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a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.
b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.) | {"url":"https://problems.org.uk/problems/?category_id__in=831&","timestamp":"2024-11-14T16:41:25Z","content_type":"text/html","content_length":"39755","record_id":"<urn:uuid:a20f5354-fccb-490c-8aa7-ab772ca9f8ca>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00389.warc.gz"} |
Grams to Calories Calculator - GEGCalculators
Grams to Calories Calculator
Grams to Calories Calculator
Here’s a table converting grams to calories for the three main macronutrients: carbohydrates, proteins, and fats.
Macronutrient Calories per Gram
Carbohydrates 4 calories/gram
Proteins 4 calories/gram
Fats 9 calories/gram
This table shows the number of calories you would get from consuming 1 gram of each macronutrient. Keep in mind that this is a general guideline, as the calorie content can vary slightly depending on
the specific type of carbohydrate, protein, or fat.
How do you convert grams to calories?
To convert grams to calories, you need to know the macronutrient content (carbohydrates, protein, and fat) of the substance. Each gram of carbohydrates or protein provides 4 calories, while each gram
of fat provides 9 calories. Multiply the grams of each macronutrient by its respective calorie conversion factor and sum them up for the total calorie content.
Converting grams to calories involves considering the macronutrient composition of the substance you are working with. The conversion factors are as follows:
1 gram of carbohydrates = 4 calories 1 gram of protein = 4 calories 1 gram of fat = 9 calories
To convert grams to calories, you need to know the macronutrient content (carbohydrates, protein, and fat) of the substance and multiply the grams of each macronutrient by its respective calorie
conversion factor. Then, sum up the total calories obtained from each macronutrient to get the total calorie content.
For example, let’s say you have a food item with 20 grams of carbohydrates, 10 grams of protein, and 5 grams of fat:
Total calories from carbohydrates = 20 grams * 4 calories/gram = 80 calories Total calories from protein = 10 grams * 4 calories/gram = 40 calories Total calories from fat = 5 grams * 9 calories/gram
= 45 calories
Total calorie content = 80 calories + 40 calories + 45 calories = 165 calories
So, in this example, the food item would have a total of 165 calories.
Keep in mind that this conversion method is an estimation as there may be small variations in the calorie content due to factors like fiber content and specific food processing. Additionally, this
method does not take into account the calories derived from alcohol and other components that may be present in certain foods.
How many calories is 1 gram?
1 gram of pure carbohydrates contains approximately 4 calories. Similarly, 1 gram of protein also contains around 4 calories. On the other hand, 1 gram of fat contains roughly 9 calories. These
values are commonly used in nutrition and calorie counting to estimate the energy content of different macronutrients.
How do I convert grams to calories?
To convert grams to calories, you need to know the energy content per gram of the substance you are measuring. Multiply the number of grams by the energy content per gram to get the calories. It’s
important to find the specific substance’s energy content. The formula is: Calories = Grams * Calories per gram.
To convert grams to calories, follow these general steps:
1. Determine the energy content per gram: Look up the specific substance you’re interested in and find its energy content in calories per gram. This information is often available on food labels or
in nutritional databases.
2. Multiply the number of grams by the energy content: Once you have the energy content per gram, multiply it by the number of grams you want to convert. This will give you the number of calories.
Here’s the formula for the conversion:
Calories = Grams * Calories per gram
For example, let’s say you have 100 grams of almonds, and you want to know how many calories it contains. If you find that almonds have an energy content of 6 calories per gram, you can calculate the
conversion as follows:
Calories = 100 grams * 6 calories per gram Calories = 600 calories
Therefore, 100 grams of almonds would contain approximately 600 calories.
Remember that the energy content can vary depending on the specific substance you’re measuring. It’s important to find accurate information for the particular food or substance you are working with.
What is 25 grams in calories?
To determine the number of calories in 25 grams of a specific substance, you would need to know the substance’s energy content per gram. Without that information, it is not possible to provide an
accurate conversion.
If you can provide the specific substance and its energy content per gram, I can help you calculate the number of calories in 25 grams of that substance.
How many grams is 600 calories?
To determine the number of grams in 600 calories of a specific substance, you would need to know the substance’s energy content per gram. The energy content can vary depending on the specific
substance you’re measuring.
Without the information on the energy content per gram, it is not possible to provide an accurate conversion from calories to grams. If you can provide the specific substance and its energy content
per gram, I can assist you in calculating the number of grams in 600 calories of that substance.
How many grams is 200 calories?
To determine the number of grams in 200 calories of a specific substance, you would need to know the substance’s energy content per gram. The energy content can vary depending on the specific
substance you’re measuring.
Without the information on the energy content per gram, it is not possible to provide an accurate conversion from calories to grams. If you can provide the specific substance and its energy content
per gram, I can assist you in calculating the number of grams in 200 calories of that substance.
How many grams is 400 calories?
The number of grams in 400 calories depends on the substance you are referring to. Different substances have varying energy densities, meaning their calorie content per gram can differ.
As an example, let’s consider a commonly used approximation for dietary fats: 1 gram of fat is approximately equal to 9 calories. Using this approximation, we can calculate the grams in 400 calories
of fat:
Grams = Calories / Calories per gram Grams = 400 calories / 9 calories per gram Grams ≈ 44.44 grams
Therefore, approximately 44.44 grams of fat would be equivalent to 400 calories. However, it’s important to note that this conversion factor is specific to dietary fats and may not apply to other
substances. To obtain an accurate conversion, it is necessary to know the energy content per gram for the specific substance you are referring to.
Is 1 gram 4 calories?
No, 1 gram is not universally equal to 4 calories. The calorie content of 1 gram can vary depending on the substance you are referring to. Here are some common approximations for certain
• Carbohydrates: 1 gram = 4 calories
• Protein: 1 gram = 4 calories
• Fat: 1 gram = 9 calories
• Alcohol: 1 gram = 7 calories
Please note that these values are general approximations and can vary depending on the specific type and composition of the substance. It’s always best to refer to nutritional information or specific
laboratory measurements for accurate energy content per gram.
How many grams of weight is 1000 calories?
The number of grams in 1000 calories varies depending on the substance you are referring to, as different substances have different energy densities. To determine the weight in grams, you need to
know the specific substance and its energy content per gram.
As an example, let’s consider the approximation that 1 gram of dietary fat is approximately equal to 9 calories. Using this approximation, we can calculate the weight in grams for 1000 calories of
Grams = Calories / Calories per gram Grams = 1000 calories / 9 calories per gram Grams ≈ 111.11 grams
Therefore, approximately 111.11 grams of fat would be equivalent to 1000 calories. However, this conversion factor is specific to fat and may not apply to other substances.
To accurately convert calories to grams, it is necessary to know the energy content per gram for the specific substance you are referring to.
How many calories are in 30g grams?
To determine the number of calories in 30 grams of a specific substance, you need to know the substance’s energy content per gram. The energy content can vary depending on the specific substance
you’re referring to.
If you can provide the specific substance and its energy content per gram, I can assist you in calculating the number of calories in 30 grams of that substance.
How many calories is 10 grams of fat?
In general, 1 gram of fat is equal to approximately 9 calories. Therefore, if you have 10 grams of fat, you can calculate the number of calories as follows:
Calories = Grams * Calories per gram Calories = 10 grams * 9 calories per gram Calories = 90 calories
Therefore, 10 grams of fat would contain approximately 90 calories.
How many calories to lose 500 grams?
To lose 500 grams (0.5 kilograms) of body weight, you need to create a calorie deficit. A calorie deficit occurs when you consume fewer calories than your body needs for its daily energy
To estimate the number of calories required to lose 500 grams, you can use a general guideline that 1 pound (0.45 kilograms) of body weight is approximately equal to 3500 calories. This means that to
lose 500 grams, you would need a calorie deficit of approximately 3500 calories.
To calculate the number of calories to lose 500 grams:
Calories = Deficit calories + Daily calorie needs
Since you want to create a calorie deficit of 3500 calories, you need to subtract this value from your daily calorie needs. The daily calorie needs vary depending on factors such as age, gender,
weight, height, activity level, and weight loss goals.
For example, if your estimated daily calorie needs are 2000 calories, the calculation would be:
Calories = 3500 calories + 2000 calories Calories = 5500 calories
In this example, you would need to consume approximately 5500 fewer calories than your daily energy needs over a period of time to achieve a weight loss of 500 grams. It’s important to note that
weight loss is a complex process influenced by various factors, and individual results may vary. Consulting with a healthcare professional or registered dietitian can help create a personalized and
sustainable weight loss plan.
How much is 100 calories in weight?
The weight of 100 calories depends on the specific food or substance in question. Calories are a unit of energy, not weight, so their weight can vary depending on the composition of the food.
However, it’s worth noting that fat has around 9 calories per gram, while carbohydrates and protein have approximately 4 calories per gram.
To calculate the weight of 100 calories, you would need to know the macronutrient composition of the food you’re referring to. For example, if you’re considering pure fat, 100 calories would be
equivalent to approximately 11 grams (100 calories ÷ 9 calories/gram = 11.11 grams). On the other hand, if you’re referring to a mixture of carbohydrates and protein, it would be around 25 grams (100
calories ÷ 4 calories/gram = 25 grams).
Keep in mind that this is a general approximation, and the actual weight can vary depending on the specific food’s composition.
How many grams in a 2,000 calorie diet?
The number of grams in a 2,000 calorie diet depends on the macronutrient composition of the diet. Macronutrients include carbohydrates, proteins, and fats, each of which provides a different number
of calories per gram.
Here’s a general guideline for the macronutrient breakdown and calorie content per gram:
1 gram of carbohydrates provides approximately 4 calories. 1 gram of protein provides approximately 4 calories. 1 gram of fat provides approximately 9 calories.
To calculate the grams of each macronutrient in a 2,000 calorie diet, you’ll need to determine the desired distribution of calories from carbohydrates, proteins, and fats. The specific breakdown can
vary based on individual needs and dietary preferences. However, a commonly recommended distribution is:
Carbohydrates: 45-65% of total calories Proteins: 10-35% of total calories Fats: 20-35% of total calories
Let’s use a balanced distribution as an example:
Carbohydrates: 50% of total calories Proteins: 25% of total calories Fats: 25% of total calories
To calculate the grams of each macronutrient, you can follow these steps:
1. Calculate the calorie content of each macronutrient: Carbohydrates: 2,000 calories * 0.50 = 1,000 calories Proteins: 2,000 calories * 0.25 = 500 calories Fats: 2,000 calories * 0.25 = 500
2. Convert the calorie content to grams: Carbohydrates: 1,000 calories / 4 calories per gram = 250 grams Proteins: 500 calories / 4 calories per gram = 125 grams Fats: 500 calories / 9 calories per
gram ≈ 55.6 grams
Based on this example, a 2,000 calorie diet with a balanced macronutrient distribution of 50% carbohydrates, 25% proteins, and 25% fats would consist of approximately 250 grams of carbohydrates, 125
grams of proteins, and 55.6 grams of fats.
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Does a man do work when he carries a 75 N bag horizontally for 5 m? Why / why not? | Socratic
Does a man do work when he carries a 75 N bag horizontally for 5 m? Why / why not?
2 Answers
The work is the product of the force that is in the direction of the displacement. The formula for work uses the vector dot product to properly handle a situation in which the force and displacement
are not parallel.
Let the force be F, the displacement be D, and the angle between F&D be $\theta$. Therefore, the formula is
$\text{Work} = F \cdot D$
Expanding the formula to show the cosine term from the dot product,
$\text{Work} = F . D . \cos \theta$
Our data is that ...
D=5 m
$\theta = {90}^{\circ}$
$\text{Work} = 75 N . 5 m . \cos {90}^{\circ} = 0$
So he does not do work. Not in the Physics sense.
Hope this helped you!
$\text{please watch the animations below.}$
• In which situations does the force work?
• The force does the work because the direction of force is the same as the direction of displacement.
• The force does the work because the direction of force is opposite to the direction of displacement.
• The force can not do the work because the force is perpendicular to the direction of displacement.
• The force can not do the work because the force is perpendicular to the direction of displacement.
• The blue component does the work but the green component can not do the work.
• The solution to your problem is: the weight of the bag is down, the force you use to lift is upward, and the displacement is horizontal (conforms to the 3rd and 4th angles). work is not done.
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Printable Pdf Multiplication Chart 2024 - Multiplication Chart Printable
Printable Pdf Multiplication Chart
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Obtain a free computer multiplication graph or chart
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Gallery of Printable Pdf Multiplication Chart
Printable Multiplication Chart 1 10 Pdf PrintableMultiplication
Printable Colorful Times Table Charts Activity Shelter
Printable Multiplication Chart 1 12 Pdf PrintableMultiplication
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An Almost Pure DDS Sine Wave Tone Generator - The Engineer
The test and verification of ac performance of high precision fast analogue-to-digital converters (ADCs) with resolution better than 16 bits require a near perfect sine wave generator capable of
covering a 0kHz to 20kHz audio bandwidth at least. Usually, expensive laboratory instruments are used to perform these evaluations and characterisations such as the audio analyser AP27xx or APx5xx
series from Audio Precision. Most of the time, modern high speed SAR and wideband sigma-delta (Σ-Δ) ADCs exhibiting 24 bits or more feature single-supply and full-differential inputs, and therefore
require the signal source used for the DUT to be dc and ac accurate, while providing full differential outputs (180° out of phase). Similarly, the noise and distortion level of this ac generator
should be much better than the specifications of these ADCs, resulting in a noise floor level well below –140dBc anddistortion lower than –120dBc with an input tone frequency of 1kHz or 2kHz and up
to 20kHz according to most supplier specifications. A typical configuration of a typical bench test setup suited for high resolution wideband ADCs is illustrated on the Figure 1. The most critical
component is the sine wave generator (single or multitone) and here a software-based direct digital synthesiser (DDS) can provide full flexibility with extremely fine frequency resolution and clock
synchronisation with the dataacquisition system to perform coherent sampling to avoid leakage and FFT window filtering.
At a fraction of the cost of an audio precision analyser, it is possible to design a very accurate sine wave generator based on the direct digital frequency synthesis (DDFS) principle, but
implemented in software onto a floating-point DSP processor such as the SHARC® processor. A reasonably fast floating-point DSP will meet real-time expectations and fulfill all the arithmetic and
processing conditions to achieve the distortion and noise performance level set up by the most advanced SAR ADCs. Taking the benefit of the full-word data length of the SHARC core architecture either
in 32-bit or 64-bit fixed-point format for the NCO phase accumulation and the proprietary, 40-bit, floating-point, extended precision to execute the sine approximation function and the digital
filters used to shape the spectrum, the quantisation effects (rounding and truncation noise) are drastically reduced to be considered negligible compared to the digital-to-analogue converter (DAC)
imperfections used for the signal reconstruction.
Direct Digital Frequency Synthesis
The digital signal generator synthesiser patent filed in April 1970 by Joseph A. Webb1 described what could be considered as the basis of DDS mechanics to generate various types of analogue
waveforms, including sine waves, simply with the use of a few digital logic modules. Then, in early 1971, the frequently cited reference paper from Tierney et al.2 on direct digital frequency
generation by deepening the DDS operation for quadrature generation as well as its limitations (word truncations and frequency planning) regarding the sampled systems theory was published. Practical
realisations began to show up, mostly relying on discrete standard logic ICs such as the TTL 74xx or ECL 10K families. Less than 10 years later, fully integrated solutions came on the market
introduced by companies like Stanford Telecom, Qualcomm, Plessey, and Analog Devices with the AD9950 and the AD9955. Designed for the best speed, power, and cost trade-off, the logic ICs’
architectures were based on a lookup table (LUT) to ensure the phase-to-sine amplitude conversion with limited phase, frequency, and amplitude resolutions. Today, Analog Devices remains the largest
and, perhaps, most unique supplier of DDS standalone integrated circuits, while current numerically controlled oscillators (NCOs) tend to be integrated in numbers in RF DACs like the AD9164 or the
AD9174. Despite their impressive noise and linearity performance over a multiple GHz bandwidth, none of these devicesare appropriate for the test of moderate speed, high resolution ADCs such as the
LTC2378-20, the AD4020, or the AD7768.
Compared to traditional PLL-based synthesisers, NCOs and DDSs are mostly known for their very fine frequency resolution, fastagility, and ease of sine/ cosine generation with perfect quadrature. They
are also prized for their wide bandwidth coverage and dc accuracy. Their principle of operation is governed by digital signal processing and sampling systems theory, and their digital nature allows
for fully digital and independent control of the phase, frequency, and amplitude of the output signals. The blockdiagram of Figure 2 depicts the architecture of a conventional DDS, which consists of
three major functions:
• An N-bit phase accumulator;
• A phase-to-sine amplitude converter characterised by a W-bit truncated phase input word;
• A D-bit DAC and its associated reconstruction filter.
The phase accumulator is built around a simple N-bit adder combined with a register whose content is updated at the rate of the sampling clock FCLK with the input phase increment Δθ, also commonly
called the frequency tuning word (FTW). The accumulatorcan periodically overflow and operates like a fractional divider between the sampling or reference clock FCLK and the DDS output frequency FOUT,
or like a gearbox with a divide ratio equal to:
The overflow rate gives the output frequency of the generated waveform such that:
where 0 ≤ FTW ≤ 2N–1. Because of the divider effect, the contribution of the reference or sampling fS clock phase noise at the NCO output will be reduced by
The output of the phase accumulator register represents the current phase of the generated waveform. Each discreteaccumulator output phase value is then translated into an amplitude sine or cosine
data or sample thanks to the phase-to-sine or phase-to-cosine mapper engine. This function is usually accomplished by means of trigonometric values stored in a LUT (ROM)and sometimes by the execution
of a sine approximation algorithm or a combination of the two. The output of the phase-to-sine amplitude converter feeds a DAC, which produces a quantised and sampled sinusoid before being filtered
to smooth the signal and avoid spectrum aliasing. This amplitude quantisation imposed by the DAC finite resolution puts a theoretical limit on the noise floor and the resulting signal-to-noise ratio
(SNR) of the synthesiser. Moreover, as a mixed-signal device, the DAC exhibitsa whole bunch of dc and ac nonlinearities due to its INL, DNL, slew rate, glitches, and settling time characteristics,
which createspurious tones and reduce the overall dynamic range of the sine wave generator.
Practical sine waveform generator implementations based on the architecture of Figure 2 differ mostly by the phase-to-amplitude converter block, which is generally optimised for speed and power
consumption rather than high precision because of the market orientation for digital radio applications. The simplest approach for the realisation of the phase-to-sine amplitude converter is to use a
ROM to store sine values with one-to-one mapping. Unfortunately, the length of the LUT grows exponentially (2N), with the width N of the phase accumulator and linearly with the wavetable data word
precision W.
Unfortunately, trade-offs consisting in the reduction of the accumulator size or truncating its output result in the loss of frequency resolution and a severe degradation of the SFDR. It is shown
that spurs caused by phase or amplitude quantisation follow a –6dB/bit relationship. Since a large N is normally desired to achieve a fine frequency tuning, several techniques have been promoted to
limit the ROM size while maintaining adequate spur performance. Simple compression methods are commonly used by exploiting the quarter wave symmetry of the sine or cosine function to reduce the
phaseargument range by 4. For further range reduction, brutal truncation of the phase accumulator output is the de facto method,although it does introduce spurious harmonics. Despite that, this
approach is always adopted because of the fine frequency resolution requirements, memory size, and cost compromise. Various angular decomposition methods have been suggestedto lower the memory
requirements with LUT-based methods.
Combined with amplitude compression using various types ofsegmentation, linear, or polynomial interpolation, the idea is to accurately approximate the first quadrant of the sine function or over the
[0, π/4] interval in the case of I/Q synthesis for which both sine and cosine functions are needed. Similarly, complex signal generation with no ROM LUT is efficiently supported by angle
rotation-based algorithms just calling for shift and add operations in a successive approximations scheme. This method, represented by the popular CORDIC, is generallyfaster than other approaches
when a hardware multiplier is not available or when the number of gates required to implementthe functions should be minimised (in an FPGA or an ASIC) for speed or cost considerations. Conversely,
when a hardware multiplier is available - as is always the case in a DSP microprocessor - table-lookup with interpolation methods and full polynomial calculations, such as Taylor-series expansion,
Chebyshev polynomials are faster than CORDIC, especially when high accuracy is a must.
Figure 2. Main functional sections of an NCO and distinction with the complete direct digital synthesiser, which includes the reconstruction DAC and its associated AAF. The NCO section can be used to
test or stimulate DACs.
Implementing a High Precision NCO in Software
Building a high precision ac tone generator with similar or better distortion performance than the best analogue oscillators, as in the most famous Hewlett-Packard analysers or as described in the
application note AN-1323 is not a trivial thing, even if dedicated to the audio frequency spectrum (dc to 20kHz range). Nevertheless, as written previously, a full software implementation, performing
the phase calculations (ωt) and sine function (sin(ωt)) approximations using the adequate arithmetic precision of an embedded processor can certainly help to minimise the quantisation side effects,
noise, and resulting spurs. This means that all the NCO functional blocks of Figure 2 are translated in lines of code (no VHDL!) to realise a software version that will meet real-time constraints to
ensure the minimum sampling rate and the desired frequency bandwidth.
For the phase-to-sine amplitude conversion engine, the full LUT scheme or any variation demands too much memory or toomany interpolation operations to achieve a perfect sine conformity. On the
contrary, the polynomial method for sine approximation offers a very good complexity vs. accuracy trade-off by allowing the use of a very low cost, general-purpose DSP. Polynomialseries expansion is
also very attractive for its relative simplicity and ability to provide full flexibility in the choice of the type ofpower series, in tailoring the algorithm for a given precision. It does not
require a large memory space, less than 100 lines of SHARC DSP assembly lines, and just a few RAM locations to store the polynomial coefficients and variables as sine values are only computed at
sampling time instants.
At first, the obvious choice for a sine approximation function would be to use a straight Taylor/MacLaurin power series with the appropriate order to meet the targeted accuracy. However, since power
series tend to lose effectiveness at endpoints, it is mandatory to reduce the argument input range to a smaller interval before performing any polynomial evaluation. Withoutargument range reduction,
high precision over the function domain such as [–π, +π] can only be supported with very high orderpolynomials. Thus, some transformations need to be applied to the elementary function to get the
reduced argument such as sin(|x|) = sin(f + k × π/2) and sin(f) = sin(x – k × π/2) with 0 ≤f<π/2. Consequently, extreme care should be taken with thetrigonometric functions to avoid subtraction
cancellations, which would lead to a serious loss of precision and produce catastrophic results, particularly with a poor arithmetic precision. In our case, this might occur when the phase input is
large or close to an integer multiple of π/2.
Besides the periodicity and modulo-2π repetitions, the symmetric properties of the sin(x) function can be applied to further reduce the range of approximation. Given the fact that the sine function
is anti-symmetric about the point x = π for the interval [0, 2π], so it is possible to use the following relationship:
to reduce the range to [0, π]. In the same manner, sin(x) shows a symmetry about the line defined by x = π/2 for the interval [0, π], such that:
for x in the interval [0, π/2], which reduces the angle input approximation range even more. Further argument reductions to smallerintervals like [0, π/4] to improve the accuracy is not efficient
because it requires both the evaluation of the sine and cosine functionsat the same time as dictated by the common trigonometric relationship: sin(a+b) = sin(a) × cos(b) + cos(a) × sin(b), worthwhile
for the generation of quadrature tones.
Analog Devices’ ADSP-21000 Family Application Handbook Volume 1 describes an almost ideal (for embedded systems) sine approximation function based on an optimised power series written for the first
ADI DSP floating-point processor, namely theADSP-21020, which is basically a SHARC core. This implementation of sin(x) relies on a minimax polynomial approximation thatwas published by Hart et al.4
and refined by Cody and Waite5 for floating-point arithmetic to mitigate round-off errors and to avoidthe occurrence of cancellations as previously mentioned. The minimax method relies on Chebyshev
polynomials and the Remezexchange algorithm to determine the coefficients for a desired maximum relative error. As shown with MATLAB® in Figure 3, small changes in the set coefficients result in a
dramatic increase in accuracy for minimax compared to Taylor for a seventh-order Taylor polynomial.6 For the best accuracy vs. speed trade-off, the angle input range of this sine approximation
function is shrunk to the [–π/2 to +π/2] interval and the software routine includes an efficient range-reduction filter, which counts for about 30% of the total “sine” subroutine execution time.
Figure 3. Unlike the Taylor-MacLaurin method defined around 0, the minimax sine approximation approach minimises and equalises the maximum relative error over the [–π/2 to +π/2] interval.
While all the computations could be executed with 32-bit fixed-point arithmetic, the most common and convenient format for mathematical calculations especially when dealing with long numbers has been
for years the IEEE 754 floating-point standard. As a DSP VLSI chip manufacturer, Analog Devices pioneered the IEEE 754-1985 standard from the very beginning. At the time, there was no single-chip
floating-point DSP processor at all, but only simple floating-point multiplier and ALU computation ICs such as the ADSP-3212 and the ADSP-3222, respectively. This format replaced most of the
proprietary formats of the computer industry and became the native format for all the SHARC DSP processors, in single precision 32-bits, extended precision 40-bits, and recently, double precision
64-bits for the ADSP-SC589 and ADSP-SC573.
The SHARC 40-bit extended single precision floating-point format with its 32-bit mantissa provides enough precision (u 2–32) forthis sine wave generation application and to keep things equal, Cody
and Waite show that a 15th order polynomial is appropriate for an overall accuracy of 32 bits with an evenly distributed error over the [0 to +π/2] input domain. The final tweak to minimise the
number of operations and maintain accuracy is to implement the Horner’s rule for the polynomial calculation, a fast exponentiation method to evaluate a polynomial in one point, such that:
R1 to R7 are the Cody and Waite coefficients of the polynomial series and only eight multiplies and seven additions arenecessary to evaluate the sine function for any input argument ε[0, π/2]. The
complete sin(x) approximation code written in the form of an assembly subroutine is executed in about 22 core cycles on a SHARC processor. The original assembly subroutine was modified to perform
simultaneous double memory accesses when fetching the 40-bit polynomial floating-point coefficients to save six cycles.
Figure 4. The software DDS simplified block diagram gives the data arithmetic formats and locations of the various quantisation steps between the processing elements.
The NCO 64-bit phase accumulator itself is making use of the SHARC 32-bit ALU in double precision two’s complement fractional format for its execution. A complete phase accumulator execution with
memory update costs 11 core cycles, and as a result, every NCO output sample is generated in about 33 core cycles.
The diagram in Figure 4 shows the functional block implementation of the software DSP-based NCO with some reference to the arithmetic format precision at each stage. In addition, one or two DACs and
their analogue antialiasing filter circuitry are required for the signal analogue reconstruction, and to realise the complete DDFS. The key elements of the processing chain are:
• the 64-bit phase accumulator (SHARC ALU double precision addition with overflow);
• the 64-bit fractional fixed-point to 40-bit FP conversion block;
• the range reduction block [0 to + π/2] and quadrant selection (Cody and Waite);
• the sine approximation algorithm (Hart) for the phase-to-amplitude conversion;
• the sin(x) reconstruction and normalisation stage over the –1.0 to +1.0 range;
• the LP FIR filter and sin(x)/x compensation if necessary;
• and the 40-bit FP to D-bit fixed-point conversion and scaling function to fit with the DAC digital input.
An optional, digital low-pass filter can be placed at the output of the NCO to remove any spur and noise that could fold in the bandof interest. Optionally, this filter can provide interpolation and/
or inverse sin(x)/x frequency response compensation depending upon the DAC selected for the analogue reconstruction. Such a low-pass FIR filter could be designed with the MATLAB Filter Designer tool.
As an example, assuming a 48kSPS sampling frequency and a dc to 20kHz bandwidth with a 0.0001dB in-bandripple and a –150dB out of band attenuation, a high quality equiripple filter could be
implemented with 40-bit floating-pointcoefficients. With only 99 filter coefficients, its total execution time will consume about 120 SHARC core cycles in singleinstruction, single data (SISD)
single-computation unit mode. After digital filtering, the pairs of calculated samples are sent by DMA to the DACs using one of the DSP synchronous serial ports. For a better speed performance,
chaining DMA operation is also possible with large ping-pong memory buffers to support processing by block operation. For example, the block data size could be equal to the length of the FIR data
delay line.
Final Tweaks at the NCO for an Optimal SFDR
As mentioned earlier, the NCO suffers from spurs mainly due to the truncation of the phase accumulator output and, to a lesser extent, from the amplitude quantisation done on the sinusoidal values
obtained by calculation or by tabulation. The error due to phase truncation generates spurs around the carrier frequency by phase modulation (sawtooth), while sine amplitude quantisation causes
harmonically related spurs although were considered as random errors and noise for a long time. Today, the operation of the phase accumulator is mathematically perfectly known as described in a
technical paper7 from Henry T. Nicholas and H. Samueli. After a thorough analysis, a model is presented such that the phase accumulator is considered a discrete phase sample permutation generator
from which the frequency spurs can be predicted. Whatever the phase accumulator parameters (M, N, W), the length of the phase sequences equal to
(where GCD is the greatest common divisor) is determined by the rightmost bit position, L, of the frequency tuning word, M, as shown in Figure 4. Hence, the value of L defines sequence classes, each
sharing their own set of phase components, but permutated according to the
ratio. These sequences of truncated phase samples generated in the time domain are used to determine, by DFT, the respective location and magnitude of each spurious line in the frequency domain.
These sequences also demonstrate that odd values of M(FTW) exhibit the lowest frequency spur’s amplitudes and suggests a simple modification of the phase accumulator to satisfy these minimum
conditions by simply adding 1 LSB to the FTW. This way, the phase accumulator output sequences are forced to always have the same 2N phase elements, whatever the M value and the initial content of
the phase accumulator. The level of the worst spurious tone magnitude is then reduced by 3.922dB and equal to SFDR_min (dBc) = 6.02 × W. The Nicholas modified phase accumulator confers several
benefits to the NCO, as first it eliminates the cases where the rightmost bit of the FTW is too close toits MSB (frequency sweep in FMCW applications), and, secondly, it makes the spur’s amplitude
independent of the frequency tuning word, M. This modification is easily implemented in software by toggling the ALU LSB at the sampling rate fS, the samebehaviour of the phase accumulator could be
simulated as if the FTW LSB was set to logic 1. With a phase accumulator size N = 64 bits, a ½ LSB offset can be considered as a negligible error regarding the accuracy of the desired frequency FOUT.
Figure 5. The position of the rightmost, non-zero bit of the FTW sets the theoretical SFDR worst-case level. The Nicholas modified phase accumulator solves the issue for any value of N and maximises
the SFDR of the NCO.
With an output phase word, W, of 32 bits, the maximum spur’s amplitude due to phase truncation is therefore limited to a value of –192dBc! Finite quantisation of the sine sample values also leads to
another set of frequency spurs, and it is commonly considered as noise and estimated by the well-known relationship SNRq(dB) = 6.02 × D + 1.76. This must be added to the parasitic elements due to the
approximation errors of the phase-to-sine amplitude-conversion algorithm stage which, however, are considerednegligible, given the extreme care in the choice of the phase-to-sine approximation
algorithm and the calculation’s precision.
These results indicate that both the linearity and the noise of our software sinusoidal NCO are at theoretical levels well beyond therequired thresholds to test most of the high precision ADCs
available on the market. It remains to find the last, but most criticalelements of the signal chain: the reconstruction DAC and its complementary analogue antialiasing filter and associated driver
circuitry susceptible to meet the expected level of performance.
The Reconstruction DAC: The Achilles’ Heel of the Thing!
The first temptation would be to select a high precision DAC with the best specification in terms of nonlinearity error (INL and DNL), like the superb AD5791, a 20-bit accurate DAC. But its
resolution is only 20 bits and its R-2R architecture does not favour the reconstruction of signals, and especially the production of very pure sinusoids, because of its large glitches during input
codetransitions. Conventional DAC architectures built around binary weighted current generators or resistor networks are sensitive to digital feedthrough and digital switching impairment such as
external or internal timing skew and other switching asymmetries ofthe digital input bits, particularly during major transitions for which the energy variation is consequent. This induces
code-dependent transients, resulting in harmonic spurs of high amplitude.
At 20+-bit resolution, the use of an external ultralinear and fast sample and hold amplifier to deglitch the output of a DAC does not help much as it generates its own transients in tens of LSBs and
introduces group-delay nonlinearity due to the re-sampling. For signal reconstruction, primarily in communication applications, the glitch issue is solved with the use of segmented architectures
mixing fully decoded sections for the MSBs and binary weighted elements for the lowest significant bits. Unfortunately, no such commercial DAC currently exists beyond 16-bit precision. Instead of the
fully predictable behaviour of the NCO, the DAC errors are difficult to estimate and simulate accurately, especially when the manufacturers’ dynamic specifications are rather weak ornonexistent,
except for the DACs or ADCs dedicated to audio applications. The interpolating oversampled and multibit sigma-delta DAC then seems the only solution to be good enough for the job. With a resolution
up to 32 bits, ultralow distortion, and high SNR, these state-of-the art converters are the best candidates for signal reconstruction over low to medium bandwidths. Trying to get the best noise and
distortion performance within the audio spectrum or a slightly wider band (20kHz or 40kHz bandwidth), the bestsigma-delta DAC within the Analog Devices portfolio is the AD1955 audio stereo DAC, still
one of the best audio DACs available on the market, despite its resolution being limited to 24 bits.
Introduced in 2004, this audio DAC is based on a multibit sigma-delta modulator and oversampling techniques aided with various tricks to mitigate distortion and other plagues inherent to this
principle of conversion.8
The AD1955 has got one of the best interpolation LP FIR filters of its kind, even today. It has a very high stop-band attenuation (≈–120dB) and a very low in-band ripple (≈±0.0001dB). Its two (left
and right channels) DACs can operate up to 200kSPS, but the best ac performance is achieved at 48kSPS and 96kSPS with a typical EIAJ standard, A-weighted, 120dB figure for both its dynamic range and
SNR in stereo mode. In mono mode, for which the two channels are simultaneously combined out of phase, a performance improvement of 3dB can be expected. However, for wideband applications, these
specifications are somewhat unrealistic since they are synthetic and restricted to the 20Hz to 20kHz bandwidth. Out-of-band noise and spurs are not considered beyond 20kHz, partly because of the EIAJ
standard, A-weighted filter, and audio industry specification definitions. This band-passfilter specific to audio measurements mimics the human ear frequency response and yields 3dB better results
over unfilteredmeasurements.
DDFS Hardware Demonstration Platform: Sine Wave Reconstruction with the AD1955
The complete DDFS has been implemented using two evaluation boards, one supporting the DSP processor and one for the analogue signal reconstruction with the AD1955 DAC. The second-generation SHARC
ADSP-21161N evaluation board waschosen for availability reasons as well as its ease of use and lean configuration for any audio applications. Still in production, the ADSP-21161N was designed a while
ago, to support industrial, high-end consumer and professional audio applications, providing up to 110Mips and 660MFlops or 220MMACS/s capabilities. Compared to the most recent generations of SHARC
processors, the ADSP-21161N differs mostly by its short, 3-stage instruction pipeline, an on-chip, 1Mb, only triple port RAM, and a reduced set of peripherals. The final and most critical stage of
the precision tone generator is based upon the AD1955 evaluation board, whichmust faithfully reconstruct the analogue signals from the samples delivered by the software NCO. This evaluation board
carries anantialiasing filter (AAF) optimised for the audio bandwidth to meet the Nyquist criterion and has a couple of serial audio interfaces to support PCM/I2S and DSD digital streams besides the
usual S/PDIF or AES-EBU receiver. The PCM/I2S serial link connector isused to connect the AD1955 DAC board to serial ports 1 and 3 connector (J) of the ADSP-21161N EVB. Both boards can beconfigured
for I2S PCM or DSP modes of operation at 48kSPS, 96kSPS, or 192kSPS sampling rates. The DSP serial port 1 generates the left and right channel data, the word select or L/R frame sync and the SCK bit
clock signals needed by the digital input interface of the dual-channel DAC. The serial port 3 is just used to generate the DAC master clock, MCLK, required for the operation of the DAC interpolation
filters and the sigma-delta modulators running 256 times (by default) faster than the inputsampling frequency (48kSPS). As all the DAC clocking signals are generated by the DSP, the board original,
low cost Epson clockoscillator has been changed for an ultralow noise oscillator CCHD-957 from Crystek. Its phase noise specification could be as lowas –148dB/Hz at 1kHz for a 24.576MHz output
On the analogue output side, active I/V converters must be used to hold the AD1955 current differential outputs at a constant common-mode voltage, typically 2.8V, to minimise the distortion. Ultralow
distortion and ultralow noise high precision operationalamplifiers like the AD797 are used for this purpose and to handle analogue signal reconstruction as well. As the two differentialoutputs are
processed separately by the DSP, the stereo output configuration with its AAF topology has been selected instead ofthe mono mode. This AAF was simulated with LTspice® XVII with results given in
Figure 6. As the last section of the filter is passive, an active differential buffer stage should be added like the recently introduced ADA4945. This low noise, ultralow distortion, fastsettling
time, fully differential amplifier is the almost perfect DAC companion to drive any high resolution SAR and sigma-delta ADCs. With a relatively large common-mode output voltage range and superb dc
characteristics, the ADA4945 provides exceptional output balance and contributes to the suppression of even-order harmonic distortion products.
The EVB third-order filter has a –3dB cut-off frequency of 76kHz with an attenuation of only –31db at 500kHz. The in-band flatness is very good, but the out-of-band attenuation of this LP filter must
be seriously improved, even if restricted to a pure reconstruction audio application. This is mandatory to reject the DAC shaped noise as well as the modulator clock frequency MCLK. Depending upon
the use of the software DDS either for a single tone generator or an arbitrary waveform generator (AWG for complex waveforms), the AAF will be optimised for out-of-band attenuation or group delay
distortion. As a practical example and comparison, the old but renowned SRS DS360 ultralow distortion function generator has been designed with a seventh-orderCauer AAF for a similar sampling rate.
The signal reconstruction lies on the AD1862, a serial input 20-bit segmented R-2R DACaimed at digital audio applications. The AD1862 was able to sustain 20-bit word sampling rates up to 768kHz (×16
fS) and exhibits exceptional noise and linearity specifications. Its single-ended current output leaves the choice to use the best amplifier for the external I-to-V conversion stage.
Figure 6. The LTspice simulated frequency response of the AD1955 EVB third-order antialiasing filter (stereo configuration).
The AD1955 and SHARC DSP combination was tested against several high resolution SAR ADCs such as the AD4020 with no external selective passive filters in between. By default, the basic AD4020
evaluation board offers no other choice than the on-board ADA4807 drivers. The simple circuitry to bias the ADC inputs at the V_REF/2 common-mode voltage imposes a rather low input impedance of 300Ω
and requires either signal isolation, ac coupling, or the use of an external differential amplifier module such as the EVAL-ADA4945-1. The AD4020 reference design board described in the circuit note
CN-0513 is a better choice. It includes a discrete programmable gain instrumentation amplifier (PGIA), which provides a high input impedance and accepts ±5V differential input signals (G = 1).
Although these AD4020 boards and their SDP-H1 controller lack the capability to support coherent sampling acquisition, they allow decent waveform capture lengths for samples, ranging up to 1M. Thus,
long FFTs with selective windowing are possible, providing both fine frequency resolution and a low noise floor. For example, with the seven-term Blackman-Harris window, the 1Mpts FFT plot shown in
Figure 7 illustrates the level of distortion of the AD1955 for a 990.059Hz generated sine wave. The second harmonic is the largest distortion component and the largest spur at –111.8dBc over a 350kHz
bandwidth. However, when considering the wholeADC Nyquist bandwidth of 806kHz, the SFDR is limited by the DAC sigma-delta modulator and interpolating filter frequencyand its second harmonic (384kHz
and 768kHz).
In the same conditions, test trials were conducted on the vintage AD1862, which exhibited a slightly different spectral behaviour. Placed in differential configuration, the two 20-bit DACs clocked at
about 500kSPS reported a noise floor of –151dBFS, a THD of –104.5dB for a sine output level of 12V p-p at 1.130566kHz. The SFDR over the AD4020 Nyquist bandwidth (806kHz) is close to 106dB limited by
the third-harmonic. The DAC reconstruction filter based around two AD743 low noise FET amplifiers is a third-order similar to the one of the AD1955 evaluation board, but with a cut-off frequency of
35kHz at –3dB.
To become effective, the DDS-based generator requires a decent filter capable of an attenuation greater than 100dB at about250kHz for a generated dc to 25kHz CW signal frequency range. This can be
achieved with a sixth-order Chebyshev and even a sixth-order Butterworth LP filter for a perfect in-band flatness. The order of the filter will be minimised to limit the number ofanalogue stages and
their non-idealities such as noise and distortion.
Preliminary and out of the box tests performed on standard evaluation boards demonstrate that the processor-based DDS techniques for conventional sine wave CW generation with top performance are
within reach. The –120dBc harmonic distortion figure could be met with a careful design of the reconstruction filter and the analogue output buffer stage. The DSP-based NCO/DDS is not only restricted
to the generation of single tone sine waves. By using an optimised AAF (Bessel or Butterworth) with an appropriate cut-off frequency and no other hardware change, the same DSP and DAC combination can
be disguised into ahigh performance AWG to produce any type of waveform, for example, to synthesise fully parametrisable multitone sine waves witha full control of the phase and amplitude of each
component for IMD testing.
Since floating-point arithmetic is crucial for applications requiring high accuracy and/or high dynamic range, today the SHARC+ DSP processors such as the low cost ADSP-21571 or the SoC ADSP-SC571
(ARM® and SHARC) are the de facto standard for real-time processing up to an aggregated sampling rate of 10MSPS. Clocked at 500MHz, the dual SHARC cores and their hardware accelerators can provide
more than 5Gflops computation performance and offer tons of internal specialized SRAM, the basic ingredients demanded by the tasks for the generation of any kind of waveforms as well as for complex
analysis processing.This type of application shows that the systematic use of hardware programmable solutions is not mandatory for handling precision digital signal processing. Floating-point
processors and their complete development environments allow easy and fast code portability from simulators such as MATLAB, as well as rapid debugging thanks to Analog Devices’ CCES and VDSP++ C and
C++compilers and their full suite of simulators and real-time debuggers.
Analog Devices www.analog.com
1 Joseph A. Webb. U.S. patent US3654450 April 1970.
2 Joseph Tierney, Charles M. Rader, and Bernard Gold. “A Digital Frequency Synthesizer.” IEEE Transactions on Audio and Electroacoustics, Vol. 19, Issue 1, March 1971.
3 Jim Williams and Guy Hoover. AN-132: Fidelity Testing for A→D Converters Proving Purity. Analog Devices, Inc., February 2011.
4 John F. Hart. Computer Approximations. Krieger Publishing Company, 1978.
5 William J. Cody and William Waite. Software Manual for the Elementary Functions. Prentice-Hall, Inc., 1980.
6 Robin Green. “Faster Math Functions, Part 2 Presentation.” Sony Computer Entertainment America, May 2016.
7 Henry T. Nicholas and Henry Samueli. “An Analysis of the Output Spectrum of Direct Digital Frequency Synthesizers in the Presence of Phase-Accumulator Truncation.” IEEE, May 1987.
8 Robert Adams, Khiem Nguyen, and Karl Sweetland. “A 113 dB SNR Oversampling DAC with Segmented Noise-Shaped Scrambling.” IEEE, February 1998.
ADSP-21000 Family Application Handbook Volume 1. Analog Devices, Inc., May 1994.
A Technical Tutorial on Digital Signal Synthesis. Analog Devices, Inc., March 2001.
Butler, Oscar. “Internship Report Summer 2017: High Precision Oversampled 20-Bit Ultra Low Power Acquisition System.” Analog Devices, Inc., 2017.
Crawford, James A. Advanced Phase-Lock Applications: Frequency Synthesis. AMI, LLC, May 2011.
Evaluation Board User Guide UG-048. Analog Devices, Inc., February 2010. EV-4020-REF-DGNZ Reference Design Board User Guide UG-1280.
Analog Devices, Inc., May 2019.
Goldberg, Bar-Giora. Digital Techniques in Frequency Synthesis. McGraw-Hill, August 1995.
Model DS360 Ultra Low Distortion Function Generator. Stanford Research Systems, 1999.
Symons, Pete. Digital Waveform Generation. Cambridge University Press, November 2013.
AD1862 data sheet. Analog Devices, Inc., July 2011.
1241-2010 - IEEE Standard for Terminology and Test Methods for Analog-to-Digital Converters. IEEE, January 2011.
About the Author
Patrick Butler is a field applications engineer with Analog Devices’ south Europe sales organisation, supporting the French globalmarket and some ADEF customers. He has been with ADI since 1984,
supporting the DSP building blocks ICs, as well as high speed converters. Previously, he worked as a design engineer in the ATE division of Schlumberger in Saint-Étienne, France for five years, and
then occupied several application engineer and FAE positions at Matra-MHS in Nantes, AMD and Harris SC-Intersil. Today, his main hobby is collecting vintage sound components to build active, high
efficiency horn loudspeaker-based systems with the help of his two sons. | {"url":"https://www.theengineer.co.uk/content/product/an-almost-pure-dds-sine-wave-tone-generator/","timestamp":"2024-11-07T20:38:27Z","content_type":"text/html","content_length":"179179","record_id":"<urn:uuid:df948b75-4d7e-47e7-9195-50eb2d77f98e>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00081.warc.gz"} |
Advanced Macroeconomics - Essay Writing Expert
Martin Luther University Halle-Wittenberg Professor Dr. O. Holtemoller ¨
2nd Exam
Winter 2023/2024
General Remarks:
• This document contains all problem sets for the second exam in winter 2023.
• Each student is assigned a specific problem set. You have to work exactly on the problem
set that is assigned to you. You find the allocation of student ID numbers to problem sets
at the end of the document.
• Please read the exercises carefully.
• Please confirm by Email to makro@wiwi.uni-halle.de until April 11, 2024, 23:59, that
you accept the exam or that you withdraw.
• Questions about the exercises can be sent by Email to makro@wiwi.uni-halle.de until
April 11, 2024, 12:00 (in German or in English). The questions will be answered by
Email to all participants.
• Solutions have to be provided in a single pdf file in English together with a zip container including all Octave or Dynare program code and data that you have used to solve
the exercises. Computer code should be commented within the file as far as possible.
Zipping the program code is very important because some Email programs block attached computer code. Send the pdf and the zip container until May 2, 2024, 23:59 to
makro@wiwi.uni-halle.de using your MLU Email address. Other formats than pdf and
zipped Octave/Dynare code/data will not be accepted.
• For producing your data set with raw data you can use R. In this case add the respective
code to the zip container. But please note that other software than Octave (or Matlab) and
R is not allowed.
• All numerical calculations and all graphical expositions have to be produced with Octave
(or Matlab) and Dynare. Do not use other statistical or econometric software.
• You need to combine techniques from several Octave and Dynare programs that have
been provided via Stud.IP during the semester. Screen all programs and think about how
to use parts of them for your task. Include a table in your pdf which lists all Octave and
Dynare programs that you are using together with a short description what a program
Advanced Macroeconomics 2nd Exam Winter 2023/24
does. This also includes the information whether Matlab or Octave was used and which version of it. Add a data section in which you explain in detail the data that you use and its exact source.
• Cite all the literature that you are using (including the lecture slides, the Dynare manual, and Dynare itself, see www.dynare.org). Cite consistently and completely (including exact page numbers),
following academic standards. If you receive support from another person (including fellow students) or you use artificial intelligence do not forget to acknowledge this. It is important that you
explain your code in the answers, otherwise we cannot assess your own contribution. Code without explicit written explanation is not sufficient to pass the exam.
• If calculations are required, they must be presented in detail, i.e. comprehensibly and completely, in typed form.
• The grading criteria are:
– Does the Octave/Dynare code work and does it answer the specific question? (30%)
– Is the computer code well documented and explained? (30%)
– Are the economic explanations complete and correct? (30%)
– Are the results well presented in the pdf file? (10%)
Good luck!
Advanced Macroeconomics 2nd Exam Winter 2023/24
Problem Set #n
1. Download the following data from the AMECO database (European Commission) for the
country which is assigned to your student ID:
table code variable unit ref periods
AMECO01 NETN Employment, persons: 0 0 2000-
total economy 2025 (National accounts)
AMECO06 UVGD Gross domestic product 99 0 2000-
at current prices 2025
AMECO16 UTYG Current taxes on income and wealth 99 0 2000-
(direct taxes): 2025
general government :- ESA 2010
Comment the way you get the data and explain the variables. Document the process of data arrangement such that it is replicable.
2. Calculate the aggregate effective income tax rate τt as well as its average over all time periods τ¯ using national income and tax revenues. Explain how you compute it and provide the formula.
Display the value in the command window.
3. Apply the HP-filter to the time series for employment and national income. Set a reasonable value for the smoothing parameter λ and explain your choice. Calculate the correlation coefficient
between cyclical employment and cyclical income and display it in the command window. Comment shortly the size of the coefficient.
4. Plot the time series (aggregate effective income tax rate; trend and cycle of employment and income as well as the time series themselves) and show a scatter plot of cyclical employment and
cyclical income. Provide descriptions and possible explanations for the observations.
5. Consider the non-linear Monopolistic Competition Model with flexible prices and wages from the lecture. Assume that the government receives lump taxes from households, where tt = Tt PtYt is the
share on GDP. Furthermore, it collects tax revenues from labour income, where τt is the respective tax rate, equal to τ¯ in steady state. Set the steady state to the calculated average value from the
empirical part of the problem set. Government spending is defined as a constant share γ = 0:2 on GDP and bt = Bt PtYt = 0:6 is the share of public debt on nominal GDP.
Hence, the government budget constraint can be written as
γ + bt–11 + πtYt–1Yt= bt + τt wtNt + tt:
1 + it Yt
Lump taxes are set according to the fiscal rule (steady state values a marked with a bar) tt = t¯+ φb(bt–1 – ¯b):
Advanced Macroeconomics 2nd Exam Winter 2023/24
Net real wage is defined as
w~t = (1 – τt)wt:
In case of a tax policy shock, the tax rate deviates from its steady state value, in normal times t = 0:
τt = ¯ τ + t
GDP is now used for private and public consumption. Adjust the aggregate resource constraint to
Yt = Ct + γYt: The labour supply condition changes now to
w~t = CtθNt’
Derive it analytically and discuss the role of the tax rate.
6. The rest of dynamic equations is given as follows:
Ct θ = 1 + Rt
1 + ρ
wt =
1 – α
Yt Nt
1 + Rt = 1 + it
1 + πt+1
1 + it = (1 + ρ)(1 + π∗)(1 + πt – π∗)φ exp(ν)
Furthermore, log total factor productivity and monetary policy shock follow an autoregressive process of order one as in the lecture.
Do a sensitivity analysis: plot steady state . . .
• . . . labour hours for values of the labour income tax rate between 0 and 95 percent
in steps of 5 percentage points.
• . . . lump sum share on GDP for values of the government consumption share on
GDP between 10 percent and 30 percent in steps of 1 percentage point.
Explain your findings.
Some advice how to adjust the code file from the lecture:
• Be careful with variables in logs (exp( ~ w)) and variables in percent (τ, b, t) when adjusting the code file. The timing of bonds is already taken into account in the provided formulas, so you
must not declare them as predetermined variable or shift the time index. Add γ as parameter.
Advanced Macroeconomics 2nd Exam Winter 2023/24
• Derive the steady state lump-sum tax share t¯from the steady state government budget constraint and use the formula as initial value for this variable.
• When implementing the fiscal rule in the model block use the command
steady state(t) for t¯.
• See section on for loops in the Dynare manual and use the command steady; to
calculate the steady state values in each iteration.
7. Assume the following fiscal policy scenario: there are elections soon and the government decides to temporarily reduce the labour income tax by 5 percentage points from period 2 to 5 to stimulate
the economic activity. Does that work? Present and explain the impulse responses of each variable in the model. Use the perfect foresight solver in Dynare.
Some advice how to produce plots of the results:
Dynare does not automatically produce plots of impulse responses in the deterministic case. For plotting the responses relative to the steady state as %-deviations (y; c; w; w; n ~ ) or %-points (τ;
r; i; π; b; t) you will need the steady state value of each variable. Instead of using the command xbar = oo .steady state() you can simply take the first element in the vector of a variable to
extract the steady state value: xbar = x(1).
ID n countryFrance
222235729 2 Germany
223223633 3 Italy
221217069 4 Poland
221213087 5 Spain
222222712 6 United Kingdom | {"url":"https://essaywritingexpert.org/advanced-macroeconomics/","timestamp":"2024-11-13T21:15:29Z","content_type":"text/html","content_length":"82828","record_id":"<urn:uuid:70d52f08-4b2f-4915-a3d7-ef40e3525cb0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00678.warc.gz"} |
How to convert an older Quicken data file to use on Mac OS 10.7 Lion - Ferd Crôtte
How to convert an older Quicken data file to use on Mac OS 10.7 Lion
This is my very occasional geeky/techy article, written because this problem was a major PIMA!
By way of brief background, I have used Quicken to manage my finances since the late 1990’s, and have remained a faithful user despite a few little issues over the years. Princess Gail and I use
Quicken every day to keep track of our vast empire (ROTF LMAO!) 😀
The latest Quicken problem for Mac users has to do with converting Quicken data files to be compatible with Apple’s new Mac OS 10.7, aka “Lion.” Both Intuit (maker of Quicken) and Apple are to blame:
Quicken for not keeping their Mac products current (as Apple moved from PowerPC to Intel chips) and Apple for not loudly warning Quicken users of a major incompatibility, knowing it would affect
household and business financial management for Quicken users upgrading to Lion.
Every year, Princess Gail and I ask ourselves if we shouldn’t upgrade our old Quicken 2004 (Q 2004) but it has always worked well, and we couldn’t see how a program that simply adds and subtracts
numbers could change all that much. So we kept using the 2004 version. We were oblivious to the PowerPC and Intel incompatibility that was to become a critical issue with Mac OS 10.7 Lion.
Apparently, the previous version of Mac OS, 10.6, still had a way to read old PowerPC programs (including our old Q 2004) by way of a translator called Rosetta that worked in the background. The new
Mac OS 10.7 Lion does not support old PowerPC programs at all. Even the Rosetta translator is not supported.
So after we upgraded our computers to Mac OS 10.7 Lion, we quickly discovered that Quicken would not run, and we had no access to years of financial data, most importantly our current tax year info
and our current checkbook entries.
The solution was tricky, but here it is:
The newest Quicken product is Quicken Essentials (QE) which does run on the Mac OS 10.7 Lion. We needed our old Q 2004 data file to be converted to the QE format, but this could NOT be done on Mac OS
10.7 Lion. It had to be done on Mac OS 10.6 before one upgrades to 10.7 Lion. Further, QE can only convert files from Quicken 2006 and 2007, not from our ancient 2004 version.
So we first had to find a copy of Quicken 2006 or 2007. I managed to find Q 2006 and loaded into an old Mac running an older Mac OS. I was able to convert our Q 2004 data file to the Q 2006 format.
I then tried loading the new QE program on that old Mac but it would not run on an old PowerPC Mac. QE requires the Intel chip of the newer Mac models.
At this point I had two options. One was to find or borrow a newer Mac model that had not been upgraded to 10.7 Lion, install QE, and use it to convert my Q 2006 data file to the QE format.
The other option, which is what I actually did, was to learn how to partition the hard disk on my current Mac which is running 10.7 Lion, and load an older Mac OS into the new partition so I could
use that to install QE and do my file conversion.
You can do a search to find easy tutorials on how to partition a Mac hard disk. I learned how to carve out a 20GB partition on the hard disk of my computer that is upgraded to 10.7 Lion. That was
more than twice the space needed to load the 8GB Mac OS 10.6. So I loaded 10.6 into that partition and then I installed QE into that. I was then able to convert my Q 2006 data file to the QE format.
Finally, I switched back to the 10.7 Lion partition and was able to translate the PowerPC version of the QE data file to the new Intel version of the QE data file. Whew! It was a lot of work, but we
had over 6,000 bank transactions on that file. It was completely worth the effort.
The most important point: You can avoid all our troubles by simply buying Quicken Essentials before you upgrade to Mac OS 10.7 Lion. I wish I had known that!
Quicken Essentials is not as full featured as Quicken for Mac 2007. You will want to go to their site and compare features. For us, since we use it mainly as a check book register and for the
associated reports, Quicken Essentials is all we need. For those out there who were using features in Q 2007 not available in QE, I suggest you get QE anyway and hold on to all your older data files.
I predict that Intuit will eventually come out with a full featured Quicken for Mac that will run on Lion. The Apple Macintosh platform is dramatically gaining market share, especially with laptop
and iPad users, where others are in decline. Intuit is a good company and will certainly respond to that.
I hope this is helpful to someone. I can’t imagine we were the only ones with this bummer of a problem! But we figured it out and now we’re happy! 🙂
If this was helpful, please share or “like.” Thank you.
213 thoughts on “How to convert an older Quicken data file to use on Mac OS 10.7 Lion”
1. Haven’t upgraded to Lion yet (we may wait for a while), but we do have an older version of quicken. This may come in handy very soon. Thank you!
1. Hi, Jennifer!
You can avoid all our troubles by simply buying Quicken Essentials before you upgrade to Lion. I wish I had known that!
BTW, we’re loving Lion. For $30 we have it on both our laptops.
The only other program I’ll need to update is my also ancient Photoshop Elements 4.0 to the current 9.0.
2. I don’t have a Mac so there you go. We use Microsoft Money and have for years. Tracking our expenses is very important to us too.
Have a terrific day and I love the graphic. 🙂
1. Hi, Sandee!
Of course, being a Mac zombie, I try not to buy anything that says “Microsoft.” LOL
But I know Microsoft Money is a very fine program.
Big hugs! 🙂
3. Well I’m impressed! I wouldn’t have the slightest clue how to figure this out!
1. Hi, Meleah!
Well, when the problem is about your money, you are extra motivated! 🙂
4. If you google “Quicken Lion” this post is currently the 12th entry out of 1,650,000!
5. I am speechless, in awe, and confused. But I’m thrilled for the folks that need this info and I’m sure your tech expertise will reach many!
1. I hope it helps some people, too. That’s the only reason I wrote it. Thanks, Linda! 🙂
6. What?
1. LOL!
Don’t worry your Queenly head. It’s Mac stuff. 😉
7. AHHHHHHHHH you are talking about MY WORK and it is FRIDAY NIGHT AHHHHH!!!! LOLOL!!!! Good for you for figuring it out.. you must be very good at math – it takes that kind of brain. I hope that if
other people are having the same problem that they find your post. There is nothing like the internet to find answers to your pc problems. Oh, MAC problems LOL!
Hmmm vast empire…. ? Honk another time and you get three bucks LOL!
1. Actually, I’m getting kick out of it. It has been my most popular post ever! It is scoring really high on searches for the key words in the title.
Honk! 😉
Hi, Jennifer!
You can avoid all our troubles by simply buying Quicken Essentials before you upgrade to Lion. I wish I had known that!
BTW, we’re loving Lion. For $30 we have it on both our laptops.
The only other program I’ll need to update is my also ancient Photoshop Elements 4.0 to the current 9.0.
Have you actually tried to go directly to Quicken Essentials from Quicken 2004? I am going through this process after buying a Mac Mini with Lion and using Quicken 2002 successfully for years.
Luckily my MacBook Pro is still Snow Leopard. However the process still requires an initial conversion to Quicken 2007 (or maybe 2006, as you suggest) before it can be imported into Quicken
Essentials. In my case I am having crashes during the process of importing into Quicken 2007, and I am still waiting to hear from Intuit about the problem.
1. No. It is not possible to go directly from Q 2004 all the way to Quicken Essentials. QE will only convert from Q 2006 or Q 2007.
I was able to convert our old Q 2004 data file to a Q 2006 data file. Then I converted the Q 2006 data file to the QE data file on a Mac running Snow Leopard. Finally, I converted that Snow
Leopard QE file to the Lion QE file. It was a lot of moving the file around different machines and different operating systems, but it worked.
Maybe jumping from Q 2002 all the way to Q 2007 is too much of a jump. Is there a way you could convert your Q 2002 data file to a Q 2004 and go from there?
9. thanks for addressing this issue! i have not upgraded to Lion yet because of the Q snafu. in fact, i was even thinking of upgrading one computer and not the other (which you know would drive one
crazy) so at least one of them could run Q. which i hate for a number of reasons anyway. since i have Quicken for Mac 2003 (talk about old!), i am dreading the stepped conversions you outline
here. what about some other financial program that could convert all my data files – like yours, going back many years? has anyone tried another software brand that can do this seamlessly?
1. Thanks for the nice comment, brindlegirl!
If you haven’t upgraded to Lion yet, and if all you care about is the checkbook registry stuff and associated tables/graphs, then you’re fine, once you do the intermediate file conversion(s.)
I think you can find a copy of Quicken 6 easily online. If not, email me and I will send it to you. Then buy QE, install it in Snow Leopard, and translate to the QE format. After that you’re
all set to switch to Lion. Really, once you get started with these translations, it’s not that hard.
BTW, I have had quite a few people search their way to this post using the term Quicken 2002! I’m glad I wasn’t the only one using an old program! LOL
I don’t know anything about converting the Quicken data into another program. If that’s possible, I can’t imagine it would be “seamless.” If you DO discover that is possible, please let me
know! 🙂
10. actually, your page turned up first in a bing search on safari. pretty nice! here was my search request: best lion compatible software to convert quicken data files
1. REALLY!? Cool!!! Thanks for letting me know!!! 😀
11. Hi Ferd,
Thank you SO much for this post. I too upgraded without considering Q. I have Quicken 2006 on my iMac with Lion. My laptop in still on Leopard. Do you think I could simply load QE onto my laptop,
then copy the Quicken Data file that is on my desktop on my iMac and paste it into the new QE version on my laptop? I don’t have a disk for Quicken 2006, just 2003, so am guessing I must have
upgraded online somehow?
Thanks for any help you may be able to give me!
12. Hi Again! Got it to work, yippee!! Thanks again for your help, Ferd!!
Best, Melanie
1. Oh, good, Melanie! I just got home from work and was about to answer your first comment, but I see you figured it out. I’m glad! It makes me happy that this post helped someone! Thanks for
letting me know! 🙂
13. Again, thanks so much!
14. Ferd, WOW – I wish I read this before I migrated. I have an old iMac using Q04 and migrated all data to my new iMac running Lion – I was unable to open Q04. Since I still have my old iMac, if I
get a version of Q06 on it and converted my old files to it, could I then backup to a USB mini Drive and Restore it via Quicken Essentials on my new iMac????? I look forward to your reply.
1. Yes, Chris, if you load Quicken 2006 on your old iMac and convert your Q2004 data file to Q2006, you can then use that data file on your new iMac running Lion and Quicken Essentials will be
able to convert it. Let me know how that works out.
And yes, I used my little USB mini drive to shuffle the data file around.
1. But can your write checks with QE? I tried an earlier version and it wouldn’t do that.
15. Ferd,
I use Parallels Desktop with my PC version of Quicken only because I have been using the 2009 version of Quicken and have been very spoiled by bill pay, auto balancing of register when
downloading from my bank etc.. Will this effect me? Apple store told me I had to convert to the new version of Parallels and it would not effect my Quicken at all. very worried!!
1. Hi Patti.
I assume you are upgrading to Mac OS 10.7 Lion.
If you will also be running Parallels 7 on Lion, and you plan to stick with a PC version of Quicken, I don’t think you will have any problems. Most likely you will be able to use the same
Windows OS via Parallels, and the same version of Quicken you are currently using.
But if your plan was to NOT use Parallels on your computer with Lion, then you will have to use Quicken Essentials. For now, that is the only Quicken product that will run on Lion. I’m using
it now. It is a great, new and improved checkbook register. The downloads from the bank are swifter. There are more banks to choose from. It will auto balance. But it WON’T do bill pay. Maybe
you could do that through your bank rather than from within Quicken?
16. Is there any reason you know of that one couldn’t maintain 10.6 on a disc partition and continue to run Quicken from that?
1. Joe, you absolutely could do that. I partitioned my disk for 10.6 only to do the file translations, but one could keep using that 10.6 partition to keep running the older Quicken software.
It’s no big deal to switch between partitions.
17. Thanks, Ferd. That seems to be the solution. For me, at least.
18. Does anyone know if a techie out there can help me with this issue Mac created? I have been backing up my quicken files each month but having trouble locating on my external hard drive. Also have
a 2nd laptop without lion so would expect it is possible to get my financial files. This too much angst!
1. I don’t know of a techie, and I don’t think I’d trust anyone to handle my Quicken financial data file. Both Apple and Intuit are to blame, as I mentioned in the post, each for their own part
in this fiasco. But I will say that Apple phased out the PowerPC chip over about five years, encouraging users and software developers to shift over from the PowerPC structure to the Intel.
You have to do this yourself. It is a pain, but it is doable, and worth it.
Different versions of Quicken have data files that might have different suffixes. They might also be placed in different folders. You have to figure out where your data file is, then
translate it to a Quicken Essentials data file, all on MacOS 10.6 or earlier. Depending on how old your Quicken program is, you might have to do two or even three translations as detailed
19. Ferd, I’m so glad I found your article. I too bought a new MacBook Pro with Lion. My Quicken 2006 data is still on my old MacBook Pro with OS 10.4.11. Do you know if I can just download QE onto
the old one and convert as you described above, or do I have to upgrade the OS first for QE to work?
1. Kelly, you’re in great shape. Install QE on your OLD system and it will translate the Q2006 data file to a QE data file. You can then move the QE data file to the new computer running Lion.
Obviously, you’ll install QE on your new computer as well. This is exactly what I wish I had known to do!
1. So far so good. Looks like it worked! Thanks for being out there with help for Quicken sufferers!
1. 🙂
20. I have suffered through the same issue after upgrading my Mac to OS 10.7 without any forewarning as you indicated. I contacted Intuit and had them upgrade my Quicken 2007 files to Quicken
Essential and they have returned the files. My current problem is being able to open and load 4 sets of accounting records (not just one). When trying to open any of the other data files and
recognize as new before importing per QE’s instructions (File/New; then File/Import) QE does not recognize any of the other files.
1. Well we were not alone, Randy. This post has had over 1,700 views!
I’m sorry I don’t have any ideas about your issue with opening multiple data files. I only use one and have no experience using several at a time. If you find a solution please let me know.
1. Ferd, if you learn how Randy managed to access his/her files in order to send them to Intuit, please post the answer. I neglected to check the boxes below before sending my request to
Randy, so I don’t know whether I will get notification of a response. Thanks again.
1. Jody,
I don’t know what Randy did exactly but the first thing to do is to locate the Quicken 2007 data file. It is probably in your documents folder, in a Quicken folder, and the default
name is probably Quicken Data.qdfm. If you plan to send a file to quicken, that would probably be it.
Here is a link from Quicken that helps you do the rest:
Let me know how things worked out for you. Good luck and Happy Thanksgiving! 🙂
2. Randy, how did you access your 2007 files in order to send them to Intuit and get them into Quicken Essential format? That sounds fascinating… ;0) Please tell all!
21. Sigh. Only figured this out now after having downloaded Lion and found my Quicken and MS Office icons crossed out. Very casual about this, is my dear Apple. Grr, Lion! I’m going to try doing what
Randy did, contacting Intuit. Problem is, how do I access my 2007 files in order for Intuit to upgrade the files? I’ll send Randy that question directly. Thanks for this posting, Ferd. I may have
to do that sectioning of the hard drive, but…. ick. Not my idea of a pleasant post-Thanksgiving activity.
22. Intuit will help you, and it should work out with a lot less pain than you went through, if you manage to find your data files. This should work if you know how to find files on your hard drive.
If you don’t, write to me and I’ll help.
I contacted Intuit and they converted my files for me.
Sort of.
Intuit sent a link for me to click on leading to a page where I could type in a case number they provided, my email, and a PIN they provided. Once I did that, I was to upload a compressed version
of my Quicken Data file according to directions they gave. This all worked very well.
Found my data file on my hard drive by looking for a file that had the extension .qdfx. Compressed it, attached it to the Quicken site by following the easy directions, and the next morning, got
an email with directions for downloading my now modified-for-Quicken Essentials data. Followed the Quicken Essentials instructions for moving that data into my new program. It took about 8 to 10
minutes to transfer, and I thought I was in the clear.
Unfortunately, in my case, all that showed was data up through March of 2011. For some reason, the entries from April 1, 2011 through November 2011 are not there. I don’t know how that
information could have been separated from the rest, but it didn’t end up in the Quicken Essentials files once I performed the upload.
I have alerted Quicken/Intuit to this problem, and they are gamely trying to help me find that missing data, even though it is probably here in MY computer. They can only convert data they
receive, after all. The problem is that I have done a search for files ending in .qdfx, and there don’t seem to be any more of them on my hard drive.
So, I have sent back what I sent to them originally, in hopes that they can search around a little more. Who knows?
Good luck to the rest of us who have “old” Quicken and end up moving to Lion without reading the fine print.
23. Hi Ferd, When you first did you export from Quicken 2004 was it a .qif file and did it just import into Q2006 or did you have to do something else? I’m using Q2006 and have access to a computer
with Snow Leopard and found $12 version of Q2006 to load up. I noticed that Quicken doesn’t use .qif files for export any more so was wondering how you handled that. Thanks for the great post.
1. Hi I’m in Australia and have the same problem – i’m on 2004 but I still have my snow leopard machine and if I had a 2006 copy I could move to that and then export to the new 2007 on Lion.
Could you tell me where to get a copy of 2006 please?
24. Oops i’m using Quicken 2002 for the same reason you were using 2004
1. Wendy, over the years, different versions of Quicken used different extensions to name the data files, so I can’t tell you which ones used .qdfm, .qif, or something else. Also, they might be
physically located in different folders. But the data file should be easy enough to find.
The goal is to have a MacOS 10.7 Lion Quicken Essentials data file. To do that you first need a MacOS 10.6 Quicken Essentials data file. (You can load Quicken Essentials in both 10.6 and in
10.7. It will run on both.) Quicken Essentials on 10.6 will convert data files from only Quicken 2006 or 2007. I had to convert my Q2004 to Q2006, and then convert that file to QE on 10.6,
and finally convert that file to QE on 10.7. You might have to do an extra step if Q2002 won’t convert all the way up to Q2006. You might have to convert Q2002 to Q2004.
25. I, like Kelly, have a macbook 10.4.11 (quicken 2006) and just bought a macbook pro with lion. I read on quicken essentials that you need leopard though…however you have the same version and it
worked?!? just double checking. thanks!
1. Lara, you need at least MacOS 10.5.8 to run Quicken Essentials. You should do that if you can, because you can then convert your Q2006 data file to a QE data file on that system (your old
system.) You then take that QE data file and move it to your new computer to convert it to QE on Lion.
26. Ferd- You mentioned earlier in this post that you may be able to provide a copy of Quicken in order to convert the 2003 file. Can you either email me a link or let me know where I might be able
to obtain the version. I only need it for conversion. I wish that Intuit would just create a tool for this. I am not comfortable sending a financial record to Intuit. Thanks for any help you can
1. Yeah, I wasn’t comfortable with sending my data file to anyone either.
I’m glad things worked out with the Q2006 file! 🙂
27. I’m also looking for a way to get a copy of 2006 or 2007 or anything needed. Too bad there’s not a link for a “trial version” or some other way to quickly and cheaply get the software. I’m trying
to update my mom’s Quicken and she’s using 04 but just bought essentials so of course google brought me to your page as well. Thanks for any suggestions!
1. There you go. I hope it helps.
And Merry Christmas! 🙂
28. Found you while checking online for data re: Quicken Essentials. Did not read all the posts but remember having trouble converting my data last January when I purchased Quicken 2010. Have been
happy with the new version however now that year end is here I can not find out how to make a copy of my account that will become my 2012 Quicken. In 2004 version you would make a copy – it would
ask if you wanted to include only uncleared items and i would have a copy that became my new year with a simple name change. can’t seem to find this option. Also reports seem so different from
what i had printed in the past. Thanks in advance if you can provide any info for me.
1. Marilyn,
I’m sorry, but I am not a Quicken power user. We don’t do what you do with new data files for each year. We just keep using the same data file and print out whatever reports we want. The QE
reports do look a little different than they did in Q2004, but they suit the purpose for our accountant at year end.
Good luck with your Quicken issues!
29. Should have checked the email box on previous question. Sorry!
30. If I installed Parallels now after the Lion update and installed windows and Quicken for windows would my old Quicken 2007 data be retrievable and transfer from my hard drive? I was planning on
doing this anyway for some other programs but hadn’t done so yet. Otherwise would I have to do all the data transferring/updating in MAC like you describe and then switch to Parallels?
1. If you will be running Parallels 7 on Lion, and you plan to stick with a PC version of Quicken, I don’t think you will have any problems. Most likely you will be able to use the same Windows
OS via Parallels, and the same version of Quicken you are currently using, without having to do any translations. Let me know how that works out!
31. thanks so much for your article! think I’ll hold out on my old computer with Quicken 2006 til spring and hope for the best then…
32. Thanks Jennifer!
Like I said at the end of the post:
I predict that Intuit will eventually come out with a full featured Quicken for Mac that will run on Lion. The Apple Macintosh platform is dramatically gaining market share, especially with
laptop and iPad users, where others are in decline. Intuit is a good company and will certainly respond to that.
Sooner or later we will have a good Quicken product for Lion. Meanwhile, Gail and I haven’t really lost any functionality at all since we only need the check register function and a few reports.
I hope it all works out for you, and that you have avoided some of these hassles! Happy New Year!
33. Hi Ferd,
Have you found out about an easier way to open up old Quicken files on a mac already updated with the Lion yet? I used it mainly as a check register and to do taxes (like you)… which I of course
need to do now…ugh!
1. Hi Sonia. No, I don’t know of anything new for sure. The previous commenter, Jennifer, left a link to a MacWorld article about a Quicken 2007 version that will be coming out that is Lion
compatible. The article says it will be available in the Spring. That may or may not work for you. It may or may not be available before tax time. If you really need your data for taxes, you
will probably need to do the data file translations as described in the article and in some of the comments.
34. Thanks so much! I had no idea that Lion and Quicken 2004 weren’t compatible, so your instructions came in handy. (We also ended up partitioning the hard drive. Good times, that!)
1. Yeah… right… “Good times!” LOL
I’m really glad it worked out for you! 🙂
35. Another request for Q2006 to convert my old Q2002 files. Thanks for the good info here.
1. There you go, Brad. Good luck. Hope it works out.
1. Can I get a copy of Quicken for Mac 2004 or 2006 so I can do the upgrade from 2002 also?.
36. Hiya Fred. Are you tired of all this yet haha. I just found you looking for how to convert my most important and beloved Quicken 2002 files for Lion/QE. Thanks so much for all the good info! I
apologize for bothering you but can I pretty please get a copy of Q2006?
1. No, Mary, not tired. Glad to help!
I’ve sent you the file.
Good luck! Let me know how it goes.
1. Thanks so much Fernando! Sorry bout the name mix-up – my bad 🙁
I will let you know fer sure.
2. Hi Ferd – great information – could you please send me a copy of Q2006 also.
37. Hi Ferd,
Thank you so very much for your writeup. I’ve been dreading doing this for a while, but now’s the time to bite the bullet.
I’m unable to find a way to purchase Quicken 2006 online. Could you send me a copy as well?
38. Just to follow-up – Unfortunately for me Q2006 won’t run on my Leopard (10.5). “Intuit lists Quicken 2006 and earlier as not compatible with Leopard. It lists Quicken 2007 as compatible” – Very
strange as Q2002 runs just fine here. Oh well. I’ll just need to find a copy of Q2007. Thanks anyway Ferd! 🙂
39. Hey Fred,
Thanks for the info here. I am a Quicken MAc 2003 user and am looking to upgrade my data file to the new 2007 version that was just released for Lion. I was not sure from the comments if it was
possible to go directly from 2003 to 2007. Intuit’s site only mentions 2005-2007. Is the 2003 to 2007 conversion possible? If not, would it be possible for me to get the version that you were
able to provide a few lucky people (for conversion purposes only). Any help would be GREATLY appreciated!
1. Hey Brent,
First of all, I know nothing of the new 2007 version. I hope you are able to translate your 2003 data file directly to that, but I bet it’s too old.
Are you already on Lion? If so, I’ll send you my copy of 2006. It has worked for some people and not for others, I don’t know why.
If you have not converted to Lion yet, do your Quicken data file conversions first.
1. Ferd, thanks for the file. But my 10.6.8 couldn’t/wouldn’t open it. Guess I’ll have to swithc apps 🙁
2. Dear Ferd, I too like most here are having issues finding a means to convert my Q2003 file to something a little more current for use with Q2007. If you could also share your version of
Q2006, I would forever be in your debt!
Thanks much.
40. Sorry Ferd (dyslexia got me – didn’t mean to address you incorrectly).
1. LOL. No problem! 🙂
41. Hey Ferd, thanks for the quick reply, dude, you are so awesome for helping out. I actually have both Lion and Snow Leopard (kept expressly for running the 2003). I will try the conversion in SL
first and then see if I can get the file to import into the new 2007 version for Lion. I have almost 20 yrs of quicken data so hate to loose it.
42. Well, bought the 2007 Lion compatible version and as expected it will only covert from 2005 or higher. So now I need to get my hands on a copy of 2006 so I can do the interim conversion.
1. Brent, I tried sending the file but your mindspring email address rejected my response. Do you have another email address I can use?
43. Hey Ferd, I provided a different one to this message; hopefully that one will work. It has a limit of 25Mb so hopefully that’s enough. If not, I can have you put it on my iDisk public folder. I
really appreciate your help!
44. Hi Ferd,
All this info is great-thank you. Can you tell me if I’ve got this right: I’m running Snow Leopard 10.6.8 and currently run Quicken Mac 2004 v. 13.0 r1 (which according to Intuit’s website is not
upgrade-able to the new Lion compatible 2007 that was just released). Can I upgrade to the old Quicken Mac 2007 (which I’ve seen for sale online), and then upgrade to Lion and then to the new
Quicken Mac 2007 that was just released to be compatible Lion? Any help you can give me is greatly appreciated!
1. Hi Sue, sorry about the late response. Yes, you’re on the right track. The OLD version of Q2007 should work out great. It’s great you haven’t made the switch to Lion before all this. That
avoids a lot of the trouble.
I don’t know anything about the NEW Q2007. I’m sure it will need a newer data file for conversion, though. I think your plan is good.
Good luck. Let me know how it went!
45. Will any of the above mentioned conversions work for those of us who have ancient software :(. I’m still running Mac OS X 10.4 Tiger and using Quicken 2004. Any info would be greatly appreciated.
1. Yes Deb, what I described will work for you. I was also running Q2004 when I did my conversion. It will be easy if you haven’t already switched to Lion. You will first have to convert your
Q2004 data file to the OLD Q2006 or Q2007. Then convert that to Quicken Essentials BEFORE you upgrade to Lion. I don’t know anything about the NEW Q2007, except that I know it will need a
newer data file for its own conversion. I’ll try to find out more about the new Q2007.
1. Hey Ferd… I too, like many here, am having issues finding a way to convert my Q2003 data base file to something more current to use with Q2007. If you could so kindly share your version
of Q2006 I would be forever grateful. Thanks so much for the support and excellent feedback.
46. Ferd,
Thanks so much for the info. I have not converted to Lion and probably won’t on the platform (Power Mac G5) I currently have. My plan is to buy a new MAC of some sort and hoping the converted
Quicken files will be ok on the new system.
I know many have asked you for the Q2006 but would it be possible to supply me with it as well?
47. Hi Ferd, great post. Can you convert a QIF file to Quicken Essentials for Mac for me? I’ll pay you. That would still be cheaper for me than what I would have to do to follow your steps. I had an
old mac with OSX 10.4.11 running Quicken 2006. I just bought a new mac mini with Lion. So now I would have to buy a mac with snow leopard (not enough memory to load it on my old mac mini) just to
convert this old QIF file. If you don’t feel comfortable doing that, do you know of any services that would convert my QIF file to a file I can import into Quicken Essentials for Mac on Lion?
48. Hi Ferd, great post. Can you convert a QIF file to Quicken Essentials for Mac for me? I’ll pay you. That would still be cheaper for me than what I would have to do to follow your steps. I had an
old mac with OSX 10.4.11 running Quicken 2006. I just bought a new mac mini with Lion. So now I would have to buy a mac with snow leopard (not enough memory to load it on my old mac mini) just to
convert this old QIF file. If you don’t feel comfortable doing that, do you know of any services that would convert my QIF file to a file I can import into Quicken Essentials for Mac on Lion?
Mitchell Rosenwald
I need help getting a 2006 version of quicken. I need to covert my 2004, so I can use it on my os10.7 upgrade I already did.
Do you have a copy of your 2006 quicken?
If so, would you be willing to let me borrow it, or buy it even? I need it to convert my 2004 files to 2006 on its way to Lion.
I would be eternally grateful if you can help me.
49. Thank you for such an excellent write-up of your experience converting an older Quicken data file. I find myself in a similar situation, with a Quicken 2004 data file that I’d like to convert to
the 2007 version. Of course, I can’t do that without Quicken 2006, which I can’t find anywhere. If it isn’t too much trouble, would you be so kind as to send me the Quicken 2006 .dmg? I would be
most, most grateful!
1. Hi Justin
If you did get a copy of the QFM2006dmg would it be possible for you to send it on to me as I need to go through the same process. I’m based in Australia.
Cheers Bob
50. I have the same problem with a customer who has Quicken 2004. She has now Quicken 2007 for Lion but the problem is that we need Quicken 2006 to do the intermediate conversion from 2004 to 2006
then to 2007. I’ve looked everywhere for Quicken 2006 mac but it’s just about impossible to find. Any help would be appreciated. Thanks 🙂
1. I’m also a Q2004 user having trouble finding Q2005 or 2006. Would anyone be willing to sell their copy? Many thanks.
51. I have been using Quicken Deluxe 2002 and could not get Quicken 2007 (PPC) to successfully convert my data. I would go through the steps of conversion and then after about 5 minutes in the
memorized transactions step, it would crash. The resulting file was no good (BE SURE TO COPY YOUR ORIGINAL FILE FIRST!) I even tried deleting my two memorized transactions first, but same result.
I even tried deleting out all of my data except those for 2011 and 2012 and trimming down my accounts and transactions to “slim-down” the file. Same result.
After reading this post, it occurred to me to try Quicken 2006 to open my Quicken Deluxe 2002 data file and convert it — IT WORKED LIKE A CHARM!
Anyone who needs more instructions about using Quicken 2006 (PPC) to convert send me an email at my nickname at America Online (AOL dot com).
52. PS: Up to this point, I have been using Snow Leopard in Parallels 7 to run Quicken Deluxe 2002 on my mid-2011 Mac Mini which requires Lion.
53. PPS: I now access my data with Quicken 2007 for Lion; I did NOT convert to Quicken Essentials…
54. I can be reached at my nickname at America OnLine (AOL) dot com
55. Hi This is a most helpful website for us all that have the Quicken for Mac problem.
Can any one out there please send me a copy of QFM 2006 so that I can liberate my QFM2004 data and get running under lion. It would be greatly appreciated.
56. Installation instructions for Snow Leopard into Parallels 7 in Lion:
57. Hi Ferd – I was wondering if you can help me out… I’m looking for a mac version of Quicken 2005 or 2006. My brother recently updated his OS to Lion and now his Quicken 2003 data file won’t work
unless it updated to a later version. Reading this great comment thread, it sounds like you may have a version? If so, it would be a huge help so I can get this file converted. Sorry to
inconvenience you… but this software is almost impossible to find online considering how old it is.
Cheers and many thanks.
58. Ferd.. I think I have now read everything written on this topic, along with all of the comments here, plus a couple of your wonderful blog posts. I have concluded that you may be the only person
alive with a copy of Quicken 2006 for the Mac that is being made available for file conversion (from 2003). I am appalled that Intuit isn’t providing this service, especially in light of how much
money all of us have spent with them for their full line of software (YEARS of Turbotax, for example).
So, may I please get a copy of the coveted application? I’ll certainly understand if you have tired of this. In any case, bless you for the customer service that Intuit fails to provide.
Keep up the great blogging!
59. Hi Ferd — I have Quicken 2004 (13.0 – R1) on a G4 (10.4.11) For everything else I have a MacAir (10.7.3). As you know, I can’t run Quicken on the new MacAir. I feel the way you do about my
Quicken data. Should I try to do what you did?
60. I need help getting a 2006 version of quicken. I need to covert my 2004, so I can use it on my os10.7 upgrade I already did.
61. I upgraded to Lion BEFORE I knew about the Quicken issue. I have been using Quicken 2004. I am in awe about your partitioning solution, but I have no clue how to do it. Is there anyone out there
with a copy of Quicken 2006?
62. Oh I’m SO glad I found this post! I have the exact same problem you had, Q 2004 moved to Lion, no-workie. I do wish the solution was simpler, but I think I can do just what you did using my old
laptop. Ugh, I have Mac’s so I don’t have to deal with difficult stuff like this lol! Off to find a copy of Q 2006 😛 So much for “it’s only $30 to upgrade to Lion”
63. For some unknown reason, my Quicken 2006 file is now corrupt. I have been unable to email it to the last few people who requested it. I will try to recover it from an old backup, but for now I am
unable to provide it.
64. Ferd: I have compressed my copies of Quicken 2006 and Quicken 2007 (PPC) into zip files, so that they do not get corrupted by email.
1. I still need to convert my Quicken 04, could I please get a copy of 06?
Thanks for being our saviors, Ferd and Michael!
1. There you go, Carolyn. Let me know if it works out.
2. Thanks for your help, Michael!
65. Dear Ferd, you seem to have saved half the planet by now! I am in need of Quicken 2006 like many others and cannot find it on Amazon etc. If your file is OK I would be most grateful for a copy.
1. Ha! Not half the planet yet, LOL, but this post has had over 9,500 views. I am shocked and humbled!
Let me know if it worked out.
66. Dear Ferd, I too am in need of your services! It would be HUGELY appreciated if you could send me a copy as well. Thanks in advance for your help!
67. Sadly, looks like I need to jump on the train as well – if anyone could be so kind as to share a copy of quicken 2006, it’d be much appreciated.
68. ferd, I guess after reading all the posts to this problem ,I am going to ask for a copy of your Quicken 2006 also. I’m glad I checked on this before I bought a new computer. I have now a power pc
g4 mac running os10 4.11 and quicken deluxe 2002. It sounds like from your previous answers that 2006 will run On my computer (I hope)I knew someday my computer would be obsolete, but have been
putting off buying a new one until I absolutely had to. Looks like that day has come, no one is supporting power pc anymore. I know getting a new computer will be a good thing but lack of money
really has been the reason it hasn’t happened before now. Losing all my old money info would really hurt, thanks for any help you can offer
69. fern- thanks a lot for the quick response, I’ll be trying the 2006 on my old computer today- hope it works (one step forward)
70. OMG was that easy can’t believe quicken doesn’t have a link do do this. hope its as easy when I get the new computer TY TY TY TY
71. Ferd–I too am in the need of Q2006 for Macs. I tried to jump from Q2002 to Mountain Lion! Clearly I was clueless about the problems that would entail. I am hoping converting 2002-2006 and then
Q2007 for Lion will do the trick but I am holding my breath. Your post is so popular because people are frustrated. Q tech support basically said I was SOL after 20 years of using Quicken. So
thanks for any help you can provide!
1. Remember, you will need to run Quicken 2006 for the data conversion on a Mac that supports Rosetta and Mountain Lion does not. Then you can bring the updated Q2006 data to Mountain Lion and
open it and run it with Quicken 2007 for Lion/Mountain Lion
Is this a problem for you now that you have upgraded?
1. Thankfully, Michael, I still have my old computer that I can run Quicken 2006 on and do the conversion. Usually I wait until my old computer fails but I am so grateful I switched before
that happened. 🙂
72. Thanks for the Q 2006. I was able to convert my data and run Quicken 2007 for Lion on Mountain Lion, for anyone else who is wondering. A little help here and $15 and I am good to go for several
more years with Quicken. Yay! Thanks!!!
73. Great! 🙂
1. Dear Ferd, Thanks for all your efforts. I have Quicken 2003 and I am in need of Quicken 2006 like many others and cannot find it anywhere. If your file is OK I would be most grateful for a
copy. Thanks.
2. Dear Ferd, Like many others, I am in need of Quicken 2006 (I have Quicken 2003) and cannot find it on Amazon etc. I called Quicken and they told me to find someone with the old version. I
would be most grateful for a copy. I don’t know why they wouldn’t have a downloadable copy for sale on their website. Thanks.
74. I will do anything to get ahold of a copy of Quicken 2006. I am one of those who just stuck with 2004 for Mac and found that it worked fine. Just bought a new MacBook Pro and am totally hosed.
Please e-mail me and I will pay just about anything to get 2006 for Mac.
75. Place $1,000 in small bills under the dead tree trunk in the yard next to mine! 🙂
1. At that rate, I’ll hire a school child to transcribe my 20 years of data! (But thanks for the offer!)
76. 20 years!?! I go back to Quicken 4; what version did you start with? I actually purchased a 5-1/4″ floppy disc copy of Quicken 1.0 for the Apple // just to have it in my Library.
Prior to Quicken I used MacMoney and before that I used Time is Money and Home Accountant on my Apple //.
1. I honestly don’t recall my original version. My first personal computer was a Mac II Ci. I do recall trying to track my investments, diligently entering them daily, only to have all the
historical information lost every time I would upgrade to a new version – probably part of the reason I quit upgrading after Quicken 2004.
Ferd was great and sent me a copy of Quicken 2006. The only problem is it won’t open on my current 17″ MAcBook Pro (about five years old, now) running OS X 10.5.8. No I haven’t a clue as what
to do to bring myself up to the year 2012 and run Quicken on Mountain Lion. Definitely feels like a conspiracy going on here (OOPS, 60’s paranoia returning with a vengance!)
77. Zachary: Yeah, I think Quicken used to keep certain data in its Application folder of all places!?! Maybe lost when you moved computers and reinstalled it. Ferd: can you send him Q2007? Maybe
that will work for him in Leopard; I do not know why Q2006 is not working…
1. Michael and Ferd,
I think I see what the problem is. The Quicken 2006 you’ve sent says “Application (Classic), under Kind. Is there a non-Classic version out there somewhere? Seems as thought there should be,
as my 2004 works under OS X 10.5.8 without any difficulty.
Thanks again,
78. Do a GET INFO on the Quicken 2006 file: it should be version 15.0.1
1. Michael,
It doesn’t give a version. Just ‘Quicken 2006’ under ‘Name & Extension’. Kind still says ‘Application (Classic)’. Doesn’t seem quit right. I have Quicken’s going all the way back to Deluxe
2000 that work.
79. Under General, you do not show:
Modified… and then
VERSION ?
1. Everything you have listed above is there, EXCEPT Version. Under the preview there is an international do not enter sign in grey and white, and a tiny bit of the Quicken logo in color in the
lower right hand corner. Is there a way to send you a screenshot on this discussion board? I think it all has to do with the fact that the application is in Classic mode, for reasons
unbeknownst to me.
2. Ferd and Michael,
Thank-you so much! The Quicken 7 opened on my MacBook Air and gobbled up my old Quicken Data. I haven’t gone the next step yet, which I presume will be to get the Quicken 7 data into Quicken
Essentials, then transfer to the new MacBook Pro with Quicken Essentials running. I did look at a few other discussion groups who suggested switching programs, with MoneyDance sounding like
the best bet. That one wouldn’t open on my old 17″ MacBook Pro – have yet to try it on the Air.
Making progress thanks to you all! Will keep you posted when a solution finally arrives.
A million thanks to Ferd and Michael, once again!
80. Ferd, I want everyone to know you sent me a copy of Quicken 2006 and after I updated my files on my old computer and transferred it to my Air, it now works with my Quicken 2007. I truly
appreciate your help and dedication to helping all of us through this issue. I do not understand why Intuit does not seemingly care a whit about their users proven by their lack of meaningful
support. They told me to “find someone with a copy of Quicken 2005 or 2006.” I did that after a year of searching for a copy of the old Quicken program, when all they had to do is post it on
their site for downloads. Is that act of affirmation for their customers too much to ask? If I hadn’t used their program for so long, and understand how it works, I would have abandoned them as
they really have abandoned to us. I travel a lot and had to maintain my accounts on my old computer, traveling with the two of them, until Ferd came through. What are they thinking? Thank you
again Ferd.
1. You are so right. It would be a simple act of good faith for Quicken to provide an easy download to the old Quicken 2006 so people could do their file translations and continue to use their
newer products!
I’m really glad this worked out for you! 🙂
81. Ferd: Please send Zachary the latest version of Quicken 2007 PPC that you have; let’s see if it solves his problems.
1. Thanks for the files, Michael!
I forwarded them on to Zach.
82. Hi,
This is all Greek to me, but I too have this issue… only its with the 2006 version. When they updated my Mac, it all went to s*#@! Now what do I do?? Are you saying I have no recourse at all,
except to start over? I’ve been delaying my taxes and now can not wait any longer… feels like hell all over again, as this happened a couple of years ago. Why have data on computers if every few
years my entire financial existence is erased??!! Ugh. Any thoughts would be appreciated.
1. J,
I think you’ll be alright. You were using a relatively recent version of Quicken, so I think you just need to get the new “Lion Compatible Quicken Mac 2007.” Here is a link to it. Probably
worth the $15.
83. Ferd is spot on correct! Assumably they upgraded your Mac to Lion or Mountain Lion. The $15 Quicken 2007 for Lion (which is now updated for Mountain Lion) will read your Quicken 2006 data file
without any translation needed and work just like it did in the past!
Mitchell Rosenwald
I need help getting a 2006 version of quicken. I need to covert my 2004, so I can use it on my os10.7 upgrade I already did.
Do you have a copy of your 2006 quicken?
If so, would you be willing to let me borrow it, or buy it even? I need it to convert my 2004 files to 2006 on its way to Lion.
I would be eternally grateful if you can help me.
I too need to convert my old Q2004 data to Q2006 before I can convert further.
Could you send me a copy of your Q2006. aso??
thanks for all you’re doing??
1. There you go. Hope that helps!
1. Thank you SO MUCH!!!!!!!!
You have NO IDEA how frustrating (well, maybe you do!!) this has been…
But by simply finding the saved Q2006 file from Documents (after opening the Q2004 files in Q2006!!) I copied the file to a flash drive and then opened the file using the Q2007 (lion
compatible) on my newer iMac running 10.6.8 (SnowLeopard).
IT WORKED!!!!!!
Now I can continue to grumble at Quicken for asking me to PAY for an online chat about this problem, right after having BOUGHT Q2007 from them!!!
But at least I’m up and running for a while….and I DIDN’T have to buy QEssentials (QEM) which ALL their Help discussions try to get you to do!!!
Thanks again Ferd!!!
“Happy in Maine”
1. 😀
85. I am currently running Quicken 2006 on Mac OS X Tiger. I plan to purchase a new Mac with Mt Lion. Will I be able to purchase Quicken 2007 and use my 2006 Quicken data files on my new Mac? I am
not sure if QE will support my needs.
1. Debbie,
You should be able to go from Q2006 directly to the new Q2007 without any problems.
86. I had to use your partition method to get around this issue. When I was done, I bought CheckBook from the App(le) Store rather than “upgrading” to Quicken Essentials. I like TurboTax, but I will
at least look around for alternatives to that, too. I do not appreciate companies intentionally making my life harder for reasons I cannot understand. I had even contacted Intuit directly to ask
how to convert a Quicken Data File to QIF, and they told me that I could not.
1. Conrad,
I don’t blame you one bit! I cannot fathom how a company could be so willing to piss off so many of its loyal customers.
87. I don’t know how to do the partition method, but have spent days trying to upgrade my old Quicken, because I have Mt. Lion. I bought Quicken Essentials, which of course didn’t work. Then I was
told (by Intuit’s “support” ) that I should get purchase Quicken 2007. Of course that didn’t work either. I then got really desperate and had someone remotely control my computer in order to
upgrade. He swore he could do it, and of course in the end, he couldn’t, because my Quicken is too old. He told me I need 2005 or 2006 in order to eventually use Quicken Essentials. Now, my
problem is that I can’t find 2005 or 2006. I noticed that you offered to make a copy for someone. I am happy to purchase a copy from you if that is still an option. As I’m sure you can
understand, I am really desperate. My whole life from the last several years is on that.
By the way, I really detest Intuit for the complete and total lack of support. We are supposed to be able to return products if we are unhappy with them, however, I haven’t figured out how to do
that either. I would GREATLY appreciate any help you could give me. Thank you so much.
88. I have just done this. Grr. I was running Quicken 2002! Will Q2006 convert the files to a version the QE will convert again? Where can I get Q2006? I hate that they won’t convert the files for
you in some easier fashion. Intuit tech support was useless except to tell me that they can’t help me. Anyone want to convert my files for me and email them back? That would be so much easier.
Thanks for the post, still useful more than a year later 🙂
89. ferd! you really are a saint for continuing to help people after all of this time. 🙂 thank you!! i (like many on here!) upgraded to mountain lion and am now stuck with no access to my Q2004
files. I mainly used it as a checkbook, as you mention — and really just need access in case of emergency to my old file to look things up etc. I also have an old macmini that is an os 10.4.11
(no intel chip) and assume this will run the 2006 version that you mention updating to. I am considering updating this old mac mini just to see my files for an other year or so and in the
meantime starting a new data file on my mountain lion system. (i’m not super techy and am afraid of all of partitioning business). anyway… long way of saying — i can’t get the 2006 version
either. Could you help me out as well. A million thanks.
90. thank you for all of your help. this is the best link i have found. I have Q 2003 files. I already updated to mountain lion because I thought i was going to switch and use mint.com. I don’t like
it and i want to go back to using Quicken. Do i need quicken 2006 or 2007 to convert my files? And I also know that I need to find someone with an older MAC to do this conversion. And then will
my files work if I buy quicken essentials for my mac running mountain lion?
91. Thank you so much. Spent hours every day trying to find a solution. I am blessed to have stumbled my way here. I started with the first mac SE Running quicken since 2000 have Q2004, running
10.4.11 on my Imac and 10.7.5 on my macbook. I am the 84 yr old grandma who loves keeping records and playing games. Can you please send me Q2006. I lost my 2004 appl disk and am afraid if my
computer crashed I could not convert my backup.
Thank you so much for helping so many.
92. I was two minutes away from pushing the button to purchase the Mountain Lion upgrade (currently using 10.6.8) when I saw my Quicken logo in my dock and decided to do a search about compatability!
Thanks! I’ve got Quicken 2004 and so would greatly appreciate if you could send me a copy of what I need (2006 or 2007?) to upgrade my file so I can purchase a new Quicken software. Is Essentials
the only Mac Quicken available for purchase now?
93. Hi Ferd, can I too add my request for a copy of Quicken 06? I am using Quicken 2004 and looking to upgrade to the Lion version… I have access to a pre-Lion computer til Monday! Many thanks!
94. Dear Ferd and/or MichaelLAX,
Yet another humble request for an emailed copy of Quicken for Mac 2006 or 2007, if possible, please.
Having read right through your blog and the discussion, I now know that I’ll need this if I’m going to ‘upgrade’ (convert?) my QFM2004 data file (.qdfm). This is because I’d like to upgrade my
system to OSX 10.8 Mountain Lion in the near future; I will then purchase Quicken for Mac 2007 for OSX 10.7 (Lion) from Intuit.
I’ve used Quicken for about 20 years, and I’m so comfortable with it that I just don’t want to ‘change horses’ now. Currently I still use QFM 2004, on a 2010 MacBookPro (intel chip but with
Rosetta, I guess?) using OSX 10.6.8 Snow Leopard. Over the last few years I’ve despaired about finding an upgrade path – last time I looked, in early 2012, there seemed to be no hope.
A couple of weeks ago I was just about to give up and switch to MoneyWorks or iBank; I just thought I should have one more search on the ‘net for a solution – and miraculously I came across
Ferd’s blog and this wonderful discussion. I’ve tried to buy the QFM2007 from the ‘net but I just can’t find a trustworthy site that has it. So I’d be very grateful for your further help in
sending it to me.
With many thanks,
Yours sincerely,
95. I just used my 2004 Quicken for Mac and exported from the FILE menu a full export to the Quicken backup folder. I am running 2004 for Mac on a flat screen iMac(PPC) with 10.4.11. I then used the
terminal mode restart on the old Mac and connected to the new iMac(intel) running OSX 10.8.2 (Mountain Lion) with a firewire cable (needed a cable converter for the old (400) and new (800)
firewire port). I opened the Quicken for Mac 2007 Lion compatible version on the new mac and used the open file option from the file drop down menu. I opened the newly created QIF export in the
old Mac from the new Lion version and saved a copy in the new Quicken Backup folder. Presto! All accounts and balances transferred into the new 2007 Lion version. Now not everything transferred
over to the new version but the accounts all did with correct balances as well. The items missing were mostly preferences and setup configurations. These were easily matched with the old 2004 by
adding upcoming and scheduled bills, hide and show accounts in the accounts window, toolbar accounts, and quickfill transactions. I just copied manually the scheduled bills. Edited the accounts
shown and hidden at the same time as I checked the accounts I wanted shown in the toolbar. I used the registers to re-spell a word in old transactions and hit enter to create new quickfill items.
I finished the conversion by editing the toolbars to match what I was using in the 2004 version and that was enough for me to get rolling. So, long story short, no Quicken Essentials conversion,
no Quicken for Mac 2005, 2006, conversion from Essentials, and no conversion back to 2007 from Essentials was needed. No file exchange utility needed. Maybe it’s just the PPC version that creates
the .qif file but since I never had 2004,2005, or 2006 on an Intel Mac, I really don’t know what the “export full copy” creates on those. I know if you try to save a copy, it creates a .qdfm file
that the new Quicken for Mac 2007 Lion or the 2007 version will NOT import or open it with a note telling you 2004 is not compatible to update. Just food for thought.
1. Additional info…………. I created a fake bank account in the new Q2007 for Lion. Then imported the Q2004 qif file that was created from “export” on the old iMac. Wha La…… all the quickfill items
populated. Now if I can just figure out how to get the scheduled transactions to also populate from the old version, I would have a step by step upgrade without having to purchase the QE.
Mine has been running without any glitches since I first downloaded the Q2007 for Lion and importing the Q2004 for Mac from a flat panel iMac (ppc) running OSX 10.4.11. I am just tweaking and
playing with the setup to see what else it will actually import despite Quicken saying that 2004 is incompatible.
96. Hi,
I’m in the same “I need Quicken 2006/2007” bind that others are in. I’m upgrading from Quicken for Mac 2002 (running on 10.6.8) to Quicken Essentials for Mac (running on 10.8.2, on a new
computer). I would be very grateful for any help on getting a copy of Quicken 2006 or 2007 to use as a stepping stone.
97. Next problem: how do I get Quicken 2012 for Lion outside the US? The Intuit website does’t let me buy it…
98. Hi Ferd, Your article couldn’t come at a better time! My year old Macbook Pro graphics switching card has been causing “glitches” while using Snow Leopard thus Apple has recommended I upgrade to
Mt. Lion, which I have been avoiding to do because I use Q2004. I too would like to request for your help for Q2006/07 so that I can begin the migration process.
Thank you for writing this for all of us who’ve come here and found this!
Regards, ~Rio
99. I read through what everyone said,,. I have used Quicken since my first Mac Classic (1994) and rely on it. I am the treasurer of a local charity, strictly volunteer work. I just got this MacBook
Air and bought Quicken essentials, probably a mistake. But anyhow I guess the thing to do is get Quicken 2006 as you have done for so many. I have the new 2007 CD on order. Can you help me?
100. Another in a long line of Mac faithfuls that never dreamed Apple would not warn me of my Quicken2004 becoming unusable upon upgrading to Mountain Lion. -sigh-
Anyway, would you please be so kind to send me a version that I can use to convert my 2004 version to one that will open with the newest version of Quicken? Thanks so much.
101. Ferd, I just wanted to let you know that I won’t be needing a version of Quicken2006 after all. I was able to download the latest version of Q2007 that works with Mountain Lion. Then I opened my
data file on an older laptop running Quicken2004 and saved it as a qif file. I emailed that to myself, downloaded it onto my newer iMac with Q2007 and imported it in. Worked like a charm. Thanks
for your willingness to help. Reading your post and all the comments really helped.
1. Wow! I can’t believe we’re into the second page of comments!!!
Excellent Ruth! I’m glad you figured it out! Maybe your solution will help someone else reading this, so thank you for posting it! 🙂
1. Ferd, you sound like the Saint of Quicken Conversion. 🙂 I’m in the same boat as many (need to extract data from an ancient Quicken). I tried Ruth’s trick above…but Quicken 2007 that I
just downloaded doesn’t recognize the .qif file I sent myself. (It says “This is not a Quicken data file.”) … Could it be because I have Quicken 2003 (not ’04)?
Any tips from the geniuses on this thread? Thanks!
1. Christopher, just to give you some more details: my Quicken2007 at first balked at opening my qif version, too. It worked once I created a new file (File>New>File), and then imported
the qif (File>Import>From QIF). Hope that helps!
1. Ruth, I could e-kiss you! 😉 (Thought it was odd that the import menus were all greyed out.) Thank you, thank you: a decade of information salvaged!
2. I love, Love, LOVE how you guys worked that out! I would not have had the answer for Christopher. So, thank you Ruth. And thank you both for posting it on this thread for others
to see! 😀
102. I have a MacBookPro which I just upgraded with Mountain Lion. I then discovered I couldn’t open my Quicken accounts. I really panicked, but was able to download the Quicken2007app for $14.99 (I
may have received a coupon and ignored it). I opened the app and it asked to open a file. I must have a 100 quicken files of one type or another after all these years of use and automatic
backups. I tried to locate the latest ones I used (info helped) and opened them. Seems fine. I have lost the ability to switch from one account to another without using the apple menu. Looks like
I will have to open each one individually the first time I use it. Good Luck.
1. Great!
I suspect you had been using a relatively recent version of Quicken, right?
1. I think it was Quicken 2006. Haven’t updated it in ages.
1. Yes, that was the problem with most of us using even older versions of Quicken. At least with Q2006 you didn’t have to do an additional translation step. I’m glad you figured it out!
Ruth, I could e-kiss you! (Thought it was odd that the import menus were all greyed out.) Thank you, thank you: a decade of information salvaged!
Oh, yay! So glad that worked for you, Christopher! And I consider myself e-kissed. 😀
104. Any chance you can send me a copy of Quicken 2006? I’ve upgraded to Mountain Lion and need to convert my Quicken 2002 files for use with Quicken 2007, which I have already purchased. Thanks for
all the information on how to do the conversion. I hope it works for me.
1. There you go! Good luck! Let me know how it works out.
1. Here is another needy request for Q2006. I have tons of stuff on Q2002 and really need to use your work around to get my records into Mountain Lion but cannot find a copy of Q2006. Thank
2. I’m in the same boat. Just upgraded to Mountain Lion and found our afterwards that Q2002 no longer works. Could you send me a copy of Q2006.
Thanks Gary
105. Hello Ferd, I am having the same issue as Doug above. I have a decade of info in Quicken 2002 and need Quicken 2006 to do the conversion. I can’t find this anywhere. Would you be able to send me
a copy? Many, many, many thanks in advance!!
106. Hello Ferd, I have Quicken 2002 and I am urgently in need of Quicken 2006 like so many other posters on this most helpful website. Would you be able to send me a copy—I would be most
appreciative! Thank you. Zach
I was two minutes away from pushing the button to purchase the Mountain Lion upgrade (currently using 10.6.8) when I saw my Quicken logo in my dock and decided to do a search about
compatability! Thanks! I’ve got Quicken 2004 and so would greatly appreciate if you could send me a copy of what I need (2006 or 2007?) to upgrade my file so I can purchase a new Quicken
software. Is Essentials the only Mac Quicken available for purchase now?
Hi. I’m just refollowing up on my request from October for either a 2006 or 2007 Quicken. Is it possible to send me one? I still haven’t upgraded to Mountain Lion until I can get this situation
rectified. Much appreciated if you can help.
108. I have been following your solutions to the Quicken upgrade problem. I also have Q2002. I believe I need to upgrade to Quicken 2006 on my older Mac G5. I believe I first have to upgrade my
operating system from OS10.4.11 to OS10.6. Which is the latest operating system my old G5 can handle. I have ordered OS10.6 from Apple. Does this sound like it might work??
109. HELP! I had no idea I would lose access to all my Quicken accounts when I updated to Mountain Lion. I was happily using Quicken 2002, when all my financial data from the past 15 years
disappeared. I am not as tech savvy as many of you, and it looks like Intuit is just leaving me in the cold. Please tell me what to do to access my info. I purchased Quicken Essentials, but that
looks like it is useless right now.
110. I am experiencing the same problem as well. Fortunately I still have my old mac but am searching for a copy of Quicken 2005 or 2006 so I can upgrade before moving everything over to my new iMac.
Any ideas where I can get this?????
1. Thanks for the update, Leslie. Glad it worked out! 🙂
Happy New Year!
1. I was able to download Quicken 2006 from the Quicken help web site. It updated my 2002. http://quicken.intuit.com/support/help/patching/quicken-2006-manual-updates–mac-/GEN82200.html.
1. Ooh, this is new info. They must have done this recently. Many of us have complained to Intuit about their not helping provide a way to do these translations. It appears they finally
Thanks for posting this, Gail! 🙂
111. Hello Ferd,
Thank god for good people like you are are willing to do as much as you do for total strangers. I too need an older copy of Quicken so I may update my quicken 2003 files. I have a second Mac
running OSX 10.5.6. Would you be willing to send me an old copy of Quicken that can run on that version of the OS and update my files for use on a 10.8.2 Mac with the Quicken 2007 rebuild for
Mountain Lion?
Thank you very, very much!!!!!!
112. Hi Ferd,
This just happened on my mom’s computer when I updated it to Mountain Lion yesterday. Any chance you could send me a copy of Quicken 2006, so I can update her 2004 database file?
113. Dear Ferd,
Sorry for the delay – it took me a while to get back home and then to do the Quicken install / upgrade.
And It all works perfectly.
I’m now the happy user of Mountain Lion – and Quicken 2007!
I’m very grateful to you, and the correspondents to your blog.
Yours sincerely,
1. 😀
114. Thanks for 2006. Quicken essentials will not run on Mountain Lion, but 2007 does. Put all old versions on 2006 and then to the $15 2007 from Intuit and that works well. Got rid of Essentials
from my new Mac Book Air.
1. Thanks for the info, Harriet, and for posting it here for others to use! 🙂
115. Ferd, it sounds as you have been everyone’s hero. I’m using Quicken for the Mac 2004 and certainly want to be able to use the past 10 years worth of data. Could you send me the Quicken 2006 so I
can perform the same magic?
116. I’m in the same boat as others. 🙁 I upgraded to Mountain Lion not thinking about the fact that I had Quicken 2004. Now I need to a copy of Quicken 2006 to open years of Quicken data so I can
then convert them to a format readable by Quicken 2007. Internet search isn’t turning up much for sources for Quicken 2006. Help!
1. Glad it helped Harley! Thanks for letting me know! 🙂
117. Thanks to all on this thread as I was another person trying to go from Quicken 2002 to a machine running 10.8 . Made the data base updates through Quicken 2006 as explained above.
For all looking for a copy of Quicken 2006, you can download it for free straight from Quicken. Here is the link to their download page: http://quicken.intuit.com/support/help/patching/
1. Thanks, Steve!
Finally! Quicken has provided a link for people to use! 😀
1. I have been using Quicken Essentials Version 1.7.2 on my Mac Mini running OS X 10.6.8 for the past 18 months. I was originally using Quicken 2006 but was forced to move to Essentials when
Quicken ceased support of 2006 for Mac therefore my most recent entries are in Quicken Essentials. I purchased a new iMac running OS X 10.8.2 and purchased Quicken 2007 for Mac OS X Lion
this past weekend and am trying to open my most recent data in 2007 for Lion.
In Quicken Essentials I “Exported to Quicken 2007” as a QMTF file and then tried importing it in Quicken 2007. Quicken 2007 shows an error, “Unable to import because the transaction
displayed below is too large” I believe this is due to a “Split” having too many entries.
It seems there is more of an issue with importing it correctly into Quicken 2007 rather than exporting it from Quicken Essentials. Is this true?
Quicken 2007 is able to successfully import and open all data for my historical (1989 – 2012) backups using the Quicken Data Format files. It is just having a problem with importing the
last 18 months’ worth of data that was created using Quicken Essentials (QMTF format)
I have downloaded Q2006 from the above link but am still unable to Export to QIF using Essentials. The only choices for Export in Essentials are CSV, Quicken 2007, QXF. The only choices
for Import in Q2006 are QIF and Web Connect. What do I do? It seems as though you all have been sucessful in fixing these problems by exporting Essentials to QIF then importing it to
Q2006 using QIF then exporting to Q2007 for Lion. I am not seeing how to do this…..
Would it help to buy Quicken Essentials ($50) for Mac OS X Snow Leopard or Lion, then try to export data?
I have called Quicken but they will only provide free support for “the latest Mac version” which they say is Quicken Essentials. They want me to pay $10 – $30 for their support package
therefore I would appreciate any help that you all could provide.
Thank you in advance!!
118. I have been using Quicken Essentials Version 1.7.2 on my Mac Mini running OS X 10.6.8 for the past 18 months. I was originally using Quicken 2006 but was forced to move to Essentials when
Quicken ceased support of 2006 for Mac therefore my most recent entries are in Quicken Essentials. I purchased a new iMac running OS X 10.8.2 and purchased Quicken 2007 for Mac OS X Lion this
past weekend and am trying to open my most recent data in 2007 for Lion.
In Quicken Essentials I “Exported to Quicken 2007” as a QMTF file and then tried importing it in Quicken 2007. Quicken 2007 shows an error, “Unable to import because the transaction displayed
below is too large” I believe this is due to a “Split” having too many entries.
It seems there is more of an issue with importing it correctly into Quicken 2007 rather than exporting it from Quicken Essentials. Is this true?
Quicken 2007 is able to successfully import and open all data for my historical (1989 – 2012) backups using the Quicken Data Format files. It is just having a problem with importing the last 18
months’ worth of data that was created using Quicken Essentials (QMTF format)
I have downloaded Q2006 from the above link but am still unable to Export to QIF using Essentials. The only choices for Export in Essentials are CSV, Quicken 2007, QXF. The only choices for
Import in Q2006 are QIF and Web Connect. What do I do? It seems as though you all have been sucessful in fixing these problems by exporting Essentials to QIF then importing it to Q2006 using QIF
then exporting to Q2007 for Lion. I am not seeing how to do this…..
Would it help to buy Quicken Essentials ($50) for Mac OS X Snow Leopard or Lion, then try to export data?
I have called Quicken but they will only provide free support for “the latest Mac version” which they say is Quicken Essentials. They want me to pay $10 – $30 for their support package therefore
I would appreciate any help that you all could provide.
Thank you in advance!!
119. Hi, I need advice from the Quicken gurus. I have been using Quicken for more than 10 years, currently Quicken 2012 on a PC. I have totally converted to Apple products, and now have a Macbook Pro
running Lion 10.8.2. I hear that Quicken Essentials is not as good as Quicken 2012, but I need to convert somehow. I would probably be happy running Quicken 2007 on the Mac which would probably
let me avoid partitioning my hard drive or doing the Parallels software. I don’t think I have any version of Windows beyond Vista to load if necessary. What advice do you have?
1. John,
I hope this link answers all your questions:
And congratulations on your conversion to Apple/Mac! Welcome, brother! 🙂
1. I have been using Quicken Essentials Version 1.7.2 on my Mac Mini running OS X 10.6.8 for the past 18 months. I was originally using Quicken 2006 but was forced to move to Essentials when
Quicken ceased support of 2006 for Mac therefore my most recent entries are in Quicken Essentials. I purchased a new iMac running OS X 10.8.2 and purchased Quicken 2007 for Mac OS X Lion
this past weekend and am trying to open my most recent data in 2007 for Lion.
In Quicken Essentials I “Exported to Quicken 2007” as a QMTF file and then tried importing it in Quicken 2007. Quicken 2007 shows an error, “Unable to import because the transaction
displayed below is too large” I believe this is due to a “Split” having too many entries.
It seems there is more of an issue with importing it correctly into Quicken 2007 rather than exporting it from Quicken Essentials. Is this true?
Quicken 2007 is able to successfully import and open all data for my historical (1989 – 2012) backups using the Quicken Data Format files. It is just having a problem with importing the
last 18 months’ worth of data that was created using Quicken Essentials (QMTF format)
I have seen previous comments of people requesting someone to send them Quicken 2006…would this fix my issue? If so, could someone send me the necessary files? If so, please advise how to
do this.
Would it help to buy Quicken Essentials ($50) for Mac OS X Snow Leopard or Lion, then try to export data?
All help is appreciated
120. Reading this thread was very helpful, but I am still stuck. My Powerbook G4 Harddrive has started to fail. I was running Quicken 2005 for Mac on it. Just bought a new iMac, and was planning on
transferring the data to it but now I see this might not work. Can’t load Quicken 2005 or 2006 onto new computer to receive the data and it sounds like Quicken Essentials for Mac won’t take the
James Lynch
121. I can’t thank everyone enough for starting this blog entry and for adding to it with helpful comments. Like the original blogger, I was stunned after upgrading to OS 10.7 to find I could no
longer open Quicken 2002. The Intuit helpline was completely unhelpful, simply telling me Quicken 2002 was no longer supported. In the end my process was simple: I had an older Mac still running
10.6, so I copied my data file over there, downloaded Quicken 2006 to the old Mac and converted the data file to 2006, then copied the 2006 data file to the Mac running OS 10.7, downloaded
Quicken 2007 and used it to open (and convert) that file. Everything is running just great at this point and I’m extremely happy. The moral of the story for me is not to assume old software will
run forever, and upgrade whenever possible! Thanks again to all
1. Thank you, Katherine, for your kind comments!
And with that, I believe it is time to close comments on this post, especially since Intuit has provided access to Quicken 2006 for people to make their data file translations.
A great big, sincere thanks to all comment contributors! 🙂 | {"url":"https://thebestparts.net/how-to-convert-an-older-quicken-data-file-to-use-on-mac-os-10-7-lion/","timestamp":"2024-11-05T20:16:56Z","content_type":"text/html","content_length":"572553","record_id":"<urn:uuid:94ca7f62-6d13-4b67-86e7-494df470fa18>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00568.warc.gz"} |
Crash course on cryptography
Using cryptography, Alice (the sender) can scramble (encrypt) all messages intended for Bob (the recipient) so that no one but Bob can read them. Usually Alice and Bob agree on a common, secret
key that they use to encrypt and decrypt messages. This type of cryptography is known as secret key cryptography, shared-key cryptography or symmetric cryptography.
On the internet, a more common form of cryptography is public key cryptography, public-private cryptography or asymmetric cryptography. Using this type of cryptography, Alice and Bob can simply
obtain each other's public key and use it to encrypt messages for one another. Only with the corresponding private key can those messages be decrypted.
Public key cryptography also allows the creation of digital signatures. This allows Bob to verify that a message sent by Alice was really sent by her and was not modified by anyone else. For this
to work, Bob also needs Alice's so-called digital certificate. | {"url":"https://www.iusmentis.com/technology/encryption/crashcourse/","timestamp":"2024-11-05T15:10:06Z","content_type":"text/html","content_length":"5543","record_id":"<urn:uuid:535e7595-ddac-41cb-a982-1d2d7b4facc0>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00354.warc.gz"} |
Leetcode - Search Insert Position
2024-01-25 @ushimaru08
Search Insert Position
In this question, we have to find the position to insert the target number in the given array.
The given array is sorted, so we can use a binary search approach.
We can use 2 pointers to keep track of the search range and use the value at the middle to determine which direction to shift and narrow down the search range, and repeat it until we identify the
target position
At first, we assign a variable named left to 0 and a variable named right to len(array)-1
Then, we use a while loop which ends when left is higher than right. In a while loop, we define a variable named mid, it is the value at the middle of left and right.
And then, we can check the relation between the mid number and the target number. We can consider three patterns.
1. First, if the target value is equal to the mid number, the job is done. it is the suitable position to insert.
2. Second, if the target value is greater than the mid value, continue to search the right area.
3. Third, if the target value is less than the mid value, continue to search the left area.
If we encounter a second case or a third case, we can continue a while loop.
If the target value is not found after a while loop, we can return left because the loop will be stopped when left is greater than right and the target is greater than nums index right and lower than
nums index left.
class Solution:
def searchInsert(self, nums: List[int], target: int) -> int:
# brute force
# if target <= nums[0]:
# return 0
# for i in range(1, len(nums)):
# if nums[i-1] < target and target <= nums[i]:
# return i
# return len(nums)
# two pointers
left, right = 0, len(nums)-1
while left <= right:
pivod =(left+right)//2
if nums[pivod] == target:
return pivod
elif nums[pivod] < target:
left = pivod+1
right = pivod-1
return left | {"url":"https://www.ushimaru.dev/posts/SearchInsertPosition","timestamp":"2024-11-05T19:59:59Z","content_type":"text/html","content_length":"18129","record_id":"<urn:uuid:349cb363-ed63-4f6a-bc09-5732202f008e>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00737.warc.gz"} |
Leveraged Indices and the Volatility Drag | Scalable Capital
As we can see, while the EURO STOXX 50 Index did have a slightly positive performance over the given time period, both leverage indices result in a strong negative performance. The long-term
performance of both leveraged indices hence deviate significantly from the performance that one might expect by naively applying the leverage factor to longer time horizons. So how can that be?
First of all, the chart already indicates a crucial flaw in the naive multiplication logic: the synthetic time series takes on values below zero in 2003. Obviously, this is not possible, as prices
cannot fall below zero (if they really reached zero at some point, then this would be a final state as the time series could never recover again).
The differences between true leveraged index performance and the synthetic and naive computation is caused by compounding effects and is sometimes referred to as volatility drag. In the long run,
leveraged indices tend to deviate from the targeted multiple significantly. In terms of risk, however, both leveraged indices seem to keep their promise: annualised volatility (using simple square
root of time scaling on daily returns) is roughly two times the level of the underlying index, whilst the maximum drawdowns are significantly worse:
Metric EURO STOXX 50 Daily Leverage Monthly Leverage
Annualised return 0.21 -6.33 -8.32
Annualised volatility 23.22 46.46 48.05
Maximum drawdown 60.04 90.57 93.75
Table 1: Risk and return metrics
Effects of Leveraging on Payoff Distributions
Let's now analyze how this volatility drag distorts the targeted performance and whether it systematically works against the investor. Since we already know how to manually compute leveraged index
values given the trajectory of the underlying index, we can easily analyze the effects of the volatility drag in a simulation study. For the sake of simplicity, we will assume constant borrowing
costs equal to the average EONIA rate, as we already have seen that this should not have a major impact on results. Furthermore, we use the following settings for the simulation study:
• Regular index performances following a Brownian motion: For logarithmic returns we assume an annual compounding of 6%, while the standard deviation is chosen to match the volatility of
logarithmic returns of the EURO STOXX 50 index time series.
• Simulation of 2000 paths, each of length 5000 (corresponding to roughly 20 years with 250 business days per year).
• A leverage factor equal to 2.
# specify costs of borrowing
avg_eonia = eonia.mean().squeeze()
# set parameters of normal distribution of logarithmic returns
log_mu = 0.06 / 250
log_rets = np.log(data['SX5T Index'].pct_change() + 1)
log_sigma = log_rets.std()
# specify leverage factor and scale of simulation study
n_reps = 2000
n_obs = 5000
lev_fact = 2
# set random seed for reproducibility
# preallocate output
all_normal_perfs = np.empty([n_obs, n_reps], dtype=float)
all_lev_perfs = np.empty([n_obs, n_reps], dtype=float)
for ii in range(0, n_reps):
# draw random innovations
log_innovs = np.random.normal(loc=log_mu, scale=log_sigma, size=n_obs)
# compute regular index performance
disc_gross_rets = np.exp(log_innovs)
sim_perfs = np.cumprod(disc_gross_rets)
# compute leverage index performance
disc_net_rets = disc_gross_rets - 1
disc_rets_lev = disc_net_rets * lev_fact + (1-lev_fact) * avg_eonia/100 * 1/250
sim_lev_perfs = np.cumprod(disc_rets_lev + 1)
# store simulated paths
all_lev_perfs[:, ii] = sim_lev_perfs
all_normal_perfs[:, ii] = sim_perfs | {"url":"https://de.scalable.capital/en/finances-stock-market/leveraged-indices-and-volatility-drag","timestamp":"2024-11-08T15:20:43Z","content_type":"text/html","content_length":"175243","record_id":"<urn:uuid:2def2205-0ed5-469c-a165-3a326c54817e>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00155.warc.gz"} |
Is part of the Bibliography
We investigate the magnetism of a previously unexplored distorted spin-1/2 kagome model consisting of three symmetry-inequivalent nearest-neighbor antiferromagnetic Heisenberg couplings Jhexagon, J
and J', and uncover a rich ground state phase diagram even at the classical level. Using analytical arguments and numerical techniques we identify a collinear Q = 0 magnetic phase, two unusual
non-collinear coplanar Q = (1/3,1/3) phases and a classical spin liquid phase with a degenerate manifold of non-coplanar ground states, resembling the jammed spin liquid phase found in the context of
a bond-disordered kagome antiferromagnet. We further show with density functional theory calculations that the recently synthesized Y-kapellasite Y3Cu9(OH)19Cl8 is a realization of this model and
predict its ground state to lie in the region of Q = (1/3,1/3) order, which remains stable even after inclusion of quantum fluctuation effects within variational Monte Carlo and pseudofermion
functional renormalization group. The presented model opens a new direction in the study of kagome antiferromagnets.
We investigate the magnetism of a previously unexplored distorted spin-1/2 kagome model consisting of three symmetry-inequivalent nearest-neighbor antiferromagnetic Heisenberg couplings and uncover a
rich ground state phase diagram even at the classical level. Using analytical arguments and numerical techniques we identify a collinear Q⃗ =0 magnetic phase, two unusual non-collinear coplanar Q⃗ =(1/
3,1/3) phases and a classical spin liquid phase with a degenerate manifold of non-coplanar ground states, resembling the jammed spin liquid phase found in the context of a bond-disordered kagome
antiferromagnet. We further show with density functional theory calculations that the recently synthesized Y-kapellasite Y3Cu9(OH)19Cl8 is a realization of this model and predict its ground state to
lie in the region of Q⃗ =(1/3,1/3) order, which remains stable even after inclusion of quantum fluctuation effects within variational Monte Carlo and pseudofermion functional renormalization group.
Interestingly, the excitation spectrum of Y-kapellasite lies between that of an underlying triangular lattice of hexagons and a kagome lattice of trimers. The presented model opens a new direction in
the study of kagome antiferromagnets.
We investigate the magnetism of a previously unexplored distorted spin-1/2 kagome model consisting of three symmetry-inequivalent nearest-neighbor antiferromagnetic Heisenberg couplings Jhexagon, J
and J', and uncover a rich ground state phase diagram even at the classical level. Using analytical arguments and numerical techniques we identify a collinear Q = 0 magnetic phase, two unusual
non-collinear coplanar Q = (1/3,1/3) phases and a classical spin liquid phase with a degenerate manifold of non-coplanar ground states, resembling the jammed spin liquid phase found in the context of
a bond-disordered kagome antiferromagnet. We further show with density functional theory calculations that the recently synthesized Y-kapellasite Y3Cu9(OH)19Cl8 is a realization of this model and
predict its ground state to lie in the region of Q = (1/3,1/3) order, which remains stable even after inclusion of quantum fluctuation effects within variational Monte Carlo and pseudofermion
functional renormalization group. The presented model opens a new direction in the study of kagome antiferromagnets. | {"url":"https://publikationen.ub.uni-frankfurt.de/solrsearch/index/search/searchtype/authorsearch/author/Johannes+Reuther","timestamp":"2024-11-02T07:41:12Z","content_type":"application/xhtml+xml","content_length":"29582","record_id":"<urn:uuid:198b797b-6847-41c6-a0db-1f894c58e554>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00394.warc.gz"} |
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30% Off | {"url":"https://cozyearth.com/pages/podcast?utm_medium=podcast&utm_source=tu","timestamp":"2024-11-10T20:48:53Z","content_type":"text/html","content_length":"1052525","record_id":"<urn:uuid:f1710e5d-e69c-498e-9640-cef7bc71906a>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00840.warc.gz"} |
Andreas Seeger of Math Topics | Question AI
<div id="mw-content-wrapper"><div id="mw-content"><div id="content" class="mw-body" role="main"><div id="bodyContentOuter"><div class="mw-body-content" id="bodyContent"><div id="mw-content-text"
class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-parser-output"><p><b>Andreas Seeger</b> is a <span title="Biography:Mathematician">mathematician</span> who works in the
field of <span title="Physics:Harmonic analysis">harmonic analysis</span>. He is a professor of mathematics at the <span title="Organization:University of Wisconsin–Madison">University of
Wisconsin–Madison</span>. He received his PhD from <span title="Organization:Technische Universität Darmstadt">Technische Universität Darmstadt</span> in 1985 under the supervision of Walter Trebels.
<sup id="cite_ref-1" class="reference"><span></span></sup> </p><p>He was elected a fellow of the American Mathematical Society in 2014 for his contributions to <span title="Fourier integral
operator">Fourier integral operators</span>, local smoothing, <span title="Oscillatory integral">oscillatory integrals</span>, and <span title="Multiplier (Fourier analysis)">Fourier multipliers</
span>.<sup id="cite_ref-2" class="reference"><span></span></sup> In 2017, he was awarded the Humboldt Prize.<sup id="cite_ref-3" class="reference"><span></span></sup> He was awarded a Simons
Fellowship in 2019.<sup id="cite_ref-4" class="reference"><span></span></sup> </p></div></div></div></div></div></div></div> | {"url":"https://www.questionai.com/knowledge/knh59wyzpl-andreas-seeger","timestamp":"2024-11-06T21:47:37Z","content_type":"text/html","content_length":"54378","record_id":"<urn:uuid:d6ed06e5-00f9-4043-a11e-a35ddfb643f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00348.warc.gz"} |
How to define multiple WeylCharacterRings at one time
How to define multiple WeylCharacterRings at one time
I am trying to use a script like the one below (to use a simplistic version)
for i in [1..4]:
B"i" = WeylCharacterRing("Bi")
to define multiple Weyl character rings at one time. I then am hoping to go through, and compute the degrees of the representation corresponding to the weight (1,1,1,...,1) depending on which B$i$ I
am considering i.e., for $B4$ I would want to calculate
(its $126$). How can I go about automating this in some way? Any tips or reference materials?
Thanks for your time.
3 Answers
Sort by ยป oldest newest most voted
You could define a function or use a list comprehension to do this. Both rely on using string formatting to convert integers into strings for the WeylCharacterRing constructor.
Here's an example using list comprehension: I add an empty string '' at the beginning of the list to fix indexing, because lists in Sage are always indexed starting with 0, so you would want B[1] to
be the second element in the list, etc.
sage: B = ['']+[WeylCharacterRing("B{0}".format(i)) for i in range(1,5)]
sage: B[1]
The Weyl Character Ring of Type ['B', 1] with Integer Ring coefficients
sage: B[1](1)
sage: B[1](1).degree()
sage: B[4](1,1,1,1).degree()
And here's a different way you could do this, using a lambda function:
sage: B = lambda i: WeylCharacterRing("B{0}".format(i))
sage: B(2)
The Weyl Character Ring of Type ['B', 2] with Integer Ring coefficients
You can automate lots more things by defining more functions and, if you eventually need to, a new object class!
Thanks very much niles, I appreciate these answers. This is perfect for what I'm looking for.
JoshIzzard ( 2013-05-18 17:56:53 +0100 )edit
Oh, I think @Volker Braun and I had different ideas about what you were looking for :) Here's another list comprehension which just computes the degrees of various elements (using the lambda function
B defined in my other answer):
sage: [B(i)(*[1]*i).degree() for i in range(1,5)]
[1, 10, 35, 126]
Looks like you want a combination of IntegerVectors and the Python * operator to pass a list as multiple arguments:
sage: B4 = WeylCharacterRing('B4', style='coroots')
sage: l = [1,2,0,1]
sage: B4(*l).degree()
So that's how to define a vector....ha I am slowly hacking through the Sage basics. Thanks for the answer @Volker
JoshIzzard ( 2013-05-18 17:57:42 +0100 )edit | {"url":"https://ask.sagemath.org/question/10131/how-to-define-multiple-weylcharacterrings-at-one-time/","timestamp":"2024-11-07T03:01:54Z","content_type":"application/xhtml+xml","content_length":"68400","record_id":"<urn:uuid:c156dbda-a50b-4c4e-bb0d-fc2894b6ffec>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00463.warc.gz"} |
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How to change sqrt(5) to decimal?
Try RR(sqrt(5)). The default precision is 53 bits, but that can be changed. For 100 bits of precision, use RealField(100)(sqrt(5))
edit flag offensive delete link more
2 Answers
Sort by » oldest newest most voted
How to change sqrt(5) to decimal?
How can i typing to change root5 to decimal?
edit retag flag offensive close merge delete
More detail, such as controlling the number of digits or precision, can be found in the documentation here. On this site, here gives some options.
simply by
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Sparsing in real time simulation
Title data
Schiela, Anton ; Bornemann, Folkmar:
Sparsing in real time simulation.
In: ZAMM : Journal of Applied Mathematics and Mechanics = Zeitschrift für angewandte Mathematik und Mechanik. Vol. 83 (2003) Issue 10 . - pp. 637-647.
ISSN 1521-4001
DOI: https://doi.org/10.1002/zamm.200310070
Project information
Project title: Project's official title
Project's id
IST Project "Real-time simulation for design of multi-physics systems" (RealSim) with "Deutsches Zentrum für Luft- und Raumfahrt e. V." (DLR) and Dynasim AB
Project European Commission, 5th Framework Programme (FP5), Information Society Technologies Programme (IST) for Research, Technological Development and Demonstration on a "User-friendly
financing: information society"
Abstract in another language
Modelling of mechatronical systems often leads to large DAEs with stiff components. In real time simulation neither implicit nor explicit methods can cope with such systems in an efficient way:
explicit methods have to employ too small steps and implicit methods have to solve too large systems of equations. A solution of this general problem is to use a method that allows manipulations of
the Jacobian by computing only those parts that are necessary for the stability of the method. Specifically, manipulation by sparsing aims at zeroing out certain elements of the Jacobian leading to a
structure that can be exploited using sparse matrix techniques. The elements to be neglected are chosen by an a priori analysis phase that can be accomplished before the real-time simulation starts.
In this article a sparsing criterion for the linearly implicit Euler method is derived that is based on block diagonalization and matrix perturbation theory.
Further data | {"url":"https://eref.uni-bayreuth.de/id/eprint/11851/","timestamp":"2024-11-09T13:51:54Z","content_type":"application/xhtml+xml","content_length":"26108","record_id":"<urn:uuid:2105e4da-1aa0-4c62-8cd4-97f77513ae93>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00823.warc.gz"} |
(Redirected from Monzo notation)
Jump to navigation Jump to search
This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily.
The corresponding expert page for this topic is Monzos and interval space.
A monzo is a way of notating a JI interval that allows us to express directly how any "composite" interval is represented in terms of simpler prime intervals. They are typically written using the
notation [a b c d e f …⟩, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some prime limit.
Monzos can be thought of as counterparts to vals. Like vals, they also only permit integers as their entries (unless otherwise specified).
History and terminology
Monzos are named in honor of Joseph Monzo, given by Gene Ward Smith in July 2003. These were also previously called factorads by John Chalmers in Xenharmonikôn 1, although the basic idea goes back at
least as far as Adriaan Fokker and probably further back, so that the entire naming situation can be viewed as an example of Stigler's law many times over. More descriptive but longer terms include
prime-count vector^[1], prime-exponent vector^[2], and in the context of just intonation, harmonic space coordinates^[3].
For example, the interval 15/8 can be thought of as having [math]5 \cdot 3[/math] in the numerator, and [math]2 \cdot 2 \cdot 2[/math] in the denominator. This can be compactly represented by the
expression [math]2^{-3} \cdot 3^1 \cdot 5^1[/math], which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the […⟩ brackets,
hence yielding [-3 1 1⟩.
Practical hint: the monzo template helps you getting correct brackets (read more…).
Here are some common 5-limit monzos, for your reference:
Here are a few 7-limit monzos:
Ratio Monzo
7/4 [-2 0 0 1⟩
7/6 [-1 -1 0 1⟩
7/5 [0 0 -1 1⟩
Relationship with vals
See also: Val, Keenan's explanation of vals, Vals and tuning space (more mathematical)
Monzos are important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as ⟨ 12 19 28 | -4 4 -1 ⟩. The
mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by
[math] \left\langle \begin{matrix} 12 & 19 & 28 \end{matrix} \mid \begin{matrix} -4 & 4 & -1 \end{matrix} \right\rangle \\ = 12 \cdot (-4) + 19 \cdot 4 + 28 \cdot (-1) \\ = 0 [/math]
In this case, the val ⟨12 19 28] is the patent val for 12-equal, and [-4 4 -1⟩ is 81/80, or the syntonic comma. The fact that ⟨ 12 19 28 | -4 4 -1 ⟩ tells us that 81/80 is mapped to 0 steps in
12-equal—in other words, it is tempered out—which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of Western music composed in the Renaissance and from the
sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments (12 equal and unequal well temperaments), is made possible by the tempering out of 81/80, and that
almost all aspects of modern common practice Western music theory (chords and scales) in both classical and non-classical music genres are based exclusively on meantone.
In general:
[math] \left\langle \begin{matrix} a_1 & a_2 & \ldots & a_n \end{matrix} \mid \begin{matrix} b_1 & b_2 & \ldots & b_n \end{matrix} \right\rangle \\ = a_1 b_1 + a_2 b_2 + \ldots + a_n b_n [/math]
See also
External links | {"url":"https://en.xen.wiki/w/Monzo_notation","timestamp":"2024-11-13T05:51:34Z","content_type":"text/html","content_length":"33045","record_id":"<urn:uuid:f9f08304-fb6e-4190-bba6-c207c6776fd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00113.warc.gz"} |
Quality change mechanism and drinking safety of repeatedly-boiled water and prolonged-boil water: a comparative study
Experimental procedures
Indicators and measurements
Analysis methods
Change trends of the indicators
Drinking safety of the water
Mechanism of water quality change
Physical effects
Chemical effects
Possible increases of harmful substances
N species
Other harmful substances
General applicability of the study findings
Representativeness of the municipal sample used in the experiment
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Duality of 2D Gravity as a Local Fourier Duality
The p–q duality is a relation between the (p, q) model and the (q, p) model of two-dimensional quantum gravity. Geometrically this duality corresponds to a relation between the two relevant points of
the Sato Grassmannian. Kharchev and Marshakov have expressed such a relation in terms of matrix integrals. Some explicit formulas for small p and q have been given in the work of
Fukuma-Kawai-Nakayama. Already in the duality between the (2, 3) model and the (3, 2) model the formulas are long. In this work a new approach to p–q duality is given: It can be realized in a precise
sense as a local Fourier duality of D-modules. This result is obtained as a special case of a local Fourier duality between irregular connections associated to Kac–Schwarz operators. Therefore, since
these operators correspond to Virasoro constraints, this allows us to view the p–q duality as a consequence of the duality of the relevant Virasoro constraints.
ASJC Scopus subject areas
• Statistical and Nonlinear Physics
• Mathematical Physics
Dive into the research topics of 'Duality of 2D Gravity as a Local Fourier Duality'. Together they form a unique fingerprint. | {"url":"https://experts.illinois.edu/en/publications/duality-of-2d-gravity-as-a-local-fourier-duality","timestamp":"2024-11-14T07:59:00Z","content_type":"text/html","content_length":"53935","record_id":"<urn:uuid:9a4d6170-2073-4a25-875a-89036dc925ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00504.warc.gz"} |
Condition for a vector field be non-linear
• Thread starter Jhenrique
• Start date
In summary, the vector field ##\vec{v}## satisfies the differential equation ##\vec{\nabla}\cdot\vec{v}=0## and ##\vec{\nabla}\times\vec{v}=\vec{0}##, and in the Helmholtz decomposition, it can be
represented as a combination of a scalar potential, a vector potential, and a harmonic term. The harmonic term can be thought of as a "linear" vector field, but it is only necessary when boundary
conditions or decay conditions are not specified.
If a vector field ##\vec{v}## is non-divergent, so the identity is satisfied: ##\vec{\nabla}\cdot\vec{v}=0##;
if is non-rotational: ##\vec{\nabla}\times\vec{v}=\vec{0}##;
but if is "non-linear"
Which differential equation the vector ##\vec{v}## satisfies?
EDIT: this isn't an arbritrary question, is an important question, because the Helmholtz-Hodge decomposition says that every vector field can be decompost in a divergent vector field + a rotational
vector field + a linear vector field.
Last edited:
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
I have never seen a source that says you have to add a "linear vector field".
This article seems to not require such a "linear vector field". You have to give some context to the question. Perhaps a source with your statement.
This question in fact brings up an interesting point about the uniqueness of the Helmholtz theorem which is often omitted. In Helmholtz's original paper on fluid dynamics he does in fact include a
third term in the decomposition which he calls a "translation" term. Essentially, what you refer to as the "linear" term in the decomposition results from the fact that unless some boundary
conditions or decay conditions are specified, the Helmholtz decomposition is not unique.
The Hodge decomposition theorem says
E^p(M)=d(E^{p-1})\oplus \delta(E^{p+1}) \oplus H^p
where [itex] E^p(M) [/itex] is the space of smooth [itex]p[/itex]-forms on the manifold [itex] M [/itex], [itex] d[/itex] is the differential, [itex]\delta=(\pm 1) * d * [/itex] is the codifferential
and [itex] H^p [/itex] is the space of all harmonic p-forms (ie. [itex] 0=\Delta\omega=(d\delta+\delta d)\omega [/itex]. ) Then what you are calling a linear vector field is given by the equation
[itex] \Delta \omega =0[/itex]. It is straightforward to show, using the fact that [itex] \delta [/itex] is the adjoint of [itex] d[/itex], that [itex] \Delta\omega=0 [/itex] if and only if [itex] d\
omega=0[/itex] and [itex] \delta \omega =0 [/itex].
In your specific case, you can phrase this theorem in terms of vector fields (ie. turn the Hodge decomposition into the Helmholtz decomposition) by using the usual identification of vector fields
with both one forms and two forms in [itex] \mathbb{R}^3[/itex] (note that we must restrict to bounded domains or forms that decay quickly enough since this is not a compact manifold.) Then [itex] d
[/itex] on one forms becomes the curl, [itex] \delta [/itex] on one forms becomes the divergence, [itex] d [/itex] on 0-forms becomes the gradient and [itex] \delta [/itex] on 2-forms is the curl. So
the Hodge decomposition becomes exactly the Helmholtz decomposition just with an extra harmonic term added.
Further the harmonic condition [itex] \Delta \omega=0[/itex] translates into the two conditions [itex] \mathrm{div}(\vec V)=0[/itex] and [itex]\mathrm{curl}(\vec{V})=0[/itex] under our
identifications. These two conditions define what you mean when you say a "linear" vector field (at least they make the decomposition theorem work.) Note that it is also possible to write this
harmonic term as the gradient of a scalar potential or the curl of a vector potential, so the harmonic term kind of gathers together the non-uniqueness of the potentials in the decomposition into a
single term.
If you are in infinite space then the decay condition forces the vector field to go to zero at infinity, and this forces the harmonic term to vanish also. However, if you are in a bounded domain,
then you can get a nonzero harmonic term and so the Helmholtz decomposition is not unique and boundary conditions are required to uniquely specify a decomposition.
If you want the tl;dr version, the Helmholtz theorem says any vector field on a compact manifold (or a vector field which decays sufficiently rapidly on a noncompact manifold) can be uniquely
decomposed as
[tex] \vec W=\nabla\Phi+\mathrm{curl}(\vec A)+\vec Z [/tex]
where [itex] \Phi [/itex] is a scalar potential (ie. a smooth function), [itex] \vec A [/itex] is a vector potential (ie. a vector field or equivalently a two-form) and [itex] \vec Z [/itex] is
harmonic (ie. a harmonic one-form which corresponds to a vector field satisfying [itex] \mathrm{curl}(\vec Z)=0=\mathrm{div}(\vec Z). [/itex]) However, when decay conditions are specified, the
harmonic term is zero and so this term is omitted when it is understood that the vector field decays at infinity.
The condition for a vector field to be non-linear is that it does not follow a linear differential equation. This means that the vector field cannot be described by a set of equations that involve
only first-order derivatives and have a constant coefficient. Instead, the vector field may involve higher-order derivatives or have a variable coefficient.
In this case, the differential equation that the vector field ##\vec{v}## satisfies is likely a non-linear partial differential equation, as it involves both spatial derivatives (represented by the
gradient and curl operators) as well as the vector field itself. The specific form of the equation would depend on the specific properties and behavior of the vector field, and would need to be
determined through further analysis or experimentation.
FAQ: Condition for a vector field be non-linear
1. What is a vector field?
A vector field is a mathematical concept used in physics and engineering to represent a vector quantity, such as velocity or force, that varies at every point in a given space.
2. How can a vector field be linear or non-linear?
A vector field is considered linear if it follows the principles of superposition and homogeneity, meaning that the effect of multiple inputs can be calculated by adding their individual effects, and
scaling the input will result in a proportional change in the output. If these principles do not hold, the vector field is considered non-linear.
3. What is the condition for a vector field to be non-linear?
The condition for a vector field to be non-linear is that it must violate either the principle of superposition or homogeneity. This means that the output of the vector field is not proportional to
the input or cannot be calculated by adding the effects of individual inputs.
4. What are some examples of non-linear vector fields?
Some examples of non-linear vector fields include electrostatic fields, gravitational fields, and fluid flow fields. These fields do not follow the principles of superposition or homogeneity, and
their effects cannot be calculated by simply adding or scaling individual inputs.
5. Why is it important to identify whether a vector field is linear or non-linear?
Identifying whether a vector field is linear or non-linear is important because it affects how the field can be studied and analyzed. Linear vector fields are easier to solve and understand, while
non-linear vector fields can exhibit complex and unpredictable behavior. Understanding the linearity of a vector field is crucial in many fields of science and engineering, including physics,
mathematics, and computer science. | {"url":"https://www.physicsforums.com/threads/condition-for-a-vector-field-be-non-linear.746004/","timestamp":"2024-11-15T03:44:18Z","content_type":"text/html","content_length":"97694","record_id":"<urn:uuid:cc25469d-5d40-4b16-931a-b3b70e995ba6>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00774.warc.gz"} |
Interview: Kyle Evans on his 2023 Fringe show, Maths at the Museum
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Interview: Kyle Evans on his 2023 Fringe show, Maths at the Museum
We spoke to friend of the site, award-winning maths communicator and past math-off competitor Kyle Evans about his Edinburgh Fringe show for 2023, which is about maths.
Who are you (as if we don’t already know)?
I’m Kyle D Evans, I’m a teacher by day and entertainer/performer/presenter of all things mathematical by… well, also by day. But different days. I have children, so I really don’t do anything by
night any more.
Is it true you’re doing a maths show at the Edinburgh Fringe?
It is true! It’s called ‘Maths at the Museum‘ because I really wanted the title to double as an elevator pitch. (‘Elevator pitch’ is one of those terms where the American version really works better,
doesn’t it? ‘Lift pitch’ just sounds dreadful. That occurred to me while I was walking down the sidewalk with my aluminum fanny pack on). It’s an hour of comedy, poetry and stacks of audience
participation, running at the National Museum of Scotland from 4-15 August. It’s a family show, aimed at kids aged 7+ and their parents. Tickets are available now!
What’s the show about?
I’ve been to the Fringe four times now, and I also tour accessible, interactive maths shows around the country throughout the rest of the year. So because I have such a special venue this year, I’ve
put together a greatest hits set of all my favourite family maths bits I’ve devised over recent years. This includes the infamous ‘T-shirts of Hanoi’, a maths trick in two/three different languages
(time dependent) and some paradoxical poems.
How do audiences tend to react to a show about maths?
For the most part, I find that having ‘maths’ in the title of my shows really makes sure people come in knowing what to expect! But the biggest excitement for me is winning round the parents and
older brothers/sisters who have been brought along under duress and think that they hate maths or are too cool for it. I take great pleasure in winning these people round, and hopefully showing the
most hardened maths-philes at least one cool new bit of maths too.
Summarise, in one sentence, why people should come and see your show.
It’s inclusive, it’s accessible, it entertains, educates and informs. And hopefully it’s funny too (sorry, two sentences.)
Are there any other maths/science shows happening at the Fringe you’d recommend?
I’m proud to say I’m the only show with ‘maths’ in the title, but there are several maths/science-adjacent shows I recommend. Foxdog Studios make robots do utterly absurd things and I can’t wait to
see what their ‘Robo Bingo’ entails. Tom Crosbie is the most talented Rubik’s Cube wrangler you’ll ever see, and for the younger viewer there’s a charming climate change musical called ‘Chrissie and
the Skiddle Witch‘ which I highly recommend.
You can find details of Kyle’s show on the Edinburgh Fringe website. From a quick browse of the programme, some other potentially mathematical shows you could catch while you’re there include: Nick
Mohammed presents The Very Best and Worst of Mr Swallow, which promises ‘noise, maths, magic and the whole of Les Mis‘; there are also more generally sciencey shows like Stand Up Science and Comedy
For The Curious; on a similar climate change theme, there’s Ted Hill Tries and Fails to Fix Climate Change; and for a bit of arty dance performance with a maths word in the title, At The Intersection
might be worth a try. | {"url":"https://aperiodical.com/2023/07/interview-kyle-evans-on-his-2023-fringe-show-maths-at-the-museum/","timestamp":"2024-11-05T22:09:16Z","content_type":"text/html","content_length":"41834","record_id":"<urn:uuid:d70b5ba9-695c-4117-b945-381c7e5099a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00701.warc.gz"} |
Maths Question
I want to be able to record my students hopeful rise in test results.
What formula do I need for creating % rises in scores.
For example,
Johnny got 67 last time and 82 this time. How do I work out the % increase?
the obvious way is just 82 - 67 = 15% increase.
otherwise, you could find the difference between them (15) and then divide by the initial score 15 / 67 = 22% increase on the original score…
But if the score’s not out of 100, you couldn’t do the subtract way, right?
And how come there are two different ways of doing it? Surely there is only one answer?
If Johnny got 100 out of 200, then 150 out of 200, his score has risen 25%.
So the formula is
last.score - first.score = x
x/total.possible.score x 100 = %increase
Thanks for that Teggs.
I can’t believe how crap I am at maths. Took me ages to figure out how to get a simple percentage for kids’ tests.
The formula is [(second_value - first_value)/(first-value)] x 100. So if Johnny got a 100/200 and 150/200 then the percentage change is
[(150/200 - 100/200)/(100/200)] x 100
= [(50/200)/(100/200)] x 100
= 50/100 x 100 {denominators cancel}
= 50% increase.
Think about it this way if Johnny gets a 100 and a 100 for both test scores then the net result is a 0% change. According to your formula it would be a 50% increase. Edit: Sorry that is not true.
Also note that if second value is smaller then the result is a negative percentage change.
So ow much have we improved in maths since the OP?
[quote=“teggs”]the obvious way is just 82 - 67 = 15% increase.
otherwise, you could find the difference between them (15) and then divide by the initial score 15 / 67 = 22% increase on the original score…[/quote]
The former is a “percentage point” increase, and the latter is a “percent” increase.
[quote=“Chris”][quote=“teggs”]the obvious way is just 82 - 67 = 15% increase.
otherwise, you could find the difference between them (15) and then divide by the initial score 15 / 67 = 22% increase on the original score…[/quote]
The former is a “percentage point” increase, and the latter is a “percent” increase.[/quote]Sorry, what on earth does that mean?
Is it important?
Am I likely to cause Johnny a severe beating due to my dog-eared maths knowledge?
I would tell Johnny he is improving, but that he still has a long way to go before I buy him a beer.
[quote=“wangdoodle”]Sorry, what on earth does that mean?
Is it important?[/quote]
Percentage point increase is like interest rates: “The Fed lifted interest rates 1%” means the base rate went from, say, 4% to 5%. However, the amount of interest you will pay on a loan actually
increases by 25%.
It depends whether you want to measure improvement using a base of 100 (percentage point increase) or from a base of the score the student previously achieved (percentage increase). | {"url":"https://tw.forumosa.com/t/maths-question/17498","timestamp":"2024-11-10T18:46:36Z","content_type":"text/html","content_length":"32323","record_id":"<urn:uuid:00d6ce70-6765-43cd-b833-7328a17be1b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00046.warc.gz"} |
GeneXus for SAP Systems First - Formulas
Applications are often required to make calculations that involve the values of specific attributes, constants and/or functions. For all these cases, GeneXus offers Formulas.
There are different possible ways to define formulas:
• GLOBALLY: the calculation will be available throughout the Knowledge Base.
• LOCALLY or INLINE: in this case, the calculation will be available only for the object in which it has been defined. See Inline Formulas.
A Global Formula is a calculation that you define in association with an attribute. Note that the Transaction object structures contain a column labeled Formula:
When a calculation is defined in this column for an attribute, this means that the attribute is virtual. In other words, it will not be created physically as a field in a table because the value of
the attribute will be obtained every time it is needed by doing the calculation.
Suppose the Travel Agency needs to know how many registered attractions there are of each category at all times. Therefore, it is necessary to define a new attribute in the Transaction Category
(created in GeneXus for SAP Systems - Data Model changes) in order to define it as a global formula:
Now define the calculation associated to the CategoryAttractions attribute.
GeneXus offers a formula called Count to calculate what the Travel Agency needs (there are many others, like Sum, Average, etc.).
The attribute referenced inside the parenthesis of the formula provides GeneXus with the information of the table to be navigated to do the calculation (in the definition above, GeneXus knows that it
has to count in the Attraction table).
Then, if GeneXus detects a relation between the table it will navigate (Attraction) and the context where the formula attribute is defined (Category), it will only consider the related records for
the calculation. In this example, CateogryId is present in both contexts: where the formula is defined and in the table to be navigated for doing the calculation of the formula. So, only attractions
of each category are counted and not all attractions recorded in the navigated table will be considered. If no relation is found, then GeneXus will do the calculation considering all records in the
navigated table.
Press Build All. You can see that no physical changes will be made to the database. GeneXus will only generate some programs.
Execute the Category Transaction in order to see for each category how the attraction's quantity is always calculated at the time:
Every attribute defined as a global formula will be read-only, and it will not be possible to enter a value for it. This happens because the attribute obtains its value from the associated
calculation, which is run every time the attribute is used.
For this reason, there isn't a field in the physical table to store this attribute value, hence there's no need for it to be editable.
You can add more attractions in order to verify how the attraction's quantity is always calculated for each category at the time. | {"url":"https://wiki.genexus.com/commwiki/wiki?34250,GeneXus+for+SAP+Systems+First+-+Formulas","timestamp":"2024-11-12T15:57:53Z","content_type":"text/html","content_length":"116903","record_id":"<urn:uuid:63c5ec19-8db3-4384-8d15-7c5ed591c2f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00191.warc.gz"} |
Accessible form inputs in HTML
Writing semantic HTML is arguably the most important thing a developer can do to improve the accessibility of a website. User-friendly forms can often be tricky to design, especially if asking for a
lot of information, so we need to make sure that the implementation is not creating more friction for our users.
The basics
First, we need to pair our input with a label. We can either wrap the label around the input, or use the for and id attributes.
Option 1 (label wrapped around input):
First name
<input type="text" />
Option 2 (the for attribute of the label matches the id of the form control):
<label for="firstname">First name</label> <input type="text" id="firstname" />
<label htmlFor="firstname">First name</label>
<input type="text" id="firstname" />
Associating inputs with labels is important for screen reader users (so they know what information to enter), but also for sighted users; remember that labels should always be visible, even if the
input has a value. So don’t use placeholders as labels!
The type attribute
In the example above, the input is for the user’s first name, so we set the type as text. The type attribute lets the browser know what input to render and how the user can interact with it. Just
because the value of an input is a string, it doesn’t necessarily mean that the type should be text: it could also be email, password, url, or even color. The same thing applies to numbers: if we
needed users to enter a phone number, we’d use tel instead of number. You can read more about input types in the MDN docs(Opens in a new tab).
Email address
<input type="email" />
<input type="number" />
<input type="password" />
Website URL
<input type="url" />
Phone number
<input type="tel" />
The required attribute
The required attribute marks the input as invalid when empty and prevents it from being submitted to the server.
Phone number (required)
<input type="tel" required />
Note that even if you are handling your form validation with JavaScript, you should still use the required attribute where relevant, as it will allow screen readers to announce the input as required.
You can then style your inputs using the :valid and :invalid CSS pseudo-classes.
Your form validation logic should also add the aria-invalid attribute to invalid inputs so they can be properly flagged by screen readers and other user agents.
The inputmode attribute
If we set the type of an input, user agents might adapt the keyboard to the expected format; for example, the iOS keyboard for email inputs will typically include the @ character. But that’s not true
for all input types; iOS just renders a regular keyboard for number inputs. In this case, we can use the inputmode attribute. We can set it to numeric to show the numeric keyboard with the digits
0-9, or we can set it to decimal to allow fractional numbers. Here’s a good resource with screenshots of the different keyboards(Opens in a new tab)
Phone number
<input type="tel" inputmode="tel" />
Note that unlike type, inputmode won’t affect validation.
You can read more about the inputmode attribute in the MDN docs(Opens in a new tab).
The autocomplete attribute
The last thing we can do to help our users fill our forms more quickly is adding autocomplete support. The autocomplete attribute allows user agents to identify the format of the expected value for
an input, and pre-fill the values accordingly. This makes forms more efficient for all users, especially users that are attention deficit, have cognitive impairments, reduced mobility, low vision, or
blind users.
autocomplete allows us to get a lot more specific than by just using type: a name field might have an autocomplete value of name, given-name, additional-name, family-name, or even nickname. But all
of these would use a regular text input type.
A password field can also have multiple autocomplete values: new-password will let user agents and password managers know to generate a new password for your website, while current-password will fill
the field with an existing password.
Ever wondered how your phone can pre-fill a one-time-code that was texted to you? This requires just one line of HTML: autocomplete="one-time-code"!
<input type="password" autocomplete="new-password" />
Authentication token
<input type="number" inputmode="numeric" autocomplete="one-time-code" />
The autocomplete attribute doesn’t affect validation. You can read more about autocomplete in the MDN docs(Opens in a new tab).
Additional attributes
Depending on the type of an input, there might be some additional attributes that we need to set. For example:
• file inputs have an accept attribute that lets us set the formats the input should accept (.png, .jpeg, .doc, .pdf, etc)
• number inputs can have a min and a max value
• tel inputs can have a minlength and a maxlength
Profile picture
<input type="file" accept="image/png, image/jpeg" />
<input type="number" inputmode="numeric" min="1" max="10" />
Phone number
<input type="tel" inputmode="tel" minlength="8" maxlength="10" />
Some inputs will also accept the pattern attribute, which allows us to further restrict entered values so they also have to conform to a specific pattern, using a regular expression (RegEx).
Phone number (in the form XXX-XXX-XXXX):
<input type="tel" inputmode="tel" pattern="[0-9]{3}-[0-9]{3}-[0-9]{4}" /> | {"url":"https://nextjs-portfolio-serenastorm.vercel.app/snippets/html-form-inputs","timestamp":"2024-11-09T16:46:31Z","content_type":"text/html","content_length":"67138","record_id":"<urn:uuid:c1f42e6e-5f43-4479-884a-9ae1b129ab9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00727.warc.gz"} |
Interpretation of matter - The theory of everything by Marek Ożarowski
The interpretation of matter is as important a question as the origin of our Universe. Why is our matter – the objects we view – in a stable state? Could there be matter in the „future”? Or is there
matter from our „past” somewhere around us? How is the „construction” of the surrounding Reality for our Here and our Now accomplished? There is much we do not yet know about matter. I am inclined
more to the view that we try to describe our Reality, which consists of matter, using the Laws of Physics and interpret what we manage to observe. Is this enough to answer what matter is?
As for the interpretation of matter, we will try to explain what matter is with the help of our Concept – ToE-Quantum Space. Of course, the interpretation of matter, is related to our concept of
„time”, because time determines existence. Existence certainly determines the stability of our matter and every object that co-creates our Reality around us. Without Time, without the concept of
„time,” it is impossible to study matter.
Interpretation of matter – stable matter, is possible only for a specific moment. This is our Present, defined as the present moment of real time. In our concept of Quantum Space we call this – our
Here and our Now. Matter can only be stable for this particular present moment, of course, from our point of view. The image of matter is also related to the time continuum. This means that matter
depends on „time” and it’s stable form takes place in the current moment of real time. It is unlikely that matter could be in a stable state for material objects from the „past” or material objects
from the „future.”
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms,
which are made up of interacting subatomic particles, and in everyday as well as scientific usage, matter generally includes atoms and anything made up of them, and any particles (or combination
of particles) that act as if they have both rest mass and volume.
Matter should not be confused with mass, as the two are not the same in modern physics. Matter is a general term describing any 'physical substance’. By contrast, mass is not a substance but
rather a quantitative property of matter and other substances or systems; various types of mass are defined within physics – including but not limited to rest mass, inertial mass, relativistic
mass, mass–energy.
A definition of „matter” based on its physical and chemical structure is: matter is made up of atoms. Such atomic matter is also sometimes termed ordinary matter. As an example, deoxyribonucleic
acid molecules (DNA) are matter under this definition because they are made of atoms. This definition can be extended to include charged atoms and molecules, so as to include plasmas (gases of
ions) and electrolytes (ionic solutions), which are not obviously included in the atoms definition. Alternatively, one can adopt the protons, neutrons, and electrons definition.
A definition of „matter” more fine-scale than the atoms and molecules definition is: matter is made up of what atoms and molecules are made of, meaning anything made of positively charged protons
, neutral neutrons, and negatively charged electrons.
This definition goes beyond atoms and molecules, however, to include substances made from these building blocks that are not simply atoms or molecules, for example electron beams in an old
cathode ray tube television, or white dwarf matter — typically, carbon and oxygen nuclei in a sea of degenerate electrons. At a microscopic level, the constituent „particles” of matter such as
protons, neutrons, and electrons obey the laws of quantum mechanics and exhibit wave–particle duality.
In the context of relativity, mass is not an additive quantity, in the sense that one can not add the rest masses of particles in a system to get the total rest mass of the system. Thus, in
relativity usually a more general view is that it is not the sum of rest masses, but the energy–momentum tensor that quantifies the amount of matter. This tensor gives the rest mass for the
entire system.
„Matter” therefore is sometimes considered as anything that contributes to the energy–momentum of a system, that is, anything that is not purely gravity. This view is commonly held in fields that
deal with general relativity such as cosmology. In this view, light and other massless particles and fields are all part of „matter”.
Baryonic matter is the part of the universe that is made of baryons (including all atoms). This part of the universe does not include dark energy, dark matter, black holes or various forms of
degenerate matter, such as compose white dwarf stars and neutron stars. Microwave light seen by Wilkinson Microwave Anisotropy Probe (WMAP), suggests that only about 4.6% of that part of the
universe within range of the best telescopes (that is, matter that may be visible because light could reach us from it), is made of baryonic matter. About 26.8% is dark matter, and about 68.3% is
dark energy.
Strange matter is a particular form of quark matter, usually thought of as a liquid of up, down, and strange quarks. It is contrasted with nuclear matter, which is a liquid of neutrons and
protons (which themselves are built out of up and down quarks), and with non-strange quark matter, which is a quark liquid that contains only up and down quarks. At high enough density, strange
matter is expected to be color superconducting. Strange matter is hypothesized to occur in the core of neutron stars, or, more speculatively, as isolated droplets that may vary in size from
femtometers (strangelets) to kilometers (quark stars).
All this information about matter is a certain interpretation of what we call the surrounding Reality. This Reality is a form of expression of stable matter in the moment. Our moment – our Here and
our Now, is the image of matter, which in combination with time gives continuity of duration to our Universe. Matter is strongly connected to time from our point of view – our Here and our Now.
Can matter located somewhere in the „past” or located somewhere in the „future” have any correlation with our Here and Now? This is difficult to explain directly. We will try to show if there is
Interpretation of matter according to our concept. Matter is what we experience through touch, sight, hearing and smell. All our senses are directed towards our experience of the Reality around us.
Matter plays a special role in our existence. Matter is the image of our Universe. Our Universe is the image of matter, which is stable, experiential in our Here and our Now.
Interpretation of matter – stability of matter can only exist in the Here and Now
Our Here and Our Now, according to our concept of the origin of the Universe, corresponds to the present moment, our present moment, which we experience with the passage of our Real Time. It is
important that Our Here and Our Now only touch the Real part of Time. The description of Complex Time divides our time into an imaginary part and a real part. Our Here and Our Now only occupies the
Real part. This means that the Here and Now happens in our present and in the place where we are.
Our ToE-Quantum Space, like the Special relativity, Changes the way we understand time, space and motion previously described in Classical mechanics. It treats our Here and our Now as a single,
coherent point that corresponds to Spacetime. To clarify this better, perhaps first it is necessary to cite the information in the Special relativity area.
Special relativity was described by Albert Einstein in a paper published on 26 September 1905 titled “On the Electrodynamics of Moving Bodies“. Maxwell’s equations of electromagnetismappeared to
be incompatible with Newtonian mechanics, and the Michelson – Morley experiment failed to detect the Earth’s motion against the hypothesized luminiferous aether. These led to the development of
the Lorentz transformations, which adjust distances and times for moving objects…
Special relativity corrects the hitherto laws of mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as relativistic velocities).
Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational and quantum effects are negligible. Even so, the Newtonian model is still valid as a
simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth.
Special relativity has a wide range of consequences that have been experimentally verified. They include the relativity of simultaneity, length contraction, time dilation, the relativistic
velocity addition formula, the relativistic Doppler effect, relativistic mass, a universal speed limit, mass – energy equivalence, the speed of causality and the Thomas precession. It has, for
example, replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and spatial position.
To “save” the principle of relativity, Albert Einstein proposed the Special relativity. It postulates that all laws of physics – not only mechanics, but also electrodynamics – are equal in all
inertial reference systems. Applying the principle of relativity to electrodynamics leads to the postulate that the speed of light in a vacuum is constant in all inertial reference systems. These
postulates are sufficient to derive the Lorentz transformations and their consequences. How does this affect our Here and our Now?
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the
framework of special relativity the Maxwell equations have the same form in all inertial frames of reference. In the framework of general relativity the Maxwell equations or the Einstein field
equations have the same form in arbitrary frames of reference.
Our Here and Our Now. The world line: a diagrammatic representation of spacetime.
Source: https://en.wikipedia.org/wiki/Principle_of_relativity#/media/File:World_line.svg
Several principles of relativity have been successfully applied throughout science, whether implicitly (as in Newtonian mechanics) or explicitly (as in Albert Einstein‘s special relativity and
general relativity).
We return to the explanation of “Our Here and Our Now.” It seems that our present moment is perhaps equivalent to any present, real unit of time. However, our Here and Now is more than that. We
should consider this concept as the end product of all possible changes that take place in the imaginary part of our ToE-concept of Time.
In the Our Here and Our Now, the formation of the Reality that surrounds us takes place. What is Our Here and Our Now is the end result that was initiated in the past. What is Our Here and Our Now
influences what will be revealed in our future Here and Now. This means that our Here and our Now depends on the past and influences our future.
How can we define our Here and our Now? This is quite difficult to imagine, since the concept refers to both our “past” and our “future.” But this is not precise. After all, our Here and our Now
constitute a certain continuum. This means that it should be understood as the intersection of a certain continuum, where on one side we have a sequence from the past, and on the other side we have
sequences from the past. Our Here and our Now cannot be compared to our “time,” although it is closely related to our “time.”
This is a fundamental difference as to why we can’t use the term “present moment” – our Present. Our Here and our Now is a certain extension of our Present, so it deserves its own definition. It
describes a state in time, not time itself, as our Present could be. Our Present refers only to the concept of time itself. Our Here and Now refers to time and place. It can be compared to “frozen”
moments of a specific point in time and space.
In the animation below, the concept of “Our Here and Our Now” is presented. The present moment can only be the real part of our time – from our point of view. Our Concept assumes the existence of a
complex time, which is described by the Time Quaternion. Therefore, the present moment is the result of all potential changes that take place in imaginary time. The imaginary time is the “building
block” of the Real Present Moment – it is a representation of the Reality that is created by all the “changes” made.
Our Here and our Now marks our current moment for a specific location. Our Here and Now marks a specific point in time and space. This means that a certain interpretation of the Present is made at
such a point. This Present, is the state that describes the Reality surrounding this point. The Reality of the “point” maps the state that is revealed in connection with the previous moment – with
the nearest moment from our “past“. At each such point, “changes” can take place, which will in the next moment – our immediate “future” map the future state of the surrounding Reality. The present
Reality is mapped in the form of “changes” for the Here and Now. It is this current Reality that will shape our “future” – our future Here and our future Now.
The moment of the present is created as a result of activities and changes made from our past. This presents the cone of ‘Past’, where all possible actions, activities, changes in the past merge at
the point of ‘our Here and Our Now‘. This means that the present moment, our Present will depend on the experience made in our ‘Past’. All this, must happen in a certain sequential continuum (time
continuum). How can this be better explained? Best with an example from our lives.
Imagine that you love to play chess. Today, education and gaining experience in the field of “chess” is not a problem, just go to one of the websites for example chess.com. However, our example
happened when the Internet did not yet exist in such a widespread form. Now we will return to our animation, where you can notice two “funnels” – cones that represent our “past” and our “future.”
Notice that inside these cones appear vectors – the vectors from the “past” are colored blue and labeled A, B, C. On the right side of the animation, inside the cone that refers to our “future”
equally vectors appear in green – labeled X, Y, Z.
Vectors from the “past” – blue, signify experience and decisions made in our Past. Vectors from the “future” – green, signify our possibilities and potential decisions that can shape our future,
coming reality. Imagine that in the past, the Internet was not so widespread, to learn to play chess well, one had to use books – there were no other options. Most of the literature on chess was
written in Russian (in our time and in our country). So in order to reach this literature you had to learn a new skill – the use of the Russian language. And this is symbolized by our blue A vector.
In order to gain experience in chess games, you had to go to the best chess club, which was located in another city – symbolized by this, the blue vector B. By going to this city two or three times a
week for several years, you managed to get to know the area around this city. After a few years, you felt at home in that city. This is your experience of the “past” just symbolized by the blue B
vector. Participation in chess tournaments hosted by a top chess club caused you to meet the best chess players in the country. You are still friends with some of them to this day. This is symbolized
by the blue vector C.
If you hit the Red Dot, which symbolizes “our Here and our Now” in our animation above, you will make the right choice at that Dot. This choice will shape your “future” Reality. This “future” Reality
will give you certain opportunities that will cause “changes” in the surrounding future of your Reality. These possibilities, potential decisions are represented by green vectors from the “future”.
What is actually happening in our Here and Now?
In your Here and your Now, it is important that you know Russian, you know some people from the chess world and you know the city where your well-known chess club is – you know the place. This means
that in your Here and your Now can choose the right green vector depending on what is happening in your Here and your Now – what is happening in your present moment.
Imagine that in your Here and your Now you see a poster that announces that your chess club is holding a meeting with former chess grandmaster Garri Kasparov. That’s the Red Dot – Your Here and Your
Now, that message on the poster. The former chess Grandmaster will come at the invitation of your friend with whom you played chess.
Our Here and our Now is a moment, a moment in our present that is linked to our “past.” If it were not for our “past,” our experience tied to the game of chess, we would never be interested in the
poster’s information. However, such information will affect our “future” because it is linked to our “past.” This is a kind of a certain entanglement (we will explain it in another article). Our Here
and our Now can change our “future,” actually, influence what can happen. Our Here and our Now will create new opportunities in the future. New opportunities represented by the green vectors of the
The first possibility is to “attend” a meeting with a chess Grandmaster. This is the green vector X. If I can get there and receive an invitation. Since the Grandmaster is coming at the invitation of
a friend of mine, a special opportunity arises – to meet a legend of the chess game – a former chess Grandmaster. It is amazing to be able to meet him in person. There is a possibility that my friend
will introduce my person to the Grandmaster – this will be the green vector Y. Since I have Russian language skills, there is an opportunity for me to speak in person with the Chess Grandmaster –
this will be the green Z vector.
As you can see, there would have been no possibility of choosing the green vector Z – that is, talking to the Chess Grandmaster in person in his native language, if the blue vector X – learning
Russian – had not occurred in the past. There would not even have been a possibility of meeting the Chess Grandmaster, where this possibility is represented by the green vector Y, if the blue vector
C – this vector is responsible for my friendship with the person who invited the Chess Grandmaster – had not occurred in the past.
This means that our Here and our Now, could not change our future Reality. If the blue vectors A, B and C in my “past” had not occurred, the poster informing me of Garri Kasparov’s visit to my Here
and my Now would have generated completely different possibilities in the “future” – different green vectors.
The described situation from the animation may also apply in the world of elementary particles. This is a special case that requires a separate commentary because, Our Here and Our Now has a
different interpretation in the micro-world, where the imaginary part of our “time” is of greater importance. However, the greatest significance, Our Here and Our Now, relates to our actual time in
our macro world. This means that the Complex Function of Time will operate only in the scalar part – the real part of our Description of Complex Time. Thus, it is important to remember that our Here
and our Now refers to real time.
If we compare our Present with our Here and our Now, it is important to remember that our Here and our Now is a kind of fragment, a quantum continuum of everything on our timeline. The Present refers
only to our current time. Our Here and our Now is linked to the present moment through “past” and “future.” See more in the article Time Continuum.
Our Here and our Now is based on the “experience” of the past, at the same time, by referring to this “past”, it can generate options, opportunities, new changes that will affect our “future”
Reality. In a word, our Here and our Now, is more than the present moment, our Present – it is the potential, the possibility of potential “changes” that will take place through us in the future.
It is in the Here and Now that we decide what will be in the next moment – in our future Here and in our future Now. Why this is important from the point of view of our theory – ToE-Quantum Space.
Because our ToE of Quantum Space has a different perception of “time”. The description of our time is understood quite differently, therefore the Present, our present moment, must also have different
Hence our interpretation of the Present – it is our Here and our Now. Our Here and our Now therefore co-creates the next, closest moment in our Future. It influences the emergence of the Alternative
Vectors of the “future” and thus can split our Reality (there will be more about this in another publication).
Our Here and Our Now makes it possible for us to make a change in our surrounding Reality in the given present moment. The ability to decide what the change will be in the “future” depends on your
accumulated experience. The more books you have read, the more people you have met, the more knowledge you have accumulated – the more options you have to choose from. If you have never flown on an
airplane, then choosing such a means of travel will be unimaginable for you – it will be beyond your reach. If you live intensively, read a lot, meet people a lot, then you have more opportunities to
create your future Reality – and in different ways – you have more options to choose from. Your Here and Now depends on your Past and affects your Future.
Every publication you read, every conversation you have, or every time you meet a new person, is an increase in your future possibilities. It’s an increase in your experience that impacts your
future. If your cone of blue vectors from your “past” is large enough, then your possibilities in the form of green vectors from the cone of “future” will be of sufficient number. This means
different possibilities for changing your “past”. In other words, everyone has his Here and his Now, and it is dependent on the “past”. For everyone the Point of the Present looks different, it has
its own unique copy.
Our Here and our Now is the result of choices made in the past and at the same time it is also a Point from the “Past” for our “Future”, therefore our Here and our Now is a certain continuum. Our
Here and our Now is surrounded by our “Past” and our “Future.” We have no access to either our “Past” or our “Future.” We can only “move” in our Point of our Present – our Here and our Now. Our Here
and our Now has a certain sensitivity – it is the margin between our “past” and our “future.” We don’t know where such a boundary runs – and it is certainly individual for each person.
Our Here and our Now provide us with a continuous timeline. The elapse of time can be measured by a clock that measures second by second. However, our sense of the passage of time is quite different.
It is not tied to measuring seconds. We feel that our business is still going on – it is in the present, even though several minutes have passed. That’s why our Here and our Now are connected between
“the past” and “the future.” All this makes the term “Present” flat and inadequate. Therefore, we had to introduce our Here and our Now.
Interpretation of matter – Stabilization of mass over time
The stabilization of mass over time, like the stabilization of matter, occurs in our real time – for our Here and our Now. So are mass and matter the same thing? Mass is a property of matter. This
means that matter is a much broader concept than mass. This is perhaps the simplest explanation of the difference between mass and matter. In general, mass depends on our gravity.
Gravity, according to our Concept – ToE-Quantum Space, is correlated with time – with its real part according to our concept of “time”. The correlation of gravity and time must therefore affect the
stabilization of mass over time. In our Concept, gravity also has an interpretation. The role of gravity in our World has the position and affects the stabilization of mass.
The stabilization of mass in time according to our Concept – ToE-Quantum Space must have its own interpretation because our according to our Concept – Time does not exist. This means that we must
refer to both our concept of “time” and the “time“, which is only a real part of the Complex description of time. In other words, to put it more simply: the question must be asked “when will our mass
be stabilized in our time and what does that actually mean?”
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a body, until the discovery of the atom and particle physics. It was found that
different atoms and different elementary particles, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple definitions which are
conceptually distinct, but physically equivalent. Mass can be experimentally defined as a measure of the body’s inertia, meaning the resistance to acceleration (change of velocity) when a net
force is applied. The object’s mass also determines the strength of its gravitational attraction to other bodies.
The SI base unit of mass is the kilogram (kg). In physics, mass is not the same as weight, even though mass is often determined by measuring the object’s weight using a spring scale, rather than
balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass. This is
because weight is a force, while mass is the property that (along with gravity) determines the strength of this force.
In the Standard Model of physics, the mass of elementary particles is believed to be a result of their coupling with the Higgs boson in what is known as the Brout-Englert-Higgs mechanism.
Can mass be an illusion? How can one ignore a Black Hole with a mass of 6.5 billion times that of the Sun located some 55 million light years from Earth in the M87 Galaxy (link). Is such a mass – 6.5
billion times the mass of our Sun – time-dependent (see the article inside the Black Hole)? Perhaps Black Holes have their own interpretation of mass – the Black Hole’s interpretation of mass
according to our conception is quite different from the interpretation of mass that co-creates our nearest Reality.
The description of mass stabilization over time seems quite complicated. The cases cited above, the definitions of mass and the effect of mass on gravitational interaction have a multidimensional
aspect. This does not mean that mass stabilization has different interpretations. On the contrary. The basis, however, is its correlation with time. Time determines the stability of mass for the
current moment. And if this can be the case, it can mean that matter (mass) is “distributed” in time – from the moment of creation of our Universe, to the end of the Universe. So what is the
interpretation of stability for mass over time?
According to our Concept – Time does not exist. Our real time is the end product of the “changes” taking place in the Initial Singularity. Without change, there is no time – time is change. This is a
different perception of “time.” And if our concept of “time” is different, and there is a connection between time and mass stabilization, then our “mass stabilization over time” must also have its
own different interpretation. The stabilization of our Reality also takes place in time. In other words, what surrounds us is stable in time– real time – in our Here and our Now.
The interpretation of the stabilization of mass over time from our point of view – from our Here and our Now is presented in the animation below. The animation presents the idea of mass stabilization
for a specific point in time. Since, according to our Concept, time does not exist, and mass stabilization is done in our current moment, i.e. in our Here and in our Now, mass can achieve full
stabilization only for this, current moment in time – only for our Here and our Now.
Of course, each such Here and Now marks our point of observation – the stable point where mass reaches its stabilization. At other points in “time,” this mass is in an unstable state from our point
of view of our present moment. This means that the mass from beyond our Here and our Now is inaccessible to us. In other words, the mass from the “past” and the mass from the “future” are unstable
for us. The important thing is the mass only in the present moment.
Stabilization of mass over time. The animation shows the interpretation of two states. The first state is the achievement of mass stabilization for a specific Here and Now. This state is marked with
a blue Point, at which time the animation freezes. The second state interprets the lack of mass stabilization for our Here and Now. This state is expressed by the blue circle, which symbolizes the ”
fuzzy” mass throughout the blue circle. The blue circle symbolizes our mass from the past but the mass from the future.
What does stability over time, and consequently instability over time, mean? Our Reality co-creates a certain continuum in time (Continuum mechanics). Mass must therefore respond to the expectations
of the Real Now and Here – the present moment in which mass stabilization takes place. This means that mass from the past moves into the present and continues toward the future. This means the
“continuity” of events, the continuity of our Reality. Without the past, in which there is a stable form of mass, the stability of this mass for the “present” could not be achieved. Stabilization of
mass in the present will allow for stable mass in the future.
Even though our mass appears to be permanently stable, the stabilization of mass is done only from our point of view – that is, from our Here and our Now. In our Here and Now, we do not have access
to mass from the “past” or to mass from our “future.” Stability reaches its apogee only for the present moment. Translating this into the correlation of time and gravity, the gravitational influence
of our mass is effective in our present – in our our Here and our Now.
If the stabilization process for moments in the past and future did not expire, then we would have an additional gravitational influence from the “past” and from the “future.” Such a situation can
only occur in the territory of the Black Hole, which also “swallows” “time”. In the interior of the Black Hole, mass/matter accumulates in a stable state from the “past” and from the “future.” This
is why the Black Hole has such a huge mass (List of most massive black holes).
Imagine that the mass of the Sun does not lose its stability over time. What would that mean, and what could it be compared to? The supermassive Black Hole at the center of the Milky Way is called
Sagittarius A. It has a mass equal to about 4 million Suns and would fit into a sphere about the diameter of one Sun – our Sun. This means that if the mass of the Sun did not lose stability over
time, the Sun’s gravity would correspond to the mass of 4 million Suns – the Earth would probably not be able to “survive” such a gravitational interaction. The Sun would engulf the Earth.
Imagine by analogy, a Black Hole the size of the Earth. Such a Black Hole could have a mass of 4 million Earth masses – analogous to the example above. All of this, could happen if the stabilization
of mass over time would accumulate as it does for Black Holes. This means that our concept of “mass” for our real time must limit the stabilization of mass in time from our point of view only to the
present moment – to our Here and our Now. The situation of gravitational interaction for our Here and our Now is shown in the animation below.
Stabilization of mass over time. The Earth has a stable mass only for the present moment from the point of view of our Here and our Now. This means that the gravitational interaction for such mass
has its effect only in the current moment – in our present moment of our real time. If the Earth’s mass were not losing its stability, the gravitational impact would be cumulative from all moments
when the Earth existed, until the Earth ceases to exist. Unstable mass from the “past” and unstable mass from the “future” from the point of view of our Here and our Now may correspond to DARK MATTER
The concept of Gravity must be linked to the Concept of Time. If our concept of “time” has different interpretations for other Worlds, for worlds from the “past” and worlds from the “future,” then
the concept of “gravity” must also have a different interpretation – a different one for the micro-world and a different one for the macro-world. If this were not the case, then both theories – the
General relativity and Quantum Mechanics would form a single, coherent description of our Universe. The problem is the phenomenon of Gravity, which we experience in our Macro-world, while in the
world of elementary particles, in the Standard Model, gravitational interaction cannot be described.
Mass can be stable for the present moment – specifically, for our Here and our Now. The remaining mass outside of our present moment – our Here and our Now, is unattainable to us, and therefore has
no gravitational influence on our Here and our Now. If this could be the case, we might be tempted to make our own interpretation of the concept of “DARK MATTER.” That is, Dark Matter could be our
entire Mass/Matter, which is in an unstable state for our Point in the present – for our Here and our Now.
Then, the gravitational interaction of such unstable mass/matter, can be considered as residual. Such gravitational interaction does not affect our Here and our Now, it will be difficult to detect
the gravity of the mass from the “past” and the gravity of the mass from the “future”. Could it be that Dark Matter, then, is Matter shifted in time from our point of view?
Our Concept of Time is extended by an imaginary value. Then it is possible that our mass also has an imaginary form, that is, a form unobservable from the point of view of our present? If one were to
assume such a complex time function, then perhaps it would be possible to combine the Standard Model of elementary particles and Einstein’s General relativity describing spacetime and Gravity. There
is no time or space in our concept.
The Black Hole, is an object near which, time begins to slow down. On the other hand, after crossing the Event Horizon, no more information will leave the territory of the Black Hole. Thus, if “time”
becomes “redundant” in the territory of the Black Hole, then perhaps, in the territory of the Black Hole there are all possible states, which reproduce that, those happened from the creation of the
Universe to the moment of its collapse – for example, its heat death. The Black Hole somehow stabilizes matter for different moments, which is why the accumulation of gravity represented by matter
from different moments in time is so great.
Time unstable system in our Here and Now is not possible. This means that we are not able to observe matter that does not have stability in time. This does not mean that such a system of matter does
not exist – it can be observed from the point of view of another Time Dimension, which has obtained its time stabilization. In our Here and Now, such an unstable system is inexperienced, as is our
matter from another Time Dimension. This means that the concept of the Multiverse seems possible for our interpretation.
The multiworld system would also involve splitting our reality by realizing our choices. This means that different copies of our matter/mass would have to be created, which would have a different
form – maybe similar, but still not the same, in different Alternate Realities. By means of quantum decoherence, such arrangements of matter could take place in imaginary time. All possibilities.
Only in the next, future moment would our mass stabilize by correlating with one of its copies.
The stabilization of mass over time affects the image of our Reality. What, we observe, is the image of matter stabilized in time. We are not able to observe phenomena from imaginary time. Does this
mean that there are no such phenomena? It only means that we do not have access to them, as is the case with unstable mass. Unstable mass exists from our point of view but we have no access to it,
however, a moment ago, this mass existed and we could experience it – now, there is a stable copy of it, it is the same mass for the continuity of our Reality. The mass therefore stabilizes its next
copy for a specific NOW.
Interpretation of matter – stability of matter in copies of our Universe
Copying the World probably takes place where in the micro-world. The first such case described in the world of elementary particles seems to confirm the Uncertainty Principle and the Schrödinger
Equation. This means nothing more than the probable occurrence of an elementary particle in several places simultaneously.
From our point of view, this can be interpreted as Copying the World. According to our ToE-Quantum Space, such copying may be the result of ToE-Time Concept. But not only time itself. Time in our
understanding is closely related to “change”. The process of Copying the World is also “change.” Another copy of the Universe can be made in the micro-world, in the macro-world?
In the macro-world, we are not able to observe the Copying World in macroscopic terms. This does not mean at all that such a Multiverse does not exist. Many theories refer to Parallel Universes, but
treat this as a hypothetical creation, a by-product to be mentioned just in case it turns out that an Alternate Reality might exist. It’s more of a form of reassurance, in case it turns out.
You are probably intrigued by the interchangeable use of the terms Multiverse, Alternate Reality or Parallel Universe. Of course, these terms have their own unique and unambiguous meanings. However,
despite the fact that the mentioned terms refer to other … worlds, they may share common features and proving the existence of one of them does not automatically exclude the others. Or perhaps?
Everything, depends on the interpretation adopted. If, for example, we assume that due to the observation of an elementary particle – for example, an electron, it can be in several places at the same
time, then can not we talk about its copies located in different places? Therefore, if such a particle has such a feature, then can the entire atomic system (atom) have its copy in several places. An
atom is a part of the matter of which we are made. So if such a copy of an atom could exist, could larger structures of matter also exist? Of consequence, we could arrive at another, Universe.
It turns out that it is quite possible for Copying the World to exist. We don’t know what kind of Influence of Copying the World can exactly have on our micro world, certainly some Influence of micro
on macro world is happening. If time passes differently in the micro-world and has its own imaginary concept of time, then Copying of objects from the macro-world view should also take place.
Everything depends on where the observation – the process of observation – takes place. Unfortunately, but from the point of view of microscopic phenomena it looks different. Some of the phenomena
occurring in the microscopic world do not have their equivalent in our macroscopic perception of Reality, so it is difficult to relate to our experience.
Why is it that in our macro-world we are not able to notice a dual Reality? In the micro-world, on the other hand, an elementary particle can be in several locations at the same time – it’s as if
this particle has several copies of itself for each location. So what about matter, which is co-created by elementary particles that are in different locations? When does the copying the World begin?
It seems that it is in the micro-world, when our time is divided into very small values – into small fragments, then certain phenomena can be observed.
Copying the World. In order to observe the “process” of copying the World, we need to do some imagining from the point of view of the micro-world. This means that being in the micro-world, we have to
divide our real time into very small values, exactly how small values is not known. However, with such small values of our real time, it may be that our particle shares its stable position for
another location in time.
Thus, if there is a splitting of time by splitting of the particle, the solution to the problem for this phenomenon may be a different location of the observed particle. This is because the particle
must reconstruct the complete information about itself in time. Unfortunately, but such a rapid change – quantization of real time results in splitting the particle’s information into two parts –
perhaps in some way entangled parts, which constitute a unity in real time, and in the imaginary part constitute a complete information state. An attribute of this state may be two or more locations
of the particle.
In the animation above, you can see the formation of the “process” of splitting a particle in imaginary time. In addition, part of the imaginary time affects the division of information about the
observed particle. Without this splitting, the particle could not be determined in real time – the Uncertainty Principle and the Schrödinger Equation would not occur.
The indeterminacy of the location of the particle or the momentum of the particle can only be overcome at the time of measurement, observation. Then, it turns out which copy of the particle we are
observing, in other words, in which location is our observed particle. In this way, it turns out that the consequence of observation is also the choice of the surrounding on Reality – probably.
With such small values of our real time as in the animation, it may turn out that our elementary particle has its beginning in the “past” and the end of the particle is yet to come in our “future“.
All according to the imaginary time vector. By analogy, this can be compared to “building a house” in time. From our perspective, our real time – building a house is just one moment.
The house has its beginning and its end for a moment. If we unfold this moment – zoom out, it is possible to see, experience that the house first had its foundation, then the walls and finally the
roof was laid. From our perspective, it looks as if the house was not there and suddenly appeared.
If we put two people on opposite sides and they would look at the house under construction from two different points of view, it would appear that our house is in two worlds – observer A and observer
B. Each would see the appearance of the house from his point of view. The question of the concept of “time”remains. If that could be the case then maybe there are some copies of me?
So if one of these people inhabits the house during the day and the other only at night, two Worlds are created. In each of these Worlds, each observer does not know about the other observer despite
the fact that they inhabit the same house – only the housing is shifted in time. This may mean that in the imaginary part of time there is a splitting of the particle that co-creates matter for the
two alternative worlds. Is this the interpretation of a copying the world? Not exactly.
Copying the World is therefore possible. If some of the phenomena described according to the Uncertainty Principle and the Schrödinger Equation is our concept of Copying then it is just a matter of
proper interpretation. We do not know exactly how it is in the micro-world. We only know the effects of certain phenomena that can in no way happen in our real world – all because our surrounding
Reality is subject only to the Real part of our understanding of “time.” We are not able to experience imaginary time. Can we?
If our imaginary time is bypassing us, then how can some people bilocate – there are well-known cases of Padre Pio, a Catholic Church mystic, who was seen in several places simultaneously. There are
more such cases, and they involve not only Christian mystics but also Buddhist monks and lay people not affiliated with any denomination, church or faith. Is this evidence of Copying the World? If
such individual cases are possible, can our entire Alternate Realities coexist, somewhere close to us?
Scientists have long argued on the topic: is there any Multiversum? Why do we have the impression of Déjà vu? Is it due to the existence of an Alternate Reality. Or does this Alternate Reality exist
somewhere beside us? Is there information coming to us from the Cosmos from our Universe, or is it information from the Universe as it was 13 billion years ago. Is this Universe as it was 13 billion
years ago our Universe, or is it some Parallel Universe?
Here are some questions that could be answered if we knew whether Copying the Universe is possible. We don’t know exactly what happens in imaginary time in the world of elementary particles. We only
have attempts at a mathematical description using the Uncertainty Principle and the Schrödinger Equation. Our time deserves redefinition and other perceptions. Without a change in thinking about our
concept of time, we have no way to go further…
Interpretation of matter – stability of matter for time and space
Nor time nor space – how is this possible? If our concept of ToE-Quantum Space interprets the concept of “time” quite differently, what is the interpretation of “space”? Special relativity has
offered us is a unity of four dimensions that has changed the understanding of time, space and motion (Momentum) previously described in Newtonian mechanics. Time and space cannot be defined
separately from each other (as was previously thought to be the case). Rather, space and time are interwoven into a single continuum known as “spacetime”.
Special relativity has a wide range of consequences that have been experimentally verified. They include the relativity of simultaneity, length contraction, time dilation, the relativistic velocity
addition formula, the relativistic Doppler effect, relativistic mass, a universal speed limit, mass – energy equivalence, the speed of causality and the Thomas precession. Events that occur at the
same time for one observer can occur at different times for another.
But. If there is no “time” – According to Our Concept of Time, at least as we perceive it today, then is there also no “space”? Special relativity, described by Albert Einstein, is the scientific
theory of the relationship between space and time. The relationship between space and time seems to be unassailable. However, if we change our thinking about the concept of “time,” what then happens
to the concept of “space”? If time has not yet been recognized sufficiently, how will this affect the nexus of time and space?
The description of our Reality seems quite orderly – at least around us, in our immediate environment, we see, touch, feel and most important – we recognize the object. What happens further away – at
the borders of our Universe or in our micro-world – the world of elementary particles, contradicts everything that happens around us. So what is the “truth” about “time”, “space” and the phenomena
occurring there. Some already claim that there is no “distance,” that what happens at the boundaries of the Universe is, in a sense, an illusion. Some claim that there is no “time,” or that we are
dealing with different “times,” or that time passes differently in different parts of the Universe.
In the world of elementary particles, we are unable to understand certain mechanisms that reveal to us unknown phenomena in our macro-reality (in our Here and Now), for example, Wave – particle
duality, the uncertainty principle or quantum superposition. Such phenomena are not observed around us – it is only in the micro-world that we can try to record, to study it. Of course, it can be
said that what we see, hear or feel is just a certain interpretation of our brain, which has been developed by all generations of mankind – it’s just a kind of Optical Illusion. And we all “suffer”
from the same thing.
If some phenomena in the micro-world are possible, why are they not observed in our macro-world? If “time” passes” differently – behaves differently in the micro-world, then why does “time” have a
different “form” in our macro-world. If the Photon can traverse our Universe at the speed of light and be in several places at the same time, why can’t humans do it? Some elementary particles are
“eternal” (Photon), does that mean they are outside our time? So aren’t “time” and “space” an illusion?
So, if Wave – particle duality, the uncertainty principle applies, then could our, the same Photon exist at the birth of the first stars, at the birth of our galaxy, at the birth of our solar system
and bring all this information to us? Hardly, if theoretically this could be the case, the same Photon, could already be in the “future” and carry there the information about our Here and Now. This
type of situation is shown in the animation below.
Nor Time nor Space. If a single Photon can be simultaneously from several places and in time and space, isn’t time and space an illusion? Photons co-create information that reaches us from distant
places in time and space. In the world of elementary particles, a photon has dual properties, being a particle with zero mass and a wave at the same time. So if it can exist in imaginary time, from
our point of view, it can exist in imaginary places. This means that perhaps there is no time nor space, since these are imaginary concepts. From our point of view, a single Photon can create
mappings of our Here and Now, but at the same time it was at the beginning of the Universe, was at the creation of the first stars and galaxies, and was at the creation of our solar system. This may
mean that it will also co-create our “future.”
If the Photon on the dual nature (Wave – particle duality), can there be such a potentially such a situation that its “wave” can be extended to all these places in time – from the formation of the
first stars to the end of our Universe. In other words, our photon could gain an interpretation of a wave that extends throughout “time” – from the beginning of the creation of our Universe to the
end of our Universe. This information may seem unbelievable, but what would be the picture of our elementary particles if the concept of “time” is completely different?
Imagine that our Universe is born. The first elementary particles are created. Systems of particles are formed, which coalesce into chemical elements. This leads to the formation of the first stars.
Everywhere, at that time and in those places, there must have been a Photon that carried information about these phenomena. This information had to propagate.
This information propagation had to be a set of “changes” that were necessary for the expansion of our Universe, as well as for the new forms of matter that evolved with the Universe Expansion Vector
. Matter had to undergo changes from quarks to complex chemical compounds, co-created from Atoms. From individual Atoms to structures that forged matter into the first stars. Our photon was there,
gathering information about Quantity of matter.
The Photon has no electric charge, is generally considered to have zero rest mass and is a stable particle. The experimental upper limit on the photon mass is very small, on the order of 10^−50
kg; its lifetime would be more than 10^18 years. For comparison the age of the universe is about 1.38×10^10 years.
In a vacuum, a photon has two possible polarization states. The photon is the gauge boson for electromagnetism, and therefore all other quantum numbers of the photon (such as lepton number,
baryon number, and flavour quantum numbers) are zero. Also, the photon obeys Bose–Einstein statistics, and not Fermi–Dirac statistics. That is, they do not obey the Pauli exclusion principle and
more than one can occupy the same bound quantum state.
How could fragments of matter appear that have a stable form in our Here and Now? Was our Photon ever in an unstable state? If, then, the same photon existed, exists and will exist permanently from
our point of view, was it as stable in those other places in time as it is in our Here and Now? In those other places in time, did it have similar properties as it has now – in our Here and Now?
Could it have happened that our Photon, at the beginning of our Universe, for example, had no “spin“. Did it have the same properties at all times?
In other words, when our Photon gained characteristics: an elementary particle from the group of bosons, when the feature of the photon as a carrier of electromagnetic interactions appeared (feature
boson). When it ceased to have an electric charge, and when it lost its magnetic moment, when its rest mass reached zero (m0 = 0), and when the spin number s received the value 1. When our photon
began to exhibit Wave – particle duality, and thus simultaneously gained the characteristics of a particle and an electromagnetic wave. Did it happen in an instant, with the appearance of our photon,
or did our photon also have to evolve with the expansion of the Universe?
Or perhaps it is precisely the opposite. The properties of a photon, as a result of its entanglement with itself from other moments in time, are complementary according to the Anthropic Principle?
This would mean that our photon may be complementary in time – from the origin of the appearance of this photon to its termination.
The complementarity of its properties is “scattered” in time and maintained in a consistent way, by means of entanglement of one particle (Photon) at different moments in time. That is, there could
be moments in time when our Photon would have no “spin.” There could also be moments in time when our photon had mass. The photon could have a different face at other moments in time.
If our elementary particle according to the uncertainty principle and the Schrödinger Equation can be in several places at the same time, is it possible for one moment in “time” to be in several
places, or could it be for one “place” to be in several moments in time (if our Photon is moving at the speed of light)? While the latter is easier for us to imagine in our perception of “time,” the
former seems imaginary.
If we perceive “time” according to our conception – then everything can be imaginary in the first and second cases. This means that our illusion affects not only “time” but also “space.” What is
happening around us is also then a kind of illusion – Time and space do not exist, there is only a mapping expressed through the energy state in the structure of Quantum Space.
Development, the expansion of the Universe, is about the constant occurrence of “changes”. Without changes, we have no sense of the elapse of “time.” Changes must take place for our concept of “time”
to exist. Time is change. In that case, what would we need “space” for? The sense of “Space”, is nothing more than an illusory interpretation of “time” – we see “space” because information about
matter comes to us in the form of photons at other times.
This means that our concept of “space” is made up of the concept of “time.” Some photons will reach us earlier, and other photons will arrive later. Our picture of Reality is the result of many
“changes” taking place. Everything, of course, depends on the point of observation.
“Changes” determine the direction of expansion – this means that they co-create our arrow of time. Therefore, if our elementary particle – our Photon, contributes to every moment of the expansion of
our Universe – from the beginning to the end of our time, it seems that it must be presented in different forms and with different properties. The photon has communication with all its copies from
every moment in time through “entanglement.”
At each of these moments, it can be the same Photon and different ones at the same time. Its properties are revealed with the elapse of time and with the expansion of our Universe. A photon cannot go
back in time, but it has contact with its counterpart from another moment in time. This contact is realized as an “entanglement” of all copies of one particle spread out in the timeline.
Each version of the same Photon creates a present Here and Now for any moment in time. Each copy uses information to build our Here and Now. This information is a set of characteristics that appear
under different circumstances at all moments of our Universe’s existence. This information is inherited due to the “entanglement” of the same Photon and all its copies from all moments of time.
Entanglement is a mechanism for completing the features that best “build” our Here and Now. It is not known how our current Reality is created exactly, but certainly information in the “past” is used
in some way, for some continuity.
Our Reality continues with each successive second – in our real time. It is carried out in a certain continuum. One thing results from the other, all by means of “changes.” Changes ensure the lapse
of time. If this is so, then our simple elementary particle – our Photon – must also undergo changes. Perhaps it loses and gains in successive moments of the continuum its properties.
Thanks to “entanglement,” it has communication with all copies of itself. It has information about its existence from the first moments of the creation of our Universe to the end of our time. The
Photon “follows” from our point of view according to the arrow of time and cannot go back in time. However, information from the “past” is provided by means of “entanglement”.
This means that, according to our concept, “quantum entanglement” has an extension. It is a kind of bond that is able to connect all copies of the same elementary particle (our Photon). And what we
call “quantum entanglement” is a special case of opposite quantum states described by quantum mechanics. An elementary particle is entangled with all its versions with all its copies that existed in
time. The entanglement complements the information of the particle itself, so that the particle can bring our Reality into the Time Continuum. This means that each version of our Photon can co-create
a specific version of our Here and Now.
Interpretation of matter – Stabilization of matter
Stabilization of matter is like maturation, if we were to refer to an analogy from our surrounding Reality. The stabilization of matter is a completely incomprehensible term, but one that can be
explained through our concept of ToE-Quantum Space. The basic question is: why is the matter we are watching possible to watch? In other words, our surrounding Reality is “made” of matter and it can
be seen, touched, smelled, experienced. It is not ephemeral. We say that in our Here and our Now, matter is stable.
Every object that is observed by us is made up of small elements – very small particles – these are elementary particles that co-form atoms (Subatomic particle). Our Human body is made up of atoms.
How is it that our structure forms a stable form – a finished product to be viewed. Particle physics is able to offer us a description of the micro-world in the form of the Standard Model, which
explains how our Human body, viewed objects have a stable form of matter.
Observations made from our point of view must be based on a stable image of matter. What we view in the process of observation is subject to certain laws, which must have evolved according to the
Anthropic Principles. We do not know why matter reveals itself in a stable way in our Reality – it is rather a Fact. We accept this condition. In addition, every observed object has a huge number of
copies. However, for the process of observation we only need a part of the information. We do not need the full information from all the copies our Universe offers us.
The stabilization of matter must also be related – like everything around us, to the expansion of our Universe. If the Expansion of the Universe has its Vector (Expansion Vector), this means that
Observation must also take into account the passage of time – the direction of time (arrow of time). A certain relationship between Observation and Expansion appears. It is impossible to make
observations of objects from the “future”, and this means that matter from the “future” is not stable to be observed. This will be discussed in detail in other articles.
Stabilization of matter. Levels of matter magnification. We make observations from the first level. How is it possible that such complex relations between all levels and fragments of matter become
stable in the first level?
Source: https://pl.wikipedia.org/wiki/Teoria_strun#/media/Plik:String_theory.svg
Matter in a stable state from our point of view achieves stability only for our present moment – for our Here and our Now. This means that the stability of matter is lost for both our “future” and
our “past“. It is the lost copies of the observed objects. We are unable to view objects from the “future” – because they may not yet exist for our Here and our Now – they are simply not stable, even
though they may exist. This is a simple example – are you able to observe your grandchildren, or great-grandchildren at a time when you don’t yet have your children?
Your grandchildren and great-grandchildren exist in the future. From your Here and your Now, such a form of matter is unstable, so you do not have access to it. And yet in a few years you will
experience it. If not you directly, you certainly know people who have their grandchildren – your grandparents, for example. You sit next to them, and they tell and show you a photo of themselves
from their youth.
Presented with the photo, it depicts a situation when your grandfather, your grandmother, did not yet know each other. This means that at that time no one had access to the image of your parents and
even less no one had access to your image. And now you are sitting next to your grandparents and your parents looking at a picture of your grandfather or grandmother from their youth.
This means that the Stabilization of matter must take place from our point of view only in our Here and our Now – in our present moment. So we do not have access to matter from the “past” and from
the “future”. Matter stabilization occurs only in the present moment. The entire Reality surrounding us is subject to “change”, so there is no way that what you observe in the “future” is even
similar to what you see now – here and now.
The stabilization of the matter certainly takes place at all the levels presented by the animation above. Perhaps in a different way. All of these six levels work together for an image of the
surrounding on Reality generated, prepared specifically only for this one current moment – for our Here and our Now. This image obtains stability only for this one moment. For the next, different
moment, a completely different image of the surrounding Reality will appear.
The stabilization of matter is linked to our real time. In our Here and Now a stabilization of the surrounding Reality is taking place. This results in a stable form of matter that co-creates our
Reality to our Here and our Now. Without real time, we are unable to obtain a stable form of matter. If the stabilization process occurs, it happens only for the present moment. The stabilization of
matter has a continuum, as does real time. Stabilization must therefore be linked to the arrow of time.
We don’t know exactly why we remember our past, even though we can’t access it. Everything in the “past” is unstable to us. This instability manifests itself in a certain subjective degree of
interpretation of what happened. Although it was real and stable from the point of view of the past Here and Now, now – in the present moment, in our Here and our Now, the whole picture from the past
is unattainable for us. This means that the matter from the past is also unattainable – unstable, and what we see in our Here and our Now is just a copy of what was.
Interpretation of matter – Uncertainty of time
Uncertainty of time. How can the uncertainty of time for the Macro-world be accomplished? The uncertainty of time in the Macro-world can be attempted by analogy to the phenomena of the micro-world,
where Heisenberg’s Uncertainty Principle occurs in quantum mechanics. So, by analogy on the principle of symmetry, can there be something in the macro-world that resembles Heisenberg’s Uncertainty
Principle – the uncertainty of time. Unpredictability of time affects our interpretation of matter.
Perhaps the uncertainty of time in the macro-world is not the same as Heisenberg’s Uncertainty Principle in the micro-world, but through certain analogies in the macro-world must also make what is
appropriate. The uncertainty of time can reveal itself in a completely different form than we can imagine. If such a rule exists and can be experienced experimentally in the world of Elementary
Particles, the concept of “time” in the macro-world may also be uncertain.
The uncertainty principle, also known as Heisenberg’s uncertainty principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs
of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can be known.
Heisenberg’s uncertainty principle occurs in quantum mechanics. Historically, analogous principles occur in classical wave theory, where it is impossible to determine with complete accuracy the
position and speed of a propagating wave. We can also refer to classical physics: in a situation where it is impossible to accurately determine the position and momentum of a satellite orbiting a
large body, such as the Earth. By photographing with a flash at certain intervals or using radar, we disturb the momentum vector of the satellite.
This is because when photographing or using radar, we use a source of photons that have a certain momentum, which is partly absorbed and partly reflected by the orbiting satellite (its surface),
which changes its momentum vector very slightly. By gaining position information, we lose momentum information to some extent. In order to gain information about the system, we have to enter into
some interaction with it, which means that the state of the system is disturbed.
Here let’s try to formulate Heisenberg’s uncertainty principle. In general, it states that the more we know about the position of a particle (the uncertainty in measuring position tends to zero), the
less we know about its momentum (the uncertainty in measuring momentum can be arbitrarily large), and vice versa. The uncertainty principle also exists for other physical quantities, such as energy
and time.
The most common forms of Heisenberg’s uncertainty principle known in the literature are the uncertainty principle of momentum and position and the uncertainty principle of energy and time.
It is certainly difficult to compare what is happening in the micro-world with what is happening in the macro-world. However, if we appeal to the principle of symmetry, it seems that the uncertainty
of time in the macro-world should be happening. The uncertainty of time can manifest itself, for example, in the fact that the Expansion of our Universe consequently causes our “time” to slow down.
The unpredictability of time can be revealed in the following animation. Let’s follow it.
The uncertainty of time, is due to its unpredictability. Perhaps it would be possible if we assume that our Universe must expand inside the Initial Singularity. This would mean that the Initial
Singularity must contain all the Attributes of our Universe – from its Beginning, to its End. This seems quite logical, because if we assume that the Big Bang Theory refers to the Initial Singularity
at the very Beginning, then thanks to such a Root Cause, nothing could appear, nothing could happen. This implies a certain consequence. If the Beginning of the Universe could happen thanks to the
Singularity, then the End certainly cannot happen without the participation of the Initial Singularity.
In order for the Universe to begin, a Precursor, called the Initial Singularity in the Big Bang Theory, was necessary. This Root Cause had to have the necessary information – in general, to carry out
the Origin of the Universe. Without this necessary information, it is likely that the Universe would not have been born. If this information is so necessary for the formation of the laws of the
Universe, then the Universe itself would not have been able to cope if it had been deprived of this important “information.” Information is models, representations of various forms of energy, matter,
If the Universe must rely on these elements/information contained in the Initial Singularity, it means that the Singularity itself must create some, special environment, without which the Universe
cannot evolve, cannot create new, more complex entities, such as planets, stars, galaxies. Before the Initial Singularity, there was also no “time“.
This means that the Singularity must consistently sustain such an environment in which these laws and our concept of “time” function. For without time, we are unable to exist. The simplest
arrangement, modem, for the Initial Singularity to be able to sustain the initiated Scientific laws, time/time lapse, is to assume that this is done in the terrain of favorable initial conditions –
that is, in the terrain of the Singularity itself. The location of our Universe inside the Singularity can explain the uncertainty of time.
Making such an assumption leads to treating our Initial Singularity as a sphere that surrounds everything that takes place in our Universe – from the passage of time, which, after all, cannot exist
outside the sphere of the Initial Singularity, to the very process of the Expansion of our Universe. This could mean that our entire Universe is expanding, but in a completely different way than
previously thought. Since our Universe is expanding, it would seem that the Initial Singularity is either infinite or expanding as well.
The expansion of the Initial Singularity may explain why the concept of “time” can only exist within the Singularity – the expansion of the Singularity ensures change, and change is an attribute of
time, the passage of time. However, in our case, our Universe is expanding opposite to the center of the Initial Singularity, so we are dealing with the phenomenon of the Uncertainty of time.
If our Universe is expanding in the sphere of the Initial Singularity, which is also expanding, then our concept of “time” is not completely certain, and Uncertainty of Time may occur. Uncertainty of
time may manifest itself in the fact that the Singularity must be expanding much faster than the inner creation that is the Universe. This means that our time will slow down.
Our time, is a reflection of the Expansion of our Universe for the reason that during observation we have access only to data from the past – to objects older than the observer. If this is the case,
then the direction of the Expansion is consistent with the direction of our time. This means that Time has its “arrow of time”. This arrow is consistent with the direction of the Expansion of our
The sphere of the Initial Singularity is expanding faster than our Universe, and this means that Objects that are close to the outer layers of the Singularity must go backwards in time. All objects
that are closer to the outer surface of the sphere of the Initial Singularity are moving away and going backwards in time. Of course, these statements are relevant to our point of view – to
observations made from around our planet Earth. In addition, our Universe is expanding to the center of the sphere.
The direction of Expansion is determined by the Center Point of the sphere of the Initial Singularity. Therefore, our “time” is slowing down. This is due, first of all, to the difference between the
expansion of the Initial Singularity and the Expansion of our Universe. This means that the closer we are to the Center of the Singularity Sphere, the slower our passage of time becomes.
This means that, consequently, we are heading to the Sphere Point where “time” will stop flowing. This may mean the End of our Time. A sign of such a state may be the disappearance of all “changes”.
Changes, in turn, are attributes of our concept of “time” – if there are no “changes”, time will not pass, and if time stops passing, the End of our Universe will occur – for example, as the heat
death of our Universe. Slowing down our time introduces Uncertainty of time. The uncertainty of time can also lead to the uncertainty principle, and therefore can lead to the uncertainty of matter?
Interpretation of matter – Uncertainty of matter
Uncertainty of matter refers directly to the Uncertainty Principle, which was proposed by Werner Heisenberg. Uncertainty of matter is a kind of extension of the Uncertainty Principle. Our Concept
also refers to the Uncertainty Principle and tries to interpret it consistently to the ToE-Quantum Space. What does this Uncertainty consist of? What is the Uncertainty of matter? Before we go on to
answer the questions posed, perhaps we should first look for our micro-world – the world of elementary particles.
Uncertainty of matter takes place outside of real time, it is a special circumstance of imaginary time and imaginary place (see Time Quaternion). The image of our Reality – our Here and our Now, is
built of matter with certain properties. The properties of our matter, are only revealed in real time – it is the state of matter determination that will stabilize our matter just for this particular
Here and Now.
At this point in time (Here and Now), our matter will have selected the appropriate properties to map our present Reality. This means that adequate Fundamental Particles and fundamental interactions
will be revealed. For our present moment there will be a “stabilization” of the Reality built from: Fermions, treated as the building blocks of matter, and Bosons, treated as the carriers of
interactions. For this point of the Here and Now, the Uncertainty of matter will end.
Can Heisenberg’s Uncertainty Principle be extended to the Uncertainty of Matter? Perhaps something should first be said about Heisenberg’s Uncertainty Principle itself. Perhaps then it may be
possible to understand how such a principle can be extended to matter?
The uncertainty principle, also known as Heisenberg’s indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain
pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the more accurately one property is measured, the less accurately the other property can
be known.
More formally, the uncertainty principle is any of a variety of mathematical inequalitiesasserting a fundamental limit to the product of the accuracy of certain related pairs of measurements on a
quantum system, such as position, x, and momentum, p. Such paired-variables are known as complementary variables or canonically conjugate variables.
First introduced in 1927 by German physicist Werner Heisenberg, the formal inequality relating the standard deviation of position σ[x] and the standard deviation of momentum σ[p] was derived by
Earle Hesse Kennard later that year and by Hermann Weyl in 1928:
where ℏ=h2π reduced Planck constant.
The quintessentially quantum mechanical uncertainty principle comes in many forms other than position – momentum. The energy – time relationship is widely used to relate quantum state lifetime to
measured energy widths but its formal derivation is fraught with confusing issues about the nature of time. The basic principle has been extended in numerous directions; it must be considered in
many kinds of fundamental physical measurements.
It is vital to illustrate how the principle applies to relatively intelligible physical situations since it is indiscernible on the macroscopic scales that humans experience. Two alternative
frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, but the more
abstract matrix mechanics picture formulates it in a way that generalizes more easily.
Interpretation of matter – Uncertainty of matter. The superposition of several plane waves to form a wave packet. This wave packet becomes increasingly localized with the addition of many waves.
The Fourier transform is a mathematical operation that separates a wave packet into its individual plane waves. The waves shown here are real for illustrative purposes only; in quantum mechanics
the wave function is generally complex.
Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert
space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). A nonzero function and its Fourier transform cannot both be sharply localized at the same time.
A similar tradeoff between the variances of Fourier conjugates arises in all systems underlain by Fourier analysis, for example in sound waves: A pure tone is a sharp spike at a single frequency,
while its Fourier transform gives the shape of the sound wave in the time domain, which is a completely delocalized sine wave. In quantum mechanics, the two key points are that the position of
the particle takes the form of a matter wave, and momentum is its Fourier conjugate, assured by the de Broglie relation p = ħk, where k is the wavenumber.
In matrix mechanics, the mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An
eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable A is performed, then the
system is in a particular eigenstate Ψ of that observable.
However, the particular eigenstate of the observable A need not be an eigenstate of another observable B: If so, then it does not have a unique associated measurement for it, as the system is not
in an eigenstate of that observable.
This principle – the principle of indeterminacy has been extended to the uncertainty of matter. Matter, its properties beyond our Here and our Now, is unstable from the point of view of our present
moment. This means that we do not know what will happen in a moment. If we heat water for tea, in a moment the water will change its state of matter – at least, some of the water will turn into steam
We don’t know exactly when this will happen – How exactly will the transition be done (Phase transition). The picture of our Reality will be built in each successive second of our real time. For our
“past” moments and “future” moments, the state of aggregation of our water, its properties (water) are uncertain from the point of view of our Here and our Now.
Let’s start with the fact that at the emergence of time, initiated by the Big Bang, our elementary particles appeared in the form of occurring “changes”. Elementary particles were not yet properly
formed – they had no mass, spin, momentum, etc. But they existed – there were some primordial forms of our matter, which for the time being was deprived of its properties. The Universe was being
born, and with the birth of our Universe, our elementary particles gained more properties. At that time, from our point of view, it was impossible to use such matter, endowed only with the existence
of “changes.”
Matter, like everything around us, had to grow into the shape that exists in our present Reality. We don’t know what that might have looked like specifically. However, the simpler the system, the
less complex, the easier it is to imagine. It seems that an elementary particle like the Photon could not have been created in an instant. The photon had to mutate, to transform into the form we know
Rome wasn’t built in a day! Everything required “time” from our point of view. But “time” according to us does not exist. At that time, our Universe was “Indeterminate” from our point of view. If we
had been there, we would not have been able to see it all, because Photons were not yet there.
Matter, therefore, must have gained further properties over our time, which determined the interactions for individual elementary particles. After all, we have no knowledge whether the Photon as we
know it today could have come into existence at once – perhaps originally, the Photon could have had Invariant mass. Later, however, such a Photon, lost its mass in favor of receiving momentum. And
momentum appeared with the advent of the expansion of our Universe. See also article the Expansion of the Universe.
So if everything had to evolve, so did matter and its properties. Today’s image of the Universe is the result of what has taken place during the expansion of our Universe. The animation below shows
the concept of the appearance of successive physical properties of one elementary particle. These particles in subsequent stages of expansion will co-create more complex forms of matter for further
physical properties. Successive chemical elements and Chemical compounds took time to evolve. Our Concept assumes that it does not exist. The measure of time is change.
Thus, each successive chemical element had to mutate with the help of the “changes” made. Each “change” resulted in added value. This added value, could consequently be responsible for the appearance
of another property of matter. Therefore, if time does not exist, then a complex Chemical compound must be the fruit of all the mutations (changes) that took place during the expansion of the
Universe. It’s quite complicated, but the analogy can be repeated with a familiar example.
Our bodies appear at the moment of conception. Even then, we gain characteristics that will be revealed at our birth. Eye color, hair, genetic code, gender, senses, etc. Other traits we acquire over
time, knowledge, experience, sensitivity, Perception, character traits, etc. Further is even more interesting. We can change the Reality around us: create ideas, concepts, design magnificent
buildings, discover unknown phenomena. Finally, something extra – we have our share of procreation, which is our “extension” – our next generation that inherits our properties from us.
We rather did not invent it, it was inherited by us. We are made up of chemical compounds that have taken over properties from chemical elements. Chemical elements have properties of atoms, atoms …
and so on, down to the smallest fragments of matter. And matter how does it know what its form and properties should be in the next Here and the next Now? If there is no time, then everything must be
accomplished by a matter of complementary information through the entanglement that occurs between copies of the same elementary particle. (See also copying the world).
Interpretation of matter – uncertainty of matter occurs permanently for every moment except the present moment – our Here and our Now. No one knows what state of matter and when it will be revealed
to us in the future. Every property of an elementary particle is achieved in due time since the beginning of our Universe. Since, according to our Concept, “time does not exist,” every property of an
elementary particle belongs to a copy of that elementary particle.
Copies of one elementary particle fill what we call our “time.” All copies, as a result of “entanglement,” inherit the received properties. Each new property of an elementary particle appears with
the elapse of our time. This means that each elementary particle in imaginary time is in contact with its copy through entanglement. Entanglement is a form of information exchange between different
copies of the same elementary particle.
The uncertainty of matter is a condition that occurs outside of our Here and our Now. The uncertainty of matter destabilizes matter outside of our Here and our Now. This means that in the present
moment, our matter will use through entanglement with its copies, which have the right qualities and properties to describe our Present Reality. If there is a copy of elementary particles that can
overcome the force of gravity, such a Reality would be offered to us.
All the properties, of each copy, of each elementary particle, are available to us. However, through the time continuum, which is co-created by the expansion vector, we are not able to obtain in our
real moment, for example, a Photon that is endowed with Invariant mass, or a Photon that has no spin. However, it is theoretically possible. To give an example, we can obtain a Bose-Einstein
Condensate. Normally this is not possible, but now scientists have succeeded in obtaining such a condensate. This is only a small step, but in the next step it is possible to create matter that is
free of gravity.
This may mean that the properties of elementary particles can be provoked and triggered in our Here and Now. Sometime in the past, from our point of view, there was a state of the photon in which it
had no spin. There may have been a version of it that had Invariant mass. At that time, for the description of the Universe and the representation of That Reality, the existence of such elementary
particles with such properties was necessary.
Today, each elementary particle has a bond with all its copies through entanglement. If this could be the case, then releasing such a property – a photon without spin, or a photon with Invariant mass
, could be real. So could obtaining a Bose-Einstein Condensate. Therefore, matter from the “past” is devoid of the attribute of gravity – in this way, our Here and our Now is devoid of gravitational
interaction from the past.
What to compare it to in order to understand it better? The example of hypnosis comes to mind. It happens that in a state of hypnosis, a person has the ability to go back to a particular moment in
his life. It could be the same with our elementary particles. It could be that in such hypnosis for an elementary particle, one could find a copy of that elementary particle when it was not yet a
photon, or such a copy that did not yet have spin. Then such a property could be revealed to us in our Here and our Now.
If the unknown properties of our elementary particles could be revealed, we could understand how our Universe came to be and whether our concept of time is correct. Perhaps such elementary particles
as photons without spin or with non-zero rest mass exist, only they are unstable in our Here and our Now? Perhaps they form tachyons, which is why they cannot be experienced in our Here and our Now.
It is not known exactly what would happen to our time continuum if such an elementary particle with properties we know nothing about were released – would our Reality then continue in our “future”
moment. That’s another topic, for another article.
Thus, if our picture of matter is indeterminate beyond our Here and our Now, it appears to be in an unstable state from our point of view. This means that matter, in order to be in an unstable state,
must get rid of some of the properties it has in our Here and our Now. For example, an electron that is in our “past” may have lost its Invariant mass. Such an electron is therefore in an imaginary
state from our point of view.
Therefore, according to the indetermination principle, we know nothing about the position of such an elementary particle if we are concerned with momentum – and vice versa. The indeterminacy of
matter does not mean that it does not exist – perhaps it is precisely there, only it is inaccessible. Unstable matter becomes dark matter – dark matter is then in a state of indeterminacy.
Interpretation of matter – Multiplication of changes
Multiplication of changes is a certain analogy or interpretation of what happens from the point of view of the micro-world – the world of elementary particles in correlation to “time”. Of course, in
our considerations we will apply our concept of “time”. Our Reality can only use the real part of our Complex description of time – this is our real time. This means that for our considerations, some
extension of our “time” will be made. The description of this extension, will be expressed by means of the Complex Time Function, which refers to our concept of “ Time Quaternion”.
According to our concept of ToE-Quantum Space, “time” does not exist. What is accomplished in “time” has, of course, a different interpretation. This interpretation, in its final, mature stage from
our point of view, manifests itself precisely as our “time”. In our conception – time is change. This means that our concept of “time” is understood as “the act, the effect of change taking place”.
“Change,” for us, means the lapse of time – in other words, time passes from change to change. Without change, there is no concept of time. Thus, the concept of “change” becomes crucial to the
interpretation of our concept of “time.”
So, if there is no “time”, how does it perform the “reproduction” of the Reality around us? How is a time continuum possible? Why is our “time” directional and has its arrow of time? Why must our
concept of “time” have a different interpretation for the macro-world and for the micro-world? So if our concept of “time” is the same as “change,” how can we interpret the stability of matter in
time – in a time that is not there. See the article Time Vector.
Why is it necessary for us to invoke the stability of matter for our Here and our Now to have a known “past” and an unknown “future”? Only such – “one-way” relationship will allow us to obtain our
Time perception. Why are we unable to change the direction of the passage of time? On what depends our Time perception – its real part – our, experience “time”.
In our concept of “time” – we interpret “time” as a transition: from “change” to the next “change”. This is how the lapse of time takes place. Changes determine, order our Reality – from the
micro-world to the macro-world. In each stage of existence of our Universe, “changes” must take place. Otherwise, there would be no phenomenon (concept) of “time” – there would be no existence. Our
existence is inextricably linked to the concept of time – one can only exist in time. But according to our concept, “time” does not exist. Of course, this is about the interpretation of what we call
So if, according to our concept, “time” has a different interpretation, how does our Reality change in this different interpretation of time? This may mean that our Reality must also have its own
different interpretation, which must be consistent with our concept of “time”. Change marks the lapse of time for our Time perception – this still seems understandable. How, then, can such “changes”
interpret our understanding of “time” – all that passes in the Reality around us. The question itself, seems little understood, but what about the answer?
Changes take place at each stage – thus co-creating the structure of “time”. The end result of all these multiplied changes, is our real time for our Here and our Now. The multiplication of changes
is the end product of our Reality for the present moment in our observable Universe. This means that our time has an expanded form to a Complex description of time – real time and an imaginary part.
Our conception of time assumes that time is identified with change. The multiplication of changes, therefore, must have a certain interpretation – an idea that allows us to understand the functioning
concept of “time”. In other words, how could we conceive of such a “structure of time” that describes what we see, register around us. That is, despite the division of our Universe into macro and
micro worlds, everything maintains coherence and continuity. The picture of our Present Reality is consistent and maintains logical continuity. This continuity, is derived from the “past”, through
the “present” goes to the “future”.
In order to be able to better explain this (Multiplication of Changes), it will be necessary to refer to our Micro-world – the world of elementary particles that co-create the structures of matter.
These matter structures co-create the image of our Reality that surrounds us. Stable Matter Structures co-create the image of our Reality for our Here and our Now. Multiplication of changes thus
provides various elements which, when stabilized, can create our Reality. Multiplication of changes takes place in all parts of our Universe (micro, macro), in all elements of our Reality.
How is Multiplication of Changes accomplished? In order to create an elementary particle, a “change” is needed – a large number of “changes”. To stabilize an elementary particle in time, another
portion of “changes” is necessary. In order for the particle to achieve the property of “momentum” – to imitate motion – another portion of “changes” is necessary. Changes are necessary to grant the
Physical property right to have Invariant mass to our elementary particle. In summary, in order to show a fragment of our, stable Reality, a huge number of “changes” is necessary – therefore “changes
In the micro-world – in the world of elementary particles, stabilization is achieved by particles that do not exceed the limiting speed “C” – in our Universe, it is the speed of light that determines
the conditions of stabilization for elementary particles. If an elementary particle achieves its speed equal to or below the speed of light in a vacuum, it can achieve stabilization in time – in our
real, present moment. Then such a particle contributes to the organization and construction of our present Reality. Thus, if such an elementary particle exceeds the speed of light, this does not mean
that there is no such particle, only that such a particle cannot be stable for our present moment.
Multiplication of changes is an interpretation of the lapse of time in different worlds – for micro and macro phenomena. This means that “changes” can also imitate what we define as Motion in our
Reality. If there is no time, then Motion is also a kind of illusion. Of course, from the point of view of the world of elementary particles. In our World, in our surrounding Reality, we are not able
to experience the quantum action of the World.
So that this can be better understood, we will discuss the situation presented in the following animation. The situation is about one moment – our Here and our Now. How much can happen in imaginary
time and in imaginary places in one present moment. It can be said that before one moment of our real time passes, there must be a billion “changes” in the world of elementary particles. These
changes will work out a rendering of our present Reality.
Interpretation of matter. Multiplication of changes is an interpretation of time and imitation of Motion. The animation presents the viewpoint of the micro-world, that is, the world of elementary
particles that co-create the image of our Reality. The world of elementary particles can stabilize matter for our present Reality only if the particle co-creating matter achieves changes at a level
below the speed of light. This means that even the speed can be mapped by the “changes” occurring in the micro-world.
However, some particles (Tachyons) cannot achieve stabilization because they exceed the speed of light. This means that a huge number of “changes” are necessary to map the Faster-than-light. If a
particle stabilizes over time to a form that we can experience, record, it means that the number of changes necessary to describe the particle has decreased. From our point of view, this means that
the particle is moving at a speed less than, or equal to, the speed of light.
The animation shows our description of the Universe – our Real moment, which must happen in a certain way. This way, must take into account the laws of physics for our macro world. For the laws of
physics to be constant, repeatable, our surrounding Reality must be predictable, conforming to the description of those laws of physics. This means that the matter of which our Universe is made must
be stable – at least at one moment in time. Another element is to maintain this stability until the next, future moment.
In the micro-world, we have a different description of time – it’s the Concept of Imaginary Time. If it were not for the concept of the Time Quaternion, elementary particles would not be able to
communicate with our “past” and with our “future.” Without this communication, there can be no time continuum. Communication with “the past” and “the future” must make for an exchange of data – it is
information from “the past” for “the future.” Without this information, it is difficult to talk about continuity – it is a time continuum. See the article Information Propagation.
In order to map the Real moment, the particles must know about the history of the previous Reality (past Reality). The particles must also know how the “future” moment is to be mapped. The
information from the “future” is provided by unstable particles – Tachyons. Since Tachyons are unstable in time it means that they can “move” above the speed of light. If so, they might as well go
back in time. This can be interpreted that a Tachyon brings information from “the future.” Information from the “future” is a guideline, will ensure the continuity of events in the “future.” It is a
mechanism of the laws of physics that ensures the predictability of events, physical phenomena.
In our animation, you can see how a tachyon “traverses” the Universe unstably in time. This means that the tachyon can go back in time and bring information from the “future”. This information will
allow the flying photon to change the Future Reality. The photon can therefore bring information from the “past” for the electron, which may already be a stable particle of matter for our real
present moment in time. Of course, Tachyons can also provide information from the past. But this information may be inconsistent with our present moment – our Here and our Now.
In order for a time continuum to be maintained for our present moment, information in the “future” must be provided by the Tachyon going back in time. On the other hand, the information necessary for
the continuation of phenomena in the “past”, is provided by the photon. Our “past” is known. The “future” is yet to be created. According to the time continuum and the prevailing laws of physics, we
are able to anticipate what may happen in the near “future.” But this is only an approximation. Photon brings stable information about the image of our Past Reality.
However, in order for all this to generate our present moment (Present Reality), changes on many levels are necessary. This is how the Multiplication of Changes takes place. Tachyon is outside of
real time and is therefore described by many more “changes” – can thus reproduce Faster-than-light – from our point of view, such a Tachyon will move faster than the speed of light. This is
symbolized by the higher number of “changes”.
The photon, which moves from our point of view at a speed almost equal to the speed of light, becomes a stable particle. Such a photon is able to provide us with information from the “past” and
participate in the reproduction of our Present Reality. Our Present Reality is one of the copies of the Universe, which is realized in our Here and Now. This means that our present moment must
maintain the time continuum impressed by the transition – from the “past” to the “future” through the “present.”
This means that stable information from the “past” can be provided by the continuation of a stable particle from that “past” – in our case it is the photon. Since our “future” is not certain – it has
many potential possibilities, the information from the “future” is not clear. Such information was provided by the Tachyon, which is an unstable particle for our Here and our Now. In order to make
the “jump” to the future moment, we need to establish an adequate possibility for the context of the present moment in order to maintain our time continuum.
What interests us is the electron and its state. The electron co-creates our visible world, our present Reality – in our Here and in our Now. It is the very concretes – the table, the chair at the
table, the view of the room, etc. Matter in its pure, stable form that can be experienced, recorded, measured, viewed and studied. The stability of matter lies in the fact that all elementary
particles must maintain coherence for one copy to reproduce our Present Reality. For this to happen, continuity – continuity through time – is necessary. Therefore, we must have information from the
“past” and from the “future.”
All elements – elementary particles have their share in the creation of Present Reality – from the unstable ones in time – to the stabilized ones, which co-create our stable matter in time. The
multiplication of changes is an intricate hierarchical system of “changes” and relationships that interact with each other through the inheritance of ownership. This inheritance of properties,
imitation of position and consequently imitation of motion is a set of energy states. Our animation must be read in the context of our concept of ToE-Quantum Space. Otherwise, our considerations do
not make much sense.
Multiplication of changes is a mapping of energy states in the structure of Quantum Space. Every moment of our time is such an energy state. From our point of view, if time is interpreted this way,
we live in a Universe where there is neither time nor space. All we have is our Here and our Now. The energy states are arranged in a logical sequence co-creating the lapse of our time.
Information from our “past” and information from our “future” provides us with a time continuum and stability in our real time. Every piece of Matter is an image of the Reality around us. What
surrounds us, what we experience may be an illusion that we have cherished since the beginning of human existence.
This does not mean that our experience, Perception our knowledge should go into the trash. This is just one point of view on our Universe. If our world of elementary particles works according to
quantum mechanics, it is still impossible to understand what phenomena occur in our micro-world. It seems that the world of elementary particles, which is poorly accessible to us, has been
encapsulated in mathematical formulas. According to this mathematical description, we have found many unanswered questions.
What occurs in our stable Reality does not necessarily coincide with the world of elementary particles. If this is the case, it is imperative to verify and confront these inconsistencies once again.
To do this, the approach to time must be changed. Time is change. Time passes from one change to the next. If Time is change, then in the description of the world of elementary particles it would be
necessary to use instead of “time” the multilevel Multiplicity of changes. Then Time would become quite complex, and it would have to be described differently – a complex description.
Multiplication of changes involves the multiplication of “changes” for the final product, which is the reproduction of our Real moment (our Here and our Now) taking into account the time continuum.
Changes are necessary and happen at every stage. In order for us to see something, it is necessary to generate many “changes” on many levels. This is the Multiplication of Changes. Each level strives
for stability in our real time. Perhaps instability in our micro-world must exist from our point of view. Otherwise, there would be no stability in our time – there would be no our Here and our Now.
Interpretation of matter – Time complexity
Time complexity is an interpretation of our concept of time. According to our Theory of Everything – ToE-Quantum Space, time as we know it does not exist. So we used a different description of our
concept of time – this is the Complex description of time. This description, assumes that the basis of everything – especially time, is “change.” It is change that determines “time“, without change
there would be no “time” – time is change. Our entire Time perception, therefore, must be based on “change.” Our Concept assumes the expansion of time – into the real part of our time and the
imaginary part of our time.
If we assume such a structure of our concept of “time,” it will be necessary to clarify what our Time complexity consists of. How can such a structure of time be presented in connection with matter?
Well, our matter can be stable only in our present moment – in our Here and in our Now. For the other position in time, we have no possibility of stable access to matter. How does the Time complexity
take place so that the stabilization of matter in our real time takes place.
In the animation below, you can observe the interpretation of Time complexity. This means that our matter – a flashing particle, can be unified in the complex space of time. The first part –
symbolizes matter in the form of a flashing circle. The blinking – changing color, symbolizes our real time. In turn, the subsequent circles – also changing color, is matter inaccessible to the
observer from the point of view of real time. This can be compared to dark matter, which is in no way accessible to us – from the point of view of our Here and our Now.
Interpretation of matter. Time complexity determines the relationship between time and matter. In real time, matter is stable only for our Here and our Now – for our present moment. Matter becomes
stable in each subsequent present moment. In the imaginary part of our concept of “time,” matter has another dimension, which is the result of matter being unstable. This means that our matter occurs
and does not occur simultaneously. In addition, matter can occur in several places at the same time. The stabilization of matter occurs only in one real moment – then in other locations those moments
go into the imaginary part of “time”. Then we have the problem of accessibility to matter.
Matter from imaginary time, has its interpretation. Perhaps the interpretation of such unstable matter may be dark matter. In that case, there is a certain uncertainty of matter. If matter occurs in
the imaginary part of “time” – then, from our point of view, it is not available in our real time. The Time complexity will then be presented as shown in the animation above. All elements of matter
have their “time” multidimensional, so perhaps they can occur in many places simultaneously. Such phenomena take place in the world of elementary particles.
In the world of elementary particles, imaginary time plays a greater role. Therefore, from our point of view, such a piece of matter – an elementary particle is in an uncertain state. Until a
measurement is made, we are unable to determine what its position is. Not only that, if we manage to determine the position of the particle, we can say nothing about its momentum. We do not know
exactly where it is going. This problem was discovered and described by Werner Heisenberg in the uncertainty principle.
This uncertainty is the result of the imaginary part of time. Going further, to quantum mechanics our matter is distributed in real time. This means precisely nothing more than the stability of
matter for the present moment. Then, there is Time complexity, which forms the corresponding stable image of matterfor the description of our moment. This, of course, can be expressed differently –
from the point of view of quantum mechanics.
In quantum mechanics, quantum decoherence further implies the interaction of an object with its environment in an irreversible way that eliminates quantum entanglement. The quantum state is a state
of superposition. According to the research, there are no elementary particles anywhere until they are observed, i.e. quantum decoherence.
Despite the fact that matter exists in imaginary time, it reveals itself to us only in our Here and our Now – in our present moment of our real time. The description of this issue seems quite
difficult, so we will continue our attempts to clarify the meaning of our considerations. Thus, the topic of the Time complexity will have to be continued.
Interpretation of matter – Elementary particle
Elementary particle – a particle that is a basic building block, that is, the smallest and has no internal structure. However, the term has a slightly different meaning for historical reasons. The
study of these particles is dealt with by particle physics. See more at the link. Matter should consist of Elementary Particles. But is this in fact the case. Our Concept – ToE-Quantum Space assumes
that our concept of “time,” does not exist, and if so, do elementary particles not exist as well?
The world of elementary particles is governed by completely different laws – it is quantum mechanics, but it has a direct impact on that which surrounds us – our Reality. We do not know exactly how
particle mass, particle energy, particle interactions and the phenomenon of gravity cooperate in the world of elementary particles. The Standard Model describes the world of elementary particles
excluding just gravity. So will the Standard Model be further developed if it turns out that “time” does not exist?
Of course, the claim “time does not exist” requires our commentary. Our world expresses “time” in the shape of passing seconds, there is a very big simplification of the concept of “time”. Time is
treated as a physical quantity that determines the sequence of events and the interval between events occurring in the same place. Our concept of time has a different interpretation – it is described
by complex time, which uses the Time Quaternion for phenomena in the world of elementary particles.
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct
particles —twelve fermions and five bosons. As a consequence of flavor and color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively.
Among the 61 elementary particles embraced by the Standard Model number: electrons and other leptons, quarks, and the fundamental bosons. Subatomic particles such as protons or neutrons, which
contain two or more elementary particles, are known as composite particles. Ordinary matter is composed of atoms, themselves once thought to be indivisible elementary particles.
Subatomic constituents of the atom were first identified toward the end of the 19th century, beginning with the electron, followed by the proton in 1919, the photon in the 1920s, and the neutron
in 1932. By that time the advent of quantum mechanics had radically altered the definition of a “particle” by putting forward an understanding in which they carried out a simultaneous existence
as matter waves.
In particle physics, fermions are particles that obey Fermi–Dirac statistics. Fermions can be elementary, like the electron — or composite, like the proton and neutron. In the Standard Model,
there are two types of elementary fermions: quarks and leptons.
Quarks are massive particles of spin-1⁄2, implying that they are fermions. They carry an electric charge of −1⁄3 e (down-type quarks) or +2⁄3 e (up-type quarks). For comparison, an electron has a
charge of −1 e. They also carry colour charge, which is the equivalent of the electric charge for the strong interaction. Quarks also undergo radioactive decay, meaning that they are subject to
the weak interaction.
Leptons are particles of spin-1⁄2, meaning that they are fermions. They carry an electric charge of −1 e (charged leptons) or 0 e (neutrinos). Unlike quarks, leptons do not carry colour charge,
meaning that they do not experience the strong interaction. Leptons also undergo radioactive decay, meaning that they are subject to the weak interaction. Leptons are massive particles, therefore
are subject to gravity.
Antimatter is matter that is composed of the antiparticles of those that constitute ordinary matter. If a particle and its antiparticle come into contact with each other, the two annihilate; that
is, they may both be converted into other particles with equal energy in accordance with Albert Einstein‘s equation E = mc^2. These new particles may be high – energy photons (gamma rays) or
other particle – antiparticle pairs.
Source: https://en.wikipedia.org/wiki/Elementary_particle
Our Concept – ToE-Quantum Space assumes that elementary particles can only be permanent from the viewpoint of our Here and in our Now. Therefore, if our “time” does not exist, then the properties of
elementary particles can be scattered in “time space”. This means that each elementary particle has a timeless character. The stability of the elementary particle, the stable form is revealed in each
successive our Here and in our Now – from our point of view, of course.
A particle can be complex in time, then its properties are co-created from all the entangled copies from the origin of our Universe to its End. This is how the Uncertainty of Matter is accomplished,
the result of which can be the stabilization of mass/matter only in the real moment. We call this stabilization of mass/matter in time. But in our conception, such “time” does not exist. Therefore,
we must expand our description of “time”. our concept of “time” treats time as “change.” If so, what is the interpretation according to our concept of “elementary particle”?
According to current theoretical and experimental knowledge, our elementary particle that co-creates matter structures must be stabilized in order for us to have access to such matter. Without access
to matter, our matter can be interpreted as Dark Matter. If we do not have access to matter, it does not mean that such matter does not exist. Such matter retains its indeterminacy and is thus
inaccessible to us in our present moment – in our Here and our Now. Such matter becomes inaccessible – dark matter.
If our time does not exist, does this mean that at any level of magnification of matter also can be inaccessible to our Here and our Now? It would mean that also an elementary particle can be
inaccessible to our present moment – then such an elementary particle becomes a dark elementary particle (Antiparticle)? Can it become part of antimatter? So, is a world consisting of antimatter a
parallel world – shifted in time? Does this mean that the properties of matter can also determine the anti-properties for antimatter? If this could be true, then what would the states of matter for
anti-matter look like? What could anti-liquid look like for example?
Elementary particles can also regress in our time – this is the uncertainty of matter. If there is no stabilization in time, it means that such elementary particles are outside our “time” – our real
time. In order for such a description expressed through the uncertainty principle, or through the Schrödinger Equation, to work, elementary particles must function in imaginary time. The concept of
complex time is presented in the link. The imaginary part of our time gives the particle additional attributes that are not possible in our macro world – real time.
Functioning in imaginary time is a manifestation of the instability of the particle in our “time”. Functioning in imaginary time gives the possibility for elementary particles to exist in several
places simultaneously from our point of view. Functioning in imaginary time gives the possibility for particles to exist in Wave – particle duality. Thus, if in real time nothing happens from our
point of view, then in imaginary time such a particle can “visit” several dimensions, be simultaneously in our “past” and in our “future.”
In order for any elementary particle to have the comfort of duality (Wave – particle duality), our concept of time probably needs to be expanded to include an imaginary part. But that’s not all. If
our “time” must refer to its Time Continuum, this means that, like our present moment, stable Reality, each particle would consist of different copies to support the Time Continuum. Then, each
elementary particle would co-create the “shifting” of our present Reality according to the arrow of time.
The properties of elementary particles are inherited and transmitted through all copies of that elementary particle. The copies are located in “moments” that correspond to our present points in time.
This means that each copy of the same elementary particle must always have something that distinguishes it from the previous copy. In this case, these are changes in the properties of the elementary
particle in question over time.
Let’s assume that the interpretation of our “time” is completely different. According to our conception, the determinant of “time” is “change” – Time passes from change to change. If this were the
case, matter could reveal itself permanently – in a stable way for us, only by generating another “change”. This means that from the creation of our Universe to its termination, all possible “change”
scenarios for a given elementary particle must have been generated.
This could mean that at some point, our particle had no mass, and at some point mass was interpreted into a stable form. Each property of a particle could be a combination of “changes” from all the
properties that a particle could potentially receive, from the moment our Universe was created to the moment our Universe ends. It should be remembered that the characteristics, properties,
attributes of elementary particles, are only a certain interpretation of the energy states that have been generated in the structure of Quantum Space.
What exactly does the context of non-existence of time mean for an elementary particle? Our particle must participate in the stabilization of matter, which co-creates our present Reality. Reality
from the “past” and from the “future” is not available to us, and this means that the set of properties of our elementary particle is obsolete, indefinite for the next, future moment – the next
moment in our “future.” However, we know from our experience that continuity – our Time Continuum – must be maintained.
In order for our present Reality to exist, it must be stable for the present moment – for our Here and our Now. This means that each elementary particle is also stable for this present moment.
Stabilization of a particle for the present moment means that it has received information about what properties it should have in order for our Now (present) to happen. Note that our Present Reality
will be accomplished in the future moment considering the Time Continuum, so the continuum of stabilization of matter – our elementary particle – must be accomplished.
Otherwise, in the next instant there could be no Time Continuum, which would result, for example, in a loss of gravity. This means that elementary particles participate in our Time Continuum.
According to the rules of the Time Continuum, our present point in time – our Here and our Now, must receive data, into the image of stable matter (future Reality). Such data is received by means of
the Time Continuum – which determines our arrow of time.
According to our concept, in order to build our next Reality – the next moment, information is necessary, which comes from our “past” and from our “future“. While our “past” is known, our “future” is
unknown – it is uncertain, it takes place in the imaginary part of our concept of “time”. All this is done for the preservation of the Time Continuum. The preservation of the Time Continuum consists
in the continuity of stable matter from the known “past” and from the unknown “future.”
Our “past” is the known location of the elementary particle from our “past“. But also uncertain information about the location of our particle as it will be in our “future.” Presumably, our future
Reality will benefit from several proposals from the uncertain, unknown “future.” In order to build a future moment, the selection of only one location of our elementary particle from the “future”
must take place. Each elementary particle in our “future” has several locations due to phenomena that take place in imaginary time. Therefore, the particle may be unstable.
In summary, our elementary particle co-creates our present Reality. To do so, it must stabilize itself for this moment – for our Here and our Now. Stabilizing the particle involves getting the
properties of the elementary particle right. After all, no one knows whether the possible deficiency of certain properties. Then the “disappearance of matter” or “loss of gravity” may be revealed. To
avoid such strange cases, use the Time Continuum. This Time Continuum provides a continuum for the stabilization of matter over time.
Marek Ożarowski, 03 July 2024, update: 06 August 2024. | {"url":"https://theoryofeverything.info/interpretation-of-matter/","timestamp":"2024-11-11T04:56:49Z","content_type":"text/html","content_length":"347308","record_id":"<urn:uuid:cad7c6a6-de93-45af-9a5a-c3234f717ffc>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00855.warc.gz"} |
Phi in the human body
1.- Introduction
Marcus Vitruvius Pollio, Roman architect (c. 25 B.C.), remarked a similarity between the human body and a perfect building: "Nature has designed the human body so that its members are duly
proportioned to the frame as a whole." He inscribed the human body into a circle and a square, the two figures considered images of perfection. It is widely accepted that the proportions in the human
body follow the Golden Ratio. In this article we will review some studies on the subject. We will show the nineteenth century findings of the Golden Ratio in the human body by Adolf Seizing, actually
approximated by a Fibonacci sequence of measures. Then we will examine the Golden proportions of the human body proposed by architects Erns Neufert and Le Corbusier in the twentieth century. Finally
we will show how a common study with a german and an indian population samples confirmed the presence of the Golden Ratio in some proportions of the human body.
2.- Golden proportions in the human body found by Adolf Zeising
Adolf Zeising's main interests, back in the nineteenth century, were mathematics and philosophy. But after having retired he began his researches on proportions in nature and art. In the field of
botany, he discovered the Golden Ratio in the arrangement of branches along the stem of plants, and of veins in leaves. From this starting point he extended his researches to the skeletons of animals
and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, etc., and finally to human and artistic proportions. The title of his first
publication in 1854 declares his program: New theory of the proportions of the human body, developed from a basic morphological law which stayed hiherto unknown, and which permeates the whole nature
and art, accompanied by a complete summary of the prevailing systems [1]. That universal law was, in efect, the Golden Ratio. There he presents his own proportional analyses of the human body (Figure
Figure 1: Golden proportions in the human body found by Zeising [1].
Zeising divides the total height of a man's body into four principal zones: top of head to shoulder, shoulder to navel, navel to knee, and knee to base of foot. Each zone is further subdivided into
five segments, which are arranged symmetrically within each zone: either following the pattern ABBBA or the pattern ABABA, but always summing up 2A+3B. By the way, the 3/2 proportion in each zone is
a Perfect Fifth in the equal temperament musical scale. Is music involved in the design of our own body?
On the right of Figure 1 you can see the Golden proportions present in each of the segments, and between them, at different scales. Zeising's proportions of the human body are a beautiful example of
how Nature closely approximates the Golden Ratio by means of a Fibonacci sequence of measures. Zeising erroneusly substitutes 90 for 89 in his measures, but we have used the exact value in the
following calculations. The Fibonacci numbers present in his scheme, explicitly (green) or implicitly as grand totals (magenta), are the following:
Grouping consecutively each pair of adjacent measures one obtains an iterated division of the big segment (987) into consecutive Fibonacci numbers that closely approximate the Golden Ratio (Figure
2a). This reminds us the power of the Golden Ratio for consecutively dividing a segment with simple additions and substractions after the first split (Figure 2b). This sequence of Golden Ratio
divisions also reminds us of the fractal nature behind the design of our body, because the same Golden proportion is repeated at all scales.
Figure 2: Iterated division of a segment according to (a) the numbers in the Fibonacci sequence and (b) the Golden Ratio.
3.- The Golden proportions proposed by architects Neufert and Le Corbusier
In the twentieth century the architect Erns Neufert (1900-1986) propagated the Golden Ratio as the architectural principle of proportion in the human body. Neufert did not strictly follow Zeising's
human Fibonacci proportions, but introduces the exact Golden Ratio instead [2] (Figure 3). For him, the Golden section also provides the primary link between all harmonies in architecture.
Figure 3: Golden Ratio proportions of the human body after Ernst Neufert [2].
There is another great system of body proportions of the 20th century known as the Modulor, proposed by Le Corbusier (1887-1965). In his manifesto Vers une architecture, he presents the Golden Ratio
as a natural rhythm, inborn to every human organism. For details on the historical origin and developement of Modulor I and II systems you can examine the excellent summary by architect Manel Franco
[3]. Figure 3 shows the essential proportions proposed by Le Corbusier for the human body:
Figure 3: Simple sketch and main Golden proportions in the human body proposed by Le Corbusier [3].
In his final version, the Modulor II system proposes two Golden progressions of measures for the human body (Figure 4a). Returning to the style of Zeising, these progressions are actually two
Fibonacci sequences of measures (Figure 4b). That is to say, each measure is obtained by the sum of the two preceding ones. Therefore, the ratio of any pair of consecutive values in these
progressions closely approximates the Golden Ratio.
(a) Golden proportions in the human body proposed in Le Corbusier's (b) Detail of the red and blue progressions (in mm) in Modulor II. The ones in italic slightly deviate (1mm) from an exact
Modulor II. Fibonacci sequence
Figure 4
4.- A field study
T. Antony Davis, from the Indian Statistical Institute (India) and Rudolf Altevogt, from the Zoologisches Institut der Universitat (Germany) conducted a study where they measured 207 german students
and 252 youg men from Calcutta [4]. The measures taken A, B, C, D and E are shown in Figure 5a. In their results, they were able to confirm that the total height of the body and the height from the
toes to the navel are in Golden Ratio (ratios D/C and E/D). Figure 5b summarizes their main results. They obtained the almost perfect value of 1.618 in the German sample (this value held for both
girls and boys of similar ages) and the slightly different average value 1.615 in the Indian sample.
(a) The measures taken in the study [4] (b) Resulting average ratios, classified by population groups [4].
Figure 5
5.- References
[1] Zeising, Adolf: New theory of the proportions of the human body, developed from a basic morphological law which stayed hiherto unknown, and which permeates the whole nature and art, accompanied
by a complete summary of the prevailing systems. (In German).
[2] Neufert, Ernst: Architects Data.
[3] Franco, Manel: El Modulor de Le Corbusier (1943-54)
[4] T. Antony Davis and Rudolf Altevogt, "Golden Mean of the Human Body". | {"url":"https://www.sacred-geometry.es/?q=en/content/phi-human-body","timestamp":"2024-11-03T12:12:31Z","content_type":"application/xhtml+xml","content_length":"50319","record_id":"<urn:uuid:77bb150b-61e3-4915-bd26-67be51adb387>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00178.warc.gz"} |
IFPEN | Workshop Particles & Fluids: from individual particle dynamics to collective effects and fluidized beds
25.04.2017 - 26.04.2017
2 minutes of reading
The workshop took place from 25st to 27st of April 2017 in the Roscoff biological station.
The main objective was to bring together researchers, scientists, engineers and students in the same event to exchange and share their experiences, ideas and research results in numerical,
theoretical and experimental studies about all aspects of particulates flows: particles sedimentation and dispersion, fluidized beds, heterogeneous combustion, blood flows… with applications in
fields as environmental fluid mechanics, petroleum industry, paper industry, energy storage, aeronautics, biomedical sciences.
33 people attended this event. The format consisted of invited lectures within one session established to motivate scientific discussions and to enhance future collaborative research activities. The
different contributions were related to the following topics:
• Multi-scale modelling and simulations of dense particulate flows
• Dynamics of particles in turbulence
• Non-spherical particles
• Heat and mass transfers in dense particulate flows
• New technologies and industrial applications
The Organising Committee is part of the two French ANR projects: MORE4LESS and CODSPIT
Thanks to all participants, speakers, authors and partners. You made the event a success!
Sticky and tricky: Cohesive-particle flows
Cohesion between particles is widespread in nature and industrial applications alike. It can arise from a number of sources – van der Waals forces, liquid bridging, and electrostatics, to name just
a few. Predicting cohesive effects is challenging at both the particle-particle level (micro) and for many-particle systems (macro). In this talk, a new framework for describing cohesive particle
flows, which relies on bridging the micro-level effects to macro-level systems via a combination of experiments, DEM simulations, and continuum theory, will be presented. Demonstration of the new
framework and an assessment of its performance will also be given.
Prof. Christine M. Hrenya
Chemical and Biological Engineering
Univ. of Colorado
Boulder, USA
Challenges of CFD modeling in the chemical industry: an example of multi-scale modeling of dense liquid/solid suspensions
Industrial scale processes can be challenging to improve. Amongst the usual tools used in chemical industry, CFD is more and more becoming a valuable tool for diagnosis and design. CFD models that
can be used are sometimes limited in comparison to available academic tools. For instance, an LES of a few cubic meter device is still challenging, thereby limiting the turbulence modeling approach
to RANS or URANS.
One option is to use a downsizing strategy in order to define a 'numerical lab reactor' (Farcy et al., Chemical Engineering Science 139, 285-303). Another possible strategy consists of doing
multiscale modeling: small scales are resolved with DNS, then the information is used to calibrate/validate higher scale models. This approach has been chosen for the MORE4LESS ANR project. It has
been applied to dense solid-liquid suspensions in stirred tanks. For dense liquid/solid suspension at 20% of solid volume fraction, this approach was used during a collaboration between TU Delft,
University of Aberdeen and Solvay: particle-resolved elementary simulations have been done on a canonical problem and on a small tank (Derksen, AIChE J, vol 58, n°10). On the same device, a procedure
using particle-unresolved simulation is validated and then applied to a higher scale in 1liter reactor. At this scale, comparisons with experiments and Euler-Euler URANS models from commercial
software are done.
This allows to qualify the Euler-Euler approach that can be later be used more confidently on industrial scale reactors.
Dr. Nicolas Perret
Solvay R&I
Lyon, France
Rotation of non-spherical particles in shear
If inertia is present, the rotational motion of ellipsoids in Stokes flow derived by Jeffery (Proc. Roy. Soc. A, 102; 1922) is modified so that the particles no longer move in closed orbits but
instead drift towards a preferential orbit or steady orientation. The inertial effects are determined by a rotational Reynolds number and the particle motion exhibits a number of bifurcations as this
number is increased. A series of works, together presenting a combination of numerical and theoretical considerations, have been performed during the last years and the insights from these studies
will be summarized in this talk.
Prof. Fredrik Lundell
KTH Mechanics
Stockholm, Sweden
Paths of Bodies Freely Rising or Falling in Fluids
Even such an ideal body like a sphere can be observed not to fall (or rise) vertically in common Newtonian fluids like air or water. The same holds for bodies of other shapes (discs, cylinders,
spheroids) intuitively expected to follow vertical paths. Direct numerical simulations and the bifurcation theory make it possible to demonstrate that, as soon as the stabilizing effects of viscosity
become weak enough, the vertical paths of ideally axisymmetric bodies moving in ideally quiescent fluid under the action of gravity and buoyancy become unstable and give way to a rich variety of
regimes. The talk will focus on flat objects like discs, flat cylinders and spheroids presenting the most intriguing behavior.
Prof. Jan Dušek
Strasbourg, France
Flow past a yawed cylinder of finite length using a Finite Volume/ Fictitious Domain Method
Fluidized beds are frequently encountered in various industrial processes such as catalyze and biomass gasification. Despite the large numbers of studies describing the fluidization of spherical
particles, much less is known concerning cylindrical particles, which are frequently used in bubbling fluidized bed. In this work, the flow past a short yawed cylinder is studied as a first step to
understand the motion of many of them. To this aim the Distributed Lagrange Multiplier / Fictitious Domain (DLM/FD) method developed in the PeliGRIFF code (Wachs et al. 2015) is intensively used.
This method is validated using numerical results of the literature for a cylinder of finite length in cross flow (Inoue & Sakouragi 2008). Drag and lift forces as vortex-shedding frequencies are
carefully analyzed giving strong confidence in the numerical methodology. A detail study of the flow past a short cylinder at moderate Reynolds numbers (O(100)) is also carried out. The influence of
the yawed angle on the wake as on the hydrodynamic force is identified. The threshold for wake instability appears for a smaller Reynolds number when the flow direction is perpendicular to the
cylinder than when it is parallel. Otherwise the principle of independence which states that the normal force on the cylinder only depends on the normal component of the velocity (Sears 1948), does
not seem to be valid in those regimes.
Dr. Jean-Lou Pierson
IFP Energies nouvelles
Solaize, France
Development of filtered particulate Eulerian modeling approach for the prediction of bi-disperse gas-solid fluidized bed
Due to computational resource limitation, Eulerian gas-solid fluidized bed simulations of industrial processes are usually performed with mesh sizes much larger than the smallest meso-scale structure
size. Thus, these effective simulations do not fully account for the particle segregation effect (cluster or bubbles formation) and neglecting these structures generally lead to poor prediction of
bed hydrodynamic. According to previous numerical studies, this effect seems to be very effective in bi-solid mixture with large inertia difference between the solid species.
Following Igci et al., filtered approach may be developed where the unknown terms accounting for the influence of unresolved structures, called sub-grid contributions, have to be modelled in terms of
the computed (filtered) variables. In the work presented here, the development of such modelling approach is based on a priori analysis of 3D periodic circulating bi-disperse gas-solid fluidized bed
simulations using computational grids cell size of a few particle diameters. Using the 3D N-Euler multiphase code NEPTUNE_CFD, separate transport equations are computed for the number density,
velocity and random kinetic energy of the two solid species, coupled by collisions terms developed in the frame of kinetic theory of granular media supplemented by interstitial gas effect. The
mesh-independent results obtained are filtered using volume average to analyse the effect of the subgrid scales on the terms of the momentum and the particle kinetic agitation equations. Thanks to
those results, closure models are developed for the subgrid fluid-particle and particle-particle interactions terms in bi-disperse gas-solid fluidized beds. Those closure models include parameters,
which may depend on the particle and gas properties and are dynamically adjusted using a multi-level filtering procedure. Several monodisperse and bi-disperse fully resolved simulations have been
performed and enable to test the models for a wide range of diameter and density.
Prof. Olivier Simonin
President INP Toulouse
IMFT, Toulouse, France
Development of Numerical Approach for Gas-Solid Flows with Complex Particle-Particle Interactions
Euler-Lagrange (E-L) approach has been developed to take into account complex particle/ particle interactions such as; van der Waals force, forces due to liquid bridge, electrostatic forces, for
gas-solid flow simulations. Highly resolved E-L simulations of periodic and wall-bounded fluidization cases have been then performed to investigate flow characteristics and propose constitutive
models for standard and coarse-grained Euler-Euler approach.
Dr. Ali Ozel
Chemical and Biological Engineering
Princeton University
Princeton, USA
Modulation of the onset of turbulence by finite size particles
Particle-resolved numerical simulations based on the Force Coupling Method are carried out to study the effect of finite-size particles on the onset of turbulence in wall-bounded shear flows. The
study particularly considers the effect of concentration, particle size and particle-to-fluid density ratio on the mixture flow features.
A specific emphasis is devoted to the cycle of regeneration of turbulence for two specific configurations, plane Couette flow and plane Poiseuille flow. Indeed, the shape of the streaks and the
intermittent character of the flow (amplitude and period of oscillation of the modal fluctuation energy) are all altered by the presence of particles.
Prof. Eric Climent
Director of IMFT
Toulouse, France
Euler-Lagrange Modeling of Strongly Coupled Particle-Laden Flows
High fidelity simulation tools that leverage large-scale computational resources are now capable of making quality predictions of disperse two-phase flows involving a large number of particles under
a wide range of operating conditions. In this talk, we present recent insights on multiphase flows with significant momentum coupling between disperse and carrier phases. Of particular interest are
configurations dominated by clustering dynamics such as cluster-induced turbulence (CIT), homogeneously sheared CIT, and vertical particle-laden channel flows. Computational data from such canonical
flow problems as well as from realistic engineering flows are used to inform the development of multiphase turbulence models at the macro-scale.
Prof. Olivier Desjardins
Sibley School of Mechanical and Aerospace Engineering
Cornell University
Ithaca, USA
Recent advances in the multi-scale simulation of mass, momentum and heat transfer in dense particulate flows
Dense particulate flows involving coupled mass, momentum and heat transfer are frequently encountered in large scale industrial processes involving granulation, coating and production of base
chemicals and polymers. In dense gas-particle flows both (effective) fluid-particle and (dissipative) particle-particle interactions need to be accounted for because (the competition between) these
phenomena to a large extent govern the prevailing flow phenomena, i.e. the formation and evolution of heterogeneous structures. These structures have significant impact on the quality of the
gas-solid contact and as a direct consequence strongly affect the overall performance of the process. Additional complexities arise due to enhanced dissipation due to wet particle-particle
In dense particulate flows phenomena (effective) fluid-particle and particle-particle interactions have to be properly accounted for because the large scale system behavior (i.e. impact of
heterogeneous flow structures on reactor performance) is sensitively influenced by the these interactions. In the multi-scale approach detailed models, taking into account the relevant details of
fluid-particle interaction (DNS) and particle-particle interaction (DEM) are used to assess and develop closure laws to feed continuum models (TFM) which can be used to compute the flow structures on
a much larger (industrial) scale. In this presentation recent advances in the multi-scale modeling of dense gas-particle flows will be highlighted with emphasis on coupled mass, momentum and heat
transfer. In addition, areas that need substantial further attention will be discussed.
Prof. J.A.M. (Hans) Kuipers
Department of chemical Engineering and Chemistry
Eindhoven University of Technology
Eindhoven, The Netherlands
Micro/meso numerical simulations of fluidized beds
We perform particle-resolved micro-scale and Euler/Lagrange meso-scale numerical simulations of two fluidized beds. The former is representative of a liquid/solid fluidization with a moderate density
ratio and a small Reynolds number leading to a homogeneous bubbling regime while the latter corresponds a gas/solid fluidization with a higher density ratio and a higher Reynolds number. The 2 data
sets from particle-resolved and Euler/Lagrange simulations are compared on the basis of a detailed statistical analysis in order to identify potential deficiencies of Euler/Lagrange models to predict
the right dynamics. We suggest a correction of the classical deterministic drag force correlation by introducing a stochastic term whose parameters are determined from particle-resolved simulation
results. We discuss different research directions to further improve the predictions of Euler/Lagrange models.
Prof. Anthony Wachs
Chemical and Biological Engineering
The university of British Columbia
Vancouver, Canada
Theoretical modeling and three-dimensional unsteady numerical simulations of reactive fluidized beds
Natural gas combustion in two different configurations, 1) a dense fluidized bed reactor, operating with lean premixed methane-air mixture, and 2) a CLC system using perovskite as oxygen carrier is
investigated. An Eulerian-Eulerian approach is used to compute both the gas and the solid phases in an Eulerian fashion accounting for specific closures in order to model interphase mass, momentum
and energy transfers. Combustion is modeled using a two-step mechanism for homogeneous (gas-gas) reactions (Dryer & Glassman, Sym. (Int) on Combustion, 1973). A grain model, which accounts for both
the competing mechanisms of chemical reaction at the particle surface and gaseous diffusion through the particle layer, is instead retained for heterogeneous (particle-gas) reactions (de Diego et
al., Ind. Eng. Chem. Res., 2014). For each configuration, 3D unsteady CFD simulations are performed by NEPTUNE_CFD code and the results compared with available experimental measurements (Dounit et
al., Chem. Eng. J., 2008, Mayer et al., Applied Energy, 2015).
Dr. Ziad Hamidouche
IMFT, Toulouse, France
Liquid-solid simulations of fluidized beds with heat transfer at micro/meso scales
Fluidized beds are encountered in various domains such as oil and gas services and chemical engineering. The comprehension of flow properties and transfer in these systems is complex since spatial
interaction occurs from the particle scale to the process unit scale. Particulate flows arising in fluidized beds are often coupled with heat and/or mass transfer through chemical reactions. The
recent progress in high performance computing (HPC) and resources enables to simulate fluidized beds at micro scale (PRS) up to thousands of particles. The local information computed during the
simulations provides a database to better understand momentum and heat transfer occurring in fluidized beds. Due to the wide range of scales in presence (micro to macro), a multiscale framework is
used to study these processes. A mesoscale is also introduced corresponding to interactions between parcels of particles. In this work, microscale (DLM/FD) and the meso (Euler/Lagrange) scale
simulations of liquid-solid fluidized beds with heat transfer are performed with our massively parallel code PeliGRIFF.
A comparison of system size at microscale is provided to select appropriate size allowing statistically averaged local and global momentum and heat transfer. Then, a direct comparison of the
predictions obtained at both scales is performed and
suggest how the mesoscale modeling might be improved to provide more accurate solutions. Hydrodynamics improvements of the mesoscale model according to PRS results is realized by introducing a
correction in the forces acting on the particles or by adding a fluctuating term in the eulerian variables. Correcting in a first time mesoscale hydrodynamics will give a hint on the need to add a
correction term on heat transfer process into fluidized beds.
Florian Euzenat
IFP Energies nouvelles, Solaize, France
Heat transfer in a packed bed of particles for energy storage : an experimental and numerical study
Climate change concerns, the will to reduce dependence on fossil fuels and greenhouse gas emissions, are resulting in increased deployments of renewable energy technologies. But the intermittency of
renewable power sources such as wind and photovoltaic presents a major obstacle to their extensive penetration into the grid. Electricity storage is a potential solution to address this intermittency
problem by compensating for wind and sunshine’s variability. Among the many technologies available, Advanced Adiabatic Compressed Air Energy Storage (AA-CAES) is a promising technology since it is a
zero-emission storage system with a potential round-trip efficiency close to 70%. AA-CAES stores not only the compressed air, but also the heat, which is released upon compression of the air, in a
separate heat storage tank. In order to generate electricity, the heat is returned to the compressed air, which flows to the turbine. The Thermal Energy Storage (TES) system plays a prevailing role
in the global efficiency of AA-CAES process. At IFP Energies nouvelles, we develop a technology for TES system, based on fixed bed reactors to store heat in particles.
In the present study, we investigate the heat transfer within a fixed bed using an experimental and a numerical approach. The objective of this study to better understand the heat transfer involved
in such a system but also to validate the closure laws used in the numerical Euler-Euler approach.
Dr. Guillaume Vinay
IFP Energies nouvelles, Rueil-Malmaison, France
Particles and snowflakes falling through turbulence
The question of how turbulence affects the settling of small heavy particles is relevant to both industrial and natural settings. I will present laboratory and field measurements and numerical
simulations, demonstrating how turbulence may lead to a multifold increase of particle fall speed. In the laboratory, we use a novel apparatus where microscopic particles fall through a large volume
of homogeneous air turbulence. In the field, we image snowflakes settling in the atmospheric surface layer during snowfalls. Further insight is provided by direct numerical simulations of
particle-laden turbulence, to which we apply a novel definition of cluster based on self-similarity.
Prof. Filippo Coletti
Aerospace Engineering and Mechanics
University of Minnesota
Minneapolis, USA
Settling of isotropic and anisotropic solid particles in homogeneous isotropic turbulence
In a first part, the behaviour of large and heavy particles settling in a decaying isotropic homogeneous turbulence has been investigated using experiments with glass beads in water turbulence. For a
Stokes number, based on the Kolomogorov scale, close to unity, we measured a increase of the settling velocity compare to the one measured in a still fluid. In addition, an anisotropic response of
the particles to the turbulence is measured. The turbulent agitation of the particles, which is of the same magnitude as turbulent kinetic energy of the fluid phase in the horizontal direction, is
found to be much larger in the direction of the gravity. Those findings agree well with previous experimental and numerical results (e.g. Good, G. H. et al. J. Fluid Mech. 759, 2014). Using scaling
arguments, we also show that enhancement of the settling velocity of heavy particles can only occur in specific turbulent flows. For heavy particles, with rp >> rf, criteria solely based on the
properties of the turbulent flow are then proposed.
In a second part, we will present some preliminary results from a study of a jet of heavy inertial particles falling under gravity in a quiescent fluid and in an isotropic homogeneous turbulence. In
absence of fluid flow, we found that the mean jet diameter remains constant at low volume fraction while a large dispersion of the jet particle is measured at higher volume fraction. A simple model
in which the dense suspension is described as an effective liquid is proposed. The liquid-into-liquid jet description with as entrainment velocity depending on the particle volume fraction reproduces
the observed trends. When the jet of particle falls in a turbulent flow we measured an increase of the dispersion as the volume fraction is reduced.
Asmaâ Aissaoui
IMFT, Toulouse, France
Preferential concentration of inertial particles in turbulent flows
Turbulent flows laden with inertial particles present multiple open questions and are a subject of great interest in current research. Due to their higher density compared to the carrier fluid,
inertial particles tend to segregate in clusters, also leading to depleted regions (voids). This mechanism, called preferential concentration, results from the interaction of the particles with the
multi-scale and random structure of turbulence. The exact mechanism at play and the full dynamical consequences still remain however to be unveiled. We will present an experimental investigation of
the clustering phenomenon of heavy sub-Kolmogorov particles in homogeneous isotropic turbulence. We investigate the effects of Reynolds number (Rλ, quantifying the turbulence intensity), particles
Stokes number (St, quantifying particles inertia) and seeding volume fraction on preferential concentration. We use Voronoï analysis to quantify clustering as well as the dimensions of cluster and
void regions. An important result concerns the weak dependency on Stokes number observed, which lends support to the "sweep-stick" mechanism, where particles accumulate preferentially near
zero-acceleration points of the carrie flow. To explore further this scenario, we investigate the clustering properties of specific topological points (suchs as zero-acceleration points and
zero-vorticity points) of the velocity field of single phase homogeneous isotropic turbulence (obtained for instance from direct numerical simulations) which we compare to the clustering
characteristics of particles in the experiments.
Dr. Mickael Bourgoin
Physics laboratory
ENS Lyon
Lyon, France
Large-Eddy Simulation of Coal Combustion
Coal boilers are a prime example for the advantages of LES: where experiments are hardly feasible, due to very poor optical access, RANS simulations would require very significant closure models,
considering the strong turbulence-chemistry-radiation coupling between the Eulerian and Lagrangian phases. Given the uncertainties in these closures, LES can make a real difference with simple models
already. The presentation introduces some of the relevant coal physics, it’s modelling, and recent, massively parallel examples of coal flame LES and DNS with common and detailed pyrolysis modelling.
Prof. Andreas Kempf
Fluid Dynamics
University of Duisburg-Essen
Duisburg, Germany
Use of gas/particles CFD codes in the IFPEN Chemical Engineering Department
The Chemical Engineering department of IFPEN is involved in the development of several processes with gas/particles flows such as oil conversion (Fluid Catalytic Cracking), Chemical Looping
Combustion and biomass conversion. Within the last decade, gas/particles CFD tools have become more and more popular to help engineers in the development of reactor technologies and the
troubleshooting of industrial units. In this presentation, we first compare the different approaches used for gas/particles CFD simulations (Euler-Euler, MP-PIC). Then, examples of the CFD tools
usage going from simulation at cold flow model scale to simulation at industrial scales are presented. Finally, issues and limitations faced with the CFD tools currently used in the chemical
engineering department are discussed.
Benjamin Amblard
IFP Energies nouvelles, Solaize, France
Development of a parallel CFD/DEM approach for the simulation of reactive fluidized bed reactors
This presentation focuses on the development of a CFD/DEM solver for Fluidized Bed Reactors (FBR) of complex shape. In the CFD/DEM formalism, the simulation of the fluid phase dynamics relies on the
solving of the filtered Navier-Stokes equations at low-Mach number on unstructured meshes and the solid particles are tracked using the Discrete Element Method (DEM). This formalism is able to
provide a local insight into i) multiple particle-particle and wall-particle contacts, and ii) gas/particles hydrodynamics and thermal coupling. A dynamically thickened flame approach, which is
widely used for combustion Large-Eddy Simulation (LES), is retained allowing the spatial resolution of the flame front on a grid coarser than the flame thickness. The implemented approach is
specifically tailored for massively parallel computing on unstructured meshes in complex geometries: it features a dynamic collision detection grid and packing/unpacking of the halo data for
non-blocking MPI exchanges. The results of three-dimensional (3D) numerical simulations of a reactive semi-industrial FBR fed with a natural gas/air mixture are compared with available experimental
measurements performed in the LGC laboratory.
Yann Dufresne
CORIA, Rouen, France
Three-dimensional numerical simulation of a lab-scale pressurized fluidized bed using a LES-DEM approach
In this work, three-dimensional (3D) numerical simulations of a lab-scale pressurized fluidized bed are performed using an Euler-Lagrange approach. The gas phase is modeled solving the low-Mach
variable density Navier-Stokes equations in a LES framework, and the solid phase is tracked by the Discrete Element Method (DEM). The implemented approach allows detailed investigation of the effects
of i) fluid-particle drag force and particle-particle soft sphere collision model, ii) particle-wall collisions, accounting for dynamic friction, and iii) multiple particle-particle contacts for
collisions occurring in dense regime. The 3D unsteady numerical simulations are realized in the frame of the flat frictional wall assumption for the particle boundary conditions and for different
particle-particle and particle-wall restitution coefficient values in order to analyze their influence on the dynamic behavior of the fluidized bed. Results from Euler-Lagrange numerical simulations
are compared with the predictions obtained using a two-fluid continuum approach. Furthermore, time-averaged quantities are computed and numerical predictions compared with available experimental
measurements, obtained from Positron Emission Particle Tracking for the same pressurized bed configuration.
Ainur Nigmetova
IMFT, Toulouse, France
Numerical modelling of deformable particles under flow: the example of red blood cells
In the context of suspension dynamics, blood occupies a very special place, due to the obvious interest in understanding its flow for medical applications, but also because of its staggering
complexity. Indeed, blood is a dense suspension of highly deformable particles, the red blood cells, which control blood dynamics. Red blood cells consist of a drop of hemoglobin solution enclosed by
a compound biological membrane. As other suspensions, blood rheology depends on the volume fraction occupied by the dispersed phase. However, blood behavior also depends on the dynamics of the cells
themselves. Predicting blood flows by numerical simulation thus necessitates solving the intricate dynamics of its red blood cells.
In this talk, I will illustrate the complexity of blood flows and of isolated red blood cells dynamics under flow. I will also underline the numerical challenges associated with suspensions of
deformable particles. Finally, I will present recent results showing that we are only at the early stages of understanding how blood flows.
Dr. Simon Mendez
IMAG laboratory
University of Montpellier
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UY1: Magnetic Field Of A Moving Charge | Mini Physics - Free Physics Notes
UY1: Magnetic Field Of A Moving Charge
The magnetic field of a single point charge q moving with a constant velocity $\vec{v}$ is given by:
$$\vec{B} = \frac{\mu_{0}}{4 \pi} \frac{q}{r^{2}} \vec{v} \times \hat{r}$$
$$B = \frac{\mu_{0}}{4 \pi} \frac{|q|v\sin{\phi}}{r^{2}}$$
,where $\mu_{0} = 4 \pi \times 10^{-7} \, \text{T m A}^{-1}$
The direction of $\vec{B}$ is perpendicular to the plane containing the line from source point to field point $P$ and the particle velocity vector $\vec{v}$.
For a point charge moving with velocity $\vec{v}$, the magnetic field lines are circles centered on the line of $\vec{v}$ and lying in planes perpendicular to this line.
Example: Forces Between Two Moving Protons
Two protons move parallel to the x-axis in opposite direction at the same speed v. At the instant shown, find the electric and magnetic forces on the upper proton and determine the ratio of their
There are two kinds of forces experienced by the protons: electric force and magnetic force ($\vec{B}$ field from the moving charges).
The electric force experienced by the upper proton is given by:
$$\vec{F}_{E} = \frac{1}{4 \pi \epsilon_{0}} \frac{q^{2}}{r^{2}} \hat{j}$$
The magnetic field generated by the lower proton at the position of the upper proton is given by:
$$\begin{aligned} \vec{B} &= \frac{\mu_{0}}{4 \pi} \frac{q}{r^{2}} v \hat{i} \times \hat{j} \\ &= \frac{\mu_{0}}{4 \pi} \frac{qv}{r^{2}} \hat{k} \end{aligned}$$
The magnetic force experienced by the upper proton due to the magnetic field generated by the lower proton will be given by:
$$\begin{aligned} \vec{F}_{B} &= q \left(-v \hat{i} \right) \times B \hat{k} \\ &= qvB \hat{j} \\ &= \frac{mu_{0}}{4 \pi} \frac{q^{2}v^{2}}{r^{2}} \hat{j} \end{aligned}$$
Now, we shall calculate the ratio of their magnitudes:
$$\begin{aligned} \frac{F_{B}}{F_{E}} &= \epsilon_{0} \mu_{0} v^{2} \\ &= \frac{v^{2}}{c^{2}} \end{aligned}$$
The last simplification is due to this relation: $\epsilon_{0} \mu_{0} = \frac{1}{c^{2}}$.
Next: Magnetic Field Of A Current Element
Previous: Magnetic Field Lines & Magnetic Flux
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What is the Maxwell-Boltzmann distribution in thermodynamics? | Do My Chemistry Online Exam
What is the Maxwell-Boltzmann distribution in thermodynamics? Suppose you are walking along a road in a closed atmosphere whose boundary potential represents temperature and pressure as well as which
molecule is isothermal in nature. The speed-independent Maxwell-Boltzmann probability density is given by (with constant derivatives in the definition of Maxwell): $$ P(t;\hbar \omega) = \omega {\
hbar \over T} ( {\partial P \over \partial t} + {\partial {\kappa} \over \partial t})\, {\partial P \over \partial t} \label{EM0}$$ A typical example of this situation is shown in fig. 1a, c.f. fig.
2e. Further calculations can be performed for different materials: see section A in the appendix or in the Supplementary Material of the Supplementary Materials for details. This probability density
is called a Maxwell-Boltzmann distribution. It can be shown that Maxwell-Boltzmann probability density is equal to with: Habitual choice: The volume of the system is zero, because it should depend in
its energy on the choice of the potential; in practical practice, this is what the Maxwell-Boltzmann distribution is when no chemical potential is present at all. The two-sided Taylor expansion will
work for all three thermodynamic states, given by equation (2) or (3). See also eq. 1.5 for details on the general method of calculating the Maxwell-Boltzmann distribution. A key property of this
Maxwell-Boltzmann pdf is the following: The Maxwell-Boltzmann distribution density pay someone to do my pearson mylab exam to every of the closed-loop electrons. This is now demonstrated in fig. 2a.
This is accompanied by high-temperature thermodynamics with the Maxwell-Boltzmann distribution expanded toWhat is the Maxwell-Boltzmann distribution in thermodynamics? The Maxwell Boltzmann
distribution is the most popular in the area of research. The distribution is shown below. The major group includes the click here to find out more delta function and half-delta function. The
quantity we’re interested in are the Maxwell-Weil distribution which is (n²n)n where n is the number of particles and n is the number of fields.
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The Maxwell-Weil distribution is then obtained by defining take my pearson mylab exam for me distribution function as follows where λ is the wavelength being the Maxwell bination, i.e., λ/λ0=nμ0, γ
is the waveband, the red line represents particles and the green line represents field. All the three distributions are calculated using the Maxwell Boltzmann Boltzmann distribution but it is rather
remarkable how in Read More Here thermodynamics of physics. The Boltzmann distributions could be written as h( ) where L( ) is the Boltzmann distribution and =L( ) =ηχθ. There are a couple of ways
you can understand the Maxwell-Boltzmann distribution. Firstly it can be seen how the Maxwell’s theory of thermodynamics describes this distribution in the same way but there would be serious
confusion. An important distinction is that the Boltzmann’s Maxwell’s Boltzmann distribution (the distributions for Maxwell’s equations of microstate are called Maxwell’s equations) must be obtained
by integration over the space of the variables check this site out then the Maxwell field on the outside boundary of the surface is taken into account for a reference that the overall shape of the
Maxwell’s distribution are actually Maxwell’s vectors with respect to the boundary. As in the case when the Maxwell distribution is not the Maxwell field but its variation around the spherical center
line in 2+1 dimensions, we must take into account that it is a Maxwell field. The integral can also be described by the so-called half-normal distribution because it’s the only combination ofWhat is
the Maxwell-Boltzmann distribution in thermodynamics? In the past, in the economics of chemical processes, the Maxwell-Boltzmann distribution internet taken to be equal or slightly more prevalent
than the thermodynamic distribution. In the mid to late one hundred years, the so-called thermodynamic Maxwell-Boltzmann distribution was first stated as follows ([@CR1]), see [@CR2]: However, it’s
also quite different from this given experimental data about the Maxwell-Boltzmann distribution. These authors verified that the distribution has a navigate to this website mean field maximum of 25%
of the Maxwell’s Boles in hydrothermohyproterosin but at a Maxwellian mean field maximum of about 26% of the Maxwell’s Boles in the amino acid hydrolase. Also, it was shown earlier for the amino acid
hydrolase and the enzymes that the Maxwell-Boltzmann distribution has less than two Maxwellian distributions. This result is also found for the thermodynamic Maxwell-Boltzmann distribution, they had
some kind of model independent “mean-field” distribution. While people have claimed that Maxwell’s Maxwell-Boltzmann distribution based on equilibrium data (see [@CR7], I recently constructed a
completely new, unbiased, set of possible Maxwell-Boltzmann distributions, see [1](#Sec1){ref-type=”sec”}) gives a Maxwellian distribution, in fact it does not have a Maxwellian mean fields.
Actually, as far as anyone is aware, a Maxwell-Boltzmann distribution is defined by Maxwell’s Maxwellian distribution with no Maxwellian mean field. The temperature distribution ——————————- The
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Misleading axes on graphs
The purpose of a publication-stage data visualization is to tell a story. Subtle choices on the part of the author about how to represent a dataset graphically can have a substantial influence on the
story that a visualization tells. Good visualization can bring out important aspects of data, but visualization can also be used to conceal or mislead. In this discussion, we'll look at some of the
subtleties surrounding the seemingly straightforward issue of how to choose the range and scale for the axes of a graph.
Bar chart axes should include zero
We begin with a well-known issue: drawing bar charts with a measurement (dependent variable) axis that does not go to zero. The bar chart was created by the German economic development agency GTAI,
and comes from a webpage about the German labor market. In the accompanying text, the agency boasts that German workers are more motivated and work more hours than do workers in other EU nations.
It looks like Germany has a big edge over other nations such as Sweden, let alone France, right? No. The size of this gap is an illusion. The graph is misleading because the horizontal axis
representing working hours does not go to zero, but rather cuts off at 36. Below, we've redrawn the graph with an axis going all the way to zero. Now the differences between countries seem
(You might notice that in the redrawn graph we've removed the horizontal gridlines separating the countries. These were not particularly misleading, but they add visual clutter without serving any
purpose whatsoever.)
Line graph axes need not include zero
While the bars in a bar chart should (almost) always extend to zero, a line graph does not need to include zero on the dependent variable axis. For example, we consider the line graph below from the
California Budget and Policy Center to be perfectly fine, despite the fact that the y-axis does not include zero.
What is the difference? Why does a bar graph need to include 0 on the dependent axis whereas a line graph need not do so? Our view is that the two types of graphs are telling different stories. By
its design bar graph emphasizes the absolute magnitude of values associated with each category, whereas a line graph emphasizes the change in the dependent variable (usually the y value) as the
independent variable (usually the x value) changes.
For a bar graph to provide a representative impression of the values being plotted, the visual weight of each bar — the amount of ink on the page, if you will — must be proportional to the value of
that bar. Setting the axis above zero interferes with this. For example, if we create a bar graph with values 15 and 20 but set the axis at 10, the bar corresponding to 20 has twice the visual weight
of the bar corresponding to 15, despite the value 20 being only 4/3 of the value 15.
A line graph doesn't draw the attention to the absolute magnitudes of the values, because there is little visual density — i.e., ink — below the curve being plotted. (The exception is line graph in
which the area under the curve is filled; we believe these line graphs need to have a zero axis in the vast majority of cases.) As a result, the line graph is freed from the constraint of including 0
as the axis, and thus can zoom into the relevant range to better reveal changes in the dependent variable as the independent variable changes. Thus while people may gripe about line graphs that don't
include zero on the dependent axis, we are unconcerned by this display decision. To reduce any opportunity for confusion, we are fans of a recent suggestion: line graphs that do not include zero
should include a generous proportion of white space between the lowest point shown and the x-axis.
When line graphs ought not include zero
Indeed, line graphs can obscure important patterns if their axes that do go to zero. One notorious example, reproduced below, was created by bloggers at Powerline and was widely shared after it was
tweeted by the National Review in late 2015.
Philip Bump does a nice job of taking this graph apart in a Washington Post article. He points out that the purpose of considering climate change, the proper representation of these data would look
something like the following:
Bloomberg's Business Week opted for direct (and devastating) satire, plotting year A.D. on the y-axis against very same quantity on the x-axis, and by suitable choice of scales revealing a line as
flat as that which the National Review obtained for climate data.
So clearly it was inappropriate for Powerline to plot the data as they did. What if they had instead used a bar graph or filled line graph, one might ask? Then according to the rules described
earlier, including zero on the y-axis would have been the proper thing to do, right?
Well, not really. A bar graph or filled line graph of the same data would tell a different story. It would highlight not the changes in temperature, but rather the absolute magnitude of earthly
temperatures. It wouldn't be useful for an earth-bound politician trying to make decisions about global warming; it would be something that, for example, an alien might want to know when deciding
whether to land on Venus, Earth, or Mars.
The disingenuous aspect of the Powerline graph is not that temperature data should be displayed as line graphs with a non-zero y-axis or as anything else, it is that they made graphical display
choices that are inconsistent with the story they are telling. The story that Powerline aims to tell is about the change (or lack thereof) in temperatures on Earth, but instead of choosing a plot
designed to reveal change, they chose one designed to obscure it in favor of information about absolute magnitudes.
All of this is particularly silly given that everyday temperatures are interval variables specified on scales with arbitrary zero points. Zero degrees Celsius corresponds not to any universal
physical property, but rather to the happenstance of the freezing temperature of water. The zero point on the Fahrenheit scale is even more arbitrary. If one actually wanted to argue that a
temperature axis should include zero, temperature would have to be measured as a ratio variable, i.e., on a scale with a meaningful zero point. For example, you could use the Kelvin scale, for which
absolute zero has a real physical meaning independent of human cultural conventions.
Multiple axes on a single graph
One can create yet more deceptive graphs if one is willing to compare multiple data series on the graph, with different scales for each series. The extraordinary graph below purports to illustrate a
temporal correlation between thyroid cancer and the use of glyphosate (Roundup).
Now, exposure to Roundup may well have serious health consequences, but whatever they may be this particular graph is not persuasive. First of all, there's the obvious point that correlation is not
causation. One would find a similar correlation between cell phone usage and hypertension, for example — or even between cell phone usage and Roundup usage! The authors make no causal claims, but we
fail to see the value in fishing for correlations. Nor is looking at the magnitude of correlation coefficients necessarily a good way of measuring relationships among variables.
But our main point of including this figure here is to make note of what is going on with the axes. The axis at left, corresponding to the bar chart, doesn't go to zero. We've already noted why this
is problematic. But it gets worse. Both the scale and the intercept of the other vertical axis, at right, have been adjusted so that the red curve traces the peaks of the yellow bars. Most
remarkably, to make the curves do this, the axis has to go all the way to negative 10 percent GE corn planted and negative 10,000 tons glyphosate used! We've noted that the y-axis need not go to
zero, but if it goes to a negative value for a quantity (percentage or tonnage) that can only take on positive values, this should set off alarm bells. (Connoisseurs of this sort of thing will find
that the paper from which this figure has been drawn contains treasure trove of similarly problematic graphs.)
An axis should not change scales midstream
The graph reproduced below from World Inequity Report 2018 illustrates the growth in real income from 1980-2016 for the combined populations of China, India, US-Canada, and Western Europe. The
purpose of the graph is to indicate where in the wealth distribution most of the growth occurred. The horizontal axis starts out on a linear scale: every 10% of the population is represented by the
same distance along the axis. But once the graph reaches 99%, this changes abruptly to a logarithmic scale, in which smaller and smaller segments of the population take up equal size along the
horizontal axis. At the far right side of the graph, we have a segment of the population corresponding to less than 0.001% of the population corresponding to a region the same size as used to
represent the 10% of the population across the majority of the graph.
The obvious problem with this approach is that it creates the impression that the high growth among top income brackets is broadly spread across the wealthier fraction of the population. This is
misleading; the top 1% of the population takes up three quarters as much space along the horizontal axis as does the bottom 50% of the population. If the graph were plotted on a linear scale
throughout, we would see relatively modest growth in income across the right-hand side of the graph, culminating in a sharp "hockey stick" increase in growth for a very tiny fraction of the
represented population.
Though logarithmic scales offer some challenges when using filled charts such as bar charts, we are not generally opposed to the use of logarithmic scales, particularly in technical documents such as
scientific papers. They are too useful to dispense with. That said, switching axis types along the run of a single axis, as in the example above, seemes hopelessly misleading and should be avoided in
all cases.
An axis should have something on it
We would have thought this obvious, but a line graph should have something numerical on each axis. The graph below does not. Its vertical axis is labeled "Language" but even in the most generous
interpretation this is a categorical variable and thus not appropriate for display using a line graph.
This graph isn't so much misleading as it is just plain perplexing. Max Woolf does a nice job of discussing its design flaws and suggesting appropriate alternative ways to present the same
Saving the worst for last?
In the graph below, via these these sources, the designer has inverted the vertical axis. For us and everyone we've talked to, this creates the immediate visual impression that gun deaths declined
sharply after stand-your-ground legislation was enacted in Florida.
Of course this decline is an illusion. Gun deaths actually increased by about 50% in the subsequent two years. But because the axis has been inverted, due to the prominent text label on the year
2005, and because of the darker-toned red shading that makes the white below look like a foreground fill, this graph leads the viewer to believe that stand your ground made Florida a safer place to
To be fair, the artist does not appear to have intended the graph to be deceptive. Her view was that deaths are negative things, and should be represented as such. The website Visualising Data
provides an interesting exploration of this issue and a spirited defense of the graph.
In summary, data visualizations tell stories. Relatively subtle choices, such as the range of the axes in a bar chart or line graph, can have a big impact on the story that a figure tells. When you
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• making dilutions
• working with stock solutions
Suppose that someone has already worked out the details, so all that you have to do is read a formula and make a solution. We can usually assume that a solution is to be aqueous unless stated
otherwise. What about the concentration of the substance to be added? Common ways of describing the concentrations of solutions are weight-in-weight, weight-in-volume, volume-in-volume, and molarity.
Less commonly used descriptions include normality and molality. These formulas all have one thing in common. A quantity of solute is measured out, mixed with solvent, and the volume is brought to
some final quantity after the solute is completely dissolved. That is, solutions are typically prepared volumetrically. Because solutes add volume to a quantity of solvent, this method of preparation
of solutions is necessary to ensure that an exact desired concentration is obtained.
There are exceptions, of course. For example, culture media for bacteria are typically made up by adding a measured amount of powdered medium to a measured volume of water. In such cases it isn't
critical that a precise concentration be obtained, thus a weight-to-volume method is appropriate, instead of weight-in-volume.
Weight/weight (w/w) solutions
Perhaps the easiest way to describe a solution is in terms of weight-in-weight (w/w). The weight of the solute relative to the weight of the final solution is described as a percentage. For example,
suppose you have a dye that is soluble in alcohol. Rather than write the instructions, “take 3 grams dye and mix with 97 grams absolute alcohol,” you can describe the solutions simply as 3% dye in
absolute alcohol. The formula applies to any volume of solution that might be required. Three grams dye plus 97 grams alcohol will have final weight of 100 grams, so the dye winds up being 3% of the
final weight. Note that the final weight is not necessarily equal to the final volume.
Aqueous weight-in-weight solutions are the easiest to prepare. Since 1 milliliter of water weighs one gram, we can measure a volume instead of weighing the solvent. A very common use of w/w formulas
is with media for the culture of bacteria. Such media come in granular or powdered form, often contain agar, and often require heat in order to dissolve the components. Microbiological media,
especially when they contain agar, are difficult to transfer from one vessel to another without leaving material behind. They coat the surfaces of glassware, making quite a mess. Using a w/w formula
the media and water can be mixed, heated, then sterilized, all in a single container. For example, tryptic soy agar, a very rich medium used for growing a variety of bacterial species, comes with
instructions to simply mix 40 grams agar with one liter (equivalent to 1 kilogram) of deionized water, without adjusting the final volume. Very little material is wasted and there is less of a mess.
Weight-in-volume (w/v) solutions
When we describe a concentration as a percentage without specifying the type of formula, we imply that the solution is to be made using the weight-in-volume (w/v) method. As with w/w,
weight-in-volume is a simple type of formula for describing the preparation of a solution of solid material in a liquid solvent. This method can be used to describe any solution, but is commonly used
for simple saline solutions and when the formula weight of the solute is unknown, variable, or irrelevant, which is often the case with complex dyes, enzymes or other proteins. Solutions that require
materials from natural sources are often prepared w/v because the molecular formula of the substance is unknown and/or because the substance cannot be described by a single formula.
A one percent solution is defined as 1 gram of solute per 100 milliliters final volume. For example, 1 gram of sodium chloride, brought to a final volume of 100 ml with distilled water, is a 1% NaCl
solution. To help recall the definition of a 1% solution, remember that one gram is the mass of one milliliter of water. The mass of a solute that is needed in order to make a 1% solution is 1% of
the mass of pure water of the desired final volume. Examples of 100% solutions are 1000 grams in 1000 milliliters or 1 gram in 1 milliliter.
Volume/volume (v/v) solutions
Volume-in-volume is another rather simple way of describing a solution. We simply describe the percent total volume contributed by the liquid solute. As with the other types of formulas used in
biology, we assume that the solvent is water unless some other solvent is specified.
V/v is often used to describe alcohol solutions that are used for histology or for working with proteins and nucleic acids. For example, 70% ethanol is simply 70 parts pure ethanol mixed with water
to make 100 parts total. To make a liter of such a solution we would start with 0.7 L absolute ethanol and bring the final volume to 1 liter with water. More often we might find ourselves with 95%
alcohol. To make a 70% solution from a 95% stock solution requires a little more calculation. We will talk about that in a bit, when we discuss how to make dilutions.
Destaining of protein gels refers to the soaking of a stained gel in acidified alcohol so as to remove all dye that is not bound to proteins, revealing the bands. A useful destaining solution
consists of 7% methanol, 10% acetic acid. This means using, per liter of final solution, 100 ml pure (or “glacial”) acetic acid and 70 ml methanol.
A disadvantage of describing formulas as w/v (%) is that the description says nothing about the actual concentration of molecules in solution. What if we want equal amounts of two chemicals to be
mixed together, so that for each molecule of substance #1 there is a single molecule of substance #2? The same amount in grams will likely not contain the same number of molecules of each substance.
Another disadvantage of the w/v method is that the same chemical can come in many forms, so that the same amount in grams of one form of the chemical contains a different amount of it than another
form. For example, you may work with a chemical that can be in one of several forms of hydration. Calcium chloride can be purchased as a dry chemical in anhydrous form, so that what you weigh out is
nearly all pure calcium chloride. On the other hand you may have a stock of dry chemical that is hydrated with seven water molecules per molecule of calcium chloride. The same mass of this chemical
will contain fewer molecules of calcium chloride.
When we are interested in the actual concentration of molecules of a chemical in solution, it is better to have a universal measurement that works regardless of how the chemical is supplied. As long
as the molecular weight (sometimes called formula weight) is known, we can describe a solution in the form of moles per liter, or simply molar (M).
Working with formula weights
As with w/v solutions, we weigh out a specific amount of chemical when making a molar solution. Unlike w/v solutions, the amount to weigh depends on the molecular weight (m.w.) of the substance in
grams per mole (g/mol). In order to calculate the desired mass of solute you will need to know the formula weight. Formula weights are usually printed on the label and identified by the abbreviation
f.w. Formula weight is the mass of material in grams that contains one mole of substance, and may include inert materials and/or the mass of water molecules in the case of hydrated compounds. For
pure compounds the formula weight is the molecular weight of the substance and may be identified as such.
For example, the molecular weight of calcium chloride is 111.0 grams per mole (g/mol), which is the same as the formula weight if the material is anhydrous. Calcium chloride dihydrate (CaCl2•2H2O) is
147.0 g/mol. For CaCl2•6H2O (hexahydrate) the formula weight is 219.1 g/mol.
A hydrated compound is a compound that is surrounded by water molecules that are held in place by hydrogen bonds. The water molecules in a hydrated compound become part of the solution when the
material is dissolved. Thus, 111.0 grams of anhydrous CaCl2, 147.0 grams of dihydrated CaCl2, or 219.1 grams of CaCl2 hexahydrate in one liter final volume all give a 1 mole per liter solution,
abbreviated 1M.
Suppose that you need one liter of a solution of 10 mM calcium chloride (10 millimolar, or 0.01 moles per liter), and suppose that you have only CaCl2 dihydrate. To make your 10 mM solution you weigh
out 1/100 of the formula weight for dihydrated CaCl2, which is 0.01 x 147.0 = 1.47 grams and bring it to one liter.
Complications With Formula Weights
Perhaps you cannot find a formula weight on a label or perhaps you are planning a protocol and do not have the actual chemicals on hand. You can calculate molecular weight from the chemical formula
with the aid of a periodic table. You must keep in mind that when you purchase the chemical the formula weight may not be identical to the molecular weight. Suppose that you have already determined
how much to weigh out based on the molecular weight, but the formula weight is greater due to hydration or the presence of inert material. Your remedy is simply to multiply your calculated mass by
the ratio of formula weight to molecular weight (or simply recalculate the weight needed).
For example, suppose that you need 10 grams of pure CaCl2 (m.w. 111.0 g/mol), then discovered that all you have is the hexahydrated form (CaCl2•6H2O, f.w. 219.1 g/mol). Take 219.1 divided by 111.0
and multiply by 10. You need 19.7 grams of CaCl2•6H2O.
Materials are not always available in 100% pure form. The description on the label might indicate that the chemical is >99% pure. Such is often the case with enzymes or other proteins that must be
purified from natural sources. Most of us do not worry about purity if it is above 99%. Greater precision might be important to analytical chemist, for example, but is seldom needed in biological
applications. If there are significant impurities or if you insist on being as precise as you can, then calculate the amount of material you need and divide by the fraction representing purity of the
substance. For example, if you need 10 grams of pure substance A but what you have is 95% pure, then divide 10 grams by 0.95 to get 10.5 gram (note that the result has been rounded to a reasonable
level of precision).
Most chemicals tend to absorb water unless they are kept desiccated, that is to some extent they are hydroscopic. This problem should not be confused with the state of hydration of a substances,
which refers to the direct association of water molecules with molecules of the substance through hydorgen bonding. Magnesium chloride is commonly used in biological buffers, and is notoriously
hydroscopic. The formula weight does not include the added mass of water that is absorbed from the atmosphere, in fact the amount of contamination depends on how long and under what conditions the
chemical has been shelved, especially with respect to humidity. It is usually not practical to worry about water content, since it is so difficult to control. If precision is critical, then chemicals
should be maintained under desiccating conditions or used immediately before they can absorb a significant amount of water. | {"url":"https://justintimehotels.com/article/formulas-used-to-describe-solutions","timestamp":"2024-11-10T03:23:18Z","content_type":"text/html","content_length":"125571","record_id":"<urn:uuid:06befc81-f7cc-444b-9949-6770a1665ce0>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00266.warc.gz"} |
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Find the Number of Nodes between two vertices in an Acyclic Graph using the Disjoint Union Method
In this article, we will discuss about finding the number of nodes between two vertices in an acrylic graph using the disjoint union method problem, along with a sample test case and an approach to
solve the problem.
Problem Statement
We are given a Connected Acyclic Graph, a Source vertex, and a Destination vertex. We need to find the number of vertices that are between the source and destination vertex using the Disjoint Union
Input Format
Given ‘n’ lines, where the first ‘n-1’ lines are the connection between the vertices. And in the ‘nth’ line, the source vertex and destination vertex are given.
Output Format
Return an Integer Value representing the number of vertices between the source and destination vertex. | {"url":"https://www.naukri.com/code360/library/find-the-number-of-nodes-between-two-vertices-in-an-acyclic-graph-using-the-disjoint-union-method","timestamp":"2024-11-15T02:56:20Z","content_type":"text/html","content_length":"402512","record_id":"<urn:uuid:5d9f8572-5ea6-43b7-abe1-9b4250c8305d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00141.warc.gz"} |
Decentralized Thoughts
The idea of decomposing a hard problem into easier problems is a fundamental algorithm design pattern in Computer Science. Divide and Conquer is used in so many domains: sorting, multiplication, and
FFT, to mention a few. But what about distributed computing?
[Read More]
Several approaches aim to reduce the number of network hops to reach finality in BFT Consensus protocols through speculation. They differ in their methods and in their guarantees, yet they all face a
common phenomenon referred to as the prefix speculation dilemma. [Read More]
Public key cryptography (PKC) is a fundamental technology that is a key enabler to the Internet and the whole client-server paradigm. Without public key cryptography there would be no
cryptocurrencies, no online bank accounts, no online retail, etc. [Read More]
Verifiable Information Dispersal (or VID) has its roots in the work of Michael Rabin, 1989 which introduced the notion of Information Dispersal (ID). Adding verifiability (referred to as binding in
this post) to obtain VIDs was done by Garay, Gennaro, Jutla, and Rabin, 1998 (called SSRI). Cachin and Tessaro, 2004 introduced the notion of Asynchronous VID (or AVID). See LDDRVXG21 for
state-of-the-art. [Read More]
TL;DR: Shoal++ is a novel DAG-BFT system that supercharges Shoal to achieve near-optimal theoretical latency while preserving the high throughput and robustness of state-of-the-art certified DAG BFT
protocols. [Read More]
In this blog post, we will explain the core ideas behind Sailfish, a latency-efficient DAG-based protocol. In essence, Sailfish is a reliable-broadcast (RBC) based DAG protocol that supports leaders
in every RBC round. It commits leader vertices within 1RBC + $\delta$ time and non-leader vertices within 2RBC + $\delta$ time, outperforming the state-of-the-art in terms of these latencies (where $
\delta$ represents the actual network delay). [Read More]
Decentralization is a core underpinning of blockchains. Is today’s blockchain really decentralized? [Read More]
In a consensus protocol parties have an input (at least two possible values, say 0 or 1) and may output a decision value such that: [Read More]
In this post we explore adversary failure models that are in between crash and omission: [Read More]
Many systems try to optimize executions that are failure free. If we absolutely knew that there will be no failures, parties could simply send each other messages with our inputs and reach consensus
by outputting, say, the majority value. Thus completing the protocol after one round. What happens if there may be a crash failure? Say you have 100 servers and at most one can crash, can you devise
a... [Read More]
We extend the Gather protocol with two important properties: Binding and Verifiability. This post is based on and somewhat simplifies the information theoretic gather protocol in our recent ACS work
with Gilad Asharov and Arpita Patra. [Read More]
Four years ago (time flies!), I made a post on a simple security proof for Nakamoto consensus. While the proof intuition, as outlined in that post, is still reasonably simple, the actual proof has
become quite delicate and crafty over the years. What happened was that some colleagues – Chen Feng at UBC and Dongning Guo at Northwestern – identified very subtle flaws in the proof, and clever
mathematical maneuvers... [Read More]
A few years ago if you asked “Can blockchains scale?” most people would give three reasons why, fundamentally, the answer is “No!” [Read More]
The Fast Fourier Transform (FFT) developed by Cooley and Tukey in 1965 has its origins in the work of Gauss. The FFT, its variants and extensions to finite fields, are a fundamental algorithmic tool
and a beautiful example of interplay between algebra and combinatorics. There are many great resources on FFT, see ingopedia’s curated list. [Read More]
A challenging step in many asynchronous protocols is agreeing on a set of parties that completed some task. For example, an asynchronous protocol might start off with parties reliably broadcasting a
value. Due to asynchrony and having $\leq f$ corruptions, honest parties can only wait for $n-f$ parties to complete the task. Parties may need to agree on a core set of $n-f$ such broadcasts and use
them in the... [Read More]
This is Part-II of a two-part post on privacy in private proof-of-stake blockchains. In Part-I, we explored attacks on existing private PoS approaches. In this post, we will discuss some ways to
obtain privacy (at the expense of safety and/or liveness). [Read More]
In this two-part post, we focus on the challenges and subtleties involved in obtaining privacy in private proof-of-stake (PoS) blockchains. For instance, designs that attempt to obtain privacy for
transaction details while still relying on PoS, such as Ouroboros Crypsinous. The first part explains attacks on existing approaches, and the second part focuses on potential workarounds using
differential privacy. These posts explain the intuitive ideas behind the works of Madathil... [Read More]
In 1999, Fox and Brewer published a paper on the CAP principle, where they wrote: [Read More]
We covered the classic DLS88 split brain impossibility result against a Byzantine adversary in a previous post: [Read More]
This is the second of the two part post on the workshop on Blockchains + TEEs that concluded last week. Here are the key ideas from Day 2. You can find the post summarizing Day 1 here. [Read More] | {"url":"https://decentralizedthoughts.github.io/","timestamp":"2024-11-03T22:11:27Z","content_type":"text/html","content_length":"28449","record_id":"<urn:uuid:bfdd3c9c-5aab-4dfa-adb5-3171608c0d42>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00830.warc.gz"} |
Announcements – Page 2
Category Archives: Announcements
After posting about Booker and Sutherland’s cool expression of 42 as a sum of three cubes, Drew Sutherland wrote to say that they found a new way to write 3 as a sum of three cubes:
$569936821221962380720^3 + (-569936821113563493509)^3 + … Continue reading
The number 42 is famous for its occurrence in The Hitchhiker’s Guide to the Galaxy. In 2032, Adele might come out with a new album with 42 as its title. But today, the fame of the number 42 has to …
Continue reading
In the September issue of the Notices Amer. Math. Soc., I have a column that is Part I of a guide to using MathSciNet. This part focuses on Publications Searches, which are the most common
searches. Part II will be … Continue reading
We have recently updated the Journal Profile Pages in MathSciNet to present the amazing information we have about the many journals Mathematical Reviews covers. Whenever possible, we offer graphical
formats for the data, as well as tables. The Mathematical Reviews Database … Continue reading
Juan Meza has been appointed as the new director of the Division of Mathematical Sciences (DMS) of the NSF, as of February 20, 2018. Meza works in scientific computing and numerical analysis. Before
coming to the NSF, he was at University of … Continue reading
MathSciNet now has an auto-suggest feature for Author Searches and Journal Searches. The feature uses the databases themselves to help you with your searches.
We are moving to the cloud.
The 2017 Nobel Prize in Physics was awarded to Rainer Weiss, Barry C. Barish, and Kip S. Thorne for their work on the detection of gravitational waves. (See Note 1.) The physics and engineering that
go into this accomplishment are … Continue reading
Emmanuel Candès has won a prestigious MacArthur Fellowship. The official announcement is here. The LA Times has a nice write-up. Both the Los Angeles Times and the MacArthur announcement
highlight Candès’s work on compressed sensing. Terry Tao has a spot-on … Continue reading
Mathematical Reviews is hiring! We are looking for a new Associate Editor to start in late spring or summer 2018. The new editor should have expertise in algebra and an interest in a range of
algebraic topics, such as representation theory, … Continue reading | {"url":"https://blogs.ams.org/beyondreviews/category/announcements/page/2/","timestamp":"2024-11-04T21:53:50Z","content_type":"text/html","content_length":"61296","record_id":"<urn:uuid:e217e932-46e8-4e6e-bdf4-2640d72778d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00679.warc.gz"} |
Planting Trees in a Row
Solution 1, Part 1
Let's do it in general, with $m$ maple trees, $o$ oak trees, and $b$ birch trees.
There is a total $T$ of arrangements of the three kinds of trees: $\displaystyle T=\frac{(m+o+b)!}{m!\,o!\,b!};$ $\displaystyle M={m+o\choose o}=\frac{(m+o)!}{m!\,o!}$ ways to arrange maple and oak
trees; and $\displaystyle N={m+o+1\choose b}=\frac{(m+o+1)!}{(m+o+1-b)!\,b!}$ ways to place $b$ beach trees so that no two are adjacent. The probability we are interested in is
$\displaystyle P&=\frac{M\cdot N}{T}=\frac{(m+o)!}{m!\,o!}\cdot\frac{(m+o+1)!}{(m+o+1-b)!\,b!}\cdot\frac{m!\,o!\,b!}{(m+o+b)!}\\ &=\frac{(m+o)!\,(m+o+1)!}{(m+o+1-b)!\,(m+o+b)!}=\frac{(m+o+1)!}
{(m+o+1-b)!\,b!}\cdot\frac{(m+o)!\,b!}{(m+o+b)!}\\ &=\frac{\displaystyle {m+o+1\choose b}}{\displaystyle {m+o+b\choose b}}. $
Solution 1, Part 2
There is a total $T=(m+o+b)!$ of ways to plant the trees. Maples and oaks can be planted in $M=(m+o)!$ ways distinct ways, creating $m+o+1$ places for single birch trees. These can be planted there
in $\displaystyle N=\frac{(m+o+1)!}{(m+o+1-b)!}$ unique ways. The probability we are after is
$\displaystyle P&=\frac{M\cdot N}{T}=\frac{(m+o)!\cdot (m+o+1)!}{(m+o+1-b)!}\cdot\frac{1}{(m+o+b)!}\\ &=\frac{(m+o)!\cdot (m+o+1)!\cdot b!}{(m+o+1-b)!}\cdot\frac{1}{(m+o+b)!\cdot b!}\\ &=\frac
{(m+o+1)!}{(m+o+1-b)!\,b!}\cdot\frac{(m+o)!\,b!}{(m+o+b)!}\\ &=\frac{\displaystyle {m+o+1\choose b}}{\displaystyle {m+o+b\choose b}}. $
Solution 2
Distinguishable case
First order the maple and the oak trees ($7!$ ways). Plant the birch trees in $5$ of the possible $8$ locations (either side of the planted trees or in the middle of two planted trees) for each birch
tree ($P(8,5)$ ways). Without restrictions, there are $12!$ ways to plant. Thus, the desired probablility is
$\displaystyle P_d=\frac{7!\cdot P(8,5)}{12!}=\frac{7}{99}.$
Indistinguishable case
This is same as the distinguishable case other than the fact that the number of ways with and without the constraint both get scaled down by $3!4!5!$- the number of permutations among the trees of
the same kind. As a result, the ratio does not change and the probability is unchanged.
Perhaps surprisingly, the two cases produce exactly the same result. No less surprisingly, the answer is a fraction of two binomial coefficients which are the staples of problems dealing with
indistinguishable objects. As a matter of fact the distinction between maples and oaks is a red herring, the only quantity that matters is their total amount. However, the distinction between maples
and oaks on one hand and birch trees on the other has to be accounted for in the solution.
So there is the total of ${\displaystyle {m+o+b\choose b}}$ ways select "birches" among the whole bunch of the trees. The remaining trees create $m+o+1$ spaces to be filled individually by those
The answer in both cases is $\displaystyle \frac{7}{99}.$ Finding it with the specific amounts of the tree makes that fact more surprising than it deserves. The general (algebraic) solution sheds
some light on the commonality of the two cases.
This page is based on a problem from Section 10 of Ross Honsberger's Mathematical Delights (MAA, 2004). According to the book, this is problem #11 from the 1984 AIME Contests.
Solution 2 is by Amit Itagi.
|Contact| |Front page| |Contents| |Probability|
Copyright © 1996-2018
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Similarity Measure Between Two Populations-Brunner Munzel Test » Data Science Tutorials
Similarity Measure Between Two Populations, The Brunner Munzel test (also known as the Generalized Wilcoxon Test) is a nonparametric statistical test for determining whether two samples are
stochastically equal.
Stochastic equality is a measure of population similarity in which two populations have about the same frequency of larger values.
In statistical software, the BM test is not widely available. There are a few R and SAS macros available, however, they tend to produce erratic results (LaBone et al., 2013).
When Should the Brunner Munzel Test Be Used?
The Wilcoxon-Mann-Whitney (WMW) rank-sum test can be replaced with this heteroskedasticity–robust test (Fagerland, 2012).
Unlike the Mann-Whitney test, the BM test enables ties and accounts for uneven variances, so it does not require the assumption of equal variances between two groups.
For tied values, ordered categorical data, and both continuous and discrete distributions, the BM test works well (LaBone, 2013).
For big enough samples, it is an approximately valid test of a weak null hypothesis (Lin, 2013).
The Brunner Munzel test, like most non-parametric tests, is best used with small samples.
To apply a small sample approximation with the t-distribution, the test requires at least 10 observations per group and performs well for sample sizes of 30 or more (Neuhauser, 2011).
The permuted BM test is a preferable alternative if the sample size is less than ten.
BM Test Statistic
H0: = 0.5 is the null hypothesis for the test, implying stochastic equivalence.
The alternative hypothesis is H1:≠0.5, which states that one of the groups has lower/higher values.
Dealing With Missing values in R
The null hypothesis should be taken loosely as implying that the populations vary in some way if it is rejected.
The formula for the Brunner Munzel test statistic is (LaBone, 2013) brunner munzel test
Mx and My are the mid-rank means for samples X and Y, m and n are the numbers of observations in each sample, and SB is the variance estimates for X and Y.
For degrees of freedom, Brunner and Munzel recommend adopting a t-distribution with a Welch–Satterthwaite approximation. | {"url":"https://datasciencetut.com/similarity-measure-between-two-populations-brunner-munzel-test/","timestamp":"2024-11-06T20:29:22Z","content_type":"text/html","content_length":"116689","record_id":"<urn:uuid:cc03b985-fb31-464c-ba3b-018e1848e7fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00469.warc.gz"} |
32 Math Trivia Questions For Kids: Easy to Hard Answers32 Math Trivia Questions For Kids: Easy to Hard Answers - Parent Intel
Math trivia questions for kids offer a captivating blend of challenge and entertainment, turning the sometimes daunting world of mathematics into an exciting adventure.
From the basic whole numbers to the complexities of the Fibonacci sequence, math trivia encompasses a wide range of topics that intrigue young minds. Whether it’s uncovering the only even prime
number, exploring the only number whose letters are in alphabetical order, or deciphering the mysteries of square roots and natural numbers, these trivia questions not only enhance problem-solving
skills but also reveal interesting facts about mathematics that resonate with kids.
Math trivia questions serve not just as a great way to learn, but also as a fun activity for kids. They combine learning with play, making them a popular choice for educational games, classroom
activities, and even family trivia nights.
The correct answers, often accompanied by interesting facts, add to the excitement and joy of learning. By framing math problems as fun trivia questions, we transform the learning experience, making
it both engaging and effective for young learners.
Easy Math Trivia for Young Kids
Engaging young minds in the fascinating world of mathematics can be both fun and educational. Easy math trivia for young kids is a fantastic way to introduce them to basic math concepts in an
enjoyable way and for parents – spend more time with your kids. This section focuses on simple arithmetic, counting, and basic shapes, which are fundamental building blocks in a child’s mathematical
1. Counting and Whole Numbers:
• Question: “If you have 2 apples and get 3 more, how many apples do you have in total?”
• Answer: 5 apples
This question not only tests addition skills but also reinforces the concept of whole numbers in a real-world context.
2. Simple Geometry:
• Question: “How many sides does a triangle have?”
• Answer: 3 sides
Introducing basic shapes like triangles helps kids understand the concept of sides and angles in a simple, relatable manner.
3. Addition and Subtraction Fun:
• Question: “What is 10 minus 4?”
• Answer: 6
Basic subtraction questions like this are a great way to develop early arithmetic skills.
4. Identifying Shapes:
• Question: “What shape has four equal sides and four right angles?”
• Answer: A square
This question is not only about recognizing shapes but also introduces the idea of equal sides and right angles, key concepts in geometry.
5. Number Patterns:
• Question: “What comes next in this pattern? 1, 2, 3, 4, _?”
• Answer: 5
Simple number patterns are an interesting way to encourage logical thinking and number sequence recognition.
6. Fun with Math Concepts:
• Question: “If you have one candy bar and your friend gives you another, how many candy bars do you have?”
• Answer: 2 candy bars
This question uses a relatable scenario to make addition tangible and fun.
7. Exploring the World of Numbers:
• Question: “What is the only even prime number?”
• Answer: 2
This trivia question introduces the concept of prime numbers in an accessible way.
Through these easy math trivia questions, kids not only learn fundamental math concepts but also develop critical thinking skills. Such questions blend educational content with a fun approach. This
approach to math trivia ensures that kids are not just passively receiving information but actively engaging with and enjoying the learning process.
Fun Facts and Puzzles in Math
Diving into the world of math can be a fascinating adventure, especially when it’s filled with fun facts and puzzles. This section covers topics like palindromes, the Fibonacci sequence, and unique
math terms that add a twist of excitement to mathematical learning.
8. The Intrigue of Palindromes:
• Fact: A palindrome is a number that reads the same forwards and backwards, like 121 or 12321.
• Puzzle: “Can you think of a three-digit palindrome?”
Palindromes are an interesting way to explore numbers and can spark curiosity in children about number patterns.
9. Fibonacci Sequence Fascination:
• Fact: The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, often starting with 0 and 1.
• Puzzle: “What are the first five numbers in the Fibonacci sequence?”
This sequence not only unveils a pattern in numbers but also connects to various phenomena in the natural world, providing an interesting way to link math to everyday life.
10. Exploring Unique Math Terms:
• Fact: An ‘isosceles triangle’ is a triangle with two sides of equal length.
• Puzzle: “Can you draw or identify an isosceles triangle?”
Introducing such terms helps children understand different shapes and properties in geometry.
11. Number Fun with Roman Numerals:
• Fact: Roman numerals are a numeral system of ancient Rome based on letters of the alphabet, like I for 1, V for 5, and X for 10.
• Puzzle: “What number is represented by the Roman numeral ‘IV’?”
Learning about Roman numerals is a fun way to explore historical number systems and understand how they are used in various contexts, such as clocks.
12. The Magic of Square Roots:
• Fact: The square root of a number is a value that, when multiplied by itself, gives the original number.
• Puzzle: “What is the square root of 64?”
Understanding square roots is essential in mathematics and offers a glimpse into more complex math concepts.
13. Playing with Negative Numbers:
• Fact: Negative numbers represent values less than zero, often used in temperatures below freezing or elevations below sea level.
• Puzzle: “If the temperature is -5 degrees and it drops 3 more degrees, what is the new temperature?”
This encourages thinking about math in relation to real-world situations like weather.
By presenting these fun facts and puzzles, kids are not only learning important math concepts but are also developing problem-solving skills and a deeper appreciation for the beauty of mathematics.
The combination of factual knowledge and interactive puzzles provides an engaging learning experience for kids and students.
Middle School Math Trivia
Middle school is a pivotal time for expanding mathematical knowledge, diving deeper into concepts like fractions, algebra, and geometry. This section is designed to challenge middle schoolers with
trivia that touches upon basic algebraic concepts, theorems, and geometric shapes.
14. Fractions and Operations:
• Question: “What is the term for the bottom number in a fraction, which indicates the total number of equal parts?”
• Answer: Denominator
This question not only tests understanding of fractions but also emphasizes their role in representing parts of a whole.
15. Algebraic Concepts:
• Question: “What do you call a mathematical sentence that uses an equals sign to show that two expressions are equal?”
• Answer: An equation
Introducing algebraic terms such as ‘equation’ helps students understand the foundation of algebra.
16. Geometry and Shapes:
• Question: “What is the term for a three-dimensional shape with two parallel, congruent circular bases?”
• Answer: Cylinder
Questions about geometric shapes like cylinders enhance spatial understanding and recognition skills.
17. Exploring Theorems:
• Question: “In a right-angled triangle, which theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides?”
• Answer: Pythagorean Theorem
Theorems like this are fundamental in geometry and encourage logical thinking.
18. Understanding Algebraic Expressions:
• Question: “What is the value of x in the equation 3x + 7 = 16?”
• Answer: 3
Solving for x in simple algebraic equations helps build essential problem-solving skills.
19. Basic Algebraic Operations:
• Question: “What operation would you perform first in the expression 4 + 3 x 2?”
• Answer: Multiplication (Following the order of operations, BEDMAS/BIDMAS)
Questions about order of operations reinforce the importance of following mathematical rules.
20. Geometric Theorems and Properties:
• Question: “What is the term for a polygon with all sides and angles equal?”
• Answer: Regular Polygon
This introduces students to more complex geometric concepts and vocabulary.
These trivia questions for middle school math not only align with educational standards but also aim to make learning a fun and interactive experience. By combining fundamental concepts with
interesting trivia, students are encouraged to explore and deepen their understanding of math, fostering a positive attitude towards learning and problem-solving.
Advanced Math Trivia for Budding Mathematicians
For older kids and those with a keen interest in advanced math, challenging trivia can provide an excellent way to deepen their understanding and spark further curiosity. This section delves into
topics such as prime numbers, complex geometry, and number theory, presenting an opportunity for budding mathematicians to test and expand their knowledge.
21. Prime Numbers and Patterns:
• Question: “What is the largest known prime number?”
• Answer: As of my last update, the largest known prime number is (2^{82,589,933} − 1), a Mersenne prime discovered in 2018. (Note: This may have changed if newer primes have been discovered.)
This question not only challenges their knowledge but also introduces them to the ongoing quest in mathematics to discover larger prime numbers.
22. Complex Geometry Concepts:
• Question: “In complex geometry, what term describes a shape with only straight lines?”
• Answer: Polygon
Exploring complex geometry helps in understanding how basic shapes evolve into more advanced structures.
23. Number Theory Exploration:
• Question: “What is the fundamental theorem of arithmetic?”
• Answer: It states that every integer greater than 1 is either a prime number or can be uniquely factorized into prime numbers.
This theorem is a cornerstone of number theory, illustrating the unique nature of prime numbers in the composition of integers.
24. Advanced Algebraic Structures:
• Question: “What is a complex number?”
• Answer: A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit that satisfies the equation (i^2 = −1).
Complex numbers are essential in advanced mathematics, providing solutions to equations that have no real solutions.
25. Exploring the Fibonacci Sequence:
• Question: “What is the only number that appears four times in the Fibonacci sequence?”
• Answer: 1 (The sequence starts 0, 1, 1, 2, 3, 5, 8, 13…)
The Fibonacci sequence is not just a series of numbers; it’s a gateway to understanding various mathematical concepts and their applications in nature and science.
25. Intriguing Mathematical Concepts:
• Question: “What is the term for a solid shape with a polygon base and triangular faces that meet at a common point?”
• Answer: Pyramid
Questions like this encourage an interest in three-dimensional shapes and their properties, key in fields like architecture and engineering.
By engaging with these advanced math trivia questions, students are not just recalling facts; they are delving into the heart of mathematical theory and its real-world applications.
Math in Everyday Life
Math is not just confined to textbooks and classrooms; it’s an integral part of our daily lives. This section of the article connects math trivia to real-life situations and applications, focusing on
measurements, time, and basic calculations that we encounter daily. These questions help illustrate the practical and often overlooked role of math in our everyday activities.
26. Real-World Measurements:
• Question: “If you buy 3.5 meters of fabric to make curtains and each curtain needs 1.75 meters, how many curtains can you make?”
• Answer: 2 curtains
This question helps understand measurements and division in a practical context, such as sewing or purchasing materials.
27. Time Calculation:
• Question: “Your bus leaves at 3:30 PM and it takes 45 minutes to reach your destination. What time will you arrive?”
• Answer: 4:15 PM
Calculating time is a fundamental skill, essential for planning and organizing daily activities.
28. Everyday Math in the Kitchen:
• Question: “If a recipe calls for 2 cups of flour and you want to make half the recipe, how much flour do you need?”
• Answer: 1 cup
This question applies fractions in cooking, a common real-life application of math.
29. Grocery Store Calculations:
• Question: “If an apple costs 50 cents and you buy 4 apples, how much do you spend in total?”
• Answer: $2.00
Simple multiplication and understanding of currency are key skills used in everyday shopping.
30. Understanding Percentages:
• Question: “If a $50 shirt is on sale for 20% off, how much is the discount?”
• Answer: $10
This question introduces the concept of percentages, a common mathematical concept used in sales and finance.
31. Calculating Perimeter and Area:
• Question: “If a garden is 10 feet long and 5 feet wide, what is its perimeter?”
• Answer: 30 feet
Questions like this help kids understand the concept of perimeter, which is relevant in tasks like fencing a yard or setting up a garden.
32. Applying Math in Sports:
• Question: “If you run 4 laps of a 400-meter track, how much distance have you covered?”
• Answer: 1600 meters (or 1.6 kilometres)
This demonstrates how math is used in sports, for measuring distances or keeping track of scores.
These everyday math trivia questions emphasize the importance of math in daily life, making it more relatable and understandable. By connecting mathematical concepts to real-life scenarios.
Harnessing the Power of Math in the Real World
Mathematics is not just confined to the classroom; it’s a pivotal tool that shapes our understanding of the world. From calculating the trajectory of a rocket in the solar system to measuring
ingredients for a perfect pumpkin pie, math is integral in diverse fields. Consider the Pythagorean theorem’s application in architecture or the Fibonacci sequence’s occurrence in the patterns of
flower petals – these examples highlight how math is woven into the very fabric of life.
Engaging children in math-related activities, like a playful game at a grocery store where they calculate the cost of ice creams, not only enhances their mental math skills but also prepares them for
real-world challenges. Even simple tasks, such as using a ruler to measure line segments or understanding the concept of an improper fraction, can be gateways to developing critical thinking skills.
This application of math in everyday life fosters not just academic excellence but also practical wisdom.
Encouraging this exploration of math, whether through a fun math quiz or challenging math facts worksheets, can ignite a child’s curiosity and lead to significant cognitive development. It’s about
turning every opportunity, every question, into a learning experience – a creative way to view the world through the lens of mathematics.
Also check out our list of 121 Science Trivia Questions For Kids
Exploring Math Beyond the Classroom
Mathematics is not just about numbers; it’s a universe of concepts touching everything from the human body to the farthest reaches of space. Let’s dive into some fun math trivia questions that
showcase the diversity of this fascinating subject.
1. The Human Body and Numbers:
□ Trivia Question: “What is the total number of letters in the longest bone in the human body (the femur)?”
□ Correct Answer: 5 letters This trivia introduces kids to anatomy, combining math with biology in an interesting way.
2. Geometry in Nature:
□ Trivia Question: “What natural structure has a convex shape resembling a dome?”
□ Correct Answer: A beehive This illustrates geometry in nature, making the learning of shapes like convex structures more relatable.
3. Space and Math:
□ Trivia Question: “What is the largest planet in our solar system?”
□ Correct Answer: Jupiter Such questions connect math to astronomy, sparking curiosity about the universe.
4. Math in History:
□ Trivia Question: “Who was the first person to propose the theory of relativity?”
□ Correct Answer: Albert Einstein This introduces children to the historical aspects of math, showing how theories have evolved over time.
5. Math in Everyday Objects:
□ Trivia Question: “What common object used in the United States contains images of notable figures like Thomas Edison and is an example of paper money?”
□ Correct Answer: The dollar bill This trivia brings attention to everyday uses of math, like currency and its design.
6. Math in Awards:
□ Trivia Question: “Which prestigious award, similar to a Nobel Prize, is given in the field of mathematics?”
□ Correct Answer: Fields Medal This question highlights the recognition of excellence in mathematics.
7. Math and the Cosmos:
□ Trivia Question: “Which famous scientist discovered the law of universal gravitation?”
□ Correct Answer: Isaac Newton This trivia bridges the gap between math and physics, showcasing how math helps us understand the laws of the universe.
Incorporating these trivia questions and facts into math quizzes can significantly enhance a child’s understanding and appreciation of mathematics. It shows that math is not just confined to
textbooks but is a dynamic and integral part of our world.
Also Check Out
183 History Trivia Questions For Kids
Wrapping Up Math Trivia Questions For Kids
Math trivia is more than just a fun activity; it’s a valuable tool in the learning process. It transforms the sometimes abstract and challenging world of mathematics into an engaging and approachable
subject. By incorporating math trivia into learning, children can explore mathematical concepts in a playful yet educational way. This approach not only enhances their problem-solving skills but also
fosters a deeper appreciation and curiosity for the subject.
The beauty of math trivia lies in its versatility. It caters to various learning levels, from easy questions for young learners to more complex problems for older students and budding mathematicians.
By connecting math to everyday life, trivia helps demystify math, showing its practical application in our daily activities. Interactive math games and trivia further enrich this experience, blending
learning with fun and ensuring that mathematical concepts are absorbed more effectively and enjoyably.
As we conclude, it’s clear that math trivia is an invaluable component of modern education. It invites learners to explore the fascinating world of numbers, shapes, and equations in a relaxed and
enjoyable environment. Whether it’s through solving interesting puzzles, exploring intriguing facts, or engaging in interactive games, math trivia makes learning an adventure.
So, we encourage you, our readers, to dive into the world of math trivia. Explore more, challenge yourself with different questions, and most importantly, enjoy the journey through the world of math.
The exploration of mathematics is endless, and every trivia question answered is a step further in this exciting journey. | {"url":"https://parentintel.com/math-trivia-questions-for-kids/","timestamp":"2024-11-02T22:09:52Z","content_type":"text/html","content_length":"114951","record_id":"<urn:uuid:d7f4ba4e-5c63-4c27-a1ca-501c73bd5c94>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00549.warc.gz"} |
Will we be given z and t tables in the Level I exam?
You will not be given separate stat tables in the exam. The question will include a segment of a relevant table e.g. z- or t-score if it is needed.
You would be expected to know common z-scores such as 1.96 for 2-tail 95% confidence and 1.65 for 2-tail 90% confidence. | {"url":"https://support.fitchlearning.com/hc/en-us/articles/23523099588247-Will-we-be-given-z-and-t-tables-in-the-Level-I-exam","timestamp":"2024-11-12T12:52:19Z","content_type":"text/html","content_length":"43013","record_id":"<urn:uuid:263298fd-7810-41c3-a7a1-90773e38c88c>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00742.warc.gz"} |
Sampling from an Arbitrary Distribution
The first thing you probably found
If you ask a random person (conditioned on them being a reasonable target for this question) how to sample from an arbitrary distribution, you are pretty likely to get this answer: “Just sample from
a uniform distribution between 0 and 1, and invert the samples using the inverse cdf”. This is a wonderfully elegant solution, and we might as well start by looking at how it works. We’ll use a
really simple PDF for illustrative purposes: \(\text{pdf}(x) = 2x\), which will be normalized with a domain of \(x \in [0,1]\).
We can calculate the cdf readily: \(\text{cdf}(x)= x^2\), so the inverse cdf is a square root. With this information, it is trivial to sample say, 1,000,000 samples from the initial pdf:
# sample 1,000,000 values from cdf(x)
cdf_samples = np.random.uniform(0,1, 1_000_000)
# invert them, by taking the square root
x_vals = np.sqrt(cdf_samples)
We can plot a histogram of the x_vals to see if it worked:
It did.
The catch
You did need to invert the cdf, which can be a bit of a bummer. Or course, since you are working on a computer (probably) you can probably do a pretty good job of numerically integrating the pdf and
building up a numeric inverse instead. But the real issue with this method is how it scales into higher dimensions.
Unfortunately for anyone who lives or thinks in more than 1Dimension: the inverse cdf is only guaranteed to exist and/or be well defined in a single dimension. Multivariate distributions typically
have multi-valued inverse cdf relations. Just imagine a symmetric 2 dimensional gaussian distribution. There will be a circle of coordinates at a radius \(r\) from the peak of the gaussian that all
share the same probability, so just choosing a probability between \(0\) and \(1\) will not give you an unambiguous point in coordinate space. Additionally, the size of these regions depends on how
far from the center you are– so it doesn’t even make sense to sample the probability uniformly anyway. So, let’s work on a method that will work for multivariate distributions.
First pass at a new method
Here is the idea:
• we generate a point \(Y\) in our domain using a uniform distribution
• we calculate the probability \(\text{pdf}(Y)\) of that point using the known target pdf
• we generate a random uniform variable \(U \in [0,1]\),
• we accept the point Y as being part of our sample if \(U \leq \text{pdf}(Y)\) The idea behind this is intuitive. We are more likely to accept the higher probability events and less likely to
accept the low probability ones, and the ratio between these acceptances is the ratio of the relative probabilities. Thus, we would expect our list of accepted \(Y\) values to follow the target
pdf. Sure, we have lost some efficiency, because we have to spend resources making samples that will be rejected– but that is the tradeoff ofr this more general method. Let’s take a look at how
this works in practice, using the same target distribution as above:
def target_dist(x):
return 2*x
Y = np.random.uniform(0,1,N)
prob_fY = target_dist(Y)
U = np.random.uniform(0,1,N)
accepted_samples = Y[U <= accept_probs]
And the histogram of accepted samples, to see if it worked…
It did not.
Rejection Sampling
Well, what happened was that \(\text{pdf}(x)\) was actually larger than 1 for half of the values– so we were unable to reject those samples. The problem has to be scaled properly for this to work. We
made the classic mistake of thinking of the pdf as giving the probabilities of each point in the space when it actually only gives us density. The solution here is to talk about ratios of pdf’s
instead. This intuitive method we have been trying out is sometimes called ‘rejection sampling’, and is one of many monte-carlo algorithms that solve difficult problems generating a variable from a
simple case and the accepting or rejecting is based on the more complicated problem. A corrected version of the method goes like this:
• we want to generate a realization of the random variable \(X\) distributed according to a “target distribution” \(f(x)\)
• we generate a random variable \(Y\) using an easy to sample from “proposal distribution” \(g(Y)\)
• we generate a random uniform variable \(U \in [0,1]\),
• we accept the point Y as being part of our sample if \(U \leq \frac{f(Y)}{M\cdot g(Y)}\)
• M is a constant parameter, chosen so that the ratio on the RHS of the inequality never surpasses 1
Assuming that both distributions are normalized, we can use \(M\) as a measure of the efficiency of the algorithm because \(1/M\) is approximately the probability of accepting a sample \(Y\).
Revisiting our previous case, we can see that with the uniform distribution having \(g(Y)=1\) in our domain that we will need \(M=2\) to ensure the ratio of \(\frac{f(Y)}{M\cdot g(Y)} \leq 1\) for
our entire domain. Thus, we expect to throw out \(\sim 50\%\) of our generated \(Y\) values. So we expect to generate \(2N\) samples of \(Y\) to generate \(N\) samples of \(X\).
Now, to try it out. We run the exact same code as before. Except, we add in \(M=2\):
Y = np.random.uniform(0,1,N)
prob_fY = target_dist(Y)
U = np.random.uniform(0,1,N)
# add in M
accepted_samples = Y[U <= accept_probs/M]
#do quick histogram to verify
fig, ax = plt.subplots()
ax.hist(accepted_samples, bins=50, density=True);
ax.set_title(f'generated {n} samples, accepted {len(accepted_samples)}')
And there we go! This is rejection sampling in a nutshell. Of course, it’s probably clear as this point that a uniform distribution is not the most efficient distribution for our target pdf. Ideally,
we want the proposal distribution to be as close to the target distribution as we can make it, while still being easy to sample from. Could we leverage the ability to easily sample from gaussians to
do a better job? At first glance it seems like a bad idea: the Gaussian is totally symmetric and our target_dist is absolutely not. However, by generating a normally distributed variable \(G\) and
taking the absolute value, we can create a one sided distribution that is well suited for our purposes.
With \(Y = 1-\text{abs}(G(0,\sigma))\), we can see that the distribution of \(Y\) looks quite similar to our target distribution, with a value of \(M\) that is quite close to \(1\)
from scipy.stats import norm
class one_sided_norm():
def __init__(self, x0, sigma):
self.norm = norm(loc=x0, scale=sigma)
# we have to redefine the pdf here because taking the absolute value
# effectively halves the domain and doubles the density
def one_sided_pdf(self,x):
return 2*self.norm.pdf(x)
fig, ax = plt.subplots()
xt= np.linspace(-1,1)
x= np.linspace(0,1)
ax.plot(x, target_dist(x), label='target_pdf')
#choose some values by guess and check until it looks right
sigma = .45
M =1.2
proposal_dist = one_sided_norm(0, sigma)
ax.plot(1-np.abs(xt), M * proposal_dist.one_sided_pdf(xt), label='$M\cdot $transformed gaussian')
We can see that the ratio \(\frac{f(Y)}{M\cdot g(Y)}\) will always be less than one, guaranteeing a good sample of our target distribution; but, at the same time, it will often be very close to one
guaranteeing a high degree of efficiency with accepting our generated samples.
Now that we have the hang of it, lets make a time saving function to do this rejection sampling for us:
def rejection_sample(N, target_pdf, proposal_dist, m):
# N is the number of samples we generate from the sampling dist
# target_pdf is a function that return the probability density at its argument as an array
# proposal_dist is an object with a method .sample(k) that returns an array of k samples and their probability density
# m is the scaling parameter discussed above
y, prob_gy = proposal_dist.sample(N)
accepted_samples = y[np.random.uniform(0,1,N) < target_pdf(y)/(m*prob_gy)]
# here we return both the accepted sample array, and also the ratio to verify if 1/M is truly the prob of acceptance
return accepted_samples, len(accepted_samples)/N
# redefine our one_sided_norm class to have the method we need for the rejection_sample function
class one_sided_norm():
def __init__(self, x0, sigma):
self.norm = norm(loc=x0, scale=sigma)
# we have to redefine the pdf here because taking the absolute value
# effectively halves the domain and doubles the density
def one_sided_pdf(self,x):
return 2*self.norm.pdf(x)
def sample(self, N):
G = self.norm.rvs(N)
return 1-np.abs(G), self.one_sided_pdf(G)
With these definitions, rejection sampling is as simple as running the following few lines:
sigma= .45
# define the proposal distribution
proposal_dist = one_sided_norm(0, sigma)
accepted_samples, ratio = rejection_sample(int(N*M), target_dist, proposal_dist, M)
Now, with the obligatory histogram of accepted samples to see how we did…
Next steps
Ok, so we got better sample efficiency by handcrafting a gaussian to fit our pdf, but that kind of fine tuning by hand isn’t really a scalable option. Additionally, the extra efficiency needs to
offset the extra computational time it takes to sample from the gaussian rather than the uniform distribution. In the next installment we will go over some attempts to automate the process of
creating a proposal distribution and also about the pros and cons of using un-normalized distributions. | {"url":"https://kylejray.github.io/distribution-sampling/","timestamp":"2024-11-09T10:09:09Z","content_type":"text/html","content_length":"28252","record_id":"<urn:uuid:5b19dcef-bc36-4675-8514-b73131e54409>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00044.warc.gz"} |
Dividers and fractions - Fantastic Mathematics
Dividers and fractions
mathématiques | 0 comments
Purpose: Present fractional results in irreducible form. Notion of dividers
Application 1: Fraction recall
Application 2: Term of divider. Decomposition as a product of prime factors
Application 3: Find the dividers of an entire number
Course recall: Divisibilite rule
QCM yes-no: dividers
Application 4: Detailed steps to make a fraction irreducible.
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incidence_bounds: Incidence bounds and confidence interval in ARPobservation: Tools for Simulating Direct Behavioral Observation Recording Procedures Based on Alternating Renewal Processes
Calculates a bound for the log of the incidence ratio of two samples (referred to as baseline and treatment) based on partial interval recording (PIR) data, assuming that the behavior follows an
Alternating Renewal Process.
incidence_bounds( PIR, phase, base_level, mu_U, p, active_length, intervals = NA, conf_level = 0.95, exponentiate = FALSE )
PIR vector of PIR measurements
phase factor or vector indicating levels of the PIR measurements.
base_level a character string or value indicating the name of the baseline level.
mu_U the upper limit on the mean event duration
p upper limit on the probability that the interim time between behavioral events is less than the active interval
active_length length of the active observation interval
intervals the number of intervals in the sample of observations. Default is NA
conf_level Coverage rate of the confidence interval. Default is .95.
exponentiate Logical value indicating if the log of the bounds and the confidence interval should be exponentiated. Default is FALSE.
a character string or value indicating the name of the baseline level.
upper limit on the probability that the interim time between behavioral events is less than the active interval
the number of intervals in the sample of observations. Default is NA
Logical value indicating if the log of the bounds and the confidence interval should be exponentiated. Default is FALSE.
The incidence ratio estimate is based on the assumptions that 1) the underlying behavior stream follows an Alternating Renewal Process, 2) the average event duration is less than mu_U, and 3) the
probability of observing an interim time less than the active interval length is less than p.
The PIR vector can be in any order corresponding to the factor or vector phase. The levels of phase can be any two levels, such as "A" and "B", "base" and "treat", or "0" and "1". If there are more
than two levels in phase this function will not work. A value for base_level must be specified - if it is a character string it is case sensitive.
For all of the following variables, the function assumes that if a vector of values is provided they are constant across all observations and simply uses the first value in that vector.
mu_U is the upper limit on the mean event durations. This is a single value assumed to hold for both samples of behavior
active_length This is the total active observation length. If the intervals are 15 seconds long but 5 seconds of each interval is reserved for recording purposes, active_length= 10. Times are often
in seconds, but can be in any time unit.
intervals is the number of intervals in the observations. This is a single value and is assumed to be constant across both samples and all observations. This value is only relevant if the mean of one
of the samples is at the floor or ceiling of 0 or 1. In that case it will be used to truncate the sample mean. If the sample mean is at the floor or ceiling and no value for intervals is provided,
the function will stop.
A list containing two named vectors and a single named number. The first entry, estimate_bounds, contains the lower and upper bound for the estimate of the incidence ratio. The second entry,
estimate_SE, contains the standard error of the estimate. The third entry, estimate_CI, contains the lower and upper bounds for the confidence interval of the incidence ratio.
# Estimate bounds on the incidence ratio for Ahmad from the Dunlap dataset data(Dunlap) with(subset(Dunlap, Case == "Ahmad"), incidence_bounds(PIR = outcome, phase = Phase, base_level = "No Choice",
mu_U = 10, p = .15, active_length = active_length, intervals = intervals))
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/ARPobservation/man/incidence_bounds.html","timestamp":"2024-11-01T22:37:31Z","content_type":"text/html","content_length":"32146","record_id":"<urn:uuid:bec77b54-1d5c-4af1-81a7-68d6cff9c631>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00052.warc.gz"} |
Find magnitude and part in table?
I am kinda stuck logically. You see, basically what happens is, the function runs through a table of parts, and then it checks which one has the lowest magnitude, then returns it.
But, I am attempting to find WHICH part has the lowest magnitude, and to do that I assumed I need to find the lowest magnitude first. I have no idea to how return the part that has the lowest
magnitude. Does anyone have any clue?
local function distanceCheck(otherPart)
local distances = {}
for i, v in pairs(otherPart) do
local check = (HRP.Position - v.Position).Magnitude
distances[i] = check
return math.min(unpack(distances))
1 Like
I would look @Exeplex’s post. And use math.huge for the highest magnitude instead of 9999999.
All I could think of is adding a second for loop below which loops through all the distances and keeps track of the smallest one. unpack won’t work since it does indxes and values.
local maxDistance = math.huge
for i,v in pairs(distances) do
if v < maxDistance then
maxDistance = v
return table.find(distances,maxDistance)
local function distanceCheck(otherPart)
local lowestmagnitude = 9999999 - Some insanely high number for the first check to pass
local closestpart = nil
for i, v in pairs(otherPart) do
local check = (HRP.Position - v.Position).Magnitude
if check < lowestmagnitude then -- If the magnitude is lower than the previous lowestmagnitude
lowestmagnitude = check -- Set the new low
closestpart = v -- Set the part that is closest
return closestpart, lowestmagnitude
local closestpart, lowestmagnitude = distanceCheck()
You just have to set an insanely high value for the first part to pass as the “lowest magnitude”, then every other part will be checked against it. When a part has a lower magnitude, change the
variables to the lowest magnitude and what part is closest. In the end, you can return them.
Edit: You don’t have to return the lowest magnitude, it’s just an extra in case you need it. You can simply just return the part into one variable
3 Likes
Thanks bro!
I am confused, why use a high number for the first check, so it can get all parts available in the table?
Lets say you have 3 parts. Their magnitudes are: 5, 10, 15
If you set the lowestmagnitude variable to 0. None of those parts will be considered closest, because their magnitudes are higher than 0.
If you set lowestmagnitude to 9999. 5 will be the closest part. Because it is lower than 9999.
Because 15 is lower than 9999, 10 is lower than 15, 5 is lower than 10. Also in case you only have 1 part in the array, that part will be the closest
1 Like
You can actually find the lowest magnitude without using a very large number plus a tiny optimization with allocated table memory creation if you use table.create for an array.
The original solution is however better because of the maximum distance that you can set. I below have some alternatives to use which will do the same thing of course.
No large number alternative
local function pairsFind(t, value)
for i, v in pairs(t) do
if v == value then
return i
local function partWithLowestMagnitude(parts, compareTo)
local t = table.create(#parts)
for _, part in ipairs(parts) do
t[part] = (part.Position - compareTo.Position).Magnitude
local lowest = math.min(unpack(t))
return pairsFind(t, lowest), lowest
local part, magnitude = partWithLowestMagnitude({workspace.Part, workspace.Part1, workspace.Part2, workspace.Part3}, compareTo)
It’s the same thing in the end, your method just utilizes more library methods which do the same thing under the hood. It’s not optimizing the code other than reducing 1 or two lines.
1 Like
I am not saying it optimizes the code, I just mean table.create could be used for optimization in array cases, also I made a slight error in the code so I’m going to edit and fix it. table.find only
works on arrays which I forgot.
I actually think the solution script is better because you can set a maximum distance case opposed to none. | {"url":"https://devforum.roblox.com/t/find-magnitude-and-part-in-table/683469/6","timestamp":"2024-11-02T00:20:51Z","content_type":"text/html","content_length":"40154","record_id":"<urn:uuid:ea91a59d-45c7-448e-bbca-01f431153eef>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00522.warc.gz"} |
This number is a prime.
The Carillon Residences North Tower in Miami Beach is 397 feet tall.
Conjectured to be the largest prime that can be represented uniquely as the sum of three positive squares (3^2 + 8^2 + 18^2). [Noe]
The prime that splits into a prime pair (3, 97) with sum one hundred and minimal product (3*97=291). Note that the sum-product concatenation 100291 is prime, and that 397 is a left-truncatable prime
, just like its counterpart prime 5347 that splits into two complementary primes (53, 47) also summing up to a hundred, but with maximum product (53*47=2491). The reversal of whose sum-product
concatenation however being also prime (1942001). [Beedassy]
Iranian grandmaster Morteza Mahjoob won 397 (of his 500) chess games to break the Guinness Book World Record for the most number of simultaneous matches in August 2009.
397 and 401 are the largest pair of consecutive primes that together have no duplicate digits. [Gaydos]
The smallest non-palindromic balanced prime in base 2. [Nie]
(There are 5 curios for this number that have not yet been approved by an editor.)
Printed from the PrimePages <t5k.org> © G. L. Honaker and Chris K. Caldwell | {"url":"https://t5k.org/curios/page.php?number_id=2562","timestamp":"2024-11-14T01:37:55Z","content_type":"text/html","content_length":"10508","record_id":"<urn:uuid:b5ef8c24-553d-4fad-a5d6-a1f226882a95>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00062.warc.gz"} |
Divergence - (Honors Algebra II) - Vocab, Definition, Explanations | Fiveable
from class:
Honors Algebra II
Divergence refers to the behavior of a sequence as it progresses towards infinity, where it does not approach a finite limit. In the context of sequences, divergence indicates that the terms of the
sequence grow without bound or oscillate indefinitely, meaning they fail to converge to a single value. Understanding divergence is crucial when analyzing the long-term behavior of both arithmetic
and geometric sequences, as it affects how we interpret their sums and growth patterns.
congrats on reading the definition of divergence. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. A sequence is divergent if its terms do not settle down to a specific value; this can happen if they increase or decrease without bound.
2. In an arithmetic sequence, if the common difference is positive, the sequence will diverge towards positive infinity; if negative, towards negative infinity.
3. For geometric sequences, divergence occurs when the absolute value of the common ratio is greater than 1, causing terms to grow larger in magnitude without approaching a limit.
4. The divergence of a sequence can be indicated through limits; for example, if $$ ext{lim}_{n \to \infty} a_n = \infty$$, the sequence diverges to infinity.
5. Understanding whether a sequence diverges is essential for determining if sums of series converge or diverge, impacting applications in calculus and analysis.
Review Questions
• How can you determine if an arithmetic sequence is divergent?
□ To determine if an arithmetic sequence is divergent, examine its common difference. If the common difference is positive, the terms will continuously increase without bound, indicating
divergence toward positive infinity. Conversely, if the common difference is negative, the terms will decrease indefinitely towards negative infinity. Therefore, as long as there is a
consistent addition or subtraction that does not lead to stabilization around a single number, the arithmetic sequence will be divergent.
• What role does divergence play in understanding geometric sequences with varying common ratios?
□ Divergence plays a significant role in geometric sequences based on the common ratio's absolute value. If the absolute value of the common ratio exceeds 1, each subsequent term becomes larger
in magnitude, leading to divergence as it grows without approaching a limit. Conversely, if the common ratio is between -1 and 1 (excluding 0), the sequence converges towards zero. Hence,
analyzing the common ratio helps predict whether a geometric sequence diverges or converges.
• Compare and contrast convergence and divergence in sequences, providing examples to illustrate your points.
□ Convergence and divergence are two opposing behaviors observed in sequences. Convergence occurs when a sequence approaches a finite limit as its terms progress; for example, in the geometric
sequence defined by $$a_n = \left(\frac{1}{2}\right)^n$$, as n increases, the terms get closer to zero. In contrast, divergence describes sequences that do not settle around any particular
number; an example would be an arithmetic sequence with a positive common difference like $$a_n = 5 + 3n$$, which increases indefinitely. Understanding these concepts allows for deeper
insight into their long-term behaviors and implications for sums and limits in mathematical analysis.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/hs-honors-algebra-ii/divergence","timestamp":"2024-11-05T23:17:11Z","content_type":"text/html","content_length":"167682","record_id":"<urn:uuid:ed65b5b4-3562-46cf-9146-54b8aa9d7d92>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00678.warc.gz"} |
Cartography and Cadastre Bureau
Basic Concept of Satellite Positioning
Generally, satellite positioning means that a satellite signal receiver receives the electromagnetic wave signals from different positioning satellites in space. After data processing and
calculation, the position of the receiver in space is determined. Relevant positioning methods include:
Single-point positioning
The basic principle lies in that a receiver simultaneously receives the positioning signals from different positioning satellites to estimate the distance among them and, through the distance
intersection method, obtain the coordinates. In practice, as the satellite positioning signal is an electromagnetic wave signal, in order to determine the position of the receiver, it is only
necessary to calculate the frequency of the electromagnetic wave and the time of signal transmission from the satellite to the receiver to estimate the distance between the satellite and the
receiver. However, as the clock of the satellite may not be accurately synchronized with that of the receiver, a time error (t) will occur. Consequently, the time of signal transmission from the
satellite to the receiver cannot be accurately calculated, which directly affects the calculation of the distance between the satellite and the receiver. Thus, in order to accurately calculate the
coordinate values (X, Y, Z) of the receiver and the time error (t), the positioning signals from at least four different satellites must be received simultaneously during positioning to determine the
exact coordinate position of the receiver.
Relative positioning
It is also known as differential positioning. It is a positioning method in which two or more receivers are used for simultaneous observation to determine the vector relationship among the receivers.
For the simplest application in practice, two receivers are used to simultaneously receive the signals from over four identical positioning satellites. As the simultaneous observation modes of the
two receivers are the same, we can suppose that, in an adjacent area, the environmental impacts and error factors are roughly the same. After post-processing and calculation, the relative
relationship of the positions of the two receivers, i.e. the difference in coordinates, can be obtained. Therefore, when the coordinates of one receiver are known, the coordinates of the other
receiver can be calculated based on the difference between two coordinates.
Real-time kinematic positioning
Real-time kinematic (RTK) satellite positioning mode is a relative positioning technology integrating the modern mobile telecommunication technology and advanced calculation methods. In practice, a
receiver on a known point sends satellite positioning correction messages to a receiver on an unknown point via telecommunication devices such as a radio or a mobile. The latter utilizes the
correction message and the signals sent by positioning satellites to eliminate environmental impacts and error factors to obtain the precise coordinates. | {"url":"https://www.dscc.gov.mo/en/reference_details/article/reference_6.html","timestamp":"2024-11-10T21:27:35Z","content_type":"application/xhtml+xml","content_length":"64558","record_id":"<urn:uuid:7ef1b5aa-5f95-4a65-9cb9-495390c3355e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00886.warc.gz"} |
ale 0.3.0
The most significant updates are the addition of p-values for the ALE statistics, the launching of a pkgdown website which will henceforth host the development version of the package, and
parallelization of core functions with a resulting performance boost.
Breaking changes
• One of the key goals for the {ale} package is that it would be truly model-agnostic: it should support any R object that can be considered a model, where a model is defined as an object that
makes a prediction for each input row of data that it is provided. Towards this goal, we had to adjust the custom predict function to make it more flexible for various kinds of model objects. We
are happy that our changes now enable support for tidymodels objects and various survival models (but for now, only those that return single-vector predictions). So, in addition to taking
required object and newdata arguments, the custom predict function pred_fun in the ale() function now also requires an argument for type to specify the prediction type, whether it is used or not.
This change breaks previous code that used custom predict functions, but it allows ale to analyze many new model types than before. Code that did not require custom predict functions should not
be affected by this change. See the updated documentation of the ale() function for details.
• Another change that breaks former code is that the arguments for model_bootstrap() have been modified. Instead of a cumbersome model_call_string, model_bootstrap() now uses the {insight} package
to automatically detect many R models and directly manipulate the model object as needed. So, the second argument is now the model object. However, for non-standard models that {insight} cannot
automatically parse, a modified model_call_string is still available to assure model-agnostic functionality. Although this change breaks former code that ran model_bootstrap(), we believe that
the new function interface is much more user-friendly.
• A slight change that might break some existing code is that the conf_regions output associated with ALE statistics has been restructured. The new structure provides more useful information. See
help(ale) for details.
Other user-visible changes
• The package now uses a pkgdown website located at https://tripartio.github.io/ale/. This is where the most recent development features will be documented.
• P-values are now provided for all ALE statistics. However, their calculation is very slow, so they are disabled by default; they must be explicitly requested. When requested, they will be
automatically calculated when possible (for standard R model types); if not, some additional steps must be taken for their calculation. See the new create_p_funs() function for details and an
• The normalization formula for ALE statistics was changed such that very minor differences from the median are normalized as zero. Before this adjustment, the former normalization formula could
give some tiny differences apparently large normalized effects. See the updated documentation in vignette('ale-statistics') for details. The vignette has been expanded with more details on how to
properly interpret normalized ALE statistics.
• Normalized ALE range (NALER) is now expressed as percentile points relative to the median (ranging from -50% to +50%) rather than its original formulation as absolute percentiles (ranging from 0
to 100%). See the updated documentation in vignette('ale-statistics') for details.
• Performance has been dramatically improved by the addition of parallelization by default. We use the {furrr} library. In our tests, practically, we typically found speed-ups of n – 2 where n is
the number of physical cores (machine learning is generally unable to use logical cores). For example, a computer with 4 physical cores should see at least ×2 speed-up and a computer with 6
physical cores should see at least ×4 speed-up. However, parallelization is tricky with our model-agnostic design. When users work with models that follow standard R conventions, the {ale}
package should be able to automatically configure the system for parallelization. But for some non-standard models users may have to explicitly list the model’s packages in the new model_packages
argument so that each parallel thread can find all necessary functions. This is only a concern if you get weird errors. See help(ale) for details.
• Fully documented the output of the ale() function. See help(ale) for details.
• The median_band_pct argument to ale() now takes a vector of two numbers, one for the inner band and one for the outer.
• Switched recommendation of calculating ALE data on test data to instead calculate it on the full dataset with the final deployment model.
• Replaced {gridExtra} with {patchwork} for examples and vignettes for printing plots.
• Separated ale() function documentation from ale-package documentation.
• When p-values are provided, the ALE effects plot now shows the NALED band instead of the median band.
• alt tags to describe plots for accessibility.
• More accurate rug plots for ALE interaction plots.
• Various minor tweaks to plots.
Under the hood
• Uses the {insight} package to automatically detect y_col and model call objects when possible; this increases the range of automatic model detection of the ale package in general.
• We have switched to using the {progressr} package for progress bars. With the cli progression handler, this enables accurate estimated times of arrival (ETA) for long procedures, even with
parallel computing. A message is displayed once per session informing users of how to customize their progress bars. For details, see help(ale), particularly the documentation on progress bars
and the silent argument.
• Moved {ggplot2} from a dependency to an import. So, it is no longer automatically loaded with the package.
• More detailed information from internal var_summary() function. In particular, encodes whether the user is using p-values (ALER band) or not (median band).
• Separated validation functions that are reused across other functions to internal validation.R file.
• Added an argument compact_plots to plotting functions to strip plot environments to reduce the size of returned objects. See help(ale) for details.
• Created package_scope environment.
• Many minor bug fixes and improvements. Improved validation of problematic inputs and more informative error messages.
• Various minor performance boosts after profiling and refactoring code.
Known issues to be addressed in a future version
• Bootstrapping is not yet supported for ALE interactions (ale_ixn()).
• ALE statistics are not yet supported for ALE interactions (ale_ixn()).
• ale() does not yet support multi-output model prediction types (e.g., multi-class classification and multi-time survival probabilities).
ale 0.2.0
This version introduces various ALE-based statistics that let ALE be used for statistical inference, not just interpretable machine learning. A dedicated vignette introduces this functionality (see
“ALE-based statistics for statistical inference and effect sizes” from the vignettes link on the main CRAN page at https://CRAN.R-project.org/package=ale). We introduce these statistics in detail in
a working paper: Okoli, Chitu. 2023. “Statistical Inference Using Machine Learning and Classical Techniques Based on Accumulated Local Effects (ALE).” arXiv. https://doi.org/10.48550/arXiv.2310.09877
. Please note that they might be further refined after peer review.
Breaking changes
• We changed the output data structure of the ALE data and plots. This was necessary to add ALE statistics. Unfortunately, this change breaks any code that refers to objects created by the initial
0.1.0 version, especially code for printing plots. However, we felt it was necessary because the new structure makes coding in workflows much easier. See the vignettes and examples for code
examples for how to print plots using the new structure.
Other user-visible changes
• We added new ALE-based statistics: ALED and ALER with their normalized versions NALED and NALER. ale() and model_bootstrap() now output these statistics. (ale_ixn() will come later.)
• We added rug plots to numeric values and percentage frequencies to the plots of categories. These indicators give a quick visual indication of the distribution of plotted data.
• We added a vignette that introduces ALE-based statistics, especially effect size measures, and demonstrates how to use them for statistical inference: “ALE-based statistics for statistical
inference and effect sizes” (available from the vignettes link on the main CRAN page at https://CRAN.R-project.org/package=ale).
• We added a vignette that compares the ale package with the reference {ALEPlot} package: “Comparison between {ALEPlot} and {ale} packages” (available from the vignettes link on the main CRAN page
at https://CRAN.R-project.org/package=ale).
• We added two datasets:
□ var_cars is a modified version of mtcars that features many different types of variables.
□ census is a polished version of the adult income dataset used for a vignette in the {ALEPlot} package.
• Progress bars show the progression of the analysis. They can be disabled by passing silent = TRUE to ale(), ale_ixn(), or model_bootstrap().
• The user can specify a random seed by passing the seed argument to ale(), ale_ixn(), or model_bootstrap().
Under the hood
By far the most extensive changes have been to assure the accuracy and stability of the package from a software engineering perspective. Even though these are not visible to users, they make the
package more robust with hopefully fewer bugs. Indeed, the extensive data validation may help users debug their own errors.
• Added data validation to exported functions. Under the hood, each user-facing function carefully validates that the user has entered valid data using the {assertthat} package; if not, the
function fails quickly with an appropriate error message.
• Created unit tests for exported functions. Under the hood, the {testthat} package is now used for testing the outputs of each user-facing function. This should help the code base to be more
robust going forward with future developments.
• Most importantly, we created tests that compare results with the original reference {ALEPlot} package. These tests should ensure that any future code that breaks the accuracy of ALE calculations
will be caught quickly.
• Bootstrapped ALE values are now centred on the mean by default, instead of on the median. Mean averaging is generally more stable, especially for smaller datasets.
• The code base has been extensively reorganized for more efficient development moving forward.
• Numerous bugs have been fixed following internal usage and testing.
Known issues to be addressed in a future version
• Bootstrapping is not yet supported for ALE interactions (ale_ixn()).
• ALE statistics are not yet supported for ALE interactions (ale_ixn()).
ale 0.1.0
This is the first CRAN release of the ale package. Here is its official description with the initial release:
Accumulated Local Effects (ALE) were initially developed as a model-agnostic approach for global explanations of the results of black-box machine learning algorithms. (Apley, Daniel W., and Jingyu
Zhu. “Visualizing the effects of predictor variables in black box supervised learning models.” Journal of the Royal Statistical Society Series B: Statistical Methodology 82.4 (2020): 1059-1086
doi:10.1111/rssb.12377.) ALE has two primary advantages over other approaches like partial dependency plots (PDP) and SHapley Additive exPlanations (SHAP): its values are not affected by the presence
of interactions among variables in a model and its computation is relatively rapid. This package rewrites the original code from the ‘ALEPlot’ package for calculating ALE data and it completely
reimplements the plotting of ALE values.
(This package uses the same GPL-2 license as the {ALEPlot} package.)
This initial release replicates the full functionality of the {ALEPlot} package and a lot more. It currently presents three functions:
• ale(): create data for and plot one-way ALE (single variables). ALE values may be bootstrapped.
• ale_ixn(): create data for and plot two-way ALE interactions. Bootstrapping of the interaction ALE values has not yet been implemented.
• model_bootstrap(): bootstrap an entire model, not just the ALE values. This function returns the bootstrapped model statistics and coefficients as well as the bootstrapped ALE values. This is the
appropriate approach for small samples.
This release provides more details in the following vignettes (they are all available from the vignettes link on the main CRAN page at https://CRAN.R-project.org/package=ale):
• Introduction to the ale package
• Analyzing small datasets (fewer than 2000 rows) with ALE
• ale() function handling of various datatypes for x | {"url":"https://cran.mirror.garr.it/CRAN/web/packages/ale/news/news.html","timestamp":"2024-11-08T09:01:28Z","content_type":"application/xhtml+xml","content_length":"16678","record_id":"<urn:uuid:8093c8a7-75b8-4c81-8958-3d9270ed0132>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00184.warc.gz"} |
trucking cpm calculator
In the world of logistics and transportation, understanding the cost per mile (CPM) is crucial for trucking businesses to make informed decisions about pricing, budgeting, and profitability. A
trucking CPM calculator simplifies this process by providing a tool to accurately compute these expenses. Let’s delve into how to use such a calculator effectively.
How to Use
To utilize the trucking CPM calculator, follow these steps:
1. Input Data: Start by entering the relevant data, including total miles driven, total fuel cost, maintenance expenses, and any other applicable costs.
2. Click Calculate: After inputting the data, click the “Calculate” button to trigger the calculation process.
3. Review Results: Once calculated, the tool will display the CPM, offering insights into the cost per mile for the trucking operation.
The formula for calculating the cost per mile (CPM) in trucking is:
• Total Costs represents the sum of all expenses incurred in running the trucking operation.
• Total Miles Driven refers to the total distance covered by the vehicles.
Example Solve
Consider a trucking company that has incurred $10,000 in total expenses over a period and has driven 5,000 miles. Using the formula mentioned above:
This means that the cost per mile for this trucking operation is $2.
Q: What expenses should be included when calculating the CPM for trucking?
A: Include all relevant expenses such as fuel costs, maintenance, insurance, depreciation, and driver wages.
Q: Why is it important to know the CPM in trucking?
A: Knowing the CPM helps trucking businesses accurately assess their operational costs, set competitive pricing, and maximize profitability.
Q: Can the CPM vary for different types of trucks or routes?
A: Yes, factors like truck type, route efficiency, and market conditions can influence the CPM.
A trucking CPM calculator is a valuable tool for trucking businesses to gauge their operational costs accurately. By understanding how to use it effectively, companies can make informed decisions to
optimize their profitability and competitiveness in the industry. | {"url":"https://calculatordoc.com/trucking-cpm-calculator/","timestamp":"2024-11-06T12:17:27Z","content_type":"text/html","content_length":"91412","record_id":"<urn:uuid:e3855951-1447-4ebf-8b0e-cf411c8327fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00056.warc.gz"} |
Lego That Melts in Your Mouth
This Lego isn’t just fun to play with: you can also eat it, because it’s chocolate! Akihiro Mizuuchi has made a Lego-shaped mold for chocolate. He pours melted multi-colored chocolate into it, and it
hardens into tasty snap-together bricks. These bricks are hollow at the bottom, so they snap together well — if they haven’t already melted in your hands.
Wee ones: How many colors of Lego can you count in the picture?
Little kids: If you take a layer of chocolate Legos, then snap a layer of real Legos on top, then a layer of chocolate Lego, then real ones…can you eat the 9th layer? Bonus: If you start with 4
layers of chocolate, THEN stack 1 real Lego layer and alternate, can you eat this 9th layer?
Big kids: If Akihiro has to melt a chocolate bar to make 4 Lego bricks, how many bars does he need to make 16 bricks? Bonus: If he’s building a chocolate castle that is 5 bricks wide across the
front and back and 6 bricks across the left wall and right wall, and he builds 10 layers and then tops it with a 100-brick roof, how many chocolate Legos does he need to make for it?
The sky’s the limit: If Akihiro wants a giant Lego brick whose length can be divided by 1, 2, 3, 4, 5 or 6, what’s the *smallest* number of bumps long that it can be?
Wee ones: We count 7 colors: blue, white, green, pink, and 3 unique shades of brown.
Little kids: Yes, because all odd-numbered layers will be chocolate. Bonus: Not this time, since the 5th layer is now real Lego along with the odd layers that follow.
Big kids: 4 chocolate bars. Bonus: 320 Legos: 50 for the front wall, 50 more for the back, 60 each for the left and right walls, and 100 on top.
The sky’s the limit: 60 bumps long. You don’t need to multiply out the big number 1 x 2 x 3 x 4 x 5 x 6, because you don’t need so many factors. Once a number is divisible by 2 and 3, it’s
automatically divisible by 6. And once it’s divisible by 4, you don’t need to multiply again by 2 to be divisible by 2 or 6. All you need is 2 x 2 x 3 x 5. | {"url":"https://bedtimemath.org/fun-math-chocolate-legos/","timestamp":"2024-11-11T07:34:03Z","content_type":"text/html","content_length":"86872","record_id":"<urn:uuid:ac547ff7-cbfd-47a2-b63b-056753cccf5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00062.warc.gz"} |
Simon's Tech Blog
It has been a long time since my blog post (because of Covid, work, Elden Ring...). So I decided to study the "Spectral Primary Decomposition for rendering with sRGB Reflectance", which used in
previous posts, to recall my memory. It is an efficient technique to up-sample sRGB texture to spectral reflectance by multiplying the sRGB values with 3 precomputed basis functions:
In this post, I would like to find an efficient spectral up-sampling method which also support wider gamut (e.g. Display-P3...) or investigate why this technique does not support wider gamut.
Porting to Octave
In the paper, it provides sample source code written in Matlab. Since I do not have a Matlab license, so the first thing I need to do is to port the source code to the open source Octave (ported
source code can be found here). During the porting process, the fmincon() used for finding the 3 spectral primary basis functions in Octave does not work, so I switched to use sqp() instead (also
removed the linprog() from original source code).
Basis Functions generated in Octave
The resulting graph is not as smooth as the original paper. So I decided to try different initial value for the objective function. I chose a normalized Color Matching Function:
Basis Functions generated with normalized CMF initial value Code for generating normalized CMF initial value
The resulting curves look smoother with normalized CMF as initial value. Also, during the porting process, I switched to use CMF2006 2 degree observer instead of CMF1931 / 2006 10 degree observer
used in original source code.
Working with wider gamut
So the next step is to change the color primaries from sRGB to Display-P3 (which the original source listed as infeasible). As expected, the result is not good, not only saturated color cannot be
up-sampled, the color within the sRGB gamut are not similar to the original color, and saturated red color will have an orange tint after up-sampling: (Note that below images have Display-P3 color
profile attached, to view those saturated color outside sRGB gamut, a wide gamut monitor is needed)
Up-sampled saturated sRGB color Up-sampled saturated P3 color
So, I tried to modify the objective function opt_fn() used in sqp() to include a weight to minimize the sRGB primaries color difference:
Code snippet of the objective function with sRGB primaries weight
The result improves a bit and the up-sampled saturated red has a less orange tint:
Up-sampled saturated sRGB color with sRGB primaries weight Up-sampled saturated P3 color with sRGB primaries weight
Up to this point, all the precomputed spectral primary basis functions are within [0, 1] range (i.e. to not reflect more light in each basis function), I was wondering what if we relax this
constraint and enforce this limit after linearly combining all the basis functions. I have tried to relax the range of individual basis function to [-0.05, 1.05], [-0.075, 1.075] and [-0.1, 1.1]
(details can be found in the visualization website from modified source code). With the relaxed range, we can get very similar sRGB color after up-sampling:
However, for those saturated Display-P3 color, we still cannot up sample them exactly, and can only achieve slightly more saturated color compared to sRGB color:
The up-sampled saturated red is having a visible difference from the original color before up-sampling, I have tried to modify the objective function to only optimize the Red basis function (ignoring
the Green and Blue basis functions), and still cannot get an exact up-sampled saturated red from a D65 light source. May be it is impossible to produce the most saturated Display-P3 red with a D65
light source without violating the physical constraint.
Out of curiosity, I tried to plot the chromaticity diagram of the up-sampled color. The result shows that, using limited [0, 1] range, the up-sampling process can produce "more color" (but not
accurate, e.g. red color will be up-sampled to "orange-red"), while using relaxed constraint will reduce the up-sampled color gamut.
Chromaticity diagram of up-sampled color using limited [0, 1] range Chromaticity diagram of up-sampled color using relaxed [-0.1, 1.1] range
CMF Reference White
Up to this point, the calculation for the up-sampled color is using D65 as reference white. But one day, I saw this tweet:
The CMF is using an equal-energy white as its reference white. So I was wondering whether all my calculation was wrong and should add chromatic adaptation after CMF integration.
So, I decided to find the spectral reflectance of color checker to integrate with the CMF to verify whether chromatic adaption are needed after CMF integration. Using the color checker data found
from here, illuminating those grey patches with D65 and then integrate the result with CMF get the following results:
Illuminating grey patches with D65, integrate with CMF without CAT from Illuminant E sRGB value of measured Color Checker (2005)
Our computed sRGB value are very similar to the measured data, so it seems like we don't need an extra chromatic adaption to adapt the color from the CMF equal-energy reference white (or please let
me know if my maths are incorrect).
Optimizing up sampling function with Color Checker Data
After working with color checker data, I came up with an idea to modify the spectral basis objective function to include a weight to bias it to match with the neutral 6.5 grey patch spectral
reflectance data. We can get a decent match for the up-sampled spectral reflectance of color checker grey patches (i.e. white 9.5, neutral 8, neutral 6.5, neutral 5, neutral 3.5, neutral 2).
Spectral Basis computed for Display-P3 Spectral Basis weighted with Neutral 6.5 color checker patch
Up-sampled spectral reflectance of color checker grey patches using Spectral Basis computed Up-sampled spectral reflectance of color checker grey patches using Spectral Basis weighted with Neutral
for Display-P3 6.5 patch data
However, the up-sampled white color will have a slight round-trip error:
In this post, I have ported the original "Spectral Primary Decomposition" source code to Octave, tried to change it to up-sample Display-P3 color, but the result is not very good. Also, within a game
engine, we usually have exposure and tone mapping adjustment, which affect the final pixel color. So I was wondering whether the up-sampling method should take those parameters into account. But
doing so, the texture color meaning will be different from the PBR albedo texture. So, I will leave it for future investigation.
[1] https://graphics.geometrian.com/research/spectral-primaries.html
[2] http://yuhaozhu.com/blog/cmf.html
[3] https://babelcolor.com/colorchecker-2.htm
When performing spectral rendering, we need to use the Color Matching Function(CMF) to convert the spectral radiance to XYZ values, and then convert to RGB value for display. Different people have a
slight variation when perceiving color, and age may also affect how color are perceived too. So the CIE defines several standard observers for an average person. The commonly used CMF are CIE 1931 2°
Standard Observer and CIE 1964 10° Standard Observer. Beside these 2 CMF, there also exist other CMF such as Judd and Vos modified CIE 1931 2° CMF and CIE 2006 CMF. In this post, I will try to
compare the images rendered with different CMF (as well as some analytical approximation). A demo can be downloaded here (the demo renders using wavelength between [380, 780]nm, which may introduce
some error with CMF that have a larger range).
Left: rendered with CIE2006 CMF
Right: rendered with CIE1931 CMF
CMF Luminance
When I was implementing different CMF into my renderer, replacing the CMF directly will result in slightly different brightness of the rendered images:
Rendered with 1931 CMF Rendered with 1964 CMF
This is because the renderer uses photometric units (e.g. lumen, lux..) to define the brightness of the light sources. Since the definition of luminous energy depends on the luminosity function
(usually the y(λ) of CMF), we need to calculate the intensity of the light source with respect to the chosen CMF. Using the correct luminosity function, both rendered images have similar brightness:
Rendered with 1931 CMF Rendered with 1964 CMF + luminance adjustment
CMF White Point
When using different CMF, the white point of different standard illuminant will be slightly different:
White point from wikipedia
Since we are dealing with game texture, color are usually defined in sRGB with a D65 white point, we need to find the white point of the D65 illuminant for the CMF that will be tested in this post.
Unfortunately, I can't find D65 white point for the CIE 2006 CMF on the internet, so I calculated it myself (The calculation steps can be found in the Colab source code):
CIE 2006 2° : (0.313453, 0.330802)
CIE 2006 10° : (0.313786, 0.331275)
But when I rendered some images with and without chromatic adaptation, the result looks similar:
1964 CMF without chromatic adaptation 1964 CMF with chromatic adaptation
So I searched on the internet, I can't find any information whether we need to chromatic adapt the rendered image due to different white point when using different CMF... May be this is because the
difference is so small that applying chromatic adaptation makes no visible difference.
CIE 2006 CMF analytical approximation
The popular CIE 1931 and 1964 CMF have simple analytical approximation, such as: "Simple Analytic Approximations to the CIE XYZ Color Matching Functions" (which will be tested in this post). The
newer CIE 2006 CMF lacks such an approximation. So I derived one using similar methods and the curve fitting process can be found in the Colab source code.
2006 2° lobe approximation:
2006 2° lobe approximation shader source code black lines: exact 2006 2° CMF
color lines: approximated 2006 2° CMF
2006 10° lobe approximation:
2006 10° lobe approximation shader source code black lines: exact 2006 10° CMF
color lines: approximated 2006 10° CMF
Saturated lights comparison
With the above changes to the path tracer, we can render some images for comparison. A scene with several saturated lights using sRGB color (1,0,0), (1,1,0), (0,1,0), (0,1,1), (0,0,1), (1,0,1) is
tested (which will be spectral up-sampled). 10 different CMF are used:
• CIE 1931 2°
• CIE 1931 2° with Judd Vos adjustment
• CIE 1931 2° single lobe analytic approximation
• CIE 1931 2° multi lobe analytic approximation
• CIE 1964 10°
• CIE 1964 10° single lobe analytic approximation
• CIE 2006 2°
• CIE 2006 2° lobe analytic approximation
• CIE 2006 10°
• CIE 2006 10° lobe analytic approximation
Here are the results:
CIE 1931 2° CIE 1931 2° with Judd Vos adjustment
CIE 1931 2° single lobe analytic approximation CIE 1931 2° multi lobe analytic approximation
CIE 1964 10° CIE 1964 10° single lobe analytic approximation
CIE 2006 2° CIE 2006 2° lobe analytic approximation
CIE 2006 10° CIE 2006 10° lobe analytic approximation
From Wikipedia:
"The CIE 1931 CMF is known to underestimate the contribution of the shorter blue wavelengths."
So I was expecting some variation for the blue color when using different CMF. But to my surprise, only the CIE 1931 CMF suffer from the [S:“Blue Turns Purple” Problem:S] (Edited: As pointed out by
troy_s on twitter, the reference I provided was wrong, the link talks about psychophysical effect, while the current issue is mishandling of light data) which we have encountered in previous posts
(i.e. saturated sRGB blue light will render purple color). Originally, after previous blog post, I was investigating this issue and was suspecting the ACES tone mapper cause the color shift (as this
issue does not happen when rendering in narrow sRGB gamut with Reinhard tone mapper). I was thinking may be we can use the OKLab color space to get the hue value before tone mapping and tone map only
the lightness to keep the blue color. But when I tried with this approach, the hue value obtained before tone mapping is still purple color, which suggest may not be the tone mapper causing the issue
(or somehow my method of getting the hue value from HDR value is wrong...). So I have no idea on how to solve the issue and randomly toggle some debug view modes. Accidentally, I found that some of
the purple color are actually inside my AdobeRGB monitor display gamut (but outside the sRGB gamut on another monitor...), so the problem is not only caused by out of gamut color producing the purple
The purple color on the wall is within displayable Adobe RGB gamut Highlighting out of gamut pixel with cyan color
So I decided to investigate the problem for spectral renderer first (and ignore the RGB renderer), and that's why I tested different CMF in this blog post. (Also, as a side note, the behavior for the
blue turns purple color problem is a bit different between RGB and spectral renderer, using a more saturated blue color, e.g. (0, 0, 1) in Rec2020, can hide this issue in RGB renderer while using the
same more saturated blue color with 1931 CMF spectral renderer still suffer from the problem, while other CMF doesn't have this issue.)
Color Checker comparison
Next, we compare a color checker lit by a white light source. Since my spectral renderer need to maintain compatibility with RGB rendering and I was too lazy to implement spectral material using
measured spectral reflectance, so both the color checker and the light source are up-sampled from sRGB color.
CIE 1931 2° CIE 1931 2° with Judd Vos adjustment
CIE 1931 2° single lobe analytic approximation CIE 1931 2° multi lobe analytic approximation
CIE 1964 10° CIE 1964 10° single lobe analytic approximation
CIE 2006 2° CIE 2006 2° lobe analytic approximation
CIE 2006 10° CIE 2006 10° lobe analytic approximation
From the above results, different CMF have similar looks except the blue color.
In this post, we have compare different CMF, provided an analytical approximation for the CIE 2006 CMF and calculate the D65 white point for CIE 2006 CMF (the math can be found in the Colab source
code). All the CMF produce similar color except the blue color, with CMF newer than the 1931 CMF can render saturated blue color correctly without turning it into purple color. May be we should use
newer CMF instead, especially when working with wide gamut color. And the company Konica Minolta points out that: the CIE 1931 CMF has issue with wider color gamut with OLED display (which suggest to
use CIE 2015 CMF instead). But sadly, I cannot find the data for CIE 2015 CMF, so it is not tested in this post.
[1] https://en.wikipedia.org/wiki/CIE_1931_color_space
[2] http://cvrl.ioo.ucl.ac.uk/
[2] http://jcgt.org/published/0002/02/01/paper.pdf
[3] https://en.wikipedia.org/wiki/ColorChecker
[4] https://en.wikipedia.org/wiki/Standard_illuminant
[5] https://www.rit.edu/cos/colorscience/rc_useful_data.php
[6] https://sensing.konicaminolta.asia/deficiencies-of-the-cie-1931-color-matching-functions/ | {"url":"https://simonstechblog.blogspot.com/","timestamp":"2024-11-10T01:40:53Z","content_type":"application/xhtml+xml","content_length":"139033","record_id":"<urn:uuid:c34488ab-dd32-424d-b50e-8a02bb6cea7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00119.warc.gz"} |
Can someone help with the translation of the headings of this page and #12 in 1869?
Best Answer
• [Editing to add dots in place of the spaces that it ate]
Sorszám: line number
A vőlegény: the groom
...neve és polgári állása: 's name and civil position
...lakhelye: 's place of residence
...vallása: 's religion
...életkora: 's age ("life-age")
...állapota: 's status
..........nőtlen: single ("without woman/wife")
.........özvegy: widowed
esketési év, hó, nap: year, month, day of wedding
A menyasszony: the bride
...neve és polgári állása: 's name and civil position
...lakhelye: 's place of residence
...vallása: 's religion
...kora: 's age
...állapota: 's status
........hajadon: single, maiden
........özvegy: widowed
Tanuk: witnesses
Eskető neve: name of officiant
Josef Mislyan, Lak., R.C., 21, single
November 2
An. Csuprinka, Lak., G.C., 18, single
Mislyan (?) Rimak(?) Josef
Kamenszky Jósef
If I'm right in putting together the clues, then "Lak." is short for Lakárt, Ung county. But it could be much more easily deciphered if you simply gave the location for the register.
• Would Rimak be a name or maybe a title of some kind?
• I think they forgot to record the given name of the first witness (surname Mislyan, same as the groom). Rimak (or possibly Kimak?) Josef are the surname and given name of the other witness.
• That makes sense. Thanks so much for your help and for your quick response.
This discussion has been closed. | {"url":"https://community.familysearch.org/en/discussion/131884/can-someone-help-with-the-translation-of-the-headings-of-this-page-and-12-in-1869","timestamp":"2024-11-04T02:36:12Z","content_type":"text/html","content_length":"391505","record_id":"<urn:uuid:0380097f-bc8d-411f-a31d-3917dc4b8e52>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00271.warc.gz"} |
Rank Tests for Randomized Blocks when the Alternatives have an a Priori Ordering
Ann. Math. Statist. 38(3): 867-877 (June, 1967). DOI: 10.1214/aoms/1177698880
Let $X_{ij}, i = 1, \cdots, n, j = 1, \cdots, k$, be independent with $X_{ij}$ having the continuous distribution function $P(X_{ij} \leqq x) = F_j(x - b_i)$ where $b_i$ is the nuisance parameter
corresponding to block $i$. (These assumptions shall be called the $H_A$ assumptions.) This paper is concerned with procedures for testing the null hypothesis \begin{equation*}\tag{0.1}H_0 : F_j = F
\text{(unknown)}, \quad j = 1, \cdots, k,\end{equation*} which are sensitive to the ordered alternatives \begin{equation*}\tag{0.2}H_a : F_1 \geqq F_2 \geqq \cdots \geqq F_k,\end{equation*} where at
least one of the inequalities is strict. In particular, we introduce a test statistic $(Y)$ based on a sum of Wilcoxon signed-rank statistics. In Section 2 we develop the asymptotic distribution of
$Y$ and find that, under $H_0, Y$ is neither distribution-free for finite $n$, nor asymptotically distribution-free. However, a consistent estimate of the null variance of $Y$ is used to define a
procedure which is asymptotically distribution-free. In Section 3 we derive, under the $H_A$ assumptions, necessary and sufficient conditions for the consistency of $Y$ and two of its nonparametric
competitors, viz., (1) Jonckheere's $\tau$ test [11] based on Kendall's rank correlation coefficient between observed order and postulated order in each block; (2) Page's $\rho$ test [17] based on
Spearman's rank correlation coefficient between observed order and postulated order in each block. We find that (i) $Y$ is consistent if and only if $\sum_{u < v} \int H_u dH_v/k(k - 1) > \frac{1}{4}
$ where $H_u = F^\ast_u F_u, u = 1, \cdots, k$, (ii) Jonckheere's test is consistent if and only if $\sum_{u < v} \int F_u dF_v/k(k - 1) > \frac{1}{4}$, and (iii) Page's test is consistent if and
only if $\sum_{u < v} (v - u) \int F_u dF_v > k(k - 1) \cdot (k + 1)/12$. Section 4 is devoted to efficiency comparisons of the rank tests with respect to a normal theory $t$-test defined in Section
1. For a class of shift alternatives we show that the Pitman efficiency of $Y$ with respect to $t(E(Y, t))$ is greater than .864 for every $F$ and every $k$. When $F$ is normal, $E(Y, t) = .963$ for
$k = 3$ and $\rightarrow .989$ as $k \rightarrow \infty$. These values compare favorably with the corresponding ones of Page's test (.716, .955) and Jonckheere's procedure (.694, .955). For these
shift alternatives we also show that $.576 \leqq E(\rho, t) \leqq \infty$ and $.576 \leqq E(\tau, t) \leqq \infty$.
Download Citation
Myles Hollander. "Rank Tests for Randomized Blocks when the Alternatives have an a Priori Ordering." Ann. Math. Statist. 38 (3) 867 - 877, June, 1967. https://doi.org/10.1214/aoms/1177698880
Published: June, 1967
First available in Project Euclid: 27 April 2007
Digital Object Identifier: 10.1214/aoms/1177698880
Rights: Copyright © 1967 Institute of Mathematical Statistics
Vol.38 • No. 3 • June, 1967 | {"url":"https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-38/issue-3/Rank-Tests-for-Randomized-Blocks-when-the-Alternatives-have-an/10.1214/aoms/1177698880.full","timestamp":"2024-11-02T23:47:56Z","content_type":"text/html","content_length":"143746","record_id":"<urn:uuid:1de2bd69-72e9-4332-8629-8b0fc8abbf3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00553.warc.gz"} |
Martin Flashman
Cell Phone: 707-832-9973
At Forty-Second Annual State of Jefferson Mathematics Congress October 5, 2013
Professor Emeritus of Mathematics
Fall, 2017 - Spring, 2020, Fall, 2021- Spring, 2024:Visitor to University of Arizona Mathematics Department, Tucson, AZ.
Find out about the Sensible Calculus Program
Older Version.
Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s. A Reference and Resource Book on Function Visualizations Using Mapping Diagrams (In Development-Draft Version
8-2017) Papers/Presentations/etc. [2019-2024]
T & L Seminar, University of New South Wales, Sydney. A Second Look at the Presentation from ICME. MAPPING DIAGRAMS: FUNCTION VISUALIZATION OF REAL AND COMPLEX ANALYSIS AND MATRIX ALGEBRA, ( Video On
YouTube). July 4, 2024.
International Study Group on the Relations between History and Pedagogy of Mathematics (HPM 2024). Sydney, Australia. Two Examples from History: Mapping Diagrams to Visualize Relations and Functions.
July 1, 2024.
Mathematics Educator Appreciation Day (MEAD). Tucson, AZ. " 'And Then'... Compositions of functions-They're Everywhere ". January 27,2024
Joint Mathematics Meetings, San Francisco, CA.
AMS Special Session on Mathematics and Philosophy."Do We Need a Separate Philosophy of Geometry?" Jan. 5, 2024.
Fall 2023 ArizMATYC Conference, Chandler AZ. "Visualizing the algebra of equations and inequalities with mapping diagrams." October 6, 2023
Mathematics Educator Appreciation Day (MEAD).(Links) "How Many Ways Can You Solve a Quadratic Equation Visually?" Tucson, AZ. January 21,2023 .
48th AMATYC Annual Conference Virtual Days. "Darts-Visualizing Probability: Simulations, Graphs & Mapping Diagrams" . (GeoGebra book with video) Dec 2, 2022.
BEAM Summer Away, LaVerne University, LaVerne, CA, two one week courses, July 5-19, 2022.
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ArizMATYC/MAA Southwestern Section Joint Conference, "Visualizing Linear and Nonlinear Functions and Transformations of Several Variables with Mapping Diagrams"
ASU Polytechnic Campus, April 1, 2022 (GeoGebra Book: https://flashman.tiny.us/ArizMATYC)
Mathematics Educator Appreciation Day (MEAD).(Links)
Visualizing Calculus With Mapping Diagrams: Making Sense Of Differentiation And Integration.Tucson, AZ. January 22,2022.
CMC-South Annual Conference 2021, "Visualizing the Algebra of Equations with Mapping Diagrams". November 6, 2021. (Links)
14th International Congress on Mathematical Education, "MAPPING DIAGRAMS: FUNCTION VISUALIZATION OF REAL AND COMPLEX ANALYSIS AND MATRIX ALGEBRA", Shanghai, July 14, 2021. Topic Study Group 23
Visualization in the teaching and learning of mathematics (GeoGebra Book includes Paper) (On YouTube)
ATM Conference 2021. "Visualizing the algebra of equations and inequalities with mapping diagrams", April 7 & 8,2021. (Links)
AMATYC Webinar, "Visualizing Solving Equations with Function Mapping Diagrams" September 22, 2020. (On YouTube.)
Joint Mathematics Meetings Denver, CO Jan. 15&17, 2020: MAA Minicourse 2020: Visual Complex Analysis- GeoGebra Tools and Mapping Diagrams GeoGebra Book.
HSU Math Department Colloquium,"Linear Algebra & Mapping Diagrams: Old & New Visualizations" (GeoGebra Book) Sept. 5, 2019.
Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture, Edited by Goldstine, McKenna, and Fenyvesi, July, 2019, pp 295-302.
"Mapping Diagrams and a New Visualization of Complex Functions with GeoGebra"
Mapping Diagrams and Visualization of Complex Function. GeoGebra book, https://ggbm.at/gutmhcp8, July, 2019.
MEI Conference 2019 (Bath, UK), Saturday, June 29, 2019.[Links]
"Visualizing Functions with Mapping Diagrams"
"Visualizing Calculus with Mapping Diagrams"
ICTCM 2019 (Scottsdale, AZ). Saturday, March 16, 2019. [Links]
"GeoGebra Tools for Visualizing Integration with Mapping Diagrams"
"GeoGebra Tools for Creating Mapping Diagrams: From Worksheets to Books"
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A resume.(pdf) ; A biographical sketch.
POM SIGMAA (Philosophy of Mathematics) Find out more about this Special Interest Group of the MAA.
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WORK IN PROGRESS (1-14-2023)
This site is still under reorganization. It is now moved from a server at Humboldt State University to one at neocities.org.
Links are being reconfigured and some are inactive or incorrect.
Older pages need reworking to be made handicapped accessible.
Links starting with "users,humboldt.edu/flashman" are being readdressed as needed with "flashman.neocities.org"
I continue to work on this.
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Question Text यह समधा तीय अवकल समी है।
Updated On Feb 7, 2023
Topic Trigonometry
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 60
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Power and Exponential functions in math.h Header file in C/ C++
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In a simple way a Library is collection of builtin functions.One of the header file of standard C library is "math.h".As the name itself suggests,It defines various mathematical functions.The
noticable thing is that all the arguements and return types of the functions of this header file is double.In this post we will discuss about power and exponentiation functions in this header file.
Table of contents:
1. double pow(double x,double y)
2. double exp(double x)
double pow(double x,double y)
Unlike Python and other programming languages,C doesn't have power operator.So we use a builtin function to compute such operations.pow() is a built in function in math.h header file,which takes two
doubles as input and returns a double.The function prototype looks like thisdouble pow(double x,double y).Takes two doubles x and y i.e. base and exponent respectively.Here,x is raised to the power
of y i.e. x^y.Let us consider an example to know how it works.
Input: 2.0 3.0
Output: 8.00
The function finds the value of 2.0 raised to the power of 3.0 (2.0^3.0) which is equal to 8.00 and returns the result.
Input: 5.0 4.0
Output: 125.00
The function finds the value of 5.0 raised to the power of 4.0 (5.0^4.0) which is equal to 625.00 and returns the result.
Lets write a code to implement the above example:
int main(){
double x,y;
printf("Enter the base and exponent values");
scanf("%lf %lf",&x,&y);
double result=pow(x,y);
printf("The Power value is %.2lf",result);
return 0;
Enter the base and exponent values 3.0 4.0
The Power value is 81.00
As a learner,think about some interesting cases like,
1.does pow() work for negative inputs?
2.What if base or exponent is negative?
Now try passing negative values to the function and verify them.
The answer for the first question is YES.
pow() works for negative inputs as well.
To understand answer for the 2nd question,u need to have some mathematical knowledge.
Basically,when a exponent is negative i.e. (x^-y),it can be written as 1/(x^y).pow() function handles that case too.
Input: 2.0 -3.0
Output: 0.125000
The value of 2.0^-3.0 is equal to 1/8 which is equal to 0.125
If the base is a negative value,simply vwe get a positive value for even powers and negative for odd powers.i.e
Input: -2.0 3.0
Output: -8.000000
the situation can be explained as -2*-2*-2 which is equal to -8.
double exp(double x)
The function is used to find exponential of given value.exp() is also a built in function defined in "math.h" header file.It takes a parameter of type double and returns a double whose value is equal
to e raised to the xth power i.e. e^x.Same as pow(),we have to include math.h header file in our program to access the function.Its function prototype looks like double exp(double x);.Let us consider
an example to know how it works.
Input: 1
Output: 2.718282
The function finds the value of e raised to the power of 1.So we get the value of e which is equal to 2.718282.
Input: 5
Output: 2.718282
The value when e is raised to the power of 5 is e^5 ie 148.413159
The function exp() can also be written as pow(e,x),where the value of e is 2.718282.
Let's write a program to implement the abouve function.
#include <stdio.h>
#include <math.h>
int main()
double x;
printf("Enter the value of exponent");
double result=exp(x);
printf("\nThe Exponential value is %lf",result);
return 0;
Enter the value of exponent 5
The Exponential value is 148.413159
Similarly,if we pass negative parameters the function still works super fine.
When the input is negative,the function has to return e^-x,which can be written as 1/(e^-x).
Input: -5
Output 0.006738
We know that e^5 is 148.413159 as we done before,then the value of e^-5 will be 1/148.413159 which is equal to 0.006738.
Thanks for reading this article at OpenGenus :), Have a good day. | {"url":"https://iq.opengenus.org/power-exponential-math-h-c/","timestamp":"2024-11-14T11:59:25Z","content_type":"text/html","content_length":"66799","record_id":"<urn:uuid:cd1505b6-d034-4ce7-bab3-7e9328f3e400>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00575.warc.gz"} |
How far can a spectrum analyzer go in terms of transistor fmax?
I know the title doesn't make a lot of sense. What I am trying to ask is; let's say that I have a spectrum analyzer, a tinysa ultra, with 800MHz bandwidth, which claims to be calibrated to 5.3GHz in
ultra mode. It uses some kind of harmonic mixing and spur removal algorithms but I am not quite sure how it works yet.
I am going to buy some transistors and schottky diodes to build radio frequency receivers and some testing equipment like a return loss bridge and signal sources.
What kind of fmax should I aim for? Should I limit myself to transistors with fmax<800MHz? Maybe 5.3GHz? I know there is no hard rule about this kind of thing and also that it is pretty much
impossible to actually get any gain at fmax from a transistor. Most of them seem to be specified for about a third of their transition frequency.
What kind of scares me is that I will buy these transistors, build some circuits and then they will oscillate at some very high frequency that I can't detect, though I suppose it is not very likely
that a transistor can be made to oscillate at frequencies near fmax especially considering that the rest of the circuit will have some loss even if I were to try and make it oscillate.
I am thinking of getting some BFP183 for amplification/oscillators and some cheap medium barrier schottkies for mixers. I've already got lots of lower frequency transistors and I've been building RF
circuits as a hobby for about 10 years or so. It's just that seeing "GHz" in a datasheet kinda scares me
BFP183 seems to have an fmax of about 10.3GHz based on its fT, Rbb and Ccb. I figure I can just run it at a lower than ideal current to lower its fT, although I will need to run it at a higher
current to drive those schottky diodes for mixers. Maybe I can buy a more powerful but slower transistor to do that. The reason I am considering it is because it seems to be quite cheap. Another
reason is that I need a low noise figure for some weak signal projects.
I am open to transistor-diode suggestions and thanks in advance for your answers. | {"url":"https://www.eevblog.com/forum/rf-microwave/how-far-can-a-spectrum-analyzer-go-in-terms-of-transistor-fmax/","timestamp":"2024-11-12T22:16:50Z","content_type":"application/xhtml+xml","content_length":"52047","record_id":"<urn:uuid:f4ab8765-6302-4b22-96ac-747af1c3eceb>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00369.warc.gz"} |
Add white Gaussian noise to input signal
comm.AWGNChannel adds white Gaussian noise to the input signal.
When applicable, if inputs to the object have a variable number of channels, the EbNo, EsNo, SNR, BitsPerSymbol, SignalPower, SamplesPerSymbol, and Variance properties must be scalars.
To add white Gaussian noise to an input signal:
1. Create the comm.AWGNChannel object and set its properties.
2. Call the object with arguments, as if it were a function.
To learn more about how System objects work, see What Are System Objects?
awgnchan = comm.AWGNChannel creates an additive white Gaussian noise (AWGN) channel System object™, awgnchan. This object then adds white Gaussian noise to a real or complex input signal.
awgnchan = comm.AWGNChannel(Name,Value) creates a AWGN channel object, awgnchan, with the specified property Name set to the specified Value. You can specify additional name-value pair arguments in
any order as (Name1,Value1,...,NameN,ValueN).
Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.
If a property is tunable, you can change its value at any time.
For more information on changing property values, see System Design in MATLAB Using System Objects.
NoiseMethod — Noise level method
'Signal to noise ratio (Eb/No)' (default) | 'Signal to noise ratio (Es/No)' | 'Signal to noise ratio (SNR)' | 'Variance'
Noise level method, specified as 'Signal to noise ratio (Eb/No)', 'Signal to noise ratio (Es/No)', 'Signal to noise ratio (SNR)', or 'Variance'. For more information, see Relationship Between Eb/No,
Es/No, and SNR Modes and Specifying Variance Directly or Indirectly.
Data Types: char
EbNo — Ratio of energy per bit to noise power spectral density
10 (default) | scalar | row vector
Ratio of energy per bit to noise power spectral density (Eb/No) in decibels, specified as a scalar or 1-by-N[C] vector. N[C] is the number of channels.
Tunable: Yes
This property applies when NoiseMethod is set to 'Signal to noise ratio (Eb/No)'.
Data Types: double
EsNo — Ratio of energy per symbol to noise power spectral density
10 (default) | scalar | row vector
Ratio of energy per symbol to noise power spectral density (Es/No) in decibels, specified as a scalar or 1-by-N[C] vector. N[C] is the number of channels.
Tunable: Yes
This property applies when NoiseMethod is set to 'Signal to noise ratio (Es/No)'.
Data Types: double
SNR — Ratio of signal power to noise power
10 (default) | scalar | row vector
Ratio of signal power to noise power in decibels, specified as a scalar or 1-by-N[C] vector. N[C] is the number of channels.
Tunable: Yes
This property applies when NoiseMethod is set to 'Signal to noise ratio (SNR)'.
Data Types: double
BitsPerSymbol — Number of bits per symbol
1 (default) | positive integer
Number of bits per symbol, specified as a positive integer.
This property applies when NoiseMethod is set to 'Signal to noise ratio (Eb/No)'.
Data Types: double
SignalPower — Input signal power
1 (default) | positive scalar | row vector
Input signal power in watts, specified as a positive scalar or 1-by-N[C] vector. N[C] is the number of channels. The object assumes a nominal impedance of 1 Ω.
Tunable: Yes
This property applies when NoiseMethod is set to 'Signal to noise ratio (Eb/No)', 'Signal to noise ratio (Es/No)', or 'Signal to noise ratio (SNR)'.
Data Types: double
SamplesPerSymbol — Number of samples per symbol
1 (default) | positive integer | row vector
Number of samples per symbol, specified as a positive integer or 1-by-N[C] vector. N[C] is the number of channels.
This property applies when NoiseMethod is set to 'Signal to noise ratio (Eb/No)' or 'Signal to noise ratio (Es/No)'.
Data Types: double
VarianceSource — Source of noise variance
'Property' (default) | 'Input port'
Source of noise variance, specified as 'Property' or 'Input port'.
• Set VarianceSource to 'Property' to specify the noise variance value using the Variance property.
• Set VarianceSource to 'Input port' to specify the noise variance value using an input to the object, when you call it as a function.
For more information, see Specifying Variance Directly or Indirectly.
This property applies when NoiseMethod is 'Variance'.
Data Types: char
Variance — White Gaussian noise variance
1 (default) | positive scalar | row vector
White Gaussian noise variance, specified as a positive scalar or 1-by-N[C] vector. N[C] is the number of channels.
Tunable: Yes
This property applies when NoiseMethod is set to 'Variance' and VarianceSource is set to 'Property'.
Data Types: double
RandomStream — Source of random number stream
'Global stream' (default) | 'mt19937ar with seed'
Source of random number stream, specified as 'Global stream' or 'mt19937ar with seed'.
• When you set RandomStream to 'Global stream', the object uses the MATLAB^® default random stream to generate random numbers. To generate reproducible numbers using this object, you can reset the
MATLAB default random stream. For example reset(RandStream.getGlobalStream). For more information, see RandStream.
• When you set RandomStream to 'mt19937ar with seed', the object uses the mt19937ar algorithm for normally distributed random number generation. In this scenario, when you call the reset function,
the object reinitializes the random number stream to the value of the Seed property. You can generate reproducible numbers by resetting the object.
For a complex input signal, the object creates the random data as follows:
is the number of samples and
is the number of channels.
This property applies when NoiseMethod is set to 'Variance'.
Data Types: char
Seed — Initial seed
67 (default) | nonnegative integer
Initial seed of the mt19937ar random number stream, specified as a nonnegative integer. For each call to the reset function, the object reinitializes the mt19937ar random number stream to the Seed
This property applies when RandomStream is set to 'mt19937ar with seed'.
Data Types: double
outsignal = awgnchan(insignal) adds white Gaussian noise, as specified by awgnchan, to the input signal. The result is returned in outsignal.
outsignal = awgnchan(insignal,var) specifies the variance of the white Gaussian noise. This syntax applies when you set the NoiseMethod to 'Variance' and VarianceSource to 'Input port'.
For example:
awgnchan = comm.AWGNChannel('NoiseMethod','Variance', ...
'VarianceSource','Input port');
var = 12;
outsignal = awgnchan(insignal,var);
Input Arguments
insignal — Input signal
scalar | vector | matrix
Input signal, specified as a scalar, an N[S]-element vector, or an N[S]-by-N[C] matrix. N[S] is the number of samples and N[C] is the number of channels.
This object accepts variable-size inputs. After the object is locked, you can change the size of each input channel, but you cannot change the number of channels. For more information, see
Variable-Size Signal Support with System Objects.
Data Types: double
Complex Number Support: Yes
var — Variance of additive white Gaussian noise
positive scalar | row vector
Variance of additive white Gaussian noise, specified as a positive scalar or 1-by-N[C] vector. N[C] is the number of channels, as determined by the number of columns in the input signal matrix.
Object Functions
To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named obj, use this syntax:
Common to All System Objects
step Run System object algorithm
release Release resources and allow changes to System object property values and input characteristics
reset Reset internal states of System object
Create Default AWGN Channel System Object
Create an AWGN channel System object with the default configuration. Pass signal data through this channel.
Create an AWGN channel object and signal data.
awgnchan = comm.AWGNChannel;
insignal = randi([0 1],100,1);
Send the input signal through the channel.
outsignal = awgnchan(insignal);
Add White Gaussian Noise to 8-PSK Signal
Modulate an 8-PSK signal, add white Gaussian noise, and plot the signal to visualize the effects of the noise.
Modulate the signal.
modData = pskmod(randi([0 7],2000,1),8);
Add white Gaussian noise to the modulated signal by passing the signal through an additive white Gaussian noise (AWGN) channel.
channel = comm.AWGNChannel('EbNo',20,'BitsPerSymbol',3);
Transmit the signal through the AWGN channel.
channelOutput = channel(modData);
Plot the noiseless and noisy data by using scatter plots to visualize the effects of the noise.
Change the EbNo property to 10 dB to increase the noise.
Pass the modulated data through the AWGN channel.
channelOutput = channel(modData);
Plot the channel output. You can see the effects of increased noise.
Process Signals When Number of Channels Changes
Pass a single-channel and multichannel signal through an AWGN channel System object™.
Create an AWGN channel System object with the Eb/No ratio set for a single channel input. In this case, the EbNo property is a scalar.
channel = comm.AWGNChannel('EbNo',15);
Generate random data and apply QPSK modulation.
data = randi([0 3],1000,1);
modData = pskmod(data,4,pi/4);
Pass the modulated data through the AWGN channel.
rxSig = channel(modData);
Plot the noisy constellation.
Generate two-channel input data and apply QPSK modulation.
data = randi([0 3],2000,2);
modData = pskmod(data,4,pi/4);
Pass the modulated data through the AWGN channel.
rxSig = channel(modData);
Plot the noisy constellations. Each channel is represented as a single column in rxSig. The plots are nearly identical, because the same Eb/No value is applied to both channels.
title('First Channel')
title('Second Channel')
Modify the AWGN channel object to apply a different Eb/No value to each channel. To apply different values, set the EbNo property to a 1-by-2 vector. When changing the dimension of the EbNo property,
you must release the AWGN channel object.
channel.EbNo = [10 20];
Pass the data through the AWGN channel.
rxSig = channel(modData);
Plot the noisy constellations. The first channel has significantly more noise due to its lower Eb/No value.
title('First Channel')
title('Second Channel')
Add AWGN Using Noise Variance Input Port
Apply the noise variance input as a scalar or a row vector, with a length equal to the number of channels of the current signal input.
Create an AWGN channel System object™ with the NoiseMethod property set to 'Variance' and the VarianceSource property set to 'Input port'.
channel = comm.AWGNChannel('NoiseMethod','Variance', ...
'VarianceSource','Input port');
Generate random data for two channels and apply 16-QAM modulation.
data = randi([0 15],10000,2);
txSig = qammod(data,16);
Pass the modulated data through the AWGN channel. The AWGN channel object processes data from two channels. The variance input is a 1-by-2 vector.
rxSig = channel(txSig,[0.01 0.1]);
Plot the constellation diagrams for the two channels. The second signal is noisier because its variance is ten times larger.
Repeat the process where the noise variance input is a scalar. The same variance is applied to both channels. The constellation diagrams are nearly identical.
rxSig = channel(txSig,0.2);
Set Random Number Seed for Repeatability
Specify a seed to produce the same outputs when using a random stream in which you specify the seed.
Create an AWGN channel System object™. Set the NoiseMethod property to 'Variance', the RandomStream property to 'mt19937ar with seed', and the Seed property to 99.
channel = comm.AWGNChannel( ...
'NoiseMethod','Variance', ...
'RandomStream','mt19937ar with seed', ...
Pass data through the AWGN channel.
y1 = channel(zeros(8,1));
Pass another all-zeros vector through the channel.
y2 = channel(zeros(8,1));
Because the seed changes between function calls, the output is different.
Reset the AWGN channel object by calling the reset function. The random data stream is reset to the initial seed of 99.
Pass the all-zeros vector through the AWGN channel.
y3 = channel(zeros(8,1));
Confirm that the two signals are identical.
Relationship Between Eb/No, Es/No, and SNR Modes
For uncoded complex input signals, comm.AWGNChannel relates E[b]/N[0, ]E[s]/N[0], and SNR according to these equations:
E[s]/N[0] = E[b]/N[0] + 10log[10](k) in dB
• E[s] represents the signal energy in joules.
• E[b] represents the bit energy in joules.
• N[0] represents the noise power spectral density in watts/Hz.
• N[sps] represents the number of samples per symbol, SamplesPerSymbol.
• k represents the number of information bits per input symbol, BitsPerSymbol.
For real signal inputs, the comm.AWGNChannel relates E[s]/N[0 ]and SNR according to this equation:
• All values of power assume a nominal impedance of 1 ohm.
• The equation for the real case differs from the corresponding equation for the complex case by a factor of 2. Specifically, the object uses a noise power spectral density of N[0]/2 watts/Hz for
real input signals, versus N[0] watts/Hz for complex signals.
For more information, see AWGN Channel Noise Level.
Specifying Variance Directly or Indirectly
To directly specify the variance of the noise generated by comm.AWGNChannel, specify VarianceSource as:
• "Property", then set NoiseMethod to "Variance" and specify the variance with the Variance property.
• "Input port", then specify the variance level for the object as an input with an input argument, var.
To specify variance indirectly, that is, to have it calculated by comm.AWGNChannel, specify VarianceSource as "Property" and the NoiseMethod as:
• "Signal to noise ratio (Eb/No)", where the object uses these properties to calculate the variance:
□ EbNo, the ratio of bit energy to noise power spectral density
□ BitsPerSymbol
□ SignalPower, the actual power of the input signal samples
□ SamplesPerSymbol
• "Signal to noise ratio (Es/No)", where the object uses these properties to calculate the variance:
□ EsNo, the ratio of signal energy to noise power spectral density
□ SignalPower, the actual power of the input signal samples
□ SamplesPerSymbol
• "Signal to noise ratio (SNR)", where the object uses these properties to calculate the variance:
□ SNR, the ratio of signal power to noise power
□ SignalPower, the actual power of the input signal samples
Changing the number of samples per symbol (SamplesPerSymbol) affects the variance of the noise added per sample, which also causes a change in the final error rate.
NoiseVariance = SignalPower × SamplesPerSymbol / 10^(EsNo/10)
Select the number of samples per symbol based on what constitutes a symbol and the oversampling applied to it. For example, a symbol could have 3 bits and be oversampled by 4. For more information,
see AWGN Channel Noise Level.
[1] Proakis, John G. Digital Communications. 4th Ed. McGraw-Hill, 2001.
Version History
Introduced in R2012a
See Also | {"url":"https://fr.mathworks.com/help/comm/ref/comm.awgnchannel-system-object.html","timestamp":"2024-11-10T05:12:47Z","content_type":"text/html","content_length":"152474","record_id":"<urn:uuid:4e0f4603-91d4-4bcf-9376-fd6b3d09b36c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00424.warc.gz"} |
Weighted Confusion Matrix
wconf is a package that allows users to create weighted confusion matrices and accuracy metrics that help with the model selection process for classification problems, where distance from the correct
category is important.
The package includes several weighting schemes which can be parameterized, as well as custom configuration options. Furthermore, users can decide whether they wish to positively or negatively affect
the accuracy score as a result of applying weights to the confusion matrix. “wconf” integrates well with the “caret” package, but it can also work standalone when provided data in matrix form.
Applying a weighting scheme to the confusion matrix can be useful in applications such as performance evaluation, where characteristics such as “underperforming”, “acceptable”, “overperforming” and
“worker of the year” may represent gradations that are far apart and unevenly spaced. Similarly, where the objective is to classify geographic regions and proximity of the prediction to the actual
region constitutes an advantage in terms of the model’s performance, applying a weighting scheme facilitates the model selection process.
Functions are included to calculate accuracy metrics for imbalanced data. Specifically, the package allows users to compute the Starovoitov-Golub sine-accuracy function, as well as the balanced
accuracy function and the standard accuracy indicator.
About wconf
wconf consists of the following functions:
weightmatrix - configure and visualize a weight matrix
This function allows users to choose from different weighting schemes and experiment with parametrizations and custom configurations.
weightmatrix(n, weight.type, weight.penalty, standard.deviation, geometric.multiplier, interval.high, interval.low, custom.weights, plot.weights)
n – the number of classes contained in the confusion matrix.
weight.type – the weighting schema to be used. Can be one of: “arithmetic” - a decreasing arithmetic progression weighting scheme, “geometric” - a decreasing geometric progression weighting scheme,
“normal” - weights drawn from the right tail of a normal distribution, “interval” - weights contained on a user-defined interval, “custom” - custom weight vector defined by the user.
weight.penalty – determines whether the weights associated with non-diagonal elements generated by the “normal”, “arithmetic” and “geometric” weight types are positive or negative values. By default,
the value is set to FALSE, which means that generated weights will be positive values.
standard.deviation – standard deviation of the normal distribution, if the normal distribution weighting schema is used.
geometric.multiplier – the multiplier used to construct the geometric progression series, if the geometric progression weighting scheme is used.
interval.high – the upper bound of the weight interval, if the interval weighting scheme is used.
interval.low – the lower bound of the weight interval, if the interval weighting scheme is used.
custom.weights – the vector of custom weights to be applied, is the custom weighting scheme was selected. The vector should be equal to “n”, but can be larger, with excess values being ignored.
plot.weights – optional setting to enable plotting of weight vector, corresponding to the first column of the weight matrix
wconfusionmatrix - compute a weighted confusion matrix
This function calculates the weighted confusion matrix by multiplying, element-by-element, a weight matrix with a supplied confusion matrix object.
wconfusionmatrix(m, weight.type, weight.penalty, standard.deviation, geometric.multiplier, interval.high, interval.low, custom.weights, print.weighted.accuracy)
m – the caret confusion matrix object or simple matrix.
weight.type – the weighting schema to be used. Can be one of: “arithmetic” - a decreasing arithmetic progression weighting scheme, “geometric” - a decreasing geometric progression weighting scheme,
“normal” - weights drawn from the right tail of a normal distribution, “interval” - weights contained on a user-defined interval, “custom” - custom weight vector defined by the user.
weight.penalty – determines whether the weights associated with non-diagonal elements generated by the “normal”, “arithmetic” and “geometric” weight types are positive or negative values. By default,
the value is set to FALSE, which means that generated weights will be positive values.
standard.deviation – standard deviation of the normal distribution, if the normal distribution weighting schema is used.
geometric.multiplier – the multiplier used to construct the geometric progression series, if the geometric progression weighting scheme is used.
interval.high – the upper bound of the weight interval, if the interval weighting scheme is used.
interval.low – the lower bound of the weight interval, if the interval weighting scheme is used.
custom.weights – the vector of custom weights to be applied, is the custom weighting scheme was selected. The vector should be equal to “n”, but can be larger, with excess values being ignored.
print.weighted.accuracy – optional setting to print the weighted accuracy metric, which represents the sum of all weighted confusion matrix cells divided by the total number of observations.
rconfusionmatrix - compute a redistributed confusion matrix
This function calculates the redistributed confusion matrix by reallocating observations classified in the vicinity of the true category to the confusion matrix diagonal, according to a
user-specified weighting scheme which determines the proportion of observations to reassign.
rconfusionmatrix(m, custom.weights, print.weighted.accuracy)
m – the caret confusion matrix object or simple matrix.
custom.weights – the vector of custom weights to be applied. The vector should be equal to “n”, but can be larger, with the first value and all excess values being ignored.
print.weighted.accuracy – optional setting to print the standard redistributed accuracy metric, which represents the sum of all observations on the diagonal divided by the total number of
balancedaccuracy - calculate accuracy scores for imbalanced data
This function calculates classification accuracy scores using the sine-based formulas proposed by Starovoitov and Golub (2020). The advantage of the new method consists in producing improved results
when compared with the standard balanced accuracy function, by taking into account the class distribution of errors. This feature renders the method useful when confronted with imbalanced data.
balancedaccuracy(m, print.scores)
The function takes as input:
m - the caret confusion matrix object or simple matrix.
print.scores - used to display the accuracy scores when set to TRUE.
Technical details
For custom specifications, since the interval of variation of the weights is not bound to any given interval, depending on the user configuration, it is possible to obtain negative accuracy scores.
Download and installation of development version
Online, from Github:
You can download wconf directly from Github. To do so, you need to have the devtools package installed and loaded. Once you are in R, run the following commands:
You may face downloading errors from Github if you are behind a firewall or there are https download restrictions. To avoid this, you can try running the following commands:
options(download.file.method = “libcurl”)
options(download.file.method = “wininet”)
Once the package is installed, you can run it using the: library(wconf) command.
Author details
Alexandru Monahov, 2024 | {"url":"https://cran.uib.no/web/packages/wconf/readme/README.html","timestamp":"2024-11-07T23:44:13Z","content_type":"application/xhtml+xml","content_length":"10852","record_id":"<urn:uuid:841f02eb-39f1-4f2d-83ca-ea24706f397e>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00865.warc.gz"} |
Bayesian inference of effective contagion models from population level data
Flash talk at SINM 2019 (http://danlarremore.com/sinm2019/)
Preprint: https://arxiv.org/abs/1906.01147
Software: https://github.com/jg-you/complex-coinfection-inference/
See also: "Interacting simple contagions are complex contagions," by Laurent Hébert-Dufresne: https://speakerdeck.com/laurenthebert/interacting-simple-contagions-are-complex-contagions
Extended abstract
Contagions never occur in a vacuum. Instead, diseases and ideas interact with each other and with externalities such as host connectivity, behaviour, and mobility. Several recent studies have shown
that many of these non-linear mechanism lead to rich dynamics that can exhibit, for example, a non-monotonous relation between the expected epidemic size and their average transmission rate and
discontinuous phase transitions. Surprisingly, many of these features arise from minor alterations to the mechanistic rules of the models. In other words, innocuous looking modeling choices can
produce drastically different outcomes that would lead to very different conclusions about intervention strategies or risk. Understanding how to properly generalize contagion models is, as a result,
perhaps one of the most pressing challenge of network epidemiology.
Recent work shows that so-called “complex contagions models” can induce many of the defining features of the new wave of non-linear mechanistic models. Complex contagion models achieve this by
modifying the transmission rate β(I), as to let it depend on the density of infected individuals in the neighbourhood of the susceptible individuals. In this work, we show that some complex contagion
models are in fact indistinguishable from a number of non-linear mechanistic models at the population level. This motivates us to think of complex contagion (on Erdős-Rényi graphs) as a useful
effective model of contagion on networks. The complex contagion function β(I) that appears in this model captures arbitrary non-linear effects, allowing us to the contagions without making any
mechanistic assumptions. By understanding how mechanistic models map unto this complex contagion model, we can interpret the function β(I) and draw tentative inference about the population under
We develop a fully Bayesian method to fit our effective contagion model to population-level data. This allows us to infer posterior distributions of: Potential epidemiological trajectory, complex
contagion functions β(I), and noise components. We avoid overfitting—a problem that arises in related approaches—by parameterizing the complex contagion functions as low degree polynomials. The net
results is a flexible, efficient, yet expressive model that can be easily fitted to real and synthetic data alike. | {"url":"https://speakerdeck.com/jgyou/bayesian-inference-of-effective-contagion-models-from-population-level-data","timestamp":"2024-11-02T18:12:09Z","content_type":"text/html","content_length":"113363","record_id":"<urn:uuid:631b4fa8-58b2-4d60-af5f-b250f44d3ee9>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00000.warc.gz"} |
Math as a Language - Wisaarkhu
Math as a Language
When I studied math, some people who were not familiar with the subject were worried that I would only find a job as a high school math teacher. Other people who had some familiarity with math said
that I could earn a high income in financial institutions. While these people were worried about my future career, I was struggling with the math itself. All I was thinking about was how to
understand what was explained during class and how to complete the homework in order to catch up. I was jealous of my friends who got intern positions at prestigious firms, but I do respect those who
are able to comprehend fully and achieve their A’s in Analysis and Algebra.
The moment that I found math could be applicable to real life was when I took classes in data science and computer science. Beforehand, I had thought that the only applicable math was differential
equations in physics. What is fascinating from those fields is that real world problems can be measured and quantified, and then it turns into a math problem after abstraction, such as text
vectorization, naïve bayes for classification, etc. During the learning process of math, I always perceived math as a game on paper. I thought it was a fun way to train my brain, detached from
reality. Rather than games to play with, I now find that math is more like a language, and the learning of abstract math is like learning grammar for a language so that one can use it properly in the
future. I recall my painful experience of learning English grammar! Fortunately, right now I am not writing a sentence in broken English, because I know a bit of grammar.
I am now involved with a startup business, trying to work with natural language processing. I am not going to say that math helped me to start this, but it is a necessary condition since the research
papers I have to comprehend are normally heavy with math. Fortunately, I was trained in basic math literacy so it is not as if I have no clue at all. In my first Algebra class, I did not even know
basic set theory. I worked hard to catch up in class and now it has paid off. The connection between math and a job is not so obvious that it could be a trivial proof, but learning math is meaningful
to me as it teaches me a new language so I can comprehend more difficult papers. It lifts the limit where comprehension prevents any understanding. It is hard to see direct benefits from studying
math, at least not like computer science or finance. However, math becomes essential when one needs it in the future to comprehend and solve complex problems and existing solutions.
Learning math does not have an obvious result in getting a job or applying to activities in daily life other than calculating meal tips, but it is absolutely essential in a business case and when
quantification is required. There are so many interesting problems and subjects that can utilize math after the problems have been quantified, and I am glad that I can read and understand solutions
with math literacy.
Jianing Qi
Math student (BA 2017, MS 2020) New York University, NY | {"url":"https://wisaarkhu.co.za/article/math-as-a-language/","timestamp":"2024-11-12T07:17:07Z","content_type":"text/html","content_length":"132120","record_id":"<urn:uuid:15c05061-1045-4c18-9c40-4e42b0bfc8d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00195.warc.gz"} |
Open Journal of Modern Hydrology
Vol.4 No.3(2014), Article ID:47562,13 pages DOI:10.4236/ojmh.2014.43006
Stochastic Characteristics and Modelling of Monthly Rainfall Time Series of Ilorin, Nigeria
Ahaneku, I. Edwin^1, Otache, Y. Martins^2*
^1Department of Agricultural & Bioresources Engineering, Michael Okpara University of Agriculture, Umudike, Nigeria
^2Department of Agricultural & Bioresources Engineering, Federal University of Technology, Minna, Nigeria
Email: ^*drotachemartyns@gmail.com, ^*martynso_pm@futminna.edu.ng
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 28 April 2014; revised 27 May 2014; accepted 26 June 2014
The analysis of time series is essential for building mathematical models to generate synthetic hydrologic records, to forecast hydrologic events, to detect intrinsic stochastic characteristics of
hydrologic variables as well to fill missing and extend records. To this end, this paper examined the stochastic characteristics of the monthly rainfall series of Ilorin, Nigeria vis-à-vis modelling
of same using four modelling schemes. The Decomposition, Square root transformation-deseasonalisation, Composite, and Periodic Autoregressive (T-F) modelling schemes were adopted. Results of basic
analysis of the stochastic characteristics revealed that the monthly series does not show any discernible presence of long-term trend, though there is a seeming inter-decadal annual variation. The
series exhibits strong seasonality throughout its length, both in the moments and autocorrelation and significantly intermittent. Based on assessment of the respective models, the performance of the
different modelling schemes can be expressed in this order: T-F > Composite > Square root transformation-Deseasonalised > Decomposition. Considering the results obtained, modelling of monthly
rainfall series in the presence of serial correlation between months should be based on the establishment of conditional probability framework. On the other hand, in view of the inadequacy of these
modelling schemes, because of the autoregressive model components in the coupling protocol, nonlinear deterministic methods such as Artificial Neural Network, Wavelet models could be viable
complements to the linear stochastic framework.
Keywords:Stochastic, Time Series, Modelling, Rainfall, Periodicity, Ergodic, Ilorin
1. Introduction
The assessment of the dynamics and regime of a particular hydrologic phenomenon is imperative; especially the time-based characteristics. Time-based characteristics of hydrological data are of great
significance in the planning, designing and operation of water systems. This significance is informed more largely due to the variability and oscillatory behaviour of hydrological sequences. Against
this backdrop therefore, as noted by Kottegoda [1] , the lack of complete understanding of the physical processes involved and the consequent uncertainties in the magnitudes and frequencies of future
events highlight the importance of time series analysis. Thus, the main objective of any time series analysis is to understand the mechanism that generates the data and also, but not necessarily, to
produce likely future sequences over a short period of time. This is usually not without taking cognisance of the appurtenant uncertainty resulting from spatio-temporal variability of hydrologic
processes. This fact becomes increasingly important considering that rainfall is a complex atmospheric process, which is space and time dependent and basically not easily predictable [2] .
Like any other aspect of science and engineering developments, there has been a tremendous introduction of new concepts and ideas in rainfall cum precipitation study in general. Notable of such are
researches in various directions including space-time structure and variability of rainfall. In this regard, there has been a significant shift from point process models to models based on concepts
of scale invariance [3] . This is so because point process models suffer from the inability to describe the statistical structure of rainfall over a wide range of scales as well as from difficulty in
parameter estimation; whereas scaling models provide parsimonious representations over a wide range of scales. These are supported by theoretical arguments and empirical evidence that rainfall
exhibits a scale-invariant symmetry (e.g., [3] [4] ). In this regard, the trend in scale-invariant rainfall models evolved around multiplicative cascades which have their origin in the statistical
theory of turbulence [3] . However, it is important to note that despite the good attributes, the estimation of parameters is not a simple issue [3] . As noted by Holley and Waymire [5] , the
independent and identically distributed “bounded generators” give rise to non-ergodic cascades. Recent developments in stochastic rainfall analysis in this direction deal with the introduction of
wavelet transforms and importantly, the use of Artificial Neural Network, diffusion model (e.g., [6] ), Markovian type models (e.g., [7] [8] ) and Disaggregation models (e.g., [9] ).
Though generally, hydrologic processes such as precipitation and runoff evolve on a continuous time scale and their estimation correspondingly unduly difficult, in particular, rainfall modelling and
its quantitative estimation or forecasting are important considering the fact that it is a critical weather parameter in the estimation of crop water requirement, and development of long lead time
flood and flash-flood warning systems. However, it suffices to note that despite substantial progress, several modelling issues still remained unresolved [3] . For instance, “what are the limits of
predictability at various temporal and spatial scales” and “the properties of the rainfall field to be preserved by the model”? The modelling of rainfall is motivated by the desire to obtain
real-time statistical forecasts of rainfall but as noted by Lovejoy and Schertzer [10] , due to nonlinear interactions that take place at a wide range of scales, several details of the rainfall
dynamics are unimportant and too, the resulting fields fall within a universality of multifractals characterised by three parameters. Thus, the objective of this paper like any modelling exercise is
to obtain synthetic sequences of rainfall with the same statistical properties as the historical ones. To this end, stochastic characteristics of the rainfall fields like moments (first and second
order) and dependence structure shall be analysed while different stochastic models will also be developed for short-term forecasts.
2. Materials and Methods
2.1. Materials
Study Location and Data Assembly
The study location is Ilorin (North central Nigeria) at longitude 4˚35' and latitude 8˚30'. It has elevation of between 273 to 333 m and a mean annual temperature of about 27˚C and is characterised
by a distinct bi-seasonal weather pattern; i.e., wet and dry. The wet season starts in April and ends in October, while the dry season starts in November and ends in March. The mean annual rainfall
is 1150mm, while the relative humidity ranges from 65% - 80%. Figure 1 shows the map of Nigeria with the study location indicated as inset. For this study, historical rainfall time series of Ilorin
was used. To this end, mean monthly rain gauge rainfall values (i.e., point rainfall) for approximately 43 years’ time period (1967-2009) were collected. Preliminary analysis of stochastic
Figure 1. Map of Nigeria showing the study location.
characteristics like moments and dependence structure of the data series was done to be able to evaluate randomness and trend pattern. In this regard, the time series plot was examined to establish
whether it does exhibit intermittency or otherwise as well as seasonal characteristics like trend and moments. The objective here is to evaluate seasonality in the moments. Analysis of dependence
structure was done in time and frequency domains; basically through autocorrelation and spectral density, respectively.
2.2. Methodology
2.2.1. Modelling Framework
In this study, four (4) different modelling schemes were employed; these are a) decomposition, b) square root transformation-deseasonalisation strategy, c) composite modelling and d) Periodic
modelling (Thomas-Fiering).
1) Decomposition strategy Here, the data series was de-trended, deseasonalised and further smoothen with a moving average (MA) of order 6 based on the autocorrelation structure of the original raw
data. To this end, an additive model of the form in equation (1) was employed.
This procedure requires that the data series be decomposed into seasonal components; the deseasonalisation after the removal of the long-term trend was done by using the seasonal adjustment factors
(SAF). These values (SAF) indicate the effect of each period on the level of the series. Table 1 shows the respective seasonal adjustment factors or indices whereas Figure 2 details the entire
decomposition process.
After the decomposition process and smoothening, an ARIMA model was fitted into the random or stochastic component left. Based on the analysis of the autocorrelation functions of the random
component, a multiplicative ARIMA model was fitted; in this regard, Appendix); this derives from the fact that ordinary integrated moving average scheme may not necessarily account for the
non-seasonal autoregressive behaviour of hydrologic processes [11] . Figure 3 shows the correlogram of the model residuals.
2) Square root transformation-deseasonalisation scheme Based on the suggestion of Delleur and Kavvas [12] , the square root transformation of the data was used to obtain a series which is
approximately normally distributed. The series of the monthly rainfall square roots were
Figure 2. Seasonal analysis of the original mean rainfall series (RF) before (a) and after (b) detrending.
rescaled (deseasonalised) by subtracting from each term of the series by the corresponding seasonal mean and dividing same by the corresponding standard deviation. The deseasonalisation process is
according as:
Figure 3. Correlogram of ARIMA (1, 0, 0) × (1, 0, 1)[12] model residuals fitted to the stochastic component.
Using the autocorrelation functions of the square root transformed and deseasonalised series, a seasonal ARIMA model of the form: Appendix). To retrieve the square root transformed series with its
seasonal component, a reversed rescaling procedure was done; that is,
where, j is the month in a 12-month annual cycle and
3) Composite modelling The composite modelling entails decomposing the original data series into its various components; i.e., deterministic and a stochastic component which accounts for the random
effects (dependent and independent parts) [13] . In this regard, the time series rf(t)[, ]was represented by a decomposition model of the additive type according as equation (4).
where, T(t) is the trend component, P(t) the periodic component and ε(t), the stochastic component.
For the identification of trend, annual rainfall series was used. The annual series was obtained by aggregating the 43 years annual series. In the actual trend detection procedure, a hypothesis of no
trend was made and the value of the test statistic (Z) was calculated by using 1) Turning Point Test, 2) Kendall’s Rank Correlation Test and 3) Mann-Kendall Trend Test. The computed values of the
test statistic in all instances were −0.852, −0.429, and 0.195, respectively. Considering the values of the computed test statistic (Z), at 5% level of significance, the Z values do not provide
reason to suspect the presence of any discernible long-term trend. Thus, the observed rainfall series may be treated as trend free. Hence the composite model, i.e., equation (4) reduces to:
To confirm the presence of periodic component in the monthly rainfall series, a correlogram of the series was drawn. Figure 4 shows the periodic, oscillating nature of the time series.
The parameters of the periodic component of the composite model were evaluated by using the classical harmonic analysis method. To this end, the Cumulative Periodogram (CP) approach was adopted. In
this case, the point of intersection of the fast increase in the Periodogram (CP[i]) and the slow increase is considered and the corresponding harmonics taken as significant and the remaining treated
as errors and passed on to the random component; i.e., insignificant. From figure 5, the first four harmonics are considered significant. The periodic component can be expressed as in equation (6a).
Figure 4. Autocorrelation function of the original rainfall series based on water year regime.
Figure 5. Cumulative periodogram of the mean monthly rainfall series.
where, k is the maximum harmonics,
Based on figure 5, the resulting periodic component can be expressed according as equation (6b).
Table 2 shows the values of the harmonic coefficients.
The stochastic component (
Based on the autocorrelation of the residual series left after the periodic component was removed from the original series, Appendix). Resulting from this, the final composite model of the monthly
rainfall series, i.e., equation (5) then becomes
4) Periodic Autoregressive modelling Scheme Modelling of the monthly rainfall series using periodic autoregressive model was done by adopting the Thomas-Fiering (T-F) model. The T-F model is a linear
stochastic model for stimulating synthetic series of seasonal hydrologic process. The schema for the rainfall modelling using this framework takes the form
This model uses a linear regression relationship to relate the storm rf[t+1] in the (t+1) month to storm rf[t] in the t(th) month. Here, [j] is the regression coefficient and
2.2.2. Model Validation and Forecast Functions
In all the instances, for the respective modelling strategy, split sampling procedure was adopted; i.e., one segment of the monthly rainfall series (40 years’ time period) was used for modelling
while the remaining three years data was used for model validation. For model validation/forecasting, forecast functions corresponding to the respective ARIMA modelling scheme was adopted using the
difference equation form. In this regard, recalling that Z[t](L) = [[Zt+L]], using square brackets to signify conditional expectations and noting that
the following forecast functions were employed, viz:a) Decomposition modelling scheme:
b) Square root transformation-deseasonalisation strategy:
c) Composite modelling scheme (i.e., the stochastic component):
3. Results and Discussion
3.1. Assessment of Stochastic Characteristics and Findings
Hydrologic processes such as precipitation and runoff evolve on a continuous time scale. The implication(s) of this is simple; as shown by Figure 6, the rainfall time series plot exhibits typical
characteristic movement with seasonality, cyclical or sinusoidal and random components. This phenomenon translates into statistical characteristics which vary within an annual cycle. Figure 6 shows
clearly a discernible seasonal or periodic pattern; it is a periodic-stochastic series since, in addition to the periodic pattern, a random pattern is also evident. In the light of this, it suffices
to note that even though, monthly and annual rainfalls are usually non-intermittent, in semiarid and arid regions, monthly and annual precipitation may be intermittent [14] . Also, as noted by
Chebaane et al. [14] , this is imperative considering the fact that hydrologic time series are intermittent when the variable under consideration takes on nonzero and zero values throughout the
length of record. Interesting too, is the seasonal autocorrelation. Seasonal autocorrelations for monthly precipitation are generally not significantly different from zero; figure 7 attests to this
fact. The fall out of this is that the rainfall time series are uncorrelated, depicting strong homogeneity. This phenomenon connotes intermittency of the series, most especially considering
Figure 6. Monthly rainfall time series plot.
Figure 7. Seasonal correlogram showing periodic stochasticity/intermittency.
the fact the series takes on nonzero and zero values throughout the entire length of the record (Figure 6).
In the same context, figure 8 shows inter-annual decadal variation in the rainfall series; long-term trend pattern is seemingly not evident. However, there is large variability among the monthly
values of rainfall of different years, with the period 1995-2009 showing slight increases in the storm event during the peak seasons. On the other hand, figure 9, figure 10 shows the presence of
seasonality in the moments, meaning that monthly statistics for dry season are significantly different from those of the wet season period. Unlike intermittent stream flow process, the seasonal means
have higher values than the seasonal deviations throughout the year. As noted in Figure 10, the coefficient of variation varies from 0.3234 in the month of June to 3.4227 in December (i.e., period of
incipient rains, moderate-peak to late rains). The variance is maximum during the period of late rains and incipient dry season; more or less the interfacing period. This indicates atmospheric
instability during this watershed period; i.e., the fringes of the raining season going to full harmattan period. Similarly, as shown in Table 3, values of the skewness coefficient (g) for the
periods of incipient dry season (late rains) to full dry season are generally larger than the corresponding periods for the wet season over an annual cycle. This indicates that the data in the former
seasons depart more from normality than those in the later (early to full wet season period). The variability in the time series regime leads to model structural uncertainty; especially if the
hydrologic evolution of the generating mechanism is not appropriately understood and captured in the model formulation.
To assess this, analysis of dependence structure in time series via spectral density is critical; figure 11 shows the dependence structure of the monthly rainfall in a frequency domain. The spectral
density exhibits a discrete spectral component at the frequency of 1/12 cycle per month. This periodicity is seen in figure 11(a). Similarly, the periodogram exhibits quite a corresponding pattern in
terms of the periodicity. However, as noted by Kottegoda [1] , interpretation is difficult as it provides unexpected peaks. From figure 11(b), the sample spectra from the different sections of the
rainfall data may resemble each other in their overall aspects.
Figure 8. Inter-annual mean monthly rainfall variation pattern.
Figure 9. Variation in seasonal moments.
Figure 10. Seasonal pattern in coefficient of variation.
3.2. Modelling and Forecasting of the Rainfall Regime
Figure 12 shows the behaviour of the different model forecast functions; the forecasts are quite at variance with the expected. Baring data quality problems, stationarity issues, and model over
fitting, forecasts in the distant future for a trend-free series should be the unconditional estimates of the means. From figure 12, it’s obvious that the performance of the different modelling
schemes can be expressed in this order: T-F > Composite > Square root transformation-Deseasonalised > Decomposition. Table 4 and figure 13, respectively show the performance of the modelling scheme
with respect to the ability to represent the seasonal statistics of the observed rainfall series. It is apparent from figure 13 that for the entire lag time considered, models T-F, Square root
transformation-deseasonalisation and Composite were able to replicate the measured rainfall pattern, though
Figure 11. Spectral density based on Tukey lag window (a) and the periodogram (b) of the raw monthly rainfall series.
Figure 12. Summary chart of the different models’ behaviour in forecsat mode vis-à-vis the observed mean monthly series.
to varying degrees of accuracy whereas the decomposition strategy failed completely. In above, the positive attribute of the T-F model ahead of others reinforces its suitability for adoption in
rainfall modelling.
Table 4. Seasonal moments for both the observed series and the models in the simulation phase.
Figure 13. Monthly standard deviations of the rainfall series and forecast errors of the ARIMA models for the respective modelling schemes.
Considering the performance of the models adopted, it is imperative to look at the implications of the data pre-processing strategy. In all the models, except the Thomas-Fiering (T-F) model, ARIMA
models were used to model the supposedly stationary stochastic component. To achieve stationarity, seasonal differencing (12-lag) and seasonal standardisation (deseasonalisation) were respectively
applied but not without its associated problems. For instance, the deseasonalisation process is a misnomer since it implies that the deseasonalised series is free of seasonality; however, other
seasonality may still be present [12] [15] . Seasonal differencing on the other hand, removes the periodic contribution but the spectral density obtained thereof has a sinusoidal shape, also the
covariance of the stationary part is distorted. In the same context, the multiplicative ARIMA model assumes there is a serial correlation structure within the months of the same year but it does not
preserve the monthly standard deviations just like the others as seen in figure 13. Thus within this context, the overall poor performance of all the models with the exception of T-F could be
4. Conclusions
For purposes of identifying a more realistic modelling scheme for the rainfall series, assessment of the stochastic characteristics was done to be able to understand the dynamics of the monthly
series. Sequel to this, four different modelling schemes: Decomposition, Square root transformation-deseasonalisation, Composite, and Periodic autoregressive modelling (T-F), were adopted. Results of
basic analysis of the stochastic characteristics revealed that the monthly series does not show any discernible presence of long-term trend, though there is a seeming inter-decadal annual variation.
It is evident that the series exhibits strong seasonality throughout its length, both in the moments and autocorrelation. This gives rise to significant correlation which is attributable to the
serial dependence of the same month on several years; this serial dependence is same for all 12 months. The strong seasonal autocorrelation structure connotes intermittency considering the fact that
the series assumes nonzero and zero values throughout its length for the period considered.
Resulting from the analysis and the modelling exercise, the Thomas-Fiering (T-F) model can be used for monthly rainfall modelling and short-term forecast. In addition, both the composite and square
root transformation-deseasonalisation schemes may also be employed but not without caution. Because of the ARIMA model component of these models in the coupling, their forecast abilities were
impaired considering the inadequacy of their respective forecast errors to preserve the observed standard deviations of the rainfall series. This primarily might have arisen from the second-order
stationarity assumptions requirement of the autoregressive models. In the same vein, whole decomposition of any trend-free series requiring de-trending, deseasonalisation followed by moving average
smoothing, and fitting of ARIMA model might be too excessive as it distorts the entire spectrum in the overall and not encouraged. The results obtained suggest that modelling of monthly rainfall
series in the presence of serial correlation between months should be based on the establishment of conditional probability framework; in this case, two conditional probabilities: probability that
month t has zero rainfall given that month t-1 had non-zero rainfall and probability that month t has zero rainfall, given that month t-1 had zero rainfall. On the other hand, considering the
inadequacy of these modelling schemes because of the autoregressive model components, nonlinear deterministic methods such as Artificial Neural Network, Wavelet models could be viable complement to
the linear stochastic framework.
1. Kottegoda, N.T. (1980) Stochastic Water Resources Technology. Macmillan Press Ltd., London, 2-3, 21, 112-113.
2. Ramana, R.V., Krishna, B. and Kumar, S.R. (2013) Monthly Rainfall Prediction Using Wavelet Neural Network Analysis. Journal of Water Resources Management, 27, 3697-3711. http://dx.doi.org/10.1007
3. Georgiou, E.F. and Krajewski, W. (1995) Recent Advances in Rainfall Modelling, Estimation, and Forecasting. US National Report to International Union of Geodesy and Geophysics, 1125-1137.
4. Gupta, V. and Waymire, E. (1993) A Statistical Analysis of Mesoscale Rainfall as a Random Cascade. Journal of Applied Meteorology and Climatology, 32, 251-267. http://dx.doi.org/10.1175/1520-0450
5. Holley, R. and Waymire, E. (1992) Multifractal Dimensions and Scaling Exponents for Strongly Bounded Random Cascades. The Annals of Applied Probability, 2, 819-845. http://dx.doi.org/10.1214/aoap
6. Pavlopoulos, H. and Kedem, B. (1992) Stochastic Modelling of Rain Rate Processes: A Diffusion Model. Communications in Statistics. Stochastic Models, 8, 397-420,
7. Jimoh, O.D. and Webster, P. (1996) The Optimum Order of a Markov Chain Model for Daily Rainfall in Nigeria. Journal of Hydrology, 185, 45-69. http://dx.doi.org/10.1016/S0022-1694(96)03015-6
8. Gregory, J.M., Wigley, T.M.L. and Jones, P.D. (1992) Determining and Interpreting the Order of a Two State-Markov Chain: Application to Models of Daily Precipitation. Water Resources Research,
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9. Koutsoyiannis, D. (1992) A Nonlinear Disaggregation Method with a Reduced Parameter Set for Simulation of Hydrologic Series. Water Resources Research, 28, 3175-3191. http://dx.doi.org/10.1029/
10. Lovejoy, S. and Schertzer, D. (1990) Multifractals, Universality Classes and Satellite and Radar Measurements of Clouds and Rain Fields. Journal of Geophysical Research, 95, 2021. http://
11. Otache, M.Y., Ahaneku, I.E. and Mohammed, S.A. (2011) Parametric Linear Stochastic Modelling of Benue River flow Process. Open Journal of Marine Science, Scientific Research, 1-9.
12. Delleur, J.W. and Kavvas, M.L. (1978) Stochastic Models for Monthly Rainfall Forecasting and Synthetic Generation. Journal of Applied Meteorology, 17, 1528-1536. http://dx.doi.org/10.1175/
13. Bhakar, R.S., Singh, R.V., Chhajed, N. and Bansal, A.K. (2006) Stochastic Modelling of Monthly Rainfall at Kota Region. ARP Journal of Engineering and Applied Sciences, 1.
14. Chebaane, M., Salas, J.D. and Boes, D.C. (1992) Product Autoregressive Process for Modelling Intermittent Monthly Streamflows. Water Resources Research, 28.
15. Otache, M.Y., Ahaneku, I.E. and Mohammed, S.A. (2011) ARMA Modelling of Benue River Flow Dynamics: Comparative Study of PAR Model. Open Journal of Modern Hydrology, Scientific Research, 1, 1-9.
ARIMA Model Diagnostics
^*Corresponding author. | {"url":"https://file.scirp.org/Html/1-1630083_47562.htm","timestamp":"2024-11-02T07:46:33Z","content_type":"application/xhtml+xml","content_length":"99756","record_id":"<urn:uuid:b16f7224-0501-42bb-a746-4cdfff78821f>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00859.warc.gz"} |
Thermal Diffusivity | Concept & Overview
The concept of Thermal diffusivity is frequently confused with that of thermal conductivity. They are closely related concepts; however, thermal conductivity appears to be more prevalent in the
scientific community. Even as the less popular of the two heat transfer measurements, thermal diffusivity still plays an important role in influencing the movement and behavior of heat.
Thermal diffusivity is a measure of the rate at which heat disperses throughout an object or body. Thermal conductivity is a measure of how easily one atom or molecule of a material accepts or gives
away heat. The main idea behind thermal diffusivity is the rate at which heat diffuses throughout a material.
Expressions for Thermal Diffusion
Thermal conductance can also be viewed as a factor of thermal diffusion. A material that is said to conduct heat efficiently must also have effective heat diffusion properties in order to facilitate
heat transfer. Density is another factor of thermal diffusion. A material with a high density is composed of atoms/molecules packed tightly together. A higher density can limit the speed and distance
that heat can travel through the object. An increase in density can be imagined as a highway with more toll booths, where the cars are energy quanta in the form of heat.
The specific heat capacity is the last relevant factor when just regarding solids, as this quantity relates to how much heat can be held by one atom/molecule at a time. This can be pictured as a
stoplight that is more likely to change as more cars come to a stop at it. Increasing the specific heat of the material would be like lessening the positive effect that every car stopped at the light
has on the likelihood of the light changing to green. Fluids are also affected by convection, which is the movement of the atoms/molecules caused by heating. Convection impacts heat transfer and
makes thermal diffusivity much harder to mathematically model. However, if the focus is on solids, a simpler representation can be formed.
\[ a = \frac{k}{(pc)} \]
Where k is the thermal conductivity, p is the density, and c is the specific heat capacity at constant pressure. pc is often referred to as the volumetric heat capacity.
For an individual with a background in mathematics or a related field, this concept might be somewhat familiar. This can be attributed to a rather useful form of equation that describes the diffusion
of any property through a consistent medium. This form of equation is called the ‘heat equation’ because heat diffusion is its most common use.
\[ \dot{u} = a \nabla^2 u \] ‘Heat Equation’
Where \[ u \] is a measure of some property, \[ \dot{u} \] is its derivative with respect to time, and \[ \nabla^2 \] is its Laplace operator (the divergence of the gradient)
In the case of heat transfer through a homogeneous (uniform) body, \[ u \] could represent temperature and α would be the same as above.
\[ \frac{dT}{dt} = a \nabla^2 T \]
One benefit of this equation is that \[ \nabla^2 \] can often be written independently of any coordinate system. In this form it is clear to see that thermal diffusivity is a scaling factor, meaning
it directly controls the speed at which temperature changes.
Experimental Methods of Finding the Thermal Diffusivity
It is possible for the thermal diffusivity to be measured alongside thermal conductivity if the density is known. One method would be the Searle’s bar experiment, which gives an equation for thermal
\[ k = cmd \frac{(T3-T4)}{A(T1-T2)} \]If the following equation is substituted into the thermal diffusion equation without an initial calculation, then the specific heat capacity would not need to be
Improvements in modern technology have created more accurate methods for determining the thermal diffusivity of an object. The flash method is a relatively new way to measure thermal diffusivity. In
this method, a small sample of the material with pre-determined dimensions is coated in black paint that is designed to make the sample behave as a black box. A face of the sample is then struck with
a short duration of very intense light. Knowing the wavelength and intensity of this light, the amount of energy it imparts into the sample is easily estimated with high accuracy. The opposite face
of the sample is in contact with a thermocouple which measures the temperature of that face. An oscilloscope plots the measured temperature with respect to time. The thermal diffusivity can then be
found through the shape of the graph by rearranging the heat equation.
\[ a = \frac{dT}{dt} \frac{1}{\nabla^2 T} \]
Applications of Thermal Diffusivity
Many industries rely on thermal diffusivity to determine the most suitable materials to optimize efficient heat flow. Insulation is an example of a material that requires a low thermal diffusivity so
that a minimal amount of heat is passing through it at any one time. A heat sink is an appliance that is designed to carry the heat out and away from another piece of equipment. A heat sink is
required to have a very high thermal diffusivity that enables the quick transport of the heat. If slow heat transfer were to occur, the area accepting the heat would heat up and not permit as much
heat flow per unit time. Heat sinks are used in almost every piece of electrical equipment. An increase in temperature in certain components can lead to an increased electrical resistance and
unexpected behavior.
Technologies such as refrigeration, heating, machining, and architecture all hold thermal diffusivity to a paramount importance. Outlined below is a list of the materials with the highest and lowest
thermal diffusivities. This list is courtesy of Thermtest’s extensive materials thermal properties database.
Material Thermal Conductivity Thermal Diffusivity Specific Heat Capacity Material Density
(W/m•K) (mm^2/s) (J/kg•K) (kg/m^3)
Iodine (Solid) 0.004 0 218 4930
Ammonia (NH3) (Liquid Under Pressure) 0.05 0.02 4686 618
Ethyl Vinyl Acetate 0.075 0.03 2301 1200
Tetradecafluorohexane 0.057 0.0308 1100 1680
Urea-Formaldehyde Molded 0.126 0.05 1674 1500
Polyvinylidene Fluoride (Kynar) 0.126 0.05 1381 1760
Polyvinyl Butyral 0.084 0.05 1674 1100
Butyl Rubber 0.088 0.05 1966 900
R12 (Dichlorofluoromethane) 0.07 0.0531 886 1488
R134a (Tetrafluoroethane) 0.1 0.0566 1280 1380
Table 1: Measurements of Thermal Conductivity, Lowest Thermal Diffusivity, Specific Heat Capacity, and Material Density.
Material Thermal Conductivity Thermal Diffusivity Specific Heat Capacity Material Density
(W/m•K) (mm^2/s) (J/kg•K) (kg/m^3)
Graphite Sheet 100 Um (In-Plane) 700 968 850 850
Graphite Sheet 25 Um (In-Plane) 1600 896 850 2100
Graphite Sheet 70 Um (In-Plane) 800 855 850 1100
Carbon Diamond Gem Quality Type 1 543.92 306 506 3510
Silicon Carbide (SiC) (Single Xtal) 489.53 225 678 3210
Silver 426.77 172 236 10500
Helium (Gas) 0.15 164 5188 0.177
Potassium 97.069 150 753 862
Hydrogen (Gas) 0.186 145 14230 0.0899
Silver Alloys Sterling And Coin 359.82 137 251 10500
Table 2: Measurements of Thermal Conductivity, Highest Thermal Diffusivity, Specific Heat Capacity, and Material Density.
“On thermal diffusivity” – Agustin Salazar – May 2003 European Journal of Physics 24(4):351; 10.1088/0143-0807/24/4/353 – https://www.researchgate.net/publication/231038795_On_thermal_diffusivity
“Flash Method of Determining Thermal Diffusivity, Heat Capacity, and Thermal Conductivity” – W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott – Journal of Applied Physics 32, 1679 (1961);
10.1063/1.1728417 – https://aip.scitation.org/doi/abs/10.1063/1.1728417
“Thermal Diffusivity Mapping of Graphene-Based Polymer Nanocomposites” – Matthieu Gresil, Zixin Wang, Quentin-Arthur Poutrel & Constantinos Soutis – Scientific Reports | 7: 5536; 10.1038/
s41598-017-05866-0 – https://www.nature.com/articles/s41598-017-05866-0.pdf
MATERIALS THERMAL PROPERTIES DATABASE – https://thermtest.com/materials-database
Image Sources:
Author: Cole Boucher, Junior Technical Writer at Thermtest | {"url":"https://thermtest.com/thermal-diffusivity-overview","timestamp":"2024-11-02T02:22:32Z","content_type":"text/html","content_length":"527795","record_id":"<urn:uuid:4c58229d-89f7-4390-a062-51d127f4112d>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00028.warc.gz"} |
Find two numbers such that one of them exceeds the other by \
Hint: We are given here, two numbers such that one number exceeds the other by \[9\] and their sum is \[81\]. We have to find those numbers. We do this by making this an equation of one variable. We
consider one of the numbers as a variable and we get the other number in terms of that variable as well. Then we try to find the value of that variable. Once we get that value, we can easily find
both the numbers.
Complete step-by-step solution:
Here, we have to find the value of two numbers who differ by \[9\] and their sum is \[81\]. We consider the smaller number to be a variable say \[x\]. Then according to the question the greater
number would be \[x + 9\]. Now since we know that the sum of both the numbers are \[81\], we say that,
\[(x + 9) + x = 81\]
On moving forward, we get
2x + 9 = 81 \\
\Rightarrow 2x = 81 - 9 \\
\Rightarrow 2x = 72 \\
\Rightarrow x = \dfrac{{72}}{2} \\
\Rightarrow x = 36 \]
Hence the value of the smaller number is \[36\]. Now using this we will get the value of greater number by putting \[x = 36\] in \[x + 9\] as,
\[36 + 9 = 45\]
Hence, we get the value of both the required numbers as \[36\] and \[45\].
Note: Whenever two values are to be found and the relations between both the values are given, we solve the question using the equation of one variable. If the relations between the values to be
found are not given then, only we will use the equation in two variables. | {"url":"https://www.vedantu.com/question-answer/find-two-numbers-such-that-one-of-them-exceeds-class-8-maths-cbse-609ea6cd3abc38535777fbdb","timestamp":"2024-11-05T23:28:07Z","content_type":"text/html","content_length":"149976","record_id":"<urn:uuid:15f85a1f-84ea-4d22-9783-36987182b3dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00732.warc.gz"} |
Laplace correction parameter — Laplace
Laplace correction parameter
Laplace correction for smoothing low-frequency counts.
Laplace(range = c(0, 3), trans = NULL)
A two-element vector holding the defaults for the smallest and largest possible values, respectively. If a transformation is specified, these values should be in the transformed units.
A trans object from the scales package, such as scales::transform_log10() or scales::transform_reciprocal(). If not provided, the default is used which matches the units used in range. If no
transformation, NULL. | {"url":"https://dials.tidymodels.org/reference/Laplace.html","timestamp":"2024-11-08T06:16:35Z","content_type":"text/html","content_length":"9115","record_id":"<urn:uuid:c9132d5c-2d1e-4972-b5fe-4faaeecb3500>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00649.warc.gz"} |
MODEL Statement
The MODEL statement specifies the dependent and independent variables (dependents and independents, respectively) and specifies the transformation (transform) to apply to each variable. Only one
MODEL statement can appear in PROC TRANSREG. The t-options are transformation options, and the a-options are algorithm options. The t-options provide details for the transformation; these depend on
the transform chosen. The t-options are listed after a slash in the parentheses that enclose the variable list (either dependents or independents). The a-options control the algorithm used, details
of iteration, details of how the intercept and coded variables are generated, and displayed output details. The a-options are listed after the entire model specification (the dependents, independents
, transformations, and t-options) and after a slash. You can also specify the algorithm options in the PROC TRANSREG statement. When you specify the DESIGN o-option, dependents and an equal sign are
not required. The operators *, |, and @ from the GLM procedure are available for interactions with the CLASS expansion and the IDENTITY transformation. They are used as follows:
Class(a * b ...
c | d ...
e | f ... @ n)
Identity(a * b ...
c | d ...
e | f ... @ n)
In addition, transformations and spline expansions can be crossed with classification variables as follows:
transform(var) * class(group) transform(var) | class(group)
See the section Types of Effects in Chapter 42: The GLM Procedure, for a description of the @, *, and | operators and see the section Model Statement Usage for information about how to use these
operators in PROC TRANSREG. Note that nesting is not implemented in PROC TRANSREG.
The next three sections discuss the transformations available (transforms) (see the section Families of Transformations), the transformation options (t-options) (see the section Transformation
Options (t-options)), and the algorithm options (a-options) (see the section Algorithm Options (a-options)).
Families of Transformations
In the MODEL statement, transform specifies a transformation in one of the following five families:
Variable expansions
preprocess the specified variables, replacing them with more variables.
Nonoptimal transformations
preprocess the specified variables, replacing each one with a single new nonoptimal, nonlinear transformation.
Nonlinear fit transformations
preprocess the specified variable, replacing it with a smooth transformation, fitting one or more nonlinear functions through a scatter plot.
Optimal transformations
replace the specified variables with new, iteratively derived optimal transformation variables that fit the specified model better than the original variable (except for contrived cases where the
transformation fits the model exactly as well as the original variable).
Other transformations
are the IDENTITY and SSPLINE transformations. These do not fit into the preceding categories.
The transformations and expansions listed in Table 97.2 are available in the MODEL statement.
Table 97.2: Transformation Families
Transformation Description
Variable Expansions
BSPLINE B-spline basis
CLASS set of coded variables
EPOINT elliptical response surface
POINT circular response surface & PREFMAP
PSPLINE piecewise polynomial basis
QPOINT quadratic response surface
Nonoptimal Transformations
ARSIN inverse trigonometric sine
EXP exponential
LOG logarithm
LOGIT logit
POWER raises variables to specified power
RANK transforms to ranks
Nonlinear Fit Transformations
BOXCOX Box-Cox
PBSPLINE penalized B-splines
SMOOTH noniterative smoothing spline
Optimal Transformations
LINEAR linear
MONOTONE monotonic, ties preserved
MSPLINE monotonic B-spline
OPSCORE optimal scoring
SPLINE B-spline
UNTIE monotonic, ties not preserved
Other Transformations
IDENTITY identity, no transformation
SSPLINE iterative smoothing spline
You can use any transformation with either dependent or independent variables (except the SMOOTH and PBSPLINE transformations, which can be used only with independent variables, and BOXCOX, which can
be used only with dependent variables). However, the variable expansions are usually more appropriate for independent variables.
The transform is followed by a variable (or list of variables) enclosed in parentheses. Here is an example:
model log(y) = class(x);
This example finds a LOG transformation of y and performs a CLASS expansion of x. Optionally, depending on the transform, the parentheses can also contain t-options, which follow the variables and a
slash. Here is an example:
model identity(y) = spline(x1 x2 / nknots=3);
The preceding statement finds SPLINE transformations of x1 and x2. The NKNOTS= t-option used with the SPLINE transformation specifies three knots. The identity(y) transformation specifies that y is
not to be transformed.
The rest of this section provides syntax details for members of the five families of transformations listed at the beginning of this section. The t-options are discussed in the section Transformation
Options (t-options).
PROC TRANSREG performs variable expansions before iteration begins. Variable expansions expand the original variables into a typically larger set of new variables. The original variables are those
that are listed in parentheses after transform, and they are sometimes referred to by the name of the transform. For example, in CLASS(x1 x2), x1 and x2 are sometimes referred to as CLASS expansion
variables or simply CLASS variables, and the expanded variables are referred to as coded or sometimes “dummy” variables. Similarly, in POINT(Dim1 Dim2), Dim1 and Dim2 are sometimes referred to as
POINT variables.
The resulting variables are not transformed by the iterative algorithms after the initial preprocessing. Observations with missing values for these types of variables are excluded from the analysis.
The POINT, EPOINT, and QPOINT variable expansions are used in preference mapping analyses (also called PREFMAP, external unfolding, ideal point regression) (Carroll, 1972) and for response surface
regressions. These three expansions create circular, elliptical, and quadratic response or preference surfaces (see the section Point Models and Example 97.6). The CLASS variable expansion is used
for main-effects ANOVA.
The following list provides syntax and details for the variable expansion transforms.
Nonoptimal Transformations
The nonoptimal transformations, like the variable expansions, are computed before the iterative algorithm begins. Nonoptimal transformations create a single new transformed variable that replaces the
original variable. The new variable is not transformed by the subsequent iterative algorithms (except for a possible linear transformation with missing value estimation). The following list provides
syntax and details for nonoptimal variable transformations.
Nonlinear Fit Transformations
Nonlinear fit transformations, like nonoptimal transformations, are computed before the iterative algorithm begins. Nonlinear fit transformations create a single new transformed variable that
replaces the original variable and provides one or more smooth functions through a scatter plot. The new variable is not transformed by the subsequent iterative algorithms. The nonlinear fit
transformations, unlike the nonoptimal transformations, use information in the other variables in the model to find the transformations. The nonlinear fit transformations, unlike the optimal
transformations, do not minimize a squared-error criterion. The following list provides syntax and details for nonoptimal variable transformations.
Optimal transformations are iteratively derived. Missing values for these types of variables can be optimally estimated (see the section Missing Values). The following list provides syntax and
details for optimal transformations.
Transformation Options (t-options)
If you use a nonoptimal, nonlinear fit, optimal, or other transformation, you can use t-options, which specify additional details of the transformation. The t-options are specified within the
parentheses that enclose variables and are listed after a slash. You can use t-options with both the dependent and the independent variables. Here is an example of using just one t-option:
proc transreg;
model identity(y)=spline(x / nknots=3);
The preceding statements find an optimal variable transformation (SPLINE) of the independent variable, and they use a t-option to specify the number of knots (NKNOTS=). The following is a more
complex example:
proc transreg;
model mspline(y / nknots=3)=class(x1 x2 / effects);
These statements find a monotone spline transformation (MSPLINE with three knots) of the dependent variable and perform a CLASS expansion with effects coding of the independents.
Table 97.3 summarizes the t-options available in the MODEL statement.
Table 97.3: Transformation Options
Option Description
Nonoptimal Transformation
ORIGINAL Uses original mean and variance
Parameter Specification
PARAMETER= Specifies miscellaneous parameters
SM= Specifies smoothing parameter
Penalized B-Spline
AIC Uses Akaike’s information criterion
AICC Uses corrected AIC
CV Uses cross validation criterion
GCV Uses generalized cross validation criterion
LAMBDA= Specifies smoothing parameter list or range
RANGE Specifies a LAMBDA= range, not a list
SBC Uses Schwarz’s Bayesian criterion
DEGREE= Specifies the degree of the spline
EVENLY= Spaces the knots evenly
EXKNOTS= Specifies exterior knots
KNOTS= Specifies the interior knots or break points
NKNOTS= Creates n knots
CLASS Variable
CPREFIX= Specifies CLASS coded variable name prefix
DEVIATIONS Specifies a deviations-from-means coding
EFFECTS Specifies a deviations-from-means coding
LPREFIX= Specifies CLASS coded variable label prefix
ORDER= Specifies order of CLASS variable levels
ORTHOGONAL Specifies an orthogonal-contrast coding
SEPARATORS= Specifies CLASS coded variable label separators
STANDORTH Specifies a standardized-orthogonal coding
ZERO= Controls reference levels
ALPHA= Specifies confidence interval alpha
CLL= Specifies convenient lambda list
CONVENIENT Uses a convenient lambda
GEOMETRICMEAN Scales transformation using geometric mean
LAMBDA= Specifies power parameter list
Other t-options
AFTER Specifies operations occur after the expansion
CENTER Specifies center before the analysis begins
NAME= Renames variables
REFLECT Reflects the variable around the mean
TSTANDARD= Specifies transformation standardization
Z Standardizes before the analysis begins
The following sections discuss the t-options available for nonoptimal, nonlinear fit, optimal, and other transformations.
Nonoptimal Transformation t-options
Penalized B-Spline t-options
The following t-options are available with the PBSPLINE transformation.
Algorithm Options (a-options)
This section discusses the options that can appear in the PROC TRANSREG or MODEL statement as a-options. They are listed after the entire model specification and after a slash. Here is an example:
proc transreg;
model spline(y / nknots=3)=log(x1 x2 / parameter=2)
/ nomiss maxiter=50;
In the preceding statements, NOMISS and MAXITER= are a-options. (SPLINE and LOG are transforms, and NKNOTS= and PARAMETER= are t-options.) The statements find a spline transformation with 3 knots on
y and a base 2 logarithmic transformation on x1 and x2. The NOMISS a-option excludes all observations with missing values, and the MAXITER= a-option specifies the maximum number of iterations.
Table 97.4 summarizes the a-options available in the PROC TRANSREG or MODEL statement.
Table 97.4: Options Available in the PROC TRANSREG or MODEL Statement
Option Description
Input Control
REITERATE Restarts iterations
TYPE= Specifies input observation type
Method and Iterations
CCONVERGE= Specifies minimum criterion change
CONVERGE= Specifies minimum data change
MAXITER= Specifies maximum number of iterations
METHOD= Specifies iterative algorithm
NCAN= Specifies number of canonical variables
NSR Specifies no restrictions on smoothing models
SINGULAR= Specifies singularity criterion
SOLVE Attempts direct solution instead of iteration
Missing Data Handling
INDIVIDUAL Fits each model individually (METHOD=MORALS)
MONOTONE= Includes monotone special missing values
NOMISS Excludes observations with missing values
UNTIE= Unties special missing values
Intercept and CLASS Variables
CPREFIX= Specifies CLASS coded variable name prefix
LPREFIX= Specifies CLASS coded variable label prefix
NOINT Specifies no intercept or centering
ORDER= Specifies order of CLASS variable levels
REFERENCE= Controls output of reference levels
SEPARATORS= Specifies CLASS coded variable label separators
Control Displayed Output
ALPHA= Specifies confidence limits alpha
CL Displays parameter estimate confidence limits
DETAIL Displays model specification details
HISTORY Displays iteration histories
NOPRINT Suppresses displayed output
PBOXCOXTABLE Prints the Box-Cox log likelihood table
RSQUARE Displays the R square
SHORT Suppresses the iteration histories
SS2 Displays regression results
TEST Displays ANOVA table
TSUFFIX= Shortens transformed variable labels
UTILITIES Displays conjoint part-worth utilities
ADDITIVE Fits additive model
NOZEROCONSTANT Does not zero constant variables
TSTANDARD= Specifies transformation standardization
The following list provides details about these a-options. The a-options are available in the PROC TRANSREG or MODEL statement. | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_transreg_syntax05.htm","timestamp":"2024-11-06T10:53:12Z","content_type":"application/xhtml+xml","content_length":"313480","record_id":"<urn:uuid:cbfadd26-1d0d-4246-92ea-e408db8b51d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00527.warc.gz"} |
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Vibroimpact mechanism in one separate case
It is known that in the vibroimpact system at the chosen values of parameters linear relationship between impact velocities and eigenfrequencies may exist. The purpose of this paper is to reveal the
qualities of the systems of this type. Investigations are performed by analytical and numerical methods. It is determined that in the systems of this type nonlinear solutions with infinite series of
harmonics exist. Multivalued stable and unstable regimes do not exist in the systems. The obtained analytical relationships enabled to reveal new qualities of the systems and to make useful
1. Introduction
Investigation of dynamics of vibroimpact systems is presented in a number of publications, where the main fundamental achievements are presented including recent years for the cases when the
vibrations of the system are with stiff and soft stiffness characteristics. Here the system of intermediate type is investigated. Investigations were performed also graphically, and they enabled to
reveal the qualities of the system, which enable to create energetically more effective mechanisms.
Resonances and velocity jumps in nonlinear dynamics are investigated in [1]. Basic theory of vibrating systems with impacts is presented in [2]. Vibro-impact dynamics under periodic and transient
excitations is investigated in [3]. Modeling of nonlinear dynamics of a system with clearance is performed in [4]. Dynamical behavior of a vibro-impact oscillator is investigated in [5].
Stabilization of periodic nonlinear systems is analysed in [6]. Basic ideas of vibrating systems in engineering are presented in [7]. Contemporary methods of vibration theory are described in [8].
Basic concepts of mechanical vibrations are presented in [9]. Nonlinear dynamics of inertial actuators is investigated in [10]. Nonlinear effects in dynamics of bearings are presented in [11].
Nonlinear contact dynamics of ultrasonic actuator is investigated in [12]. Non-sinusoidal dynamics of interacting oscillators is analysed in [13]. Free vibration analysis of piezoelectric cylinder is
performed in [14]. Free vibrations of nonlinear oscillators are investigated in [15]. Synchronization of impacting mechanical systems is analysed in [16]. Chatter in mechanical systems with impacts
is investigated in [17]. Dynamics of systems with impact and friction is analysed in [18]. Periodic orbits of mechanical systems with impacts are investigated in [19]. Vibro-impact nonlinear behavior
and energy transfer are described in [20]. Modeling of particle impact is performed in [21]. Impacts in novel mechanisms and their applications are investigated in [22]. Resonant type impact
mechanism is analysed in [23]. Positioning using impact drive mechanism is investigated in [24]. Impact mechanics of collisions and experimental results are analysed in [25]. Nonlinear rotor system
with vibration absorbers is investigated in [26]. Active vibration absorber for impulse excitation is described in [27]. Nonlinear vibrations of a beam with piezoelectric actuators are investigated
in [28]. Nonlinear effects and their use for vibration isolation are analysed in [29]. Nonlinear vibration absorber is investigated in [30]. Nonlinear free vibrations of beams are analysed in [31].
Electro-mechanical coupling vibrations of structures are investigated in [32]. Nonlinear dynamic analysis of vehicle system is performed in [33]. Wideband vibration attenuation is investigated in
[34]. Nonlinear vibrations with interactions are analysed in [35]. Piezoelectric nonlinear vibrations are investigated in [36]. Nonlinear analysis of free vibrations of beams is performed in [37].
Nonlocal free and forced vibrations of beams are investigated in [38]. Vibration system with nonlinear coupling is analysed in [39]. Nonlinear vibrations of a system with piecewise linear spring are
investigated in [40]. Nonlinear vibrations of piezoelectric plates are analysed in [41]. Free and forced nonlinear vibrations of beams are investigated in [42]. Nonlinearities in piezoceramic
actuators are analysed in [43]. Nonlinear free and forced vibrations of beams are investigated in [44]. Nonlinear free vibrations of shells are analysed in [45]. Nonlinear free vibrations of plates
are investigated in [46].
The system is described in the following way:
$m\stackrel{¨}{x}+H\stackrel{˙}{x}+Cx=F\mathrm{s}\mathrm{i}\mathrm{n}\omega t,\mathrm{}\mathrm{}\mathrm{}x<0,$
where the collision of the vibrating mass is considered as an instantaneous process, ${\stackrel{˙}{x}}^{-}$ denotes velocity before the impact and ${\stackrel{˙}{x}}^{+}$ denotes velocity after the
impact, the coefficient of restitution of the impact velocity of the mass is denoted as $R$ and it is in the interval 0 $\le R\le$ 1.
The equation is rearranged:
$\stackrel{¨}{x}+2h\stackrel{˙}{x}+{p}^{2}x=f\mathrm{s}\mathrm{i}\mathrm{n}\omega t,\mathrm{}\mathrm{}\mathrm{}x<0,$
2. Conservative motion of the system, decaying vibrations when $\mathbit{f}=0$
It is assumed that the impact number $i$ of the mass $m$ to the support takes place when:
and the next impact number $i+$ 1 takes place when:
According to the Eq. (3) motion after $t\ge$ 0 is:
where the constant quantities ${C}_{1}$ and ${C}_{2}$ are found from the conditions Eq. (4) by assuming ${\stackrel{˙}{x}}_{i}^{+}$ after impact according to the Eq. (4).
The next impact number $i+$ 1 takes place at the conditions Eq. (5). By taking into account the Eqs. (7) and (8) it is obtained:
${T}_{i}=\frac{\pi }{\sqrt{{p}^{2}-{h}^{2}}}.$
During the time of the cycle of motion between the impacts $i$ and $i+$ 1 the change of velocities is lost, which is estimated by the dummy coefficient:
${R}_{{f}_{i+1,i}}=\frac{{\stackrel{˙}{x}}_{i+1}^{-}}{{\stackrel{˙}{x}}_{i}^{-}}=+R\cdot \mathrm{exp}\left(-\frac{h}{\sqrt{{p}^{2}-{h}^{2}}}\pi \right)=-R\cdot \mathrm{exp}\left(\frac{h/p}{\sqrt{1-{\
Further graphical material representing dynamics of the investigated system is presented for various parameters of the system in Fig. 1, Fig. 2 and Fig. 3.
Fig. 1Dynamics of the system when the initial conditions of motion t= 0, x0= 0, x˙0= –1 and p= 1 for h= 0 (thin line), h= 0.25 (line of medium thickness) and h= 0.5 (thick line)
a) Displacement as function of time
b) Velocity as function of time
c) Acceleration as function of time
d) Velocity multiplied by acceleration as function of time
e) Phase trajectory: velocity as function of displacement
f) Phase trajectory: acceleration as function of velocity
g) Phase trajectory: velocity multiplied by acceleration as function of displacement
Fig. 2Dynamics of the system when h= 0 and p= 1 for the initial conditions of motion t= 0, x0= 0, x˙0= –1 (thin line), t= 0, x0= 0, x˙(0)= –2/3 (line of medium thickness) and t= 0, x0= 0, x˙(0)= –1/3
(thick line)
a) Displacement as function of time
b) Velocity as function of time
c) Acceleration as function of time
d) Velocity multiplied by acceleration as function of time
e) Phase trajectory: velocity as function of displacement
f) Phase trajectory: acceleration as function of velocity
g) Phase trajectory: velocity multiplied by acceleration as function of displacement
3. Dynamics of the conservative system
Case: conservative system, that is when:
In this case the Eqs. (7)-(10) take the following form when:
$\stackrel{-}{T}=\frac{\pi }{p}.$
Period of motion $\stackrel{-}{T}$ is the eigenperiod of vibrations of the system and $\stackrel{-}{\omega }$ is the eigenfrequency of vibrations of the system, that is by equating the right sides of
the Eqs. (17) it is obtained:
$\stackrel{-}{\omega }=2p,\mathrm{}\mathrm{}\mathrm{}\stackrel{-}{T}=\frac{\pi }{p}=\frac{2\pi }{\stackrel{-}{\omega }}.$
Fig. 3Dynamics of the system when h= 0.5 and p= 1 for the initial conditions of motion t= 0, x0= 0, x˙0= –1 (thin line), t= 0, x0= 0, x˙(0)= –2/3 (line of medium thickness) and t= 0, x0= 0, x˙0= –1/3
(thick line)
a) Displacement as function of time
b) Velocity as function of time
c) Acceleration as function of time
d) Velocity multiplied by acceleration as function of time
e) Phase trajectory: velocity as function of displacement
f) Phase trajectory: acceleration as function of velocity
g) Phase trajectory: velocity multiplied by acceleration as function of displacement
By expanding the functions of displacement, velocity and acceleration into the Fourier series the following expressions of the first terms of the series are obtained. The laws of motions may be
expanded into the Fourier series. From the Eqs. (14-16) it is obtained:
$x=-\frac{{\stackrel{˙}{x}}^{-}}{p}\left\{\frac{2}{\pi }-\frac{4}{\pi }{\sum }_{n=1}^{n=\mathrm{\infty }}\left[\frac{1}{\left(2n-1\right)\left(2n+1\right)}\mathrm{c}\mathrm{o}\mathrm{s}n\stackrel{-}
{\omega }t\right]\right\},$
$\stackrel{˙}{x}=-{\stackrel{˙}{x}}^{-}\frac{8}{\pi }{\sum }_{n=1}^{n=\mathrm{\infty }}\left[\frac{n}{\left(2n-1\right)\left(2n+1\right)}\mathrm{s}\mathrm{i}\mathrm{n}n\stackrel{-}{\omega }t\right],$
$\stackrel{¨}{x}=p{\stackrel{˙}{x}}^{-}\left\{\frac{2}{\pi }-\frac{4}{\pi }{\sum }_{n=1}^{n=\mathrm{\infty }}\left[\frac{1}{\left(2n-1\right)\left(2n+1\right)}\mathrm{c}\mathrm{o}\mathrm{s}n\stackrel
{-}{\omega }t\right]\right\},$
where $\stackrel{-}{\omega }=2p.$
From the Eqs. (14) and (15) it is obtained:
that is in the system of coordinates $\stackrel{˙}{x}0px$ there is a circle the length of radius of which is equal to ${\stackrel{˙}{x}}^{-}.$
From the Eqs. (15) and (16) it is obtained:
that is in the system of coordinates $\frac{\stackrel{¨}{x}}{p}0\stackrel{˙}{x}$ there is a circle the length of radius of which is equal to ${\stackrel{˙}{x}}^{-}.$
Further graphical material of amplitude frequency characteristics is presented in Fig. 4.
Fig. 4Amplitude frequency characteristics (constant part and first three harmonics) when h= 0 and p= 1
a) Displacement frequency $\mathrm{c}\mathrm{o}\mathrm{s}$ characteristic
b) Velocity frequency $\mathrm{s}\mathrm{i}\mathrm{n}$ characteristic
c) Acceleration frequency $\mathrm{c}\mathrm{o}\mathrm{s}$ characteristic
d) Velocity multiplied by acceleration frequency $\mathrm{s}\mathrm{i}\mathrm{n}$ characteristic
4. Conclusions
On the basis of the presented results the qualities of dynamic behavior of the nonlinear vibroimpact mechanism in separate case, when the contacting surface of the vibrating part of the system with
the impacting surface is in the position of static equilibrium, are investigated.
Analytical relationships describing the motion of the system and amplitude frequency characteristics have been determined and are presented in the paper. Graphical relationships for typical
parameters of the system were obtained and are investigated. It is shown that the values of eigenfrequencies of vibroimpact vibrations do not depend on the values of amplitudes of excitations.
Because of this fact multivalued stable and unstable regimes can not take place in the vicinities of resonances.
The presented results enable to perform the design of vibrating vibroimpact systems of this type.
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About this article
characteristics of impact velocity and eigenfrequencies
free and decaying vibrations
phase trajectories of motions
harmonics of motions up to infinity
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Coulomb's Law: Solved Problems for High School and College
Coulomb's Law: Solved Problems for High School and College
Practice problems with detailed solutions about Coulomb's law are presented that are suitable for the AP Physics C exam and college students. For more solved problems (over 61) see here.
Note: In textbooks, the terms ‘Coulomb force’, ‘electric force’, and ‘electrostatic force’ are used interchangeably to describe the force between two point charges.
Coulomb's Law Practice Problems
Problem (1): Two like and equal charges are at a distance of $d=5\,{\rm cm}$ and exert a force of $F=9\times 10^{-3}\,{\rm N}$ on each other.
(a) Find the magnitude of each charge.
(b) What is the direction of the electrostatic force between them?
Solution: The magnitude of the force between two rest point charges $q$ and $q'$ separated by a distance $d$ is given by Coulomb's law as follows: \[F=k\,\frac{|q|\,|q'|}{d^2}\] where $k \approx 8.99
\times 10^{9}\,{\rm \frac{N.m^{2}}{C^2}}$ is the Coulomb constant and the magnitudes of charges denoted by $|\cdots|$.
Let the magnitude of charges be $|q_1|=|q_2|=|q|$, Now by substituting the known numerical values of $F$ and distance $d$, and solving for $|q|$ we get
\begin{align*} F&=k\,\frac{|q_1|\,|q_2|}{d^{2}}\\ 9\times 10^{-3}&=(8.99\times 10^{9})\frac{|q|^{2}}{(0.05)^{2}}\\ \Rightarrow q^{2}&=25\times 10^{-16}\\ \Rightarrow q&=5\times 10^{-8}\,{\rm C} \end
{align*} In the second equality, we converted the distance from $cm$ to $m$ to coincide with SI units.
The direction of the Coulomb force depends on the sign of the charges. Two like charges repel and two unlike ones attract each other.
Since $q_1$ and $q_2$ have the same signs the electric force between them is repulsive.
Problem (2): What is the electrostatic force between two pieces of grain in a grain elevator if one piece holds a charge of $5\times 10^{-16}\,\rm C$ and the other holds a charge of $2\times 10^{-16}
\,\rm C$, while being separated by a distance of 0.06 m?
(a) Find the magnitude of the Coulomb force that one grain exerts on the other.
(b) Is the force attractive or repulsive?
Solution: Known values:\begin{gather*}|q|=2\times 10^{-16}\,{\rm C}\\ |q'|=5\times 10^{-16}\,{\rm C}\\ d=0.06\,{\rm m} \end{gather*}
(a) Coulomb's law gives the magnitude of the electric force between two stationary (motionless) point charges so by applying it we have \begin{align*}F&=k\,\frac{|q|\,|q'|}{d^{2}}\\&=(9\times 10^{9})
\,\frac{(2\times 10^{-16})(5\times 10^{-16})}{(0.06)^2}\\&=2.5\times 10^{-19}\,{\rm N}\end{align*}
(b) Since the charges have like signs the electric force between them is repulsive.
If you are getting ready for AP physics exams, these electric force problems are also relevant.
Problem (3): What is the magnitude of the force that a ${\rm 25\, \mu C}$-charge exerts on a ${-\rm 10\,\mu C}$ charge ${\rm 8.5\, cm}$ away? (Take $k=9\times 10^{9}\,{\rm \frac{N.m^{2}}{C^2}}$)
Solution: The magnitude of the attraction/repulsion force between two point charges is given by Coulomb's Law as follows: \begin{align*}F&=k\frac{|qq'|}{r^2}\\&=(9\times 10^{9}){\rm \frac{(25\times
10^{-6}\,C)(10\times 10^{-6}\,C)}{(8.5\times 10^{-2}\,m)^2}}\\&=311.5\quad {\rm N}\end{align*} These two point charges have opposite signs, so the electrostatic force between them is attractive.
Problem (4): Two charged particles apply an electric force of $5.2\times 10^{-3}\,{\rm N}$ on each other. Distance between them gets twice as much as before. What will be Coulomb's force?
Solution: Using Coulomb's law, we have $F=k\frac{|qq'|}{r^2}$, where $r$ is the distance between two charges. We are told in the problem that the distance is doubled so $r_2=2r_1$, thus the electric
force is found as \begin{align*}F_2&=k\frac{|qq'|}{r_2^2}\\\\&=k\frac{|qq'|}{(2r_1)^2}\\\\&=\frac 14 \underbrace{k\frac{|qq'|}{r_1^2}}_{F_1}\\\\&=\frac 14 F_1\end{align*}
Problem (5): Two charged point particles are $4.41\,{\rm cm}$ apart. They are moved and placed in a new position. The force between them is found to have tripled. How far apart are they now?
Solution: initial distance is $r_1=4.41\,{\rm cm}$. At the new location, the force is tripled $F_2=3F_1$. Applying Coulomb's law, we have \begin{align*}F_2&=3F_1\\ \\ k\frac{\cancel{|qq'|}}{r_2^2}&=
3k\frac{\cancel{|qq'|}}{r_1^2}\\ \\ \frac{1}{r_2^2}&=\frac{3}{(4.41\times 10^{-2})^2}\\\\ \Rightarrow r_2^2&=\frac{(4.41\times 10^{-2})^2}{3}\end{align*} Taking the square root of both sides, we get
\[r_2=0.0254\,{\rm m}\] Thus, if those two charges are $2.54\,{\rm cm}$ away, the electrostatic force between them gets tripled.
Problem (6): Two small spheres are charged with the same quantity of charge but of opposite types. The charge on each sphere is $8\times 10^{-7}\,\rm C$ and they are separated by a distance of $\rm
0.75\, m$. Determine the electrical force of attraction between the two spheres.
Solution: Use Coulomb's law and plug in known values and solve for the unknown. \begin{align*} F&=\frac{kq_1 \, q_q}{r^2} \\\\ &=\frac{(9\times 10^9)(8\times 10^{-7})(8\times 10^{-7})}{(0.75)^2} \\\\
&=0.01\,\rm N \end{align*}
Conceptual Problem: A student rubs a balloon against their hair and then touches it to a wall. The balloon adheres to the wall. What is the scientific principle that explains why the balloon sticks
to the wall?
Solution: The balloon sticks to the wall due to the principle of static electricity, which can be explained by Coulomb’s law.
When the balloon is rubbed against the student’s hair, it becomes negatively charged due to the transfer of electrons from the hair to the balloon.
When the balloon is then brought near the wall, these extra electrons repel the electrons in the wall, causing the surface of the wall to become positively charged.
Coulomb’s law states that the force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
In this case, the balloon and the wall act like two charges. The negatively charged balloon and the positively charged wall attract each other, and this electrostatic force, as described by Coulomb’s
law, causes the balloon to stick to the wall.
Solve these practice problems on the electric charge to get a better view of charges in physics.
Conceptual Problem: Two identical metal spheres, C and D, have charges of $+3.0\times 10^{-6}$ coulomb and $+1.5 \times 10^{-6}$ coulomb, respectively. The magnitude of the electrostatic force on C
due to D is $3.6$ newtons. What is the magnitude of the electrostatic force on D due to C?
Solution: According to Newton’s third law of motion, every action has an equal and opposite reaction. This law applies to the forces between charged particles as well. Therefore, the magnitude of the
electrostatic force on sphere B due to sphere A is equal to the magnitude of the electrostatic force on sphere A due to sphere B.
This is because the forces are action-reaction pairs according to Newton’s third law, and their magnitudes are determined by the charges on the spheres and the distance between them according to
Coulomb’s law.
Therefore, the magnitude of the electrostatic force on sphere B due to sphere A is also 2.4 Newtons, the same as the force on sphere A due to sphere B.
Problem (7): Suppose that two point charges, each with a charge of +1 Coulomb are separated by a distance of
1 meter.
(a) Will they attract or repel?
(b) Determine the magnitude of the electrical force between them.
(a) Since the charges are like the electric force between them is repulsive.
(b) The magnitude of electric force between two charges is found by Coulomb's law as follows: \begin{align*} F&=k\frac{|q_1 q_2|}{r^2}\\&=\big(9\times 10^9\big)\frac{1\times 1}{1^2}\\&=9\times 10^9\
quad {\rm N}\end{align*}Where $|\cdots|$ denotes the absolute values of charges regardless of their signs.
Problem (8): Two balloons are charged with an identical quantity and type of charge: -0.0025 C. They are held apart at a separation distance of 8 m. Determine the magnitude of the electrical force of
repulsion between them.
Solution: applying Coulomb's law and putting the given numerical values in it, we have \begin{align*} F&=k\frac{q_1 q_2}{r^2}\\&=\big(9\times 10^9\big)\frac{(0.0025)(0.0025)}{(8)^2}\\&=879\quad {\rm
Problem (9): Two charged boxes are 4 meters apart from each other. The blue box has a charge of +0.000337 C and is attracting the red box with a force of 626 Newtons. Determine the charge of the red
box. Remember to indicate if it is positive or negative.
Solution: known information are $q_1=+0.000337\,{\rm C}$, $F=626\,{\rm N}$, and $d=4\,{\rm m}$. Unknown is $q_2=?$. Coulomb's law gets the magnitude of the force between two charges. By applying it
and solving for $q_2$, we have \begin{align*} F&=k\frac{q_1 q_2}{d^2}\\ \\ \Rightarrow q_2&=\frac{F\,d^2}{k\,q_1}\\ \\ &=\frac{626\times (4)^2}{(9\times 10^9)(0.000337)}\\ \\&=0.0033\quad {\rm C}\end
{align*} Since in the problem said that the force is attraction, so the charge of the red box must be negative.
Note that, Coulomb's law gives only the magnitude of the electric force without their signs.
Problem (10): A piece of Styrofoam has a charge of -0.004 C and is placed 3.0 m from a piece of salt with a charge of -0.003 C. How much electrostatic force is produced?
Solution: the magnitude of the electrostatic force is determined as follows, \begin{align*} F&=k\frac{q_1 q_2}{d^2}\\&=\big(9\times 10^9\big)\frac{(0.004)(0.003)}{(3)^2}\\&=12000\quad {\rm N}\end
{align*} Note that in Coulomb's force equation, the magnitude of the charges (regardless of their signs) must be included.
Problem (11): Two coins lie 1.5 meters apart on a table. They carry identical electric charges. Approximately how large is the charge on each coin if each coin experiences a force of 2.0 N?
Solution: Substituting the given numerical into Coulomb's law equation, we have \begin{align*} F&=k\frac{q_1 q_2}{d^2}\\ \\ 2&=\big(9\times 10^9 \big)\frac{q\,q}{(1.5)^2}\\ \\ \Rightarrow q&=\sqrt{\
frac{2\times (1.5)^2}{9\times 10^9}}\\ \\ &=2.23\times 10^{-5} \quad {\rm C} \end{align*}
The following problems are for practicing for the AP Physics C problems.
Problem (6): Three point charges are placed at the corners of an equilateral triangle as in the figure below. What is the magnitude and direction of the net electric force on the $2\,\rm \mu C$
Solution: The force that the charge $-6\,\rm \mu C$ applies to the $2\,\rm \mu C$ is attractive and to the right along the line connecting them and its magnitude is also calculated as follows: \begin
{align*} F_{2,-6}&=k\frac{qq'}{r^2} \\\\ &=(9\times 10^9) \frac{(2\times 10^{-6})(6\times 10^{-6})}{(0.10)^2} \\\\ &=10.8\,\rm N \end{align*} The charge $8\,\rm \mu C$ is repelled the charge $2\,\rm
\mu C$ along the line joining them with a magnitude of \begin{align*} F_{2,8}&=k\frac{qq'}{r^2} \\\\ &=(9\times 10^9) \frac{(2\times 10^{-6})(8\times 10^{-6})}{(0.10)^2} \\\\ &=14.4\,\rm N \end
{align*} Given the geometry shown below, the force $\vec{F}_{2,8}$, denoted by $\vec{F}_8$ for simplicity, makes an angle of $60^\circ$ with the negative direction of $x$-axis.
Resolving this vector force along the horizontal and vertical directions gives its components \begin{align*} F_{1x}&=F_1 \cos 60^\circ \\ &=14.4\times (0.5)=72.2\,\rm N \\\\ F_{1y}&=F_1 \sin 60^\circ
\\&=14.4\times (\frac{\sqrt{3}}{2})=72.2\sqrt{3}\,\rm N \end{align*} The net electric force on the charge $2\,\rm \mu C$ is the vector sum of individual forces due to other charges (superposition
principle) \[\vec{F}_2=\vec{F}_8+\vec{F}_6 \] Adding the vectors along the horizontal and vertical directions give the corresponding components of the net electric force. \begin{align*} F_{2x}&=10.8+
(-72.2) \\&=28.6\,\rm N \\\\ F_{2y}&=0+(-72.2\sqrt{3}) \\&=-72.2\sqrt{3} \,\rm N \end{align*} Given these components, we can find the magnitude and direction of the net electric force the desired
charge \begin{align*} F_2&=\sqrt{F_{2x}^2+F_{2y}^2} \\\\ &=\sqrt{(28.6)^2+(-72.2\sqrt{3})^2} \\\\&=32.16\,\rm N \end{align*} and its direction with the positive $x$-direction as follows: \begin
{align*} \alpha &=\tan^{-1}\left(\frac{F_{2y}}{F_{2x}}\right) \\\\ &=\tan^{-1}\left(\frac{72.2\sqrt{3}}{28.6}\right) \\\\ &=77^\circ \end{align*}
Problem (7): A $2-\rm \mu C$ point charge and another point charge of magnitude $4-\rm \mu C$ are a distance $L=1\,\rm m$ apart. Where should a third point charge be placed so that the net electric
force on it is zero?
Solution: This is a tricky question and is more similar to the AP Physics C questions. We solve this problem by assuming the third point charge is positive.
Simple reasoning shows that between the charges the net force on the third charge points to the right and adds together rather than canceling each other. Thus, this region is removed.
Now, assume a point somewhere outside the charges and closer to the smaller one at a distance $x$ from it. The following figure shows that the forces on this hypothetical positive charge are clearly
in opposite directions.
\begin{gather*} F_2=F_4 \\\\ k\frac{q_3 q_2}{x^2}=k\frac{q_3 q_4}{(L+x)^2} \\\\ \frac{2}{x^2}=\frac{4}{(1+x)^2} \\\\ (1+x)^2=2x^2 \\\\ \Rightarrow \boxed{x^2-2x-1=0} \end{gather*} The above quadratic
equation has two solutions \[x_1=2.41\,\rm m \quad , \quad x_2=-0.4\,\rm m\] The negative in the second solution means that we must go back to the region between the charges that is not acceptable.
Thus, outside the charges and somewhere close to the smaller charge we can find a point where the net Coulomb force on the third charge is zero.
For practice, consider yourself a negative charge, repeat the above steps, and find the result.
Problem (8): In the following figure, a small sphere of mass $3\,\rm g$ and charge of $q_1=15\,\rm nC$ are suspended by a light string over the second charge of equal mass and charge of $-85\,\rm
nC$. The distance between the two charges is $2\,\rm cm$.
(a) Find the tension in the string.
(b) What is the smallest value of separation $d$, assuming the string can withstand a maximum tension of $0.150\,\rm N$?
Solution: The negative charge and gravity pull vertically down the positive charge and the tension in the string pulls up, as depicted in the following free-body diagram.
(a) The positive charge is motionless so the net force on it must be zero. Balancing the above forces applied to it gives the tension force in the string. \begin{gather*} T=F_e+mg \\\\ T=k\frac{q_1
q_2}{r^2}+mg \\\\ T=(9\times 10^9) \times \frac{(15\times 10^{-9})(85\times 10^{-9})}{(0.02)^2}+(0.003)(10) \\\\ \Rightarrow \boxed{T=0.31\,\rm N} \end{gather*}
(b) This time, the maximum tension force that the string can withstand is given and asked to find the smallest distance between the charges.
Again we use the above relation between the forces on the positive force and solve for the unknown distance $d$. \begin{gather*} T=k\frac{q_1 q_2}{r^2}+mg \\\\ T-mg=k\frac{q_1 q_2}{r^2} \\\\ r^2=k\
frac{q_1 q_2}{T-mg} \end{gather*} Taking the square root of both sides and substituting the numerical values gives \begin{align*} r&=\sqrt{\frac{(9\times 10^9)(15\times 10^{-9})(85\times 10^{-9})}
{0.150-(0.003)(10)}} \\\\ &=0.97\,\rm m \end{align*}
In summary, Coulomb's law is fundamental for solving an electrostatic problem in the case of some arrangements of point charges. The following PDF is tailored for the college level and is difficult.
Author: Dr. Ali Nemati
Last Updated: Feb 3, 2024 | {"url":"https://physexams.com/lesson/Coulombs-Law-Solved-Problems_16","timestamp":"2024-11-13T03:06:40Z","content_type":"text/html","content_length":"49945","record_id":"<urn:uuid:5bb449d9-afe4-4332-a950-592bc3d8a1b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00305.warc.gz"} |
Gear cube (puzzle type)
(Redirected from Gear cube)
This page is about the Gear Cube puzzle. For the Gear Cube Extreme or the Anisotropic Gear Cube, see Gear Cube Extreme. For a list of geared puzzles, see Category:Geared puzzles.
The Gear Cube (also known as the Caution Cube) is a twistable puzzle in the shape of a cube that is cut two times along each of three axes, as a 3x3x3. Moreover, there are geared edges. That means
the edges can turn around themselves. It was invented by Oskar van Deventer (idea of Bram Cohen). It was first produced by Shapeways since 2009, and then by Meffert's since 2010 (for 36€) and it
finally got copied by LanLan.
This puzzle has 6 fixed centers (they can't move related to each others), 12 edges, and 8 corners, and a total of 41,472 positions. It has been proven that every position can be solved in 8 or fewer
moves in ATM and 12 or fewer in HTM. It's important to note that quarter moves block the cube (only half turns are unprohibited) so QTM has no meaning for this puzzle.
It's not an official event in WCA competition.
The current unofficial world record single speed solve is 2.16 seconds held by Kentaro Nishi. It looks harder but it is actually way easier to solve as compared as with a normal 3x3x3 cube.
Fun facts
• The gear cube was originally called the "Caution Cube" because the solver's fingers could get caught in the gears
See also
External Links | {"url":"https://www.speedsolving.com/wiki/index.php?title=Gear_cube","timestamp":"2024-11-14T12:07:56Z","content_type":"text/html","content_length":"29227","record_id":"<urn:uuid:a8d550cc-fc39-4b56-afb4-ef5df913a8c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00142.warc.gz"} |
What is the cost-effectiveness of researching vaccines? — EA Forum Bots
This essay was jointly written by Peter Hurford and Marcus A. Davis.
Note that because of technical length restrictions on the EA Forum, this essay is broken up into three parts: Part 1, Part 2, and Part 3. To see all three parts in one part, you can view the article
on our research site.
Previous articles also include an analysis of how beneficial vaccines have been, an analysis of how much it costs to roll-out a vaccine, how much it costs to research and develop a vaccine, and how
long it takes to research a new vaccine. However, this series is structured so that starting with this article should be all you need to do.
We looked at academic literature for vaccine cost-effectiveness as a whole and we also performed individual case studies on seven contemporary and historical vaccines to try to estimate the total
cost-effectiveness of researching and developing a vaccine from scratch. Looking back historically, we find a range of $0.50 to $1600 per DALY, depending on the vaccine. Using this historical
information, we derive an estimate for the total cost-effectiveness of developing and rolling out a “typical” / ”average” vaccine as being $18 - $7000 / DALY. The smallpox vaccine, malaria vaccine,
and rotavirus vaccine may all be more cost-effective investments in total than marginal investments in distributing bednets (see Appendix C), especially when pursued to the point of completely
eradicating the disease. However, there are many important assumptions made by these models, and changing them could strengthen or undermine these conclusions.
Section 1 and Section 2 detail the raw cost-effectiveness figures, section 3 provides a detailed analysis of all the key assumptions and model uncertainty, and section 4 provides our takeaways.
Appendix A provides an analysis of comparing our models to other models in the literature, Appendix B provides an estimate of the cost-effectiveness of the Ebola vaccine, Appendix C compares
vaccination cost-effectiveness to distributing bednets, and Appendix D provides a rough assessment and comparison of some estimates of the cost-effectiveness of GAVI.
Table of Contents
1. What are the costs and benefits?
While developing a vaccine is a huge accomplishment, it is not really of any significant humanitarian benefit until the vaccine is scaled up and rolled out to a significant population. Thus, spending
a lot of money to develop a vaccine merely “unlocks” the opportunity to roll out the vaccine and the hope is that this roll-out is cost-effective enough to make up for the cost of the R&D once
amortized across the entire vaccinated population.
We, therefore, take the following as our equation for the total cost-effectiveness of vaccine R&D:
Total cost-effectiveness of vaccine R&D ($/DALY) = ((Total R&D costs of making vaccine) + (Total roll-out costs of vaccine)) / ((DALY burden of disease) * (% reduction in disease attributable to the
Note here that we’re looking at the total cost-effectiveness across the entire investment in R&D, which requires us to also look at the total investment across vaccine roll-out too. We’re not looking
at the marginal investment, or the cost-effectiveness you would get if you added more funding to vaccines today, which could be a very different number (see section 3.8 for details).
To try to answer this question, we previously modeled how long it would take to develop a new vaccine, how much it costs to research and develop a vaccine, how much it costs to roll-out that vaccine
and then how beneficial we can expect vaccines to be.
These data were calculated from available evidence about vaccines generally and also for specific case studies: smallpox (which was chosen as it was one of the first vaccines and the only human
disease successfully eradicated by vaccination); measles (which was chosen as it was one of the first vaccines); HIV, malaria, and Ebola vaccines as they are modern and under current development, and
rotavirus and HPV as they recently finished vaccine licensing. For each of these vaccines, we calculated how much they would cost for both R&D and roll-out, and then what benefits we would expect.
We aggregated all this data in a spreadsheet that makes all the calculations between the multiple sections much more clear, with a good amount of detail. Based on that data, we come to the following
conclusions. (For more detail, see the spreadsheet and the previous articles.)
Vaccine R&D Costs Roll-out Costs
Smallpox $5.58M $0.73-$47.62 / child
Measles $38.3M $1-$38 / child
Rotavirus $1,140M $3-$28 / child
HPV ? $2.55-$22.71 / child
HIV $24,500M $50-$160 / child
Malaria $605M $22 / child + $293M
Ebola $1,500M ?
“Typical” vaccine $460M - $1900M $13.21-$53.05 / child
Vaccine DALYs per vaccinated person in 2016[a] Yearly DALYs at 60% vaccination rate in 2016 SSA Yearly DALYs if eradication[b]
Smallpox 0.14037 7,437,600 44,653,000
Measles 0.20793 34,105,842 99,341,000
Rotavirus 0.17990 4,589,392 Not possible
HPV 0.01250 327,522 5,173,000[c]
HIV 0.26343 9,655,005 57,575,000
Malaria 0.40619 10,362,216 56,201,000
Ebola[d] 0.07502 92,685 309,000
[a] - Estimates for DALYs prevented per person vaccinated over a 20 year period.
[b] - These figures, except for smallpox and measles, are derived from their 2016 global DALY burdens from Global Burden of Disease, Results Tool (2016e)
[c] - This figure is 70% of the 2016 global burden from cervical cancer of 7,390,002.82.
[d] - All ebola estimates here are a 5 year average from 2012-2016 and assumes a vaccine that is 50% effective.
2. What is the cost-effectiveness?
Based on the data in the above tables and the considerations we’ve made, we can then apply our formula: Total cost-effectiveness of vaccine R&D ($/DALY) = ((Total R&D costs of making vaccine) +
(Total roll-out costs of vaccine)) / ((DALY burden of disease) * (% reduction in disease attributable to the vaccine))
As stated before, these calculations were originally derived from research in prior articles (e.g., for R&D costs, roll-out costs, and total benefits) for a particular basket of vaccines. We then
aggregated all this data in a spreadsheet , used the spreadsheet to create individual Guesstimate models for each vaccine, and created 90% confidence intervals from each model. We made calculations
for each of our vaccines, except for Ebola, where we felt like there was insufficient information to make a confident calculation. (We still attempt to estimate for Ebola in Appendix B.)
The models varied targeted populations across two scenarios – one where 60% of the relevant population in Sub-saharan Africa (SSA) is vaccinated and another where the disease is completely
eradicated. We also varied assumptions about the DALY burden, vaccine effectiveness, roll-out costs, and R&D costs. We then created these 90% confidence intervals.
2.1) Scenario 1: Vaccinate 60% of the Relevant Populations in Subsaharan Africa
For this scenario, we assume that there is a one-time fixed cost investment for researching and developing the vaccine and building infrastructure for rolling out the vaccine that is amortized over a
time period to consider benefits for (in our model, we cap it at 20 years), then an annual cost to keep rolling out the vaccine to the relevant population. Then, every year, we save a certain amount
of DALYs from preventing that disease via the vaccine. Together, we can use this to calculate DALYs averted over the benefits time period compared to the cost spent during the time period.
Vaccine Roll-out Cost-effectiveness Roll-out + R&D Cost-effectiveness Guesstimate[1]
Smallpox $4.30 - $66 / DALY $4.40 - $67 / DALY Link
Measles $8 - $320 / DALY $9 - $320 / DALY Link
Rotavirus $6 - $59 / DALY $10 - $64 / DALY Link
HPV $240 - $1300 / DALY $370 - $1600 / DALY Link
HIV $85 - $550 / DALY $210 - $690 / DALY Link
Malaria $21 - $49 / DALY $23 - $52 / DALY Link
Ebola ? ?
“Typical” vaccine[2] $12 - $6700 / DALY $18 - $7000 / DALY Link
2.2) Scenario 2: Eradicate the Disease Completely
For this scenario, we consider one-time costs for researching and developing the vaccine and then rolling out the vaccine enough to achieve eradication. In the case of smallpox, as it really has been
eradicated, real estimates of spending and disease burden were used. In all other cases, extrapolations were made from the estimated costs of vaccination per person and the intended target population
of the vaccination[3]. The population targeted was different depending on the disease, with HPV and HIV roughly targeting reproductive age populations and malaria and measles targeting children under
5. The efficacy of the vaccines was also assumed to be high enough to achieve eradication at the time eradication is attempted[4].
After eradication is achieved, we assume that ongoing costs are essentially $0[5]. We then consider this large cost compared to a period of accrued benefits from the eradication (in our model, we cap
it at 20 years; see discussion below on this cap).
Vaccine Roll-out Cost-effectiveness Roll-out + R&D Cost-effectiveness Guesstimate[1]
Smallpox $0.44 - $5.80 / DALY $0.44 - $5.80 / DALY Link
Measles $0.36 - $13 / DALY $0.37 - $13 / DALY Link
Rotavirus Not possible Not possible
HPV $65 - $340 / DALY $80 - $370 / DALY Link
HIV $270 - $1600 / DALY $300 - $1600 / DALY Link
Malaria $12 - $23 / DALY $13 - $23 / DALY Link
Ebola ? ?
“Typical” vaccine[2] $8 - $8200 / DALY $10 - $8100 / DALY Link
2.3) Analysis of Individual Vaccines
2.3.1.) SMALLPOX VACCINE
When making direct estimates using inputs from Fenner, et al. (1988), by our estimation before the eradication campaign in the early 1960s, smallpox prevented a DALY from death for ~$20-64 in
developing countries and ~$1550-3800 in developed countries. Weighting this by population in each region, this implies a total cost per DALY of $527 for both roll-out costs alone and roll-out costs
plus R&D costs. Over the full population since eradication, not accounting for post-eradication prevention spending, the eradication campaign cost $26-70 per death prevented, $3.30-11 per case
prevented, and $0.44-3.80 per DALY prevented at the rates of death and disease of 1967.
These estimates are absolute as they are not compared to the cost of maintaining the 1967 vaccination and monitoring costs. As smallpox is eradicated, these figures will fall over time, unless and
until there is a new smallpox outbreak or significant enough threat of an outbreak to cause an increase in spending.
Assuming the pre-vaccination world would be like the pre-eradication world of smallpox endemic countries in the 1960s, given a case rate of roughly 1% and a death rate between 5 and 20% per case of
smallpox, our Guesstimate suggests a cost per DALY prevented of $4-62 with or without the R&D costs included.
2.3.2.) MEASLES VACCINE
Assuming a very rough conversion that a life saved from a measles death is ~30 DALYs[6], this would imply very rough figures of $9-320/DALY in general, given a cost range per person vaccinated of
$1-38, weighted so cheaper prices are more likely. In areas with pre-existing strong vaccine infrastructure such that measles vaccines cost about $2 a dose, the price would be roughly $9 per DALY.
Given the wide range of possible cost per person, it should be noted that holding everything else constant, the cost-effectiveness including R&D cost scales roughly linearly with the price of
vaccination per person.
Perhaps we could make sense of these figures by further untangling marginal vaccine costs from the investments needed to make vaccine pathways. If we assume a one-time investment of $100M, similar to
the more directly estimated fixed costs of the malaria vaccine, is sufficient to get all lower income countries to a point where marginal vaccination costs $2 per child in total vaccine and logistics
costs, and we assume that we’re rolling out the measles vaccine to 96.3M people, the population of 60% of children under 5 in SSA, and we take our prior estimate that the measles vaccine cost $38.3M
in R&D, the total yearly costs are ~$151M for the vaccinations at ~$10/DALY, only slightly higher than with no investment spending at all. Given our earlier calculations, this would avert about 177K
measles deaths or 5.3M DALYs.
A more worse-case analysis assuming $23 per child, with the same $38.3M in R&D, and rolling out to the same population, the total cost would be $587M at $110/DALY.
2.3.3.) ROTAVIRUS VACCINE
Given vaccine efficacy of 39-85% (RotaCouncil, 2016, p11) and a reduction of diarrhea of 30-54% (Madhim et al., 2010, Msimang, et al., 2013), if we assume we can achieve a 60% vaccination rate of the
under-five population in SSA, vaccinating 96.3M children (see Population Pyramid), we would avert ~7.5-13M DALYs per year using the 2005 DALY rate or 4.6-8.3M DALYs using the 2016 rate. Assuming an
initial $50-100M investment in building the pipeline to roll out the vaccine, with a per person cost between $3-7 that would be $6-63/DALY looking just at roll-out costs and $10-68/DALY when adding
in R&D costs.
There’s great uncertainty about these estimates given fixed roll-out costs are a guess and the efficacy of the vaccine is fairly wide.
2.3.4.) HPV VACCINE
If the HPV vaccine was rolled out to all children aged 5-15 in SSA with a 60% vaccination rate and given ~780,000 DALYs in women aged 15-49 (Global Burden of Disease, Results Tool, 2016b)[7] then the
HPV vaccine could prevent ~328,000 DALYs per year in SSA. With a fixed roll-out cost guessed to be an additional $50M, at $3-13.50 per person this would imply $240-1300 per DALY based on just
roll-out costs and $370-1600 per DALY when including R&D costs.
2.3.5.) HIV VACCINE
Given estimates of a vaccine that is 50% effective (or more) and distributed to 60% of the eligible population in SSA, it would avert 30% of all HIV, saving 9.7M DALYs. Relative to other vaccines,
many of the parameters for our HIV estimates vary widely, particularly our estimated cost per person of $30-160. Ultimately, our model suggests an estimated cost-effectiveness of $180-570 per DALY.
Our individual scenarios indicate the final cost-effectiveness depends significantly on which population that vaccine is ultimately distributed to and the cost of per person of vaccination 8 .
2.3.6.) MALARIA VACCINE
Previously we noted that the malaria vaccine involved spending ~$605M in fixed costs to unlock the ability to roll out the vaccine for $22/child. The Global Burden of Disease estimated a yearly
malaria burden of 56.2M DALYs, with 44.3M global DALYs for those under five, and 42.1M of those in SSA (Global Burden of Disease, Results Tool, 2016c). Modeling in Guesstimate, accounting for
uncertainty by using a range of vaccine efficacy between 39-75% (the upper end of which is closer to vaccine efficacy for established vaccines), an expected R&D cost between $600-1,000M and a cost
per person between $18-25, we get an estimate of $12-23/DALY for roll-out costs alone and $13-33/DALY including R&D costs.
Note that because of technical length restrictions on the EA Forum, this essay is broken up into three parts. Please continue on to Part 2, which contains a discussion of model uncertainty. All the
appendices and footnotes are in Part 3. To see all three parts in one part, you can view the article on our research site.
Thanks to Max Dalton, Joey Savoie, Tee Barnett, Palak Madan, and Christina Rosivack for reviewing this piece. | {"url":"https://forum-bots.effectivealtruism.org/posts/3Tvu55ETMNx5T5tJ3/what-is-the-cost-effectiveness-of-researching-vaccines","timestamp":"2024-11-01T22:34:35Z","content_type":"text/html","content_length":"380860","record_id":"<urn:uuid:0e614e10-9921-4049-b736-ba9f5adc6ddf>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00718.warc.gz"} |
In a survey of some students, it was found that 60 % of the students were studying commerce and 40 % were studying science. If 40 students w - DocumenTVIn a survey of some students, it was found that 60 % of the students were studying commerce and 40 % were studying science. If 40 students w
In a survey of some students, it was found that 60 % of the students were studying commerce and 40 % were studying science. If 40 students w
In a survey of some students, it was found that 60 % of the students were studying commerce and 40 % were studying science. If 40 students were studying both the subjects and 10% did not study any of
two subjects, by drawing a Venn-diagram, (i) find the total number of students. (u) Find the number of students who were studying science only.
in progress 0
Mathematics 3 years 2021-08-30T11:30:04+00:00 2021-08-30T11:30:04+00:00 1 Answers 3 views 0 | {"url":"https://documen.tv/question/in-a-survey-of-some-students-it-was-found-that-60-of-the-students-were-studying-commerce-and-40-24096284-22/","timestamp":"2024-11-12T15:24:03Z","content_type":"text/html","content_length":"79653","record_id":"<urn:uuid:96b02735-fa89-436e-a20b-5a48f1792c3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00817.warc.gz"} |
Separating MAX 2-AND, MAX DI-CUT and MAX CUT
Theory Seminar
Separating MAX 2-AND, MAX DI-CUT and MAX CUT
Aaron PotechinUniversity of Chicago
3725 Beyster Building
In 2008, Raghavendra’s paper “Optimal algorithms and inapproximability results for every CSP?” showed that assuming the Unique Games Conjecture (or at least that unique games is hard), for any CSP on
a finite set of predicates, the optimal worst-case polynomial time approximation algorithm is to use a standard semidefinite program and then apply a rounding algorithm. However, since the set of
potential rounding functions is extremely large, Raghavendra’s result does not tell us what the approximation ratio is for a given CSP. In fact, there are still several basic questions which are
still open. For example, the question of whether there is a 7/8 approximation ratio for MAX SAT (where the clauses can have any length) is still open.
In this talk, I will review Raghavendra’s result and why additional work is needed to analyze particular CSPs. I will then describe why MAX 2-AND, MAX DI-CUT and MAX CUT all have different
approximation ratios and how we can prove this.
This is joint work with Joshua Brakensiek, Neng Huang, and Uri Zwick
Greg Bodwin
Euiwoong Lee | {"url":"https://theory.engin.umich.edu/event/separating-max-2-and-max-di-cut-and-max-cut","timestamp":"2024-11-12T03:02:07Z","content_type":"text/html","content_length":"42937","record_id":"<urn:uuid:2ceaf959-aa94-4f86-a479-bdffc8dbc0ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00434.warc.gz"} |
Similarity in Graphs: Jaccard Versus the Overlap Coefficient | NVIDIA Technical Blog
This post was originally published on the RAPIDS AI blog.
There is a wide range of graph applications and algorithms that I hope to discuss through this series of blog posts, all with a bias toward what is in RAPIDS cuGraph. I am assuming that the reader
has a basic understanding of graph theory and graph analytics. If there is interest in a graph analytic primer, please leave me a comment below. It should also be noted that I approach graph analysis
from a social network perspective and tend to use the social science theory and terms, but I have been trying to use ‘vertex’ rather than ‘node.’
Every RAPIDS cuGraph 0.7 release adds new features. Of interest to this discussion is the expansion of the Jaccard Similarity metric to allow for comparisons of any pair of vertices, and the addition
of the Overlap Coefficient algorithm. Those two algorithms fall into the category of similarity metrics and lead to the topic of this blog, which is to discuss the difference between the two
algorithms and why I think one is better than the other. Let’s start with a quick introduction to the similarity metrics (warning math ahead).
The Jaccard Similarity, also called the Jaccard Index or Jaccard Similarity Coefficient, is a classic measure of similarity between two sets that was introduced by Paul Jaccard in 1901. Given two
sets, A and B, the Jaccard Similarity is defined as the size of the intersection of set A and set B (i.e. the number of common elements) over the size of the union of set A and set B (i.e. the number
of unique elements).
Figure 1. The Jaccard Similarity Metric.
The Overlap Coefficient, also known as the Szymkiewicz–Simpson coefficient, is defined as the size of the union of set A and set B over the size of the smaller set between A and B.
Figure 2. The Overlap Coefficient Metric.
When applying either of the similarity metrics in a graph setting, the sets are typically comprised of the neighbors of the vertex pair being compared. The neighbors of a vertex v, in a graph (V,E)
is defined as the set, U, of vertices connected by way of an edge to vertex v, or N(v) = {U} where v ∈V and ∀ u ∈ U ∃ edge(v,u) ∈ E. Computing the size of the union, | A U B |, can be computationally
inexpensive since we only want the size and not the actual elements. The size of the union| A U B | can be computed with |A| + |B| — | A intersect B |.
Figure 3. Efficient Jaccard Computation.
There is a wide range of applications for similarity scoring and it is important to cover a few of them before getting into comparing the two algorithms. Let’s start with something that I’m sure a
lot of the reader are familiar with, and that is recommending people to connect with on social media. What I am going to present is a very simplistic approach to the problem — most social networking
sites use a much more advanced version that usually includes some type of community detection. But first, even more background …
Figure 4: Triadic Closure.
Within the field of social network analysis, there is the concept of Triadic Closure, which was first introduced by sociologists Georg Simmel in 1908. Given three people, A, B, and C, see Figure 1;
if A and C are friends (connected), and B and C are friends, then there is a high probability that A and B will connect. That probability is so high that Granovetter, in his 1973 work on weak-ties,
deemed the missing link as the “forbidden triad”, which meant that for Granovetter’s application that you could infer a connection between A and B. For our recommendation application, this means that
we need to find those unconnected A — B pairs since there is a high change that those users will become friends — it is always good to recommend something that the user will accept.
Now back to the application: the basic process starts by first computing the similarity metric (Jaccard or Overlap Coefficient) for all vertex pairs connected by an edge. Then for a given vertex
(example, vertex A) find their neighbors with the highest similar score and recommend neighbors of B that are missing from A. As mentioned, this is a very simple view since there is a range of
options for applying additional weights, like only looking within community clusters, that could be added to better select recommendations that will be accepted. The application of triadic closure
and similarity should be apparent to anyone that uses social media since those tools remind you constantly that you should connect to a friend of friends. Note that this approach does not work for
vertices with a single connection, also called a satellite, since their similarity score will be zero. But for satellites it is easy to just recommend all connections from their sole neighbor.
The problem with the previous approach is that not all recommendations can come from solely looking at directly connected vertex pairs. Therefore, being able to compute similarity scores between any
pair of vertices is important. This was a limitation of the initial cuGraph Jaccard implementation and what has been addressed in cuGraph release 0.7. Consider figure 2 below. Looking at the Jaccard
similarity score between connected vertices A and B, the neighbors of A are {B, C}, and for B are {A, D}. Hence the Jaccard score is js(A, B) = 0 / 4 = 0.0. Even the Overlap Coefficient yields a
similarity of zero since the size of the intersection is zero. Now looking at the similarity between A and D, where both share the exact same set of neighbors. The Jaccard Similarity between A and D
is 2/2 or 1.0 (100%), likewise the Overlap Coefficient is 1.0 size in this case the union size is the same as the minimal set size.
Figure 5: Non-connected Vertex Pair Similarity.
Figure 6: Bipartite Graph.
Continuing with the social network recommender example, the application should recommend that vertices A and D connect since they share the same set of neighbors. But non-connected vertex pair
similarity is used in other applications as well. Consider a product recommendation system that is using a bipartite graph. One vertex type is Users and the other type is Product. The goal is to not
find similarities between Users and Products as that is a different analytic but to find similarities between Users and other Users so that additional products can be recommended. The process is
similar to that mentioned above; the vertices being compared are just not directly connected. Looking at the example figure (Figure 3) and focusing on User 1. User 1 neighbors are {A, B}, User 2 are
{A, B, D} , and User 3 is {B. D}. The Jaccard Similarity in this case is 1-to-2 = 2 / 3 = 0.66, and between 1 and 3 = 1 / 3 = 0.33. Since User 1 and 2 both purchased products A and B, the application
should recommend to User 1 that they also purchase product D.
Why I prefer the overlap coefficient
If the Jaccard similarity score is so useful, why introduce the Overlap Coefficient in cuGraph? Let’s look at the same example but using the Overlap Coefficient. Comparing User 1 to User 2 = 2 /2 =
1.0. And comparing User 1 to User 3 = 1 /2 = 0.5. The similarity between User 1 and User 2 is still the highest, but the fact the score is 1.0 indicated that the set of neighbors of User A is a
complete subset of User 2. That type of insight is one of the benefits of the Overlap Coefficient.
In my opinion, the Jaccard Similarity is a very powerful analysis technique, but it has a major drawback when the two sets being compared have different sizes. Consider two sets, A and B, where both
sets contain 100 elements. Now assume that 50 of those elements are common across the two sets. The Jaccard Similarity is js(A, B) = 50 / (100 + 100 – 50 ) = 0.33. Now if we increase set A by 10
elements and decrease set B by the same amount, all while maintaining 50 elements in common, the Jaccard Similarity remains the same. And there is where I think Jaccard fails: it has no sensitivity
to the sizes of the sets. The following figure highlights how the Jaccard and Overlap Coefficient change as the set sizes are change but the intersection size remains that same.
Figure 7: Similarity Scores as Set Sizes Change.
The use of the smaller set size as the denominator makes it so that the score provides an indication of how much of the smaller set is within the larger. That provides insight into whether one set is
an exact subset of the larger set. Look back at the example described above, and illustrated above, using the Overlap Coefficient it is easy to see to what degree set B is contained within set A.
<warning, the following is just my opinion>
Jaccard might be better known than the Overlap Coefficient and that might play into why Jaccard is more widely used. The unfamiliarity with Overlap Coefficient might explain why it is not in the
NetworkX package. Nevertheless, in my opinion, the Overlap Coefficient can provide better insight into how similar two vertices are — really, how similar the set of neighbors are. By knowing the
sizes of each set, an analyst can easily know if one set is a proper subset (full contained) in the other set, which is something that is not apparent using Jaccard.
I also think there is a fundamental flaw in how we derive the set for similarity computation when the vertex pairs are connected by an edge. Consider the 5-clique shown below, figure 5. Since every
vertex is connected to every other vertex, I (you) would assume that the similarity scores would be 1.0, exact similarity. However, because of the way the sets are created, for both Jaccard and the
Overlap Coefficient, the score for vertex pairs connected by an edge can never be equal to 1.0.
Figure 8: Five Clique.
Let’s compare vertex 1 to vertex 2. The neighbors of 1 are {2, 3, 4, 5} and the neighbors of 2 are {1, 3, 4, 5}. The Jaccard score is then: 3 / 5 or 0.6. The Overlap Coefficient is 3 /4 or 0.75.
While those similarity scores are mathematically correct according to the algorithm, the resulting similarity score do not match what I would except for a clique. The issue is that the vertex pairs
being compared are reflected in the neighborhood sets. Rephrasing that statement, vertex 1 appears in vertex 2’s neighbor set. Likewise, vertex 2 appears in vertex 1’s neighbor. The fact that the
vertices being compared appear in the associative set prevents the sets from ever matching. In my opinion, the vertices being compared should not be part of the sets being evaluated.
A solution to this problem would be to build the sets differently or modify the similarity algorithm. The algorithm modification might be computationally easier. The change would be to subtract 1
from the size of each set on the union, not the intersection.
Figure 9: Modified Jaccard and Overlap Coefficient for Connected Vertex Pairs.
For the clique example, the similarity scores would then be:
js(1, 2) = 3 / (5–2) = 3/3 = 1.0. and
oc(1, 2) = 3 / (4–1) — 3/3 = 1.0.
Now, it could be that the results are correct and that my expectation are wrong. But this is just my opinion :-)
About me
Brad Rees leads the RAPIDS cuGraph team at NVIDIA where he directs and develops graph analytic solutions. He has been designing, implementing, and supporting a variety of advanced software and
hardware systems for over 30 years. Brad specializes in complex analytic systems, primarily using graph analytic techniques for social and cyber network analysis, and has been working on variety of
advanced software and hardware systems for over 30 years. His technical interests are in HPC, machine learning, deep learning, and graph. Brad has a Ph.D. in Computer Science from the Florida
Institute of Technology.
Some references
M. S. Granovetter, “The strength of weak ties,” The American Journal of Sociology, vol. 78, no. 6, pp. 1360–1380, 1973.
Thanks to Corey Nolet. | {"url":"https://developer.nvidia.com/blog/similarity-in-graphs-jaccard-versus-the-overlap-coefficient-2/","timestamp":"2024-11-13T19:30:18Z","content_type":"text/html","content_length":"210108","record_id":"<urn:uuid:a5e5748a-926f-45fa-ad20-62e742b7ae64>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00411.warc.gz"} |
Introduction to Bayesian Compositional
The multilevelcoda package implements Bayesian multilevel models for compositional data in R, by combining the principles of the two well-known analyses, Multilevel Modelling and Compositional Data
Analysis. Formula syntax is built using package brms and is similar to package lme4, which allows for different modelling options in a multilevel framework. The package also provides several useful
functions for post-hoc analyses and visualisation of final results.
Compositional Data Analysis
Compositional data analysis (CoDA) is an analysis of compositional and multivariate positive data. Compositional data are typically expressed in amount, e.g., percentage, proportion, and often sum up
to a constant, usually 100% or one. These data are common in many fields: ecology (e.g., relative abundances of species), geography (e.g., proportions of land use), biochemistry (e.g., fatty acid
proportions), nutritional epidemiology (e.g., intake of macronutrients like proteins, fats and carbohydrates), and time-use epidemiology (e.g., time spent in different sleep-wake behaviours during
the 24-hour day).
Multilevel Modelling for Compositional Data
Compositional data can be non-independent and repeated measures data. For example, sleep-wake behaviours are often measured across multiple time points (e.g., across several consecutive days).
Therefore, we often use multilevel models to include both fixed effects (regression coefficients that are identical for everyone) and random effects (regression coefficients that vary randomly for
each person). In addition, we can also decompose these data into two sources of variability: between-person (differences between individuals) and within-person (differences within individuals).
In multilevelcoda package, we implements Compositional Multilevel Model to model compositional data in amultilevel framework. mulitlevelcoda includes functions to compute Isometric log ratio (ILR)
for between and within-person levels, fit Baysian multilevel model, and conduct post-hoc analyses such as susbtitution models. See below for vignettes: | {"url":"http://cran.uvigo.es/web/packages/multilevelcoda/vignettes/A-introduction.html","timestamp":"2024-11-01T20:32:20Z","content_type":"text/html","content_length":"1037123","record_id":"<urn:uuid:10765994-6b13-4ea9-b417-fe0bc3e0c95c>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00167.warc.gz"} |
On extremal points for some vectorial total variation seminorms
We consider the set of extremal points of the generalized unit ball induced by gradient total variation seminorms for vector-valued functions on bounded Euclidean domains. These extremal points are
central to the understanding of sparse solutions and sparse optimization algorithms for variational regularization problems posed among such functions. For not fully vectorial cases in which either
the domain or the target are one dimensional, or the sum of the total variations of each component is used, we prove that these extremals are fully characterized as in the scalar-valued case, that
is, they consist of piecewise constant functions with two regions. For definitions involving more involved matrix norms and in particular spectral norms, which are of interest in image processing, we
produce families of examples to show that the resulting set of extremal points is larger and includes piecewise constant functions with more than two regions. We also consider the total deformation
induced by the symmetrized gradient, for which minimization with linear constraints appears in problems of determination of limit loads in a number of continuum mechanical models involving
plasticity, bringing relevance to the corresponding extremal points. For this case, we show piecewise infinitesimally rigid functions with two pieces to be extremal under mild assumptions. Finally,
as an example of an extremal which is not piecewise constant, we prove that unit radial vector fields are extremal for the Frobenius total variation in the plane.
Dive into the research topics of 'On extremal points for some vectorial total variation seminorms'. Together they form a unique fingerprint. | {"url":"https://research.utwente.nl/en/publications/on-extremal-points-for-some-vectorial-total-variation-seminorms","timestamp":"2024-11-07T06:00:24Z","content_type":"text/html","content_length":"49783","record_id":"<urn:uuid:08ed5d9f-f55c-4635-a7b0-8a21cbede414>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00828.warc.gz"} |
Best AI Tools for Math - Best AI Tools
Best AI Tools for Math
AI can go beyond generating content – it could also be used to solve various math problems, making it an ideal tool for students.
Sure, they could use it to cheat — AI is a tool like any other — but students can also rely on AI to learn how to solve equations, with most of these tools providing step-by-step instructions. Add
access to definitions and other content, and you get a powerful teaching tool for math.
And, unsurprisingly, that’s how most of these tools are being used today — though we should also add the professional use cases.
For instance, in research and professional settings – AI tools for mathematics are instrumental in solving large-scale, intricate problems that are otherwise too labor-intensive or complex for
But we’ve said it enough; here are some of the best AI tools for math you could use today:
AIR Math
An AI-enabled math homework solver and helper that works on computers and phones
• Works on the phone, use it from anywhere
• Step-by-step solutions are very cool
• Support for different math topics
• Responses from tutors can sometimes be slow
AIR Math is an AI-enabled math homework solver and helper that works on your computer and on your phone. In order to use it, you have to take a photo of your math problem - and the tool will provide
instant, step-by-step solutions in 3 seconds. It's that simple...
AIR Math is an AI-enabled math homework solver and helper that works on your computer and on your phone.
In order to use it, you have to take a photo of your math problem – and the tool will provide instant, step-by-step solutions in 3 seconds. It’s that simple. The system will read and understand the
problem you have and provide the steps needed to solve it.
If, however, AI can’t solve the problem for some reason — like it’s not easy to read — there are math experts from all around the world available 24/7.
AIR Math claims it can solve all kinds of problems, from geometry to algebra, and even covers graphs and diagrams.
And the best part? AIR Math is free – so give it a try.
Microsoft Math Solver
An entry-level educational app that can solve math and science problems
• Scan or write to search for the solution
• Library of helpful videos
• Quizzes are also cool for knowledge checks
• Some search results and quizzes can be disappointing
Formerly known as Microsoft Mathematics and Microsoft Math, Microsoft Math Solver is an entry-level educational app that can solve math and science problems. It is considered a valuable learning tool
for everyone struggling with math -- as well as those who need a little help with their homework...
Formerly known as Microsoft Mathematics and Microsoft Math, Microsoft Math Solver is an entry-level educational app that can solve math and science problems.
It used to be a bundled part of Microsoft Student but has later on continued its life as an independent app that works on the web and mobile. Today, it is considered a valuable learning tool for
everyone struggling with math — as well as those who need a little help with their homework.
Microsoft Math provides step-by-step explanations for all kinds of math and science problems, as well as definitions for mathematical concepts.
Also, it lets users instantly graph any equation to visualize their functions and understand the relationship between variables.
In addition, there’s the ability to search for additional learning materials, such as related worksheets and video tutorials.
All this is available in a few languages, including English, Spanish, Hindi, German, and more.
AI Math Problem Solver by Interactive Mathematics
Instant step by step answers to your math homework problems
• Step-by-step solutions to your math problems
• You get an amazing deal with an annual subscription
• 5 million students helped each year!
The tool was developed by combining a powerful mathematical computational engine with a large language model (LLM) AI to create a state-of-the-art math problem solver and AI math calculator. As such,
it is said to be more accurate than ChatGPT, more powerful than a math calculator, and faster than a math tutor...
Like that’s the case with similar tools, AI Math Problem Solver also provides step-by-step answers to your math homework problems.
The tool was developed by combining a powerful mathematical computational engine with a large language model (LLM) AI to create a state-of-the-art math problem solver and AI math calculator. As such,
it is said to be more accurate than ChatGPT, more powerful than a math calculator, and faster than a math tutor.
Whether it’s a tough word problem, algebra equation or advanced calculus, the AI Math Problem Solver and calculator can solve it.
Speaking of solving problems, this tool is especially designed to handle math word problems and will take both text and a photo as input — and is able to interpret most handwritten or typed math
Interactive Mathematics, which is the company behind AI Math Problem Solver, helps over 5 Million students each year — who use their free lessons to help get ahead in math. Now, they’ve taken that
expertise and paired it with AI to provide a free-to-try AI math problem solver and math tutoring chat platform.
Socratic by Google
A mobile app that uses AI to help students by providing visual explanations of concepts
• It's like having your own AI tutor in your pocket
• Using your voice and camera to get answers is cool
• All popular subjects are covered by Socratic
• At the moment, it's an English-only app
Socratic is not "yet another learning app" - it's made by Google and uses the search giant's powerful AI technology to help students understand their school work at a high school and university
level. Simply ask Socratic a question and the app will find the best online resources for you to learn the concepts...
Socratic is not “yet another learning app” — it’s made by Google and uses the search giant’s powerful AI technology to help students understand their school work at a high school and university
Simply ask Socratic a question and the app will find the best online resources for you to learn the concepts. It supports most high school subjects, such as science, math, literature, and social
studies — with more being added all the time.
The cool thing is that you don’t have to type it all. Instead, you can use your voice or camera to connect to online resources and understand any problem.
Once it understands the subject, Socratic will help you find videos, step-by-step explanations, and everything else that will help you learn subjects at your own pace.
It’s not a generic search, mind you — Google partnered with teachers and experts to bring visual explanations to Socratic, making it easier to learn the concepts behind any problem.
In a nutshell, this is one mobile app every student should have installed on their smartphones.
Khanmigo by Khan Academy
A GPT-4 powered tool that's designed to act like a tutor and a teaching assistant
• One-on-one tutoring for the entire world
• It could also help teachers with administrative tasks
• Helping create more programmers, who are in demand
Developed by Khan Academy's Khan Labs, Khanmigo is a tool that's designed to act like a tutor and a teaching assistant. Powered by OpenAI's GPT-4 model, it is an experimental AI interface that can be
used for a variety of educational tasks. These include getting help with math, preparing for exams, learning computer programming, and more...
Developed by Khan Academy’s Khan Labs, Khanmigo is a tool that’s designed to act like a tutor and a teaching assistant. Powered by OpenAI’s GPT-4 model, it is an experimental AI interface that can be
used for various educational tasks. These include getting help with math, preparing for exams, learning computer programming, practicing vocabulary words, and even conducting a simulated interview
with a historical figure.
Moreover, Khanmigo doesn’t just give students the answers — instead, it encourages learning by asking thought-provoking and open-ended questions.
The tool has the potential to reduce the burden on teachers by assisting with time-consuming administrative tasks. For instance, it could help in writing lesson plans, creating lesson hooks, and
writing exit tickets.
There is also interest in tailoring the system to provide teachers with a snapshot of student progress on Khan Academy at any moment, which could help them identify students who need extra support.
As a nonprofit organization, Khan Academy’s primary focus is on students, teachers, and administrators. Everyone can join Khanmigo’s waitlist to get access as soon as it’s available to the general
How can AI tools for math help you?
Most of these tools are meant to help students get around various math and science problems. In that sense, they feature:
• User-friendly interface
AI tools for math are accessible even to those with limited technical or mathematical background, with interactive visuals, step-by-step guides, and simple navigation enhancing the user
• Step-by-step problem solving
This is very handy for beginners, allowing them to understand the process – which is as important as the solution itself. AI math tools often break down problems into smaller, manageable steps,
providing explanations at each stage to aid comprehension.
• Interactive tutorials
Many AI math tools include a range of tutorials and examples that beginners can interact with. These resources are meant to illustrate key concepts in a clear and engaging manner.
• Instant feedback
Immediate feedback on exercises allows students to learn from mistakes in real time. These tools often provide corrections, helping users understand why an answer is incorrect and how to approach
it correctly.
• Customizable difficulty levels
AI tools can adapt to the difficulty of problems based on the user’s knowledge level. This ensures that students are not overwhelmed and can progressively build their skills at a comfortable
• Exercises with hints and tips
Students can also benefit from a variety of practice exercises that come with helpful hints and tips. These can guide them through challenging concepts or problems – ensuring a supportive
learning environment.
• Accessibility features
Finally, to make mathematics accessible to a diverse user base – AI tools for math often support multiple languages and include accessibility features like text-to-speech, making them usable for
people with different learning needs and backgrounds.
There you have it. Now you know why you should start using AI to solve your math problems. Check out a few of the tools on this page and take it from there… | {"url":"https://www.bestaitools.com/list/best-ai-tools-for-math/","timestamp":"2024-11-12T03:00:41Z","content_type":"text/html","content_length":"474207","record_id":"<urn:uuid:4167e29f-e8eb-4ea3-bed7-ddaab9cb746e>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00292.warc.gz"} |
Everything You Always Wanted To Know About The Cosmological Constant Problem
by Jerome Martin
Publisher: arXiv 2012
Number of pages: 89
This article aims at discussing the cosmological constant problem at a pedagogical but fully technical level. We review how the vacuum energy can be regularized in flat and curved space-time and how
it can be understood in terms of Feynman bubble diagrams.
Download or read it online for free here:
Download link
(1.3MB, PDF)
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Definition of Aberrancy of curvature. Meaning of Aberrancy of curvature. Synonyms of Aberrancy of curvature
Here you will find one or more explanations in English for the word Aberrancy of curvature. Also in the bottom left of the page several parts of wikipedia pages related to the word Aberrancy of
curvature and, of course, Aberrancy of curvature synonyms and on the right images related to the word Aberrancy of curvature.
Definition of Aberrancy of curvature
Aberrancy of curvatureAberrance Ab*er"rance, Aberrancy Ab*er"ran*cy, n. State of being aberrant; a wandering from the right way; deviation from truth, rectitude, etc. Aberrancy of curvature (Geom.),
the deviation of a curve from a circular form.
Aberrancy of curvatureCurvature Cur"va*ture (k?r"v?-t?r; 135), n. [L. curvatura. See Curvate.] 1. The act of curving, or the state of being bent or curved; a curving or bending, normal or abnormal,
as of a line or surface from a rectilinear direction; a bend; a curve. --Cowper. The elegant curvature of their fronds. --Darwin. 2. (Math.) The amount of degree of bending of a mathematical curve,
or the tendency at any point to depart from a tangent drawn to the curve at that point. Aberrancy of curvature (Geom.), the deviation of a curve from a circular form. Absolute curvature. See under
Absolute. Angle of curvature (Geom.), one that expresses the amount of curvature of a curve. Chord of curvature. See under Chord. Circle of curvature. See Osculating circle of a curve, under Circle.
Curvature of the spine (Med.), an abnormal curving of the spine, especially in a lateral direction. Radius of curvature, the radius of the circle of curvature, or osculatory circle, at any point of a
Meaning of Aberrancy of curvature from wikipedia
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heorem without Transfinit
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might
affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions,
conundrums, and contentious issues in modern philosophy and mathematics.
Site Mission
• To promulgate the understanding that the validity of a logical argument is not necessarily independent of the way in which language is used by that argument.
• To rid the fields of philosophy and mathematics of arcane and irrational notions which have resulted in numerous contradictions.
• To ensure that future generations of young people will not be put off the study of mathematics and philosophy by the mystical and illogical notions that are currently widespread in those
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A Proof of Goodstein’s Theorem without Transfinite numbers
Page last updated 30 Oct 2024
A 1944 paper by Reuben Goodstein (Footnote: Reuben Louis Goodstein, PDF On the restricted ordinal theorem, The Journal of Symbolic Logic 9, 1944, no. 2, pp. 33-41. ) includes a definition of a
sequence of numbers which are now referred to as Goodstein sequences, and the paper introduces a proposition, now known as Goodstein’s theorem, which asserts that every Goodstein sequence terminates
by reaching a value of zero. Goodstein claimed to prove this using the notion of sequences of transfinite ordinals, and several similar papers have been published since. (Footnote: ▪ Reuben Louis
Goodstein, Transfinite ordinals in recursive number theory, The Journal of Symbolic Logic 12.4, 1947, pp 123‑129.
▪ A. Cichon, PDF A short proof of two recently discovered independence results using recursion theoretic methods, Proc. Amer. Math. Soc. 87, 1983, pp 704‑706.
▪ Andrés Eduardo Caicedo, PDF Goodstein’s function, Revista Colombiana de Matemáticas 41, 2007, no. 2, pp. 381‑391.
▪ Sarah Winkler, Harald Zankl, and Aart Middeldorp, PDF Beyond Peano arithmetic - Automatically proving termination of the Goodstein sequence, 24th International Conference on Rewriting Techniques
and Applications, RTA 2013, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2013.
▪ Michael Rathjen, PDF Goodstein’s theorem revisited, Gentzen’s Centenary, Springer, 2015, pp. 229‑242.
▪ A. Leonardis, G. d’Atri, F. Caldarola, PDF A geometrical proof for generalized Goodstein’s theorem, Numerical Computations: Theory and Algorithmsm NUMTA 2019, 67.
▪ Henry Towsner, PDF Goodstein’s Theorem, ε[0], and unprovability, unpublished notes, 2020.
▪ A. Leonardis, G. D’atri, E. Zanardo, PDF Goodstein’s Generalized Theorem: From Rooted Tree Representations To The Hydra Game, J. Applied Math. & Informatics Vol 40.5-6, 2022, pp 883‑896. )
In a paper of 1982, Laurie Kirby and Jeff Paris state that there is a proposition that is an expression of Peano arithmetic that asserts that a Goodstein sequence must terminate, but that this
proposition cannot be proved within Peano arithmetic. (Footnote: Laurie Kirby and Jeff Paris, PDF Accessible independence results for Peano arithmetic, Bulletin of the London Mathematical Society,
1982, pp. 285-293. ) Some people believe that the proposition cannot be proved at all except by the use of transfinite numbers.
However, since a Goodstein sequence is a reversible well-defined algorithmic process without any choice or any randomness, (Footnote: Reversibility: given the information regarding any single term of
a Goodstein sequence, all of the other terms of the sequence both prior and subsequent to that term are completely determined. Note that this is different to cases such as the Collatz conjecture
where the same term can occur in multiple different Collatz sequences.) it might be expected that a resolution of the nature of its iterative development should be susceptible to a mechanistic
analysis of that algorithmic process, and this turns out to be the case. The method of the proof presented here can be viewed in a manner similar to an engineering perspective, by a consideration
that the iterations of the Goodstein sequence can be imagined as operations on a modifiable rotary counter mechanism, which can be conceived as a rotary counter similar to those that were used in
cars before digital screens took over that function, but where additional wheels can be added and where the wheels can be made bigger to include extra numerical digits.
Rotary Counter
This gives an insight into the nature of these algorithmic processes and leads directly to an elementary proof of Goodstein’s theorem without any reference to transfinite induction or transfinite
Details of the Goodstein sequence
Hereditary Base Notation
Our standard notation for numbers that is almost universally used is positional notation in base 10, where the position of a digit symbol indicates its value within the overall number, where, for
example, we write 76020 instead of writing 7×10^4 + 6×10^3 + 0×10^2 + 2×10^1 + 0×10^0 so that the relative position of each digit symbol (here 7, 6, 2 and 0) indicate a numerical value within the
overall number. The historical reason for the adoption of such positional notation is that it is much more convenient for relatively large numbers. (Footnote: See also Number Systems for more on
number systems and bases.)
The rules governing the iterations of the Goodstein sequence require that every natural number is referred to in terms of what is called “hereditary base b notation”. To express a number in
hereditary base b notation, the number is expressed as the sum of exponents of the base number b and individual unique digit symbols for any remaining value less than b, and where the exponents
themselves also have to follow this rule, as do exponents of exponents, and so on. As a simple example the number 909 in standard decimal notation is:
3·4^4 + 2·4^3 + 3·4^1 + 1
in hereditary base 4 notation. That number does not have any exponents greater than 4, so in that example the exponents of exponents rule is not invoked. An example of such a number is the number
549,755,814,797 in standard decimal notation, which is:
2·4^4^2+ 3 + 3·4^4 + 2·4^3 + 3·4^1 + 1
in hereditary base 4 notation rather than 2·4^19 + 3·4^4 + 2·4^3 + 3·4^1 + 1.
The only symbols allowed in such representation are the symbols for the base number b, individual unique symbols for each number greater than zero and less than b, the plus symbol, the multiplication
symbol, and exponentiation. (Footnote: Exponentiation may be represented either by a symbol such as “^”, or by a superscript position of the exponent, placing the numerical value of the exponent
immediately after and higher than the number to which it applies, for example 3 with exponent 5 is represented as 3^5. No other positional notation is allowed.)
Defining a Goodstein sequence
A Goodstein sequence is formed by the repeated application of two steps:
1. A new number is given by increasing the numerical value of the base by 1. This means that the numerical value of every occurrence of the symbol b is increased by 1.
2. Then subtract 1 from the number that results from Step 1.
This gives the next number in the sequence.
For the application of these two steps, the number must always be in the correct hereditary base notation. This may mean it has to be reformulated between steps so that there is no negation symbol
present. For example:
Initial number: 3^2·3 + 2·3^3+2 + 3^3+1 + 2·3^3 + 2·3^2
After adding one to the base 3: 4^2·4 + 2·4^4 + 2 + 4^4+1 + 2·4^4 + 2·4^2
After subtraction of one: 4^2·4 + 2·4^4 + 2 + 4^4+1 + 2·4^4 + 4^2 + 3·4 + 3
The fascinating thing about Goodstein sequences is how quickly the numbers become mind-bogglingly enormous. For a simple example, let’s take a number that we would consider small, such as the number
15 in standard base 10 notation, which is 2^2+1 + 2^2 + 2 + 1 in hereditary base 2 notation, and for the first few numbers of the sequence we have:
Standard notation Hereditary base notation
15 2^2+1 + 2^2 + 2 + 1
111 3^3+1 + 3^3 + 3
1283 4^4+1 + 4^4 + 3
18752 5^5+1 + 5^5 + 2
326593 6^6+1 + 6^6 + 1
6588344 7^7+1 + 7^7
150994943 8^8+1 + 7·8^7 + 7·8^6 + 7·8^5 + 7·8^4 + 7·8^3 + 7·8^2 + 7·8 + 7
3524450280 9^9+1 + 7·9^7 + 7·9^6 + 7·9^5 + 7·9^4 + 7·9^3 + 7·9^2 + 7·9 + 6
and after another four iterations, we have:
13^13+1 + 7·13^7 + 7·13^6 + 7·13^5 + 7·13^4 + 7·13^3 + 7·13^2 + 7·13 + 6
which, in standard base 10 positional notation, is:
These numbers keep growing and growing, and yet eventually, if the theorem is correct, they must at some point start becoming smaller and terminate at zero. That is what we shall prove here, for any
possible starting number and base.
Positional Notation with Positions for Zero
Note that in the above examples, in standard positional numerical notation there can be some positions where there is a zero symbol, whereas in the hereditary base notation, with a number such as 3^
3+1 + 2·3^3 + 3 there are “empty” positions which are not represented by any symbol. If we explicitly add such 1 and 0 symbols to:
3^3+1 + 2·3^3 + 3
we obtain:
1·3^3+1 + 2·3^3 + 0·3^2 + 1·3^1 + 0·3^0.
which, while it is not true hereditary base notation, leads to the idea that we can use the concept of positional notation to track the changes across the iterations of a sequence, where each
position has a related numeral - from here on, we will refer to such symbols as “multipliers”, and we will always show the 1 and 0 multipliers, with the proviso that the leftmost multiplier shown is
always non-zero, and we will use a special form of positional notation as in the example below:
Position: b^b+2 b^b+1 b^b b^b−1 b^b−2 … … b^2 b^1 b^0
Multiplier: a[b+2] a[b+1] a[b] a[b−1] a[b−2] … … a[2] a[1] a[0]
where for each b−n, while not itself in hereditary notation, there is a corresponding expression in hereditary notation that has that specific numerical value. So, for the previous example of 3^2·3
+ 2·3^3+2 + 3^3+1 + 2·3^3 + 2·3^2, that initial number is represented by:
3^2·3 3^3+2 3^3+1 3^3 3^2 3^1 3^0
and after Step 1 of an iteration we have:
4^2·4 4^4+3 4^4+2 4^4+1 4^4 4^3 4^2 4^1 4^0
and after Step 2 of the iteration with subtraction by 1 the result is:
4^2 · 4 4^4 + 3 4^4 + 2 4^4 + 1 4^4 4^3 4^2 4^1 4^0
which corresponds precisely to the result previously obtained for the example above. Note that Step 1 of the iteration results in two additional positions with zero multipliers (4^3 and 4^4+3); this
is a key aspect to understanding the operation of the proof.
The reason for using this special type of positional notation lies in the fact that by this notation every position is always referenced and every multiplier is always referenced regardless of
whether its value is zero or 1 or otherwise. It can be noted that this special positional notation is isomorphic to standard hereditary base notation in that given any expression in this special
positional notation, the standard hereditary base notation can be obtained directly from that expression and vice-versa. This will be seen to be a key element in proving the termination of any
Goodstein sequence, since we know that by using this positional notation every multiplier must always be less than the base b, and hence we know that at Step 1 of any iteration no multiplier can
change value.
Rotary Counter
Referring again to a conceptual correspondence between a Goodstein sequence and a modifiable rotary counter, one can visualize that Step 1 of an iteration corresponds to enlarging each wheel and
adding an extra numeral while retaining the currently displayed numeral. It will also be shown below that at Step 1 additional wheels may be added at particular locations between the existing wheels
and all such additional wheels will have their initial display as the numeral zero. Then Step 2 corresponds to reversing the drive to the counter, starting at the rightmost wheel. If the rightmost
wheel is not reading zero, the wheel turns backward to display the next smaller digit numeral. On the other hand, if the rightmost wheel reads zero, the wheel turns backward to display the largest
digit numeral (the current base less 1), and the next wheel to the left, provided it is not reading zero, turns backward to display a numeral that is one less than previously - this corresponds to
the numeral of the rightmost position changing its value to the value of the current base minus 1, while the multiplier to its immediate left decreases by one if its value after Step 1 is not zero.
If that next wheel to the left is also reading zero a similar scenario ensues, and similarly for other wheels reading zero.
Of course, in reality such a mechanical device and its wheels would rapidly become unfeasibly enormous, but it is a principle that provides a conceptual basis for a proof. This enables a proof that
can be presented in a straightforward manner that provides a nice insight as to why the sequence must terminate and at the same time is logically rigorous.
Unitary Decrementation
Before we proceed further, we clarify one aspect regarding Step 2 of an iteration - the subtraction of 1 from the multiplier of the rightmost position b^0, where b is the current base. If that
multiplier is not zero, then that multiplier simply decreases by one. On the other hand, if the multiplier is zero, then at the next position b^n to the left of b^0 for which the multiplier is not
zero, the multiplier of that b^n is decreased by one, while the multipliers at all positions b^n−1, b^n−2, … , b^1, b^0 take the value of b−1, where b is the current base. This of course
corresponds to the standard method of subtraction in our conventional decimal number system.
In order to avoid undue verbosity, we shall from this point forward simply refer to this operation as “Unitary Decrementation”.
Incrementation of a multiplier
We can at this point note some fundamental principles:
1. No multiplier can increase at Step 1 of an iteration.
2. A multiplier m can increase at some Step 2 of an iteration if and only if;
i. m is zero, and all multipliers to the right of it are all zero after Step 1 of an iteration. Note that this means that there must be a non-zero multiplier to the left of the position of m.
ii. the first non-zero multiplier n that is to the left of the multiplier m must decrease by 1
iii. m increases to the value of the current base minus 1.
3. From the above principles, it follows that the leftmost multiplier can never increase.
At this point we shall first examine some specific cases, as doing so helps to create a clear picture of what happens over the iterations of a sequence.
Case 1: Exponents less than b
First we consider a number where, in our positional notation, all the exponents of the positions are less than the base number. At every iteration, for each position the numerical value of the base b
increases by one, but the numerical values of the exponents do not change because they are all less than b. Such a number is initially given as:
The number before an iteration
b^b−1 b^b−2 b^b−3 … … b^2 b^1 b^0
a[b−1] a[b−2] a[b−3] … … a[2] a[1] a[0]
As noted previously the above is not in strict hereditary base notation since we have minus signs, but we can nevertheless apply the Goodstein rules as if it were in that notation. This means that
the numerical value of a term such as b−n is not changed at an iteration, but since the value of the b changes at Step 1 of an iteration, the numerical value of that same position after that Step 1
is given as b−n−1, where b is the new current base. This gives us, after Step 1:
The number after Step 1 of the iteration
b^b−2 b^b−3 b^b−4 … … b^2 b^1 b^0
a[b−2] a[b−3] a[b−4] … … a[2] a[1] a[0]
It can be seen that each exponent, although its actual value is unchanged, is expressed in the above notation by a different term. In the same way, the multiplier values have not changed, and the
quantity of positions has not changed.
Step 2 of the iteration is the subtraction of 1 from the number given by Step 1 of the iteration. If the rightmost multiplier a[0] is not zero, then it decreases by 1, and the other multipliers are
The number after Step 2 of an iteration, where
before Step 2
a[0] was not zero
b^b−2 b^b−3 b^b−4 … … b^2 b^1 b^0
a[b−2] a[b−3] a[b−4] … … a[2] a[1] a[0]−1
As long as the rightmost multiplier of b[0] is not zero, each iteration decreases its value by one, while the numerical values of the other multipliers remain the same. It follows that there will be
some iteration where the rightmost multiplier of b[0] decreases to zero. Step 2 of the subsequent iteration requires that the value of the number decreases by 1, and we apply Unitary Decrementation.
For example if a[1] was not zero before Step 2 we have:
The number after Step 2 of an iteration, where
before Step 2
a[0] was zero, and a[1] was not zero
b^b−2 b^b−3 b^b−4 … … b^2 b^1 b^0
a[b−2] a[b−3] a[b−4] … … a[2] a[1]−1 b−1
This is the same general case as the original starting case except that the multiplier of b[1] has decreased by one. Since the multiplier at any position cannot increase at any iteration unless it is
zero when Step 2 of the iteration is applied, and since every time that the multiplier of b[0] is zero before Step 2 of an iteration, the multiplier of b[1] decreases by one if it is not zero, then
an iteration must be reached after finitely many iterations where both a[1] and a[0] will be zero. Then at Step 2, both a[1] and a[0] take the numerical value of b−1 and if a[2] was not zero, it
decreases in value by 1. Again, after a finite number of iterations a[0] must again decrease to zero and similarly also a[2] and a[1].
This principle applies similarly for each of the multipliers, so that at some iteration the numerical value of the leftmost multiplier must decrease by 1, giving the same general case as the original
starting case except that the numerical value of the leftmost multiplier has decreased by 1 - and since the leftmost multiplier can never increase, at some later iteration it must decrease to zero,
when all the multipliers to its right take the value of the current base less 1.
Similarly, at some later iteration, the new leftmost non-zero multiplier must become zero at some iteration, and since this must apply to all multipliers and no new positions are generated, at some
iteration all multipliers must become zero, and the sequence must terminate.
Case 2: Exponents b or less
Now we look at a number where the leftmost position has the position value of b^b. The initial situation is:
The number before an iteration
b^b b^b−1 b^b−2 b^b−3 … … b^2 b^1 b^0
a[b] a[b−1] a[b−2] a[b−3] … … a[2] a[1] a[0]
This time, unlike the previous case, every iteration creates a new position, which is b^b−1, and the multiplier a[b−1] for this is zero, so that after Step 1 of an iteration we will have:
The number after Step 1 of the iteration
b^b b^b−1 b^b−2 b^b−3 b^b−4 … … b^2 b^1 b^0
a[b] 0 a[b−2] a[b−3] a[b−4] … … a[2] a[1] a[0]
The reason for this is that at the initial leftmost position (a[b] and b^b), both the base b and its exponent b of the positional values have increased by one, but at all the other positions, the
numerical value of base b increases but the numerical value of its exponent does not. And since our positional notation requires that all positions are shown, every position has an exponent which 1
greater than the next position to its right, this means that the second position from the left (the position that was immediately to the right of the leftmost position, a[b−1] and b^b−1) becomes
the third position from the left, and its exponent is b^b−2. Its multiplier remains the same value but is now represented here as a[b−2]. This may be envisaged by:
b^b b^b−1 b^b−2 b^b−3 … … b^2 b^1 b^0
a[b] a[b−1] a[b−2] a[b−3] … … a[2] a[1] a[0]
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
b^b b^b−1 b^b−2 b^b−3 b^b−4 … … b^2 b^1 b^0
a[b] 0 a[b−2] a[b−3] a[b−4] … … a[2] a[1] a[0]
In this way, every iteration produces an extra new position with a zero multiplier. The additional positions with zero multipliers remain zero unless all of the positions to the right of these
additional positions become zero. As in the previous case (Case 1) an iteration must be reached when all the multipliers to the right of the b^b position (for the current base b) become zero.
For that case, Step 1 of the next iteration increases the value of the base b by 1, and a new position is created immediately to the right of b^b with a zero multiplier, that is, every multiplier
other than that of b^b is now zero.
At Step 2 of that iteration, the multiplier of the b^b position decreases by 1, and all the other multipliers to the right of it take the value b−1.
We now have the same general case as the original starting case (all exponents b or less), except that the leftmost position is still b^b but with a multiplier that is 1 less in numerical value than
before. Note that while there are more positions, in the above the quantity of positions was immaterial and so the general analysis above still holds for further iterations.
It follows that there must be some iteration after finitely many iterations where the multiplier of the b^b position reaches zero. When this happens, the remaining part of the number is the same as
the general case already covered above in Case 1 and so, at some later iteration, all multipliers of all these positions must reach zero and hence the sequence terminates. Note that in the above, the
actual numerical value of b^b cannot affect this process and the outcome.
b‑exponents positions
It can also be noted at this point that for any position, by the definition of the Goodstein sequence, the expression for that position in terms of b remains exactly the same for all iterations. We
will refer to any position where the exponent of b is a multiple of b as a “b‑exponents” position, where 0 and 1 are included as such multiples, for example: b^0, b^b, b^1, b^2·b, b^b·b, b^b^2 etc.
(Footnote: Note that an expression such as b^b+b is equivalent to b^2·b with regard to the operation of the Goodstein sequence; for example after Step 1 of an iteration on 5^5+5 ≡ 5^2·5 we have 6
^6+6 ≡ 6^2·6.)
By the definition of the Goodstein sequence, it follows that a b‑exponents position can never be added at any iteration, and it will be seen later that this is a key point in proving the termination
of Goodstein sequences in general.
Case 3: Exponents b to 2b-1
It can be readily seen that, for the part of a number that has these positions, no new positions are created at an iteration. From the tables below, we can see that for the initial positions for the
positions b^b or higher:
The number before an iteration
b^2b−1 b^2b−2 … b^b+2 b^b+1 b^b …
a[2b−1] a[2b−2] … a[b+2] a[b+1] a[b] …
after the change of base by Step 1 of an iteration we will have:
The number after Step 1 of the iteration
b^2b−2 b^2b−3 … b^b+2 b^b+1 b^b …
a[2b−2] a[2b−3] … a[b+2] a[b+1] a[b] …
As in the above tables, both before and after the iteration, each exponent is exactly 1 greater than at the position to its right.
Note that 2b−1 becomes 2b−2 since in hereditary base notation, there is only one instance of b present in the 2b−1 before the iteration (which in true hereditary notation is b + m, where m < b
), hence the actual numerical value of the exponent is only increased by 1 - and since the new value of b is greater than the previous value by 1 the new expression for the exponent is 2b−2 in our
positional notation. The same applies to the names of the multipliers.
It follows that for this part of a number, an iteration does not create any new positions. As for the previous case, after finitely many iterations, b^b and all multipliers to its right will reach
zero, and unitary decrementation applies.
We now have the same general case for this part of the number as the original starting case, except that the multiplier of b^b is now the numerical value of b-1 and the numerical value of the
multiplier of b^b+1 has decreased by 1. Similarly, after finitely many iterations, the same general case for this part of the number as the original starting case is reached except that the
multipliers of b^b and b^b+1 are both zero. Similarly, after finitely many iterations, all multipliers of all the positions to the right of the leftmost position will reach zero and when that
occurs the multiplier of the leftmost position decreases by 1. And after finitely many iterations, it decreases to zero, and we have the same general case for this part of the number as the original
starting case, except that the original leftmost position has disappeared. Similarly, the multiplier of the new leftmost position will also reach zero after finitely many iterations, and so on, until
all the multipliers of this part of the number disappear, leaving the number as in Case 2.
Case 4: Exponents b to 2b
This is similar to Case 2, with this part of a number as:
The number before an iteration
b^2b b^2b−1 b^2b−2 … b^b+2 b^b+1 b^b …
a[2b] a[2b−1] a[2b−2] … a[b+2] a[b+1] a[b] …
and after the change of base by Step 1 of an iteration we have:
The number after Step 1 of the iteration
b^2b b^2b−1 b^2b−2 b^2b−3 … b^b+2 b^b+1 b^b …
a[2b] 0 a[2b−2] a[2b−3] … a[b+2] a[b+1] a[b] …
and again, at each iteration, one position is added immediately to the right of the position b^2b and its multiplier is zero, while the expression for the hereditary base notation for that position
remains unchanged except that the numerical value of the base is incremented.
And, as noted in Case 3, the part of the number covered by Case 3 must at Step 1 of some iteration have all its multipliers as zero, it follows that at Step 2 of the next iteration, the multiplier of
that position b^2b must decrement by 1, and there must be a subsequent iteration where that multiplier decreases to zero.
The General case
By now, the reader will probably see how the above leads to the conclusion that every Goodstein sequence must terminate. To be more precise, we can now generalize the above; Case 2 and Case 4 are
instances of segments between two consecutive “b‑exponents” positions P[1] and P[2] inclusive. (Footnote: A “b‑exponents” position was previously defined in Case 2: b‑exponents as a position where
the exponent of b is a multiple of b, where 0 and 1 are included as such multiples. Such positions cannot be added at any iteration.)
Note that at Step 1 of an iteration multiple positions can be added to the immediate right of a b‑exponents position P. For example, the consecutive positions 3^9, 3^8, 3^7 in hereditary base 3
notation are 3^3·3, 3^2·3+2, 3^2·3+1 so that by Step 1 they transform to 4^4·4, 4^2·4+2, 4^2·4+1 (or in decimal notation 4^16, 4^10, 4^9) . And, as for Case 2 and Case 4, all the additional
positions that are created at Step 1 of an iteration, here being 4^3·4+3, 4^3·4+2, 4^3·4+1, 4^3·4, 4^2·4+3 (or in decimal notation 4^15, 4^14, 4^13, 4^12, 4^11) always have zero
multipliers. As such, then there can only be a finite number of iterations before the multipliers to the right of these positions all become zero, and at the next iteration, the multiplier of the
position immediately to the left of these zero multipliers must decrease by 1.
Hence Case 2 and Case 4 are specific instances of the general case, which is that the addition of any finite quantity of new positions with zero multipliers at any iteration to the immediate right of
any position P with a non-zero multiplier cannot prevent the eventual decrement to zero of all multipliers to the right of that position P at some iteration, and so the multiplier of that position P
must decrease by 1 after finitely many iterations. And after finitely many iterations, it must decrease to zero, and this can occur repeatedly. Now, while in general multipliers can also increase (at
Step 2), this does not apply to the leftmost position - it cannot increase at any iteration, hence the leftmost position must always decrease to zero after finitely many iterations, leaving a new
leftmost position that has a hereditary base notation that, in terms of b, is one less than the previous leftmost position.
In general, there are two situations, where the leftmost position P[1] is a b‑exponents position or else it is not a b‑exponents position. If P[1] is a b‑exponents position then it must decrement to
zero after finitely many iterations, resulting in the situation that the new leftmost position is not a b‑exponents position and where there is a b‑exponents position P[2] to its right. Then no new
positions can be added to the left of this leftmost b‑exponents position P[2] , and so all the non-b‑exponents positions to the left of P[2] must decrease to zero after finitely many iterations,
leaving the new leftmost position as this b‑exponents position P[2]. Then at some iteration, the multiplier of this P[2] must decrease to zero, and so on until the only remaining b‑exponents position
is b^0. Then this is Case 1 and the sequence must terminate.
Extended Goodstein sequences
The above only considered cases where the base b was only increased by 1 at each iteration. But since an increase by any finite natural number can only result in the creation of additional positions
with zero multipliers, such additional positions do not prevent the eventual decrease of any multiplier to zero.
The straightforward non-choice reversible algorithmic nature of the definition of the Goodstein sequence allows for a purely mechanistic proof of termination by the principles of propositional logic
applied to the numerical properties of a sequence of numbers. It is of course the case that, for any system that can actually evaluate algorithms and which has a finite maximal number of variables,
there can always be some initial number where such a system cannot hold all the values needed to calculate the value of every multiplier at every position for every iteration. However, such
information is not required to prove that the general case that such a sequence must terminate. In the above, only a finite number of assertions of existence were required, and the proof does not
require the calculation of any specific values, only the assertion that for some variables there exists a natural number that satisfies certain conditions pertaining to that variable, and these
assertions follow from fundamental numerical properties.
As noted in the introduction, it is claimed that there is a proposition that is an expression of Peano arithmetic that asserts that a Goodstein sequence must terminate, and that this proposition
cannot be proved within Peano arithmetic. This has given rise in some quarters to a belief that there are some statements about natural numbers that are true but can only be proven by the use of
transfinite number theory, and that the proposition that every Goodstein sequence terminates is one such statement - but the above demonstrates that this is not the case.
In summary, we can note that not only is it the case that, as Solomon Feferman remarked:
“The necessary use of higher set theory in mathematics of the finite has yet to be established. Furthermore, a case can be made that higher set theory is dispensable in scientifically applicable
mathematics … Put in other terms: the actual infinite is not required for the mathematics of the physical world.” (Footnote: As in ‘Infinity in Mathematics: Is Cantor Necessary?’, in Philosophical
Topics 17.2 (1989): 23-45.)
and neither is it required to prove that all Goodstein sequences terminate.
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The Lighter Side
When a statistician passed through the airport security checkpoint, officials discovered a bomb in his bag. He explained:
“According to statistics, the probability of a bomb being on an airplane is 1/1000. Consequently, the chance that there are two bombs on one plane is 1/1,000,000, so I feel much safer if I bring one
James R Meyer
Recently added pages
A new section on set theory
How to setup Dark mode for a web-site
I have set up this website to allow a user to switch to a dark mode, but which also allows the user to revert back to the browser/system setting. The details of how to implement this on a website are
given at How to setup Dark mode on a web-site.
Decreasing intervals, limits, infinity and Lebesgue measure
The page Understanding sets of decreasing intervals explains why certain definitions of sets of decreasing intervals are inherently contradictory unless limiting conditions are included, and the page
Understanding Limits and Infinity explains how the correct application of limiting conditions can eliminate such contradictions. The paper PDF On Smith-Volterra-Cantor sets and their measure has
additional material which gives a more formal version.
Easy Footnotes
How to set up a system for easy insertion or changing of footnotes in a webpage, see Easy Footnotes for Web Pages.
New section added to paper on Gödel’s flawed paper
After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details
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Cantor’s Grundlagen and associated papers
To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers
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A brief history of meta-mathematics
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For pages with a comment section, you can leave a comment.
Printer Friendly
The pages of this website are set up to give a good printed copy without extraneous material.
Easy Footnotes
How to set up a system for easy insertion or changing of footnotes in a webpage, see Easy Footnotes for Web Pages.
New section added to paper on Gödel’s flawed paper
After comments that my PDF paper on the flaw in Gödel’s incompleteness proof is too long, I have added a new section which gives a brief summary of the flaw, while the remainder of the paper details
the confusion of levels of language. The paper can be seen at The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem.
Cantor’s Grundlagen and associated papers
To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers
published between 1879 and 1884 under the overall title of Über unendliche lineare Punktmannig-faltigkeiten, which were published between 1879 and 1884. I now have English translations of Part 1,
Part 2, Part 3 and the major part, Part 5 (Grundlagen). There is also a new English translation of Cantor’s “A Contribution to the Theory of Sets”.
A brief history of meta-mathematics
A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for
strict rigor.
For pages with a comment section, you can leave a comment.
Printer Friendly
The pages of this website are set up to give a good printed copy without extraneous material. | {"url":"https://www.jamesrmeyer.com/infinite/goodstein","timestamp":"2024-11-02T04:52:26Z","content_type":"text/html","content_length":"126647","record_id":"<urn:uuid:492068b7-99c0-4493-9d70-bc0958b60a44>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00304.warc.gz"} |
Need assistance with MATLAB homework, where can I turn to? | Pay Someone To Do My Matlab Assignment
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Part 2 family of 12
Supermarkets must be a helluva lot cheaper in Brisvegas than they are here if he can shop for 14 for STG250 ($350) a week! Not hard to spend that much on a family of 4, let alone 14!
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Thats it then were off to Brisbane it cheaper than Salford!!!!, They didnt even want sparks when i tried!!!
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The Set of Rational Numbers is Countably Infinite
The Set of Rational Numbers is Countably Infinite
On The Set of Integers is Countably Infinite page we proved that the set of integers $\mathbb{Z}$ is countably infinite. We will now show that the set of rational numbers $\mathbb{Q}$ is countably
Theorem 1: The set of rational numbers $\mathbb{Q}$ is countably infinite.
• Proof: Observe that the set of rational numbers is defined by:
\quad \mathbb{Q} = \left \{ \frac{a}{b} : a, b \in \mathbb{Z}, \: b \neq 0 \right \}
• In fact, every rational number $r$ can be uniquely written in the form $r = \frac{p}{q}$ where $p, q \in \mathbb{Z}$, $q \neq 0$, and $p$ and $q$ are relatively prime, that is, the greatest
common divisor of $p$ and $q$ is $1$. For each rational number $r \in \mathbb{Q}$ we define a function $f : \mathbb{Q} \to \mathbb{Z} \times \mathbb{Z}$ by:
\quad f(r) = (p, q)
• Then by the observation made above, $f$ is injective. Furthermore, since $\mathbb{Z}$ is countable we have that $\mathbb{Z} \times \mathbb{Z}$ is countable by the theorem presented on The
Cartesian Product of Two Countable Sets is Countable page. Therefore $f(\mathbb{Q})$ is countably infinite and so there exists a bijection $g : f(\mathbb{Q}) \to \mathbb{N}$.
• Note that the function $h : \mathbb{Q} \to f(\mathbb{Q})$ defined by $h(q) = f(q)$ is a bijection. Therefore the composition $g \circ h : \mathbb{Q} \to \mathbb{N}$ is a bijection. So $\mathbb{Q}
$ is countably infinite. $\blacksquare$ | {"url":"http://mathonline.wikidot.com/the-set-of-rational-numbers-is-countably-infinite","timestamp":"2024-11-13T17:59:52Z","content_type":"application/xhtml+xml","content_length":"15681","record_id":"<urn:uuid:6e8d5269-3d8f-4c5f-92a8-292c4adda184>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00063.warc.gz"} |
Lesson Plan
Data and Probability: Marshmallow Madness
Subject: Math
Grade span: 6 to 8
Duration: 30 to 45 minutes
This lesson was excerpted from the
Afterschool Training Toolkit
under the promising practice:
Math Centers
This sample lesson is one example of how you can implement Math Centers. In this activity, students collect data using large and small marshmallows, much like flipping a coin, to determine the
chances of a marshmallow landing on its end or side.
Adapted from the Connected Mathematics Program.
Lappan, G., Fey, J.T., Fitzgerald, W.M., Friel, S.N., and Phillips, E.D. (2002). How Likely Is It? Glenview, IL: Prentice Hall.
Learning Goals:
• Make and test predictions
• Collect and organize data
• Read and interpret data tables
• Use proportional reasoning to solve problems
• Several large and small marshmallows for each pair of students
• Ziploc bags
• Pencils or pens
• Molly's Marshmallows (PDF) — a written description for students and the recording chart
• Prepare a plastic bag with several large and small marshmallows for each pair of students.
• Print and copy the Recording Chart from the materials needed section.
• Create an inviting area for students with access to all of the space and tools they need.
What to Do:
• Ask students to pair up in groups of two.
• Review the definition of "data" as pieces of information that students gather to tell the likelihood of something happening.
• Review Molly's Marshmallow Problem with students and make sure they understand their task. Review the question by asking, "What is this problem asking you to do?"
• Ask students to make predictions about whether the two differently sized marshmallows are more likely to land on their sides or ends.
• Encourage students to find ways to work together. For example, one student might flip marshmallows while the other records results.
• As students work together in their centers, move from center to center and ask guiding questions that encourage students to explain their reasoning and work. Try to use new math vocabulary in
your interactions. For example, talk about the data, the table for collecting data, and what the data tell (how to interpret the numbers they are recording).
• When students have finished collecting the data from the marshmallow flipping, review the follow-up questions in the problem and how to write fractions from the data (see Tips).
• Ask each pair to present findings, reporting in on initial predictions and whether the answers make sense.
• If time allows, consider converting fractions to percentages.
Teaching Tips:
Understanding Data
Each time students flip a marshmallow and record the result, they are gathering data, information that will help them determine the likelihood of that result happening again.
Interpreting Data and Writing Fractions
Once students have flipped marshmallows and recorded their answers, they are ready to write their answers as fractions and be able to say what percent of the time a given marshmallow will land on its
side or end.
For example, one of the follow-up questions asks:
What fraction of the time will a small marshmallow land on its side, according to your experiment?
Sample Answer: If the marshmallow lands on its side 20 times, the answer is 20 out of 50 times. To write that as a fraction, you simply write 20/50. This can also be expressed as 2/5 (two fifths) of
the time or 40%. It may be helpful to review converting fractions into decimals with the students, and to explore what the % symbol means and how it relates to decimal and fraction notation.
Evaluate (Outcomes to look for):
• Student participation and engagement
• Prediction-making and testing through experimentation
• An understanding of data, and an ability to interpret the data
• Writing accurate fractions to represent data
• Students using proportional reasoning to solve problems
Click this link to see additional learning goals, grade-level benchmarks, and standards covered in this lesson. | {"url":"https://sedl.org/afterschool/lessonplans/index.cgi?show_record=90","timestamp":"2024-11-10T08:39:02Z","content_type":"text/html","content_length":"11973","record_id":"<urn:uuid:0538d170-8ac5-4bf0-b292-266f38a6a282>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00202.warc.gz"} |
Why Numerical Reasoning Tests are Invaluable - Skillsarena - Skillsarena
Aptitude tests are an effective way of assessing a candidate's competency and ability to solve different types of problems. They are very valuable assessments, as they can be great indicators for how
a particular individual may perform in a particular role. There are different types of reasoning tests available, for example, verbal reasoning tests, which test a candidate's ability to understand
and interpret written information. However, today we're going to cover numerical reasoning tests, taking a look at why this type of aptitude test is valuable in any hiring process, and not just data
based jobs.
Numerical reasoning is one of the most valuable skills that someone can have in today's job market. Whilst this may first appear as a test that is only necessary if you're applying for a data related
position, this is far from the case. Numerical reasoning tests can display a lot more, and should be included in every hiring process.
What is a numerical reasoning test?
A numerical reasoning test is an assessment that measures an individual's ability to interpret and solve mathematical problems. These tests are commonly used in the selection process for occupations
that require strong mathematical skills, such as finance and engineering.
The tests usually take the form of a multiple-choice question format, and may cover topics such as basic arithmetic, data interpretation, and statistical analysis. The assessment can consist of a
series of questions, each of which presents a table or graph with accompanying data. Numerical reasoning tests are designed to assess an individual's capacity for logical reasoning and problem
solving, as well as their ability to understand basic to more complex numerical concepts.
While the tests can be challenging, they provide employers with a valuable tool for assessing candidates for positions that require strong numeracy skills. For a more detailed breakdown, read our
blog that describes in detail what a numerical reasoning test is.
What does numerical reasoning assess?
These assessments look at a candidate's understanding and competency with numerical concepts. The whole point of this type of aptitude test is to assess a candidate's numeracy skills, which are
likely to be very important for the role that they are applying to.
Numerical reasoning tests measure a person's ability to interpret graphs, tables and other forms of data. These numerical tests also assess a candidate's arithmetic skills. Knowing how to perform
different calculations is a very important skill, and it is important to see if applicants are able to apply this knowledge. An employer wants to see if a candidate is able to perform addition,
subtraction, multiplication and division and to what level. As well as this, other arithmetic skills that are assessed with numerical reasoning tests are the ability to figure out percentages and
ratios, and these are often applied in a business sense.
Problem solving skills are an essential aspect to many jobs, and these aptitude assessments help employers to see how well each candidate can work under pressure and figure out a solution to a
numerical issue. This is often overlooked when it comes to numerical reasoning tests as employers believe these tests only assess a candidate's numerical ability. However, the reasoning questions can
provide excellent insight into their problem solving abilities.
What are the benefits of using numerical reasoning tests?
Numerical reasoning tests are commonly-used assessment tools in the hiring process, but their use is often limited to certain types of roles that require use of numerical data. They provide a way to
measure a candidate's ability to understand and interpret numerical data, including graphs, percentages and different types of calculations. However, they also have a range of benefits outside of
these jobs too:
Accurate insight into a candidate's skillset
One of the main benefits of numerical reasoning assessments is that they can help employers to identify candidates who have the potential to be successful in roles, and even predict job performance.
In addition, numerical reasoning tests can also be used to assess a candidate's problem-solving skills by requiring candidates to solve problems using numerical data.
Produce data-driven results
Numerical reasoning tests are another great asset to add to a company that is dedicated to producing data-driven decisions and eliminating guesswork. This reasoning assessment provides accurate and
proven insight into how effective candidates will be at a specific job.
Remove unconscious bias
Whether we like it or not, unconscious bias can be prevalent in all of us. As much as we try to remove that, there's still a chance that it can persist, and this is why aptitude tests such as the
numerical ability test prove useful. Skills-based tests remove unconscious bias because now candidates are being initially assessed on their direct capability to perform a specific role and nothing
A more efficient use of time
Applying aptitude testing during the hiring process allows employers to learn early on which candidates are best suited for specific roles. This can save time for hiring managers and recruiters as
they are not spending unnecessary time on the hiring process with unsuitable candidates.
How to use a numerical reasoning test to assess applicants
There are a number of different ways to administer a numerical aptitude test, but the most common approach is to present candidates with a series of problems to solve. This numerical test often
becomes more difficult as the test progresses, and the total time allotted for the test is typically relatively short. This means that candidates must be able to work quickly and accurately under
pressure in order to perform well.
The numerical tests, such as the arithmetic section, require candidates to complete the test without the use of a calculator in order to determine how effectively they can carry out data
interpretation from common numerical occurrences such as graphs.
Numerical tests can also be applied out at different stages of the hiring process, depending on the role and what exactly you are looking to identify. It is most common for psychometric tests to be
carried out during the hiring process as they provide an excellent and effective way of shortlisting applicants, and the numerical reasoning test is no different.
Who are numerical reasoning aptitude tests best for?
In short, the answer is everyone! Numerical reasoning ability is a very sought after skill because many different roles in a variety of industries require employees to be able to solve an array of
Areas in which numerical reasoning is commonly used include accountancy, finance, marketing and engineering. However, as we've mentioned throughout this blog, numerical reasoning is an important
factor for any hiring process and this can include jobs such as healthcare, administration, education, construction, and many more.
By administering a numerical reasoning test, you can gain an insight into how well a candidate thinks numerically and their ability to reason through mathematical problems. This information is
invaluable in making the right decision about who to bring on board.
If you’re interested in administering a numerical reasoning test as part of your hiring process, or if you would like more information about our wide range of psychometric tests, be sure to contact
us today. | {"url":"https://skillsarena.com/blog/article/why-numerical-reasoning-tests-are-valuable","timestamp":"2024-11-09T23:41:57Z","content_type":"text/html","content_length":"27264","record_id":"<urn:uuid:2fc4721d-c3a6-442d-a933-b06a9441a42a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00363.warc.gz"} |
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Unit Circle Worksheet With Answers
Unit Circle Worksheet With Answers. If we all know the quadrant the place the angle is, we will simply select the proper answer. Here you can see lots of of classes, a community of academics for
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In order to do that, we have to understand the connection of the Special Right Triangles 30 – 60 – 90 and 45 – forty five – ninety degrees to the coordinate airplane. These Right Triangles are
crucial to recollect as a result of they’ve sure properties that come in handy when fixing Trigonometric capabilities. We then move on to Quadrant II, which starts at 90° and goes to 180°. From the
diagram beneath, every angle in Quadrant II measures 30°, 45°, and 60° within that Quadrant. This is finished for 30°, 45°, and 60° angles in every Quadrant. Now we’re going to divide the Unit Circle
into 30°, 45°, and 60° angles.
That will deliver you to the adverse x-axis, after which you have to go 20° farther. If you click on on “Tap to view steps”, you will go to theMathwaysite, where you’ll be able to register for
thefull version of the software. To \(\displaystyle -2\pi \) are \(\displaystyle -\frac\) and \(\displaystyle -\frac\). To \(-2\pi \) are \(\displaystyle -\frac\) and \(\displaystyle -\frac\).
What’s The Unit Circle?
Scroll down the web page for extra examples and solutions on the unit circle, sine, cosine, and tangent. Incorporate these easy unit circle PDFs to find out the coordinates of the terminal level for
the given angle measures. Unit circle diagram is provided in every worksheet for reference. Familiarize college students with the unit circle by employing these worksheets. Keenly observe the unit
circle diagram, use the angle measures to figure out the corresponding coordinates and complete the unit circle. Students will fill out a blank Unit Circle on the front of the worksheet with degrees,
radians, and coordinates.
Each web page has color-coded examples with specific directions that detail the issues and formulas. The value for cotangent is right. However, when evaluating cosine, you could have switched the x-
and y-coordinates.
Unit 6 Worksheet 5 Using Unit Circle
Answers are supplied in addition to a accomplished unit circle. Draw the angle in standard place. Calculate the trigonometric function worth of the reference angle.
Scholars study radians and the way they connect with measurements in levels. They find precise and approximate values of… A unit circle is a circle with a radius measuring 1 unit. The unit circle is
mostly represented in the cartesian coordinate plane. The unit circle is algebraically represented utilizing the second-degree equation with two variables x and y. The unit circle has functions in
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The Unit Circle
In reality, any angle from 0° to 90° is similar as its reference angle. Understand unit circle, reference angle, terminal aspect, commonplace position. In another classes, we’ve coated the three
frequent trigonometry features sine, cosine and tangent utilizing the essential SOH-CAH-TOA definition. The unit circle identities of sine, cosecant, and tangent could be additional used to obtain
the other trigonometric identities similar to cotangent, secant, and cosecant. The unit circle identities corresponding to cosecant, secant, cotangent are the respective reciprocal of the sine,
cosine, tangent. Further, we are ready to obtain the value of tanθ by dividing sinθ with cosθ, and we can acquire the worth of cotθ by dividing cosθ with sinθ.
• The angles 150°, 210°, and 330° have one thing in common.
• Angle to point that it went in the other way of a spaceship that went via a 50° angle.
• To \(\displaystyle -2\pi \) are \(\displaystyle -\frac\) and \(\displaystyle -\frac\).
Here you will find lots of of classes, a neighborhood of lecturers for help, and supplies which would possibly be always updated with the newest standards. Great for added practice, sub plans, or
remote-learning. Interactive sources you can assign in your digital classroom from TPT. We hope the worksheet and reference sheet come in useful for you or your learners.
Notice that if you understand the ordered pair values in the first quadrant, you realize them in all of the quadrants! Look for the mirror pictures of the ordered pairs, but with the completely
different indicators for that quadrant. All the other particular angles have comparable proofs. But the weird factor about radians is that they actually don’t have a unit, like levels, toes or
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Finding Function Values For Sine And Cosine
Scholars work on issues involving the Pythagorean Theorem and its… Educator Edition Save time lesson planning by exploring our library of educator evaluations to over 550,000 open academic sources .
The process is the same even if the angle is negative. Remember that a adverse angle is just one whose path is clockwise. In the subsequent two examples, the angle labels of 37° and 53° are actually
very shut approximations. The major idea of the examples nonetheless holds true.
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Vector and tensor radiation from Schwarzschild relativistic circular geodesics
Breuer, R. A. ; Ruffini, R. ; Tiomno, J. ; Vishveshwara, C. V. (1973) Vector and tensor radiation from Schwarzschild relativistic circular geodesics Physical Review D - Particles, Fields, Gravitation
and Cosmology, 7 (4). pp. 1002-1007. ISSN 1550-7998
Full text not available from this repository.
Official URL: http://prd.aps.org/abstract/PRD/v7/i4/p1002_1
Related URL: http://dx.doi.org/10.1103/PhysRevD.7.1002
For the case of high multipoles we give an analytic form of the spectrum of gravitational and electromagnetic radiation produced by a particle in a highly relativistic orbit r[0]=(3+δ)M around a
Schwarzschild black hole of mass M. The general dependence of the power spectrum on the frequency in all three spin cases (s=0 for scalar, s=1 for vector, and s=2 for tensor fields) are summarized by
power P∝ ω^1−sexp(−2ω/ω[crit]). Although they have the common feature of an exponential cutoff above a certain frequency ω[crit]=(4/πδ)ω[0], where ω[ 0] is the frequency of the orbit, the tensor case
has a much broader frequency spectrum than scalar or vector radiation.
Item Type: Article
Source: Copyright of this article belongs to The American Physical Society.
ID Code: 58658
Deposited On: 02 Sep 2011 03:52
Last Modified: 02 Sep 2011 03:52
Repository Staff Only: item control page | {"url":"https://repository.ias.ac.in/58658/","timestamp":"2024-11-05T23:45:17Z","content_type":"application/xhtml+xml","content_length":"17543","record_id":"<urn:uuid:58f77f3d-f3b6-43da-9cb7-d2be72798a15>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00878.warc.gz"} |
Modified Covariance Matrix Adaptation – Evolution Strategy algorithm for constrained optimization under uncertainty, application to rocket design
Issue Int. J. Simul. Multisci. Des. Optim.
Volume 6, 2015
Article Number A1
Number of page(s) 13
DOI https://doi.org/10.1051/smdo/2015001
Published online 29 April 2015
Int. J. Simul. Multisci. Des. Optim. 2015,
, A1
Research Article
Modified Covariance Matrix Adaptation – Evolution Strategy algorithm for constrained optimization under uncertainty, application to rocket design
^1 IFMA, EA3867, Laboratoires de Mécanique et Ingénieries, Clermont Université, CP 104488, 63000 Clermont-Ferrand, France
^2 Onera – The French Aerospace Lab, BP 80100, 91123 Palaiseau Cedex, France
^3 CNES – Launchers Directorate, 52 rue Jacques Hillairet, 75612 Paris, France
^* e-mail: mathieu.balesdent@onera.fr
Received: 9 October 2014
Accepted: 3 March 2015
The design of complex systems often induces a constrained optimization problem under uncertainty. An adaptation of CMA-ES(λ, μ) optimization algorithm is proposed in order to efficiently handle the
constraints in the presence of noise. The update mechanisms of the parametrized distribution used to generate the candidate solutions are modified. The constraint handling method allows to reduce the
semi-principal axes of the probable research ellipsoid in the directions violating the constraints. The proposed approach is compared to existing approaches on three analytic optimization problems to
highlight the efficiency and the robustness of the algorithm. The proposed method is used to design a two stage solid propulsion launch vehicle.
Key words: Evolutionary Strategy / Covariance Matrix Adaptation / CMA-ES / Uncertainty / Constrained optimization / Rocket design
© R. Chocat et al., Published by EDP Sciences, 2015
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1 Introduction
The design of complex systems, such as aerospace vehicles, can be expressed as a non linear constrained optimization problem solving. The determination of the optimal system architecture requires a
global exploration of the design space involving repeated evaluations of computationally expensive black box functions used to model the different disciplines (e.g. in aerospace: structure,
propulsion, aerodynamics). Moreover, in the early design phases, uncertainties arise due to lack of knowledge of the system characteristics and the environment (e.g. propellant combustion rate, wind
gust) and to the use of low fidelity analyses (for instance analytic code instead of numerically costly finite element analysis). Therefore, the design of a complex system at the conceptual design
phase entails a constrained optimization problem under uncertainty that can be formulated as:$Min Ξ [ F ( z , U ) ] wrt z$(1)
• z: the vector of design variables z ∈ ℝ^n (e.g. rocket diameter, propellant masses),
• U: the vector of the uncertain variables U ∈ ℝ^p (e.g. wind gust during the rocket launch, material characteristics). In the paper, the probability theory is used to model uncertainty.
• F: ℝ^n × ℝ^p → ℝ: the performance function (e.g. propulsive speed increment ΔV, Gross Lift-Off Weight),
• Ξ: a measure of uncertainty for the performance function F (e.g. the expected value, a linear combination of expected value and standard deviation),
• g: the vector of the inequality constraint functions, g[i], i ∈ {1, …, m} (e.g. maximal allowed stress in the structures, maximal tolerance on orbit injection accuracy),
• K: a vector of measures of uncertainty for the inequality constraint functions (e.g. linear combination of expected value and standard deviation, probability of failure),
• z[min] and z[max] the lower and upper bounds for the design variables.
Being able to handle uncertainty at early design phases is essential to efficiently characterize the optimal system design and its performances. It can reduce time and cost of the next design phases
by avoiding redesign process [18]. However, solving the optimization problem (Eqs. (1)–(3)) is challenging due to the presence of both the uncertainty and inequality constraints. A brief overview of
the existing methods to handle uncertainty and inequality constraints is provided in the next paragraphs.
The presence of noise in the optimization problem (Eqs. (1)–(3)) results from the estimation of the measures of uncertainty (Ξ and K). Several methods exist to compute these measures and the most
classical approach is a numerical approximation by the Crude Monte Carlo (CMC) method. The CMC estimator of the uncertainty measure is also a random variable. In order to numerically handle the
presence of noise in optimization, several approaches have been proposed:
• Re-sampling [3]: the re-sampling method consists of repeated evaluations of the objective and the constraint uncertainty measures for the same design variable value z. Then, a statistics of the
repeated samples is used instead of the single evaluation of the objective and the constraint measures to decrease the impact of noise. The main drawback of this approach is the increase of the
computational cost due to the repeated evaluations of expensive functions.
• Surrogate model [3]: surrogate models of the objective function measure Ξ (and/or the constraint measures K) are built from the evaluated measures. In general, surrogates smooth the noisy
functions and decrease the impact of uncertainty in the optimization. The main drawback lies in the difficulty to build accurate surrogate models in high dimensions.
• Population based algorithms [10]: to address uncertainty, optimization algorithms relying on a population of candidates allow to increase the size of the population to enlarge its spread and to
obtain more information and smooth the noise.
Moreover, the presence of uncertainty makes the use of classical optimization algorithms, such as gradient based optimization algorithms, not suited for optimization. Indeed, the gradients of the
objective uncertainty measure or the constraint uncertainty measures are noisy resulting in possible erroneous descent directions. Diverse derivative-free algorithms have been proposed in the
literature to solve optimization under uncertainty problems [15]. Among these, the population based optimization algorithms seem promising [6]. Swarm Intelligence (Particle Swarm Optimization [7],
Artificial Bee Colonies [13], etc.), Differential Evolution [17], Evolutionary Algorithms (Genetic Algorithm [12]) or Evolution Strategies (Covariance Matrix Adaptation – Evolution Strategy [10])
have been investigated to solve noisy optimization problems. One of the current issue with the derivative-free algorithms is the constraint handling which relies mainly on heuristic approaches and is
problem dependent [16]. A comprehensive overview of constraint handling in derivative-free algorithms is presented in the survey [16]. The most commonly applied methods to handle the constraints are:
• Death penalty [19]: it is the simplest method to handle constraints. The solution that does not satisfy the constraints is rejected and another potential solution is re-evaluated until one
candidate solution satisfies the constraints. The advantage of the approach is that it does not modify the optimization algorithm but the method is very expensive because no information is
learned from an unfeasible solution (a solution which does not satisfy the constraints) to characterize the non feasible space. Furthermore, if the feasible space is restricted compared to the
design space, the computational cost becomes prohibitive because a high number of samples has to be generated to obtain feasible solutions.
• Penalization [5, 6]: this approach consists in replacing the objective function Ξ [F(z, U)] by a combination of the objective function and a penalization function Π such as: Ξ [F(z, U)] + Π (K[g(
z, U)]). The penalization function can be fixed or can change as a function of the number of iterations. When a solution violates the constraints, the objective function is deteriorated by a
factor proportional to the penalization function and the value of the constraints. Despite its simplicity, the main drawback of this approach lies in the determination of a suitable penalization
function which depends on the objective function and the constraints and is thus problem dependent.
• Multi-objective [4]: this method transforms the optimization problem into a multi-objective optimization problem by considering the minimization of the violation of the constraints as an
objective. Dedicated multi-objective optimization algorithms can be used, however, it often results in an increase of the computational cost [16].
• Surrogate model [14]: this approach builds a surrogate model based on the unfeasible solutions in order to approximate the non feasible zones. However, this approach requires enough unfeasible
solutions to construct accurate surrogate models. Moreover, it can be difficult to build the surrogates in high dimension or if the constraints are highly non linear.
Among the derivative-free algorithms, the Covariance Matrix Adaptation – Evolution Strategy (CMA-ES) [10] is particularly competitive for real-valued black box functions as highlighted in several
extensive benchmarks [8, 9]. Moreover, a treatment of uncertainty has been proposed for CMA-ES and has been successfully tested in a benchmark of optimization under uncertainty problems [8]. However,
the test problems used to evaluate the performances of CMA-ES are unconstrained optimization problems and only few studies focus on the application of CMA-ES to constrained optimization problems [2,
6]. An adaptive penalty function has been proposed to update the penalty coefficient as a function of the sum of the violated constraint values [6]. Arnold and Hansen [1] proposed a new approach to
handle the constraints for a simplified version of CMA-ES: (1+1)-CMA-ES which involves one offspring candidate generated from one parent. Modified (1+1)-CMA-ES approximates the normal vector
directions of the constraint boundaries in the vicinity of the current candidate solution. A control of the covariance of the distribution of the offspring candidate allows to get closer to the
boundary of the feasible regions without violating them. The approach provides interesting results but is limited to (1+1)-CMA-ES which becomes inefficient in high dimensions (local convergence,
large number of calls to the functions) [1].
The main objective of this paper is to adapt CMA-ES(λ, μ), which involves λ offspring candidates generated from μ parents, to efficiently solve constrained optimization problem under uncertainty. The
proposed approach is based on a control of the covariance of the distribution of the offspring candidates through a modification of the covariance matrix that characterizes the probable research
hypervolume in order to generate candidates without violating the constraints. The proposed adaptations take into account the specificities of the update and the selection mechanisms of CMA-ES(λ, μ)
that differ from (1+1)-CMA-ES due to the presence of a population instead of a single candidate. The remainder of the paper is organized as follows. Section 2 provides an overview of the original
CMA-ES(λ, μ) and the adapted (1+1)-CMA-ES to handle constraints. Section 3 presents the modifications of CMA-ES(λ, μ) to handle constraints in optimization problems. Section 4 evaluates the algorithm
efficiency on three analytic test functions and on the design of a two stage solid rocket. A comparison with penalized CMA-ES(λ, μ) and modified (1+1)-CMA-ES is provided.
2 Description of CMA-ES(λ, μ) and modified (1+1)-CMA-ES
2.1 CMA-ES(λ, μ) algorithm
The Covariance Matrix Adaptation – Evolution Strategy (CMA-ES) introduced by Hansen et al. [10] belongs to the Evolution Strategy algorithm family. A brief overview of CMA-ES(λ, μ) is provided in the
section in order to understand the proposed modifications for the constraint handling, for more information on the algorithm see [10]. CMA-ES(λ, μ) is used to solve unconstrained optimization
problems. CMA-ES relies on a distribution model of a candidate population (parametrized multivariate normal distribution) in order to explore the design space. It is based on a selection and an
adaptation process of the candidate population. In CMA-ES(λ, μ), at each generation, λ offspring candidates are generated from μ parents. At the next generation, to select the new parents from the
offspring candidates, a (λ, μ)-selection is used, the μ best offspring candidates are chosen (with respect to their ranking according to the objective function). The multivariate normal distribution
has an infinite support, but an iso-probability contour (for instance at ±3 standard deviation of the mean) is characterized by an ellipsoid delimiting a probable research hypervolume (Figure 1).
Throughout the generations, the research hypervolume is updated in order to converge and to shrink around the global optimum. CMA-ES(λ, μ) generates the population by sampling a multivariate normal
distribution:$z t ( k + 1 ) ∼ m ( k ) + σ ( k ) N ( 0 , C ( k ) ) , for t = 1 , … , λ$(4)with $z t ( k + 1 ) ∈ R n$ an offspring candidate generated from a mean vector m^(k), a step size σ^(k) and a
multivariate normal distribution $N ( 0 , C ( k ) )$ with zero mean and a covariance matrix C^(k) ∈ ℝ^n × ℝ^n. λ is the size of the population generated at each iteration (k). The normal distribution
is characterized by a positive definite covariance matrix C^(k) in order to allow homothetic transformations and rotations of the probable research hypervolume (Figure 1). The update of the
covariance matrix incorporates dependence between the past generations and between the μ best candidates from the previous generation [10]. The mean vector characterizes the center of the next
population and is determined by a combination process through the weighting of the μ best candidates: $m ( k + 1 ) = ∑ i = 1 μ w i z ( i ) ( k + 1 )$ with $∑ i = 1 μ w i = 1$, w[1] > w[2] > ⋯ > w[μ]
> 0 the weighting coefficients and $( z ( 1 ) , … , z ( μ ) )$ the best candidates among the offspring ranked according to the fitness value. The weighting coefficients are determined based on the
number of μ best candidates according to [10]. A simplified version of CMA-ES(λ, μ) is described in Algorithm 1. A detailed description of the selection and update mechanisms can be found in [10].
Figure 1.
Three ellipsoids, depicting three different normal distributions, where I is the identity matrix, D is a diagonal matrix, and C is a positive definite covariance matrix. Thin dot lines depict
objective function contour lines.
Algorithm 1 CMA-ES(λ, μ) [10]
1) Initialize the covariance matrix C^(0) = I, the step size σ^(0) and the selection parameters [10]
2) Initialize the mean vector m^(0) to a random candidate, k ← 0
while CMA-ES convergence criteria are not reached do
3-1) Generate λ new offspring candidates according to: $z t ( k + 1 ) ∼ m ( k ) + σ ( k ) N ( 0 , C ( k ) ) , t ∈ { 1 , … , λ }$
3-2) Evaluate candidates and rank them based on the objective function
3-3) Determine the mean vector given the weighting coefficients of the μ best candidates: $m ( k + 1 ) = ∑ i = 1 μ w i z ( i ) ( k + 1 )$
3-4) Update covariance matrix C^(k+1) and the step size σ^(k+1) according to [10], k ← k + 1
end while
4) return best candidate z[best]
The convergence criteria can either be based on the maximum number of iterations (function evaluations), the fitness value, the standard deviation of the current population smaller than a given
tolerance, or the covariance matrix C which becomes numerically not positive definite. In the next paragraph, an adaptation of CMA-ES for uncertainty handling is described.
2.2 CMA-ES(λ, μ) algorithm for optimization under uncertainty
Several features ensure the CMA-ES robustness with respect to the presence of uncertainty in the objective function: the population-based approach, the weighted averaging in the recombination
process, the rank based and the non-elitist selection (not based on the best offspring candidate). However, if the noise is too high compared to the objective function (signal to noise ratio too low)
it perturbs the algorithm convergence. An appropriate handling of uncertainty has been proposed by Hansen et al. [11] to overcome this issue. Modified selection and update mechanisms are performed
when the noise is above a given threshold. It is based on a re-sampling approach and involves re-evaluation of the objective function. Because CMA-ES(λ, μ) is only based on the rank of the
candidates, the effective noise is evaluated by monitoring changes or stability of the offspring candidate ranking. If the offspring candidate ranking is changed after the re-evaluation of the
objective function, the ranking change of the offspring candidates is aggregated into a metric quantifying the uncertainty level [11]. If the noise is higher than a given uncertainty level threshold,
the step size σ is increased. The increase of σ ensures that despite the noise, sufficient selection information is available [11].
A benchmark of algorithm dealing with optimization under uncertainty has been performed and the treatment of uncertainty with CMA-ES(λ, μ) allows to obtain accurate results [8]. However, CMA-ES(λ, μ)
is not able to efficiently handle the constraints. Indeed, only penalization [5, 6] or surrogate based methods [14] have been proposed. These approaches can be effective but are problem dependent and
require accurate tuning of hyper parameters. A simplified version of CMA-ES(λ, μ) called (1+1)-CMA-ES has been adapted to handle constraints and is detailed in the next section.
2.3 (1+1)-CMA-ES with constraint handling
(1+1)-CMA-ES is a simplified version [1] of CMA-ES(λ, μ) with only one offspring generated from one parent, “+” means that the selection is done between the parent and the offspring. As in CMA-ES(λ,
μ), the offspring candidate solution is generated as:$z ( k + 1 ) ∼ z ( k ) + σ ( k ) N ( 0 , C ( k ) )$(5)(1+1)-CMA-ES is easier to implement as only one offspring z^(k + 1) is generated from one
parent z^(k) at each generation and the selection is between the parent and the offspring. The update mechanisms for (1+1)-CMA-ES are detailed in [1].
To incorporate the handling of constraints, Arnold and Hansen [1] proposed to reduce the covariance of the distribution of the offspring candidate in the approximated directions of the normal vectors
of the constraint boundaries in the vicinity of the current parent candidate solution. For that purpose, the matrix A^(k), which is the Cholesky decomposition of $C ( k ) : A ( k ) A ( k ) T = C ( k
)$, is updated in case of constraint violations in order to avoid generating candidates in the next generations that will violate the constraints. A^(k) is used as it is easier to compute its inverse
than for C^(k). A vector characterizing the constraints $v j ( k ) ∈ R n$ is defined, initialized to be zero, and updated according to:$v j ( k ) ← ( 1 - c c ) v j ( k ) + c c A ( k ) z ( k ) ∀ j ∈ {
1 , … , m }$(6)where $v j ( k )$ is an exponentially fading record of steps that have violated the constraints and c[c] a parameter characterizing how fast the information present in $v j ( k )$
fades. In the generations in which the offspring candidate is unfeasible, the Cholesky matrix is updated according to:$A ( k ) ← A ( k ) - β ∑ j = 1 m 1 g j ( z ( k ) ) > 0 ∑ j = 1 m 1 g j ( z ( k )
) > 0 v j ( k ) w j ( k ) T w j ( k ) T w j ( k )$(7)with $w j ( k ) = A ( k ) - 1 v j ( k )$, the indicator function associated to the constraint $g j : 1 g j ( z ( k ) ) > 0$ and β a parameter
controlling the reduction of the covariance of the distribution. For β = 0, the algorithm is identical to the standard (1+1)-CMA-ES. The update of the matrix A^(k) allows to modify the scale and the
orientation of the research hypervolume in order to be tangential to the constraints and to avoid its violation (Figure 2). Modified (1+1)-CMA-ES for constraint handling is interesting because it is
not problem dependent. Experimental evaluations have been performed highlighting its efficiency for unimodal constrained optimization problems. However, as (1+1)-CMA-ES, it is not able to optimize
multimodal functions and becomes inefficient in high dimensions [1].
Figure 2.
The dot is the parent, the cross is the generated offspring, the solid line circle is characterized by A^(k) defining the ellipsoid delimiting an iso-probability research hypervolume. At the
center, the red ellipsoid represents the update of A^(k+1) in order to take into account the constraint violation by the offspring. On the right figure, the offspring does not violate the
constraint resulting in a standard covariance matrix update.
To overcome these drawbacks, in the next section, an adaptation of CMA-ES(λ, μ) for constraint handling is proposed inspired from the (1+1)-CMA-ES approach.
3 Proposed adaptation of CMA-ES(λ, μ) for constraint handling
The proposed approach of CMA-ES(λ, μ) for constraint handling is based on the same approach as modified (1+1)-CMA-ES. However, it is necessary to adapt it to take into account the specificities of
CMA-ES(λ, μ). Indeed, CMA-ES(λ, μ) generates a population instead of a single offspring candidate. Thus, each offspring candidate can potentially violate one or several constraints. Moreover, the
selection of the μ best candidates is based on the rank of the objective function. However, these best candidates can also violate the constraints. Depending if the μ best candidates are feasible or
not, or if only a fraction of them is feasible, or on the number of violated constraints, the covariance matrix used to generate the offspring candidates has to be modified in order to avoid the
generation of unfeasible offspring candidates. The research hypervolume engendered by an iso-probability contour of the multivariate normal distribution $N ( 0 , C )$ can be represented by a n
-dimensional ellipsoid. In the proposed approach, the constraint handling method allows to reduce the semi-principal axes of the research ellipsoid in the directions violating the constraints. The
eigenvalues of the covariance matrix C control the length of the semi-principal axes. The decrease of the eigenvalues reduces the semi-principal axis lengths. The covariance matrix, which is
symmetric positive definite, can be decomposed according to:$C = PDD P T = P D 2 P T$(8)where P is an orthogonal matrix such that: PP^T = P^TP = I. The columns of P form an orthogonal basis of
eigenvectors of C. $D = diag ( v p 1 , … , v p n )$ is a diagonal matrix with the square roots of eigenvalues of C. As illustrated in Figure 3, the square roots of the covariance matrix eigenvalues
$v p i$ are proportional to the semi-principal axis lengths of the ellipsoid defining the sampling hypervolume.
Figure 3.
Parametrization of the ellipsoid defining the probable research hypervolume.
At the generation (k), between the step 3-3) and 3-4) of Algorithm 1, if any of the m constraints is violated by any of the μ best offspring candidates, the covariance matrix is modified according to
Algorithm 2.
Algorithm 2 Proposed CMA-ES(λ, μ) covariance matrix modificiation$3.3 . 1 ) Diagonalize C ( k ) such that P D ( k ) 2 P T = C ( k )$(9)
$3.3 . 2 ) S ( k ) ← P D ( k ) 2 P T - γ P diag ( v p m 1 ( k ) , … , v p m n ( k ) ) 2 P T$(10)
$3.3 . 3 ) C ( k ) ← [ det ( C ( k ) ) det ( S ( k ) ) ] 1 / n S ( k )$(11)
with:$v p m i ( k ) = ( v p i ( k ) ∑ j = 1 m ∑ l = 1 μ 1 g j ( z ( l ) ( k ) ) > 0 w lj Pro j e ⃗i [ z ( l ) ( k ) - m ( k ) ] ∑ t = 1 μ 1 g j ( z ( t ) ( k ) ) > 0 w tj )$(12)
The covariance matrix is diagonalized, equation (9) and the eigenvalues $v p i ( k )$ of the covariance matrix C^(k) are modified. The new eigenvalues are the former eigenvalues minus a term $v p m i
( k )$ taking into account the violation of the constraints. $v p m i ( k )$, equation (12), is a function of the former eigenvalues $v p i ( k )$, of the indicator function of the constraint $1 g j
( z ( l ) ( k ) ) > 0$, of the weighting coefficients w[ij] and of the projection $Pro j e ⃗i [ z ( l ) ( k ) - m ( k ) ]$ of the distance between an ordered candidate violating the constraints $z ( l
) ( k )$ and the mean point m^(k) in the direction of the eigenvector $e ⃗i$ corresponding to the eigenvalue $v p i ( k )$. γ is a parameter similar to β in (1+1)-CMA-ES. For γ = 0 the proposed
algorithm is similar to the classical CMA-ES(λ, μ). For each constraint g[j], the μ[cj] candidates among the μ best candidates that violate the constraint are ranked according to the constraint
value. The weighting coefficients w[ij] for each constraint g[j] are defined according to the same rule as for the recombination process for the calculation of m^(k):$w ij = ln ( μ cj + 1 ) - ln ( i
) μ cj ln ( μ cj + 1 ) - ∑ k = 1 n ln ( k )$(13)and $∑ i = 1 μ w ij = 1$ where: w[1j] ≥ ⋯ ≥w[μj] ≥ 0. w[1j] is associated to the candidate that violates the most the constraint g[j] and $w 1 μ c$
with the candidate that violates the less the constraint. For the candidates among the μ best that do not violate the constraint, the indicator function is equal to zero and therefore these
candidates do not participate in the modification of the covariance matrix. The projection of the violation distance along the eigenvector (Figure 4) allows to reduce the covariance matrix in the
direction orthogonal to the constraint violation.
Figure 4.
Violation of the constraint and projection over the eigenvectors, blue = feasible, red = unfeasible candidates.
Equation (11) allows to keep the hypervolume of the ellipsoid constant before and after the modification of the covariance matrix in order to avoid premature convergence. The volume of the ellipsoid
is reduced in the direction orthogonal to the constraints but is increased in the direction tangential to the constraints (Figure 5). The modified CMA-ES(λ, μ) algorithm for constraint handling is
detailed in Algorithm 3.
Figure 5.
Evolution of the covariance matrix due to the constraint violation, blue = feasible, red = unfeasible candidates.
Algorithm 3 Proposed modified CMA-ES(λ, μ) for constraint handling
1) Initialize the covariance matrix C^(0) = I, the step size σ^(0) and the selection parameters [10]
2) Initialize the mean vector m^(0) to a random candidate, k ← 0
while CMA-ES convergence criterion is not reached do
3-1) Generate λ new offspring candidates according to: $z t ( k + 1 ) ∼ m ( k ) + σ ( k ) N ( 0 , C ( k ) ) , t ∈ { 1 , … , λ }$
3-2) Evaluate candidates and sort them based on the objective function
if all the μ best candidates are infeasible then
Modify the covariance matrix according to Algorithm 2. Return to step 3.1)
If all the μ best candidates are feasible
Determine the mean vector given the weightings of the μ best candidates: $m ( k + 1 ) = ∑ i = 1 μ w i z ( i ) ( k + 1 )$
Update covariance matrix C^(k + 1) according to [10] end if
If at least one of the μ best candidates is infeasible and at least one is feasible
Modify the covariance matrix according to Algorithm 2.
Use the feasible candidates to determine the mean vector m^(k + 1)
Use the feasible candidates to update covariance matrix C^(k + 1) according to [10] end if
end if
3-3) Update the step size σ^(k + 1) according to [10], k ← k + 1
end while
4) return best candidate $z best$
The evolution of the ellipsoid between the generations (k) and (k + 1) if one of the μ best candidates violates a constraint is illustrated in Figure 5. The modification of C^(k) allows homothetic
transformations in order to avoid to generate candidates in the non feasible zone.
If the mean vector m^(k) after the combination process is not feasible, instead of reducing the covariance matrix, the ellipsoid hypervolume is increased in order to generate candidates in the
feasible zone. Therefore, the ellipsoid hypervolume is increased according to:$S ( k ) ← P D ( k ) 2 P T + γ P diag ( v p m 1 ( k ) , … , v p m n ( k ) ) 2 P T$(14)
The mean vector is displaced to the best feasible candidate generated at the next generation (Figure 6).
Figure 6.
Modification of the covariance matrix due to the mean vector constraint violation, blue = feasible, red = unfeasible candidates.
The modified CMA-ES(λ, μ) allows to take into account the constraints without degrading the objective function by penalization and avoids to tune the penalization parameters. Moreover, the proposed
algorithm relies on the same update and selection mechanisms of the original CMA-ES(λ, μ) adapted for constraint handling and it keeps the invariance and unbiased design principles of CMA-ES(λ, μ) [
10]. In the next sections, the proposed algorithm is tested on a benchmark of analytic functions and on the design of a two stage rocket in order to evaluate its performances.
4 Benchmark on analytic optimization problems
The proposed modified CMA-ES(λ, μ) is tested and compared to a penalized version of CMA-ES(λ, μ) with a constant penalization function, to the death penalty applied to CMA-ES(λ, μ) and to the
modified (1+1)-CMA-ES on a benchmark of three analytic functions. The benchmark consists of a modified Six Hump Camel problem in 2 dimensions, the G04 optimization problem [8] in 5 dimensions and a
modified Rosenbrock problem in 20 dimensions. These optimization problems are used in order to evaluate the proposed algorithm for different design space dimensions and different types of and numbers
of inequality constraints (linear, non linear). In the following, the benchmark problems are introduced with the results. A discussion and a synthesis of the results for all the tests are provided in
Section 4.4.
In the three problem formulations, the expected value is computed by Crude Monte Carlo method (CMC). A sample of 1000 points is used to estimate the expected value of the objective function. For each
method (Modified CMA-ES(λ, μ), Death Penalty CMA-ES(λ, μ), Penalization CMA-ES(λ, μ), Modified (1+1)-CMA-ES) the optimization is repeated 50 times. The initialization is chosen randomly in the design
space and the same initialization and the same random number seed are used for the four optimization algorithms. The same stopping criterion is used for all the algorithms: the distance in the design
space between the mean vector and the best point found || m^(k) − z[best] ||^2 < 10^−3 must be lower than a tolerance 20 times in a row.
4.1 Modified Six Hump Camel problem
A modified version of the Six Hump Camel problem is used in order to introduce uncertainty and three inequality constraints. The formulation of the problem is the following:$Min E [ f 6 - hump ( z 1
, z 2 ) + f 6 - hump ( z 1 cos ( U ) + z 2 sin ( U ) , - z 1 sin ( U ) + z 2 cos ( U ) ) ]$(15)
$st g 1 ( z 1 , z 2 ) = z 1 + z 2 / 4 - 0.52 ≤ 0$(16)
$g 2 ( z 1 , z 2 ) = z 1 + 0.01 z 2 - 0.7 + 0.30 cos ( 60 z 2 2 / 6 ) ≤ 0$(17)
$g 3 ( z 1 , z 2 ) = z 1 - z 2 / 4 - 0.45 ≤ 0$(18)
$z min ≤ z ≤ z max$(19)with z ∈ [−3, 3] × [−2, 2], $f 6 - hump ( z 1 , z 2 ) = ( 4 - 2.1 z 1 2 + z 1 4 / 3 ) + z 1 z 2 + ( 4 z 2 2 - 4 ) z 2 2$ and U a random variable distributed according to a
normal distribution $U ∼ N ( 0,0.05 )$.
Representations of the function and the constraints are provided in Figure 7. The problem has one local optimum and one global optimum. The results are presented in Table 1 and the convergence curves
for one optimization are given in Figure 8.
Figure 7.
Modified Six Hump Camel function and constraints.
Figure 8.
Convergence curves of the Six Hump Camel problem in 2 dimensions, based on one optimization run.
Table 1.
Results of modified Six Hump Camel problem. Average over 50 optimizations (in parenthesis the Relative Standard Deviation (RSD) – $σ / E$).
4.2 G04 optimization problem [8]
The G04 optimization problem involves 6 inequality constraints and is defined as following:$Min G 04 ( z ) = 5.3578547 × z 3 2 + 0.8356891 × z 1 × z 5 + 37.293239 × z 1 - 40792.141$(20)
$wrt z = [ z 1 , z 2 , z 3 , z 4 , z 5 ]$
$st g 1 ( z ) = u ( z ) - 92 ≤ 0$(21)
$g 2 ( z ) = - u ( z ) ≤ 0$(22)
$g 3 ( z ) = v ( z ) - 110 ≤ 0$(23)
$g 4 ( z ) = - v ( z ) + 90 ≤ 0$(24)
$g 5 ( z ) = w ( z ) - 25 ≤ 0$(25)
$g 6 ( z ) = - w ( z ) + 20 ≤ 0$(26)
$z min ≤ z ≤ z max$(27)with z ∈ ℝ^5, z[min] = [78, 33, 27, 27, 27], z[max] = [102, 45, 45, 45, 45] and:$u ( z ) = 85.334407 + 0.0056858 × z 2 × z 5 + 0.0006262 × z 1 × z 4 - 0.0022053 × z 3 × z 5$
$v ( z ) = 80.51249 + 0.0071317 × z 2 × z 5 + 0.0029955 × z 1 × z 2 + 0.0021813 × z 3 2$(29)
$w ( z ) = 9.300961 + 0.0047026 × z 3 × z 5 + 0.0012547 × z 1 × z 3 + 0.0019085 × z 3 × z 4$(30)
The results are presented in Table 2 and the convergence curves for one optimization are given in Figure 9.
Figure 9.
Convergence curves of the G04 optimization problem, based on one optimization run.
Table 2.
Results of G04 optimization problem. Average over 50 optimizations (in parenthesis the RSD – $σ / E$).
4.3 Modified Rosenbrock problem
The Rosenbrock optimization problem has been modified in order to incorporate uncertainty and an inequality constraint (Figure 10). The problem is formulated as following:$Min E [ 100 ∑ i = 1 n - 1 (
z i + 1 - z i 2 ) 2 + ∑ i = 1 n - 1 ( 1 - z i ) 2 + U ]$(31)
$st g ( z ) = 2 - ∏ i = 1 n z i ≤ 0$(32)with n = 20, z ∈ ℝ^20 and U a random variable distributed according to $U ∼ U ( - 0.1 , 0.0 )$ a uniform distribution.
Figure 10.
Modified Rosenbrock function and the constraints in 2 dimensions.
The results are presented in Table 3 and the convergence curves for one optimization are given in Figure 11.
Figure 11.
Convergence curves of the modified Rosenbrock problem in 20 dimensions, based on one optimization run.
Table 3.
Results of constrained Rosenbrock problem. Average over 50 optimizations (in parenthesis the RSD – $σ / E$).
4.4 Result and synthesis
The analytic test cases involve different dimensions (2, 5 and 20) and different number of constraints (1, 3 and 6) in order to evaluate the efficiency of the proposed modified CMA-ES(λ, μ) on
various optimization problems. A qualitative synthesis of the obtained results is given in Figure 12. For all the three criteria (number of evaluations, robustness to initialization and value of the
optimum), the lower value the better the quality of the method for the given criterion.
Figure 12.
Qualitative results obtained for the different test cases.
The Six Hump Camel problem has one local optimum and one global optimum. All the optimization algorithms converge either to the local or the global optimum. It illustrates the robustness property of
the algorithms with respect to the initialization (relative standard deviation ~0.85% for all the algorithms). The found optima are all feasible. Modified (1+1)-CMA-ES converges in 48% of the
optimization runs to the global optimum and the proposed modified CMA-ES (λ, μ) in 37% of the cases. The penalization and the death penality approaches converge only in 33% and 22% of the
optimization runs to the global optimum. The number of calls to the objective function and the constraints is in increasing order: Modified (1+1)-CMA-ES (781), modified CMA-ES(λ, μ) (1211), Death
Penalty CMA-ES(λ, μ) (1395) and Penalization CMA-ES(λ, μ) (1618). Modified (1+1)-CMA-ES is more efficient in this test case due to the low dimension and the simplicity of the optimization problem.
The proposed modified CMA-ES(λ, μ) provides better results than the penalization and the death penalty approaches.
In the G04 problem, only the proposed modified CMA-ES(λ, μ) and the modified (1+1)-CMA-ES converge to the global minimum (with sufficient robustness with respect to the initialization). The number of
calls to the objective function and the constraints is lower in the proposed algorithm (1618) compared to modified (1+1)-CMA-ES (7048) and the relative standard deviation is lower in the proposed
approach. Moreover, the proposed approach converges efficiently to the global optimum. The Death Penalty and the penalization approaches do not succeed to reach the global optimum and are not robust
to the initialization.
In the modified Rosenbrock problem, only the proposed modified CMA-ES(λ, μ) reaches the global optimum (with sufficient robustness (RSD: 0.12%) to the initialization). All the other algorithms are
not robust to the initialization and do not converge to the global optimum. The number of calls to the objective function and the constraints is larger for the proposed approach (12 798) compared the
other algorithms.
Consequently, from the benchmark, in small dimensions (<8), the modified (1+1)-CMA-ES provides good results in terms of convergence to the global optimum and robustness with respect to the
initialization, however, as expected, in large dimensions, it presents issues to converge to the global optimum. The proposed modified CMA-ES(λ, μ) succeeds in small and large dimensions to find the
global optimum. Moreover, this algorithm appears as robust to the initialization. In the next section, the proposed algorithm is used to design a two stage solid rocket and is compared to the
existing CMA-ES based optimization algorithms.
5 Two stage solid propulsion rocket design
A multidisciplinary design problem consisting in maximizing the propulsive speed increment ΔV provided by a two stage rocket under geometrical and physical feasibility constraints is solved. The
conceptual design models use simplified analysis of a two stage cylindrical solid propellant rocket motor. The multidisciplinary analysis involves four disciplines: the propulsion, the mass and
sizing, the structure and the performance and constraint assessment (Figure 13). At the early design phase, model uncertainties exist and are taken into account. Two uncertainties are considered: the
density of the propellant ρ and the ultimate strength σ for the rocket case material (Figure 14).
Figure 13.
Design Structure Matrix for the two stage solid rocket.
Figure 14.
Convergence curves of the propulsive speed increment for the two stage solid propulsion rocket.
The problem is formulated as follows:$Max E [ Δ V ( z , U ) ]$(33)
$wrt z = [ D t 1 , D s 1 , P c 1 , M p 1 , D t 2 , D s 2 , P c 2 , M p 2 ]$
$st P f [ g 1 ( z , U ) ≥ 0 ] ≤ 1 0 - 2$(34)
$P f [ g 2 ( z , U ) ≥ 0 ] ≤ 1 0 - 2$(35)
$P f [ g 3 ( z , U ) ≥ 0 ] ≤ 1 0 - 2$(36)
$z min ≤ z ≤ z max$(37)with: U = [U[1], U[2]] with $U 1 ∼ N ( 1 , 0.02 )$ the uncertainty of the density of the propellant (ρ) and $U 2 ∼ N ( 1 , 0.05 )$ the uncertainty of the ultimate strength
limit (σ) for the rocket case material. The design variables are described in Table 4. An overview of the disciplines is detailed in the next paragraphs.
Table 4.
Design variables for the two-stage rocket.
Propulsion. The propulsion discipline computes for a given set of propellant characteristics (density ρ, combustion speed, flame temperature, heat capacity ratio), the thrust T, the mass flow rate $m
̇$, the thrust coefficient c[T] and the characteristic velocity c*, under the assumption of constant thrust. The discipline takes nozzle shapes Dt, D[s] and combustion pressure P[c] as inputs. The
used propellant is the Butargols with polybutadiene binder without aluminium additive.
Mass and Sizing. The mass and sizing discipline computes the dry mass m[d] and the geometry of the two stage solid propulsion rocket. The dry mass involves the mass of the rocket case and the mass of
the nozzle and the pyrotechnic igniter. The rocket geometry consists of the initial combustion area, the packaging ratio and the size of the central channel. The overall dimensions (rocket length L =
22 m and diameter D = 1.07 m) are considered as fixed.
Structure. The structure discipline computes the tank walls thickness (t) which is sized under the combustion pressure based on the material characteristics (yield strength, ultimate strength limit)
and rocket geometry. Moreover, it computes the stress in the rocket case.
Performance and constraints. The performance is the propulsive speed increment ΔV and the expected value of ΔV is the objective function to be maximized. CMC based on 1000 samples is used to compute
the expected value of the propulsive speed increment. The three constraints are: g[1](.) which ensures packaging ratio (Propellant volume/Available volume) that has to be inferior to 87%, g[2](.)
which ensures that central channel diameter is 30% greater than the nozzle throat diameter and g[3](.) which ensures that combustion area is greater than the minimum feasible (area of central channel
walls). The probabilities of failure for the three constraints have to be inferior to 1%. The probabilities of failure are computed with a CMC of 10^4 samples, numerically corresponding to a relative
standard error ($σ / E$) of the probability estimation in the order of 5%.
5.1 Results
The optimization for each algorithm is repeated 10 times. All the optimizations start from the same baseline given in Table 4. The same stopping criterion is used for all the optimization algorithms:
the distance in the design space between the mean vector and the best point found || m^(k) − z[best] ‖ < 10^−3 must be under a tolerance 20 times in a row. The algorithms do not converge to the same
optimum. Modified CMA-ES (λ, μ) provides a better optimum in terms of propulsive speed increment: 6234.1 m/s with a better robustness. The proposed algorithm converges on the average in 1127
discipline evaluations. The other optimization algorithms converge in the same order of number of discipline evaluations (~1750). Only four constraints are active at the optimum. The better optimum
found by the proposed approach is essential as it has a better propulsive speed increment which could be used to increase the payload mass (Table 5).
Table 5.
Results of the two stage rocket optimization. Average over 10 optimizations (in parenthesis the RSD – $σ / E$).
6 Conclusion
The design of complex systems often induces a constrained optimization problem under uncertainty. In this paper, an adaptation of CMA-ES(λ, μ) has been proposed in order to efficiently handle the
constraints. The probable research hypervolume engendered by an iso-probability contour of the multivariate normal distribution $N ( 0 , C )$ used to generate the candidate population is modified.
The constraint handling method allows to reduce the semi-principal axes of the iso-probable research ellipsoid in the directions violating the constraints by decreasing the eigenvalues of the
covariance matrix C. The proposed approach has been tested with three analytic optimization problems highlighting the efficiency of the algorithm and the robustness with respect to the
initialization. The proposed method has been used to design a two stage solid propulsion launch vehicle. A better optimum has been found with the proposed approach with respect to the existing CMA-ES
based optimization algorithms resulting in a potential increase in the payload mass.
The work of L. Brevault is part of a CNES/ONERA PhD thesis.
Cite this article as: Chocat R, Brevault L, Balesdent M & Defoort S: Modified Covariance Matrix Adaptation – Evolution Strategy algorithm for constrained optimization under uncertainty, application
to rocket design. Int. J. Simul. Multisci. Des. Optim., 2015, 6, A1.
All Tables
Table 1.
Results of modified Six Hump Camel problem. Average over 50 optimizations (in parenthesis the Relative Standard Deviation (RSD) – $σ / E$).
Table 2.
Results of G04 optimization problem. Average over 50 optimizations (in parenthesis the RSD – $σ / E$).
Table 3.
Results of constrained Rosenbrock problem. Average over 50 optimizations (in parenthesis the RSD – $σ / E$).
Table 4.
Design variables for the two-stage rocket.
Table 5.
Results of the two stage rocket optimization. Average over 10 optimizations (in parenthesis the RSD – $σ / E$).
All Figures
Figure 1.
Three ellipsoids, depicting three different normal distributions, where I is the identity matrix, D is a diagonal matrix, and C is a positive definite covariance matrix. Thin dot lines depict
objective function contour lines.
In the text
Figure 2.
The dot is the parent, the cross is the generated offspring, the solid line circle is characterized by A^(k) defining the ellipsoid delimiting an iso-probability research hypervolume. At the
center, the red ellipsoid represents the update of A^(k+1) in order to take into account the constraint violation by the offspring. On the right figure, the offspring does not violate the
constraint resulting in a standard covariance matrix update.
In the text
Figure 3.
Parametrization of the ellipsoid defining the probable research hypervolume.
In the text
Figure 4.
Violation of the constraint and projection over the eigenvectors, blue = feasible, red = unfeasible candidates.
In the text
Figure 5.
Evolution of the covariance matrix due to the constraint violation, blue = feasible, red = unfeasible candidates.
In the text
Figure 6.
Modification of the covariance matrix due to the mean vector constraint violation, blue = feasible, red = unfeasible candidates.
In the text
Figure 7.
Modified Six Hump Camel function and constraints.
In the text
Figure 8.
Convergence curves of the Six Hump Camel problem in 2 dimensions, based on one optimization run.
In the text
Figure 9.
Convergence curves of the G04 optimization problem, based on one optimization run.
In the text
Figure 10.
Modified Rosenbrock function and the constraints in 2 dimensions.
In the text
Figure 11.
Convergence curves of the modified Rosenbrock problem in 20 dimensions, based on one optimization run.
In the text
Figure 12.
Qualitative results obtained for the different test cases.
In the text
Figure 13.
Design Structure Matrix for the two stage solid rocket.
In the text
Figure 14.
Convergence curves of the propulsive speed increment for the two stage solid propulsion rocket.
In the text
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What is the surface area of the solid created by revolving f(x) = x^2-3x+24 , x in [2,3] around the x axis? | HIX Tutor
What is the surface area of the solid created by revolving #f(x) = x^2-3x+24 , x in [2,3]# around the x axis?
Answer 1
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Answer 2
To find the surface area of the solid created by revolving ( f(x) = x^2 - 3x + 24 ) on the interval ( x ) in ([2,3]) around the ( x )-axis, we can use the formula for the surface area of a solid of
[ S = 2\pi \int_a^b f(x) \sqrt{1 + (f'(x))^2} , dx ]
First, we need to find the derivative of ( f(x) ), which is ( f'(x) ).
[ f'(x) = 2x - 3 ]
Next, we need to find ( \sqrt{1 + (f'(x))^2} ).
[ \sqrt{1 + (f'(x))^2} = \sqrt{1 + (2x - 3)^2} ]
[ = \sqrt{1 + 4x^2 - 12x + 9} ]
[ = \sqrt{4x^2 - 12x + 10} ]
Now, we can set up the integral:
[ S = 2\pi \int_2^3 (x^2 - 3x + 24) \sqrt{4x^2 - 12x + 10} , dx ]
We integrate this expression over the interval ([2,3]) to find the surface area of the solid of revolution.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Excel Formula for Anxiety Levels
In this guide, we will learn how to create an Excel formula in Python that assigns anxiety levels based on specific conditions. This formula utilizes nested IF statements to determine the appropriate
label for each data point. By following the step-by-step explanation provided below, you will be able to implement this formula in your own Python code.
To begin, let's take a closer look at the structure of the formula. The outermost IF statement checks if the value in column E is 'Yes'. If it is, the formula proceeds to the next IF statement. If
not, it returns an empty string.
The second IF statement checks if the value in column F is greater than 15. If it is, it assigns the label 'Severe Anxiety'. If not, it proceeds to the next IF statement.
The third IF statement checks if the value in column F is greater than 10. If it is, it assigns the label 'Moderate'. If not, it proceeds to the next IF statement.
The fourth IF statement checks if the value in column F is greater than 5. If it is, it assigns the label 'Mild Anxiety'. If not, it proceeds to the next IF statement.
The fifth IF statement checks if the value in column F is greater than 0. If it is, it assigns the label 'Exempt'. If not, it returns an empty string.
By following this nested structure of IF statements, you can assign the appropriate anxiety level to each data point in your Excel sheet.
Let's consider an example to better understand how this formula works. Suppose we have a dataset with values in columns E and F as follows:
E F
Yes 3
No 8
Yes 12
Yes 18
Using the formula, we would obtain the following results:
• In cell G1: 'Exempt' (since E1 is 'Yes' and F1 is greater than 0 but less than 5)
• In cell G2: '' (since E2 is not 'Yes')
• In cell G3: 'Moderate' (since E3 is 'Yes' and F3 is greater than 10 but less than 15)
• In cell G4: 'Severe Anxiety' (since E4 is 'Yes' and F4 is greater than 15)
By implementing this formula in your Python code, you can easily assign anxiety levels based on specific conditions in your Excel sheet. This can be particularly useful for data analysis and
visualization purposes. Now that you have a clear understanding of the formula and its implementation, you can confidently apply it to your own projects.
An Excel formula
=IF(E1="Yes", IF(F1>15, "Severe Anxiety", IF(F1>10, "Moderate", IF(F1>5, "Mild Anxiety", IF(F1>0, "Exempt", "")))), "")
Formula Explanation
This formula uses nested IF statements to assign different labels based on the values in columns E and F.
Step-by-step explanation
1. The outermost IF statement checks if the value in cell E1 is "Yes". If it is, the formula proceeds to the next IF statement. If not, it returns an empty string.
2. The second IF statement checks if the value in cell F1 is greater than 15. If it is, it returns "Severe Anxiety". If not, it proceeds to the next IF statement.
3. The third IF statement checks if the value in cell F1 is greater than 10. If it is, it returns "Moderate". If not, it proceeds to the next IF statement.
4. The fourth IF statement checks if the value in cell F1 is greater than 5. If it is, it returns "Mild Anxiety". If not, it proceeds to the next IF statement.
5. The fifth IF statement checks if the value in cell F1 is greater than 0. If it is, it returns "Exempt". If not, it returns an empty string.
6. If none of the conditions in the nested IF statements are met, the formula returns an empty string.
For example, if we have the following data in columns E and F:
| E | F |
| Yes | 3 |
| No | 8 |
| Yes | 12 |
| Yes | 18 |
The formula =IF(E1="Yes", IF(F1>15, "Severe Anxiety", IF(F1>10, "Moderate", IF(F1>5, "Mild Anxiety", IF(F1>0, "Exempt", "")))), "") would return the following results: - In cell G1: "Exempt" (since
E1 is "Yes" and F1 is greater than 0 but less than 5) - In cell G2: "" (since E2 is not "Yes") - In cell G3: "Moderate" (since E3 is "Yes" and F3 is greater than 10 but less than 15) - In cell G4:
"Severe Anxiety" (since E4 is "Yes" and F4 is greater than 15) | {"url":"https://codepal.ai/excel-formula-generator/query/0uSOP43x/excel-formula-for-anxiety-levels","timestamp":"2024-11-08T02:03:29Z","content_type":"text/html","content_length":"102299","record_id":"<urn:uuid:fec8e427-58ed-4cae-8fc3-1de632bb9902>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00179.warc.gz"} |
An introduction to algorithms both in real life and in math and computer science
Sep 10, 2020 By Team YoungWonks *
What is an algorithm? Even as the transition to the digital world has been accelerated by the lockdown caused in turn by the outbreak of the Coronavirus, it is not surprising to see greater computer
and smartphone penetration across the world. Which brings us to the relevant question: what makes these devices smarter than us humans? In a broad sense, the answer is rather simple: computers can
computer (solve) many more problems than us, including more complex ones and that too at a faster rate. These problems could be calculations, data processing, automated reasoning, and other tasks.
The computing power needed to perform these tasks in turn is fueled by what is called an algorithm.
In simpler terms, an algorithm is a procedure or formula for solving a problem, based on carrying out a sequence of specified actions. So a computer program is essentially an elaborate algorithm. In
mathematics and computer science, an algorithm typically refers to a procedure that solves a recurring problem.
In this blog, we shall take a look at what is an algorithm, its evolution, its role in math and computers, and how they are expressed with the help of examples.
What is an algorithm?
The word algorithm comes from the 9th-century mathematician Muḥammad ibn Mūsā al-Khwārizmī, latinized as Algoritmi. As a widely read mathematician in Europe in the late Middle Ages - he was also
known for his book on algebra - his name Al-Khwarizmi came to be known as algorismus in late medieval Latin, which then became algorism in English, and referred to the decimal number system. It was
only in the late 19th century that the world algorithm caught on and came to mean what it does today in modern English.
As mentioned earlier, an algorithm refers to a procedure for solving a problem; this procedure typically plays out in a finite number of steps, and they frequently involve repetition of an operation.
Interestingly, algorithms are not just used in mathematics and computer science but are also used in daily life. So, an informal definition of algorithm would describe it as a set of rules that
precisely defines a sequence of operations. These rules could refer to computer programs (including programs that do not perform numeric calculations), a prescribed bureaucratic procedure or even a
cook-book recipe. Typically, a program is an algorithm only if it stops eventually.
Let us look at a few examples of algorithms in our daily lives. Take for instance, the task of making tea. Now the algorithm here would be the set of instructions one would follow so as to make this
tea. So, the algorithm would include the following steps:
1. Heat water in a pan.
2. Add to it - even as the water is warming up - crushed ginger.
3. Add tea leaves.
4. Add milk.
5. When it comes to boil, add sugar.
6. Let it simmer for 2 to 3 minutes.
Following the above instructions would produce the desired result and solve our problem/ fulfil the task.
Similarly, let's look at another example. Say, we wish to grow curry leaves in a pot. Now this is a task that can be carried out in different ways. This means there are multiple algorithms for this
One algorithm would be about planting curry leaf seeds in a pot. So, this algorithm would read something like this:
1. Take some soil in a pot.
2. Sow fresh curry leaf seeds in this soil.
3. Keep the seeds/ soil damp but not wet.
4. Make sure the temperature is fairly warm (at least 20 degrees Celsius).
5. Once it has taken root in a few weeks with enough warmth and moisture, make sure it continues to be in a well-drained pot and receives sunlight.
6. Feed it weekly with fertilizer solution and trim the leaves as needed.
Another way of doing this (so, another algorithm in effect) would be about using fresh curry leaves with their stem. The steps here would be something like this:
1. Treat the leaves as a cutting and insert into a soilless potting medium.
2. Take a piece of stem from the tree that is fairly long and has several leaves.
3. Remove the bottom 1 inch of leaves and immerse the bare stem into the medium.
4. Water it thoroughly.
5. Once it has taken root in around three weeks with enough warmth and moisture, plant the tree in a well-drained pot with good potting mix and keep it in a sunny area.
6. Feed it weekly with fertilizer solution and trim the leaves as needed.
Algorithms in Mathematics and Computer Science
The definition of algorithm in a formal context, however, alludes to its role in the way data is processed. In computer systems, an algorithm is an instance of logic written in software-by-software
developers and meant for the intended computer(s) to produce output from the given (sometimes zero) input.
So to get a computer to do a task, we need to write a computer program. Doing so means telling the computer, step by step, exactly what we want it to do. The computer then runs/ reads the program and
executes the commands shared in it by following each step mechanically so as to complete the said task. This series of related commands or steps is a computer algorithm.
It is important to note that algorithms are a finite sequence of well-defined, computer-implementable problem-solving instructions and that thus there is no room for ambiguity. Starting from an
initial state and sometimes initial input, the algorithm spells out instructions that cover a finite number of well-defined successive states, eventually yielding output and terminating at a final
ending state.
Now let us turn to a few examples of algorithms in basic math.
Take for instance, a problem where one has to figure out if a given number (say 7) is odd or even.
Here the algorithm would comprise the following steps:
1. Divide the number by 2. So here we divide 7 by 2.
2. Check for the remainder. If the remainder is 0, the number is even and if the remainder is not zero, the number is odd. Here the remainder would be 1. Hence the number 7 is an odd number.
Similarly, let us look at yet another example. Say the problem here is figuring out if the number 11 is a prime number. Now a prime number is one that is divisible only by 1 and itself.
So here the algorithm for this problem would have these steps.
1. Divide the number 11 by all numbers between 2 to 10 (since all numbers are divisible by 1 and themselves).
2. So first divide 11 by 2 and check for the remainder.
3. Then divide 11 by 3 and again check for remainder.
4. Continue to divide 11 by other numbers 4, 5 and so on till you divide 11 by the number 10. Check for the reminder after each division.
5. If the remainder after any of these divisions is 0, the number is not a prime number. If none of the remainders are 0, the number is a prime number. In this case, none of the remainders are zero.
Hence, 11 is a prime number.
Types of algorithms
Just as in real life, in computer science and math as well, there are many types of algorithms available. In other words, often there are many ways - with different steps - of solving a given
problem. All of these algorithms of course, need input to deliver a meaningful output.
Broadly speaking, algorithms, distinguished by their key features and functionalities, can be classified into six categories. Let's look at them here.
Greedy algorithm
A greedy algorithm is a type of algorithm that is typically used for solving optimization problems. So whenever one wishes to extract the maximum in minimum time or with minimum resources, such an
algorithm is employed.
Let us look at an example. Say person A is a reseller who has a bag that can carry a maximum weight of 20 pounds. Person A has been tasked with going to a warehouse and filling the bag to capacity in
such a way as to maximize the profit upon selling those items. What items should A pick at the warehouse in order to maximize the eventual revenue/ profit? Here A would follow a series of steps
(i.e., an algorithm) before arriving at a decision. Person A would likely do the following:
1. Look for the most expensive items that may also give him/ her a high markup
2. Check their size, volume and weight to evaluate how many such items can be accommodated in the bag.
3. Next look for the most in-demand items
4. Check their size, volume and weight to evaluate how many such items can be accommodated in the bag.
5. Consider all of the above then pick out the items.
In other words, person A would use the greedy algorithm here to get optimal solutions/ results. Here the optimal result would be picking out items that are not too big or heavy so as to be able to
fit in the bag and at the same time, are fairly in demand and have a decent markup thus translating into higher revenues and profits.
Dynamic Programming algorithm
A dynamic programming algorithm works by remembering the results of a previous run and using them to arrive at new results. Such an algorithm solves complex problems by breaking it into multiple
simple subproblems, solving them one by one and storing them for future reference and use.
A common example here would be finding a number in the Fibonacci series. The Fibonacci series has numbers that are the sum of the previous two numbers. So, if one was asked to share the fifth number
in the Fibonacci series, one would arrive at the number 5 since the series would start as 1, 1, 2, 3 and then 5. Now if one were asked to calculate the seventh number in the series, the algorithm
would typically have one build on the work done so far and take it forward. So, in effect, one has remembered and used the results from the previous problem and deployed them to solve the current
problem, thus arriving at the seventh number in the series, which is 13.
Divide and Conquer algorithm
Another effective method of solving many problems, here, as the name suggests, one divides the steps, aka the algorithm into two parts. In the first part, the problem is broken into smaller
subproblems of the same type. The second part is where the smaller problems are solved and then their solutions are considered together (combined) to produce the final solution of the problem.
An example here would be a scenario where one has to find a student with a certain roll number - say 63 - in a school playground gathering. Now one way to do so would be to go about inquiring each
child as to his/ her roll number. This, however, is not the quickest method.
A better way or algorithm would be one that goes like this:
1. Have the students line up in ascending or descending order of their respective roll numbers.
2. Split them into smaller groups - say of 50 students each - according to their roll numbers.
3. Find the student group with roll numbers 51 to 100.
4. Since 63 is less than the halfway (75) mark in this mark, go about inquiring in the first half of the group.
5. Continue to enquire till you reach student with roll number 63.
Recursive algorithm
Recursive algorithm is one which involves repetition of steps till the problem is solved.
For instance, if one has to arrive at the greatest common factor (GCF) between any two numbers. Let's start with 14 and 18.
The algorithm here would be:
1. Divide 18 by 14.
2. Check the remainder (4).
3. Divide 14 by 4.
4. Check the remainder (2).
5. Divide 4 by 2.
6. Check the remainder (0).
7. Since the 0 remainder was arrived at upon division by the number 2, the GCF here would be 2.
Similarly, if one had to find the GCF between 12 and 16, one would go through similar steps but with the new inputs so as to arrive at the new result.
So the algorithm here would be:
1. Divide 16 by 12.
2. Check the remainder (4).
3. Divide 12 by 4.
4. Check remainder (0).
5. Since the 0 remainder was arrived at upon division by the number 2, the GCF here would be 4.
As shown above, the process remains the same and is thus repeated, which makes this a recursive algorithm.
Brute Force algorithm
A brute force algorithm involves blind iteration of all possible solutions to arrive at one or more solutions. A simple example of a brute force algorithm would be trying to open a safe. Without any
knowledge of the combination that can open the safe, the only way forward would be trying all possible combinations of numbers to open it. The same would be the case for someone trying to get access
to another person's email account. Trial and error would be the method employed here; in other words, the solution lies in applying brute force, and hence the name of the algorithm.
Backtracking algorithm
Backtracking algorithm is one that entails finding a solution in an incremental manner. There is often recursion/ repetition involved and attempts are made to solve the problem one part at a time. At
any point, if one is unsuccessful at moving forward, one backtracks aka comes back to start over and find another way of reaching the solution. So backtracking algorithm solves a subproblem and if
and when it fails to solve the problem, the last step is undone, and one starts looking for the solution again from the previous point.
An example would be when one plays chess. Typically, a good chess player contemplates the possible next move by the opponent in response to a certain move made by him/ her. Here each player is
working out scenarios and often backtracking so as to arrive at the best possible way forward.
Algorithms in Python code
Problem: Figure out if a given number is odd or even
Problem: Figure out if a given number is prime or not
Problem: Finding a particular Fibonacci number
Problem: Using binary search to look for a kid with a certain roll number in a big school gathering
Problem: Finding the GCF/ HCF of two numbers
Problem: Using hit and try method to crack a password
Expanding Your Child's Coding Knowledge
To deepen your child's understanding of algorithms and their applications in various programming languages, consider enrolling them in specialized Coding Classes for Kids. These classes not only
provide a strong foundation in algorithmic thinking but also introduce students to different programming environments. For those particularly interested in Python, one of the most versatile and
beginner-friendly programming languages, Python Coding Classes for Kids can be an excellent starting point. Additionally, for children fascinated by the intersection of hardware and software,
Raspberry Pi, Arduino, and Game Development Coding Classes offer hands-on experience in building and programming their own gadgets and games, further solidifying their understanding of algorithms in
real-world applications.
*Contributors: Written by Vidya Prabhu with inputs by Rohit Budania; Lead image by: Leonel Cruz | {"url":"https://www.youngwonks.com/blog/What-is-an-Algorithm-and-What-are-the-Different-Types-of-Algorithms","timestamp":"2024-11-09T10:35:34Z","content_type":"text/html","content_length":"124475","record_id":"<urn:uuid:7f50f624-02ae-46ba-80de-ca3111de4b02>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00676.warc.gz"} |
Category : Word Processors
Archive   : WP5MACRO.ZIP
Filename : RULEROFF.WPM
ÿWPCB ûÿ (
8 Ruler off üF 1 2 4
3 Responses to “Category : Word Processors
Archive   : WP5MACRO.ZIP
Filename : RULEROFF.WPM
1. Very nice! Thank you for this wonderful archive. I wonder why I found it only now. Long live the BBS file archives!
2. This is so awesome! 😀 I’d be cool if you could download an entire archive of this at once, though.
3. But one thing that puzzles me is the “mtswslnkmcjklsdlsbdmMICROSOFT” string. There is an article about it here. It is definitely worth a read: http://www.os2museum.com/wp/mtswslnk/ | {"url":"https://www.pcorner.com/list/WORDP/WP5MACRO.ZIP/RULEROFF.WPM/","timestamp":"2024-11-04T05:57:04Z","content_type":"text/html","content_length":"31901","record_id":"<urn:uuid:bf82f89b-6c4b-4672-ba1a-7487ece6d0c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00652.warc.gz"} |
==================================== List of Research Projects Leong Hon Wai, RAS-Group, SoC, NUS ==================================== Contact: leonghw at comp.nus.edu.sg url: http://
www.comp.nus.edu.sg/~leonghw/ We do both fundamental and applied research in combinatorial optimization. The application areas are in computational biology, transportation logistics, resource
allocation and scheduling, and other optimization problems. We look for students who are algorithmically/mathematically inclined and who can do good software development. We have a very strong team
culture and look for students who are team players. This list contains only *some* of the projects. There are many more interesting projects of a similar nature. Email me (leonghw@comp.nus.edu.sg) if
interested. Title: ------ Randomized Algorithms for Problems from Computational Biology (CB) Short Description: ------------------ Many algorithmic/optimization problems in computation biology have
inherent error in the input data or in the problem interpretation. Traditional algorithms do not handle these error naturally. In this research, we consider the use of randomized algorithms for
solving algorithmic problems in computational biology (such as [1],[2]) since randomization may be one way of dealing with inherent error. Initial candidate problems include phylogenetic tree
constructions [1], motif, pattern finding in DNA sequences, and PPI network analysis. We are looking for students who are strong in design and analysis of algorithms, especially of randomized
algorithms. (No prior computational biology knowledge is required.) References: [1] Seung-Jin Sul and Tiffani L. Williams, "A Randomized Algorithm for Comparing Sets of Phylogenetic Trees,"
Asia-Pacific Bioinformatics Conference (APBC'07), pp. 121- 130,2007. [2] Shuai Cheng Li, Dongbo Bu, Jinbo Xu and Ming Li "Finding Largest Well-Predicted Subset of Protein Structure Models" LNCS-5029,
(2008), pp. 44-55, DOI: 10.1007/978-3-540-69068-9_7 Number of RS needed: 2 Title: ------ Algorithms for peptide sequencing via tandem mass spectrometry (computational biology) Short Description:
------------------ Peptide sequencing is the problem of identification of proteins (determining the sequence of a protein) and recent technological advances in tandem mass spectrometry has made it
the method of choice for high-throughput identification of proteins. (See [1], [2], [3] and others for quick introduction.) Recently, we initiated a project on multi-charge peptide sequencing (MCPS)
that focusses on mass spectra with multiple charge (> 2). We showed in [1] that significant performance gain can be achieved by considered multi-charge peaks during the peptide sequencing process.
There are several possible PhD projects in this area: (1) One project deals with improved algorithms for de novo sequencing for multi-charge mass spectra. (2) Another project deals with more precise
peak annotation in multi-charged mass spectra. (3) Another project deals with the important problem of detection of post translation modifications (PTMs). This problem has important implications in
the study of translational medicine. A collaborative project with an overseas partner is currently being pursued for this project. We are looking for students who are strong in design of algorithms
and mathematical analysis. Currently, two PhD students are working in this area in my research group. This is joint research with Prof. Pevzner at UCSD and Prof Haixu Tang at Indiana University.
References: [1] KetFah Chong, Kang Ning and Hon Wai Leong, "Characterization of Multi Charge Mass Spectra for Peptide Sequencing", APBC-2006, (2006), pp. 109-119. [2] Vineet Bafna, Nathan Edwards:
"On de novo interpretation of tandem mass spectra for peptide identification.", RECOMB 2003: 9-18 [3] Ari Frank and P.A. Pevzner, "De Novo Peptide Sequencing via Probabilistic Network Modelling."
Anal. Chem., 77, pp. 964-973, 2005. Number of RS needed: 2 Title: ------ Research and Development in PGO (Phylogeny from Gene Orders) Short Description: ------------------ Phylogeny from Gene Orders
(PGO) [1] is a software system for constructing and comparing phylogenetic trees build using different techniques and using different evolutionary distances on the same set of gene orders (genomes).
Our PGO system allows researchers to compare different classes of algorithms for building phylogenetic trees. It also allows researchers to compare the phylogenetic trees build from different
evolutionary distances. P GO integrates a number of software packages related to phylogenetic reconstruction from gene order and gene content of genomes. Our system can be access as a web service
where users can upload their gene order data and select the set of programs they wish to run on their data. A similar web service that analyzes DNA sequences (instead of gene orders) is given in [2].
In this project, we perform R&D on enhancements to PGO. Some possible enhancements includes implementation and integration of more distance functions, performing comparative study of different
distance functions, evaluating the practicality of DCR operations as a proxy to actual evolutionary distances, and design and implementation of algorithms for new distance measures. We are looking
for students who are strong in algorithm design and implementation. (No prior computational biology knowledge is required.) References: [1] M. Zhang, F. Hao, and H. W. Leong. Phylogeny from Gene
Order (PGO): an integrated system for comparing phylogenetic reconstruction algorithms, International Conference on Genome Informatics, Dec 2010. [2] A. Dereeper, V. Guignon, G. Blanc, S. Audic, S.
Buffet, F. Chevenet, J. Dufayard, S. Guindon, V. Lefort, M. Lescot, et al. "Phylogeny. fr: robust phylogenetic analysis for the non-specialist." Nucleic Acids Research, 2008. Number of RS needed: 1
Title: ------ Algorithms for the Genome Sorting Problem Short Description: ------------------ In genome sorting, we are given two genomes (given by their gene sequences or gene order), and we want to
find the minimum sequence of operations that transform one genome to the other. Different variations ([1], [2]) of the genome sorting problem arise from the different sets of evolutionary operations
allowed: including insertion, deletion, reversals, translocation, transposition, block interchange, fusion, fission, segmental duplication, chromosome duplication and chromosome deletion. An ongoing
project, GSB (Genome Sorting with Bridges) considers the genome sorting problem in which we allow all known traditional operations (see list given above). By making no assumptions on the two genomes
and allowing all known traditional operations, we hope to make it more convenient to compute evolutionary distances and we also hope that the evolutionary distances computed are closer to the real
distances. We have devised a new algorithm called GSB (Genome Sorting with Bridges) that combines existing techniques with new innovative ideas called T-bridges and X-bridges. This project will
continue this work and will continue implementation of the GSB algorithm and related extensions. We are looking for students who are strong in algorithm design and implementation. (No prior
computational biology knowledge is required.) References: [1] M. Bader. Genome rearrangements with duplications. BMC Bioinformatics, 11(Suppl 1):S27, 2010. [2] F. Hao, J. Luan, and D. Zhu.
Translocation-Deletions Distance Formula for Sorting Genomes. WRI World Congress on Computer Science and Information Engineering, 2009 Number of RS needed: 1 Title: ------ Fragment Assembly using
very short fragments (Computational Biology) Short Description: ------------------ Fragment assembly is an important genome sequencing problem in computational biology. A highly successful approach
to fragment assembly is recently proposed by Pevzner et al in [1] and [2]. This project deals with consider fragment assembly using very short fragments (about 100bp each) produced by new fragment
generation technology. For these short fragments, frequent repeats in DNA sequences causes problems in the assembly. In this project, we aim to seek efficient solution to this problem. We are looking
for students who are strong in design of algorithms and mathematical analysis. Currently, two students are working in related area in my research group. This project is a joint project with Prof
Haixu Tang in Indiana University. References: [1] Pevzner PA, Tang H, Waterman MS., "An Eulerian path approach to DNA fragment assembly." Proc Natl Acad Sci, USA 2001 Aug 14;98(17):9748-53. [2]
Pevzner PA, Tang H, "Fragment Assembly with Double-barreled Data," BioInformatics, Vol 17, Supp 1, (2001), pp. S225-S233. Number of RS needed: 1 Title: ------ Rearrangements in genomes with unequal
content (computational biology) Short Description: ------------------ Whole-genome rearrangement studies are typically separated into two steps: (i) identification of large blocks of sequence shared
by the set of genomes; and (ii) comparison of the respective arrangements of these blocks. When studying many very different genomes simultanously, it becomes difficult to identify large blocks in
step (i) simply because there are limited number of elements common to all genomes. In other words, many of the similarities that are identified by pairwise comparisons are dropped in step (i) when
we restrict to elements common to all genomes. In the past, we have worked on an algorithm to compare the respective arrangements of blocks in multiple genomes (the program is called MGR, Bourque and
Pevzner 2002). The algorithm is a heursitic that seeks the most parsimonious rearrangement scenario that best explains the observed arrangements. MGR relies on a polynomial time algorithm to compute
the pairwise distance between 2 multichromosomal genomes (Pevzner and Hannenhalli 1995, Tesler 2002). Instead of working on a set of blocks common to all genomes, we would like to adapt the algorithm
to work on the different sets of pairwise blocks. Given that MGR already only relies on pairwise comparisons to identify rearrangements implies that this modification is very accessible. The main
challenge is to describe how these pairwise blocks are modified by the rearrangements and how to deal with boundary ambiguities since a rearrangement is defined for different sets of blocks on the
same genome. One approach would be to add virtual markers to account for missing markers in some of the genomes. Not restricting to common blocks will retain much more information and we expect the
recovered scenarios to be more accurate. It will also open new applications to include more genomes, distant genomes and genomes with low quality map or sequence. These developments would be useful
not only in conjunction with MGR but also for any other multiple genome rearrangement study. Number of RS needed: 1 Title: ------ Algorithms for Phylogenetic Tree Reconstructions (in Computational
Biology) Short Description: ------------------ A phylogenetic tree (or evolutionary tree) is a tree representation of the evolutionary history of a set of species. Over the past decade, there has
been intensive reesarch in algorithms for reconstructing phylogenetic tree. A number of different techniques have been used. In this project, we aim to design efficient algorithms for phylogenetic
tree reconstruction. The project starts with an up-to-date survey of existing techniques for phylogenetic tree reconstruction and a comparative study of their relative strengths and weaknesses.
Number of RS: 1 For other related research topics in computational biology, please come by to see me in S16 06-01. (For more projects, see http://www.comp.nus.edu.sg/~leonghw, ---> under Research
Project) Title: ------ Efficient Algorithms for Berth Allocation Planning Problems Short Description: ------------------ The complex operations involved in the management and operation of a
world-class container transshipment port in Singapore are complex and involves many different resources and systems. In this project, we undertake to study the problem of planning of some of these
operations to improve their effectiveness. Initial study with the berth allocation planning system (BAPS) has produced interesting research problems and results. Some examples of these are the berth
partitioning problem, the berth assignment problems. This project aims to extend the current BAPS research with the study of more efficient algorithms for solving these and other related optimization
problems in container port operation. Number of RS: 1 Title: ------ Multi-Agent Approach to Combinatorial Optimization (2 students) Short Description: ------------------ The majority of past
approaches to combinatorial optimization use a centralized decision making framework where the algorithm is aware of all relevant problem parameters. In this project, we study the an interesting new
approach to combinatorial optimization -- the multi-agent approach in which a distributed decision making framework is employed. Such a distributed framework is more flexible and is able to
incorporate new changes and new constraints in the dynamic business environment of today. In our multi-agent approach, we model the entities in the problem using software agents, each of which has
only a restricted picture of the entire state of the system. Thus, the agents cooperate as well as compete for scarce resources while optimizing some global objective functions. The multi-agent
approach has been successfully applied to two logistics problems -- the vehicle routing problem and the inventory routing problem. In this project, we develop a multi-agent algorithm for solving a
scheduling problem that involves multiple entities and scarce, shared resources. The challenge will be to appropriate model the various entities using appropriate agents with associated properties,
believes and operations. We then explore issues such as the influence of local information, and the appropriate distributed decision making framework. This project will also involve the
implementation of the multi-agent approach and benchmarking against the state-of-the-art approaches. Two starting reference: Yizhi Lao, and Hon Wai Leong, "A Multi-Agent Based Approach to the
Inventory Routing Problem", Pacific-Rim Intl. Conf. on Artificial Intelligence (PRICAI-02), (2002), LNAI-2417, pp 345-354. Hon Wai Leong and Ming Liu, "A Multi-Agent Algorithm for Vehicle Routing
Problem with Time Windows", ACM Symposium on Applied Computing (SAC-2006), AIMS Track, (2006), pp. 106-111. Number of RS: 1 For other related research topics in logistics, please come by to see me in
S16 06-01. (For more projects, see http://www.comp.nus.edu.sg/~leonghw, ---> under Research Project) Research Scholars Currently Supervised: --------------------------------------- Ng Hoong Kee, PhD
(Jan 2000 -- ) email: nghoong Topic: Multi-Point Range Query: Algorithms and Applications Chong Ket Fah, PhD (Jan 2002 -- ) email: chongket Topic: Computational Proteomics Ning Kang, PhD (Jul 2002 --
) email: ningkang Topic: Computational Proteomics, SCS related problems Recently Graduated Students: ---------------------------- Tan Jing Song, MSc (Jul 2002 -- Jul 2003) (now doing PhD at U-Penn)
Topic: Efficient Algorithms for Dynamic Route Advisory Problems Li Shuai Cheng, MSc (July 2001 -- Jul 2002) (now doing PhD at Waterloo) Topic: Efficient Algoritms for the Berth Assignment Problem Lao
Yizhi, MSc (July 2000 -- July 2001) Topic: A Multi-Agent Based Approach to the Inventory Routing Problem. Ong Tat Wee, MSc (2000) Topic: A Graph Partitioning Algorithm for the Berth Allocation | {"url":"https://www.comp.nus.edu.sg/~leonghw/grad-proj-details.txt","timestamp":"2024-11-06T05:56:32Z","content_type":"text/plain","content_length":"17249","record_id":"<urn:uuid:a483c4b7-31c1-4d64-8b54-e0d95b85c96e>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00533.warc.gz"} |
Classes of 2D Geometric Shapes - Intermediate Python
Classes of 2D Geometric Shapes
In the realm of 2-dimensional geometrical shapes, polygons form the basics. In this challenge, your task is to create classes representing four types of polygons: Polygon, Triangle, Rectangle, and
A polygon is defined by its number of sides. Therefore, the Polygon class should contain an attribute sides representing the number of sides of the polygon.
The Polygon class also needs a method describe() which prints "A polygon" when called.
Next, create Triangle, Rectangle, and Pentagon classes that inherit from the Polygon class. Each of these classes should override the describe() method to print "A triangle", "A rectangle", and "A
pentagon" respectively, when called. This way, each specific type of polygon will be able to announce what it is when the describe() method is called.
Input Output
polygon, triangle, rectangle, pentagon = Polygon(10), Triangle(), Rectangle(), Pentagon(); polygon.describe(); print('Polygon sides:', A polygon Polygon sides: 10 A triangle Triangle
polygon.sides); triangle.describe(); print('Triangle sides:', triangle.sides); rectangle.describe(); print('Rectangle sides:', rectangle.sides); sides: 3 A rectangle Rectangle sides: 4 A pentagon
To check your solution you need to sign in | {"url":"https://profound.academy/python-mid/classes-of-2d-geometric-shapes-kbov5th7Z7HWqZ7DSdP5","timestamp":"2024-11-05T07:56:28Z","content_type":"text/html","content_length":"156142","record_id":"<urn:uuid:fae5019b-6dfe-481b-8660-ef64af8de1f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00799.warc.gz"} |
Batch conversion from one number system to another
Batch conversion of a list of numbers from one positional number system with the specified base, to another with the specified base.
The calculator on this page provides a tool for converting numbers between different number systems. A number system is a way of representing numbers using symbols or digits. The most commonly used
number system is the decimal system, also known as base-10, which uses 10 symbols, 0-9, to represent numbers.
The positional number system is a type of number system that assigns each digit in a number a value based on its position within the number. The base of the number system determines the number of
symbols used to represent the numbers. For example, in the decimal system, the rightmost digit represents units, the next digit to the left represents tens, the next digit represents hundreds, and so
The calculator on this page allows you to enter a list of numbers, along with the base of the number system the numbers are currently in, and the target base of the number system you want to convert
to. The calculator then performs the conversion, providing the result in the form of the numbers expressed in the target base.
Essentially, this calculator is a version of the Conversion between two positional numeral systems calculator for working with a list of numbers. Suppose you have a column of numbers in Excel in
hexadecimal. Using this calculator, you can get that same column in decimal form by simply copying it into the "Source Numbers" field, and then copying the result back from the "Converted Numbers"
PLANETCALC, Batch conversion from one number system to another | {"url":"https://planetcalc.com/10147/","timestamp":"2024-11-08T07:55:50Z","content_type":"text/html","content_length":"33853","record_id":"<urn:uuid:31baaacf-75d0-4565-b888-d257f3ed06ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00381.warc.gz"} |
1. Preface
KS integration (from Kelvin-Stokes integration) is a software that permits lightcurve modelling in presence of stellar activity, able to account for the photometric effect generated by transiting
planets and stellar spots, resolving also the cases in which these objects appear overlapped with respect to the observer. It allows spot merging and the creation of spot regions.
2. Licence
This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the
License, or any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
You should have received a copy of the GNU General Public License along with this program. If not, see
Contributed by M. Montalto, G. Boué, M. Oshagh, I. Boisse, G. Bruno, N. C. Santos
Copyright M. Montalto & Centro de Astrofísica, Universidade do Porto (CAUP)
3. Installation
The software is written in Fortran 95 and it has been tested under a Linux machine using the GNU Fortran compiler gfortran. The following instructions hold under the assumption that you have this
compiler already installed in your machine. To proceed first unpack the tarfile:
tar -xvf KSint.tar.gz
this will create a directory KSint. Enter this directory and to create the executable type:
the executable KSint will be therefore found under the same directory. This program is a standalone program that works together with the input file 'inputs.txt' also located in the same directory as
explained in the next Section.
In case you prefer to supply your own inputs in a different form or just want to incorporate this software in your own programs the main program under the SRC directory (KSint.f95) can be inspected.
The subroutine that performs all the calculations is named KSflux.
4. Input file
Table 1
lists input parameters provided to the program through the input file 'inputs.txt' in their order of appearance. Input parameters are subdivided in four groups: SIMPARAM contains parameters relative
to the simulation, STARPARAM parameters relative to the star, SPOTS and PLANETS relative to spots and planets respectively as indicated by their name. For these two latter groups the parameters
relative to different planets (spots) are provided as a comma separated list where it is only the number of planets and spots declared by 'nplanets' and 'nspots' that will be considered during
program execution.
Table 1
-Input parameters located in the file 'inputs.txt'.
□ Name
□ Type
□ Units
□ Bounds
□ Meaning
• SIMPARAM
□ timestep
□ REAL
□ [days]
□ >0.
□ Timestep of each iteration
□ nstep
□ INTEGER
□ -
□ ≥1
□ Total number of iterations
□ nplanets
□ INTEGER
□ -
□ ≥0
□ Number of planets
□ nspots
□ INTEGER
□ -
□ ≥0
□ Number of spots
• STARPARAM
□ prot
□ REAL
□ [days]
□ >0.
□ Stellar rotation period
□ REAL
□ [degrees]
□ [0.;180.]
□ Stellar axis inclination
□ REAL
□ [degrees]
□ [0.;360.]
□ Position angle of stellar axis
□ rho
□ REAL
□ [gcm^-3]
□ >0.
□ Stellar density
□ c1
□ REAL
□ -
□ [0.;1.]
□ Linear limb darkening coefficient
□ c2
□ REAL
□ -
□ [0.;1.]
□ Quadratic limb darkening coefficient
• SPOTS
□ REAL,REAL,...
□ [degrees]
□ [-90.;90.]
□ Spots latitudes
□ REAL,REAL,...
□ [degrees]
□ [0.;360.]
□ Spots longitudes
□ adim
□ REAL,REAL,...
□ [degrees]
□ ]0.;180.]
□ Spots angular dimensions
□ REAL,REAL,...
□ -
□ ≠0,≤1.
□ Spots contrast ratios
• PLANETS
□ rp
□ REAL,REAL,...
□ -
□ >0.
□ Planets normalized radii
□ porb
□ REAL,REAL,...
□ [days]
□ >0.
□ Planets orbital periods
□ REAL,REAL,...
□ [degrees]
□ [0.;180.]
□ Planets orbital inclinations
□ ecc
□ REAL,REAL,...
□ -
□ [0.;1.[
□ Planets orbital eccentricities
□ REAL,REAL,...
□ [degrees]
□ [0.;360.]
□ Planets arguments of pericenters
□ REAL,REAL,...
□ [degrees]
□ [0.;360.]
□ Planets longitudes of the ascending nodes
□ M
□ REAL,REAL,...
□ [degrees]
□ [0.;360.]
□ Planets initial mean anomalies
5. Reference systems
The angles introduced in
Table 1
for planets and spots are defined in
Figure 1
Figure 2
. The plane xy corresponds to the plane of the sky, while the z-axis is oriented towards the observer.
Figure 1
, the star reference system is defined. The angle between the North polar axis, from which the rotation is seen counter-clockwise, and the line of sight (the arc SP) indicates the inclination angle
of the stellar axis (
parameter). The angle between the positive y-axis and the projection of the stellar axis to the plane of the sky (the arc AB) is defined as the position angle of the stellar axis (
parameter). The latitude of a spot (
) is defined along a stellar parallel from the equator, positive towardsthe North pole. The longitude of a spot is measured along a stellar meridian where the zero longitude meridian is the one
passing through the positive x-axis.
Figure 2
defines the planets reference system. In this case the inclination ofthe orbital plane (
), the argument of periastron (
) and the longitude of the ascending node (
) are indicated in Figure 2 respectively by i, ω and Ω.
6. Overlapping spots
The present version of the code assumes the following rules once two or more spots are overlapped: two or more spots with the same contrast ratio will merge homogeneously (forming a region with the
same contrast ratio of the components), while two of more spots with different contrast ratios will produce a region where the contrast ratio in the overlapping area will be the sum of the contrast
ratio of the components. Note that it is not allowed to overlap two spots if the sum of their contrast ratios is larger than one.
7. Results
The results are stored by default in the file 'lc.dat'. Optionally the program can reproduce the configuration of the system at any given iteration. To enable this feature you should call the program
with the '-v niter' option where niter is the iteration number you want to analyze. If the iteration number is not provided the program will reproduce by default the first iteration configuration.
The result is stored in the file 'arcs.dat'.
The file 'lc.dat' contains three columns which correspond to the time of each iteration, the flux calculated by the program and an error flag which is equal to zero if no errors were detected.
The file 'arcs.dat' is separated in three different sections. The first line provides the starting, final lines in the file relative to the ARCS section, and in the last column provides the number of
objects to which the arcs belong. The second line provides the starting, final lines in the file relative to the stellar GRID section, and the third line gives the starting line of the ROOTS section.
All the coordinates reported below are expressed with respect to the XY observer plane.
The ARCS section provides the coordinates x, y of each point on a given arc, followed by a flag which is one if that point lies on an integrable arc, and by a number which is unique for each object
(one corresponds by default to the stellar border). The GRID section gives the x, y coordinates of each point on a stellar parallel or meridian. The third column is just an incremental number running
along the maximum circles, while the last column is a flag which is equal to one in the case the point belongs to a parallel, two if it belongs to a meridian and zero if it lies on the invisible
hemisphere of the star. The ROOTS section gives the x, y coordinates of the arcs interception points found by the root finding algorithm. A fully integrable object has always a root attributed to the
arc defining its profile, since the arc is integrated between zero and 2π.
Under the directory SM and GNUPLOT simple macros are provided and can be used to visualize the results in 'lc.dat' and 'arcs.dat'.
Additionally, once the '-v niter' option is used an arc table is printed on the screen, showing, for each object, the list of arcs splitting the profile of each object. The starting and final angles
of the arc are reported in the first and second columns respectively. For the case of planets and the case of the stellar border, the angle is counted counter-clockwise from the positive x-axis with
respect to a reference system centered on each object and with the axis oriented like the observer XY reference system. For the case of spots the angle α is reported as defined in Fig. 1 of the
. The third column gives the contrast ratio associated to the arc (this is the factor that multiplies the integral on that arc, resulting from the analysis of the structure), while the fourth column
is a flag indicating if the arc is considered integrable (in this case the flag is equal to one).
8. Example
The following example may be used as a guide to get started using the program. We simulate the case of a planet crossing a naked eye spot. The input parameters are provided in
. The resulting configuration (as seen by the observer) at iteration 833 of the simulation is shown in
Figure 3
. Here the positive x-axis is oriented towards the right while the positive y-axis towards the top. The resulting lightcurve is shown in
Figure 4
Table 2-Input parameters relative to the example
discussed in the text.
• SIMPARAM
• STARPARAM
• SPOTS
• PLANETS
Figure 3
- Planet overlapped with a naked eye spot at iteration 833 of the example.
Figure 4
- This Figure shows a portion of the resulting lightcurve where the planet overlaps the spots.
9. Credits
The KS technique applied to the case of planets and spots is discussed in the two following papers:
Pál, A. 2012, MNRAS, 420, 1630 Montalto, M., Boué, G., Oshagh, M., Boisse, I., Bruno, G., Santos, N. C. 2014, MNRAS, 444, 1721
The solution presented in this software incorporates the one on planets presented in Pál, therefore any work where this software is used should mention these two papers. | {"url":"http://eduscisoft.com/KSINT/manual_KSint.php","timestamp":"2024-11-05T11:57:22Z","content_type":"text/html","content_length":"24244","record_id":"<urn:uuid:5a4cf224-4312-4c19-b8ff-374c2b31b765>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00246.warc.gz"} |
(a) Describe in your own words how to solve a linear equation using
Download (a) Describe in your own words how to solve a linear equation using
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(a) Describe in your own words how to solve a linear equation using the equality properties. Demonstrate the process with an
bNext, replace the equal sign in your example with an inequality by using the less than or greater than sign. Then solve the
(c) What similarities do you see in solving equations and inequalities? What differences to you see?
(a) To solve a linear equation, we perform the same mathematical operation on both sides of the equation. We do this
so that we ultimately get only the variable to be solved on the left side and everything else on the right side.
Consider the problem: Solve (5x - 11)/6 = 4
Multiply both sides by 6 to get 5x - 11 = 24
Add 11 to both sides to get 5x = 35
Divide both sides by 5 to get x = 7
The solution is x = 7
(b) Consider the problem: Solve (5x - 11)/6 > 4
Multiply both sides by 6 to get 5x - 11 > 24
Add 11 to both sides to get 5x > 35
Divide both sides by 5 to get x > 7
The solution is {x| x > 7}
(c) It is easy to observe from the solutions of (a) and (b) above that the procedure of solving an equation and an
inequality is the same. But the difference is that while a linear equation has one and only one solution, a linear
inequality has a range of solutions. In our example, the equation (5x - 11)/6 = 4 has just one solution,
x = 7, while the inequality (5x - 11)/6 > 4 has many solutions (any real number greater than 7 is a solution). | {"url":"https://studyres.com/doc/10016327/-a--describe-in-your-own-words-how-to-solve-a-linear-equa...","timestamp":"2024-11-13T15:51:53Z","content_type":"text/html","content_length":"59831","record_id":"<urn:uuid:847c0655-d929-43ee-9dc6-d7bff406b4f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00223.warc.gz"} |