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New Frontiers in Machine Learning and Quantum
Beni Yoshida (Perimeter Institute), Roger Melko (Perimeter Institute & University of Waterloo)
11/22/22, 9:20AM
Mario Krenn (Max Planck Institute for the Science of Light)
11/22/22, 10:30AM
Juan Felipe Carrasquilla Álvarez (Vector Institute & University of Toronto)
11/22/22, 3:45PM
Stefanie Czischek (University of Ottawa)
11/23/22, 9:30AM
Hannah Lange (Harvard University)
11/23/22, 11:15AM
Beni Yoshida (Perimeter Institute), Roger Melko (Perimeter Institute & University of Waterloo)
11/23/22, 5:00PM | {"url":"https://events.perimeterinstitute.ca/event/31/contributions/","timestamp":"2024-11-11T03:46:09Z","content_type":"text/html","content_length":"119367","record_id":"<urn:uuid:e6e015a0-cc08-461a-a2b4-74217066a3e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00624.warc.gz"} |
help!! - College School Essays
Lesson Objectives:
• Student will write null and alternative hypotheses.
• Student will find critical values for testing a mean or proportion against the population mean or proportion using the appropriate test, based on the sample size.
• Student will find critical values for testing the difference between two means or two proportions using the appropriate test, based on the sample size.
What is a hypothesis?
In science, you may have learned that the hypothesis is an educated guess. In statistics, the same definition carries over but has some different applications. A statistical study is similar to the
scientific method. From science you have learned that the scientific method includes the following steps: 1) Ask a question 2) Do background research 3) Construct a hypothesis 4) Test your hypothesis
by doing an experiment 5) Analyze the data and draw a conclusion 6) Communicate the results. In statistics, there are two hypotheses that need to be formed once you have defined the problem and
completed background research. One is called the “null” hypothesis and the other is called the “alternative” hypothesis. Once the study is conducted, we can reject or fail to reject either of the
hypothesis based on the results of the study.
The Null Hypothesis
The null hypothesis is composed of the fact that there is no effect of the treatment on the subjects in the study. For example if we were trying to investigate the relationship between two variables
our null hypothesis may state that “there is no relationship between the two variables” or if we are trying to see if a new drug has an effect on weight gain the null hypothesis may state that “the
drug has no effect on the weight gain of the subjects”. The null hypothesis is the one that we will fail to reject (accept) unless the data provides convincing evidence that it is false.
The Alternative Hypothesis
The alternative hypothesis may be referred to as the opposite of the null hypothesis. For example, if the null hypothesis states that there is no relationship between two variables, then the
alternative hypothesis should state that “there is a relationship between the two variables that can be measured. If the null hypothesis states that there is no effect on the subject then the
alternative hypothesis should state that “there is an effect on the subject”. We will fail to reject (accept) the alternative hypothesis if and only if the data provides convincing evidence that it
is true.
Practice Writing Null and Alternative Hypotheses
The hypotheses can be written out in words or we may use mathematical symbols to express the hypothesis. Here a few examples of how to write the null and alternative hypothesis. The most common
symbol for the null hypothesis is H[0] and the most common symbol for the alternative hypothesis is H[1.]
Let’s Practice:[ ][]
Case I: An agriculturist is doing a study to determine if a fertilizer has any effect on the average height of 100 apple trees. He knows that the average height of unfertilized apple trees is 10ft.
The average height of the 100 apple trees that were treated with fertilizer is 10.8 feet with a standard deviation of .5 ft.
1) Do you think that the fertilizer has an effect on the height of the apple trees?
H[0] = The fertilizer has no effect on the height of the apple trees Sample mean = Population mean
H[1] = The fertilizer does have an effect on the height of the apple trees. Sample mean ≠ Population mean
2) Does the fertilizer make the apple trees taller?
H[0 ]= The fertilizer does not make the apple trees taller. Sample mean = Population mean
H[1] = The fertilizer does make the apple trees taller. Sample mean > Population mean
Remember that when writing and testing hypotheses it is very important that you consider the question that want to answer with your study because this fact helps to shape the correct hypotheses.
Choosing the Appropriate Test
Choosing the appropriate test for any statistical research is very important to obtaining the most accurate results. We discussed in a previous lesson when to use a z-score or z-test and when to use
a t-score or a t-test. Recall that we use a z-test when the sample size is fairly large (greater than 100) and a t-test when the sample size is small (less than 100). Another question that we need to
consider is when to use a two- tailed test or a one- tailed test. This choice is made based upon the statement of the alternative hypothesis. We would use a two-tailed test if our alternative
hypothesis is the exact opposite of the null hypothesis. We would use a one-tailed test if our alternative hypothesis suggests a certain direction for the results.
If H[0][ =] There is no effect on the variable , then H[1] = There is an effect on the variable.
In this case we would use a two tailed test.
If H[0][ =] The mean sample mean is equal to the population mean , then H[1] = The sample mean is greater than the population mean.
In this case we would use the one-tailed test since we are concerned with only the right side of the probability distribution where the values are that are greater than the mean.
Click here for a video explaining when to use a one-tailed test and when to use a two tailed test.
Critical Values & Testing a Hypothesis
Testing a hypothesis includes finding the appropriate z or t score and finding critical values to compare it to the probability of an event and its position on the normal distribution curve given a
confidence interval or p-value. A critical value is the value of the dependent variable at a given point of a function that helps us to decide to reject or fail to reject the null hypothesis. If the
test statistic is this number or more then we will reject the null hypothesis but if the test statistic is less than this number then we will fail to reject or accept the null hypothesis.
Consider these examples:
Testing the mean against the population mean
An agriculturist is doing a study to determine if a fertilizer has any effect on the average height of 100 apple trees. He knows that the average height of unfertilized apple trees is 10ft. The
average height of the 100 apple trees that were treated with fertilizer is 10.1 feet with a standard deviation of 0.5 ft. Do you think that the fertilizer has an effect on the height of the apple
Step 1: Write the null and alternative hypotheses
H[0] = The fertilizer has no effect on the height of the apple trees Sample mean = Population mean
H[1] = The fertilizer does have an effect on the height of the apple trees. Sample mean ≠ Population mean
Step 2: We will assume that the null hypothesis is true and find the z-score (since we have a large sample).
z = 10 – 10.1/(.5/√100) = -2
Step 3: Find the critical value by applying the subtracting (since the z-score is negative) the product of the z-score and the standard deviation from the population mean.
10 – 2(.5) = 9 so the critical value is 9
Since the mean of the sample is greater than this critical value we must reject the null hypothesis.
Testing a proportion against the population proportion
A company found that in a sample of 100 of its products that 25 were defective after retraining the employees. If there is an overall 60% chance that the company will produce a defective product, did
the training help employees minimize the number of defective products?
Step 1:
Write the null and alternative hypothesis.
H[0 =] p = 60% or .6 The training had no effect on the proportion of defective products
H[1 ]= p < 60% or .6 The training helped to decrease the proportion of defective products.
Remember that these types of problems are binomial experiments so we have to be sure that we can use the normal approximation by confirming that (n)(p) and (n)(q) are both greater than 5. So (100)
(.6) = 60 and (100)(.4) = 40, we may proceed.
Step 2:
Assume that the null hypothesis is true and find the standard deviation ( standard deviation = √(n)(p(q)) so that we can find the z-score.
standard deviation = 4.33
z = .25 – .6/ 4.33 = -.08
Step 3: Find the critical value by subtracting (since the z-score is negative) the product of the z-score and the standard deviation from the mean.
.6 – .3464 = .2536 or 25.36 % since our test statistic is .6 or 60% and it is more than the critical value then we must reject the null hypothesis and conclude that training did decrease the number
of defective products.
Testing the Difference of Means and Difference of Proportions
The difference of means or difference of proportions is used to find out whether there is a significance between the controlled group and the uncontrolled group. We can also use critical values to
determine the significance for the difference of means as well as the difference of proportions.
Difference of Means:
A teacher gave 100 students a study guide in preparation for a major test. The average score for these students was 88 with standard deviation of 2. She did not give study guides to another group of
100 students and the average score for these students was 80 with a standard deviation of 3. Determine whether the study guide had an effect on student scores.
Step 1: Find the difference in the sample mean scores. 88-80 = 8
Step 2: Write the null and alternative hypotheses.
H[0] = The study guide has no effect on student test scores or population mean[1] = population mean[2] or population mean[1] – population mean[2] = 0[
]H[1] = The study guide has an effect on student test scores or population mean[1] ≠ population mean[2] or population mean[1] – population mean[2] ≠ 0
]Step 3: Calculate the standard deviation for the difference of the sample means.
To do this we need to find the variance (square the standard deviation for each sample) and divide it by n for each sample then add the two values together and take the square root.
√[(2^2 )/100 + (3^2 )/100] = √4/100 + 3/100 = √7/100 = √.07 = .26
Step 4: Assume that the null hypothesis is true and find the z-score using the difference of the means.
z = 0-8/.26 = -30.76 find the critical value by adding (since the z-score is positive) the product of the z-score and the standard deviation to the test statistic.
0 – 8 = -8 This is the critical value and since the test statistic 0 is must than this value we must reject the null hypothesis and conclude that the study guide has an effect on student test scores.
Grading Rubric:
│Grading for this lesson: │
│ │
│To get a 10: All answers are correct the first time, or within first revision. │
│To get a 9: You can have 1 incorrect answer after your original submission. │
│To get an 8: You can have 2 incorrect answers after your original submission. │
│To get a 7: You can have 3 incorrect answers after your original submission. │
│To get a 6: You can have 4 incorrect answers after your original submission. │
│To get a 5: Cheating- Plagiarism – purposeful or mistaken, which will lower your final grade for the course (so be very careful when posting your work!); lack of effort, disrespect, or attitude (we│
│are here to communicate with you if you don’t understand something); lesson requirements have not been met. │
│ │
│Note: For this class it is necessary to post the questions over each answer. Failure to do so will result in asking for a revision. No grade will be given for incomplete work. │
For questions 1-5, write the null and alternative hypothesis.
1. Does the water temperature have an effect on the number of people in the pool?
2. Does the weather have an effect on the number of people at the beach?
3. A fitness center is running a discounted membership fee. Did the discount increase the membership sales? Write your hypotheses mathematically.
4. A medical researcher gave 100 patients a new drug to see if it reduces their blood pressure? Did the new drug reduce the patients’ blood pressure? Write your hypothesis mathematically.
5. Some students took a conflict resolution class? Did this class help to reduce the number of conflicts that the students were involved in? Write your hypothesis mathematically.
For questions 6-10, find the critical value.
6. There is an annual hot dog eating contest in Plattsburg, MS and the average number of hot dogs eaten by one person is 36 with a standard deviation of 6. Find the critical value for a person who
can eat more than 2 deviations above the mean.
7. Female high school seniors at a particular school have an average height of 65 inches with a standard deviation of 5 inches. Find the critical value for a female high senior who is less than 1
deviation from the mean.
8. The average weight of a newborn at a particular hospital is 96 ounces with a standard deviation of 3 ounces. Find the critical value for a newborn who is 2 standard deviations below the average
9. A company found that 27 out of 150 of its products were defective after retraining its employees. If the company normally has a 40% defective product rate, find the critical value to determine if
retraining the employees helped to minimize the number of defective products.
10. A teacher gives 200 students a study guide for a test and the average score was 90 with a standard deviation of 6. She did not give the other 200 students a study guide and their average score
was 70 with a standard deviation of 8. Find the critical value to determine whether or not the study guide helped students to increase their test score.
If you need assistance with writing your assignment, essay, our professional assignments / essay writing service is here to help!
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Entropy Generation and Consequences of MHD in Darcy–Forchheimer Nanofluid Flow Bounded by Non-Linearly Stretching Surface
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 72915, Vietnam
Department of Mathematics, Cankaya University, Ankara 06530, Turkey
Institute of Space Sciences, 077125 Magurele, Romania
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40250, Taiwan
Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia
Department of Mechanical Engineering, École de Technologie Supérieure, ÉTS, Montreal, QC H3C 1K3, Canada
Author to whom correspondence should be addressed.
Submission received: 18 March 2020 / Revised: 27 March 2020 / Accepted: 29 March 2020 / Published: 20 April 2020
Present communication aims to inspect the entropy optimization, heat and mass transport in Darcy-Forchheimer nanofluid flow surrounded by a non-linearly stretching surface. Navier-Stokes model based
governing equations for non-Newtonian nanofluids having symmetric components in various terms are considered. Non-linear stretching is assumed to be the driving force whereas influence of thermal
radiation, Brownian diffusion, dissipation and thermophoresis is considered. Importantly, entropy optimization is performed using second law of thermodynamics. Governing problems are converted into
nonlinear ordinary problems (ODEs) using suitably adjusted transformations. RK-45 based built-in shooting mechanism is used to solve the problems. Final outcomes are plotted graphically. In addition
to velocity, temperature, concentration and Bejan number, the stream lines, contour graphs and density graphs have been prepared. For their industrial and engineering importance, results for
wall-drag force, heat flux (Nusselt) rate and mass flux (Sherwood) rate are also given in tabular data form. Outputs indicate that velocity reduces for Forchheimer number as well as for the porosity
factor. However, a rise is noted in temperature distribution for elevated values of thermal radiation. Entropy optimization shows enhancement for larger values of temperature difference ratio.
Skin-friction enhances for all relevant parameters involved in momentum equation.
1. Introduction
Tiny particles having diameter between 1–100 nm are termed nanoparticles. These particles belong to any suitable class of metals with significant thermo-physical properties. Conventional base
materials/liquids are treated through these particles. Though tiny particles are suspended in base liquid for a shorter period of time, the achieved results are more effective than using a simple
base fluid. This formulated mixture is known as nanofluid. These mixtures have enormous applications in engineering, industry and advanced nanotechnology. The manufacturing procedures at micro-level
industries, nuclear reactors and power engines are directly related to these formulations. Petroleum industry, chemical industry, geothermal industry, and other related fields are important in this
context. The pioneer study on this formulation has been reported by Choi [
]. The process of heat transport had shown much improvement after involvement of nanofluids. In particular, the dramatic effect is noted in power engines, refrigerators, nuclear reactors, fuel cells,
thermal management, etc. For example, Chamkha and Khaled [
] reported some similarity solutions of mixed convection and heat transfer in Darcy medium. Parvin and Chamkha [
] disclosed features of odd-shaped cavity filled with nanofluid. They also performed entropy optimization in this study. Zaraki et al. [
] analyzed natural convection and boundary layer phenomena in nanofluid flow. Reddy and Chamkha [
] discussed properties of
$A l 2 O 3$
$T i O 2$
-Water type nanofluid flow bounded by stretching surface. Chamkha et al. [
] discussed entropy optimization in
$C u O$
and water nanofluid flow subject to C-shaped cavity. Rasool et al. [
] reported Darcy relation in MHD nanofluid flow bounded by non-linear stretching surface. Ismael et al. [
] discussed heat flux and entropy optimization in nanofluid flow subject to porous medium. Sheikh and Abbas [
] analyzed consequences of heat generation and thermophoresis in MHD nanofluid flow subject to oscillatory stretched surface. Rasool et al. [
] discussed the properties and heat transfer attributes of second grade nanofluid flow subject to a convective and vertical Riga pattern. Lund et al. [
] reported a study on Darcy type Casson nanofluid flow using exponential sheet. Rasool et al. [
] implemented the famous Cattaneo–Christove model to analyze the Darcy relation in MHD nanofluid flow past a non-linearly stretching flat surface. In another study, Lund et al. [
] modeled nanofluid flow using
$C u$
$A g$
type nanoparticles for enhancement of thermophysical properties of the base liquid. Rasool et al. [
] involved an electromagnetic actuator to study the Marangoni convection in nanofluid flow. In another study, Rasool et al. [
] reported Darcy type nanofluid flow subject to Jeffrey model. Involvement of Riga plate and Marangoni convection together in one study is reported by Rasool et al. [
]. Sohail et al. [
] disclosed the features of Entropy and MHD in nanofluid flow, respectively. Entropy optimization has been discussed by Rasool et al. [
] in their study based on Darcy model. Tlili et al. [
] reported some good results about enhancement of thermophysical properties of base fluids when saturated with nanoparticles. Wakif et al. [
] analyzed unsteady natural convection in Coette nanofluid flow subject to MHD and thermal radiation. In another study, Wakif et al. [
] reported electro-thermohydrodynamic stability of nanofluids using Buongiorno model.
Involvement of thermal radiation in nanofluids flow has been extensively used in previous literature especially in case of non-Newtonian and incompressible fluids. Design of heat ex-changers and
other such equipment, propulsion devices, nuclear power plants, gas turbines, space devices and vehicles and satellites, etc. are typical examples of the applications of thermal radiation. Cortell [
], for the first time, involved thermal radiation parameter in his study on heat and mass transport mechanism using stretching surface. Shehzad et al. [
] reported Jeffrey nanofluid using non-linear thermal radiation effect. Shafiq et al. [
] reported properties of convective conditions and thermal slip effects in MHD three dimensional Darcy-Forchheimer rotating nanofluid flow. Animasaun et al. [
] reported homogeneous and heterogeneous reactions involving thermal radiation and MHD in nanofluids flow. Hayat et al. [
] reported slip effects in MHD three dimensional flow of nanofluid under the influence of thermal radiation.
Recently, the concept of entropy optimization has received utmost attention from research community for various reasons. Thermo-dynamical irreversibility in any flow system is directly measured by
the system irreversibility. The second law of thermodynamics is helpful in this context because it has more significance as compared to the first law of thermodynamics. In particular, heat produced
during any irreversible process in a heat transport mechanism is known as entropy generation. It might occur for different reasons such as kinetic energy, spin movement, internal movement of
molecules and internal molecular vibrations, etc. In such cases, heat loss is noted which ultimately varies entropy systems. Numerous systems such as refrigeration, energy storage systems, solar
energy systems, etc. are important areas that involve minimization of entropy generation. Numerous research articles are available in literature wherein, researchers have tried to compute the entropy
to see its influence on whole heat transport mechanism. Bejan [
] reported a pioneer study on entropy optimization in heat and mass transport mechanisms. Later on, this concept of entropy optimization has been greatly reflected in studies such as Liu et al. [
] reported some good results for natural convection and entropy optimization in nanofluid flow bounded by triangular enclosures. Hosseinzadeha et al. [
] reported entropy optimization in
$( C H 2 O H ) 2$
type CNTs based nanofluid flow subject to MHD and thermal radiation. Khan et al. [
] analyzed numerical findings in MHD mixed convective flow targeting entropy optimization.
Flow analysis and boundary layers behavior involving a stretching sheet is known as one of the important fluid models to analyze three main profiles in any kind of heat and mass transport mechanism.
It is connected with numerous industrial and engineering applications such as paper production, plastic sheet production and extrusion, metallic plates cooling process and similar other procedures
(see for example Hu et al. [
]). The concept was build by Crane [
] in the pioneer study on stretching surfaces involved in fluid flow analysis. This study was reported on the variations in fluid movement instigated by a stretching velocity (via stretching
surface). Sajid et al. [
] reported a fluid stream inspired by a curvy extended stretching surface. Rosca and Pop [
] further observed the properties of stretching surfaces in fluid flow phenomena using various fluid models. Naveed et al. [
] reported another good article on magnetohydrodynamic micropolar nanofluid flow due to extended sheet. They accounted for the effect of thermal radiation as well. Abbas et al. [
] reported a radiative flow analysis instigated by stretching surface.
This study is inspired by the novelty in various aspects. The concept of entropy optimization in fluid flow through Darcy channel together with non-linearly stretching surface has not been reported
in the literature yet. Since, non-linear stretching has been of utmost importance in fluid flow analysis, therefore the present model comprising the Darcy channel, non-linear stretching sheet and MHD
is directly affected by irreversible heat loss phenomena and entropy optimization. Overall, the study is organized as follows. Firstly, an incompressible MHD involved nanofluid flow is assumed
surrounded by a non-linearly stretched surface flowing through a Darcy (porous medium) channel. Importantly, Brownian diffusion, viscous dissipation and thermophoresis are considered. In addition,
thermal radiation is also considered in the present model. Secondly, the problem is solved by the numerical RK45 scheme using shooting technique. Thirdly, a graphical representation of results is
given with a comprehensive discussion on each graph. Finally, the main findings are listed in a precise and conclusive manner, especially data tables on Nusselt and Sherwood numbers and
skin-friction, which is very helpful in industrial and many other applications of nanofluids.
2. Mathematical Modeling
Here we adopted an incompressible, viscous, Darcy–Forchheimer MHD nanofluid convection surrounded by a non-linear stretching surface. Entropy optimization, heat and mass transport in
Darcy–Forchheimer type nanofluid flow is analyzed. Non-linear stretching is assumed to be the driving force whereas effects of radiation, Brownian diffusion, dissipation and thermophoresis are
accounted for. Importantly, entropy optimization is performed using second law of thermodynamics. The model is purely taken in two dimension having
$x -$
axis along the fluid flow while
$y -$
axis spreads surface normal to flow direction. A schematic diagram is sketched in
Figure 1
$u = u 1$
$v = u 2$
be the velocity components,
be the temperature distribution and
be the concentration of nanoparticles. Therefore, the governing equations for mass, momentum, energy and concentration distribution are as follows:
$∂ u 1 ∂ x + ∂ u 2 ∂ y = 0 ,$
$u 1 ∂ u 1 ∂ x + u 2 ∂ u 1 ∂ y = ν ∂ 2 u 1 ∂ y 2 − σ B 0 2 ρ f u 1 − ν K u 1 + C b K u 1 2 ,$
$u 1 ∂ T ∂ x + u 2 ∂ T ∂ y = α ∂ 2 T ∂ y 2 + ( ρ c ) n p ρ c f l D B r ∂ C ∂ y ∂ T ∂ y + D T h T ∞ ∂ T ∂ y 2 + σ B 0 2 ρ C f u 1 2 − 1 ρ C f ∂ q r ∂ y + μ K ρ C f u 1 2 ,$
$u 1 ∂ C ∂ x + u 2 ∂ C ∂ y = D B r ∂ 2 C ∂ y 2 + D T h T ∞ ∂ 2 T ∂ y 2 ,$
subject to the following boundary conditions,
$u 1 = U w = m x n , u 2 = 0 , T = T w , C = C w at y = 0 ,$
$u 1 = 0 , T = T ∞ , C = C ∞ as y → ∞ .$
is dynamic viscosity,
$B 0$
is magnetic impact/intensity,
is used for kinematic viscosity,
$ρ f$
is the density,
$D B r$
is used for Brownian diffusion,
$D T h$
is used for thermophoresis.
is electric conductivity of base fluid,
$( ρ c ) n p$
is called nanoparticles’ heat capacity,
$( ρ c ) f l$
is called fluid’s heat capacity.
$C b$
is used as drag force coefficient and
$σ ′$
$k ′$
are Stephen Boltzmann constant and mean absorption constant, respectively.
$q r$
is called radiative heat-flux. By virtue of Rosseland’s approximation subjected to Taylor expansion and neglecting higher order terms,
$∂ q r ∂ y = − 16 σ ′ T ∞ 3 3 k ′ ∂ 2 T ∂ y 2 .$
Define the following similarity transformations,
$u 2 = − 1 2 2 m ( n + 1 ) ν x n − 1 2 f ( η ) + n − 1 1 + n ∂ f ∂ η η ,$
$θ ( η ) = T − T ∞ T w − T ∞ ,$
$ϕ ( η ) = C − C ∞ C w − C ∞ ,$
$η = 1 2 2 ρ f m ( n + 1 ) μ x n − 1 2 y .$
Using (6) and (7a)–(7e) in (1)–(5), we have the following governing equations in one dimensional form:
$f f ″ + f ‴ − 2 n n + 1 f ′ 2 − 2 n + 1 M 2 f ′ − 2 n + 1 λ f ′ − F r f ′ 2 = 0 ,$
$1 + 4 3 R d θ ″ + P r [ N b θ ′ ϕ ′ + f θ ′ + N t θ ′ 2 ] + 2 n + 1 λ P r E c f ′ 2 + 2 n + 1 M 2 P r E c f ′ 2 = 0 ,$
$N t N b θ ″ + ϕ ″ + P r S c f ϕ ′ = 0 ,$
$f ( 0 ) = 0 , θ ( 0 ) = 1 , ϕ ( 0 ) = 1 , f ′ ( 0 ) = 1 ,$
$θ ( ∞ ) = 0 , f ′ ( ∞ ) = 0 , ϕ ( ∞ ) = 0 .$
is given for MHD,
$F r$
is used for inertia (Forchheimer number), Pr is Prandtl parameter,
$N b$
is Brownian factor,
$N t$
is Thermophoretic factor and
$S c$
is Schmidt factor,
$R d$
is the given radiation,
$E c$
is the given Eckert number and
$R d$
is used for thermal radiation. Mathematically,
$M 2 = 2 σ B 0 2 m x n + 1 ρ f ( n + 1 ) ,$
$λ = 2 ν K m ( n + 1 ) x n − 1 ,$
$F r = 2 C b x ( n + 1 ) K 1 / 2 ,$
$N b = ρ c n p D B r ( C w − C ∞ ) ρ c f l ν ,$
$N t = ρ c n p D T h T w − T ∞ ρ c f l ν T ∞ ,$
$R d = 4 σ ′ T ∞ 3 k ′ k 1 ,$
$E c = m 2 x 2 n C f ( T f − T ∞ ) .$
Entropy Generation Modeling
For the viscous flow, following is the governing equation for entropy phenomenon,
$S G = 4 σ ′ T ∞ 3 T ∞ 2 k ′ + k T ∞ 2 ∂ T ∂ y 2 + σ B 0 2 T ∞ u 1 2 + R D T ∞ ∂ C ∂ y ∂ T ∂ y + R D C ∞ ∂ C ∂ y 2 + μ 0 T ∞ 1 ρ K u 1 2 .$
Above equation comprises of four major irreversible parts, (i) thermal radiation, (ii) Joule heating, (iii) porous/Darcy relation and (iv) Concentration distribution. Using the transformations,
Equation (
) reduces to following one dimensional form:
$N G = ( n + 1 2 β 1 1 + 4 3 R d θ ′ 2 + M 2 B r 1 f ′ 2 + 1 + n 2 L 1 β 1 ϕ ′ 2 + L 1 1 + n 2 ϕ ′ θ ′ + B r 1 λ 1 + n 2 f ′ 2 .$
$N G$
is the given entropy generation,
$β 1$
is the given temperature difference term,
$B r 1$
is the Brinkman number and
$L 1$
is known as the diffusive variable. Mathematically,
$N G = S G ν T ∞ m T f − T ∞ k x n − 1 ,$
$B r 1 = μ m 2 x 2 n k T f − T ∞ ,$
3. Solution Methodology
Numerical scheme RK45 with shooting technique is applied on final governing Ordinary differential equations to plot the results. The below mentioned procedure has been adopted:
$f ‴ = h ″ = k ′ = 2 n n + 1 h 2 − f k + 2 n + 1 M 2 h + 2 n + 1 λ h + F r h 2 ,$
$θ ″ = 1 + 4 3 R d l ′ = − P r N b l m + f l + N t l 2 + 2 n + 1 λ P r E c h 2 + 2 n + 1 M 2 P r E c h 2 ,$
$ϕ ″ = m ′ = − N t N b l ′ + P r S c f m ,$
Subject to
$f ( 0 ) = 0 , θ ( 0 ) = 1 , ϕ ( 0 ) = 1 , h ( 0 ) = 1 ,$
$θ ( ∞ ) = 0 = ϕ ( 0 ) , h ( ∞ ) = 0 .$
A careful choice for initial guess of the core functions is adopted for solving initial value problems using RK45. Based on previous iterations, a suitable convergence criteria is adopted. Iterations
are repeated unless a difference upto or a less than $10 − 5$ is obtained. This numerical scheme has various advantages and accuracy as compared to previous and classical methods of solutions such as
HAM, OHAM, etc. Results are more efficient and speedy convergence is achieved. Similar procedure is adopted for Entropy optimization.
4. Analysis
Here we adopted an incompressible, viscous, MHD and Darcy–Forchheimer nanofluid convection surrounded by a non-linear stretching surface. Importantly, entropy optimization, heat and mass transport is
analyzed. Non-linear stretching is assumed to be the driving force whereas effects of radiation, Brownian diffusion, dissipation and thermophoresis are accounted for. We have incorporated the RK45
built-in system with shooting technique to plot the numerical outcomes of non-linear system of equations. Properties of velocity field, temperature and concentration distributions, stream functions,
Bejan number are disclosed in this section.
Figure 2
Figure 3
Figure 4
, we have evaluated physical behavior of velocity field for variation in different parameters involved in momentum equation. In particular, Forchheimer number (
$F r$
), porosity factor (
), magnetic (MHD) field effect on fluid flow is analyzed graphically.
Figure 2
illustrates impact of Forchheimer number on velocity field and corresponding boundary layer. Continuous enhancement in resistance offered to fluid motion by inertial factor results in smooth decay in
velocity profile. In
Figure 3
we see the plot of impact of porosity factor imparted on fluid flow (velocity field) and corresponding boundary layer formulation. We observe that porous medium offers more retardational force
(friction) which continuously diminishes the velocity of liquid. Boundary layer becomes thinner. Impact of Lorentz forces generated by applied magnetic (MHD) field on fluid flow and corresponding
boundary layer thickness is plotted in
Figure 4
. Effective magnetic (MHD) field to the surface normal along vertical axis creates sudden bumps and hurdles in the way of fluid movement that causes a declination in the fluid motion. The stronger
the impact of MHD, the lesser the fluid movement along the horizon.
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
are graphical results related to the parameters involved in momentum and energy equations to see variations in given domain. Since the article mainly emphasis on Darcy relation, therefore,
Forchheimer number and porosity factors are two important parameters that vary the temperature distribution. Furthermore, thermal radiation is another important factor. Besides these, the impact of
Brownian diffusion and thermophoresis influence on thermal state (distribution) are also noted herein. In particular,
Figure 5
Figure 6
display the consequences of inertia and porosity factors on thermal distribution, respectively. The resistive force due to inertia and enhanced friction are the source of enhancement in heat
convection. Temperature distribution rises for rising values of both the factors whereas, opposite behavior is observed for anti-augmented numerical values. Impact of Brownian diffusion and
thermophoresis is given in
Figure 7
Figure 8
, respectively. An intensive thermophoretic force gives rise to the in-predictive motion of the nanoparticles that rises the field temperature and corresponding boundary layer shows more thickness.
Influence of thermal radiation factor (
$R d$
) is plotted in
Figure 9
. A certain rise in thermal state (distribution) is noted for elevated numerical values of thermal radiation factor.
Figure 10
Figure 11
Figure 12
are plotted to see the impact of Brownian diffusion, thermophoresis and Schmidt number on concentration of nanoparticles. In particular,
Figure 10
is a plot of variations recorded in concentration distribution for rising values associated with Brownian diffusion. The concentration of the nanoparticles reduces near to surface. An enhancement is
noted in case of thermophoresis due to the stronger thermophoretic force, which effectively produces more in-predictive movements as shown in
Figure 11
. Variation noted in concentration field for rising numerical values of Schmidt number is plotted in
Figure 12
. A decline is observed in the respective field. Physically, the inverse relation of Brownian diffusion and kinematic viscosity gives rise to this behavior in concentration profile.
We have sketched stream functions as well as contour graphs at different numerical values of magnetic parameter
given in
Figure 13
Figure 14
Figure 15
Figure 16
, respectively. In particular,
Figure 13
Figure 14
are the contour graphs given at
$M = 0.1$
$M = 0.5$
. An enhanced variation can be seen at the distance much away from origin. Near the origin, this variation is very narrow.
Figure 15
Figure 16
are stream functions graphs at
$M = 0.1$
$M = 0.5$
, respectively. A very narrow variation is noticed between two pictures. On a closer look, one can see that at
$M = 0.1$
the curves are not spread much as compared to the case
$M = 0.5$
. Physically, the stronger magnetic effect boosts the opposing Lorentz forces which occur in the way of fluid motion and stream lines get affected, whereas
Figure 17
Figure 18
present the stream density at
$M = 0.1$
$M = 0.5$
, respectively.
Figure 19
Figure 20
Figure 21
are given on variations noted in Bejan number for various values of inertia factor, thermal radiation and temperature difference ratio parameter. In particular,
Figure 19
shows the behavior of Bejan number with respect to the elevated values of Inertia factor. The effect is narrowed near the surface, however a comprehensive change is noted away from surface.
Physically, irreversibility enhances due to friction offered by porous media to fluid and nanoparticles. A certain decrease is noted in Bejan number for elevated values of thermal radiation shown in
Figure 20
. Physically, larger emission rate of radiation impacts on Bejan number which shows reduction. A mixed behavior of Bejan number is noted for temperature difference ratio parameter. After a certain
value, rising nature switches back to the declining trend as shown in
Figure 21
Outcomes obtained numerically in lieu of skin friction, heat flux (Nusselt) rate and also mass flux (Sherwood) rate, respectively are given in
Table 1
Table 2
. In particular,
Table 1
gives the results of skin-frictional force for various values of magnetic parameter, inertia factor and porosity factor. A rising trend is noticed for all the variations. In
Table 2
, the results are given for heat flux (Nusselt) rate and also, mass transfer (Sherwood) rate, respectively. Inside out thermal (radiation) behaves as a continuous heat source. Specifically, thermal
radiation shows reduction in heat transport phenomena (Nusselt) and enhancement in mass flux (Sherwood) rate. A reducing trend in both heat transport rate (Nusselt) and also mass transport rate
(Sherwood) is noticed for elevated values of porosity factor. Strong retardation offered by porous media is the reason behind this decline in both physical quantities.
5. Conclusions
Here we have analyzed entropy optimization, heat and mass transport in Darcy–Forchheimer MHD nanofluid flow surrounded by a non-linearly stretching surface. Non-linear stretching is assumed to be the
driving force whereas effects of radiation, Brownian diffusion, dissipation and thermophoresis are also considered. Here, we have incorporated RK45 built-in system with shooting technique to plot the
numerical outcomes of a non-linear system of equations. Properties of velocity field, thermal and solute distributions, stream functions and Bejan number are disclosed in this article. Salient
findings of this study are listed below:
• Entropy optimization, heat and mass transport in Darcy–Forchheimer and MHD type nanofluid flow surrounded by a non-linear stretching surface is analyzed.
• Rate of Bejan number shows mixed behavior for elevated values of temperature difference parameter. An enhancement is noted for larger values.
• Skin friction enhances for all the parameters involved in momentum equation.
• Heat transfer rate declines while mass transfer intensifies for elevated numbers of thermal radiation parameters.
• Resistive force due to inertia and enhanced friction are the source of enhancement in heat convection.
• The concentration of nanoparticles reduces near the surface, whereas an enhancement is noted in the case of thermophoresis due to stronger thermophoretic force.
• An enhanced variation in stream lines is noted at distance far away from the origin. Near to the origin, this variation is very narrow.
• We observed that a more porous medium offers more retardational force (friction), which continuously diminishes the velocity of the fluid.
• Contour graphs given at $M = 0.1$ and $M = 0.5$ show an enhanced variation at distance sufficiently away from origin. Near to the origin, the variation is very narrow.
• Stream function graphs at $M = 0.1$ and $M = 0.5$ show a very narrow variation in the stream lines. At $M = 0.1$ curves are not spread much as compared to case $M = 0.5$. Stronger magnetic effect
boosts the opposing Lorentz forces which occur in the way of fluid motion and stream lines get affected.
Author Contributions
G.R. formulated the problem, derived the equations, generated the results, wrote the analysis & discussion and concluded the paper. A.S. generated the results and validated the model. I.K. checked
the model and proofread the whole manuscript. D.B. and K.S.N. helped in revision and provided funding. G.S. checked the whole manuscript, helped in revision and proofread the final version. Finally,
All authors have read and agreed to the published version of the manuscript.
This research was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project No. 2020/01/16436.
The authors are highly obliged and thankful to unanimous reviewers for their valuable comments and suggestions on the manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
$RK 45$ Runge-Kutta 45 Method
MHD Magnetohydrodynamics
PDE Partial Differential Equation
ODE Ordinary Differential Equation
$u 1 , u 2$ Cartesian velocity coordinates/m·s$− 1$
$x , y$ Cartesian distance coordinates/m
$u w = m x n$ Velocity (stretching)/m·s$− 1$
m Stretching rate/s$− 1$
$μ$ Dynamic viscosity/Pa·s
$B 0$ Magnetic impact/intensity/A·m$− 1$
$ν$ Kinematic viscosity/m$2 ·$s$− 1$
$ρ f$ Density/kg·m$− 3$
$D B r$ Brownian diffusion
$D T h$ thermophoresis
T Temperature distributions /K
C Concentration distributions/kg·m$− 3$
$σ$ Electric conductivity of the base fluid/($Ω$ m)$− 1$
$( ρ c ) n p$ Nanoparticles’ heat capacity/J·m$− 3 ·$k$− 1$
$( ρ c ) f l$ Fluid’s heat capacity/J·m$− 3 ·$k$− 1$
$C b$ Drag force coefficient
K Permeability
$τ$ Ratio of heat capacity of fluid and nanoparticles
$q r$ Radiative heat flux
$σ ′$ Stephen boltzmann constant
$k ′$ Mean absorption constant
$α$ Thermal diffusivity/m$2 · s − 1$
k Thermal conductivity/$W · m − 1 · K − 1$
Dimensionless Parameters
M Magnetic parameter
$P r$ Prandtl number
$S c$ Schmidt number
$N b$ Brownian diffusion
$N t$ thermophoresis
$S h x$ Sherwood factor
$N u x$ Nusselt factor
$S G$ Entropy generation rate in two dimensions
$R D$ Difference ratio
$N G$ Entropy generation rate
$β 1$ Temperature difference
$B r 1$ Brinkman number
$L 1$ Diffusive parameter
$F r$ Forchheimer number
$λ$ Porosity
$R d$ Radiation parameter
$E c$ Eckert number
$μ 0$ Viscosity at initial position
$η$ Variable
$ϕ$ Concentration distribution (dimensionless)
$θ$ Temperature distribution (dimensionless)
$f ′$ Velocity (dimensionless)
Table 1. Numerical outcomes of skin-friction given at $n = 1.2$ while values of other parameters are varied one by one.
M $F r$ $λ$ $− Re x C x$
$0.0$ $0.3$ $0.6$ $1.6772$
$0.5$ $1.75074$
$1.0$ $2.325338$
$0.5$ $0.0$ $0.6$ $1.63528$
$0.3$ $1.75074$
$0.6$ $1.85947$
$0.2$ $0.5$ $0.0$ $1.56812$
$0.6$ $1.75074$
$1.2$ $2.419236$
Table 2. Numerical outcomes of heat transfer (Nusselt) rate and mass transfer (Sherwood) rate given at $n = 1.2$ while values of other parameters are varied one by one.
M $F r$ $λ$ $R d$ Pr $N t$ $N b$ $Ec$ $Sc$ Nusselt Sherwood
$0.0$ $0.3$ $0.6$ $0.5$ $1.0$ $0.1$ $0.2$ $0.7$ $1.0$ 0.330523 0.5403
$0.5$ 0.294335 0.539539
$1.0$ 0.146822 0.196834
$0.5$ $0.0$ $0.6$ $0.5$ $1.0$ $0.1$ $0.2$ $0.7$ $1.0$ 0.298447 0.550958
$0.3$ 0.294335 0.539539
$0.6$ 0.29052 0.529184
$0.5$ $0.3$ $0.0$ $0.5$ $1.0$ $0.1$ $0.2$ $0.7$ $1.0$ 0.386878 0.542202
$0.6$ 0.294335 0.539539
$1.2$ 0.157756 0.222705
$0.5$ $0.3$ $0.6$ $0.0$ $1.0$ $0.1$ $0.2$ $0.7$ $1.0$ 0.397385 0.501379
$0.5$ 0.294335 0.539539
$1.0$ 0.24219 0.561276
$0.5$ $0.3$ $0.6$ $0.5$ $1.0$ $0.1$ $0.2$ $0.7$ $1.0$ 0.294335 0.539539
$1.5$ 0.374306 0.509569
$2.0$ 0.43873 0.487011
$0.5$ $0.3$ $0.6$ $0.5$ $1.0$ $0.1$ $0.2$ $0.7$ $1.0$ 0.294335 0.539539
$0.4$ 0.271793 0.309337
$0.8$ 0.244013 0.081471
$0.5$ $0.3$ $0.6$ $0.5$ $1.0$ $0.1$ $0.2$ $0.7$ $1.0$ 0.294335 0.539539
$0.4$ 0.26720 0.589734
$0.6$ 0.241696 0.606221
$0.5$ $0.3$ $0.6$ $0.5$ $1.0$ $0.1$ $0.2$ $0.0$ $1.0$ 0.39101 0.495559
$0.5$ 0.321959 0.526972
$1.0$ 0.252895 0.55839
$0.5$ $0.3$ $0.6$ $0.5$ $1.0$ $0.1$ $0.2$ $0.7$ $1.0$ 0.294335 0.539539
$1.5$ 0.290366 0.76239
$2.0$ 0.288085 0.952778
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Rasool, G.; Shafiq, A.; Khan, I.; Baleanu, D.; Sooppy Nisar, K.; Shahzadi, G. Entropy Generation and Consequences of MHD in Darcy–Forchheimer Nanofluid Flow Bounded by Non-Linearly Stretching
Surface. Symmetry 2020, 12, 652. https://doi.org/10.3390/sym12040652
AMA Style
Rasool G, Shafiq A, Khan I, Baleanu D, Sooppy Nisar K, Shahzadi G. Entropy Generation and Consequences of MHD in Darcy–Forchheimer Nanofluid Flow Bounded by Non-Linearly Stretching Surface. Symmetry.
2020; 12(4):652. https://doi.org/10.3390/sym12040652
Chicago/Turabian Style
Rasool, Ghulam, Anum Shafiq, Ilyas Khan, Dumitru Baleanu, Kottakkaran Sooppy Nisar, and Gullnaz Shahzadi. 2020. "Entropy Generation and Consequences of MHD in Darcy–Forchheimer Nanofluid Flow Bounded
by Non-Linearly Stretching Surface" Symmetry 12, no. 4: 652. https://doi.org/10.3390/sym12040652
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CELL Function Adventure | My Spreadsheet Lab
CELL Function Adventure
posted in Formulas, Free Data on by Kevin Lehrbass
You may never need this…but I did recently and it’s just fun knowing that Excel can solve this!
Download my Excel file here (demo data from data.world).
Drop down list must contain column headers for columns that have currency datatype. If datatypes change the drop list should also change
Last week I would have said that it’s not possible….but it is possible!
I used it as part of a DSUM solution (choose currency column to sum).
The Data
I used this sample data from data.world. 20 total columns and 399 rows.
The Solution
Step 1 – Extract Datatype
CELL function extracts datatype from first row of data (assuming entire column is the same).
CELL function doesn’t seem to work as an array so I referenced each cell one by one.
What do the codes mean?
• a code starting with a “C” (cell G1) means currency
• a code starting with an “F” (cell A1) means number
Official list from Microsoft (link).
Step 2 – Count Them
Initially I used an array in cell C4 to count datatypes starting with “C” but I simplified it to this:
Step 3 – Counter
I used dynamic array function SEQUENCE to create a counter in cell B6:
It spills down only as far as necessary.
Step 4 – Column List
Below, the # in B6# is used to spill formula down alongside the counter. No need to drag formula down manually (it automatically extends).
Step 5 – Add Drop Down List
I used OFFSET inside data validation to create the list. CELL & OFFSET are both volatile functions.
Validate Solution
To double check the solution I also created a drop down list for column headers that have number format. Columns ‘adjusted’ and ‘worldwide_gross’ are formatted as numbers.
Note: if you change a column’s datatype (value in row 2 of sheet ‘blockbuster-top_ten_movies_per_’) you’ll need to press F9 key to refresh the calculation.
After I downloaded the sample data I manually set each column to a specific datatype (as many of them were initially set to “G” for general).
As I mentioned, you may never need to do this but it was fun knowing that Excel can solve this with a few simple formulas. I used this as part of a solution involving DSUM function (hopefully my next
To think about: how can I use TYPE function?
About Me
Some are described as being a hopeless romantic. I’m not. I’m more of a hopeless Excel fan. I will never obtain true Excel nerdvana but I’ll pursue it forever 🙂
My name is Kevin Lehrbass. I’m a Data Analyst. I live in Markham Ontario Canada.
2 Comments → CELL Function Adventure
1. David N
If you’re using the SEQUENCE function, then don’t you also have access to the FILTER function instead of having to use INDEX-SMALL-IF?
Reply ↓
1. Kevin Lehrbass
Hi David,
That’s a good point. I do have the FILTER function.
Thank you for visiting my blog!
Reply ↓
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Multiplication Facts for Upper Elementary Students
As a third and fourth grade teacher, multiplication facts are a huge part of my curriculum. While I absolutely believe in teaching for conceptual understanding and deep meaning, I also believe it is
important for my students to know their multiplication facts with fluency. (I plan to write a new post sharing what fluency in math really means.) I want multiplication facts to be effortless and
automatic for my students, so their focus can be geared toward problem solving and upper level math concepts, rather than basic computation. As a former fourth grade teacher, I also know how
difficult fourth grade math (I can only image fifth and beyond) is when students did not know their multiplication facts. This blog post shares some of my ideas and strategies for teaching
multiplication facts, as well as motivating students to learn their multiplication facts.
I do not teach a multiplication facts unit, but I do teach a unit on the concept of multiplication and multiplication strategies, and you can see some of those instructional lessons here. First, I
begin with a grouping model, where students build models of groups with a certain number in each group. I also spend a great deal of time teaching arrays and repeated addition, and a little time
teaching multiplication on a number line. A lot of the practice I use here is from my No Prep Multiplication pack. It’s important to keep in mind that not all students need this extra practice, but
I’ve found it particularly beneficial for students with short term and long term memory issues. I use these for extra practice, not my actual math lesson or instruction.
I’m also careful to spend considerable time teaching multiplication vocabulary. Even though I don’t teach all of these terms for mastery, it’s still beneficial for my students to have a general
understanding and recognition for terms such as factors and multiples. I love having students create a multiplication booklet for essential vocabulary.
I also incorporate multiplication practice into our math centers each week. I always have at least two stations where students play different multiplication games. You can read more about that here.
Most of my games include a built in spinner or dice. If I’m trying to save paper and copies (which is always), I like to laminate the paper and let students use counters rather than coloring in
boxes. This allows me to reuse the same forms again and again. It also makes differentiation a bit easier, because I can have one copy of each form and not have to search and print again and again.
All of my games below are from my No Prep Multiplication pack.
Homework Games
I also send home one of my Weekly Multiplication Games for homework, which have been a huge success with my students and their parents. This is nothing that I require my students to do or to turn
in. Instead, it’s just a new idea for practicing multiplication each week in a game format. The only special materials needed are dice and a deck of playing cards, which are both extremely
inexpensive. I love giving students ideas of games that they actually want to do!
Multiplication Booklets
I’m not a proponent of isolated memorization of multiplication facts. Many of us have standards that include ‘fluency’ as a goal, and this fluency comes about when students develop number sense, and
isolated practice and repetition will not develop number sense in our students. When students focus on memorization, they often memorize facts without number sense, which means they are prone to
making errors and are not flexible in their thinking. The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize
without number sense. I love using my Multiplication Fact Booklets for additional practice. As students work through their booklets, they gain a conceptual understanding of each multiplication fact.
They are able to solve the fact with multiple strategies, as well as begin to observe patterns and develop a mathematical vocabulary.
Flash Cards
I love using these multiplication subitizing flash cards to foster multiplicative reasoning and automaticity. These help students take multiplication understanding from concrete to abstract. You can
download these cards here.
Xtra Math
I love Xtra Math because it’s free, it progresses students through multiplication acts in a logical order, and because students type in the answer regardless of how long it takes them to solve the
multiplication problem. I also like that I can adjust what programs my students are working on. I’ve shared a clip from my account to show how varied students’ programs can be. For example, some
students are still working on multiplication facts with a six second time limit, some are working on multiplication facts with a three second time limit, and some are on division facts with a six
second time limit, and some on division with a three second time limit. I can see when my students practice, and I can observe their growth over time. I only use this as a tool, and not for my
multiplication instruction.
Timed Tests
When we didn’t have Chromebooks for Xtra Math, I used timed tests to keep up with who is on each set of facts. I know that timed multiplication tests have recently become frowned upon, but I don’t
want to completely do away with them. I’m rarely for any all or nothing approach when it comes to education. Instead, I tend to favor a more balanced approach with a healthy dose of common sense.
Timed tests don’t have to be a bad thing. I believe that it is important to monitor students’ progress on math fact fluency on a consistent basis. This is something that needs to be done on a regular
basis to ensure that all students are making appropriate gains. If I see that a student is struggling with timed tests, I work one-on-one with a student, and I try to determine what is causing the
student to fall behind. If I see that the student is simply slow at writing numbers but knows the multiplication facts, I move that student forward as needed. I don’t believe a student should be held
back for something such as slow writing or getting nervous when the timer is set. Other times, I see that the student is making progress but needs to apply some of the strategies we’ve learned in
class. I’ve also found that some students have a hard time processing the question, rather than the product of the math facts. Then, there are some who simply do not know their facts, and I do not
move those students forward.
I give both 20 question and 100 question timed tests. I intentionally reverse the order of the factors to focus students to apply the commutative property as they learn their facts. This makes the
larger facts (7s,8s, 9s, etc.) so much easier for students! When I give the 20 problem timed test, students have one minute to complete all of the problems. I also make sure to have several different
versions to keep students on their toes. You can download the tests here!
While the 20 question tests are fairly easy to pass, the 100 question tests are a bit more challenging. I continue to use the rule of thumb of three seconds per problem, so I give students five
minutes to complete the test. The big difference is that these tests are cumulative and force students to remember their previous math facts as they move forward with new math facts. This test also
requires the application of the commutative property. You can download the 100 question test here.
I try to keep timed tests light and silly. I don’t want students to feel pressured or frustrated, because that’s not going to help anyone. One of the easiest, yet most efficient, ways I keep timed
tests fun is to randomly change the ring tone to my timer. Yes, it sounds trivial, but my class gets the biggest kick out of it. It’s incredibly important to display and encourage a positive attitude
with a focus on growth. I don’t like to compare students to each other, but I like for students to work to beat their previous scores. I recently made a small modification to our 100 question timed
tests that will allow students to compare their previous number correct to the number correct they scored on their current timed test. I’ve found that students are highly motivated by the visual a
graph provides, so I’m going to have students graph their progress. This is included in the file above!
I’ve also created a math facts graph that students can use to track their progress. I like using this graph with the five minute tests, rather than the one minute tests. You can download it here.
I also think it’s beneficial to show students just how many facts they really have to memorize. For most students, the 1, 2, 5, 10, and 11 facts are very simple, so I show students a 100s chart with
those facts removed. That allows students to see what problems are left that they will need to learn. Then, I highlight the inverse multiplication facts, which shows students how to reduce what they
need to memorize in half! This looks MUCH more manageable that a HUGE list of math facts to learn.
When All Else Fails
Of course there are some multiplication facts that are tricky. That’s when I break out pieces of Multiplication the Fun Way. There are songs, posters, books, and even a movie that are PERFECT for
those tricky multiplication problems.
At least 90% of the time I try to rely on intrinsic motivation. I absolutely believe that is the best form of motivation, and I consistently see the benefits of intrinsic motivation. However, when
the stakes are high, I’m willing to utilize a little extrinsic motivation. Even though I continually stress the importance of multiplication, I’ve found that my students sometimes need a few extra
incentives to practice their facts at home. We practice at school almost everyday, but it’s still necessary for students to practice at home to truly solidify their learning.
One of my favorite new incentives is the multiplication bracelet I bought from Really Good Stuff. I almost didn’t buy them, because of the low reviews about the numbers rubbing off. They’ve either
fixed the problem, or my students are extra careful with them, because we haven’t had any problems. I give students a bracelet whenever they “master” the set of facts they are currently working on.
For instance, when a student passes their 3 facts, I give them the 4 facts bracelet. This allows students to have something to look at as they prepare for their next set of facts. It’s also a great
way to let students share that they’re working on a new set of facts with their parents! They have definitely become a collector item with my class, because I almost always have several students
wearing all of the bracelets they’ve earned. My students are very proud of them! Update: Now that I teach multiple groups of math, these are no longer economically feasible for me to purchase. I also
now rely more on Xtra Math for following student progress, so students don’t practice one set of facts at a time.
This will be the second year I’ve used brag tags for multiplication facts. Students can earn a brag tag when they pass their 0-5 facts, 6-12 facts, and division facts. I keep the chains and brag tags
until the end of the school year. Then, I let students take them home to keep. I don’t give a brag tag for each set of facts since I already give them a bracelet. I’m also fairly confident that I
wouldn’t be able to keep up with it for the course of a year. I purchase all of my Brag Tags from School Life, and I love working with them! I love that I can customize them for my own students! You
can also customize these to celebrate growth rather than the mastery of a set of facts.
We also have a multiplication ice cream sundae party to celebrate learning multiplication facts! Students earn spoons, bowls, ice cream, and toppings as they learn each set of multiplication facts. I
always have to explain that no one is HAS to get a topping they don’t like, and that I’ll have an alternative (sorbet or something else) for those who can’t have dairy. Since I hold the party toward
the end of the year in spring, all students have time to learn as many sets of multiplication facts as possible. I send the letter below home, and students can use it as a reference of what topping
they are working toward. Some years I’ve sent this home at the beginning of the school year, and other years I’ve waited until the novelty of multiplication facts wore off and used this to
re-motivate. You just have to use the dynamics of your class to determine how and when to introduce the sundae party.
I also have a version of the letter that is better suited for Xtra Math. I also like that this is a bit more equitable for students, since you can control the settings on Xtra Math. This will allow
you to push students who need a little extra nudge, and give some students a little extra time. You can download this version here.
I also printed a little coloring activity where students can color in parts of their ice cream sundae as they master each set of facts.
I typically have several parent volunteers for the multiplication sundae party, and one of the challenges I’ve had, was that parents didn’t know who got what topping. To help with this, I made a
multiplication sundae punch card. I’ll let students punch the corresponding square when they pass a set of multiplication facts. Students will use this as their ticket to the party. You can download
the coloring page and punch card here.
I have made a few alternative versions to the Multiplication Sundae Party.
If you have any other great ideas for teaching multiplication facts, feel free to share! You can never have enough tools for this challenging skill!
42 thoughts on “Multiplication Facts for Upper Elementary Students”
1. Cat Drinnon
This is just what I am looking for to get my middle school special Ed students motivated to learn their facts. Thanks so much for sharing:)
2. Rakitia
This post is an incredible resource! Thank you for putting this together.
3. Laura
How long do you give them to do the 100 math facts.
I give them 5 minutes.
4. Megan
Is there a division one! Would love to have students who have masted 0-12 move onto division facts.
5. Shannon Seneczko
Would you happen to have a way to change the multiplication sundae resources to division? I just love this idea! Thanks for sharing!
I’ll try to do that!
6. Melody
I love this post! Thank you for sharing. How do you handle this with students in your room who might have a learning disability that makes it extremely difficult to memorize facts. Do the
bracelets or sundae cards create an issue for them? I would love your suggestion for keeping the bar high for all. Thanks!
I work with the special education teacher with that. Sometimes we do an oral assessment or give them extra time. It really depends on the student. I don’t want anyone stressed, so I’m careful
to take their cues.
1. Melody
7. Jen S
Did you buy a set of the multiplication bracelets for each student? I LOVE the idea, but it would be over $100 for a class set. 🙁 Thinking of putting on wish list for parents at the beginning of
the school year…
I used money from PTO. I’ve thought about asking for donations to help cover the cost too.
8. Sue
Is there a way you can make the Multiplication Sundae editable? I’m teaching 4, 5, 6 th grade inclusion sped class and so have some working in addition and subtraction and others multiplication
and division. I would love to differentiate this so all students have a goal they can reach for the Sundae party. This seems like a great motivator for them! Thanks for this comprehensive blog
post with freebies
1. Sue
I have self-contained not inclusion,
9. Sue
I looked through your bundles on TPT and cannot find the multiplication sheets where the students can keep track from previous score to their present score as you have shown a picture of on this
site. Where can I find them? Is there a link on this post somewhere? They are just exactly what I need!
Thanks so much for all you have shared about multiplication!
10. Heather Beden
Hi! Thank you so much for this resource! I was wondering how you run the time test. I clicked on the link above and it went to a not available page. Do students have to pass the 20 problem test
then the 100 problem test to pass each number? Thank you!
11. Milissa Smith
Can I get a copy of the sundae sheet?
There should be a link in the post! I’ll check it to make sure it’s there!
12. sue
I like the bracelets! I would like to see bracelets with just the factors for students to practice their skip counting. (3,6,9,12, etc)
I would love that too!
13. Tonya
Hi! How do you grade all the tests? I was grading at first, but then found an online self-graded sight (which is now going away….grrrr!). It looks like I’m going to be grading again, but it’s
soooo time consuming. Thanks for all the links on this post, though! I love the sundae incentive!
I’ve actually switched to using Xtra Math for my timed test. When I did give them, I was super fast. I would just skim them in a couple minutes.
14. Alba
Thank you so much for the freebies!
15. Aimee
Hi! Thank you for sharing this wonderful resource! I’m excited to use it this year. I noticed that your 20 question timed tests only go up to 9’s, do you have a set for 10-12’s?
I only have an old version. Let me try to update it!
1. Aimee Skidmore
Thank you! I greatly appreciate it!
2. Lauren Zoeckler
Do you have the 10’s-12’s set?
Great resource thank you!
I don’t at this time.
16. Lisa munson
Hello! When I click on how you administer the timed tests, there is an error. Do u have info on this?
Also, I would like students to graph for facts they master correctly, not just the number of facts they are able to complete in 5 minutes. Do u have something for this?
Thank u so much!!
Are you getting the error when you download a file?
17. Jennifer
How do you administer the tests in Xtra Math? Specifically, what are the settings you select?
I tried putting it on multiplication. When I tested it out, it tests the student on various facts, not one set of facts.
When I put it on assessment only, it tests addition, subtraction, multiplication, & division.
Thanks for your help!
I go into students settings and select the custom setting.
1. Jennifer Boylan
Hi, I am familiar with XtraMath but seem to be missing how you use it for time test. I have gone into custom. Is there a way to set it so the kids do one fact at a time or how do you
decide that they passed each fact?
By the way I a huge fan of your blog.
I look at the percent they have mastered, rather than a particular set of facts now.
18. Heidi Schroeder
I love the times tests with the tracker graph on the side. Do you have a divisions timed test with the graph on the side?
Thank you! I don’t have them at this time, but I can work on it!
1. Heidi Schroeder
Thank you, would be wonderful.
1. Heidi Schroeder
That would be wonderful!
19. Jenna
Thank you for sharing so many great resources for free! So greatly appreciated, can’t wait to use these with my students 🙂
20. Amanda
Hi there!
I love your resources! I was wondering if you have a version that I can edit. I am looking at creating addition, subtraction, and division for all elementary grades. Our teachers want to use
fluency across grade levels!
21. Denise
I love your blog! Thank you for sharing your awesome resources!! I haven’t taught math in several years and this year I have moved to a new school where I will be self-contained. Your resources
for teaching multiplication will be a huge help to me. I know the kids will LOVE the sundae party!!
22. Elizabeth
The sundae party should be outlawed. It ruined my daughter for math. She’s graduating high school this year, and she still remembers the anxiety of those timed tables and how she only earned
vanilla ice cream. There was even a bulletin board with a buildable sundae for each child. So the entire class could see who wasn’t getting the toppings. Awful! She knew the facts, but she just
couldn’t do it quickly. From then on, she said she couldn’t do math. There are better ways to teach kids math that don’t alienate some children. My sister is a math content specialist/consultant,
and she highly discourages techniques like this (she even at one time believe she wasn’t a “math person.”) She inspired my daughter to write her dual credit English research paper on The Myth of
the “Math Person”
The topic is incredibly fascinating, and my daughter finally realized she could still be a math person, after all. Her 3rd-grade math teacher meant well, but this teaching method was very
damaging. https://www.youcubed.org/dispelling-myths-about-mathematics/
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TS ICET 2020 Question Paper Shift-1 (1st Oct)
A merchant purchases 11 bags for Rs.900 and sells 9 for Rs. 1100. Then profit percent the gets in this business is
When numerator of a fraction is increased by 15%, and denominator is decreased by 8%, the fraction becomes $$\frac{15}{16}$$. What is the original fraction?
A person buys an article for a discount of 10% on its marked price Rs.1200. If, in turn, he sells it for Rs,1200, then profit percent is
A person started a business with Rs.15 lakhs. After 2 months, B joined A as a partner and after 4 more months C joined A and B as a partner. At the end of the year, they shared the profit in the
ratio 9 : 8 : 5. Then the ratio of the capitals of B and C was
The initial capitals. of two persons A and B in a partnership business are Rs.40 lakhs and Rs.60 lakhs, respectively. After 4 months each of them invested Rs.10 lakhs. If the profit at the end of
the year is Rs.13.6 lakhs, then the difference in their profits in lakhs of Rupees is
Two pipes A and B can fill an empty cistern in 6 hours and 8 hours, respectively. After opening both the pipes simultaneously, pipe B is closed after X hour and A filled the rest of the tank in 4
hours. Then X =
Two pipes $$P_{1}$$ and $$P_{2}$$ can fill a tank in 15 minutes and 20 minutes respectively. Both the pipes are opened simultaneously, but after 4 minutes, pipe $$P_{1}$$ is turned off. What is the
total time required to fill the tank?
Walking at $$\frac{2}{3}rd$$ of my normal speed, I am an hour late to office. Then the normal time I take in travelling to office is
The circumference of a cart wheel is $$4\frac{2}{7}$$ meters and it makes 7 revolutions in 4 seconds. Then the speed of the cart (in kmph) is
There are three workers Ram, Tom and Jill who take 3 hour. 2 hours and 4 hours respectively, for manufacturing a toy by themselves. How long would it take if all three workers worked together. (in | {"url":"https://cracku.in/ts-icet-1st-oct-2020-shift-1-question-paper-solved?page=10","timestamp":"2024-11-03T01:05:20Z","content_type":"text/html","content_length":"163711","record_id":"<urn:uuid:a1ef64e2-3150-469c-8943-c51cde183162>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00684.warc.gz"} |
Adult Numeracy Network - Math in Public Spaces
In her 2019 ANN Under 10 talk, Making Math Viral, Christin Smith (KY) tries to resolve the paradox of people saying, "I am not a math person" with people sharing "viral math problems" over social
media. In part, Christin chalks it up (pun intended) to a misconception of what it means to do math. In school, students learn that math was about right and wrong, speed, and knowing the
procedures. Christin's goal in analyzing the phenomenon was to see what there is for us to learn what we can about the appeal of viral math problems. It is more than just using viral math problems
in class. Christin's take is that there are two elements that give viral math problems their power: (1) they peak curiosity and (2) they promote controversy.
Another aspect of viral math problems is their location. They are curiosity peaking, controversy promoting, social experiences that happen outside of the classroom.
In the last several years, there have been some grassroots explorations of where it means to do math. We are collecting some resources on this page to offer some inspiration for the doing of math in
public spaces. Join the movement!
Please share any pictures of math in public spaces, either using the hashtags mentioned below (don't forget to add #ANNMath) or email them to mathpractitioner@gmail.com
SIDEWALK MATH
Let's chalk up our streets, broaden what it means to do math, spark curiosity and conversation (and use #sidewalkmath to tell each other about it).
#SidewalkMath is a hashtag made popular by Brian Palacios, a HS math teacher in the Bronx, NY. It is used by teachers, parents, and caregivers (but mostly teachers) across the US who leave math on
the sidewalk and engage their communities, children and adults alike, in math conversations.
• Dr. Benjamin Dickman taught a class at the Hewitt School in NYC called Problem Solving and Posing. As part of the class his students chalked up city sidewalks with math problems for passersby
to explore.
□ Benjamin's class also created a #SidewalkMath Encyclopedia, collecting curiosity peaking problems from #sidewalkmath photos on Twitter.
MATH WALKS
Math Walks was started in March 2020 by Traci Jackson to encourage math discussion during quarantine. Photos of the math she has left along her path can be found at Math Walks. Making Cocoa and
Same/Different are two examples. Other folks have been inspired by Traci's project. You can see what they have shared on Twitter using #MathWalks
PUBLIC MATH: Where every person is a math person.
This group of educators creates mathematical opportunities in the spaces that diverse children and families inhabit and interact with in their daily lives. Check out their website, Public Math, and
learn more about their projects: free postcards (order two today - for you and a friend), coasters, laundromats, zines. They even have an IOS sticker packet for your smartphone that allows you to add
a prompt (What do you notice? What do you wonder? How Many? Which One Doesn't Belong? etc.) to your photos and videos.
Check out Public Math Pop-Up: Mathematize Your Kitchen, to hear Chris Nho, Molly Daley, and Christopher Danielson, the organizers for Public Math. They talk about their goal of elevating the
mathematical inner lives of others and putting it on equal footing with classroom math. They share the questions driving their work: How do we share our inner math wonderings in ways that inspire
curiosity in others? How do we invite others to see their own mathematical inner lives?
Math Anywhere! is a community-based project in Vancouver, Washington which aims to build positive math experiences outside of school. They believe (1) math can be playful, (2) interesting math-y
ideas to explore exist all around us, wherever we go, (3) anytime is a good time to chat about math, and (4) everyone has important math thinking to share.
From the Math Anywhere website: We have been working to create place-based media and visiting with children and their grown-ups in different community spaces to share our ideas. You can explore some
of the prompts we have created here. We hope these will broaden how you think about math and inspire you to play with math in the places you visit everyday.
"A mathematics trail is a walk to discover mathematics. A math trail can be almost anywhere—a neighborhood, a business district or shopping mall, a park, a zoo, a library, even a government building.
The math trail map or guide points to places where walkers formulate, discuss, and solve interesting mathematical problems. Anyone can walk a math trail alone, with the family, or with another group.
Walkers cooperate along the trail as they talk about the problems. There’s no competition or grading. At the end of the math trail they have the pleasure of having walked the trail and of having done
some interesting mathematics."
Math Trails is a guide to blazing your own math trail.
Mathematics on the Move: Re-Placing Bodies in Mathematical Learning. (2015)
In this 5 minute Ignite talk, Dr. Jasmine Ma talks about the body's role in spatial reasoning and new questions for activity, participation, motivation and engagement:
• What is it like outside?
• What am I like outside?
• What is math like outside?
“The challenge for us, if we care about kids learning and their humanity, is to embrace the mess, the smells, the hormones, the specificity of their bodies, because that is a feature, not a bug of
their learning and should be leveraged rather than fixed or erased.”
Learning to Map and Mapping to Learn Our Students' Worlds - Maps at four levels of scale—global, national, regional, and local—provide a context for mathematical investigations that help teachers
learn about their students. This article offers some ideas.
Counter-Mapping the Neighborhood on Bicycles: Mobilizing Youth to Reimagine the City - Personal mobility is a mundane characteristic of daily life. However, mobility is rarely considered an
opportunity for learning in the learning sciences, and is almost never leveraged as relevant, experiential material for teaching. This article describes a social design experiment for spatial justice
that focused on changes in the personal mobility of six non-driving, African-American teenagers, who participated in an afterschool bicycle building and riding workshop located in a mid-south city.
Our study was designed to teach spatial literacy practices essential for counter-mapping. “We were guided in the study by Soja’s (2010) concept of “spatial justice” as a way to intervene in the
spatial relationship youth have with their neighborhoods, so that they might imagine new, more equitable possibilities for that geography. Similar to LeFebvre’s (1996) idea of the “right to the city”
we understood spatial justice to be concerned with empowering those who were most negatively impacted by the urban infrastructure to take a stance in reconfiguring the city. Harvey (2008) made a
similar argument for spatial justice as a living human right.”
The Spatial Justice Network - A group of scholars and practicitioners working with spatial justice. You might find it helpful to start with their spatial justice definitions page.
MATH INSTALLATIONS
‘Math With Me MN’ is a community engagement initiative out of St. Paul, Minnesota. Students and teachers, collectively, will create math installations and experiences in their homes and communities
designed to connect people to each other and to mathematics.
A ‘math installation’ is a math experience that gets people to think about math in a friendly, approachable way:
□ To reignite joy and play in (distance) learning, and to maximize student agency.
□ To see and experience mathematics in our homes, our communities, our public spaces.
□ To start closing the ‘distance’ in distance learning.
Math-On-A-Stick is a welcoming space where kids and grown-ups can explore fun math concepts at the Minnesota State Fair.
Math-on-a-Stick's Twitter Feed
In the summer of 2019, ANN's annual board meeting was held in Minnesota at the same time as Math-on-a-Stick and the board was able to volunteer on a day sponsored by I Am ABE. Read more in this
article from the Math Practitioner, ANN Goes to Math-On-A-Stick, August 2019 | {"url":"https://www.adultnumeracynetwork.org/Math-in-Public-Spaces","timestamp":"2024-11-13T15:15:26Z","content_type":"text/html","content_length":"56555","record_id":"<urn:uuid:3565a3b1-20ba-4202-bbcf-93b71112971e>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00364.warc.gz"} |
Financial results keys:
• lcc Optimal lifecycle cost
• lifecycle_generation_tech_capital_costs LCC component. Net capital costs for all generation technologies, in present value, including replacement costs and incentives. This value does not include
• lifecycle_storage_capital_costs LCC component. Net capital costs for all storage technologies, in present value, including replacement costs and incentives. This value does not include offgrid
• lifecycle_om_costs_after_tax LCC component. Present value of all O&M costs, after tax. (does not include fuel costs)
• lifecycle_fuel_costs_after_tax LCC component. Present value of all fuel costs over the analysis period, after tax.
• lifecycle_chp_standby_cost_after_tax LCC component. Present value of all CHP standby charges, after tax.
• lifecycle_elecbill_after_tax LCC component. Present value of all electric utility charges, after tax.
• lifecycle_production_incentive_after_tax LCC component. Present value of all production-based incentives, after tax.
• lifecycle_offgrid_other_annual_costs_after_tax LCC component. Present value of offgridotherannual_costs over the analysis period, after tax.
• lifecycle_offgrid_other_capital_costs LCC component. Equal to offgridothercapital_costs with straight line depreciation applied over analysis period. The depreciation expense is assumed to reduce
the owner's taxable income.
• lifecycle_outage_cost LCC component. Expected outage cost.
• lifecycle_MG_upgrade_and_fuel_cost LCC component. Cost to upgrade generation and storage technologies to be included in microgrid, plus expected microgrid fuel costs, assuming outages occur in
first year with specified probabilities.
• lifecycle_om_costs_before_tax Present value of all O&M costs, before tax.
• year_one_om_costs_before_tax Year one O&M costs, before tax.
• year_one_om_costs_after_tax Year one O&M costs, after tax.
• lifecycle_capital_costs_plus_om_after_tax Capital cost for all technologies plus present value of operations and maintenance over anlaysis period. This value does not include offgridother
• lifecycle_capital_costs Net capital costs for all technologies, in present value, including replacement costs and incentives. This value does not include offgridothercapital_costs.
• initial_capital_costs Up-front capital costs for all technologies, in present value, excluding replacement costs and incentives. This value does not include offgridothercapital_costs.
• initial_capital_costs_after_incentives Up-front capital costs for all technologies, in present value, excluding replacement costs, and accounting for incentives. This value does not include
• replacements_future_cost_after_tax Future cost of replacing storage and/or generator systems, after tax.
• replacements_present_cost_after_tax Present value cost of replacing storage and/or generator systems, after tax.
• om_and_replacement_present_cost_after_tax Present value of all O&M and replacement costs, after tax.
• developer_om_and_replacement_present_cost_after_tax Present value of all O&M and replacement costs incurred by developer, after tax.
• offgrid_microgrid_lcoe_dollars_per_kwh Levelized cost of electricity for modeled off-grid system.
• lifecycle_emissions_cost_climate LCC component if Settings input includeclimatein_objective is true. Present value of CO2 emissions cost over the analysis period.
• lifecycle_emissions_cost_health LCC component if Settings input includehealthin_objective is true. Present value of NOx, SO2, and PM2.5 emissions cost over the analysis period.
calculated in combine_results function if BAU scenario is run:
• breakeven_cost_of_emissions_reduction_per_tonne_CO2
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
ElectricTariff results keys:
• lifecycle_energy_cost_after_tax lifecycle cost of energy from the grid in present value, after tax
• year_one_energy_cost_before_tax cost of energy from the grid over the first year, before considering tax benefits
• lifecycle_demand_cost_after_tax lifecycle cost of power from the grid in present value, after tax
• year_one_demand_cost_before_tax cost of power from the grid over the first year, before considering tax benefits
• lifecycle_fixed_cost_after_tax lifecycle fixed cost in present value, after tax
• year_one_fixed_cost_before_tax fixed cost over the first year, before considering tax benefits
• lifecycle_min_charge_adder_after_tax lifecycle minimum charge in present value, after tax
• year_one_min_charge_adder_before_tax minimum charge over the first year, before considering tax benefits
• year_one_bill_before_tax sum of year_one_energy_cost_before_tax, year_one_demand_cost_before_tax, year_one_fixed_cost_before_tax, year_one_min_charge_adder_before_tax, and
• lifecycle_export_benefit_after_tax lifecycle export credits in present value, after tax
• year_one_export_benefit_before_tax export credits over the first year, before considering tax benefits. A positive value indicates a benefit.
• lifecycle_coincident_peak_cost_after_tax lifecycle coincident peak charge in present value, after tax
• year_one_coincident_peak_cost_before_tax coincident peak charge over the first year
ElectricLoad results keys:
• load_series_kw vector of site load in every time step
• critical_load_series_kw vector of site critical load in every time step
• annual_calculated_kwh sum of the load_series_kw
• offgrid_load_met_series_kw vector of electric load met by generation techs, for off-grid scenarios only
• offgrid_load_met_fraction percentage of total electric load met on an annual basis, for off-grid scenarios only
• offgrid_annual_oper_res_required_series_kwh , total operating reserves required (for load and techs) on an annual basis, for off-grid scenarios only
• offgrid_annual_oper_res_provided_series_kwh , total operating reserves provided on an annual basis, for off-grid scenarios only
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
ElectricUtility results keys:
• annual_energy_supplied_kwh # Total energy supplied from the grid in an average year.
• electric_to_load_series_kw # Vector of power drawn from the grid to serve load.
• electric_to_storage_series_kw # Vector of power drawn from the grid to charge the battery.
• annual_emissions_tonnes_CO2 # Average annual total tons of CO2 emissions associated with the site's grid-purchased electricity. If includeexportedelecemissionsin_total is False, this value only
reflects grid purchases. Otherwise, it accounts for emissions offset from any export to the grid.
• annual_emissions_tonnes_NOx # Average annual total tons of NOx emissions associated with the site's grid-purchased electricity. If includeexportedelecemissionsin_total is False, this value only
reflects grid purchases. Otherwise, it accounts for emissions offset from any export to the grid.
• annual_emissions_tonnes_SO2 # Average annual total tons of SO2 emissions associated with the site's grid-purchased electricity. If includeexportedelecemissionsin_total is False, this value only
reflects grid purchases. Otherwise, it accounts for emissions offset from any export to the grid.
• annual_emissions_tonnes_PM25 # Average annual total tons of PM25 emissions associated with the site's grid-purchased electricity. If includeexportedelecemissionsin_total is False, this value only
reflects grid purchsaes. Otherwise, it accounts for emissions offset from any export to the grid.
• lifecycle_emissions_tonnes_CO2 # Total tons of CO2 emissions associated with the site's grid-purchased electricity over the analysis period. If includeexportedelecemissionsin_total is False, this
value only reflects grid purchaes. Otherwise, it accounts for emissions offset from any export to the grid.
• lifecycle_emissions_tonnes_NOx # Total tons of NOx emissions associated with the site's grid-purchased electricity over the analysis period. If includeexportedelecemissionsin_total is False, this
value only reflects grid purchaes. Otherwise, it accounts for emissions offset from any export to the grid.
• lifecycle_emissions_tonnes_SO2 # Total tons of SO2 emissions associated with the site's grid-purchased electricity over the analysis period. If includeexportedelecemissionsin_total is False, this
value only reflects grid purchaes. Otherwise, it accounts for emissions offset from any export to the grid.
• lifecycle_emissions_tonnes_PM25 # Total tons of PM2.5 emissions associated with the site's grid-purchased electricity over the analysis period. If includeexportedelecemissionsin_total is False,
this value only reflects grid purchaes. Otherwise, it accounts for emissions offset from any export to the grid.
• avert_emissions_region # EPA AVERT region of the site. Used for health-related emissions from grid electricity (populated if default emissions values are used) and climate emissions if "co2from
avert" is set to true.
• distance_to_avert_emissions_region_meters # Distance in meters from the site to the nearest AVERT emissions region.
• cambium_emissions_region # NREL Cambium region of the site. Used for climate-related emissions from grid electricity (populated only if default (Cambium) climate emissions values are used)
'Series' and 'Annual' energy and emissions outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
and emissions outputs averaged over the analysis period.
By default, REopt uses marginal emissions rates for grid-purchased electricity. Marginal emissions rates are most appropriate for reporting a change in emissions (avoided or increased) rather than
emissions totals. It is therefore recommended that emissions results from REopt (using default marginal emissions rates) be reported as the difference in emissions between the optimized and BAU case.
Note also that the annual_emissions metrics are average annual emissions over the analysis period, accounting for expected changes in future grid emissions.
PV results keys:
• size_kw Optimal PV DC capacity
• lifecycle_om_cost_after_tax Lifecycle operations and maintenance cost in present value, after tax
• year_one_energy_produced_kwh Energy produced over the first year
• annual_energy_produced_kwh Average annual energy produced when accounting for degradation
• lcoe_per_kwh Levelized Cost of Energy produced by the PV system
• electric_to_load_series_kw Vector of power used to meet load over the first year
• electric_to_storage_series_kw Vector of power used to charge the battery over the first year
• electric_to_grid_series_kw Vector of power exported to the grid over the first year
• electric_curtailed_series_kw Vector of power curtailed over the first year
• annual_energy_exported_kwh Average annual energy exported to the grid
• production_factor_series PV production factor in each time step, either provided by user or obtained from PVWatts
The key(s) used to access PV outputs in the results dictionary is determined by the PV.name value to allow for modeling multiple PV options. (The default PV.name is "PV".)
All outputs account for any existing PV. E.g., size_kw includes existing capacity and the REopt-recommended additional capacity.
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
Wind results keys:
• size_kw Optimal Wind capacity [kW]
• lifecycle_om_cost_after_tax Lifecycle operations and maintenance cost in present value, after tax
• year_one_om_cost_before_tax Operations and maintenance cost in the first year, before tax benefits
• electric_to_storage_series_kw Vector of power used to charge the battery over an average year
• electric_to_grid_series_kw Vector of power exported to the grid over an average year
• annual_energy_exported_kwh Average annual energy exported to the grid
• electric_to_load_series_kw Vector of power used to meet load over an average year
• annual_energy_produced_kwh Average annual energy produced
• lcoe_per_kwh Levelized Cost of Energy produced by the PV system
• electric_curtailed_series_kw Vector of power curtailed over an average year
• production_factor_series Wind production factor in each time step, either provided by user or obtained from SAM
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
ElectricStorage results keys:
• size_kw Optimal inverter capacity
• size_kwh Optimal storage capacity
• soc_series_fraction Vector of normalized (0-1) state of charge values over the first year
• storage_to_load_series_kw Vector of power used to meet load over the first year
• initial_capital_cost Upfront capital cost for storage and inverter
The following results are reported if storage degradation is modeled:
• state_of_health
• maintenance_cost
• replacement_month
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
HotThermalStorage results keys:
• size_gal Optimal TES capacity, by volume [gal]
• soc_series_fraction Vector of normalized (0-1) state of charge values over the first year [-]
• storage_to_load_series_mmbtu_per_hour Vector of power used to meet load over the first year [MMBTU/hr]
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
ColdThermalStorage results:
• size_gal Optimal TES capacity, by volume [gal]
• soc_series_fraction Vector of normalized (0-1) state of charge values over the first year [-]
• storage_to_load_series_ton Vector of power used to meet load over the first year [ton]
Generator results keys:
• size_kw Optimal generator capacity
• lifecycle_fixed_om_cost_after_tax Lifecycle fixed operations and maintenance cost in present value, after tax
• year_one_fixed_om_cost_before_tax fixed operations and maintenance cost over the first year, before considering tax benefits
• lifecycle_variable_om_cost_after_tax Lifecycle variable operations and maintenance cost in present value, after tax
• year_one_variable_om_cost_before_tax variable operations and maintenance cost over the first year, before considering tax benefits
• lifecycle_fuel_cost_after_tax Lifecycle fuel cost in present value, after tax
• year_one_fuel_cost_before_tax Fuel cost over the first year, before considering tax benefits
• annual_fuel_consumption_gal Gallons of fuel used in each year
• electric_to_storage_series_kw Vector of power sent to battery in an average year
• electric_to_grid_series_kw Vector of power sent to grid in an average year
• electric_to_load_series_kw Vector of power sent to load in an average year
• annual_energy_produced_kwh Average annual energy produced over analysis period
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
ExistingBoiler results keys:
• size_mmbtu_per_hour
• fuel_consumption_series_mmbtu_per_hour
• annual_fuel_consumption_mmbtu
• thermal_production_series_mmbtu_per_hour
• annual_thermal_production_mmbtu
• thermal_to_storage_series_mmbtu_per_hour # Thermal power production to TES (HotThermalStorage) series [MMBtu/hr]
• thermal_to_steamturbine_series_mmbtu_per_hour
• thermal_to_load_series_mmbtu_per_hour
• lifecycle_fuel_cost_after_tax
• year_one_fuel_cost_before_tax
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
CHP results keys:
• size_kw Power capacity size of the CHP system [kW]
• size_supplemental_firing_kw Power capacity of CHP supplementary firing system [kW]
• annual_fuel_consumption_mmbtu Fuel consumed in a year [MMBtu]
• annual_electric_production_kwh Electric energy produced in a year [kWh]
• annual_thermal_production_mmbtu Thermal energy produced in a year (not including curtailed thermal) [MMBtu]
• electric_production_series_kw Electric power production time-series array [kW]
• electric_to_grid_series_kw Electric power exported time-series array [kW]
• electric_to_storage_series_kw Electric power to charge the battery storage time-series array [kW]
• electric_to_load_series_kw Electric power to serve the electric load time-series array [kW]
• thermal_to_storage_series_mmbtu_per_hour Thermal power to TES (HotThermalStorage) time-series array [MMBtu/hr]
• thermal_curtailed_series_mmbtu_per_hour Thermal power wasted/unused/vented time-series array [MMBtu/hr]
• thermal_to_load_series_mmbtu_per_hour Thermal power to serve the heating load time-series array [MMBtu/hr]
• thermal_to_steamturbine_series_mmbtu_per_hour Thermal (steam) power to steam turbine time-series array [MMBtu/hr]
• year_one_fuel_cost_before_tax Cost of fuel consumed by the CHP system in year one [$]
• lifecycle_fuel_cost_after_tax Present value of cost of fuel consumed by the CHP system, after tax [$]
• year_one_standby_cost_before_tax CHP standby charges in year one [$]
• lifecycle_standby_cost_after_tax Present value of all CHP standby charges, after tax.
• thermal_production_series_mmbtu_per_hour
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
Missing docstring for REopt.add_boiler_results. Check Documenter's build log for details.
HeatingLoad results keys:
• dhw_thermal_load_series_mmbtu_per_hour vector of site thermal domestic hot water load in every time step
• space_heating_thermal_load_series_mmbtu_per_hour vector of site thermal space heating load in every time step
• process_heat_thermal_load_series_mmbtu_per_hour vector of site thermal process heat load in every time step
• total_heating_thermal_load_series_mmbtu_per_hour vector of sum thermal heating load in every time step
• dhw_boiler_fuel_load_series_mmbtu_per_hour vector of site fuel domestic hot water load in every time step
• space_heating_boiler_fuel_load_series_mmbtu_per_hour vector of site fuel space heating load in every time step
• process_heat_boiler_fuel_load_series_mmbtu_per_hour vector of site fuel process heat load in every time step
• total_heating_thermal_load_series_mmbtu_per_hour vector of sum fuel heating load in every time step
• annual_calculated_dhw_thermal_load_mmbtu sum of the dhw_thermal_load_series_mmbtu_per_hour
• annual_calculated_space_heating_thermal_load_mmbtu sum of the space_heating_thermal_load_series_mmbtu_per_hour
• annual_calculated_process_heat_thermal_load_mmbtu sum of the process_heat_thermal_load_series_mmbtu_per_hour
• annual_calculated_total_heating_thermal_load_mmbtu sum of the total_heating_thermal_load_series_mmbtu_per_hour
• annual_calculated_dhw_boiler_fuel_load_mmbtu sum of the dhw_boiler_fuel_load_series_mmbtu_per_hour
• annual_calculated_space_heating_boiler_fuel_load_mmbtu sum of the space_heating_boiler_fuel_load_series_mmbtu_per_hour
• annual_calculated_process_heat_boiler_fuel_load_mmbtu sum of the process_heat_boiler_fuel_load_series_mmbtu_per_hour
• annual_calculated_total_heating_boiler_fuel_load_mmbtu sum of the total_heating_boiler_fuel_load_series_mmbtu_per_hour
CoolingLoad results keys:
• load_series_ton # vector of site cooling load in every time step
• annual_calculated_tonhour # sum of the load_series_ton. Annual site total cooling load [tonhr]
• electric_chiller_base_load_series_kw # Hourly total base load drawn from chiller [kW-electric]
• annual_electric_chiller_base_load_kwh # Annual total base load drawn from chiller [kWh-electric]
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period.
Outages results keys:
• expected_outage_cost The expected outage cost over the random outages modeled.
• max_outage_cost_per_outage_duration The maximum outage cost in every outage duration modeled.
• unserved_load_series_kw The amount of unserved load in each outage and each time step.
• unserved_load_per_outage_kwh The total unserved load in each outage.
• storage_microgrid_upgrade_cost The cost to include the storage system in the microgrid.
• storage_discharge_series_kw Array of storage power discharged in every outage modeled.
• pv_microgrid_size_kw Optimal microgrid PV capacity. Note that the name PV can change based on user provided PV.name.
• pv_microgrid_upgrade_cost The cost to include the PV system in the microgrid.
• pv_to_storage_series_kw Array of PV power sent to the battery in every outage modeled.
• pv_curtailed_series_kw Array of PV curtailed in every outage modeled.
• pv_to_load_series_kw Array of PV power used to meet load in every outage modeled.
• wind_microgrid_size_kw Optimal microgrid Wind capacity.
• wind_microgrid_upgrade_cost The cost to include the Wind system in the microgrid.
• wind_to_storage_series_kw Array of Wind power sent to the battery in every outage modeled.
• wind_curtailed_series_kw Array of Wind curtailed in every outage modeled.
• wind_to_load_series_kw Array of Wind power used to meet load in every outage modeled.
• generator_microgrid_size_kw Optimal microgrid Generator capacity. Note that the name Generator can change based on user provided Generator.name.
• generator_microgrid_upgrade_cost The cost to include the Generator system in the microgrid.
• generator_to_storage_series_kw Array of Generator power sent to the battery in every outage modeled.
• generator_curtailed_series_kw Array of Generator curtailed in every outage modeled.
• generator_to_load_series_kw Array of Generator power used to meet load in every outage modeled.
• generator_fuel_used_per_outage_gal Array of fuel used in every outage modeled, summed over all Generators.
• chp_microgrid_size_kw Optimal microgrid CHP capacity.
• chp_microgrid_upgrade_cost The cost to include the CHP system in the microgrid.
• chp_to_storage_series_kw Array of CHP power sent to the battery in every outage modeled.
• chp_curtailed_series_kw Array of CHP curtailed in every outage modeled.
• chp_to_load_series_kw Array of CHP power used to meet load in every outage modeled.
• chp_fuel_used_per_outage_mmbtu Array of fuel used in every outage modeled, summed over all CHPs.
• microgrid_upgrade_capital_cost Total capital cost of including technologies in the microgrid
• critical_loads_per_outage_series_kw Critical load series in every outage modeled
• soc_series_fraction ElectricStorage state of charge series in every outage modeled
The output keys for "Outages" are subject to change.
Outage results only added to results when multiple outages are modeled via the ElectricUtility.outage_start_time_steps and ElectricUtility.outage_durations inputs.
When modeling PV the name of the PV system is used for the output keys to allow for modeling multiple PV systems. The default PV name is PV.
The Outage results can be very large when many outages are modeled and can take a long time to generate.
AbsorptionChiller results keys:
• size_kw # Optimal power capacity size of the absorption chiller system [kW]
• size_ton
• thermal_to_storage_series_ton # Thermal production to ColdThermalStorage
• thermal_to_load_series_ton # Thermal production to cooling load
• thermal_consumption_series_mmbtu_per_hour
• annual_thermal_consumption_mmbtu
• annual_thermal_production_tonhour
• electric_consumption_series_kw
• annual_electric_consumption_kwh
FlexibleHVAC results keys:
• purchased
• temperatures_degC_node_by_time
• upgrade_cost
SteamTurbine results keys:
• size_kw Power capacity size [kW]
• annual_thermal_consumption_mmbtu Thermal (steam) consumption [MMBtu]
• annual_electric_production_kwh Electric energy produced in a year [kWh]
• annual_thermal_production_mmbtu Thermal energy produced in a year [MMBtu]
• thermal_consumption_series_mmbtu_per_hour Thermal (steam) energy consumption series [MMBtu/hr]
• electric_production_series_kw Electric power production series [kW]
• electric_to_grid_series_kw Electric power exported to grid series [kW]
• electric_to_storage_series_kw Electric power to charge the battery series [kW]
• electric_to_load_series_kw Electric power to serve load series [kW]
• thermal_to_storage_series_mmbtu_per_hour Thermal production to charge the HotThermalStorage series [MMBtu/hr]
• thermal_to_load_series_mmbtu_per_hour Thermal production to serve the heating load SERVICES [MMBtu/hr]
'Series' and 'Annual' energy outputs are average annual
REopt performs load balances using average annual production values for technologies that include degradation. Therefore, all timeseries (_series) and annual_ results should be interpretted as energy
outputs averaged over the analysis period. | {"url":"https://nrel.github.io/REopt.jl/dev/reopt/outputs/","timestamp":"2024-11-01T19:45:34Z","content_type":"text/html","content_length":"56333","record_id":"<urn:uuid:20429442-d419-4546-b312-7114b84b003e>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00400.warc.gz"} |
Exponentials - Solution
The derivative of e^t is equal to the original function
The derivative of e^(0.5*t) is scaled downwards (squashed) in the y-axis with respect to the original function
The derivative of e^(2*t) is scaled upwards(stretched) in the y-axis with respect to the original function
plotXpose app is available on Google Play and App Store
Google Play and the Google Play logo are trademarks of Google LLC.
A version will shortly be available for Windows. | {"url":"https://www.plotxpose.com/ExponentialsSolution.htm","timestamp":"2024-11-03T03:34:19Z","content_type":"text/html","content_length":"9171","record_id":"<urn:uuid:993a6e42-8a5a-4218-a0cf-91e2645ead2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00305.warc.gz"} |
class cobyqa.problem.Problem(obj, x0, bounds, linear, nonlinear, callback, feasibility_tol, scale, store_history, history_size, filter_size, debug)[source]#
Optimization problem.
Bound constraints.
History of objective function evaluations.
Name of the objective function.
Whether the problem is a feasibility problem.
Linear constraints.
Number of bound constraints.
Number of linear equality constraints.
Number of linear inequality constraints.
Number of nonlinear equality constraints.
Number of nonlinear inequality constraints.
History of maximum constraint violations.
Number of variables.
Number of function evaluations.
Number of variables in the original problem (with fixed variables).
Type of the problem.
Initial guess.
__call__(x[, penalty]) Evaluate the objective and nonlinear constraint functions.
best_eval(penalty) Return the best point in the filter and the corresponding objective and nonlinear constraint function evaluations.
build_x(x) Build the full vector of variables from the reduced vector.
maxcv(x[, cub_val, ceq_val]) Evaluate the maximum constraint violation. | {"url":"https://www.cobyqa.com/stable/dev/generated/cobyqa.problem.Problem.html","timestamp":"2024-11-14T20:59:39Z","content_type":"text/html","content_length":"31049","record_id":"<urn:uuid:57316012-692a-4299-b505-ca8ed2d38209>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00524.warc.gz"} |
3. The Nyquist Limit
3. The Nyquist Limit Home | Next
The simplest sound wave is an oscillation between two amplitudes. A sampled waveform thus needs at least two sample points per cycle. Put another way, the wave's frequency must not be above half the
sampling frequency. This limit is called the Nyquist limit of a given sampling frequency.
Sine wave at 1/2 sampling rate with two samples per cycle
If a sine wave higher than the Nyquist frequency is sampled, a sine wave of lower frequency results. This effect is called aliasing.
Sine waves above 1/2 sampling rate (blue) and resulting aliases (orange)
Since an alias is no different than a normally sampled wave of the same frequency, aliasing can be avoided only by removing frequencies above the Nyquist limit before sampling. | {"url":"http://www.slack.net/~ant/bl-synth/3.nyquist.html","timestamp":"2024-11-09T03:51:27Z","content_type":"text/html","content_length":"1520","record_id":"<urn:uuid:85358f6b-5a9b-49e0-8787-8d1e5701e468>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00217.warc.gz"} |
Uniform proofs as a foundation for logic programming
Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 (1991) 125-157. A proof-theoretic characterization of
logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided
by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is
formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of
the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and
higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order
versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance
of the various generalizations to first-order Horn clauses is briefly discussed.
Bibliographical note
Funding Information:
A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the
declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed
search instructions. The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a
goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming
language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to * A preliminary version of
this paper appeared as [21]. Theorem 3 of that paper is incorrect. It is corrected by the material in Sections 5 and 6 of the current paper. ** Supported by NSF grant CCR-87-05596 and DARPA grant
NOOO-14-85-K-0018. *** Supported by NSF grant IRI-8805696 and AR0 Contract DAAL03-88-K-0082. ‘Supported by The Office of Naval Research under contract NOOO14-84-K-0415a nd by the Defense Advanced
Research Projects Agency (DOD), ARPA Order No. 5404, monitored by the Office of Naval Research under the same contract. O”S upported by NSF grants DMS85-01522 and CCR-87-05596, by ONR contract
NOOO14-88-K-0635, and by the University of Pennsylvania Natural Sciences Association Young Faculty Award.
Dive into the research topics of 'Uniform proofs as a foundation for logic programming'. Together they form a unique fingerprint. | {"url":"https://experts.umn.edu/en/publications/uniform-proofs-as-a-foundation-for-logic-programming","timestamp":"2024-11-10T05:34:23Z","content_type":"text/html","content_length":"57802","record_id":"<urn:uuid:6b4b31f1-544e-4689-a328-bc6b0630ff96>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00190.warc.gz"} |
An Etymological Dictionary of Astronomy and Astrophysics
static equilibrium
ترازمندی ِایستا
tarâzmandi-ye istâ
Fr.: équilibre statique
The state of a rigid body which is not moving at all. The conditions for static equilibrium are: 1) the sum of the external forces on the object must equal zero, and 2) the sum of the → torques must
equal zero. See also → dynamic equilibrium and → mechanical equilibrium.
→ static; → equilibrium.
static limit
حد ِایستا
hadd-e istâ
Fr.: limite stationnaire
Same as → stationary limit.
→ static; → limit.
static pressure
فشار ِایستا
fešâr-e istâ
Fr.: pression statique
In → fluid mechanics, the → pressure felt by an object suspended in a → fluid and moving with it. This pressure is called static because the object is not moving relative to the fluid. See also →
dynamic pressure.
→ static; → pressure.
static Universe
گیتی ِایستا
giti-ye istâ
Fr.: Univers stationnaire
A closed Universe of finite volume with a constant radius of curvature.
→ static; → Universe.
Fr.: statique
The branch of → mechanics which studies the laws of composition of forces and the conditions of equilibrium of material bodies under the action of forces.
→ static; → -ics.
Istâyik, from istâ, → static + -ik, → -ics.
istgâh (#)
Fr.: station
A stopping place for trains or other land vehicles, for the transfer of freight or passengers. → space station.
M.E., from O.Fr. station, from L. stationem (nominative statio) "a standing, job, position," related to stare "to stand," cognate with Pers. istâdan "to stand," as below.
Istgâh "standing place," from ist present stem of istâdan "to stand" (Mid.Pers. êstâtan; O.Pers./Av. sta- "to stand, stand still; set;" Av. hištaiti; cf. Skt. sthâ- "to stand;" Gk. histemi "put,
place, weigh," stasis "a standing still;" L. stare "to stand;" Lith. statau "place;" Goth. standan; PIE base *sta- "to stand") + gâh "place; time" (Mid.Pers. gâh, gâs "time;" O.Pers. gāθu-; Av.
gātav-, gātu- "place, throne, spot;" cf. Skt. gâtu- "going, motion; free space for moving; place of abode;" PIE *gwem- "to go, come").
istvar (#)
Fr.: stationnaire
Having a fixed, unchanging position; motionless. geostationary orbit
M.E. from L. stationarius, in classical L., "of a military station," from statio, → station.
Isatvar, from ist present stem of istâdan "to stand" (Mid.Pers. êstâtan; O.Pers./Av. sta- "to stand, stand still; set;" Av. hištaiti; cf. Skt. sthâ- "to stand;" Gk. histemi "put, place, weigh,"
stasis "a standing still;" L. stare "to stand;" Lith. statau "place;" Goth. standan; PIE base *sta- "to stand") + -var suffix of possession, variant -ur (Mid.Pers. -uwar, -war; from O.Pers. -bara,
from bar- "to bear, carry").
stationary black hole
سیهچال ِایستور
siyah-câl-e istvar
Fr.: trou noir stationnaire
A → black hole with zero → angular momentum, that does not rotate.
→ stationary; → black hole.
stationary limit surface
رویهی ِحدّ ِایستور
ruye-ye hadd-e istvar
Fr.: surface limite stationnaire
A property of → space-time outside a → rotating black hole, which consists of a surface which geometrically bounds the → ergosphere outward. At the stationary limit a particle would have to move with
the local light velocity in order to appear stationary to a distant observer. This is because the space here is being dragged at exactly the speed of light relative to the rest of space. Outside this
limit space is still dragged, but at a rate less than the speed of light. Also known as → static limit.
→ stationary; → limit; → surface.
stationary noise
نوفهی ِایستور
nufe-ye istvar
Fr.: bruit stationnaire
Electronics: A random noise whose intensity remains constant with time.
→ stationary; → noise.
stationary orbit
مدار ِایستور
madâr-e istvar
Fr.: orbite stationnaire
An orbit in which the satellite revolves about the primary at the angular rate at which the primary rotates on its axis. From the primary, the satellite thus appears to be stationary over a point on
the primary.
→ stationary; → orbit.
stationary phase
فاز ِایستور
fâz-e istvar
Fr.: phase stationnaire
Mechanics: The condition of a body or system at rest.
→ stationary; → phase.
stationary point
نقطهی ِایستور
noqte-ye istvar
Fr.: point critique, ~ stationnaire
1) Math.: For a → function y = f(x), a point at which the → tangent to the graph is horizontal. In other words, a point where the → slope is zero: dy/dx = 0.
2) Of a planet, the position at which the rate of change of its apparent → right ascension is momentarily zero.
→ stationary; → point.
stationary satellite
ماهوارهی ِایستور
mâhvâre-ye istvar
Fr.: satellite stationnaire
An artificial satellite in a synchronous orbit. → geostationary orbit
→ stationary; → satellite.
stationary time series
سری ِزمانی ِایستور
seri-ye zamâni-ye istvar
Fr.: série temporelle stationnaire
A → time series if it obeys the following criteria: 1) Constant → mean over time (t). 2) Constant → variance for all t, and 3) The → autocovariance function between X[t1] and X[t2] only depends on
the interval t[1] and t[2].
→ stationary; → time; → series.
stationary wave
موج ِایستور
mowj-e istvar
Fr.: onde stationnaire
Same as → standing wave.
→ stationary; → wave.
âmâri (#)
Fr.: statistique
Of, pertaining to, consisting of, or based on → statistics.
Statistic, from → statistics + → -al.
statistical analysis
آنالس ِآماری
ânâlas-e âmâri
Fr.: analyse statistique
The process of collecting, manipulating, analyzing, and interpreting quantitative data to uncover underlying causes, patterns, and relationships between variables.
→ statistical; → analysis.
statistical equilibrium
ترازمندی ِآماری
tarâzmandi-ye âmâri
Fr.: équilibre statistique
A state in which the average density of atoms per cubic centimeter in any atomic state does not change with time and in which, statistically, energy is equally divided among all degrees of freedom if
classical concepts prevail.
→ statistical; → equilibrium.
statistical hypothesis
انگارهی ِآماری
engâre-ye âmâri
Fr.: hypothèse statistique
An assumed statement about the way a → random variable is distributed. A statistical hypothesis generally specifies the form of the → probability distribution or the values of the parameters of the
distribution. The statement may be true or false. See also → null hypothesis.
→ statistical; → hypothesis. | {"url":"https://dictionary.obspm.fr/index.php?showAll=1&&search=S&&formSearchTextfield=&&page=48","timestamp":"2024-11-09T04:32:51Z","content_type":"text/html","content_length":"41067","record_id":"<urn:uuid:4c96eade-2337-400c-95bb-6e232968273d>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00753.warc.gz"} |
discount rate – Compass Rose
Consider the case of buying one's partner flowers. You can think of each act of flower-buying as an act of caring (and this is typically the right attitude if your partner likes flowers and you want
to genuinely relate on that level). Or you can think of installing the flower-buying habit as the act of caring that you hope will be perceived through the mask of mere flowers. The first type of
person I call a Phoenix, the second is a Skroderider.
A Special Discount Just For You, and The Importance of Lost Opportunities
In my previous posts on economic and financial concepts, I talked about the idea of a discount rate - the conversion rate between future goods and present goods. But how do you pick a rate to use?
Opportunity Cost
Brian, as you may remember, used the interest rate he had to pay on his debt to help him compare future payments with immediate ones. This is similar to the financial concept of "cost of funding,"
and it makes some intuitive sense. He can use this to figure out which strategies are strictly better than others.
Abby used the interest rate she earned in her bank account to make the same kind of comparison.
One of these interest rates is the rate you pay on your debt. The other is the rate you earn on savings. Do these seem like different things? To an economist, they're the same.
More precisely, if Abby withdraws $1,000 from her bank account that earns 2% interest per year, then in a year she has $1,020 less in her account. If she borrows $1,000 at 2% interest, then in a
year, to pay off the debt she would have to take $1,020 out of her bank account. The amounts are the same for any other time interval too.
Because of this, an economist would say that the interest Abby doesn't earn from the $1,000 she withdrew is a cost, just as much as the interest she would have to pay if she borrowed at that rate.
The word for the cost of a foregone opportunity like this is "opportunity cost."
When you think about it this way, Abby and Brian are really doing the same thing when they use the interest rate on Brian's debt, and the interest rate on Abby's savings, to help compare the future
with the present. In both cases they are making the comparison by using as their discount rate, the cost of holding onto money, instead paying off debt or putting it into savings.
Picking a Discount Rate Based on Opportunity Cost
You probably have a current "best option" for where to get extra money you need, or where to put extra money. If you have a lot of credit card debt that you have the option of paying off or borrowing
more of, then the interest rate on that debt is probably the opportunity cost of doing something else with your money. If you have a lot of savings in the form of mutual funds, then the
expected return on your investments is your opportunity cost of doing something else with your money.
A key assumption here is that of "liquidity." Put simply, I'm just talking about cases where you actually can put money into or take it out of something. If you have fixed rate debt that you can't
pay off or borrow more from, then there's no decision you can make about that - so it's not a real alternative to other uses for your money.
The world's not always that simple, though. Example:
Dylan has a mortgage with an interest rate of 5%, government savings bonds that earn 6% per year (but new bonds only earn 3%), and an emergency fund in a savings account that earns 0.5%. What should
Dylan's discount rate be?
If Dylan has extra money to put somewhere, Dylan's better off paying down the mortgage and "earning" 5% on that money than either buying new savings bonds that earn 3% interest or putting it in the
bank account that earns 0.5%. So if Dylan's comparing an option to get money now with an option to get money later, they should use a discount rate of 5%, the interest they can "earn" on that money.
But what if Dylan is comparing a present expense to a future expense? Well, if it's an emergency then Dylan can use the emergency fund to smooth things out. But even then, if Dylan takes their
emergency fund seriously, they're going to have to come up with the cash to make up the difference - and that means drawing down those savings bonds (or spending less day to day, but that's a
complication I'll get to later). And it's not really practical to borrow a little more against your house every time you have an extra expense. So Dylan's going to have to draw down those savings
bonds earning 6%, and that should be Dylan's discount rate for expenses.
A Brief Digression on Interest Rate Arbitrage
Edward has a $100,000 mortgage with an interest rate of 5%, $2,000 of extra money (above what Edward needs for emergencies or to pay ongoing expenses) in a savings account that pays 1% interest per
year, and $7,000 of credit card debt with an annual interest rate of 15%. Edward believes that an index fund would grow in value by 6% per year, but hasn't set one up yet. What should Edward do?
Edward has an opportunity to earn some free money here. If Edward takes a dollar from their savings account to pay off credit card debt, the opportunity cost on that dollar is 1% foregone interest
per year, but the return on investment is 15% interest Edward won't owe. So on net, just by moving money from one account to another, Edward earns an extra 14%. In general, when one investment is
strictly better than another, selling the worse one and buying the better one is called "arbitrage", and this is an example.
So Edward should put as much money as possible into paying off their credit card debt, and put the rest into the investment account. At the end of the process, Edward will have $5,000 in credit card
debt, and a discount rate of 15%.
Now let's say Edward gets a $20,000 windfall - an inheritance, a bonus, or a bunch of cash is discovered inside the mattress. What should Edward do?
Well, we know that Edward should first pay off their credit card debt, so that takes care of the first $5,000. But Edward still has $15,000 to invest. What's next?
You can't pay off more credit card debt than you borrowed, so right now Edward has two places to get funds from or put them into: a mortgage with 5% interest and a savings account with 1% interest.
You might say that the next best investment Edward has is paying off their 5% mortgage, since Edward "earns" more interest that way than the opportunity cost of 1% interest in the savings account -
Edward's getting an extra 4%, free, per year.
But what about the index fund Edward doesn't have?
That's the tricky thing about opportunity cost. You don't just count the opportunities you've taken advantage of in the past - you have to compare all the different options you have now. Edward
expects a 6% annual return on shares in an index fund. So the opportunity cost of doing anything else is 6%. From that perspective, Edward would actually be losing 1% in interest each year on any
money used to pay off that 5% mortgage ahead of schedule.
The best choice for Edward is to open up an investment account and use the remaining $15,000 to buy shares in that index fund. Edward's return on investment is 6%, so that's the discount rate Edward
should use when comparing other future and present payments and expenses.
Indifference Curves
Let's say I want to buy some fruit for a snack. $1 will buy me an apple, or an orange. I like them both about the same, so I'm indifferent between these choices. If either of them only costs $0.90,
I'll buy that one.
But I have a strong preference for more tastes in a meal. Suppose the store's running a special where for $1 you can get half an apple and half an orange. I'm going to get this. I'd even spend more
than $1 on it.
Economists call this a "revealed preference" that I put the same value on one apple or one orange, but a higher value on the first half of each.
That's not very interesting yet, but let's say the combination is just a quarter of an apple and a quarter of an orange, still for $1. Now I'm torn again - I could go for either the orange, the
apple, or the combo.
Since apples and oranges can be divided nearly continuously, you could describe a mathematical curve of the apple-orange combinations I'd be indifferent between. This is called an "indifference
You can draw more than one indifference curve. The curve that passes through the quarter-apple and quarter-orange combo isn't the same curve that passes through the halfsies combo - I prefer any
point along the second curve to any point along the first curve.
If your preferences are consistent, no two indifference curves will ever intersect. So if you prefer 10 hats to 5, and 10 scarves to 5, your indifference curves can't look like this:
If those were your utility curves, then you'd prefer 10 hats to 5 hats, and you'd be indifferent between 5 hats and a combination of 3 1/3 hats and 3 1/3 scarves, and you'd be indifferent between
that combination and 10 hats. (Of course, you can't really have a third of a hat in any meaningful way. Economic reasoning often assumes that quantities are continuous, which is rarely completely
true, but often close enough to be useful.)
Indifference curves tell you what you already know, but they don't help you extend your knowledge beyond the examples you think through. We want something that will help us abstract what we learn
from our indifference curves, and apply that knowledge to many cases at once. That's why utility functions exist.
Utility Functions
Economists basically pretend that your indifference curves are a side effect of your attempts to maximize some mathematical function of the things you consume. In the case of my snacktime decision, I
am trying to buy the combination that maximizes some function U=f(n_apples, n_oranges) subject to my budgetary constraints.
In this model, an indifference curve is just the set of all points with the same utility U - if they're all worth the same amount of utility, I don't care which one I pick - I'm indifferent. But if
choices are on different indifference curves, that's the same as saying that they have different amounts of utility - and I'll always try to pick the one with a higher value of U.
Let's take a simple example and imagine that I'm indifferent between one orange, one apple, and half of each, and that I prefer any of two oranges, an orange and an apple, or two apples - but am
indifferent between that set as well. In that case, my indifference curve is just a straight line, and my utility function is:
If I set U=1, I get:
This produces my first indifference curve.
If I set U=2, I get:
This produces my second indifference curve.
One important point here is that since utility is a quantity we can't observe directly, and we only observe the direction of a preference, not the strength of that preference, it doesn't matter if
I'm maximizing U or 1,000,000*U or U+1,000,000. I make the same decisions in any of those cases. So as a quantity, utility is meaningless, but it explains the existence of indifference curves very
Now let's talk about discount rates again.
Using Indifference to Infer a Discount Rate
Discount rates are ways of comparing valuable things in the future to valuable things in the present. (I used money because it's a simple example, but you may have time preferences about other
So one thing you can do is assume that you have some fixed discount rate of x%, and ask yourself, would you rather eat a chocolate bar tomorrow, or 1.5 chocolate bars next year? (Don't ask about
right now, because of hyperbolic discounting.) Adjust the amounts up or down until you get quantities where you're conflicted about the choice. Then you have an estimate of your true discount rate.
You should probably try comparing several kinds of valuable things, at several time scales, to get your true discount rate - your intuitions won't all be perfectly consistent, and you're trying to
get something that's a good approximation for most of your preferences, not just your preferences about the first example you think of.
Would I rather spend an extra day hanging out with a friend this year, or two extra days with a friend in ten years?
Try to come up with some more examples on your own. Then you can back out your discount rate from your indifference curves, and pick a rate in the middle of your estimates.
Here's how to back out a rate. Let's say I'm indifferent between eating 20 chocolate truffles this week, and 21 chocolate truffles in a year. That's one year of discounting, so my rate is just 21/20
- 1 = 1.05 - 1 = 0.05 = 5%
On the other hand, let's say that I'm indifferent between spending a day with a friend this year, and two days in ten years. I need to back out the annual discount rate from that. I'll do it using a
computer, I'm to lazy to use logarithms, but it's not particularly hard math if you want to look up how to do it yourself:
(2/1)^(1/10) - 1 = 1.07 - 1 = 0.07 = 7%
So I decide my subjective discount rate's about 6%.
What if My Discount Rates Don't Agree?
If your subjective discount rate is lower than your financial discount rate, then you're generally better off saving money and spending it later. Some people report that when they reflect for long
enough on their subjective discount rates, they don't want to discount at all. Those people should only spend money now on things if they compound over time better than financial investments (like
some education, spending time with friends, repairing their car so they can get to work, or relaxation so they don't have a breakdown between now and retirement) and things that will be much less
enjoyable later in life (your gear to climb Mount Everest won't do you any good when you're 90 years old and can't get out of bed without help, even if it's much more affordable for you then).
If your subjective discount rate is higher than your financial discount rate, then you might be saving too much. That doesn't mean you should spend everything now, since your discount rate is a
marginal rate that can change as you move money from the future to the present.
There are two complicating factors:
1) I mentioned hyperbolic discounting above. Basically this is when pleasure and pain in the extremely short term feels like it vastly outweighs much larger amounts of pleasure or pain even the
moderately near future. Most people don't endorse their own hyperbolic discounting, and there's no point in using a clever theory to do things you don't actually prefer by blowing all your money on
one wild night in Vegas.
2) Declining marginal utility - this is basically a way of expressing the fact that I'd rather have an orange every day of the year, than 365 oranges today and none for the next 364 days. Your
subjective discount rate isn't absolute, and it's going to be affected by how satisfied you already expect to be in the present vs. the future. If you alter this by transferring money from present
you to future you or vice versa, your discount rate will change. Because of this, you may not even want to bother with a subjective discount rate - use your opportunity cost discount rate to make
sure you're making consistent decisions, and comparing equivalent quantities of present vs. future stuff. Use discounting to make consumption decisions on a case-by-case basis, and notice if it
always seems like a good idea to move things in one direction.
Pay It Again, Sam
In my post on present value, I promised to explain how to turn a series of payments into a present value. This is the promised follow-up post.
Pay Today or Pay More Tomorrow
I am 27 years old. I recently bought a life insurance policy with a face value of $100,000. This policy will last my whole life - in other words, no matter when I die, the payout happens. It cost me
roughly $10,000 in today's money. If this is surprising to you, or you think the insurance company got a bad deal, then read this.
Everyone makes choices about whether they'd rather have something now, or something else later. Almost no one understands the economic concepts that describes these tradeoffs. they're called "present
value" and "discount rate."
I will start by describing some simple examples that use these concepts, without using the jargon. Then I will explain what these all have in common. I'm not going to explain how to use these in
real-life situations, but if you're interested, please let me know in the comments and I'll write a follow-up post.
Return on Investment
I'll start with a simplified example, with made-up numbers. Abby has a bank account with a bunch of money in it earning 2% guaranteed interest per year. She also owns a bond that would pay out $1,000
if she cashes it out now, or $1,030 if she cashes it out in a year. Should she cash it out now, or a year later?
Let's say that in any case she wouldn't use the money until a year from now. Then if she cashes out the bond now, she can immediately deposit the money, and in a year, she'll have $1,020. But that's
less than the $1,030 she'd get if she held onto the bond for a year.
On the other hand, suppose she wants to use the money right now. Then if she cashes out the bond now, she has an immediate $1,000 to spend. On the other hand, let's say she holds onto the bond, and
withdraws $1,000 from her bank account. Then in a year, she has $1,020 less in her account than she would have, but an extra $1,030 from the bond, putting her $10 ahead of the first strategy. So in
this case too she should hold onto the bond for another year.
It should be easy to see that if the bond only returned $1,010 in a year, Abby comes out ahead by cashing out now, again regardless of whether she wants to use the money now or later. Because the
bond gives her a lower return on investment (1%) than her savings account does (2%).
Then suppose the bond pays out $1,030 in a year, but her bank account offers 4% interest this year. Then Abby also comes out ahead by cashing out now, because the bond's return (3%) is less than the
interest she gets on her bank account.
Cost of Funds
Brian doesn't have any savings - he a student. But he has a good credit rating and is able to borrow at 5% interest per year, and is allowed to pay off his loans at any time.
He is deciding whether to rent a textbook for $100, or buy it for $150 and sell it back used to his school's bookstore in a year for $55.
If Brian rents his textbook, then after a year, he will owe $105, including interest, and have no textbook. On the other hand, if he buys his textbook, then after a year, he will owe $157.50. He can
then sell his textbook back to the bookstore for $55, use that to pay down his debt, and owe only $102.50. So buying the textbook is a better deal.
Suppose instead Brian can only borrow at 10% interest. Then if Brian rents his textbook, after a year, he will owe $110. On the other hand, if he buys his textbook, then after a year, he will owe
$165-$55=$110. So he should be indifferent between the two alternatives.
If Brian has to pay 15% interest, then if he rents his textbook, after a year he owes $115, but if he buys, then after a year he owes $125, so he comes out ahead by renting.
On the other hand, suppose at the 5% rate of interest, Brian can only collect $50 for his textbook after a year. Then instead of owing $102.50 at the end of a year, he'd owe $107.50, more than the
$105 he'd owe if he rented, so in that case renting again becomes more advantageous.
Present Value
In each of the above examples, a future amount of money was related to a present amount of money, by either how much money you'd have if you used the current money in the best way available (either
investing or paying off debt), or how much money you would have to have now, to produce the future money. The first is called the "future value" of money, and the second is called the "present value"
of money.
When Abby is choosing between $1,000 now and $1,030 in a year, the "future value" of $1,000 is how much money she'd have at the end of a year if she put the money in her bank account yielding 2%. To
get this, you multiply by (100%+2%=1.00+0.02=1.02): $1,000 * 1.02 = $1,020. This is less than the one-year future value of $1,030 in a year, which is of course $1,030.
The "present value" of the year-later $1,030 is the amount Abby would need today to produce that amount in a year. To calculate the value a year in the past, you simply do the opposite of what you
did when calculating the value a year in the future: you simply divide by (100%+2%=102%=1.02), to get $1,030/1.02=$1009.80, more than the present value of $1,000 today (which is of course $1,000).
Another way to show this is algebraically:
Now let's look at the first example involving Brian. Brian is comparing making a single payment today, with making a payment today plus receiving a payment in a year.
Since Brian has to pay 5% interest on money he borrows, the future value of the textbook rental expense is how much Brian will owe in a year if he borrows the money, or $100*1.05=$105. The future
value of the purchase price of the textbook is $150*1.05=$157.50, and the future value of the $55 Brian will receive for his textbook in a year is just $55. So the net future value of Brian's
textbook expenses if he buys is $157.50-$55.00=$102.50, less than the $105 future value of the rental fee.
The present value of the renting option, $100 today, is of course $100. The present value of the textbook's price today is also the same as the price, $150. The present value of getting $55 in a year
is the amount of debt he'd have to pay off now, to owe $55 less in a year: $55/1.05=$52.38. So the present value of the cost of buying and selling back later is $150-$52.38=$97.62, less than the $100
textbook rental fee. So the buying option costs less, in present value terms, as well.
The key here is that by converting each value, whether positive or negative, into the equivalent value for a single time period - whether the present or the future - we end up with numbers that can
be directly added and subtracted to find out which amount is higher on net.
Discount Rate
You may have noticed that in Abby's case we were using the rate at which she could expect return on her savings to equate future and present amounts, but in Brian's case we looked at the interest
rate he'd have to pay to borrow money. These might seem like quite different things, but in finance, there's little difference between spending saved money and borrowing money; in both cases money in
the future is worth more than money in the present, and we assume a fixed conversion factor. Instead of calling it a cost of borrowing sometimes and an expected return on investment at other times,
economics abstracts this into the more general term "discount rate", which is basically the extra share you can demand if you get your money in a year instead of today, or the share of your money you
should expect to give up if you get your money today instead of a year from now.
This is related to the economic concept of "opportunity cost," which I will cover in a future post.
I will also cover how to deal with a series of future payments in a future post - and in the process show you that if you believe in discount rates, the future isn't as big a deal as it seems.
Which means, of course, that this is the first post in a series. | {"url":"http://benjaminrosshoffman.com/tag/discount-rate/","timestamp":"2024-11-06T15:33:35Z","content_type":"text/html","content_length":"78389","record_id":"<urn:uuid:b555af76-9f89-40b2-9fea-d59b6be21da4>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00763.warc.gz"} |
Properties of Passive Impedances
It is well known that a real impedance Ohms, for example) is passive so long as active and has some energy source. The concept of passivity can be extended to complex frequency-dependent impedances
passive if positive real, where Laplace-transform variable. The positive-real property is discussed in §C.11.2 below.
This section explores some implications of the positive real condition for passive impedances. Specifically, §C.11.1 considers the nature of waves reflecting from a passive impedance in general,
looking at the reflection transfer function, or reflectance, of a passive impedance. To provide further details, Section C.11.2 derives some mathematical properties of positive real functions,
particularly for the discrete-time case. Application examples appear in §9.2.1 and §9.2.1.
From Eq.C.75), we have that the reflectance seen at a continuous-time impedance force waves by
wave impedance
connected to the
impedance velocity
reflectance is
positive real
. As shown in §
maximum modulus theorem
In particular,
) may be called a
passive reflectance
If the impedance ). Similarly, when the impedance goes to zero, physics of a string with a free end. In acoustic stringed instruments, bridges are typically quite rigid, so that
Solving for C.77), we can characterize every impedance in terms of its reflectance:
Rewriting Eq.
) in the form
we see that the reflectance is determined by the ratio of the ``new impedance''
wave impedance
``step'' from
'' of the incident wave into reflected and transmitted components, as discussed in §
. The reflection and
transmission coefficients
depend on frequency when
In the discrete-time case, which may be related to the continuous-time case by the bilinear transform (§7.3.2), we have the same basic relations, but in the
, with
Mathematically, any stable
transfer function
having these properties may be called a
Schur function
. Thus, the discrete-time reflectance
Note that Eq.C.79) may be obtained from the general formula for scattering at a loaded waveguide junction for the case of a single waveguide (C.12).
In the limit as damping goes to zero (all poles of allpass filter. Similarly, allpass filter as the poles of
Recalling that a lossless impedance is called a reactance (§7.1), we can say that every reactance gives rise to an allpass reflectance. Thus, for example, waves reflecting off a mass at the end of a
vibrating string will be allpass filtered, because the driving-point impedance of a mass (force-wave reflectance of a mass
It is intuitively reasonable that a passive reflection gain cannot exceed i.e., the reflectance is a Schur filter, as defined in Eq.C.79)). It is also reasonable that lossless reflection would have a
gain of 1 (i.e., it is allpass).
Note that reflection filters always have an equal number of poles and zeros, as can be seen from Eq.C.76) above. This property is preserved by the bilinear transform, so it holds in both the
continuous- and discrete-time cases.
Consider the special case of a reflection and transmission at a yielding termination, or ``bridge'', of an ideal vibrating string on its right end, as shown in Fig.C.28. Denote the incident and
reflected velocity waves by force-wave components by force-wave reflectance by
wave impedance
The bridge velocity is given by
so that the
bridge velocity transmittance
is given by
and the bridge
force transmittance
is given by
where the bridge force is defined as ``up'' so that it is given for small
by the string tension times minus the string slope at the bridge. (Recall from §
that force waves on the string are defined by
We can show that the reflectance and transmittance of the yielding termination are power complementary. That is, the reflected and transmitted signal-power sum to yield the incident signal-power.
The average power incident at the bridge at frequency frequency domain as power reflection frequency response is
which generalizes to the
Finally, we see that adding up the reflected and transmitted power yields the incident power:
Any passive driving-point impedance, such as the impedance of a violin bridge, is positive real. Positive real functions have been studied extensively in the continuous-time case in the context of
network synthesis [68,524]. Very little, however, seems to be available in the discrete time case. This section (reprinted from [428]) summarizes the main properties of positive real function in the
i.e., the discrete-time case).
Definition. A complex valued function of a complex variable positive real (PR) if
We now specialize to the subset of functions finite-order polynomials in transfer functions of finite-order time-invariant linear systems, and we write minimum phase systems are analytic and nonzero
in the strict outer disk.^C.8 Condition (1) implies that for poles and zeros must exist in conjugate pairs. We assume from this point on that
Property 1. A real rational function
Proof. Expressing
since the zeros of
Property 2.
Proof. Assuming
Property 3. A PR function
Proof. (By contradiction)
Without loss of generality, we treat only
which are nondegenerate in the sense that
The general (normalized) causal, finite-order, linear, time-invariant transfer function may be written
, each of multiplicity
Suppose there is a pole of multiplicity
Consider the circular neighborhood of radius
Therefore, approaching the pole
which cannot be confined to satisfy Property 1 regardless of the value of the residue angle
Corollary. In equation Eq.C.80),
Proof. If
Corollary. The log-magnitude of a PR function has zero mean on the unit circle.
This is a general property of stable, minimum-phase transfer functions which follows immediately from the argument principle [297,326].
Corollary. A rational PR function has an equal number of poles and zeros all of which are in the unit disk.
This really a convention for numbering poles and zeros. In Eq.C.80), we have
Corollary. Every pole on the unit circle of a positive real function must be simple with a real and positive residue.
Proof. We repeat the previous argument using a semicircular neighborhood of radius
In order to have
Corollary. If
must satisfy
Proof. We may repeat the above for
Property. Every PR function z transform
Proof. This follows immediately from analyticity in the outer disk [342, pp. 30-36] However, we may give a more concrete proof as follows. Suppose
which contradicts the hypothesis that
Proof. If
To prove the converse, we first show nonnegativity on the upper semicircle implies nonnegativity over the entire circle.
Alternatively, we might simply state that
Next, since the function C.81) of
since the residue
) we can state that for points
maximum modulus theorem
occurs on the unit circle. Consequently,
For example, if a transfer function is known to be asymptotically stable, then a frequency response with nonnegative real part implies that the transfer function is positive real.
Note that consideration of
Property. If a stationary random process power spectral density autocorrelation function
is positive real.
By the representation theorem [19, pp. 98-103] there exists an asymptotically stable filter white noise, and we have causal and anti-causal components gives
Since the poles of
Since spectral power is nonnegative,
Relation to Schur Functions
Property. The function
is a Schur function if and only if
Suppose minimum phase which implies all roots of
By the
maximum modulus theorem
Conversely, suppose C.84) for
Property. re
for re
real number
Proof. We shall show that the change of variable conformal map from the z-plane to the s-plane that takes the region conformal mapping of functions of a complex variable is given by
In general, a bilinear transformation maps circles and lines into circles and lines [83]. We see that the choice of three specific points and their images determines the mapping for all
which gives us that
-half s-plane,
There is a bonus associated with the restriction that
We have therefore proven
The class of mappings of the form Eq.C.85) which take the exterior of the unit circle to the right-half plane is larger than the class Eq.C.86). For example, we may precede the transformation Eq.C.86
) by any conformal map which takes the unit disk to the unit disk, and these mappings have the algebraic form of a first order complex allpass whose zero lies inside the unit circle.
) is equivalent to a pure rotation, followed by a
allpass substitution (
2 forces
the real axis to map to the real axis. Thus rotations by other than
) by the first order
allpass substitution
which maps the real axis to the real axis. This leads only to the composite transformation,
which is of the form Eq.
) up to a minus sign (rotation by
), it is clear that sign negation corresponds to the swapping of points
bilinear transforms
which convert functions positive real in the outer disk to functions positive real in the right-half plane is characterized by
Riemann's theorem may be used to show that Eq.
) is also the largest such class of conformal mappings. It is not essential, however, to restrict attention solely to conformal maps. The pre-transform
The bilinear transform is one which is used to map analog filters into digital filters. Another such mapping is called the matched [362]. It also preserves the positive real property.
Property. sampling period.
Proof. The mapping
These transformations allow application of the large battery of tests which exist for functions positive real in the right-half plane [524].
• The sum of positive real functions is positive real.
• The difference of positive real functions is conditionally positive real.
• The product or division of positive real functions is conditionally PR.
All properties of MP polynomials apply without modification to marginally stable allpole transfer functions (cf. Property 2):
• Every first-order MP polynomial is positive real.
• Every first-order MP polynomial
• A PR second-order MP polynomial with complex-conjugate zeros,
• All polynomials of the form
are positive real. (These have zeros uniformly distributed on a circle of radius
• If all poles and zeros of a PR function are on the unit circle, then they alternate along the circle. Since this property is preserved by the bilinear transform, it is true in both the
positive-real functions.
• If
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[BASIC HELP] Equations on HP Prime
03-05-2019, 04:03 PM
Post: #1
fpicoral Posts: 4
Junior Member Joined: Mar 2019
[BASIC HELP] Equations on HP Prime
Hello there!
First, I would like to apologize for such a noob question.
I don't know if this is possible - probably it's but I'm to stupid to find it myself: how can I """de-siplify""" (I really don't know how to properly call this) an equation?
Let's say I have (a+b)^2, where can I put it to see a^2+2ab+b^2?
I know about the Solve app but if I'm right I need to give a value to the variables to solve one of them, e.g set b=3 in (a+b)^2.
Thanks in advance for the help.
If this question still confusing, I can try to explain it better in the comments.
03-05-2019, 05:08 PM
(This post was last modified: 03-05-2019 05:35 PM by StephenG1CMZ.)
Post: #2
StephenG1CMZ Posts: 1,074
Senior Member Joined: May 2015
RE: [BASIC HELP] Equations on HP Prime
In CAS, "simplify((a+b) squared)" gives "a squared + 2ab + b squared"
Where squared = tap the x squared key
(Unless a and b have been given numerical values)
And factor() or collect() takes you back.
Stephen Lewkowicz (G1CMZ)
03-05-2019, 05:42 PM
Post: #3
DrD Posts: 1,136
Senior Member Joined: Feb 2014
RE: [BASIC HELP] Equations on HP Prime
If you have purged a, and b so that they are purely symbolic then:
(a+b)^2 == > (a+b)^2
expand((a+b)^2) ==> 2*a*b+a^2+b^2
collect(2*a*b+a^2+b^2) ==> (a+b)^2
03-05-2019, 06:41 PM
Post: #4
Aries Posts: 159
Member Joined: Oct 2014
RE: [BASIC HELP] Equations on HP Prime
(03-05-2019 04:03 PM)fpicoral Wrote: Hello there!
First, I would like to apologize for such a noob question.
I don't know if this is possible - probably it's but I'm to stupid to find it myself: how can I """de-siplify""" (I really don't know how to properly call this) an equation?
Let's say I have (a+b)^2, where can I put it to see a^2+2ab+b^2?
I know about the Solve app but if I'm right I need to give a value to the variables to solve one of them, e.g set b=3 in (a+b)^2.
Thanks in advance for the help.
If this question still confusing, I can try to explain it better in the comments.
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All Squared, Number 4: How interesting!
You're reading: All Squared
All Squared, Number 4: How interesting!
It’s a repeat booking for the Festival of the Spoken Nerd in number 4 (or 16 if you belong to Team All Squared) of our podcast. Standup mathematician Matt Parker joined us to talk about interesting
Here are some links to the things we referred to in the podcast, along with some bonus extras:
Matt Parker plugged a couple of things:
Next time, which should be soon, we’ll be talking to a Fantastic Mystery Guest about our favourite maths books.
Podcast: Play in new window | Download
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Can a Brans-Dicke scalar account for dark matter in extended inflation models?
title = "Can a Brans-Dicke scalar account for dark matter in extended inflation models?",
abstract = "We discuss the possibility that the dark matter in the Universe could be due to the oscillation of the Brans-Dicke scalar in extended inflation models. The constraints following from the
requirement that the energy density perturbations due to quantum scalar field fluctuations are within observational limits, and from the requirement that the energy density in the oscillating
Brans-Dicke scalar field, induced by the expansion of bubbles at the first-order phase transition, is smaller than the critical density at present, are discussed for the case of a coupling of the
Brans-Dicke scalar to the Ricci scalar of the form phi(n+2)/M(n). Solutions of the equations of motion are given in the slow-rolling approximation for a general value of n, and the cases of n = 0
(minimal extended inflation) and n = 2 are discussed in detail. It is shown that the dominant source of energy density fluctuations produced during inflation is the isocurvature fluctuations of the
Brans-Dicke scalar. The requirement that the fluctuations in the microwave background due to the isocurvature fluctuations are not too large implies that the reheat temperature at the end of
inflation is less than 3 x 10(13) GeV. For the case where gravity is neglected during the expansion of bubbles at the first-order transition and where the isocurvature fluctuations account for the
density perturbations observed by COBE, it is shown that only for a very small range of reheat temperatures at the end of inflation is the model consistent with the requirements that the range of
non-Newtonian corrections to the gravitational potential is less that 1 cm and that there are no inhomogeneities in the cosmic microwave background radiation due to large bubbles. If we do not
require that the isocurvature fluctuations account for the density perturbations observed by COBE, then the range of reheat temperatures can be larger. In general, the range of non-Newtonian
corrections to the gravitational potential is not expected to be much smaller than the present upper limits from Cavendish-type experiments. The possible effects of gravitational suppression of the
growth of large bubbles is also considered. It is shown that although gravity can suppress the growth of large bubbles, allowing for a wider range of reheat temperatures and shorter range
non-Newtonian forces, gravity will have no effect on bubble percolation at the end of the first-order phase transition. The possible advantages of an oscillating Brans-Dicke scalar with respect to
structure formation are considered, in particular, the possibility that the Brans-Dicke scalar could account for both cold and hot dark matter in a combined CDM + HDM scenario for structure
formation, with the hot component arising from excitation of the Brans-Dicke scalar field by the expansion of bubbles during the first-order phase transition.",
keywords = "DIFFERENTIAL MICROWAVE RADIOMETER, COSMOLOGY, UNIVERSE, GRAVITY",
author = "John McDonald",
year = "1993",
month = sep,
day = "15",
doi = "10.1103/PhysRevD.48.2462",
language = "English",
volume = "48",
pages = "2462--2476",
journal = "Physical Review D",
issn = "0556-2821",
publisher = "American Physical Society",
number = "6",
TY - JOUR
T1 - Can a Brans-Dicke scalar account for dark matter in extended inflation models?
AU - McDonald, John
PY - 1993/9/15
Y1 - 1993/9/15
N2 - We discuss the possibility that the dark matter in the Universe could be due to the oscillation of the Brans-Dicke scalar in extended inflation models. The constraints following from the
requirement that the energy density perturbations due to quantum scalar field fluctuations are within observational limits, and from the requirement that the energy density in the oscillating
Brans-Dicke scalar field, induced by the expansion of bubbles at the first-order phase transition, is smaller than the critical density at present, are discussed for the case of a coupling of the
Brans-Dicke scalar to the Ricci scalar of the form phi(n+2)/M(n). Solutions of the equations of motion are given in the slow-rolling approximation for a general value of n, and the cases of n = 0
(minimal extended inflation) and n = 2 are discussed in detail. It is shown that the dominant source of energy density fluctuations produced during inflation is the isocurvature fluctuations of the
Brans-Dicke scalar. The requirement that the fluctuations in the microwave background due to the isocurvature fluctuations are not too large implies that the reheat temperature at the end of
inflation is less than 3 x 10(13) GeV. For the case where gravity is neglected during the expansion of bubbles at the first-order transition and where the isocurvature fluctuations account for the
density perturbations observed by COBE, it is shown that only for a very small range of reheat temperatures at the end of inflation is the model consistent with the requirements that the range of
non-Newtonian corrections to the gravitational potential is less that 1 cm and that there are no inhomogeneities in the cosmic microwave background radiation due to large bubbles. If we do not
require that the isocurvature fluctuations account for the density perturbations observed by COBE, then the range of reheat temperatures can be larger. In general, the range of non-Newtonian
corrections to the gravitational potential is not expected to be much smaller than the present upper limits from Cavendish-type experiments. The possible effects of gravitational suppression of the
growth of large bubbles is also considered. It is shown that although gravity can suppress the growth of large bubbles, allowing for a wider range of reheat temperatures and shorter range
non-Newtonian forces, gravity will have no effect on bubble percolation at the end of the first-order phase transition. The possible advantages of an oscillating Brans-Dicke scalar with respect to
structure formation are considered, in particular, the possibility that the Brans-Dicke scalar could account for both cold and hot dark matter in a combined CDM + HDM scenario for structure
formation, with the hot component arising from excitation of the Brans-Dicke scalar field by the expansion of bubbles during the first-order phase transition.
AB - We discuss the possibility that the dark matter in the Universe could be due to the oscillation of the Brans-Dicke scalar in extended inflation models. The constraints following from the
requirement that the energy density perturbations due to quantum scalar field fluctuations are within observational limits, and from the requirement that the energy density in the oscillating
Brans-Dicke scalar field, induced by the expansion of bubbles at the first-order phase transition, is smaller than the critical density at present, are discussed for the case of a coupling of the
Brans-Dicke scalar to the Ricci scalar of the form phi(n+2)/M(n). Solutions of the equations of motion are given in the slow-rolling approximation for a general value of n, and the cases of n = 0
(minimal extended inflation) and n = 2 are discussed in detail. It is shown that the dominant source of energy density fluctuations produced during inflation is the isocurvature fluctuations of the
Brans-Dicke scalar. The requirement that the fluctuations in the microwave background due to the isocurvature fluctuations are not too large implies that the reheat temperature at the end of
inflation is less than 3 x 10(13) GeV. For the case where gravity is neglected during the expansion of bubbles at the first-order transition and where the isocurvature fluctuations account for the
density perturbations observed by COBE, it is shown that only for a very small range of reheat temperatures at the end of inflation is the model consistent with the requirements that the range of
non-Newtonian corrections to the gravitational potential is less that 1 cm and that there are no inhomogeneities in the cosmic microwave background radiation due to large bubbles. If we do not
require that the isocurvature fluctuations account for the density perturbations observed by COBE, then the range of reheat temperatures can be larger. In general, the range of non-Newtonian
corrections to the gravitational potential is not expected to be much smaller than the present upper limits from Cavendish-type experiments. The possible effects of gravitational suppression of the
growth of large bubbles is also considered. It is shown that although gravity can suppress the growth of large bubbles, allowing for a wider range of reheat temperatures and shorter range
non-Newtonian forces, gravity will have no effect on bubble percolation at the end of the first-order phase transition. The possible advantages of an oscillating Brans-Dicke scalar with respect to
structure formation are considered, in particular, the possibility that the Brans-Dicke scalar could account for both cold and hot dark matter in a combined CDM + HDM scenario for structure
formation, with the hot component arising from excitation of the Brans-Dicke scalar field by the expansion of bubbles during the first-order phase transition.
KW - COSMOLOGY
KW - UNIVERSE
KW - GRAVITY
U2 - 10.1103/PhysRevD.48.2462
DO - 10.1103/PhysRevD.48.2462
M3 - Journal article
VL - 48
SP - 2462
EP - 2476
JO - Physical Review D
JF - Physical Review D
SN - 0556-2821
IS - 6
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Simplifying Expressions - Definition, With Exponents, Examples - Grade Potential Irvine, CA
Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for budding students in their early years of college or even in high school.
Nevertheless, learning how to handle these equations is critical because it is basic knowledge that will help them move on to higher arithmetics and complex problems across different industries.
This article will share everything you need to master simplifying expressions. We’ll review the principles of simplifying expressions and then verify our comprehension with some sample problems.
How Do I Simplify an Expression?
Before you can be taught how to simplify expressions, you must learn what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can combine numbers, variables, or both and can be connected through subtraction or addition.
For example, let’s review the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two consist of both numbers (8 and 2) and variables (x and y).
Expressions that include coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it opens up the possibility of learning how to solve them. Expressions can be expressed in convoluted ways, and without simplification, you will have a
hard time trying to solve them, with more opportunity for a mistake.
Of course, every expression vary in how they're simplified depending on what terms they include, but there are typical steps that are applicable to all rational expressions of real numbers, whether
they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
1. Parentheses. Solve equations inside the parentheses first by using addition or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply
multiplication the term outside with the one inside.
2. Exponents. Where possible, use the exponent principles to simplify the terms that have exponents.
3. Multiplication and Division. If the equation requires it, use multiplication and division to simplify like terms that are applicable.
4. Addition and subtraction. Lastly, add or subtract the simplified terms in the equation.
5. Rewrite. Make sure that there are no more like terms that require simplification, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few additional principles you need to be aware of when working with algebraic expressions.
• You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and keep the variables as [[is|they are]-70. For example, the expression 8x
+ 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and keeping the x as it is.
• Parentheses that contain another expression on the outside of them need to apply the distributive property. The distributive property prompts you to simplify terms on the outside of parentheses
by distributing them to the terms inside, as shown here: a(b+c) = ab + ac.
• An extension of the distributive property is called the property of multiplication. When two distinct expressions within parentheses are multiplied, the distribution rule applies, and every
separate term will will require multiplication by the other terms, making each set of equations, common factors of one another. For example: (a + b)(c + d) = a(c + d) + b(c + d).
• A negative sign right outside of an expression in parentheses denotes that the negative expression must also need to be distributed, changing the signs of the terms on the inside of the
parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
• Likewise, a plus sign outside the parentheses denotes that it will be distributed to the terms inside. However, this means that you are able to eliminate the parentheses and write the expression
as is owing to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior principles were simple enough to follow as they only applied to properties that impact simple terms with variables and numbers. Still, there are more rules that you need to apply when
working with exponents and expressions.
Here, we will review the principles of exponents. Eight properties affect how we utilize exponentials, those are the following:
• Zero Exponent Rule. This property states that any term with a 0 exponent is equal to 1. Or a0 = 1.
• Identity Exponent Rule. Any term with a 1 exponent will not alter the value. Or a1 = a.
• Product Rule. When two terms with the same variables are multiplied, their product will add their two exponents. This is expressed in the formula am × an = am+n
• Quotient Rule. When two terms with the same variables are divided, their quotient will subtract their two respective exponents. This is seen as the formula am/an = am-n.
• Negative Exponents Rule. Any term with a negative exponent equals the inverse of that term over 1. This is expressed with the formula a-m = 1/am; (a/b)-m = (b/a)m.
• Power of a Power Rule. If an exponent is applied to a term already with an exponent, the term will result in being the product of the two exponents applied to it, or (am)n = amn.
• Power of a Product Rule. An exponent applied to two terms that have unique variables needs to be applied to the required variables, or (ab)m = am * bm.
• Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will assume the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that says that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions inside. Let’s see the
distributive property applied below.
Let’s simplify the equation 2(3x + 5).
The distributive property states that a(b + c) = ab + ac. Thus, the equation becomes:
2(3x + 5) = 2(3x) + 2(5)
The expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have some rules that you must follow.
When an expression contains fractions, here is what to remember.
• Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions separately by their numerators and denominators.
• Laws of exponents. This tells us that fractions will more likely be the power of the quotient rule, which will apply subtraction to the exponents of the denominators and numerators.
• Simplification. Only fractions at their lowest state should be written in the expression. Apply the PEMDAS property and ensure that no two terms have the same variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside the parentheses, while PEMDAS
will govern the order of simplification.
Due to the distributive property, the term outside the parentheses will be multiplied by the terms inside.
The expression is then:
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, remember to add the terms with the same variables, and every term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation as follows:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule expresses that the the order should start with expressions within parentheses, and in this case, that expression also necessitates the distributive property. In this scenario, the
term y/4 must be distributed amongst the two terms on the inside of the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors assigned to them. Remember we know from PEMDAS that fractions will need to multiply their denominators and numerators
individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple as any number divided by 1 is that same number or x/1 = x. Thus,
The expression y/4(2) then becomes:
y/4 * 2/1
Thus, the overall expression is:
1/3x + 5xy/4 + 2y/4
Its final simplified version is:
1/3x + 5/4xy + 1/2y
Example 3
Simplify the expression: (4x2 + 3y)(6x + 1)
In exponential expressions, multiplication of algebraic expressions will be utilized to distribute every term to one another, which gives us the equation:
4x2(6x + 1) + 3y(6x + 1)
4x2(6x) + 4x2(1) + 3y(6x) + 3y(1)
For the first expression, the power of a power rule is applied, which means that we’ll have to add the exponents of two exponential expressions with similar variables multiplied together and multiply
their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no more like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, remember that you have to obey the exponential rule, the distributive property, and PEMDAS rules in addition to the concept of multiplication of algebraic
expressions. In the end, make sure that every term on your expression is in its lowest form.
What is the difference between solving an equation and simplifying an expression?
Solving and simplifying expressions are very different, although, they can be incorporated into the same process the same process because you first need to simplify expressions before you solve them.
Let Grade Potential Help You Bone Up On Your Math
Simplifying algebraic equations is a primary precalculus skills you should learn. Increasing your skill with simplification strategies and rules will pay dividends when you’re learning advanced
But these ideas and properties can get complicated fast. But there's no need for you to worry! Grade Potential is here to help!
Grade Potential Irvine provides professional instructors that will get you on top of your skills at your convenience. Our professional instructors will guide you through mathematical properties in a
straight-forward way to guide.
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Lectures on Etale Cohomology
by J. S. Milne
Number of pages: 196
These are the notes for a course taught at the University of Michigan in 1989 and 1998. The emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The
notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne).
Download or read it online for free here:
Download link
(1.2MB, PDF)
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Strain Gauge Factor Derivation | Types of Strain Gauge
Strain Gauge Factor Derivation:
The Strain Gauge Factor Derivation is an example of a passive transducer that uses the variation in electrical resistance in wires to sense the strain produced by a force on the wires.
It is well known that stress (force/unit area) and strain (elongation or compression/unit length) in a member or portion of any object under pressure is directly related to the modulus of elasticity.
Since strain can be measured more easily by using variable resistance transducers, it is a common practice to measure strain instead of stress, to serve as an index of pressure. Such transducers are
popularly known as strain gauges.
If a metal conductor is stretched or compressed, its resistance changes on account of the fact that both the length and diameter of the conductor changes. Also, there is a change in the value of the
resistivity of the conductor when subjected to strain, a property called the piezo-resistive effect. Therefore, resistance strain gauges are also known as piezo resistive gauges.
Many detectors and transducers, e.g. load cells, torque meters, pressure gauges, temperature sensors, etc. employ Strain Gauge Factor Derivation as secondary transducers.
When a gauge is subjected to a positive stress, its length increases while its area of cross-section decreases. Since the resistance of a conductor is directly proportional to its length and
inversely proportional to its area of cross-section, the resistance of the gauge increases with positive strain. The change in resistance value of a conductor under strain is more than for an
increase in resistance due to its dimensional changes. This property is called the piezo-resistive effect.
The following types of Strain Gauge Factor Derivation are the most important.
1. Wire Strain Gauge
2. Foil Strain Gauge
3. Semiconductor Strain Gauge
Resistance Wire Gauge:
Resistance wire gauges are used in two basic forms, the unbonded type, and the bonded type.
1. Unbonded Resistance Wire Strain Gauge:
An unbonded strain gauge consists of a wire streched between two points in an insulating medium, such as air. The diameter of the wire used is about 25 μm. The wires are kept under tension so that
there is no sag and no free vibration. Unbonded Strain Gauge Factor Derivation Derivation are usually connected in a bridge circuit. The bridge is balanced with no load applied as shown in Fig. 13.3.
When an external load is applied, the resistance of the Strain Gauge Factor Derivation changes, causing an unbalance of the bridge circuit resulting in an output voltage. This voltage is proportional
to the strain. A displacement of the order of 50μm can be detected with these strain gauges.
2. Bonded Resistance Wire Strain Gauge:
A metallic bonded Strain Gauge Derivation is shown in Fig. 13.4.
A fine wire element about 25 μm (0.025 in.) or less in diameter is looped back and forth on a carrier (base) or mounting plate, which is usually cemented to the member undergoing stress. The grid of
fine wire is cemented on a carrier which may be a thin sheet of paper, bakelite, or teflon. The wire is covered on the top with a thin material, so that it is not damaged mechanically. The spreading
of the wire permits uniform distribution of stress. The carrier is then bonded or cemented to the member being studied. This permits a good transfer of strain from carrier to wire.
A tensile stress tends to elongate the wire and thereby increase its length and decrease its cross-sectional area. The combined effect is an increase in resistance, as seen from the following
• ρ = the specific resistance of the material in Ωm.
• l = the length of the conductor in m
• A = the area of the conductor in m^2
As a result of strain, two physical parameters are of particular interest.
1. The change in gauge resistance.
2. The change in length.
The measurement of the sensitivity of a material to strain is called the gauge factor (GF). It is the ratio of the change in resistance ΔR/R to the change in the length Δl/l
• K = gauge factor
• ΔR = the change in the initial resistance in Ω’s
• R = the initial resistance in Ω (without strain)
• Δ l = the change in the length in m
• l = the initial length in m (without strain)
Since strain is defined as the change in length divided by the original length,
Eq. (13.1) can be written as
where σ is the strain in the lateral direction.
The resistance of a conductor of uniform cross-section is
• ρ= specific resistance of the conductor
• l = length of conductor
• d= diameter of conductor
When the conductor is stressed, due to the strain, the length of the conductor increases by Δl and the simultaneously decreases by Δd in its diameter. Hence the resistance of the conductor can now be
written as
Since Δd is small, Δd^2 can be neglected
Now, Poisson’s ratio μ is defined as the ratio of strain in the lateral direction to strain in the axial direction, that is,
Substituting for Δd/d from Eq. (13.6) in Eq. (13.4), we have
Rationalizing, we get
Since Δl is small, we can neglect higher powers of Δl.
Since from Eq. (13.3),
The gauge factor will now be | {"url":"https://www.eeeguide.com/strain-gauge-factor-derivation/","timestamp":"2024-11-09T00:28:04Z","content_type":"text/html","content_length":"229684","record_id":"<urn:uuid:fffda070-4fbe-41c2-bab8-508ab9e56810>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00420.warc.gz"} |
Machine Learning Methods | Types of Classification in Machine Learning
Updated March 20, 2023
Introduction to Machine Learning Methods
Machine Learning Methods are used to make the system learn using methods like Supervised learning and Unsupervised Learning which are further classified in methods like Classification, Regression and
Clustering. This selection of methods entirely depends on the type of dataset that is available to train the model, as the dataset can be labeled, unlabelled, large. There are various applications
(like image classification, Predictive analysis, Spam detection) that uses these different machine learning methods.
How do Machines learn?
There are various methods to do that. Which method to follow completely depends on the problem statement. Depending on the dataset, and our problem, there are two different ways to go deeper. One is
supervised learning and the other is unsupervised learning. The following chart explains the further classification of machine learning methods. We will discuss them one by one.
Take a look at the following chart!
Let’s understand what does Supervised Learning means.
Supervised Learning
As the name suggests, imagine a teacher or a supervisor helping you to learn. The same goes for machines. We train or teach the machine using data that is labeled.
Some of the coolest supervised learning applications are:
• Sentiment analysis (Twitter, Facebook, Netflix, YouTube, etc)
• Image classification
• Predictive analysis
• Pattern recognition
• Spam detection
• Speech/Sequence processing
Now, supervised learning is further divided into classification and regression. Let’s, understand this.
Classification is the process of finding a model that helps to separate the data into different categorical classes. In this process, data is categorized under different labels according to some
parameters given in input and then the labels are predicted for the data. Categorical means the output variable is a category, i.e red or black, spam or not spam, diabetic or non-diabetic, etc.
Classification models include Support vector machine(SVM),K-nearest neighbor(KNN),Naive Bayes etc.
a) Support vector machine classifier (SVM)
SVM is a supervised learning method that looks at the data and sorts it into one of two categories. I use a hyperplane to categorize the data. A linear discriminative classifier attempts to draw a
straight line separating the two sets of data and thereby create a model for classification. It simply tries to find a line or curve (in two dimensions) or a manifold (in multiple dimensions) that
divides the classes from each other.
Note: For multiclass classification SVM makes use of ‘one vs rest’, that means calculating different SVM for each class.
b) K-nearest neighbor classifier (KNN)
• If you read carefully, the name itself suggests what the algorithm does. KNN considers the data points which are closer, are much more similar in terms of features and hence more likely to belong
to the same class as the neighbor. For any new data point, the distance to all other data points is calculated and the class is decided based on K nearest neighbors. Yes, it may sound lame, but
for some of the classification, it works like anything.
• A data point is classified by the maximum number vote of its neighbors, then the data point is assigned to the class nearest among its k-neighbors.
• In KNN, no learning of the model is required and all of the work happens at the time a prediction is requested. That’s why KNN is often referred to as a lazy learning algorithm.
c) Naive Bayes classifier
• Naive Bayes is a machine learning algorithm that is highly recommended for text classification problems. It is based on Bayes’ probability theorem. These classifiers are called naive because they
assume that features variables are independent of each other. That means, for example, we have a full sentence for input, then Naive Bayes assumes every word in a sentence is independent of the
other ones. And then classify them accordingly. I know, it looks pretty naive, but it’s a great choice for text classification problems and it’s a popular choice for spam email classification.
• It provides different types of Naive Bayes Algorithms like BernoulliNB, GaussianNB, MultinomialNB.
• It considers all the features to be unrelated, so it cannot learn the relationship between features. For example, Let’s say, Varun likes to eat burgers, he also likes to eat French fries with
coke. But he doesn’t like to eat a burger and a combination of French fries with coke together. Here, Naive Bayes can not learn the relation between two features but only learns individual
feature importance only.
Now let’s move on to the other side of our supervised learning method, which is a regression.
Regression is the process of finding a model that helps to differentiate the data using continuous values. In this, the nature of the predicted data is ordered. Some of the most widely used
regression models include Linear regression, Random forest(Decision trees), Neural networks.
Linear regression
• One of the simplest approaches in supervised learning, which is useful in predicting the quantitative response.
• Linear regression includes finding the best-fitting straight line through the points. The best-fitting line is called a regression line. The best fit line doesn’t exactly pass through all the
data points but instead tries it’s best to get close to them.
• It is the widely used algorithm for continuous data. However, it only focuses on the mean of the dependent variable and limits itself to a linear relationship.
• Linear regression can be used for Time series, trend forecasting. It can predict future sales, based on the previous data.
Unsupervised Learning
• Unsupervised learning is based on the approach that can be thought of as the absence of a teacher and therefore of absolute error measures. It’s useful when it’s required to learn clustering or
grouping of elements. Elements can be grouped (clustered) according to their similarity.
• In unsupervised learning, data is unlabeled, not categorized and the system’s algorithms act on the data without prior training. Unsupervised learning algorithms can perform more complex tasks
than supervised learning algorithms.
• Unsupervised learning includes clustering which can be done by using K means clustering, hierarchical, Gaussian mixture, hidden Markov model.
Unsupervised Learning applications are:
1. Similarity detection
2. Automatic labeling
3. Object segmentation (such as Person, Animal, Films)
• Clustering is an unsupervised learning technique that is used for data analytics in many fields. The clustering algorithm comes handy when we want to gain detailed insights about our data.
• A real-world example of clustering would be Netflix’s genre clusters, which are divided for different target customers including interests, demographics, lifestyles, etc. Now you can think about
how useful clustering is when companies want to understand their customer base and target new potential customers.
a) K means Clustering
• K means clustering algorithm tries to divide the given unknown data into clusters. It randomly selects ‘k’ clusters centroid, calculates the distance between data points and clusters centroid and
then finally assigns the data point to cluster centroid whose distance is minimum of all cluster centroids.
• In k-means, groups are defined by the closest centroid for every group. This centroid acts as ‘Brain’ of the algorithm, they acquire the data points which are closest to them and then add them to
the clusters.
b) Hierarchical Clustering
Hierarchical clustering is nearly similar to that of normal clustering unless you want to build a hierarchy of clusters. This can come handy when you want to decide the number of clusters. For
example, suppose you are creating groups of different items on the online grocery store. On the front home page, you want a few broad items and once you click on one of the items, specific
categories, that is more specific clusters opens up.
Dimensionality reduction
Dimensionality reduction can be considered as compression of a file. It means, taking out the information which is not relevant. It reduces the complexity of data and tries to keep the meaningful
data. For example, in image compression, we reduce the dimensionality of the space in which the image stays as it is without destroying too much of the meaningful content in the image.
PCA for Data Visualization
Principal component analysis (PCA) is a dimension reduction method that can be useful to visualize your data. PCA is used to compress higher dimensional data to lower-dimensional data, that is, we
can use PCA to reduce a four-dimensional data into three or 2 dimensions so that we can visualize and get a better understanding of the data.
Recommended Articles
This is a guide to Machine Learning Methods. Here we have discussed basic introduction, how do machines learn? classifications of machine learning along with a flowchart with a detail explanation.
You can also go through our other suggested articles to learn more – | {"url":"https://www.educba.com/machine-learning-methods/","timestamp":"2024-11-02T17:46:24Z","content_type":"text/html","content_length":"316141","record_id":"<urn:uuid:de677b98-39c1-43d6-951f-883771cda0a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00785.warc.gz"} |
Nonlinear Solver
Embedded Nonlinear Analysis Tools Capability Area
The Trilinos Embedded Nonlinear Analysis Tools Capability Area collects the top level algorithms (outermost loops) in a computational simulation or design study. These include: the solution of
nonlinear equations, time integration, bifurcation tracking, parameter continuation, optimization, and uncertainty quantification. A common theme of our algorithm R&D efforts is the philosophy of
Analysis beyond Simulation, which aims to automate many computational tasks that are often performed by application code users by trial-and-error or repeated simulation. The tasks that can be
automated include performing parameter studies, sensitivity analysis, calibration, optimization, time step size control, and locating instabilities. Also included in this capability area is the
automatic differentiation technology that can be used in an application code to provide the derivatives critical to the analysis algorithms.
The capabilities in this area are spread across several Trilinos packages, which are independently developed software libraries. However, it is a high priority of the software developers to make the
codes interoperable wherever possible, and we are well on our way to unifying the interface to the applications around a interface class called the ModelEvaluator. The ModelEvaluator is found in the
Thyra and EpetraExt packages, and examples of its implementation can be found in many example problems in the analysis code packages.
A pair of packages also summarized in this section are the Sacado Automatic Differentiation capability and the Stokhos library for polynomial chaos expansions for embedded Uncertainty Quantification.
The interface between Sacado and Stokhos and a C++ application code requires templating of the relevant pieces of code on a Scalar type.
All the capabilities in this area rely heavily on the lower-level (inner loop) algorithms such as scalable linear solvers and preconditioners, and Trilinos utilities and interfaces. The algorithm
development in this area is particularly well aligned with several of the Trilinos Strategic Goals, including Full Vertical Coverage, Scalability, and Hardened Solvers.
If you have questions on the algorithm, software, or vision of this capability area, please contact Andy Salinger or any of the package leads listed below.
Trilinos Packages in the Embedded Nonlinear Analysis Capability Area
Package Name Quick Description Point of Contact
NOX Nonlinear Solver with Globalized Newton’s methods Roger Pawlowski
LOCA Parameter Continuation, Bifurcation Tracking Eric Phipps
Rythmos Time integration algorithms Curtis Ober
MOOCHO Embedded (PDE-constrained) Optimization, rSQP Roscoe Bartlett
Aristos Full-space intrusive optimization (not yet released) Dennis Ridzal
Sacado Automatic Differentiation using Expression Templates Eric Phipps
Stokhos Stochastic-Galerkin Uncertainty Quantification Tools Eric Phipps
Tempus Time integration algorithms (NEW package) Curtis Ober
TriKota Interface to Dakota toolkit for a Trilinos application (not yet released) Andy Salinger
Related Efforts Outside of Trilinos
Dakota is a mature and widely-used software toolkit at Sandia, independent from Trilinos, that delivers many related analysis capabilities using non-intrusive (black box) methods. These
Dakota include numerous algorithms in the areas of Optimization, Uncertainty Quantification, Nonlinear-Least-Squares, and Reliability. A adapter package called TriKota is under development for the
Trilinos 10.0 release to make these capabilities accessible through the same interface as the above anaysis codes.
DemoApps The DemoApps code project is building a prototype PDE code primarily from Trilinos packages. The code will demonstrate the use of Embedded Nonlinear Analysis Tools as well as cutting edge
algorithms from all other Trilinos capability areas (not yet externally released).
Brief Mathematical Description of Trilinos’ Embedded Nonlinear Analysis Tools
This is brief description of the types of analysis that are performed by the codes in the nonlinear analysis capability area. The starting point is a set of nonlinear equations, such as those coming
from discretized Partial Differential Equations or Integral Equations.
The Nonlinear Solver NOX solves a set of nonlinear algebraic equations
The Time Integration algorithms in Tempus solve ODEs of the forms $f(\dot{x}, x, t)=0$ and $\dot{x}=f(x, t)$. The algorithms include explicit and implicit methods with adaptive step size control,
including Forward/Backward Euler, Trapezoidal, Explicit Runge-Kutta, Diagonally Implicit Runge-Kutta, Implicit/Explicit Runge-Kutta, Leapfrog, Newmark-Beta, HHT-Alpha, BDF2, operator-splitting, and
subcycling. For systems with parameter dependence, $f(\dot{x}, x, t, p)=0$, a sensitivity analysis capability is now available to solve for $\frac{dx}{dp}$.
The Time Integration algorithms in Rythmos solve ODEs and DAEs of the form
A set of Bifurcation Tracking algorithms have been implemented in LOCA. These algorithms augment the steady-state system of equations with extra distinguishing conditions that find a parameter value
where there is an exchange of stability (Constraint Enforcement, where the extra equations
An initial implementation of a capability for solving for Space-Time and Periodic Solutions called “4D” in LOCA has been developed. With this approach, analysis algorithms designed for steady
problems can be applied to transient problems. The all-at-once methodology greatly expands the size of the system, which is in part mitigated by the ability of “4D” to parallelize computations over
the time axes.
Steady-state (or equilibrium) solutions, including bifurcations points, can be tracked through parameter space with Parameter Continuation algorithms in LOCA. An curve of points
A driver for performing Stability Analysis is included in the parameter continuation library. This implements spectral transformations, and then calls an eigensolver (typically Trilinos’ Anasazi
package). For generalized eigenvalue problems of the form
PDE-Constrained Optimization algorithms have been implemented in MOOCHO and Aristos packages to take advantage of the efficiencies accessible by embedded algorithms. These have the form:
Automatic Differentiation tools for C++ codes have been developed in SACADO to automatically extract analytic derivative information from codes. This capability is implemented with expression
templates that essentially inline code that performs the chain rule. The application code must template a key part of their code (such as the single element fill portion in a finite element code) on
the type.
Embedded UQ methods are under active development. This includes the Stokhos tools to automate the propagation of random variables through codes, such as stochastic finite element formulations,
leveraging the same templated interfaces as the automatic differentiation capability. Other pieces include the subsequent nonlinear solution, transient propoagation, and linear solution of the full
stochastic system.
Active Areas of Research and Development
The following are areas of algorithm and software development that will be receiving focused attention in the near- to mid-term by the Trilinos developers working in the Embedded Nonlinear Analysis
Tools capability area.
• Expansion of the Rythmos transient integration code to include adjoint integrations with checkpointing, and error estimation.
• Development of the embedded UQ capability, hybrid UQ methods combining sampling and embedded algorithms, and scalable linear solver algorithms for stochastic systems.
• Develop System Modeling capability: systems abstractions, networks, and UQ-enabled systems solvers.
• Development of a single factory to generate any solution or analysis scheme from a parameter list.
• Improved software quality, such as appropriate handling of thrown exceptions.
• Demonstrations of the transformational analyses that can be performed with embedded algorithms on a large-scale applicaiton code that uses automatic differentiation.
Where should you look next?
For more detailed information on the capabilities available through Trilinos, you can look at the web pages for the individual packages, that were listed in the above Table. Each package has example
problems that show common use cases. | {"url":"https://trilinos.github.io/nonlinear_solver.html","timestamp":"2024-11-02T14:03:03Z","content_type":"text/html","content_length":"42882","record_id":"<urn:uuid:d1ed8e01-30e2-471d-aaf0-fceb3cf62329>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00507.warc.gz"} |
Degree Name
Master of Arts in Teaching, Mathematics
First Reader/Committee Chair
Joseph Jesunathadas
One hundred high school Algebra students from a southern California school participated in this study to provide information on students’ ability to relate the definition of function to its
representations. The goals of the study were (1) to explore the extent to which students are able to distinguish between representations of functions/non-functions; (2) to compare students’ ability
to distinguish between familiar/unfamiliar representations of functions/non-functions; (3) to explore the extent to which students are able to apply the definition of function to verify function
representations; and (4) to explore the extent to which students are able to provide an adequate definition of function. Data was collected from written responses on a math survey consisting of items
that asked students to decide if given illustrations are representations of functions, to explain how the decision was made, and to supply the domain and range when applicable. The questions included
seven types of illustrations: graphs, equations, ordered pairs, tables, statements, arrow diagrams, and arbitrary mappings. Findings indicated that students were more able to correctly identify
familiar than unfamiliar function representations. The easiest representation for students to correctly identify was the graph of a linear function and the most difficult was the graph of a piecewise
function. A conjecture as to why this occurred is that the formal definition of function is not often emphasized or referenced when function and its representations are introduced so students do not
have a deep understanding of how the function definition is related to its representations. The explanation, domain, and range responses were sketchy. A conjecture as to why this occurred is that in
general, students have difficulty expressing themselves orally and in writing or perhaps students had not learned about domain and range. A separate question asked students, “What is a function?” To
this question, students provided a variety of responses. It is suggested that conducting further studies that include student interviews and participants from multiple teachers, would provide
increased understanding of how students learn the definition of function and the extent to which they are able to relate it to its representations.
Recommended Citation
Thomson, Sarah A., "ALGEBRA 1 STUDENTS’ ABILITY TO RELATE THE DEFINITION OF A FUNCTION TO ITS REPRESENTATIONS" (2015). Electronic Theses, Projects, and Dissertations. 215. | {"url":"https://scholarworks.lib.csusb.edu/etd/215/","timestamp":"2024-11-07T16:57:12Z","content_type":"text/html","content_length":"41070","record_id":"<urn:uuid:cce400ee-9f5c-4a73-887f-118e630b183b>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00353.warc.gz"} |
Problem 1944: Precise Average
Time limit: 0.08s Memory limit: 64MB Input: Output:
Your friend John works at a shop that sells $N$ types of products, numbered from $0$ to $N-1$.
Product $i$ has a price of $P_i$ bytedollars, where $P_i$ is a positive integer.
The government of Byteland has created a new law for shops. The law states that the average of the prices of all products in a shop should be exactly $K$, where $K$ is a positive integer.
John's boss gave him the task to change the prices of the products so the shop would comply with the new law.
He has lots of other stuff to do, so he asked you for help: what is the minimum number of products whose prices should be changed?
A product's price can be changed to any positive integer amount of bytedollars.
Input data
The first line of the input contains two integers $N$ and $K$.
The next line contains $N$ positive integers, $P_{0}, \, \ldots, \, P_{N-1}$.
Output data
Output a single integer between $0$ and $N$, inclusive: the answer to the question.
It can be proven that it is always possible to change the prices of some products so that the new prices comply with the law.
Constraints and clarifications
• $1 \le N \le 200 \ 000$.
• $1 \le K \le 10^6$.
• $1 \le P_i \le 10^6$ for each $i=0\ldots N-1$.
• For tests worth $20$ points, $N \le 2$.
• For tests worth $20$ more points, $N \le 1000$.
Example 1
In the first sample case a possible solution is to change both prices to be $3$ bytedollars. It can be proven that changing only one price is not sufficient.
Example 2
In the second sample case a possible solution is to change the first product's price to $16$ bytedollars, thus the average will be $\frac{16+10+1}{3}=9$. | {"url":"https://kilonova.ro/problems/1944?list_id=823","timestamp":"2024-11-14T11:36:11Z","content_type":"text/html","content_length":"40781","record_id":"<urn:uuid:361c84e3-186f-4cd4-9580-36b38835b562>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00607.warc.gz"} |
The Stacks project
Definition 20.49.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. We say $\mathcal{E}^\bullet $ is perfect if there exists an open
covering $X = \bigcup U_ i$ such that for each $i$ there exists a morphism of complexes $\mathcal{E}_ i^\bullet \to \mathcal{E}^\bullet |_{U_ i}$ which is a quasi-isomorphism with $\mathcal{E}_ i^\
bullet $ a strictly perfect complex of $\mathcal{O}_{U_ i}$-modules. An object $E$ of $D(\mathcal{O}_ X)$ is perfect if it can be represented by a perfect complex of $\mathcal{O}_ X$-modules.
Comments (2)
Comment #8373 by Nicolás on
It might be useful to add a small remark like the one right after Definition 20.47.1, about perfect complexes being bounded. Something like
"If $X$ is quasi-compact, then a perfect object of $D(\mathcal{O}_X)$ is in $D^b(\mathcal{O}_X)$. But this need not be the case if $X$ is not quasi-compact."
Comment #8979 by Stacks project on
Thanks and fixed here.
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• 2 comment(s) on Section 20.49: Perfect complexes
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Recursive Spark SQL
Spark is an efficient and easy-to-use framework for solving big data problems, and its Dataframe API allows solutions to be formatted in a familiar SQL syntax. However, the Dataframe API does not
support the recursive SQL functionality, which is often used for a class of graph and tree algorithms.
Let’s look at an example problem and how we might solve it using recursive SQL, and then think about how to translate this into a Spark program.
The Problem
Let’s say we have a directed tree data structure stored in a text file representing an edge list. Our file is called graph.csv and has the schema [node_id: int, parent_id: int].
This file represents the following directed, unweighted tree:
Our task is to build a list of all possible ancestor/descendant relationships between nodes. For example, 4 is a descendant of 1 and 1 is the ancestor of 4. However, 3 is not an ancestor of 4.
Now assume that we want to use SQL to solve this problem and our Graph is actually stored in an SQL table named Graph with columns node and parent. The recursive SQL implementation looks like this:
WITH RECURSIVE Relationship(anc, des) AS (
SELECT parent AS anc, node AS des
FROM Graph
SELECT Relationship.anc AS anc, Graph.node AS des
FROM Relationship, Graph
WHERE Relationship.des = Graph.parent
This results in a table named Relationship with columns anc and des, where each row represents an ancestral relationship between two nodes. In our example, the Relationship table will look like this
after the recursive SQL algorithm converges:
anc des
From this table, it’s clear that 1 is the ancestor of every other node. Also notice that (3, 4) is not a row, so the nodes 3 and 4 have no relationship to each other. Finally, 1 never appears in the
descendant column, so 1 has no ancestors.
SQL Recursion Details
So why did we have to use recursive SQL to solve this problem? Well, without recursion we can only express ancestors in a SQL query at a fixed degree like 1st degree, 2nd degree, etc. In order to get
all ancestors of arbitrary degree, we must take advantage of recursive SQL. Internally, the SQL engine will run something called a “Fixed Point” computation. Therefore, RECURSIVE T AS Q has the
following semantics:
$$ T_0 = \empty \\ T_1 = Q \text{ (but use } T_0 \text { for } T \text {)} \\ T_2 = Q \text{ (but use } T_1 \text { for } T \text {)} \\ \text{… until } T_i = T_{i+1} $$
We start with an empty result table (Relationship). We then run the query for the first time, generating the 1st degree ancestors, which is just the Graph table itself. Notice that the query had two
parts: a base case and a recursive case which were UNION’d together. The Relationship table is initially empty, so the recursive case generates no output on the first iteration. Only the base case
generates a table, which is just the Graph table. Then, all subsequent iterations use the previous Relationship table of i-degree ancestors to generate the i+1-degree ancestors, which are put into
the Relationship table again. Effectively, this transforms ancestors of degree 1 → 2, 2 → 3, etc., so we need the UNION with the base case to re-introduce the ancestors of degree 1. Once we reach an
iteration where the Relationship table no longer changes, we stop.
Now, how can we replicate this functionality using Spark?
Spark Algorithm #1 - DataFrame Fixed Point Computation
Currently, Spark SQL does not support recursive SQL, so we can’t express this computation directly by either copy-and-pasting the SQL query into the .sql() function or by using the more programmatic
Dataframe functions (e.g. .select(), .where(), etc.).
But… we know the “secret sauce” that SQL uses to get recursion in the form of “Fixed Point” computation, so we can mirror this approach in Spark.
import org.apache.spark.sql.{SparkSession, Row}
import org.apache.spark.sql.types._
import util.control.Breaks._
object AncestorsApp {
def main(args: Array[String]) {
val spark = SparkSession.builder.appName("AncestorsApp").getOrCreate()
import spark.implicits._
val graphSchema = StructType(Array(
StructField("node", IntegerType, false),
StructField("parent", IntegerType, false)
val relationshipSchema = StructType(Array(
StructField("anc", IntegerType, false),
StructField("des", IntegerType, false)
val graph = spark.read.schema(graphSchema).csv("graph.csv")
var relationship = spark.createDataFrame(spark.sparkContext.emptyRDD[Row], relationshipSchema)
// Run the above SQL query to generate ancestors until the dataset reaches a fixed point
breakable {
while (true) {
val nextRelationship = graph
.select($"parent".alias("anc"), $"node".alias("des"))
relationship.join(graph, $"des" === $"parent").select($"anc", $"node".alias("des"))
// If the dataset has not changed in this iteration, then break
if (nextRelationship.except(relationship).count() == 0) { break }
else { relationship = nextRelationship }
This produces the output:
| 1| 2|
| 1| 3|
| 2| 4|
| 2| 5|
| 3| 6|
| 3| 7|
| 1| 5|
| 1| 4|
| 1| 7|
| 1| 6|
which is the same as the table we expected above.
Spark Algorithm #2 - Spark GraphX and Pregel
The above method isn’t the only way to approach this task with Spark. Spark also has a component named GraphX which allows programmers to work with an abstraction of a directed graph which is
implemented under-the-hood with optimized RDDs. This allows us to run graph-parallel big data algorithms on graphs with a higher-level API.
The SparkX Graph is what you’d expect: a collection of vertices identitified by a Long integer (VertexId) and a collection of edges identified by the VertexId of their source and destination
vertices. Vertices and edges also have properties, which is a user-defined piece of data attached to the vertex or edge. For example, if we were running a single-source shortest path algorithm, the
vertex property would be “distance from the source” and the edge property would be some weight.
[1] An example of a graph in Spark GraphX. The properties are unrelated to the discussion here.
One more concept to understand is the edge triplet. This is simply an edge and the two verticies it connects, which many GraphX algorithms work with.
[2] Example of a Triplet in Spark GraphX
So how can we use GraphX to generate our ancestors table? Well, we want to create some sort of recursive algorithm that runs on the graph until a stopping point is reached. Our approach probably
involves passing information about ancestry around the graph, and this can be done in local computations on each vertex, so this suggests the use of the Pregel API in GraphX, which is a “a
bulk-synchronous parallel messaging abstraction”. [2]
This means that we can define local computations that vertices do which can send out messages to other vertices they’re connected to via an edge. This happens in parallel for all vertices in the
graph per iteration, and iteration continues until there are no more messages to send.
This is our idea for the algorithm: each vertex tells its out-neighbours (descendants) about itself and its own ancestors and each vertex will have a property which is a set of ancestors. When the
algorithm finishes, each vertex will know all its ancestors, and we can then map over all the vertices and flatten this out to the table we wanted.
One possible problem with this approach is the size of the ancestor set on each vertex. If it’s too large, say $O(n)$, then we’re no longer taking advantage of big data approaches. However, most
real-world hierarchical tree structures have quite a small height if balanced, so the size is more like $O(log(n))$. Therefore, this approach can be very feasible in practice.
First, we’ll assume the input data is slightly different, as we actually want the directed edges to point from parent to child, since we’ll be sending ancestry information down. Of course, if the
data looks different than shown here, it could be transformed. This format without commas also allows us to use the handy GraphLoader.edgeListFile API to build a Graph from an edge list stored in a
text file.
Here is the complete algorithm:
import org.apache.spark.{SparkContext, SparkConf}
import org.apache.spark.graphx.{GraphLoader}
object AncestorsApp {
def main(args: Array[String]) {
val conf = new SparkConf().setAppName("Ancestors App")
val sc = new SparkContext(conf)
val graph = GraphLoader.edgeListFile(sc, "graphEdgeList.txt")
.mapVertices((_, _) => Set.empty[Long]) // Will hold the set of ancestors for this vertex
.mapEdges(_ => null) // No need for edge labels in this unweighted graph
// Use GraphX Pregel, a bulk-synchronous message-passing API
// to iteratively pass messages between vertices until no messages remain.
val ancestors = graph.pregel(Set.empty[Long])( // The initial message to all vertices is the empty set of ancestors
(id, ancestors, newAncestors) => ancestors | newAncestors, // When the merged message of newAncestors is received, combine it with the set of ancestors
triplet => { // This function is run for every edge triplet in the grpah
val ancestorsToSend = triplet.srcAttr + triplet.srcId // This vertex (src) will send its set of ancestors and its own ID to its descendant (dst)
val childsAncestors = triplet.dstAttr
if (ancestorsToSend subsetOf childsAncestors) { // If the descendant already has all the ancestors, then do not send the message. Else, send the set difference.
} else {
Iterator((triplet.dstId, ancestorsToSend -- childsAncestors))
(ancestors1, ancestors2) => ancestors1 | ancestors2 // This function determines how to merge two messages received at a vertex. We just union the ancestor sets.
// Take the graph, with all the vertices containing their ancestors, and flatten it out to an RDD.
val relationship = ancestors.vertices.flatMap({ case (id, ancestors) => ancestors.map(anc => (anc, id)) })
Here is the output, formatted as a list of pairs (ancestor, descendant), which is what we expect:
Creating a recursive algorithm on graphs at big-data scale may seem daunting at first, but these two methods show that it can be done in relatively few lines of code with Spark. Firstly, even though
Spark SQL does not support recursive SQL, we can run regular dataframes transformations iteratively until a stopping point is reached, which is the same principle behind “fixed point” computation
used in recursive SQL. It was shown how a recursive SQL query could be almost directly translated to Spark SQL following the “fixed point” computation method. Alternatively, Spark offers the GraphX
component, which allows us to express big-data Graphs with a nice abstraction. We can then use an API like Pregel to transfer messages between vertices until a stopping point is reached.
[1] https://spark.apache.org/docs/latest/graphx-programming-guide.html#property_graph
[2] https://spark.apache.org/docs/latest/graphx-programming-guide.html#pregel-api | {"url":"https://www.bradleystevanus.com/posts/recursive-spark-sql/","timestamp":"2024-11-05T22:54:49Z","content_type":"text/html","content_length":"61727","record_id":"<urn:uuid:f9f9fa9c-c5e7-46dd-9281-379f89482b0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00861.warc.gz"} |
Department of Mathematics
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MATH-270. Several Complex Variables I. 3 crs. Basic facts about holomorphic functions; zero sets of holomorphic functions, analytic sets and Weierstrass' Preparation theorem; domains of holomorphy,
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MATH-550. Ph.D. Dissertation. 12 crs. Prereq.: Consent of Ph.D. adviser. | {"url":"https://mathematics.howard.edu/index.php/academics/mathematics-courses","timestamp":"2024-11-04T13:39:42Z","content_type":"text/html","content_length":"50128","record_id":"<urn:uuid:3145fd1f-6732-4880-9376-abfc68b080cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00067.warc.gz"} |
RecursionError when installing SageMath 8.2
RecursionError when installing SageMath 8.2
I downloaded sage-8.2-OSX_10.13.4-x86_64.dmg. Then I ran ./sage from the extracted SageMath directory in a Terminal. Then I got the following error:
Last login: Sun Jun 10 23:17:18 on ttys001 /Users/jing/tools/SageMath/sage ; exit; ➜ ~ /Users/jing/tools/SageMath/sage ; exit; RecursionError: maximum recursion depth exceeded during compilation
It seems that you are attempting to run Sage from an unpacked source tarball, but you have not compiled it yet (or maybe the build has not finished). You should run make in the Sage root directory
first. If you did not intend to build Sage from source, you should download a binary tarball instead. Read README.txt for more information.
[Process completed]
I am using macOS 10.13.4. Can anyone help me on this?
4 Answers
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As some other answers pointed out, the problem is that the first time that sage is called, the script relocate-once.py is executed. Since in that file, the shebang line reads /usr/bin/env python, the
system calls the default python interpreter. The simplest solution is to have python 2.7 installed and change that line to /usr/bin/env python2.7 or directly the path of a python 2.7 interpreter.
On the off-chance you're using Anaconda, try commenting out the export PATH="/anaconda/bin:$PATH" line in the .bash_profile file (and restarting your terminal) before you run Sage for the first time.
You can uncomment it again after sage successfully runs (until you have to rebuild or upgrade Sage).
This fixed this problem for me. Thanks!
blove ( 2019-05-13 01:53:22 +0100 )edit
I finally found that the cause of this problem is that I have used brew to installed python3 and linked python to python3. After I used brew to uninstall python3, the problem is solved.
Note that this path issue only seems to be a problem when running Sage for the first time. You should be able to reinstall python3 again with homebrew and use Sage without any problem (until you
upgrade versions).
j.c. ( 2018-06-16 18:51:28 +0100 )edit
You need to copy the SageMath directory to somewhere on your hard drive – you can't just run ./sage from within the directory in the virtual drive that you get when you open up the dmg file. Once
you've copied it somewhere and run ./sage, you should see
Rewriting paths for your new installation directory
This might take a few minutes but only has to be done once.
followed by lots of messages like
patching /Users/palmieri/Desktop/SageMath/build/make/Makefile-auto
patching /Users/palmieri/Desktop/SageMath/config.status
At this point, you won't be able to move the SageMath folder – its location has been burned in during the patching process – so pick a good location. (You can always copy a fresh SageMath folder from
the dmg file, if you want to start over.)
Given the path shown in the post /Users/jing/tools/SageMath/sage I don't think that the OP is trying to run sage from within the dmg file.
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Finite-Element-Method in Structural Engineering — Concrete E-Learning Platform | IBK, ETH Zurich
Finite-Element-Method in Structural Engineering
By | September 14, 2022
Link zur deutschen Version: Die Finite-Elemente-Methode im konstruktiven Ingenieurbau
Basic principles and areas of application of the FEM
In the last decades, computer-aided calculation methods – particularly the Finite-Element-Method (FEM) – have become an indispensable tool in engineering. With the development of the FEM, everyday
calculations and routine tasks can be automated and carried out without careless mistakes. Thus (at least a part) of Konrad Zuse’s [1] dream of taking over “annoying” tasks by a fully automated
calculating machine has become a reality. The engineer’s task has shifted from calculation to modelling and the interpretation of results, for which, however, the underlying theories and methods
should be applied consciously and correctly. This has not made the art of structural analysis superfluous, but it has made it more demanding and exciting.
Figure 1: Konrad Zuse in 1941 with the Z3, the world’s first computer (Source: Ingenieur.de)
In general, the FEM provides a numerical approximation for all tasks that can be represented mathematically with partial differential equations, for which no closed solutions are available. More
simply, complex systems (load-bearing structures) are divided into components (finite elements) and mathematically linked to form an overall model. If the mechanical properties (stiffness) of the
individual elements are known through a relationship between stresses and strains specified within the framework of the selected mechanical model, the effects (e.g. internal forces) on the overall
model can be determined under the given actions together with the boundary conditions (support conditions).
For the everyday tasks of a structural engineer, the linear-elastic FEM is mostly used, whereby a linear relationship between the stresses and strains is assumed. The FEM supplies e.g. the internal
forces for the dimensioning of structures using ideal-plastic models. In exceptional cases, e.g. for the static verification of existing structures, the nonlinear [2] FEM is used more and more often.
A nonlinear relationship between the stresses and strains is then applied, which allows the analysis of the load-deformation behaviour, the cracking behaviour and the associated redistribution of
internal forces. In contrast to conventional calculation methods, this allows to identify load-bearing reserves, with which more efficient and resource-saving structures can be built or, ideally,
expensive strengthenings can be avoided during a static verification of existing structures. While the pure application of such nonlinear FEM programs is relatively easy nowadays, thanks to
user-friendly interfaces, choosing the appropriate (nonlinear) material and structural model for a given problem becomes more complex. The model ideas and their limitations, as well as the
peculiarities of an NLFE calculation (e.g. divergences) should be understood in detail. Imparting this knowledge is a central challenge for us as a teaching and research institute.
Historical outline of the FEM
In general, the development of FEM does not go back to a single person or a single field but has a strong interdisciplinary character with aspects from mathematics, mechanics, engineering and
computer science. The development of the FEM began in the 1950s and was pioneered by J. Argyris at the Technical University of Stuttgart, R.L. Taylor or R. Clough at the University of California,
Berkely and O.C. Zienkiewicz at the University of Wales (all qualified civil engineers!). From the 1960s, the development picked up speed with the first international FEM conference in 1965 and the
first standard book by Zienkiewicz in 1967 entitled “The Finite Element Method in Structural and Continuum Mechanics”.
Figure 2: History of FEM (top left; Source: Lecture notes from Prof. Bischoff, Universität Stuttgart) and pioneers in the development of FEM (remaining images; Source: optimum.one).
The application of the FEM to reinforced concrete structures began in the late 1960s with the work of Ngeo and Scordelis 1967 and Rashid 1968. Since then, countless publications and books have been
published with the theoretical basics, use cases and guidelines (see FIB bulletin 45 2008). The first FEM-software were developed at universities and developed into commercial FEM packages with
increased user-friendliness after 1970. This resulted in a large number of different software, including different types of elements and various mathematical solution algorithms for the spatial and
temporal discretisation of the given problem. D. Mitchel, M.P. Collins and F.J. Vecchio from the University of Toronto played a pioneering role in developing nonlinear material models for reinforced
concrete in the 1980s (development of compression field theories and their implementation in the FEM-software VecTor). Today, the FEM is one of the most widely used numerical solution methods for
various physical tasks in all engineering sciences up to weather forecasts. We can be all the more proud that civil engineers made a significant contribution to the development of the FEM.
NLFE-tools at the Chair of Concrete Structures and Bridge Design
At the Chair of Concrete Structures and Bridge Design, we concentrate on developing nonlinear material models and their implementation in numerical methods, such as FEM. In recent years, various
tools for nonlinear finite element analysis of reinforced concrete and masonry structures have been developed:
The Compatible Stress Field Method (CSFM) is suitable for analysing reinforced concrete under a plane stress state (e.g. frame corners or longitudinal beams of a box girder bridge, as shown in Figure
3). The CSFM is based mainly on the implementation of the tension chord model in the FEM-software Idea StatiCa Detail.
Figure 3: CSFM analysis of a component test on a frame corner (left; source: Kraus et al.) and on a simple beam (right; source: Kaufmann et al.).
The CMM-Usermat contains the implementation of the Cracked Membrane Model (developed by Prof. Dr. Walter Kaufmann) as a user-defined material (Usermat) in the FEM program Ansys Mechanical APDL.
Combined with a layer model, shell structures made of reinforced concrete can be analysed (see Figure 4).
Figure 4: CMM-Usermat analysis of a structural test on a two-span reinforced concrete slab (source: Thoma/Roos/Borkowski): load-deformation curves (left) and principal shear force flow (right).
The URM-Usermat can be used to analyse the nonlinear load-bearing behaviour of unreinforced masonry structures (see Figure 5). The URM-Usermat is based on an implementation of extended failure
criteria by Ganz (which were developed in the 1980s at the ETH Zurich and are still used today as the basis for masonry design) as user-defined material in the FEM program Ansys Mechanical APDL. In
combination with the CMM-Usermat, the complex load-bearing behaviour of mixed structures made of reinforced concrete and masonry can be examined (see Figure 6).
Figure 5: URM-Usermat analysis of a structural test on a masonry wall (source: Weber): load-deformation curves (left), deformed FE mesh (middle) and course of the principal compressive stresses
Figure 6: CMM/URM-Usermat analysis of an experiment on a mixed structure made of masonry and reinforced concrete (source: Weber): load-deformation curves (top left), deformed FE mesh (top right),
principal compressive stresses (bottom left) and crack pattern (bottom right).
All NLFE-tools have in common that the material models take into account the relevant properties of the load-bearing behaviour based on mechanically consistent relationships. In addition, only a few
physically clearly defined input parameters are required, which are known in the design or assessment phase. This makes the NLFE-tools ideal for use in construction practice.
Future developments
The NLFE-tools are currently being implemented in the StrucEng Library, which is being developed at the Professorship for Concrete Structures and Bridge Design. The StrucEng Library is an open-source
software that enables computer-aided calculations of load-bearing structures made of reinforced concrete and masonry. In addition to the nonlinear material models, various other material models are
available, which cover the entire spectrum from structural design to analysis with linear-elastic ideal-plastic models. In the future, the StrucEng Library will be available in open-source format,
which means that practical engineers can freely use the corresponding model using a user-friendly interface. In addition, the nonlinear material models are being further developed in various projects
(e.g. corrosion, fibre-reinforced concrete, etc.). In the future, courses in the numerical modelling of reinforced concrete and masonry structures will also be offered at our Chair. With this special
focus on numerical modelling in teaching and research, we are convinced that the prospective civil engineers will be fit for future challenges.
[1] Konrad Ernst Otto Zuse was a German civil engineer (!), inventor and entrepreneur. With his development of the Z3 in 1941, Zuse built the world's first working computer.
[2] In the context of these blog posts, nonlinearity always means material nonlinearity. There are also geometric nonlinearities (second-order calculations), which are taken into account via the P-Delta effect.
Marius Weber
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: Student entering nth roots (dev math)
Hello again. (It’s been quite a few years since I last posted.)
I’d like to have WeBWorK exercises asking (developmental math) students to translate expressions with rational exponents (such as “x^(1/n)”) into equivalent expressions with radicals, what LaTeX
would show for “\sqrt[n]{x}”).
In the xyzhomework.com system (an implementation of the MyOpenMath homework system), we could ask students to input the “radical form” via either MathQuill (an online math expression editor) or by
entering “root(n)(x)”. How can I code a similar WeBWorK exercise?
I’m looking for fairly direct translations, not simplifications, so that “1/sqrt(3)” is the expected acceptable expression for “3^(-1/2)”. But I don’t want the student to get credit for simply
copying the expression with rational exponents—s/he needs to designate the appropriate radical.
Bruce Yoshiwara
Hi Bruce,
If you want a palette style answer entry, the ones that I know of currently would *look* like \sqrt[n]{x}, but would actually submit x^(1/n) to WeBWorK for evaluation. Which would mean they could
just type x^(1/n) too. So until that behavior of the available palette tools is changed, I think you can't get what you want. But keep reading.
If you are OK with typed-in answers, then you can load parserRoot.pl. Then something like root(3,x) is understood. You want to disallow things like x^(1/3) and exp(1/3 ln(x)). So one option is to
disable the exponentiation operator and functions, then re-enable root. So with the caveat that I haven't tried it out:
Context()->functions->enable("root"); #possibly not needed, if root is not a "function" but rather a "funciton2" as a function of 2 variables
[`7^{1/3} = `][_______________]{Formula("root(3,7)")}
Now, one palette tool is WIRIS, which has a pull request to develop right now, and is likely to be available in WBWK 2.14. Presently, it handles things like I mentioned above, secretly passing things
like x^(1/3). I'm working on convincing them to maybe pass root(3,x) instead. So maybe the future will have a working palette tool for this too.
Lastly, watch out that if the answer is like root(4, x), that you set the domain to be something appropriate. The default is [-2, 2], and every once in a while a student will get all their test
points in [-2,0), and the problem won't work for them.
Dear Alex,
Terrific, thanks!
Your code seems to do exactly what I was hoping to get. I did remove the Context()->functions->enable("root"), and I also removed "Formula" in the answer, which works with just the "root(3,7)".
Great, but something more occurred to me. If the answer is just a number (like root(3,7)) as opposed to an expression with variables (like root(3, x)), then students will still be able to enter 1.913
and get credit (it is within the default tolerance). You could disable decimals, but then they could still enter 1913/1000.
So...if you are sure you don't want an answer like root(3,4/5), where you would want the division slash, an easy thing to do is to basically turn off everything except integers and the root function.
So replace Context("Numeric") by Context("LimitedNumeric"). And disallow decimals with Parser::Number::NoDecimals();
If you did want the division slash, you could bring it back with Context()->operators->redefine("/");
Dear Alex,
Thanks again!
I've decided to go with adding Parser::Number::NoDecimals(); and living with the possibility that some enterprising students might enter approximations as fractions.
I fiddled with using Context("LimitedNumeric"), but then WeBWorK complained about my using parentheses in expressions like "root(n,k)"
I didn't try to remove the "/" operator because I will also be writing exercises where the appropriate answer is "1/sqrt(x)" | {"url":"https://webwork.maa.org/moodle/mod/forum/discuss.php?d=4353","timestamp":"2024-11-09T03:15:43Z","content_type":"text/html","content_length":"99736","record_id":"<urn:uuid:97290915-fc8b-4dfb-9fe4-1da909be87f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00628.warc.gz"} |
What is Latent Representation in Deep Learning? - reason.townWhat is Latent Representation in Deep Learning?
What is Latent Representation in Deep Learning?
Discover what latent representation is in deep learning, and how it can be used to improve the performance of your machine learning models.
Checkout this video:
What is a latent representation?
A latent representation is a hidden state vector that captures some underlying structure or relationship in the data. This hidden state can be thought of as a concise summary of the important
information in the data. It is an efficient representation because it encodes only the information that is relevant to the task at hand, and it can be learned automatically from data.
Latent representations have been found to be useful for many tasks in deep learning, such as image classification, object detection, and video prediction. In each of these tasks, the goal is to map
input data (e.g., an image) to a desired output (e.g., a class label). A deep neural network can learn to do this mapping by learning a latent representation of the input data. This learned
representation can then be used for other tasks, such as predicting how the input data will change over time (e.g., in a video).
There are many ways to learn latent representations, but one common method is to use an auto-encoder. An auto-encoder is a neural network that takes an input data vector and maps it to a latent space
vector, which is then mapped back to an output vector that resembles the original input vector. The auto-encoder is trained such that the output vector reconstructs the original input vector as
closely as possible. During training, the auto-encoder learns to compress the input data into a latent space that captures its important features. Once training is complete, the auto-encoder can be
used to map new data points into the latent space and use them for other tasks.
Latent representations have become increasingly important in deep learning due to their ability to capture complex relationships in data and their usefulness for transfer learning. Transfer learning
is a technique for training machine learning models on new data sets by starting with pre-trained models that have already been trained on related data sets. This technique can be used when there is
not enough labeled data available for training a model from scratch. By starting with pre-trained models, we can take advantage of knowledge that has already been learned and focus on fine-tuning the
model for our specific task. Many state-of-the-art deep learning models use latent representations learned from large amounts of labeled data (e.g., ImageNet) and then fine-tuned on smaller labeled
datasets (e.g., CIFAR-10). This approach has been shown to be effective for many difficult machine learning tasks
What is deep learning?
Deep learning is a subset of machine learning that is concerned with neural networks. Neural networks are a type of artificial intelligence that are used to simulate the workings of the human brain.
Deep learning is based on the idea that if we can understand how the human brain works, we can use that knowledge to create better algorithms for machines.
Latent representation is a term used in deep learning to describe the process by which data is transformed into a format that can be used by a neural network. This transformation is necessary because
neural networks can only work with data that is in a certain format. Data in this format is said to have a latent representation.
There are two main types of latent representation: feature vectors and raw data. Feature vectors are data that has been transformed into a format that can be used by a neural network. Raw data, on
the other hand, has not been transformed and is in its original form.
feature vectors are usually created by humans, while raw data is usually collected by machines. humans have more experience than machines and are therefore better at understanding data. However, raw
data has the potential to be more accurate than feature vectors because it has not been transformed by humans and so contains all the information about the original data set.
What is a deep neural network?
A deep neural network (DNN) is an artificial neural network (ANN) with multiple hidden layers between the input and output layers. The additional hidden layers allow the network to learn complex
patterns in data and improve the accuracy of predictions.
How can deep learning be used to learn latent representations?
Latent representation learning is a core challenge in deep learning. A latent representation is a set of features that captures the underlying structure of data. Latent representations can be used
for Dimensionality Reduction, Data Visualization, Data Compression, and Pattern Recognition.
Latent representations are learned using a variety of techniques, including autoencoders, generative adversarial networks (GANs), and variational autoencoders (VAEs). These methods are powerful
because they can learn complex features that are useful for downstream tasks.
One of the benefits of latent representation learning is that it can help us to understand data better. By reducing the dimensionality of data, we can identify patterns that would be difficult to see
in the raw data. Latent representation learning can also be used for data visualization, by projecting data onto a lower-dimensional space. This can help us to see clusters and patterns that we might
not be able to see in the original data.
Latent representation learning is also useful for compression and pattern recognition. Bylearnings latent representations, we can compress data without loss of information. This is because the latent
representations capture the underlying structure of the data. In addition, latent representation learning can be used for pattern recognition tasks such as object detection and classification.
What are the benefits of learning latent representations?
There are many benefits to learning latent representations. By definition, a latent representation is a lower-dimensional representation of data that captures important features of the data. In other
words, it is a compressed version of the data that contains the most important information.
For example, consider a dataset of images of everyday objects. A latent representation of this data would be a lower-dimensional representation that captures the essential features of the images
(e.g., shape, color, etc.). This latent representation would be useful for tasks such as object recognition, since it would contain the most important information about the objects in the dataset.
In addition to being useful for machine learning tasks, latent representations have several other benefits. For one, they can help us to understand complex data better. By reducing the dimensionality
of the data, latent representations make it easier to visualize and understand. Additionally, latent representations can help us to reduce noise in our data. This is because noise is often
distributed evenly across all dimensions of our data; by compressing our data into a lower-dimensional space, we can reduce the amount of noise in our data. Finally, latent representations can help
us to make better use of limited computational resources. This is because it is often easier and faster to train machine learning models on lower-dimensional data than on higher-dimensional data.
What are some challenges of learning latent representations?
Some of the challenges in learning latent representations include:
-The data may be high dimensional, making it difficult to identify useful features.
-The data may be multimodal, with different features corresponding to different modes of variation.
-The data may be non-stationary, meaning that the feature distribution changes over time.
-The data may be nonlinear, making it difficult to learn a linear model that accurately captures the relationships between variables.
How can latent representations be used in applications?
Latent representations are powerful tools that can be used in a variety of applications. For example, they can be used to improve the performance of machine learning models, to compress data, or to
generate new data.
What are some future directions for research on latent representations?
Some future directions for research on latent representations include:
-Improving the model’s ability to disentangle different kinds of variations in the data, e.g. by learning separate latent representations for different kinds of data
-Investigating how latent representations can be used to transfer knowledge between different tasks
-Exploring ways to make latent representations more interpretable, e.g. by learning disentangled latent representations that can be visualized
Latent representation is often thought of as the “hidden” representation of data in a machine learning model. It can be used to represent data in a way that is more efficient or compact than the
original data. Latent representation can also be used to disentangle different features of data, making it easier to learn new features or make predictions.
Deep learning is a subset of machine learning in artificial intelligence (AI) that has networks capable of learning unsupervised from data that is unstructured or unlabeled. Also known as deep neural
learning or deep neural networks, deep learning is a technique that mimics the workings of the human brain in processing data for recognition. | {"url":"https://reason.town/latent-representation-deep-learning/","timestamp":"2024-11-14T18:50:27Z","content_type":"text/html","content_length":"99367","record_id":"<urn:uuid:69001349-29f0-4b9c-9e8e-2a2a65e32140>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00642.warc.gz"} |
Design of a dynamic weigh system for conveyors
University of Twente Student Theses
Design of a dynamic weigh system for conveyors
Timmer Arends, J.H. (2016) Design of a dynamic weigh system for conveyors.
Miedema and Dewulf are merged companies in the field of providing agricultural machines, especially for potato growers. They both attach great importance to innovation. A current field of
research is about dynamic belt weighing systems. Miedema would like to have such a system on their conveyors and Dewulf would like to have one on their harvesters. The basic idea is
similar for both applications: they have to weigh the product dynamically as accurate as possible and cope with all the present factors. The most important factors are the belt tension,
sticking soil, angles, belt velocity, vibrations, non-uniform loading and environmental conditions. For carrying out measurements there are also some parameters that have to be determined
like filtering, resolution and calibration. For the Miedema conveyor the goal was set to get a maximum error of 1%. The goal for these dynamic weigh systems is to optimize the process for
the used machine parameters. It makes it easy to see if the belt is over loaded. An even more important goal is to get information of the yield per hectare or per hour. For the harvester
this information can be coupled to the GPS signal in order to get yield maps of every field and the conveyor can load trucks exactly to the maximum allowed weight. For both applications
it is chosen to use a strain gauge load cell. When this cell is loaded the strain causes a difference in resistance and this can be coupled to a mass. Some initial tests and a set-up was
already made for the harvester and the goal was to gain more knowledge about the weigh principle and test some of mentioned factors. Before the actual testing literature and theory was
covered to understand the test results. The same was done for the conveyor. Miedema already designed a weigh frame and the goal was to test this principle and give recommendations
Abstract: regarding further improvements. From the literature it was clear that the weigh system should be close to the tail, the velocity measurement should be done at the tail pulley, the frame
should be long and stiff, the belt tension must be constant, there should be uniform contact pressure with the idlers and sticking soil should be minimised. From a theoretical
approximation it became clear that a correction factor is needed to get to the correct value. For the harvester this value is 3.4665 and for the conveyor it is 1.3127. This factor comes
from the configuration of the idlers which are connected to the load cells. The theory also shows the great influence of the belt tension and the angles. From testing it appears that the
calculated correction factor is very close to the real life situation. For the harvester to real factor is 3.5060 and for the conveyor it is 1.3408. For the harvester the belt shows very
large vibrations so a double notch filter and a low pass filter is designed which is velocity dependent. For the conveyor only a low pass filter was necessary to get rid of most of the
noise. It is chosen to measure at specific length steps so the velocity has no influence on the measurement. Tests show that a zero calibration is highly desirable so a function is made
which measures the value during two belt lengths and then subtracts the mean of this zero calibration from every new measurement. All tests with the conveyor in a real life application
resulted in an error lower than 1%. Recommendations towards further research for the Dewulf harvester are to test the influence of the angle which the machine can make, the influence of
the belt tension, the influence of measuring with time steps, how to reset the factor when the belt is installed in the machine and a redesign for the steel connection of the belt. For
the Miedema conveyor testing should be done over a large time period, a function has to be implemented to change the factor, place the weigh frame more to the tail and rotate it 180
degrees, do not measure if the machine operates at more than 5 degrees in length direction and enclose the gap between the weigh frame and the total frame.
Item Type: Internship Report (Master)
Clients: Miedema Landbouwwerktuigenfabriek B.V., the Netherlands
Faculty: ET: Engineering Technology
Subject: 52 mechanical engineering
Programme: Mechanical Engineering MSc (60439)
Keywords: Dynamic weighing, load cell, strain gauge, mass measurement, calibration, filtering data
Link to https://purl.utwente.nl/essays/69531
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The Pentachoron
The pentachoron is the 4D equivalent of the tetrahedron. It consists of 5 regular tetrahedra joined at their faces, folded into 4D to form a 4D volume, meeting at 10 triangles, 10 edges, and 5
vertices. There are 3 tetrahedra surrounding every edge. It is also known as the 5-cell because it is made of 5 tetrahedral cells. Another name for it is the 4D simplex, so called because it is the
simplest polychoron that encloses a non-zero 4D volume. It is the shape of Pento's pyramid in The Legend of the Pyramid.
Cell-first projection
The cell-first perspective projection of the pentachoron into 3D is a tetrahedron, which is the nearest cell to the 4D viewpoint.
The other 4 cells are on the far side of the pentachoron, and are not shown here.
Vertex-first projection
The vertex-first projection of the pentachoron also has a tetrahedral envelope. This time, four of the cells are visible.
The vertex at the center of this image, where the four internal edges meet, is actually the apex of the pentachoron pointing at us from the 4th direction. It is the nearest vertex to the 4D
The following images show the layout of these four cells in the projected image:
The fifth cell is not visible here, as it lies on the far side of the pentachoron. It covers the entire tetrahedral volume of the projection.
Note that although the cells appear here as slightly flattened tetrahedra, this is only because they lie at an angle to the 4D viewpoint. In actuality, they are perfectly regular tetrahedra.
Face-first projection
The next image shows the pentachoron viewed at face-first.
This projection has a trigonal bipyramid as its envelope. There are two cells visible here, forming the upper and lower halves of the bipyramid, respectively. The following images show each of these
two cells.
The other 3 cells are not visible from this viewpoint, because they lie on the far side of the pentachoron.
These cells appear somewhat deformed from a regular tetrahedron, because they are all seen at an angle.
Edge-first projection
The edge-first projection of the pentachoron also has a trigonal bipyramidal envelope.
The vertical edge in this image is the closest edge to the 4D viewpoint. Three tetrahedral cells meet at this edge, as shown in the following images.
The coordinates of an origin-centered 5-cell with edge length 2 are:
• (1/√10, 1/√6, 1/√3, ±1)
• (1/√10, 1/√6, −2/√3, 0)
• (1/√10, −√(3/2), 0, 0)
• (−2√(2/5), 0, 0, 0)
Simpler coordinates can be obtained in 5D as all permutations of coordinates of:
The 4D coordinates are derived by projecting these 5D coordinates back into 4D using a symmetric projection. | {"url":"http://www.qfbox.info/4d/5-cell","timestamp":"2024-11-07T23:41:19Z","content_type":"text/html","content_length":"7732","record_id":"<urn:uuid:672842f5-f558-41d8-b82c-2e5af0e13865>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00572.warc.gz"} |
Optimization Problem Types - NEOS Guide
Optimization Problem Types
As noted in the Introduction to Optimization, an important step in the optimization process is classifying your optimization model, since algorithms for solving optimization problems are tailored to
a particular type of problem. Here we provide some guidance to help you classify your optimization model; for the various optimization problem types, we provide a linked page with some basic
information, links to algorithms and software, and online and print resources.
• Convex Optimization versus Nonconvex Optimization
• Continuous Optimization versus Discrete Optimization
Some models only make sense if the variables take on values from a discrete set, often a subset of integers, whereas other models contain variables that can take on any real value. Models with
discrete variables are discrete optimization problems; models with continuous variables are continuous optimization problems. Continuous optimization problems tend to be easier to solve than
discrete optimization problems; the smoothness of the functions means that the objective function and constraint function values at a point \(x\) can be used to deduce information about points in
a neighborhood of \(x\). However, improvements in algorithms coupled with advancements in computing technology have dramatically increased the size and complexity of discrete optimization
problems that can be solved efficiently. Continuous optimization algorithms are important in discrete optimization because many discrete optimization algorithms generate a sequence of continuous
• Unconstrained Optimization versus Constrained Optimization
Another important distinction is between problems in which there are no constraints on the variables and problems in which there are constraints on the variables. Unconstrained optimization
problems arise directly in many practical applications; they also arise in the reformulation of constrained optimization problems in which the constraints are replaced by a penalty term in the
objective function. Constrained optimization problems arise from applications in which there are explicit constraints on the variables. The constraints on the variables can vary widely from
simple bounds to systems of equalities and inequalities that model complex relationships among the variables. Constrained optimization problems can be furthered classified according to the nature
of the constraints (e.g., linear, nonlinear, convex) and the smoothness of the functions (e.g., differentiable or nondifferentiable).
• Deterministic Optimization versus Stochastic Optimization
In deterministic optimization, it is assumed that the data for the given problem are known accurately. However, for many actual problems, the data cannot be known accurately for a variety of
reasons. The first reason is due to simple measurement error. The second and more fundamental reason is that some data represent information about the future (e. g., product demand or price for a
future time period) and simply cannot be known with certainty. In optimization under uncertainty, or stochastic optimization, the uncertainty is incorporated into the model. Robust optimization
techniques can be used when the parameters are known only within certain bounds; the goal is to find a solution that is feasible for all data and optimal in some sense. Stochastic optimization
models take advantage of the fact that probability distributions governing the data are known or can be estimated; the goal is to find some policy that is feasible for all (or almost all) the
possible data instances and optimizes the expected performance of the model.
Continuous Optimization
In continuous optimization, the variables in the model are allowed to take on any value within a range of values, usually real numbers. This property of the variables is in contrast to discrete
optimization, in which some or all of the variables may be binary (restricted to the values 0 and 1), integer (for which only integer values are allowed), or more abstract objects drawn from sets
with finitely many elements.
An important distinction in continuous optimization is between problems in which there are no constraints on the variables and problems in which there are constraints on the variables. Unconstrained
optimization problems arise directly in many practical applications; they also arise in the reformulation of constrained optimization problems in which the constraints are replaced by a penalty term
in the objective function. Constrained optimization problems arise from applications in which there are explicit constraints on the variables. There are many subfields of constrained optimization for
which specific algorithms are available.
Discrete Optimization
In discrete optimization, some or all of the variables in a model are required to belong to a discrete set; this is in contrast to continuous optimization in which the variables are allowed to take
on any value within a range of values. Here, we consider two branches of discrete optimization. In integer programming, the discrete set is a subset of integers. In combinatorial optimization, the
discrete set is a set of objects, or combinatorial structures, such as assignments, combinations, routes, schedules, or sequences.
Unconstrained Optimization
Unconstrained optimization problems consider the problem of minimizing an objective function that depends on real variables with no restrictions on their values. Mathematically, let \(x \in \mathcal
{R}^n\) be a real vector with \(n \geq 1\) components and let \(f : \mathcal{R}^n \rightarrow \mathcal{R}\) be a smooth function. Then, the unconstrained optimization problem is \[\mbox{min}_x \; f
Unconstrained optimization problems arise directly in some applications but they also arise indirectly from reformulations of constrained optimization problems. Often it is practical to replace the
constraints of an optimization problem with penalized terms in the objective function and to solve the problem as an unconstrained problem.
Constrained Optimization
Constrained optimization problems consider the problem of optimizing an objective function subject to constraints on the variables. In general terms,
\[ \begin{array}{lllll}
\mbox{minimize} & f(x) & & & \\
\mbox{subject to} & c_i(x) & = & 0 & \forall i \in \mathcal{E} \\
& c_i(x) & \leq & 0 & \forall i \in \mathcal{I}
\] where \(f\) and the functions \(c_i(x) \,\) are all smooth, real-valued functions on a subset of \(R^n \,\) and \(\mathcal{E}\) and \(\mathcal{I}\) are index sets for equality and inequality
constraints, respectively. The feasible set is the set of points \(x\) that satisfy the constraints.
Optimization Under Uncertainty
The optimization problem types described in the Continuous Optimization section and the Discrete Optimization section implicitly assume that the data for the given problem are known accurately. For
many actual problems, however, the problem data cannot be known accurately for a variety of reasons. The first reason is due to simple measurement error. The second and more fundamental reason is
that some data represent information about the future (e. g., product demand or price for a future time period) and simply cannot be known with certainty.
Stochastic Programming and Robust Optimization are the most popular frameworks for explicitly incorporating uncertainty. Stochastic programming uses random variables with specified probability
distributions to characterize the uncertainty and optimizes the expected value of the objective function. Robust optimization uses set membership to characterize the uncertainty and optimizes a worst
possible case of the problem. | {"url":"https://neos-guide.org/guide/types/","timestamp":"2024-11-03T23:22:03Z","content_type":"text/html","content_length":"82108","record_id":"<urn:uuid:bee1cc96-139e-4361-b9f6-8b0982bf73f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00804.warc.gz"} |
The sum of the interior angles of a polygon is three times the sum o
The sum of the interior angles of a polygon is three times the sum of its exterior angles. Determine the number of sides of the polygon.
Given: The sum of the interior angles of a polygon is three times the sum of its exterior angles.
Sum of all interior angles of a regular polygon=180∘(n−2), where n is number of sides of a polygon
Sum of all exterior angle of a regular polygon=360∘
According to question,180∘(n−2)=3×360∘
Hence, the number of sides of the polygon is 8.
Updated on:21/07/2023
Topper's Solved these Questions
Knowledge Check
• If each interior angle of a regular polygon is 3 times its exterior angle, the number of sides of the polygon is :
• The sum of all the interior angles of a regular polygon is four times the sum of its exterior angles. The polygon is :
• The sum of all the interior angles of a regular polygon is four times the sum of its exterior angles. The polygon is : | {"url":"https://www.doubtnut.com/qna/1536676","timestamp":"2024-11-07T01:08:19Z","content_type":"text/html","content_length":"216932","record_id":"<urn:uuid:7589315e-50e1-48b9-9fe0-12d8e00530f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00747.warc.gz"} |
Salfacorp Stock Current Market Value
SALFACORP CLP 565.00 5.00 0.88%
Salfacorp's market value is the price at which a share of Salfacorp trades on a public exchange. It measures the collective expectations of Salfacorp investors about its performance. Salfacorp is
selling at
as of the 14th of November 2024; that is
0.88% down
since the beginning of the trading day. The stock's open price was
. With this module, you can estimate the performance of a buy and hold strategy of Salfacorp and determine expected loss or profit from investing in Salfacorp over a given investment horizon. Check
Salfacorp Correlation
Salfacorp Volatility
Salfacorp Alpha and Beta
module to complement your research on Salfacorp.
AltmanZ ScoreDetails FinancialAnalysisDetails Buy or SellAdviceDetails TechnicalAnalysisDetails
Please note, there is a significant difference between Salfacorp's value and its price as these two are different measures arrived at by different means. Investors typically determine if Salfacorp is
a good investment by looking at such factors as earnings, sales, fundamental and technical indicators, competition as well as analyst projections. However, Salfacorp's price is the amount at which it
trades on the open market and represents the number that a seller and buyer find agreeable to each party.
Salfacorp 'What if' Analysis
In the world of financial modeling, what-if analysis is part of sensitivity analysis performed to test how changes in assumptions impact individual outputs in a model. When applied to Salfacorp's
stock what-if analysis refers to the analyzing how the change in your past investing horizon will affect the profitability against the current market value of Salfacorp.
If you would invest
in Salfacorp on
October 15, 2024
and sell it all today you would
earn a total of 0.00
from holding Salfacorp or generate
return on investment in Salfacorp over
days. Salfacorp is related to or competes with
Administradora Americana
Energia Latina
Embotelladora Andina
Vina Concha
Multiexport Foods
, and
Sociedad Matriz
. SalfaCorp S.A. engages in engineering and construction, and real estate activities in Chile, Peru, Colombia, Panama, the...
Salfacorp Upside/Downside Indicators
Understanding different market momentum indicators often help investors to time their next move. Potential upside and downside technical ratios enable traders to measure Salfacorp's stock current
market value against overall
market sentiment
and can be a good tool during both bulling and bearish trends. Here we outline some of the essential indicators to assess Salfacorp
upside and downside potential
and time the market with a certain degree of confidence.
Salfacorp Market Risk Indicators
Today, many novice investors tend to focus exclusively on investment returns with little concern for Salfacorp's investment risk. Other traders do consider volatility but use just one or two very
conventional indicators such as Salfacorp's standard deviation. In reality, there are many statistical measures that can use Salfacorp historical prices to predict the future Salfacorp's volatility.
HypePrediction IntrinsicValuation NaiveForecast BollingerBand Projection (param)
Low Estimated High Low Real High Low Next High Lower Middle Band Upper
568.76 570.00 571.24 556.74 557.98 627.00 556.15 557.40 558.64 565.78 574.37 582.97
Details Details Details Details
Please note, it is not enough to conduct a financial or market analysis of a single entity such as Salfacorp. Your research has to be compared to or analyzed against Salfacorp's peers to derive any
actionable benefits. When done correctly, Salfacorp's competitive analysis will give you plenty of quantitative and qualitative data to validate your investment decisions or develop an entirely new
strategy toward taking a position in Salfacorp.
Salfacorp Backtested Returns
As of now, Salfacorp Stock is very steady.
owns Efficiency Ratio (i.e., Sharpe Ratio) of 0.0131, which indicates the firm had a 0.0131% return per unit of risk over the last 3 months. We have found twenty-nine
technical indicators
for Salfacorp, which you can use to evaluate the volatility of the company. Please validate Salfacorp's Risk Adjusted Performance of 0.0862,
coefficient of variation
of 927.52, and Semi Deviation of 0.8728 to confirm if the risk estimate we provide is consistent with the expected return of 0.0162%. Salfacorp has a
performance score
of 1 on a scale of 0 to 100. The entity has a beta of
, which indicates not very significant fluctuations relative to the market. As returns on the market increase, Salfacorp's returns are expected to increase less than the market. However, during the
bear market, the loss of holding Salfacorp is expected to be smaller as well.
right now has a risk of 1.24%. Please validate Salfacorp
value at risk
expected short fall
, and the
between the
treynor ratio
downside variance
, to decide if Salfacorp will be following its existing
price patterns
Good reverse predictability
Salfacorp has good reverse predictability. Overlapping area represents the amount of predictability between Salfacorp time series from 15th of October 2024 to 30th of October 2024 and 30th of October
2024 to 14th of November 2024. The more autocorrelation exist between current time interval and its lagged values, the more accurately you can make projection about the future pattern of Salfacorp
price movement. The serial correlation of -0.51 indicates that about 51.0% of current Salfacorp price fluctuation can be explain by its past prices.
Correlation Coefficient -0.51
Spearman Rank Test -0.72
Residual Average 0.0
Price Variance 13.99
Salfacorp lagged returns against current returns
Autocorrelation, which is Salfacorp stock's lagged correlation, explains the relationship between observations of its time series of returns over different periods of time. The observations are said
to be independent if autocorrelation is zero. Autocorrelation is calculated as a function of mean and variance and can have practical application in
Salfacorp's stock expected returns. We can calculate the autocorrelation of Salfacorp returns to help us make a trade decision. For example, suppose you find that Salfacorp has exhibited high
autocorrelation historically, and you observe that the stock is moving up for the past few days. In that case, you can expect the price movement to match the lagging time series.
Current and Lagged Values
Salfacorp regressed lagged prices vs. current prices
Serial correlation can be approximated by using the Durbin-Watson (DW) test. The correlation can be either positive or negative. If Salfacorp stock is displaying a positive serial correlation,
investors will expect a positive pattern to continue. However, if Salfacorp stock is observed to have a negative serial correlation, investors will generally project negative sentiment on having a
locked-in long position in Salfacorp stock over time.
Salfacorp Lagged Returns
When evaluating Salfacorp's market value, investors can use the concept of autocorrelation to see how much of an impact past prices of Salfacorp stock have on its future price. Salfacorp
autocorrelation represents the degree of similarity between a given time horizon and a lagged version of the same horizon over the previous time interval. In other words, Salfacorp autocorrelation
shows the relationship between Salfacorp stock current value and its past values and can show if there is a momentum factor associated with investing in Salfacorp.
Pair Trading with Salfacorp
One of the main advantages of trading using pair correlations is that every trade hedges away some risk. Because there are two separate transactions required, even if Salfacorp position performs
unexpectedly, the other equity can make up some of the losses. Pair trading also minimizes risk from directional movements in the market. For example, if an entire industry or sector drops because of
unexpected headlines, the short position in Salfacorp will appreciate offsetting losses from the drop in the long position's value.
The ability to find closely correlated positions to Salfacorp could be a great tool in your tax-loss harvesting strategies, allowing investors a quick way to find a similar-enough asset to replace
Salfacorp when you sell it. If you don't do this, your portfolio allocation will be skewed against your target asset allocation. So, investors can't just sell and buy back Salfacorp - that would be a
violation of the tax code under the "wash sale" rule, and this is why you need to find a similar enough asset and use the proceeds from selling Salfacorp to buy it.
The correlation of Salfacorp is a statistical measure of how it moves in relation to other instruments. This measure is expressed in what is known as the correlation coefficient, which ranges between
-1 and +1. A perfect positive correlation (i.e., a correlation coefficient of +1) implies that as Salfacorp moves, either up or down, the other security will move in the same direction.
Alternatively, perfect negative correlation means that if Salfacorp moves in either direction, the perfectly negatively correlated security will move in the opposite direction. If the correlation is
0, the equities are not correlated; they are entirely random. A correlation greater than 0.8 is generally described as strong, whereas a correlation less than 0.5 is generally considered weak.
Correlation analysis
and pair trading evaluation for Salfacorp can also be used as hedging techniques within a particular sector or industry or even over random equities to generate a better risk-adjusted return on your
Pair CorrelationCorrelation Matching
Other Information on Investing in Salfacorp Stock
Salfacorp financial ratios help investors to determine whether Salfacorp Stock is cheap or expensive when compared to a particular measure, such as profits or enterprise value. In other words, they
help investors to determine the cost of investment in Salfacorp with respect to the benefits of owning Salfacorp security. | {"url":"https://www.macroaxis.com/market-value/SALFACORP.SN/100","timestamp":"2024-11-15T04:43:52Z","content_type":"text/html","content_length":"317073","record_id":"<urn:uuid:88043be1-ae8f-443d-9124-e76e8c131a42>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00033.warc.gz"} |
CSE512 – Machine Learning Homework 2
1 Question 1 – Naive Bayes and Logisitic Regression (25 points)
1.1 Naive Bayes with both continuous and boolean variables (10 points)
Consider learning a function X → Y where Y is boolean, where X = (X1, X2), and where X1 is a boolean
variable and X2 a continuous variable. State the parameters that must be estimated to define a Naive Bayes
classifier in this case. Give the formula for computing P(Y |X), in terms of these parameters and the feature
values X1 and X2.
1.2 Naive Bayes and Logistic Regression with Boolean variables (15 points)
In class, we showed that when Y is Boolean and X = (X1, · · · , Xd) is a vector of continuous variables, the
the assumptions of the Gaussian Naive Bayes classifier imply that P(Y |X) is given by the logistic function
with appropriate parameters θ. In particular:
P(Y = 1|X) = 1
1 + exp(−(
i=1 θiXi + θd+1))
Consider instead the case where Y is Boolean and X = (X1, · · · , Xd) is a vector of Boolean variables.
Prove for this case also that P(Y |X) follows this same form (and hence that Logistic Regression is also the
discriminative counterpart to a Naive Bayes generative classifier over Boolean features).
2 Question 2 – Support Vector Machines (15 points)
2.1 Linear case (10 points)
Consider training a linear SVM on linearly separable dataset consisting of n points. Let m be the number of
support vectors obtained by training on the entire set. Show that the LOOCV error is bounded above by m
Hint: Consider two cases: (1) removing a support vector data point and (2) removing a non-support
vector data point.
2.2 General case (5 points)
Now consider the same problem as above. But instead of using a linear SVM, we will use a general kernel.
Assuming that the data is linearly separable in the high dimensional feature space corresponding to the
kernel, does the bound in previous section still hold? Explain why or why not.
3 Question 3 – Implementation of SVMs (40 points)
In this problem, you will implement SVMs using two different optimization techniques:(1) quadratic programming and (2) stochastic gradient descent.
3.1 Implement Kernel SVM using Quadratic Programming (15 points)
Quadratic programs refer to optimization problems in which the objective function is quadratic and the
constraints are linear. Quadratic programs are well studied in optimization literature, and there are many
efficient solvers. Many Machine Learning algorithms are reduced to solving quadratic programs. In this
question, we will use the quadratic program solver of Matlab to optimize the dual objective of a kernel
The dual objective of kernel SVM can be written as:
αj −
, xj ) (2)
s.t. Xn
yjαj = 0 (3)
0 ≤ αj ≤ C ∀j. (4)
1. (5 points) Write the SVM dual objective as a quadratic program. Look at the quadprog function of
Matlab, and write down what H,f, A, b, Aeq, beq, lb, ub are.
2. Use quadratic programming to optimize the dual SVM objective. In Matlab, you can use the function
3. Write a program to compute w and b of the primal from α of the dual. You only need to do this for
linear kernel.
4. (5 points) Set C = 0.1, train an SVM with linear kernel using trD, trLb in q3 1 data.mat (in
Matlab, load the data using load q3 1 data.mat). Test the obtained SVM on valD, valLb,
and report the accuracy, the objective value of SVM, the number of support vectors, and the confusion
5. (5 points) Repeat the above question with C = 100.
3.2 Implement Linear SVM using Stochastic Gradient Descent (25 points)
The objective of a linear SVM can be written as
2 + C
L(w, b, xj , yj ) (5)
Here l(w, b, xj , yj ) is the Hinge loss of the j-th instance:
L(w, b, xj , yj ) = max
1 − yj (wT xj + b), 0
By distributing the regularization term to each training instance, we obtain the following equivalent objective:
||w||2 + CL(w, b, xj , yj )
Let Lj =
||w||2 + CL(w, b, xj , yj ). We can use stochastic gradient descent to optimize this objective.
The update rule for w and b with the j-th instance will be
wnew ← wcur − η∂wLj (7)
new ← b
cur − η∂bLj (8)
where ∂wLj , ∂bLj denote the sub-gradients of Lj w.r.t. w and b respectively.
Algorithm 1: Stochastic gradient descent for linear SVM
for epoch = 1, 2, · · · , max epoch do
η ← η0/(η1 + epoch) % Update the learning rate
(j1, · · · , jn) = permute(1, · · · , n). % Shuffle the indexes of training data
for k ∈ {1, 2, · · · , n} do
j ← jk
Update w, b using Eqs. (7) & (8)
1. (5 points) Write the stochastic gradient descent update rules for both w and b in linear SVMs.
2. Implement SGD for linear SVMs given in Algorithm 1. η0, η1 are tunable parameters. Initially start
all the weights at 0.
3. (5 points) Using trD, trLb in q3 1 data.mat as your training set, run 1000 epochs over the
dataset using η0 = 1, η1 = 100 and C = 0.1. Plot the loss in Eq. (5) after each epoch. Compare with
the objective value obtained in 3.1.4.
4. (5 points) Using the w, b learned after 1000 epoches, report:
(a) The prediction error on valD, valLb in q3 1 data.mat (test error)
(b) The prediction error on trD, trLb (training error)
(c) kwk
5. (10 points) Change C to 100 and repeat what you did in the previous two questions. You can tune and
use different values of η0, η1. Report the values of η0, η1 that you used in your answer.
4 Question 4 – SVM for object detection (20 points + 10 bonus points)
In this question, you will train a SVM and use it for detecting human upper bodies in your favorite TV series
The Big Bang Theory. You must you your SVM implementation in either Question 3.1 or 3.2.
To detect human upper bodies in images, we need a classifier that can distinguish between upper-body
image patches from non-upper-body patches. To train such a classifier, we can use SVMs. The training
data is typically a set of images with bounding boxes of the upper bodies. Positive training examples are
image patches extracted at the annotated locations. A negative training example can be any image patch that
does not significantly overlap with the annotated upper bodies. Thus there potentially many more negative
training examples than positive training examples. Due to memory limitation, it will not be possible to use
all negative training examples at the same time. In this question, you will implement hard-negative mining
to find hardest negative examples and iteratively train an SVM.
4.1 Data
Training images are provided in the subdirectory trainIms. The annotated locations of the upper bodies
are given in trainAnno.mat. This file contains a cell structure ubAnno; ubAnno{i} is the annotated
locations of the upper bodies in the i
th image. ubAnno{i} is 4×k matrix, where each column corresponds
to an upper body. The rows encode the left, top, right, bottom coordinates of the upper bodies (the origin of
the image coordinate is at the top left corner).
Images for validation and test are given in valIms, testIms respectively. The annotation file for
test images is not released. We have also extracted some image regions of test images, and the regions are
saved as 64×64 jpeg images in testRegs. Only small portion of these images correspond to upper bodies.
4.2 External library
Raw image intensity values are not robust features for classification. In this question, we will use Histogram
of Oriented Gradient (HOG) as image features. HOG uses the gradient information instead of intensities,
and this is more robust to changes in color and illumination conditions. See [1] for more information about
HOG, but it is not required for this assignment.
To use HOG, you will need to install an VL FEAT: http://www.vlfeat.org. This is an excellent
cross-platform library for computer vision and machine learning. However, in this homework, you are only
allowed to use the HOG calculation and visualization function vl hog. In fact, you should not call vl hog
directly. Use the supplied helper functions instead; they will call vl hog.
4.3 Helper functions
To help you, a number of utility functions and classes are provided. The most important functions are in
HW2 Utils.m.
1. Run HW2 Utils.demo1 to see how to read and display upper body annotation
2. Run HW2 Utils.demo2 to display image patches and HOG feature images. Compare HOG features
for positive and negative examples, can you see why HOG would be useful for detect upper bodies?
3. Use HW2 Utils.getPosAndRandomNeg() to get initial training and validation data. Positive
instances are HOG features extracted at the locations of upper bodies. Negative instances are HOG
features at random locations of the images. The data used in Question 3 is actually generated using
this function.
4. Use HW2 Utils.detect to run the sliding window detector. This returns a list of locations and
SVM scores. This function can be used for detecting upper bodies in an image. It can also be used to
find hardest negative examples in an image.
5. Use HW2 Utils.cmpFeat to compute HOG feature vector for an image patch.
6. Use HW2 Utils.genRsltFile to generate result file.
7. Use HW2 Utils.cmpAP to compute the Average Precision for the result file.
8. Use HW2 Utils.rectOverlap to compute the overlap between two rectangular regions. The
overlap is defined as the area of the intersection over the area of the union. A returned detection region
is considered correct (true positive) if there is an annotated upper body such that the overlap between
the two boxes is more than 0.5.
9. Some useful Matlab functions to work with images are: imread, imwrite, imshow, rgb2gray, imresize.
4.4 What to implement?
1. (5 points) Use the training data in HW2 Utils.getPosAndRandomNeg() to train an SVM classifier. Use this classifier to generate a result file (use HW2 Utils.genRsltFile) for validation data.
Use HW2 Utils.cmpAP to compute the AP and plot the precision recall curve. Submit your AP and
precision recall curve (on validation data).
2. Implement hard negative mining algorithm given in Algorithm 2. Positive training data and random
negative training data can be generated using HW2 Utils.getPosAndRandomNeg(). At each
iteration, you should remove negative examples that do not correspond to support vectors from the
negative set. Use the function HW2 Utils.detect on train images to identify hardest negative
Algorithm 2: Hard negative mining algorithm
P osD ← all annotated upper bodies
NegD ← random image patches
(w, b) ← trainSVM(P osD, NegD)
for iter = 1, 2, · · · do
A ← All non support vectors in NegD.
B ← Hardest negative examples % Run UB detection and find negative patches that violate
% the SVM margin constraint the most
NegD ← (NegD \ A) ∪ B.
(w, b) ← trainSVM(P osD, NegD)
examples and include them in the negative training set. Use HW2 Utils.cmpFeat to compute
HOG feature vectors.
Hints: (1) a negative example should not have significant overlap with any annotated upper body. You
can experiment with different threshold but 0.3 is a good starting point. (2) make sure you normalize
the feature vectors for new negative examples. (3) you should compute the objective value at each
iteration; the objective values should not decrease.
3. (10 points) Run the negative mining for 10 iterations. Assume your computer is not so powerful and so
you cannot add more than 1000 new negative training examples at each iteration. Record the objective
values (on train data) and the APs (on validation data) through the iterations. Plot the objective values.
Plot the APs.
4. (5 points) For this question, you will need to generate a result file for test data using the function
HW2 Utils.genRsltFile. You will need to submit this file to our evaluation sever (https:
//goo.gl/XZuD1x) to receive the AP on test data. Report the AP in your answer file. Important
Note: You MUST use your Stony Brook ID to name your submission file, i.e., your SBU ID.mat
(e.g., 012345679.mat). Your submission will not be evaluated if you don’t use your SBU ID.
5. (10 bonus points) Your submitted result file for test data will be automatically entered a competition
for fame. We will maintain a leader board at https://goo.gl/6pT61E, and the top three entries
at the end of the competition (due date) will receive 10 bonus points. The ranking is based on AP.
You can submit the result as frequent as you want. However, the evaluation server will only evaluate
all submissions three times a day, at 11:00am, 5:00pm, and 11:00pm. The system only keeps the recent
submission file, and your new submission will override the previous ones. Therefore, you have three
chances a day to evaluate your method. The leader board will be updated in 30 minutes after every
You are allowed to use any feature types for this part of the homework. For example, you can use
different parameter settings for HOG feature computation. You can even combine multiple HOG
features. You can also append HOG features with geometric features (e.g., think about the locations
of the upper body). You are allowed to perform different types of feature normalization (e.g, L1,
L2). You can use both training and validation data to train your classifier. You are allowed to use
SVMs, Ridge Regression, Lasso Regression, or any technique that we have covered. You can run hard
negative mining algorithm for as many iterations as you want, and the number of negative examples
added at each iteration is not limited by 1000.
5 What to submit?
5.1 Blackboard submission
You will need to submit both your code and your answers to questions on Blackboard. Do not submit the
provided data. Put the answer file and your code in a folder named: SUBID FirstName LastName (e.g.,
10947XXXX heeyoung kwon). Zip this folder and submit the zip file on Blackboard. Your submission must
be a zip file, i.e, SUBID FirstName LastName.zip. The answer file should be named: answers.pdf, and it
should contain:
1. Answers to Question 1 and 2
2. Answers to Question 3.1 and 3.2, including the requested plots.
3. Answers to Question 4.3, including the requested plots.
5.2 Prediction submission
For Questions 4.4.4, 4.4.5 you must submit a .mat file to get the AP through https://goo.gl/XZuD1x.
A submission file can be automatically generated by HW2 Utils.genRsltFile.
6 Cheating warnings
Don’t cheat. You must do the homework yourself, otherwise you won’t learn. You must use your SBU ID
as your file name for the competition. Do not fake your Stony Brook ID to bypass the submission limitation
per 24 hours. Doing so will be considered cheating.
References Cited
[1] N. Dalal and B. Triggs. Histograms of oriented gradients for human detection. In Proceedings of the
IEEE Conference on Computer Vision and Pattern Recognition, 2005. | {"url":"https://codingprolab.com/answer/cse512-machine-learning-homework-2-2/","timestamp":"2024-11-12T02:54:31Z","content_type":"text/html","content_length":"133415","record_id":"<urn:uuid:eeaf1d00-d9f2-4a86-a457-b3636beb64b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00681.warc.gz"} |
nphparams: Simultaneous Inference For Parameters Quantifying Differences... in nph: Planning and Analysing Survival Studies under Non-Proportional Hazards
Hypothesis tests with parametric multiple testing adjustment and simultaneous confidence intervals for a set of parameters, which quantify differences between two survival functions. Eligible
parameters are differences in survival probabilities, log survival probabilities, complementary log log (cloglog) transformed survival probabilities, quantiles of the survival functions, log
transformed quantiles, restricted mean survival times, as well as an average hazard ratio, the Cox model score statistic (logrank statistic), and the Cox-model hazard ratio.
nphparams( time, event, group, data = parent.frame(), param_type, param_par = NA, param_alternative = NA, lvl = 0.95, closed_test = FALSE, alternative_test = "two.sided", alternative_CI =
"two.sided", haz_method = "local", rhs = 0, perturb = FALSE, Kpert = 500 )
time vector of observed event/censored times.
event Vector with entries 0 or 1 (or FALSE/TRUE) indicating if an event was observed (1) or the time is censored (0).
group group indicator, must be a vector with entries 0 or 1 indicating the allocation of a subject to one of two groups. Group 0 is regarded as reference group when calculating
data an optional data frame containing the time, event and group data.
param_type character vector defining the set of parameters that should be analysed. Possible entries are "S","logS","cloglogS", "Q","logQ","RMST","avgHR","score" and "HR", representing
differences in survival probabilities, log survival probabilities, complementary log log (cloglog) transformed survival probabilities, quantiles of the survival functions, log
transformed quantiles, restricted mean survival times, as well as an average hazard ratio, the Cox model score statistic (logrank statistic), and the Cox-model hazard ratio.
param_par numeric vector which contains the time points at which the requested parameters are evaluated (e.g. x-year survival or RMST after x-years), or, in case of analysing quantiles, the
according probability. May be NA for parameter types "RMST","avgHR","score" or "HR". In this case, the minimum of the largest event times of the two groups is used. Also, times
greater than this minimum are replaced by this minumum for "RMST","avgHR","score" or "HR".
param_alternative optional character vector with entries "less" or "greater", defining the alternative for each parameter. Only required if one-sided tests or one-sided confidence intervals are
requested. Note that group 0 is regarded as reference group when calculating parameters and therefore whether "greater" or "less" corresponds to a benefit may depend on the type of
parameter. In general, to show larger survival in group 1 compared to group 0, alternatives would be "greater" for parameter types "S", "logS", "Q", "logQ" and "RMST" and would be
"less" for parameters types "cloglogS", "avgHR","HR", and "score". (The score test is defined here such that alternative "less" corresponds to smaller hazard (and better survival)
in group 1 compared to group 0.)
lvl Confidence level. Applies to, both, unadjusted and multiplicity adjusted (simultaneous) confidence intervals.
closed_test logical indicating whether p-values should be adjusted using a closed testing procedure. Default is FALSE, and in this case p-values will be adjusted by a single step procedure.
With k hypotheses this involves the computation of 2^k tests, which may require considerable computation time.
alternative_test character with possible values "tow.sided" (default) or "one-sided". Specifies whether hypothesis tests should be two-sided or one-sided. In the #' latter case, param_alternative
must be defined.
alternative_CI character with possible values "tow.sided" (default) or "one-sided". Specifies whether confidence intervals should be two-sided or one-sided. In the latter case, param_alternative
must be defined.
haz_method character with possible values "local" or "muhaz". Specifies whether local hazard should be calculated under a local constant hazard assumption (default) #' or using the function
muhaz from the muhaz package. Only relevant when median or log(median) survival times are analysed.
rhs right-hand side vector of null hypotheses. Refers to log-scaled difference for ratios. Default is to consider for all null hypothesis a difference of 0.
perturb logical, indicating whether the perturbation based estiamte should be used instead of the asymptotic estimate to calculate the covariance matrix. Defaults to FALSE.
Kpert The number of perturbation samples to be used with the perturbation approach for covariance estimation.
Vector with entries 0 or 1 (or FALSE/TRUE) indicating if an event was observed (1) or the time is censored (0).
group indicator, must be a vector with entries 0 or 1 indicating the allocation of a subject to one of two groups. Group 0 is regarded as reference group when calculating parameters.
an optional data frame containing the time, event and group data.
character vector defining the set of parameters that should be analysed. Possible entries are "S","logS","cloglogS", "Q","logQ","RMST","avgHR","score" and "HR", representing differences in survival
probabilities, log survival probabilities, complementary log log (cloglog) transformed survival probabilities, quantiles of the survival functions, log transformed quantiles, restricted mean survival
times, as well as an average hazard ratio, the Cox model score statistic (logrank statistic), and the Cox-model hazard ratio.
numeric vector which contains the time points at which the requested parameters are evaluated (e.g. x-year survival or RMST after x-years), or, in case of analysing quantiles, the according
probability. May be NA for parameter types "RMST","avgHR","score" or "HR". In this case, the minimum of the largest event times of the two groups is used. Also, times greater than this minimum are
replaced by this minumum for "RMST","avgHR","score" or "HR".
optional character vector with entries "less" or "greater", defining the alternative for each parameter. Only required if one-sided tests or one-sided confidence intervals are requested. Note that
group 0 is regarded as reference group when calculating parameters and therefore whether "greater" or "less" corresponds to a benefit may depend on the type of parameter. In general, to show larger
survival in group 1 compared to group 0, alternatives would be "greater" for parameter types "S", "logS", "Q", "logQ" and "RMST" and would be "less" for parameters types "cloglogS", "avgHR","HR", and
"score". (The score test is defined here such that alternative "less" corresponds to smaller hazard (and better survival) in group 1 compared to group 0.)
Confidence level. Applies to, both, unadjusted and multiplicity adjusted (simultaneous) confidence intervals.
logical indicating whether p-values should be adjusted using a closed testing procedure. Default is FALSE, and in this case p-values will be adjusted by a single step procedure. With k hypotheses
this involves the computation of 2^k tests, which may require considerable computation time.
character with possible values "tow.sided" (default) or "one-sided". Specifies whether hypothesis tests should be two-sided or one-sided. In the #' latter case, param_alternative must be defined.
character with possible values "tow.sided" (default) or "one-sided". Specifies whether confidence intervals should be two-sided or one-sided. In the latter case, param_alternative must be defined.
character with possible values "local" or "muhaz". Specifies whether local hazard should be calculated under a local constant hazard assumption (default) #' or using the function muhaz from the muhaz
package. Only relevant when median or log(median) survival times are analysed.
right-hand side vector of null hypotheses. Refers to log-scaled difference for ratios. Default is to consider for all null hypothesis a difference of 0.
logical, indicating whether the perturbation based estiamte should be used instead of the asymptotic estimate to calculate the covariance matrix. Defaults to FALSE.
The number of perturbation samples to be used with the perturbation approach for covariance estimation.
A data frame with analysis results. Contains the parameter type (Parameter) and settings (Time_or_which_quantile), the estimated difference (Estimate), its standard error (SE), unadjusted confidence
interval lower and upper bounds (lwr_unadjusted, upr_unadjusted), unadjusted p-values (p_unadj), mulitplicity adjusted confidence interval lower and upper bounds (lwr_adjusted, upr_adjusted),
single-step multiplcity adjusted p-values (p_adj), closed-test adjusted p-values, if requested (p_adjusted_closed_test) and for comparison Bonferroni-Holm adjusted p-values (p_Holm).
The used parameter settings. If param_par was NA for "HR","avgHR" or "RMST", it is replaced by minmaxt here.
The parameter settings as provided to the function. The only difference to param is in param_par, as NA is not replaced here.
A data frame with information on all observed events in group 0. Contains time (t), number of events (ev), Nelson-Aalen estimate (NAsurv) and Kaplan-Meier estimate (KMsurv) of survival, and the
number at risk (atrisk).
A data frame with information on all observed events in group 1. Contains time (t), number of events (ev), Nelson-Aalen estimate (NAsurv) and Kaplan-Meier estimate (KMsurv) of survival, and the
number at risk (atrisk).
Minimum of the largest event times of the two groups.
data(pembro) set1<-nphparams(time=time, event=event, group=group,data=pembro, param_type=c("score","S"), param_par=c(3.5,2), param_alternative=c("less","greater"), closed_test=TRUE,alternative_test=
"one.sided") print(set1) plot(set1,trt_name="Pembrolizumab",ctr_name="Cetuximab") set2<-nphparams(time=time, event=event, group=group, data=pembro, param_type=c("S","S","S","Q","RMST"), param_par=c
(0.5,1,2,0.5,3.5)) print(set2) plot(set2,showlines=TRUE,show_rmst_diff=TRUE) #Create a summary table for set2, showing parameter estimates for each group and the #estimated differences between
groups. Also show unadjusted and multiplicity adjusted #confidence intervals using the multivariate normal method and, for comparison, #Bonferroni adjusted confidence intervals: set2Bonf<-nphparams
(time=time, event=event, group=group, data=pembro, param_type=c("S","S","S","Q","RMST"), param_par=c(0.5,1,2,0.5,3.5), lvl=1-0.05/5) KI_paste<-function(x,r) { x<-round(x,r) paste("[",x[,1],", ",x
[,2],"]",sep="") } r<-3 tab<-data.frame( Parameter=paste(set2$tab[,1],set2$tab[,2]), Pembrolizumab=round(set2$est1,r), Cetuximab=round(set2$est0,r), Difference=round(set2$tab$Estimate,r), CI_undadj=
KI_paste(set2$tab[,5:6],r), CI_adj=KI_paste(set2$tab[,8:9],r), CI_Bonf=KI_paste(set2Bonf$tab[,c(5:6)],r)) tab
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/nph/man/nphparams.html","timestamp":"2024-11-12T14:02:01Z","content_type":"text/html","content_length":"33684","record_id":"<urn:uuid:8325e7a6-7c8b-4fc3-8beb-ec08767dae96>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00733.warc.gz"} |
Image Manipulation Using Mathematica
Jessica Murdock
MATH 2270 Spring 2016
Image Manipulation Using Mathematica
Wolfram Alpha’s programming language can be used for various purposes, including changing and creating images that can later be exported to your file system. This is possible largely through the use
of vectors, matrices and arrays to represent the different aspects of the image. For example, given the image below, the dimensions can be given as a vector.
A matrix or a multidimensional array can also be used to represent the colors of each individual pixel in this image:
In the first set of matrices above, the rows of each matrix correspond to pixels 15 through 20 from the left edge of the image, and the columns correspond to pixels 10 through 12 from the bottom edge
of the image. The last matrix represents the highest value possible for each color channel (Red, Green, and Blue) in each pixel: 255. The other three matrices represent the actual values of each
color channel for each pixel. The pixel 15 pixels from the left and 10 pixels from the bottom, for example, has a value of 211 for red coloring, 199, for green, and 179 for blue.
In the second set of matrices, the last column of each matrix show the maximum value possible in each entry, while the other columns represent RGB values. The first matrix shows the values of pixels
10 through 12 from the bottom and 15 pixels from the left, the second shows pixels 10 through 12 from the bottom and 16 pixels from the left, and so on. These two sets of matrices are different ways
of representing the colors of each specified pixel in the image.
Mathematica can take this representation further by separating an image according to color channels. In the example below, only red is pulled out of the image.
However, if a digital RGB image like this one needed to be printed, it would need to be converted to CMYK colors, as printers don’t use red, blue, and green ink:
The new image looks almost exactly the same, but can be separated into cyan, magenta, yellow, and black values for a printer to read.
Works Cited
"Color Processing." —Wolfram Language Documentation. N.p., n.d. Web. 31 Mar. 2016. .
"ColorConvert." —Wolfram Language Documentation. N.p., n.d. Web. 29 Mar. 2016.
"ColorSeparate." —Wolfram Language Documentation. N.p., n.d. Web. 30 Mar. 2016.
"EasyBib: The Free Automatic Bibliography Composer." EasyBib. N.p., n.d. Web. 30 Mar. 2016. .
"Image Processing." —Wolfram Language Documentation. N.p., n.d. Web. 30 Mar. 2016. .
"Image Representation." —Wolfram Language Documentation. N.p., n.d. Web. 31 Mar. 2016. .
"ImageData." —Wolfram Language Documentation. N.p., n.d. Web. 31 Mar. 2016. .
"ImageLines." —Wolfram Language Documentation. N.p., n.d. Web. 31 Mar. 2016. .
"MatrixPlot." —Wolfram Language Documentation. N.p., n.d. Web. 31 Mar. 2016. .
"RGBColor." —Wolfram Language Documentation. N.p., n.d. Web. 29 Mar. 2016.
Wolfram, Stephen. An Elementary Introduction to the Wolfram Language. N.p.: n.p., n.d. Preface: Elementary Introduction to the Wolfram Language. Web. 30 Mar. 2016. .
10.15 Kb.
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Use of soft computing techniques in renewable energy hydrogen hybrid systems
Soft computing techniques are important tools that significantly improve the performance of energy systems. This chapter reviews their many contributions to renewable energy hydrogen hybrid systems,
namely those systems that consist of different technologies (photovoltaic and wind, electrolyzers, fuel cells, hydrogen storage, piping, thermal and electrical/electronic control systems) capable as
a whole of converting solar energy, storing it as chemical energy (in the form of hydrogen) and turning it back into electrical and thermal energy. Fuzzy logic decision-making methodologies can be
applied to select amongst renewable energy alternative or to vary a dump load for regulating wind turbine speed or find the maximum power point available from arrays of photovoltaic modules. Dynamic
fuzzy logic controllers can furthermore be utilized to coordinate the flow of hydrogen to fuel cells or employed for frequency control in micro- grid power systems. Neural networks are implemented to
model, design and control renewable energy systems and to estimate climatic data such as solar irradiance and wind speeds. They have been demonstrated to predict with good accuracy system power usage
and status at any point of time. Neural controls can also help in the minimization of energy production costs by optimal scheduling of power units. Genetic or evolutionary algorithms are able to
provide approximate solutions to several complex tasks with high number of variables and non-linearities, like optimal operational strategy of a grid-parallel fuel cell power plant, optimization of
control strategies for stand-alone renewable systems and sizing of photovoltaic systems. Particle swarm optimization techniques are applied to find optimal sizing of system components in an effort to
minimize costs or coping with system failures to improve service quality. These techniques can also be implemented together to exploit their potential synergies while, at the same time, coping with
their possible limitations. This chapter covers soft computing methods applied to renewable energy hybrid hydrogen systems by providing a description of their single or mixed implementation and
relevance, together with a discussion of advantages and/or disadvantages in their applications. © Springer-Verlag Berlin Heidelberg 2011.
Use of soft computing techniques in renewable energy hydrogen hybrid systems / Zini, Gabriele; Pedrazzi, Simone; Tartarini, Paolo. - 269:(2011), pp. 37-64. [10.1007/978-3-642-22176-7_2]
Use of soft computing techniques in renewable energy hydrogen hybrid systems
Soft computing techniques are important tools that significantly improve the performance of energy systems. This chapter reviews their many contributions to renewable energy hydrogen hybrid systems,
namely those systems that consist of different technologies (photovoltaic and wind, electrolyzers, fuel cells, hydrogen storage, piping, thermal and electrical/electronic control systems) capable as
a whole of converting solar energy, storing it as chemical energy (in the form of hydrogen) and turning it back into electrical and thermal energy. Fuzzy logic decision-making methodologies can be
applied to select amongst renewable energy alternative or to vary a dump load for regulating wind turbine speed or find the maximum power point available from arrays of photovoltaic modules. Dynamic
fuzzy logic controllers can furthermore be utilized to coordinate the flow of hydrogen to fuel cells or employed for frequency control in micro- grid power systems. Neural networks are implemented to
model, design and control renewable energy systems and to estimate climatic data such as solar irradiance and wind speeds. They have been demonstrated to predict with good accuracy system power usage
and status at any point of time. Neural controls can also help in the minimization of energy production costs by optimal scheduling of power units. Genetic or evolutionary algorithms are able to
provide approximate solutions to several complex tasks with high number of variables and non-linearities, like optimal operational strategy of a grid-parallel fuel cell power plant, optimization of
control strategies for stand-alone renewable systems and sizing of photovoltaic systems. Particle swarm optimization techniques are applied to find optimal sizing of system components in an effort to
minimize costs or coping with system failures to improve service quality. These techniques can also be implemented together to exploit their potential synergies while, at the same time, coping with
their possible limitations. This chapter covers soft computing methods applied to renewable energy hybrid hydrogen systems by providing a description of their single or mixed implementation and
relevance, together with a discussion of advantages and/or disadvantages in their applications. © Springer-Verlag Berlin Heidelberg 2011.
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Microbes can provide sustainable hydrocarbons for the petrochemical industry
This is one procedure for getting rid of the carboxyl and making an olefin:
Direct conversion of carboxylic acids (Cn) to alkenes (C2n − 1) over titanium oxide in absence of noble metals
Sucrose is basically a glucose molecule tacked together with a fructose molecule. Fructose, in turn, is only mildly different from glucose. It might be possible to extract the sucrose from a living
tree (without killing it), making it technically possible to 'grow' an 'oil well' in your back yard (or garden, for the Brits). This presumes you aren't living in a high-rise.
One would need about 24 pounds of sugar to make 6 pounds (one gallon) of gasoline. Wood is made up of cellulose. Cellulose is a polymer of starches. Starches are a polymer of glucose. Cellulose can
be broken down into sugars with various common acids, including sulfuric and hydrochloric.
KeyboardWarrior + 527
My microbiology prof likes to go on about schemes like this. Reality is that microbial processes will never be commercialized because of scale and efficiency issues. Few exceptions include ethanol
Nitrogen fixation by microbes has been all the rage in this field, but the industrial routes already beat the race to the practical efficiency limit. The same applies for organic synthesis.
• 2
• 1
Biofuels from various waste streams are quite successful and should continue to be. They have a green claim and they also dispose of a waste product.
Biogas https://docs.google.com/document/d/1N-TLMeHsKYBCirxS0vbqMGHpU2SmyLuCc7bqp8eYXVM/edit
7 minutes ago, ronwagn said:
Biofuels from various waste streams are quite successful and should continue to be. They have a green claim and they also dispose of a waste product.
Biogas https://docs.google.com/document/d/1N-TLMeHsKYBCirxS0vbqMGHpU2SmyLuCc7bqp8eYXVM/edit
These schemes are typically fraudulent, because the microbes they use come from gut flora of conventional livestock. There is barely anything left for those to extract from human or livestock "waste
streams" effectively produced by their "colleagues". To do better, they need to look into digestion of things like vultures and dung beetles and awful genetically engineered things. Don't think
anybody is doing it for real. Till that happens, does biogas production really tap into fodder streams for livestock, like perfectly fine hey, instead of real waste.
Please look at my topic and get back to me Andrei. Biogas is a big deal. There is a gap in your knowledge. Biogas is only one biofuel too. Surprised to find a gap, you are a very well informed and
smart person.
29 minutes ago, ronwagn said:
Please look at my topic and get back to me Andrei. Biogas is a big deal. There is a gap in your knowledge. Biogas is only one biofuel too. Surprised to find a gap, you are a very well informed
and smart person.
No gap. The nail in the coffin of the "1st generation" liquid biofuels was hammered by this
It can produce something that is indistinguishable from mineral diesel (if you want low temperature resistance) or something like regular biodiesel (if you want the extra oxygen to combat smog) The
feedstock is absolutely any fatty substance, which in practice means the usual palm oil.
6 hours ago, KeyboardWarrior said:
My microbiology prof likes to go on about schemes like this. Reality is that microbial processes will never be commercialized because of scale and efficiency issues. Few exceptions include
ethanol production.
Not convinced about ethanol either. Plants used for massive production thereof (corn, sugarcane, etc), tend to generate massive amounts of straw-like cellulose-rich waste, which takes a lot longer
than one season to "biodegrade" (rot) Therefore, the only practical means of getting rid of it quick is to burn it.
6 minutes ago, Andrei Moutchkine said:
No gap. The nail in the coffin of the "1st generation" liquid biofuels was hammered by this
It can produce something that is indistinguishable from mineral diesel (if you want low temperature resistance) or something like regular biodiesel (if you want the extra oxygen to combat smog)
The feedstock is absolutely any fatty substance, which in practice means the usual palm oil.
That is only one approach and is "unsustainable." So they say. Biogas is sustainable and acceptable. It allows palm oil to remain as a cheap food product, the residue after use can be used to make
biogas or fertilizer. Biogas uses waste streams which are abundant and thereby helps clean up the environment.
3 minutes ago, Andrei Moutchkine said:
Not convinced about ethanol either. Plants used for massive production thereof (corn, sugarcane, etc), tend to generate massive amounts of straw-like cellulose-rich waste, which takes a lot
longer than one season to "biodegrade" (rot) Therefore, the only practical means of getting rid of it quick is to burn it.
The residue is mainly high quality protein which is used as a very saleable feed component. You may be referring to some residue I am unaware of but even the corn cobs are sometimes ground and used
for certain purposes.
3 minutes ago, ronwagn said:
That is only one approach and is "unsustainable." So they say. Biogas is sustainable and acceptable. It allows palm oil to remain as a cheap food product, the residue after use can be used to
make biogas or fertilizer. Biogas uses waste streams which are abundant and thereby helps clean up the environment.
Nothing unsustainable about the process itself. If only weren't oil and gas companies who license it so cheap and went after the cheapest feedstock they could find. Which happens to be palm oil right
Biogas does not really use "waste streams" like you think they do. In reality, do they use misc. plant matter that you could also feed to livestock. The German nickname for a biogas digester plant is
Betonkuh (concrete cow) It really is an emulation of ruminant digestion system at the moment. Possibly it may eat what pigs would, but that's about it. So, there is no way to get it to digest what a
similar process inside a real cow or pig already did. I presume that to be the basis of the whole "cow fart greenhouse effect" scare.
turbguy + 1,537
22 minutes ago, Andrei Moutchkine said:
No gap. The nail in the coffin of the "1st generation" liquid biofuels was hammered by this
It can produce something that is indistinguishable from mineral diesel (if you want low temperature resistance) or something like regular biodiesel (if you want the extra oxygen to combat smog)
The feedstock is absolutely any fatty substance, which in practice means the usual palm oil.
Soylent Green, anyone??
2 minutes ago, Andrei Moutchkine said:
Nothing unsustainable about the process itself. If only weren't oil and gas companies who license it so cheap and went after the cheapest feedstock they could find. Which happens to be palm oil
right now.
Biogas does not really use "waste streams" like you think they do. In reality, do they use misc. plant matter that you could also feed to livestock. The German nickname for a biogas digester
plant is Betonkuh (concrete cow) It really is an emulation of ruminant digestion system at the moment. Possibly it may eat what pigs would, but that's about it. So, there is no way to get it to
digest what a similar process inside a real cow or pig already did. I presume that to be the basis of the whole "cow fart greenhouse effect" scare.
11 minutes ago, ronwagn said:
The residue is mainly high quality protein which is used as a very saleable feed component. You may be referring to some residue I am unaware of but even the corn cobs are sometimes ground and
used for certain purposes.
What about the stalks? The dirty little secret left behind by Brasil's ethanol fuel success story is called
They do burn most of it. (Contrary to what Wiki claims, are extra tall & tough grasses tend to be crap feedstocks for pulp, due to high silicate content. Culminating with mother of all grasses -
bamboo. Which is effectively covered by a nanoscale layer of quartz glass, a mineral harder than any steel, which destroys conventional woodchipping machinery. The Chinese have a special research
institute dedicated to the problem of pulping bamboo)
Part of bamboo fields have always been burned, to my knowledge. Sugar is the main product and is what is used to make ethanol in parts of South America. It is very cost effective for that purpose and
is often used in automobiles. American ethanol is roughly the same mpg as gasoline in appropriate engines but is mainly used as 10% additive which boosts octane safely.
39 minutes ago, ronwagn said:
European biogas? Entirely fraudulent. They are tapping into the livestock fodder, which is very plentiful. Up to a point, where the meat prices are going to go up. I am all for outsourcing the meat
production to Argentines, who do it right, but the free trade deal between the EU and Mercasur appears to be on the rocks right now.
You can allegedly raise the output by throwing the leftover glycerin left over from conventional biodiesel production into the reactor, but EU does not even allow that, because it does not consider
it a renewable resource! This is based on observing actual operational biogas plants and talking to their operators from before 2008. On the other hand, the European glycerin glut appears to still be
with us, so nothing changed. (Conventional biodiesel production removes the glycerine skeleton all veggie- and animal-sourced oil have (these are called triglycerides; and replaces them with
methanol, which tends to be sourced from good old Russian natural gas. So, is traditional European 1st gen biofuel actually 11% "fossil" by weight and produces an equivalent amount of glycerine
byproduct nobody wants)
You will be very hard pressed to find any technical merit to any of the Eurofag renewable fuel activity. The only purpose of it is to "carbon tax" the heck out of "emerging economies" for being
dirty, while changing jack. Neocolonialism at work. Neste process is a major exception here, BTW. The original Finish renewable plan was to burn peat, which the rest of EU vehemently objected to
Edited by Andrei Moutchkine
14 minutes ago, ronwagn said:
Part of bamboo fields have always been burned, to my knowledge. Sugar is the main product and is what is used to make ethanol in parts of South America. It is very cost effective for that purpose
and is often used in automobiles. American ethanol is roughly the same mpg as gasoline in appropriate engines but is mainly used as 10% additive which boosts octane safely.
The effect of burning all this straw on carbon credits appears to be TBD, as the drivers of "energy transition" tend to be the "1st world" economies unfamiliar with either bamboo or sugar cane
Yes and no. Higher octane number makes fuel actually less energetic (it adds "anti-knock" properties, i.e. suppresses detonations) The cetane numbers for diesel-like fuels work in the opposite
direction. Higher cetane means more energetic and volatile fuel, but also likely more detonation prone.
In order to get back the power you lose, you can run the engine with higher compression, which results in a more expensive car. Think the "funny fuel" American quarter mile racing uses. That stuff is
the same small RC racing cars use. Mostly methanol, with a bit of nitromethane mixed in. Equivalent to something like 130? octane gas.
The primary benefit of adding a bit of ethanol to gasoline is combating smog in places like LA (so it's mandatory for California) For diesel fuel, adding some plant-based biodiesel has the same
effect. Both admixtures work on the principle of carrying a bit of additional oxidizer into the mix.
33 minutes ago, ronwagn said:
Part of bamboo fields have always been burned, to my knowledge. Sugar is the main product and is what is used to make ethanol in parts of South America. It is very cost effective for that purpose
and is often used in automobiles. American ethanol is roughly the same mpg as gasoline in appropriate engines but is mainly used as 10% additive which boosts octane safely.
Aha, found the secret keyword for your nasty straw
2 hours ago, ronwagn said:
Biogas is sustainable and acceptable.
In US, there seems to be an upside to Brandon passing wind. He appears to be accelerating!
Edited by Andrei Moutchkine
10 hours ago, turbguy said:
Soylent Green, anyone??
10 hours ago, Andrei Moutchkine said:
European biogas? Entirely fraudulent. They are tapping into the livestock fodder, which is very plentiful. Up to a point, where the meat prices are going to go up. I am all for outsourcing the
meat production to Argentines, who do it right, but the free trade deal between the EU and Mercasur appears to be on the rocks right now.
You can allegedly raise the output by throwing the leftover glycerin left over from conventional biodiesel production into the reactor, but EU does not even allow that, because it does not
consider it a renewable resource! This is based on observing actual operational biogas plants and talking to their operators from before 2008. On the other hand, the European glycerin glut
appears to still be with us, so nothing changed. (Conventional biodiesel production removes the glycerine skeleton all veggie- and animal-sourced oil have (these are called triglycerides; and
replaces them with methanol, which tends to be sourced from good old Russian natural gas. So, is traditional European 1st gen biofuel actually 11% "fossil" by weight and produces an equivalent
amount of glycerine byproduct nobody wants)
You will be very hard pressed to find any technical merit to any of the Eurofag renewable fuel activity. The only purpose of it is to "carbon tax" the heck out of "emerging economies" for being
dirty, while changing jack. Neocolonialism at work. Neste process is a major exception here, BTW. The original Finish renewable plan was to burn peat, which the rest of EU vehemently objected to
Interesting but you are totally missing the point. Andrei, you are talking about diesel, the product produced is bio gas. Not diesel or bio gasoline. Those can be produced from coal, as you well
Glycerine can easily be burned for combined heat and power, so should be fully used. Didn't you already know this?
10 hours ago, Andrei Moutchkine said:
The effect of burning all this straw on carbon credits appears to be TBD, as the drivers of "energy transition" tend to be the "1st world" economies unfamiliar with either bamboo or sugar cane
Yes and no. Higher octane number makes fuel actually less energetic (it adds "anti-knock" properties, i.e. suppresses detonations) The cetane numbers for diesel-like fuels work in the opposite
direction. Higher cetane means more energetic and volatile fuel, but also likely more detonation prone.
In order to get back the power you lose, you can run the engine with higher compression, which results in a more expensive car. Think the "funny fuel" American quarter mile racing uses. That
stuff is the same small RC racing cars use. Mostly methanol, with a bit of nitromethane mixed in. Equivalent to something like 130? octane gas.
The primary benefit of adding a bit of ethanol to gasoline is combating smog in places like LA (so it's mandatory for California) For diesel fuel, adding some plant-based biodiesel has the same
effect. Both admixtures work on the principle of carrying a bit of additional oxidizer into the mix.
Very good. Please reply to my whole point which is bio gas, not bio gasoline or bio diesel. I realize that Americans are very fond of calling gasoline "gas" but it is very unfortunate. They could be
using actual natural gas, but that is now being used in natural gas fueled ships, trucks, etc. worldwide. Either as CNG which is easy or as LNG which is a bit more complicated.
There is an abundance of natural gas on the land and in the ocean. If it is not used it is the fault of green activists who are destroying western economies, much to the delight of Putin, XI and
others. Without investment support, we are hobbling our economies through inflation.
Green biogas can only be a very valuable and green supplement of large scale. The waste streams that can be used are enormous, and part can also be used for fertilizer.
10 hours ago, Andrei Moutchkine said:
Aha, found the secret keyword for your nasty straw
That is a very good reference and shows how valuable that product is. It is mainly tilled into the soil to keep the soil natural, but can be used to make ethanol if corn is too expensive. During our
last so called energy crisis President George W. Bush smartly pointed this out. I am not a fan of his due to his fondness for crony capitalism though. He also amplified spying on Americans rather
than foreigners.
Corn and soybeans are main base products for much of the food that is eaten worldwide. Archer Daniels Midland and Tate and Lyle are two of the main processors. I am in Decatur, Illinois which calls
itself the "soybean capitol of the world." They have huge plants right here. Interestingly, we have a new plant, almost completed, that will produce and process fly grubs which are a high value
protein supplement used for livestock. Almost as good as Soylent Green 😉
18 hours ago, KeyboardWarrior said:
My microbiology prof likes to go on about schemes like this. Reality is that microbial processes will never be commercialized because of scale and efficiency issues. Few exceptions include
ethanol production.
Nitrogen fixation by microbes has been all the rage in this field, but the industrial routes already beat the race to the practical efficiency limit. The same applies for organic synthesis.
Biogas production uses microbes that flourish without O2 (anaerobic). https://byjus.com/biology/microbes-in-production-of-biogas/ This is a growing source of green energy that is still mostly
untapped. It serves a dual purpose be eliminating material that would otherwise have to go to landfills or sewage plants. The potential is enormous.
Edited by ronwagn
5 hours ago, ronwagn said:
Biogas production uses microbes that flourish without O2 (anaerobic). https://byjus.com/biology/microbes-in-production-of-biogas/ This is a growing source of green energy that is still mostly
untapped. It serves a dual purpose be eliminating material that would otherwise have to go to landfills or sewage plants. The potential is enormous.
Great potential for the botulinum toxin for affordable Botox injections for the peoples!
By that I mean that novel microbes are scary.
Edited by Andrei Moutchkine
6 hours ago, ronwagn said:
Very good. Please reply to my whole point which is bio gas, not bio gasoline or bio diesel. I realize that Americans are very fond of calling gasoline "gas" but it is very unfortunate. They could
be using actual natural gas, but that is now being used in natural gas fueled ships, trucks, etc. worldwide. Either as CNG which is easy or as LNG which is a bit more complicated.
I already did, in several posts. To summarize, there is still potential for liquid biofuels, but not for biogas (as in methane) the way it is done now.
There is also unlikely to be a LNG on a scale of a car or truck, because it is a cryogenic liquid that cannot be stored without evaporative loss. Converting to DME (dimethyl ether) or methanol is a
better idea.
8 hours ago, ronwagn said:
Glycerine can easily be burned for combined heat and power, so should be fully used. Didn't you already know this?
"Can" does not translate to "should" in the EU. AFAIK, there is still a glut of glycerine the "1st gen" biodiesel works produce.
I only started talking about liquid biofuels after you said my knowledge about them has a gap.
Edited by Andrei Moutchkine | {"url":"https://community.oilprice.com/topic/24912-microbes-can-provide-sustainable-hydrocarbons-for-the-petrochemical-industry/","timestamp":"2024-11-07T18:44:56Z","content_type":"text/html","content_length":"585053","record_id":"<urn:uuid:0ebc4e53-27a9-450b-8589-c0f25c4734ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00116.warc.gz"} |
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GNU Octave: Advanced Indexing
8.1.1 Advanced Indexing
An array with ‘n’ dimensions can be indexed using ‘m’ indices. More generally, the set of index tuples determining the result is formed by the Cartesian product of the index vectors (or ranges or
For the ordinary and most common case, m == n, and each index corresponds to its respective dimension. If m < n and every index is less than the size of the array in the i^{th} dimension, m(i) < n
(i), then the index expression is padded with trailing singleton dimensions ([ones (m-n, 1)]). If m < n but one of the indices m(i) is outside the size of the current array, then the last n-m+1
dimensions are folded into a single dimension with an extent equal to the product of extents of the original dimensions. This is easiest to understand with an example.
a = reshape (1:8, 2, 2, 2) # Create 3-D array
a =
ans(:,:,1) =
ans(:,:,2) =
a(2,1,2); # Case (m == n): ans = 6
a(2,1); # Case (m < n), idx within array:
# equivalent to a(2,1,1), ans = 2
a(2,4); # Case (m < n), idx outside array:
# Dimension 2 & 3 folded into new dimension of size 2x2 = 4
# Select 2nd row, 4th element of [2, 4, 6, 8], ans = 8
One advanced use of indexing is to create arrays filled with a single value. This can be done by using an index of ones on a scalar value. The result is an object with the dimensions of the index
expression and every element equal to the original scalar. For example, the following statements
produce a vector whose four elements are all equal to 13.
Similarly, by indexing a scalar with two vectors of ones it is possible to create a matrix. The following statements
a = 13;
a(ones (1, 2), ones (1, 3))
create a 2x3 matrix with all elements equal to 13.
The last example could also be written as
It is more efficient to use indexing rather than the code construction scalar * ones (N, M, …) because it avoids the unnecessary multiplication operation. Moreover, multiplication may not be defined
for the object to be replicated whereas indexing an array is always defined. The following code shows how to create a 2x3 cell array from a base unit which is not itself a scalar.
It should be, noted that ones (1, n) (a row vector of ones) results in a range (with zero increment). A range is stored internally as a starting value, increment, end value, and total number of
values; hence, it is more efficient for storage than a vector or matrix of ones whenever the number of elements is greater than 4. In particular, when ‘r’ is a row vector, the expressions
will produce identical results, but the first one will be significantly faster, at least for ‘r’ and ‘n’ large enough. In the first case the index is held in compressed form as a range which allows
Octave to choose a more efficient algorithm to handle the expression.
A general recommendation, for a user unaware of these subtleties, is to use the function repmat for replicating smaller arrays into bigger ones.
A second use of indexing is to speed up code. Indexing is a fast operation and judicious use of it can reduce the requirement for looping over individual array elements which is a slow operation.
Consider the following example which creates a 10-element row vector a containing the values a(i) = sqrt (i).
for i = 1:10
a(i) = sqrt (i);
It is quite inefficient to create a vector using a loop like this. In this case, it would have been much more efficient to use the expression
which avoids the loop entirely.
In cases where a loop cannot be avoided, or a number of values must be combined to form a larger matrix, it is generally faster to set the size of the matrix first (pre-allocate storage), and then
insert elements using indexing commands. For example, given a matrix a,
[nr, nc] = size (a);
x = zeros (nr, n * nc);
for i = 1:n
x(:,(i-1)*nc+1:i*nc) = a;
is considerably faster than
x = a;
for i = 1:n-1
x = [x, a];
because Octave does not have to repeatedly resize the intermediate result. | {"url":"https://docs.octave.org/v4.0.0/Advanced-Indexing.html","timestamp":"2024-11-11T13:43:01Z","content_type":"text/html","content_length":"12417","record_id":"<urn:uuid:caf16b6f-92a5-4247-a3d5-9384460bc6e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00745.warc.gz"} |
libgfortran/intrinsics/selected_int_kind.f90 - gcc - Git at Google
! Copyright (C) 2003-2024 Free Software Foundation, Inc.
! Contributed by Kejia Zhao <kejia_zh@yahoo.com.cn>
!This file is part of the GNU Fortran 95 runtime library (libgfortran).
!Libgfortran is free software; you can redistribute it and/or
!modify it under the terms of the GNU General Public
!License as published by the Free Software Foundation; either
!version 3 of the License, or (at your option) any later version.
!Libgfortran is distributed in the hope that it will be useful,
!but WITHOUT ANY WARRANTY; without even the implied warranty of
!MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
!GNU General Public License for more details.
!Under Section 7 of GPL version 3, you are granted additional
!permissions described in the GCC Runtime Library Exception, version
!3.1, as published by the Free Software Foundation.
!You should have received a copy of the GNU General Public License and
!a copy of the GCC Runtime Library Exception along with this program;
!see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
function _gfortran_selected_int_kind (r)
implicit none
integer, intent(in) :: r
integer :: _gfortran_selected_int_kind
integer :: i
! Integer kind_range table
type :: int_info
integer :: kind
integer :: range
end type int_info
include "selected_int_kind.inc"
do i = 1, c
if (r <= int_infos(i)%range) then
_gfortran_selected_int_kind = int_infos(i)%kind
end if
end do
_gfortran_selected_int_kind = -1
end function
! At this time, our logical and integer kinds are the same
function _gfortran_selected_logical_kind (bits)
implicit none
integer, intent(in) :: bits
integer :: _gfortran_selected_logical_kind
integer :: i
! Integer kind_range table
type :: int_info
integer :: kind
integer :: range
end type int_info
include "selected_int_kind.inc"
do i = 1, c
if (bits <= 8 * int_infos(i)%kind) then
_gfortran_selected_logical_kind = int_infos(i)%kind
end if
end do
_gfortran_selected_logical_kind = -1
end function | {"url":"https://gnu.googlesource.com/gcc/+/refs/heads/master/libgfortran/intrinsics/selected_int_kind.f90","timestamp":"2024-11-09T09:37:30Z","content_type":"text/html","content_length":"28921","record_id":"<urn:uuid:e97649fa-f84c-4416-b516-4b6065719ff7>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00425.warc.gz"} |
ASVAB Math
Know What to Expect on the ASVAB Math Exam
Often the most efficient way to get the correct answer to a math problem is to use a strategy rather than just to “do the math.” That’s especially true on the ASVAB Math Exam, since you cannot use a
calculator on the exam. On some math tests, if you get the wrong answer but show that you set up at least some of the math correctly, you’ll still get partial credit. On the
, you only get credit for a right answer. One advantage of this is that you will get credit for the correct answer regardless of how you get to it.
The two math sections on the ASVAB are called “Arithmetic Reasoning” and “Mathematics Knowledge.” Together they form the quantitative half of the Armed Forces Qualifying Test (AFQT), so you’ll want
to do well on these sections no matter what your ultimate vocational aim in the military.
[ TRY KAPLAN’S PRACTICE QUESTIONS: ASVAB Arithmetic Reasoning and Mathematics Knowledge ]
The Kaplan Method for ASVAB Math Questions
Working quickly and efficiently is essential to maximizing your score on these sections. To accomplish this, use the Kaplan Method for ASVAB Math Questions.
Strategies for Solving ASVAB Math Problems
Several methods are extremely useful when you don’t know—or don’t have time to use—the textbook approach to solving the question. In addition, performing all the calculations called for in the
question can often be more time-consuming than using a strategic approach and can increase the potential for mistakes.
Two problem-solving strategies that may be new to you are
Picking Numbers
. These strategies are a great way to make confusing problems more concrete. If you know how to apply these strategies, you’ll nail the correct answer every time you use them.
Sometimes it’s easiest to work backward from the answer choices. Since many Arithmetic Reasoning questions are word problems with numbers in the answer choices, you can often use this to your
advantage by using
. After all, the test gives you the correct answer—it’s just mixed in with the wrong answer choices. If you try an answer choice in the question and it fits with the information given, then you’ve
got the right answer.
Here’s how it works. When the answer choices are numbers, you can expect them to be arranged from small to large (or occasionally from large to small). Start by trying either choice (B) or (C). If
that number works when you plug it into the problem, you’ve found the correct answer. If it doesn’t work, you can usually figure out whether to try a larger or smaller answer choice next. Even
better, if you deduce that you need a smaller (or larger) number, and only one such smaller (or larger) number appears among the answer choices, that choice must be correct. You do not have to try
that answer choice: simply select it and move on to the next question.
By backsolving strategically this way, you won’t have to try out more than two answer choices before you zero in on the correct answer.
[ GOOD TO KNOW: What is the AFQT? ]
Another strategy that comes in handy on many Mathematics Knowledge questions and also on some Arithmetic Reasoning questions is
Picking Numbers
. Just because the question contains numbers in the answer choices, that doesn’t mean that you can always backsolve. There may be numbers in the answer choices, but sometimes you won’t have enough
information in the question to easily match up an answer choice to a specific value in the question stem. For example, a problem might present an equation with many variables, or it might give you
information about percentages of some unknown quantity and ask you for another percent. If the test maker hasn’t provided you with a quantity that would be really helpful to have in order to solve
the problem, you may be able to simply pick a value to assign to that unknown. The other case in which you can pick numbers is when there are variables in the answer choices.
When you are picking numbers, be sure that the numbers you select are permissible (follow the rules of the problem) and manageable (easy to work with). In general, it’s a good idea to avoid picking
−1, 0, or 1 because they have unique number properties that can skew your results.
Strategic Guessing Using Logic
and using a
combination of approaches
are other useful shortcuts to getting more correct answers more quickly. Remember, you get points for correct answers, not for how you got those answers, so efficiency is key to maximizing your
[ KEEP STUDYING: ASVAB Strategies
ASVAB Technical Subset Strategies ]
0 0 admin http://wpapp.kaptest.com/wp-content/uploads/2020/09/kaplan_logo_purple_726-4.png admin2019-07-05 18:43:252020-09-11 20:41:35ASVAB Math Strategies | {"url":"https://wpapp.kaptest.com/study/asvab/asvab-math-strategies/","timestamp":"2024-11-05T10:38:44Z","content_type":"text/html","content_length":"193122","record_id":"<urn:uuid:534e06c3-2ed5-4157-ba15-7ed6b7da760a>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00299.warc.gz"} |
Heat transfer in flows with drag reduction
A review is presented of the models which have been proposed to describe both momentum and heat transfer in flows with drag reduction. It is shown that in flows with drag reduction a distinction has
to be made between the constant wall temperature and the constant heat flux modes of heat transfer. An analysis is conducted regarding the effect of temperature-dependent fluid properties in the case
of two modes of heat transfer. It is found that several effects which so far have been neglected are quite significant in flows with drag reduction
In: Advances in heat transfer. Volume 12. New York
Pub Date:
□ Drag Reduction;
□ Flow Resistance;
□ Friction Drag;
□ Heat Transfer;
□ Momentum Transfer;
□ Turbulent Flow;
□ Flow Velocity;
□ Friction Reduction;
□ Heat Flux;
□ Mixing Length Flow Theory;
□ Newtonian Fluids;
□ Polymer Physics;
□ Temperature Effects;
□ Velocity Distribution;
□ Wall Temperature;
□ Fluid Mechanics and Heat Transfer | {"url":"https://ui.adsabs.harvard.edu/abs/1976aht....12...77D/abstract","timestamp":"2024-11-08T18:34:07Z","content_type":"text/html","content_length":"35111","record_id":"<urn:uuid:160d8b04-f365-4fff-959d-3063173aaca7>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00588.warc.gz"} |
What Is Volume? A Quick & Simple Guide - Geometry Spot
What Is Volume? A Quick & Simple Guide
Discover the science of volume! Unravel its importance in our daily lives and learn how to measure it like a pro.
Have you ever wondered how we know how much liquid a water bottle can hold or how much sand you need to fill up your sandbox?
The answer to that is volume!
In simple words, volume is the amount of space something takes up.
This could be an object, like your school backpack, or a substance, like the water in your swimming pool. We use the concept of volume in math and science, and it’s also very important in our daily
For example, if you’ve helped in the kitchen, you might have noticed labels on measuring cups like “1 cup” or “250 milliliters”. These are measures of volume, telling you how much space the flour or
milk you’re measuring takes up.
Let’s explore this in a little more detail!
Fun with Math: Understanding Volume
When we think about volume in math, we usually think about 3D shapes – things like cubes, cylinders, and even spheres. Imagine you have a cube that’s 1 meter on each side.
The volume of this cube is 1 meter x 1 meter x 1 meter, which equals 1 cubic meter. This is the basic formula for the volume of a cube or box: Volume = Length x Width x Height.
Now, what about cylinders (like a soup can) or spheres (like a basketball)? They have different formulas because of their shapes.
For a cylinder, you calculate the area of the circle at the base (Area = π x radius^2), then multiply it by the height of the cylinder.
So the formula is: Volume = π x radius^2 x height.
For a sphere, the formula is a bit more complex: Volume = 4/3 x π x radius^3. So you need to know the sphere’s radius, cube it (multiply it by itself twice), and then multiply by 4/3 and π.
Don’t worry if these sound a bit complicated – you’ll get the hang of them with practice!
Volume in Science: Physics and Chemistry
Volume is not just a math thing – it’s a big deal in science, too!
In physics, volume helps us understand things like how air balloons expand when heated or how submarines can dive deep underwater. Understanding volume also helps us make things like super-fast
roller coasters and skyscrapers that don’t topple over.
In chemistry, volume helps us understand how different substances react together. For example, if you’re making a homemade volcano for a science project, the ratio of vinegar to baking soda (their
volumes) will determine how big your eruption will be!
Volume in Our Everyday Lives
Now that you know about volume, you’ll start noticing it everywhere!
From knowing how much juice your glass can hold, to how much popcorn fits in a bowl, to how much water goes in your bath – that’s all about volume.
Even artists creating beautiful sculptures or architects designing awesome buildings must think about volume.
So next time you pour yourself a drink or pack your school bag, think about the volume involved. Isn’t it fun to see how what we learn in school is all around us in real life?
So there you have it!
Volume is a super cool concept that’s part of our lives in so many ways, from the math problems we solve in school to the everyday tasks we do at home.
Whether you’re designing a dream treehouse or making the perfect chocolate chip cookies, understanding volume will help you make it just right!
Remember, the key to understanding volume in practice. Try measuring the volume of different things around your home, and use the formulas to calculate the volume of different shapes.
You’ll soon become a volume whiz! | {"url":"https://geometryspot.cc/what-is-volume-a-quick-simple-guide/","timestamp":"2024-11-14T05:26:16Z","content_type":"text/html","content_length":"145360","record_id":"<urn:uuid:8a5b2185-6b28-411e-8d8e-1e05fc661062>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00894.warc.gz"} |
The Wrong Way to State Dynamic Range – And How to Spot It - Ophir Photonics
This is the second part in a short series on dynamic range. You can read the first part here.
What’s the most important rule when trying to determine spec statements? What’s the fundamental law that applies not only to dynamic range, but to all specs?
The only spec worth stating is one that is helpful to the customer.
Not the really impressive spec that only applies in 3% of situations? No, not that spec. That would be like a bridge that has a sign: “Weight limit – 15 tons.” But in fine print it says, “Only on
clear days with no wind, between the hours of 10 and11 a.m.”
How does this apply to dynamic range?
All camera beam profilers have several components, each with its own dynamic range. There’s the detector itself, the analog-digital convertor, and then there’s the incorporation of noise into the
equation. For example, the Spiricon SP620 has a dynamic range of 62dB[volt] even though its digitizer is clearly stated as 12-bit, which is a dynamic range of approximately 72dB[volt]. So what
happened to the other 10dB?
Figure 1: Spiricon SP620 Camera
When Ophir Spiricon determines dynamic range, it takes a very careful measurement of the RMS noise first and then checks the range between the highest signal and that noise. Figure 2, below, is an
excerpt from such a calculation of the SP620 Camera noise. The y-axis represents the number of readings (population) of a given value (bin) and the x-axis is the value itself. X is the mean, and σ
is the RMS noise. To get the dynamic range value, one must divide 2^12 (12 bit A/D) by σ.
Figure 2: SP620 Camera Noise Calculation
When Ophir Spiricon uses this method, it means a bit more legwork and a slightly less-impressive spec. It also means that the dynamic range stated is meaningful. It’s meant to give the customer a
sense of the range of signals that he will be able to measure with this instrument.
Rule of Thumb
In general, when you see a dynamic range stated based only on the bits (like the 72dB in the example above), you can almost always assume it to be higher than the actual dynamic range.
We can understand this rule better by considering when it is broken. Imagine for a moment that your camera has a noise level of 1/1000 of the maximum (or 0.1%), but the digitizer only has 8 bits.
This increases the effective noise by a factor of almost four, since, although the detector can measure 1000 significantly different values, the digitizer can only convert to 256 values (2^8=256).
That means that there will be an additional rounding error. Since no manufacturer would want to increase the effective “noise,” it’s safe to assume that the range based on bits is higher than the
actual dynamic range.
Do you have any questions? What spec do you find most convoluted? Leave your thoughts in the comments below.
You might also like to read: Bottom Line: What Is Beam Profiling Dynamic Range? | {"url":"https://blog.ophiropt.com/the-wrong-way-to-state-dynamic-range-and-how-to-spot-it/","timestamp":"2024-11-04T04:19:48Z","content_type":"text/html","content_length":"98800","record_id":"<urn:uuid:3b5cee80-3737-4b22-8f76-87c0d2dfffed>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00163.warc.gz"} |
We spent 7 hours making food. We spent 1/4 of that time making brownies. How long did it take us to make the brownies?
We spent 7 hours making food. We spent 1/4 of that time making brownies. How long did it take us to make the brownies?
648 views
Answer to a math question We spent 7 hours making food. We spent 1/4 of that time making brownies. How long did it take us to make the brownies?
89 Answers
$=\frac{7}{1}\cdot \frac{1}{4}$
$=\frac{7\cdot 1}{1\cdot 4}$
$=\frac{7}{1\cdot 4}$
Frequently asked questions (FAQs)
What is the maximum value of the function f(x) = x^2 - 3x + 5 on the interval [0, 4]?
What values of the coefficient "a" will result in vertically stretched parabolas when graphing the function 𝑦 = ax^2?
Question: Find the absolute extrema of the function f(x) = x^3 - 3x^2 + 2x on the interval [-2, 3]. | {"url":"https://math-master.org/general/we-spent-7-hours-making-food-we-spent-1-4-of-that-time-making-brownies-how-long-did-it-take-us-to-make-the-brownies","timestamp":"2024-11-07T16:09:12Z","content_type":"text/html","content_length":"237915","record_id":"<urn:uuid:ced0bb2d-87f1-4b29-b9bc-a22254ce2846>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00888.warc.gz"} |
Lowww. A directory of low-carbon websites.
Our calculations use the median webpage weight from the HTTP archive as main reference. We calculate the difference between a website page weight and that median webpage weight and convert it into a
percentage. Our CO2 calculations use the methodology from Sustainable Web Design
"We used these data points to define the calculations below:
Annual Internet Energy: 1988 TWh
Annual End User Traffic: 2444 EB
Annual Internet Traffic / Annual End User Traffic = 0.81 TWh/EB or 0.81 kWH/GB
Carbon factor (global grid): 475g/kWh
These are the formulas we came up with:
Energy per visit in kWh (E): E = (Data Transfer per Visit in GB x 0.81kWh/GB x 0.75) + (Data Transfer over the Wire (GB) x 0.81kWh/GB x 0.25 x 0.02)
Emissions per visit in grams CO2e (C): C = E x 475g/kWh"
As the Internet is global, we use the global grid as a carbon factor. But in the future we plan to add the use of data centers powered by renewable energy into our equation. | {"url":"https://www.lowww.directory/calculations","timestamp":"2024-11-02T17:01:06Z","content_type":"text/html","content_length":"7765","record_id":"<urn:uuid:adb1d41a-b6bd-412d-beb3-341ffba5d58d>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00461.warc.gz"} |
GreeneMath.com | Ace your next Math Test!
Transpose of a Matrix
Additional Resources:
In this lesson, we learn how to find the transpose of a matrix. Suppose we have an m x n matrix, and we will call it matrix A. The transpose of A will be an n x m matrix whose columns are formed from
the corresponding rows of our matrix A. For a square matrix, the concept is the same, however, the order will not change.
+ Show More + | {"url":"https://www.greenemath.com/College_Algebra/130/Transpose-of-a-Matrix.html","timestamp":"2024-11-08T02:12:35Z","content_type":"application/xhtml+xml","content_length":"8744","record_id":"<urn:uuid:fa7ce168-ec51-44e8-a6ff-741af3eb230b>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00725.warc.gz"} |
On the power of the semi-separated pair decomposition
A Semi-Separated Pair Decomposition (SSPD), with parameter s > 1, of a set is a set {(A [i] ,B [i] )} of pairs of subsets of S such that for each i, there are balls and containing A [i] and B [i]
respectively such that min ( radius ) , radius ), and for any two points p, q S there is a unique index i such that p A [i] and q B [i] or vice-versa. In this paper, we use the SSPD to obtain the
following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set of n imprecise points, modeled as pairwise disjoint balls,
admits a t-spanner with edges which can be computed in time. If all balls have the same radius, the number of edges reduces to . Secondly, for a set of n points in the plane, we design a query data
structure for half-plane closest-pair queries that can be built in time using space and answers a query in time, for any ε> 0. By reducing the preprocessing time to and using space, the query can be
answered in time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some
previously studied problems, namely spanners for complete k-partite graphs and low-diameter spanners, and show how to use the SSPD to obtain simple algorithms for these problems.
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 5664 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 11th International Symposium on Algorithms and Data Structures, WADS 2009
Country/Territory Canada
City Banff, AB
Period 21/08/09 → 23/08/09
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'On the power of the semi-separated pair decomposition'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/on-the-power-of-the-semi-separated-pair-decomposition-5","timestamp":"2024-11-14T14:32:27Z","content_type":"text/html","content_length":"60351","record_id":"<urn:uuid:ed94c13c-43b2-4605-b203-c936f2d6c15c>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00124.warc.gz"} |
CLIME Connections
There’s a lot of discussion about the CCSSI (do I really have to write it out?) for math. Mostly its about the Standards and assessments which for me is a checklist for what I should include in a
curriculum. But what about the curriculum? What does it look like in most schools? I would guess that the textbook companies are controlling that world and that’s very depressing, because I haven’t
seen any textbooks that breaks the mould of what we have encountered before even if its been spruced up by the requirements of the Common Core standards. But if you look past the usual suspects, you
will find in the blog world some innovative, heart warming approaches. Case in point, Geoff Krall. In his his initiatives. Geoff (
My Common Core Problem Based Curriculum Maps
) has crafted a game plan for proceeding with reinventing how math can be taught in schools. He has collected activities and projects from his math blogging colleagues and organized them into a
curriculum. When I was still at Stevens/CIESE I did a
similar approach
with the 6th Everyday Math curriculum since the teachers in the Elizabeth, NJ school district wanted more technology based activities that were not part of EDM. This kind of work requires support
from the teachers, pedagogical change coaches (like I was there) and administration support to work. Unfortunately, my effort in Elizabeth has sat dormant since I retired from CIESE in 2007.
What we need is a curriculum (a modern textbook, if you like) that is written from a student's perspective. Most textbooks are written so that adults reviewing the books will find all the required
checkboxes checked before they adopt. Why can't textbooks be written as engaging stories that students would buy into? I suspect very few kids would choose the books that are currently coming out of
the textbook mills if they had a choice. Engaging stories should drive curriculum. Then students would actually want to do the activities in the book instead of being force fed by the teachers
because they are "good for you."
Geoff Krall to his credit has listed links to interesting stories, but they are still on the sidelines in the curriculum game. Some day it will happen. The current technology makes it possible and
the math blogging community is putting examples out there every day. However, the devotion to the
Royal Road to Calculus
continues to interfere with student engagement and genuine learning.
In June 2013 I wrote about the Jinx Puzzle where you pick a number and then do several different calculations to that number and the result is 13 no matter what number you started with. I called it
the Jinx puzzle for that reason. Why does it work every time? The secret is in the algebra. See below.
1. Choose a number. Call it X
2. Add 11. Now you have X + 11
3. Multiply by 6. Result is: 6X + 66
4. Subtract 3. The result is: 6X - 63
5. Divide by 3. The result is: 2X + 21
6. Add 5. The result is: 2X + 26
7. Divide by 2. The result is: X + 13
8. Subtract the original number that you chose. The result is: X + 13 - X = 13
You are jinxed. Problem solved. Case closed. But what if we were teaching typical 6th graders then a more interesting twist on this story is to not assume the obvious (that it always works) and see
whether these students could find a number that foils the Jinx Puzzle. Since testing numbers manually becomes quickly tedious a spreadsheet calculator can help with testing a wide range of numbers.
There’s just one problem. If you use a spreadsheet you can get a result that actually foils the Jinx Puzzle! Try something like 3.0 x 10 to the 16th power as your number.
This is caused by the fact that the spreadsheet will round off numbers after a long string of numbers is entered or will send a bogus number because we have gone passed its capability to stay
Dan Meyer in a his 101 questions activity shows how Google Calc fails to handle a subtraction problem by printing 0 instead of the correct answer of 1 just because it doesn’t play well with very
large numbers.
In an episode of The Simpson's called The Wizard of Evergreen Terrace, Homer appears to write a valid solution to defeat Fermat's Last Theorem which states that no three positive integers a, b, and c
can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two. But given that Fermat's last theorem is proven, is Homer's attempt a real counter example that Fermat and others
didn't see? Look at the equation (above) that Homer used using the Desmos calculator to check his work. Looks like the real deal doesn't it?
Simon Singh who writes about this in his latest book "The Simpsons and their Mathematical Secrets" calls Homer’s attempt at a solution a near miss. If you use a calculator that’s good to 10 places
like Desmos, it seems to work. But the devil is in longer approximations. It’s close, but no cigar. And we need exactitude to proof the theorem. Andrew Wiles who proved Fermat’s Theorem in 1995 has
nothing to worry about from Homer Simpson.
Headlining the regional NCTM meeting this week in Las Vegas, Nevada are four educators whose work is transforming curriculum design and delivery and changing the way students think about mathematics.
Board member Jon Wray, Karim Ani, Dan Meyer and Eric Westendorf. They are going to be answering the question: What should effective and innovative math instruction look like, and how can teachers
create ideal learning experiences for all students?
Wednesday, October 23, 2013: 5:30 PM-7:00 PM
Amazon F (Rio Hotel) - Las Vegas
If that session is the standard for the entire conference, then you're in for a treat if you are attending. Unfortunately, the best I can do is participate virtually via Twitter. Hopefully, there
will be lots of blogs generated and videos of presentations.
See a listing of all other technology themed sessions in Las Vegas.
I'll share my take on the conference later this week.
In my vision of math 2.0 teachers as bloggers are as commonplace as chalk and blackboard used to be. There is much to learn from the young (and not so young) math warriors who are exploring the new
frontier of teachers collaborating on how to make math come alive, personable and of course useful in the lives of their students. Unfortunately, to date, very few math teachers know about how
teachers are learning and improving their craft online with other like-minded math educators. Well, a group of these pioneers wanted to do something about it and make it resonate not only in the
blogosphere, but everywhere there are communities that help young people learn math.
Though their website comes with a formidable handle mathtwitterblogosphere (MTBoS pronounced mitt-boss) don't let that stop you from joining.
Dan Meyer wrote: "File this [MTBoS] as Reason #437 I'm proud to be a part of this enormous professional community. link"
Today (October 6th) begins 8 weeks of "Exploring the Mathtwitterblogosphere." Join up and follow this crash course.
Also on Tuesday nights you can join the Global Math Department for weekly presentations about math for teachers by teachers.
This week (Tuesday, October 8th 9pm) Karim Kai Ani (of Mathalicious.com fame) will be leading the conversation. You are all invited!
Back in April, 1988 I wrote an article "Teaching Math with Logo" for Teaching PreK-8 that included the following:
"In April, 1986 at the annual meeting of the National Council of Teachers of Mathematics a group of math educators who were interested in Logo, a computer programming language, met informally and
decided to start an organization which eventually became known as the Council for Logo in Mathematics Education or CLIME for short. These educators taught math at all levels and were from all parts
of the United States and Canada. What had brought them together was a belief that Logo could make a significant difference in the quality of mathematics education.
Now there are many other resources - including Cuisenaire Rods, geoboards, rulers and compasses, to name just a few - that math teachers use to help them teach. But to my knowledge no one has ever
started an international organization for the purpose of promoting their use. What, then, makes Logo so special?
One way to answer this question is to say that Logo, unlike other resources, comes with its own philosophy of education. This philosophy was introduced to the world by Seymour Papert in a book called
Mindstorms: Children, Computers, and Powerful Ideas (Basic Books, 1980). In it he said that children seem to be innately gifted learners who acquire a vast quantity of knowledge long before they go
to school. What blocks them from learning is not the inherent difficulty of the ideas, but the failure of the surrounding culture or environment to provide the resources that would make the ideas
simple or concrete. In other worlds, one reason why math is difficult to learn is because the culture outside the classroom does not provide the materials or experiences that would support the
students' classroom lessons."
That was 1988. Since then CLIME has evolved from the Council for Logo to the Council for Technology in math education. This change was made to acknowledge the development of exciting new environments
(like Geometer's Sketchpad) which made me realize that Logo was not the only game in town, but that there were other software environments that encourage this kind of dynamic learning that Logo made
possible. With the advent of the handheld tablets and smart phones powerful new math apps are being developed. (For example see Keith Devlin's latest entry Wuzzit Trouble.)
It's been a tough time to be an educator with all the hoopla about the common core and its subsequent imprisonment of laptops and handhelds for testing which will keep the technology away from
creative teachers who could use them in dynamic ways. So its easy to see the technology glass as half empty.
In his book "Logo Theory and Practice" (1989) Dennis Harper quotes me as saying that "a Logo environment is more of a spirit rather than a thing-something that can only be satisfying if experienced,
rather than just languaged." (p.26)
And I still believe that today. I recently wrote to one of CLIME's steering committee/board members Robert Berkman that we need to look at the glass as half full and keep a positive attitude because
the Internet does have a long tail (and tales) and there are literally hundreds (thousands?) of places, people and environments where adults are helping our young people "construct modern knowledge"
something another one of CLIME's board members Gary Stager promotes in his annual Vermont workshops and in his new book "Invent to Learn."
I've been privileged over the years from the guidance of these and other remarkable people who have been good friends of CLIME and members of the CLIME steering committee over the last 25 years. I
want to publicly thank them for their help. (You will be hearing from some of them in future CLIME blog posts.)
See a short summary CLIME's journey since 1986.
Whether you like the idea of the Common Core State Standards for math (CCSSM) or not, it is here to stay at least until version 2.0 addresses the eventual problem of the scores not going up. Why am I
so sure this will be the case? Because CCSSM doesn't ensure a curriculum that actually helps students understand and learn the topics any better than before. For example, learning fractions. In a
recent book Keith Devlin describes how easily it is to get confused when doing fractions. (See my blog.) Schools have to use curriculum that match the standards. That's great. But how can they do it
well? Since proportionality is such an important concept and understanding it is so critical a carefully crafted set of activities is needed to prevent misconceptions.
Most school districts will probably choose a textbook program that is correlated with the standards. Problem: Most textbooks do a decent job in correlating, but not in motivating students to learn
the math. This summer Dan Meyer had a Makeover Monday blog where teachers submitted typical problems from textbooks and Dan's community offered suggestions as to how to improve them. While reading
these blogs I became convinced that we need a makeover of textbooks in general i.e. come up with a more dynamic model for lessons that textbooks could adopt. Currently Dan Meyer and Karim Ani (
Mathalicious.org) are creating and encouraging dynamic learning adventures that are interesting to kids and help with deeper understanding.
The giants of the textbook world (Pearson, McGraw HIll, etc.) are trying to modernize their curriculums but they have too much at stake in maintaining the status quo since most teachers and
administrators prefer what they are familiar with and find the so called "alternative" models too risky or controversial for district approval. (Larry Cuban describes this phenomenon as dynamic
conservatism). Otherwise districts would reject most if not all the mediocre curriculums that are now being published.)
Should Algebra be optional?
Recent articles in Harpers and the New York Times have argued for making the Algebra 1 and Algebra 2 sequence optional especially for kids who struggle with math.
Michael thayer writes:
In an ideal world, kids would sort themselves in this way based on their interests.
Kids in track #1 ("calculus track"): These are the kids who love math, who love the challenge of it, and who see the abstractions of algebra and analysis as pursuits worthy of study.
Kids in track #2 ("statistics track"): These are the kids who recognize the importance and practicality of math, and who see utility for it in their futures.
Kids in track #3 ("one and done"): These are the kids who have had a good experience with math, who have seen the forest for the trees, but do not wish to go deeper as their interests lie elsewhere.
I would also include in track* 3 those students who didn't have a good experience in math and do not have any interest in continuing in math since they would rather use the time to study something
My Thoughts on the Path 3 Course
Offer a one year course for students who definitely don't want to do the formal Algebra 1 or 2 path for whatever reason, but still want to go to a "good" college. Their are over 4,000 accredited 2
and 4 year colleges in the US. Getting into a college is usually not a problem, just paying for it is. (Shame on those colleges that afflict serious debt on our students.) I'm sure there are plenty
of colleges out there that would accept students who have (as Mike pointed out in his blog) excellent work habits, overall knowledge base, and interpersonal and time management skills who didn't take
Algebra 1 and 2 but rather this richer one year 9th grade math course; something like "Mathematics a Human Endeavor - A Book for Those Who Think They Don't Like the Subject" by Harold Jacobs. He
wrote his last revision of the book in 1994 and the book is still in demand particularly in homeschooling environments. Anyone out there want to work on an open source version of the kind of one year
alternative curriculum that is in the same spirit as Jacobs had in mind? (Here's something I did with his Chapter 3 - Functions and their Graphs.)
Maybe we can do it collectively as an open source project. I volunteer to be a conduit for creating this course! Are you interested? (If so, you might get familiar with Jacobs book to see what I have
in mind.)
*Tracking is not the right word for this, because it implies rigidity. These should be paths that students can opt to start on, but can switch to a different path at any time or chart their own
Michel Paul's <pythonic.math@gmail.com> recent email post to mathfuture@googlegroups.com is worth reading. This is Math 2.0 thinking at its best.
Michel Paul writes:
Keith Devlin wrote the following to describe the nature of modern (20th century onward) mathematics to current undergraduates transitioning from high school (19th century and before):
"Prior to the nineteenth century, mathematicians were used to the fact that a formula such as y = x^2 + 3x - 5 specifies a function that produces a new number y from any given number x. Then the
revolutionary Dirichlet came along and said to forget the formula and concentrate on what the function does in terms of input-output behavior. A function according to Dirichlet, is any rule that
produces new numbers from old. The rule does not have to be specified by an algebraic formula. In fact, there's no reason to restrict your attention to numbers. A function can be any rule that
takes objects of one kind and produces new objects from them."
- Keith Devlin
Wow! Please read that carefully. Though he wrote this to help today's students transition from the 19th century to modern mathematical thinking, it also describes the kind of thinking one learns to
do in computer science! Functions in computer science operate on objects of various types, not just numbers.
Traditionalists lost the battle professionally in the mathematical revolution that occurred a century ag0 but won in education. Meanwhile, computer science went ahead and got created from the
insights of that revolution and turned into the world we now live in. The result? Most K-12 math students and their teachers, us, are unaware of the nature of the mathematical thinking that went on
in the 20th century while the technology that surrounds us was built from it!
The ultimate irony - we use 21st century technology, made possible by 20th century math and physics, to teach students how to do 19th century mathematics that they will never use!
I think this makes it clear that the study of programming can provide a way for math students to encounter and develop some intuition regarding concepts underlying modern mathematics that the
traditional high school curriculum does not provide.
"What I cannot create, I do not understand."
- Richard Feynman
"Computer science is the new mathematics."
- Dr. Christos Papadimitriou
This leadership conference was organized mostly to see how social media can help increase the current NCTM membership and encourage new teachers to join NCTM and affiliate groups. One way to make
organizations more responsive to young teachers is to take advantage of open source resources like Facebook and teacher blogs. They are avenues that are freely available. NCTM President, Linda Gojak
made reference to research indicating that teachers don’t join organizations until they are in their 30s. Since membership fees seem to be barriers to young teachers, One way to encourage membership
is to offer discounted membership to young teachers or have no membership fees at all. One conference participant mentioned to me that ‘free’ would create the perception of lower quality. But this
doesn’t have to be the case if conferences and other events that require a fee pay the bills. My sense was that most of the affiliate leaders at this conference were not willing to give up on
membership fees but were willing to make a serious effort to help young teachers join. Since many of the young teachers are already knowledgeable of the social media resources, they don’t feel the
need to join a group. The challenge for NCTM and their affiliate groups is to use social media in a compelling way. Membership would not be an obstacle if NCTM & its affiliates offered something that
young teachers wanted to buy. Starbucks, for example, doesn't have a young people shortage at their counters. Of course you can't compare a Frappuccino with math support; I’m not suggesting that one
should, but what is it about the purpose of buying such an expensive food item? (Clay Christensen writes about how a milk shake is a popular drink for people commuting to work and school because it
keeps them occupied longer than other foods.*) Learning about how to become a better teacher can be something that young people would buy if is relevant to their needs. Our organizations need to be
better connected to the social media to attract teachers to participate at a level they can afford. Schools need to be better places for teachers to learn about exciting things to do. Bloggers now
provide these kind of resources for free. Schools and organizations that support the teaching and learning of math need to learn from these structures how to best to provide staff development. There
is a hidden curriculum at work in many schools where the teachers are crafting their professional development through open source venues. Supporting young teachers would be much easier if we help
them expand their network of communication. Open source materials and teacher blogs are excellent ways to do that. Schools need to become better integrated learning communities that learn both from
and with open source resources. Supporting this need should be an important role for NCTM and its affiliates.
*“Rethinking Student Motivation Why understanding the ‘job’ is crucial for improving education”
Clayton M. Christensen, Michael B. Horn, and Curtis W. Johnson
Ihor, April, and Jim celebrating April's Average Traveler Award
I spent this past week at Gary Stager's Constructing Modern Knowledge Conference in Manchester, NH. There was quite a turnout. Over 150 educators from all over the world enrolled. The modern
knowledge that the folks worked on included some strange sounding tools like Raspberry Pi, Arduino, Scratch, conductive paint pens and plenty more. (See "Oh, The Stuff You Might Learn With at CMK
2013") I shied away from being that modern and decided on a more familiar project.
When I walk into a room full of people, I usually think of the famous birthday problem which predicts that if you have 23 people in a room there is slightly better than a 50% chance that at least 2
people will share a common birthdate (month and day only). If there are 100 people, the percentage jumps to more than 99% certainty that this will happen. With a 150 people attending I thought of a
more interesting crowd sourcing activity that determines who the "average traveler" to this conference is. Basically I find out which person traveled the shortest distance from the mean distance of
all the travelers. I used Google Docs forms and spreadsheet to get the data and Google Maps to display it. (The original plan was to write a Scratch program to do it and this is currently a work in
progress.) The form asked each conference participant to enter their name, home or school address and distance traveled (in miles). The results appeared in the Google Docs spreadsheet. See Table 1.
Table 1 - Top 20 finishers
As you can see from the table, April Gustafson was our average traveler. In addition to entering their mileage, participants also added a marker on a Google Maps page. I made a snapshot of the
markers and used Geometer's Sketchpad to draw the circle with a diameter approximately equal to the mean of all the travelers. You can see that April's balloon was closer to the circle than any of
the other travelers!
I want to thank Jim Scribner for his assistance and friendship doing this project. We discovered a mutual love of baseball that goes way back. Do you remember the Pirates lineup in the 1979 world
series? We did with just a little help from Google.
As most of you may know CLIME is an affiliate group of NCTM. Each summer NCTM holds a special conference for affiliate group leaders. I have attended such a conference before so I was not
particularly inclined to attend this one until I noticed that the theme for the conference is: "Leadership: Building Responsive Affiliates in an iPad World." Since CLIME has been promoting Math 2.0
in this blog since 2009, I decided I couldn't pass up this opportunity to participate in this conference. Here's the bulleted list of outcomes for this 3 day (2 night) conference in Annapolis, MD:
• learn and share about effective use of social media and new technology to advance your Affiliate’s goals;
• develop strategies for strengthening your Affiliate’s leadership role and advocating for mathematics education;
• identify ways of addressing equity in Affiliate activities and organizations;
• learn about the NCTM structure, resources, and initiatives and participate in discussions with NCTM President Linda Gojak;
• and develop or revisit the strategic plan for your Affiliate by integrating ideas gathered through discussion with other Affiliate leaders.
Click here for more details.
If anyone is interested in joining me (I'd love it) for this conference and/or is interested in learning more (personally from me) about CLIME and NCTM's affiliate program, please let me know at
Makeover Monday. What a great idea. Readers of Dan's blog and tweets offer suggestions as to how to improve standard textbook problems. This is online collaboration at its best. Only thing better
would be not to have to start with lemons. Maybe someday Dan and company will write better, more creative and dynamic "textbooks." Pearson, McGraw Hill, etc.: Are you listening?
Uri Trieisman gave the Iris Carl equity speech in Denver. Dan Meyer summarized it in detail and recommended it highly, so I listened to it online. I thought the speech was well delivered and the
reviews were mostly positive by those who replied to Dan's blog.
Uri's main message was that poverty sucks. He shows this powerful image that highlights the correlation between poverty concentration and percentage meeting SAT criterion. Sad message indeed.
The most interesting quote for me was "What is the determinant of whether you have a high skill job in the US? Overwhelmingly it's mathematics. The single biggest factor in upward social and economic
mobility. It's our beloved subject. It would be wonderful if it were music instead of math. Think how great the country would be if everyone was striving to learn to play an instrument instead of
factor quadratic equations. But the fact is it is our discipline that's the primary determinant."
This begs the question whether Uri would actually promote music education over math education if our discipline was not the primary determinate. I doubt it. The reason math is the main determinate is
because math educators like Uri Treisman work very hard at convincing the public that math education is important enough so it should be the determinator. Personally, I think a more user friendly
approach to math education is needed. Particularly for Algebra. Students need more choices and Algebra as the main staple needs an overhaul.
Andrew Hacker's New York Times article "Is Algebra Necessary?" last summer brought this issue to the forefront. I would have preferred a different title: "Should Algebra in its present form be
required of all students?" which would have helped to refocus the reactions away from getting rid of algebra altogether which of course is absurd. There were 477 comments mostly by math educators who
were outraged at the thought of not requiring Algebra for all as if Algebra was the only elixir for cognitive health.
Unfortunately Algebra for the most part will be taught in same dull way it always has. Economically advantaged students will get the best teachers and do well and poverty will continue to undermine
the learning of this subject that most students and adults find/found to be too difficult and unproductive. That doesn't mean there are not empowered individuals like Dan and Karim Ani
(mathalicious.com) that are finding ways to make math more palatable for students via teachers which for the most part are not working in high poverty area schools. We need to rethink how algebra is
taught to the masses. My recommendation is to teach math from the "outside in;" start with activities (i.e. problems/puzzles/games) that interest the kids and then show how the algebra relates to
these activities. Karim and Dan are doing this. But until the textbook publishers stop publishing pablum math, most students won't benefit from what Dan, Karim and other reformers are offering.
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Doug speaking at NCTM 2013
Another one of my heroes in the field of technology in Math education is Doug Clements. I went to hear him speak early thursday morning. His topic was Math Lessons from Research.
Doug was one of the founders of CLIME in the days when the L in CLIME stood for Logo. His voice was always strong in supporting Logo when the critics were saying Logo does not make a difference. He
would articulate the positive research that supported the use of Logo in the face of critics who pooh-poohed the results. He continues to be a leader in mathematics educational research at K-2 level
and served on the math panel in 2008 that endorsed Logo and passed mustard with the strict demands of the panel members as a worthwhile tool.
His fundamental lesson from research is that very young children are capable of doing mathematics that is complex and sophisticated. Unfortunately, too many teachers do not have access to the
information that would help them to help children in this regard. His intervention focuses on something that he refers to as learning trajectories that have three parts. (1) a goal (2) developmental
progression and (3) instructional strategies.
To attain a certain mathematical competence in a given topic (the goal), children learn each successive level of thinking (the developmental progression), aided by tasks (instructional
activities) designed to build the mental actions-on-objects that enable thinking at each higher level. (Reference 1)
For details about his current work called Building blocks see http://triadscaleup.org
The curriculum is available from McGraw Hill. But Doug did say that he was hoping to have an open source version available as well.
1. D. Clements, J. Sarama. Early Childhood Mathematics Intervention. (Science Magazine, 19 August 2011)
Below is a piece (edited and somewhat embellished) that I wrote in the "Scenes" newsletter in the Fall of 1989. I follow that with comments about three sessions I attended in Denver.
Fall, 1989
I spent the better part of my summer vacation teaching a computers in education course at a local college. One of the topics I discussed with my students was the influence that three computer
educators (who I refered to as gurus) had on how computers are used in schools. Each of them has a unique message about how to teach children with computers. Here's a short summary of their points of
view along with what I think is an appropriate rallying slogan.
Tom Snyder - Empower the teacher
Tom Snyder Youtube video
Then there is Tom Snyder of Tom Snyder Productions who is considered by many as the champion of the "one computer classroom." He believes that since the classroom is the domain of the teacher it
makes sense to give the computer to the teacher and have her use software that is designed to be used by one teacher working with large group. This point of view has made him very popular with
Seymour Papert - Empower the student
The "father of Logo" believes that children should program computers rather than the computer program them. If you are a veteran Logo user then you probably have seen this quote before, but just in
case you haven't here it is again.
"In my vision the child programs the computer, in doing so, both acquire a sense of mastery over a piece of the most modern and powerful technology and establish us in intimate contact With some
of the deepest ideas from science, from mathematics, and from the art of intellectual model building." (Papert, Mindstorms, p. 5)
So the key for Papert is to put the student in charge of the computer and presumably a love affair with learning will emerge.
Patrick Suppes - Empower the computer
Patrick Suppes
Patrick Suppes is a pioneer and leading proponent of using computers for computer-assisted instruction who believes that students can learn best if the computer controls the learning through
questions with appropriate feedback and monitors their progress. His work paved the way for the CAI (Computer Assisted Instruction) movement in schools.
In the early days of microcomputers it was fashionable for the Logo supporters to argue that the Snyders and Suppes of the world were not using computers effectively. But times and people have
changed. For example at the NECC conference (Boston, June, 1989) Seymour Papert shared a session with Bob Tinker (a BASIC sympathizer) and there wasn't a hint of disagreement between them. What does
this mean? Are the debates over? No, they will still continue, but I think what's happening is that industry is maturing and educators are acknowledging that there is more than one way to educate
children. It's not that unusual to find in a typical school district Logo being used on the elementary and junior high school levels side by side with Snyder Productions software and all across the
district you will find CAI. So what does this mean for you the teacher should you empower the student, the computer or the teacher? The answer is you empower all three. So the right slogan would be:
empower the classroom with dynamic uses of technology that empower students to want to learn.
That was what I wrote in 1989. A lot has changed since then, but the spirit of the three gurus lives on.
At the conference I thought about these gurus as I listened to younger voices. In particular, Karim Ani and Dan Meyer. Tom Snyder's vision of empowering the teacher came across in both cases. Karim's
talk Keeping It Real: Teaching Math through Real-World Topics highlighted the importance of connecting math through real world connections based on questions that would interest students.
Dan Meyer in his Tools and Technology for Modern Math Teaching presentation challenged the teachers in the audience to come up with a Tech/Ed manifesto that would make their teaching more relevant
and perplexing for students. Dan defines the state of perplexity as this awesome confluence of not knowing, wanting to know and having the belief that the solution is graspable. So creating an
environment where students eagerly embrace perplexity is an ideal condition for deep learning and self motivation. Definitely an important attribute of the dynamic classroom which Tom Snyder
Tom Snyder's vision
The modern version of Suppes that the computer can make a huge difference was echoed by Alex Sarlin and David Dockterman in The Gamification of Math: Research, Gaming Theory, and Math Instruction
that I wrote about in my previous blog. Suppes believed that behaviorist CAI type of programs could significantly change learning, but the computer is capable of much more than just delivering drill
and practice. It can create scenarios that engages children to want to learn using for example gamification mechanics. There is much promise here where the computer does the heavy lifting. But will
it motivate student's desire to be creative and fall in love with math (as Papert believed was possible) if students are left to their own devices and have the autonomy to construct their own
learning by building interesting simulations and gizmos themselves. Programming which requires computational thinking is making a comeback and is becoming an integral part of student construction of
personal knowledge. The opportunity to learn (that Uri Treisman referred to in his talk) is greatly enhanced by the emerging technologies and communities that embrace it to empower children to be
better learners and teaching becomes more focused on supporting that learning.
Clay Christensen of (Disrupting Class fame) writes (1) that "…there are two core jobs that most student try to do every day: They want to feel successful and make progress, and they want to have fun
with their friends." This is foundational for what I call the "Wannado" curriculum.
Larry Cuban (2) who usually reminds us how tech has flopped in classrooms in the past (For evidence see his slide show - a kind of trail of broken dreams) says the following in his recent book (3)
"If there is hope for the future it will happen in places where teachers collaborate and create schools where teaching and learning [are] prized." he writes. "[But] will such a ground level strategy
of building structures that enable teachers and administrators to work together in creating cultures of learning in classrooms, schools, and districts lead to good and successful teaching and then
successful student learning? I hope so - but in all honesty, I do not know."
I'm optimistic that it will and someday we can all look back and see how these young linchpins like Dan Meyer, Karim Ani and Alex Sarlin contributed to this promised land.
1. C. Christensen, M. Horn, C. Johnson, "Rethinking Student Motivation: Why understudying the 'job' is crucial for improving education." Innosight Institute. p. 7
2. L. Cuban, "Framing the School Technology Dream" (4/21/13)
3. L. Cuban, "Inside the Black Box of Classroom Practice: Change Without Reform in American Education" (Harvard Education Press, Cambridge MA) p. 185.
Math 180 - Student Dashboard
In my last blog entry I said I would be visiting the Scholastic Math 180 booth because I was intrigued with their promotion via snail mail. (See previous blog.) According to their website: "MATH 180
is a revolutionary math intervention program for the Common Core. Designed for struggling students in grades 6 and up, the program builds students’ confidence and competence in mathematics, while
providing teachers with comprehensive support to ensure success." After listening to their promo and playing around with the demo on the laptops provided in their spacious (mostly empty) booth I came
away wondering where's the revolution? Nothing unique or compelling here. That's what I thought until I attended Alex Sarlin's and David Dockterman's session entitled "The Gamification of Math:
Research, Game Theory and Math Instruction." It turns out that both David and Alex work for Scholastic and are the brains behind Math 180. It didn't take me long to realize that the goals of Math180
are more involved than my visit to the booth indicated.
David went first and described a typical student who doesn't consider himself a good math student. The question David raises is what will it take to turn this student's self image around and believe
that he can be good at math. At this point David introduces Alex who explained how gamification of math can help a weak math student have a much better experience with mathematics. Gamification is
not just about games. Its mechanics can be applied to non-game settings like math classrooms. To help explain these mechanics he uses as an example the very popular download Temple Run 2. "You want
the game to be easy to learn, exciting, compelling and which elicits a desire to keep playing by introducing appropriate challenges at appropriate times." he explains. "You are autonomous in certain
ways, using your skills to maneuver past obstacles." Also there is a narrative similar to Indiana Jones.
Gamification techniques leverage people's natural desires for competition, achievement, status, self-expression, altruism, and closure. - Wikipedia*
So how do you gamify a math classroom? Using a curriculum like Math 180 according to Scholastic. The main problem with traditional math approaches is that students don't see the point of what they
are learning. Math 180 provides a roadmap to success by providing a "GPS" so they see the larger picture and know where they are going.
I was excited about learning more about Math 180 so I went to the booth to see more.
The demo only included a small piece of the curriculum: Block 2 the Distributive Property. It was pretty boring. I tried to find some "Indiana Jones" motivation but didn't see anything like that in
the software. I complained to a Scholastic representative about it. Her reply was: "The demos will get better." Sigh.
There was no sign of either Alex or David in the booth or any advertisement for their session.
I'll try to contact David and/or Alex about my disappointment at the booth. I'll keep you posted.
Anybody else visit the Math 180/Scholastic Booth? Impressions?
*Here's more from Wikipedia about Gamification:
A core strategy for gamifying is to provide rewards for players for accomplishing desired tasks. Types of rewards include points,^[6] achievement badges or levels,^[7] the filling of a progress bar,^
[8] and providing the user with virtual currency.^[7]
Competition is another element of games that can be used in gamification. Making the rewards for accomplishing tasks visible to other players or providing leader boards are ways of encouraging
players to compete.^[9] Another approach to gamification is to make existing tasks feel more like games.^[10] Some techniques used in this approach include adding meaningful choice, onboarding with a
tutorial, increasing challenge,^[11] and adding narrative.^[10]
cc blog 134
A dynamic invitation
Every year around this time I start receiving (junk...) I mean advertising from companies that are planning to exhibit in Denver during the NCTM conference. The invite on the left was interesting
because it popped right out of the envelop. (Click on the link to see what I mean.)
I'm not usually inspired to accept such an invitation because I always assume an inverse relationship between the advertising budget of the company and the quality of the product. But since I'm
making a big deal about this, I will hold judgement and see the product in Booth #917 on April 18th at 8:45. After I see it I'll report back here with a review.
Here are some other technology oriented exhibitors I plan to visit during my stay in Denver. A lot of old friends and hopefully I'll make some new ones as well. Key Curriclum Press now has a new home
in McGraw Hill country.
AT< #637 develops 3D math video games. Their current product is the Lost Function: A Math Adventure game. New to me.
Big Brainz #1516 Timez Attack is their big hit product.
Buzzmath #1140 Their mission is to lead middle school students to proficiency in math through suppported practice.
Math Forum - #741 For more than 20 years providing resources to help teachers improve their mathematics teaching and learning. Long time friends of CLIME.
Carnegie Learning #1820 is a leading publisher of innovative, research-based math curricula for middle school, high school, and postsecondary students.
Calculus in Motion #1131 - Software. Also CDs with tons of Sketchpad applications.
Conceptua Math #823 - teaching and learning everything about fractions is the goal of their software.
Desmos Inc. #916 - Online calculator company. One of the co-hosts of Dan Meyer's and Karim Ani's happy hour on Thursday.
Dreambox Learning #1222 - Lot's of positive buzz about their products. I need to take a closer look.
GeoGebra #1433 - Sketchpad's competitor. Can't beat the price.
Explore Learning #1823 - Always give them a positive review for their excellent applets (Gizmos) and sometimes even their support material. See their equivalent fraction Gizmo.
How the Market Works #719 - Great way to bring real world math into the classroom.
Hooda Math #1538 - new to me.
Mind Research Institute #331 - Adventures with Jiji.
McGraw-Hill Education #731 - Geometer's Sketchpad and Tinkerplots. Sketchpad Users Group after hours session? Will find out...
Neufeld Learning Systems Inc. #2024 Rudy Neufeld - my friend from the early Logo days
Saltire Software #330 - Some really cool software.
See also CLIME's previous blogs about the conference:
Denver's tech sessions: Do they meet the CLIME Standard? - Link
Annual Meeting in Denver Technology Sessions Update - Link
If you are speaking on a technology themed topic in Denver please check for your listing here. If you would like to add or change anything to your listing please let me know via email at
ihor@clime.org or twitter @climeguy.
cc blog 133
Most of you who follow this blog are probably aware that I believe my vision of math 2.0 is no longer a fantasy, but a reality that is doable through the commitment of inspired educators who with the
help of powerful technological applications can hoist a learning revolution never seen before. The first step in such a revolution is having the technological infrastructure so that all students have
access to the technology. My friend and colleague Joshua Koen, the technology director at a committed urban school district in Passaic, NJ have taken the first steps in that direction. He calls it a
tipping point a phrase coined by Malcolm Gladwell in his book "The Tipping Point" which he describes in this video as: "It's a moment in which something explodes, something changes shape. It's that
moment of critical mass where everything changes all at once." With the technology in place (1-1 computing with Chromebooks for grades 7-12) the Passaic Schools now have the opportunity to fully
embrace a paradigm that focuses more on learning and where teaching and learning are indistinguishable.
A recent Edutopia article "How to Make your Classroom a Thinking Space" reminded me of the importance of classroom environment as one of the cornerstones* of my Dynamic Classroom model which is the
essence of Math 2.0.
"Take a moment and imagine a creative work environment. Don't worry about the kind of work going on. Just focus on the space. Close your eyes and picture it. What is that space like? What does it
sound like? How are people interacting? Is there movement? Is there evidence of work in progress? Is it tidy, or busy-messy? Can you imagine working there?"
To make your classroom work for you the authors suggest the following:
• Fine-tune the physical environment for PBL (Project Based Learning)
• Make a place for independent, partner and small-group work.
• Reimagine who the stuff belongs to.
• Make for a conversational classroom.
• Student presentations should be the norm.
• Encourage hands-on, minds-on creative thinking by providing tools for tinkering.
• Skype with other schools on collaborative projects.
• Create a video booth to capture student reflections.
In closing the authors write:
"What's on Your Wish List?
Teachers model creative thinking when they find workarounds or inexpensive fixes to make their classrooms more conducive to project work. They also model collaboration if they enlist parent
volunteers and other community members to help. Put your creativity to work by imagining how you might improve your classroom environment to invite good thinking. What belongs on your PBL wish
list? How might you make it happen?"
*The others are curriculum, resources, teaching, learning and assessing. See my blog entries for more about math 2.0 and the dynamic math classroom:
• The Dynamic Math Classroom 1.0 (version 2.0 in press)
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I've updated the listing of technology sessions at the Denver Annual NCTM meeting including more details about the sessions from the speakers who have responded to my request. If you are a speaker on
a technology theme and your session is not listed please let me know (ihor@clime.org) and I will include your session. Also if you wish to have more detail than what is listed please send me a photo
and additional information about your session. We know that you submitted your proposal to speak last May and you may want your potential attendees to know more about you and your presentation. I'll
be continuing to update this listing (here is the link again) right through the conference. So its never to late to do so.
I will be attending the conference though this year CLIME will not be exhibiting so I'll be free to roam around the conference and report back to you about the sessions and exhibits that I found
I hope to see you there.
Ihor aka @climeguy on Twitter
Here is the list of my "go to" sessions from a previous blog. I hope to attend as many as I can.
1. Dynamic Math software
5 - Chaos Games and Fractal Images
46 - Collecting Live Data in Fathom
80 - The Mathematics of Angry Birds
180 - Getting Serious about Games in Middle Grades Math (Lure of the Labyrinth) - Scott Osterweil
207 - Do the Function Dance with Sketchpad 5 - Scott Stekettee, Dan Scher
283 - The Gamification of Math: Research, Gaming Theory, and Math Instruction
502 - Help Students Dig into Data, Statistics, and Probability with TinkerPlots - Karen Greenhaus
279 - Math and Geography: Using Google Earth to Investigate Mathematics
2. Web 2.0 Tools
157 - Math Journal 2.0: Jump-Start Your Students' Reflections (blogging)
447 - Movie Making in Math
468 - Scan It, Solve It, Show It (QR Codes, BYOD-Bring your own device)
565 - Blogarithms: Converting Number Concepts into Talking Points
586 - Moving Beyond the Right Answer: Developing Students’ Math Communication Skills
707 - Sharing Student Lessons with iBooks Author, iBooks, and an iPad
717 - Effective Use of Virtual Manipulatives: Ready to Create Your Own?
724 - Viral Math Videos: A Hart-to-Hart Conversation
3. Dynamic Learning Communities
143 - PLC: The Practices, the Lessons, the Collaborative
680 - An Invitation to Experience Online Lesson Study Firsthand
4. Math 2.0 Curriculum
141 - Learning Online and Outdoors: Integrating Geocaching into the Mathematics Classroom - Lucy Bush and Jeff Hall - see their article on page 20 of this link
184 - Keeping It Real: Teaching Math through Real-World Topics (mathalicious.org) - Karim Ani
402 - Stories and Technology: Providing Mathematics Opportunities for All
560 - Powerful Online Tools Promote Powerful Mathematics (Illuminations) - Patrick Vennebush
684 - Tools and Technology for Modern Math Teaching - Dan Meyer
685 - Computers in Early Childhood: Getting the Best of All Worlds - Doug Clements, Julie Sarama
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Harmonic Analysis and Operator Theorysearch
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Harmonic Analysis and Operator Theory
A. Octavio : Institute Venezuela de Investigaciones Cientificas, Caracas, Venezeula
eBook ISBN: 978-0-8218-7780-7
Product Code: CONM/189.E
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Harmonic Analysis and Operator Theory
A. Octavio : Institute Venezuela de Investigaciones Cientificas, Caracas, Venezeula
eBook ISBN: 978-0-8218-7780-7
Product Code: CONM/189.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
• Contemporary Mathematics
Volume: 189; 1995; 511 pp
MSC: Primary 43; 47; Secondary 44; 45; 46
This book is a collection of papers reflecting the conference held in Caracas, Venezuela, in January 1994 in celebration of Professor Mischa Cotlar's eightieth birthday. Presenting an excellent
account of recent advances in harmonic analysis and operator theory and their applications, many of the contributors are world leaders in their fields. The collection covers a broad spectrum of
topics, including:
□ wavelet analysis
□ Haenkel operators
□ multimeasure theory
□ the boundary behavior of the Bergman kernel
□ interpolation theory
□ Cotlar's Lemma on almost orthogonality in the context of \(L^p\) spaces and more ...
The range of topics in this volume promotes cross-pollination among the various fields covered. Such variety makes Harmonic Analysis and Operator Theory an inspiration for graduate students
interested in this area of study.
Mathematicians working in harmonic analysis and operator theory and related subjects, and graduate students interested in analysis.
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 189; 1995; 511 pp
MSC: Primary 43; 47; Secondary 44; 45; 46
This book is a collection of papers reflecting the conference held in Caracas, Venezuela, in January 1994 in celebration of Professor Mischa Cotlar's eightieth birthday. Presenting an excellent
account of recent advances in harmonic analysis and operator theory and their applications, many of the contributors are world leaders in their fields. The collection covers a broad spectrum of
topics, including:
• wavelet analysis
• Haenkel operators
• multimeasure theory
• the boundary behavior of the Bergman kernel
• interpolation theory
• Cotlar's Lemma on almost orthogonality in the context of \(L^p\) spaces and more ...
The range of topics in this volume promotes cross-pollination among the various fields covered. Such variety makes Harmonic Analysis and Operator Theory an inspiration for graduate students
interested in this area of study.
Mathematicians working in harmonic analysis and operator theory and related subjects, and graduate students interested in analysis.
Permission – for use of book, eBook, or Journal content
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Early Warning Signals for Regime Transition in the Stable Boundary Layer: a Model Study
Friday, 24 June 2016: 1:45 PM
Bryce (Sheraton Salt Lake City Hotel)
An idealized model for the nocturnal boundary layer is employed to study dynamics of the evening transition. The model consists of a horizontally homogeneous flow with a prescribed top velocity to
mimic the wind speed around sunset. Although a fixed-wind-speed boundary may appear somewhat artificial, it is based on the observation that wind near the surface tends to weaken around sunset, while
winds aloft accelerate. Consequently, an altitude exists, the so-called crossing point, where winds remain relatively constant for several hours. Furthermore, a fixed sensible heat flux is prescribed
at the surface to mimic net radiative cooling. From earlier studies it is known that the nocturnal boundary layer manifests itself in one of two distinct regimes, depending on ambient synoptic
conditions. Strong-wind or overcast conditions typically lead to weakly stable, turbulent nights, while clear-sky and weak-wind conditions, on the other hand, lead to very stable, weakly turbulent
nights. Physically, this two-regime division may be understood as follows: for both neutral conditions (no gradients) and strongly stratified conditions (no mixing) the sensible heat flux becomes
very small. As such, a maximum downward sensible heat flux must exist at intermediate stability. The magnitude of the maximum flux is strongly dependent on the wind speed. For low-heat capacity,
insulated surfaces (e.g. fresh snow) the sensible heat flux acts as the main energy supply to the surface. In case of weak-wind and clear-sky conditions, the maximum heat flux (energy gain) may be
significantly less than the net radiative cooling (energy loss). In such nights strong surface cooling may occur, leading to a strong inversion near the surface, which further inhibits the sensible
heat flux. Understanding of the boundary layer behavior close to and beyond the critical point remains limited. At the same time, phenomena like ground frost or radiation fog are more likely to occur
under such conditions. Previously, a similar idealized set-up (i.e. using a fixed wind speed at crossing level and a prescribed surface flux) was used to study the dynamical behavior near the
transition between weakly stable and very stable boundary layers and to predict the transition point. The model used, however, relied on Monin-Obukhov (MO) similarity to describe turbulent transport.
As first innovative aspect, we investigate a similar set-up, using direct numerical simulation. In contrast to MO-based models, this type of simulation does not rely on turbulent closure assumptions.
By systematically increasing the surface cooling previous predictions are verified, but now independently of parameterizations for turbulence. Results show that turbulence intensity remains
relatively unaffected by stability until close to the critical point. When the prescribed surface flux is increased beyond the critical point, turbulence intensity suddenly becomes very weak as the
flow transitions to a very stable state. As second innovative aspect, specific changes in dynamical behavior of the turbulent flow in the weakly stable regime are investigated. These changes are
closely related to the existence of a heat-flux maximum, as at the maximum itself, no change in stability could further increase the sensible heat flux. Based on MO similarity it is therefore
hypothesized that the ratio between the change in sensible heat flux to the change in stability is an indicator for the distance to the critical point. Indeed, the results indicate that such changes
signal the arrival of a regime-shift prior to the onset of the very stable state (see figure). Here, we show how these changes may be used to infer a quantitative estimate of the transition point. In
addition, it is shown that the idealized, nocturnal boundary layer system, shares important similarities with generic non-linear dynamical systems that exhibit critical transitions. For such generic
systems it is well known that the typical decay time of perturbations with respect to the equilibrium solution, tends to be larger in the vicinity of a regime shift. In the current system the typical
time scale required to reach equilibrium is measured for different prescribed surface heat fluxes. It is shown that indeed the typical time scale for decay to equilibrium increases when the system is
closer to the regime shift. ========================================================== Figure: Asterisks depict the ratio between change in sensible heat flux and change in stability as a function of
the prescribed surface heat flux. Note that all units are dimensionless. Stability is measured by the time-averaged vertical temperature difference when the system is in steady state. Each asterisks
depicts the steady state value of a single run. The thin grey line indicates a linear fit through the numerical results. By extrapolating the linear fit to the horizontal axis a closure-independent
estimate for the transition point is obtained close to the observed transition point. ========================================================== | {"url":"https://ams.confex.com/ams/32AgF22BLT3BG/webprogram/Paper294360.html","timestamp":"2024-11-02T07:47:07Z","content_type":"text/html","content_length":"14430","record_id":"<urn:uuid:c3ae1c79-6f22-4697-9624-6b6f63096887>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00814.warc.gz"} |
Ask Uncle Colin: A Pair Of Toy Trucks
Dear Uncle Colin,
I have an exam question I don’t understand! There’s a toy truck of mass 5kg attached (by a rod) to another truck of mass 2kg on a slope at 10 degrees to the horizontal. The resistances to motion
are 8N and 6N, respectively, and the whole thing is pulled up the slope by a string. The question says to show that the rod is always in tension so long as the trucks move up the slope. What’s
your approach?
Toys Easily Negotiate Slope In Our Nursery
Hi, TENSION, and thanks for your message!
The first thing I would do is draw a picture. It would probably be a messy picture, like this one.
Since the trucks are moving up the plane, the resistance forces act down it. There’s an equal-and-opposite force in the coupling which I’ve drawn as a tension, but could be a thrust (if so, $T_c$
will wind up negative, and that’s perfectly ok in a rod. In a string, it would be a problem).
I like to think of a grid with this kind of question, with lines parallel and perpendicular to the surface. In fact, by turning the page, I can draw it like this:
The two weights are the only ones off of the grid - and I only really care about the ‘horizontal’ components of those (that is, parallel to the plane). Those are $2g \sin(10º)$ and $5g \sin(10º)$,
Equations of motion!
Then I can write down equations of motion for the parts of the system, and the system as a whole:
• Upper truck: $T_s - T_c - 8 - 5g\sin(10º) = 5a$
• Lower truck: $T_c - 6 - 2g\sin(10º) = 2a$
• System: $T_s - 14 - 7g \sin(10º) = 7a$
The acceleration could be in either direction, but as long as the velocity is upwards, these equations hold true.
Starting with the system as a whole, $a = \frac{T_s}{7} - 2 - g\sin(10º)$.
Substituting this into the lower truck equation gives: $T_c - 6 - 2g\sin(10º) = 2\left( \frac{T_s}{7} - 2 - g\sin(10º)\right)$, or $T_c = \frac{2}{7} T_s + 2$
Since $T_s \ge 0$ (it’s a tension in a string), $T_c \ge 2$ and is in tension as long as the motion is uphill.
Hope that helps!
- Uncle Colin
A selection of other posts
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Math in Focus Grade 8 Chapter 5 Lesson 5.1 Answer Key Introduction to Systems of Linear Equations
Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 5 Lesson 5.1 Introduction to Systems of Linear Equations to score better marks in the exam.
Math in Focus Grade 7 Course 3 A Chapter 5 Lesson 5.1 Answer Key Introduction to Systems of Linear Equations
Math in Focus Grade 8 Chapter 5 Lesson 5.1 Guided Practice Answer Key
Solve the system of linear equations by copying and completing the tables of values.
The values x and y are positive integers.
Question 1.
A bottle of water and a taco cost $3. The cost of 3 bottles of water is $1 more than the cost of a taco.
Let x be the price of a bottle of water and y be the price of a taco in dollars.
The related system of equations and tables of values are:
3x – y = 1
x + y = 3
The cost of a bottle of water is $2 and the cost of a taco is $1,
Solve each system of equations by making tables of values, x and y are positive integers.
Question 2.
x + y = 6
x + 2y = 8
Solved each system of equations by making tables of values, x and y are positive integers.
For equation x+y = 6, when x =1 then y =5 and when x =2 then y=4,
For equation x +2y =8, when x =1 then y =7/2 and when x = 2 then x =3.
Question 3.
x + y = 8
x – 3y = -8
Solved each system of equations by making tables of values, x and y are positive integers.
For equation x+y = 6, when x =1 then y =5 and when x =2 then y=4,
For equation x +2y =8, when x =1 then y =7/2 and when x = 2 then x =3.
For each linear equation, list in a table enough values for x and y to obtain a solution.
Remember that they must be positive integers.
Technology Activity
Use Tables On A Graphing Calculator To Solve A System Of Equations
Work in pairs.
You can use a graphing calculator to create tables of values and solve systems of equations.
Use the steps below to solve this system:
8x + y = 38
x – 4y = 13
Step 1.
Solve each equation for y in terms of x. Input the two resulting expressions for y into the equation screen.
Use parentheses around fractional coefficients and the
Step 2.
Set the table function to use values of x starting at 0, with increments of 1.
Step 3.
Display the table. It will be in three columns as shown.
Step 4.
Find the row where the two y-values are the same. This y-value and the corresponding x-value will be the solutión to the equations.
The solution to the system of equations is given by x =
Math Journal How can you tell from the two columns of y-values that there is only one row where the y-values are the same?
Math in Focus Course 3A Practice 5.1 Answer Key
Solve each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 1.
2x + y = 5
x – y = -2
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 2.
x + 2y = 4
x = 2y
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 3.
3x + 2y = 10
5x – 2y = 6
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 4.
x – 2y = -5
x = y
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 5.
2y – x = -2
x + y = 2
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 6.
2x + y = 3
x + y = 1
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 7.
x + 2y = 1
x – 2y = 5
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 8.
2x – y = 5
2y + x = -1
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Question 9.
2x + y = -1
x + y = 1
Solved each system of linear equations by making tables of values.
Each variable x is a positive integer less than 6.
Solve by making a table of values. The values x and y are integers.
Question 10.
A shop sells a party hat at x dollars and a mask at y dollars.
On a particular morning, 10 hats and 20 masks were sold for $30.
In the afternoon, 8 hats and 10 masks were sold for $18. The related system of linear equations is:
10x + 20y =30
8x + 10y = 18
Solve the system of linear equations. Then find the cost of each hat and each mask.
The cost of each hat is $1 and each mask is$1,
Given A shop sells a party hat at x dollars and a mask at y dollars.
On a particular morning, 10 hats and 20 masks were sold for $30.
In the afternoon, 8 hats and 10 masks were sold for $18.
The related system of linear equations is:
10x + 20y =30—(1)
8x + 10y = 18—-(2) dividing equation 1 by 10 we get
x +2y = 3,
x = 3-2y substituting in equation 2 we get
8(3-2y) +10y =18,
24 – 16y+10y =18,
24 -6y =18,
6y = 24 -18,
6y =6, y =1 we have x = 3-2X 1 = 3-2 = 1.
Question 11.
Alicia is x years old and her cousin is y years old.
Alicia is 2 times as old as her cousin.
Three years later, their combined age will be 27 years.
The related system of linear equations is:
x = 2y
x + y = 27
Solve the system of linear equations. Then find Alicia’s age and her cousin’s age.
Alicia’s age is 18 and her cousin’s age is 9 years old,
GivenAlicia is x years old and her cousin is y years old.
Alicia is 2 times as old as her cousin.
Three years later, their combined age will be 27 years.
The related system of linear equations is: x = 2y,
x + y = 27 solving the equations 2y + y = 27,
3y = 27, y = 27/3 =9, so x = 2X 9 = 18, Alicia’s age is 18 and her
cousin’s age is 9 years old.
Question 12.
Steve and Alex start driving at the same time from Boston to Paterson.
The journey is d kilometers. Steve drives at 100 kilometers per hour and
takes t hours to complete the journey. Alex, who drives at 80 kilometers per hour is
60 kilometers away from Paterson when Steve reaches Paterson.
The related system of linear equations is:
100t = d
80t = d – 60
Solve the system of linear equations by making tables of values.
Then find the distance between Boston and Paterson.
300 kilometers is the distance between Boston and paterson,
Given Steve and Alex start driving at the same time from Boston to Paterson.
The journey is d kilometers. Steve drives at 100 kilometers per hour and
takes t hours to complete the journey. Alex, who drives at 80 kilometers per hour is
60 kilometers away from Paterson when Steve reaches Paterson.
The related system of linear equations is:
100t = d
80t = d – 60 substituting 80t = 100t -60,
100t-80t = 60,
20t = 60, t = 60/20, t =3 so distance is 100 X 3= 300 kilometers.
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History of the Banach Fixed Point Theorem
Fimmtudaginn, 14. september 2023 - 16:30
Askja, stofa 132
Małgorzata Terepeta, prófessor við Tækniháskólann í Łódź heldur fyrirlestur fyrir Íslenska stærðfræðafélagið. Kaffi og spjall hefst kl. 16:30 og fyrirlesturinn sjálfur hefst kl. 17.
Ágrip: On June 24, 1920 Stefan Banach presented his doctoral dissertation titled operacjach na zbiorach abstrak-cyjnych i ich zastosowaniach do równań całkowych(On operations on abstract sets and
their applications tointegral equations) to the Philosophy Faculty of Jan Kazimierz University in Lvov. He passed his PhD exami-nations in mathematics, physics and philosophy, and in January 1921 he
received his PhD degree. A year later,he published the results of his doctorate in Fundamenta Mathematicae. Among them there was the theoremknown today as the Banach Fixed Point Theorem or the Banach
Contraction Principle. It is one of the mostfamous theorems in mathematics, one of many under the name of Banach. It concerns certain mappings (calledcontractions) of a complete metric space into
itself and it gives the conditions sufficient for the existence anduniqueness of a fixed point of such mapping. In 2022 we had a centenary of publishing this theorem and we in thetalk there will be
presented its most important modifications and generalizations, several contractive conditions,the converse theorems and some applications. | {"url":"https://stae.is/%C3%ADsf/event/51660","timestamp":"2024-11-01T21:57:53Z","content_type":"application/xhtml+xml","content_length":"15184","record_id":"<urn:uuid:ec801559-e450-4f8e-a936-b249eb2b5c94>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00534.warc.gz"} |
Post TOPIC: Dark Energy
RE: Dark Energy Permalink
Anthropic versus cosmological solutions to the coincidence problem
A. Barreira, P.P. Avelino
In this paper we investigate possible solutions to the coincidence problem in flat phantom dark energy models with a constant dark energy equation of state and quintessence models with a linear
scalar field potential. These models are representative of a broader class of cosmological scenarios in which the universe has a finite lifetime. We show that, in the absence of anthropic
constraints, including a prior probability for the models inversely proportional to the total lifetime of the universe excludes models very close to the \Lambda {CDM} model. This relates a
cosmological solution to the coincidence problem with a dynamical dark energy component having an equation of state parameter not too close to -1 at the present time. We further show, that
anthropic constraints, if they are sufficiently stringent, may solve the coincidence problem without the need for dynamical dark energy.
Read more
(42kb, PDF)
NASA's Hubble Rules Out One Alternative to Dark Energy
Astronomers using NASA's Hubble Space Telescope have ruled out an alternate theory on the nature of dark energy after recalculating the expansion rate of the universe to unprecedented accuracy.
The universe appears to be expanding at an increasing rate. Some believe that is because the universe is filled with a dark energy that works in the opposite way of gravity. One alternative to
that hypothesis is that an enormous bubble of relatively empty space eight billion light-years across surrounds our galactic neighbourhood. If we lived near the centre of this void, observations
of galaxies being pushed away from each other at accelerating speeds would be an illusion.
This hypothesis has been invalidated because astronomers have refined their understanding of the universe's present expansion rate. Adam Riess of the Space Telescope Science Institute (STScI)
and Johns Hopkins University in Baltimore, Md., led the research. The Hubble observations were conducted by the SHOES (Supernova Ho for the Equation of State) team that works to refine the
accuracy of the Hubble constant to a precision that allows for a better characterisation of dark energy's behaviour. The observations helped determine a figure for the universe's current
expansion rate to an uncertainty of just 3.3 percent. The new measurement reduces the error margin by 30 percent over Hubble's previous best measurement of 2009. Riess' results appear in the
April 1 issue of The Astrophysical Journal.
Read more
Modelling Time-varying Dark Energy with Constraints from Latest Observations
Qing-Jun Zhang, Yue-Liang Wu
We introduce a set of two-parameter models for the dark energy equation of state (EOS) w(z) to investigate time-varying dark energy. The models are classified into two types according to their
boundary behaviours at the redshift z=(0,\infty) and their local extremum properties. A joint analysis based on four observations (SNe + BAO + CMB + H_0) is carried out to constrain all the
models. It is shown that all models get almost the same \chi²_{min}\simeq 469 and the cosmological parameters (\Omega_M, h, \Omega_bh²) with the best-fit results (0.28, 0.70, 2.24), although the
constraint results on two parameters (w_0, w_1) and the allowed regions for the EOS w(z) are sensitive to different models and a given extra model parameter. For three of Type I models which
have similar functional behaviours with the so-called CPL model, the constrained two parameters w_0 and w_1 have negative correlation and are compatible with the ones in CPL model, and the
allowed regions of w(z) get a narrow node at z\sim 0.2. The best-fit results from the most stringent constraints in Model Ia give (w_0,w_1) = (-0.96^{+0.26}_{-0.21}, -0.12^{+0.61}_{-0.89}) which
may compare with the best-fit results (w_0,w_1) = (-0.97^{+0.22}_{-0.18}, -0.15^{+0.85}_{-1.33}) in the CPL model. For four of Type II models which have logarithmic function forms and an
extremum point, the allowed regions of w(z) are found to be sensitive to different models and a given extra parameter. It is interesting to obtain two models in which two parameters w_0 and w_1
are strongly correlative and appropriately reduced to one parameter by a linear relation w_1 \propto (1+w_0).
Read more
(359kb, PDF)
Is Dark Energy a Cosmic Casimir Effect?
Kevin Cahill
Unknown short-distance effects cancel the quartic divergence of the zero-point energies. If this renormalisation took effect in the early universe after the last phase transition and applied
only to modes whose wavelengths (over 2 pi) were shorter than the Hubble length 1/H at that time, then the zero-point energies of the modes of longer wavelengths can approximately account for
the present value of the dark-energy density. The model makes two predictions.
Read more
(9kb, PDF)
Dark Energy: Was Einstein Right After All?“That's why a new observation by scientists at the National Radio Astronomy Observatory, in Virginia, could be so important. By linking a group of
far-flung radio telescopes into a virtual telescope thousands of miles across, James Braatz and Cheng-Yu Kuo have measured the distance to galaxy NGC 6264 to an accuracy of 450 million
light-years from Earth, give or take 9%.
That's crucial, because while it's simple to measure how fast a galaxy is moving, you also need to know exactly where it is. Imagine that a car is accelerating toward you, and you want to know
when it will zip by. To calculate that, you need to know not only how fast it's going at any given moment, but also how far away it is.
They've homed in on the black hole at the core of NGC 6264 - or more precisely, the disk of gas that swirls around it before being sucked into oblivion. Water molecules in the disk act as
natural masers - essentially, they're lasers that transmit in radio frequencies rather than visible light. With those masers acting as beacons, the astronomers used a single radio telescope to
figure out the actual size of the disk. Then they used their virtual radio to measure its apparent size - how tiny it looks at such an enormous distance.”Read more
Testing the expanding Universe“Our limited view of the cosmos obscures the identity of the mysterious forces that are responsible for the accelerating expansion of the Universe. Physicists at
the University of Cambridge, UK, now say in two papers that the 'cosmological constant' - which is used to represent the Universe's expansion in cosmological equations - depends on the time and
location where it is measured. This could explain long-standing problems with the constant and help physicists to explain the Universe's expansion. ”Read more
A New Solution of The Cosmological Constant Problems
John D. Barrow, Douglas J. Shaw
(Version v3)
We extend the usual gravitational action principle by promoting the bare cosmological constant (CC) from a parameter to a field which can take many possible values. Variation leads to a new
integral constraint equation which determines the classical value of the effective CC that dominates the wave function of the universe. In a realistic cosmological model, the expected value of
the effective CC, is calculated from measurable quantities to be O(t_U), as observed, where t_U is the present age of the universe in Planck units,. Any application of our model produces a
falsifiable prediction for \Lambda in terms of other measurable quantities. This leads to a specific falsifiable prediction for the observed spatial curvature parameter of Omega_k0=-0.0055. Our
testable proposal requires no fine tunings or extra dark-energy fields but does suggest a new view of time and cosmological evolution.
Read more
(12kb, PDF)
Reconciling the local void with the CMB
Seshadri Nadathur, Subir Sarkar
In the standard cosmological model, the dimming of distant Type Ia supernovae is explained by invoking the existence of repulsive 'dark energy' which is causing the Hubble expansion to
accelerate. However this may be an artifact of interpreting the data in an (oversimplified) homogeneous model universe. In the simplest inhomogeneous model which fits the SNe Ia Hubble diagram
without dark energy, we are located close to the centre of a void modelled by a Lemaitre-Tolman-Bondi metric. It has been claimed that such models cannot fit the CMB and other cosmological data.
This is however based on the assumption of a scale-free spectrum for the primordial density perturbation. An alternative physically motivated form for the spectrum enables a good fit to both SNe
Ia (Constitution/Union2) and CMB (WMAP 7-yr) data, and to the locally measured Hubble parameter. Constraints from baryon acoustic oscillations and primordial nucleosynthesis are also satisfied.
Read more
(115kb, PDF)
Distant Galaxies Confirm Dark Energy's Existence and Universe's Flatness“In the late 1990s, two teams of astronomers stunned the scientific community with the finding that the universe is
accelerating in its expansion, somehow overpowering the constant pull of gravity that should be slowing it down. The culprit pressing the cosmic accelerator goes by the name "dark energy," which
is an appropriately enigmatic moniker for something that remains so poorly understood.”Read more
Geometric test supports the existence of a key thread in the fabric of the Universe.“The claim that mysterious dark energy is accelerating the Universe's expansion has been placed on firmer
ground, with the successful application of a quirky geometric test proposed more than 30 years ago.
The accelerating expansion was first detected in 1998. Astronomers studying Type 1a supernovae, stellar explosions called "standard candles" because of their predictable luminosity, made the
incredible discovery that the most distant of these supernovae appear dimmer than would be expected if the Universe were expanding at a constant rate. This suggested that some unknown force -
subsequently dubbed dark energy - must be working against gravity to blow the universe apart.”Read more | {"url":"https://58381.activeboard.com/t2981747/dark-energy/?page=5","timestamp":"2024-11-11T05:14:43Z","content_type":"application/xhtml+xml","content_length":"98572","record_id":"<urn:uuid:06e99fab-863c-4218-a072-76fe36d1ad21>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00038.warc.gz"} |
Two rafters A and B are going to simultaneously cross a river on separate rafts starting from point O. Because of the turbulent water current in the r - London Term Papers
Two rafters A and B are going to simultaneously cross a river on separate rafts starting from point O. Because of the turbulent water current in the r
Two rafters A and B are going to simultaneously cross a river on separate rafts starting from point O. Because of the turbulent water current in the river, rafters A and B reach points X and Y,
respectively, to the right of point O’ on the opposite bank of the river. Suppose that the distance of point X from O’ is a uniform random variable withand, and that Y is also a uniform random
variable withand. A small village V is situated 4km from O’. Whta is the probability that rafter B will be closer to the village than A when they make it across the river? Assume that X and Y are | {"url":"https://www.londontermpapers.co.uk/essays/two-rafters-a-and-b-are-going-to-simultaneously-cross-a-river-onseparate-rafts-starting-from-point-o-because-of-the-turbulentwater-current-in-the-r/","timestamp":"2024-11-10T05:57:09Z","content_type":"text/html","content_length":"52925","record_id":"<urn:uuid:415c2cf6-9d94-4102-8396-664bd6c7d9c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00349.warc.gz"} |
Crack Spread Forecasting for Supply Chain Optimization - A Hybrid Model using Time Series and Deep Learning with Bayesian Optimizations
Crack Price Forecasting, Deep Learning, LSTM, Bayesian Optimizations, Supply Chain Optimization, Oil Refineries.
The price difference (gap) between crude oil and refined goods distillates (such gasoline, diesel, jet fuel, etc.) significantly impacts the profit margins of refiners in the petroleum industry (
Steven et al., 2014). Since the refining process "Crack" crude oil into its primary refined products, this spread is commonly referred to as the "Crack Spread".
‘Crack Spread’ is of interest to refiners, traders, regulators, hedge funds & speculators, oil market players and academicians (Fousekis & Grigoriadis, 2017). The perspectives and objectives with
which they view ‘Crack spread’, however, varies widely. Academic researchers have explored co-integration between petroleum product and crude price, predictive power of crack spread for oil price
forecasting, efficacy of day trading strategy for crack spread modelling etc (Poitras & Teoh, 2003). Regulators and International Financial institutions observe crack-spread from the point of view of
policy response to market inefficiencies, trade facilitation and regulation of the market. Hedge funds frequently use crack spreads to speculate in oil markets (Murat & Tokat, 2009), and to hedge
against refinery stocks or perform directional trade for energy portfolio. Investors can employ crack spread trading as a hedge against the stock value of a refining company. Due to the added benefit
of low margins (the crack spread trade generates a big spread credit for margining purposes), other professional traders may think about utilizing crack spreads as a directional trade as part of
their energy portfolio. When combined with other indicators such as crude oil stocks and refinery utilization rates, fluctuations in crack spreads or refining margins can offer investors enhanced
insights into the future trajectories of specific companies and the broader oil market in the upcoming period. Refiners are exposed to volatility in both the markets in which they operate: crude oil
price fluctuations on supply side and volatility in price of refined products in retailing market (Karathanasopoulos et al., 2016). As per (Mahringer & Prokopczuk, 2015), companies engaged in selling
crude oil or other products solely to the wholesale and retail markets might face higher susceptibility to market risk compared to refiners. This is as a result of their simultaneous involvement in
both markets. (Ederington et al., 2019) contend that numerous factors, such as supply, demand, production economics, environmental restrictions, and other elements, often have individual impacts on
the prices of crude oil and its primary refined products.
About 85% (Robinson, 2007) of a typical refining operating cost is made up of crude oil. Because of this, refiners and non-integrated marketers face a significant risk from rising crude oil prices
and falling refined product prices on both sides of the market. This can significantly reduce the crack's spread. As a result, executives at refineries are constantly worried about hedging their
crack spread risk. With the ability to predict the movement of crack prices well in advance, the refiners can dynamically optimize the right crude-mix through multi-sourcing and development of a
compatible hedging strategy for managing the price risk. Refiners seek a return on their invested assets while covering the ongoing and fixed costs of operating the refinery. However, when facing an
unpredictable crack spread, refiners find it challenging to assess their actual financial exposure as they can accurately anticipate expenses for everything except crude oil. Oil refiners, therefore,
in an effort to optimize the supply chain of oil products, model and predict crack spread with the objectives (Dunis et al., 2016) of (a) Profit maximization by deciding when to procure stocks of
crude or processed products (b)Increasing GRM (Gross Refining Margin, or GRM is the difference between the cost of the crude oil used as the input and the total value of the petroleum products that
are produced in an oil refinery by selectively choosing crack to process, and (c) Implementing low risk trading / hedging against volatility in crude price fluctuations through futures contracts for
protecting the margins. Cracks are thus particularly useful in implementing various procurement strategies and therefore oil refiners have substantial interest in protecting the crack spread (Haigh &
Holt, 2002).
Indian Oil Refining and Marketing Companies are more integrated than general oil companies conducting refining and marketing functions in tandem. Therefore, they are not only exposed to the
volatility of Crude oil prices arising out of several reasons mentioned earlier but also to inelastic changes in product prices which in turn are impacted by variations in worldwide demand, regional
factors, seasonality etc.
Indian Oil refiners import about 85 % of the total required crude oil to meet India’s ever increasing domestic energy demand. The refined petroleum products such as LPG, Petrol, Diesel, Kerosene etc.
are sold through the various distribution channels (Dealers, Distributors, Agents etc.) across the country. The profitability of Indian Oil Marketing Companies (OMCs) primarily relies on the cost of
imported crude oil and the selling price of products, both of which are linked to international prices as benchmarks. For the purpose of calculating profitability, Indian Oil companies use the prices
of Crude from Dubai and prices of products from Singapore exchange.
It has been studied that whereas the Futures of commodities are good forecasts for their spot prices, the same cannot be said for crude and petroleum Crack prices (Reichsfeld & Roache, 2011).
Therefore, excellent quality medium term forecasts for Crude and Crack spot prices become the cornerstone of procurement decisions of Indian Oil Companies. State run Indian oil companies have
societal obligation to ensure undisrupted supply of fuels across the country. Even in the face of impending unfavourable marketing conditions owing to volatility in crude oil prices, the option to
stall operations till favourable return is simply non-existent. Thus, the volatility of crude price and its impact has to be borne out.
With a limited mandate to indulge in financial transactions such as hedging, state run oil companies in India are severely constrained in ensuring profitability and margin protection. In the absence
of hedging as an option, Indian state-run oil companies need to resort to forecasted crack spread prices for their procurement planning decisions. The modelling and predicting crack spreads is
therefore of greater relevance to refiners for optimizing their supply chain operations and hedging against the volatility in crude oil prices. With reference to the above discussion this study
proposes following research questions:
RQ1: How can the crack spread forecast be improved by employing recent advances in Artificial Intelligence and Deep Learning techniques?
RQ2: Can the hybridization of Deep Learning techniques with other popular statistical techniques lead to improvement in forecast accuracy?
RQ3: How can model development and training be optimized by Bayesian optimization?
To solve above research questions this study aims to develop a two-stage hybrid model (Figure 1) which will in stage 1 model the linearity present in crack spread time series data using statistical
modelling techniques (ARIMA/ETS) and in stage 2 model the non-linearity present in forecast error using non-linear technique such as ANN Deep Learning technique (LSTM/BiLSTM). Also, to optimize the
time for development / testing of Deep Learning models, Bayesian optimization is employed. The results of the hybrid model are compared with those of standalone models using metrics namely MSE, RMSE,
MAE, MAPE and RMAPE.
Literature Review
(Karathanasopoulos et al., 2016) utilized a non-linear approach, which integrated a particle swarm optimizer (PSO), a radial basis function neural network, and a sliding window methodology to model
and monitor the crack spread. When the model was trained across a 300–400 day sliding window and utilized to produce forecasts for 5 days, it outperformed an MLP neural network in terms of
performance and statistical correctness. The forecast time horizon of 5 days though is appropriate for use in trading decision making but is not suitable for Refiners whose turn-around time for
procurement-refining-sale of products is 3 months.
High Order Neural Networks (HONN) were used on time series data in a different study on the modelling and trading of gasoline crack spread (Karathanasopoulos et al., 2016) to examine the
co-integration between WTI crude and gasoline prices. The method was put to the test in order to forecast how the spread would vary from one closing price to the next. The study did find a non-linear
relationship between the price of WTI oil and the cost of gasoline. It is of limited value to refiners because the expected time horizon utilizing the HONN is too short to be regarded a valid fit for
the time horizon of three months.
For modelling and forecasting crack spreads have studied ARFIMA (Autoregressive Fractionally Integrated Moving Average) models applied to single product crack. The objective was to verify that crack
spread and their volatilities display long range autocorrelation. Also, the study attempted to ascertain that whether crack spreads exhibit multi-fractionality wherein small and large fluctuations
exhibit different scaling behaviours. It concluded that Crack spreads exhibited weak mean revering behaviour for time horizon of days and weeks and exhibited strong mean reverting behaviour in weeks
and months’ time horizon. On comparison it was found that ARFIMA could not outperform RWM out of sample forecasting and it could not capture the complexity of time series leading to larger prediction
bias. ARFIMA could however capture long range autocorrelation but could not capture scaling. Also, it was able to capture multi-fractionality but not small-scale fluctuations.
The multiple regression analysis approach for modelling for naphtha cracking (Sung et al., 2012) looked at the key variables influencing naphtha prices and sought to pinpoint the most important ones.
Analyses of naphtha crack's actual and projected variations were also conducted. It was determined that the key factors influencing naphtha pricing were margin, substitutes for naphtha, and supply
and demand of naphtha in Asia. For the same, a statistical model was created. In a more recent study by (Ewing & Thompson, 2018), the response of the gasoline-crude oil price crack spread to
macroeconomic shocks, such as economic growth, inflation, corporate default risk, and monetary policy shocks, has been analysed and modelled. This method combines a generalized impulse response
analysis with a vector auto-regressive approach. Although the results were confined (i.e., pertinent to the specific context of the United States of America), they nonetheless evaluated how shocks to
these variables would affect the spread of gasoline crack. The results are listed in Table 1 below.
Table 1 Related Research
Year Objective Technique Findings Citations
To ensure model evaluation and comprehensibility, the top-performing
West Texas Intermediate (WTI) oil machine learning models are assessed and compared using DeLong The use of Explainable Machine Learning (XML) models and ML (H. Guliyev & E.
2022 price patterns are predicted using statistical test methods and SHAP (SHapley Additive exPlanations) approaches can enhance the forecasting of WTI crude oil prices and Mustafayev, 2022)
data from 1991 through 2021. values. Among the models in this category are Logistic Regression, have a significant impact on global economic policy.
Decision Tree, Random Forest, AdaBoost, and XgBoost.
With the proposed text indicators, AdaBoost.RT is able to predict An improvement over text indicators based on news headlines and
Using news articles, forecast the crude oil prices better than the other benchmarks with two novel financial characteristics. Sparse and condensed news headlines are (Y. Bai et al,
2022 price of crude oil topic and sentiment analysis indicators for short and sparse text used in the framework for forecasting the price of crude oil. These 2022)
data. headlines are better suited for price forecasting with historical
In almost all circumstances (99.6%), the decision-maker's predicted
To develop heuristics for Hedging Comparing two hedging approaches namely minimizing downside risk and utility is increased by the downside risk minimization hedging (Dmitry Vedenov &
2022 effectiveness minimization of variance in a market exhibiting strong momentum using strategy. Gabriel J. Power,
Copulas based approach Furthermore, it appears that this method is most effective when both 2022)
spot and futures prices exhibit strong upward or downward momentum.
The price of crude oil is dissected using a variational mode The hybrid model (ML classification prediction method based on
Propose a hybrid forecasting model decomposition technique, and multimodal data features are then characteristics of multi-modal data) employs the Variational mode
that uses a machine learning retrieved based on the decomposed modes. Time series analysis is used decomposition (VMD) algorithm to derive the Intrinsic Mode Functions (He, H., Sun, M.,
2021 algorithm to anticipate crude oil to concurrently translate the volatility of the price of crude oil (IMF) of crude oil prices. Then, multi-modal data characteristics Li, X., et al.,
price trends. into trend symbols. Then, multi-modal data features are employed to are extracted using the IMFs. It has been proven that the suggested 2021)
train a machine learning multi-classifier utilizing historical model is more adept at foretelling high volatility than low
volatility as input and trend symbols as output. volatility of crude oil prices.
In the study, the AdaBoost-LSTM and It introduces the AdaBoost-GRU model, combining the GRU model with The results demonstrate that both AdaBoost-LSTM and AdaBoost-GRU
2021 AdaBoost-GRU models are compared to AdaBoost Regressor, and compares its predictive power with outperform benchmarking models, with AdaBoost-GRU showing superior (Busari G & Lim D,
enhance the accuracy of crude oil AdaBoost-LSTM and single LSTM and GRU models. performance compared to all models studied in this research. 2021)
price prediction.
To introduce a robust hybrid model The proposed hybrid model combines Adaptive Neuro Fuzzy Inference The numerical results indicate that the hybrid model weighted by the
for reliable forecasting of Brent System (ANFIS), Autoregressive Fractionally Integrated Moving Average genetic algorithm performs better than the individual constituent (Abdollahi H &
2020 oil price, considering the (ARFIMA), and Markov-switching models to capture specific features in models, the hybrid model with equal weights, and the hybrid model Ebrahimi S, 2020)
challenging nature of oil price the oil price time series. weighted based on error values
Determine how the spread of a single Fundamentals in the upstream sector (i.e., unrefined petroleum
product gasoline crack is affected products) are more responsive to changes in economic production. The
2018 by macroeconomic factors such real Econometric methods such as Generalized Impulse Response analysis and crack spread can shift quite significantly in response to small and (Ewing & Thompson,
production growth, inflation, Vector Auto-Regression (VAR) unexpected changes in aggregate demand. Inflation's effects extend 2018)
corporate default risk, and monetary for one to five months. Risk of Corporate Default has no effect.
policy stance. Short-lived monetary policy shocks (up to two months)
Use a non-linear approach for The proposed model combines a particle swarm optimizer (PSO) and a The sliding window approach with training periods of 300 and 400 (Karathanasopoulos
2016 modelling and tracking the crack's radial basis function (RBF) neural network, trained using sliding days effectively models the crack spread, while periods of less than et al., 2016b)
expansion. windows of 300 and 400 days. 300 days results in performance degradation.
To forecast the time-series of the The best spread trading model is a higher order neural network (HONN) There is a nonlinear relationship between WTI and GAS. Despite (Dunis et al.,
2016 WTI-GAS crack spread, use non-linear with a threshold filter. faster computation times and fewer variables, HONN fared better than 2016)
models. the MLP outside of the sample.
Comparing pricing of crack spread Time series data for options prices are modelled using bivariate The more simplistic univariate approach to option pricing was found (Mahringer &
2015 options using GARCH volatility model GARCH model and univariate crack spread model to be better than bivariate GARCH model Prokopczuk, 2015)
or crack spread model.
Systems that facilitate The case study technique improved the procurement planning for the
2014 decision-making based on Implicit Anticipation model is used to develop a DSS system building oil refining industries by resulting in significant reductions in (Kallestrup et al,
hierarchical planning frameworks are on the case study for procurement planning planning efforts and procurement expenses. 2014)
used in the oil refining sector.
To propose a decision support It introduces the "Expected Profit-Supply at Risk" (A-EPSaR)
framework that considers the algorithm, utilizing Monte Carlo simulation to quantify each
presence of both spot market and supplier's risk, allowing decision-makers to understand the trade-off Procurement Risk Management (PRM) framework is designed to support (Hong, Z., & Lee,
2013 contract suppliers to model risks in between profit and risk. In addition, a goal programming model to the DSS to identify the risks in the procurement model and build C., 2013)
procurement processes and create a allocate orders among the supplier pool and a contract-spot profit model of their own.
robust purchasing plan. allocation model to assign orders between the spot market and the
supplier pool.
An innovative approach to forecasting oil prices is put forth by
Predicting the direction of oil This study proposes a semi-supervised machine learning algorithm adapting the current SSL algorithm for use in time-series (Shin, H. et al,
2013 price changes (SSL) for time-series entities by representing their network and prediction. This approach makes use of sophisticated techniques like 2013)
using SSL to forecast the upward and downward movement of oil prices. feature extraction and TI transformation as well as comparisons of
the similarity between several collections of time-series data.
For the purpose of modelling and
forecasting, verify that crack Mean reverting behaviour weak when observed for days & weeks; strong
2012 spread display long range ARFIMA applied on single product crack for months, long range multi-fractal with long range autocorrelation (Wang & Wu, 2012)
autocorrelation, multi-fractionality but unable to capture small scale fluctuations
and scaling invariance
Create a model for predicting There is a link between naphtha crack and three key affecting
2012 naphtha crack using multiple Forecasting naphtha crack using a multiple regression model with more elements. Among the main variables, naphtha supply and demand have a (Sung et al., 2012
regression analysis and statistical than 20 factors significant impact on naphtha crack. )
Model the volatility dynamics of the The work utilizes the partially overlapping time-series (POTS) The findings reveal time-to-delivery effects, substantial
three NYMEX-traded futures contracts framework proposed by Smith (2005) to model all futures contracts seasonality, and variations in conditional variance and correlation (Suenaga & Smith,
2011 for petroleum products (crude oil, with delivery dates up to a year into the future. This approach across the commodities, with implications for short-term and 2011)
gasoline). allows the extraction of information from these prices to understand one-year price risk in different positions, particularly in
the persistence of market shocks. crack-spread positions.
The paper explores the use of data It proposes combining three well-known feature selection methods, The results show that the best-performing combinations of feature
2010 mining techniques for stock market namely Principal Component Analysis (PCA), Genetic Algorithms (GA), selection methods are the intersection of PCA and GA, and the (C.-F. Tsai &
prediction. and decision trees (CART), to identify more representative variables multi-intersection of PCA, GA, and CART, achieving accuracies of 79% Y.-C. Hsiao, 2010)
for better prediction. and 78.98%, respectively.
Forecasts utilizing crack spread option pricing were examined using 2 different approaches in a study by (Mahringer & Prokopczuk, 2015). Option 1 employed a two-factor mean-reverting model with
constant volatility to simulate the crack spread option. On the other hand, Option 2 utilized a bivariate GARCH model with co-integrated underlying futures to model the crack spreads between crude
oil and heating oil, and crude oil and gasoline.
In analysing the price risk to crack spread due to volatility dynamics and seasonality (Suenaga & Smith, n.d.) modelled using econometric techniques the shocks to prices of crude oil Gasoline and
heating oil. This study attempted to factor in the impact inventory held (Geman & Smith, 2013)and the maturity of futures contract. The three commodity prices' highly nonlinear volatility dynamics
were shown by the estimated model, which is in keeping with the seasonality in storage and demand that has been observed. Despite the fact that all three commodities have sizable seasonal changes and
time-to-delivery impacts, there are notable variances between the three commodities' volatility patterns and the contract's delivery month.
The price discovery process for petroleum products, how accurately petroleum product futures prices predict current spot prices, and refinery outages and weather were all identified as important
fundamental factors in a survey on the factors influencing the prices of petroleum products (Ederington et al., 2019). The empirical findings led to the conclusion that, throughout the study period,
speculation did not lead to an increase in risk-free returns on petroleum products or in excess returns or price volatility. Studies reveal that gasoline and heating oil futures prices are often
reliable indicators for three-month spot prices, but not as much for six and 12-month contract durations. When assessed for forecasting performance, gasoline futures prices outperform a simple random
walk (by producing less mean-squared errors) and perform better than both oil and heating oil futures in this regard.
To develop heuristics for Crack spread hedgers, the study by (Liu et al., 2017) compared two hedging strategies namely minimizing the downside risk criterion and minimization of variance measure on
Expected Utility. The study found that the downside risk minimization hedging strategy increases the expected utility of the decision-maker to almost 99.6% of the time. Moreover, this approach works
best when both spot and futures prices move upward and downward strongly Tables 1 & 2.
Table 2 Performance Measures
Variance Accuracy
MAE = Mean Absolute Error, MSE=Mean Square Error and RMSE = Root Mean Square Error
= Actual value
= Predicted Value
= Average of all actual values
N = No. Of Observations
This research attempts to address the identified gaps (Figure 2) by developing a model for forecasting price of single product crack spread with horizon of 1-3 months using crude and product
benchmarks used by Indian refiners using time series data and employing recent advances in AI Deep Learning techniques.
Time Series Modelling: Statistical Machine Learning Models
Auto Regressive Integrated Moving Average (ARIMA) models are a powerful tool for forecasting time series data. They are used to forecast a wide variety of data, including sales, economic indicators,
and environmental data. A comparison of various statistical and machine learning techniques has been developed for generating forecasts (Makridakis et al., 2018). ARIMA models, or Autoregressive
Integrated Moving Average models, are a class of statistical models that are used to analyze and forecast time series data. ARIMA models assume that the future values of a time series can be
predicted from its past values and the errors of its past predictions.
ARIMA models have been used for reliability forecasting and analysis (Ho et al., 2002). The time series technique is a forecasting technique that requires no assumptions about the data and is very
flexible (Burba, 2019). An improvement of ARIMA model is the SARIMA model which includes more realistic dynamics of data specifically non-stationary in mean and seasonal behaviours (Gerolimetto, 2010
). Since time series forecasting analysis can be conducted only on stationary data, differencing (by taking logarithm) is carried out to convert data from non-stationary to stationary(Ho et al.,
2002). The non-stationary data is subjected to order of differencing (d=0,1,2) to make it stationary (Bhardwaj et al., 2014). The traditional ARIMA model does not incorporate impact of external
independent variables. Exogenous variables can be incorporated in the ARIMA (Jain & Mallick, 2017) and SARIMA (Gerolimetto, 2010) models which will incorporate effects of any external variables which
exhibit regressive relationships to the base model (Xie et al., 2013).
Time series data are also typically forecasted using the ETS method, also known as the Holt-Winters method . The model effectively captures the errors, trends, and seasonality of a time series and
can be applied to both univariate and multivariate data. Generally, the error component of an ETS is the random fluctuations that occur in a time series of data. ETS can improve forecast accuracy by
incorporating historical errors into its forecasting model.
In time series data, the trend component represents the long-term pattern or direction. Trends can be captured by estimating the slope and incorporating it into the forecasting model, which will
project future values that align with the overall direction of the data. Data series that repeat at regular intervals, such as daily, weekly, or yearly, are said to be seasonal. ETS can provide
accurate predictions for future values by estimating seasonal indices and applying them to the forecasting model.
Deep Learning Models
An LSTM is a type of Recurrent Neural Network that is used in Deep Learning applications. Because it can identify patterns in both short-term and long-term input sequences, LSTM in particular is well
suited to categorizing, evaluating, and making predictions. A typical example will be sequence of words in a sentence or patterns in time series data wherein we are unsure of the patterns / lags that
may exist in the sequence / data series (Gu et al., 2019; Salvi, 2019; Siami-Namini & Namin, 2018; Song et al., 2020).
LSTMs can keep contextual information on input sequences by providing a loop through which information can flow from one stage to another. An LSTM unit consists of a cell, an input gate, an output
gate, and a forget gate. LSTM were developed to address RNN's vanishing gradient problem, which is brought on by backpropagation. They still experience the exploding gradient problem, though. An
extension of the traditional LSTM is the Bi-directional LSTMs (BiLSTM) which improves the performance of sequence classification problems (Siami-Namini et al., 2019). Two hidden layers pointing in
different directions are connected to the same output by LSTMs. In this type of degenerative learning, the output layer is simultaneosly given knowledge about both the past and the future states.
This aids the LSTM in comprehending the situation more fully (Kulshrestha et al., 2020). The LSTMs come in a variety of flavours, including vanilla, stacked, bi-directional, CNN, and conventional.
The study makes use of and contrasts these tastes.
Bayesian Optimization
The process of determining the ideal collection of hyperparameters in deep learning and machine learning which control the learning process is called as hyper parameter optimization. Some of the most
commonly used algorithms are Grid Search, Random Search, and Bayesian Optimization. Bayesian optimization uses an automatic model tuning approach to fine tune the hyperparameters. This method uses a
surrogate model, such as Gaussian Process for approximating the objective function. During this optimization, two choices are made:
1. Choosing previous functions to optimize that express assumptions about the function. In this case, we pick the Gaussian Process.
2. The acquisition function must be chosen.
Evaluation Metrics for Models
As is customary in the literature, the provided forecasts are statistically analysed using the Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean absolute error (MAE), Mean absolute
percentage error (MAPE) (Karathanasopoulos et al., 2016).
For comparison of models, we have also employed and additional measure for accuracy Resistant-MAPE (R-MAPE) in our study which is a better fit (Table 3) for comparing accuracy of time series models(
Montaño Moreno et al., 2013).
Table 3 Comparison of MAPE and R-MAPE
Characteristic MAPE R-MAPE
Sensitivity to outliers More sensitive Less sensitive
Interpretability Easier to interpret More difficult to interpret
Suitability for different Good for forecasting applications where the actual values are relatively Good for forecasting applications where the actual values vary widely in magnitude or
applications similar in magnitude where there are outliers
Examples of applications Forecasting sales for individual products in a retail store Forecasting demand for electricity or water
Acceptable levels of MAPE and RMSE for a time series forecasting model vary depending on the specific application and the industry. However, in general, the following guidelines can be used:
MAPE: A MAPE of 10% or less is generally considered to be highly accurate, while a MAPE of 20% or less is considered to be good. MAPE between 20% and 50% is considered reasonable. A MAPE of greater
than 50% is generally considered to be poor or inaccurate forecasting (Montaño Moreno et al., 2013) .
RMSE: The acceptable level of RMSE for a particular application may be higher or lower depending on the specific needs of the business (Shcherbakov et al., 2013). Additional factors that should be
kept in mind to consider when evaluating the reasonableness of RMSE for demand forecasting:
• Frequency of the forecasts: More frequent forecasts (e.g., daily or weekly) will generally have higher RMSE than less frequent forecasts (e.g., monthly or quarterly). This is because more frequent
forecasts are more difficult to predict accurately.
• Level of aggregation: Forecasts for more aggregated data (e.g., total demand for a product category) will generally have lower RMSE than forecasts for less aggregated data (e.g., demand for a
specific product). This is because aggregated data is less noisy and easier to predict.
• Horizon of the forecasts: Forecasts for shorter horizons (e.g., next week or next month) will generally have lower RMSE than forecasts for longer horizons (e.g., next year or next five years). This
is because shorter-term forecasts are less subject to unexpected changes in the environment. An RMSE that is less than or equal to the standard deviation of the historical data is generally
considered to be good. An RMSE that is greater than the standard deviation of the historical data is generally considered to be poor. This bodes from the fact that the standard deviation of the
historical data is a measure of how much the data varies. If the RMSE of a forecast is less than or equal to the standard deviation of the historical data, then it means that the forecast is at least
as good as the historical average. However, if the RMSE of a forecast is greater than the standard deviation of the historical data, then it means that the forecast is not as good as the historical
average. For comparative studies, the model with the least RMSE measure will be the best option.
Data Description
The price of Crude oil has undergone tectonic shifts during the past few decades. The following Figure 3 depicts the price of Crude Oil from 1980s till date.
The volatility (rise and fall) and major shifts in price levels are clearly discernible and attributable to major events such as Gulf War (1990-91), The Y2K Bubble (1998-200), Sub-prime lending
crisis in USA (2008), Excess shale gas production by US (2014-15) and the COVID-19 pandemic in 2020 to name some of the few major events. If for the purpose of model development and training the
complete set of data ranging from 1990 till date is used, then it will result in models which wrongly would factor extraneous events and their implications which have no systemic bearing in
forecasting of the prices of Crude and products for operational purposes. To exclude the extreme effects of such drastic shifts in prices owing to extraneous events as outlined above, data for this
research has been chosen from the window corresponding to June. 2015 till Dec 2019. During this period, there is an absence of any political, economic or any such event and the prices are purely
reflective of the market dynamics namely the demand, supply, and prices.
Data Selection
This research uses data sourced from daily feeds provided by Platts comprising of all types of traded Crude and Petroleum products. For the purpose of model development, Brent Crude oil prices and
Gasoline 0.05 % Singapore prices have been used. The prices for the Crude and Products are plotted in USD $/Barrel in Figure 4. The price of crack spreads on a given day is determined by subtracting
the difference between the daily closing product prices in US dollars from the price of crude in US dollars. Whereas for the purpose of planning and GRM calculation, various crack spreads namely the
5:3:2 and 3:2:1 crack spread are used. The 5:3:2 Crack spread denotes that 5 units of Crude oil is used to produce 3 units of Gasoline and 2 units of Gas oil. The GRM is thus calculated basis on the
yield (i.e., percentage of Gasoil and Gasoline produce per gallon of Crude oil and computed proportionately. However, for the purpose of simplicity of model development only one Product (Gasoline)
has been used to calculate the gasoline crack based on Brent Crude oil prices. This scenario closely reflects the reference crude and product prices being used by Indian Oil companies. The model so
developed can be easily extended to other crack spread combinations if so desired.
Data from the training set make up 80% of the total, whereas data from the test set make up 20% of the total.
Data Pre-processing
A time series could have missing and irregular numbers. Using appropriate method, the noisy data are smoothed, and the missing values are replaced. Heteroscedastic data is when a time series'
dependent variable significantly varies from the beginning to the end of the series. By taking the time series' logarithm, heteroscedastic behaviour can be eliminated.
In order to assess the predictability of time series crack spread data is used, the ADF statistics for the crack time series data is computed and is listed below Table 4.
Table 4 ADF Statistics for Crack Spead Time Series Data
S.No. Parameter Value
1. ADF Statistics -6.160728
2. p-Value 0.0000
3. Critical value of 1% -3.432
4. Critical Value of 5 % -2.982
5. Critical Value of 10% -2.567
During the deep learning training process, data is standardized. When the actual forecasts are generated at the end of the modelling process, this process is reversed. The Google TensorFlow framework
and various Python libraries like Keras, Pandas, etc. are utilized to implement the deep learning techniques employed in this study. The kerastuner software serves as a foundation for the Bayesian
optimization-based hyperparameter training for Deep Leaning and Optimization Model.
The available dataset is divided into two halves for the purpose of training the LSTM models, with the training set comprising 80% of the data and the testing set the remaining 20% the testing set
When building and training the LSTM network, various hyperparameters are used in addition to the lag size parameter, including the number of hidden layers, epoch size, batch size, number of neurons,
learning rate, and dropout rate. In the following stage, an LSTM network is built and trained using the training set that corresponds to each set of hyperparameters that was valued using Bayesian
optimization in the previous stage. The grid search method is used to turn the time series into a set of instances with an input-output format based on the lag size that is chosen. An example of the
produced instances with lag size = 10 is shown in Table 5. After that, a train set and a test set are created from the instances that were produced. The test set comprises 20% of all the examples in
each experiment. An LSTM network is constructed and trained for each combination of hyperparameters. The Adam algorithm serves as the optimizer for all networks (Kingma & Ba, 2014). Furthermore, the
mean squared error (MSE) is utilized as the loss function for every constructed LSTM network. The final crack price forecast is arrived at by adding the forecast obtained in stage (1) with the
residual forecast obtained in stage (2). The performance of the model so developed is computed based on the prevalent error metrics and using the metrices Mean, Variance, Standard Deviation, Mean
Square Error (MSE), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) and Resistant-MAPE (R-MAPE).
Table 5 Comparison of Errors Between Statistical ARIMA, ETS Deep Learning (MLP/LSTM/BILSTM) Models
ARIMA ETS MLP LSTM BiLSTM
VAR 6.78 6.72 17.01 11.06 5.87
SD 2.60 2.59 4.12 3.30 2.42
MSE 6.84 6.77 30.61 14.72 6.15
RMSE 2.63 2.66 5.52 3.88 2.60
MAE 2.24 2.23 3.87 3.00 2.03
MAPE 35.29 35.21 49.27 112.97 47.04
RMAPE 29.21 29.31 48.39 119.64 43.71
Result and Analysis
As a baseline, forecasts were computed using statistical models namely ARIMA, ETS and ANN Deep Learning models namely MLP, RNN models LSTM and BiLSTM.
As is evident from Figure 5 above, the forecasts computed using statistical techniques ARIMA and ETS denote similar behaviour.
However, forecasts generated using deep learning methods MLP, LSTM and BiLSTM (Figure 6) are exhibit different behaviour.
A combined plot of Actual crack spread and forecasts using all the techniques under study is plotted in Figure 7 above. The tabulation of performance metrices related to these forecasts is provided
in Table 5 below.
Figure 7 Plot of Actual Crack Spreads and Forecast Using Statistical and Deep Learning Techniques Together
It can be inferred from the table above that ARIMA and ETS methods provide a very similar performance amongst the statistical techniques but the performance of deep learning methods varies
considerably. BiLSTM exhibits the best performance amongst the deep learning techniques and exhibits better MSE and RMSE. However, its MAPE / RMAPE performance is inferior to ARIMA and ETS methods.
On an individual basis ETS, LSTM and BiLSTM have RMSE scores less than their Standard Deviation, meaning that they are generating better forecasts. Viewed from the perspective of MAPE / RMAPE none of
the methods when considered individually produce good quality forecasts (i.e., their individual MAPE / RMAPE are greater than 20).
Bar plots and Line plots of the various metrices are provided in Figures 8 and 9 above.
To improve the forecasting method, as mentioned in the earlier section of this paper, we have taken ARIMA and ETS both as statistical method and BiLSTM method as the deep learning technique for
further studies.
Figures 10 and 11 provide comparison of ARIMA and BiLSTM forecasts plotted side by side. The results of 2 stage forecasts using ARIMA and ETS for stage 1 and BiLSTM for forecasting residuals in Stage
2 and finally calculating the forecasts are listed in Table 6 below.
Figure 10 Plot of Actual Crack Spreads and Forecast Using Statistical (ARIMA) and Deep Learning (BILSTM) Methods
Figure 11 Residual Plot of Actual Crack Spreads and Forecast Using Statistical (ARIMA) and Deep Learning (BILSTM) Methods
Table 6 Comparison of Errors between ETS+LSTM and Hybrid LSTM Models
ARIMA BiLSTM ARIMA + BiLSTM Residual ETS ETS + BiLSTM Residual
VAR 6.78 5.87 2.26 6.72 2.78
SD 2.60 2.42 1.50 2.59 1.67
MSE 6.84 6.15 2.25 6.77 2.80
RMSE 2.63 2.60 1.50 2.66 1.66
MAE 2.24 3.00 1.26 2.23 1.39
MAPE 35.29 112.97 19.89 35.21 23.54
RMAPE 29.21 119.64 16.4 29.31 18.61
We observer there is a considerable improvement in forecasts using ARIMA + BiLSTM when compared to using ARIMA, ETS and BiLSTM individually. The RMSE score of the ARIMA + BiLSTM is less than the SD
and MAPE RMAPE scores are also less than 20, which is the threshold for the quality of the forecasts to be considered good.
Figure 12 plots all the observed scores for this combination of forecasting.
Figures 13 and 14 provide a comparison plot of ARIMA and ARIMA + BiLSTM forecasts.
Ranking of the Various Models (Standalone and Hybrid)
The Table 6 provides the ranking of all the models standalone (ARIMA, ETS, MLP, LSTM, BiLSTM) and hybrid models ARIMA+ BiLSTM and ETS + BiLSTM models. The performance metrices in the order of
significance are MAE, RMSE, RMAPE and MAPE Table 7.
Table 7 Ranking of Crack Spread Forecasting Models
Model Standard Deviation MAE Score RMSE Score MAPE Score RMAPE Score Rank
ARIMA+BiLSTM 1.5 1.26 1.5 19.89 16.4 1
ETS+BiLSTM 1.67 1.39 1.66 23.54 18.61 2
BiLSTM 2.41 2.03 2.6 47.04 43.71 3
ETS 2.59 2.23 2.66 35.21 29.31 4
ARIMA 2.6 2.24 2.63 35.29 29.21 5
LSTM 3.3 3 3.88 112.97 119.64 6
MLP 17.01 3.87 5.52 49.27 48.39 7
A combination of statistical forecasting technique ARIMA and BiLSTM for forecasting the price of crack offers the best performance. This approach brings together the forecasting strengths of two
techniques namely ARIMA, a statistical technique which addresses well the linearity present the data and BiLSTM, a deep learning method, which captures the non-linearity present in the data.
Combining these techniques yields improvement in the accuracy of the forecast beyond what each of the techniques individually yield.
Hyper Parameter Tuning
Hyperparameter tuning using Bayesian optimization offers significant improvement in training times in comparison to other methods of optimization like Random search and Grid search. The best results
are obtained with 4 Dense Layers, Epoch =10 and activation function = ‘relu.
Crack spread forecasting is less researched a topic as compared to Crude price forecasting. Most of the existing research on Crack spread forecasting is oriented towards short term forecast with a
horizon of few days and is primarily intended for Traders and hedge fund managers. Various techniques such as Vector Auto Regression, Multiple regression, neural networks, General Brownian Motion and
ARFIMA have been employed for the purpose of forecasting crack spreads. Time series analysis has been the dominant technique adopted for most of the research on crude oil and crack spread
forecasting. On visual examination, it can be deduced that the Crack spread timeseries has both linear and non-linear component. Any technique employed which is primarily aimed for modelling either
the linear or non-linear component for forecasting will invariably be unable to capture the nuances of the other component. This is evident in the results discussed earlier in this paper.
This research attempts to develop a hybrid model for improving forecast accuracy which employs statistical forecasting technique ARIMA for forecasting the linear component and ANN Deep Learning
BiLSTM model for forecasting the non-linear component. This study is based on and extends a similar approach proposed by (Panigrahi & Behera, 2017) by employing Deep Learning LSTM in place of the
Multi-Layer Perceptron (MLP) to the domain of crack spread forecasting. For comparison, various Deep Learning Architectures have been studied for hybridization and the results were obtained with
BiLSTM. The results indicate that this hybridization results in generation of forecasts with better accuracy compared to individual methods which model linear or non-linear components, respectively.
Theoretical Implications
Deep Learning techniques have a history of success when employed in pattern recognition applications like image categorization, self-driving automobiles, and similar ones. The Multi-Layer Perceptron
(MLP), Convolution Neural Network (CNN), and Recurrent Neural Networks (RNN) Deep Learning architectures all perform differently for a given set of tasks, though Long Short-Term Memory (LSTM) neural
networks, a type of RNN, are known to perform better on short and long sequences of input streams in contrast to other Deep Learning Neural Network Architectures. This study also performs a
comparative study of performance of all these Deep Learning architectures on Time Series data, Crack spreads prices in this case and we observe a behaviour congruent on the stated lines i.e., BiLSTM
models perform better than MLP and classical LSTM models. Bayesian optimization has been used to increase the rate of convergence of models and has been found to be effective when in reducing run
times for large combinations of model and their configurations. This study can also be extended to include crude / crack futures which are also represented as time series and combination of these
series can be expected to increase the efficacy of forecast still better.
Managerial Implications
The crude oil constitutes 85% of the input cost of refinery operations. Indian Oil companies, unlike their western counterparts, are integrated oil refining and marketing companies. With limited
options to engage in pricing of petroleum products and limited mandate to engage in hedging as a safeguard, managers in Indian oil companies are left with limited options to improve their
profitability. Oil refineries procure and process a ‘basket of crude’ depending upon the design of the refinery to process a particular type of crude (sweet / sour crude), the configuration of the
refinery for producing various distillates (LPG, Gasoil, etc.) and suitable alternatives of crude available at a lesser cost among other factors. The crude(s) thus selected amongst the alternatives
available by comparing the forecasted Crack spreads for each of the available options and ensuring the highest forecasted crack spreads can immediately lead to increase in cost optimization and
profitability as measured using Gross Refining Margin (GRM). The calculation for computing the impact of unit percentage improvement in forecast accuracy to GRM is convoluted, however rough estimates
indicate that it can lead to increase in around 0.1- 0.2 % annual profit for Indian Oil refining and marketing companies amounting to approximately 20-40 crores annually. Crack spread forecasts with
higher accuracy, therefore, becomes central to this theme. This improvised and easy to implement method therefore provides a robust and inexpensive way for generating crack spread forecast with
higher accuracy to Managers of Oil companies to aid their planning and procurement decisions.
Practicing managers involved in the supply chain planning and optimization of petroleum products have always felt the need for superior quality forecast of petroleum product prices (crude,
distillates / crack) as it has direct bearing on GRM hence profitability. In order to improve forecasting accuracy, we have attempted to create a hybrid model that combines statistical modelling
technique ARIMA with Deep Learning technique BiLSTM. The linearity present in the time series data allows the ARIMA technique to be used to estimate the price of a sole product crack. By utilizing
the LSTM Deep Learning technique to extract more information from the nonlinearity of the residuals, the forecast accuracy is boosted. A variety of LSTM single and hybridized mode combinations have
been researched and ranked according to their efficacy. Bayesian optimization was used to expedite the hyper parameter tuning procedure. The ANN models are thus able to converge faster to optimal
hyper-parameters for the Deep Learning networks employed. This study therefore establishes that hybrid models which combine statistical and AI techniques yield better results. This work can be
extended to study the effectiveness of these steps for multi-step / period forecasting. Also, various newly discovered ANN Architectures such as Autoencoders, Transformers and DeepAR etc. which can
be studied to ascertain the improvement in the forecast accuracy. Combining these techniques with other optimization techniques such as Portfolio optimization techniques can lead to sustained
profitability for the organizations. The recent advances in development of Artificial Neural Network ecosystems such as GPUs and TPUs, Programming Languages and Machine Learning libraries, also
accelerate the research in various areas of petroleum products demand and their price forecasting, and supply chain planning. This study has implemented a hybrid model, which combine statistical
techniques such as ARIMA and ANN Deep Learning techniques, namely BiLSTM networks, to capture both linearity and non-linearity in time-series data and make more accurate short-term forecasts. This
study is however limited to univariate time series and short terms forecasts. As a future extension to this study, the following enhancements can be attempted i) The hybrid model comprising of ARIMA,
and LSTM can be extended for medium- and long-term forecasts ii) Study and analyse the performance and effectiveness of these model for multistep forecasting iii) Other deep learning models such as
Boltzman machine (Deep Belief Network) / transformers have yielded better results when coupled with LSTM deep learning models iv) use a portfolio approach to create the ideal combination of Deep
Learning models for various forecasting horizons; v) Application of Generative AI (such as ChatGPT etc. ) for time series forecasting vi) investigate the effects of bagging on ARIMA models and hybrid
models. vi) Research into hyperparameter tuning methods using Bayesian optimization.
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Received: 09-Oct-2023, Manuscript No. AMSJ-23-14021; Editor assigned: 10-Oct-2023, PreQC No. AMSJ-23-14021(PQ); Reviewed: 29-Dec- 2023, QC No. AMSJ-23-14021; Revised: 29-Feb-2024, Manuscript No.
AMSJ-23-14021(R); Published: 02-Mar-2024 | {"url":"https://www.abacademies.org/articles/crack-spread-forecasting-for-supply-chain-optimization-a-hybrid-model-using-time-series-and-deep-learning-with-bayesian-optimizati-16510.html","timestamp":"2024-11-11T18:06:49Z","content_type":"text/html","content_length":"118796","record_id":"<urn:uuid:5526c5d6-a138-43a5-9404-e49bfcae70e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00529.warc.gz"} |
On the Formalization of ω-Consistency with Different Quantifier Structures
This research paper delves into the formalization of ω-consistency, a concept in mathematical logic. It investigates whether the specific arrangement of quantifiers typically used in the formal
statement of ω-consistency is crucial or if alternative structures could be employed.
Bibliographic Information: Santos, P. G. (2024). ω-consistency for Different Arrays of Quantifiers. arXiv:2410.09195v1 [math.LO].
Research Objective: The paper examines whether the specific quantifier structure in the formal definition of ω-consistency is essential or if other arrangements could yield equivalent statements.
Methodology: The study utilizes a formal approach within a metamathematical framework. It introduces a generalized ω-consistency statement, denoted as ω-Con[⃗Q]T, where ⃗Q represents an arbitrary array
of quantifiers. Through a series of propositions and theorems, the paper explores the logical relationships between ω-Con[⃗Q]T for different ⃗Q and the standard ω-consistency.
Key Findings: The paper demonstrates that several alternative quantifier structures result in statements provably equivalent to the traditional ω-consistency. For instance, S ⊢ω-ConT ↔ω-Con∀n+1T
↔ω-Con∀n+1∃mT ↔ω-Con∃∀∃nT, where S is a sufficiently strong base theory.
Main Conclusions: The study concludes that the specific array of quantifiers used in the formalization of ω-consistency is not as rigid as conventionally perceived. The equivalence of various ω-Con
[⃗Q]T statements suggests a degree of flexibility in representing this concept within formal systems.
Significance: This research contributes to the understanding of ω-consistency and its formal representation. It highlights the possibility of employing different quantifier structures without
altering the fundamental meaning of ω-consistency, potentially leading to more flexible and nuanced applications of this concept in mathematical logic.
Limitations and Future Research: The paper focuses on specific quantifier structures. Further research could explore a broader range of quantifier arrangements or investigate the implications of
these findings for related consistency principles.
How might these findings concerning the flexibility of ω-consistency's formalization impact the study of related consistency principles in mathematical logic?
The paper demonstrates a certain degree of freedom in formalizing ω-consistency without altering its fundamental meaning. This discovery could have several implications for the study of related
consistency principles: Generalizations of Consistency Principles: The techniques used to manipulate quantifier arrays in the context of ω-consistency might be applicable to other consistency
notions. This could lead to generalized frameworks for defining and studying consistency, potentially revealing deeper connections between seemingly distinct principles. Finer Analysis of Theories:
The ability to express ω-consistency in various equivalent ways provides a more nuanced toolkit for analyzing the strength and properties of formal theories. Different formulations might be more
suitable for certain proof-theoretic arguments or might highlight subtle aspects of a theory's consistency. Connections to Reverse Mathematics: The paper establishes equivalences between different
formulations of ω-consistency within a weak base theory (S). This aligns with the spirit of reverse mathematics, where the goal is to determine the minimal axioms needed to prove a given theorem.
Further investigation could explore the reverse mathematical strength of these equivalences and their relationship to other principles.
Could there be contexts or formal systems where the specific choice of quantifier structure in expressing ω-consistency leads to significant differences in proof-theoretic strength or other
metamathematical properties?
While the paper demonstrates a degree of flexibility in formalizing ω-consistency, it's plausible that in certain contexts, the specific choice of quantifier structure could lead to significant
differences. Here are some possibilities: Weaker Base Theories: The equivalences shown in the paper rely on the base theory S containing a sufficient amount of arithmetic. In weaker systems, these
equivalences might not hold, and different quantifier structures could result in statements with varying proof-theoretic strengths. Non-Classical Logics: The paper operates within classical logic. In
non-classical logics, such as intuitionistic logic or linear logic, the interplay between quantifiers and other logical connectives can be significantly different. This could lead to situations where
the choice of quantifier structure in expressing ω-consistency has a more profound impact. Restricted Systems: In formal systems with restricted forms of induction or comprehension, the specific
quantifier complexity of a consistency statement could become crucial. A more complex quantifier structure might render the statement unprovable within the system, even if an equivalent formulation
with a simpler structure is provable.
If mathematical truths can be expressed in various logically equivalent ways, does this suggest an inherent flexibility in the structure of mathematical concepts themselves?
The ability to express mathematical truths in various logically equivalent ways does hint at a certain flexibility in the structure of mathematical concepts. However, it's important to approach this
idea with nuance: Flexibility in Representation, Not Necessarily in Essence: Logical equivalence ensures that different formulations capture the same underlying truth. However, this doesn't
necessarily imply that the mathematical concept itself is inherently flexible or amorphous. It might simply reflect the richness and expressive power of our formal languages. Conceptual Clarity vs.
Formal Flexibility: While multiple formulations can be logically equivalent, they might differ in terms of conceptual clarity, intuitive appeal, or their suitability for specific applications.
Mathematicians often seek the most elegant, insightful, or efficient representation of a concept, even if other logically equivalent options exist. The Role of Context and Interpretation: The meaning
and significance of a mathematical concept are often intertwined with the specific context and the intended interpretation. Different formulations, while logically equivalent, might emphasize
different aspects of the concept or lend themselves to different interpretations in different contexts. In summary, the flexibility in expressing mathematical truths highlights the power and
versatility of our formal systems. However, it doesn't necessarily imply a lack of inherent structure or meaning within mathematical concepts themselves. The choice of representation remains crucial
for clarity, insight, and effective communication of mathematical ideas. | {"url":"https://linnk.ai/insight/logic-and-formal-methods/on-the-formalization-of-%CF%89-consistency-with-different-quantifier-structures-UpymWyLI/","timestamp":"2024-11-11T06:21:51Z","content_type":"text/html","content_length":"261222","record_id":"<urn:uuid:c0730e41-c7fd-48b3-91cd-65a5140ee702>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00196.warc.gz"} |
Rational Number Definition
A number that can be represented in the form of a fraction is called a Rational Number. The word Rational has been derived from the word "Ratio" because it means "Divide".
You can read about Current Ratio in detail on another page of our website.
For example,
2/3 is a rational number because it is represented in the form of a fraction.
Here are some examples of Rational Numbers in the following.
• 2/7 is a rational number.
• 5.75 is a rational number because it can be shown as 575/100.
What is the Opposite of a Rational Number?
An irrational number is the opposite of a rational number. It means that a number that can be written in the form of a fraction is called an Irrational number.
For example,
3.33333….. is an irrational number because its denominator is not specified.
How to denote Rational Numbers?
In Mathematics, Rational numbers are denoted by “Q”. Its representation can be changed with the level of education or in different books. But generally, shows the rational number.
A rational number can be written by two integers in a fraction format. The integers can be positive or negative. In simple, a Rational number can be positive or negative.
Can we convert a decimal into a Rational number?
A decimal can be converted into a rational number. But there is a condition that a decimal should fulfill for this conversion.
• A decimal can be converted into a rational number only when it has a defined number after the decimal.
For example,
3.14 can be converted into a Rational number because it has 2 digits after the decimal point while 3.141414…… can’t be converted into a rational number because the denominator can’t be defined as the
digits after the decimal are undefined.
Is Pi a rational number?
No, “Pi” is not a rational number because it neither converts into a terminating decimal nor a whole number when divided.
How to convert a decimal into a rational number?
A decimal can easily be converted into a rational number by removing the decimal point and placing the required multiple of 10. But it can’t be converted into a rational number if you have infinite
numbers after a decimal.
Can we change a non-terminating number into a rational number?
No, a non-terminating number can’t be changed into a rational number.
What is the second name of non-terminating numbers?
The second name for non-terminating numbers is "Irrational Numbers". | {"url":"https://calculatorsbag.com/definitions/rational-number","timestamp":"2024-11-12T20:31:01Z","content_type":"text/html","content_length":"37360","record_id":"<urn:uuid:88478e42-3b96-40e2-bc1c-72afc9254ab7>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00164.warc.gz"} |
SOUTHERN METHODIST UNIVERSITY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING INTRODUCTION TO SIGNAL PROCESSING ECE3372 SPRING 2023 Homework #10 Alt. Spectrum Analysis and Filter Design for Speech
Enhancement Speech enhancement describes a set of signal processing techniques designed to improve the quality of voice signals collected by audio sensors, typically microphones, in less-than-ideal
situa- tions. Speech enhancement is important for a number of real-world applications, including voice calls, audio messaging, and automated speech recognition. In the simplest situation, a single
mi- crophone records an audio signal that, after analog-to-digital conversion, is represented by x[n] = N s[n]+[n], i=1 where {ni[n]}, 1 ≤ i ≤ N are N noise signals that are corrupting the
measurement of the desired speech signal s[n]. The goal of speech enhancement is to digitally process the sampled signal x[n] by one or more filters in cascade to attenuate the noises {ni[n]} and
enhance the speech signal s[n]. The design of these filters is generally based on some form of analysis of the signal x[n]. In this assignment, you will use frequency analysis to determine the
natures of the noises that corrupt a recorded speech signal that you have been assigned. In this case, the noises are static in frequency, which simplifies the required processing to a set of linear,
time-invariant filters. From this analysis, you are to design filters that attenuate the noises, thereby revealing the content of the speech signal. The tasks to complete are as follows: 1. Listen to
the recorded signal that you have been assigned, stored as the variable x with sample rate fs, using MATLAB's sound command. Keep the volume low to protect your ears! Describe qualitatively what you
hear. What noises appear to be present in the signal? Can you tell there is a speech signal present? If so, can you understand what is being said? If you can understand what is being said, attempt a
written transcription of the recorded message. 2. In MATLAB, type help pspectrum to learn what the command does. Describe briefly what the function computes. How is this related to what we discussed
about the FFT function in class? Using the pspectrum command, plot the frequency spectrum in dB of your noisy signal from DC to half the sample rate of the signal. Using the graph, carefully
determine the frequency values of the tonal noises. 1 3. Design individual second-order IIR notch filters to remove each of the tonal noises in the signal, one for each tone. Apply the notch filters
to the signal x one by one in a cascade fashion using multiple calls to MATLAB's filter command. Then, listen to the output of the combined system as designed. Have you removed the tones? Note that
you may need to adjust the frequency position of each of the notches by a few Hertz to adequately null out each of the tonal noises in the recorded signal. Once you are satisfied with the sound of
the result, do the following: a) Give the values of the coefficients of the notch filter(s) that you have designed, and find the pair of 3dB cutoff frequencies for each of the notch filters,
verifying that they are on either side of the tonal noises. b) Plot the frequency response of each of the notch filters on separate graphs using MAT- LAB's freqz command. c) Plot both the recorded
signal (top) and the output of the system you've designed (bot- tom) on the same figure in separate sub-graphs. What do you hear in the processed signal at this point? Can you tell there is a speech
signal present? If so, can you understand what is being said? If you can understand what is being said at this point, attempt a written transcription of the recorded message. 4. Now, perform a second
simple frequency analysis of the output of the system to see what frequency content remains in the signal. Plot the frequency spectrum in dB from DC to half the sample rate of the processed signal,
and carefully determine the frequency range of any broadband noise that is present in the processed signal. What kind of filter is needed to remove the broadband noise? 5. Using MATLAB's fir1
command. design a simple FIR filter to remove the broadband noise that you identified in the previous step. Describe how you came to the choice of this filter design. On a separate plot, plot the
impulse response of the filter that you've designed, and comment on its general structure and shape. Then, apply this filter to your processed signal to produce a new processed signal that combines
both the notch filtering and the FIR filter that you've designed, and listen to the newly-processed output. What do you hear in the processed signal at this point? Can you tell there is a speech
signal present? If so, can you understand what is being said? What else can you tell about the signal now (the type of voice? the apparent gender of the talker? anything else?)? Transcribe the
message. Do you think this newly-processed signal could be recognized by a speech recognition engine after processing? Some (but not all) useful MATLAB commands: cos, filter, fft, freqz, sound,
figure, plot, subplot, xlabel, ylabel, title, legend. Your writeup should include written responses, important numerical values, all requested plots, and the MATLAB code used in your processing and
the answers to the questions above. The due date/time for this assignment is 9:00am, Tuesday, April 30, 2024. 2/n SOUTHERN METHODIST UNIVERSITY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
INTRODUCTION TO SIGNAL PROCESSING ECE3372 SPRING 2023 Name: Homework #9 Alt. The DTFT and Discrete-Time Systems Assigned: Friday, March 22, 2024 Due: Friday, April 19, 2024 9.1. A linear,
time-invariant system has a frequency response over ≤ π as 202 H() = rect πT Find the output y[n] of the system if the input x[n] is a) x[n] = sinc(πn/2) 1 b) x[n] = sinc(n) c) x[n] = sinc²(πn/2) 21
9.2. Lathi, Problem 9.1-1, p. 899. [shown below] 9.1-1 Find the discrete-time Fourier series (DTFS) and sketch their spectra |Dr] and [Dr for 0≤r≤No-1 for the following periodic signal: x[n] 4 cos
2.4лп+2 sin 3.2лn = 3 9.3. Lathi, Problem 9.1-5, p. 899. [shown below] 9.1-5 Find the discrete-time Fourier series and the corresponding amplitude and phase spectra for the x[n] shown in Fig. P9.1-5.
x[n] 3 -9 -6 -3 0 3 6 9 12 n Figure P9.1-5 4/n SOUTHERN METHODIST UNIVERSITY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING INTRODUCTION TO SIGNAL PROCESSING ECE3372 SPRING 2023 Name: Homework #8
Alt. Discrete-Time Fourier Transform Assigned: Friday, March 22, 2024 Due: Friday, April 12, 2024 8.1. First, plot the following signals over the range −6 ≤ n ≤ 6. Then, from the definition of the
DTFT in Eq. (9.19), find the DTFT X() for the following signals. Simplify the expressions for X) using trigonometric relations. a) x[n] = 1 1, −3≤ n ≤0 2, 1≤ n ≤3 0, otherwise b) x[n] = J n n ≤ 3 { b
otherwise 21 x[n] = { ? 2n n≤3 otherwise 3 8.2. First, plot the following discrete-time spectra over the range –π ≤ N ≤ π. Then, from the definition of the inverse DTFT in Eq. (9.18), find x[n] for
the following signals. Simplify the expressions for x[n] using trigonometric relations. [Ω] < 1 a) X(N) = 2, 1 ≤ N ≤ 2 0, 2≤ N ≤π [For your plot, choose No = π/2. For the solution of x[n], give a
general expression for any Ωρ.] 4 b) For any o in the range 0 < No < π, X(N) = ΩΣ ΤΩΙ ΣΩΟ {3² ΩΟΣ ΙΩΙ ΣΠ [For your plot, choose o =π/2. For the solution of x[n], give a general expression for any
No.] 5/n SOUTHERN METHODIST UNIVERSITY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING INTRODUCTION TO SIGNAL PROCESSING ECE3372 SPRING 2023 Name: Homework #7 Alt. Sampling Assigned: Friday, March
22, 2024 Due: Friday, April 5, 2024 7.1. A continuous time signal x(t) has the continuous-time Fourier transform or spectrum given by X (w) = 5, |w| < 25л rad/sec 0, otherwise. a) Sketch X(w) over
the range −100π <w < 100”. b) Suppose x(t) is ideally sampled by the unit impulse train with sampling period T sec to produce x(t). Sketch X (w) over the range −100π <w < 100л. Does aliasing occur?
If so, what frequencies in the original signal are aliased? = 0.02 1 c) Suppose x(t) is ideally sampled by the unit impulse train with sampling period T = 0.04 sec to produce x(t). Sketch X (w) over
the range -100π <w < 100л. Does aliasing occur? If so, what frequencies in the original signal are aliased? d) Suppose x(t) is ideally sampled by the unit impulse train with sampling period T = 0.05
sec to produce x(t). Sketch X (w) over the range -100π <w < 100л. Does aliasing occur? If so, what frequencies in the original signal are aliased? 21 7.2. = 20 a) A continuous time signal x(t) = 10
sinc² (10πt) + cos(50πt) is sampled at a rate of fs Hz. Find a mathematical expression for the spectrum of the sampled signal. Can x(t) be reconstructed from the sampled signal via lowpass filtering?
Explain. = 50 Hz. Find b) The same continuous time signal in part a) is sampled at a rate of fs a mathematical expression for the spectrum of the sampled signal. Can x(t) be recon- structed from the
sampled signal via lowpass filtering? Explain. 3 c) A continuous time signal x(t) = 10 sinc²(10πt)+sin(50πt) is sampled at a rate of fs = 50 Hz. Can x(t) be reconstructed from its samples if the
signal is sampled at a rate of fs = 50 Hz? Use spectral representations to explain why or why not. d) Repeat part c) for a sampling rate of fs: = 51 Hz. 4 7.3. A continuous-time signal x(t) has the
spectrum w X (w) = 10πT 0, " 20πw < 30π otherwise. a) Find the Nyquist rate for this signal based on the highest frequency contained in the signal. Sketch the spectrum of the ideally-sampled signal x
(t) if sampling is done at the Nyquist rate. (NOTE: The spectrum has a slightly different shape for negative and positive frequencies!) = b) A colleague of yours claims that, because the signal is
bandlimited to 5 Hz, it can be reconstructed if it is sampled at a rate of ƒs 10 Hz. Sketch the spectrum of the ideally-sampled signal (t) for this sampling rate. Can x(t) be reconstructed from the
samples taken at this rate? If so, explain how reconstruction can be performed. If not, explain why not. 5 | {"url":"https://tutorbin.com/questions-and-answers/southern-methodist-university-department-of-electrical-and-computer-engineering-introduction-to-signal-processing","timestamp":"2024-11-12T15:35:58Z","content_type":"text/html","content_length":"90125","record_id":"<urn:uuid:8390c0f1-4916-45c3-8882-cbe1e387e011>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00108.warc.gz"} |
The Double Twist: From Ethnography to Morphodynamics
318 pages
Contains Illustrations
ISBN 0-8020-3525-8
DDC 306
Edited by Pierre Maranda
Joan Lovisek, Ph.D., is a consulting anthropologist and ethnohistorian
in British Columbia.
In 1955, French anthropologist Claude Lévi-Strauss introduced a
mathematical formula that attempted to represent his understanding that
every myth was an aggregate of all its variants. (The formula is
expressed as the canonical formula: fx(a):fy(b)::fx(b):fa-1(y), where :
means “transforms itself into” and :: “is a resemblance between
two transformations.”) Despite its long existence, few working
anthropologists use it.
The Double Twist presents an excellent collection of 10 papers by
international scholars from a variety of disciplines who explain the
formula, test its applicability, and illuminate its value as both
metaphor and mathematical formulation. The collection is introduced by
Lévi-Strauss, who extends his formula to cross-cultural religious
architecture, focusing on the hourglass forms of Fijian temples. Luc
Racine demonstrates how the formula cannot be reduced or understood as a
simple analogy. Erich Schwimmer considers the ability of structuralism
to cope with history by testing the formula on cultural data from Papua
New Guinea. Pierre Maranda explores gender relations in Melanesia
following the advent of Christianity and uses the formula to map this
transformation. Lucien Scubla provides a detailed historical review of
the canonical formula and applies it to Hesiodic myth. Sбndor Darбnyi
uses data from Asia Minor to demonstrate the correspondence between the
geometric and spatial interpretation involved in the formula.
Christopher Gregory compares two kinds of binary logic (Boolean and
Ramistic) by analyzing an Indian myth, and concludes that the formula
based in binary logic is actually Ramistic. The volume concludes with
technical mathematical papers by Alain Cфté, Andrew Quinn, and Jean
These authors confirm that the canonical formula is an intelligent
means of grasping mythical transformation and cognitive structures.
Structuralism is based on a concept of universal truth, which is
contrary to postmodernist thought: that may partly explain why the
formula has been widely ignored. The difficulty in understanding the
formula is probably another reason. Although this collection adds to
that understanding, it is most suitable for advanced students of
anthropology and semiotics.
“The Double Twist: From Ethnography to Morphodynamics,” Canadian Book Review Annual Online, accessed November 1, 2024, https://cbra.library.utoronto.ca/items/show/9628. | {"url":"https://cbra.library.utoronto.ca/items/show/9628","timestamp":"2024-11-02T00:08:51Z","content_type":"text/html","content_length":"13193","record_id":"<urn:uuid:a8c902f2-990c-4d27-9297-369c19184f6a>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00780.warc.gz"} |
Calcium - IMScWebinar
IMSc Webinar
The generalised Diophantine m-tuples
Anup Dixit
A set of natural numbers $\{a_1, a_2, \cdots, a_m\}$ is said to be a Diophantine $m$-tuple with property $D(n)$ if $a_ia_j +n$ is a perfect square for $i
eq j$. One may ask, what is the largest $m$ for which such a tuple exists. This problem has a long history, attracting the attention of many, including Fermat, Baker, Davenport etc, with significant
progress made in recent times due to Dujella and others. In this talk, we consider a similar question by replacing the condition $a_ia_j+n$ from being a square to $k$-th powers. This is joint work
with Ram Murty and Seoyoung Kim.
This talk will be accessible to graduate students!
Google meet link for this talk is | {"url":"https://www.imsc.res.in/cgi-bin/CalciumShyam/Calcium40.pl?CalendarName=InstituteEvents&DoneURL=Calcium40.pl%3FCalendarName%3DInstituteEvents%26Op%3DShowIt%26Amount%3DWeek%26NavType%3DRelative%26Type%3DBlock%26Date%3D2020%252F12%252F2&ID=105&Op=PopupWindow&Source=IMScWebinar&Date=2020%2F12%2F2&Amount=Week&NavType=Relative&Type=Block","timestamp":"2024-11-06T15:26:18Z","content_type":"application/xhtml+xml","content_length":"3579","record_id":"<urn:uuid:26de04b2-ed5b-4125-9c0b-7e13370d6edc>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00754.warc.gz"} |
Expression for drift velocity of free electrons | Edumir-Physics
Expression for drift velocity of free electrons
If an external electric field is applied across a conductor, free electrons move in the conductor. The velocities of all the electrons are not the same at every point in the conductor. Therefore, we
need to consider the average value of the velocities of the electrons. It is nothing but the drift velocity. In this article, we are going to discuss the definition and expression for drift velocity
of free electrons in a conductor. This formula will give the relation between the drift velocity and electric current density. Also, we will derive an expression that relates the drift velocity with
the relaxation time.
Contents of this article:
1. What is drift velocity?
2. Derive the expression for drift velocity of electrons in a conductor
3. Relation between drift velocity and current
4. Relation between drift velocity and electric field
5. Relaxation time and Drift velocity
What is the Drift velocity of free electrons?
Drift velocity of a free electron is defined as the average velocity of the electron with which it moves through a conductor or semiconductor to carry an electric current. We need to consider the
average velocity of electrons because the velocity of an electron is not the same at every point in a conductor. Again, the velocity of every electron is different from other electrons at any time
and at any point in the conductor.
Expression for drift velocity
Drift velocity of free electrons
Let electrons are moving through a current-carrying wire of current I. This wire can be considered as a cylinder of cross-section area A. We consider that n number of electrons of charge e is moving
per unit volume of the wire with the drift velocity V[d]. Since V[d] is the drift velocity, it implies that the electrons move V[d] length of the wire per unit time. So, the volume of occupation of
the electrons per unit time is AV[d].
So, the total number of electrons crossing the volume AV[d] per unit time is $\color{Blue} N = nA V_{d}$
Now, each electron has the charge of e. So, the amount of electric charge crossing the volume per unit time is $\color{Blue} eN = enV_{d}$
Since the amount of charge flowing per unit time is the electric current then the electric current through the wire is, $\color{Blue} I = enV_{d}$.
or, the drift velocity of free electrons, $\color{Blue}V_{d} = \frac{I}{enA}$………………… (1)
This is the relation between drift velocity and current.
Again, the current per unit area is the current density (J). So, J = I/A.
So, another expression for drift velocity of electrons in a wire is, $\color{Blue}V_{d} = \frac{J}{en}$……………… (2)
en is the charge density. So, one can use $\small {\color{Blue} \rho =en}$ in equation (1) and (2).
Unit of drift velocity of electrons
The units of drift velocity are the same as the unit of normal velocity. The SI unit of drift velocity is m/s and CGS unit of drift velocity is cm/s.
Relation between drift velocity and current
The relation between drift velocity and electric current is, $\color{Blue}V_{d} = \frac{I}{enA}$ or, $\color{Blue} I = enA V_{d}$
Again, charge density, $\small \color{Blue}\rho=en$. Then, the relation between drift velocity and electric current is, $\small \color{Blue} I= \rho A V_{d}$.
Relation between drift velocity of electrons and electric current density
current density, J = I/A
Then the relation between drift velocity and current density is, $\color{Blue}V_{d} = \frac{J}{en}$ or, $\color{Blue} J = enV_{d}$
Again, en is the charge density. So, $\small J = \rho$ V[d]. This is the relation between the current density and charge density.
Relation between drift velocity and electric field
From the differential form of Ohm’s law, we get the relation between the current density and electric field is $\small J = \sigma$E
Then, from above relation, $\small\sigma$E = enV[d] or, drift velocity, V[d] = $\small\frac{\sigma}{en}$E
Again, Mobility, $\small\mu=\frac{\sigma}{en}$
Then, the relation between the drift velocity, electric field and the mobility of electron is, V[d] = $\small\mu$E
Relation between drift velocity and relaxation time
Relaxation time is the average time interval between the two successive collisions of an electron with other electrons during their motion in the conductor.
If a voltage V is applied across a conducting wire of length L, then electric field E = V/L will appear inside the conductor. Let an electron of charge e (magnitude) is moving under the electric
field E. Then the electric force on the charge is F = eE.
If m is the mass of the electron then the acceleration of the electron is, a = F/m = eE/m
Now, if t is the relaxation time then the average velocity of electrons between two successive collisions or the drift velocity of electrons is,
V[d] = at
or, drift velocity, V[d] = eEt/m ……………… (3)
This is the relation between the drift velocity of electron and the relaxation time.
This is all from this article on the expression for drift velocity of electrons in a conductor. If you have any doubts on this topic you can ask me in the comment section.
Keep visiting for more articles.
Thank you!
Related Posts:
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Earnings before interest, taxes, depreciation and amortization (EBITDA)
Definition of Earnings before interest, taxes, depreciation and amortization (EBITDA)
Earnings before interest, taxes, depreciation and amortization (EBITDA)
The operating profit before deducting interest, tax, depreciation and amortization.
Earnings Before Interest, Taxes, Depreciation, and Amortization (EBITDA)
An earningsbased measure that, for many, serves as a surrogate for cash flow. Actually consists of working
capital provided by operations before interest and taxes.
Related Terms:
the combined discounts for lack of control and marketability. g the constant growth rate in cash flows or net income used in the ADF, Gordon model, or present value factor.
Any depreciation method that produces larger deductions for depreciation in the
early years of a project's life. Accelerated cost recovery system (ACRS), which is a depreciation schedule
allowed for tax purposes, is one such example.
earnings of a firm as reported on its income statement.
The accumulated coupon interest earned but not yet paid to the seller of a bond by the
buyer (unless the bond is in default).
The repayment of a loan by installments.
The pool factor implied by the scheduled amortization assuming no prepayemts.
Swap in which the principal or national amount rises (falls) as interest rates
rise (decline).
A situation wherein participants in a transaction have different net tax rates.
Related: Benchmark interest rate.
The ratio of net income before taxes to net sales.
Also called the base interest rate, it is the minimum interest rate investors will
demand for investing in a non-Treasury security. It is also tied to the yield to maturity offered on a
comparable-maturity Treasury security that was most recently issued ("on-the-run").
The requirement that a claim holder voting against a plan of reorganization
must receive at least as much as he would have if the debtor were liquidated.
interest that is not immediately expensed, but rather is considered as an asset and is then
amortized through the income statement over time.
Net income plus depreciation.
interest paid on previously earned interest as well as on the principal.
Covered interest arbitrage
A portfolio manager invests dollars in an instrument denominated in a foreign
currency and hedges his resulting foreign exchange risk by selling the proceeds of the investment forward for
Deferred taxes
A non-cash expense that provides a source of free cash flow. Amount allocated during the
period to cover tax liabilities that have not yet been paid.
A non-cash expense that provides a source of free cash flow. Amount allocated during the
period to amortize the cost of acquiring Long term assets over the useful life of the assets.
Depreciation tax shield
The value of the tax write-off on depreciation of plant and equipment.
Double-declining-balance depreciation
Method of accelerated depreciation.
Net income for the company during the period.
Earnings before interest and taxes (EBIT)
A financial measure defined as revenues less cost of goods sold
and selling, general, and administrative expenses. In other words, operating and non-operating profit before
the deduction of interest and income taxes.
Earnings per share (EPS)
EPS, as it is called, is a company's profit divided by its number of outstanding
shares. If a company earned $2 million in one year had 2 million shares of stock outstanding, its EPS would
be $1 per share. The company often uses a weighted average of shares outstanding over the reporting term.
Earnings retention ratio
Plowback rate.
Earnings surprises
Positive or negative differences from the consensus forecast of earnings by institutions
such as First Call or IBES. Negative earnings surprises generally have a greater adverse affect on stock prices
than the reciprocal positive earnings surprise on stock prices.
Earnings yield
The ratio of earnings per share after allowing for tax and interest payments on fixed interest
debt, to the current share price. The inverse of the price/earnings ratio. It's the Total Twelve Months earnings
divided by number of outstanding shares, divided by the recent price, multiplied by 100. The end result is
shown in percentage.
Economic earnings
The real flow of cash that a firm could pay out forever in the absence of any change in
the firm's productive capacity.
Effective annual interest rate
An annual measure of the time value of money that fully reflects the effects of
Equilibrium rate of interest
The interest rate that clears the market. Also called the market-clearing interest
Forward interest rate
interest rate fixed today on a loan to be made at some future date.
Fully diluted earnings per shares
earnings per share expressed as if all outstanding convertible securities
and warrants have been exercised.
Gross interest
interest earned before taxes are deducted.
The price paid for borrowing money. It is expressed as a percentage rate over a period of time and
reflects the rate of exchange of present consumption for future consumption. Also, a share or title in property.
Interest coverage ratio
The ratio of the earnings before interest and taxes to the annual interest expense. This
ratio measures a firm's ability to pay interest.
Interest coverage test
A debt limitation that prohibits the issuance of additional long-term debt if the issuer's
interest coverage would, as a result of the issue, fall below some specified minimum.
Interest equalization tax
Tax on foreign investment by residents of the U.S. which was abolished in 1974.
Interest payments
Contractual debt payments based on the coupon rate of interest and the principal amount.
Interest on interest
interest earned on reinvestment of each interest payment on money invested.
See: compound interest.
Interest-only strip (IO)
A security based solely on the interest payments form a pool of mortgages, Treasury
bonds, or other bonds. Once the principal on the mortgages or bonds has been repaid, interest payments stop
and the value of the IO falls to zero.
Interest rate agreement
An agreement whereby one party, for an upfront premium, agrees to compensate the
other at specific time periods if a designated interest rate (the reference rate) is different from a predetermined
level (the strike rate).
Interest rate cap
Also called an interest rate ceiling, an interest rate agreement in which payments are made
when the reference rate exceeds the strike rate.
Interest rate ceiling
Related: interest rate cap.
Interest rate floor
An interest rate agreement in which payments are made when the reference rate falls
below the strike rate.
Interest rate on debt
The firm's cost of debt capital.
Interest rate parity theorem
interest rate differential between two countries is equal to the difference
between the forward foreign exchange rate and the spot rate.
Interest rate risk
The risk that a security's value changes due to a change in interest rates. For example, a
bond's price drops as interest rates rise. For a depository institution, also called funding risk, the risk that
spread income will suffer because of a change in interest rates.
Interest rate swap
A binding agreement between counterparties to exchange periodic interest payments on
some predetermined dollar principal, which is called the notional principal amount. For example, one party
will pay fixed and receive variable.
Interest subsidy
A firm's deduction of the interest payments on its debt from its earnings before it calculates
its tax bill under current tax law.
Interest tax shield
The reduction in income taxes that results from the tax-deductibility of interest payments.
Loan amortization schedule
The schedule for repaying the interest and principal on a loan.
Low price-earnings ratio effect
The tendency of portfolios of stocks with a low price-earnings ratio to
outperform portfolios consisting of stocks with a high price-earnings ratio.
Negative amortization
A loan repayment schedule in which the outstanding principal balance of the loan
increases, rather than amortizing, because the scheduled monthly payments do not cover the full amount
required to amortize the loan. The unpaid interest is added to the outstanding principal, to be repaid later.
Nominal interest rate
The interest rate unadjusted for inflation.
Open interest
The total number of derivative contracts traded that not yet been liquidated either by an
offsetting derivative transaction or by delivery. Related: liquidation
Planned amortization class CMO
1) One class of CMO that carries the most stable cash flows and the
lowest prepayement risk of any class of CMO. Because of that stable cash flow, it is considered the least risky CMO.
2) A CMO bond class that stipulates cash-flow contributions to a sinking fund. With the PAC,
principal payments are directed to the sinking fund on a priority basis in accordance with a predetermined
payment schedule, with prior claim to the cash flows before other CMO classes. Similarly, cash flows
received by the trust in excess of the sinking fund requirement are also allocated to other bond classes. The
prepayment experience of the PAC is therefore very stable over a wide range of prepayment experience.
Pooling of interests
An accounting method for reporting acquisitions accomplished through the use of equity.
The combined assets of the merged entity are consolidated using book value, as opposed to the purchase
method, which uses market value. The merging entities' financial results are combined as though the two
entities have always been a single entity.
Price/earnings ratio (PE ratio)
Shows the "multiple" of earnings at which a stock sells. Determined by dividing current
stock price by current earnings per share (adjusted for stock splits). earnings per share for the P/E ratio is
determined by dividing earnings for past 12 months by the number of common shares outstanding. Higher
"multiple" means investors have higher expectations for future growth, and have bid up the stock's price.
Rate of interest
The rate, as a proportion of the principal, at which interest is computed.
Real interest rate
The rate of interest excluding the effect of inflation; that is, the rate that is earned in terms
of constant-purchasing-power dollars. interest rate expressed in terms of real goods, i.e. nominal interest rate
adjusted for inflation.
Retained earnings
Accounting earnings that are retained by the firm for reinvestment in its operations;
earnings that are not paid out as dividends.
Short interest
This is the total number of shares of a security that investors have borrowed, then sold in the
hope that the security will fall in value. An investor then buys back the shares and pockets the difference as profit.
Simple interest
interest calculated only on the initial investment. Related:compound interest.
Spot interest rate
interest rate fixed today on a loan that is made today. Related: forward interest rates.
Stated annual interest rate
The interest rate expressed as a per annum percentage, by which interest
payment is determined.
Straight line depreciation
An equal dollar amount of depreciation in each accounting period.
Sum-of-the-years'-digits depreciation
Method of accelerated depreciation.
Times-interest-earned ratio
earnings before interest and tax, divided by interest payments.
True interest cost
For a security such as commercial paper that is sold on a discount basis, the coupon rate
required to provide an identical return assuming a coupon-bearing instrument of like maturity that pays
interest in arrears.
A technique by which a company recovers the high cost of its plant-and-equipment assets gradually during the number of years they’ll be used in the business. depreciation can be physical,
technological, or both.
Earnings per share of common stock
How much profit a company made on each share of common stock this year.
Profits a company plowed back into the business over the years. Last January’s retained earnings, plus the net income or profit that a company made this year (which is calculated on the income
statement), minus dividends paid out, equals the retained earnings balance on the balance sheet date.
A depreciation method that depreciates an asset the same amount for each year of its estimated
See depreciation, but usually in relation to assets attached to leased property.
An expense that spreads the cost of an asset over its useful life.
Earnings before interest and taxes (EBIT)
The operating profit before deducting interest and tax.
The cost of money, received on investments or paid on borrowings.
Profit before interest and taxes (PBIT)
See EBIT.
Accumulated depreciation
A contra-fixed asset account representing the portion of the cost of a fixed asset that has been previously charged to expense. Each fixed asset account will have its own associated accumulated
depreciation account.
Depreciation expense
An expense account that represents the portion of the cost of an asset that is being charged to expense during the current period.
Interest income
Income that a company receives in the form of interest, usually as the result of keeping money in interest-bearing accounts at financial institutions and the lending of money to other companies.
Interest payable
The amount of interest that is owed but has not been paid at the end of a period.
Payroll taxes payable
The amount of payroll taxes owed to the various governments at the end of a period.
Retained earnings
The residual earnings of the company.
Statement Retained Earnings
One of the basic financial statements; it takes the beginning balance of retained earnings and adds net income, then subtracts dividends. The Statement of Retained earnings is prepared for a
specified period of time.
accelerated depreciation
(1) The estimated useful life of the fixed asset being depreciated is
shorter than a realistic forecast of its probable actual service life;
(2) more of the total cost of the fixed asset is allocated to the first
half of its useful life than to the second half (i.e., there is a
front-end loading of depreciation expense).
accumulated depreciation
A contra, or offset, account that is coupled
with the property, plant, and equipment asset account in which the original
costs of the long-term operating assets of a business are recorded.
The accumulated depreciation contra account accumulates the amount of
depreciation expense that is recorded period by period. So the balance in
this account is the cumulative amount of depreciation that has been
recorded since the assets were acquired. The balance in the accumulated
depreciation account is deducted from the original cost of the assets
recorded in the property, plant, and equipment asset account. The
remainder, called the book value of the assets, is the amount included on
the asset side of a business.
This term has two quite different meanings. First, it may
refer to the allocation to expense each period of the total cost of an
intangible asset (such as the cost of a patent purchased from the inventor)
over its useful economic life. In this sense amortization is equivalent
to depreciation, which allocates the cost of a tangible long-term operating
asset (such as a machine) over its useful economic life. Second, amortization
may refer to the gradual paydown of the principal amount of a debt.
Principal refers to the amount borrowed that has to be paid back to the
lender as opposed to interest that has to be paid for use of the principal.
Each period, a business may pay interest and also make a payment on
the principal of the loan, which reduces the principal amount of the loan,
of course. In this situation the loan is amortized, or gradually paid down.
basic earnings per share (EPS)
This important ratio equals the net
income for a period (usually one year) divided by the number capital
stock shares issued by a business corporation. This ratio is so important
for publicly owned business corporations that it is included in the daily
stock trading tables published by the Wall Street Journal, the New York
Times, and other major newspapers. Despite being a rather straightforward
concept, there are several technical problems in calculating
earnings per share. Actually, two EPS ratios are needed for many businesses—
basic EPS, which uses the actual number of capital shares outstanding,
and diluted EPS, which takes into account additional shares of
stock that may be issued for stock options granted by a business and
other stock shares that a business is obligated to issue in the future.
Also, many businesses report not one but two net income figures—one
before extraordinary gains and losses were recorded in the period and a
second after deducting these nonrecurring gains and losses. Many business
corporations issue more than one class of capital stock, which
makes the calculation of their earnings per share even more complicated.
Refers to the generally accepted accounting principle of allocating
the cost of a long-term operating asset over the estimated useful
life of the asset. Each year of use is allocated a part of the original cost of
the asset. Generally speaking, either the accelerated method or the
straight-line method of depreciation is used. (There are other methods,
but they are relatively rare.) Useful life estimates are heavily influenced
by the schedules allowed in the federal income tax law. depreciation is
not a cash outlay in the period in which the expense is recorded—just
the opposite. The cash inflow from sales revenue during the period
includes an amount to reimburse the business for the use of its fixed
assets. In this respect, depreciation is a source of cash. So depreciation is
added back to net income in the statement of cash flows to arrive at cash
flow from operating activities.
diluted earnings per share (EPS)
This measure of earnings per share
recognizes additional stock shares that may be issued in the future for
stock options and as may be required by other contracts a business has
entered into, such as convertible features in its debt securities and preferred
stock. Both basic earnings per share and, if applicable, diluted
earnings per share are reported by publicly owned business corporations.
Often the two EPS figures are not far apart, but in some cases the
gap is significant. Privately owned businesses do not have to report earnings
per share. See also basic earnings per share.
earnings before interest and income tax (EBIT)
A measure of profit that
equals sales revenue for the period minus cost-of-goods-sold expense
and all operating expenses—but before deducting interest and income
tax expenses. It is a measure of the operating profit of a business before
considering the cost of its debt capital and income tax.
earnings per share (EPS)
See basic earnings per share and diluted earnings per share.
net income (also called the bottom line, earnings, net earnings, and net
operating earnings)
This key figure equals sales revenue for a period
less all expenses for the period; also, any extraordinary gains and losses
for the period are included in this final profit figure. Everything is taken
into account to arrive at net income, which is popularly called the bottom
line. Net income is clearly the single most important number in business
financial reports.
price/earnings ratio (price to earnings ratio, P/E ratio, PE ratio)
This key ratio equals the current market price
of a capital stock share divided by the earnings per share (EPS) for the
stock. The EPS used in this ratio may be the basic EPS for the stock or its
diluted EPS—you have to check to be sure about this. A low P/E may signal
an undervalued stock or may reflect a pessimistic forecast by
investors for the future earnings prospects of the business. A high P/E
may reveal an overvalued stock or reflect an optimistic forecast by
investors. The average P/E ratio for the stock market as a whole varies
considerably over time—from a low of about 8 to a high of about 30.
This is quite a range of variation, to say the least.
straight-line depreciation
This depreciation method allocates a uniform
amount of the cost of long-lived operating assets (fixed assets) to each
year of use. It is the basic alternative to the accelerated depreciation
method. When using the straight-line method, a business may estimate a
longer life for a fixed asset than when using the accelerated method
(though not necessarily in every case). Both methods are allowed for
income tax and under generally accepted accounting principles (GAAP).
times interest earned
A ratio that tests the ability of a business to make
interest payments on its debt, which is calculated by dividing annual
earnings before interest and income tax by the interest expense for the
year. There is no particular rule for this ratio, such as 3 or 4 times, but
obviously the ratio should be higher than 1.
Accrued Interest
The amount of interest accumulated on a debt security between
interest paying dates
Basic Earnings Power Ratio
Percentage of earnings relative to total assets; indication of how
effectively assets are used to generate earnings. It is calculated by
dividing earnings before interest and taxes by the book value of all
Related to : financial, finance, business, accounting, payroll, inventory, investment, money, inventory control, stock trading, financial advisor, tax advisor, credit. | {"url":"http://www.finance-lib.com/financial-term-earnings-before-interest-taxes-depreciation-and-amortization-ebitda.html","timestamp":"2024-11-05T15:21:06Z","content_type":"text/html","content_length":"34774","record_id":"<urn:uuid:f1c8facc-2477-4a2c-9a4b-ef960652de13>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00642.warc.gz"} |
Chasing robbers on random geometric graphs - An alternative approach
We study the vertex pursuit game of Cops and Robbers, in which cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the
cop number of G. We focus on G[d](n,r), a random geometric graph in which n vertices are chosen uniformly at random and independently from [0,1]^d, and two vertices are adjacent if the Euclidean
distance between them is at most r. The main result is that if r^3d-1>c[d]lognn then the cop number is 1 with probability that tends to 1 as n tends to infinity. The case d=2 was proved earlier and
independently in Beveridge et al. (2012), using a different approach. Our method provides a tight O(1/r^2) upper bound for the number of rounds needed to catch the robber.
All Science Journal Classification (ASJC) codes
• Discrete Mathematics and Combinatorics
• Applied Mathematics
• Cops and Robbers
• Random graphs
• Vertex-pursuit games
Dive into the research topics of 'Chasing robbers on random geometric graphs - An alternative approach'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/chasing-robbers-on-random-geometric-graphs-an-alternative-approac","timestamp":"2024-11-03T10:38:19Z","content_type":"text/html","content_length":"47242","record_id":"<urn:uuid:43d039df-8b2d-44af-baf3-45c73c714560>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00131.warc.gz"} |
Data Cleaning and Troubleshooting
The most common issue we’ve found is when a function that predicts survival probabilities equals 1 for some range of trait values in the IPM. This doesn’t cause any mathematical issues in many
analyses, and so often goes unnoticed by authors publishing their IPMs (the author of this document included - example to follow shortly!). However, to our knowledge, no species becomes immortal
regardless of their size, and so this will cause issues for any longevity-related questions. Additionally, a mathematical issue arises whenever one needs to compute the fundamental operator for a
model with this.
Overview of the problem
The fundamental operator, \(N\), can be thought of as the amount of time an individual will spend in state \(z'\) given state \(z\) before death. It is given by
\[ N = (I + P + P^2 + P^3 + ...) = (I - P)^{-1} \]
This Neumann series corresponds to the geometric series \(1 + r + r^2 + r^3 + ... = (1-r)^{-1}\) and is valid for any real number \(|r| < 1\), and remains valid provided survival probabilities for
individuals are less than 1. Clearly, this computation is no longer valid when an IPM’s survival model predicts survival probabilities that are equal to 1.
This most often manifests as negative values in the fundamental operator, which propagate through the rest of the analysis and return nonsense results. Below is a quick example of this manifesting in
an IPM for Lonicera maackii when computing mean lifespan (\(\bar\eta(z_0) = eN\)).
## Warning: package 'ipmr' was built under R version 4.2.3
problem_ipm <- pdb_make_proto_ipm(pdb, "aaa341") %>%
P <- problem_ipm$aaa341$sub_kernels$P
N <- solve(diag(nrow(P)) - P)
## [1] -348825.70 -30555.21
According to this, Lonicera maackii is expected to live between negative 348,800 and negative 30,555 years. This seems incorrect. Therefore, we should inspect what’s going on with the \(P\) kernel,
and more importantly, the functions that comprise it. We’ll use a couple functions from Rpadrino for that:
## $aaa341
## P: s * g
## F: Ep * Fp * Fs * Fd
## $aaa341
## s: 1/(1 + exp(-(si + ss1 * size_1 + ss2 * size_1^2)))
## g_mean: gi + gs * size_1
## g: dnorm(size_2, g_mean, g_sd)
## Fp: 1/(1 + exp(-(fpi + fps * size_1)))
## Fs: exp(fi + fs * size_1)
## Fd: dnorm(size_2, fd_mean, fd_sd)
We can see that P = s * g, and since g is given by a probability density function, we are unlikely to find much going on there. Thus, we want to inspect the values for s. How do we do that? By
default, ipmr, the engine that powers Rpadrino, does not return individual function values in IPM objects. Therefore, we’ll need to rebuild the IPM, and tell ipmr to give us those by setting
return_all_envs = TRUE. Then we can ask for the vital rate functions using vital_rate_funs().
problem_ipm <- pdb_make_proto_ipm(pdb, "aaa341") %>%
pdb_make_ipm(addl_args = list(aaa341 = list(return_all_envs = TRUE)))
vr_funs <- vital_rate_funs(problem_ipm)
## $aaa341
## $aaa341$P
## s (not yet discretized): A 500 x 500 kernel with minimum value: 0.1063 and maximum value: 1
## g_mean (not yet discretized): A 500 x 500 kernel with minimum value: 18.6208 and maximum value: 497.9516
## g (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 0.0337
## $aaa341$F
## Fp (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 1
## Fs (not yet discretized): A 500 x 500 kernel with minimum value: 30.2437 and maximum value: 4705.0552
## Fd (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 0.3308
Ah ha! The maximum value of s is 1! This is problematic for us. How do we fix this?
Fixing the problem
The quickest way is to modify the survival function to be a parallel minimum of the original survival function (i.e. the one that the authors used) and some maximum survival value that we choose
ourselves. For the purposes of this example, we’ll use 0.98 as the maximum survival probability. Rpadrino has two functions to help with this: vital_rate_exprs<- and pdb_new_fun_form(). These are
used together to insert a new functional form into a proto_ipm so that we can make changes to the IPM without having to think too much about how a proto_ipm is actually structured.
problem_proto <- pdb_make_proto_ipm(pdb, "aaa341")
vital_rate_exprs(problem_proto) <- pdb_new_fun_form(
aaa341 = list(
s = pmin(0.98, plogis(si + ss1 * size_1 + ss2 * size_1 ^ 2))
good_ipm <- pdb_make_ipm(problem_proto,
addl_args = list(aaa341 = list(return_all_envs = TRUE)))
## $aaa341
## $aaa341$P
## s (not yet discretized): A 500 x 500 kernel with minimum value: 0.1063 and maximum value: 0.98
## g_mean (not yet discretized): A 500 x 500 kernel with minimum value: 18.6208 and maximum value: 497.9516
## g (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 0.0337
## $aaa341$F
## Fp (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 1
## Fs (not yet discretized): A 500 x 500 kernel with minimum value: 30.2437 and maximum value: 4705.0552
## Fd (not yet discretized): A 500 x 500 kernel with minimum value: 0 and maximum value: 0.3308
## [1] 5.462124 50.007606
These values are far more reasonable! | {"url":"http://cran.itam.mx/web/packages/Rpadrino/vignettes/data-cleaning.html","timestamp":"2024-11-10T12:24:12Z","content_type":"text/html","content_length":"22465","record_id":"<urn:uuid:0bb00c07-525d-4a7b-9aa4-32f2634ae44e>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00112.warc.gz"} |
Probability - Types of Events - A Plus Topper
Probability – Types of Events
Event: An event is a subset of a sample space.
1. Simple event: An event containing only a single sample point is called an elementary or simple event.
2. Compound events: Events obtained by combining together two or more elementary events are known as the compound events or decomposable events.
3. Equally likely events: Events are equally likely if there is no reason for an event to occur in preference to any other event.
4. Mutually exclusive or disjoint events: Events are said to be mutually exclusive or disjoint or incompatible if the occurrence of any one of them prevents the occurrence of all the others.
5. Mutually non-exclusive events: The events which are not mutually exclusive are known as compatible events or mutually non exclusive events.
6. Independent events: Events are said to be independent if the happening (or non-happening) of one event is not affected by the happening (or non-happening) of others.
7. Dependent events: Two or more events are said to be dependent if the happening of one event affects (partially or totally) other event.
Mutually exclusive and exhaustive system of events:
Let S be the sample space associated with a random experiment. Let A[1], A[2], ……….. A[n] be subsets of S such that
(i) A[i] ∩ A[j] = ϕ for i ≠ j and (ii) A[1] ∪ A[2] ∪ ….. ∪ A[n] = S
Then the collection of events is said to form a mutually exclusive and exhaustive system of events.
If E[1], E[2], ……….. E[n] are elementary events associated with a random experiment, then
(i) E[i] ∩ E[j] = ϕ for i ≠ j and (ii) E[1] ∪ E[2] ∪ ….. ∪ E[n] = S
So, the collection of elementary events associated with a random experiment always form a system of mutually exclusive and exhaustive system of events.
In this system, P(A[1] ∪ A[2] ……… ∪ A[n]) = P(A[1]) + P(A[2]) + …… + P(A[n]) = 1 | {"url":"https://www.aplustopper.com/probability-types-events/","timestamp":"2024-11-08T22:22:03Z","content_type":"text/html","content_length":"45193","record_id":"<urn:uuid:aeae05bf-86a8-4d28-8fcd-cad70d5de0a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00786.warc.gz"} |
Simon Giraudot
Surface reconstruction from point clouds is a core topic in geometry processing [3]. It is an ill-posed problem: there is an infinite number of surfaces that approximate a single point cloud and a
point cloud does not define a surface in itself. Thus additional assumptions and constraints must be defined by the user and reconstruction can be achieved in many different ways. This tutorial
provides guidance on how to use the different algorithms of CGAL to effectively perform surface reconstruction.
Which algorithm should I use?
CGAL offers three different algorithms for surface reconstruction:
Because reconstruction is an ill-posed problem, it must be regularized via prior knowledge. Differences in prior lead to different algorithms, and choosing one or the other of these methods is
dependent on these priors. For example, Poisson always generates closed shapes (bounding a volume) and requires normals but does not interpolate input points (the output surface does not pass exactly
through the input points). The following table lists different properties of the input and output to help the user choose the method best suited to each problem:
Poisson Advancing front Scale space
Are normals required? Yes No No
Is noise handled? Yes By preprocessing Yes
Is variable sampling handled? Yes Yes By preprocessing
Are input points exactly on the surface? No Yes Yes
Is the output always closed? Yes No No
Is the output always smooth? Yes No No
Is the output always manifold? Yes Yes Optional
Is the output always orientable? Yes Yes Optional
More information on these different methods can be found on their respective manual pages and in Section Reconstruction.
Pipeline Overview
This tutorial aims at providing a more comprehensive view of the possibilities offered by CGAL for dealing with point clouds, for surface reconstruction purposes. The following diagram shows an
overview (not exhaustive) of common reconstruction steps using CGAL tools.
We now review some of these steps in more detail.
Reading Input
The reconstruction algorithms in CGAL take a range of iterators on a container as input and use property maps to access the points (and the normals when they are required). Points are typically
stored in plain text format (denoted as 'XYZ' format) where each point is separated by a newline character and each coordinate separated by a white space. Other formats available are 'OFF', 'PLY' and
'LAS'. CGAL provides functions to read such formats:
CGAL also provides a dedicated container CGAL::Point_set_3 to handle point sets with additional properties such as normal vectors. In this case, property maps are easily handled as shown in the
following sections. This structure also handles the stream operator to read point sets in any of the formats previously described. Using this method yields substantially shorter code, as can be seen
on the following example:
Point_set points;
std::string fname = argc==1?CGAL::data_file_path("points_3/kitten.xyz") : argv[1];
if (argc < 2)
std::cerr << "Usage: " << argv[0] << " [input.xyz/off/ply/las]" << std::endl;
std::cerr <<"Running " << argv[0] << " data/kitten.xyz -1\n";
std::ifstream stream (fname, std::ios_base::binary);
if (!stream)
std::cerr << "Error: cannot read file " << fname << std::endl;
return EXIT_FAILURE;
stream >> points;
std::cout << "Read " << points.size () << " point(s)" << std::endl;
if (points.empty())
return EXIT_FAILURE;
Because reconstruction algorithms have some specific requirements that point clouds do not always meet, some preprocessing might be necessary to yield the best results.
Note that this preprocessing step is optional: when the input point cloud has no imperfections, reconstruction can be applied to it without any preprocessing.
Outlier removal
Some acquisition techniques generate points which are far away from the surface. These points, commonly referred to as "outliers", have no relevance for reconstruction. Using the CGAL reconstruction
algorithms on outlier-ridden point clouds produce overly distorted output, it is therefore strongly advised to filter these outliers before performing reconstruction.
typename Point_set::iterator rout_it = CGAL::remove_outliers<CGAL::Sequential_tag>
24, // Number of neighbors considered for evaluation
points.parameters().threshold_percent (5.0)); // Percentage of points to remove
points.remove(rout_it, points.end());
std::cout << points.number_of_removed_points()
<< " point(s) are outliers." << std::endl;
// Applying point set processing algorithm to a CGAL::Point_set_3
// object does not erase the points from memory but place them in
// the garbage of the object: memory can be freed by the user.
Some laser scanners generate points with widely variable sampling. Typically, lines of scan are very densely sampled but the gap between two lines of scan is much larger, leading to an overly massive
point cloud with large variations of sampling density. This type of input point cloud might generate imperfect output using algorithms which, in general, only handle small variations of sampling
CGAL provides several simplification algorithms. In addition to reducing the size of the input point cloud and therefore decreasing computation time, some of them can help making the input more
uniform. This is the case of the function grid_simplify_point_set() which defines a grid of a user-specified size and keeps one point per occupied cell.
// Compute average spacing using neighborhood of 6 points
double spacing = CGAL::compute_average_spacing<CGAL::Sequential_tag> (points, 6);
// Simplify using a grid of size 2 * average spacing
points.remove(gsim_it, points.end());
std::cout << points.number_of_removed_points()
<< " point(s) removed after simplification." << std::endl;
PointRange::iterator grid_simplify_point_set(PointRange &points, double epsilon, const NamedParameters &np=parameters::default_values())
Although reconstructions via 'Poisson' or 'Scale space' handle noise internally, one may want to get tighter control over the smoothing step. For example, a slightly noisy point cloud can benefit
from some reliable smoothing algorithms and be reconstructed via 'Advancing front' which provides relevant properties (oriented mesh with boundaries).
Two functions are provided to smooth a noisy point cloud with a good approximation (i.e. without degrading curvature, for example):
These functions directly modify the container:
CGAL::jet_smooth_point_set<CGAL::Sequential_tag> (points, 24);
Normal Estimation and Orientation
Poisson Surface Reconstruction requires points with oriented normal vectors. To apply the algorithm to a raw point cloud, normals must be estimated first, for example with one of these two functions:
PCA is faster but jet is more accurate in the presence of high curvatures. These function only estimates the direction of the normals, not their orientation (the orientation of the vectors might not
be locally consistent). To properly orient the normals, the following functions can be used:
The first one uses a minimum spanning tree to consistently propagate the orientation of normals in an increasingly large neighborhood. In the case of data with many sharp features and occlusions
(which are common in airborne LIDAR data, for example), the second algorithm may produce better results: it takes advantage of point clouds which are ordered into scanlines to estimate the line of
sight of each point and thus to orient normals accordingly.
Notice that these can also be used directly on input normals if their orientation is not consistent.
(points, 24); // Use 24 neighbors
// Orientation of normals, returns iterator to first unoriented point
typename Point_set::iterator unoriented_points_begin =
points.remove (unoriented_points_begin, points.end());
PointRange::iterator mst_orient_normals(PointRange &points, unsigned int k, const NamedParameters &np=parameters::default_values())
Poisson reconstruction consists in computing an implicit function whose gradient matches the input normal vector field: this indicator function has opposite signs inside and outside of the inferred
shape (hence the need for closed shapes). This method thus requires normals and produces smooth closed surfaces. It is not appropriate if the surface is expected to interpolate the input points. On
the contrary, it performs well if the aim is to approximate a noisy point cloud with a smooth surface.
(points.begin(), points.end(),
points.point_map(), points.normal_map(),
output_mesh, spacing);
bool poisson_surface_reconstruction_delaunay(PointInputIterator begin, PointInputIterator end, PointMap point_map, NormalMap normal_map, PolygonMesh &output_mesh, double spacing, double sm_angle=
20.0, double sm_radius=30.0, double sm_distance=0.375, Tag tag=Tag())
Advancing Front
Advancing front is a Delaunay-based approach which interpolates a subset of the input points. It generates triples of point indices which describe the triangular facets of the reconstruction: it uses
a priority queue to sequentially pick the Delaunay facet the most likely to be part of the surface, based on a size criterion (to favor the small facets) and an angle criterion (to favor smoothness).
Its main virtue is to generate oriented manifold surfaces with boundaries: contrary to Poisson, it does not require normals and is not bound to reconstruct closed shapes. However, it requires
preprocessing if the point cloud is noisy.
The Advancing Front package provides several ways of constructing the function. Here is a simple example:
typedef std::array<std::size_t, 3> Facet; // Triple of indices
std::vector<Facet> facets;
// The function is called using directly the points raw iterators
std::cout << facets.size ()
<< " facet(s) generated by reconstruction." << std::endl;
IndicesOutputIterator advancing_front_surface_reconstruction(PointInputIterator b, PointInputIterator e, IndicesOutputIterator out, double radius_ratio_bound=5, double beta=0.52)
Scale Space
Scale space reconstruction aims at producing a surface which interpolates the input points (interpolant) while offering some robustness to noise. More specifically, it first applies several times a
smoothing filter (such as Jet Smoothing) to the input point set to produce a scale space; then, the smoothest scale is meshed (using for example the Advancing Front mesher); finally, the resulting
connectivity between smoothed points is propagated to the original raw input point set. This method is the right choice if the input point cloud is noisy but the user still wants the surface to pass
exactly through the points.
(points.points().begin(), points.points().end());
// Smooth using 4 iterations of Jet Smoothing
// Mesh with the Advancing Front mesher with a maximum facet length of 0.5
Output and Postprocessing
Each of these methods produce a triangle mesh stored in different ways. If this output mesh is hampered by defects such as holes or self-intersections, CGAL provide several algorithms to post-process
it (hole filling, remeshing, etc.) in the package Polygon Mesh Processing.
We do not discuss these functions here as there are many postprocessing possibilities whose relevance strongly depends on the user's expectations on the output mesh.
The mesh (postprocessed or not) can easily be saved in the PLY format (here, using the binary variant):
std::ofstream f ("out_poisson.ply", std::ios_base::binary);
f.close ();
Mode set_binary_mode(std::ios &s)
bool write_PLY(std::ostream &os, const Surface_mesh< P > &sm, const std::string &comments, const NamedParameters &np=parameters::default_values())
A polygon soup can also be saved in the OFF format by iterating on the points and faces:
std::ofstream f ("out_sp.off");
f << "OFF" << std::endl << points.size () << " "
<< reconstruct.number_of_facets() << " 0" << std::endl;
for (Point_set::Index idx : points)
f << points.point (idx) << std::endl;
for (const auto& facet : CGAL::make_range (reconstruct.facets_begin(), reconstruct.facets_end()))
f << "3 "<< facet[0] << " " << facet[1] << " " << facet[2] << std::endl;
f.close ();
Finally, if the polygon soup can be converted into a polygon mesh, it can also be saved directly in the OFF format using the stream operator:
// copy points for random access
std::vector<Point_3> vertices;
vertices.reserve (points.size());
std::copy (points.points().begin(), points.points().end(), std::back_inserter (vertices));
std::ofstream f ("out_af.off");
f << output_mesh;
f.close ();
void polygon_soup_to_polygon_mesh(const PointRange &points, const PolygonRange &polygons, PolygonMesh &out, const NamedParameters_PS &np_ps=parameters::default_values(), const NamedParameters_PM &
Full Code Example
All the code snippets used in this tutorial can be assembled to create a full algorithm pipeline (provided the correct includes are used). We give a full code example which achieves all the steps
described in this tutorial. The reconstruction method can be selected by the user at runtime with the second argument.
Point_set points;
std::ifstream stream (fname, std::ios_base::binary);
stream >> points;
points.remove(rout_it, points.end());
std::cout << points.number_of_removed_points()
points.remove(gsim_it, points.end());
std::cout << points.number_of_removed_points()
CGAL::jet_smooth_point_set<CGAL::Sequential_tag> (points, 24);
= argc==1? -1 : (argc < 3 ? 0 : atoi(argv[2]));
points.remove (unoriented_points_begin, points.end());
(points.begin(), points.end(),
points.point_map(), points.normal_map(),
output_mesh, spacing);
f.close ();
std::vector<Facet> facets;
std::cout << facets.size ()
std::vector<Point_3> vertices;
vertices.reserve (points.size());
std::copy (points.points().begin(), points.points().end(), std::back_inserter (vertices));
f << output_mesh;
f.close ();
(points.points().begin(), points.points().end());
f << points.point (idx) << std::endl;
f.close ();
Full Pipeline Images
The following figures show a full reconstruction pipeline applied to a bear statue (courtesy EPFL Computer Graphics and Geometry Laboratory [5]). Two mesh processing algorithms (hole filling and
isotropic remeshing) are also applied (refer to the chapter Polygon Mesh Processing for more information). | {"url":"https://cgal.geometryfactory.com/CGAL/doc/master/Manual/tuto_reconstruction.html","timestamp":"2024-11-09T00:38:09Z","content_type":"application/xhtml+xml","content_length":"58166","record_id":"<urn:uuid:af5ebf6c-4af8-4c41-b8db-a1b6083ad706>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00010.warc.gz"} |
Physics - Online Tutor, Practice Problems & Exam Prep
Small changes in the length of an object can be measured using a strain gauge sensor, which is a wire that when undeformed has length ℓ₀ , cross-sectional area A₀, and resistance R₀. This sensor is
rigidly affixed to the object’s surface, aligning its length in the direction in which length changes are to be measured. As the object deforms, the length of the wire sensor changes by Δℓ, and the
resulting change ΔR in the sensor’s resistance is measured. Assuming that as the solid wire is deformed to a length ℓ, its density and volume remain constant (only approximately valid), show that the
strain ( = Δℓ / ℓ₀ ) of the wire sensor, and thus of the object to which it is attached, is approximately ΔR / 2R₀ . | {"url":"https://www.pearson.com/channels/physics/explore/resistors-and-dc-circuits/solving-resistor-circuits?chapterId=8fc5c6a5","timestamp":"2024-11-07T18:32:54Z","content_type":"text/html","content_length":"461683","record_id":"<urn:uuid:fabd7bc1-95ef-404c-a3b4-a948aad7b359>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00086.warc.gz"} |
Compound Interest Calculator [Daily, Monthly & Yearly] | CoinSwitch
Interest rates and investment returns are closely related. Simple interest, as we know, is interest earned on the principal amount invested. Compound interest kicks in when your interest is
reinvested and you get interest on your principal amount along with the accumulated interest. The interest on interest feature makes compound interest attractive but makes it difficult to calculate
as the principal amount keeps increasing. Use our compound interest calculator above to learn how much you will earn as compound interest during your investment tenure.
This blog post aims to simplify compound interest calculations for you. You just need to key in the input variables to calculate the interest amount accrued on your investment. Read on as we analyze
the tool in detail.
Compound interest calculator explained
The compound interest calculator is a tool designed to ease compound interest calculations on your investment. All you need to do is plug in the numbers that the calculator field prompts you to enter
and the tool will do the job for you.
Understanding how compound interest calculator works
The CoinSwitch compound interest calculator works on four variables:
1. The principal or initial investment amount
The principal amount is the sum that you will start your investment with. All your compound interest calculations will be based on your principal amount.
2. The compounding frequency
The interest you will earn on your investment will depend on the compounding frequency. Some commonly utilized compounding frequencies are annual, semi-annual, quarterly, or monthly.
3. The rate of interest
This is the rate at which your investment will grow. The annual rate of interest will be mentioned in the investment scheme documents.
4. Investment period
This variable determines the time frame of your investment. The number of compoundings will depend on the investment tenure.
Once you input these variables, the compound interest calculator will compute the total amount receivable at the end of the investment period and the total interest you will receive on your
principal—all in just a few seconds.
The formula to calculate compound interest
A reliable and easy way to calculate compound interest is by using the calculator. But in case you want to test your math acumen, you can calculate compound interest using the formula below:
A = Compound Interest
P = Principal Amount
r = Interest Rate
n = Compounding Frequency per year
t = Number of Investment Years
Let’s understand the concept of compounding with the help of an example:
Ms. B is looking to invest ₹10,000 for 5 years at a 10% rate of interest compounded annually. The calculation will look something like this::
Year 1 = 10,000*10/100
= 1000
The first-year or first-period interest calculator will be the same as simple interest calculations. When calculating the compound interest, the difference in calculation will start from the second
year or the second compound period.
For the second year, the principal amount for Ms. B will be ₹10,000 plus the interest accrued from the previous year. That is ₹11,000.
Year 2 = ₹11,000*10/100
= ₹1100
Similarly, the principal amount for the third year will be Rs. 12,100.
This is how compound interest is calculated. While these calculations look easy for short periods with few compounding periods, complications set in when the tenure is long with multiple compounding
periods in a year.
A compound interest calculator can help calculate the receivable amount based on the variables fed in the calculator fields.
How the compound interest calculator can benefit you
Using a compound interest calculator can be beneficial in many ways. Let’s discuss some of them.
Money is a serious matter and this makes the accuracy of calculations all the more important. As demonstrated above, calculating compound interest manually is tedious. The complexity of calculations
can lead to human errors, defeating the whole purpose.
A compound interest calculator helps you tide over this problem. It not only makes the math super easy, it consistently reduces the chances of errors. That makes it the more reliable option.
Calculating compound interest manually is time-consuming. A compound interest calculator on the other hand can calculate the interest amount as well as the receivable amount within seconds. A
well-designed compound interest calculator can help you save time while ensuring reliability and accuracy.
• Sharpens financial decision-making
In practical terms, you can use a compound interest calculator like the one above to evaluate investment products such as fixed deposits. It can also help you understand how your investment will grow
with time at a given rate of interest and compounding frequency. This can help you closely align your investments with your financial goals. Essentially, the tool gives you the flexibility to do some
trial and error before you set out to make your investment.
Small wonder the magic of compounding is one of the classic investment lessons in finance. Compound interest has the potential to make your investment grow by leaps and bounds. What’s more, the
compound interest calculator is simple, accurate, and reliable. | {"url":"https://coinswitch.co/calculators/compound-interest-calculator","timestamp":"2024-11-02T04:56:19Z","content_type":"text/html","content_length":"105342","record_id":"<urn:uuid:10260fe2-93e7-4309-b2f4-89da3449bfe8>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00367.warc.gz"} |
Orbital Optimized Density Functional Theory for Electronic Excited States
Q-Chem Webinar 60
Orbital Optimized Density Functional Theory for Electronic Excited States
Presented by Diptarka Hait on
Diptarka Hait received his S.B. degree in Chemistry and Physics from MIT in 2016, working in the group of Prof. Troy Van Voorhis and was introduced to the world of OO-DFT. He is currently a PhD
candidate in Prof. Martin Head-Gordon's group at the UC Berkeley. He is interested in the development of quantum chemistry methods and their application to problems of interest to the experimental
community, with specific emphasis on the use of DFT beyond ground state energetics.
Density Functional Theory-based modeling of electronic excited states is of importance for the investigation of the photophysical/photochemical properties and spectroscopic characterization of large
systems. The widely-used linear response Time-Dependent DFT (TDDFT) approach is not effective at modeling many types of excited states, including charge-transfer states, doubly excited states and
core-level excitations.
In this webinar, I will discuss the use of state-specific Orbital Optimized (OO) DFT approaches as an alternative to TDDFT for electronic excited states. I will first discuss the motivation and
theory behind such approaches, along with some challenges faced by such methods both historically and at present. Subsequently, I will present the Square Gradient Minimization (SGM) algorithm for
reliable and efficient excited state orbital optimization, which has been implemented in Q-Chem. In particular, SGM permits use of Restricted Open-shell Kohn-Sham (ROKS) for modeling arbitrary
singlet excited states. ROKS/SGM can thus be used to compute core-excitation energies, with the modern SCAN functional yielding ~ 0.3 eV error vs experiment (compared to ~0.1 eV uncertainty in the
experimental values themselves). The use of ROKS/SGM in computing Near-Edge X-ray Absorption Fine Structure (NEXAFS) will be discussed. Time permitting, I would also touch upon the recent
implementation of the X2C relativistic model in Q-Chem, which (among other things) can be used to accurately compute NEXAFS of elements as heavy as Cr.
Supporting Material | {"url":"https://q-chem.com/webinars/60/","timestamp":"2024-11-13T19:28:42Z","content_type":"text/html","content_length":"45169","record_id":"<urn:uuid:b330392e-6805-4664-9757-ea5551a797d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00862.warc.gz"} |
Nth Tribonacci Number
Let's solve the Nth Tribonacci Number problem using Dynamic Programming.
Tribonacci numbers are a sequence of numbers where each number is the sum of the three preceding numbers. Your task is to find the $n^{th}$ Tribonacci number.
The Tribonacci sequence is defined as:
$T_0 = 0,\space T_1 = 1,\space T_2 = 1$, and $\space T_n = T_{n-1} + T_{n-2} + T_{n-3}, \space$ for $n >= 3$
The input number, n, is a non-negative integer.
Let’s say you have to find the fifth Tribonacci number in the sequence. From the sequence defined above, we know that $T_0 = 0, T_1 = 1, T_2 = 1$. The sequence will be:
$0, 1, 1, 2, 4, 7$
Therefore, the fifth term will be 7.
• $0 \leq$ n $\leq 73$
• The answer is guaranteed to fit within a 64-bit integer, i.e., answer $\leq 2 ^{63} - 1$
Level up your interview prep. Join Educative to access 80+ hands-on prep courses. | {"url":"https://www.educative.io/courses/grokking-dynamic-programming-interview-python/nth-tribonacci-number","timestamp":"2024-11-11T17:16:44Z","content_type":"text/html","content_length":"818973","record_id":"<urn:uuid:3ab9f17b-d746-42a8-92bb-b287be978693>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00580.warc.gz"} |
cheev - Linux Manuals (3)
cheev (3) - Linux Manuals
cheev.f -
subroutine cheev (JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK, INFO)
CHEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
Function/Subroutine Documentation
subroutine cheev (characterJOBZ, characterUPLO, integerN, complex, dimension( lda, * )A, integerLDA, real, dimension( * )W, complex, dimension( * )WORK, integerLWORK, real, dimension( * )RWORK,
CHEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices
CHEEV computes all eigenvalues and, optionally, eigenvectors of a
complex Hermitian matrix A.
JOBZ is CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
A is COMPLEX array, dimension (LDA, N)
On entry, the Hermitian matrix A. If UPLO = 'U', the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If UPLO = 'L',
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if JOBZ = 'V', then if INFO = 0, A contains the
orthonormal eigenvectors of the matrix A.
If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
or the upper triangle (if UPLO='U') of A, including the
diagonal, is destroyed.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
W is REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,2*N-1).
For optimal efficiency, LWORK >= (NB+1)*N,
where NB is the blocksize for CHETRD returned by ILAENV.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
RWORK is REAL array, dimension (max(1, 3*N-2))
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the algorithm failed to converge; i
off-diagonal elements of an intermediate tridiagonal
form did not converge to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 140 of file cheev.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-cheev/","timestamp":"2024-11-05T12:04:32Z","content_type":"text/html","content_length":"10244","record_id":"<urn:uuid:71b0dc46-ad9d-4d50-ad3b-5ba708bbde66>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00707.warc.gz"} |
Possible Isolation Number of a Matrix Over Nonnegative Integers
Journal/Book Title/Conference
Czechoslovak Mathematical Journal
Springer Berlin Heidelberg
Publication Date
Let ℤ[+] be the semiring of all nonnegative integers and A an m × n matrix over ℤ[+]. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation
number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number
of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation
number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number
is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.
Recommended Citation
Beasley, LeRoy B.; Jun, Young Bae; and Song, Seok-Zun, "Possible Isolation Number of a Matrix Over Nonnegative Integers" (2018). Mathematics and Statistics Faculty Publications. Paper 232. | {"url":"https://digitalcommons.usu.edu/mathsci_facpub/232/","timestamp":"2024-11-02T18:00:30Z","content_type":"text/html","content_length":"41431","record_id":"<urn:uuid:ad869d7c-676b-44ab-a64e-50370fc20c36>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00328.warc.gz"} |
Lesson 6
Introducing Double Number Line Diagrams
Let’s use number lines to represent equivalent ratios.
6.1: Number Talk: Adjusting Another Factor
Find the value of each product mentally.
\((4.5)\boldcdot 4\)
\((4.5)\boldcdot 8\)
\(\frac{1} {10}\boldcdot 65\)
\(\frac{2} {10}\boldcdot 65\)
6.2: Drink Mix on a Double Number Line
The other day, we made drink mixtures by mixing 4 teaspoons of powdered drink mix for every cup of water. Here are two ways to represent multiple batches of this recipe:
1. How can we tell that \(4:1\) and \(12:3\) are equivalent ratios?
2. How are these representations the same? How are these representations different?
3. How many teaspoons of drink mix should be used with 3 cups of water?
4. How many cups of water should be used with 16 teaspoons of drink mix?
5. What numbers should go in the empty boxes on the double number line diagram? What do these numbers mean?
Recall that a perfect square is a number of objects that can be arranged into a square. For example, 9 is a perfect square because 9 objects can be arranged into 3 rows of 3. 16 is also a perfect
square, because 16 objects can be arranged into 4 rows of 4. In contrast, 12 is not a perfect square because you can’t arrange 12 objects into a square.
1. How many whole numbers starting with 1 and ending with 100 are perfect squares?
2. What about whole numbers starting with 1 and ending with 1,000?
6.3: Blue Paint on a Double Number Line
Here is a diagram showing Elena’s recipe for light blue paint.
1. Complete the double number line diagram to show the amounts of white paint and blue paint in different-sized batches of light blue paint.
2. Compare your double number line diagram with your partner. Discuss your thinking. If needed, revise your diagram.
3. How many cups of white paint should Elena mix with 12 tablespoons of blue paint? How many batches would this make?
4. How many tablespoons of blue paint should Elena mix with 6 cups of white paint? How many batches would this make?
5. Use your double number line diagram to find another amount of white paint and blue paint that would make the same shade of light blue paint.
6. How do you know that these mixtures would make the same shade of light blue paint?
You can use a double number line diagram to find many equivalent ratios. For example, a recipe for fizzy juice says, “Mix 5 cups of cranberry juice with 2 cups of soda water.” The ratio of cranberry
juice to soda water is \(5:2\). Multiplying both ingredients by the same number creates equivalent ratios.
This double number line shows that the ratio \(20:8\) is equivalent to \(5:2\). If you mix 20 cups of cranberry juice with 8 cups of soda water, it makes 4 times as much fizzy juice that tastes the
same as the original recipe.
• double number line diagram
A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. The locations of the tick marks match on both number lines. The tick marks labeled 0 line up, but
the other numbers are usually different. | {"url":"https://im.kendallhunt.com/MS/students/1/2/6/index.html","timestamp":"2024-11-12T17:10:48Z","content_type":"text/html","content_length":"84403","record_id":"<urn:uuid:a7ef0a7b-38fc-4b0a-ace1-c9d16cb452d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00855.warc.gz"} |
Re: [Inkscape-devel] Mask calculation error.
29 Nov 2010 29 Nov '10
9:18 a.m.
Krzysztof KosiĆ ski wrote:
In this case it doesn't matter because the input values are limited to 0...255. We can use any multiplier we want - as long as the coeffs sum up to the value we use as the divisor, it will work
correctly. There are many other cases where it won't, precisely because of the reason you mentioned (e.g. in blending, or when alpha is used as a coefficient) and then we can't avoid the
Well, I was speaking about avoiding the final multiplication and the use of floats: if you want to get 255 as final maximum starting from 256 (or 512, or 2^n), you either have to divide by 1,00392...
(or 2,00784..., or some other float) or first you have to multiply by 255 and then shift-divide (by 8 or 9). Then, why not implicitly mutliply by 2^n in each partial coefficient so you sum up to
255*2^n, which are multiplications you must do and in this case can turn into integer multiplications rather than floats and without loss of final precision? If you sum up to 32768 (16 bits!) and
then divide by 128 (>>7) and your result is '1' you get 256 that is over the limit of 255. But if you sum up to 32767 (or better 32640, both 15 bits so we could use one more bit for precision) you
just have to >>7 and you're done. No floats, no final multiplitcation: just the final scale already taken into consideration in all preceeding calculations. And the difference between the two is only
in considering 256 or 255 (or better, 2^n rather than (2^n)-1) as maximum. Indeed, it's really in this case that matters.
This way of doing calculations is typical of microcontollers and DSPs where performances do count a lot because you either need to be really fast (but really!) or you don't have enough resources for
your application but you have to make it work anyway :) My opinion is that Inkscape could be faster than it is, or at least give the feeling that it is. Maybe some more attention to these details
(because they are details, but many details make a whole) could help to improve performances.
Regards. Luca
-- View this message in context:
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Change in Enthalpy Calculator - Calculator Wow
Change in Enthalpy Calculator
Change in Enthalpy Calculator
Enthalpy is a crucial concept in thermodynamics, representing the total heat content of a system. Calculating the change in enthalpy (ΔH) can be complex, but with this interactive calculator, you can
quickly determine it. Simply input the change in internal energy (ΔU), change in pressure (ΔP), and change in volume (ΔV), and let the calculator do the rest.
How to Use the Change in Enthalpy Calculator
Calculating the change in enthalpy involves the following formula:
ΔH = ΔU + Δ(PV)
• ΔH = Change in Enthalpy (in Joules, J)
• ΔU = Change in Internal Energy (in Joules, J)
• ΔP = Change in Pressure (in Pascals, Pa)
• ΔV = Change in Volume (in cubic meters, m³)
Follow these steps to calculate ΔH:
1. Enter the value for ΔU in the input field labeled “Change in Internal Energy (J).”
2. Enter the value for ΔP in the input field labeled “Change in Pressure (Pa).”
3. Enter the value for ΔV in the input field labeled “Change in Volume (m³).”
4. Click the “Calculate ΔH” button.
Change in enthalpy is a crucial concept in thermodynamics, and understanding how to calculate it is essential for various scientific and engineering applications. With our Change in Enthalpy
Calculator and the knowledge provided in this article, you are well-equipped to tackle problems involving enthalpy changes in processes occurring at constant pressure. | {"url":"https://calculatorwow.com/change-in-enthalpy-calculator/","timestamp":"2024-11-07T01:10:02Z","content_type":"text/html","content_length":"62234","record_id":"<urn:uuid:f525eae0-ab0e-42bb-b85e-e1c584be9b47>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00531.warc.gz"} |
EECS 349 Homework 1 solution
Problem 1 (2 points): A typical machine learning task is supervised learning of a classification function (aka a concept): give example x the right class label l. Assume there is an unknown target
concept c(x), whose domain is a set of unique examples X, called the instance space, and whose range is a set of unique labels L. The task is to learn that function. A learner does this by testing
hypotheses. A hypothesis h(x) is also function whose domain is X and whose range is L. If the set of hypotheses a learner is able to consider is the same as the set of possible concepts, then the set
of unique hypotheses (the hypothesis space H) is identical to the set of unique concepts (the concept space C). For this question, assume H = C. In supervised learning, exploration of the hypothesis
space is guided by the use of a finite data set, D that contains examples from X with known labels from the target concept c(x). Since we can only measure success with respect to the data, we reframe
the task of learning c(x) to that of finding a hypothesis consistent with the data: ∀x∈D, c(x)=h(x)
Definition: Two functions f1 and f2 are distinguishable, given D, if they differ in their labeling of at least one of the examples in D.
Definition: A set of hypotheses is distinguishable, given D, iff ALL pairs of hypotheses in the set are distinguishable given D. Call HD a largest set of distinguishable hypotheses, given D.
A) (1/2 point) Assume that X, L and D are all finite and given (i.e. they are all fixed). Is there one unique HD? Explain your reasoning.
B) (1/2 point) Let the size of the data and the label set be drawn from the counting numbers (i.e. |D|,|L|∈{1,2,3…}). Formulate the size of HD, as a function of |L| and |D|. Explain your reasoning.
For parts C and D assume the following. The label set L is {0,1}. The size of the data set |D| = 100. The size of the instance space |X| = 200. Assume a learner able to consider a maximal set of
distinguishable hypotheses HD.
C) (1/2 point) Assume the learner is able to consider 10^9 (one billion) hypotheses per second. In the worst case, how long would it have to work to find a hypothesis h that is indistinguishable from
the target function c, given D? Would this be a reasonable time to wait? Explain your reasoning. State your assumptions.
D) (1/2 point) Assume the learner HAS found a hypothesis h that is indistinguishable from the target concept c , given D. What is the probability that hypothesis h is also indistinguishable from the
target concept c , given X? Explain your reasoning. State your assumptions.
EECS 349 (Machine Learning) Homework 1
Problem 2 (1 point): Given the following: Instances X: cars described by following attributes x is a tuple: | {"url":"https://jarviscodinghub.com/product/eecs-349-homework-1-solution/","timestamp":"2024-11-03T09:08:45Z","content_type":"text/html","content_length":"114539","record_id":"<urn:uuid:38f89b0d-f7e6-4963-b511-fbb07b7fe9fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00389.warc.gz"} |
The Core Key on math websites Revealed
Engaging animated learning videos, video games, quizzes, and actions to encourage children on their unique learning path. This website how to make a class for outschool offers e-textbooks, answer
keys, video lessons, and printables. Topics embrace algebra 1 and 2, geometry, and trigonometry.
In 2019 more than 2,9 million distinctive users visited the site. Math planet is a web-based resource the place one can research math at no cost. Take our high school math programs
inPre-algebra,Algebra 1,Algebra 2andGeometry. Providing mathematical training to high-performing grade-school students all around the globe. Dedicated to gifted students in math and science, the
place customers be taught by solving fascinating issues of accelerating difficulties. Upper-level math college students will respect the no-frills data that’s easy to search out on this site.
An extensive assortment of film and TV clips during which arithmetic seems, posted by Oliver Knill of Harvard University. Nine movies of stereographic projections in 3 and four dimensions, complex
numbers, Hopf fibration, and proof. Social places like Mathematics Stack Exchange and Reddit even have robust math communities. Let’s finish with the 20th website that goes back to the History of
Math. It will not teach you any level of math, but a look at its evolution helps place everything in context.
• Engaging animated learning movies, games, quizzes, and actions to encourage children on their distinctive learning path.
• The web site focuses on concept and data and provides academic workouts instantly following the lesson.
• Upper-level math students will recognize the no-frills data that’s straightforward to search out on this web site.
We’ve skipped any website that focuses so much on concept and history, as it’s more important to follow with numbers rather than reading about numbers. The algebra part lets you broaden, issue or
simplify virtually any expression you select. It additionally has instructions for splitting fractions into partial fractions, combining several fractions into one and cancelling common elements
inside a fraction. This platform permits teachers to create technology-enhanced on-line math assessments from a huge question financial institution. Formerly often identified as Mathalicious, this
web site offers supplemental math classes.
Unknown Details About math websites Made Known
Build youngsters quantity sense whereas they have fun with a variety of different math video games, all out there online. For instance, guests can access a page about angle measurement. It covers
subjects corresponding to levels, radians and minutes whereas that includes an interactive protractor software. Further down, the web page covers associated topics and customary questions. Math Open
Reference additionally has tools such as graphing and scientific calculators. No type of registration is required, letting students simply entry math questions ranging from counting to evaluating
exponents of unfavorable numbers.
UseSuperKidsto create custom worksheets, allowing you to effectively preview, review and supplement your lessons. LoadTeacherVision’s math pageto access sources that, among different uses, join math
with different topics. A sequence of essays on a broad range of topics, corresponding to voting, bin-packing and networks.
How My math for kids Saves Me Time
Algebra is a critical topic, and it’s often referred to as the “gatekeeper” for all of the different levels and a prerequisite to comprehending other levels. Mattecentrum is a Swedish non for revenue
https://www.topschoolreviews.com/outschool-review member group founded in 2008 in Sweden. Since then, the center has been offering free assist in arithmetic to all who research math.
The 5-Second Trick For math websites for kids
Figure This is a site designed to encourage families to practice math collectively. It contains fun and engaging math video games and high-quality challenges. Two customers play a recreation by which
each player tries to attach 4 game pieces in a row . The trainer chooses how a lot time each participant has to reply, the extent of problem, and the type of math drawback. These interactive math
websites provide students with instruction and unbiased practice.
After algebra, the following step in the proper direction towards studying math can be geometry. Some say geometry, which is the research of shapes, should be taken earlier than algebra 2, however
the order is entirely as a lot as you. Arabiska.matteboken.se has the same theoretical and counting workout routines as above, however are at present lacking video lessons . The material covers the
start phase grades 3-9 and Matte 1, 2 & 3 for the higher secondary school.
The greatest method to learn something in math is to know the method to get to a solution. Learning pre-algebra must also be enjoyable and informative, however theory and data ought to begin to
appear at this degree. Getting to study math free of charge can appear too good to be true. There are plenty of sources and websites that may assist you to be taught or relearn maths from
fundamentals to advanced levels. Matteboken.seis an entire Swedish math book with principle, video lessons and counting exercises. Matteboken.se is out there in Swedish and an Arabic model. | {"url":"https://www.ark.com.mx/the-core-key-on-math-websites-revealed/","timestamp":"2024-11-04T08:50:13Z","content_type":"text/html","content_length":"100762","record_id":"<urn:uuid:c63b8621-ef34-4065-b6cb-cad30de43537>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00521.warc.gz"} |
The Physical Universe – Part 4
Quantum mechanics.
Continuing our essay series on the basic laws of physics, this essay draws again from Jim Al-Khalili’s book The World According to Physics.
Al-Khalili describes that, at very small scales even below the scales of atoms, yet another, much different physics is at work, namely quantum mechanics:
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Quantum mechanics is seen, quite rightly, as the most fascinating, yet at the same time most mind-boggling and frustrating scientific theory ever devised by humankind … "[Q]uantum mechanics as
the most powerful and important theory in all of science. After all, it is the foundation on which much of physics and chemistry is built, and it has revolutionised our understanding of how the
world is built from the tiniest of building blocks … The first major theoretical breakthrough—the concept of the ‘quantum’—was made by the German physicist Max Planck. In a lecture in December
1900, he proposed the revolutionary idea that the heat energy radiated by a warm body is linked to the frequency at which its atoms vibrated, and consequently that this radiated heat is ‘lumpy’
rather than continuous, emitted as discrete packets of energy, which became known as quanta. Within a few years, Einstein had proposed that it wasn’t just Planck’s radiation that was emitted in
lumps, but that all electromagnetic radiation, including light, came in discrete quanta. We now refer to a single quantum of light—a particle of light energy—as a photon … Particles of matter,
such as electrons, can exhibit a wavelike nature, too. This general notion, tested and confirmed for almost a century now, is known as wave-particle duality and is one of the central ideas of
quantum mechanics. This does not mean that an electron is both a particle and a wave at the same time—but rather that, if we set up an experiment to test the particle-like nature of electrons, we
find that they do indeed behave like particles. But if we then set up another experiment to test if electrons have wavelike properties (such as diffraction or refraction or wave interference), we
see them behaving like waves. It’s just that we cannot carry out an experiment that would show both the wave and particle nature of electrons at the same time.
The odd phenomenon that both the wave nature and the particle nature of matter cannot be measured at the same time has resulted in a situation in which the location and speed of a subatomic particle
can only be predicted in terms of probability. The problem of measurement in quantum mechanics is closely tied to the probabilistic nature of quantum systems. A quantum system exists in a
“superposition” of possible states until it is measured, at which point it “collapses” into one of its possible states with certain probabilities. This probabilistic behavior arises from the inherent
uncertainty and indeterminacy of quantum mechanics. The Heisenberg Uncertainty Principle is a fundamental concept that describes the inherent limitations in simultaneously knowing the precise
position and momentum of a particle. Formulated by Werner Heisenberg, the principle highlights that the more precisely we know a particle's position, the less precisely we can know its momentum, and
vice versa. This intrinsic uncertainty is not due to measurement limitations but is a fundamental property of nature at the quantum level:
This balance between how much we can simultaneously know about an electron’s particle nature (its position in space) and its wave nature (how fast it is travelling) is governed by Heisenberg’s
uncertainty principle, which is regarded as one of the most important ideas in the whole of science and a foundation stone of quantum mechanics. The uncertainty principle puts a limit on what we
can measure and observe, but many people, even physicists, are prone to misunderstanding what this means. Despite what you will find in physics textbooks, the formalism of quantum mechanics does
not state that an electron cannot have a definite position and a definite speed at the same time, only that we cannot know both quantities at the same time. A related common misunderstanding is
that humans must play some kind of crucial role in quantum mechanics: that our consciousness can influence the quantum world, or even bring it into existence when we measure it. This is nonsense.
Our universe, all the way down to its elementary building blocks at the quantum scale, existed long before life began on Earth—it wasn’t sitting in some fuzzy limbo state waiting for us to come
along, measure it, and make it real. By the mid-1920s, physicists were beginning to realise that the concept of quantisation is more general than just the ‘lumpiness’ of light or the ‘waviness’
of matter. Many physical properties, familiar to us as continuous, are, in fact, discrete (digital rather than analog) once you zoom down to the subatomic scale. For example, the electrons bound
within atoms are ‘quantised’ in the sense that they can only have certain specific energies and never energies in between these discrete values. Without this property, electrons would
continuously leak energy while orbiting the nucleus, meaning that atoms would not be stable and complex matter, including life, could not exist. According to nineteenth-century (pre-quantum)
electromagnetic theory, negatively charged electrons should spiral inwards towards the atom’s positively charged nucleus. But their quantised energy states prevents this from happening. Certain
quantum rules also define which energy states the electrons occupy and how they arrange themselves within atoms. As such, the rules of quantum mechanics dictate how atoms can bind together to
make molecules, making quantum mechanics the foundation of the whole of chemistry. Electrons are able to jump between energy states by emitting or absorbing the correct amount of energy. They can
drop to a lower state by emitting a quantum of electromagnetic energy (a photon) of exactly the same value as the difference in energies between the two states involved. Likewise, they can jump
to a higher state by absorbing a photon of the appropriate energy. The sub-microscopic world, down at the scale of atoms and smaller, therefore behaves very differently from our familiar everyday
world. When we describe the dynamics of something like a pendulum or tennis ball, or a bicycle or a planet, we are dealing with systems comprising many trillions of atoms, which are far removed
from the fuzziness of the quantum realm. This allows us to study the way these objects behave using classical mechanics and Newton’s equations of motion, the solutions of which are an object’s
precise location, energy or state of motion, all knowable simultaneously at any given moment in time … Typically, we would solve Schrödinger’s equation to calculate a quantity called the wave
function, which describes not the way an individual particle move along definite path, but the way its ‘quantum state’ evolves in time. [The wave function contains all the information about the
system and can be used to calculate probabilities.] The wave function can describe the state of a single particle or group of particles and has a value that provides us with the probability of,
say, finding an electron with any given set of properties or location in space if we were to measure that property. The fact that the wave function has value at more than one point in space is
often wrongly taken to mean that the electron itself is physically smeared out across space when we are not measuring it. But quantum mechanics does not tell us what the electron is doing when we
are not looking—only what we should expect to see when we do look … Despite its tremendous success, if we dig a little deeper into what quantum mechanics tells us about the microcosm, we could
easily lose our minds. We ask ourselves, ‘But how can it be so? What am I not ‘getting’?’ The truth is, no one really knows for sure. We do not even know if there is any more to ‘get’. Physicists
have tended to use terms like ‘strange’, ‘weird’, or ‘counterintuitive’ to describe the quantum world. For, despite the theory being powerfully accurate and mathematically logical, its numbers,
symbols and predictive power are a facade hiding a reality we find difficult to reconcile with our mundane, commonsense view of the everyday world. There is, however, a way out of this
predicament. Since quantum mechanics describes the subatomic world so remarkably well, and since it is built on such a complete and powerful mathematical framework, it turns out we can manage
just fine by learning how to use its rules in order to make predictions about the world and to harness it to develop technologies that rely on those rules, leaving the hand-wringing and
head-shaking to the philosophers.
Sean Corroll, a wonderful science explainer from John Hopkins University, explores more of the “deep meaning” aspects of quantum physics in his Great Courses lecture series The Many Hidden Worlds of
Quantum Mechanics:
Discussions about the foundations of quantum mechanics often center on the measurement problem. But another important issue is the reality problem: Does the wave function represent reality, or is
it merely a useful way of making predictions for observations? In the Copenhagen interpretation [the interpretation given by scientists Niels Bohr, Werner Heisenberg, and Max Born], the wave
function does not represent reality. But there are other formulations of quantum mechanics—many-worlds being one of them—where the wave function does represent reality.
In classical mechanics, the state of a particle is not specified by only giving its position, x. You also need to specify the momentum, p, which is mass times velocity: p = mv. So, if you know
where a particle is in classical mechanics but don’t know how fast it’s moving, you can’t predict anything about where it’s going to be next. You don’t know what direction it’s moving in. But in
quantum mechanics, the wave function doesn’t depend on position and momentum separately—it only depends on the position. So, in quantum mechanics—as a successor theory to classical
mechanics—where is the information about momentum in the quantum wave function? How do you know what you’re going to observe about the momentum, p? The answer is that you can also define a wave
function that depends on the momentum. You can call it Ψ(p) as opposed to Ψ(x). Ψ(x) is the position wave function, whereas Ψ(p) is the momentum wave function. And the momentum wave function will
tell you the probability of measuring the momentum to have any particular value in the usual way: You take the square of the modulus (that is, the amplitude squared), and that gives you the
probability … [Y]ou can talk separately about the position wave function, which tells you the probability of any position measurement, and the momentum wave function, which tells you the
probability of a momentum measurement. The trick is that the momentum wave function is not defined separately from the position wave function. In classical mechanics, position and momentum are
independent. In quantum mechanics, the position wave function is equivalent to the momentum wave function. There is a well-defined mathematical way of going back and forth, and it’s called the
Fourier transform. Neither the position wave function nor the momentum wave function by itself is fundamental. There is a single thing—call it the quantum state— that’s an abstract state of
description, and it’s just that the quantum state can be expressed equally well as a wave function depending on position or as a wave function depending on momentum. Classically, knowing the
position of a particle tells you nothing about its momentum, and vice versa, whereas quantum mechanically, there’s a single quantum state that tells you everything there is to know about both
position and momentum. How does that work? The answer is there’s an interesting relationship between these two ways of expressing the quantum state: the position wave function and the momentum
wave function. If the position wave function is highly localized—if Ψ(x) is almost entirely concentrated around some particular point in space such that you know that if you measure, that’s
probably where you’re going to see it—then the Fourier transform, or the map from position wave functions to momentum wave functions, tells you that the momentum wave function will be spread out
all over. If you were to measure the position of a particle with a highly localized wave function, you’d have a pretty good idea ahead of time where you’re going to find it. But if you take that
same particle with the same wave function, and want to measure its momentum, you have no idea what answer you would get. There are many different possibilities, all of which would have
substantial probability for getting that answer. Likewise, the other way around works just as well. If you have a momentum wave function that is localized around some value—so you know
essentially what you would get if you measured the velocity or momentum of the particle—then the position version of the wave function is spread out all over, so you have no idea what position
you would measure for that kind of particle. There are also wave functions that are neither exactly localized in position or momentum—they’re compromises. They are somewhat compact with respect
to position and somewhat compact with respect to momentum. In that case, there’s uncertainty about what you would measure if you measured either the position or the momentum.
None of these options lets you say that if you measure either one, you can predict carefully what you are going to observe. This feature is the heart of the uncertainty principle: You can have a
wave function that is localized in position, and you can have a wave function that is localized in momentum. You cannot have both at the same time. There is an unavoidable trade-off in how
compact a wave function can be in both position and momentum at once. Notice that it has not been stated that in quantum mechanics, measuring the position disturbs the system and leaves its
momentum uncertain, or vice versa. It’s true that when you measure a system, you can collapse the wave function and change it—but that’s not the fundamental point of the uncertainty principle.
The uncertainty principle is not a statement about measurements, though it might imply things about measurements. It’s a statement about what kinds of wave functions can possibly exist. You can
measure either position or momentum. That’s your choice as an experimentalist. And if you’re a good experimentalist, you’re going to get a well-defined answer: You’re going to see the electron
having a position or see it having a momentum, depending on your experiment. But there are no wave functions where both quantities are precisely defined at the same time ahead of time. So, is the
wave function all there is? That’s the reality problem. Or does an electron really have a well-defined position and momentum even before you measure it, but somehow quantum mechanics doesn’t know
how to describe those real-world quantities?
Regarding what we can describe through experiments that have been conducted, Carroll writes:
One good way to get into these issues is to consider yet another famous thought experiment in quantum mechanics: the double-slit experiment. Picture a tub of water. If the water is perfectly
still and you poke it, ripples will emanate outward from where you poked it. They’ll move in a circular pattern, getting farther and farther away. Then, imagine that you put a barrier into the
water some distance away from where you start the ripples. If there is a single slit in the barrier, then some waves are going to pass through. On the other side of the barrier, the waves will
again emanate from the slit in another semicircular pattern. Now imagine that there are two slits in the barrier. Then, two circular patterns of ripples will emerge on the other side. And you’ve
learned that when you have waves, you can have interference. So, these ripples coming out of the two slits will interfere. At some points, they will reinforce each other— points where the two
ripples are moving in the same direction—and at some points, they will cancel each other out. Next, consider doing this with light rather than water. Take a beam of light pointed at a thin slit
in a barrier and set up a detector on the other side. If you start with a single slit, then because light has wavelike properties, you will see the light spread out a bit. That’s diffraction. It
won’t quite be a semicircle because the light has some momentum in that direction. Then, imagine you have a barrier with two slits that are pretty close together—not much farther apart than the
wavelength of the light. Then, at a detector placed on the opposite side, you see an interference pattern: bright bands with dark bands in between. Given the distances from the source of light to
the slits to the detector, the light waves will interfere either constructively (build up) or destructively (cancel each other out). This version of the double-slit experiment was done by Thomas
Young in the 19th century to demonstrate the wave nature of light.
If you instead send particles through the slits, some of the particles will hit the barrier and then stop, while others will pass through the slit. But if they get lucky enough to pass through
the slit, they’re just going to continue in a straight line. Classically, particles do not diffract; they keep moving unless they hit something. Mostly, a particle detector on the other side of
the two slits will basically see the slits: It will measure a bunch of detected particles in a single location for a single slit. If there are two slits, there will be two bunches in two
locations. You don’t get interference patterns by sending particles through double slits. Particles don’t interfere with each other. Particles have a location in space; they do not spread out
everywhere. Returning to the double-slit experiment using light, imagine you make the intensity of the light beam extremely small. When you make the light that dim, you eventually stop seeing a
continuous image at the detector. Rather, you start detecting individual flashes at specific locations. This is because you’ve reached the realm of individual photons. What you thought for bright
light was a wave passing through the slits, you now realize that when the light gets very dim, it’s actually a collection of individual photons. So, a sensitive photodetector—a machine that
discovers photons when they get hit by photons—can measure photons one at a time. High-intensity light shows wavelike interference patterns, but low-intensity light behaves like point-like
particles. How can that be reconciled? The answer comes if you look at the individual flashes from low-intensity light. Any individual photon leaves a flash, but if you allow those flashes to
accumulate over time, the flashes are detected one by one by the detector— but they do not congregate around just two points, one for each slit. Rather, the photon dots that show up in the
detector cluster in high-intensity bands separated by low-intensity empty regions and then fall away if you get far away from where the slits are pointing. In other words, photons do exhibit
interference patterns—even when you’re viewing them one at a time as individual particles. And the same behavior holds for electrons. This was a thought experiment for a long time, but the
experiment can actually be done … [I]t wasn’t until 2012 that physicists could do the experiment in a laboratory. If you send a single electron through two slits, the detector on the other side
will see a dot like a particle. But—just like light—if you keep sending electrons through one at a time, many dots will accumulate, and that accumulation of dots will look like an interference
If classical physics were right and electrons were just particles, this is very hard to make sense of. You’re sending electrons through one at a time, yet their accumulated effect is an
interference pattern that seems wavelike. This means that each electron must somehow be interfering with itself. So, it knows not to go to the places where the interference bands are dark; it
knows to probably go where the interference bands are light. And if you insist on thinking of the electron as an ordinary particle—which you shouldn’t—that electron either went through one slit
or the other one. A particle cannot go through both. So, how did it interfere with itself? Within quantum mechanics, the answer is that the electron is a wave function. When it’s passing through
two slits, there is absolutely something wavelike. Wave functions have no trouble interfering with themselves. And then it’s only when the electron is finally detected on the other side that it
looks like a particle. That’s the short motto for quantum mechanics: It’s a wave when you’re not looking at it; it looks like a particle when you do. If you wanted to stubbornly insist that the
electron went through either one slit or the other, you could presumably find out by placing small detectors near the slits to measure which slit the electron goes through. For every electron you
fire at the slits, you detect that it goes through one slit—not two. It looks like a particle, and then it continues on to be detected on the other side. When you detect it, it always looks like
a particle when you observe it. But if you send many electrons through and monitor which slit they go through, each electron only goes through one or the other each time. And on the other side,
where you have the main detector, the interference pattern goes away. When you try to figure out which path the electron takes, you force it to take one or the other. The wave function collapses
onto whichever slit you observed it to go through. So, your observation has fundamentally changed the state of the system. The double-slit experiment seems to have implications for the question
of whether the wave function represents something real or whether it should be thought of as something epistemic. Philosophers use the word epistemic to refer to knowledge or the ability to make
What could possibly cause the wave function to “collapse”? That is a fundamental, as-yet-unanswered question. “Decoherence theory” offers a general explanation by suggesting that interactions with
the environment cause the wave function to appear to collapse. That is, when a quantum system interacts with its larger environment, the superpositions of the system become “entangled” with the
states of the environment. This interaction leads to the loss of coherence between the components of the superposition (“decoherence”), making the system appear classical and thus effectively
collapsing the wave function. But no one has been able to pinpoint exactly at which point a surrounding environment can “overwhelm” a quantum system and thereby cause it to behave under the rules of
classical physics. (The Copenhagen interpretation of this phenomenon posits that the wave function collapse is a fundamental process that occurs during measurement, but offers no causal explanation
of why that happens. On the other hand, the “Many-worlds” interpretation of this phenomenon posits that all possible outcomes of a quantum measurement actually occur, each in a separate, branching
universe, such that there is no collapse, but instead the universe splits into multiple copies. Pretty wild.)
Carroll (himself a proponent of the “Many-worlds” interpretation) continues:
The double-slit experiment implies that the electron isn’t simply described by a wave. It seems to legitimately act like a wave when you’re not observing it. It’s not just that individual
electrons go through one slit or the other and you just don’t know. The correct thing to say is that the wave function of the electron goes through both slits. You need to be able to say that
because that’s how the electron is able to interfere with itself—by being a wave when it goes through those two slits. At least at face value, this kind of behavior is what makes people think
that the wave function should be thought of as a real thing, not just a way of making predictions. A physical, actual wave has no problem interfering with itself. But something that you just
think of as a tool for making predictions— how does that interfere with itself? Bohr, Heisenberg, and colleagues decided that wave functions should not be thought of as directly representing
reality. The fact that you don’t observe wave functions directly lends that view some credence. If the wave functions were the real physical stuff of the world, why can’t you just measure them?
Why do they keep collapsing every time you look? Many intelligent physicists today still don’t think of wave functions as representing reality. But quantum mechanics is confusing. You should be
open to different possibilities, and phenomena like the double-slit experiment suggest that you should at least take seriously the possibility that the wave function does represent reality. This
is called the realist position about the wave function.
We can’t yet know (and may never know) what exactly explains the causal mechanism behind the results we see from quantum mechanics. But humankind has come a long way in our understanding, as will be
explored in the next essay in this series.
Links to other essays in this series: Part 1; Part 2; Part 3; Part 4; Part 5; Part 6.
Big Picture by Paul Taylor is a reader-supported publication. To receive new posts and support my work, consider becoming a free or paid subscriber. | {"url":"https://paultaylor.substack.com/p/the-physical-universe-part-4","timestamp":"2024-11-03T00:50:59Z","content_type":"text/html","content_length":"217979","record_id":"<urn:uuid:b349a0f3-a186-4058-9dc5-dcf734290dbc>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00584.warc.gz"} |
For a real number t, let M(t) denote the maximum value of f(x) = 4x^2 - x + t for 11 \le x \le 1. What is the smallest possible value of M(t)?
MeIdHunter Oct 30, 2024
To find the minimum value of M(t) with the function f(x) = 4x^2 - x + t on the interval 11 ≤ x ≤ 1, we need to determine the optimal value of f(x) in this interval. Doing the calculation, we see that
M(t) depends on t. To relax after solving math problems, you can play Slope Game, an interesting game that helps improve reflexes and thinking!
PaulineMahone Oct 31, 2024 | {"url":"https://web2.0calc.com/questions/algebra_54513","timestamp":"2024-11-11T19:30:10Z","content_type":"text/html","content_length":"20535","record_id":"<urn:uuid:ba86363e-4e70-4cc4-8756-b1307ada0cba>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00848.warc.gz"} |
The Saint Petersburg Paradox
by Andrew Boyd
Today, a paradox for the generations. The University of Houston's College of Engineering presents this series about the machines that make our civilization run, and the people whose ingenuity created
Eighteenth century mathematician Daniel Bernoulli looked at his equation and saw a problem. The math was simple. It was the implications about human behavior that had him puzzled.
Portrait of Daniel Bernoulli (1700-1782) Wikipedia Image
Bernoulli introduced his problem in a journal of the Imperial Academy of Science of Saint Petersburg, after which it came to be known as the Saint Petersburg Paradox. And like many good paradoxes it
involves a game of chance. It's a great game — you're guaranteed to win money. The only question's how much.
I'll start the game by putting two dollars in a pot. Next, I'll flip a coin. If the coin comes up heads, you get the two dollars and the game's over. However, if the flip is tails, I'll double the
amount in the pot and we'll flip again. That's all there is to it. Heads, you win what's in the pot. Tails, and I double the pot. It's truly a case of heads you win, tails I lose. So to be fair, I'm
going to add a small twist. I'm going to charge a price to play the game. And here's the question: what price would you be willing to pay?
Let's think about it. You're guaranteed to win at least two dollars, so presumably you'd be willing to pay at least that much to play. But there's a chance you could walk away with hundreds — even
thousands. Would you be willing to pay five dollars? Ten dollars? More?
With a very simple calculation, the mathematician Bernoulli pointed out the unsettling fact that no matter what the price to play, the expected value of winning is in your favor. Whether I asked a
hundred dollars or a thousand or any other amount, the game is fairer than any game you'll find in a casino. But if I asked a price of a million dollars, would you pay to play? I imagine not.
And that was Bernoulli's point. Even favorable bets aren't always perceived as good, something that was implicit in much of the early work on games of chance. Bernoulli reasoned that winning $100
when you're flat broke means more than winning $100 when you have vast sums stashed in your bank account. Our perceived value, or utility, of additional money decreases the more we have.
It's a common sense idea, but Bernoulli formalized it in the language of mathematics. And the concept, known as decreasing marginal utility, is now as fundamental in modern economic theory as supply
and demand curves. And as a result the Saint Petersburg Paradox isn't quite as paradoxical as it once was.
But Bernoulli didn't close the book on this time-honored conundrum. Generations of scholars have contributed to the discussion, including such distinguished names as Euler, Cournot, Arrow, Keynes,
Samuelson, and von Neumann. It seems there are always new and interesting ways to look at a good paradox, and new and interesting ways to explain human behavior — or, at least, to try.
I'm Andy Boyd at the University of Houston, where we're interested in the way inventive minds work.
(Theme music)
The expected value of a bet is the amount of money a person can expect to win 'on average.' It's calculated by multiplying the money won on each possible outcome, weighting it by the probability of
that outcome, and adding the numbers together. For example, in American roulette, with 18 black numbers, 18 red numbers, and 2 green numbers ' for a total of 38 numbers in all ' a one dollar bet on
black has an expected value of
(18/38) x $1 + (20/38) x (-$1) = -$2/38
which implies that on average you can expect to win -$2/38 per $1 wagered.
Recognizing that the probability the first head comes on the first coin toss is 1/2, the first head on the second toss is 1/4, the first head on the third toss is 1/8, and so on, the expected value
of the money a person will win from the game outlined in the essay is
(1/2) x $2 + (1/4) x $4 + (1/8) x $8 + ...
= 1 + 1 + 1 + ...
= ∞
The payouts rise so quickly that the expected value of the money won is infinite. Thus, no matter what finite amount you pay to play the game, the expected value of the winnings remains infinite.
D. Bernoulli (1954) [1738]. 'Exposition of a New Theory on the Measurement of Risk.' Econometrica, 22, pp. 23'36.
B. Hayden and M. Platt (2009). 'The Mean, the Median, and the Saint Petersburg Paradox.' The Journal of Judgment and Decision Making, 4(4), pp. 256-272.
R. Martin. The Saint Petersburg Paradox. From the Stanford Encyclopedia of Philosophy website: http://plato.stanford.edu/entries/paradox-stpetersburg/. Accessed December 2, 2014.
This episode first aired on December 3, 2014. | {"url":"https://engines.egr.uh.edu/episode/2980","timestamp":"2024-11-09T19:36:37Z","content_type":"text/html","content_length":"33460","record_id":"<urn:uuid:22eaf2ba-8454-4045-a488-73156ec3f406>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00220.warc.gz"} |
Search Results
In this lesson, students compare and contrast shapes using attribute blocks. Because the equilateral triangle (a triangle with congruent sides and congruent angles) is the most common example used in
textbooks and other reference materials, this is an important opportunity for you to help students realize that other triangles exist and that triangles can have angles of different measures. The
lesson is designed to accommodate multiple learning styles and intelligences.
In this lesson, students continue to discuss attributes of triangles. They trace and draw triangles individually. Students recognize objects in their environment that are shaped like triangles and
explain to the class how they recognized the shape.
Students use appropriate vocabulary to describe shapes to their classmates. Students focus on the properties of shapes to develop mental images of objects from descriptors. They create multiple
representations of triangles using geoboards, string, and crayons and paper.
A nursery rhyme provides a context for using the number 2. Students make groups of two, write the numeral 2, and record a group of two on a personal recording chart.
In this lesson, students construct sets of three, compare them with sets of two, and write the numeral 3. They also show a set of three on their recording chart.
After reviewing the numbers 2 and 3, students construct and identify sets of one. They compare sets of one, two, and three objects and record a set of three in chart form.
Students explore the number 4. They make sets of four, write the numeral 4, and compare sets of four to sets of one, two, and three.
Students construct sets of up to five items, write the numeral 5, identify sets of five, and record "5" on a chart. They also play a game that requires recognizing the numerals to 5. This lesson
provides opportunities for connecting mathematics with music.
Students explore sets of zero items and practice writing the numbers 0 through 5. Students count back from five, identify sets of up to five items, and record "0" on a chart. They also construct sets
of up to five items.
A game encourages students to find the sums of two one-digit numbers. Students explore commutativity and examine patterns on an addition table. They then use a personal addition chart to record and
keep track of known facts. | {"url":"https://illuminations.nctm.org/Search.aspx?view=search&&page=5","timestamp":"2024-11-11T04:17:44Z","content_type":"application/xhtml+xml","content_length":"66341","record_id":"<urn:uuid:0c88aedc-4c7d-41f3-81d0-a41b89c4a409>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00215.warc.gz"} |
The Complexity of Periodic Energy Minimisation
The computational complexity of pairwise energy minimisation of N points in real space is a longstanding open problem. The idea of the potential intractability of the problem was supported by a lack
of progress in finding efficient algorithms, even when restricted the integer grid approximation. In this paper we provide a firm answer to the problem on Zd by showing that for a large class of
pairwise energy functions the problem of periodic energy minimisation is NP-hard if the size of the period (known as a unit cell) is fixed, and is undecidable otherwise. We do so by introducing an
abstraction of pairwise average energy minimisation as a mathematical problem, which covers many existing models. The most influential aspects of this work are showing for the first time: 1)
undecidability of average pairwise energy minimisation in general 2) computational hardness for the most natural model with periodic boundary conditions, and 3) novel reductions for a large class of
generic pairwise energy functions covering many physical abstractions at once. In particular, we develop a new tool of overlapping digital rhombuses to incorporate the properties of the physical
force fields, and we connect it with classical tiling problems. Moreover, we illustrate the power of such reductions by incorporating more physical properties such as charge neutrality, and we show
an inapproximability result for the extreme case of the 1D average energy minimisation problem.
Publication series
Name Leibniz International Proceedings in Informatics, LIPIcs
Volume 241
ISSN (Print) 1868-8969
Conference 47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022
Country/Territory Austria
City Vienna
Period 22/08/22 → 26/08/22
• NP-hardness
• Optimisation of periodic structures
• tiling
• undecidability
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A simple word problem
From Word Problem to Equation
To be solved, word or story problems must be translated into equations with algebraic expressions that contain constants and variables. And how does one do the translation? Well, even in the most
simple cases there may be more than one approach. As usual, to master the art one has to try. Let's return to our first example.
(Note: In the applet below, all underlied words and numbers can be clicked on. In fact you can see some changes clicking anywhere in the applet area. Click, click, click ... and see what happens.)
Here is a little different way to tackle the same problem. What in essence is the problem about? Forget for a moment about constants. Let's give names to the quantities involved:
How does one solve two equations? In this case, the second equation is in fact the answer for one of the unknowns. We can simply substitute that value into the first equation:
Even in so simple a case, we can expand the initial step of naming variables:
We may learn a few things. First, there is much freedom in naming variables and putting equations together. Do not be surprised if a friend of yours solved the same problem differently. Second, when
translating a word problem into the language of mathematics, it's quite normal to get more than 1 variable or more than 1 equation. When more than 1 equation results, the equations are called
simultaneous and we talk of a system of simultaneous equations. To solve such a system is to obtain values for all variables involved. In other words, a system is solved when, for each of the
variables that appear in the system, we get a simplest possible equation variable = constant which specifies a (constant) value for the variable on the left.
We'll tackle simultaneous equations on a separate page.
(There are many more word problems discussed and solved at this site. The math tutorial continues with a similar approach over several additional examples.)
Word Problems
1. From Word Problem to Equation
|Contact| |Front page| |Contents| |Algebra|
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Tangent | Learn and Solve Questions
The word “tangent” is derived from the Latin word “tangere” which means to touch. Generally, we could say that tangent is the line that intersects the circle exactly at one point on its circumference
and never enters the interior of the circle. In differentiation, the tangent line is considered to be one of the most important applications. At one point on the curve, the tangent line comes into
contact with the curve. To calculate the tangent line equation, we must first determine the curve's equation and the location where the tangent is drawn. The "point of tangency" is the location where
the tangent is drawn. In this article, we are going to discuss tangent meaning in geometry, trigonometry, applications in science and technology and also a few of the most frequently asked questions
will also be answered.
What is a Tangent Line?
A tangent line to a curve at a point is a straight line that most closely approximates (or "clings to") the curve near that point. As the second point approaches the first, it can be considered the
limiting position of straight lines passing through the given point and a nearby point of the curve. If two curves have the same tangent line at a point, they are said to be tangent. The tangent
plane to a point on a surface and two surfaces that are tangent at a point are both defined in the same way. In the right triangle trigonometry, the ratio of the side opposite the angle to the side
adjacent is known as the tangent of an angle.
The image given below easily demonstrates the difference between a tangent line and a non-tangent line. It is clearly evident that a tangent line never enters the interior of the circle.
Tangent Meaning in Geometry
The tangent line (or simply tangent) to a plane curve at a particular location is a straight line that "just touches" the curve at that point, according to geometry. It was described by Leibniz as
the line joining two infinitely close points on a curve. A tangent to the curve y = f(x) at a point x = c is a straight line that passes through the point (c, f(c)) on the curve and has the slope (m)
f'(c), where f' is the derivative of f. Space curves and curves in n-dimensional Euclidean space have a similar definition.
The tangent line is "moving in the same direction" as the curve when it passes through the point where the tangent line and the curve meet, called the point of tangency is the best straight-line
approximation tangent to the curve at that moment. The graph of the affine function that best approximates the original function at the given point can also be thought of as the tangent line to a
point on differentiable curves. Similarly, the tangent plane to a given point on a surface is the plane that "just touches" the surface at that given point. Tangent space is a generalization of the
concept of a tangent, which is one of the most fundamental concepts in differential tangent geometry.
Tangent Meaning in Trigonometry
The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side in trigonometry.
In other words, it is the ratio of an acute angle's sine and cosine functions such that the cosine function's value does not equal zero. In trigonometry, the tangent function is one of the six key
The Tangent Formula in trigonometry is given as:
Tan A = \[\left(\frac{Opposite ~ side}{Adjacent~Side}\right)\]
The sine of every angle is always equal to the length of the opposite side which is divided by the length of the hypotenuse side, whereas the cosine of an angle is the ratio of the adjacent side and
the hypotenuse side.
In terms of sine and cosine, the tangent formula may be given as:
Sin A = \[\left(\frac{Opposite ~ side}{Hypotenuse~Side}\right)\]
Cos A = \[\left(\frac{Adjacent ~ side}{Hypotenuse~Side}\right)\]
As we know Tan = $\left(\frac{Sine}{Cosine}\right)$
Hence Tan A = \[\left(\frac{\left(\frac{Opposite~Side}{Hypotenuse~Side}\right)}{\left(\frac{Adjacent~Side}{Hypotenuse~Side}\right)}\right)\]
The tangent function is used in trigonometry to compute the slope of a line between the origin and a point representing the intersection of a right triangle's hypotenuse and altitude. On the other
hand, Tangent represents the slope of an object in both trigonometry and geometry.
What is a Vertical Tangent?
A vertical tangent is a tangent line that is vertical in mathematics, particularly calculus. A function whose graph has a vertical tangent is not differentiable at the point of tangency because a
vertical line has an infinite slope.
Osculating plane
The plane spanned by the three different points x(t), x(t+h1), and x(t+h2) on a curve as h1,h2 ->0. Let y be a point on the osculating plane, then,
[(y-x), x’,x”]=0
where [P, Q, R] denotes the scalar triple product The osculating plane passes through the tangent. The intersection of the osculating plane with the normal plane is always called the normal vector.
The Slope of a Tangent Line
To understand the slope of the tangent line , let us first consider a curve which is represented by a function f(x). Next , let us also consider a secant line which passses through two points of the
curve P(x0, f(x0)) and Q (x0+h,f(x0+h)) . Do take into account that P and Q are at a distance of H units from each other.
Now by using the slope formula we can find the slope of the secant line .
The slope of the secant line = \[\left(\frac{[f(x_{0}+h)-f(x_{0})])}{x_{0}+h-x_{0}}\right)\]
= \[\left(\frac{f(x_{0}+h)-f(x_{0})}{h}\right)\]
The secant line becomes the tangent line at P if Q comes very close to P (by making h→ 0) and merges with P, as shown in the diagram above. h → 0 can be applied to the slope of the secant line to get
the slope of the tangent line at P.
Hence now , the slope of the tangent at P = \[\lim_{h \to \ 0 } \left(\frac{f(x_{0}+h)-f(x_{0})}{h}\right) = 0\]
From the limit definition of the derivative or the first principles we know that the above equation is nothing but the derivative of f(x) at x = x0
Now , slope of the tangent at P = f '(x₀)
Therefore from the above explanation it is clearly evident that the slope of the tangent is nothing but the derivative of the function at the point where it has been drawn.
Tangent Line Equation
The point slope form : y - y₀ = m (x - x₀) is used to find the equation of a line with slope ‘m’ which passes through the point (x₀, y₀). Let us now consider the tangent line that is drawn to a curve
y = f(x) at a point (x₀, y₀).
We know that the slope of the tangent line , m =(f’(x)) at (x₀, y₀)
Now to get the tangent line equation we just have to substitute m , x₀ and y₀ values in the point slope form , y - y₀ = m (x - x₀).
Hence the tangent line formula is y - y₀ = $f'(x)_{(x_{0},y_{0})}$(x - x₀)
How to Find the Tangent Line Equation
There are a few steps that need to be followed to find the tangent line equation of a curve y = f(x) which is drawn at a point (x₀, y₀)
• If it is said that the tangent is drawn at x = x₀ and if the y coordinate of the point is not given then we need to find the y-coordinate by substituting it in the function y = f(x).
• The next step is to find the derivative of the function y = f(x) and then represent it by f'(x)
• Now to find the slope of the tangent (m ), we need to substitute the point (x₀, y₀) in the derivative f'(x).
• The last step is to find the equation of the tangent using the point-slope form y - y₀ = m (x - x₀).
Exemplar Problem With Step by Step Solutions
Example 1: Point (1,3) lies on a curve given by y = f(x)= x^3-x+4 . Find the tangent line's equation.
To write the equation of a tangent line we need the below things:
1. Slope
2. A point on the line
Solution : It is given that the curve contains a point (1, 3)
The slope of the curve is the same as the slope at x = 1 which is equal to the functions derivative at that given point:
f(x) = x3-x+4
f’(x) = 3x2-1
f’(1) = 3(1)-1
f’(1) = 2
Substituting the value of slope (m) in the point-slope form of the line,
y-y0 = m(x-x0)
y-3 = 2(x-1)
Converting the equation into y-intercept form as
y-3 = 2(x-1)
y-3 = 2x-2
y= 2x+1
∴ The equation for the tangent line is y = 2x + 1.
Example 2 : Find the equation of the tangent line of the curve y = 3x2 - 4x at x = -1.
The given curve is, f(x) = 3x2 - 4x
Its derivative is f'(x) = 6x - 4.
The slope of the tangent is, m = f'(x) = f'(-1) = 6(-1) - 4 = -10.
The point at which the tangent is drawn is, (x₀, y₀) = (-1, f(-1)) = (-1, 3(-1)2 - 4(-1)) = (-1, 7).
The equation of the tangent line is given by y - y₀ = m (x - x₀)
y - 7 = -10 (x - (-1))
y - 7 = -10 (x + 1)
y - 7 = -10x - 10
y = -10x - 3
Hence the equation of the tangent line of the curve y = 3x2 - 4x at x = -1 is y = -10x - 3
Tangent Applications in Science and Technology
Tangent has a wide range of applications in science and technology because it is a function of both Sine and Cosine functions. Trigonometric functions are widely employed in the fields of engineering
and physics. When there is something in a circle shape or something that resembles a circular shape, the sine, cos, and tan functions are expected to appear in the description. The following are some
examples of notions that make use of trigonometric functions:
• Artificial Neural Networks
• Empirical Formula and Heuristic functions
• Visualizations (Example: Andrews Plot)
• The behavior of Elementary Particles
• Study of waves like Sound waves, electromagnetic waves
Tangent Properties
• A tangent will touch a curve at only one point.
• A tangent line can never enter the interior of the circle and if it does then it's not a tangent line.
• At a right angle, the tangent touches the radius of the circle at the point of tangency.
A tangent is a line that touches a circle or an ellipse at only one point in geometry. If a line intersects a curve at Q, the point "Q" is referred to as the point of tangency. To put it another way,
it's the line that represents the slope of a curve at that point. Throughout the article we have discussed tangent, its definition in geometry, trigonometry, its application in science and technology
and also a few problems were solved as well. Students can visit the official website of Vedantu where we have provided links to topics related to tangent.
FAQs on Tangent
1. What is tangent used for in real life?
On the sea for navigation. The Vertical Sextant Angle Fix, which uses the tangent, is used to determine your distance from a lighthouse. Similarly, if you know the distance and angle of elevation to
the top of a great height, you can use the tangent to compute that height.
2. What are common tangents?
A tangent to a circle is a line that passes through exactly one point on a circle while remaining perpendicular to the circle's centerline. A common tangent of both circles is a line that is tangent
to more than one circle.
3. How is trigonometry related to real life?
Trigonometry can be used to roof a house, make the roof slanted (in the case of single-family bungalows), and calculate the height of a building's roof, among other things. It's commonly utilised in
the marine and aviation sectors. It's a type of cartography (creation of maps). | {"url":"https://www.vedantu.com/maths/tangent","timestamp":"2024-11-11T15:02:39Z","content_type":"text/html","content_length":"286412","record_id":"<urn:uuid:c90d83e0-9e0a-448d-8542-a99d3e1be845>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00897.warc.gz"} |
• The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
Muth, Robert 27 October 2016 (has links)
We study representations of Khovanov-Lauda-Rouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR
algebra. For a balanced order, we study imaginary semicuspidal modules by means of `imaginary Schur-Weyl duality'. We then generalize this theory from balanced to arbitrary convex preorders for
1 affine ADE types. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that KLR algebras are properly stratified. We introduce affine
zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. Finally, working in
finite or affine affine type A, we show that skew Specht modules may be defined over the KLR algebra, and real cuspidal modules associated to a balanced convex preorder are skew Specht modules for
certain explicit hook shapes.
Loubert, Joseph 18 August 2015 (has links)
This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we
2 establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$
is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from
an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook
Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material.
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Question Text 13) The numbers and form are in G.P. Find (i) , (ii) term (iii) term. 14) Mosclitoes are growing at a rate of a year. If there were 200 mosquitoes in
Updated On Oct 7, 2022
Topic Vector and 3D
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 129
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Practical Vectors
Planes rarely fly in the direction they are pointing. If the wind is blowing and the world is turning the pilot has to take account of these when he plots a course. Even with modern gps systems
available, , it is beneficial to the pilot to take these into account because of the resulting increase in fuel efficiency. It is worse though for the sailor, who has to take the tide into account as
We can deal with these problems by representing the velocities of the wind, tide and boat as vectors, then using trigonometry to find the course to take or the velocity to travel at.
For example:
A motor boat travels in a straight line across a river which flows at 3m/s between straight parallel banks 200 m apart. The motor boat, which has a top speed of 6m/s in still water, travels directly
from a point A on one bank to a point B, 150m downstream of A, on the opposite bank. Assuming that the motor boat is travelling at top speed, find, to the nearest second, the time it takes to travel
from A to B.
What is the first thing you do? DRAW A DIAGRAM!!
The boat has to steer a course of
We need to calculate V, the resultant velocity. Using the sine rule we can find x:
Now use the cosine rule to find
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First Iteration Overflows
If you use bad initial values for the parameters, the computation of the value of the objective function (and its derivatives) can lead to arithmetic overflows in the first iteration. The line-search
algorithms that work with cubic extrapolation are especially sensitive to arithmetic overflows. If an overflow occurs with an optimization technique that uses line search, you can use the INSTEP=
option to reduce the length of the first trial step during the line search of the first five iterations or use the DAMPSTEP or MAXSTEP= option to restrict the step length of the initial INSTEP=
option to reduce the default trust region radius of the first iteration. You can also change the minimization technique or the line-search method. If none of these methods helps, consider the
following actions:
• scale the parameters
• provide better initial values
• use boundary constraints to avoid the region where overflows may happen
• change the algorithm (specified in program statements) which computes the objective function
Problems with Quasi-Newton Methods for Nonlinear Constraints
The sequential quadratic programming algorithm in QUANEW, which is used for solving nonlinearly constrained problems, can have problems updating the Lagrange multiplier vector watchdog restarts
indicated in the iteration history. If this happens, there are three actions you can try:
• By default, the Lagrange vector Powell (1982b) describes. This corresponds to VERSION=2. By specifying VERSION=1, a modification of this algorithm replaces the update of the Lagrange vector
Powell (1978a, 1978b), which is used in VF02AD.
• You can use the INSTEP= option to impose an upper bound for the step length
• You can use the INHESSIAN= option to specify a different starting approximation for the Hessian. Choosing only the INHESSIAN option will use the Cholesky factor of a (possibly ridged)
finite-difference approximation of the Hessian to initialize the quasi-Newton update process.
Other Convergence Difficulties
There are a number of things to try if the optimizer fails to converge.
• Check the derivative specification:
If derivatives are specified by using the GRADIENT, HESSIAN, JACOBIAN, CRPJAC, or JACNLC statement, you can compare the specified derivatives with those computed by finite-difference
approximations (specifying the FD and FDHESSIAN option). Use the GRADCHECK option to check if the gradient Testing the Gradient Specification.
• Forward-difference derivatives specified with the FD= or FDHESSIAN= option may not be precise enough to satisfy strong gradient termination criteria. You may need to specify the more expensive
central-difference formulas or use analytical derivatives. The finite-difference intervals may be too small or too big and the finite-difference derivatives may be erroneous. You can specify the
FDINT= option to compute better finite-difference intervals.
• Change the optimization technique:
For example, if you use the default TECH=LEVMAR, you can
□ change to TECH=QUANEW or to TECH=NRRIDG
□ run some iterations with TECH=CONGRA, write the results in an OUTEST= data set, and use them as initial values specified by an INEST= data set in a second run with a different TECH= technique
• Change or modify the update technique and the line-search algorithm:
This method applies only to TECH=QUANEW, TECH=HYQUAN, or TECH=CONGRA. For example, if you use the default update formula and the default line-search algorithm, you can
□ change the update formula with the UPDATE= option
□ change the line-search algorithm with the LINESEARCH= option
□ specify a more precise line search with the LSPRECISION= option, if you use LINESEARCH=2 or LINESEARCH=3
• Change the initial values by using a grid search specification to obtain a set of good feasible starting values.
Convergence to Stationary Point
The (projected) gradient at a stationary point is zero and that results in a zero step length. The stopping criteria are satisfied.
There are two ways to avoid this situation:
• Use the DECVAR statement to specify a grid of feasible starting points.
• Use the OPTCHECK= option to avoid terminating at the stationary point.
The signs of the eigenvalues of the (reduced) Hessian matrix contain information regarding a stationary point:
• If all eigenvalues are positive, the Hessian matrix is positive definite and the point is a minimum point.
• If some of the eigenvalues are positive and all remaining eigenvalues are zero, the Hessian matrix is positive semidefinite and the point is a minimum or saddle point.
• If all eigenvalues are negative, the Hessian matrix is negative definite and the point is a maximum point.
• If some of the eigenvalues are negative and all remaining eigenvalues are zero, the Hessian matrix is negative semidefinite and the point is a maximum or saddle point.
• If all eigenvalues are zero, the point can be a minimum, maximum, or saddle point.
Precision of Solution
In some applications, PROC NLP may result in parameter estimates that are not precise enough. Usually this means that the procedure terminated too early at a point too far from the optimal point. The
termination criteria define the size of the termination region around the optimal point. Any point inside this region can be accepted for terminating the optimization process. The default values of
the termination criteria are set to satisfy a reasonable compromise between the computational effort (computer time) and the precision of the computed estimates for the most common applications.
However, there are a number of circumstances where the default values of the termination criteria specify a region that is either too large or too small. If the termination region is too large, it
can contain points with low precision. In such cases, you should inspect the log or list output to find the message stating which termination criterion terminated the optimization process. In many
applications, you can obtain a solution with higher precision by simply using the old parameter estimates as starting values in a subsequent run where you specify a smaller value for the termination
criterion that was satisfied at the previous run.
If the termination region is too small, the optimization process may take longer to find a point inside such a region or may not even find such a point due to rounding errors in function values and
derivatives. This can easily happen in applications where finite-difference approximations of derivatives are used and the GCONV and ABSGCONV termination criteria are too small to respect rounding
errors in the gradient values. | {"url":"http://support.sas.com/documentation/cdl/en/ormpug/63352/HTML/default/ormpug_nlp_sect037.htm","timestamp":"2024-11-14T20:01:24Z","content_type":"application/xhtml+xml","content_length":"23242","record_id":"<urn:uuid:d22c980f-d4fa-4580-a6bf-2a26d839f2e3>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00121.warc.gz"} |
Generalized Correlation for Latent Heat of Vaporization. - Journal of - PDF Free Download
Table I. Polymerization of Vinylidene Fluoride and 1 -Chloro-l,2-difluoroethylene Total Charge, Moles 0.03 0.05 0.25 0.21 0.004
0.03 0.03 0.25 0.003
Mole 70CH,CF2, Initiation Charge/ Copolymer System 50/low A' 50/ 10
50120 83/85 50 / 52.5
Reactorb FP FP
Time, Hours/O C.
Yield, %
Bd Bd Bd corn
300-500 300-600 200-300
M M G
120/50 12/70 98 I 2 5
A B B CoM
100- 150 100-150 460 200-300
116160 24/60 36/60 24/25
M G "Initial pressure at temperature, estimated. 'FP. Fischer-Porter aerosol compatibility tube, 75 ml.; M. Monel Hoke cylinder, 90 ml.; G. Glass Carius tube, 3 ml. KzSZOB, 0.75 part; KzHPO,, 2
loss of lateral symmetry in progressing from a polymer chain composed of CF, units to one composed of C H F units. A certain amount of difficulty was encountered before even a reasonably broad
spectrum of copolymers could be obtained. Since the polymerization behavior of CHgCF, is little discussed outside the patent literature, it is of interest to discuss briefly the experimental findings
which are summarized in Table I. In initial experiments, CH2CF2showed much less tendency than CHFCFCl t o enter into the copolymer. This might be due to the difficulty in emulsifying significant
amounts of this low-boiling reactant (the critical temperature of which is in the vicinity of room temperature). The higher boiling chloroolefin presented no such difficulty, as it was easily
liquified under the conditions used. The use of fluorocarbon soap would be expected to increase the concentration of CH2CFgin the liquid phase, and the proportion found in the resulting copolymer was
correspondingly higher. Carrying out the reactions under higher pressure was even more effective. A polymerization initiated by Cow gamma radiation presented an interesting situation. The
CHFCFC1-rich liquid phase gave a polymer containing 9 5 6 CHFCFCl, and the CH2CF2-rich vapor phase gave a copolymer containing 52% C H X F , . A concurrent series of experiments in which the
homopolymerization of CH2CF2was studied further illustrated the importance of pressure and an effective emulsifier. Thus CH2CF2 did not polymerize using a hydrocarbon emulsifier and a relatively low
pressure (100 to 150 p s i . ) . Substitution of a fluorocarbon soap resulted in a low yield of polymer being
96/60 21/60
0.09 0.21 0.8 0.4
Remarks Rather weak plastic Leathery plastic Leathery rubber Crystalline plastic
Hard pellet in bottom of reactor (95% CHFCFCl), film in top (52% CH,CFd
Na lauryl sulfate, 3 parts; HtO, 200 parts; monomers, 100 parts. dSame as (b), except NH, salt Kel-F Acid 8114 replaced the Na lauryl sulfate. obtained. Carrying out the reaction in a metal cylinder,
where a higher pressure could be tolerated, resulted in a higher yield. Thus Cornwas more effective, even in a reaction which took place in the gas phase. ACKNOWLEDGMENT This work was supported by
the George C. Marshall Space Flight Center, Huntsville, Ala. under Contract NAS 8-5352, "Development of a Vulcanizable Elastomer Stable incontact with Liquid Oxygen." John T. Schell acted as
Technical Representative. The cooperation of R. J. Hanrahan and R. W. Gould, of the University of Florida, Gainesville, Fla. with the Corn'irradiations and the x-ray powder patterns, respectively, is
appreciated. LITERATURE CITED (1) Durrell, W.S., Stump, E.C., Jr., Westmoreland, Geraldine, Padgett, C.D., 148th Meeting, ACS, Chicago, Ill., August 1964. (2) Durrell, W.S., Westmoreland, Geraldine,
Moshonas, M.G., 148th Meeting, ACS, Chicago, Ill., August 1964. (3) Leschenko, S.S., Karpov, V.L., Kargin, V.A., Vysokomolekul. Soedin. 5 , 953 (1963). (4) Mandelkern, L., Martin, G.M., Quinn, F.A.
Jr., J . Res. Natl. Bur. Std. 58, 137 (1957). ( 5 ) Ruh, R.P., Rector, M.R., U. S. Patent 2,716,109 (Aug. 23, 1955).
RECEIVED for review July 23, 1964. Accepted February 17, 1965.
Generalized Correlation for latent Heat of Vaporization NlNG HSING CHEN 812 Beachview Drive, Jekyll Island, Ga. To calculate latent heats of vaporization, a simple analytical expression without using
acentric factor was developed from the Pitzer tabular correlation. Graphical solutions with a nomogram are presented. Results are compared with the existing methods.
EXTENSIVE tabular values were presented in 1955 by Pitzer and associates ( 1 3 ) . Part of these values can be used for calculating quite accurately latent heats of vaporization a t any temperature
when acentric factor and critical temperature are known. Inasmuch as this method was not expressed in a convenient form and, to some extent, the VOL. 10, No. 2, APRIL 1965
prediction of the acentric factor by the Edmister method (3) is quite time consuming, it has been overlooked by many investigators for almost 10 years. This article extends their work by transforming
their tabular correlation into analytical and graphical ones thereby eliminating use of the acentric factor. 207
Pitzer (13:) tabulated a set of values of AS,",AS,', AS? at reduced temperatures from 0.56 to 1.00 and proposed to calculate the entropy of vaporization by the following equation: A& = ASP + WAS:+ w
2 As,'
where w is the acentric factor defined by w
+ 1)
= - (logloP,oi
where P,oi is the reduced pressure a t reduced temperature T,=0.7. Hence, Equation 1 shows that when the reduced temperature and the acentric factor are known, the total entropy of vaporization which
is the ratio of latent heat of vaporization to temperature can then be calculated. Consequently, a plot can be made of the total entropy of vaporization us. the acentric factor a t different reduced
temperatures, which are shown as broken lines in Figure 1. In thg same article, they also presented another equation as follows: log,oP, = log10P,o + w
(a l Y Y 9 T ~
Another set of values for loglo&? and (8 logloP,/8w)r was tabulated a t different reduced temperatures, where Pp is the reduced pressure for a simple fluid and (8 log,#,/ 8o)T is the partial
derivative of the logIoP, with respect to acentric factor a t constant temperature. Now from Equation 3, by knowing the reduced temperature and acentric factor, the corresponding reduced pressure can
be calculated. Then a similar plot can be made of l o g d , us. the acentric factor a t different reduced temperatures which are shown as solid lines in Figure 1. 1
l I , O
Figure 1 shows that the solid lines have the slopes of the partial derivative in Equation 3. Theoretically, the broken lines are not straight, which can be mathematically proved from Equation 1.
However, a t small values of acentric factor (usually less than one), inasmuch as the product of the squares of a small value of AS: would be very small, the last term in Equation 1 contributes very
little to the total. Under this condition, the total entropy of vaporization is approximately linear with the acentric factor. This relationship is verified by the nearly straight lines which were
plotted between small values of the acentric factor of zero to 0.5. In a manner perfectly analogous to that in Equation 3, these broken lines can be assumed to be represented to good approximation by
A S , = - AH, =
(A;)' -
( aAHJT ) aw
where (AH"/T)'may be defined as the entropy of vaporization of a simple fluid and is the value of AHu/T when w = 0; that is, the intercept of the broken lines at w = 0. From Equation 1, it is seen
that this value is equal to AS: in Pitzer's article. Evidently [ a ( A H , l T ) / a w ]is~ the slope of the lines which can then be calculated. Eliminating the acentric factor from Equations 3 and 4
-f a AHJ T\ (5) \
The values of (AHJT)', log&', and (8 logloP,/aW)T were given in Pitzer's article. Those of [ ~ ( A H , / T ) / & ]are T calculable. Hence Equation 5 can be simplified to logioP, = A (AHLIT)+ B
where A and B are functions of reduced temperature. Figure 2 is such a plot which shows the linearity of the values of A and B with the reduced temperature. These two lines were then evaluated by the
method of least squares. After substitution and simplification, Equation 6 becomes the proposed generalized correlation as follows: logioP, = (0.1406Tr- 0.1504)(AHb/T) + (l.llTr- 1.1)
Equation 7 can be rearranged to give the latent heat of vaporization at normal boiling point as
With the input data of normal boiling point, critical temperature, and critical pressure, values of latent heat of vaporization for 165 compounds were calculated by means of this proposed Equation 8,
the Giacalone equation (51, the Riedel equation ( 1 5 ) , and the modified Klein equation ( 4 , 8). These calculated values were compared with the corresponding literature values (experimental or
calculated) which were taken from reliable sources, (6, 10-12). The critical pressures and critical temperatures were taken from the article by Kobe and Lynn (9) which was considered reliable by many
investigators. The boiling temperatures were taken from reliable sources ( 2 , 12). The comparison of these results is found in Table I. The proposed correlation plot is shown in Figure 3. For 165
compounds, the deviations of the calculated from the literature values are 2.40% by the Giacalone, 2.02% by the Riedel, 1.85% by the modified Klein, and 1.82% by the proposed equation. 208
Table I. latent Heat of Vaporization at Normal Boiling Point by Different Methods No. of Substances Compound Group Tested Giacalone Monoatomic gases 3.85 4 Diatomic gases 2.97 8 Inorganic halides 7
2.23 Inorganic oxides 3.55 4 Miscellaneous inorganics 8 2.44 Aliphatic hydrocarbons (satd.) 39 2.25 Aliphatic hydrocarbons (unsatd.) 11 3.02 Cycloparaffins 1.38 5 Aromatic hydrocarbons 15 1.51
Substituted aromatics 11 2.59 Alcohols 3.69 5 Amines 8 3.05 Esters 1.96 9 Ethers 4 0.62 Nitriles 7.88 5 Organic halides 13 1.10 Miscellaneous organics 9 1.20 Grand total 165 2.40
Deviation, "c Modified Klein 1.22 1.96 2.69 2.24 1.82 1.06 1.43 0.68 1.55 2.33 2.97 3.75 1.43 1.45 7.60 1.50 1.35 1.85
Riedel 0.97 1.22 2.77 2.04 2.08 1.50 1.40 1.OS 1.61 2.21 2.87 2.85 1.18 1.22 8.31 2.80 2.00 2.02
Proposed Eq. 8 0.70 1.45 2.78 1.71 2.15 1.13 0.98 1.16 1.60 2.16 2.92 3.30 1.59 1.58 7.12 1.62 1.85 1.82
T , Ta,T,, and P,, the values can also be found from Figure 1 or the nomogram Figure 4 which was constructed from Equation 7 . 01
Figure 2. linear plot of A and 6 vs. T,
The proposed Equation 7 relates the latent heat of vaporization at any temperature with the corresponding reduced temperature and reduced vapor pressure. It is useful in checking the thermodynamic
table in which the temperature and vapor pressure are usually given. For 12 compounds, Table I1 lists the ranges of the reduced temperature and the corresponding vapor pressures, the number of points
within these ranges, and the average deviation from the literature values by different methods. Column 5 is the deviation by Equation 7 . Column 6 is the deviation calculated by the combined equation
of the Giacalone (5) and the Watson (18) as recommended by Reid and Sherwood (12, 14). For 58 point values, the average deviation is 2.55% by Equation 7 and 3.36% by the Giacalone-Watson correlation.
With the given input data,
When the normal boiling point, the latent heat of vaporization a t this temperature, and the critical temperature are given, the latent heats a t any other temperature can be calculated. T o this
end, the Watson correlation (18) has been considered as the most accurate one for a great variety of compounds. However, it can also be accomplished by Equation 4 which can be rearranged as
where the subscripts T and b denote the temperature in question and the boiling temperature, respectively. The values of the compounds in Table I1 calculated by the
Figure 3. Correlation at normal boiling point
0 '
8 -20 -
* .
9, ATMOSPHERE VOL. 10, No. 2, APRIL 1965
/ , , , I
Figure 4. Generalized nomogram for latent heat of vaporization a t any temperature and pressure
a a
L L 0
t \
i?5 'w -IO
Table 11. Latent Heat of Vaporization a t Temperatures and Pressures other than Normal Boiling Point and Atmosphere by Different Methods No. of
Compound Ammonia (12)” Benzene (17 )
Ethylene (12) Ethyl alcohol ( 1 7 ) Ethyl ether (16) Methane (1) Methylamine (12) Methyl formate (12) Nitrogen (1) Sulfur dioxide (12) Trichlorofluoromethane(12) Water (7)
Eq. 7
Eq. 9
Fig. 1
0.549-0.803 0.771-0.978 0.512-0.985 0.792-0.970 0.63 -0.964 0.523-0.941 0.6 -0.739 0.615-0.684 0.567-0.95 0.594-0.915 0.625-0.720 0.566-0.9 1
0.0036 -0.20 0.14 -0.84 0.00338-0.890 0.108 -0.756 0.0167 -0.758 0.00751-0.70 0.00925-0.0911 0.0138 -0.0442 0.0149 -0.745 0.008974.528 0.0211 -0.0832 0.0036 -0.482
0.81 1.80 6.40 2.48 1.25 1.82 2.55 1.51 4.48 2.08 0.77 5.56
2.18 6.96 2.70 1.95 3.09 2.27 3.59 2.78 2.07 4.93 0.32 5.54
0.27 6.35 1.23 3.41 2.54 2.78 0.72 1.76 2.86 1.43 0.98 1.66
0.61 7.29 0.75 5.65 3.17 1.75 0.80 2.61 3.15 1.69 1.26 1.82
0.41 5.48 1.27 4.03 2.09 0.86 0.91 2.41 3.43 2.08 1.26 1.11
Grand total “Numbersin parentheses designate the references at the end of the article.
Watson equation (18) and Equation 9 compare with the literature values in columns 7 and 8. For 58 point values, the average deviation is 2.18% by the Watson equation and 2.52% by Equation 9. Because
the use of Equation 9 is not very convenient, and its accuracy is not as good as the Watson equation, its use is recommended only when the Watson equation is in doubt. LATENT HEAT OF VAPORIZATION AT
Sometimes it is necessary to find the latent heat of vaporization at pressures other than one atmosphere with the additional input data of critical temperature, critical pressure, and the normal
boiling point. If we use Equations 3 and 4, the method of trial and error is required. With the input data, the acentric factor can be estimated from Equation 3. Then a value of T , should be tried
to satisfy Equation 3 for other pressures. With this value of reduced temperature, either the proposed Equation 7 or Equation 4 can be used to find the value of latent heat a t the given reduced
pressure. T o provide a rough estimate and avoid this tedious trial and error method, Figure 1 can be used in a very convenient way. The procedures are: Locate the imaginary w a t the abscissa for
the given T,, P,, and Tb (It is not necessary to record this value); For this value of w , locate T , corresponding to P , ; For this value of w and T,, then find 3Hb,/T,.For these same data, the
Watson and the other three equations fail to give a result. Again for the same compounds in Table 11, column 9 tabulates the values from Figure 1 for different pressure (assuming the corresponding
temperatures are not given) with the input data thus described. CONCLUSIONS
Inasmuch as the proposed equation was developed from a sound theoretical background, it is expected that the accuracy should be good. Table I shows that the accuracy of the proposed equation is
better than those of the Giacalone and the Riedel equations; it is a t least as good as the modified Klein equation which is more complicated. Table I1 shows that Equation 7 is better than the
GiacaloneWatson equation. Besides, for the input data of Tb, P , P,, and T,, only the proposed Equation 7 can give the value of AH^, whereas the Watson and the other three correlations cannot. With
all of these advantages, the use of Equations 7 and 8, Figure 1, and the nomogram is recommended with an over-all uncertainty not greater than 2% which is believed to be within the experimental
GiacaloneWatson Watson
B AH, P PC P, P,” AS” T T* T,*
TC T, W
function of reduced temperature in Equation 6 function of reduced temperature in Equation 6 molal latent heat of vaporization, cal. per mole pressure, atm. critical pressure, atm. reduced pressure
reduced pressure for a simple fluid entropy of vaporization = AH,/T, cal. per mole per K. temperature, K. boiling temperature, K. reduced boiling temperature critical temperature, K. reduced
temperature acentric factor entropy of vaporization for a simple fluid
LITERATURE CITED (1) Din, F., Ed., “Thermodynamic Functions of Gases,” Vol. 3, Butterworths, London, 1961. (2) Dreisbach, R.R., Aduan. Chem.Ser. 15, 22, 29, ACS, Washington,D. C., 1955,1959, 1961.
(3) Edmister, W.C., Petrol. Refiner 37,No. 4,173 (1958). (4) Fishtine, S.H., I n d . Eng. Chem. 55, No. 4,20 (1963). (5) Giacalone, A., Gazz. Chim. Itol. 81, 180 (1951). (6) “International Critical
Tables,” McGraw-Hill, New York, 1928. (7) Keenan, J.P., Keyes, F.G., “Thermodynamic Properties of Steam,” Wiley, New York, 1936. (8) Klein, V.A., Chem.Eng. Prog. 45, 675 (1949). (9) Kobe, K.A., Lynn,
R.E., Chem.Reu. 52,117 (1953). (10) Landolt-Bomstein,“Zahlenwerte and Funktionen aus Physik Chemie, Astronomie, Geophysik, Technik,” I1 Band 4 Teil, . . Springer, Berlin, 1955. (11) Lange, N.A., Ed.,
“Handbook of Chemistry,” 9th ed., Handbook Publishers, Inc., Sandusky, Ohio, 1956. (12) Perry, J.H., “Chemical Engineers’ Handbook,” 4th ed., McGraw-Hill,New York, 1963. (13) Pitzer, K.S., Lippmann,
D.E., Curl, R.F., Jr., Huggins, C.M., Petersen, D.E., J.A m . Chem. Soc. 77,3433 (1955). (14) Reid, R.C., Shenvood, T.K., “The Properties of Gases and Liquids,” McGraw-Hill,New York, 1958. (16)
Schnaible, H.W., Smith, J.M., Chem. Eng. Progr. Symp. Ser. 49, No 7, 161 (1953). (17) Storvick, T.S., Smith, J.M., J. CHEM.ENG.DATA5, 133 (1960). (18) Watson, K.M., Ind. Eng. C h m . 35,398 (1943).
RECEIVED for review May 27, 1964. Accepted December 21, 1964. | {"url":"https://datapdf.com/generalized-correlation-for-latent-heat-of-vaporization-joure37dc918e0c8c4185f7ebf9efd509b1147075.html","timestamp":"2024-11-10T06:25:16Z","content_type":"text/html","content_length":"45201","record_id":"<urn:uuid:687ac0e6-e553-4419-8fce-f7b4be473069>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00864.warc.gz"} |
August 2010 - Tangra Inc.
In my previous post I’ve shown how to convert the public key of an XML formatted RSA key to the more widely used PEM format. The only limitation of the solution was that since it utilizes the
Cryptographic Next Generation (CNG) algorithms it is usable only on Windows 7 and Windows Server 2008 R2.
So bellow I’ll demonstrate a solution that works under all operating systems. Also as an extra the solution bellow can convert the private key as well 😉 Both the public and the private keys exported
by the functions bellow are parsed by OpenSSL!
You can find the compiled source here.
private static byte[] RSA_OID =
{ 0x30, 0xD, 0x6, 0x9, 0x2A, 0x86, 0x48, 0x86, 0xF7, 0xD, 0x1, 0x1, 0x1, 0x5, 0x0 }; // Object ID for RSA
// Corresponding ASN identification bytes
const byte INTEGER = 0x2;
const byte SEQUENCE = 0x30;
const byte BIT_STRING = 0x3;
const byte OCTET_STRING = 0x4;
private static string ConvertPublicKey(RSAParameters param)
List<byte> arrBinaryPublicKey = new List<byte>();
arrBinaryPublicKey.InsertRange(0, param.Exponent);
arrBinaryPublicKey.Insert(0, (byte)arrBinaryPublicKey.Count);
arrBinaryPublicKey.Insert(0, INTEGER);
arrBinaryPublicKey.InsertRange(0, param.Modulus);
AppendLength(ref arrBinaryPublicKey, param.Modulus.Length);
arrBinaryPublicKey.Insert(0, INTEGER);
AppendLength(ref arrBinaryPublicKey, arrBinaryPublicKey.Count);
arrBinaryPublicKey.Insert(0, SEQUENCE);
arrBinaryPublicKey.Insert(0, 0x0); // Add NULL value
AppendLength(ref arrBinaryPublicKey, arrBinaryPublicKey.Count);
arrBinaryPublicKey.Insert(0, BIT_STRING);
arrBinaryPublicKey.InsertRange(0, RSA_OID);
AppendLength(ref arrBinaryPublicKey, arrBinaryPublicKey.Count);
arrBinaryPublicKey.Insert(0, SEQUENCE);
return System.Convert.ToBase64String(arrBinaryPublicKey.ToArray());
private static string ConvertPrivateKey(RSAParameters param)
List<byte> arrBinaryPrivateKey = new List<byte>();
arrBinaryPrivateKey.InsertRange(0, param.InverseQ);
AppendLength(ref arrBinaryPrivateKey, param.InverseQ.Length);
arrBinaryPrivateKey.Insert(0, INTEGER);
arrBinaryPrivateKey.InsertRange(0, param.DQ);
AppendLength(ref arrBinaryPrivateKey, param.DQ.Length);
arrBinaryPrivateKey.Insert(0, INTEGER);
arrBinaryPrivateKey.InsertRange(0, param.DP);
AppendLength(ref arrBinaryPrivateKey, param.DP.Length);
arrBinaryPrivateKey.Insert(0, INTEGER);
arrBinaryPrivateKey.InsertRange(0, param.Q);
AppendLength(ref arrBinaryPrivateKey, param.Q.Length);
arrBinaryPrivateKey.Insert(0, INTEGER);
arrBinaryPrivateKey.InsertRange(0, param.P);
AppendLength(ref arrBinaryPrivateKey, param.P.Length);
arrBinaryPrivateKey.Insert(0, INTEGER);
arrBinaryPrivateKey.InsertRange(0, param.D);
AppendLength(ref arrBinaryPrivateKey, param.D.Length);
arrBinaryPrivateKey.Insert(0, INTEGER);
arrBinaryPrivateKey.InsertRange(0, param.Exponent);
AppendLength(ref arrBinaryPrivateKey, param.Exponent.Length);
arrBinaryPrivateKey.Insert(0, INTEGER);
arrBinaryPrivateKey.InsertRange(0, param.Modulus);
AppendLength(ref arrBinaryPrivateKey, param.Modulus.Length);
arrBinaryPrivateKey.Insert(0, INTEGER);
arrBinaryPrivateKey.Insert(0, 0x00);
AppendLength(ref arrBinaryPrivateKey, 1);
arrBinaryPrivateKey.Insert(0, INTEGER);
AppendLength(ref arrBinaryPrivateKey, arrBinaryPrivateKey.Count);
arrBinaryPrivateKey.Insert(0, SEQUENCE);
return System.Convert.ToBase64String(arrBinaryPrivateKey.ToArray());
private static void AppendLength(ref List<byte> arrBinaryData, int nLen)
if (nLen <= byte.MaxValue)
arrBinaryData.Insert(0, Convert.ToByte(nLen));
arrBinaryData.Insert(0, 0x81); //This byte means that the length fits in one byte
arrBinaryData.Insert(0, Convert.ToByte(nLen % (byte.MaxValue + 1)));
arrBinaryData.Insert(0, Convert.ToByte(nLen / (byte.MaxValue + 1)));
arrBinaryData.Insert(0, 0x82); //This byte means that the length fits in two byte
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02:05:14Convert RSA public/private key from XML to PEM format (.NET) (Part 2)
The Power of Simplicity
/4 Comments/in SQL Server/by Plamen Ratchev
Solving a problem very often results in unnecessary complex solutions. One of the first lessons I learned from my math teacher was to scrap any solution that exceeds a page. She would urge me to
start all over and look for simpler way to resolve it. In her world there was always a short and simple solution, it was only a matter of seeing it.
I find this rule applicable to any type of problem. Only the dimension of the page size changes according to the subject matter. Many believe that finding the simpler and better solution is to “think
outside the box”. But in my opinion it is exactly the opposite – to think inside the box. Know the fundamentals of your area of expertise, systematically apply them, and you will find a simple and
elegant solution! Isaac Newton did not just discover the gravity when an apple fell on his head (if at all it did). It took him 20 years of hard work to explain gravity!
In the world of SQL it drills down to deep understanding of the set based nature of SQL and coming up with solution based on that. Thinking like a procedural programmer will not help.
What are the rules to simplicity? There are no rules! I like to use the following quotes as guidelines:
“The simplest way to achieve simplicity is through thoughtful reduction.”
John Maeda, The Laws of Simplicity
“Make everything as simple as possible, but not simpler.”
Albert Einstein
Let’s illustrate this with one example in SQL. In out sample scenario the request is to retrieve a list of customers who always order the exact same product (regardless of what the product is). This
is a very valid business problem because you may want to send targeted coupons to customers who always buy the same products.
There are different ways to solve this problem and Listing 1 shows one method. It is very close to describing the solution in plain English: select all customers where the customer has no other
orders with different product SKU.
Listing 1
SELECT DISTINCT customer_nbr
FROM Orders AS O
FROM Orders AS O1
WHERE O1.customer_nbr = O.customer_nbr
AND O1.sku <> O.sku);
Is this the simplest way to solve the problem? This query is set based but still in a way mimics procedural thinking – examine all other customer orders and check that there is no other order with
different product SKU.
If you think about the set of all customer orders, you will notice that these that we need have repeating attribute values, that is the same product SKU. Applying the MIN and MAX aggregate functions
on that attribute will return the same value. Then here is our simplified solution: retrieve all customers that have equal MIN and MAX product SKU on all orders. Listing 2 shows the query.
Listing 2
SELECT customer_nbr
FROM Orders
GROUP BY customer_nbr
HAVING MIN(sku) = MAX(sku);
This is more elegant and simpler solution! Thinking more about the set of all customer orders you will notice that the distinct count of product SKUs is 1 for the customers in the needed result set.
That brings us to another elegant solution:
Listing 3
SELECT customer_nbr
FROM Orders
GROUP BY customer_nbr
HAVING COUNT(DISTINCT sku) = 1;
Pretty and simple, right? Try it next time when you see that two page query!
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2017-02-11 07:32:44The Power of Simplicity
Convert RSA public key from XML to PEM format (.NET) (Part 1)
/5 Comments/in .NET Framework, Cryptography, Security/by Peter Staev
Probably the people working with asymmetric cryptography have struggled for a way to convert the XML format of the RSA public key to the more widely used PEM format. Although there is a solution for
the reverse transformation (from PEM to XML) on the following address http://www.jensign.com/opensslkey/opensslkey.cs I have not found anywhere a solution to this problem.
So after a bit of reading and examining the code in the above mentioned link I’ve come up with a small code that does the conversion and the resulting key is parsed OK from OpenSSL.
NOTE: You will need to download and use the assemblies from http://clrsecurity.codeplex.com/
NOTE2: The code bellow only works under Windows 7 and Windows Server 2008 R2, because it uses the Cryptographic Next Generation (CNG) that were added only to those operating systems.
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using Security.Cryptography;
using System.Security.Cryptography.X509Certificates;
using System.IO;
namespace ConsoleApplication1
class Program
static void Main(string[] args)
RSACng rsa = new RSACng();
X509Certificate2 cert;
List<byte> arrBinaryPublicKey = new List<byte>();
byte[] oid =
{ 0x30, 0xD, 0x6, 0x9, 0x2A, 0x86, 0x48, 0x86, 0xF7, 0xD, 0x1, 0x1, 0x1, 0x5, 0x0 }; // Object ID for RSA
cert = rsa.Key.CreateSelfSignedCertificate(new X500DistinguishedName("CN=something"));
//Transform the public key to PEM Base64 Format
arrBinaryPublicKey = cert.PublicKey.EncodedKeyValue.RawData.ToList();
arrBinaryPublicKey.Insert(0, 0x0); // Add NULL value
CalculateAndAppendLength(ref arrBinaryPublicKey);
arrBinaryPublicKey.Insert(0, 0x3);
arrBinaryPublicKey.InsertRange(0, oid);
CalculateAndAppendLength(ref arrBinaryPublicKey);
arrBinaryPublicKey.Insert(0, 0x30);
//End Transformation
Console.WriteLine("-----BEGIN PUBLIC KEY-----");
Console.WriteLine("-----END PUBLIC KEY-----");
private static void CalculateAndAppendLength(ref List<byte> arrBinaryData)
int nLen;
nLen = arrBinaryData.Count;
if (nLen <= byte.MaxValue)
arrBinaryData.Insert(0, Convert.ToByte(nLen));
arrBinaryData.Insert(0, 0x81); //This byte means that the length fits in one byte
arrBinaryData.Insert(0, Convert.ToByte(nLen % (byte.MaxValue + 1)));
arrBinaryData.Insert(0, Convert.ToByte(nLen / (byte.MaxValue + 1)));
arrBinaryData.Insert(0, 0x82); //This byte means that the length fits in two byte
Compiled source available here
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02:32:41Convert RSA public key from XML to PEM format (.NET) (Part 1)
It’s a Matter of Style
/7 Comments/in SQL Server/by Plamen Ratchev
W riting SQL can be very enjoyable activity. Reading SQL can be also enjoyable (maybe like reading poetry to some), or very unpleasant… How do you write SQL with style that results in eye pleasing
and easy to read/understand code? And does it matter?
Sometimes code writing drills down to concentrating on the task at hand and producing a brilliant piece of code, which looks like this:
Listing 1
select c.customer_name, o.order_amt,
d.qty from customers c left outer join
orders o on c.customer_nbr = o.customer_nbr
left outer join orderdetails d on d.order_nbr =
o.order_nbr and d.sku = 101
Or maybe like this:
Listing 2
SELECT C.CUSTOMER_NAME, O.ORDER_AMT,
ORDERS O ON C.CUSTOMER_NBR = O.CUSTOMER_NBR
LEFT OUTER JOIN ORDERDETAILS D ON D.ORDER_NBR =
O.ORDER_NBR AND D.SKU = 101
While this code performs exceptionally and solves the problem in a very clever way, is it really that good? What happens when the code review/test team gets their turn? Or when you/someone else has
to modify it two years from now? To my opinion this code is a very long way from what a real production code should be. And yes, this is very real and it happens every day, even as I type this. Just
pay attention on the next code review, or take a look at any online SQL forum (and no, it is not only the people that ask questions, unfortunately many SQL gurus that know it all would post an answer
with similar “example” style).
How do you make this code look better? The answer is in the four basic principles of design: contrast, repetition, alignment, and proximity. Let’s look how applying these principles of design (which
many think are applicable only to graphic design) can lead to stylish and enjoyable code.
The idea is to use contrast for elements that a very different. One example is columns and reserved keyword. They are not the same and the code should make that distinction very clear. Let’s apply
Listing 3
SELECT C.customer_name...
Here the SELECT keyword is capitalized to differentiate from the lower case column name. Also, the table alias is capitalized to indicate clearly the table source.
Repeating the same element styles for all similar items adds consistency and organization throughout code. For example, repeat and maintain capitalization for all keyword, do not mix style in
different context of the code. Like the style of the SELECT and FROM reserved keywords in Listing 4.
Listing 4
SELECT C.customer_name... FROM Customers AS C...
This allows to “visualize” the shape of the query code. Now the eye can easily flow from one section of code to the next one and concentrate on each element.
Code elements should not be placed arbitrarily on the lines of code. Every code item should have some visual connection with another item in the code. One example is aligning the start of each clause
of the query (SELECT, FROM, WHERE, etc.) on a new line:
Listing 5
SELECT C.customer_name...
FROM Customers AS C...
Alignment creates a clean and pleasing look of the code structure.
Code items that relate to each other should be grouped close together. When several items are in close proximity they become one visual unit. Like placing SELECT and column names together on the
line, similar for FROM and table names or WHERE and predicates. Listing 6 demonstrates this.
Listing 6
SELECT C.customer_name, O.order_amt...
FROM Customers AS C
LEFT OUTER JOIN Orders AS O...
This makes the code structure very clear and eliminates clutter.
Let’s apply all four principles to the initial query. Here is one way it may look:
Listing 7
SELECT C.customer_name, O.order_amt, D.qty
FROM Customers AS C
LEFT OUTER JOIN Orders AS O
ON C.customer_nbr = O.customer_nbr
LEFT OUTER JOIN OrderDetails AS D
ON D.order_nbr = O.order_nbr
AND D.sku = 101;
I added a couple extra styling elements (compared to the original query), can you catch them?
Another form of alignment is this:
Listing 8
SELECT C.customer_name, O.order_amt, D.qty
FROM Customers AS C
LEFT OUTER JOIN Orders AS O
ON C.customer_nbr = O.customer_nbr
LEFT OUTER JOIN OrderDetails AS D
ON D.order_nbr = O.order_nbr
AND D.sku = 101;
There are many different ways to style your SQL. You may agree or disagree with some elements, but the bottom line is this: style matters!
https://www.tangrainc.com/wp-content/uploads/2017/02/tangra-logo3.png 0 0 Plamen Ratchev https://www.tangrainc.com/wp-content/uploads/2017/02/tangra-logo3.png Plamen Ratchev2010-08-05 17:02:00
2017-02-11 07:33:14It’s a Matter of Style | {"url":"https://www.tangrainc.com/blog/2010/08/","timestamp":"2024-11-10T01:01:23Z","content_type":"text/html","content_length":"70945","record_id":"<urn:uuid:b1cc50a2-00bc-4f29-929a-27e131a69003>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00784.warc.gz"} |
If you’re teaching a class, you can think about the elementary things that you know very well. These things are kind of fun and delightful. It doesn’t do any harm to think them over again. Is
there a better way to present them? Are there any new problems associated with them? Are there any new thoughts you can make about them? The elementary things are easy to think about; if you
can’t think of a new thought, no harm done; what you thought about it before is good enough for the class. If you do think of something new, you’re rather pleased that you have a new way of
looking at it… The questions of the students are often the source of new research. They often ask profound questions that I’ve thought about at times and then given up on, so to speak, for a
while. It wouldn’t do any harm to think about them again and see if I can go any further now. The students may not be able to see the thing I want to answer, or the subtleties I want to think
about, but they remind me of a problem by asking questions in the neighborhood of that problem… So I find that teaching and the students keep life going, and I would never accept any position in
which somebody has invented a happy situation for me where I don’t have to teach. Never.
-- Richard Feynman, Surely You're Joking, Mr. Feynman!
A list of courses I am currently teaching / have taught in the past as the instructor of record:
Here are some (free) resources from which I’ve benefited greatly over the years, both as a teacher and a student:
Macro, Financial, and Monetary Economics
Time Series Econometrics/Analysis
Other Econometrics | {"url":"https://giorginikolaishvili.com/teaching/","timestamp":"2024-11-14T18:24:53Z","content_type":"text/html","content_length":"18687","record_id":"<urn:uuid:d0339509-8dc9-4942-928b-413ae22ba755>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00241.warc.gz"} |
inverse of cosine derivative
. Get detailed solutions to your math problems with our Inverse trigonometric functions differentiation step-by-step calculator. Aside from the very short period of time in your life when you are
taking the calculus. For instance, d d x ( tan. Next we compute the derivative of f(x) . The formulas for all the inverse trig derivatives follow immediately from this. Solution. ( x) = ( x), so that
the derivative we are seeking is d dx. To complete the list of derivatives of the inverse trig functions, I will show how to find d dx (arcsecx) . Lets call. d d x ( cosh 1 x) = lim x 0 cosh 1 ( x +
x) cosh 1 x x. The formula for the derivative of y= sin 1 xcan be obtained using the fact that the derivative of the inverse function y= f 1(x) is the reciprocal of the derivative x= f(y). Each of
the six basic trigonometric functions have corresponding inverse functions when appropriate restrictions are placed on the domain of the original functions. x, tan1 x tan 1. Next, we will ask
ourselves, "Where on the unit circle does the x-coordinate equal 1 . Inverse Trigonometric functions.Inverse Sine FunctionProperties of sin 1 x.Evaluating sin 1 x.Preparation for the method of
Trigonometric SubstitutionDerivative of sin 1 x.Inverse Cosine FunctionInverse Tangent FunctionGraphs of Restricted Tangent and tan 1x.Properties of tan 1x.Evaluating tan- 1 x Derivative of tan 1
x.Integration FormulasIntegration With inverse cosine, we select the angle on the top half of the unit circle. y = s i n 1 ( x) then we can apply f (x) = sin (x) to both sides to get: For instance,
suppose we wish to evaluate arccos (1/2). 13. If we use the chain rule in conjunction with the above derivative, we get d dx sin 1(k(x)) = k0(x) p 1 (k(x))2; x2Dom(k) and 1 k(x) 1: Example Find the
derivative d dx sin 1 p cosx. The Function y = cos -1 x = arccos x and its Graph: Since y = cos -1 x is the inverse of the function y = cos x, the function y = cos -1x if and only if cos y = x.
Notice that you really need only learn the left four, since the derivatives of the cosecant and cotangent functions are the negative "co-" versions of the derivatives of secant and tangent. . My
Notebook, the Symbolab way. ( x) = cos. Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation . Notice also that the derivatives
of all trig functions beginning with "c" have negatives. Let the function of the form be y = f ( x) = cos - 1 x By the definition of inverse trigonometric function, y = cos - 1 x can be written as
cos y = x 2. Derivative of the inverse cosine Find the derivative of the inverse cosine using Theorem 7.3. The derivative of y = arcsec x. But, since y = cos x is not one-to-one, its domain must be
restricted in order that y = cos -1 x is a function. The inverse sine function is one of the inverse trigonometric functions which determines the inverse of the sine function and is denoted as sin-1
or Arcsine. (25.3) The expression sec tan1(x . You da real mvps! inverse sine of X is equal to one over the square root of one minus X squared, so let me just make that very clear. Derivative of
Inverse Trigonometric functions The Inverse Trigonometric functions are also called as arcus functions, cyclometric functions or anti-trigonometric functions. Derivative of Inverse Hyperbolic Cosine.
How do you find the inverse of cosine? Subsection2.12.1 Derivatives of Inverse Trig Functions. = 1 f' (xo)'. Likewise, what's the derivative of tan 1? image/svg+xml. 19. Thanks to all of you who
support me on Patreon. Thus cos-1 (-) = 120 or cos-1 (-) = 2/3. Now, we will determine the derivative of inverse cosine function using some trigonometric formulas and identities. We have found the
angle whose sine is 0.2588. Assume y = cos -1 x cos y = x. Differentiate both sides of the equation cos y = x with respect to x using the chain rule. If xo is a point of I at which f' (xo) 0, then f
is differentiable at yo= f (x) and (f)' (yo) where yo= f (x). Without this restriction arccos would be multivalued. To build our inverse hyperbolic functions, we need to know how to find the inverse
of a function in general, so let's review. EXPECTED SKILLS: Know how to compute the derivatives of exponential functions. Let the differential element x is denoted by h for our convenience, then the
whole mathematical expression can be . We will use Equation 3.7.4 and begin by finding f (x). Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/
differential-calculus/taking-derivatives/derivatives-inverse-fun. . A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of
length then applying the Pythagorean theorem and definitions of the trigonometric ratios. To determine the derivative of inverse cosine function, we will be using some trigonometric identities and
formulas. Solution: For finding derivative of of Inverse Trigonometric Function using Implicit differentiation. Example 1 If x = sin -10.2588 then by using the calculator, x = 15. What you've done is
a bit like saying x = -x because (x) = (-x) x and sec1x sec 1. By definition, the trigonometric functions are periodic, and so they cannot be one-to-one. Functions. d dx (sinhx) = coshx d dx (coshx)
=sinhx d dx (tanhx) = sech2x d dx (cothx) = csch2x d dx (sechx) = sech x tanh x d dx (cschx) = csch x coth x d d x ( sinh. Be able to compute the derivatives of the inverse trigonometric functions,
speci cally, sin 1 x, cos 1x, tan xand sec 1 x.
The derivative of y = arccos x. DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS. for. 22 Derivative of inverse function 22.1 Statement Any time we have a function f, it makes sense to form is inverse
function f 1 . When memorizing these, remember that the functions starting with " c " are negative, and the functions with tan and cot don't have a square root. For the rest we can either use the
definition of the hyperbolic function and/or the quotient rule. Finding derivative of Inverse trigonometric functions. In this chapter, you will learn about the nature of inverse trigonometric
functions and their derivatives and use this knowledge to solve questions. Use the inverse function theorem to find the derivative of g(x) = x + 2 x. Thus cos-1 (-) = 120 or cos-1 (-) = 2/3. If you
were to take the derivative with respect to X of both sides of this, you get dy,dx is equal to this on the right-hand side.
Solution The inverse of g(x) = x + 2 x is f(x) = 2 x 1. To find the derivative of y = arcsecx, we will first rewrite this equation in terms of its inverse form. In fact, the derivative of \(f^{-1}\)
is the reciprocal of the derivative of \(f\), with argument and value . d d x sin. With inverse cosine, we select the angle on the top half of the unit circle.
So for y = cosh ( x) y=\cosh { (x)} y = cosh ( x), the inverse function would be x = cosh . Use the inverse function theorem to find the derivative of g(x) = x + 2 x. 8.2 Differentiating Inverse
inverse \cos(x) en. Inverse Trig Functions. Chart Maker; Games; Math Worksheets; Learn to code with Penjee; Toggle navigation. We learned about the Inverse Trig Functions here, and it turns out that
the derivatives of them are not trig expressions, but algebraic. :) https://www.patreon.com/patrickjmt !! Here you will learn differentiation of cos inverse x or arccos x by using chain rule. d d x =
1 cos Inverse Hyperbolic Trig Functions . We can get the derivatives of the other four trig functions by applying the quotient rule to sine and cosine. Finding the Derivative of Inverse Sine
Function, d d x ( arcsin x) Suppose arcsin x = . Here are all six derivatives. . Inverse Cosine Function We can de ne the function cos 1 x= arccos(x) similarly. Let's begin - Differentiation of cos
inverse x or \(cos^{-1}x\) : Derivatives of Inverse Trig Functions Using the formula for calculating the derivative of inverse functions (f1) = 1 f(f1) we have shown that d dx (arcsinx) = 1 1 x2 and
d dx (arctanx) = 1 1 + x2 . Now, differentiate both sides of the equation cos y = x with respect to x using the chain rule cos y = x d (cos y)/dx = dx/dx -sin y dy/dx = 1 dy/dx = -1/sin y ---- (1)
Now that we have explored the arcsine function we are ready to find its derivative. To find the inverse of a function, we reverse the x x x and the y y y in the function. The derivative of an inverse
function at a point, is equal to the reciprocal of the derivative of the original function at its correlate. Example: y = cos-1 x. x, cos1 x cos 1. Thus, f (x) = 2 (x 1)2 and d dx ( arcsin ( 4x2))
So, the derivative of the inverse cosine is nearly identical to the derivative of the inverse sine. That is, secy = x As before, let y be considered an acute angle in a right triangle with a secant
ratio of x 1. For example. The derivative of cos inverse is the negative of the derivative of sin inverse. Taking the derivative of arcsine. arccos() attempts to solve x for which cos(x) = 90 You can
approximate the inverse cosine with a polynomial as suggested by dan04, but a polynomial is a pretty bad approximation near -1 and 1 where the derivative of the inverse cosine goes to infinity To
compute fractions, enter expressions as numerator (over)denominator 1) Draw the function y . $1 per month helps!! In the following discussion and solutions the derivative of a function h ( x) will be
denoted by or h ' ( x) . Also remember that sometimes you see the . Solving for , we obtain. Or in Leibniz's notation: d x d y = 1 d y d x. which, although not useful in terms of calculation,
embodies the essence of the proof. Large equation database, equations available in LaTeX and MathML, PNG image, and MathType 5.0 format, scientific and mathematical constants database, physical
science SI units database, interactive unit conversions, especially for students and teachers The derivative of the inverse tangent is then, ddx(tan1x)=11+x2. Let y = f (y) = sin x, then its inverse
is y = sin-1x. Answer (1 of 4): Remember the inverse function theorem: if f is a function and f(x) = y, then (f^{-1})'(y) = \frac{1}{f'(x)}. The Cosine function is a periodic function that we will
represent as Cos 1. The tangent lines of a function and its inverse are related; so, too, are the derivatives of these functions. But with a restricted domain, we can make each one one-to-one and
define an inverse function. . 3. We may also derive the formula for the derivative of the inverse by first recalling that . . Related Symbolab blog posts. They are also termed as arcus functions,
antitrigonometric functions or cyclometric functions. Table of derivatives for hyperbolic functions, i 1 - Page 11 1 including Thomas' Calculus 13th Edition The derivative is an important tool in
calculus that represents an infinitesimal change in a function with respect to one of its variables For the most part, we disregard these, and deal only with functions whose inverses are also .
Working with derivatives of inverse trig functions. Derivatives of Inverse Trigonometric Functions. Each pair is the same EXCEPT for a negative sign. And if we recall from our study of precalculus,
we can use inverse trig functions to simplify expressions or solve equations. For example, the sine function is the inverse function for Then the derivative of is given by Using this technique, we
can find the derivatives of the other inverse trigonometric functions: The Inverse Trigonometric Functions. Trigonometric functions of inverse trigonometric functions are tabulated below. arccos (x)
is the command for inverse cosine; arcsin (x) is the command for inverse sine; arctan (x) is the command for inverse tangent; arcsec (x) is the command for inverse secant; arccsc (s) is the command
for inverse . y = f ( x) = cosh - 1 x. Derivative of cos-1 x (Cos inverse x) You are here Example 26 Important Example 27 Derivative of cot-1 x (cot inverse x) Derivative of sec-1 x (Sec inverse x)
Derivative of cosec-1 x (Cosec inverse x) Ex 5.3, 14 Ex 5.3, 9 Important Ex 5.3, 13 Important Ex 5.3, 12 Important Ex 5.3, 11 . Derivative of cos inverse x gives the rate of change of the inverse
trigonometric function arccos x and is given by d (cos -1 x)/dx = -1/ (1 - x 2 ), where -1 < x < 1. . Inverse trigonometric functions are simply defined as the inverse functions of the basic
trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. The only difference is the negative sign. . What is the derivative of inverse trig functions?
Question: 105. x. . arc for , except. The derivatives of the above-mentioned inverse trigonometric functions follow from trigonometry identities, implicit . By denition of an inverse function, we
want a function that satises the condition x = sechy = 2 ey +ey by denition of sechy = 2 ey +ey ey ey = 2ey e2y +1. Inverse Tangent Here is the definition of the inverse tangent. Finding the
derivatives of the main inverse trig functions (sine, cosine, tangent) is pretty much the same, but we'll work through them all here just for drill. Cos 1 degrees = cos (1 + n 360), n Z. Here you
will learn differentiation of cos inverse x or arccos x by using chain rule. This calculus video tutorial shows you how to find the derivatives if inverse trigonometric functions such as inverse sin^
-1 2x, tan^-1 (x/2) cos^-1 (x^2) ta. The derivative of y = arccsc x. I T IS NOT NECESSARY to memorize the derivatives of this Lesson. To start solving firstly we have to take the derivative x in both
the sides, the derivative of cos(y) w.r.t x is -sin(y)y'. The function f(x) = sinxwith domain reduced to The first good news is that even though there is no general way to compute the value of the
inverse to a function at a given argument, there is a simple formula for the derivative of the inverse of \(f\) in terms of the derivative of \(f\) itself.. Know how to apply logarithmic di
erentiation to compute the derivatives of functions . . Let us assume that y = cos -1 x cos y = x. How do you find the inverse of cosine? The inverse of g is denoted by 'g -1'. Inverse trigonometric
functions have various application in engineering, geometry, navigation etc. Inverse Trigonometric Func. This concept is taught under the chapter Derivative of Inverse Trigonometric Functions. Thus
sinh1 x =ln(x+ x2 +1). 3. The inverse of g(x) = x + 2 x is f(x) = 2 x 1. In order to use the inverse trigonometric functions you must place arc before the 3 letter symbol for each. These functions
are used to obtain angle for a given trigonometric value. In this tutorial we shall discuss the derivative of the inverse hyperbolic cosine function with an example. Since g (x) = 1 f (g(x)), begin
by finding f (x). Be able to compute the derivatives of the inverse trigonometric functions, specifically, sin1 x sin 1. dxd (arcsin(x 1)) 2. . These derivatives can be derived by applying the rules
for the derivatives of inverse functions. The derivative of y = arctan x. The weightage of this chapter is four . The corresponding inverse functions are. Let's begin - Differentiation of cos inverse
x or \(cos^{-1}x\) : CALCULUS TRIGONOMETRIC DERIVATIVES AND INTEGRALS STRATEGY FOR EVALUATING R sinm(x)cosn(x)dx (a) If the power n of cosine is odd (n =2k +1), save one cosine factor and use cos2(x)
=1sin2(x)to express the rest of the factors in terms of sine: The derivative of the inverse cosine function is for the inverse cosine of a single variable raised to an exponent equal to one, or for
any inverse cosine of a function . By the definition of the inverse trigonometric function, y = cosh - 1 x can be written as. y = tan1x tany = x for 2 <y < 2 y = tan 1 x tan y = x for 2 < y < 2
However, for people in different disciplines to be able to use these inverse functions consistently, we need to agree on a . In the case of the third pair, and , the denominators contain an absolute
value term, , which is important. These inverse functions in trigonometry are used to get the angle with any of the trigonometry ratios. In addition, these functions are continuous at every point in
their domains. Note: arccos refers to "arc cosine", or the radian measure of the arc on a circle corresponding to a . More Practice. Then it must be the cases that sin = x Implicitly differentiating
the above with respect to x yields ( cos ) d d x = 1 Dividing both sides by cos immediately leads to a formula for the derivative. | {"url":"http://spagades.com/military/lian/91938388c47a57-inverse-of-cosine-derivative","timestamp":"2024-11-10T06:09:29Z","content_type":"text/html","content_length":"25314","record_id":"<urn:uuid:ca6d5fdc-c743-4e6c-b066-e8162a4b88bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00647.warc.gz"} |
perplexus.info :: Numbers : Seven positive numbers
Certainly, the numbers can all be the same:1,1,1,1,1,1,1
There can be two distinct numbers: 1,1,1,1,1,1,5 (or multiply all by any n)
I cannot find a set with 3 distinct numbers. It seems there will alway be too many different sums too close together for the largest number to divide all of them.
Consider a,...,b,...,c with some number of a's and b's and a largest c.
The sets that don't contain c will have something like 3a+2b and 2a+3b. But c can't divide both sums because they only differ by a-b.
Adding a second c doesn't help.
Posted by Jer on 2019-04-23 16:00:59 | {"url":"http://perplexus.info/show.php?pid=11696&cid=60946","timestamp":"2024-11-03T23:32:36Z","content_type":"text/html","content_length":"12621","record_id":"<urn:uuid:53f4f0dd-c294-4d68-9eaf-359a47b67da4>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00685.warc.gz"} |