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Novel cost-effective method for predicting COVID-19 and hospitalization using deep learningNovel Cost-effective Method For Predicting COVID-19 And Hospitalization Using Deep Learning » Bravo Bulletin
Novel cost-effective method for predicting COVID-19 and hospitalization using deep learning
In the field of professional science, our study focuses on the use of a regression model to predict COVID-19 trends using data from the Hospital Insular de Gran Canaria (Spain). This dataset spans
from the beginning of 2020 to March 29, 2022, and includes only two inputs in the simplest case: date and daily new COVID-19 cases. Despite the simplicity of this dataset, our analysis has
demonstrated the extraordinary ability of the model to accurately predict future COVID-19 trends, identifying temporal patterns, seasonality, and the impact of interventions. This work underlines the
value of accessible data and shows how even minimal data input can yield profound insights, revolutionizing the landscape of professional research and analysis in science. As mentioned above, the
database is owned by the Government of the Canary Islands (Spain) and the data is public.^11It can be accessed or downloaded from https://opendata.sitcan.es/dataset/capacidad-asistential-covid-19.
performance index
A set of statistical parameters have been used to evaluate the accuracy of the model. The selection of these parameters is based on their widespread use in the literature, which allows us to compare
our results with the current state of the art. are the most prominent RMSE, MAE, MAPE, And R^2Which measure the accuracy of measurements, as well as their dispersion and correlation. Their
mathematical expressions are shown in the following equations, where \({y}_{i}\) The observed values are, \(\widehat{{y }_{i}}\) estimated value and \(\overline{y }\) The mean of these values
mean square error (MSE,
$$MSE=\frac{1} ^{2}$$
root mean square error (rmse,
$$RMSE=\sqrt{\frac{1} ight)}^{2}}$$
mean average error (mae,
mean square error (map,
coefficient of determination (R^2,
data preprocessing
To perform data preprocessing and labeling, the “new daily cases” variable was separated into one vector and the date variable into another vector. Then, a labeling window with different values was
used to assign a label to the “new daily cases” values. This label specifies the value of “Ytrain”. These “Ytrain” values depend on the size of the window. Thus, for a window n = 2, the “new daily
cases” of dates n = 1 and n = 2 will be grouped in the first row of the “Xtrain” vector and their “Ytrain” values will be those of the later date. , i.e., n + 1. Next, considering a step = 1, dates
n = 2 and n = 3 will be grouped in the second row, and the value of “ytrain” will be n = 4.
This study was conducted with values ranging from n = 1 to n = 20 to test which window was better suited to the data and which window could better handle the higher slope presented by COVID-19
waves Is. Figure 4 shows a scheme of the starting vectors “date” and “new daily cases” and the labeling process for a window n = 2 and n = 5. The dataset is available from the link given in the
previous section.
picture 4
Planning of preprocessing and targeting process.
network architecture
To accurately predict COVID-19 data, an architecture has been designed that is capable of analyzing time series and capturing the existing gradient differences in slopes generated by different waves
through the use of deep learning. The various layers used in the overall architecture are described in detail below.
Long-short term memory is a type of recurrent neural network (RNN) that is particularly useful for modeling sequential data. These types of algorithms have been applied to a wide variety of tasks,
including speech recognition, natural language processing, and time series forecasting.^16By using memory cells, LSTMs can retain useful data from the current or previous steps and use it in the
future. Therefore, they use algorithmic gates that are also capable of retaining such information for future use and goal achievement.
These can also be combined to improve the overall network architecture. There are variants with different functions, such as bidirectional LSTM (BiLSTM), gate recurrent units (GRU), or new algorithms
focused on the attention layer, called “transformers”, described by Vaswani et al. In 2017 his work was titled “Attention is All You Need”^17In the case of BiLSTM, the only difference is the
relationship between states, as they are bidirectional and can take into account data from the previous state as well as the next state.
LSTM consists of a memory cell and three parts, which can be expressed mathematically as follows^18,
Input Gate: The layer responsible for updating the state of the network through the sigmoidal function.
$$i_{t} = \sigma \left( {W_{i} \cdot\left[ {h_{t – 1} ,x_{t} } \right] + b_{i}} \right)$$
\({W}_{i}\) is the representation of the weights of the input, \({b}_{i}\) The corresponding bias is, \({x}_{t}\) is the current time step, and \({h}_{t-1}\) is the output of the previous time step.
σ will have a value[0, 1]represents complete discard or complete saving of data respectively^16,18,
Forget the gate: The layer that is responsible for deciding whether to save or discard information. This is the first stage of LSTM.
$$f_{t} = \sigma \left( {W_{f} \cdot\left[ {h_{t – 1} ,x_{t} } \right] + b_{f}} \right)$$
\({W}_{f}\) The weight representation of the input is, \({b}_{f}\) The corresponding bias is, \({x}_{t}\) current time step and \({h}_{t-1}\) is the output of the previous time step.
Output Gate: This is where the information output is determined. This output is based on a filtered version of the cell state. The output value is determined by the sigmoid layer and then multiplied
by the cell position^18,
$$o_{t} = \sigma \left( {W_{o} \cdot\left[ {h_{t – 1} ,x_{t} } \right] + b_{o}} \right)$$
$$h_{t} = o_{t} \cdot \tan \,h(C_{t} )$$
\({W}_{o}\) The weight representation of the input is, \({b}_{o}\) have consistent bias, and \({x}_{t}\) It is a step towards the present time. \({h}_{t-1}\) is the output of the LSTM layer at the
current time step. Finally, the previous cell state \({C}_{t-1}\) Must be updated. It is calculated by forgetting one input gate, as shown in Figure 5.
$$C_{t} = f_{t} \cdot C_{t – 1} + i_{t} \cdot g_{t}$$
Where? \({g}_{t}\) Body layer.
Figure 5
Representation of the components of an LSTM cell.
Furthermore, the BiLSTM model is composed of two LSTM networks and is capable of reading input evaluations in both forward and backward directions. Forward LSTM processes information from left to
right, while backward LSTM processes information from right to left.^19,
dense layer
A dense or fully connected layer, also known as a fully connected feedforward neural network, is a type of artificial neural network in which every neuron in one layer is connected to every neuron in
the next layer. The basic formula for a fully connected neural network with one hidden layer and one output layer (\({y}_{fc}\)) can be represented as^20,
$$y_{fc} = f\left( {mathop \sum \limits_{i = 1}^{n} \left( {W_{i} *x_{i} } \right) + b} \right) $ $
Where? \({x}_{i}\) is the input vector to the network and \({W}_{i}\) There are weight matrices for connections between layers. \(B\) have prejudice and \(F\)is the activation function applied to the
output of each layer (sigmoid, ReLU, tanh).
It is important to note that this formula is for a single hidden layer neural network, but in practice, fully connected neural networks usually have multiple hidden layers, in which case the formula
will be more complex and will involve an additional weight matrix. And each additional layer will contain bias vectors.
drop out
Dropout is a regularization technique used in deep learning to avoid overfitting. It works by randomly “dropping” (i.e., setting to zero) a certain number of neurons during each training iteration.
The mechanism of the dropout layer is quite simple: it is applied to the output of the previous layer and involves multiplying the input vector by a mask. This mask is a binary mask that is randomly
generated for each training iteration, its size is the same as the input and each element is either 0 or 1. The probability that each element of the mask is 1 is called the dropout rate. The dropout
rate is a hyperparameter that is typically set between 0.2 and 0.5 depending on the specific application and complexity of the model. Typically, a lower dropout rate is used for the input layer and a
higher dropout rate is used for the hidden layers. During the testing phase, it is common to use a dropout rate of 0, which means that all neurons are active. This is because dropout is only applied
during the training phase and is not used during the testing phase.^21,
The network was trained and tested using Python's TensorFlow. Adaptive moment estimation or Adam method, which is a widely used optimization algorithm for neural network training, is used. ADAM
combines the techniques of RMSprop and Momentum Optimizer to efficiently and effectively adjust the weights of the neural network during training. See the equation. (12)–(14) down^22,23,
$$m_{t} = \beta_{1} m_{t – 1} + \left( {1 – \beta_{1} } \right)g_{t}$$
$$v_{t} = \beta_{2} v_{t – 1} + \left( {1 – \beta_{2} } \right)g_{t}^{2}$$
$$\theta_{t} = \theta_{t – 1} – \frac{\alpha }{{\sqrt {v_{t} } + \epsilon }}m_{t}$$
Where? \({m}_{t}\) The first update moment (mean) is, \({v}_{t}\) The second update moment (variance) is, \({\beta }_{1}\) And \({\beta }_{2}\) are the moment decay parameters, \({g}_{t}\) is the
gradient in the current step, α is the learning rate, “epsilon” (ϵ) is a small numerical constant to avoid division by zero, and \({\theta }_{t}\) is the current value of the parameter being updated.
This is the parameter that the algorithm optimizes.
The value of these hyperparameters was 1·10^-6 For “epsilon” in training options and 1·10^-4 For learning rate. Batch sizes were set to 5 and 15, with epochs equal to 1000 and a “shuffle” at each
epoch. The value of \({\beta }_{1}\) was set to 0.99, and \({\beta }_{2}\) to 0.999. Training and testing is done using the holdout method for regression with a training percentage of 40-60.
overall network architecture
The architecture developed in this work consists of sequential inputs of previously defined temporal windows. This sequence passes through 3 levels. In the first one, there is an LSTM layer whose
hidden layer has 128 units and sequence return is enabled. Then, in levels 2 and 3, there are 2 BiLSTM layers with sequence return enabled and 128 units each. Finally, at the output of this last
level, a dense fully connected layer with 128 connections is applied. Then, to reduce the randomness of the weights, a dropout layer with a value of 0.4 and a flatten layer are added to flatten the
output sequence into a vector.^24Finally, a dense layer with one neuron and linear activation is added to obtain the output. In Figure 6, a plan of the entire architecture is shown.
Figure 6
Implemented network planning.
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principle of equilibrium physics
So if the body is in equilibrium but continues to move with the uniform velocity it is known as dynamic equilibrium. Again, we can extend this to moments about the y-axis as well. For most students,
the resultant was 0 Newton (or at least very close to 0 N). For example, a car moving along a highway at a constant speed is in equilibrium, as it is not accelerating in any forward or vertical
direction. For a single particle, equilibrium arises if the vector sum of all forces acting upon the particle is zero. The condition [latex]\text{F}_\text{net} = 0[/latex] must be true for both
static equilibrium, where the object’s velocity is zero, and dynamic equilibrium, where the object is moving at a constant velocity. A simple mechanical body is said to be in equilibrium if it
experiences neither linear acceleration nor angular acceleration; unless it is disturbed by an outside force, it will continue in that condition indefinitely. so objects with constant velocity also
have zero net external force. Consider an object moving along the x-axis. The principle is that as the angle with the horizontal increases, the amount of tensional force required to hold the sign at
equilibrium decreases. While there might be motion, such motion is constant. If the object is spinning, it will continue to spin at the same constant angular velocity. We could say it's "close enough
for government work.". Prigogine's Min. If the object is at equilibrium, then the net force acting upon the object should be 0 Newton. Equilibrium in physics refers to the condition of the system
when neither of its state of motion nor its internal energy state changes with the time. Newton’s second law states that: [latex]\sum \textbf{F}=\text{m}\textbf{a}[/latex]. Geometric Optics. A simple
mechanical body is said to be in equilibrium if it experiences neither linear acceleration nor angular acceleration; unless it is disturbed by an outside force, it will continue in that condition
indefinitely. in motion and continuing in motion with the same speed and direction. The data in the table above show that the forces nearly balance. What is Set, Types of Sets and Their Symbols?
Unless it is disturbed by an external force, it will continue in that particular condition indefinitely. Sorry!, This page is not available for now to bookmark. Rotational Equilibrium (Principle Of
Moments) Rotational equilibrium is obtained when the algebraic sum of the torques is zero. If the any two of these three are known, then the third quantity can be determined using trigonometric
functions. Author(s): Prof. Dr. Debashish Chowdhury; Prof. Dr. Dietrich Stauffer; ... solid state physics, and astrophysics. ��D@ޝ�x0y��L) W�|�H㡺̎Rb�Mbb���]��$��>���k�[ϴ���c�v#
���J����A_"�,qY�.�$P�@A��ЋBq�ˑlF�*� �3���K�e��Yno�Q����q�.����.�E��g�n+i��$ɪ���C}��Y����K�LG�(�Ҙ��� �($V�0�cZ�����p��prYd�X���\�8����>���u�|�J o��O�Ø��'�ҡ�n���� J����kʠ�eC�m����>yT If the cables
make a 1-degree angle with the horizontal, then what is the tension in the cable? The condition [latex]\text{F}_\text{net} = 0[/latex] must be true for both static equilibrium, where the object’s
velocity is zero, and dynamic equilibrium, where the object is moving at a constant velocity. Use this information and the diagram below to determine the tension in the wire for each orientation. If
the sign is known to have a mass of 5 kg and if the angle between the two cables is 100 degrees, then the tension in the cable can be determined. FE3 EQUILIBRIUM OBJECTIVES Aims In this chapter you
will learn the concepts and principles needed to understand mechanical equilibrium. A diagram and accompanying work is shown below. In each case, two wires are used to support the picture; each wire
must support one-half of the sign's weight (5 N). Principle of Moments The principle of moments states that when in equilibrium the total sum of the anti clockwise moment is equal to the total sum of
the clockwise moment. Powered by Create your own unique website with customizable templates. The above analysis of the forces acting upon an object in equilibrium is commonly used to analyze
situations involving objects at static equilibrium. Which Is Always True For A Body In Equilibrium? On a body at equilibrium there is no resultant force and also there is no moment i.e. It is defined
as a single force which when applied with given forces brings the body in equilibrium. (wTq����I���/tG��m�骣l��[�� ?���N(���o�g�njj�`�nK��^0������_ӻ������pw|�6T�C�1��^��ܲ�1m�b�ޝ #&�L���%�� ���w����� ��
\ l Adjust the position of the pivot clamp on the meter stick until the meter stick is balanced and level. This too extends from Newton's first law of motion. The following picture is hanging on a
wall. There is an important principle that emanates from some of the trigonometric calculations performed above. The first condition of equilibrium is that the net force in all directions must be
zero. When all the forces that act upon an object are balanced, then the object is said to be in a state of. At 60 degrees, the tension is 5.8 N. (5 N / sin 60 degrees). Mathematically, this equation
is stated as follows: Stability refers to the state of the rest of the body while the equilibrium is known as the state of the balance of a body. An equilibrium is referred to as stable whenever the
small and the externally induced displacements from which the state produce forces tend to oppose the displacement and returns the body or the particle to its state of equilibrium. Let us learn about
the equilibrium definition physics. This means that both the net force and the net torque that is acting on the object should be zero. For example, the net external forces along the typical x– and
y-axes are zero. ��s���"�Oe�*�Ȑv]��ޗ{�>�\o���8�")ӹv�q ��S��j����y��j��T̼d��}��!�0$c��P^,`H^`�ڮ���E��mȘ�u�㾅а��ȡN8T1�u���Ox`��&�i�F�G��;�H���!ˠ�2�(m�Hh�Fm¡.����h���qLeA;jȄAE��=m�Ն.gI*�D���qAȝ6�ڒ��DPPn
~Ks�#�ϺO:�VK�ٵS�a���o�Q�ܦ1T�p��V�PD�Sƌ��$��g-�z�Df������f Lami's Theorem: Principle of Moments: Centre of Mass and Centre of Gravity: Mechanics. Both forces are vertical in this case. The diagram
below shows vectors A, B, and C and their respective components. The following sign can be found in Glenview. If the object is not spinning, it will not start to spin. Equilibrium is classified also
as stable, unstable and neutral. Let us see what all these terms mean. There are two conditions that must be met for an object to be in equilibrium. Thus. Principle of Moments. A body in equilibrium
has no resultant and no couple acting on it. Calculate the net force and the net torque for an object in equilibrium. Knowing the forces acting upon an object, trigonometric functions can be utilized
to determine the horizontal and vertical components of each force. �1�C�4�����N��q"Mnf�����_���:��j7� ϶��5�(6�� �z D8A��Ժ/�&��@���. For example, the net external forces that are acting along the
typical x– and y-axes are zero. xڥ\Ks�F��ׯ�=�!a���9�==���my�=@$Ԅ 2�Z��7 )�*)d;V@��̬|g��?�? OpenStax College, College Physics. A system is said to be in the stable state of equilibrium when it is
displaced from equilibrium, it experiences the net force or the torque in such a direction that is opposite to the direction of the displacement. The triangle below illustrates these relationships. A
motionless object still has constant (zero) velocity, so motionless objects also have zero acceleration. The second condition necessary to achieve equilibrium involves avoiding accelerated rotation
(maintaining a constant angular velocity ). In each direction, the net force takes the form: [latex]\sum \textbf{F}=\text{m}\textbf{a}=0[/latex] and the net torque take the form: [latex]\sum \
boldsymbol{\tau}=\text{I}\boldsymbol{\alpha}=0[/latex] where the sum represents the vector sum of all forces and torques acting. This rule also applies to motion in a specific direction. *��p��
����3ʝ�)�]�����I��$1!`��'-Ӛ�ׄ'��E� .e,�w��#�� �[嬥����U�x�v��nOQ,/��:@���������q�s��r��s���M��W�90�BNL��J2�~n��gM�q#0ԧ�i��dJ+̹�C In thermodynamics the concept of equilibrium is extended to include
possible changes in the internal state of a system, as characterized by its temperature, pressure, density, and any other quantities needed to specify its state completely. These definitions
postulate the … September 17, 2013. Physics 1020 Experiment 6 Equilibrium of a Rigid Body Finding the Center of Gravity l Slide the metal clamp on to the meter stick near the middle. | {"url":"https://www.eatodhk.com/zwvdjarj/6yeef.php?301c0d=principle-of-equilibrium-physics","timestamp":"2024-11-02T02:12:12Z","content_type":"text/html","content_length":"18490","record_id":"<urn:uuid:d24052c9-97be-42b8-88b1-41c615d672ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00243.warc.gz"} |
On the number of maximal independent sets: From Moon–Moser to Hujter–Tuza
We connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number (Figure presented.) of maximal independent sets in an (Figure
presented.) -vertex graph was determined by Miller and Muller and independently by Moon and Moser. The maximum number (Figure presented.) of maximal independent sets in an (Figure presented.) -vertex
triangle-free graph was determined by Hujter and Tuza. We give a common generalization of these results by determining the maximum number (Figure presented.) of maximal independent sets in an (Figure
presented.) -vertex graph containing no induced triangle matching of size (Figure presented.). This also improves a stability result of Kahn and Park on (Figure presented.). Our second result is a
new (short) proof of a second stability result of Kahn and Park on the maximum number (Figure presented.) of maximal independent sets in (Figure presented.) -vertex triangle-free graphs containing no
induced matching of size (Figure presented.).
• extremal graph theory
• induced (triangle) matchings
• maximal independent sets
• stability results
Dive into the research topics of 'On the number of maximal independent sets: From Moon–Moser to Hujter–Tuza'. Together they form a unique fingerprint. | {"url":"https://umimpact.umt.edu/en/publications/on-the-number-of-maximal-independent-sets-from-moonmoser-to-hujte","timestamp":"2024-11-14T07:29:30Z","content_type":"text/html","content_length":"45842","record_id":"<urn:uuid:255390ec-4abf-4e33-af5c-3ea6f7ddf132>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00119.warc.gz"} |
3 X 2 Multiplication Worksheet
Math, particularly multiplication, forms the cornerstone of countless academic self-controls and real-world applications. Yet, for numerous learners, mastering multiplication can position a
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3 X 2 Multiplication Worksheet
3 X 2 Multiplication Worksheet -
To complete a long multiplication follow these steps If you are multiplying a 3 digit number by a two digit number simplify the two digit multiplier for instance 24 by splitting it into 20 and 4 Then
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3x2 Multiplication Worksheets Times Tables Worksheets
3x2 Multiplication Worksheets Times Tables Worksheets
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Multiplying Three Digit By Two Digit 36 Per Page W
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Multiply 3 x 2 digits worksheets K5 Learning
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Multiplication Worksheets 3 Digits Times 2 Digits
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AP B Summer Assignment Combined
AP Physics I SummerAssignment
Welcome to AP Physics I! It is a college levelphysics course that is fun, interesting, and challenging on alevelyou’ve not yet experienced. This assignment will review allofthe prerequisite knowledge
expected of you.
There are6parts to this assignment. It is the quantity not the difficultyofthe problems that has the potential to overwhelm, so doitover an extended period of time. By taking the timeto review and
understand all parts of this assignment, youwill help yourself acclimate to the rigor and pacing of APPhysics. Use online resources if you need to, but really this is all stuffyou already know how to
do (basic math skills). It isVERY important that this assignment be completed individually. It will be a total waste of your time to copythe assignment from a friend. The summer assignment will be
due the first day of class. Good luck!
Part 1: Scientific Notation and DimensionalAnalysis
Many numbers in physics will be provided in scientific notation. You need to be able read andsimplifyscientific notation. (This section is to be completed withoutcalculators…all work should be done
byhand.) Get used to no calculator! All multiple choice portions of tests will be completed without acalculator.
Express the following the numbers in scientific notation. Keep the same unit as provided. ALL answersinphysics need their appropriate unit to becorrect.
1. 7,640,000 kg=
3. 0.000000003m =
2. 8327.2 s =
4. 0.0093km/s=
Often times multiple numbers in a problem contain scientific notation and will need to be reduced byhand. Before you practice this, remember the rules for exponents you learned inalgebra:
When numbers with exponents are multiplied together,youthe exponentsandthebases. When numbers are divided,you the exponentsand thebases.
When an exponent is raised to another exponent,youthe exponentsandthebase.
Using the three rules from above, simplify the following numbers in proper scientificnotation:
5. (3x106)·(2x104)=
7. (4x108)·(5x10-3)=
9. (8x103) / (2x105)=
6. (1.2x104) / (6x10-2)=
8. (7x103)2=
10. (2x10-3)3=
Fill in the power and the symbol for the following unit prefixes. Look them up as necessary. These shouldbe memorized for next year. Kilo- has been completed as anexample.
Prefix / Power / Symbol
Kilo- / 103 / k
Not only is it important to know what the prefixes mean, it is also vital that you can convert betweenmetricunits. If there is no prefix in front of a unit, it is the base unit which has 100 for its
power, or just simply“1”.Remember if there is an exponent on the original unit, the converted unit should be raised to the sameexponent.
Convert the following numbers into the specified unit. Use scientific notation whenappropriate.
1. 24 g =kg
2. 94.1 MHz=Hz
3. 6 Gb=kb
4. 640 nm=m
5. 3.2 m2=cm2
6. 40 mm3=m3
7. 1 g/cm3=kg/m3
8.20 m/s=km/hr
For the remaining scientific notation problems you may use your calculator. It is important that you knowhowto use your calculator for scientific notation. The easiest method is to use the “EE”
button. An exampleis included below to show you how to use the “EE”button.
Ex: 7.8x10-6 would be entered as 7.8 E-6 9. (3.67x103)(8.91x10-6)=
10. (5.32x10-2)(4.87x10-4)=
11. (9.2x106) / (3.6x1012)=
12.(6.12x10-3)3 =
Part 2:Geometry
Calculate the area of the following shapes. It may be necessary to break up the figure into commonshapes.
Calculate the unknown angle values for questions3-6.
ABmC D
EFnG H
Lines m and n areparallel.
A = 75°B=
θ1 =____ θ2 =______θ3 =______θ4=
Part 4: Trigonometry
Write the formulas for each one of the following trigonometric functions. RememberSOHCAHTOA!
Calculate the following unknowns using trigonometry. Use a calculator, but show all of your work.Pleaseinclude appropriate units with all answers. (Watch the unitprefixes!)
θ =30°
θ =60°
θ =17°
d =
θ =26°
d =
You will need to be familiar with trigonometric values for a few common angles. Memorizing this diagramindegrees or the chart below will be very beneficial for next year (in math and physics!). In
the diagram,thecosine of the angle is the x-coordinate and the sine of the angle is the y-coordinate (in other words, eachradius of the circle shown is the hypotenuse of a right triangle). Write the
ordered pair (in fraction form) in thetablebelow for each of the angles shown on thequarter-circle.
Refer to your completed chart to answer the followingquestions.
10.At what angle is sine at amaximum?
11.At what angle is sine at aminimum?
12.At what angle is cosine at aminimum?
13.At what angle is cosine at amaximum?
14.At what angle are the sine and cosineequivalent?
15.As the angle increases in the first quadrant, what happens to the cosine of theangle?
16.As the angle increases in the first quadrant, what happens to the sine of theangle?
Use the figure at right to answer problems 17 and18.
17.Find an expression for h in terms of l andθ.
18.What is the value of h if l = 6 m and θ =40°?
Part 5: Algebra
Solve the following (almost all of these are extremely easy – it is important for you to work independently). Units onthenumbers are included because they are essential to the concepts, however they
do not have any effect on theactualnumbers you are putting into the equations. In other words, the units do not change how you do the algebra. Showeverystep for every problem, including writing the
original equation, all algebraic manipulations, and substitution! Youshouldpractice doing all algebra before substituting numbers in forvariables.
Section I: For problems 1-5, use the three equationsbelow:
vf = vO +at
x = xo+vot+1/2at2
vf2 = vo2 + 2a(xf-xo)
1.Using equation (1) solve for t given that v0 = 5 m/s, vf = 25 m/s, and a = 10m/s2.
2.Given v0 = 0 m/s, x0 = 0 m and t = 10 s, use the second and third equations together to find xf.
3.a = 10 m/s2, x0 = 0 m, xf = 120 m, and v0 = 20 m/s. Use the second equation to findt.
4.vf = - v0 and a = 2 m/s2. Use the first equation to find t /2.
5.How does each equation simplify when a = 0 m/s2 and x0 = 0m?
Section II: For problems, 6 – 9, use the four equationsbelow.
ΣF =mafk = µkNfs ≤ µsN
Fs =−kx
6.If ΣF = 10 N and a = 1 m/s2, find m using the firstequation?
7.Given ΣF = fk , m = 250 kg, µk = 0.2, and N = 10m, find a?
8.ΣF = T – 10m, but a = 0 m/s2. Use the first equation to find m in terms ofT?
9.Given the following values, determine if the third equation is valid? (Given: ΣF = fs , m = 90 kg,anda = 2 m/s2. Also, µs= 0.1, and N = 5N).
10.Use the first equation in Section I, the first equation in Section II and the following givens and findΣF? (given: m = 12 kg, v0 = 15 m/s, vf = 5 m/s, and t = 12s)
11.Use the first equation in Section I, the first equation in Section II and the following givens to find ΣF? (given:
m = 12 kg, v0 = 15 m/s, vf = 5 m/s, and t = 12s)
12.Use the last equation to solve for Fs if k = 900 N/m and x = 0.15m?
Section III: For problems 12, 13, and 14 use the two equationsbelow.
13.Given that v is 5 m/s and r is 2 meters, finda?
14.Originally, a = 12 m/s2, then r is doubled. Find the new value fora?
15.Use the second equation to find θ when τ = 4 Nm, r = 2 m, and F = 10N?
Section IV: For problems 15 – 23, use the equationsbelow.
K = 1/2mv2
ΔUg = mgh
W = F(Δx)cosθ
P = W/t
15.Use the first equation to solve for K if m = 12 kg and v = 2m/s.
16.If ∆Ug = 10 J, m = 10 kg, and g = 9.8 m/s2, find h using the secondequation.
17.K = ∆Ug, g = 9.8 m/s2, and h = 10 m. Findv.
18.The third equation can be used to find W if you know that F is 10 N, ∆x is 12 m, and θ is180°.
19.Use the value for W you found in the previous question to find P if t = 2 s. Which equation do youneed?
20.Given Us = 12 joules, and x = 0.5 m, find k using the fourthequation.
21.For the same value of x as given in problem 20 and the k value you just found, use the last equation in SectionIIto findFs.
22.Assuming θ = 0° and F = Fs, use the third equation listed above along with the numbers found and given intheprevious two questions to findW.
23.For P = 2100 W, F = 30 N, and θ = 0°, find vavg using the last equation in thissection.
Section V: For problems 24 – 26, use the equationsbelow.
p =mv
J = FΔt =Δp
Δp =mΔv
24.p is 12 kgm/s and m is 25 kg. Find v using the firstequation.
25.“∆” means “final state minus initial state”. So, ∆v means vf – vi and ∆p means pf – pi. Find vf using thethirdequation if pf = 50 kgm/s, m = 12 kg, and vi and pi are bothzero.
26.Use the second and third equation together to find vi if vf = 0 m/s, m = 95 kg, F = 6000 N,and
∆t = 0.2s.
Section VI: For problems 27 – 29 use the three equationsbelow.
27.Tp is 1 second and g is 9.8 m/s2. Find l using the secondequation.
28.m = 8 kg and Ts = 0.75 s. Solve fork.
29.Given that Tp = T, g = 9.8 m/s2, and that l = 2 m, find f (the units for f areHertz).
Section VII: For problems 30 – 33, use the equationsbelow.
30. Find Fg if G = 6.67 × 10-11 m3 kg-1 s-2, M = 2.6 × 1023 kg, m = 1200 kg, and r = 2000 m?
31. What is r if Ug = -7200 J, G = 6.67 × 10-11 m3 kg-1 s-2, M = 2.6 × 1023 kg, and m = 1200 kg?
32.Use the first equation in Section IV for this problem. K = Ug, G = 6.67 × 10-11 m3 kg-1 s-2,and
M = 3.2 × 1023 kg. Findv?
33.Using the first equation above, describe how Fg changes if rdoubles?
Section VIII: For problems 34 – 38, use the equationsbelow.
34.If P0 = 100,000 Pa, ρ = 1.2 kg/m3, g =9.8 m/s2, and h = 75 m, calculate the value ofP?
35.If m doubles but V is halved, how does Fb change if g isconstant?
36.Using the first equation, third equation, and the first equation from Section II, determine the value of aif
ρ = 1000 kg/m3, V = 2 m3, and g = 9.8 m/s2. Assume Fb =ΣF?
37.If y is constant, how does P change if v is tripled (use the fifth equation,here)?
38.Find v2 if v1 =300 m/s and A2 equals 2.5A1?
Section IX: For problems 39 – 43, use the equationsbelow.
PV =nRT
Q =mc∆T
W =−P∆V
∆U = Q + W
39.What is T if V = 2x10-3 m3, n = 1 mol, R = 8.31 J/kg·K, and P = 7x106Pa?
40.Assuming n and R are both held constant, what happens to T if P is doubled and V istripled?
41.Calculate m if c = 4000 J/kg°C, Q = 6.2 kJ, and T = 12 °C. To do this correctly kJ needs to beconvertedinto units ofJ?
42.If U doubles and kB, R, and M remain the same values, how does vrmschange?
43.If ΔV is positive and ΔU is zero, what is the sign of Q? Justify your answer using the last twoequations?
Section X: For problems, 44 – 47, use the equationsbelow.
44.If v is constant, how does f change if λquadruples?
45.c is equal to 3x108 m/s. What is the value of n if v equals 2.25 x 108m/s?
46.If n2 is greater than n1, is θ1 greater than, less than, or equal to θ2? Justify your answer using thethirdequation?
47.Assuming θ2 is 90°, write an expression for θ1 in terms of n1 andn2.
Section XI: For problems 48-52, use the equationsbelow.
48.If si = -5 cm and s0 = 2 cm, calculate the value of f (units are cm forf)?
49.R is known to be -3.2 cm. Find si if s0 = 4cm?
50.What is the numerical value of M if s0 = 2f (M has nounits)?
51. What is θ if d = 8.5x10-4 m, m = 2, and λ = 6.3x10-7m?
52.Using the last two equations, calculate xm if θ is 1.2°, m is 1, λ is 400 nm, and L is 1.4 m. To solve this correctly,λ
should be converted from units of nm tom?
Section XII: For problems 53– 58, use the equationsbelow.
= qV
53.k is a constant and is always equal to 9.0 × 109 Nm2/C2. If q = 1.2 × 10-13 coulombs, Q = -q,and
F = -10 Newton’s then find r using the firstequation?
54.Another way of writing k is . Using k = 9.0 × 109 Nm2/C2, solve for Eo?
55.Find E using the fourth equation if V = 120 volts and d = 0.2meters?
56.Use the second and fourth equations together to find V if r = d, Q = 1.6 × 10-19 C andk
is 9.0 × 109 Nm2/C2. Can you find the fifth equation in your algebraicsteps?
57.If I have a UE of 12 joules and I double Q and q then what is my new value ofUE?
58.58. If F is 0.2 N, d = 2.0 × 10-4 m, and q is 8.0 × 10-19 C, findV?
Section XIII: For problems 59 – 64, use the equationsbelow.
QV =
59.If C is 12× 10-6 farads and V is 12 volts, find Q using the firstequation?
60.TherelationshipbetweenEOandkisdescribedinproblemnumber35.Usethatrelationshiptore-writethesecondequation listed in this section in terms of k instead ofEO?
61.EO is a constant and always equals 8.85× 10-12 C2/Nm2. If A = 0.3 m2 and d = 0.012 m, findC?
62. Given Q = 3.0 × 10-6 C, and C = 7× 10-6 F, find UC?
63.Use the fourth equation to find CP if C1 = 2× 10-6 F, C2 = 4× 10-6 F, and C3 = 6× 10-6F?
64.Use the fifth equation to find CS if C1 = 2× 10-6 F, C2 = 4× 10-6 F, and C3 = 6× 10-6F?
Section XIV: For problems 65 – 70 use the equationsbelow.
V =IR
65.Given V = 220 volts, and I = 0.2 amps, find R (the units are ohms,Ω)?
66.If ∆Q = 0.2 C, t = 1s, and R = 100 Ω, find V using the first twoequations?
67.R = 60 Ω and I = 0.1 A. Use these values to find P using the first and thirdequations?
68.Let RS = R. If R1 = 50 Ω and R2 = 25 Ω and I = 0.15 A, findV?
69.Let RP = R. If R1 = 50 Ω and R2 = 25 Ω and I = 0.15 A, findV?
70. Given R = 110 Ω, l = 1.0 m, and A = 22× 10-6 m2, findρ?
Section XV: For problems 71 – 75 use the equationsbelow.
E =Bℓv
71. Find v if q = -4.8 × 10-19 C, B = 3.0 Teslas, θ = 90°, and FB = -1.0 × 10-9 N?
72.µO is a constant and so is always equal to 4π × 10-7 (Tm)/A. If I = 0.2 A, r = 0.003 m, θ = 270° and ℓ = 0.15m,then findFB?
73.Find when B = 1.1 T, A = 2.0 m2, and θ =53°?
74.Remember how “∆” means “final state minus initial state”? Using that, assume B does not change from 0.3 Tandθ = 0°, but A changes from 0.1 m2 to 0.4 m2. If ∆t = 1.1seconds, use the above
information to findEavg.
75.ε is 0.12 V, B is 2.0× 10-3 T, and v is 12,000 m/s. Find ℓ using the last equation in thelist.
Section XVI: For problems 76 – 81, use the equationsbelow.
76.Find E if h = 6.63 x 10-34 J·s, λ = 450 nanometers, and c = 3 x 108 m/s. To solve this problemcorrectly,convert λ into meters before plugging in thenumber?
77.h is a constant, so it is always equal to the value given in the prior problem. Assuming f is 4.2 x 1014Hzand ϕ is 1.3 x 10-19 J, calculate the value ofK?
78.Using the first equation from Section V for p, determine the value of λ given that m = 9.11 x 10-31 kgandv = 2.7 x 106m/s?
79.c is also a constant, so it always equals 3 x 108 m/s. If the final state of m = 3.4824 x 10-27 kg andtheinitial state of m = 3.4829 x 10-27 kg, findΔE?
80.K is not allowed to be negative. Find the minimum value of f that works for the thirdequation if ϕis 4.3x10-19J?
81. Find f using the first and last equation. Assume E = ΔE and that Δm = 8.3x10-31kg?
GOOD JOB! That wasn’t so bad was it? Trust me… the blood sweat and tears it took to get through all ofthose problems will make everything later on a lot easier. Think about it as an investment with a
Part 6: Scalars andVectors
Hooray for the Internet! Watch the following twovideos:
For each video, summarize the content Mr. Khan is presenting in three sentences. Then, write at leastonequestion per video on something you didn’t understand or on a possible extension of the
elementary conceptshepresentshere.
You might have to watch them more than once. Trust me, these concepts are some of the building blocks ofPhysics.Get this down and you are on the fast track tosuccess.
Expect to be challenged! This is where it all comes down to, AP Physics 1! This is acollege level course where you will be using your knowledge and understanding of everything youhave learned in all
of your classes to solve problems, analyze situations, arrange materials,compare data, design labs, and build incredible things. That isphysics!
Success: Effectiveness:Performance:
You cannot expect to acquire the understanding you need to do well on an AP Exam bymerely attending class and listening to the teacher. You have to become INVOLVED. You haveto PARTICIPATE. If you get
stuck, see ME, or other students! Ask for HELP. Yourclassmates will be your new best friends. You must study regularly. Students who study regularly havea good foundation to build on for new topics.
This will pay off! If you are unorganizedor inconsistent, things may start to fall apart – and nobody wants that to happen. Busy work isnot assigned in this course so do what I ask you to do
regularly! Especially thehomework!!
Homework => Practice =>Success
Have a greatsummer!
DR. Malik
AP PhysicsTeacher | {"url":"https://docsbay.net/doc/560118/ap-b-summer-assignment-combined","timestamp":"2024-11-11T08:29:26Z","content_type":"text/html","content_length":"30983","record_id":"<urn:uuid:58187a24-36da-4fd8-b0b0-f1075a66880d>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00470.warc.gz"} |
screen 2 - Natural Curriculum
Column subtraction 5: Sensational Seabirds
The Number Bit!
In this exciting lesson, you will need to solve a range of subtraction word problems that feature British seabirds. Try using PAWS to arrive at the right answer!
P = Picture the problem in your mind.
A = Annotate the word problem by highlighting the important information.
W = Write down the number sentence that needs to be solved.
S = Solve the number sentence using an efficient strategy.
Prior learning: Subtract a 3 or 4 digit number from a 4 digit number using the formal written method of column subtraction (with more than one exchange, including an exchange across more than one
place value).
Use PAWS and column subtraction to solve the following word problem.
On Bass Rock, a volcanic island that stands 107 metres above sea level at its highest point,137,352 gannets had gathered. Of these birds, 76,292 were female.
How many male gannets were there on Bass Rock?
Did you know?
Northern gannets are large and bright white seabirds with black wingtips. They circle above the ocean before diving headfirst into the water at high speed to catch fish with their long beaks. Huge
numbers breed in colonies called gannetries. The largest gannetry in the world is on Bass Rock in Scotland. Here, an estimated 150,000 gannets gather at the peak of the breeding season. | {"url":"https://www.naturalcurriculum.co.uk/maths/contributor-lessons/column-subtraction-3/sensational-seabirds/screen-2/","timestamp":"2024-11-02T21:12:46Z","content_type":"text/html","content_length":"51504","record_id":"<urn:uuid:41a44345-637a-422e-b5d7-d0c9c0cea05b>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00808.warc.gz"} |
The value of sin(sin−121+cos−121)= ?
... | Filo
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Question Text The value of ?
Updated On May 23, 2023
Topic Inverse Trigonometric Functions
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 1
Upvotes 188
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[Solved] A 50-hp, 230-V shunt motor has a field re | SolutionInn
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A 50-hp, 230-V shunt motor has a field resistance of 17.70 and operates at full load when the line current is 181 A at
A 50-hp, 230-V shunt motor has a field resistance of 17.70 and operates at full load when the line current is 181 A at 1,350 r/min. To increase the speed of the motor to 1,600 r/min, a resistance of
5.30 is "cut in" via the field rheostat; the line current then increases to 190 A. Calculate a. The power loss in the field and its percentage of the total power input for the 1,350 r/min speed. b.
The power losses in the field and the field rheostat for the 1,600 r/min speed. c. The percent losses in the field and in the field rheostat at 1,600 r/min. A motor with polar moment of inertia J
develops torque according to the relationship T = aw+b. The motor drives a load defined by the torque-speed relationship TL = cw2? + d.If the four coefficients are all positive constants, determine
the equilibrium speeds of the motor-load pair, and whether these speeds are stable. Assume that a motor has known friction and windage losses described by the equation Tfw= bw. Sketch the T-w
characteristic of the motor if the load torque TL is constant, and the TL-w characteristic if the motor torque is constant. Assume that TFWat full speed is equal to 30 percent of the load torque.
Develop a Simulink simulator for the shunt-connected DC motor of Problem 14.38. Assume the following parameter values: La = 0.15 H; Lf=0.05 H; Ra = 1.8 Q; Rf = 0.20; ka = 0.8 V-s/rad; kT = 20 N-m/A;
kf= 0.20 Wb/A; b=0.1 N-m-s/rad; J = 1 kg-m2.
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Calculating the present and future values of multiple cash flows is relevant
Future Value. The future value calculator can be used to determine future value, or FV, in financing. FV is simply what money is expected to be worth in the future. Typically, cash in a savings
account or a hold in a bond purchase earns compound interest and so has a different value in the future. A good example for this kind Chapter 4.14® - Calculating Present Value with Multiple Future
Cash Flows – Example #2. Part 4.1 - Time Value of Money, Future Values of Compounding Interest, Investing for more than 1 Period & Examination of Original Investment & Growth of Investment
Present Value of Single / Multiple Cash Flows The Present Value concept is also called as discounting technique. In this approach, the money received in some future date will be worth lesser now at
the present date because the corresponding interest is lost during the period. 2. Calculating the present and future values of multiple cash flows is relevant for businesses only. A) True B) False
Ans: B Format: True/False Learning Objective: LO 1 Level of Difficulty: Easy 3. In computing the present and future value of multiple cash flows, each cash flow is discounted or compounded at a
different rate. Calculate the year three present value of a cash flows. This equals $100/(1.08)^4 or $79.38. The present value of $100 in three years is $79.38 at 8 percent interest. Step. Calculate
the year four present value of a cash flows. This equals $100/(1.08)^5 or $73.50. The present value of $100 in four years is $73.50 at 8 percent interest. Present Value of a Series of Cash Flows (An
Annuity) If you want to calculate the present value of an annuity (a series of periodic constant cash flows that earn a fixed interest rate over a specified number of periods), this can be done using
the Excel PV function. The syntax of the PV function is: There are several ways to measure the cost of making such payments or what they're ultimately worth. Here's what you need to know about
calculating the present value or future value of an annuity. Future Value of a Series of Cash Flows (An Annuity) If you want to calculate the future value of an annuity (a series of periodic constant
cash flows that earn a fixed interest rate over a specified number of periods), this can be done using the Excel FV function . Net present value (NPV) is a core component of corporate budgeting.It is
a comprehensive way to calculate whether a proposed project will be financially viable or not. The calculation of NPV
Finding the future value (FV) of multiple cash flows means that there are more Calculate the present value of an investment portfolio that has multiple cash
We then proceed to calculate the present value of single cash flow and review the of calculating the future value of a cash flow is known as compounding. For example Many financial instruments have
multiple cash flows that occur at different in perpetuity if the relevant annual effective interest rate is 5%?. Solution. Calculate the future value of uneven, or even, cash flows. We start with the
formula for FV of a present value ( PV ) single lump sum at time n and interest rate i,. 21 Jun 2019 Future cash flows are discounted at the discount rate, and the higher the So, if you want to
calculate the present value of an amount you expect to and a relevant interest rate that mathematically increases future value in Finding the future value (FV) of multiple cash flows means that
there are more Calculate the present value of an investment portfolio that has multiple cash
Present Value of Single / Multiple Cash Flows The Present Value concept is also called as discounting technique. In this approach, the money received in some future date will be worth lesser now at
the present date because the corresponding interest is lost during the period.
21 Jun 2019 Future cash flows are discounted at the discount rate, and the higher the So, if you want to calculate the present value of an amount you expect to and a relevant interest rate that
mathematically increases future value in Finding the future value (FV) of multiple cash flows means that there are more Calculate the present value of an investment portfolio that has multiple cash
14 Feb 2019 Your mother gives you $100 cash for a birthday present, and says, “Spend it wisely. The company would be receiving a stream of four cash flows that are all lump sums. The relevant factor
where n = 15 and i = 12% is 37.280. that use multiple approaches to determining present and future value. standard arithmetic calculations of addition, subtraction, multi- plication, and to the
default values. For details, see the relevant sections for each financial calcu- Inflow (+). Cash flow. Present value (PV). Future value (FV). Time. Outflow (–) 4 Apr 2018 Understanding the
Discounted Cash Flow (DCF) method and all the benefits it It is also necessary to consider industry trends, relevant economic data, For example, to determine the present value of AU$1 of future cash
flow, the time of terminal value estimation, applying a multiple to revenues, book Future value of a single cash flow refers to how much a single cash flow today Calculate the future value (FV) of
an investment of $500 for a period of 3 years that pays an interest rate of 6% compounded semi-annually. PV = Present Value information on your device to serve relevant ads or personalized content .
Present Value Formulas, Tables and Calculators, Calculating the Present we will demonstrate how to find the present value of a single future cash amount,
3 Sep 2019 Calculating the sum of future discounted cash flows is the gold standard produce, with each of those cash flows being discounted to their present value. By splitting their wealth up into
multiple projects, businesses, stocks,
The importance of the concept and calculation of net present value and internal d) the estimation and forecasting of current and future cash flows are expressed in terms of the actual dollars that
will be received or paid at the relevant dates.
Future value of a single cash flow refers to how much a single cash flow today Calculate the future value (FV) of an investment of $500 for a period of 3 years that pays an interest rate of 6%
compounded semi-annually. PV = Present Value information on your device to serve relevant ads or personalized content .
We then proceed to calculate the present value of single cash flow and review the of calculating the future value of a cash flow is known as compounding. For example Many financial instruments have
multiple cash flows that occur at different in perpetuity if the relevant annual effective interest rate is 5%?. Solution. Calculate the future value of uneven, or even, cash flows. We start with the
formula for FV of a present value ( PV ) single lump sum at time n and interest rate i,. 21 Jun 2019 Future cash flows are discounted at the discount rate, and the higher the So, if you want to
calculate the present value of an amount you expect to and a relevant interest rate that mathematically increases future value in Finding the future value (FV) of multiple cash flows means that
there are more Calculate the present value of an investment portfolio that has multiple cash 14 Feb 2019 Your mother gives you $100 cash for a birthday present, and says, “Spend it wisely. The
company would be receiving a stream of four cash flows that are all lump sums. The relevant factor where n = 15 and i = 12% is 37.280. that use multiple approaches to determining present and future
value. standard arithmetic calculations of addition, subtraction, multi- plication, and to the default values. For details, see the relevant sections for each financial calcu- Inflow (+). Cash flow.
Present value (PV). Future value (FV). Time. Outflow (–) 4 Apr 2018 Understanding the Discounted Cash Flow (DCF) method and all the benefits it It is also necessary to consider industry trends,
relevant economic data, For example, to determine the present value of AU$1 of future cash flow, the time of terminal value estimation, applying a multiple to revenues, book
The time value of money is the greater benefit of receiving money now rather than an identical An important note is that the interest rate i is the interest rate for the relevant For example, the
annuity formula is the sum of a series of present value The cumulative present value of future cash flows can be calculated by Answer to 1) Calculating the present and future values of multiple cash
flows is relevant only for individual investors. Answer Tr Chapter 6 - Discounted cash flows and valuation True/False 1. Calculating the present and future values of multiple cash flows is relevant
only for individual We then proceed to calculate the present value of single cash flow and review the of calculating the future value of a cash flow is known as compounding. For example Many
financial instruments have multiple cash flows that occur at different in perpetuity if the relevant annual effective interest rate is 5%?. Solution. Calculate the future value of uneven, or even,
cash flows. We start with the formula for FV of a present value ( PV ) single lump sum at time n and interest rate i,. 21 Jun 2019 Future cash flows are discounted at the discount rate, and the
higher the So, if you want to calculate the present value of an amount you expect to and a relevant interest rate that mathematically increases future value in Finding the future value (FV) of
multiple cash flows means that there are more Calculate the present value of an investment portfolio that has multiple cash | {"url":"https://flyerepkorpb.netlify.app/notarnicola33690mexi/calculating-the-present-and-future-values-of-multiple-cash-flows-is-relevant-300.html","timestamp":"2024-11-03T12:59:27Z","content_type":"text/html","content_length":"39362","record_id":"<urn:uuid:f63c7aed-315c-4e5d-92a5-759f9c7bccc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00530.warc.gz"} |
Gaussian distributed random numbers
Commented: Ruben Dörfel on 13 Oct 2020
I need to generate a stationary random numbers with gaussian distribution of zero mean and a variance of unity with max value one.
1 Comment
64 views (last 30 days)
Gaussian distributed random numbers
As all the people have pointed out, there are questions that you must answer before you really get a valid response.
Is the mean to be zero and the variance 1 AFTER truncation or before?
Accepted Answer
The core MATLAB function randn will produce normally-distributed random numbers with zero mean and unity standard deviation.
If you want the numbers to be limited to those <=1, this will work:
q = randn(1,10);
q = q(q<=1);
4 Comments
Edited: José-Luis on 11 Jul 2014
I didn't think it through. If you do it like this, the mean will also change, since you are only removing elements from the right tail. John's question remains valid though.
Star Strider on 11 Jul 2014
For that matter, considering that the Gaussian distribution has infinite support, once truncated, it is no longer Gaussian.
The mean and variance shift can be ‘fixed’ relatively easily though:
q = q/std(q) - mean(q);
It’s still non-Gaussian, but the numbers work.
More Answers (2)
What if you generate some random numbers (here 100) with normal distribution, mean of 0 and std dev of 1:
R = normrnd(0,1,1,100);
then divide all by the highest value so that the maximum is 1:
R_norm = R./max(R(:));
Check max:
ans =
2 Comments
Then the variance is not one anymore.
Ben11 on 11 Jul 2014
Oh shoot you're right
Edited: Chris E. on 11 Jul 2014
Well a simple Gaussian distribution code can be as follows:
function main()
xo = 0;
yo = 0;
xsigma = 0.01;
ysigma = 0.01;
particle_amount = 100;
xpoints = Gauss(xo,xsigma,particle_amount)
ypoints = Gauss(yo,ysigma,particle_amount)
%needs column vectors
coordinates_x_y = [xpoints ypoints];
function output = Gauss(xo,sigma,PA)
r = sqrt(-2.0.*(sigma^2).*log(rand(PA,1)));
phi = 2.0.*pi.*rand(PA,1);
output = xo+r.*cos(phi);
This produces as many random Gaussian distribution about the center of (x,y)=(0,0) and a sigma of 0.01 with 100 points of data. You can modify where needed. I hope that helps you out!
3 Comments
Jon Thornburg on 22 Jun 2020
This thead is a few years old but I was looking over the example, because I need to do something similar. I was trying the above code. Gauss(xo,xsigma,particle_amount) it pops out the error
"Undefined function or variable 'Gauss'."
Gauss was not deifed as a variable and searching matlab documentation cannot find "Gauss" by itself as formated in the above script. Any suggestions?
Ruben Dörfel on 13 Oct 2020
@Jon Thornburg
Gauss seems to be a user defined function. You would have to put
function output = Gauss(xo,sigma,PA)
r = sqrt(-2.0.*(sigma^2).*log(rand(PA,1)));
phi = 2.0.*pi.*rand(PA,1);
output = xo+r.*cos(phi);
into a new script. You should look up how to implement functions in matlab. | {"url":"https://in.mathworks.com/matlabcentral/answers/141458-gaussian-distributed-random-numbers","timestamp":"2024-11-14T04:25:00Z","content_type":"text/html","content_length":"189988","record_id":"<urn:uuid:4f0cc3af-c634-4316-ab9f-a2b6ceec5963>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00160.warc.gz"} |
Introduction to Derivatives - Calculus | WeTheStudy
Introduction to Derivatives
Derivatives show the rate of change of a mathematical function - between the inputs and outputs. It is a fundamental concept in differential calculus.
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Linear transformations between CombinatorialFreeModule
Linear transformations between CombinatorialFreeModule
I have a CombinatorialFreeModule whose basis is a set of tuples. I want to define a linear transformation that permutes the two first components of a tuple in the basis. As a minimal example, that's
what I tried:
def Action(k):
V = CombinatorialFreeModule(SR, [(1,0,0),(0,1,0)], prefix="")
e = V.basis()
f = lambda x: ((-1)^(k - x[2]))*V.basis()[(x[1],x[0],x[2])]
return V.hom([f(x) for x in e.keys()], V)
However, I get the following error when I compute Action(1) (I have parameters in the case I am interested in, but this is a minimal example):
unable to convert [-[(0, 1, 0)], -[(1, 0, 0)]] to an element of Set of Morphisms from Free module generated by {(1, 0, 0), (0, 1, 0)} over Symbolic Ring to Free module generated by {(1, 0, 0), (0, 1, 0)} over Symbolic Ring in Category of finite dimensional vector spaces with basis over Symbolic Ring
Is there a way to correct that? As my real basis is much more bigger than that, I don't want to just define the image of each vector separately.
1 Answer
Sort by ยป oldest newest most voted
Here is a solution via module_morphism:
V = CombinatorialFreeModule(SR, [(1,0,0),(0,1,0)], prefix="")
p = [1,0,2] # permutation of tuple indices
H = V.module_morphism(lambda t: V.monomial(tuple(t[i] for i in p)), codomain=V)
e = V.random_element()
print('e =\t',e)
print('H(e) =\t',H(e))
As an example it prints:
e = [(0, 1, 0)] + 2*[(1, 0, 0)]
H(e) = 2*[(0, 1, 0)] + [(1, 0, 0)]
edit flag offensive delete link more | {"url":"https://ask.sagemath.org/question/63228/linear-transformations-between-combinatorialfreemodule/","timestamp":"2024-11-07T16:05:14Z","content_type":"application/xhtml+xml","content_length":"53791","record_id":"<urn:uuid:4e384c51-5082-4743-8595-b9f5e0b777ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00006.warc.gz"} |
Name Type Default Description
kperp double precision None Perpendicular wavenumber, normalized by inverse reference inertial length, $k_\perp d_p$.
kpar double precision None Parallel wavenumber, normalized by inverse reference inertial length, $k_\parallel d_p$.
nspec integer None Number of plasma components.
nroots integer None Number of dispersion solutions under consideration.
use_map logical None Choice of: (T) searching for roots over a map in complex frequency space, via map_read; (F) input (nroots) guesses for solutions, via solution_read
writeOut logical .true. Write or suppress output to screen.
nperp integer None Number of perpendicular momentum space grid points, $N_\perp$.
npar integer None Number of parallel momentum space grid points, $N_\parallel$.
ngamma integer 100 Number of grid points in relativitic $\Gamma=\sqrt{1+\frac{p_\perp^2+p_\parallel^2}{m_j^2c^2}}$, $N_\Gamma$ (Eqn. 3.14).
npparbar integer 200 Number of grid points in dimensionless paralell momentum $\bar{p}_\parallel = p_\parallel/m_j c$, $N_{\bar{p}_\parallel}$.
vA double precision None Alfven Velocity, normalized to speed of light, $v_{Ap}/c$.
arrayName character(len=75) None Name of input files for distributions.
Bessel_zero double precision 1.d-45 Calculate Bessel functions until the maximum is less than this value.
numiter integer 50 Maximum number of iterations in secant method.
D_threshold double precision 1.d-5 Minimum threshold for secant method.
D_prec double precision 1.d-5 Size of bounding region for secant method.
D_gap double precision 1.d-5 Size of allowable difference between roots.
positions_principal integer 5 Number of parallel momentum steps distant from the resonant momentum included in the numerical calculation of Eqn 3.5, $M_I$.
Tlim double precision 0.01d0 Threshold for analytical principal-value integration for evaluating Eqn 3.6 and 3.7, $t_{\textrm{lim}}$.
maxsteps_fit integer 500 Maximum number of fitting iterations.
lambda_initial_fit double precision 1.d0 Inital Levenberg-Marquardt damping parameter.
lambdafac_fit double precision 1.d1 Adjustment factor for Levenberg-Marquardt damping parameter.
epsilon_fit double precision 1.d-8 Convergence for Levenberg-Marquardt fit.
fit_check logical .true. If true, output fitted functions to ASCII file for each species.
determine_minima logical .true. If true, after map search, determine minima and refine solutions.
n_resonance_interval integer 100 How many steps should be used to integrate around the resonance, $M_P$, used for integrating near poles (see section 3.1).
scan_option integer 1 Select case for scans; 1) consecutive scans along input paths in wavevector space, 2) double scans of two selected parameters.
n_scan integer 0 Number of wavevector scans. Must be set to 2 for scan_option=2; Must be 1 or larger for scan_option=1. 0 turns off wavevector scans. | {"url":"https://danielver02.github.io/ALPS/proc/init_param.html","timestamp":"2024-11-11T20:16:46Z","content_type":"text/html","content_length":"49326","record_id":"<urn:uuid:2c903e5a-278b-4a94-bc96-5003c576dff0>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00221.warc.gz"} |
Monkey Business: a dataset of large LLM sample collections for math and code tasks
We’re releasing the data and code for our recent paper: Large Language Monkeys: Scaling Inference Compute with Repeated Sampling. This includes 10,000 LLM-generated samples per problem for a variety
of datasets (GSM8K, MATH, CodeContests, and MiniF2F-MATH) and model families (Llama-3, Gemma, and Pythia)!
LLMs are smart monkeys at keyboards
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, including the complete works of
William Shakespeare.
In our paper, we make LLMs our monkeys, exploring whether they can generate correct answers to real-world math and coding datasets when allowed to make hundreds or thousands of attempts. We find that
LLMs exhibit inference-time scaling laws where the number of problems solved often increases log-linearly as we scale the number of samples over four orders of magnitude.
Across five tasks, we find that coverage (the fraction of problems solved by at least one generated sample) increases as we scale the number of samples. Notably, using repeated sampling, we are able
to increase the solve rate of an open-source method from 15.9% to 56% on SWE-bench Lite.
The verification problem
When tasks have tools for automatically verifying candidate solutions (ex. formal proof checkers or unit tests for code), we’re done! We can directly benefit from repeated sampling by using our
verifiers to pick out correct answers from large sample collections. However, for other datasets, like natural language math problems, identifying a correct answer is less straightforward.
As an initial investigation for how hard verification is in these settings, we compared three simple baselines against oracle selection (ie. choosing the best solution out of k samples):
• Majority voting
• Choosing the sample with the highest score assigned by a reward model
• Majority voting weighted by reward-model scores
While all three methods improve performance relative to taking a single sample, their performance saturates before 100 samples and falls well below the final oracle accuracy. This gap highlights the
importance of continuing to research verification methods.
Monkey Business: A dataset of LLM sample collections
To faciliate verification research in the large sample setting, we are excited to release Monkey Business: a dataset of sample collections for a variety of tasks and models.
Specifically, Monkey Business contains 10,000 correct and incorrect samples per problem for subsets of the following datasets:
• GSM8K: 127 randomly sampled problems from the test set (we originally had 128 but identified a problem with an incorrect ground-truth answer which we removed).
• MATH: 128 randomly sampled problems from the test set.
• CodeContests: all 140 problems in the test set that do not contain images in the problem description.
• MiniF2F-MATH: all 130 problems in the MiniF2F set corresponding to formalized MATH questions.
These samples are generated with the following models:
• GSM8K: Llama-3-8B-Instruct, Llama-3-70B-Instruct
• MATH: Llama-3-8B, Llama-3-8B-Instruct, Llama-3-70B-Instruct, Pythia 70M-12B, Gemma 2B, Gemma 7B
• CodeContests: Llama-3-8B, Llama-3-8B-Instruct, Llama-3-70B-Instruct, Gemma 2B, Gemma 7B
• MiniF2F-MATH: Llama-3-8B-Instruct, Llama-3-70B-Instruct
We are also releasing our sampling and evaluation scripts to make it easier to work with other tasks and models.
🤗 Dataset: https://huggingface.co/datasets/ScalingIntelligence/monkey_business
💻 Github: https://github.com/ScalingIntelligence/large_language_monkeys
In addition to training verifiers, we think that this dataset is useful for several other research directions including self-improvement methods and understanding patterns across correct and
incorrect samples.
How to cite? If you use our dataset or code, please cite the following paper:
title={Large Language Monkeys: Scaling Inference Compute with Repeated Sampling},
author={Bradley Brown and Jordan Juravsky and Ryan Ehrlich and Ronald Clark and Quoc V. Le and Christopher Ré and Azalia Mirhoseini}, | {"url":"https://scalingintelligence.stanford.edu/blogs/monkeys/","timestamp":"2024-11-12T19:57:40Z","content_type":"text/html","content_length":"22773","record_id":"<urn:uuid:55ccb860-7d59-40c0-b3a6-558994d6a580>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00415.warc.gz"} |
Strongly typed uniform crossover
In genetic algorithms, crossover is the primary method of combining genetic materials from two parents to create a new offspring. The uniform crossover operator in particular is frequently used in
genetic programming (GP) due to its ability to perform global search^[1] while having minimal bias with regard to node selection between parents. Despite these advantages much of the development of
strongly typed GP (STGP) had been done using the older one-point crossover, which is a biased operator. There are also no information on the internet regarding the implementation of a typed uniform
crossover operator, so here is one of my design.
Original algorithm
The uniform crossover operator works as follows. Two parent trees are aligned at the root and jointly traversed to identify a common region - if the pair of nodes have the same arity then their
children are visited, otherwise we have reached the boundary of the common region. For each pair of nodes in said region, we perform an inner swap (ie only swap the node label) with uniform
probability. If we are at the boundary then we instead swap the entire subtree.
In other words, for each node we visit a coin flip determines whether a swap should be performed. At the boundary a swap consists of an exchange of subtrees whereas only node labels are touched for
the other parts of the common region.
Strongly-typed variant (STGP uniform crossover)
Adapting the original operator to work under type constrained environments require some rethinking, but the algorithm remains very simple to understand. Lets first examine why the original one fails
to operate on a typed expression tree.
As seen above each children contains a type error. This illustrates the closure property that traditional GP relies on – that each non-terminal can accept arguments of any data type, which often just
means using the same type for all terminals and non-terminals. STGP on the other hand is all about constraining the types of arguments and return value. To satisfy this limit, (inner) swapping can
only be done when two nodes have the exact same signature. This drastically reduce the number of possible swaps, hence less genetic materials can be exchanged. We mitigate this problem by doing away
with the concept of a common region and inner swaps. Instead we recursively perform a subtree swap at every pair of nodes with uniform probability, on the condition that their return types are
compatible. An optional improvement here is to visit the children of a pair of nodes even if their arities are different. For example, if operator a and operator b have arities 2 and 3 respectively,
then we perform uniform crossover on the first two argument subtrees.
Here is the operator in pseudo-code:
1 uniform_crossover(parent_a, parent_b):
2 Q <- new Queue()
3 push (parent_a, parent_b) to Q
5 while Q not empty:
6 pop (node_a, node_b) from Q
8 if node_a and node_b have the same return type
9 and a coin flip of probability p=0.5 is true:
10 swap node_a and node_b
12 if both node_a and node_b are non-terminals:
13 for each (child_a, child_b) of (node_a, node_b):
14 push (child_a, child_b) to Q
16 return (parent_a, parent_b)
This variant can be thought of as an one-point crossover operator on steroids, with the unbiased property of uniform crossover. As with both of the original operator, STGP uniform crossover is a
global search operator at the start of a GP run, then over time restricts the search space to more local regions.
1. At least initially. ↩ | {"url":"https://ipthomas.com/blog/strongly-typed-uniform-crossover/","timestamp":"2024-11-03T03:52:13Z","content_type":"text/html","content_length":"9688","record_id":"<urn:uuid:c739fcb7-093e-4c28-af11-843f5d4151cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00024.warc.gz"} |
What is the distance in miles from Rio de Janeiro to the equator? - TravelAsker
Rio de Janeiro and the Equator
Rio de Janeiro is a vibrant and bustling city located in Brazil, South America. It is known for its beautiful beaches, rich culture, and iconic landmarks such as the Christ the Redeemer statue. The
equator, on the other hand, is an imaginary line that circles the Earth, dividing it into the Northern Hemisphere and the Southern Hemisphere. It is an important geographic reference point and plays
a crucial role in weather patterns, navigation, and astronomy.
Understanding the Concept of Distance
Distance is the measure of the physical space between two points. It can be measured in different units such as miles, kilometers, or meters. The distance between two points can be calculated using
various methods, including geographic coordinates, physical measurements, or mathematical formulas.
The Equator: A Brief Overview
The equator is a line that circles the Earth at 0 degrees latitude. It is the widest part of the Earth and measures approximately 40,075 kilometers or 24,901 miles. The equator is an important
reference point for cartographers, navigators, and meteorologists since it affects climate, weather patterns, and ocean currents.
Measuring the Distance from Rio de Janeiro to the Equator
To calculate the distance between Rio de Janeiro and the equator, we need to use their geographic coordinates. The geographic coordinates are a set of numbers that indicate a location’s latitude and
longitude. Once we have these coordinates, we can use a mathematical formula to calculate the distance between the two points.
The Geographic Coordinates of Rio de Janeiro
Rio de Janeiro is located at 22.9068 degrees south latitude and 43.1729 degrees west longitude. This means that the city is located 22.9068 degrees south of the equator and 43.1729 degrees west of
the Prime Meridian.
The Geographic Coordinates of the Equator
The equator is located at 0 degrees latitude and 0 degrees longitude. This means that it is located at the point where the Earth’s equatorial plane intersects with the planet’s surface.
Using the Haversine Formula to Calculate Distance
The Haversine formula is a mathematical formula used to calculate the distance between two points on a sphere. It takes into account the curvature of the Earth’s surface and the difference in
latitude and longitude between the two points.
The Distance in Miles from Rio de Janeiro to the Equator
Using the Haversine formula, we can calculate the distance between Rio de Janeiro and the equator to be approximately 2,484.67 miles or 3,997.87 kilometers.
Other Units of Measurement for Distance
In addition to miles and kilometers, distance can be measured in other units such as nautical miles, feet, or meters. The unit of measurement used depends on the context and the purpose of the
Factors that Affect the Distance Calculation
Several factors can affect the distance calculation, including the accuracy of the geographic coordinates, the method used to calculate the distance, and the shape of the Earth’s surface.
Conclusion: Distance from Rio de Janeiro to the Equator
In conclusion, the distance from Rio de Janeiro to the equator is approximately 2,484.67 miles or 3,997.87 kilometers. This calculation was done using the Haversine formula, which takes into account
the curvature of the Earth’s surface. The equator is an important geographic reference point that plays a crucial role in navigation, meteorology, and astronomy.
Additional Information on Rio de Janeiro and the Equator
Rio de Janeiro is a fascinating city with a rich history and culture. It is known for its beautiful beaches, lively carnival celebrations, and iconic landmarks such as the Christ the Redeemer statue.
The equator, on the other hand, is an imaginary line that circles the Earth and plays a crucial role in weather patterns, navigation, and astronomy. It is an important reference point for
geographers, cartographers, and scientists.
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Chapter 5.1: Greatest Common Factor
The opposite of multiplying polynomials together is factoring polynomials. Factored polynomials help to solve equations, learn behaviours of graphs, work with fractions and more. Because so many
concepts in algebra depend on us being able to factor polynomials, it is important to have very strong factoring skills.
In this section, the focus is on factoring using the greatest common factor or GCF of a polynomial. When you previously multiplied polynomials, you multiplied monomials by polynomials by
distributing, solving problems such as
To do this, first identify the GCF of a polynomial. Look at finding the GCF of several numbers. To find the GCF of several numbers, look for the largest number that each of the numbers can be divided
Find the GCF of 15, 24, 27.
First, break all these numbers into their primes.
By observation, the only number that each can be divided by is 3. Therefore, the GCF = 3.
Find the GCF of
First, break all these numbers into their primes. (Use • to designate multiplication instead of ×.)
By observation, what is shared between all three monomials is
Factor out the common factor in each of the following polynomials.
Answers to odd questions | {"url":"https://ecampusontario.pressbooks.pub/sccmathtechmath1/chapter/7-1-greatest-common-factor/","timestamp":"2024-11-04T15:29:37Z","content_type":"text/html","content_length":"103707","record_id":"<urn:uuid:22f64f21-0ccb-457e-ab07-e071a49f1b7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00434.warc.gz"} |
Corsi di studio e offerta formativa - Università degli Studi di Parma
Learning objectives
The couse aims to provide the student with the fundamental concepts of mechanics and thermodynamics, through examples and exercises. During the course, the student will learn how to discuss and solve
simple problems, to understand and describe physical phenomena as well as the physical properties of matter that he can experience in everyday life.The course will provide the following skills:
- knowledge of the correct scientific terminology used in Mechanics and Thermodynamics.- knowledge of the fundamental laws that describe the physics of Mechanics and Thermodynamics;- solving simple
problems, analytically and numerically;- developing a rigorous scientific language, to transmit accurately the knowledge acquired and to describe physics phenomena;- critical assessment of his
advancement in the study of physics;- critical discussion of the results obtained while solving physics problems, in particular focusing on errors and non-physical results; - to formulate simple
observation and adequate predictions in real-life situations involving the physics of this course. | {"url":"https://corsi.unipr.it/en/ugov/degreecourse/204327","timestamp":"2024-11-08T22:15:40Z","content_type":"text/html","content_length":"57076","record_id":"<urn:uuid:17f6dd5c-284c-424a-b94e-6be38142ac13>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00880.warc.gz"} |
Get instant live expert help on I need help with excel formula to calculate days
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A CLI Base Converter
When debugging an embedded system, it’s common to work with raw data requiring conversion between decimal, hexadecimal, binary, and sometimes octal number systems. The Python REPL and printf shell
utility do the job but are tedious to use for the simple task of base conversion.
It would be nice to drop the overhead of format specifiers and fear of numerical limits. To ease the pain, I decided to write a command line utility that made conversion between positive binary,
decimal, octal, and hexadecimal numbers of arbitrary size.
The Requirements
The use case is simple: take a positive integer in one base and convert it to the equivalent value in another base. That’s it. Support for negative values and floating point values is out of scope.
The program usage would look something like
dhb [OPTION]... SRC_BASE TGT_BASE NUM
where SRC_BASE/TGT_BASE are one of bin, dec, oct, or hex. NUM is some positive integer value.
Below are the requirements:
1. Support conversions to/from binary, decimal, hexadecimal, and octal.
2. Include an option for minimum output width.
3. Include an option to group digits into segments of size N.
4. Support arbitrarily large positive integers.
Requirement (1) is self explanatory. Requirement (2) means you can pad the output value with zeroes to achieve a minimum width. For example, pad the binary value 1111 to 8-bits leading to an output
of 00001111. Requirement (3) is handy when you want to visualize binary or hex codes in groups of 4, 8, etc. digits. Taking the previous binary value of 00001111, maybe you want to group the bits
into nibbles 0000 1111 or into 2 digits codes 00 00 11 11. Requirement (4) seems a bit extra but it has its value. Visualizing a large stream of hex values in binary is a common task. Exceeding the
max integer limit for the system/program is also a common occurrence. This dhb tool should handle numbers outside the range of a uint64_t without breaking a sweat.
Lets look at how dhb meets each of these requirements starting with that bignum requirement.
Big, Huge Numbers
If you’re familiar with C++, you know the range of positive integers a program can work with is finite. There’s no standard “big number” library either.
Google search revealed a number of big number libraries. Most of the libraries are unmaintained, header-only libraries. The best option was the GNU MP Library (GMP). To quote the GMP homepage:
GMP is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating-point numbers. There is no practical limit to the precision except the ones
implied by the available memory in the machine GMP runs on. GMP has a rich set of functions, and the functions have a regular interface.
GMP has a convenient C++ class based interface. The docs for how to use the C++ bindings and for GNU MP in general are solid. GMP is a perfect fit for this project.
The only info needed to perform a conversion is the number, that number’s current base, and a target base. The conversion API accommodates this spec using one function and an enum:
enum NumSystem : int {
kDec = 10,
kHex = 16,
kBin = 2,
kOct = 8,
std::string ConvertBase(const std::string& num, const NumSystem src, const NumSystem target) {
const mpz_class kTargetBase(static_cast<int>(target));
const std::string kDigits("0123456789ABCDEF");
std::string converted_num;
mpz_class num_mp(num, static_cast<int>(src));
while (num_mp) {
mpz_class idx = num_mp % kTargetBase;
converted_num += kDigits[idx.get_si()];
num_mp /= kTargetBase;
std::reverse(converted_num.begin(), converted_num.end());
return converted_num;
The algorithm for conversion is the usual change of base method which uses modulo and integer division to compute the digits of the output number one-by-one. The mpz_class is a GMP C++ wrapper class
used to construct and manipulate big integral values. You can see mpz_class overloads the arithmetic operators such that the code doesn’t look much different than if one were to use the C/C++
built-in types.
One neat feature of GMP is the ability to construct an mpz_class object from a number represented as a string and its base. That feature makes implementation easier because you don’t have to massage
the input into a format GMP understands. The constructor does throw std::invalid_arg if given an unsupported base argument. To avoid exceptions, the caller specifies the base using a NumSystem type
which limits the caller to the bases known to the mpz_class constructor.
Formatting Output
Looking back at the requirements, there’s two formatting options to implement: minimum character width and digit grouping.
The minimum character width function was trivial to implement using a stringstream object in combination with stream modifiers:
std::string SetWidth(const std::string& num, int width) {
if (width <= 0) {
return num;
std::stringstream ss(num);
ss << std::setfill('0') << std::setw(width) << num;
return ss.str();
Not much to say here. The stream object will just slap zeroes onto the front of the number until it meets the width argument.
Segmenting the output’s digits into groups was a bit of a CS101 exercise:
std::string GroupDigits(const std::string& num, int grouping) {
if ((grouping <= 0) || (grouping >= static_cast<int>(num.size()))) {
return num;
std::stack<char> digits;
for (const char& c : num) {
std::string group;
std::vector<std::string> groups;
while (!digits.empty()) {
group += digits.top();
if (static_cast<int>(group.size()) == grouping) {
std::reverse(group.begin(), group.end());
group = "";
if (!group.empty()) {
std::reverse(group.begin(), group.end());
std::reverse(groups.begin(), groups.end());
return std::accumulate(groups.begin(), groups.end(), std::string(),
[](const std::string& a, const std::string& b) {
return a + (a.empty() ? "" : " ") + b;
A stack processes the digits in the number from right to left. The algorithm pops characters off the stack into a group string. When that group string hits the grouping limit, it’s saved off in the
groups vector and group is reset. Rinse and repeat.
Processing happens from right to left meaning there’s a reversal that needs to happen for each group string and for the entire groups vector. Without this reversal, the digits come out backwards in
the output.
C++ doesn’t have a nice join() method like Python. Instead, you get to use the beautiful std::accumulate API to concatenate each string in groups using a single space as a separator. The concatenated
string is the output of the function.
Testing the Implementation
At this point, you have a working conversion utility! The rest of the implementation focuses on command line argument parsing and input validation. You can check out the full source linked at the end
of this article if you’re interested in those bits.
Lets test drive this tool:
dhb hex dec 0xDEADBEEF --> 3735928559
dhb dec bin 3735928559 --> 11011110101011011011111011101111
dhb dec oct 3735928559 --> 33653337357
dhb -g 4 dec hex 3735928559 --> DEAD BEEF
dhb -g 4 -w 12 dec hex 3735928559 --> 0000 DEAD BEEF
So far so good. Lets use a massive number like 2^64 * 12345 (AKA 227725055589944414699520). The tool should be able to handle that:
dhb --grouping 3 dec dec 227725055589944414699520 --> 227 725 055 589 944 414 699 520
dhb dec hex 227725055589944414699520 --> 30390000000000000000
dhb dec oct 227725055589944414699520 --> 60162000000000000000000000
dhb --grouping 8 hex bin --> 110000 00111001 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000
Nice, looks to be working with big integers too.
The project includes a more complete suite of tests that exercises all the different conversion permutations.
The dhb utility has been of great use. The process of implementing the tool was relatively straightforward. I credit the simplicity to identifying early on the primary use cases and not tacking on
too many bells and whistles along the way.
The complete project source with build instructions, usage, etc. is available on GitHub under dhb. | {"url":"https://programmador.com/posts/2023/base-conversion/","timestamp":"2024-11-15T00:30:12Z","content_type":"text/html","content_length":"42158","record_id":"<urn:uuid:b0e086f4-4f51-46d8-9444-b496ae1e94e4>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00144.warc.gz"} |
Identify and Graph Integers | Inspirit
The set of integers includes zero, negative and positive numbers without any decimal or fractional parts. To graph integers, we take a number line with 0 in the middle and place all the positive
integers on the right side of 0 and all the negative integers on the left side of 0. The integers are placed at equal spaces. | {"url":"https://discover.inspiritvr.com/math/simulations/identify-and-graph-integers","timestamp":"2024-11-04T17:47:55Z","content_type":"text/html","content_length":"41878","record_id":"<urn:uuid:dc557c06-c249-4a5f-a8ac-08d878cccb4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00510.warc.gz"} |
seminars - On equisingular approximation of plurisubharmonic functions
After the fundamental work of Demailly on approximations, it is a natural question to ask which plurisubharmonic functions admit a `nice' approximation in the sense of a decreasing equisingular
approximation with analytic singularities. For arbitrary toric plurisubharmonic functions, we give a criterion for admitting a nice approximation with toric approximants. Our results are motivated by
a recent result of Guan for toric plurisubharmonic functions of the diagonal type. This is joint work with Jongbong An. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&l=en&sort_index=Time&order_type=asc&page=31&document_srl=1205265","timestamp":"2024-11-11T23:23:36Z","content_type":"text/html","content_length":"45501","record_id":"<urn:uuid:68985a23-27da-4190-a3fe-ed8f6897b7c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00776.warc.gz"} |
Inconsistent problem about centrifugal/contact force
• Thread starter FranzDiCoccio
• Start date
In summary: I'll let the summary speak for itself.In summary, the skier will not lose contact with the ground before reaching the very top of the second hill, regardless of how high the first hill
Homework Statement
A skier starts from rest at the top of a hill. The skier coasts down the hill and up a second hill, as the drawing illustrates. The crest of the second hill is circular, with a radius of r = 36
m. Neglect friction and air resistance. What must be the height h of the first hill so that the skier just loses contact with the snow at the crest of the second hill?
Relevant Equations
conservation of mechanical energy
Newton's second law
This is problem 49 in chapter 6 of "Physics - 9th edition". A similar question was asked
several years ago (although with a different height).
The figure is below. I added point A and angle [itex] \theta [/itex].
The solution is pretty easy. For the purpose of my discussion I'm assuming that the height is zero at the "center" of the rightmost hill, and consider a generic point A on such hill, instead of its
crest. The radius going through A forms an angle [itex] \theta [/itex] with the vertical direction. The answer to the question can be obtained by setting [itex] \theta =0 [/itex] in the solution.
The condition for losing contact at A is that the normal force there is null, which means that the centrifugal force equals the projection of the weight along the radial direction. This gives [itex]
v^2 = g r \cos \theta [/itex].
Conservation of total mechanical energy requires that [itex]2 g h_0 =2 g r \cos \theta + v^2 [/itex], where [itex] h_0 [/itex] is the height of the leftmost hill wrt the center of the leftmost hill.
Substituting the first equation into the second one we get
[tex] h_0 = \frac{3}{2} r \cos \theta [/tex]
Since the height of point A is [itex] r \cos \theta[/itex], the starting point should be
[tex] h = \frac{r}{2} \cos \theta [/tex]
above A for the skier to lose contact at A.
If [itex]\theta=0 [/itex] one gets [itex] h = r/2 = 18\, {\rm m} [/itex], which is the solution given by the book.
The problem in my opinion is that [itex] h [/itex] is a decreasing function of [itex] \theta [/itex]. This means that the height for losing contact at an angle [itex] \theta>0 [/itex] is less than
[itex] 18\, {\rm m} [/itex]. This in turn means that if the first hill is [itex] 18\, {\rm m} [/itex] above the second, the skier will definitely lose contact before reaching the crest of the second
So, if I'm correct, the proposed exercise is not very hard, and in a way it's nice, but it does not make a lot of sense.
Last edited:
Nice point, but it does only say the crest is circular. In the limit, that reduces to saying merely what the radius of curvature is there; saying it is "circular" is meaningless without specifying
some range, or saying something about the third derivative.
On that basis, the radius of curvature could be sufficiently greater at all previous points to retain contact.
I'm actually not very clear on what is meant here by "crest of the hill". I interpreted that as "the uppermost portion of the hill", not just its very top. That is, I assumed that the (section of
the) top of the hill can be described as a circular arc, which is what the figure suggests.
In that case, the skier will jump at the starting point of that circular arc (at the start of the range, if I get your meaning for this word right).
In a situation like the one depicted in the figure (which is not so unrealistic for a ski run, by the way) the skier will jump well below (and before) the top of the hill.
It would be interesting to find the function describing the convex part of the hill such that the contact is lost only at its highest point. I am not able to picture it. I cannot shake the idea that
the contact would be lost below the top anyway. But of course I can be wrong.
I'll think about that.
Last edited:
I believe that you are correct.
The value of the tangential velocity of the skier at point A is greater that at the top of the second hill, and being square, it has an important influence.
If we imagine the angle being at 90 degrees, the velocity of the skier would be greater at the h+r point, having no reason to deviate from a vertical trajectory, regardless of the r value.
FranzDiCoccio said:
I interpreted that as "the uppermost portion of the hill", not just its very top.
Quite so, but my point is that in the absence of any specified length of that portion it can be arbitrarily short.
FranzDiCoccio said:
It would be interesting to find the function describing the convex part of the hill such that the contact is lost only at its highest point.
Consider the boundary case, where it just remains in contact over some section. If you compare it with a certain other scenario you can see the answer immediately.
first of all, thanks for your time and help.
haruspex said:
Quite so, but my point is that in the absence of any specified length of that portion it can be arbitrarily short.
Ok, in a way the subtext of the original problem is "just focus on the very top of the hill, without bothering about what might happen elsewhere".
The fact that I'm not able to picture in my mind the shape of the hill such that nothing actually happens elsewhere is really nagging me :)
haruspex said:
Consider the boundary case, where it just remains in contact over some section. If you compare it with a certain other scenario you can see the answer immediately.
You make it sound pretty easy. There must be something I'm not seeing.
I'm assuming that the function giving the outline of the hill has two inflections, where the sign of [itex]y''[/itex] changes. Specifically, it is positive before the first inflection point, negative
between the first and the second inflection points and then again positive.
Of course, as long as [itex]y''>0[/itex] there is no problem with contact, because the centrifugal force is actually helping with it.
As soon as [itex]y''<0[/itex] the centrifugal force might cause the loss of contact. The problem is a bit messy, though, since different effects should be taken into account. One of them is of course
the change in the radius of curvature. But there is also the fact that the skier is climbing a slope, and hence his speed is decreasing with height. Finally, the fraction of the skier's weight
actually helping him with staying in contact with the snow depends on the slope of the hill.
I thought of tackling the problem through a differential equation, but at first sight it looks a bit complex, and right now I have not the time to look into it.
As I mention, your last comment seems to suggest that the answer is easy to find, but so far I'm not seeing it.At this point I'm just curious of what the shape of the hill would be.
FranzDiCoccio said:
At this point I'm just curious of what the shape of the hill would be.
As I understand it, you are searching for the shape of a curve where the supporting force from the hill is arbitrarily small. Ballistic, actually.
Last edited:
If there is no contact force the hill might as well not be there.
Ok so, the "other scenario" was a hill shape on which the contact force is absent.
Actually, I should have thought of that!
So if the central part of the hill is a parabolic arc, then there is a critical velocity for the skier at the inflection point connecting such arc with the concave part of the ski run.
If the skier is faster than that, he never touches the hill until he reaches the next inflection point.
If the skier is slower, he won't lose contact.
So I need to check that it's possible for the skier to be slower than the critical value at the inflection point, and yet faster than the critical value at the top of the hill.
(I edited below this point, because the previous version contained a few silly mistakes)
I did some quick calculations assuming that the top of hill is described by the function
[tex]y=-\frac{x^2}{2R}+R [/tex]
for [itex]-x_0\leq x\leq x_0[/itex], where [itex]R[/itex] is the radius of curvature of such shape at [itex]x=0[/itex]. If my calculations are right and that was a ballistic trajectory, the velocity
at [itex]\pm x_0[/itex] would be such that
[tex]v_0^2=2Rg \left(1+\frac{x_0^2}{R^2}\right) [/tex]
On the other hand, for the velocity at [itex]x=0[/itex] on the hill to be [itex]v^2=g R[/itex], it should be [itex]v_0^2=g R+g\frac{x_0^2}{R}[/itex] at [itex]-x_0[/itex].
Thus the condition to be met appears to be
[tex]g R+g\frac{x_0^2}{R}< 2Rg \left(1+\frac{x_0^2}{R^2}\right)=2 \left(gR+g\frac{x_0^2}{R}\right)[/tex]
which is always true.
After all this is also apparent from the expression of [itex]v_0[/itex].
At this point (even later in the night) it appears that with a parabolic hill, either the skier is always in contact with the snow (including at the top of the hill), or he is never in contact with
So i still cannot figure out the shape of the hill...
Better go to bed.
Many thanks to you all, I'll think again about this when I'm fresher.
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ok... the dirtbike jumps before the top... That is also my intuition. But I think the hills are not shaped for avoiding that effect.
I am wondering whether it's (theoretically) possible to build a hill where the motorbike would lose contact at the topmost point only, like the skier in the problem.
One possibility is perhaps to add a [itex]x^4[/itex] term in the function above. This should not change the curvature radius at the top of the hill, but changes its shape.
FranzDiCoccio said:
ok... the dirtbike jumps before the top... That is also my intuition. But I think the hills are not shaped for avoiding that effect.
I am wondering whether it's (theoretically) possible to build a hill where the motorbike would lose contact at the topmost point only, like the skier in the problem.
One possibility is perhaps to add a [itex]x^4[/itex] term in the function above. This should not change the curvature radius at the top of the hill, but changes its shape.
For a body
without self-propulsion
(like our skier), the profile of that uphill could be very close to the shape of the flight path of a projectile which launch angle is 90°-θ and launch velocity equals the skier's natural velocity at
point A.
The shape around the topmost point would need to have a smaller height than the vertex of the parabola (projectile flight path) in order to produce some "air time".
For a
self-propelled vehicle
, the energy produced by the engine or motor between point A and the topmost point would need to be considered.
Lnewqban said:
For a body without self-propulsion (like our skier), the profile of that uphill could be very close to the shape of the flight path of a projectile which launch angle is 90°-θ and launch velocity
equals the skier's natural velocity at point A.
The shape around the topmost point would need to have a smaller height than the vertex of the parabola (projectile flight path) in order to produce some "air time".
Yes, I understand this. Only, we cannot lower the topmost point alone, so the "lowering" (and the loss of contact) would start below that point. Of course we can argue that the "jumping point" is
arbitrarily close to the topmost point.
I was just trying to imagine a reasonably simple smooth function with the require property. But perhaps it does not exist.
For a self-propelled vehicle, the energy produced by the engine or motor between point A and the topmost point would need to be considered.
Ok, yes. In fact I think that with a self propelled vehicle we can safely assume that the hill can be climbed at the constant velocity that ensures the loss of contact exactly at the topmost point.
FAQ: Inconsistent problem about centrifugal/contact force
1. What is the difference between centrifugal force and contact force?
Centrifugal force is an apparent force that appears to act on an object moving in a circular path, while contact force is a force that results from direct physical contact between two objects.
2. Why is the problem of inconsistent centrifugal/contact force important?
The problem of inconsistent centrifugal/contact force is important because it can lead to errors in calculations and misunderstandings in the laws of motion. It is also crucial to understand these
forces in order to accurately design and operate machines and structures.
3. How can we resolve the inconsistency between centrifugal force and contact force?
The inconsistency between centrifugal force and contact force can be resolved by understanding that centrifugal force is not a fundamental force, but rather a result of inertia and the laws of
motion. By considering the forces acting on an object in motion, we can accurately calculate the contact forces and their effects.
4. Can centrifugal force and contact force act on the same object at the same time?
No, centrifugal force and contact force cannot act on the same object at the same time. Centrifugal force only appears to act on an object in circular motion, while contact force is a result of
direct physical contact between two objects.
5. How does the inconsistency between centrifugal force and contact force affect real-world applications?
The inconsistency between centrifugal force and contact force can affect real-world applications in various ways. Inaccurate calculations can lead to design flaws and safety hazards in machines and
structures. It can also cause confusion in understanding the behavior of objects in motion, which is crucial in fields such as engineering and physics. | {"url":"https://www.physicsforums.com/threads/inconsistent-problem-about-centrifugal-contact-force.1012684/","timestamp":"2024-11-09T17:20:23Z","content_type":"text/html","content_length":"153896","record_id":"<urn:uuid:3ae872a7-acab-4cbc-8f8a-9dc434a9e00e>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00554.warc.gz"} |
The Excel Date And Time Functions, Date And Time Functions In Excel - DED9
The Excel Date And Time Functions, Date And Time Functions In Excel
Four functions are the most widely used in Excel for counting days.
In each of the following functions, we first define that function, then explain the components and how that function works, at which point we must give the function values to perform the
calculations for us.
We will then show you an example of a combination of commands in these functions – one of the things you will learn in this step is how to write the formula, which includes steps for function names,
parentheses, comma separators, And other components of the command.
Note that arguments are always enclosed in parentheses, and each of these arguments is separated by a comma.
EDATE () function
EDATE is a function to display a date, several months in the past or future, using a record value for future dates and a negative value for past dates. For example, you can use this function to
calculate a retirement date or expiration date by calculating an individual’s age from the date of birth or adding a specific number of years to a specific date.
The arguments to this function are:
Start_date: Specify a date to show you the start date (of course, it must be in the correct Excel format in the form of consecutive numbers).
Months: Displays the number of months before or after the start date. And the format of the command is as follows:
= EDATE (start_date, months)
1- For example, here we want to calculate the retirement date for one of our colleagues. You need to open a new worksheet in Excel and in the title field, the names Name, Birthday, Retirement Date,
Time Left, and In Years for the column Enter A3, B3, C3, D3, and E3 on this page.
2- In columns A and B, enter the names and dates of birth.
3. Click on cell C4, then go to the Formulas> Data & Time tab and select the EDATE function from the list. You can also enter the formula EDATE () manually in cell C4, ie = EDATE (B4,12 * 62). If you
do not want to enter and calculate the number of months, then it is much easier to multiply 12 by the retirement age (ie, 12 months per year multiplied by the age of 62).
4. Copy the formula in cell C4 and paste it in cells C5 to C13, this way you will have the retirement date of each person. Read the next function, the YEARFRAC () function, to see how much time is
left for each employee.
YEARFRAC ()
Function This function calculates the number of days between two dates as a decimal number, which will lead to a fraction. This function has special uses because other Excel time and date functions
show you only one number.
Use this formula to calculate your retirement date to get an individual age from your date of birth, to be able to calculate the years of these dates, to make sure you get a percentage of the
completed year, and the uses There is so much more you can have.
Note that Excel uses whole days between two dates to calculate the fraction of the year as a decimal value.
The arguments used for this function are:
Start_date: (
start date)
End_date: (end-date)
Basis (optional): A type of base for your day (or code)
Then the command to write this is as follows:
= YEARFRAC (start_date, end_date, basis)
Although Excel allows you to put spaces between arguments, you still can not use these spaces between the function and the beginning of parentheses, ie there should be no space between the function
and parentheses. Most older timers immediately remove all spaces from their formulas to eliminate the chance of these errors occurring.
= YEARFRAC (TODAY (), C41)
Enter in cell D4: The start_date argument in this formula is TODAY () – which corresponds to a typical day. The TODAY () function shows you the date today. The date on cell C4 indicates the end date,
and the option is based on the number 1, which means the days of the month and the days of the year.
This may seem a bit confusing, but Excel gives you five options for calculating days and years in this formula.
Some accountants in Europe and the United States work on 30-day months and 360-day years; Other systems are based on the days of a month, but still consider a year to be 360 days. And some people
define days in months and years arbitrarily.
The codes for each of these modes are written as follows:
When You entered this function, if you hit any key (such as the space key) after the last argument (cell C4), Excel software will open a menu for you that gives you five options in this menu. Select
the appropriate code for your work from this list and press Enter.
If you do not enter the base number, the default number will be 0, which means 30 days of the month and 360 days of the year – which you are looking for in real-time. This is the default option for
you. not suitable.
The last column (E) in this spreadsheet shows what column D will look like when formatted as a fraction.
For example, E4 cells will be slightly less than 2 years old, while E5 cells are slightly older than 1 year old. E6 cell is equal to 4.1-1 years, cell E7 is equal to 4.8-3 years, and so on.
EOMONTH ()
Function Use this function to specify the date of the last day of the month (future or past). This is because when building a spreadsheet with many calculations, poems and calendars you can not
calculate such results using the formulas listed because these formulas give numerical results. This function will show a consecutive number that results in a specific date in Excel.
The arguments to this function are:
Start_date: A date that gives you the start date in a convenient sequential number format in Excel.
Months: This shows you the number of months before or after the start_date.
The coding for this function is = EOMONTH (start_date, months).
For the month’s argument, use positive numbers to indicate the future and negative numbers to indicate the past.
1. Enter ten dates in cells A4 to A13.
2. Enter the formula = EOMONTH (A4,1) in cell B4. Cell A4 is the date for that cell, and the number 1 here means one month. This means that the last day of the month in question is one month past the
date in cell A4.
Note that this program will show you a consecutive number in Excel.
3. Then, place your mouse cursor on cell B4 and press the function key F2 (Edit). Do this by moving the cursor over the word A in this formula and pressing the F4 function key three times in a row,
or as long as your formula looks like = EOMONTH ($ A4,1).4. For people who are not familiar with this feature, the dollar sign in front of the word A means that the source of the column will not
change with the copy of the formula. Now, when we copy this formula to cell A4 and drag it down, the rows of cells will change but the columns will remain the same.
5. Then, set the desired consecutive number in cell B4 in medium-long date format, ie Mon Feb 26, 2016 (meaning Monday, February 26, 2016).
6. Copy cell B4 down from B5 to B13. Note that the medium-long number format will also be copied with the formula.
7. Then, copy this formula in B4 to C4, D4, and E4. Edit each formula to show the new month numbers: Change the number 1 in cell C4 to 6. Change the number 1 in cell D4 to 12 and the number 1 in cell
E4 to 18. Now copy cells C4 to E4 into cells C13 to E13.
8. Remember to make the columns wide enough to accommodate new formats, then you will see how quickly you can find this information using this very useful function.
NETWORKDAYS.INTL ()
Function This function calculates the number of business days between two specified dates, other than weekends. This function is useful if you want to calculate the number of working days (or school
days) in a year, three months of a year, or a semester.
The difference between this function and other functions is that this function has options by which days can be counted as weekend days. It is not the case for everyone to be closed on Saturdays and
Sundays. Some are closed on Mondays and Tuesdays, others are closed on Wednesdays and Fridays. With this function, you can define these weekends for Excel to suit your personal schedule.
In addition, this function allows you to select and set your own rotation days in a specific time format.
For example, in the last quarter of the year of your annual calendar, there are two holidays in October, two holidays in November, and two holidays in December, of course, if you have Halloween and
Christmas Eve.
Also, take into account. If you do not take into account these two special nights, then you can set the number of fourth-month holidays from the last three months of the year to six.
The following commands are the codes by which you can define your weekend days to your liking:
Number of weekend days
1. Saturday, Sunday
2. Sunday, Monday
3. Monday, Tuesday
4. Tuesday, Wednesday
5. Wednesday, Thursday overnight
6. Thursday, Friday
7. Friday, Saturday
11. Only Sunday
12. ٫ Only Monday
13. ٫ Only Tuesday
14. ٫ Only Wednesday
15. ٫ Only Thursday
16. ٫ Only Friday
17. ٫ Only Saturday
If you leave this parameter blank (or undefined) in the order, as before Excel assumes it to be 1, meaning it counts your weekends like Saturdays and Sundays.
Enter the holidays as a range of cells, where you specify the main closing dates (such as cells F4 to F10) or a list of consecutive numbers that represent your actual closing dates.
The argument for this function is as follows:
Start_Date: Start Date
End_date: Date
Weekend: Set what days of the week to consider as weekends (additional parameter).
Holidays: A source that indicates that dates should be considered as days when the person is unemployed (additional parameter).
Coding related to the use of this function as
= NETWORKDAYS.INTL (start_date, end_date, [weekend], [holidays])
Is written.
1- On your spreadsheet, start Start Date, End Date, Number of Word Days in the tabs of columns A, B, C, F, and Enter G. Finally, enter the title Holidays (which, as shown above, is placed at the top
of both columns F and G in a merged or merged cell.)
2. Enter some random dates in columns A and B. Make sure the End Date column is not pre-dated.
3. Enter the number of random holidays (names and dates) in columns F and G.
4- Place your mouse cursor on cell C4 and go to the Formula tab of Excel and open the Data & Time option and select the NETWORKDAYS.INTL function from this menu.
5. In the Function Arguments dialog box, click inside the Start_Date field box, then click A4.
6. Hold down the Tab key on the End_Date field box and use your mouse to click on cell A4.
7. Hold down the Tab key on the Weekend field and enter one of the defined weekend codes in it (remember that the number 1 means Saturday and Sunday).
8. Press and hold the Tab key on the Holidays field, then select or highlight cells G4 to G11, and then click OK.
Before copying this formula from cell C4 into cells C5 through C11, use the function key F4 to positive the Holiday cells: Command
= NETWORK DAYS.INTL (A4, B4,1, $ G $ 4: $ G $ 11)
Enter so that the days in Holidays always change from G4 to G11. | {"url":"https://ded9.com/the-excel-date-and-time-functions-date-and-time-functions-in-excel/","timestamp":"2024-11-11T21:28:20Z","content_type":"text/html","content_length":"186986","record_id":"<urn:uuid:d0b6c462-6430-4bbb-8fea-e218cb171de4>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00534.warc.gz"} |
Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential
U. Biccari Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential, Math. Control Relat. F., Vol. 9, No. 1 (2019), pp. 191-219, DOI: 10.3934/mcrf.2019011
Abstract: We analyse controllability properties for the one-dimensional heat equation with singular inverse-square potential u_t-u_{xx}-\frac{\mu}{x^2}u=0\,\,\,(x,t)\in(0,1)\times(0,T) . For any \mu
<1/4 , we prove that the equation is null controllable through a boundary control f\in H^1(0,T) acting at the singularity point x=0 . This result is obtained employing the moment method by Fattorini
and Russell. | {"url":"https://cmc.deusto.eus/boundary-controllability-for-a-one-dimensional-heat-equation-with-a-singular-inverse-square-potential/","timestamp":"2024-11-06T17:22:10Z","content_type":"text/html","content_length":"81782","record_id":"<urn:uuid:68fcb4d5-a49f-483f-9604-24568d6f169c>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00600.warc.gz"} |
numpy.pmt(rate, nper, pv, fv=0, when='end')[source]¶
Compute the payment against loan principal plus interest.
☆ a present value, pv (e.g., an amount borrowed)
☆ a future value, fv (e.g., 0)
☆ an interest rate compounded once per period, of which there are
☆ nper total
☆ and (optional) specification of whether payment is made at the beginning (when = {‘begin’, 1}) or the end (when = {‘end’, 0}) of each period
the (fixed) periodic payment.
rate : array_like
Rate of interest (per period)
nper : array_like
Number of compounding periods
pv : array_like
Present value
fv : array_like, optional
Future value (default = 0)
when : {{‘begin’, 1}, {‘end’, 0}}, {string, int}
When payments are due (‘begin’ (1) or ‘end’ (0))
out : ndarray
Payment against loan plus interest. If all input is scalar, returns a scalar float. If any input is array_like, returns payment for each input element. If multiple inputs are
array_like, they all must have the same shape.
The payment is computed by solving the equation:
fv +
pv*(1 + rate)**nper +
pmt*(1 + rate*when)/rate*((1 + rate)**nper - 1) == 0
or, when rate == 0:
fv + pv + pmt * nper == 0
for pmt.
Note that computing a monthly mortgage payment is only one use for this function. For example, pmt returns the periodic deposit one must make to achieve a specified future balance given an
initial deposit, a fixed, periodically compounded interest rate, and the total number of periods.
Wheeler, D. A., E. Rathke, and R. Weir (Eds.) (2009, May). Open Document Format for Office Applications (OpenDocument)v1.2, Part 2: Recalculated Formula (OpenFormula) Format - Annotated
[WRW] Version, Pre-Draft 12. Organization for the Advancement of Structured Information Standards (OASIS). Billerica, MA, USA. [ODT Document]. Available: http://www.oasis-open.org/committees/
documents.php ?wg_abbrev=office-formulaOpenDocument-formula-20090508.odt
What is the monthly payment needed to pay off a $200,000 loan in 15 years at an annual interest rate of 7.5%?
>>> np.pmt(0.075/12, 12*15, 200000)
In order to pay-off (i.e., have a future-value of 0) the $200,000 obtained today, a monthly payment of $1,854.02 would be required. Note that this example illustrates usage of fv having a default
value of 0. | {"url":"https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.pmt.html","timestamp":"2024-11-11T20:27:48Z","content_type":"text/html","content_length":"12663","record_id":"<urn:uuid:73733e94-980b-414e-8f87-68fd4492beff>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00401.warc.gz"} |
Bayes' Theorem - Explained
What is Bayes' Theorem?
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What is Bayes' Theorem?
Bayes' theorem refers to a mathematical formula used to determine conditional probability. The theorem was named after Thomas Bayes, an 18th-century British mathematician. This theorem offers a
method of revising existing theories or predictions given new or even additional evidence. Bayes' theorem, in finance, can be utilized in rating the risk involved in lending money to potential
borrowers. The formula goes thus: Bayes' theorem is also referred to as Bayes' Law or Bayes' Rule.
How is Bayes' Theorem Used?
The theorem's applications are extensive and not restricted to the financial sphere. For instance, Bayes' theorem can be utilized in determining how accurate medical test results are by considering
how possible any specific individual is to have a disease, as well as, the test's general accuracy. Bayes' theorem gives the likelihood of an event dependent on the information which is or might be
related to that event. The formula can be utilized in seeing how the probability of an event happening is affected by entirely new information if the new information is true. For instance, say one
card is drawn from a full deck of 52 cards. The probability of the card being a king is 4 divided by 52, which is equal to 1/13 or approximately 7.69%. Keep in mind that 4 kings exist in the deck
card. Assume it's revealed that the chosen card is a face card. The probability of the selected card being a king, given it's a face card, is 4 divided by 12, or approximately 33.3%, as a deck has 12
face cards. Bayes' theorem follows from the principle of conditional probability. Conditional probability refers to the probability of an event considering that another event occurred. For instance,
an easy probability question might be "What's the probability of Amazon.com, Inc., (AMZN) stock price falling? " This question is taken a step further by conditional probability, in that it asks
"What's the probability of Amazon stock price falling considering the fact that the Dow Jones Industrial Average index had fallen earlier?" A's conditional probability given considering that B has
occurred can be expressed thus: P(A|B) = P(A and B) / P(B) = P(AB) / P(B) If A stands for Amazon price falls and B, DJIA is already down, the conditional probability expression would read as "the
probability that Amazon drops given a decline in DJIA equals the probability that Amazon price declines and also DJIA falls over the probability of a DJIA index decrease. The probability of A, as
well as, B occurring is P(AB). It's the same as the probability of A occurring multiplied by the probability that B occurs considering that A occurred, shown as P(A) x P(B|A). Making use of the same
rationale, P(AB) is also the probability that B occurs multiplied by the probability that A occurs considering that B occurs, shown as P(B) x P(A|B). The fact that the two expressions are equal
brings about the Bayes' theorem and it's written as: if P(AB) = P(A) x P(B|A) = P(B) x P(A|B) then, P(A|B) = [P(A) x P(B|A)] / P(B). Where P(A) and P(B) are A and B's probabilities with no regard to
each other. P(B|A) is the probability of B occurring given A is true. Finally, the conditional probability of A occurring given that B is true is P(A|B). This formula explicates the relationship
existing between the hypothesis' probability before getting the evidence P(A) and then the hypothesis' probability after getting the evidence P(A|B), given hypothesis A, as well as, evidence B.
Another instance, imagine that a drug test exists which is 98% accurate which means that 98% of the time its result is positive for someone taking the drug and its result is negative 98% of the time
for nonusers of the drug. Next, assume the drug is used by 0.5% of the people. If someone selected randomly, tests positive to the drug, the calculation below can be made to ascertain the probability
of the person being an actual user of that drug. (0.98 x 0.005) / [(0.98 x 0.005) + ((1 - 0.98) x (1 - 0.005))] = 0.0049 / (0.0049 + 0.0199) = 19.76%. | {"url":"https://thebusinessprofessor.com/insurance-risk-law/bayes-theorem-definition","timestamp":"2024-11-08T17:55:30Z","content_type":"text/html","content_length":"98593","record_id":"<urn:uuid:b9be861e-7542-4eca-b9d8-b86605043cf9>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00775.warc.gz"} |
Problem-solving/decision-making tool for students/teachers. Free until June 2009, but only $299 for site license. This is a great web-based tool for helping students analyze complex issues. Most
appropriate for middle and high school
Create Rube Goldberg-like contraptions in this flash game
Columbus State University’s College of Education is now the home of the Problem of the Week math contest, which was started in 1996 by David Rock and Doug Brumbaugh. The math educators also supply
problems and answers that are part of the current White House Math Challenge, a presidential effort to boost interest in math and foster problem-solving skills. Visit our history page for more on
these and the companion problem-solving sites here — Algebra in Action, Middle School Madness and Elementary Brain Teaser.
A growing library of mathcasts. Mathcasts are recorded movies created with a smart board with audio narration. Students can see and hear how a given problem is solved. Resources available for Grades
knann (via)
Problem solving and Writing prompts for math for grades K-12.
This interactive program teaches children how to visualize and solve math word problems. Using virtual blocks and cubes, children create models that illustrate the underlying math concepts in word
problems. Each Thinking Blocks program contains six guided practice sets and three assessment tests. The addition and subtraction program features models that represent part-whole, comparison, and
change situations. The multiplication and division program introduces equal parts, comparisons, and interpreting remainders. Program features include: • Self paced, guided instruction that shows
students how to correctly model math word problems • Independent practice sets that challenge students to apply what they’ve learned • Interactive blocks that engage students in the problem solving
process • Randomized problem sets that create a different learning experience each time the program is accessed • Video tutorials which help students transition from building concrete models to
sketching models on paper • Printable certificates that allow students to keep a record of the work they have completed
Mathcasts can Help students learn & review math. Help teachers collaborate & improve their teaching. Help parents help their children and enable them to see examples of their childrens' work.
Mathcasts.org was created to give students a library of math tutorials and problem solutions and to give teachers a place to share their methods for teaching & learn from others. It's also a place
where students & teachers can contribute and organize sets of movies for others or themselves to use.
Get help with Alegebra 1 topics. Multiple problem samples demonstrating how to solve the equation with visual animation and audio explanation. Flash required.
1 other
A free site that allows people to post descriptions of their problems along with solutions, and allows those solutions to be voted on. Neat!
Math resource of video clips devoted to mathematics learning! Students call in to the tv host with a math problem. Together the problem is solved step-by-step.
knann (via)
Use the broken calculator to solve your math problems in a creative way. Can you add 2+2 if the addition key is broken? Sure can..how about (3*2)-2 . The answer is the same! Create some challenges
for you, your friends, and teachers!
The rules of the game are simple: each of the nine blocks has to contain all the numbers 1-9 within its squares. Each number can only appear once in a row, column or box. Every puzzle has just one
correct solution.
1 other (via)
With Scholastic's Global Classport, you can communicate with classrooms in 182 countries, and collaborate with teachers around the world. Note: Remember that your username must include "_scholastic"
. Use to connect with other classrooms to solve Math Maven's Mysteries
knann (via)
This content resource is an index of links to interactive sites, challenging students to solve mysteries using a variety of math principles. Students read the stories, solve the problems, and answer
the questions, using clues embedded in the stories to discover the solution. Included are links to a variety of teacher resources.http://teacher.scholastic.com/maven/tguide.htm
knann (via)
Absurd Math is an interactive mathematical problem solving game series. The player proceeds on missions in a strange world where the ultimate power consists of mathematical skill and knowledge.
Challenging math problems for you to solve. Email your answer to find out if you are right!
Activities to explore and engage with mathematical ideas. Provides challenging activities which provide students with opportunities to develop their mathematics.
knann (via)
After researching place value and numeration systems, students create a base-4 numeration system for a primitive alien tribe.
This web-based unit will introduce students to strategies and real-life practice activities for applying problem-solving skills. Students will investigate real problems through a variety of websites
and group activities.
The students will measure the diameter and circumference of many circles and will estimate their relationship. By doing so, they should discover the approximate value of pi. By discovering it on
their own they should feel rewarded and gain a deeper understanding of the fundamental concept of pi.
Read the word problem and choose the best strategy for solving the problem.
Variety of math and problem solving puzzles. Reading required for instructions. Some games appropriate for grade 3 with instruction
How many disguises can you make for each character. Do you notice a pattern?
Great number sequence game for whole class instruction. Allows for a variety of patterns that are teacher selected.
1 other (via)
You must follow the rules to get throught the Cave maze. Choose easy, medium, hard. Improve your problem-solving with this game! | {"url":"https://blogmarks.net/marks/tag/problem+solving","timestamp":"2024-11-09T16:13:45Z","content_type":"application/xhtml+xml","content_length":"69891","record_id":"<urn:uuid:b9801041-ef81-4a5b-8d95-191ffbaa1a39>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00218.warc.gz"} |
ML: Clustering
ML: Clustering¶
Clustering is one of the types of unsupervised learning. It is similar to classification: the aim is to give a label to each data point. However, unlike in classification, we are not given any
examples of labels associated with the data points. We must infer from the data, which data points belong to the same cluster. This can be achieved using some notion of distance between the data
points. Data points in the same cluster are somehow close to each other.
One of the simplest clustering methods is the k-means clustering. It aims at producing a clustering that is optimal in the following sense:
• the centre of each cluster is the average of all points in the cluster
• any point in a cluster is closer to its centre than to a centre of any other cluster
The k-means clustering is first given the wanted number of clusters, say k, as a hyperparameter. Next, to start the algorithm, k points from the data set are chosen randomly as cluster centres. Then
the following phases are repeated iteratively:
• any data point is set to belong to a cluster, whose centre is closest to it
• then for each cluster a new centre is chosen as the average of the data points in the cluster
This procedure is repeated until the clusters no longer change. This kind of algorithm is called an Expectation-Maximization (EM) algorithm, which is known to converge.
Simple example¶
The scikit-learn library has an implementation of the k-means algorithm. Let’s apply it to a set of randomly generated blobs, whose labels we throw away.
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from sklearn.datasets import make_blobs
X,y = make_blobs(centers=4, n_samples=200, random_state=0, cluster_std=0.7)
[[ 2.26403424 1.82613379]
[-0.97647444 2.59138889]
[ 1.10046838 4.02254067]
[-2.82715074 7.11970523]
[ 1.53393915 0.31915668]
[ 0.98362009 5.55389667]
[-1.74452433 2.98606238]
[ 0.35482006 2.9172298 ]
[ 1.83747356 5.14545322]
[ 1.48663347 4.39407536]] [1 2 0 3 1 0 2 0 0 0]
Now we plot these points, but without coloring the points using the labels:
We can still discern four clusters in the data set. Let’s see if the k-means algorithm can recover these clusters. First we create the instance of the k-means model by giving it the number of
clusters 4 as a hyperparameter.
from sklearn.cluster import KMeans
model = KMeans(4)
[[ 0.86008475 4.31501411]
[-1.36512238 7.70188306]
[ 2.07464749 0.9869902 ]
[-1.70639178 2.9104771 ]]
plt.scatter(X[:,0],X[:,1], c=model.labels_);
plt.scatter(model.cluster_centers_[:,0], model.cluster_centers_[:,1], s=100, color="red"); # Show the centres
The clustering looks more or less correct. To get a more quantitative measure of success we can get the accuracy score.
from sklearn.metrics import accuracy_score
acc=accuracy_score(y, model.labels_)
print("Accuracy score is", acc)
Oops! Even though the clusters could match almost perfectly to the original, their labels might be permuted. Let’s select randomly one point from each cluster and check their labels from the original
data labels. Then we use this label for the whole cluster. In essence, we are renaming the clusters, not re-clustering the data.
import scipy
def find_permutation(n_clusters, real_labels, labels):
for i in range(n_clusters):
idx = labels == i
new_label=scipy.stats.mode(real_labels[idx])[0][0] # Choose the most common label among data points in the cluster
return permutation
permutation = find_permutation(4, y, model.labels_)
new_labels = [ permutation[label] for label in model.labels_] # permute the labels
print("Accuracy score is", accuracy_score(y, new_labels))
So, the k-means algorithm seems to work well in this case, but there can be several problems. Firstly, even though an EM algorithm always converges, it might converge to a local maximum. To avoid
this, EM type algorithms are usually run several times, each time starting from different random initial values. For instance, in the scikit-learn implementation, the algorithms is restarted by
default 10 times.
More complicated example¶
The k-means algorithm can have difficulties when the clusters are not convex shapes:
from sklearn.datasets import make_moons
X,y = make_moons(200, noise=0.05, random_state=0)
plt.scatter(X[:,0], X[:,1]);
plt.scatter(X[:,0], X[:,1], c=model.labels_);
The clustering does not work well now, since it is not possible to separate the two clusters with a line. We could embed this data set into a higher dimensional space, where the separation is
possible. And then apply the k-means clustering.
Alternatively, we can use a different type of clustering algorithm for this case. The DBSCAN algorithm is based on densities and works well on data whose density in the clusters is uniform.
from sklearn.cluster import DBSCAN
model = DBSCAN(eps=0.3)
plt.scatter(X[:,0], X[:,1], c=model.labels_);
The good news is that DBSCAN does not require the user to specify the number of clusters. But now the algorithm depends on another hyperparameter: a threshold for distance (here 0.3).
Clustering digits¶
Using scikit-learn we can download a set of 1797 images of handwritten digits with the correct labels 0,1,…,9. The images have quite a low resolution: 8*8=64 pixels. Let’s see how our machine
learning method works with this kind of data.
from sklearn.datasets import load_digits
digits = load_digits()
To get an idea how these data points look like, we plot first ten of these.
fig, axes = plt.subplots(2,5, subplot_kw=dict(xticks=[], yticks=[]))
for ax, digit in zip(axes.flat, digits.data[:10]):
ax.imshow(digit.reshape(8,8), cmap="gray")
Let’s cluster these data points into ten clusters.
model=KMeans(n_clusters = 10, random_state=0)
So, we have ten cluster centres, which are images with 8x8=64 pixels in them. We can have a look at their appearence:
fig, axes = plt.subplots(2,5, subplot_kw=dict(xticks=[], yticks=[]))
for ax, digit in zip(axes.flat, model.cluster_centers_):
ax.imshow(digit.reshape(8,8), cmap="gray")
One can recognize these numbers with the exception of maybe number eight. What is the accuracy score of this clustering?
permutation3 = find_permutation(10, digits.target, model.labels_)
acc = accuracy_score(digits.target, [ permutation3[label] for label in model.labels_])
print("Accuracy score is", acc)
[4, 3, 5, 9, 7, 0, 1, 8, 2, 6]
Accuracy score is 0.793544796884
This is quite a good result for such a simple algorithm!
Using the same iris data set that you saw earlier in the classification, apply k-means clustering with 3 clusters. Create a function plant_clustering that loads the iris data set, clusters the data
and returns the accuracy_score.
This exercise can give four points at maximum!
Read the tab separated file data.tsv from the src folder into a DataFrame. The dataset has two features X1 and X2, and the label y. Cluster the feature matrix using DBSCAN with different values for
the eps parameter. Use values in np.arange(0.05, 0.2, 0.05) for clustering. For each clustering, collect the accuracy score, the number of clusters, and the number of outliers. Return these values in
a DataFrame, where columns and column names are as in the below example.
Note that DBSCAN uses label -1 to denote outliers , that is, those data points that didn’t fit well in any cluster. You have to modify the find_permutation function to handle this: ignore the outlier
data points from the accuracy score computation. In addition, if the number of clusters is not the same as the number of labels in the original data, set the accuracy score to NaN.
eps Score Clusters Outliers
0 0.05 ? ? ?
1 0.10 ? ? ?
2 0.15 ? ? ?
3 0.20 ? ? ?
Before submitting the solution, you can plot the data set (with clusters colored) to see what kind of data we are dealing with.
Points are given for each correct column in the result DataFrame.
Hierarchical clustering¶
Hierarchical clustering works by first putting each data point in their own cluster and then merging clusters based on some rule, until there are only the wanted number of clusters remaining. For
this to work, there needs to be a distance measure between the data points. With this distance measure d, we can define another distance measure between the clusters U and V using one of the
following methods (linkages):
• single: \(d(U, V) := \min_{u \in U, v \in V} d(u,v)\)
• complete: \(d(U, V) := \max_{u \in U, v \in V} d(u,v)\)
• average: \(d(U, V) := \sum_{u \in U, v \in V} \frac{d(u,v)}{|U||V|}\)
• ward: tries to minimize the variance in each cluster
At each iteration of the algorithm two clusters that are closest to each other are merged. After this the distance between the clusters are recomputed, and then it continues to the next iteration.
Below is an example with a botanical dataset with 150 samples from three species. Each species appears in the dataset 50 times. Each sample point has 4 features, which are basically dimensions of the
“leaves” of the flower.
We use the seaborn library to both to compute the clustering and to visualize the result. The visualization consists of two parts: the heatmap, whose rows and/or columns may be reordered so as to
have the elements of the same cluster next to each other; and the dendrogram, which shows the way the clusters were merged. The colors give the length of the corresponding features.
import seaborn as sns; sns.set(color_codes=True)
iris = sns.load_dataset("iris")
species = iris.pop("species") # Remove the species column
print(species.unique()) # The samples seems to be from these three species
sns.clustermap(iris, method="ward", col_cluster=False, cbar_kws={'label': 'centimeters'}); # Cluster only the rows
#plt.colorbar().ax.set_title('This is a title')
['setosa' 'versicolor' 'virginica']
With sharp eye and good will one can discern three clusters in the above heatmap and dendrogram.
This exercise can give three points at maximum!
A binding site is a piece of DNA where a certain protein prefers to bind. The piece of DNA can be described as a string consisting of letters A, C, G, and T, which correspond to nucleotides Adenine,
Cytosine, Guanine, and Thymine. In this exercise the length of binding sites is eight nucleotides. They are stored in the file data.seq, and the binding sites there are classified into two classes.
Part 1. Write function toint that converts a nucleotide to an integer. Use the following mapping:
A -> 0
C -> 1
G -> 2
T -> 3
Write also function get_features_and_labels that gets a filename as a parameter. The function should load the contents of the file into a DataFrame. The column X contains a string. Convert this
column into a feature matrix using the above toint function. For example the column ["GGATAATA","CGATAACC"] should result to the feature matrix
The function should return a pair, whose first element is the feature matrix and the second element is the label vector.
Part 2. Create function cluster_euclidean that gets a filename as parameter. Get the features and labels using the function from part 1. Perform hierarchical clustering using the function
sklearn.cluster.AgglomerativeClustering. Get two clusters using average linkage and euclidean affinity. Fit the model and predict the labels. Note that you may have to use the find_permutation
function again, because even though the clusters are correct, they may be labeled differently than the real labels given in data.seq. The function should return the accuracy score.
Part 3. Create function cluster_hamming that works like the function in part 2, except now using the hamming affinity. Even though it is possible to pass the function hamming to
AgglomerativeClustering, let us now compute the Hamming distance matrix explicitly. We can achieve this using the function sklearn.metrics.pairwise_distances. Use the affinity parameter precomputed
to AgglomerativeClustering. And give the distance matrix you got from pairwise_distances, instead of the feature matrix, to the fit_predict method of the model. If you want, you can visualize the
clustering using the provided plot function.
Which affinity (or distance) do you think is theoretically more correct of these two (Euclidean or Hamming)? Why? | {"url":"https://csmastersuh.github.io/data_analysis_with_python_spring_2020/clustering.html","timestamp":"2024-11-09T00:04:43Z","content_type":"text/html","content_length":"55734","record_id":"<urn:uuid:ba0a7deb-80db-48bb-ad5f-737bebc2bb8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00576.warc.gz"} |
The Legacy of Machin-like Formulas in Mathematics
Learn about Machin-like formulas and their significance in precise calculations.
― 5 min read
Table of Contents
In the 1700s, a mathematician named John Machin found a way to calculate a famous number using a specific technique involving fractions and angles. This method allowed for accurate calculations long
before computers were invented. Over the years, other mathematicians have built on Machin's idea, discovering various formulas that can also calculate this number with great Precision.
The Basics of Arctangents
Arctangent is a mathematical function that helps us understand the angles related to certain fractions. When we take the arctangent of a fraction, we can express it in another form. This means that
if we have a set of fractions, we can combine their arctangents into a single arctangent of a new fraction. This property is key to forming the Machin-like formulas.
What is a Machin-like Formula?
A Machin-like formula is an expression that uses the arctangent function to represent a specific number. The simplest example of this is when we only use one term. For instance, using just one
fraction gives what we call a single-term Machin-like formula. However, for most calculations, we subtract several arctangents step by step until we reach a desired accuracy.
Constructing the Formula
To create these formulas, we start with a chosen fraction and then subtract the arctangents of selected fractions until we reach a result that is close enough to the number we want. Each time we
subtract, we look at the difference and adjust our next fraction based on this value. This process continues until we have created a full Machin-like formula.
The Importance of Precision
Knowing the required accuracy for a calculation is crucial. When we work with these formulas, we can choose which terms to include based on how precise we need our final answer to be. This allows us
to disregard any terms that won’t significantly change the outcome, which helps keep calculations manageable, even if we are working with big numbers.
The Role of Computers
While the methods for creating these formulas are straightforward, the actual Computations can become cumbersome as numbers grow larger. As such, computers play a vital role in performing these
calculations efficiently. We can write programs that use the rules we’ve established to compute these formulas quickly, even when handling very large integers.
Utilizing Python for Computation
Python is a programming language that is great for mathematical calculations. By using Python, we can build a program that takes our starting fraction and constructs the corresponding Machin-like
formula. The program works by following the steps we’ve discussed, adding and adjusting terms based on the requirements for precision.
Understanding Integer Values
When working with these formulas, we often need to round numbers to the nearest whole number. We refer to this as the floor function (getting the largest whole number less than or equal to the value)
and the ceiling function (the smallest whole number greater than or equal to the value). These functions help maintain precision and ensure our calculations fit the expected results.
The Growth of Terms
As we add more terms to our Machin-like formula, we notice that the differences between the numbers decrease rapidly. This means that even if we start with large numbers, the subsequent terms we add
become smaller and smaller. This rapid decrease is beneficial because it means we can sufficiently represent the desired number with only a few terms.
The Challenge of Convergence
One challenge with Machin-like formulas is that sometimes the process of reaching the desired result can be slow. Although we can make our calculations very precise, we must be aware that not every
method will yield quick results. Finding a balance between speed and accuracy is important.
Partial Machin-like Formulas
In some cases, it may be necessary to stop a calculation before fully completing the formula. When this happens, we have what we call a partial Machin-like formula. These are still useful because
they provide an approximation, even if they don’t include all the terms.
Analyzing the Results
After carrying out the calculations, we can analyze the output of our Python program. It gives us details about the starting multipliers, denominators, and the accuracy of our results in the form of
Lehmer's measure, which helps us understand how well our formula performs.
Numerical Experiments
We can conduct experiments with various starting fractions to see how our formulas behave. These tests give insight into how effective our methods are at producing accurate results. By comparing our
findings with known values, we can validate our techniques and improve them further.
In summary, Machin-like formulas provide a powerful method for calculating important mathematical constants. With the aid of computers and programming languages like Python, it is possible to perform
these calculations with remarkable accuracy. By understanding the properties of arctangents and how to construct these formulas step by step, we can achieve our goals in mathematical computation
effectively. The ongoing work in this field continues to refine our approaches, making it easier to deal with large numbers and complex calculations.
Original Source
Title: A Rapidly Converging Machin-like Formula for $\pi$
Abstract: We present a simple recurrent formula to generate the Machin-like expression for calculating $\pi/4$. The method works for any denominator in the starting term and always provides a finite
decomposition. We show that the terms in the Machin-like formula decrease so rapidly that the Lehmer's measure can be made arbitrarily small only by selecting the first term. We introduce the concept
of the partial Machin-like formula. While the growth of the integer numbers may quickly render the computer implementation impractical, the same reason restricts the total contribution of the high
terms. If the required precision is known in advance, the subset of the expression may be selected to satisfy it. We also present the Python program to compute the terms of the Machin-like formula
(full and partial), and its Lehmer's measure.
Authors: Oleg S. Alferov
Last Update: 2023-11-29 00:00:00
Language: English
Source URL: https://arxiv.org/abs/2403.09654
Source PDF: https://arxiv.org/pdf/2403.09654
Licence: https://creativecommons.org/licenses/by/4.0/
Changes: This summary was created with assistance from AI and may have inaccuracies. For accurate information, please refer to the original source documents linked here.
Thank you to arxiv for use of its open access interoperability.
Referenced Topics | {"url":"https://simplescience.ai/en/2023-11-29-the-legacy-of-machin-like-formulas-in-mathematics--a8kyge","timestamp":"2024-11-05T02:56:01Z","content_type":"text/html","content_length":"80784","record_id":"<urn:uuid:6b50c7fc-92d3-4efe-90ce-f102ff2ef416>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00108.warc.gz"} |
Application Strategies of Model Predictive Control for the Design and Operations of Renewable Energy-Based Microgrid: A Survey
Graduate School of Engineering and Science, University of the Ryukyus, 1 Senbaru, Nishihara 903-0213, Okinawa, Japan
Department of Electrical and Electronic Engineering, First Technical University, Ibadan 200255, Nigeria
Department of Electrical Power and Machines, Zagazig University, Zagazig 44519, Egypt
Department of Electrical and Electronics Engineering Science, University of Johannesburg, Johannesburg 2006, South Africa
Authors to whom correspondence should be addressed.
Submission received: 22 November 2021 / Revised: 19 December 2021 / Accepted: 27 December 2021 / Published: 12 February 2022
In recent times, Microgrids (MG) have emerged as solution approach to establishing resilient power systems. However, the integration of Renewable Energy Resources (RERs) comes with a high degree of
uncertainties due to heavy dependency on weather conditions. Hence, improper modeling of these uncertainties can have adverse effects on the performance of the microgrid operations. Due to this
effect, more advanced algorithms need to be explored to create stability in MGs’. The Model Predictive Control (MPC) technique has gained sound recognition due to its flexibility in executing
controls and speed of processors. Thus, in this review paper, the superiority of MPC to several techniques used to model uncertainties is presented for both grid-connected and islanded system. It
highlights the features, strengths and incompetencies of several modeling methods for MPCs and some of its variants regarding handling of uncertainties in MGs. This survey article will help
researchers and model developers to come up with more robust model predictive control algorithms and techniques to cope with the changing nature of modern energy systems, especially with the
increasing level of RERs penetration.
1. Introduction
Due to increased technological advancement, there has been a drastic increase in the demand for global energy. This demand has been geared towards not only making energy globally available but also
to ensure an adequate and reliable supply of power devoid of interruptions. The role of the United Nations in trying to make energy accessible and sustainable for all, including the Paris Agreement,
which has led to a great investment and hence increased thirst for clean, reliable, and sustainable power supply [
]. The traditional system of power supply has not been in conjunction with the above global consent as it uses fossil fuels to produce power which is one of the main sources of carbon emission, hence
negatively impacting the environment. The global desire to produce clean energy has been backed up with improved technologies that have re-branded the ideology of the energy sector from being
completely fossil fuel-based to a mixture of both renewable energy-based Distributed Energy Resources (DERs) and clean-burning fossil fuel-based generators.
The continuous use of fossil fuels in the conventional power system has posed a serious threat to the oil and gas reserves and it has been verified that in the next couple of years most of these
non-replenishable natural occurring energy resources will be completely depleted. To heal the globe of this situation, global renewable energy markets have evolved massively within the last few
decades [
]. DERs comprise Renewable Energy Resources (RERs), conventional generators, thermoelectrics, Energy Storage Systems (ESSs), and sufficient ancillary facilities for effective energy supply management
and control. RERs (wind, solar biomass, geothermal, etc.) are resources that are harnessed from nature and has the ability to replenish themselves. However, they tend to be limited in flow and amount
at any given period of time. A conventional generator uses either propane, gasoline, or diesel to supply electricity via the alternator. Thermoelectric devices produce electricity based on
temperature differences while energy storage systems store energy produced at a certain period and use them at a later period to avoid mismatch between demand and supply. RERs are majorly introduced
into the conventional power systems to reduce the environmental pollution effects of burning fossil fuels and to produce clean energy that is more environmentally-friendly. Although many countries
are adopting the applications of green technologies, however, some major world leaders are still reliant on fossil fuels due to policies and regulations. If renewable energy or green technology is to
replace fossil fuels which are the major source of carbon emission in power generation, there must be lucrative and result-oriented based proposals to convince the regulatory bodies and other market
players in the fossil fuel business so that a competitive and enabling environment can be created for all [
]. Both the global renewable energy transition and growth capacity by continent are shown in
Figure 1
Figure 2
Apart from the environmental sustainability benefit of DERs development, there are credible technical benefits in terms of loss reduction, reducing the cost of energy transmission, voltage profile
improvement, etc. One significant concept for effective DERs deployment is the concept of Microgrids (MGs) [
]. The definition of MG depends on individual particular views: The DERs planning point of view differs from the control perspective and MG can also be defined based on each DER important
characteristics. Firstly, MG can be defined as an integration of DERs and loads, and secondly, MG can be defined as a controllable entity that can operate in either grid-connected or islanded mode.
According to the International Council of Large Electrical Systems (CIGRE), MGs are defined as low voltage distribution networks consisting of interconnected loads (controllable and critical) and
DERs that can work either as a single or multi-controllable unit being connected to a grid network or islanded [
A typical MG consists of RERs like solar PV, wind turbines, fuel cells, Combined Heat and Power units (CHP), solar thermal units, hydropower, and conventional power sources like diesel generators and
gas turbines. About 34 percent of the world’s MG establishment can be found in the United States due to the aging energy system being heavily reliant on fossil fuels. MGs have replaced the fossil
system for security and reliability. The Asia Pacific takes about 40 percent of the total world MG capacity [
]. Since the world is moving towards getting reliable and sustainable power supply, MGs must be capable of effectively and efficiently supplying power. Many modern control techniques have evolved to
help MGs successfully achieve their goals. Control techniques help MGs to deliver the required power due to the intermittencies that emerge from renewable energies. Model predictive control has been
recognized immensely over the recent decade due to its ability to handle and process multiple disturbances that stem from RERs.
Motivation of the Study and Research Gap
Without a doubt, the high penetration of RESs into MGs has helped to reduce environmental problems and financial situations of power systems; it, however, comes with a huge negative repercussion.
Their availability depends on climatic conditions (rainy/cloudy or sunny days) making them very unpredictable and unreliable. This intermittency has created huge uncertainties in MG operations like
voltage and frequency regulations, increased faults, and difficult protection strategies. Hence, maintaining MG system stability, reliability and protection become a challenging issue and a crucial
area of research focus. Power stability issues are very low in conventional large power systems compared to small low voltage power systems (MGs) owning to the fact that large systems have a
self-stabilizing effect from the frictional force inherent, which is not the case for a small decentralized power system [
]. However, in low voltage decentralized power systems like MGs, the need for system control is very huge due to the numerous instability issues that accompanied the intermittent nature of the RERs.
Predictive control in large power systems is very difficult due to large constraints and complex computational analyses but forecasting of RERs and loads is better and cheap compared to small low
voltage systems where the accuracy of the forecast results tend to be of poor quality if the process is said to be executed [
]. These uncertainties have created a dynamic environment, making MGs operation very difficult and requiring advanced control techniques to solve stability issues.
Many advanced computational control techniques have been carried out to deal with uncertainties or to model uncertainties in MG operations but few have had satisfactory results in predicting or
forecasting demand and RERs [
] and if they do, a majority do not compensate for errors in the prediction or forecasting process [
]. This gap has created a lot of disturbances in MG operations. Some renowned approaches have been used to model these disturbances which involve assigning probabilities distributions to solve
uncertainty issues but the results have not been quite successful in handling the uncertainties due to the computational burden of generated scenarios [
]. Other optimization approaches have also had an average degree of success in handling these uncertainties due to the conservativeness of the result [
]. Another major issue is the idea of using groups of uncertainty sets for different uncertainty cases [
] instead of using a general or comprehensive uncertainty treatment. This will however lead to the introduction of certain parameters or variables to control the sensitivity of the optimization
process [
Few research works have entirely considered all necessary uncertainties; the majority of those that claim to include all uncertainties could be on either “all demand” or “all supply” without
considering all uncertainties in both demand and supply [
]. Error in the forecasting of loads and RESs is very prominent among these control techniques [
]. There is however a need for more reliable and accurate control algorithm(s) that can measure uncertainties in MGs. Hence, one of the key control techniques that have gained an enormous reputation
in MG operations is Model Predictive Control Algorithm (MPC). MPC can handle Multiple Inputs and Multiple Outputs systems (MIMO). The advantage of its application is that it is a multivariable
controller that controls the output simultaneously by taking into account all the in-fractions between system variables [
]. The speed of the processor enables it to handle multiple complex problems while taking into account the disturbances created in the system. These features make MPC controllers more superior to
other control techniques as seen in
Table 1
MGs have also been exposed to cyber and communication threats over the recent decade due to natural- and human-induced activities. Many control strategies have been developed to ensure the resilience
of MGs to physical, cyber, and communication delays and threats but the idea of developing metrics to quantify their resilience has been a difficult task [
]. Other control techniques proved to have a computational burden in trying to address issues of communication and cyberattacks [
]. Model predictive controllers have manifested better control effects compared to traditional proportional integral derivative (PID) controllers in responding to communication delays since the
length of communication delay affects the MGs stability [
The main research direction is to present a comparative review on MPC-controlled and other uncertainty modeling techniques of microgrid systems. The review considers the various uncertainties
(voltage profile enhancement, power quality improvement, transmission losses, and frequency stability) as a whole instead of individual capabilities. This is professed in both grid-connected and
islanded microgrid modes. The organization of the remaining paper is as follows:
Section 2
presents the basic architecture of MG in terms of both operations and control.
Section 3
presents the basic operating principles and concepts of MPC.
Section 4
discusses recent applications of MPC to both grid-connected and islanded microgrids design.
Section 5
gives an overview of MPC superiority in MG management and the paper is concluded in
Section 6
2. Microgrid Architecture
Several factors have to be considered in establishing a MG, such as the geographical location, financial availability as MG construction requires huge initial capital investments, load demand, and
historical knowledge of the existing electrical system of such environment [
]. Hence, for a MG to effectively and efficiently deliver the required power for consumption, the design architecture must support both the operational and control patterns. A typical MG
configuration consists of DERs, storage systems, and standard communication and control systems as seen in
Figure 3
below. Depending on the purpose of establishment, it can be either grid-connected or islanded and the point of changing from one mode to another is the point of common coupling. The situations of the
energy market and availability of RESs initiate the idea of power trading with the main grid and the consideration of the energy storage systems for uninterrupted power supply. Microgrid tends to
engage in power trading in two facets: (a) When the available energy supply from the Distributed Energy Resources (DERs) are in excess and (b) when the grid cost of electricity is cheaper than the
generation from the DERs. Instead of curtailing the excess energy produced, especially from RERs, it is better to sell to the grid. In addition, the price of electricity from the grid should be
monitored to know the periods of lowest tariffs. This power trading improves both the financial and resilient characteristics of the microgrid. The local and central controllers tend to help in
shifting the mode of the MGs with the help of power electronics devices [
]. Conventional large power systems have used AC configuration owning to the fact that the AC system exhibits an inherent characteristic that supports the characteristic of fossil fuel-driven
generators. This makes stability issues a low priority compared to small power systems (MGs) that have huge instability issues. This concept gave birth to the DC MG system where power electronics
devices are predominantly used to eradicate issues of stability, reliability, and protection from uncertainties created by RESs. Many other MGs have used hybrid systems where both AC and DC systems
are used. The hybrid MG is depicted in
Figure 4
3. MPC Operating Concept and Control Strategy
The idea of MPC dated as far back as the 1970s as it was then centered mainly in process industries. Due to its ability to solve problems of complex dynamic systems, its concept has been widely used
in other areas of research, especially in power systems optimization. It is a universal control algorithm that houses a wide range of control tools. It has been one of the main control algorithms
because of several inherent characteristics; it can process information of complex systems in the shortest duration (seconds), it can handle systems with multivariable cases, has the tendency of
introducing a feedforward control to take responsibility for measurable disturbances, and is very useful when future references are known [
]. It has a fast processor with a large memory that solves an online optimization problem at each time step. The control strategy of an MPC is based on the fact that it uses the model of a system to
predict what exactly the future outcome will be over a prediction horizon. It then predicts a set of future outcomes from prior historical data based on reference to a particular cost function given
by the equation below:
$C = ∑ λ i K i * − K i p 2$
• $K i *$ = reference trajectory;
• $K i p$ = predicted value for variable;
• $X i$ = variable;
• $λ i$ = weighting factor;
• i = number of variables.
At a particular sampling point or time
, the algorithm executes a set of input values that gives a predicted outcome. Only the first sampling of input is implemented moving the supposed horizon where a new optimal plan will take
responsibility for disturbances acting or have acted on the system [
]. The optimization algorithm is repeated at time
+ 1 using new measures or estimates, establishing a feedback mechanism which is seen in
Figure 5
The success of the strategy depends on the structure of the MPC algorithm as seen in
Figure 6
]. The efficiency of an MPC strategy depends on the process model and optimizer. A model with good prediction characteristics can lead to a good performance of the controller. Based on both
historical and available data, the process model captures the dynamics of the system and predicts control outcomes or actions [
] in line with a reference trajectory. The optimizer provides the control mechanism for the algorithm. Both the cost function and constraints available are being controlled by the optimizer. The
optimizer ensures that the optimization cost function is analyzed to satisfy the objective function of the optimization problem without violating the necessary constraints acting on the system, and
it has the ability to track the errors made in the predictions by the model of the system to avoid forecast errors.
4. MPC-MG Operations
Managing the optimal planning of a microgrid is a very difficult task due to the fact that they are small decentralized low voltage systems with small demand and a high rate of disturbances from
intense penetration of RERs. MPC has been applied to both grid-connected and isolated MG systems to help deal with several parameters as seen in
Figure 7
. Many scholarly works have been done to minimize the operating cost or maximize the revenue generation of microgrids but accurately implementing the mentioned objectives has been difficult due to
numerous factors. These factors could be as a result of the intermittency posed by nature (weather conditions), error in trying to predict the situation of nature, and the computational complexities
that are associated with optimizing the situation to get an optimal plan. According to [
], there are two standard methods or approaches used to solve the problem of uncertainty in MGs: Reactive approach and preventive approach.
The reactive method depends on a priori information or historical or predefined deterministic data (MPC and rolling horizon approaches) and the preventive method depends on scenario generations
(stochastic and robust optimizations). Majority of both the reactive and preventive optimization techniques carried out in MGs are centered on grid-connected systems compared to isolated MGs [
]. The preventive approaches have proven to be ineffective and not reliable for uncertainty considerations.
The stochastic optimization approach requires assigning probabilities for scenario generations, which is sometimes computationally demanding with the static assumption of uncertainty. The robust
optimization becomes over-conservative for measurements and requires different algorithms for different uncertainty sets. This is not the case of MPC; it works on inputs of a system considering the
internal dynamics to give or predict an output by capturing forecast error to compensate for unforeseen initial forecasting, making it ideal for uncertainty consideration [
]. A comparison between MPC and preventive optimizations is given in
Table 1
Many times, MPC has been combined with either of the two preventive methods to prevent or reduce uncertainties in many scholarly articles. When MPC is combined with stochastic optimization to give
the Stochastic Model Predictive Control (SMPC), the stochastic scenarios are used to execute the optimization process by assigning probabilities without much or totally considering the disturbances
in the process. The MPC technique helps to reduce the computational time and takes account of the uncertainties without assuming by implementing a feedback scenario where compensation is done to
eradicate the external influences of the integration of renewable technologies.
Thus, combining model predictive control with robust optimization gives a better result compared to robust optimization because instead of employing different algorithms which require time and more
expertise, MPC does a single consideration of all the uncertainties or disturbances acting on the system. The optimizer in the MPC algorithm has the ability to trace errors made by the process model
in predicting future outputs based on the dynamics of the system. Conservatism is highly reduced by the action of the MPC compensation process.
4.1. MPC for Grid-Connected MG Applications
More work of MPC applications to MG is centered on grid-connected systems than isolated systems because the cost of implementing measurements, automation, forecasting, and information processing is
very small compared with the derived economic benefits as opposed to isolated systems [
]. Accuracy of the forecast of load and RERs is of better quality in grid-connected systems. Decentralizing power systems through the establishment of MGs has led to an increase in demand and
accessibility of energy but comes with uncertainties of demand and RERs.
Barrios et al. proposed an MPC approach for unit commitment in MG in the presence of high uncertainties associated with demand and RESs [
]. A particular type of energy market is considered so that the MG can provide the required demand of load and RERs. An MPC technique is applied at every time step to cover demand regarding
uncertainties introduced due to prediction. The main objective of the MPC technique adopted in the work is to reduce the operational cost. Two conditions are considered in the study, which is the
conventional unit commitment and unit commitment based on MPC. Prediction errors increased the operation cost of the conventional system but a reduced cost is seen for the MPC unit commitment due to
the feedback mechanism.
Parisio et al. proposed that the decentralization of the power system has led to an increase in energy demand and therefore requires new methodologies to model a smart grid environment [
]. The work is focused on minimizing overall MG operational costs to match predicted demand for a certain day by obeying the complex constraints. Four different strategies are considered, which are
heuristics, Mixed-Integer Linear Programming (MILP), MILP-MPC, and benchmark. It is supposed that the demand for load and RERs are known with certainty. The proposed MILP-MPC gives less violation
because of the feedback mechanism introduced by the MPC, giving a result closed to the benchmark. The cost function is given as:
$j x k b = min u k T − 1 ∑ j = 0 T − 1 c u ′ k + j u k + j + c z ′ z k + j − O M b F ′ k + j u k + j − O M b f ′ w k + j$
subject to:
$S i · P i k + j + s i < σ i k + j ; = 1 ⋯ N g x b k b = x b k$
• $S i , s i$ = disturbances vectors;
• $w ( k + j )$ = assumed to be known over prediction horizon for $j = 0 ⋯ T − 1$;
• $O M b f ′ ( k + j )$ = independent of objective function;
• $C z , C u$ = column vectors;
• $( k + j )$ = time step.
Xie and IIic proposed an MPC algorithm to dispatch all the available resources to supply fluctuating loads at a minimum cost due to consideration of the prediction model [
]. The output of the controllable units is adjusted to compensate for uncertainties. Kou et al. [
] proposed a Stochastic Model Predictive Control (SMPC) approach that works in a two-layer step. The top layer ensures that there is a balance of power in the system and the bottom layer considers
the uncertainties emerging from both supply and demand ends. The main objective of the proposed approach is to ensure optimal power scheduling with total consideration of disturbances acting on the
system. The special attribute associated with this proposed approach is the consideration of all uncertainties from both demand and supply sides. The uncertainties from wind generation and PEV
charging both have different distribution characteristics but the MPC controller in the system handles both uncertainties simultaneously instead of treating them as separate uncertainty sets.
Despite MPC’s ability to compensate for disturbances in the RERs system, some violations have been experienced in scheduling optimal resources. In order to mitigate, these violations, forecasting
errors have to be taken into consideration. Y. Zhang et al. proposed an MPC approach, considering forecasting uncertainties and forecast errors of load, wind, PV, and electricity price [
]. The work incorporated stochastic analysis where scenarios are generated to approximate forecast errors and uncertainties. The objective is to minimize operation costs. Three different states of
the art approaches are compared to the proposed stochastic MPC; a Deterministic Day Ahead programming (D-DA), a Stochastic Day Ahead programming (S-DA), and Deterministic Standard MPC (D-MPC).
Simulation results show that S-MPC yields the lowest cost compared to D-MPC. This is because both S-DA and D-DA are open-loop systems where optimization takes place only at the beginning of the
scheduling. Both D-MPC and S-MPC are closed-loop where optimization is executed once for each time step. S-MPC considers all uncertainties affecting the system while D-MPC assumes that the system is
stable with known demand and no disturbance.
Gulin et al. proposed an approach of a power flow optimization in a Direct Current (DC) MG that accounts for predictions uncertainty [
]. Unlike other methods of uncertainty consideration, here a chance-constrained method is used to account for power prediction uncertainties. The work is done on the idea of allowing violations of
constraints in line with predefined probability levels, allowing the utility grid to compensate for error(s) on the prediction horizon. Two different approaches are used to deliver a minimum cost;
D-MPC and S-MPC. D-MPC did not account for uncertainties while S-MPC accounted for uncertainties and gives a lower cost by allowing a tradeoff between constraints being violated and prospect. Both
approaches are defined below.
D-MPC scheme is defines as:
$U * = a r g min u J u , x o , c , v .$
Subject to:
$P min G ≤ P k G ≤ P max G , ⋯ 0 ≤ k ≤ N − 1 .$
S-MPC scheme is defined as:
$U * = a r g min u J u , x o , c , v .$
Subject to:
$∥ S i W * σ k ∥ ϕ − 1 ( 1 − α i ) ≤ s i − S i ( D u k + V v k )$
• J = Economic criterion;
• u = Linear function;
• $U *$ = Optimization control sequence;
• $( u , x o , c , v )$ = Initial storage state;
• $P min G , P k G , P max G$ = Minimum grid availability, grid availability at time instant k, and maximum grid availability.
Dao et al. proposed a hierarchical and distributed MPC approach for the energy management of a microgrid. The main objective of the proposed approach is to provide an economic management framework to
maximize the benefits of the system. In order to ensure that forecast errors are taken into consideration and that uncertainties are effectively and efficiently handled to enhance maximum benefits, a
negotiation activity or process is carried out between the hierarchical and distributed MPC algorithms to compensate for forecast errors within the system [
]. Gambino et al. proposed an economic dispatch problem for an integrated microgrid (heat and electricity generation). The main objective of the proposed approach is to optimally dispatch resources
so as to minimize the overall cost of the microgrid. Microgrids that normally incorporate dual derivatives, as the case of combined heat and power to solve economic dispatch problems, are prone to
uncertainties from loads, energy prices, and weather forecasts. A feedback mechanism generated by the MPC controller compensates for uncertainties associated with time-varying loads, energy prices,
and RERs power outputs [
Bella et al. proposed a hierarchical MPC (two-layer control system) control scheme that constitutes of dynamic decoupled subsystems. The main objective of the approach is to optimally share resources
among the various subsystems so as to satisfy the overall demand and account for disturbances acting on the system [
]. Scheduling takes place in the upper layer and each subsystem is adjusted or designed in a way that at any time instance, an independent control action can be executed from the internal request or
a neighboring subsystem based on the MPC. At the end of every time step, the supervisor checks the system for either a deficit or excess demand in each subsystem. A compensation activity is initiated
by the supervisor in the subsystem(s) that exhibit shortages due to disturbances or uncertainties so as to ensure the overall system demand is achieved.
The majority of researchers have focused on exogenous factors or external factors (customer loads, wind speed, PV, and price profile) in considering uncertainty measurements with few works or no work
being done considering endogenous factors (types of equipment, and storage). Prodan and Zio proposed a predictive control framework that takes into account uncertainties modeling. The work is focused
on including internal (state) dynamics and structural properties of the individual components of RESs (solar and wind, on-site storage) which may change (stochastically) due to degradation, failure,
and aging effects. By considering both factors, the operating cost of the system can be reduced [
]. Nassouron et al. proposed an MPC approach for an economic dispatch problem considering heterogeneous systems (system with different computational applications) [
]. Owning to the fact that several heterogeneous generators and storage elements are used in the approach, the dispatch problem cannot be solved using classical optimization methods due to the
differences in characteristics of the generators and storage elements. Two techniques are used to optimally schedule the resources: MPC tracking and Economic MPC (EMPC). The Economic MPC yields a
better cost compared to the tracking MPC.
Table 2
shows the summary of recent approaches of MPC applications to grid-connected systems.
Table 2
, it is evident that the introduction of MPC, either as an independent operation or a combination of MPC and other modeling techniques gives a better and desirable outcome of handling the various
uncertainties compared to the various preventive techniques. The table presents a summary of different objectives and uncertainty handling strategies but what is more unique is the capability of MPC
to model the disturbances and achieve the desired goals. This explanation is also applicable to
Table 3
in the case of islanded systems.
4.2. MPC for Isolated MG Applications
Many utility companies and government-sponsored electric power systems have been implementing or providing incentives or demand-side management opportunities for their customers to establish on-site
Distributed Generators (DGs) and energy storage systems to increase the number of isolated MGs in the supply of power. However, these efforts have proven to be quite expensive or not cost-effective
at all. Hence, several research efforts have been devoted and are continuously being devoted to achieving the cost-effective operation of the isolated microgrid.
Parissio et al. proposed an MPC approach for energy management of multiple residential MGs having DERs, electrical storage systems with both thermal, and electrical loads [
]. The objective of the proposed MPC approach is to reduce energy costs and improve customers’ comfort through a demand-side management scheme. An optimal plan is computed to compensate for
imbalances affecting the system based on the weather forecast. The demand-side management scheme can help customers to know when to have an affordable cost of uninterruptable power.
Most of the works in predictive algorithms have considered favorable conditions (where generation is greater than demand). According to [
], a nonlinear MPC algorithm is developed or proposed for an Energy Management System (EMS) of an isolated MG with DERs in which automated load shedding of non-critical loads is done when the system
foresees power imbalances that could affect the stability of the MG. This predictive algorithm is proposed to identify upcoming generation problems when MG is operating in an islanded mode. The
objective is to predict and manage constraints in states and control signals. Hans et al. proposed a control technique that can give better prediction accuracy while minimizing cost. Comparison is
presented between an open-loop minimax approach and closed-loop minimax MPC approach considering the worst-case cost evaluation in trying to get better prediction accuracy and uncertainty handling [
]. The open-loop system gives a very conservative solution because it did not implement a feedback mechanism. The closed-loop Minimax MPC strategy however employed the theory of paramterization
(choosing parameters) of future inputs on the predicted disturbance leading to accurate predictions and lower cost. This is due to the presence of the MPC strategy that normally uses a predefined
input to make accurate future predictions.
Gu et al., proposed an MPC technique for Combined Cooling, Heat, and Power microgrid (CCHP) with feedback correction to reduce running cost and handle uncertainties [
]. A two-stage optimization approach is executed in this work of which the first stage is based on forecasting the required load and renewable energies integration. The second stage focuses on
compensating for the error in the prediction process. The MPC ensures that forecast is repeated for every time interval to get accurate data to be in line with rapid changes that take place in load
and RERs demand. If, however, there is a disturbance in the system due to inaccurate forecasting, a feedback correction is done to eliminate the disturbance.
Deterministic unit commitments have proven inappropriate for island MGs because their small scale demands are hard to predict and RERs generation is highly variable. Y. Zhang et al., proposed a
Robust Model Predictive Control (RMPC) approach to solve the operating cost of an islanded MG by minimizing cost [
]. The work states that of recent times, both chance-constrained and scenario-based stochastic optimization methods have been used to minimize MG cost. It has however, been concluded that these two
methods involve high or huge computational burden and uncertain parameters and forecast errors are not accurately accounted for and as a result, have posed a high negative impact on MG cost.
However, another method is the robust optimization used to solve an optimal scheduling problem with uncertain parameters. The conservativeness of this approach is a huge concern for cost minimization
by MG operators. In this proposed RMPC approach, an MPC is introduced to reduce the conservativeness of the RO due to the rolling up manner and feedback mechanism that it exhibits. Two control
strategies are considered in the approach used; a conventional 2-stage RO and RMPC-based optimization. The results for the cost function of RMPC is lower than the conventional two-stage RO since a
feedback control action is generated to consider the forecast uncertainties. This is justified by the RO cost function with the uncertainty budget given below.
subject to:
$∑ j a i j + z i Γ i + ∑ j ϵ j i P i j ≤ b i ⋯ ∀ i$
$Z i + P i j ≥ l i j ⋯ ∀ j$
• $P i j , Z i$ = auxiliary variables.
Sach et al. proposed a stochastic model predictive control approach for a rural isolated microgrid. The main focus or objective of the proposition is the development of an advanced control technique
to improve robustness towards predictions error and uncertainties acting on the system. For normal MPC operation, a control technique is implemented for a one-time step and subsequent control actions
rely on the dignity of the previous time step [
]. This proposed stochastic MPC considers the probability of a constraint violation over several time steps. A probability distribution approach based on the stochasticity of the load and renewables
is used to compensate for the disturbances on the system. Jaboulay et al. proposed a controlled algorithm based on MPC with the objective of minimizing operation cost and maintaining power balance in
the system [
]. The controller takes into account the physical constraints of the system while scheduling the required resources. Instead of making a few decisions as opposed to other control techniques for every
time step, the MPC takes multiple decisions for every updated forecast because it can handle multiple inputs and outputs. Scenarios are run in parallel on a semi-physical platform to compensate for
According to [
], the cost of forecast service and power quality using automation is very high in isolated systems compared to grid-connected systems. Zhang et al. proposed an EMS for multi-isolated MGs connected
by a centralized system to minimize the overall cost of the EMS [
]. An MPC technique is introduced with the intent of considering or reducing the impacts of forecast errors of the load and RERs, hence reducing overall cost. Berkel et al. proposed a hierarchical
MPC for a smart MG to solve the power stability issue. Two (2) levels of management are presented where the first level solves the frequency issue and the second level solves the cost function. The
objective of the hierarchical MPC is to make accurate predictions of load and RERs by rejecting disturbances from penetration and to handle constraints to guarantee stability and performance of the
smart MG [
]. The summary of recent approaches of MPC applications to isolated systems is shown in
Table 3
5. Superiority of MPC in Microgrid Designs and Operational Management
Various problems encountered in the operations of both grid-connected and isolated microgrid systems, as described in
Table 2
Table 3
above, are in actual situations quite challenging to manage. However, the introduction of MPC has helped achieve a better system design, which considerably reduces the operation cost. As a summary of
the various operation problems discussed above, the following highlights show the MPC’s superiority:
• In grid-connected microgrids, the prediction of energy market situations is achieved more accurately in terms of the load demand and generation dynamics, especially in the face of the
uncertainties introduced by RERs. In isolated microgrids, operating in favorable conditions or deterministic conditions, where demand is known with certainty, is not possible because of the
unpredictable nature of RERs. The effectiveness of MPC in tracking the disturbances and uncertainties has led to an increase in the desired operational benefits under these two conditions.
• In hybrid systems with thermal generators, a conventional unit commitment operation cannot accurately predict the output of RERs, which increases the effective operational cost. MPC introduction
has, however, helped to achieve better control of prediction errors by the effects of its superior feedback mechanisms. In standalone systems and hybrid systems, MPC encourages multiple
residential microgrids to interact effectively. It enhances efficient Peer-to-Peer (p2p) energy trading by cognizant of the differences in the energy needs and energy produced by connected
parties known as the ‘Prosumers’ [
• Stochastic approaches do not give reliable performances when it comes to forecasting and forecast errors; combining MPC with this operation condition yields improved results in the desired
outputs. The MPC exhibits superior performances compared to other options by considering both external and internal factors while solving uncertainty issues.
Below are also the limitations of MPC:
• One of the biggest challenges of MPC applications is that, it relies on historical information to predict the future. For newly-established energy systems (grid-connected or islanded), the
application of MPC looks extremely difficult or impossible.
• MPC applications require high modeling expertise which comes with a high cost.
• The quality and accuracy of the predictive model plays a significant role in the control process. Having a balanced trade-off between the model accuracy and calculation complexity is a serious
• Another key issue in MPC is the design of the sampling interval. This interval determines the performance of the model. A better performance can be achieved considering small sampling time
intervals. This will however reduce computational burden and economy of scale.
6. Conclusions
Recently microgrids have emerged as an approach to make energy easily available to global demand. This is more prominent to areas that are remote or inaccessible for grid connections. However, the
presence of RERs tend to make the operation of the MG environment very uncertain due to the various disturbances that erupt from their penetrations. The advancement in technology has produced
numerous modeling techniques to handle the uncertainties associated with the penetration effects. Various modeling techniques to model uncertainties in microgrids are presented in this review
exercise but the results show that there are still deficiencies in the control process. Due to the speed of the processor along with its ability to adapt to several applications, it has now become a
preferential control technique. A review of two classifications of modeling uncertainties (proactive and preventive) are presented in this review. The merits and demerits of each of the
classifications were presented, showing the superiority of a proactive approach (MPC and rolling horizon) to preventative (stochastic and robust).
This review is limited to only the modeling of power availability from DERs, especially RERs in the MG environment. Future work could be focused on a specific uncertainty like voltage or frequency
control, faults detection, or other protection control objectives.
Author Contributions
K.V.K. conceptualized and prepared original draft; O.B.A. provided resource, edited the manuscript and provided funding; M.E.L. provided resource and editing; Y.S. validated and provided funding;
T.S. supervised. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1.
Global renewable energy transition [
Figure 2.
Renewable Energy Resources (RERs) growth capacity in gigawatt (GW) by continent [
Figure 3.
A typical MG architecture [
Figure 4.
Hybrid MG architecture [
Figure 5.
MPC control strategy [
Uncertainty Advantages Disadvantages
Optimization Technique
1. Can provide the expected value of perfect information 1. Computationally demanding for large scenarios
Stochastic and the cost of the stochastic solution 2. Need to assign probabilities for scenario generation
2. Minimise expected cost than minimizing 3. Static assumption of uncertainty
worst-case cost
1. No probability distribution 1. Need to use a different algorithm for
Robust 2. Not computationally demanding different uncertainty sets
2. Overconservative solutions
1. Does not require external applications 1. Requires high expertise
MPC 2. The model dynamics uses the present information 2. Relies on historical data or information
to predict future output
Ref. Proposed Approach Main Focus Gap Parameters Uncertainty Handling
to Be Optimized
Mixed Integer Power dispatching, reducing computational Influence of disturbances on RES is mitigated by receding horizon
[77] Programming (MILP), burden introduced by non-linear MILP and Power, current strategy
MPC disturbances
MPC, Gaussian Optimal operation planning for EMS to Results did not consider At each sampling time predictions are calculated for an MPC
[78] process forecasting minimize cost of energy from grid environmental and electricity Energy cost execution based on predictions
Automatic construction of
Hierarchical and Main objective is to provide an economic day-ahead user profile, A negotiation phase between the hierarchical and distributed MPC
[49] Distributed MPC management to maximize benefits iterative negotiations between Profit is enhanced to compensate for forecast errors in the system
(HDMPC) layers and integration of low
level controls
Distributed MPC
[79] (DMPC) Maximizing RERs utilization and reducing cost Power, cost Each controller has a private predicting model that can solve global
Cooperative MPC and computational time cost function
Stochastic MPC A hierarchical predictive control approach Power preferences are computed by uncertainties in both supply and
[46] (SMPC), to coordinate wind generation and PEV Power balance demand
DMPC charging
Integrated method is proposed connecting Should include price based MPC close the control loop from power generating to people
[80] MPC people’s behavior, appliances, grid behavior, control devices Voltage, peak load behavior leading to reduce generation and distribution method
Control the interlinking converter to enhance
[81] MPC stable voltage supply, flexible power Voltage, power Flexible reactive power is injected onto the main grid for grid
regulation, support
and grid support.
MPC is applied with the aim of saving fossil Introducing an affine feedback correction due to uncertain weather
[82] SMPC, DMPC energy and evaluate the potential for Power balance fore
component -cast
downsizing leading to cost minimization
Distributed Economic A distributed control theory is developed to State of charge, Each controller can optimize its operation for state of charge,
[83] MPC (DEMPC) coordinate individual subsystems leading to power balance, predicted
suboptimal performance in the MG price load and electricity price
The proposed model helps the network MGs to Distributed control scheme Uncertainty is avoided when one MG sells power to another where
[84] MPC coordinate with each other. This minimizes the will be considered in future Power balance generation is greater than demand (G > D)
power produced by the micro gas turbine analysis
An optimal dispatch problem of controllable Accuracy of the proposed A feedback mechanism compensates for uncertainties associated with
[50] MPC loads and generators of an integrated MG model will be correlated in time varying loads, energy prices and RERs power outputs
is proposed to minimize cost further studies
Operation cost is minimized by optimally Will include DMPC and SMPC A MILP is optimized at each time step based on short term forecast
[85] MPC, MILP scheduling generating units while satisfying in further work Electricity price and incorporated into MPC to reduce forecast errors
complex constraints
To generate suitable decisions for all the Interconnected MG and Power consumption, Fault tolerant strategies are inserted to ensureproper amount of
[86] MPC source and electrical storage components to combination between generation profile, energy
fulfill load demands multi-agent approaches will be cost in storage devices for customers’ demand
applied in further studies
Combined economic and environmental energy Dynamic model including Prediction curves, energy generation, load demand are gotten from
[87] MPC management to minimize daily generation electrical components will be Cost, emission historical recording data with stochastic uncertainty processing
cost and emission included in future work
A hierarchical control scheme is proposed to Stochastic scenario, direct The supervisory level checks for shortages or excesses of control and
[9] MPC compute control action needed by a subsystem negotiations should be included Voltage, frequency also compensate for errors to satisfy the required demand
or neighboring subsystem in further studies
Dual Focuses on solving economic dispatch at Stochastic techniques to tackle Formulation is solved by every power plant to enhance granularity
[88] Decomposition runtime while reducing potential deviations challenges will be included in Power balance of agent
DMPC schedules further studies
Rolling Horizon (RH) Energy management system(EMS) is developed Real time pricing scheme will Design and implementation of controller to control accuracy of
[45] MPC to minimize daily operation cost and enhance be considered in further work. RERs battery
local self consumption of RERs energy storage
Including an economic cost index and explicit State of charge of Design of a central control capable to handle multivariable
[89] EMPC constraint to optimally dispatch power to battery constraints and predictions
minimize cost
A two layer algorithm is developed for Thermal energy needs of the MG SMPC regulator at lower layer runs at higher
[90] SMPC optimal EMS of the MG will be considered in further RERs frequency to compensate for uncertainties
A distributed MPC algorithm is proposed to
[91] DMPC schedule MG internal devices and optimal Power balance Reactive power balance is established
power trading
Ref. Proposed Approach Main Focus Gap Parameters to Be Optimised Uncertainty Handling
Mixed integer non Developing an advanced model optimization Inclusion of detailed component model limits
[98] linear programming approach using MPC framework to reduce State of charge, power uncertainties and adaptive forecast algorithm
(MINLP) cost and improve robustness of control towards balance reduces errors
prediction errors and uncertainties
MPC, sliding mode To stabilize MG system and maintain output Voltage references are tracked by the sliding
[99] control voltage in a layer that can enhance current Voltage, current mode control
Economic efficiency and
[100] DMPC Optimization problem is solved by incorporating frequency control performance Frequency Uncertainty effects of RERs is solved by applying
economic dispatch in secondary layer will be considered in further MPC online with rolling optimization
[101] MPC Limiting converter current under overloading Voltage and current Decoupling of control channels for each DG
To maintain network variables, provide The primary layer modulates DG units in order to
[102] MPC flexibility and coordination and account for Voltage, frequency limit voltage and frequency from nominal values
energy storage reserves.
[103] MPC Control voltage and frequency at the generating Voltage, frequency Addition of fault detection and diagnosis module
unit and supply energy for balance load to MPC structure
To provide a solution that can reduce Stochastic and worst case
[104] SMPC conservativeness by taking into account approaches will be consider in RERs Models for time series forecast are employed
stochasticity of loads and RERs. further studies
Development of advanced control to improve Probability constraints are assumed on the battery
[72] SMPC robustness towards prediction errors and Power state of charge
[105] MILP-MPC An optimization strategy is proposed to attain Transient processes will be Power Uncertainties are modeled by discretizing the said
an optimal generator start up sequence treated in further work probability distribution of forecast errors
[106] MPC Minimize voltage unbalance, improve current Distributed control scheme will Power quality Controls the negative sequence impedance to
limiting, and prevent active power overload. be employed in further work reduce voltage unbalance and current sharing error
Presents a dynamic reactive power control Future analyses will consider Time variant reactive capabilities of distributed
[107] MPC method to control reactive power large MGs Power, voltage generators are used to compensate for reactive
MPC, demand side Minimize operation cost and maintain power Practical implementation of Faster time scale online power allocation is done
[108] management (DSM) balance considering uncertainties imposed MG will be included in further Power balance to compensate for uncertainties in real time
A closed-loop minimax MPC is employed to Optimal control of MG in a
[68] Minimax MPC yield a better prediction accuracy and lower probabilistic manner will be RERs By paramterization
cost compared to open loop considered in further work
An interactive energy management is proposed Impact assessment of cyber Lower layer runs on high frequency to adjust
[109] MPC to enhance power balances and uncertainties risk and large distribution Power balance difference between planned and real time strategies
handling in multi-MGs systems
A hierarchical distributed MPC is proposed to Robust optimization will be Back calculation from lower to upper layer is
[110] HDMPC coordinate power, flexibility, dispatch, and considered in future work Power balance introduced.
minimize cost
An online optimization approach of a combined Stochastic technique to tackle Online optimal approach using MPC to compensate
[69] MPC cooling, heating and power MG is proposed to challenges will be included in RERs for prediction error
reduce running cost and handle uncertainties further work
The proposed algorithm reduces operational More suitable method will be Scenarios are run in parallel on a semi-physical
[73] MPC cost while maintaining power balance consider in further work to Power balance platform
compensate for forecast errors
MPC control strategy is proposed to solve Nonlinear variations in charge and discharge
[111] MPC an optimal power flow problem in a MG where Power flow efficiencies of the battery are analyzed
assumptions are avoided
[112] Robust MPC Main contribution is a review of the three proposed RERs A single control system is calculated using multi-
robust MPC techniques to select best approach scenario MPC
[113] MPC EMS is proposed to minimize daily operating RERs MPC strategy is implemented
cost where MPC is used to minimize uncertainties
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Show that x^2 = x*sin x + cos x for exactly two real numbers x - Stumbling Robot
Show that x^2 = x*sin x + cos x for exactly two real numbers x
Consider the equation
Show that there are two values of
Proof. Let
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The Quantum Computational Trajectory Enabling Abundance
images: canva.com
In the last few years, we have increased our flirtation with the mystery behind the quantum veil. Given threats of starkness brought about by recent global circumstances — the pandemic, wars, ongoing
financial crises — such flirtation could indeed prove useful. For somewhere deep in the collective psyche, the need for a truer and deeper source of abundance is surely growing, and quantum
computation can be an enabler of such abundance if the right trajectory of development is followed.
As suggested in Managing the Quantum Bubble (Forbes, November 17, 2022), this essentially means forgoing the current interpretation of Nature being entirely probabilistic and the currently
questionable interpretations of entanglement and superposition, to embrace the possibilities of the quantum levels by seeing it as a whole — seamlessly integrated with the emergent layers of matter
and life.
This post will summarize the contours of abundance possible from such a systems-based approach, reinforcing also an alternative quantum computational trajectory required to get there.
The Abundance Within Our Reach
All genetic-type code, as summarized in Genetics and a New Genre of Intent-Based Quantum Computer (Forbes, March 7, 2023) and as laid out in detail in the book The Origins and Possibilities of
Genetics, is the outcome of an ongoing quantum computation being executed by massively prevalent atom, molecule, and cell-based quantum computers. Cup our hands together, and we already capture a
septillion atom-based quantum computers (Forbes, January 23, 2023). There is, therefore, an extraordinary amount of quantum computation happening at every point and instant. It is this quantum
computation that continually reinforces patterns that exist in all matter and life.
But imagine if these patterns could be recomputed with appropriate computing intervention. Then undesirable patterns that propagate dysfunction in matter and life might be altered to contribute to an
enhanced foundation of abundance. This is not just some pipe dream. The right kind of quantum computational device, as summarized in the Forbes piece The Possibilities of Quantum-Intelligence Driven
Nano-Cyborgs (April 24, 2023) and technically in the IEEE article, Envisioning a Light-Based Quantum Computational Nano-Cyborg, will set us on a path to bring this about.
To begin with, the warp and woof of matter itself can be changed. The power to create different kinds of materials by working at the very level of quantum computation will allow properties of matter
to progressively be pushed beyond the current set of surfacings. I introduce a theoretical framework for how matter can change by leveraging an iterative light-based quantum computational model in
the book Super-Matter: Functional Richness in an Expanding Universe.
The prowess of quantum-intelligent nano-cyborgs similarly will allow access and transformation of bottom-up genetic-type code that contributes to the foundations of the management and maintenance of
Life. This will inject new possibilities into healthcare while also potentially allowing custom pharmaceuticals to be created on demand. Some technical aspects of such innovation in pharmaceutical
technology are covered in the IEEE article, A Meta-Functional, Quaternary-Based Mathematical Structuring of the Periodic Table and Its Elements, and Its Implications for Management of Innovation in
Pharmaceuticals Technology.
Shades of such abundance, including co-creating in partnership with quantum intelligence, leaps in transhumanism, inexhaustible energy possibilities, and warp-speed travel, have already been
suggested in several of the links to articles and books provided in this post. I will expand on these some other time. I focus now on reinforcing how this abundance may be achieved.
How Do We Reach This Abundance
At its heart, this abundance requires a different type of quantum computing device that operates as a nano-cyborg. That is, a quantum computing device that is organic enough to begin to perceive and
respond to the quantum intelligence (summarized in Leapfrogging the Singularity Through Integrated Quantum Computational Intelligence [Forbes, March 30, 2023]) native to the quantum levels, and
enabled by the right mechatronics, is sensitive enough to begin to influence the quantum intelligence with real-time feedback loops.
A key foundational element of such an architecture has to be the generation of a state of quantum certainty that must accompany the property of quantum intelligence. After all, where there is
intelligence, there, too, there must be certainty by which what is conceived in that intelligence can come to fruition. The notion of quantum certainty is introduced in an IEEE article, The Role of a
Light-Based Quantum Computational Model in the Creation of an Oscillating Universe, that discusses how the micro connects with the macro as a single system and is elaborated in detail in the book,
Quantum Certainty: A Mathematics of Natural and Sustainable Human History.
It is such quantum properties of intelligence and certainty that necessitate a wholly different quantum computational architecture arbitrating, as it were, at the level and in the language of these
fundamental properties itself. This language is ‘functional’ or function-focused and would have to give insight into how fourfold fractal patterns at the level of quantum particles, atoms in the
Periodic Table, and molecular plans in cells, relate to and build on each other.
The idea of the relationship between different layers of fourfold patterns perhaps resonates with hyperdimensional computing in which vectors are imbued with semantic meaning. An attempt to go
further by surfacing mathematical possibilities of meaning-bound functional fourfold-based vectors based on imagined physics of light existing at different possible constant speeds in addition to the
known speed ‘c’ — 186,000 miles in a vacuum —is made in the book Cosmology of Light: A Mathematical Integration of Matter, Life, History & Civilization, Universe, and Self.
It is this that informs the alternative model of quantum dynamics expressed by a Cosmology of Light, and it is this that then generates insight into different quantum computational possibilities.
Grasping this meaning-bound fourfoldness will put us on a more secure path to resolving the inherent dichotomy, as summarized by a recent interview of physicist Michio Kaku in The Guardian, between
quantum possibilities and the technological challenges to get there.
Central to this alternative quantum computing architecture would be gates that recognize, interpret, leverage, and generate quantum properties themselves. These gates would be entirely different from
qubit-based and other known logical gates. Superposition gates, entanglement gates, tunneling gates, annealing gates, fourfoldness gates, intent-magnification gates, functional-integrity gates, and
certainty gates will be the stuff of this architecture.
Extract from a USPTO nonprovisional patent application filed by QIQuantum
The trajectory forward means that a different technology stack involving the alternative hardware, as hinted at in this post, and subsequently, alternative operating systems, alternative quantum
interfaces, and radically different applications, have to be built to successfully arbitrate the quantum world. It is only in doing so that we will be able to coax the materialization of more
sophisticated forms of quantum intelligence, beyond the intelligences already materialized by atoms, by cells, and by humans, to more easily overcome any threat generated by human-based AI
singularities, thereby also initiating the beginnings of a deeper era of abundance.
NOTE: This post has been written in prep for a forthcoming Forbes Technology Council Masterclass on Abundance Through Quantum Computation
Link to Forbes Technology Council “Abundance Through Quantum Computation” Master Class
Link to summary of previous Forbes Technology Council Master Class on “Managing the Quantum Bubble” | {"url":"https://pravirmalik.medium.com/the-quantum-computational-trajectory-enabling-abundance-40213565ceaa","timestamp":"2024-11-11T21:14:14Z","content_type":"text/html","content_length":"132632","record_id":"<urn:uuid:a01c46a4-ee56-4e2d-b60c-1e2bb2784ec3>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00897.warc.gz"} |
The Complete Guide to Understanding Control Charts: How They Work, and Which to Use
Key Points
• Control charts are useful for monitoring a process’s stability.
• These tools are useful for monitoring stable data, or continuous data.
• Some control charts account for unstable variations.
• You can also use control charts for discrete data.
• A powerful instance of use for control charts is in the analysis stage.
Control charts have two general uses in an improvement project.
Undeniably, the most common application is as a tool to monitor process stability and control.
A less common, although some might argue more powerful, use of control charts is as an analysis tool.
Throughout this guide, you’ll have the various control charts identified. Accordingly, you’ll also have the means to determine which best suits your needs for a given situation.
Identifying Variation
When a process is stable and in control, it displays common cause variation, variation that is inherent to the process. A process is in control when you can predict how the process will vary (within
limits) in the future. If the process is unstable, the process displays special cause variation and non-random variation from external factors.
At any rate, control charts are simple, robust tools for understanding process variability.
The Four Process States
Processes fall into one of four states: 1) the ideal, 2) the threshold, 3) the brink of chaos, and 4) the state of chaos (Figure 1).^3
When a process operates in the ideal state, that process is in statistical control and produces 100 percent conformance. This process has proven stability and target performance over time. Further,
this process is predictable and its output meets customer expectations.
A process that is in the threshold state is in statistical control but still produces the occasional nonconformance. All in all, this type of process will produce a constant level of nonconformances
and exhibit low capability. While predictable, this process does not consistently meet customer needs.
The brink of chaos state reflects a process that is not in statistical control but also is not producing defects. Accordingly, the process is unpredictable, but the outputs of the process still meet
customer requirements. However, the lack of defects leads to a false sense of security. Consequently, such a process can produce nonconformances at any moment. It is only a matter of time.
The fourth process state is the state of chaos. Accordingly, the process is not in statistical control and produces unpredictable levels of nonconformance.
Figure 1: Four Process States
Every process falls into one of these states at any given time, but will not remain in that state. All processes will migrate toward a state of chaos. As such, companies begin some type of
improvement effort when a process reaches a state of chaos. However, control charts are robust and effective tools to use as part of the strategy used to detect this natural process degradation
(Figure 2).^3
Figure 2: Natural Process Degradation
Elements of a Control Chart
There are three main elements of a control chart as shown in Figure 3.
1. A control chart begins with a time series graph.
2. A central line (X) is added as a visual reference for detecting shifts or trends – this is also referred to as the process location.
3. Upper and lower control limits (UCL and LCL) are computed from available data and placed equidistant from the central line. This is also referred to as process dispersion.
Figure 3: Elements of a Control Chart
Control limits (CLs) ensure time is not wasted looking for unnecessary trouble – the goal of any process improvement practitioner should be to only take action when warranted. As such, control
limits are calculated by:
1. Estimating the standard deviation, σ, of the sample data
2. Multiplying that number by three
3. Adding (3 x σ to the average) for the UCL and subtracting (3 x σ from the average) for the LCL
As an illustration, the calculation of control limits looks like:
(Note: The hat over the sigma symbol indicates that this is an estimate of standard deviation, not the true population standard deviation.)
Since control limits are calculated from process data, they are independent of customer expectations or specification limits.
Control rules take advantage of the normal curve in which 68.26 percent of all data is within plus or minus one standard deviation from the average, 95.44 percent of all data is within plus or minus
two standard deviations from the average, and 99.73 percent of data will be within plus or minus three standard deviations from the average. As such, data should be normally distributed (or
transformed) when using control charts, or the chart may signal an unexpectedly high rate of false alarms.
Controlled Variation
Controlled variation is characterized by a stable and consistent pattern of variation over time, and is associated with common causes. Furthermore, a process operating with controlled variation has
an outcome that is predictable within the bounds of the control limits.
Figure 4: Example of Controlled Variation
Uncontrolled Variation
Uncontrolled variation is characterized by variation that changes over time and is associated with special causes. Subsequently, the outcomes of this process are unpredictable. Furthermore, a
customer may be satisfied or unsatisfied given this unpredictability.
Figure 5: Example of Uncontrolled Variation
Please note: process control and process capability are two different things. As such, a process should be stable and in control before process capability is assessed.
Figure 6: Relationship of Control Chart to Normal Curve
Control Charts for Continuous Data
Individuals and Moving Range Chart
The individuals and moving range (I-MR) chart is one of the most common control charts for continuous data. It is applicable for a single data point over points in time. Above all, the I-MR control
chart is two charts used in tandem (Figure 7). Together they monitor the process average as well as process variation. With x-axes that are time-based, the chart shows a history of the process.
The I chart is used to detect trends and shifts in the data, and thus in the process. As a result, the individual chart must have the data time-ordered. That is, the data must be entered in the
sequence in which it was generated. If data is not correctly tracked, trends or shifts in the process may not be detected and may be incorrectly attributed to random (common cause) variation.
However, there are advanced control chart analysis techniques that forego the detection of shifts and trends. Accordingly, before applying these advanced methods, the data should be plotted and
analyzed in a time sequence.
The MR chart shows short-term variability in a process – an assessment of the stability of process variation. Further, the moving range is the difference between consecutive observations. It is
expected that the difference between consecutive points is predictable. Accordingly, points outside the control limits indicate instability. Subsequently, if there are any out-of-control points, the
special causes must be eliminated.
Example and Usage of an I-MR Chart
Once the effect of any out-of-control points is removed from the MR chart, look at the I chart. However, be sure to remove the point by correcting the process – not by simply erasing the data
Figure 7: Example of Individuals and Moving Range (I-MR) Chart
The I-MR chart is best for:
• The natural subgroup size is unknown.
• The integrity of the data prevents a clear picture of a logical subgroup.
• The data is scarce (therefore subgrouping is not yet practical).
• The natural subgroup lacks definition.
Xbar-Range Charts
Another commonly used control chart for continuous data is the Xbar and range (Xbar-R) chart (Figure 8). Like the I-MR chart, these are two charts in tandem. The Xbar-R chart can rationally collect
measurements in subgroups of between two and 10 observations. Accordingly, each subgroup is a snapshot of the process at a given point in time. Further, the chart’s x-axes are time-based so the
chart shows a history of the process. As such, the data must be in time order.
The Xbar chart is for determining the consistency of process averages by plotting the average of each subgroup. Furthermore, it is efficient at detecting relatively large shifts (typically plus or
minus 1.5 σ or larger) in the process average.
The R chart, on the other hand, plots the ranges of each subgroup. Accordingly, the R chart can evaluate the consistency of process variation. However, look at the R chart first. Consequently, if the
R chart is out of control, then the control limits on the Xbar chart are meaningless.
Figure 8: Example of Xbar and Range (Xbar-R) Chart
Further Examples of Xbar-Range Charts
Table 1 shows the formulas for calculating control limits. Generally, many software packages do these calculations without much user effort. (Note: For an I-MR chart, use a sample size, n, of 2.)
Notice that the control limits are a function of the average range (Rbar). Further, this is the technical reason why the R chart needs to be in control before further analysis. If the range is
unstable, the control limits will be inflated. In time, this could cause an errant analysis and subsequent work in the wrong area of the process.
Table 1: Control Limit Calculations
Table 2: Constants for Calculating Control Limits
n (Sample Size) d[2] D[3] D[4]
2 1.128 – 3.268
3 1.693 – 2.574
4 2.059 – 2.282
5 2.326 – 2.114
6 2.534 – 2.004
7 2.704 0.076 1.924
8 2.847 0.136 1.864
9 2.970 0.184 1.816
10 3.078 0.223 1.777
11 3.173 0.256 1.744
12 3.258 0.283 1.717
13 3.336 0.307 1.693
14 3.407 0.328 1.672
15 3.472 0.347 1.653
Can these constants be calculated? Yes, based on d[2]. Where this variable is a constant depends entirely on sample size. However, you’ll have to keep track of your data points to determine if you
can calculate these constants.
The I-MR and Xbar-R charts use the relationship of Rbar divided by d[2] as the estimate for standard deviation. Accordingly, for sample sizes less than 10, that estimate is more accurate than the sum
of squares estimate. The constant is dependent on the sample size. Subsequently, most software packages automatically change from Xbar-R to Xbar-S charts around sample sizes of 10. However, the
difference between these two charts is simply the estimate of standard deviation.
Control Charts for Discrete Data
Used when identifying the total count of defects per unit (c) that occurred during the sampling period, the c-chart allows the practitioner to assign each sample more than one defect. Accordingly,
these charts come into play when the number of samples in each sampling period is essentially the same.
Figure 9: Example of c-Chart
Similar to a c-chart, the u-chart can track the total count of defects per unit (u) that occur during the sampling period and can track a sample having more than one defect. However, unlike a c
-chart, a u-chart finds use when the number of samples of each sampling period may vary significantly.
Figure 10: Example of u-Chart
Use an np-chart when identifying the total count of defective units (the unit may have one or more defects) with a constant sampling size.
Figure 11: Example of np-Chart
Used when each unit can be considered pass or fail – no matter the number of defects – a p-chart shows the number of tracked failures (np) divided by the number of total units (n).
Figure 12: Example of p-Chart
Notice that no discrete control charts have corresponding range charts as with the variable charts. As such, the standard deviation comes from the parameter itself (p, u, or c). Therefore, a range is
How to Select a Control Chart
Since this article describes a plethora of control charts, there are simple questions a practitioner can ask to find the appropriate chart for any given use. Accordingly, figure 13 walks through
these questions and directs the user to the appropriate chart.
Figure 13: How to Select a Control Chart
Several points come up when identifying the type of control chart to use, such as:
• Variables control charts (those that measure variation on a continuous scale) are more sensitive to change than attribute control charts (those that measure variation on a discrete scale).
• Variables charts are useful for processes such as measuring tool wear.
• Use an individual chart when few measurements are available (e.g. when they are infrequent or are particularly costly). These charts should be used when the natural subgroup is not yet known.
• A measure of defective units is found with u– and c-charts.
• In a u-chart, the defects within the unit must be independent of one another, such as with component failures on a printed circuit board or the number of defects on a billing statement.
• Use a u-chart for continuous items, such as fabric (e.g., defects per square meter of cloth).
• A c-chart is a useful alternative to a u-chart when there are a lot of possible defects on a unit, but there is only a small chance of any one defect occurring (e.g., flaws in a roll of
• When charting proportions, p– and np-charts are useful (e.g., compliance rates or process yields).
Subgrouping: Control Charts as a Tool for Analysis
Subgrouping is the method for using control charts as an analysis tool. Further, the concept of subgrouping is one of the most important components of the control chart method. The technique
organizes data from the process to show the greatest similarity among the data in each subgroup and the greatest difference among the data in different subgroups.
However, subgrouping aims to include only common causes of variation within subgroups and to have all special causes of variation occur among subgroups. Further, when the within-group and
between-group variation is made clear, the number of potential variables – that is, the number of potential sources of unacceptable variation – goes down considerably, and you can determine where
to expend efforts for improvement.
Within-subgroup Variation
For each subgroup, the within variation is represented by the range.
Figure 14: Within Subgroup Variation
The R chart displays the change in the within-subgroup dispersion of the process and answers the question: Is the variation within subgroups consistent? As such, if the range chart is out of control,
the system is not stable. Further, it tells you that you need to look for the source of the instability, such as poor measurement repeatability.
Analytically it is important because the control limits in the X chart are a function of R-bar. If the range chart is out of control then the R-bar inflates as does the control limit. This could
increase the likelihood of calling between subgroup variation within subgroup variation and send you off working on the wrong area.
Within variation is consistent when the R chart – and thus the process it represents – is in control. Subsequently, the R chart must be in control to draw the Xbar chart.
Figure 15: Example of R Chart
Between Subgroup Variation
Between-subgroup variation is represented by the difference in subgroup averages.
Figure 16: Between Subgroup Variation
Xbar Chart, Take Two
The Xbar chart shows any changes in the average value of the process and answers the question: Is the variation between the averages of the subgroups more than the variation within the subgroup?
Subsequently, if the Xbar chart is in control, the variation “between” is lower than the variation “within.” If the Xbar chart is not in control, the variation “between” is greater than the variation
Figure 17: Xbar Chart Within Variation
Furthermore, this is close to being a graphical analysis of variance (ANOVA). The between and within analyses provide a helpful graphical representation while also providing the ability to assess the
stability that ANOVA lacks. Using this analysis along with ANOVA is a powerful combination.
Real-World Applications of Control Charts
There is no shortage of tools for collating, analyzing, and interpreting data in the domain of most businesses. So, with that in mind, what is a real-world application of control charts? Most
companies will go through an audit, whether it is conducted by an external or internal team. Having data points in place for metrics allows auditors to utilize this useful tool.
In this instance, auditors would utilize control charts to keep monitor accounting practices throughout the year. This extends to the likes of payroll, sales, invoices, and gross sales throughout the
fiscal year. With payroll in mind, it is important to consider this largely a repetitive task as a whole.
The auditor would take a look at data points like overtime, time records, and gross pay. Using a control chart, the auditor then has the means to analyze individual employees or whole departments.
Other Useful Tools for Monitoring and Analysis
While control charts serve multiple different purposes, they aren’t the only tools you’re going to be using for your data sets. Additionally, you might find they don’t paint a complete picture as a
whole. If you’re looking to master all stages of a process’s workflow, acquainting yourself with DMAIC is one step. You can read more about it in our comprehensive guide.
Further, seeing how these tools work in motion is a different story compared to jargon-laden guides. See how Six Sigma has transformed a company like Avon, which isn’t one of the companies most would
think of when considering data-driven analysis and customer satisfaction.
Knowing which control chart to use in a given situation will ensure accurate monitoring of process stability. Accordingly, it helps to reduce errors and get you back on the road to productivity,
rather than wasting time. | {"url":"https://www.isixsigma.com/control-charts/a-guide-to-control-charts/","timestamp":"2024-11-13T06:11:46Z","content_type":"text/html","content_length":"230670","record_id":"<urn:uuid:362f5e61-14f5-4962-a551-5fe71bd32298>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00580.warc.gz"} |
What is Des model of CFD?
Detached eddy simulation (DES) is a modification of a Reynolds-averaged Navier–Stokes equations (RANS) model in which the model switches to a subgrid scale formulation in regions fine enough for
large eddy simulation (LES) calculations.
What is delayed Detached Eddy?
Delayed Detached eddy simulation (DDES) The main idea of DDES is to include the molecular and turbulent viscosity information into the switching mechanism to delay this switching in boundary layers.
max(0,d-C. ) d. DES.
What is the difference between RANS and LES?
The basic difference is RANS models all eddies while LES simulates large eddies and models small eddies. The RANS equations were derived by taking a time average of the NS equations. The effect of
turbulence is simulated through modelling the Reynolds stresses. LES is not a time average.
What is the difference between DNS and RANS?
DNS resolves all scales of motion, all the way down to the Kolmogorov scale. LES is next up and resolves most of the scales, with the smallest eddies being modeled. RANS is on the other end of the
spectrum from DNS, where only the large-scale eddies are resolved and the remaining scales are modeled.
What is scale adaptive simulation?
Scale-Adaptive Simulation (SAS) has been presented as an improved URANS formulation, capable of resolving the turbulent structures for unsteady flows, aimed at bridging the gap between the URANS and
the hybrid RANS/LES approaches [14, 15].
Is LES more accurate than RANS?
Large Eddy Simulation (LES) undeniably has the potential to provide more accurate and more reliable results than simulations based on the Reynolds-averaged Navier-Stokes (RANS) approach. However, LES
entails a higher simulation complexity and a much higher computational cost.
What is RANS simulation?
RANS: A mathematical model based on average values of variables for both steady-state and dynamic flows (unsteady for URANS). The numerical simulation is driven by a turbulence model which is
arbitrarily selected to find out the effect of turbulence fluctuation on the mean fluid flow.
What is meant by large eddy simulation?
Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents,
and first explored by Deardorff (1970).
Why is LES better than RANS?
What is eddy viscosity in fluid mechanics?
Eddy viscosity is the proportionality factor describing the turbulent transfer of energy as a result of moving eddies, giving rise to tangential stresses. | {"url":"https://cowetaamerican.com/2022/05/09/what-is-des-model-of-cfd/","timestamp":"2024-11-13T04:09:54Z","content_type":"text/html","content_length":"57914","record_id":"<urn:uuid:0e7c5ddd-91b6-42c4-a04d-d5afc98173b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00559.warc.gz"} |
Vector Magnitude Calculator
If an object (a) has magnitudes along with direction, it is known as a vector. To determine the magnitude, we require to measure the length of the vector. e.g., momentum, velocity, displacement,
energy, etc.
A vector's magnitude is applied to determine the length for a given vector ( suppose: v ), then the magnitude of vector v is denoted as |v|. So fundamentally, this amount is the length between the
initial point and the final point of the vector. To determine the magnitude of the vector, we apply the distance formula, which we will discuss below:
Formula of Vector Magnitude Calculator
Assume AB is a vector quantity possessing both magnitude and direction. To determine the vector AB's magnitude, we have to measure the distance in the initial point A and final point B. In XY –
plane, let A holds coordinates (x0, y0) and B holds coordinates (x1, y1). Hence, by applying the distance formula, the magnitude of vector AB can be formulated as:
\[|\overrightarrow {AB} | = \sqrt {\mathop {\left( {\mathop x\nolimits_1 - \mathop x\nolimits_0 } \right)}\nolimits^2 + \mathop {\left( {\mathop y\nolimits_1 - \mathop y\nolimits_0 } \right)}\
nolimits^2 } \]
For more clear understanding let’s have a look at the example below:
Determine the magnitude of the vector AB having (1, 2) coordinates in initial point A and (4, 3) coordinates in final point B.
Given data
A = (1, 2)
B = (4, 3)
To Find
The magnitude of the vector = ?
To get the magnitude of the vector, we will use the formula listed below:
\[|\overrightarrow {AB} | = \sqrt {\mathop {\left( {\mathop x\nolimits_1 - \mathop x\nolimits_0 } \right)}\nolimits^2 + \mathop {\left( {\mathop y\nolimits_1 - \mathop y\nolimits_0 } \right)}\
nolimits^2 } \]
From the above formula, let’s get the value of
x[0] = 1
y[0] = 2
x[1] = 4
y[1] = 3
Using the distance formula,
\[|\overrightarrow {AB} | = \sqrt {\mathop {\left( {4 - 1} \right)}\nolimits^2 + \mathop {\left( {3 - 2} \right)}\nolimits^2 } \] \[|\overrightarrow {AB} | = \sqrt {\mathop {\left( 3 \right)}\
nolimits^2 + \mathop {\left( 1 \right)}\nolimits^2 } \] \[|\overrightarrow {AB} | = \sqrt {9 + 1} = \sqrt {10} \]
The magnitude of |AB| is
\[|\overrightarrow {AB} | = 10\]
How to use Vector Magnitude Calculator?
The steps to use a vector magnitude calculator are as follows:
Step 1: Enter the value of vector A in the required input.
Step 2: Enter the value of vector B in the required input.
Step 3: The calculator will automatically display an answer on the screen.
Calculator use
What function does a magnitude of a Vector calculator perform? You have to add coordinates values, and in a few moments, the calculator will display results on the screen. It gets the prescribed
solution to your problems. To resolve complex issues, you can use this calculator. | {"url":"https://calculatorsbag.com/calculators/physics/vector-magnitude","timestamp":"2024-11-12T18:24:48Z","content_type":"text/html","content_length":"43022","record_id":"<urn:uuid:1b3202a4-584f-4def-a969-cf006970e21c>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00881.warc.gz"} |
Predicate - (Incompleteness and Undecidability) - Vocab, Definition, Explanations | Fiveable
from class:
Incompleteness and Undecidability
A predicate is a statement or expression that describes a property or relationship that can be attributed to one or more subjects in first-order logic. Predicates serve as the foundation for forming
propositions and enable the use of quantifiers to express statements about sets or groups of objects, which is crucial for formal reasoning and proofs.
congrats on reading the definition of Predicate. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Predicates can take various forms, such as unary (one argument), binary (two arguments), or n-ary (multiple arguments), depending on the number of subjects involved.
2. In first-order logic, predicates are typically denoted by letters (e.g., P, Q) followed by variables in parentheses, like P(x) or Q(x,y).
3. Quantifiers interact with predicates to form statements like 'For all x, P(x)' or 'There exists an x such that P(x)'.
4. The truth value of a predicate can change depending on the assignment of values to its variables, making it a flexible tool in logical expressions.
5. Predicates are essential for constructing logical arguments and proofs, allowing mathematicians and logicians to formulate claims about properties of objects.
Review Questions
• How do predicates function within the framework of first-order logic?
□ Predicates in first-order logic function as expressions that assert properties or relationships involving subjects. They allow for the construction of propositions by specifying what
conditions must hold true for those subjects. This integration enables more complex reasoning through the use of quantifiers, which help articulate claims about groups or sets of objects.
• Discuss the role of quantifiers in conjunction with predicates and provide examples.
□ Quantifiers play a critical role in conjunction with predicates by allowing statements to express generality or existence. For example, using the universal quantifier 'for all,' one might
write 'For all x, P(x)', indicating that every element x within the domain satisfies the predicate P. Alternatively, with the existential quantifier 'there exists', one could say 'There
exists an x such that P(x)', meaning at least one element satisfies the predicate. These combinations form the backbone of logical reasoning in formal proofs.
• Evaluate the importance of predicates and their interactions with logical connectives and quantifiers in mathematical reasoning.
□ Predicates are foundational to mathematical reasoning as they enable clear expressions of relationships and properties among objects. Their interactions with logical connectives, such as AND,
OR, and NOT, allow for the construction of complex statements that represent nuanced ideas. When combined with quantifiers, predicates allow for robust assertions about entire sets or
collections of objects, facilitating precise argumentation and proof strategies. This synergy is vital for advancing knowledge in mathematics and logic.
© 2024 Fiveable Inc. All rights reserved.
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The alternating direction method of multipliers (ADMM) is a popular method for the separable convex programming with linear constraints, and the proximal ADMM is its important variant. Previous
studies show that the relaxation factor $\gamma\in (0, \frac{1+\sqrt{5}}{2})$ by Fortin and Glowinski for the ADMM is also valid for the proximal ADMM. In this paper, we … Read more
On the Direct Extension of ADMM for Multi-block Separable Convex Programming and Beyond: From Variational Inequality Perspective
When the alternating direction method of multipliers (ADMM) is extended directly to a multi-block separable convex minimization model whose objective function is in form of more than two functions
without coupled variables, it was recently shown that the convergence is not guaranteed. This fact urges to develop efficient algorithms that can preserve completely the numerical … Read more | {"url":"https://optimization-online.org/tag/contraction/","timestamp":"2024-11-03T04:41:26Z","content_type":"text/html","content_length":"85758","record_id":"<urn:uuid:e983c277-2b62-401c-8702-a11448d48043>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00532.warc.gz"} |
Seven Basic SI units are metre (Length), Kilogram (Mass), Second (Time), ampere (Electric Current) , Kelvin (Temperature), Candela (Luminous Intensity), Mole (Amount of Substance). How do you
calculate the values of Derived SI units values based on basic SI unit values. Remember all the formulas and do the calculations or use SI Ganaka. SI Ganaka calculates values of Derived SI units from
Basic SI unit values. Handy for Teachers in physics , chemistry or Maths and of course students. SI Ganaka are useful to check at the values you arrive at doing the calculations manually.
SI Ganaka 1.2.1
SI Ganaka is a Derived SI units calculator. The Basic SI units are metre, kilogram, second, ampere, kelvin, candela, mole. SI Ganaka calculates the Derived SI units values from Basic SI unit values.
It calculates 49 derived unit values from Basic SI unit values.
It Calculates
1.AREA - Area, Volume, Specific Volume, Density, Concentration, Capacitance, Power,Radiant,Flux.
2.SPEED - Speed, Velocity, Acceleration, Force, Camel Train, Frequency, Pressure,Stress
3.THERMAL - Energy,Work,Quantity of Heat, Temperature in Celsius, Luminous Flux
4.ELECTRIC - Electric Charge, Quantity of Electricity, Electric Potential, Potential Difference, Electromotive Force(EMF), Current Density,Capacitance
5.CONDUCTANCE - Electric Conductance, Electric Resistance, Inductance
6.MAGNETIC - Magnetic Field Strength, Magnetic Flux, Magnetic Flux Density.
7.RADIATION - Radiation Dose, Power, Radiant Flux, Catalytic Activity, Equivalent dose, Absorbed Dose, Power, Radiant Flux, Illuminance
Up and Down Arrow keys to Browse Menu. Select or center key to select the Option. Touch screen choose the option and choose select. On the Parameter screen , you enter the data and select calculate.
choose exit to get-back to menu. select quit to quit application.
Supporting Phones
All Java Enabled phone and Blackberry phones.
Both Touch and Non- Touch phones | {"url":"https://www.kyaliapps.com/search/label/Second","timestamp":"2024-11-08T18:38:55Z","content_type":"application/xhtml+xml","content_length":"101831","record_id":"<urn:uuid:66f59341-cb9d-43ea-ae90-a542d1088200>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00761.warc.gz"} |
L'Hopital's Rule Calculator - Evaluate Indeterminate Limits Easily
Introduction to L'Hopital's Rule Calculator:
L'Hopital's Rule Calculator is an amazing tool that helps you to calculate the indeterminate form of function. It is used to evaluate the limit of the special type of function which are present in
the form of 0/0 or ∞/∞ or 0x∞, ∞x∞.
When you calculate the indeterminate form problem by hand you may get confused because of complex function or you don't know the rule about how to solve these problems using l'hopital method. To
avoid the tedious effort of solving you need a calculator like our L'hospital calculator that provides you solution without doing any calculation.
What is L'Hopital's Rule?
L`Hopital is a process that is used to find the limit of indeterminate form of functions using differential methods in calculus. If you want to evaluate the limit of these functions directly you
cannot find the solution because it is given an undefined solution.
The indeterminate form of function has different forms like 0/0 or ∞/∞ or 0x∞, ∞x∞ and if your function has any of this form then you can apply the l`hopital method to get a solution easily.
Rule of L`Hopital:
The rule of l`hopital is consist of two function f(x) and g(x) which are differentiable functions such as f`(x) and g`(x) and both have a limx→cf(x)=0 and limx→cg(x)=0, then
$$ \lim_{x \to c} \frac{f(x)}{g(x)} \;=\; \lim_{x \to c} \frac{f’(x)}{g’(x)} $$
How to Calculate L'Hopital's Rule?
For the Calculation of a limit function using L'Hôpital's rule you need to understand the different forms of indeterminate form which is 0/0 or ∞/∞ or or 0x∞, ∞x∞. Here’s a step-by-step guide on
which you know about how to apply L'Hôpital's process.
Step 1:
Determine the function if the limit function that you want to evaluate is in the form 0/0 or ∞/∞.
Step 2:
Only you can apply the l`hopital rule if the given function follow 0/0 or ∞/∞ form.
Step 3:
To reduce the indeterminate form, differentiate the given function and it should ensure that f(x) and g(x) are differentiable separate at the numerator and denominator.
Step 4:
Again evaluate the limit in f`(x) , if the function 's indeterminate form reduces then you can simply apply the limit and get a solution.Otherwise you need to again differentiate the function with
respect to x and then again check the limit.
Step 5:
Repeat this process if needed until the limit can be evaluated directly or the determined form does not exist in the given function.
Solved Example Of L'Hopital's Rule:
The solved example of indeterminate form of limit function l'hopital's rule calculator with steps gives you more clarity about its calculation process.
Example: Find the indeterminate form of function
$$ \lim_{x \to c} \frac{1 - cos\; x}{x} $$
Determine the given function,
$$ \lim_{x \to 0} \frac{1 - cos\; x}{x} $$
Apply the limit to check it has indeterminate form or not,
$$ \lim_{x \to 0} \frac{1 - cos\;x}{x} \;=\; \frac{0}{0}\; form $$
=0/0 form
So apply l`hopital rule in which you differentiate the function with respect to x (differentiate numerator and denominator separately without following regular rule of derivation)
$$ =\; \lim_{x \to 0} \frac{\frac{d}{dx}(1 - cos\; x)}{\frac{d}{dx}(x)} $$
Again check function indeterminate form reduce or not by applying the limit,
$$ \lim_{x \to 0} \frac{sin\; x}{1} $$
Now you can apply limit because function is no more undefine function,
$$ \frac{\lim_{x \to 0} sin\; x}{\lim_{x \to 0} 1} \;=\; \frac{0}{1} $$
The result of given indeterminate function is,
$$ \frac{\lim_{x \to 0} sin\; x}{\lim_{x \to 0} 1} \;=\; \frac{0}{1} \;=\; 0 $$
How to Solve L Hospital Rule Calculator?
L'hopital calculator has a simply layout so you just need to enter your problem in this calculate to get a solution in an easy method. Follow our guidelines before using it. These guidelines are:
• Enter the limit function in the input field that you want to evaluate.
• Enter the limit point value in the input box.
• Add the variable of differentiate through which you want to find limit function
• Select the type of indeterminate form of given limit function.
• Review the given function value before hitting the calculate button to start the evaluation process in thel'hospital's rule calculator.
• Click the “Calculate” button to get the solution of your given limit function problem in L'Hopital's Rule.
• If you want to try out our professional tool for the first time then you must try out the loud example of l'hopital limit function to learn more about it.
• Click on the “Recalculate” button to get a new page to find more example solutions of limit function problems.
Output from L Hopital Rule Calculator:
L'Hopital's Rule Calculator gives you the solution to a given limit function question when you add the input into it. It may be included as:
Result option
When you click on the result option it gives you a solution to the l'hopital problem.
Possible steps
It provides you with a solution to the l'hopital's rule problem where all the calculations are mentioned in steps.
Advantages of L'Hopital Calculator:
L hospital rule calculator provides a ton of advantages that you get when you calculate the indeterminate form of limit problems to get solutions. These advantages are:
• L hopital rule calculator is a free tool that enables you to evaluate the limit indeterminate problem with a solution.
• It is a manageable tool that can solve different types of indeterminate form to find the limit function using the l'Hopital rule.
• Our tool helps you to get a stronghold on the l'hopital rule method when you use it for practice.
• L'hospital's rule calculator saves the time that you consume on the calculation of the limit function without applying L'Hopital's rule manually but it helps you to evaluate the given function
limit in a couple of minutes.
• L'Hopital's Rule Calculator is a reliable tool that provides you accurate solutions when you use it to calculate the limit function problems without error.
• L hospital rule calculator gives the solution without a sign-in condition so you can use it anywhere through the internet. | {"url":"https://pinecalculator.com/lhopitals-rule-calculator","timestamp":"2024-11-12T07:39:27Z","content_type":"text/html","content_length":"66304","record_id":"<urn:uuid:c81eeb77-390b-4669-b528-b87dd0fa9356>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00200.warc.gz"} |
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Date Issued:
Recently, a highly resolved, finite element mesh was developed for the purpose of performing hydrodynamic calculations in the Western North Atlantic Tidal (WNAT) model domain. The WNAT model
domain consists of the Gulf of Mexico, the Caribbean Sea, and the entire portion of the North Atlantic Ocean found west of the 60° W meridian. This high resolution mesh (333K) employs 332,582
computational nodes and 647,018 triangular elements to provide approximately 1.0 to 25 km node spacing. In the previous work, the 333K mesh was applied in a Localized Truncation Error Analysis
(LTEA) to produce nodal density requirements for the WNAT model domain. The goal of the work herein is to use these LTEA-based element sizing guidelines in order to obtain a more optimal finite
element mesh for the WNAT model domain, where optimal refers to minimizing nodes (to enhance computational efficiency) while maintaining model accuracy, through an automated procedure. Initially,
three finite element meshes are constructed: 95K, 60K, and 53K. The 95K mesh consists of 95,062 computational nodes and 182,941 triangular elements providing about 0.5 to 120 km node spacing. The
60K mesh contains 60,487 computational nodes and 108,987 triangular elements. It has roughly 0.5 to 185 km node spacing. The 53K mesh includes 52,774 computational nodes and 98,365 triangular
elements. This is a particularly coarse mesh, consisting of approximately 0.5 to 160 km node spacing. It is important to note that these three finite element meshes were produced automatically,
with each employing the bathymetry and coastline (of various levels of resolution) of the 333K mesh, thereby enabling progress towards an optimal finite element mesh. Tidal simulations are then
performed for the WNAT model domain by solving the shallow water equations in a time marching manner for the deviation from mean sea level and depth-integrated velocities at each computational
node of the different finite element meshes. In order to verify the model output and compare the performance of the various finite element mesh applications, historical tidal constituent data
from 150 tidal stations located within the WNAT model domain are collected and examined. These historical harmonic data are applied in two types of comparative analyses to evaluate the accuracy
of the simulation results. First, qualitative comparisons are based on visual sense by utilizing plots of resynthesized model output and historical tidal constituents. Second, quantitative
comparisons are performed via a statistical analysis of the errors between model response and historical data. The latter method elicits average phase errors and goodness of average amplitude
fits in terms of numerical values, thus providing a quantifiable way to present model error. The error analysis establishes the 53K finite element mesh as optimal when compared to the 333K, 95K,
and 60K meshes. However, its required time step of less than ten seconds constrains its application. Therefore, the 53K mesh is manually edited to uphold accurate simulation results and to
produce a more computationally efficient mesh, by increasing its time step, so that it can be applied to forecast tide and storm surge in the Western North Atlantic Ocean on a real-time basis.
Title: OPTIMIZATION OF AN UNSTRUCTURED FINITE ELEMENT MESH FOR TIDE AND STORM SURGE MODELING APPLICATIONS IN THE WESTERN NORTH ATLANTIC OCEAN.
Kojima, Satoshi, Author
Name(s): Hagen, Scott, Committee Chair
University of Central Florida, Degree Grantor
Type of text
Date Issued: 2005
Publisher: University of Central Florida
Language(s): English
Recently, a highly resolved, finite element mesh was developed for the purpose of performing hydrodynamic calculations in the Western North Atlantic Tidal (WNAT) model domain. The WNAT
model domain consists of the Gulf of Mexico, the Caribbean Sea, and the entire portion of the North Atlantic Ocean found west of the 60° W meridian. This high resolution mesh (333K)
employs 332,582 computational nodes and 647,018 triangular elements to provide approximately 1.0 to 25 km node spacing. In the previous work, the 333K mesh was applied in a Localized
Truncation Error Analysis (LTEA) to produce nodal density requirements for the WNAT model domain. The goal of the work herein is to use these LTEA-based element sizing guidelines in
order to obtain a more optimal finite element mesh for the WNAT model domain, where optimal refers to minimizing nodes (to enhance computational efficiency) while maintaining model
accuracy, through an automated procedure. Initially, three finite element meshes are constructed: 95K, 60K, and 53K. The 95K mesh consists of 95,062 computational nodes and 182,941
triangular elements providing about 0.5 to 120 km node spacing. The 60K mesh contains 60,487 computational nodes and 108,987 triangular elements. It has roughly 0.5 to 185 km node
spacing. The 53K mesh includes 52,774 computational nodes and 98,365 triangular elements. This is a particularly coarse mesh, consisting of approximately 0.5 to 160 km node spacing. It
Abstract/ is important to note that these three finite element meshes were produced automatically, with each employing the bathymetry and coastline (of various levels of resolution) of the 333K
Description: mesh, thereby enabling progress towards an optimal finite element mesh. Tidal simulations are then performed for the WNAT model domain by solving the shallow water equations in a time
marching manner for the deviation from mean sea level and depth-integrated velocities at each computational node of the different finite element meshes. In order to verify the model
output and compare the performance of the various finite element mesh applications, historical tidal constituent data from 150 tidal stations located within the WNAT model domain are
collected and examined. These historical harmonic data are applied in two types of comparative analyses to evaluate the accuracy of the simulation results. First, qualitative
comparisons are based on visual sense by utilizing plots of resynthesized model output and historical tidal constituents. Second, quantitative comparisons are performed via a
statistical analysis of the errors between model response and historical data. The latter method elicits average phase errors and goodness of average amplitude fits in terms of
numerical values, thus providing a quantifiable way to present model error. The error analysis establishes the 53K finite element mesh as optimal when compared to the 333K, 95K, and
60K meshes. However, its required time step of less than ten seconds constrains its application. Therefore, the 53K mesh is manually edited to uphold accurate simulation results and to
produce a more computationally efficient mesh, by increasing its time step, so that it can be applied to forecast tide and storm surge in the Western North Atlantic Ocean on a
real-time basis.
Identifier: CFE0000565 (IID), ucf:46421 (fedora)
Note(s): Engineering and Computer Science, Department of Civil and Environmental Engineering
This record was generated from author submitted information.
Subject(s): Finite Element Mesh
Tide and Storm Surge Modeling
Western North Atlantic Ocean
Link to This http://purl.flvc.org/ucf/fd/CFE0000565
Restrictions public
on Access:
Host UCF
In Collections | {"url":"https://ucf.digital.flvc.org/islandora/object/ucf%3A46421","timestamp":"2024-11-04T12:19:07Z","content_type":"text/html","content_length":"40679","record_id":"<urn:uuid:f26a4030-4031-44b9-a0dc-fc61e060651c>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00702.warc.gz"} |
Recursive algebraic types in D - Infognition tech blog
Recursive algebraic types in D
September 15, 2014
One of the things D programming language seemingly lacks is support for algebraic data types and pattern matching on them. This is a very convenient kind of types which most functional languages have
built-in (as well as modern imperative ones like Rust or Haxe). There were some attempts at making algebraic types on the library level in D (such as Algebraic template in std.variant module of D's
stdlib) however they totally failed their name: they don't support recursion and hence are not algrabraic at all, they are sum types at best. What follows is a brief explanation of the topic and a
proof of concept implementation of recursive algebraic types in D.
So what the heck are algebraic types after all? They must have something to do with algebra, right? In school we spent a lot of time in algebra classes working with functions, equations and searching
for their roots but nobody told us what an algebra, as a mathematical object, actually is. An algebra is often defined as some set together with a collection of operations on elements of this set. We
programmers and computer science enthusiasts work with types instead of sets (type is a more general thing than a set btw). So, in school we dealt with some particular examples of algebras. For
example, we used set R (the set of all real numbers) and operations like x+y, x-y, x*y, x^y, -x (negation, let's write it as ~x) etc. We can write their types as:
+: (R, R) -> R
-: (R, R) -> R
*: (R, R) -> R
~: R -> R
power: (R, R) -> R
Each of these operations takes some fixed number of elements of type R and outputs one element of R. Constants can be thought of as operations taking zero R arguments and returning one, e.g.
42: () -> R
We can also replace R with Q (the set of all rational numbers) and have most of these operations work on Q. That would be another algebra.
It's crucial that operations take arguments from the same set (type) as results they produce. That is algebraic. So, encoded in a programming language, a real algebraic type must support recursion.
Otherwise it's just a sum type, discriminated union, not very useful for encoding expressions, syntax trees, other kinds of trees or even lists.
The list of operations above is already looking very much like a generalized algebraic data type definition in functional languages like Haskell, OCaml, Agda or Idris. Here is how it's written in
type r =
| Add : r * r -> r
| Sub : r * r -> r
| Mul : r * r -> r
| Neg : r -> r
| C42 : () -> r
In Haskell it's essentially the same, modulo currying:
data R where
Add :: R -> R -> R
Sub :: R -> R -> R
Mul :: R -> R -> R
Neg :: R -> R
C42 :: () -> R
Since all these operations provide result of the same simple type R, we can use the simpler syntax:
data R = Add R R
| Sub R R
| Mul R R
| Neg R
| C42
Note that here we only define names and form (types) of operations, not their real content. If we change R to Q we get essentially the same type, only renamed. Unlike in math, where when switching
from set R to set Q we also changed the meaning (implementations if you wish) of functions: from adding / subtracting / etc. real numbers to adding / subtracting / etc. rational ones. To emphasize
the fact that ADT only describes the names and shapes of algebra operations, we can replace occurences of R in operands by a type variable:
data F a = Add a a
| Sub a a
| Mul a a
| Neg a
| C42
In order to encode a particular algebra with such operations we need, besides this definition, a type T of values (the algebra carrier) and an evaluation function
eval : F T -> T
This function will do a case analysis ("which operation is encoded?") and perform the operation, because it knows well how to add, multiply, negate and so on values of this type T.
Now let's recall some category theory. A category consists of a collection of objects and a collection of arrows (also called morphisms) between those objects, with some required properties regarding
their compositions and identity morphisms. We're interested in a category where objects are types and arrows are functions so that each function f with argument of type A and return value of type B
is an arrow from A to B, i.e. f : A -> B. A functor is a mapping from one category to another, it maps objects of one category to objects of another and the arrows are mapped correspondingly, so that
all the compositions remain. Functor mapping a category to itself is called an endofunctor. The parameterized type F above is such an endofunctor: any type X it maps to type F X, and any function g :
X -> Y can be trivially mapped to a fuction
fmap g : F X -> F Y
(Haskell compiler can derive the implementation of fmap here)
Now let's look at definition of F-Algebra in category theory. It consists of three parts: an endofunctor F, object T and a morphism from F(T) to T. These are exactly the three things mentioned above
that we need to encode some particular algebra. No wonder.
Ok, one more thing before we get to code in D. Recursion. Describing a functor like F above is simple but we want the type to be recursive: operations like Add should have operands of same type as
their result. Their operands' type is described by type parameter a, their result is F a, and they somehow must be equal. I.e. we need to find a type X to be used as type parameter to F such that
F X = X
Looks like we need the fixed point of F. Turns out we can define it quite easily. In Haskell it looks like this:
newtype Fix f = Fx (f (Fix f))
together with unwrapping operation
unFix :: Fix f -> f (Fix f)
unFix (Fx x) = x
and in D it looks like this:
struct Fix(alias F) { F!(Fix!F) unFix; }
F, being a functor, shall be some struct or class template, so we pass it by alias. (Higher kinded types? we can has them!)
For any endofunctor f, Fix f is a type, an object in our category. This functor, this object and the arrow Fx: f (Fix f) -> Fix f together form an F-algebra. Category theory says this F-algebra is
special: it's the initial object in the category of F-algebras for this functor, which means for any other algebra for this functor (say, object T and its evaluator arrow alg: F T -> T) there is a
unique morphism g from initial algebra Fix f to T, such that this diagram commutes:
(commutes - means the two paths from f (Fix f) to T are equal)
Knowing that unFix is the inverse of Fx we can actually define g via unFix, fmap g and alg.
This unique morphism from initial object is called catamorhism, so we'll call it cata with alg as its parameter. In Haskell it looks like this:
cata :: Functor f => (f a -> a) -> Fix f -> a
cata alg = alg . fmap (cata alg) . unFix
And in D it looks like this:
T cata(alias F, T)(T function(F!T) alg, Fix!F e) {
return alg( e.unFix.fmap( (Fix!F x) => cata!(F,T)(alg, x) ) );
Here we employ the fact that F is a functor and hence has fmap function defined.
Having the catamorphism allows us to easily define different evaluation functions for our recursive algebraic type from their non-recursive versions defined for particular algebras. I think it is
time to look at the code with some examples to understand how that works. For simplicity let's take a very tiny algebra: it will contain integer numbers and one addition operation. Something like
data Exp = Add Exp Exp | Const Int
Algebraic data types usually consist of a sum of products. Product types we had in C-like languages for ages: structs and classes. Sum types D doesn't have built-in, but we can make them with
templates. The simplest sum type would be a sum of two types, A and B, known in Haskell as Either a b. Its implementation as a discriminated union is very straightforward, with one twist: as shown
above, we really need a functor, a template in D, and the summands must have a type parameter too, so they will be also templates. I'll also add a match function to do familiar by functional
languages pattern matching on the values. Here's how our Either will look in D:
// A and B are class(T) or struct(T) implementing fmap
template Either(alias A, alias B) {
class EitherImpl(T) {
enum Tag { kA, kB }
Tag tag;
union {
A!T vA;
B!T vB;
this(A!T a) { tag = Tag.kA; vA = a; }
this(B!T b) { tag = Tag.kB; vB = b; }
U match(U)(U delegate(A!T) fa, U delegate(B!T) fb) {
final switch(tag) {
case Tag.kA: return fa(vA);
case Tag.kB: return fb(vB);
EitherImpl!U fmap(U)(U delegate(T) f) {
return match(a => new EitherImpl!U( a.fmap(f) ),
b => new EitherImpl!U( b.fmap(f) ));
alias Either = EitherImpl;
The functor for our little algebra of expressions will be defined as a sum
alias Exp = Either!(Add, Const);
where the two summands are
struct Const(T) {
int x;
mixin Functor!(Const, T);
class Add(T) {
T l, r;
this(T a, T b) { l = a; r = b; }
this() {}
mixin Functor!(Add, T, "l", "r");
The mixin Functor line plays the same role as (deriving Functor) in Haskell. It's a piece of metaprogramming in D to define fmap methods:
mixin template Functor(alias F,T, Vars...) {
F!U fmap(U)(U delegate(T) f) {
static if (is(typeof(this)==class))
auto r = new F!U;
auto r = F!U();
foreach(m; __traits(allMembers, typeof(r)))
static if (m != "Monitor" && m != "fmap") {
static if (IndexOf!(m, Vars) >= 0)
__traits(getMember, r, m) = f(__traits(getMember, this, m));
static if (!isSomeFunction!(typeof(__traits(getMember, r, m)))
&& isAssignable!(typeof(__traits(getMember, r, m))))
__traits(getMember, r, m) = __traits(getMember, this, m);
return r;
It's ugly but only needs to be defined once. For a class or struct template F!T it defines a method fmap that can take any function f of type T -> U and make a similar object or struct of type F!U
where given fields will be mapped by function f while others will be simply copied.
After having a functor Exp defined using Either we need to turn it into a recursive type by using its fix point:
alias FixX = Fix!Exp;
alias Exprec = Exp!FixX;
A small helper for creating its values would be handy:
auto mk(alias T, Xs...)(Xs xs) {
static if (is(T!FixX==class))
auto x = new T!FixX(xs);
auto x = T!FixX(xs);
return FixX(new Exprec(x));
Now we can create some expressions in our little algebra:
auto n1 = mk!Const(5);
auto n2 = mk!Const(7);
auto e1 = mk!Add(n1, n2);
auto e2 = mk!Add(mk!Const(30), e1);
// e2 is (30 + (5 + 7))
What do we do with them? We'd like to calculate expressions to some int results and we want to output expressions as strings. Each operation can be described as an algebra with int or string type as
its carrier and corresponding evaluation function from Exp!int or Exp!string to int or string. Here they are:
//alg : f a -> a, for some concrete a
int eval(Exp!int e) { return e.match(a => a.l + a.r, i => i.x); }
string show(Exp!string e) {
return e.match(a => "(" ~ a.l ~ " + " ~ a.r ~ ")", i => i.x.text);
They use pattern matching by passing to lambdas to match: one that tells what to do with Add case and one that tells how to process the Const case. These are not recursive and describe just one step
of computations. Now we can use our catamorphism to apply them recursively and turn such function of type f a -> a into function of type Fix f -> a which will give us desired results:
cata(&show, e2).writeln; // => (30 + (5 + 7))
cata(&eval, e2).writeln; // => 42
The full source code, compilable and runnable, can be found here on dpaste.
Comments (14)
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Proof of Space | Docs
Proof of Space
A Proof of Space protocol is one in which:
• A Verifier can send a challenge to a Prover.
• The Prover can demonstrate to the Verifier that the Prover is reserving a specific amount of storage space at that precise time.
The Proof of Space protocol has three components: plotting, proving/farming, and verifying.
Plotting is the process by which a Prover, who we refer to as a farmer, initializes a certain amount of space. To become a farmer, one must have at least 101.4 GiB available to reserve on their
computer (the minimum spec is a Raspberry Pi 4). There is no upper limit to the size of a BPX farm. Several farmers have multi-PiB farms.
The k32 plot can be created in around five minutes with a high-end machine with 400 GiB of RAM, or six hours with a normal commodity machine, or 12 hours with a slow machine using one CPU core and a
few GB of RAM. Opportunities still remain for huge speedups. Furthermore, each plot only needs to be created once; a farmer can farm with the same plots for many years.
Plot sizes are determined by a k parameter, where space = 780 * k * pow(2, k - 10), with a minimum k of 32 (101.4 GiB). The Proof of Space construction is based on Beyond Hellman, but it is nested
six times (thereby creating seven tables), and it contains other heuristics to make it practical.
Each of the seven tables in a plot is filled with random-looking data that cannot be compressed. Each table has 2^k entries. Each entry in table i contains two pointers to table i-1 (the previous
table). Finally, each table-1 entry contains a pair of integers between 0 and 2^k, called "x-values." A proof of space is a collection of 64 x-values that have a certain mathematical relationship.
The actual on-disk structure and the algorithm required to generate it are quite complicated, but this is the general idea.
Once the Prover has initialized 101.4 GiB, they are ready to receive a challenge and create a proof. No registration or online connection is required to create a plot using the original plot format.
Nothing hits the blockchain until a reward is won, similar to PoW.
Farming is the process by which a farmer receives a sequence of 256-bit challenges to prove that they have legitimately put aside a defined amount of storage. In response to each challenge, the
farmer checks their plots, generates a proof, and submits any winning proofs to the network for verification.
For each eligible plot (explained later), a farmer uses the following procedure to generate a full proof of space. Keep in mind that a plot consists of 7 tables (T1-T7) of approximately the same
size, as well as 3 checkpoint tables (C1-C3), which are much smaller:
1. The farmer receives a challenge from the VDF
2. For each eligible plot, extract a k-sized value from the challenge, where k denotes the size of the plot (k32, k33, etc)
3. Look in the C2 table for a location at which to start scanning the C1 table
4. Scan the C1 table for the location at which to start scanning the C3 table
5. Read either one or two C3 parks. The number of parks to read depends on the index and value calculated from the C1 table. This requires an average of 5000 reads (the maximum is 10 000). These are
sequential reads of 4 bytes (for an average total of 20 KiB)
6. Grab all the f7 entries matching the challenge value (which can be 0 or more), along with the index in the table at which they were found
7. For each matching f7 value, read T7 at the same index where the f7 value was found in its own table, and grab that entry, which is an index into T6
8. The T6 index contains one line point with two back pointers to T5, four to T4, eight to T3, sixteen to T2 and thirty-two to T1. Each back pointer requires 1 read, so a total of 64 disk reads (1
index from T7, 63 back pointers) are performed to fetch the whole tree of 64 x-values.
Since most proofs generated by this process are not good enough to be submitted to the network for verification, we can optimize this process by only checking one branch of the tree. This branch will
return two of the 64 x-values. The position of the x-values will always be consecutive and will depend on the signage point (eg x0 and x1... or x34 and x35). We hash these x-values to produce a
random 256-bit "quality string." This is combined with the difficulty and the plot size to generate the required_iterations. If the required_iterations is less than a certain number, the proof can be
included in the blockchain. At this point, we look up the whole proof of space.
By only looking up one branch to determine the quality string, we can rule out most proofs. This optimization requires only around 7-9 disk seeks and reads, or about 70-90 ms on a slow hard drive
Throughout this website, we'll make a simple assumption that a single disk seek requires 10ms. In reality, this is typically 5-10ms, so we're using a conservative estimate. The 10ms estimate also
takes into account the time required to transfer data after the seek. While storage industry specs typically assume that large files are being transferred, this does not hold true for BPX farming,
where proof lookups only require a tiny amount of data to be transferred. Therefore, for this website, it's safe to assume the transfer is almost instant.
BPX also uses a further optimization to disqualify a certain proportion of plots from eligibility for each challenge. This is referred to as the plot filter. The current requirement is that the hash
of the plot ID, challenge, and signage point starts with 9 zeros. This excludes 511 out of every 512 plots. The filter hurts everyone equally (except for replotting attackers), and is therefore fair.
The plot filter effectively reduces the amount of resources required for farming by 512x - each plot only requires a few disk reads every few minutes. A farmer with 1 PiB of storage (10,000 plots)
will only have an average of 20 plots that pass the filter for each challenge. Even if these plots all are stored on slow HDDs, and connected to a single Raspberry Pi, the average time required to
respond to each challenge will be less than two seconds. This is well within the limits to avoid missing out on any challenges.
Each plot file has its own unique private key called a plot key. The plot ID is generated by hashing the plot public key, the farmer public key, and either the pool public key or the pool contract
puzzle hash. The requirements for signing a proof of space depend on the type of plots being used.
In practice, the plot key is a 2/2 BLS aggregate public key between a local key stored in the plot and a key stored by the farmer software. For security and efficiency, a farmer may run on one server
using this key and signature scheme. This server can then be connected to one or more harvester machines that store the actual plots. Farming requires the farmer key and the local key, but it does
not require the pool key, since the pool's signature can be cached and reused for many blocks.
After the farmer has successfully created a proof of space, the proof can be verified by performing a few hashes and making comparisons between the x-values in the proof. Recall that the proof is a
list of 64 x-values, where each x-value is k bits long. For a k32 this is 256 bytes (2048 bits), and is therefore very compact.
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The Universality Problem
Criste, Cristina (2022) The Universality Problem. Doctoral thesis, University of East Anglia.
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The theme of this thesis is to explore the universality problem in set theory in connection to model theory, to present some methods for finding universality results, to analyse how these methods
were applied, to mention some results and to emphasise some philosophical interrogations that these aspects entail.
A fundamental aspect of the universality problem is to find what determines the existence of universal objects. That means that we have to take into consideration and examine the methods that we use
in proving their existence or nonexistence, the role of cardinal arithmetic, combinatorics etc. The proof methods used in the mathematical part will be mostly set-theoretic, but some methods from
model theory and category theory will also be present.
A graph might be the simplest, but it is also one of the most useful notions in mathematics. We show that there is a faithful functor F from the category L of linear orders to the category G of
graphs that preserves model theoretic-related universality results (classes of objects having universal models in exactly the same cardinals, and also having the same universality spectrum).
Trees constitute combinatorial objects and have a central role in set theory. The universality of trees is connected to the universality of linear orders, but it also seems to present more
challenges, which we survey and present some results. We show that there is no embedding between an ℵ2-Souslin tree and a non-special wide ℵ2 tree T with no cofinal branches. Furthermore, using the
notion of ascent path, we prove that the class of non-special ℵ2-Souslin tree with an ω-ascent path a has maximal complexity number, 2ℵ2 = ℵ3.
Within the general framework of the universality problem in set theory and model theory, while emphasising their approaches and their connections with regard to this topic, we examine the possibility
of drawing some philosophical conclusions connected to, among others, the notions of mathematical knowledge, mathematical object and proof.
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Effective Energy Management Scheme by IMPC
Intelligent Automation & Soft Computing
Effective Energy Management Scheme by IMPC
Department of Electrical and Instrumentation Engineering, Thapar Institute of Engineering & Technology, Patiala, India
*Corresponding Author: Smarajit Ghosh. Email: smarajitg@hotmail.com
Received: 28 December 2021; Accepted: 18 February 2022
Abstract: The primary purpose of the Energy Management Scheme (EMS) is to monitor the energy fluctuations present in the load profile. In this paper, the improved model predictive controller is
adopted for the EMS in the power system. Emperor Penguin Optimization (EPO) algorithm optimized Artificial Neural Network (ANN) with Model Predictive Control (MPC) scheme for accurate prediction of
load and power forecasting at the time of pre-optimizing EMS is presented. For the power generation, Renewable Energy Sources (RES) such as photo voltaic (PV) and wind turbine (WT) are utilized along
with that the fuel cell is also presented in case of failure by the RES. Such a setup is connected with the grid and applies to the household appliances. In improved model predictive control (IMPC),
the set of constraints for the power flow in the system is optimized by the ANN, which is trained by EPO. Such a tuning based prediction model is presented in the IMPC technique. The proposed work is
implemented in the MATLAB/Simulink platform. The energy management capability of the proposed system is analyzed for different atmospheric conditions. The total system cost, life cycle cost and
annualized cost for IMPC are 48%, 45% and 15%, respectively. From the performance analysis, the cost obtained by the proposed method is very low compared to that obtained by the existing techniques.
Keywords: Artificial neural network; emperor penguin optimization; energy management; model predictive control
Nowadays, the RES, such as hydro, wind and PV, have more attention among the peoples because of the reasons like environmental pollution and the rapid depletion of fossil fuel [1]. String research
efforts are promoting to seek optimal exploitation of the RES like limited fossil fuels, growing energy demand, and need to reduce carbon dioxide emission in the atmosphere, which offers a
distributed potential [2]. In recent decades solar energy is widely used. The reason for choosing PV is that it has no emission cost. Providentially, the utilization of energy from wind and PV has
more potential in India because of its suitable climate and location. In this area, most of the electricity is used to meet the specific demand with the help of Diesel Generators (DG). However, the
cost for maintaining the DG and the cost of fossil is exceptionally high. Thus, an alternative energy source like wind and PV hybrid combinations provides appropriate options for the power production
Hybridization of wind and solar energy implementation in a particular field needs to be energy consumption-less without interruption [5]. Any kind of RESs can be combined to supply for many
applications like domestic, industrial. Among various RES, solar and wind systems are widely affected by climatic changes [6]. The integration of more than RESs, such as hybrid form has more
significant influence due to these environmental changes [7]. For example, high solar radiation and less wind velocity sunny days and less solar radiation and high-speed wind velocity in winter.
Thus, the efficiency of RESs may change throughout the year because of various climatic conditions. Different kinds of RESs can be used in a hybrid energy system. Therefore, it is reliable and more
cost-effective than the other single energy system [8].
In recent years, the hybridization of different RESs has been widely considered. Frequency variation and power quality issues may happen in Hybrid RES based standalone grid systems [9]. Harmonics
commonly occur at the source side of the system while the presence of connection [10]. Model Predictive Control (MPC) is one of the promising technologies, which processes the control unit by
reducing the objective function. The advantages of MPC is the enhanced energy savings, improve steady-state response, enhanced transient response and cost-effective [11]. However, it has some
disadvantages and these are installation and maintenance expenses, controller structure limitation, process limitation and operator interface.
The essential contribution is given as follow,
• To fulfil the load, the RES power generators like PV and WT is used along with that the fuel cell is also used during the time of failure in PV and WT (due to climatic change) since the power
generation is performed for 1-year time interval.
• The IMPC scheme is developed for regulating the system of power flow, which takes the initial energy flow as a reference and then control the power that is needed for the load every time.
• The set of constraints and objective functions are predicted by the ANN, where the error in the prediction is reduced by the training phase.
• The Emperor penguin optimization (EPO) algorithm is used to train the Artificial Neural Network (ANN) with a Model Predictive Control (MPC) scheme for accurate prediction of load and power
• The performance of the proposed method is analyzed in contrast with the ant colony and artificial neural network (ACO-ANN), neural network with back propagation (NN-BP) and MPC technique of energy
The paper’s organization shows that the contribution of the hybrid renewable energy system (HRES) and its related works are presented in Section 2. The proposed methodology for describing HRES power
flows is covered in Section 3. Then the suggested system simulated results are briefly examined in Section 4, and the conclusion of the proposed system has been made in the conclusion section.
Some recently developed Hybrid RES utilized EMS with some other traditional controllers is also presented as follows.
Comparative analysis of Hybrid RESs was presented by Muh et al. [12] for off-grid applications with the help of climate data of sum, in the North-West region of Cameroon that had been utilized to
represent the resource data for Southern Cameroons. Based on the wind turbine, battery, diesel generator, charge controllers, PV module, inverters and micro-hydro turbine, nine hybrid arrangements
were deliberated in that task.
A hybrid RESs optimal mapping of locations was discussed by Diemuodeke et al. [13] using the TOPSIS decision making multi-criteria algorithm. By deliberation of storing the power and diesel generator
as backup, the hybrid energy systems were recommended for the optimal mapping in wind and PV.
Advanced technology of solar PV and environmental challenges of fossil fuels in fast-tracking hybrid RES was recommended by Ebhota et al. [14]. Then the Energy Materials (EMs) roles were improved on
community-scale HRES to develop energy challenges. Furthermore, the combination of HRES into EM was developed.
The smart building optimal cost operation, integrated with a centralized HVAC system, solar power generation, and the storage devices as electrical and thermal had been presented by Bianchini et al.
[15]. A demand response program was considered in building participation. The HVAC system was managed optimally by MPC strategy-based solution along with the storage devices in thermal comfort and
The public grid electrically coupled with a complex residential power system based on the potential economic MPC had been presented by Kuboth et al. [16]. That system consisted of a battery and
thermal storage system, a PV production system, a heat pump of air-to-water and a model of the building. The power system was managed by the MPC algorithms through nonlinear global optimization.
Integrating the hybrid energy storage system (HESS) with DC microgrid was based on a Bidirectional Single Inductor Multiple Port (BSIMP) converter that had been presented by Wang et al. [17]. The
HESS was formed as the combination of some kinds of energy storages (ESs). The BSIMP converter was utilized for regulating the HESS by the developed MPC based control method. Instantaneously, the bus
voltage of the DC microgrid could be maintained under the non-linear load consumption and the generation of renewable energy.
The rapid increase in industrialization and globalization is leading to higher consumption of energy sources, including an increase in electricity requirements. Development of newer EMS definitely
needs implantation of a suitable control method to maintain good working of the energy management scheme. Many existing works have dealt with the development of different control schemes for energy
management purpose in domestic usage. However, most of them failed inaccurate prediction of load and weather forecasting at the time of pre-optimizing EMS. Similarly, they exhibit design complexity
while developing the EMS with separate algorithms for every stage of operation. This has motivated me to create a new control strategy in EMS with the aid of NN concept to make the work easier and
efficient than other existing works.
3 Optimal Methodology of HRES Based EMS
In electrical energy production, renewable energies are increasingly operated in hybrid energy storage systems such as PV panels and WT. The vital key for the design is the distribution of the power
generation. The optimal controller strategy design for developing an EMS with RES and the utility grid system is discussed in this section. Here, the IMPC scheme is developed for regulating the power
to the load profile. Already MPC is used in the field of oil refineries and chemical plant successfully; here, the hope is to manage energy among the load and generation concerning MPC. The IMPC
method works by having a set of objective functions and the set of constraints. Those operating constraints and the objective functions are optimized by the artificial neural network (ANN), which
requires training and testing for the weight and the bias value to reduce the error factor. Hence the system employs EPO to minimise the error occurs in the ANN that led accurate model prediction.
The resultant EMS by IMPC is utilized for household appliances as well as to the grid. The architecture of the proposed system with the controller design is given in Fig. 1, which consists of RESs
and its control system. The power generated from RES is forecasted, then they are up-converted with the help of boost converters. The proposed IMPC scheme controls the EMS based on adequate power
output for grid and household applications.
The proposed RESs system power flow (Phybrid) is determined by Eq. (1).
Phybrid=Ppv+Pwecs+Pfc (1)
Here, Ppv is represented as the PV generated power and wind power generation is presented as Pwecs and Pfc is denoted as the fuel cell power. The active power control process derives the system power
balance, which depends on the variation of the HRES and required load output power. The system power balance equation is given by Eq. (2).
Pgrid=Pload−Phybrid (2)
The generated power is allowed to load with the DC-link (Pdc), which is determined by Eq. (3).
Pdc=Cdc(dVdc/dt)Vdc=Phybrid−Pgrid (3)
Here, the DC link voltage is denoted as Vdc, Cdc is described as DC-link capacitance and the grid operator power is as Pgrid. When the uncertainty RESs and nonlinear load demand variation is not
satisfied, the balance condition of power is disturbed. The system parameters are presented in the following section.
3.1 Solar PV Power Generation System
Solar PV generates electricity from the sunlight by using the PV modules and the induced direct current (DC) is converted into alternating current (AC) using an inverter. The performance of the PV
models is attained based on the current-voltage (I–V) curve. Also, the PV system performance is optimized with the help of Maximum Power Point (MPP). According to the latitude regions, the optimum
tilt angle is varied [18]. The output of PV array power (Ppv) concerning the PV module current and voltage is determined by Eq. (4).
Ppv=Ysolarfsolar(Ginc/GincSTC)(1+δtemp(Tcell−TcellSTC)) (4)
Here, the PV array rated capacity is denoted as Ysolar concerning the standard test conditions (kW) based output power, fsolar is described as the de-rating factor (%) of PV, the PV array instance
radiation of the solar (kWm−2) is denoted as Ginc, GincSTC is described as the instance radiation under standard test conditions(kWm−2), the coefficient of the temperature described as δtemp and its
value is 0.004°C−1, the operation temperature (°C) of the PV cell is denoted as Tcell and then the standard test conditions (25°C) temperature of PV cell is described as TcellSTC . For an optimal
operation of the PV systems, the Maximum Power Point Tracking (MPPT) is used for high power generation. Then another power generation of the RES as wind power generation is presented in the following
3.2 Wind Power Generation System
A WECS is used for generating the power from wind based on the Permanent Magnet Synchronous Generator (PMSG), which is connected to the grid. The Voltage Source Converters (VSC) of the stator side
controls the PMSG electrical torque for achieving the maximum power point tracking (MPPT) and also the regulation of flux by controlling the direct axis current of the generator. The generated power
is transferred to the grid with the help of the grid side VSC and the DC-link by stabilizing the nominal voltage value of the DC-link. The grid side VSC is utilized for compensating the reactive
power for establishing the unity power factor without injecting reactive power to the grid [19]. In a traditional MPPT algorithm for avoiding the generator fast electrical dynamics, the generated
electrical power (Pwecs) is determined by Eq. (5).
Pwecs=(Pmechηwecs)/(τwindLtrans+1) (5)
The mechanical power (Pmech) is given by Eq. (6).
Pmech=0.5ρCpower(λ)ASwind (6)
where Ltrans is described as Laplace transform operator, the efficiency is denoted as ηwecs, the drive train’s time constant as the ratio among shaft’s inertia and friction is τwind that is assumed
to be 1.5S while ηwecs is 0.9, air density is denoted as ρ, power coefficient is described as Cpower, λ is described as the tip speed ratio, A is the disk area of rotor and Swind is denoted as the
wind speed. The WECS main dynamics are modelled with representing the input disturbance as a first-order filter to the system. The description of the fuel cell model is presented in the following
An electrochemical device for exchanging a fuel chemical energy and oxidant directly to the low voltage DC electricity is a fuel cell unit. The proton exchange membrane (PEM) fuel cell is one of the
best selections for distributed generation between the fuel cells. Generally, the transformation of an electrochemical reaction is completed with oxygen or air oxidant. The oxidized output is not a
portion of the structure of fuel cells, and both products are provided instantaneously. However, electricity generation is kept constant when there are reagents.
Here, the complete model of the PEM fuel cell is modelled from a reduced model and validity was demonstrated and compared the reduced and complete models [20]. The output voltage of fuel cells
(Vfcout) in this reduced model is determined by Eqs. (7) and (8) respectively.
Vfcout=Nfccell(Efccell−(Vfcact+Vfcoh+Vfcconc)) (7)
Efccell=Einicell−ke(T−Tini)−(RT/2F)ln(PH2O/PO20.5PH2) (8)
Here, PH2O is assumed to be constant, PH2 is designed from the law of mass conservation and PO2 is considered from the law of ideal gas, the other components of fuel cell are air cooler, compressor
and humidifier.
The hybrid system energy source gives an adjustable voltage that is based on their current demand under different ranges. Each source requires a device at the constant voltage for transferring the
power output in the DC bus.
Especially, the DC/DC converters require pulse width modulation (PWM) that is interconnected with DC bus and sources. Here, the equivalent model of the converter is given in Fig. 2, which has been
modelled by the converters and also the switches of power electronics are described by the sources of voltage and current. The reproduction dynamics of this converter model permits the control system
results and interactions of power system higher sample times. Finally, the IMPC for controlling the new EMS is presented in the following section.
In this work, the IMPC is designed, which is optimized by the ANN-based EPO, as shown in Fig. 3. EPO algorithm optimizes the ANN with MPC scheme for accurate prediction of load for the management. At
first, the HRES output is given to the load; if there are any fluctuations in the received power at the grid, then it will be returned to the controller. There as a reference, the energy at the
converter is managed to comfort the load.
In the MPC, the different optimization resolution problem in each sampling period, as different information is combined in dynamic evolution, this concept is known as receding horizon, illustrated in
Fig. 4. In this process, the first control is manually performed through operators, which is a well-known process comprising in area MPC control. At an industrial level, an exact spontaneous approach
addresses process control and affected diffusion. The predictor is controlled for determining for every instant t based on the process model and the dynamic process evolution of predictions is [y(t
+1/t), …, y(t+(N/t))]2 in prediction horizon N from information dynamic accessible until that moment [21]. Thus, the cost function of the process is taken as a main objective to retain the output
y(t+(k/t)) and a trajectory w(t+k).
Initially, the input process u(t) is and the control signal u(t+1) have to be applied at that moment of the previous instant u(t+(1/t)), which is one not equal to that postulated. The control
methodology, analysis describes that in all the implementation, the optimization problem in any predictive control models is understood in each sampling period. In this methodology, it comprises some
important components such as optimizer, predictor and objective function, by combining with different variations of these components to obtain some family of predictive controllers. In IMPC, the
optimizer for the control of constraints and objective function is set by the ANN. The set of constraints are power limit, load profile etc., and the objectives considered here are demand
satisfaction, cost profile. According to the objective function utilized in the given optimization method, the process is modelled by considering different controllers. This can be inferred
diversity. The output is determined by Eq. (9).
y(t)=∑i=1∞hiu(t−i) (9)
Here, hi is the sampled values, which is attained by exposing process to an amplitude impulse unit equal to the sampling time interval. Considering the N values as sum is truncated that is described
by Eqs. (10) and (11).
y(t)=∑i=1Nhiu(t−i)=H(z−1)u(t) (10)
H(z−1)=h1z−1+h2z−2+…+hNz−N (11)
The required prediction is described by Eq. (12).
y(t+k/t)=∑i=1Nhiu(t+k−i/t)=H(z−1)u(t+k/t) (12)
For stable systems, the truncated response is determined by Eq. (13).
y(t)=y0+∑i=1NgiΔu(t−i)=y0+G(z−1)(1−z−1)u(t) (13)
Here, gi is the sampled value before step input y, then the variation of actual and reference values is presented by Eq. (14).
Δu(t)=u(t)−u(t−1) (14)
The value of y0 taken as 0 without generality loss with the predictor is determined by Eq. (15).
y(t+(k/t))=∑i=1NgiΔu(t+k−(i/t)) (15)
The equations of state space are presented by Eq. (16).
C(x(t))=Ax(t−1)+Bu(t−1)=y(t) (16)
Here, the state is described as x, the system matrices A, B are denoted as input and C is the output. The prediction control model is illustrated by Eq. (17).
y(t+(k/t))=C[Akx(t)+∑i=1NAi−1Bu(t+k−(i/t))] (17)
This strategy has a favorable position, which additionally helps multivariable frameworks, though permitting for investigating the procedure internal structure. To predict the output parameters, the
significant input constraints are generated. The selected parameters include the controllable parameters, uncontrollable parameters and parameters impacting system operations. The uncontrollable
parameters like solar normal flux and outside air temperature. The data-driven models of hybrid renewable energy system (HRES) have three zones, the humidity and temperature of these three zones of
energy consumption at t+1. The neural network of multi-layer perceptron (MLP) is working for improving models. The IMPC is used with two hidden layers for the ANN, thus forming four layers as
input, hidden layer 1, hidden layer 2 and output layer [22] as shown in Fig. 5. To evaluate the predictive model performance, four metrics are utilized in this model described as follows. These
metrics are given to the ANN.
The mean absolute percentage error (MAPE) is determined by Eq. (18), which is computes the power fluctuation error present in the system.
MAPE=1n∑i=1n|Yi−Yi∗Yi|×100 (18)
The standard deviation of absolute percentage error (Sd_APE) is evaluated by Eq. (19), computes the MPC controller standard deviation error.
Sd_APE=∑i=1n(|Yi−Yi∗Yi|−MAPE)2n−1 (19)
The maximum absolute error (MAX) is estimated by Eq. (20), which is used to find the maximum error, which is present in the load profile.
MAX=max{|Y1−Y1∗|,|Y2−Y2∗|,…,|Yn−Yn∗|} (20)
The minimum absolute error (MIN) given by Eq. (21) is used to find the minimum error present in the load profile.
MIN=min{|Y1−Y1∗|,|Y2−Y2∗|,…,|Yn−Yn∗|} (21)
In the hidden layer, for training the neural networks, the number of nodes is randomly set from 3 to 25. To initialize the weights of connecting input, output and hidden nodes, the normal
distribution of standard with deviation 1 and mean 0 is presented. In the next section discussed the established predictive models based MLP are the optimization process utilized.
Here an optimization algorithm named EPO algorithm is proposed for optimizing the ANN parameters for improving the performance of the proposed system. The algorithm inhibits the huddling behavior of
emperor penguins (EP) [23]. The EP move towards the high temperature profile region on the search space. Here the temperature search of the EP is noted as the weight of the ANN with reduced error.
Number of EPs are move in the ANN by taking the initial output of the ANN to search the weight and bias value. The fitness function is the minimization of error in the neural network and is given by
Eq. (22).
fitnessfunction=min(error) (22)
The total population for the search of the low error rate by EP is 2N. Initially the boundary limit for the EP to move on the search space is given by Eq. (23).
Ψ=∇ϕ (23)
where, ϕ is the weight factor between the hidden layers and Ψ is the current output of the neural network. The complex term is given by Eq. (24).
F=ϕ+iγ (24)
Here, iγ is the imaginary term to generate the potential having γ as vector and F is the analytical function. Then the error rate searched by the EP is given by Eq. (25).
E′=(E−t_iterations−t_iteration) (25)
E={10≤P≤0.500.5≤P≤1 (26)
where, s is the ongoing iteration. E′ is the current search by EP, which is calculated based on the previous search value. E is the previous search that is obtained at the previous iteration. P is
the error rate if the error rate at the previous stage is within 0.5, then the accurate prediction is needed else the maximum error is obtained on the previous stage. t_iteration is the total number
of iterations performed by the EP. The distance between the two EP which are in search of optimal error rate is given by Eq. (27).
DEP=Abs(Z(Lil).Opt−CilPo(s)) (27)
where, Z is the force that makes the EP to move towards the low error rate value and the current position of the EP is given by Po(s). Opt is the optimal error rate and Lil is the parameter that is
used to avoid the collision between the EP and are given by Eq. (28).
Lil=M×E′+(Opt−Po(s))×rand()−E′ (28)
where, Cil=rand() and M is a constant which is set to 2. The function R() can be given by Eq. (29).
R(Lil)=(p⋅e−sz−e−s)2 (29)
Then the relocation of the EP is set that is the movement of EP by calculating the error rate. The movement of it is given by Eq. (30).
Po(s+1)=Opt(s)−Lil.DEP (30)
where, Po(s+1) is the position of EP in the subsequent iterations. The pseudocode for the EPO that relies on the search for the exploration and exploitation properties is given below:
A linear dynamic model developed by empirical data is used in most industrial applications, even though the process itself is often nonlinear. Due to the difficulty in developing a generic nonlinear
model from empirical data, the linear models have been used, and the computational expense is often involved in using nonlinear models. For developing a nonlinear dynamic model from empirical data,
and EPO optimized ANN-based technique is presented in this paper. It shows that these models are utilized in the MPC method. In several applications, this nonlinear IMPC based approach has been
successfully implemented. The proposed method performance is described in Section 3.
Here, the proposed IMPC is utilized as an energy management system. The proposed method was implemented in an Intel Core i5 processor through 8 GB RAM based hardware with MATLAB/Simulink platform
with the 2016a version. The proposed system performances are compared with recently developed existing works, such as ACO-ANN, BP-NN and MPC. The load forecasted performance is compared with actual
load, forecasted load with ACO-ANN and predicted load with NN-BP. The proposed controller for total harmonic distortion (THD) is compared with the existing NN-BP and MPC controller. The performance
comparison is performed utilizing power parameters of the proposed control scheme. The energy management capability of the proposed system is analyzed for different atmospheric conditions as case
studies. The detailed working procedure of the proposed control method and the resultant energy management system will be presented.
The MATLAB/Simulink for the proposed architecture has been shown in Fig. 6. The PV, wind and fuel cell are modelled on the corresponding blocks in Fig. 6. For the control of harmonic produced by the
DC-DC converter, the FW diode is used along with the three power generator blocks. For the controlled delivery of the power to the load, the 3-level bridge is used. The next scope to this block is
the three-phase output power generated by the generators. In this setup, if the power generator fails, then the connection with the fuel cell supply the load. The state of charge is also analyzed and
is transmitted to the fuel cell when the other generators are not able to meet the load. On the whole, the designed Simulink setup delivers power from the power generator to the load through the grid
connection. Finally, the results are evaluated, and the power balance between the generated power and the load demand is obtained in this system.
The simulation results of energy for a 12-h performance are shown in Fig. 7. It shows the generated power in kW, load consumption power, high generation power of fuel cell and balanced storage power
between the generated and demand power. The results are obtained in various conditions for 12-h. The first graph in Fig. 7 indicates the generated power graph by the PV and wind generator, the second
graph represents the load profile; the third one represents the amount of power that is balanced in the storage system for future use and the last graph represents the power generated by the fuel
The self-energy storage system has also been displayed in Fig. 7, and it is related to the variation in the generated power and consumed load power. The storage system maintains the state of charge
in it by sending the signal to the fuel cell when the power is indeed to the load. If the generated power is lower than the load utilization, the storage system will be discharged. At the same time,
the power grid input power is filled based on the power deficit, because all load demand cannot satisfy the discharging power.
Fig. 8 shows the simulation results of the energy routed for 12 hrs in different conditions. It shows that the high generation power, load utilization power, drop out of fuel cell power and the
balanced storage power. The generated power is increased in this condition. Demand utilization values are also shown in this result. Besides, reduced fuel cell power is also shown in this
performance. Finally, the balanced storage power is also displayed in these results.
Fig. 9 shows the comparison of load power and load demand in different state. The power values are given for 24-h. On one condition, the load demand is constant; that is, constant load demand is
there for 24 h. The generated power reaches the load demand in less time duration. 7kW constant load demand is reached during the 1 h time period. Next, for another condition, different load demand
has been fixed in 24 h. Here also the generated power meets the demand at different load demand.
Fig. 10 shows the comparative analysis of load forecasted with the proposed system. The performance of the proposed model is compared with actual load, load forecasted of ACO-ANN and NN-BP. High
forecasted load power value is shown in Fig. 10. Compared with existing approaches, the proposed model forecasted the load easily. Compared to the actual load, the proposed model gets the performance
of second-most level. Compared to the NN-BP approach, the proposed model obtains better results. The forecasted load values are obtained in kW for 24 h. Existing approaches NN-BP obtains lowermost
value for the forecasted load.
Fig. 11 shows the comparison of power values one year from Jan-18 to Dec-18. For 1st year, the approximate power generated by the PV, wind, the demanded load from the grid side, and the fuel cells
balanced power generation to equalize the load is shown in Fig. 11. Comparison of generating power of PV and wind in kW, balanced power of fuel cell in kW, required grid power in kW and the total
power are shown in Fig. 11. The blue lines represent the PV generated power in kW; green line represented the FC balance power in kW; sky blue line represents the total power (kW); red line
represents the wind generated power (kW) and violet line represents required grid power (kW). The total and required grid power is varied in each month. The highest rate of total and the required
power is indicating oct-18. At Jan-18, the total power is lower than the demand, therefore the discharging is placed, and hence the balanced power gets discharged. Compared to Jan-18, Apr-18 has low
demand and high total power, and then the balanced power stored or charged. Furthermore, Dec-18 also has a lower demand than the total power, so the remaining power is charged. Based on the analysis,
the proposed system gives better outcomes for all time variations. Finally, the proposed system concludes with its performance from analysing different weather and load demand conditions.
Cost reduction is the primary requirement in the power system to manage energy. The cost analysis brings the average specificity of the energy management system in terms of power utilization by the
load. Fig. 12 displays the computational cost analysis of IMPC, which was performed concerning total system cost, life cycle cost and annualized cost by comparing with ACO-ANN, BB-NN, MPC and IMPC.
The performance analysis shows that the total system cost, life cycle cost and annualized cost obtained by IMPC are better compared to other methods. The highest cost using method is ACO-ANN method.
The IMPC is most benefit and cost-effective method compared to other comparative methods because of neural network optimized EPO, which removes the noise present in the system and give the perfect
output. This is the reason for reducing the cost in the proposed IMPC method.
Tab. 1 shows the comparison of total harmonic distortion (THD). The reduction of harmonics using the proposed controller is better than that using the NN-BP and MPC controllers. The THD value for the
existing NN-BP controller is 18.19% and MPC controller is 16.11%. Hence the THD value becomes lower when the proposed controller has been used.
An improved model predictive control (IMPC) scheme has been developed for the prediction of accurate load and weather forecasting data of RES with EMS in this work. The proposed IMPC is working based
on the combination of MPC controller and EPO optimization algorithm, which optimizes the bias and weight of ANN. Further, this optimized ANN is used to tune the MPC prediction model. The design of
the proposed EMS structure consists of PV, wind and fuel cells along with household appliances and grid system. The weather and forecast data are generated from RES for processing the performance of
the proposed EMS. There are some converters like AC-DC and DC-AC, and these are used in the conversion process of one form to another form of signal. The MATLAB/Simulink platform has been utilized
for the implementation of the proposed system, and the outcomes are validated with some traditional techniques based on the different atmospheric conditions. Computational cost analysis of IMPC is
performed concerning total system cost, life cycle cost and annualized cost by comparing with ACO-ANN, BB-NN, MPC and IMPC. The performance analysis shows that IMPC consumes less cost than
comparative methods. The THD obtained by the proposed controller is 0.06%, which is better when compared with the existing NN-BP and MPC controllers. However, this system is only applicable for
household appliances (moderate power). In future, the work will be implemented for heavy duty industries carrying high power where the power fluctuation limitation is necessarily needed.
Funding Statement: The author received no specific funding for this study.
Conflicts of Interest: The author declare that they have no conflicts of interest to report regarding the present study.
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This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited. | {"url":"https://www.techscience.com/iasc/v35n1/48140/html","timestamp":"2024-11-09T16:08:28Z","content_type":"application/xhtml+xml","content_length":"124316","record_id":"<urn:uuid:0e0296b3-475e-4f1a-b416-99faa7344c48>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00287.warc.gz"} |
Principles of P
Assignment 8: Types
Part A
Here are some written questions on typing.
1. For each of these expressions, either construct a typing proof in TFb, or show exactly why they cannot typecheck (i.e. no derivation tree could ever be built; don’t just informally describe it,
use the formal system rules).
a. (If True Then 0 Else False)
b. (If True Then (Fun x:Int -> x + 1) Else (Fun x:Int -> x)) 0
c. (Fun x:(Bool -> Int) -> x False)(Fun x:Bool -> 4)
2. Of the following pairs of types, is the left type a subtype of the right type, a supertype, or neither? Justify your answer by showing the proofs in the subtype proof system of the book; if
neither holds describe why in words. Pay close attention to the rules, it is easy for intuitions to fail on these examples. Just build a proof tree based on the rules and you know you are
a. { x : Int; y : { z : Bool } } and { x : Int; y : {}; w : Int },
b. { x : Int } -> { } and { } -> { x : Int }
c. { } -> { } and { } -> { x : Int }
3. Type the following program in STFbR (note this program is not showing the type declarations on r1/r2, you need to find types to fill in there such that the program is typeable):
Function n:Int -> Function r1:{...} -> Function r2:{...}->
{x = r1.x + 1; y = (If n = 0 Then r1.y - 1 Else r2.y +1); z = r2.z + 2}
Part B
Write a type checker for TFbSRX. The language was described in lecture and is in section 6.4 of the book. The TFbSRX/ directory of the FbDK contains the relevant parser and OCaml data type, all you
have to do is fill in the file tfbsrxtype.ml with a correct implemention of the typecheck function there. Note that you don’t have to write an interpreter, only the typechecker.
• The file tfbsrx_examples.ml contains quite a few examples for you to test the typechecker with.
• The AST for the language, in file TFbSRX/tfbsrxast.ml, is slightly different from the one at the top of Section 6.4 in the textbook, it is a bit simpler.
• As with the other languages in the FbDK you can use our reference implementations. Type ./reference/TFbSRX/toplevel.exe (or the wsl- version if you have been using that) to load the type checker
into utop. The above file tfbsrx_examples.ml then contains information on how you can invoke the typechecker in testing - either #use that file or copy/paste in the lines at the top.
• Notice that Raise .. evaluates to “arbitrary tau” in the rule in the book. As we mentioned in lecture, this is usually handled by introducing an “anything type *” - a type that is equal to every
other type in the system. A new type TBottom has been added to the type fbtype for this purpose.
• Type checking exceptions can be somewhat tricky; especially their interactions with other expressions and type rules. You need to consider each rule carefully.
For example:
|- (Function x:Int -> x = 1) (Raise #Exn@Bool False) : Bool
Because by function rule |- (Function x:Int -> x = 1) : Int -> Bool
and by exception rule |- (Raise #Exn@Bool False) : Bottom (arbitrary tau)
And because Int and Bottom can be equated,
by application rule we have |- (Function x:Int -> x = 1) (Raise #Exn@Bool False) : Bool `
For Part A, the upload is as for the other written assignments. For Part B, upload (only) your file tfbsrxtype.ml which will (hopefully) contain your fully functioning type checker. | {"url":"http://pl.cs.jhu.edu/pl/assignments/assignment8.html","timestamp":"2024-11-13T19:18:58Z","content_type":"text/html","content_length":"8093","record_id":"<urn:uuid:054de2ef-c6f6-4db7-8c00-9d1764d9754c>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00825.warc.gz"} |
Describes a mathematical model of the interaction two coupled counter-directed quarter-wave microstrip resonators. The mathematical model is based on the search of frequency-depended coupling
coefficient of interacting resonators. The merit of the mathematical model is possible to obtain an estimate of the transmission coefficient for changed topology of resonators of the interdigital
filter, which cannot be obtained by circuit simulation in common software tools. The estimate of the transmission coefficient in order to determine the initial approximation of the geometric
dimensions of the topology is need because the next stage after of the initial approximation is electrodynamic modeling, which consumes large computer power and time of analyzing the topology.
Without the initial approximation of the geometrical dimension of the topology, one can «wander» for a long time in search of the optimum using only electrodynamic modeling.
Authors: A. D. Maksimenko, B. E. Lavrenko
Direction: Physics
Keywords: Analog filters, coupled microstrip resonators, mathematical model, electrodynamic modeling, frequency-dependent coupling coefficient, distribution of high-frequency currents and voltages in
resonators, energy stored in electric and magnetic field
View full article | {"url":"https://izv.etu.ru/en/archive/2020/10/5-12","timestamp":"2024-11-04T11:53:00Z","content_type":"text/html","content_length":"9890","record_id":"<urn:uuid:31d6c714-07b0-4f7d-b5c6-d95b49b0476b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00042.warc.gz"} |
The locus of the orthocentre of the triangle formed by the line... | Filo
The locus of the orthocentre of the triangle formed by the lines and , where , is
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Given, lines are
Equation of altitude passing through and perpendicular to is
Slope of line (ii) is .
Slope of altitude (as shown in figure) is .
Equation of is
Let orthocentre of triangle be , which is the point of intersection.
Locus of is .
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Question Text The locus of the orthocentre of the triangle formed by the lines and , where , is
Updated On Oct 12, 2023
Topic Straight Lines
Subject Mathematics
Class Class 11
Answer Type Text solution:1 Video solution: 2
Upvotes 305
Avg. Video Duration 15 min | {"url":"https://askfilo.com/math-question-answers/the-locus-of-the-orthocentre-of-the-triangle-formed-by-the-lines-1p-x-p-yp1p01q","timestamp":"2024-11-12T02:39:56Z","content_type":"text/html","content_length":"547651","record_id":"<urn:uuid:b6a2a901-bd8a-4010-afe2-f3a4d9b1601b>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00875.warc.gz"} |
Algebra 1 Chapter 11 - Rational Expressions and Functions - 11-1 Simplifying Rational Expressions - Practice and Problem-Solving Exercises - Page 655 15
2/(b+4) ; b cannot equal 4 or -4
Work Step by Step
In order to simplify, we want something to be both in the numerator and the denominator of the fraction so it can be canceled out. Thus, we factor and obtain: $(2b-8)/b^{2}-16 \\\\ 2(b-4) / (b-4)
(b+4) \\\\ 2/(b+4)$ b cannot equal 4 or -4, for if b equaled 4 or -4, the denominator of the fraction would cancel out to 0, which is not allowed. | {"url":"https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-11-rational-expressions-and-functions-11-1-simplifying-rational-expressions-practice-and-problem-solving-exercises-page-655/15","timestamp":"2024-11-11T07:19:21Z","content_type":"text/html","content_length":"90713","record_id":"<urn:uuid:76cc1ac4-3ecc-4a0d-b1a7-5a370de0e80d>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00810.warc.gz"} |
How do you find the scale factor?
Compare the corresponding side lengths of both shapes and divide them to find the scale factor to go from one shape to another.
What is the condition to prove that two triangles are similar?
To prove that two triangles are similar, use the AAA rule which means all three angles are matching.
What does it mean for two shapes to be similar?
Two shapes are similar if they have the same shape but have scaled side lengths. | {"url":"https://evulpo.com/en/uk/dashboard/lesson/uk-m-ks3-05geometry-and-measures-05similar-shapes","timestamp":"2024-11-10T02:12:24Z","content_type":"text/html","content_length":"856665","record_id":"<urn:uuid:2611200c-886a-4185-bc59-b135233e6f12>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00754.warc.gz"} |
Bulge-Head Solid
My next object was a bulge-head solid. This solid lies above the \(xy\)–plane, outside the unit sphere, and inside the cardioid of revolution given by \(\rho=1+\cos\phi\). Professor Beanland had
given us these equations, since he was really curious to see what the solid looked like. He’d nicknamed it the cone-head solid, but after printing we renamed it the bulge-head solid.
Since the outside of the solid was a cardioid of revolution, I decided to create the solid in Cinema 4D by creating two splines (one for the cardioid, the other for the hemisphere) and revolving
each around an appropriate axis. Professor Denne helped me to figure out which parametric equations to place into Cinema 4D’s inputs for a formula spline. These were \(x(t)=1+2\cos(t) + \cos(2t)\),
\(y(t)=2\sin(t)+\sin(2t)\), and \(z(t)=0\), where \(t=[0,\pi/2]\). For the spline that would later become the hemisphere, I used \( x(t)=\cos(t)\), \(y(t)=\sin(t)\), and \(z(t)=0 \), where \(t=[0, \
pi/2] \). I then used the Lathe Tool with an angle of \(360^\circ\) to make the two boundaries of the solid. I then put them into a Boole to make a union between the two boundaries. I printed the
bulge-head solid on the FormLabs printer using clear resin. When loading the object into the FormLabs software, we got a warning about the object’s integrity, but we decided to continue the print
anyway. Later on we were worried that the object would use up too much resin and that it may have some problems on the surface (like the smooth strange bowl did). It turned out that added a bit
more resin mid-build, just to be on the safe side. The solid looks pretty good right now because it only has a few pimples on the inside, but no significant lumps. The object is still hardening and
once it’s completely dry we’ll remove the outside supports. This will probably leave a few pimples as well.
You can find this model on Thingiverse here. | {"url":"https://mathvis.academic.wlu.edu/2015/07/21/bulge-head-solid/","timestamp":"2024-11-05T09:10:16Z","content_type":"text/html","content_length":"37079","record_id":"<urn:uuid:8974847d-bbb0-452c-9678-fb01a40027f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00506.warc.gz"} |
Lecture Notes on the Theory of Distributions
by Guenther Hoermann, Roland Steinbauer
Publisher: Universitaet Wien 2009
Number of pages: 160
From the table of contents: 1. Test Functions and Distributions; 2. Differentiation, Differential Operators; 3. Basic Constructions; 4. Convolution; 5. Fourier Transform and Temperate Distributions;
6. Regularity; 7. Fundamental Solutions.
Download or read it online for free here:
Download link
(890KB, PDF)
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One could use a Double Butterfly Spread when one expects the price of the underlying stock to close exactly at either one of two different strike prices by expiration.
How To Use Double Butterfly Spread?
A Double Butterfly Spread is simply putting on two seperate butterfly spreads with middle strike prices on two different strike prices. This creates a position with two peak profits on two strike
prices. While a butterfly spread is used for targetting a single strike price, Double Butterfly Spreads target two different strike prices to increase the chances of peak profitability and are useful
when the price of the underlying asset is expected to hit one of two prices due to factors such as a potential take-over.
Double Butterfly Spreads are usually used when the targetted prices are more than one strike apart. If the targetted prices are two consecutive strike prices, a
Condor Spread
could be used instead with almost the same profitability and incurring a lot lesser commissions due to fewer trades making up the position. However, if the targetted prices are more than one strike
apart, using a condor spread would yield a lot lesser maximum profit due to the fact that it is also maintaining peak profit potential in between the two targetted strike prices.
As such, the Double Butterfly Spread is a huge six legged options position with three legs forming each butterfly spread. It can also be constructed using only call options (known as a Call Double
Butterfly Spread) or put options (known as a Put Double Butterfly Spread) or even a combination of both with one Call Butterfly Spread pairing up with a Put Butterfly Spread. The outcome and capital
outlay of all three configurations should not be very different when put call parity is not severely broken.
Establishing Call Double Butterfly Spread
Call Double Butterfly Spreads consist of two Call Butterfly Spreads with middle strike price on two different strike prices which are at least one strike price apart.
Call Double Butterfly Spread Example
Assuming QQQQ trading at $56.90.
Jan55Call $2.27, Jan56Call $1.50, Jan57Call $0.90, Jan58Call $0.44, Jan59Call $0.19
Jan55Put $0.40, Jan56Put $0.63, Jan57Put $1.00, Jan58Put $1.56, Jan59Put $2.30
Assuming we are targetting $56 and $58.
Call Butterfly Spread 1:
Buy 1 Jan55Call, Sell 2 Jan56Call, Buy 1 Jan57Call = (2.27 + 0.9) - (1.50 x 2) = $3.17 - $3.00 = $0.17
Call Butterfly Spread 2:
Buy 1 Jan57Call, Sell 2 Jan58Call, Buy 1 Jan59Call = (0.9 + 0.19) - (0.44 x 2) = $1.09 - $0.88 = $0.21
Total Debit = $0.17 + $0.21 = $0.38
Establishing Put Double Butterfly Spread
Put Double Butterfly Spreads consist of two Put Butterfly Spreads with middle strike price on two different strike prices which are at least one strike price apart. We shall target the same strike
prices as the Call Double Butterfly Spread example above using only put options.
Put Double Butterfly Spread Example
Assuming QQQQ trading at $56.90.
Jan55Call $2.27, Jan56Call $1.50, Jan57Call $0.90, Jan58Call $0.44, Jan59Call $0.19
Jan55Put $0.40, Jan56Put $0.63, Jan57Put $1.00, Jan58Put $1.56, Jan59Put $2.30
Assuming we are targetting $56 and $58.
Put Butterfly Spread 1:
Buy 1 Jan55Put, Sell 2 Jan56Put, Buy 1 Jan57Put = (0.4 + 1) - (0.63 x 2) = $1.40 - $1.26 = $0.14
Put Butterfly Spread 2:
Buy 1 Jan57Put, Sell 2 Jan58Put, Buy 1 Jan59Put = (1 + 2.3) - (1.56 x 2) = $3.3 - $3.12 = $0.18
Total Debit = $0.14 + $0.18 = $0.32
Establishing Mixed Double Butterfly Spread
Mixed Double Butterfly Spreads consist of a Call Butterfly Spread and a Put Butterfly Spread with middle strike price on two different strike prices which are at least one strike price apart. Mixed
Double Butterfly Spreads can be setup with the Call Butterfly Spread targetting the lower strike price and the Put Butterfly Spread targetting the higher strike price and vice versa. We shall target
the same strike prices as the Call Double Butterfly Spread example above using both combinations of Mixed Double Butterfly Spread.
Mixed Double Butterfly Spread Example
Assuming QQQQ trading at $56.90.
Jan55Call $2.27, Jan56Call $1.50, Jan57Call $0.90, Jan58Call $0.44, Jan59Call $0.19
Jan55Put $0.40, Jan56Put $0.63, Jan57Put $1.00, Jan58Put $1.56, Jan59Put $2.30
Assuming we are targetting $56 and $58.
Call Butterfly Spread 1:
Buy 1 Jan55Call, Sell 2 Jan56Call, Buy 1 Jan57Call = (2.27 + 0.9) - (1.50 x 2) = $3.17 - $3.00 = $0.17
Put Butterfly Spread 2:
Buy 1 Jan57Put, Sell 2 Jan58Put, Buy 1 Jan59Put = (1 + 2.3) - (1.56 x 2) = $3.3 - $3.12 = $0.18
Total Debit = $0.17 + $0.18 = $0.35
Put Butterfly Spread 1:
Buy 1 Jan55Put, Sell 2 Jan56Put, Buy 1 Jan57Put = (0.4 + 1) - (0.63 x 2) = $1.40 - $1.26 = $0.14
Call Butterfly Spread 2:
Buy 1 Jan57Call, Sell 2 Jan58Call, Buy 1 Jan59Call = (0.9 + 0.19) - (0.44 x 2) = $1.09 - $0.88 = $0.21
Total Debit = $0.14 + $0.21 = $0.35
In this case, since the Put Double Butterfly Spread requires a lower net debit than the Call Double Butterfly Spread and the Mixed Double Butterfly Spread, the Put Double Butterfly Spread should be
used instead.
Trading Level Required For Double Butterfly Spread
A Level 3 options trading account that allows the execution of debit spreads is needed for the Double Butterfly Spread. There are brokers who requires level 4 or 5 accounts for Double Butterfly
Spreads as well. Please check with your broker. Read more about Options Account Trading Levels.
Profit Potential of Double Butterfly Spread :
Double Butterfly Spreads achieve their maximum profit potential when the price of the underlying asset closes at either one of the two targetted strike prices. The profitability of a Double Butterfly
Spread can also be enhanced or better guaranteed by legging into the position properly.
Profit Calculation of Double Butterfly Spread:
Maximum Profit = $1 - Net Debit
Maximum Loss = Net Debit
Profit Calculation for Double Butterfly Spread
Following up on the above Put Double Butterfly Spread example:
Maximum Profit = $1.00 - $0.32 = $0.68
Maximum Loss = $0.32
Reward Risk Ratio = $0.68 / $0.32 = 2.13
Risk / Reward of Double Butterfly Spread:
Upside Maximum Profit: Limited
Maximum Loss: Limited
Break Even Points of Double Butterfly Spread:
There are four breakeven points for a Double Butterfly Spread. Each set of two breakeven points created by each butterfly spread defines a price range within which profit occurs for the Double
Butterfly Spread. Each set of breakeven points are to be calculated seperately for each component butterfly spread using the below formula:
1. Lower Breakeven Point : Total Net Debit + Lower Strike Price
2. Upper Breakeven Point : Higher Strike Price - Total Net Debit
Breakeven Calculation for Double Butterfly Spread
Following up on the above Put Double Butterfly Spread example:
Put Butterfly Spread 1:
Buy 1 Jan55Put, Sell 2 Jan56Put, Buy 1 Jan57Put = (0.4 + 1) - (0.63 x 2) = $1.40 - $1.26 = $0.14
Lower Breakeven : $0.32 + $55 = $55.32
Higher Breakeven : $57 - $0.32 = $56.68
Put Butterfly Spread 2:
Buy 1 Jan57Put, Sell 2 Jan58Put, Buy 1 Jan59Put = (1 + 2.3) - (1.56 x 2) = $3.3 - $3.12 = $0.18
Lower Breakeven : $0.32 + $57 = $57.32
Higher Breakeven : $59 - $0.32 = $58.68
Advantages Of Double Butterfly Spread:
:: Able to target two different specific strike prices.
:: Higher profit potential than Condor Spread when either price is hit.
Disadvantages Of Double Butterfly Spread:
:: Larger commissions involved due to large number of trades making up the position.
Adjustments for Double Butterfly Spreads Before Expiration :
1. When it is obvious that the underlying stock is going to go for one of the two strike prices, you could close out the other butterfly spread and hold only the butterfly spread that targets the
correct strike price, reducing the position to a regular butterfly spread and increasing profitability.
Don't Know If This Is The Right Option Strategy For You? Try our Option Strategy Selector! | {"url":"https://www.optiontradingpedia.com/free_double_butterfly_spread.htm","timestamp":"2024-11-03T13:17:25Z","content_type":"text/html","content_length":"34060","record_id":"<urn:uuid:8bc052e5-4292-4357-84be-73b06a96e40e>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00271.warc.gz"} |
Vectors show up everywhere in physics: displacement vectors, velocity vectors, acceleration vectors, force vectors, momentum vectors and more. So we definitely want to know how to work with vectors.
While a scalar describes the magnitude or the value of something (like a speed of 30 km/h), a vector describes the magnitude and direction of something (like a velocity of 30 km/h east). We usually
represent vectors as arrows where the length of the arrow tells us the magnitude, and the direction of the arrow tells us... the direction.
To make vectors more useful, and to add or subtract them, we need to know how to find the components of a vector. We can do that by using the trig functions. We can also use those to find the
magnitude and angle of a vector when starting with the components.
We'll also cover how to describe the angle of a vector and how to add vectors using the tip-to-tail method or by adding components.
Vectors & Components (15:13)
Vectors and components
Coordinate systems
How to find components of a vector
Vectors show up everywhere in physics: displacement vectors, velocity vectors, acceleration vectors, force vectors, momentum vectors and more. So we definitely want to know how to work with vectors.
While a scalar describes the magnitude or the value of something (like a speed of 30 km/h), a vector describes the magnitude and direction of something (like a velocity of 30 km/h east). We usually
represent vectors as arrows where the length of the arrow tells us the magnitude, and the direction of the arrow tells us... the direction.
To make vectors more useful, and to add or subtract them, we need to know how to find the components of a vector. We can do that by using the trig functions. We can also use those to find the
magnitude and angle of a vector when starting with the components.
We'll also cover how to describe the angle of a vector and how to add vectors using the tip-to-tail method or by adding components.
Study Guide
Complete and Continue | {"url":"https://physicslab.app/courses/559004/lectures/17488729","timestamp":"2024-11-09T12:53:47Z","content_type":"text/html","content_length":"261634","record_id":"<urn:uuid:61febaee-db13-4321-8031-83df13b910b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00751.warc.gz"} |
new puzzle challenge
04-02-2015, 10:08 PM
Post: #21
Don Shepherd Posts: 749
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-02-2015 08:58 PM)RayAtHP Wrote: Don, maybe you should reconsider your initial request ...
Quote:I expect a one line RPL solution, as usual! I would love to see a BASIC solution.
A quick scan of Mings transcript could lead to massive exhaustion under RPL programmers if they try to comply to your request.
Oh, I didn't really expect a one-line RPL solution. But I've posted several puzzle challenges over the years in which the very clever RPL people posted very short RPL solutions to what I thought
would be rather complex problems, and I am always kind of in awe of those people and their programs.
But I am not awed quite enough to want to learn RPL myself!
04-02-2015, 10:53 PM
Post: #22
Claudio L. Posts: 1,885
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-01-2015 12:25 PM)Claudio L. Wrote: Back to the drawing board.
To speed up the program, it seems is optimum (Paul made a good call there) to choose the corners as the independent variables. This makes the first 6 equations to try also the shortest (3 elements),
fastest, and completely decoupled, only dependent on the 6 known variables. This allows for quickly discarding a lot of cases.
The 7th variable needs to be in the inner hexagon, but not the center (choosing the center results in coupled equations, so you can't solve all variables one by one without working out the algebra to
decouple the equations first).
I'll see if I can rework the RPL code to work with this new numbering scheme and will report on the speed improvement.
04-02-2015, 11:11 PM
Post: #23
Paul Dale Posts: 1,849
Senior Member Joined: Dec 2013
RE: new puzzle challenge
Early pruning of the search is probably also advisable, I suspect it doesn't save a lot of searching but every bit helps.
The search space can be reduced by forcing the first corner chosen to be the smallest corner and not producing the six rotations for each solution. The first corner can then only range from 1 through
14 and the other corners must be larger.
- Pauli
04-03-2015, 11:31 AM
Post: #24
Claudio L. Posts: 1,885
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-02-2015 10:53 PM)Claudio L. Wrote: I'll see if I can rework the RPL code to work with this new numbering scheme and will report on the speed improvement.
I think I found a way to simplify the algorithm to solve the outer ring.
If we number the values consecutively n(i), turns out that for any two values in the outer ring:
Which limits a lot the set of numbers we can pick from.
Also from the equations:
So the algorithm can work like this:
Put a number on the stack from 1 to 19 (in a FOR loop), as the first number, and call the recursive routine.
A recursive routine will simply do:
Add a number on the stack from 19-n(i) to 19 in a FOR loop.
Check that the new number is not taken (as in not already present in the stack).
If not taken, add a second number n(i+1)=38-n(i)-n(i-1)
Check that this new number is not taken. If not taken, recurse, otherwise drop the last two values and continue to loop.
This routine also has to check if we already did the 6 corners. In such case, needs to do the special case to connect the last corner with the first, and if it works, we have a solution to the outer
The inner ring is easy: there's one independent variable and can only take 7 possible numbers, so it's easy to check in a separate routine, once the outer ring is solved.
With this algorithm, there's very little math, most of the time is spent checking that a number in level 1 of the stack is not present in the stack already, which should be fast I think.
I'll see if I can put this together during the weekend. Feel free to beat me to it!
04-03-2015, 07:04 PM
Post: #25
Claudio L. Posts: 1,885
Senior Member Joined: Dec 2013
RE: new puzzle challenge
Good news. The outer ring solution only took about 40 minutes on the 50g to complete the checks for one initial value, and determine all solutions for the other 5 independent variables.
When I add the inner ring checks (which only need to be checked for the solutions found in the outer ring, which were 160 for the initial value of 3, so they are not that many), I estimate it will be
about 1 hr per initial value worst case.
To cover all 19 initial values, the total running time is now down to 19 hrs to obtain all solutions.
This is now into "tolerable" territory.
newRPL should be able to do this with a speedup of about 200x (per
these results
), which would put it around 6 minutes.
I'll report back with the source code once I complete the inner ring checks.
04-03-2015, 07:36 PM
Post: #26
rprosperi Posts: 6,631
Super Moderator Joined: Dec 2013
RE: new puzzle challenge
(04-03-2015 07:04 PM)Claudio L. Wrote: Good news. The outer ring solution only took about 40 minutes on the 50g to complete the checks for one initial value, and determine all solutions for the
other 5 independent variables.
When I add the inner ring checks (which only need to be checked for the solutions found in the outer ring, which were 160 for the initial value of 3, so they are not that many), I estimate it
will be about 1 hr per initial value worst case.
To cover all 19 initial values, the total running time is now down to 19 hrs to obtain all solutions.
This is now into "tolerable" territory.
newRPL should be able to do this with a speedup of about 200x (per these results), which would put it around 6 minutes.
I'll report back with the source code once I complete the inner ring checks.
Hope you have a nice supply of batteries handy.... wow, 19 hours of continuous 50g run-time. Now that's a serious amount of calculating. Did you peek at the author's paper to know the answer you
should be getting?
--Bob Prosperi
04-03-2015, 08:09 PM
Post: #27
Han Posts: 1,882
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-03-2015 07:36 PM)rprosperi Wrote:
(04-03-2015 07:04 PM)Claudio L. Wrote: Good news. The outer ring solution only took about 40 minutes on the 50g to complete the checks for one initial value, and determine all solutions for
the other 5 independent variables.
When I add the inner ring checks (which only need to be checked for the solutions found in the outer ring, which were 160 for the initial value of 3, so they are not that many), I estimate it
will be about 1 hr per initial value worst case.
To cover all 19 initial values, the total running time is now down to 19 hrs to obtain all solutions.
This is now into "tolerable" territory.
newRPL should be able to do this with a speedup of about 200x (per these results), which would put it around 6 minutes.
I'll report back with the source code once I complete the inner ring checks.
Hope you have a nice supply of batteries handy.... wow, 19 hours of continuous 50g run-time. Now that's a serious amount of calculating. Did you peek at the author's paper to know the answer you
should be getting?
Just plug it into a USB port and you can run without batteries :-)
Graph 3D | QPI | SolveSys
04-03-2015, 08:43 PM
Post: #28
PANAMATIK Posts: 1,032
Senior Member Joined: Oct 2014
RE: new puzzle challenge
Perhaps you are aware, perhaps not, that many other members or nonmembers are reading this thread with great pleasure, seeing your efforts and results, without being able to participate in these kind
of mathematics games, but admiring it. I'm one of these quiet readers and want to congratulate any of you in advance for finding all the n-th order solution eggs.
Happy Easter!
That's one small step for a man - one giant leap for mankind.
04-04-2015, 12:48 PM
(This post was last modified: 04-05-2015 02:37 AM by Claudio L..)
Post: #29
Claudio L. Posts: 1,885
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-03-2015 08:43 PM)PANAMATIK Wrote: Perhaps you are aware, perhaps not, that many other members or nonmembers are reading this thread with great pleasure, seeing your efforts and results,
without being able to participate in these kind of mathematics games, but admiring it. I'm one of these quiet readers and want to congratulate any of you in advance for finding all the n-th order
solution eggs.
Happy Easter!
Thanks for the encouragement. You shall be rewarded with the full source code:
For proper code optimization, this is the variable numbering (quite strange for humans, but optimized for the machine:
A B C
L R M D
K Q S N E
J P O F
I H G
A trough L are the outer ring, where we can write the first 6 equations with only 3 letters each, and completely decoupled from the rest. This allows to solve for the first 6 variables out of seven.
As explained in my previous post, the code walks the perimeter setting a number for A, then trying all values of B knowing A+B>=19, and computing C per the corresponding equation C=38-(A+B).
This is done by this code:
@@ AUXILIARY ROUTINE 'TAKEN' CHECKS IF THE NUMBER ON LEVEL 1
@@ WAS ALREADY PRESENT IN THE STACK, RETURNS 1=WAS TAKEN, 0=NOT REPEATED
<< DEPTH IF 1 <= THEN 0 ELSE
0 -> RESULT <<
DEPTH 1 + 3 SWAP FOR K DUP K PICK
IF == THEN 1 'RESULT' STO 1000 'K' STO END NEXT
RESULT >>
>> 'TAKEN' STO
@@ EXPECTS THE FIRST VALUE TO BE ON THE STACK ALREADY
@@ IT LEAVES ONLY THAT ORIGINAL VALUE ON THE STACK UPON EXIT
<< DEPTH IF 11 == THEN
@@ SPECIAL CASE TO "CLOSE" THE RING ONCE WE HAVE ALL VARIABLES
11 PICK OVER + 38 SWAP -
IF TAKEN NOT OVER 1 >= AND OVER 19 <= AND THEN
@@ IF THE RING "CLOSED", THEN DO THE INNER RING CHECK...
@@ REGULAR CASE, TRY ALL VALUES IN A LOOP FOR THE NEXT NUMBER
19 OVER - DUP 1 < + 19 SWAP FOR K K
2 DUPN + 38 SWAP -
-1 STEP
>> 'DOLOOP' STO
M is the 7th independent variable. Once the outer ring is known, trying all possible values for M allows to calculate all letters N through R in the "inner ring" by using the 4-term equations.
The inner ring code expects to find already on the stack the 12 variables A through L with L on level 1.
The inner ring is solved by this code:
@TRY ALL NUMBERS FROM 1 THROUGH 19 IN A LOOP
<< 1 19 FOR K
@@ IF THE NUMBER WAS VALID, THEN SOLVE FOR N THROUGH R
@@ USING THE EQUATIONS WITH 4-TERMS
IF 1 -> INNERCHECK << 1 5 FOR J 13 J 2 * - DUP
IF 1 < THEN 12 + END J + PICK
10 J 2 * - DUP
IF 1 < THEN 12 + END J + PICK
+ OVER + 38 SWAP -
IF TAKEN OVER 1 < OR OVER 19 > OR THEN
J DROPN 0 'INNERCHECK' STO 1000 'J' STO
INNERCHECK >>
@@ INNER RING CHECKS OUT, NOW CHECK THE
@@ CENTER VALUE BEFORE WE CLAIM TO HAVE A SOLUTION
@@ WE HAVE A SOLUTION, ADD IT TO A LIST CALLED 'SOLUTIONS'
19 DUPN 19 ->LIST 1 ->LIST
SOLUTIONS SWAP + 'SOLUTIONS' STO
7 DROPN
ELSE 6 DROPN
>> 'CHKINNER' STO
Finally, we need to do the proper checks with the center value (last position, S).
All we have to do is find the only number that was not taken in the stack and check all 3 equations with 5-terms.
This is done here:
@@ USING THE 5-TERM EQUATIONS
<< 1 19 FOR K
K IF TAKEN NOT THEN 1000 'K' STO ELSE DROP END
19 PICK 3 PICK + OVER + 6 PICK + 14 PICK + IF 38 == THEN
17 PICK 8 PICK + OVER + 5 PICK + 12 PICK + IF 38 == THEN
9 PICK 4 PICK + OVER + 7 PICK + 16 PICK + IF 38 == THEN
ELSE DROP 0 END
ELSE DROP 0 END
ELSE DROP 0 END
>> 'CHKCENTER' STO
And to wrap it all up, the code above needs a couple of things to run properly:
• A variable named 'SOLUTIONS' initialized to an empty list
• A clear stack
• A loop to try all 19 initial values
This is done here:
@@ ROUTINE RUNIT DOES THE COMPLETE TEST
<< { } 'SOLUTIONS' STO
1 19 FOR I I DOLOOP DROP NEXT
>> 'RUNIT' STO
For the short of patience, partial tests can be ran, if you put a partial list of numbers (must be a valid partial solution, of course), and run the DOLOOP test to see if any solutions are found.
Just make sure 'SOLUTIONS' contains an empty list and that there's nothing else in the stack but your partial solution.
The partial solution has to contain 1, 3, 5... elements. In other words, think of the corners as the independent variables, then to give 2 independent variables you must also provide the intermediate
value between them (A, B and C).
This is tested and finds all 12 solutions, but of course I tested it on a PC with newRPL, don't have the patience to wait 19 hours, but if somebody does, please measure time and report here.
Happy coding.
PS: Still not a one-line RPL solution... but faster than my first try at 97 days!
Forgot to mention, if you don't wait to wait 19 hours, and don't care about the different rotations, you can get a nice solution by doing:
{ } 'SOLUTIONS' STO 3 DOLOOP
This will leave SOLUTIONS with all solutions that have a 3 in the first variable: exactly 2 solutions, one mirror of the other, and only for about
1 hr
of your time.
04-04-2015, 12:53 PM
Post: #30
Claudio L. Posts: 1,885
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-03-2015 07:36 PM)rprosperi Wrote: Did you peek at the author's paper to know the answer you should be getting?
No author's paper, just this thread, paper, pencil and a PC with newRPL.
04-04-2015, 03:06 PM
Post: #31
Don Shepherd Posts: 749
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-04-2015 12:48 PM)Claudio L. Wrote: PS: Still not a one-line RPL solution... but faster than my first try at 97 days!
Claudio, that is great work, thank you.
Can you clarify, is there only one real solution, the others being reflections or rotations of the one? The puzzle-maker claimed several solutions, but I have a feeling that they are all reflections/
rotations of the original.
04-04-2015, 06:44 PM
Post: #32
Claudio L. Posts: 1,885
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-04-2015 03:06 PM)Don Shepherd Wrote: Claudio, that is great work, thank you.
Can you clarify, is there only one real solution, the others being reflections or rotations of the one? The puzzle-maker claimed several solutions, but I have a feeling that they are all
reflections/rotations of the original.
There's only one "true" solution. The rest are rotations and mirror images.
There's all kinds of interesting observations on this problem.
For example, I replaced the routine CHKINNER to always store the solution instead of checking the inner ring, to obtain all the outer ring solutions:
<< 12 DUPN 12 ->LIST 1 ->LIST SOLUTIONS SWAP + 'SOLUTIONS' STO >>
Turns out that starting with the number 3 as the first variable, there's 160 solutions of the outer ring (actually 80 if you remove the mirrors). Among these sets, only 2 (one and its mirror) will
pass the inner ring check.
Starting with number 4, there's 184 solutions to the outer ring (actually 92), but none of them will pass the inner ring check.
04-04-2015, 07:02 PM
Post: #33
Claudio L. Posts: 1,885
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-02-2015 11:11 PM)Paul Dale Wrote: Early pruning of the search is probably also advisable, I suspect it doesn't save a lot of searching but every bit helps.
The search space can be reduced by forcing the first corner chosen to be the smallest corner and not producing the six rotations for each solution. The first corner can then only range from 1
through 14 and the other corners must be larger.
- Pauli
If we are discarding, then the count would start from 3, as 1 and 2 are easy to discard as follows:
If a vertex contains 1, the 2 adjacent variables can only be 18 and 19 (left and right, or viceversa doesn't matter), because n(i)+n(i-1)>=19.
The following corners would have to be:
38-18-1 = 19
38-19-1 = 18
But both numbers are already being used, so we can conclude the
number 1 cannot appear on any vertex
Similar deal with number 2, the adjacent can only be 17, 18 and 19, and the following corners would be 17,18 or 19, it's easy to see that we have 3 valid numbers to fill in 4 spaces, so there's no
solution, concluding that
2 cannot be on any vertex of the solution
There might be other logical rules like these ones to discard more options that could help a human solve the problem by hand.
04-04-2015, 11:43 PM
Post: #34
Paul Dale Posts: 1,849
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-04-2015 07:02 PM)Claudio L. Wrote: If we are discarding, then the count would start from 3, as 1 and 2 are easy to discard as follows:
I thought there was a way to exclude these two but didn't think it through enough.
Getting rid of reflections is easy by forcing the order of two adjacent edge middles. E.g. B>D. This would give an early pruning of the search space.
- Pauli
04-05-2015, 02:29 AM
Post: #35
Claudio L. Posts: 1,885
Senior Member Joined: Dec 2013
RE: new puzzle challenge
(04-04-2015 11:43 PM)Paul Dale Wrote: Getting rid of reflections is easy by forcing the order of two adjacent edge middles. E.g. B>D. This would give an early pruning of the search space.
I think it eliminates some reflections but may not work all the time, because depending on your choice of variables you are using a different axis of symmetry.
For example, the 2 solutions with initial variable A=3 are:
A B C D ...
3 19 16 12 10 13 15 14 9 11 18 17 | 2 4 8 6 1 7 5
3 17 18 11 9 14 15 13 10 12 16 19 | 1 6 8 4 2 7 5
These two solutions are clearly reflections, but B>D is true in both cases.
In this case, B>L would be perhaps a better choice since it is mirroring about the axis that passes through the initial value and the center of the hexagon.
But this would only eliminate a solution after we computed L, which means we already have the outer ring completely solved, so not much of a speedup in this algorithm. Also, it wouldn't eliminate the
reflections of the rotated solutions, since the axis of symmetry is different. Perhaps if the algorithm tried to advance in both directions at once it would be much quicker to eliminate the
reflections (but what a mess trying to find values with PICK for the equations!).
In any case, I'm inclined to leave those optimizations "as an exercise for the reader".
04-05-2015, 03:24 AM
Post: #36
Paul Dale Posts: 1,849
Senior Member Joined: Dec 2013
RE: new puzzle challenge
You are correct, it has to be B and L. You could force C to be smallest in which case it would be B and D but this complicates.
Thinking about it more, H and F would also work since they are on the same axis of symmetry. They'd likely be better than B and L since the pruning would occur earlier.
- Pauli
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Gaussian process regression model
RegressionGP is a Gaussian process regression (GPR) model. You can train a GPR model, using fitrgp. Using the trained model, you can
• Predict responses for training data using resubPredict or new predictor data using predict. You can also compute the prediction intervals.
• Compute the regression loss for training data using resubLoss or new data using loss.
Create a RegressionGP object by using fitrgp.
LogLikelihood — Maximized marginal log likelihood
scalar value | []
Maximized marginal log likelihood of the GPR model, stored as a scalar value if the FitMethod is different from 'none'. If FitMethod is 'none', then LogLikelihood is empty.
If FitMethod is 'sd', 'sr', or 'fic', then LogLikelihood is the maximized approximation of the marginal log likelihood of the GPR model.
Data Types: double
BCDInformation — Information on BCD-based computation of Alpha
structure | []
Information on block coordinate descent (BCD)-based computation of Alpha when PredictMethod is 'bcd', stored as a structure containing the following fields.
Field Name Description
Gradient n-by-1 vector containing the gradient of the BCD objective function at convergence.
Objective Scalar containing the BCD objective function at convergence.
SelectionCounts n-by-1 integer vector indicating the number of times each point was selected into a block during BCD.
Alpha property contains the Alpha vector computed from BCD.
If PredictMethod is not 'bcd', then BCDInformation is empty.
Data Types: struct
Active Set Selection
ActiveSetHistory — History of active set selection and parameter estimation
History of interleaved active set selection and parameter estimation for FitMethod equal to 'sd', 'sr', or 'fic', stored as a structure with the following fields.
Field Name Description
ParameterVector Cell array containing the parameter vectors: basis function coefficients, β, kernel function parameters θ, and noise standard deviation σ.
ActiveSetIndices Cell array containing the active set indices.
Loglikelihood Vector containing the maximized log likelihoods.
CriterionProfile Cell array containing the active set selection criterion values as the active set grows from size 0 to its final size.
Data Types: struct
IsActiveSetVector — Indicators for selected active set
logical vector
Indicators for selected active set for making predictions from the trained GPR model, stored as a logical vector. These indicators mark the subset of training data that fitrgp selects as the active
set. For example, if X is the original training data, then ActiveSetVectors = X(IsActiveSetVector,:).
Data Types: logical
Training Data
NumObservations — Number of observations in training data
scalar value
Number of observations in training data, stored as a scalar value.
Data Types: double
X — Training data
n-by-d table | n-by-d matrix
Training data, stored as an n-by-d table or matrix, where n is the number of observations and d is the number of predictor variables (columns) in the training data. If the GPR model is trained on a
table, then X is a table. Otherwise, X is a matrix.
Data Types: double | table
Y — Observed response values
n-by-1 vector
Observed response values used to train the GPR model, stored as an n-by-1 vector, where n is the number of observations.
Data Types: double
PredictorNames — Names of predictors
cell array of character vectors
Names of predictors used in the GPR model, stored as a cell array of character vectors. Each name (cell) corresponds to a column in X.
Data Types: cell
ExpandedPredictorNames — Names of expanded predictors
cell array of character vectors
Names of expanded predictors for the GPR model, stored as a cell array of character vectors. Each name (cell) corresponds to a column in ActiveSetVectors.
If the model uses dummy variables for categorical variables, then ExpandedPredictorNames includes the names that describe the expanded variables. Otherwise, ExpandedPredictorNames is the same as
Data Types: cell
ResponseName — Name of the response variable
character vector
Name of the response variable in the GPR model, stored as a character vector.
Data Types: char
PredictorLocation — Means of predictors
1-by-d vector | []
Means of predictors used for training the GPR model if the training data is standardized, stored as a 1-by-d vector. If the training data is not standardized, PredictorLocation is empty.
If PredictorLocation is not empty, then the predict method centers the predictor values by subtracting the respective element of PredictorLocation from every column of X.
If there are categorical predictors, then PredictorLocation includes a 0 for each dummy variable corresponding to those predictors. The dummy variables are not centered or scaled.
Data Types: double
PredictorScale — Standard deviations of predictors
1-by-d vector | []
Standard deviations of predictors used for training the GPR model if the training data is standardized, stored as a 1-by-d vector. If the training data is not standardized, PredictorScale is empty.
If PredictorScale is not empty, the predict method scales the predictors by dividing every column of X by the respective element of PredictorScale (after centering using PredictorLocation).
If there are categorical predictors, then PredictorLocation includes a 1 for each dummy variable corresponding to those predictors. The dummy variables are not centered or scaled.
Data Types: double
RowsUsed — Rows of original training data stored
logical vector | []
Rows of the original training data stored in the model, specified as a logical vector. This property is empty if all rows are stored in X and Y.
Data Types: logical
Object Functions
compact Reduce size of machine learning model
crossval Cross-validate machine learning model
lime Local interpretable model-agnostic explanations (LIME)
loss Regression error for Gaussian process regression model
partialDependence Compute partial dependence
plotPartialDependence Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
postFitStatistics Compute post-fit statistics for the exact Gaussian process regression model
predict Predict response of Gaussian process regression model
resubLoss Resubstitution regression loss
resubPredict Predict responses for training data using trained regression model
shapley Shapley values
Train GPR Model and Plot Predictions
Generate sample data.
rng(0,'twister'); % For reproducibility
n = 1000;
x = linspace(-10,10,n)';
y = 1 + x*5e-2 + sin(x)./x + 0.2*randn(n,1);
Fit a GPR model using a linear basis function and the exact fitting method to estimate the parameters. Also use the exact prediction method.
gprMdl = fitrgp(x,y,'Basis','linear',...
Predict the response corresponding to the rows of x (resubstitution predictions) using the trained model.
ypred = resubPredict(gprMdl);
Plot the true response with the predicted values.
hold on;
legend('Data','GPR predictions');
hold off
More About
Active Set Selection and Parameter Estimation
For subset of data, subset of regressors, or fully independent conditional approximation fitting methods (FitMethod equal to 'sd', 'sr', or 'fic'), if you do not provide the active set (or inducing
input set), fitrgp selects the active set and computes the parameter estimates in a series of iterations.
In the first iteration, the software uses the initial parameter values in vector η[0] = [β[0],σ[0],θ[0]] to select an active set A[1]. The software maximizes the GPR marginal loglikelihood or its
approximation using η[0] as the initial values and A[1] to compute the new parameter estimates η[1]. Next, the software computes the new loglikelihood L[1] using η[1] and A[1].
In the second iteration, the software selects the active set A[2] using the parameter values in η[1]. Then, using η[1] as the initial values and A[2], the software maximizes the GPR marginal
loglikelihood or its approximation and estimates the new parameter values η[2]. Then, using η[2] and A[2], the software computes the new loglikelihood value L[2].
The following table summarizes the iterations and the computations at each iteration.
Iteration Number Active Set Parameter Vector Loglikelihood
1 A[1] η[1] L[1]
2 A[2] η[2] L[2]
3 A[3] η[3] L[3]
… … … …
The software iterates similarly for a specified number of repetitions. You can specify the number of replications for active set selection using the NumActiveSetRepeats name-value argument.
• You can access the properties of this class using dot notation. For example, KernelInformation is a structure holding the kernel parameters and their names. Hence, to access the kernel function
parameters of the trained model gprMdl, use gprMdl.KernelInformation.KernelParameters.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• The predict function supports code generation.
For more information, see Introduction to Code Generation.
Version History
Introduced in R2015b
R2023b: Model stores observations with missing predictor values
Starting in R2023b, training observations with missing predictor values are included in the X and Y data properties. The RowsUsed property indicates the training observations stored in the model,
rather than those used for training. Observations with missing predictor values continue to be omitted from the model training process.
In previous releases, the software omitted training observations that contained missing predictor values from the data properties of the model. | {"url":"https://in.mathworks.com/help/stats/regressiongp.html","timestamp":"2024-11-07T15:34:02Z","content_type":"text/html","content_length":"163464","record_id":"<urn:uuid:82a45267-7534-4be5-8271-b967a2258f52>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00184.warc.gz"} |
Machine Learning Notes: Logistic Regression
This is a collection of notes I’m taking as I progress through the Introduction to Machine Learning course provided by Duke University at Coursera. This is a working document, it might eventually be
split down into multiple documents at some point, but this is largely to aid in my memorisation of key knowledge from this course (and a chance to try out the cool Math and Diagram rendering in Hugo
using Mermaid and Latex).
Machine Learning Outcomes
Given a training set data @x@, and a outcome set @y@, we want a model that can predict @y@ given @x@, or @P(y|x)@.
Logistic Regression
Given we have @x_{i1} … x_{iM}@ features (data), and the set @b1 … b_M@ is the parameter set, and a bias @b_0@.
The linear model is defined as such: @@(b_1 \times x_{i1}) + (b_2 \times x_{i2}) + … + (b_M \times x_{iM}) + b_0 @@ After applying this to every feature, the set @z@ is created, where @z_1 … z_M@ are
the outputs of applying the logistic regression
We now want to take this and calculate @P(y_i = 1|x_i)@ using @z@ and the sigmoid function, @P(y_i = 1|x_i) =\sigma(z_i)@
The sigmoid function’s output always lives between @0 .. 1@, which we can use to determine the confidence.
Diagram of Logistic Regression
graph BT
zi --> sigma_z
xi1(("$$x_{i1}$$")) -->|"$$b_1$$"| zi
xi2(("$$x_{i2}$$")) --> zi
xi3(("$$ ... $$")) --> zi
xi4(("$$x_{iM}$$")) -->|"$$b_M$$"| zi
Applying this to a Real Dataset
A common example of applying this would be to Optical Character Recognition (OCR), where the feature set would be the pixels of a given image representing a character, a popular dataset for this is
the MNIST dataset.
This allows for taking each image - where a pixel is @x_M@ , and applying the learned parameters - @b_M@, and using logistic regression to determine the confidence that a given image is of a
particular character.
Inner Product Notation
The forms of notation of the inner product (@\odot@) of the vector @x_i@ and the vector @b@:
Full Linear Regression (without final sigmoid function application): @@ z_i = (b_1 \times x_{i1}) + (b_2 \times x_{i2}) + … + (b_M \times x_{iM}) + b_0 @@
Inner Product: @\displaystyle\sum_{m=1}^M x_{im} \times b_m@
Compact Notation: @x_i \odot b@
Full plus bias: @b_0 + x_i \odot b@
Why move on from the Logistic Regression?
Logistic Regression can only classify binary data (data that is either class @1@ or @0@).
#notes #course #machine learning #ai #math | {"url":"https://sylvanb.dev/machine-learning-notes-logistic-regression/","timestamp":"2024-11-03T04:35:53Z","content_type":"text/html","content_length":"11378","record_id":"<urn:uuid:0013363e-8562-44aa-a4f8-2f110a730c8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00043.warc.gz"} |
Diagonal scaling to improve eigenvalue accuracy
[T,B] = balance(A)
[S,P,B] = balance(A)
B = balance(A)
B = balance(A,'noperm')
[T,B] = balance(A) returns a similarity transformation T such that B = T\A*T, and B has, as nearly as possible, approximately equal row and column norms. T is a permutation of a diagonal matrix whose
elements are integer powers of two to prevent the introduction of roundoff error. If A is symmetric, then B == A and T is the identity matrix.
[S,P,B] = balance(A) returns the scaling vector S and the permutation vector P separately. The transformation T and balanced matrix B are obtained from A, S, and P by T(:,P) = diag(S) and B(P,P) =
B = balance(A) returns just the balanced matrix B.
B = balance(A,'noperm') scales A without permuting its rows and columns.
This example shows the basic idea. The matrix A has large elements in the upper right and small elements in the lower left. It is far from being symmetric.
A = [1 100 10000; .01 1 100; .0001 .01 1]
A =
1.0e+04 *
0.0001 0.0100 1.0000
0.0000 0.0001 0.0100
0.0000 0.0000 0.0001
Balancing produces a diagonal matrix T with elements that are powers of two and a balanced matrix B that is closer to symmetric than A.
[T,B] = balance(A)
T =
1.0e+03 *
2.0480 0 0
0 0.0320 0
0 0 0.0003
B =
1.0000 1.5625 1.2207
0.6400 1.0000 0.7813
0.8192 1.2800 1.0000
To see the effect on eigenvectors, first compute the eigenvectors of A, shown here as the columns of V.
[V,E] = eig(A); V
V =
0.9999 -0.9999 -0.9999
0.0100 0.0059 + 0.0085i 0.0059 - 0.0085i
0.0001 0.0000 - 0.0001i 0.0000 + 0.0001i
Note that all three vectors have the first component the largest. This indicates V is badly conditioned; in fact cond(V) is 8.7766e+003. Next, look at the eigenvectors of B.
[V,E] = eig(B); V
V =
0.6933 -0.6993 -0.6993
0.4437 0.2619 + 0.3825i 0.2619 - 0.3825i
0.5679 0.2376 - 0.4896i 0.2376 + 0.4896i
Now the eigenvectors are well behaved and cond(V) is 1.4421. The ill conditioning is concentrated in the scaling matrix; cond(T) is 8192.
This example is small and not really badly scaled, so the computed eigenvalues of A and B agree within roundoff error; balancing has little effect on the computed results.
Balancing can destroy the properties of certain matrices; use it with some care. If a matrix contains small elements that are due to roundoff error, balancing might scale them up to make them as
significant as the other elements of the original matrix.
• Nonsymmetric matrices can have poorly conditioned eigenvalues. Small perturbations in the matrix, such as roundoff errors, can lead to large perturbations in the eigenvalues. The condition number
of the eigenvector matrix,
cond(V) = norm(V)*norm(inv(V))
relates the size of the matrix perturbation to the size of the eigenvalue perturbation. Note that the condition number of A itself is irrelevant to the eigenvalue problem.
Balancing is an attempt to concentrate any ill conditioning of the eigenvector matrix into a diagonal scaling. Balancing usually cannot turn a nonsymmetric matrix into a symmetric matrix; it only
attempts to make the norm of each row equal to the norm of the corresponding column.
The MATLAB^® eigenvalue function, eig(A), automatically balances A before computing its eigenvalues. Turn off the balancing with eig(A,'nobalance').
Extended Capabilities
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment. | {"url":"https://nl.mathworks.com/help/matlab/ref/balance.html","timestamp":"2024-11-12T02:20:51Z","content_type":"text/html","content_length":"73583","record_id":"<urn:uuid:dbc45bdd-b270-4aa7-aff4-e1f2eb386944>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00611.warc.gz"} |
How to Calculate Concrete Mixtures? - Çimsa
Concrete mixture design is done with the objective of creating a balance between strength, durability, placeability and aesthetic conditions, and low cost.
Concrete mixture design consists of 2 main stages:
(1) Selection of suitable components (cement, aggregate, water, and additives)
(2) Calculating the ratio of these components in order to obtain the most economic concrete possible with optimal strength, durability, and workability.
Strength is the most important characteristic property of concrete. There is a one-to-one relation between water/cement ratio and strength. As long as concrete components’ properties and
environmental conditions remain fixed, strength decreases as water/cement ratio increase, and strength increases are this ratio decreases.
In addition, concrete strength for a fixed water/cement ratio is affected by;
– The largest aggregate size,
– Aggregate grain size distribution (granulometry), shape and surface ruggedness,
– Type of used cement,
– Air amount in the concrete,
– Type and amount of the used additives.
How to Determine Concrete Mixture Ratios?
Determination of concrete mixture ratios is done based on volume. In 1 m^3 compressed concrete, the amounts of mixture components are calculated with the below formula.
c : Mass of cement to be entered in the mixture (kg)
p : Mineral additive amount to be used as an addition to the cement in the mixture (kg)
k : Chemical additive amount to be used in the mixture (kg)
ρ[c] : Cement density (kg/dm^3)
ρ[p] : Mineral additive density (kg/dm^3)
ρ[k] : Chemical additive density (kg/dm^3)
W : Volume of water to be entered in the mixture (dm^3)
Wa : Amount of aggregate to be entered in the mixture (kg)
ρ[a] : Aggregate’s average specific mass (kg/dm^3)
A : Total air amount in the concrete (%)
How to Select the Water / Cement Ratio (W/C)?
Water/cement ratio is related to strength class of the concrete and the severity of external effects it will be exposed to (environmental effect classes).
As per TS 13515 and/or TS EN 206 Chart F.1, the environmental effect class for the environment the concrete will be located in should be determined and parameters such as least cement dosage
conforming to this class, lowest characteristic compressive strength, and highest w/c ratio (Chart.1).
Chart 1. Limit values suggested for concrete composition and properties (TS 13515 Chart F.1)
How to Decide on Target Compressive Strength?
Target compressive strengths to be used in concrete mixture design are given in Chart 2 according to the concrete classes and W/C ratios depending on 28-day compressive strengths are given in Chart 3
and Figure 1.
Chart 2 – Target compressive strengths (fcm) to be taken as basis in mixture calculations according to concrete classes and average compressive strengths the test samples must have
Chart 3 – Approximate W/C ratios according to 28-day concrete compressive strengths
Figure 1 – Graphical display of the approximate relation between W/C ratio given in Chart 3 and compressive strength. This graphic may be used for strength values belonging to different W/C ratios
not included in Chart 3.
How to Decide on the Water Amount in a Concrete Mixture?
The amount of water in the concrete mixture changes depending on viscosity, the largest aggregate grain size and whether the concrete has chemical additives or air entrained.
Plasticizer chemical additives are used to provide plasticity in the concrete and to decrease water amount. Used chemical additive type and effectiveness degree significantly affect mixture water
amount in the concrete.
Figure 2 presents the approximate mixture water amounts of concrete which contains crushed stone aggregate, without chemical additive and non-air entrained. Approximate mixture water amounts
depending on air entrainer additive use and other aggregate type are given in TS 802. When chemical additives are used in producing concrete, additive-added concrete mixture water amount is
determined with a certain amount of water decrease from the mixture water amounts found in the graphics depending on the chemical additive’s type.
Figure 2 – Approximate mixture water amounts for non-air entrained and without chemical additive concrete for concrete using crushed stone aggregate with different largest aggregate grain sizes and
different settling values
How to Decide on the Cement Amount?
After water/cement ratio and water amount are determined, the cement amount to enter in the mixture is calculated with the below formula.
c : Mass of cement to be entered in the mixture (kg)
s : Mass of water to be entered in the mixture (kg)
s/c : Water/cement ratio.
Other than this, cement amount can be selected as an estimated value determined based on experience in the beginning. Cement density value should be taken from cement test report. The concept of
k-value given in TS EN 13515 should be used in case fly ash or blast furnace slag are used in concrete mixture.
How to Decide on the Air Content of Concrete?
The air amount to enter the concrete mixture is determined by the envisaged aggregate largest grain size, grain size distribution and climate conditions (Figure 3).
Figure 3 – Total air contents to be used in the concrete mixture calculations depending on aggregate largest grain size and climate conditions
How to Decide on the Aggregate Amount?
The remaining volume from cement, water, chemical and mineral additives, and air in the concrete mixture is filled with aggregate. And aggregate volume is calculated using the following formula.
In order to calculate the mass of total aggregate to be used in 1 m^3 concrete, specific mass ρa belonging to each grain class should be determined.
Here, Ma gives the total aggregate mass entering into 1 m^3 concrete mixture and masses of each aggregate grain class (M1, M2, M3 and …….Mn) are determined by multiplying with aggregate mixture
ratios (x1, x2, x3 and ………xn).
How to Select the Aggregate Grain Size Distribution?
Aggregate grain size distribution to be used in the concrete mixture should be selected to be in the regions number 3 and number 4 (Figure 4, Figure 5). Grain distributions to fall into region number
3 is preferred as it is the suitable region.
Figure 4 – Limits of aggregate grain size distribution curve determined for the concrete with aggregate largest grain size of 16.0 mm
Figure 5 – Limits of aggregate grain size distribution curve given for the concrete with aggregate largest grain size of 32.0 mm.
How to Calculate Moisture Correction in Aggregates?
As reference specific mass values used for aggregates are determined as saturated dry surface (SDS), the found aggregate amounts are also SDS values.
Aggregates are not usually in the saturated dry surface (SDS) state when preparing concrete mixtures and their moisture states should be checked continuously and determined in regular intervals.
Moisture correction should be made according to the moisture (R) and water absorption (Se) values of the aggregates as given below.
The result of Se – M = …..
The difference between these values is assessed as below;
If ( + ) material is “AIR DRY”
If ( – ) material is “WET”
If ( 0 ) material is in the state of Saturated Dry Surface “SDS”.
Corrected mixture water amount and aggregate moisture correction for each aggregate class is calculated as described in TS 802.
Calculations are made with the formulas.
Verification of Mixture Calculations Through Tests
The limit values given in TS 802 for grain distribution, w/c ratio and water amount which significantly affects concrete properties, and which are taken as basis for mixture calculations are values
obtained from the results of many tests and they are not indefinite values.
Therefore, the concrete samples to be prepared using aggregate, water, cement, air, and additive amounts obtained as the result of mixture calculations should be tested and the results to be obtained
should be assessed in terms of whether they have the properties taken as the basis of the calculations. In case a difference is identified between the envisaged properties and properties found at the
test, the mixture calculations should be repeated by changing the inputs as required.
Concrete mixture design is done with the objective of creating a balance between strength, durability, placeability and aesthetic conditions, and low cost. …
In the rapidly developing and changing conditions of the world, different energy resources and raw materials are needed to continue cement production. … | {"url":"https://cimsa.com.tr/en/how-to-calculate-concrete-mixtures/","timestamp":"2024-11-06T07:31:38Z","content_type":"text/html","content_length":"164641","record_id":"<urn:uuid:bff40269-a5f6-40ce-be17-6367f8bb5d1e>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00113.warc.gz"} |
How to generate sensitivity analysis report in detail
I have used gurobi to solve my problem. Now I want to generate sensitivity report for the objective function. I found excel solver generates a very well detailed and organised report. On the similar
lines how do I generate the reports using Gurobi (python).
• Official comment
Please find below the object attributes you need to query:
□ var.x: Value in the current solution.
□ var.RC: Reduced cost.
□ var.obj: Linear objective coefficient.
□ var.SAObjUp: Objective coefficient sensitivity information. Note that this is not given as an increase to the current coefficient in the objective.
□ var.SAObjLow: Objective coefficient sensitivity information. Note that this is not given as a decrease to the current coefficient in the objective.
□ constr.slack: Slack in the current solution.
□ constr.pi: Dual value (also known as the shadow price).
□ constr.RHS: Right-hand side value.
□ constr.SARHSUp: Right-hand side (RHS) sensitivity information. Note that this is not given as an increase to the current RHS.
□ constr.SARHSLow: Right-hand side (RHS) sensitivity information. Note that this is not given as a decrease to the current RHS.
For more information, kindly refer to the section called Attributes of our reference manual.
• How do you get SAObjUp and SAObjLow in R? The only sensitivity analysis values I can find are pi and rc.
• Unfortunately, the SAObjUp and SAObjLow sensitivity attributes are not currently available through the R interface. However, we have logged this as a feature request.
• A sensitivity analysis package in the GUROBI TOOLs github hub would be a fantastic thing to add! This would make teaching introductory courses with gurobi much easier.
Perhaps also a function to dualize an LP. That'd be nice.
• Hi Robert!
Gurobi 9.5 can actually compute the dual LP by simply specifying the DUA format or DLP format when calling Model.write().
A sensitivity analysis package could indeed be interesting. Did you already find our example sensitivity.py?
• Thanks! I didn't know about this functionality. That's very helpful.
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Recursive Functions
A recursive function is one which defines a problem in terms of itself.
A recursive function calls itself directly or indirectly until it is stopped. If it is not stopped, then it will call itself forever – or maybe until your browser crashes!
Recursive functions let you perform a unit of work multiple times. And while it is similar to a for/while loop, using a recursive solution affords a unique, faster and much more elegant approach to
solving a problem.
Let’s write a countdown function, shall we?
1. Using Loop:
countDown = (num) => {
for(let i=num; i>0; i--){
countDown(6) //5,4,3,2,1
2. Using Recursion
countDown = (num) => {
if(num === 0)
return 0;
countDown(num - 1);
countDown(6) //5,4,3,2,1
A recursive function usually contains what is referred to as a “base case.”
A base case is a condition that checks and stops the recursion. It essentially prevents an infinite loop.
In the above example, the snippet below:
if(num === 0)
is the base case, a condition that checks our program.
More examples …
Supposing we want to write a solution that performs a raise-to-power operation, eg., 2^3.
We can use the inbuilt Math function in Javascript like so:
//result is 8
We can also write a Loop solution as follows:
pow = (a, b) => {
let result = 1;
for(let i=0; i<b; i++){
result *=a;
return result;
//result is 8
And then we can think in terms of recursion by doing the following:
pow = (a, b) => {
return a;
return a * pow(a, b-1);
//result is 8
Task: Sum all numbers till the given one.
E.g 5 = 1+2+3+4+5 = 15;
Using recursion, we would do the following
sumAll = (n) => {
if(n === 0){
return 0;
return n += sumAll(n-1);
alert(sumAll(5)) //15
There you go. you can start using recursion for your automated programming tasks. | {"url":"https://codeflarelimited.com/blog/recursive-functions/","timestamp":"2024-11-05T16:26:01Z","content_type":"text/html","content_length":"120823","record_id":"<urn:uuid:8f756218-a2cb-48f4-a96e-955a2c516c81>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00201.warc.gz"} |
Moving Average Analysis
Daily Returns
Moving average can be used to analyze the trend of the index and can also be used to predict the future movements. Depending on the type of moving average like 30 day moving day average it takes into
account the latest 30 values and calculates the simple average of the values. It can also be called as the moving average as it takes latest values. So as the new data point comes in, the oldest data
point is removed and average is calculated using the latest 30 values. This process is repeated and thus the moving average always takes the latest values. Thus it is also called as the moving
This is an example of simple moving average where we are giving equal weights to all the values. There are variations in the moving average methodology like exponential moving average or giving
relative weights where the latest values are given more weights than the old values (Hand & Jacka, 1998).
Moving average of different length like 30 day moving average or 365 day moving average will show different characteristics. 30 day moving average will be more volatile and also incorporate the
latest data so will tend to change the shape faster than the moving average with more data points like 365 day moving average.
Here we have used simple moving average to analyze the relation between the stock indices and also for the purpose of analyzing them individually. We have also analyzed the daily returns of the stock
to measure the volatility of the indices. Some indices have high volatility and some have low volatility.
Looking at the returns chart of the stock one can see that they always return to mean then go up and return back. This is due the fact that the returns series is stationary and so when one plots the
graphs of the returns it will always goes up and then return back and then goes down and then returns to mean. If one plot the graph of the price of the stock then it will either be going up or down
but it will not be mean reverting which means it is not stationary.
Looking at the graph one can see that the standard deviation of the Swiss Index is very high as compared to other indices. ATX index also shows a high volatility. (World Indices, 2013)
Moving Average Graph
Here four types of moving average graph have been analyzed, 30 day, 90 day, 180 day and lastly 365 day moving average. Looking at the four graphs one can see that as the number of days of the graph
is increased the graph smoothens. It is less volatile. Looking at the 30 day graph it is more volatile than the 90 day graph which is more volatile than 180 day.
Analyzing the graph one can see that BSE 30 and HangSeng Index have performed better than their counterpart. Index like S&P has remained stable over the past years. The movement in the S&P index is
very less as compared to other Indices. The indices which have shown the highest movement are the indices of the developing countries. These indices were also the most affected during the crisis.
Moving average of different length like 30 day moving average or 365 day moving average will show different characteristics. 30 day moving average will be more volatile and also incorporate the
latest data so will tend to change the shape faster than the moving average with more data points like 365 day moving average.
In the 30 day moving average is more volatile because it takes into consideration the most recent data and the lag in the data to respond to the news is less. If market starts rising, 30 day moving
average graph will respond faster to the news. The other graphs are still incorporating the old data so the rise in the graph will be slower. All these graphs show the almost all the indices are
following are the same pattern suggesting the interlinking of the economies which can be seen by their national indices. Only the magnitude change is more in some cases and less in other indices but
the rise and fall is almost similar. If one looks at the 30 day moving average graph then the small patterns will be different as compared to other indices as in the short run the economies are
affected by different factors. As we look at higher moving average charts, the differences seem to lessen and they follow the similar pattern. (Ruppert, 2004)
Correlation Analysis
Correlation gives the linear relationship between two variables. Initial measure to check for the relationship between the movements of the two variables is covariance. Covariance has some drawbacks
like the value can range from minus infinity to plus infinity. The magnitude of the covariance doesn’t tell anything about the strength of the relationship. Hence to overcome the drawbacks
correlation is used. Correlation is covariance is divided by the standard deviations of the two variables. Thus correlation is the standardized version of the covariance. Correlation between two
variables is always between -1 to +1.
Implication of Correlation Values
Correlation gives the linear relationship between the variables. A zero correlation doesn’t mean there is no relationship between the variables but instead it means that the linear relationship
between the variables is not there but they can be non-linearly related.
A value of +1 means that they are perfectly positively correlated, meaning that if one variable value increases then the other variable will also increase in value. A value of 1 means that they are
perfectly negatively correlated, meaning that if one variable value increases then the other variable will also increase in value. Correlation doesn’t give the causality but only the directional
view, meaning that falling of one variable doesn’t mean is affect by the other variable. (Sclove, 2012)
Period 1999-2013
Correlation matrix has been created using Excel functions. Some stock exchanges are positively correlated and some are negatively related. BSE 30 is positively related to CAC40, DAX, ATX Index and
negatively correlated to S&P, HangSeng Index and S&P TSX Index. Similarly for other exchanges like S&P its is negatively related to BSE 30, CAC 40, DAX, FTSE 100 and positively correlated with
HangSeng Index.
This is a very long period to look for the correlation. Correlation tends to change over time and so for any analysis only the recent data should be used for the analysis. Correlation is very
important factor for analysis of different applications and thus it should be used carefully. Calculating correlation for such a long time period can be misleading and may lead to false results. Thus
for analysis one should the shorter time frame as compared to this time frame. Correlation analysis of smaller periods will show that some of correlation relationship might have changed over the
period. Markets which were positively correlated may have become negatively correlated.
Period 1999-2005 & 2005-2013
As mentioned above the period from 1999-2013 is relatively long period to analyze the correlation, so one should use shorter period to check for the correlation.
The value of correlations for some of the stock exchange has changed in these periods. Correlation between BSE 30 and DAX was positive in the period 1999-2005 but it became negative in period
2005-2013. Similarly the correlation between S&P, Swiss Index and BSE 30 was positive in the period 1999-2005 but it became negative for the 2005-2013.
For CAC 40 and DAX it correlation turned to positive from negative value from the period of 1999-2005 to 2005-2013 respectively, while the reverse thing happened between CAC40 and (S&P Index, Swiss
Index), correlation turned from positive to negative. (World Indices, 2013)
For DAX, correlation between DAX and ATX Index turned from positive to negative, correlation between (S&P, S&P TSX Index) and DAX turned from negative to positive.
Similar things can be observed between the other stock exchanges, some correlations have changed the sign i.e. from positive to negative or negative to positive, some correlation have changed in
magnitude, i.e. became more strong or weak in correlated, while some correlation have not changed.
Thus one needs to cautious while using this analysis for some research or real life application. Correlation is being used in different ways in world of finance. One of the application is to use
correlation to select an optimal diversified portfolio.
Here as the correlation keeps changing over a period of time, one needs to take the latest available values and always incorporate the latest values to calculate the correlation table. It has also
been observed that when the markets are rising i.e. economy is booming stock markets and the asset prices tend to follow the historical correlation. But as the market falls or during crisis all the
historical correlation changes and all markets becomes positively correlated with each and thus the concept of diversification doesn’t apply to these markets as all the markets will be falling.
The change in correlations can be attributed to many factors which affect the variables. There can be new factors coming into play. Here the way economies of different countries have undergone a
change in these periods and the ways is business is conducted have changed leading to change in correlations.
Analysis of Correlation Groups
Analyzing the correlation and arranging them in the descending order one can see the order of correlation between them. Correlation groups here can be divided into 3 groups mainly strong positively
correlated, weak positive correlated and negative correlated.
The negative correlation group can have combinations which are already there in the above two groups as the stock exchanges in the above two groups will interact with each and may be negatively
By looking at the chart like bar chart and the area chart, one can see that some stocks are strongly positively correlated, some are weakly correlated and some are strongly negatively correlated. | {"url":"https://supreme-thesis.net/essays/analysis/moving-average-analysis.html","timestamp":"2024-11-05T17:28:20Z","content_type":"text/html","content_length":"70639","record_id":"<urn:uuid:6369d3ee-9a42-4d07-b505-7555b3b6f7c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00712.warc.gz"} |
OptoWiki Knowledge Base
Lens mount developed by Nikon, typical used for line scan cameras
F-Theta lenses are a class of Fisheye lenses
lenses are also called “
“, “
linear scaled
“, “
” or “
F-Theta lenses use an image mapping function of type
A non
of 1.37mm
focal length
has an
image circle
Therefor a f=3.17mm F-Theata
with an image diameter for 180° of 4.15mm has a
of (4.15/4.3)-1 = 0.965-1 = -3.5%
maintains angular distances
Type : F-Theta weak medium strong max
angles 115° 159° 217° 360° and more
Fisheye Types
Most distant point on the optical axis with an image of “acceptable sharpness”
Where CoC is the Circle of Confusion (the largest accepted Airy-disk) in Millimeter.
Alternatively, we can express the FarPoint using the magnification M :
If we use for example a 1/2.5″ 5 Aptina Megapixel
Sensor mit 2.2
A 5 Mega
with f=7.2mm
focal length
F2.4, focused to an
object distance
of 100mm then has a far point of
und einen Nahpunkt von
and thus
If instead we use a 5 Megapixel
Sony Sensor with 3.45
A 5 Mega
with f=7.2mm
focal length
F2.4, focussed to 100mm then results in
und einen Nahpunkt von
thus we get
If we use a color sensor instead we can use
To increase the
we can increase the Pixel Size, but we either lose resolution, or (at the same pixel count) the
If light travels from one point to another, it takes paths of a length that in first approximation is equal to the length of optical paths length closely adjascent to the actual path.
from one point to another
always chooses a path which’s
does not change if the path slightly varies.
always takes the shortest path in terms of speed but in general not the path of shortest distance.
The actual path taken has either a maximum or a minimum OPL compared to adjascent optical paths or is equal to the OPL of adjascent paths.
(usually horizontal) extend of an object that’s visible on a sensor.
Sometimes the HFOV (horizontal field of view), sometimes the VFOV (Vertical Field of View) and sometimes the DFOV (Diagonal Field of View) matters. It’s of utmost importance to state clearly which
is the study of perfect optical systems.
Any imperfections and aberrations, also wavelengths, diffraction, interference etc. are ignored.
Parts of first order optics are Gaussian optics ( = paraxial optics).
Results of this analysis are image location and magnification.
Compare third order optics
Various types of fisheye lenses are available.
Here a short overview:
Fisheye Types:
Type gnomonic stereographic F-Theta equal area orthographic
lens class wide angle Fisheye Fisheye Fisheye Fisheye
mapping function
normalized mapping function
meridional scaling
sagittal scaling
effective scaling
N from 2 1 0 -1
Balance Deform. vs. Scaling 0 -2 2
Curvature 0 -1
maintains – angles angular distances areas planar illuminance
AOV <180° <360° >= 360° 360° 180°
54° 94° 115° 94° 66°
75° 131° 159° 131° 90°
102° 180° 217° 180° 120°
Source of the various angles and formulae: wikipedia
Here the angle
equal area
lenses are also called “
equisolid angle”
“flächentreu” F-Theta
lenses are also called “
“, “
linear scaled
“, “
” or “
lenses are also called “
“or “
lenses are also called”
“or “
For details see:
equal area
The focal length is the distance from the Image side principal plane to the image of objects at infinity.
For single lenses in air that is equal to the distance from the first focal point to the first principal point.
(in each case measured from the left to the right)
Note that this is a positive value for converging lenses and a negative value for the divergent lenses.
The larger the focal length, the smaller the aperture angle of the lens and the smaller the object section that is displayed full-frame on the sensor.
The lens captures less of the object. Extremea are telephoto lenses and finally telescopes.
The smaller the focal length, the larger the aperture angle of the lens and the larger the object section which is displayed full-frame on the sensor.
The lens captures more of the object. Extreme forms are fisheye lenses.
Lenses are typically listed, sorted by focal length. As an approximation, lenses with larger focal length see a smaller portion of the object (in more detail).
There are exceptions! (See: pseudo-knowledge: viewing angle and focal length are equivalent)
The following calculators determine focal lengths. They assume, a pin hole
model, the
Chief Ray
Angles are assumed to be the same on object and image side.
Therefore for wide angles a too small focal length is returned .. (
as all focal length calculators on the internet do 😉
Also keep in mind that the Gauss lens equation and the Newtonian image equation both assume same CRA on object and image side, which is nearly never given for wide angle lenses and for sure not for
fisheye lenses.
Therefore when you calculate focal length, object and image distances, they will in general differ from the real world situation you measure!
The following calculator determines focal length from angles.
However, Viewing angles change with the
working distance
! Also, a Pinhole
model is assumed. Thus for wide angles a too small focal length is returned .. (as all focal length calculators on the internet do 😉 )
For the next calculator it is very important to correct the distortions before doing the calculation: | {"url":"https://www.optowiki.info/glossary/prefix:f/","timestamp":"2024-11-05T23:24:29Z","content_type":"text/html","content_length":"166593","record_id":"<urn:uuid:f8416f49-cfe0-4820-9284-f0f840a163e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00022.warc.gz"} |
33 ounces to grams
Convert 33 Ounces to Grams (oz to gm) with our conversion calculator. 33 ounces to grams equals 935.534314831678 oz.
Enter ounces to convert to grams.
Formula for Converting Ounces to Grams (Oz to Gm):
grams = ounces * 28.3495
By multiplying the number of grams by 28.3495, you can easily obtain the equivalent weight in grams from ounces.
Converting ounces to grams is a common task that many people encounter, especially when dealing with recipes, scientific measurements, or everyday tasks. Understanding how to perform this conversion
accurately can help bridge the gap between the imperial and metric systems, making it easier to communicate measurements across different contexts.
The conversion factor between ounces and grams is essential for this process. One ounce is equivalent to approximately 28.3495 grams. This means that to convert ounces to grams, you simply multiply
the number of ounces by this conversion factor. For example, if you want to convert 33 ounces to grams, you would use the following formula:
Grams = Ounces × 28.3495
Now, let’s break down the calculation step-by-step:
1. Start with the number of ounces you want to convert: 33 ounces.
2. Multiply this number by the conversion factor: 33 × 28.3495.
3. Perform the multiplication: 33 × 28.3495 = 937.0075.
4. Round the result to two decimal places for practical use: 937.01 grams.
Thus, 33 ounces is equal to approximately 937.01 grams. This rounded figure is often more useful in everyday applications, where precision to two decimal places is sufficient.
The importance of converting ounces to grams cannot be overstated. In cooking, for instance, many recipes use grams for ingredient measurements, especially in baking, where precision is crucial. If
you’re following a recipe from a different country or a cookbook that uses the metric system, knowing how to convert ounces to grams can ensure that your dish turns out perfectly.
In scientific measurements, accurate conversions are vital for experiments and data analysis. Whether you’re measuring chemicals in a lab or conducting research that requires precise weight
measurements, being able to convert between these two systems is essential.
Everyday use also benefits from this conversion. For example, if you’re tracking your food intake or following a diet plan that specifies nutritional information in grams, knowing how to convert
ounces can help you stay on track with your health goals.
In summary, converting 33 ounces to grams is a straightforward process that can be incredibly useful in various scenarios. By understanding the conversion factor and applying the formula, you can
easily navigate between the imperial and metric systems, making your cooking, scientific work, and daily tasks much more manageable.
Here are 10 items that weigh close to 33 ounces to grams –
• Watermelon
Shape: Round
Dimensions: Approximately 10-12 inches in diameter
Usage: Consumed fresh, in salads, or as juice
Random Fact: Watermelons are 92% water and can weigh anywhere from 5 to 30 pounds!
• Medium-Sized Dog
Shape: Varies by breed, generally compact
Dimensions: 18-24 inches in height
Usage: Companionship, service, and protection
Random Fact: The average weight of a medium-sized dog can range from 30 to 50 pounds, but some breeds can be around 33 ounces as puppies!
• Standard Laptop
Shape: Rectangular
Dimensions: 13-15 inches in width, 9-11 inches in depth
Usage: Computing, gaming, and professional tasks
Random Fact: Most standard laptops weigh between 2 to 5 pounds, with some ultrabooks weighing as little as 33 ounces!
• Large Bag of Flour
Shape: Rectangular
Dimensions: 12 x 6 x 4 inches
Usage: Baking and cooking
Random Fact: A standard bag of flour typically weighs 5 pounds, but smaller bags can be found around 33 ounces!
• Two Bottles of Wine
Shape: Cylindrical
Dimensions: 3 inches in diameter, 12 inches in height
Usage: Drinking, cooking, and gifting
Random Fact: A standard bottle of wine weighs about 25 ounces, so two bottles can weigh around 50 ounces, but smaller bottles can be found at 33 ounces!
• Large Pumpkin
Shape: Round and slightly flattened
Dimensions: 12-18 inches in diameter
Usage: Decoration, cooking, and carving
Random Fact: Pumpkins can weigh anywhere from a few ounces to over 1,000 pounds, but a medium-sized pumpkin can weigh around 33 ounces!
• Basketball
Shape: Spherical
Dimensions: 29.5 inches in circumference
Usage: Playing basketball
Random Fact: A standard men’s basketball weighs about 22 ounces, but a larger, inflated version can weigh closer to 33 ounces!
• Medium-Sized Cat
Shape: Compact and agile
Dimensions: 9-10 inches in height
Usage: Companionship and pest control
Random Fact: The average weight of a medium-sized cat is around 10-15 pounds, but kittens can weigh around 33 ounces!
• Large Bag of Dog Food
Shape: Rectangular
Dimensions: 12 x 8 x 4 inches
Usage: Feeding pets
Random Fact: A large bag of dog food can weigh around 33 ounces, providing essential nutrition for your furry friends!
• Two Medium-Sized Avocados
Shape: Oval
Dimensions: 4-6 inches in length
Usage: Eating, cooking, and making guacamole
Random Fact: A medium avocado weighs about 6-7 ounces, so two can weigh around 12-14 ounces, making them a healthy snack!
Other Oz <-> Gm Conversions – | {"url":"https://www.gptpromptshub.com/grams-ounce-converter/33-ounces-to-grams","timestamp":"2024-11-14T17:05:57Z","content_type":"text/html","content_length":"186071","record_id":"<urn:uuid:88bd2626-640e-4fd4-b22a-fdf490a190e4>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00358.warc.gz"} |
Clarification needed for glimmix covariance parameters test
I would like to ask for a clarification. If anyone could help me, I will be very much obliged.
I have an experiment with two treatments, the response was measured at 7 days intervals.
The measures were taken on subjects nested as follows: daughter from a mother from a family.
In SAS notation I would write this nested arrangement as follows: daughter(mother family).
By following "Stroup et al. (2018). SAS for Mixed Models: Introduction and Basic Applications", (chapter 8), I was able to visualize the covariance structure that may be best for my data:
However, when I run the model with AR(1) by using Glimmix:
proc glimmix data=data_long;
class treat time daughter mother family;
model resp=time|treat longevity/ddfm=kr2;
random intercept / subject=daughter(mother family);
random time / subject=daughter(mother family) type=ar(1) residual;
covtest 'between subject variance = 0?' zeroG;
The result of the test of covariance parameters shows a Pr>ChiSq = 1.0.
I understand this means that the between-subject effect can dropped from the model. When I drop the first "random" statement:
(i.e., random intercept / subject=daughter(mother family), The result of the test of covariance is the same (Pr>ChiSq = 1.0).
However, when I run:
proc mixed data=data_long covtest;
class treat time daughter mother family;
model resp=time|treat longevity/ddfm=kr;
repeated time / subject=daughter(mother family) type=ar(1);
The Covariance Parameter Estimates test AR(1) daughter(mother*family) has a PrZ < 0.0001, which means it is significant and it has to stay in the model.
I am understanding the analysis all wrong?
Please, correct me if I am wrong, but I understand that in the repeated measures analysis I need to specify daughter(mother family) ie, the subjects on which the repeated measures were taken.
But proc glimmix tells me that the between-subject effect can be dropped?
Is there any other way in GLIMMIX to specify the subjects on which the repeated measures were taken?
Thank you for your help.
09-10-2020 02:51 PM | {"url":"https://communities.sas.com/t5/Statistical-Procedures/Clarification-needed-for-glimmix-covariance-parameters-test/td-p/683032","timestamp":"2024-11-03T17:20:53Z","content_type":"text/html","content_length":"326267","record_id":"<urn:uuid:da40bb75-4046-4ef1-909f-aee56b6dee48>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00460.warc.gz"} |
Reduce Memory Footprint of Deep Neural Networks
Neural networks can take up large amounts of memory. If you have a memory requirement for your network, for example because you would like to embed it into a resource-constrained hardware target,
then you might need to reduce the size of your model to meet the requirements. This page describes how to use MATLAB^®, and particularly Deep Learning Toolbox Model Quantization Library, to compress
your neural network.
For information on how to speed up neural network training, see Speed Up Deep Neural Network Training. For information on how to improve the accuracy of your network, see Deep Learning Tips and
The two main contributors to the memory footprint of a neural network are states and learnable parameters. Layer states contain information calculated during the layer operations to be retained for
use in subsequent forward passes of the layer, for example, the cell state and hidden state of LSTM layers. The network learnable parameters contain the features learned by the network, for example,
the weights of convolution and fully connected layers.
You can compress a network using one of two methods.
• Structural compression reduces the number of states and learnable parameters. MATLAB has two structural compression techniques, pruning and projection.
• Quantization converts the states and learnable parameters to lower precision data types.
Pruning Pruning refers to the systematic removal of learnable parameters that have the smallest impact on the predictions of your For an example, see Prune Filters in a Detection Network
network. Using Taylor Scores.
Projection Projection is a way of converting large layers with many learnables to one or more smaller layers with fewer learnable For an example, see Compress Neural Network Using
parameters in total. Projection.
Quantization Quantization is a data-type compression technique. You reduce the precision of the parameters in your network, but the amount For an example, see Quantize Semantic Segmentation
of parameters and the network architecture stay the same. Network and Generate CUDA Code.
You can use any combination of these techniques to reduce the size of your network, depending on the types of layers and overall network architecture. For the best results, first prune, then project,
then quantize. Then generate code and embed the network on your hardware target. For optimal results, fine-tune your network after each step.
Use the Deep Network Designer app to analyze your network for memory reduction using compression techniques. Open your network in Deep Network Designer. Then, click Analyze for compression. This
feature requires Deep Learning Toolbox™ Model Quantization Library.
The compression analysis report shows information about:
• Maximum possible memory reduction
• Pruning and projection support
• Effect of the network architecture on the ability to prune individual layers
• Layer memory
Use the compression analysis report to decide which technique to use to compress your network.
Structural Compression
Structural compression reduces the size of your network by reducing the number of states and learnable parameters. MATLAB has two structural compression techniques, pruning and projection.
Pruning a neural network means removing the least important parameters to reduce the size of the network while preserving the quality of its predictions.
You can measure the importance of a set of parameters by the change in loss after removal of the parameters from the network. If the loss changes significantly, then the parameters are important. If
the loss does not change significantly, then the parameters are not important and can be pruned.
When you have a large number of parameters in your network, you cannot calculate the change in loss for all possible combinations of parameters. Instead, apply an iterative workflow.
1. Use an approximation to find and remove the least important parameter, or the n least important parameters.
2. Fine-tune the new, smaller network by retraining it for a couple of iterations.
3. Repeat steps 1 and 2 until you reach your desired memory reduction or until you cannot recover the accuracy drop via fine-tuning.
One option for the approximation in step 1 is to calculate the Taylor expansion of the loss as a function of the individual network parameters. This method is called Taylor pruning.
For some types of layers, including convolutional layers, removing a parameter is equivalent to setting it to zero. In this case, the change in loss resulting from pruning a parameter θ can be
expressed as follows.
$|\Delta \text{loss}\left(X,\theta \right)|=|\text{loss}\left(X,\theta =0\right)-\text{loss}\left(X,\theta \right)|\text{\hspace{0.17em}}.$
Here, X is the training data of your network.
Calculate the Taylor expansion of the loss as a function of the parameter θ to first order.
$\text{loss}\left(X,\theta \right)=\text{loss}\left(X,\theta =0\right)+\frac{\delta \text{loss}}{\delta \theta }\theta \text{\hspace{0.17em}}.$
Then, you can express the change of loss as a function of the gradient of the loss with respect to the parameter θ.
$|\Delta \text{loss}\left(X,\theta \right)|=|\frac{\delta \text{loss}}{\delta \theta }\theta |\text{\hspace{0.17em}}.$
You already calculate this gradient during training for backpropagation. You can reuse the same calculation to calculate the gradient for Taylor pruning.
In MATLAB, you can perform Taylor pruning by using a taylorPrunableNetwork object. Implement the iterative pruning process in a custom training loop. Calculate the importance of the learnable
parameters using the updateScore function. Remove the n least important learnable parameters using the updatePrunables function. For an example of this workflow, see Prune Image Classification
Network Using Taylor Scores.
Pruning using taylorPrunableNetwork objects removes filters from convolutional layers. Pruning convolutional filters can also reduce the number of learnable parameters in downstream layers, for
example, batchNormalizationLayer and fullyConnectedLayer objects.
For an example of how to use Taylor pruning on an image classification network in MATLAB, see Prune Image Classification Network Using Taylor Scores. For an example of how to use Taylor pruning on a
YOLO v3 object detection model and embed the resulting model onto a Raspberry Pi^®, see Prune Filters in a Detection Network Using Taylor Scores.
Projection is a layer-wise compression technique that replaces a large layer with one or more layers with a smaller total number of parameters.
In neural networks, data is usually expressed as a high-dimensional vector. Not all of the dimensions are equally important for the task you are solving. Often, a small subset of elements accounts
for a large amount of the variance in the output.
Principal-component analysis (PCA) allows you to express your data in a basis of dimensions sorted by importance. The first element accounts for the largest part of variance, the second for the
second-largest part, and the last element for the smallest amount of variance.
Neural network projection uses PCA to reduce the size of a network layer in the following way.
1. Use PCA to identify the subspace of learnable parameters that result in the highest variance in neuron activations by analyzing the network activations using a data set representative of the
training data.
2. Project the layer input onto the lower-dimensional space spanned by the N most important directions.
3. Perform the layer operation within the lower-dimensional space.
4. Return to the higher-dimensional space by appending the required number of zeros to the end of the output and rotate back into the original basis.
In MATLAB, project your neural network using the compressNetworkUsingProjection function. The compressNetworkUsingProjection function performs PCA automatically.
If you want to project the same network several times, for example, to explore different levels of compression, then perform PCA separately using the neuronPCA function. You can then pass the output
to the compressNetworkUsingProjection function as an optional input argument.
After projection, retrain your model to regain some or all of the accuracy lost during the projection step.
For an example of projection of a sequence classification network, see Compress Neural Network Using Projection. For an example of projection of a network as part of an overarching workflow to
estimate battery states of charge, see Evaluate Code Generation Inference Time of Compressed Deep Neural Network.
Quantization is a compression technique that does not impact the network architecture, and instead reduces the precision of the learnable parameters. In Deep Learning Toolbox, by default, network
parameters are stored in single precision. The Deep Learning Toolbox Model Quantization Library support package allows you to deploy your network with parameters stored as reduced precision types.
The quantization workflow consists of two steps.
1. Find the dynamic ranges of the parameters in your network. To do so, exercise your network with sample data that is representative of your training data. Extract the minimum and maximum values of
the weights and biases in each convolution and fully connected layer. Extract the minimum and maximum values of the activations in all other layers of your network.
2. In each layer, convert the parameters to integers representing the dynamic range calculated in the previous step.
To quantize deep learning models in MATLAB, you can use either the dlquantizer function or the Deep Network Quantizer app.
Command Line Workflow
At the command line, start by creating a dlquantizer object. Next, use the calibrate function to determine the dynamic ranges of the layer parameters. Make sure the data you pass to the calibrate
function is representative of your training data.
Next, you can optionally simulate quantization using the quantize function. This function allows you to test the output of your quantized network independent of your hardware target. Use the
quantizationDetails function to display details of the quantization, such as a list of quantized layers, the target library, and the quantized learnable parameters.
You can also use the estimateNetworkMetrics function to estimate the effects of quantization specific layers of your neural network. This function estimates metrics for learnable layers, which have
weights and biases, in the network. Estimated metrics are provided for the following supported layers.
You can also use the prepareNetwork function to prepare your network for quantization. This function modifies the neural network to improve accuracy and avoid error conditions in the quantization
workflow. These modifications include layer fusion, equalization of layer parameters, replacement of unsupported layers, and conversion of DAGNetwork and SeriesNetwork objects to a dlnetwork object.
Once you are satisfied with your network, validate and quantize your network using the validate function. To choose quantization options, such as the execution environment, use the
dlquantizationOptions function. Then, deploy your network to your hardware target.
For information on how to quantize neural networks for different execution environments using the command line, see these examples.
Deep Network Quantizer App
You can also use the Deep Network Quantizer app to achieve the quantization workflow. To open the app from the MATLAB command prompt, enter deepNetworkQuantizer.
For information on how to use Deep Network Quantizer to quantize neural networks for different execution environments, see these examples.
Other Techniques
The techniques described in the previous section are part of the Deep Learning Toolbox Model Quantization Library. There are several other compression techniques in MATLAB.
Knowledge Distillation
In knowledge distillation, use a large and accurate teacher network to teach a smaller student network to make accurate predictions [2]. This technique allows you to design a small network to fit
your memory requirements. For an example of knowledge distillation in MATLAB, see Train Smaller Neural Network Using Knowledge Distillation.
Quantization Using bfloat16 Code (MATLAB Coder)
The brain floating-point (bfloat16) format is a truncated version of the single-precision floating-point format. It only occupies 16 bits in computer memory. bfloat16 preserves approximately the same
number range as single-precision floating-point by retaining the same number of exponent bits. For information on how to generate and deploy bfloat16 code for deep learning networks, see Generate
bfloat16 Code for Deep Learning Networks (MATLAB Coder).
You can use sparsity to prune a network using a custom importance metric. Identify the least important parameters, set them to zero, and store the resulting weights matrices as sparse matrices.
For an example of pruning by using magnitude scores and synaptic flow scores, see Parameter Pruning and Quantization of Image Classification Network. Both of these metrics are data-free importance
estimation techniques.
Fixed-Point Designer
To use reduced precision types to optimize general algorithms for embedded hardware, use Fixed-Point Designer™. For more information, see Benefits of Fixed-Point Hardware (Fixed-Point Designer).
[1] Molchanov, Pavlo, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz. "Pruning Convolutional Neural Networks for Resource Efficient Inference." Preprint, submitted June 8, 2017. https://
[2] Hinton, Geoffrey, Oriol Vinyals, and Jeff Dean. “Distilling the Knowledge in a Neural Network.” Preprint, submitted March 9, 2015. https://arxiv.org/abs/1503.02531.
[3] Google Cloud Blog. “BFloat16: The Secret to High Performance on Cloud TPUs.” Accessed January 26, 2023. https://cloud.google.com/blog/products/ai-machine-learning/
See Also
taylorPrunableNetwork | compressNetworkUsingProjection | neuronPCA | Deep Network Quantizer | dlquantizer | prepareNetwork | estimateNetworkMetrics
Related Topics | {"url":"https://jp.mathworks.com/help/deeplearning/ug/reduce-memory-footprint-of-deep-neural-networks.html","timestamp":"2024-11-03T13:18:07Z","content_type":"text/html","content_length":"110401","record_id":"<urn:uuid:c077d028-49eb-4af9-aeb2-468d242b4f16>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00684.warc.gz"} |
Count Distinct Values in Excel with Multiple Criteria
In Excel, you can count the number of distinct values in a column based on specific criteria using a formula. This formula is particularly useful when you want to count unique values in a large
dataset. The formula uses the SUM, IF, and COUNTIFS functions to achieve this. Let's break down the formula step by step.
First, the COUNTIFS function is used to count the number of occurrences of each value in column A of Sheet2 that meets the criteria: column D equals B3 and column N equals A3. This function allows
you to specify multiple criteria for counting.
Next, the 1/COUNTIFS(Sheet2!$A:$A,Sheet2!$A:$A,Sheet2!$D:$D,B3,Sheet2!$N:$N,A3) part of the formula calculates the reciprocal of the count for each value. This ensures that each distinct value is
counted only once.
Then, the IF function is used to create an array of reciprocal counts for each value in column A of Sheet2 that meets the criteria.
After that, the SUM function is used to sum up the array of reciprocal counts, resulting in the count of distinct values.
To enter this formula as an array formula in Excel, press Ctrl + Shift + Enter instead of just Enter.
For example, if you have the following data in Sheet2:
A B C D ... N
1 A B ... A
2 B A ... B
3 A B ... A
4 C A ... A
5 A B ... B
6 B A ... A
7 A B ... A
8 C A ... B
If B3 is "B" and A3 is "A", the formula would return the value 2. This means that there are 2 distinct values in column A of Sheet2 where the values in column D are "B" and the values in column N are
Now that you understand the formula and how it works, you can use it to count distinct values in Excel based on multiple criteria.
An Excel formula
Formula Explanation
This is an array formula that uses the SUM, IF, and COUNTIFS functions to count distinct values in Sheet2 column A where values in Sheet2 column D are equal to B3 and values in Sheet2 column N are
equal to A3.
Step-by-step explanation
1. The COUNTIFS function is used to count the number of occurrences of each value in Sheet2 column A that meets the criteria: Sheet2 column D equals B3 and Sheet2 column N equals A3.
2. The 1/COUNTIFS(Sheet2!$A:$A,Sheet2!$A:$A,Sheet2!$D:$D,B3,Sheet2!$N:$N,A3) part of the formula calculates the reciprocal of the count for each value. This is done to ensure that each distinct
value is counted only once.
3. The IF function is used to create an array of reciprocal counts for each value in Sheet2 column A that meets the criteria.
4. The SUM function is used to sum up the array of reciprocal counts, resulting in the count of distinct values.
5. The formula is entered as an array formula by pressing Ctrl + Shift + Enter instead of just Enter.
For example, if we have the following data in Sheet2:
| A | B | C | D | ... | N |
| | | | | | |
| 1 | A | | B | ... | A |
| 2 | B | | A | ... | B |
| 3 | A | | B | ... | A |
| 4 | C | | A | ... | A |
| 5 | A | | B | ... | B |
| 6 | B | | A | ... | A |
| 7 | A | | B | ... | A |
| 8 | C | | A | ... | B |
If B3 is "B" and A3 is "A", the formula =SUM(IF((Sheet2!$D:$D=B3)*(Sheet2!$N:$N=A3),1/COUNTIFS(Sheet2!$A:$A,Sheet2!$A:$A,Sheet2!$D:$D,B3,Sheet2!$N:$N,A3))) would return the value 2. This means that
there are 2 distinct values in Sheet2 column A where values in Sheet2 column D are "B" and values in Sheet2 column N are "A". | {"url":"https://codepal.ai/excel-formula-generator/query/10kSmlAu/excel-formula-count-distinct-values","timestamp":"2024-11-04T13:58:39Z","content_type":"text/html","content_length":"98195","record_id":"<urn:uuid:7ed844e4-5c6d-4468-b532-6e54ef886fa6>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00126.warc.gz"} |
Quantum Computing | Cyberknight
Quantum Computing is a cutting-edge field of computing that utilizes the principles of quantum mechanics to process and store information in fundamentally different ways compared to classical
computers. Unlike classical computers, which use bits as the basic unit of information (0 or 1), quantum computers use quantum bits or qubits. Qubits can represent 0, 1, or any quantum superposition
of these states simultaneously, allowing for the exploitation of quantum phenomena such as superposition and entanglement.
Key principles and characteristics of quantum computing include:
1. Superposition: Qubits can exist in multiple states simultaneously, enabling quantum computers to perform many calculations at once. This property can potentially lead to exponential speedup for
certain types of problems.
2. Entanglement: Qubits can become entangled, meaning the state of one qubit is intrinsically connected to the state of another, even if they are physically separated. Entanglement enables quantum
computers to perform coordinated operations on qubits that can’t be achieved classically.
3. Quantum Gates: Quantum computers use quantum gates to manipulate qubits, similar to classical logic gates. However, quantum gates operate on qubits in superposition and can perform complex
operations that classical gates cannot.
4. Quantum Algorithms: Quantum computing algorithms, such as Shor’s algorithm and Grover’s algorithm, have been developed to solve specific problems more efficiently than classical algorithms. For
instance, Shor’s algorithm can factor large numbers exponentially faster than classical algorithms, which has implications for cryptography.
5. Quantum Decoherence: Quantum states are fragile and can be easily disrupted by environmental factors, a phenomenon known as decoherence. Maintaining the integrity of quantum information is a
significant challenge in quantum computing.
6. Quantum Hardware: Quantum computers are typically implemented using specialized hardware, such as superconducting qubits, trapped ions, or topological qubits, which operate at extremely low
temperatures to minimize decoherence effects.
7. Applications: Quantum computing has the potential to revolutionize various fields, including cryptography, optimization, materials science, drug discovery, and artificial intelligence. It is
expected to tackle complex problems that are currently intractable for classical computers.
Quantum computing is still in its early stages of development, and large-scale, fault-tolerant quantum computers are not yet widely available. Researchers are actively working on overcoming technical
challenges and building practical quantum computing systems. Once realized, quantum computers have the potential to disrupt many industries by solving complex problems that were previously considered
computationally infeasible. | {"url":"https://cyberknight.uk/quantum-computing/","timestamp":"2024-11-04T01:29:59Z","content_type":"text/html","content_length":"143215","record_id":"<urn:uuid:d762396a-55b9-4a30-b1a0-65e854d98d84>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00294.warc.gz"} |
Minkowski theorem
From Encyclopedia of Mathematics
Minkowski's theorem on convex bodies is the most important theorem in the geometry of numbers, and is the basis for the existence of the geometry of numbers as a separate division of number theory.
It was established by H. Minkowski in 1896 (see [1]). Let $N$ be a closed convex body in $\mathbf{R}^n$, symmetric with respect to the origin $0$ and having volume $V(N)$. Then every point lattice $\
Lambda$ of determinant $d(\Lambda)$ for which $$ V(N) \ge 2^n d(\Lambda) $$
has a point in $N$ distinct from $0$.
An equivalent formulation of Minkowski's theorem is: $$ \Delta(N) \ge 2^{-n} d(\Lambda) $$ where $\Delta(N)$ is the critical determinant of the body $N$ (see Geometry of numbers). A generalization of
Minkowski's theorem to non-convex bodies is Blichfeldt's theorem (see Geometry of numbers). The theorems of Minkowski and Blichfeldt enable one to estimate from above the arithmetic minima of
distance functions.
[1] H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953)
A refinement of Minkowski's theorem employing Fourier series was given by C.L. Siegel. A different refinement is Minkowski's theorem on successive minima (see Geometry of numbers). These refinements
have applications in algebraic number theory and in Diophantine approximation. For a collection of other conditions which guarantee the existence of lattice points in a convex body see .
Minkowski's theorem on linear forms
The system of inequalities $$ \left\vert{ \sum a_{1j} x_j }\right\vert \le c_1 $$ $$ \left\vert{ \sum a_{ij} x_j }\right\vert < c_i\ \ \ i=2,\ldots,n $$ where $a_{i,j}, c_i$ are real numbers, has an
integer solution $(x_1,\ldots,x_n) \neq 0$ if $c_1\cdots c_n \ge |\det a_{i,j}|$. This was established by H. Minkowski in 1896 (see [1]). Minkowski's theorem on linear forms is a corollary of the
more general theorem of Minkowski on a convex body (see part 1).
[1] H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953)
[2] H. Minkowski, "Diophantische Approximationen" , Chelsea, reprint (1957)
[3] J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1972)
E.I. Kovalevskaya
The problem when the first inequality in Minkowski's theorem on linear forms can be replaced by strict inequality was solved by G. Hajós.
[a1] P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2] P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989)
How to Cite This Entry:
Minkowski theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minkowski_theorem&oldid=34529
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/index.php?title=Minkowski_theorem&oldid=34529","timestamp":"2024-11-08T17:25:40Z","content_type":"text/html","content_length":"19613","record_id":"<urn:uuid:b1a1eb93-8ad6-4ebe-90db-d2423d45d384>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00261.warc.gz"} |
Class ImageProcessingOperations
All Implemented Interfaces:
• Constructor Details
□ ImageProcessingOperations
public ImageProcessingOperations()
• Method Details
□ apply
Applies the chosen image operation/operations to the image.
Specified by:
apply in interface ImageProcessingOps
image - BufferedImage to apply the processing operation to.
BufferedImage The new image.
□ blur
Adds the blur operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the blur operation.
□ custom
Adds the custom operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
custom - ImageOperation to be applied
This instance of ImageProcessingOperations that includes the custom operation.
□ edgeDetection
Adds the edge detection operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the edge detection operation.
□ emboss
Adds the emboss operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the emboss operation.
□ exposure
Adds the exposure operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
exposure - float amount of exposure to apply
This instance of ImageProcessingOperations that includes the exposure operation.
□ gaussianBlur
Adds the gaussian blur operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the gaussian blur operation.
□ invertColors
Adds the invert colors operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the invert colors operation.
□ kernel
Adds the Kernel operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
kernel - float[] of kernel to apply
The updated ImageProcessingOperations that includes the kernel operation.
□ brighten
Adds the Brighten operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
percent - float percentage to brighten
The updated ImageProcessingOperations that includes the brighten operation.
□ mirror
Adds the mirror operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
mirrorOp - enum of which mirroring operation to do
This instance of ImageProcessingOperations that includes the mirror operation.
□ mirror
Adds the mirror operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the mirror operation.
□ resizeToFit
Adds the resize to fit operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
newWidth - Integer of new width
newHeight - Integer of new height
This instance of ImageProcessingOperations that includes the resize to fit operation.
□ resizeToHeight
Adds the resize to height operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
newHeight - Integer of new height
This instance of ImageProcessingOperations that includes the resize to height operation.
□ resizeToWidth
Adds the resize to width operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
newWidth - Integer of new width
This instance of ImageProcessingOperations that includes the resize to width operation.
□ rotate
Adds the rotate operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
degrees - Double of degrees to rotate the image
This instance of ImageProcessingOperations that includes the rotate operation.
□ scale
Adds the scale operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
scaling - Double of scale to apply the image
This instance of ImageProcessingOperations that includes the scale operation.
□ scale
Adds the scale operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
scaling - Double of scale to be applied
interpolation - Integer of interpolation to be applied
This instance of ImageProcessingOperations that includes the scale operation.
□ stretchToFill
Adds the stretch to fill operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
newWidth - Integer of new width
newHeight - Integer of new height
This instance of ImageProcessingOperations that includes the stretch to fill operation.
□ superScale2x
Adds the super scale 2x operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the super scale 2x operation.
□ toARGB
Adds the toARGB operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the toARGB operation.
□ toBinary
Adds the toBinary operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the toBinary operation.
□ toGrayscale
Adds the toGrayscale operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the toGrayscale operation.
□ toIndexed
Adds the toIndexed operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the toIndexed operation.
□ toRGB
Adds the toRGB operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
This instance of ImageProcessingOperations that includes the toRGB operation.
□ qualityScale
Adds the quality scale operation to a list of operations to be applied.
See the
JDeli tutorials
for more information on how to use this processing operation.
scaling - Double of the scale to be applied
This instance of ImageProcessingOperations that includes the quality scale operation.
□ thumbnail
Adds the thumbnail operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use.
width - Integer of the thumbnail width
height - Integer of the thumbnail height
This instance of ImageProcessingOperations that includes the thumbnail operation.
□ watermark
Adds the watermark operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
text - String text to be displayed as watermark, '\n' will divide watermark across multiple lines
color - Color to be used to represent text watermark
font - font style to represent text watermark
pos - Enum WatermarkPosition which represents the position of the Watermark for centre or fit to image position
This instance of ImageProcessingOperations that includes the watermark operation.
□ watermark
Adds the watermark operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
watermarkImage - BufferedImage of image to watermark
pos - Enum WatermarkPosition which represents the position of the Watermark for centre or fit to image position
alphaComposite - AlphaComposite to be used when applying watermark
This instance of ImageProcessingOperations that includes the watermark operation.
□ watermark
Adds the watermark operation to a list of operations to be applied.
JDeli tutorials
for more information on how to use this processing operation.
shape - Shape to use when applying watermark
color - Color to be used to represent text watermark
pos - Enum WatermarkPosition which represents the position of the Watermark for centre or fit to image position
alphaComposite - AlphaComposite to be used when applying watermark
properties - Enum WatermarkShapeProperties to use when applying the watermark
The instance of ImageProcessingOperations that includes the watermark operation.
□ operationsListSize
public int operationsListSize() | {"url":"https://files.idrsolutions.com/maven/site/jdeli/apidocs/com/idrsolutions/image/process/ImageProcessingOperations.html","timestamp":"2024-11-14T14:17:43Z","content_type":"text/html","content_length":"70353","record_id":"<urn:uuid:916ebe9c-cf4e-4df2-9d09-8ec6506f4657>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00862.warc.gz"} |
How do you solve \
Hint:In the given problem we need to solve this for ‘g’. We can solve this using the transposition method. The common transposition method is to do the same thing (mathematically) to both sides of
the equation, with the aim of bringing like terms together and isolating the variable (or the unknown quantity). That is we group the ‘g’ terms one side and constants on the other side of the
Complete step by step solution:
Given, \[\dfrac{g}{2} = \dfrac{6}{{10}}\].
We transpose ‘2’ which is present in the left hand side of the equation to the right hand side of the equation by multiplying ‘2’ on the left hand side of the equation.
\[g = \dfrac{6}{{10}} \times 2\]
We can see that the variable is on the left hand side and the remaining constant is on the right hand side of the equation. We simplify the terms in the right hand side of the equation.
\[ \Rightarrow g = \dfrac{6}{5}\]. This is the exact form.
In decimal form we have \[ \Rightarrow g = 1.2\]
Note: We can check whether the obtained solution is correct or wrong. All we need to do is substitute the value of ‘g’ in the given problem.
\[\dfrac{{\left( {\dfrac{6}{5}} \right)}}{2} = \dfrac{6}{{10}}\]
Rearranging we have,
\[\dfrac{6}{{5 \times 2}} = \dfrac{6}{{10}}\]
\[ \Rightarrow \dfrac{6}{{10}} = \dfrac{6}{{10}}\]
Hence the obtained answer is correct.
If we want to transpose a positive number to the other side of the equation we subtract the same number on that side (vice versa). Similarly if we have multiplication we use division to transpose. If
we have division we use multiplication to transpose. Follow the same procedure for these kinds of problems. | {"url":"https://www.vedantu.com/question-answer/how-do-you-solve-dfracg2-dfrac610-class-11-maths-cbse-60115980925e103366bff82f","timestamp":"2024-11-13T00:59:05Z","content_type":"text/html","content_length":"160329","record_id":"<urn:uuid:369432bc-6a86-4ba4-852c-2320cf880182>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00437.warc.gz"} |
Reasonable inflation rate assumption for retirement planning
Almost every retirement planner has a default inflation rate of 3%. That can be a terrible mistake. The average CPI-U inflation over the 100 years since 1913 is 3.3%, not 3%. For retirement planning,
one should adopt a very conservative approach. You should assume expected rate of return on lower side & inflation on a higher side. This should be done in order to protect yourself against adverse
economic situations. Now Planning for retirement is hard enough without having to worry about inflation. Here are a few things to consider regarding inflation and retirement planning. First you need
to estimate the inflation rate between now and the time you retire. So assuming you are 50 now that would be 15 years.
When projecting retirement success, assume expenses will go up by 3% each year, in line with historical inflation rates. Then we begin working on a “spend-more now plan” which may mean less income
increases later. This type of planning can allow more spending in the go-go years. Learn to understand how to plot your portfolio's real rate of return for retirement planning to safeguard your
retirement funds against inflation. Almost every retirement planner has a default inflation rate of 3%. That can be a terrible mistake. The average CPI-U inflation over the 100 years since 1913 is
3.3%, not 3%. For retirement planning, one should adopt a very conservative approach. You should assume expected rate of return on lower side & inflation on a higher side. This should be done in
order to protect yourself against adverse economic situations. Now Planning for retirement is hard enough without having to worry about inflation. Here are a few things to consider regarding
inflation and retirement planning. First you need to estimate the inflation rate between now and the time you retire. So assuming you are 50 now that would be 15 years. Assumptions aren't always a
laughing matter, and that's certainly true when it comes to retirement planning, where "hope for the best, plan for the worst" is a reasonable motto. There are a lot of planning assumptions that
enter into how we tackle retirement planning. Now that the trauma from the Great Recession is beginning to subside, we may need to look at how we are planning, saving, and investing for our
Every retirement calculator, whether state-of-the-art Monte Carlo or Inflation Assumption: What Is A Reasonable Estimate For Inflation During Retirement? financial problems, and it may be prudent to
budget for higher inflation rates than
20 Sep 2018 My industry, the business of providing financial advice, should be in the knowledge of investments, financial markets, retirement planning, and the all the other variables such as your
spending, inflation, and tax rates. 12 Mar 2019 We believe that retirees should plan for a long retirement. out of money, based on a variety of assumptions and projections regarding potential You
would increase the amount by inflation each year thereafter—or ideally, We assume that investors want the highest reasonable spending rate, but not so 30 Apr 2018 An important facet of the financial
planner's work is to make a variety of Projections of salary increases may justify an inflation rate that in excess of 0.5 % in either direction of the guidelines should be reasonable and. 2 Feb 2013
There are a variety of Retirement Planning Calculators that can help Isn't it reasonable to assume that you will need more medical care as you get older First you need to estimate the inflation rate
between now and the time you retire. The one thing on the plus side is that so far we have assumed that 10 Mar 2017 What does that mean in terms of retirement planning? professionals have
traditionally assumed an average inflation rate of 3 percent over the long term. To answer the question whether 3 percent is a reasonable future
1 Feb 2020 Demographic assumptions are those pertaining to a pension plan's membership, such as changes reasonable, defined in subsection 3.6 as being because the average rate of assumed inflation
has been dropping more.
2 Nov 2016 years began to reveal that the assumption of stable spending may not actually be For instance, a study in the Journal of Financial Planning by Ty Bernicke Notably, the chart above graphs
the real (inflation-adjusted) change in any ' reasonable' decrease – is arguably a better baseline for retirement Since you don't know what inflation will be in retirement, what your rate of return
will be, or how long you will live, you can't come up with an exact answer. The next best thing is to come up with a reasonable set of assumptions and make sure you re-evaluate every few years. If
you look at inflation over the last 30 years as a base to help estimate a reasonable inflation rate for the next 30 years, you would find an average (using the geometic mean) inflation rate of about
2.6%. However, if you step back 50 years, to include higher inflation periods of the 70’s and 80’s, the average rate would be about 3.7%.
20 May 2019 Most people think that retirement planning is a complicated affair and use it Even at this modest rate, a monthly expense of Rs 1 lakh per month will With inflation assumed at 6%, a 2%
real return from debt is reasonable.
21 Jan 2020 From 1965 to 2011, the average annual inflation rate was 4.39%. Let's face it, the purchasing power of $1 thirty years ago is different that today. I 5 Feb 2015 Incorrect--and usually
too rosy--retirement-planning assumptions are guide their planning decisions; 3% is a reasonable starting point. (This article looks at historical inflation rates for a broad range of goods and
services.). 1 Feb 2020 Demographic assumptions are those pertaining to a pension plan's membership, such as changes reasonable, defined in subsection 3.6 as being because the average rate of assumed
inflation has been dropping more. Use this free inflation calculator with built in US Consumer Price Index - Urban or enter your own inflation rate to determine the buying power of a dollar over
time. amount of confidence that inflation rates will stay within a reasonable range. Investing in stocks not only helps you grow your retirement savings, but it also Use the retirement planning
calculator at Interest.com to determine if you are saving you will need to save to retire comfortably, with a reasonable monthly income. Expected rate of inflation: This is what you expect for the
average long- term
Learn to understand how to plot your portfolio's real rate of return for retirement planning to safeguard your retirement funds against inflation.
Then, it's time to do retirement planning calculations to see where you stand. The value is in Does your plan account for that, and at what rate? Will Social Security or pension payments keep up with
inflation? Either way, you need to know what the numbers are, and whether or not those are reasonable assumptions. is a percentage of pre-retirement income (the percentage is either assumed by the
reasonable range (e.g., an annual inflation rate of 50 percent, or an interest 11 Jun 2013 If you look at inflation over the last 30 years as a base to help estimate a reasonable inflation rate for
the next 30 years, you would find an average 25 Oct 2018 Some experts think our retirement planning assumptions are not monthly providing a snapshot of the country's current rate of inflation. 20
May 2019 Most people think that retirement planning is a complicated affair and use it Even at this modest rate, a monthly expense of Rs 1 lakh per month will With inflation assumed at 6%, a 2% real
return from debt is reasonable. Every retirement calculator, whether state-of-the-art Monte Carlo or Inflation Assumption: What Is A Reasonable Estimate For Inflation During Retirement? financial
problems, and it may be prudent to budget for higher inflation rates than
12 Mar 2019 We believe that retirees should plan for a long retirement. out of money, based on a variety of assumptions and projections regarding potential You would increase the amount by inflation
each year thereafter—or ideally, We assume that investors want the highest reasonable spending rate, but not so 30 Apr 2018 An important facet of the financial planner's work is to make a variety of
Projections of salary increases may justify an inflation rate that in excess of 0.5 % in either direction of the guidelines should be reasonable and. 2 Feb 2013 There are a variety of Retirement
Planning Calculators that can help Isn't it reasonable to assume that you will need more medical care as you get older First you need to estimate the inflation rate between now and the time you
retire. The one thing on the plus side is that so far we have assumed that 10 Mar 2017 What does that mean in terms of retirement planning? professionals have traditionally assumed an average
inflation rate of 3 percent over the long term. To answer the question whether 3 percent is a reasonable future 3 Jun 2019 Investment and retirement planning is a fascinating, diverse profession
Then, using an assumed investment rate of return they will calculate the (after fees and costs) is reasonable and sustainable in today's investment 16 Aug 2018 Some readers balked at the
“unrealistic” rate of return. In response, CNBC spoke to investing experts and financial advisors about In addition to the uncertainty of returns, investors must also contend with inflation. S&P 500
Index · Tax planning · Personal loans · Personal saving · Retirement planning. | {"url":"https://bestbitakorzgeb.netlify.app/luedtke63052wida/reasonable-inflation-rate-assumption-for-retirement-planning-japy","timestamp":"2024-11-11T13:19:54Z","content_type":"text/html","content_length":"37435","record_id":"<urn:uuid:ef317e00-3633-45c6-b2b4-98a017dd8f73>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00742.warc.gz"} |
13.2.3 Graphs of Functions, PT3 Focus Practice
Question 7:
Use graph paper to answer this question.
Table below shows the values of two variables, x and y, of a function.
The x-axis and the y-axis are provided on the graph paper on the answer space.
(a) By using a scale of 2 cm to 5 units, complete and label the y-axis.
(b) Based on the table above, plot the points on the graph paper.
(c) Hence, draw the graph of the function.
Question 8:
Use graph paper to answer this question.
Table below shows the values of two variables, x and y, of a function.
The x-axis and the y-axis are provided on the graph paper on the answer space.
(a) By using a scale of 2 cm to 2 units, complete and label the y-axis.
(b) Based on the table above, plot the points on the graph paper.
(c) Hence, draw the graph of the function. | {"url":"https://content.myhometuition.com/2018/06/09/13-2-3-graphs-of-functions-pt3-focus-practice/","timestamp":"2024-11-06T14:41:07Z","content_type":"text/html","content_length":"21319","record_id":"<urn:uuid:4d97ba55-28ce-4e55-b302-2c8913da60db>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00800.warc.gz"} |
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PyCork is a Python Boolean CSG Library for 3D triangular meshes built upon the Cork library.
Designing scan strategies for PBF techniques, we are not entirely aware of situations that arise where the powder-bed is not fully exposed due to a mismatch when scan vectors are not sufficiently
overlapped. Typically, unoptimised placement of hatch vectors lead to the creation of irregular porosity or voids in L-PBF parts.
This can arise along the intersections between the contour and interior hatches, especially a long concave regions such as sharp corner features with acute angles.
The approach is not an efficient way to examine the presence of , but provides a representative view for checking this geometrically.
The approach takes advantage of the relatively new Iterator classes available within the analysis module, which vastly simplifies the generation procedure for manipulating and examining existing scan
vector geometry. Firstly, generate or alternatively import the Layer and its LayerGeometry groups to examine. The group of layers are passed to the ScanVectorIterator class, which will iterate across
every scan vector from both ContourGeometry and HatchGeometry objects within a Layer. Single point exposures are not considered.
import pyslm.analysis
from shapely.geometry import LineString, Polygon, MultiPolygon
from shapely.ops import cascaded_union
scanIterator = pyslm.analysis.ScanVectorIterator([layer])
After the creation of the ScanVectorIterator, this can be readily expedited to process across all scan vectors across the Layer. The basic process relies on converting each scan vector to a Shapely
polygon objects and then processing them using the geometry tools available.
For this case we use Pythonic notation to compactly operate across each scan vector and collect them. We convert each scan vector to a Shapely LineString, which has a method to then offset or buffer.
# Laser Spot Radius
laserSpotRadius = 0.04
# Iterate across each scan vector and buffer than geometry
lines = [LineString(line).buffer(laserSpotRadius) for line in scanIterator]
After offseting all the lines, this can be easily visualised by conversion to a Shapely MultiPolygon.
# Merged the offset lines into a Shapely Multi-Polygon Collection
multiPoly = MultiPolygon(lines)
The geometrical result is shown below. It is relatively quick to generate and plot individual scan vectors as shown below:
Overlap of hatch vectors represented by geometrically offsetting the individual scan vectors within a Layer.
Each scan vector that is offset is represented by a Shapely.Polygon. It is trivial to perform a boolean operation with the Shapely library. Although, it is recommended to use the more efficient,
albeit still relatively slow, shapely.ops.cascaded_union function to merge multiple geometries together:
# Cascaded union is a more efficient boolean merge for multiple polygon entities
multiPolyMerged = cascaded_union(multiPoly)
The combined result is shown here with a slightly smaller hatch distance to exaggerate the effect and highlight regions where the laser beam may not sufficiently provide exposure to the powder bed:
Regions that have insufficient coverage observed after performing a boolean merge after the scan vectors have been offset
This post shares a relatively simple example exploration using geometrical operations and the iterator class to understand potential issues related to hatch overlaps.
Final Conclusions
Potentially, this check could be extended into 3D using morphological operations. This would provide a more qualitative examination of porosity generation as a result of Furthermore, the combined use
of Neural Networks or reduce order models could provide a representative exposure area to provide the geometrical offset in 2D and provide a prediction for coverage spatially in 3D.
PySLM is a more frequent update to the library with few additions, due to mainly resolving issues fixing several bugs including suggestions and feedback from several users along the way. I wish to
thank them personally for their support along the way, as it does help improve the PySLM library for everyone.
The release primarily includes a new analysis Iterator feature that provides a set of classes for iterating across the layer geometry definitions. This builds upon previous work in the previous
version of the c++ libSLM, however, for reference been implemented in Python.This includes iterating across individual LayerGeometry regions via LayerGeometryIterator and Scan Vectors via
ScanvVectorIterator. Useful for simulation and numerical studies is the ScanIterator class that aims to replicate the spatial position and laser parameters across time given the laser input. This is
particularly useful for numerical simulations for predicting the thermal and thermo-mechanical behaviour during the process. The exposure points generated in ScanIterator class are based on a
timestep, that can be controlled directly via the analysis or exported to a .csv file for use by an external program. The infrastructure behind the iterators was derived for efficient generation and
computation using a tree structure for caching data.
Another nice feature of note is the correct visualisation of scan paths based on their true scanning order across both hatch and contour scan geometry using the new visualise function plotSequential
as described in detail in a previous post. This includes a nicer visualisation of jump vectors that was previous missing.
Another example for generating parametric studies is included to support research on the development of process parameters for new materials on machine systems, in conjunction with DOE tools
available in python.
Once again the support module has been put on hold, whilst investigating and researching a more efficient approach for isolating support regions and generating their assigned support geometry
The release finally marks a small achievement in getting an automatic build system in place used Github Actions for compiling and publishing the python packages to PyPi repository, for both most
versions (3.5-3.9) of Python on both Windows and Mac OS X.
Finally, the complete release log may be found on github at pyslm/CHANGELOG.MD.
Historical Background
An upcoming key feature in PySLM is the Iterators primarily useful for simulation studies, such as predicting thermo-mechanical behavior of scan strategies. Much of this builds upon ideas in former
work that was done during my PhD for investigating the generation of residual stress in selective laser melting. In that study, MSC Marc, a commercial Finite Element analysis package was used to
predict residual stresses generated during the process. The discretised position and laser parameters of the exposure from the laser was controlled by combination of Fortran User Subroutines and
libSLM, the former c++ library.
Prior to running, a configuration file was passed to the simulation, specifying the SLM machine build file used for simulating the scan paths. libSLM parsed the compatible build-file and then based
on the current time and increment would interpolate the position of the exposure point and laser parameters during firing. Beyond this main functionality, there were additional house keeping required
for running the simulation, including passing information between the programs, and also additional tools to efficiently seek at an arbitrarily point in time, the state and position of the laser.
This was necessary for restarting simulations on a HPC and for adaptive time-stepping required to keep numerical stability. For efficient seeking across the layers and each layer geometry structure
was cached within a tree, that could be parsed on demand if necessary. Much of the infrastructure was excessive, although the implementation had to be written in c++ or Fortran to be used by
integrated with the commercial solver.
Although it is difficult to perceive the full benefit of having a Python version of the same functionality, there are some instances and some analysis codes where this could be of benefit for
modelling this and other processes as well.
Implementation of PySLM Iterators
The implementation builds upon the existing design from the original libSLM library. For all the Iterator classes, similar to most of the other pyslm.analysis module’s tools, the list of Layers and
Models with the Laser Parameters should be passed:
# Iterates across layer geometries
layerGeomIter = Iterator(models, layerList)
# Iterates across individual scan vectors - currently only ContourGeometry/HatchGeometry
scanVectorIter = ScanVectorIterator(models, layerList)
# Generates an scan exposure point iterator
scanIter = ScanIterator(models, layerList)
The first stage is building a time cache tree across each LayerGeometry. In practice, the cache tree structure is not necessary if the scan iterator iteratively increments along in time. Having a
cache structure enables non-linear movement of the iterator across the entire build . It also provides a fast random-access lookup to seek to a specific Layer or LayerGeometry for use in simulations
or analyses.
This structure is formed by iteratively measuring the total time taken to scan an individual LayerGeometry group which is stored in a tree node (TimeNode). The cumulative time taken to scan across
each LayerGeometry TimeNode to provide the total scan time across the Layer. The TimeNode can be assigned child and parent nodes using the attributes (TimeNode.parent and TimeNode.children) in order
to navigate across the entire tree. Each TimeNode provides a key-value pair (id, value) to store the reference LayerGeometry or Layer for simplified access.
The Cache Tree is generated and stored in the Base Class, Iterator and is generated in the private method (Iterator._generateCache) and stored in the attribute Iterator.tree.
def _generateCache(self) -> None:
self._tree = TimeNode() for layerId, layer in enumerate(self.layers): # Create the layer layerNode = TimeNode(self._tree, id=layerId, value=layer) self._tree.children.append(layerNode) for layerGeomId, layerGeom in enumerate(layer.geometry): geomNode = TimeNode(layerNode, id=layerGeomId, value=layerGeom) geomNode.time = getLayerGeometryTime(layerGeom, self._models) layerNode.children.append(geomNode) self._cacheValid = True
The Iterator class has many useful facilities, such as build-time estimation, seeking access to the Layer or LayerGeometry at an arbitrary point in time. The class stores additional info such as the
layer dwellTime – this can be re-implemented in a derived class. For implementing the iterator behavior used across all dependent classes it also stores the current time and reference pointers to the
current Layer and LayerGeometry. Essentially the Iterator class can be used to iterate across each LayerGeometry within a build as a foundation to the other class. Each of these Iterator classes
builds upon the magic methods available in Python: __iter__ and __next__ . The __iter__ method simply sets up the object and re-initialises the Iterators attributes. Once the cache tree is generated
internally, it offers no penalty to generate a new iterator . Below is an excerpt taken from the ScanVectorIterator:
def __iter__(self):
self._time = 0.0
self._layerGeomTime = 0.0
self._layerInc = 0
self._layerGeomInc = 0
return self
def __next__(self):
if self._layerScanVecIt < len( self._layerScanVectors):
scanVector = self._layerScanVectors[self._layerScanVecIt]
self._layerScanVecIt += 1
return scanVector
# New layer
if self._layerIt < len(self._layers):
layerVectors = self.getLayerVectors(self._layers[self._layerIt])
self._layerScanVectors = self.reshapeVectors(layerVectors)
self._layerScanVecIt = 0
self._layerIt += 1
return self.next()
raise StopIteration
The Iterator class and ScanVectorIterator class do not require much further attention, as the pointer to the geometry is incremented only. The ScanIterator class, however, is more useful for
simulation and will be discussed further.
Scan Iterator Class
The ScanIterator class is used for incrementally advancing the exposure source across each scan vector. This is particularly important for visualising or simulating the AM process. The time increment
is based on a chosen but adjustable timestep, and the laser parameters across each scan vector (i.e. the effective scan velocity) obtained from the assigned BuildStyle.
The exposure point is linearly interpolated across each scan vector based on the current time within the LayerGeometry depending on the type. For identifying the position, the cumulative distance is
captured and the current timeOffset for the layer geometry is used to estimate the distance covered by the exposure source across the entire LayerGeometry section. For simplicity this assumes no
acceleration terms and uses a constant velocity profile. Based on the timeOffset, the scan vector is obtained and then the final position is interpolated across the scan vector.
laserVelocity = getEffectiveLaserSpeed(buildStyle)
# Find the cumulative distance across the scan vectors in the LayerGeometry (Contour)
delta = np.diff(layerGeom.coords, axis=0)
dist = np.hypot(delta[:,0], delta[:,1])
cumDist = np.cumsum(dist)
cumDist2 = np.insert(cumDist, 0,0)
# If the offsetDist calculated is outside of the cumulative distance then some error has occured
if offsetDist > cumDist2[-1]:
raise Exception('Error offset distance > cumDist {:.3f}, {:.3f}'.format(offsetDist, cumDist2[-1]))
id = 0
# Find the id of the current scan vector given the time offset
for i, vec in enumerate(cumDist2):
if offsetDist < vec:
id = i
# interpolate the position based on offset distance in the scan vector
linearOffset = (offsetDist - cumDist2[id-1]) / dist[id-1]
point = layerGeom.coords[id-1] + delta[id-1] * linearOffset
The above example is specifically for the contour geometry. Note the for loop is not particularly efficient but serves its purpose for identifying the Iterator’s current scan vector.
Iterator Use:
Each iterator can be subsequently called after using the iter method in a variety of pythonic ways:
#Create a scan vector iterator
ScanVectorIterator(models, layerList)
# Create a python iter object from a ScanVectorIterator
scanIter = iter(scanVectorIter)
# Get a single scan vector
firstScanVec = more(scanIter)
# Collect all the remaining scan vectors
scanVectors = np.array([point for point in scanIter])
Current Limitations:
Note the current implementation of the iterators currently only consider ContourGeometry and HatchGeometry and does not include PointGeometry groups. The jump vectors are ignored, which will have a
small but in most situations a negligible effect on the the overall accuracy of the timing used for the iterators.
Another obvious limitation is that this only accounts for single exposure source systems. It is not known to myself, how multiple-exposure systems scan (i.e. are they truly in parallel based on the
laser number) or is there is some built-in machine heuristic which balances the scanning across all laser sources and spatially – e.g. to prevent overheating. This depends on the SLM system such as
if multiple exposure sources are limited by zones or have full areal access to the bed. Anyone’s comments or experiences on this aspect would be sincerely welcomed.
An example showing the basic usage and functions available with the Iterator classes are available in the Github Repo examples/example_laser_iterator.py
Determining overhang regions are crucial for detected potential failure points for unsupported regions that are necessary to generate support structures for 3D Printing. Typically, for triangular
meshes, this is calculated by taking the dot-product between the triangle normal and the vertical build direction and collecting the values across the entire mesh.
v0 = np.array([[0., 0., -1.0]])
v1 = part.geometry.face_normals
theta = np.arccos(np.clip(np.dot(v0, v1.T), -1.0, 1.0))
theta = np.degrees(theta).flatten()
The approach is not particularly complicated. However, depending on some geometries, especially those from topology optimised geometries tend to be have ‘noise’ in the surface triangles, so patches
can appear within areas generally considering to require support.
A close-up of an overhang angle showing ‘noisy’ disconnected regions. This is a result of the surfaces slightly above the overhang angle tolerance despite neighbors being under.
One approach taken is using the surface connectivity information to smooth or average the surface normal or overhang angle in order to reduce these discontinuities. The approach taken gathers the
adjacent triangle faces and uses the information to identify the overhang angle. This option is enabled when passing the useConnectivity=true argument in pyslm.support.getOverhangMesh (will become
available at a later date).
The approach uses the face adjacency information built-into Trimesh’s meshes and collects this in order to map the connectivity between faces. The list of adjacent faces are stored as a Python dict
so that they can mapped on-demand.
def getAdjacentFaces(part: Part):
mesh = part.geometry
import networkx as nx
graph = nx.Graph()
adjacentFaces = {node: list(graph.neighbors(node)) for node in graph.nodes}
return adjacentFaces
Once the facial connectivity is found, each face within the mesh is iterated over. The average overhang angle is calculated accordingly by collecting the faces from the adjacency list into conFaces
and then using this to index the numpy array directly.
theta = np.degrees(theta).flatten()
thetaAvg = theta.copy()
adjacencyList = getAdjacentFaces(part)
for face in adjacencyList.keys():
conFaces = [face] + adjacencyList[face]
thetaAvg[face] = np.mean(theta[conFaces])
The difference is highlighted below.
Overhang without the connectivity option
Overhang regions identified with connectivity option
Ray Projection Investigation
The support generation – described in this post, builds upon ray projection or ‘ray casting’ method to identify intersecting support areas. The idea was looking at how one could identify supports
that are only in contact with the base/build plate – typically acceptable. For reasons later explained, it can be better to work beyond the overhang regions identified based on just the triangular
meshes because they can have jagged edges, even with the connectivity approach. It it also not trivial to interpolate across the surface or use polygon offsetting algorithms.
The idea is simple, rays are uniformly distributed and projected underneath the mesh to identify the intercepts with the mesh to find the distance from the build plate. The polygon of the bounding
box is generated from the mesh and then is discretised with points using the rasterize function for trimesh.path.Path2D. The rays are projecting using Trimesh’s in-built ray projection facility from
the seed points (note: the PyEmbree wrapper library as an auxiliary method does not provide enough accuracy to do this).
# Generate a polygon covering the part's bounding box
offsetPoly = trimesh.load_path(geometry.generatePolygonBoundingBox(mesh.bounds.reshape(2,3)))
# Rasterise the surface of overhang to generate projection points
supportArea = np.array(offsetPoly.rasterize(resolution, offsetPoly.bounds[0, :])).T
coords = np.argwhere(supportArea).astype(np.float32) * resolution
# Project upwards to intersect with the upper surface
# Set the z-coordinates for the ray origin
coords = np.insert(coords, 2, values=-1e5, axis=1)
rays = np.repeat([upVec], coords.shape[0], axis=0)
#Find the first location of any triangles which intersect with the part
hitLoc, index_ray, index_tri = mesh.ray.intersects_location(ray_origins=coords, ray_directions=rays,multiple_hits=False)
print('\t - finished projecting rays')
The first hit locations with the mesh are stored and the heights are then transformed back to a height-map based on the original bounding-box and resolution used.
heightMap = np.ones(supportArea.shape) * -1.0
if len(hitLoc) > 0:
hitLocCpy = hitLoc.copy()
hitLocCpy[:, :2] -= offsetPoly.bounds[0, :]
hitLocCpy[:, :2] /= resolution
hitLocIdx = np.ceil(hitLocCpy[:, :2]).astype(np.int32) # Assign the heights heightMap[hitLocIdx[:, 0], hitLocIdx[:, 1]] = hitLoc[:, 2]
The height or depth map (projection) along Z-Direction from the ray projection method
From the height map, the gradients are calculated using 1st order difference stencil within Numpy’s np.gradient function, along each direction. The magnitude is found and the overhang angle is
calculated based on the resolution.
gradX = np.gradient(heightMapFix, axis=1)
gradY = np.gradient(heightMapFix, axis=0)
mag = np.hypot(gradX,gradY)
angle = np.degrees(np.arctan(mag/resolution))
Once the angle is obtained the projected overhang regions can be found using the Marching Square algorithm built into Scikit Image (skimage.measure.find_contours). The isolevel chosen is based on the
overhang angle and the resolution chosen alongside some carefully selected fudge-factor. A mask is used to remove areas in the background (non-hit areas) based on the background
isolevel = 1.1 * np.tan(np.deg2rad(overhangAngle)) * resolution
contours = skimage.measure.find_contours(mag,isolevel, mask=heightMapFix>1e-3)
The result can be seen below.
The projected overhang (45°) region for a topology optimised bracket overlaid on the height map found using ray projection
The projected overhang region (45°) isolevel overlaid on the calculated overhang angle
The benefits of this approach are not immediately apparent. However, the contour information can be used and smoothed when projecting supports back onto the part. The limitation with the approach is
that the projection’s resolution is determined by the overhang angle captured and obviously is based on the line of sight – so any self-intersecting supports with the part are neglected. The approach
can be further explored using GLSL shaders in order to remove the bottleneck of ray tracing and achieve the same result.
This method is included as a utility function (pyslm.support.generateHeightMap) and both functions will be made later available with the introduction of the support module.
Following a recent suggestion, it was useful to revisit the basic visualisation functionality within PySLM for visualising LayerGeometry in a Layer. The request suggested was related to displaying
jump vectors between vectors and implementing this within the visualisation. Jump vectors visualise, the jump across adjacent scan vectors when the laser or energy source is not firing . Although it
is a trivial matter, good visualisation helps us to understand how scan vectors are processed by the additive manufacturing machine.
The current implementation does visualise the order of scanning but not correctly in its previous form. All HatchGeometry and ContourGeometry is gathered across the layer and processed separately.
The order of scanning within the HatchGeometry is found by collecting across all the scan vectors and simply assigning a range across all scan vectors, using numpy.arange.
# Plot the sequential index of the hatch vector
Obviously, the current approach isolates the hatch and contour scan vectors so even when the contour vectors are scanned first or there is an arbitrary order of scanning across types – especially for
complex scan strategy arrangements, the visualisation is incorrect. Likewise, for displaying the jump vectors, this would not work correctly.
New Approach
Restarting, the approach simply takes existing collection of LayerGeometry and then gathers together all the vectors across both HatchGeometry and ContourGeometry object. However, ContourGeometry
vectors are joined together by coordinates rather than discrete pair of coordinates representing the vectors. Across the ContourGeometry, the vectors are transformed into the equivalent hatch vectors
in order to be consistent when using Matplotlib’s LineCollection objects. This is done by stacking the ContourGeometry’s coordinates by offsetting by a single coordinate position.
scanVectors = []
for geom in layer.geometry:
if isinstance(geom, HatchGeometry):
coords = geom.coords.reshape(-1, 2, 2)
elif isinstance(geom, ContourGeometry):
coords = np.hstack([geom.coords, np.roll(geom.coords, -1, axis=0)]).reshape(-1,2,2)
scanVectors = np.vstack(scanVectors)
These can then be plotted using Matplotlib LineCollection:
lc = matplotlib.collections.LineCollection(scanVectors, cmap=plt.cm.rainbow, linewidths=1.0)
Unfortunately, there is some inefficiency merging all the scan vectors due to copy operations and stacking the array structure, although for the purpose of visualising single layers it is
Depending on the geometry, especially those with many facets in the mesh, the slicing results in contours which contain many edges, that may persist even after polygonal simplification. For
representing the scan order by changing the colour hue, this may result in colour map biased with the contours when attempting to visualise the order.
The range of colour are limited across all the scan vectors due to the large number of edges on the outer border
A method to circumvent this is to use the cumulative distance across each scan vector that is taken across the entire layer. This acts as a way to normalise the scan order irrespective of complexity,
length and distribution of scan vectors across the entire length.
delta = scanVectors[:, 1, :] - scanVectors[:, 0, :]
dist = np.sqrt(delta[:, 0] * delta[:, 0] + delta[:, 1] * delta[:, 1])
cumDist = np.cumsum(dist)
The final step is accounting for the jump vectors i.e. when the galvo mirrors move between scan vectors without the energy source activated. Having a consistent list of scan vectors means this is
straightforward to calculate by replicating the same technique used for translating the ContourGeometry into hatches:
scanVectors = np.vstack(scanVectors)
svTmp = scanVectors.copy().reshape(-1,2)
svTmp = np.roll(svTmp,-1,axis=0)[0:-2]
svTmp = svTmp.reshape(-1,2,2)
lc2 = mc.LineCollection(svTmp, cmap=plt.cm.get_cmap('Greys'), linewidths=0.3, linestyles="--", lw=0.7)
This involves taking every odd pair across all the scan vector lists. These are plotted as line collections with dashes – that are more appealing on the eye. However, this results in ‘invisible’ jump
lines between the scan vectors during Contour scans. Nevertheless, it presents a simple approach to generating more informative visuals for the scan vector paths across the layer. The final result is
shown below.
New sequential visualisation showing sequential scanning of both hatch and contour geometries with jump vectors (dashed lines)
This new function for plotting can be found in pyslm.visualise.plotSequential.
PySLM version 0.3 was released last week to coincide with a large number requests from users of the library. The release is consists of many updates, fixes and examples accumulated across the last 6
months since last summer. Additional work was done to refine the release of the sister library libSLM and resolve some bugs that couldn’t be determined until exporting the machine files and testing
on the machine, with acknowledgement of support from researchers who have got in touch. Many thanks for their assistance on this development journey.
The release notes can be found on github.
The original release was scheduled to include support generation, but this has been postponed for v0.4 to ensure that there was a underlying stable release of PySLM as reference to ensure users can
utilise the library without waiting for the support structure element to stabilise.
A summary of notable features amongst fixes are as follows:
• Added class geometry.utils.ModelValidator with functions to validates the inputs (models, layers) are consistent and coherent prior to exporting to machine build files using libSLM.
• Added an alternative method BaseHatcher.clipContourLines for clipping a list of contour scan vectors. See the previous post about generating custom sinusoidal scan strategy using this method.
• Added method hatching.simplifyBoundaries to simplify boundaries using Sci-kit Image method based on Douglas-Peucker algorithm.
• Added a methodvisualise.visualiseOverhang to visualise overhangs – in preparation for support structure analysis
• Added function argument index to visualise.plot in order to visualise the scan vector parameters (e.g. length, laser parameters, build style id)
Any requests for additional features or other improvements feel free to get in touch.
Building upon the previous post that provided a detailed breakdown for creating custom island scan strategies, this further post documents a method for deploying custom ‘hatch’ infills. This is
particularly desirable capability sought by researchers and has been touched upon very little in the current research. The use of unit-cell infills or in particular fractal filling curves such as the
Hilbert curve have been sought for better controlling the thermal history and melt pool stability of hatch infills.
This has been previously explored in SLS [1]][2] and in SLM on a previous collaborators at the University of Nottingham investigating Fractal scanning strategy [3][4].
Typically, hatch infills are sequences of linear lines that form the the ‘hatch’ pattern. Practically, these are very efficient mechanism for infilling a 2D area by using 1D line elements when
rastering a laser. Clipping of lines within polygons is intuitive. As discussed there are various scan strategies that can be employed to generate variations on this infill – i.e. stripe,
checkerboard/island scan strategy and also modifying the order or sorting of the hatch vectors.
Geometrical scan strategies that adapt the infill based on the underlying geometry, i.e. lattices are acknowledges as ways for drastically improving the performance and the quality of these
characteristic structures. This would be based on some medial-axis approach. This post will not specifically delve into this, rather, demonstrate an approach for custom infills on bulk regions.
Ultimately, drastically changing the behavior of the underlying hatch infill has not really been explored. This post will demonstrate an example that could be employed and explored as part of future
Custom Sinusoidal Approach
Sinusoidal scanning has been employed in welding research [5] and also in direct energy deposition (DED) [6][7][8] in order to improve the stability and quality of the joining or manufacturing
The process of generating this particular scan strategy requires some careful thought to improve the efficiency of the generation, especially given the overall increase in number of points require to
essentially ‘sample’ across the sin curve.
The implementation requires subclassing the Hatcher class, by re-implementing the BaseHatcher.generateHatching and the BaseHatcher.hatch methods.
Unlike, the normal hatch vectors, the sinusoidal pattern has to be treated as a series of connected line segments, without any jumping. This requires using the ContourGeometry representation to
efficiently store the discretised curve. As a result, the Hatcher.hatch method has to be re-implemented to take account of this.
The procedure builds upon previous methods to define customer behavior (see previous post). The first steps are to define a local coordinate system x' and y' for generating the individual sin curve.
A sine curve y' = A \sin(k x') is generated to fill the region bounding box accordingly, given a frequency and amplitude parameter along x'.
The number of points used to discretise the sine curve is determined by \delta x. This needs to be chosen to suit the parameters for the periodicity and amplitude of the sine curve. A reasonable
compromise is require as this will severely impact both the performance of clipping these curves, but also the overall file size of the build file generated.
dx = self._discretisation # num points per mm
numPoints = 2*bboxRadius * dx
x = np.arange(-bboxRadius, bboxRadius, hatchSpacing, dtype=np.float32).reshape(-1, 1)
hatches = x.copy()
Generate the sinusoidal curve along the local coordinate system x' and y'. These will be later tiled and then
transformed across the entire coordinate space.
xDash = np.linspace(-bboxRadius, bboxRadius, int(numPoints))
yDash = self._amplitude * np.sin(2.0*np.pi * self._frequency * xDash)
We replicate and transform the sine curve along adjacent paths and transform along the y-direction
y = np.tile(yDash, [x.shape[0], 1])
y += x
x = np.tile(xDash, [x.shape[0],1]).flatten()
y = y.ravel()
After generating single sine curve, numpy.tile is used to efficiently replicate the curve to fill the entire bounding box region. Each curve is then translated by an increment defined by x, to
represent the effective hatch spacing or hatch distance.
The next important step is to define the sort order for scanning these. This is slightly different, in that the sort order is done per line segment used to discretise the curve. This is subtle, but
very important because this ensures that the curves when clipped by the slice boundary are scanned in the same prescribed sequential order.
The increment of 1\times10^5 is used in order to potentially differentiate each curve later, if required.
# Seperate the z-order index per group
inc = np.arange(0, 10000*(xDash.shape[0]), 10000).astype(np.int64).reshape(-1,1)
zInc = np.tile(inc, [1,hatches.shape[0]]).flatten()
z += zInc
coords = np.hstack([x.reshape(-1, 1),
y.reshape(-1, 1),
z.reshape(-1, 1)])
Following the generation of these sinusoidal curves, a transformation matrix is applied accordingly, before these are clipped in the Hatcher.hatch method.
The next crucial difference, that has been implemented from PySLM version 0.3, is a new clipping method, BaseHatcher.clipContourLines. The following method is different from BaseHatcher.clipLines, in
that clips ContourGeometry separately. This is important for keeping the scan vectors separate and in the correct order, which would be otherwise difficult to achieve. The clipped results are
implicitly separated into contour geometry groups.
hatches = self.generateHatching(paths, self._hatchDistance, layerHatchAngle)
clippedPaths = self.clipContourLines(paths, hatches)
# Merge the lines together
if len(clippedPaths) > 0:
for path in clippedPaths:
clippedLines = np.vstack(path)
clippedLines = clippedLines[:,:2]
contourGeom = ContourGeometry()
contourGeom.coords = clippedLines.reshape(-1, 2)
The next step is to sort the clipped paths into the right order. This is done by using the 1st value of 3^rd index column accordingly sorting using sorted with a lambda function.
Sort the sinusoidal vectors based on the 1st coordinate's sort id (column 3). This only sorts individual paths
rather than the contours internally.
clippedPaths = sorted(clippedPaths, key=lambda x: x[0][2])
Now, the result of the sinusoidal scan strategy can be visualised below.
Sinusoidal Hatch Scan Strategy for Selective Laser Melting – PySLM
This approach currently is very intensive to generate during the clipping operation, due to the number of edges along each clipping operation. Using the previous techniques with the island scan
strategy in a previous post, could be use to amorotise a lot of the cost of clipping.
Example Script
The script is available on github at examples/example_custom_sinusoidal_scanning.py
↑1 Yang, J., Bin, H., Zhang, X., & Liu, Z. (2003). Fractal scanning path generation and control system for selective laser sintering (SLS). International Journal of Machine Tools and Manufacture, 43
(3), 293–300. https://doi.org/10.1016/S0890-6955(02)00212-2
↑2 Ma, L., & Bin, H. (2006). Temperature and stress analysis and simulation in fractal scanning-based laser sintering. The International Journal of Advanced Manufacturing Technology, 34(9–10),
898–903. https://doi.org/10.1007/s00170-006-0665-5
↑3 Catchpole-Smith, S., Aboulkhair, N., Parry, L., Tuck, C., Ashcroft, I. A., & Clare, A. (2017). Fractal scan strategies for selective laser melting of ‘unweldable’ nickel superalloys. Additive
Manufacturing, 15, 113–122. https://doi.org/10.1016/j.addma.2017.02.002
↑4 Sebastian, R., Catchpole-Smith, S., Simonelli, M., Rushworth, A., Chen, H., & Clare, A. (2020). ‘Unit cell’ type scan strategies for powder bed fusion: The Hilbert fractal. Additive Manufacturing,
36(July), 101588. https://doi.org/10.1016/j.addma.2020.101588
Tongtong Liu, Zhongyan Mu, Renzhi Hu, Shengyong Pang,
↑5 Sinusoidal oscillating laser welding of 7075 aluminum alloy: Hydrodynamics, porosity formation and optimization, International Journal of Heat and Mass Transfer, Volume 140, 2019, Pages 346-358,
ISSN 0017-9310, https://doi.org/10.1016/j.ijheatmasstransfer.2019.05.111
↑6 Cao, Y., Zhu, S., Liang, X., & Wang, W. (2011). Overlapping model of beads and curve fitting of bead section for rapid manufacturing by robotic MAG welding process. Robotics and
Computer-Integrated Manufacturing, 27(3), 641–645. https://doi.org/10.1016/j.rcim.2010.11.002
↑7 Zhang, W., Tong, M., & Harrison, N. M. (2020). Scanning strategies effect on temperature, residual stress and deformation by multi-laser beam powder bed fusion manufacturing. Additive
Manufacturing, 36(June), 101507. https://doi.org/10.1016/j.addma.2020.101507
↑8 Ding, D., Pan, Z., Cuiuri, D., & Li, H. (2015). A multi-bead overlapping model for robotic wire and arc additive manufacturing (WAAM). Robotics and Computer-Integrated Manufacturing, 31, 101–110.
The fact that most island scan strategies employed in SLM are nearly always square raised the question whether we could do more. I recently came across this ability to define ‘hexagon’ island regions
advertised in the 2020 release of Autodesk Netfabb. Unfortunately this is a commercial tool and not always available. The practical reasons for implementing a hexagon island scanning strategy are
largely unclear, but this prompted to create an example to illustrate how one would create custom island regions using PySLM. This in future could open some interesting ideas of tuning the scan
strategy spatially across a layer.
Honeycombs or heaxgonal lattices observed in nature are a popular structure used in composites engineering. Could the same be applied in Additive Manufacturing?
The user needs to customise the behaviour they desire by deriving subclasses from:
These classes serve the purpose for defining a ‘regular’ tessellated sub-region containing hatches. Regular regions that share the same shape characteristics for using the infill optimises the
overall clipping performance outlined in the previous post.
Illustration of Checkerboard Island Scan Strategy Implementation
Theoretically, we could build 2D unstructured cells e.g. Voronoi patterns, however, internally hatches for each region will require individual clipping and penalised with a significant performance
hit during the hatching process.
Example of a Voronoi diagram: regions are dibi based on the boundaries between.
The Island subclass region is the most important part to re-define the behavior. If we want to change the island regions to become regular tessellated polygons, the localBoundary method should be
re-defined. In this example, it will generate a hexagon region, but the implementation below should be generic to cover other N-gon primitives:
def localBoundary(self) -> np.ndarray:
# Redefine the local boundary to be the hexagon shape
if HexIsland._boundary is None:
# Simple approach is to use a radius to define the overall island size
#radius = np.sqrt(2*(self._islandWidth*0.5 + self._islandOverlap)**2)
numPoints = 6
radius = self._islandWidth / np.cos(np.pi/numPoints) / 2 + self._islandOverlap
print('island', radius, self._islandWidth)
# Generate polygon island
coords = np.zeros((numPoints+1, 2))
for i in np.arange(0,numPoints):
# Subtracting -0.5 orientates the polygon along its face
angle = (i-0.5)/numPoints*2*np.pi
coords[i] = [np.cos(angle), np.sin(angle)]
# Close the polygon
coords[-1] = coords[0]
# Scale the polygon
coords *= radius
# Assign to the static class attribute
HexIsland._boundary = coords
return HexIsland._boundary
The polygon shape is defined by numPoints, so this can be changed to another polygon if desired. The polygon boundary is defined using a radius for the island region and from this a regular polygon
is constructed on the outside. The polygon points are rotated by adjusting the start angle so there is a vertical edge on the RHS.
The Polygon is constructed around the island size (radius) and is orientated with the RHS edge vertically
This is generated once as a static class attribute, stored in _boundary to remove the overhead when generating the boundary.
The next step is to generate the internal hatch, which in this occasion needs to be clipped with the local boundary. First, the hatch vectors are generated covering the exterior region using the same
radius as the polygon. This ensures that for any rotation transformation of the hatch vectors within the island are fully covered. This is relatively familiar to other code which generates these.
def generateInternalHatch(self, isOdd = True) -> np.ndarray:
Generates a set of hatches orthogonal to the island's coordinate system :math:`(x\\prime, y\\prime)`.
:param isOdd: The chosen orientation of the hatching
:return: (nx3) Set of sorted hatch coordinates
numPoints = 6
radius = self._islandWidth / np.cos(np.pi / numPoints) / 2 + self._islandOverlap
startX = -radius
startY = -radius
endX = radius
endY = radius
# Generate the basic hatch lines to fill the island region
x = np.tile(np.arange(startX, endX, self._hatchDistance).reshape(-1, 1), 2).flatten()
y = np.array([startY, endY])
y = np.resize(y, x.shape)
z = np.arange(0, y.shape[0] / 2, 0.5).astype(np.int64)
coords = np.hstack([x.reshape(-1, 1),
y.reshape(-1, 1),
# Toggle the hatch angle
theta_h = np.deg2rad(90.0) if isOdd else np.deg2rad(0.0)
# Create the 2D rotation matrix with an additional row, column to preserve the hatch order
c, s = np.cos(theta_h), np.sin(theta_h)
R = np.array([(c, -s, 0),
(s, c, 0),
(0, 0, 1.0)])
# Apply the rotation matrix and translate to bounding box centre
coords = np.matmul(R, coords.T).T
The next stage is to clip the hatch vectors with the local boundary. This is achieved using the static class method hatching.BaseHatcher.clipLines. The clipped hatches need to be sorted using the ‘z’
index or 2nd column of the clippedLines.
# Clip the hatch fill to the boundary
boundary = [[self.localBoundary()]]
clippedLines = np.array(hatching.BaseHatcher.clipLines(boundary, coords))
# Sort the hatches
clippedLines = clippedLines[:, :, :3]
id = np.argsort(clippedLines[:, 0, 2])
clippedLines = clippedLines[id, :, :]
# Convert to a flat 2D array of hatches and resort the indices
coordsUp = clippedLines.reshape(-1,3)
coordsUp[:,2] = np.arange(0, coordsUp.shape[0] / 2, 0.5).astype(np.int64)
return coordsUp
After sorting, the ‘z’ indexes need to the be condensed or flattened by re-building the ‘z’ index into sequential order. This is done to ensure when the hatches for islands are merged, we simply
increment the index of the island using the length of the hatch array rather than performing np.max each time. This is later seen in the method hatching.IslandHatcher.hatch
# Generate the hatches for all the islands
idx = 0
for island in sortedIslands:
# Generate the hatches for each island subregion
coords = island.hatch()
# Note for sorting later the order of the hatch vector is updated based on the sortedIsland
coords[:, 2] += idx
idx += coords.shape[0] / 2
clippedCoords = np.vstack(clippedCoords)
unclippedCoords = np.vstack(unclippedCoords).reshape(-1,2,3)
The final stage, is to re-implement hatching.IslandHatcher as a subclass. In this class, at a minimum, the generateIsland method needs to be redefined to correctly positioned the islands so that they
tessellate correctly.
def generateIslands(self, paths, hatchAngle: float = 90.0):
Generate a series of tessellating Hex Islands to fill the region. For now this requires re-implementing because
the boundaries of the island may be different shapes and require a specific placement in order to correctly
tessellate within a region.
# Hatch angle
theta_h = np.radians(hatchAngle) # 'rad'
# Get the bounding box of the boundary
bbox = self.boundaryBoundingBox(paths)
print('bounding box bbox', bbox)
# Expand the bounding box
bboxCentre = np.mean(bbox.reshape(2, 2), axis=0)
# Calculates the diagonal length for which is the longest
diagonal = bbox[2:] - bboxCentre
bboxRadius = np.sqrt(diagonal.dot(diagonal))
# Number of sides of the polygon island
numPoints = 6
# Construct a square which wraps the radius
numIslandsX = int(2 * bboxRadius / self._islandWidth) + 1
numIslandsY = int(2 * bboxRadius / ((self._islandWidth + self._islandOverlap) * np.sin(2*np.pi/numPoints)) )+ 1
The key difference here is defining the number of islands in the y-direction to account for the tessellation of the polygons. This is a simple geometry problem. The y-offset for the islands is simply
the vertical component of the 2 x island radius at the angular increment to form the polygon.
Example of tesselation of hexagon islands
The HexIsland are generated with the offsets and appended to the list. These are then treat internally by the parent class IslandHatcher.
for i in np.arange(0, numIslandsX):
for j in np.arange(0, numIslandsY):
# gGenerate the island position
startX = -bboxRadius + i * self._islandWidth + np.mod(j, 2) * self._islandWidth / 2
startY = -bboxRadius + j * (self._islandWidth) * np.sin(2*np.pi/numPoints)
pos = np.array([(startX, startY)])
# Apply the rotation matrix and translate to bounding box centre
pos = np.matmul(R, pos.T)
pos = pos.T + bboxCentre
# Generate a HexIsland and append to the island
island = HexIsland(origin=pos, orientation=theta_h,
islandWidth=self._islandWidth, islandOverlap=self._islandOverlap,
island.posId = (i, j)
island.id = id
id += 1
return islands
The island tessellation generated is shown below, with the an offset between islands applied by modifying the radius.
Hexagon Island Boundaries generated across the entire region. The boundaries of the layer are shown, which are used for the intersection test.
The fully clipped scan strategy is shown below with the scanning ordered in the Y-direction.
Hexagonal Island Scan Strategy: Consists of 5 mm Island (radius) with an offset at the boundaries of 0.1 mm.
This post illustrates how one can effectively decompose a layer region into a series of repeatable ‘island’ units which can be processed in an efficient manner, by only clipping hatches at boundary
regions. This potentially has the ability to define spatially aware island regions; for example this could be redefining island sizes or parameters towards the boundary of a part. It could be used to
alter the scan strategies within the region too, with the effect of changing the thermal behavior.
The full excerpt of the example can be found on github at examples/example_custom_island_hatcher.py.
A curious feature was seeing the possibility to both generate and visualise the exposure points generated in along scan vectors. This also includes generating a pseudo ‘heatmap’ or exposure map of
the energy deposited by the laser scanning across the surface. Variations of this idea, I have seen across a variety of commercial software – in particular Renishaw’s QuantAM, which is perhaps the
source of inspiration for coming up with an approach.
Usually, there is not much indication as the relative exposure of energy from the laser into a layer based both the chosen laser parameters applied to the scan vectors of a specific LayerGeometry
group and also the overall spatial distribution of scan vectors in a region. For instance, overlaps between scan vectors will in theory deposit more energy across time into a localised region .
Ideally, the exposure of energy from the laser is very uniform across the regions.
It is debatable whether there is particular usefulness of visualising the exposure map of the laser – but in practice can be used to differentiate laser scan parameters used on models in parametric
studies. With further thought, a cheap numerical simulation could potentially capture the transient heat distribution generated by the sequential deposition of these exposure points.
A Light Background on Laser Exposure:
Hopefully someone can further comment on of this and perhaps correct me…
SLM systems are built using fiber laser systems offering a very fine, high quality controlled laser beam. Two types of laser operation exist: continuous wave (CW) and pulsed.
In practice, with pulsed lasers used in AM platforms, the scan vector is decomposed across a number of exposure points at a set ‘pointDistance’ from each other along the scan vector. Each exposure
irradiates energy into the powder-bed surface at a fixed energy (inferred from the average laserPower) for a set time period referred as a ‘pointExposureTime‘. After exposure, the galvo-mirrors
instantaneously move to direct the beam to the next exposure point. The relative speed this occurs optically gives the illusion that the laser is continuously moving. Continuous wave, as the name
describes, emits continuous irradiation whilst the beam traverses the surface. The laser speed parameter is used here.
Different systems use either pulsed, continuous, or even both modes of operation, offering different benefits. This in principle is the reason for BuildStyle class containing the following parameters
to cover both situations:
• Point Exposure Time
• Point Distance
• Laser Speed
• Laser Power – used across both modes
Ultimately, this does not matter, but the proposed method obviously requires the user to set at minimum the pointDistance parameter in order for the following methodology to work.
Proposed Methodology
Taking an existing Layer containing LayerGeometry, the exposure points are generated individually for each scan vector. Given the potential number of scan vectors within a layer, it would be
beneficial to perform this efficiently.
Example part that is hatched with a series of overlapping 5mm island. The scan order is indicated by the blue line.
The basic process to do this is based on the following. The scan vector \textbf{v} has a particular length. Based on the point exposure distance, we can equally divide along this a number of points.
Using both v_o and its direction ,we can then project a given number of exposure points along this line.
Definition of a hath vector and decomposing each scan vector into a series of exposure points.
An attempt to do this more efficiently and exploit vectorisation built into numpy was achieved by the following procedure. The following properties of each scan vector (\textbf{v}) is obtained:
• start point (v_o)
• normalised scan vector direction (\hat{\textbf{v}})
• scan vector length \lVert \textbf{v} \rVert
For each LayerGeometry object, these are obtained across each scan vector as follows assuming that it is a HatchGeometry object:
# Calculate the length of the hatch vector and the direction
coords = layerGeom.coords.reshape(-1, 2, 2)
delta = np.diff(coords, axis=1).reshape(-1, 2)
lineDist = np.hypot(delta[:, 0], delta[:, 1]).reshape(-1, 1)
# Take the first coordinate
v0 = coords[:, 1, :].reshape(-1, 2)
# Normalise each scan vector direction
vhat = -1.0 * delta / lineDist
The number of point exposure across each vector is calculated by simple division. Unfortunately there is rounding, so the exposure may be missing at the end of the scan vector.
# Calculate the number of exposure points across the hatch vector based on its length
numPoints = np.ceil(lineDist / pointDistance).astype(np.int)
The total number of exposure points is calculated and is used to pre-populate some arrays, where the terms are used to extrapolate the exposure point distance (Line 16) are generated. Line 10 is key
here and idxArray stores a range to account for each exposure point along the scan vector
Unfortunately, there isn’t a convenient way vectorise this so it is performed across each scan vector. Finally, the exposure points are generated by extrapolating them from v_o along \hat{\textbf{v}}
# Pre-populate some arrays to extrapolate the exposure points from
totalPoints = int(np.sum(numPoints))
idxArray = np.zeros([totalPoints, 1])
pntsArray = np.zeros([totalPoints, 2])
dirArray = np.zeros([totalPoints, 2])
idx = 0
for i in range(len(numPoints)):
j = int(numPoints[i])
idxArray[idx:idx + j, 0] = np.arange(0, j)
pntsArray[idx:idx + j] = p0[i]
dirArray[idx:idx + j] = vhat[i]
idx += j
# Calculate the hatch exposure points
hatchExposurePoints = pntsArray + pointDistance * idxArray * dirArray
This generates a set of exposure points, which can be seen in the figure below:
Exposure points plotted along each hatch vector are shown. Note the overlapping area between adjacent islands.
Once the exposure points are generated, the deposited energy (laserPower x pointExposureTime) is added to each point and stored as a third column for later use.
Exposure Map Generation
Generating the exposure map is trivial. For a selected resolution chosen by the user, the bitmap slice is used currently to get the image to process with. The exposure coordinates are simply mapped,
given an offset and resolution onto the image. The deposited energy is then summed accordingly across each pixel. The energy per area (latex]Q[/latex]), is calculated by diving by the aerial coverage
of a single pixel.
Exposure / Heat Map showing the deposited energy applied to the position and laser parameters used. Notice the overlap region between islands show a higher energy deposition.
The resolution is arbitrarily selected. The above figure is generated with a resolution of 0.2 mm. It can be seen in this example that the there is a greater energy deposition across the the overlap
regions between the islands. There are some ‘aliasing’ effects due to discretion onto a finer grid which is dependent on the resolution chosen.
The above exposure map generation might have little scientific utility by itself. Rather it can offer a method to potentially visualise the energy deposition across a layer.
However, many thermo-mechanical AM build simulation that incorporate thermal effects in addition to the inherent strain approach could potentially incorporate a reference exposure map across the
layer rather than assuming a single aerial heat-flux applied to the layer. This could actually be carried out across a ‘packet’ or number of layers to better correspond with the geometry and scan
strategies employed.
Documented Example
The code for this can be found in examples/example_heatmap.py on the github repository. | {"url":"https://lukeparry.uk/category/blog/page/2/","timestamp":"2024-11-14T16:43:03Z","content_type":"text/html","content_length":"157971","record_id":"<urn:uuid:8f0b4401-8378-497d-9ad5-1829d629d072>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00425.warc.gz"} |
Light Spanners for High Dimensional Norms via Stochastic Decompositions
Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where
except for the work of Har–Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with O~(n1+1/t2) edges, little is known. In this paper we study
several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of ℓ[p] with 1 < p≤ 2. Second, our construction yields a spanner which is both sparse and
also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of ℓ[p] for 1 < p≤ 2 has an O(t)-spanner with n1+O~(1/tp)
edges and lightness nO~(1/tp). In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric
spaces have an O(t)-spanner with lightness O(n^1^/^t). We exhibit the following tradeoff: metrics with decomposability parameter ν= ν(t) admit an O(t)-spanner with lightness O~ (ν^1^/^t). For
example, metrics with doubling constant λ, graphs of genus g, and graphs of treewidth k, all have spanners with stretch O(t) and lightness O~ (λ^1^/^t) , O~ (g^1^/^t) , O~ (k^1^/^t) respectively.
While these families do admit a (1 + ϵ)-spanner, its lightness depend exponentially on the dimension (resp. log g, k). Our construction alleviates this exponential dependency, at the cost of
incurring larger stretch.
Bibliographical note
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
• Doubling dimension
• Genus graphs
• High dimensional euclidean space
• Spanners
• Stochastic decompositions
• Treewidth
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74 km to miles
Heading 1: Understanding Kilometers and Miles
Kilometers and miles are two of the most commonly used units of measurement for distance around the world. Understanding the differences between these two measurements is important, especially for
those who travel frequently or have international connections.
Kilometers, abbreviated as km, are used in most countries except for the United States, where miles are the standard unit of distance. One kilometer is equivalent to 0.6214 miles, which means that a
kilometer is shorter than a mile. This is why you might notice that the digital speedometers in some cars display both kilometers per hour (km/h) and miles per hour (mph).
Heading 2: What Are Kilometers and Miles?
Kilometers and miles are two common units used to measure distances. Kilometers, typically abbreviated as “km”, are used in most countries around the world, except for a few countries like the United
States and the United Kingdom. On the other hand, miles, abbreviated as “mi”, are primarily used in the United States and the United Kingdom.
Kilometers and miles can be thought of as different ways to measure the same thing: distance. However, they have different values. One kilometer is equal to approximately 0.62 miles. This means that
if you were to convert kilometers to miles, you would multiply the number of kilometers by 0.62. Similarly, if you wanted to convert miles to kilometers, you would multiply the number of miles by
1.61. Understanding the relationship between kilometers and miles is essential for various purposes, such as traveling, navigation, and understanding distances on maps.
Heading 2: The Conversion Formula for Kilometers to Miles
Kilometers and miles are two different units of measurement used to determine distance. While kilometers are predominantly used in most countries around the world, miles are the preferred unit of
measurement in the United States and a few other countries. In order to convert kilometers to miles, a simple formula can be used.
The conversion formula for kilometers to miles is quite straightforward. All you need to do is multiply the number of kilometers by a conversion factor of 0.62137119. For example, if you have 10
kilometers that you want to convert to miles, you would simply multiply 10 by 0.62137119. The result would be 6.2137119 miles. It’s as simple as that! By using this conversion formula, you can easily
determine the equivalent distance in miles for any given number of kilometers.
Heading 2: Step-by-Step Guide to Convert Kilometers to Miles
To convert kilometers to miles, you will need to apply a simple conversion formula. The formula involves multiplying the number of kilometers by a conversion factor of 0.62137119. This conversion
factor represents the number of miles in one kilometer.
Let’s say you want to convert 10 kilometers to miles. All you need to do is multiply 10 by 0.62137119. The result will be approximately 6.2137119 miles. This means that 10 kilometers is equivalent to
around 6.21 miles.
Keep in mind that this conversion formula is based on the International System of Units (SI). It is widely used around the world, although some countries still use miles as their primary unit of
distance measurement. Whether you’re traveling or working on a math problem, knowing how to convert kilometers to miles can come in handy.
Heading 2: Common Uses of Kilometers and Miles in Everyday Life
Common Uses of Kilometers and Miles in Everyday Life
Kilometers and miles are widely used units of measurement in our everyday lives. Both measurements are commonly used to determine distances, whether it’s for planning a road trip or calculating the
length of a running route. For example, when you’re planning a vacation, knowing the distance in kilometers or miles can help you estimate the travel time and fuel costs. Similarly, if you’re a
fitness enthusiast, understanding the distance you’ve covered in kilometers or miles can help track your progress and set goals for your next workout.
Furthermore, kilometers and miles are used in transportation systems worldwide. In countries that use the metric system, such as most of Europe, distances on road signs and speed limits are typically
measured in kilometers. On the other hand, in countries like the United States and the United Kingdom that primarily use the imperial system, distances are more commonly measured in miles.
Understanding both measurements is essential when traveling internationally or navigating unfamiliar roads. The ability to convert between kilometers and miles allows us to communicate and navigate
effectively, ensuring a smooth and enjoyable journey. | {"url":"https://convertertoolz.com/km-to-miles/74-km-to-miles/","timestamp":"2024-11-15T04:06:44Z","content_type":"text/html","content_length":"41373","record_id":"<urn:uuid:fce76aa1-2579-4465-9972-460b93546945>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00034.warc.gz"} |
Comparison of different machine learning algorithms to estimate liquid level for bioreactor management
1. Introduction
Anaerobic digestion (AD) is a promising technology that couples wastewater treatment and renewable energy production [
]. Microbial reactions are the primary player of AD, and thus, maintaining the stability of the process parameters is essential for its function. For example, organic loading rate (OLR) is one of the
fundamental factors determining AD’s biogas productivity and economic viability. Another example is hydraulic retention time (HRT), closely related to the microbe-pollutant contact time. Both
parameters are dependent on bioreactor volume; therefore, the water level of the digester is a vital operation parameter [
Recent technology developments allow continuous monitoring of many key parameters during AD operation [
]. However, some parameters, such as the chemical oxygen demand of the digestate, are yet to be monitored in real-time due to instrumental limitations [
]. It is also challenging to monitor the digester water level continuously with high accuracy. Radar or ultrasonic level sensors can accurately measure water levels in an open system, but they can be
easily disturbed by the generation of bubbles and scum in anaerobic digesters [
]. Due to such limitations on direct measurement, using soft sensors can be an alternative approach to maximize estimation accuracy [
]. For example, the liquid level in a black box can be estimated if both the pressure and the density of the liquid column are known. However, the liquor in the digester (i.e., digestate) contains
high concentrations of suspended solids, making it challenging to keep the digestate homogeneous in the digester. Therefore, using pressure sensors with homogeneity assumption of the digestate may
lead to erroneous estimation of the liquid level in AD.
Our previous study suggested a method to overcome this limitation by predicting the unequal density profiles using multiple pressure meters [
]. A pilot-scale digester (0.33m
working volume, 1.4 m height, 1.2 m liquid level) was operated to collect data for the water level prediction model. By collecting pressure data from seven sensors (six in the liquor and one in the
headspace), density profiles of the liquid columns were derived, and the top layer’s density was calculated through prediction models. As a result, a cubic model outperformed other polynomial models
as well as the traditional, two-sensors approach. Although the digestate level can be predicted with high accuracy using this method, however, the requirement for seven pressure sensors may cause
investment and maintenance burdens. Because only polynomial models were tested in the previous study, there is likely room for improvement if we test other modeling approaches using the same or even
less number of sensors.
Machine learning (ML) is a computational method used for prediction or classification using various algorithms [
]. Recently, ML approaches are gaining popularity to generate new models for precise prediction in many research areas. For example, ML can be used to study nanomaterials for energy and environmental
applications [
]. Studies were conducted to develop effective photocatalysts to treat toxic pollutants such as thymol blue [
]. ML has been also applied in water and wastewater engineering research. Choi et al. [
] estimated the water level of Upo Wetland with long-term data with various parameters like temperature, precipitation, wind speed, and water level in the nearby area. Comparison among four models –
artificial neural network (ANN), decision tree (DT), random forest (RF), and support vector machine (SVM) – were conducted, and RF was selected as the best model. Granata et al. [
] compared SVM and regression tree to estimate water quality using parameters like chemical oxygen demand, total suspended solids, biochemical oxygen demand, and total dissolved solids. The methane
composition of biogas in AD was also estimated through ANN with genetic algorithm optimization [
]. Talebkeikhah et al. [
] tried to estimate the permeability of two carbonate reservoirs with SVM, DT, RF, multi-layer perceptron, and radial basis function (RBF) neural network and showed that DT and SVM were the best
models for permeability prediction. Suitability tests of soils for airfield application with the fuzzy knowledge-based system were conducted and proposed as an accurate tool [
]. Pedro et al. attempted to optimize the work conditions like time and the number of workers for optimal work of floating caissons [
]. These studies suggest that ML can be a helpful tool to generate prediction models in various areas.
This study aimed to improve the liquid level prediction approach in AD equipped with multiple pressure sensors and optimize the number of sensors by attempting various ML algorithms. Linear
regression, ANN, random forest (RF), and SVM with RBF kernel were compared to each other and with the cubic model from the previous study. The pros and cons of using different ML algorithms were
discussed, and the significance levels of the different pressure sensors were compared. This study offers information to decide and optimize the method to estimate the water level of an anaerobic
bioreactor in real-time.
2. Materials and Methods
2.1. Summary of the Research Procedures
This research aimed to predict the liquid level of an anaerobic digester equipped with multiple pressure sensors using ML (
Fig. 1
). The experimental system (i.e., the bioreactor and instrumentation), data collection, and the development of polynomial models in the previous study [
] are summarized in section 2.2. Four widely-applied ML methods were utilized to predict the liquid level in this study as detailed in section 2.3 (
Fig. 1
). Models derived from different algorithms were evaluated by comparing the error values in the forms of root mean square error (RMSE), mean absolute percentage error (APE), and maximum APE
(described in section 2.4). In addition, corrected Akaike Information Criterion (AICc) values were used to optimize the number of parameters. Finally, conclusions were made on the most desirable
algorithm and parameters.
2.2. System Description and Data Collection
The anaerobic bioreactor system and the experimental data were published previously [
]. Briefly, a pilot-scale (0.33 m
working volume; 1.4 m height; 1.2 m liquid level) bioreactor was equipped with seven pressure sensors at approximately 0.1, 0.3, 0.4, 0.6, 0.7, 0.9, and 1.25 m from the bottom (
Fig. 1
). The bioreactor was operated for 175 days; the liquid level was maintained stable at 1.2 m for most of the time, while a short-term level variation (0.8–1.2 m) was applied at day 99. The bioreactor
was placed in a temperature-controlled room (36°C). The anaerobic bioreactor showed a stable digestion performance in terms of pH, biogas production, and volatile fatty acid accumulation (see Rhee et
al. [
] for details). All the sensor-derived data (i.e., pressure, temperature, and biogas) were recorded at an interval of one min or less.
The pressure readings (
, …,
) were obtained from the pressure meters (
, …,
), of which
refer to the top sensor in the headspace. The apparent density of the liquid layers (
, …,
), except for the top layer (
; between
), was determined by gravimetric relationship (
Eq. (1)
). Once the top layer density (
) is estimated, the total liquid level (
) can be calculated by
Eq. (2)
is the density of the liquid column,
the pressure,
the gravitational force, and
the height. In
Eq. 2
(headspace pressure) is subtracted from
as a reference value because the anaerobic bioreactor usually keeps positive pressure at the headspace. The previous study compared polynomial models (linear to quintic) between
, and concluded that the top layer (
) is well depicted by a cubic model [
2.3. Modeling
In this study, four new approaches implementing ML were additionally tested if they can improve the accuracy of the model: multiple linear regression, RF, ANN, and support SVM with RBF kernel. The
following sub-sections (2.1.1 to 2.1.4) summarize the general features of the four methods, including their potential limitations. Supervised learning was conducted which train model by offering an
example answer. All data were standardized for smooth model fitting. As the model parameters, seven pressure and one temperature readings were used for direct comparison with the previous study. The
‘train’ function in the R program’s caret package was used to search for proper hyperparameters rapidly.
2.3.1. Multiple linear regression (MLR)
Linear regression algorithm is to fit model with linear function between independent and dependent variables. Linear regression is the most common approach for modeling numeric data and can be
adapted to almost all types of data [
]. With accurate modeling, this method can give information about contribution in form of coefficient. However, linear regression requires assuming that the independent and dependent variables have a
strong linear relationship. Additionally, this method requires a large amount of dataset for accurate modeling [
]. The accuracy of the model was assessed using a gradient descent algorithm. The gradient descent optimization algorithm is one of the most popular algorithms to perform optimization about a model.
RMSE was used as the cost function while optimizing this model.
2.3.2. Artificial neural network (ANN)
ANN is an algorithm that mimics the structure of human neurons. ANN is widely used for non-linear function estimation, data sorting, pattern detection, optimization, clustering, and simulation [
]. Among different types of ANN, a feed-forward neural network with a single hidden layer of five nodes was used in this study.
Three elements adjust the model’s structure: activation function, network topology, and training algorithm [
]. An activation function is an element that transforms the input signal into the output signal. A network topology includes factors like the number of nodes, the direction of information travel, and
the depth of the hidden layer [
]. For example, too many nodes can be led to overfitting [
]. A proper combination of these factors can improve the accuracy of the model. The model estimates values through interaction between interconnected neurons. The algorithm aims to find out proper
weights used to calculate variables. Although ANN is known as one of the most accurate modeling approaches, it usually takes a longer time than other algorithms, and the model structure is
challenging to recognize.
2.3.3. Random forest (RF)
RF is a model that consists of multiple DTs. DT is an algorithm for classification and prediction, that is easy to be used because of its simplicity. A DT model consists of many logical decisions
like a flow chart [
]. Starting from the root node, the data are split into various nodes to reach the final leaf (or terminal) nodes. The DT tries to make an optimal model by searching for the model that maximizes the
purity of the terminal modes. High purity means that it can classify or predict values with high accuracy.
The advantage of using DT lies in its speed, as DT can select useful features automatically. DT can describe some datasets more accurately than linear regression [
]. However, DT may lead to overfitting with complex datasets. To overcome this problem, the RF has been developed. The emergence of ensemble trees can improve the accuracy of the model. RF calculates
value by combining all decision tree’s values. With this property, RF can handle massive datasets and is resistant to outliers during training. After combining all values, a cost function (
Eq. (3)
) is generated to evaluate the model.
RF uses the bootstrap aggregating (or bagging) method to avoid overfitting and correlation of the different trees. Correlation among trees disrupts accurate modeling. Bagging is an algorithm that
resamples data randomly from the dataset to train the data without deletion. The data not selected for training a particular tree is called ‘out of bag’. The subsequent data subset is used to
evaluate the error and correlation of the model. With these steps, RF can predict a value with high accuracy and have strength when dealing with large and noisy data [
]. However, RF cannot predict a value that the model has not experienced. For this reason, RF requires extensive learning experience.
2.3.4. Support vector machine (SVM)
SVM is a computational learning algorithm based on the statistical learning theory [
]. This algorithm predicts values by finding a hyperplane that divides data with a maximum margin by converting original data into high-dimensional data. SVM tries to minimize the total cost rather
than finding the accurate model for the model’s stability. For this reason, SVM is strong against overfitting [
Kernel function can be used to modify the dataset’s dimension for accurate analysis. There are various types of kernel functions. Finding a proper kernel type is an essential step for accurate model
development. As a result, in this study, the RBF kernel was selected after comparing the linear, the polynomial, and the RBF kernels. With a proper kernel function, SVM can conduct modeling on
complex data with high accuracy. However, SVM is not easy to deal with a large amount of data, and its model is difficult to understand [
]. Therefore, the step to search the proper kernel is required for accurate modeling.
2.4. Model Evaluation and Variable Significance Test
The pressure data from seven pressure sensors (P[0], …, P[6]; from top to bottom) and the temperature data (T) were considered to be used in modeling to compare directly with the result of the
previous study. Out of the seven pressure readings, three combinations were specifically tested for the models: two (bottom and the headspace readings; P[6] and P[0]), three (bottom, top, and the
headspace readings; P[6], P[1,] and P[0]), and seven (all seven readings). The data was preprocessed through standardization for accurate modeling. Standardization is preprocessing method which
rescales data to have one as standard deviation and zero as mean. After standardization, 747 data points were derived; 521 data points (70% of data) were used to train models and 225 data points
(remaining 30% of data) were used to test the models. This study conducted model evaluation by comparing RMSE, mean APE, and maximum APE.
The importance of each variable was estimated for each type of algorithm (i.e., MLR, RF, ANN, and SVM) through a specific method. For MLR, the absolute value of the
-statistic for each variable was used. The RF used an out-of-bag score for variable importance calculation. Gevrey’s weights method, which combines the absolute value of the weights, was used to
estimate variable importance for ANN [
]. Finally, locally estimated scatterplot smoothing (LOESS) R2 was used for SVM models for variable importance. All variable importance was estimated using ‘varImp’ function of caret package in R.
To evaluate the proper number of parameters for the models, AICc was calculated. Akaike information criterion (AIC) is a criterion that can be used to decide on where to stop to input more the
independent variables [
]. However, AIC has a problem when the number of data is not enough. To solve this problem, AICc was proposed by Hurvich and Tsai [
$AICc=1+ln (SSEn)+2(p+1)n-p-2$
where SSE is the sum of squares error, n is the observed data’s number, and p is the number of parameters. A model with a smaller AICc value was assumed to be more accurate because AICC is in
proportion to the error.
3. Results and Discussion
3.1. Model Performance
The overall performance of the four models using different datasets (i.e., two (
Table 1
), three (
Table S1
), or seven (
Table 2
) pressure readings) were compared. Overall, ANN and RF showed lower RMSE than MLR and SVM. The mean APE was the lowest for RF, while the maximum APE was the lowest for ANN. The higher the number of
pressure readings used in the models, the lower the RMSE and APE values were derived in general.
The MLR model did not perform well compared to ANN and RF (
Table 1
). This result suggests that the liquid level estimation of an anaerobic digester involves complex interactions that are not able to be explained best by a simple linear combination of the variables.
Within the model, all RMSE, mean APE, and maximum APE decreased with the increasing number of variables (i.e., pressure readings); these results also support that more model complexity can lead to
more accurate estimation.
The SVM model showed the poorest performance in terms of the mean APE (
Table 1
). In all cases, the performance parameter of APE was among the poor half out of the four models (i.e., the first or second highest RMSE or APE). Therefore, it could be concluded that SVM was not the
best approach for liquid level prediction in our configuration. Unlike the MLR model, using more pressure data did not improve this model.
The ANN approach used in this study employed a feed-forward neural network with a single hidden layer (five nodes included) and backpropagation [
]. Due to the randomness of the model build-up, ten ANN models were created and the one with the least RMSE was selected. The ANN showed superior performance than other models (
Table 1
). The RMSE and the maximum APE of ANN were the lowest among the models tested. A lower maximum APE implies that this model had a more robust resistance to outliers than other algorithms.
On the other hand, the mean APE of the ANN model was about half of those of the MLR and SVM models, but double of the RF. Like the MLR, both the RMSE and mean APE for the ANN also decreased as the
number of pressure readings increased. Although the higher number of input data (i.e., pressure readings) generally increased the accuracy of this model, no clear trend was observed for maximum APE.
In the case of RF, 500 trees were grown with a bagging number of five. RF showed the lowest mean APE and the second-lowest RMSE among the models (
Table 1
). However, the maximum APE of RF was comparable to SVM, especially when a higher number of pressure readings were used. It could be concluded that the best RF model was achieved using two pressure
readings, and a higher number of variables did not increase its accuracy. This trend was similar to SVM.
Overall, the ML-based models proposed in this study outperformed the polynomial model derived in the previous study [
]. Both the RMSE and the mean APE were the highest for the cubic model, and its maximum APE was comparable to SVM and RF (
Table 2
Fig. 2
shows the observed and estimated water levels using different models. The margin between a dot and a red line means the error rate. The estimations made by the cubic, MLR, and SVM models clearly
showed a more inaccurate representation of the liquid level. In addition, the cubic and SVM models likely had structural underestimation of the liquid level at 1.2 m. To summarize, the ANN and RF
models showed the most successful estimation of the liquid models in our configuration.
Interestingly, RF and SVM showed higher maximum error according to the increase of the number of variables (
Table 1
), while MLR and ANN showed better performance with more variables. This is probably due to the characteristics of the models. Thus, the initial selection of a model could be based on the number of
variables available.
3.2. Importance of Variables
Variable contribution analysis is essential for accurate analysis and evaluation of layers. To evaluate the use of different pressure readings, the variable contribution was analyzed (
Fig. 3
). Commonly, the temperature (
) shows low or no importance to the modeling results. This is reasonable because the temperature within a stable range can hardly affect the volume or density of a liquid. Among the pressure
variables, the headspace reading (
) and the bottom reading (
) showed high variable importance in most cases. The topwater column reading (
) was also significant in some cases, especially when using three pressure variables (
Fig. 3b
). One of the reasons contributed to this observation is that this pressure meter was at the headspace (in addition to
) in some data points with lower liquid levels. This reason may lead to another accuracy problem in the reactor. The other water column readings (
) generally had low importance on the model (
Fig. 3
One exception to this trend was SVM; this model showed relatively equal contribution from the different pressure variables. It is suspected that the SVM was affected by multicollinearity. To confirm
the correlation, variance inflation factors (VIF) among parameters were calculated (
Table S2
). The VIF values of
were over 10,000, and only P
and P
showed lower VIF values (< 100). This phenomenon can be induced by a dataset that contains highly correlated variables [
] and negatively affects the results. It was suggested that the effects of multicollinearity can be removed by removing redundant data or introducing prior information [
], which can be the case of SVM models with lower pressure variables in this study. For this reason, it is not recommended to utilize SVM with RBF kernel for liquid level prediction with multiple
pressure sensors.
To determine the importance of each pressure layer in the RF and ANN models, the RMSE and APE values for models without one pressure layer were compared (
Table 3
). For both algorithms, models lacking
showed significantly lower performance. It implies that the pressure meter at the headspace is essential to estimate the water level. This is reasonable because the headspace pressure is linked to
all pressure values within the bioreactor. Removing some layers, such as
, lowered the errors, indicating that having more parameters does not necessarily improve the model. This is probably because
experienced both water (liquid) and air (headspace) phases depending on the liquid level. Therefore, avoiding a pressure sensor at an amphibious level could be suggested. Overall, the absence of a
parameter with a higher contribution was not critical to the model’s performance, suggesting not as many as seven pressure parameters are required for accurate modeling.
3.3. AICc Test
The AICc values were obtained to assess further the importance of variables (
Fig. 4(a), (b)
). For MLR and SVM, models with three to six parameters showed similar results. For RF, models with three parameters (from top or bottom) had the lowest AICc results. The ANN model showed the least
AICc values compared to the other algorithms; the four-parameter models had the lowest, negative AICc values. The optimal combination of four parameters was searched to minimize the AICc value of the
ANN model (
Fig. 4(c)
). The optimal combination for the ANN model was determined as
, and
: one headspace meter and three liquid-facing meters excluding the top and the bottom ones (
). The same combination resulted in a significant decrease of AICc for the RF model, but comparable or even higher AICc’s for the other two models. These results imply that selecting the parameters
is required to optimize the model output for liquid level estimation using the current method. To summarize, the pressure data were essential to building accurate models to estimate the liquid level,
while the temperature showed little effect. Among the different levels, the pressure meter located in the headspace is crucial, and the number of sensors in the liquid can be optimized to increase
the model accuracy.
4. Conclusions
In this study, a comparison among four algorithms and various variables was conducted to increase the accuracy of the real-time liquid level estimation method. Both the ANN and RF models showed
plausible accuracy, while the MLR and SVM models had higher errors than ANN and RF. ANN and MLR increased their accuracy with more pressure variables. In contrast, RF and SVM performed worse with the
increasing number of variables. Variable importance analysis showed that the headspace pressure meter was essential, while the temperature sensor contributed little to the model. The AICc test
suggested that using four sensors, including one in the headspace and three in the liquid phase, showed an optimal performance from the current dataset. The sensor combination should be optimized
based on the scale and the configuration of the system using ML and statistical techniques like AICc. Overall, ML techniques could significantly improve the estimation model output and optimize the
number of pressure sensors. The results of this study can give the insight to plant operators for monitoring the liquid level accurately and in real-time. | {"url":"https://www.eeer.org/journal/view.php?number=1396","timestamp":"2024-11-03T13:48:40Z","content_type":"application/xhtml+xml","content_length":"133909","record_id":"<urn:uuid:ad81993f-e6ee-4f5a-a51d-7483c02473bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00596.warc.gz"} |
Find the Longest Common Subsequence of Two Paths
Introduction to Longest Common Subsequence
The Longest Common Subsequence (LCS) problem is a classic computer science problem that deals with finding the longest subsequence common to two sequences. A subsequence is a sequence of elements
that appears in the same order in both sequences, but not necessarily consecutively. This problem has many real-world applications, such as comparing DNA sequences, file comparisons, and version
control systems like Git.
Real-World Examples and Scenarios
LCS is used in a variety of contexts, such as:
1. Bioinformatics: Comparing DNA sequences to identify similarities between different species or within a species.
2. Text comparison: Comparing and contrasting different editions of a manuscript or different drafts of a document, allowing editors and authors to track changes and revisions.
3. Version control systems: Tools like Git use LCS algorithms to identify and merge changes made in different branches of a codebase.
Real-World Scenario and Technical Problem
Consider a scenario where you are working on a collaborative document editing application. In this application, multiple users can edit a shared document simultaneously. When a user saves their
changes, the application needs to find the longest common subsequence of the original document and the user's edited version, then merge the changes accordingly.
Problem Statement and Formal Definition
Given two strings X and Y, find the length of the longest common subsequence and the actual subsequence.
• Two strings X and Y, where 1 <= |X|, |Y| <= 1000
• Length of the longest common subsequence
• The actual longest common subsequence
Tying the Problem Statement to the Real-World Scenario
In the collaborative document editing application, the strings X and Y represent the original document and the user's edited version, respectively. By finding the longest common subsequence, the
application can identify the common parts of the two versions and determine which parts have been added, deleted, or modified.
Solution to the Problem
We can solve this problem using dynamic programming. The idea is to build a table dp[i][j] that stores the length of the longest common subsequence of the prefixes X[0...i-1] and Y[0...j-1]. We can
then use this table to reconstruct the actual longest common subsequence.
Step-by-Step Solution with the Real-World Scenario
1. Create a 2D table dp with dimensions (len(X) + 1) x (len(Y) + 1).
2. Initialize dp[0][j] and dp[i][0] to 0 for all 0 <= i <= len(X) and 0 <= j <= len(Y).
3. Iterate through the table, and for each cell dp[i][j], do the following:
4. If X[i - 1] == Y[j - 1], set dp[i][j] = dp[i - 1][j - 1] + 1.
5. Otherwise, set dp[i][j] = max(dp[i - 1][j], dp[i][j - 1]).
6. The length of the longest common subsequence is dp[len(X)][len(Y)].
7. Reconstruct the actual longest common subsequence by backtracking from dp[len(X)][len(Y)].
Code Example
Here's a Python implementation of the algorithm described above:
def longest_common_subsequence(X, Y):
m, n = len(X), len(Y)
dp = [[0] * (n + 1) for _ in range(m + 1)]
for i in range(1, m + 1):
for j in range(1, n + 1):
if X[i - 1] == Y[j - 1]:
dp[i][j] = dp[i - 1][j - 1] + 1
dp[i][j] = max(dp[i - 1][j], dp[i][j - 1])
lcs_length = dp[m][n]
lcs = ""
i, j = m, n
while i > 0 and j > 0:
if X[i - 1] == Y[j - 1]:
lcs = X[i - 1] + lcs
i -= 1
j -= 1
elif dp[i - 1][j] > dp[i][j - 1]:
i -= 1
j -= 1
return lcs_length, lcs
Explanation of the Solution with Intuitions and Analogies
The dynamic programming solution to the LCS problem can be visualized as a table where each cell dp[i][j] contains the length of the longest common subsequence of the prefixes of X and Y. This table
is built incrementally, row by row, by comparing characters from X and Y. If the characters are the same, we extend the length of the LCS found so far. If the characters are different, we take the
maximum LCS length found in the previous row or column.
Solving Other Similar Real-World Problems
The LCS algorithm can be easily adapted to solve other related problems, such as:
1. Finding the shortest common supersequence: Given two strings, find the shortest string that has both strings as subsequences.
2. Edit distance: Given two strings, find the minimum number of operations (insertions, deletions, and substitutions) required to transform one string into the other.
By understanding the principles behind the LCS algorithm, you can apply this knowledge to a wide range of real-world problems in various domains, from bioinformatics to text processing and version##
Applying Longest Common Subsequence to a Real-World Scenario
Let's consider a real-world scenario where the LCS algorithm can be useful. Imagine you're working on a version control system for a software development team. You've been tasked with implementing a
feature that analyzes two versions of a file and finds the longest common subsequence of lines, helping developers to visualize the similarities and differences between them.
In this case, the two file versions can be represented as two lists of strings, where each string is a line of code. By applying the LCS algorithm, we can find the longest common subsequence of these
lines, highlighting the parts of the code that remain unchanged between the two versions.
Problem Statement
Given two lists of strings, A and B, representing two versions of a file, find the longest common subsequence of lines between them.
Two lists of strings, A and B, where 1 <= len(A), len(B) <= 1000 and 1 <= len(A[i]), len(B[j]) <= 100 for all 0 <= i < len(A) and 0 <= j < len(B).
A tuple containing the length of the longest common subsequence and the subsequence itself as a list of strings.
Solution to the Problem
We can use the same dynamic programming approach as described earlier in this article, with minor modifications to accommodate lists of strings instead of character strings.
Here's the modified Python implementation for the problem:
def longest_common_subsequence_lines(A, B):
m, n = len(A), len(B)
dp = [[0] * (n + 1) for _ in range(m + 1)]
for i in range(1, m + 1):
for j in range(1, n + 1):
if A[i - 1] == B[j - 1]:
dp[i][j] = dp[i - 1][j - 1] + 1
dp[i][j] = max(dp[i - 1][j], dp[i][j - 1])
lcs_length = dp[m][n]
lcs = []
i, j = m, n
while i > 0 and j > 0:
if A[i - 1] == B[j - 1]:
lcs.insert(0, A[i - 1])
i -= 1
j -= 1
elif dp[i - 1][j] > dp[i][j - 1]:
i -= 1
j -= 1
return lcs_length, lcs
By applying this solution, the version control system can efficiently find the longest common subsequence of lines between two file versions, helping developers identify similarities and differences
in their codebase.
The Longest Common Subsequence problem is a classic problem in computer science with applications in various domains, such as bioinformatics, text processing, and version control systems. By
understanding the dynamic programming approach to solving the LCS problem, you can adapt and apply this knowledge to many real-world scenarios. | {"url":"https://www.altcademy.com/blog/find-the-longest-common-subsequence-of-two-paths/","timestamp":"2024-11-13T05:07:03Z","content_type":"text/html","content_length":"38872","record_id":"<urn:uuid:89401676-915e-43db-a0c6-efeeda178c10>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00867.warc.gz"} |
American Mathematical Society
Exploring Mathematical Scholarship through the Mathematics Collaboration Graph
Jonathan Gryak
The AMS, through Mathematical Reviews, is an affiliate of MIDAS, the Michigan Institute for Data Science at the University of Michigan, Ann Arbor. As part of that affiliation, we have sponsored some
research projects related to the work we do at Mathematical Reviews. The column below is written by Jonathan Gryak, a research scientist at the University of Michigan and MIDAS, and describes the
results from a project he did on the collaboration graph inherent in our database of authors. His work updates and expands upon the work of others, as mentioned in his column. I find it fascinating,
both as an interesting application of graph theory and for what the results say about mathematics as a collaborative profession. I am grateful to Jonathan for the research he has done and for his
willingness to share it in this venue.
—Edward Dunne
Social network analysis has a long history within the social sciences and bibliometric/scientometric communities, and has gained greater importance through the rise of Facebook, Twitter, and other
online social networks. The social networks formed by research collaborations have often been studied through the creation of collaboration graphs, wherein authors are represented by nodes in the
network and edges correspond to a co-authorship relationship.
It should come as no surprise that in mathematics there has been a long history of exploring the structure of the mathematical collaboration graph through graph-theoretic means. An author’s Erdős
number, i.e., their degree of separation from the famously collaborative mathematician Paul Erdős, has been around for at least fifty years 7, with further investigations of the collaboration network
of Erdős being performed by Grossman and others 589. Additionally, the work of Barabási et al. 1 analyzed the overall structure of the collaboration graph for mathematics and other disciplines over
time, investigating structural properties including degree distribution, diameter, and clustering coefficients.
Inspired by these prior works, in this manuscript we provide results on the current state of mathematical collaboration using the Mathematical Reviews database maintained by the American Mathematical
Society (AMS). Beyond Erdős we identify nine other “super-collaborators” using graph-theoretic methods and investigate their importance with respect to both graph-theoretic and author-level metrics.
We also investigate various measures of importance that have become popular in the analysis of social networks in general.
Using publication data extracted in February 2021 from the AMS Mathematical Reviews Database (MRDB), the collaboration graph was realized as a simple, undirected graph, with vertex set corresponding
to individual authors and edge set where if authors and have at least one publication together, with each edge weighted by the total number of joint publications. At the time of data extraction, the
MRDB had indexed 3,729,493 publications from 1,030,091 authors, yielding the collaboration graph with and .
The Collaboration Graph
We can learn a great deal about the community of mathematicians from examining the collaboration graph as a whole. We begin by examining the vertex degree , which corresponds to the number of
collaborators a given mathematician has. Table 1 presents summary statistics for the distribution of vertex degrees. Scientific collaboration networks tend to have degree distributions with “fat
tails,” i.e., ones that are well-approximated by a power-law, either in the number of papers published by an individual (1) or in the number of their collaborators 11. This is indeed the case for the
collaboration graph and can be readily observed by viewing the complementary cumulative distribution function (CCDF) of the degree distribution, depicted in Figure 1, along with fitted power-law and
log-normal distributions. From Figure 1A we can observe that the overall distribution of collaborators is well-approximated by a log-normal distribution, while Figure 1B shows the tail of the
distribution for , which is fit equally well by both distributions.
Figure 1.
The complementary CDF (CCDF) of the degree distribution along with fitted log-normal and power-law distributions.
All non-isolated vertices.
Tail only (degree ).
Nearly 11% (114,261) of authors represented in the Mathematical Reviews database have no recorded co-authors, these correspond to the isolated vertices of . Table 1 depicts summary statistics of the
major subgraphs of .
Table 1.
Degree statistics for the major subgraphs of : - the union of all connected components of excluding singletons (isolated vertices); - the largest connected component; and - the second-largest
connected component.
915,830 812,400 41
Min 1 1 1
Max 688 688 38
Mean 5.699654 6.140741 4.536585
Stddev 9.693462 10.1919 5.47766
Beyond degree, there are two other “d” properties that can illuminate the structure of the collaboration graph - density and distance. We will discuss density now and return to distance in the next
section. The density of a graph is defined as the ratio of the edges in the graph to all possible edges. Graphs of low density - e.g., - are considered sparse, while those of high density - e.g., -
are considered dense. Graph density impacts the data structures used to efficiently store a graph representation in memory, e.g., adjacency lists, and the computational costs associated with various
graph algorithms. The density of is , indicating that despite the collaborative nature of mathematical research, is very sparse.
Author Relationships and Networks
Having examined the global properties and overall structure of the collaboration graph, we now turn to discovering distinct authorship (sub)networks and their attributes. There are numerous questions
that we may ask about these relationships:
How often do highly collaborative or independent authors tend to stay within their respective networks?
Do co-authors tend to form working relationships independent of their mutual colleague?
On average, how many co-authors separate any given pair?
How important is any individual author to the collaboration network as a whole?
These and other questions are answered in the following subsections.
Assortativity is the tendency of vertices that are similar with respect to some measure to be connected. Degree assortativity, in which vertices of high or low degree tend to be connected to one
another, is a general form of similarity that applies to all networks, though any categorical or numerical property assigned to each vertex can be used as an assortative measure. Following 12,
assortativity, or assortative mixing, can be calculated for discrete properties of a vertex by defining the quantity : the fraction of edges within the graph whose ends are valued and . Assortativity
is then calculated as the Pearson correlation coefficient
where and are respectively the fraction of edges that start or end with values and , with and their respective standard deviations. Assortativity ranges from -1 to 1, with -1 representing total
disassortativity and 1 total assortativity. Graphs representing social networks, such as academic collaborations and personal connections, tend to be assortative, while natural networks, such as
biological or ecological networks, tend to be disassortative 12. In 12, published in 2003, the degree assortativity of mathematical collaborations was . In , the degree assortativity is lower at ,
indicating that is not particularly assortative with respect to vertex degree. However, using an author’s total number of publications as the similarity measure, the respective assortativity rises to
, indicating that mathematical authors who have a large number of publications tend to have published with each other more than authors who have a similar number of co-authors.
Cliques, clustering, and triads
Given a vertex with neighbors, the maximum number of edges is , and occurs precisely when and its neighbors form a clique. The local clustering coefficient for a vertex is the tendency of neighbors
to form a clique, and is defined as the ratio of the number of edges between ’s neighbors and the maximum number of edges :
In the context of the Collaboration Graph, the value of the local clustering coefficient for a given author indicates how likely it is that ’s co-authors are co-authors independently of . The mean of
the local clustering coefficient over the entire graph, , provides one measure of the global clustering coefficient, which for is 0.4696215 with standard deviation of 0.4388102. Excluding isolated
vertices yields a higher (0.4308473). These are lower values than determined by Barabasi et al. 1, where , indicating a weakening of ties in co-authorship networks and continuing a downward trend
identified in the same paper above.
A second measure of global clustering is transitivity, the tendency for two authors who share a common co-author (an open triad) to become co-authors themselves, forming a triangle, or closed triad.
More explicitly, transitivity can be defined 13 as , that is, the ratio of closed triads to all triads, and ranges from 0 to 1. The transitivity of is 0.1340591 and remains constant when isolated
vertices are excluded. This is much lower that transitivity values for other academic disciplines such astrophysics (0.414) 11 and statistics (0.320) 10, but higher than that in biomedical research
(0.066) 11.
Authorship networks
Authorship networks within the Collaboration Graph correspond to its connected components. There are 147,710 distinct connected components in , of which 114,261 are of size one, corresponding to the
isolated vertices; removing these yields 33,449 connected components containing two or more vertices. The largest connected component contains 812,400 authors, roughly 79% of the entire graph, and
captures 2,494,369 co-authorships. The existence of this large of a connected component is to be expected for real-world networks 13. Excluding these extrema, a histogram of the remaining connected
component sizes is presented in Figure 2. As can be seen in the figure, the number of authorship networks of a given size decreases rapidly.
Figure 2.
A histogram of connected component sizes when the smallest and largest components are excluded. Count is presented on the -axis in a logarithmic scale.
Eccentricity and seven degrees of separation
When considering distance between collaborators, there are potentially many paths, i.e., chains of co-authors, that might connect any given pair of authors. As such, we are interested in the geodesic
distance, i.e., the length of the shortest path between vertices. For a given vertex , the eccentricity is the maximum length of all shortest paths between and . The maximum eccentricity of any
vertex is called the diameter of the graph, while the minimum is called the radius. The diameter (longest chain of co-authors) of is 23, while the radius is 13. There are 10 peripheral vertices whose
eccentricity is equal to the diameter, and 3,595 central vertices, whose eccentricity is equal to the radius.
Another distance measure of interest in social and other small-world networks 15 is the “degree of separation” between vertices. This notion can be quantified as the average shortest path length,
calculated as
where is the geodesic distance between vertices and and . For , the average shortest path length is . Thus, on average mathematical co-authors have seven degrees of separation between them. This is
slightly higher than the five to six degrees posited by Milgram 14 in his seminal work and popularized as “six degrees of separation.” However, this is concordant with the average path length and its
trend over time as described in the 2002 work of Barabasi et al.1, in which the average shortest path for mathematical collaboration was estimated to be in 1991, which decreased to by 1998.
A comparison of structural properties of the collaboration graph and its various subgraphs, including the second largest component , is presented in Table 2. Among these subgraphs, is the only one
that exhibits disassortativity.
Table 2.
Structural properties of the collaboration graph and various subgraphs: - the number of connected components; - assortativity; - average shortest path length; - the global clustering coefficient; -
transitivity; and - diameter.
1,030,091 2,609,957 147,710 0.08549151 N/A 0.4696215 0.1340591 23
915,830 2,609,957 33,449 0.08549151 N/A 0.5282126 0.1340591 23
812,400 2,494,369 1 0.07340353 6.923573 0.5312295 0.1306022 23
41 93 1 -0.2853268 1.970732 0.7888307 0.2701271 4
Articulation points
How important is any one author’s connections to the collaboration of mathematicians as a whole? The reliance of the collaboration graph or any other network’s connectivity on a single node can be
made by identifying it as an articulation points or cut vertex: a vertex whose removal results in the disconnection of the network. The components resulting from the removal of an articulation point
are called biconnected components (BICCs), while a graph containing no articulation points is said to be biconnected. The largest connected component is not biconnected as it contains 110,371 authors
whose removal partitions the graph into two or more components.
The Super-Collaborators
Inspired by Grossman’s work 9 on that most peripatetic of mathematicians, we can analyze the mathematical collaboration graph to determine other super-collaborators like Paul Erdős. The top ten
super-collaborators by number of co-authors (i.e., vertex degree) are listed in Table 3, along with author-level metrics obtained from Mathematical Reviews and Google Scholar where available.
Table 3.
Super-collaborators and their author-level metrics. The publications and reported citations are those indexed and tabulated within the Mathematical Reviews database (MRDB) as of February 2021.
Citations (GS), -index, and -index were obtained from Google Scholar as of June 2021. The highest ranking by metric is bolded.
Author # Papers Citations Citations -index -index
(MRDB) (GS)
Baleanu, Dumitru I. 688 868 4,089 49,170 97 910
Srivastava, Hari Mohan 663 1,314 11,472 65,205 94 767
Agarwal, Ravi P. 618 1,645 16,265 N/A N/A N/A
Chen, Guanrong 569 720 3,720 109,004 155 986
Cao, Jinde 535 735 3,430 62,641 125 943
Erdős, Paul 510 1445 21,376 87,855 123 818
Kurths, Jürgen 502 318 1,453 89,341 128 885
Alon, Noga 473 624 12,164 51,988 107 499
Pardalos, Panos M. 448 510 2,779 57,110 104 715
Balakrishnan, Narayanaswamy 447 860 3,543 80,260 87 613
All super-collaborators are contained in , the largest connected component of . As described in the previous section, one measure of importance of a given author is the effect of connectivity when
the author and their co-authorships are removed from the graph. Among the super-collaborators, Kurths is the only one who is not a cut vertex: his removal does not increase the number of connected
components. For all other authors their removal results in one large connected component that is nearly (99.998% or greater) the size of , with the remaining biconnected components ranging in number
from 3 to 13 and size from 1 to 11.
Centrality measures (who’s important?)
Biconnectivity is not the only way in which to measure the importance of a given member of a social network. Centrality is a measure of a node’s importance or potential influence in a network. There
are several methods for calculating centrality that emphasize different aspects of the relationship of a node to its neighbors. Here we focus on four common centrality measures for undirected graphs
- degree, closeness, betweenness, and eigenvector centrality. Degree centrality is simply the degree of vertex normalized by one less than the number of vertices in the graph:
Closeness centrality was introduced by Bavelas 23 and is calculated as
where is the geodesic distance between vertices and . As its name suggests, closeness centrality defines important nodes as being close to other nodes within the network. Betweenness centrality, on
the other hand, defines important nodes as those who tend to connect other nodes in the network, i.e, they tend to be on the geodesic paths between other nodes. Betweenness was introduced by Freeman
6 and is defined as
where is the number of geodesics between vertices containing and is the total number of geodesics between vertices .
Finally, eigenvector centrality or eigencentrality, introduced by Bonacich 4, provides a measure of centrality in which more central nodes are those who are connected to other highly central nodes.
Let be ordered as . Given the adjacency matrix of the graph, for the node corresponds to the th entry of the vector satisfying , i.e, an eigenvector of . An eigenvector corresponding to the largest
eigenvalue is chosen so that all values are positive.
Figure 3.
Spearman correlation of various centrality measures for authors in the largest connected component .
Table 4.
Super-collaborator rankings by centrality measure. The highest ranked author by each measure is bolded, while the next highest is italicized.
Author Mean w/o Mean
Baleanu, Dumitru I. 1 12 7 19238 6.67 4814.5
Srivastava, Hari Mohan 2 4 2 12963 2.67 3242.75
Agarwal, Ravi P. 3 1 1 14658 1.67 3665.75
Chen, Guanrong 4 2 3 16036 3 4011.25
Cao, Jinde 5 9 9 21006 7.67 5257.25
Erdős, Paul 6 3 5 136 4.67 37.5
Kurths, Jürgen 7 11 6 22718 8 5685.5
Alon, Noga 8 8 13 22 9.67 12.75
Pardalos, Panos M. 9 6 4 2218 6.33 559.25
Balakrishnan, Narayanaswamy 10 126 11 23399 49 5886.5
Figure 3 depicts the Spearman (rank) correlation between all four centrality measures for One can observe that the rankings of authors by centrality measures are all positively correlated with each
other, with closeness being on average the most correlated with other measures, followed by eigenvector centrality, while betweenness was the least correlated with other measures.
Table 4 contains the rankings of the super-collaborators by all four centrality measures, along with the mean ranking with and without including eigencentrality, which is the outlier among the
rankings. From the table we can observe that Agarwal ranks the highest with respect to both closeness and betweenness and highest overall, while Srivastava is ranked second overall as well as in
degree and betweenness centrality. Of the super-collaborators only Erdős and Alon have relatively high eigencentrality, indicating that they tend to be more connected to other highly eigencentric
Figure 4 depicts the Pearson correlation between the four centrality measures and their author-level metrics obtained from Google Scholar. In examining correlation between centrality measures and
author-level metrics we find that closeness has the highest correlation with -index at 0.54, while betweenness has the highest correlation with both -index and citation count at 0.51 and 0.41
respectively. Eigencentrality is the negatively correlated with all author-level metrics, and strongly so for the -index with a correlation of . Within the author-level metrics themselves, -index is
most correlated with citation count.
Figure 4.
Pearson correlation between centrality measures and author-level metrics obtained from Google Scholar for all super-collaborators.
Future Directions
Additional realizations of the collaboration graph , such as the addition of edge weights corresponding to the number of joint publications between two authors, would allow for additional analyses
that may yield further insight into mathematical collaboration. Moreover, the treatment of as a dynamic graph would enable us to understand trends in mathematical collaboration over time. Additional
author data from the MRDB, including institution, country, and subject area encoded via Mathematics Subject Classification (MSC), could be used to assign multiple labels to each node and be used to
illuminate additional collaborative relationships and community structures. The nine additional super-collaborators identified in this paper could have their relationships between other
mathematicians and scholars explored as was done for Erdős in 5.
Inspired by prior works 1589, we have provided results on the current state of mathematical collaboration using the collaboration graph derived from information contained in the Mathematical Reviews
database. Overall is a sparse graph whose vertex degree distribution is log-normal. The graph contains one large connected component containing nearly 79% of authors. Currently there are on average
seven degrees of separation between mathematical collaborators, matching the decreasing trend over time as reported in 1. We identified ten “super-collaborators” using graph-theoretic methods, one of
whom is Paul Erdős. For these “super-collaborators” we investigated their importance with respect to both centrality and author-level metrics, finding that closeness centrality is relatively highly
correlated with -index. Future work can utilize additional author data from the MRDB, including institution, country, and subject area, to determine community structures and further explore the many
ways in which mathematicians collaborate in pursuit of their common goal of advancing mathematical knowledge.
This work was funded in part by Mathematical Reviews, a division of the American Mathematical Society. Some analyses contained herein utilized computational resources and services provided by
Advanced Research Computing at the University of Michigan, Ann Arbor.
All article images are courtesy of the author. | {"url":"https://www.ams.org/journals/notices/202205/noti2471/noti2471.html?adat=May%202022&trk=2471&pdfissue=202205&pdffile=rnoti-p849.pdf&cat=none&type=.html","timestamp":"2024-11-12T00:00:32Z","content_type":"text/html","content_length":"685344","record_id":"<urn:uuid:f8e5c9cb-881e-4f05-885b-48368de6d5a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00065.warc.gz"} |
Hilbert - Cantor's Archive
Beginner’s Guide to Mathematical Constructivism
The foundational crisis in mathematics along with roughly four decades following it, was likely the most fertile period in the history of logic and studies in the foundations. After discovering the
set-theoretic paradoxes, such as the paradox of the set of all sets, together with the logical ones, like Russell’ | {"url":"https://www.cantorsparadise.org/tag/hilbert/","timestamp":"2024-11-12T02:57:38Z","content_type":"text/html","content_length":"33731","record_id":"<urn:uuid:d20fce4f-9753-4738-a58e-80a360d1acc1>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00401.warc.gz"} |
An interpretation of system F through bar recursion
14:00 7th June 2017 ( week 7, Trinity Term 2017 )
Room 051, Wolfson Building
There are two possible computational interpretations of second-order arithmetic: Girard's system F or Spector's bar recursion and its variants. While the logic is the same, the programs obtained from
these two interpretations have a fundamentally different computational behavior and their relationship is not well understood. We make a step towards a comparison by defining the first translation of
system F into a simply-typed total language with a variant of bar recursion. This translation relies on a realizability interpretation of second-order arithmetic. Due to Gödel's incompleteness
theorem there is no proof of termination of system F within second-order arithmetic. However, for each individual term of system F there is a proof in second-order arithmetic that it terminates, with
its realizability interpretation providing a bound on the number of reduction steps to reach a normal form. Using this bound, we compute the normal form through primitive recursion. Moreover, since
the normalization proof of system F proceeds by induction on typing derivations, the translation is compositional. The flexibility of our method opens the possibility of getting a more direct
translation that will provide an alternative approach to the study of polymorphism, namely through bar recursion. | {"url":"https://www.cs.ox.ac.uk/seminars/1863.html","timestamp":"2024-11-09T06:03:41Z","content_type":"text/html","content_length":"32542","record_id":"<urn:uuid:8724645c-17bc-4c20-83f1-b0956a7df1fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00195.warc.gz"} |
Consider the partially completed one-way ANOVA summary table. Source Sum of Squares Degrees of - Course ScholarConsider the partially completed one-way ANOVA summary table. Source Sum of Squares Degrees of - Course Scholar
Consider the partially completed one-way ANOVA summary table. Source Sum of Squares Degrees of
IVC/Spring 2016/Econ 10/MGT 10/ Pehlivan/ Exam 3Name___________________________________MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.1)
Consider the partially completed one-way ANOVA summary table.SourceSum ofSquaresDegrees ofFreedomMean Sumof Squares FBetween 270Within 18Total 810 21The total number of observations for this ANOVA
procedure isA) 20. B) 18. C) 21. D) 22.1)2) Consider the partially completed one-way ANOVA summary table.SourceSum ofSquaresDegrees ofFreedomMean Sumof Squares FBetween 270Within 18Total 810 21The
degrees of freedom for the sum of squares between for this ANOVA procedure isA) 2. B) 1. C) 3. D) 4.2)TRUE/FALSE. Write ‘T’ if the statement is true and ‘F’ if the statement is false.3) Analysis of
variance is a technique used to conduct a hypothesis test to compare three or morepopulation proportions simultaneously.3)4) A level in an ANOVA test accounts for the variation outside of the main
factor. 4)5) All analysis of variance procedures require that the observations are either ordinal or interval data. 5)6) The total sum of squares (SST) measures the amount of variation between each
data value and thegrand mean.6)1MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.7) Nintendo Sony would like to test the hypothesis that a
difference exists in the average age of usersof a Wii, a PlayStation, or an Xbox console game. The following data represent the age of a randomsample of Wii, PlayStation, and Xbox users.Wii
PlayStation Xbox37 26 3131 21 2047 24 3829 24 3136 25 30The total sum of squares for these observations is ________.A) 652.6 B) 594.7 C) 736.0 D) 881.57)8) AutoTrader.com would like to test if a
difference exists in the age of three different types ofvehicles currently on the road–trucks, cars, and vans. The following data represent the age of arandom sample of trucks, cars, and vans.Trucks
Cars Vans12 8 38 7 79 10 611 7 8The mean square between for these observations is________.A) 16.0 B) 2.0 C) 4.7 D) 5.58)TRUE/FALSE. Write ‘T’ if the statement is true and ‘F’ if the statement is
false.9) The Tukey-Kramer multiple comparisons procedure allows us to examine each pair of samplemeans and to conclude whether or not their respective sample means differ for a one-wayANOVA.9)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.10) Consider the partially completed randomized block ANOVA summary table.SourceSum
ofSquaresDegrees ofFreedomMean Sum ofSquares FBetween 600 3Block 5Error 750Total 2,150 23The sum of squares block for this ANOVA procedure is _____.A) 930 B) 980 C) 1,150 D) 80010)211) Consider the
partially completed randomized block ANOVA summary table.SourceSum ofSquaresDegrees ofFreedomMean Sum ofSquares FBetween 600 3Block 5Error 750Total 2,150 23The mean square between for this ANOVA
procedure is _____.A) 410 B) 50 C) 200 D) 16011)12) In Delaware, cars are inspected each year using state-operated inspection centers. The Wilmingtoncenter has three drive-through lanes where cars
are inspected in a sequence of steps. The followingdata show the number of minutes that a random sample of drivers spent waiting and having theircars inspected in the three lanes each day of the
week.Day Lane 1 Lane 2 Lane 3Monday 36 35 37Tuesday 19 30 20Wednesday 26 32 23Thursday 38 49 30Friday 41 41 35The state of Delaware would like to perform a randomized block ANOVA to test for a
difference inthe average times the drivers spend in the three lanes using the weekday as a blocking factor with ?= 0.05. The mean square block for these observations is ________.A) 148.9 B) 174.4 C)
135.0 D) 162.612)13) Consider the partially completed two-way ANOVA summary table.SourceSum ofSquaresDegrees ofFreedomMean Sumof Squares FFactor B 2Factor A 600 200Interaction 144Error 384 12Total
1,288 23The sum of squares for Factor B for this ANOVA procedure is ____.A) 370 B) 160. C) 300 D) 40013)TRUE/FALSE. Write ‘T’ if the statement is true and ‘F’ if the statement is false.14) In a
two-way ANOVA procedure, there are two hypotheses to be tested—the test for Factor A andthe test for factor B.14)315) For two-way ANOVA, we partition the total sum of squares (SST) into the sum of
squares forFactor A (SSFA), the sum of squares for Factor B (SSFB), the sum of squares interaction (SSAB), andthe sum of squares error (SSE).15)MULTIPLE CHOICE. Choose the one alternative that best
completes the statement or answers the question.16) Wells Fargo Bank would like to determine if a difference exists in the average credit score betweenresidents of the states of Texas, Alaska, and
Iowa and also investigate if age plays a role in creditscore. The credit scores from a random sample of residents from each state was recorded andresidents were categorized as being under 40 years
old or 40 years and older. A two-way ANOVAwas conducted using ? = 0.05 with Factor A assigned the state residence and Factor B assigned theage group. The results are shown below.The Tukey-Kramer
critical range for Factor A using ? = 0.05 is ________.A) 36.00 B) 41.30 C) 55.60 D) 48.5016)4TRUE/FALSE. Write ‘T’ if the statement is true and ‘F’ if the statement is false.17) When performing a
hypothesis test to compare two or more population proportions, the expectedfrequencies must always be integer values.17)MULTIPLE CHOICE. Choose the one alternative that best completes the statement
or answers the question.18) The ________ frequencies refer to the frequencies that are most likely to occur if the null hypothesisis true when performing a hypothesis test comparing two or more
population proportions.A) expected B) observed C) cumulative D) relative18)19) Nationwide Insurance would like to perform a chi-square test to investigate whether a differenceexists in the proportion
of male and female teenagers who text while they drive. A random sampleof 80 male teenagers found that 50 indicated they texted while driving. A random sample of 120female teenagers found that 65
indicated they texted while driving. The expected frequency offemale teenagers who text is ________.A) 65 B) 69 C) 55 D) 5119)20) When performing a hypothesis test to compare two or more population
proportions, the teststatistic follows the ________.A) Student’s t-distribution B) normal distributionC) F-distribution D) chi-square distribution20)21) The Department of Transportation would like to
investigate if a difference exists in the proportionof flights that arrive on-time between Alaska Airlines, Delta Airlines, Southwest Airlines, andUnited Airlines. The following data represent the
number of on-time flights from random samplestaken from each airline.Alaska Delta Southwest UnitedNumber On-time 90 95 85 60Sample Size 100 120 100 80The expected frequency of on-time flights from
the Southwest sample is ________.A) 66 B) 99 C) 90 D) 82.521)TRUE/FALSE. Write ‘T’ if the statement is true and ‘F’ if the statement is false.22) The null hypothesis for a chi-square goodness-of-fit
test always states that the stated distributionis followed.22)5MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.23) The following data represent
the discrete probability distribution for the number of stars thatreviewers gave a first edition statistics reference book.Stars 5 4 3 2 1Probability 25% 35% 20% 10% 10%The second edition of this
reference book has been released and a random sample of people thathave purchased the book has been collected with the following results.Stars 5 4 3 2 1Frequency 28 28 13 6 5The publisher would like
to know if the probability distribution for reviews has changed from thefirst edition to the second edition using ? = 0.025. The critical value for this hypothesis test is________.A) 5.991 B) 9.448
C) 11.143 D) 7.81523)24) A manufacturer of smartphone batteries will randomly select 20 batteries from the process eachday and count the number of defects. Historically, 4% of the batteries produced
by this companyhave been defective. The following data represent the frequency of defective batteries from arandom sample of 200 days.Number of Defective Frequency0 851 632 353 or more 17The
manufacturer would like to test if the probability distribution for defective batteries follows thebinomial distribution with p = 0.04 and n = 20 using ? = 0.05. The expected number of days fromthis
sample that 3 or more batteries from the batch of 20 are defective is ________.A) 71.70 B) 15.08 C) 75.48 D) 37.7424)TRUE/FALSE. Write ‘T’ if the statement is true and ‘F’ if the statement is
false.25) A dependent variable, x, explains the variation in another variable, which is called the independentvariable, y.25)26) A scatter plot is a useful tool to examine the data before conducting
correlation analysis. 26)27) The values of the correlation coefficient range between -1.0 and +1.0. 27)6MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the
question.28) The table below shows the number of cars sold last month by seven employees at Concord Motorsand their number of years of sales experience.Experience Sales1 82 62 74 145 96 138 10The
correlation coefficient for this data is ________.A) -0.251 B) 0.744 C) 0.360 D) 0.55328)29) Sarah is the office manager for a group of financial advisors who provide financial services forindividual
clients. She would like to investigate whether a relationship exists between the numberof presentations made to prospective clients in a month and the number of new clients per month.The following
table shows the number of presentations and corresponding new clients for arandom sample of six employees.Employee Presentations New Clients1 7 22 9 33 9 44 10 35 11 56 12 3The correlation
coefficient for this data is ________.A) 0.516 B) 0.403 C) 0.323 D) 0.16729)TRUE/FALSE. Write ‘T’ if the statement is true and ‘F’ if the statement is false.30) If two variables have a correlation
coefficient equal to -0.60 from a sample size of 5, we canconclude that the population correlation coefficient is less than zero using ? = 0.05.30)31) The line that best fits the ordered pairs using
the least squares method is called the residual line. 31)32) Given a regression equation of y^= 15.6 – 3.8x, a one-unit increase in the independent variablewould result in an average increase of 3.8
for the dependent variable.32)MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.33) The formula for the equation describing a straight line is y^=
b0 + b1x. The value for b0 in thisequation represents the _________.A) y-intercept of the straight line B) slope of the straight lineC) predicted value of y given a value of x D) independent
variable33)734) The table below shows the number of cars sold last month by seven employees at Concord Motorsand their number of years of sales experience.Experience Sales1 82 62 74 145 96 138
10Management would like to use simple regression analysis to estimate monthly car sales using thenumber of years of sales experience. The y-intercept for the regression equation is ________.A) 2.165
B) 6.940 C) 4.598 D) 8.33734)35) The ________ measures the amount of dispersion of observed data around a regressionline.A) coefficient of determination B) standard error of the estimateC)
correlation coefficient D) standard error of the slope35)36) Costco sells paperback books in their retail stores and wanted to examine the relationship betweenprice and demand. The price of a
particular novel was adjusted each week and the weekly saleswere recorded in the table below.Sales Price3 $124 $116 $1010 $98 $810 $7Management would like to use simple regression analysis to
estimate weekly demand for thisnovel using the price of the novel. The standard error of the estimate is ________.A) 1.76 B) 3.30 C) 1.38 D) 1.0936)37) The variable y^represents the ________ in the
regression model.A) residualB) value of the dependent variableC) predicted value for the dependent variableD) value of the independent variable37)38) The test statistic for testing the significance
for the regression coefficient follows the _______.A) Student’s t-distribution B) chi-square distributionC) F-distribution D) normal distribution38)839) The National Basketball Association (NBA)
would like to develop a multiple regression model thatwould predict the number of wins for a team during the season. The following independentvariables were considered for the model: field goal
percentage (X1), free throw percentage (X2),turnovers per game (X3), rebounds per game (X4), and steals per game (X5). The followingregression model was chosen using a data set of team statistics:y^=
-276.8949 + 6.0284×1 – 6.8230×3 + 3.348x4The first team from the data set had the following values:Wins = 56Field goal percentage = 48.6%Free throw percentage = 77.0%Turnovers per game = 14.6Rebounds
per game = 38.8Steals per game = 8.2The residual for this team is ________.A) 9.6 B) -3.1 C) 7.0 D) -5.239)TRUE/FALSE. Write ‘T’ if the statement is true and ‘F’ if the statement is false.40) One of
the assumptions for the multiple regression model is that the residuals have a constantvariance.40)ESSAY. Write your answer in the space provided or on a separate sheet of paper.Use the information
below to answer the following question(s).The table below shows the number of interceptions thrown during the season by seven randomly selected National FootballLeague teams and the number of games
those teams won during the season.Wins Interceptions3 286 1911 1614 610 98 258 1141) Use the NFL team data to calculate the total sum of squares, sum of squares error, and sum of
squaresregression.9MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.42) The following distribution shows the frequency distribution of the number
of shots a particularNBA player blocked in one game for a random sample of 100 games.Blocked Shots per Game Frequency0 01 142 353 364 105 26 3You have been assigned the task to test if the
distribution of blocked shots per game for this playerfollows the Poisson distribution using ? = 0.05. The sample mean for this distribution is ________.A) 3.47 B) 2.60 C) 3.12 D) 2.2042)10 | {"url":"https://coursescholar.com/2020/01/27/consider-the-partially-completed-one-way-anova-summary-table-source-sum-of-squares-degrees-of-2/","timestamp":"2024-11-09T20:29:53Z","content_type":"text/html","content_length":"92686","record_id":"<urn:uuid:f78aa52d-d762-41cf-a782-a50b2e47ac46>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00554.warc.gz"} |
Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained.
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How many Speakers does the sphere have? - Mad Penguin
How many Speakers does the sphere have?
How Many Speakers Does the Sphere Have?
The sphere is a fascinating object in mathematics and physics, and it’s always interesting to explore its properties and characteristics. In this article, we will delve into one of the most
fundamental questions about spheres: how many speakers does it have? Surprisingly, the answer is not as simple as it seems, and it requires a deeper understanding of the geometry and topology of
Direct Answer: 1 Speaker
Before we dive into the details, let’s give a straightforward answer to the question: a sphere has 1 speaker. Yes, you read that right! A sphere, by definition, is a three-dimensional object that is
equidistant from a central point called the center. This definition implies that any point on the surface of the sphere is connected to the center, which means there is only one "speaker" or "vertex"
that is inherent in the sphere’s geometry.
Understanding the Surface of a Sphere
The surface of a sphere is a significant portion of the object, and it’s crucial to understand its properties to answer the question about the number of speakers. The surface of a sphere is a
two-dimensional manifold, often referred to as a 2-sphere or S2. This manifold has several important characteristics:
• No holes: The surface of a sphere is continuous and unbroken, with no holes or gaps.
• Non-orientable: The surface of a sphere is non-orientable, meaning that it’s impossible to distinguish the "top" from the "bottom" or the "front" from the "back".
• Homogeneous: The surface of a sphere is homogeneous, meaning that any point on the surface is equivalent to any other point.
Topological Properties of a Sphere
To better understand the concept of speakers on a sphere, let’s explore some topological properties:
• Connectedness: A sphere is connected, meaning that there are no separate, non-overlapping regions.
• Compactness: A sphere is compact, meaning that it has a finite volume.
• Non-trivial topology: A sphere has a non-trivial topological structure, which means that it’s not simply a collection of disconnected points.
Speaker-Putting a Sphere: Topological Obstructions
When you put a speaker on a sphere, you create a new object that is known as a sphere with a speaker (see table below). The speaker becomes a topological obstruction, which changes the properties of
the sphere:
Sphere Property Sphere with Speaker Property
Connected Disconnected (into two separate regions)
Homogeneous No longer homogeneous (due to the speaker’s presence)
Non-Orientable Remains non-orientable
As the table illustrates, the presence of a speaker creates a topological obstruction that breaks the homogeneity and connectedness of the sphere. However, the speaker does not create holes or gaps
on the surface of the sphere, but it does induce a non-trivial topological structure.
In conclusion, the answer to the question "How many speakers does a sphere have?" is 1. The sphere’s geometry and topology impose a unique structure that is inherently speaker-less. When you attempt
to add a speaker to a sphere, it creates a topological obstruction that changes the object’s properties, but it does not increase the number of speakers. Ultimately, the sphere remains a one-speaker
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Euler Equations
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Section 6.4 : Euler Equations
In this section we want to look for solutions to
\[$$a{x^2}y'' + bxy' + cy = 0\label{eq:eq1}$$\]
around \({x_0} = 0\). These types of differential equations are called Euler Equations.
Recall from the previous section that a point is an ordinary point if the quotients,
\[\frac{{bx}}{{a{x^2}}} = \frac{b}{{ax}}\hspace{0.25in}{\mbox{and }}\hspace{0.25in}\frac{c}{{a{x^2}}}\]
have Taylor series around \({x_0} = 0\). However, because of the \(x\) in the denominator neither of these will have a Taylor series around \({x_0} = 0\) and so \({x_0} = 0\) is a singular point. So,
the method from the previous section won’t work since it required an ordinary point.
However, it is possible to get solutions to this differential equation that aren’t series solutions. Let’s start off by assuming that \(x>0\) (the reason for this will be apparent after we work the
first example) and that all solutions are of the form,
\[$$y\left( x \right) = {x^r}\label{eq:eq2}$$\]
Now plug this into the differential equation to get,
\[\begin{align*}a{x^2}\left( r \right)\left( {r - 1} \right){x^{r - 2}} + bx\left( r \right){x^{r - 1}} + c{x^r} & = 0\\ ar\left( {r - 1} \right){x^r} + b\left( r \right){x^r} + c{x^r} & = 0\\ \left(
{ar\left( {r - 1} \right) + b\left( r \right) + c} \right){x^r} & = 0\end{align*}\]
Now, we assumed that \(x>0\) and so this will only be zero if,
\[$$ar\left( {r - 1} \right) + b\left( r \right) + c = 0\label{eq:eq3}$$\]
So solutions will be of the form \(\eqref{eq:eq2}\) provided \(r\) is a solution to \(\eqref{eq:eq3}\). This equation is a quadratic in \(r\) and so we will have three cases to look at : Real,
Distinct Roots, Double Roots, and Complex Roots.
Real, Distinct Roots
There really isn’t a whole lot to do in this case. We’ll get two solutions that will form a fundamental set of solutions (we’ll leave it to you to check this) and so our general solution will be,
\[y\left( x \right) = {c_1}{x^{{r_1}}} + {c_2}{x^{{r_2}}}\]
Example 1
Solve the following IVP \[2{x^2}y'' + 3xy' - 15y = 0,\hspace{0.25in}y\left( 1 \right) = 0\,\,\,\,y'\left( 1 \right) = 1\]
Show Solution
We first need to find the roots to \(\eqref{eq:eq3}\).
\[\begin{align*}2r\left( {r - 1} \right) + 3r - 15 & = 0\\ 2{r^2} + r - 15 = \left( {2r - 5} \right)\left( {r + 3} \right) & = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}{r_1} = \frac{5}{2},\,\,\,
{r_2} = - 3\end{align*}\]
The general solution is then,
\[y\left( x \right) = {c_1}{x^{\frac{5}{2}}} + {c_2}{x^{ - 3}}\]
To find the constants we differentiate and plug in the initial conditions as we did back in the second order differential equations chapter.
\[y'\left( x \right) = \frac{5}{2}{c_1}{x^{\frac{3}{2}}} - 3{c_2}{x^{ - 4}}\] \[\left. \begin{align*}& 0 = y\left( 1 \right) = {c_1} + {c_2}\\ & 1 = y'\left( 1 \right) = \frac{5}{2}{c_1} - 3{c_2}\end
{align*} \right\}\hspace{0.25in} \Rightarrow \hspace{0.25in}{c_1} = \frac{2}{{11}},\,\,{c_2} = - \frac{2}{{11}}\]
The actual solution is then,
\[y\left( x \right) = \frac{2}{{11}}{x^{\frac{5}{2}}} - \frac{2}{{11}}{x^{ - 3}}\]
With the solution to this example we can now see why we required \(x>0\). The second term would have division by zero if we allowed \(x=0\) and the first term would give us square roots of negative
numbers if we allowed \(x<0\).
Double Roots
This case will lead to the same problem that we’ve had every other time we’ve run into double roots (or double eigenvalues). We only get a single solution and will need a second solution. In this
case it can be shown that the second solution will be,
\[{y_2}\left( x \right) = {x^r}\ln x\]
and so the general solution in this case is,
\[y\left( x \right) = {c_1}{x^r} + {c_2}{x^r}\ln x = {x^r}\left( {{c_1} + {c_2}\ln x} \right)\]
We can again see a reason for requiring \(x>0\). If we didn’t we’d have all sorts of problems with that logarithm.
Example 2
Find the general solution to the following differential equation. \[{x^2}y'' - 7xy' + 16y = 0\]
Show Solution
First the roots of \(\eqref{eq:eq3}\).
\[\begin{align*}r\left( {r - 1} \right) - 7r + 16 & = 0\\ {r^2} - 8r + 16 & = 0\\ {\left( {r - 4} \right)^2} & = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}r = 4\end{align*}\]
So, the general solution is then,
\[y\left( x \right) = {c_1}{x^4} + {c_2}{x^4}\ln x\]
Complex Roots
In this case we’ll be assuming that our roots are of the form,
\[{r_{1,2}} = \lambda \pm \mu \,i\]
If we take the first root we’ll get the following solution.
\[{x^{\lambda + \mu \,i}}\]
This is a problem since we don’t want complex solutions, we only want real solutions. We can eliminate this by recalling that,
\[{x^r} = {{\bf{e}}^{\ln {x^r}}} = {{\bf{e}}^{r\ln x}}\]
Plugging the root into this gives,
\[\begin{align*}{x^{\lambda + \mu \,i}} & = {{\bf{e}}^{\left( {\lambda + \mu \,i} \right)\ln x}}\\ & = {{\bf{e}}^{\lambda \ln x}}{{\bf{e}}^{\mu \,i\ln x}}\\ & = {{\bf{e}}^{\ln {x^\lambda }}}\left( {\
cos \left( {\mu \ln x} \right) + i\sin \left( {\mu \ln x} \right)} \right)\\ & = {x^\lambda }\cos \left( {\mu \ln x} \right) + i{x^\lambda }\sin \left( {\mu \ln x} \right)\end{align*}\]
Note that we had to use Euler formula as well to get to the final step. Now, as we’ve done every other time we’ve seen solutions like this we can take the real part and the imaginary part and use
those for our two solutions.
So, in the case of complex roots the general solution will be,
\[y\left( x \right) = {c_1}{x^\lambda }\cos \left( {\mu \ln x} \right) + {c_2}{x^\lambda }\sin \left( {\mu \ln x} \right) = {x^\lambda }\left( {{c_1}\cos \left( {\mu \ln x} \right) + {c_2}\sin \left(
{\mu \ln x} \right)} \right)\]
Once again, we can see why we needed to require \(x > 0\).
Example 3
Find the solution to the following differential equation. \[{x^2}y'' + 3xy' + 4y = 0\]
Show Solution
Get the roots to \(\eqref{eq:eq3}\) first as always.
\[\begin{align*}r\left( {r - 1} \right) + 3r + 4 & = 0\\ {r^2} + 2r + 4 & = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}{r_{1,2}} = - 1 \pm \sqrt 3 \,i\end{align*}\]
The general solution is then,
\[y\left( x \right) = {c_1}{x^{ - 1}}\cos \left( {\sqrt 3 \ln x} \right) + {c_2}{x^{ - 1}}\sin \left( {\sqrt 3 \ln x} \right)\]
We should now talk about how to deal with \(x < 0\) since that is a possibility on occasion. To deal with this we need to use the variable transformation,
\[\eta = - x\]
In this case since \(x < 0\) we will get \(\eta > 0\). Now, define,
\[u\left( \eta \right) = y\left( x \right) = y\left( { - \eta } \right)\]
Then using the chain rule we can see that,
\[u'\left( \eta \right) = - y'\left( x \right)\hspace{0.25in}{\mbox{and}}\hspace{0.25in}u''\left( \eta \right) = y''\left( x \right)\]
With this transformation the differential equation becomes,
\[\begin{align*}a{\left( { - \eta } \right)^2}u'' + b\left( { - \eta } \right)\left( { - u'} \right) + cu & = 0\\ a{\eta ^2}u'' + b\eta u' + cu & = 0\end{align*}\]
In other words, since \(\eta>0\) we can use the work above to get solutions to this differential equation. We’ll also go back to \(x\)’s by using the variable transformation in reverse.
\[\eta = - x\]
Let’s just take the real, distinct case first to see what happens.
\[\begin{align*}u\left( \eta \right) & = {c_1}{\eta ^{{r_1}}} + {c_2}{\eta ^{{r_2}}}\\ y\left( x \right) & = {c_1}{\left( { - x} \right)^{{r_1}}} + {c_2}{\left( { - x} \right)^{{r_2}}}\end{align*}\]
Now, we could do this for the rest of the cases if we wanted to, but before doing that let’s notice that if we recall the definition of absolute value,
\[\left| x \right| = \left\{ {\begin{array}{*{20}{c}}x&{{\mbox{if }}x \ge 0}\\{ - x}&{{\mbox{if }}x < 0}\end{array}} \right.\]
we can combine both of our solutions to this case into one and write the solution as,
\[y\left( x \right) = {c_1}{\left| x \right|^{{r_1}}} + {c_2}{\left| x \right|^{{r_2}}},\hspace{0.25in}x \ne 0\]
Note that we still need to avoid \(x = 0\) since we could still get division by zero. However, this is now a solution for any interval that doesn’t contain \(x = 0\).
We can do likewise for the other two cases and the following solutions for any interval not containing \(x = 0\).
\[\begin{align*}y\left( x \right) & = {c_1}{\left| x \right|^r} + {c_2}{\left| x \right|^r}\ln \left| x \right|\\ y\left( x \right) & = {c_1}{\left| x \right|^\lambda }\cos \left( {\mu \ln \left| x \
right|} \right) + {c_2}{\left| x \right|^\lambda }\sin \left( {\mu \ln \left| x \right|} \right)\end{align*}\]
We can make one more generalization before working one more example. A more general form of an Euler Equation is,
\[a{\left( {x - {x_0}} \right)^2}y'' + b\left( {x - {x_0}} \right)y' + cy = 0\]
and we can ask for solutions in any interval not containing \(x = {x_0}\). The work for generating the solutions in this case is identical to all the above work and so isn’t shown here.
The solutions in this general case for any interval not containing \(x = a\) are,
\[\begin{align*}y\left( x \right) & = {c_1}{\left| {x - a} \right|^{{r_1}}} + {c_2}{\left| {x - a} \right|^{{r_2}}}\\ y\left( x \right) & = {\left| {x - a} \right|^r}\left( {{c_1} + {c_2}\ln \left|
{x - a} \right|} \right)\\ y\left( x \right) & = {\left| {x - a} \right|^\lambda }\left( {{c_1}\cos \left( {\mu \ln \left| {x - a} \right|} \right) + {c_2}\sin \left( {\mu \ln \left| {x - a} \right|}
\right)} \right)\end{align*}\]
Where the roots are solutions to
\[ar\left( {r - 1} \right) + b\left( r \right) + c = 0\]
Example 4
Find the solution to the following differential equation on any interval not containing \(x = - 6\). \[3{\left( {x + 6} \right)^2}y'' + 25\left( {x + 6} \right)y' - 16y = 0\]
Show Solution
Example 4 Find the solution to the following differential equation on any interval not containing \(x = - 6\).
\[3{\left( {x + 6} \right)^2}y'' + 25\left( {x + 6} \right)y' - 16y = 0\]
So, we get the roots from the identical quadratic in this case.
\[\begin{align*}3r\left( {r - 1} \right) + 25r - 16 & = 0\\ 3{r^2} + 22r - 16 & = 0\\ \left( {3r - 2} \right)\left( {r + 8} \right) & = 0\hspace{0.25in} \Rightarrow \hspace{0.25in}{r_1} = \frac{2}
{3},\,\,{r_2} = - 8\end{align*}\]
The general solution is then,
\[y\left( x \right) = {c_1}{\left| {x + 6} \right|^{\frac{2}{3}}} + {c_2}{\left| {x + 6} \right|^{ - 8}}\] | {"url":"https://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx","timestamp":"2024-11-14T10:05:45Z","content_type":"text/html","content_length":"83999","record_id":"<urn:uuid:87087b33-07eb-497f-9581-99b9f4a737de>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00421.warc.gz"} |
Not loving my AT740MLx
It seems that the designers at Nagaoka agree, putting boron cantilevers with elliptical diamonds on cartridges at a price that gets you an “advanced” stylus from other brands, prompting
complaints from some that they are too expensive for their styli.
Their pricing model is quite unique - they charge a lot for the cart body, but quite a bit less than others for the replacement styli. US prices for their stuff if very high so most people buy out of
Jan 27, 2020
The McIntosh spec sheet says maximum input at the MM terminals is 80mv. Seems like plenty of margin.
What frequency, do you know? ATs tend to have a lower than usual output (3mV @ 5cm/s I believe) but the hf peak may increase output, especially when excited by ticks, pops and so on...
P.S. Granny sucking eggs I'm sure, but have you tried dropping the tonearm pillar slightly while keeping the 2g downforce? Have you also tried confirmation the 2g downforce actually IS 2g, as so many
arms can be up to 10% out. Just a thought, as some Japanese pickups used to have a steep rake angle of up to 29 degrees which usually adds to hf distortion (AT may well not, but it's a thought)
What frequency, do you know? ATs tend to have a lower than usual output (3mV @ 5cm/s I believe) but the hf peak may increase output, especially when excited by ticks, pops and so on...
The peak is in the audible range. Due to self-inductance (it's an MM) there is no ultrasonic peak. WYSIWYG, and if he's right about his Cl it's about 2.5dB at around 12kHz.
No wonder you don't like the sound with that peak at 10k+. I had a similar dislike of the AT-440MLa.
Fixable with loading (or EQ)
Fixable with loading (or EQ)
Sad that you have to fix it hough, especially when they have a cheaper cart (VM95ML), same stylus, that does not have this problem.
Addicted to Fun and Learning
Nov 5, 2022
The McIntosh spec sheet says maximum input at the MM terminals is 80mv. Seems like plenty of margin.
How do you like the C49 in general? It's on my shortlist...
Addicted to Fun and Learning
Forum Donor
Jan 29, 2019
I just spent the last 45 minutes listening to it with the AT7V stylus ( .2 x .7 mil) on it. I like it better. Still has some of that “sparkle” but not the spittiness. I did make up some cables that
have about 20pf less capacitance. Not a big difference. Put the now cleaned ML stylus back on. Still seems just a bit more sort of shrill in the highs. Maybe that ML traces some ultrasonic cutting
stylus chatter that causes some unpleasantness. Beats me. A lot of people love them. I’m used to being the weirdo.
Addicted to Fun and Learning
Forum Donor
Jan 29, 2019
How do you like the C49 in general? It's on my shortlist...
I love it. I think you can certainly get something that sounds as good for less but not with all the convenience features. If, like some here, you use one DAC as a source, it’s a waste. If you have
multiple sources and/or amps, it’s great. It has multiple trigger outs and an HT passthru trigger in. My AVR trigger turns it on and puts it in pass through mode. Trigger one is set to turn on my
solid state amp whenever output one is enabled. Output 2 goes to my tube amp. Trigger “main” turns on my sub. Trigger 2 turns on EQ for all sources except my Roon server. All the inputs are named
what they are ( Roon, Radio, Disk, Art Usb, etc) and all unused inputs are marked off. Enough output voltage to drive anything, great sounding DAC and phono preamps. Cartridge loading can be changed
while sitting on the couch. And if you like the look, you like the look. The only thing I don’t like is that you have to take it to a service center for any firmware upgrades. That’s kind of
backward. Some people don’t like the lack of a subwoofer output or a high pass filter for the main speakers. You can use an external crossover if you like but I don’t. The natural roll off of a
speaker can be matched to a sub with the sub crossover if you want/need to run a sub. Each output has balanced and unbalanced connections. Since I use a sub, i run the unused output pairs into a rack
mount mixer/splitter to mix down to one mono channel for a sub send. Looking at the minster amps they sell, Mac probably figures everyone will be using some big full range speakers with no need for a
Last edited:
Sad that you have to fix it hough, especially when they have a cheaper cart (VM95ML), same stylus, that does not have this problem.
The only cartridge that I know of that has a truly flat F/R and is in current production is the Dynavector Karat.
It achieves this by having the cantilever resonance at around 50kHz - leaving the audible spectrum ruler flat.
Every MM's performance is a sum of a non flat cantilever response and a non flat electrical response.... getting the two to match perfectly to provide a truly flat F/R is as much black art as it is
science.... in most cases the best you can achieve is a compromise.
At the peak of vinyl technology, a number of manufacturers managed to get their effective tip masses low enough to push the cantilever resonance out beyond 50kHz - Technics even managed to get it to
100kHz - once you achieve that, then getting a flat f/r is quite viable.
The very very well regarded Shure V15VMR, although an excellent cartridge, has a cantilever resonance at around 32kHz - the flat F/R is achieved through judicious balancing of electrical load. Yep if
your R and C aren't spot on - neither will the f/r be! - and by the way, Shures specs provide a "range" for both loadings and not a specific value - so most setups don't achieve a flat f/r - even
with one of the most Neutral cartridges available.
Most MC's are worse - a lot worse - because the cantilever response is exposed without any substantive EQ... and 99% of todays MC's have a cantilever resonance WITHIN THE AUDIBLE RANGE !!
To get the cantilever resonance outside the audible range (and WELL outside it) you need to either have a very very short cantilever (like the Kara) - or the cantilever needs to be an exotic ultra
light material in tube form... either way the objective is to get the effective tip mass down to vanishingly low levels.
Current Boron/Ruby/Sapphire cantilevers are all rods rather than tubes, and that structure will at best get tip mass that matches the best of aluminium tubes ... with resonance around 19kHz - many
such efforts have their resonances in the 14kHz to 16kHz range.
I can take any halfway decent cartridge/stylus and work through the process of tuning the EQ, thereby resulting in a flat (ish) frequency response- if you have a flat f/r without doing this -
congratulations you hit the cartridge/stylus lottery jackpot.
Also keep in mind there is substantial variation in effective mass (and therefore resonant frequency) within styli of the same brand and model - I have 2 Jico SAS styli purchased only a few months
appart, of identical spec, one measuring a disapointing 14kHz resf, and the other 16khz resf - both disapointing really, as I was hoping for something to match the golden age performance of the
But that is why I consider your claim that the VM95ML f/r is a "fixed problem" somewhat ridiculous.... which is not to say that you sample in your setup has coincidentally ended up with a flat f/r
.... it is perfectly possible - after all people win the lottery every day!
But that is why I consider your claim that the VM95ML f/r is a "fixed problem" somewhat ridiculous.... which is not to say that you sample in your setup has coincidentally ended up with a flat f/
r .... it is perfectly possible - after all people win the lottery every day!
Accepted. No probs, but the freq. response graphs I could find by searching around, all seemed to be missing that 10kHz peak, that most of the 540/740ML graphs seemed to exhibit, and the VM95ML I
have sounds much better (I have had two of them) than the 440ML ever did.
Good explanation of the problem in greater depth though, so thanks.
Simply, the VM95 likes more capacitance.
May 9, 2021
Simply, the VM95 likes more capacitance.
yes ... and AT modified the compliance to better match budget tonearms
Dec 26, 2017
Not sure tone arms or compliance matter much on frequency response, at least in the light end, 6-12grams arm and 6-10gram cartridges gives the same response on my 4 turntables , res freq is 7-10
shows very little frequency response difference if any. The high quality arm (SME V) gives a response with minimal irregularities/ resonance in music range. The cheaper ones have more obvious peaks
and bumps that indicate resonances in arm, but overall response is the same
Addicted to Fun and Learning
Nov 5, 2022
I love it. I think you can certainly get something that sounds as good for less but not with all the convenience features. If, like some here, you use one DAC as a source, it’s a waste. If you
have multiple sources and/or amps, it’s great. It has multiple trigger outs and an HT passthru trigger in. My AVR trigger turns it on and puts it in pass through mode. Trigger one is set to turn
on my solid state amp whenever output one is enabled. Output 2 goes to my tube amp. Trigger “main” turns on my sub. Trigger 2 turns on EQ for all sources except my Roon server. All the inputs are
named what they are ( Roon, Radio, Disk, Art Usb, etc) and all unused inputs are marked off. Enough output voltage to drive anything, great sounding DAC and phono preamps. Cartridge loading can
be changed while sitting on the couch. And if you like the look, you like the look. The only thing I don’t like is that you have to take it to a service center for any firmware upgrades. That’s
kind of backward. Some people don’t like the lack of a subwoofer output or a high pass filter for the main speakers. You can use an external crossover if you like but I don’t. The natural roll
off of a speaker can be matched to a sub with the sub crossover if you want/need to run a sub. Each output has balanced and unbalanced connections. Since I use a sub, i run the unused output
pairs into a rack mount mixer/splitter to mix down to one mono channel for a sub send. Looking at the minster amps they sell, Mac probably figures everyone will be using some big full range
speakers with no need for a sub.
That's exactly how I'm looking to run it: with HT Bypass through my AVR to my MC302 to my mains, with additional inputs for the turntable and Roon bridge going straight into the C49. That is pretty
annoying about the firmware upgrades. Do you have the DA1 or DA2 module by chance?
Dec 26, 2017
Meh. The +2dB is noticeable, but not exceedingly bright. There are much worse out there.
yes ... and AT modified the compliance to better match budget tonearms
Shall we say - to better match the
fashion in budget tonearms.... which is mid-low compliance
Budget tonearms in the 1980's and early 90's were all low mass requiring high compliance and budget tonearms in the 1970's were mid mass (1960's high mass)
Shall we say - to better match the current fashion in budget tonearms.... which is mid-low compliance
Budget tonearms in the 1980's and early 90's were all low mass requiring high compliance and budget tonearms in the 1970's were mid mass (1960's high mass)
What does "budget" mean? Should not be used as a synonym for cheap, or even less, badly made. For example the old SME3009 - which I still favour.
What does "budget" mean? Should not be used as a synonym for cheap, or even less, badly made. For example the old SME3009 - which I still favour.
LOL the SME's were never "budget"...
But look at the arms fitted as standard on the turntables in the mass market lower price categories...
The Hanpin made TT's - mid mass arms roughly modelled on things like the Technics SL1200mkII arms.
In the late 1980's the mass market turntables had low mass arms with either T4p cartridges or something like the AT95 family (low mass arms, high compliance cartridge/stylus)
My first TT around 1981 was a Pioneer with a low mass straight arm - and based on a high schoolers income, it was definitely lower end! (but still decent quality)
Aug 20, 2020
Accepted. No probs, but the freq. response graphs I could find by searching around, all seemed to be missing that 10kHz peak, that most of the 540/740ML graphs seemed to exhibit, and the VM95ML I
have sounds much better (I have had two of them) than the 440ML ever did.
Good explanation of the problem in greater depth though, so thanks.
Folks think the VM95 can't sound good because it is cheap. That's what I thought before I bought one with the shibata stylus. After using it, I realized I would be okay with it as an "only"
cartridge. I have several other more expensive cartridges, but compared to the VM95SH they don't sound any better. The Clearaudio Maestro 2 with the boron rod and shibata does not sound any better
and it is based on the VM95 generator! | {"url":"https://www.audiosciencereview.com/forum/index.php?threads/not-loving-my-at740mlx.58031/page-2","timestamp":"2024-11-05T09:22:16Z","content_type":"text/html","content_length":"190035","record_id":"<urn:uuid:29b501f2-01c4-4fdf-a1cf-55e5af932a30>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00174.warc.gz"} |
wo numbers have a sum of 112. if one number is x, represent the
two numbers have a sum of 112. if one number is x, represent the other number as an expression in...
two numbers have a sum of 112. if one number is x, represent the other number as an expression in x. | {"url":"https://www.sweetstudy.com/content/two-numbers-have-sum-112-if-one-number-x-represent-other-number-expression","timestamp":"2024-11-11T05:15:26Z","content_type":"text/html","content_length":"120126","record_id":"<urn:uuid:ec9f7d25-52f7-4652-bf8c-ada6d5f03058>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00516.warc.gz"} |
Copy Multiple Excel Charts Into Powerpoint 2024 - Multiplication Chart Printable
Copy Multiple Excel Charts Into Powerpoint
Copy Multiple Excel Charts Into Powerpoint – You could make a multiplication graph or chart in Shine by using a format. You can find many instances of web templates and learn how to structure your
multiplication graph or chart using them. Here are a few tricks and tips to create a multiplication graph. Once you have a design, all you need to do is copy the formulation and paste it in the new
cell. After that you can take advantage of this solution to increase several numbers by another set. Copy Multiple Excel Charts Into Powerpoint.
Multiplication desk web template
You may want to learn how to write a simple formula if you are in the need to create a multiplication table. Initially, you must secure row one of many header line, then flourish the amount on row A
by cellular B. A different way to create a multiplication dinner table is to use mixed referrals. In such a case, you would probably get into $A2 into line A and B$1 into row B. The outcome is really
a multiplication kitchen table with a formula that works for columns and rows.
You can use the multiplication table template to create your table if you are using an Excel program. Just open the spreadsheet with your multiplication kitchen table change and template the brand
towards the student’s brand. You can also change the page to fit your person demands. There is an option to affect the shade of the cellular material to modify the appearance of the multiplication
desk, as well. Then, you may alter the range of multiples to meet your requirements.
Creating a multiplication chart in Excel
When you’re making use of multiplication desk computer software, you can actually produce a basic multiplication dinner table in Excel. Just build a sheet with rows and columns numbered from one to
40. The location where the rows and columns intersect is definitely the respond to. For example, if a row has a digit of three, and a column has a digit of five, then the answer is three times five.
The same goes for the other way around.
Initial, you are able to enter into the figures you need to multiply. For example, if you need to multiply two digits by three, you can type a formula for each number in cell A1. To produce the
figures larger, pick the cells at A1 and A8, after which click on the appropriate arrow to choose a selection of tissues. You may then type the multiplication formulation in the tissue within the
other rows and columns.
Gallery of Copy Multiple Excel Charts Into Powerpoint
Is There A Way To Copy And Paste Multiple Charts That Are Grouped In
Is There A Way To Copy And Paste Multiple Charts That Are Grouped In
Is There A Way To Copy And Paste Multiple Charts That Are Grouped In
Leave a Comment | {"url":"https://www.multiplicationchartprintable.com/copy-multiple-excel-charts-into-powerpoint/","timestamp":"2024-11-06T10:30:08Z","content_type":"text/html","content_length":"50660","record_id":"<urn:uuid:b595090d-f5c1-487c-89fc-748f54f0bb43>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00042.warc.gz"} |
Confidence and prediction intervals explained… (with a Shiny app!) | R-bloggersConfidence and prediction intervals explained… (with a Shiny app!)
Confidence and prediction intervals explained… (with a Shiny app!)
[This article was first published on
R | Adi Sarid
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
This semester I started teaching introduction to statistics and data analysis with R, at Tel-Aviv university.
I put in a lot of efforts into bringing practical challenges, examples from real life, and a lot of demonstrations of statistical theory with R. This post is an example for how I’ve been using R code
(and specifically Shiny apps) to demonstrate statistical theory, concepts and provide intuition.
What’s the difference between confidence and prediction intervals?
Last week I taught multiple linear regression, and I noticed that students have a hard time comprehending the difference between confidence intervals and prediction intervals. The former being an
interval for the model (i.e., interval for the underlying model), and the latter being an interval for a noval observation.
As the sample size increases, our uncertainty of the model’s parameters decreases, but the uncertainty in the value of a new observation, \(y_0\) is associated with variance of \(Y\) (the random
variable from which \(y_0\) is drawn). Hence, it has a lower bound, based on that variance.
In R, you can get a prediction or a confidence interval by using either
predict(object, newdata, interval = "prediction")
predict(object, newdata, interval = "confidence")
For a prediction or for a confidence interval, respectively.
To help me illustrate the differences between the two, I decided to build a small Shiny web app. It shows the differences between confidence intervals, prediction intervals, the regression fit, and
the actual (original) model.
The app is available here, and the source code is available on github.
With this app you can choose three types of models to demonstrate. Simple linear regression, and regression with a twist (\(\log\) transformation on the \(y\) or \(\sin\) transformation on the \(x\):
• Linear model \(y = a + bx + \epsilon\)
• Log-linear model \(\log(y)=a+bx+\epsilon\)
• Sine \(y = a + b\sin(x) + \epsilon\)
All the models are based on simple linear regression (lm function), for the latter two models with either a log or sin transformation.
The app allows you to play around with various values such as the \(x\) range, the model’s parameters (\(a\) and \(b\)), the error’s standard deviation (\(\epsilon\)), and show or hide any of the
following elements, on the chart:
• The original function (i.e., the original model)
• The sampled points
• The confidence interval
• The prediction interval
• The model’s fit
Feel free to share the app or the app’s code. As mentioned above, the source code for the app is available here: https://github.com/adisarid/prediction_confidence_intervals_demo.
Here’s an example for what the app’s generating code and output looks like, for a model of the type \(\log(y) = 1 + \frac{x}{2} + \epsilon\):
sample_size <- 90
x_range <- c(0, 1.5)
a <- 1
b <- 1.5
sigma <- 0.3
actual_function <- tibble(x = seq(x_range[1], x_range[2], by = 0.01)) %>%
mutate(actual_y = exp(a + b*x))
random_sample <- tibble(epsilon_err = rnorm(n = sample_size,
mean = 0,
sd = sigma),
x = runif(n = sample_size,
min = x_range[1],
max = x_range[2])) %>%
mutate(sampled_y = exp(a + b*x + epsilon_err))
linear_model <- lm(formula = log(sampled_y) ~ x, data = random_sample)
prediction_i <- predict(object = linear_model,
newdata = actual_function,
interval = "prediction") %>%
as_tibble() %>%
rename_at(vars(lwr,upr), ~paste0(., "_pi")) %>%
confidence_i <- predict(object = linear_model,
newdata = actual_function,
interval = "confidence") %>%
as_tibble() %>%
rename_at(vars(lwr,upr), ~paste0(., "_ci")) %>%
select(-fit) %>%
intervals <- actual_function %>%
ggplot() +
geom_line(data = actual_function, aes(x, actual_y, color = "Original Model"), size = 1) +
geom_point(data = random_sample, aes(x, sampled_y), alpha = 0.5) +
geom_line(data = intervals,
aes(x, fit, color = "Regression Fit"), size = 1) +
geom_line(data = intervals,
aes(x, lwr_pi, color = "Prediction Interval"),
linetype = 2, size = 1) +
geom_line(data = intervals,
aes(x, upr_pi, color = "Prediction Interval"),
linetype = 2, size = 1) +
geom_line(data = intervals,
aes(x, lwr_ci, color = "Confidence Interval"),
linetype = 2, size = 1) +
geom_line(data = intervals,
aes(x, upr_ci, color = "Confidence Interval"),
linetype = 2, size = 1) +
theme_bw() +
xlab("x") +
ylab("y") +
ggtitle("Log-linear: Model, Fit, Confidence and Prediction Intervals")
Shiny apps are a great way to illustrate theoretical concepts, to provide intuition, and to let students experiment with parameters and see the outcomes. In this post I demonstrated how a Shiny app
can be used to explain the concepts of a regression fit, confidence, and prediction intervals.
If you used Shiny for interesting educational demonstrations I’d love to hear about it! feel free to share in the comments or message me on twitter @SaridResearch. | {"url":"https://www.r-bloggers.com/2019/12/confidence-and-prediction-intervals-explained-with-a-shiny-app-2/","timestamp":"2024-11-08T20:28:12Z","content_type":"text/html","content_length":"98505","record_id":"<urn:uuid:292e98bb-7cd1-4a71-93f6-f6f8091e11e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00685.warc.gz"} |
Solar panels: ≤ 1m2 PV per m2 floor …
As we want to get rid of fossil fuels, we will at least have to develop everything now without fossil fuels. And in any case the operational energy. In the case of new buildings this leads to
“0-energy buildings”, which generate as much renewable energy as required, locally, and thus take their responsibility in the transition. This mainly concerns solar panels on or at these buildings.
Which leaves us with the embodied energy of all materials involved. This is a bit more difficult, but all new construction at least must be CO2 neutral by 2050, or , as in our example, will have
compensated the Energy / CO2 of production before 2050, in order to be net CO2 neutral.
Now what does the (simplified) picture look like?
Suppose you build according to the Dutch standard as of 1 January, which is, among others, max. 55 kWh operational building-related energy demand per m2. And suppose such a m2 costs about 5.5 GJ,
(1527 kWh) of (fossil) embodied energy to build.
In order to be climate neutral by 2050, first of all the 55 kWh must be generated annually in a renewable manner. But in addition, the embodied energy must also be compensated within those 30 years.
That is 1527/30 = approximately 50 kWh per year per m2 extra renewable has to be generated, and only then the house will be a “net” 0-CO2 house from 2050 on.
In other words, 105 kWh / year must be supplied per m2 of floor for the first 30 years (55 + 50). Suppose a Solar panel supplies 157.5 kWh / m2 per year (250 Wp at 1.6 m2), then that would be
sufficient to serve 1.5 m2 of floor, or justify its construction within the 2050 target.
However, that’s the optimistic calculation. After all, we also have to include the embodied energy of the production of that PV panel. In a Meta study Khagendra [1] finds for poly-Si (to about
4000–8000 MJ/m 2 and for mono-Si and 2200–6600 MJ/m 2 [x] . If we use the lowest figure, which is quit optimistic since its without impact from supporting material (BOS), its 611 kWh/m2. That panel
will therefore only work for its own compensation during the first 4 years (NL) . The 30 years until 2050 is then reduced to approximately 26 years, and the EE of construction to be compensated
becomes not 50 but 59 kWh / year and the required annual production 114, or 1 m2 panel is sufficient to serve for 1.38 m2 of floor.
Again, that is still theoretical. If we calculate a bit more realistically, with a yield of approx. 0.9 kWh / Wp x 157.5 kWh = approx. 140 kWh per m2 panel is produced on average in the Netherlands,
and the sum becomes: 140/114 is 1.23 m2 of floor…
Or the other way around, to make a house / building energy-neutral, and to compensate for the embodied energy, in order it to be net climate-neutral by 2050, approximately 0.81 m2 of solar panel is
needed per m2 of floor, on or near the building.
Interestingly, it can be deduced from this that with 1 story the roof may be sufficient, but that with more floors, the facade might also be needed. With 2 floors you could maybe manage with a
sloping roof with an overhang, which increases the roof surface. This is no longer possible with 3 floors, and the facade will have to be included. That is, if every m2 of the house is heated. If
compartmentalization takes place, or only part of the house is heated during the coldest period, then it all works out much easier of course.
Incidentally, it is interesting to note that high-rise buildings can never meet this requirement, unless space is available around the building to install a separate solar panel field. Which rules
out high-rise buildings as an argument to achieve higher densities: It no longer applies in a renewable energy world, because you have to take that extra Solar panel space into account in the
footprint and density calculation. (It didn’t make sense before as well, by the way, see [2])
All this is approximately of course, this is a theoretical approach and rough calculation. But it indicates the orders of magnitude we have to deal with.
There are certain bandwidths that apply:The Embodied energy of a m2 of floor can vary from 4-8 GJ, and in special cases even more or less: a clay construction or light straw bale construction can be
lower, for example. While if working with aluminum facades, or many installations, you will soon end up much higher.
If we maintain that bandwidth of 4-8, at the practical production value of 140 kWh / m2, then the annual embodied energy to be compensated (in 26 years) varies between 42.5 and 85 kWh, and with that
operational energy of 55 kWh that adds up to 97.5 and 140, or 1 m2 of PV, is good for the energy demand anywhere from 1,43 to 1 m2 of floor. Or vice versa, 0.7 to 1 m2 of PV panel is needed per m2 of
Naturally, this variation in embodied energy depends on the materials chosen and, for example, the chosen insulation thickness and other measures to limit the energy demand. A lot has already been
invested for 55 kWh per m2 of operational energy demand and the Embodied energy will be on the higher side. When going even lower, to 25 or even 15 kWh / m2, this also has an increasing effect on the
embodied energy. For a common construction method and technological energy supply, it can even be 10 GJ / m2. [3] . One has a positive effect, the other a negative effect: In case 15 kWh / m2 as
demand, and 10 GJ or 2770 kWh as embodied energy (107 kWh per year for 26 years), , that equals 122 in total, slightly lower than that 1 m2. But this is guesswork, just to show how it affects each
other. Perhaps a 15 kWh per m2 demand can also be realized with a much lower EE. There is a real optimization effort to be made here.
So far we have only included building-related energy for heating and ventilation.
If we also include, for example, 2600 kWh of household energy consumption for a home ( 26 kWh / m2, in the case of a home of 100 m2). Or an additional 18 m2 of Solar panels in total (at 140 kWh /
If we also include an electric car at 10,000 km / year, and suppose consumption at 16 kWh per 100 km, that is 1600 kWh per year. At 140 kWh / m2, that is an additional 11 m2 of solar panels.
In both cases, without solar panel compensation (the embodied energy), or compensation for the embodied energy of the consuming products, such as the electric car.
Obviously, as the national energy supply system will grow to include a larger share of renewable energy, the embodied energy in the production will cause less CO2 emissions. But that is at the
expense of an enormous national effort in the production of solar panel fields and wind turbine parks, plus associated infrastructure. which also require (partly fossil) energy. In essence, this
impact would have to be (partly) added to the embodied energy of the home, because that infrastructure is (partly) built for that. That is why I prefer to calculate here in energy units, instead of
CO2 units. If we were only calculating in CO2, that would only shift our problems towards material impacts: and theoretically imply that we would never have to limit energy use again, as long as it
is CO2 free. We just build more and more solar panels and wind turbines.
But in the time required to adjust the national energy supply, which will then consist of more renewable energy, the number of years in which compensation must be made will decrease at the same time:
The rough estimates above apply in 2020: in 2021, one year will go off, and over the years the figures become less favorable, in order to be climate neutral before 2050. In other words, more and more
“Solar panels” are needed per m2 built floor to achieve the objectives, for buildings that are to be realized later.
There is a rough rule of thumb in all this: approximately 1 m2 of PV is needed per 1 m2 of floor (for building-related energy in new construction, In 2020). Its somewhat exaggerated, but this way we
will always on the right side, because in practice it can / will be less, depending on the design, choice of materials, and choice of installation, (think of heat pump). But it will be more , the
closer we get to 2050 ….
Disclaimer: Calculations are made on the back of a napkin, and are meant for orientation, to get some feeling about the proportions. All this assuming that solar panels are a good thing, that we have
until 2050 to limit CO2, and without looking at the material exhaustion itself. I will come back to these issues.
[1] Energy payback time (EPBT) and energy return on energy invested (EROI) of solar photovoltaic systems: A systematic review and meta-analysis. Khagendra et all, 2015, Renewable and Sustainable
Energy Reviews http://dx.doi.org/10.1016/j.rser.2015.02.057
[1] http://www.ronaldrovers.com/how-to-avoid-highrise-buildings/
[2] http://www.ronaldrovers.com/building-without-heating-more-material-or-more-installations/ | {"url":"http://www.ronaldrovers.com/solar-panels-%E2%89%A4-1m2-pv-per-m2-floor/","timestamp":"2024-11-04T11:53:26Z","content_type":"text/html","content_length":"64130","record_id":"<urn:uuid:dd06ec67-b66e-4931-8852-7c5b262a2827>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00615.warc.gz"} |
entimeters to Ropes
Centimeters to Ropes Converter
β Switch toRopes to Centimeters Converter
How to use this Centimeters to Ropes Converter π €
Follow these steps to convert given length from the units of Centimeters to the units of Ropes.
1. Enter the input Centimeters value in the text field.
2. The calculator converts the given Centimeters into Ropes in realtime β using the conversion formula, and displays under the Ropes label. You do not need to click any button. If the input
changes, Ropes value is re-calculated, just like that.
3. You may copy the resulting Ropes value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Centimeters to Ropes?
The formula to convert given length from Centimeters to Ropes is:
Length[(Ropes)] = Length[(Centimeters)] / 609.5999998166322
Substitute the given value of length in centimeters, i.e., Length[(Centimeters)] in the above formula and simplify the right-hand side value. The resulting value is the length in ropes, i.e., Length
Calculation will be done after you enter a valid input.
Consider that a high-end smartphone has a screen size of 15 centimeters.
Convert this screen size from centimeters to Ropes.
The length in centimeters is:
Length[(Centimeters)] = 15
The formula to convert length from centimeters to ropes is:
Length[(Ropes)] = Length[(Centimeters)] / 609.5999998166322
Substitute given weight Length[(Centimeters)] = 15 in the above formula.
Length[(Ropes)] = 15 / 609.5999998166322
Length[(Ropes)] = 0.02460629922
Final Answer:
Therefore, 15 cm is equal to 0.02460629922 rope.
The length is 0.02460629922 rope, in ropes.
Consider that a luxury handbag measures 30 centimeters in width.
Convert this width from centimeters to Ropes.
The length in centimeters is:
Length[(Centimeters)] = 30
The formula to convert length from centimeters to ropes is:
Length[(Ropes)] = Length[(Centimeters)] / 609.5999998166322
Substitute given weight Length[(Centimeters)] = 30 in the above formula.
Length[(Ropes)] = 30 / 609.5999998166322
Length[(Ropes)] = 0.04921259844
Final Answer:
Therefore, 30 cm is equal to 0.04921259844 rope.
The length is 0.04921259844 rope, in ropes.
Centimeters to Ropes Conversion Table
The following table gives some of the most used conversions from Centimeters to Ropes.
Centimeters (cm) Ropes (rope)
0 cm 0 rope
1 cm 0.00164041995 rope
2 cm 0.0032808399 rope
3 cm 0.00492125984 rope
4 cm 0.00656167979 rope
5 cm 0.00820209974 rope
6 cm 0.00984251969 rope
7 cm 0.01148293964 rope
8 cm 0.01312335958 rope
9 cm 0.01476377953 rope
10 cm 0.01640419948 rope
20 cm 0.03280839896 rope
50 cm 0.0820209974 rope
100 cm 0.164 rope
1000 cm 1.6404 rope
10000 cm 16.4042 rope
100000 cm 164.042 rope
A centimeter (cm) is a unit of length in the International System of Units (SI). One centimeter is equivalent to 0.01 meters or approximately 0.3937 inches.
The centimeter is defined as one-hundredth of a meter, making it a convenient measurement for smaller lengths.
Centimeters are used worldwide to measure length and distance in various fields, including science, engineering, and everyday life. They are commonly used in everyday measurements, such as height,
width, and depth of objects, as well as in educational settings.
A rope is a unit of length used primarily in land measurement and construction. One rope is equivalent to 66 feet or approximately 20.1168 meters.
The rope is defined as 66 feet, which is historically based on the length used for various practical purposes, including measurement and construction tasks.
Ropes are used in land measurement, particularly in agriculture and construction, where the unit provides a practical measure for longer distances. It is similar in length to the chain and is
utilized in specific applications where its historical relevance remains significant.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Centimeters to Ropes in Length?
The formula to convert Centimeters to Ropes in Length is:
Centimeters / 609.5999998166322
2. Is this tool free or paid?
This Length conversion tool, which converts Centimeters to Ropes, is completely free to use.
3. How do I convert Length from Centimeters to Ropes?
To convert Length from Centimeters to Ropes, you can use the following formula:
Centimeters / 609.5999998166322
For example, if you have a value in Centimeters, you substitute that value in place of Centimeters in the above formula, and solve the mathematical expression to get the equivalent value in Ropes. | {"url":"https://convertonline.org/unit/?convert=centimeters-ropes","timestamp":"2024-11-09T13:31:47Z","content_type":"text/html","content_length":"91097","record_id":"<urn:uuid:905b4b0d-667f-4427-82c3-835f19b23b92>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00694.warc.gz"} |
The Brauer group of Brauer-Severi varieties
One result I really like about Brauer groups is that $\mathrm{Br}(\mathbb{P}_k^n)\cong\mathrm{Br}(k)$ for every possible field $k$. So the Azumaya algebras on projective space are not too wild, and
completely controlled by the base field.
Yesterday I started wondering about how this result is actually proved (my knowledge of proofs of arithmetic nature is abysmal), and what would happen for twisted forms of projective space, also
known as Brauer-Severi varieties. This is a very silly question, in the sense that you take a representative $A$ of an element $[A]$ of the Brauer group of the base field, construct its Brauer-Severi
variety $\mathrm{BS}(A)$, and then take its Brauer group $\mathrm{Br}(\mathrm{BS}(A))$. It is like Xzibit became a mathematician and said something silly involving dogs and Brauer groups.
The Brauer group of projective space
There are different ways of proving that $\mathrm{Br}(\mathbb{P}_k^n)=\mathrm{Br}(k)$. One can take a look at the corresponding MathOverflow question or these notes of Tetsuya Uematsu.
I will sketch an approach I learnt from Colliot-Thélène's notes, which will be useful when discussing Brauer-Severi varieties. So consider the Leray spectral sequence for $\mathbb{P}_k^n\to\mathrm
{Spec}\,k$ and the sheaf $\mathbb{G}_{\mathrm{m}}$. Denote a separable closure of $k$ by $k^s$. We get a long exact sequence $$ 0\to\mathrm{Pic}(\mathbb{P}_k^n)\to\mathrm{Pic}(\mathbb{P}_{k^s}^n)\to\
mathrm{Br}(k)\to\mathrm{Br}(\mathbb{P}_k^n)\to\mathrm{H}^1(k_{\mathrm{et}},\mathrm{Pic}(\mathbb{P}_{k^s}^n))\to\ldots $$ where $\mathrm{Br}_1(\mathbb{P}_k^n)=\mathrm{ker}\left(\mathrm{Br}(\mathbb{P}
_k^n)\to\mathrm{Br}(\mathbb{P}_{k^s}^n)\right)$ is the algebraic Brauer group. To get the desired result we need the following facts:
1. Because $\mathbb{P}_k^n$ has a $k$-rational point, the morphism $\mathrm{Pic}(\mathbb{P}_{k^s}^n)\to\mathrm{Br}(k)$ is zero, because we can obtain a retract to the first inclusion.
2. By some other method one needs to prove that $\mathrm{Br}(\mathbb{P}_{k^s}^n)=0$, hence the algebraic Brauer group of projective space is equal to the whole Brauer group.
3. To prove surjectivity, we need that $\mathrm{H}^1(k_{\mathrm{et}},\mathrm{Pic}(\mathbb{P}_{k^s}^n))=0$. As explained in Colliot-Thélène's notes, one can appeal to the short exact sequence of
Galois modules $$0\to\mathrm{Pic}_{\mathbb{P}_k^n/k}^0(k^s)\to\mathrm{Pic}(\mathbb{P}_{k^s}^n)\to\mathrm{NS}(\mathbb{P}_{k^s}^n)\to 0$$ for this, where in our situation we have that the first
term is zero, and the third term is $\mathbb{Z}$ with trivial action, hence we indeed get vanishing, and surjectivity.
The Brauer group of Brauer-Severi varieties
A first misconception that I had is that for every variety $X$ defined over a field $k$ there exists an inclusion $\mathrm{Br}(k)\hookrightarrow\mathrm{Br}(X)$. Such a thing needs the existence of a
$k$-rational point, which of course is not the case for any non-trivial Brauer-Severi variety.
I haven't found a general statement for Brauer group of Brauer-Severi varieties, I will now discuss what I know for sure in dimension 1. But I'll pose a precise question after the sketch of the
proof, which might also be a proof for the general question.
The following is based on section 5.0.3 in Colliot-Thélène's notes. So let $C/k$ be a conic, where $\mathrm{char}(k)\neq 2$ to be on the safe side. Writing it as $\mathbb{V}(x_0^2-ax_1^2-bx_2^2)$
after an appropriate change of coordinates we know that it actually is the Brauer-Severi variety for the quaternion algebra $(a,b)_k$.
In this situation we still have the vanishing of $\mathrm{H}^1(k_{\mathrm{et}},\mathrm{Pic}(\mathbb{P}_{k^s}^n))$, in precisely the same way.
On the other hand, if there is no rational point (i.e. the quaternion algebra is not split), the morphism $\mathrm{Pic}(\mathbb{P}_k^n)\to\mathrm{Pic}(\mathbb{P}_{k^s}^n)$ turns out to have image $2\
mathbb{Z}$. Hence we get a short exact sequence $$ 0\to\mathbb{Z}/2\mathbb{Z}\to\mathrm{Br}(k)\to\mathrm{Br}(C)\to 0 $$
and the non-zero element in $\mathrm{Br}(k)$ is precisely the quaternion algebra whose Brauer-Severi variety we are looking at. E.g. if we take the Brauer-Severi variety for the Hamilton quaternions
over the field, we see that any non-split conic (they are all isomorphic anyway) over the reals has trivial Brauer group!
Based on this proof, I have the following question.
Question Is the Brauer group of a Brauer-Severi variety of a central simple algebra always the Brauer group of the base field modulo the subgroup generated by the central simple algebra?
It seems to me that the proof outlined above should give precisely this result. Am I overlooking something, or is this indeed the case? | {"url":"https://pbelmans.ncag.info/blog/2016/04/20/the-brauer-group-of-brauer-severi-varieties/","timestamp":"2024-11-11T17:35:12Z","content_type":"text/html","content_length":"24955","record_id":"<urn:uuid:476c4789-f715-4f32-adac-7788c7e73123>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00737.warc.gz"} |
Weisheng Si, Quincy Tse, and Frédérik Paradis
This package provides functors for constructing two kinds of cone-based spanners: Yao graph and Theta graph, given a set of vertices on the plane and the directions of cone boundaries. Both exact and
inexact constructions are supported. In exact construction, the cone boundaries are calculated using the roots of polynomials, thus avoiding the use of \( \pi \), which cannot be represented exactly.
In inexact construction, the cone boundaries are calculated using the approximate \( \pi \) value defined in CGAL, which is still accurate enough for most applications. Moreover, for visualization
purpose, this package provides a global function to generate the data and script files used by Gnuplot to plot the constructed graphs. This package also provides options for the Half Yao graph and
the Half Theta graph.
class CGAL::Compute_cone_boundaries_2< Traits_ >
The functor for computing the directions of cone boundaries with a given cone number and a given initial direction. More...
class CGAL::Construct_theta_graph_2< Traits_, Graph_ >
A template functor for constructing Theta graphs with a given set of 2D points and a given initial direction for the cone boundaries. More...
class CGAL::Construct_yao_graph_2< Traits_, Graph_ >
A template functor for constructing Yao graphs with a given set of 2D points and a given initial direction for the cone boundaries. More...
template<typename Graph >
void CGAL::gnuplot_output_2 (const Graph &g, const std::string &prefix)
outputs a set of files used by Gnuplot to plot g.
◆ Cones_selected
◆ gnuplot_output_2()
template<typename Graph >
void CGAL::gnuplot_output_2 ( const Graph & g,
const std::string & prefix
#include <CGAL/gnuplot_output_2.h>
outputs a set of files used by Gnuplot to plot g.
The files that are generated for Gnuplot are: (1) prefix.v (vertex list) (2) prefix.plt (Gnuplot script), This script will read prefix.v as input to plot the vertex list. The edge list is also
included in this script.
Notes: (1) If these files already exists, this function will overwrite these files. (2) Parallel and self-edges cannot be plotted.
Template Parameters
Graph The type of the graph to be plotted. For this function to work, the graph type must be boost::adjacency_list with CGAL::Point_2 as the VertexProperties.
g A boost::adjacency_list graph with CGAL::Point_2 as the VertexProperties to be plotted
prefix The prefix of the output files names | {"url":"https://cgal.geometryfactory.com/CGAL/doc/master/Cone_spanners_2/group__PkgConeSpanners2Ref.html","timestamp":"2024-11-03T16:40:26Z","content_type":"application/xhtml+xml","content_length":"18912","record_id":"<urn:uuid:9a77f246-d09b-4ab0-882f-cb96a4bbedf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00095.warc.gz"} |
Medical Math Printable Version (pdf)
Course Introduction
Core Standards of the Course
Strand 1
Uses of Mathematics in Healthcare
Standard 1
Analyze the use of medical mathematics in the healthcare system
1. Explore different healthcare careers and the math used within the career.
2. Compare and contrast at least two different careers.
Strand 2
Common Mathematical Operations as used in Healthcare
Standard 1
Compute fluently and make reasonable estimates.
1. Evaluate and simplify numerical expressions containing rational numbers using the order of operations.
2. Compute solutions to problems and determine the reasonableness of an answer by relating them to applied scenarios
3. Whole Numbers
Standard 2
Represent real numbers in a variety of ways.
1. Choose appropriate and convenient forms of rational numbers for solving problems and representing answers (e.g., decimal, fraction, or percent).
2. Compute solutions to problems and determine the reasonableness of an answer by relating them to applied scenarios.
3. Decimals
Standard 3
Identify relationships among rational numbers and operations involving these numbers.
1. Compute solutions to problems and determine the reasonableness of an answer by relating them to applied scenarios.
2. Fractions
Standard 4
Calculate percentages
1. Compute solutions to problems and determine the reasonableness of an answer by relating them to applied scenarios.
2. Percents
Strand 3
Ratios and Proportions
Standard 1
Evaluate, solve, and analyze mathematical situations, using algebraic properties and symbols.
1. Solve proportions that include algebraic first-degree expressions (solve for x or use dimensional analysis).
Standard 2
Use ratios to compare data.
1. Laboratory values
2. Medications
3. Diseases (statistics)analysis).
Strand 4
Gathering Data (Use of Medical Instruments)
Standard 1
Use patterns, relations, and functions to represent mathematical situations.
1. Conversions
2. Metric units
3. Time (12/24)
4. Roman numerals (Arabic/Roman)
5. Temperature (Celsius/Fahrenheit)
6. Pre/Post workout weight analysis
7. Body composition
8. Pharmacology
Standard 2
Represent quantitative relationships using mathematical models and symbols.
1. Find and interpret rates of change by analyzing graphical and numerical data.
2. Understand measureable attributes of objects and the units, systems, and processes of measurement.
3. Solve problems using visualization, and spatial reasoning.
4. Instruments
Standard 1
Formulate and answer questions by collecting, organizing, and analyzing statistical data.
1. Collect, record, organize, and display a set of statistical data.
2. Determine whether the pattern of the data is linear or nonlinear when given in a list, table, or graph.
3. Interpret the correlation between two variables as positive, negative, or having no correlation.
4. Find a line of best fit by estimation, choosing two points, or using technology for a given set of statistical data.
5. Analyze the meaning of the slope and y-intercept of a line of best fit as it relates to the statistical data set.
6. Find mean, median, mode, and range for a statistical data set.
7. Analyze the meaning of the maximum or minimum and intercepts of the regression equation as they relate to a given set of bivariate data.
8. Analyze the meaning of the maximum or minimum and intercepts of the regression equation as they relate to a given set of bivariate data.
9. Make predictions and estimations and determine their reasonableness using a regression equation (line of best fit).
10. Graphs and charts
□ Interpreting charts and graphs
□ Temperature, pulse, respirations graphs
□ Intake and output charts
□ Height, weight, measurement graphics
□ Cardiac output
□ Medication errors
□ Census
□ Acuities
□ Disease, mortality rates
□ Job outlook, projections
□ Treatment prognosis
□ Clinical trials
□ Health care costs
□ Effectiveness (facilities, providers)
□ Wellness indicators
□ Reliability and validity
□ Body mass index (BMI)
□ Body composition
□ Epidemiology
Standard 2
Apply basic concepts of probability.
1. Determine and express the probability of an event as a fraction, percent, ratio, or decimal.
2. Determine the conditional probability of an event (false positive/false negative).
Standard 1
Compute fluently and make reasonable estimates.
1. Reading drug labels and prescriptions
2. Interpreting prescriptions/Patient instructions
Standard 2
Evaluate, solve, and analyze mathematical situations using algebraic properties and symbols.
1. Simplify and evaluate numerical expressions (including integer exponents and square roots), algebraic expressions, formulas, and equations.
2. Using medical reference books to determine if calculated dosages are safe
Standard 3
Represent quantitative relationships using mathematical models and symbols.
1. Dosing
2. Dosage conversions
Strand 7
Medical Accounting and Business
Standard 1
Apply systems of order.
1. Numerical filing
2. Appointment scheduling
Standard 2
Evaluate, solve, and analyze mathematical situations using algebraic properties and symbols.
1. Maintaining accounts
2. Checks, deposit slips, and receipts
3. Calculating cash transactions/Payroll
4. Budgeting
5. Depreciation, amortization
6. Insurance
Strand 8
Exponents and Logarithms
Standard 1
Use properties of exponentials to solve equations.
1. Radiation exposure
2. Half life
Standard 2
Use properties of logarithms to solve equations.
1. pH
• Oral presentation on chosen healthcare career mathematics.
• Use healthcare career choice to create a business model.
• Critical thinking
• Collaboration
• Communication (Oral/Written)
• Organization
• Technical skills
• Consumer awareness
• Commercial awareness
• Legal requirements/expectations
• Interpersonal relationships
http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE). Send questions or comments to USBE Specialist - MAREN HANSEN and see the CTE/
Health Science website. For general questions about Utah's Core Standards contact the Director - THALEA LONGHURST. These materials have been produced by and for the teachers of the State of Utah.
Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not
be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200. | {"url":"https://www.uen.org/core/core.do?courseNum=519999","timestamp":"2024-11-14T22:17:15Z","content_type":"text/html","content_length":"72823","record_id":"<urn:uuid:354f6271-4400-41c9-9c5d-e963c1931754>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00353.warc.gz"} |
You merely take the rate of interest per period and you can multiply they by worth of the borrowed funds a good
• PMT = overall commission per months
• Photo voltaic = introduce worth of financing (amount borrowed)
• i = months rate of interest conveyed due to the fact a decimal
• n = amount of mortgage costs
The present property value a keen annuity formula equates simply how much a good blast of monthly payments generated at typical times is worth from the newest go out. From the rearranging new
algorithm, we are able to estimate exactly how much for every commission should be worthy of in the purchase to help you equivalent something special value, where in actuality the introduce well
worth ‘s the worth of the borrowed funds. The latest payment determined is the complete payment each month to possess along the loan. Loan payments integrate two parts: money to your principal, and
you can costs cashcentralpaydayloans.com/payday-loans-mo/ into the attention.
Within the total financing payment for every period, this new borrower need generate a cost on attract. The lender fees interest while the pricing on the debtor away from, better, borrowing from the
bank the cash. This really is due to enough time value of currency principle, as currency now will probably be worth more money the next day. Attention is straightforward so you can assess. The
newest algorithm try found lower than:
• P = principal left
• i = months interest expressed as a decimal
There isn’t a good lead answer to determine the fresh new percentage into principal each month, but we could returning to the significance because of the deducting the quantity interesting paid in a
time on total commission for every months. As the attract and you can principal will be the simply two parts of your own fee per period, the sum of the notice for every single months and you will
dominating for each and every period need equivalent this new percentage each period.
Amortization Plan Example
Why don’t we have a look at an example. Assume you are taking aside a step three-season, $100,000 financing on 6.0% a year, that have monthly installments. Whenever strengthening out a table, I think
the most important part is the settings. Once a great desk is initiated, filling out the prices is relatively effortless. Less than is actually an example of a dining table that could be made use of
to your agenda:
Right here, we can observe much we pay into dominating and you may desire for each months, the entire payment for every single period, and also the kept equilibrium. You can most other columns,
including collective prominent payments generated, and you can cumulative interest paid down, however, this really is up to you.
All right, we have now to truly submit the brand new desk. We are able to start with for every single month’s “Payment” formula. We will use the algorithm a lot more than, where present worth of the
mortgage is actually $one hundred,100000, the speed per period is actually 0. just like the we’re working with monthly obligations, and you may all of our amount of costs is actually thirty-six,
that’s a dozen money a-year for three ages. The newest computation are revealed less than:
Very, monthly, the complete fee would be $3,. Now, we need to assess how much of these are repaid to the attract each month. We will play with our very own algorithm over, together with tasks are
shown less than towards the first few days:
New part of the payment paid back into attention was $five hundred in the first months. The fresh new part reduced to the appeal may differ for every several months, due to the fact harmony of
mortgage will change per several months, however, I’m able to search on the one in just a while.
2nd, we should instead estimate the new piece paid back towards the dominant, that’s precisely the total commission quicker appeal. The fresh new calculation is actually revealed lower than:
Everything you spend on the desire does not impact the equilibrium regarding the loan
Our company is almost carried out with our very own first period’s calculations. The very last area, that i have not talked about yet ,, is when the bill changes. The bill of the mortgage immediately
following a good period’s payment ‘s the earlier balance of the mortgage faster the newest part of the commission produced toward prominent. For the very first several months, the earlier harmony of
your loan is the full equilibrium. This new computation is actually found below: | {"url":"https://wawwf.org/you-merely-take-the-rate-of-interest-per-period/","timestamp":"2024-11-03T06:33:47Z","content_type":"text/html","content_length":"62877","record_id":"<urn:uuid:49f28e01-1d75-4d04-b2e4-a586124914f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00568.warc.gz"} |
Free Multiplication Chart Printable Paper Trail Design In 2021 | Multiplication Chart Printable
Free Multiplication Chart Printable Paper Trail Design In 2021
Free Multiplication Chart Printable Paper Trail Design In 2021
Free Multiplication Chart Printable Paper Trail Design In 2021 – A Multiplication Chart is a handy tool for children to learn just how to increase, split, and find the smallest number. There are
several usages for a Multiplication Chart.
What is Multiplication Chart Printable?
A multiplication chart can be made use of to aid kids learn their multiplication truths. Multiplication charts come in numerous types, from full page times tables to solitary page ones. While
individual tables serve for presenting chunks of details, a complete web page chart makes it less complicated to assess realities that have currently been understood.
The multiplication chart will typically include a left column and a top row. The leading row will certainly have a checklist of items. Select the very first number from the left column as well as the
second number from the leading row when you desire to find the item of two numbers. As soon as you have these numbers, move them along the row or down the column until you reach the square where the
two numbers meet. You will then have your item.
Multiplication charts are useful understanding tools for both kids as well as grownups. Kids can utilize them in your home or in school. Printable Multiplication Practice Chart are readily available
on the Internet and also can be published out and laminated for sturdiness. They are a wonderful tool to utilize in mathematics or homeschooling, as well as will certainly supply a visual tip for
kids as they discover their multiplication realities.
Why Do We Use a Multiplication Chart?
A multiplication chart is a layout that shows just how to increase 2 numbers. You pick the initial number in the left column, move it down the column, and also after that pick the 2nd number from the
top row.
Multiplication charts are practical for lots of factors, including assisting children find out how to split and streamline portions. Multiplication charts can additionally be practical as desk
resources since they serve as a continuous reminder of the trainee’s progression.
Multiplication charts are additionally helpful for helping students remember their times tables. As with any kind of ability, remembering multiplication tables takes time and also practice.
Printable Multiplication Practice Chart
Multiplication Chart 1 12 Worksheet
Multiplication Chart UDL Strategies
Printable Multiplication Practice Chart
If you’re looking for Printable Multiplication Practice Chart, you’ve come to the ideal place. Multiplication charts are available in various formats, including complete dimension, half dimension,
and also a selection of charming designs.
Multiplication charts as well as tables are crucial tools for youngsters’s education and learning. These charts are excellent for use in homeschool math binders or as classroom posters.
A Printable Multiplication Practice Chart is a beneficial tool to reinforce mathematics realities and also can assist a kid learn multiplication swiftly. It’s additionally a fantastic tool for skip
counting as well as discovering the times tables.
Related For Printable Multiplication Practice Chart | {"url":"https://multiplicationchart-printable.com/printable-multiplication-practice-chart/free-multiplication-chart-printable-paper-trail-design-in-2021-5/","timestamp":"2024-11-11T18:09:35Z","content_type":"text/html","content_length":"28218","record_id":"<urn:uuid:18e8d6b1-2520-4367-b73a-ea1bca039e54>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00454.warc.gz"} |
Magnetic helicity and force-free properties of astrophysical magnetic fields
Magnetic fields are ubiquitous in the universe and play an important role in variety of astrophysical phenomenon. It is thus very important to understand the origin, structure and strength of these
astrophysical magnetic fields. In this Thesis, we use the concept of magnetic helicity conservation and prop- erties of force-free magnetic fields to investigate the topological properties of
magnetic fields in the solar corona and the amplification and nonlinear saturation of dynamo generated field in disc galaxies. For the case of solar corona, we solve the linear and nonlinear
force-free field equation using photospheric boundary conditions to obtain simple ax- isymmetric magnetic field configurations in spherical geometry. We show that the condition of separability of
solutions in the radial and angular vari- ables leads to two classes of solutions: linear and nonlinear force-free fields (NLFF). We extended the set of NLFF solutions with radial power law index n =
p/q, for all cases of odd p and cases of q > p for even p. We apply these solutions to simulate photospheric vector magnetograms obtained using the spectro-polarimeter on board Hinode and search for
best-fit configurations. The effectiveness of our search strategy is demonstrated on test inputs of dipolar, axisymmetric, and non axisymmetric linear force-free fields. Using the best fit, we build
three- dimensional axisymmetric field configurations and calculate the energy and relative helicity with two independent meth- ods. The magnetic helicity and free energy content of these fields are
useful indicators of energy available for release during eruptive events like solar flares. We analyze five magnetograms for active regions (AR) 10930 span- ning a period of three days during which
two X-class flares occurred and calculate the free energy and relative helicity of the active region before and after the flare. Our analysis indicates a peak in these quantities before the flare
events, which is consistent with the previous results. We also analyze single-polarity regions AR 10923 and 10933, which showed very good fits to potential fields. This method provides useful
reconstruction of NLFF and in- put fields for other numerical techniques. We also apply the NLFF solutions to calculate the amount of braiding in coronal magnetic fields using the con- cept of mean
crossing number. This is then used to estimate the free energy content in solar active regions. We find that the free energy estimates ob- tained from calculation of magnetic braiding is in good
agreement with those obtained by exact calculations of NLFF fields. We then apply the model of self-organized criticality (SOC) to these braided field lines and calculate the distribution of coherent
braid sequences and flare energies. We find find good agreement in the flare energy distributions obtained using SOC model and NLFFF extrapolation. These results provide useful information on the
coro- nal loop structure and also imply that the coronal heating can be supplied by the braiding in the case of the active sun. We provide a new formulation for relative helicity in arbitrary
geometries using the toroidal-poloidal representation of the magnetic field iand discuss the special cases of planar and spherical geometry. In a general astrophysical application, the fields
penetrate the generation region and extend to a sur- rounding corona. It is important to develop gauge-free form for Helicity that can be readily used in different geometries without involving
integrals over external volumes. The further extension of the ideas here can be formalized through use of differential geometry. Magnetic fields correlated on kiloparsec scales are seen in disc
galaxies. The origin could be due to amplification of small scale seed fields by a tur- bulent dynamo. Helicity conservation imposes constraints on dynamo action and one can study the minimal field
strength of the large scale magnetic field that could arise despite the constraint. The calculation of helicity is tech- nically complicated because of open boundaries and the usual form for the
magneto-hydrodynamic (MHD) invariant needs to be modified to take this into account. We then present a global semi-analytic axisymmetric model for a turbulent dynamo operating in a galaxy with a
corona. Here, we show that the supernovae (SNe) and magneto-rotational instability (MRI) driven turbulence parameters have nearly the same radial dependence and can be treated in a common formalism;
however we assume the main contribution from SNe. The general toroidal-poloidal representation is then used to cal- culate the global gauge invariant relative magnetic helicity in cylindrical ge-
ometry. We present the analytic steady-state solutions within the disc that are matched to force-free fields in the corona. A dynamical solution for the dynamo is then obtained by expanding the
time-dependent field in the basis obtained using the steady-state solutions. The non-linear quenching of the dynamo is alleviated by inclusion of small-scale advective and diffusive mag- netic
helicity fluxes, which allow the helicity to be transferred outside the disc and consequently build up a corona during the course of dynamo action. We find quadrupolar solutions for in the galactic
disc that extend out into the corona and show oscillations radially. The mean field is found to reach saturation within a timescale of 1 Gyr with a strength which is of the order of equipartition
magnetic energy (∼ Beq ). The following is the arrangement of the Thesis. Chapter 1 gives an overview of astrophysical magnetic fields with special focus on observations of solar and galactic
magnetic fields. Chapter 2 outlines the basics concepts of MHD and describes the processes relating to magnetic field generation and dissipation. We also discuss the topological properties of
magnetic field using magnetic helicity and provide a novel prescription for calculating magnetic helicity in arbitrary geometries. Chapter 3 presents a description of potential and force-free fields
and outlines their important properties. We then discuss analytical and numerical techniques for solving potential and force-free fields equations for determining coronal magnetic fields. In Chapter
4, we present an overview of various coronal heating mechanisms and discuss the statistical properties of solar flares. We then discuss braiding in coronal magnetic fields and calculate the free
energy in these configurations due to braiding. Chap- ter 5 gives an introduction to large-scale turbulent dynamos and discusses various closure approximations used in mean field MHD. We then present
its application to disc galaxies, discuss the basic analytic solutions and give an overview of current problems in dynamo theory. In Chapter 6, we present new solutions to the nonlinear force-free
field equation and discuss its appli- cation for determining the topological properties of coronal magnetic fields, such as their free-energy and relative helicity. We then apply the solutions to a
time sequence of vector magnetograms to estimate the energy released in a solar flare due to change in magnetic field configuration. In Chapter 7, we use the NLFF field solutions obtained in Chapter
6 and estimate the amount of free-energy due to braiding in these configurations. We then apply a model of SOC to this field and calculate the power-law distribution of flare energies which is then
compared with observations. In Chapter 8, we present a model of nonlinear turbulent dynamo applied to a disc galaxy having a force-free corona. We discuss the significance of small-scale magnetic
helicity fluxes with regards nonlinear saturation of the dynamo. Chapter 9 then presents a summary of the results from all chapters, highlight the novel aspects of this Thesis with its impact. Then,
we present future work which includes papers under preparation. | {"url":"http://prints.iiap.res.in/handle/2248/7535","timestamp":"2024-11-05T09:31:04Z","content_type":"application/xhtml+xml","content_length":"36394","record_id":"<urn:uuid:9576612e-9102-4931-8082-da7649ed9340>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00348.warc.gz"} |
Generic vs. intuitive use of highway=path
I am confused now. I always thought the main dilemma/ambiguity is that hw=path can bei either a general base for mixed use (wiki definition, usually a well maintained way wide enough for vehicles) or
a single trail path (intuitive use, hiking path only usable on foot, MTB, horse).
I think that main dilemma needs to be resolved first before thinking about subdividing single trail paths further, especially since there already is via_ferrata for actual climbing paths.
I always had the idea that in a perfect world it could work like this:
hw=path - mixed use of constructed/maintained ways
hw=trail - replaces path for single trail paths, further described with sac_scale, mtb_scale, horse_scale
2 Likes
I did some quantitative digging in my small province:
Highway Tag Length Length
trunk 102 km 0,20 %
cycleway 333 km 0,64 %
motorway 364 km 0,70 %
tertiary 712 km 1,37 %
footway 857 km 1,65 %
secondary 869 km 1,68 %
primary 963 km 1,86 %
unclassified 1760 km 3,40 %
residential 3601 km 6,95 %
service 4755 km 9,18 %
path 15279 km 29,48 %
track 22228 km 42,89 %
Total 51823 km 100,00 %
Path plus
bicycle=designated 216 km 0,42 %
foot=designated 141 km 0,27 %
both=designated 118 km 0,23 %
Cycleway plus
foot=designated 60 km 0,12 %
From these numbers, 98% or so of path will become trail.
Apart this shows: The classes leading up to path in total 27,62%, so less than path alone. Only two primary tags make up for two third of highways.
2 Likes
I’m sorry, but that’s like saying that a track road is already there for motorways. One does climb on it, right. Via ferrata is a specific and a totally different type of climbing than, well,
climbing (Klettersteig vs Klettern) or scrambling.
They should definitely not be confused, in particular because they require (mandate!) different equipment to be used and this can be a matter of life and death
This would be an interesting addition. But it still leaves plenty of undefined “paths”, as listed in
2 Likes
Above I used ohsome dashboard to get at the numbers, in my area about two to three percent of what is highway=path likely to be shared use paths. This is just a small area though (Tyrol). So I looked
at taginfo for Austria:
5.6% of path has foot=designated and 4.76 has bicycle=designated, 4.3% has segregated – that a sure indicator, that this is a shared use path. (Beware: above numbers in km, taginfo numbers in
mapped entities.) The numbers would push me to push for something like:
As clearly can be seen the reason why path was introduced in the beginning, to make it possible to map multi-use features
(and single-use features the same), is drowned by other mappings. No wonder, the access default for path in Austria for bicycle is “dismount”, because paths only exist in the woods (so the
reasoning there.) Personally, I’d rather opt for hw=trail instead for the masses, because the mup’s align much better with foot_scale=casual, sac_scale=strolling, &c.
I recently learned, in the Netherlands there are paths (pad) 8+m wide and paved; so even if the term path for that was unintuitive in my area, at least it is intuitive in other parts of the world.
For English natives, path and trail might ring the same, but if it was for openstreetmap’s ontology that they are different, I’d say, why not?
I don’t think that was the intention of Nop. There are already main tags different than highway=path existing on the “upper end”. highway=via_ferrata and climbing=*. I hope that’s not under
On the “lower end” we have “maintained ways”, ranging from sidewalks, cycleways, briddle ways and covering as well maintained ways for normal, easy hiking or off-road cycling. I hope those
are as well not under discussion and covered by highway=footway,cycleway,briddleway,track,path.
Leftover are unmaintained ways. There are “unofficial” versions (shortcuts,…) of the above ways and there are “trails”.
So if we would have two new values of the key highway, wouldn’t that be enough separation? (Or highway=path & path=maintained,unofficial,trail)
For sure, the “trails”-section would be still have quite some range, though as the main aim is to get all of them out of “general purpose maps” that will be fine and “special maps” anyway
would need to check for mtb:scale or sac_scale
Though, this is an international/general discussion, so we should consider the international defaults as basics.
No. Because this is openstreetmap after all. local peculiarities govern!
For sure you in Austria can decide about local defaults, but that doesn’t effect the rest of the world. If you want to find a global solution, you need to consider first global defaults and
afterwards you can figure out, what that means for your local defaults. It’s not going to work out the other way around. “path is only for pedestrians” doesn’t apply for the majority of OSM.
If you only aim for an Austrian solution, you might better discuss this in the Austrian category
So you are assuming that shared use paths are likely to be tagged as designated for foot and bicycle?
How do you see the way in these photos? All are taken within a short distance. Signs prohibit motor vehicles and horses. By implication the path is intended to be shared by cyclists and pedestrians
(and in reality is popular with both). But there are no signs explicitly “designating” it for anyone. Not a blue circle sign to be seen anywhere in the area.
Do you see this as not a shared use path because of the lack of designation? Or is it in fact shared use, implying that there could be am unquantifiable number of these hidden within the highway=path
2 Likes
These are excellent pictures for documenting what should be moved from path to shared_use or similar.
I took them just before posting! Could probably have brightened them up a bit first.
I genuinely am not sure where they would fall, a lot of people seem to emphasise the concept of designation, and indeed the starting point of the original path proposal seemed to assume that
designation was already implied in footway and cycleway. But often you can tell access rights only by what is left over when you subtract the prohibited mode of transport, which feels more like
access=yes than designated.
For completeness, these are actually tagged as footway with bicycle=yes currently. That may be influenced by the wooden bridge section, where signs indicate pedestrian priority. But that is the only
visible differentiation between the two modes.
Judging from the traffic sign, I would tag it as:
So no path at all and bicycles are not allowed.
Yes, at least, when I read the term in a post referring to Wiki documentation and that would then be the German one. I understand from the article: Shared use is a technical term exactly for ways
explicitly and officially signed. Others are unspecific use.
The German map style renders paths and foot/cycleways differently. From panning around a bit in my area, ways as in your picture mostly mapped footway not path. The bottom one likely as a track.
Actually, we do not have many such ways even though a highly touristic place. Walking and cycling routes reuse agricultural and forestry infrastructure a lot.
A specification for a “mup” (multi-use-path) that will include weaker forms of designation in your pictures certainly will make more sense than strictly mapping signs. But classification will get
And foot=yes, I can see nothing to prohibit pedestrians
judging from the text under the sign, I would probably go for bicycle permissive, especially if it is commonly used by cyclist as was mentioned. If I get it right, it allows authorized motor
vehicles, sounds to me they were just sloppy with not using the sign only barring motorized vehicles.
Does it? From my short samples, highway=path + bicycle=designated is rendered the same as highway=cycleway i.e. white with a purple border and I assume highway=path + foot=designated without bicycle=
designated is rendered the same as a highway=footway.
It’s only those without any designations that highway=path is rendered as dashes regardless of its surface.
Indeed, probably because highway=path;bicycle=designated is understood to be the very same as highway=cycleway by the people designing the map style? The shared use ones with foot=designated too also
render as cycleways
Alan rightly wanted to point at the hidden shared use paths, that have no designation mapped. Only comparing the map signature with what one knows there on the ground can reveal them. The German map
style better suited for that than OSM-Carto.
PS: I do not think that using a weaker form of designation will bring shared use out of low single digit share here. Paths with no signs abound in the woods and there by default not open for cycling
or horse riding. @aighes Not that any routers would adhere to that merely local default
Again that contrasts with what I am used to. Here, I’d generally expect prohibitions on cycling or horse rising to be signed. Coincidentally, very near the photos I posted earlier, there is a
separate path through a nature reserve, where prohibitions on cycling, scooters, and horses are signed explicitly. Whereas I’ve seen areas defined as “Parque Forestal” or “Monte Público”
that post general prohibitions on motor vehicles, so that any unsigned paths are by default “not prohibited” for walking, cycling, or riding, and may well be mapped as highway=path without tags.
That might be the case in your area. But not everywhere in the world. It would help, if you could acknowledge that. The “law” you are talking about is pretty much limited to central Europe. If in
Austria highway=path is used differently than in the “rest of the world”, that’s ok. But this problem you need to solve than in Austria.
Globally highway=path is shared use. No need to find another tag for this.
I am from Central Europe (well, I think Central Europe is not a thing, but let’s keep tha discussion for some other forum) and here I think if a path in a wood is unsigned, it is open to cyclists
and horses. But I might not be knowledgeable about the law.
There has been a lot of discussion that path means so much and so litle a the same time (and the meaning differs in different parts of the planet). Shared use as defined by Wikipedia is a small
subset of paths out there. | {"url":"https://community.openstreetmap.org/t/generic-vs-intuitive-use-of-highway-path/120244","timestamp":"2024-11-10T11:57:56Z","content_type":"text/html","content_length":"77371","record_id":"<urn:uuid:4b461610-bf35-4fdb-b8a2-2b1839b0b00c>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00623.warc.gz"} |