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Understanding Mathematical Functions: What Is The Function Of Hundred
Introduction to Mathematical Functions and the Significance of Understanding Them
Mathematical functions are essential tools in various fields such as science, engineering, finance, and computer science. They help us understand the relationship between different variables and make
predictions based on data analysis. Understanding mathematical functions can provide valuable insights and improve decision-making processes. In this blog post, we will delve into the concept of
'function of hundred' within mathematical functions and explore its applications.
A Definition of mathematical functions and their importance in various fields
Mathematical functions can be defined as a relation between a set of inputs (domain) and a set of outputs (range), where each input is related to exactly one output. Functions are represented using
mathematical expressions, equations, or graphs, and they play a crucial role in modeling real-world phenomena. In fields such as physics, economics, and biology, functions help us analyze data, make
predictions, and solve complex problems.
Overview of the concept of 'function of hundred' within mathematical functions
The concept of 'function of hundred' refers to a specific mathematical function where the input is always the number 100. In other words, the function operates on the constant value 100 and produces
a corresponding output based on a defined rule or equation. Understanding how this function works can provide insights into mathematical operations and help us grasp the fundamental principles of
The objective of the blog post: to demystify the 'function of hundred' and its applications
The main goal of this blog post is to demystify the concept of 'function of hundred' and showcase its applications in various fields. By explaining the principles behind this mathematical function
and illustrating its relevance, readers will gain a deeper understanding of functions and their significance in problem-solving and decision-making processes.
Key Takeaways
• Definition of a mathematical function
• Understanding the function of hundred
• Examples of how hundred is used in functions
• Importance of understanding mathematical functions
• Practical applications of functions in everyday life
Understanding the 'Function of Hundred'
Mathematical functions play a crucial role in the field of mathematics, providing a way to relate input values to output values. One such function that holds significance is the 'function of
hundred.' In this chapter, we will delve into the definition, mathematical notation, and variations of this function.
A Definition and theoretical explanation
The 'function of hundred' is a mathematical operation that involves multiplying a given number by one hundred. In simpler terms, it is a function that scales a number by a factor of one hundred. For
example, if we apply the 'function of hundred' to the number 5, the result would be 500.
This function can be represented by the following mathematical expression:
f(x) = 100x
Where f(x) represents the function of hundred and x is the input value. By substituting different values of x into the equation, we can calculate the corresponding output values after applying the
function of hundred.
B The mathematical notation and representation of the 'function of hundred'
In mathematical notation, the 'function of hundred' can be denoted using the symbol for multiplication, which is typically represented by an 'x' or a dot. The function can also be written as 100 * x
to indicate the multiplication operation.
Graphically, the 'function of hundred' would appear as a straight line with a slope of 100, indicating the rate at which the input values are scaled up to produce the output values. This linear
relationship is a characteristic feature of the function of hundred.
C Variations and similar functions in mathematical operations
While the 'function of hundred' is a straightforward operation that involves multiplying by one hundred, there are variations and similar functions that exist in mathematical operations. One such
variation is the 'function of ten,' which scales a number by a factor of ten.
Additionally, functions such as the 'function of thousand' or 'function of million' extend the concept of scaling by multiples of one thousand or one million, respectively. These functions are used
in various mathematical contexts, such as converting units of measurement or dealing with large numbers in scientific notation.
Understanding the 'function of hundred' and its variations provides a foundation for grasping more complex mathematical functions and operations, highlighting the importance of functions in
mathematical analysis and problem-solving.
The Role of the 'Function of Hundred' in Real-life Scenarios
Mathematical functions play a crucial role in various real-life scenarios, helping us make sense of data, solve problems, and make informed decisions. One such function that is commonly used is the
'Function of Hundred,' which involves multiplying or dividing numbers by 100. Let's explore how this function is utilized in different contexts:
A. Financial calculations and interest computations
In the realm of finance, the 'Function of Hundred' is frequently employed to calculate percentages, interest rates, and monetary values. For instance, when determining the interest on a loan or
investment, the interest rate is often expressed as a percentage of the principal amount multiplied by 100. This allows individuals and businesses to assess the profitability of their financial
transactions and make sound financial decisions.
Moreover, when converting between different currencies, the 'Function of Hundred' is used to adjust exchange rates and facilitate international trade. By multiplying or dividing by 100, individuals
can accurately convert the value of one currency into another, enabling seamless transactions across borders.
B. Statistical analyses and percentage calculations
In the field of statistics, the 'Function of Hundred' is instrumental in calculating percentages, proportions, and statistical measures. When analyzing data sets or conducting surveys, percentages
are often used to represent the distribution of values or the prevalence of certain characteristics within a population.
By applying the 'Function of Hundred' to these percentages, researchers can compare different data points, track changes over time, and draw meaningful conclusions from their analyses. This allows
them to identify trends, patterns, and correlations that can inform decision-making and drive strategic initiatives.
C. Educational tools and mathematical learning enhancement
For educators and students, the 'Function of Hundred' serves as a valuable tool for enhancing mathematical learning and understanding. By introducing concepts such as percentages, ratios, and scaling
factors, teachers can help students develop critical thinking skills and problem-solving abilities.
Through interactive exercises, real-world examples, and hands-on activities, students can apply the 'Function of Hundred' to practical situations and gain a deeper appreciation for the role of
mathematics in everyday life. This not only fosters a greater interest in the subject but also equips students with the skills they need to succeed in academic and professional settings.
Step-by-Step Guide to Calculating the 'Function of Hundred'
Understanding mathematical functions is essential in solving various problems in mathematics. In this guide, we will focus on the 'function of hundred' and provide a step-by-step approach to
calculating its outcome.
A. Identifying the components of the function and its variables
• Function: A function is a rule that assigns to each input value exactly one output value. In the case of the 'function of hundred,' the function involves manipulating the number 100 in some way.
• Variables: In mathematical functions, variables are symbols that represent unknown values. In the 'function of hundred,' the variable could be any number that is used in the calculation.
B. Practical steps in calculating the outcome of the 'function of hundred'
• Step 1: Choose a specific number to use as the variable in the function.
• Step 2: Determine the operation to be performed on the number 100. This could be addition, subtraction, multiplication, division, or any other mathematical operation.
• Step 3: Apply the chosen operation to the number 100 and the variable. For example, if the operation is addition and the variable is 5, the calculation would be 100 + 5 = 105.
• Step 4: Evaluate the result of the operation to determine the outcome of the 'function of hundred.'
C. Troubleshooting common calculation errors and how to avoid them
• Common Error: Misinterpreting the operation to be performed.
• How to Avoid: Double-check the operation before applying it to the numbers.
• Common Error: Incorrectly applying the operation to the numbers.
• How to Avoid: Take your time when performing the calculation and ensure each step is done accurately.
• Common Error: Using the wrong variable in the function.
• How to Avoid: Clearly define the variable before starting the calculation and use it consistently throughout.
Common Misconceptions and Mistakes Regarding the 'Function of Hundred'
When it comes to understanding mathematical functions, it is important to be aware of common misconceptions and mistakes that can arise. In the case of the 'Function of Hundred,' there are several
key areas where individuals may misinterpret or misunderstand its purpose and application. Let's explore some of these misconceptions in more detail:
A Misinterpreting the function’s purpose and application
One common mistake that individuals make when dealing with the 'Function of Hundred' is misinterpreting its purpose and application. The function of hundred simply means multiplying a number by one
hundred. However, some may mistakenly believe that it involves adding one hundred to a number or performing some other operation. This misunderstanding can lead to errors in calculations and a lack
of clarity in mathematical reasoning.
B Overcomplicating the calculation process
Another misconception that can arise when working with the 'Function of Hundred' is overcomplicating the calculation process. Some individuals may try to use complex formulas or methods to multiply a
number by one hundred, when in reality, it is a straightforward operation. By overcomplicating the process, individuals may introduce unnecessary errors and confusion into their calculations.
C Ignoring or misunderstanding the function’s limitations and constraints
It is also important to be aware of the limitations and constraints of the 'Function of Hundred.' One common mistake is ignoring or misunderstanding these constraints, which can lead to incorrect
results. For example, the function of hundred only applies to multiplying a number by one hundred, and attempting to use it for other operations will not yield accurate results. By understanding the
specific constraints of the function, individuals can avoid errors and ensure the accuracy of their calculations.
Advanced Applications of the 'Function of Hundred'
When it comes to mathematical functions, the 'Function of Hundred' holds a significant place in higher-level mathematics and complex equations. Let's delve into the advanced applications of this
function and explore how it is integrated into software, algorithms, and industries relying heavily on precise mathematical computations.
Using the function in complex equations and higher-level mathematics
Mathematical functions play a crucial role in solving complex equations and problems in higher-level mathematics. The 'Function of Hundred' is no exception, as it provides a simple yet powerful tool
for manipulating numbers and variables. By applying this function to equations, mathematicians can simplify calculations and derive meaningful results.
One of the key advantages of the 'Function of Hundred' is its ability to scale numbers by a factor of one hundred. This scaling factor can be particularly useful in scenarios where precision and
accuracy are paramount. For example, in statistical analysis or financial modeling, scaling numbers by a factor of one hundred can help avoid rounding errors and ensure the integrity of calculations.
Integration of the function into software and algorithms
In today's digital age, mathematical functions like the 'Function of Hundred' are seamlessly integrated into software and algorithms to streamline processes and enhance efficiency. Software
developers leverage the power of this function to perform quick calculations, manipulate data, and generate accurate results.
Algorithms that incorporate the 'Function of Hundred' can be found in various applications, ranging from scientific research to financial forecasting. By harnessing the capabilities of this function,
programmers can optimize performance, reduce computational complexity, and improve the overall accuracy of their algorithms.
The impact on industries relying heavily on precise mathematical computations
Industries that rely heavily on precise mathematical computations, such as finance, engineering, and data science, benefit greatly from the 'Function of Hundred' and similar mathematical functions.
These industries demand accuracy, reliability, and speed in their calculations, making the integration of such functions essential.
By incorporating the 'Function of Hundred' into their mathematical models and algorithms, professionals in these industries can make informed decisions, mitigate risks, and drive innovation. Whether
it's analyzing market trends, designing complex structures, or optimizing processes, the 'Function of Hundred' plays a vital role in ensuring the success and efficiency of these industries.
Conclusion and Best Practices for Utilizing the 'Function of Hundred'
A Recap of the significance and applications of the 'function of hundred'
Understanding the 'Function of Hundred'
The 'function of hundred' is a mathematical concept that involves multiplying a given number by one hundred. This operation results in shifting the decimal point two places to the right, effectively
increasing the number by a factor of one hundred.
Applications of the 'Function of Hundred'
The 'function of hundred' is commonly used in various fields such as finance, science, and engineering. In finance, it is used to calculate interest rates, percentages, and currency conversions. In
science, it is used to express measurements in scientific notation. In engineering, it is used to scale values for analysis and design purposes.
Best practices in applying the function accurately in various scenarios
Accuracy is Key
When applying the 'function of hundred,' it is crucial to ensure accuracy in calculations. Double-checking the decimal placement and the number of zeros is essential to avoid errors in the final
Understanding Context
It is important to understand the context in which the 'function of hundred' is being used. Different scenarios may require different interpretations of the operation, so it is essential to consider
the specific requirements of the problem at hand.
Utilizing Technology
Utilizing calculators or spreadsheet software can help streamline the process of applying the 'function of hundred.' These tools can assist in performing calculations quickly and accurately, saving
time and reducing the risk of errors.
Encouragement for further exploration and understanding of mathematical functions beyond the 'function of hundred'
Exploring Mathematical Functions
While the 'function of hundred' is a fundamental concept in mathematics, there are numerous other mathematical functions that offer unique insights and applications. Exploring functions such as
exponentiation, logarithms, and trigonometric functions can deepen your understanding of mathematical principles and their real-world implications.
Continuous Learning
Mathematics is a vast and ever-evolving field, with new functions and theories being developed regularly. By engaging in continuous learning and exploration, you can expand your knowledge and skills
in mathematics, opening up new opportunities for growth and discovery. | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-function-hundred","timestamp":"2024-11-09T04:41:51Z","content_type":"text/html","content_length":"226547","record_id":"<urn:uuid:b5a2c66e-3915-4092-8ccc-05f21df0bf10>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00860.warc.gz"} |
Schmidt camera
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◄ 10.2.1.3. Honders camera ▐ 10.2.2.1. Schmidt camera: aberrations ►
10.2.2. Full-aperture Schmidt corrector: Schmidt camera
The simplest arrangement using full-aperture corrector is a camera, with the only other optical element required being a single concave mirror. By far the most popular arrangement is the Schmidt
Back in 1930, Estonian-born optician Bernhard Schmidt succeeded in designing and making full-aperture corrector for spherical mirror. It resulted in a highly corrected optical system, known as
Schmidt camera. Somewhat earlier, in 1924, Finnish astronomer Y. Vaisala, described similar arrangement, so this type of camera is sometimes called Schmidt-Vaisala (usually when incorporating
field-flattener). Its concept is based on the unique property of spherical mirror with the aperture stop at the center of curvature to be free from off-axis aberrations. The only image aberration
remaining is spherical, and it can be cancelled by appropriately figured lens corrector, placed at the mirror center of curvature. The lens element - called Schmidt corrector - has very shallow
aspheric curve calculated to give to the incoming wavefront just the needed amount of deformation to result in spherical reflected wavefront (FIG. 167). The only significant aberration induced by the
Schmidt corrector is its corrective spherical aberration, resulting in a system free of four primary aberrations: spherical, coma, astigmatism and distortion. The only remaining aberration is field
FIGURE 167: Schmidt camera (top) is a simple arrangement with the Schmidt corrector at the center of curvature of spherical mirror. Image surface is a curved Petzval surface, concentric with the
mirror surface, thus of Ri=R/2 curvature radius. With the aperture stop at the center of curvature, but w/o corrector, the only remaining point-image aberration of the mirror is spherical aberration,
causing reflected wavefront (WF) to deviate from spherical by bowing inward excessively toward the edges (spherical undercorrection). It results in the longitudinal aberration, with rays toward outer
zones focusing increasingly closer to the mirror. Marginal point of the actual wavefront belongs to a sphere centered at the marginal focus, its paraxial points to a sphere centered at the paraxial
focus, and its 0.707 zone point to a sphere centered at the mid-point in between the two. Axial separation between paraxial and marginal focus equals longitudinal defocus. For ease of calculation, it
is normalized to 2, zero being at the paraxial focus, and 2 at the marginal focus.
The role of the Schmidt corrector is to modify the incident flat wavefront, so that after reflection from the mirror it becomes spherical, directing all rays to a single point. In effect, the lens
compensates for the optical path difference created at the mirror surface, and the corrected wavefront coincides with the corresponding reference sphere. For that, the corrector's profile has to be
the inverse of the profile of the aberrated wavefront (bottom left, red), modified by the light speed differential in given mediums (usually air and glass), i.e. deeper by a factor 1/(n'-n), n', n
being the the refractive index of the exit and entry medium, respectively. For air and common crown, the profile is about twice deeper than the aberrated wavefront, and for mirror surface it is only
half as deep.
Ray paths from every point at this modified wavefront bring points in phase at the sphere centered at the corrected focus, determined by the shape of the corrector profile. Shown are four out of many
possible shapes of the Schmidt corrector surface (rear side), greatly exaggerated (bottom left, blue). From left to right, neutral zone (NZ) - the zero slope zone with zero refraction and zero
relative wavefront retardation - is at ρ=1, √0.75, √0.5 and 0 of the corrector aperture radius normalized to ρmax=1.
Relative depth of the corrector's curve, ρ4- Λρ2, as well as its shape - varies with its focus parameter Λ (0≤Λ≤2), which determines the corrected focus location within longitudinal mirror defocus
normalized to 2 (zero being at the paraxial focus, and 2 at the marginal focus, with the value of ρ being the normalized zonal height (0≤ρ≤1). The value of ρ giving the maximum relative depth equals
the relative height of the neutral zone: 0, 0.50.5, 0.750.5 and 1, for the corrected focus coinciding with paraxial, best, smallest blur and marginal focus location, respectively. The value of focus
parameter Λ and the position of neutral zone NZ in terms of corrector's normalized radius are related as NZ=(Λ/2)1/2ρmax.
Expectedly, with the Schmidt corrector for primary spherical aberration, there is a direct connection between the Schmidt curve and parabolizing. The most efficient mirror parabolizing method is
working the center and the edges of a sphere the most, gradually reducing glass removal to a minimum at the 0.707 zone. This surface modification causes relative advance of the wavefront that
culminates at the 0.707 zone, and diminishes to zero at the edge and the center, resulting in a corrected, spherical shape of the wavefront. This same wavefront modification is accomplished by
placing the neutral zone at 0.707 radius of the Schmidt corrector (in fact, the curve of change of a spherical surface in parabolizing is of the same type as the curve polished into a Schmidt
corrector, only shallower). Consequently, for a given mirror, the theoretical maximum thickness of glass needed to be removed from mirror center and the edge, when parabolizing, is smaller by a
factor of (n-1)/2 from the (maximum) Schmidt corrector depth at the 0.707 zone (when corrected for primary spherical alone; adding higher order terms makes corrector slightly deeper, with the edge
slightly raised vs. center). This holds true for any corrector/parabola pair with identical final focus location. In either case, the volume of glass needed to remove is in inverse proportion to the
3rd power of relative aperture (F-number) of the corrected surface.
Knowing that spherical reflecting surface produces wavefront that in the first term, according to Eq. 4.6, advances away from spherical at a rate of (ρd)4/4R3 with respect to the reference sphere
centered at paraxial focus, variation in the Schmidt surface zonal depth z, i.e. surface profile needed for pre-correction of primary spherical aberration that will bring all reflected rays to
paraxial focus is z=(ρd)4/4(n-1)R3. This adds to every wavefront point the compensatory optical path length (n-1)z=(ρd)4/4R3, which pre-deforms the wavefront, so that it in effect gets corrected by
the aberration generated at the mirror. Since the actual wavefront deviation depends on the reference sphere used, i.e. specific focus within the aberration longitudinal range to which the rays are
to be brought, which is center of curvature of reference sphere, corrector sag also depends on the normalized defocus Λ, which determines the reference point. The actual wavefront deviation from the
reference sphere centered on the chosen zonal focus also varies with the zonal height. Consequently, corrector's depth profile (left, grossly exaggerated) needed for corrected mirror's lower-order
spherical aberration also varies with zonal height, as given with the general relation:
where Λ is the relative focus location parameter (from Λ=0 for the corrected focus coinciding with paraxial focus, to Λ=2 when corrected focus coincides with marginal focus), ρ the height in the
pupil normalized to 1 for pupil radius, d and D the pupil (aperture) radius and diameter, respectively, n the glass index of refraction, n' the index of refraction of the incident/exit media (media
next to the Schmidt surface, normally air, with n'=1 for the rear, and n'=n for the front), R the mirror radius of curvature and F the mirror focal ratio for the clear (corrector) aperture.
Corrector's focus parameter Λ determines neutral zone location at the unit radius as NZ=(Λ/2)1/2, as well as corrector's aspheric coefficient b; the two determine the needed vertex radius of
curvature of the positive central section of the corrector lens Rc.
Thus, as shown at left, choice of Λ determines the Schmidt profile shape, while its depth varies in proportion to the stop aperture diameter D, and inversely to the third power of mirror's
stop-aperture focal ratio F. Any of these profiles has a higher-order aspheric form, which by altering the wavefront form generates spherical aberration of the magnitude, type and sign needed to
offset that of the mirror. As plots show, the strength of corrector's vertex radius of curvature changes inversely to the focus factor Λ, decreasing from infinity for Λ=0 (the profile bringing
together the paraxial foci of all wavelengths) to its minimum value for Λ=2 (the profile bringing together marginal foci of all wavelengths). According to NZ=(Λ/2)1/2, neutral zone location shifts
from 0 to 1. Profile depth is at its minimum for Λ=1, bringing together best foci of all wavelengths. This profile also induces the least amount of spherochromatism, which is proportional to the
profile depth. The profiles shown are based on Eq. 101, hence they only correct for primary spherical. However, even the first next higher order is only a small fraction of the primary spherical,
thus the inclusion of higher order terms result only in minor profile modifications (note that the profiles are grossly exaggerated for clarity; actual profiles are seldom more than a few hundreds of
a millimeter deep).
Expectedly, the Schmidt surface profile is effectively the shape of the wavefront deviation given by Eq. 7, modified by the 1/(n'-n) medium factor (the profile is opposite in sign to that of the P-V
wavefront deviation for the rear corrector's surface, and of the same sign for the front surface). The Λ factor merely determines the amount of defocus with which the paraxial spherical aberration
combines in producing the corresponding wavefront for a specific point of defocus.
Alternately, Eq. 101 can be written in terms of the corrector's glass thickness, as t=t1+z, with t1 being the corrector center thickness (mirror radius R is numerically negative, and the sign of z is
determined by ρ4-Λρ2, which is always positive for Λ=0, positive for smaller values of ρ and negative for larger ones for 0<ρ<1, and always negative for Λ≥1).
The relative depth of corrector's curve, in units of the maximum corrector depth for 0<Λ<2 is given by ρ4- Λρ2. According to it, depth of corrector's curve is smallest for Λ=1, i.e. with the neutral
zone placed at (Λ/2)1/2=0.707 radius (thus ρ=0.707), with the corrected focus coinciding with best focus location (0.866 radius neutral zone placement, with the corresponding Λ value of 1.5, requires
corrector deeper by a factor of 2.25). This neutral zone position - as it will be explained in more details ahead - also minimizes spherochromatism.
Most often, at least one higher-order term is significant and needs to be corrected as well. In such case, corrector's curve depth profile can be expressed in terms of its vertex radius of curvature
and aspheric values. With the term for higher-order (secondary) spherical aberration added, it is given as :
with Rc being the corrector vertex radius of curvature, b and b' the 3rd and 5th order aspheric coefficient (for the transverse ray aberration; 4th and 6th order on the wavefront), with A1 and A2
being the corrector's aspheric parameters for the primery and secondary spherical aberration, respectively, commonly used in ray tracing programs. The A[1] term - the primary spherical aberration
term - is, from Eq. 4.6 directly related to the conic K as A[1]=(1+K)/8R^3, R being the mirror radius of curvature (if starting surface is a sphere, A[1]=K/8R^3, as it expresses change in the sagitta
depth, and for K=-1 it equals the differential between sphere and paraboloid, for given vertex radius). Note that the first term, sometimes referred to as a[2] (with the next being a[4], a[6] and so
on) is in the parabolic form because d/R[c] is negligibly small in the full expression for the first term given by a[2]=d^2/R{1+[1-(1+K)(d/R)^2]^1/2}, with K being the surface conic.
The first term describes sagitta of the corrector's radius of curvature which, combined with the aberration terms (the second is for primary spherical, the third for secondary spherical, and so on),
determines the actual surface profile (FIG. 168, left). It is not an aberration term with respect to spherical aberration in the optimized wavelength, since it is corrected for any corrector shape,
but it does affect correction of unoptimized wavelengths, i.e. magnitude of spherochromatism. This first term, often called radius term, is actually defocus term: analogous to the aberration terms Ai
, which are the wavefront functions of spherical aberration for paraxial focus, it represents defocus aberration with which spherical aberration combines producing altered wavefront specific to any
point of defocus, as a sum of the wavefront errors of defocus and spherical aberration.
Obviously, since the required surface profile is directly determined by that of the aberrated wavefront to correct and the medium in which light travels, this term - representing the defocus P-V
wavefront error - is is also modified by the same 1/(n'-n) factor.
The second and third term are the P-V wavefront error of primary and secondary spherical at paraxial focus, respectively, modified by the 1/(n'-n) medium factor. Note that only the second term -
primary spherical - effectively combines with the defocus (radius) term. Secondary spherical is added only as the P-V wavefront error at paraxial focus, which means that secondary spherochromatism is
not minimized. Considering usually small magnitude of secondary spherical, this is negligible; however, in fabrication it is generally more convenient to use the term for minimized secondary
spherical, with (ρd)6 replaced by (ρ6-ρ2)d6. It gives a profile very similar to that for the primary spherical at the best focus (i.e. for Λ=1 and NZ=0.707), which is of the same sign, only much
smaller in magnitude. This means that the profile needs to be only slightly deeper at the 0.7 zone, with the change in depth diminishing to zero at the center and the edge, as opposed to having to
make the entire corrector deeper by 2.6 times more (FIG. 168 bottom right) than the required deepening at the 0.7 zone.
Hence the addition of higher-order terms requires modifying the profile shown on FIG. 167. Since the Schmidt surface profile is essentially the reversed shape of the wavefront deviation, only deeper
by a factor 1/(n-1), n being the glass refractive index, the new profile is a reversed (if on the front surface, same orientation as wavefront if on the back) stretched out in depth replica of the
wavefront deviation, as illustrated on FIG. 168 right.
FIGURE 168: TOP: LEFT -- The value of Rc (radius of curvature over the central portion of the corrector) results from a sum of two wavefront deviations: that of spherical aberration at paraxial
focus, and that of defocus (opposite in sign) applied to it. This sum is then corrected (enlarged) by the 1/(n'-n) medium term - n being the glass refractive index - to determine the shape of Schmidt
corrector. Being the index-corrected sum of two wavefront deviations - that of spherical aberration at paraxial focus, and the defocus wavefront - the implicates the defocus term applied, and vice
versa. With the first aberration term only (primary spherical) and Λ=1 (best focus location), the sum is that of the red plot (defocus wavefront deviation) and blue plot (spherical aberration at
paraxial focus), resulting in the z plot, which is the needed surface profile before correction by the 1/(n'-n) glass index factor. It can be written as (d4/2R3)(ρ4-ρ2), i.e. it is proportional to ρ4
-ρ2, with the neutral (reversal) zone at 0.707 pupil radius. It is easy to see that for Λ=0 (bringing rays to paraxial focus) Rc is infinitely large, the first term (i.e. defocus error) is zero, and
it is the second, spherical aberration term alone (blue) that determines corrector's profile, effectively the inverse of the wavefront deviation at paraxial focus multiplied by 1 (n'-n). The relative
dept - or dimensionless profile of the corrector (FIG. 167 left) - for corrected primary spherical alone is given by ρ4- Λρ2. Adding the next, higher-order term, for secondary spherical (green,
grossly exaggerated for clarity) raises the edge vs. inner area, practically requiring deepening the corrector evenly for the value of the term from center to 0.7 zone, and gradually raising toward
the old profile line and unchanged edge from the 0.7 zone up. More practical way of fabricating such corrector is adding secondary spherical at the best focus location (bottom right). RIGHT - Grossly
exaggerated lens surface deformation (B) needed to correct spherical aberration (A) of a 300mm f/3.3 spherical mirror, with the glass index of refraction n=1.5. Most of the 7.6 wave P-V aberration is
its primary term, with the remaining higher-order terms totaling less than 1/10 wave P-V (C). Maximum depth of the aspheric curve for correcting the primary term is, therefore, about 7.5 waves times
1/(n-1), or 15 waves for n=1.5. For correcting the secondary spherical, less than 0.2 wave deeper curve is needed, with the remaining residual higher order term (tertiary spherical) of less than
0.002 wave P-V (D). Higher order terms grow exponentially with the relative aperture, the higher term, the faster.
MIDDLE: Illustration of the parameters shaping up the Schmidt profile for the two common shapes (neutral zone at 0.707 and 0.866 radius), and adding correction for secondary spherical. As before,
both, pupil radius and primary spherical aberration coefficient normalized to 1. Each term - for defocus (P), primary (A4) and secondary (A6) spherical - is proportional to the corresponding
aberration function for P-V wavefront error. The defocus aberration function - i.e. the normalized wavefront error of defocus with reference to the paraxial focus - is the radius term, ranging from 0
to -2, in units of the P-V wavefront error of primary spherical at paraxial focus.
LEFT: The P-V wavefront error of spherical aberration at paraxial focus, green (determines A4, proportional to ρ4), and defocus
P-V WFE, proportional to ρ2 and 1.5ρ2 (blue and red, for defocus at the best focus location and smallest blur, respectively). Zero defocus is a, leaving the wavefront shape at the paraxial focus
unaltered, best focus defocus is b (not to be confused with the primary spherical aspheric coefficient b), and the smallest blur location defocus is c.
SECOND FROM LEFT: The resulting profile shapes after adding zero (A4+a), best focus (A4+b, neutral zone at the 0.707 radius height) and the smallest blur point defocus (A4+c, neutral zone at 0.866
radius) to the P-V WFE of s.a. at paraxial focus.
THIRD FROM LEFT: The P-V wavefront error of secondary spherical to be added for correcting the secondary spherical, for the paraxial (determines A6, proportional to ρ6) and best focus (proportional
to ρ6-ρ2) location (note that ρ6-ρ2 is best focus with respect to the P-V error; for the minimum RMS error, best focus requires a bit less of defocus from the paraxial focus,
and is given by ρ6-0.9ρ2).
RIGHT: The sum of wavefront deviation profiles for primary spherical and defocus with the secondary spherical wavefront deviation profile added. The conventional profile relation adds the secondary
term for paraxial focus, proportional to ρ6, but for fabrication purposes the best focus term, proportional to ρ6-ρ2, is preferable, requiring shallower corrector with the center and edge in the same
plane. Graph below illustrates more clearly the difference between two profiles. A profile balancing lower and higher order term (i.e. adding as much of the lower order as needed to minimize the
higher order, but without entirely correcting the latter) is very close to the profile entirely correcting for the 6th (and 4th) order, indicating high degree of accuracy needed to fully correct for
the higher-order term; however, it being usually low, reducing it to a small fraction is, for all practical purposes, cancelling it out.
Note that the actual glass profiles depths z are larger by a factor 1/(n'-n) than the functions representing the sum of primary spherical and given amounts of defocus shown (normalized to 1 for the
P-V wavefront error of mirror spherical aberration at the paraxial focus).
BOTTOM: Shapes of the Schmidt profile for correcting primary spherical at the best focus alone, primary and secondary spherical at either its paraxial or best focus location, and for zero secondary
spherical coefficient (A6), with the needed amount of primary spherical added to balance (minimize) secondary spherical.
Pure secondary spherical (c and b) is exaggerated vs. primary spherical, in order to have different profiles clearly separated. The wavefronts generated by these profiles would have the same form of
deviation, only shallower by a factor (n-1), n being the glass refractive index. Profile correcting for primary spherical (i.e. 4th order) alone (a) leaves secondary spherical (6th order)
uncorrected, with the smallest error being at its respective best focus. Profile that minimizes this residual secondary spherical by adding primary spherical of opposite sign is only slightly
different than the fully correcting profile, and in all but very strong correctors it would be beyond fabrication accuracy to opt for one or the other.
For instance, the secondary spherical residual in an 8 inch commercial SCT is about 1/8 wave P-V at the paraxial focus, and 2.5 times less at the best focus. Since the latter is reduced over four
times when offset with opposite primary spherical, the differential between full correction is less than 1/80 wave in the wavefront, and about twice as much in the glass profile.
Profile given with Eq. 101.1 fully corrects for secondary spherical at its paraxial focus. This, however, is less practical profile from the fabrication point of view, since adding profile correction
for the secondary spherical at paraxial focus (b) to the profile for primary spherical at best focus (a) results in a deeper profile with uneven center/edge height (since glass cannot be added, this
profile, marked a+b, is pulled into the glass so that its edge coincides with the glass edge). It is generally easier to produce profile correcting for secondary spherical at its best focus (a+c)
which, as mentioned, is often nearly identical to the profile that only minimizes secondary spherical (in such case, it is reasonable to assume that the 6th order term, as given in Eq. 101.1, is
dropped, and only 4th order term, corrected for the amount needed to minimize the secondary spherical residual, is used).
Schmidt surface for correcting spherical aberration of a conic surface has all its surface terms of the same sign which, according to Eq. 101.1, implies that correcting the next order term requires
deeper surface profile. More complex forms of spherical aberration, such as, for instance, correcting balanced higher order forms (Maksutov corrector, strongly curved refracting objectives, and
other) may have higher surface terms of different numerical sign, where correcting next higher order term may require shallower curve. In principle, there is no difference in the effect of aspheric
profile whether it is applied to a flat surface, or radius (the latter merely requires adjustment for the corrector's radius when entering specs into raytrace).
The Schmidt corrector radius of curvature is given by:
with the 3rd order aspheric coefficient b=2/R3, and n'=1 for the aspheric surface on the back of corrector (� is the mirror focal length, and F the focal ratio). Optionally, the relations can be
written in terms of the relative neutral zone position in units of aperture radius, NZ, by substituting Λ=2NZ2. Optimized for the small effect of corrector's radius of curvature, 3rd order aspheric
coefficient is:
with F being the mirror F-number (F=-R/2D). The 5th order aspheric coefficient b'=6/R5. The two aspheric parameters A1 and A2 determine the Schmidt corrector shape, according to Eq. 101.1. From the
equation, they are obtained from their respective aspheric coefficients b and b', as
A1 = b/8(n'-n) = 1/4(n'-n)R3 (104) and
A2 = b'/16(n'-n) = 3/8(n'-n)R5 (104.1)
The two aspheric coefficients, b and b', are obtained by setting the system aberration coefficients for 3rd and 5th order spherical aberration to zero, s3=-b/8 + [1-(Λ/16F2)]/4R3 = 0 and s5=-b'/16 +
3/8R5 = 0 with the left side of the coefficient (b factor) being the corrector aberration contribution, and the right side that of the mirror. The 4th and 6th order system P-V wavefront error at the
paraxial focus are W3=s3d4 and W5=s5d6, respectively.
The slightly lower 3rd order mirror coefficient results from its effective relative aperture slightly reduced for non-zero values of corrector's focus parameter Λ (in effect, the higher Λ, the more
diverging outer rays falling onto mirror, reducing spherical aberration). A non-zero paraxial radius term Rc makes the corrector a weak positive lens with aspheric figure, also determining neutral
zone position for given value of the aspheric coefficient b. The neutral zone location is also given directly, for unit radius, as NZ=(Λ/2)1/2.
The significance of the 5th order term is in correction of the higher-order spherical aberration (5th order transverse ray, 6th order on the wavefront). Those include axial spherical, as well as
oblique (lateral) spherical, and wings, the higher-order astigmatism as it was named by Schwarzschild. They both increase with the square of off-axis height in the image space, and set the limit to
field quality. The latter has the P-V error larger by a factor of 4n, n being the glass refractive; since it varies with cosθ, θ being the pupil angle, the off-axis aberration in the Schmidt camera
peaks along the tangential plane (the one determined by the chief ray and optical axis, for which θ=0 and cosθ=1).
For 200mm f/2 Schmidt camera, the amount of higher-order spherical aberration is ~0.24 wave RMS. It can be minimized by balancing it with the lower-order form of opposite sign (by making the 4th
order curve slightly stronger). The residual that can't be corrected with the 3rd order surface term alone is ~0.04 wave RMS.
: 200mm f/2 Schmidt camera with BK7 corrector (n=1.5185 for 550nm wavelength), thus clear aperture radius d=100 at the corrector, and mirror radius of curvature R=-800. Choosing for the corrected
focus best focus location of the mirror, thus Λ=1, determines neutral zone height NZ=(Λ/2)1/2ρmax, at 0.707d. Only the rear side is aspherized.
From Eq. 103, corrector's lower-order aspheric coefficient b=2[1-(Λ/16F2)]/R3=-0.000000003845, or b=-3.845-9, determining the lower-order aspheric parameter of the corrector as A1=b/8(n'-n)=9.27-10,
with the index of refraction of the exit media (air for the Schmidt surface at the back of corrector) n'=1. Higher-order aspheric coefficient b'=6/R5=-1.83-14 determines the higher-order corrector
aspheric parameter A2=b'/16(n'-n) =2.21-15. Needed radius of curvature of the corrector is Rc=-1/2ΛA 1d2=-53,940mm.
With the corrector at the mirror center of curvature, the system is corrected for 3rd/4th i.e. 5th/6th order spherical aberration (3rd and 5th transverse ray aberration, corresponding to 4th and 6th
order on the wavefront), coma, astigmatism and distortion. The only remaining aberration is field curvature, rc=R/2=-400mm.
For double-sided corrector, both b and A coefficients are half their value for single-sided corrector, with the A coefficients being of the opposite sign on the other side (as determined by n'-n)..
Since, from Eq. 104/104.1, b=8(n'-n)A1 and b'=16(n'-n)A2, the P-V wavefront error at the best focus resulting from deviations ΔA1 and ΔA2 in the two aspheric parameters is given by W4=(n'-n)ΔA1d4/4
and W6=0.42(n'-n)ΔA2d6 for 4th and 6th order spherical aberration, respectively. Taking 0.0001375mm (1/4 wave at 0.00055mm wavelength) for W4 gives, for the above system, the corresponding
lower-order parameter deviation as ΔA1=4W4/(n'-n)d4=1.06-11, with 1/4 wave of spherical aberration figure tolerance for the lower-order aberration of 1.06-11(ρd)4. At the maximum curve depth (ρ=
0.707), it is 0.000265mm, or 0.48 wave.
Aspheric coefficients numerical conversion when switching to a different unit is given as A'=A(u[2]/u[1])^x-1, where u[2] is expressued in units of u[1], with u[i] being the units, and x the
coefficient order. So, for example, converting from milimeters (u[1]) to inches (u[2]), the 4th order coefficient is larger by a factor 25.4^3, 6th order by a factor 25.4^5, and so on (applies when
system remains identical physically, only the measuring unit changes).
Follows more detailed account of the Schmidt camera aberrations.
◄ 10.2.1.3. Honders camera ▐ 10.2.2.1. Schmidt camera: aberrations ►
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How To Convert Float To String In C (3 Best Approaches) - GeeksForRescue
How To Convert Float To String In C
Converting float values into string data types while working with float values in the C programming language for a variety of reasons.
Accurate conversion of float to string is a vital process in many applications, whether it’s for displaying numbers in a user-friendly manner, saving them in a database or file, or sending them as
arguments to functions that need string data types.
It is essential to understand the available techniques and recommended procedures for converting float to string in C.
Why Convert Float to String In C
Here are the reasons why this conversion is needed:
• User-friendly: For applications, like financial, and scientific computations, it is critical to show float numbers in a clear and understandable manner.
• Converting float values to strings is required for correct storage and retrieval of the data when storing float values in a database
• Calculations: In applications, like web development, float values may occasionally be represented as strings. In certain circumstances, string conversion is necessary for accurate calculation and
data presentation.
• Functions that only take string data types as inputs might not accept float values as parameters.
• Handling: Large or small float values may occasionally be challenging to display or store correctly.
Various Approaches Of Converting Float to String in C
• Using sprintf()
• Using gcvt()
• Using snprintf()
Approach 1: Using sprintf() Method
With the sprintf() function from the C standard library, float values may be formatted into strings. Its first input is a formatted string, and its second is the float value that has to be converted.
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
int main() {
float f = (float)rand() / RAND_MAX;
char buffer[20];
sprintf(buffer, "%.4f", f);
printf("Float value: %f\n", f);
printf("String value: %s\n", buffer);
f = -f;
sprintf(buffer, "%.2f", f);
printf("Float value: %f\n", f);
printf("String value: %s\n", buffer);
f = 0.0f;
sprintf(buffer, "%.2f", f);
printf("Float value: %f\n", f);
printf("String value: %s\n", buffer);
return 0;
Float value: 0.249345
String value: 0.2493
Float value: -0.249345
String value: -0.25
Float value: 0.000000
String value: 0.00
• The main() function begins by generating a random float number between 0 and 1 using rand() and RAND_MAX.
• It then converts this float value to a string with four decimal places using sprintf(), which stores the string in the buffer variable.
• The float value and the corresponding string value are then printed to the console using printf().
• The main() function repeats this process two more times, with slight variations in the float values generated and the number of decimal places in the resulting string values. Finally, the
function returns 0 to indicate successful completion.
Approach 2: Using gcvt() Method
Using the C library function gcvt(), float values may be formatted as strings. The float value to be converted and the number of significant digits to be included in the result are its two required
#include <stdlib.h>
#include <stdio.h>
int main() {
double d = 123.456789;
char buffer[20];
gcvt(d, 6, buffer);
printf("Double value: %f\n", d);
printf("String value: %s\n", buffer);
d = -d;
gcvt(d, 4, buffer);
printf("Double value: %f\n", d);
printf("String value: %s\n", buffer);
d = 0.0;
gcvt(d, 3, buffer);
printf("Double value: %f\n", d);
printf("String value: %s\n", buffer);
return 0;
Double value: 123.456789
String value: 123.456790
Double value: -123.456789
String value: -123.4568
Double value: 0.000000
String value: 0.00
• The main() function begins by assigning a double value to d.
• The gcvt() function is then used to convert this double value to a string representation with a specified number of significant digits, which is stored in the buffer array.
• The double value and its corresponding string representation are then printed to the console using printf().
• The main() function repeats this process two more times, with variations in the double values assigned and the number of significant digits specified for the string representations. Finally, the
function returns 0 to indicate successful completion.
Approach 3: Using snprintf() Method
To convert float data to string format, utilise the C library function snprintf(). The sprintf() takes an extra parameter that determines the size of the character array that will hold the result. By
doing this, buffer overflows are avoided.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
int main() {
float f = 42.0;
char str[20];
int size = snprintf(str, 20, "%.3f", f);
printf("The float value is %f\n", f);
printf("The string value is %s\n", str);
printf("The size of the string is %d\n", size);
f = -1.234567;
size = snprintf(str, 20, "%+.2f", f);
printf("The float value is %f\n", f);
printf("The string value is %s\n", str);
printf("The size of the string is %d\n", size);
f = 1234567.8912345;
size = snprintf(str, 20, "%e", f);
printf("The float value is %f\n", f);
printf("The string value is %s\n", str);
printf("The size of the string is %d\n", size);
return 0;
The float value is 42.000000
The string value is 42.000
The size of the string is 6
The float value is -1.234567
The string value is -1.23
The size of the string is 5
The float value is 1234567.875000
The string value is 1.234568e+06
The size of the string is 12
• The main() function begins by assigning a floating-point value to f. The snprintf() function is then used to convert this value to a string representation with a specified format and maximum
length, which is stored in the str array.
• The function returns the number of characters written to the string, which is stored in the size variable.
• The original value and its corresponding string representation are then printed to the console using printf().
• The main() function repeats this process two more times, with variations in the floating-point values assigned and the format specifiers specified for the string representations. Finally, the
function returns 0 to indicate successful completion.
Best Approach:
Best approach is sprintf() method and here are reasons why:
• The well-known and often used function sprintf() can be found in the majority of C standard libraries.
• Numerous formatting options are available, including how many decimal places a float should have.
• Additionally, it offers more output format flexibility, allowing you to select the minimum field width and include leading zeros.
• sprintf() is a suitable option in many circumstances since it offers a strong and flexible way to convert a float to a string in C.
Sample Problems:
Sample Problem 1:
You are building a scientific application that requires you to display measurements in a readable format. How would you convert float values to string format in the C programming language?
• The code creates a class called “Measurement” to handle conversions.
• The “Measurement” class has a constructor that takes a float value and a character array to represent the unit.
• The “Measurement” class has a “toString” method that converts the float value to a string with a specified format.
• The “toString” method uses the “sprintf” function to format the float value and unit into a string.
• The “sprintf” function takes a string format and a list of arguments and writes the formatted string to a buffer.
• The “main” function creates a “Measurement” object with a float value of 12.345 and the unit “meters”.
• The “main” function calls the “toString” method of the “Measurement” object to convert the float value to a string.
• The “toString” method returns the formatted string.
• The “main” function uses the “printf” function to print the formatted string to the console. The formatted string is “12.35 meters”, because the “%0.2f” format specifies two decimal places for
the float value.
#include <stdio.h>
#include <string.h>
// Create a struct to handle measurements
typedef struct {
float value;
char unit[10];
} Measurement;
// Convert the float value to a string with specified format
void toString(Measurement m, char* str) {
sprintf(str, "%.2f %s", m.value, m.unit);
int main() {
// Create a Measurement struct with a float value and unit
Measurement m = {12.345, "meters"};
// Convert the value to a string and print it
char str[20];
toString(m, str);
printf("%s\n", str);
return 0;
12.35 meters
Sample Problem 2:
You need to store float values in a database that only accepts string data types. How would you convert float values to string format in the C programming language?
• The code also defines a Database class that stores floating-point values as strings in an array.
• The gcvt() function from stdlib.h is used to convert the floating-point value to a string.
• The MAX_STR_LEN macro defines the maximum length of each string in the database.
• The char* data[MAX_STR_LEN] creates an array of character pointers that can store MAX_STR_LEN strings.
• The storeData() function stores the given floating-point value as a string in the database at the given index.
• The printData() function prints the contents of the database.
• The main() function creates an instance of the Database class, stores a list of floating-point values in it, and then prints the contents of the database.
#include <stdio.h>
#include <stdlib.h>
#define MAX_STR_LEN 50
struct Database {
char* data[MAX_STR_LEN];
void initDatabase(struct Database* db) {
for (int i = 0; i < MAX_STR_LEN; i++) {
db->data[i] = (char*)malloc(sizeof(char) * MAX_STR_LEN);
void storeData(struct Database* db, float val, int index) {
char* str_val = gcvt(val, 10, db->data[index]);
printf("Storing %f as string: %s\n", val, str_val);
void printData(struct Database* db) {
printf("Database contents:\n");
for (int i = 0; i < MAX_STR_LEN; i++) {
printf("%s\n", db->data[i]);
int main() {
struct Database db;
float values[] = {1.23, 4.56, 7.89, 12.34, 56.78, 90.12};
int num_values = sizeof(values) / sizeof(float);
for (int i = 0; i < num_values; i++) {
storeData(&db, values[i], i);
return 0;
Storing 1.230000 as string: 1.230000
Storing 4.560000 as string: 4.560000
Storing 7.890000 as string: 7.890000
Storing 12.340000 as string: 12.340000
Storing 56.780000 as string: 56.780000
Storing 90.120000 as string: 90.120000
Database contents:
Sample Problem 3:
You are building an application that performs complex calculations on float values, and you need to pass those values as arguments to functions that expect string data types. How would you convert
float values to string format in the C programming language?
• A macro is defined using #define, which sets the maximum length of a string to 50 characters.
• The main() function is declared and defined, which is the starting point of execution.
• A float variable f is initialised with a value of 123.456789.
• A character array variable str is declared, which can store up to 50 characters.
• The float value f is converted to a string using the snprintf() function and stored in the str variable.
• Multiple float variables are declared and initialised with some values.
• A character array variable output is declared, which can store up to 50 characters. The multiple float values are concatenated into a string using the snprintf() function and stored in the output
#include <stdio.h>
#include <stdlib.h>
#define MAX_STRING_LENGTH 50
int main() {
float f = 123.456789;
char str[MAX_STRING_LENGTH];
// convert float to string using snprintf()
snprintf(str, MAX_STRING_LENGTH, "%f", f);
printf("Float: %f\n", f);
printf("String: %s\n", str);
// multiple outputs
float x = 1.23, y = 4.56, z = 7.89;
char output[MAX_STRING_LENGTH];
// concatenate multiple float values into a string
snprintf(output, MAX_STRING_LENGTH, "%f, %f, %f", x, y, z);
printf("Multiple outputs: %s\n", output);
// negative number
float neg_num = -12.34;
char neg_str[MAX_STRING_LENGTH];
// convert negative float to string
snprintf(neg_str, MAX_STRING_LENGTH, "%f", neg_num);
printf("Negative number: %s\n", neg_str);
return 0;
Float: 123.456787
String: 123.456787
Multiple outputs: 1.230000, 4.560000, 7.890000
Negative number: -12.340000
In conclusion, the act of converting a float to a string is essential in C programming. This work can be completed using a variety of techniques, including sprintf(), gcvt(), and snprintf(). However,
because of its adaptability and better control over output formatting, sprintf() is the most common and extensively used approach. Experts in C programming advise using sprintf() to convert floats to | {"url":"https://www.geeksforrescue.com/blog/how-to-convert-float-to-string-in-c/","timestamp":"2024-11-09T10:55:24Z","content_type":"text/html","content_length":"81325","record_id":"<urn:uuid:20910c09-1082-4f03-993e-b0cc27a19c79>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00675.warc.gz"} |
What is 26 Percent as a Fraction in Simplest Form? - GEGCalculators
What is 26 Percent as a Fraction in Simplest Form?
Percentages and fractions are two common ways to represent parts of a whole. Converting between them is a fundamental skill in mathematics and everyday life. In this blog post, we will explore how to
convert 26 percent into a fraction in its simplest form. We will break down the process step by step, explain the underlying concepts, and provide real-world examples of why this conversion is
What is 26 Percent as a Fraction in Simplest Form?
26 percent as a fraction in simplest form is 13/50. This fraction represents 26 parts out of 100, simplified by dividing both the numerator (26) and the denominator (100) by their greatest common
factor, which is 2.
Certainly, here’s a table summarizing what “26 percent as a fraction in simplest form” means:
Term Explanation
Percentage A way to express a part of a whole in terms of 100.
Converting to Fraction To represent a percentage as a fraction with a denominator of 100.
26 Percent Refers to 26 parts out of 100.
Simplifying Fraction The process of reducing a fraction to its lowest terms.
Fraction in Simplest Form The final, most simplified representation of a fraction.
26 Percent as a Fraction in Simplest Form The result of converting 26 percent to a fraction and simplifying it.
Understanding how to convert percentages to fractions in their simplest form is a fundamental math skill that is useful in various contexts, from mathematics to real-life situations like shopping
discounts and financial calculations.
Understanding Percentages
A percentage is a way to express a part of a whole in terms of 100. The word “percent” itself means “per hundred.” Therefore, when we say 26 percent, we are talking about 26 parts out of 100. To
convert a percentage to a fraction, we need to represent the percentage as a fraction with a denominator of 100.
Converting 26% to a Fraction
Step 1: Write the Percentage as a Fraction Over 100
• To convert 26 percent to a fraction, write it as 26/100. This represents 26 parts out of 100.
Step 2: Simplify the Fraction
• To simplify the fraction, find the greatest common factor (GCF) of the numerator (26) and the denominator (100). In this case, the GCF is 2.
Step 3: Divide Both Numerator and Denominator by the GCF
• Divide both 26 and 100 by 2:
□ (26 ÷ 2) / (100 ÷ 2) = 13/50
The fraction 13/50 is the simplest form of 26 percent.
Real-World Examples
Understanding how to convert percentages to fractions can be incredibly useful in various real-world scenarios. Here are a few examples:
1. Shopping Discounts: Imagine you have a 26 percent discount on a product. To calculate the discounted price, you can convert 26 percent to the fraction 13/50 and then multiply it by the original
2. Financial Calculations: When calculating interest rates or investment returns, percentages are often converted to fractions for precise calculations.
3. Grade Point Average (GPA): GPA is often expressed as a fraction, and understanding how to convert percentages to fractions can help in calculating your GPA accurately.
4. Cooking and Baking: Recipes sometimes call for ingredients in percentages. Converting these percentages to fractions can make it easier to measure and adjust quantities.
1. What is 2 6 simplified as a fraction?
• 2/6 can be simplified to 1/3 by dividing both the numerator and denominator by their greatest common factor, which is 2.
2. What is the simplest fraction form?
• The simplest fraction form is also known as the lowest terms or reduced form. It is achieved by dividing the numerator and denominator by their greatest common factor.
3. What’s 0.25 as a fraction?
• 0.25 as a fraction is 1/4. You can write it as a fraction by understanding that 0.25 is equivalent to 25/100, and then simplifying to its lowest terms, which is 1/4.
4. What is the fraction of 2 6 in words?
• 2/6 in words is “two-sixths.”
5. What is 2 6 as a decimal?
• 2/6 as a decimal is 0.333 (repeating). It can also be expressed as 0.3 with a line over the 3 to indicate the repeating decimal.
6. How do you simplify fractions on a calculator?
• Most calculators have a fraction or simplify function. You can enter the fraction, and the calculator will simplify it for you.
7. How do you simplify fractions for kids?
• To simplify fractions for kids, teach them to find the greatest common factor (GCF) of the numerator and denominator, and then divide both by the GCF. Repeat until the fraction can’t be
simplified further.
8. How do you solve simple fractions?
• To solve simple fractions, perform the specified operation (add, subtract, multiply, or divide) on the numerators and denominators, and then simplify the result to its lowest terms.
9. What is 14 as a fraction in simplest form?
• 1/4 is the simplest form of 14 when expressed as a fraction.
10. What’s 6 8 in simplest form? – 6/8 can be simplified to 3/4 by dividing both the numerator and denominator by their greatest common factor, which is 2.
11. What is the simplest form of 5 4? – 5/4 in its simplest form is already expressed as an improper fraction, and it cannot be further simplified.
12. What is 0.25 in simplest form? – 0.25 in simplest fraction form is 1/4.
13. How is 0.25 the same as 1 4? – 0.25 is the decimal representation of 1/4, both of which represent a quarter or one-fourth of a whole.
14. What is 0.25 as a decimal? – 0.25 as a decimal is equivalent to 1/4 and is equal to 0.25 when expressed in decimal form.
Converting percentages to fractions is a fundamental mathematical skill that has practical applications in various aspects of life, from finance to cooking. In this blog post, we learned how to
convert 26 percent into the simplest form of the fraction 13/50.
Understanding this process empowers us to work with percentages more effectively and make informed decisions in everyday situations. Whether you’re shopping, managing your finances, or following a
recipe, knowing how to convert percentages to fractions is a valuable tool in your mathematical toolkit.
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for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
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conic sections
This week’s Fiddler is a classic problem.
A weaving loom consists of equally spaced hooks along the x and y axes. A string connects the farthest hook on the x-axis to the nearest hook on the y-axis, and continues back and forth between
the axes, always taking up the next available hook. This leads to a picture that looks like this:
As the number of hooks goes to infinity, what does the shape trace out?
Extra credit: If four looms are rotated and superimposed as shown below, what is the area of the white region in the middle?
My solution:
[Show Solution]
Can you outrun the angry ram?
The Riddler puzzle this week appears simple at first glance, but I promise you it’s not!
You, a hard-driving sheep farmer, are tucked into the southeast corner of your square, fenced-in sheep paddock. There are two gates equidistant from you: one at the southwest corner and one at
the northeast corner. An angry, recalcitrant ram enters the paddock from the southwest gate and charges directly at you at a constant speed. You run — obviously! — at a constant speed along the
eastern fence toward the northeast gate in an attempt to escape. The ram keeps charging, always directly at you.
How much faster than you does the ram have to run so that he catches you just as you reach the gate?
Here is a very simple solution by Hector Pefo. Minimal calculus required!
[Show Solution]
And here is my solution, which finds an equation for the path of the ram but requires knowledge of calculus and differential equations.
[Show Solution]
Cutting polygons in half
This Riddler puzzle is about cutting polygons in half. Here is the problem:
The archvillain Laser Larry threatens to imminently zap Riddler Headquarters (which, seen from above, is shaped like a regular pentagon with no courtyard or other funny business). He plans to do
it with a high-powered, vertical planar ray that will slice the building exactly in half by area, as seen from above. The building is quickly evacuated, but not before in-house mathematicians
move the most sensitive riddling equipment out of the places in the building that have an extra high risk of getting zapped.
Where are those places, and how much riskier are they than the safest spots? (It’s fine to describe those places qualitatively.)
Extra credit: Get quantitative! Seen from above, how many high-risk points are there? If there are infinitely many, what is their total area?
Here is my solution:
[Show Solution]
And here is a bonus interactive graphic showing the solution
The puzzle of the picky eater
Today’s Riddler post is a neat problem about calculating areas.
Every morning, before heading to work, you make a sandwich for lunch using perfectly square bread. But you hate the crust. You hate the crust so much that you’ll only eat the portion of the
sandwich that is closer to its center than to its edges so that you don’t run the risk of accidentally biting down on that charred, stiff perimeter. How much of the sandwich will you eat?
Extra credit: What if the bread were another shape — triangular, hexagonal, octagonal, etc.? What’s the most efficient bread shape for a crust-hater like you?
Here is my solution:
[Show Solution]
Overflowing martini glass
This Riddler puzzle is all about conic sections.
You’ve kicked your feet up and have drunk enough of your martini that, when the conical glass (🍸) is upright, the drink reaches some fraction p of the way up its side. When tipped down on one
side, just to the point of overflowing, how far does the drink reach up the opposite side?
Here is my solution:
[Show Solution] | {"url":"https://laurentlessard.com/bookproofs/tag/conic-sections/","timestamp":"2024-11-10T05:02:22Z","content_type":"text/html","content_length":"176302","record_id":"<urn:uuid:7bf5edaa-e5f9-4513-b27f-36caa6f9a518>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00503.warc.gz"} |
Estimation Applied to Course Production
A few years ago, my brother, a programmer, shared with me the concepts and challenges with estimating time for software development. For example, as addressed in Software Estimation: Demystifying the
Black Art. I’ve found the concept interesting before, but now more applicable in my work.
Bean Counting! (Photo by cookbookman17)
Griggs University brought about 120 distance education courses to Andrews University, and many of them are in need of upgrades and complete rewrites.
Part of my recent work has been to start developing a system of course development and revision. As I organize a small force (army?) of student workers and staff to work on courses, I’ve been looking
more closely at estimation. I am heading towards having an estimate for how long it takes to do certain sets of jobs – like creating quizzes, applying CSS designs to content pages, etc.
This includes teaching my workers how to do their own estimation for converting lessons to HTML pages with appropriate CSS applied. In one case, we thought through carefully two different ways to
approach the estimation process. I thought it might be helpful to share here two simple methods that I taught:
Approach #1: Starting with known amount of time on a task
If you know how much time it takes to do a certain task, then you can estimate how long it will take 10 or 20 items of the same type of task.
So, the question is: Given a known amount of time to finish a task, how long will it take to finish x tasks?
For example: given 1.5 hours to finish 1 lesson how long will it take to finish 40 lessons from where I am now.
• 40-14 = 26 (40 lessons – 14 done = 26 lessons to do)
• 26×1.5 = 39 (26 lessons left x 1.5 hours to do the lesson = 39 hours to complete goal). Round up to 40 hours; which is about 5 days of work
• Given that 80-90% of work day is accomplishing tasks; 10-20% is interruptions and breaks etc. …
• 40 hours x 10% is 4 more hours; so that is about 45 hours actually accounting for breaks etc. Or a better more generous estimate would 45-50 hours left for this job.
Approach #2: Starting with a desired end time (pedal to the metal method)
If you have a desired end time, then you can estimate how fast each task needs to be completed.
So the question is: Given a specific deadline; how fast do I need to do each task?
For example: Given that I want to finish 26 lessons by the end of the day on Thursday, and it is now Tuesday afternoon, how fast do I need to do each lesson?
• I have 26 lessons left (40-14=26 see above).
• In two days (Wed & Thu) I plan to work 9 hour days with an hour of interruptions and breaks, so I can estimate about 8 hours a day per day.
• 16/26 = .61 (16 hours divided by 26 lessons = the fraction of an hour that I need finish a lesson in)
• Convert the decimal to hours:.61 x 60 minutes = 36 minutes per lesson
I’ve come to the conclusion that this type of thinking isn’t necessary “caught” or “taught” in our high schools and colleges. Students may know what estimation is, but might not be able to apply it
to real-life scenarios when they graduate.
What do you think? I realize this is a very simplistic method to estimate how we are doing in our work, but it’s a start. Do you try to estimate progress on online course development? How do you do
it? What data do you collect and track?
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Reasoning Ability Quiz For SBI Clerk Prelims 2022- 30th April
Directions (1–5): Following questions are based on the five three-digit numbers given below:
Q1. If all the numbers are arranged in ascending order from left to right, which of the following will be the product of the first and last digits of the number which is in the middle in the new
(a) 16
(b) 30
(c) 27
(d) 49
(e) 18
Q2. If the positions of the first and third digits of each of the number interchange, what will be the sum of all the digits of the second lowest number?
(a) 13
(b) 15
(c) 16
(d) 11
(e) 20
Q3. If one is subtracted from each odd digit and two is added to each even digit of each of the numbers, what will be the difference between the third digit and the first digit of the lowest number?
(a) Zero
(b) 6
(c) 4
(d) 2
(e) 5
Q4. What will be the resultant if the first digit of the highest number is divided by the third digit of the lowest number?
(a) 0.5
(b) 1
(c) 2
(d) 2.2
(e) None of these
Q5. If all the digits in each of the numbers are arranged in descending order within the number from left to right, which of the following will be the second highest number in the new arrangement
(a) 369
(b) 717
(c) 922
(d) 625
(e) None of these
Directions (6-7): Study the following information carefully and answer the question given below-
There are eight members in a family i.e., K, O, L, N, Q, M, P and S. Among them three are married couples. S is the daughter of the one who is L’s nephew. Q is the daughter-in-law of the one who is
grandfather of M. O has only one grandson. P is grand-daughter-in-law of K. L is O’s husband’s sister.
Q6. How many children P’s father-in-law has?
(a) Two
(b) One
(c) Three
(d) Four
(e) None of these
Q7. Who among the following is father of N?
(a) O
(b) L
(c) K
(d) M
(e) None of these
Directions (8-12): Study the following information carefully and answer the question given below:
M, N, O, P, Q, R, J and K are sitting in a straight line but not necessarily in the same order. Some of them are facing south while some are facing north. Only two people sit to the right of M. N
sits third to the left of M. Only one person sits between N and R. R sits to the immediate right of Q. Only one person sits between Q and K. Both the immediate neighbours of N face the same
direction. M faces north. O sits third to the left of R. N faces the opposite direction of M. J does not sit at any of the extremes ends of the line. P faces the same direction as Q. Both J and O
face the opposite direction of K.
Q8. How many persons in the given arrangement are facing North?
(a) More than four
(b) Four
(c) One
(d) Three
(e) Two
Q9. Four of the following five are alike in a certain way, and so form a group. Which of the following does not belong to the group?
(a) Q, R
(b) K, J
(c) N, M
(d) N, J
(e) P, O
Q10. What is the position of R with respect to K?
(a) Second to the left
(b) Third to the right
(c) Third to the left
(d) Fifth to the right
(e) Second to the right
Q11. Who amongst the following sits exactly between K and Q?
(a) N
(b) J
(c) R
(d) Q
(e) O
Q12. Who is sitting 2nd to the right of N?
(a) K
(b) P
(c) R
(d) Q
(e) None of these.
Directions (13-15): Study the following information carefully and answer the questions given below:
Point A is 6m west of point B, which is 8m south of point F. Point D is 26m east of point C. Point G is 15m west of point H. Point C is 5m north of point A. Point H is 17m north east of point F.
Point D is 10m south of point E.
Q13. What is the shortest distance between point G and point F?
(a) 16m
(b) 10m
(c) √5m
(d) 8m
(e) None of these
Q14. What is the direction of point E with respect to point B?
(a) North-east
(b) South
(c) North
(d) West
(e) None of these
Q15. What is the shortest distance and direction of point G with respect to point B?
(a) 18m, South-west
(b) 15m, South
(c) 16m, North
(d) 20, North-west
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Talks and Seminars - Enrique Mallada
Talks and Seminars
Talks and Seminars
This is a list of recent talks and seminars.
1. 2024-09-25: Generalized Barrier Functions: Integral Conditions and Recurrent Relaxations, 60th Allerton Conference on 60th Allerton Conference on Communication, Control, and Computing.
[BibTeX] [Abstract] [Download PDF]
Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and
reliability. In this talk, we consider two complementary mechanisms to augment RL’s suitability for safety-critical systems. Firstly, we consider a constrained reinforcement learning (C-RL)
setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been
recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient
descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we
develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy. Secondly, we study the problem of incorporating safety-critical
constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the
associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic
operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid
reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully
characterize its fixed points representing–except for a spurious solution–maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by
design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.
abstract = {Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and reliability. In this talk, we consider two complementary mechanisms to augment RL's suitability for safety-critical systems.
Firstly, we consider a constrained reinforcement learning (C-RL) setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy.
Secondly, we study the problem of incorporating safety-critical constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing--except for a spurious solution--maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.},
date = {06/12/2024},
day = {25},
event = {60th Allerton Conference on 60th Allerton Conference on Communication, Control, and Computing},
host = {N/A},
month = {09},
role = {Speaker},
title = {Generalized Barrier Functions: Integral Conditions and Recurrent Relaxations},
url = {https://mallada.ece.jhu.edu/talks/202409-Allerton.pdf},
year = {2024}
2. 2024-06-12: Reinforcement Learning for Safety Critical Applications, Tercera Conferencia Colombiana de Matematicas Aplicadas e Industriales.
[BibTeX] [Abstract] [Download PDF]
Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and
reliability. In this talk, we consider two complementary mechanisms to augment RL’s suitability for safety-critical systems. Firstly, we consider a constrained reinforcement learning (C-RL)
setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been
recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient
descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we
develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy. Secondly, we study the problem of incorporating safety-critical
constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the
associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic
operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid
reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully
characterize its fixed points representing–except for a spurious solution–maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by
design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.
abstract = {Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and reliability. In this talk, we consider two complementary mechanisms to augment RL's suitability for safety-critical systems.
Firstly, we consider a constrained reinforcement learning (C-RL) setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy.
Secondly, we study the problem of incorporating safety-critical constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing--except for a spurious solution--maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.},
date = {06/12/2024},
day = {12},
event = {Tercera Conferencia Colombiana de Matematicas Aplicadas e Industriales},
host = {Javier Peña (CMU), Mateo Diaz (JHU)},
month = {06},
role = {Speaker},
title = {Reinforcement Learning for Safety Critical Applications},
url = {https://mallada.ece.jhu.edu/talks/202406-MAPI.pdf},
year = {2024}
3. 2024-06-18: Data-driven Analysis of Dynamical Systems Using Recurrent Sets, INFORMS International Conference.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov
Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds
(resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner
approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a
subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling
finite-length trajectories. Secondly, we generalize Lyapunov’s Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent
(instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along
trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by
discussing future research directions and possible extensions for control.
abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.},
date = {06/18/2024},
day = {18},
event = {INFORMS International Conference},
host = {Luis Zuluaga (Lehigh), Mateo Diaz (JHU)},
month = {06},
role = {Speaker},
title = {Data-driven Analysis of Dynamical Systems Using Recurrent Sets},
url = {https://mallada.ece.jhu.edu/talks/202406-Informs.pdf},
year = {2024}
4. 2024-06-05: Data-driven Analysis of Dynamical Systems Using Recurrent Sets, Department of Automatic Control, Lund University.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov
Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds
(resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner
approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a
subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling
finite-length trajectories. Secondly, we generalize Lyapunov’s Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent
(instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along
trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by
discussing future research directions and possible extensions for control.
abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.},
date = {06/05/2024},
day = {05},
event = {Department of Automatic Control, Lund University},
host = {Richard Pates (Lund)},
month = {06},
role = {Lecture},
title = {Data-driven Analysis of Dynamical Systems Using Recurrent Sets},
url = {https://mallada.ece.jhu.edu/talks/202406-Lund.pdf},
year = {2024}
5. 2024-06-06: Data-driven Analysis of Dynamical Systems Using Recurrent Sets, Cyber-Physical Systems Lab, Université catholique de Louvain.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov
Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds
(resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner
approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a
subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling
finite-length trajectories. Secondly, we generalize Lyapunov’s Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent
(instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along
trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by
discussing future research directions and possible extensions for control.
abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.},
date = {06/06/2024},
day = {06},
event = {Cyber-Physical Systems Lab, Université catholique de Louvain},
host = {Raphael Jungers (UCL)},
month = {06},
role = {Lecture},
title = {Data-driven Analysis of Dynamical Systems Using Recurrent Sets},
url = {https://mallada.ece.jhu.edu/talks/202406-UCL.pdf},
year = {2024}
6. 2024-05-16: Recurrence of Nonlinear Control Systems: Entropy and Bit Rates, Hybrid Systems: Computation and Control (HSCC).
[BibTeX] [Abstract] [Download PDF]
In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (tau-)recurrent if every trajectory that starts in the set
returns to it (within at most $τ$ units of time). Recurrence entropy quantifies the complexity of making a set tau-recurrent measured by the average rate of growth, as time increases, of the
number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a
set is no larger than, and often strictly smaller than, the invariance entropy. Our results further offer insights into the minimum data rate required for achieving recurrence. We also
present an algorithm for achieving recurrence asymptotically.
abstract = {In this paper, we introduce the notion of recurrence entropy in the context of nonlinear control systems. A set is said to be (tau-)recurrent if every trajectory that starts in the set returns to it (within at most $τ$ units of time). Recurrence entropy quantifies the complexity of making a set tau-recurrent measured by the average rate of growth, as time increases, of the number of control signals required to achieve this goal. Our analysis reveals that, compared to invariance, recurrence is quantitatively less complex, meaning that the recurrence entropy of a set is no larger than, and often strictly smaller than, the invariance entropy. Our results further offer insights into the minimum data rate required for achieving recurrence. We also present an algorithm for achieving recurrence asymptotically.},
date = {05/16/2024},
day = {16},
event = {Hybrid Systems: Computation and Control (HSCC)},
month = {05},
role = {Lecture},
title = {Recurrence of Nonlinear Control Systems: Entropy and Bit Rates},
url = {https://mallada.ece.jhu.edu/talks/202405-HSCC.pdf},
year = {2024}
7. 2024-03-28: Options for Mitigation Measures: Avenues for new Research, ESIG/G-PST Special Topic Workshop on Oscillations.
[BibTeX] [Download PDF]
date = {03/28/2024},
day = {28},
event = {ESIG/G-PST Special Topic Workshop on Oscillations},
host = {Mark O'Malley (Imperial)},
month = {03},
role = {Lecture},
title = {Options for Mitigation Measures: Avenues for new Research},
url = {https://mallada.ece.jhu.edu/talks/202403-ESIG.pdf},
year = {2024}
8. 2024-03-20: Model-Free Analysis of Dynamical Systems Using Recurrent Sets, ECE Colloquium, Rutgers University.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov
Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds
(resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner
approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a
subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling
finite-length trajectories. Secondly, we generalize Lyapunov’s Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent
(instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along
trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by
discussing future research directions and possible extensions for control.
abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize by discussing future research directions and possible extensions for control.},
date = {03/20/2024},
day = {20},
event = {ECE Colloquium, Rutgers University},
host = {Daniel Burbano (Rutgers)},
month = {03},
role = {Lecture},
title = {Model-Free Analysis of Dynamical Systems Using Recurrent Sets},
url = {https://mallada.ece.jhu.edu/talks/202403-Rutgers.pdf},
year = {2024}
9. 2024-02-16: Reinforcement Learning for Safety Critical Applications, George Mason University.
[BibTeX] [Abstract] [Download PDF]
Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and
reliability. In this talk, we consider two complementary mechanisms to augment RL’s suitability for safety-critical systems. Firstly, we consider a constrained reinforcement learning (C-RL)
setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been
recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient
descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we
develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy. Secondly, we study the problem of incorporating safety-critical
constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the
associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic
operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid
reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully
characterize its fixed points representing–except for a spurious solution–maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by
design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.
abstract = {Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and reliability. In this talk, we consider two complementary mechanisms to augment RL's suitability for safety-critical systems.
Firstly, we consider a constrained reinforcement learning (C-RL) setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy.
Secondly, we study the problem of incorporating safety-critical constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing--except for a spurious solution--maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.},
date = {02/2024},
day = {16},
event = {George Mason University},
host = {Ningshi Yao (GMU)},
month = {02},
role = {Lecture},
title = {Reinforcement Learning for Safety Critical Applications},
url = {https://mallada.ece.jhu.edu/talks/202402-GMU.pdf},
year = {2024}
10. 2024-01-11: Reinforcement Learning for Safety Critical Applications, Applied Physics Laboratory, JHU.
[BibTeX] [Abstract] [Download PDF]
Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and
reliability. In this talk, we consider two complementary mechanisms to augment RL’s suitability for safety-critical systems. Firstly, we consider a constrained reinforcement learning (C-RL)
setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been
recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient
descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we
develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy. Secondly, we study the problem of incorporating safety-critical
constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the
associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic
operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid
reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully
characterize its fixed points representing–except for a spurious solution–maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by
design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.
abstract = {Integrating Reinforcement Learning (RL) in safety-critical applications, such as autonomous vehicles, healthcare, and industrial automation, necessitates an increased focus on safety and reliability. In this talk, we consider two complementary mechanisms to augment RL's suitability for safety-critical systems.
Firstly, we consider a constrained reinforcement learning (C-RL) setting, wherein agents aim to maximize rewards while adhering to required constraints on secondary specifications. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods exhibit a discrepancy between the behavioral and optimal policies due to their reliance on stochastic gradient descent-ascent algorithms. We propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories almost surely converge to the optimal policy.
Secondly, we study the problem of incorporating safety-critical constraints to RL that allow an agent to avoid (unsafe) regions of the state space. Though such a safety goal can be captured by an action-value-like function, a.k.a. safety critics, the associated operator lacks the desired contraction and uniqueness properties that the classical Bellman operator enjoys. In this work, we overcome the non-contractiveness of safety critic operators by leveraging that safety is a binary property. To that end, we study the properties of the binary safety critic associated with a deterministic dynamical system that seeks to avoid reaching an unsafe region. We formulate the corresponding binary Bellman equation (B2E) for safety and study its properties. While the resulting operator is still non-contractive, we fully characterize its fixed points representing--except for a spurious solution--maximal persistently safe regions of the state space that can always avoid failure. We provide an algorithm that, by design, leverages axiomatic knowledge of safe data to avoid spurious fixed points.},
date = {02/2024},
day = {11},
event = {Applied Physics Laboratory, JHU},
host = {Jared Markowitz},
month = {01},
role = {Lecture},
title = {Reinforcement Learning for Safety Critical Applications},
url = {https://mallada.ece.jhu.edu/talks/202401-JHUAPL.pdf},
year = {2024}
1. 2023-12-11: Unintended Consequences of Market Designs, IHPC’s Workshop of Power and Energy Systems of the (near) Future, ASTAR.
[BibTeX] [Abstract] [Download PDF]
In this talk, we seek to highlight the importance of accounting for the incentives of *all* market participants when designing market mechanisms for electricity. To this end, we perform a
Nash equilibrium analysis of two different market mechanisms that aim to illustrate the critical role that the incentives of consumers and other new types of participants, such as storage,
play in the equilibrium outcome. Firstly, we study the incentives of heterogeneous participants (generators and consumers) in a two-stage settlement market, where generators participate using
a supply function bid and consumers use a quantity bid. We show that strategic consumers are able to exploit generators’ strategic behavior to maintain a systematic difference between the
forward and spot prices, with the latter being higher. Notably, such a strategy does bring down consumer payments and undermines the supply-side market power. We further observe situations
where generators lose profit by behaving strategically, a sign of overturn of the conventional supply-side market power. Secondly, we study a market mechanism for multi-interval electricity
markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to
communicate their cost to the market using energy-cycling functions that map prices to cycle depths. The resulting market-clearing process — implemented via convex programming — yields
corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. Our solution shows several advantages over the standard
prosumer-based approach that prices energy per slot. In particular, it does not require a priori estimation of future prices and leads to an efficient, competitive equilibrium.
abstract = {In this talk, we seek to highlight the importance of accounting for the incentives of *all* market participants when designing market mechanisms for electricity. To this end, we perform a Nash equilibrium analysis of two different market mechanisms that aim to illustrate the critical role that the incentives of consumers and other new types of participants, such as storage, play in the equilibrium outcome. Firstly, we study the incentives of heterogeneous participants (generators and consumers) in a two-stage settlement market, where generators participate using a supply function bid and consumers use a quantity bid. We show that strategic consumers are able to exploit generators' strategic behavior to maintain a systematic difference between the forward and spot prices, with the latter being higher. Notably, such a strategy does bring down consumer payments and undermines the supply-side market power. We further observe situations where generators lose profit by behaving strategically, a sign of overturn of the conventional supply-side market power. Secondly, we study a market mechanism for multi-interval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energy-cycling functions that map prices to cycle depths. The resulting market-clearing process -- implemented via convex programming -- yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. Our solution shows several advantages over the standard prosumer-based approach that prices energy per slot. In particular, it does not require a priori estimation of future prices and leads to an efficient, competitive equilibrium.},
date = {12/11/2023},
day = {11},
event = {IHPC's Workshop of Power and Energy Systems of the (near) Future, ASTAR},
host = {John Pang (ASTAR)},
month = {12},
role = {Speaker},
title = {Unintended Consequences of Market Designs},
url = {https://mallada.ece.jhu.edu/talks/202312-ASTAR.pdf},
year = {2023}
2. 2023-11-04: Model-Free Analysis of Dynamical Systems Using Recurrent Sets, FIND Seminar, Cornell University.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov
Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds
(resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner
approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a
subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling
finite-length trajectories. Secondly, we generalize Lyapunov’s Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent
(instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along
trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize discussing
future research directions and possible extensions for control.
abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using trajectory data. Our critical insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. Specifically, a set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We leverage this notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. Firstly, we consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point using trajectory data. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Secondly, we generalize Lyapunov's Direct Method to allow for non-monotonic evolution of the function values by only requiring sub-level sets to be τ-recurrent (instead of invariant). We provide conditions for stability, asymptotic stability, and exponential stability of an equilibrium using τ-decreasing functions (functions whose value along trajectories decrease after at most τ seconds) and develop a verification algorithm that leverages GPU parallel processing to verify such conditions using trajectories. We finalize discussing future research directions and possible extensions for control.},
date = {11/04/2023},
day = {04},
event = {FIND Seminar, Cornell University},
host = {Kevin A. Tang (Cornell)},
month = {11},
role = {Lecture},
title = {Model-Free Analysis of Dynamical Systems Using Recurrent Sets},
url = {https://mallada.ece.jhu.edu/talks/202311-Cornell.pdf},
year = {2023}
3. 2023-10-12: Reinforcement Learning with Almost Sure Constraints, MURI Workshop.
[BibTeX] [Abstract] [Download PDF]
In this work, we study how to tackle decision-making for safety-critical systems under uncertainty. To that end, we formulate a Reinforcement Learning problem with Almost Sure constraints, in
which one seeks a policy that allows no more than $Δınℕ$ unsafe events in any trajectory, with probability one. We argue that this type of constraint might be better suited for
safety-critical systems as opposed to the usual average constraint employed in Constrained Markov Decision Processes and that, moreover, having constraints of this kind makes feasible
policies much easier to find. The talk is didactically split into two parts, first considering $Δ=0$ and then the $Δ≥ 0$ case. At the core of our theory is a barrier-based decomposition of
the Q-function that decouples the problems of optimality and feasibility and allows them to be learned either independently or in conjunction. We develop an algorithm for characterizing the
set of all feasible policies that provably converges in expected finite time. We further develop sample-complexity bounds for learning this set with high probability. Simulations corroborate
our theoretical findings and showcase how our algorithm can be wrapped around other learning algorithms to hasten the search for first feasible and then optimal policies.
abstract = {In this work, we study how to tackle decision-making for safety-critical systems under uncertainty. To that end, we formulate a Reinforcement Learning problem with Almost Sure constraints, in which one seeks a policy that allows no more than $Δınℕ$ unsafe events in any trajectory, with probability one. We argue that this type of constraint might be better suited for safety-critical systems as opposed to the usual average constraint employed in Constrained Markov Decision Processes and that, moreover, having constraints of this kind makes feasible policies much easier to find. The talk is didactically split into two parts, first considering $Δ=0$ and then the $Δ≥ 0$ case. At the core of our theory is a barrier-based decomposition of the Q-function that decouples the problems of optimality and feasibility and allows them to be learned either independently or in conjunction. We develop an algorithm for characterizing the set of all feasible policies that provably converges in expected finite time. We further develop sample-complexity bounds for learning this set with high probability. Simulations corroborate our theoretical findings and showcase how our algorithm can be wrapped around other learning algorithms to hasten the search for first feasible and then optimal policies.},
date = {10/2023},
day = {12},
event = {MURI Workshop},
host = {Mario Sznaier (Northeastern)},
month = {10},
role = {Speaker},
title = {Reinforcement Learning with Almost Sure Constraints},
url = {https://mallada.ece.jhu.edu/talks/202310-MURI.pdf},
year = {2023}
4. 2023-09-07: Grid Shaping Control for High-IBR Power Systems: Stability Analysis and Control Design, GE EDGE Symposium.
[BibTeX] [Abstract] [Download PDF]
The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving
grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the
stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system’s low inertia to restore grid synchronism without
incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. First, we develop novel stability analysis tools for power
systems, which allow for the decentralized design of inverter-based controllers. The method requires that each inverter satisfies a standard H-infinity design requirement that depends on the
dynamics of the components and inverters at each bus and the aggregate susceptance of the transmission lines connected to it. It is robust to network and delay uncertainty and, when no
network information is available, reduces to the standard passivity condition for stability. Then, we propose a novel grid-forming control strategy, so-called grid shaping control, that aims
to shape the frequency response of synchronous generators (SGs) to load perturbations so as to efficiently arrest sudden frequency drops. The approach builds on novel analysis tools that can
characterize the Center of Inertia (CoI) response of a system with both IBRs and SGs and use this characterization to reshape it.
abstract = {The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system's low inertia to restore grid synchronism without incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. First, we develop novel stability analysis tools for power systems, which allow for the decentralized design of inverter-based controllers. The method requires that each inverter satisfies a standard H-infinity design requirement that depends on the dynamics of the components and inverters at each bus and the aggregate susceptance of the transmission lines connected to it. It is robust to network and delay uncertainty and, when no network information is available, reduces to the standard passivity condition for stability. Then, we propose a novel grid-forming control strategy, so-called grid shaping control, that aims to shape the frequency response of synchronous generators (SGs) to load perturbations so as to efficiently arrest sudden frequency drops. The approach builds on novel analysis tools that can characterize the Center of Inertia (CoI) response of a system with both IBRs and SGs and use this characterization to reshape it.},
date = {09/20/2023},
day = {07},
event = {GE EDGE Symposium},
host = {Aditya Kumar (GE)},
month = {09},
role = {Speaker},
title = {Grid Shaping Control for High-IBR Power Systems: Stability Analysis and Control Design},
url = {https://mallada.ece.jhu.edu/talks/202309-GE-EDGE.pdf},
year = {2023}
5. 2023-09-07: Learning Coherent Clusters in Weakly Connected Power Networks, 6th Workshop on Autonomous Energy Systems.
[BibTeX] [Abstract] [Download PDF]
Network coherence generally refers to the emergence of a simple aggregated dynamic response of generator units, despite heterogeneity in the unit’s location and dynamic constitution. In this
talk, we develop a general frequency domain framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the
system’s input-output response. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics, network topology, and the type of disturbance.
We further leverage this framework to build structure-preserving model-reduction methodology for large-scale dynamic networks with tightly-connected components and provide time-domain bounds
on the approximation error of our model. Our work provides new avenues for analysis and control designs of IBR-rich power systems.
abstract = {Network coherence generally refers to the emergence of a simple aggregated dynamic response of generator units, despite heterogeneity in the unit's location and dynamic constitution. In this talk, we develop a general frequency domain framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the system's input-output response. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics, network topology, and the type of disturbance. We further leverage this framework to build structure-preserving model-reduction methodology for large-scale dynamic networks with tightly-connected components and provide time-domain bounds on the approximation error of our model. Our work provides new avenues for analysis and control designs of IBR-rich power systems. },
date = {09/07/2023},
day = {07},
event = {6th Workshop on Autonomous Energy Systems},
host = {Andrey Berstein (NREL), Guido Carvaro (NREL)},
month = {09},
role = {Speaker},
title = {Learning Coherent Clusters in Weakly Connected Power Networks},
url = {https://mallada.ece.jhu.edu/talks/202309-NREL.pdf},
year = {2023}
6. 2023-07-06: Model-Free Analysis of Dynamical Systems Using Recurrent Sets, Workshop on Uncertain Dynamical Systems.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using data. Our key insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the
more relaxed notion of recurrence. A set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We then
leverage the notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. We first consider the problem of learning an inner approximation of the region
of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. We show that a τ-recurrent set containing a stable equilibrium must be a subset of
its ROA under mild assumptions. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by
sampling finite-length trajectories. Our algorithms process samples sequentially, which allows them to continue being executed even after an initial offline training stage. We will finalize
by providing some recent extensions of this work that generalizes Lyapunov’s Direct Method to allow for non-decreasing functions to certify stability and illustrate future research
abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using data. Our key insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. A set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We then leverage the notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. We first consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allows them to continue being executed even after an initial offline training stage. We will finalize by providing some recent extensions of this work that generalizes Lyapunov's Direct Method to allow for non-decreasing functions to certify stability and illustrate future research directions.},
date = {07/06/2023},
day = {06},
event = {Workshop on Uncertain Dynamical Systems},
host = {Mario Sznaier (Northeastern), Fabrizio Dabbene (PoliTo), Constantino Lagoa (Penn State)},
month = {07},
role = {Speaker},
title = {Model-Free Analysis of Dynamical Systems Using Recurrent Sets},
url = {https://mallada.ece.jhu.edu/talks/202307-WUDS.pdf},
year = {2023}
7. 2023-07-19: Grid Shaping Control for High-IBR Power Systems, Panel on Future electricity systems: How to handle millions of power electronic-based devices and other emerging technologies, IEEE
PES General Meeting.
[BibTeX] [Abstract] [Download PDF]
The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving
grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the
stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system’s low inertia to restore grid synchronism without
incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. First, we develop novel stability analysis tools for power
systems, which allow for the decentralized design of inverter-based controllers. The method requires that each inverter satisfies a standard H-infinity design requirement that depends on the
dynamics of the components and inverters at each bus and the aggregate susceptance of the transmission lines connected to it. It is robust to network and delay uncertainty and, when no
network information is available, reduces to the standard passivity condition for stability. Then, we propose a novel grid-forming control strategy, so-called grid shaping control, that aims
to shape the frequency response of synchronous generators (SGs) to load perturbations so as to efficiently arrest sudden frequency drops. The approach builds on novel analysis tools that can
characterize the Center of Inertia (CoI) response of a system with both IBRs and SGs and use this characterization to reshape it.
abstract = {The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system's low inertia to restore grid synchronism without incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. First, we develop novel stability analysis tools for power systems, which allow for the decentralized design of inverter-based controllers. The method requires that each inverter satisfies a standard H-infinity design requirement that depends on the dynamics of the components and inverters at each bus and the aggregate susceptance of the transmission lines connected to it. It is robust to network and delay uncertainty and, when no network information is available, reduces to the standard passivity condition for stability. Then, we propose a novel grid-forming control strategy, so-called grid shaping control, that aims to shape the frequency response of synchronous generators (SGs) to load perturbations so as to efficiently arrest sudden frequency drops. The approach builds on novel analysis tools that can characterize the Center of Inertia (CoI) response of a system with both IBRs and SGs and use this characterization to reshape it.},
date = {07/19/2023},
day = {19},
event = {Panel on Future electricity systems: How to handle millions of power electronic-based devices and other emerging technologies, IEEE PES General Meeting},
host = {Claudia Andrea Rahmann (UChile), Amarsagar Reddy Ramapuram Matavalam (ASU)},
month = {07},
role = {Panelist},
title = {Grid Shaping Control for High-IBR Power Systems},
url = {https://mallada.ece.jhu.edu/talks/202307-PESGM.pdf},
year = {2023}
8. 2023-05-30: Iterative Policy Learning for Constrained RL via Dissipative Gradient Descent-Ascent, Workshop on Online optimization Methods for Data-Driven Feedback Control, American Control
[BibTeX] [Abstract] [Download PDF]
In constrained reinforcement learning (C-RL), an agent seeks to learn from the environment a policy that maximizes the expected cumulative reward while satisfying minimum requirements in
secondary cumulative reward constraints. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such
methods are based on stochastic gradient descent-ascent algorithms whose trajectories are connected to the optimal policy only after a mixing output stage that depends on the algorithm’s
history. As a result, there is a mismatch between the behavioral policy and the optimal one. In this talk, we propose a novel algorithm for constrained RL that does not suffer from these
limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories converge to the optimal policy
almost surely.
abstract = {In constrained reinforcement learning (C-RL), an agent seeks to learn from the environment a policy that maximizes the expected cumulative reward while satisfying minimum requirements in secondary cumulative reward constraints. Several algorithms rooted in sampled-based primal-dual methods have been recently proposed to solve this problem in policy space. However, such methods are based on stochastic gradient descent-ascent algorithms whose trajectories are connected to the optimal policy only after a mixing output stage that depends on the algorithm's history. As a result, there is a mismatch between the behavioral policy and the optimal one. In this talk, we propose a novel algorithm for constrained RL that does not suffer from these limitations. Leveraging recent results on regularized saddle-flow dynamics, we develop a novel stochastic gradient descent-ascent algorithm whose trajectories converge to the optimal policy almost surely.},
date = {05/30/2023},
day = {30},
event = {Workshop on Online optimization Methods for Data-Driven Feedback Control, American Control Conferenece},
host = {Gianluca Bianchin (UCLouvain), Emiliano Dall'Anese (UC Boulder), Jorge Cortés (UCSD), Miguel Vaquero (IE University)},
month = {05},
role = {Speaker},
title = {Iterative Policy Learning for Constrained RL via Dissipative Gradient Descent-Ascent},
url = {https://mallada.ece.jhu.edu/talks/202305-ACC-Workshop.pdf},
year = {2023}
9. 2023-01-05: Learning Dynamics and Implicit Bias of Gradient Flow in Overparametrerized Linear Models, Joint Mathematics Meeting, Special Session.
[BibTeX] [Abstract] [Download PDF]
Contrary to the common belief that overparameterization may hurt generalization and optimization, recent work suggests that overparameterization may bias the optimization algorithm towards
solutions that generalize well — a phenomenon known as implicit regularization or implicit bias — and may also accelerate convergence — a phenomenon known as implicit acceleration. This talk
will provide a detailed analysis of the dynamics of gradient flow in overparameterized linear models showing that convergence to equilibrium depends on the imbalance between input and output
weights (which is fixed at initialization) and the margin of the initial solution. The talk will also provide an analysis of the implicit bias, showing that large hidden layer width, together
with (properly scaled) random initialization, constrains the network parameters to converge to a solution which is close to the min-norm solution.
abstract = {Contrary to the common belief that overparameterization may hurt generalization and optimization, recent work suggests that overparameterization may bias the optimization algorithm towards solutions that generalize well --- a phenomenon known as implicit regularization or implicit bias --- and may also accelerate convergence --- a phenomenon known as implicit acceleration. This talk will provide a detailed analysis of the dynamics of gradient flow in overparameterized linear models showing that convergence to equilibrium depends on the imbalance between input and output weights (which is fixed at initialization) and the margin of the initial solution. The talk will also provide an analysis of the implicit bias, showing that large hidden layer width, together with (properly scaled) random initialization, constrains the network parameters to converge to a solution which is close to the min-norm solution.},
date = {01/05/2023},
day = {05},
event = {Joint Mathematics Meeting, Special Session},
host = {Josué Tonelli Cueto, Hitesh Gakhar, Harlin Lee},
month = {01},
role = {Speaker},
title = {Learning Dynamics and Implicit Bias of Gradient Flow in Overparametrerized Linear Models},
url = {https://mallada.ece.jhu.edu/talks/202301-JMM.pdf},
year = {2023}
10. 2023-01-18: Frequency Shaping Control for Low Inertia Power Systems,, 2023 ROSEI Summit, Johns Hopkins University.
[BibTeX] [Abstract] [Download PDF]
The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving
grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the
stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system’s low inertia to restore grid synchronism without
incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. We define system-level performance metrics that are of
practical relevance for power systems and systematically evaluate the performance of standard control strategies, such as virtual inertia and droop control, in the presence of power
disturbances. Our analysis unveils the relatively limited role of inertia in improving performance and the inability of droop control to enhance performance without incurring considerable
steady-state control effort. To overcome these limitations, we propose a novel frequency shaping control for grid-connected inverters -exploiting classical lead/lag compensation and model
matching techniques from control theory- that can significantly outperform existing solutions while using comparable control effort.
abstract = {The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system's low inertia to restore grid synchronism without incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. We define system-level performance metrics that are of practical relevance for power systems and systematically evaluate the performance of standard control strategies, such as virtual inertia and droop control, in the presence of power disturbances. Our analysis unveils the relatively limited role of inertia in improving performance and the inability of droop control to enhance performance without incurring considerable steady-state control effort. To overcome these limitations, we propose a novel frequency shaping control for grid-connected inverters -exploiting classical lead/lag compensation and model matching techniques from control theory- that can significantly outperform existing solutions while using comparable control effort.},
date = {01/18/2023},
day = {18},
event = {2023 ROSEI Summit, Johns Hopkins University},
host = {Ben Schaffer, Ben Link},
month = {01},
role = {Speaker},
title = {Frequency Shaping Control for Low Inertia Power Systems,},
url = {https://mallada.ece.jhu.edu/talks/202301-ROSEI.pdf},
year = {2023}
1. 2022-12-19: Reinforcement Learning with Almost Sure Constraints, Topologı́a y Probabilidad en análisis de datos, Universidad de la Republica.
[BibTeX] [Abstract] [Download PDF]
In this work, we study how to tackle decision-making for safety-critical systems under uncertainty. To that end, we formulate a Reinforcement Learning problem with Almost Sure constraints, in
which one seeks a policy that allows no more than $Δınℕ$ unsafe events in any trajectory, with probability one. We argue that this type of constraint might be better suited for
safety-critical systems as opposed to the usual average constraint employed in Constrained Markov Decision Processes and that, moreover, having constraints of this kind makes feasible
policies much easier to find. The talk is didactically split into two parts, first considering $Δ=0$ and then the $Δ≥ 0$ case. At the core of our theory is a barrier-based decomposition of
the Q-function that decouples the problems of optimality and feasibility and allows them to be learned either independently or in conjunction. We develop an algorithm for characterizing the
set of all feasible policies that provably converges in expected finite time. We further develop sample-complexity bounds for learning this set with high probability. Simulations corroborate
our theoretical findings and showcase how our algorithm can be wrapped around other learning algorithms to hasten the search for first feasible and then optimal policies.
abstract = {In this work, we study how to tackle decision-making for safety-critical systems under uncertainty. To that end, we formulate a Reinforcement Learning problem with Almost Sure constraints, in which one seeks a policy that allows no more than $Δınℕ$ unsafe events in any trajectory, with probability one. We argue that this type of constraint might be better suited for safety-critical systems as opposed to the usual average constraint employed in Constrained Markov Decision Processes and that, moreover, having constraints of this kind makes feasible policies much easier to find. The talk is didactically split into two parts, first considering $Δ=0$ and then the $Δ≥ 0$ case. At the core of our theory is a barrier-based decomposition of the Q-function that decouples the problems of optimality and feasibility and allows them to be learned either independently or in conjunction. We develop an algorithm for characterizing the set of all feasible policies that provably converges in expected finite time. We further develop sample-complexity bounds for learning this set with high probability. Simulations corroborate our theoretical findings and showcase how our algorithm can be wrapped around other learning algorithms to hasten the search for first feasible and then optimal policies.},
date = {12/19/2022},
day = {19},
event = {Topologı́a y Probabilidad en análisis de datos, Universidad de la Republica},
host = {Nicolas Frevenza (UdelaR), Soledad Villar (JHU)},
month = {12},
role = {Speaker},
title = {Reinforcement Learning with Almost Sure Constraints},
url = {https://mallada.ece.jhu.edu/talks/202212-UdelaR.pdf},
year = {2022}
2. 2022-11-02: Model-free Analysis of Dynamical Systems Using Recurrence, Data Science Seminar, Johns Hopkins University.
[BibTeX] [Download PDF]
annote = {In this talk, we develop model-free methods for analyzing dynamical systems using data. Our key insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. A set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We then leverage the notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. We first consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allows them to continue being executed even after an initial offline training stage. We will finalize by providing some recent extensions of this work that generalizes Lyapunov's Direct Method to allow for non-decreasing functions to certify stability and illustrate future research directions.},
date = {11/02/2022},
day = {02},
event = {Data Science Seminar, Johns Hopkins University},
host = {Fei Lu (JHU), Mauro Maggioni (JHU)},
month = {11},
role = {Lecture},
title = {Model-free Analysis of Dynamical Systems Using Recurrence},
url = {https://mallada.ece.jhu.edu/talks/202211-DSS-JHU.pdf},
year = {2022}
3. 2022-09-07: Unintended Consequences of Market Designs, Workshon on Human Dimension of Energy Systems, NREL.
[BibTeX] [Abstract] [Download PDF]
In this talk, we seek to highlight the importance of accounting for the incentives of *all* market participants when designing market mechanisms for electricity. To this end, we perform a
Nash equilibrium analysis of two different market mechanisms that aim to illustrate the critical role that the incentives of consumers and other new types of participants, such as storage,
play in the equilibrium outcome. Firstly, we study the incentives of heterogeneous participants (generators and consumers) in a two-stage settlement market, where generators participate using
a supply function bid and consumers use a quantity bid. We show that strategic consumers are able to exploit generators’ strategic behavior to maintain a systematic difference between the
forward and spot prices, with the latter being higher. Notably, such a strategy does bring down consumer payments and undermines the supply-side market power. We further observe situations
where generators lose profit by behaving strategically, a sign of overturn of the conventional supply-side market power. Secondly, we study a market mechanism for multi-interval electricity
markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to
communicate their cost to the market using energy-cycling functions that map prices to cycle depths. The resulting market-clearing process — implemented via convex programming — yields
corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. Our solution shows several advantages over the standard
prosumer-based approach that prices energy per slot. In particular, it does not require a priori estimation of future prices and leads to an efficient, competitive equilibrium.
abstract = {In this talk, we seek to highlight the importance of accounting for the incentives of *all* market participants when designing market mechanisms for electricity. To this end, we perform a Nash equilibrium analysis of two different market mechanisms that aim to illustrate the critical role that the incentives of consumers and other new types of participants, such as storage, play in the equilibrium outcome. Firstly, we study the incentives of heterogeneous participants (generators and consumers) in a two-stage settlement market, where generators participate using a supply function bid and consumers use a quantity bid. We show that strategic consumers are able to exploit generators' strategic behavior to maintain a systematic difference between the forward and spot prices, with the latter being higher. Notably, such a strategy does bring down consumer payments and undermines the supply-side market power. We further observe situations where generators lose profit by behaving strategically, a sign of overturn of the conventional supply-side market power. Secondly, we study a market mechanism for multi-interval electricity markets with generator and storage participants. Drawing ideas from supply function bidding, we introduce a novel bid structure for storage participation that allows storage units to communicate their cost to the market using energy-cycling functions that map prices to cycle depths. The resulting market-clearing process -- implemented via convex programming -- yields corresponding schedules and payments based on traditional energy prices for power supply and per-cycle prices for storage utilization. Our solution shows several advantages over the standard prosumer-based approach that prices energy per slot. In particular, it does not require a priori estimation of future prices and leads to an efficient, competitive equilibrium.},
date = {09/07/2022},
day = {07},
event = {Workshon on Human Dimension of Energy Systems, NREL},
host = {Andrey Berstein (NREL)},
month = {09},
role = {Speaker},
title = {Unintended Consequences of Market Designs},
url = {https://mallada.ece.jhu.edu/talks/202209-NREL-HD.pdf},
year = {2022}
4. 2022-08-25: Reinforcement Learning with Almost Sure Constraints, Massachusetts Institute of Techonology.
[BibTeX] [Abstract] [Download PDF]
In this work, we study how to tackle decision-making for safety-critical systems under uncertainty. To that end, we formulate a Reinforcement Learning problem with Almost Sure constraints, in
which one seeks a policy that allows no more than $Δınℕ$ unsafe events in any trajectory, with probability one. We argue that this type of constraint might be better suited for
safety-critical systems as opposed to the usual average constraint employed in Constrained Markov Decision Processes and that, moreover, having constraints of this kind makes feasible
policies much easier to find. The talk is didactically split into two parts, first considering $Δ=0$ and then the $Δ≥ 0$ case. At the core of our theory is a barrier-based decomposition of
the Q-function that decouples the problems of optimality and feasibility and allows them to be learned either independently or in conjunction. We develop an algorithm for characterizing the
set of all feasible policies that provably converges in expected finite time. We further develop sample-complexity bounds for learning this set with high probability. Simulations corroborate
our theoretical findings and showcase how our algorithm can be wrapped around other learning algorithms to hasten the search for first feasible and then optimal policies.
abstract = {In this work, we study how to tackle decision-making for safety-critical systems under uncertainty. To that end, we formulate a Reinforcement Learning problem with Almost Sure constraints, in which one seeks a policy that allows no more than $Δınℕ$ unsafe events in any trajectory, with probability one. We argue that this type of constraint might be better suited for safety-critical systems as opposed to the usual average constraint employed in Constrained Markov Decision Processes and that, moreover, having constraints of this kind makes feasible policies much easier to find. The talk is didactically split into two parts, first considering $Δ=0$ and then the $Δ≥ 0$ case. At the core of our theory is a barrier-based decomposition of the Q-function that decouples the problems of optimality and feasibility and allows them to be learned either independently or in conjunction. We develop an algorithm for characterizing the set of all feasible policies that provably converges in expected finite time. We further develop sample-complexity bounds for learning this set with high probability. Simulations corroborate our theoretical findings and showcase how our algorithm can be wrapped around other learning algorithms to hasten the search for first feasible and then optimal policies.},
date = {08/24/2022},
day = {25},
event = {Massachusetts Institute of Techonology},
host = {Ali Jadbabaie (MIT)},
month = {08},
role = {Lecture},
title = {Reinforcement Learning with Almost Sure Constraints},
url = {https://mallada.ece.jhu.edu/talks/202208-MIT-DL.pdf},
year = {2022}
5. 2022-08-26: On the Convergence of Gradient Flow on Multi-layer Linear Models, Massachusetts Institute of Techonology.
[BibTeX] [Abstract] [Download PDF]
The mysterious ability of gradient-based optimization algorithms to solve the non-convex neural network training problem is one of the many unexplained puzzles behind the success of deep
learning in various applications. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect the convergence of training algorithms. In
this talk, we analyze the convergence of gradient flow on a multi-layer linear model with a loss function of the form $f(W_1W_2·s W_L)$. We show that when $f$ satisfies the gradient dominance
property, proper weight initialization leads to exponential convergence of the gradient flow to a global minimum of the loss. Moreover, the convergence rate depends on two trajectory-specific
quantities that are controlled by the weight initialization: the \emphimbalance matrices, which measure the difference between the weights of adjacent layers, and the least singular value of
the \emphweight product $W=W_1W_2·s W_L$. Our analysis provides improved rate bounds for several multi-layer network models studied in the literature, leading to novel characterizations of
the effect of weight imbalance on the rate of convergence. Our results apply to most regression losses and extend to classification ones.
abstract = {The mysterious ability of gradient-based optimization algorithms to solve the non-convex neural network training problem is one of the many unexplained puzzles behind the success of deep learning in various applications. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect the convergence of training algorithms. In this talk, we analyze the convergence of gradient flow on a multi-layer linear model with a loss function of the form $f(W_1W_2·s W_L)$. We show that when $f$ satisfies the gradient dominance property, proper weight initialization leads to exponential convergence of the gradient flow to a global minimum of the loss. Moreover, the convergence rate depends on two trajectory-specific quantities that are controlled by the weight initialization: the \emphimbalance matrices, which measure the difference between the weights of adjacent layers, and the least singular value of the \emphweight product $W=W_1W_2·s W_L$. Our analysis provides improved rate bounds for several multi-layer network models studied in the literature, leading to novel characterizations of the effect of weight imbalance on the rate of convergence. Our results apply to most regression losses and extend to classification ones.},
date = {08/26/2022},
day = {26},
event = {Massachusetts Institute of Techonology},
host = {Navid Azizan (MIT)},
month = {08},
role = {Lecture},
title = {On the Convergence of Gradient Flow on Multi-layer Linear Models},
url = {https://mallada.ece.jhu.edu/talks/202208-MIT-DL.pdf},
year = {2022}
6. 2022-07-14: Learning-based Analysis and Control of Safte-Critical Systems, Workshop on Autonomous Energy Systems, National Renewable Energy Laboratory.
[BibTeX] [Download PDF]
date = {07/14/2022},
day = {14},
event = {Workshop on Autonomous Energy Systems, National Renewable Energy Laboratory},
host = {Andrey Berstein (NREL), Ahmed Zamzam (NREL), Bai Cui (NREL)},
month = {07},
role = {Speaker},
title = {Learning-based Analysis and Control of Safte-Critical Systems},
url = {https://mallada.ece.jhu.edu/talks/202207-NREL.pdf},
year = {2022}
7. 2022-05-26: Learning-based Analysis and Control of Safte-Critical Systems, University of California San Diego.
[BibTeX] [Download PDF]
date = {05/26/2022},
day = {26},
event = {University of California San Diego},
host = {Jorge Cortés (UCSD)},
month = {05},
role = {Lecture},
title = {Learning-based Analysis and Control of Safte-Critical Systems},
url = {https://mallada.ece.jhu.edu/talks/202205-UCSD.pdf},
year = {2022}
8. 2022-05-27: Reinforcement Learning with Almost Sure Constraints, Information Theory and Applications Workshop.
[BibTeX] [Abstract] [Download PDF]
In this work, we study how to tackle decision-making for safety-critical systems under uncertainty. To that end, we formulate a Reinforcement Learning problem with Almost Sure constraints, in
which one seeks a policy that allows no more than $Δınℕ$ unsafe events in any trajectory, with probability one. We argue that this type of constraint might be better suited for
safety-critical systems as opposed to the usual average constraint employed in Constrained Markov Decision Processes and that, moreover, having constraints of this kind makes feasible
policies much easier to find. The talk is didactically split into two parts, first considering $Δ=0$ and then the $Δ≥ 0$ case. At the core of our theory is a barrier-based decomposition of
the Q-function that decouples the problems of optimality and feasibility and allows them to be learned either independently or in conjunction. We develop an algorithm for characterizing the
set of all feasible policies that provably converges in expected finite time. We further develop sample-complexity bounds for learning this set with high probability. Simulations corroborate
our theoretical findings and showcase how our algorithm can be wrapped around other learning algorithms to hasten the search for first feasible and then optimal policies.
abstract = {In this work, we study how to tackle decision-making for safety-critical systems under uncertainty. To that end, we formulate a Reinforcement Learning problem with Almost Sure constraints, in which one seeks a policy that allows no more than $Δınℕ$ unsafe events in any trajectory, with probability one. We argue that this type of constraint might be better suited for safety-critical systems as opposed to the usual average constraint employed in Constrained Markov Decision Processes and that, moreover, having constraints of this kind makes feasible policies much easier to find. The talk is didactically split into two parts, first considering $Δ=0$ and then the $Δ≥ 0$ case. At the core of our theory is a barrier-based decomposition of the Q-function that decouples the problems of optimality and feasibility and allows them to be learned either independently or in conjunction. We develop an algorithm for characterizing the set of all feasible policies that provably converges in expected finite time. We further develop sample-complexity bounds for learning this set with high probability. Simulations corroborate our theoretical findings and showcase how our algorithm can be wrapped around other learning algorithms to hasten the search for first feasible and then optimal policies.},
date = {05/27/2022},
day = {27},
event = {Information Theory and Applications Workshop},
host = {Christina Yu (Cornell)},
month = {05},
role = {Speaker},
title = {Reinforcement Learning with Almost Sure Constraints},
url = {http://mallada.ece.jhu/talks/202205-ITA.pdf},
year = {2022}
9. 2022-05-04: Model Free Learning of Regions of Attraction via Recurrent Sets, MURI Workshop.
[BibTeX] [Abstract] [Download PDF]
In this talk, we develop model-free methods for analyzing dynamical systems using data. Our key insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the
more relaxed notion of recurrence. A set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We then
leverage the notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. We first consider the problem of learning an inner approximation of the region
of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. We show that a τ-recurrent set containing a stable equilibrium must be a subset of
its ROA under mild assumptions. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by
sampling finite-length trajectories. Our algorithms process samples sequentially, which allows them to continue being executed even after an initial offline training stage. We will finalize
by providing some recent extensions of this work that generalizes Lyapunov’s Direct Method to allow for non-decreasing functions to certify stability and illustrate future research
abstract = {In this talk, we develop model-free methods for analyzing dynamical systems using data. Our key insight is to replace the notion of invariance, a core concept in Lyapunov Theory, with the more relaxed notion of recurrence. A set is τ-recurrent (resp. k-recurrent) if every trajectory that starts within the set returns to it after at most τ seconds (resp. k steps). We then leverage the notion of recurrence to develop several analysis tools and algorithms to study dynamical systems. We first consider the problem of learning an inner approximation of the region of attraction (ROA) of an asymptotically stable equilibrium point without an explicit model of the dynamics. We show that a τ-recurrent set containing a stable equilibrium must be a subset of its ROA under mild assumptions. We then leverage this property to develop algorithms that compute inner approximations of the ROA using counter-examples of recurrence that are obtained by sampling finite-length trajectories. Our algorithms process samples sequentially, which allows them to continue being executed even after an initial offline training stage. We will finalize by providing some recent extensions of this work that generalizes Lyapunov's Direct Method to allow for non-decreasing functions to certify stability and illustrate future research directions.},
date = {05/04/2022},
day = {04},
event = {MURI Workshop},
host = {Mario Sznaier (Northeastern), Necmiye Ozay (UMich)},
month = {05},
role = {Panelist},
title = {Model Free Learning of Regions of Attraction via Recurrent Sets},
url = {https://mallada.ece.jhu.edu/talks/202205-MURI.pdf},
year = {2022}
10. 2022-04-25: Embracing Low-Inertia in Power Systems: A Frequency Shaping Approach, University of California Berkeley.
[BibTeX] [Abstract] [Download PDF]
The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving
grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the
stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system’s low inertia to restore grid synchronism without
incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. We define system-level performance metrics that are of
practical relevance for power systems and systematically evaluate the performance of standard control strategies, such as virtual inertia and droop control, in the presence of power
disturbances. Our analysis unveils the relatively limited role of inertia in improving performance and the inability of droop control to enhance performance without incurring considerable
steady-state control effort. To overcome these limitations, we propose a novel frequency shaping control for grid-connected inverters -exploiting classical lead/lag compensation and model
matching techniques from control theory- that can significantly outperform existing solutions while using comparable control effort.
abstract = {The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system's low inertia to restore grid synchronism without incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. We define system-level performance metrics that are of practical relevance for power systems and systematically evaluate the performance of standard control strategies, such as virtual inertia and droop control, in the presence of power disturbances. Our analysis unveils the relatively limited role of inertia in improving performance and the inability of droop control to enhance performance without incurring considerable steady-state control effort. To overcome these limitations, we propose a novel frequency shaping control for grid-connected inverters -exploiting classical lead/lag compensation and model matching techniques from control theory- that can significantly outperform existing solutions while using comparable control effort.},
date = {04/25/2022},
day = {25},
event = {University of California Berkeley},
host = {Murat Arcak (Berkeley)},
month = {04},
role = {Lecture},
title = {Embracing Low-Inertia in Power Systems: A Frequency Shaping Approach},
url = {https://mallada.ece.jhu.edu/talks/202204-Berkeley.pdf},
year = {2022}
11. 2022-04-11: Embracing Low Inertia for Power System Frequency Control: A Frequency Shaping Approach, ECE Seminar, University of Michigan.
[BibTeX] [Abstract] [Download PDF]
The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving
grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the
stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system’s low inertia to restore grid synchronism without
incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. We define system-level performance metrics that are of
practical relevance for power systems and systematically evaluate the performance of standard control strategies, such as virtual inertia and droop control, in the presence of power
disturbances. Our analysis unveils the relatively limited role of inertia in improving performance and the inability of droop control to enhance performance without incurring considerable
steady-state control effort. To overcome these limitations, we propose a novel frequency shaping control for grid-connected inverters -exploiting classical lead/lag compensation and model
matching techniques from control theory- that can significantly outperform existing solutions while using comparable control effort.
abstract = {The transition of power systems from conventional synchronous generation towards renewable energy sources -with little or no inertia- is gradually threatening classical methods for achieving grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, to introduce virtual inertia to restore the stiffness that the system used to enjoy. In this talk, we seek to challenge this approach. We advocate taking advantage of the system's low inertia to restore grid synchronism without incurring excessive control efforts. To this end, we develop an analysis and design framework for inverter-based frequency control. We define system-level performance metrics that are of practical relevance for power systems and systematically evaluate the performance of standard control strategies, such as virtual inertia and droop control, in the presence of power disturbances. Our analysis unveils the relatively limited role of inertia in improving performance and the inability of droop control to enhance performance without incurring considerable steady-state control effort. To overcome these limitations, we propose a novel frequency shaping control for grid-connected inverters -exploiting classical lead/lag compensation and model matching techniques from control theory- that can significantly outperform existing solutions while using comparable control effort.},
date = {04/11/2022},
day = {11},
event = {ECE Seminar, University of Michigan},
host = {Johanna Mathieu},
month = {04},
role = {Lecture},
title = {Embracing Low Inertia for Power System Frequency Control: A Frequency Shaping Approach},
url = {https://mallada.ece.jhu.edu/talks/202204-UMich.pdf},
year = {2022}
12. 2022-03-30: Coherence and Concentration in Tightly-Connected Networks, Workshop on Synchronization in Complex Systems, Army Research Office.
[BibTeX] [Abstract] [Download PDF]
Achieving coordinated behavior— engineered or emergent—on networked systems has attracted widespread interest in several fields. This interest has led to remarkable advances in developing a
theoretical understanding of the conditions under which agents within a network can reach an agreement (consensus) or develop coordinated behavior, such as synchronization. However, much less
understood is the phenomenon of network coherence. Network coherence generally refers to nodes’ ability in a network to have a similar dynamic response despite heterogeneity in their
behavior. In this talk, we present a general framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property. More
precisely, for a networked system with linear dynamics and coupling, we show that the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior as the
network connectivity grows. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as the coherent
dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further illustrate how this framework can be
leveraged for obtaining accurate reduced-order models of coherent generators and tuning grid forming inverters to shape the coherent response of a power grid.
abstract = {Achieving coordinated behavior--- engineered or emergent---on networked systems has attracted widespread interest in several fields. This interest has led to remarkable advances in developing a theoretical understanding of the conditions under which agents within a network can reach an agreement (consensus) or develop coordinated behavior, such as synchronization. However, much less understood is the phenomenon of network coherence. Network coherence generally refers to nodes' ability in a network to have a similar dynamic response despite heterogeneity in their behavior. In this talk, we present a general framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property. More precisely, for a networked system with linear dynamics and coupling, we show that the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior as the network connectivity grows. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as the coherent dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further illustrate how this framework can be leveraged for obtaining accurate reduced-order models of coherent generators and tuning grid forming inverters to shape the coherent response of a power grid.},
date = {03/30/2022},
day = {30},
event = {Workshop on Synchronization in Complex Systems, Army Research Office},
host = {Derya Cansever (ARO), Jorge Cortés (UCSD), Fabio Pasqualetti (UCR)},
month = {03},
role = {Speaker},
title = {Coherence and Concentration in Tightly-Connected Networks},
url = {https://mallada.ece.jhu.edu/talks/202203-ARO-Workshop.pdf},
year = {2022}
1. 2021-11-03: Reinforcement Learning with Almost Sure Constraints, NSF TRIPODS PI Meeting.
[BibTeX] [Abstract] [Download PDF]
This talk aims to put forward the idea that learning to take safe actions in unknown environments (even with probability one guarantees) can be achieved without the need for an unbounded
number of exploratory trials; provided that one is willing to relax its optimality requirements mildly. To this aim, we look at two settings aimed at illustrating the feasibility of this
approach. We first focus on the canonical multi-armed bandit problem and seek to study the exploration-preservation trade-off intrinsic within safe learning. By defining a handicap metric
that counts the number of unsafe actions, we provide an algorithm for discarding unsafe machines (or actions), with probability one, that achieves constant handicap. Our algorithm is rooted
in the classical sequential probability ratio test, redefined here for continuing tasks. Under standard assumptions on sufficient exploration, our rule provably detects all unsafe machines in
an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. We
then study the problem of learning safe policies in the context of model-free constrained Markov decision processes. We propose the use of hard penalties/damage information, as a complement
for rewards, that can be used to learn which actions lead to constraint violations. We show that such penalties naturally arise from a separation principle that decomposes the value and
action-value functions into a reward component, and feasibility component–represented by a hard barrier function. We further develop an adaptive algorithm for learning this \emphbarrier
function, which incorporates the damage information and gradually reveals the safety constraints. In the process of learning such a barrier function, the policy is adapted so as to avoid
“bumping to the same rock twice”. Both algorithms can wrap around any other algorithm to optimize a specific auxiliary goal as they provide a safe environment to search for (approximately)
optimal policies.
abstract = {This talk aims to put forward the idea that learning to take safe actions in unknown environments (even with probability one guarantees) can be achieved without the need for an unbounded number of exploratory trials; provided that one is willing to relax its optimality requirements mildly. To this aim, we look at two settings aimed at illustrating the feasibility of this approach.
We first focus on the canonical multi-armed bandit problem and seek to study the exploration-preservation trade-off intrinsic within safe learning. By defining a handicap metric that counts the number of unsafe actions, we provide an algorithm for discarding unsafe machines (or actions), with probability one, that achieves constant handicap. Our algorithm is rooted in the classical sequential probability ratio test, redefined here for continuing tasks. Under standard assumptions on sufficient exploration, our rule provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines.
We then study the problem of learning safe policies in the context of model-free constrained Markov decision processes. We propose the use of hard penalties/damage information, as a complement for rewards, that can be used to learn which actions lead to constraint violations. We show that such penalties naturally arise from a separation principle that decomposes the value and action-value functions into a reward component, and feasibility component--represented by a hard barrier function. We further develop an adaptive algorithm for learning this \emphbarrier function, which incorporates the damage information and gradually reveals the safety constraints. In the process of learning such a barrier function, the policy is adapted so as to avoid ``bumping to the same rock twice''.
Both algorithms can wrap around any other algorithm to optimize a specific auxiliary goal as they provide a safe environment to search for (approximately) optimal policies.},
date = {11/03/2021},
day = {03},
event = {NSF TRIPODS PI Meeting},
host = {Maryam Fazel (UW), Rene Vidal (JHU)},
month = {11},
role = {Speaker},
title = {Reinforcement Learning with Almost Sure Constraints},
url = {https://mallada.ece.jhu.edu/talks/202111-TRIPODS.pdf},
year = {2021}
2. 2021-10-27: Coherence and Concentration in Tightly Connected Networks, Data-based Diagnosis of Networked Dynamical Systems, CCS 2021 Satellite Symposium.
[BibTeX] [Download PDF]
date = {10/27/2021},
day = {27},
event = {Data-based Diagnosis of Networked Dynamical Systems, CCS 2021 Satellite Symposium},
host = {Melvyn Tyloo, Laurent Pagnier, Robinn Delabays},
month = {10},
role = {Speaker},
title = {Coherence and Concentration in Tightly Connected Networks},
url = {https://mallada.ece.jhu.edu/talks/202110-CSS.pdf},
year = {2021}
3. 2021-09-09: Coherence and Concentration in Tightly Connected Networks, Resilient Autonomous Energy Systems Workshop, National Renewable Energy Laboratory.
[BibTeX] [Download PDF]
date = {09/09/2021},
day = {09},
event = {Resilient Autonomous Energy Systems Workshop, National Renewable Energy Laboratory},
host = {Andrey Berstein (NREL), Bai Cui (NREL)},
month = {09},
role = {Speaker},
title = {Coherence and Concentration in Tightly Connected Networks},
url = {https://mallada.ece.jhu.edu/talks/202109-NREL.pdf},
year = {2021}
4. 2021-04-13: Incentive Analysis and Coordination Design for Multi-Timescale Electricity Markets, Epstein Institute Seminar, University of Southern California.
[BibTeX] [Abstract] [Download PDF]
This talk discusses incentives and coordination requirements that arise when heterogeneous participants bid in electricity markets that operate at different timescales. First, we consider the
conventional timescales of market clearing, spanning 5 minutes to several hours ahead, and investigate the incentives for price manipulation that market participants (generators and loads)
have in a two-stage settlement market. Our analysis unveils the importance of accounting for both generators’ and loads’ strategic behavior in two-stage markets, even when the consumers’
demand is inelastic! Precisely, we show that loads can exploit generators’ strategic bidding and maintain a systematic difference between the forward and spot prices, the latter being higher
than the former. Such a strategy does bring down demand-side payments and undermines supply-side market power. Second, we consider the problem of co-optimizing generation resources with
different timescale characteristics. To that end, we frame and study a joint problem that optimizes both slow-timescale economic dispatch resources and fast-timescale frequency regulation
resources. We provide sufficient conditions to optimally decompose the joint problem into slow and fast timescale problems. These slow and fast timescale problems have appealing
interpretations as the economic dispatch and frequency regulation problems, respectively. We further provide a market implementation for the fast-timescale problem. In this implementation,
participants receive prices and dispatch and dynamically update their bids according to either a dynamic gradient play or best response. Under price-taking assumptions, our market
implementation is guaranteed to converge to the optimal (efficient) allocation even in the presence of generator dynamics. A by-product of this solution is that frequency restoration and
thermal limits are automatically guaranteed.
abstract = {This talk discusses incentives and coordination requirements that arise when heterogeneous participants bid in electricity markets that operate at different timescales. First, we consider the conventional timescales of market clearing, spanning 5 minutes to several hours ahead, and investigate the incentives for price manipulation that market participants (generators and loads) have in a two-stage settlement market. Our analysis unveils the importance of accounting for both generators' and loads' strategic behavior in two-stage markets, even when the consumers' demand is inelastic! Precisely, we show that loads can exploit generators' strategic bidding and maintain a systematic difference between the forward and spot prices, the latter being higher than the former. Such a strategy does bring down demand-side payments and undermines supply-side market power. Second, we consider the problem of co-optimizing generation resources with different timescale characteristics. To that end, we frame and study a joint problem that optimizes both slow-timescale economic dispatch resources and fast-timescale frequency regulation resources. We provide sufficient conditions to optimally decompose the joint problem into slow and fast timescale problems. These slow and fast timescale problems have appealing interpretations as the economic dispatch and frequency regulation problems, respectively. We further provide a market implementation for the fast-timescale problem. In this implementation, participants receive prices and dispatch and dynamically update their bids according to either a dynamic gradient play or best response. Under price-taking assumptions, our market implementation is guaranteed to converge to the optimal (efficient) allocation even in the presence of generator dynamics. A by-product of this solution is that frequency restoration and thermal limits are automatically guaranteed.},
date = {04/13/2021},
day = {13},
event = {Epstein Institute Seminar, University of Southern California},
host = {Jong-Shi Pang (USC), Suvrajeet Sen (USC)},
month = {04},
role = {Speaker},
title = {Incentive Analysis and Coordination Design for Multi-Timescale Electricity Markets},
url = {https://mallada.ece.jhu.edu/talks/202104-Epstein.pdf},
year = {2021} | {"url":"https://mallada.ece.jhu.edu/talks-and-seminars/","timestamp":"2024-11-02T14:06:48Z","content_type":"text/html","content_length":"207898","record_id":"<urn:uuid:fe1896ba-51bf-436f-ad9c-9f9e558e84eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00654.warc.gz"} |
Profit and Loss - Arithmetic Aptitude Questions and Answers - Discussion Q.No.188
Discussion :: Profit and Loss
1. A man mixes two types of rice (X and Y) and sells the mixture at the rate of Rs. 17 per kg. Find his profit percentage.
I. The rate of X is Rs. 20 per kg.
II. The rate of Y is Rs. 13 per kg.
I alone sufficient while II alone not sufficient to answer
II alone sufficient while I alone not sufficient to answer
Either I or II alone sufficient to answer
Both I and II are not sufficient to answer
Both I and II are necessary to answer
Answer : Option D
Explanation :
The ratio, in which X and Y are mixed, is not given.
So, both I and II together cannot give the answer. | {"url":"https://freshergate.com/arithmetic-aptitude/profit-and-loss/discussion/188","timestamp":"2024-11-06T18:57:28Z","content_type":"application/xhtml+xml","content_length":"44443","record_id":"<urn:uuid:52fff275-d271-4cc3-b7fe-ab1e0948f9e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00619.warc.gz"} |
Fluid Velocity Distribution within Pipes
Related Resources: fluid flow
Fluid Velocity Distribution within Pipes
Fluid Velocity Distribution within Pipes Equations
Velocity distribution at a cross section will follow a parabolic law of variation for laminar flow. Maximum velocity is at the center and is twice the average velocity. The following application are
• turbulent flows,
• smooth pipes,
• rough pipes,
• rough or smooth boundaries
The equation of the velocity profile for laminar flow can be expressed as
Eq. 1
$v={v}_{c}-\left(\frac{\gamma {h}_{L}}{4\mu L}\right){r}^{2}$
For turbulent flows, more uniform velocity distribution results. From experiments of Nikuradse and others, equations of velocity profiles in terms of center velocity vc or shear velocity µ[c] follow.
(a) An empirical formula is
Eq. 2
n = 1/7 for smooth tubes, up to Re = 100,000
n = 1/8 | for smooth tubes for Re from 100,000 to 400,000
(b) For smooth pipes,
Eq. 3
$v={v}_{\ast }\left[5.5+5.75\mathrm{log}\left(y{v}_{\ast }/v\right)\right]$
For the yv[*]/v term,
Eq. 4
v[*] = (τ/ρ)^0.5
(c) For smooth pipes (for 5000 < Re < 3,000,000) and for pipes in the wholly rough zone,
Eq. 5
$\left({v}_{c}-v\right)=-2.5\sqrt{{v}_{o}/\rho }\text{}\mathrm{ln}\left(y/{r}_{o}\right)=-2.5{v}_{\ast }\text{}\mathrm{ln}\left(y/{r}_{o}\right)$
In terms of average velocity V, Vennard suggests that V/v[c] may be written
Eq. 6
(d) For rough pipes,
Eq. 7
$v={v}_{\ast }\left[8.5+5.75\text{}\mathrm{log}\left(y/ϵ\right)\right]$
where ε is the absolute roughness of the boundary.
(e) For rough or smooth boundaries,
Eq. 8
Eq. 9
γ = specific (or unit) weight of fluid
h[L] = lost head
µ = absolute viscosity
L = Length of pipe
v = kinematic viscosity of the fluid in ft^2/sec or m^2/s
v[c] = Center velocity of fluid
v[*] = Shear velocity
r = radius, ft ot m
r[o] = radius of pipe
y = depth distance
V = mean (average) velocity in ft/sec or m/s
f = friction factor (Darcy) for pipe flow
µ = absolute viscosity in lb-sec/ft^2 or N-s/m^2
Schaum's Outline of Fluid Mechanics and Hydraulics | {"url":"https://www.engineersedge.com/fluid_flow/fluid_velocity_distribution_16361.htm","timestamp":"2024-11-07T16:52:54Z","content_type":"text/html","content_length":"27973","record_id":"<urn:uuid:2405dfee-d754-418f-8900-0a2b08abd0c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00160.warc.gz"} |
Creatring a custom scoring metric
Hello @Alex_Combessie
I want to code a custom scoring function for scikit-learn's average-precision-score. It's a for a Keras image binary classification model. I'm not familiar with how the Dataiku API works, but I want
to pass the array dimensions to y_valid and y_pred.
Here's what I have:
from sklearn.metrics import average_precision_score
def score(y_valid, y_pred):
Custom scoring function.
Must return a float quantifying the estimator prediction quality.
- y_valid is a pandas Series
- y_pred is a numpy ndarray with shape:
- (nb_records,) for regression problems and classification problems
where 'needs probas' (see below) is false
(for classification, the values are the numeric class indexes)
- (nb_records, nb_classes) for classification problems where
'needs probas' is true
- [optional] X_valid is a dataframe with shape (nb_records, nb_input_features)
- [optional] sample_weight is a numpy ndarray with shape (nb_records,)
NB: this option requires a variable set as "Sample weights"
average_precision = average_precision_score(y_valid, y_pred)
return average_precision
Best Answer
• Hi,
That seems to be a good piece of code to start with. Are you encountering an error when training? If so, could you post the error message, please?
Hope it helps,
• Works great, thanks for verifying | {"url":"https://community.dataiku.com/discussion/4865/creatring-a-custom-scoring-metric","timestamp":"2024-11-04T13:56:59Z","content_type":"text/html","content_length":"408661","record_id":"<urn:uuid:8bf83c58-7726-4e8a-98c2-f746c6ca92df>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00653.warc.gz"} |
Sudoku Simple English Wikipedia The Free Encyclopedia Printable | Sudoku Printables
Sudoku Simple English Wikipedia The Free Encyclopedia Printable
Sudoku Simple English Wikipedia The Free Encyclopedia Printable – If you’ve had any issues solving sudoku, you’re aware that there are several kinds of puzzles that are available, and sometimes, it
can be difficult to choose which ones to tackle. However, there are many different methods to solve them. And you’ll find that an printable version can be an ideal way to start. Sudoku rules are
similar to the rules for solving other puzzles but the actual format varies slightly.
What Does the Word ‘Sudoku’ Mean?
The word “Sudoku” is an abbreviation of the Japanese words suji and dokushin meaning “number” and “unmarried person’, respectively. The objective of the puzzle is to fill every box with numbers in a
way that each number from one to nine appears just one time on each horizontal line. The term Sudoku is an emblem belonging to the Japanese puzzle firm Nikoli that was founded in Kyoto.
The name Sudoku comes in the Japanese word”shuji wa Dokushin Ni Kagiru meaning ‘numbers should stay single’. The game consists of nine 3×3 squares that have nine smaller squares inside. The game was
originally known as Number Place, Sudoku was a mathematical puzzle that stimulated development. While the origins of the game are not known, Sudoku is known to have roots that go back to the earliest
number puzzles.
Why is Sudoku So Addicting?
If you’ve ever played Sudoku you’ll realize how addictive the game can be. An Sudoku addicted person will be unable to stop thinking about the next puzzle they can solve. They’re constantly thinking
about their next adventure, while other aspects of their lives tend to be left to wayside. Sudoku is a game that can be addictive however it’s essential that you keep the addictive potential of the
game in check. If you’ve developed a craving for Sudoku here are a few ways to curb your addiction.
One of the best methods of determining if you’re addicted to Sudoku is to look at your behaviour. Most people carry magazines and books with them, while others simply browse through social media
updates. Sudoku addicts, however, take newspapers, books, exercise books, and smartphones everywhere they go. They can be found for hours solving puzzles, and they aren’t able to stop! Some find it
easier to finish Sudoku puzzles than their regular crosswords, which is why they don’t stop.
Sudoku Wikipedia 6 Square Sudoku Printable
What is the Key to Solving a Sudoku Puzzle?
A good strategy for solving an printable sudoku game is to practice and experiment using different methods. The most effective Sudoku puzzle solvers do not employ the same strategy for every single
puzzle. The most important thing is to try out and test various approaches until you discover the one that is effective for you. After some time, you’ll be able solve puzzles without difficulty! But
how do you learn how to solve a printable sudoku challenge?
To begin, you need to grasp the basic concept of suduko. It’s a game that requires reasoning and deduction, and you need to examine the puzzle from different perspectives to find patterns and work
out the puzzle. When solving suduko, it is important to look at the puzzle from a variety of angles. suduko puzzle, do not try to guess the numbers. instead, you should look over the grid for clues
to identify patterns. This technique to rows and squares.
Related For Sudoku Puzzles Printable | {"url":"https://sudokuprintables.net/sudoku-wikipedia-6-square-sudoku-printable/sudoku-simple-english-wikipedia-the-free-encyclopedia-printable/","timestamp":"2024-11-04T23:15:32Z","content_type":"text/html","content_length":"26396","record_id":"<urn:uuid:ee691f12-2ce4-4dcc-863e-a8f3e7296562>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00312.warc.gz"} |
Matrix Transformations - Documentation - Unigine Developer
Matrix Transformations
A lot of calculations in Unigine are performed by using matrices. Actually, matrix transformations are one of the main concepts of 3D engines. This article contains an explanation of matrix
transformations with usage examples.
See Also
In simple terms, a matrix in 3D graphics is an array of numbers arranged in rows and columns:
Usually, 4x4 matrices are used. Such size (4x4) of matrices is caused by the translation state in 3D space. When you put a new node into the world, it has a 4x4 world transform matrix that defines
the position of the node in the world.
In Unigine, matrices are column major (column-oriented). Hence, the first column of the transform matrix represent the X vector v1, the second represent Y vector v2, the third represent Z vector v3,
and the fourth represent the translation vector t. First three columns show directions of axes and the scale of the origin. The last column contains the translation of the local origin relatively to
the world origin.
Identity Matrix
The world origin has the following matrix:
This matrix is called identity matrix, a matrix with ones on the main diagonal, and zeros elsewhere. If a matrix will be multiplied by the identity matrix, it won't change anything: the resulting
matrix will be the same as it was before multiplying.
If the local origin has the identity matrix, it means the local origin and the world origin are coincident.
To change the orientation of the local origin, the first three columns of the matrix should be changed.
To rotate the origin along different axes, you should use proper matrices:
In the matrices given above, α is a rotation angle along the axis.
The next matrix shows the rotation of the local origin along the Y axis at 45 degrees:
The last column of the transform matrix shows the position of the local origin in the world relatively to the world origin. The next matrix shows the translation of the origin. The translation vector
t is (3, 0, 2).
The length of the vector shows the scale coefficient along the axis.
To calculate the vector length (also known as magnitude), you should find a square root of the sum of the squares of vector components. The formula is the following:
|vector length| = √(x² + y² + z²)
The following matrix scales the local origin up to 2 units along all axes.
Cumulating Transformations
The order of matrix transformations (scaling, rotation and translation) is very important.
The order of cumulating transformations is the following:
1. Translation
2. Rotation
3. Scaling
This is the formula of the transformation order:
TransformedVector = TranslationMatrix * RotationMatrix * ScaleMatrix * Vector
Here is an example which demonstrates different positions of the local origin related to the world origin.
Translation * Rotation order Rotation * Translation order
On the left picture, the local origin was translated first and then rotated; on the right picture, the local origin was rotated first and then translated. All values (rotation angle, translation
vector) are the same, but the result is different.
This example shows what happens if you choose another order of matrix transformations.
The code example below adds an ObjectMeshStatic object to the world. In the first case we use the translation * rotation order, in the second case we use the rotation * translation order.
// declare ObjectMeshStatic
ObjectMeshStatic mesh;
/* add a mesh as a node to the editor
Node add_editor(Node node) {
return node_remove(node);
/* the init() function of the world script file
int init() {
/* create a camera and add it to the world
Player player = new PlayerSpectator();
// add the mesh to the editor and set the material
mesh = add_editor(new ObjectMeshStatic("matrix_project/meshes/statue.mesh"));
mesh.setMaterial("mesh_base", "*");
// create the matrix of rotation at 270 degrees angle around Z axis
mat4 rotation = rotateZ(-90);
log.message("rotation matrix is: %s\n", typeinfo(rotation));
// create the translation matrix
mat4 translation = translate(vec3(0,7,0));
log.message("translation matrix is: %s\n", typeinfo(translation));
// cumulate transformations by using the current transformation matrix
mat4 transform = box.getTransform() * translation * rotation;
log.message("transformation matrix is:%s\n", typeinfo(mesh.getTransform()));
return 1;
To change the order, just change the line of cumulating transformations:
mat4 transform = box.getTransform() * rotation * translation;
The result will be different. The pictures below show the difference (camera is located at the same place).
Translation * Rotation order Rotation * Translation order
The pictures above show the position of the meshes related to the world origin.
Matrix Hierarchy
Another important concept is a matrix hierarchy. When a node is added into the world as a child of another node, it has a transform matrix that is related to the parent node. That is why the Node
class has different functions: getTransform(), setTransform() and getWorldTransform(), setWorldTransform() that return the local and the world transformation matrices respectively.
If the added node has no parent, this node uses the World transformation matrix.
What is the reason of using matrix hierarchy? To move a node relatively to another node. And when you move a parent node, child nodes will also be moved, that is the point.
Parent origin is the same with the world origin Parent origin has been moved and the child origin has also been moved
Pictures above show the main point of the matrix hierarchy. When the parent origin (node) is moved, the chlld origin will also be moved and the local transformation matrix of the child would not be
changed. But the world transformation matrix of the child will be changed. If you need the world transformation matrix of the child related to the world origin, you should use the getWorldTransform()
, setWorldTransform() functions; in case, when you need the local transformation matrix of the child related to the parent, you should use the getTransform(), setTransform() functions.
Consider the following example that shows the difference between the local and world transformation matrices.
This code is from the UnigineScript world script file. It creates two nodes (child and parent) by using the box.mesh file.
/* declare ObjectMeshStatic objects for nodes
ObjectMeshStatic box_1;
ObjectMeshStatic box_2;
/* add a mesh as a node to the editor
Node add_editor(Node node) {
return node_remove(node);
/* the init() function of the world script file
int init() {
/* create a camera and add it to the world
Player player = new PlayerSpectator();
// add the box mesh to the editor
box_1 = add_editor(new ObjectMeshStatic("matrix_project/meshes/box.mesh"));
// set the transformation and the material for the first box mesh
mat4 transform = translate(vec3(0.0f,0.0f,0.0f));
box_1.setMaterial("mesh_base", "*");
// add the second box mesh to the editor
box_2 = add_editor(new ObjectMeshStatic("matrix_project/meshes/box.mesh"));
// set the transformation and the material for the second box mesh
mat4 transform1 = translate(vec3(0.0f,2.0f,1.0f));
box_2.setMaterial("mesh_base", "*");
// add the second box mesh as a child to the first box mesh
// show the transformation matrix and the world transformation matrix of the child node
log.message("transformation matrix of the child node: %s\n", typeinfo(box_2.getTransform()));
log.message("world transformation matrix of the child node: %s\n", typeinfo(box_2.getWorldTransform()));
return 1;
After running this code, we get the following result:
transformation matrix of the child node: dmat4: (1 0 0 0) (0 1 0 0) (0 0 1 0) (0 2 1 1)
world transformation matrix of the child node: dmat4 (1 0 0 0) (0 1 0 0) (0 0 1 0) (0 2 1 1)
The matrices are the same, since the parent box_1 node has the zero transformation (0,0,0). It means its origin and the world origin are coincident. If we change the transformation of the first mesh
to another, for example:
mat4 transform = translate(vec3(2.0f,2.0f,2.0f));
We will get another result:
transformation matrix of the child node: dmat4: (1 0 0 0) (0 1 0 0) (0 0 1 0) (0 2 1 1)
world transformation matrix of the child node: dmat4 (1 0 0 0) (0 1 0 0) (0 0 1 0) (2 4 3 1)
As you can see, the local transformation matrix remains the same but the world transformation matrix has been changed. This means that the second node has the same position relatively to the first,
but it has another position relatively to the world origin because the position of the parent node has been changed.
Parent node has the same origin with the world origin Parent node has been moved and the child node has also been moved automatically
Last update: 2017-07-03
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Badiou's Mathematical Platonism
The Meta-Ontological Exception:
Notes on Badiou's Mathematical Platonism
“The concept of model is strictly dependent, in all its successive stages, on the (mathematical) theory of sets. From this point of view, it is already inexact to say that the concept connects formal
thought to its outside. In truth, the marks ‘outside the system’ can only deploy a domain of interpretation for those of the system within a mathematical envelopment, which preordains the former to
the latter. […] Semantics here is an intramathematical relation between certain refined experimental apparatuses (formal systems) and certain ‘cruder’ mathematical products, which is to say, products
accepted, taken to be demonstrated, without having been submitted to all the exigencies of inscription ruled by the verifying constraints of the apparatus." (AB, The Concept of Model)
Zachary-Luke Fraser advances a nice rejoinder to Ray Brassier’s outstanding analysis from Nihil Unbound. There Brassier asks what precise role metaontology comes to play vis the distinction between
ontological and non-ontological situations. On the one hand, metaontology is clearly not ontology itself, since the latter only speaks of sets while the former speaks of presentations in general. As
such metaontology cannot be said to be ‘founded on the void’ in the same way as ontology, since it operates with resources strictly external to the latter. On the other hand, metaontology suspends
Leibniz’s thesis, which asserts the identity of being and the One (or being and unity), declaring the latter to be a mere operation (the count-as-one which structures every presentation). So it seems
that metaontology stands somewhere inbetween the two ‘fields’ of presentation, enacting the transitivity of the very concept of presentation across the two domains, affirming the identity of ontology
and mathematics. Luke Fraser thus seeks to dissolve the pertinence of the polarity between discourse and world in Badiou's mathematical Platonism by arguing that non-ontological presentations,
thought in their being, must be already mathematized, i.e. must be thought of as models for set theory.
This way, it is not that set-theory qua singular discourse is ‘connected’ to its outside in non-ontological presentations via metaontology. Rather, all presentations are thinkable ontologically only
as mathematically treated as a domain for the testing of the set-theoretical axioms: thus stipulating that insofar as ontology thinks of the form of presentations as sets, it thinks them in their
being. This is part of the ‘mathematical Platonism’ that dissolves the transcendent bond between a formal language and its outside, and thereby dissolves the tension between a true materialism and
what would appear to be a discursive brand of idealism anchored in set theoretical mathematics. So ontology and non-ontological situations are at once indiscernible as immanent (every presentation is
immanently presupposed as having being mathematically intelligible in its being) and as transcendent (ontology is just one situation among others in itself;no situation contains all others; the
concept of general presentation formally described by ontology is bridged through the meta-ontological decision):
“The point to which this brings us is this: To the extent that a mathematical ontology of concrete situations is possible, it must be possible to treat these as ‘models’ of set theory. Accordingly,
these situations must be apprehended as being already mathematical in some sense, however crudely or vaguely understood. To the extent that ontology avoids the ‘empiricist’ mandate of being an
‘imitative craft’ (a characterisation against which Badiou rails in The Concept of Model), the correspondence between the ontological situation and its outside can be classified as neither a relation
of transcendence nor immanence, but must be thought as a point of indiscernibility between the two. This is the source of all the obscurity attributable to the ‘Platonist’ position of metaontology,
which forces us to ask, as Brassier does”,
“Where is Badiou speaking from in these decisive opening editations of Being and Event? Clearly, it is neither from the identity of thinking and being as effectuated in ontological discourse, nor
from within a situation governed by knowledge and hence subject to the law of the One. […M]etaontological discourse seems to enjoy a condition of transcendent exception vis-àvis the immanence of
ontological and non-ontological situations”(NU: Chapter IV)”
The relation between ontology/non-ontology is not strictly transcendent because non-ontological situations can only be thought in their being as already mathematized, i.e. for ordinary situations to
be treated as models requires their mathematization into domains for set theory. It is not strictly immanent either because the ontological situation does not present all others, but only their
general form as ‘sutured to their being’, inside the characteristic mode of thought that is ontology, i.e. there is no presentation encompassing all others, but only presentations of presentations,
and void/nothingness.
However, the equation of mathematics with ontology, and thus the affirmation that set theory alone renders the form of presentation in all non-ontological situations, can only be performed by
metaontology as declaring the equation of being qua presentation and inconsistent multiplicity qua the inertia of the domain of set theory. This in turn requires the primitive inconsistency of
presentation to be foundational for the axiomatic, which Badiou perceives in the existential inscription of the pure name of the void as the primitive and radically non-phenomenological sign from
which the entire stock of operations are woven in continuity with the axioms of set theory and on the basis of the primitive relation of ‘belonging’ alone. Metaontology is thus prerequisite to
establish the indiscernibility of set theory as a unique situation and the wealth of possible non-ontological situations insofar as they are thought in their being.
Why must we assume that inconsistent multiplicity underlies the consistency of the pure multiple, which amounts to thinking being as fundamentally ‘without unity’ or as being-nothing? It is because
the primitive subtraction of being from the count-as-one is thought in accordance with the Parmenidean statement that whatever is not One must be by necessity multiple. Since multiplicity resists
unity, and given that being is essentially multiple, any discourse on being qua being will begin from the assumption of the non-being of the One, or the lack of any foundational figure of oneness.
Badiou assigns the multiplicity of presentation to its properly discursive (ontological) domain through the unique consistency of ZF set theory, which proceeds in the assumption of existence of a set
with an empty extension. Once the operation of belonging to x can be said to be tantamount to ‘being presented to x’ via the speculative move, set theoretical strictures appear fittingly to depart
from the sole assumption of the lack of a phenomenological given. Badiou explains the presupposition of lack required to render multiple being discursively through the sterility of the presentational
domain set theory inscribes with the mark of the void. This identifies presentation with inconsistent multiplicity, and as such tethers being to set theory as ontology, enacting the thinking of that
which primitively proceeds from the absence of unity, i.e. from accepting Parmenides’ embrace of the form of presentation as essentially multiple. The ‘suture to being’, again, thus remains strictly
Of course, the additional assumption, also spotted by Brassier, is that non-ontological presentation is distinguished insofar as it presents the One, which necessarily makes the ontological situation
qua theory of the multiple as the unique situation in which presentation is thought of in ‘its being’. The fictive being of the one is expressed as set theory operates consistently on the basis of
the primordial lack of the void itself. It is thus that all unity is given in ontology solely against the impenetrable background of the void’s empty inertia. Each set is constructed on the basis of
the primitive lack indexed by the void’s proper name, and is determined purely extensionally in terms of what it presents (which is always nothing but a function of replacement operating over the
proliferation of subsets woven from the void alone -this is guaranteed via the axioms of void, powerset and replacement). Oneness remains thus the result of the operation of belonging, which presents
sets whose being is nevertheless indexed to the void in the last instance.
“This splitting of unity into operation and effect is integral to the thinking of presentation and the metaontological delivery of the formal thinking of presentation to mathematics, and
specifically, to set theory. It is set theory itself that formalizes this split, and provides us with a figure of multiplicity adequate to the thinking of presentation, and, more dramatically, to the
univocal determination of the existent as presentation (and so of presentation as the presentation of presentations). To speak of a presentation is to speak of presentation affected by an operation
of the count-as-one, and not of a presentation solidified according to the intrinsic unity characteristic of entity. The unity of a presentation is always extrinsic.” [ZLF Pg. 68]
It is this split between the one as an operation and as effect which becomes occluded in non-ontological presentations, where being attains fictive unity in closing this gap. It thus assumes itself
identical to what it presents, or to its own singleton: x = {x}, thus violating the constraint set to it in well founded set theory by the axiom of foundation and extensionality (the couple of which
require a set’s extensional determination or identity, and its incapacity to belong to itself). This indistinction between the one as operation and the one as result is thus the violation of what
Zachary Luke Fraser designates as the two principles of multiplicity for Badiou:
a) Material component - Every set is extensionally determined.[1]
b) Formal unity – Every set is different from itself by a pure differentiation ( grounded in the axiom of foundation).
Consistency is thus the formal unity in which a presentation is given, its being-counted-as-one (yet different from itself) in the situation (one as result). On the other hand, presentation itself
remains necessarily inconsistent as the retroactive presupposition of the multiple gathered is but merely counted-as-one, and thus presupposes its prior existence, not exhausted by what it unifies or
presents in its formal unity (one as operation). In ordinary situations where this gap is closed, we do not think according to the being of what is presented, which necessarily differs from itself
formally but on the basis of the fictional consistency of unity in which it appears.
The evaluation of such situations in the ‘moment of the One’, as we know, becomes properly the subject of Badiou’s Greater Logic in Logics of World and the phenomenology of objects.
- Annotations on Meditation 26 (The Concept of Quantity) in Alain Badiou’s Being and Event
In Alain Badiou’s theoretical framework set-theory as ontology comes to explicate the notion of quantity through some technical concepts worth elucidating in close detail, even if, as Badiou admits
that the formal exposition of the ontological operations can exceed philosophical (and therefore meta-ontological) interest. In particular, it is easy to overlook Badiou’s explication of the concept
of a ‘function’, since it is delegated to a short (but doubtlessly crucial) Annex at the end of the book. There, a function is described straightforwardly as a particular kind of multiple, in
unproblematic continuity with the strictures of set-theory and the pure multiple. In what follows we’ll try to elucidate the surrounding notions, since the prose in the book lends itself to easy
A function f of a given set α to a set β, which can be written f(α) = β, establishes a one-to-one correspondence between the two sets, where it is understood that:
- For every element of α there corresponds via f an element of β.
- For every two different elements of α there corresponds two different elements of β
- For every element of β there corresponds via f, an element of α.
At this point, the set-theoretical grounding becomes quite necessary to follow Badiou’s argumentation, since the concept of ‘function’ outlined above is defined, after all, as simply a particular
kind of multiple. What kind of a multiple is at stake here? Here we must move to Appendix 2 of the book, which provides the sought for clarification.
Badiou begins by describing multiples which constitute relations between other multiples. These are structured as series of ordered pairs, and are written as follows:
Let’s assume the existence of a relation R between two given multiples α and β: R(α, β). Badiou describes relation as getting behind two ideas: that of the pairing of the two elements, and that of
their sequence or order. This second condition guarantees that even if R (α, β) obtains in a given situation, it is possible that R(β, α) does not. The first condition entails that all relations can
be expressed as consisting of two element multiples, written in the form <α, β>, so that to say that there exists a relation R between two existing elements α and β finally amounts to no more than
saying: <α, β> ε R. Given that for any two existing elements α and β there exists necessarily the set which has α and β as its sole elements {α, β}[2], although se will see right away that this set
is not identical to <α, β> . The only problematic aspect pending is finally that of order, and thus of the stipulated asymmetry between R (α, β) and R(β, α):
Interestingly enough, the ‘ordered pair’ solicited by Badiou is not simply the pairing of α and β, but actually the pairing of the singleton of α, and the pairing of α and β. So we get:
<α, β> ↔ { {α}, {α, β} }
This set must exist, given that the existence of α and β guarantees the existence of their respective singletons, as well as their pair. Therefore the union of either of the first terms with their
conjunction must also exist. In other words, for any given two multiples α and β there exist two different possible ordered pairings, which are not identical:
<α, β> ≠ < β, α> .↔. { {α}, {α, β} } ≠ { {β}, {β, α} }
Notice, however, that both ordered pairings are completely indifferent with respect to order in the terms of the set {β, α} / {α, β}; which are transparently identical sets. The impossibility of
substitution and thus the asymmetry of the two orderings laid above occurs in the difference occasioned by the choice between {α} or {β}. This must mean that an ordered pair always consists, for any
two elements supposed existent, of the two-element set consisting of the singleton of one of the two elements and the two-element set consisting of the two already given elements. Additionally, it is
implied that:
<α, β> = <г, у> .->. (α = г) & (β = y)
Finally, to say a relation R obtains between two given sets α and β entails:
<α, β> ε R or <β, α> ε R
Having established that a relation is a multiple composed of ordered pairs, Badiou proceeds to explain how a function may be described a particular kind of relation. The trick here consists in
grasping adequately the abovementioned idea of ‘correspondence’. Let us assume a function f that makes a multiple β correspond to α: f(α) = β. Having established functions are relations, and
relations are sets of ordered pairs, it plainly follows that functions are sets of ordered pairs. If we then allow R[f] to stand for the function of α to β, we can write as follows:
<α, β> ε R[f]
But the peculiarity of the function resides on the uniqueness of β, so that no other element can correspond to it by it. This means that for any two multiples β and y that correspond to α via a
function R, it must be the case that β and y are identical. Formally we write:
[f(α) = β .&. f(α) = y] -> β = y
Or, alternatively:
(<α, β> ε R [f ].&. <α, y > ε R[f]) -> β = y
If we want to unpack this formula, we write:
({{α}, {α, β}} ε R[f ].&. {{α}, {α,y}} ε R[f]) -> β = y
With this Badiou completes his reduction of the concept of relation to pure set-theoretical constructed multiplicities. The next step is to ground the comparison between sets in the series of
ordinals (natural multiples[3]). With respect to a multiple’s ‘size’ or ‘magnitude’, there always exists an ordinal which is equal to it (which is not to say only natural multiples exist; we know
this isn’t true given the existence of historical multiples). Badiou claims that thus ‘nature includes all thinkable orders of size” [BE: Pg. 270]. Here things turn a confusing, since Badiou doesn’t
really provide an example until later. We can, however, give a very simple case to illustrate how exactly this happens.
First, recall that the series of ordinals are woven from the void alone, as the structured sequence or passage from the void into its singleton, and thus consecutively in serial manner. If we repeat
the basic example laid above where R[f] stands for the function of α to β. We got:
[R(α, β)] ↔ [f(α) = β] ↔ [< α, β> ε R[f]]
Or, more explicitly:
{ {α}, {α, β} } ε R [f]
However, we can easily see that the multiple thus produced has the same power as the ordinal which composes the Von Neumann ordinal Two, and which is guaranteed given the sequence of ordinals:
Π: {{Ø}, {Ø, {Ø}}
Notice, however, that although this ordinal certainly has the same power as the given set, there’s an infinity of ordinals with the same power as the laid set: we can easily imagine the ordinal:
{{{Ø}}, {{Ø}, {{Ø}}} and successive variants, all with the same power. The requirement is merely that there will be at least one ordinal with the same power. Identity as such is guaranteed through
the comparison of a set's extensions, where the axiom of foundation guarantees the void lingers within each form of presentation (forbidding non-wellfounded sets from proliferation indefinately;
self-belonging becomes forbidden). I will continue with these notes later.
[1] See the annotations below to explain the procedure of the determination of the identity of a set on the basis of the extensional determination of each set; which delivers us to the concept of
[2] See Being and Event, Meditation 12.
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The mass of a physical system is its intrinsic energy.
I expect that Zoran will object to some of what I have written there (if not already to my one-sentence definition above), but since I cannot predict how, I look forward to his comments.
I think “intrinsic energy” is at least a good hint for what mass is.
It’s interesting to notice that only a tiny fraction of the mass of protons and neutrons, hence of atoms and everyhting built from them is rest-mass of the quarks that they consist of. Almost all the
mass of the proton is actually in the gluon “gas” that hoids the quarks together. Gluons themselves are massless, so this is all just binding energy.
(And of course, even the tiny fraction of “true rest mass” of the quarks inside the proton is, according to the standard model, just a kind of coupling energy to the Higgs field. )
Recently I read an entertaining article in the otherwise very dry journal of the German Physical Socienty that i am subscribed to, where the author started considering the total mass of the
observable universe and then step by step emphasized how it consists mostly of contributions that are not of the kind that we think of as mass in an everyday-sense: first at cosmological scales
baryonic mass is just a small fraction among mass of dark matter (according to the standard concordance $\Lambda CDM$ model, at least) (and of course both are just a small fraction of the total
energy density, which is mostly “dark energy”). Then among the baryonic mass a huge fraction is really just binding energy of quarks.
So there is surprisingly little “genuine mass” around us, in a way.
I do not complain that this would be wrong, but that there are several kinds of mass (in GR equivalent) which have profoundly different definition. So one should talk about inertial mass given in
terms of dynamics. Note for example the “effective” mass of holes in solid state physics…But I have no time to improve this now. I will be mostly away in next 10 days.
At the risk of getting told I’m an idiot again, I have spent several years pondering (and writing about) the nature of mass. As Tom Moore as pointed out, the only self-consistent definition of mass
that can be applied both to point-particles as well as to systems of particles, is that “mass” is the magnitude of the four-momentum vector. He has a very elegant “proof” of (argument for?) this that
gets overlooked because it is in an introductory textbook. Please note, however, that he set out to write this textbook precisely so that students could learn the nuances of concepts from the very
beginning instead of teaching them “watered-down” versions of things and correcting them later.
I once made this argument to Frank Wilczek in a debate that was published in Physics Today but he missed my point.
Edit: I will admit that there are some issues with this in GR, though they are not intractable issues and certainly don’t make this definition any worse than others. | {"url":"https://nforum.ncatlab.org/discussion/1459/mass/","timestamp":"2024-11-11T04:12:13Z","content_type":"application/xhtml+xml","content_length":"45982","record_id":"<urn:uuid:c98df956-b3e7-42f3-94c0-29b538c908f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00199.warc.gz"} |
Lectures on Kinetic Theory | Download book PDF
Kinetic Theory and Magnetohydrodynamics of Plasmas
The PDF covers the following topics related to Kinetic Theory and Magnetohydrodynamics of Plasmas : Kinetic Description of a Plasma, Equilibrium and Fluctuations, Linear Theory: Plasma Waves, Landau
Damping and Kinetic Instablities, Energy, Entropy, Free Energy, Heating, Irreversibility and Phase Mixing, General Kinetic Stability Theory, Nonlinear Theory: Two Pretty Nuggets, Quasilinear Theory,
Kinetics of Quasiparticles, Langmuir Turbulence, Stochastic Echo and Phase-Space Turbulence, MHD Equations, MHD in a Straight Magnetic Field, MHD Relaxation, MHD Stability and Instabilities.
Author(s): Alexander A. Schekochihin
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How to solve a SQUARE-1 in 6 Steps | Easy to follow Beginners Steps
The Square-1 puzzle is unique in that its layers—top, middle, and bottom - are split diagonally, allowing for a "slice" move. This middle layer is a divider, and due to its simple structure solving
it isn't as crucial. Most of the solving process focuses on the top and bottom layers.
At a beginner level the Square-1 is solved in the following 6 Steps:
All the images are shown in pairs. The left side shows the TOP and the right side shows the BOTTOM.
STEP 1 - Get the puzzle into a Cube Shape.
The goal of the first part is to get
all eight edge pieces next to each other
in the top layer.
1. Hold the puzzle so that the middle layer (with two pieces) is facing you. The small square in the middle layer should be on the left side.
2. The middle layer allows for a "slice move", where the right-hand side of the puzzle can be rotated 180 degrees (a half-turn). This is how you’ll move pieces between the top and bottom layers.
3. Here’s the trick: The left side doesn't move, so store your edges there. Move pieces from the right hand, bottom layer to the top layer with the “slice”.
4. Keep most of the edges on the top layer on the left side. Rotate the bottom layer to put edges in the right side. Then slice to move the bottom edges to the top alongside the ones you already
This step is intuitive; you can do it for sure but it does take a little practice. The following examples show you how to get the last couple of edges in place:
This first example is if you have seven edges together and one is left between two corners.
The two example below is when you have five edges together and one edge separate in the top and two edges together in the bottom layer.
Holding the Puzzle:
Top Layer Orientation: The top layer should have eight edges, meaning it's almost a full circle. Adjust the top layer so that there are four edges either side of the slice as indicated above.
Bottom Layer Orientation: The bottom layer should have six corners, with three corners on each side of the slice as indicated above.
Middle Layer Setup: Ensure that the middle layer has two sides (edges) facing you, with the smaller square (a piece that is thinner than the others) positioned on the left side.
Perform the following algorithm to turn the Sq-1 into a cube:
STEP 2 - Orient Corners (CO)
In this step we will swap two corners until we have four Yellow corners in the Top and four White corners in the bottom.
Setup for the Swap:
Top Layer (Yellow): Rotate the top layer so that there’s a white corner in the front right position.
Bottom Layer (White): Rotate the bottom layer so that a yellow corner is in the front right position.
This setup places both corners you want to swap in the front right positions of the top and bottom layers, respectively.
To swap these two corners, use the following algorithm:
After performing this algorithm, the two corners (white and yellow) will be swapped between the top and bottom layers, helping you move toward solving the puzzle! Repeat this process until you have
ALL four YELLOW corners in the top layer.
STEP 3 - Orient Edges (EO)
The goal of this step is to move all the Yellow edges in the TOP face, whilst maintaining the square shape for the top and bottom layers.
Setup for the Swap:
Top Layer (Yellow): Rotate the top layer so that you have a WHITE edge in the right position.
Bottom Layer (White): Rotate the bottom layer so that you have a YELLOW edge in the back position.
Perform the algorithm below to swap the two edges
Repeat this process until you have all four YELLOW edges in the top layer.
STEP 4 - Permute Corners (CP)
In this step, you're working on swapping two corners in the top layer of your Square-1 to place all corners in their correct positions. Here’s how to handle the corner swaps depending on whether or
not you have "headlights" (two matching side-colored corners).
Setup for the Corner Swap:
If you have two corners with matching side colors ("headlights"):Rotate the top layer so that the two matching corners (the "headlights") are positioned on the back side of the puzzle.
If you do NOT have any matching corners ("no headlights"):Perform the algorithm below. After completing it, you should have one set of "headlights.”
Once you have one set of headlights, rotate the top layer so that these "headlights" are positioned on the back side of the puzzle.
Once the Top layer is permuted then SWAP the top and bottom layers with this algorithm: / (6,6) / . Slice, Turn the top and bottom layers 180 degrees, Slice.
Repeat the process above for the bottom layer, which is now on top.
STEP 5 - Permute Edges (EP
You're now working on swapping two edges in the top and bottom layers simultaneously to either solve all edges or leave only two edges to swap.
Setup for Swapping Edges:
Top Layer (Yellow):Rotate the top layer so that the edge on the right matches the corners on the front of the puzzle.
Bottom Layer (White):Rotate the bottom layer so that the edge on the left matches the corners on the front of the puzzle.
Repeat this process until either: All edges are solved, or Only two edges remain unsolved in the entire cube.
If you find yourself with three unsolved edges, you will need to perform the algorithm twice: First, solve one edge. This will also swap two solved edges in the opposite layer, but they will be
swapped back in the second step.
Then, you'll be left with two edge pairs to swap. Perform the Setup step above before executing the algorithm below again.
Repeat this process until you have either all edges solved OR only two edges for the whole cube swapped.
STEP 6 - Fix Parity
If you only have two edges swapped then you have edge parity.
Hold the cube with the two swapped edges in the top layer. on the front and right sides.
The steps to swap only two edges is:
/ (-3,0) / (0,3) / (0,-3) / (0,3) / (2,0) / (0,2) / (-2,0) / (4,0) / (0,-2) / (0,2) / (-1,4) / (0,-3) /
ADDITIONAL - Fix Misaligned middle layer
To fix the misaligned middle layer of the Square-1 without disturbing the already solved top and bottom layers, you can follow these specific steps. These can be performed at any stage of the solve
without affecting the progress you've made in solving the top and bottom.
When solving a Square-1 puzzle, there are times when you need to swap the top and bottom layers to bring the puzzle into a better state for solving, particularly during corner permutation. Swapping
the layers is helpful when the corners of both layers need to be moved or swapped without disturbing their order or orientation.
Here’s an algorithm that will swap the top and bottom layers of the Square-1 puzzle without messing up their positions: | {"url":"https://in.speedcube.com.au/blogs/speedcubing-solutions/how-to-solve-a-square-1-in-6-steps-easy-to-follow-beginners-steps","timestamp":"2024-11-09T14:14:32Z","content_type":"text/html","content_length":"319405","record_id":"<urn:uuid:4ac700bd-73f8-472b-b217-29a6e311e33c>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00632.warc.gz"} |
Blocked Schur Algorithms for Computing the Matrix Square Root
Deadman, Edvin and Higham, Nicholas J. and Ralha, Rui (2013) Blocked Schur Algorithms for Computing the Matrix Square Root. Lecture Notes in Computer Science, 7782. pp. 171-182.
This is the latest version of this item.
Download (281kB)
The Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using either standard
blocking or recursive blocking the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show
significant speedups over the point algorithm, both in the square root phase and in the algorithm as a whole. In parallel implementations, recursive blocking is found to provide better performance
than standard blocking when the parallelism comes only from threaded BLAS, but the reverse is true when parallelism is explicitly expressed using OpenMP. The excellent numerical stability of the
point algorithm is shown to be preserved by blocking. These results are extended to the real Schur method. Blocking is also shown to be effective for multiplying triangular matrices.
Available Versions of this Item
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Using LaTeX to Add Mathematical Expressions to Plot Titles - 3DCAD.news
Wouldn’t it be great if you could include mathematical expressions in plot titles? As of version 6.2 of the COMSOL Multiphysics^® software, it’s possible to do so using LaTeX, which is a
high-quality, document-preparation system for mathematical and scientific typesetting. In version 6.2, it is also easy to create multiline plot titles. In this blog post, we will explore these new
About LaTeX
LaTeX supports visually appealing and comprehensive typesetting of complex mathematical expressions. Previous versions of COMSOL Multiphysics^® supported LaTeX segments in reports, and with the
extended support introduced in version 6.2, plot titles can now also be enriched with mathematical content.
The LaTeX system contains a large number of elements for creating mathematical expressions, including:
• Greek letters and other special characters, including Unicode characters
• Mathematical symbols and operators
• Delimiters and spaces
• Mathematical function names
• Special mathematical typesetting for fractions and roots
• Text and font elements such as superscript, subscript, overlining, underlining, and italics
The COMSOL Multiphysics Reference Manual includes lists of all the supported LaTeX commands. All of the commands start with a backslash. For example, \alpha is used for the Greek character \alpha.
Adding LaTeX Segments to Plot Titles
It is easy to add mathematical expressions in a plot title using LaTeX. The process is as follows:
1. Locate the Title section in the Settings window for a plot group or plot.
2. In the title settings, choose Manual from the Title type list.
3. In the Title field, add the text with one or more LaTeX segments.
To add LaTeX commands in the Title field, enclose them in either \[ and \] or /$ and /$, invoking one of two LaTeX modes:
1. Use \[ and \] around the LaTeX mathematical expression for the display math mode. The display math mode uses more vertical space for the math expression, but it doesn’t add a new line like when
using display math mode for standard LaTeX.
2. Use /$ and /$ around the LaTeX mathematical expression for the inline math mode. The inline math mode is meant to be included inline in the text (in standard LaTeX) and has a more compact
presentation. The compact presentation may make this mode less suitable for large mathematical expressions.
Conversely, you can convert part of a LaTeX command to be formatted in a different mode type, such as in cases where you want to mix the LaTeX formatting types but don’t want to break up and rewrite
the expressions. For instance, to convert a LaTeX command from inline math mode to display math mode, use the special command \displaystyle, and use \textstyle to convert from display math mode to
inline math mode.
In the following examples, we’ll take a look at how to add equations and other mathematical expressions to plot titles.
Examples of Math Expressions
Adding Magnetic Field Equations
When plotting the solution of field equations computed using COMSOL Multiphysics^®, it can be of interest to include those equations in the plot title. As an example, we’ll use the Submarine Cable 8a
— Inductive Effects 3D model from the AC/DC Module Application Library. It uses the Magnetic Fields interface, solving for the magnetic potential A and its components. The solved equations include:
\nabla \times \textbf{H} = \textbf{J}\\
\textbf{B} =\nabla \times \textbf{A}\\
\textbf{J} = \sigma\textbf{E} + j\omega\textbf{D} + \textbf{J}_{e}\\
\textbf{E} = -j\omega\textbf{A}.
To create a multiline plot title such as this in the COMSOL Desktop^®, simply start a new line after each expression (using the Enter or Return key, depending on your keyboard). For this example,
enter the title as:
Magnetic field equations:
\[\nabla \times \textbf{H} = \textbf{J}\]
\[\textbf{B} =\nabla \times \textbf{A}\]
\[\textbf{J} = \sigma\textbf{E} + j\omega\textbf{D} + \textbf{J}_{e}\]
\[\textbf{E} = -j\omega\textbf{A}\]
There are some characteristics to note about this title:
• The \nabla command creates a nabla symbol (\nabla).
• The \textbf command makes the text appear with a bold font to symbolize a vector quantity.
• The _ (underscore) symbol makes the subsequent characters appear in subscript.
• The title appears on five lines since four line breaks were entered in the Title field.
• These equations are written in the display math mode, though in this case there isn’t a notable difference between the display and inline math modes.
In the image below, you can see how the expression appears as a title.
A plot of inductive effects in a submarine cable with the magnetic field equations as the plot title.
Adding Function Limits
A type of mathematical expression that requires some vertical space is the limit expression, which describes the value of a mathematical function as its function argument approaches some limit. Such
limits are common in calculus courses and can be formulated like the following example:
Show that \displaystyle\lim_{x\to 0}{\frac{e^x-1}{2x}} and \displaystyle\lim_{x\to 0}{\frac{e^x}{2}} are equal to 1/2.
The second limit expression is mathematically easy: By inserting 0 for x in the numerator, you get a value of 0.5, as expected. The first limit expression is trickier because inserting 0 for both
cases of x results in an undefined 0 divided by 0. Still, when x approaches 0 in the limit, the function value can approach a number that is well defined. In mathematics, you could use l’Hôpital’s
rule, which states that the limit of the quotient of two functions is the same as the limit of the quotient of those functions’ derivatives. Using differentiation rules from calculus, you can see
that the second limit’s numerator and denominator are the derivatives of the first limit’s numerator and denominator, respectively, which shows that the limit in the first case is also 1/2. In the
COMSOL Multiphysics^® plot shown below, we have plotted the two functions in an interval that starts with a value close to 0. It seems evident from the plot that both functions approach 0.5 as the
function argument approaches 0.
To add the title shown in the image, use the following text:
Show that \[\lim_{x\to 0}{\frac{e^x-1}{2x}} \textrm{ and} \lim_{x\to 0}{\frac{e^x}{2}}={\frac{1}{2}}\]
Here is some notable information about this title:
• Only one math expression segment using the display math mode has been used. To get the “and” to appear as normal text in Roman font, the \textrm command is used. An alternative would be to use
two mathematical expressions, with “and” surrounded by spaces between those expressions.
• The \lim and \frac commands in LaTeX create a limit and a fraction, respectively. The \to command in the limit’s subscript is a right-pointing arrow.
• The limit with subscript x \to 0 takes up some vertical space, so it’s best to use the display math mode with /[ and \]. If you use the inline math mode, the subscript arrow doesn’t fit
underneath \lim, as can be seen here:
To make this particular title look good using the inline math mode, you can enclose the \lim_{x\to 0} part using the \displaystyle command to convert it to the display mode: \displaystyle{\lim_{x\to
0}}. It then looks similar to the title in the plot below.
This plot uses a title that shows a mathematical expression with a statement that the plot shows to be correct, as both curves approach 0.5 when x approaches 0.
Concluding Thoughts
In this blog post, we have shown a couple of examples of how you can use LaTeX in plot titles to enhance them with mathematical expressions. We have also discussed how to create multiline plot
titles. With all of the LaTeX commands at your disposal, the possibilities to include equations and other mathematical content in plot titles are endless. We hope that these examples can serve as
inspiration for plot titles in your own modeling projects. | {"url":"https://3dcad.news/using-latex-to-add-mathematical-expressions-to-plot-titles/","timestamp":"2024-11-08T04:46:33Z","content_type":"text/html","content_length":"196254","record_id":"<urn:uuid:d44a7448-d91b-4b3a-a495-54cca44dd50a>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00414.warc.gz"} |
Quantum effects on the inner (Cauchy) horizon of rotating black holes
All black holes in the Universe are believed to be rotating. This poses interesting questions, since rotating black hole solutions of Einstein’s equations of General Relativity possess a so-called
Cauchy horizon in their interior, which threatens the predictability of Einstein’s theory. However, these exact solutions may not model sufficiently accurately black holes in Nature, which have
classical matter in their neighbourhood and, furthermore, are inevitably surrounded by a quantum vacuum (which is responsible for Hawking radiation). On the classical side, it has been found that the
Cauchy horizons of some black holes become irregular under classical field perturbations whereas the Cauchy horizons of other black holes (e.g., in a Universe with a positive cosmological constant)
seem to remain regular. On the quantum side, effects on Cauchy horizons due to quantum fields are believed to be generally stronger than those due to classical fields. In this talk, we will review
some results on the linear stability of Cauchy horizons of black holes and we will present recent results on semiclassical effects due to a quantum field on the Cauchy horizon of a rotating (Kerr)
black hole. In particular, we will show that the (renormalized) fluxes from a quantum scalar field generically diverge on the Cauchy horizon of a Kerr black hole that is evaporating via the emission
of Hawking radiation.
Mardi, 28 mars, 2023 - 14:00 to 15:00
Nom/Prénom // Last name/First name:
Institut für Theoretische Physik, Universität Leipzig
Equipe(s) organisatrice(s) / Organizing team(s): | {"url":"https://apc.u-paris.fr/APC_CS/fr/quantum-effects-inner-cauchy-horizon-rotating-black-holes","timestamp":"2024-11-03T04:32:39Z","content_type":"text/html","content_length":"33373","record_id":"<urn:uuid:4207a739-6c6f-44bd-b9df-6e3ed3a1e6b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00525.warc.gz"} |
A Class Was Interested In The Amount Of Time It Takes Them To Travel To And From School. They Gathered
Answer:a box plot i did the exam
Step-by-step explanation: i did the exam.
Answer: The best graphical representation for their data would be a box plot. A box plot, also known as a box and whisker plot, is a useful tool for displaying the distribution of a dataset,
including the minimum, first quartile, median, third quartile, and maximum values. This type of plot is particularly useful for comparing the spread of different sets of data, which would be useful
for the class to compare the time it takes different students to travel to and from school.
Step-by-step explanation:
Using the normal distribution, there is a 0.2148 = 21.48% probability that the sum of the 40 values is less than 7,100.
Normal Probability Distribution
The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score
how many standard deviations the measure is above or below the mean. Looking at the z-score table, the
associated with this z-score is found, which is the
of X.By the
Central Limit Theorem
, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For this problem, these parameters are given as follows:
[tex]\mu = 180, \sigma = 20, n = 40, s = \frac{20}{\sqrt{40}} = 3.1623[/tex]
A sum of 7100 is equivalent to a sample mean of 7100/40 = 177.5, which means that the probability is the p-value of Z when X = 177.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{177.5 - 180}{3.1623}[/tex]
Z = -0.79
Z = -0.79 has a p-value of 0.2148.
There is a 0.2148 = 21.48% probability that the sum of the 40 values is less than 7,100.
More can be learned about the normal distribution at https://brainly.com/question/28135235 | {"url":"https://cjp.edu.py/quiz-answers/a-class-was-interested-in-the-amount-of-time-it-takes-them-t-1djb.html","timestamp":"2024-11-09T07:27:23Z","content_type":"text/html","content_length":"82825","record_id":"<urn:uuid:999bf958-5afb-4d81-bb22-2fe0bc4982b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00230.warc.gz"} |
Advanced Mathematics
Topic outline
• Introduction
As we saw it in senior 4, trigonometry studies relationship involving lengths and angles of a triangle. The techniques in trigonometry are used for finding relevance in navigation particularly
satellite systems and astronomy, naval and aviation industries, oceanography, land surveying and in cartography (creation of maps). Now, those are the scientific applications of the concepts in
trigonometry, but most of the mathematics we study would seem (on the surface) to have little real-life application. Trigonometry is really relevant in our day to day activities.
1.1. Trigonometric formulae
1.1.1. Addition and subtraction formulae
From Activity 1.1, the addition and subtraction formulae are:
Addition and subtraction formulae are useful when finding trigonometric number of some angles.
The given information produces the triangle shown below.
Note the signs associated with a and b. The Pythagorean Theorem is used to find the hypotenuse.
Files: 2
• Introduction
Consider a scientist doing an experiment; he/she is collecting data every day. Let 1 u be the data collected the first day, 2 u be the data collected the second day, be the data collected the
third day, and so on…, and n u be the data collected after n days. Clearly, we are generating a set of numbers with a very special characteristic. There is an order in the numbers; that is, we
actually have the first number, the second number and so on. A sequence is a set of real numbers with a natural order.
• Introduction
People such as scientists, sociologists and town planners are often more concerned with the rate at which a particular quantity is growing than with its current size. The director of education is
more concerned with the rate of at which the school population is increasing or decreasing than with what the population is now, because he/she has to plan for the future and ensure that there
are enough(and not too many) school places available to meet demand each year. The scientists may need to know the rate at which a colony of bacteria is growing rather than how many of the
bacteria exists at this moment, or the rate at which a liquid is cooling rather than the temperature of the liquid now, or the rate at which a radioactive material is decaying rather than how
many atoms currently exist.
One thing that each of these populations has in common is that their rate of increase is proportional to the size of the population at any time. Exponential and logarithmic equations are really
relevant in our day to day activities. The above events show us the areas where this unit finds use in our daily activities.
• Introduction
We know how to solve linear equations and quadratic equations, either by factorising, by formula or by completing the square. In some instances, it may be almost impossible to use an exact method
to solve an equation for example, 0 sin1 = −− θθ precisely. In such cases, we may be able to use other techniques which give good approximations to the solution. In this unit, we reconsider such
approximations in a more formal way.
• Introduction
The techniques in trigonometry are used for finding relevance in navigation particularly satellite systems and astronomy, naval and aviation industries, oceanography, land surveying, and in
cartography (creation of maps). Now those are the scientific applications of the concepts in trigonometry, but most of the mathematics we study would seem (on the surface) to have little
real-life application. Trigonometry is really relevant in our day to day activities. In this unit, we will see how we can use trigonometry to resolve problems we might encounter.
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context.
To put it really simple, vectors are basically all about directions and magnitudes. These are critical in basically all situations.
In physics, vectors are often used to describe forces, and forces are added in the same way as vectors.
For example, in Classical Mechanics: Block sliding down a ramp, you need to calculate the force of gravity (a vector down), the normal force (a vector perpendicular to the ramp), and a friction
force (a vector opposite the direction of motion).
• Introduction
A matrix is a rectangular arrangement of numbers, expressions, symbols which are arranged in rows and columns.
Matrices play a virtual role in the projection of a three dimensional image into a two dimensional image. Matrices and their inverse are used by programmers for coding or encrypting a message.
Matrices are applied in the study of electrical circuits, quantum mechanics and optics. A message is made as a sequence of numbers in a binary format for communication and it follows code theory
for solving. Hence with the help of matrices, those equations are solved. Matrices are used for taking seismic surveys.
• My goals
By the end of this unit, I will be able to:
□ plot points in three dimensions.
□ find equation of straight lines in three dimensions.
□ find equation of planes in three dimensions.
□ position of lines and planes in space.
□ find equation of sphere.
In 2-Dimensions, the position of a point is determined by two coordinates x and y. However, in 3-Dimensions the position of point determined by three coordinates x, y, z obtained with reference
to three straight lines (x-axis , y-axis and z-axis respectively) intersecting at right angles.
In the plane, a line is determined by a point and a number giving the slope of the line. However, in 3-dimensional space, a line is determined by a point and a direction given by a parallel
vector, called the direction vector of the line.
In a 2-dimensional coordinate system, there were three possibilities when considering two lines: intersecting lines, parallel lines and the two were actually the same line but in 3-dimensional
space. There is one more possibility: Two lines may be skew, which means they do not intersect, but are not parallel.
In space, a plane is determined by a point and two direction vectors which form a basis (linearly independent vectors).
Advanced Mathematics Learner’s Book Five
Sphere is the locus of a point in space which remains at a constant distance called the radius from a fixed point called the centre of the sphere.
8.1. Points in 3 dimensions
8.1.1. Location of a point in space
Activity 8.1:
Consider the point ()2,3,5A in space, on a piece of paper
1. Copy the following figure
2. From x-coordinate 2, draw a line parallel to y-axis.
3. From y-coordinate 3, draw another line parallel to x-axis.
4. Now you have a point of intersection of two lines, let us call it P. From this point, draw another line parallel to z-axis and another joining this point and origin of coordinates which is
line OP.
5. From z-coordinate, draw another line parallel to the line OP.
6. Draw another line parallel to z-axis and passing through point P.
Suppose that we need to represent the point
Let us see it using a box
8.1.2. Coordinates of a midpoint of a segment and centroid of a geometric figure
8.2. Straight lines in 3 dimensions
8.2.1. Equations of lines
In the plane, a line is determined by a point and a number giving the slope of the line. In 3-dimensional space, a line is determined by a point and a direction given by a parallel vector, called
the direction vector of the line. We will denote lines by capital letters such as L, M,...
One of the methods of finding this shortest distance is to write the parametric form of any point of each given line. Next, find the vector joining the points in parametric form which will be the
vector in the direction of the common perpendicular of both lines. Now, the dot product of this vector and the direction vector of each line must be zero. This will help us to find the value of
parameters and hence two points (one on the first line and another on the second line). The common perpendicular of the two lines passes through these two points. Then, the distance between these
two points is the required shortest distance between the two lines.
Using this method, we can find the equation of the common perpendicular since we have two points where this common perpendicular passes.
Note that if two lines intersect (not skew lines), the shortest distance is zero.
A line L is perpendicular to plane α if and only if each direction vector of L is perpendicular to each direction vector of α or the scalar product of direction vector of the line and the
direction vector of the plane is zero.
In this case, the direction vector of the line is perpendicular to the plane and is said to be the
vector of the plane.
that the normal vector of the plane can be found by finding the vector product of its two non proportional direction vectors.
Thus, the angle between the given plane and the given line is 67.8 degrees.
b) Angle between two planes
It is important to choose the correct angle here. It is defined as the angle between two lines, one in each plane, so that they are at right angles to the line of intersection of the two planes
(like the angle between the tops of the pages of an open book).
When finding the image of a point P with respect to the plane α , we need to find the line, say L, through point P and perpendicular to the plane α .
The next is to find the intersection of line L and plane α , say N. Now, if Q is the image of P, the point N is the midpoint of PQ. From this, we can find the coordinate of Q.
Similarly, if we need the image of a line, we will need the parametric form of any point on the line and then find its image using the same method. The image will be in parametric form.
Now, replacing the parameter by any two chosen values in the obtained image, we will get two points. From these two points, we can find the equations of the line which will be the image of the
given line.
• My goals
By the end of this unit, I will be able to:
□ find measures of central tendency in two quantitative variables.
□ find measures of variability in two quantitative variables.
□ determine the linear regression line of a given series.
□ calculate a linear correlation coefficient
Descriptive statistics is a set of brief descriptive coefficients that summarises a given data set, which can either be a representation of the entire population or sample. Data may be
qualitative such as sex, color and so on or quantitative represented by numerical quantity such as height, mass, time and so on.
The measures used to describe the data are measures of central tendency and measures of variability or dispersion. Until now, we know how to determine the measures of central tendency in one
variable. In this unit, we will use those measures in two quantitative variables known as double series.
In statistics, double series include technique of analyzing data in two variables, when focus on the relationship between a dependent variable-y and an independent variable-x. The linear
regression method will be used in this unit. The estimation target is a function of the independent variable called the regression function which will be a function of a straight line.
Descriptive statistics provide useful summary of security returns when performing empirical and analytical analysis, as they provide historical account of return behavior. Although past
information is useful in any analysis, one should always consider the expectations of future events. Some variables are discrete, others are continuous. If the variable can take only certain
values, for example, the number of apples on a tree, then the variable is discrete. If however, the variable can take any decimal value (in some range), for example, the heights of the children
in a school, then the variables are continuous. In this unit, we will consider discrete variables.
9.1. Covariance
Activity 9.1
Complete the following table
What can you get from the following expressions:
In case of two variables, say x and y, there is another important result called covariance of x and y, denoted (x,y)
The covariance of variables x and y is a measure of how these two variables change together. If the greater values of one variable mainly correspond with the greater values of the other variable,
and the same holds for the smaller values, i.e. the variables tend to show similar behavior, the covariance is positive. In the opposite case, when the greater values of one variable mainly
correspond to the smaller values of the other, i.e. the variables tend to show opposite behavior, the covariance is negative. If covariance is zero, the variables are said to be uncorrelated,
meaning that there is no linear relationship between them.
Therefore, the sign of covariance shows the tendency in the linear relationship between the variables. The magnitude of covariance is not easy to interpret.
Developing this formula, we have
Example 9.1
Find the covariance of x and y in the following data sets
We have
Example 9.2
Find the covariance of the following distribution
Convert the double entry into a simple table and compute the arithmetic means | {"url":"https://elearning.reb.rw/course/view.php?id=500","timestamp":"2024-11-07T20:50:35Z","content_type":"text/html","content_length":"194868","record_id":"<urn:uuid:26c03c79-6ec5-496b-ac1d-a432ba6948d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00435.warc.gz"} |
IAP: conduit stream fusion
Both the changes described in this blog post, and in the previous blog post, are now merged to the master branch of conduit, and have been released to Hackage as conduit 1.2.0. That doesn't indicate
stream fusion is complete (far from it!). Rather, the optimizations we have so far are valuable enough that I want them to be available immediately, and future stream fusion work is highly unlikely
to introduce further breaking changes. Having the code on Hackage will hopefully also make it easier for others to participate in the discussion around this code.
Stream fusion
Last time, I talked about applying the codensity transform to speed up conduit. This greatly increases performance when performing many monadic binds. However, this does nothing to help us with
speeding up the "categorical composition" of conduit, where we connect two components of a pipeline together so the output from the first flows into the second. conduit usually refers to this as
fusion, but given the topic at hand (stream fusion), I think that nomenclature will become confusing. So let's stick to categorical composition, even though conduit isn't actually a category.
Duncan Coutts, Roman Leshchinskiy and Don Stewart wrote the stream fusion paper, and that technique has become integral to getting high performance in the vector and text packages. The paper is well
worth the read, but for those unfamiliar with the technique, let me give a very brief summary:
• GHC is very good at optimising non-recursive functions.
• We express all of our streaming functions has a combination of some internal state, and a function to step over that state.
• Stepping either indicates that the stream is complete, there's a new value and a new state, or there's a new state without a new value (this last case helps avoid recursion for a number of
functions like filter).
• A stream transformers (like map) takes a Stream as input and produces a new Stream as output.
• The final consuming functions, like fold, are the only place where recursion happens. This allows all other components of the pipeline to be inlined, rewritten to more efficient formats, and
optimized by GHC.
Let's see how this looks compared to conduit.
Data types
I'm going to slightly rename data types from stream fusion to avoid conflicts with existing conduit names. I'm also going to add an extra type parameter to represent the final return value of a
stream; this is a concept that exists in conduit, but not common stream fusion.
data Step s o r
= Emit s o
| Skip s
| Stop r
data Stream m o r = forall s. Stream
(s -> m (Step s o r))
(m s)
The Step datatype takes three parameters. s is the internal state used by the stream, o is the type of the stream of values it generates, and r is the final result value. The Stream datatype uses an
existential to hide away that internal state. It then consists of a step function that takes a state and gives us a new Step, as well as an initial state value (which is a monadic action, for cases
where we want to do some initialization when starting a stream).
Let's look at some functions to get a feel for what this programming style looks like:
enumFromToS_int :: (Integral a, Monad m) => a -> a -> Stream m a ()
enumFromToS_int !x0 !y =
Stream step (return x0)
step x | x <= y = return $ Emit (x + 1) x
| otherwise = return $ Stop ()
This function generates a stream of integral values from x0 to y. The internal state is the current value to be emitted. If the current value is less than or equal to y, we emit our current value,
and update our state to be the next value. Otherwise, we stop.
We can also write a function that transforms an existing stream. mapS is likely the simplest example of this:
mapS :: Monad m => (a -> b) -> Stream m a r -> Stream m b r
mapS f (Stream step ms0) =
Stream step' ms0
step' s = do
res <- step s
return $ case res of
Stop r -> Stop r
Emit s' a -> Emit s' (f a)
Skip s' -> Skip s'
The trick here is to make a function from one Stream to another. We unpack the input Stream constructor to get the input step and state functions. Since mapS has no state of its own, we simply keep
the input state unmodified. We then provide our modified step' function. This calls the input step function, and any time it sees an Emit, applies the user-provided f function to the emitted value.
Finally, let's consider the consumption of a stream with a strict left fold:
foldS :: Monad m => (b -> a -> b) -> b -> Stream m a () -> m b
foldS f b0 (Stream step ms0) =
ms0 >>= loop b0
loop !b s = do
res <- step s
case res of
Stop () -> return b
Skip s' -> loop b s'
Emit s' a -> loop (f b a) s'
We unpack the input Stream constructor again, get the initial state, and then loop. Each loop, we run the input step function.
Match and mismatch with conduit
There's a simple, straightforward conversion from a Stream to a Source:
toSource :: Monad m => Stream m a () -> Producer m a
toSource (Stream step ms0) =
lift ms0 >>= loop
loop s = do
res <- lift $ step s
case res of
Stop () -> return ()
Skip s' -> loop s'
Emit s' a -> yield a >> loop s'
We extract the state, and then loop over it, calling yield for each emitted value. And ignoring finalizers for the moment, there's even a way to convert a Source into a Stream:
fromSource :: Monad m => Source m a -> Stream m a ()
fromSource (ConduitM src0) =
Stream step (return $ src0 Done)
step (Done ()) = return $ Stop ()
step (Leftover p ()) = return $ Skip p
step (NeedInput _ p) = return $ Skip $ p ()
step (PipeM mp) = liftM Skip mp
step (HaveOutput p _finalizer o) = return $ Emit p o
Unfortunately, there's no straightforward conversion for Conduits (transformers) and Sinks (consumers). There's simply a mismatch in the conduit world- which is fully continuation based- to the
stream world- where the upstream is provided in an encapsulated value. I did find a few representations that mostly work, but the performance characteristics are terrible.
If anyone has insights into this that I missed, please contact me, as this could have an important impact on the future of stream fusion in conduit. But for the remainder of this blog post, I will
continue under the assumption that only Source and Stream can be efficiently converted.
Once I accepted that I wouldn't be able to convert a stream transformation into a conduit transformation, I was left with a simple approach to start working on fusion: have two representations of
each function we want to be able to fuse. The first representation would use normal conduit code, and the second would be streaming. This looks like:
data StreamConduit i o m r = StreamConduit
(ConduitM i o m r)
(Stream m i () -> Stream m o r)
Notice that the second field uses the stream fusion concept of a Stream-transforming function. At first, this may seem like it doesn't properly address Sources and Sinks, since the former doesn't
have an input Stream, and the latter results in a single output value, not a Stream. However, those are really just special cases of the more general form used here. For Sources, we provide an empty
input stream, and for Sinks, we continue executing the Stream until we get a Stop constructor with the final result. You can see both of these in the implementation of the connectStream function
(whose purpose I'll explain in a moment):
connectStream :: Monad m
=> StreamConduit () i m ()
-> StreamConduit i Void m r
-> m r
connectStream (StreamConduit _ stream) (StreamConduit _ f) =
run $ f $ stream $ Stream emptyStep (return ())
emptyStep _ = return $ Stop ()
run (Stream step ms0) =
ms0 >>= loop
loop s = do
res <- step s
case res of
Stop r -> return r
Skip s' -> loop s'
Emit _ o -> absurd o
Notice how we've created an empty Stream using emptyStep and a dummy () state. And on the run side, we loop through the results. The type system (via the Void datatype) prevents the possibility of a
meaningful Emit constructor, and we witness this with the absurd function. For Stop we return the final value, and Skip implies another loop.
Fusing StreamConduit
Assuming we have some functions that use StreamConduit, how do we get things to fuse? We still need all of our functions to have a ConduitM type signature, so we start off with a function to convert
a StreamConduit into a ConduitM:
unstream :: StreamConduit i o m r -> ConduitM i o m r
unstream (StreamConduit c _) = c
Note that we hold off on any inlining until simplification phase 0. This is vital to our next few rewrite rules, which is where all the magic happens.
The next thing we want to be able to do is categorically compose two StreamConduits together. This is easy to do, since a StreamConduit is made up of ConduitMs which compose via the =$= operator, and
Stream transformers, which compose via normal function composition. This results in a function:
fuseStream :: Monad m
=> StreamConduit a b m ()
-> StreamConduit b c m r
-> StreamConduit a c m r
fuseStream (StreamConduit a x) (StreamConduit b y) = StreamConduit (a =$= b) (y . x)
That's very logical, but still not magical. The final trick is a rewrite rule:
We're telling GHC that, if we see a composition of two streamable conduits, then we can compose the stream versions of them and get the same result. But this isn't enough yet; unstream will still end
up throwing away the stream version. We now need to deal with running these things. The first case we'll handle is connecting two streamable conduits, which is where the connectStream function from
above comes into play. If you go back and look at that code, you'll see that the ConduitM fields are never used. All that's left is telling GHC to use connectStream when appropriate:
The next case we'll handle is when we connect a streamable source to a non-streamable sink. This is less efficient than the previous case, since it still requires allocating ConduitM constructors,
and doesn't expose as many opportunities for GHC to inline and optimize our code. However, it's still better than nothing:
connectStream1 :: Monad m
=> StreamConduit () i m ()
-> ConduitM i Void m r
-> m r
connectStream1 (StreamConduit _ fstream) (ConduitM sink0) =
case fstream $ Stream (const $ return $ Stop ()) (return ()) of
Stream step ms0 ->
let loop _ (Done r) _ = return r
loop ls (PipeM mp) s = mp >>= flip (loop ls) s
loop ls (Leftover p l) s = loop (l:ls) p s
loop _ (HaveOutput _ _ o) _ = absurd o
loop (l:ls) (NeedInput p _) s = loop ls (p l) s
loop [] (NeedInput p c) s = do
res <- step s
case res of
Stop () -> loop [] (c ()) s
Skip s' -> loop [] (NeedInput p c) s'
Emit s' i -> loop [] (p i) s'
in ms0 >>= loop [] (sink0 Done)
There's a third case that's worth considering: a streamable sink and non-streamable source. However, I ran into two problems when implementing such a rewrite rule:
• GHC did not end up firing the rule.
• There are some corner cases regarding finalizers that need to be dealt with. In our previous examples, the upstream was always a stream, which has no concept of finalizers. But when the upstream
is a conduit, we need to make sure to call them appropriately.
So for now, fusion only works for cases where all of the functions can by fused, or all of the functions before the $$ operator can be fused. Otherwise, we'll revert to the normal performance of
conduit code.
I took the benchmarks from our previous blog post and modified them slightly. The biggest addition was including an example of enumFromTo =$= map =$= map =$= fold, which really stresses out the
fusion capabilities, and demonstrates the performance gap stream fusion offers.
The other thing to note is that, in the "before fusion" benchmarks, the sum results are skewed by the fact that we have the overly eager rewrite rules for enumFromTo $$ fold (for more information,
see the previous blog post). For the "after fusion" benchmarks, there are no special-case rewrite rules in place. Instead, the results you're seeing are actual artifacts of having a proper fusion
framework in place. In other words, you can expect this to translate into real-world speedups.
You can compare before fusion and after fusion. Let me provide a few select comparisons:
Benchmark Low level or vector Before fusion After fusion Speedup
map + sum 5.95us 636us 5.96us 99%
monte carlo 3.45ms 5.34ms 3.70ms 71%
sliding window size 10, Seq 1.53ms 1.89ms 1.53ms 21%
sliding vector size 10, unboxed 2.25ms 4.05ms 2.33ms 42%
Note at the map + sum benchmark is very extreme, since the inner loop is doing very cheap work, so the conduit overhead dominated the analysis.
Streamifying a conduit
Here's an example of making a conduit function stream fusion-compliant, using the map function:
mapC :: Monad m => (a -> b) -> Conduit a m b
mapC f = awaitForever $ yield . f
mapS :: Monad m => (a -> b) -> Stream m a r -> Stream m b r
mapS f (Stream step ms0) =
Stream step' ms0
step' s = do
res <- step s
return $ case res of
Stop r -> Stop r
Emit s' a -> Emit s' (f a)
Skip s' -> Skip s'
map :: Monad m => (a -> b) -> Conduit a m b
map = mapC
Notice the three steps here:
• Define a pure-conduit implementation (mapC), which looks just like conduit 1.1's map function.
• Define a pure-stream implementation (mapS), which looks very similar to vector's mapS.
• Define map, which by default simply reexposes mapC. But then, use an INLINE statement to delay inlining until simplification phase 0, and use a rewrite rule to rewrite map in terms of unstream
and our two helper functions mapC and mapS.
While tedious, this is all we need to do for each function to expose it to the fusion framework.
Vector vs conduit, mapM style
Overall, vector has been both the inspiration for the work I've done here, and the bar I've used to compare against, since it is generally the fastest implementation you can get in Haskell (and tends
to be high-level code to boot). However, there seems to be one workflow where conduit drastically outperforms vector: chaining together monadic transformations.
I put together a benchmark which does the same enumFromTo+map+sum benchmark I demonstrated previously. But this time, I have four versions: vector with pure functions, vector with IO functions,
conduit with pure functions, and conduit with IO functions. You can see the results here, the important takeaway is:
• Pure is always faster, since it exposes more optimizations to GHC.
• vector and conduit pure are almost identical, at 57.7us and 58.1us.
• Monadic conduit code does have a slowdown (86.3us). However, monadic vector code has a drastic slowdown (305us), presumably because monadic binds defeat its fusion framework.
So there seems to be at least one workflow for which conduit's fusion framework can outperform even vector!
The biggest downside to this implementation of stream fusion is that we need to write all of our algorithms twice. This can possibly be mitigated by having a few helper functions in place, and
implementing others in terms of those. For example, mapM_ can be implemented in terms foldM.
There's one exception to this: using the streamSource function, we can convert a Stream into a Source without having to write our algorithm twice. However, due to differences in how monadic actions
are performed between Stream and Conduit, this could introduce a performance degredation for pure Sources. We can work around that with a special case function streamSourcePure for the Identity monad
as a base.
Getting good performance
In order to take advantage of the new stream fusion framework, try to follow these guidelines:
• Use fusion functions whenever possible. Explicit usage of await and yield will immediately kick you back to non-fusion (the same as explicit pattern matching defeats list fusion).
• If you absolutely cannot use an existing fusion function, consider writing your own fusion variant.
• When mixing fusion and non-fusion, put as many fusion functions as possible together with the $= operator before the connect operator $$.
Next steps
Even though this work is now publicly available on Hackage, there's still a lot of work to be done. This falls into three main categories:
• Continue rewriting core library functions in streaming style. Michael Sloan has been working on a lot of these functions, and we're hoping to have almost all the combinators from
Data.Conduit.List and Data.Conduit.Combinators done soon.
• Research why rewrite rules and inlining don't play nicely together. In a number of places, we've had to explicitly use rewrite rules to force fusion to happen, when theoretically inlining should
have taken care of it for us.
• Look into any possible alternative formulations of stream fusion that provide better code reuse or more reliable rewrite rule firing.
Community assistance on all three points, but especially 2 and 3, are much appreciated!
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Safe Haskell Safe-Inferred
This module establishes a class hierarchy that captures the interfaces of Par monads. There are two layers: simple futures (ParFuture) and full IVars (ParIVar). All Par monads are expected to
implement the former, some also implement the latter.
For more documentation of the programming model, see
class Monad m => ParFuture future m | m -> future whereSource
ParFuture captures the class of Par monads which support futures. This level of functionality subsumes par/pseq and is similar to the Control.Parallel.Strategies.Eval monad.
A minimal implementation consists of spawn_ and get. However, for monads that are also a member of ParIVar it is typical to simply define spawn in terms of fork, new, and put.
spawn :: NFData a => m a -> m (future a)Source
Create a potentially-parallel computation, and return a future (or promise) that can be used to query the result of the forked computataion.
spawn p = do
r <- new
fork (p >>= put r)
return r
spawn_ :: m a -> m (future a)Source
Like spawn, but the result is only head-strict, not fully-strict.
get :: future a -> m aSource
Wait for the result of a future, and then return it.
spawnP :: NFData a => a -> m (future a)Source
Spawn a pure (rather than monadic) computation. Fully-strict.
spawnP = spawn . return
class ParFuture ivar m => ParIVar ivar m | m -> ivar whereSource
ParIVar builds on futures by adding full anyone-writes, anyone-reads IVars. These are more expressive but may not be supported by all distributed schedulers.
A minimal implementation consists of fork, put_, and new.
fork :: m () -> m ()Source
Forks a computation to happen in parallel. The forked computation may exchange values with other computations using IVars.
new :: m (ivar a)Source
put :: NFData a => ivar a -> a -> m ()Source
put a value into a IVar. Multiple puts to the same IVar are not allowed, and result in a runtime error.
put fully evaluates its argument, which therefore must be an instance of NFData. The idea is that this forces the work to happen when we expect it, rather than being passed to the consumer of the
IVar and performed later, which often results in less parallelism than expected.
Sometimes partial strictness is more appropriate: see put_.
put_ :: ivar a -> a -> m ()Source
like put, but only head-strict rather than fully-strict.
newFull :: NFData a => a -> m (ivar a)Source
creates a new IVar that contains a value
newFull_ :: a -> m (ivar a)Source
creates a new IVar that contains a value (head-strict only)
class NFData a
A class of types that can be fully evaluated.
NFData Bool
NFData Char
NFData Double
NFData Float
NFData Int
NFData Int8
NFData Int16
NFData Int32
NFData Int64
NFData Integer
NFData Word
NFData Word8
NFData Word16
NFData Word32
NFData Word64
NFData ()
NFData Version
NFData a => NFData [a]
(Integral a, NFData a) => NFData (Ratio a)
NFData (Fixed a)
(RealFloat a, NFData a) => NFData (Complex a)
NFData a => NFData (Maybe a)
This instance is for convenience and consistency with seq. This assumes that WHNF is
NFData (a -> b) equivalent to NF for functions.
(NFData a, NFData b) => NFData (Either a b)
(NFData a, NFData b) => NFData (a, b)
(Ix a, NFData a, NFData b) => NFData (Array a b)
(NFData a, NFData b, NFData c) => NFData (a, b, c)
(NFData a, NFData b, NFData c, NFData d) => NFData (a, b, c, d)
(NFData a1, NFData a2, NFData a3, NFData a4, NFData a5) => NFData (a1, a2, a3, a4, a5)
(NFData a1, NFData a2, NFData a3, NFData a4, NFData a5, NFData a6) => NFData (a1, a2, a3, a4, a5, a6)
(NFData a1, NFData a2, NFData a3, NFData a4, NFData a5, NFData a6, NFData a7) => NFData (a1, a2, a3, a4, a5,
a6, a7)
(NFData a1, NFData a2, NFData a3, NFData a4, NFData a5, NFData a6, NFData a7, NFData a8) => NFData (a1, a2,
a3, a4, a5, a6, a7, a8)
(NFData a1, NFData a2, NFData a3, NFData a4, NFData a5, NFData a6, NFData a7, NFData a8, NFData a9) => NFData
(a1, a2, a3, a4, a5, a6, a7, a8, a9) | {"url":"http://hackage.haskell.org/package/abstract-par-0.3.3/docs/Control-Monad-Par-Class.html","timestamp":"2024-11-12T19:21:47Z","content_type":"application/xhtml+xml","content_length":"21380","record_id":"<urn:uuid:6ad0e48c-22fb-4474-ab6d-eadae3789905>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00154.warc.gz"} |
Nozzle Design and Optimization in context of specific impulse to thrust
30 Aug 2024
Title: Nozzle Design and Optimization for Specific Impulse to Thrust: A Review of Recent Advances
The design and optimization of nozzles are crucial aspects in the development of efficient propulsion systems, particularly in the context of specific impulse to thrust (Isp). This article provides a
comprehensive review of recent advances in nozzle design and optimization techniques, focusing on the relationship between Isp and thrust. Theoretical formulations and numerical examples are
presented to illustrate the concepts.
The specific impulse (Isp) is a fundamental parameter in rocket propulsion, defined as the total impulse per unit mass flow rate [1]. It is a measure of the efficiency of a propulsion system, with
higher values indicating better performance. Thrust, on the other hand, is the force exerted by the exhaust gases expelled from the nozzle. The optimization of nozzles to achieve high Isp and thrust
is a critical aspect in the design of efficient rocket engines.
Nozzle Design Formulations:
The design of a nozzle involves the determination of its shape and size to maximize Isp while minimizing thrust losses. The following formulations provide a foundation for understanding nozzle
1. Isentropic Expansion: The isentropic expansion of gases through a nozzle can be modeled using the following equation [2]:
p2 = p1 * (γ / γ + 1)^(γ+1)/(γ+1)
where p1 and p2 are the stagnation and exit pressures, respectively, and γ is the adiabatic index.
1. Nozzle Area Ratio: The nozzle area ratio (A2/A1) is a critical parameter in determining Isp. A higher area ratio can lead to increased Isp, but may also result in reduced thrust [3]:
Isp ∝ (A2/A1)^(-1/2)
where A1 and A2 are the nozzle entrance and exit areas, respectively.
Optimization Techniques:
Several optimization techniques have been employed to optimize nozzle design for high Isp and thrust. These include:
1. Gradient-Based Optimization: Gradient-based methods, such as the conjugate gradient algorithm [4], can be used to minimize the difference between desired and actual Isp values.
2. Evolutionary Algorithms: Evolutionary algorithms, such as genetic algorithms [5], can be employed to search for optimal nozzle shapes and sizes that maximize Isp while minimizing thrust losses.
Numerical Examples:
To illustrate the concepts, consider a simple nozzle design problem:
Suppose we want to design a nozzle with an exit Mach number of 2.5 and a stagnation pressure of 1000 kPa. Using the isentropic expansion equation (1), we can calculate the exit pressure as:
p2 = 1000 * (2.5 / 3.5)^(3.5/2.5) ≈ 234.8 kPa
Next, using the nozzle area ratio formulation (2), we can determine the optimal area ratio for maximum Isp:
A2/A1 = (234.8 / 1000)^(-1/2) ≈ 0.65
In conclusion, nozzle design and optimization are critical aspects in the development of efficient propulsion systems. Theoretical formulations and numerical examples have been presented to
illustrate the concepts. Future research directions include the application of advanced optimization techniques and the consideration of non-isentropic flow effects.
[1] Sutton, G. P., & Biblarz, O. (2018). Rocket Propulsion Elements. Wiley.
[2] Anderson, J. D. (2017). Fundamentals of Aerodynamics. McGraw-Hill Education.
[3] Glassman, A. J. (2006). Rocket Nozzle Design and Optimization. Journal of Propulsion and Power, 22(4), 741-748.
[4] Powell, M. J. D. (1971). Gradient Methods for Nonlinearly Constrained Optimization. Journal of the Institute of Mathematics and Its Applications, 8(2), 147-162.
[5] Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley Professional.
ASCII Art:
/ \
/ \
| NOZZLE |
| |
| Isp |
| |
| THRUST |
Related articles for ‘specific impulse to thrust’ :
• Reading: Nozzle Design and Optimization in context of specific impulse to thrust
Calculators for ‘specific impulse to thrust’ | {"url":"https://blog.truegeometry.com/tutorials/education/c842dc7e14c5ce2e8fbf6fd9fd0dfbbd/JSON_TO_ARTCL_Nozzle_Design_and_Optimization_in_context_of_specific_impulse_to_t.html","timestamp":"2024-11-03T16:17:10Z","content_type":"text/html","content_length":"18271","record_id":"<urn:uuid:ea6ceb19-0534-40ea-abd2-932aa92b29a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00477.warc.gz"} |
Learn Fibonacci – Bare Metal JavaScript: The JavaScript Virtual Machine
Check out a free preview of the full Bare Metal JavaScript: The JavaScript Virtual Machine course
The "Fibonacci" Lesson is part of the full, Bare Metal JavaScript: The JavaScript Virtual Machine course featured in this preview video. Here's what you'd learn in this lesson:
Miško discusses the transition from low-level machine language to assembly language in computer programming, how assembly language provides a more human-readable and manageable way to work with
machine-level instructions. The concept of subroutine calls and recursion using an example of implementing the Fibonacci sequence in assembly language is also covered in this segment.
Transcript from the "Fibonacci" Lesson
>> Now, at this point you should be looking to yourself and saying, like, this is insanity. Every time I change program, I have to change all these values, right? And so you need a compiler, pretty
quickly you're like, I need a compiler. And what the compiler allows you to do is to say things like, hey, into a0 I wanna load the address of the data section.
And I'm just gonna put a label where the data section is, right. And so when the compiler runs it like does the math and figures it out and like yeah that's 37 and just puts it over there, right. You
don't have to do that because you'll just go crazy always changing these values right like imagine every time you add the feature to the program you have to go back and like recompute all these
things it's just insanity.
So that's basically the difference between assembly and machine language, machine language is just numbers, right? Assembly are for each number we look up its corresponding name on the instruction
name of the instruction, so we say load ae instead of 37. And instead of random number, it puts an address, we can put an actual text label in there, and so we know that if we add things, shift
things around, the compiler will recompute all these things so you don't have to worry about it, right?
And it also makes it much easier for you to read, because now you'll be like, yeah, so a0 gets the address. A1 gets zero then we call the gosubroutine which is the increment. Then we look at array
location one increment two, increment three increment halt, right, much easier to read, right?
Okay, what does the subroutine label do? Well, the label is increment and what it does is it reads from the memory. It increments the value, stores it back in there. And by the way, the subroutine
automatically produces the return instruction and anything else that you need to kinda not worry about it.
And then you just have your label for data section and label for Stack. And you know that things get initialized, and everything works, right? This is assembly right here. A little more complicated,
but fundamentally that's what it's. Okay, let's implement Fibonacci. So the way Fibonacci works is again, we have to load the data location at zero, and we call the Fibonacci.
Go to the subroutine Fibonacci and then we store the result inside of location one and so what is that location zero. So location zero is four so we're asking for the fourth Fibonacci number, right.
Okay, so how does it work? The subroutine has a label notice. We save, say, save R1, meaning that in the process of executing this particular code, we're gonna destroy the value of R1.
What it means is that we are gonna have to push R1 on a stack. The way this works is you push R1 on a stack, then you do whatever set of instructions you want. And then on the end of it, you pop it
out of the stack, right?
So the stack not only can store your program counter information, it can also store arbitrary data, including local variables that but that's an advanced concept that we haven't we're not getting
into yet. And so this is an example of calling convention, right. We have a deal that when I call a subroutine and when I come back out of it the R1 is going to have the parameter I'm passing in or
the result value.
So the R1 is used both as an input and an output. Sorry, R0, not R1. And I really don't want you to mess with R1, that is not for you. So if you need to use R1 for some internal things, make sure you
save it so that it's available later.
And then we can do a comparator instruction. Now remember, at the beginning I said the R0 is little special because when you write into the R0 you set flags. And so what set flags operation does is
it says there is a set of bits and says if the value you set is zero, then we'll set a bit if the value is negative, then we'll set a bid if its value is positive, we set a bid, right?
And what that allows you to do, is to run a compare instruction. And what compare instruction basically is, a subtraction where you throw away the result. And so you compare it to zero, in other
words, you subtract it from zero, and the only time that it's zero is if it's the same number, right?
And if you get a negative number, then you know you're less than. If you get a positive then you're greater than etc. And this is basically how you do an if statement. So you do a subtraction which
sets a set of flags. And then based on the flags you have special instructions that basically say jump.
So, the way that works is that you run your expression which modifies the flags. Then you wanna run your condition in this particular case, it is jump if less than or equal to. And then the condition
then has to have a label of where you wanna go to the then location.
So you have to have a label for then and you have to have a label for end and so now this if condition either ends up in this location or this location. And then, obviously, if you end up in the
else, then you have to jump out of it so you don't double run the end of it, so you have to have a label end in here as well.
So you can see how an if statement is actually a bunch of jumps that happen in order to kind of figure out what's going on for the CPU. Does that make sense, or was it too fast?
>> A little fast.
>> A little fast, okay, so basically you run an expression in.
So let me just actually do this next to each other. You run a comparator expression which basically is just a subtraction. And the side effect of subtraction is that you set flags in the flags
register, okay. These flags are things like, is it zero, is it a positive number, is it a negative number, was it an overflow, is it a carryover, like there's a bunch of different flags that exist in
the system.
So this instruction that you run, so R0 is special that whatever value you write into R0, again, depends on architecture but typically R0 is special that it ends up affecting the flags register. So
by subtracting the R0 from some number unit, we can then see if that number was bigger, smaller equal to whatever, right?
And so now based on that you wanna do a conditional jump. In other words, you wanna modify the programme counter to some other address, but only if as positive or negative or equal to, right? And so
this is the conditional jump that says hey jump but only if this was true.
If it was not true then don't jump, right? So right after the condition you have to give it an address of where exactly you should be jumping, so you make a branch, right? You're either gonna jump or
you're not gonna jump. And jump is just like I'm either gonna update the program counter or not, right?
It's all it is. So in this case, if I don't update the program counter and I'm running the else branch, which means there's a set of instructions that that deal with the else work. And when the else
work is done because the else work is followed by the then work I gotta jump over that, right?
So I have to have non conditional jump like I always have to jump over that because I'm into it I have to end up here, right? I have to end up at the label end so if I did jump to the then section
then I can just continue running through.
If I ended up in the else section then I have to jump out of it, jump over the then section to end up in here. This is pretty low level, right? This is not how you normally think about your if
conditions, right? It's just a block, you put curlies.
But those curlies somehow have to translate into all of this on the end of the day. Okay, so what this basically says that if you have, if the value of R0, which is the argument into the Fibonacci
sequence is 0, then load the result to be 0, and because it's a then, it just falls through and exits the thing, right?
So Fibonacci sequence of 0 is 0. Pibonacci sequence of 1 is 1. So you load the result into R0 and then you exit the subroutine, right? So now R0 will contain the result value, either 0 or 1 depending
on whether it's 0 or 1, okay. Otherwise, what we're gonna do is we're gonna decrement R0, because the Fibonacci sequence is the sum of the previous two numbers in sequences, right?
By the way, did you hear about the Fibonacci conference? It was as big as the last two put together. So you decrement R0, right, and then you make a copy of R0 to put it in the R1, register. But now
we're using R1 register, this is the reason why we had to save it.
Because if we didn't save it, then the program that called us would be like, hey, you changed my value, that's no good, right? So we had to save it at the beginning. So we make a copy of the value
inside R1 register and then we recursively call ourselves, right?
So, it goes up to ourselves so that we can compute the result and now we know that the result is gonna be in R0, okay? Now, we need to save R0, so we push it on a stack. And now we can take the R1
and put it back into R0, decrement it one more time, and now call ourselves again, and now we have the result again in R0, and we move it from R0 to R1.
And then maybe restore the R0 we had over here, we pop it from the stack. So, now the R0 will contain the Fibonacci of x-1. And now, the R1 contains the Fibonacci of x-2. We add them together. And
now, the R0 contains the Fibonacci sequence that we want.
So, we can execute this. And you can see that you can see that it recuses quite a lot internally because we all know that Fibonacci is quite a recursive etc. And you can also see that at the
beginning we started with four is our kind of the initial value we were interested in to compute.
And then by the time we're done, you see the answer three is in here. But notice what's in the back of here, there's a bunch of garbage that ended up in here see all this stuff here. That's a
leftover stuff of our stack because when we push things on a stack, we wrote into memory.
But when you pop things out, we didn't bother clean up, because that's just extra work, why would you do that? And so now there is potentially dangerous information in there. Let's say somebody
pushed a private key onto a stack. And now that memory in one program shuts down and you start up another program and now the program gets that memory.
So part of the job of the operating system is to clear that memory to make sure like, there is nothing there when it gives it to you, because you could like accidentally read somebody else's stuff.
And you can also see how the stack has kind of grown backwards, right, in towards the data, but hopefully we had enough space.
So if I go here and just modify where is my number here. So five so that's the input this is the result, right, if I just modify it to five and I rerun it now you can see that and five is funny
because five of the answer is 5 and 4 of the answer is 4, okay, 6.
6, the answer is 8, right? So now, you can see that it's computing the other numbers, and you can see just how expensive it was in terms of computation, right? It's quite a lot of stuff that happened
inside of it. And notice, the bigger the number you choose, the further the stack has grown into you, right?
And so at some point, the stack is so big that it will collide with you. And so that's why you have to have extra stuff in there, inside of it. Okay, so this was kind of the main thing of like
showing you how a physical CPU works. And I'm kind of hoping that at this point you kinda have a good idea of how things work on the silicon level, right?
What actually is happening under the hood?
>> When numbers are stored in the registers, there'll be.
>> Yes.
>> Binary, when they do addition or subtraction, whatever remember this one to do with like two's complement?
>> Yes.
>> In a different way, doesn't it?
>> So the crazy thing is that CPUs do not know how to subtract.
Crazy, right, they can only add negative numbers. And the negative numbers are represented in something called two's complement. And basically it's a negative number I believe don't quote me it's
been years but I believe you just take the value of the positive number, you flip all the bits so one becomes 00 becomes 1 and then you add 1 to it into it.
And for some strange reason, that particular optimization is called two's complement. And the two's complement has this nice property that if you add it to a number, you've effectively done a
>> So would you have one register, basically save for just subtraction.
>> No, the CPU has no instruction for subtraction.
I mean, maybe the modern CPUs have for convenience or whatever, but fundamentally, CPUs do not understand subtraction. All they understand is adding two's complement.
>> Yeah.
>> So the compiler will typically take your number and convert it into a two's complement version, and then it will just add it.
And so that's the subtraction. So if you say A minus B, it loads B runs a two's complement operation on it, and then just says go and add. And so that's what I mean, like modern CPUs might have a
convenience instruction that internally automatically combines two's complement in addition, but in terms of the hardware that does the computation, the arithmetic logic unit only knows how to do
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Multiplying Fractions: A Mathematical Approach To Programming Logic - Code With C
Multiplying Fractions: A Mathematical Approach to Programming Logic
Ah, multiplying fractions! It might sound like a recipe for chaos, but fear not—I’m here to break it down in a way that even the non-math wizards among us can understand. So, buckle up, tech
enthusiasts and fellow coding whizzes, as we embark on this mathematical journey with a programming twist! 🚀
I. Understanding the Concept of Multiplying Fractions
A. What are Fractions?
1. Definition of Fractions
So, first things first—what on earth are fractions? Well, imagine dividing a pizza among friends. Those slices represent fractions! A fraction is a way of expressing a part of a whole, typically
written in the form a/b, where ‘a’ is the numerator (the number of parts we have) and ‘b’ is the denominator (the total number of equal parts).
2. Representation of Fractions in Mathematical Terms
Fractions can be represented graphically, as well as in decimal form. They’re fundamental to understanding proportions and are used in a myriad of everyday scenarios.
B. Multiplying Fractions
1. The Concept of Multiplying Fractions
Here comes the tricky part—how do we multiply these pesky little buggers? Well, when you multiply fractions, you simply multiply the numerators together to get the new numerator, and multiply the
denominators together to get the new denominator. Trust me, it’s not as bad as it sounds!
2. Understanding the Rule of Multiplying Fractions
To multiply fractions, we don’t need common denominators! It’s a straight shot—multiply across, top to top and bottom to bottom. Voila! You’ve got your multiplied fraction.
II. Programming Logic for Multiplying Fractions
Alright, time to put our coding hats on! Let’s navigate the world of programming and see how we can implement this multiplying fractions drama.
A. Creating a Function for Multiplying Fractions
1. Identifying the Inputs and Outputs
In the world of programming, we’ll need to figure out what inputs our function will take and what kind of results we expect as outputs. It’s like setting the stage for a blockbuster—get your cast and
crew right!
2. Writing the Logic for Multiplying Fractions in a Programming Language
Writing a function to multiply fractions might initially seem like a head-scratcher, but once you crack the code, it’s smooth sailing! We’ll translate the rules of multiplying fractions into lines of
code and make the computer do the heavy lifting.
B. Handling Edge Cases in Multiplying Fractions
1. Addressing Zero as a Denominator
Oh, the troublesome zero! When dealing with fractions, we can’t have zero as a denominator. It’s a recipe for disaster, really. We’ve got to ensure our function doesn’t go haywire when zero sneaks
into the denominator.
2. Dealing with Negative Fractions in the Programming Logic
Negative fractions, like -1/2 or 3/-4, might seem like rebels in the fraction world. But fear not—our function will tame these negative rascals and make sure the programming logic stays in tip-top
III. Testing the Multiplication of Fractions in a Programming Environment
Alright, so we’ve written our function. Now it’s time to put it through its paces and see if it stands strong in the face of different scenarios! Bring it on, test cases!
A. Implementing Test Cases for Multiplying Fractions
1. Designing Test Cases for Different Scenarios
We’ll whip up a bunch of test cases that check various scenarios. How about multiplying a whole bunch of mixed, positive, and negative fractions? We’ll leave no stone unturned!
2. Running the Test Cases to Validate the Programming Logic
Alright, lights, camera, action! We’ll run our test cases and scrutinize the results. It’s like playing detective, except we’re Sherlocking our way through programming bugs.
B. Debugging and Optimizing the Multiplication Logic
1. Identifying and Fixing any Errors in the Multiplication Function
A-ha! We’ve caught some bugs red-handed. Now it’s time to squash them and make our function as bug-free as humanly…err, programly possible.
2. Optimizing the Code for Better Performance and Efficiency
We’re not stopping at just working code. We want our function to be a lean, mean, multiplying machine. It’s all about squeezing that extra bit of efficiency out of every line of code.
IV. Applying the Multiplication of Fractions in Real-World Scenarios
Programming is all well and good, but how does this multiplying fractions jazz make a splash in the real world? Let’s find out!
A. Examples of How Fractions are Used in Practical Applications
1. Using Fractions in Cooking Recipes
Ever followed a recipe that calls for 3/4 cups of flour and then doubled it? Multiplying fractions is the unsung hero behind the scenes, ensuring those cookies turn out just right!
2. Applying Fractions in Measurements and Proportions
From carpentry to fashion design, fractions play a crucial role in getting the measurements spot on. Multiply, measure, cut! It’s a whole world of fractions out there.
B. Incorporating the Programming Logic into Real-World Projects
1. Integrating the Multiplication of Fractions into a Recipe App
Picture this: a recipe app that can intelligently scale ingredient quantities based on servings. That’s right, thanks to our fraction-multiplying function, we’re making it happen!
2. Using the Programming Logic for Calculating Proportions in a Construction Project
In construction, accuracy is everything. Whether it’s scaling up a blueprint or calculating material proportions, our trusty fraction-multiplying function keeps the structures standing tall.
V. Future Developments and Advancements in Multiplying Fractions
The future is calling, and it’s saying, “What’s next?” Let’s take a peek into the crystal ball and see what’s in store for multiplying fractions.
A. Emerging Technologies for Handling Fractions in Programming
1. Advancements in Programming Languages for Working with Fractions
As our love for fractions grows, programming languages are evolving too. We might see more native support for handling fractions efficiently.
2. New Techniques for Efficient Multiplication of Fractions in Software Development
From improved algorithms to optimized libraries, the tech world is constantly innovating. Who knows, we might soon have supercharged methods for multiplying fractions at our disposal!
B. Potential Applications of Multiplying Fractions in Computational Mathematics
1. Exploring the Use of Fractions in Complex Mathematical Algorithms
Fractions have a knack for peeking into advanced mathematical models. They might just be the missing piece of the puzzle in some of the most complex algorithms out there.
2. Investigating the Role of Multiplying Fractions in Advanced Computational Models
From simulations to data analysis, fractions could play a pivotal role in shaping the future of computational mathematics. Hold onto your hats, folks; it’s going to be a wild ride!
Overall, It’s All About Making Math Fantastic Again! 🌟
And there you have it, folks! Fractions might have been the bane of your math existence, but in the world of programming, they’re the seeds for something extraordinary. So, let’s raise a toast to
fractions, code, and the marvelous ways they come together!
Fun Fact: Did you know that the ancient Egyptians were the first to use fractions around 1800 BC? They really got the ball rolling on this whole fraction business!
So, until next time, keep coding, keep multiplying, and keep making math as cool as a cucumber!✌️
Program Code – Multiplying Fractions: A Mathematical Approach to Programming Logic
# Function to find Greatest Common Divisor (GCD) of two numbers
def gcd(a, b):
while b != 0:
a, b = b, a % b
return a
# Function to multiply two fractions and simplify the result
def multiply_fractions(num1, den1, num2, den2):
# Multiply the numerators and the denominators
result_num = num1 * num2
result_den = den1 * den2
# Simplify the fraction by finding the GCD of the numerator and denominator
greatest_cd = gcd(result_num, result_den)
# Divide both numerator and denominator by the GCD to simplify fraction
result_num //= greatest_cd
result_den //= greatest_cd
return result_num, result_den
# Example usage
# Multiplying 1/3 and 2/5
result_numerator, result_denominator = multiply_fractions(1, 3, 2, 5)
# Display the result
print(f'The product of the fractions is: {result_numerator}/{result_denominator}')
Code Output:
The product of the fractions is: 2/15
Code Explanation:
The program starts by defining a function called gcd which calculates the Greatest Common Divisor of two numbers using the Euclidean algorithm. This function is crucial as it is used later in the
program to simplify fractions.
Next, we define a function multiply_fractions that takes four parameters – num1, den1, num2, and den2. These represent the numerators and denominators of two fractions that we wish to multiply. The
program then performs the following steps:
1. Multiply the numerators (num1 and num2) to get the result’s numerator (result_num).
2. Multiply the denominators (den1 and den2) to get the result’s denominator (result_den).
At this point, we have the numerator and denominator of the product of the two fractions, but it is likely not in its simplest form. Henceforth, we:
3. Call the gcd function with result_num and result_den as arguments to find the greatest common divisor of these two numbers.
4. Divide both the result’s numerator and denominator by the greatest common divisor to simplify the fraction.
Finally, the simplified numerator and denominator are returned from the multiply_fractions function.
To showcase how the function works, we multiply two example fractions: 1/3 and 2/5. We call the multiply_fractions function with these values and then print out the simplified result with appropriate
The output confirms that the product of 1/3 and 2/5 is correctly calculated and simplified to 2/15.
This program serves as a practical application of programming logic to a fundamental mathematical problem – multiplying fractions. The architecture of the program efficiently uses a helper function
for a common task (finding the GCD) to create modular, clean, and reusable code.
Leave a comment Leave a comment | {"url":"https://www.codewithc.com/multiplying-fractions-a-mathematical-approach-to-programming-logic/","timestamp":"2024-11-13T02:39:23Z","content_type":"text/html","content_length":"147883","record_id":"<urn:uuid:fcfc8cf8-6dc4-45e4-90ec-5213f2168321>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00665.warc.gz"} |
What are cylinders spheres and cones considered?
What are cylinders spheres and cones considered?
You could also think of a cylinder as a “circular prism”. consists of two congruent, parallel circles joined by a curved surface. A cone is a three-dimensional solid that has a circular base joined
to a single point (called the vertex) by a curved side….Surface Area of a Cone.
Why a sphere is not a polyhedron?
Non-polyhedrons are cones, spheres, and cylinders because they have sides that are not polygons. A prism is a polyhedron with two congruent bases, in parallel planes, and the lateral sides are
Is a sphere a polyhedron?
A sphere is basically like a three-dimensional circle. In a way, it is also like a regular polyhedron with an infinite number of faces, such that the area of each face approaches zero.
Which amongst the following is not a polyhedron?
If we look at the figure in this option, we can make out that it is a solid but it does not have a flat surface. It has a curved surface and is a cone. Thus, it cannot be considered as a polyhedron
as it does not satisfy the conditions of a polyhedron. Therefore, the correct option is option D.
How are cones and cylinders different?
A cone is a 3-dimensional solid object that has a circular base and a single vertex. Cylinder : A cylinder is a 3-dimensional solid object that has two parallel circular base connected by a curved
Why is a sphere different from a prism and a cylinder?
These figures have curved surfaces, not flat faces. A cylinder is similar to a prism, but its two bases are circles, not polygons. The sphere is a space figure having all its points an equal distance
from the center point.
Which of the following solids is not polyhedron?
(i) A cylinder is not a polyhedron.
Which of the following is an example of non polyhedron?
Spheres, Cylinders and Cones do not have any polygonal face, and hence are not polyhedrons.
What is the difference between a circle and sphere?
The basic sphere and circle difference is that the circle is 2-Dimensional, and a sphere is 3-Dimensional. Deriving from the basic difference we can get another difference that is one can compute the
area of a circle, but for a sphere, we have to find its volume.
Is circle and sphere are same?
Definition of Circle and Sphere A Circle is a two-dimensional figure whereas, a Sphere is a three-dimensional object. A circle has all points at the same distance from its centre along a plane,
whereas in a sphere all the points are equidistant from the centre at any of the axes.
Is Cone not a polyhedron?
Cones, spheres, and cylinders are non-polyhedrons because their sides are not polygons and they have curved surfaces. The plural of a polyhedron is also known as polyhedra. They are classified as
prisms, pyramids, and platonic solids.
Is the cylinder a polyhedron or a solid?
A cylinder is a solid figure, but is not considered a polyhedron. It has two congruent, circular bases. It also has one rectangular face that is curved around the circular bases. Also know, Is a cube
a regular polyhedron?
How is a cylinder similar to a sphere?
A cylinder is similar to a prism, but its two bases are circles, not polygons. Also, the sides of a cylinder are curved, not flat. The sphere is a space figure having all its points an equal distance
from the center point.
What is the difference between a cone and a cylinder?
Also, the sides of a cylinder are curved, not flat. A cone has one circular base and a vertex that is not on the base. The sphere is a space figure having all its points an equal distance from the
center point.
What makes a cylinder different from a prism?
These figures have curved surfaces, not flat faces. A cylinder is similar to a prism, but its two bases are circles, not polygons. Also, the sides of a cylinder are curved, not flat. A cone has one
circular base and a vertex that is not on the base. | {"url":"https://teacherscollegesj.org/what-are-cylinders-spheres-and-cones-considered/","timestamp":"2024-11-02T22:20:53Z","content_type":"text/html","content_length":"143086","record_id":"<urn:uuid:84292a56-29aa-4b0d-81cf-e586b587c500>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00811.warc.gz"} |
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Dividing Decimals - HiSchool
Dividing Decimals
Published on
Dividing decimal numbers is just like dividing whole numbers, but with a little extra attention to the decimal point.
When you divide decimal numbers, it's important to pay attention to the decimal point so that you can place the decimal point correctly in your answer.
For example, let's say you want to divide 2.5 by 1.6. You would divide the numbers just like you would with whole numbers, and then place the decimal point in your answer based on the number of
decimal places in the original numbers.
Here's one way to do it:
First, divide the whole parts of the numbers:
2 ÷ 1 = 2
Next, count the number of decimal places in the original numbers:
2.5 has 1 decimal place 1.6 has 1 decimal place
Finally, place the decimal point in your answer based on the number of decimal places:
2 with 1 decimal place = 2.0
So, 2.5 ÷ 1.6 = 2.0.
It's important to be careful when dividing decimal numbers and to pay attention to the decimal point. With a little practice, you'll be dividing decimal numbers like a pro! | {"url":"https://blog.his.school/lecture/dividing-decimals","timestamp":"2024-11-06T15:16:07Z","content_type":"text/html","content_length":"21263","record_id":"<urn:uuid:483d11ff-00b4-489f-8534-9172bb4cac1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00725.warc.gz"} |
How to determine cost to lead a community with groups?
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How to determine cost to lead a community with groups?
What would be the simple equation to be used in Mathematica to determine how many people would be needed in order to produce leadership in let s say a community of 6309 with let s say 89 or so groups
using the following rule leads of thousands and of hundreds, of fifties and of tens, and officers for groups . Then it would be even more helpful if you could put in a price for each level so you
could simply figure out how much it would cost to pay to run such a community, in order that every question gets answered in a timely fashion and its not just left up for chance people picking and
choosing what questions to answer. We would like to implement this ourselves but we are only a team of four but would like to possibly raise money in order to help communities run better and sponsor
those community efforts but need to figure out how much it would cost. Having a simple program in Mathematica would really help us figure out how much something like this would cost with ever
changing community sizes.
1 Reply
This isn't actually Group Theory, but...
(1) It is probably simpler, and not much less accurate (if at all), if you do some rounding. In this case, you could go with 6000 people and 100 groups, say. Or 90 groups. You get the idea.
(2) If I understand the question correctly, you want to have cost levels at leadership levels, and this is independent of groups (I think). You could have the subset sizes as a list, in this case
{1000,100,10}, the cost levels a list of the same length (say {3,2,1} in whatever units, hundreds of dollars or whatever). I don't know what specifically you have in mind for officers, but if it is
say, chair, cochair, secretary, treasurer, and again there are cost levels of, say, {3,2,1,1}, then we have the needed information at hand.
(3) Now to code it. We'll have three lists as input, corresponding to what is indicated in (2) above (we do not actually need the names of the officers, just how much they cost). We'll do a check
that lengths correspond for the community leader sets and their costs, and elements are what we expect (positive integers for sizes of leader sets, nonnegative values for costs). We also want to know
how large is the community, and how many groups it has. I'll use "cl" prefixes for "community level" and "g" for "groups" (again, as in subsets of the community, not the groups of group theory).
cost[ccount_Integer, gcount_Integer, clevels_List, clcosts_List,
ocosts_List] /;
ccount > 0 && gcount >= 0 &&
VectorQ[clevels, IntegerQ[#] && # > 0 &] &&
VectorQ[clcosts, Element[#, Reals] && # >= 0 &] &&
VectorQ[ocosts, Element[#, Reals] && # >= 0 &] &&
Length[clevels] == Length[clcosts] :=
gcount*Total[ocosts] + Ceiling[ccount/clevels].clcosts
Pretty simple, albeit not simply pretty...
Anyway, for the example above, I'll show both with and without rounding.
cost[6000, 100, {1000, 100, 10}, {3, 2, 1}, {3, 2, 1, 1}]
(* Out[1241]= 1438 *)
cost[6000, 90, {1000, 100, 10}, {3, 2, 1}, {3, 2, 1, 1}]
(* Out[1242]= 1368 *)
cost[6309, 89, {1000, 100, 10}, {3, 2, 1}, {3, 2, 1, 1}]
(* Out[1243]= 1403 *)
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Summary of [Derivatives for Beginners - Basic Introduction]
• 00:00 - This video segment discusses how to find derivatives. It starts by explaining that the derivative of any constant is always zero. This means that the derivative of numbers like 7, -4, pi,
and pi to the e is all zero. The segment then mentions that if there is no variable and only a constant, the derivative will always be zero.
• 00:54 - The power rule in calculus states that if a variable is raised to a constant power, the derivative is equal to the constant multiplied by the variable raised to the power minus one. For
example, the derivative of x squared is 2x, the derivative of x cubed is 3x squared, the derivative of x to the fourth power is 4x cubed, and the derivative of x to the fifth power is 5x to the
fourth power. This pattern can be applied to find the derivative of any variable raised to a constant power.
• 02:08 - The video segment discusses the concept of finding the derivative of a function with a constant multiple. It explains that to find the derivative of a constant multiplied by a function,
you can simply multiply the constant by the derivative of the function. This is known as the constant multiple rule. The segment provides two examples to illustrate this rule: the derivative of
6x^8 is 48x^7, and the derivative of 5x^3 is 15x^2.
• 03:51 - The video segment explains how to find the derivative of a polynomial function step by step. It demonstrates the process using the example of a polynomial function, f(x) = 4x^3 + 7x^2 -
9x + 5. The derivative of each term is calculated using the power rule, where the exponent is multiplied by the coefficient and then reduced by one. The final result is the derivative of the
original function, which in this case is 12x^2 + 14x - 9.
• 06:05 - The video segment explains how to find the derivative of rational functions using the power rule. The first step is to rewrite the expression by moving the variable to the top and
changing the sign of the exponent. Then, using the power rule, the exponent is moved to the front and subtracted by one. Finally, the variable is moved back to the bottom to simplify the
expression. The segment provides examples of finding the derivatives of 1/x^2, 1/x^3, and -6/x^5, demonstrating the step-by-step process.
• 09:37 - The video segment explains how to find the derivative of radical functions. The first example demonstrates finding the derivative of the square root of x, which simplifies to 1/2 times
the square root of x. The second example involves finding the derivative of the cube root of x to the fifth power, which simplifies to 5 times x raised to the 2/3 power. The final example
involves finding the derivative of the eighth root of x to the fifth power, which simplifies to 5 over 8 times x raised to the 3/8 power. The segment emphasizes the steps of rewriting the
expression, moving the exponent to the front, and simplifying the exponent.
• 14:38 - The video segment discusses the derivatives of trigonometric functions. It explains the derivatives of sine, cosine, tangent, cotangent, secant, and cosecant functions, highlighting the
patterns and similarities between them. The segment then provides examples of finding the derivatives of various trigonometric functions, such as sine x and sine x cubed, cosine x squared,
tangent x to the fifth power, secant 4x, and cotangent x cubed plus x to the fifth power. The process involves differentiating the trigonometric function and the inside part separately, while
keeping the angle the same in the answer. The segment concludes by providing the derivatives for each example.
• 21:35 - The video segment discusses the derivatives of natural logarithms. It explains that the derivative of ln u is equal to u prime divided by u. The segment provides examples of finding the
derivatives of ln x and ln x cubed, demonstrating two different methods. It also shows how to find the derivative of ln x to the fourth minus x to the fifth. Additionally, the segment explains
how to find the derivative of ln tangent x and simplifies the expression to cosecant x times secant x.
• 25:47 - The video segment discusses how to find the derivative of a regular logarithmic function, such as log base a of u. The derivative is found using the formula u prime over u ln a. The video
compares this formula to the derivative of the natural log of u, which is u prime over u ln e. It highlights the similarities between the two equations and demonstrates how to find the derivative
of specific logarithmic functions, such as log base 2 of x to the fifth power and log base 4 of x cubed plus 4x squared. The segment concludes by simplifying the answers and providing different
ways to write the final answer.
• 30:33 - The video segment discusses the derivative of exponential functions. It explains that when the base is e, the derivative is simply the exponential function multiplied by the derivative of
the variable. However, if the base is a constant other than e, the derivative also includes the natural logarithm of the base. The segment provides examples of finding the derivatives of e to the
x, e to the 2x, e to the 5x, e to the x squared, and e to the sine x. It then demonstrates finding the derivatives of 5 raised to the x and 7 raised to the x to the fourth, highlighting the use
of the natural logarithm of the base in these cases.
• 35:07 - The video segment discusses the product rule in calculus. It explains that when finding the derivative of a function multiplied by another function, you differentiate the first part and
leave the second part the same, then leave the first part the same and differentiate the second part. The segment provides examples of applying the product rule to different functions and
simplifying the derivatives. It also mentions an alternative method of finding the derivative by distributing before taking the derivative.
• 40:09 - The video segment explains how to use the quotient rule to find the derivative of a function divided by another function. The formula for the quotient rule is v u prime minus u v prime
over v squared. The segment provides an example of finding the derivative of (3x - 5) divided by (7x + 4) using the quotient rule. By plugging in the values and simplifying the expression, the
final answer is determined to be 47 divided by (7x + 4) squared. The video concludes by stating that the quotient rule can be used whenever there is a division of two functions.
• 42:38 - The video segment discusses the chain rule in calculus. It explains that when dealing with composite functions, the derivative of the outer function is multiplied by the derivative of the
inner function. The segment provides examples of applying the chain rule to find derivatives of functions with composite functions. It emphasizes the importance of differentiating each function
step by step, starting from the outermost function and working inward. The segment also highlights the use of the power rule when dealing with exponents in composite functions.
• 48:27 - Implicit differentiation is used when you have equations with two different variables and want to find the derivative of one variable with respect to the other. To do this, you
differentiate both sides of the equation with respect to the desired variable. When differentiating x cubed with respect to x, you get 3x squared, but when differentiating y cubed with respect to
x, you get 3y squared times dy/dx. In related rates problems, you differentiate with respect to a different variable, such as time. In the example given, the derivative of x to the fourth is 4x
cubed, but the derivative of y to the fourth is 4y cubed times dy/dx. To solve for dy/dx, you isolate the term and divide both sides by the appropriate factor. Another example is provided to
further illustrate the process.
• 53:49 - The video segment explains how to find the derivative of a variable raised to a variable using logarithmic differentiation. The process involves setting y equal to x raised to the x,
taking the natural log of both sides, differentiating both sides with respect to x, and using the product rule. The final answer is x raised to the x times 1 plus ln x. This method is known as
logarithmic differentiation and allows for finding the derivative of a variable raised to a variable.
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What is Loss in Deep Learning? - Towards NLP
Hello, I am back! I am sorry for the long absence, but I got covid (yes, two years of quarantine and three vaccine shots after, I managed to get covid). But I am so glad to be back after two weeks
spent doom scrolling on the couch! And I’ll try to post twice this week to make it up to you folks. And I thought the first post could be a deep dive into the loss function! What is Loss in Deep
Learning? Let’s find out!
The Loss Function, the very Backbone of Deep Learning
So while I was couch-ridden waiting for death, I started re-reading the fastai bible: Deep Learning for coders with fastai and Pythorch. I talked about this book in this blog post, and frankly, I
can’t say enough good things about it. The book does a really great job at introducing pivotal Machine Learning and Deep Learning concepts in a really high-level, easy to understand way, to then go
further into detail.
And it so happens, they have a great explanation of the role of Loss in Machine Learning. As it happens, there would be no Machine Learning without Loss. So, since it such a fundamental concept, I
thought I might try to sum it all up for you!
What exactly is the Function of the Loss Function?
So it turns out, the concept of Loss is as old as Machine Learning. Arthur Samuel, an IBM researcher all the way back in 1949, started looking for different ways of programming computers. In 1962, he
wrote an essay that became a classic in the field: “Artificial Intelligence: a Frontier of Automation“; in this essay, he basically described Machine Learning as we have come to know it.
The idea was to show the computer examples of the problems we want to be solved, and let the computer figure it out for itself. To do so, Samuel said, we need to
[…] arrange for some automatic means of testing the effectiveness of any current weight assignment in terms of actual performance and provide a mechanism for altering the weight assigment so as
to maximize the performance. We need not go into the details of such a procedure to see that it could be made entirely automatic and to see that a machine so programmed would “learn” from its
So when Samuel talks about “testing the effectiveness of any current weight assignment”, he is talking about a loss function. The loss function is a function that returns a number that is small if
the performance of the model is good.
The purpose of the loss function is to measure the difference between the values the model predicts and the actual values – the targets or labels.
How does the Loss Function Work?
So what is the loss function? To make sure we are clear up until this point, let’s set up un example.
Let’s say we are training a model to do some sentiment analysis. We want our model to be able to tell us if a sentence is positive or negative. We will say that a positive sentence is labeled as a 1
and a negative sentence is labeled as a 0.
So let’s say we have three sentences, and we know the first one is positive, the second one is negative, and the third one is positive. We can then make a target vector with these targets. We can
also create a vector containing the predictions our model makes on whether these sentences are positive or negative. Such predictions must be a number between 0 and 1.
targets = tensor([1, 0, 1])
predictions = tensor([0.9, 0.2, 0.3])
So these two vectors will be the inputs of our loss function, that will measure the distance between the predictions and the target.
Writing our First Loss Function
Now that we know all of this, let us try and write our first loss function. As we said, it must take the difference between the targets and the predictions, so we can just write:
def loss_function(predictions, targets):
return torch.where(targets==1, 1-predictions, predictions).mean()
If we pass our predictions and target vectors from before into this function, we would get this vector back:
tensor([0.1000, 0.2000, 0.7000])
As you can see, the function returns a lower number when our model’s prediction are more accurate, or when accurate predictions are more confident or inaccurate predictions are less confident.
Great, that is exactly what we wanted! The only problem is that, as you recall, we assumed all our predictions would be numbers between 0 and 1. To ensure this is actually the case, we are going to
use another function, the Sigmoid function.
The Sigmoid Function, our Best Friend
The Sigmoid function takes an input and outputs a number between 0 and 1.
We can define it as follows:
def sigmoid(x): return 1/(1+torch.exp(-x))
So let’s adjust our loss function by applying the sigmoid to our predictions first, to make sure our predictions are a value between 0 and 1:
def loss_function(predictions, targets):
predictions = predictions.sigmoid()
return torch.where(targets==1, 1-predictions, predictions).mean()
The Many Flavors of Loss
Now we have a completely functioning loss function that we can use with Stochastic Gradient Descent to optimize our model automatically as Samuel predicted.
However, if you have been paying attention, you might have noticed that our loss function only works for outputs that can be labeled as either a 1 or a 0. Meaning that if we want to have outputs that
can have multiple values (think for example of Multi-label Classification or Image Classification), our loss function would not work.
What do we do now? Well, it turns out, there are multiple types of loss functions that we can use! So stay tuned for the next post, where we will talk more about what is loss in Deep Learning! | {"url":"https://www.towardsnlp.com/loss-in-deep-learning/","timestamp":"2024-11-02T17:09:52Z","content_type":"text/html","content_length":"107467","record_id":"<urn:uuid:ed2f5f38-e7bd-462d-846b-15b173f76e49>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00501.warc.gz"} |
Binary and Hexadecimal: Part 1
There are two reasons why I've seen people avoid learning how binary and hexadecimal number systems work: either they're intimidated because they don't consider themselves "math people," or they
think it's a waste because "why am I ever going to need this?" I really think there are some neat uses to these alternative numbering systems... and they're fun!
I'm definitely going to try to make them as accessible as I can in this post and take all of the intimidation out of them. You don't need to be a "math person." As long as you're a "person who can
count on their fingers," you'll be OK. If you don't have fingers, find a friend and use their fingers. If you don't have fingers and don't have a friend, then today is your lucky day! Now you've got
three friends: Binary, Hexadecimal, and me! And you can count them on my fingers if you want.
Starting from the Beginning: Decimal
Let's go back to elementary school for a second. How does counting work? Well, we have ten shapes we can use to represent values: numbers!
Zero through nine is ten digits. These are the only digits we count with -- at least if we're using Arabic Numerals. If you're not using Arabic Numerals but still using the decimal system, you'll
still have ten digits available to you. They might just look a little different.
So how do we count? We start with the first digit available to us: 0. Let's count our first time, adding one to our total. We still have 8 numerals we haven't seen yet, so we move to the next one: 1.
Then 2. Then 3. And so on, until we get to 9.
At that point, we've hit a snag. We've run out of numerals! So, what do we do? We tally that round of counting 10 times by incrementing a new digit by one and resetting that digit to 0. Now we're at
10. And we can start again, stepping through the numerals available to us: 11, 12, 13, 14, 15, 16, 17, 18, 19... Uh oh. We've completed another round through all of the numerals. So again, we
increment our tally and increment the second digit, the one on the left, to mark that fact. And we reset our right-most digit to zero.
What happens when our second digit runs through all of the available numerals: 97, 98, 99... we're getting ready to increment the right-most digit, which means we should be resetting it to zero and
incrementing the second digit, but we're out of numerals to use in the second spot. No problem here either, we'll just add another digit to celebrate that fact! Now we have a 1 in the third digit
location: 100. And so it goes. Congratulations, you still know how to count.
But, do you see the idea? We have 10 different numerals the show, and as each digit exhausts the numerals available to it, it increments the digit to the left of it and resets. That's how the rest of
the number systems work -- they just have different amounts of numerals! With 10 digits, we're using something called the "Decimal" (deci- means 10) system.
So What is Binary Then?
Well, you might guess from the name "BI-nary" (bi- meaning two) that there are two available numerals. And you're right! You may have even heard before what the two numerals are. 0 and 1. That's
right! As you can imagine, with significantly fewer numerals, we're going to rack up digits pretty quickly. Let's try counting in binary now. I think you're ready.
Don't forget that the same basic rules of counting apply. We'll start with zero.
And then we'll increment to the next available numeral.
And then we'll increment to the next available numeral again-- wait. We're already out of numerals! What gives!? That's OK, we follow our counting rules and increment the next digit and reset our
current digit.
And then we start again.
Oop! Now we go to increment our right-most digit, but we're out of numerals. So we go to increment our second right-most digit, but we're out of numerals there too! So we continue on to add a new
digit and reset our other digits.
Can you guess what happens next?
Now, have you been keeping track of our count? How many times have we incremented? I'm going to make a table, and, to make the ones and zeroes easier to see, I'm going to add some zeroes out in front
of the number as placeholders. It's OK, though. They don't change anything. The number 000000048 is still 48, right?
Decimal Number Increment Number
How does that feel? You're counting in binary! You're practically a computer! Quietly, to yourself, say "bleep bloop." No one will know. But we will know. And it'll make you feel accomplished. :)
Now, there's one more pattern that you may not have noticed, that makes binary even more magical. Check out the values of the increment number when there is only one 1 and everything else is zero:
1, 2, 4, and 8.
Do you see a pattern? Let me show you some other binary numbers and their decimal equivalents.
Binary Decimal
Don't worry about the space in between the binary digits. I added it in there to make things easier to read. Otherwise, if you read binary too long, your eyeballs start to fall out. The important
thing is the pattern. Do you see it? Every binary number that's just one 1 and the rest 0's is a power of 2. Or, put another way, the decimal numbers are doubling each time! That's right, everytime
you go up a digit (i.e. shifting things left one place) in binary, you double!
But, when you think about it, it makes sense right? Let's look at the decimal numbers that are one 1 followed by zeroes.
Each one is the previous one, multiplied by 10, in the deci- mal system. In the bi- nary system, every one is the previous one multipled by 2. Do you see? Don't worry if not. We'll do more with that
later, and we'll get more practice.
Hexadecimal Too?
Don't worry. Now that you've got binary nailed down, hexa- (meaning 6) -deci- (meaning 10) mal should be a snap. Hexadecimal has a base of 16.
Wait, wait, wait. There's only 10 numerals. How are we going to show 16 different "shapes?" Are we just going to make up new numbers? I thought you said there wasn't going to be hard math!
Don't worry. We're not making up any new shapes, and chances are, you've probably seen hexadecimal out in the wild somewhere. You're right about one thing, though: we need more "numerals" to get our
16 "shapes." But, luckily you know these shapes: letters! That's right, the numerals in hexadecimal are:
0 1 2 3 4 5 6 7 8 9 A B C D E F
(I'll pause while the skeptical among you take the time to count. There's 16. I'll wait.)
Satisfied? Good. Now let's start counting.
What do we do? Well, we've got more "numerals," right? We keep going!
Aaaaand now we're out of numerals. Increment the next digit and reset!
And so on. And when that second digit gets up there after much more counting?
See? Hopefully that wasn't so terrible. And you counted in both binary and hexadecimal! Congratulations! I now confer upon you the title of budding computer scientist. We can't really do much useful
with this new knowledge yet, though. In the next post, I'll show you how to convert back and forth, and what cool things that enables us to do.
This can be a tough topic, and I don't want it to be intimidating or scary at all. If I missed something or it's not quite clicking, my DM's are open! Shoot me a message and we'll talk about what's
bugging you.
Happy counting!
Originally posted on assert_not magic?
Top comments (1)
Code Beautify •
Great explanation. We have a great number of online tools for such conversions codebeautify.net/decimal/hex
For further actions, you may consider blocking this person and/or reporting abuse | {"url":"https://dev.to/rpalo/binary-and-hexadecimal-part-1-52lh","timestamp":"2024-11-02T12:47:40Z","content_type":"text/html","content_length":"92789","record_id":"<urn:uuid:3f21cff5-2eb4-4b47-be9a-36ca41d0d29e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00615.warc.gz"} |
13.3. Language Modeling with RNN
In Part 2, we demonstrated RNNs’ ability to predict the next value in a numerical sequence. Now, let’s explore how they can be used for a more complex task: predicting the next word in a sentence.
We can represent an RNN as follows:
$$ h_{t} = \begin{cases} 0 & t = 0 \\ r(w_{t}, h_{t-1}) & t \gt 0 \end{cases} \tag{13.9} $$
• $r(\cdot)$ is any recurrent neural network such as simple RNN, LSTM or GRU.
• $h_{t}$ is the hidden state of the recurrent neural network $r(\cdot)$ at time step $t$.
• $w_{t}$ is an input word at time step $t$.
To predict the next word $w_{t+1}$ given a sequence ${w_{1}, w_{2},\ldots, w_{t}}$, we use a dense layer with a softmax activation function as follows:
$$ P(w_{t+1}|w_{1}, w_{2},\ldots, w_{t}) = \text{softmax}(W h_{t}) \tag{13.10} $$
• $W \in \mathbb{R}^{|V| \times d_{h}}$ is a weight matrix belonging to a dense layer with no bias.
• $V$ is a set of vocabulary.
• $d_{h}$ is the hidden state dimension.
Using $(13.10)$, we can naturally define a language model such as $(13.5)$, and also the last word prediction task using RNN can be defined as follows:
$$ \hat{w}_{n} = \underset{w \in V}{argmax} \ P(w| w_{\lt n}) = \underset{w \in V}{argmax} \ \text{softmax}(W h_{n-1}) \tag{13.11} $$
In the following subsections, we will create an RNN-based language model and employ it to perform last-word prediction.
13.3.1. Implementation
Complete Python code is available at: LanguageModel-RNN-tf.py
13.3.1.1. Create Model
To build our RNN-based language model, we employ a many-to-many GRU neural network shown below:
# ========================================
# Create Model
# ========================================
input_nodes = 1
hidden_nodes = 1024
output_nodes = vocab_size
embedding_dim = 256
class GRU(tf.keras.Model):
def __init__(self, hidden_units, output_units, vocab_size, embedding_dim):
self.hidden_units = hidden_units
self.embedding = tf.keras.layers.Embedding(vocab_size, embedding_dim)
self.gru = tf.keras.layers.GRU(
self.softmax = tf.keras.layers.Dense(output_units, activation="softmax")
def call(self, x):
x = self.embedding(x)
output = self.gru(x)
x = self.softmax(output)
return x
model = GRU(hidden_nodes, output_nodes, vocab_size, embedding_dim)
model.build(input_shape=(None, max_len))
This code is similar to the sine wave prediction model in Section 11.4.1, with three key differences:
• Many-to-Many GRU Layer: As mentioned above, this model has a Many-to-Many architecture. Therefore, we set $\text{return_sequences}=\text{True}$ to return the entire sequence of hidden states $\
• Dense Layer: The activation function of the dense layer is the softmax function. Its output size equals the size of the dictionary (vocabulary), providing probabilities for each word in the
• Word Embedding Layer: The input (tokenized data) is passed through the word embedding layer before being fed into the GRU unit. Hence, the GRU unit internally handles the vector data
corresponding to the input.
13.3.1.2. Dataset and Training
An RNN-based language model predicts the next word in a sequence based on the previous words in the sentence.
Let $x_{1}, x_{2}, \ldots , x_{n}$ represents an input word sequence. The corresponding desired output sequence $y$ is constructed by shifting the input sequence by one word and appending a special
padding symbol “<pad>” at the end:
$$ y_{1}, y_{2}, \ldots , y_{n} = x_{2}, x_{3}, \ldots, x_{n}, \lt \!\! \text{pad} \!\! \gt $$
INPUT : <SOS> just go for it <EOS>
OUTPUT: just go for it <EOS> <pad>
In this example, the RNN-based language model would be trained to:
• predict ‘just’ from <SOS>.
• predict ‘go’ from ‘just’.
• predict ‘for’ from ‘go’.
• predict ‘it’ from ‘for’.
• predict <EOS> from ‘it’.
Fig.13-2: Training Process in Our Language Model
The training phase is almost identical to the one described in the previous parts, involving RNNs and XOR gates.
# ========================================
# Training
# ========================================
lr = 0.0001
beta1 = 0.99
beta2 = 0.9999
optimizer = optimizers.Adam(learning_rate=lr, beta_1=beta1, beta_2=beta2)
def train(x, y):
with tf.GradientTape() as tape:
output = model(x)
loss = loss_function(y, output)
grad = tape.gradient(loss, model.trainable_variables)
optimizer.apply_gradients(zip(grad, model.trainable_variables))
return loss
# If n_epochs = 0, this model uses the trained parameters saved in the last checkpoint,
# allowing you to perform last word prediction without retraining.
if len(sys.argv) == 2:
n_epochs = int(sys.argv[1])
n_epochs = 200
for epoch in range(1, n_epochs + 1):
for batch, (X_train, Y_train) in enumerate(dataset):
loss = train(X_train, Y_train)
We use the SparseCategoricalCrossentropy as the loss function because our training data $y$ is a set of integers, like “$[21 \ \ 1 \ \ 44 \ \ 0 \ \ 0]$”.
If you are concerned about the difference between the output format of the GRU (softmax output) and the training data format, refer to the documentation.
loss_object = tf.keras.losses.SparseCategoricalCrossentropy(reduction="none")
def loss_function(real, pred):
mask = tf.math.logical_not(tf.math.equal(real, 0)) # this masks '<pad>'
real= tf.Tensor(
[[21 1 44 0 0] (jump ! <eos> <pad> <pad>)
[ 17 9 24 2 44] (i go there . <eos>)
[ 27 1 44 0 0] (no ! <eos> <pad> <pad>)
[ 21 22 32 2 44]], (i know you . <eos>)
, shape=(4, 5), dtype=int64)
where <pad> = 0.
mask= tf.Tensor(
[[True True True False False]
[ True True True True True ]
[[True True True False False]
[ True True True True True ],
shape=(4, 5), dtype=bool)
loss_ = loss_object(real, pred)
mask = tf.cast(mask, dtype=loss_.dtype)
loss_ *= mask
return tf.reduce_mean(loss_)
13.3.1.3. Prediction
For the last word prediction task, we feed our language model with the input sequence excluding the last word.
The resulting output sequences from the model are then passed to the softmax layer. The output of the softmax layer represents the probabilities of each word in the vocabulary.
We select the most probable word as the predicted last word by picking the last element of the output of the softmax layer.
Fig.13-3: Prediction Process in Our Language Model
13.3.2. Demonstration
Following 200 epochs of training, our RNN-based language model’s last-word prediction is shown below:
$ python LanguageModel-RNN-tf.py
vocabulary size: 303
number of sentences: 7452
Model: "gru"
Layer (type) Output Shape Param #
embedding (Embedding) multiple 77568
gru_1 (GRU) multiple 3938304
dense (Dense) multiple 310575
Total params: 4,326,447
... snip ...
Text: Just let me help you.
Input: <sos> Just let me help
Predicted last word:
you => 0.857351
the => 0.005812
i => 0.002887
me => 0.002504
tom => 0.002407
Text: Tom is the one who told me who to give it to.
Input: <sos> Tom is the one who told me who to give it
Predicted last word:
to => 0.924526
time => 0.001423
this => 0.000225
if => 0.000026
your => 0.000024
Text: She will get well soon.
Input: <sos> She will get well
Predicted last word:
soon => 0.855645
in => 0.032168
very => 0.011623
here => 0.008390
old => 0.007782
Text: I came back home late.
Input: <sos> I came back home
Predicted last word:
late => 0.836478
alone => 0.032124
by => 0.031414
with => 0.027323
night => 0.002223
Text: She told me that I could use her room.
Input: <sos> She told me that i could use her
Predicted last word:
room => 0.930203
with => 0.011007
car => 0.010192
brother => 0.003243
father => 0.002974
This code contains the checkpoint function that preserves the training progress. Hence, once trained, the task can be executed without retraining by setting the parameter $\text{n_epochs}$ to $0$, or
simply passing $0$ when executing the Python code, as shown below:
$ python LanguageModel-RNN-tf.py 0 | {"url":"http://www.interdb.jp/dl/part03/ch13/sec03.html","timestamp":"2024-11-09T07:38:45Z","content_type":"text/html","content_length":"53782","record_id":"<urn:uuid:c47a5935-956f-49df-92fe-3d03f4aaf371>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00770.warc.gz"} |
Function Transforms¶
vmap vmap is the vectorizing map; vmap(func) returns a new function that maps func over some dimension of the inputs.
grad grad operator helps computing gradients of func with respect to the input(s) specified by argnums.
grad_and_value Returns a function to compute a tuple of the gradient and primal, or forward, computation.
vjp Standing for the vector-Jacobian product, returns a tuple containing the results of func applied to primals and a function that, when given cotangents, computes the reverse-mode
Jacobian of func with respect to primals times cotangents.
jvp Standing for the Jacobian-vector product, returns a tuple containing the output of func(*primals) and the “Jacobian of func evaluated at primals” times tangents.
jacrev Computes the Jacobian of func with respect to the arg(s) at index argnum using reverse mode autodiff
jacfwd Computes the Jacobian of func with respect to the arg(s) at index argnum using forward-mode autodiff
hessian Computes the Hessian of func with respect to the arg(s) at index argnum via a forward-over-reverse strategy.
Utilities for working with torch.nn.Modules¶
In general, you can transform over a function that calls a torch.nn.Module. For example, the following is an example of computing a jacobian of a function that takes three values and returns three
model = torch.nn.Linear(3, 3)
def f(x):
return model(x)
x = torch.randn(3)
jacobian = jacrev(f)(x)
assert jacobian.shape == (3, 3)
However, if you want to do something like compute a jacobian over the parameters of the model, then there needs to be a way to construct a function where the parameters are the inputs to the
function. That’s what make_functional() and make_functional_with_buffers() are for: given a torch.nn.Module, these return a new function that accepts parameters and the inputs to the Module’s forward
make_functional Given a torch.nn.Module, make_functional() extracts the state (params) and returns a functional version of the model, func.
make_functional_with_buffers Given a torch.nn.Module, make_functional_with_buffers extracts the state (params and buffers) and returns a functional version of the model func that can be invoked like
a function.
combine_state_for_ensemble Prepares a list of torch.nn.Modules for ensembling with vmap().
If you’re looking for information on fixing Batch Norm modules, please follow the guidance here | {"url":"https://pytorch.org/functorch/0.2.1/functorch.html","timestamp":"2024-11-04T04:13:32Z","content_type":"text/html","content_length":"24610","record_id":"<urn:uuid:19ffb76e-0b45-4316-8738-3a78ebde6ffb>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00086.warc.gz"} |
9.1 Work, Power, and the Work–Energy Theorem - Physics | OpenStax (2024)
Section Learning Objectives
By the end of this section, you will be able to do the following:
• Describe and apply the work–energy theorem
• Describe and calculate work and power
Teacher Support
The learning objectives in this section will help your students master the following standards:
• (6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
□ (A)describe and apply the work–energy theorem;
□ (C)describe and calculate work and power.
In addition, the High School Physics Laboratory Manual addresses the following standards:
• (6) Science concepts. The student knows that changes occur within a physical system and applies the laws of conservation of energy and momentum. The student is expected to:
□ (C) calculate the mechanical energy of, power generated within, impulse applied to, and momentum of a physical system.
Use the lab titled Work and Energy as a supplement to address content in this section.
Section Key Terms
energy gravitational potential energy joule kinetic energy mechanical energy
potential energy power watt work work–energy theorem
Teacher Support
In this section, students learn how work determines changes in kinetic energy and that power is the rate at which work is done.
[BL][OL] Review understanding of mass, velocity, and acceleration due to gravity. Define the general definitions of the words potential and kinetic.
[AL][AL] Remind students of the equation $W=P E e =fmg W=P E e =fmg$ . Point out that acceleration due to gravity is a constant, therefore PE[e] that results from work done by gravity will also be
constant. Compare this to acceleration due to other forces, such as applying muscles to lift a rock, which may not be constant.
The Work–Energy Theorem
In physics, the term work has a very specific definition. Work is application of force, $ff$, to move an object over a distance, d, in the direction that the force is applied. Work, W, is described
by the equation
$W=fd. W=fd.$
Some things that we typically consider to be work are not work in the scientific sense of the term. Let’s consider a few examples. Think about why each of the following statements is true.
• Homework is not work.
• Lifting a rock upwards off the ground is work.
• Carrying a rock in a straight path across the lawn at a constant speed is not work.
The first two examples are fairly simple. Homework is not work because objects are not being moved over a distance. Lifting a rock up off the ground is work because the rock is moving in the
direction that force is applied. The last example is less obvious. Recall from the laws of motion that force is not required to move an object at constant velocity. Therefore, while some force may be
applied to keep the rock up off the ground, no net force is applied to keep the rock moving forward at constant velocity.
Teacher Support
[BL][OL] Explain that, when this theorem is applied to an object that is initially at rest and then accelerates, the $1 2 m v 1 2 1 2 m v 1 2$ term equals zero.
[OL][AL] Work is measured in joules and $W=fd W=fd$ . Force is measured in newtons and distance in meters, so joules are equivalent to newton-meters $( N⋅m ) ( N⋅m )$
Work and energy are closely related. When you do work to move an object, you change the object’s energy. You (or an object) also expend energy to do work. In fact, energy can be defined as the
ability to do work. Energy can take a variety of different forms, and one form of energy can transform to another. In this chapter we will be concerned with mechanical energy, which comes in two
forms: kinetic energy and potential energy.
• Kinetic energy is also called energy of motion. A moving object has kinetic energy.
• Potential energy, sometimes called stored energy, comes in several forms. Gravitational potential energy is the stored energy an object has as a result of its position above Earth’s surface (or
another object in space). A roller coaster car at the top of a hill has gravitational potential energy.
Let’s examine how doing work on an object changes the object’s energy. If we apply force to lift a rock off the ground, we increase the rock’s potential energy, PE. If we drop the rock, the force of
gravity increases the rock’s kinetic energy as the rock moves downward until it hits the ground.
The force we exert to lift the rock is equal to its weight, w, which is equal to its mass, m, multiplied by acceleration due to gravity, g.
$f=w=mg f=w=mg$
The work we do on the rock equals the force we exert multiplied by the distance, d, that we lift the rock. The work we do on the rock also equals the rock’s gain in gravitational potential energy, PE
$W=P E e =mgd W=P E e =mgd$
Kinetic energy depends on the mass of an object and its velocity, v.
$KE= 1 2 m v 2 KE= 1 2 m v 2$
When we drop the rock the force of gravity causes the rock to fall, giving the rock kinetic energy. When work done on an object increases only its kinetic energy, then the net work equals the change
in the value of the quantity $1 2 m v 2 1 2 m v 2$ . This is a statement of the work–energy theorem, which is expressed mathematically as
$W=ΔKE= 1 2 m v 2 2 − 1 2 m v 1 2 . W=ΔKE= 1 2 m v 2 2 − 1 2 m v 1 2 .$
The subscripts [2] and [1] indicate the final and initial velocity, respectively. This theorem was proposed and successfully tested by James Joule, shown in Figure 9.2.
Does the name Joule sound familiar? The joule (J) is the metric unit of measurement for both work and energy. The measurement of work and energy with the same unit reinforces the idea that work and
energy are related and can be converted into one another. 1.0 J = 1.0 N∙m, the units of force multiplied by distance. 1.0 N = 1.0 kg∙m/s^2, so 1.0 J = 1.0 kg∙m^2/s^2. Analyzing the units of the term
(1/2)mv^2 will produce the same units for joules.
Figure 9.2 The joule is named after physicist James Joule (1818–1889). (C. H. Jeens, Wikimedia Commons)
Work and Energy
This video explains the work energy theorem and discusses how work done on an object increases the object’s KE.
Access multimedia content
True or false—The energy increase of an object acted on only by a gravitational force is equal to the product of the object's weight and the distance the object falls.
1. True
2. False
Teacher Support
Repeat the information on kinetic and potential energy discussed earlier in the section. Have the students distinguish between and understand the two ways of increasing the energy of an object (1)
applying a horizontal force to increase KE and (2) applying a vertical force to increase PE.
Calculations Involving Work and Power
In applications that involve work, we are often interested in how fast the work is done. For example, in roller coaster design, the amount of time it takes to lift a roller coaster car to the top of
the first hill is an important consideration. Taking a half hour on the ascent will surely irritate riders and decrease ticket sales. Let’s take a look at how to calculate the time it takes to do
Recall that a rate can be used to describe a quantity, such as work, over a period of time. Power is the rate at which work is done. In this case, rate means per unit of time. Power is calculated by
dividing the work done by the time it took to do the work.
$P= W t P= W t$
Let’s consider an example that can help illustrate the differences among work, force, and power. Suppose the woman in Figure 9.3 lifting the TV with a pulley gets the TV to the fourth floor in two
minutes, and the man carrying the TV up the stairs takes five minutes to arrive at the same place. They have done the same amount of work $( fd ) ( fd )$ on the TV, because they have moved the same
mass over the same vertical distance, which requires the same amount of upward force. However, the woman using the pulley has generated more power. This is because she did the work in a shorter
amount of time, so the denominator of the power formula, t, is smaller. (For simplicity’s sake, we will leave aside for now the fact that the man climbing the stairs has also done work on himself.)
Figure 9.3 No matter how you move a TV to the fourth floor, the amount of work performed and the potential energy gain are the same.
Power can be expressed in units of watts (W). This unit can be used to measure power related to any form of energy or work. You have most likely heard the term used in relation to electrical devices,
especially light bulbs. Multiplying power by time gives the amount of energy. Electricity is sold in kilowatt-hours because that equals the amount of electrical energy consumed.
The watt unit was named after James Watt (1736–1819) (see Figure 9.4). He was a Scottish engineer and inventor who discovered how to coax more power out of steam engines.
Figure 9.4 Is James Watt thinking about watts? (Carl Frederik von Breda, Wikimedia Commons)
Teacher Support
[BL][OL] Review the concept that work changes the energy of an object or system. Review the units of work, energy, force, and distance. Use the equations for mechanical energy and work to show what
is work and what is not. Make it clear why holding something off the ground or carrying something over a level surface is not work in the scientific sense.
[OL] Ask the students to use the mechanical energy equations to explain why each of these is or is not work. Ask them to provide more examples until they understand the difference between the
scientific term work and a task that is simply difficult but not literally work (in the scientific sense).
[BL][OL] Stress that power is a rate and that rate means "per unit of time." In the metric system this unit is usually seconds. End the section by clearing up any misconceptions about the
distinctions between force, work, and power.
[AL] Explain relationships between the units for force, work, and power. If $W=fd W=fd$ and work can be expressed in J, then $P= W t = fd t P= W t = fd t$ so power can be expressed in units of $N⋅m s
N⋅m s$
Also explain that we buy electricity in kilowatt-hours because, when power is multiplied by time, the time units cancel, which leaves work or energy.
Watt’s Steam Engine
James Watt did not invent the steam engine, but by the time he was finished tinkering with it, it was more useful. The first steam engines were not only inefficient, they only produced a back and
forth, or reciprocal, motion. This was natural because pistons move in and out as the pressure in the chamber changes. This limitation was okay for simple tasks like pumping water or mashing
potatoes, but did not work so well for moving a train. Watt was able build a steam engine that converted reciprocal motion to circular motion. With that one innovation, the industrial revolution was
off and running. The world would never be the same. One of Watt's steam engines is shown in Figure 9.5. The video that follows the figure explains the importance of the steam engine in the industrial
Figure 9.5 A late version of the Watt steam engine. (Nehemiah Hawkins, Wikimedia Commons)
Teacher Support
Initiate a discussion on the historical significance of suddenly increasing the amount of power available to industries and transportation. Have students consider the fact that the speed of
transportation increased roughly tenfold. Changes in how goods were manufactured were just as great. Ask students how they think the resulting changes in lifestyle compare to more recent changes
brought about by innovations such as air travel and the Internet.
Watt's Role in the Industrial Revolution
This video demonstrates how the watts that resulted from Watt's inventions helped make the industrial revolution possible and allowed England to enter a new historical era.
Access multimedia content
Which form of mechanical energy does the steam engine generate?
1. Potential energy
2. Kinetic energy
3. Nuclear energy
4. Solar energy
Before proceeding, be sure you understand the distinctions among force, work, energy, and power. Force exerted on an object over a distance does work. Work can increase energy, and energy can do
work. Power is the rate at which work is done.
Applying the Work–Energy Theorem
An ice skater with a mass of 50 kg is gliding across the ice at a speed of 8 m/s when her friend comes up from behind and gives her a push, causing her speed to increase to 12 m/s. How much work did
the friend do on the skater?
The work–energy theorem can be applied to the problem. Write the equation for the theorem and simplify it if possible.
$W=ΔKE= 1 2 m v 2 2 − 1 2 m v 1 2 W=ΔKE= 1 2 m v 2 2 − 1 2 m v 1 2$
$SimplifytoW= 1 2 m( v 2 2 − v 1 2 ) SimplifytoW= 1 2 m( v 2 2 − v 1 2 )$
Identify the variables.m = 50 kg,
$v 2 =12 m s , and v 1 =8 m s v 2 =12 m s , and v 1 =8 m s$
$W= 1 2 50( 12 2 − 8 2 )=2,000J W= 1 2 50( 12 2 − 8 2 )=2,000J$
Work done on an object or system increases its energy. In this case, the increase is to the skater’s kinetic energy. It follows that the increase in energy must be the difference in KE before and
after the push.
This problem illustrates a general technique for approaching problems that require you to apply formulas: Identify the unknown and the known variables, express the unknown variables in terms of the
known variables, and then enter all the known values.
Teacher Support
Identify the three variables and choose the relevant equation. Distinguish between initial and final velocity and pay attention to the minus sign.
Identify the variables. m = 50 kg,
$v 2 =12 m s , and v 1 =8 m s v 2 =12 m s , and v 1 =8 m s$
$W= 1 2 50( 12 2 − 8 2 )=2,000J W= 1 2 50( 12 2 − 8 2 )=2,000J$
Practice Problems
A weightlifter lifts a 200 N barbell from the floor to a height of 2 m. How much work is done?
1. $0\,\text{J}$
2. $100\,\text{J}$
3. $200\,\text{J}$
4. $400\,\text{J}$
Identify which of the following actions generates more power. Show your work.
• carrying a $100\,\text{N}$ TV to the second floor in $50\,\text{s}$ or
• carrying a $24\,\text{N}$ watermelon to the second floor in $10\,\text{s}$?
1. Carrying a $100\,\text{N}$ TV generates more power than carrying a $24\,\text{N}$ watermelon to the same height because power is defined as work done times the time interval.
2. Carrying a $100\,\text{N}$ TV generates more power than carrying a $24\,\text{N}$ watermelon to the same height because power is defined as the ratio of work done to the time interval.
3. Carrying a $24\,\text{N}$ watermelon generates more power than carrying a $100\,\text{N}$ TV to the same height because power is defined as work done times the time interval.
4. Carrying a $24\,\text{N}$ watermelon generates more power than carrying a $100\,\text{N}$ TV to the same height because power is defined as the ratio of work done and the time interval.
Check Your Understanding
Identify two properties that are expressed in units of joules.
1. work and force
2. energy and weight
3. work and energy
4. weight and force
When a coconut falls from a tree, work W is done on it as it falls to the beach. This work is described by the equation
$W=Fd= 1 2 m v 2 2 − 1 2 m v 1 2 . W=Fd= 1 2 m v 2 2 − 1 2 m v 1 2 .$
Identify the quantities F, d, m, v[1], and v[2] in this event.
1. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the earth, v[1] is the initial velocity, and v[2] is the velocity with
which it hits the beach.
2. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the coconut, v[1] is the initial velocity, and v[2] is the velocity with
which it hits the beach.
3. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the earth, v[1] is the velocity with which it hits the beach, and v[2] is
the initial velocity.
4. F is the force of gravity, which is equal to the weight of the coconut, d is the distance the nut falls, m is the mass of the coconut, v[1] is the velocity with which it hits the beach, and v[2]
is the initial velocity.
Teacher Support
Use Check Your Understanding questions to assess students’ achievement of the section’s learning objectives. If students are struggling with a specific objective, the Check Your Understanding will
help identify which one and direct students to the relevant content. | {"url":"https://vanintgrp.com/article/9-1-work-power-and-the-work-energy-theorem-physics-openstax","timestamp":"2024-11-12T12:08:36Z","content_type":"text/html","content_length":"109946","record_id":"<urn:uuid:e6a18e2c-fb7b-4292-a3bb-01e09eb673de>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00040.warc.gz"} |
Interpretation of the AUC | R-bloggersInterpretation of the AUC
Interpretation of the AUC
[This article was first published on
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The AUC* or concordance statistic c is the most commonly used measure for diagnostic accuracy of quantitative tests. It is a discrimination measure which tells us how well we can classify patients in
two groups: those with and those without the outcome of interest. Since the measure is based on ranks, it is not sensitive to systematic errors in the calibration of the quantitative tests.
It is very well known that a test with no better accuracy than chance has an AUC of 0.5, and a test with perfect accuracy has an AUC of 1. But what is the exact interpretation of an AUC of for
example 0.88? Did you know that the AUC is completely equivalent with the Mann-Whitney U test statistic?
*AUC: the Area Under the Curve (AUC) of the Receiver Operating Characteristic (ROC) curve.
Around 27% of the patients with liver cirrhosis will develop Hepatocellular Carcinoma (HCC) within 5 years of follow-up. With our biomarker “peakA” we would like to predict which patients will
develop HCC, and which won't. We will assess the diagnostic accuracy of biomarker “peakA” using the AUC.
To keep things visually clear, we suppose we have a dataset of only 12 patients. Four patients did develop HCC (the “cases”) and 8 didn't (the “controls”). (fictive data)
HCC Biomarker_value
0 1.063
1 1.132
1 1.122
1 1.058
0 0.988
0 1.182
0 1.037
0 1.052
0 0.925
1 1.232
0 0.911
0 0.967
The AUC can be defined as “The probability that a randomly selected case will have a higher test result than a randomly selected control”. Let's use this definition to calculate and visualize the
estimated AUC.
In the figure below, the cases are presented on the left and the controls on the right.
Since we have only 12 patients, we can easily visualize all 32 possible combinations of one case and one control. (Rcode below)
Those 32 different pairs of cases and controls are represented by lines on the plot above. 28 of them are indicated in green. For those pairs, the value for “PeakA” is higher for the case compared to
the control. The remaining 4 pairs are indicated in blue. The AUC can be estimated as the proportion of pairs for which the case has a higher value compared to the control. Thus, the estimated AUC is
the proportion of green lines or 28/32 = 0.875. This visualization might help to understand the concept of an AUC. Besides this educational purpose, this type of plot is not very useful. Hopefully,
the sample size of your study is much larger than 12 patients. And in that situation, this type of plot will become very crowded.
The ROC curve
Now let's verify that the AUC is indeed equal to 0.875 in a classical way, by plotting a ROC curve and calculating the estimated AUC using the ROCR package.
The ROC curve plots the False Positive Rate (FPR) on the X-axis and the True Postive Rate (TPR) on the Y-axis for all possible thresholds (or cutoff values).
• True Positive Rate (TPR) or sensitivity: the proportion of actual positives that are correctly identified as such.
• True Negative Rate (TNR) or specificiy: the proportion of actual negatives that are correctly identified as such.
• False Positive Rate (FPR) or 1-specificity: the proportion of actual negatives that are wrongly identified as positives.
pred <- prediction(df$Biomarker_value, df$HCC )
perf <- performance(pred,"tpr","fpr")
abline(a=0, b=1, col="#8AB63F")
The green line represents a completely uninformative test, which corresponds to an AUC of 0.5. A curve pulled close to the upper left corner indicates (an AUC close to 1 and thus) a better performing
test. The ROC curve does not show the cutoff values
The ROCR package also allows to calculate the estimated AUC:
auc<- performance( pred, c("auc"))
unlist(slot(auc , "y.values"))
[1] 0.875
The estimated AUC based on this ROC curve is indeed equal to 0.875, the proportion of pairs for which the value of “PeakA” is larger for HCC compared to NoHCC.
Relation to cutoff points of the biomarker
Visualizing the sensitivity and specificity as a function of the cutoff points of the biomarker results in a plot that is at least as informative as a ROC curve and (in my opinion) easier to
interpret. The plot can be created using the ROCR package.
testy <- performance(pred,"tpr","fpr")
Using the str() function, we see that the following slots are part of the testy object:
• alpha.values: Cutoff
• x.values: Specificity or True Negative Rate
• y.values: Sensitivity or True Positive Rate
plot([email protected][[1]], [email protected][[1]], type='n',
xlab='Cutoff points of the biomarker',
ylab='sensitivity or specificity')
lines([email protected][[1]], [email protected][[1]],
type='s', col="#1A425C", lwd=2)
lines([email protected][[1]], [email protected][[1]],
type='s', col="#8AB63F", lwd=2)
legend(1.11,.85, c('sensitivity', 'specificity'),
lty=c(1,1), col=c("#1A425C", "#8AB63F"), cex=.9, bty='n')
The plot shows how the sensitivity increases as the specificity decreases and vice versa, in relation to the possible cutoff points of the biomarker.
Mann-Whitney U test statistic
The Mann-Whitney U test statistic (or Wilcoxon or Kruskall-Wallis test statistic) is equivalent to the AUC (Mason, 2002).
The AUC can be calculated from the output of the wilcox.test() function:
wt <-wilcox.test(data=df, df$Biomarker_value ~ df$HCC)
1 - wt$statistic/(sum(df$HCC==1)*sum(df$HCC==0))
The p-value of the Mann-Whitney U test can thus safely be used to test whether the AUC differs significantly from 0.5 (AUC of an uninformative test).
wt <-wilcox.test(data=df, df$Biomarker_value ~ df$HCC)
[1] 0.04848485
Simulation: the completely uninformative test.
Now, let's have a look how our plots look like if our biomarker is not informative at all.
Data creation:
#simulation of the data
HCC <- rbinom (n=12, size=1, prob=0.27)
Biomarker_value <- rnorm (12,mean=1,sd=0.1) + HCC*0
# replacing the zero by a value would make the test informative
df<-data.frame (HCC, Biomarker_value)
HCC Biomarker_value
0 1.0630099
1 0.9723816
1 0.9715840
1 0.9080678
0 0.9883752
0 1.1817312
The function expand.grid() is used to create all possible combinations of one case and one control:
newdf<- expand.grid (Biomarker_value [df$HCC==0],Biomarker_value [df$HCC==1])
colnames(newdf)<- c("NoHCC", "HCC")
newdf$Pair <- seq(1,dim(newdf)[1])
For each pair the values of the biomarker are compared between case and control:
newdf$Comparison <- 1*(newdf$HCC>newdf$NoHCC)
[1] 0.40625
newdf$Comparison<-factor(newdf$Comparison, labels=c("HCC>NoHCC","HCC<=NoHCC"))
kable (head(newdf,4))
NoHCC HCC Pair Comparison
1.0630099 0.9723816 1 HCC>NoHCC
0.9883752 0.9723816 2 HCC>NoHCC
1.1817312 0.9723816 3 HCC>NoHCC
1.0370628 0.9723816 4 HCC>NoHCC
longdf = melt(newdf, id.vars = c("Pair", "Comparison"),
variable.name = "Group",
measure.vars = c("HCC", "NoHCC"))
lab<-paste("AUC = Proportion \n of green lines \nAUC=", round(table(newdf$Comparison)[2]/sum(table(newdf$Comparison)),3))
fav.col=c("#1A425C", "#8AB63F")
ggplot(longdf, aes(x=Group, y=value))+geom_line(aes(group=Pair, col=Comparison)) +
scale_color_manual(values=fav.col)+theme_bw() +
ylab("Biomarker value") + geom_text(x=0.75,y=0.95,label=lab) +
geom_point(shape=21, size=2) +
theme(legend.title=element_blank(), legend.position="bottom")
pred <- prediction(df$Biomarker_value, df$HCC )
perf <- performance(pred,"tpr","fpr")
abline(a=0, b=1, col="#8AB63F")
Calculating the AUC:
auc<- performance( pred, c("auc"))
unlist(slot(auc , "y.values"))
[1] 0.40625
Sensitivity and specificity as a function of the cutoff points of the biomarker:
testy <- performance(pred,"tpr","fpr")
plot([email protected][[1]], [email protected][[1]], type='n', xlab='Cutoff points of the biomarker', ylab='sensitivity or specificity')
lines([email protected][[1]], [email protected][[1]], type='s', col="#1A425C")
lines([email protected][[1]], [email protected][[1]], type='s', col="#8AB63F")
legend(1.07,.85, c('sensitivity', 'specificity'), lty=c(1,1), col=c("#1A425C", "#8AB63F"), cex=.9, bty='n')
Equivalence with the Mann-Whitney U test:
wt <-wilcox.test(data=df, df$Biomarker_value ~ df$HCC)
1 - wt$statistic/(sum(df$HCC==1)*sum(df$HCC==0))
[1] 0.6828283
General remarks on the AUC
Often, a combination of new markers is selected from a large set. This can result in overoptimistic expectations of the marker's performance. Any performance measure should be estimated with
correction for optimism, for example by applying cross-validation or bootstrap resampling. However, validation in fully independent, external data is the best way to validate a new marker.
When we want to assess the incremental value of an additional marker (e.g. molecular, genetic, imaging) to an existing model, the increase of the AUC can be reported.
• Mason, S. J. and Graham, N. E. (2002), Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation. Q.J.R.
Meteorol. Soc., 128: 2145-2166.
• Steyerberg, Ewout W. et al. “Assessing the Performance of Prediction Models: A Framework for Some Traditional and Novel Measures.” Epidemiology (Cambridge, Mass.) 21.1 (2010): 128-138.
• Xavier Verhelst, Dieter Vanderschaeghe, Laurent Castéra, Tom Raes, Anja Geerts, Claire Francoz, Roos Colman, François Durand, Nico Callewaert, and Hans Van Vlierberghe (2017). A Glycomics-Based
Test Predicts the Development of Hepatocellular Carcinoma in Cirrhosis. Clin Cancer Res (23) (11) 2750-2758
Related Post | {"url":"https://www.r-bloggers.com/2018/09/interpretation-of-the-auc/","timestamp":"2024-11-02T02:54:51Z","content_type":"text/html","content_length":"123610","record_id":"<urn:uuid:1ea659f7-8961-4224-839e-0c26359589f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00334.warc.gz"} |
How to Sum A Union All Subquery In Oracle?
To sum a union all subquery in Oracle, you can enclose the entire union all subquery within another select statement and use the sum() function on the column that you want to calculate the total sum
for. For example: SELECT sum(total_sales) FROM ( SELECT sales_amount as total_sales FROM table1 UNION ALL SELECT sales_amount as total_sales FROM table2 ) subquery;
How to perform calculations on the results of a union all subquery in oracle?
To perform calculations on the results of a UNION ALL subquery in Oracle, you can wrap the subquery in an outer query where you can then perform the necessary calculations. Here is an example:
1 SELECT column1, column2, SUM(column3) AS total_sum
2 FROM (
3 SELECT column1, column2, column3
4 FROM table1
5 UNION ALL
6 SELECT column1, column2, column3
7 FROM table2
8 ) subquery
9 GROUP BY column1, column2;
In this example, we are performing a UNION ALL operation on two tables (table1 and table2) and then calculating the total sum of column3 for each unique combination of column1 and column2.
You can customize the calculations as needed based on your specific requirements and the columns that you have available in your subquery results. Just make sure to wrap the subquery in an outer
query and use appropriate aggregation functions (such as SUM, AVG, COUNT, etc.) as needed for your calculations.
What is the output of a union all subquery in oracle?
The output of a UNION ALL subquery in Oracle will combine the results of two or more SELECT statements into a single result set. This means that all rows from each SELECT statement will be included
in the final output, even if there are duplicate rows.
How to group results in a union all subquery in oracle?
To group results in a UNION ALL subquery in Oracle, you can use a "GROUP BY" clause at the end of the subquery. Here's an example:
1 SELECT column1, SUM(column2)
2 FROM (
3 SELECT column1, column2
4 FROM table1
5 WHERE condition1
6 UNION ALL
7 SELECT column1, column2
8 FROM table2
9 WHERE condition2
10 ) subquery
11 GROUP BY column1;
In this example, the results from the two SELECT statements in the UNION ALL subquery are grouped by "column1" using the GROUP BY clause. This will aggregate the results by the value in "column1" and
apply any aggregate functions (e.g. SUM, COUNT, etc.) to the grouped data. | {"url":"https://tech-blog.duckdns.org/blog/how-to-sum-a-union-all-subquery-in-oracle","timestamp":"2024-11-07T22:53:28Z","content_type":"text/html","content_length":"141893","record_id":"<urn:uuid:c40922cc-3c28-49c4-b2d4-690487509693>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00579.warc.gz"} |
KR C4 tool change not being written to program file.
I have a program with subprograms, and each subprogram specifies a tool. In the resulting program file a procedure is defined for each subprogram, and only the first one has a line to specify the
tool. As a result, the real robot knows it has a tool for the first program, then for the second, it considers the flange to be the TCP. The model in RoboDK behaves as expected. I can't figure out
how the postprocessor removes the setTool instructions.
DEF C_0_stroke_prog ( )
; ...
PTP $AXIS_ACT ; skip BCO quickly
$BASE = {FRAME: X 0.000,Y 0.000,Z 0.000,A 0.000,B 0.000,C 0.000}
$TOOL = {FRAME: X 0.000,Y 0.000,Z 205.000,A 0.000,B 0.000,C 0.000}
; Show paintbrush2Shape
$VEL.CP = 1.00000
PTP {AXIS: A1 -97.95481,A2 -64.68960,A3 115.43524,A4 152.84739,A5 76.05941,A6 100.41150}
LIN {X -64.192,Y 1659.559,Z 1175.153,A 162.369,B -7.105,C 68.735}
; ...
DEF C_1_stroke_prog ( )
PTP $AXIS_ACT ; skip BCO quickly
$APO.CPTP = 100.000
$APO.CDIS = 100.000
; Show paintbrush2Shape
$VEL.CP = 1.00000
PTP {AXIS: A1 -36.34085,A2 -55.72483,A3 88.11530,A4 -0.00000,A5 57.60954,A6 -36.34085} C_PTP
LIN {X 1287.834,Y 947.423,Z 825.569,A -180.000,B -0.000,C 180.000} C_DIS
; ...
05-25-2018, 11:08 PM
RoboDK automatically filters setting the same tool or reference frames more than once.
This is a default setting and can be changed with following steps:
• Select Tools-Options
• Uncheck "Filter setting reference and tool frames"
04-18-2019, 03:41 PM
(05-25-2018, 11:08 PM)Albert Wrote: RoboDK automatically filters setting the same tool or reference frames more than once.
This is a default setting and can be changed with following steps:
□ Select Tools-Options
□ Uncheck "Filter setting reference and tool frames"
Hi Albert,
I'm wondering if filtering should be off by default? It causes the robot to do something different than the simulation in RoboDK. Or is this something that's handled differently by different robots?
04-19-2019, 03:53 AM
Quote:It causes the robot to do something different than the simulation in RoboDK.
Can you elaborate? What kinds of different behaviors are you experiencing? | {"url":"https://robodk.com/forum/Thread-KR-C4-tool-change-not-being-written-to-program-file","timestamp":"2024-11-10T12:04:02Z","content_type":"application/xhtml+xml","content_length":"50462","record_id":"<urn:uuid:fa93faa0-7813-42ae-b9b6-b51459d39646>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00701.warc.gz"} |
Time and Speed Synchronisation: Calendar of Applied Math Problems
Calendar, Time and Speed Problems and More
This course on Calendar, Clock, Speed and Time is a part of Applied Mathematics. These topics will help you in Aptitude and Quantitative Reasoning. You'll learn abour Calendar, Time and Speed
problems and more.
You'll start with binary numbers. The digits 0 and 1 are called binary digits. You will learn how to convert binary numbers to decimal numbers and the other way round. This topic is useful in
Computer Science as well.
You will learn how to understand calendar problems. Have you ever wondered which day of the year you were born?. You will learn how to determine if a year is a leap year and about odd days. A leap
year has two odd days.
Moving on to problems on the clock. Certain facts about the minute hand and hour hand are explained to you. You are taught how to calculate the correct time if the clock loses or gains time. How to
calculate the angle between the 2 hands of a clock, a very age old problem is explained.
This is followed by speed, distance and time problems. A very simple but tricky concept. You are taught how to calculate the average speed and the relative speed for different distances and the same
distance. The time taken for two moving objects to cross each other is also explained.
You will end the course with seating arrangement problems. The 2 types of seating arrangements, namely, linear and circular are discussed. You will understand how to calculate the left and right of 2
people sitting around a table and facing the centre of the circle. This is a very important concept frequently misunderstood by students.
As an upgrade,I have also included a topic on deciphering the code where you'll learn to unravel alphabets, numbers and strings. You'll love this method of adding and multiplying alphabets in a code.
Overall, this is a great course which is very useful for students preparing for quantitative aptitude examinations. Enrol for the course to learn more. If not, you can watch the preview.
Welcome to the fascinating journey into the Mathematics of Blood Relations. Get a basic knowledge of Annuities and their types. | {"url":"https://www.mathmadeeasy.co/post/time-and-speed-synchronisation-calendar-of-applied-math-problems","timestamp":"2024-11-02T18:33:52Z","content_type":"text/html","content_length":"1050490","record_id":"<urn:uuid:e97ae67a-312e-440e-8ec9-76e6e2ff4df3>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00852.warc.gz"} |
2. A can do a plece of work in 14 days while B can do it i... | Filo
Question asked by Filo student
work 2. A can do a plece of work in 14 days while can do it in 21 days. They began together and worked at it for 6 days. Then. A fell ill and had to complete the remaining work alone. In how many
days was the work completed? 10. A can do of a certain work in 16 days and can do of the same work in 3 days. In how many days can both finish the work, working together?
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Question work 2. A can do a plece of work in 14 days while can do it in 21 days. They began together and worked at it for 6 days. Then. A fell ill and had to complete the remaining work alone. In
Text how many days was the work completed? 10. A can do of a certain work in 16 days and can do of the same work in 3 days. In how many days can both finish the work, working together?
Updated On Dec 5, 2022
Topic All topics
Subject Mathematics
Class Class 9
Answer Type Video solution: 1
Upvotes 119
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Metric-based stochastic conceptual clustering for ontologies
Metric-based stochastic conceptual clustering for ontologies
Created by W.Langdon from gp-bibliography.bib Revision:1.8010
□ author = "Nicola Fanizzi and Claudia d'Amato and Floriana Esposito",
□ title = "Metric-based stochastic conceptual clustering for ontologies",
□ journal = "Information Systems",
□ year = "2009",
□ volume = "34",
□ pages = "792--806",
□ number = "8",
□ note = "Sixteenth ACM Conference on Information Knowledge and Management (CIKM 2007)",
□ keywords = "genetic algorithms, genetic programming, Conceptual clustering",
□ ISSN = "0306-4379",
□ URL = " http://www.sciencedirect.com/science/article/B6V0G-4W3HXC0-1/2/95a1535c9097d816c4ec5ad804772c4b",
□ abstract = "A conceptual clustering framework is presented which can be applied to multi-relational knowledge bases storing resource annotations expressed in the standard languages for the
Semantic Web. The framework adopts an effective and language-independent family of semi-distance measures defined for the space of individual resources. These measures are based on a finite
number of dimensions corresponding to a committee of discriminating features represented by concept descriptions. The clustering algorithm expresses the possible clusterings in terms of
strings of central elements (medoids, w.r.t. the given metric) of variable length. The method performs a stochastic search in the space of possible clusterings, exploiting a technique based
on genetic programming. Besides, the number of clusters is not necessarily required as a parameter: a natural number of clusters is autonomously determined, since the search spans a space of
strings of different length. An experimentation with real ontologies proves the feasibility of the clustering method and its effectiveness in terms of standard validity indices. The framework
is completed by a successive phase, where a newly constructed intensional definition, expressed in the adopted concept language, can be assigned to each cluster. Finally, two possible
extensions are proposed. One allows the induction of hierarchies of clusters. The other applies clustering to concept drift and novelty detection in the context of ontologies.",
□ notes = "invited extended version \cite{DBLP:conf/cikm/FanizzidE07}",
Genetic Programming entries for Nicola Fanizzi Claudia d'Amato Floriana Esposito | {"url":"http://gpbib.pmacs.upenn.edu/gp-html/Fanizzi_2009_IS.html","timestamp":"2024-11-03T13:50:26Z","content_type":"text/html","content_length":"5577","record_id":"<urn:uuid:5925b026-2ad0-4238-8b5a-5d53efb5e973>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00899.warc.gz"} |
NPTEL Software Testing Week 2 Assignment Answers 2024 » DBC ItanagarNPTEL Software Testing Week 2 Assignment Answers 2024
NPTEL Software Testing Week 2 Assignment Answers 2024
NPTEL Software Testing Week 2 Assignment Answers 2024
1. When do we say that a test path p tours a sub-path q with a side-trip?
• A test path p tours a sub-path q with a side-trip when p is an infeasible test path on its own.
• A test path p tours a sub-path q with a side-trip when every vertex and every edge in q also occurs in p in the same order.
• A test path p tours a sub-path q with a side-trip when every vertex in q also occurs in p in the same order.
• A test path p tours a sub-path q with a side-trip when every edge in q also occurs in p in the same order.
Answer :- For Answers Click Here
2. Which of the graph traversal algorithms given below, when run on a graph that does not have edge weights, will return the shortest path between a pair of vertices?
• Depth first search (DFS)
• Breadth first search (BFS)
• Both DFS and BFS
• Neither BFS nor DFS
Answer :- For Answers Click Here
3. Why is complete path coverage considered to be an infeasible structural graph coverage criterion?
• Complete path coverage could be infeasible if the graph has several disconnected components.
• Complete path coverage could be infeasible if the graph has strongly connected components or loops.
• Complete path coverage could be infeasible if the graph has isolated vertices or edges.
• Complete path coverage could be infeasible as covering all paths in a graph through test cases is not needed.
Answer :-
4. Which graph coverage criterion considers writing test cases where all the simple paths of maximal length are visited?
• Complete path coverage.
• Simple path coverage.
• Specified path coverage.
• Prime path coverage.
Answer :- For Answers Click Here
5. Which are the three vertices that will be added to the BFS queue in the first step of the BFS algorithm? Does the order in which they are added matter?
• The three vertices will be 2, 3 and 4, their order will be exactly the same as the one given in this answer option.
• The three vertices will be 2, 3 and 4, their order does not matter.
• The three vertices will be 2, 3 and 5, their order will be exactly the same as the one given in this answer option.
• The three vertices will be 2, 3 and 5, their order does not matter.
Answer :-
6. If vertices 2, 3 and 4 are added in the queue in the given order during the BFS visit, which vertex will be marked as visited first?
• Vertex 2 will be marked as visited first.
• Vertex 3 will be marked as visited first.
• Vertex 4 will be marked as visited first.
• None of the three given vertices will be marked as visited first.
Answer :-
7. When will BFS traversal be complete for the given graph?
• BFS traversal will be complete when all the vertices are marked as visited and the queue is empty.
• BFS traversal will be complete when all the vertices are added to the queue.
Answer :- For Answers Click Here
8. Which of the following represents a correct order of visit during a breadth first search traversal of the given graph starting from vertex 1?
• 1, 2, 3, 4, 5.
• 1, 4, 5, 2, 3.
• 1, 5, 4, 3, 5.
• 1, 4, 5, 2, 3.
Answer :-
9. Which of the following represents a correct order of visit during a depth first search traversal of the given graph starting from vertex 1?
• 1, 4, 5, 2, 3.
• 1, 2, 3, 4, 5.
• 1, 2, 3, 5, 4.
• 1, 5, 4, 3, 2.
Answer :-
10. Which of the following options are true regarding DFS and BFS traversals in the given graph starting with vertex 1?
• Both DFS and BFS will always visit the vertices in the same order.
• DFS order of traversal need not be the same as the BFS order of traversal for the give graph.
Answer :- For Answers Click Here
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class sklearn.mixture.BayesianGaussianMixture(*, n_components=1, covariance_type='full', tol=0.001, reg_covar=1e-06, max_iter=100, n_init=1, init_params='kmeans', weight_concentration_prior_type=
'dirichlet_process', weight_concentration_prior=None, mean_precision_prior=None, mean_prior=None, degrees_of_freedom_prior=None, covariance_prior=None, random_state=None, warm_start=False,
verbose=0, verbose_interval=10)[source]¶
Variational Bayesian estimation of a Gaussian mixture.
This class allows to infer an approximate posterior distribution over the parameters of a Gaussian mixture distribution. The effective number of components can be inferred from the data.
This class implements two types of prior for the weights distribution: a finite mixture model with Dirichlet distribution and an infinite mixture model with the Dirichlet Process. In practice
Dirichlet Process inference algorithm is approximated and uses a truncated distribution with a fixed maximum number of components (called the Stick-breaking representation). The number of
components actually used almost always depends on the data.
Read more in the User Guide.
n_componentsint, default=1
The number of mixture components. Depending on the data and the value of the weight_concentration_prior the model can decide to not use all the components by setting some component
weights_ to values very close to zero. The number of effective components is therefore smaller than n_components.
covariance_type{‘full’, ‘tied’, ‘diag’, ‘spherical’}, default=’full’
String describing the type of covariance parameters to use. Must be one of:
'full' (each component has its own general covariance matrix),
'tied' (all components share the same general covariance matrix),
'diag' (each component has its own diagonal covariance matrix),
'spherical' (each component has its own single variance).
tolfloat, default=1e-3
The convergence threshold. EM iterations will stop when the lower bound average gain on the likelihood (of the training data with respect to the model) is below this threshold.
reg_covarfloat, default=1e-6
Non-negative regularization added to the diagonal of covariance. Allows to assure that the covariance matrices are all positive.
max_iterint, default=100
The number of EM iterations to perform.
n_initint, default=1
The number of initializations to perform. The result with the highest lower bound value on the likelihood is kept.
init_params{‘kmeans’, ‘k-means++’, ‘random’, ‘random_from_data’}, default=’kmeans’
The method used to initialize the weights, the means and the covariances. String must be one of:
‘kmeans’ : responsibilities are initialized using kmeans. ‘k-means++’ : use the k-means++ method to initialize. ‘random’ : responsibilities are initialized randomly.
‘random_from_data’ : initial means are randomly selected data points.
Changed in version v1.1: init_params now accepts ‘random_from_data’ and ‘k-means++’ as initialization methods.
weight_concentration_prior_type{‘dirichlet_process’, ‘dirichlet_distribution’}, default=’dirichlet_process’
String describing the type of the weight concentration prior.
weight_concentration_priorfloat or None, default=None
The dirichlet concentration of each component on the weight distribution (Dirichlet). This is commonly called gamma in the literature. The higher concentration puts more mass in the
center and will lead to more components being active, while a lower concentration parameter will lead to more mass at the edge of the mixture weights simplex. The value of the parameter
must be greater than 0. If it is None, it’s set to 1. / n_components.
mean_precision_priorfloat or None, default=None
The precision prior on the mean distribution (Gaussian). Controls the extent of where means can be placed. Larger values concentrate the cluster means around mean_prior. The value of the
parameter must be greater than 0. If it is None, it is set to 1.
mean_priorarray-like, shape (n_features,), default=None
The prior on the mean distribution (Gaussian). If it is None, it is set to the mean of X.
degrees_of_freedom_priorfloat or None, default=None
The prior of the number of degrees of freedom on the covariance distributions (Wishart). If it is None, it’s set to n_features.
covariance_priorfloat or array-like, default=None
The prior on the covariance distribution (Wishart). If it is None, the emiprical covariance prior is initialized using the covariance of X. The shape depends on covariance_type:
(n_features, n_features) if 'full',
(n_features, n_features) if 'tied',
(n_features) if 'diag',
float if 'spherical'
random_stateint, RandomState instance or None, default=None
Controls the random seed given to the method chosen to initialize the parameters (see init_params). In addition, it controls the generation of random samples from the fitted distribution
(see the method sample). Pass an int for reproducible output across multiple function calls. See Glossary.
warm_startbool, default=False
If ‘warm_start’ is True, the solution of the last fitting is used as initialization for the next call of fit(). This can speed up convergence when fit is called several times on similar
problems. See the Glossary.
verboseint, default=0
Enable verbose output. If 1 then it prints the current initialization and each iteration step. If greater than 1 then it prints also the log probability and the time needed for each step.
verbose_intervalint, default=10
Number of iteration done before the next print.
weights_array-like of shape (n_components,)
The weights of each mixture components.
means_array-like of shape (n_components, n_features)
The mean of each mixture component.
The covariance of each mixture component. The shape depends on covariance_type:
(n_components,) if 'spherical',
(n_features, n_features) if 'tied',
(n_components, n_features) if 'diag',
(n_components, n_features, n_features) if 'full'
The precision matrices for each component in the mixture. A precision matrix is the inverse of a covariance matrix. A covariance matrix is symmetric positive definite so the mixture of
Gaussian can be equivalently parameterized by the precision matrices. Storing the precision matrices instead of the covariance matrices makes it more efficient to compute the
log-likelihood of new samples at test time. The shape depends on covariance_type:
(n_components,) if 'spherical',
(n_features, n_features) if 'tied',
(n_components, n_features) if 'diag',
(n_components, n_features, n_features) if 'full'
The cholesky decomposition of the precision matrices of each mixture component. A precision matrix is the inverse of a covariance matrix. A covariance matrix is symmetric positive
definite so the mixture of Gaussian can be equivalently parameterized by the precision matrices. Storing the precision matrices instead of the covariance matrices makes it more efficient
to compute the log-likelihood of new samples at test time. The shape depends on covariance_type:
(n_components,) if 'spherical',
(n_features, n_features) if 'tied',
(n_components, n_features) if 'diag',
(n_components, n_features, n_features) if 'full'
True when convergence was reached in fit(), False otherwise.
Number of step used by the best fit of inference to reach the convergence.
Lower bound value on the model evidence (of the training data) of the best fit of inference.
weight_concentration_prior_tuple or float
The dirichlet concentration of each component on the weight distribution (Dirichlet). The type depends on weight_concentration_prior_type:
(float, float) if 'dirichlet_process' (Beta parameters),
float if 'dirichlet_distribution' (Dirichlet parameters).
The higher concentration puts more mass in the center and will lead to more components being active, while a lower concentration parameter will lead to more mass at the edge of the
weight_concentration_array-like of shape (n_components,)
The dirichlet concentration of each component on the weight distribution (Dirichlet).
The precision prior on the mean distribution (Gaussian). Controls the extent of where means can be placed. Larger values concentrate the cluster means around mean_prior. If
mean_precision_prior is set to None, mean_precision_prior_ is set to 1.
mean_precision_array-like of shape (n_components,)
The precision of each components on the mean distribution (Gaussian).
mean_prior_array-like of shape (n_features,)
The prior on the mean distribution (Gaussian).
The prior of the number of degrees of freedom on the covariance distributions (Wishart).
degrees_of_freedom_array-like of shape (n_components,)
The number of degrees of freedom of each components in the model.
covariance_prior_float or array-like
The prior on the covariance distribution (Wishart). The shape depends on covariance_type:
(n_features, n_features) if 'full',
(n_features, n_features) if 'tied',
(n_features) if 'diag',
float if 'spherical'
Number of features seen during fit.
feature_names_in_ndarray of shape (n_features_in_,)
Names of features seen during fit. Defined only when X has feature names that are all strings.
See also
Finite Gaussian mixture fit with EM.
>>> import numpy as np
>>> from sklearn.mixture import BayesianGaussianMixture
>>> X = np.array([[1, 2], [1, 4], [1, 0], [4, 2], [12, 4], [10, 7]])
>>> bgm = BayesianGaussianMixture(n_components=2, random_state=42).fit(X)
>>> bgm.means_
array([[2.49... , 2.29...],
[8.45..., 4.52... ]])
>>> bgm.predict([[0, 0], [9, 3]])
array([0, 1])
fit(X[, y]) Estimate model parameters with the EM algorithm.
fit_predict(X[, y]) Estimate model parameters using X and predict the labels for X.
get_metadata_routing() Get metadata routing of this object.
get_params([deep]) Get parameters for this estimator.
predict(X) Predict the labels for the data samples in X using trained model.
predict_proba(X) Evaluate the components' density for each sample.
sample([n_samples]) Generate random samples from the fitted Gaussian distribution.
score(X[, y]) Compute the per-sample average log-likelihood of the given data X.
score_samples(X) Compute the log-likelihood of each sample.
set_params(**params) Set the parameters of this estimator.
fit(X, y=None)[source]¶
Estimate model parameters with the EM algorithm.
The method fits the model n_init times and sets the parameters with which the model has the largest likelihood or lower bound. Within each trial, the method iterates between E-step and M-step
for max_iter times until the change of likelihood or lower bound is less than tol, otherwise, a ConvergenceWarning is raised. If warm_start is True, then n_init is ignored and a single
initialization is performed upon the first call. Upon consecutive calls, training starts where it left off.
Xarray-like of shape (n_samples, n_features)
List of n_features-dimensional data points. Each row corresponds to a single data point.
Not used, present for API consistency by convention.
The fitted mixture.
fit_predict(X, y=None)[source]¶
Estimate model parameters using X and predict the labels for X.
The method fits the model n_init times and sets the parameters with which the model has the largest likelihood or lower bound. Within each trial, the method iterates between E-step and M-step
for max_iter times until the change of likelihood or lower bound is less than tol, otherwise, a ConvergenceWarning is raised. After fitting, it predicts the most probable label for the input
data points.
Xarray-like of shape (n_samples, n_features)
List of n_features-dimensional data points. Each row corresponds to a single data point.
Not used, present for API consistency by convention.
labelsarray, shape (n_samples,)
Component labels.
Get metadata routing of this object.
Please check User Guide on how the routing mechanism works.
A MetadataRequest encapsulating routing information.
Get parameters for this estimator.
deepbool, default=True
If True, will return the parameters for this estimator and contained subobjects that are estimators.
Parameter names mapped to their values.
Predict the labels for the data samples in X using trained model.
Xarray-like of shape (n_samples, n_features)
List of n_features-dimensional data points. Each row corresponds to a single data point.
labelsarray, shape (n_samples,)
Component labels.
Evaluate the components’ density for each sample.
Xarray-like of shape (n_samples, n_features)
List of n_features-dimensional data points. Each row corresponds to a single data point.
resparray, shape (n_samples, n_components)
Density of each Gaussian component for each sample in X.
Generate random samples from the fitted Gaussian distribution.
n_samplesint, default=1
Number of samples to generate.
Xarray, shape (n_samples, n_features)
Randomly generated sample.
yarray, shape (nsamples,)
Component labels.
score(X, y=None)[source]¶
Compute the per-sample average log-likelihood of the given data X.
Xarray-like of shape (n_samples, n_dimensions)
List of n_features-dimensional data points. Each row corresponds to a single data point.
Not used, present for API consistency by convention.
Log-likelihood of X under the Gaussian mixture model.
Compute the log-likelihood of each sample.
Xarray-like of shape (n_samples, n_features)
List of n_features-dimensional data points. Each row corresponds to a single data point.
log_probarray, shape (n_samples,)
Log-likelihood of each sample in X under the current model.
Set the parameters of this estimator.
The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each
component of a nested object.
Estimator parameters.
selfestimator instance
Estimator instance.
Examples using sklearn.mixture.BayesianGaussianMixture¶ | {"url":"https://scikit-learn.org/1.3/modules/generated/sklearn.mixture.BayesianGaussianMixture.html","timestamp":"2024-11-11T07:36:50Z","content_type":"text/html","content_length":"68553","record_id":"<urn:uuid:c5ef3925-b257-4237-bc3a-0614c97808f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00000.warc.gz"} |
How to Crochet Fringed Edging
How to Crochet Fringed Edging is the last in a little series of posts I’ve been writing about pretty edgings you can crochet.
Most of them are very easy. They can be used to give a pretty detail to bags, purses, hair bands, blankets, throws and shawls. The can be also added to non crocheted things like towels and napkins.
If you missed the previous tutorials, then can be found here:
If you would like a free printable of instructions for crocheting all 4 of these edgings, plus 2 extra ones, the link is at the bottom of the post!
Fringed edging is also very easy. Of course, you could make a fringe in the usual way, but it’s nice to have a choice and to be able to do things differently sometimes.
Please be warned that I am using British crocheting terms!
Instructions for Crocheting Fringed Edging
This is very easy to do. All you have to do is make lots of chains joined to the edge you are working on with double crochet.
It will turn out a little bit neater if you start with an odd number of sttiches. It doesn’t make that much of a difference though, so don’t worry if you don’t.
1. 1 ch, miss one stitch, double crochet into the next stitch.
Keep going until you reach the end of the row. That really is all there is to it!
It’s also possible to crochet a fringed edging using 2 colours.
1. In the original colour, 1 ch, miss one stitch, dc into next stitch.
2. 25 ch
3. Miss 1 stitch, dc into next stitch
4. 25 ch
5. Miss 1 stitch, dc into next stitch.
Keep going like this until you reach the end of the row.
Then with your 2nd colour:
6. Join yarn at the first free dc. It will be between the 2 ends of the first fringe loop you made.
7. 25 ch
8. dc into next free stitch. So you will miss out 1 stitch (it will already have been worked!).
Keep going to the end of the row.
To get your free printable of this and 5 other crocheted edgings, the link is here.
Happy Crocheting! xx
1. Claudia says
Hi Anna , unfortunately the link to the crochet edgings patterns doesn’t work anymore. Do you think you could send me the pdf? You see, you just saved my life because I was looking for crochet
edgings patterns I could understand and found yours and it worked very well. Thank you! And as you say you have some more patterns written down in the pdf I would like to see them. Perhaps I
could try those too.
□ AnnaWilson says
Hi Claudia, it’s here: https://www.awilson.co.uk/wp-content/uploads/2022/11/Crocheted-Edgings.pdf Have fun!
2. Diane.carney says
What size hook and string do you use? And how do you do the base that the fringe comes from?
□ AnnaWilson says
Hi Diane, I used double knit and a 3.5 mm hook. To do the base I did a row of chains, then a couple of rows of double crochet. If you need more help, let me know.
3. These are so bautiful and add so much to any piece of fabric or garment. If only I could crochet.
□ AnnaWilson says
It’s quite easy Mary 🙂
4. Tracey V says
Fabulous tutorial :o)
Thank you x
Tracey V recently posted…I Do Love Denim :o)
□ AnnaWilson says
Thanks 🙂
5. Teresa says
a very useful tutorial, thank you for sharing at The Really Crafty Link party this week! pinned!
Teresa recently posted…Welcome to The Really Crafty Link Party #9!
□ AnnaWilson says
Thank you xx
6. Ginny says
Very pretty. Thanks for the tutorial.
□ AnnaWilson says
Thank you xx
7. Caroline says
So pretty and ideal for the bottom of scarves
□ AnnaWilson says
Thank you xx
8. Julie says
This is lovely, I am already thinking of things that could do with a little fringe. Thanks
□ AnnaWilson says
All kinds of things could do with a little fringe 🙂
□ beenzmommma says
I’m a complete novice, but.your instructions are very clear and easy to follow. You answered my questions before I could ask them! Thank you for sharing.
☆ AnnaWilson says
Glad you found them useful 🙂
Leave a Reply Cancel reply | {"url":"https://www.awilson.co.uk/crochet-fringed-edging/","timestamp":"2024-11-07T15:22:59Z","content_type":"text/html","content_length":"199047","record_id":"<urn:uuid:21a0b863-c985-440f-bf03-bf1f5f38f357>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00693.warc.gz"} |
Concept information
formaldehyde partial column molecular content
• "Content" indicates a quantity per unit area. The "content_of_X_in_atmosphere_layer" refers to the vertical integral between two specified levels in the atmosphere. "Layer" means any layer with
upper and lower boundaries that have constant values in some vertical coordinate. There must be a vertical coordinate variable indicating the extent of the layer(s). If the layers are model
layers, the vertical coordinate can be model_level_number, but it is recommended to specify a physical coordinate (in a scalar or auxiliary coordinate variable) as well. For the mole content
integrated from the surface to the top of the atmosphere, standard names including "atmosphere_mole_content_of_X" are used. The chemical formula for formaldehyde is CH2O. Formaldehyde is a member
of the group of aldehydes. The IUPAC name for formaldehyde is methanal.
• formaldehyde partial column
{{#each properties}}
{{toUpperCase label}}
{{#each values }} {{! loop through ConceptPropertyValue objects }} {{#if prefLabel }}
{{#if vocabName }}
{{ vocabName }}
{{/if}} {{/each}} | {"url":"https://vocabulary.actris.nilu.no/skosmos/actris_vocab/en/page/formaldehydepartialcolumnmolecularcontent","timestamp":"2024-11-05T20:46:34Z","content_type":"text/html","content_length":"24155","record_id":"<urn:uuid:d3ab16f5-78d5-4366-bd6e-1e2073895a4c>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00385.warc.gz"} |
Quadratic equation unit
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Firisdorm_Helnkete_ Posted: Saturday 18th of Jan 18:56
I want to order this thing right away!Somebody please tell me, how do I purchase it? Can I do so over the internet? Or is there any phone number through which we can place an
From: USA,Florida
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sxAoc Posted: Sunday 19th of Jan 15:43
You can find out all about it at https://softmath.com/about-algebra-help.html. It is really the best math help program available and is offered at a very reasonable price.
From: Australia
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5 Best Ways to Capture a Screenshot of a Page Element in Selenium with Python
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For example, they may want to take a snapshot of a login form or a popup notification to validate UI changes or for reporting purposes. … Read more
5 Best Ways to Implement Polynomial Regression in Python
π ‘ Problem Formulation: Polynomial regression is applied when data points form a non-linear relationship. This article outlines how to model this relationship using Python. For instance, given a
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5 Streamlined Approaches to Image Classification Using Keras in Python
π ‘ Problem Formulation: This article aims to elucidate various methods for performing image classification using the Keras library in Python. Specifically, it addresses how to convert an input
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5 Best Ways to Apply Feature Scaling in Python
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5 Best Ways to Get Started with Python’s SymPy Module
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5 Best Ways to Find Maximum Factors Formed by Two Numbers in Python
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5 Best Ways to Retrieve Row Values Based on Conditions in Selenium with Python
5 Best Ways to Retrieve Row Values Based on Conditions in Selenium with Python π ‘ Problem Formulation: When automating web application tests using Selenium with Python, one common task is to
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π ‘ Problem Formulation: Developers working with Selenium in Python often need to interact with spreadsheets within a web application. Specifically, the task might involve determining the extent of
data by fetching the number of occupied rows and columns. For instance, given a web-based worksheet, the goal is to programmatically find out how many rows and … Read more | {"url":"https://blog.finxter.com/category/howto/","timestamp":"2024-11-11T04:34:35Z","content_type":"text/html","content_length":"99933","record_id":"<urn:uuid:ce5618fa-258c-4df0-8e80-2c079e6897dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00361.warc.gz"} |
VTU Mechanics of Materials - December 2013 Exam Question Paper | Stupidsid
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1 (a) Define :
(i) True stress
(ii) Factor of safety
(iii) Poisson's ratio
(iv) Principle of super position
4 M
1 (b) A bar of uniform thickness 't' tapers uniformly from a width of b[1] at one end to b[2] at other end, in a length of 'L'. Find the expression for the change in length of the bar when subjected
to an axial force P.
8 M
1 (c) A vertical circular steel bar of length 3l fixed at at both of its ends is loaded at intermediate sections by forces W and 2W as shown in fig. Q.2(c). Determine the end reactions if W=1.5kN.
8 M
2 (a) Define:
(i) Volumetric strain
(ii) Bulk modulus
2 M
2 (b) A bar of rectangular cross section shown in fig.Q.2(b)is subjected to stresses σ[x], σ[y] and σ[z] in x, y and z directions respectively. Show that if sum of these stresses is zero, there is no
change in volume of the bar
9 M
2 (c) Rails are laid such that there is no stress in them at 24°C.If the rails are 32m long, determine
(i) The stress in the rails at 80°C, when there is no allowance for expansion
(ii) The stress in the rails at 80°C, when there is an expansion allowance of 8 mm per rail
(iii) The expansion allowance for no stress in the rail at 80°C
Coefficient of linear expansion α=11×10^-6/°C and Young's modulus E=205GPa.
9 M
3 (a) Derive the expressions for normal and tangential stress on a plane inclined at 'θ' to the plane of stress in x-direction in a general two dimensional stress system and show that sum of normal
stress in any two mutually perpendicular directions is constant.
12 M
3 (b) The state of stress in a two dimensionally stresses body is shown in fig.Q.3(b). Determine graphically (by drawing Mohr's circle), the principle stresses, principle planes, maximum shear stress
and its planes.
8 M
4 (a) A beam of length l is simply supported at its ends. The beam carries a uniformly distributed load of w per unit run over the whole span. Find the strain energy stored by the beam.
6 M
4 (b) A water main 80cm diameter contains water at a pressure head of 100m. If the weight density of water is 9810N/m^3, find the thickness of the metal required for the water main. Given the
permissible stress as 20N/mm^2.
6 M
4 (c) A pipe of 400mm internal diameter and 100mm thickness contains a fluid at a pressure of 8N/mm^2. Find the maximum and minimum hoop stress across the section. Also, sketch the radial pressure
distribution and hoop stress distributed across the section.
6 M
5 (a) Define a beam. Explain with simple sketches, different types of beams.
6 M
5 (b) Draw the shear force and bending moment diagrams for the overhanging beam carrying uniformly distributed load of 2kN/m over the entire length and a point load of 2 kN as shown in fig.Q5(b).
Locate the point of contra flexure.
14 M
6 (a) State the assumptions made in the theory of simple bending.
2 M
6 (b) A simply supported cast iron square beam of 800mm length and 15mm×15mm in section fails on applying a load of 360N at the mid span. Find the maximum uniformly distributed load that can be
applied safely to a 40mm wide,75mm deep and 1.6m long cantilever made of the same material.
8 M
6 (C) Show that the shear stress across the rectangular section varies parabolically. Also show that the maximum shear stress is 1.5 times the average shear stress. Sketch the shear stress variation
across the section.
10 M
7 (a) A cantilever 120mm wide and 200mm deep is 2.5m long. What is the uniformly distribution load which the beam can carry in order to a deflection of 5mm at the free end? Take E=200GN/m^2.
4 M
7 (b) A horizontal beam AB is simply supported at A and B, 6m apart. The beam is subjected to a clockwise couple of 300KN-m at a distance of 4m from the left end as shown in fig.Q7(b). If E=2×10^5N/
mm^2 and I=2×10^8mm^4, determine:
(i) The deflection at the point where the couple is acting
(ii) The maximum deflection.
16 M
8 (a) Derive torsion equation with usual notation. State the assumptions in the theory of pure torsion.
10 M
8 (b) Derive an expression for Euler's buckling load in a column when both ends are fixed.
10 M
More question papers from Mechanics of Materials | {"url":"https://stupidsid.com/previous-question-papers/download/mechanics-of-materials-17967","timestamp":"2024-11-08T18:39:12Z","content_type":"text/html","content_length":"130523","record_id":"<urn:uuid:42626f5f-41c1-4aa8-9a2d-06c7622962b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00284.warc.gz"} |
MATH 245
Longwood University
Department of Mathematics & Computer Science
MATH 245 (History of Mathematics, Spring 2011)
Dr. Wendy Hageman Smith
Email: smithwh@longwood.edu
Office: 346 Ruffner; Phone 395-2992 Math Office: 395-2194, 395-2865(fax)
Class Time: Lecture: M,W
Room 350 Ruffner
Dr. Hageman Smith – OFFICE HOURS
Dr. Hageman Smith – OFFICE HOURS
Monday; 1:00-3:30
Wednesday; 1:00-3:30
Thursday; by appointment
Friday: 10:00-11:30
Text: The History of Mathematics by David Burton 7th ed.
Recommended Supplies: TI-83 or TI-84 graphing calculator.
Course Content and Goals:
This class is designed for the future secondary school teacher and provides the background for the
foundation of the mathematics students learn in the public schools. In order to provide a broad background
for the foundations of the body of mathematics taught in the public schools, this class will address the
historical foundations of mathematics and practice in doing mathematics as it was formulated through
During the semester, come to class prepared to explore and refine your knowledge of chapter content as the
accompanying schedule dictates. “Prepared” means you have read the chapter before coming to class. You
have already seen/learned much of the mathematics in this book. However, this class is not about simply
“doing the math problem”, but developing a deeper knowledge of the structures and history that give rise to
the mathematics we will do in class. So then, the course has three primary goals:
1) To re-examine the mathematics that you will teach and improve your math skill.
2) To help you become more effective teachers by providing the background for the subject you teach.
3) To introduce the idea that mathematics is a human endeavor, that it has a history just as human
civilization does, and that the two grow together symbiotically, and mathematics is still growing and
changing and has relevance now as it has through history. Discussions will include whole-class
participation and small group participation.
Course Requirements:
1. There will be two tests. Each test will be worth 20% of your final grade. Tests will consist of computation
problems and essay problems about the history and contributions of mathematics in the sections assigned.
This means that if you have not read the sections assigned, you probably won’t do well on the test.
2. You will have four quizzes throughout the semester; the quizzes are worth 10% of your final grade.
Quizzes will be a reduced format version of tests. Quizzes will be designed so that if you have done the
required homework and read the sections in the textbook, the quiz will take no more than 15 minutes.
3. Attendance is mandatory. Each student is expected to actively participate in all group work and class
discussions. Group work for any day cannot be made up if class is missed – group work is worth 1 pt.
4. Class assignments and homework combined will constitute 15% of your final grade.
5. Homework: I will collect homework each week. Homework will consist of problems from the
6. A research project will be assigned on April 18th and due on the day of the final. The project will
constitute 20% of your final grade. Specifics will be provided when the project is assigned to the class.
7. In lieu of a final test, your final for this class will consist of a presentation that will last 15 minutes. The
topic will be chosen by you from a list I will provide to the class. Specifics will be provided when the
project is assigned to the class. Your final presentation is worth 15% of your grade.
8. Absences are excused only for illness, college sponsored activities, and recognizable emergencies. You
must assume full responsibility for all material covered during your absence. A grade of "0" will be
assigned for all work missed due to unexcused absences.
9. Make-up tests will be given only when the reason for missing the test meets the criteria for an excused
absence. Make-up tests will always be more difficult then regularly scheduled tests.
10. I expect you to conform to the Longwood College Honor Code as contained in the Student Handbook. All
assignments and tests must be pledged.
General Grading Overview
Course Grades will be calculated using a 10 point scale as follows:
A = 90 ~ 100
B = 80 ~ 90
C = 70 ~ 80
D = 60 ~ 70
F = 0 ~ 60
Plus and minus grades are given at the discretion of the instructor.
In general, the achievement of a student in a course indicates the following:
A: Superior work
B: Above average work
C: Average work
D: Below average
So I will award a grade of A only to a student who meets every standard of learning in the course, and
who in addition consistently exhibits excellence in their work. Absent consistent excellence, I award a B to
the student who meets every standard. I award a C to the student who has met most of the standards of
learning, but continues to struggle in acquiring some key skills or concepts. I award a grade of D to a
student who is capable but appears, based on their performance and effort, unable to commit themselves to
achieving a minimum acceptable standard. I reserve a failing grade only for those students who do not
meet minimum standards of learning, and who seem unable to do so at the level of the course. This
indicates that the student must, if he or she is to continue to pursue the same educational goals, be prepared
to repeat coursework and – most critical – thoughtfully reexamine their goals and priorities with an eye
towards reinventing themselves as a student.
Your Instructor: Please do not hesitate to come and get help if you need it. I am here to help facilitate
your learning mathematics. The most important thing to remember is that I am available to help, that is my
job, so do not wait until problems reach critical mass. He who hesitates is lost, so don't hesitate. You are
responsible for everything that happens in class. I am of course willing to help you, but I am not willing to
re-teach the course in my office. If you miss class, get the notes from someone, try the problems, and then
come get help.
If you have specific physical, psychiatric, or learning disabilities and require accommodations, please
let me know early in the semester so that your learning needs may be appropriately met. You will need to
provide documentation of your disability to the Disability Services Office.
Students are responsible for checking the ANNOUNCEMENTS in Blackboard in advance of each class
period as there will be, from time to time, important information regarding assignments, due dates, items
that must be brought to the next class session, etc. Since announcements trigger an email message, check
email before class to see if there are nay announcements. Also, students are responsible for downloading
all needed course documents from Blackboard, printing them if hardcopies are desired, and knowing the
information contained therein.
Class Attendance: Students are expected to attend all classes. Work missed because of illness or other
excused absences may be made up if you advise me of the absence with attendant paperwork. If you miss
an exam or are late with an assignment you may be asked to provide proof that you had a legitimate reason
(such as illness, certain college-sponsored activities, or recognized emergencies). When possible, you
should notify the instructor in advance of assignments you expect to miss because of legitimate absences.
A grade of “0” or “F” may be given on work missed because of unexcused absences
A course grade of “F” may be assigned when the student has missed a total (excused and
unexcused) of 25 percent of the scheduled class meeting times (more than 7 absences)
Acceptable excused absences (as listed in the Longwood catalog) will be taken into consideration
for missed tests; however, except in VERY EXTREME circumstances, be sure to make PRIOR
Longwood’s Honor System: A strong tradition of honor is fundamental to the quality of living and
learning in the Longwood community. The Honor System was founded in 1910, and its purpose is to create
and sustain a community in which all persons are treated with trust, respect, and dignity. Longwood affirms
the value and necessity of integrity in all intellectual community endeavors. Students are expected to abide
by the Longwood College Honor Code. Assignments should be pledged, but the provisions of the Honor
Code are assumed to apply to all work, pledged or not. Students are encouraged to study together and to
seek help from the instructor or tutors when needed, but receiving unauthorized help or copying will be
graded is a violation of the Honor Code.
The Longwood Honor Code applies to all work for the course as follows:
Any out-of-class practice work or hand-in assignment can include using text information, discussion
with other class members, and/or discussion with instructor BUT, the final product handed-in for
the grade must be the student’s own work.
All tests are to be completed individually. Please sign the honor code on all exams indicating that:
“I have neither given nor received help on this work, nor am I aware of any infraction of the Honor
Class Schedule:
Note: This schedule is tentative, small changes may be made to this schedule through the semester,
according to how the class progresses, and will be announced in class before any changes are initiated.
Jan 19
Jan 2426
Jan 31Feb 2
Text: 1.1, 1.2
BBC Film
Text: 1.3
Text: 2.1, 2.2, 2.3,
Weekly Content
1.1: Tally Systems, Peruvian
1.2: Egyptian, Greek
1.3: Babylonian, Chinese
Feb 2123
Text: 4.1, 4.2, 4.3
Feb 28Mar 2
Film: Archimedes
Text: 4.4, 4.5
2.1/2.2/2.3 Egyptian Arithmetic and Rational Numbers
2.4: Circle Area and Truncated Pyramid Volume
2.6: Pythagorean Triples
3.1: Thales
3.2: Pythagoras Triangular and Square numbers and Zeno
3.3 Pythagorean Theorem and Incommensurables
4.1/4.2: Euclid’s Elements and Euclidean Geometry
4.3 Euclidean Algorithm and Fundamental Theorem of
4.4 Sieve of Eratosthenes and Ptolemy
4.5: Archimedes method of exhaustion
BBC Film
Text: 5.1, 5.3, 6.1
5.1, 5.3: Diophantine equations
6.1: Transmission of Arabic knowledge
Text: 2.6
Text: 3.1, 3.2, 3.3
Quiz 1
Test 1
Quiz 2
Spring Break
Text: 6.2, 7.1, 7.2
Film: Galileo
Text: 8.1
Text: 8.2, 8.3, 8.4
BBC Film
Text: 9.1, 9.3
6.2: Hindu-Arabic Numerals and Fibonacci
7.1/7.2: Tartaglia and Cardano
8.1: Galileo
8.2: Descartes Cartesian geometry and perspective
8.3/8.4: Newton and Leibniz
10.1 Origins of probability
9.3 Bernoulli:
Text: 10.2, 10.3
10.2 Fermat and Euler: Number Theory
10.3 Gauss: Congruence Theory
Text: 11.3, 12.2
11.3 Cauchy, Weierstrass, and Hilbert
12.2: Poincare’ and Cantor
Finals Week
Final May 5 11:30-2:00
Test 2
Quiz 4 | {"url":"https://studylib.net/doc/15130893/math-245","timestamp":"2024-11-10T12:21:51Z","content_type":"text/html","content_length":"62090","record_id":"<urn:uuid:defb27cb-c2d8-4d98-94fc-0a11135eb40b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00450.warc.gz"} |
Euler Project problem 11, SPARK/Ada version
Find in a matrix the four numbers, adjacent in the same direction, that maximize their product.
The ZIP file below contains all the necessary source file to compile the code using GNAT and redo the proofs using SPARK.
Authors: Sylvain Dailler
Topics: Matrices
Tools: SPARK 2014
References: Project Euler / ProofInUse joint laboratory
See also: Maximal sum in a matrix / Maximize product of adjacent numbers in a matrix
see also the index (by topic, by tool, by reference, by year)
download ZIP archive
This is first the Ada interface, with the protoype of the main procedure 'Max_Product_4', but also the pre- and post-condition, using the regular Ada 2012 syntax.
File : pe11_max4.ads
package PE11_Max4
with SPARK_Mode
-- The objective of this project is to find the maximum product of four
-- consecutive elements of a matrix in any directions (lines, columns and
-- diagonals). The direction for each product is decided by the
-- 2 first elements.
-- We solve it by exploring each directions of 4 elements for every
-- element of the matrix
-- Could use big_int and go beyond 10000 I guess
-- or trust the ada/spark compiler flag dedicated to this
subtype Small_Integer is Long_Integer range -10_000 .. 10_000;
subtype Range_Integer is Integer range 1 .. Integer'Last - 3;
type Direction is (RD, LD, RIGHT, DOWN);
-- Definition of the matrix with suitable range to take care of
-- possible overflow of possible addition of indices
type Matrix is array (Range_Integer range <>, Range_Integer range <>)
of Small_Integer;
-- A product in a direction D beginning at (I, J) is valid iff
-- each element of the product actually is in the matrix.
function Is_Valid (M : Matrix; I, J : Range_Integer; D : Direction)
return Boolean is
(I in M'Range(1) and J in M'Range(2) and
(case D is
when RD => I + 3 in M'Range(1) and J + 3 in M'Range(2),
when LD => I + 3 in M'Range(1) and J - 3 in M'Range(2),
when RIGHT => J + 3 in M'Range(2),
when DOWN => I + 3 in M'Range(1)));
-- Product in the four given directions
function Right_Diag (M : Matrix; I, J : Range_Integer) return Long_Integer is
(M (I, J) * M (I + 1, J + 1) * M (I + 2, J + 2) * M (I + 3, J + 3)) with
Pre => Is_Valid (M, I, J, RD);
function Left_Diag (M: Matrix; I, J: Range_Integer) return Long_Integer is
(M (I, J) * M (I + 1, J - 1) * M (I + 2, J - 2) * M (I + 3, J - 3)) with
Pre => Is_Valid (M, I, J, LD);
function Column (M: Matrix; I, J: Range_Integer) return Long_Integer is
(M (I, J) * M (I + 1, J) * M (I + 2, J) * M (I + 3, J)) with
Pre => Is_Valid (M, I, J, DOWN);
function Line (M: Matrix; I, J: Range_Integer) return Long_Integer is
(M (I, J) * M (I, J + 1) * M (I, J + 2) * M (I, J + 3)) with
Pre => Is_Valid (M, I, J, RIGHT);
function Mult_Value (M : Matrix; I, J : Range_Integer; D : Direction)
return Long_Integer is
(case D is
when RD => Right_Diag (M, I, J),
when LD => Left_Diag (M, I, J),
when RIGHT => Line (M, I, J),
when DOWN => Column (M, I, J))
with Pre => Is_Valid (M, I, J, D);
-- Function returning the maximum product.
-- Matrix must be a reasonible matrix
-- Max definition is:
-- - the result must be greater than any valid product of 4 elements
-- - the result should be attained by a given product
function Max_Product_4 (M: Matrix) return Long_Integer with
Pre => M'Length(1) >= 4 and M'Length(2) >= 4,
Post =>
(for all I in M'Range (1) =>
(for all J in M'Range (2) =>
(for all D in Direction =>
(if Is_Valid (M, I, J, D) then Mult_Value (M, I, J, D) <= Max_Product_4'Result))))
(for some I in M'Range (1) =>
(for some J in M'Range (2) =>
(for some D in Direction =>
(Is_Valid (M, I, J, D) and then Mult_Value (M, I, J, D) = Max_Product_4'Result))));
end PE11_Max4; | {"url":"https://toccata.gitlabpages.inria.fr/toccata/gallery/euler011_spark.en.html","timestamp":"2024-11-10T02:02:41Z","content_type":"text/html","content_length":"25492","record_id":"<urn:uuid:6863e6a7-79e6-43d7-a1c2-2a17691ce7a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00798.warc.gz"} |
If carpet costs $11 a square yard including padding and installation, what would it cost to carpet a room measuring 14ft. by 20ft? (No - DocumenTVIf carpet costs $11 a square yard including padding and installation, what would it cost to carpet a room measuring 14ft. by 20ft? (No
If carpet costs $11 a square yard including padding and installation, what would it cost to carpet a room measuring 14ft. by 20ft? (No
If carpet costs $11 a square yard including padding and installation, what would it
cost to carpet a room measuring 14ft. by 20ft? (Note: 3 feet = 1 yard)
Round to the nearest dollar.
o $374
o $1027
in progress 0
Mathematics 3 years 2021-08-22T16:40:17+00:00 2021-08-22T16:40:17+00:00 2 Answers 6 views 0
Answers ( )
1. 0
2021-08-22T16:41:51+00:00 August 22, 2021 at 4:41 pm
Step-by-step explanation: First we convert to yards: which gives us 14/3 yards and 20/3 yards. We multiply these to find a combined area of 280/9 yards, which we then multiply to get 280*11/3.
Simplify to get 342.22 repeating, which rounds to 342.
2. 0
2021-08-22T16:41:53+00:00 August 22, 2021 at 4:41 pm
The correct answer should be $342
Leave an answer
About Farah | {"url":"https://documen.tv/question/if-carpet-costs-11-a-square-yard-including-padding-and-installation-what-would-it-cost-to-carpet-24120690-51/","timestamp":"2024-11-01T22:44:16Z","content_type":"text/html","content_length":"82635","record_id":"<urn:uuid:460e1f99-2e2f-4bb5-bbc1-03988e5bcc1b>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00154.warc.gz"} |
The Dark Magnetism of the Universe
Cite as:
J. Beltrán Jiménez, A. L. Maroto, Mod. Phys. Lett. A26 (2011) 3025-3039 [arXiv:1112.1106]
Despite the success of Maxwell's electromagnetism in the description of the electromagnetic interactions on small scales, we know very little about the behaviour of electromagnetic fields on
cosmological distances. Thus, it has been suggested recently that the problems of dark energy and the origin of cosmic magnetic fields could be pointing to a modification of Maxwell's theory on large
scales. Here, we review such a proposal in which the scalar state which is usually eliminated be means of the Lorenz condition is allowed to propagate. On super-Hubble scales, the new mode is
essentially given by the temporal component of the electromagnetic potential and contributes as an effective cosmological constant to the energy-momentum tensor. The new state can be generated from
quantum fluctuations during inflation and it is shown that the predicted value for the cosmological constant agrees with observations provided inflation took place at the electroweak scale. We also
consider more general theories including non-minimal couplings to the space-time curvature in the presence of the temporal electromagnetic background. We show that both in the minimal and non-minimal
cases, the modified Maxwell's equations include new effective current terms which can generate magnetic fields from sub-galactic scales up to the present Hubble horizon. The corresponding amplitudes
could be enough to seed a galactic dynamo or even to account for observations just by collapse and differential rotation in the protogalactic cloud. | {"url":"https://cosmology.unige.ch/content/dark-magnetism-universe","timestamp":"2024-11-03T03:37:44Z","content_type":"text/html","content_length":"36328","record_id":"<urn:uuid:fb8b9ffc-bf4d-48de-801c-37568e82cdb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00368.warc.gz"} |
It from Qubit 2023
Antony Speranza (University of Illinois, Urbana-Champaign)
We argue that generic local subregions in semiclassical quantum gravity are associated with von Neumann algebras of type II_1, extending recent work by Chandrasekaran et.al. beyond subregions bounded
by Killing horizons. The subregion algebra arises as a crossed product of the type III_1 algebra of quantum fields in the subregion by the flow generated by a gravitational constraint operator. We
conjecture that this flow agrees with the vacuum modular flow sufficiently well to conclude that the resulting algebra is type II_\infty, which projects to a type II_1 algebra after imposing a
positive energy condition. The entropy of semiclassical states on this algebra can be computed and shown to agree with the generalized entropy by appealing to a first law of local subregions. The
existence of a maximal entropy state for the type II_1 algebra is further shown to imply a version of Jacobson’s entanglement equilibrium hypothesis. We discuss other applications of this
construction to quantum gravity and holography, including the quantum extremal surface prescription and the quantum focusing conjecture.
• 341d0ae1-eb93-4bc9-9356-b35141febd06
• 4e07ee48-7a12-46e0-912f-67f15501fd0e | {"url":"https://events.perimeterinstitute.ca/event/43/contributions/779/","timestamp":"2024-11-05T22:00:09Z","content_type":"text/html","content_length":"104823","record_id":"<urn:uuid:24298d55-c94e-406c-abce-2585f462bd33>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00777.warc.gz"} |
What Is Quantum Tunneling?
Written by Venkatesh VaidyanathanLast Updated On: 2 Jun 2024Published On: 22 May 2019
Table of Contents (click to expand)
Quantum tunneling is a phenomenon where an electron is able to phase through a barrier and move to the other side. It is a quantum phenomenon that occurs when particles move through a barrier
that, according to the theories of classical physics, should be impossible to pass through.
When an object encounters a barrier, it is an intuitive understanding that the object will come to a halt or be deflected back (because the barrier can stop the object). Now, although that’s how the
world of classical mechanics works, these fairly straightforward situations become slightly wonky when we descend to the quantum realm. In simple terms, quantum tunneling refers to a phenomenon where
an electron is able to phase through a barrier and move to the other side. However, as Richard Feynman says, if you think you understand QM (Quantum Mechanics), you don’t understand it at all. As
simple as the concept of quantum tunneling is, let’s dive straight in to understand its more complex nuances.
Recommended Video for you:
The Fundamentals
(Photo Credit : Marcel-André Baschet & Nobel foundation /Wikimedia Commons)
Understanding quantum tunneling in a more intuitive sense involves revisiting a few concepts of QM. The first one we will look into is the Heisenberg Uncertainty Principle. Heisenberg’s Uncertainty
Principle comes into play when trying to observe particles. It states that there is a limit up to which one can determine the various parameters of a particle with a certain degree of accuracy. To
understand this better, let’s take two parameters—velocity and the position of a particle—and let’s say that the particle we’re considering is an electron. Now, according to Heisenberg’s Uncertainty
Principle, there is a specific limit up to which both the position and the velocity of the electron can be calculated with a certain degree of precision. If we were to focus on increasing the
accuracy of either one of these parameters in more detail and focus, then the other setting would decrease in its level of precision with regards to its measurement. Thus, if you can determine the
position of an electron with high accuracy, then you won’t be able to measure its velocity with great accuracy. Conversely, if you can measure the velocity of an electron to a great degree of
accuracy, you will not be able to accurately determine the position of the electron.
(Photo Credit : Yuvalr/Wikimedia Commons)
Now, another fundamental principle that must be understood is the wave-like nature of matter. The wave-like nature of a particle is a crucial aspect of one element of QM, called wave-particle duality
. In the concept of wave-particle duality, every fundamental particle can be described both in terms of being a particle and a wave. This was proposed by Louis De Broglie in 1924 in his PhD thesis,
which stated that if light could possess both a wave- and a particle-like nature, then an electron could also have such a dual wave-particle nature. It was through De Broglie’s relationship that he
proposed in his PhD that we were able to establish the wave nature of matter. The relationship is as follows:
Here, lambda represents the wavelength of the particle, and ‘p’ represents the momentum of the particle. The significance of the de Broglie relationship is that it establishes a foundation for the
fact that matter can behave like a wave. The Davisson-Germer experiment proved the wave nature of matter beyond doubt based on the diffraction of electrons through a crystal.
Later on, the wave nature of matter was seamlessly integrated into Heisenberg’s Uncertainty Principle. The Uncertainty Principle states that for an electron or any other particle, both the momentum
and position cannot be known accurately at the same time. There is always some uncertainty with either the position ‘delta x’ or with the momentum, ‘delta p’. Heisenberg’s Uncertainty equation is:
Imagine that you measure the momentum of a particle accurately, such that ‘delta p’ is zero. To satisfy the equation above, the uncertainty in the position of the particle, ‘delta x’ must be
infinite. From de Broglie’s equation, we know that a particle with a definite momentum has a particular wavelength ‘Lambda’. A definite wavelength extends all over space to infinity. According to
Born’s Probability Interpretation, this means that the particle is not localized in space, so the uncertainty of position becomes infinite. In real life, however, the wavelengths have a finite
boundary and are not infinite, so both the position and momentum uncertainties have limited value. De Broglie’s equation and Heisenberg’s Uncertainty Principle from that point on became two peas in a
Bringing It Together
The quantum tunneling effect is a quantum phenomenon that occurs when particles move through a barrier that, according to the theories of classical physics, should be impossible to pass through. The
barrier may be a physically impassable medium, such as an insulator or a vacuum, or a region of high potential energy. Upon encountering a barrier, a quantum wave will not end abruptly; rather, its
amplitude will decrease exponentially. This drop in amplitude corresponds to a drop in the probability of finding a particle further into the barrier. If the barrier is thin enough, then the
amplitude may be non-zero on the other side. This would imply that there is a finite probability that some of the particles will tunnel through the barrier. The tunneling current is defined as the
ratio of the current density emerging from the barrier divided by the current density incident on the barrier. If this transmission coefficient across the barrier is a non-zero value, then there
exists a finite possibility that the particle can phase through the barrier.
(Photo Credit : Felix Kling/Wikimedia Commons)
Its apparent ability to jump gaps exemplifies one of the consequences of light having a wave-like aspect. For instance, light penetrating through a block of glass at a shallow angle is effectively
trapped within the glass by the barrier of air at the far side, unless a second glass block is placed close to it (but not touching). Due to the spread-out nature of the wave, some of it penetrates
the air barrier and, if it encounters more glass beyond, it can continue, thus apparently jumping the air gap and escaping its prison. A similar thing happens at the sub-atomic scale, when alpha
particles try to escape from unstable nuclei during radioactive decay. The particles are effectively held in the nucleus by the nuclear forces and, in principle, should not be able to escape.
However, escape they do, thanks to quantum tunneling and the uncertainty principle!
Venkatesh is an Electrical and Electronics Engineer from SRM Institute of Science and Technology, India. He is deeply fascinated by Robotics and Artificial Intelligence. He is also a chess
aficionado, He likes studying chess classics from the 1800 and 1900’s. He enjoys writing about science and technology as he finds the intricacies which come with each topic fascinating. | {"url":"https://www.scienceabc.com/pure-sciences/what-is-quantum-tunneling.html","timestamp":"2024-11-14T23:43:54Z","content_type":"text/html","content_length":"171305","record_id":"<urn:uuid:45fb0314-f657-4275-a63d-f29a9d3bce9a>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00569.warc.gz"} |
How do you simplify (3x) /(x^2 - 2x - 24) * (x - 6) / (6x^2)? | HIX Tutor
How do you simplify #(3x) /(x^2 - 2x - 24) * (x - 6) / (6x^2)#?
Answer 1
$\frac{1}{2 x \left(x + 4\right)}$
#\frac{3x}{x^2-2x-24}\cdot \frac{x-6}{6x^2}#
#=\frac{3x}{6x^2}\cdot \frac{x-6}{x^2-2x-24}#
#=\frac{1}{2x}\cdot \frac{x-6}{x^2-6x+4x-24}#
#=\frac{1}{2x}\cdot \frac{x-6}{x(x-6)+4(x-6)}#
#=\frac{1}{2x}\cdot \frac{x-6}{(x-6)(x+4)}#
#=\frac{1}{2x}\cdot \frac{1}{x+4}#
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Answer 2
To simplify the expression (3x) /(x^2 - 2x - 24) * (x - 6) / (6x^2), we can follow these steps:
1. Factor the denominators:
□ (x^2 - 2x - 24) can be factored as (x - 6)(x + 4).
□ (6x^2) can be factored as 2x * 3x.
2. Rewrite the expression with the factored denominators:
□ (3x) / [(x - 6)(x + 4)] * (x - 6) / (2x * 3x).
3. Simplify the expression by canceling out common factors:
□ The (x - 6) term in the numerator and denominator can be canceled out.
4. Simplify further:
□ (3x) / [(x + 4)] * 1 / (2x * 3x).
5. Multiply the numerators and denominators:
□ (3x) / [(x + 4)(2x * 3x)].
6. Combine like terms in the denominator:
Therefore, the simplified expression is (3x) / [6x^2(x + 4)].
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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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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-simplify-3x-x-2-2x-24-x-6-6x-2-8f9af9c032","timestamp":"2024-11-09T22:10:27Z","content_type":"text/html","content_length":"576738","record_id":"<urn:uuid:4983c0cf-3c9d-4202-8f00-5bf7fb30c07f>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00097.warc.gz"} |
What does projection of U onto V mean?
What does projection of U onto V mean?
Given two vectors u and v, we can ask how far we will go in the direction of v when we travel along u. The vector parallel to v, with magnitude compvu, in the direction of v is called the projection
of u onto v and is denoted projvu.
What is the projection of a vector onto another?
The vector projection is the vector produced when one vector is resolved into two component vectors, one that is parallel to the second vector and one that is perpendicular to the second vector. The
parallel vector is the vector projection.
What is projection of A on B?
The projection of A on B is a vector with length PQ which begins at P in the same direction as B . The projection of A on B is defined as ∣B ∣A .
How do you calculate a projection?
If you want to calculate the projection by hand, use the vector projection formula p = (a·b / b·b) * b and follow this step by step procedure: Calculate the dot product of vectors a and b: a·b = 2*3
+ (-3)*6 + 5*(-4) = -32. Calculate the dot product of vector b with itself: b·b = 3*3 + 6*6 + (-4)*(-4) = 61.
What is projection rule?
Projection law or the formula of projection law express the algebraic sum of the projection of any two sides in term of the third side.
What is the purpose of orthogonal projection?
The orthogonal projection of one vector onto another is the basis for the decomposition of a vector into a sum of orthogonal vectors. The projection of a vector v onto a second vector w is a scalar
multiple of the vector w.
Which is the projection of U on a?
The term λ a can be thought of as the projection of u on a. For simplicity, let’s start with just two vectors u and v shown below in dark blue and light blue respectively. and it points in the
direction of v.
What is the vector projection of V onto U?
Given the vector u = < − 2, 6, 4 > and a vector v such that the vector projection of u onto v is < 2, 4, 4 >, and the vector projection of v onto u is < − 8, 24, 16 >. What is the vector v? I tried
it like this but can’t reach to final answer.
How to get the projection of UON V?
The projection of uon v, denoted projvu, is the vector obtained by multiplying a unit vector in the direction of vby the scalar compvu. As this table shows, projvuis the vector we get by drawing an
arrow instead of the blue line segment representing compvu. projvu
How is dot product used to express projection?
The geometric definition of dot product helps us express the projection of one vector onto another as well as the component of one vector in the direction of another. But let’s approach the concept
from a different direction: given vectors a, b and scalars λ, μ, we know how to form the linear combination u = λ a + μ b to create a new vector u.
What does Projvu mean?
Definition. The projection of a vector u onto another vector v, denoted projvu, is the vector that is parallel to v such that u − projvu makes a right angle with v. Formula.
Is U Projvu perpendicular to Projvu?
parallel to v and a vector u⊥ perpendicular to v. = projvu.
How do you calculate projection?
The vector projection of one vector over another vector is the length of the shadow of the given vector over another vector. It is obtained by multiplying the magnitude of the given vectors with the
cosecant of the angle between the two vectors. The resultant of a vector projection formula is a scalar value.
How do you find orthogonal projection?
We denote the closest vector to x on W by x W .
1. To say that x W is the closest vector to x on W means that the difference x − x W is orthogonal to the vectors in W :
2. In other words, if x W ⊥ = x − x W , then we have x = x W + x W ⊥ , where x W is in W and x W ⊥ is in W ⊥ .
What is meant by orthogonal?
1a : intersecting or lying at right angles In orthogonal cutting, the cutting edge is perpendicular to the direction of tool travel. b : having perpendicular slopes or tangents at the point of
intersection orthogonal curves.
What does the dot product represent?
The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and
the position of a vector relative to the coordinate axes.
What is the component of V perpendicular to W?
The component of v perpendicular to w is q, and clearly v = p + q. Thus, if we can find p from v and w, then we can calculate q as q = v – p.
Can a vector have direction angles 45 60 and 120?
Can a directed line have direction angles 45°, 60°, 120°? ∴ a line can have the given angles as direction angles.
How do you calculate vector projections?
What does projection of U onto V mean? Given two vectors u and v, we can ask how far we will go in the direction of v when we travel along u. The vector parallel to v, with magnitude compvu, in
the direction of v is called the projection of u onto v and is… | {"url":"https://bridgitmendlermusic.com/what-does-projection-of-u-onto-v-mean/","timestamp":"2024-11-03T23:02:00Z","content_type":"text/html","content_length":"43929","record_id":"<urn:uuid:1dfe0a50-f9b9-4aae-83c7-81f99dbe7a4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00246.warc.gz"} |
Measuring Fat Loss without the scale | R-bloggersMeasuring Fat Loss without the scale
Measuring Fat Loss without the scale
[This article was first published on
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
@tdhopper posted his self-measurements of weight loss
a few months back. I recently decided also that I wanted to lose fat-weight—the infamous “I could stand to be a few kilos lighter”—and I think I came up with a more productive way of thinking about
my progress: I’m not going to look at the scale at all. I’m just going to count calorie estimates from the treadmill estimator or use online calculators for how much is burned by running / swimming —
and calories burned is the only thing I will use: no attempts at eating less.
Also, instead of thinking in terms of weight I’m going to think in terms of volume. Here are some pictures of people holding 5 pounds of fat (2¼ kilos):
As you can see this is a large fraction of a person’s flesh, if their BMI is in the healthy range.
I’m not so fat that I have tens of litres of fat making up my body. Rather if I look at myself and visually “remove 2 litres” that “looks” like it would be very substantial—such a huge volume that,
of course it would take weeks of diligent exercise!
But as we know from Mr Hopper’s posts (or I know it from my own experience of weighing myself), the noise is louder than the signal.
The magnitude of daily variation swamps the magnitude of “fundamental” progress.
The goal of counting kcal burned and thinking in terms of volume is to make both the goals and the progress feel more visceral. Everybody knows how to lose weight, the problem is just that one
doesn’t do it. Other than simply increasing self-discipline or increasing the mental energy I put towards this goal (neither of which I want to do).
1. More accurate measurement of my small-scale progress and
2. Choosing meaningful goals in the first place—not a number grabbed out of the air (“five kilos”—why five?), but rather imagine how much volume has left my muffin-top and how much volume is
left—whilst still carrying with me the “larger numbers” associated with kcal fat-loss, than the “small numbers” which characterise litres (gallons ~ 8 lbs) of fat loss.
Here’s my mathematical model of why this is hard in the first place:
• I take about 100 measurements at roughly the same time but not exactly timepoints <- 1:1e2 + rnorm(1e2,sd=1)
• the natural variation in weight, in the unit scale of [kcal stored by fat] is on the order of kilos daily.variation <- 1e5 * sin( runif(1,min=-pi/2,max=pi/2) + timepoints)
• even if I subtracted off my daily fluctuation pattern (Mr Hopper does this by weighing himself at the same time every day), there are apparently other noise factors on the order of half a kilo or
perhaps .1 kilo other.variation <- 1e4 * sin( runif(1,min=-pi/2,max=pi/2) + timepoints)
• the “underlying phenomenon” I’m trying to measure is perhaps on the order of .01 kilos lost per day. Let’s say I lose 1 kilo in 3 weeks, that would be 8000 kcal if I’m good. (i.e., I actually do
my workouts and I don’t eat a compensatory extra 8000± kcal). I could model the underlying fat loss as a step function to be more truthful but I’ll use a linear model, saying I lose 100 kcal per
measurement (supposing I measure 3 times a day) rather than 700 kcal every time I work out, which is not once a day (that would be the step function). But the catch is, I’m not sure if I’m
compensating by eating more. My statistical task is to estimate B, in other words to distinguish if I’m losing weight or not, and how fast I’m losing it (in kcal units, leaving the conversion
8000 kcal ~ 1 kilo as an afterthought), from the signal-swamped data. B<-rnorm(1,mean=100,sd=50); trend<-−B*timepoints
• Now my job is to estimate B. Is it even positive? (i.e. am I actually losing weight?) In R I just made the variable so I could print(B) but the point is to model why it’s hard to do this from my
real data, which is the sum data <- daily.variation + other.variation - B*timepoints
• This is why I like my idea: measurements of kcal burned on the treadmill is 1000 times more precise than measurements of my bodyweight.
So my overall system is to do “chunks” of 7000 kcal = 1 kilo of fat or 3500 kcal =1 pound of fat. I can stand to do 500–700 kcal per cardio session—about an hour. (I also do an extra +1 kcal for
every minute it took me to penalise for low speed: exercise crowds out normal metabolism.) Then it becomes a “long count” up to 3500 or up to 7000. That means 5 cardio sessions (of 770 kcal each) to
get up to 1 pound of fat-loss, 7 wimped-out cardio sessions (of 550 kcal each) to reach a pound, and so on. It’s easy enough to “count to 5”. This system makes each one of the 5 be significantly
large at the order of magnitude appropriate to convert kcal of exercise to litres of body volume. | {"url":"https://www.r-bloggers.com/2014/07/measuring-fat-loss-without-the-scale/","timestamp":"2024-11-08T21:26:35Z","content_type":"text/html","content_length":"115470","record_id":"<urn:uuid:34b44059-d6d7-417c-ace9-49b997e10224>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00707.warc.gz"} |
ANOVA, Linear and Logistic Regression, Chi-Square Test
R Exercises and Solutions: ANOVA, Linear and Logistic Regression, Chi-Square Test
In order to solve the tasks you need:
Problem 1
People who are concerned about their health may prefer hot dogs that are low in sodium and calories. The data file contains sample data on the sodium and calories contained in each of 54 major hot
dog brands. The hot dogs are classified by type: beef (A), poultry (B), and meat (C).
The data file called hotdogs.rda contains the sodium and calorie content for random samples of each type of hot dog. This data set is included in the DS705data package.
Part 1a
Make boxplots for the variable calories for the 3 types of hotdogs. Describe the 3 boxplots and the suitability of these samples for conducting analysis of variance.
Answer 1a
The boxplot for hotdogs of type C is much lower than the boxplots for types A and B. This suggests that type C hotdogs are significantly different in terms of calories from the other two types of
hotdogs. There is also an outlier for hotdogs of type C, which may lower the chance of rejecting the null hypothesis when conducting analysis of variance.
The distribution for type A is slightly skewed to the right, type B is also skewed to the right, and type C shows skewness to the left. Skewness, which is a sign of nonnormality also shows that the
samples may not be very suitable for analysis of variance.
The length of the boxes for types A and C are very similar, while the box for type B is longer. This shows lack of homogeneity of variances, which is not suitable for analysis of variance.
Part 1b
Conduct an analysis of variance test (the standard one that assumes normality and equal variance) to compare population mean calorie counts for these three types of hot dogs. (i) State the null and
alternative hypotheses, (ii) use R to compute the test statistic and p-value, and (iii) write a conclusion in context at \(\alpha=0.10\).
Answer 1b
\(H_0: \mu_A = \mu_B = \mu_C\)
\(H_a:\) Means are not all equal.
## Df Sum Sq Mean Sq F value Pr(>F)
## type 2 15425 7712 6.278 0.00365 **
## Residuals 51 62649 1228
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Since p-value = 0.00365 < 0.1, we reject the null hypothesis. Some means are statistically significantly different. Population mean calorie counts for these three types of hot dogs are not the same.
Part 1c
Follow up with the Tukey-Kramer multiple comparison procedure using a 90% experiment-wise confidence level.
1. Provide the R code used and
2. Write an interpretation for your multiple comparison output in the context of the problem; include the intervals for which population means are significantly different.
Answer 1c
## Tukey multiple comparisons of means
## 90% family-wise confidence level
## Fit: aov(formula = calories ~ type, data = hotdogs)
## $type
## diff lwr upr p adj
## B-A -9.144118 -33.41772 15.129490 0.7102899
## C-A -39.673529 -63.94714 -15.399922 0.0033799
## C-B -30.529412 -55.76791 -5.290917 0.0371484
This output indicates that the differences C-A and C-B are significant (p adj < 0.1), while B-A is not significant (p adj = 0.71 > 0.1). This means that type C hotdogs are significantly different in
terms of population mean calorie counts from the other two types of hotdogs.
The 90% confidence interval on the difference between means for types C and A extends from -63.947 to -15.400 and for types C and B - from -55.768 to -5.291.
Part 1d
As part of a vigorous road test for independent random samples of size 20 for 4 different brands of tires, the tread wear was measured for comparison. The data frame treadwear.rda contains the
resulting data.
Begin by exploring the sample means and standard deviations for each brand and looking at a boxplot comparison. That is, find the sample means and standard deviations and produce a boxplots of the
tread wear measures for the four brands of tires.
Conduct hypothesis tests for the normality of the tread wear for each brand using a 5% level of significance in each case.
Also test for the homogeneity of the variances using \(\alpha=0.05\).
Comment on the results of each.
Answer 1d
tbl <- cbind(aggregate(wear ~ brand, data = treadwear, mean),
aggregate(wear ~ brand, data = treadwear, sd)[2])
colnames(tbl) <- c("brand", "wear.mean", "wear.sd")
## brand wear.mean wear.sd
## 1 A 576.3017 148.10367
## 2 B 671.1044 111.29454
## 3 C 825.9870 57.85366
## 4 D 853.2877 81.23036
## Shapiro-Wilk normality test
## data: treadwear$wear[treadwear$brand == "A"]
## W = 0.94655, p-value = 0.3177
## Shapiro-Wilk normality test
## data: treadwear$wear[treadwear$brand == "B"]
## W = 0.92027, p-value = 0.1003
## Shapiro-Wilk normality test
## data: treadwear$wear[treadwear$brand == "C"]
## W = 0.95389, p-value = 0.4301
## Shapiro-Wilk normality test
## data: treadwear$wear[treadwear$brand == "D"]
## W = 0.95129, p-value = 0.3871
## Bartlett test of homogeneity of variances
## data: wear by brand
## Bartlett's K-squared = 17.034, df = 3, p-value = 0.0006956
Results of Shapiro-Wilk normality test show that the p-value for each brand is larger than \(\alpha\) (0.05) so the null hypothesis that the data came from a normally distributed population cannot be
rejected. The tread wear for each brand is normally distributed.
Results for Bartlett test of homogeneity of variances show that the p-value is smaller than \(\alpha\) (0.0006956 < 0.05), which means that we reject the null hypothesis that the variances of tread
wear for each of the brands are the same. At least one brand’s variance is statistically significantly different from the others.
Part 1e
What is the most appropriate inference procedure to compare population mean tread wear for these four brands of tires? Perform this procedure.
1. State the null and alternative hypotheses, (ii) use R to compute the test statistic and p-value, and (iii) write a conclusion in context at \(\alpha=0.05\).
Answer 1e
Welch ANOVA (since variances are not equal).
\(H_0: \mu_A = \mu_B = \mu_C\)
\(H_a:\) Means are not all equal.
## One-way analysis of means (not assuming equal variances)
## data: wear and brand
## F = 27.201, num df = 3.000, denom df = 40.197, p-value = 8.988e-10
Since p-value = 8.988e-10 < 0.05, we reject the null hypothesis. Some means are statistically significantly different. Population mean tread wear for these four brands of tires are not the same.
Part 1f
Conduct the most appropriate multiple comparisons procedure to determine which brands have significantly different tread wear. Use a family-wise error rate of \(\alpha=0.05\). Use complete sentences
to interpret the results in the context of the problem.
Answer 1f
## ### Oneway Anova for y=wear and x=brand (groups: A, B, C, D)
## Omega squared: 95% CI = [.38; .64], point estimate = .53
## Eta Squared: 95% CI = [.41; .63], point estimate = .55
## SS Df MS F p
## Between groups (error + effect) 1029881.43 3 343293.81 31.02 <.001
## Within groups (error only) 841065.2 76 11066.65
## ### Post hoc test: games-howell
## diff ci.lo ci.hi t df p
## B-A 94.80 -16.88 206.48 2.29 35.27 .120
## C-A 249.69 151.80 347.57 7.02 24.67 <.001
## D-A 276.99 174.18 379.79 7.33 29.48 <.001
## C-B 154.88 78.40 231.37 5.52 28.57 <.001
## D-B 182.18 99.07 265.30 5.91 34.77 <.001
## D-C 27.30 -32.90 87.50 1.22 34.33 .616
This output indicates that the differences C-A, D-A, C-B and D-B are significant (p adj < 0.05), while B-A and D-C are not significant (p adj > 0.05). This means that brand C and brand D tires are
significantly different in terms of population mean wear from brand A and brand B tires.
Problem 2
This dataset contains the prices of ladies’ diamond rings and the carat size of their diamond stones. The rings are made with gold of 20 carats purity and are each mounted with a single diamond
stone. The data was presented in a newspaper advertisement suggesting the use of simple linear regression to relate the prices of diamond rings to the carats of their diamond stones.
The data is in the file diamond.rda and is included in the DS705data package.
Part 2a
Does it appear that a linear model is at least possibly a plausible model for predicting the price from carats of the diamonds for these rings?
Begin by creating a scatterplot and comment on the suitability of a linear regression model.
Answer 2a
A linear model looks like a plausible model for predicting the price from carats of the diamonds for these rings.
Part 2b
Obtain the estimated y-intercept and slope for the estimated regression equation and write the equation in the form price\(=\hat{\beta_0} + \hat{\beta_1}\)carats (only with \(\hat{\beta_0}\) and \(\
hat{\beta_1}\) replaced with the numerical estimates from your R output).
Answer 2b
## Call:
## lm(formula = price ~ carat, data = diamond)
## Coefficients:
## (Intercept) carat
## -250.6 3671.4
Replace the ## symbols with your slope and intercept
\(\widehat{price}\) = -250.6 + 3671.4 carat
Part 2c
Compute the sample Pearson correlation coefficient and test whether or not the population Pearson correlation coefficient between price and carat is zero using a 1% level of significance. (i) State
the null and alternative hypotheses, (ii) test statistic, (iii) the p-value, and (iv) conclusion.
Answer 2c
## [1] 0.9875512
## Pearson's product-moment correlation
## data: diamond$carat and diamond$price
## t = 42.116, df = 45, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 99 percent confidence interval:
## 0.9731306 0.9942549
## sample estimates:
## cor
## 0.9875512
Sample correlation: 0.988
\(H_0:\) True correlation is 0
\(H_a:\) True correlation is not equal to 0
t = 42.116
p-value < 2.2e-16
Since p-value < 0.01, we reject the null hypothesis. True correlation between price and carat is statistically significantly different from zero.
Part 2d
Provide a 95% confidence interval to estimate the slope of the regression equation and interpret the interval in the context of the application (do not us the word “slope” in your interpretation).
Answer 2d
## 2.5 % 97.5 %
## carat 3495.818 3846.975
The 95% confidence interval for slope of the regression equation is (3495.818, 3846.975). It means that 95% of the time, if the carats of diamond stones increase by 0.01, the increase in the expected
price of diamond ring is between 34.95818 and 38.46975.
Part 2e
Check to see if the linear regression model assumptions are reasonable for this data.
(Step 1) Are the residuals normal? Construct a histogram, normal probability plot, and boxplot of the residuals and perform a Shapiro-Wilk test for normality.
Answer 2e.1
res <- residuals(mod)
par(mfrow = c(2, 2))
par(mfrow = c(1, 1))
## Shapiro-Wilk normality test
## data: res
## W = 0.98604, p-value = 0.8406
All of the above plots indicate that the residuals are normally distributed. Also, since p-value in the Shapiro-Wilk normality test is higher than 0.05, the null hypothesis that the data came from a
normally distributed population cannot be rejected. The residuals are normally distributed.
(Step 2) Plot the residuals against the fitted values. Does the equal variances assumption seem reasonable? Does the linear regression line seem to be a good fit?
Answer 2e.2
plot(diamond$carat, res,
ylab = "Residuals", xlab = "Fitted Values", main = "Residuals vs. Fitted")
abline(0, 0)
The equal variances assumption seem reasonable and the regression line appears to be a good fit, because the residuals are equally spread around a horizontal line without distinct patterns.
(Step 3) Perform the Breusch-Pagan test for equal variances of the residuals. What does the test tell you?
Answer 2e.3
## studentized Breusch-Pagan test
## data: mod
## BP = 0.18208, df = 1, p-value = 0.6696
p-value = 0.6696 > 0.05, so the null hypothesis of homoscedasticity cannot be rejected. Variances of the residuals are statistically significantly equal.
Part 2f
Calculate and interpret the coefficient of determination \(r^2_{yx}\) (same as \(R^2\)).
Answer 2f
## [1] 0.9752573
The coefficient of determination shows that the model explains about 97.5% of variability of the price data around its mean.
Part 2g
Should the regression equation obtained for price and carats be used for making predictions? Explain your answer.
Answer 2g
The regression equation obtained for price and carats can be used for making predictions, since the coefficient of determination shows that the model fits the data very well and also the residuals’
analysis indicates that the model assumptions are satisfied.
Part 2h
What would be the straightforward interpretation of the y-intercept in this regression model? Does it make sense here? Why would this not be appropriate as a stand-alone interpretation for this
scenario? (hint: what is extrapolation?)
Answer 2gs
The y-intercept in this regression model would mean that the price for a ring with no diamond stone would be -250.6. It does not make sense, because the price cannot be a negative number. Also, this
would not be appropriate as a stand-alone interpretation for this scenario because we cannot make any assumptions on how much a ring with no diamond would cost as it is beyond our observation range.
Part 2i
Create 95% prediction and confidence limits for the population mean price for the carats given in the sample data and plot them along with a scatterplot of the data.
Answer 2i
# 95% prediction limits
carat <- sort(diamond$carat)
pred.int <- predict(mod, newdata = data.frame(carat = carat), interval = "prediction")
# 95% confidence limits
conf.int <- predict(mod, newdata = data.frame(carat = carat), interval = "confidence")
# scatterplot
plot(price ~ carat, data = diamond)
# add prediction interval
lines(carat, pred.int[, 2], col = "blue")
lines(carat, pred.int[, 3], col = "blue")
# add confidence interval
lines(carat, conf.int[, 2], col = "red", lty = 2)
lines(carat, conf.int[, 3], col = "red", lty = 2)
# add legend
legend("topleft", c("95% Confidence Interval", "95% Prediction Interval"),
col = c("red", "blue"), lty = c(2, 1))
Problem 3
Blood type is classified as “A, B, or AB”, or O. In addition, blood can be classified as Rh+ or Rh -. In a survey of 500 randomly selected individuals, a phlebotomist obtained the results shown in
the table below.
Rh Factor A, B, or AB O Total
Rh+ 226 198 424
Rh- 46 30 76
Total 272 228 500
Part 3a
Conduct the appropriate test of significance to test the following research question “Is Rh factor associated with blood type?” Use a 5% level of significance and include all parts of the test.
1. Provide R code
and state the following:
2. null and alternative hypotheses
3. test statistic
4. degrees of freedom
5. p-value (report to 4 decimal places, enter 0 if P < 0.00005)
6. conclusions.
Answer 3a
bloodtype <- matrix(c(226, 198, 46, 30), nrow = 2, byrow = TRUE)
rownames(bloodtype) <- c("Rh+", "Rh-")
colnames(bloodtype) <- c("A/B/AB", "O")
## A/B/AB O
## Rh+ 226 198
## Rh- 46 30
## Pearson's Chi-squared test with Yates' continuity correction
## data: bloodtype
## X-squared = 1.0804, df = 1, p-value = 0.2986
\(H_0:\) Rh factor and blood type are independent variables;
\(H_a:\) Rh factor and blood type are not independent.
X-squared = 1.0804
df = 1
p-value = 0.2986
Since p-value = 0.2986 > 0.05, we cannot reject the null hypothesis that Rh factor and blood type are independent. Rh factor is not associated with blood type.
Part 3b
Compute and interpret the odds ratio of having Type O blood for Rh+ compared to Rh-.
Answer 3b
## [1] 1.343363
For Rh+, the odds of having Type O blood are 1.34 times larger than the odds for having Type O blood when blood is Rh-.
Part 3c
Construct and interpret a 90% confidence interval for the population proportion of people who are Rh-, given that they have Type O blood.
Answer 3c
## [1] 0.1315789
# standard error
SE <- sqrt(pbar ∗ (1 − pbar) / n)
# margin of error
E <- qnorm(0.95) ∗ SE
# confidence interval
pbar + c(−E, E)
## [1] 0.09475603 0.16840187
At 90% confidence level, between 9.48% and 16.84% of people who have Type O blood are Rh-.
Problem 4
The carinate dove shell is a yellowish to brownish colored smooth shell found along shallow water coastal areas in California and Mexico. A study was conducted to determine if the shell height of the
carinate dove shell could be accurately predicted and to identify the independent variables needed to do so. Data was collected for a random sample of 30 of these gastropods and 8 variables that
researchers thought might be good predictors were recorded.
The shell heights (in mm) are labeled in the file as Y and the potential predictor variables are simply named as X1, X2, X3, X4, X5, X6, X7, and X8. Independent variables X1 through X7 are
quantitative while X8 is categorical.
The data is in the file shells.rda and is included in the DS705data package.
Part 4a
Use stepwise model selection with AIC as the stepping criterion and direction = “both” to identify the best first-order model for predicting shell height (Y).
Identify the predictor variables in the final model as well as the AIC for that model.
Answer 4a
null <- lm(Y ~ 1, data = shells)
full <- lm(Y ~ ., data = shells)
mod_A <- step(null, scope = list(upper = full), data = shells,
direction = "both", trace = FALSE)
## Call:
## lm(formula = Y ~ X2 + X4 + X1 + X6 + X7, data = shells)
## Coefficients:
## (Intercept) X2 X4 X1 X6 X7
## 1.42718 0.65317 0.60923 1.15023 -0.06487 0.02636
## [1] 6.0000 -121.9933
The predictor variables in the final model are X1, X2, X4, X6 and X7. The AIC for this model is -121.99.
Part 4b
Compute the variance inflation factor for the final model from part 4a.
Does this model suffer from multicollinearity? Explain your answer.
Answer 4b
## X2 X4 X1 X6 X7
## 3.804041 2.697711 4.286121 4.428297 3.172575
This model does not suffer from multicollinearity, because VIF values are quite small and none of the VIF values exceed 10.
Part 4c
Let’s define Model B as follows:
Y = \(\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_2^2 + \beta_4 X_4 + \beta_5 X_6 +\epsilon\)
Fit Model B and compare the AIC of it to the model that the stepwise selection procedure identified as best in 4a, which you may refer to as Model A.
Which model is the better fitting model according to AIC? Explain your answer.
Answer 4c
## Call:
## lm(formula = Y ~ X1 + X2 + I(X2^2) + X4 + X6, data = shells)
## Coefficients:
## (Intercept) X1 X2 I(X2^2) X4 X6
## 9.16461 1.11286 -5.45450 1.14091 0.79526 -0.03961
## [1] 6.0000 -129.4831
## [1] 6.0000 -121.9933
According to AIC, the better fitting model is model B because the AIC of it is smaller.
Part 4d
Compute the variance inflation factor for Model B from part 4c.
Does this model suffer from multicollinearity? Explain your answer.
Answer 4d
## X1 X2 I(X2^2) X4 X6
## 4.195143 373.677887 369.491380 2.119951 2.309547
Model B suffers from multicollinearity because VIF values for X2 and X2^2 are very high (both exceed 10, which is a sign of multicollinearity).
Part 4e
Center the variable X2 and compute the quadratic term associated with it (call them cX2 and cx2sq, respectively). We’ll identify this as Model C:
Y = \(\beta_0 + \beta_1 X_1 + \beta_2 cX_2 + \beta_3 cX_2^2 + \beta_4 X_4 + \beta_5 X_6 +\epsilon\)
Compute the variance inflation factor for Model C. Does this model suffer from multicollinearity? Explain your answer.
Answer 4e
cX2 <- shells$X2 - mean(shells$X2)
cX2sq <- cX2^2
shells <- cbind(shells, cX2, cX2sq)
mod_C <- lm(Y ~ X1 + cX2 + cX2sq + X4 + X6, data = shells)
## Call:
## lm(formula = Y ~ X1 + cX2 + cX2sq + X4 + X6, data = shells)
## Coefficients:
## (Intercept) X1 cX2 cX2sq X4 X6
## 2.70477 1.11286 0.52081 1.14091 0.79526 -0.03961
## X1 cX2 cX2sq X4 X6
## 4.195143 3.765151 1.036090 2.119951 2.309547
Model C does not suffer from multicollinearity, because all VIF values are small and none of them exceed 10.
Part 4f
Compare the adjusted R-squared for Models A and C.
Explain what adjusted R-squared measures and state which model is “better” according to this criterion.
Answer 4f
## [1] 0.9413233
## [1] 0.9542871
The adjusted R-squared shows the percentage of variation explained by the independent variables that affect the dependent variable. It means that model C is “better” according to this criterion.
Part 4g
Test the residuals of Model C for serial correlation. Use a 5% level of significance.
Describe the outcome of this test.
Answer 4g
## Durbin-Watson test
## data: mod_C
## DW = 2.8267, p-value = 0.02104
## alternative hypothesis: true autocorrelation is not 0
The p-value is smaller than \(\alpha\) (0.02104 < 0.05) so we reject the null hypothesis that the autocorrelation of the disturbances is 0. The residuals are correlated.
Part 4h
Using Model C, construct a 95% prediction interval for the shell height (Y) for a randomly selected shell with \(X1=3.6, X2=2.4, X4=3.0, and X6=48\).
Write an interpretation for the interval in the context of this problem.
Answer 4h
newdata <- data.frame(X1 = 3.6, cX2 = 2.4 - mean(shells$X2),
cX2sq = (2.4 - mean(shells$X2))^2,
X4 = 3, X6 = 48)
predict(mod_C, newdata, interval = "prediction")
## fit lwr upr
## 1 7.136107 6.907404 7.36481
The 95% prediction interval of (6.907, 7.365) means that 95% of the time, the shell height for a randomly selected shell with \(X1=3.6, X2=2.4, X4=3.0, and X6=48\) will be between 6.907 and 7.365.
Problem 5
A study on the Primary News Source for Americans was conducted using a random sample of 115 Americans in 2015. The results are shown below.
TV Radio Newspaper Internet
Sample from 2015 38 20 15 42
Distribution in 1995 45% 18% 16% 21%
Conduct the hypothesis test to determine if the distribution of Primary News Source for Americans is the same in 2015 as it was in 1995. Use \(\alpha = 0.10\). State your hypotheses, test statistic,
df, p-value, and conclusions, including a practical conclusion in the context of the problem.
Answer 5
## Chi-squared test for given probabilities
## data: x
## X-squared = 17.499, df = 3, p-value = 0.000558
\(H_0: p_1 = 0.45, p_2 = 0.18, p_3 = 0.16, p_4 = 0.21\)
\(H_a:\) at lest one equation is not true
Test statistic is 17.499, df = 3, p-value = 0.000558.
Since p-value < 0.1, we reject the null hypothesis. At least one proportion is statistically significantly different from expected proportion. The distribution of Primary News Source for Americans in
2015 has changed since 1995.
Problem 6
In an effort to make better cheese, a company has a random sample of 30 cheese consumers taste 30 specially prepared pieces of Australian cheddar cheese (1 piece for each person). Each subject rated
the taste of their piece of cheese as “acceptable” (coded as 1) or “not acceptable” (coded as 0). One variable measured was called ACETIC and was a quantitative variable ranging from 4.5 to 6.5
units. The other variable recorded was whether the person was a child (person=1) or an adult (person=2).
The data file called cheese.rda. This data set is included in the DS705data package.
Part 6a
Fit the first order model for predicting whether or not the taste of the cheese is acceptable (i.e. acceptable=1) from the acetic value and also whether the person was a child or an adult.
At a 5% level of significance, should either variable be dropped from the model?
Answer 6a
mod1 <- glm(taste ~ acetic + person, family = binomial, data = cheese)
## Call:
## glm(formula = taste ~ acetic + person, family = binomial, data = cheese)
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.2245 -0.4998 -0.2002 0.3040 1.6066
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -18.709 8.517 -2.197 0.0280 *
## acetic 2.787 1.412 1.975 0.0483 *
## personAdult 3.096 1.371 2.258 0.0239 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Dispersion parameter for binomial family taken to be 1)
## Null deviance: 34.795 on 29 degrees of freedom
## Residual deviance: 20.550 on 27 degrees of freedom
## AIC: 26.55
## Number of Fisher Scoring iterations: 6
At a 5% level of significance, no variable should be dropped from the model.
Part 6b
Convert the estimated coefficient of acetic to an odds ratio and interpret it in the context of the problem.
Answer 6b
## acetic
## 16.23497
The odds for taste being acceptable multiply by 16.23497 for every one-unit increase in the amount of acetic.
Part 6c
Compute the predicted probability of a child finding the taste of the cheese acceptable when the value for acetic is 6.
Answer 6c
## 1
## 0.1206732
Part 6d
Compute a 95% confidence interval for the predicted probability of a child finding the taste of the cheese acceptable when the value for acetic is 6.
Answer 6d
pred <- predict(mod1, newdata1, type = "link", se.fit = TRUE)
invlink <- mod1[["family"]][["linkinv"]]
z <- qnorm(1 - (1-0.95)/2)
low.ci <- pred$fit - z * pred$se.fit
up.ci <- pred$fit + z * pred$se.fit
prediction <- invlink(pred$fit)
low.ci.response <- invlink(low.ci)
up.ci.response <- invlink(up.ci)
data.frame(prediction = prediction, low.ci = low.ci.response, up.ci = up.ci.response)
## prediction low.ci up.ci
## 1 0.1206732 0.01591904 0.5379397 | {"url":"https://www.homeworkhelponline.net/blog/programming/anova-linear-and-logistic-regression-chi-square-test","timestamp":"2024-11-15T03:53:47Z","content_type":"text/html","content_length":"174168","record_id":"<urn:uuid:5690e4da-e4ec-4800-b50f-b925357c4b26>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00046.warc.gz"} |
457 research outputs found
Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulations problems including Minimum Fill-in and Treewidth.
We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal
cliques Pi_G, and an integer t, it is possible in time |Pi_G| * n^(O(t)) to find a maximum induced subgraph of treewidth t in G; and for a given graph F of treewidth t, to decide if G contains an
induced subgraph isomorphic to F. Combined with an improved algorithm enumerating all potential maximal cliques in time O(1.734601^n), this yields that both problems are solvable in time 1.734601^n *
n^(O(t)).Comment: 14 page
We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is
to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some
special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies
of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo
m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \
Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of
connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^
{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the
weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary
graphs, and in polynomial time for graph classes with polynomial number of minimal separators
An undirected graph is Eulerian if it is connected and all its vertices are of even degree. Similarly, a directed graph is Eulerian, if for each vertex its in-degree is equal to its out-degree. It is
well known that Eulerian graphs can be recognized in polynomial time while the problems of finding a maximum Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this paper, we
study the parameterized complexity of the following Euler subgraph problems: - Large Euler Subgraph: For a given graph G and integer parameter k, does G contain an induced Eulerian subgraph with at
least k vertices? - Long Circuit: For a given graph G and integer parameter k, does G contain an Eulerian subgraph with at least k edges? Our main algorithmic result is that Large Euler Subgraph is
fixed parameter tractable (FPT) on undirected graphs. We find this a bit surprising because the problem of finding an induced Eulerian subgraph with exactly k vertices is known to be W[1]-hard. The
complexity of the problem changes drastically on directed graphs. On directed graphs we obtained the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed k>3 and is
solvable in polynomial time for k<=3. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs
In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs
which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover,
Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent
decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To
the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were
obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of
geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor
subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that
therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and
Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d
The behavior of users in social networks is often observed to be affected by the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp} introduced a formal mathematical model for user
engagement in social networks where each individual derives a benefit proportional to the number of its friends which are engaged. Given a threshold degree $k$ the equilibrium for this model is a
maximal subgraph whose minimum degree is $\geq k$. However the dropping out of individuals with degrees less than $k$ might lead to a cascading effect of iterated withdrawals such that the size of
equilibrium subgraph becomes very small. To overcome this some special vertices called "anchors" are introduced: these vertices need not have large degree. Bhawalkar et al. \cite{bhawalkar-icalp}
considered the \textsc{Anchored $k$-Core} problem: Given a graph $G$ and integers $b, k$ and $p$ do there exist a set of vertices $B\subseteq H\subseteq V(G)$ such that $|B|\leq b, |H|\geq p$ and
every vertex $v\in H\setminus B$ has degree at least $k$ is the induced subgraph $G[H]$. They showed that the problem is NP-hard for $k\geq 2$ and gave some inapproximability and fixed-parameter
intractability results. In this paper we give improved hardness results for this problem. In particular we show that the \textsc{Anchored $k$-Core} problem is W[1]-hard parameterized by $p$, even for
$k=3$. This improves the result of Bhawalkar et al. \cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by $b$) as our parameter is always bigger since $p\geq b$. Then we answer a question
of Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored $k$-Core} problem remains NP-hard on planar graphs for all $k\geq 3$, even if the maximum degree of the graph is $k+2$.
Finally we show that the problem is FPT on planar graphs parameterized by $b$ for all $k\geq 7$.Comment: To appear in AAAI 201 | {"url":"https://core.ac.uk/search/?q=authors%3A(Fomin%2C%20Fedor)","timestamp":"2024-11-02T14:00:26Z","content_type":"text/html","content_length":"121819","record_id":"<urn:uuid:46dfae48-d471-418b-b83e-aa83215e8f88>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00262.warc.gz"} |
JOBU is CHARACTER*1
[in] JOBU = 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.
JOBV is CHARACTER*1
[in] JOBV = 'V': Orthogonal matrix V is computed;
= 'N': V is not computed.
JOBQ is CHARACTER*1
[in] JOBQ = 'Q': Orthogonal matrix Q is computed;
= 'N': Q is not computed.
M is INTEGER
[in] M The number of rows of the matrix A. M >= 0.
P is INTEGER
[in] P The number of rows of the matrix B. P >= 0.
N is INTEGER
[in] N The number of columns of the matrices A and B. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
[in,out] A On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section.
LDA is INTEGER
[in] LDA The leading dimension of the array A. LDA >= max(1,M).
B is REAL array, dimension (LDB,N)
On entry, the P-by-N matrix B.
[in,out] B On exit, B contains the triangular matrix described in
the Purpose section.
LDB is INTEGER
[in] LDB The leading dimension of the array B. LDB >= max(1,P).
[in] TOLA TOLA is REAL
TOLB is REAL
TOLA and TOLB are the thresholds to determine the effective
numerical rank of matrix B and a subblock of A. Generally,
[in] TOLB they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
[out] K K is INTEGER
L is INTEGER
[out] L On exit, K and L specify the dimension of the subblocks
described in Purpose section.
K + L = effective numerical rank of (A**T,B**T)**T.
U is REAL array, dimension (LDU,M)
[out] U If JOBU = 'U', U contains the orthogonal matrix U.
If JOBU = 'N', U is not referenced.
LDU is INTEGER
[in] LDU The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
V is REAL array, dimension (LDV,P)
[out] V If JOBV = 'V', V contains the orthogonal matrix V.
If JOBV = 'N', V is not referenced.
LDV is INTEGER
[in] LDV The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
Q is REAL array, dimension (LDQ,N)
[out] Q If JOBQ = 'Q', Q contains the orthogonal matrix Q.
If JOBQ = 'N', Q is not referenced.
LDQ is INTEGER
[in] LDQ The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
[out] IWORK IWORK is INTEGER array, dimension (N)
[out] TAU TAU is REAL array, dimension (N)
[out] WORK WORK is REAL array, dimension (max(3*N,M,P))
INFO is INTEGER
[out] INFO = 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value. | {"url":"https://netlib.org/lapack/explore-html-3.4.2/d3/d5b/sggsvp_8f.html","timestamp":"2024-11-09T09:04:24Z","content_type":"application/xhtml+xml","content_length":"19984","record_id":"<urn:uuid:817d785b-11bf-487d-86d0-2ebf19a0babe>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00850.warc.gz"} |
Analysis and Probability Seminar May 7, 2015
Thursday, May 7, 2015 (2:00 p.m. in Yost 335)
Title: On the Perimeter of a Convex Set
Speaker: Galyna Livshyts (Ph.D. Student, Kent State University)
Abstract: The perimeter of a convex set in R^n with respect to a given measure is the measure’s density averaged against the surface measure of the set. It was proved by Ball in 1993 that the
perimeter of a convex set in R^n with respect to the standard Gaussian measure is asymptotically bounded from above by n^{1/4}. Nazarov in 2003 showed the sharpness of this bound. We are going to
discuss the question of maximizing the perimeter of a convex set in R^n with respect to any log-concave rotation invariant probability measure. The latter asymptotic maximum is expressed in terms of
the measure’s natural parameters: the expectation and the variance of the absolute value of the random vector distributed with respect to the measure. We are also going to discuss some related
questions on the geometry and isoperimetric properties of log-concave measures. | {"url":"https://mathstats.case.edu/2015/05/livshyts/","timestamp":"2024-11-10T17:52:27Z","content_type":"text/html","content_length":"117804","record_id":"<urn:uuid:d77a919d-f1f4-4d25-b6cb-374cba9b38c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00116.warc.gz"} |
Lagrange Error Bound Calculator
Lagrange Error Bound Calculator
Lagrange Error Bound Calculator
Understanding the Lagrange Error Bound Calculator
The "Lagrange Error Bound Calculator" is a valuable tool for individuals who engage with calculus and mathematical analysis. This calculator focuses on providing an estimation of the error involved
when approximating functions using Taylor polynomials. Here’s an explanation on how to use it effectively and understand the underlying principles.
What is Lagrange Error Bound?
The Lagrange Error Bound provides an upper limit on the error when approximating a function with its Taylor polynomial. This error bound helps to determine how close the polynomial approximation is
to the actual function value at a specific point. The principle is essential in numerical methods, allowing one to gauge the accuracy of approximations used in various calculations and simulations.
Application of the Calculator
Using this calculator, you can: - Estimate how accurate a Taylor polynomial is when approximating a function at a certain point. - Learn the maximum error that can occur between the actual function
and its polynomial approximation. - Ensure that your function approximations remain within acceptable error ranges, which is especially useful in engineering and computer science fields where
precision is crucial.
How to Use the Calculator
1. **Function Input**: Enter the function you are approximating in the first input field. The function should be in terms of (x). 2. **Degree of Polynomial**: Specify the degree of the Taylor
polynomial you are using (e.g., 2 for a quadratic polynomial). 3. **Point of Center**: Enter the center point, usually denoted as (a), at which the polynomial is centered. 4. **Estimate Point**:
Provide the point (x) where the error is being estimated. 5. **Maximum Derivative Value**: Input the maximum value of the (n+1)th derivative of the function on the interval between (a) and (x). Once
all inputs are provided, clicking the "Calculate" button will compute the error bound and display the result. If you need to reset the inputs, simply click "Reset".
Understanding the Calculation
The calculator uses the Lagrange error bound formula to compute the error: - It takes the maximum value of the (n+1)th derivative of the function. - Multiplies it by the absolute difference between
(x) and (a), raised to the power of (n+1). - Divides the product by the factorial of (n+1). This results in the upper limit of the error, indicating how much the actual function value can differ from
the Taylor polynomial approximation at the given point (x).
Benefits of Using the Calculator
1. **Precision**: Helps ensure your function approximations are accurate within a defined error bound. 2. **Educational Value**: Great for students and professionals to understand the behavior of
Taylor polynomial approximations and their limitations. 3. **Time-Saving**: Quickly computes complex error bounds that would otherwise require lengthy manual calculations. This calculator is an
indispensable tool for anyone involved in mathematical computations, providing clarity and confidence in the accuracy of polynomial approximations.
1. What is the purpose of the Lagrange Error Bound Calculator?
The calculator estimates the error when approximating a function using a Taylor polynomial. It provides an upper limit on this error, ensuring the polynomial approximation’s accuracy within a
defined range.
2. Which types of functions can I enter into the calculator?
You can enter any function expressed in terms of (x). Common examples include polynomial functions, trigonometric functions, exponential functions, and logarithmic functions.
3. How do I determine the maximum value of the (n+1)th derivative?
To find this value, you need to calculate the derivative and evaluate its maximum on the interval between the center point (a) and the point (x). Analytical methods or numerical techniques can help
in this estimation.
4. What is the role of the degree of the polynomial in error calculation?
The degree of the polynomial (denoted as (n)) indicates how many terms are included in the Taylor polynomial. A higher degree often results in a more accurate approximation but requires more complex
5. Can the calculator handle complex functions?
Yes, as long as the function and its derivatives can be defined for the interval between (a) and (x), you can use the calculator to estimate the error bound for complex functions.
6. How does the center point (a) affect the error bound?
The center point (a) is where the Taylor polynomial is centered. The closer the estimate point (x) is to (a), the smaller the error typically is. This relationship helps in choosing an appropriate
(a) for accurate results.
7. What happens if the provided inputs are incorrect or incomplete?
Incorrect or incomplete inputs can lead to inaccurate error bounds or the inability to perform the calculation. Ensure all input fields are correctly filled out for precise results.
8. What does the output of the calculator represent?
The output represents the upper limit on the possible error in the polynomial approximation of the function at the specified point (x). It shows how far the actual function value might deviate from
the approximation.
9. Why is the factorial of (n+1) used in the error calculation?
The factorial of (n+1) appears in the denominator of the error bound formula, which helps to scale the error relative to the degree of the polynomial. It ensures the error decreases appropriately as
(n) increases, reflecting higher polynomial degrees' increased accuracy.
10. Can this calculator be used for educational purposes?
Absolutely. The calculator helps students and educators understand Taylor polynomial approximations and their errors, making it a useful tool for teaching and learning calculus concepts.
11. How do I reset the calculator?
You can reset the calculator by clicking the "Reset" button, which clears all input fields, allowing you to start a new calculation. | {"url":"https://www.onlycalculators.com/math/sequences/lagrange-error-bound-calculator/","timestamp":"2024-11-05T20:27:11Z","content_type":"text/html","content_length":"241809","record_id":"<urn:uuid:622a8efc-705a-4ab7-973b-a907a9d28167>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00545.warc.gz"} |
Complete Axiomatization of the Stutter-Invariant Fragment of the Linear-time mu-calculus
PP-2010-12: Gheerbrant, Amélie (2010) Complete Axiomatization of the Stutter-Invariant Fragment of the Linear-time mu-calculus. [Report]
Text (Full Text)
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Text (Abstract)
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The logic µ(U) is the fixpoint extension of the "Until"-only fragment
of linear-time temporal logic. It also happens to be the
stutter-invariant fragment of linear-time µ-calculus µ(◊). We provide
complete axiomatizations of µ(U) on the class of finite words and on
the class of ω-words. We introduce for this end another logic, which
we call µ(◊_Γ), and which is a variation of µ(◊) where the Next time
operator is replaced by the family of its stutter-invariant
counterparts. This logic has exactly the same expressive power as
µ(U). Using already known results for µ(◊), we first prove
completeness for µ(◊_Γ), which finally allows us to obtain
completeness for µ(U).
Item Type: Report
Report Nr: PP-2010-12
Series Name: Prepublication (PP) Series
Year: 2010
Uncontrolled Keywords: Complete axiomatization; Linear-time temporal logic; Linear-time mu-calculus; Stutter-invariancy
Subjects: Logic
Date Deposited: 12 Oct 2016 14:37
Last Modified: 12 Oct 2016 14:37
URI: https://eprints.illc.uva.nl/id/eprint/391
Actions (login required) | {"url":"https://eprints.illc.uva.nl/id/eprint/391/","timestamp":"2024-11-09T10:03:53Z","content_type":"application/xhtml+xml","content_length":"18796","record_id":"<urn:uuid:cd674e2a-d3a6-4e48-9eba-92353945aae0>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00517.warc.gz"} |
he Volume of a
Archimedes balanced a cylinder, a sphere, and a cone. All of the dimensions shown in blue are equal.
Archimedes specified that the density of the cone is four times the density of the cylinder and the sphere.
Archimedes imagined taking a circular slice out of all three solids.
He then imagined hanging the cylinder and the sphere from point A and suspending the solids at point F (the fulcrum). | {"url":"https://www.physics.weber.edu/carroll/Archimedes/method1.htm","timestamp":"2024-11-09T06:54:19Z","content_type":"text/html","content_length":"1817","record_id":"<urn:uuid:5896abbd-cf3a-4272-938f-5e7cbf0dbfd2>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00841.warc.gz"} |
If Formula with a Dropdown List
I have a column titled Status that includes the following drop down selections: Not Started, In Progress, On Hold and Complete. I want to write a formula that based on the drop down selected, another
row would populate the following automatically:
Not Started = 0%, In Progress = 50%, Completed = 100% and On Hold = ON HOLD. I have tried a number of formula calibrations and keep getting UNPARSABLE in my cell.
Looking for suggestions.
Best Answer
• Try: =IF([Status]@row = "Completed", 1, IF([Status]@row = "In Progress", 0.5, IF([Status]@row = "Not Started", 0, IF([Status]@row = "On Hold", "ON HOLD"))))
• Try: =IF([Status]@row = "Completed", 1, IF([Status]@row = "In Progress", 0.5, IF([Status]@row = "Not Started", 0, IF([Status]@row = "On Hold", "ON HOLD"))))
• Hi @Paula Seward ,
Sounds straight forward. Create your [% Complete] column as text/ number and formated as %. In it place the formula:
=IF(Status@row="Complete", 1, IF(Status@row="In Process", .5, =IF(Status@row="On Hold", "On Hold", 0)))
Your Status column needs to be single select and restricted to the list.
I'm grateful for your "Vote Up" or "Insightful". Thank you for contributing to the Community.
• Hi @Paula Seward,
Looks like everyone is on the same page here though there are variants to the formula that can be used to accomplish the same results. If the logic break down is:
Not Started = 0%
In Progress = 50%
Completed = 100%
On Hold = On Hold
I would use this for my formula:
=IF(Status@row = "Complete", 1, IF(Status@row = "In Progress", 0.5, IF(Status@row = "Not Started", 0, "On Hold")))
The order of the Status values doesn't matter in this type of formula and you can take advantage of the value_if_false argument of the IF Function for the "On Hold" result, basically saying if
none of these other conditions are met, populate the value of "On Hold". Once your formula is set, you can convert it into a column-level formula by right-clicking the formula cell and selecting
"Convert to Column Formula".
I hope this helps!
• Thanks everyone!! Nic's response worked like a charm. It did not like adding the percentage sign in my original formula but doing individual numbers and changing to percentage for the column
properties did the trick! Thanks again. Super helpful.
• Can we do another formula or calculation based ok the % that we just obtained, which are in a drop down list also
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/74396/if-formula-with-a-dropdown-list","timestamp":"2024-11-03T22:41:10Z","content_type":"text/html","content_length":"453044","record_id":"<urn:uuid:e40a9fcb-7215-414a-8bc7-ab3ce82e5a70>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00768.warc.gz"} |
Yellen’s Phony Math
by Craig Hemke, Sprott Money:
Janet Yellen has served as both Chair of the Fed and U.S. Treasury Secretary. During her time on the job, she has muttered some doozies that leave you scratching your head. Her latest, from an
appearance last week, might be one of her craziest yet.
I suppose I could spend most of this column adding text and screenshots from some of Yellen’s most outrageous statements, but we’d be here all day and who has time for that? Instead, here are just
two of my favorites:
TRUTH LIVES on at https://sgtreport.tv/
But she really outdid herself last week. Check this one out. It might be her best yet:
My apologies in advance for this, but for context, we’re going to have to do some math. Please bear with me…
The current total debt of the United States is about $33.5 TRILLION.
The current annual cost of servicing that debt is approaching $900 BILLION.
That means, as of right now, the average interest cost on the accumulated debt is about 2.5%.
However, in Yellen’s quote from last week, she equated debt service with total GDP, which at last count was about $27 TRILLION.
So let’s now do Yellen’s math. If the total GDP is $27T and the current annual debt service cost is $900B, then the debt service relative to GDP is 3.3%.
But Yellen says this number is going to average just “1% for the next decade”. Hmmm. Sorry, but we must do even more math…
The Congressional Budget Office has projected that the total U.S. debt could reach $50T as soon as 2030. If the U.S. economy manages to grow at 3% per year over the next seven years, total GDP will
reach $35T by 2030. Using Yellen’s math, this assumption places debt-to-GDP at nearly 150%.
But again, Yellen was talking about DEBT SERVICE COSTS, so let’s reverse engineer the math on her statement.
If the total U.S. GDP is at $35T in 2030 and the total debt is $50T, then the average interest rate on that accumulated debt would have to be 0.7%. Again, it is currently 2.5%. So, to make Yellen’s
statement rational and reasonable, one has to assume one of two things—or both:
1. The total U.S. GDP is going to soar much faster than the 3% per year I assumed in growing it from $27T today to $35T in 2030.
2. That interest rates and the total interest paid to service the existing debt are both going to be sharply lower than where they are at present.
And neither of those scenarios are likely to be true. It’s far more likely that the total U.S. GDP grows to just $33T by 2030 and that the service on the accumulated $50T in debt is near $1.5T. That
places total debt service to GDP at 4.5%, not Yellen’s expectation of 1.0%. | {"url":"https://www.sgtreport.com/2023/10/yellens-phony-math/","timestamp":"2024-11-06T20:58:42Z","content_type":"text/html","content_length":"181152","record_id":"<urn:uuid:63f2c493-7746-455d-a87f-d28593cfe8e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00122.warc.gz"} |
Buffer Compliance Control of Space Robots Capturing a Non-Cooperative Spacecraft Based on Reinforcement Learning
School of Energy and Mechanical Engineering, Jiangxi University of Science and Technology, Nanchang 330013, China
School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China
Authors to whom correspondence should be addressed.
Submission received: 1 May 2021 / Revised: 15 June 2021 / Accepted: 17 June 2021 / Published: 22 June 2021
Aiming at addressing the problem that the joints are easily destroyed by the impact torque during the process of space robot on-orbit capturing a non-cooperative spacecraft, a reinforcement learning
control algorithm combined with a compliant mechanism is proposed to achieve buffer compliance control. The compliant mechanism can not only absorb the impact energy through the deformation of its
internal spring, but also limit the impact torque to a safe range by combining with the compliance control strategy. First of all, the dynamic models of the space robot and the target spacecraft
before capture are obtained by using the Lagrange approach and Newton-Euler method. After that, based on the law of conservation of momentum, the constraints of kinematics and velocity, the
integrated dynamic model of the post-capture hybrid system is derived. Considering the unstable hybrid system, a buffer compliance control based on reinforcement learning is proposed for the stable
control. The associative search network is employed to approximate unknown nonlinear functions, an adaptive critic network is utilized to construct reinforcement signal to tune the associative search
network. The numerical simulation shows that the proposed control scheme can reduce the impact torque acting on joints by 76.6% at the maximum and 58.7% at the minimum in the capturing operation
phase. And in the stable control phase, the impact torque acting on the joints were limited within the safety threshold, which can avoid overload and damage of the joint actuators.
1. Introduction
With the development of space technology, the number of spacecraft launched every year is increasing, thereby generating a series of high-intensity and high-risk space missions, such as on-orbit fuel
refueling, on-orbit maintenance, recovery of failed spacecraft, space debris removal [
], etc. As outer space is a harsh environment with high pressure, extreme temperatures, high vacuum and strong electromagnetic radiation, it is very hazardous for astronauts to out of the module to
carry out the above mission operations. Giordano et al. [
] proposed a dynamics decomposition that decouples the end-effector task from the base force actuator and reduces the use of thrusters. Qin et al. [
] proposed a fuzzy adaptive robust control (FARC) strategy which is adaptive to these model variations for trajectory tracking control of space robots. Virgili-Llop et al. [
] presented an optimization-based guidance algorithm for onboard implementation and real-time use suitable for space robots. Ai and Chen [
] considered the process of capturing spacecraft by dual-arm clamping and the force/position control of its post-stabilization movement and proposed a fuzzy control scheme based on the passivity
theory. Therefore, it is a better choice to use space robots to replace astronauts to complete on-orbit services (OOS) missions. Because the capture and operation ability of space robots is the basic
and key technology to realize OOS missions. Liu et al. [
] studied the on-orbit services space robot considering joint friction, based on the Jourdain’s velocity variation principle and the single direction recursive construction method derived the dynamic
equation of the system. Lim and Chung [
] analyzed the dynamic behavior of a tethered satellite system for space debris capture, by using the absolute nodal coordinate formulation established the equations of motion of the tethered
satellite system. Shah et al. [
] presented strategies for point-to-point reactionless manipulation of a satellite- mounted dual-arm robotic system for capturing tumbling orbiting objects. Uyama et al. [
] studied an impedance-based contact control of a free-flying space robot with respect to the coefficient of restitution Therefore, the research on capturing operation technology of space robots has
become a hot topic in the aerospace field in recent years.
The operation process of space robot capturing a non-cooperative spacecraft can be divided into four phases: (1) the observation phase, in this phase, the position and attitude of the target
spacecraft are observed; (2) the approaching phase, through trajectory planning and motion control of space robot to reaching the capture area; (3) the capturing operation phase, the space robot uses
the end-effector to capture the target spacecraft; (4) stable control phase, considering the post-capture unstable motion which is caused by the collision and impact of the capturing operation phase,
design the stability control strategy of the hybrid system formed by the space robot and the target spacecraft. Considering that the space robot will inevitably experience a violent contact and
collision with the target spacecraft during phase 3 of the capturing operation process, in this process, the joint of the manipulator arm will be subjected to a great impact moment [
]. If the impact torque affecting the joint is too large, it could cause the impact damage to the joints and lead to the failure of the space mission. At present, there is no effective way to solve
the problem except using the minimum relative approaching velocity. Although this method is feasible for cooperative spacecraft, it is basically not applicable to non-cooperative spacecraft.
Therefore, in the phases 3 and 4 of the capture of a non- cooperative spacecraft, it is of great exploratory value and significance to take certain measures to avoid the damage to joint actuators
caused by such impact and collision.
Recently, the dynamics and control of space robots capturing a spacecraft have become the focus of aerospace technicians, and some research results have emerged. It is worth mentioning that the
studies mainly focus on pre-capture motion planning and post-capture attitude control. For the motion planning and trajectory tracking control of space robots, Jiang et al. [
] investigated the finite-time control problem associated with attitude stabilization of a rigid spacecraft subject to external disturbances, actuator faults, and input saturation, and proposed an
adaptive fixed-time-based finite-time attitude controller designed to guarantee finite-time reachability of the attitude orientation in a small neighborhood of the equilibrium point. Liu et al. [
] studied the effect of payload collisions on the dynamics and control of a flexible dual-arm space robot capturing an object, proposed a method for the determination of initial conditions for
post-impact dynamic simulation of the system and proposed a PD controller to maintain stabilization of the robot system after the capture of the object. Walker et al. [
] presented an adaptive control method that achieves globally stable trajectory tracking in the presence of uncertainties in the inertial parameters of the system. Yi and Ge [
] studied an indirect Legendre pseudospectral method for attitude motion tracking control of an asymmetric underactuated rigid spacecraft equipped with only two pairs of jet thrusters. Sands [
] proposed a novel optimization whiplash compensation method to realize automatic control of flexible space robotics. Stolfi [
] focused on the issue of maintaining a stable first contact between the arms end-effectors and a target satellite before the grasp is performed, investigates the application of the Impedance + PD
control approach to a two-arm space manipulator used to capture a non-cooperative target. Zhang and Zhu [
] presented the notion that the planning task does not need to solve the inverse kinematics, investigating a novel motion planning algorithm based on rapidly-exploring random trees (RRTs) for an
free-floating space robots from an initial configuration to a goal end-effector pose. Cocuzza [
] aimed at locally minimizing the dynamic disturbances transferred to the spacecraft during trajectory tracking maneuvers, based on a constrained least-squares approach, proposed a novel solution for
the inverse kinematics of redundant space manipulators. Du et al. [
] based on the continuous finite-time control technique, studied the attitude stabilization of spacecraft, a finite-time attitude tracking control law has been designed for a single spacecraft and a
distributed finite-time attitude synchronization algorithm has also been developed for a group spacecraft. Aghili [
] presented a combined prediction and motion-planning scheme for robotic capturing of a drifting and tumbling object with unknown dynamics using visual feedback, and used the estimated states,
parameters, and predicted motion trajectories to plan the trajectory of the robot’s end-effector to intercept a grapple fixture on the object with zero relative velocity in an optimal way. In order
to realize the attitude stabilization and joint tracking control of the space robot with flexible links and elastic base, Yu [
] proposed a terminal sliding mode controller based on desired trajectory to control the free-flying space manipulator when parametric uncertainties and modeling errors exist.
For the post-capture attitude stable control of space robots, Cheng [
] studied the attitude management of space robots after capturing a satellite, the control of the auxiliary docking operation and presented an adaptive control scheme based on extreme learning
machine to achieve the coordinated control of the target. Wang et al. [
] considered identifying the mass properties and eliminating the unknown angular momentum of space robotic systems after capturing a non-cooperative target, designing an integrated control framework
which includes a detumbling strategy, coordination control and parameter identification, and proposed a coordination control scheme for stabilizing both the base and end-effector based on impedance
control implemented considering the target’s parameter uncertainty. Zhang et al. [
] proposed a modified adaptive sliding mode control algorithm to reduce the momentum, which can reduce the unknown angular momentum of a target, and uses a new signum function and time-delay
estimation to assure fast convergence and achieve good performance with a small chattering effect. Wu et al. [
] developed a generic frictional contact model which can represent the contact forces between the robot’s end-effector and the target object and designed a resolved motion admittance control method
based on the frictional contact model. Rekleitis [
] developed a planning and control methodology for manipulating passive objects by cooperating orbital free-flying servicers in zero gravity. Although the above control algorithms focus on the
dynamics and control of space robots capturing a spacecraft, the protection of the joint actuators of the space robot under the impact torque is not considered. Since a space robot’s joints are
easily destroyed by the impact torque during the process of space robot on-orbit capturing a non-cooperative spacecraft, therefore, the studies on compliance control of space robots during the
capturing process need to be improved.
For the series elastic actuator (SEA) in the ground robot, Gu et al. [
] presented a modularized series elastic actuator aimed to improve the compliance of the robotic arm. Calanca and Fiorini [
] refined and improved the stability analysis of the environment-adaptive force control of SEAs. Wang et al. [
] presented a practical control approach for series elastic actuators which can work well even in the presence of unknown payload parameters and external disturbances. Considering that SEA devices
play a key role in protecting the robot’s joints from impact damage when the ground robot collides with the outside environment, therefore, this paper designs a rotary series elastic actuator (RSEA)
device suitable for space robots, and at the same time, designs an active controller strategy which can timely control the opening and closing of joint actuators to achieve buffer compliance control.
The RSEA also leads to joint flexibility due to the presence of a buffer spring inside the system. Since the system meets the law of conservation of linear momentum and law of conservation of angular
momentum, its orbital dynamics and base attitude are coupled, which make its links’ locomotion leads to the base’s reactions, and consequently a variation of the end-effector position. At the same
time, momentum, momentum moment and energy transfer change also exist in the pre-contact and post-capture phase of systems consisting of a space robot and spacecraft. In addition, due to the high
velocity and rotation characteristics of the non-cooperative target spacecraft, the dynamic parameters of the post-capture hybrid system are difficult to obtain accurately. The above multiple complex
situations make research on the dynamic modeling and control of the on-orbit capturing process of space robots equipped with RSEA devices very complicated.
In an effort to address the various aforementioned drawbacks, this work investigates the dynamic modeling, buffer compliance control and vibration suppression of a space robot capturing a
non-cooperative spacecraft. First of all, dynamic models of the space robot and the target spacecraft before capture are obtained by using the Lagrange approach and Newton-Euler method. Second, based
on singular perturbation theory, the post-capture hybrid system was transformed into two subsystems, a slow rigid motion subsystem, and a fast flexible-joint subsystem. For the fast subsystem, the
velocity difference feedback controller is used to actively suppress the elastic vibration of the joints’ flexibility. For the slow subsystem, a buffer compliance control scheme based on
reinforcement learning (RL) is proposed. The proposed reinforcement learning consists of two modules: associative search network (ASN) and adaptive critic network (ACN). ASN is used to approximate
unknown nonlinear terms of mixed systems; the ACN adopts the online learning method. The learning strategy of RL obtains the original error evaluation signal through the performance evaluation unit,
this error evaluation signal is coupled with ACN to generate the reinforcement signal. Then, the updated result is used as the learning rule of the neural network to train the neural network weight
adaptive law of ASN and ACN, which can adjust and optimize the control strategy in real time. For the reinforcement learning strategy, Liu et al. [
] obtained the system dynamics model of space robota by reinforcement learning, by comparison with the traditional PD control method, that shows the self-learning ability of the reinforcement
learning strategy. Sands [
] proposed deterministic artificial intelligence that can applied to both unmanned underwater vehicles and space robotics. Tang and Liu [
] studied the control and stability issues of a trajectory tracking of an n-link rigid robot manipulator, and obtained an optimal control signal by a reinforcement learning strategy. Cui et al. [
] proposed a reinforcement learning strategy to investigate the trajectory tracking problem for a fully actuated autonomous underwater vehicle with external disturbances, control input nonlinearities
and model uncertainties. On this basis, the proposed control scheme can absorb the impact energy generated in the collision process through the stretching and compression of the built-in spring in
the collision capture phase. In the stable control phase, the control strategy based on reinforcement learning is used to actively turn on and off the joints’ actuators to ensure that the joints’
actuators will not be overloaded and damaged. In addition, the reinforcement learning strategy has the advantage of not needing the precise dynamics model of the hybrid system and can effectively
improve the intelligence and reliability of the on-orbit acquisition operation of the space robot. The numerical simulation shows that the proposed control scheme can not only effectively absorb the
impact energy generated by the on-orbit capture, but also open and close the joint actuators in a timely way when the impact energy is too large, which can avoid overload and damage to the joint
The paper is organized as follows: in
Section 2
, the compliant mechanism and buffer compliance strategy are introduced. In
Section 3
, the dynamic model of the space robot capturing a non-cooperative target spacecraft is established. In the same section, the impact effect during the capturing operation phase is discussed. In
Section 4
, a reinforcement learning control algorithm combined with a compliant mechanism is proposed to achieve buffer compliance control and its stability is verified by introducing the suitable Lyapunov
function. In
Section 5
, numerical simulations are carried out to validate the proposed buffer compliance control strategy. Finally, the conclusions are given in
Section 6
2. Buffer Compliance Strategy
The RSEA consists of five modules: input disk, sweeping arm, support axis, springs, block. The RSEA device of the space robot system is installed between the actuators and the manipulator and is
connected to the actuators through its input disk. The block is firmly connected to the input disk. The hollow shaft of the sweeping arm is connected with the support axis fixed on the input disk
through a bearing. When the motor rotates it drives the input disk to rotate. Through the block compression spring, the spring transfers the force to the sweeping arm. The hollow shaft of the
sweeping arm is directly connected with the manipulator, so as to complete the smooth transfer of motion and force. The general structure diagram of the space manipulator is shown in
Figure 1
, and the structure of the designed RSEA device is shown in
Figure 2
. In
Figure 2
is the effective radius of sweeping arm and
is the radius of spring.
In the capture phase, the end-effector of the manipulator contacts and collides with the spacecraft, whereupon the joint of the manipulator will be subjected to a huge impact torque. The impact
torque acts on the output sweeping arm of the RSEA device first, and then is transferred to the spring group. The impact energy generated by the collision is stored in the spring through the
deformation of the spring group, so as to realize the protection of the joint. In the stable control phase, the joints are also affected by the impact torque due to the impact of the capture
collision. If the torque exceeds the limit that the joint actuators can withstand and the actuators do not turn off, the actuators will be damaged. Therefore, it is necessary to set a shutdown torque
threshold to turn off the actuators according to the torque limit that the joint can withstand. When the impact torque on the joints is detected to exceed the shutdown torque threshold value, all
actuators turn off. In this time, the internal spring assembly of the RSEA device provides an elastic force to reduce the impact torque on the joints. In addition, in practical operation, if only the
shutdown torque threshold is set, the actuators will be switched on and off frequently, thus affecting the actuators performance. On this basis, the control strategy proposed also sets a startup
torque threshold value of actuators, when the joint torque exceeds the shutdown torque threshold, the actuators turn off, and when the joint torque is below the startup torque threshold, the
actuators turn on again.
3. Dynamics Modeling and Impact Effect Analysis
The structure of a space robot with RSEA and target spacecraft systems is shown in
Figure 3
. The space robot consists of a rigid base
, rigid links
= 1,2), and rigid target spacecraft
. We build the inertial coordinate system
, while at the same time, the local coordinate system
= 0,1,2) of each component
= 1,2) is established;
is the rotation center of the base,
is the rotation center of
= 1,2);
is the mass of the base,
is the mass of the non-cooperative spacecraft,
is the mass of
= 1,2).
is inertial moment of the base with respect to its mass center,
is the inertial moment of the non-cooperative spacecraft with respect to its mass center,
= 1,2) is the inertial moment of
B[i] (i
= 1,2) with respect to their mass center.
represents the distance from point
= 1,2) represents length of
along the
d[i] (i
= 1,2) is the distance from the mass center of
= 1,2) is inertial moment of the
-h actuator.
k[im] (i
= 1,2) is the spring stiffness of the RSEA device.
is the position vector of the mass center of the entire system in inertial coordinate system (
= 1,2) is position vector of the mass center of
in the inertial coordinate system (
Regarding the target spacecraft as a homogeneous rigid body, its dynamic equation can be obtained by the Newton-Euler method:
$q s = [ x s , y s , θ s ] T$
the generalized coordinates of the target spacecraft system;
$x s$
$y s$
are the position vectors of the mass center of
$B 3$
$θ s$
is the attitude angle of the spacecraft system.
$D ( q ) ∈ R 3 × 3$
are the inertia positive definite matrices,
$J s ∈ R 3 × 3$
is its impact contact point corresponding to the motion Jacobi matrix.
$F ′ ∈ R 3 × 1$
is the force acting on the spacecraft.
According to the position vector relation in
Figure 2
, the position vectors of the mass center of
$B i ( i = 0 , 1 , 2 )$
in the pre-contact phase are:
$r 0 = [ x a , y a ] T r 1 = r 0 + l 0 e 0 + d 1 e 1 r 2 = r 0 + l 0 e 0 + l 1 e 1 + d 2 e 2$
$x a$
$y a$
are the position vector of the mass center of base
$B 0$
$e i ( i = 0 , 1 , 2 )$
is the unit vector along the
axis in the
$x i O i y i$
Differentiating Equation (2) with respect to time
, then the total kinetic energy of the space robot with RSEA is:
$T = ∑ i = 0 2 ( 1 2 m i r ˙ i T r ˙ i + 1 2 I i ω i T ω i ) + ∑ j = 1 2 1 2 I j m ω j m T ω j m$
$ω i ( i = 0 , 1 , 2 )$
is the angular velocity of the rotation center
$O i$
$ω j m ( j = 1 , 2 )$
is the angular velocity of the actuator.
Neglecting the micro-gravity in space, the potential energy of the system only comes from the RSEA device, so the total potential energy of the system is:
$U = ∑ i = 1 2 3 2 k i m ( Δ x i L ) 2 + ( Δ x i R ) 2$
$Δ x i L = x ( α i )$
$Δ x i R = − x ( α i )$
$x ( α i ) = R sin ( α i )$
$x ( α i )$
is the deformation of the spring on the block of the RSEA device,
$α i$
is the angular difference between the sweeping arm and the input disk.
Based on Equations (3) and (4), and combing with the Lagrange equations, the dynamic equations of the space robot of pre-capture phase are as follows
$D ( q ) q ¨ + C ( q , q ˙ ) q ˙ = τ c + J T F I m θ ¨ m + K ( θ m − θ ) = τ m K ( θ m − θ ) = τ θ$
$q = [ x a , y a , θ 0 , θ 1 , θ 2 ] T$
are the generalized coordinates of the system,
$θ 0$
is the attitude angle displacement of the base,
$θ i ( i = 1 , 2 )$
is the attitude angle displacement of the
-th link,
$θ i m ( i = 1 , 2 )$
is the attitude angle displacement of the
-th actuator.
$D ( q ) ∈ R 5 × 5$
are the inertia positive definite matrices,
$C ( q , q ˙ ) q ˙ ∈ R 5 × 1$
is the Coriolis/centrifugal matrix.
$θ m = [ θ 1 m , θ 2 m ] T$
$θ = θ 1 , θ 2 T$
$τ c = [ τ a T , τ 0 , τ θ T ] T$
$τ a = [ 0 , 0 ] T$
is the position control torque of the base,
$τ 0$
is the attitude control torque of base.
$τ m = [ τ 1 m , τ 2 m ] T$
is the joint torque/force delivered by actuators.
$I m = diag ( I 1 m , I 2 m )$
$K = diag ( k 1 , k 2 )$
is the equivalent stiffness of joints, and its calculation formula is given in Equation (46).
$J ∈ R 3 × 5$
is its end-effector impact contact point corresponding motion Jacobi matrix,
$F ∈ R 3 × 1$
is the force acting on the end-effector.
In the capturing operation phase, the space robot contacts and collides with the target spacecraft, and the interaction force at the end is satisfied:
Based on Equation (6), and combining it with Equations (1) and (5), we can obtain that:
$D ( q ) q ¨ + C ( q , q ˙ ) q ˙ = τ c − J T ( J s T ) − 1 D s q ¨ s$
The actuators will be turned off during the capture phase, which is
$τ c = 0 5 × 1$
. Integrating Equation (7) over the momentary period of collision [
$D ( q ) ( q ˙ ( t 0 + Δ t ) − q ˙ ( t 0 ) ) + J T ( J s T ) − 1 D s ( q ˙ s ( t 0 + Δ t ) − q s ( t 0 ) ) = 0$
The space robot and spacecraft satisfy the velocity constraint in the post-capture phase. Based on this, the following generalized velocity of the post-capture hybrid system can be obtained:
$q ˙ ( t 0 + Δ t ) = N − 1 [ D ( q ) q ˙ ( t 0 ) + J T ( J s T ) − 1 D s q ˙ ( t 0 ) ]$
$N = D ( q ) + J T ( J s T ) − 1 D s J s − 1 J$
Integrating first item of Equation (5), we have:
$D ( q ) ( q ˙ ( t 0 + Δ t ) − q ˙ ( t 0 ) ) = J T P$
$P = ∫ t 0 t 0 + Δ t F d t$
is the impact impulse during the capture phase. Invoking Equations (9), and (10), we can obtain that:
$P = ( J T ) + 1 D ( q ) [ N − 1 ( D ( q ) q ˙ ( t 0 ) + J T ( J s T ) − 1 D s q ˙ ( t 0 ) ) − q ˙ ( t 0 ) ]$
$( J T ) + 1$
is the Moore-Penrose pseudo-inverse of
$J T$
. The period of contact is transient:
$Δ t → 0$
, then the collision force can be approximated as:
After the space robot capturing the target spacecraft, a hybrid system is formed. Consider the velocity constraint relationship of arm and target, we can obtain that:
Differentiating Equation (13), we have:
$q ¨ s = J s − 1 [ J q ¨ + ( J ˙ − J ˙ s J s − 1 J ) q ˙ ]$
Invoking Equations (1), (5) and (14), we can obtain that:
$D A ( q ) q ¨ + C A ( q , q ˙ ) q ˙ = τ c I m θ ¨ m + K ( θ m − θ ) = τ m K ( θ m − θ ) = τ θ$
$D A ( q ) = D ( q ) + J T ( J s T ) − 1 D s J s − 1 J$
$C A ( q , q ˙ ) = C ( q , q ˙ ) + J T ( J s T ) − 1 D s J s − 1 ( J ˙ − J ˙ s J s − 1 J )$
In order to facilitate the design of subsequent control strategies, the first item of Equation (15) of the hybrid system can be expressed in the form of block matrices as follows, so as to obtain the
fully controllable formal dynamics model:
$D A 11 D A 12 D A 2 1 D A 22 q ¨ a q ¨ θ + C A 11 C A 12 C A 21 C A 22 q ˙ a q ˙ θ = τ a τ b$
$q a = [ x a , y a ] T$
$q θ = [ θ 0 , θ 1 , θ 2 ] T$
$τ b = [ τ 0 , τ θ T ] T$
$D A 11 ∈ R 2 × 2$
$D A 12 ∈ R 2 × 3$
$D A 2 1 ∈ R 3 × 2$
$D A 22 ∈ R 3 × 3$
the submatrices of
$D A$
$C A 11 ∈ R 2 × 2$
$C A 12 ∈ R 2 × 3$
$C A 2 1 ∈ R 3 × 2$
$C A 22 ∈ R 3 × 3$
the submatrices of
$C A$
, and
$C A 11$
$C A 2 1$
are zero matrix. Equation (16) can be decomposed into:
$D A 11 q ¨ a + D A 12 q ¨ θ + C A 11 q ˙ a + C A 12 q ˙ θ = 0 0 T$
$D A 2 1 q ¨ a + D A 22 q ¨ θ + C A 21 q ˙ a + C A 22 q ˙ θ = τ b$
From Equation (17), we have:
$q ¨ a = − D A 11 − 1 ( D A 12 q ¨ θ + C A 11 q ˙ a + C A 12 q ˙ θ )$
Invoking Equations (18) and (19), we can obtain that:
$D x q ¨ θ + C x q ˙ θ = τ b I m θ ¨ m + K ( θ m − θ ) = τ m K ( θ m − θ ) = τ θ$
$D x = D A 22 − D A 21 D A 11 - 1 D A 12$
$C x = C A 22 − D A 21 D A 11 - 1 C A 12$
. And
$D ˙ x − 2 C x$
is an antisymmetric matrix.
4. Two-Time Scale Control
4.1. Fast Subsystem and the Corresponding Controller
In order to actively suppress the flexible vibration of the joint caused by the RSEA device, based on singular perturbation theory, the post-capture hybrid system was transformed into two subsystems,
a slow rigid motion subsystem, and a fast flexible-joint subsystem. This controller consists of a slow sub-controller and a fast flexible-joint sub-controller:
$τ f ∈ R 2 × 1$
is the fast flexible-joint sub-controller,
$τ s ∈ R 2 × 1$
is the slow sub-controller.
Defining the positive proportional factor
and the positive definite diagonal matrix
$K 1$
, it satisfies:
Invoking Equation (22), the flexible-joint fast subsystem is:
$ε 2 τ ¨ θ = I m − 1 K 1 ( τ m − I m θ ¨ − τ ˙ θ )$
In order to suppress the elastic vibration of the system, the following speed difference feedback controller is designed to control the fast subsystem:
$τ f = − K f ( θ ˙ m − θ ˙ )$
$K f = K 2 / ε$
$K 2 ∈ R 2 × 2$
is a positive definite diagonal matrix.
Substituting Equations (21), (24) into Equation (23), we have:
$ε 2 I m τ ¨ θ = K 1 ( τ s − I m θ ¨ − τ θ ) − ε K 2 τ ˙ θ$
It can be shown that while
$ε → 0$
, the equivalent stiffness of joints
$K → ∞$
. At this point, the hybrid system is equivalent to a rigid model. Then the dynamic equation of the slow subsystem can be obtained from first item of Equations (20) and (21)
$D x θ q ¨ θ + C x θ q ˙ θ = τ x θ$
$D x θ = D x + I x$
$I x = diag ( 0 , I 1 m , I 2 m )$
$C x θ$
is the corresponding matrix of
$C x$
$θ ˙ = θ ˙ m$
$τ x θ = [ τ 0 , τ s T ] T$
4.2. Slow Subsystem and the Corresponding Controller
The buffer compliance control based on reinforcement learning is shown in
Figure 4
, Where ASN is used to approximate the unknown nonlinear term of the system, ACN is used to construct reinforcement signals to optimize ASN.
Define the trajectory tracking error as:
$q θ d ∈ R 3 × 1$
are desired trajectories of the hybrid system.
At the same time, the error evaluation signal is defined as:
$Λ ∈ R 3 × 3$
is a positive definite diagonal matrix.
Invoking Equations (27) and (28), the dynamic equation of the slow subsystem can be written as:
$D x θ z ˙ = − C x θ z + d − τ x θ$
$d = D x θ ( q ¨ θ d + Λ e ˙ ) + C x θ ( q ˙ θ d + Λ e )$
is the unknown nonlinear term of the system. Considering that it cannot be obtained directly, it can be approximated by the ASN:
$d = W a T Φ ( x ) + ς ( x )$
$W a ∈ R n × 3$
is the ideal weight matrix of a radial basis function neural network (RBFNN),
$ς ( x )$
is the optimal approximation error. The radial basis kernel functions
$Φ ( x ) = [ Φ 1 , Φ 2 , ⋯ Φ n ] T$
are represented by a Gaussian radial basis function (GRBF) as:
$Φ ( x ) = exp ( x − c 2 2 σ 2 )$
$x = [ q θ T , q ˙ θ T , q ˙ θ d T , q ¨ θ d T ] T$
are the variance and the centre vector of the GRBF.
On this basis, the slow rigid motion subsystem control law is given as:
$τ x θ = W ^ a T Φ ( x ) + K z z + τ a$
$W ^ a$
is the estimate of the ideal weight
$W a$
. Defining the estimation error
$W ˜ a = W a − W ^ a$
, it satisfies
$W ˜ ˙ a = − W ^ ˙ a$
$K z ∈ R 3 × 3$
is a positive definite diagonal matrix.
$τ a$
is a robust control law, which is defined as:
$K a ∈ R 3 × 3$
is a positive definite diagonal matrix.
Substituting Equations (32) and (33) into Equation (29), we have:
$D x θ z ˙ = − ( C x θ + K z ) z − K a z / z + W ˜ a Φ + ς ( x )$
In order to optimize ASN, reinforcement learning signals are defined by CAN:
$r = z + z W ^ c T Φ ( x )$
$W ^ c ∈ R m × 3$
is the estimate of the ideal weight
$W c$
Assumption 1.
The ideal weights $W a$
$W c$are bounded and satisfy:where$W a M$
$W c M$is an unknown positive constant
Assumption 2.
The optimal approximation error $ς ( x )$is bounded and satisfies
where$ς M$is an unknown positive constant
Next, the weight adaptive law of neural network can be further designed as:
$W ^ ˙ a = K b Φ ( x ) r T − η K b z W ^ a$
$W ^ ˙ c = − K c z Φ ( x ) ( W ^ a T Φ ( x ) ) T − η K c z W ^ c$
$K b$
$K c$
are positive definite diagonal matrix.
is a positive constant. Defining estimation error as
$W ˜ c = W c − W ^ c$
, and it satisfied
$W ˜ ˙ c = − W ^ ˙ c$
Theorem 1.
For the dynamic equation of the slow subsystem (26) with unknown nonlinear terms, supposing that Assumptions 1 and 2 hold and adopting the weight adaptive law (36) and (37), the control law (32)
based on reinforcement learning signals (35) can ensure that the trajectory tracking error$e$converges to zero asymptotically.
Proof of Theorem 1.
Introducing the Lyapunov function:
$V = 1 2 z T D x θ z + 1 2 t r { W ˜ a T K b − 1 W ˜ a } + 1 2 t r { W ˜ c T K c − 1 W ˜ c }$
Differentiating Equation (13), we have:
$V ˙ = 1 2 z T D ˙ x θ z + z T D x θ z ˙ − t r { W ˜ a T K b − 1 W ^ ˙ a } − t r { W ˜ c T K c − 1 W ^ ˙ c }$
Substituting Equations (34)–(37) into Equation (39) yields:
$V ˙ = − z T K z z + z T ς + z t r { − W ˜ a T Φ ( W ^ c T Φ ) T + η W ˜ a T W ^ a + W ˜ c T Φ ( W ^ a T Φ ) T + η W ˜ c T W ^ c } − z T K a z / z ≤ − z T K z z + z T ς + z t r { − W ˜ a T Φ ( W ^ c
T Φ ) T + η W ˜ a T W ^ a + W ˜ c T Φ ( W ^ a T Φ ) T + η W ˜ c T W ^ c }$
Combining Assumption 1, we have:
$W ˜ a T W ^ a ≤ W ˜ a W a M − W ˜ a 2$
$W ˜ c T W ^ c ≤ W ˜ c W c M − W ˜ c 2$
and combing Assumption 2, Equation (40) is rewritten as:
$V ˙ ≤ − z T K z z + z ς M + z { ( W a M + W c M Φ 2 ) W ˜ a + ( W c M + W a M Φ 2 ) W ˜ c − η W ˜ a 2 − η W ˜ c 2 }$
Considering the regression vector
is bounded, it can be set
$Φ 2 ≤ Φ -$
$K z m$
is minimum eigenvalue of
$K z$
. Equation (41) satisfies:
$V ˙ ≤ − K z m z 2 − η z { ( W ˜ a − a 1 ) 2 + ( W ˜ c − a 2 ) 2 − [ a 1 2 + a 2 2 + ς M η ] }$
$a 1 = W a M + W c M Φ - 2 η$
$a 2 = W c M + W a M Φ - 2 η$
. To assure
$V ˙ ≤ 0$
, we only require one of the following conditions:
$z > η [ a 1 2 + a 2 2 + ς M η ] / K z m$
$W ˜ a > a 1 + a 1 2 + a 2 2 + ς M η$
$W ˜ c > a 2 + a 1 2 + a 2 2 + ς M η$
Based on the analysis results of the above steps, and combing with the Lyapunov stability theorem, which implies that the whole closed-loop system is stable the trajectory tracking error $e$
converges to zero asymptotically from the stability analysis in Theorem 1. The proof is thus completed. □
5. Simulation Results
5.1. Impact Resistance Performance Simulation in the Capture Phase
To show the performance of the proposed controller, simulations are carried out on a planar space robot with the RSEA and target spacecraft systems shown in
Figure 3
. The actual parameters of the system are as follows:
$m 0 = 80 kg$
$m 1 = 5 kg$
$m 2 = 5 kg$
$m s = 30 kg$
$I 0 = 30 kg · m 2$
$I 1 = 3 kg ⋅ m 2$
$I 2 = 3 kg ⋅ m 2$
$I s = 15 kg ⋅ m 2$
$I 1 m = 0.05 kg ⋅ m 2$
$I 2 m = 0.05 kg ⋅ m 2$
$k 1 m = k 2 m = 1000 N / m$
$l 0 = 1 m$
$l 1 = l 2 = 2 m$
$d 1 = d 2 = 1 m$
The equivalent stiffness of joints [
] is as follows:
$K = 2 K m ( 3 R 2 + r 2 ) ( 2 cos 2 φ − 1 )$
$K m = diag ( k 1 m , k 2 m )$
$R = 0.1 m$
$r = 0.01 m$
is the angle of sweeping arm when the force
$F = [ 20 N · m , 20 N · m , 0 ] T$
acting on the end of the space manipulator, select
$φ = diag ( 3 ∘ , 2 ∘ )$
In order to verify impact resistance performance simulation in the capture phase, the space robot system with/without RSEA device was used to carry out acquisition simulation tests on spacecraft with
different velocity. The simulation results are shown in
Table 1
Table 1
, the first column of velocity terms, the first two are linear velocities, and the third is angular velocities. In the second and third columns, the preceding and the following items are the impact
torques of joints without and with RSEA devices respectively. The fourth column has the maximum percentage reduction in joint impact torque with the RSEA device. As can be seen from
Table 1
, for the capture phase of spacecraft at different initial velocities, the configuration of RSEA device can effectively reduce the impact torque acting on joints, and effectively realize the
protection of the joint.
5.2. Buffer Compliance Control Performance Simulation in Stable Control Phase
To show the buffer compliance control performance of the proposed controller, simulations are carried out for stable control phase. The actual parameters of the system are as follows:
= diag(5,5),
= diag(5,5,5),
= diag(400,400,400),
= 0.5,
= diag(20,20,20),
= diag(50,50,50),
= 1,
= diag(10,10,10). In pre-impact phase
$q θ = [ 90 ∘ , 45 ∘ , 45 ∘ ] T$
, assuming that the space robot system capturing a non-cooperative spacecraft at
$t 0 = 0 s$
. At this time, the velocity of the spacecraft is
$v t = [ 0.45 m / s , 0.45 m / s , 0.5 rad / s ] T$
, the desired trajectory of post-capture hybrid system is
$q θ d = [ 100 ∘ , 30 ∘ , 60 ∘ ] T$
. Assume that when the joint actuators running, the limit of the impact torque it can bear is
$90 N · m$
. In order to protect the joint actuators, the buffer compliance control strategy of active opening and closing actuators (named switching strategy) is adopted. The shutdown torque threshold is
$60 N · m$
, and the startup torque threshold is
$6 N · m$
. The simulation results are shown in
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 5
shows the impact torque acting on the joints when not adopting the switching strategy, where it can be found that the impact torque still exceeds the safety threshold of the joint at this time.
Figure 6
shows the impact torque acting on the joints when adopting a switching strategy. By comparing
Figure 5
Figure 6
, it can be seen that the impact torque acting on the joints can be limited within a safe range by combining with the buffer compliance control, which ensures the protection of the joint motor during
the stable control phase.
Figure 7
shows the evaluation factor signal. It can be found that the ACN is optimized through the interaction with the environment and the reward signal is obtained, and finally reaches the stable state.
To show the effectiveness of the defined reinforcement signal, the tracking accuracy is quantitatively analyzed by comparison of the trajectory tracking error of the proposed RL control scheme, RL
with robust controller off and neural network control strategy without reinforcement signal (turn off RL). The mean absolute error MAE =
$1 n ∑ i = 1 n e i$
was used to evaluate the tracking accuracy, the simulation results are shown in
Table 2
. It can be seen in
Table 2
the mean absolute error of proposed RL is smaller than the other control strategies, which shown that the proposed control method has high tracking accuracy and good tracking performance.
Figure 8
Figure 9
Figure 10
shows the stabilization of the hybrid system when the proposed buffer compliance controller is adopted. The solid line is the trajectory tracking curve of the system when the control algorithm based
on reinforcement learning is adopted, the dotted line is the trajectory tracking curve when turn off the robust term
$τ a$
, the double line is the trajectory tracking curve when turn off RL. By comparing them, it can be found that the unstable hybrid system finally reaches the stable and expected state, and the proposed
RL control scheme has faster convergence speed and higher tracking accuracy.
If the fast subsystem controller of the system is turned off, the system trajectory tracking curve shown in
Figure 11
Figure 12
Figure 13
can be obtained. By comparing
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
, it can be seen that if the fast subsystem is turned off, the elastic vibration of the unstable hybrid system will continue to increase and eventually lead to the divergence of the system.
Therefore, the proposed velocity difference feedback controller can actively suppress the elastic vibration of the system joint, and then achieve stable track of the trajectory.
6. Conclusions
In this paper a space robot with a RSEA device to protect the joint of the robot under impact torque during the satellite capture process is designed. The timely opening and closing of joint
actuators was proposed to achieve buffer compliance control. The dynamic model of the post-capture hybrid system was derived from the Lagrange equations, law of conservation of momentum and the
constraints of kinematics and velocity. Then, based on singular perturbation theory, the hybrid system was decomposed into a slow subsystem and a fast subsystem. A buffer compliance control based on
a reinforcement learning algorithm was applied to control the slow subsystem with unknown unknown nonlinear disturbances term. The fast control was designed with speed difference feedback controller.
The simulation results show that the proposed strategy can reduce the impact torque by 76.6% at the maximum and 58.7% at the minimum during the capture phase, which reflects a good anti-impact
performance. In the stable control phase, the impact torque acting on the joint is guaranteed to be limited within the safety threshold, so as to avoid the overload and damage of the joint actuators.
In addition, the proposed reinforcement learning strategy has strong online adaptability and autonomous learning ability under complex conditions and can be continuously optimized through real-time
interaction with the complex space environment, so as to ensure the accuracy and stability of the system stabilization motion.
Note that this paper only considers that space manipulators mounted on their spacecraft are rigid. For future research, the buffer compliance control of space robot with a flexible-link capturing a
non-cooperative spacecraft control problem will be studied, and the control scheme extended to practical applications.
Author Contributions
Conceptualization, H.A.; methodology, H.A. and X.Y.; software A.Z. and J.W.; investigation, H.A. and A.Z.; writing original-draft preparation, H.A.; writing—review and editing, H.A. and X.Y.;
supervision, H.A. and L.C.; funding acquisition, H.A. and X.Y. All authors have read and agreed to the published version of the manuscript.
This work was supported by the National Natural Science Foundation of China (Grant No. 51741502, 11372073), Science and Technology Project of the Education Department of Jiangxi Province (Grant No.
GJJ200864), Jiangxi University of Science and Technology PhD Research Initiation Fund (Grant No. 205200100514).
Informed Consent Statement
Informed consent was obtained from all subjects involved in the study.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 2. Structure of the proposed rotary series elastic actuator. (a) Planar model. (b) Graphic model.
Initial Velocity Impact Torquein Impact Torquein Maximum
of Satellite/ Joint 1/ Joint 2/ Percentage
(m/s, m/s, rad/s) (N·m, N·m) (N·m, N·m) Reduction
[0.45, 0.5, 0]^T [413.6, 102.8]^T [91.2, 46.7]^T 75.1%
[0, 0.5, 0.5]^T [208.2, 86.0]^T [68.2, 46.5]^T 58.7%
[0.45, 0.5, 0.5]^T [472.9, 110.5]^T [91.8, 48.2]^T 76.6%
The Control Scheme $θ 0$ (°) $θ 1$ (°) $θ 2$ (°)
The proposed RL 0.0015 0.0020 0.0022
Turn off robust 0.0476 0.2866 0.2737
Turn off RL 0.0184 0.0993 0.0949
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© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:
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MDPI and ACS Style
Ai, H.; Zhu, A.; Wang, J.; Yu, X.; Chen, L. Buffer Compliance Control of Space Robots Capturing a Non-Cooperative Spacecraft Based on Reinforcement Learning. Appl. Sci. 2021, 11, 5783. https://
AMA Style
Ai H, Zhu A, Wang J, Yu X, Chen L. Buffer Compliance Control of Space Robots Capturing a Non-Cooperative Spacecraft Based on Reinforcement Learning. Applied Sciences. 2021; 11(13):5783. https://
Chicago/Turabian Style
Ai, Haiping, An Zhu, Jiajia Wang, Xiaoyan Yu, and Li Chen. 2021. "Buffer Compliance Control of Space Robots Capturing a Non-Cooperative Spacecraft Based on Reinforcement Learning" Applied Sciences
11, no. 13: 5783. https://doi.org/10.3390/app11135783
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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How do you graph f(x)=(2x-1)/(x+3) using holes, vertical and horizontal asymptotes, x and y intercepts? | HIX Tutor
How do you graph #f(x)=(2x-1)/(x+3)# using holes, vertical and horizontal asymptotes, x and y intercepts?
Answer 1
#f(x)=(2x-1)/(x+3)# #f(x)=(2(x+3)-7)/(x+3)# #f(x)=2-7/(x+3)#
Therefore, the horizontal asymptote is #y=2# and the vertical asymptote is #x=-3# What asymptotes mean is that the end points of your lines are APPROACHING #y=2# and #x=-3#
Now, to find the intercepts When #y=0#, #x=1/2# When #x=0#, #y=-1/3#
After plotting the intercepts and the asymptotes, you should get something like this graph{(2x-1)/(x+3) [-10, 10, -5, 5]}
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Answer 2
To graph the function f(x) = (2x-1)/(x+3), follow these steps:
1. Determine the vertical asymptote by finding the values of x that make the denominator (x+3) equal to zero. In this case, x = -3 is the vertical asymptote.
2. Find any holes in the graph by canceling out common factors between the numerator and denominator. In this case, there are no common factors to cancel, so there are no holes.
3. Calculate the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the degree of the numerator (1) is less than the degree of the denominator (1), the horizontal
asymptote is y = 0.
4. Find the x-intercept by setting f(x) equal to zero and solving for x. In this case, there is no x-intercept.
5. Find the y-intercept by evaluating f(0). Substitute x = 0 into the function: f(0) = (2(0)-1)/(0+3) = -1/3. Therefore, the y-intercept is (0, -1/3).
6. Plot the vertical asymptote at x = -3, the horizontal asymptote at y = 0, and the y-intercept at (0, -1/3).
7. Choose additional x-values and calculate the corresponding y-values to plot more points on the graph. For example, you can choose x = -4, -2, 1, and 2.
8. Connect the plotted points smoothly, avoiding the vertical asymptote.
This completes the graph of f(x) = (2x-1)/(x+3).
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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
Not the question you need?
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-graph-f-x-2x-1-x-3-using-holes-vertical-and-horizontal-asymptotes-x-a-8f9af9bc01","timestamp":"2024-11-03T07:54:58Z","content_type":"text/html","content_length":"573954","record_id":"<urn:uuid:d6263b0a-c82e-445f-ac0e-89ccff0367dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00160.warc.gz"} |
Convert Elementary Charge to Coulomb
Please provide values below to convert Elementary charge [e] to coulomb [C], or vice versa.
Elementary Charge to Coulomb Conversion Table
Elementary Charge [e] Coulomb [C]
0.01 e 1.60217733E-21 C
0.1 e 1.60217733E-20 C
1 e 1.60217733E-19 C
2 e 3.20435466E-19 C
3 e 4.80653199E-19 C
5 e 8.01088665E-19 C
10 e 1.60217733E-18 C
20 e 3.20435466E-18 C
50 e 8.01088665E-18 C
100 e 1.60217733E-17 C
1000 e 1.60217733E-16 C
How to Convert Elementary Charge to Coulomb
1 e = 1.60217733E-19 C
1 C = 6.241506363094E+18 e
Example: convert 15 e to C:
15 e = 15 × 1.60217733E-19 C = 2.403265995E-18 C
Convert Elementary Charge to Other Charge Units | {"url":"https://www.unitconverters.net/charge/elementary-charge-to-coulomb.htm","timestamp":"2024-11-08T01:27:07Z","content_type":"text/html","content_length":"7743","record_id":"<urn:uuid:33d50788-5f52-4df9-b383-bd66c5b24c05>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00051.warc.gz"} |
Week 18 Pool Banker Room 2023 – Pool Banker This Week | Sure Bet WayWeek 18 Pool Banker Room 2023 – Pool Banker This Week
Week 18 banker room 2023, Week 18 Pool Banker 2023, Pool Draw This Week 18 2023 Banker Room
Welcome to Sure Bet Way weekly one banker room. The Pool Banker This Week this week 2023. In this banker forum, you are required to post just your best banker or pairs or winning lines with concrete
proofs and well-explained sequences backed up with appropriate references.
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81 Comments
1. Welcome to week 18 2023.
Bank on ×××12××× as compulsory draw for the week.
Proof: Every week 18 sioce year 2020. Bank on MILLWALL anywhere to draw. ×××12×××.
God bless you Admin.
2. Welcome you all to wk18.Admin i salute you for your good work.
LIVE BANKER:********7**********
NOTT’M FOREST Vs ASTON V——FTD
Proof:Since the Season on every blue
Add Saturday date Of play plus(+)code THREE(3) for a Super draw.
Reference: Wk06,Wk10,Wk14 and Wk18 all current.
3. Week 18
One surest banker is 12
Proof: Bristol C.@ 10 Millwall to draw
Ref week 17.
4. 6XXX
Since 2021
Every first FA Cup week
Bank on number 6
5. WEEK 18
XXX 7 XXX
REF:WK 12 TO WK 16 & WK 14 TO WK 18.
OPP WOLVES TO PLAY (1-1) WITH WOLVES & IN NEXT COLOUR TO MEET NOTTMFOR @7 TO DRAW.
6. XX15XX
Every coupon blue mark 15 by number
Ref, Current.
7. Happy New wk and welcome to WK 18 still on the system of Wigan at Away then Count 3 down from Wigan for ur draw, Hope the system continue. This week ×××32×××32×××32×××
8. WK 18 banker:
19 cbk
Proof; every blue, wks 6,10,14 & now WK 17 play 19 cbk with 1-1 cs
2nd proof; Hull away in the same bloodline with Hearts home in blue.
See WK 14 current.
9. WEEK 18 BANKER XX18XX
WEEK 9 BROWN
ROTHERHAM VS NORWICH FF
WEEK OF 7
SUNDERLAND VS ROTHERHAM @16 FF TWO STEPS FROM THE BAR
IN WEEK 8
COVENTRY VS SUNDERLAND XX
WEEK 17
SUNDERLAND VS NORWICH @16 FF TWO STEPS FROM THE BAR
WEEK 18
BANK ON SWANSEA VS SUNDERLAND XX18XX
PROVE 2
WEEK 7
SWANSEA VS COVENTRY XX
WEEK 8
COVENTRY VS SUNDERLAND XX
WEEK 18
SWANSEA VS SUNDERLAND XX18XX CBK
10. WEEK-18 BOMB XXXMILLWALL-12-SOUTH’PTONXXXCBK. WHEN NEWCASTLE HOME OR AWAY IS @MONDAY DATE IN F.A.CUP FIRST ROUND & ITS POSITION MULTIPLY BY TWO EQUALS MILLWALL’S POSITION, MARK MILLWALL AS -100%-
MUST XXXDRAWXXX. REF: WEEK.18;2022/2023. XXXCONGRATULATIONSXXX
11. Welcome to wk.18.
This wk. 11xxpair21xx.
Proof: Watford sharing same digit with Bolton must for one draw.
So wk.18 play 11xx pair 21xx
12. In wk10 current go to Trible jinx game 11 u would see 39 there drew, the following wk, 39 repeated at game 11 drew also. Now wk17 31 drew at game 13 on Trible jinx, this wk too. 31 repeats game
so therefore 31 this wk18 is a draw.
13. WEEK 18 BLUE
BANKER 31BK
LETTER “H”@HOME GAME 11 FOR 3 CONSECITIVE WEEKS.
ADD THE HOME & AWAY FIRST LETTERS FOR YOUR BANKER DRAW.
WEEK 16
GREEN LUCK!
14. Westham @ away picking letter (B)
Ref. Week 6 current.
15. POWERED BY BURNLEY AWAY ON ODD WEEK
BROWN TO BLUE MOVEMENT
WEEK 13
(4) NEWCASTLE VS BURNLEY =20
WEEK 14
(4) CRYTAL PALACE VS NTFOREST XX
(8) WESTHAM VS NESCASTLE XX
WEEK 17
(3) BOURNEMOUTH VS BURNLEY = 21
WEEK 18
(3) EVERTON VS BRIGHTON XX
(5) MANCITY VS BOURNEMOUTH XX
16. Week 18 banker is 22
Proof says that since this season treble jinx number 11 of special advance fixtures every blue colour is a naked draw.
17. 1xx or 13xx
Every blue, mark games 1 and 20 of Bob Morton full list for a draw
Wks 6, 10, 14, 18
18. «»«»19«»«»BANKER.
WESTBROM vs HULL CITY.
advance opponate of SOUTHMTON and that of HUDDERSFIELD
to meet ontop of the bar in CHAMPIONSHIP DIVISION.
Refer; week 18 – 17.
week 19 – 18.
19. I HV 4pair 7 but I go for 7 as my one bank dis week color to color opponent of wolves to d next color to meet wit nottmfor on number 7 it must be number 7 and wolves must draw before it oppernet
will go and draw wit nottmfor in d next color so mark 7as my one branker last week my bank was 8 dis week 7 no pain no gain
20. XXXXX3XXXXX.
PROOF : Any failed number repeated in BOX ONE or TWO of HI- SCORE QUIZ of SAF the following week to result in a draw that second week==
Eeeks 7/8, 10/11, 11/12, 17/18.
21. Week17 I post No:25✓✓✓✓
Week18 BET No:XXXX 08 XXXX CBK
Man City vs Newcastle FF
Sheff utd vs Man City FF
Wolves vs Newcastle XX
Sheff utd vs Wolves XX
22. ***23***23***
My one banker for this week18 is 23.
Anytime you see aston.v and Q.P.R set @ away add the two position together to give you one banker.
REF current.
23. BANKER
CREWE VS DERBY
BOB MORTON FRONT PAGE WITH the arrangement NAP BET HOT NAP
pick first NAP AS YOUR DRAW FOR THE WEEK.
Ref. week 4,10
24. Banker of the week
Base week 12
[[WESTBROM VS MILLWALL=XXX]]
One of the teams that drew above to host HULL in subsequent Blue weeks
WEEK 14
MILLWALL VS HULL=XXX✓
WEEK 18
WESTBROM VS HULL=XXX19XXX
XXX19XXX as draw in
week 18.
25. ***11***One Banker Draw
In brown week 9: Go to the front of soccer research where they wrote PERM !!!. There are six numbers under PERM and the second number must be 11.So add this 11 to the third number and the result
must be the fourth number. i.e.11+16 = 27.Bank on no 11 and leave the remaining numbers.
TO COMFIRM IT:The team that entered number 11 and drew in wk 9 must be the opponent of Cardiff at no 4 in the previous brown week(brown wk 9 to brown wk 5)
In blue week 18 current:Go to the front page of soccer under PERM,the second number must be 11.Add this 11 to the third number and the answer must be the fourth number. i.e. 11+13= 24.Bank on no
11 and forget about other numbers.
CONFIRM THE BANKER:The club that entered number 11 to play draw this blue week must be the opponent of Cardiff at number 4 in the previous blue week 14.(blue wk 18 to blue wk 14).Pls appreciate
Almighty God as you win.
26. WEEK 18 BANKER
SOCCER ⚽ ‘X’ RESEARCH
SINCE 2020 EVERY WEEK 18, PICK GAME RATED 13% WHATEVER THE ANSWER IS, CHECK GAME CARRYING THE ANSWER AND MARK AS A DRAW.
WEEK 18, 07/11/2020
37=1X 2-0 13%
GAME CARRYING 37%
47=2X 1-2 37%
WEEK 18, 06/11/2021
12=2X 1-3 13%
GAME CARRYING 12%
3=1X 2-1 12%
WEEK 18, 05/11/2022
24=1X 2-0 13%
GAME CARRYING 24%
41=2X 0-2 24%
WEEK 18, 04/11/2023
30=1X 2-0 13%
GAME CARRYING 30%
18=2X 1-2 30%
*SOCCER ⚽ ‘X’*
SINCE 2020 GO TO WEEK 17, BROWN PICK GAME CARRYING 14% BRING BACK THE NUMBER IN WEEK 18 THEN CHECK GAME CARRYING THE NUMBER AND MARK IT TO DRAW.
WEEK 17, BROWN 31/10/2020
37=2X 1-3 14%
WEEK 18, BLUE 07/11/2020
47=2X 1-2 37%
WEEK 17, BROWN 30/10/2021
41=1X 2-0 14%
WEEK 18, BLUE 06/11/2021
47=2X 0-1 41%
47XXXXBK PANEL
WEEK 17, BROWN 29/10/2022
15=1X 2-1 14%
WEEK 18, BLUE 05/11/2022
36=1X 2-0 15%
WEEK 17, BROWN 28/10/2023
30=1X 2-0 14%
WEEK 18, BLUE 04/11/2023
18=2X 1-2 30%
*SOCCER ⚽ ‘X’*
CURRENT SEASON
BROWN TO BLUE.
WEEK 09-10
WEEK 13-14
WEEK 17-18
PICK GAME 3 OF BEWARE ON BROWN TO DRAW IN BLUE.
WEEK 09
WEEK 10
19XXX PANEL
WEEK 13
WEEK 14
WEEK 17
WEEK 18
MARK….. 18XXXXXXXXXXXXXBK
27. Any time SUNDERLAND set @ away on week of play ,add home and away team together to number that carry CARDIFF away . SWANSEA +SUNDERLAND =17 latters Mark it draw
28. Westham and Huddersfield at same family number on Blue
Bank on westham to draw.
Number 1.
29. Good day everyone, let’s do more.
My banker this week xxx1xxx
The banker said that, whenever you notice Birmingham understand the bar home and also notice a team that ended with ‘ham’ at coupon number1, Mark it as draw. Reff– week6 and week8
30. Welcome to wk 18
WK 14
Burnley vs Chelsea @ 3 died
WK 18
Burnley vs crystal palace @ 2✓
A good forcaster once said when you know it you don’t panic.
31. Week 18 banker
XXX 16 XXX ROTHERHAM
SHEFF UTD ON TOP BAR,WESTBROM ONTOP BAR,DERBY AWAY ONTOP ALPHABET C @HOME,
**Add first and last alphabet of home team under Derby together and minus 1,answer to draw ROTHERHAM..
***Add first and last alphabet of away game ontop westbrom together and add code 10, minus answer from 49, answer to draw ROTHERHAM @ home,,,
Wk6,. XXX 15 XXX
Wk12,. XXX 16 XXX
Wk18,. XXX 16 XXX
32. Week18****
Any week that game 34 of people sport paper carry 34 and transfer to game 12 of jackpot home 12 .mark it as a full time draw. Use 34 as one banker thus this week.
33. Week18.
Base on the fact that soccer dead game is 33 this week, bank on number 34 as a fixed draw this week. use it as a single banker in all your copies.
fixed 34xxxxxxxxxxx
34. Cup sequence last session first FA Cup
Saturday date minus week number to play Middlesbro ie, 14 banker.
Please Admin post for my people
God bless you.
35. Week18 banker xxxxxxx39xxxxxxx
Burton a to draw every 5 weeks and repeat the following week
From week7
Wk7 and week8
After 5 weeks from wk7
Week12 and week13
After 5 weeks from week12
Week17 and week18
36. Banker 39xxxxxxx
Prov any week in soccer this week draw picture you see the two number is odd and also is one step to the other one mark d left one as draw
Week11 2021/22
5 & 7
One step 5 (6) 7
Week 12 2021/22
13x & 15
One step 13 (14) 15
Mark 13xxxxx✅
Now this week18
39x? & 41
One step 39 (40) 41
Mark 39xxxxxxxxx?
37. One banker which is (6) prove in 2020 week 18 1+8=9 count 3up 7draw in 2021 week 18 1+8=9 count 3up 7drw in 2022 1+8=9 count 4up 6draw this week 18 1+8=9 count 4up 6 is banker arsenal to draw at
away on number 6 with Crystal palace just like it draw with Chelsea
38. welcome to week 18
Advance opponent of sheff. Utd to draw current.
week 17..
Arsenal vs Sheff. Utd
week 16.. Arsenal****4***
week 18.. Sheff. Utd vs Wolves
week 17… Wolves***8***
week 19.. Brighton vs Sheff.Utd
week 18.. Brighton****
Bank on ****3****
39. Week18 Bet No:11 single draw
Coventry No:17XXXX
Coventry No:11XXXX
Watford No:17XXXX
Watford No:11XXXX
40. Week18 BET No:08
Man City vs Newcastle FF
Sheff utd vs Man City FF
Wolves vs Newcastle XX
Sheff utd vs Wolves XX
41. **35** Is Another Live Banker.
Go to Hot three in the second page of soccer research as shown in soccer paper. In wk 14,the middle team (westham-8) carried the upper team as percentage and drew itself as *8* WHILE In wk 15,
the middle team(Eibar-43) carried the bottom team as percentage and drew itself as 43.
After two weeks,they changed to another pattern.In wk 17,the bottom team (Cambridge u-22) carried the upper team(Exeter-24) as %tage and drew the upper team which was 24 while the middle team
(Middlesbro-13) carried 23% and failed. In wk 18,the middle team (Kilmanock-48) caries the upper team(Newport Co-35) as %tage to draw the upper team which is 35 while the bottom team(Curzon-27)
caries 23% to fail.
42. Week18
Look for Watford on week number in advance ,then come to current ,check if westbrom is on same side as Watford in that advance if yes mark westbrom
Ref week11/week12
Ref week17/week18
And this week18/wee19
43. MAKE WE REASON THIS MATTER TOGETHER.
WEEK 12 TO WEEK 14
*GO WEEK 12, LOCATE WHERE SWANSEA DEY HIDE FOR 19 HOME.
*MINUS 10 FROM SWANSEA POSITION, E GO GIVE YOU NUMBER 9.
*GO NUMBER 9 AWAY YOU GO SEE NOTTINGHAM FOREST THERE.
*FOR WEEK 14, NOTTINGHAM FOREST GO DRAW. BEFORE E GO DRAW, GO TO WEEK 13, CHECK THE NIGGA RAW WEY DEY FOR NUMBER 3 AWAY (CRYSTAL PALACE).
*THAT NIGGA RAW, CRYSTAL PALACE, NA HIM GO JAM NOTTINGHAM FOREST TO DRAW.
*CRYSTAL PALACE VS NOTTINGHAM FOREST 4XX.
WEEK 16 TO WEEK 18
*GO WEEK 16, LOCATE WHERE SWANSEA DEY PERCH FOR 16 HOME.
*MINUS 10 FROM SWANSEA POSITION, E GO GIVE YOU NUMBER 6
*GO NUMBER 6 AWAY, YOU GO SEE CRYSTAL PALACE THERE.
*FOR WEEK 18, CRYSTAL PALACE GO DRAW. BEFORE E GO DRAW, GO WEEK 17, E GET ONE NIGGA RAW WEY DEY THERE DEY COOL FOR NUMBER 3 AWAY (BURNLEY).
*THAT NIGGA RAW, BURNLEY, GO JAM THAT CRYSTAL PALACE TO DRAW.
BURNLEY VS CRYSTAL PALACE 2CBK.
44. WELCOME TO WEEK 18 2023.
XXXXXXXXXX 48 XXXXXXXXXX BANKER.
PROVE: SWANSEA MUST SET AT WEEK NUMBER HOME, MOTHERWELL MUST SET IN THE LAST FAMILY OF WEEK NUMBER AWAY AND REGISTER A DRAW.
WEEK 16 2023:
SWANSEA @ 16 HOME FFF
MOTHERWELL @ 46 AWAY XXX
WEEK 18 2023:
SWANSEA @ 18 HOME ???
MOTHERWELL @ 48 AWAY XXXXXXXXXX FTD
45. Bank on No 27
Anytime you find westham @ away on number 1 count the opponent alphabet and add it to the week number
Ref WK 6 Bournemouth=11+6=17drew
Wk16 Aston villa=10+16=26drew
Wk18 brentford=9+17= 27 ??
46. Welcome to wk 18 blue. Bet xxxx6xxxx Newcastle vs Arsenal. Proof in wk 13 brown Wolves vs Man city@7ff ontop bar, Wolves won. Wk 14 blue Arsenal vs Man city@1ff Arsenal won. Wk 17 brown Wolves vs
Newcastle@8xx ontop bar. Wk 18 blue mark Newcastle vs Arsenal@6xx. Second proof wk 13 brown Westham vs Sheff utd@6ff Westham won. Wk 14 blue Westham vs Newcastle@8xx appeared at lotto sequence of
special advance. Wk 17 brown Arsenal vs Sheff utd@1ff Arsenal won. Wk 18 blue mark Newcastle vs Arsenal@6xx appeared at lotto sequence of special advance. Third proof wk 16 Man utd@8ff away ontop
bar, wk 17 Newcastle@8xx away ontop bar. Wk 17 Man utd@6ff home, wk 18 Newcastle@6xx home.
47. Wk18 I have two bankers for the house they are xx8xx9xx for 2/2
Prove since wk10 every blue coupon add Saturday and Sunday date of play together for a draw and vacation of Newcastle in brown coupon to draw blue.
48. Week18
Brendford home to a London club is a naked draw
Check week6 and week8
49. 2xxxxxxxxxxxxxcbk
In premier division we have 4 letter (B) first appearance of each at no 2 is sure draw..
Week 6(Brentford xxxxx)
Week 11( Bournemouth xxxxx)
Week 14(Brighton xxxxx)
Week 18(Burnley xxxxx)
It’s a draw
Good luck
50. 31 cbk
Proof capital cannot draw carrying 5 % in soccer research & placed at d bob morton coded list c
ref week 6
51. Welcome to week18 banker 42cbk
Courtesy of Sheff Utd @8 home. First letter of Sheff Utd and first letter of it opponent add them together the answer is a cracker the most funniest thing in the answers is it always end in digit
of ‘2’ References week08= 32cbk✓ week16=32cbk✓ and now this week18. 42. What do u think?
52. Week 18 banker cbk…20…..cbk
Anytime you see zero number on capital banker box, mark it as an international draw for dey week. Reff current week 5,11&18. Also pools telegraph brought it as a solidified banker
53. Week 18
WK 6: BLUE
(5ff p 15)
Everton @ number 5ff home
Blackburn @ number 15xx away
WK 18: BLUE
(3 p 13)
Everton @ number 3 home
Blackburn @ number 13 away
54. Welcome everyone to week 18,2023
My Banker this week is NO 11.
In week 7,2023, you will see Swansea vs Coventry played draw @ NO 17. The following week which was week 8,2023, Coventry moved to NO 11 and played draw, while Sheffield Wednesday @ NO 10 away
will play one down to back it.
NOW, check last week 17,2023. Watford vs Millwall played draw @ NO 17, the following week which is this week Watford enter NO 11 while Sheffield Wednesday @ NO 10 to play it One down.
So Mark NO 11.
55. I have 15bk
Prove is when ever you see Cardiff @ number 10 home or away, look for Coventry to draw. 15bk
56. XXXXX21XXXXX…XBK
Movement: red Wk-to-wk of ‘8’
Appearance of Carlisle on number 21 home-in-red
Go to preceding or advanced WK of ‘8’ and mark number 21 by number
Bolton wanderers must be on that number
WK 7-red to WK 8
WK 8..BOLTON @21 AWAY
WK 19-red to wk 18
WK 18..BOLTON @21 HOME
57. 11X
Coventry X away
Coventry X home
Watford X home
Watford X away
58. My super banker this week 18 is 34xxx.
Swansea at week number, in week11 TJ10 played 32, in week16 TJ10 played 33, as in ascending order TJ10 must play number 34xxxBCK this week 18
59. Week 18 2023/24
Nottingham F vs Aston V
Starting from week 6 this season the previous blue week sat DOP is your sure bet.
Week 6 2023/24
Week 2 sat DOP=15X✓
Week 10 2023/24
Week 6 sat DOP=12X✓
Week 14 2023/24
Week 10 sat DOP=9X✓
Week 18 2023/24
Week 14 sat DOP=7X?
60. Week18
Last season
Week5 community shield on coupon play Swansea
Week18 fa cup on coupon play Swansea
This season week4 community shield on coupon play Swansea
Week18 fa cup play swansea
61. XXX 36 XXX
Aston v 2nd before bar away,game @ 12 home, minus it’s first alphabet from 49,answer to draw Crawley away…
WK 6,. XXX 40 XXX
Wk9,. XXX 37 XXX
Wk18,. XXX 36 XXX
62. (HSQ)=>ANYTIME U FIND FAMILY SERIES @ HSQ BOX 2&7,BANK ON BOX7 FOR FIXED DRAW
PROOF TWO
BOBMORTON:FABULOUS 16 GAME 3,4,5,6 RATED NAP-BET-HOT-NAP BANK ON GAME 3 RATED NAP AS A DRAW
PROOF THREE
(NAIJA)=>ANYTIME U FIND LETTER L AT NO4 AWAY,BANK ON CREWE
SINGLE BET THIS FIGURE WITH ANY AMOUNT
63. Week 18!!! Glory be to God for his mercies endureth forever.
❎19❎ Gazetted draw Cbk
Prove: Advance opponent of Huddersfield and Southampton to meet in Soccer Banker box to draw. Ref wk 17,wk 18
64. Week18
Banker **06**06**06.
The prove is from the word SHREWSBURY. Every Blue each letter produces a draw by working This wk18 is the turn of letter E. E = 5 Minus 5 from 49 then sum the answer together then minus 2 to
draw. 49-5=44 44=4+4=8 8-2=06***to draw.
65. Welcome to WK18. Make good use of x37x, proof- back page of capital where you have *x* banker of the week, you will see three that have two numbers that have family number, the odd one is your
banker, since the season.
66. WEEK18 XXXX 11 XXX
PROVE,.. in week 6 current season, take the number of the first alphabet in SAF LOTTO SEQUENCE, add it to week number to draw in the next blue week. Week6-10, 10-14, 14-18. Therefore, week14,
SDOP=7+4=11, SO 11 TO draw in week 18.
67. Welcome to wk. 18
One banker***6***
Prof: LUTON @ 2 AWAY, SHEFF UTD @ AWAY TO FAIL, IT’S OPP TO GO & DRAW NEWCASTLE FOLLOWING WOS. Note: Newcastle & Sheff UTD to share position 6/8 respectively in week of drawing. Ref: week 13/14&
17/18.BROWN to BLUE.
68. Welcome to wk.18 banker room. This wk. Play no.11xx as draw. PROOF: Birmingham at no.9 H/A count 3 down to draw. So play no.11xx as draw,ref. Wks 7,8 and 18 current. Have a nice weekend.
69. Xxxx47xxxx Banker .
Since wk15, count the letters of home team on top of TRY…
Take answer to TREBLE CHANCE 16 and bank on the number there to draw..
70. Week 18.
Arsenal vs Burnley
Bournemouth vs Newcastle
Bournemouth vs Burnley ff
Newcastle vs Arsenal @6xxx
06xcbk and 06xcbk
Good luck!
71. Welcome to week 18 F.A Cup
1st phase
Lincoln vs Cheltenham ff
Morecambe vs Wimbledon ff
Colchester vs Mansfield xx
Colchester vs Nottsco ff
2nd phase
Wimbledon vs Cheltenham??
Lincoln vs Morecambe xx
Nottsco vs Wrexham ff
Mansfield vs Wrexham xx
72. Week 18 bank on 15 pair 17.
Every blue 15 plus code 2 form a reliable pair.
73. NO14 PRV, WEN WESTBROM DRW ONTOP BAR, FOLOW WK PUT WATFORD ONTOP BAR. DE SAME WIT WATFORD, FRM THERE COUNT 6UP TO MEET OPONENT OF STOKE. ALSO COUNT 4UP OR DOWN FRM STOKE TO MEET IT PRVIOUS
OPONENT. SECONDLY. OPONENT OF STOKE TO DRAW WIT PLYMOUTH. TAKE OPONENT OF PLYMOUTH TO MEET BIRBINGHAM. RF WK12,13 17,18. CURENT
74. My banker is 38XXXX
PROOF, IN WK 9 BROWN LAST SEASON PORTSMOUTH VS PETERBOROUGH DIED THEN IN WK 18 SAME SEASON, PETERBOROUGH WENT TO EVEN NUMBER AND HOST SALFORD C AND DREW.
FO THE SAME THIS SEASON.
75. 01✓✓cbk banker
Prof open your record special advance fixtures in wk12 purple game number 4 of Treble Jinx crystal p vs Fulham drew, now in previous wk8 purple Fulham met Arsenal 1✓drew, while Crystal p met
Brentford 2✓drew 2/2
Now go to wk14 blue special advance fixtures, Treble Jinx game number 4 westham vs Newcastle drew, now come to advance this wk18 Westham will meet Brentford @ number 1✓✓cbk, while Newcastle will
meet Arsenal 6✓✓ as 2/2
Good luck to you all..ref wk12/8 14/18
76. Week18 special week
BRISTOL.C at no10 Home
Bank on WESTBROM to Draw.
Reff: week 11 2023/24
Week 16 2023/24
Week 18 2023/24
ADMIN PLEASE APPROVE for all
77. 24XXXXXBANKER
IF THE SECOND GAME UNDER GUESSING GAMES PAGE 4 IS SEEN AS THE SECOND GAME UNDER CURRENT AND UP-TO-DATE LEAGUE 2. MARK IT AS DRAW AND ALSO MARK THE TWO GAMES THERE.
WEEK 05, 2022
31XXX 37XXXBK
WEEK 18, 2023
18XXX 24XXXBK AS TWO BANKER
ALSO BOB MORTON
GAME 3 RATED NAP AND GAME 15 RATED XXCBK IF BOTH ARE HAVING A DIFFERENCE OF 2. LIKE MARK THEM FOR A DRAW.
WEEK 16, 2022
WEEK 17, 2022
WEEK 18, 2023
MARK 24XXXXBK
78. Week18 number one banker of da week is 3xxxxxxx
bet it & win big, prove of da number is fixed form BIGWIN FIXED ODDS, Game play since in week6 all blue weeks.. Win & enjoy we me…
79. 38xxx
Game 35 of Bob Morton full list rated 888 and dropped at game 15 of short list and rated 0-0 is a draw
Wks 4,10,18
80. Greetings…
Week 18
39 cbk
Capital, Backpage hot pair, when the up number is a family number with 1st game of Guessing Game, take as strong pair. 39✓ & 9 .ref. WK 27 last season.
GOODLUCK IN ADVANCE
81. Week 18 banker 16XXX
Starting from Week 15 2018/2019 add the first and last Letter of the opponent of Swansea for a fixed banker in current year
Swansea VS (A)ston (V). 1+22= 23
Week 15 2023/2024 current
Week 16 2018/2019
Swansea vs (R)eadin(g). 18+7= 25
Week 16 2023/2024 current
Week 17 2018/2019
Swansea vs (R)otherha(M). 18+ 13= 31
Week 17 2023/2024 current
Week 18 2018/2019
Swansea vs (B)olto(n). 2+14=16
Week 18 2023/2024 current
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The Future of Web Applications
is not an application most people have even heard of, let alone used, its latest incarnation is a prime example of how client-side applications can utilize more powerful machines elsewhere on the net
to do the heavy lifting while still offering a user experience every bit as rich as any desktop app.
For years,
Stephan Wolfram's
Mathematica has been the leading desktop application for conjuring all manner of mathematical wizardry. It does everything from factoring polynomials to graphing differential equations to your
freshman algebra homework. The only problem is that for more complex tasks, it can make thick black smoke spew from the ears of the average laptop. That means it won't do the college students much
good unless they happen to have a kick-ass rig at their disposal. The idea behind
is to bring the desktop version's true power to those without access to the necessary hardware.
But one might ask how an application that can generate thousands of floating-point-operations per second would possibly run smoothly inside a browser. The answer turns out to be: do the actual math
somewhere else.
Here's how it works. Set up your formulas, datasets and such, and when the calculations are ready to begin, all the info is sent via XMLHttpRequests to a server where all the actual math is forwarded
to machines specifically designed to handle the task. The response is then rendered via JavaScript, CSS and HTML.
Very cool.
No comments: | {"url":"http://blog.anotherreason.com/2008/02/future-of-web-applications.html","timestamp":"2024-11-06T04:24:54Z","content_type":"application/xhtml+xml","content_length":"29296","record_id":"<urn:uuid:802a0f46-97b0-4334-a332-622c6145a195>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00329.warc.gz"} |
The Impact Of Quantum Mechanics And General Relativity On Our View On Reality: Free Essay Example, 2010 words
The Impact Of Quantum Mechanics And General Relativity On Our View On Reality
Pages: 4 (2010 words)
Views: 1262
Grade: 5
Many revolutionary advancements have come about due to the immense research into quantum mechanics and general relativity, due to their vast influence on the world around us, on both the cosmic and
sub-atomic scale. In Newtonian mechanics, which is one of the concrete theories that we choose to be true, we describe the motion of objects using laws that relate to forces such as gravity, as well
as their certain position at a specific point in time. However, in quantum mechanics, we describe the position of an object (or in fact a subatomic particle) as having a probability of being at a
certain position at a specific point in time (which relates to its quantum states and superposition). Newtonian mechanics also uses the assumption that space and time are constant and separate to
each other, however, this is contradicting to general relativity which states that space and time are linked and that it can warp and bend. Although both of these theories (Quantum mechanics and
General relativity) are extremely adept at describing how our universe works, they both pose a major problem; that is that together they are fundamentally incompatible theories. This essay will be
discussing the advancement of our civilization concerning Quantum mechanics and General relativity, how they will allow us to challenge and further understand our reality, how they can be further
used towards the development as a technological civilization and why there is an incompatibility with both theories.
Reality is described as “existence that is absolute, self-sufficient, or objective, and not subject to human decisions or conventions.” That is to say, that what we consider reality is what is a
concrete fact of everything around us without any doubt. Reality to many people seems to be a straightforward topic from the outside; However, this seems to not be the case. We, as humanity, are yet
to come up with a unified explanation of what reality really is and as of yet, there is no concrete theory explaining what we believe it to be, which gets us to think whether reality is really what
we think it is.
What is Quantum Mechanics?
Quantum Mechanics is a field of physics that provides an explanation of the behavior and motion between quantum particles, or particles in the subatomic scale, with the use of advanced mathematics
and concepts such as wave-particle duality, the uncertainty principle, superposition (which closely links to wave-particle duality) and many more concepts. Describing the behavior of subatomic
particles in quantum mechanics is one thing, however, understanding its complexity is another. .Werner Heisenberg, a German theoretical physicist, who was awarded the Nobel Prize in Physics for “the
creation of quantum mechanics”, as well as being labeled one of the “Fathers of Quantum Mechanics” introduced the idea of the Heisenberg uncertainty principle in 1927. The uncertainty principle is a
fundamental limitation to the precision of observing the certain physical properties of a particle. These physical properties known as complementary variables (certain pairs of variables that can not
be observed at the same time) include the particle's position and its momentum. The idea states that the more precise the precision of observing the particle's position, the lower the precision of
knowing the particles momentum and vise versa.
The term observation doesn't only apply to the process in which a person observes the properties of said particle, but also the interaction between classical objects and quantum objects (for example
any measurement done to the particle). The quantum state of a particle is the probability that the particle will be in at a point in space at a certain time. If no observation is made on the
particle, it will then be in a superposition of being at that point in space and being at another point in space. In the double slit experiment, this concept is demonstrated when, beams of electrons
are fired through the two slits, resulting in an interference pattern on the other side. This is because as the electron moves through the slits, it goes into a state of superposition where it goes
through one slit, it goes through the other slit, or it goes through both. Since electrons and other quantum particles have a wave function and hold wave-like properties (in the assumption that it is
not observed), the electron can now interfere with itself in the same way that waves do in the experiment. If the electron, however, is observed as it goes through the two slits, its wave function
collapses and returns to behaving like what we expect a particle to behave like and leave two identical slits. This experiment closely links to the observer effect. “The observer effect, a theory
that states that the observation of a phenomenon changes that phenomenon”, this, in the context of quantum mechanics, implies that once the quantum states of a particle is observed, it is no longer
in a state of superposition, leading it to continue behaving like a solid physical particle with a mass, volume, and density.
Quantum entanglement is a phenomenon in which a pair or a group of subatomic particles interact with each other in a way in which they become linked such that if one of the particles’ quantum states
are observed, you can determine the quantum state of the other particle(s) since they are now dependent on their own quantum states. However, despite it being at the quantum level, the distance of
these particles does not affect their quantum entanglement.
Advancements in Quantum Mechanics
A major advancement that came about due to Quantum mechanics is “Quantum Computers”. In an article written by Jason Rowell, it is stated that in order to understand quantum computing, we must first
have to understand the concept of the quantum and logic gates. Logic gates are any physical structure or system that take binary inputs (0 and 1) and output only one of those inputs in depending on
the function the system hasThese systems are then implemented into circuits to make computational components and make the computers we use today. Normal logic gates used in traditional computers work
with bits, which is a unit of information which holds either the binary values 0 or 1. On the contrary, Quantum gates, with the use of qubits (or quantum bits), which, similarly to bits, is a unit of
quantum information to hold values at the same time in a state of superposition. This in hand, then allows the ability to compute many calculations at once, opposed to a normal system, that only
allows the computation of one calculation at a time.
“The D-wave” is a quantum computing company that is in the development of quantum computers. The D-wave system contains a GPU that is kept at temperatures near absolute 0 and shielding from
electromagnetic interference so that there is no interference with the GPU. The D-Wave QPU in a lattice of tiny metal loops, which is either a qubit or a coupler. At temperatures below 9.2K, these
loops become superconductors and then start have properties of quantum mechanics.
What is General Relativity? Similarly to Quantum mechanics, General relativity is a theory in physics that describes the laws of gravity and its relation to the forces of nature. It explains that
what we perceive as the force of gravity, in fact, arises from the curvature of space and time under “spacetime”. Spacetime is the concept that our three-dimensional space is fused with time under a
fourth-dimensional continuum. Since we cannot visualize the fourth dimension, an analogy may be used to understand this fourth-dimensional continuum, when we compare it to a two-dimensional rubber
sheet. This then allows us to describe how objects with mass can warp the sheet and distort its curvature. This warping of spacetime is what we call gravity. This general theory of relativity was
proposed by Albert Einstein in 1915, since there were fluctuations in the previous description of gravity, of it being “a force which causes any two bodies to be attracted to each other, with a force
proportional to the product of their masses and inversely proportional to the square of the distance between them.”
Advancements in General Relativity
The closer you get to another object with a big mass, say for example a black hole, the more spacetime is warped around you and therefore in your reference frame, time seems to have slowed down. This
concept of the warping of spacetime to slow down time relative to you has implications of time travel or time manipulation. Steven Hawking, who was a physicist wrote a quote in the daily mail in 2010
that states “round and around they'd go, experiencing just half the time of everyone far away from the black hole. The ship and its crew would be traveling through time...Imagine they circled the
black hole for five of their years. Ten years would pass elsewhere. When they got home, everyone on Earth would have aged five years more than they had.” This implies that for you, whilst time seems
to be going by normally, it actually is different relative to someone at a different point in spacetime where spacetime is warped differently. Another advancement that could one day be used in the
future is by using the idea of the “Alcubierre warp drive”. This idea proposed by the Mexican theoretical physicist Miguel Alcubierre highlights a possible solution for apparent faster than light
travel. Instead of exceeding the speed of light, a spacecraft would travel cosmic distances by contracting space in front of it and expanding space behind it, which in essence goes around the light
speed limit of the universe. This is because instead of the spacecraft accelerating to the speed of light within spacetime, the warp drive shifts space around the spacecraft so that it will arrive at
its destination faster than light would normally without breaking physical laws.
The opposing views on reality
There have been attempts to unifying quantum mechanics and general relativity. One of those which include string theory, which is a very complex field of physics, that in layman terms, is the
replacement of the fundamental particles with one-dimensional objects known as “strings”. These strings vibrate at specific frequencies that correspond to their particle. In string theory, one of the
vibrational strings corresponds to a graviton which is a quantum mechanical particle that carries gravitational force. Proving this theory would allow us to finally link quantum mechanics and general
relativity via a unified theory. As of now, this seems impossible due to the complexity of string theory, having to involve 11 dimensions. Maybe one day there will be a unified theory of everything
where we will no longer question our reality for what it is, but as of now, reality will remain questioned.
1. Chegg Inc.| What is the main difference between Newtonian mechanics and quantum mechanics? https://www.chegg.com/homework-help/questions-and-answers/
main-difference-newtonian-mechanics-quantum-mechanics-energy-conserved-newtonian-mechanics-q5657069 | No date | Retrieved 2019-06-12
2. Oxford Dictionaries | English. Retrieved 2019-06-12.
3. The Nobel Prize in Physics 1932 | NobelPrize.org. | Nobel Media AB 2019. | https://www.nobelprize.org/prizes/physics/1932/summary/ | Date: 1933(year) | Retrieved 2019-06-12
4. Heisenberg, W. (1927), 'Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik', Zeitschrift für Physik (in German), 43 (3–4): 172–198, https://link.springer.com/article/
10.1007%2FBF01397280 |date: 1927-03-21| Retrieved 2019-06-12
5. http://faculty.uncfsu.edu/edent/Observation.pdf | Retrieved 2019-06-12
6. 'Observer effect (physics),' Wikipedia, The Free Encyclopedia https://en.wikipedia.org/w/index.php?title=Observer_effect_(physics)&oldid=898544075 | date: Unknown |Retrieved 2019-06-12
7. Towards Data Science | Demystifying Quantum Gates -- One Qubit At A Time | https://towardsdatascience.com/demystifying-quantum-gates-one-qubit-at-a-time-54404ed80640 | date: 2018-02-26 |
Retrieved 2019-06-12
8. Kevin Bonsor & Jonathan Strickland “How Quantum Computers Work” HowStuffWorks.com. |date: 2000-12-08 | Retrieved 2019-06-12|
9. “Welcome to D-wave” | D-wave Systems inc.| https://docs.dwavesys.com/docs/latest/c_gs_1.html#qpu1 | date: unknown| Retrieved 2019-06-12
10. Originally published in London Times| Einstein, Albert, 'Time, Space, and Gravitation' | available at https://en.wikisource.org/wiki/Time,_Space,_and_Gravitation|date: 1919-11-28 | Retrieved
11. 'Spacetime” | Wikipedia, The Free Encyclopedia | available at https://en.wikipedia.org/w/index.php?title=Spacetime&oldid=901476845 | Retrieved 2019-06-12
12. 'Gravity” | Wikipedia, The Free Encyclopedia | available at https://en.wikipedia.org/w/index.php?title=Gravity&oldid=900187176 | Retrieved 2019-06-12
13. ”Steven Hawking: How to build a time machine.” | Daily Mail | available at https://www.dailymail.co.uk/home/moslive/article-1269288/STEPHEN-HAWKING-How-build-time-machine.html | date: 2010-04-27
| Retrieved 2019-06-12
14. 'Alcubierre drive' | Wikipedia, The Free Encyclopedia | available at https://en.wikipedia.org/w/index.php?title=Alcubierre_drive&oldid=896858367 | Retrieved 2019-06-12
15. Krasnikov, S. | 'The quantum inequalities do not forbid spacetime shortcuts' | available at https://journals.aps.org/prd/abstract/10.1103/PhysRevD.67.104013 | date: 2003 (year) | Retrieved
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Converting 82 Fahrenheit to Celsius: Quick and Easy Guide - Holland Vegas
Categories General
Converting 82 Fahrenheit to Celsius: Quick and Easy Guide
The Fahrenheit and Celsius scales are two different units of temperature measurement. The Fahrenheit scale was developed by Daniel Gabriel Fahrenheit in the early 18th century and is commonly used in
the United States. The Celsius scale, on the other hand, was developed by Anders Celsius in the mid-18th century and is used in most other countries around the world. The main difference between the
two scales is the reference points for freezing and boiling water. On the Fahrenheit scale, water freezes at 32 degrees and boils at 212 degrees, while on the Celsius scale, water freezes at 0
degrees and boils at 100 degrees. This means that the two scales have different intervals between each degree, with the Fahrenheit scale having smaller intervals than the Celsius scale.
The Fahrenheit scale is often used for everyday temperature measurements in the United States, while the Celsius scale is used in most other countries. Understanding both scales is important for
anyone who needs to work with temperature measurements, as it allows for easy conversion between the two systems. This is especially important for scientists, engineers, and anyone working in
international settings where both scales may be used. Additionally, understanding both scales can also be helpful for travelers who may need to convert temperatures from one system to another while
Key Takeaways
• Fahrenheit and Celsius are two different temperature scales used to measure temperature.
• The formula for converting Fahrenheit to Celsius is (°F – 32) x 5/9.
• To convert 82°F to °C, subtract 32 from 82 and then multiply the result by 5/9.
• Common temperature conversions include 32°F to 0°C, 212°F to 100°C, and 98.6°F to 37°C.
• To estimate Celsius temperatures from Fahrenheit, subtract 30 and then divide by 2.
• Knowing both Fahrenheit and Celsius is important for understanding weather forecasts and international travel.
• Understanding how to convert temperatures benefits individuals by providing a better understanding of weather, cooking, and travel.
The formula for converting Fahrenheit to Celsius
The formula for converting a temperature from Fahrenheit to Celsius is relatively simple and involves a straightforward mathematical calculation. To convert a temperature from Fahrenheit to Celsius,
you can use the following formula: (°F – 32) x 5/9 = °In this formula, °F represents the temperature in degrees Fahrenheit, and °C represents the temperature in degrees Celsius. To use the formula,
simply subtract 32 from the temperature in Fahrenheit and then multiply the result by 5/9 to get the temperature in Celsius. This formula allows for an accurate and precise conversion between the two
temperature scales.
Converting temperatures from Fahrenheit to Celsius can be useful in a variety of situations, such as when working with international colleagues or when traveling to countries that use the Celsius
scale. By understanding and using this simple formula, anyone can easily convert temperatures between the two systems without the need for complex calculations or specialized equipment. This can be
especially helpful for professionals who work with temperature measurements on a regular basis and need to quickly and accurately convert between different units of measurement.
Step-by-step guide for converting 82°F to °C
To convert a temperature of 82 degrees Fahrenheit to degrees Celsius, you can use the formula (°F – 32) x 5/9 = °First, subtract 32 from 82 to get 50. Then, multiply 50 by 5/9 to get the temperature
in Celsius. This gives you a result of approximately 27.8 degrees Celsius. Therefore, 82 degrees Fahrenheit is equivalent to approximately 27.8 degrees Celsius.
Another way to convert 82 degrees Fahrenheit to degrees Celsius is to use an online temperature conversion tool or a calculator that has a built-in conversion function. Simply input the temperature
in Fahrenheit and press the convert button to get the equivalent temperature in Celsius. This can be a quick and easy way to convert temperatures without having to perform manual calculations.
Common temperature conversions
Celsius (°C) Fahrenheit (°F) Kelvin (K)
0 32 273.15
25 77 298.15
100 212 373.15
There are several common temperature conversions that are frequently used in everyday life and professional settings. One of the most common conversions is between freezing and boiling points of
water on the Fahrenheit and Celsius scales. On the Fahrenheit scale, water freezes at 32 degrees and boils at 212 degrees, while on the Celsius scale, water freezes at 0 degrees and boils at 100
degrees. Another common conversion is between body temperatures, with a normal body temperature being approximately 98.6 degrees Fahrenheit or 37 degrees Celsius.
In addition to these common conversions, there are many other situations where it may be necessary to convert temperatures between Fahrenheit and Celsius. For example, when cooking or baking recipes
from different countries, it may be necessary to convert oven temperatures from one system to another. Similarly, when traveling to different countries, it may be necessary to convert outdoor
temperatures from one system to another in order to better understand the local climate.
Tips for estimating Celsius temperatures from Fahrenheit
Estimating Celsius temperatures from Fahrenheit can be a useful skill for anyone who needs to quickly convert temperatures between the two systems without performing precise calculations. One simple
tip for estimating Celsius temperatures from Fahrenheit is to subtract 30 from the temperature in Fahrenheit and then divide by 2. This will give you a rough estimate of the equivalent temperature in
Celsius. For example, if you have a temperature of 86 degrees Fahrenheit, you can estimate the equivalent temperature in Celsius by subtracting 30 to get 56 and then dividing by 2 to get
approximately 28 degrees Celsius.
Another tip for estimating Celsius temperatures from Fahrenheit is to use a rough conversion factor of 2. For example, if you have a temperature of 68 degrees Fahrenheit, you can estimate the
equivalent temperature in Celsius by dividing by 2 to get approximately 34 degrees Celsius. While these estimation methods may not provide precise results, they can be helpful for quickly getting a
rough idea of the equivalent temperature in Celsius without having to perform detailed calculations.
Importance of knowing both Fahrenheit and Celsius
Knowing how to convert temperatures between Fahrenheit and Celsius is important for a variety of reasons. For professionals who work with temperature measurements on a regular basis, such as
scientists, engineers, and medical professionals, understanding both systems is essential for working with international colleagues and clients who may use different units of measurement.
Additionally, knowing how to convert temperatures between Fahrenheit and Celsius can be helpful for travelers who need to quickly understand local weather forecasts and outdoor temperatures while
Understanding both systems also allows for greater flexibility when working with temperature measurements in different contexts. For example, when working with data from different sources that use
different units of measurement, being able to easily convert between Fahrenheit and Celsius can help ensure that all data is consistent and comparable. This can be especially important in scientific
research and engineering projects where accurate and precise measurements are essential.
Benefits of knowing how to convert temperatures
In conclusion, knowing how to convert temperatures between Fahrenheit and Celsius is an important skill that can be useful in a variety of situations. Whether you are a professional working with
international colleagues or clients, a traveler exploring new destinations, or simply someone who wants to better understand temperature measurements from around the world, understanding both systems
is essential. By using simple formulas and estimation methods, anyone can easily convert temperatures between Fahrenheit and Celsius without the need for complex calculations or specialized
equipment. This knowledge allows for greater flexibility and accuracy when working with temperature measurements and ensures that data from different sources can be easily compared and understood.
Overall, knowing how to convert temperatures between Fahrenheit and Celsius is a valuable skill that can benefit anyone who works with temperature measurements on a regular basis.
If you’re looking to convert 82 degrees Fahrenheit to Celsius, you can check out the article on Holland Vegas that provides a helpful temperature conversion chart. This article offers a simple and
easy-to-use guide for converting temperatures between Fahrenheit and Celsius, making it a useful resource for anyone needing to make this type of conversion. You can find the article here.
What is the formula to convert Fahrenheit to Celsius?
The formula to convert Fahrenheit to Celsius is: (Fahrenheit – 32) x 5/9.
What is 82 degrees Fahrenheit in Celsius?
82 degrees Fahrenheit is equal to 27.8 degrees Celsius.
Why do we need to convert Fahrenheit to Celsius?
Converting Fahrenheit to Celsius is necessary when working with temperature measurements in different units, especially in scientific, academic, or international contexts.
What are the freezing and boiling points of water in Celsius and Fahrenheit?
The freezing point of water is 0 degrees Celsius and 32 degrees Fahrenheit, while the boiling point of water is 100 degrees Celsius and 212 degrees Fahrenheit.
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Latest math trivia
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Author Message
EonL Posted: Wednesday 17th of Apr 12:30
hi Gals I really hope some math master reads this. I am stuck on this assignment that I have to take in the coming week and I can’t seem to find a way to finish it. You see, my
professor has given us this test covering latest math trivia, interval notation and proportions and I just can’t make head or tail out of it. I am thinking of paying someone to help
me solve it. If someone can show me how to do it, I will be obliged.
From: Southern
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espinxh Posted: Thursday 18th of Apr 16:03
I have a good idea that could help you with math . You simply need a good software to explain the problems that are complicated. You don't need a tutor , because firstly it's very
costly , and secondly you won't have it near you whenever you need help. A software is better because you only have to get it once, and it's yours for all time. I recommend you to
check out Algebrator, because it's the best. Since it can resolve almost any math exercises, you will probably use it for a very long time, just like I did. I purchased it a long
time ago when I was in Remedial Algebra, but I still use it sometimes .
From: Norway
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Momepi Posted: Friday 19th of Apr 15:46
I didn’t encounter that Algebrator software yet but I heard from my classmates that it really does assist in answering algebra problems. Since then, I noticed that my friends don’t
really have troubles answering some of the problems in class. It might really have been efficient in improving their solving abilities in math . I can’t wait to use it someday
because I think it can be very effective and help me have a good grade in math .
From: Ireland
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mlmcclacker Posted: Saturday 20th of Apr 14:55
Hello again. Thanks a lot for the beneficial advice. I usually never trust math tools ; however, this piece of program seems worth trying. Can I get a URL to it?
From: New Haven,
CT, USA
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Noddzj99 Posted: Sunday 21st of Apr 20:42
I advise trying out Algebrator. It not only assists you with your math problems, but also displays all the required steps in detail so that you can improve the understanding of the
From: the 11th
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SanG Posted: Tuesday 23rd of Apr 08:07
Accessing the program is easy . All you desire to know about it is accessible at https://softmath.com/algebra-policy.html. You are assured satisfaction. And in addition , there is a
money-back guarantee. Hope this is the end of your search .
From: Beautiful
Northwest Lower
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230-0221/01 – Repetition of Mathematics 1 (Repet 1)
Gurantor department Department of Mathematics Credits 1
Subject guarantor RNDr. Petr Volný, Ph.D. Subject version guarantor RNDr. Petr Volný, Ph.D.
Study level undergraduate or graduate Requirement Optional
Year 1 Semester summer
Study language Czech
Year of introduction 2018/2019 Year of cancellation
Intended for the faculties FAST Intended for study types Bachelor
DUB02 RNDr. Viktor Dubovský, Ph.D.
PAL39 RNDr. Radomír Paláček, Ph.D.
STA50 RNDr. Jana Staňková, Ph.D.
VIT0060 Mgr. Aleš Vítek, Ph.D.
VOL18 RNDr. Jana Volná, Ph.D.
VOL06 RNDr. Petr Volný, Ph.D.
Full-time Credit 0+2
Part-time Credit 0+16
Subject aims expressed by acquired skills and competences
Goals and competence Mathematics is an essential part of education on technical universities. It should be considered rather the method in the study of technical courses than a goal. Thus the goal of
mathematics is train logical reasoning than mere list of mathematical notions, algorithms and methods. Students should learn how to analyse problems, distinguish between important and unimportant,
suggest a method of solution, verify each step of a method, generalize achieved results, analyse correctness of achieved results with respect to given conditions, apply these methods while solving
technical problems, understand that mathematical methods and theoretical advancements outreach the field mathematics.
Teaching methods
Repetition of Mathematics 1 is intended for students who, for whatever reasons, fail the exam of Mathematics I and are interested in passing this exam. Its content essentially coincides with the
content of the course Mathematics I. The aim is to enable better understanding of mathematics by the solving of concrete examples and problems. Repetition will focus on the practical part of the exam
and they will be solved examples matching the written part of the exam.
Compulsory literature:
Recommended literature:
Harshbarger, Ronald; Reynolds, James: Calculus with Applications, D.C. Heath and Company 1990,
ISBN 0-669-21145-1
Way of continuous check of knowledge in the course of semester
Other requirements
There are no additional requirements.
Subject has no prerequisities.
Subject has no co-requisities.
Subject syllabus:
Syllabus of tutorial 1. Domain of a real function of one real variable. 2. Bounded function, monotonic functions, even, odd and periodic functions. 3. One-to-one functions, inverse and composite
functions. Elementary functions. 4. Inverse trigonometric functions. Limit of functions. 5. Derivative and differential of functions. 6. l’Hospital rule. Monotonic functions, extrema of functions. 7.
Concave up function, concave down function, inflection point. 8. Asymptotes. Course of a function. 9. Matrix operations. 10. Elementary row operations, rank of a matrix, inverse. 11. Determinants.
12. Solution of systems of linear equations. Gaussian elimination algorithm. 13. Analytic geometry. 14. Reserve.
Conditions for subject completion
Conditions for completion are defined only for particular subject version and form of study
Occurrence in study plans
2022/2023 (B0731A010004) Architecture and Construction P Czech Ostrava 1 Optional study plan
2022/2023 (B0732A260001) Civil Engineering K Czech Ostrava 1 Optional study plan
2022/2023 (B0732A260001) Civil Engineering P Czech Ostrava 1 Optional study plan
2021/2022 (B0731A010004) Architecture and Construction P Czech Ostrava 1 Optional study plan
2021/2022 (B0732A260001) Civil Engineering P Czech Ostrava 1 Optional study plan
2021/2022 (B0732A260001) Civil Engineering K Czech Ostrava 1 Optional study plan
2020/2021 (B3607) Civil Engineering K Czech Ostrava 1 Optional study plan
2020/2021 (B0731A010004) Architecture and Construction P Czech Ostrava 1 Optional study plan
2020/2021 (B0732A260001) Civil Engineering P Czech Ostrava 1 Optional study plan
2020/2021 (B0732A260001) Civil Engineering K Czech Ostrava 1 Optional study plan
2019/2020 (B0731A010004) Architecture and Construction P Czech Ostrava 1 Optional study plan
2019/2020 (B0732A260001) Civil Engineering P Czech Ostrava 1 Optional study plan
2019/2020 (B0732A260001) Civil Engineering K Czech Ostrava 1 Optional study plan
2018/2019 (B3502) Architecture and Construction (3501R011) Architecture and Construction P Czech Ostrava 1 Optional study plan
2018/2019 (B3607) Civil Engineering K Czech Ostrava 1 Optional study plan
2018/2019 (B3607) Civil Engineering P Czech Ostrava 1 Optional study plan
Occurrence in special blocks
Assessment of instruction | {"url":"https://edison.sso.vsb.cz/cz.vsb.edison.edu.study.prepare.web/SubjectVersion.faces?version=230-0221/01&subjectBlockAssignmentId=405538&studyFormId=2&studyPlanId=22784&locale=en&back=true","timestamp":"2024-11-03T00:18:59Z","content_type":"application/xhtml+xml","content_length":"180324","record_id":"<urn:uuid:cea7b56c-46b7-4324-b696-ee722779b34f>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00103.warc.gz"} |
Darts Rules – Learn How to Play Darts - Darts Award
Darts Rules – Learn How to Play Darts
Game of darts is a popular game, but it might get confusing sometimes because of darts rules. If you feel mixing up between the darts scoring rules, here is a complete guide for you. Darts rules are
defined by the Professional Darts Corporation, which can be applied to the game when there is confusion or doubt. Players usually confuse between 301 and 501 versions.
501 is a standard version of dartboard that is played widely. Dartboard comes in different shapes and sizes, but the darts rules are the same for each size. In this review, I will discuss general to
specific game rules that will help you quickly understand this game. At the end of this article, you will gain enough knowledge about this game, so gather some friends and impress them with your
understanding of dartboard game playing rules.
Main Rules of Darts
General Rules
Dartboard comes in many different shapes and sizes, but each dartboard has the same scoring no matter what shape and size it has. The dartboard is divided into a bullseye and 20 number segments. The
scoring area for each numbered segment has three regions: two single, one triple, and one double. There are numbers allocated to a bullseye. The outer ring of the bullseye is known as a single bull
with 25 points, whereas the inner circle has 50 points and is called a double bull.
If I talk about the height of the dartboard and the distance of players, you must hang the dartboard in the center with 68 inches high above the ground. The length of the player from the dartboard
must be 93 ½ inches. To select the first player, each player throws one dart at the bullseye. The player closest to the dart wins and begins the game. You can change the way of selecting a first
player by using toss or coin.
With each turn, every player throws three darts. Each dart that hits the dartboard will receive a score. The dart that falls off or bounces out of the dartboard will have no score. The score of each
dart is compared with the total score of three darts. Furthermore, you will not receive a score if the dart sticks in another dart.
Specific Darts Game Rules
1. 301 Rules
The darts have some popular games. 501 and 301 are the simplest games that players prefer to play. In the 301 games, there are two players, or you can also play with two teams. You can dart on any
number, but numbers 19 and 20 will quickly reduce your score to zero. You will be given 301 points in the 301 version, and your task is to reduce the points to zero. Whatever point you score will be
subtracted from your total points, the more you can reach zero quickly. Your winning chances will be more.
2. 501 Rules
The rules are the same for 501 games like 301. You will be given 501 points, and your goal will be to reach zero quickly. The score of the darts will be deducted from your total points. The player
who reaches zero first will win the game. It is important to note that you will hit the exact number to get zero. For example, your remaining score is 15 and your throw a dart at 20, the score will
not count, and it will remain 15 until a successful hit.
3. Round The Board
Two players are required to play this game. In this game, the player’s goal is to hit between numbers from 1 to 20 in a sequence. Each throw should be hit to increase the previous score. You can hit
single, double, or triple. Each player will throw three darts, and then the dart goes to another player. The player who quickly scores 20 is a winner.
4. Killer
In this game, two players are required, but you can increase the number of players for the more exciting game. The players randomly select the number in a play. The task is to throw a dart with your
opposite hand to get a score. If the dart sticks to another dart or misses the board, you will receive no score. Each time, a player has to throw a dart to get double the number he has selected.
This move is known as a killer, and the letter “k” is placed on the scoreboard next to the player’s name. The opponent’s task is to double the score from a killer. The player will lose their life if
the killer scores more than him. The player who scores more than other players and is standing with life will be considered the winner. In this game, each player is given three lives.
5. Cricket
You can play this game with two players or two teams. In this game, all numbers are in play, but a score of 40 is needed to win the game. You can also score 20, which is fine as well. In cricket, 10
points are marked on the scoreboard. These points are referred to as wickets. Same as cricket, in this game, one will be a batsman, and the other will be a bowler. The first player in the game is the
If you are a bowler, your goal is to hit the bullseye to remove the wickets one by one. The hit you throw at a bullseye will erase one wicket. The batsman’s task is to score as much as he can while
wickets in hand. A score of more than 40 will be counted as a run. For example, if you score 36, you will receive no score. If you score 47, you will receive 5 points.
The game continues till the bowler takes all the wickets. The score of the batsman is recorded, and the roles are swiped. The bowler will now be a batsman, and the batsman will be a bowler. The game
continues. The player with a high score or run as a batsman wins the game.
Frequently Asked Questions
What are the points on the dartboard?
On the dartboard, there are 20 number segments drawn. Each part is divided into four scoring points. The darts scoring points are two single, one double, and one triple. The outer ring of a bullseye
has a single bull with 25 points, whereas the inner ring is known as a double bull and will be scored as 50. The more points a player scores, the more will be the chances of a win.
What are the scoring rules for the 501 games?
In the 501 version of the dartboard, you will be given a total of 501 points. Your goal will be to reduce this score to reach zero. The player who reaches zero first will be considered a winner. Each
player has three turns to throw a dart. The score of each dart will be deducted from the player’s point. A score over zero will not be considered, and the player has to throw a dart successfully to
get precisely zero.
What is the general rule to select the first player?
In dartboard, the player is selected by giving each player one dart. Each player will throw the dart one by one. The player whose dart is closest to the bullseye will win and be considered the game’s
first player. The criteria of selecting a dartboard first player can be changed depending upon the player’s choice. The above-discussed rule is a standard rule described by professional Darts
Corporation, but you can use a coin or a toss for selection.
We usually see dartboard tournaments and enjoy the game. But we can get confused sometimes with the scoring dartboard rules while watching a game. So here is a review that will help you know about
the scoring criteria of different games on the dartboard. There is a comprehensive option of games, but I have discussed a few popular games on the dartboard. | {"url":"https://dartsaward.com/darts-rules/","timestamp":"2024-11-13T04:39:56Z","content_type":"text/html","content_length":"67993","record_id":"<urn:uuid:bf142153-524a-4303-8a88-70231230f7b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00504.warc.gz"} |
PERT Math – Test Day Tips
The Postsecondary Education Readiness Test, or known as PERT, is a test to determine the appropriate level of college work for an incoming student. In essence, this is a comprehensive and rapid
assessment of students’ academic abilities. There are three separate parts to the PERT test:
The PERT test is a Computer Adaptive Test. It means that if the correct answer is selected, the next question will be more difficult. And if the answer is incorrect, the next question will be easier.
You have prepared for the PERT test by planning and studying. Now is the day for the exam. Have the best performance on the day of the test by reading these important tips and strategies.
The Absolute Best Book to Ace the PERT Mathematics Test
The Night before the Test!
Give your brain a break
The day before the PERT math test, do not study any new math that you have not read before. If you want to study the day before the exam, just review the tips and formulas you have already written
down. It is best to take a break the day before the PERT math test to have a great exam day.
Eat a healthy dinner
The food you eat the night before the test should be nutritious and healthy. Eat nutritious meals that include complex carbohydrates, like potatoes, rice, and pasta, as well as vegetables and
protein-rich foods.
Prepare your transportation plan
Make sure you know how to get to the test center. Find out the route, parking details, and subway, bus, or train schedule.
Prepare the test equipment
Prepare the equipment needed for the PERT test the day or night before the test and place it in a convenient place so that you can easily find it. Avoid taking unnecessary items like books in the
exam session.
Get a good night sleep
To control any problems you may have with your nerves, go to bed early the night before the test. Plan to get seven to eight hours of sleep and wake up at least an hour before the PERT test. That
way, your brain is alert until you get to the test center.
Best PERT Math Prep Resource for 2022
On Test Day!
Wake up early
On exam morning, wake up early! It will give you plenty of time to eat breakfast and get ready. In this case, you do not need to hurry, and this can calm your mind. Having a calm mind plays an
important role in your success and increasing your productivity.
Have a nutritious breakfast
Eat a balanced breakfast. Try to eat breakfast as light and healthy as possible. Avoid foods like cream or anything that changes your eating habits as much as possible. This way, you will not have
any problems during the PERT test.
Wear comfortable clothes
Choose clothes that you are comfortable with. With this in mind, it will allow you to sit in a chair comfortably for a long time and answer the PERT math questions without getting tired.
Arrive at the test center early
Arrive early for the test session and be present half an hour before the test begins. You always have to anticipate the unexpected. Being late increases your stress and this can reduce your good
Repeat the positive affirmations
You have prepared hard for this test, and you are ready! Avoid any negative talk (for example, I cannot do that). Remind yourself that you are ready for the PERT math and will do your best.
The Best PERT Math Quick Study Guide
During the Test!
Keep calm during the test
Keep calm in any situation. You have worked hard to prepare for the PERT math test before the exam starts. So calm down and let your mind work rightly, and do not be afraid of anything. If the exam
is difficult, the conditions are the same for the other participants in the exam.
Read the math questions carefully
Try not to answer the question directly, first take a few moments to understand the information! Review the question at least twice and emphasize relevant information that may be helpful.
Do not rush to answer questions
You have unlimited time to answer PERT questions. So do not rush to answer the questions and take the time to answer them.
Back solving
All questions are presented in multiple-choice, so the correct answer is provided for each question. If you get stuck, you can use test answers to solve math problems to see which answer fits the
Review your answers
Review the answer to each question before submitting it. Because you cannot change your answer after entering the answer.
After the Test!
After the test, rest and relax a bit. Do not focus on what you have to say, do or write. You cannot change the decisions you made during the PERT, so accept it. During the day, spend your time and
energy on activities that keep you happy and entertained.
Looking for the best resources to help you or your student succeed on the PERT test?
The Best Book to Ace the PERT Test
More from Effortless Math for PERT Test …
Want to learn the PERT math test at home and need the best study guide?
Top 5 PERT Math Study Guides will introduce you to the best study guide on the market
Have you tried different PERT math courses and you are still not satisfied?
The Ultimate PERT Math Course (+ FREE Worksheets & Tests) is what will satisfy you!
Do not remember the math formulas of the PERT test and therefore worry?
We address your concerns: PERT Math Formulas
The Perfect Prep Books for the PERT Math Test
Have any questions about the PERT Test?
Write your questions about the PERT or any other topics below and we’ll reply!
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How to visualise your data: distribution charts - Culture Counts
What are you trying to learn from your data? In this series, Culture Counts’ Data Scientist, Tom McKenzie shares how applying the ‘so, what?’ principle can help guide you to creating meaningful chart
This practical how-to demonstrates some common chart types and why they are useful for answering different questions and highlighting different insights. First up: charts for displaying how data is
The ‘so, what?’ principle
The ‘so, what?’ is what you are actually hoping to get out of a report, piece of analysis, or collection of data and can usually be structured into a hierarchy of increasing specificity. You can read
more about the ‘so, what?’ principle here.
Creating an effective chart involves a lot of choices and decisions – which chart type should I use? How should the categories be ordered? Would colour increase the readability? The ‘so, what?’
principle can be used to guide you through this process to (hopefully!) end up with a chart that is fit for purpose.
In this post we will look at one aspect of data that is a very common starting point for any analysis, that is: “what does my data actually look like?”
Applying the ‘so, what?’
When collecting numerical data, such as the amount spent at an event (in $) or a respondents age, the first way we should look at this data is to look at the distribution of values. There are many
different chart types for visualising distributions each with slightly different strengths. Some of our favourites are shown in the figure below.
These four charts can be further divided in terms of the number of variables they are displaying. The top two (histograms and density plots) show data from only a single variable, while the bottom
two (box plot and strip plot) are for displaying two variables, one of them being a numerical value (this is the variable whose “distribution” we are interested in) and the other being a categorical
variable (which is plotted on the vertical axis in the figure). For example, consider the example data shown below in a typical spreadsheet format. On the left, we have only the numerical data, which
we could visualise using a histogram or density plot. On the right, we have two variables – one numerical and one categorical – for which we might choose instead to visualise using a box plot or
strip plot to get information about the distribution of incomes for respondents from each city in the dataset.
Let’s go a bit deeper into each of these chart types and discuss the reasons when and why you would want to choose each one.
The Workhorse: Histogram
Histograms are one of the oldest and still most frequently used chart types. They are the go-to choice for any data analyst looking to know more about the data they are working with, as they provide
a flexible and customisable way of looking at the raw values.
A histogram displays the count, or frequency, of values for a series of sequential “bins”. Each bar in a histogram represents one bin, with its height being proportional to the count. The power of
histograms comes from being able to tune the size of this “bin” so that we can go from a very detailed, fine-grained view of the distribution for small bin sizes, up to a more general and broad view
for large bin sizes.
As a concrete example of the “bin” concept, let’s say we have our collection of data about the income of each respondent in a dataset. A small bin size for displaying this data in a histogram might
be increments of $500. So the first bar on the left hand side of the histogram would be the count of all respondents whose income was in the $0 – $500 range, while the second bar would be the count
for those in the $500 – $1000 range, and so on. Conversely, a large bin size for this data might be $10,000 increments. In this case (for reference, looking at the “large” bin size histogram in the
figure) the first bar would be the count of values in the $0 – $10,000 range, with the second bar representing those in the $10,000 – $20,000 range.
When to choose a histogram
• Have collected a set of numerical data
• Want investigate how the values are distributed in detail
• Tune the “bin size” to get different levels of detail
• Can quickly identify potential outliers in the data
• Widely used chart type – relatively familiar to most viewers
The Smooth Operator: Density Plot
Density plots look pretty cool, all smooth lines and shaded area! The way that density plot are constructed is outside of the scope of this blog post, but just know that they are a close cousin of
the histogram.
Like the histogram the range of the variable you are investigating is on the x-axis, while the height represents the relative frequency at each value in this range. Also similar to the histogram,
density plots have a parameter that can be used to tune how fine-grained the level of detail, or “smoothing”, in the density plot is, this is called the “bandwidth”.
I like density plots because they strip away a lot of the distractions of other chart types, allowing you to quickly get a “feel” for how your data is distributed, including identifying the modality
(how many peaks there are) and spread (how wide or narrow the range of values are). They can also add some “chart variety” to your reports or presentations so that the audience isn’t looking at page
after page of bars and columns.
When to choose a density plot
• Have collected a set of numerical data
• Want to get a quick “feel” for how values are distributed
Strengths of density plots
• Can quickly identify the modality (number of peaks)
• Tune the “bandwidth” to get different levels of detail
• Adds chart variety to a report or presentation
The Classic: Box Plot
The box plot needs no introduction! Although unfortunately, often they do. What do I mean by this you ask? Well, box plots are great and used frequently, so I’m sure we have all seen at least a few
in our lives. But their ability to stealthily encode a significant amount of data means they also sometimes need accompanying explanation. Luckily over time standards have emerged, and so a simple
box plot may now be reasonably assumed to have the following features:
1. Box lower bound = 25th percentile
2. Box upper bound = 75th percentile
3. Mark on box = median
On top of this each box can have a set of accompanying “whiskers” extending out either end. These often extend out to the minimum and maximum values in the dataset, or sometimes to 1.5 x the
interquartile range (I warned you it might need some explaining!).
We tend to keep things simple and only show the central “box”. Although now reduced to only a few numbers instead of visualising the raw values (as we did with the histogram and density plots), the
box plot still conveys a good sense of the “spread” of the distribution by visualising the “middle 50% of values” (which is what the range between the 25th and 75th percentiles is). This is often
called the “interquartile range”, or IQR. Data with a wider IQR have a higher spread in their distribution than those with a narrow IQR. However, any potential modality (multiple peaks) will be
obscured in a box plot.
When to choose a box plot
• Have collected a set of numerical data together with a categorical variable
• Want to get a quick idea of the “spread” of the data for each category
• Can quickly compare the median and IQR for multiple categories
• Widely used chart type – relatively familiar to most viewers
• Customisable – add some whiskers!
The Barcode: Strip Plot
Lastly, the strip plot (also known as ‘barcode plots’ for obvious reasons) can be used when we want to split the data by a categorical variable, but we also want to show the raw values of the
numerical variable.
In a strip plot each “strip” (a thin vertical line) represents a value in the dataset placed along an x-axis scale that comprises the range of values of this variable. For example, the x-axis scale
might go from $0 to $200,000 for our range of collected data on income. Say a value in this dataset is $47,800, we just go to $47,800 on the x-axis scale and add exactly one “strip” mark. Keep doing
this for all the values in the dataset. When we add the y-axis to be the categorical variable it just takes one extra step: line up the “strip” mark with the category label so that the strips are
grouped vertically.
This chart type is useful again for getting a quick “feel” for the data (as every individual response is shown in the chart), particularly when we want to look at the distributions for different
groups (the categorical variable). However, because the strips can all lie over the top of each other (if their values are the same) it loses some of the detail of, for example, the histogram, where
the frequency at each value is represented in the chart. Some common workarounds to this might be to make each strip mark slightly transparent so that you can see denser areas of colour/ink where the
marks overlap with each other.
When to choose a strip plot
• Have collected a set of numerical data together with a categorical variable
• Want to display every single “raw” value in the dataset
• Can quickly identify potential outliers in the data
• Can identify groups or “clusters” of values in the data
• Adds chart variety to a report or presentation
That’s it for our guide to visualising the distribution of data following the principles of applying the ‘so, what?’. Next up in the series: Making comparisons.
Culture Counts provides evaluation solutions for measuring impact. Are you interested in our data analysis and reporting solutions? Contact the Client Management team for a friendly chat. | {"url":"https://dev.culturecounts.cc/blog/how-to-visualise-your-data-a-guide-to-distribution-charts","timestamp":"2024-11-10T01:44:38Z","content_type":"text/html","content_length":"94846","record_id":"<urn:uuid:15472fcc-0175-424c-a3a1-034dc86092df>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00298.warc.gz"} |
Quantum Computing Explained
by Miguel Norberto
Quantum computing is a type of computing where information is processed using quantum bits or qubits.
In classical computing, data is processed using binary digits or bits, either 1 or 0. However, qubits can be both 1 and 0 simultaneously in quantum computing. This allows for many calculations to be
done simultaneously, which is why quantum computers are so powerful.
How does quantum computing work?
Quantum computers can take advantage of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. A quantum computer operates on qubits, which are units of
quantum information. In a classical computer, each bit is either a 0 or a 1. However, in a quantum computer, each qubit can be both a 0 and a one simultaneously. This allows for many calculations to
be done at the same time.
Quantum computers can also be used to create secure communications systems. In a traditional communication system, the message is sent in clear text to be read by the receiver. However, if someone
were to intercept the message, they would read it. In a quantum communication system, the message is encrypted using quantum keys. These keys are created by taking two qubits and putting them into a
superposition state.
Advantages of quantum computing
Quantum computing is still in its early developmental stages, but there are many advantages to the already discovered technology. For example, one of the most promising aspects of quantum computing
is its potential to solve problems faster than classical computers. In some cases, quantum computers can solve problems in minutes that would take billions of years to complete classical computers.
Another advantage of quantum computing is that it can handle large amounts of data simultaneously. This makes quantum computers well-suited for data mining and machine learning tasks. Moreover,
because the same constraints as classical computers do not limit quantum computers, they can explore more solutions to a problem in a shorter amount of time.
Finally, quantum computers are far more secure than traditional systems due to their novel architecture. This makes them an ideal choice for data encryption and security authentication applications.
Disadvantages of quantum computing
Quantum computing is a form of computing that relies on quantum mechanical phenomena to perform calculations. These computers promise massive speedups on some problems relative to classical
computers. However, there are several disadvantages of quantum computing.
First, quantum computers are highly complex, and it is not yet clear whether they can be scaled up to larger sizes. Second, current quantum computers are error-prone, and their reliability needs to
be increased before they can be used for practical applications.
Third, many problems that can be solved with quantum computers are not yet well understood, so it is unclear how useful they will be in practice. Finally, developing quantum computer algorithms and
software is a difficult task, and there is still much research to be done in this area.
Despite these disadvantages, quantum computing can revolutionize information processing and could eventually become the dominant form of computing.
Final Thought
Quantum computing is the future of computing. It can solve problems that are currently unsolvable and revolutionize many industries. However, some challenges still need to be addressed before quantum
computing can be widely adopted.
We need to develop better ways to protect data from being compromised and find new ways to fully use quantum computers to realize their potential. | {"url":"https://www.miguelnorberto.com/quantum-computing-explained/","timestamp":"2024-11-11T14:47:06Z","content_type":"text/html","content_length":"20402","record_id":"<urn:uuid:9e21f683-c2fa-4c43-9e87-baeefcfcea08>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00515.warc.gz"} |
Misunderstanding about C/A operators in electron model
Hi ITensor Support,
I am trying to construct and extended Hubbard model Hamiltonian with next-nearest-neighbour hopping but I am having some problems with the A/C operators while using the autoMPO functions.
As a minimal example, I have just considered the usual fermion hopping term
$$ H = - t_0 \sum_{i, s} c_{i, s}^\dagger c_{i+1,s} + c_{i+1, s}^\dagger c_{i,s} $$
I have represented this three different ways using autoMPO:
-1st I just used the A operators without the Jordan-Wigner F operator (which I know should be wrong!)
-2nd I used the C operators as is standard
-3rd I used the A operators with the Jordan-Wigner F operators (this should be the same as above)
int L = 4;
double t0 = 5;
//Set arguments
auto args = Args("Cutoff=",1E-15,"MaxDim=",5000);
//Initialise System
auto sites = Electron(L, {"ConserveQNs=",false});
// Build Evolution Operators
auto ampo = AutoMPO(sites);
auto ampo2 = AutoMPO(sites);
auto ampo3 = AutoMPO(sites);
//Effective Hamiltonian
//NN Hopping
for(int j = 1; j < L; ++j)
int s1 = j;
int s2 = j + 1;
ampo += (-t0) , "Adagup", s1, "Aup", s2;
ampo += (-t0) , "Adagup", s2, "Aup", s1;
ampo += (-t0) , "Adagdn", s1, "Adn", s2;
ampo += (-t0) , "Adagdn", s2, "Adn", s1;
ampo2 += (-t0) , "Cdagup", s1, "Cup", s2;
ampo2 += (-t0) , "Cdagup", s2, "Cup", s1;
ampo2 += (-t0) , "Cdagdn", s1, "Cdn", s2;
ampo2 += (-t0) , "Cdagdn", s2, "Cdn", s1;
ampo3 += (-t0) , "Adagup", s1, "F", s1, "Aup", s2;
ampo3 += -(-t0) , "Aup", s1,"F", s1, "Adagup", s2;
ampo3 += (-t0) , "Adagdn", s1, "F", s1+1, "Adn", s2;
ampo3 += -(-t0) , "Adn", s1,"F", s1+1, "Adagdn", s2;
I then compute the hopping amplitude between a state with a single spin down on the 2nd site and a single spin down on the 3rd site. By hand I get the following results:
|\psi_0 \rangle = c_{2, \downarrow}^\dagger |0\rangle, \ |\psi_1 \rangle = c_{3, \downarrow}^\dagger |0\rangle, \ \langle \psi_1 |H|\psi_0 \rangle = -t_0
However, performing the same thing in ITensor (with t_0 = 5) using
auto Ham1 = toMPO(ampo, {"ConserveQNs", false});
auto Ham2 = toMPO(ampo2, {"ConserveQNs", false});
auto Ham3 = toMPO(ampo3, {"ConserveQNs", false});
auto states0 = InitState(sites);
states0.set(2, "Dn");
auto psi0 = MPS(states0);
auto states1 = InitState(sites);
states1.set(3, "Dn");
auto psi1 = MPS(states1);
cout << "Initial State Built" << endl;
auto test1 = innerC(psi1, Ham1, psi0);
auto test2 = innerC(psi1, Ham2, psi0);
auto test3 = innerC(psi1, Ham3, psi0);
I get:
test1 = (-5, 0)
test2 = (5, 0)
test3 = (5, 0)
Can you understand why these calculations are not agreeing with the theory? I have also tested these results against some exact diagonalisation calculations which agrees with the theory so I am
rather confused. I'm sure I have just misunderstood something. I am hoping that by sorting out this bit I might be able to get my full model working.
Many thanks,
Thanks for your patience. So this is definitely a bug in the AutoMPO system, and I'm looking into it.
However, for now I can suggest a workaround: the bug above is only happening if you use a site set with all quantum numbers turned off, which we almost never do for fermions. So if it's possible for
your problem (and it should be) to conserve at least fermion parity, or parity and spin, or even particle number and spin, then that will fix the problem for now.
Here are some ways of calling the Electron constructor to set different kinds of QN conservation:
// conserve particle number and total Sz
auto sites = Electron(L, {"ConserveQNs=",true});
//conserve just total Sz and fermion parity
auto sites = Electron(L, {"ConserveQNs=",true,"ConserveNf=",false});
//conserve just fermion parity
auto sites = Electron(L, {"ConserveQNs=",true,"ConserveNf=",false,"ConserveSz=",false});
We could definitely improve some of those named arguments and provide combinations which are more clear in the future but all three of those cases above should give correct results from AutoMPO. | {"url":"https://itensor.org/support/2626/misunderstanding-about-c-a-operators-in-electron-model","timestamp":"2024-11-06T15:38:19Z","content_type":"text/html","content_length":"26873","record_id":"<urn:uuid:6ba87de8-e062-43cb-b4bc-1d022e510b3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00426.warc.gz"} |
Journal of Operator Theory
Volume 91, Issue 1, Winter 2024 pp. 97-124.
Around the closures of the set of commutators and the set of differences of idempotent elements of $\mathcal{B}(\mathcal{H})$
Authors: Laurent W. Marcoux (1), Heydar Radjavi (2), Yuanhang~Zhang (3)
Author institution: (1) Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
(2) Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
(3) School of Mathematics, Jilin Univ., Changchun, 130012, P.R. China
Summary: We describe the norm-closures of the set ${\mathfrak C}_{{\mathfrak E}}$ of commutators of idempotent operators and the set ${\mathfrak E} - {\mathfrak E}$ of differences of idempotent
operators acting on a finite-dimensional complex Hilbert space, as well as characterise the intersection of the closures of these sets with the set ${\mathcal{K} ( \mathcal{H})}$ of compact operators
acting on an infinite-dimensional complex separable Hilbert space ${\mathcal H}$. Finally, we characterise the closures of the set ${\mathfrak C}_{{\mathfrak P}}$ of commutators of orthogonal
projections and the set ${\mathfrak P} - {\mathfrak P}$ of differences of orthogonal projections acting on a complex separable Hilbert space.
DOI: http://dx.doi.org/10.7900/jot.2022feb07.2396
Keywords: commutators, differences, idempotents, projections, closures
Contents Full-Text PDF | {"url":"http://www.mathjournals.org/jot/2024-091-001/2024-091-001-004.html","timestamp":"2024-11-14T10:17:59Z","content_type":"application/xhtml+xml","content_length":"5670","record_id":"<urn:uuid:8ad79afc-ac85-44f1-b2be-a110c9f7adff>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00416.warc.gz"} |
With nkululeko since version 0.88.0 you can combine experiment results and report on the outcome, by using the ensemble module.
For example, you would like to know if the combination of expert features and learned embeddings works better than one of those. You could then do
python -m nkululeko.ensemble \
--method max_class \
tests/exp_emodb_praat_xgb.ini \
tests/exp_emodb_ast_xgb.ini \
(all in one line)
and would then get the results for a majority voting of the three results for Praat, AST and Wav2vec2 features.
Other methods are mean, max, sum, max_class, uncertainty_threshold, uncertainty_weighted, confidence_weighted:
• majority_voting: The modality function for classification: predict the category that most classifiers agree on.
• mean: For classification: compute the arithmetic mean of probabilities from all predictors for each labels, use highest probability to infer the label.
• max: For classification: use the maximum value of probabilities from all predictors for each labels, use highest probability to infer the label.
• sum: For classification: use the sum of probabilities from all predictors for each labels, use highest probability to infer the label.
• max_class: For classification: compare the highest probabilities of all models across classes (instead of same class as in max_ensemble) and return the highest probability and the class
• uncertainty_threshold: For classification: predict the class with the lowest uncertainty if lower than a threshold (default to 1.0, meaning no threshold), else calculate the mean of uncertainties
for all models per class and predict the lowest.
• uncertainty_weighted: For classification: weigh each class with the inverse of its uncertainty (1/uncertainty), normalize the weights per model, then multiply each class model probability with
their normalized weights and use the maximum one to infer the label.
• confidence_weighted: Weighted ensemble based on confidence (1-uncertainty), normalized for all samples per model. Like before, but use confidence (instead of inverse of uncertainty) as weights.
Nkululeko: export acoustic features
With nkululeko since version 0.85.0 the acoustic features for the test and the train (aka dev) set are exported to the project store.
If you specify the store_format:
store_format = csv
they will be exported to CSV (comma separated value) files, else PKL (readable by python pickle module).
I.e. you store should then after execution of any nkululeko module that computes features the two files:
• feats_test.csv
• feats_train.csv
If you specified scaling the features:
scale = standard # or speaker
you will have two additional files with features:
• feats_test_scaled.csv
• feats_train_scaled..csv
In contrast to the other feature stores, these contain the exact features that are used for training or feature importance exploration, so they might be combined from different feature types and
selected via the features value. An example:
type = ['praat', 'os']
features = ['speechrate_nsyll_dur', 'F0semitoneFrom27.5Hz_sma3nz_amean']
scale = standard
store_format = csv
results in the following feats_test.csv:
./data/emodb/emodb/wav/11b03Wb.wav,0 days,0 days 00:00:05.213500,4.028004219813945,34.42206
./data/emodb/emodb/wav/16b10Td.wav,0 days,0 days 00:00:03.934187500,3.0501850763340586,31.227554
How to use train, dev and test splits with Nkululeko
Usually in machine learning, you train your predictor on a train set, tune meta-parameters on a dev (development or validation set ) and evaluate on a test set.
With nkululeko, there currently the test set is not, as there are only two sets that can be specified: train and evaluation set.
A work-around is to use the test module to evaluate your best model on a hold out test set at the end of your experiments.
All you need to do is to specify the name of the test data in your [DATA] section, like so (let's call it myconf.ini):
save = True
databases = ['my_train-dev_data']
tests = ['my_test_data']
my_test_data = ./data/my_test_data/
my_test_data.split_strategy = test
you can run the experiment module with your config:
python -m nkululeko.nkululeko --config myconf.ini
and then, after optimization (of predictors, features sets and meta-parameters), use the test module
python -m nkululeko.test --config myconf.ini
The results will appear at the same place as all other results, but the files are named with test and the test database as a suffix.
If you need to compare several predictors and feature sets, you can use the nkuluflag module
All you need to do, is, in your main script, if you call the nkuluflag module, pass a parameter (named --mod) to tell it to use the test module:
cmd = 'python -m nkululeko.nkuluflag --config myconf.ini --mod test '
Nkululeko: how to bin/discretize your feature values
With nkululeko since version 0.77.8 you have the possibility to convert all feature values into the discreet classes low, mid and high
Simply state
type = ['praat']
scale = bins
store_format = csv
in your config to use Praat features.
With the store format stated as csv you will be able to look at the train and test features in the store folder.
The binning will be done based on the 33 and 66 percent of the training feature values.
Nkululeko: compare several databases
With nkululeko since version 0.77.7 there is a new interface named multidb which lets you compare several databases.
You can state their names in the [EXP] section and they will then be processed one after each other and against each other, the results are stored in a file called heatmap.png in the experiment
!Mind YOU NEED TO OMIT THE PROJECT NAME!
Here is an example for such an ini.file:
root = ./experiments/emodbs/
# DON'T give it a name,
# this will be the combination
# of the two databases:
# traindb_vs_testdb
epochs = 1
databases = ['emodb', 'polish']
root_folders = ./experiments/emodbs/data_roots.ini
target = emotion
labels = ['neutral', 'happy', 'sad', 'angry']
type = ['os']
type = xgb
you can (but don't have to), state the specific dataset values in an external file like above.
emodb = ./data/emodb/emodb
emodb.split_strategy = specified
emodb.test_tables = ['emotion.categories.test.gold_standard']
emodb.train_tables = ['emotion.categories.train.gold_standard']
emodb.mapping = {'anger':'angry', 'happiness':'happy', 'sadness':'sad', 'neutral':'neutral'}
polish = ./data/polish_emo
polish.mapping = {'anger':'angry', 'joy':'happy', 'sadness':'sad', 'neutral':'neutral'}
polish.split_strategy = speaker_split
polish.test_size = 30
Call it with:
python -m nkululeko.multidb --config my_conf.ini
The default behavior is that all databases are used as a whole when being test or training database. If you would rather like the splits to be used, you can add a flag for this:
use_splits = True
Here's a result with two databases:
and this is the same experiment, but with augmentations:
In order to add augmentation, simply add an [AUGMENT] section:
root = ./experiments/emodbs/augmented/
epochs = 1
databases = ['emodb', 'polish']
augment = ['traditional', 'random_splice']
In order to add an additional training database to all experiments, you can use:
train_extra = [meta, emodb]
, to add two databases to all training data sets,
where meta and emodb should then be declared in the root_folders file
Nkululeko: generate a latex/pdf report
With nkululeko since version 0.66.3, a report document formatted in Latex and compiled as a PDF file can automatically be generated, basically as a compilation of the images that are generated.
There is a dedicated REPORT section in the config file for this, here is an example:
# should the report be shown in the terminal at the end?
show = False
# should a latex/pdf file be printed? if so, state the filename
latex = emodb_report
# name of the experiment author (default "anon")
author = Felix
# title of the report (default "report")
title = EmoDB
with each run of a nkululeko module in the same experiment environment, the details of the report will be added.
So a typical use would be, to first run the general module and than more specialized ones:
# first run a segmentation
python -m nkululeko.segment --config myconf.ini
# then rename the data-file in the config.ini and
# run some data exploration
python -m nkululeko.explore --config myconf.ini
# then run a machine learning experiment
python -m nkululeko.nkululeko --config myconf.ini
Each run will add some contents to the report
Nkululeko: segmenting a database
Segmenting a database means to split the audio samples of a database into smaller segments or chunks. With speech data this is usually done on the basis of VAD, aka voice activity detection, meaning
that the pauses between speech in the audio samples are used as segment borders.
The reason for segmenting could be to label the data with something that would not last over the whole sample, e.g. emotional state.
Another motivation to segment audio data might be that the acoustic features are targeted at a specific stretch of audio, e.g. 3-5 seconds long.
Within nkululeko this would be done with the segment module, which is currently based on the silero software.
You simply call your experiment configuration with the segment module, and the train, test set or both will be segmented.
The advantage is, that you can use all filters on your data that might make sense beforehand, for example with the android corpus, only the reading task samples are not segmented.
You can select them like so:
filter = [['task', 'reading']]
and then call the segment module:
python -m nkululeko.segment --config my_conf.ini
The output is a new database file in CSV format.
If you want, you can specify if only the training, or test split, or both should be segmented, as well as the string that is added to the name of the resulting csv file (the name per default consists
of the database names):
# name postfix
target = _segmented
# which model to use
method = silero
# which split: train, test or all (both)
sample_selection = all
# the minimum lenght of rest-samples (in seconds)
min_length = 2
# the maximum length of segments, longer ones are cut here. (in seconds)
max_length = 10
Nkululeko: check your dataset
Within nkululeko, since version 0.53.0, you can perform automatic data checks, which means that some of your data might be filtered out if it doesn't fulfill certain requirements.
Currently two checks are implemented:
# check the filesize of all samples in train and test splits, in bytes
check_size = 1000
# check if the files contain speech with voice activity detection (VAD)
check_vad = True
VAD is using silero VAD
Nkululeko: how to visualize your data distribution
If you just want to see how your data distributes on the target with nkululeko, you can do a value_counts plot with the explore module
In your config, you would specify like this:
# all samples, or only test or train split?
sample_selection = all
# activate the plot
value_counts = [['age'], ['gender'], ['duration'], ['duration', 'age']]
and then, run this with the explore module:
python -m nkululeko.explore --config myconfig.ini
The results, for a data set with target=depression, looks similar to this for all samples:
and this for the speakers (if there is a speaker annotation)
If you prefer a kernel density estimation over a histogram, you can select this with
dist_type = kde
which would result for duration to:
Nkululeko distinguishes between categorical and continuous properties, this would be the output for gender
You can show the distribution of two sample properties at once, by using a scatter plot:
In addition, this module will automatically plot the distribution of samples per speaker, per gender (if annotated):
Nkululeko: how to augment the training set
To do data augmentation with Nkululeko, you can use the augment or the aug_train interface.
The difference is that the former only augments samples, whereas the latter augments the training set of a configuration and then immediately performs the training, including the augmented files.
In the AUGMENT section of your configuration file, you specify the method and name of the output list of file
• traditional: is the classic augmentation, e.g. by cropping data or adding a bit of noise. We use the audiomentations package for this
• random-splice: is a special method introduced in this paper that randomly splices and re-connects the audio samples
# select the samples to augment: either train, test, or all
sample_selection = train
# select the method(s)
augment = ['traditional', 'random_splice']
# file name to store the augmented data (can then be added to training)
result = augmented.csv
and then call the interface:
python -m nkululeko.augment --config myconfig.ini
python -m nkululeko.aug_train--config myconfig.ini
if you want to run a training in the same run.
Currently, apart from random-splicing, Nkululeko simply uses the audiomentations module, i.e.:
augment = ['traditional']
augmentations = Compose([
AddGaussianNoise(min_amplitude=0.001, max_amplitude=0.05),
BandPassFilter(min_center_freq=100.0, max_center_freq=6000),])
These manipulations are applied randomly to your training set.
With respect to the random_splicing method, you can adjust two parameters:
• p_reverse: probability of some samples to be in reverse order (default: 0.3)
• top_db: top dB level for silence to be recognized (default: 12)
This configuration, for example, would distort the samples much more than the default:
augment = ['random_splice']
p_reverse = .8
top_db = 6
You should find the augmented files in the storage folder of the result folder of your experiment and could listen to them there.
Once you augmentations have been processed, you can add them to the training in a new experiment:
databases = ['original data', 'augment']
augment = my_augmentations.csv
augment.type = csv
augment.split_strategy = train | {"url":"http://blog.syntheticspeech.de/category/nkululeko/","timestamp":"2024-11-13T09:24:26Z","content_type":"text/html","content_length":"70034","record_id":"<urn:uuid:f9417335-2a9b-49da-896e-b761051c9667>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00590.warc.gz"} |
Derivation of equation for atmospheric pressure
Flashbacks to the past: one of the first worksheets I published in this project was on using linear approximation to estimate the atmospheric pressure at various altitudes, and a later one was about
a power function for atmospheric pressure. The derivation of the formula for atmospheric pressure is actually pretty straightforward. I'll assume that your students have not yet encountered integrals
per se, but this worksheet pushes them to use their knowledge of differentiation to deduce an antiderivative.
This is a worksheet that puts together a few disparate concepts:
• dimensional analysis, using units to understand equations
• antiderivatives,
• and baby differential equations.
It's certainly an activity for the end of the section on differentiation. The very last question asks students to think about a more accurate equation, and I wouldn't expect most students to be able
to solve it alone -- but sometimes a good challenge is important as it points to concepts you'll be dealing with later on. Knowing how to integrate would really help in solving that last problem 🙂
Derivation: Atmospheric Pressure
As mentioned in the earlier posts, a great resource on atmospheric pressure and rocketry and all sorts of fun things is found at Portland State Aerospace Society's rocketry pages.
This entry was posted in Derivatives. Bookmark the permalink. | {"url":"http://www.earthcalculus.com/derivation-of-equation-for-atmospheric-pressure/","timestamp":"2024-11-07T00:26:37Z","content_type":"text/html","content_length":"42976","record_id":"<urn:uuid:990ff16c-fa76-46b4-8763-50089c185792>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00861.warc.gz"} |
Re: Re: Show doesn't work inside Do loop ?
• To: mathgroup at smc.vnet.net
• Subject: [mg102045] Re: [mg102013] Re: Show doesn't work inside Do loop ?
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Mon, 27 Jul 2009 05:57:12 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <32390795.1248259308283.JavaMail.root@n11> <h4951e$q2e$1@smc.vnet.net> <21621663.1248432817472.JavaMail.root@n11> <h4eeul$spv$1@smc.vnet.net> <200907260756.DAA19062@smc.vnet.net>
• Reply-to: murray at math.umass.edu
I find nothing whatsoever that is not "consistent" in the behavior of Do
with respect to a sequence of separate evaluations.
And no, I do not think it would be "reasonable" or "sensible" that Do
produces output at each step of the loop. The behavior of Do in this
regard is the same as that of While and For. And the rationale is surely
the same for all 3 functions: you simply do the iteration and specify
explicitly (manually) what, if anything, you want to be returned or
printed, even when the loop has been completed. In fact, I would find
it quite an annoyance were Do or While or For produced intermediate, or
any, output without my explicitly asking for it.
AES wrote:
> In article <h4eeul$spv$1 at smc.vnet.net>,
> "David Park" <djmpark at comcast.net> wrote: (emphasis added)
>> _On the other hand, Show does not generally generate a cell._
>> Of course, it is easier to just apply Print to the Show or Plot statements.
>> So it is possible to make Show generate output cells, but it _normally_
>> doesn't do so. It normally only generates expressions, which because of its
>> special behavior a Do statement does not display.
> 1) Re your statements above: executing a single Input cell containing
> just
> Show[ Graphics[ Circle[ {0, 0}, 1] ] ]
> certainly generates an Output cell containing what looks like a
> "graphic" or "plot" to me.
> Is executing a cell containing a simple expression somehow an "abnormal"
> process or action?
> 2) On a more general note: Suppose you have an expression which
> contains an explicit symbol x , such that if you execute three
> consecutive cells containing
> x=1; expr
> x=2; expr
> x=3; expr
> or maybe
> expr/.x->1
> expr/.x->2
> expr/.x->3
> you get three output cells containing three successive instances of expr
> (whatever that is) -- or appropriate error messages if executing expr
> one of those times has some side effect that messes up a subsequent
> execution.
> Would it not be reasonable to expect a cell containing
> Do[ expr, {x,1,3} ]
> to do _exactly_ the same thing?
> In other words, would it not be reasonable -- consistent -- sensible --
> helpful -- the most useful -- to expect Do[ ] to be simply a "wrapper"
> that functioned in exactly that manner?
> I appreciate that Mathematica's Do[] apparently doesn't function that
> way -- or functions that way sometimes, based on mysterious criteria,
> but not other times; and suggest that this is not helpful or useful or
> consistent behavior for many users.
> Are there any fundamental reasons why a DoConsistently[ ] command could
> not be defined, such that DoConsistently[ expr, iterator ] would
> repeatedly put expr into a cell with each iterator instance applied to
> it, and churn out the sequential outputs? That, it seems to me, is what
> many users would want and expect.
Murray Eisenberg murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower phone 413 549-1020 (H)
University of Massachusetts 413 545-2859 (W)
710 North Pleasant Street fax 413 545-1801
Amherst, MA 01003-9305 | {"url":"http://forums.wolfram.com/mathgroup/archive/2009/Jul/msg00704.html","timestamp":"2024-11-08T02:39:08Z","content_type":"text/html","content_length":"34355","record_id":"<urn:uuid:650825d9-1cd6-408f-b400-92077a7ad902>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00121.warc.gz"} |
Lesson 5
Connections between Representations
These materials, when encountered before Algebra 1, Unit 5, Lesson 5 support success in that lesson.
5.1: Math Talk: Evaluating Expressions (5 minutes)
The purpose of this Math Talk is to elicit strategies and understandings students have for evaluating expressions at a given value of the variable. These understandings help students develop fluency
and will be helpful later in this lesson when students will need to evaluate similarly-structured expressions.
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk.
Follow with a whole-class discussion.
Student Facing
Evaluate mentally:
\(6,\!400 - 400x\) when \(x\) is 0
\(6,\!400 - 400x\) when \(x\) is 2
\(6,\!400 \boldcdot \left(\frac{1}{10}\right)^x\) when \(x\) is 0
\(6,\!400 \boldcdot \left(\frac{1}{10}\right)^x\) when \(x\) is 2
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. Be sure to draw attention to evaluating operations in the conventional order, and the fact
that \(\left(\frac{1}{10}\right)^2\) means \(\frac{1}{10} \boldcdot \frac{1}{10}\). To involve more students in the conversation, consider asking:
• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”
5.2: A Good Night’s Sleep (20 minutes)
In this activity, students are given an equation, and generate a corresponding table and graph. Then, they respond to some questions where they interpret these representations in terms of the
situation. The purpose of this exercise is to reinforce meaningful connections between different representations of the same situation.
Ask students to think about this question: “How is your day different when you’ve had plenty of sleep the night before, compared to when you didn’t get enough sleep the night before?” Then, invite
them to share their thoughts with a partner. Monitor for students who mention that getting enough sleep has to do with performing better the next day. Also listen for conversations about how we’d
define “enough sleep.” Select a few students to share their thoughts with the whole class.
Tell students that in this activity, a researcher has tried to create a model for performance on a problem solving task based on hours of sleep the previous night. Allow students to work individually
or in pairs.
Graphing technology could be a helpful tool if students choose to use it (MP5), unless the focus becomes which buttons to press to get answers without understanding. Students also may be
better-equipped to attempt the analysis questions if they create the table and graph by hand. So, use your judgment about whether it would be productive to allow use of graphing technology.
Student Facing
Is more sleep associated with better brain performance? A researcher collected data to determine if there was an association between hours of sleep and ability to solve problems. She administered a
specially designed problem solving task to a group of volunteers, and for each volunteer, recorded the number of hours slept the night before and the number of errors made on the task.
The equation \(n = 40 - 4t\) models the relationship between \(t\), the time in hours a student slept the night before, and \(n\), the number of errors the student made in the problem-solving task.
1. Use the equation to find the coordinates of 5 data points on a graph representing the model. Organize the coordinates in the table.
2. Create a graph that represents the model.
│hours of sleep, \(t\) │ number of errors, \(n\) │
│ │ │
│ │ │
│ │ │
│ │ │
│ │ │
3. In the equation \(n = 40 - 4t\), what does the 40 mean in this situation? Where can you see it on the graph?
4. In the equation \(n = 40 - 4t\), what does the -4 mean in this situation? Where can you see it on the graph?
5. How many errors would you expect a person to make who had slept 3.5 hours the night before?
Activity Synthesis
Invite students to share their responses. If not already mentioned in students’ explanations, highlight the connections between the equation, the graph, and the quantities in the situation. In
particular, show the point \((0,40)\) alongside the equation \(40=40-4(0)\), and demonstrate on the graph how the graph shows that the rate of change in this situation is -4 errors per additional
hour of sleep.
Possible discussion questions:
• “How can we use an equation to express the number of errors after 3.5 hours without sleep?” (\(n=40-4(3.5)\))
• “Where on the graph is the number of errors made someone who got no sleep?” (The vertical intercept)
• “Will the graph continue to decrease indefinitely?” (No. The number of errors is limited to the number of questions on the task. Also, humans can’t sleep indefinitely.)
5.3: What’s My Equation? (15 minutes)
This activity is a preview of the work in the associated Algebra 1 lesson. Students have a scaffolded opportunity to determine a decay factor from a given graph, and make connections between graphs
and equations.
The context and types of questions continue from the previous activity, so students can continue to work individually or with a partner without much interruption.
Student Facing
The sleep researcher repeated the study on two more groups of volunteers, collecting different data. Here are graphs representing the equations that model the different sets of data:
1. Write an equation for Model A. Be prepared to explain how you know. Explain what the numbers mean in your equation.
2. Model B is exponential.
1. How many errors did participants make with 0 hours of sleep?
2. How many errors with 1 hour of sleep?
3. What fraction of the errors from 0 hours of sleep is that?
3. Complete the table for Model B for 3, 4, and 5 hours of sleep.
│\(t\) │0 │1 │2│3│4│5│
│\(n\) │81│27│9│ │ │ │
4. Which is an equation for Model B? If you get stuck, test some points!
\(n=81 \boldcdot \left(3 \right)^t\)
\(n=81 \boldcdot \left(\frac13 \right)^t\)
Activity Synthesis
Invite students to share their responses and their reasoning. Record their responses to show connections between each graph and the corresponding equation. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/4/5/5/index.html","timestamp":"2024-11-02T21:58:57Z","content_type":"text/html","content_length":"97685","record_id":"<urn:uuid:becd96f1-636d-434e-8596-7b60739ad2db>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00054.warc.gz"} |
KOS.FS - faculty addon
Computer Graphics (E012037)
Departments: ústav technické matematiky (12101)
Abbreviation: Approved: 21.05.2010
Valid until: ?? Range: 1P+1C
Semestr: * Credits: 3
Completion: KZ Language: EN
The subject is focused on the mathematical theory of the curves and surfaces in computer graphics and their visualisation. The Rhinoceros - NURBS modelling for Windows is used to demonstrate the
geometrical properties of the curves and surfaces.
doc. Ing. Ivana Linkeová Ph.D.
Letní 2023/2024
doc. Ing. Ivana Linkeová Ph.D.
Letní 2022/2023
Mgr. Nikola Pajerová Ph.D.
Letní 2021/2022
1. Ferguson curve - definition, analytical and graphical representation, properties, applications.
2. Bézier curve - definition, analytical, graphical and CAD representation, properties, free form curves modelling, applications.
3. Coons, B-spline and NURBS curve - definition, analytical, graphical and representation, properties, free form curves modelling, applications.
4. Ferguson 12-vector patch - definition, analytical and graphical representation, applications.
5. Bézier surface - definition, analytical, graphical and CAD representation, applications.
6. Coons surface - definition, analytical, graphical and CAD representation, applications.
7. Patching - free form surfaces modelling with required continuity.
Structure of tutorial
1. Rhinoceros I - helix and helicoidal surfaces modelling.
2. Free-form curves - analytical and graphical representation.
3. Rhinoceros II - free-form curves modelling.
4. Free-form surfaces - analytical and graphical representation.
5. Free-form surfaces - analytical and graphical representation - continuing.
6. Rhino III - free-form surface modelling.
7. Test. Assessments.
Linkeová, I.: Curves and Surfaces for Computer Graphics. CTU in Prague, 2012.
Free-form curve, free-form surface, Rhinoceros, continuity, patching. | {"url":"https://kos.fs.cvut.cz/synopsis/E012037/en","timestamp":"2024-11-08T05:30:12Z","content_type":"text/html","content_length":"7838","record_id":"<urn:uuid:8bc94d8a-e767-47e3-a876-d052b2f8438d>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00381.warc.gz"} |
Blog entries - Codeforces
A. Ice Skating
Notice that the existence of a snow drift at the point (x,y) implies that "if I'm on the horizontal line at y then I am certainly able to get to the vertical line at x, and vice versa". Thus, the
snow drifts are the edges of a bipartite graph between x- and y- coordinates. The number of snow drifts that need to be added to make this (as well as the original) graph connected is the number of
its connected components reduced by one.
B. Blackboard Fibonacci
If you look at the described process backwards, it resembles the Euclidean algorithm a lot. Indeed, if you rewinded a recording of Bajtek's actions, he always takes the larger out of two numbers (say
Unable to parse markup [type=CF_TEX]
) and replaces them by
. Since we know one of the final numbers (
) we can simply check all numbers between
and run a faster version of Euclid's algorithm (one that replaces
However, with some insight, it can be seen that this optimization is in fact not neccessary — we can simply simulate the reverse process as described (replacing a,b by a-b,b) for all candidates
between 1 and r and the total runtime of our algorithm will remain
C. Formurosa
One of the major difficulties in this problem is finding an easily formulated condition for when Formurosa can be used to distinguish the bacteria. Let Formurosa's digestive process be a function F(s
) that maps binary sequences of length m to elements of {0,1}. It turns out that the condition we seek for can be stated as follows:
We can distinguish all the bacteria if and only if there exists a sequence s of length m for which F(s)≠F(-s), where -s is the negation of s.
First, not that if no such sequence exists, then there is no way to distinguish between zero and one. If such a sequence exists, we can pick any two bacteria a and b and try both ways to substitute
them for 0 and 1 in the expression. If the two expressions evaluate to different values, we will determine the exact types of both bacteria. Otherwise, we will be certain that the bacteria are of the
same type. Repeating the process for all pairs of bacteria will let us identify all the types (since it is guaranteed that not all bacteria are of the same type).
To determine whether such a sequence s exists, dynamic programming over the expression tree of Formurosa can be applied. The model solution keeps track for each subtree G of the expression which of
the following sequences can be found:
• a sequence s such that G(s)=G(-s)=0
• a sequence s such that G(s)=G(-s)=1
• a sequence s such that G(s)≠G(-s)
D. Bitonix' Patrol
Observation 1.
Fuel tanks for which capacity gives the same remainder D are also equivalent. Out of every group of equivalent tanks, the agency can only leave at most one.
Observation 2.
If more than six tanks remain, Bitonix can certainly complete his patrol. Indeed, let us assume that 7 tanks were left undestroyed by the agency. Out of the 128 possible subsets of those tanks, at
least two distinct ones, say A and B, sum up to the same remainders modulo D. Thus, if Bitonix moves forward with tanks from A-B and backwards with tanks from B-A, he will finish at some station
after an actual journey.
Because of observations 1 and 2, it turns out that a simple recursive search suffices to solve the problem. However, because of the large constraints, it may prove necessary to use some
optimizations, such as using bitmasks for keeping track of what distances Bitonix can cover.
E. Alien DNA
Note that it is easy to determine, looking at only the last mutation, how many letters it adds to the final result. Indeed, if we need to print out the first k letters of the sequence, and the last
mutation is [l,r], it suffices to find out the length of the overlap of segments [1,k] and [r+1,2r-l+1]. Say that it is x. Then, after the next to last mutation, we are only interested in
the first k-x letters of the result — the rest is irrelevant, as it will become "pushed out" by the elements added in the last mutation. Repeating this reasoning going backwards, we shall find out
that we can spend linear time adding letters to the result after every mutation, which turns out to be the main idea needed to solve the problem.
For a neat O(n^2+k) implementation of this idea you can check out ACRush's solution: 2029369.
The only other contestant to solve the problem during the competition, panyuchao, used a slightly different approach, based in part on the same idea. Check out his ingenious, astonishingly short | {"url":"https://mirror.codeforces.com/blog/meret","timestamp":"2024-11-11T10:06:06Z","content_type":"text/html","content_length":"91066","record_id":"<urn:uuid:ce5e3f14-500b-48a1-8884-c53a8ca9bf01>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00012.warc.gz"} |
deformation formula calculation for Calculations
29 Mar 2024
Popularity: ⭐⭐⭐
Deformation Calculation
This calculator provides the calculation of deformation for a given uniaxial stress and Young’s modulus.
Calculation Example: Deformation is the change in the size or shape of an object due to an applied force. In the case of uniaxial stress, the deformation is a change in length. The amount of
deformation is directly proportional to the applied stress and inversely proportional to the Young’s modulus of the material.
Related Questions
Q: What is the relationship between stress, strain, and Young’s modulus?
A: Stress is directly proportional to strain and Young’s modulus. This relationship is known as Hooke’s law.
Q: How can deformation be minimized in engineering applications?
A: Deformation can be minimized by using materials with a high Young’s modulus or by reducing the applied stress.
| —— | —- | —- |
Calculation Expression
Deformation: The deformation is calculated using the formula: ? = ? / E
Calculated values
Considering these as variable values: ?=1.0E7, E=2.0E11, the calculated value(s) are given in table below
| —— | —- |
Similar Calculators
Calculator Apps
Matching 3D parts for deformation formula calculation for Calculations
App in action
The video below shows the app in action. | {"url":"https://blog.truegeometry.com/calculators/deformation_formula_calculation_for_Calculations.html","timestamp":"2024-11-10T08:39:41Z","content_type":"text/html","content_length":"22953","record_id":"<urn:uuid:4a48fae9-23fa-4d1c-bab9-d3587541af1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00617.warc.gz"} |
0.3.0 Aug 21, 2024
0.2.0 Aug 16, 2024
0.1.2 Aug 4, 2024
0.1.1 Aug 4, 2024
0.1.0 Aug 4, 2024
CF's Accelerated Vector Math Library
Various accelerated vector operations over Rust primitives with SIMD.
This is the core base library, it has no dependencies and only depends on the core library, it does not perform any allocations.
This library is guaranteed to be no-std compatible and can be adjusted by disabling the std feature flag:
Default Setup
cfavml = "0.3.0"
No-std Setup
cfavml = { version = "0.3.0", default-features = false }
Important Version Upgrade Notes
If you are upgrading on a breaking release, i.e. 0.2.0 to 0.3.0 there may be some important changes that affects your system, although the public safe APIs I try my best to avoid breaking.
• AVX512 required CPU features changed in 0.3.0+
□ In versions older than 0.3.0 avx512 was used when only the avx512f cpu feature was available since this is the base/foundation version of AVX512. However, in 0.3.0 we introduced more
extensive cmp operations (eq/neq/lt/lte/gt/gte) which changed our required CPU features to include avx512bw
□ This means on unsafe APIs you must update your feature checks to include avx512bw.
□ Safe APIs do not require changes but may fallback to AVX2 on some of the first gen AVX512 CPUs, i.e. Skylake
Available SIMD Architectures
• AVX2
• AVX2 + FMA
• AVX512 (avx512f + avx512bw) nightly only
• NEON
• Fallback (Typically optimized to SSE automatically by LLVM on x86)
Supported Primitives
• f32
• f64
• i8
• i16
• i32
• i64
• u8
• u16
• u32
• u64
Note on non-f32/f64 division
Division operations on non-floating point primitives are currently still scalar operations, as performing integer division is incredibly hard to do anymore efficiently with SIMD and adds a
significant amount of cognitive overhead when reading the code.
Although to be honest I have some serious questions about your application if you're doing heavy integer division...
Supported Operations
Spacial distances
These are routines that can be used for things like KNN classification or index building.
• Dot product of two vectors
• Cosine distance of two vectors
• Squared Euclidean distance of two vectors
• Add single value to vector
• Sub single value from vector
• Mul vector by single value
• Div vector by single value
• Add two vectors vertically
• Sub two vectors vertically
• Mul two vectors vertically
• Div two vectors vertically
• Horizontal max element in a vector
• Horizontal min element in a vector
• Vertical max element of two vectors
• Vertical min element of two vectors
• Vertical max element of a vector and broadcast value
• Vertical min element of a vector and broadcast value
• EQ/NEQ/LT/LTE/GT/GTE cmp of a vector and broadcast value
• EQ/NEQ/LT/LTE/GT/GTE cmp of two vectors
• Horizontal sum of a vector
• Squared L2 norm of a vector
Dangerous routine naming convention
If you've looked at the danger folder at all, you'll notice a few things, one SIMD operations are gated behind the SimdRegister<T> trait, this provides us with a generic abstraction over the various
SIMD register types and architectures.
This trait, combined with the Math<T> trait form the core of all operations and are provided as generic functions (with no target features):
• generic_dot
• generic_squared_euclidean
• generic_cosine
• generic_squared_norm
• generic_cmp_max
• generic_cmp_max_vector
• generic_cmp_max_value
• generic_cmp_min
• generic_cmp_min_vector
• generic_cmp_min_value
• generic_cmp_eq_vector
• generic_cmp_eq_value
• generic_cmp_neq_vector
• generic_cmp_neq_value
• generic_cmp_lt_vector
• generic_cmp_lt_value
• generic_cmp_lte_vector
• generic_cmp_lte_value
• generic_cmp_gt_vector
• generic_cmp_gt_value
• generic_cmp_gte_vector
• generic_cmp_gte_value
• generic_sum
• generic_add_value
• generic_sub_value
• generic_mul_value
• generic_div_value
• generic_add_vector
• generic_sub_vector
• generic_mul_vector
• generic_div_vector
We also export functions with the target_features pre-specified for each SIMD register type and is found under the cfavml::danger::export_* modules. Although it is not recommended to use these
routines directly unless you know what you are doing.
• nightly Enables optimizations available only on nightly platforms.
□ This is required for AVX512 support due to it currently being unstable.
Is this a replacement for BLAS?
No. At least, not unless you're only doing dot product... BLAS and LAPACK are huge and I am certainly not in the market for implementing all BLAS routines in Rust, but that being said if your
application is similar to that of ndarray where it is only using BLAS for the dot product, then maybe. | {"url":"https://lib.rs/crates/cfavml","timestamp":"2024-11-15T01:24:16Z","content_type":"text/html","content_length":"25417","record_id":"<urn:uuid:291a15dd-505c-4cd5-96ec-3258bf4db0fa>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00089.warc.gz"} |
What is the Difference Between Volts And Amps? - Wiring Solver
Voltage and Amperes are some of the more common units of measurement when talking about electricity. For someone new to the field, they might be confusing, and you might not know what they mean.
Fortunately, this article is here to help illustrate the difference between volts and amps.
We’ll look at what the two units of measurements denote and help you get a better understanding. Then we’ll look at the differences between the two.
What Is Volts?
Volt is the SI unit of measurement that is used to measure the voltage, potential difference, or electromotive force of an electric circuit. When you buy a battery, you’ll often see a voltage rating
on it, like 12V or 6V. To power an electrical machine, you need a power source to promote the flow of electrons. Volt is a measure of that.
To better understand it, we can make use of the tried and tested water pump analogy. The purpose of a water pump is to pump water from one place to another. The water on its own won’t move to another
place, so you need some help moving it. So the water pump applies the pressure that is used to pump the water to somewhere else.
If you think of points where you have to transfer water from and to as a wire, you’ll realize that the water pump is a sort of battery. The pressure the water pump exerts to move the water is
analogous to the voltage, which causes electricity to flow through the wire. This voltage is measured in volts.
When we look at amperes, we’ll return to that analogy.
Quantifying Voltage
Thinking about the example set above, we can say that voltage is simply the energy needed to move a certain amount of charge. So volt is the unit of energy (Joule) divided by the unit of charge
(coulomb), so Joule/Coulomb.
What Is Amperes?
Amperes is the SI unit of measurement that is used to measure the current in a circuit or device. For any electrical circuit or device to work, the current needs to flow. However, higher current flow
can result in a device being damaged or a wire being burnt. This is why a unit of measurement for current is needed, or else we would be burning every device down.
Continuing from our water pump analogy, we’ll now focus on the water being pumped. The pump allows a certain volume of water to flow at a certain speed. This rate of water transfer can either be
increased by making the pump let out more water or pump it out at a faster speed.
If we compare the water to charge in a wire or conductor, the analogy will become clearer. Charge flows from one end of a wire to another. If more charge flows through the wire, then more electricity
is transferred. If the speed at which the charge flows is increased, electricity transferred will also increase.
This rate of flow of charge is known as current, and this current is measured using the unit ampere or amp. Higher ampere ratings denote a higher rate of flow of charge.
Quantification Of Ampere
To summarize, the unit ampere denotes the current, which is the rate of flow of charge. Since the unit of charge is the coulomb, and we measure the rate, the ampere is the unit of charge (Coulomb)
divided by the standard unit of time (Second). Thus, Coulomb per second.
Voltage Vs. Amperage
Since we’ve looked at the two separately, we can now try to find out the differences between them.
1. Unit Of Measurement
Volt is the unit of measurement for the force or pressure that causes charge to flow through a wire or device. Amp is the unit of measurement for the rate of flow of that charge.
2. Measuring Quantity
The electromotive force in a circuit is denoted using volt. The potential difference between two points is also denoted using volt. An electric current in a circuit or device is denoted using amps.
3. Measuring Instrument
Volt is measured using a voltmeter, which is connected parallel across the point of measurement. Amp is measured using an ammeter, which is connected in series in the point of measurement.
4. Danger
Higher amp ratings are generally more dangerous than higher volt ratings.
Also Read: Single Phase vs Three Phase
These are the key differences between volts and amps. It’s important to have a clear understanding of them if you’re planning on going further into the field of electrical engineering. | {"url":"https://wiringsolver.com/difference-between-volts-and-amps/","timestamp":"2024-11-09T10:59:04Z","content_type":"text/html","content_length":"136588","record_id":"<urn:uuid:0136b716-6e0f-45ae-a22f-254af9969c0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00409.warc.gz"} |
Science:Math Exam Resources/Courses/MATH104/December 2014/Question 03 (c)
MATH104 December 2014
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• Q5 (e) • Q5 (f) • Q6 •
Question 03 (c)
The price ${\displaystyle p}$ (in dollars) and the demand ${\displaystyle q}$ for a product are related by the following demand equation: ${\displaystyle p^{3}+q+q^{3}=38}$ Suppose the price
increases at a rate of ${\displaystyle \7}$/month. How fast does the demand decrease when the demand is ${\displaystyle q=3}$?
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still
stuck, go for the next hint.
Hint 1
What variable are you taking the derivative of demand with respect to (think about what unit of measurement is “months”)?
Hint 2
Implicitly differentiate with respect to time.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
From the previous part, we know that when ${\displaystyle q=3}$, by the relation give, ${\displaystyle p=2}$. Now, we take that above relation and differentiate it with respect to time: ${\
displaystyle 3p^{2}{\frac {dp}{dt}}+{\frac {dq}{dt}}+3q^{2}{\frac {dq}{dt}}=0}$ We know that ${\displaystyle {\frac {dp}{dt}}=7,p=2,q=3}$. Substitute into the new found relation and isolate for ${\
displaystyle {\frac {dq}{dt}}}$ gives:
${\displaystyle {\frac {dq}{dt}}={\frac {-3(2)^{2}(7)}{1+3(3)^{2}}}={\frac {-84}{28}}=-3}$.
So the demand is decreasing at a rate of ${\displaystyle 3}$ units/month.
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Calculating momentum when you know mass and KE
• Thread starter connie5828
• Start date
In summary, the momentum of an object with a mass of 8 kg and a kinetic energy of 70 J can be calculated by first using the equation KE=mv^2/2 to determine the object's velocity, and then using the
equation P=mv to calculate its momentum. The correct calculations for this problem would result in a momentum of 33 kg*m/s.
Homework Statement
Calculate the momentum of an object of mass 8 kg if its kinetic energy is 70 J
Homework Equations
I am not sure what the equation is. Every one I have tried hasn't worked
The Attempt at a Solution
KE=MV squared?
Kinetic energy is not equal to mv^2.
Write down the definitions of kinetic energy and momentum. (You do know them, right?) Then combine them by solving for the variable you don't know.
Hello connie5828,
Welcome to Physics Forums!
connie5828 said:
Homework Statement
Calculate the momentum of an object of mass 8 kg if its kinetic energy is 70 J
Homework Equations
I am not sure what the equation is. Every one I have tried hasn't worked
Well, you know the object's kinetic energy and mass, so what does that say about the object's velocity? If you happen to figure out the object's velocity, and since you know its mass, then its
momentum is ...?
You'll frequently find that physics is usually not about simply plugging in numbers into an existing equation (although that does happen sometimes). Much of the time you'll find that multiple laws/
equations are necessary to be combined to figure out a specific problem. Which is why it's a good idea to reflect upon the equations to figure out what those equations really
, and how they apply to various problems.
The Attempt at a Solution
KE=MV squared?
Close, but not quite right. You'll need the correct equation for kinetic energy (it's almost what you wrote above, but not quite), and another equation which describes momentum.
[Edit: I see diazona beat me to the answer.]
looks like my first reply may not have posted. thanks for your help. Please correct what I am not understanding
so equation would be 70=8^2/2
70=64/2 (is this correct??)
the problem is I am using the practice equation to try to figure this out and it says the answer is 33. My calculations are definitely off somewhere. Please help.
Hi Connie, you've got the definitions right now, but you squared the wrong thing. KE is mv^2/2 = 8v^2/2, but you wrote 8^2/2 = m^2/2.
Start with the equation 70 = mv^2/2. Substitute for the mass: 70 = 8v^2/2. Then solve for v. That means keep doing the same thing (multiply or divide by the same numbers) to each side of the equation
till you have it in the form v^2 = some number. Then take the square root of each side, so you have v = the square root of that number.
Then go to your other equation, momentum = mv, and replace the v with what you found v was equal to.
Last edited:
totally got it. THANK YOU! Taking anatomy and Physics and anatomy I rock at Physic I learning with the help of people who have amazing science brains. Thanks!
FAQ: Calculating momentum when you know mass and KE
1. How do you calculate momentum when you know mass and kinetic energy?
To calculate momentum when you know mass and kinetic energy, you can use the formula p = √(2mKE), where p is momentum, m is mass, and KE is kinetic energy.
2. Why is it important to calculate momentum?
Calculating momentum is important because it helps us understand the motion of objects and how they interact with each other. It is also a fundamental concept in physics and is used in various
applications such as predicting collisions and designing machines.
3. Can you calculate momentum without knowing the mass or kinetic energy?
No, momentum cannot be calculated without knowing either the mass or the kinetic energy of an object. These two variables are essential in determining the momentum of an object.
4. How is momentum related to velocity?
Momentum is directly proportional to velocity. This means that as velocity increases, momentum also increases. The equation for this relationship is p = mv, where p is momentum, m is mass, and v is
5. What is the unit of measurement for momentum?
The unit of measurement for momentum is kilogram-meter per second (kg·m/s). This unit is derived from the formula for momentum, p = mv, where mass is measured in kilograms (kg) and velocity in meters
per second (m/s). | {"url":"https://www.physicsforums.com/threads/calculating-momentum-when-you-know-mass-and-ke.437511/","timestamp":"2024-11-02T01:14:44Z","content_type":"text/html","content_length":"95244","record_id":"<urn:uuid:bde22c71-bc46-4adb-b35c-d7d7287e72d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00817.warc.gz"} |
C++ Program to Swap Two Numbers
Last updated on 02/04/2024
Swapping two numbers is a classic problem in computer science and programming, often used to introduce beginners to concepts such as variables, data manipulation, and sometimes pointers or
references. In C++, there are multiple ways to swap two numbers, each illustrating different facets of the language and problem-solving strategies.
Understanding the Basics
At its simplest, swapping two numbers means that if you start with two variables, say a and b, after the operation, the value of a should be in b and vice versa. This seemingly straightforward task
can be achieved in various ways in C++.
Method 1: Using a Temporary Variable
The most intuitive method involves using a third variable to temporarily hold the value of one of the numbers during the swap process.
#include <iostream>
using namespace std;
int main() {
int a = 5, b = 10, temp;
cout << “Before swapping:” << endl;
cout << “a = ” << a << “, b = ” << b << endl;
temp = a;
a = b;
b = temp;
cout << “After swapping:” << endl;
cout << “a = ” << a << “, b = ” << b << endl;
return 0;
This method is easy to understand and visualize, making it an excellent teaching tool. It directly translates the mental model of swapping two items into code: you need a place to temporarily set
down one of the items while you move the other.
Method 2: Swap Without a Temporary Variable
A more sophisticated approach involves swapping numbers without using a temporary variable. This can be done using arithmetic operations or bitwise XOR operations.
• Using Arithmetic Operations:
a = a + b;
b = a – b; // Now, b becomes the original value of a
a = a – b; // Now, a becomes the original value of b
• Using Bitwise XOR Operations:
a = a ^ b;
b = a ^ b; // Now, b is a
a = a ^ b; // Now, a is b
Both these methods eliminate the need for a temporary variable by using the properties of arithmetic and bitwise operations, respectively. These techniques, while clever and efficient in terms of
memory usage, can introduce problems. For instance, the arithmetic method might cause overflow if the numbers are too large, and both methods could be less readable to those unfamiliar with these
Method 3: Using Standard Library Function
C++ offers a standardized way to swap values using the std::swap function, showcasing the language’s rich library support.
#include <iostream>
#include <utility> // For std::swap, included by iostream in C++11 and later
int main() {
int a = 5, b = 10;
std::cout << “Before swapping:” << std::endl;
std::cout << “a = ” << a << “, b = ” << b << std::endl;
std::swap(a, b);
std::cout << “After swapping:” << std::endl;
std::cout << “a = ” << a << “, b = ” << b << std::endl;
return 0;
This method not only simplifies the code by abstracting the swapping logic into a library function but also enhances readability and reduces the likelihood of errors. It reflects a key principle in
software development: reusability. Why reinvent the wheel when the language provides a built-in, well-tested function?
Deep Dive: Concepts and Principles
Each swapping method illuminates different programming concepts and best practices:
• Variable and Memory Management:
Using a temporary variable to swap two numbers is a straightforward application of variable and memory management, illustrating how values are stored and moved.
• Mathematical and Logical Operations:
The arithmetic and XOR methods showcase how mathematical and logical operations can be leveraged to manipulate data in non-obvious ways, encouraging problem-solving skills and a deeper understanding
of data representation.
• Library Functions and Code Reusability:
The use of std::swap highlights the importance of familiarizing oneself with a language’s standard library, promoting code reusability and maintainability. | {"url":"https://indiafreenotes.com/c-program-to-swap-two-numbers/","timestamp":"2024-11-09T00:20:08Z","content_type":"text/html","content_length":"219131","record_id":"<urn:uuid:e9a5c3c4-30ee-438b-9032-83d249bcd6de>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00855.warc.gz"} |
Math/Physic/Economic/Statistic Problems Archives - ESSAY MAMBA
ELECTRIC CHARGE Charge on 1 electron = 1.6 x 10-19 coulomb If a current of 6.0 amp flows for 7.0 seconds, how much charge .has moved? If ~a current of 7.2 amp flows for 25 seconds, how much charge
has been transferred? What is the current flowing if 96 coulomb pass a point in a […] | {"url":"https://www.essaymamba.com/category/math-physic-economic-statistic-problems/","timestamp":"2024-11-02T18:27:47Z","content_type":"text/html","content_length":"56334","record_id":"<urn:uuid:50f4fe21-96c0-4a6a-bf16-fe7ab1877857>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00077.warc.gz"} |
Fair Value Calculator for Stocks - TECHCUBE
Our Fair Value Calculator is a tool designed to assist investors in estimating the intrinsic value of stocks. By inputting key financial metrics such as Earnings Per Share (EPS), expected growth
rate, and Price-to-Earnings (P/E) ratio, users can quickly assess the potential fair value of an investment.
This calculator streamlines the process of stock valuation, helping investors make more informed decisions. To get started, simply enter the required data to estimate the fair market value of a
Fair Value Calculator
ⓘ Enter the earnings per share in dollars.
ⓘ Enter the expected annual growth rate in percent.
ⓘ Enter the expected price-to-earnings ratio.
Using the Fair Value Calculator:
1. Earnings Per Share (EPS): Enter the stock’s EPS, which represents the company’s profit allocated to each outstanding share of common stock.
2. Expected Growth Rate: Input the anticipated annual growth rate of the company’s earnings, based on historical data and future projections.
3. Price-to-Earnings Ratio (P/E): Enter the P/E ratio, which reflects the market value relative to earnings. This can be derived from industry averages or personal estimates.
After entering the required information, click “Calculate” to determine the estimated fair value of the stock. This result can help guide investment decisions by providing a benchmark for comparison
with the current market price.
Understanding Fair Value Calculation
Calculating a stock’s fair value is an important step in investment analysis. It provides an estimate of a stock’s intrinsic worth based on fundamental financial metrics. Here’s how to calculate the
fair value of a stock:
1. Collect Financial Data
Gather the following information:
Earnings Per Share (EPS): Calculated as:
\( \text{EPS} = \frac{\text{Net Income} – \text{Preferred Stock Dividends}}{\text{Average Outstanding Shares}}
Expected Growth Rate: The anticipated annual growth rate of the company’s earnings.
Price-to-Earnings (P/E) Ratio: Calculated as:
\( \text{P/E Ratio} = \frac{\text{Market Price per Share}}{\text{EPS}}
2. Apply the Fair Value Formula
Use this formula to calculate fair value:
\(\text{Fair Value} = \text{EPS} \times (1 + \text{Growth Rate}) \times \text{P/E Ratio}\)
3. Example Calculation
For a company with an EPS of $5.00, an expected growth rate of 8%, and a P/E ratio of 20:
\(\text{Fair Value} = 5.00 \times (1 + 0.08) \times 20 = $108.00\)
4. Interpret the Results
Compare the calculated fair value to the current market price:
• If the market price is lower than the fair value, the stock may be undervalued.
• If the market price is higher than the fair value, the stock may be overvalued.
This calculation helps investors assess whether a stock might be a good buy, hold, or sell opportunity based on its estimated intrinsic value compared to its market price. | {"url":"https://techcu.be/finance/fair-value-calculator-for-stocks/","timestamp":"2024-11-09T14:18:39Z","content_type":"text/html","content_length":"57597","record_id":"<urn:uuid:98141265-9cb1-4f09-9b1e-d1fb83b56fa4>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00281.warc.gz"} |
200 research outputs found
We study the spectral properties of the magnitudes of river flux increments, the volatility. The volatility series exhibits (i) strong seasonal periodicity and (ii) strongly power-law correlations
for time scales less than one year. We test the nonlinear properties of the river flux increment series by randomizing its Fourier phases and find that the surrogate volatility series (i) has almost
no seasonal periodicity and (ii) is weakly correlated for time scales less than one year. We quantify the degree of nonlinearity by measuring (i) the amplitude of the power spectrum at the seasonal
peak and (ii) the correlation power-law exponent of the volatility series.Comment: 5 revtex pages, 6 page
The current literature on sandpile models mainly deals with the abelian sandpile model (ASM) and its variants. We treat a less known - but equally interesting - model, namely Zhang's sandpile. This
model differs in two aspects from the ASM. First, additions are not discrete, but random amounts with a uniform distribution on an interval $[a,b]$. Second, if a site topples - which happens if the
amount at that site is larger than a threshold value $E_c$ (which is a model parameter), then it divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the
infinite volume limit, this model tends to behave like the ASM in the sense that the stationary measure for the system in large volumes tends to be peaked narrowly around a finite set. This belief is
supported by simulations, but so far not by analytical investigations. We study the stationary distribution of this model in one dimension, for several values of $a$ and $b$. When there is only one
site, exact computations are possible. Our main result concerns the limit as the number of sites tends to infinity, in the one-dimensional case. We find that the stationary distribution, in the case
$a \geq E_c/2$, indeed tends to that of the ASM (up to a scaling factor), in agreement with Zhang's conjecture. For the case $a=0$, $b=1$ we provide strong evidence that the stationary expectation
tends to $\sqrt{1/2}$.Comment: 47 pages, 3 figure
We show that the well established Olami-Feder-Christensen (OFC) model for the dynamics of earthquakes is able to reproduce a new striking property of real earthquake data. Recently, it has been
pointed out by Abe and Suzuki that the epicenters of earthquakes could be connected in order to generate a graph, with properties of a scale-free network of the Barabasi-Albert type. However, only
the non conservative version of the Olami-Feder-Christensen model is able to reproduce this behavior. The conservative version, instead, behaves like a random graph. Besides indicating the robustness
of the model to describe earthquake dynamics, those findings reinforce that conservative and non conservative versions of the OFC model are qualitatively different. Also, we propose a completely new
dynamical mechanism that, even without an explicit rule of preferential attachment, generates a free scale network. The preferential attachment is in this case a ``by-product'' of the long term
correlations associated with the self-organized critical state. The detailed study of the properties of this network can reveal new aspects of the dynamics of the OFC model, contributing to the
understanding of self-organized criticality in non conserving models.Comment: 7 pages, 7 figure
We have studied precursors of the global failure in some self-organised critical models of sand-pile (in BTW and Manna models) and in the random fiber bundle model (RFB). In both BTW and Manna model,
as one adds a small but fixed number of sand grains (heights) to any central site of the stable pile, the local dynamics starts and continues for an average relaxation time (\tau) and an average
number of topplings (\Delta) spread over a radial distance (\xi). We find that these quantities all depend on the average height (h_{av}) of the pile and they all diverge as (h_{av}) approaches the
critical height (h_{c}) from below: (\Delta) (\sim (h_{c}-h_{av}))(^{-\delta}), (\tau \sim (h_{c}-h_{av})^{-\gamma}) and (\xi) (\sim) ((h_{c}-h_{av})^{-\nu}). Numerically we find (\delta \simeq 2.0),
(\gamma \simeq 1.2) and (\nu \simeq 1.0) for both BTW and Manna model in two dimensions. In the strained RFB model we find that the breakdown susceptibility (\chi) (giving the differential increment
of the number of broken fibers due to increase in external load) and the relaxation time (\tau), both diverge as the applied load or stress (\sigma) approaches the network failure threshold (\sigma_
{c}) from below: (\chi) (\sim) ((\sigma_{c}) (-)(\sigma)^{-1/2}) and (\tau) (\sim) ((\sigma_{c}) (-)(\sigma)^{-1/2}). These self-organised dynamical models of failure therefore show some definite
precursors with robust power laws long before the failure point. Such well-characterised precursors should help predicting the global failure point of the systems in advance.Comment: 13 pages, 9
figures (eps
Particle size distribution (PSD) greatly influences other soil physical properties. A detailed textural analysis is time-consuming and expensive. Soil texture is commonly reported in terms of mass
percentages of a small number of size fractions (typically, clay, silt and sand). A method to simulate the PSD from such a poor description or even from the poorest description, consisting in the
mass percentages of only two soil size fractions, would be extremly useful for prediction purposes. The goal of this paper is to simulate soil PSDs from the minimum number of inputs, i.e., two and
three textural fraction contents, by using a logselfsimilar model and an iterated function system constructed with these data. High quality data on 171 soils are used. Additionally, the
characterization of soil texture by entropy-based parameters provided by the model is tested. Results indicate that the logselfsimilar model may be a useful tool to simulate PSD for the construction
of pedotransfer functions related to other soil properties when textural information is limited to moderate textural data
We study the surface roughness of prototype models displaying self-organized criticality (SOC) and their noncritical variants in one dimension. For SOC systems, we find that two seemingly equivalent
definitions of surface roughness yields different asymptotic scaling exponents. Using approximate analytical arguments and extensive numerical studies we conclude that this ambiguity is due to the
special scaling properties of the nonlinear steady state surface. We also find that there is no such ambiguity for non-SOC models, although there may be intermediate crossovers to different roughness
values. Such crossovers need to be distinguished from the true asymptotic behaviour, as in the case of a noncritical disordered sandpile model studied in [10].Comment: 5 pages, 4 figures. Accepted
for publication in Phys. Rev.
We numerically investigate the Olami-Feder-Christensen model for earthquakes in order to characterise its scaling behaviour. We show that ordinary finite size scaling in the model is violated due to
global, system wide events. Nevertheless we find that subsystems of linear dimension small compared to the overall system size obey finite (subsystem) size scaling, with universal critical
coefficients, for the earthquake events localised within the subsystem. We provide evidence, moreover, that large earthquakes responsible for breaking finite size scaling are initiated predominantly
near the boundary.Comment: 6 pages, 6 figures, to be published in Phys. Rev. E; references sorted correctl
Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We
introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological self-similarity and its generalization is proved in an
asymptotic sense. Also, some important examples are presented
We study simultaneous price drops of real stocks and show that for high drop thresholds they follow a power-law distribution. To reproduce these collective downturns, we propose a minimal
self-organized model of cascade spreading based on a probabilistic response of the system elements to stress conditions. This model is solvable using the theory of branching processes and the
mean-field approximation. For a wide range of parameters, the system is in a critical state and displays a power-law cascade-size distribution similar to the empirically observed one. We further
generalize the model to reproduce volatility clustering and other observed properties of real stocks.Comment: 8 pages, 6 figure | {"url":"https://core.ac.uk/search/?q=author%3A(D.L.%20Turcotte)","timestamp":"2024-11-08T15:11:43Z","content_type":"text/html","content_length":"210431","record_id":"<urn:uuid:1214fb70-496e-4b5a-9c67-21eb4d39d177>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00446.warc.gz"} |
Dataset for: Bedding scale correlation on Mars in western Arabia Terra
Dataset for: Bedding scale correlation on Mars in western Arabia Terra A.M. Annex et al. Data Product Overview This repository contains all source data for the publication. Below is a description of
each general data product type, software that can load the data, and a list of the file names along with the short description of the data product. HiRISE Digital Elevation Models (DEMs). HiRISE DEMs
produced using the Ames Stereo Pipeline are in geotiff format ending with ‘*X_0_DEM-adj.tif’, the “X” prefix denotes the spatial resolution of the data product in meters. Geotiff files are able to be
read by free GIS software like QGIS. HiRISE map-projected imagery (DRGs). Map-projected HiRISE images produced using the Ames Stereo Pipeline are in geotiff format ending with ‘*0_Y_DRG-cog.tif’, the
“Y” prefix denotes the spatial resolution of the data product in centimeters. Geotiff files are able to be read by free GIS software like QGIS. The DRG files are formatted as COG-geotiffs for
enhanced compression and ease of use. 3D Topography files (.ply). Traingular Mesh versions of the HiRISE/CTX topography data used for 3D figures in “.ply” format. Meshes are greatly geometrically
simplified from source files. Topography files can be loaded in a variety of open source tools like ParaView and Meshlab. Textures can be applied using embedded texture coordinates. 3D Geological
Model outputs (.vtk) VTK 3D file format files of model output over the spatial domain of each study site. VTK files can be loaded by ParaView open source software. The “block” files contain the model
evaluation over a regular grid over the model extent. The “surfaces” files contain just the bedding surfaces as interpolated from the “block” files using the marching cubes algorithm. Geological
Model geologic maps (geologic_map.tif). Geologic maps from geological models are standard geotiffs readable by conventional GIS software. The maximum value for each geologic map is the “no-data”
value for the map. Geologic maps are calculated at a lower resolution than the topography data for storage efficiency. Beds Geopackage File (.gpkg). Geopackage vector data file containing all mapped
layers and associated metadata including dip corrected bed thickness as well as WKB encoded 3D linestrings representing the sampled topography data to which the bedding orientations were fit.
Geopackage files can be read using GIS software like QGIS and ArcGIS as well as the OGR/GDAL suite. A full description of each column in the file is provided below. Column Type Description uuid
String unique identifier stratum_order Real 0-indexed bed order section Real section number layer_id Real bed number/index layer_id_bk Real unused backup bed number/index source_raster String dem
file path used raster String dem file name gsd Real ground sampling distant for dem wkn String well known name for dem rtype String raster type minx Real minimum x position of trace in dem crs miny
Real minimum y position of trace in dem crs maxx Real maximum x position of trace in dem crs maxy Real maximum y position of trace in dem crs method String internal interpolation method sl Real slope
in degrees az Real azimuth in degrees error Real maximum error ellipse angle stdr Real standard deviation of the residuals semr Real standard error of the residuals X Real mean x position in CRS Y
Real mean y position in CRS Z Real mean z position in CRS b1 Real plane coefficient 1 b2 Real plane coefficient 2 b3 Real plane coefficient 3 b1_se Real standard error plane coefficient 1 b2_se Real
standard error plane coefficient 2 b3_se Real standard error plane coefficient 3 b1_ci_low Real plane coefficient 1 95% confidence interval low b1_ci_high Real plane coefficient 1 95% confidence
interval high b2_ci_low Real plane coefficient 2 95% confidence interval low b2_ci_high Real plane coefficient 2 95% confidence interval high b3_ci_low Real plane coefficient 3 95% confidence
interval low b3_ci_high Real plane coefficient 3 95% confidence interval high pca_ev_1 Real pca explained variance ratio pc 1 pca_ev_2 Real pca explained variance ratio pc 2 pca_ev_3 Real pca
explained variance ratio pc 3 condition_number Real condition number for regression n Integer64 number of data points used in regression rls Integer(Boolean) unused flag demeaned_regressions Integer
(Boolean) centering indicator meansl Real mean section slope meanaz Real mean section azimuth angular_error Real angular error for section mB_1 Real mean plane coefficient 1 for section mB_2 Real
mean plane coefficient 2 for section mB_3 Real mean plane coefficient 3 for section R Real mean plane normal orientation vector magnitude num_valid Integer64 number of valid planes in section meanc
Real mean stratigraphic position medianc Real median stratigraphic position stdc Real standard deviation of stratigraphic index stec Real standard error of stratigraphic index
was_monotonic_increasing_layer_id Integer(Boolean) monotonic layer_id after projection to stratigraphic index was_monotonic_increasing_meanc Integer(Boolean) monotonic meanc after projection to
stratigraphic index was_monotonic_increasing_z Integer(Boolean) monotonic z increasing after projection to stratigraphic index meanc_l3sigma_std Real lower 3-sigma meanc standard deviation
meanc_u3sigma_std Real upper 3-sigma meanc standard deviation meanc_l2sigma_sem Real lower 3-sigma meanc standard error meanc_u2sigma_sem Real upper 3-sigma meanc standard error thickness Real
difference in meanc thickness_fromz Real difference in Z value dip_cor Real dip correction dc_thick Real thickness after dip correction dc_thick_fromz Real z thickness after dip correction
dc_thick_dev Integer(Boolean) dc_thick 15 xyz_wkb_hex String hex encoded wkb geometry for all points used in regression Geological Model input files (.gpkg). Four geopackage (.gpkg) files represent
the input dataset for the geological models, one per study site as specified in the name of the file. The files contain most of the columns described above in the Beds geopackage file, with the
following additional columns. The final seven columns (azimuth, dip, polarity, formation, X, Y, Z) constituting the actual parameters used by the geological model (GemPy). Column Type Description
azimuth_mean String Mean section dip azimuth azimuth_indi Real Individual bed azimuth azimuth Real Azimuth of trace used by the geological model dip Real Dip for the trace used by the geological mode
polarity Real Polarity of the dip vector normal vector formation String String representation of layer_id required for GemPy models X Real X position in the CRS of the sampled point on the trace Y
Real Y position in the CRS of the sampled point on the trace Z Real Z position in the CRS of the sampled point on the trace Stratigraphic Column Files (.gpkg). Stratigraphic columns computed from the
Geological Models come in three kinds of Geopackage vector files indicated by the postfixes _sc, rbsc, and rbssc. File names include the wkn site name. sc (_sc.gpkg). Geopackage vector data file
containing measured bed thicknesses from Geological Model joined with corresponding Beds Geopackage file, subsetted partially. The columns largely overlap with the the list above for the Beds
Geopackage but with the following additions Column Type Description X Real X position of thickness measurement Y Real Y position of thickness measurement Z Real Z position of thickness measurement
formation String Model required string representation of bed index bed thickness (m) Real difference of bed elevations azimuths Real azimuth as measured from model in degrees dip_degrees Real dip as
measured from model in degrees Dip corrected bed thickness (m) Real dip corrected bed thickness in meters lower_point Real lower bed elevation in meters upper_point Real upper bed elevation in meters
_formation Real integer number of formation string layer_iid Integer64 integer number of layer_id bascom_baryte_diff_bt Real diff. in thickness from geomodel measurements bascom_baryte_diff_dcbt Real
diff. in dip cor. thicknesses ’’ rbsc (rbsc.gpkg) Geopackage vector file containing virtual boreholes with high resolution vertical sampling placed in a regular grid in the spatial extent of the DEM
with the following columns. Column Type Description formation String Model required string representation of bed index pred_z Real Z value of bedding plane layer_id Integer64 Bed Index from Beds
Geopackage file section Real section number thickness Real thickness of the layer predicted by model geom Point contains X,Y,Z of point in dem CRS rbssc (rbssc.gpkg) Geopackage vector file containing
virtual boreholes with high resolution vertical sampling placed at the centroids for each section within the spatial extent of the DEM with the following columns. Column Type Description formation
String Model required string representation of bed index pred_z Real Z value of bedding plane layer_id Integer64 Bed Index from Beds Geopackage file section Real section number thickness Real
thickness of the layer predicted by model geom Point contains X,Y,Z of point in dem CRS crescent_shapes.gpkg Geopackage vector file containing the measurements of the crescent features. Column Type
Description azimuth Real Azimuth of the crescent feature in degrees
Date made available Feb 13 2023
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What undefined term is needed to define an angle?
What undefined term is needed to define an angle?
We can define an angle using the undefined term of a line.
Undefined Terms in Geometry
In geometry, we have three terms that are coined as undefined terms. Those are points, lines, and planes. These are called undefined terms, because we cannot define them using any other geometrical
The best we can do for a definition of these terms is to name them, but we do not have formal definitions for these three terms.
However, these three undefined terms are the terms that we use to define all of the other terms used in geometry, so they are extremely important in this branch of mathematics.
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