9, but are varied and interesting enough to get you started.
If you'd like an age-guessing routine that has the precision of math, but without the appearance of math (or even use of a calculator), I think you'll enjoy the next post, which will be the final
installment in our series on how to guess people's ages.
When it comes to age guessing, very few people think of the calculator feats such as the one in the previous post.
The first thing that usually comes to mind is carnival age-guessers. In this post, we'll take a closer look at age-guessing as a skill.
The best tips I've found on determining someone's age are in the article How to Guess Ages More Accurately. Since men tend to put less effort into hiding their age than women, here are a few extra
tips on guessing a man's age.
Just knowing these tips isn't of much good without practice. Thankfully, there are several sites where you can practice guessing the age of random people:
• How Old Are You?
• Guess my Age
• Match>Age
Even though carnival age-guessers aren't having you put any numbers in a calculator, they're still able to use some very subtle math tricks. For example, instead of advertising that they'll hit your
exact age, you'll usually see a margin of error such as, I'll guess your age within 3 years! That sounds quite close to most people.
If you think about it, however, a 3-year margin of error really isn't that close. If someone is 35, a guess of anywhere from 32 to 38 would be considered correct. In other words, all they have to do
is be within the decade you were born in, and they'll be considered correct. The more experience they have, the smaller margin of error they can offer. For example, professional age-guesser Lee
Bennett used an impressive 1 year margin of error.
Even more central to an age-guesser's actual purpose is the simple economics of the situation. Let's assume that the cost to have the carny make a guess is $3, and the cost per stuffed animal to the
carnival is $.25 (since they buy them in bulk). If we assume the guess is wrong every time, perhaps to keep every customer flattered, they're making an 1100% profit on each prize!
As the guesser becomes more skillful, the profit margin goes up! If we assume the age-guesser can correctly guess the ages of 4 out of 5 people (an 80% success rate), then that's 5 people times $3/
person or $15 they're taking in. Only 1 wrong guess out of those 5 means that they're giving up $.25 for every $15 they take in, a staggering 5900% profit margin!
So, when it comes down to it, age-guessing as a skill is all about the margin of error and the profit margin. And that's assuming they don't employ standard scams like writing two ages and then
covering up the one that's farther away, using magician's techniques to write down a close answer after you state your age, or simply pickpocketing your wallet and looking at your ID.
Guessing ages is a skill, but only ever an approximate one at best. The mathematical approaches, as we've seen, offer precision. Perhaps the best approach is to develop the skill of age-guessing, and
use math in a way that doesn't detract from the skill.
That's the approach we'll start developing in the next post in this series.
Back in 2008, I wrote a post about guessing ages. Unfortunately, it was several approaches compacted into one long post and lacked clarity, as a few readers have noted.
I've decided it's time to update the post. I'll break age-guessing up across several posts in an effort to improve the clarity, as well.
In this post, I'll start with the methods for finding someone's age using purely mathematical methods.
The first type of mathematical age trick that usually comes to mind is the algebraic type, such as the kind listed on this page under Guess Your Age. In the first, you have the person put their age
in a calculator, triple it, add 1, triple it again, add their age again, and then show you the result. While the process of performing (((x * 3) + 1) * 3) + x looks complicated, it's just a long way
around of getting them to multiply their age by 10 and add 3 (See the alternate forms section).
The second trick, in which they multiply their age by 7 and then by 1,443, isn't so much mysterious as it is surprising and amusing. 7 * 1,443 = 10101, so any 2-digit number multiplied by that is of
course going to repeat itself 3 times.
In the original Age Guessing post, I also linked to this age plus a secret number approach, which explains it's own algebra, and these two algebraic approaches, one of which breaks up the age into
two different numbers, and the other that makes use of the year the person was born.
These types of tricks can be very impressive for an audience unfamiliar with the basic concept of algebra, and can also be a great way to introduce new students to algebra. Anyone beyond that stage,
even if they can't work it out at the moment, will recognize that there's some simple pattern that will get you the answer. Since this is the case, perhaps there's a mathematical approach that is
more deceptive.
A deceptive approach that's long been a favorite of magicians is one known as the Age Cards. You can find an interactive version of it at this link. Look for your age in each group. If you see your
age in a given group, click the checkbox for that group. Once you've checked all six groups for your age, and clicked where appropriate, click on the CALC button. The computer will tell you your age!
It works simply by adding up the smallest number (the one on the upper left corner) on each card on which the age was seen. If your age was 27, you would only click the boxes of Group One (smallest
number is 1), Group Two (2), Group Four (8), and Group Five (16). Adding 16 + 8 + 2 + 1 gives 27, so the chosen age is 27.
See if you can follow how the secret number of 38 is determined in this video:
That's how the trick is done, but why does it work that way?
The method here is better hidden than the algebraic methods because instead of using our usual base 10 numbering system, which uses the digits 0 through 9, the Age Cards trick is based on the base 2
numbering system, better known as binary, which only uses the digits 0 and 1. Working with a different number base can seem scary and confusing, but BetterExplained points out that you work with
different number bases more than you might think.
Even though binary is limited to using 0 and 1, it can represent any number our more familiar 0 through 9 system can. The PDF and the first video on Computer Science Unplugged's binary numbers page
explain how clearly and quickly. The number 27, for example, converted to binary becomes 11011. In base 10, we only need 2 places (2 tens and 7 ones) to represent the number, but in binary, we need 5
places to represent the same number (1 sixteen, 1 eight, 0 fours, 1 two, and 1 one).
How does this all relate to the Age Cards? Note that there were six Age Cards used. Each card acts like one of the places in the binary number. Note that the smallest number on each card corresponds
to one of the binary places, as well: 32, 16, 8, 4, 2, and 1.
To find out where a given number goes, we use it's binary code. As mentioned, 27 converts to 11011. We're working with 6 cards, though, so just like our regular base 10 numbering system, we can add
zeroes to the left side without changing the value. Doing this, 11011 becomes 011011.
The rightmost spot in binary is the 1s spot, and if there's a 1 there, as there is in our 27 example, we put that number on the 1 card. There's a 1 in the twos place, so we also put 27 on the 2 card.
There's a 0 in the 4s place, so we don't put 27 on the 4s card. The 1 in the 8s place and the 16s place indicate that the 8 and 16 cards do have 27 put on them. Finally, the leftmost 0 in the 32s
place tells us not to put 27 on the 32 card.
In the video above, 38 only appeared on the 32 card, the 4 card and the 2 card because 38 in binary is 100110, which only has 1s in the 32s place, the 4s place, and the 2s place. Get the idea?
The Age Cards is well-known among magicians, so even this routine could benefit from a better disguise. Fortunately, Werner Miller has come up with some very creative work on the Age Cards!
First, there's his ingenious Age Cube, which is presented as a giveaway with five magic squares on it. You ask someone who is 31 or younger (because we're only working with 4 binary places) on which
magic squares they see their age, and thanks to your secret addition of the numbers in the upper left corner of each magic square, you can magically divine their age!
His other approach comes as a webapp that works in any modern browser, and also as a Windows executable file. It's called Age Square, and builds impressive from the Age Cube. It only uses 4 binary
places, but thanks to a secret better described in the original Age Square post, it still manages to cover ages from 30 to 85! Instead of giving the age directly as an answer, the app generates a new
magic square, with their age as the total.
Divining someone's age purely using math can be interesting, but what about getting someone's age with some help from their appearance? That will be the topic of the next post in this series.
I teach quite a few fun mental challenges over in the Mental Gym.
While I teach methods in as simple and straightforward a manner as possible, there isn't always just one approach. In this post, I'll take a look at new approaches to feats in the Mental Gym.
In my tutorial on Squaring 2-Digit Numbers Mentally, I already teach two methods - a mathematical approach, and Jim Wilder's pure memory approach.
NumberSense's approach takes advantage of an algebraic pattern. The number is separated into 2 variables, a being the 10s digit and b being the 1s digit. The problem then becomes (a + b)^2, whose
expanded form shows how to make the problem easier:
Besides making the squaring of two digit numbers easier, this video also illustrates a good point about algebra. Algebra lets you see patterns of which you may not have been previously aware, and
help you see a shorter, and possibly better approach.
Another mathematical challenge I tried to simplify over in the Mental Gym was the unit circle and its associated trigonometric functions.
These lessons are especially handy for students taking trigonometry. Here's a handy approach to memorizing the unit circle, especially useful for tests, that works solely by taking advantage of
several simple patterns:
We'll wind up this post by focusing on two of the puzzles.
First, there's the Sudoku. I already link to instructions on Sudoku strategy, but if you find those hard to understand, e-How has a series of excellent instructional videos on the Sudoku-solving
techniques that you may find helpful.
In the Towers of Hanoi, the seemingly-simple task of moving disks from 1 peg to another quickly gets complicated. Here's a short, direct tutorial that helps make the solving pattern much clearer:
If you've come across an alternative way of doing any of the feats over in the Mental Gym, I'd love to hear about it in the comments!
April's snippets want to run free. They range from our usual topics like math and memory, to games, and even a little law and politics!
• If you enjoyed my previous post on Notakto, but you can't play the iPad app, Thane Plambeck has an online version you can play. Like the app, you start with one board, and work your way up to
5-board play. You can only move to the next level after winning 3 consecutive games on your current level.
• Speaking of strategy games, I've come across some new work on a classic. Ever play Hangman, and always use the same letters to guess in the same order, such as E-A-T-O-N? There is a better Hangman
strategy, described over at DataGenetics. Instead of just giving you a new strategy, though, they go the extra step and explain the detailed thinking behind it, so you can understand it more
• The Major System is a great technique for memorizing numbers, but can be challenging to learn. Over at the memory basics page, I've just added a few new resources that may help those who want to
learn it. First, I added the Great Courses free video How To Memorize Numbers, a free lecture video from their Secrets of Mental Math course which I originally mentioned in October's snippets.
Over at Math Dude :: Quick & Dirty Tips, they also have an excellent series of 3 podcasts that teach the Major System. Part 1, Part 2, and Part 3 are available on their web site, as well as from
iTunes (episodes 92 through 94).
• Vi Hart fans may remember her video Oh No, Pi Politics Again, about someone who claimed to have copyrighted music based on Pi. Writing music based on Pi is hardly a new and original idea, but the
copyright claim was used to shut down others' videos anyway. Judge Michael H. Simon was the Oregon judge who presided over this case. Read the article Can you copyright music of pi? Judge says no to
learn more about this decision, and exactly why the claim was denied.
• If you'll forgive me, I'm going to wind this post up with a little boasting. Back in January, I released Day One, my approach to speeding up and presenting the classic Day of the Week For Any Date
feat. I updated it in February to include some unusual calendar-related bonus feats, as well. I'm proud to announce that, according to Lybrary.com's hot list, Day One is currently their 3rd
best-selling magic item at this writing! The response has been simply incredible, and I'd like to thank everyone who bought it and who helped make this possible.
Backgammon giant Bob Koca was playing tic-tac-toe with his 5-year-old nephew, when the nephew whimsically suggested that they both play as X.
Being a mathematics professor, he used his knowledge to analyze this weird version of the classic game with various rules, boards, and objectives. It turns out that this all-Xs version of tic-tac-toe
is a version of our old friend Nim!
To keep the game familiar, I'll stick to the standard 3-by-3 board in this post. The rules are as follows:
• Players alternate taking turns, and neither player may pass on their turn.
• A player marks any empty space on the board with an X on their turn.
• The loser is the first person to mark an X on the board that completes a horizontal, vertical, or diagonal line of 3 Xs.
This game is known to mathematicians as neutral or impartial tic-tac-toe, but I prefer the name given to it by Thane Plambeck, who lectured on this game at G4G10: Notakto (pronounced No Tac Toe ).
As I mentioned, this is variation of Nim, more specifically a Misère version, so there must be some way to win it. I'll start, however, by explaining how to lose the game, instead:
What YOU Should NOT Do
You should start by going first, but the worst possible opening move is to place your X on any of the edge or corner squares. Why?
Because your opponent can basically mirror your moves, and this strategy will ensure that you must eventually make a line of 3 Xs, as shown in the following animation:
As you can see at the end of the animation, when the first player puts their X on an edge or corner square, and the second player mirrors them, this leaves an open diagonal line on the first player's
turn that forces them to complete a line.
I mention this strategy mainly so you can be aware of it, and make sure that it doesn't happen to you inadvertently. Should you let the other player go first and they place their first X on an edge
or corner square, knowing about this becomes a winning strategy for you!
How To Win
To assure yourself the win, you start by placing your X in the center square. To play from there, Timothy Chow discovered the answer comes with help from a chess knight!
Knowing how a chess knight moves (2 squares horizontally and 1 vertically, OR 2 squares vertically and 1 horizontally) is all you need to win.
After the other player makes their move, mark your next X a knight's move away from where their previous move. Keep using this strategy and they'll always be forced to draw the losing X. Watch the
following animation carefully, and you'll get the general idea:
When choosing your spaces using the knight's move strategy, you'll usually have more than one space that qualifies. Often, one of the spaces will complete a line of 3 Xs, while the other is safe, so
you'll always want to double check that you don't inadvertently make a losing play when you don't have to.
You can find out more about the game from Bob Koca's original discussion or the MathOverflow discussion. For a deeper look at the mathematics of Notakto, you can also read Thane Plambeck's
presentation in PDF form.
If you'd like to practice this and you have an iPad, Thane Plambeck has also developed a Notakto app which will let you practice this version, as well as more difficult versions!
There's a closely related game taught on Scam School, called Napkin Chess, which is won using a similar symmetrical strategy. It's interesting to see the similarities, even though it doesn't have a
tic-tac-toe board's discrete spaces.
I've posted about memorizing the periodic table of the elements before, but understanding is just as important.
You might think trying to understand the basics of the elements would be a chore, but it can actually be quite fun.
Surprisingly, one of the best introductions to the atom I've ever seen is not from a documentary, but an episode of WKRP in Cincinnati. In this episode, Venus is trying to help a friend whose son has
dropped out of school. In the following scene, Venus explains the basics of the atom in an effort to help get the son to go back to school:
Earlier this week, NOVA aired a special called Hunting the Elements. The full special is about 2 hours long, and I recommend you make time to watch the entire thing.
Below are two short excerpts from that special, both roughly 8 minutes long. This first one discusses why the periodic table is arranged the way it is:
This second excerpt talks about the characteristics of the atom that gives each element gets its particular properties:
For more direct learning, NOVA has provided some wonderful teaching tools, such as their Name That Element Quiz. If you have an iPad, check out the NOVA Elements app (iTunes link). It not only
includes the entire special, but also lets you play around with the elements by building atoms, putting them together in compounds, and much more!
Should you want to learn specific information about a given element, there's a great site called the Periodic Table of Videos. The periodic table on their homepage links to videos about the
corresponding element. These videos are also available on their YouTube channel.
Of course, one of the things for which Grey Matters is known is teaching how to memorize just about anything. If you've been inspired to try and memorize the periodic table, check out my 2008
Elementary post. (Being 4 years old, some of the links are no longer available, but most of them are still functional.)
Just last week, there was a gathering honoring the late Martin Gardner in Atlanta, called Gathering For Gardner 10, or G4G10 for short.
I didn't go myself, but the people who did attend have already started sharing their experiences with us.
If you're not familiar with Martin Gardner, you can see posts relating to his work right here on Grey Matters. David Suzuki, in his documentary series Nature of Things, spent one entire program on
Martin Gardner, and introduces it at an early Gathering for Gardner event:
The G4Gs are invitation-only events, and there's not much available from G4G10, the most recent get-together. There are, however, a few goodies already online.
Over on flickr, there are already many photos from G4G10 posted. Even if you don't understand the subjects of the photos themselves, they're still wondrous and amazing to behold.
One of the biggest treats from G4G10, however, has to be Colm Mulcahy's library lecture about the life and work of Martin Gardner. Regular Grey Matters readers will probably recognize Colm from his
Card Colm column and his Colm's Cards page.
Here's part 2 of the lecture:
The searchable collection of Gardner's work, which Colm mentions in the lecture, is Martin Gardner's Mathematical Games CD-ROM, and is currently available for around $40 at Amazon.com.
If you attended G4G10, or even just have any personal stories to share about Martin Gardner's influence, I'd love to hear about it in the comments.
In the US and Canada, April is national poetry month! (Sorry, Great Britain, you'll have to wait until October.)
Since memory is one of my favorite topics, I'll take a look at memorizing poetry in this post.
To most people, memorizing poetry sounds like something out of 19th century schoolhouses or 1960s beatnik coffee shops. The truth is, there are plenty of good reasons to learn to memorize poetry,
especially if it's something you want to do, as opposed to something you're being forced to do. In Five Benefits of Memorizing Poems there are the usual education reasons. If that's not enough, Ten
Reasons You Should Memorize Poetry expands on this, including some reasons that are right down my alley, including:
1) It is a brain challenge. Got a kid with a strong memory? I’ve got some long poems for you. Interested in history? Learn a poem based on a historical event or some of the poetry of that period.
For anyone seeking a way to challenge a gifted child in way that is free (!) and virtually unlimited, you’ve found it. Even copying poems down (or lines of poems) and illustrating them is a
wonderful activity for younger children.
8) It’s a great party trick. If you’re ever stuck for a spur of the moment talent, you’re in luck if you’ve got a poem in your mind you can whip out and recite from memory. It’s easy, it needs no
props, and you will not be doing the same tired trick as everyone else. Unless they read this blog.
Some of the other reasons might not seem as impressive, such as the entries about keeping us connected and being a bridge among disciplines. If you take those lightly, check out Be a Man. Read a
Poem. from the Art of Manliness site.
Once you appreciate the benefits, how do you go about doing the actual memorization? I've written quite a bit about memorizing poetry in past posts, but there are many more approaches. New
technologies make it easier to memorize than ever before. In Essay on memorizing poetry - at the gym talks about using crib sheets while exercising, although these crib sheets could be recording or
videos on a mobile device of poems you wish to learn, as well.
Mensa For Kids' A Year of Living Poetically lessons are a good selection with a great structure. The poem is presented, broken down, and once the poem is memorized, there are varying types of quizzes
to test your knowledge.
A more adult version of this same approach is used in Shmoop.com's poetry section. For example, their guide to Poe's The Raven includes not just the poem text, but an intro, a summary, an analysis, a
quiz and much more! Their poetry section also has plenty of classic choices, and is a great place to look for material.
Another good source is the book Committed to Memory: 100 Best Poems to Memorize. You can even find the full intro and a majority of the selections from this book at poets.org.
Remember, memorizing poetry should be fun. Looking for a fun short piece to memorize right away? How about this ironic choice, titled Forgetfulness by Billy Collins: | {"url":"https://headinside.blogspot.com/2012/04/","timestamp":"2024-11-11T23:47:39Z","content_type":"application/xhtml+xml","content_length":"718032","record_id":"<urn:uuid:a30f974a-61ce-4dcd-803f-c99efe96683b>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00893.warc.gz"} |
TS 6th Class Maths Solutions Chapter 13 Practical Geometry Ex 13.3
Students can practice Telangana SCERT Class 6 Maths Solutions Chapter 13 Practical Geometry Ex 13.3 to get the best methods of solving problems.
TS 6th Class Maths Solutions Chapter 13 Practical Geometry Exercise 13.3
Question 1.
Draw a line segment PQ = 5.8 cm and construct its perpendicular bisector using ruler and compasses.
Steps of construction:
1. Draw a line segment \(\overline{\mathrm{PQ}}\) = 5.8 cm.
2. Take P as centre and radius more than half of PQ draw two arcs above and below the line segment \(\overline{\mathrm{PQ}}\).
3. Take Q as centre and with the same radius, draw two more arcs intersecting the previous arcs at A and B.
4. Join A and B. This line intersects PQ at O.
5. AB is the required perpendicular bisector of the line PQ.
Question 2.
Ravi made a line segment of length 8.6 Find the length of AC and BC.
Steps of construction :
1. Draw a line segment AB = 8.6 cm.
2. Take A as centre and radius more
than half of the length AB, draw two arcs above and below the line segment AB. ,
3. Take B as cfentre and with the same radius, draw two more arcs intersecting the previous- arcs at P and Q.
4. Join PQ. This line intersects AB at C.
5. Measure AC and BC. On measuring, it is noticed that AC = BC = 4.3 cm.
Question 3.
Using ruler and compasses, draw AB = 6.4 cm. Find its mid point.
We can find the mid point of the line segment AB = 6.4 cm by drawing its perpendicular bisector.
Steps of construction:
1. Draw a line segment AB = 6.4 cm.
2. Take A as centre and radius more than half of the length AB draw two arcs above and below the line segment AB.
3. Take B as centre and with the same radius draw two more arcs intersecting the previous arcs at M and N.
4. Join MN. This line intersects AB at O. ‘O’ is the mid point of AB.
Leave a Comment | {"url":"https://tsboardsolutions.in/ts-6th-class-maths-solutions-chapter-13-ex-13-3/","timestamp":"2024-11-06T15:22:48Z","content_type":"text/html","content_length":"153670","record_id":"<urn:uuid:42cdbd58-3772-431c-93b4-ed2b4bebe699>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00592.warc.gz"} |
Compiled and Solved Problems in Geometry and Trigonometry
by Florentin Smarandache
Publisher: viXra.org 2015
ISBN-13: 9781599732992
Number of pages: 221
This book includes 255 problems of 2D and 3D Euclidean geometry plus trigonometry. The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the
end of each chapter. The book is especially a didactic material for the mathematical students and instructors.
Download or read it online for free here:
Download link
(5.3MB, PDF)
Similar books
Geometry and Billiards
Serge TabachnikovMathematical billiards describe the motion of a mass point in a domain with elastic reflections from the boundary. Billiards is not a single mathematical theory, it is rather a
mathematician’s playground where various methods are tested.
Lectures on Discrete and Polyhedral Geometry
Igor Pak
UCLAThis book is aimed to be an introduction to some of our favorite parts of the subject, covering some familiar and popular topics as well as some old, forgotten, sometimes obscure, and at times
very recent and exciting results.
Coordinate Geometry
Henry B. Fine, Henry D. Thompson
The MacMillan CompanyContents: Coordinates; The Straight Line; The Circle; The Parabola; The Ellipse; The Hyperbola; Transformation Of Coordinates; The General Equation Of The Second Degree; Sections
Of A Cone; Systems Of Conics; Tangents And Polars Of The Conic; etc.
Advanced Geometry for High Schools: Synthetic and Analytical
A.H. McDougall
Copp, ClarkContents: Theorems of Menelaus and Ceva; The Nine-Point Circle; Simpson's Line; Areas op Rectangles; Radical Axis; Medial Section; Miscellaneous Theorems; Similar and Similarly Situated
Polygons; Harmonic Ranges and Pencils; etc. | {"url":"http://www.e-booksdirectory.com/details.php?ebook=10400","timestamp":"2024-11-13T16:11:39Z","content_type":"text/html","content_length":"11467","record_id":"<urn:uuid:38f03062-c22e-401a-bd1d-89902473e25d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00762.warc.gz"} |
Suppose a circle’s radius is measured to be 3, with an error… | Wiki Cram
Suppose a circle’s radius is measured to be 3, with an error…
Suppоse а circle's rаdius is meаsured tо be 3, with an errоr of at most 0.5. What is the greatest possible error in the circumference measurement of the circle?
Suppоse а circle's rаdius is meаsured tо be 3, with an errоr of at most 0.5. What is the greatest possible error in the circumference measurement of the circle?
Suppоse а circle's rаdius is meаsured tо be 3, with an errоr of at most 0.5. What is the greatest possible error in the circumference measurement of the circle?
Suppоse а circle's rаdius is meаsured tо be 3, with an errоr of at most 0.5. What is the greatest possible error in the circumference measurement of the circle?
Suppоse а circle's rаdius is meаsured tо be 3, with an errоr of at most 0.5. What is the greatest possible error in the circumference measurement of the circle?
Suppоse а circle's rаdius is meаsured tо be 3, with an errоr of at most 0.5. What is the greatest possible error in the circumference measurement of the circle?
Suppоse а circle's rаdius is meаsured tо be 3, with an errоr of at most 0.5. What is the greatest possible error in the circumference measurement of the circle?
Suppоse а circle's rаdius is meаsured tо be 3, with an errоr of at most 0.5. What is the greatest possible error in the circumference measurement of the circle?
Whаt type оf pupаe is pictured belоw? Hint: Lоok аt the mandibles.
__________ аre the cells necessаry fоr nоurishing neurоns in cuticulаr sensilla.
Skip back to main navigation | {"url":"https://wikicram.com/suppose-a-circle-s-radius-is-measured-to-be-3-with-an-error-of-at-most-0-5-what-is-the-greatest-possible-error-in-the-circumference-measurement-of-the-circle/","timestamp":"2024-11-04T08:43:52Z","content_type":"text/html","content_length":"47356","record_id":"<urn:uuid:ee7db789-d426-44d3-a093-61c608d09024>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00129.warc.gz"} |
forum.alglib.net :: View topic - achieving ~e-10 KKT feasibility for well-scaled problem
Thank you for reading my post and or the suggestion. I've attached the trace file as well as a more detailed one.
FWIW, to show how similar the solution values are, here is the alglib solution. Please excuse the many decimal places. I generally ignore Visual Studio C++ numerical places beyond 1e-13 but I value
extreme precision in my problem.
Alglib filter sqp:
Unit-Canonical Ellipsoid 1: (-0.329176838420615, 0.94401746027854, 0.0217633576606146)
Unit-Canonical Ellipsoid 2: (0.930895021061298, 0.329232641233283, -0.158241359036905)
In general space, this corresponds to
General Ellipsoid 1: (-10.5967582185939, 206.347913283321, 6.77869831250489)
General Ellipsoid 2: (18.8893523034911, -49.9697297257856, 570.805604711329)
The Euclidean norm is: 620.237467405261
In code, of the 6 KKT condition equations, the worst feasibility is 9.7502106655156240e-06 and in my Excel "implementation" of my geometric convergence algorithm, the KKT feasibility evaluates as
0.0001294260728173 or roughly 1e-4.
In contrast, the point I find in my Excel "algorithm" differs from the ALGLIB solution by at most e-9 in unit-canonical space and e-8 in general space!
My solution:
Unit-Canonical Ellipsoid 1: (-0.32917684136241, 0.944017459191415, 0.0217633603207913)
Unit-Canonical Ellipsoid 2: (0.930895019622935, 0.329232643072075, -0.158241363672689)
In general space, this corresponds to
General Ellipsoid 1: (-10.5967582026675, 206.347913213863, 6.77869828010731)
General Ellipsoid 2: (18.8893522318179, -49.9697297427802, 570.805604707352)
The Euclidean norm is: 620.237467405261
In code, of the 6 KKT condition equations, the worst feasibility is 9.0949470177292824e-12 and in my Excel "implementation" of my geometric convergence algorithm, the KKT feasibility evaluates as
0.00000000003547029 or roughly 3.5e-11.
Does this give you any hints? I realize it may seem like polishing an already very nice apple, but I'd love to squeeze out the extra accuracy from alglib if at all possible...
File comment: trace_file("SQP.DETAILED,SQP.PROBING,PREC.F15",
logDirectory + "trace.log");
trace_15.log [168.53 KiB]
Downloaded 1365 times
File comment: trace_file("SQP",
logDirectory + "trace.log");
trace.log [47.92 KiB]
Downloaded 1437 times | {"url":"http://forum.alglib.net/viewtopic.php?f=2&t=4606&view=print","timestamp":"2024-11-03T01:13:03Z","content_type":"text/html","content_length":"19170","record_id":"<urn:uuid:e7698e7a-b235-4580-b261-23c5238a4570>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00279.warc.gz"} |
Math - Branches, Fundamentals, Important Topics, Preparation Tips
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.
Math is the science of quantity, pattern, order, structure and relation that has continuously evolved from basic practices of counting, measurements and symmetric study of shapes. It primarily
involves applying logical reasoning and quantitative computation to find optimal solutions to problems. It has been worldwide recognized as an indispensable computational tool in the field of
engineering, biology, medicine and natural sciences.
Math As A Subject
Math as a subject is an important part of the curriculum that plays a significant role in shaping a child’s future. It involves studying many useful concepts and topics relevant to practical life.
Although math is an interesting subject, students often find it boring and complex due to the way it is taught conventionally. Cuemath helps students to explore and understand fundamental concepts in
a fun and intuitive manner.
Fundamentals of Math
Fundamentals of math are the basic building blocks that help students form a solid mathematical foundation. Math learning entirely relies on the understanding of these fundamental concepts. If
children lack the basic understanding of division or subtraction, then algebra automatically becomes confusing for them. Therefore, it is imperative that children must have a crystal clear knowledge
of all fundamentals of math.
• Addition and Subtraction of Whole Numbers
• Multiplication and Division of Whole Numbers
• Exponents, Roots, and Factorization of Whole Numbers
• Introduction to Fractions and Multiplication and Division of Fractions
• Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions
• Decimals and Fractions
• Ratios and Rates
• Techniques of Estimation
• Measurement and Geometry
• Signed Numbers
• Algebraic Expressions and Equations
Branches of Mathematics
Mathematics involves complex studies of interlinked topics and several concepts that overlap each other. Generally, it can be categorized into the following branches:
Arithmetic is the most basic branch of mathematics that deals with the elementary aspects of numbers, mensuration and numerical computations. This term is derived from the Greek word ‘arithmos’ which
means number. It generally involves studying numbers and their relationships to solve problems that include the operations of addition, multiplication, subtraction, division, extraction of roots and
raise to power.
Algebra is an important and ancient branch of math that covers basic operations and symbols to represent numbers in formulas and equations. The word Algebra means the science of restoring and
balancing. Learning algebra enables students to understand many real-life phenomena around them. It is a symbolic representation of numbers and how they work together to provide structure to
equations. It forms the basis for advanced study in many fields like science, medicine, engineering, etc. It allows mathematicians to write formulas and solve maths problems more efficiently.
Geometry is the branch of math which deals with computation of various dimensions of solid shapes including height, width, area, volume, perimeter and angles. It has several useful applications from
the construction of homes to interior designs.
Trigonometry is an important branch of math that involves studying the relationship between angles, lengths, heights and distance. The applications of trigonometry can be found in many spheres
including architecture, physics, surveying, electronics, satellite navigation, astronomy and engineering.
List of Branches of Maths
│Pure Mathematics │Applied Mathematics │
│Number Theory │Calculus │
│Algebra │Statistics and Probability │
│Geometry │Set Theory │
│Arithmetic │Trigonometry │
│Combinatorics │ │
│Topology │ │
│Mathematical Analysis │ │
Important Math Topics
A thorough understanding of all important math topics will benefit students throughout their lifetime. Some of the essential math concepts that students must have an in-depth understanding of are
based on the topics listed below.
• Prime and Composite numbers
• HCF and LCM
• Basic Menstruation
• Decimal and Fractions
• Ratio and Proportion
• Geometry
• Probability
Math Calculators
Students often find math challenging due to complex mathematical calculations. Math Calculators are handy tools to resolve all such problems. It makes calculations simple and quick. With the use of
math calculators, calculations ranging from elementary arithmetic operations to complicated equations can be solved within a few seconds.
List of important calculators for students to solve problems quickly and get accurate solutions.
│Prime Factorization Calculator │LCM Calculator │
│Algebra Calculator │Exponent Calculator │
│Mean Median Mode Calculator │Place Value Calculator │
│Roots Calculator │Simple Interest Calculator │
│Scientific Notation Calculator │Arithmetic Sequence Calculator │
Important Math Formulas
Math formulas are expressions created after several years of research to help solve problems easily. Performing simple numerical operations like addition, subtraction and division are easy. However,
to solve algebraic expressions, and other complex operations we use mathematical formulas. These are quite useful in obtaining the answers precisely. Cuemath provides formulas for each math topic
along with the illustrated steps of equations for students to understand them logically.
List of important formulas that students must learn and memorize.
│Algebra Formulas │Probability Formula │
│Circle Formulas │Prime Factorization Formula │
│Trigonometry Formulas│Statistics Formulas │
│Heron's Formula │Permutation and Combination Formulas │
│Integration Formulas │Pythagorean Theorem Formula │
Tips And Tricks To Learn Math Fast
Although math is a vast subject there are some tips and tricks to learn math fast. These tips and tricks will help students improve along their math journey.
Clear All Basic: The first and foremost step in learning mathematics is to clearly understand all basics. It will not only allow you to learn math faster but will also help in establishing links
between various math topics.
Set Objectives: After clearing all basics, set goals for what you need to focus on. Once you understand your objective, start working on it. Explore various resources that can help you improve and
get well versed in those topics.
Practice Daily: Math requires daily practice, implementing a proper study routine will help in grasping concepts better.
Take Guidance: Heading in the right direction is necessary as it will ensure good results. Consider taking help from your teacher or a math tutor if you feel doubtful about topics and concepts.
FAQs on Math
What Exactly is Math?
Math is the science involving numbers, shapes and patterns which is present in almost everything around us. It helps us to derive analytical solutions to practical problems. It is applied in various
fields such as engineering, finance, physical science, etc. It has a great impact in every domain of our life and we can find many mathematics applications around us.
How to Get Better at Math?
Getting better at math requires enforcing a study routine and analyzing mistakes. Students must try to understand and rectify their mistakes through daily practice. Doing so will also help them clear
all their doubts. Students with better mathematical abilities achieve higher academic success. Thus, it is crucial to cultivate math interest in children at an early age. Cuemath’s visually-enriched
math concepts enhance a child’s interest in mathematics and make it easier to learn the subject.
Why is Math Important in Our Daily Life?
Math is highly important in our daily life as there are several applications of mathematics in real-world situations. Statistics or probability theory are examples of applied maths.
What are the Fundamentals of Mathematics?
Fundamentals of mathematics are the building blocks for a solid math foundation. Students must possess a clear knowledge of all the fundamentals of mathematics to study advanced mathematical
concepts. These fundamentals of mathematics are given below.
• Addition and Subtraction of Whole Numbers
• Multiplication and Division of Whole Numbers
• Exponents, Roots, and Factorization of Whole Numbers
• Introduction to Fractions and Multiplication and Division of Fractions
• Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions
• Decimals and Fractions
• Ratios and Rates
• Techniques of Estimation
• Measurement and Geometry
• Signed Numbers
• Algebraic Expressions and Equations
How many Branches of Mathematics do we Have?
The branches of mathematics can be broadly categorized as:
• Arithmetic: Arithmetic involves studying numbers and their relationships to solve problems that include the operations of addition, multiplication, subtraction, division, extraction of roots and
raise to power.
• Algebra: Algebra is the symbolic representation of numbers that provides structure to equations. It forms the basis for advanced study in many fields like science, medicine, engineering, etc.
• Geometry: Geometry is the calculation of various dimensions of solids including height, width, areas, volumes, perimeter and angles. It has many practical applications in architecture and other
• Trigonometry: trigonometry deals with the study of the relationship between angles, lengths, and heights. The applications of trigonometry can be found in many spheres including architecture,
physics, surveying, electronics, satellite navigation, astronomy and engineering.
What are the Most Important Math Topics?
Some of the most important math topics are prime numbers, composite numbers, BODMAS rule, geometry, probability, divisibility rules, HCF, LCM, three-dimensional shapes, basic menstruation, decimal,
fractions, ratio and proportion. An in-depth understanding of all important math topics will enable students to score well in exams.
How Math is Used in Sports?
Math is used in sports to accumulate data, study conditions and generate performance statistics which are considered for planning and optimizing the training session. The data collected in these
mathematical calculations is also helpful for taking strategic decisions based on the team’s performance.
How Maths is Related to Other Subjects?
Math is related to other subjects, especially chemistry, physics, computer science and engineering. In chemistry, mathematics is used to write and balance equations. In physics. It is applied to
calculate mass, velocity and acceleration. In computer science, math is used to build algorithms and solve problems.
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/maths/","timestamp":"2024-11-04T10:35:32Z","content_type":"text/html","content_length":"233070","record_id":"<urn:uuid:fc96f5fe-0d91-49dc-aeac-66ff4744437e>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00236.warc.gz"} |
Submatrix Concatenation in C++
Are you ready for the task? Here it is: Imagine having two different 2D matrices, A and B. Our job is to devise a C++ function—let's name it submatrixConcatenation()—which takes these two matrices as
inputs, along with the coordinates specifying submatrices within A and B. This function is expected to stitch the two chosen submatrices together, forming a new one, C. Notably, the submatrices from
A and B should have the same number of rows, and in the final matrix C, elements from A's submatrix should be on the left and those from B's submatrix on the right.
Let's visualize this with a couple of matrices.
Given the matrix A as:
Plain text
1{{1, 2, 3, 4},
2 {5, 6, 7, 8},
3 {9, 10, 11, 12}}
and the matrix B as:
Plain text
1{{11, 12, 13},
2 {14, 15, 16},
3 {17, 18, 19}}
If we select 2x2 submatrices from each (comprising the 2nd to 3rd rows and 2nd to 3rd columns from A, and 1st to 2nd rows and 1st to 2nd columns from B), their concatenation would look like:
Plain text
1{{6, 7, 11, 12},
2 {10, 11, 14, 15}} | {"url":"https://learn.codesignal.com/preview/lessons/2121","timestamp":"2024-11-01T19:24:11Z","content_type":"text/html","content_length":"157590","record_id":"<urn:uuid:d59d0340-8d82-4435-a53f-d1d1e73d060b>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00010.warc.gz"} |
Centimeters to Meters c
How do I convert centimeters to meters?
Converting centimeters to meters is a simple process that involves dividing the number of centimeters by 100. Since there are 100 centimeters in a meter, this conversion allows us to express a length
in a larger unit. To convert centimeters to meters, divide the number of centimeters by 100.
For example, if you have a length of 150 centimeters, divide 150 by 100 to get 1.5 meters. Similarly, if you have a length of 250 centimeters, divide 250 by 100 to get 2.5 meters.
What is a centimeter?
A centimeter is a unit of length in the metric system, specifically the International System of Units (SI). It is equal to one hundredth of a meter, making it a smaller unit of measurement compared
to a meter. The centimeter is commonly used for measuring small distances, such as the length of objects or the height of individuals.
In terms of conversion, one centimeter is approximately equal to 0.0328 feet. This means that if you have a measurement in centimeters and you want to convert it to feet, you would divide the number
of centimeters by 30.48. Conversely, if you have a measurement in feet and you want to convert it to centimeters, you would multiply the number of feet by 30.48.
The centimeter is a versatile unit of measurement that is widely used in various fields, including science, engineering, and everyday life. It provides a convenient and precise way to measure small
distances, allowing for accurate calculations and comparisons. Whether you are measuring the length of a pencil or determining the height of a person, the centimeter is a valuable unit that helps us
understand and quantify the world around us.
What is a meter?
A meter is a unit of length in the metric system, and it is equivalent to 100 centimeters or 1,000 millimeters. It is the base unit of length in the International System of Units (SI) and is widely
used around the world for measuring distances. The meter was originally defined as one ten-millionth of the distance from the North Pole to the equator along a meridian passing through Paris, France.
However, in 1983, the meter was redefined as the distance traveled by light in a vacuum during a specific time interval. | {"url":"https://www.metric-conversions.org/length/centimeters-to-meters.htm","timestamp":"2024-11-14T15:25:25Z","content_type":"text/html","content_length":"108816","record_id":"<urn:uuid:966d4dde-0a9c-4cd3-9fc6-b6c7b3f3aa7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00102.warc.gz"} |
Metron for Medical | Chiropractic
Note that for proper interpretation of these measures, the patient must be positionedappropriately. Here are some brief notes and comments on each of the measures:
George’s Deviation: George’s line is the curve created by connection points chosen on the posterior sides of all vertebral bodies. According to some literature (based on the “Harrison Spine Model”)
suggests that in the ideal case these points lie on a smooth curve which is a section of an ellipse with certain major-to-minor axis ratio. Metron’s “George’s Deviation” parameter is a sum of all the
offsets from this perfect elliptical arc. A perfect value would be zero (0.0) meaning that all points lie on the ideal elliptical arc.The higher the value of this parameter, the more deviation there
is from a circular arc.
Jackson’s Angle: The angle between two constructed lines: one at the posterior of of L1, and the other at the posterior of the body of L5.
Click here for complete list of measurements. | {"url":"https://www.metron-imaging.com/industries/medical-chiropractic/","timestamp":"2024-11-10T01:03:42Z","content_type":"text/html","content_length":"29166","record_id":"<urn:uuid:bf9e95cb-9617-4592-81ad-13757590e948>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00592.warc.gz"} |
Biography of Carl Friedrich Gauss - www.Notable.name
Johann Carl Friedrich Gauss, often referred to as the Prince of Mathematicians, was a German mathematician, physicist, and astronomer who made profound contributions to a wide range of fields during
the late 18th and early 19th centuries. Gauss’s exceptional mathematical talent, coupled with his keen intellect and insatiable curiosity, propelled him to become one of the most influential
mathematicians in history. His groundbreaking work revolutionized various branches of mathematics and laid the foundation for many important discoveries.
Early Life and Education: Carl Friedrich Gauss was born on April 30, 1777, in Brunswick, Germany. He was the only son of Gebhard Dietrich Gauss, a gardener and bricklayer, and Dorothea Benze. Gauss’s
extraordinary mathematical abilities were evident from an early age. According to an anecdote, when Gauss was just three years old, he corrected an error made by his father in calculating the annual
wages of the workers.
Gauss’s educational journey began at the age of seven when he attended the Brunswick Collegium Carolinum. Despite his humble background, Gauss impressed his teachers and demonstrated exceptional
aptitude in mathematics. Recognizing his talent, the Duke of Brunswick intervened to secure Gauss a scholarship to the Collegium Carolinum. There, Gauss received a comprehensive education in various
subjects, including mathematics, Latin, Greek, and physics.
Mathematical Prodigy: Gauss’s mathematical prowess became apparent during his teenage years. At the age of 14, he made a remarkable discovery when he found a way to construct a regular heptadecagon,
a polygon with 17 sides, using only a compass and a straightedge. This feat astounded his teachers and revealed his ability to solve complex mathematical problems through innovative approaches.
Gauss’s academic journey continued at the University of Göttingen, where he studied from 1795 to 1798. He initially pursued a degree in theology but quickly shifted his focus to mathematics. At
Göttingen, Gauss encountered several influential mathematicians, including Abraham Gotthelf Kästner and Johann Friedrich Pfaff, who recognized his exceptional talent and provided guidance.
Disquisitiones Arithmeticae: In 1798, Gauss published his groundbreaking book, “Disquisitiones Arithmeticae,” which established his reputation as a leading mathematician. In this seminal work, Gauss
presented profound and original insights into number theory. He introduced important concepts and theorems, including modular arithmetic, quadratic forms, and the law of quadratic reciprocity.
Gauss’s Disquisitiones Arithmeticae laid the foundation for modern number theory and became a seminal text in the field.
Least Squares Method: In addition to his work in number theory, Gauss made significant contributions to statistics and the theory of errors. He developed the method of least squares, a statistical
technique used to minimize the sum of the squares of deviations in a data set. The least squares method has wide-ranging applications in various scientific disciplines and has become a fundamental
tool in data analysis.
Celestial Mechanics and Orbit Determination: Gauss’s mathematical talents extended beyond number theory and statistics. He also made important contributions to celestial mechanics. In 1801, he
formulated the method of orbit determination, a mathematical technique that allows astronomers to calculate the orbits of celestial bodies based on observations. Gauss’s method revolutionized the
field of celestial mechanics and enabled astronomers to predict the paths of comets and planets with greater accuracy.
Gaussian Distributions and the Bell Curve: Gauss’s work in statistics led to the development of the Gaussian distribution, also known as the normal distribution or bell curve. This probability
distribution has a symmetrical shape and is characterized by its mean and standard deviation. The Gaussian distribution is widely used in statistics, economics, and many other fields to model various
phenomena, thanks to its mathematical properties and its prevalence in natural and social systems.
Geodesy and Differential Geometry: Gauss’s interests were not limited to mathematics and physics. He also made significant contributions to the field of geodesy, the science of measuring the Earth’s
shape and dimensions. Gauss developed a method called Gauss’s Curvature Theorem, which allowed for the measurement of the curvature of a surface at any given point. His work in differential geometry
laid the foundation for the modern understanding of curved spaces and became instrumental in the development of Einstein’s theory of general relativity.
Personal Life and Character: Despite his towering intellect, Gauss led a relatively private and unassuming life. He had a reserved personality and preferred solitude, spending much of his time
immersed in his work. Gauss was known for his strict work ethic and meticulous attention to detail. He maintained extensive correspondence with prominent mathematicians and scientists of his time,
exchanging ideas and collaborating on various projects.
Gauss married Johanna Osthoff in 1805, and the couple had three children. Tragically, his wife passed away in 1809, which deeply affected Gauss. He later married Friederica Wilhelmine Waldeck in
1810, with whom he had two more children.
Legacy and Honors: Gauss’s contributions to mathematics and science had a lasting impact, and his legacy continues to resonate to this day. His groundbreaking work paved the way for numerous
discoveries and provided a solid mathematical framework for many areas of study. Gauss’s influence can be seen in various fields, including number theory, statistics, celestial mechanics, geodesy,
and differential geometry.
Gauss received numerous accolades and honors during his lifetime. He was elected to several prestigious scientific societies, including the Royal Society of London and the Royal Society of Göttingen.
In 1831, he was awarded the Copley Medal, the highest honor of the Royal Society, for his work in mathematics and astronomy. Gauss’s contributions to mathematics have been recognized through various
mathematical concepts named in his honor, such as Gauss’s Law, Gauss’s Lemma, and Gauss’s Theorem.
Later Years and Death: In his later years, Gauss gradually reduced his mathematical activity but continued to work on various projects. He held the position of director of the Göttingen Observatory
from 1807 until his retirement in 1855. Gauss passed away on February 23, 1855, in Göttingen, Germany, at the age of 77.
Carl Friedrich Gauss’s genius and profound contributions to mathematics and science solidify his place as one of the greatest mathematicians of all time. His innovative ideas, rigorous methods, and
profound insights continue to inspire generations of mathematicians, scientists, and scholars. Gauss’s legacy serves as a testament to the power of human intellect, curiosity, and perseverance in
advancing our understanding of the natural world. | {"url":"https://notable.name/carl-friedrich-gauss/","timestamp":"2024-11-02T15:57:22Z","content_type":"text/html","content_length":"58907","record_id":"<urn:uuid:33c3df92-6b5a-4ffd-b714-c439feb8ce5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00191.warc.gz"} |
Global Historical Population
Global historical population data The population data starts from -1000000 BC to 1990 with the average number of people. There are several population data from the different reports such as:
Read more
Data Previews
name type description
Year number
Average number Average number of people in millions
Deevey number Number of people in millions
McEvedy and Jones 1978 number Number of people in millions
Durand Low number Number of people in millions
Durand High number Number of people in millions
Clark number Number of people in millions
Biraben number Number of people in millions
Blaxter number Number of people in millions
UN number Number of people in millions
Kremer number Number of people in millions
Global historical population data
The population data starts from -1000000 BC to 1990 with the average number of people. There are several population data from the different reports such as: Deevey,McEvedy and Jones 1978,Durand
Low,Durand High,Clark,Biraben,Blaxter,UN,Kremer.
Source: Appendix in Joel E. Cohen, How Many People Can the Earth Support?, Norton 1996, ISBN 0-393-31495-2.
Public Domain Dedication and License. | {"url":"https://datahub.io/core/population-global-historical","timestamp":"2024-11-15T01:02:30Z","content_type":"text/html","content_length":"56012","record_id":"<urn:uuid:4f8e21e8-759f-450d-89ea-e58ed4673a0c>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00194.warc.gz"} |
Properties of the Robin’s Inequality
Download PDFOpen PDF in browserCurrent version
Properties of the Robin’s Inequality
EasyChair Preprint 3708, version 10
8 pages•Date: August 27, 2020
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. Many consider
it to be the most important unsolved problem in pure mathematics. The Robin's inequality consists in $\sigma(n) < e^{\gamma } \times n \times \ln \ln n$ where $\sigma(n)$ is the divisor function and
$\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Robin's inequality is true for every natural number $n > 5040$ if and only if the Riemann hypothesis is true. We prove the Robin's
inequality is true for every natural number $n > 5040$ when $15 \nmid n$, where $15 \nmid n$ means that $n$ is not divisible by $15$. More specifically: every counterexample should be divisible by $2
^{20} \times 3^{13} \times 5^{8} \times k_{1}$ or either $2^{20} \times 3^{13} \times k_{2}$ or $2^{20} \times 5^{8} \times k_{3}$, where $2 \nmid k_{1}$, $3 \nmid k_{1}$, $5 \nmid k_{1}$, $2 \nmid
k_{2}$, $3 \nmid k_{2}$, $2 \nmid k_{3}$ and $5 \nmid k_{3}$.
Keyphrases: Divisor, inequality, number theory
Links: https://easychair.org/publications/preprint/zPPG
Download PDFOpen PDF in browserCurrent version | {"url":"https://yahootechpulse.easychair.org/publications/preprint/zPPG","timestamp":"2024-11-06T21:30:35Z","content_type":"text/html","content_length":"9023","record_id":"<urn:uuid:94a8aa55-8651-4064-b75e-4a77dce5bfc6>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00791.warc.gz"} |
Mastery Vs. Spiral Review in Math - BJU Press Blog
When looking for a homeschool math curriculum, you’ve likely seen recommendations for spiral review in math or a mastery learning approach. Homeschool families often discuss and compare these two
topics to each other as if they’re mutually exclusive. However, mastery learning describes how a teacher chooses to teach and at what pace they progress through the material. Spiral review describes
how to review material previously covered. Before searching for programs that use one method, look at what each of these concepts mean and how they work.
What is spiral review in math?
In math, spiral review is returning to review an earlier concept later to practice it. This approach keeps math concepts fresh in students’ minds. It gives them regular opportunities to use math so
they develop automaticity (the ability to answer familiar math questions automatically).
Spiral review vs. spiral learning
Spiral review reviews concepts often throughout a math program. A spiral learning approach introduces a topic early and builds on it over time. For instance, students use spiral learning when
building on addition as they progress from adding whole numbers to fractions with different denominators or negative numbers. A first-grade student who is still developing number sense is usually
ready to learn addition with whole numbers, but they are not often ready to add fractions with different denominators, a skill that requires multiplication and addition. A spiral review math program
would offer consistent opportunities to practice addition until the student is ready. When they begin learning the new skill, they should be familiar and confident with all the previous skills they
What is the mastery learning approach in mathematics?
Mastery learning, or teaching for mastery, means the educator covers one concept at a time until students can demonstrate mastery. You would not move on from a topic until students have grasped how
to use and apply what they have learned. Of course, a 6-year-old learning addition can’t truly master the concept because he or she isn’t developmentally ready to learn multiplication (a building
block on addition). So, your child would “master” addition when the process becomes automatic.
If you’re interested in the mastery approach for your students, be mindful of your students’ abilities, mental and emotional. When using a mastery approach, you may want to cover every aspect of a
topic or concept fully before progressing. However, younger students may not be ready for more difficult concepts—like division in math, or the full details of a war in history.
Mastery Learning Vs. Spiral Learning
A mastery learning approach depends on how the teacher schedules time for learning and introducing new topics. A spiral learning approach covers topics in increasing difficulty over time.
Since mastery learning and spiral learning aren’t mutually exclusive, they cannot be pitted against each other, especially in math. We often compare one against the other because of a
misunderstanding about the approaches. It’s easy to think that mastery means covering a topic once and never touching it again and that spiral means covering topics so briefly that a student never
fully grasps it.
Can you use mastery and spiral learning together?
Absolutely. Increasing instructional time (mastery approach) helps students learn, and repetition (spiral review approach) aids learning. Some students may need more of one than the other, but both
are beneficial. Some parents decide they want an approach that promises to cover each topic incrementally and thoroughly. Others want a program that promises repetition.
Is spiral curriculum or mastery learning better for math?
Math often relies on building one idea on another over time. A curriculum that doesn’t use this approach to teaching math is likely not laying a proper foundation for deep understanding of a topic.
Similarly, an educator who does not give students enough time to learn and understand concepts in math is also not preparing them to use math and to understand it at a higher level. You need both
time and repetition to encourage learning in math.
Mastery Learning and Spiral Math Curriculum for Homeschool
Searching for a curriculum that promises to use either a spiral learning approach or a mastery learning approach in mathematics can be helpful to your children. But it’s important to recognize that a
curriculum can’t always make such promises. A curriculum can’t teach for mastery because teaching for mastery depends on giving students time to learn and practice. A curriculum can direct you, the
parent, to take it slow and move with your students’ understanding. On the other hand, it won’t be able to predict when every student will get it or how much practice they’ll need.
A curriculum can design a program to build on information over time, show connections to previous concepts, and reteach in greater detail as the student develops—what is often described as a spiral
curriculum. This is the approach that BJU Press math curriculum uses to cover topics with increasing complexity. We apply a spiral review approach to make sure new learning remains top-of-mind. It
will be up to you, the parent and teacher, to determine how much time and additional practice your students need to master concepts.
1. BJU Press Writer says
Please reach out to customer service at 800-845-5731, they will answer your questions on textbooks.
2. B. Hay says
Can you send me a copy of the BJU algebra 1 and 2 to review, please? | {"url":"https://blog.bjupress.com/blog/2021/11/23/mastery-vs-spiral-review-in-math/","timestamp":"2024-11-14T08:55:52Z","content_type":"text/html","content_length":"82470","record_id":"<urn:uuid:4202bbee-aae3-409e-8585-06564349d327>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00844.warc.gz"} |
General Mathematics | Data USA
In 2022, the locations with the highest concentration of General Mathematics degree recipients are New York, NY, Irvine, CA, and Los Angeles, CA. The most common degree awarded to students studying
General Mathematics is a bachelors degree.
Information about the types of higher education institutions that grant degrees in General Mathematics and the types of students that study this field. University of Wisconsin-Madison awards the most
degrees in General Mathematics in the US, but Allen University and Indiana University-East have the highest percentage of degrees awarded in General Mathematics.
Tuition costs for General Mathematics majors are, on average, $7,128 for in-state public colleges, and $35,954 for out of state private colleges.
The most common sector, by number of institutions, that offers General Mathematics programs are Private not-for-profit, 4-year or above institutions (719 total). The most common sector, by number of
degrees awarded, is Public, 4-year or above (16,942 completions).
Institution with the Most Degrees Awarded in General Mathematics (2022)
The most common sector, by number of degrees awarded in General Mathematics, is Public, 4-year or above (16,942 completions in 2022).
The following chart shows the share of universities that offer General Mathematics programs, by the total number of completions, colored and grouped by their sector.
University of Wisconsin-Madison has the most General Mathematics degree recipients, with 385 degrees awarded in 2022.
The following bar chart shows the state tuition for the top 5 institutions with the most degrees awarded in General Mathematics.
Out of all institutions that offer General Mathematics programs and have at least 5 graduates in those programs, Allen University has the highest percentage of degrees awarded in General Mathematics,
with 11.8%.
This map shows the counties in the United States colored by the highest number of degrees awarded in General Mathematics by year.
This map shows the counties in the United States colored by the highest growth in degrees awarded for General Mathematics.
Information on the businesses and industries that employ Math & Statistics graduates and on wages and locations for those in the field.
The average salary for Math & Statistics majors is $116,385 and the most common occupations are Postsecondary teachers, Software developers, and Secondary school teachers.
The industry that employs the most Math & Statistics majors is Elementary & secondary schools, though the highest paying industry, by average wage, is Internet publishing, broadcasting & web search
This chart shows the average annual salaries of the most common occupations for Math & Statistics majors.
This map shows the public use micro areas (PUMAs) in the United States colored by the average salary of Math & Statistics majors.
Note that the census collects information tied to where people live, not where they work. It is possible that Math & Statistics majors live and work in the same place, but it is also possible that
they live and work in two different places.
The most common occupations Math & Statistics majors, by number of employees, are Postsecondary teachers, Software developers, and Secondary school teachers.
Compared to other majors, there are an unusually high number of Math & Statistics majors working as Actuaries, Miscellaneous mathematical science occupations, including mathematicians &
statisticians, and Computer and information research scientists.
The highest paid occupations by median income for Math & Statistics majors are Surgeons, Securities, commodities, & financial services sales agents, and Physicians.
The number of Math & Statistics graduates in the workforce has been growing at a rate of 2.57%, from 756,726 in 2021 to 776,185 in 2022.
The largest single share of Math & Statistics graduates go on to work as Postsecondary teachers (8.17%). This chart shows the various jobs filled by those with a major in Math & Statistics by share
of the total number of graduates.
The most common industries that employ Math & Statistics majors, by number of employees, are Elementary & secondary schools, Colleges, universities & professional schools, including junior colleges,
and Computer Systems Design.
The highest paying industries of Math & Statistics majors, by average wage, are Internet publishing, broadcasting & web search portals, Petroleum & petroleum products merchant wholesalers, and
Securities, commodities, funds, trusts & other financial investments.
The industry which employs the most Math & Statistics graduates by share is Elementary & secondary schools, followed by Colleges, universities & professional schools, including junior colleges. This
visualization shows the industries that hire those who major in Math & Statistics.
This map shows the public use micro areas (PUMAs) in the United States where there are a relatively high population of Math & Statistics majors.
Demographic information for those who earn a degree in Math & Statistics in the United States.
The average age of a person in the workforce with a degree in Math & Statistics is 43.9.
The most common degree type these workers hold is a Bachelors Degree. Male employees are more likely to hold Math & Statistics degrees, and White students are the most common race/ethnicty group
awarded degrees in Math & Statistics (12,972 students).
This chart shows distribution of ages for employees with a degree in Math & Statistics. The most common ages of employees with this major are 30 and 31 years old, which represent 2.91% and 2.85% of
the population, respectively.
The most common degree types awarded to students graduating in General Mathematics are Bachelors Degree, Associates Degree, and Masters Degree.
The most common degree types held by the working population in Math & Statistics are Bachelors Degree, Masters Degree, and Doctorate degree.
This chart shows the granted degrees by sex at the 5 institutions that graduate the most students in General Mathematics.
This chart shows the number of degrees awarded in General Mathematics for each race & ethnicity. White students earned the largest share of the degrees with this major.
This chart illustrates the differences by sex for each race & ethnicity of Bachelors Degree recipients in General Mathematics.
White Male students, who earn most of the degrees in this field, are the most common combination of race/ethnicity and sex.
There are a relatively high number of people that were born in USSR that hold Math & Statistics degrees (6.32 times more than expected), and the most common country of origin by total numbers for
non-US students earning a degree in this field is India (34,510 degree recipients).
Data on the critical and distinctive skills necessary for those working in the General Mathematics field from the Bureau of Labor Statistics. General Mathematics majors need many skills, but most
especially Reading Comprehension. The revealed comparative advantage (RCA) shows that General Mathematics majors need more than the average amount of Programming, Mathematics, and Science.
These two visualizations, one a radial chart and one a bar chart, show the same information, a rating of how necessary the following skills are for General Mathematics majors. Toggle between "value"
and "RCA" to see the absolute rating of that skill (value) and the revealed comparative advantage (RCA), or how much greater or lesser that skill's rating is than the average. The longer the bar or
the closer the line comes to the circumference of the circle, the more important that skill is. The importance of Programming is very distinctive for majors, but the Reading Comprehension, Critical
Thinking, and Mathematics are the three most important skills for people in the field. | {"url":"https://datausa.io/profile/cip/general-mathematics?redirect=true","timestamp":"2024-11-05T07:40:46Z","content_type":"text/html","content_length":"372108","record_id":"<urn:uuid:ea65482b-586d-4b77-b1d8-b3eeda1732fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00322.warc.gz"} |
New algorithms for generalized network flows for Mathematical Programming
Mathematical Programming
New algorithms for generalized network flows
This paper, of which a preliminary version appeared in ISTCS'92, is concerned with generalized network flow problems. In a generalized network, each edge e=(u, v) has a positive 'flow multiplier' ae
associated with it. The interpretation is that if a flow of xe enters the edge at node u, then a flow of ae×e exits the edge at v. The uncapacitated generalized transshipment problem (UGT) is defined
on a generalized network where demands and supplies (real numbers) are associated with the vertices and costs (real numbers) are associated with the edges. The goal is to find a flow such that the
excess or deficit at each vertex equals the desired value of the supply or demand, and the sum over the edges of the product of the cost and the flow is minimized. Adler and Cosares [Operations
Research 39 (1991) 955-960] reduced the restricted uncapacitated generalized transshipment problem, where only demand nodes are present, to a system of linear inequalities with two variables per
inequality. The algorithms presented by the authors in [SIAM Journal on Computing, to appear] result in a faster algorithm for restricted UGT. Generalized circulation is defined on a generalized
network with demands at the nodes and capacity constraints on the edges (i.e., upper bounds on the amount of flow). The goal is to find a flow such that the excesses at the nodes are proportional to
the demands and maximized. We present a new algorithm that solves the capacitated generalized flow problem by iteratively solving instances of UGT. The algorithm can be used to find an optimal flow
or an approximation thereof. When used to find a constant factor approximation, the algorithm is not only more efficient than previous algorithms but also strongly polynomial. It is believed to be
the first strongly polynomial approximation algorithm for generalized circulation. The existence of such an approximation algorithm is interesting since it is not known whether the exact problem has
a strongly polynomial algorithm. © 1994. | {"url":"https://research.ibm.com/publications/new-algorithms-for-generalized-network-flows--1","timestamp":"2024-11-03T13:24:19Z","content_type":"text/html","content_length":"68036","record_id":"<urn:uuid:5d1727ed-7f5d-405d-97e8-5ec5431637f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00883.warc.gz"} |
Chin. Phys. B
ASTROD-GW (ASTROD [astrodynamical space test of relativity using optical devices] optimized for gravitational wave detection) is a gravitational-wave mission with the aim of detecting gravitational
waves from massive black holes, extreme mass ratio inspirals (EMRIs) and galactic compact binaries together with testing relativistic gravity and probing dark energy and cosmology. Mission orbits of
the 3 spacecrafts forming a nearly equilateral triangular array are chosen to be near the Sun–Earth Lagrange points L3, L4, and L5. The 3 spacecrafts range interferometrically with one another with
arm length about 260 million kilometers. For 260 times longer arm length, the detection sensitivity of ASTROD-GW is 260 fold better than that of eLISA/NGO in the lower frequency region by assuming
the same acceleration noise. Therefore, ASTROD-GW will be a better cosmological probe. In previous papers, we have worked out the time delay interferometry (TDI) for the ecliptic formation. To
resolve the reflection ambiguity about the ecliptic plane in source position determination, we have changed the basic formation into slightly inclined formation with half-year precession-period. In
this paper, we optimize a set of 10-year inclined ASTROD-GW mission orbits numerically using ephemeris framework starting at June 21, 2035, including cases of inclination angle with 0° (no
inclination), 0.5°, 1.0°, 1.5°, 2.0°, 2.5°, and 3.0°. We simulate the time delays of the first and second generation TDI configurations for the different inclinations, and compare/analyse the
numerical results to attain the requisite sensitivity of ASTROD-GW by suppressing laser frequency noise below the secondary noises. To explicate our calculation process for different inclination
cases, we take the 1.0° as an example to show the orbit optimization and TDI simulation. | {"url":"https://cpb.iphy.ac.cn/EN/volumn/volumn_2568.shtml","timestamp":"2024-11-05T15:13:38Z","content_type":"text/html","content_length":"508292","record_id":"<urn:uuid:979b3cab-ee9e-4ed0-b421-a0effbf6914f>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00838.warc.gz"} |
Restrictions of a Function: Definition, Examples
When a function is defined on a smaller set, it’s called a restriction. Reasons to restrict functions include:
1. To formally define the domain of a function.
2. To find an inverse function. For example, trigonometric functions do not have any restrictions, so there is no inverse. However, restricting the domain to a smaller interval allows you to work
with inverse functions.
3. To analyze segments that are “better behaved” with bounded variation, better modulus of continuity or monotonicity [1].
1. Restricting Domains to Define Functions
While many functions like the exponential function or cubed root function have no restrictions, others only work on specific sets. Therefore, their formal definitions include one or more
restrictions. For example:
Restrictions of a function often happen because of some problem with algebra in the formula. For example, with rational functions, look out for places where the denominator would equal zero:
For radical functions, you can’t take square roots of negative numbers:
Function Restrictions
f(x) = √(x + 3). x ≥-3 to avoid the problem of negative numbers.
f(x) = &radic(-x) x ≥ 0.
2. Restrictions of a Function to Find an Inverse
As an example, if you set an interval to [-π/2, π/2], the sine function becomes one to one [2]:
Switching the x and y values of the three coordinates (-π/2, -1), (0, 0), (π/2, 1) gives the inverse: (-1, -π/2), (0, 0), (1, π/2).
3. Restriction of a Function to make it “Better Behaved”
It’s impossible to pin down a limit for the function sin(1/x) at zero.
Some functions, like f(x) = sin(1/x), behave badly around x = 0. Because the function isn’t defined there, the values oscillate wildly and the function is almost impossible to work with. Restricting
the domain to a smaller segment (one that doesn’t include zero!) means that the function is a little more workable.
Graph: Desmos.com
[1] Restrictions of continuous functions. Retrieved April 2, 2021 from: https://arxiv.org/abs/0711.0679
[2] Trigonometry With Restrictions. Retrieved May 2, 2021 from: https://www.alamo.edu/contentassets/35e1aad11a064ee2ae161ba2ae3b2559/trigonometric/math2412-inverse-trig-functions.pdf
Comments? Need to post a correction? Please Contact Us. | {"url":"https://www.statisticshowto.com/restrictions-of-a-function/","timestamp":"2024-11-04T06:04:54Z","content_type":"text/html","content_length":"68665","record_id":"<urn:uuid:c38c2a38-504b-454d-a549-df14fa063aeb>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00168.warc.gz"} |
Expression function only returns value for first row in TIBCO Spotfire
When using an Expression Function in a calculated column, you may see a value returned only in the first row. You may expect this to return a value for every row, in the way many built-in Spotfire
functions work.
For example, if you have the following Expression Function (Edit > Data Function Properties > Expression Function) with the name mySum():
# Define custom function using R syntax, which will be run using the Spotfire desktop client's built-in TERR engine. This example is a basic sum:
mySum <- function( input )
out <- sum( input )
# Run the function to produce the output, repeat for length of original column
output <- mySum( input1 )
And a data set like:
Then your calculated column with the expression "mySum([value])" would result in:
category value mySum
A 1 16
B 2
C 4
D 7
E 2
Think of the Expression Functions in the following way when looking to use them in a Calculated Column.
- Input: You pass into the expression function any number of columns (per its definition).
- Output: Your expression function should return one column, normally with the same number of rows as the original columns.
In the original example expression function, only a single value was returned, "16". Therefore, it was returned for the first row, and the subsequent rows were empty because there was no more output | {"url":"https://support.tibco.com/external/article?articleUrl=Tibco-KnowledgeArticle-Article-48855","timestamp":"2024-11-11T07:18:57Z","content_type":"text/html","content_length":"55438","record_id":"<urn:uuid:fb4825be-07b8-4f7b-8e20-94dd391bc64c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00021.warc.gz"} |
Lesson 5
Creating Scale Drawings
Let’s create our own scale drawings.
5.1: Number Talk: Which is Greater?
Without calculating, decide which quotient is larger.
\(11 \div 23\) or \(7 \div 13\)
\(0.63 \div 2\) or \(0.55 \div 3\)
\(15 \div \frac{1}{3}\) or \(15 \div \frac{1}{4}\)
5.2: Bedroom Floor Plan
Here is a rough sketch of Noah’s bedroom (not a scale drawing).
Noah wants to create a floor plan that is a scale drawing.
1. The actual length of Wall C is 4 m. To represent Wall C, Noah draws a segment 16 cm long. What scale is he using? Explain or show your reasoning.
2. Find another way to express the scale.
3. Discuss your thinking with your partner. How do your scales compare?
4. The actual lengths of Wall A, Wall B, and Wall D are 2.5 m, 2.75 m, and 3.75 m. Determine how long these walls will be on Noah’s scale floor plan.
5. Use the Point tool
and the Segment tool
to draw the walls of Noah's scale floor plan in the applet.
If Noah wanted to draw another floor plan on which Wall C was 20 cm, would 1 cm to 5 m be the right scale to use? Explain your reasoning.
5.3: Two Maps of Utah
A rectangle around Utah is about 270 miles wide and about 350 miles tall. The upper right corner that is missing is about 110 miles wide and about 70 miles tall.
1. Make a scale drawing of Utah where 1 centimeter represents 50 miles.
Make a scale drawing of Utah where 1 centimeter represents 75 miles.
2. How do the two drawings compare? How does the choice of scale influence the drawing?
If we want to create a scale drawing of a room's floor plan that has the scale “1 inch to 4 feet,” we can divide the actual lengths in the room (in feet) by 4 to find the corresponding lengths (in
inches) for our drawing.
Suppose the longest wall is 15 feet long. We should draw a line 3.75 inches long to represent this wall, because \(15 \div 4 = 3.75\).
There is more than one way to express this scale. These three scales are all equivalent, since they represent the same relationship between lengths on a drawing and actual lengths:
• 1 inch to 4 feet
• \(\frac12\) inch to 2 feet
• \(\frac14\) inch to 1 foot
Any of these scales can be used to find actual lengths and scaled lengths (lengths on a drawing). For instance, we can tell that, at this scale, an 8-foot long wall should be 2 inches long on the
drawing because \(\frac14 \boldcdot 8 = 2\).
The size of a scale drawing is influenced by the choice of scale. For example, here is another scale drawing of the same room using the scale 1 inch to 8 feet.
Notice this drawing is smaller than the previous one. Since one inch on this drawing represents twice as much actual distance, each side length only needs to be half as long as it was in the first
scale drawing.
• scale
A scale tells how the measurements in a scale drawing represent the actual measurements of the object.
For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac12\) inch would
represent 4 feet.
• scale drawing
A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale. | {"url":"https://im-beta.kendallhunt.com/MS_ACC/students/2/2/5/index.html","timestamp":"2024-11-03T07:39:47Z","content_type":"text/html","content_length":"91923","record_id":"<urn:uuid:01b6414e-30fb-45d5-9ae7-0012f9057688>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00896.warc.gz"} |
[EM] Beatpath for dummies
Roy royone at yahoo.com
Fri Aug 17 13:58:00 PDT 2001
I haven't seen a description of beatpath that is really easy to
visualize, so I pondered the concept a while and came up with my
own version. It's very similar to the Dodgson-like method I describe
in my last post (RE Don't ignore the margins), except instead of
eliminating by row or column total, margin entries are eliminated in
order of increasing size.
This elimination method is at least in the spirit of beatpath, and may
be functionally equivalent to it, apart from my use of run-offs.
Start with a matrix of winning margins. Eliminate Condorcet winners
and losers until none are left. Winners accumulate in top-to-bottom
ranking order; losers accumulate separately, bottom-to-top.
Keep a copy of the original matrix to use in run-offs (below).
Eliminate all occurrences of the smallest margin from the working
matrix. This eliminates the weakest paths: candidates who do not win
any contests by a larger margin are beatpath losers, because there are
no paths from them to anywhere else that are stronger than that
margin. Similarly, those who do not lose any contest by a larger
margin are beatpath winners. During this step, if you like, you can
say, "You are the weakest link. G'bye!"
Collect any losers (those with no remaining victories) and, if there
are more than one, conduct a run-off among them, using numbers from
the original matrix, to determine the order in which they will be
added to the losers. Collect any winners and do likewise. Remove
winners and losers from the working matrix.
Repeat Elimination and Collection until the working matrix is empty.
More information about the Election-Methods mailing list | {"url":"http://lists.electorama.com/pipermail/election-methods-electorama.com/2001-August/071884.html","timestamp":"2024-11-10T19:33:31Z","content_type":"text/html","content_length":"4413","record_id":"<urn:uuid:28f477c3-d992-4d7f-a5ff-fa805a3b52b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00572.warc.gz"} |
Redline Poker - Player of The Year
The POY standings are based on a formula developed by Tourney.com. The only modification is to multiply the number of points by 10 to make the values meaningful in the smaller buy-ins found in home
Their site gives a full explanation of the formula, so the description here touches on the two most important points.
First, the number of points earned in a tournament is determined according to this formula:
• B - The Amount of the Buy-in
• E - The Number of Entrants
• P - Finishing Position
For example, a player finishing 1st out of 15 players in a $25 buy-in tournament earns 62 points.
Second, the points age as time passes, so that points earned in a recent tournment are greater than points earned in an identical tournament a year ago.
After 10 months, the 62 points earned in the example above will have deteriorated to 39 points.
The aging is done by multiplying the points by a factor that decreases according to this schedule:
Age (months) Factor
< 3 1.0
3 - 6 0.875
6 - 9 0.75
9 - 12 0.625
12 - 15 0.5
15 - 18 0.375
18 - 21 0.25
21 - 24 0.125
> 24 0.0
This results in a rolling system that doesn't have to be reset, lets new players climb in the standings, and keeps past winners from resting on their laurels. | {"url":"https://www.redlinepoker.com/poyinfo","timestamp":"2024-11-08T18:38:32Z","content_type":"text/html","content_length":"4538","record_id":"<urn:uuid:32483ce0-33ad-412e-956d-08896508f775>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00779.warc.gz"} |
Permutations with Heap's Algorithm
ETOOBUSY 🚀 minimal blogging for the impatient
Permutations with Heap's Algorithm
All permutations over $N$ objects can be generated by Heap’s Algorithm.
It also seems that it’s pretty good at minimizing the number of movements of stuff around.
The good thing with the current version of the Wikipedia page (as of this writing) is the pseudo-code for an iterative (i.e. non-recursive) version of the algorithm. The following is taken from
there, without the comments and with variable names that are a bit more readable in my opinion:
procedure generate(n : integer, A : array of any):
stack : array of integer
sp : integer
for sp := 0; sp < n; sp += 1 do
stack[sp] := 0
end for
sp := 0;
while sp < n do
if stack[sp] < sp then
if sp is even then
swap(A[0], A[sp])
swap(A[c[sp]], A[sp])
end if
stack[sp] += 1
sp := 0
stack[sp] := 0
sp += 1
end if
end while
I’m particularly fond of iterative implementations, which probably makes me a bad functional programmer.
But there’s a reason for my bias, which I already talked about (I think): turning an iterative implementation into an iterator-based implementation is easier (at least for me!) and iterator-based
implementations on stuff that can grow factorially can be a very, very interesting characteristic.
Here’s a corresponding Perl implementation, where we avoid taking parameter n explicitly and the input array is passed via @_:
sub permutations {
my @indexes = 0 .. $#_;
my @stack = (0) x @indexes;
my $sp = 0;
while ($sp < @indexes) {
if ($stack[$sp] < $sp) {
my $other = $sp % 2 ? $stack[$sp] : 0;
@indexes[$sp, $other] = @indexes[$other, $sp];
$sp = 0;
else {
$stack[$sp++] = 0;
As a matter of fact, we use array @indexes to take the role of the input array A in the pseudocode, so that we avoid messing up directly with the array.
I really like how compact this implementation is! | {"url":"https://github.polettix.it/ETOOBUSY/2021/01/29/permutations-algorithm/","timestamp":"2024-11-12T22:17:10Z","content_type":"text/html","content_length":"9739","record_id":"<urn:uuid:c596ee7e-1deb-486d-af70-182d157ae370>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00407.warc.gz"} |
Introduction to Determining Volumes by Slicing
What you’ll learn to do: Find the volume of a solid of revolution using various methods
In the preceding section, we used definite integrals to find the area between two curves. In this section, we use definite integrals to find volumes of three-dimensional solids. We consider three
approaches—slicing, disks, and washers—for finding these volumes, depending on the characteristics of the solid. | {"url":"https://courses.lumenlearning.com/calculus2/chapter/solids-of-revolution/","timestamp":"2024-11-02T21:24:19Z","content_type":"text/html","content_length":"45493","record_id":"<urn:uuid:1b57861a-d5c0-4242-89e3-71e7e6cd3bef>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00649.warc.gz"} |
Trace Ratio vs. Ratio Trace for Dimensionality Reduction
A large family of algorithms for dimensionality reduction end with solving a Trace Ratio problem in the form of arg maxW Tr(WT SpW)/Tr(WT SlW)1 , which is generally transformed into the corresponding
Ratio Trace form arg maxW Tr[ (WT SlW)-1 (WT SpW) ] for obtaining a closed-form but inexact solution. In this work, an efficient iterative procedure is presented to directly solve the Trace Ratio
problem. In each step, a Trace Difference problem arg maxW Tr[WT (Sp -Sl)W] is solved with being the trace ratio value computed from the previous step. Convergence of the projection matrix W, as well
as the global optimum of the trace ratio value , are proven based on point-to-set map theories. In addition, this procedure is further extended for solving trace ratio problems with more general
constraint WT CW=I and providing exact solutions for kernel-based subspace learning problems. Extensive experiments on faces and UCI data demonstrate the high convergence speed of the proposed
solution, as w... | {"url":"https://www.sciweavers.org/publications/trace-ratio-vs-ratio-trace-dimensionality-reduction","timestamp":"2024-11-02T06:03:54Z","content_type":"application/xhtml+xml","content_length":"38993","record_id":"<urn:uuid:31560bf3-fd1d-4f9f-b6a8-609251e0b5c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00023.warc.gz"} |
SAGE Getting Started from Rob Beezer FCLA
Example GS Getting Started from Rob Beezer FCLA
Sage is a powerful system for studying and exploring many different areas of mathematics. In the next section, and the majority of the remaining section, we will include short descriptions and
examples using Sage. You can read a bit more about Sage in the Preface. If you are not already reading this in an electronic version, you may want to investigate obtaining the worksheet version of
this book, where the examples are "live" and editable. Most of your interaction with Sage will be by typing commands into a compute cell. That's a compute cell just below this paragraph. Click once
inside the compute cell and you will get a more distinctive border around it, a blinking cursor inside, plus a cute little "evaluate" link below it.
At the cursor, type 2+2 and then click on the evaluate link. Did a 4 appear below the cell? If so, you've successfully sent a command off for Sage to evaluate and you've received back the (correct)
Here's another compute cell. Try evaluating the command factorial(300).
Hmmmmm. That is quite a big integer! The slashes you see at the end of each line mean the result is continued onto the next line, since there are 615 digits in the result.
To make new compute cells, hover your mouse just above another compute cell, or just below some output from a compute cell. When you see a skinny blue bar across the width of your worksheet, click
and you will open up a new compute cell, ready for input. Note that your worksheet will remember any calculations you make, in the order you make them, no matter where you put the cells, so it is
best to stay organized and add new cells at the bottom.
Try placing your cursor just below the monstrous value of $300!$ that you have. Click on the blue bar and try another factorial computation in the new compute cell.
Each compute cell will show output due to only the very last command in the cell. Try to predict the following output before evaluating the cell.
The following compute cell will not print anything since the one command does not create output. But it will have an effect, as you can see when you execute the subsequent cell. Notice how this uses
the value of b from above. Execute this compute cell once. Exactly once. Even if it appears to do nothing. If you execute the cell twice, your credit card may be charged twice.
Now execute this cell, which will produce some output.
So b came into existence as 6. Then a cell added 50. This assumes you only executed this cell once! In the last cell we create b+20 (but do not save it) and it is this value that is output.
You can combine several commands on one line with a semi-colon. This is a great way to get multiple outputs from a compute cell. The syntax for building a matrix should be somewhat obvious when you
see the output, but if not, it is not particularly important to understand now.
Some commands in Sage are "functions" an example is factorial() above. Other commands "are methods" of an object and are like characteristics of objects, examples are .factor() and .derivative() as
methods of a function. To comment on your work, you can open up a small word-processor. Hover your mouse until you get the skinny blue bar again, but now when you click, also hold the SHIFT key at
the same time. Experiment with fonts, colors, bullet lists, etc and then click the "Save changes" button to exit. Double-click on your text if you need to go back and edit it later.
Open the word-processor again to create a new bit of text (maybe next to the empty compute cell just below). Type all of the following exactly, but do not include any backslashes that might precede
the dollar signs in the print version: Pythagorean Theorem: \$c^2=a^2+b^2\$ and save your changes. The symbols between the dollar signs are written according to the mathematical typesetting language
known as TeX -- cruise the internet to learn more about this very popular tool. (Well, it is extremely popular among mathematicians and physical scientists.)
Much of our interaction with sets will be through Sage lists. These are not really sets -- they allow duplicates, and order matters. But they are so close to sets, and so easy and powerful to use
that we will use them regularly. We will use a fun made-up list for practice, the quote marks mean the items are just text, with no special mathematical meaning. Execute these compute cells as we
work through them.
So the square brackets define the boundaries of our list, commas separate items, and we can give the list a name. To work with just one element of the list, we use the name and a pair of brackets
with an index. Notice that lists have indices that begin counting at zero. This will seem odd at first and will seem very natural later.
We can add a new creature to the zoo, it is joined up at the far right end.
We can remove a creature.
We can extract a sublist. Here we start with element 1 (the elephant) and go all the way up to, but not including, element 3 (the beetle). Again a bit odd, but it will feel natural later. For now,
notice that we are extracting two elements of the lists, exactly $3-1=2$ elements.
Often we will want to see if two lists are equal. To do that we will need to sort a list first. A function creates a new, sorted list, leaving the original alone. So we need to save the new one with
a new name.
Notice that if you run this last compute cell your zoo has changed and some commands above will not necessarily execute the same way. If you want to experiment, go all the way back to the first
creation of the zoo and start executing cells again from there with a fresh zoo.
A construction called a "list comprehension" is especially powerful, especially since it almost exactly mirrors notation we use to describe sets. Suppose we want to form the plural of the names of
the creatures in our zoo. We build a new list, based on all of the elements of our old list.
Almost like it says: we add an "s" to each animal name, for each animal in the zoo, and place them in a new list. Perfect. (Except for getting the plural of "ostrich" wrong.)
One final type of list, with numbers this time. The range() function will create lists of integers. In its simplest form an invocation like range(12) will create a list of 12 integers, starting at
zero and working up to, but not including, 12. Does this sound familiar?
Here are two other forms, that you should be able to understand by studying the examples.
There is a "Save" button in the upper-right corner of your worksheet. This will save a current copy of your worksheet that you can retrieve from within your notebook again later, though you have to
re-execute all the cells when you re-open the worksheet later.
There is also a "File" drop-down list, on the left, just above your very top compute cell (not be confused with your browser's File menu item!). You will see a choice here labeled "Save worksheet to
a file..." When you do this, you are creating a copy of your worksheet in the "sws" format (short for "Sage WorkSheet"). You can email this file, or post it on a website, for other Sage users and
they can use the "Upload"link on their main notebook page to incorporate a copy of your worksheet into their notebook.
There are other ways to share worksheets that you can experiment with, but this gives you one way to share any worksheet with anybody almost anywhere.
We have covered a lot here in this section, so come back later to pick up tidbits you might have missed. There are also many more features in the notebook that we have not covered. | {"url":"https://flashman.neocities.org/MD/diagrams/sage.GS.knowls","timestamp":"2024-11-06T02:17:25Z","content_type":"text/html","content_length":"12805","record_id":"<urn:uuid:223a1fbc-d901-44aa-ac10-adbe8d1488a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00557.warc.gz"} |
Re: [tlaplus] How to write my operator which can be used in TLA+
Thanks for your reply. I have solve the problem. With the help of Java, things become more convenient.
But unfortunately, I encountered some difficulties again, and I don’t know if there is a solution. Could you give me some advice?
As I mentioned before, I want to enumerate all possible event structure satisfying partial order.
But the raw method by enumerating all subsets and judging them is of a great time complexity.
Everything seemed to be going well, however, something unexpected happened.
What bad news ! I understand that this design makes a certain sense, but a question raises that as follows.
Suppose a set s = {1,2,3,4,5,6,7,8} and we calculate subset SUBSET (s \X s), what is the behavior of TLA+? Out of bound too? Or there is an iterator-like operation?
I know my question is a bit strange. But if you are interested, looking forward to your early relpy.
Hi Tsuna,
TLC looks primarily for this class (there is also another way to configure it) for operators overriding.
Let me know the result o/
In my work, I want to enumerate all possible event structure satisfying partial order.
However, if we write TLA+ like this:
PartialOrderSubset(s) ==
LET rels == SUBSET (s \X s)
IN {po \in rels : IsStrictPartialOrder(po, s)}
We should enumerate all subsets, it's time costing. So I'd like to define my operator by Java and override it. I forked this repository and writes my implementation. See
Update PartialOrderExt
And then, I built my CommunityModules-deps.jar and added it to TLC's or the Toolbox's TLA+ library path.
I think it will call my implementation in
instead of the original definition in
but it didn't work. Is there anything wrong of my settings?
You received this message because you are subscribed to the Google Groups "tlaplus" group.
To unsubscribe from this group and stop receiving emails from it, send an email to tlaplus+unsubscribe@xxxxxxxxxxxxxxxx.
To view this discussion on the web visit https://groups.google.com/d/msgid/tlaplus/cd111cbb-a3a4-46d8-b7a3-0b81f5c3d75bn%40googlegroups.com. | {"url":"https://discuss.tlapl.us/msg04370.html","timestamp":"2024-11-14T11:38:40Z","content_type":"text/html","content_length":"12839","record_id":"<urn:uuid:ed733e9e-0541-4223-b6d2-20f217e4fd59>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00036.warc.gz"} |
Guide to Solvers
OpenSolver supports a wide variety of solvers for use inside Excel, and several different ways in which the solver integrates with the Excel model. This page gives information about the solvers,
including its uses and limitations, to help you find the right solver for your problem.
Most of the solvers need to be given a copy of your model in a form they understand. To create this copy for linear models, OpenSolver uses an iterative process that requires one spreadsheet
re-calculation for each decision cell. This can become slow for large models.
Versions of OpenSolver released from 2015 also include a new experimental parser that directly translates the formulae in your spreadsheet into a form the solver understands. This is needed
non-linear models, and will, in the future, also be available for linear ones. We can only translate formulae that both we and the solvers understand, and so currently our parser will fail if your
model uses spreadsheet-specific formulae such as OFFSET(), INDIRECT(), INDEX() etc. However, if our parser works, then the solver can typically solve your problem quickly as it doesn’t need to
re-calculate the spreadsheet repeatedly.
Unlike all the other solvers, the NOMAD non-linear solver works directly with the spreadsheet. It tries out solutions by putting them into the spreadsheet and doing a spreadsheet recalculation. This
process is often slower and less efficient than the other non-linear solvers, but can be used to (try to) optimise even the most complicated spreadsheets, since it can work with any model regardless
of the formulae. There are some things that can slow NOMAD down, see below for more details.
Some of our solvers are linked to the NEOS Optimization Server, a cloud-based compute cluster that is free to use. OpenSolver can send your model to NEOS for solving, and then bring back the answer
when NEOS is finished. Note that any model submitted to NEOS becomes publicly visible.
More detailed information on each solver is given below.
Satalia SolveEngine
SolveEngine is a new commercial online service that we are trialling with OpenSolver; details are here.
License: Commercial
CBC (COIN-OR Branch-and-Cut) is an open-source linear and mixed-integer programming solver actively developed by COIN-OR. Questions about CBC (and bug reports) are best addressed using the CBC
Mailing List. OpenSolver lets you use all of the CBC command lines options.
License: EPL
Scheduled for OpenSolver 2.9.5 Beta.
HiGHS is an active open-source project, developing high performance solvers for linear optimisation, mixed-integer programming, and quadratic programming models. The solvers are written primarily in
C++. HiGHS is led by Julian Hall at highs.dev.
Important for Windows users:
• Some multithreading issues have been reported with the latest version of HiGHS (November 2023) when running on Windows. We have bypassed this for the time being by hardcoding the option “threads
= 1”.
• All options that are passed to this solver are written into an options file.
• The hard-coded “threads = 1” option can be overwritten by passing any other number of threads as an option to the solver as normal. If the options file contains duplicate options, it will use the
last option.
• If a different number of threads is specified, and OpenSolver stalls while solving, you need to close the highs.exe background process through Task Manager.
License: MIT
Scheduled for OpenSolver 2.9.5 Beta.
SCIP (Solving Constraint Integer Problems) is one of the fastest open-source solvers for mixed integer linear and nonlinear programming. SCIP can also provide total control over the solver settings
and solution information if the user chooses, but is built into OpenSolver for simplicity, robusticity, and speed.
License: Apache 2.0
The Gurobi Optimizer is a state-of-the-art commercial linear and mixed-integer programming solver from Gurobi Optimization Inc. It is one of the fastest solvers available for linear and integer
License: Commercial – A valid license is required to use Gurobi in OpenSolver (a free license is available for academic use). Once Gurobi is installed and activated, it will become available in
NOMAD (Nonlinear Optimization by Mesh Adaptive Direct Search) is an open source non-linear blackbox optimizer that is able to solve a variety of problem types, including general non-linear problems.
Important: There are some additional considerations to be aware of when using NOMAD:
• Performance is much poorer on models with equality constraints – it is better to use inequalities
• NOMAD works best for models with fewer variables, even if there are lots of complicated constraints. It is likely to be less effective on solving models with large numbers of variables.
• If possible try to set good both lower and upper bounds on the adjustable cells in the model so that NOMAD knows where to search for solutions.
License: GPLv3
Bonmin (Basic Open-source Nonlinear Mixed INteger programming) is an experimental open-source solver developed by COIN-OR that aims to solve mixed-integer non-linear problems, where the objective and
constraints are twice continuously differentiable functions. It uses the very successful IPOPT non-linear solver.
License: EPL
Couenne (Convex Over and Under ENvelopes for Nonlinear Estimation) is an experimental open-source solver from COIN-OR that seeks to solve mixed-integer non-linear problems with general (non-convex)
objective and constraint functions. It uses the very successful IPOPT non-linear solver.
License: EPL
The IBM ILOG CPLEX Optimizer (more commonly known simply as CPLEX) is a high-performance mathematical programming solver for linear programming, mixed-integer programming and quadratic programming.
IPOPT (via Bonmin/Couenne)
IPOPT is a very successful non-linear solver. OpenSolver makes IPOPT available via the Couenne and Bonmin solvers, both of which have IPOPT at their core. To use IPOPT for your model, we recommend
choosing Bonmin, but not specifying any integer/binary requirements.
Solver Summary
The following table summarizes the characteristics of the solvers:
Solver Linear Non-Linear Sensitivity Analysis Uses Iteration Uses Parsing Uses NEOS Advanced Only
CBC ✓ ✓ ✓
CBC using NEOS ✓ ✓ ✓
Gurobi ✓ ✓ ✓
NOMAD ✓ ✓ ✓
Bonmin ✓ ✓ ✓ ✓
Bonmin using NEOS ✓ ✓ ✓ ✓
Couenne ✓ ✓ ✓ ✓
Couenne using NEOS ✓ ✓ ✓ ✓
Satalia SolveEngine ✓ ✓
CPLEX using NEOS ✓ ✓ ✓
Guide to the columns:
• Linear/Non-Linear: Linear solvers can only be used on problems where the adjustable cells appear linearly in the problem. If this is not the case, the linear solvers are very likely to return
meaningless results. If a linear solver is used, there is the option to run a “Linearity Check” after the solve, which tries to make sure the problem was indeed linear. Note that this is not
guaranteed to identify all non-linear problems.
• Sensitivity Analysis: Solvers that support sensitivity analysis can produce a Sensitivity Report (similar to the Excel Solver) detailing the shadow prices and reduced costs of the constraints and
variables respectively.
• Uses Iteration: These solvers build the model by changing the variable cells one-by-one, building the model iteratively. This requires no knowledge of the formulae in the model, but does require
that the model is linear.
• Uses Parsing: These solvers build the model by reading the formulae in the spreadsheet. They can understand a lot of Excel formulae (linear and non-linear), but do not support some functions,
most notably functions like OFFSET, INDEX, MATCH etc.
• Uses NEOS: These solvers do not run on your machine, instead the model is sent to the NEOS Optimization Server which solve the model and send the results back to Excel. This can result in faster
solve times for the model depending on the speed of your computer, but there can be a small delay when solving on NEOS depending on the current load on the server. Once sent to NEOS the model
becomes publicly available.
• Advanced Only: These solvers are only included in the “Advanced” version of OpenSolver (which is still free and open source!). These solvers tend to be slightly more experimental than the others.
Parsing List
The following table contains some formulae the solvers that use parsing can understand:
Formula Description Notes
SUM Adds all arguments together.
PRODUCT Multiplies all arguments together.
SUMPRODUCT Multiplies corresponding components in the given arrays, and returns
the sum of those products.
SUMIF Adds values in a range if the condition is met. Not supported by NEOS solvers. Limited support when the condition relies on decision variables as we attempt to
evaluate the condition once before solving.
MIN Finds the minimum number from its arguments. Not supported by Couenne.
MAX Finds the maximum number from its arguments. Not supported by Couenne. | {"url":"https://opensolver.org/guide-to-solvers/","timestamp":"2024-11-10T08:34:30Z","content_type":"text/html","content_length":"53012","record_id":"<urn:uuid:37485c17-9f13-4075-bb26-0fbe991f4fd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00376.warc.gz"} |
Tutorial 3.1: The Perceptron
1 Tutorial 3.1: The Perceptron
Reading: Stephen Marsland: Chapter 1-3
In this tutorial we shall implement our first learning algorithm, namely a single neuron. The celebrate artificial neural networks (ANN) are built up of numerous neurons, so this tutorial is the
first step.
1.1 Problem 1: Single Neuron and Perceptron Training
In this Problem we will implement only a single neuron with the perceptron training algorithm. The neuron is depicted in Figure 1. It acts as a function, with input $\stackrel{\to }{x}=\left({x}_{1},
\dots ,{x}_{n}\right)$ on the left, and output $y$ on the right. The weights $\stackrel{\to }{w}=\left({w}_{0},\dots ,{w}_{n}\right)$ defines a particular neuron. Different neurons (as far as we are
concerned) use the same summing and thresholding functions, but they have different weights.
1.1.1 Step 1: Data Types
At the conceptual (mathematical) level, the neuron receives a real vector as input. The output is 0.0 or 1.0, which we also consider as a real number.
Discuss the following:
What data type should be used in Haskell for the output $y$?
What data type should be used in Haskell for the input $\stackrel{\to }{x}$?
What data type should be used in Haskell for the weights $\stackrel{\to }{w}$?
1.1.2 Step 2: The Neuron Type
Let us define a data type (alias) Neuron to represent a single neuron, recording all the weights.
Discuss: What information must be stored as part of the neuron?
Discuss: What types can we use to define the Neuron type?
Create a new module with the name Perceptron.
Add a definition for the Neuron type to the module.
1.1.3 Step 3: Initialisation
We need a function to create a new, pristine neuron. In a production system, this should be done randomly, but randomness is non-trivial, so we have to return to that later. For the time being, we
are happy to initialise the neuron with small constant weights (say 0.01).
Give the type declaration for the initialisation function in your module: initNeuron :: Int −> Neuron The input is the number $n$ of input values. The number of weights is $n+1$.
Add a definition for initNeuron. The return value is a list of $n+1$ numbers, each equal to 0.01.
You can start with the list [0..n] to get the right number of weights, and then use either map or list comprehension to generate a list of the same length and the right values (0.01).
Test the function in ghci. Does the function give you what you expect?
1.1.4 Step 4: Recall
The neuron as depicted in Figure 1 defines a function called recall. In Haskell it would have the following signature.
recall :: Neuron −> [Double] −> Double The function takes the neuron and the input list, and it produces a scalar output.
Looking at Figure 1, we see that recall is the composition of two functions: the summation (circle) and the thresholding (square). In Haskell, this can be written as follows: recall :: Neuron −>
[Double] −> Double
recall n = threshold . neuronSum n
It remains to implement the the threshold and neuronSum.
The threshold function is defined as $\begin{array}{lll}\hfill threshold\phantom{\rule{3.33234pt}{0ex}}x=\begin{array}{r}0\phantom{\rule{1em}{0ex}}\text{for }x<0,\\ 1\phantom{\rule{1em}{0ex}}\
text{for }x\ge 0.\end{array}& \phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\end{array}$
Implement this function in your module using guards. Use the following type declaration.
threshold :: Double −> Double
Secondly, we implement the summation (the circle node in Figure 1). Add the following to your module. neuronSum :: Neuron −> [Double] −> Double
neuronSum n is = sum $ zipWith (*) n ((−1):is)
Discuss how this function works.
What does (-1):is mean?
What does the zipWith function do?
What does the sum function do?
Test the function. Start ghci and try the following recall (initNeuron 3) [ 1.0, 0.5, −1.0 ] Do you get the expected output?
Obviously, you do not learn all that you want to know from the above test, but at least you get to check for type errors. Develop your own test, by manually defining a test neuron with other
weights, and use that in lieu of initNeuron.
1.1.5 Step 5: Training
The first step of implementing training is to update the neuron weights based on a single input/output pair. That is a function
trainOne :: Double −> [Double] −> Double
−> Neuron −> Neuron The first argument is the training factor $\eta$. The second is the input vector $\stackrel{\to }{x}$, and the third argument is the target output value $t$.
The last (fourth) argument is the old neuron $\stackrel{\to }{w}$. The output is the updated neuron ${\stackrel{\to }{w}}^{\prime }$.
The updated weights are defined as
$\begin{array}{lll}\hfill {w}_{i}^{\prime }& ={w}_{i}-\eta \left(y-t\right){x}_{i},\text{ where}\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill y& =recall\phantom{\rule{3.33234pt}{0ex}}\phantom
{\rule{3.33234pt}{0ex}}\stackrel{\to }{w}\stackrel{\to }{x}.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\end{array}$
In other words, if the actual output is different from the target output ($ye t$), then the weight is adjusted proportionally to the difference ($y-t$).
Implemente the weight update as defined above. We need a function with the following signature weightUpdate :: Double −> Double −> Double
−> Double −> Double
weightUpdate eta diff w x = We have introduced diff for the different $y-t$, the other arguments are $\eta$, $w$ and $x$ as in (2). Complete the implementation and add it to your module.
We implement the trainOne as follows: trainOne :: Double −> [Double] −> Double
−> Neuron −> Neuron
trainOne eta xs t ws = zipWith (weightUpdate eta diff)
ws ((−1):xs)
where diff = recall ws xs − t
This implementation uses zipWith and partial application of weightUpdate. Discuss the following:
What does zipWith do?
What do we mean by partial application?
What is the type of the first argument to zipWith, i.e. weightUpdate eta diff?
1.1.6 Step 6: Training on a Set of Vectors
The trainOne function uses only a single object for training. Now we need a trainSet function which uses a set of objects for training. This is a beautiful application of recursion over the list of
training objects.
The function declaration is similar to that of trainOne, except that we get a lists instead of a single input vector and a single output value. Add it to your module as follows: trainSet ::
Double −> [[Double]] −> [Double]
−> Neuron −> Neuron
Then, add the recursive case: trainSet eta (v:vs) (t:ts) n = trainSet eta vs ts
$ trainOne eta v t n
Discuss the following
What does the notation (v:vs) (and (t:ts)) mean?
What is the last argument to trainSet? What is its type? How is it computed?
How does the recursion work?
Test the function as you did with trainOne, but replace the input vector with a list of two input vectors (of you choice), each of length 3.
1.1.7 Step 7: Complete Training Function
It is usually not sufficient to run the training once only. Usually, we want to repeat the trainSet operation $T$ times. In other words, we want a function with the following signature:
train :: Int −> Double −> [[Double]] −> [Double]
−> Neuron −> Neuron The first argument is the number $T$ of iterations, while the other argumens are as they were for trainSet.
Add the function declaration to your module.
Add a base case, defining the return value for $T=0$ iterations.
Add a recursive case which uses trainSet to do a single iteration and calls train recursively to do $T-1$ more iterations.
You may look at the definition of trainSet above for an example of recursion, but remember that train recurses over an integer (the number of iterations) while trainSet recursed over a list.
Test the function using the same test data as you used for trainSet. Try both $T=2$ and $T=5$. replace the input vector with a list of two input vectors (of you choice), each of length 3.
1.1.8 Step 8: Testing
A simple test to device is to take a simple function and try to predict whether it is positive or negative. Take for instance the following:
Choose a couple of feature vectors $\left(x,y,z\right)$ (randomly or otherwise), and calculate the corresponding class label $f\left(x,y,z\right)$. This gives a training set.
Use the training set to train a neuron $n$ in GHCi.
Choose another feature vector $\left(x,y,z\right)$ and calculate $f\left(x,y,z\right)$. Use GHCi to calculate $recall\phantom{\rule{3.33234pt}{0ex}}\left(x,y,z\right)n$.
Compare the real class label $f\left(x,y,z\right)$ with the prediction obtained in GHCi. Do they match?
Repeat the last two steps a couple of times.
1.2 Problem 2: Multi-Neuron Perceptrons
1.2.1 Step 1: Data Type
Define a data type Layer to represent a multi-neuron perceptron.
What data type can be used to hold a set of neurons?
The name ‘layer’ will make sense when we advance to more complex neural networks. The perceptron consists of a single layer only, but other neural networks will be lists of layers.
1.2.2 Step 2: Initialisation
Define a function initLayer to return a perceptron (layer) where all weights in all neurons is set to some small, constant, non-zero value.
Remember arguments so that the user can choose both the number of neurons in the layer and the number of inputs. Each neuron in the layer should be created by a call to initNeuron which you defined
1.2.3 Step 3: Recall
Define a function recallLayer which does a recall for each neuron in the layer, and returns a list of output values.
1.2.4 Step 4: Training
Generalise each of the training functions trainOne, trainSet, and train for perceptrons. The training functions for perceptrons have to apply the corresponding training function for each neuron in
the layer.
1.3 Epilogue
You have just implemented your first classifier. Well done.
However, this prototype leaves much to be desired.
We cannot initialise with random weights.
We have to type in the data for training and for testing.
We only have a single layer, and not a full network.
As you can see, we have to go back and learn some more techniques in Haskell. First of all, we will learn I/O in the next tutorial, to be able to read complete data sets from file, both for training
and for testing.
7th April 2017 | {"url":"http://kerckhoffs.schaathun.net/FPIA/week03se1.html","timestamp":"2024-11-02T13:50:38Z","content_type":"application/xhtml+xml","content_length":"43531","record_id":"<urn:uuid:c731d6d8-15d6-4094-a2d4-60c5990977c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00434.warc.gz"} |
Eureka Math Grade 6 Module 6 Lesson 8 Answer Key
Engage NY Eureka Math Grade 6 Module 6 Lesson 8 Answer Key
Eureka Math Grade 6 Module 6 Lesson 8 Example Answer Key
Example 1: Comparing Two Data Distributions
Robert’s family is planning to move to either New York City or San Francisco. Robert has a cousin in San Francisco and asked her how she likes living in a climate as warm as San Francisco. She
replied that it doesn’t get very warm in San Francisco. He was surprised by her answer. Because temperature was one of the criteria he was going to use to form his opinion about where to move, he
decided to investigate the temperature distributions for New York City and San Francisco. The table below gives average temperatures (in degrees Fahrenheit) for each month for the two cities.
Exercises 1 – 2:
Use the data in the table provided in Example 1 to answer the following:
Exercise 1.
Calculate the mean of the monthly average temperatures for each city.
The mean of the monthly temperatures for New York City is 63 degrees.
The mean of the monthly temperatures for San Francisco is 64 degrees.
Exercise 2.
Recall that Robert is trying to decide where he wants to move. What is your advice to him based on comparing the means of the monthly temperatures of the two cities?
Since the means are almost the some, it looks like Robert could move to either city. Even though the question asks students to focus on the means, they might make a recommendation that takes
variability into account.
For example, they might note that even though the means for the two cities are about the same, there are some much lower and much higher monthly temperatures for New York City and use this as a basis
to suggest that Robert move to San Francisco.
Example 2: Understanding Variability
Maybe Robert should look at how spread out the New York City monthly temperature data are from the mean of the New York City monthly temperatures and how spread out the San Francisco monthly
temperature data are from the mean of the San Francisco monthly temperatures. To compare the variability of monthly temperatures between the two cities, it may be helpful to look at dot plots. The
dot plots of the monthly temperature distributions for New York City and San Francisco follow.
Dot Plot of Temperature for New York City
Dot Plot of Temperature for San Francisco
Exercises 3 – 7:
Use the dot plots above to answer the following:
Exercise 3.
Mark the location of the mean on each distribution with the balancing A symbol. How do the two distributions compare based on their means?
Place ∆ at 63 for New York City and at 64 for Son Francisco. The means are about the same.
Exercise 4.
Describe the variability of the New York City monthly temperatures from the New York City mean.
The temperatures are spread out around the mean. The temperatures range from a low of around 39 °F to a high of 85 °F.
Exercise 5.
Describe the variability of the San Francisco monthly temperatures from the San Francisco mean.
The temperatures aæ clustered around the mean. The temperatures range from a low of 57 °F to a high of 70 °F.
Exercise 6.
Compare the variability in the two distributions. Is the variability about the same, or is it different? If different, which monthly temperature distribution has more variability? Explain.
The variability is different. The variability in New York City is much greater than the variability in San Francisco.
Exercise 7.
If Robert prefers to choose the city where the temperatures vary the least from month to month, which city should he choose? Explain.
He should choose San Francisco because the temperatures vary the least, from a low of 57 °F to a high of 70 °F. New York City has temperatures with more variability, from a low of 39°F to o high of
Example 3: Considering the Mean and Variability in a Data Distribution
The mean is used to describe a typical value for the entire data distribution. Sabina asks Robert which city he thinks has the better climate. How do you think Robert responds?
He responds that they both have about the same mean but that the mean is a better measure or a more precise measure of a typical monthly temperature for San Francisco than it is for New York City.
Sabina is confused and asks him to explain what he means by this statement. How could Robert explain what he means?
The temperatures in New York City in the winter months are in the 40’s and in the summer months are in the 80’s. The mean of 63 isn’t very close to those temperatures. Therefore, the mean is not a
good indicator of a typical monthly temperature. The mean is a much better indicator of a typical monthly temperature in SanFrancisco because the variability of the temperatures there is much
Exercises 8 – 14:
Consider the following two distributions of times it takes six students to get to school in the morning and to go home from school in the afternoon.
Exercise 8.
To visualize the means and variability, draw a dot plot for each of the two distributions.
Exercise 9.
What is the mean time to get from home to school in the morning for these six students?
The mean is 14 minutes.
Exercise 10.
What is the mean time to get from school to home in the afternoon for these six students?
The mean is 14 minutes.
Exercise 11.
For which distribution does the mean give a more accurate indicator of a typical time? Explain your answer.
The morning mean is a more accurate indicator. The spread in the afternoon data is far greater than the spread in the morning data.
Distributions can be ordered according to how much the data values vary around their means. Consider the following data on the number of green jelly beans in seven bags of jelly beans from each of
five different candy manufacturers (AllGood, Best, Delight, Sweet, and Vum). The mean in each distribution is 42 green jelly beans.
Exercise 12.
Draw a dot plot of the distribution of the number of green jelly beans for each of the five candy makers. Mark the location of the mean on each distribution with the balancing A symbol.
The dot plots should each hove a balancing A symbol located at 42.
Exercise 13.
Order the candy manufacturers from the one you think has the least variability to the one with the most variability. Explain your reasoning for choosing the order.
Note: Do not be critical; answers and explanations may vary. One possible answer:
In order from least to greatest: All Good, Sweet, Vum, Delight, Best. The data points are all close to the mean for all good, which indicates it has the least variability, followed by Sweet and Yum.
The data points are spread farther from the mean for Delight and Best, which indicates they have the greatest variability.
Exercise 14.
For which company would the mean be considered a better indicator of a typical value (based on least variability)?
The mean for All Good would be the best indicator of a typical value for the distribution.
Eureka Math Grade 6 Module 6 Lesson 8 Problem Set Answer Key
Question 1.
The number of pockets in the clothes worn by seven students to school yesterday was 4, 1, 3, 4, 2, 2, 5. Today, those seven students each had three pockets in their clothes.
a. Draw one dot plot of the number of pockets data for what students wore yesterday and another dot plot for what students wore today. Be sure to use the same scale.
b. For each distribution, find the mean number of pockets worn by the seven students. Show the means on the dot plots by using the balancing symbol.
The mean of both dot plots is 3.
c. For which distribution is the mean number of pockets a better indicator of what is typical? Explain.
There is certainly variability in the data for yesterday’s distribution, whereas today’s distribution has none. The mean of 3 pockets is a better indicator (more precise) for today’s distribution.
Question 2.
The number of minutes (rounded to the nearest minute) it took to run a certain route was recorded for each of five students. The resulting data were 9, 10, 11, 14, and 16 minutes. The number of
minutes (rounded to the nearest minute) it took the five students to run a different route was also recorded, resulting in the following data: 6, 8, 12, 15, and 19 minutes.
a. Draw dot plots for the distributions of the times for the two routes. Be sure to use the same scale on both dot plots.
First Route
Second Route
b. Do the distributions have the same mean? What is the mean of each dot plot?
Yes, Both distributions have the same mean, 12 minutes.
c. In which distribution is the mean a better indicator of the typical amount of time taken to run the route? Explain.
Looking at the dot plots, the times for the second route are more varied than those for the first route. So, the mean for the first route is a better indicator (more precise) of a typical value.
Question 3.
The following table shows the prices per gallon of gasoline (in cents) at five stations across town as recorded on
Monday, Wednesday, and Friday of a certain week.
┃Day │R&C│Al’s│PB │Sam’s│Ann’s┃
┃Monday │359│358 │362│359 │362 ┃
┃Wednesday │357│365 │364│354 │360 ┃
┃Friday │350│350 │360│370 │370 ┃
a. The mean price per day for the five stations is the same for each of the three days. Without doing any calculations and simply looking at Friday’s prices, what must the mean price be?
Friday’s prices ore centered at 360 cents. The sum of the distances from 360 for values above 360 is equal to
the sum of the distances from 360 for values below 360, so the mean is 360 cents.
b. For which daily distribution is the mean a better indicator of the typical price per gallon for the five stations? Explain.
From the dot plots, the mean for Monday is the best indicator of o typical price because there is the least variability in the Monday prices.
Eureka Math Grade 6 Module 6 Lesson 8 Exit Ticket Answer Key
Question 1.
Consider the following statement: Two sets of data with the same mean will also have the same variability. Do you agree or disagree with this statement? Explain.
Answers will vary, but students should disagree with this statement. There are many examples in this lesson that could be used as the basis for an explanation.
Question 2.
Suppose the dot plot on the left shows the number of goals a boys’ soccer team has scored in 6 games so far this
season and the dot plot on the right shows the number of goals a girls’ soccer team has scored in 6 games so far this season.
a. Compute the mean number of goals for each distribution.
The mean for each is 3 goals.
b. For which distribution, if either, would the mean be considered a better indicator of a typical value? Explain your answer.
Variability in the distribution for girls is less than ¡n the distribution for boys, so the mean of 3 goals for the girls is a better indicator of a typical value.
Leave a Comment
You must be logged in to post a comment. | {"url":"https://ccssmathanswers.com/eureka-math-grade-6-module-6-lesson-8/","timestamp":"2024-11-13T01:40:22Z","content_type":"text/html","content_length":"273040","record_id":"<urn:uuid:42dc9fb2-9ef8-40ad-8f78-989c5114e30c>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00637.warc.gz"} |
mne_connectivity.phase_slope_index(data, names=None, indices=None, sfreq=6.283185307179586, mode='multitaper', fmin=None, fmax=inf, tmin=None, tmax=None, mt_bandwidth=None, mt_adaptive=False,
mt_low_bias=True, cwt_freqs=None, cwt_n_cycles=7, block_size=1000, n_jobs=1, verbose=None)[source]#
Compute the Phase Slope Index (PSI) connectivity measure.
The PSI is an effective connectivity measure, i.e., a measure which can give an indication of the direction of the information flow (causality). For two time series, and one computes the PSI
between the first and the second time series as follows
indices = (np.array([0]), np.array([1])) psi = phase_slope_index(data, indices=indices, …)
A positive value means that time series 0 is ahead of time series 1 and a negative value means the opposite.
The PSI is computed from the coherency (see spectral_connectivity_epochs), details can be found in [1].
Can also be a list/generator of array, shape =(n_signals, n_times); list/generator of SourceEstimate; or Epochs. The data from which to compute connectivity. Note that it is also possible
to combine multiple signals by providing a list of tuples, e.g., data = [(arr_0, stc_0), (arr_1, stc_1), (arr_2, stc_2)], corresponds to 3 epochs, and arr_* could be an array with the
same number of time points as stc_*.
The names of the nodes of the dataset used to compute connectivity. If ‘None’ (default), then names will be a list of integers from 0 to n_nodes. If a list of names, then it must be equal
in length to n_nodes.
Two arrays with indices of connections for which to compute connectivity. If None, all connections are computed.
The sampling frequency.
Spectrum estimation mode can be either: ‘multitaper’, ‘fourier’, or ‘cwt_morlet’.
The lower frequency of interest. Multiple bands are defined using a tuple, e.g., (8., 20.) for two bands with 8Hz and 20Hz lower freq. If None the frequency corresponding to an epoch
length of 5 cycles is used.
The upper frequency of interest. Multiple bands are dedined using a tuple, e.g. (13., 30.) for two band with 13Hz and 30Hz upper freq.
Time to start connectivity estimation.
Time to end connectivity estimation.
The bandwidth of the multitaper windowing function in Hz. Only used in ‘multitaper’ mode.
Use adaptive weights to combine the tapered spectra into PSD. Only used in ‘multitaper’ mode.
Only use tapers with more than 90 percent spectral concentration within bandwidth. Only used in ‘multitaper’ mode.
Array of frequencies of interest. Only used in ‘cwt_morlet’ mode.
Number of cycles. Fixed number or one per frequency. Only used in ‘cwt_morlet’ mode.
How many connections to compute at once (higher numbers are faster but require more memory).
How many epochs to process in parallel.
If not None, override default verbose level (see mne.verbose() for more info). If used, it should be passed as a keyword-argument only.
conninstance of Connectivity
Computed connectivity measure(s). Either a SpectralConnnectivity, or SpectroTemporalConnectivity container. The shape of each array is either (n_signals ** 2, n_bands) mode: ‘multitaper’
or ‘fourier’ (n_signals ** 2, n_bands, n_times) mode: ‘cwt_morlet’ when “indices” is None, or (n_con, n_bands) mode: ‘multitaper’ or ‘fourier’ (n_con, n_bands, n_times) mode: ‘cwt_morlet’
when “indices” is specified and “n_con = len(indices[0])”.
Examples using mne_connectivity.phase_slope_index# | {"url":"https://mne.tools/mne-connectivity/stable/generated/mne_connectivity.phase_slope_index.html","timestamp":"2024-11-11T04:49:49Z","content_type":"text/html","content_length":"42550","record_id":"<urn:uuid:c8c515f1-68f7-4564-9aef-a616498565d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00152.warc.gz"} |
Education and career
As a high school student, Sklar studied poetry at the Interlochen Arts Academy. She did her undergraduate studies at Swarthmore College, where her mother Elizabeth S. had earned a degree in English
(later becoming an English professor at Wayne State University) and her father Lawrence Sklar had taught philosophy. Jessica completed a double major in English and mathematics in 1995.^[2]^[3]
Next, Sklar moved to the University of Oregon for graduate study in mathematics, earning a master's degree in 1997 and completing her Ph.D. there in 2001.^[4] Her dissertation, Binomial Rings and
Algebras, was supervised by Frank Wylie Anderson.^[5]
She has been a faculty member in the mathematics department at Pacific Lutheran since 2001.^[2]
Combining her interests in mathematics and art she is one of 24 mathematicians and artists who make up the Mathemalchemy Team.^[6]^[7]
Selected publications
• “‘Bok bok’: exploring the game of Chicken in film,” with Jennifer F. Nordstrom. In: Handbook of the Mathematics of the Arts and Sciences. Ed. Bharath Sriraman. Springer International Publishing,
Cham, 2020.^[8]
• “‘Elegance in design’: mathematics and the works of Ted Chiang.” In: Handbook of the Mathematics of the Arts and Sciences. Ed. Bharath Sriraman. Springer International Publishing, Cham, 2020.^[8]
• “Disciple” (poem). Journal of Humanistic Mathematics 7(2) (July 2017), 418.
• First-Semester Abstract Algebra: A Structural Approach. Archived 2018-11-14 at the Wayback Machine GNU Free Documentation License, 2017.
• “A confused electrician uses Smith normal form,” with Tom Edgar. Mathematics Magazine 89(1) (2016), 3–13.
• Mathematics in Popular Culture: Essays on Appearances in Film, Literature, Games, Television and Other Media.^[9] Jefferson, NC: McFarland & Co., 2012. Editor, with Elizabeth S. Sklar.
• “The graph menagerie: abstract algebra and the Mad Veterinarian,” with G. Abrams. Mathematics Magazine 83(3) (2010), 168–179.
• “Dials and levers and glyphs, oh my! Linear algebra solutions to computer game puzzles.”Mathematics Magazine 79(5) (2006), 360–367.
• "Binomial rings.” Communications in Algebra 32(4) (2004), 1385–1399.
• “Binomial algebras.” Communications in Algebra 30(4) (2002), 1961–1978.
Sklar was a winner of the Carl B. Allendoerfer Award of the Mathematical Association of America in 2011 for her paper with Gene Abrams, The Graph Menagerie: Abstract Algebra and the Mad Veterinarian.
^[10] The paper provides a general solution to a class of lattice reduction puzzles exemplified by the following one:^[3]
"Suppose a mad veterinarian creates a transmogrifier that can convert one cat into two dogs and five mice, or one dog into three cats and three mice, or a mouse into a cat and a dog. It can also
do each of these operations in reverse. Can it, through any sequence of operations, convert two cats into a pack of dogs? How about one cat?"
She was the July 2012 Author of the Month at Ada's Technical Books in Seattle, Washgington.
1. ^ Birth year from Library of Congress catalog entry, retrieved 2018-12-02.
2. ^ ^a ^b ^c Curriculum vitae (PDF), archived from the original (PDF) on 2018-11-14, retrieved 2020-08-02
3. ^ ^a ^b Mackenzie, Dana (January 2013), "1 Plus 1 Makes Engaging Book: Mother and Daughter Bridge Generations and Disciplines", Swarthmore College Bulletin
4. ^ "Jessica Sklar", Mathematics Faculty and Staff, Pacific Lutheran University, retrieved 2020-05-03
5. ^ Jessica Sklar at the Mathematics Genealogy Project
6. ^ Mathemalchemy’s Team
7. ^ Shinn, Lora (8 January 2021), "1 By the Numbers: PLU Professor Collaborates on a New Artwork Illuminating the Beauty of Math", PLU Marketing and Communications
8. ^ ^a ^b Sriraman, Bharath, ed. (2020). Handbook of the Mathematics of the Arts and Sciences. Springerlink. doi:10.1007/978-3-319-70658-0. ISBN 978-3-319-70658-0. S2CID 241723585.
9. ^ Reviews of Mathematics in Popular Culture:
□ Ashbacher, Charles (June 2012), "Review", MAA Reviews
□ Sterling, Chris (2012), Communication Booknotes Quarterly, 43 (3): 140, doi:10.1080/10948007.2012.700870, S2CID 218575824{{citation}}: CS1 maint: untitled periodical (link)
□ Johnson, J. (September 2012), Choice Reviews: 201{{citation}}: CS1 maint: untitled periodical (link)
□ Karaali, Gizem (November–December 2013), "Review", AWM Newsletter, 43 (6): 22–25
□ Kozek, Mark (March 2014), The American Mathematical Monthly, 121 (3): 274–278, doi:10.4169/amer.math.monthly.121.03.274, S2CID 218547798{{citation}}: CS1 maint: untitled periodical (link)
□ Campbell, Paul J. (December 2014), Mathematics Magazine, 87 (5): 404–405, doi:10.4169/math.mag.87.5.404, S2CID 218543117{{citation}}: CS1 maint: untitled periodical (link)
10. ^ "MAA Awards Presented" (PDF), Mathematics People, Notices of the American Mathematical Society, 58 (10): 1464, November 2011
External links | {"url":"https://www.knowpia.com/knowpedia/Jessica_Sklar","timestamp":"2024-11-08T07:50:08Z","content_type":"text/html","content_length":"89805","record_id":"<urn:uuid:f1076d19-b977-4dd2-9793-8d34d26eb939>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00709.warc.gz"} |
, typename
, typename
class CGAL::Segment_tree_d< Data, Window, Traits >
A \( d\)-dimensional segment tree stores \( d\)-dimensional intervals and can be used to find all intervals that enclose, partially overlap, or contain a query interval, which may be a point.
A \( d\)-dimensional segment tree is constructed in \(O(n\log n^d)\) time. An inverse range query is performed in time \(O(k+{\log}^d n )\), where \( k\) is the number of reported intervals. The tree
uses \(O(n\log n^d)\) storage.
bool make_tree (In_it first, In_it last)
The tree is constructed according to the data items in the sequence between the element pointed by iterator first and iterator last.
OutputIterator window_query (Window win, OutputIterator result)
win \( =[a_1,b_1),\ldots, [a_d,b_d)\), \( a_i,b_i\in T_i\), \( 1\le i\le d\).
OutputIterator enclosing_query (Window win, OutputIterator result)
All elements that enclose the associated \( d\)-dimensional interval of win are placed in the associated sequence container of OutputIterator and returns an output iterator that
points to the last location the function wrote to.
bool is_valid ()
The tree structure is checked.
bool is_inside (Window win, Data object)
returns true, if the interval of object is contained in the interval of win, false otherwise.
bool is_anchor ()
returns false.
bool CGAL::Segment_tree_d< Data, Window, Traits >::is_valid ( )
The tree structure is checked.
For each vertex either the sublayer tree is a tree anchor, or it stores a (possibly empty) list of data items. In the first case, the sublayer tree of the vertex is checked on being valid. In the
second case, each data item is checked whether it contains the associated interval of the vertex and does not contain the associated interval of the parent vertex or not. true is returned if the tree
structure is valid, false otherwise.
Segment_tree_d< Data, Window, Traits > CGAL::Segment_tree_d< Data, Window, Traits >::s ( Tree_base< Data, Window > sublayer_tree )
A segment tree is defined, such that the subtree of each vertex is of the same type prototype sublayer_tree is.
We assume that the dimension of the tree is \( d\). This means, that sublayer_tree is a prototype of a \( d-1\)-dimensional tree. All data items of the \( d\)-dimensional segment tree have container
type Data. The query window of the tree has container type Window. Traits provides access to the corresponding data slots of container Data and Window for the \( d\)-th dimension. The traits class
Traits must at least provide all functions and type definitions described, for example, in the reference page for tree_point_traits. The template class described there is fully generic and should
fulfill the most requirements one can have. In order to generate a one-dimensional segment tree instantiate Tree_anchor<Data, Window> sublayer_tree with the same template parameters Data and Window
Segment_tree_d is defined. In order to construct a two-dimensional segment tree, create Segment_tree_d with a one-dimensional Segment_tree_d with the corresponding Traits of the first dimension.
Traits::Data==Data and Traits::Window==Window.
OutputIterator CGAL::Segment_tree_d< Data, Window, Traits >::window_query ( Window win,
OutputIterator result
win \( =[a_1,b_1),\ldots, [a_d,b_d)\), \( a_i,b_i\in T_i\), \( 1\le i\le d\).
All elements that intersect the associated \( d\)-dimensional interval of win are placed in the associated sequence container of OutputIterator and returns an output iterator that points to the last
location the function wrote to. In order to perform an inverse range query, a range query of \( \epsilon\) width has to be performed. | {"url":"https://doc.cgal.org/latest/SearchStructures/classCGAL_1_1Segment__tree__d.html","timestamp":"2024-11-04T08:09:37Z","content_type":"application/xhtml+xml","content_length":"27008","record_id":"<urn:uuid:da366022-1236-42ac-ba34-de7f85bc7370>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00494.warc.gz"} |
Kolmogorov was one of the broadest of this century's mathematicians. He laid the mathematical foundations of probability theory and the algorithmic theory of randomness and made crucial contributions
to the foundations of statistical mechanics, stochastic processes, information theory, fluid mechanics, and nonlinear dynamics . All of these areas, and their interrelationships, underlie complex
systems, as they are studied today.
Kolmogorov graduated from Moscow State University in 1925 and then became a professor there in 1931. In 1939 he was elected to the Soviet Academy of Sciences, receiving the Lenin
His work on reformulating probability started with a 1933 paper in which he built up probability theory in a rigorous way from fundamental axioms, similar to Euclid's treatment of geometry.
Kolmogorov went on to study the motion of the planets and turbulent fluid flows, later publishing two papers in 1941 on turbulence that even today are of fundamental importance.
In 1954 he developed his work on dynamical systems in relation to planetary motion, thus demonstrating the vital role of probability theory in physics and re-opening the study of apparent randomness
in deterministic systems, much along the lines originally conceived by Henri Poincare .
In 1965 he introduced the algorithmic theory of randomness via a measure of complexity, now referred to Kolmogorov Complexity . According to Kolmogorov, the complexity of an object is the length of
the shortest computer program that can reproduce the object. Random objects, in his view, were their own shortest description. Whereas, periodic sequences have low Kolmogorov complexity, given by the
length of the smallest repeating "template" sequence they contain. Kolmogorov's notion of complexity is a measure of randomness, one that is closely related to Claude Shannon 's entropy rate of an
information source.
Kolmogorov had many interests outside mathematics research, notable examples being the quantitative analysis of structure in the poetry of the Russian author Pushkin, studies of agrarian development
in 16th and 17th century Novgorod, and mathematics education. | {"url":"https://annex.exploratorium.edu/complexity/CompLexicon/kolmogorov.html","timestamp":"2024-11-08T17:26:55Z","content_type":"text/html","content_length":"5060","record_id":"<urn:uuid:eac70b57-0833-4cb1-91c2-ddbc68c7aec5>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00735.warc.gz"} |
Eat, Sleep, Autocross! A blog about all things cars and autocrossing.
One of things I like to do in my spare time is write. I am generally not that great at writing. There are plenty of more talented individuals than me out there, but what I do enjoy is sharing my
experiences and thoughts through writing.
As this week leads up to the SCCA Pro Solo Finale and Solo Nationals, I thought this space would be a great place for a “Ask Me Anything About Autocross.” I am going to give this Q&A format a try. I
would love to field questions from anyone, especially non-racing or non-car people. So… ask away!
Send your questions to me on FB, Instagram, or whatever!
Why are you a shit driver?
K: I don’t push the right pedal all the way down, and I use the middle pedal a way too much.
What is your favorite auto-x element? Least favorite? What do you want to see more/less of at an autox?
K: Most favorite to date would be the “fake walloms” as I would call it. It’s essentially a wallom where only the last cone is relevant. The cones in front don’t do anything at all. Visually
deceptive, but lots of fun. Least favorite element would be anything that requires me to slow down to 20 mph to get through. There isn’t any one element I would like to see more or less of. Maybe
more creative/unique elements.
Is it more fun/stressful/expensive to auto-x with your daily driver or a car you keep just for racing?
K: Fun – Dedicated race car. The more prepared the car, the more fun it is to drive. Stressful – This kind of depends. More prepared car requires more work to get to get ready. A street car is
essentially arrive and drive. Expensive – Definitely the more prepared car. Race tires are not cheap.
Random Thoughts: Playing Better Pool Players Makes You Better is a Lie.
Back in 2015, a friend and I decided to pick up pool (billiards) again. We both played a lot during our undergraduate days, but since moving back to Fremont and getting real jobs, pool has kind of
taken a back seat.
With the opening of California Billiards in Fremont, our desire to shoot was re-ignited. With the opening of a local pool room also came the introduction of pool leagues where there was organized
weekly play. Our enthusiasm and desire to become competitive flourished. Playing in a league allowed us more opportunities to be competitive not only at the local level, but also allowed us to “flex
our muscles” at the regional and national levels.
Over the last 3 years, as I moved from an APA Skill Level (SL) 4 to Level 7 in 9-Ball, and SL 4 to SL 6 in 8-ball, I’ve noticed several reoccurring themes. The biggest theme is that pool to us
average folks is 90% mental. The theme I want to talk about is:
Playing better opponents or in harder leagues or (insert text here) DOES NOT make you a better player.
Anyone that tells you that it does is lying to you.
Yes. Playing better people is a component of becoming better, but itself alone is not enough for you to improve. And this really is true for any other passion or hobby, whether it be sports or
Most of the improvement and growth happens outside of the match. Ask yourself these questions after your match or during practice:
Before my match, did I adequately prepare? Did I warm up? Did I dial in my shot or stroke? Did I practice enough so that I am confident in making common shots? Did I work on my weak spots? What are
my goals for this match? (Consistently run 3 balls? Make good decisions for pocketing vs. safeties? Execute key safeties? Minimize the number of mistakes I make?)
During my match, what did I do well? What didn’t I do well? Which shots did I struggle with? Which shots carried me? What were some of the critical mistakes I made? And how could I have avoided them?
Or what was a better way to go about the situation? Was I fully invested in my match? And did I give the match the best effort I had?
After my match, what can I take away regardless of winning or losing? Did I practice key shots or safeties that I either struggled with during the match, or completely missed? Did I practice and
improve on troubled areas or concepts of my game. Do I understand my mistakes? And did I make an honest effort to do any or all these things?
These are the key questions you really need to ask yourself if you want to improve. More importantly, being disciplined about critically analyzing your game and committing to practice in order to
improve is key.
There’s nothing wrong with playing pool at a casual level. You can improve a little bit by playing casually, but modest improvement at most. However, if you wanted to get to that next level, you need
to input the work that is needed into becoming a better player.
2017 Resolutions Recap
Update on my 2017 Resolutions
My goals for 2017 and in no particular order:
1. Run a 9-ball rack (pool) and improve my billiards game. I’ve actually broke and ran a 9-ball game before, but haven’t done so since my session of APA League Pool. Stretch goal: Top Gun in 9-ball
for my local APA league. There were a lot of break and runs this year. No Top Gun, but a strong showing in both 8 and 9 ball.
2. Stage 0 Weight Reduction (lose some weight) and be a little healthier. Maybe make some time and pick up Aikido again. First milestone is to lose 20 lbs. This didn’t happen. I think that I
actually gained a few lbs. This year, I’ve decided to change a few things and am working on making this a reality.
3. Be a little more selfish and spend MORE time focusing on myself and the things I am passionate about. I think I did this. It’s a work in progress.
4. Chair at least one autocross event for the 2017 Championship Series and one event for the 2017 Slush series. Done.
5. Podium at an autocross event in my class with more than 5 entries. Done. This past year has been a huge improvement. After 3 years of owning the car, I finally have a full platform to race on.
And this rear wheel drive thing is starting to click.
6. Fix the Impreza, and then fix the Porsche. Let’s not talk about this. Nothing has changed. Lol.
7. Be more supportive of my friends in their interests and passions, even if those things are no interest to me. Still trying to do this. I think I’ve gotten better at it, but I still think there is
a lot of growth that still needs to take place.
8. Be relatively debt-free by the end of 2017. I am more debt free, but still a ways to debt-free.
9. Play more Magic: the Gathering than I did in 2016. I think I played about the same amount of Magic. I am looking to play more in 2018.
Instant Pot – Vietnamese Yogurt
Sweet Yogurt.
• 1 can of sweetened condensed milk
• 1.5 cans (use the can from the condensed milk) of hot water
• 2 cans of room temp water
• about 3 spoons full of plain yogurt with active/live cultures
Mix. Mix. Mix.
Instant Pot on Yogurt for 6.5 hrs.
For Sweeter Yogurt Recipe.
• 1 can of sweetened condensed milk
• 1.5 cans (use the can from the condensed milk) of hot water
• 1 cans of room temp water
• about 3 spoons full of plain yogurt with active/live cultures
For Sweet and a Thicker Yogurt
• 1 can of sweetened condensed milk
• 1.5 cans (use the can from the condensed milk) of hot water
• 1 can of room temp water
• 1 can of 2% milk
• about 3 spoons full of plain yogurt with active/live cultures
Instant Pot – BBQ Pulled Pork
Super easy to do and great for any occasion! Recently made this for our SuperBowl gathering!
• 3-4 lbs pork butt/shoulder roast
• Slow Cooker Pulled Pork Marinade Packet
I used this: http://redforkfoods.com/products/slow-cook-sauce/barbecue-pulled-pork-slow-cook-sauce
• 2 cups of water
• BBQ Sauce
1. Cut up the roast into 1.5″-2″ cubes and put all the pork, water, and marinade packet in the liner of the Instant Pot. I think the pulled pork tastes better if you allow it to marinate overnight.
2. Close and lock the lid, making sure the sealing knob is on sealing. I set the Instant Pot to manual for 40 minutes.
3. Once the cooking is done, you can release the pressure by the quick or natural pressure release.
4. Once the pressure is released, remove the pork chunks and shred. Once shredded, place into a large bowl and add BBQ sauce and mix. Add as much or as little as you want. To help keep the pulled
pork moist, I also ladled a little bit of the marinade into the mixing bowl. As I liked my pulled pork with a little sweeter sauce, I added a little brown sugar to the mixture. Feel free to add
sriracha if you are looking for a little heat!
5. Serve with buns and coleslaw or enjoy however you like!
Understanding Probability in Pandemic Legacy: Season 2 – Part 3
Understanding Probability in Pandemic Legacy: Season 2 is a multi-part series. Links to the other parts can be found here:
Understanding Probability in Pandemic Legacy: Season 2 – Part 1
Probability of Incidents During Set-Up
Understanding Probability in Pandemic Legacy: Season 2 – Part 2
Probability of Incidents During Set-Up and How-To Use the Front 9 Calculator
Understanding Probability in Pandemic Legacy: Season 2 – Part 3
Strategies to Survive Between Set-Up and First Epidemic
Strategies to Survive Between Set-Up and First Epidemic
After the infection stage of set-up, what is the best strategy to hold the disease at bay between Set-Up and the first Epidemic?
Why only manage until the first Epidemic? Once the first Epidemic occurs, the discard is shuffled and put back at the top of the deck. Once this happens, we’ll generally have the same subset of cards
that we’ll be drawing from after each Epidemic. Right after set-up, we are still drawing from the larger unknown Infection Deck. All we can really do is cover all the cities. Once the Epidemic
happens, we’ll know the composition of the next 20 or so cards which can help us determine how many cubes, or none at all are needed on each city and the likelihood that a specific city may be drawn.
Since we already know the contents (deck size and composition) of the Infection Deck, what we do between Set-Up and the First Epidemic is largely based on what cards are revealed during Set-Up.
We can use adapt the Front 9 Calculator to better demonstrate the probabilities during the Infection step at the end of each player’s turns.
The Previous Strategy
The strategy will be similar to that of how we distributed the the supply cubes prior to set-up. We want to minimize incidents by addressing the most vulnerable cities. In the case of distribution of
set-up, we distributed the cubes like this:
1. Place one supply cube in each city that has 3 cards in the infection deck.
2. Place one supply cube in each city that has 2 cards in the infection deck.
3. Place one supply cube in each city that has 1 card in the infection deck.
4. Place a second supply cube in each city that has 3 cards in the infection deck.
5. Place a second supply cube in each city that has 2 cards in the infection deck.
6. Place a third supply cube in each city that has 3 cards in the infection deck.
We know that regardless of the deck size, the order (of probabilities) of items #1-6 will always stay true, though as the deck starts to thin, the percentages of an event happening increases.
The Adapted Formula
Deck size = N (Remaining number of cards in the Infection Deck)
Total number of cards drawn = R = 2 (Infection Rate prior to Epidemic 1)
Number of New York cards = n = 3
Number of New York Cards drawn = r = [0, 1, or 2]; r can not be 3 as the most cards that can be drawn is 2.
P(X = r) =
C(n, r) * C(N-n, R-r)
C(N, R)
In layman’s terms,
C(n, r) is the number of ways to draw r New York card from n New York cards.
C(N, R) is the number of ways to draw R (R=2) cards, from N (N = # of cards remaining in the Infection Deck).
C(N-n, R-r) refers to the number of ways to draw R-r (Non-New York) from N-n (Non-New York)
The formula is exactly the same as the Front 9. The biggest difference is the N and R values. Here is the adapted calculator.
A simple substitution of a larger or smaller number in to the Backside Calculator will confirm that the order/priority of the guidelines are still true.
After the 9 infection cards are revealed during set-up, we can use this information to work backwards to determine what cards are remaining in the Infection Deck. The Infection Deck discard is known
information. You may look at its content at any time.
Since the order of probabilities is the same, you won’t need to use the calculator. You only need to base your decision from the remaining Infection Deck subset.
Example 1: If two New York cards are revealed during set-up, then there is one New York card remaining in the Infection Deck. Looking at the N=18 chart, the probability that a New York card will be
drawn in the next Infection step is 0.1111. This can be found by looking at Row 8 for 1 remaining card in the Infection Deck, and column D, E, and F. The key cell we are looking at is D8 as E8 and F8
are impossible outcomes. (You can’t draw 2 or more cards when there is a fewer number left in the deck.
Example 2: If no New York cards were revealed during set-up. How many New York cards are left in the Infection Deck at the beginning of the game? What’s the probability of this card coming up during
the next Infect step (given there is no Epidemic)?
There are three New York cards left in the Infection Deck. Looking at Row 10, and column D, E, F. (Cells D10, E10, and F10.) F10 isn’t a possible outcome and is 0. The sum of D10 and E10 is the
probability. Probability = 0.3137.
Revised Guidelines for Game-play Between Set-Up and Epidemic 1
Here’s the revised guidelines. These guidelines are based on the strategy of keeping the most venerable cities stocked with at least one supply cubes. When all cities have at least one supply cube,
then does the city receiving a second supply cube is feasible. According to the table provided by the Backside Calculator, it will be very rare to see back-to-back city cards to come out during the
Infection step between set-up and the first Epidemic.
1. If a city has 3 Infection cards remaining in the Infection Deck, and there are no cubes on that city, place 1 supply cube on that city.
2. If a city has 2 Infection cards remaining in the Infection Deck, and there are no cubes on that city, place 1 supply cube on that city.
3. If a city has 1 Infection cards remaining in the Infection Deck, and there are no cubes on that city, place 1 supply cube on that city.
4. If a city has 3 Infection cards remaining in the Infection Deck, and there is 1 cubes on that city, place 1 supply cube on that city.
5. If a city has 2 Infection cards remaining in the Infection Deck, and there is 1 cubes on that city, place 1 supply cube on that city.
6. If a city has 3 Infection cards remaining in the Infection Deck, and there is 2 cubes on that city, place 1 supply cube on that city.
Understanding Probability in Pandemic Legacy: Season 2 – Part 2
Understanding Probability in Pandemic Legacy: Season 2 is a multi-part series. Links to the other parts can be found here:
Understanding Probability in Pandemic Legacy: Season 2 – Part 1
Probability of Incidents During Set-Up
Understanding Probability in Pandemic Legacy: Season 2 – Part 2
Probability of Incidents During Set-Up and How-To Use the Front 9 Calculator
Understanding Probability in Pandemic Legacy: Season 2 – Part 3
Strategies to Survive Between Set-Up and First Epidemic
Using the Front 9 Calculator
Use this calculator to help determine the best way to distribute supply cubes onto your game board prior to the Infection step of Set-Up.
In Part 1, we calculated the probabilities of drawing 0, 1, 2, or 3 cards of a city (Ex. New York) during the Infection step of Set-Up. In Google Sheets, using the built-in combination function [=
combin(n, r)], we are easily able to fill out a table comparing the probabilities of Cities with # of Infection Cards vs. Number of Times a City Infected During Set-Up.
The calculator is provided here. It is write-protected so it won’t provide any spoilers of the game outside of the prologue. You will need to copy it to your own Google Drive to be able to edit it
and use it.
The only cell you need to populate is B5 (cell filled green). As your Infection Deck grows or shrinks, update this cell to see the updated probabilities of an event happening.
Understanding the Values in the Front 9 Calculator
The vertical axis (the area shaded in red) represents the cities with 1, 2, or 3 city cards in the infection deck.
The horizontal axis (the shaded in blue) represents the number of times a city is infected during set-up (0, 1, 2, or 3).
The purple area represents probabilities of possible outcomes.
The gray area represents impossible outcomes (0% probability).
The sum of the outcomes of Row 1, Row 2, and Row 3 are 1 (or 100%). This is important as we will used this piece of information to work backwards and to derive context.
Example 1: What does Cell C11 represent? Infection Deck = 27. The city we are trying to calculate the probability for has only 2 cards in the Infection Deck. What is the probability that we will not
draw any infection cards in this city during the Set-Up Infection?
= P(X = 0) or 1 – P(X > 0) or 1 – P(X = 1, 2, or 3)
= C11 = 1 – (D11 + E11 + F11) = 1 – (D11 + E11 + F11)
= 43.59%
In the context of Pandemic Legacy: Season 2: If no supply cubes where placed in this city, what’s the likelihood that this city will have an incident (Infection when no supply cues are present)?
Since there are no supply cubes, both outcomes of P(X = 1) and P(X =2) will cause an incident. Therefore the probability of an incident happening is:
P(X = incident when no supply cubes are present) = P(X = 1) + P(X = 2)
= 0.4615 + 0.1026 = 0.5641
Example 2: What do Cells C12 to F12 represent? Infection Deck = 27. The city we are trying to calculate the probability for has only 3 cards in the Infection Deck. If we place a single supply cube on
this city, what is the probability that we will not have any incidents during the Set-Up Infection?
Since there is a single supply cube in the city, the only way for the city to have an incident is if two or more infection cards are revealed during set-up. There are no incident outcomes if 0 or 1
infection card for this city comes up as the single supply cube negates the effect of the 1 infection card scenario.
= P(X = incident when a city with 3 cards in the infection deck has 1 supply cube on it)
Since we know that X = 0 and X = 1 will not cause an incident; and X > 1 (X = 2 and X = 3) will cause incident, the probability of an incident happening is:
= P(X = 2) + P(X = 3) = E12 + F12
= 0.2215 + 0.0287 = 0.2502
Theory Behind the Distribution Guidelines
1. Place one supply cube in each city that has 3 cards in the infection deck.
2. Place one supply cube in each city that has 2 cards in the infection deck.
3. Place one supply cube in each city that has 1 card in the infection deck.
4. Place a second supply cube in each city that has 3 cards in the infection deck.
5. Place a second supply cube in each city that has 2 cards in the infection deck.
6. Place a third supply cube in each city that has 3 cards in the infection deck.
Understanding the Numbers Behind the Guidelines
We already know this fact: The chance of getting an infection card from a city with 3 cards in the Infection Deck is higher than that of cities with only 2 cards in the Infection Deck. We also know
that the probability that a city will be infected is higher for cities with two cards than it is for 1 cards.
Example 3: There are no available cubes to distribute during set-up. List in order the type of cities (1 card, 2 cards, or 3 cards) that are most-likely to get at least one incident.
This information can be drawn from column C in the Front 9 Calculator. If we put context to these three cells, we would get this:
The probability that a city with 1 card in the infection doesn’t get an incident is 66.67%.
The probability that a city with 2 cards in the infection doesn’t get an incident is 43.59%.
The probability that a city with 3 cards in the infection doesn’t get an incident is 27.90%.
Most Probable to Have at least One Incident
Cities with 3 Infection Cards in the Infection Deck
Cities with 2 Infection Cards in the Infection Deck
Cities with 1 Infection Card in the Infection Deck
Least Probable
This order of most to least probable will stay true as long the three types of cities have the same number of supply cubes on it. (Except when the number of supply cubes is greater or equal than the
number of card cards in the Infection Deck. In that case, the outcomes has zero probability.)
Example 5: Place a supply cube in each of the cities that have 3 cities card in the Infection Deck. List in order the type of cities (1 card, 2 cards, or 3 cards) that are most-likely to get at least
one incident.
We’ve already determine the probability of this scenario in Example 2 and know the probability is the sum of the outcomes where X > 1 (or cells E12 and F12).
Most Probable to Have at least One Incident
Cities with 2 cards and has 0 supply cubes. (Cells D11 + E11) = 0.5641
Cities with 1 cards and has 0 supply cubes. (Cell D10) = 0.3333
Cities with 3 cards and has 1 supply cube. (Cell E12 and F12) = 0.2502
Least Probable
Knowing these things, we are now able to order all the scenarios from most to least probable.
Most to Least Probable of All Scenarios
Most Probable to Have at least One Incident
Cities with 3 cards and 0 supply cubes. (Cells D12/E12/F12) = 0.7210
Cities with 2 cards and 0 supply cubes (Cells D11/E11) = 0.5641
Cities with 1 cards and has 0 supply cubes. (Cell D10) = 0.3333
Cities with 3 cards and has 1 supply cube. (Cells E12/F12) = 0.2502
Cities with 2 cards and has 1 supply cubes. (Cells E11) = 0.1026
Cities with 3 cards and has 2 supply cubes. (Cell F12) = 0.0287
Cities with 1 cards in the Infection Deck and has 1 supply cubes. = 0
Cities with 2 cards in the Infection Deck and has 2 supply cubes. = 0
Cities with 3 cards in the Infection Deck and has 3 supply cubes. = 0
Least Probable
Distributing the Supply Cubes
Using the information above, we can now determine how to best minimize the chance of incidents. In ideal situations where you have unlimited supply cubes, you would just distribute 3 supply cubes to
each 3 card city, 2 supply cubes to each 2 card city, and a single supply cube to each 1 card city for a 0% percent chance of incidents.
However, the real-game scenario limits the number of supply cubes that are available during Set-Up. In-game, you would want to prioritize the cities with the highest probability first. As you add
cubes one by one to a city, the probability of incidents occurring will decrease.
Cities with 3 cards and 0 supply cubes has the highest probability of incidents. Once a supply cube is added to the city, the probability of an incident decreases to 0.2502.
The strategy is to address the highest probability incident scenarios, not cities, first. Once that scenario is completely addressed, the next step is address the next highest probable scenario,
cities with 2 cards and 0 supply cubes (0.5641). And so on.
When the allotted supply cubes are distributed with the above priority, the board state will have the least probable set-up for incidents. Of course, this is just by the numbers. Higher probability
will happen more often. Lower probability will happen less often.
If this method doesn’t work for you, my is disclaimer is this: This method doesn’t take into account luck, misfortune, poor shuffling, and/or Acts of God.
Understanding Probability in Pandemic Legacy: Season 2 – Part 1
Oh my god. I am actually using my degree.
Before I begin, I just wanted everyone to know that I am going to try my best to not spoil any of the game other than what you might have seen in the Prologue month.
Understanding Probability in Pandemic Legacy: Season 2 is a multi-part series. Links to the other parts can be found here:
Understanding Probability in Pandemic Legacy: Season 2 – Part 1
Probability of Incidents During Set-Up
Understanding Probability in Pandemic Legacy: Season 2 – Part 2
Probability of Incidents During Set-Up and How-To Use the Front 9 Calculator
Understanding Probability in Pandemic Legacy: Season 2 – Part 3
Strategies to Survive Between Set-Up and First Epidemic
Understanding Probability in Pandemic Legacy: Season 2
Game play of the Prologue month can easily be found on YouTube and the month is often used to introduce the game play, mechanics, and work flow to new players. The prologue has no permanent effects
on the game and can be played an unlimited number of times before starting the January month.
For those of you who are in a current game of Pandemic Legacy: Season 2, you might have noticed that the game is quite a bit harder than Season 1. You aren’t alone. It seems like everyone is
struggling to meet their monthly objectives. Here are some tips.
Optimize Your Chances of Winning
1. Review the contents of the Infection Deck. Count the number of cards and see how the city infection cards are distributed. At the beginning of the game, there are 27 cards in the infection deck.
9 cities with 3 card each. This will change as the game progresses. As you are playing the game, keep track of which cards came up. This will help you determine the likelihood of future
cards.After each epidemic, before shuffling the infection deck discard and putting it on top of the infection deck, review the played cards and take notes. Again, this will help you prepare for
the upcoming infections.
2. Review the contents of the player deck. Most importantly, get an idea of how often an Epidemic will come up. If the player deck has 60 cards, and there are only 5 Epidemics in the game, you
should expect an Epidemic in each 12 card cycle.
3. Take Notes. Tally the number of players cards that you have gone through. Tally the number of cities that have come up. Tally how many turns/player cards since the last epidemic, or which
epidemic cycle you are currently on.
Solving for the Probability of Event (blank)
Now onto the probability. Why is probability so important? A core mechanic of Season 2 is keeping enough supply (gray) cubes in each city on the board to adequately protect the city from incidents,
or infection of a city when there are no stockpile cubes left. When a city gets infected, a supply cube is removed. When a city is infected and has no supply cubes left, a disease cube is placed in
the city and the incident marker is moved forward. When the game reaches the 8th incident for the game, Pandemic has won and that attempt for that month is over.
How do we determine how many stockpile cubes are adequate? How do we distribute stockpile cubes in such a way that we minimize the number of incidents occurring during the initial infection?
That’s really a great question. To best answer that, we need to make some assumptions as well as simplify the scenario a little bit.
During the set-up of the game, the players have a limited number of supply cubes to distribute among the cities connected to the grid. For the prologue month, there’s 9 cities and 36 stockpile cubes
that need to be distributed. The infection deck has 27 cards. 3 cards for each city connected to the grid. After the cubes are distributed, set-up does an initial infection by drawing 9 cards from
the infection deck and infecting the 9 revealed cities.
Solution (Kind of.)
To answer the initial question, let’s simplify even more. Let’s only concentrate on a single city, New York. Of the 27 cards in the infection deck, 3 cards are New York. During the set-up infection
(9 infection cards), what’s the likelihood that 0, 1, 2, or 3 New York cards show up? Does it make sense to place 0, 1, 2, or 3 cubes on New York to prevent incidents?
Defining some variables:
Deck size = N = 27
Total number of cards drawn = R = 9
Number of New York cards = n = 3
Number of New York Cards drawn = r = [0, 1, 2, or 3]
If we were only drawing one card, probability would be really easy to determine. P(X = Draw New York) = 3/27.
However, since we are drawing 9 infection cards, we have to factor in that New York cards can fall into any slots. We could draw a New York with the first card, or the last card, or any card in
between. In order to account for this, we will use combinations as don’t care about order.
Some other assumptions: Though the remaining 24 cards in the Infection Deck are not all the same, for this problem, we’ll just lump them together and consider them “Non-New York.”
More information on Combinations can be found here under “Combinations, Ho!”
For reference:
Combinations = nCr = C(n, r)
Solve in Google Sheets using this function: =combin(n, r)
P(X = r) =
C(n, r) * C(N-n, R-r)
C(N, R)
In layman’s terms,
C(n, r) is the number of ways to draw r New York card from n New York cards.
C(N, R) is the number of ways to draw R (R=9) cards, from N (N=27).
C(N-n, R-r) refers to the number of ways to draw R-r (Non-New York) from N-n (Non-New York)
P(X = Draw 1 New York)
P(X = Draw 1 New York) =
[Combination(of drawing 1 New York from 3 New York cards) x Combination(of drawing 8 Non New York cards from 24 Non-New York Cards)] / [Combination(of drawing 9 cards from 27)]
It should look something like this:
C(3,1) * C(24, 8)
C(27, 9)
P(X = Draw 1 New York) = 0.4708
P(X = Draw 2 New York)
Let’s try for P(X = Draw 2 New York).
C(3,2) * C(24, 7)
C(27, 9)
As you can see, only the numerator changes. C(3,2) refers to drawing 2 New York’s from a total of 3. While C(24,7) refers to the number of ways to draw the 7 remaining cards from the Non New York
P(X = Draw 2 New York) = 0.2215
P(X = Draw 3 New York)
C(3,3) * C(24, 6)
C(27, 9)
P(X = Draw 3 New York) = 0.0287
P(X = Draw 0 New York)
Just for due diligence, let’s also do P(does not draw New York).
C(3,0) * C(24, 9)
C(27, 9)
P(X = 0 New York) = 0.2790
P(X = 0) = 0.2790
P(X = 1) = 0.4708
P(X = 2) = 0.2215
P(X = 3) = 0.0287
Let’s interpret the results.
P(X = 0) = 0.2790 – Probability that New York gets exactly 0 infections
P(X = 1) = 0.4708 – Probability that New York gets exactly 1 infection
P(X = 2) = 0.2215 – Probability that New York gets exactly 2 infections
P(X = 3) = 0.0287 – Probability that New York gets exactly 3 infections
Ex. I put only supply cubes on New York. What’s the risk or likelihood that New York will have an incident during the initial infection set-up? (What’s the probability that New York will be infected
3 times during Set-Up?)
P(X > 2) = 1 – P(X > 2) = P(X = 3) = 0.0287
Ex. I put one supply cube in New York. What’s the risk or likelihood that New York will have an incident during the initial infection set-up? (What’s the probability that New York will be infected 2
or more times?)
P(X > 1) = P(X = 2, 3) = P(X = 2) + P(X = 3) = 0.2215 + 0.0287 = 0.2502
Other Scenarios
Ex. There are only two New York Infection cards in the deck. (2 New York cards and 25 Non-New York cards) What are the probabilities of P(X = 0), P(X = 1), and P(X = 2)?
Deck size = N = 27
Total number of cards drawn = R = 9
Number of New York cards = n = 2
Number of New York Cards drawn = r = [0, 1, or 2]
The set-up of the formula is the same except that we’ll be using the new numbers above to substitute them into the combinations formula.
For P(X = 0), the formula will look like this:
C(2,0) * C(25, 9)
C(27, 9)
P(X = 0) = 0.4359
Furthermore: P(X = 1) = 0.4615 and P(X = 2) = 0.1026.
Set-Up Infection Calculator in Google Docs
I’ve set up this cool little table/calculator to determine the probability that cities will become infected with 0, 1, 2, or 3 infection cubes during set-up. Just update your Infection Deck size in
the green box.
Pandemic Legacy: Season 2 – Front 9 Calculator
What’s the best way to distribute supply cubes during set-up? Using the Set-Up Infection Calculator and the combination theories above, the best guidelines are to distribute supply cubes in this
order. Continue to the next point if you still have stockpile cubes still available. Proof of this strategy can be found in Part 2.
1. Place one supply cube in each city that has 3 cards in the infection deck.
2. Place one supply cube in each city that has 2 cards in the infection deck.
3. Place one supply cube in each city that has 1 card in the infection deck.
4. Place a supply stockpile cube in each city that has 3 cards in the infection deck.
5. Place a supply stockpile cube in each city that has 2 cards in the infection deck.
Instant Pot – Beef Pho
My first attempt at beef pho. If you haven’t made pho before, I would recommend trying Pho Ga first. Neither recipe is difficult, but the Pho Ga is a little easier and a little more straight forward.
And less messy. I would also check out Cuong Can Cook’s video down below (See Source.)
Based on the recipe by Cuong Can Cook:
• ~2-2.5 lbs of beef bones (I used ~2 lbs of beef bones and ~1 lb of oxtail
• 1 yellow onion
• 1 palm sized piece of ginger
• green onion
• cilantro
• 1/2 cup of fish sauce
• 1 tablespoon of sugar
• 1 teaspoon of salt
• pho noodles
• lime wedges
• packet of pho spices
This was one of the items available at my local Asian Market. $3.00 for 5 spice packets.https://www.amazon.com/Pho-Hoa-Beef-Noodle-Spices/dp/B003GRMUCUThe filter bag becomes really fragile after
the cooking process. Use a ladle to remove the packet as to not break the spice bag during removal.
• Proteins: Sliced beef from the shabu shabu section, bo vien (beef/tendon meatballs), brisket, tendon, tripe (whatever you desire)
Cooking Directions:
• In the Instant Pot (on saute) or a stock pot on the stove, bring the beef bones/oxtail and water to a boil. Boil for 10-15 minutes. You want to boil the beef bones until it releases the fat/gunk.
Once boiled, set aside the beef bones/oxtail and dump the remaining water. With the remaining bones, rinse and gently clean with clean water.
• Over direct heat gas flame, char the whole onion and ginger. If a gas burner is not available, you can use cut the onion and ginger in half and broil in the oven.
• Place beef bones, fish sauce, sugar, charred onion and ginger, salt, and spice packet into Instant Pot and fill with water to a little under the max line in the Instant Pot. Close the lid, set
vent knob to Sealing, and use Manual (High Pressure) for 60 minutes.
• While the broth is cooking, slice green onion and cilantro into small pieces; and prepare any other items. Do this step at a later time if you are pre-making beef pho for a later time.
• After the 60 minute cooking time elapses, allow a NPR (Natural Pressure Release). If you do a Quick Release, the Instant Pot will make a mess with this recipe. Once the pressure is released,
remove the lid and carefully fish out the spice packet, onion, and ginger. The spice packet is very delicate and can break easily if you aren’t careful. If there is any fat or gunk at the top of
the broth, now is the time to remove it. Taste and season broth as needed with fish sauce and/or salt.For the best results, allow the broth to cool, then place the Instant Pot inner pot in the
refrigerator overnight. As the broth cools, the fat will form at the top and will be easier to remove.
• In a separate pot, boil some water, Using a noodle strainer if available, boil the dry pho noodles about 5 to 20 seconds until it is loose and pliable. Place into soup bowl.
• Add your proteins on top of the rice noodles.
• Garnish with additional green onion and cilantro. Pour the boiling soup broth into bowl. If the noodles are cold prior to adding the broth, consider microwaving the bowl for 30-45 secs prior to
adding the broth.
• Garnish with your favorite pho condiments (green onions, cilantro, lime, Sriracha, etc.)
5 Tips for an Easier Time at the Airport
Not sure if it is me or not, but I’ve noticed that the average person dumbs down quite a bit as soon as they arrive at the airport. Here are 5 tips to make your trip a little easier.
Tip #1: Pack light and know what is and isn’t allowed onto the airplane.
Let’s face it, people pack too much shit. Pack light and only bring what you need. You don’t need a large suitcase full of stuff for a weekend trip. The lighter you pack and the fewer bags you travel
with, the easier your airport experience will be.
For my most recent trip to Dallas, where I was expecting stay for two weeks, I packed:
7 or 8 tshirts
7 pairs of underwear
3 pairs of shorts
1 pair of khaki pants
1 light jacket
toiletries (toothbrush, small toothpaste, contact fluid, body wash, shampoo, etc.)
I was easily able to stuff all of these items into one of those half height rolling suitcases (underseat carry-on luggage). Instead of carrying two weeks worth of clothes, I decided to carry a weeks
worth and use the local washer and dryer to wash my clothes for the second week. The short of it is that you need to be smart about what and how you pack. Don’t bring more stuff than you need and
don’t bring anything you don’t need. No point in bringing a heavy jacket to Dallas in July.
If you are terrible at Tetris, consider checking out these packing cubes. These definitely help in keeping things organized.
Last note: Know what you can and can’t bring onto the airplane. TSA’s website outlines what is and isn’t allowed and what must be packed in checked bags. And don’t forget about liquids and gels and
the 3-1-1 rule.
Tip #2: Print Out Your Boarding Pass and Carry Your Government Issued Identification
You’ll need a government issued ID to get through the security check-point. A photocopy or a picture on your smart phone WILL NOT work. When I travel, I carry my drivers license and passport.
As convenient as smart phones are, it takes too much fumbling to pull up the boarding pass for the TSA officer. I opt to just print out my boarding pass. As I get to the front of the line, I have my
boarding pass and ID ready for easy passage through the first security station. You wouldn’t believe how many people are searching through their phones for their boarding pass or how many people are
fighting with the scanners to read the barcode.
Tip #3: Check-In Your Bags
If you are flying Southwest, check in your bags! First two are FREE! F-R-E-E! Why bother lugging them through the airport, through security, and on/off the plane, when you can check them in and meet
them at your destination? Afraid of losing your bags?? That’s why you put an owner information tag onto it. Bags do get misplaced time to time, but they can’t grow legs and walk away.
Your airline charges you money to check-in your bags?? I get it… I wouldn’t want to fork over money either. Check with your frequent flyer program or credit card to see if you get your first bag
checked in for free. If not, consider checking your bags in at the gate. Most airlines are more than happy to check your bags in at the gate at no charge!
If you are one of those people who like to travel with oversized luggage, don’t be that a-hole that thinks that they are special and the rules don’t apply to them. If your luggage doesn’t fit in the
carry-on baggage template, it shouldn’t be carried on.
Tip #4: Follow the Instructions at the TSA Screening
Pay attention at the security screening. Expect to remove your laptop and your toiletries from your luggage for screening. And prepare to take off your jacket and shoes. Don’t be that guy or gal that
gets to the front of the line thinking that he/she doesn’t have to do any of this. You are holding up everyone behind you.
Sub-tip: Pack your bags so that it is easy to remove your laptop and toiletries. While in line for the security screening, empty your pockets and throw everything (keys, wallet, phone, etc.) into
your carry-on bag. This will save you time from fumbling with x-ray bowls. Once you get through the metal detectors, grab your bag and make your way through the rest of screening avoiding the
Tip #5: Prepare at home and not at the airport
Do all your preparations at home. If you adequately prepared, you’ll have an easier time at the airport. Print out all the necessary documents (boarding pass, rental car confirmation, hotel info,
etc.) prior to heading to the airport. As convenient as smartphone and email are, they are not the quickest way of keeping and providing travel information. I like to use a small coupon folder/
envelope to store all my travel documents, not to mention a great place to store my receipts. | {"url":"http://www.kimete.com/","timestamp":"2024-11-01T22:43:37Z","content_type":"text/html","content_length":"154960","record_id":"<urn:uuid:624b5479-aff0-4f1e-be97-e00ed94831ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00725.warc.gz"} |
What is a vertical asymptote in calculus? + Example
What is a vertical asymptote in calculus?
1 Answer
The vertical asymptote is a place where the function is undefined and the limit of the function does not exist.
This is because as $1$ approaches the asymptote, even small shifts in the $x$-value lead to arbitrarily large fluctuations in the value of the function.
On the graph of a function $f \left(x\right)$, a vertical asymptote occurs at a point $P = \left({x}_{0} , {y}_{0}\right)$ if the limit of the function approaches $\infty$ or $- \infty$ as $x \to {x}
For a more rigorous definition, James Stewart's Calculus, ${6}^{t h}$ edition, gives us the following:
"Definition: The line x=a is called a vertical asymptote of the curve $y = f \left(x\right)$ if at least one of the following statements is true:
${\lim}_{x \to a} f \left(x\right) = \infty$
${\lim}_{x \to a} f \left(x\right) = - \infty$
${\lim}_{x \to {a}^{+}} f \left(x\right) = \infty$
${\lim}_{x \to {a}^{+}} f \left(x\right) = - \infty$
${\lim}_{x \to {a}^{-}} f \left(x\right) = \infty$
${\lim}_{x \to {a}^{-}} f \left(x\right) = - \infty$"
In the above definition, the superscript + denotes the right-hand limit of $f \left(x\right)$ as $x \to a$, and the superscript denotes the left-hand limit.
Regarding other aspects of calculus, in general, one cannot differentiate a function at its vertical asymptote (even if the function may be differentiable over a smaller domain), nor can one
integrate at this vertical asymptote, because the function is not continuous there.
As an example, consider the function $f \left(x\right) = \frac{1}{x}$.
As we approach $x = 0$ from the left or the right, $f \left(x\right)$ becomes arbitrarily negative or arbitrarily positive respectively.
In this case, two of our statements from the definition are true: specifically, the third and the sixth. Therefore, we say that:
$f \left(x\right) = \frac{1}{x}$ has a vertical asymptote at $x = 0$.
See image below.
Stewart, James. Calculus. ${6}^{t h}$ ed. Belmont: Thomson Higher Education, 2008. Print.
Impact of this question
15474 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/what-is-a-vertical-asymptote-in-calculus#108192","timestamp":"2024-11-12T20:28:26Z","content_type":"text/html","content_length":"37845","record_id":"<urn:uuid:917f83ed-4741-4865-8059-b304e8e99298>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00824.warc.gz"} |
Creator of
procedural fire effect for Pico-8 tiny cart jam
Run in browser
for tweettweetjam7
Play in browser
Recent community posts
Very cute drawings!
Looking forward to giving this a try!
Source (290 chars)
for i=1,#z do
sfx(1)for j=i,#z do
goto d
Neat! Seems like it doesn’t take suits into consideration, so there would be four cards with the value 0, right?
Full disclosure: I started working on this on Sunday, before the jam officially started, because I cannot make time to work on it during this week. This means I’ve used at most 1.5 days (<36 hours)
on this - less than the jam’s official duration would allow for.
Regardless, if you think I’m being unfair, please let me know and I’ll forfeit my entry from the voting. I don’t care about winning, I just wanted to make and publish a game for once :)
Nice idea! I found the game very hard though, only managed to save one patient with dozens of attempts xD
Is there something I can do to avoid losing the game, when the RNG gives no workable parts? | {"url":"https://itch.io/profile/thykka","timestamp":"2024-11-08T11:42:24Z","content_type":"text/html","content_length":"20345","record_id":"<urn:uuid:ebe1cad8-4e0d-4016-85dc-d9a78b4c109b>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00605.warc.gz"} |
Addressing levels in hierarchies #1
When you work with hierarchies, you often need to address a certain level of the hierarchy and that can be a little difficult.
Let me start by showing you a few examples:
In this crosstab I would like to create a calculate the difference between the 2 year in the filter.
As you can see – it’s a hierarchy of time that we are looking at on the horizontal axis.
That hierarchy is currently expanded to the quarter level – but I guess the users might expand it further (to month/day level).
However the hierarchy is expanded, I want to stick with the year level.
My time hierarchy has 4 levels: Year, Quarter, Month, Day
In terms of TARGIT syntax we can address:
• Level 0 (means the grand total of the time dimension – in this case the sum of 2019 and 2020)
• Level 1 (means year level – in this case either the sum of 2019 and the sum of 2020)
• Level 2 (means month level – in this case either the sum of Q1, Q2, Q3 and Q4)
• Level 3 (means day level) – in this case a particular day in a particular month)
This formula (Year growth)
Sum(d2(l1),0,m1)-sum(d1(l1),0,m1) means subtract 2019 from 2020 on year level.
Because of the l1 reference this will remain on year level regardless how the hierarchy is expanded.
1 comment
• Is there anything to be aware of when calculating with levels? I can get it to work on coded measures but not calculated measures.
The table contains sales data for the past 12 months, date hierarchy used is YMD(Month) thus the same as the example.
I want the total for this year only.
It gives me the correct numbers if I use specific dimensions:
c1 = (@"[2022].[January]":d-1, 0, m4) = correct
but this is very unconvenient as it requieres me to remember to change it next year.
It does not work with calculated measure m4:
m4 = returns m1 if store is older than 1 year else 0
c1 = d2(l1),0,m4) = 0 (not correct)
But if
c1 = d2(l1),0,m1) = returns turnover for 2022
which means it works, just not usable in the wanted context.
I also tried to use a different calculated measure to see what happens:
m3 = 1
c1: sum(d2(l1), 0, m3) = 1
but should be 7 so this does something, but does not calculate the total sum of 2022 for m3, looks like it takes the average rather than sum in this instance.
Any suggestions?
Please sign in to leave a comment. | {"url":"https://community.targit.com/hc/en-us/articles/360018691977-Addressing-levels-in-hierarchies-1","timestamp":"2024-11-11T13:25:03Z","content_type":"text/html","content_length":"34834","record_id":"<urn:uuid:9b90b05d-5750-45a1-96a0-e3cb80e208f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00899.warc.gz"} |
(1-12) Acoustics of a Greek amphitheater
It is shown that a surface of porous material or auditorium seats yields negative reflection at the grazing angle1). It is also mentioned2) that the reflection from the front seats at an amphitheater
is less because of their steep angle (26.3 degrees) and that its path deference from the direct sound from the stage is larger than at the grazing angle.
These discussions were done when the surface is flat. Here it is discussed on the effect of the curvature of a surface.
1) Sound reflection at the grazing angle
Reflection coefficient was defined for a surface which is not rigid and it was measured with a large panel of the surface. It was used to get the reflection of a limited dimension with the surface to
be convolved to the rigid panel of the dimension.
Thus defined reflection coefficients of two different porous surfaces are show in Fig.1. They are for a punch carpet layer and a urethane foam layer.
Fig.1 Reflection coefficients of two porous layers
It has a positive surface reflection at the normal incidence but it shows a large negative surface reflection at 80 degrees which is close to the grazing angle.
An apparent surface was given as shown in Fig.2 for uneven auditorium seats. A reflection coefficient was measured at the ear position for an audience. An incident angle is shown by θi. They are for
hard and absorptive seats in Fig.2.
Fig.2 Apparent surface of auditorium seats and a receiving point for the reflection coefficient measurement
Thus measured reflection coefficients of auditorium seats at the grazing angle for world famous concert halls are shown in Fig.3. They are shown in the time and frequency domains. Fig.4 shows the
sketches of their auditorium seats.
The first negative reflection coefficients are gathered in Fig.5 with a few additions of Japanese auditoriums.
Fig.3 Reflection coefficients of auditorium seats at six world famous concert halls in the time and frequency domains
Fig.4 Sketches of auditorium seats of the six concert halls
Fig.5 Incident angles and first surface reflections of auditorium seats at world famous concert halls and a few Japanese auditoriums
While, a Greek amphitheatre has a slope with 26.3 degrees, as shown in Fig.6. The angle refers to around 65 degrees in Fig.5, where the reflection coefficient is smaller in negative.
Fig.6 Steep slope of audience seats at a Greek amphitheater
In this way, a Greek amphitheatre has commonly the audience slope of 26.3 degrees and they get the direct sound from the stage very clear.
2) Reflection from a rigid concaved surface
To have the reflection of a concaved surface it was divided into small rectangles where each of them gets plane wave incidence and it was summed up being calculated by the Fraunhofer equation. The
reflection from each rectangle has a pair of positive and negative rectangular waves from the far field term and a trapezoid wave from the near field term as shown in the above of Fig.7.
In the below of Fig.7, it is shown how they are summed for a concave and convex surfaces. For a concave surface, the reflection from each rectangle is in phase with the one of the rectangle at the
specular reflection and they compose a pair of large and negative reflections.
For a convex surface, the positive pair remains but the negative pair is cancelled with the positive pairs of the surrounding rectangles. So it tends to remain only a positive reflection from the
rectangle of the specular reflection and it is affected by the curvature.
In fig.8, two examples are shown compared with experimental results. The successive multiple reflections are not calculated but they simulate the specular reflection well.
Fig.9 shows the calculated reflection of a concaved surface when its curvature is changed. Specular reflection is very large with the addition in phase and its spectrum is large too. In this
condition, the amplitude and spectral level are largest at the curvature Rc 70cm.
Fig.7 Reflections from a divided rectangle in the above and the addition of reflections of the surrounding rectangles to the specular reflection in the below
The reflection of a convex surface is given in the left and that of a concave surface in the right.
Fig.8 Comparison of a calculated and measured result for a concave surface
Fig.9 Reflection of a concaved surface with the change of its curvature
Thus, the negative reflection from the front seats becomes small and delayed because of the steep slope, and the received level is amplified by the curvature at the audience.
It is possible to have an impulse response at an audience with the practical dimensions of an amphitheatre. It should be convolved with the transient response of our hearing system and integrated in
the time window after absolutization.
3) Another possibility which makes amphitheatre acoustics good
It has been discussed only for the direct sound from the stage but it needs a bit of reverberation.
The impulse response for the curvature Rc 70cm in Fig.9 shows a large specular reflection with ± amplitude and a successive transient wave after them. It is given from the area surrounding the
specular reflection rectangles with delayed reflections. They are imagined to add a bit of reverberation.
Further discussion should be done after the transient response of our hearing system is convolved, comparing with the reflection at a usual auditorium seats.
4) Rectangular reflectors surrounding the stage
There are wooden rectangular panels at the three sides on the stage to support performers as can be seen in the later photos. To show the acoustical difference, I spoke or sang outside or inside the
enclosure, they recognized it very clearly.
It was clearly noticed too from the stage when audience talks back to me. It is reciprocal.
It is apparent that the rectangular enclosure gives a good support to the performers on the stage. Rich normal modes are created by the enclosure. How would it be if a ceiling is added to get richer
normal modes or the dimensions of reflectors are changed referring to the musical notes? Interesting questions are never finished.
An impulse response at an audience from a sound source on the stage of rectangular enclosure, which is the direct sound, can be calculated. The sound reflected by the front audiences can be
calculated by the convolution of the reflection coefficient and the reflection of the concaved surface. Then the impulse response at an audience is obtained by the addition of them.
The evaluation of the sound field can be done there and after the transient response of our hearing system is convolved.
It is not difficult to get such slopes in the green country NZ. It is very lucky to be able to enjoy music and/or plays in the green space under the stars. We held music gatherings ten times once in
a year in March celebrating autumn harvests. If this kind of event would be expanded, it would be wonderful. A few photo shots are given below.
1）Y. Sakurai and K. Nagata, “Practical estimation of sound reflection of a panel with a reflection coefficient” , J. Acoust. Soc. Jpn (E), 3, 1, p7-19 (1982)
2) Y.Sakurai, H.Morimoto and K.Ishida,” The reflection of sound at grazing angles by auditorium seats”, p209-227, Applied Acoustics 39(1993).
3) Y.Sakurai,” The early reflection of the impulse response in an auditorium”, p127-138, J.Acoust.Soc.Jpn.(E), 8, 4(1987).
4) Y.Sakurai, ”Sound reflection of a curved rigid panel”, p63-70, J. Acoust. Soc. Jpn(E), 2, 3(1981).
5) Y.Sakurai and H.Morimoto,” The transient response of human hearing system”, J.Acoust.Soc.Jpn.(E), 10, 4(1989).
Photos from handmade music gatherings | {"url":"https://www.ecohouse.co.nz/new/CH6/1-12.htm","timestamp":"2024-11-04T20:35:35Z","content_type":"text/html","content_length":"15749","record_id":"<urn:uuid:38f8a20c-7ae9-4dd8-9000-5fa51e46a9d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00762.warc.gz"} |
New tech reports
Soft maximum
I had a request to turn my blog posts on the soft maximum into a tech report, so here it is:
Basic properties of the soft maximum
There’s no new content here, just a little editing and more formal language. But now it can be referenced in a scholarly publication.
More random inequalities
I recently had a project that needed to compute random inequalities comparing common survival distributions (gamma, inverse gamma, Weibull, log normal) to uniform distributions. Here’s a report of
the results.
Random inequalities between survival and uniform distributions
This tech report develops analytical solutions for computing Prob(X > Y) where X and Y are independent, X has one of the distributions mentioned above, and Y is uniform over some interval. The report
includes R code to carry out the analytic expressions. It also includes R code to estimate the same inequalities by sampling for complementary validation.
Here are some other tech reports and blog posts on random inequalities.
2 thoughts on “New tech reports”
1. I had my students implement a 2-arg softmax procedure while learning Scheme a homework or two ago, and then consider how to make it variable arity. They noticed that different implementations
give different answers, but remarkably similar.
2. John,
The first line of the second page of the soft maximum paper says ‘This report call’ instead of ‘This report calls’. Love the blog though! | {"url":"https://www.johndcook.com/blog/2011/09/26/tech-reports/","timestamp":"2024-11-06T08:52:59Z","content_type":"text/html","content_length":"52611","record_id":"<urn:uuid:45deaba4-0530-4b89-b195-443a6787c39d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00160.warc.gz"} |
Part 1: Introduction to VASP
By the end of this tutorial, you will be able to:
• explain a density-functional-theory (DFT) calculation on the level of pseudocode
• create input files to run a DFT calculation for an isolated atom
• recognize the basic structure of the stdout and OUTCAR
• extract the relevant energies for molecules and atoms
• restart a DFT calculation from the previous Kohn-Sham (KS) orbitals
Let's perform a DFT calculation for a single oxygen atom in a large box in order to compute the energies of an isolated atom.
DFT is the framework in which the electronic degrees of freedom are treated. On the level of pseudocode that is
1. Given the electronic charge density, the Hamiltonian can be defined.
2. The eigenfunctions and eigenvalues of the Hamiltonian are computed.
3. The electronic charge density is updated.
4. Iterate 1.-3. until converged.
For details about the implementation in VASP go to algorithms used in VASP to calculate the electronic ground state.
VASP looks in the current directory for four main input files, i.e., POSCAR, INCAR, KPOINTS and POTCAR. The general format of each input file is explained in details in the linked articles that lead
to the VASP Wiki, but below we will discuss the particular choices for this example.
The input files to run this example are prepared at $TUTORIALS/molecules/e01_O-DFT.
Open the terminal and navigate to this examples directory. Confirm that all files are present:
cd $TUTORIALS/molecules/e01_*
Do the same in the file browser on the left hand side and open the files by double clicking.
Let us now discuss the content of the input files! First, we define the position of a single atom in a box in the POSCAR file.
O atom in a box
1.0 ! scaling parameter
8.0 0.0 0.0 ! lattice vector a(1)
0.0 8.0 0.0 ! lattice vector a(2)
0.0 0.0 8.0 ! lattice vector a(3)
1 ! number of atoms
cart ! positions in cartesian coordinates
The lattice parameters are set to be sufficiently large, i.e., 8 Å, so that no significant interaction between atoms in neighboring cells occurs.
In general, the INCAR file contains tags that control the calculation. These tags are documented on the VASP Wiki in the category and have some default value in case they are not set in the INCAR
file. These default values will start a DFT calculation. Check out the tags explicitly set in this example!
SYSTEM = O atom in a box
ISMEAR = 0 ! Gaussian smearing
Go ahead and check the meaning of the SYSTEM tag and ISMEAR = 0. Also pay attention to related tags!
Next, we define a single $\mathbf{k}$ point in the KPOINTS file.
Gamma-point only
Monkhorst Pack
One $\mathbf{k}$ point suffices to describe an isolated atom or molecule, because a wavevector $\mathbf{k}$ connects points separated by a lattice vector, i.e., a translation to a neighboring cell.
Thus, including more $\mathbf{k}$ points would describe the interaction of oxygen atoms in neighboring cells more accurately. But that interaction should be zero, if we have indeed set the box
sufficiently large to describe an isolated atom in the POSCAR file.
The following POTCAR file contains the pseudopotential and related data for oxygen. It is rather long, so please open it as a tab using the file browser, but be careful not to edit it! Here is just
the head of the POTCAR file.
PAW_PBE O 08Apr2002
parameters from PSCTR are:
VRHFIN =O: s2p4
LEXCH = PE
EATOM = 432.3788 eV, 31.7789 Ry
TITEL = PAW_PBE O 08Apr2002
LULTRA = F use ultrasoft PP ?
IUNSCR = 1 unscreen: 0-lin 1-nonlin 2-no
RPACOR = 1.200 partial core radius
POMASS = 16.000; ZVAL = 6.000 mass and valenz
The tags here do not need to be understood in detail! Only one thing might be noteworthy: In line 4, you can see the valence configuration of oxygen, and in the last line ZVAL = 6 indicates that
there are 6 valence electrons.
Let us now run VASP with the input discussed above! Open a terminal, navigate to this example's directory and run VASP by entering the following:
cd $TUTORIALS/molecules/e01_*
mpirun -np 2 vasp_std
VASP is a Fortran-based program with an executable vasp_std. It can be executed in parallel using mpirun. In order to understand how to control mpirun enter:
How to efficiently parallelize your calculation is an advanced topic.
1.3.1 stdout¶
After you run VASP, the following stdout will be printed to the terminal:
running on 2 total cores
distrk: each k-point on 2 cores, 1 groups
distr: one band on 1 cores, 2 groups
vasp.6.3.0 16May21 (build Oct 09 2021 15:55:16) complex
POSCAR found : 1 types and 1 ions
Reading from existing POTCAR
scaLAPACK will be used
Reading from existing POTCAR
LDA part: xc-table for Pade appr. of Perdew
POSCAR, INCAR and KPOINTS ok, starting setup
FFT: planning ...
WAVECAR not read
entering main loop
N E dE d eps ncg rms rms(c)
DAV: 1 0.384469864424E+02 0.38447E+02 -0.96726E+02 16 0.293E+02
DAV: 2 0.345967752349E+01 -0.34987E+02 -0.34942E+02 32 0.450E+01
DAV: 3 -0.244421465184E+00 -0.37041E+01 -0.34307E+01 16 0.308E+01
DAV: 4 -0.312489880281E+00 -0.68068E-01 -0.66911E-01 16 0.508E+00
DAV: 5 -0.313453098582E+00 -0.96322E-03 -0.96305E-03 32 0.506E-01 0.307E-01
DAV: 6 -0.314470890869E+00 -0.10178E-02 -0.17849E-03 16 0.332E-01 0.155E-01
DAV: 7 -0.314569344422E+00 -0.98454E-04 -0.23774E-04 16 0.137E-01
1 F= -.31456934E+00 E0= -.16030702E+00 d E =-.308525E+00
writing wavefunctions
Let us understand this output! After the setup, it says entering main loop. For each iteration step of the DFT calculation, that is treated by a Davidson iteration scheme, a summary is written:
tag meaning
N iteration count
E total energy
dE change of the total energy
d eps change of the eigenvalues
ncg number of optimization steps to iteratively diagonalize the Hamiltonian
rms weighted absolute value of the residual vector of the KS orbitals
rms(c) absolute value of the charge density residual vector
After convergence is reached, the final line summarizes:
tag meaning
F the total free energy
E0 the converged total energy E
d E the final change of the total energy due to the entropy T*S term discussed below
• The initial electronic charge density is read from the POTCAR file and remains fixed for the first $4$ steps. This is because the Hamiltonian is iteratively diagonalized and, thus, the KS
orbitals used to update the charge density should only be trusted after a warm-up period. Then, the charge density is updated before entering the next iteration step, i.e., not in the final
iteration step. See rms(c) column.
• The default convergence criterion is dE $< 10^{-4}$ and d eps $< 10^{-4}$.
1.3.2 OUTCAR¶
The OUTCAR file is a more detailed log of the calculation. Open the file from the file browser on the left to get a brief look!
The file is divided in parts by lines. The exact information that can be found in the OUTCAR file strongly depends on the calculation. In this example, we can find the following parts:
• Version and execution details.
• Reading of INCAR, POSCAR and POTCAR.
• Analysis of nearest-neighbor distances and symmetries.
• Tags and parameters set by INCAR, POSCAR, POTCAR and KPOINTS or by default.
• Verbose information describing the calculation.
• Information on the lattice, the $\mathbf{k}$ points and the positions.
• Information on the basis set, e.g., number of plane waves.
• Information concerning the nonlocal pseudopotential.
• Details for each DFT iteration step.
• Computed quantities, such as energies, forces, etc.
• Computational time.
On a high-performance computer, you are likely to open your OUTCAR file directly in the terminal from this example's directory, e.g., using vim:
vim opens directly in the terminal window and can be quit by entering :q+enter. To search a string, enter /mystring+enter.
Let us have a closer look at the information concerning the eigenvalues after convergence! Try to find the following information in the OUTCAR file:
E-fermi : -8.8433 XC(G=0): -0.8046 alpha+bet : -0.1463
Fermi energy: -8.8432905119
k-point 1 : 0.0000 0.0000 0.0000
band No. band energies occupation
1 -23.8440 2.00000
2 -8.9042 1.33333
3 -8.9042 1.33333
4 -8.9042 1.33333
5 -0.4679 0.00000
6 1.8630 0.00000
7 1.8630 0.00000
8 1.8630 0.00000
The Fermi energy is the value of E-fermi in eV and must be subtracted from the eigenvalues of the Hamiltonian (band energies) before interpreting these. Then, all states with negative energy are
occupied (occupation). Note that your results might slightly differ, because the chosen settings result in a low precision.
How does the DFT result for the band energies and occupation relate to the occupation of energy levels in the real physical system of an isolated oxygen atom?
Click to see the answer!
Oxygen ([He]2s$^2$2p$^4$) has six valence electrons, which correspond to two 2s electrons and four 2p electrons. From the energies we can infer that the $2s$ electrons are at
$$ -23.8439~\text{eV} + 8.8431~\text{eV} = -15.0008~\text{eV} $$
below the Fermi energy, because 2s is expected to be more tightly bound than 2p. The three 2p orbitals are degenerate and occupied by four electrons. Thus, "the forth" electron is equally likely to
be in any of the 2p orbitals, which is reflected by the occupation of band 2, 3 and 4 reading 1.33333.
Next, let us have a closer look at the information about the free energy at the end of the file:
FREE ENERGIE OF THE ION-ELECTRON SYSTEM (eV)
free energy TOTEN = -0.31456934 eV
energy without entropy= -0.00604470 energy(sigma->0) = -0.16030702
entropy T*S EENTRO = -0.30852464
in the list of energies of the last iteration.
The degeneracy of the 2p orbitals introduces an unphysical entropy (S). And SIGMA, which is the broadening of energy levels introduced by the Gaussian smearing, can be interpreted as an artificial
electronic temperature (T). A smaller value of SIGMA would reduce the entropy, but might slow down convergence. This leads to an unphysical term (T*S) in the free energy.
The reference system for which the pseudopotentials have been generated are isolated, nonspinpolarized atoms. Why is the total energy (energy without entropy) in this calculation close to zero?
Click to see the answer!
In this calculation, the total energy (energy without entropy) of the single, isolated oxygen atom is close to zero. And actually, if the box size were larger and the precision of the calculation
higher, it would go to zero. This is only because all pseudopotentials have been generated for isolated, nonspinpolarized atoms. So we are comparing the reference system with itself. Keep in mind
that, while the choice of the reference system is generally arbitrary, as soon as you do calculations for more atoms or in a smaller box, i.e., not isolated systems, the absolute values of energies
become physically meaningless. In general, only relative energies (band energies w.r.t. Fermi level, total energies of two related systems etc.) can be interpreted and have physical meaning.
1.3.3 Restarting a calculation¶
The default way of restarting a calculation is to read the KS orbitals from the WAVECAR file. Try to restart and check the stdout to confirm that the calculation rapidly converges! For a fresh
calculation, the WAVECAR file should be removed. How can you do that?
Click to see the answer!
In the terminal, navigate to this example's directory and enter:
This is most likely how you will work on a high performance computer, nevertheless you may use the file browser and delete the file by selecting to delete it from the drop down menu.
1.4 Questions¶
1. What does the tag ISMEAR = 0 mean? How is it connected to SIGMA?
2. How many $\mathbf{k}$ points are necessary to describe an isolated atom? Why?
3. How many times is the electronic charge density updated in this example? Why?
4. Where can you find the total energy in the OUTCAR file? Why is this value close to zero in this example?
5. Where can you find the eigenvalues of the Hamiltonian and the Fermi energy in the OUTCAR file?
6. Which file must be in the current directory to restart a calculation from the previous run by default?
By the end of this tutorial, you will be able to:
• switch on spin polarization to run a spin-density-functional-theory (SDFT) calculation
• infer the spin magnetization from the eigenvalues of the Hamiltonian and the occupation of the two spin components
• decide when to run a calculation with vasp_gam
Perform a SDFT calculation for an isolated oxygen atom in order to extract the spin magnetization.
In SDFT the Hamiltonian is a $2\times2$ matrix and the KS orbitals are two component vectors. Here, the two components correspond to spin up and down in the isolated oxygen atom, which are allowed to
have different eigenvalues. The interaction between spin components is effectively considered by separately computing the KS orbitals for both spin components for an effective Hamiltonian that takes
into account both, spin-up and spin-down, charge densities.
The input files to run this example are prepared at $TUTORIALS/molecules/e02_O-SDFT.
Check out the input files, i.e., POSCAR, INCAR, KPOINTS and POTCAR!
O atom in a box
1.0 ! universal scaling parameters
8.0 0.0 0.0 ! lattice vector a(1)
0.0 8.0 0.0 ! lattice vector a(2)
0.0 0.0 8.0 ! lattice vector a(3)
1 ! number of atoms
cart ! positions in cartesian coordinates
SYSTEM = O atom in a box
ISMEAR = 0 ! Gaussian smearing
ISPIN = 2 ! spin polarized calculation
Gamma-point only
Monkhorst Pack
Pseudopotential of O
Open a terminal, navigate to this examples directory and run VASP by entering the following lines into the terminal:
cd $TUTORIALS/molecules/e02_*
mpirun -np 2 vasp_gam
In case the KPOINTS file contains only the Gamma point, one can use the vasp_gam executable instead of vasp_std. This will treat some arrays as real numbers instead of complex numbers, and is thus
computationally cheaper. But mind that the assumption is only valid for calculations where only the Gamma point is considered.
Click to check that you obtain a similar standard output!
running on 2 total cores
distrk: each k-point on 2 cores, 1 groups
distr: one band on 1 cores, 2 groups
vasp.6.3.0 16May21 (build Oct 09 2021 15:55:16) complex
POSCAR found : 1 types and 1 ions
Reading from existing POTCAR
| |
| W W AA RRRRR N N II N N GGGG !!! |
| W W A A R R NN N II NN N G G !!! |
| W W A A R R N N N II N N N G !!! |
| W WW W AAAAAA RRRRR N N N II N N N G GGG ! |
| WW WW A A R R N NN II N NN G G |
| W W A A R R N N II N N GGGG !!! |
| |
| You use a magnetic or noncollinear calculation, but did not specify |
| the initial magnetic moment with the MAGMOM tag. Note that a |
| default of 1 will be used for all atoms. This ferromagnetic setup |
| may break the symmetry of the crystal, in particular it may rule |
| out finding an antiferromagnetic solution. Thence, we recommend |
| setting the initial magnetic moment manually or verifying carefully |
| that this magnetic setup is desired. |
| |
scaLAPACK will be used
Reading from existing POTCAR
LDA part: xc-table for Pade appr. of Perdew
POSCAR, INCAR and KPOINTS ok, starting setup
FFT: planning ...
WAVECAR not read
entering main loop
N E dE d eps ncg rms rms(c)
DAV: 1 0.389725260694E+02 0.38973E+02 -0.10098E+03 32 0.259E+02
DAV: 2 0.317915985579E+01 -0.35793E+02 -0.35786E+02 64 0.438E+01
DAV: 3 -0.119079350189E+01 -0.43700E+01 -0.36688E+01 32 0.328E+01
DAV: 4 -0.126191802522E+01 -0.71125E-01 -0.69187E-01 32 0.508E+00
DAV: 5 -0.126277731595E+01 -0.85929E-03 -0.85922E-03 48 0.504E-01 0.654E+00
DAV: 6 0.164099304113E+00 0.14269E+01 -0.32210E+00 32 0.894E+00 0.152E+00
DAV: 7 -0.112530565028E+01 -0.12894E+01 -0.77968E-01 32 0.398E+00 0.401E-01
DAV: 8 -0.153722308977E+01 -0.41192E+00 -0.11021E-01 48 0.146E+00 0.266E-01
DAV: 9 -0.160894087591E+01 -0.71718E-01 -0.55023E-03 32 0.412E-01 0.129E-01
DAV: 10 -0.167216457827E+01 -0.63224E-01 -0.15822E-02 32 0.576E-01 0.605E-02
DAV: 11 -0.167192460954E+01 0.23997E-03 -0.49458E-04 32 0.131E-01 0.179E-02
DAV: 12 -0.167270841708E+01 -0.78381E-03 -0.10283E-04 32 0.459E-02 0.127E-02
DAV: 13 -0.167295633802E+01 -0.24792E-03 -0.12473E-05 32 0.197E-02 0.961E-03
DAV: 14 -0.167295985216E+01 -0.35141E-05 -0.29861E-06 32 0.108E-02
1 F= -.16729599E+01 E0= -.15958287E+01 d E =-.154262E+00 mag= 1.9998
Click to see the answer!
The warning is related to the default value set for the MAGMOM tag, because we have set ISPIN = 2. Generally receiving a warning does not mean your calculation is faulty, but that you should check if
this setting was intended. Warnings should help to avoid common pitfalls or improve computational performance.
2.3.2 OUTCAR¶
Let us have a look at the eigenvalues of the two spin components in your OUTCAR. Open the file from the file browser or in the terminal using vim:
vim $TUTORIALS/molecules/e02_*/OUTCAR
Then, find the following information:
spin component 1
k-point 1 : 0.0000 0.0000 0.0000
band No. band energies occupation
1 -25.0878 1.00000
2 -10.0830 1.00000
3 -10.0830 1.00000
4 -10.0830 1.00000
5 -0.4932 0.00000
6 1.8213 0.00000
7 1.8303 0.00000
8 1.8303 0.00000
spin component 2
k-point 1 : 0.0000 0.0000 0.0000
band No. band energies occupation
1 -21.8396 1.00000
2 -7.0543 0.33333
3 -7.0543 0.33333
4 -7.0543 0.33333
5 -0.3594 0.00000
6 1.9830 0.00000
7 1.9830 0.00000
8 1.9830 0.00000
Is this system spin-polarized?
Click to see the answer!
Yes, we see that the spin components have different eigenvalues, so the SDFT ground state is spin-polarized. In agreement with Hund’s rules, first each 2p orbital is occupied with one spin direction,
e.g., spin up, as reflected by the occupation of spin component 1. Then, the "forth" 2p electron has spin down, which leads to occupation of 0.33333 in band No. 2-4 of spin component 2.
The spin magnetization is the expectation value of the magnetic moment due to the spin. In other words, it is the projection along the spin-quantization axis, e.g., the $z$ axis. Therefore, the spin
magnetization is $$ \langle \mu_{s,z} \rangle=g_s\langle\frac{1}{2} \sigma_z\rangle\mu_B, $$ where $g_s\approx 2$ is the spin g-factor, $\sigma_z$ is a Pauli matrix, and $\mu_B$ is the Bohr magneton.
What value does $\langle \mu_{s,z} \rangle$ have in this example?
Click to see the answer!
The spin component 1 has two more electrons as seen by comparing the occupation. Hence, $\langle \mu_{s,z} \rangle=2\mu_B$ and the summary, after the convergence has been reached, in the stdout
indeed states mag= 1.9999 corresponding to $\langle \mu_{s,z} \rangle$.
2.4 Questions¶
1. Which value of ISPIN switches spin-polarization off?
2. Are the band energies of spin component 1 and spin component 2 equal? How about the Fermi energy?
3. How can you compute the magnetization based on the occupation of the two spin components?
By the end of this tutorial, you will be able to:
• find the symmetry that is imposed on the electronic charge density
• explain how the imposed symmetry is extracted from the input
• consider whether the total energy of two calculations can be compared
Perform an SDFT calculation for an isolated oxygen atom with orthorhombic symmetry (D$_{2h}$) and compare it to the calculation with cubic symmetry (O$_h$).
There are $32$ crystallographic point groups, which are discrete rotational symmetries of crystals. In order to reduce the computational effort, VASP searches and takes advantage of all symmetries of
the system by default. This helps to efficiently solve large systems, but it is important to ensure that no symmetry is imposed, that the ground state is not expected to have! In particular, even
when the physical system prefers a different symmetry, the use of periodic boundary conditions may impose a symmetry on the solution, if it is not manually switched off.
For more information about the determination of symmetries in VASP read about the ISYM tag.
The input files to run this example are prepared at $TUTORIALS/molecules/e03_O-SDFT-symm. Check the input files!
The point group symmetry is determined such that it is consistent with the structure in the POSCAR file and applied to the solution if the ISYM tag has its default value. The calculation in Example 2
is the desired calculation with cubic symmetry (O$_h$), because all lattice vectors have equal length and enclose $90^\circ$ and the single atom at the origin does not break any symmetry.
O atom in a box
1.0 ! scaling parameters
7.0 0.0 0.0 ! lattice vector 1
0.0 7.5 0.0 ! lattice vector 2
0.0 0.0 8.0 ! lattice vector 3
1 ! number of atoms
Cartesian ! positions in Cartesian coordinates
0 0 0 ! position of the atom
SYSTEM = O atom in a box
ISMEAR = 0 ! Gaussian smearing
ISPIN = 2 ! spin polarized calculation
Gamma-point only
Monkhorst Pack
Pseudopotential of O
Check out the meaning of all tags used in the INCAR file on the VASP Wiki! Does this calculation include the effects of spin-orbit coupling? Or in other words, is the spin degree of freedom coupled
to spacial degrees of freedom?
Click to see the answer!
No! This calculation accounts for spin polarization by treating two spin components separately to arrive at a self-consistent solution for a non-relativistic Hamiltonian. Spin-orbit coupling is a
relativistic correction that can be switched on in VASP using the LSORBIT tag.
Consider the box defined in the POSCAR file! In contrast to the calculation in Example 2 with cubic symmetry (O$_h$), we change the lattice vectors to obey orthorhombic symmetry (D$_{2h}$). In other
words, all lattice vectors have distinct length, but still enclose $90^\circ$.
How can the input be changed to run a calculation with tetragonal symmetry ($D_{4h}$)?
Click to see the answer!
Two lattice vectors must have equal length and all enclose $90^\circ$. POSCAR
O atom in a box
1.0 ! scaling parameters
7.0 0.0 0.0 ! lattice vector 1
0.0 7.0 0.0 ! lattice vector 2
0.0 0.0 8.0 ! lattice vector 3
1 ! number of atoms
Cartesian ! positions in Cartesian coordinates
0 0 0 ! position of the atom
Run VASP with the input given in section 3.2. In the terminal enter the following:
cd $TUTORIALS/molecules/e03_*
mpirun -np 2 vasp_gam
Then, you should check the stdout, that is printed to the terminal, and the OUTCAR file.
3.3.1 OUTCAR¶
Try to find the following symmetry analysis in your OUTCAR file.
ion position nearest neighbor table
1 0.000 0.000 0.000-
LATTYP: Found a simple orthorhombic cell.
ALAT = 7.0000000000
B/A-ratio = 1.0714285714
C/A-ratio = 1.1428571429
Lattice vectors:
A1 = ( 7.0000000000, 0.0000000000, 0.0000000000)
A2 = ( 0.0000000000, 7.5000000000, 0.0000000000)
A3 = ( 0.0000000000, 0.0000000000, 8.0000000000)
Analysis of symmetry for initial positions (statically):
Subroutine PRICEL returns:
Original cell was already a primitive cell.
Routine SETGRP: Setting up the symmetry group for a
simple orthorhombic supercell.
Subroutine GETGRP returns: Found 8 space group operations
(whereof 8 operations were pure point group operations)
out of a pool of 8 trial point group operations.
The static configuration has the point symmetry D_2h.
What symmetry do you find in this example? And what did you find in Example 2?
Click to see the answer!
In this example, VASP finds the orthorhombic symmetry, i.e., point symmetry D_2h, based on the static positions. In Example 2, it found the point symmetry O$_h$.
Now, let us compare the total energy (energy without entropy) found for the orthorhombic symmetry,
energy without entropy = -1.51495945 energy(sigma->0) = -1.59207906,
to the total energy of the calculation with cubic symmetry, which was essentially zero. Which state is more stable?
Click to see the answer!
The calculation with lower symmetry is lower in energy and, thus, presumably closer to the theoretical, approximate many-body ground state. One important choice in DFT is the choice of the
exchange-correlation functional, which effectively accounts for many-body effects. Here, we used the default generalized gradient approximation (GGA), which prefers the symmetry broken solution for
most atoms.
Now, open the INCAR file of this example from the file browser, and add the tag SIGMA = 0.01. Then, run a fresh calculation! What is the result of setting SIGMA = 0.01?
Click to see an alternative way!
You will obtain:
running on 2 total cores
distrk: each k-point on 2 cores, 1 groups
distr: one band on 1 cores, 2 groups
vasp.6.2.1 16May21 (build Jun 06 2021 00:45:36) complex
DAV: 15 -0.189033389318E+01 -0.30871E-03 -0.44399E-05 48 0.517E-02 0.961E-03
DAV: 16 -0.189062353556E+01 -0.28964E-03 -0.39883E-05 40 0.293E-02 0.586E-03
DAV: 17 -0.189067302283E+01 -0.49487E-04 -0.32857E-06 48 0.132E-02
1 F= -.18906730E+01 E0= -.18906730E+01 d E =-.290707E-20 mag= 1.9998
writing wavefunctions
In other words, reducing the default value SIGMA = 0.2 to SIGMA = 0.01 causes the calculation to take more iterations until convergence is reached.
Apart from the symmetry, let us consider the effect of the SIGMA tag on the total energy. Find the following line in your OUTCAR file.
energy without entropy = -1.89067302 energy(sigma->0) = -1.89067302
The value changes significantly! This shows that, if the energy of two calculations shall be compared, both calculations must use the same value of the SIGMA tag for the Gaussian broadening.
3.4 Questions¶
1. How many space-group operations are imposed in the orthorhombic case according to your OUTCAR file?
2. How can the input be changed to run a calculation with tetragonal symmetry ($D_{4h}$)?
3. Can you compare the energies of calculations with different symmetry, if simultaneously SIGMA is changed? | {"url":"https://vasp.at/tutorials/latest/molecules/part1/","timestamp":"2024-11-05T09:07:29Z","content_type":"text/html","content_length":"86173","record_id":"<urn:uuid:08b88c8d-1135-4f30-8309-cf78dc94f96b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00495.warc.gz"} |
BNCC na educação infantil
Então, mesmo que fosse uma história com poucas personagens, eu precisava criar outras para ter a participação de todos. Muitas vezes, era necessário propor acordos com os alunos que se destacavam
mais, para que eles deixassem os demais também participarem. Dessa forma, poderão observar o funcionamento da escrita num contexto especial, como o sarau, no qual a declamação de poesias se torna
algo de grande sentido para elas. Como elas, nesse momento, já saberão os textos de cor, poderão se arriscar na busca pelo ajuste fino entre o falado e o escrito, observando as características e
regularidades do sistema de representação da escrita.
Isso aguçará a criatividade delas, bem como o senso crítico e estético e o conhecimento sobre suas próprias singularidades. Também ampliará seus repertórios, as ajudará a interpretar experiências e
vivências artísticas e a criar as suas próprias produções. Outro ponto importante para a gestão pedagógica é que a base curricular reconhece a Educação Infantil como uma etapa essencial do processo
educativo. O documento identifica objetivamente esse momento — que agora vai dos 0 aos 5 anos do ensino — como uma parte fundamental na formação da identidade e da subjetividade da criança. A BNCC
para a Educação Infantil tem o objetivo de uniformizar as atividades promovidas nas instituições de ensino com o intuito de garantir o desenvolvimento intelectual dos alunos.
Um dos momentos significativos foi a interação e a atenção que foram sendo construídas apesar das dificuldades iniciais. Outro momento significativo foi ver as crianças desenvolverem a linguagem
oral, pela fala e pelos gestos. As etapas do trabalho foram, inicialmente, a investigação e a avaliação da turma, levantando o que, de fato, os alunos já sabiam e o que ainda não sabiam. A turma do
berçário I é esperta; superadas as dificuldades no período da adaptação dos bebês, notamos avanços em seu desenvolvimento.
Adaptação e Aprendizagem e o Método Montessoriano
Na sequência, organizei rodas de leitura e rodas de conversas que funcionavam como ensaios e como meio de encorajar as crianças para a apresentação no Sarau Literário. Elas declamavam umas para as
outras, na sala de leitura ou em outras salas, e até mesmo nas “Sextas Culturais” da escola, como convidadas. Embora o projeto tenha ocorrido ao longo de três meses, ele tem potencial para se tornar
um projeto institucional, acontecendo durante todo o ano. O desenvolvimento da criança como leitora exige que a rotina pedagógica estabeleça continuidade, regularidade e condições para a aquisição do
hábito. Ela continha livros de literatura infantil, lápis de cor, desenhos dos contos literários para pintar e um pequeno relato sobre o projeto.
Viagem pedagógica – Pirenópolis
Nos últimos anos, escolas de educação básica têm passado por importantes mudanças para atender às demandas da nova Base Nacional Comum Curricular (BNCC). Sem dúvida alguma, a Base Nacional Comum
Curricular (BNCC) traz importantes desafios relativos à Educação Infantil. O documento aponta aprendizagens fundamentais que farão a diferença na formação dos estudantes na primeira etapa da Educação
Básica. Depois da etapa inicial de exploração da diversidade de materiais, é interessante organizar, no boião, materiais específicos, visando aprofundar o direito dos bebês de aprenderem e se
desenvolverem, por meios cada vez mais sofisticados de exploração dos objetos. Inspiradas nos materiais coletados e no painel da sala, as crianças também terão elementos para construir novos
materiais e acessórios para suas brincadeiras de faz-de-conta relacionadas à temática do projeto.
Ele pode se dar pela leitura realizada por um adulto de livros de literatura infantil, poemas, contos, cordéis, fábulas etc. Assim, a criança já irá se familiarizando com livros, o que despertará a
sua curiosidade e contribuirá para o desenvolvimento do gosto pela leitura. Por fim, a gestão pode ajudar os professores simplificando o trabalho deles de registro de aulas e atividades, por meio de
um sistema de gestão escolar.
Novas tendências na Educação Infantil para 2022
Assim, os jogos e brincadeiras podem ajudar no processo de construção do conhecimento, quando incluem atividades que favoreçam a troca de sugestões e opiniões das questões e criam situações para o
desenvolvimento da autonomia. Com tudo isso, parece que a escola ainda não compreendeu o valor da brincadeira, isto é, não estão levando a sério o sentido do brincar e a sua importância para o
desenvolvimento da criança. Desse modo, o apoio do professor é essencial para a construção do conhecimento, além da organização de um espaço escolar que favoreça a aprendizagem das crianças. Assim, a
organização desse espaço não é tarefa apenas do professor, mas da equipe escolar como um todo; gestor, coordenador pedagógico, entre outros que possam colaborar para o aperfeiçoamento da prática e o
desenvolvimento intelectual da criança. Ainda que brincar seja considerado um direito de toda criança, muitos pais reclamam que as crianças da pré-escola só vão para a escola brincar e não aprendem a
ler e a escrever, que os professores não ensinam conteúdos e brincam o tempo todo.
Quando percebi o maior interesse dos alunos pelas dramatizações de “O Macaco e a Viola” e de “Os Três Cabritinhos”, combinei com os alunos apresentá-las também para as outras turmas. O auge foi a
apresentação que fizemos da peça “O Macaco e a Viola” na festa do Dia das Mães. Nessas apresentações, as crianças utilizavam a própria fala, plano de aula educação infantil sem ter de decorar o texto
escrito por mim. Outro fato interessante era a facilidade que as crianças tinham para usar qualquer coisa em substituição a um objeto necessário para a peça. Pegavam, por exemplo, um pincel grande
para ser o facão do homem da carroça, um monte de lápis para a lenha ou ainda uma das mochilas como cesto.
616 comments
1. I truly appreciate this article.Really looking forward to read more. Much obliged.
2. Im thankful for the post.Really thank you! Cool.
3. A round of applause for your post. Will read on…
4. Thanks so much for the article.Really thank you! Cool.
5. I think this is a real great article post.Much thanks again. Fantastic.
6. Great article post.Really thank you! Awesome.
7. Very good article post.Much thanks again. Much obliged.
8. I really liked your article post.Really looking forward to read more. Fantastic.
9. I truly appreciate this blog post. Awesome.
10. Thanks for the article.
11. Thanks for sharing, this is a fantastic post.Really looking forward to read more. Great.
12. Great article.Thanks Again. Great.
13. Appreciate you sharing, great blog post.Thanks Again. Much obliged.
14. I cannot thank you enough for the blog.Really thank you! Cool.
15. Very informative article.Really thank you! Keep writing.
16. Thanks again for the blog article. Cool.
17. Im obliged for the article post.Thanks Again. Really Cool.
18. Hey, thanks for the blog.Much thanks again.
19. Say, you got a nice blog article.Really looking forward to read more. Really Great.
20. Really informative blog post. Great.
21. Appreciate you sharing, great post.Thanks Again. Really Cool.
22. wow, awesome blog article. Cool.
23. Very neat article post.Thanks Again. Awesome.
24. Thanks for the article. Want more.
25. I cannot thank you enough for the article post.Thanks Again. Want more.
26. Say, you got a nice blog.Really thank you! Really Cool.
27. Im grateful for the article post.Really thank you! Fantastic.
28. I really like and appreciate your blog article.Thanks Again. Keep writing.
29. Really informative blog post.Really thank you! Really Great.
30. Very informative article.Really looking forward to read more. Much obliged.
31. I appreciate you sharing this blog post. Keep writing.
32. wow, awesome blog post.Thanks Again.
33. Muchos Gracias for your article.Thanks Again. Really Cool.
34. Great, thanks for sharing this article post.Much thanks again. Fantastic.
35. Looking forward to reading more. Great blog post.Really looking forward to read more. Will read on…
36. I really like and appreciate your post.Really thank you!
37. Muchos Gracias for your article. Want more.
38. Im obliged for the blog post.Much thanks again.
39. Thanks for the blog.Much thanks again. Want more.
40. This is one awesome post.Really thank you! Awesome.
41. Im grateful for the blog post.Much thanks again. Great.
42. Im obliged for the blog. Cool.
43. Really appreciate you sharing this article post.Much thanks again. Cool.
44. Hey, thanks for the blog article.Really looking forward to read more. Awesome.
45. Looking forward to reading more. Great post.Thanks Again. Really Great.
46. Really appreciate you sharing this blog post.Thanks Again. Cool.
47. Im thankful for the post.Thanks Again. Cool.
48. Thanks so much for the article post.Really looking forward to read more. Great.
49. Thank you for your article. Really Cool.
50. Thank you ever so for you article post.Thanks Again. Really Cool.
51. Thanks a lot for the post. Want more.
52. Fantastic article post. Want more.
53. Muchos Gracias for your article post.Much thanks again. Great.
54. Im obliged for the blog. Keep writing.
55. wow, awesome blog article.Really thank you! Really Great.
56. I value the blog article.Really looking forward to read more. Really Great.
57. A round of applause for your article.Much thanks again. Really Cool.
58. Awesome article post.Really thank you!
59. This is one awesome blog post.Really looking forward to read more. Really Cool.
60. Great blog post.Really looking forward to read more. Fantastic.
61. Thanks for the article post.Really looking forward to read more.
62. This is one awesome post.Really looking forward to read more. Awesome.
63. I am so grateful for your blog article.Much thanks again. Will read on…
64. Very informative post. Want more.
65. Major thankies for the article post.Really thank you! Much obliged.
66. Great blog.Really looking forward to read more. Great.
67. Thank you for your blog.Really looking forward to read more. Will read on…
68. Muchos Gracias for your blog post.Really thank you! Will read on…
69. Really informative article. Will read on…
70. A round of applause for your blog article.Thanks Again. Awesome.
71. I truly appreciate this article post.Much thanks again. Want more.
72. Very informative blog article. Keep writing.
73. Thank you ever so for you post.Really thank you! Fantastic.
74. Thanks for the blog article.Much thanks again. Will read on…
75. Great, thanks for sharing this post.Much thanks again. Fantastic.
76. Thank you for your post.Really looking forward to read more.
77. I truly appreciate this blog article.Much thanks again.
78. Im obliged for the blog.Really looking forward to read more. Fantastic.
79. This is one awesome article post.Thanks Again. Awesome.
80. Say, you got a nice blog.Really thank you! Great.
81. I loved your article post.Much thanks again. Will read on…
82. Thanks for sharing, this is a fantastic article.Really thank you! Great.
83. This is one awesome blog post.Really thank you! Awesome.
84. I am so grateful for your blog article.Much thanks again. Awesome.
85. A round of applause for your article.Really looking forward to read more. Awesome.
86. Say, you got a nice article post.Thanks Again. Will read on…
87. Thank you ever so for you blog.Really looking forward to read more. Keep writing.
88. Thank you for your blog. Really Cool.
89. Thanks so much for the post.Really thank you! Keep writing.
90. Enjoyed every bit of your post.
91. I really enjoy the article post.Much thanks again.
92. Major thanks for the article post.Really thank you! Keep writing.
93. Fantastic blog. Cool.
94. I cannot thank you enough for the article post. Fantastic.
95. Thank you for your article post.Much thanks again. Want more.
96. Really enjoyed this article post.Really thank you! Awesome.
97. Appreciate you sharing, great article post.Really looking forward to read more. Want more.
98. A big thank you for your blog post. Really Cool.
99. Really informative article.Thanks Again.
100. Very good blog article.Really thank you! Cool.
101. Thanks again for the article.Much thanks again. Cool.
102. Fantastic article post.Thanks Again. Keep writing.
103. Really informative blog post.Thanks Again. Much obliged.
104. I cannot thank you enough for the blog post.Much thanks again. Want more.
105. I appreciate you sharing this blog post.Really looking forward to read more. Great.
106. I truly appreciate this blog post.Really thank you! Really Cool.
107. I really liked your article.Really looking forward to read more. Really Cool.
108. A round of applause for your blog post.Much thanks again. Great.
109. I really enjoy the blog article.Thanks Again. Great.
110. Great, thanks for sharing this blog. Keep writing.
111. Say, you got a nice blog.Really thank you! Awesome.
112. I value the article post.Really thank you! Really Cool.
113. A round of applause for your article. Awesome.
114. Im thankful for the post.Really looking forward to read more. Much obliged.
115. I really enjoy the article.Much thanks again. Cool.
116. Thanks a lot for the post.Really thank you!
117. Great, thanks for sharing this article. Awesome.
118. Fantastic post.Really looking forward to read more. Really Cool.
119. I appreciate you sharing this article.Really thank you! Really Great.
120. Very neat article.Really looking forward to read more. Keep writing.
121. Really appreciate you sharing this article post.Really looking forward to read more. Really Great.
122. Great, thanks for sharing this article post.Really thank you! Really Cool.
123. I really liked your blog article.Much thanks again. Cool.
124. I cannot thank you enough for the blog.Thanks Again. Keep writing.
125. Very neat article.Much thanks again. Great.
126. I am so grateful for your blog post. Awesome.
127. Say, you got a nice article post.Much thanks again. Much obliged.
128. This is one awesome blog article. Keep writing.
129. Thanks so much for the blog article.Really looking forward to read more. Really Cool.
130. Thanks for the article.Really looking forward to read more. Keep writing.
131. I truly appreciate this post.Thanks Again. Much obliged.
132. Say, you got a nice article.Thanks Again.
133. Thanks a lot for the post.Much thanks again. Really Great.
134. I really liked your article post.Really thank you! Really Great.
135. Hey, thanks for the blog article.Thanks Again. Really Cool.
136. Enjoyed every bit of your blog post. Great.
137. Very informative article.Thanks Again. Much obliged.
138. Muchos Gracias for your blog.Really thank you! Awesome.
139. Very informative article.Really thank you! Cool.
140. Great article.Thanks Again. Really Great.
141. Thanks for the blog.Really thank you! Really Cool.
142. Very neat blog post.Really looking forward to read more. Really Cool.
143. Muchos Gracias for your article.Really thank you! Great.
144. Really appreciate you sharing this article post.Thanks Again.
145. Thanks for the post.Really thank you! Great.
146. Thanks for the article.Much thanks again. Really Cool.
147. This is one awesome post.Much thanks again. Want more.
148. Really appreciate you sharing this post.Really thank you! Really Cool.
149. Very informative blog post.Really thank you! Great.
150. Im obliged for the blog article.Really looking forward to read more. Fantastic.
151. Say, you got a nice article.Really thank you! Will read on…
152. Thanks again for the article.Thanks Again. Great.
153. Appreciate you sharing, great post.Thanks Again. Really Great.
154. I loved your blog. Awesome.
155. Thanks-a-mundo for the article post.Really thank you!
156. Thank you for your article post.Really thank you!
157. I value the blog article.Really thank you! Want more.
158. Really enjoyed this blog article.Really thank you!
159. I really like and appreciate your blog article.Thanks Again. Fantastic.
160. Thanks a lot for the blog.Really thank you! Fantastic.
161. Thanks again for the blog post.Really thank you! Cool.
162. Im obliged for the blog.Really thank you! Keep writing.
163. Im grateful for the blog article.Really thank you! Much obliged.
164. Really informative blog.Really thank you! Will read on…
165. Appreciate you sharing, great post.Much thanks again. Will read on…
166. Thanks for the blog article.Really thank you! Awesome.
167. Great, thanks for sharing this blog article.Thanks Again. Really Cool.
168. I really enjoy the blog post. Cool.
169. Thanks a lot for the article post.
170. I truly appreciate this article.Thanks Again. Awesome.
171. Im obliged for the blog.Much thanks again. Much obliged.
172. Thanks-a-mundo for the blog.Thanks Again. Awesome.
173. I really liked your article.Really thank you! Fantastic.
174. I really liked your blog article.Really thank you! Cool.
175. A big thank you for your blog post.Really looking forward to read more.
176. Fantastic blog article.Much thanks again. Really Great.
177. Im obliged for the blog.Really looking forward to read more. Keep writing.
178. Im thankful for the blog article.Really thank you! Keep writing.
179. Very good article.Much thanks again. Fantastic.
180. Very neat article.Really looking forward to read more. Awesome.
181. Im thankful for the blog post.Thanks Again. Much obliged.
182. Say, you got a nice article post.Really thank you! Fantastic.
183. I appreciate you sharing this blog article.Thanks Again. Cool.
184. Very good blog post.Really thank you! Really Great.
185. I really enjoy the blog post. Will read on…
186. Really appreciate you sharing this post.Really looking forward to read more. Want more.
187. Great article.Thanks Again. Awesome.
188. Im thankful for the post.Really looking forward to read more. Much obliged.
189. Say, you got a nice blog article.Much thanks again. Cool.
190. Im grateful for the article.Really thank you! Cool.
191. Major thanks for the blog post. Fantastic.
192. Thanks for sharing, this is a fantastic blog post.Much thanks again.
193. wow, awesome article post.Really looking forward to read more. Really Great.
194. Great, thanks for sharing this article. Much obliged.
195. A round of applause for your post.Really thank you! Really Cool.
196. I think this is a real great blog post.Really thank you! Great.
197. I loved your blog post.Really looking forward to read more. Really Great.
198. Very neat blog.Really looking forward to read more. Will read on…
199. Thanks for sharing, this is a fantastic article.Thanks Again. Fantastic.
200. Im obliged for the blog.
201. Awesome blog post.Much thanks again. Really Cool.
202. Awesome blog post.
203. Appreciate you sharing, great article post.Much thanks again. Fantastic.
204. Thanks for sharing, this is a fantastic article.Much thanks again. Want more.
205. Hey, thanks for the post.Really looking forward to read more. Fantastic.
206. A big thank you for your post.Much thanks again. Really Cool.
207. I think this is a real great post.Really thank you! Will read on…
208. Major thankies for the blog post.Really thank you! Really Cool.
209. I am so grateful for your blog post.Really looking forward to read more. Want more.
210. Great, thanks for sharing this blog post.Really thank you! Really Cool.
211. Great blog post.Really looking forward to read more. Keep writing.
212. I think this is a real great blog.Much thanks again. Much obliged.
213. Im grateful for the blog article.Really thank you! Awesome.
214. I really like and appreciate your blog post. Really Cool.
215. Thanks so much for the article post.Thanks Again. Much obliged.
216. A big thank you for your post.Really thank you! Cool.
217. I appreciate you sharing this article. Really Great.
218. Im obliged for the blog post.Really looking forward to read more. Will read on…
219. Great, thanks for sharing this blog.Much thanks again. Want more.
220. Great, thanks for sharing this blog post.Really thank you! Cool.
221. Enjoyed every bit of your blog article.Really looking forward to read more. Will read on…
222. Muchos Gracias for your post.Thanks Again. Awesome.
223. This is one awesome article.Really thank you!
224. Thank you for your blog article.Much thanks again. Great.
225. Thank you for your blog article. Will read on…
226. This is one awesome blog article.Thanks Again.
227. Very good article post.Really thank you! Awesome.
228. Im thankful for the blog.Really thank you! Really Great.
229. Muchos Gracias for your post.Much thanks again. Fantastic.
230. Thanks for the post.Really thank you! Awesome.
231. Really informative article.Really looking forward to read more. Really Great.
232. Thanks a lot for the article post.Really looking forward to read more. Much obliged.
233. Thanks-a-mundo for the article post. Really Cool.
234. Hey, thanks for the blog article.Really looking forward to read more. Cool.
235. Thanks again for the blog.Really looking forward to read more. Awesome.
236. I truly appreciate this post.Thanks Again.
237. Really informative blog article.Much thanks again. Fantastic.
238. I really enjoy the blog post.Thanks Again. Great.
239. I really like and appreciate your article post.Really looking forward to read more. Awesome.
240. This is one awesome blog article.Thanks Again. Cool.
241. wow, awesome article.Really looking forward to read more. Really Great.
242. I truly appreciate this blog post.Really looking forward to read more. Cool.
243. Very neat blog.Thanks Again. Fantastic.
244. Hey, thanks for the blog post.Much thanks again. Really Cool.
245. I loved your blog article.Really looking forward to read more. Keep writing.
246. A round of applause for your blog article.Thanks Again. Cool.
247. Major thanks for the blog post.Really thank you! Fantastic.
248. Really informative blog. Really Great.
249. Fantastic blog post. Great.
250. Thanks for sharing, this is a fantastic blog.Thanks Again. Great.
251. I value the blog.Really thank you! Keep writing.
252. I cannot thank you enough for the blog article.Thanks Again. Fantastic.
253. Enjoyed every bit of your blog.Really looking forward to read more. Cool.
254. Say, you got a nice post.Really looking forward to read more. Really Great.
255. Im obliged for the blog article.Really looking forward to read more.
256. Really enjoyed this post.Really looking forward to read more. Fantastic.
257. Thanks again for the post.Really looking forward to read more. Will read on…
258. I really enjoy the article post.Really thank you! Awesome.
259. Thanks so much for the blog article.Much thanks again. Cool.
260. Major thanks for the article post.Really looking forward to read more. Cool.
261. I cannot thank you enough for the blog article.Thanks Again. Want more.
262. Awesome article.Really thank you! Want more.
263. Very informative blog post. Will read on…
264. Awesome post. Awesome.
265. Thanks so much for the article.Thanks Again. Great.
266. Major thankies for the article.Much thanks again. Awesome.
267. Really appreciate you sharing this blog post. Really Great.
268. Great, thanks for sharing this post.Really thank you! Fantastic.
269. wow, awesome blog.Really thank you! Awesome.
270. I really liked your post.Really looking forward to read more. Awesome.
271. Really enjoyed this blog post.Really looking forward to read more. Great.
272. wow, awesome article post.Really looking forward to read more. Cool.
273. Thanks a lot for the blog article.Much thanks again. Much obliged.
274. I appreciate you sharing this blog post.Really looking forward to read more. Keep writing.
275. Im grateful for the post. Fantastic.
276. I am so grateful for your blog article.Much thanks again. Awesome.
277. Thanks a lot for the blog.Really thank you! Much obliged.
278. Awesome article.Really looking forward to read more. Will read on…
279. I truly appreciate this article. Awesome.
280. I truly appreciate this post.Thanks Again. Really Great.
281. Thank you for your blog article.Really looking forward to read more. Great.
282. Looking forward to reading more. Great article post.Much thanks again. Cool.
283. I am so grateful for your blog post.Really thank you! Awesome.
284. I really enjoy the blog article.Much thanks again. Cool.
285. Really appreciate you sharing this post.Really thank you! Keep writing.
286. Very neat blog.Really thank you! Cool.
287. I truly appreciate this blog post.Really thank you! Really Great.
288. Thanks for the article.Much thanks again. Keep writing.
289. Major thankies for the blog post.Much thanks again. Really Cool.
290. Thanks for sharing, this is a fantastic post.Thanks Again. Want more.
291. Good post! We are linking to this great article on our site. Keep up the good writing.
292. I truly appreciate this blog.Really looking forward to read more. Cool.
293. Major thanks for the article.Really thank you! Want more.
294. Really appreciate you sharing this article.Much thanks again. Fantastic.
295. Thanks-a-mundo for the article post.Much thanks again. Cool.
296. After I initially commented I appear to have clicked on the -Notify me when new comments are added- checkbox and from now on whenever a comment is added I receive 4 emails with the exact same
comment. Perhaps there is a way you are able to remove me from that service? Cheers.
297. Very neat article.Really thank you!
298. I think this is a real great blog article.Thanks Again. Really Great.
299. wow, awesome article.Much thanks again. Awesome.
300. Muchos Gracias for your article.Really looking forward to read more. Awesome.
301. After exploring a few of the articles on your web page, I really like your way of blogging. I saved it to my bookmark webpage list and will be checking back soon. Please visit my website as well
and let me know how you feel.
302. I think this is a real great blog post.Really looking forward to read more. Will read on…
303. A round of applause for your blog post. Want more.
304. A round of applause for your blog article.Really looking forward to read more. Keep writing.
305. Great information. Lucky me I recently found your blog by accident (stumbleupon). I have book marked it for later!
306. Say, you got a nice blog article.Really thank you! Will read on…
307. Very neat blog article.Really thank you! Really Great.
308. Thanks for the article. Want more.
309. Really informative post. Cool.
310. I value the article.Really thank you! Much obliged.
311. Thank you for your article post. Will read on…
312. I truly appreciate this blog.Really looking forward to read more. Really Great.
313. Say, you got a nice blog.Thanks Again. Awesome.
314. Thanks for the blog article.
315. I loved your blog post.Really thank you! Great.
316. Really enjoyed this article post. Really Great.
317. Can I just say what a comfort to uncover someone that actually understands what they’re talking about over the internet. You definitely know how to bring a problem to light and make it
important. A lot more people really need to check this out and understand this side of your story. I was surprised that you’re not more popular given that you most certainly possess the gift.
318. I think this is a real great blog article.Much thanks again. Fantastic.
319. Im obliged for the article.Really looking forward to read more. Really Cool.
320. A round of applause for your blog post. Awesome.
321. Really appreciate you sharing this article.Thanks Again. Awesome.
322. Very neat article post.Thanks Again. Awesome.
323. Thanks for sharing, this is a fantastic blog. Really Cool.
324. I really enjoy the post.Really looking forward to read more. Fantastic.
325. I really like and appreciate your article post. Fantastic.
326. Very informative blog article.Thanks Again. Will read on…
327. Thanks again for the post. Fantastic.
328. Great blog post.Really thank you! Much obliged.
329. Im grateful for the blog article. Great.
330. Very neat post.Really thank you! Cool.
331. You’re so awesome! I don’t suppose I’ve read through anything like this before. So nice to find another person with a few unique thoughts on this topic. Seriously.. thank you for starting this
up. This site is something that is needed on the internet, someone with some originality.
332. Looking forward to reading more. Great blog post. Really Great.
333. Im thankful for the article post.Really looking forward to read more. Want more.
334. wow, awesome article. Awesome.
335. I would like to thank you for the efforts you’ve put in penning this blog. I really hope to see the same high-grade content from you later on as well. In truth, your creative writing abilities
has inspired me to get my own site now 😉
336. Thanks for sharing, this is a fantastic blog post. Will read on…
337. Thanks for the blog article. Really Great.
338. You ought to take part in a contest for one of the best websites on the web. I most certainly will highly recommend this site!
339. Looking forward to reading more. Great blog post.Really looking forward to read more. Awesome.
340. Fantastic article.Really looking forward to read more. Cool.
341. Appreciate you sharing, great blog article. Really Great.
342. Really appreciate you sharing this post. Much obliged.
343. Really appreciate you sharing this blog post.Thanks Again.
344. Say, you got a nice post.Really looking forward to read more. Want more.
345. There’s definately a lot to learn about this subject. I like all of the points you’ve made.
346. Looking forward to reading more. Great article. Really Great.
347. I really enjoy the blog article.Thanks Again. Really Cool.
348. Appreciate you sharing, great blog.Really looking forward to read more. Keep writing.
349. I really like and appreciate your blog article.Much thanks again. Much obliged.
350. Great, thanks for sharing this blog article. Great.
351. I loved your blog post.Really looking forward to read more. Great.
352. Spot on with this write-up, I absolutely think this web site needs far more attention. I’ll probably be returning to read more, thanks for the info.
353. Muchos Gracias for your article.Thanks Again. Really Great.
354. Thanks-a-mundo for the blog post. Awesome.
355. Thanks for sharing, this is a fantastic blog post. Cool.
356. Hey, thanks for the post.Much thanks again. Great.
357. Muchos Gracias for your blog.Much thanks again. Really Cool.
358. Fantastic blog.
359. Fantastic blog article.Much thanks again. Much obliged.
360. There is certainly a great deal to know about this subject. I really like all the points you have made.
361. Aw, this was a very nice post. Taking the time and actual effort to make a great article… but what can I say… I procrastinate a lot and never manage to get anything done.
362. Looking forward to reading more. Great blog article.Much thanks again. Fantastic.
363. Great article.
364. Thanks-a-mundo for the article.Really looking forward to read more.
365. I think this is a real great blog article.
366. Very good blog post.Much thanks again. Great.
367. Thanks again for the blog.Thanks Again. Awesome.
368. I think this is a real great post.Thanks Again. Awesome.
369. I loved your blog post.Really thank you!
370. Great article post.Much thanks again. Really Great.
371. I loved your blog article.Really thank you! Want more.
372. A big thank you for your blog. Awesome.
373. Great, thanks for sharing this blog article. Really Cool.
374. I cannot thank you enough for the post.Really thank you! Great.
375. I truly appreciate this blog post.Really looking forward to read more. Awesome.
376. Thanks for the post. Want more.
377. Enjoyed every bit of your blog article. Great.
378. I loved your article post.Really thank you! Really Great.
379. Appreciate you sharing, great article. Fantastic.
380. Very informative blog post.Much thanks again. Want more.
381. I truly appreciate this blog.Thanks Again. Great.
382. Hi there! This article could not be written any better! Looking at this article reminds me of my previous roommate! He constantly kept talking about this. I’ll send this post to him. Pretty sure
he will have a good read. I appreciate you for sharing!
383. Thanks for sharing, this is a fantastic blog article.Thanks Again. Fantastic.
384. You made some decent points there. I checked on the web to learn more about the issue and found most individuals will go along with your views on this site.
385. Thank you for your article.Really looking forward to read more. Really Cool.
386. Im grateful for the blog.Really looking forward to read more. Keep writing.
387. Enjoyed every bit of your blog article.Really looking forward to read more. Awesome.
388. Really appreciate you sharing this post.Thanks Again.
389. wow, awesome post.
390. Thanks for sharing, this is a fantastic article post.Really looking forward to read more. Want more.
391. Really informative blog article.Thanks Again. Really Great.
392. An impressive share! I have just forwarded this onto a co-worker who has been conducting a little homework on this. And he in fact ordered me dinner simply because I discovered it for him… lol.
So allow me to reword this…. Thank YOU for the meal!! But yeah, thanks for spending time to talk about this subject here on your website.
393. I think this is a real great blog article.Thanks Again. Want more.
394. I need to to thank you for this wonderful read!! I certainly enjoyed every little bit of it. I have you book-marked to check out new things you post…
395. I really like it when folks get together and share views. Great site, continue the good work!
396. Great, thanks for sharing this article.Really looking forward to read more. Awesome.
397. This excellent website really has all the information I needed about this subject and didn’t know who to ask.
398. I was extremely pleased to find this website. I need to to thank you for ones time for this wonderful read!! I definitely appreciated every little bit of it and I have you saved to fav to look
at new stuff in your site.
399. A big thank you for your article.Really thank you! Keep writing.
400. A big thank you for your post.Thanks Again. Fantastic.
401. I loved your blog article.Really thank you!
402. I blog quite often and I truly thank you for your information. The article has truly peaked my interest. I will bookmark your website and keep checking for new information about once a week. I
subscribed to your Feed as well.
403. I truly appreciate this article post. Want more.
404. Very good blog post.Much thanks again. Great.
405. Great post.
406. I would like to thank you for the efforts you’ve put in penning this website. I am hoping to view the same high-grade content by you later on as well. In fact, your creative writing abilities
has inspired me to get my very own site now 😉
407. I really enjoy the article post. Really Cool.
408. Looking forward to reading more. Great blog.Really looking forward to read more. Awesome.
409. I really like and appreciate your article post.Much thanks again. Awesome.
410. Way cool! Some very valid points! I appreciate you writing this post and the rest of the website is really good.
411. Really appreciate you sharing this blog.
412. I think this is a real great article post. Much obliged.
413. Aw, this was an extremely nice post. Taking the time and actual effort to create a top notch article… but what can I say… I hesitate a whole lot and don’t seem to get nearly anything done.
414. I think this is a real great blog post.Really looking forward to read more. Really Cool.
415. Thanks for the blog post.Thanks Again. Keep writing.
416. Thanks a lot for the post.Thanks Again. Really Great.
417. Im grateful for the post.Really thank you! Really Great.
418. Oh my goodness! Awesome article dude! Thanks, However I am going through difficulties with your RSS. I don’t understand why I cannot join it. Is there anyone else having identical RSS problems?
Anyone who knows the solution will you kindly respond? Thanx!!
419. Hey, thanks for the article post.Really looking forward to read more. Keep writing.
420. I cannot thank you enough for the post.Much thanks again. Will read on…
421. I value the article post.Much thanks again. Really Cool.
422. A round of applause for your blog. Great.
423. Great, thanks for sharing this post.Thanks Again.
424. I appreciate you sharing this blog.Really looking forward to read more. Really Great.
425. Thank you for your blog post.Really thank you!
426. I really like and appreciate your blog article.Really thank you! Really Great.
427. Im grateful for the blog post. Fantastic.
428. I think this is a real great blog article. Really Cool.
429. Thanks for sharing, this is a fantastic article post.Much thanks again.
430. Thanks for sharing, this is a fantastic article post. Will read on…
431. A big thank you for your post.Thanks Again. Really Cool.
432. A round of applause for your post.Really looking forward to read more. Really Great.
433. Hey, thanks for the article. Really Cool.
434. Very neat article.Much thanks again. Much obliged.
435. Your style is very unique compared to other folks I’ve read stuff from. Thank you for posting when you’ve got the opportunity, Guess I will just book mark this web site.
436. A big thank you for your post.Really looking forward to read more. Really Great.
437. This is a topic that is close to my heart… Cheers! Where are your contact details though?
438. I value the article post.Thanks Again. Really Great.
439. Really informative blog article.
440. Great, thanks for sharing this article post.Really thank you! Awesome.
441. Say, you got a nice blog article.Thanks Again. Great.
442. A big thank you for your article post.Really looking forward to read more. Cool.
443. This site was… how do you say it? Relevant!! Finally I have found something which helped me. Cheers.
444. A big thank you for your article post. Want more.
445. Thank you ever so for you blog article.Thanks Again. Really Cool.
446. Thanks so much for the post.Really thank you!
447. Thanks so much for the blog article. Cool.
448. I really like and appreciate your blog post.Really looking forward to read more. Awesome.
449. Im thankful for the blog post.Much thanks again. Will read on…
450. Thank you for your blog article. Awesome.
451. Enjoyed every bit of your post.Much thanks again. Keep writing.
452. Thanks for sharing, this is a fantastic article.Really looking forward to read more. Will read on…
453. Looking forward to reading more. Great blog article.Really looking forward to read more. Keep writing.
454. I cannot thank you enough for the blog article. Really Cool.
455. Thanks so much for the post.Really thank you! Will read on…
456. I want to to thank you for this great read!! I absolutely loved every bit of it. I have got you saved as a favorite to check out new stuff you post…
457. Thank you ever so for you post. Want more.
458. Great, thanks for sharing this article.Really looking forward to read more. Great.
459. I quite like reading through an article that will make people think. Also, thanks for allowing for me to comment.
460. I loved your blog post.Really looking forward to read more. Keep writing.
461. I loved your blog.Thanks Again. Will read on…
462. Great, thanks for sharing this article post.Much thanks again. Really Great.
463. Great post. Want more.
464. Wow, great post.Really thank you! Keep writing.
465. Fantastic article post.Thanks Again. Will read on…
466. Im obliged for the blog.Thanks Again. Fantastic.
467. A big thank you for your post. Really Cool.
468. I think this is a real great article.Thanks Again.
469. Very informative blog.Much thanks again. Cool.
470. Im obliged for the article. Awesome.
471. Im obliged for the article post.Much thanks again. Really Great.
472. bookmarked!!, I really like your web site.
473. wow, awesome blog article.Thanks Again. Cool.
474. I really enjoy the post.Really thank you! Really Great.
475. Really appreciate you sharing this blog post.Thanks Again. Want more.
476. Hey, thanks for the blog post.Thanks Again. Great.
477. Say, you got a nice blog post.Thanks Again.
478. Major thanks for the article post.Thanks Again. Keep writing.
479. Wow, great article.Really thank you! Cool.
480. I really like and appreciate your blog. Much obliged.
481. This is one awesome article.Much thanks again. Much obliged.
482. Very good article post.Really thank you!
483. Really informative blog.Really thank you!
484. Very good blog post.Thanks Again. Awesome.
485. Really informative blog post.Really thank you! Awesome.
486. Im grateful for the post.Thanks Again. Cool.
487. A round of applause for your article. Will read on…
488. Really appreciate you sharing this article.Really looking forward to read more. Keep writing.
489. Im thankful for the post.Really thank you! Really Cool.
490. Awesome article.Thanks Again.
491. I really like and appreciate your article post.Really looking forward to read more.
492. I really enjoy the blog article.Thanks Again. Awesome.
493. Say, you got a nice post.Thanks Again. Keep writing.
494. I think this is a real great blog. Really Great.
495. I really enjoy the article post.Really looking forward to read more.
496. Thank you for your article.Really thank you! Want more.
497. I loved your blog post.Thanks Again. Awesome.
498. Im thankful for the article. Really Cool.
499. Major thankies for the post.Much thanks again. Keep writing.
500. Thanks for sharing, this is a fantastic blog post.Really thank you!
501. I appreciate you sharing this blog article.Really looking forward to read more. Will read on…
502. Say, you got a nice blog.Really looking forward to read more. Will read on…
503. Thanks for the blog.Thanks Again. Want more.
504. Im thankful for the post.Really looking forward to read more. Fantastic.
505. Very informative blog.Much thanks again. Really Great.
506. Enjoyed every bit of your blog.Thanks Again. Really Great.
507. Major thankies for the article post. Keep writing.
508. Im obliged for the blog article.Thanks Again. Cool.
509. I really like and appreciate your blog post.Really looking forward to read more. Really Cool.
510. Very good info. Lucky me I discovered your website by chance (stumbleupon). I have bookmarked it for later.
511. Wow, great article post. Really Great.
512. I cannot thank you enough for the blog article.Thanks Again. Fantastic.
513. Im obliged for the article.Thanks Again. Much obliged.
514. Muchos Gracias for your blog post.Really thank you! Want more.
515. I appreciate you sharing this blog.Thanks Again. Will read on…
516. Thank you for your post.Really thank you! Want more.
517. Major thanks for the post.Thanks Again. Really Great.
518. Very good blog post.Thanks Again. Fantastic.
519. Looking forward to reading more. Great blog post.Really thank you! Awesome.
520. I really liked your post.Thanks Again. Great.
521. Very neat article.Much thanks again. Much obliged.
522. This is one awesome blog.Really thank you! Want more.
523. I value the blog post. Really Great.
524. Really appreciate you sharing this blog.Thanks Again. Keep writing.
525. Hey, thanks for the blog post.Really thank you!
526. I cannot thank you enough for the article post.Really looking forward to read more. Great.
527. Really appreciate you sharing this blog article.Thanks Again. Great.
528. Im grateful for the blog article.Thanks Again. Keep writing.
529. I loved your article post.Much thanks again. Want more.
530. I truly appreciate this post.Really thank you!
531. Im thankful for the blog.Really thank you! Will read on…
532. Im obliged for the blog post.Really thank you! Cool.
533. Im grateful for the article.Much thanks again. Much obliged.
534. Thank you ever so for you blog article.Really thank you! Really Cool.
535. Thanks a lot for the article post. Cool.
536. Great article post.Really thank you! Fantastic.
537. Fantastic article post. Really Great.
538. Really enjoyed this article. Great.
539. Thanks for sharing, this is a fantastic post.Really looking forward to read more. Awesome.
540. Really informative post.Really thank you!
541. Thanks-a-mundo for the blog post.Much thanks again. Really Cool.
542. Major thanks for the blog article.Much thanks again. Want more.
543. Really informative post.Much thanks again. Keep writing.
544. Thank you for your blog.Really thank you! Fantastic.
545. Great blog post. Want more.
546. Awesome blog post.Really thank you! Keep writing.
547. Appreciate you sharing, great blog article. Cool.
548. Im grateful for the blog post.Much thanks again. Want more.
549. I cannot thank you enough for the post.Much thanks again. Cool.
550. Thanks so much for the article post.Much thanks again. Much obliged.
551. Im thankful for the blog article. Want more.
552. Fantastic post.Really looking forward to read more. Want more.
553. I am so grateful for your blog. Really Cool.
554. A round of applause for your blog post. Fantastic.
555. I am so grateful for your article post.Thanks Again. Fantastic.
556. I loved your blog.Really thank you! Keep writing.
557. I really enjoy the post.Really looking forward to read more. Cool.
558. That is a really good tip particularly to those new to the blogosphere. Simple but very precise information… Thanks for sharing this one. A must read article!
559. Enjoyed every bit of your blog post.Much thanks again. Awesome.
560. Very informative post.Really looking forward to read more. Much obliged.
561. It’s nearly impossible to find educated people on this topic, but you seem like you know what you’re talking about! Thanks
562. Im grateful for the article.Much thanks again. Fantastic.
563. Wonderful post! We are linking to this great post on our website. Keep up the good writing.
564. Great article. I will be going through some of these issues as well..
565. Thanks so much for the blog article.Really thank you! Cool.
566. I am so grateful for your blog post. Will read on…
567. I loved your blog.Much thanks again. Cool.
568. Im obliged for the article post. Fantastic.
569. Thank you ever so for you post.Much thanks again. Fantastic.
570. Major thanks for the blog. Awesome.
571. Thank you for your blog.Much thanks again. Really Great.
572. I truly appreciate this article.Really thank you! Really Great.
573. Hey, thanks for the post.Much thanks again. Keep writing.
574. Wow, great post. Really Cool.
575. After I originally commented I seem to have clicked the -Notify me when new comments are added- checkbox and now each time a comment is added I receive four emails with the same comment. There
has to be an easy method you are able to remove me from that service? Cheers.
576. This is one awesome post. Keep writing.
577. Hey, thanks for the blog article.Thanks Again. Want more.
578. I really enjoy the post.Much thanks again. Great.
579. Thanks again for the article post.Much thanks again. Will read on…
580. I really enjoy the article.Much thanks again. Awesome.
581. Thanks a lot for the blog article.Much thanks again. Awesome.
582. Thanks so much for the blog article.Really thank you!
583. I loved your article. Cool.
584. Looking forward to reading more. Great blog article.Really thank you! Fantastic.
585. Enjoyed every bit of your blog article. Great.
586. Awesome blog post.Really looking forward to read more. Much obliged.
587. Really enjoyed this blog.Much thanks again. Want more.
588. Im grateful for the blog article. Will read on…
589. Thanks so much for the blog.Really thank you! Much obliged.
590. I truly appreciate this blog.Much thanks again. Keep writing.
591. Really enjoyed this article post.Really looking forward to read more. Fantastic.
592. I really like and appreciate your blog article.Much thanks again. Cool.
593. Say, you got a nice blog article.Really looking forward to read more. Great.
594. Looking forward to reading more. Great blog post.Thanks Again. Will read on…
595. A round of applause for your article.Really thank you! Much obliged.
596. Thank you for your blog article.Thanks Again. Great.
597. Great information. Lucky me I discovered your site by accident (stumbleupon). I’ve saved it for later!
598. Looking forward to reading more. Great post. Cool.
599. Enjoyed every bit of your post.Really looking forward to read more. Want more.
600. Muchos Gracias for your blog article.Really looking forward to read more. Cool.
601. Thanks so much for the blog post. Really Great.
602. Very good article. Much obliged.
603. Very informative article post.Really looking forward to read more. Fantastic.
604. Thanks again for the blog article.Really looking forward to read more. Cool.
605. Your article helped me a lot, is there any more related content? Thanks!
606. Hi! I just wish to offer you a big thumbs up for your great information you have got here on this post. I am returning to your web site for more soon.
607. Enjoyed every bit of your post. Cool.
608. wow, awesome article post.Much thanks again. Cool.
609. I cannot thank you enough for the blog.Thanks Again. Want more.
610. Hey, thanks for the post.Much thanks again. Fantastic.
611. Major thanks for the blog post.Thanks Again. Great.
612. Thanks for the post.Thanks Again. Really Cool.
613. Thanks for the blog.Thanks Again. Fantastic.
614. I truly appreciate this article. Fantastic.
615. Fantastic post.Really thank you!
616. Thanks again for the article.Thanks Again. Fantastic. | {"url":"https://aservicodaindustria.com.br/bncc-na-educacao-infantil/","timestamp":"2024-11-06T15:31:56Z","content_type":"text/html","content_length":"1014350","record_id":"<urn:uuid:ffb89fb6-0ce9-4530-b084-1c14e5a70c36>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00267.warc.gz"} |
WeBWorK Standalone Renderer
You are given the four points in the plane $A = (-5,2)$, $B = (0,-8)$, $C = (4,2)$, and $D = (8,-4)$. The graph of the function $f(x)$ consists of the three line segments $AB$, $BC$ and $CD$. Find
the integral $\displaystyle \int_{-5}^{8} f(x)\,dx$ by interpreting the integral in terms of sums and/or differences of areas of elementary figures.
$\displaystyle \int_{-5}^{8} f(x)\,dx =$ | {"url":"https://wwrenderer.libretexts.org/render-api?sourceFilePath=Library/ma122DB/set11/s5_2_29.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&answersSubmitted=0&showSummary=1&displayMode=MathJax&language=en&outputFormat=nosubmit","timestamp":"2024-11-03T10:00:35Z","content_type":"text/html","content_length":"5907","record_id":"<urn:uuid:d62c0ff5-0648-4601-bdc1-3a4d67f0e8fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00029.warc.gz"} |
Digital Math Resources
Title Description Thumbnail Curriculum Topics
Relate Fractions to Decimals
Percents and Numerical Expressions
Algebra Tiles--Expressions and Equations
Numerical Expressions
Rational Expressions
Rational Expressions, Sequences, Series, Polynomial Functions and Equations, Graphs of Quadratic Functions, Quadratic
Equations and Functions, Solving Systems of Equations, Trig Expressions and Identities, Probability, Geometric
Overview This colle Constructions with Triangles, Composite Functions, Geometric Constructions with Angles and Planes, Distance Formula,
Data Analysis, Slope, Special Functions, Trigonometric Functions, Graphs of Exponential and Logarithmic Functions,
Radical Functions and Equations, Rational Functions and Equations, Slope-Intercept Form, Coordinate Systems, Graphs of
Linear Functions, Inequalities, Matrix Operations and Midpoint Formula
Numerical and Algebraic Expressions and Numerical Expressions
Compare and Order Fractions, Fractions and Mixed Numbers and Identify and Name Fractions
Addition Facts to 25, Addition Facts to 100, Subtraction Facts to 100, Add and Subtract Fractions, Graphs of Linear
Functions, Slope-Intercept Form, Graphs of Quadratic Functions and Numerical and Algebraic Expressions
Numerical Expressions
Closed Captioned Video: Algebra Tiles: Adding Integers Using
Algebra Tiles
Video Tutorial: Algebra Tiles: Adding Integers Using Algebra Algebra Tiles--Expressions and Equations
Tiles. In this tutorial, review the basic definition of what
algebra tiles are and how they are used. Then the video focuses
on how to add integers using algebra tiles.
Closed Captioned Video: Algebra Tiles: Modeling Negative
Integers Using Algebra Tiles Description
Video Tutorial: Algebra Tiles: Modeling Negative Integers Using Algebra Tiles--Expressions and Equations
Algebra Tiles. In this tutorial, review the basic definition of
what algebra tiles are and how they are used.
Closed Captioned Video: Algebra Tiles: Modeling Positive
Integers Using Algebra Tiles Description
Video Tutorial: Algebra Tiles: Modeling Positive Integers Using Algebra Tiles--Expressions and Equations
Algebra Tiles. In this tutorial, review the basic definition of
what algebra tiles are and how they are used.
Closed Captioned Video: Algebra Tiles: Multiplying Integers
Using Algebra Tiles Description
This is part of a collection of video tutorials the topic of Algebra Tiles--Expressions and Equations
Algebra Tiles.This series of videos describes what algebra tiles
are and how they can be used to model numbers, operations,
expressions, and equations.
Closed Captioned Video: Algebra Tiles: Subtracting Integers
Using Algebra Tiles
Video Tutorial: Algebra Tiles: Subtracting Integers Using Algebra Tiles--Expressions and Equations
Algebra Tiles. In this tutorial, review the basic definition of
what algebra tiles are and how they are used. Then the video
focuses on how to subtract integers using algebra tiles.
Closed Captioned Video: Integers: Adding Integers
Video Tutorial: Integers: Adding Integers. In this video
students learn to add integers.
Numerical Expressions
This is part of a series of videos on the topic of Integers.
This includes defining integers, modeling integers, integer
operations, and integer expressions.
Closed Captioned Video: Integers: Comparing and Ordering
Video Tutorial: Integers: Comparing and Ordering Integers. In Numerical Expressions
this video students learn how to compare and order integers
using various techniques.
Closed Captioned Video: Integers: Dividing Integers
Video Tutorial: Integers: Dividing Integers. In this video
students learn to divide integers.
Numerical Expressions
This is part of a series of videos on the topic of Integers.
This includes defining integers, modeling integers, integer
operations, and integer expressions.
Closed Captioned Video: Integers: Integers and Absolute Value
Video Tutorial: Integers: Integers and Absolute Value. In this Numerical Expressions
video students continue their exploration of integers by
investigating absolute value.
Closed Captioned Video: Integers: Integers and Exponents
Video Tutorial: Integers: Integers and Exponents. In this video, Numerical Expressions
students explore the use of integers as exponents with different
Closed Captioned Video: Integers: Integers on a Number Line
This is part of a series of videos on the topic of Integers. Numerical Expressions
This includes defining integers, modeling integers, integer
operations, and integer expressions.
Closed Captioned Video: Integers: Integers on the Cartesian
Coordinate System
Video Tutorial: Integers on the Cartesian Coordinate System. In Numerical Expressions
this video, students learn to graph integer-based coordinates on
a Cartesian Coordinate plane.
Closed Captioned Video: Integers: Multiplying Integers
Video Tutorial: Integers: Multiplying Integers. In this video,
students learn to multiply integers.
Numerical Expressions
This is part of a series of videos on the topic of Integers.
This includes defining integers, modeling integers, integer
operations, and integer expressions.
Closed Captioned Video: Integers: Numerical Expressions with
Video Tutorial: Integers: Numerical Expressions with Integers. Numerical Expressions
In this video, students simplify various numerical expressions
that use integers.
Closed Captioned Video: Integers: Subtracting Integers
Video Tutorial: Integers: Subtracting Integers. In this video,
students learn to subtract integers. Numerical Expressions
This is part of a series of videos on the topic of Integers.
This includes defining integers, modeling integers, integer
operations, and integer expressions. | {"url":"https://www.media4math.com/MAFS.7.NS.1.2","timestamp":"2024-11-07T13:27:56Z","content_type":"text/html","content_length":"92355","record_id":"<urn:uuid:87743f81-25d8-4080-b5fa-998ea137ecda>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00895.warc.gz"} |
University of Washington
NumericalUniversality is a Mathematica library that contains a database, algorithms and random matrix distributions for examining two-component universality in the halting times for these algorithms.
NumericalUniversality Repository
A full wiki article is available above to describe the use of the package.
Two-component universality in the QR eigenvalue algorithm
ISTPackage is a Mathematica package that contains various routines for effectively evaluating the inverse scattering transform. Thus far it has been implemented for the Korteweg-de Vries (KdV)
equation, the focusing and defocusing modified KdV equations and the focusing and defocusing nonlinear Schrödinger (NLS) equations. It also contains a routine for the linear Schrödinger equation. The
code can be found on bitbucket:
ISTPackage Repository
The package allows for the computation of highly oscillatory solutions. The animation below includes a picture-in-picture evolution of a solution of the KdV equation. The animation tracks both the
dispersive tail moving to the left and the solitons moving to the right.
Hill's Method
The following is a Python-based implementation of Hill's method. The software includes a GTK-based GUI. This project is no longer being maintained.
Hill's Method in Python | {"url":"http://faculty.washington.edu/trogdon/software.html","timestamp":"2024-11-02T11:37:08Z","content_type":"text/html","content_length":"8391","record_id":"<urn:uuid:43d31e82-2d52-40eb-8ca6-7ba19e2df734>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00604.warc.gz"} |
All About Quantity of cement and sand is required for 100 sqft plastering
Achieving a smooth, durable and aesthetically pleasing plaster finish is a crucial aspect of any construction project. And a key component in achieving this is the right quantity of cement and sand
for plastering. Often overlooked, the correct proportion of these materials can greatly impact the strength and longevity of plaster, making it an essential factor to consider for any builder or
homeowner. In this article, we will delve into the details of how much cement and sand are required for plastering 100 sqft and provide some helpful tips to ensure a successful plastering project.
Quantity of cement and sand is required for 100 sqft plastering
Plastering is a process of applying a thin layer of plaster over a surface like walls or ceilings to provide a smooth and even finish. It is an essential step in the construction of buildings and is
done to protect the underlying surface, to improve its appearance, and provide a base for painting or other finishes.
The quantity of cement and sand required for plastering depends on various factors such as the type of surface to be plastered, the thickness of the plaster, and the type of plastering mix used. In
general, a standard mix of cement and sand in the ratio of 1:4 or 1:6 is used for plastering.
For a surface area of 100 square feet, the quantity of cement and sand needed for plastering can be calculated as follows:
Step 1: Determine the thickness of the plaster
The thickness of plaster is usually measured in millimeters (mm). A standard thickness for plastering is 12 to 15 mm. Let’s take 12 mm as the thickness for our calculation.
Step 2: Calculate the volume of plaster
The volume of plaster can be calculated by multiplying the surface area to be plastered (100 square feet in this case) with the thickness of plaster (12 mm in this case). 100 square feet = 9.29
square meters. Therefore, the volume of plaster required would be 9.29 x 0.012 = 0.111 square meters.
Step 3: Calculate the quantities of material required
As mentioned earlier, a standard mix of 1:4 or 1:6 (cement:sand) is used for plastering. Let’s take the example of 1:6 mix for our calculation.
Quantity of cement: The quantity of cement needed for a 1:6 mix is 1/7th of the volume of plaster. Therefore, the quantity of cement required would be 1/7 x 0.111 = 0.0159 cubic meters. As 1 cubic
meter of cement weighs approximately 1440 kg, the weight of cement needed would be 23 kilograms (0.0159 x 1440 = 22.9 kg).
Quantity of sand: To calculate the quantity of sand required, we need to multiply the volume of plaster (0.111 cubic meters) with the ratio of sand in the mix, i.e., 6. Therefore, the quantity of
sand would be 0.111 x 6 = 0.666 cubic meters. As 1 cubic meter of sand weighs approximately 1650 kg, the weight of sand required would be 1099 kilograms (0.666 x 1650 = 1099 kg).
Therefore, for a surface area of 100 square feet and a plaster thickness of 12 mm, the quantity of cement and sand required for plastering would be approximately 23 kilograms of cement and 1099
kilograms of sand.
It is always advisable to add an extra 5-10% of material to account for wastage during the mixing and application process. Hence, the final quantity of cement and sand required would be slightly
In conclusion, the quantity of cement and sand required for 100 square feet of plastering would depend on the thickness of the plaster and the type of mix used, but for standard 12 mm plaster with a
1:6 mix, approximately 23 kilograms of cement and 1099 kilograms of sand would be needed.
How much quantity of cement and sand is required for 100 sq ft plastering
Plastering is a crucial step in the construction process that involves covering the walls and ceilings with a smooth and even layer of cement mortar. It not only enhances the aesthetic appeal of the
building but also provides protection from moisture and other external factors.
When it comes to plastering a surface, the two main materials required are cement and sand. The amount of cement and sand needed for plastering a specific area depends on various factors such as the
thickness of the plaster, surface conditions, and wastage. In this article, we will discuss the approximate quantity of cement and sand required for plastering 100 sq ft of surface.
Quantity of Cement:
The amount of cement needed for plastering is calculated in terms of bags. Generally, one bag of cement weighs 50 kg, and for plastering, a mix of 1:6 (cement:sand) is used. Therefore, for plastering
100 sq ft of surface with a thickness of 12 mm, the following formula can be used:
Quantity of Cement (in bags) = [Area (in sq ft) x Thickness of plaster (in inches) / 12] x 0.43
= [100 x (12/12)] x 0.43
= 100 x 0.43
= 43 bags (approx.)
Quantity of Sand:
The quantity of sand needed for plastering is also calculated in terms of bags. For the sand-cement mix of 1:6, the volume of sand required is six times that of cement. Therefore, for plastering 100
sq ft of surface with a thickness of 12 mm, the following formula can be used:
Quantity of Sand (in bags) = 6 x Quantity of Cement (in bags)
= 6 x 43 bags
= 258 bags
= 1290 kg (approx.)
Note: It is important to keep in mind that the above calculations are based on an estimated wastage of 5-10%. If the surface is uneven or porous, the wastage can be higher, and the quantity of
materials required will increase accordingly.
In conclusion, for plastering 100 sq ft of surface with a thickness of 12 mm, approximately 43 bags of cement and 258 bags (1290 kg) of sand will be required. It is also essential to note that these
quantities may vary depending on the quality of materials and the mixing ratio. It is always recommended to consult a professional or use a cement quantity calculator to get an accurate estimation of
the materials needed for plastering.
How much sand & cement required for 100 square feet internal wall plastering
Plastering is an important process in the construction of a building. It is a finishing layer applied on the internal surfaces of walls and ceilings to provide a smooth and even surface. Sand and
cement are the two main materials used for internal wall plastering, and their proper proportion is crucial for the quality of the plaster.
In general, the amount of sand and cement required for 100 square feet of internal wall plastering depends on several factors such as the thickness of the plaster, the type of sand and cement used,
and the level of skill of the plasterer. However, there is a standard ratio that is commonly used in most construction projects which is 1:6. This means that for every one part of cement, six parts
of sand are required.
To calculate the exact amount of sand and cement needed for 100 square feet of internal wall plastering, the following steps can be followed:
1. Determine the thickness of the plaster – The thickness of the plaster is an important factor as it directly affects the quantity of materials required. Generally, the thickness of the plaster is
kept between 12 to 15 millimeters.
2. Calculate the volume of plaster needed – Multiply the thickness of the plaster by the area to get the volume. For example, if the thickness of the plaster is 15 mm, the volume required for 100
square feet wall will be 15 x 100 = 1500 cubic feet.
3. Calculate the quantity of cement – As mentioned, the standard ratio for sand and cement is 1:6. Therefore, the quantity of cement needed will be 1/7 x 1500 = 214.29 cubic feet.
4. Calculate the quantity of sand – The quantity of sand needed is 6/7 x 1500 = 1285.71 cubic feet.
5. Convert the volume to bags – Cement and sand are commonly sold in bags containing specific volumes. To calculate the number of bags needed, divide the volume of cement and sand by the volume of
one bag. For example, a typical bag of cement in India contains 50 kilograms, which is approximately 1.25 cubic feet. Therefore, the number of bags needed will be 214.29/1.25 = 172 bags of cement and
1280/1.25 = 1028 bags of sand.
In conclusion, for 100 square feet of internal wall plastering, approximately 172 bags of cement and 1028 bags of sand will be required. It is recommended to buy a little extra material (around 10%
to 15%) to account for any wastage during the plastering process. It is also important to note that the actual amount of materials needed may vary depending on the factors mentioned above and it is
always best to consult with a professional before starting any construction work.
Cement mortar calculation for ceiling plastering
Cement mortar is a commonly used mixture in construction for various applications, with ceiling plastering being one of them. The purpose of using cement mortar for ceiling plastering is to provide a
smooth and even surface, add strength and durability to the ceiling, and create a protective barrier against moisture and other elements.
The calculation of cement mortar for ceiling plastering involves a few key factors such as the surface area of the ceiling, the thickness of the plaster, and the ratio of cement to sand. Here is a
step-by-step guide on how to calculate the amount of cement mortar required for ceiling plastering:
Step 1: Measure the Surface Area
The first step is to measure the total surface area of the ceiling to be plastered. This can be done by multiplying the length and width of the ceiling. Make sure to take accurate measurements in
feet or meters.
Step 2: Determine the Thickness of Plaster
The next step is to determine the thickness of the plaster layer. This can vary depending on the type of ceiling and the surface condition. For standard ceilings, a thickness of 10-12mm is usually
Step 3: Calculate the Volume of Plaster
The volume of plaster needed can be calculated by multiplying the surface area with the thickness. For example, if the surface area is 100 square feet and the thickness of the plaster is 10mm
(0.01m), then the volume of plaster required would be 100 x 0.01 = 1 cubic meter.
Step 4: Determine the Ratio of Cement to Sand
The cement-sand ratio is crucial in determining the strength and workability of the plaster. The most commonly used ratio for ceiling plastering is 1:4, where one part of cement is mixed with four
parts of sand.
Step 5: Calculate the Quantity of Cement and Sand
Using the ratio determined in the previous step, the amount of cement and sand needed can be calculated. For the given example, the total quantity of cement needed would be 1/5 x 1 = 0.2 cubic meters
and the quantity of sand would be 4/5 x 1 = 0.8 cubic meters.
Step 6: Convert to Weight
Lastly, the volume of cement and sand calculated in cubic meters needs to be converted to weight in kilograms. This can be done by multiplying the volume with the density of the material. The density
of cement is 1440 kg/m3 and for sand, it is 1600 kg/m3. Therefore, the weight of cement required would be 0.2 x 1440 = 288 kg and the weight of sand would be 0.8 x 1600 = 1280 kg.
In conclusion, the calculation of cement mortar for ceiling plastering is essential to ensure proper mix proportions and optimize material usage. It is important to note that these calculations are
approximate and may vary depending on the specific requirements of the project. It is always recommended to consult a structural engineer or a construction professional for accurate calculations.
How much sand & cement required for 100 square feet ceiling plastering
Ceiling plastering is an important step in any construction or renovation project. It involves applying a layer of plaster material to the ceiling surface in order to provide a smooth and finished
look. The amount of sand and cement required for ceiling plastering will depend on various factors such as the thickness of the plaster, the size of the ceiling area, and the type of plaster mixture
used. In this article, we will discuss how much sand and cement are required for 100 square feet of ceiling plastering.
1. Calculate the area of ceiling:
Before determining the quantity of sand and cement required, it is important to calculate the total area of the ceiling that needs to be plastered. This can be easily done by multiplying the length
and width of the ceiling in feet. For example, if the ceiling is 10 feet long and 10 feet wide, the total area would be 100 square feet.
2. Determine the thickness of the plaster layer:
The next crucial step is to decide the thickness of the plaster layer. This will depend on the type of plaster mixture used and the condition of the ceiling surface. Generally, the thickness of a
ceiling plaster layer can range from 10-15 mm. For our calculation, let’s assume a thickness of 12 mm.
3. Calculate the volume of plaster:
To determine the volume of plaster, we need to multiply the area of the ceiling by the thickness of the plaster layer. In this case, the volume of plaster required would be 100 square feet x 12 mm =
1200 cubic feet.
4. Calculate the quantity of sand and cement:
The standard ratio of sand to cement for ceiling plastering is 3:1, which means for every 3 parts of sand, 1 part of cement is required. Therefore, to determine the quantity of sand and cement, we
need to multiply the total volume of plaster by the ratio of sand and cement.
Sand required = 1200 cubic feet x (3/4) = 900 cubic feet
Cement required = 1200 cubic feet x (1/4) = 300 cubic feet
5. Convert cubic feet to bags:
Sand and cement are usually measured in bags, so the final step is to convert the cubic feet into bags. One bag of cement is equal to 0.035 cubic meters, and one bag of sand is equal to 0.048 cubic
Sand required = (900 x 0.048) / 1.5 = 28.8 bags
Cement required = (300 x 0.035) / 1.25 = 8.4 bags
Therefore, for 100 square feet ceiling plastering, you would need approximately 29 bags of sand and 8 bags of cement.
Note: The above calculation is based on the assumption that the ceiling surface is in good condition and does not require any repair work. If there are any cracks or holes, you would need to add
extra sand and cement for patching before applying the plaster layer.
In conclusion, the amount of sand and cement required for ceiling plastering will vary depending on the size of the ceiling, the thickness of the plaster layer, and the type and condition of the
ceiling surface. It is always recommended to consult a professional or experienced contractor to accurately determine the quantity of materials needed for your specific project.
Cement mortar calculation for external wall plastering
Cement mortar is one of the most commonly used building materials for external wall plastering. It is a mixture of cement, sand, and water that is used to create a smooth and even surface on external
walls. The quality and durability of cement mortar can greatly impact the overall strength and stability of a building.
The calculation for mortar quantity required for external wall plastering can be done by following a few simple steps:
1. Determine the area of the external wall: The first step in calculating cement mortar quantity is to determine the area of the external wall that needs to be plastered. Measure the length and
height of the wall in meters and multiply them to get the total area in square meters (m2).
2. Determine the thickness of the plaster: The thickness of the plaster depends on the type of wall and the finish required. Typically, the thickness of plaster for external walls varies between 12
mm to 18 mm.
3. Calculate the volume of mortar: Once you have the area and thickness, you can calculate the volume of mortar required for the external wall plastering. The formula for calculating the mortar
volume is:
Volume of mortar = Area of wall x Thickness of plaster
4. Determine the proportion of cement and sand: The standard ratio for cement mortar is 1:6, which means one part cement and six parts sand by volume. This ratio can be adjusted depending on the
quality of sand available and the type of wall being plastered. For example, for a stronger plaster, you can use a ratio of 1:4.
5. Calculate the proportion of materials: To calculate the quantity of materials, you need to multiply the volume of mortar by the proportion of cement and sand. For example, if the volume of the
mortar is 0.05 m3 and the ratio is 1:6, then:
Quantity of cement = 0.05 x 1 = 0.05 m3
Quantity of sand = 0.05 x 6 = 0.3 m3
6. Convert volume into weight: The density of cement is 1400 kg/m3 and the density of sand is 1600 kg/m3. Therefore, the weight of 0.05 m3 of cement is 0.05 x 1400 = 70 kg and the weight of 0.3 m3 of
sand is 0.3 x 1600 = 480 kg.
7. Add 20% extra material: It is recommended to add 20% extra material to account for wastage during mixing and application. This means adding an extra 14 kg of cement (70 x 0.2) and 96 kg of sand
(480 x 0.2).
8. Final calculation for cement mortar: The final calculation for cement mortar quantity for external wall plastering is:
Quantity of cement = 70 + 14 = 84 kg
Quantity of sand = 480 + 96 = 576 kg
9. Determining the quantity of water: The quantity of water required for mixing the mortar will depend on the consistency of the mix. A good rule of thumb is to use 20% of the weight of the cement.
This means using 0.2 x 84 = 16.8 liters of water for the above calculation.
10. Mixing the mortar: Once you have the quantity of materials calculated, mix the cement and sand thoroughly and gradually add water to achieve the desired
How much sand & cement required for 100 square feet ceiling plastering
Ceiling plastering is an important step in any construction or renovation project. It involves applying a layer of plaster material to the ceiling surface in order to provide a smooth and finished
look. The amount of sand and cement required for ceiling plastering will depend on various factors such as the thickness of the plaster, the size of the ceiling area, and the type of plaster mixture
used. In this article, we will discuss how much sand and cement are required for 100 square feet of ceiling plastering.
1. Calculate the area of ceiling:
Before determining the quantity of sand and cement required, it is important to calculate the total area of the ceiling that needs to be plastered. This can be easily done by multiplying the length
and width of the ceiling in feet. For example, if the ceiling is 10 feet long and 10 feet wide, the total area would be 100 square feet.
2. Determine the thickness of the plaster layer:
The next crucial step is to decide the thickness of the plaster layer. This will depend on the type of plaster mixture used and the condition of the ceiling surface. Generally, the thickness of a
ceiling plaster layer can range from 10-15 mm. For our calculation, let’s assume a thickness of 12 mm.
3. Calculate the volume of plaster:
To determine the volume of plaster, we need to multiply the area of the ceiling by the thickness of the plaster layer. In this case, the volume of plaster required would be 100 square feet x 12 mm =
1200 cubic feet.
4. Calculate the quantity of sand and cement:
The standard ratio of sand to cement for ceiling plastering is 3:1, which means for every 3 parts of sand, 1 part of cement is required. Therefore, to determine the quantity of sand and cement, we
need to multiply the total volume of plaster by the ratio of sand and cement.
Sand required = 1200 cubic feet x (3/4) = 900 cubic feet
Cement required = 1200 cubic feet x (1/4) = 300 cubic feet
5. Convert cubic feet to bags:
Sand and cement are usually measured in bags, so the final step is to convert the cubic feet into bags. One bag of cement is equal to 0.035 cubic meters, and one bag of sand is equal to 0.048 cubic
Sand required = (900 x 0.048) / 1.5 = 28.8 bags
Cement required = (300 x 0.035) / 1.25 = 8.4 bags
Therefore, for 100 square feet ceiling plastering, you would need approximately 29 bags of sand and 8 bags of cement.
Note: The above calculation is based on the assumption that the ceiling surface is in good condition and does not require any repair work. If there are any cracks or holes, you would need to add
extra sand and cement for patching before applying the plaster layer.
In conclusion, the amount of sand and cement required for ceiling plastering will vary depending on the size of the ceiling, the thickness of the plaster layer, and the type and condition of the
ceiling surface. It is always recommended to consult a professional or experienced contractor to accurately determine the quantity of materials needed for your specific project.
In conclusion, understanding the quantity of cement and sand required for plastering a 100 sqft area is crucial for achieving a smooth and durable finish. By following the guidelines and calculations
provided, one can ensure that they have the right ratio of materials to achieve a high-quality plastering job. It is important to remember that the exact amount of cement and sand required may vary
depending on factors such as surface condition and plaster thickness. It is always advisable to consult a professional when in doubt to avoid any errors and ensure the best results. With the proper
knowledge and attention to detail, anyone can master the art of plastering and create beautiful and long-lasting walls.
Leave a Comment | {"url":"https://civilstep.com/all-about-quantity-of-cement-and-sand-is-required-for-100-sqft-plastering/","timestamp":"2024-11-07T13:54:22Z","content_type":"text/html","content_length":"216854","record_id":"<urn:uuid:8ae86591-bb7f-47fc-8c87-dfb6196e1562>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00273.warc.gz"} |
ES 5/17/2011 Unfilled Gaps
Good morning!
Watching 4/15 (1316.25) and 4/19 (1308.25) unfilled
for support this morning.
I'm using 27.50 as first restistance as we have
low volume
from day session but High volume from
Above there is the price spike and
VA high
, Overnight high up at 30 - 31.0....
Big area for today *********
Initial point of the downside will be 1322.50 as all the volume before that last price drop happened there...so we can extend initial support down to the
low at 21....
so 21 - 22.50 is key on downside...
Originally posted by BruceM
I'm using 27.50 as first restistance as we have low volume from day session but High volume from O/N there....
Coincidently, that's today's
fill that I am watching (27.25).
Weekly S1 1322.50 and Weekly S2 1310.75
Back at S1 (19.75), hoping for push up to
fill from here (27.25), then short to unfilled
BANK pushing up toward its
fill. It gapped down 6.24.
Yesterday was a down day and we opened below yesterday's low, so that hurts the percentages of
for info if any more downside 1318.3 (Stretch 1.61) 1313.80 (stretch 2.61). Note I have been looking more at Dow futures then SPX futures, so dont let these numbers put you off
Originally posted by BruceM
Initial point of the downside will be 1322.50 as all the volume before that last price drop happened there...so we can extend initial support down to the O/N low at 21....
so 21 - 22.50 is key on downside...
Came down and tagged that area you mentioned. Nice!
I have a higher level fib at 22.25 as well. Feeling a little more worried about seeing that 27.25 print now.
BANK did fly through its
fill, but is pulling back a bit now and
is reflecting it.
can you provide the description of the ticker you ID as $BANK?
who's bank index is it, I assume it's
as long they can keep it inside and above Mondays
lows then they should push for the 30 - 31.50
Originally posted by PAUL9
can you provide the description of the ticker you ID as $BANK?
who's bank index is it, I assume it's Esignal ticker
Yeah, sorry, $BANK is the
ticker. It's the "NASDAQ Banking index". I'm not sure of the ticker on other platforms, unfortunately.
so far we are holding above the open, the key 22.50 area and yesterdays lows......but we seem to be rejecting that 27.50...
so 3 out of 4 filters favor the long side still...if we start getting under 24.50 then we will only have 2 out of 4 in favor of longs....so
even between bears and bull filters
stuck inside hour range too.......
if we make new lows on the day I would be real careful fading down there...probably be better to look for sells then
old school analysis here:
measured move potential to 1314.
5/2 H = 1367.25
1st L 5/5 1325.25
diff = 42
sunbsequent swing H on 5/10 at 1356.50
1356.50 – 42 = 1314.50
Speaking of
fill levels, today's
fill is at 1327.25, which we have been struggling with for the past 45 min or so. | {"url":"https://www.mypivots.com/board/topic/6702/1/es-5-17-2011-unfilled-gaps","timestamp":"2024-11-06T20:32:59Z","content_type":"text/html","content_length":"32641","record_id":"<urn:uuid:c64837e2-cc07-4d07-b4b3-15d72ae16cf9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00833.warc.gz"} |