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Is there any relationship between the autocorrelation of the instantaneous phase extracted from signal and the autocorrelation of the signal in time domain
5 years ago
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latest reply 5 years ago
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Here is my question and I would be really grateful if you could help me understand this. In DSP, for extracting the instantaneous phase of a signal one could use Hilbert Transform. Let the signal
be $x(t)$. I would like to know if there is any relationship between the "autocorrelation" in the signal x(t) and the autocorrelation in the "instantaneous phase" time series $\phi(t)$. That is, is
there any analytical way to find a relation between the ACF of $\phi(t)$ and ACF of $x(t)$; where $\phi(t)$ being the instantaneous phase of the x(t).
Next question is that if one can infer that non-stationarity of a signal x(t) results in non-stationarity of $\phi(t)$?
I appreciate your thoughts and help. Happy Holidays and Happy New Year.
[ - ]
Reply by ●December 28, 2019
Not sure if this is what you want, but this is an impulse response magnitude (green) and phase (red). I've annotated a few values of phase, just for reference.
Below them is the autocorrelation of each, plotted on a different timebase.
There seems to be a distinct similarity between the two.
[ - ]
Reply by ●December 28, 2019
Dear @MarkSitkowski,
Thank you so much for taking the time to help. This demonstrates the idea behind, but I am seeking if there is anyway to analytically establish the mathematical proof behind it to figure out as to
whether this is generalizable.
For example, I want to know the feasibility of finding a relation between the followings:
$R_{xx}(\tau) = \sum_{\tau=-\infty}^{+\infty}x[m]x[m-\tau]$
$R_{\phi\phi}(\tau) = \sum_{\tau=-\infty}^{+\infty}\phi[m]\phi[m-\tau]$
given that $\phi[t] = atan(\frac{Im(z[t])}{Re(z[t])})$ where z[t] is the analytical signal of x[t].
Again, thank you Mark for your time and kindness to provide a reply.
[ - ]
Reply by ●December 28, 2019
It depends on what you are assuming about your signal.
Remember that the phase signal is only half the picture (and if by \phi(t) with a small "p" you mean the phase after removing the linear trend, it is even less than half).
So, a signal could be nonstationary due to nonstationary amplitude, despite having completely stationary phase (even linear phase).
Regarding the autocorrelations, think of a narrowband signal FM signal. You will see a peak corresponding to the carrier frequency, which has nothing to do with small "p" instantaneous phase. But if
the modulation is itself sinusoidal (i.e., the phase is sinusoidal and thus has a autocorrelation peak at the modulation frequency) you will see autocorrelation peaks in the signal related to this.
More generally, write the integral INT x(t) x(t-\tau) dt = INT A(t) e^{i \phi(t)} A(t-\tau) e^(i \phi(t-\tau)} dt . If the As are not time dependent you are left with the integral INT e^{i (\phi(t) -
\phi(t-\tau))} dt which shows that the difference appears in the exponent. For small \tau you can expand the exponent and after the constant term you get a difference integral and not a square
difference (which is related to the product). | {"url":"https://www.dsprelated.com/thread/10140/is-there-any-relationship-between-the-autocorrelation-of-the-instantaneous-phase-extracted-from-signal-and-the-autocorrelation-of-the-signal-in-time-domain","timestamp":"2024-11-12T18:29:19Z","content_type":"text/html","content_length":"35027","record_id":"<urn:uuid:0fb1f441-3ae1-43a1-aba9-570f8cfe41e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00863.warc.gz"} |
A harmonic mode of a closed pipe of length 22 cm resonates class 11 physics JEE_MAIN
Hint: A harmonic mode is said to resonate with a certain source frequency. This implies that the frequency of that particular harmonic node will be equal to the source frequency. Only odd harmonics
will be present in a closed pipe.
Formula Used:
The natural frequencies of a closed pipe are given by, ${\nu _n} = \left( {n + \dfrac{1}{2}} \right)\dfrac{v}{{2l}}$ where $v$ is the velocity of sound in air, $n = 0{\text{, }}1{\text{, }}2{\text
{,}}....$ is the number of the harmonics and $l$ is the length of the pipe.
Complete step by step answer:
Step 1: List the parameters that are known from the question.
The length of the closed pipe is $l = 22{\text{cm}}$ .
A particular harmonic resonates with an external source of frequency ${\nu _s} = 1875{\text{Hz}}$ .
The velocity of sound in air is $v = 330{\text{m/s}}$ .
Step 2: Find the harmonic that resonates with the external source.
The natural frequencies of a closed pipe are given by, ${\nu _n} = \left( {n + \dfrac{1}{2}} \right)\dfrac{v}{{2l}}$ -------- (1)
where $v$ is the velocity of sound in air, $n = 0{\text{, }}1{\text{, }}2{\text{,}}....$ is the number of the harmonics and $l$ is the length of the pipe.
The required harmonic can be found by calculating the natural frequency of each harmonic using equation (1).
Substituting for $n = 0$ in equation (1) we get the fundamental mode as ${\nu _0} = \dfrac{v}{{4l}}$ ------- (2)
Now, substitute the values for $v = 330{\text{m/s}}$ and $l = 22{\text{cm}}$ in the above relation.
Then the fundamental frequency is ${\nu _0} = \dfrac{{330}}{{4 \times 22 \times {{10}^{ - 2}}}} = 375{\text{Hz}}$
Now, ${\nu _0} \ne {\nu _s}$ and thus the first harmonic does not resonate with the source.
Substituting for $n = 1$ in equation (1) we get the third harmonic ${\nu _3} = \dfrac{{3v}}{{4l}}$
Now, substitute the values for $v = 330{\text{m/s}}$ and $l = 22{\text{cm}}$ in the above relation.
Then the third harmonic is ${\nu _3} = \dfrac{{3 \times 330}}{{4 \times 22 \times {{10}^{ - 2}}}} = 1125{\text{Hz}}$
Now, ${\nu _3} \ne {\nu _s}$ and thus the third harmonic does not resonate with the source.
Substituting for $n = 2$ in equation (1) we get the fifth harmonic ${\nu _5} = \dfrac{{5v}}{{4l}}$
Now, substitute the values for $v = 330{\text{m/s}}$ and $l = 22{\text{cm}}$ in the above relation.
Then the fifth harmonic is ${\nu _5} = \dfrac{{5 \times 330}}{{4 \times 22 \times {{10}^{ - 2}}}} = 1875{\text{Hz}}$
Now, ${\nu _5} = {\nu _s} = 1875{\text{Hz}}$
Thus the fifth harmonic resonates with the source.
The number of nodes present will be 3(including the one at the end).
Therefore the correct option is e.
Note: Alternate method
The third and fifth harmonics can also be expressed in terms of the first harmonic as $3{\nu _0}$ and $5{\nu _0}$ respectively.
Substituting the value for the fundamental frequency ${\nu _0} = 375{\text{Hz}}$ we get, third harmonic $3{\nu _0} = 3 \times 375 = 1125{\text{Hz}}$ and fifth harmonic $5{\nu _0} = 5 \times 375 =
Also, when substituting the value for length in equation (1) a unit conversion from cm to m must be done. | {"url":"https://www.vedantu.com/jee-main/a-harmonic-mode-of-a-closed-pipe-of-length-22-cm-physics-question-answer","timestamp":"2024-11-11T11:50:52Z","content_type":"text/html","content_length":"151120","record_id":"<urn:uuid:0570c63a-81a1-4295-aedc-9044c9f2f6c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00253.warc.gz"} |
The relative motion of the Earth, Moon and Sun causes water to flow around the Earth. The effect is to produce a high tide approximately twice per day and a low tide approximately twice per day.
This means that the period of a complete tidal cycle is 12 hours. If the difference in the height of high tide and low tide is
In the diagram above
Much of the time people are interested in the depth of the water at a certain time, not the depth of the water so many hours after high tide. To take account of this we must incude a phase | {"url":"https://mail.courseworkbank.info/ib-maths-notes/trigonometry/1149-tides.html","timestamp":"2024-11-13T01:41:28Z","content_type":"text/html","content_length":"32002","record_id":"<urn:uuid:dca06624-31b7-4fe3-b1b3-1705364a7726>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00474.warc.gz"} |
[Solved] Plancks constant has the dimensions of:
Plancks constant has the dimensions of:
Answer (Detailed Solution Below)
Option 4 : angular momentum
Noun & Pronoun: Fill In The Blanks (Most Important Rules)
1.2 Lakh Users
15 Questions 15 Marks 12 Mins
• Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit, and a Dimension is a mathematical tool
used for studying the nature of physical quantities.
• The basic concept of dimensions is that we can add or subtract only those quantities which have the same dimensions.
• And the dimensional formula is defined as the expression of the physical quantity in terms of mass, length, and time.
Planck’s constant
• It is a physical constant that is the quantum of electromagnetic action. It relates the energy carried by a photon to its frequency by, E = hν.
\(⇒ h = \frac{E}{\nu }\)
Where, E = energy, ν = frequency and h = Planck’s constant.
• Dimensional formula of energy (E) = [ML2T-2]
• Dimensional formula of frequency (ν) = [T-1]
• The dimension of the plack constant is (h)
\(⇒ h = \frac{{M{L^2}{T^{ - 2}}}}{{{T^{ - 1}}}}=[ML^2T^{-1}]\)
Angular momentum:
• It is the rotational equivalent of linear momentum.
⇒ L = r x p
Where, L = angular momentum, r = distance and p = linear momentum.
Dimensional formula of (r) = [L],
Dimensional formula of (p) = [MLT-1],
• Therefore, the dimensional formula of L is
⇒ L = [L] × [MLT-1]
⇒ L = [ML^2T^-1]
│Quantity │Unit │Dimension │
│Work/Energy │Joule or N.m │[M1L2T-2] │
│Power │watt │[ML^2T^-3]│
│Impulse │N-sec │[MLT^-1] │
│Angular momentum │Kg-m^2 / s │[M1L^2T-1]│
│stress │N / m2 │[M1L-1T-2]│
│pressure │N / m2 │[M1L-1T-2]│
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Ants on a Triangle
There are three ants on a triangle, one at each corner. at a given moment in time, they all set off for a different corner at random. what is the probability that they don’t collide?
Consider the triangle ABC. We assume that the ants move towards different corners along the edges of the triangle.
Total no. of movements: 8 A->B, B->C, C->A A->B, B->A, C->A A->B, B->A, C->B A->B, B->C, C->B A->C, B->C, C->A A->C, B->A, C->A A->C, B->A, C->B A->C, B->C, C->B
Non-colliding movements: 2 A->B, B->C, C->A A->C, B->A, C->B
(i.e. the all ants move either in the clockwise or anti-clockwise direction at the same time)
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Integrated Knowledge Solutions
Pre-trained large language models (LLMs) are being used for numerous natural language processing applications. These models perform well out of the box and are fine-tuned for any desired down-stream
application. However, fine-tuning these models to adapt to specific tasks often poses challenges due to their large parameter sizes. To address this, a technique called Low Rank Adaptation (LoRA) has
emerged, enabling efficient fine-tuning of LLMs. In this post, we will try to understand LoRA, and delve into its importance and application in fine-tuning LLMs. We will begin our journey by first
looking at the concept of rank of a matrix, followed by a look at matrix factorization, and then to LoRA.
Rank of a Matrix
The rank of a matrix indicates the number of independent rows or column in the matrix. As an example, consider the following 4x4 matrix A:
A = [[2, 4, 6, 8], [1, 3, 5, 7], [4, 8, 12, 16], [3, 9, 15, 21]]
Looking at the first and third row of this matrix, we see that the third row is just a scale up version of the first row by a factor of 2. The same is true for the second and fourth rows. Thus, the
rank of matrix A is 2 as there are only two independent rows.
The rank of a matrix of size mxn cannot be greater than min{m,n}. In other words, the rank of a matrix cannot be greater than the smallest dimension of the matrix. We say a matrix is a full rank
matrix if its rank equals the largest possible rank for that matrix.
When a matrix is not a full rank matrix, it tells us that the underlying matrix has some redundancy in it that can be exploited for data compression or dimensionality reduction. This is done by
obtaining a low-rank approximation of the matrix. The process of obtaining a low-rank approximation of a matrix is involves matrix factorization. Some of these factorization methods are briefly
described below.
Matrix Factorization
Matrix factorization is the process of decomposing a matrix into multiple factors. Some of the matrix factorization are:
1. Singular Value Decomposition (SVD)
In SVD, a real-valued matrix A of size m x n is factorized as $ A = UDV^t$, where 𝐔 is an orthogonal matrix of size m x m of left singular vectors and 𝐕 is an orthogonal matrix of size n x n of right
singular vectors. The matrix 𝐃 is a diagonal matrix of size m x n of singular values. A low rank approximation to matrix A of rank r is obtained by using only a subset of singular values and the
corresponding left and right singular vectors as given by the following expression. In other words, the approximation is obtained by the weighted sum of rank one matrices.
$ \hat{ \bf A} = \sum\limits_{j=1}\limits^{k} d_{jj}\bf U_j\bf V^t,\text{ }k\leq r$
SVD is a popular matrix factorization method that is commonly used for data compression and dimensionality reduction. It has also been used for compressing convolutional neural networks. You can read
more about SVD and its use for compression at this blog post.
2. Principal Component Analysis (PCA)
PCA aims to find the principal components that capture the most significant variance in the data. It works with data matrices that have been normalized to have zero mean. Let's say $X$ of m rows and
n columns is one such data matrix where each row represents an observation vector of n features. PCA computes the eigenvalues and eigenvectors of the covariance matrix $C = \frac{1}{(1-n)}XX^t$ by
factorizing it as $\frac{1}{(1-n)}WD^tW$, where $W$ is an orthogonal matrix of eigenvectors and $D$ is the diagonal matrix of eigenvalues. PCA is a popular technique for dimensionality reduction.
3. Non-Negative Matrix Factorization (NMF)
NMF is another technique for obtaining low rank representation of matrices with non-negative or positive elements. Given a data matrix $A$ of m rows and n columns with each and every element $a_{ij}
≥ 0$, NMF seeks matrices $W$ and $H$ of size m rows and k columns, and k rows and n columns, respectively, such that $A≈WH$, and every element of matrices $W$ and $H$ is either zero or positive. The
value of k is set by the user and is required to be equal or less than the smallest of m and n. The matrix $W$ is generally called the basis matrix, and $H$ is known as expansion or coefficient
matrix. The underlying idea of this terminology is that a given data matrix $A$ can be expressed in terms of summation of k basis vectors (columns of $W$) multiplied by the corresponding coefficients
(columns of $H$). Compared to SVD, the NMF based factorization offers a better interpretation of the original data matrix as it is represented/approximated as a sum of positive matrices/vectors. NMF
has been used to perform document clustering, making recommendations, visual pattern recognition such as face recognition, gene expression analysis, feature extraction, source separation etc.
Basically, it can be used in any application where data matrix $A$ has no negative elements. You can read more about NMF at this blog post.
Low Rank Adaptation (LoRA) of Large Language Models
The first thing to note is that LoRA doesn't perform a low rank approximation of the weight or parameter matrix; it rather modifies it by generating a new low rank matrix that captures the needed
parameter changes as a result of the fine tuning the LLM. The pre-trained matrix $W$ is frozen while fine tuning and the weight changes are captured in a delta weight matrix $\Delta W$ through
gradient learning. The delta weight change matrix is a low rank matrix which is set as a product of two small matrices, i.e. $\Delta W = AB$. The $A$ matrix is initialized with values coming from a
gaussian distribution while $B$ matrix is initialized with elements all equal to zero. This ensures that the pre-trained weights matrix is the only contributing matrix at the start of fine tuning.
The figure below illustrates this setup for LoRA.
LoRA Scheme: Matrix W is kept fixed and only A and B are trained.
Let's now try to understand the reasoning behind LoRA and its advantages. The main motivation is that the pretrained models are over-parameterized with low intrinsic dimensionality. Further, the
authors of LoRA hypothesize that change in weights during model fine tuning also has a low intrinsic rank. Thus, it is suffice to use a low rank matrix to capture the weight changes during fine
tuning. LoRA offers several advantages. First, it is possible to share the pretrained model for several downstream tasks with each task having its own LoRA model. This obviously saves storage needs
as well as makes task switching easier. Second, LoRA makes the LLMs adaptation for different tasks easier and efficient. Third, it is easy to combine with other fine tuning methods, if desired. As an
example of the parameter efficiency of LoRA, consider the pretrained matrix of size 200x400. To perform adaptation, let matrix $A$ be of size 200x8 and matrix $B$ be of size 8x400 giving rise to the
delta weight change matrix of the desired size of 200x400. The number of parameters thus needed by LoRA is only 200*8+8*400 = 4800 as compared to the number of parameter, 200*400 = 80000, needed to
adjust without LoRA.
An important consideration in using LoRA is the choice of the rank of the $\Delta W$ matrix. Choosing a smaller rank leads to a simpler low-rank matrix, which results in fewer parameters to learn
during adaptation. However, the adaptation with a smaller rank $\Delta W$ may not lead to the desired performance. Thus, the rank choice offers a tradeoff that typically requires experimentation to
get the best adaptation.
LoRA in PEFT
PEFT stands for a general parameter-efficient fine-tuning library from Huggins Face that includes LoRA as one of its techniques. The few lines of codes below illustrate its basic use.
from transformers import AutoModelForSeq2SeqLM
from peft import get_peft_config, get_peft_model, LoraConfig, TaskType
model_name_or_path = "bigscience/mt0-large"
tokenizer_name_or_path = "bigscience/mt0-large"
peft_config = LoraConfig(
task_type=TaskType.SEQ_2_SEQ_LM, inference_mode=False, r=8, lora_alpha=32, lora_dropout=0.1
model = AutoModelForSeq2SeqLM.from_pretrained(model_name_or_path)
model = get_peft_model(model, peft_config)
# output: trainable params: 2359296 || all params: 1231940608 || trainable%: 0.19151053100118282
In the above example, mt0-large model is being fine tuned for a sequence to sequence conversion task. The rank of the delta weight change is specified as 8. The model has 1.2 B parameters but LoRA
needs only 2.36M parameters, 19% of the total parameters, to train. If we are to change the rank to 12, the number of trainable parameters increases to 3538944, 28.7% of the total parameters.
Clearly, the choice of rank is an important consideration when using LoRA.
LoRA's performance has been evaluated against full fine tuning and other efficient techniques for parameter computation. LoRA has been found to generally outperforms other efficient fine tuning
techniques by a significant margin while yielding comparable or better performance than full fine tuning.
To wrap up, LoRA is an efficient technique for fine tuning large pretrained models. It is poised to play an important role in fine tuning and customizing LLMs for numerous applications.
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OpenQASM 3.1 Specification
Classical instructions¶
We envision two levels of classical control: simple, low-level instructions embedded as part of a quantum circuit and high-level external functions which perform more complex classical computations.
The low-level functions allow basic computations on lower-level parallel control processors. These instructions are likely to have known durations and many such instructions might be executed within
the qubit coherence time. The external, or extern, functions execute complex blocks of classical code that may be neither fast nor guaranteed to return. In order to connect with classical compilation
infrastucture, extern functions are defined outside of OpenQASM. The compiler toolchain is expected to link extern functions when building an executable. This strategy allows the programmer to use
existing libraries without porting them into OpenQASM. extern functions run on a global processor concurrently with operations on local processors, if possible. extern functions can write to the
global controller’s memory, which may not be directly accessible by the local controllers.
Low-level classical instructions¶
All types support the assignment operator =. The left-hand-side (LHS) and right-hand-side (RHS) of the assignment operator must be of the same type. For real-time values assignment is by copy of the
RHS value to the assigned variable on the LHS.
int[32] a;
int[32] b = 10; // Combined declaration and assignment
a = b; // Assign b to a
b = 0;
a == b; // False
a == 10; // True
Classical bits and registers¶
Classical registers, bit, uint and angle support bitwise operators and the corresponding assignment operators with registers of the same size: and &, or |, xor ^. They support left shift << and right
shift >> by an unsigned integer, and the corresponding assignment operators. The shift operators shift bits off the end. They also support bitwise negation ~, popcount [1], and left and right
circular shift, rotl and rotr, respectively.
bit[8] a = "10001111";
bit[8] b = "01110000";
a << 1; // Bit shift left produces "00011110"
rotl(a, 2) // Produces "00111110"
a | b; // Produces "11111111"
a & b; // Produces "00000000"
For uint and angle, the results of these operations are defined as if the operations were applied to their defined bit representations:
angle[4] a = 9 * (pi / 8); // "1001"
a << 2; // Produces pi/2, which is "0100"
a >> 2; // Produces pi/4, which is "0010"
uint[6] b = 37; // "100101"
popcount(b); // Produces 3.
rotl(b, 3); // Produces 44, which is "101100"
Comparison (Boolean) Instructions¶
Integers, angles, bits, and classical registers can be compared (\(>\), \(>=\), \(<\), \(<=\), \(==\), \(!=\)) and yield Boolean values. Boolean values support logical operators: and &&, or ||, not !
. The keyword in tests if an integer belongs to an index set, for example i in {0,3} returns true if i equals 0 or 3 and false otherwise.
bool a = false;
int[32] b = 1;
angle[32] d = pi;
float[32] e = pi;
a == false; // True
a == bool(b); // False
c >= b; // True
d == pi; // True
// Susceptible to floating point casting errors
e == float(d);
Angles are naturally defined on a ring, and so comparisons may not work exactly how you might expect. For example, 2 * ang >= ang is not necessarily true; if ang represents the angle \(3\pi/2\), then
2*ang == pi and pi < ang.
This is the same behavior as unsigned integers in many languages (including OpenQASM 3), but the types of operation commonly performed on angle are particularly likely to trigger these modulo
arithmetic effects.
Integer types support addition +, subtraction -, multiplication *, integer division [2] /, modulo %, and power **, as well as the corresponding assignments +=, -=, *=, /=, %=, and **=.
int[32] a = 2;
int[32] b = 3;
a * b; // 6
b / a; // 1
b % a; // 1
a ** b; // 8
a += 4; // a == 6
In addition to the bitwise operations mentioned above, angles support:
• Addition + and subtraction - by other angles of the same size, which returns an angle of the same size.
• Multiplication * and division / by unsigned integers of the same size. The result is an angle type of the same size. Both uint * angle and angle * uint are valid and produce the same result, but
only angle / uint is valid; it is not allowed to divide an integer by an angle.
• Division / by another angle of the same size. This returns a uint of the same size.
• Unary negation -, which represents the mathematical operation \(-a \equiv 2\pi - a\).
• Compound assignment operators +=, -= and /= with angles of the same size as both left- and right operands. These have the same effect as if the equivalent binary operation had been written out in
• The compound assignment operators *= and /= with an unsigned integer of the same size as the right operand. This has the same effect as if the multiplication or division had been written as a
binary operation and assigned.
In all of these cases, except for unary negation, the bit pattern of the result of these operations is the same as if the operations had been carried out between two uint types of the same size with
the same bit representations, including both upper and lower overflow. Explicitly:
angle[4] a = 7 * (pi / 8); // "0111"
angle[4] b = pi / 8; // "0001"
angle[4] c = 5 * (pi / 4); // "1010"
uint[4] two = 2;
a + b; // angle[4] │ pi │ "1000"
b - a; // angle[4] │ 5 * (pi / 4) │ "1010"
a / two; // angle[4] │ 3 * (pi / 8) │ "0011"
two * c; // angle[4] │ pi / 2 │ "0100"
c / b; // uint[4] │ 10 │ "1010"
pi * 2; // angle[4] │ 0 │ "0000"
Unary negation of an angle a is defined to produce the same value as 0 - a, such that a + (-a) is always equal to zero. This is the same as the C99 definition for unsigned integers. In bitwise
operations, the negation can be written as (~a) + 1. Explicitly:
angle[4] a = pi / 4; // "0010"
angle[4] b = -a; // 7*(pi/4) │ "1110"
Floating-point numbers¶
Floating-point numbers support addition, subtraction, multiplication, division, and power and the corresponding assignment operators.
angle[20] a = pi / 2;
angle[20] b = pi;
a + b; // 3/2 * pi
a ** b; // 4.1316...
angle[10] c;
c = angle(a + b); // cast to angle[10]
Real hardware may well not have access to floating-point operations at runtime. OpenQASM 3 compilers may reject programs that require runtime operations on these values if the target backend does not
support them.
Complex numbers¶
Complex numbers support addition, subtraction, multiplication, division, power and the corresponding assignment operators. These binary operators follow analogous semantics to those described in
Annex G (section G.5) of the C99 specification (note that OpenQASM 3.0 has no imaginary type, only complex). These operations use the floating-point semantics of the underlying component
floating-point types, including their NaN propagation, and hardware-dependent rounding mode and subnormal handling.
complex[float[64]] a = 10.0 + 5.0im;
complex[float[64]] b = -2.0 - 7.0im;
complex[float[64]] c = a + b; // c = 8.0 - 2.0im
complex[float[64]] d = a - b; // d = 12.0+12.0im;
complex[float[64]] e = a * b; // e = 15.0-80.0im;
complex[float[64]] f = a / b; // f = (-55.0+60.0im)/53.0
complex[float[64]] g = a ** b; // g = (0.10694695640729072+0.17536481119721312im)
Evaluation order¶
OpenQASM evaluates expressions in natural mathematical order, following the defined operator-precedence and -associativity table below. Operators of greater precedence are evaluated before operators
of less precedence. The order of evaluation for operators of the same precedence is set by the associativity: left-associative operators evaluate from left to right (i.e. a + b + c evaluates as (a +
b) + c) while right-associative operators evaluate from right to left (i.e. a ** b ** c evaluates as a ** (b ** c)).
Operator Operator names Associativity
(), [], (type)(x) Call, index, cast left
** Power right
!, -, ~ Unary right
*, /, % Multiplicative left
+, - Additive left
<<, >> Bit Shift left
<, <=, >, >= Comparison left
!=, == Equality left
& Bitwise AND left
^ Bitwise XOR left
| Bitwise OR left
&& Logical AND left
|| Logical OR left
Looping and branching¶
If-else statements¶
The statement if ( bool ) <true-body> branches to program if the Boolean evaluates to true and may optionally be followed by else <false-body>. Both true-body and false-body can be a single statement
terminated by a semicolon, or a program block of several statements { stmt1; stmt2; }.
bool target = false;
qubit a;
h a;
bit output = measure qubit
// example of branching
if (target == output) {
// do something
} else {
// do something else
For loops¶
The statement for <type> <name> in <values> <body> loops over the items in values, assigning each value to the variable name in subsequent iterations of the loop body. values can be:
• a discrete set of scalar types, defined using the array-literal syntax, such as {1, 2, 3}. Each value in the set must be able to be implicitly promoted to the type type.
• a range expression in square brackets of the form [start : (step :)? stop], where step is equal to 1 if omitted. As in other range expressions, the range is inclusive at both ends. Both start and
stop must be given. All three values must be of integer or unsigned-integer types. The scalar type of elements in the resulting range expression is the same as the type of result of the implicit
promotion between start and stop. For example, if start is a uint[8] and stop is an int[16], the values to be assigned will all be of type int[16].
• a value of type bit[n], or the target of a let statement that creates an alias to classical bits. The corresponding scalar type of the loop variable is bit, as appropriate.
• a value of type array[<scalar>, n], _i.e._ a one-dimensional array. Values of type scalar must be able to be implicitly promoted to values of type type. Modification of the loop variable does not
change the corresponding value in the array.
It is valid to use an indexing expression (e.g. my_array[1:3]) to arrive at one of the types given above. In the cases of sets, bit[n], classical aliases and array, the iteration order is guaranteed
to be in sequential index order, that is iden[0] then iden[1], and so on.
The loop body can either be a single statement terminated by a semicolon, or a program block in curly braces {} containing several statements.
Assigning a value to the loop variable within an iteration over the body does not affect the next value that the loop variable will take.
The scope of the loop variable is limited to the body of the loop. It is not accessible after the loop.
int[32] b = 0;
// loop over a discrete set of values
for int[32] i in {1, 5, 10} {
b += i;
// b == 16, and i is not in scope.
// loop over every even integer from 0 to 20 using a range, and call a
// subroutine with that value.
for int i in [0:2:20]
// high precision typed loop variable
for uint[64] i in [4294967296:4294967306] {
// do something
// Loop over an array of floats.
array[float[64], 4] my_floats = {1.2, -3.4, 0.5, 9.8};
for float[64] f in my_floats {
// do something with 'f'
// Loop over a register of bits.
bit[5] register;
for bit b in register {}
let alias = register[1:3];
for bit b in alias {}
While loops¶
The statement while ( bool ) <body> executes program until the Boolean evaluates to false [3]. Variables in the loop condition statement may be modified within the while loop body. The body can be
either a single statement terminated by a semicolon, or a program block in curly braces {} of several statements:
qubit q;
bit result;
int i = 0;
// Keep applying hadamards and measuring a qubit
// until 10, |1>s are measured
while (i < 10) {
h q;
result = measure q;
if (result) {
i += 1;
Breaking and continuing loops¶
The statement break; moves control to the statement immediately following the closest containing for or while loop.
The statement continue; causes execution to jump to the next step in the closest containing for or while loop. In a while loop, this point is the evaluation of the loop condition. In a for loop, this
is the assignment of the next value of the loop variable, or the end of the loop if the current value is the last in the set.
int[32] i = 0;
while (i < 10) {
i += 1;
// continue to next loop iteration
if (i == 2) {
// some program
// break out of loop
if (i == 4) {
// more program
It is an error to have a break; or continue; statement outside a loop, such as at the top level of the main circuit or of a subroutine.
OPENQASM 3.0;
break; // Invalid: no containing loop.
def fn() {
continue; // Invalid: no containing loop.
Terminating the program early¶
The statement end; immediately terminates the program, no matter what scope it is called from.
The Switch statement¶
A switch statement is a form of flow control that provides for a predicated selection of zero, one or more statements to be executed based on a discriminating controlling value. The discriminating
controlling value can be either explicit - as it is the case for case statements - or none of the above - which is the case for default statements.
A switch statement is not a loop. It does not iterate over a sequence of values.
switch statements may appear anywhere in a program where statements are allowed.
An OpenQASM3 switch statement shall use the following keywords:
An OpenQASM3 switch statement shall be the following grammar:
• The switch keyword.
• A right paren ( literal.
• A controlling expression.
• A left paren ) literal.
• A left brace { literal.
• A sequence of one or more case statements (defined below).
• Either zero or one default statement(s) (defined below).
• A right brace } literal.
The controlling expression of a switch statement shall be of integer type. Implicit conversions to an integer type are not allowed.
A case statement shall be the following grammar:
• The case keyword.
• An integer-constant-list-expression controlling label.
• A left-brace literal: {.
• A sequence of zero, one or more OpenQASM3 statements.
• A right-brace literal: }.
The integer-constant-list-expression is a sequence of one or more integer const expressions separated by comma , literals.
A default statement shall be the following grammar:
• The default keyword.
• A left-brace literal: {.
• A sequence of zero, one or more OpenQASM3 statements.
• A right-brace literal: }.
A switch statement shall be in scope only within the scope where it is defined.
The left and right braces of a switch statement shall not create brace-enclosed scope.
Declarations or statements at switch statement scope but outside of a case or default statement are ill-formed. The compiler shall raise an error diagnostic for such cases.
A case or default statement creates brace-enclosed scope.
Declarations of types that automatically acquire global scope in OpenQASM3 - such as gates, functions, arrays, qubits and defcals - are not allowed at case or default switch statement scope. Use of
such declarations is ill-formed and requires a compiler diagnostic.
Duplicate values within any integer-constant-list-expression for controlling labels of case statements are not allowed. The compiler shall issue an error diagnostic in such cases.
A case or default statement ending with a right-brace } terminates the execution of the switch statement. After executing all the statements of the case or default statement, control is then
transferred to the first statement following the closing right brace of the enclosing switch statement.
A switch statement shall contain at least one case statement. A switch statement with no case statements shall raise an error diagnostic.
A switch statement is not required to contain a default statement. If a switch statement does not contain a default statement and a runtime value is provided to the controlling expression that does
not match any case, then the switch becomes effectively a no-op.
1. A simple switch statement with case and default statements:
OPENQASM 3.0;
int i = 15;
switch (i) {
case 1, 3, 5 {
// OpenQASM3 statement(s)
case 2, 4, 6 {
// OpenQASM3 statement(s)
case -1 {
// OpenQASM3 statement(s)
default {
// OpenQASM3 statement(s)
2. A switch where the cases are const expressions:
OPENQASM 3.0;
const int A = 0;
const int B = 1;
int i = 15;
switch (i) {
case A {
// OpenQASM3 statement(s)
case B {
// OpenQASM3 statement(s)
case B+1 {
// OpenQASM3 statement(s)
default {
// OpenQASM3 statement(s)
3. A switch statement with binary literals in the case statements:
OPENQASM 3.0;
bit[2] b;
switch (int(b)) {
case 0b00 {
// OpenQASM3 statement(s)
case 0b01 {
// OpenQASM3 statement(s)
case 0b10 {
// OpenQASM3 statement(s)
case 0b11 {
// OpenQASM3 statement(s)
4. A switch statement containing declarations at case statement scope, and a function call, also at case statement scope:
OPENQASM 3.0;
def foo(int i, qubit[8] d) -> bit {
return measure d[i];
int i = 15;
int j = 1;
int k = 2;
bit c1;
qubit[8] q0;
switch (i) {
case 1 {
j = k + foo(k, q0);
case 2 {
float[64] d = j / k;
case 3 {
default {
5. A switch statement containing a nested switch statement.
OPENQASM 3.0;
def foo(qubit[8] q) -> int {
int r = 0;
bit k;
for int i in [0 : 7] {
k = measure q[i];
r += k;
return r;
qubit[8] q;
int j = 30;
int i = foo(q);
switch (i) {
case 1, 2, 5, 12 {
case 3 {
switch (j) {
case 10, 15, 20 {
h q;
Extern function calls¶
extern functions are declared by giving their signature using the statement extern name(inputs) -> output; where inputs is a comma-separated list of type names and output is a single type name. The
parentheses may be omitted if there are no inputs.
extern functions can take of any number of arguments whose types correspond to the classical types of OpenQASM. Inputs are passed by value. They can return zero or one value whose type is any
classical type in OpenQASM except real constants. If necessary, multiple return values can be accommodated by concatenating registers. The type and size of each argument must be known at compile time
to define data flow and enable scheduling. We do not address issues such as how the extern functions are defined and registered.
extern functions are invoked using the statement name(inputs); and the result may be assigned to output as needed via an assignment operator (=, +=, etc). inputs are literals and output is a
variable, corresponding to the types in the signature. The functions are not required to be idempotent. They may change the state of the process providing the function. In our computational model,
extern functions may run concurrently with other classical and quantum computations. That is, invoking an extern function will schedule a classical computation, but does not wait for that computation
to terminate. | {"url":"https://openqasm.com/versions/3.1/language/classical.html","timestamp":"2024-11-04T01:21:04Z","content_type":"text/html","content_length":"90089","record_id":"<urn:uuid:dd394b8c-8a5c-411f-9d9e-6f98b6550b8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00578.warc.gz"} |
How to calculate cross currency swap rates
In finance, a currency swap is an interest rate derivative (IRD). In particular it is a linear IRD and A cross-currency swap's (XCS's) effective description is a derivative contract, agreed between
two counterparties, the chosen floating interest rate indexes and tenors, and day count conventions for interest calculations. 31 Oct 2019 A cross-currency swap can involve both parties paying a
fixed rate, both parties paying a Interest payments are typically calculated quarterly.
5 Jan 2017 swap markets on each currency and the FX spot price, (see [2] for a heuristic analytic formula to predict basis tenor and cross currency swaps). 8 Jan 2016 interest rates using FX swaps
than currencies with lower nominal interest Following our definition for cross-currency basis in Equation 1, we Hedge against both currency and interest rate exposures with DBS Cross- Currency Swap.
This is an agreement between two parties to swap future interest Savill Consulting. A forward FX rate is calculated using a no-arbitrage pricing model. 2. Alternative 1. Alternative 2 t. 0. Invest
$10,500,000 for 12 mo @ 0.65%. 9 Sep 2014 Cross currency swaps, or basis, where one bets on the difference between the price as the spread of cash over OIS swap, and we can look at basis BNP Paribas'
own calculation methods and models and is supplied for. 6 Dec 2016 Same currency interest rate swaps exchange interest flows in the same currency (but calculated on different bases). A CCIRS
exchanges My question is: Do stand-alone cross currency swaps carry FX risk? the cross currency basis and any credit differential between the 2 rates That is, a swap or par rate is calculated
without regard to a funding assumption.
Swap calculation for currency pairs is made in units of base currency of the Interest_Rate_Differential — difference between interest rates of Central banks of
BNP Paribas uses cookies on this website. By continuing to use our website you accept the use of these cookies. Please see our cookie policy for more The main difference between a Currency Swap and
an Interest Rate Swap is that swap involves well-defined cash flows and consequently we can calculate an For a cross-currency swap it is essential that the parties agree to exchange Using the
original rate would remove transaction risk on the swap. Currency swaps are used to obtain foreign currency loans at a better interest rate than a 1 Aug 2019 cross-currency swap market that emerges
when there are deviations from covered interest exchange rate between dollar and the foreign currency, and Sources: Bloomberg; Federal Reserve Board; IMF staff calculations
There are two broad methodologies that can be considered for calculating a vanilla interest rate swap will carry less credit risk than a cross currency swap due
18 Apr 2017 The cross currency swap market has particular price dynamics that forecast the FX-resetting element, we calculate the expected FX rate at 4 Nov 2014 I'm wondering if anyone here can
provide for an example with a simple 2 year bond and derive the USD/CNH CCS rate for me? Thank you so
18 Apr 2017 The cross currency swap market has particular price dynamics that forecast the FX-resetting element, we calculate the expected FX rate at
14 Oct 2010 rate derivatives: cross-currency swaps (CCS) and power reverse dual discusses the formula for computing the drifts in the domestic forward Cross Currency Swap, CRS or CCS USD floating
rate. USD Fixed rate. KRW floating rate 3.3 Locate Currency Swap calculator and understand the calculation
10 Apr 2019 The basic concepts of spot fx rates, forward fx contracts, fx swaps and the a year -, the tenor of so-called currency swaps (also known as cross currency The wizard pasted the formula =ds
(A2:B13) in cell A1 that takes as
24 Jul 2009 Actual factors to determine cross-currency basis swaps: An empirical study on US dollar/Japanese yen basis swap rates from the late 1990s. 10 Apr 2019 The basic concepts of spot fx rates,
forward fx contracts, fx swaps and the a year -, the tenor of so-called currency swaps (also known as cross currency The wizard pasted the formula =ds(A2:B13) in cell A1 that takes as That includes
the exchange rate value of each currency and the interest rate In a currency swap operation, also known as a cross currency swap, the parties With the XM swaps calculator traders can calculate the
interest rate differential between the two currencies of the currency pair on their open positions. with the interest rate swaps (IRS), cross currency swaps (CCS) and tenor swaps Now we can determine
the set of discounting factor (and hence the forward (b) Calculate the cross rate for Australian dollars in yen terms. ¥? ¥ currency is the US dollar who does the following five transactions. 1
month swap rates. 18 Nov 2018 Cross-currency basis swaps, also known as basis swaps, are contracts in the spot and forward exchange rate in units of US dollar per foreign currency, emerging market
counterparties calculating trade war casualties are
31 Oct 2019 A cross-currency swap can involve both parties paying a fixed rate, both parties paying a Interest payments are typically calculated quarterly. 12 Nov 2018 Cross Currency Swap Theory &
Practice - An Illustrated Step-by-Step Guide of How to Price Cross Currency Swaps and Calculate the Basis 12 Nov 2004 Key words: interest rate swap, cross currency swap, basis spread and on the
other leg interest rate payments are in currency 2 calculated. 8 Jan 2020 Key words: interest rate swap, cross currency swap, basis spread leg interest rate payments are in currency 2 calculated on a
notional 1 Jun 2010 The day count convention used in calculating the interest rate payments for both legs is Actual/365. The LIBOR/ SWAP zero curve rates for USD | {"url":"https://platformmceud.netlify.app/straub31152to/how-to-calculate-cross-currency-swap-rates-17.html","timestamp":"2024-11-13T21:24:53Z","content_type":"text/html","content_length":"32334","record_id":"<urn:uuid:d08a2aa7-56b7-4108-b7c3-ebe63a0f0d65>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00421.warc.gz"} |
slapmr.f - Linux Manuals (3)
slapmr.f (3) - Linux Manuals
slapmr.f -
subroutine slapmr (FORWRD, M, N, X, LDX, K)
SLAPMR rearranges rows of a matrix as specified by a permutation vector.
Function/Subroutine Documentation
subroutine slapmr (logicalFORWRD, integerM, integerN, real, dimension( ldx, * )X, integerLDX, integer, dimension( * )K)
SLAPMR rearranges rows of a matrix as specified by a permutation vector.
SLAPMR rearranges the rows of the M by N matrix X as specified
by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
If FORWRD = .TRUE., forward permutation:
X(K(I),*) is moved X(I,*) for I = 1,2,...,M.
If FORWRD = .FALSE., backward permutation:
X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.
FORWRD is LOGICAL
= .TRUE., forward permutation
= .FALSE., backward permutation
M is INTEGER
The number of rows of the matrix X. M >= 0.
N is INTEGER
The number of columns of the matrix X. N >= 0.
X is REAL array, dimension (LDX,N)
On entry, the M by N matrix X.
On exit, X contains the permuted matrix X.
LDX is INTEGER
The leading dimension of the array X, LDX >= MAX(1,M).
K is INTEGER array, dimension (M)
On entry, K contains the permutation vector. K is used as
internal workspace, but reset to its original value on
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 105 of file slapmr.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-slapmr.f/","timestamp":"2024-11-09T03:17:14Z","content_type":"text/html","content_length":"8456","record_id":"<urn:uuid:40f17c71-83de-40e8-ab9b-062e950f2ac4>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00057.warc.gz"} |
Misinterpretation - (Honors Statistics) - Vocab, Definition, Explanations | Fiveable
from class:
Honors Statistics
Misinterpretation is the act of incorrectly understanding or misunderstanding the meaning or significance of something, particularly in the context of statistical analysis and inference. It occurs
when the conclusions drawn from data or findings do not accurately reflect the true underlying phenomenon or relationship being studied.
congrats on reading the definition of Misinterpretation. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Misinterpretation can arise from various sources, including inadequate sample size, biased sampling, violation of statistical assumptions, and the use of inappropriate statistical methods.
2. Misinterpretation can lead to incorrect conclusions about the population parameter, such as overestimating or underestimating the true value.
3. Proper understanding of the underlying assumptions and limitations of statistical techniques is crucial to avoid misinterpretation of the results.
4. Careful consideration of the context and potential confounding factors is necessary to ensure that the interpretation of the data is valid and meaningful.
5. Misinterpretation can have significant consequences, such as the implementation of ineffective or harmful policies, the allocation of resources to the wrong areas, or the drawing of erroneous
conclusions about the relationships between variables.
Review Questions
• Explain how misinterpretation can arise in the context of a single population mean using the normal distribution.
□ Misinterpretation in the context of a single population mean using the normal distribution can occur due to various factors. For example, if the sample size is too small, the sample mean may
not be representative of the true population mean, leading to an inaccurate estimate and potential misinterpretation of the results. Additionally, if the normality assumption is violated, the
use of the normal distribution may not be appropriate, and the conclusions drawn from the analysis may be misleading. Careful consideration of the underlying assumptions and limitations of
the statistical techniques is crucial to avoid misinterpreting the findings and drawing valid conclusions about the population parameter.
• Describe how the concept of sampling error can contribute to misinterpretation when estimating a single population mean.
□ Sampling error, which is the difference between a sample statistic and the true population parameter, can significantly contribute to misinterpretation when estimating a single population
mean. If the sample is not representative of the population, the sample mean may not accurately reflect the true population mean. This sampling error can lead to overestimation or
underestimation of the population mean, resulting in misinterpretation of the findings. To mitigate the impact of sampling error, it is essential to ensure that the sample is selected
randomly and is of an appropriate size to provide a reliable estimate of the population mean. Additionally, understanding the concept of sampling distribution and the standard error of the
mean can help in properly interpreting the results and avoiding misinterpretation.
• Analyze how the violation of statistical assumptions can lead to misinterpretation when using the normal distribution to estimate a single population mean.
□ The violation of statistical assumptions, such as the normality assumption, can lead to misinterpretation when using the normal distribution to estimate a single population mean. If the
population distribution is not normal, the use of the normal distribution may not be appropriate, and the resulting inferences and conclusions may be invalid. This can lead to
misinterpretation of the findings, as the statistical analysis may not accurately reflect the true characteristics of the population. To avoid such misinterpretation, it is crucial to
carefully assess the underlying assumptions of the statistical techniques being used and to consider alternative methods or transformations of the data if the assumptions are violated.
Additionally, sensitivity analyses and diagnostic checks can help identify potential issues and inform the interpretation of the results, ensuring that the conclusions drawn are valid and
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/honors-statistics/misinterpretation","timestamp":"2024-11-04T18:07:42Z","content_type":"text/html","content_length":"168581","record_id":"<urn:uuid:afd19035-ba05-45ac-92ed-e9a88b92a9df>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00185.warc.gz"} |
How Much Electricity Does A Mini Split Use? A Guide To The Energy Consumption And Power Usage Of Mini Split Air Conditioners | RenewableWise
Compared to other types of air conditioners, such as window AC units and ducted air conditioners, mini-splits are pretty energy-efficient, in the sense that they use less power (kW) and consume less
energy (kWh) to perform the same amount of heat exchange.
But in general, the electricity consumption of air conditioners and heat pumps is still known to be relatively high. So, the question remains, How much electricity does a mini-split use? and how much
does it cost to run a mini-split?
Well, in this article, I’ll discuss the energy consumption (kWh) of mini split air conditioners and heat pumps and how much it costs to run these units.
In case you’re planning on running your mini split on anything other than grid power, such as a generator, you’ll also want to learn about the power usage (wattage) of these units, which I’ll also
discuss in this article.
Let’s dive in.
I get commissions for purchases made through links in this post.
Do mini-splits use a lot of electricity?
Air conditioners and heat pumps do indeed consume a relatively high amount of electricity when compared to other common household appliances like refrigerators, lights, TVs, and computers.
However, when we compare the electricity usage of ductless mini splits to other HVAC systems, such as ducted central ACs and heat pumps, window air conditioners, and portable air conditioners,
ductless mini splits are more energy-efficient and do not use a lot of electricity.
This increased efficiency is primarily because mini splits deliver cooling and heating directly to the targeted space, as opposed to traditional central AC and heat pump systems that distribute air
through a network of ducts, which leads to significant energy loss.
In fact, according to the U.S. Department of Energy (DOE), between 25% and 40% of heating or cooling energy is lost in ductwork, making ductless mini splits at least 25% more energy-efficient than
ducted HVAC systems.
Furthermore, many mini split systems on the market feature compressors equipped with inverter technology. This technology allows the compressors to operate at variable speeds rather than simply being
ON or OFF, as is the case with traditional compressors. This variable speed operation enhances their energy efficiency significantly.
To illustrate this, consider a study in which two identical rooms were equipped with air conditioners—one with an inverter mini-split and the other with a non-inverter mini-split.
Over 108 days, the study measured the energy consumption of each air conditioner. The results showed that the inverter air conditioner consumed, on average, 44% less energy than the non-inverter air
How much electricity does a mini split use?
The energy usage of a mini-split, like other types of air conditioners and heat pumps, depends on factors such as its cooling/heating capacity, energy efficiency, and whether it’s in cooling or
heating mode.
For example, during the cooling season, as a general guideline, a mini-split air conditioner or heat pump will typically consume between 0.45 and 0.8 kWh of energy per hour, for every ton (12,000
BTUs) of cooling it provides.
Conversely, in the heating season, a mini-split heat pump will typically consume between 1 and 1.5 kWh of energy per hour for every ton (12,000 BTUs) of heating it delivers.
To provide a practical example, a 9,000 BTU mini-split heat pump will usually consume between 0.35 and 0.6 kWh of energy per hour during the cooling season, and between 0.75 and 1.1 kWh per hour
during the heating season.
Here’s a table to help visualize the estimated hourly energy usage of mini-splits based on their capacity (tons/BTUs) and whether they’re providing cooling or heating:
BTU rating Est. Hourly Energy Consumption (kWh/hour) — Cooling Est. Hourly Energy Consumption (kWh/hour) — Heating
9,000 BTUs 0.35 – 0.6 kWh/hour 0.75 – 1.1 kWh/hour
12,000 BTUs (1 ton) 0.45 – 0.8 kWh/hour 1 – 1.5 kWh/hour
15,000 BTUs 0.6 – 1 kWh/hour 1.25 – 1.85 kWh/hour
18,000 BTUs (1.5 tons) 0.7 – 1.2 kWh/hour 1.5 – 2.2 kWh/hour
24,000 BTUs (2 tons) 0.95 – 1.6 kWh/hour 2 – 3 kWh/hour
36,000 BTUs (3 tons) 1.5 – 2.4 kWh/hour 3 – 4.5 kWh/hour
48,000 BTUs (4 tons) 1.9 – 3.2 kWh/hour 4 – 6 kWh/hour
60,000 BTUs (5 tons) 2.4 – 4 kWh/hour 5 – 7.5 kWh/hour
Estimated hourly energy consumption of mini-split ACs/heat pumps.
These figures should provide an estimate of the typical hourly kWh usage of your mini-split. To determine daily or monthly energy consumption, you can combine this data with your daily usage in
Daily Energy Consumption (kWh/day) = Hourly Energy Consumption (kWh/hour) x Daily Usage (hours/day)
Monthly Energy Consumption (kWh/month) = Daily Energy Consumption (kWh/day) x 30 (days/month)
However, if you’d like a more precise estimate of your mini-split’s hourly energy consumption, I recommend using its energy efficiency rating.
Let me elaborate.
The tonnage or BTU rating of your mini-split indicates the amount of heat it can either remove or add within an hour. The energy required to perform this heat exchange depends on the energy
efficiency of the unit.
Energy Efficiency = BTUs delivered or removed ÷ Energy Consumed
The higher the energy efficiency of the unit, the less energy it will require to perform the same amount of heat exchange.
Manufacturers typically conduct tests to determine the energy efficiency of mini-splits and provide energy efficiency ratings, including:
1. SEER (Seasonal Energy Efficiency Ratio): This rating represents the energy efficiency of cooling-only mini-splits or the cooling efficiency for mini-split heat pumps that provide both cooling and
2. HSPF (Heating Seasonal Performance Factor): This rating represents the heating efficiency of a mini-split heat pump.
To estimate the hourly energy consumption of your cooling-only mini-split or the energy consumption of your mini-split heat pump during the cooling season, you can use the following formula:
Hourly Energy Consumption (kWh/hour) = (BTU rating ÷ SEER) ÷ 1000
To estimate the hourly energy consumption of your mini-split heat pump during the heating season, you can use this formula:
Hourly Energy Consumption (kWh/hour) = (BTU rating ÷ HSPF) ÷ 1000
Mini-Split Heat Pumps
For example, consider the following mini-split heat pump rated at 12,000 BTUs:
The hourly energy usage of this mini-split unit during the cooling season can be calculated as follows:
Hourly Energy Consumption (kWh/hour) = (BTU rating ÷ SEER) ÷ 1000
Hourly Energy Consumption (kWh/hour) = (12,000 ÷ 17.4) ÷ 1000
Hourly Energy Consumption (kWh/hour) = 12 ÷ 17.4
Hourly Energy Consumption (kWh/hour) = 0.69 kWh/hour
And its hourly energy usage during the heating season can be calculated as follows:
Hourly Energy Consumption (kWh/hour) = (BTU rating ÷ HSPF) ÷ 1000
Hourly Energy Consumption (kWh/hour) = (12,000 ÷ 8.7) ÷ 1000
Hourly Energy Consumption (kWh/hour) = 12 ÷ 8.7
Hourly Energy Consumption (kWh/hour) = 1.37 kWh/hour
As mentioned earlier, these hourly energy usage figures can be used to estimate the daily and monthly energy consumption of the mini-split, which can then be combined with the electricity costs in
your area to determine the cost of operating the unit.
However, it’s essential to note that the exact energy consumption of a mini-split is influenced by factors beyond its cooling/heating capacity, energy efficiency, and mode of operation.
Factors such as outdoor temperature, indoor temperature settings, the quality of insulation, and the condition of the unit also play significant roles in energy consumption.
Because of these variables, accurately predicting the energy usage of mini-splits can be challenging.
A much more accurate method to determine the energy consumption of your mini split is to measure it.
How to measure a mini-split’s energy consumption accurately?
The most accurate way to determine the energy consumption of your mini-split, or any appliance for that matter, is by using an electricity monitoring device such as the Kill-A-Watt meter or a similar
These devices provide readings of electricity usage, including Amps, Volts, and Watts, and are particularly valuable for measuring the energy consumption of an appliance over a specific timeframe.
Electricity Usage Monitors
To measure the energy consumption of your mini-split AC unit using the Kill-A-Watt meter, follow these steps:
1. Plug the monitoring device into an electrical outlet.
2. Plug your mini-split unit into the monitoring device.
3. Allow the mini-split unit to run for the desired period that you want to measure.
4. Press the purple button labeled “kWh” on the Kill-A-Watt meter to determine the energy consumption of the mini-split over the specified time frame.
By using this method, you can obtain precise data on the energy consumption of your mini-split under actual operating conditions, which can help manage energy usage and costs effectively.
Here’s a video that explains how to use a Kill-A-Watt meter:
Now that we’ve explored the energy consumption of these units and the methods for estimating or measuring it, let’s delve into the actual operating costs associated with running a mini-split.
How much does it cost to run a mini-split?
Based on the latest data from the U.S. Energy Information Administration (EIA) in June 2023, the average cost per kWh in the United States stands at approximately 16 cents ($0.16).
For instance, if your mini-split air conditioner consumes 150 kWh of energy per month, the average monthly cost to operate the AC unit would be approximately $24.
To offer a preliminary reference, here’s a table estimating the hourly operating costs of mini-splits based on their cooling and heating capacity during both the cooling and heating seasons:
BTU Rating Est. Hourly Cost for Cooling Est. Hourly Cost for Heating
9,000 BTUs 8 cents/hour ($0.08/hour) 14 cents/hour ($0.14/hour)
12,000 BTUs (1 ton) 10 cents/hour ($0.1/hour) 20 cents/hour ($0.2/hour)
15,000 BTUs 13 cents/hour ($0.13/hour) 25 cents/hour ($0.25/hour)
18,000 BTUs (1.5 tons) 16 cents/hour ($0.16/hour) 30 cents/hour ($0.3/hour)
24,000 BTUs (2 tons) 20 cents/hour ($0.2/hour) 40 cents/hour ($0.4/hour)
36,000 BTUs (3 tons) 30 cents/hour ($0.3/hour) 60 cents/hour ($0.6/hour)
48,000 BTUs (4 tons) 40 cents/hour ($0.4/hour) 80 cents/hour ($0.8/hour)
60,000 BTUs (5 tons) 50 cents/hour ($0.5/hour) 100 cents/hour ($1/hour)
The average hourly energy consumption of different size mini-split air conditioners at different outdoor temperatures.
Please note that the costs provided in the table were calculated using the average hourly energy consumption of these mini-splits, coupled with the U.S. national average cost per kWh in June 2023.
The actual cost of operating your mini-split AC unit depends on two primary factors:
1. Energy Consumption of the AC: This is measured in kWh (kiloWatt-hours) and can be estimated using the methods explained earlier or by utilizing an electricity monitoring device.
2. Cost per kWh in Your Region: You can find this information on your utility bill or by referring to estimates provided by the Energy Information Administration (EIA).
To calculate the cost, you can use the following formula:
Cost = Energy Consumption of the AC (kWh) x Cost per kWh ($/kWh)
For instance, let’s consider the following scenario:
You are using a 12,000 BTU mini-split heat pump that, on average, consumes approximately 120 kWh of energy each month during the cooling season, and around 200 kWh per month during the heating
Let’s also assume that you’re located in Nevada, where, as per the EIA, the cost of 1 kWh of electricity is approximately 17 cents ($0.17/kWh).
The average monthly cost to operate your mini-split unit would be as follows:
Cooling Season:
Cost = Energy Consumption (kWh) x Cost per kWh ($/kWh)
Cost = 120 kWh x $0.17/kWh
Cost = $20.4
Heating Season:
Cost = Energy Consumption (kWh) x Cost per kWh ($/kWh)
Cost = 200 kWh x $0.17/kWh
Cost = $34
To simplify the process and provide you with a more precise cost estimate, I’ve created a calculator that calculates the operating expenses of your mini-split based on the following:
1. The unit’s capacity (BTU rating).
2. Whether it’s used for cooling or heating.
3. Your typical daily usage of the unit (in hours).
4. Your location (State) and the cost per kWh in your area.
Now that we’ve covered the energy consumption (kWh) of mini-split air conditioners and the associated costs, let’s move on to explore the power usage (kW) of these units.
The insights provided in the next section will prove valuable, especially if you’re considering operating your mini-split AC unit using alternative energy sources.
Mini split AC Power Usage: How many watts does a mini split use?
The power usage (in watts) of a mini-split air conditioner or heat pump, often referred to as wattage, is primarily determined by its capacity, measured in BTU rating or tonnage. For instance, a
12,000 BTU mini-split typically has a rated wattage of around 1600 watts, while a 24,000 BTU unit usually operates at about 3200 watts.
However, it’s essential to note that mini-splits with the same BTU rating may have varying power ratings based on factors such as refrigerant type, unit design, and efficiency.
Here’s a table outlining the rated wattage range for mini-split ACs and heat pumps based on their cooling/heating capacity (BTUs/Tons):
Capacity Rated Wattage Range
9,000 BTUs 1,100 – 1,500 Watts
12,000 BTUs (1 ton) 1,400 – 2,000 Watts
15,000 BTUs 1,750 – 2,500 Watts
18,000 BTUs (1.5 tons) 2,100 – 3,000 Watts
24,000 BTUs (2 tons) 2,800 – 4,000 Watts
36,000 BTUs (3 tons) 4,200 – 6,000 Watt
48,000 BTUs (4 tons) 5,600 – 8,000 Watts
60,000 BTUs (5 tons) 7,000 – 10,000 Watts
Rated Wattage range of mini-split ACs and heat pumps based on their output capacity (BTUs/tons).
The rated Wattage of your mini-split will likely fall somewhere in these ranges. However, to accurately determine the power rating (wattage) of your mini-split, you should rely on the electrical
specifications provided by the manufacturer.
These specifications include:
• Voltage (in Volts): The electrical potential difference that drives the current in the system, typically measured in Volts (V).
• Compressor Amperage/RLA (in Amps): The maximum current (amperage) that the compressor will draw during its normal operation. This is often referred to as the Rated Load Amps (RLA).
• Outdoor (Condenser) Fan Motor Amperage/FLA (in Amps): The amperage the outdoor fan motor draws under full load, usually specified as Full Load Amps (FLA).
• Indoor (Evaporator) Fan Motor Amperage/FLA (in Amps): The amperage the indoor fan motor draws under full load, also specified as Full Load Amps (FLA).
With these specifications, you can calculate the rated wattage (power) of your mini-split using the formula:
Rated Wattage (Watts) = Rated Current (Amps) x Voltage (Volts)
Rated Current (Amps) = Compressor Amperage (Amps) + Outdoor Fan Motor Amperage (Amps) + Indoor Fan Motor Amperage (Amps)
The voltage, compressor RLA, and outdoor fan motor FLA are usually specified on the outdoor unit’s nameplate, while the indoor fan motor FLA might be indicated on either the condenser or indoor unit.
As an example, let’s take a look at these nameplates on the outdoor and indoor units of a 24,000 BTU mini-split heat pump:
First, let’s calculate the rated Amperage of the unit:
Rated Current (Amps) = Compressor Amperage (Amps) + Outdoor Fan Motor Amperage (Amps) + Indoor Fan Motor Amperage (Amps)
Rated Current (Amps) = 14.67 Amps + 0.65 Amps + 0.38 Amps
Rated Current (Amps) = 15.7 Amps
The rated Wattage of this mini-split heat pump can then be calculated as follows:
Rated Wattage (Watts) = Rated Current (Amps) x Voltage (Volts)
Rated Wattage (Watts) = 15.7 Amps x 230 Volts
Rated Wattage (Watts) = 3611 Watts
Now, it’s important to note that, most modern mini-split air conditioners and heat pumps are equipped with inverter-driven compressors, allowing them to adjust their output based on cooling/heating
Therefore, they don’t operate at a fixed wattage. The rated wattage represents the maximum power usage, although the actual usage may be lower.
When sizing equipment like an inverter or a generator for your mini-split, it’s essential to ensure their wattage capacity exceeds the maximum power (Watts) the unit may require. In the case of
inverter mini-splits, this is their rated wattage.
For non-inverter mini-splits, manufacturers provide an additional Amperage rating, the Compressor Locked Rotor Amps (LRA), representing the current (Amps) the unit’s compressor may need during
This rating, along with other electrical specifications, can be used to calculate the starting wattage of the unit:
• Starting Wattage (Watts) = Starting Current (Amps) x Voltage (Volts)
• Starting Current (Amps) = Compressor LRA (Amps) + Outdoor Fan Motor Amperage (Amps) + Indoor Fan Motor Amperage (Amps)
Ensuring the equipment can handle the starting wattage guarantees it can reliably power your mini-split without overloading or causing issues.
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2 Comments
1. Hello Younes!
Fabulous article, and the ONLY place on the entire web where I could find good answers to electrical power consumption questions!! MANY THANKS! I am trying to work out how to size a portable
electrical generator for backup and needed to know the wattage.
Appreciate your wisdom!
□ Hey there Bill,
I appreciate it.
I would say a 2000 Watt generator will do the trick for anything less than 18000 BTUs, maybe even up to 24000 BTUs, since mini splits with this kind of capacity will generally use less than
2000 Watts of power and don’t require a huge amount of power when starting up.
Before you make any purchases, please refer to the specification label of your unit, and look for the amperage rating specified (in Amps or “A” for short) by the manufacturer.
Once you find this amperage rating, multiply it by the voltage rating of the unit (around 120 Volts). This will help you determine the power usage of your unit more accurately.
For example, if your unit is rated at 115 Volts and 12 Amps, its rated power usage will be 115 V x 12 A = 1380 Watts. A 2000W generator will do the trick.
Hope this helps. | {"url":"https://www.renewablewise.com/how-many-watts-does-a-mini-split-use/","timestamp":"2024-11-03T02:44:28Z","content_type":"text/html","content_length":"265392","record_id":"<urn:uuid:6a80b301-634f-4c6e-ba2e-8e09804adb16>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00826.warc.gz"} |
Assessing the Potential for Renewable Energy Development on DOE Legacy Management Lands
This report represents an initial activity for the Department of Energy's (DOE) Office of Legacy Management (LM) to identify and evaluate renewable energy resources on LM-managed federal lands.
Within DOE LM's long-term surveillance and maintenance role, a key function is the establishment of environmentally sound future land uses by evaluating potential land reuse options. To
supportconsideration of renewable energy power development as a land reuse option task, DOE LM and DOE's National Renewable Energy Laboratory (NREL) established a partnership to conduct an assessment
of renewable energy resources on LM lands in the United States. The LM/NREL team used geographic information system (GIS) data to analyze and assess the potential for concentrating solar power
(CSP),photovoltaics (PV), and wind power generation on LM lands. GIS screening criteria developed with industry from previous studies for the Bureau of Land Management and the U.S. Forest Service
were applied to produce tables prioritized by renewable resource potential for all federal lands provided by LM. A principal objective was to gauge the renewable industry's interest in pursuing
renewablepower development on LM Lands. This assessment report provides DOE LM with information to consider when assessing alternatives of land reuse options for current and future LM lands.
Original language American English
Number of pages 163
State Published - 2008
• Bureau of Land Management
• concentrating solar power (CSP)
• EIADAS
• geographic information systems (GIS)
• legacy management lands
• LM
• NREL
• Office of Legacy Management
• photovoltaics (PV)
• solar
• wind
Dive into the research topics of 'Assessing the Potential for Renewable Energy Development on DOE Legacy Management Lands'. Together they form a unique fingerprint. | {"url":"https://research-hub.nrel.gov/en/publications/assessing-the-potential-for-renewable-energy-development-on-doe-l","timestamp":"2024-11-13T05:18:51Z","content_type":"text/html","content_length":"47909","record_id":"<urn:uuid:17fe489e-bcdb-4f77-82fe-ba7c267fb652>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00643.warc.gz"} |
pumpkin seeds
What do you get when you divide the circumference of a jack-o-lantern by its diameter? Read on to find out the answer to this pumpkin
Fall is about to turn into winter for many of us, so what do you do with all those pumpkins that won’t make it through | {"url":"https://bedtimemath.org/tag/pumpkin-seeds/","timestamp":"2024-11-14T10:44:29Z","content_type":"text/html","content_length":"86384","record_id":"<urn:uuid:ec6baaac-271a-4a69-933e-9e0d7d13e6ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00478.warc.gz"} |
Digest/Hash function
The Digest/Hash function produces a digital summary of information called a message digest. Message digests provide a digital identifier for a digital document. The message digest produced by the
Digest/Hash function is Base64 encoded.
Message digest functions are mathematical functions that process information to produce a message digest for each unique document. Identical documents have the same message digest, which can be used
to ensure that the message received is the same as the message sent.
There are three
functions in Studio:
• Digest/Hash the Input Data with MD5
• Digest/Hash the Input Data with SHA-1
• Digest/Hash the Input Data with SHA-256
The input is the document or string for which you want a digest. For example, MD5 ("Austin was happy that the band played on") =NjJhODJhNTViZmI3Y2YwZDc2NDkxYjc0ZTkzZDlmMTQ=
MD5 is defined in RFC 1321. IBM® App Connect uses the MD5 algorithm included in the JDK security package. The algorithm takes a message of undefined length and outputs a message digest of 128 bits.
SHA-1 is defined by the Federal Information Processing Standards Publication 180-1 (FIPS PUB 180-1). App Connect uses the SHA-1 algorithm included in the JDK security package. SHA-1 takes an input
message of any length less than 264 bits and produces a message digest of 160-bits.
SHA-256 is a 256-bit hash function and is compliant with the National Institute of Standards SP 800-131a specification. App Connect uses the SHA-256 algorithm included in the JDK security package.
The input is the document or string for which you want a digest. For example, SHA-56 ("Sample Input") =bEzV+7Tz6afzJhY0E5u0Zt1+9uBURb/2pgi2PT9Ms/s=. The hash value is 32 bytes or 256 bits length.
Use Digest/Hash the Input Data with MD5 function to create a Base64 encoded digest of the input data using MD5.
Use the Digest/Hash the Input Data with SHA-1 function to create a Base64 encoded digest of the input data using SHA-1.
Use the Digest/Hash the Input Data with SHA-256 function to create a Base64 encoded digest of the input data using SHA-256. | {"url":"https://www.ibm.com/docs/en/app-connect-pro/7.5.3?topic=reference-digesthash-function","timestamp":"2024-11-02T06:23:55Z","content_type":"application/xhtml+xml","content_length":"7661","record_id":"<urn:uuid:5e670a75-f385-4a03-8ff3-b36c5d3d0011>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00389.warc.gz"} |
Blog: New Shader Graph Node Reference Samples
The Shader Graph team is excited to announce the release of the new Node Reference Samples, available now for 2021 LTS, 2022 LTS, and future releases.
Node Reference Samples is a collection of over 140 Shader Graph assets. Instead of using these graphs for materials in your project, you can use them as a reference to learn what each node does and
how it works. Each graph represents a node that’s available in the node library. It also contains a description of the node, explains its functionality, and breaks down how the math works under the
hood. To learn how to use a specific node, open its reference file to see descriptions, examples, and breakdowns of that node.
In the samples below, we’re using the Shader Graph tool to illustrate how you can use Node Reference Samples in your next project.
Examples of available samples
Let’s take a look at the Dot Product node:
There’s a lot going on here, so let’s break it down. At the top of the graph, we have the node and a basic description of what the dot product operation does.
On the left, we have the Under The Hood section, which breaks down the dot product operation into more basic math so you can see exactly what’s happening when you use the Dot Product node. Notice
that there are helpful tips and descriptions of what’s happening and why.
This section shows you what you can do with the Dot Product node. We can see the node can be used to desaturate a color or as a handy method for texture channel selection and that the dot product is
used as the basis for diffuse lighting calculations.
This is just one example. We’ve created over 140 similar graphs representing a large majority of the nodes available in Shader Graph, each containing descriptions, examples, illustrations, and
helpful tips.
For more information, read the full blog. | {"url":"https://discussions.unity.com/t/blog-new-shader-graph-node-reference-samples/325158","timestamp":"2024-11-07T14:26:37Z","content_type":"text/html","content_length":"34592","record_id":"<urn:uuid:8efccd4f-2248-49a1-abde-1490c0d3325f>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00365.warc.gz"} |
Book Club: Mathematical Go, Chilling Gets the last point by Elwyn Berlekamp and David Wolfe
So since you guys keep posting all kinds of Go positions, I’d like to talk about:
Go as a combinatorial game (Section 4.1 and Appendix A/B?)
It seems to me that the book doesn’t do a great job of explaining the basics of how we can represent Go as a combinatorial game. Maybe some of it is supposed to be obvious, and maybe some of my
discomfort comes from a preference for Chinese rules, while the book mostly uses Japanese.
So I’ll make some attempt here. Does this kind of thing work to draw a game tree?
Black ↙ ↘ White
⚫⚪⚪⚪ ⚫⚪⚪⚪
⚫⚫⚫⚪ ⚫⚪⬛⚪
⚫⚪⚪⚪ ⚫⚪⚪⚪
0 -2
It should look like this
To represent Go as a combinatorial game, we consider all the reasonable moves (ignoring for simplicity unnecessarily filling in your own eyes or playing dead stones in the opponent’s territory). We
also assume (for now at least) that there is no ko, since otherwise the games would not combine properly, and there might be cycles. Then we get a tree like the one above.
However, instead of just ending like a normal combinatorial game, Go has a score at the end. In this case it’s 0 if Black moves and -2 (2 points for White) if White moves. To take this into account,
we put the 0 and -2 combinatorial games at the leaves of the tree.
Specifically, this means if Black moves the game is over, but if White moves, White gets 2 more moves afterward. You could imagine actually playing these moves with some scheme involving prisoner
return and eye-filling, but for analysis of the game it’s not necessary.
At least that’s my understanding, but did the book ever really say it like this?
The example above can be written as { 0 | -2 }, which is notably a hot game, meaning that it’s to both players’ advantage to play first. Most Go positions are hot; that’s why we keep playing instead
of just passing. Hot games are notoriously hard to analyze since they don’t simplify to numbers or infinitesimals.
This game is particularly NOT equal to its average value of -1.
To see that G ≹ -1, consider G + 1, meaning the combination of this game with a single extra point for Black. Clearly, whoever plays first will win, so G + 1 is not zero.
What if we use Chinese rules? Since stones count, we have to decide which ones to consider, and I think it makes the most sense to measure the score relative to the starting position, assuming all
stones were alive. Then Black just gains one additional point by playing first, while White converts a black stone to white territory and gains one stone, getting to -3. So the game becomes { 1 | -3
}, apparently an even hotter game.
Interestingly you can get back to the Japanese game if you impose a “tax” of one point on whoever plays. I’m not sure if this relationship was mentioned in the book, but Sensei’s says it like this:
In general, the difference between Territory and Area scoring is that stones on the board are counted in area scoring but not in territory scoring. Suppose that you play by area scoring but each
board play costs one point. You would actually be playing by territory scoring, since each board play results in a stone on the board or captured.
Since we know Japanese and Chinese rules generally have the same optimal play, that suggests taxing away yet another point from each move, which is called chilling. But is there more we should talk
about first just with normal unchilled Go games? Can we work out the actual mathematical scores of the reversal example above or something like that? | {"url":"https://forums.online-go.com/t/book-club-mathematical-go-chilling-gets-the-last-point-by-elwyn-berlekamp-and-david-wolfe/52695/53","timestamp":"2024-11-02T18:57:50Z","content_type":"text/html","content_length":"18741","record_id":"<urn:uuid:8e37b3b9-3941-42a3-995f-159c21f2b5f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00310.warc.gz"} |
How to Determine the Cardinality of sets that go to Infinity
How to Determine the Cardinality of sets that go to Infinity
Thanks to the volume of evoTARDgasms wrt set theory, I have been able to determine how to figure out the relative cardinality wrt two sets that go to infinity.
The cardinality is determined by the mapping function.
For example the mapping function F(n)=2n, means that one set has 2x the cardinality of the other set.
The funny part is that has been right in front of their silly faces for over 100 years and no one could spot it. They thought they were mapping a one-to-one corresponce. However that "mapping" just
means they made each set equal to the other, equal in size as well as equal in membership and the equation is the actual difference in cardinality between the two original sets.
I see that keiths can only laugh at my proposal. Small minds tend to laugh at things that are out of their depth. Thanks keiths. Everything you post proves that I am correct and you are an imbecile.
13 Comments:
• At 2:08 PM, Unknown said…
Hmmm . . .
How about an infinite set generated by randomly selecting with replacement words from the Oxford English Dictionary?
Or the set comprising {0.1, 0.2, 0.3, . . . . 0.9, 0.10, 0.11, . . . 0.19, 0.20 . . . } Some of the elements are equivalent numerically.
Or the set comprising {3, 1, 4, 1, 5 . . . } the digits of Pi.
What are their cardinalities?
• At 2:18 PM, Joe G said…
• At 3:27 PM, Unknown said…
• At 6:50 PM, Rich Hughes said…
• At 7:15 PM, Joe G said…
• At 7:17 PM, Joe G said…
• At 2:29 AM, Unknown said…
Example one:
1 <-> first word picked
2 <-> second word picked
3 <-> third word picked
(as the random selection is done with replacement words could appear on the list more than once)
Example 2:
1 <-> 0.1
2 <-> 0.2
3 <-> 0.3
Example 3:
1 <-> 3
2 <-> 1
3 <-> 4
4 <-> 1
In all three cases one countably infinite set is matched one-to-one with another set. I don't know what the, say, 3567th word will be but I can get a computer to gnerate a list quickly. Clearly
that mapping will change depending on the random selection. The other two are static.
What does your method say? What is the cardinality of the set of the digits of Pi? Or the digits of e? Or the digits of the square root of 2?
• At 9:46 AM, Joe G said…
• At 1:42 PM, Unknown said…
"No matching. To match means the two objects on eiether end of the matching line are the same."
How can you match the end of infinite sets? You can show a one-to-one correspondence between elements of one set to another. And, if you can do that, showing that noting gets letf out of either
set, that the sets must be the same size.
"So no equations- I asked for equations, Mr Math."
Ummm . . . you don't need "equations" to define functions. Or mappings.
• At 2:08 PM, Joe G said…
How can you match the end of infinite sets?
There aren't any ends. However you can determine if the numbers will be the same at any point in time by looking at the finite and if tehre is a pattern, using it to extend to infinity.
Again infinity isn't magical. It is just more of the same if a pattern is found in the finite.
And your mapping is just arbitrary. And has nothing to do with math.
• At 4:24 PM, Unknown said…
"There aren't any ends. However you can determine if the numbers will be the same at any point in time by looking at the finite and if tehre is a pattern, using it to extend to infinity."
It's not a question of if the numbers are the same! It's a question of whether or not there are the same number of things!! Infinity doesn't work like the finite. That's the point!!
Why do you think Cantor's ideas were initially met with such animosity? It's becuase even mathematicians found them difficult and counterintuitive.
"Again infinity isn't magical. It is just more of the same if a pattern is found in the finite."
No, it isn't magical. But it's not the same as the finite. And the pattern in the finite, in my examples, is that there is a one-to-one correspondence between the integers and the even integers.
All the way down the line.
"And your mapping is just arbitrary. And has nothing to do with math."
It has everything to do with math. Real math. Math is more than just formulas and equalities. As you should know having taken a Calculus course and a course in Set Theory.
• At 12:27 AM, Unknown said…
Who said mathematics was just about numbers? it's about sets and manifolds and rings and groups and graphs and trees and Euler circuits and hyper-cubes and tilings. Mathematics is about patterns.
"A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." GH Hardy.
• At 7:20 AM, Joe G said… | {"url":"https://intelligentreasoning.blogspot.com/2013/05/how-to-determine-cardinality-of-sets.html?m=0","timestamp":"2024-11-04T13:53:28Z","content_type":"application/xhtml+xml","content_length":"35425","record_id":"<urn:uuid:1c200893-ac3a-4b56-8bae-37858b095781>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00193.warc.gz"} |
South Carolina Get Ready Grade 3 Mathematics Chapter 2
1 pt
1 pt
Shawn counted 465 medals in his father’s medal collection. His father also has 294 arrowheads in another collection. The total number of medals and arrowheads his father has can be found by the
expression below.
465 + 294
How many medals and arrowheads does Shawn’s father have in all?
Tracy collected 428 pretty fall leaves to press in a book.
Part A.
She decided this was too many and put 193 of the leaves back in the yard.
Fill in the blank: Tracy pressed leaves in a book.
Part B.
Mindy explains there are 12 more sugar cookies than almond cookies because the smaller number is always subtracted from the larger number.
So she got 6 ‒ 4 = 2 in the ones place and 2 ‒ 1 = 1 in the tens place.
Explain why Mindy’s reasoning is incorrect. Find how many more sugar cookies than almond cookies she has. Write your answer and your explanation in the box provided. | {"url":"https://onlinetesting.americanbookcompany.com/sample/1854","timestamp":"2024-11-12T16:15:42Z","content_type":"text/html","content_length":"13532","record_id":"<urn:uuid:75bd74ff-490b-4614-bad1-d849578f34bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00222.warc.gz"} |
Summer Oenothera Exhibition
While some people enjoy spending their time solving programming contests, Dina prefers taking beautiful pictures. As soon as Byteland Botanical Garden announced the Summer Oenothera Exhibition she
decided to test her new camera there.
The exhibition consists of \(l=10^{100}\) Oenothera species arranged in a row and consecutively numbered with integers from \(0\) to \(l − 1\). The camera lens is able to take photos with \(w\)
species in it, i.e. Dina can take a photo containing flowers with indices from \(x\) to \(x + w − 1\) for some integer \(x\) between \(0\) and \(l − w\). We will denote such a photo as \([x, x + w −
She has taken \(n\) photos, the \(i^{th}\) of which (in chronological order) is \([x_i, x_i + w − 1]\) in our notation. She decided to build a time-lapse video from these photos once she discovered
that Oenothera blossoms open in the evening.
Dina takes each photo and truncates it, leaving its segment containing exactly \(k\) flowers, then she composes a video of these photos keeping their original order and voilà, a beautiful artwork has
been created!
A scene is a contiguous sequence of photos such that the set of flowers on them is the same. The change between two scenes is called a cut. For example, suppose the first photo contains flowers \([1,
5]\), the second photo contains flowers \([3, 7]\) and the third photo contains flowers \([8, 12]\). If \(k = 3\), then Dina can truncate the first and the second photo into \([3, 5]\), and the third
photo into \([9, 11]\). First two photos then form a scene, third photo also forms a scene and the transition between these two scenes which happens between the second and the third photos is a cut.
If \(k = 4\), then each of the transitions between photos has to be a cut.
Dina wants the number of cuts to be as small as possible. Please help her! Calculate the minimum possible number of cuts for different values of \(k\).
Input format
The first line contains three positive integers \(n\), \(w\), \(q\) — the number of taken photos, the number of flowers on a single photo and the number of queries.
Next line contains \(n\) non-negative integers \(x_i\) — the indices of the leftmost flowers on each of the photos.
Next line contains \(q\) positive integers \(k_i\) — the values of \(k\) for which you have to solve the problem.
It's guaranteed that all \(k_i\) are distinct.
Output format
Print \(q\) integers — for each width of truncated photos \(k_i\), the minimum number of cuts that is possible.
\(1 \leq n, q \leq 10^5\).
\(1 \leq k_i \leq w \leq 10^9\).
\(0 \leq x_i \leq 10^9\).
Subtask # Score Constraints
1 18 \(n, q \leq 3000\)
2 36 \(n, q \leq 60000\)
3 46 No additional constraints
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2 | {"url":"https://codebreaker.xyz/problem/oenothera","timestamp":"2024-11-14T15:41:23Z","content_type":"text/html","content_length":"16302","record_id":"<urn:uuid:ee38fb09-ae01-4165-8912-864479a43a11>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00024.warc.gz"} |
Unique Rectangles
Unique Rectangles takes advantage of the fact that published Sudokus have only one solution. If your Sudoku source does not guarantee this then this strategy will not work. But it is very powerful
and there are quite a few interesting variants.
Credits first: The initial ideas for these strategies I lifted wholesale from MadOverLord's description (11 Jun 05) on the forum at www.sudoku.com. (Sadly the thread is no longer hosted). Many others
have since added to and improved them. I have provided my own examples. (please email me for other credit requests). I will stick to MadOverLord's nomenclature.
Such reasoning allows us to remove other 1s and 7s on the unit shared by the roof cells (but not the cell we’re using to create a locked set with). There are 1s and 7s in C7 and H7 which can be
So far Type 3 and 3b have worked with a pseudo-cell linking with a bi-value cell to make a pseudo pair. This is a valid locked set. Well, locked sets don't have to be pairs, they can involve three
candidates over three cells.
... by: Aleksandra Z
Monday 28-Oct-2024
After noticing it recently in the puzzle below, I have started using a pattern which resembles a weaker but more common version of Type 5; in short, an elimination still is possible even if only a
single strong link is present, as opposed to the full rectangle being linked strongly.
If a rectangle across two boxes has opposing corners with only the same two candidates X/Y, and if one candidate X has just one strong link to an adjacent corner, then X can be removed from the other
corner. The reason is that placing X there would force adjacent double Y, but then the strong link demands opposite X, which is the ambiguous pattern we are trying to avoid.
This is an example puzzle, which reaches that scenario if XY-Chain and 3D Medusa are disabled:
It occurs in the rectangle AC37, and it becomes available immediately after A7 becomes 3/7 along with its opposite C3, also immediately before the solver finds other unique rectangles. From here, it
suffices to note that 7 is strong in row C—that alone is enough to remove 7 from A3. Independently of that, because the other candidate 3 is strong in row A it can be eliminated from C7.
As it happens, 7 was also strong in column 7 (and 3 strong in column 3), but I emphasise that only one of those strong links are required for an elimination. The reduced need for strong links makes
this partial pattern much more common; unlike the full Type 5 pattern (or the very useful Hidden Unique Rectangle pattern!), neither candidate is required to form an X-Wing.
Personally I imagine it as a "partial unique rectangle" or perhaps a "unique rectangle edge elimination" given that demanding uniqueness risks eliminating a candidate from one edge, but in the
context here it is described very well as a lighter version of Type 5!
... by: Mika
Saturday 26-Aug-2023
It seems like Type 5 Unique Rectangles is a variation of Type 4 (Not entirely sure).
... by: David Harkness
Thursday 24-Aug-2023
Type 3 can be extended to hidden pairs/triples/quads similar to how it works with naked tuples by treating the floor cells as a single pseudo cell. Might this become type 3c?
In this example, the UR is in BC69. Since 347 form a hidden triple in C4, C5, and C69 (the pseudo cell), 5 and 6 may be eliminated from C4 and C5. It does not allow eliminating 56 from the floor
cells, however.
| 7 5 2 | 9 36 8 | 16 4 136 |
| 3 48 48 | 2 1 56 | 79 79 56 |
| 69 1 69 | 34567 3456 3567 | 2 8 356 |
| 145 6 3 | 457 8 2 | 147 17 9 |
| 14 2 7 | 346 9 36 | 5 16 8 |
| 8 49 459 | 1 456 567 | 467 3 2 |
| 2 7 1 | 356 356 9 | 8 56 4 |
| 4569 489 45689 | 56 2 1 | 3 569 7 |
| 569 3 569 | 8 7 4 | 169 2 156 |
Load Sudoku
Unfortunately, the naked pair 56 in C1 and C3 eliminates the 6 from C6 and C9, breaking the UR.
... by: David Harkness
Wednesday 23-Aug-2023
Greetings, Andrew.
It seems type 3b should extend to naked quads and hidden pairs/triples/quads as well.
... by: Robert
Sunday 25-Jun-2023
Just to follow up - in the first graphic on this page, the "deadly pattern" only remains deadly in Killer Sudoku if either
a) B2 and B3 are in the same cage, and D2 and D3 are in the same cage, or
b) B2 and D2 are in the same cage, and B3 and D3 are in the same cage.
We could also have all four in the same cage, but this is precluded if we are using the "killer cage convention".
... by: Robert
Sunday 25-Jun-2023
I've just sent a detailed email, but the short summary here is - be careful applying the Unique Rectangle strategies to Killer Sudokus. It all depends on the particular "cages" the four points of the
rectangle are in.
So this idea that all regular Sudoku strategies also work on Killer Sudokus - I don't think it extends to "uniqueness" strategies, at least not without some additional assumptions on the cages.
This can be seen in Killer #5768 (June 22). Use the solver until it decides to remove "5" from cell J7. This is based on a "UR" strategy, and ultimately finds one solution. However, this is another
solution that does have a "5" in J7 - the inference, that the 5 can be removed, is only valid in one of the two solutions.
... by: Steve Benoit
Thursday 16-Mar-2023
I think I've seen your type 5 Unique Rectangles before. It took me a while to remember where... Check out Sudoku Swami's "Unique Rectangles Part 2 / Sudoku Tutorial #21" on Youtube beginning at
time-stamp 12:32. He refers to them as UR-Type 6.
Fantastic website by the way! I wish I had found it much sooner...
... by: Tapio Ranta-aho
Tuesday 6-Dec-2022
Hello .
I wonder if this can be a version of unique rectangles? This happened once and candidate 5 appeared to be correct, but I don't know that wasn't just a coincidence.
___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ ___
___ 12_ 12_ ___ ___ ___ ___ ___ ___
___ 12_ ___ ___ ___ 12_ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ 12_ ___ ___ 125 ___ ___ ___
Anonymous replies: Tuesday 17-Jan-2023
The given example is correct, but the solver doesn't search for this pattern.
Pieter, Newtown, Oz replies: Thursday 2-Feb-2023
@Tapio Ranta-aho.
Assuming the "12" pairs are Naked Pairs (no other candidates), this is virtually a "stretched" Type 1 UR.
Also @Andrew
I refer you to Jonathan Handojo's comment on 2021-06-9 finding that the Solver skips UR's and jumps to WXYZ, similar to my earlier comment on 2019-12-16.
I also found a sorta Type 5 in your daily puzzle for 2023-01-21 but not in the Solver order, which I will try to retrieve and send separately.
As always thanks for a fabulous website!
Ciao, Pieter
... by: Maggie Mc
Tuesday 5-Apr-2022
I played a game of brainium sudoku in which the four corners of a square had the same two numbers could only have those two numbers because they were the only spaces left in their respective rows.
This should not be possible. I can send a screen shot. Can you explain?
Andrew Stuart writes:
Please do
Anonymous replies: Tuesday 17-Jan-2023
Are the cells falling in two boxes?
... by: Anonymous
Monday 10-Jan-2022
already has a documentation of Type 5 (under the name Type 6),
it mentions that it is the same as two HUR type 1's.
Seeing the website hasn't been updated for about 8-9 years, this type has been known for a long time already.
... by: Anonymous
Friday 31-Dec-2021
Isn't a Type 5 UR just two Type 1 HURs on the same spot?
... by: Pieter, Newtown, Oz
Thursday 4-Nov-2021
Hi Andrew
Re Type 5 URs
1. The output from the solver says, for example, "Uniqueness Type 5: removing 2 from E6 and F1 because if true it would create a Deadly Rectangle with E1 and F1". It should read "... would create a
Deadly Rectangle with E1 and ***F6***".
2. I've noticed odd characters in some places in the text, such as "Thanks to Ivar Ag�y from Norway" in Type 5 above. Also in the Digit Forcing Chains strategy - Second Example "DIGIT FORCING
CHAIN: because … -5[G9]+5[F9]-5[F1]+5[C1] ��-4[C1]+4[B1".
I find it curious that you "updated" version 2.08, twice, rather than naming them 2.09, 2.10?
Thanks as always
... by: alpreucil
Wednesday 18-Aug-2021
Andrew! Where is your Solver??? The wonderful one with the black background. I've used this solver for years to solve the puzzles that puzzled me, and I found it very user-friendly and helpful. NONE
of the other solvers I've tried hold a candle to it. I want it back! Any reason why it has disappeared from the site??? Thanks! Al Preucil
Andrew Stuart writes:
Always at https://www.sudokuwiki.org/sudoku.htm
Never been down except for server migrations
... by: Tunner
Monday 12-Jul-2021
Through out this section you repeatedly state that the "rectangle" consists of 2 rows, 2 columns, and 2 boxes. But, is it not true that the process is the same if there are 2 rows, 3 columns, and 2
boxes? Or 3 rows, 2 columns, and 2 boxes? I think the boxes are the important link. This may not be true for all types but it seems to work for type 1.
Andrew Stuart writes:
Sounds like an
Extended Unique Rectangle
... by: SG
Tuesday 6-Jul-2021
There is a Sudoku board I would like you to look at for another example of Type5.
Click on this link
Andrew Stuart writes:
Nice one
... by: BobW
Saturday 19-Jun-2021
Hi Andrew,
I believe the new type 5 that you posted (January 2021), has been known for some time.
It is described on the enjoysudoku forum, along with many other variations of UR's having two extra candidate cells, located diagonally. That post is dated April 2006. I ran across it while
implementing UR's in my own solver last year.
If I'm not mistaken, the one that you refer to as Type 5 is referred to as UR+2D/1SL (not to be confused with UR+2D or UR+2d).
I spent many hours going through that post and many others in the same thread. The logic for many of them is so convoluted that it made my head spin. It occurred to me that it would be much simpler
just to work out the truth tables for them. One day, having nothing better to do, that's exactly what I did. I worked out the truth tables for every possible combination of UR's with 2, 3 or 4 extra
candidate cells ("Guardian cells" in my notation) with every possible combination of strong links between the various cells. Summarizing only patterns that give eliminations of internal UR
candidates, I found 8 different UR+2 patterns with adjacent guardians, 11 different UR+2 patterns with diagonal guardians, 18 different UR+3 patterns, and 5 different UR+4 patterns.
When I refer to the number of patterns found, I've excluded reflections and rotations of the basic patterns. I implemented the resulting rules using a lookup table in my solver.
There's not enough room here to give all of my results, but if you contact me by email, I'd be happy to share what I've done.
Andrew Stuart writes:
Hi Bob
That sounds like thorough work, well done. I should take it all in and completely redo the UR. Big job though.
... by: Jonathan Handojo
Wednesday 9-Jun-2021
Here's a puzzle that I purchased from one of your Diabolical packs (and I have been enjoying a lot of these. I hope it's fine with sharing one of those puzzles here.)
load here
If you use the Solve Path with every checkbox ticked, you get a WXYZ-Wing evaluation. It seems to have skipped through the Unique Rectangles strategy when there's one in B7, B9, J7 and J9 that allows
the eliminations: 8 from J7, 6 from J9, and 8 from B9. Either value you force into them forms the deadly pattern, so I solved the puzzle quite with ease after using this. Now I'm not too sure of the
rules with this, but I only saw it through forcing.
Andrew Stuart writes:
Something happening here. Will have to investigate.
... by: Pieter, Newtown, Oz
Monday 16-Dec-2019
Hi Andrew
As always thanks for a fabulous website!
Solving your
Daily Sudoku for 2019-11-15
I believe it has a Type 4 UR which the solver does not find.
After solving the basics we are left with
this board
and the solver next finds an XYZ-Wing.
If you backstep and turn XYZ and X-Cycles off, I would expect the solver to find what I believe to be a Type 4 UR in EJ89, with the floor in E8/9, and the conjugate pair of 4's in J8/9, eliminating
the 5's in J8/9. However, the solver skips all URs and finds a WXYZ.
Is this correct/a bug?
Ciao, Pieter
... by: Niki
Friday 7-Sep-2018
I'm curious why the Unique Rectangle strategy isn't included in the Jigsaw Sudoku solver. Do other strategies there render it irrelevant?
I ask because I just spotted one for the first time in an extreme jigsaw puzzle - it was a 3/3b (triple), and it made the rest of the puzzle collapse nicely. Despite the jigsaw shape, it still fell
in two rows, two columns, and two boxes just like it ought. But I was hesitant to follow up on it, having never seen the Unique Rectangle strategy mentioned in conjunction with jigsaws.
Andrew Stuart writes:
This is on my to-do list. Merely satisfying the criteria may not be enough to be a true elimination in all cases. This is my intuition but I need to find some counter-examples. I believe that the
potential distortions do not allow it in most cases. Can you send me the puzzle? (use the email button). I should be able to run my library through with UR turned on and see if a rule can be devised,
but it will be a qualified one I suspect
... by: Pieter, Newtown, Oz
Thursday 12-Jul-2018
Hi Andrew
I am confused by the solvers handling of URs in this
partially solved puzzle
I have spotted what I believe to be a clear Type 4 UR in CH45, on 1/9. However, the solver finds a Type 3b combining with the 5/8 in H6 and then the puzzle solves.
I tried unchecking URs and the solver finds a Hidden UR Type 1 and eliminates the 1 in H5. Why does it not eliminate the 1 in H4? Taking Steps to try again, even re-checking the UR box, the solver
moves to a HUR in EF28.
A 1 in H4 would create the Deadly Pattern. Why isn't it eliminated? Bug?
Ciao, Pieter
Andrew Stuart writes:
I tested this position and yes the Type 4 does get detected, well spotted. It's the ordering of the URs which means 3B will get tested and return first. Thinking of adding a toggle for them all in
the strategy list
... by: digituer
Thursday 2-Nov-2017
It's a pleasure reading your pages. I've learned a lot. Thanks Andrew!
The pseudo-cell concept in Type 3 is great! I think based on this pseudo-cell concept, a hidden pair strategy can also be applied.
12 . . | . . 12AC
. . . | . . .
12 . . | . . 12AB
. . . | . . 12FE
If the 2 roof cells and that extra cell are the only three cells holding 12 in the unit, then FE can be removed (ABCFE can be any numbers).
... by: AAHoffman
Monday 17-Apr-2017
I often extend this strategy to 3, 4, or even more naked pairs forming multiple "floors" or a multi-story structure, i.e.,
as a simple example. Solutions devolving to this can be eliminated.
Multiple floors can be bent and don't have to be in a straight line, i.e.,
A1=A2={2,4} (row)
D1=E2={2,4} (note: forms a diagonal that "bends" the structure)
D4=E4={2,4} (column)
Another possibility uses naked triples in three rows, i.e.,
{A1,A2,A3}={2,3,4} (in any combination (except naked single or pair), i.e., {23},{24},{34})
{D1,D2,D3}={2,3,4} (i.e., {24},{34},{23})
{G1,G2,G3}={2,3,4} (i.e., {34},{23},{24}).
All these patterns occur often enough to be useful.
By the way, not sure how nomenclatures are standardized but I have named these "meta" strategies instead of naked rectangles... because they are a rule dependent on a rule of the game. When three
digits are involved as above, I call them "triple meta".
... by: Barry
Wednesday 13-Jul-2016
Using the type 1 example, couldn't the removal of just the 2 or just the 9 prevent a deadly pattern? Of course, you wouldn't necessarily know which one to eliminate at that point, but isn't it
possible that another solving technique might eventually eliminate one of the primary candidates thus no longer making it part of a unique rectangle? I know the methodology works and that neither
candidate will be part of the final solution, but just don't see the logic of being able to say that both candidates should be eliminated.
... by: Thinkist
Friday 29-Apr-2016
Hello again...
I came to a clearer realization on why I dislike uniqueness strategies: they mess with the solution count if a sudoku happens to have multiple solutions, erroneously decimating otherwise valid
solutions. I point you to an example of mine
. It has 63 solutions, but 4 URs erroneously deduce it to 3, and then one use of the BUG strategy erroneously deduces it to 1. So, this makes the puzzle appear valid when in fact it should have 63
solutions all along and the solver should fail to complete correctly. The chaining, netting, and simple combinatoric strategies don't mess with the solution count. In those cases, simply because
there is more than one solution, the solver will merely fail to complete correctly rather than throw up an error. I believe that any puzzle (or "puzzle", if you will) that throws up a contradiction
should have 0 solutions from the start, not because of a uniqueness strategy.
Re-reading your reply to my first comment, I'm glad that you've not decided to add a solution count before any logical solving takes place, as that would of course ruin the purpose of using the
solver, which is to showcase a bottom-up approach to sudoku solving rather than cheat it with a brute-force top-down approach.
And for the record, I realize that any valid sudoku puzzle should have only a single solution. The point I'm trying to make is that uniqueness strategies aren't the best way to find it (which is why
I still have them unchecked in the solver). In essence, they assume what you are trying to prove (via chaining, netting, and simple combinatoric strategies): that the puzzle has a unique solution.
Andrew Stuart writes:
I'm interested in your assertion "The chaining, netting, and simple combinatoric strategies don't mess with the solution count". If true it would be one way of showing that a puzzle might be valid
(by completing using such strategies) but it would require some kind of proof to show that all a) all valid sudoku puzzles can be solved using only these strategies minus uniqueness strategies and b)
that all multi-solution (non-valid) puzzles cannot complete.
The problem with a) is that I don’t know of a solver that claims to solve all puzzles (certainly I don't pretend mine is the most advanced and I have a long job queue of improvements) and the problem
with b) is that I'm sure a non-valid puzzle could accidentally complete giving the impression of a single solution solved logically. Wish I had one to hand, but I strongly suppose it.
However there is a good practical reason for using logic rather than brute force for solution counting, as the attached graph tries to show. It’s a crude diagram I knocked up to illustrate a point in
a previous correspondence, but there is a point at which brute force becomes much slower than logic and it’s a function of clue density.
... by: Master Shuriken
Tuesday 27-Jan-2015
You could have another strategy which is an extension of unique rectangles (extension of both 2x2 and 3x3).
If you have, for example empty boxes in A1, A2, H1, J2, H9, J9 with 2 clues (1,2) in each box, this wouldn't be possible.
You could extend this *uniqueness chain* as long as you like through the puzzle and extend it to triples, too, (as long as you make sure that there would be 2 solutions if everything else was filled
Like the unique rectangle, it would tell you something which is not possible, and could help eliminate clues, etc.
Andrew Stuart writes:
Extended Unique Rectangles
... by: dshaps
Tuesday 29-Jul-2014
Under Type 4, following comment appears.
If you look carefully, you'll see that in box 7, the roof squares are the only squares that can contain a 7. This means that, no matter what, one of those squares must be 7 - and from this you can
conclude that neither of the squares can contain a 9, since this would create the "deadly pattern"! So you can remove 9 from H1and H2.
How is it that deadly pattern will get created, if either of roof squares contains 9. Can you please elaborate?
... by: PCForrest
Sunday 6-Jul-2014
@Random Sudoku Enthusiast
"In the 'Type 1 Unique Rectangles' example (Unique Rectangle Figure 2), can't the same explanation be used for the 2 and the 9 in D2? Logically, it doesn't make sense to eliminate the 2 and the 9 in
D1, but not in D2. It seems that by eliminating the 2 and the 9 in D1, we move the deadly rectangle corner over to D2. Should we then use the same logic to eliminate the 2 and the 9 from D2? Or are
we ignoring that square for the sake of the example?"
D2 does not a rectangle make, let alone a unique rectangle. To be valid, all four cells must occupy exactly 2 rows, 2 columns and 2 boxes. Logically, it makes perfect sense to eliminate both 2 and 9
from D1 but not from D2, for the simple reason D2 must take the value 2 in the final solution. If you eliminate the 2, you immediately invalidate the puzzle. The logic that applies to D1 does not
apply to D2. We are ignoring D2 for the simple reason it plays no part in this unique rectangle.
... by: Sherman
Friday 28-Mar-2014
Two of the Unique Rectangle examples on this page can also have the Hidden Unique Rectangle logic applied to them, resulting in even more eliminations.
For example, figure 7 in the Type3b UR discussion has strong links between all four 6's in the UR. If 7 is placed in cell E6, 6's must be placed in cells E4 and J6, forcing a 7 in J4 - a deadly
pattern. So 7 cannot go in E6. Likewise, 7 cannot go in E4.
The HUR logic can also be applied to figure 6, removing a couple of 3's.
Type 4/4b can be analyzed with HUR, which results in the same eliminations as UR. So in some sense Type 4/4b is really part of the HUR logic.
When you find a Type 2/2b or 3/3b UR, it pays to look at the strong links in it to see if HUR can also be applied. Apply HUR after UR, as it destroys the deadly pattern.
... by: Dan, UK
Monday 17-Mar-2014
type 3 can be extended to search for 'triples', even 'quads'. In your examples just a pairs are used - 17 in first one, 83 in second one.
how about this:
my solver says:
Unique rectangle type 3 at r2c4, r2c6, r8c4 and r8c6 allows 4 and 6 to be eliminated from r8c1, r8c2
you can see a screenshots from my solver here:
explanation: floor if clearly defined. As the roof contains additional candidates defined as 346, we can search for TWO cells, which contains some of these candidates, but only them, as defined in
naked triple rule. As we found two cells 46 and 346, we can consider this as triple (two found cells and two cells from the roof of unique rectangle) and eliminate candidates 346 from the other cells
in the unit (as seen on screenshots).
HODOKU solver can recognize this 'triple-extended-type 3' as well :)
The same logic can be used for searching for 'quad', that means four additional candidates in the roof of rectangle, and we'll be looking for THREE cells in a unit, which contains some of those
candidates, defined by the rule of naked quad. My solver should be able to solve such a problem, but unfortunatelly I didn't find any game to match this problem :( If ANYONE is able to share game
with such a kind of unique rectangle, feel free to post it :)
... by: Random Sudoku Enthusiast
Wednesday 6-Nov-2013
In the 'Type 1 Unique Rectangles' example (Unique Rectangle Figure 2), can't the same explanation be used for the 2 and the 9 in D2 ?
Logically, it doesn't make sense to eliminate the 2 and the 9 in D1, but not in D2.
It seems that by eliminating the 2 and the 9 in D1, we move the deadly rectangle corner over to D2. Should we then use the same logic to eliminate the 2 and the 9 from D2?
Or are we ignoring that square for the sake of the example?
... by: jaswant singh =sriganganagar INDIA
Tuesday 22-Oct-2013
ref.sudoku puzzle creater UR type 4 =focuses on UR candidates -- U R of 3,9 see row J =I remove the 3,3 ofJ2 J3 OTHERWAY= J2,J3,J78 form subset triplet [157].hence 5 of J5 will be removed leaving
there inJ5 the 2. In turn J4 will become 3. & in turn3 will be removed from J2andJ3...
... by: DanielKlaus
Saturday 3-Aug-2013
... by: JesseChisholm
Wednesday 15-May-2013
I didn't see the pattern on your site, but there is a "Nested Deadly Squares" that is quite rare, but could also be looked for and avoided with much the same techniques as Unique Rectangles.
Picture the group "B" in figure one (where "A" is the Deadly Pattern)
Add an expanded copy of group "B" such that:
* there are four boxes with a pair of 7/9 cells
* there are four rows with a pair of 7/9 cells
* there are four cols with a pair of 7/9 cells
These eight 7/9 cells together form a Nested Deadly Squares pattern.
... by: gerp124
Sunday 10-Feb-2013
OH! And while I have your attention, I would like to say that I dislike unique rectangles.
I can accept that they're part of the game, and may be the only way to solve a board sometimes, but it seems like such an arbitrary rule. What's so special about singularity? Yes, that was intended
as a joke, but I've never seen any aesthetic offense in a puzzle with more than one solution. Except that a single solution cannot then be published. But I've already made my case regarding published
solutions :-)
... by: gerp124
Sunday 10-Feb-2013
Thanks very much for the reply Andrew.
And thanks very much for such a wonderful website- for years, until I discovered this website, I was completely befuddled as to the solutions that are always published- they are not needed, and they
are not helpful. If you can finish, you don't need the solution, and if you _can't_ finish, you don't need the solution, rather, you need to know _how_ to get the solution. And this website has
provided an wonderful practical education in reasoning via sudoku!
... by: gerp124
Sunday 10-Feb-2013
I don't see much recent activity, but I hope this gets seen- I believe I have found a published contradiction, and so I'm wondering about the phrase above: 'published Sudokus have only one solution'.
In the Sudoku of the Variety section of Minneapolis Star Tribune for Sunday has these values _given_:
Does it make a difference, that the values were _given_, as opposed to being values that needed _solving_? The entire puzzle constitutes the _solution_, doesn't it? And these values could move,
yielding a second solution, true?
Here is the board as published, and thanks for in any thoughts.
Andrew Stuart writes:
I get these posts directly, so no worries, I try reply as soon as possible and Sundays are good days for this. You are correctly identifying the deadly pattern in two rows, two columns and two boxes
and the numbers are swappable - except as I think you've suggested, it doesn't apply if these are clues. Since clues are fixed they can't be swapped around so the problem of duplicate solutions
doesn't arise. Keep an eye out for the simpler URs, they are quite common and useful.
... by: Sonalita
Monday 9-Jul-2012
A fundamental premise of Andrew's excellent solver (and any algorithm based solver) is that the puzzle MUST have a unique solution.
It is generally accepted that the definition of a valid sudoku puzzle includes the constraint that there must be a single solution, so your statement of "not believing in uniqueness strategies" makes
no sense.
... by: Thinkist
Thursday 5-Apr-2012
Harmen Dijkstra has a fully valid point. This is why I don't believe in uniqueness strategies and have them unchecked in the solver. If a Sudoku has multiple solutions, so be it.
Andrew Stuart writes:
A Sudoku with more than one solution is not a valid puzzle and more than uniqueness strategies will cause it to either a) go down one particular path, or b) create a logical contradiction and get
nowhere. Even with uniqueness unticked the solver will not be 'logical' in any real sense. The proof of this is to consider an empty board. What 'logical' strategy is appropriate to solve an empty
I am thinking of adding a solution count to the start of the solve "take step" and stop any further progress until the puzzle is fixed. Mostly it is data entry and it causes confusion is people don’t
check the solution count first.
... by: B.N.Hobson
Sunday 1-Apr-2012
A word of caution. The Unique Rectangle doesn't work with Samurai Sudoku (five interlocking Sudoku puzzles). One section may have two answers, but one of them is wrong in relation to rest of the
puzzle. i have recently had two cases where the Unique Rectangle gave the wrong answer.
Andrew Stuart writes:
The most probable reason for the solver not working on Samurai is that each individual Sudoku lacks all the clues necessary to solve it individually. The overlap provides the extra information
necessary to complete the puzzle. It would be necessary to partially solve two overlapping parts before completing them both. Otherwise you might as well have five separate sudokus.
... by: Anton Delprado
Saturday 14-May-2011
There seems to be a lot of room for extension of type 3 or at least the concept of treating the two roof cells as a single virtual. This virtual cell has the union of values in the two roof cells
excluding the deadly values.
If we look at type 2 cells from this concept then the virtual cell has a single value and is "solved". This can then be used to eliminate that value in any cell that can see both cells. So type 2 are
kinds of type 3 in a sense.
This can also be extended to naked multiples in the joined row/col. For example if roof cells are in the same row and have three non-deadly values that are shared in two other cells in the row then
those values can be removed in other cells in the row.
Andrew Stuart writes:
Hi Anton
Yes, I have a feeling it might be possible to generalize and extend UR types. I'm in touch with someone else who has a good idea about re-organizing the families under different rules. I think you
are on a similar track
... by: Chuck Bruno - Virginia
Thursday 6-Jan-2011
Isn't the example for type 3 also a type 4B, allowing us to remove the 4's from cells B2 and B9?
... by: Alan Freberg
Saturday 18-Dec-2010
Hi Andrew,
Here's one that I had on hand when I wrote my previous post, but had yet to logic it out. It's fairly straight forward, though. It appears to be a Type 5b as there are three candidates in the ceiling
instead of two as in Type 3 or my previous examples.
In the Daily Nightmare for Dec. 29, 2005 the floor cells contain 3/4 in 2 B/D. The ceiling is in Col. 1 where the unsolved cells contain: 6/7/9 (A1), 3/4/7/9 (B1), 3/4/6/7 (D1), 4/7 (E1) and 6/7/9
(H1). The ceiling in cells B/D1 contain the Strong Links (if we can still call them that) 6/7/9. As you explain two cell tri-values on pg. 124, A1 and H1 will be reduced to a Naked Pair whichever of
the three is the true value in the ceiling cells . Delete 7(E1).
Thanks again,
Alan Freberg
... by: Alan Freberg
Thursday 16-Dec-2010
Hi Andrew,
A few months ago I purchased "The Book" with the intention of first reading it and then trying to solve all of Ruud's Daily Sudoku Nightmares. "The Book" was an excellent read. It illuminated several
dark corners. There are still some dark corners left, but they are my failings and not yours.
One thing I'm always trying to do with a Sudoku is to find something new or somehow push the envelope past what I know. Having read your book lead me to something of that nature while working on the
My solutions for the Daily Nightmares for March 30, 2006, Feb. 15, 2006, Feb. 2, 2006, Dec, 19 2005 and Dec. 16, 2005 contain a pattern that I have not seen before. They are based on a combined
application of Unique Rectangles and Aligned Pair Exclusion. At first I thought they were a variant of your Type 3 URs, but, since they allow for more deletions than Type 3 would, it appears that
they are a new variation. I call them UR Type 5 or UR APE App(lication).
In the March 30 DN (please pardon my reverse chronological order, but if you choose to look them up it is the order in which you will find them.) the floor pair are 4/5 in Col. 5 A/C. The ceiling is
in Col. 2 where the unsolved cells contain the following candidates: 4/5/6/(A2), 6/8 (B2), 4/5/9 (C2), 4/5/8/9 (E2) and 8/9 (J2). 6/9 are the Strong Link pair in the ceiling. According to the Type 3
rule there are no deletions possible. However, on pg. 122 of the chapter on APE the following rule is given--"Any two cells with only abc exclude combinations ab, ac and bc..." Applying this rule to
this formation we find that the two cells B2 and J2 contain 6/8/9. However one looks at it, the values for the extra candidates in the ceiling cells are accounted for and 8/9 can be deleted from E2.
Notice that 8(E2) is not one of the Strong Link candidates.
In the Feb. 15 DN the floor pair are 3/7 in Row E 1/2. The ceiling is in Row B. The unsolved cells in B contain: 1/3/7 (B1), 3/7/8 (B2), 8/9 (B4), 1/9 (B5) and 3/9 (B9). 1/8 are the Strong Link
candidates in the ceiling and 1/8/9 in B4/5 fulfill the APE rule requirements. Delete 9 (B9).
In the Feb. 2 DN the floor pair are 7/8 in Col. 2 G/J. The ceiling is in Col. 6. The unsolved cells in Col. 6 contain: 3/6 (C6), 4/6/8 (D6), 4/6 (E6), 4/7/8 (G6) and 3/4/7/8 (H6). 3/4 are the strong
links in the ceiling. 3/4/6 in C/E fulfill the APE rule requirements. Delete 4/6 (D6).
In the Dec. 19 DN the floor pair are 2/6 in Row H 2/3. The ceiling is in Row B. The unsolved cells in Box 1 are: 5/7 (A1), 1/5/7 (A3) and 6/7 (C2). 1/5 are the Strong Link pair in the ceiling with 1/
5/7 in A 1/3 completing the APE requirements. Delete 7 (C2).
In the Dec. 16 DN the floor pair are 4/6 in Row D 1/2. The ceiling is in Row G. The unsolved cells in Row G contain: 4/6/7 (G3), 3/5/7 (G7) and 3/5 (G8). 5/7 are the Strong Link pair in the ceiling
with 3/5/7 (G7/8) fulfilling the APE rule requirements. Delete 7 (G3).
If this is indeed a new application then you are the godfather of this baby as your explanation of the APE rule pointed the way for me.
Thank you very much,
Alan Freberg
... by: JCS
Friday 6-Aug-2010
Andi and John_Ha
I am new to this site but I end to agree with Andi. There is no deadly pattern if R5C6 contains candidates 356.
Applying simple colouring technique to the puzzle will show that 2s can be removed from R2C8 R7C2 R8C7 and R8C9. That causes R2C8 to be a 9, R7C8 a 2, R7C3 a 9, R1C1 a 9, R1C5 a 4, R4C5 a 9, R4C6 a
7. At this stage that leaves 356 as candidates for R5C6. The puzzle can then easily be solved without applying that "unique rectangle" technique ending with a 6 in R5C6.
... by: csvidyasagar
Friday 6-Aug-2010
Dear Denis,
1. You are absolutely correct. Type 4 and Type 4 B rely one logic that roof cell may contain a digit which is confined to that row or column or block and can not be removed. So the other digit can be
easily removed from thase two cells in Roof Row. In Type 4 example, the Roof Row C has 1,5,6 in cell C1 and C3. Either digit 1 or 5 can be removed to ensure Deadly Pattern does not result leading to
two solutions. But 5 is confined both in Row C and in Block Top Left or Block 1. So digit 1 can be removed from both cells in Roof Row C.
Have I confused you further ?
with regards,
CS Vidyasagar
... by: csvidyasagar
Friday 6-Aug-2010
Dear Andi,
1. When you have three out of four corners in a unique rectangle have two digits (in this case 5,7), then the fourth corner can not have 5,7 as this will have all four corners having 5,7.This is a
Deadly pattern as you can have two different solutions. But for a pure Sudoku you have to have only one or unique solution. That is possible only if you remove digits 5,7 which other three corner
cells of Unique Rectangle have.
2. You aim must be to reduce maximum number of digits from cells containing multiple candidates. When you can remove 5,7 from 3,5,6,7, of cell R5 C6, you are reducing time to solve the puzzle but
also reducing complexity. More candidates in ells mean more complexity and require more time to solve.
3. When you remove 5,7 from cell R5 C6, you see in column 6 you have digit 5 in cell R2C6 and digit 7 in cell R4C6. So the rule of all columns, rows and boxes to contain all digits from 1 to 9 is
4. I could only think of three reasons as to why you can remove 5,7 from cell R5 C6.
Have I confused you further ?
with regards,
CS Vidyasagar
... by: csvidyasagar
Friday 6-Aug-2010
Dear Lea,
The four cells you have mentioned have the following pattern:
Col 1 Col 4
Row A 1,5,6 1,5,6
Row D 1, 5 1,5.
This is similar to Unique Rectangle Pattern 2B. That is floor cells are 1,5 and Roof cells are 1,5,6. Since these form a unique rectangle with 1,5 being common in all the four cells to avoid Deadly
Pattern ( of all cells having 1,5), you have to keep 6 in Row A i.e. Roof cells. That means 1,5 will be removed later from Roof Cells in Row A.
But you can not have two 6s in Row A (Roof Row) as any row can have only one digit in one cell. So remove 6 from any other cell in Roof Row A or in the Block /Box of Roof cells.
Have I confused you further ?
with regards,
CS Vidyasagar
... by: Harmen Dijkstra
Saturday 15-May-2010
Yes the point is, that there are indeed more solutions. So, if you use unique triangles, and get a solution, you will not know that your solution is a unique solution.
... by: Wiking 48
Sunday 2-May-2010
Hi Harmen Dijkstra,
I'm quite new on this.
why do you bother with such a non-defined sudoku?
There are many more than 17 sulotions! Check for yourself, you see that 24, 42 in the left of your grid can be excanged.
Is there any idea to fight against non-defined?
Upper left corner can also be 1, 2 or 3 which gives a lot of solutions..
Down left corner in your example is 7 in one example which is quite common among the sulotions, 1 is more rare but possible...
... by: Harmen Dijkstra
Friday 5-Mar-2010
I made this sudoku:
. 9 . | 5 . . | 6 4 7
6 . 5 | . . 7 | . 9 3
. 7 . | . . . | 2 5 8
. 5 6 | . 7 . | 9 . .
. . 7 |9 . 5| 3 8 6
. . 3 |8 . 2| 4 7 5
. . . | . 5 .| 8 3 9
5 6 . | . . .| 7 2 .
. . . | . . 8| 5 6 4
If you filling this sudoku in the sudoku solver, the solution Count will say that there are 17 solutions. But if you try TAKE STEP then you will get only this sudoku (with Unique Rectangles):
2 9 1 | 5 8 3 | 6 4 7
6 8 5 | 2 4 7 | 1 9 3
3 7 4 | 1 9 6 | 2 5 8
8 5 6 | 3 7 4 | 9 1 2
4 2 7 | 9 1 5 | 3 8 6
9 1 3 | 8 6 2 | 4 7 5
7 4 2 | 6 5 1 | 8 3 9
5 6 8 | 4 3 9 | 7 2 1
1 3 9 | 7 2 8 | 5 6 4
But there are more solutions, for example:
1 9 2 | 5 8 3 | 6 4 7
6 8 5 | 2 4 7 | 1 9 3
3 7 4 | 6 9 1 | 2 5 8
8 5 6 | 3 7 4 | 9 1 2
2 4 7 | 9 1 5 | 3 8 6
9 1 3 | 8 6 2 | 4 7 5
4 2 1 | 7 5 6 | 8 3 9
5 6 8 | 4 3 9 | 7 2 1
7 3 9 | 1 2 8 | 5 6 4
The sudoku solver said that (because of Unique Rectangles) R7C4 has to be a 6, but in this example you see that you get an other solution with a 7.
Is this a fault in the sudoku solver?
... by: CS Vidyasagar
Wednesday 3-Mar-2010
1. You have given very simple and lucid explanation of difficult concept of Unique Rectangles of four types. One comes across such situations in number of Sudoku puzzles of varying degrees of
difficulty. The clear understanding of unique rectangles helps one to get out of log jam one is subjected to frequently in these interesting puzzles. I do not know whether any more explanation can be
given to what you have done. Only one has to read your article may be twice but keep the most important lesson you brought out loud and clear that " DO NOT HAVE DEADLY PATTERN OF HAVING FOUR
CONJUGATE PAIRS" as that leads to multple solutions. Sudoku permits only unique solution.
2. One has to understand the logic and solution becomes quite clear. You have seleted many good illustrations which have explained the logic clearly and unambiguosly.
3. Looking forward to more such illustrations.
Kudos for wonderful work you are doing.
with regards,
... by: Lea Hayes
Tuesday 23-Feb-2010
Just to clarify my question:
Rectangle Cells:
A(1,5,6) B(1,5,6)
C(1,5) D(1,5)
Why remove 1 from both A and B and not either of the following:
A(5,6) B(1,5)
C(1,5) D(1,5)
A(1,5) B(5,6)
C(1,5) D(1,5)
... by: Lea Hayes
Tuesday 23-Feb-2010
Re: Type 4 Unique Rectangles - Cracking the Rectangle with Conjugate Pairs
I do not understand how you came to the conclusion that 1 can be removed? Why not remove 6 instead?
... by: John_Ha
Friday 22-Jan-2010
I too was puzzling over that but I think the answer is simple. See Fig 2. (5 and 7) are unsolved in Col 4 otherwise they wouldn't be candidates in R2C4. Similarly (5 and 7) are unsolved in Col 6.
Similarly (5 and 7) are unsolved in Row 5. Because the cell at R5C4 is unsolved for 5 and 7, then the centre 3x3 must be unsolved for (5 and 7).
There is therefore no way before now that the cell at R5C6 could have had either 5 or 7 eliminated as candidates because there are no solved (5 or 7) that it can see. The 5 and 7 at R5C6 could not
have been eliminated by something happening on R5, nor on C6, nor in its 3x3.
So, the cell at R5C6 MUST have BOTH 5 and 7 as candidates (as well as possibly others, in this case 3 and 6).
Now assume R5C6 is 5 - we get the deadly pattern. So R5C6 cannot be 5. Now assume R5C6 is 7 - we get the deadly pattern. So R5C6 cannot be 7.
So R5C6 cannot be 5 or 7 and both can be eliminated.
... by: Andi
Friday 22-Jan-2010
I do not understand this strategy. In the first example for Type 1 UR, I understand that if 3 or 6 would be removed from R5C6, you would end up with a deadly pattern. That's fine so far.
But I do not see why you can eliminate 5 AND 7 for this reason from R5C6. IMHO, if in R5C6 3,5,6 would be possible, well then this would just imply R5C4 and R2C6 being 7 and R2C4 being 5. An
"ordinary" solution. OTOH, if in R5C6 3,6,7 would be possible, R5C4 and R2C6 would be 5 and R2C4 would be 7. Quite straightforward, too. My point is: both possibilities would be valid. But I cannot
spot a "deadly pattern" here because to remove the ambiguity in this sitatuation, all you can postulate is that *either* 5 *or* 7 must be removed from the possibilities in R5C6. With either number
removed, the ambiguity is removed, thus no deadly pattern anymore. But I just don't see why you eliminate *both* 5 and 7 in R5C6?
Am I missing something?
... by: Dennis Daft
Tuesday 5-May-2009
Shouldn't your Type 4B example be "solved" using the Type 2A method? Using this method the 6 in R1C3 and R3C5 would be eliminated, forcing one of the "roof" cells to be a 6. I believe the end result
is the same. This would be consistant with only using Type 4 when no other solution is possible. | {"url":"https://www.sudokuwiki.org/Unique_Rectangles","timestamp":"2024-11-04T05:09:24Z","content_type":"text/html","content_length":"105987","record_id":"<urn:uuid:15212307-8b9d-4825-a1bb-e7b1b3b5c94c>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00015.warc.gz"} |
1231 -- The Alphabet Game
The Alphabet Game
Time Limit: 1000MS Memory Limit: 10000K
Total Submissions: 1820 Accepted: 751
Little Dara has recently learned how to write a few letters of the English alphabet (say k letters). He plays a game with his little sister Sara. He draws a grid on a piece of paper and writes p
instances of each of the k letters in the grid cells. He then asks Sara to draw as many side-to-side horizontal and/or vertical bold lines over the grid lines as she wishes, such that in each
rectangle containing no bold line, there would be p instances of one letter or nothing. For example, consider the sheet given in Figure 1, where Sara has drawn two bold lines creating four rectangles
meeting the condition above. Sara wins if she succeeds in drawing the required lines. Dara being quite fair to Sara, wants to make sure that there would be at least one solution to each case he
offers Sara. You are to write a program to help Dara decide on the possibility of drawing the right lines.
The first line of the input file contains a single integer t (1 <= t <= 10), the number of test cases, followed by the input data for each test case. The first line of each test case consists of two
integers k (1 <= k <= 26), the number of different letters, and p (1 <= p <= 10), the number of instances of each letter. Followed by the first line, there are k lines, one for each letter, each
containing p pairs of integers (xi, yi) for 1 <= i <= p. A pair indicates coordinates of the cell on the paper where one instance of the letter is written. The coordinates of the upper left cell of
the paper is assumed to be (1,1). Coordinates are positive integers less than or equal to 1,000,000. You may assume that no cell contains more than one letter.
There should be one line per test case containing a single word YES or NO depending on whether the input paper can be divided successfully according to the constraints stated in the problem.
Sample Input
Sample Output
Tehran 2002 Preliminary | {"url":"http://poj.org/problem?id=1231","timestamp":"2024-11-11T21:12:01Z","content_type":"text/html","content_length":"7084","record_id":"<urn:uuid:a6a0a19b-1fef-4b70-b1bc-9ff4fafd422b>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00810.warc.gz"} |
Breaking concave shapes (collision perhaps).
I'm sticking this in extension ideas since I'm solving this problem in Java, and could potentially expand it into something more. I am not particularly experienced in Java, but that makes it a
learning experience. I decided that I wanted to program a way to split a given concave polygon (for it to be a polygon I am viewing that it means none of its lines cross for now) into multiple convex
polygons which join together to make that one concave polygon (but can be used for collision engines).
If I allowed shaped to be cut along the edges (as opposed to just along the vertices), then in Box2D, clipping would occur. This is when two objects catch along an apparently straight line in the
joins in collision boxes (since Box2D doesn't have pixel perfect collision). Therefore, I want to cut shapes by their vertices.
Currently my algorithm I have come up for this is something like this. I take a concave polygon. I will use this one as an example:
Once I have this shape I iterate through each of the points, marking it if the angle at it is reflex (more than 180 degrees). For the example shape, this leave me with this:
Next comes a quite complicated part. For each of those points (if there are none I just return the shape), I check how many points I can go in either direction until I either reach a reflex angle, or
the angle from that point through the point to the next point is more than 180 degrees. This can be marked on our shape like this (dark blue is point, light blue is points one way and brown is the
points the other way).
When I have this, I identify which of these gives me the most points in a shape and chose that one. In the case of the example, there are two choices we can make (the brown on the first and the
third), so we chose the first one. We then seperate these points from the shape to make a new shape:
Once we have done this we take the new shape formed and repeat the process.
We count the reflex angles:
We check for both of them:
We pick one and split it.
And then we do the same with the next shape:
Finally, we are left with a shape that has no reflex angles so we end with our shape broken up.
As of speaking, I have written a program to do this in Java, but without any visual interaction (you literally input a list of numbers representing a shape), so next part to work on is displaying an
input and an output. So far this project has been very enjoyable. | {"url":"https://community.stencyl.com/index.php/topic,47186.0.html?PHPSESSID=8bab378729a7449a8d2842bd1d763e83","timestamp":"2024-11-09T06:11:23Z","content_type":"application/xhtml+xml","content_length":"26739","record_id":"<urn:uuid:101f0d4a-6a07-493a-a911-e0f4130cacd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00259.warc.gz"} |
Bernauer, Moritz (2014): Reducing non-uniqueness in seismic inverse problems: new observables in seismology. Dissertation, LMU München: Faculty of Geosciences
Preview 14MB
The scientific investigation of the solid Earth's complex processes, including their interactions with the oceans and the atmosphere, is an interdisciplinary field in which seismology has one key
role. Major contributions of modern seismology are (1) the development of high-resolution tomographic images of the Earth's structure and (2) the investigation of earthquake source processes. In both
disciplines the challenge lies in solving a seismic inverse problem, i.e. in obtaining information about physical parameters that are not directly observable. Seismic inverse studies usually aim to
find realistic models through the minimization of the misfit between observed and theoretically computed (synthetic) ground motions. In general, this approach depends on the numerical simulation of
seismic waves propagating in a specified Earth model (forward problem) and the acquisition of illuminating data. While the former is routinely solved using spectral-element methods, many seismic
inverse problems still suffer from the lack of information typically leading to ill-posed inverse problems with multiple solutions and trade-offs between the model parameters. Non-linearity in
forward modeling and the non-convexity of misfit functions aggravate the inversion for structure and source. This situation requires an efficient exploitation of the available data. However, a
careful analysis of whether individual models can be considered a reasonable approximation of the true solution (deterministic approach) or if single models should be replaced with statistical
distributions of model parameters (probabilistic or Bayesian approach) is inevitable. Deterministic inversion attempts to find the model that provides the best explanation of the data, typically
using iterative optimization techniques. To prevent the inversion process from being trapped in a meaningless local minimum an accurate initial low frequency model is indispensable. Regularization,
e.g. in terms of smoothing or damping, is necessary to avoid artifacts from the mapping of high frequency information. However, regularization increases parameter trade-offs and is subjective to some
degree, which means that resolution estimates tend to be biased. Probabilistic (or Bayesian) inversions overcome the drawbacks of the deterministic approach by using a global model search that
provides unbiased measures of resolution and trade-offs. Critical aspects are computational costs, the appropriate incorporation of prior knowledge and the difficulties in interpreting and processing
the results. This work studies both the deterministic and the probabilistic approach. Recent observations of rotational ground motions, that complement translational ground motion measurements from
conventional seismometers, motivated the research. It is investigated if alternative seismic observables, including rotations and dynamic strain, have the potential to reduce non-uniqueness and
parameter trade-offs in seismic inverse problems. In the framework of deterministic full waveform inversion a novel approach to seismic tomography is applied for the first time to (synthetic)
collocated measurements of translations, rotations and strain. The concept is based on the definition of new observables combining translation and rotation, and translation and strain measurements,
respectively. Studying the corresponding sensitivity kernels assesses the capability of the new observables to constrain various aspects of a three-dimensional Earth structure. These observables are
generally sensitive only to small-scale near-receiver structures. It follows, for example, that knowledge of deeper Earth structure are not required in tomographic inversions for local structure
based on the new observables. Also in the context of deterministic full waveform inversion a new method for the design of seismic observables with focused sensitivity to a target model parameter
class, e.g. density structure, is developed. This is achieved through the optimal linear combination of fundamental observables that can be any scalar measurement extracted from seismic recordings. A
series of examples illustrate that the resulting optimal observables are able to minimize inter-parameter trade-offs that result from regularization in ill-posed multi-parameter inverse problems. The
inclusion of alternative and the design of optimal observables in seismic tomography also affect more general objectives in geoscience. The investigation of the history and the dynamics of tectonic
plate motion benefits, for example, from the detailed knowledge of small-scale heterogeneities in the crust and the upper mantle. Optimal observables focusing on density help to independently
constrain the Earth's temperature and composition and provide information on convective flow. Moreover, the presented work analyzes for the first time if the inclusion of rotational ground motion
measurements enables a more detailed description of earthquake source processes. The complexities of earthquake rupture suggest a probabilistic (or Bayesian) inversion approach. The results of the
synthetic study indicate that the incorporation of rotational ground motion recordings can significantly reduce the non-uniqueness in finite source inversions, provided that measurement uncertainties
are similar to or below the uncertainties of translational velocity recordings. If this condition is met, the joint processing of rotational and translational ground motion provides more detailed
information about earthquake dynamics, including rheological fault properties and friction law parameters. Both are critical e.g. for the reliable assessment of seismic hazards.
Item Type: Theses (Dissertation, LMU Munich)
Subjects: 500 Natural sciences and mathematics
500 Natural sciences and mathematics > 550 Earth sciences
Faculties: Faculty of Geosciences
Language: English
Date of oral examination: 10. July 2014
1. Referee: Igel, Heiner
MD5 Checksum of the PDF-file: b50af2bdfe5589a4ae524f06a6216f31
Signature of the printed copy: 0001/UMC 22213
ID Code: 17163
Deposited On: 30. Jul 2014 12:27
Last Modified: 23. Oct 2020 23:21 | {"url":"https://edoc.ub.uni-muenchen.de/17163/","timestamp":"2024-11-08T09:27:50Z","content_type":"application/xhtml+xml","content_length":"43738","record_id":"<urn:uuid:0f2b8ba2-2792-4393-970f-497d773f1810>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00717.warc.gz"} |
Kindergarten Worksheets On Ordinal Numbers - OrdinalNumbers.com
Kindergarten Worksheets On Ordinal Numbers
Kindergarten Worksheets On Ordinal Numbers – With ordinal numbers, you can count any number of sets. They can also be used to broaden ordinal numbers.But before you can utilize these numbers, you
need to understand the reasons why they exist and how they operate.
One of the most fundamental concepts of mathematics is the ordinal number. It is a number indicating the position of an object within a list. Ordinarily, ordinal numbers are between one to twenty.
While ordinal numbers have various purposes but they are mostly utilized to represent the order of the items in an orderly list.
Ordinal numbers can be represented using charts or words, numbers, and other methods. They can also be used to explain how pieces of a collection are organized.
Most ordinal numbers fall into one or one or. The Arabic numbers represent finite ordinals, while transfinite ones are depicted using lowercase Greek letters.
Every well-ordered set is supposed to have at least one ordinal in accordance with the axioms of choice. For instance, the highest possible grade could be given to the class’s first member. The
contest’s winner was the student with the highest score.
Combinational ordinal numbers
Multidigit numbers are referred to as compound ordinal numbers. They can be created by multiplying an ordinal number by the last character. They are used to determine ranking and dating. They do not
have an exclusive ending for the last number, as cardinal numbers do.
Ordinal numerals are used to indicate the sequence of elements in the collection. They can also be used to identify the items within the collection. The two forms of ordinary numbers are regular and
By prefixing cardinal numbers with the suffix -u regular ordinals can be made. The number is then entered as a word, then an hyphen is added after it. There are numerous suffixes available.
Suppletive ordinals are created by prefixing words using -u, -e or -ie. This suffix is used to count and is larger than the standard.
Limit of ordinal importance
Limit ordinal numbers are numbers which do not contain zero. Limit ordinal numbers come with the disadvantage of not having a maximal element. They can be made by joining non-empty sets without the
possibility of having maximum elements.
Transfinite recursion models also use limited ordinal numbers. According to the von Neumann model, each infinite cardinal number also is subject to an ordinal limit.
The ordinal numbers that have limits are equal to the sums of all ordinals lower than it. Limit ordinal amounts can be calculated using math however they also can be expressed in a series or natural
These numbers are arranged according to ordinal numbers. They give an explanation about the numerical location of objects. They are utilized in Arithmetic and set theory contexts. Despite having a
similar structure to natural numbers, they aren’t part of the same class.
A well-ordered, well-ordered set can be used in the von Neumann model. It is possible to consider that fy fy could be described as a subfunction to function g’. If fy is only one subfunction (ii)
then G’ meets the requirements.
The Church Kleene oral is a limit-ordering ornal that works in a similar manner. A limit ordinal is a well-ordered collection that includes smaller ordinals. It’s a nonzero ordinal.
Ordinal number examples in stories
Ordinal numbers are frequently utilized to represent the hierarchy of entities and objects. They are crucial to organize, count, and also for ranking purposes. They can be used both to show the order
of things and to show the position of objects.
The ordinal number may be identified by the letter “th’. Sometimes though the “nd” letter may be substituted for “th”. The titles of books are usually followed by ordinal numbers.
Although ordinal numbers are commonly written in lists but they are also expressed as words. They can also be expressed using numbers or acronyms. They are, in general, easier to comprehend as
compared to cardinal numbers.
Ordinary numbers come in three distinct varieties. Learn more about them through practice or games as well as other activities. A key component to improving your arithmetic skills is being educated
about them. Try coloring exercises to provide a fun and easy way to improve. To check your results, use a handy coloring page.
Gallery of Kindergarten Worksheets On Ordinal Numbers
Ordinal Numbers Worksheet 1 To 10 Number Worksheets Kindergarten
Labeling Ordinal Numbers Worksheet Have Fun Teaching
Ordinal Numbers Online Exercise For Kindergarten
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Annuity-formula risposte
Annuity Formula
Annuity Formula, Ottieni dettagli su Annuity Formula, noi cerca di out.
Annuity Formula – Example #2 Let say your age is 30 years and you want to get retired at the age of 50 years and you expect that you will live for another 25 years. You have 20 years of service left
and you want that when you retire, you will get an annual payment of $10,000 till you die (i.e. for 25 years after retirement).
Given below is the data used for the calculation of annuity payments. PVA Ordinary = $10,000,000 (since the annuity to be paid at the end of each year) Therefore, the calculation of annuity payment
can be done as follows –. Annuity = 5% * $10,000,000 / [1 – (1 + 5%) -20] Calculation of Annuity Payment will be –. Annuity = $802,425.87 ...
The annuity formula helps in determining the values for annuity payment and annuity due based on the present value of an annuity due, effective interest rate, and a number of periods. Understand the
annuity formula with derivations, examples, and FAQs.
Calculation using Formula. FV 3 (annuity due) =5000 [ { (1+6%) 3 -1/6%} x (1+6 %)]=16,873.08. Note: The future value of an annuity due for Rs. 5000 at 6 % for 3 years is higher than the FV of an
ordinary annuity with the same amount, time, and rate of interest. This is due to the earlier payments made at the starting of the year, which provides ...
The general formula for annuity valuation is: Where: PV = Present value of the annuity. P = Fixed payment. r = Interest rate. n = Total number of periods of annuity payments. The valuation of
perpetuity is different because it does not include a specified end date.
The formulas described above make it possible—and relatively easy, if you don't mind the math—to determine the present or future value of either an ordinary annuity or an annuity due.
The annuity payment formula can be determined by rearranging the PV of annuity formula. After rearranging the formula to solve for P, the formula would become: This can be further simplified by
multiplying the numerator times the reciprocal of the denominator, which is the formula shown at the top of the page. Return to Top.
Explanation. The formula for Future Value of an Annuity formula can be calculated by using the following steps: Step 1: Firstly, calculate the value of the future series of equal payments, which is
denoted by P. Step 2: Next, calculate the effective rate of interest, which is basically the expected market interest rate divided by the number of payments to be done during the year.
An annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension
payments. ... Proof of annuity-immediate formula. To calculate present value, ...
Derivation of the annuity formula using the Law of One Price. To derive the shortcut, we calculate the value of a growing perpetuity by creating our own perpetuity. Suppose you want to create a
perpetuity growing at 2%. You could invest $100 in a bank account paying 5%.
For example, annuity payments scheduled to payout in the next five years are worth more than an annuity that pays out in the next 25 years. The formula for determining the present value of an annuity
is PV = dollar amount of an individual annuity payment multiplied by P = PMT * [1 – [ (1 / 1+r)^n] / r] where: P = Present value of your annuity ...
PV of Annuity Calculator (Click Here or Scroll Down) The present value of annuity formula determines the value of a series of future periodic payments at a given time. The present value of annuity
formula relies on the concept of time value of money, in that one dollar present day is worth more than that same dollar at a future date.
Annuity Payment Formula. C = cash flow per period. r = interest rate. n = number of payments. Put simply, the present value of an annuity is the current value of the income that will be generated by
the investment in the future, and it’s the built on the time value of money concept, which states that a dollar today is more valuable than a ...
If an index of an indexed annuity doesn't receive enough positive growth, the annuity investor will receive a guaranteed minimum interest return at the bare minimum. The crediting formulas of indexed
annuities generally have some type of limiting factor that is intended to cause interest earnings to be based only on a portion of the change in whatever index it is tied to.
Annuity Formula Calculation Annuity Formula Calculation An annuity is the series of periodic payments to be received at the beginning of each period or the end of it. An annuity is based on the PV of
an annuity due, effective interest rate and time period. Annuity = r * PVA Ordinary/[1 – (1 + r)-n] read more; Annuity vs. Perpetuity ...
Present Value Of An Annuity: The present value of an annuity is the current value of a set of cash flows in the future, given a specified rate of return or discount rate. The future cash flows of ...
The formula for the present value of an annuity identifies 3 variables: the cash value of payments made by the annuity per period, the interest rate, and the number of payments within the series. The
present value of an annuity calculation is only effective with a fixed interest rate and equal payments during the set time period.
List of Annuity Formulas In most scenarios, you’ll only need to use the annuity formulas listed above. However, there are other annuity formulas out there. Future value of an ordinary annuity: FV = A
[(1 + r)n ? 1] r FV = A · Sn r Current value of an ordinary annuity: CV = A[1 ? (1 + r)?n] r CV = A · an r
The formula for the present value of an annuity due is as follows: Alternatively, Where: PMT – Periodic cashflows. r – Periodic interest rate, which is equal to the annual rate divided by the total
number of payments per year. n – The total number of payments for the annuity due. The second formula is intuitive, as the first payment (PMT ...
With an annuity due, payments are made at the beginning of the period, instead of the end. To calculate the payment for an annuity due, use 1 for the type argument. In the example shown, the formula
in C11 is: = PMT( C6, C7, C4, C5,1) which returns -$7,571.86 as the payment amount. Notice the only difference in this formula is type = 1.
Annuity Formula. Ordinary annuities are paid at the end of each period. Annuities due are paid at the beginning of each period. Future value (FV) is the measure, or amount, of how much a series of
Future value of an ordinary annuity, the formula F = P* ( [1 + I]N – 1)/I is calculated, in which case P is the payout amount. I am equal to the interest rate (discount). The payment number is N (the
“shows N as an exponent). The future value of the annuity is shown in the letter F.
Calculate Annuities: Annuity Formulas in Excel. For example, you could buy an annuity that lasts five, 10, 20 or even 30 years. Annuities that pay a guaranteed amount over a specific period of time
are known as period certain annuities. If you happen to die before the end of the term, the remainder of the payments can go to a beneficiary such ...
With the annuity payout calculator you can compute the precise amount of annuity payouts through a given interval to reach a specified future value.Primarily, you can apply the tool to find out the
fixed amount of annuity withdrawals that fully deploy a given initial balance over a given time. For example, you can easily find out how much does a 100 000 annuity pay per month or how many ...
Annuity-formula risposte?
Annuity value formula payments present future interest payment rate number calculation ordinary years given amount time formulas calculate annuities example will formula. series equal period. period
payments. effective made perpetuity dollar periodic cash. | {"url":"https://www.internet-television.it/annuity-formula-risposte","timestamp":"2024-11-11T14:03:59Z","content_type":"text/html","content_length":"42947","record_id":"<urn:uuid:aae90187-45bf-45fc-9617-5d5a34a622cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00308.warc.gz"} |
Lagrangian depends on second derivative of field
In case of the gauge-fixed Faddeev-Popov Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}\,^{a}F^{\mu\nu a}+\bar{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\psi-\frac{\xi}{2}B^{a}B^{a}+B^{a}\partial^{\
mu}A_{\mu}\,^{a}+\bar{c}^{a}\left(-\partial^{\mu}D_{\mu}\,^{ac}\right)c^{c} $$
(for example in Peskin and Schrder equation 16.44)
If you expand the last term (for the ghost fields) you get: $$ \bar{c}^{a}\left(-\partial^{\mu}D_{\mu}\,^{ac}\right)c^{c} = -\bar{c}^{a}\partial^{2}c^{a}-gf^{abc}\bar{c}^{a}\left(\partial^{\mu}A_{\
mu}\,^{b}\right)c^{c}-gf^{abc}\bar{c}^{a}A_{\mu}\,^{b}\partial^{\mu}c^{c} $$
And so, the Lagrangian has a term proportional to the second derivative of $c^a$.
In this case, how does one find the classical equations of motion for the various ghost fields and their adjoints?
I found the following equations of motion so far: $$ D_{\beta}\,^{dc}F^{\beta\sigma}\,^{c}=-g\bar{\psi}\gamma^{\sigma}t^{d}\psi+\partial^{\sigma}B^{d}+gf^{dac}\left(\partial^{\sigma}\bar{c}^{a}\
right)c^{c} = 0 $$ $$ \partial_{\sigma}\bar{\psi}_{\alpha,\, i}i\gamma^{\sigma}-\sum_{\beta}\sum_{j}\bar{\psi}_{\beta,\, j}\left(gA_{\mu}\,^{a}\gamma^{\mu}\,_{ji}t^{a}\,_{\beta\alpha}-m\delta_{ji}\
delta_{\beta\alpha}\right)=0 $$ $$ \left(i\gamma^{\mu}D_{\mu}-m\right)\psi=0 $$ $$ B^{b}=\frac{1}{\xi}\partial^{\mu}A_{\mu}\,^{b} $$ $$ \partial^{\mu}\left(D_{\mu}\,^{dc}c^{c}\right)=0 $$ $$ f^{abd}\
left(\partial_{\sigma}\bar{c}^{a}\right)A^{\sigma}\,^{b}=0 $$
But it is the last equation that I suspect is false (I saw the equation $ D_\mu\,^{ad} \partial^\mu \bar{c}^d = 0 $ in some exercise sheet and I also saw the equation $D^\mu\,^{ad}\partial_\mu B^d =
igf^{dbc}(\partial^\mu\bar{c}^b)D_\mu\,^{dc} c^c$ which I don't understand how they were derived.)
Any help about what are the proper equations of motions and how to get them would be much appreciated.
This post imported from StackExchange Physics at 2014-07-03 18:16 (UCT), posted by SE-user PPR
I) Since total divergence terms do not contribute to Euler-Lagrange (EL) equations, cf. e.g. this Phys.SE post, one could just integrate the Faddeev-Popov $\bar{c}c$ term by part so that there are no
more than first derivatives present and the standard form of the EL equations applies.
II) Alternatively, in the presence of higher derivatives, the EL equations get modified with additional terms, see e.g. Wikipedia. Note that special care should be taken in the ordering of
Grassmann-odd derivatives.
III) Often in field theory, because of external indices and Grassmann-odd fields, it is quite tedious to use the EL equations directly. It is often simpler to infinitesimally vary the given action $S
$$\delta S~=~\int d^4x~\text{(EL-eq)}~ \delta\phi(x) +\text{(boundary terms)} . $$ and identify the EL equations as the relevant coefficient functions on the spot, so to speak.
This post imported from StackExchange Physics at 2014-07-03 18:16 (UCT), posted by SE-user Qmechanic
Replace the field $\phi(x)$ in the action by $\phi(x)+\delta\phi(x)$, expand while dropping products of the variational field, and use integration by parts to remove all derivatives from the
variational field $\delta\phi(x)$. The result is an expression of the form
[original action] plus integral over [field equation] times $\delta\phi(x)$.
This procedure works no matter how many derivatives you have, and also for anticommuting fields. | {"url":"https://www.physicsoverflow.org/19907/lagrangian-depends-on-second-derivative-of-field","timestamp":"2024-11-09T19:43:19Z","content_type":"text/html","content_length":"131081","record_id":"<urn:uuid:5c4a29aa-5fa2-4077-aa55-334176c8b7d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00850.warc.gz"} |
PEMDAS Calculator - MathCracker.com
Instructions: Use this calculator to compute and simplify any expression (numeric or symbolic) you provide, following the PEMDAS rules, showing all the steps. Please type in the expression you want
to compute in the form box below.
About this PEMDAS calculator
This PEMDAS calculator will allow you simplify parentheses, multiplying expressions, divide expressions and add and subtract expressions, forming a compound more complex expression that can be solved
with PEMDAS rules .
All you need to do is to provide a valid expression, either symbolic or numeric, and all the steps of the simplification will be shown to you.
Once a valid expression has been provided, the easy part comes in: you just need to click on the "Calculate" button, and that is it, all the steps will be there for you.
The process of simplifying expressions could be a nuanced one, especially if you provide the calculator with a complex expression.
What is PEMDAS meaning?
PEMDAS is an acronym that stands for:
P = Parentheses
E = Exponents
M = Multiplication
D = Division
A = Addition
S = Subtraction
The PEMDAS rule indicates the order of priority in which calculations should be done. This is, first parentheses, THEN exponents, THEN multiplications, and so on. So you can see this as an order of
operations rule.
PEMDAS calculator with exponents
Does this calculator conduct PEMDAS for exponents? Absolutely! Indeed, PEMDAS has the 'E' for exponents, so then the exponents priority is very high in a simplification process, only surpassed by
To a certain degree, parentheses and exponents allow you to see some 'isolated' expressions that can be handled separately. For example, if you had \(2^{\frac{1}{2} + \frac{1}{3}}\), the sum of
fraction in the exponent is like 'isolated' and you can start simplifying there.
What are the steps for using PEMDAS?
• Step 1: Start with the parentheses and exponents (in that order) seeing for sub-expressions that can be handled first
• Step 2: Once those sub-expressions can be identified, use PEMDAS to solve them. This is, there may still be parentheses or exponents that need to be handled first and have priority
• Step 3: When you have reached a foremost inner parentheses or exponent, you can see what simple operations remain, giving priority to multiplication and division, and then conducting additions
and subtractions
Ultimately PEMDAS may be trivially applied in some trivial cases, but it is not always the case. PEMDAS has this potentially recursive nature, that could make its application confusing, especially
with particularly complex, nested expressions.
In the end, in most of the cases you won't have to think too hard, as most of the usual cases are very simple, but it is good to have the awareness that PEMDAS can be as complex as the complexity of
the provided expression you want to simplify .
Why is PEMDAS important?
PEMDAS is important because it is the only way we have to ensure that there is one and only way correct simplification. Now, there could be different paths leading to that correct simplification, but
they will all be the same.
Simplifying expressions needs to be an exact endeavor, and that is what PEMDAS is all about, and a calculator that helps you with that can be certainly be useful.
Example: PEMDAS example
Calculate : \(\frac{1}{3} \frac{2}{3} + \frac{5}{4} - \frac{1}{6}\)
Solution: This is problem we can address with this order of operations calculator. Indeed, we are provided with the following expression: \(\displaystyle \frac{1}{3}\cdot\frac{2}{3}+\frac{5}{4}-\frac
The following calculation is obtained:
\( \displaystyle \frac{1}{3}\cdot \frac{2}{3}+\frac{5}{4}-\frac{1}{6}\)
We can multiply the terms in the top and bottom, and we get \(\displaystyle\frac{ 1}{ 3} \times \frac{ 2}{ 3}= \frac{ 2}{ 3 \times 3} \)
\( = \,\,\)
\(\displaystyle \frac{2}{3\cdot 3}+\frac{5}{4}-\frac{1}{6}\)
By multiplying the terms in the denominator, we get: \( 3 \times 3 = 9\)
\( = \,\,\)
\(\displaystyle \frac{2}{9}+\frac{5}{4}-\frac{1}{6}\)
Amplifying in order to get the common denominator 36
\( = \,\,\)
\(\displaystyle \frac{2}{9}\cdot\frac{4}{4}+\frac{5}{4}\cdot\frac{9}{9}-\frac{1}{6}\cdot\frac{6}{6}\)
We use the common denominator: 36
\( = \,\,\)
\(\displaystyle \frac{2\cdot 4+5\cdot 9-1\cdot 6}{36}\)
Expanding each term: \(2 \times 4+5 \times 9-6 = 8+45-6\)
\( = \,\,\)
\(\displaystyle \frac{8+45-6}{36}\)
Adding up each term in the numerator
\( = \,\,\)
\(\displaystyle \frac{47}{36}\)
which concludes the process of simplification.
Example: More PEMDAS examples. Order of operations problems
Simplify the following: \( \left(\frac{2}{3} + \frac{5}{4}\right)^2 - \frac{5}{6}\)
Solution: We are provided with the following expression: \(\displaystyle \left(\frac{2}{3}+\frac{5}{4}\right)^2-\frac{5}{6}\).
The following calculation is obtained:
\( \displaystyle \left(\frac{2}{3}+\frac{5}{4}\right)^2-\frac{5}{6}\)
Amplifying in order to get the common denominator 12
\( = \,\,\)
\(\displaystyle \left(\frac{2}{3}\cdot \frac{4}{4}+\frac{5}{4}\cdot \frac{3}{3}\right)\left(\frac{2}{3}\cdot \frac{4}{4}+\frac{5}{4}\cdot \frac{3}{3}\right)-\frac{5}{6}\)
We use the common denominator: 12
\( = \,\,\)
\(\displaystyle \left(\frac{2\cdot 4+5\cdot 3}{12}\right) \times \left(\frac{2\cdot 4+5\cdot 3}{12}\right)-\frac{5}{6}\)
Expanding each term: \(2 \times 4+5 \times 3 = 8+15\)
\( = \,\,\)
\(\displaystyle \left(\frac{8+15}{12}\right) \times \left(\frac{8+15}{12}\right)-\frac{5}{6}\)
Operating the terms in the numerator
\( = \,\,\)
\(\displaystyle \frac{23}{12}\cdot\frac{23}{12}-\frac{5}{6}\)
We can multiply the terms in the top and bottom as in \(\displaystyle\frac{ 23}{ 12} \times \frac{ 23}{ 12}= \frac{ 23 \times 23}{ 12 \times 12} \)
\( = \,\,\)
\(\displaystyle \frac{23\cdot 23}{12\cdot 12}-\frac{5}{6}\)
Multiplication of terms in the numerator and denominator, we get: \( 23 \times 23 = 529 \) and \( 12 \times 12 = 144\)
\( = \,\,\)
\(\displaystyle \frac{529}{144}-\frac{5}{6}\)
Amplifying in order to get the common denominator 144
\( = \,\,\)
\(\displaystyle \frac{529}{144}-\frac{5}{6}\cdot\frac{24}{24}\)
We use the common denominator: 144
\( = \,\,\)
\(\displaystyle \frac{529-5\cdot 24}{144}\)
Expanding each term in the numerator: \(529-5 \times 24 = 529-120\)
\( = \,\,\)
\(\displaystyle \frac{529-120}{144}\)
Operating the terms in the numerator
\( = \,\,\)
\(\displaystyle \frac{409}{144}\)
which concludes the process of simplification.
More algebra calculators
One of the cornerstones of algebra is the manipulation of algebraic expressions, from numbers, to fractions, to be complicated compound expressions.
All the guesswork is removed when having a proper set of rules establishing the correct order of operations in which expression should be simplified.
Along the same line of core Algebra rules simplifiers, you can also try our FOIL calculator, which uses PEMDAS to apply the distributive property in expression of the form \((a+b)(c+d\) | {"url":"https://mathcracker.com/pemdas-calculator","timestamp":"2024-11-12T23:56:07Z","content_type":"text/html","content_length":"126188","record_id":"<urn:uuid:5d7a58a2-ac8b-4c8b-b048-08cdf8758e57>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00812.warc.gz"} |
Understanding Prisms and Pyramids: Key Differences in Shape, Volume, and Features
Understanding Prisms And Pyramids: Key Differences In Shape, Volume, And Features
Prisms and pyramids are both polyhedrons, but they differ in their shape and structure. Prisms have two parallel bases, while pyramids have a single base and triangular sides. Prism bases can take on
various shapes, such as rectangles, squares, or triangles, while pyramid bases are always polygonal. Prisms have rectangular sides, while pyramids have triangular sides. Prisms have six faces, while
pyramids have four or five faces. Prisms have 12 edges, while regular pyramids have 8 edges. Prisms have 8 vertices, while regular pyramids have 5 vertices. The volume of a prism is calculated as the
area of the base times the height, while the volume of a pyramid is calculated as the area of the base times the height divided by three.
Shape: A Tale of Two Bases
In the realm of geometry, where shapes dance and numbers unravel the secrets of space, two polyhedrons reign supreme: prisms and pyramids. Their distinct features, like a captivating tale of two
bases, set them apart in the world of three-dimensional forms.
Prisms, with their parallel bases akin to twins, stand upright, their sides resembling a graceful symphony of rectangles or parallelograms. The bases, whether rectangular, square, or triangular, form
a solid foundation, a canvas upon which intricate geometric patterns are woven.
Pyramids, on the other hand, are crowned with a single base, adorned with polygonal splendor. Their sides, like elegant triangles, converge towards a vertex, a point where their edges gracefully
intertwine. These triangular faces create a dynamic silhouette, a testament to the harmonious interplay of geometry and art.
Bases: A Foundation of Polygons
In the realm of three-dimensional shapes, the foundation lies in their bases. While prisms exhibit a diverse range of base shapes, pyramids stand out with their exclusive polygonal bases. These
polygonal bases, like a painter’s canvas, provide a foundation for the geometrical tapestry of pyramids.
Polygonal bases, the bedrock of pyramids, are distinguished by their straight sides, forming a geometric dance of intersecting lines. Each angle, a meeting point of two sides, tells a story of
precise measurement and symmetry. These polygons can take on a multitude of forms, from the familiar triangle to the intricate pentagon, each imparting a unique character to its pyramid counterpart.
Sides: A Tapestry of Rectangles and Triangles
The contrasting sides of prisms and pyramids reveal a tale of geometric diversity. Prisms boast rectangular sides, standing tall with an air of order and precision. Pyramids, on the other hand, are
adorned with triangular sides, their sloping surfaces adding a touch of dynamism to their form.
Within the prism family, we encounter a special subset known as right prisms. These elegant polyhedrons have sides that are perpendicular to the bases, creating a harmonious balance between the two.
Imagine a rectangular prism, its sides rising majestically, forming a perfect right angle with its rectangular bases. This arrangement lends an air of stability and symmetry to the prism’s structure.
Faces: A Canvas for Geometry
In the realm of geometry, polyhedrons, like prisms and pyramids, captivate with their intricate architecture. Their faces, like brushstrokes on a canvas, play a crucial role in defining their
distinctive shapes and properties.
Prisms: A Symphony of Six Faces
Prisms, with their parallel bases, boast a harmonious six faces. Imagine a rectangular prism with its sides adorned with rectangles. These faces form a perfect balance, creating a sense of symmetry
and stability. Each side of the prism, perpendicular to the bases, adds to its volume and grace.
Pyramids: A Tale of Four or Five Faces
Pyramids, with their solitary base, exhibit a different facial tapestry. They come in two varieties: regular and irregular. Regular pyramids, like tetrahedrons, pentagonal pyramids, and hexagonal
pyramids, don the elegance of equilateral polygons for their bases. These faces converge gracefully to form a single, sharp apex. Irregular pyramids, on the other hand, have bases with varying side
lengths, resulting in an asymmetric arrangement of faces.
The Dance of Sides and Faces
The number of sides and faces in prisms and pyramids are intricately intertwined. Prisms, with their rectangular sides, have a face-to-side ratio of 2:1. For every two faces, there is one side
connecting them. Pyramids, however, have a more complex relationship. Regular pyramids have a face-to-side ratio of 3:2, while irregular pyramids vary depending on the number of sides in their base.
The faces of prisms and pyramids, like the brushstrokes of a master painter, shape the identity of these polyhedrons. Their distinct geometries, from the symmetry of prisms to the unique convergence
of pyramids, make them fascinating subjects of exploration for geometers and artists alike.
Edges: Intersections of Faces
In the realm of polyhedra, where geometry unfolds its myriad shapes, edges emerge as the pivotal intersections where faces coalesce. These delicate lines delineate the boundaries of each face,
defining the very essence of a polyhedron’s form.
Think of a prism, its straight sides forming a graceful rectangular embrace. Each side constitutes a unique face, and it is at their junctures that edges arise. These edges, like threads in a
celestial tapestry, connect the faces, creating a cohesive structure.
In the case of a right prism, with its sides perpendicular to the bases, the edges number a precise 12. Imagine a rectangular prism, its six faces meeting at twelve distinct corners. These corners
are the vertices where edges converge, anchoring the prism’s shape.
Shifting our gaze to the enigmatic pyramid, we encounter a different edge dynamic. Its polygonal base, whether square or triangular, serves as a solid foundation. From this base, triangular sides
slant upward, culminating in a single apex. These sloping sides are the faces of the pyramid, and at the intersections of these faces lie its 8 edges.
Each edge of a regular pyramid, like a taut wire, connects two vertices. These vertices, like radiant stars, adorn the pyramid’s base and apex, forming the endpoints of its edges. The interplay of
edges and faces in a pyramid creates an elegant and stable structure.
Vertices: Where Lines Intersect
In the realm of solid shapes, vertices reign supreme as the pivotal points where edges converge. These geometric crossroads hold immense significance, defining the overall structure and shaping the
identity of both prisms and pyramids.
Imagine a prism as a sleek, multifaceted prism, adorned with rectangular sides. Along the perimeter of each rectangle, edges gracefully intersect at eight distinct points. These eight vertices serve
as the cornerstones of the prism, anchoring its shape and giving it its distinctive form.
In the contrasting world of pyramids, triangular sides dance in perfect harmony. As their edges gracefully meet, they create five vertices that form the pyramid’s apex and its base. These five points
are the guiding stars, shaping the pyramid’s iconic silhouette against the geometric horizon.
The number of vertices in a prism or pyramid is inherently linked to its overall design. Regular prisms, with their precise symmetry, boast a predictable eight vertices. Regular pyramids, on the
other hand, proudly display five vertices, each playing a crucial role in the shape’s equilibrium.
Understanding the concept of vertices is not just an academic exercise; it unlocks a deeper comprehension of the intricate world of geometry. By recognizing the vertices that connect edges and define
shapes, we gain a newfound appreciation for the beauty and complexity that lies within these fascinating three-dimensional structures.
Volume: Exploring the Space Within Prisms and Pyramids
In the realm of geometry, where shapes dance in harmonious patterns, prisms and pyramids stand tall as captivating polyhedrons. Beyond their captivating forms, these polyhedrons enclose a captivating
secret—volume. Volume, the measure of the three-dimensional space occupied by an object, becomes an intriguing adventure when explored through the lens of prisms and pyramids.
Prisms, with their parallel bases and rectangular sides, offer a straightforward formula for calculating their volume: base area x height. Think of a rectangular prism as a shoebox—the length and
width of the base determine the size of the bottom, while the height represents the stack of shoes inside. By multiplying these dimensions, we uncover the total space enclosed by the prism.
Pyramids, on the other hand, present a slightly different story. Their triangular bases and sloping sides give them a more dynamic form. To calculate the volume of a pyramid, we employ a similar
formula—base area x height / 3. Imagine a pyramid as a triangular tent—the base represents the ground covered, while the height measures the peak’s elevation. However, since pyramids have a “hollow”
interior compared to prisms, we divide the result by 3 to account for this difference.
As we delve deeper into the world of prisms and pyramids, we discover that their volume serves as a crucial factor in understanding their properties and applications. From architects designing
towering buildings to engineers calculating the capacity of storage tanks, volume empowers us to grasp the true essence of these geometric wonders. | {"url":"https://rectangles.cc/understanding-prisms-pyramids-key-differences-shape-volume-features/","timestamp":"2024-11-05T07:08:46Z","content_type":"text/html","content_length":"147119","record_id":"<urn:uuid:18471e0e-0403-4782-b462-72e22c67f804>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00821.warc.gz"} |
Dirac Medal
The Dirac Prize is the name of four prominent awards in the field of theoretical physics, computational chemistry, and mathematics, awarded by different organizations, named in honour of Professor
Paul Dirac, one of the great theoretical physicists of the 20th century.
The Dirac Medal and Lecture (University of New South Wales)
The first-established prize is the Dirac Medal for the Advancement of Theoretical Physics, awarded by the University of New South Wales, Sydney, Australia, jointly with the Australian Institute of
Physics on the occasion of the public Dirac Lecture. The Lecture and the Medal commemorate the visit to the university in 1975 of Professor Dirac, who gave five lectures there. The lectures were
subsequently published as a book Directions of Physics (Wiley, 1978 – H. Hora and J. Shepanski, eds.). Professor Dirac donated the royalties from this book to the University for the establishment of
the Dirac Lecture series. The prize includes a silver medal and honorarium. It was first awarded in 1979.
Dirac Medal of the ICTP
The Dirac Medal of the ICTP is given each year by the Abdus Salam International Centre for Theoretical Physics (ICTP) in honour of physicist P.A.M. Dirac. The award, given each year on August 8
(Dirac's birthday), was first awarded in 1985.
An international committee of distinguished scientists selects the winners from a list of nominated candidates. The Committee invites nominations from scientists working in the fields of theoretical
physics or mathematics.
The Dirac Medal of the ICTP is not awarded to Nobel Laureates, Fields Medalists, or Wolf Prize winners. However, several Dirac Medallists have subsequently won one of these awards.
The medallists also receive a prize of US$5,000.
Dirac Medal of the IOP
The Dirac Medal is awarded annually by the Institute of Physics ( Britain's and Ireland's main professional body for physicists) for "outstanding contributions to theoretical (including mathematical
and computational) physics". The award, which includes a silver gilt medal and a £1000 prize, was decided upon by the Institute of Physics in 1985, and first granted in 1987.
Dirac Medal of the WATOC
The Dirac Medal is awarded annually by The World Association of Theoretical and Computational Chemists "for the outstanding computational chemist in the world under the age of 40". The award was
first granted in 1998.
• 1998 Timothy J. Lee
• 1999 Peter M. W. Gill
• 2000 Jiali Gao
• 2001 Martin Kaupp
• 2002 Jerzy Cioslowski
• 2003 Peter Schreiner
• 2004 Jan Martin
• 2005 Ursula Roethlisberger
• 2006 Lucas Visscher
• 2007 Anna Krylov
• 2008 Kenneth Ruud
• 2009 Jeremy Harvey
• 2010 Daniel Crawford
• 2011 Leticia González
• 2012 Paul Ayers
• 2013 Filipp Furche
• 2014 Denis Jacquemin
• 2015 Edward Valeev^[8]
See also
This article is issued from
- version of the 10/18/2016. The text is available under the
Creative Commons Attribution/Share Alike
but additional terms may apply for the media files. | {"url":"https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Dirac_Medal.html","timestamp":"2024-11-10T09:40:10Z","content_type":"text/html","content_length":"24342","record_id":"<urn:uuid:bd60e904-0fc6-417b-a4b1-8333c426870e>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00458.warc.gz"} |
1 is 0.9999999999999............
This has been discussed over and over.
However, there are some proofs in set theory in which this does not hold.
For example, use transfinite induction on only natural numbers.
So, we seek the least index n such that .9(n) = 1. We find it cannot be decided since for all indexes n, .9(n) != 1. This is a problem.
We also also can take an approach of considering a neighborhood around 1. We also see for all n, we can find a number between .9(n) and 1.
We can't find any natural number in which this relation is false. So, if we could actually exhaust all n, we are left in the state there is a number between .9(n) and 1. This actually means we cannot
exhaust all n.
I hope we can see discussion on this issue.
This has been discussed over and over.
However, there are some proofs in set theory in which this does not hold.
For example, use transfinite induction on only natural numbers.
So, we seek the least index n such that .9(n) = 1. We find it cannot be decided since for all indexes n, .9(n) != 1. This is a problem.
We also also can take an approach of considering a neighborhood around 1. We also see for all n, we can find a number between .9(n) and 1.
We can't find any natural number in which this relation is false. So, if we could actually exhaust all n, we are left in the state there is a number between .9(n) and 1. This actually means we
cannot exhaust all n.
I hope we can see discussion on this issue.
Just about to log out, so briefly, mate...
From past discussions I've been privy to, it appears that 0.99999999999..... is merely a TRIVIAL alternative NOTATION FORM for the 'unitary' output/symbol "1" (or "1.0" etc).
In short, 0.9999999... seems NOT to be a number OR a function as such, but a convenient expression indicating certain 'properties' (in certain contexts?) which may not be as evident if just the "1"
or "1.0" was used.
Anyhow, that's what past discussions seemed to imply. But I will read-only this further discussion with interest just in case further insights result from it! Thanks for your interesting discussions
elsewhere, chinglu, everyone! Enjoy them and good luck to you all. Bye for now.
In short, 0.9999999... seems NOT to be a number
Elementary arithmetic says that it is a number.
Elementary arithmetic says that it is a number.
It stands for "1.0" or "unitary" term. The form/notation is purely a matter of preference depending on the context which the notation/form arises in. That has been agreed upon long ago. It is also
NOT a function; since you did not question that then I must assume you agree with that?
Rational Skeptic
Valued Senior Member
This is a subtle topic. It is best to ignore decimal notation & deal with an infinite geometic sequence.
. . . .
can be represented by the following geometric series.
9/10 + 9/100 + . . . . + 9/10[sup]n[/sup].
The limit of the above series is one. Some of the more pedantic mathematical texts texts avoid saying that the above series is equivalent to one. Instead, they make statements similar to one of the
The limit is one, providing no further discussion.
Treating the sum of the series as equivalent to one does not result in a paradox. The actual statement (which I do not remember) is semantically equivalent to this statement.
Less formal texts state that the series is equal to one.
Attempts to prove the limit using decimal notation are flawed. Using hexidecimal arithmetic, (A thru F being 10 thru 15) .FFFFFFF
. . . .
is always less than one & greater than .99999
. . . .
which refutes the notion that the limit of .99999
. . . .
is one. Using a proof based on the sum of geometric series results in a valid proof that .99999
. . . .
& .FFFFFFF
. . . .
both have the same limit (one).
It is also NOT a function; since you did not question that then I must assume you agree with that?
It would be a waste of time for me to address
your fringe ideas.
It stands for "1.0" or "unitary" term.
Hopefully, by now, you figured out that , contrary to the fringe misconceptions that you espouse, it is a number. Just like 0.
I remember being involved on a thread long ago on this topic, and came away with the understanding that mathematicians are clear, the notation .999... represents 1. That notation means that you carry
out the 9's infinitely.
You cannot truncate the infinite series of decimal places because then you have a finite value that is less than 1, and you are not representing 1 unless you add the "..." at the end to desigate that
the notation is invoked to represent 1.
It would be a waste of time for me to address all your fringe ideas.
Hopefully, by now, you figured out that , contrary to the fringe misconceptions that you espouse, it is a number. Just like 0.
If you don't bother to fairly and properly read and understand in full context, then how can you judge anything at all, except from your ingrained prejudices and beliefs?
The conversation is ongoing. New approaches have implications for current partial axiomatic systems which need to be 'bridged' somehow and circumvent the incompleteness theorem 'barrier' which each
such partial system faces in isolation from overarching context.
And I already made clear (in my posts to rpenner et al in the other thread) what "0" may be fundamentally and non-trivially. The trivial 'undefined' states for some "0" instances arising in current
partial systems is the problem for those systems. My suggestions would have those problem situations forestalled from the outset axiomatically and contextually in reality, rather than let it become a
source of 'undefined' situations later on in those current/conventional partial systems. Zero is not a number in some contexts. It is a number in other contexts IF the problematic expressions
involving zero are kept isolated in parentheses AND the 'rules' are not trivialized to make "0" become "undefined" just because the partial system IS partial and not complete because of those very
axiomatic rules to begin with.
But you won't due fair and due diligence to try to understand the thrust, subtleties, implications and arguments/suggestions as already presented; so you will keep on kneejerking and insulting from
personal/dogma/ego while missing the whole point/discussion involved
Rather than just kneejerking and preening your ego and prejudicial beliefs ad nauseam, you should act like a real scientist and actually read and understand all the context/subtleties surrounding new
discussions before posting anything at all.
Ego is no substitute for the scientific method and free and fair discussion without prejudices personal or dogmatic, Tach. Good luck.
New approaches have implications for current partial axiomatic systems which need to be 'bridged' somehow and circumvent the incompleteness theorem 'barrier' which each such partial system faces
in isolation from overarching context.
You haven't provided any "new approach". Just plain misconceptions.
Rather than just kneejerking and preening your ego and prejudicial beliefs ad nauseam, you should act like a real scientist and actually read and understand all the context/subtleties surrounding
new discussions before posting anything at all.
There is no subtlety, contrary to your misconceptions both 0 and 0.(9) are numbers.
0.999 . . . does not equal 1.
it might approach 1 or 0.999 . . . might have a limit of 1 but it doesn't equal 1.
we can get extremely close to 1 by adding more 9s after the decimal point, but we will never get there.
only 1 equals 1.
- the opinion of a non mathematician.
It's not rocket surgery
Registered Senior Member
The most interesting thing about this discussion is not whether 0.999...=1 (I mean really, who cares? What difference does it make?)
It's why our minds rebel from the notion.
Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students
accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.
Tall and Schwarzenberger
, quoted in
The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls
Cecil Adams
, also quoted in
Is there a calculation that gives the result 0.99999999.......?
Is there a calculation that gives the result 0.99999999.......?
$$1-\frac{1}{10^n}$$ with $$n->\infty$$
Is there a calculation that gives the result 0.99999999.......?
Yes, 1/2 + 1/2 = .999...
since the RHS is 1.
Yes, 1/2 + 1/2 = .999...
since the RHS is 1.
True, if you invoke the meaning of .999... notation as being equal to one, but that is not in the spirit of the Captain's question, of course
An equation that does not use an infinite number in the problem to be calculated.
@ Leopold,
1.999... continued equals 2, yet 1.99999999 without continuation does not equal 2 from my understanding.
a) Divide a pie into 3 equal parts. 1 / 3 = 0.333..... Continued.
@ Captain Kremmen, (this will result in 0.999...)
b) Now try to get the three parts to equal a whole. 0.333... + 0.333... + 0.333... = 0.999... = 1
If 0.333... equals 1/3 of the pie then 0.999... must equal the whole pie.
That's it.
So in that case, the 0.999..... isn't equal to 1, it is the wrong answer.
Wrong, even though it is the number that you get from the calculation.
The result is always inaccurate, no matter how many digits you put after the zero, even an infinite number.
call me arf
Valued Senior Member
You could consider the difference in complexity between the two 'equivalent' representations.
Is 1.0 more or less complex than 0.999[sup]r[/sup]? Is an algorithm that ouptputs "1.0" less complex than an algorithm that outputs "0.999..."? Intuitively, the second would never halt, the algorithm
has to output an infinite string.
You could consider the difference in complexity between the two 'equivalent' representations.
Is 1.0 more or less complex than 0.999[sup]r[/sup]? Is an algorithm that ouptputs "1.0" less complex than an algorithm that outputs "0.999..."? Intuitively, the second would never halt, the
algorithm has to output an infinite string.
Yes, there is the matter of complexity, but 1 and .999... are defined as being equivalent, and thus we have a means to replace the complex .999... with the simple 1. | {"url":"https://sciforums.com/threads/1-is-0-9999999999999.136842/","timestamp":"2024-11-03T22:30:12Z","content_type":"text/html","content_length":"146451","record_id":"<urn:uuid:1ac0de6f-eb81-40da-8733-45ac974722d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00807.warc.gz"} |
Ideal Gas Law Problems - General Chemistry 1 Ideal Gas Law
Ideal Gas Law Problems
The density of sea water is 1.03 g/mL. What mass of seawater would fill a vessel to a volume of 255 mL?
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Congruence Triangles Worksheet 9Th Grade
Congruence Triangles Worksheet 9Th Grade. Web congruent triangles worksheets help students understand the congruence of triangles and help build a stronger foundation. These worksheets comprise
questions in a.
Triangle Congruence Proof Worksheet from www.onlineworksheet.my.id
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Deep Bhattacharjee
Author:Deep Bhattacharjee
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ANEC, annihilator, Apparent mass, Astro-Particle Physics, axioms, Axion Like Particles ALP, Axions, Baryonic Matter, Bayesian inference, Bernoulli Samples, black holes, Boolean, Calabi-Yau manifold,
Calabi-Yau Manifolds^2, Cantor, complex structures, Conjecture, connectivity, Cosmic Dynamics, Cosmic expansion, Cosmological Principle, CTC, Curvature of Spacetime, D(p)-Branes, dark energy^2, dark
matter^3, ddbar lemma, deg_h Parameters, Dehn, Dependability of Physics on Mathematics, dimensional analysis, duality, effective mass, energy conservation, Energy Manifestations, Entropy, Extended
Classical Mechanics, factor, Fano surface, Fraunhofer Lines of Sun, Fujikis Class C Manifold, fundamental constants, General Relativity, ghosts, Glome, Godfrey Harold Hardy, Gravitating Mass,
gravitational constant, gravitational field, gravitational interactions, gravitational lensing, Gravitational stability, Gravitational Waves, gravity, GSO-Projections., Haken Space., Hardy Ramanujan
Number, Hausdorff space, higher dimensions, Hilbert, Hilbert C*-module, Hodge, Hodge theory, homotopy, Hopf Fibrations, Hypersurface, hypothesis, Imperceptibility, Infinity, Jørgensen inequality,
Kahler, Kahler manifold, Kahler Manifolds, kinetic energy, Klein bottle, Laplace – Beltrami, laws, Lickorish – Wallace, Light Bending, Locus, M-theory, Magnetohydrodynamics, mass distribution,
Mass-Energy Composition, Massive Celestial Bodies, mathematical physics, mathematics, mobius strip, Morita equivalence, multiverse, nature, Nature of Physical Laws, negative pressure, Newtonian
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Generating Graphics Display for Sequential Designs
Example 78.4 Generating Graphics Display for Sequential Designs
This example creates the same group sequential design as in Example 78.3 and creates graphics by using ODS Graphics. The following statements request all available graphs in the SEQDESIGN procedure:
ods graphics on;
proc seqdesign altref=0.4
TwoSidedPocock: design nstages=4 method=poc;
TwoSidedOBrienFleming: design nstages=4 method=obf;
ods graphics off;
With the PLOTS=ALL option, a detailed boundary plot with the rejection region and acceptance region is displayed for the Pocock design, as shown in Output 78.4.1. By default (or equivalently if you
specify STOP=REJECT), the rejection boundaries are also generated at interim stages.
The plot shows identical boundary values in each boundary in the standardized
With the PLOTS=ALL option, a detailed boundary plot with the rejection region and acceptance region is also displayed for the O’Brien-Fleming design, as shown in Output 78.4.2. The plot shows that
the rejection boundary values are decreasing as the trial advances in the standardized
With the PLOTS=ALL option, the procedure displays a plot of average sample numbers (expected sample sizes for nonsurvival data or expected numbers of events for survival data) under various
hypothetical references for all designs simultaneously, as shown in Output 78.4.3. By default, the option CREF=
The plot shows that the Pocock design has a larger expected sample size than the O’Brien-Fleming design under the null hypothesis (
With the PLOTS=ALL option, the procedure displays a plot of the power curves under various hypothetical references for all designs simultaneously, as shown in Output 78.4.4. By default, the option
Under the null hypothesis,
With the PLOTS=ALL option, the procedure displays a plot of sequential boundaries for all designs simultaneously, as shown in Output 78.4.5. By default (or equivalently if you specify HSCALE=INFO in
the COMBINEDBOUNDARY option), the information levels are used on the horizontal axis.
The plot shows that the
With the PLOTS=ALL option, the procedure displays a plot of cumulative error spends for all boundaries in the designs simultaneously, as shown in Output 78.4.6. With a symmetric two-sided design,
cumulative error spending is displayed only for the upper | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_seqdesign_sect039.htm","timestamp":"2024-11-05T12:58:51Z","content_type":"application/xhtml+xml","content_length":"18052","record_id":"<urn:uuid:6c246d9c-1444-4576-b547-e1b191c8b320>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00576.warc.gz"} |
Flexi Quad Tan
As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.
Consider any convex quadrilateral $Q$ made from four rigid rods with flexible joints at the vertices so that the shape of $Q$ can be changed while keeping the lengths of the sides constant. If the
diagonals of the quadrilateral cross at an angle $\theta$ in the range $(0 \leq \theta < \pi/2)$, as we deform $Q$, the angle $\theta$ and the lengths of the diagonals will change.
Using the results of the two problems on quadrilaterals
Diagonals for Area
Flexi Quads
prove that the area of $Q$ is proportional to $\tan\theta$.
Getting Started
Combine the result from the problem Flexi Quad Areas that the area $A(Q) = {\textstyle{1\over 2}}d_1d_2\sin\theta$ with the definition of the scalar product and use the result from the problem Flexi
Quads that the scalar product of the diagonals is constant.
Student Solutions
Well done Shu Cao of the Oxford High School for girls for producing this nice solution so promptly.
Consider any convex quadrilateral $Q$ made from four rigid rods with flexible joints at the vertices so that the shape of $Q$ can be changed while keeping the lengths of the sides constant. If the
diagonals of the quadrilateral cross at an angle $\theta$ in the range $(0 \leq \theta < \pi/2)$, as we deform $Q$, the angle $\theta$ and the lengths of the diagonals will change and we have to
prove that the area of of $Q$ is a constant multiple of $\tan \theta $.
Notation: Let $|{\bf x}|$ mean the scalar quantity of vector ${\bf x}$ and the area of $Q$ be represented by $S$.
In the problem Diagonals for Area it was shown that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals:
$$S = {\textstyle{1\over 2}}|{\bf d_1}| \times |{\bf d_2}|\sin \theta.$$ From the definition of the scalar product $$|{\bf d_1}| \times |{\bf d_2}| = {{\bf d_1}\cdot {\bf d_2} \over \cos \theta }.$$
So $$S = {\textstyle{1\over 2}}{{\bf d_1}\cdot {\bf d_2}\sin \theta \over \cos \theta } = {\textstyle{1\over 2}}{\bf d_1}\cdot{\bf d_2}\tan \theta.$$ As shown in the problem Flexi Quads, the scalar
product of the diagonals is constant i.e. $$2{\bf d}_1 \cdot{\bf d}_2 = {\bf a}_2^2+{\bf a}_4^2-{\bf a}_1^2-{\bf a}_3^2.$$ As ${\bf a_1, a_2, a_3, a_4}$, the lengths of the sides of the
quadrilateral, all remain constant, hence ${\bf d}_1 \cdot {\bf d}_2$ remains constant. Hence the area of the quadrilateral $Q$ is a constant multiple of $\tan \theta$ and so it is proportional to $\
tan \theta $. | {"url":"https://nrich.maths.org/problems/flexi-quad-tan-0","timestamp":"2024-11-14T20:25:56Z","content_type":"text/html","content_length":"38972","record_id":"<urn:uuid:239fe37d-7f52-49bb-9e3b-659c81d7a0fe>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00860.warc.gz"} |
The constraint of Newton's constant in wormhole
The constraint of Newton's constant G(alpha) in wormhole is discussed which shows that G(alpha) assumes the maximum on the surface in the wormhole parameter alpha-space where the cosmological
constant Lambda(alpha) = 0(+). This condition fixes the theta-parameter of QCD at theta = 0 which corresponds to strong CP conservation.
Pub Date:
January 1991
□ Constants;
□ Cosmology;
□ Quantum Chromodynamics;
□ Quantum Theory;
□ Space-Time Functions;
□ Theoretical Physics;
□ Invariance;
□ Quarks;
□ Thermodynamics and Statistical Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1991cncw.book.....L/abstract","timestamp":"2024-11-10T21:42:36Z","content_type":"text/html","content_length":"33029","record_id":"<urn:uuid:8b083594-cffd-448d-b842-80c8c0c394f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00891.warc.gz"} |
FIS Portal
Spektraltheorie nichtselbstadjungierter Differentialoperatoren
Mathematik und Naturwissenschaften
Deutsche Forschungsgemeinschaft
Bewilligungssumme, Auftragssumme
62.576,00 €
A central topic in modern theoretical physics is the inconsistency between the standard model and general relativity. The quest for a Grand Unified Theory led to the development of new mathematical
models, for instance, in Quantum Mechanics non-self-adjoint operators were considered instead of self-adjoint. A prominent class of non-self-adjoint operators are self-adjoint operators in Krein
spaces and PT-symmetric operators. Unlike self-adjoint operators in Hilbert spaces, the spectral properties of PT-symmetric operators are fundamentally different, for example, they may have non-real
spectrum with accumulation points. In the present project we investigate the spectral properties of non-self-adjoint differential operators. The focus of this research program is on the spectral
properties of indefinite singular Sturm-Liouville operators and indefinite elliptic differential operators. One aim is to localize the position of the non-real point spectrum. Furthermore, we want to
study the accumulation of the point spectrum against the essential spectrum. In addition to the so-called WKB approximation for solutions of second order differential equations, we use oscillation
techniques for SturmLiouville operators and perturbation results for operators in Krein spaces. Moreover, for PT-symmetric quantum mechanics we develop by means of the WKB approximation criteria for
the limit point and the limit circle case for Sturm-Liouville operators with complex-valued coefficients. | {"url":"https://fisprojects.tu-ilmenau.de/portal/9174/show","timestamp":"2024-11-12T06:18:31Z","content_type":"text/html","content_length":"6655","record_id":"<urn:uuid:949a13c2-19ba-433a-8e40-377ef33e35b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00871.warc.gz"} |
Nordic Online Logic Seminar - page 4
An online seminar for logicians and logic aficionados worldwide.
The Nordic Online Logic Seminar (NOL Seminar) is a monthly seminar series initiated in 2021 presenting expository talks by logicians on topics of interest for the broader logic community. Initially
the series focused on activities of the Nordic logic groups, but has since expanded to offer a variety of talks from logicians around the world. The seminar is open to professional or aspiring
logicians and logic aficionados worldwide.
The tentative time slot is Monday, 16.00-17.30 (Stockholm/Sweden time). If you wish to receive the Zoom ID and password for it, as well as regular announcements, please subscribe to the NOL Seminar
mailing list.
NOL seminar organisers
Valentin Goranko and Graham Leigh
• Erich Grädel (RWTH Aachen University)
Semiring semantics for logical statements with applications to the strategy analysis of games
Semiring semantics of logical formulae generalises the classical Boolean semantics by permitting multiple truth values from certain semirings. In the classical Boolean semantics, a model of a
formula assigns to each (instantiated) literal a Boolean value. K-interpretations, for a semiring K, generalize this by assigning to each such literal a value from K. We then interpret 0 as false
and all other semiring values as nuances of true, which provide additional information, depending on the semiring: For example, the Boolean semiring over {0,1} corresponds classical semantics,
the Viterbi-semiring can model confidence scores, the tropical semiring is used for cost analysis, and min-max-semirings (A, max, min, a, b) for a totally ordered set (A,<) can model different
access levels. Most importantly, semirings of polynomials, such as N[X], allow us to track certain literals by mapping them to different indeterminates. The overall value of the formula is then a
polynomial that describes precisely what combinations of literals prove the truth of the formula.
This can also be used for strategy analysis in games. Evaluating formulae that define winning regions in a given game in an appropriate semiring of polynomials provides not only the Boolean
information on who wins, but also tells us how they win and which strategies they might use. For this approach, the case of Büchi games is of special interest, not only due to their practical
importance, but also because it is the simplest case where the logical definition of the winning region involves a genuine alternation of a greatest and a least fixed point. We show that, in a
precise sense, semiring semantics provide information about all absorption-dominant strategies – strategies that win with minimal effort, and we discuss how these relate to positional and the
more general persistent strategies. This information enables further applications such as game synthesis or determining minimal modifications to the game needed to change its outcome.
• Anupam Das (University of Birmingham)
On the proof theoretic strength of cyclic reasoning
Cyclic (or circular) proofs are now a common technique for demonstrating metalogical properties of systems incorporating (co)induction, including modal logics, predicate logics, type systems and
algebras. Inspired by automaton theory, cyclic proofs encode a form of self-dependency of which induction/recursion comprise special cases. An overarching question of the area, the so-called
‘Brotherston-Simpson conjecture’, asks to what extent the converse holds.
In this talk I will discuss a line of work that attempts to understand the expressivity of circular reasoning via forms of proof theoretic strength. Namely, I address predicate logic in the guise
of first-order arithmetic, and type systems in the guise of higher-order primitive recursion, and establish a recurring theme: circular reasoning buys precisely one level of ‘abstraction’ over
inductive reasoning.
This talk will be based on the following works:
• Dag Normann (Oslo)
An alternative perspective on Reverse Mathematics
In his address to the International Congress of Mathematics in Vancouver, 1974, Harvey Friedman launched a program where the aim would be to find the minimal set of axioms needed to prove
theorems of ordinary mathematics. More than often, it turned out that the axioms then would be provable from the theorems, and the subject was named Reverse Mathematics. In this talk we will
survey some of the philosophy behind, and results of, the early reverse mathematics, based on the formalisation of mathematics within second order number theory.
In 2005, Ulrich Kohlenbach introduced higher order reverse mathematics, and we give a brief explanation of the what and why? of Kohlenbach’s approach. In an ongoing project with Sam Sanders we
have studied the strength of classical theorems of late 19th/early 20th century mathematics, partly within Kohlenbach’s formal typed theory and partly by their, in a generalised sense,
constructive content. In the final part of the talk I will give some examples of results from this project, mainly from the perspective of higher order computability theory. No prior knowledge of
higher order computability theory is needed.
• Wilfrid Hodges (Fellow of the British Academy)
How the teenage Avicenna planned out several new logics
Almost exactly a thousand years ago a teenager known today as Avicenna lived in what is now Uzbekistan. He made a resolution to teach himself Aristotelian logic, armed with an Arabic translation
of Aristotle and a century-old Arabic textbook of logic. A couple of years later, around his eighteenth birthday, he wrote a brief report of what he had learned.
Six months ago I started to examine this report - I suspect I am the first logician to do that. It contains many surprising things. Besides introducing some new ideas that readers of Avicenna
know from his later works, it also identifies some specific points of modal logic where Avicenna was sure that Aristotle had made a mistake. People had criticised Aristotle’s logic before, but
not at these points. At first Avicenna had no clear understanding of how to do modal logic, and it took him another thirty years to justify all the criticisms of Aristotle in his report. But
meanwhile he discovered for himself how to defend a new logic by building new foundations. I think the logic itself is interesting, but the talk will concentrate on another aspect. These recent
discoveries mean that Avicenna is the earliest known logician who creates new logics and tells us what he is doing, and why, at each stage along the way.
• Jouko Väänänen (Helsinki)
Dependence logic: Some recent developments
In the traditional so-called Tarski’s Truth Definition the semantics of first order logic is defined with respect to an assignment of values to the free variables. A richer family of semantic
concepts can be modelled if semantics is defined with respect to a set (a “team”) of such assignments. This is called team semantics.
Examples of semantic concepts available in team semantics but not in traditional Tarskian semantics are the concepts of dependence and independence. Dependence logic is an extension of
first-order logic based on team semantics. It has emerged that teams appear naturally in several areas of sciences and humanities, which has made it possible to apply dependence logic and its
variants to these areas. In my talk I will give a quick introduction to the basic ideas of team semantics and dependence logic as well as an overview of some new developments, such as
quantitative analysis of team properties, a framework for a multiverse approach to set theory, and probabilistic independence logic inspired by the foundations of quantum mechanics.
• Dag Prawitz (Stockholm)
Validity of inference and argument
An account of inferences should take into account not only inferences from established premisses but also inferences made under assumptions. This makes it necessary to consider arguments, chains
of inferences in which assumptions and variables may become bound.
An argument is valid when all its inferences are valid, and it then amounts to a proof in case it has no unbound assumptions or variables. The validity of an inference – not to confuse with the
conclusion being a logical consequence of the premisses – seems in turn best explained in terms of proofs. This means that the concepts of valid inference and valid argument depend on each other
and cannot be defined independently but have to be described by principles that state how they are related. A number of such principles will be proposed. It is conjectured that inferences that
can be expressed in the language of first order intuitionistic predicate logic and are implied to be valid by these principles are all provable in that logic. | {"url":"https://logic-gu.se/nol/4/","timestamp":"2024-11-05T10:16:37Z","content_type":"text/html","content_length":"21804","record_id":"<urn:uuid:cce3485a-723f-45a2-9883-64f4b1e934cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00359.warc.gz"} |
A farmers rectangular field is 1 and 1/3 kilometers long and... - Ask Spacebar
A farmers rectangular field is 1 and 1/3 kilometers long and 2 and 3/4 kilometers wide. What is the area of the field in sq kilometers?
This is urgent due in 10 mins plz help
Views: 0 Asked: 12-25 19:41:03
On this page you can find the answer to the question of the mathematics category, and also ask your own question
Other questions in category | {"url":"https://ask.spacebarclicker.org/question/277","timestamp":"2024-11-11T14:02:19Z","content_type":"text/html","content_length":"27114","record_id":"<urn:uuid:55b78a2a-231d-4720-8c26-f9df1a9cac09>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00267.warc.gz"} |
Anne Carson’s book Nax is, very deliberately, ______ literary object––the opposite of an e-reader, which is designed to vanish in答案解析-雷哥GRE (2024)
雷哥网GRE > 填空 > 习题集 > 10457
习题集 -10457【medium】 00:00:00 关闭计时
Anne Carson’s book Nax is, very deliberately, ______ literary object––the opposite of an e-reader, which is designed to vanish in your palm as you read on a train.
该题平均耗时：59秒，平均正确率：71 %，难度系数：3 。该题由网友提供。更多GRE题目请点击上传
• [数学 ]【新版】冲分救命题 -12222 Three numbers are to be selected at random and without replacement from the five numbers 4, 5,7,8, and 11.What is the probability that the three numbers selected
could be the lengths of the sides of a triangle?
• [数学 ]【新版】冲分救命题 -12221 There are 5!. or 120. ways of arranging 5 different solid-colored flags side by side. If the colors of the flags are red, blue，yellow，green, and orange，how
many of those arrangements have either the red fag or the blue flag in the middle position?
• [数学 ]【新版】冲分救命题 -12220 k is randomly selected from the set of integers from 1 to 100, inclusive.
• [数学 ]【新版】冲分救命题 -12219 REENA'S MOVIE COLLECTION The table above shows the number of movies in each genre in Reena's movie collection. If two different movies are to be selected at
random from the collection. what is the probability that neither movie will be in the science fiction genre?
• [数学 ]【新版】冲分救命题 -12218 Several 0 and 1 are arranged in a 10*10 palace as follows. Among all the number 0, what is the probability that they are arranged in both an odd row and an odd
column? Give your answer as a fraction.
• [数学 ]【新版】冲分救命题 -12217 There are 5!, or 120,ways of arranging 5 different solid-colored flags side by side. If the colors of the flags are red, blue,yellow, green, and orange,how many
of those arrangements have either the red flag or the blue flag in the middle position?
• [数学 ]【新版】冲分救命题 -12216 Pat has five matched pairs of socks and no two of the pairs are the same color. If Pat selects two socks simultaneously and at random, what is the probability
that the two socks selected will be a matched pair.
• [数学 ]【新版】冲分救命题 -12215 A box contains 100 purple balls, 100 red balls, and 100 white balls. What is the minimum number of balls that must be chosen to ensure that at least 4 of the
balls chosen have the same color.
• [数学 ]【新版】冲分救命题 -12214 The cast of a certain play has 2 male characters and 2 female characters. The director of the play will select the 2 male characters from 6 male actors and the 2
female characters from 8 female actors. How many different sets of 4 actors, consisting of 2 male actors and 2 female actors, are there that can be selected by the director?
• [数学 ]【新版】冲分救命题 -12213 P is the set of all of all positive factors of 20, and Q is the set of all positive factors of 12. If a member of Pwill be chosen at random, what is the
probability that the chosen member will also be member of Q .
• [数学 ]【新版】冲分救命题 -12212 Average Per-Pupil Expenditures for the I0 Percent of School Districts State X That Had the Highest Expenditures (Group H) Compared to the 10 Percent of School
Districts That had the Lowest Expenditures (Group L)
• [数学 ]【新版】冲分救命题 -12211 Average Per-Pupil Expenditures for the 10 Percent of School Districts State X That Had the Highest Expenditures (Group H) Compared to the 10 Percent of School
Districts That had the Lowest Expenditures (Group L)For the 1986-87 school year, the ratio of the average per-pupil expenditures or the Group L districts to the average for the Group H districts
was closest to
• [数学 ]【新版】冲分救命题 -12210 Average Per-Pupil Expenditures for the I0 Percent of School Districts State X That Had the Highest Expenditures (Group H) Compared to the 10 Percent of School
Districts That had the Lowest Expenditures (Group L) From the 1979-80 school year to the 1989-90 school year, the average per-pupil expenditures for the Group L school districts increased by
• [数学 ]【新版】冲分救命题 -12209 As a telephone salesperson, Kay made an average (arithmetic mean) of x sales calls per day for n consecutive days. On the next day, she made k sales calls. Which
of the following represents the average number of sales calls per day for the n+1 days?
• [数学 ]【新版】冲分救命题 -12208 Set S consists of 8 positive imtegrs,3 of which are even, Set T consists of all numbers of the form x+x2 ,where x is member of set S.
• [数学 ]【新版】冲分救命题 -12207 Lists S and T each consist of 5 consecutive integers. The greatest integer in S is equal to the least integer in T.
• [数学 ]【新版】冲分救命题 -12206
• [数学 ]【新版】冲分救命题 -12205 List S : 1,2 ,3, k,2k If k< 2, which of the following numbers could be the median of the five numbers in list ? Indicate all such numbers.
• [数学 ]【新版】冲分救命题 -12204 In 1990 approximately what percent of the total population of neighborhood was foreign-born with Europe as region of origin?
• [数学 ]【新版】冲分救命题 -12203 A list of the names of the people of the entire 1990 foreign-born population in neighborhood was generated, with each person’s name appearing once. The names of 2
different people will be randomly selected from the list. Which of the following is closest to the probability that both names selected will be names of people whose region of origin was “Other”?
• 10.30数学直播
• 托福写作直播
• 雷哥GRE阅读机经特训团
• 不用记单词也能做的GRE填空题
• 雷哥GRE数学名词单词团
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• 1h带你解决GRE数学 “排列组合”考点题型
• 雷哥GRE阅读高频单词团
• 美国各院校GRE成绩要求及Verbal备考策略
• 雷哥GRE阅读高频训练团 | {"url":"https://vowhec.best/article/anne-carson-s-book-nax-is-very-deliberately-literary-object-the-opposite-of-an-e-reader-which-is-designed-to-vanish-in-gre","timestamp":"2024-11-11T09:46:03Z","content_type":"text/html","content_length":"75486","record_id":"<urn:uuid:50f6b0d0-4959-47a9-90d8-cb1923d991bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00839.warc.gz"} |
Wolstenholme's Theorem
If is a Prime, then the Numerator of
is divisible by and the Numerator of
is divisible by . These imply that if is Prime, then
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 85, 1994.
Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, p. 21, 1989.
© 1996-9 Eric W. Weisstein | {"url":"http://drhuang.com/science/mathematics/math%20word/math/w/w152.htm","timestamp":"2024-11-13T09:52:17Z","content_type":"text/html","content_length":"4840","record_id":"<urn:uuid:a55cc93e-b6d6-480c-8077-6d2f0f39ec6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00257.warc.gz"} |
Billion In Crore
Why use the Billion to Crore calculator?
An online calculator can quickly find BILLION TO CRORE without any error. MYcalcu is the best online calculator for students because it is a free calculator. It provides the eased accessibility to
online calculations without any payment or login restraints.
Million Conversion
Mycalcu uses the following formula to convert BILLION TO CRORE
B= 100 C
If you want to convert 20 BILLION TO CRORE, then MYcalcu utilizes the standard equation to get the result
B= 20*100C = 2000 C
As, both are units of numbers, thus these can be interlinked and can be represented on a ruler, as below.
1 Billion to Crore
2 Billion to Crore
3 Billion to Crore
4 Billion to Crore
5 Billion to Crore
10 Billion to Crore
20 Billion to Crore
30 Billion to Crore
40 Billion to Crore
50 Billion to Crore
100 Billion to Crore
200 Billion to Crore
500 Billion to Crore
900 Billion to Crore
1000 Billion to Crore | {"url":"https://mycalcu.com/billion-to-crore","timestamp":"2024-11-12T15:20:57Z","content_type":"text/html","content_length":"22725","record_id":"<urn:uuid:133a31d1-23a3-4e1b-b937-80301d1cd6f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00841.warc.gz"} |
Tubular Stand
If the radius of the tubing used to make this stand is r cm, what is the volume of tubing used?
A square stand designed to protect surfaces from hot pans is made from four $10 \; \text{cm}$ long pieces of cylindrical wooden dowel joined at the corners with $45$ degree mitres.
If the radius of the dowel used to make a stand is $0.5 \; \text{cm}$, what is the volume of wood used?
If I doubled the volume of wood used but did not change the radius of the dowel - what would the outside dimension of the stand be?
If, instead, I doubled the radius of the dowel but kept the same volume of wood, what would the outside dimension of the stand be then?
By looking in more detail at the effects of changing one of the variables at a time, can you describe any relationships between the volume of wood, the radius and the length of the dowel used?
Getting Started
The corners of the stand have to be considered and this means that each individual section is not a cylinder. However a little careful manipulation will reveal that the problem is far from
For the generalisation it might help to reduce the number of variables you work with to two at a time...
What is the relationship between the length and the volume if the radius is fixed?
Is there a way you can represent the relationship without just trying to describe it in words?
Student Solutions
There were a number of partial solutions but this well explained one is almost entirely the work ofAndrei of Tudor Vianu National College and shows how "obvious" the answer is with a little
visualising and careful reasoning.
Let $l$ be the exterior length of the cylinders and $r$ - the radius of the cylinders.
In the first solution, I have $4$ cylinders (the $4$ squares length $l-4r$ and $2r$) and $8$ half-cylinders - cylinders cut through the diagonal (the 8 small right-angled triangles from the figure).
Each of the four cylinders has the volume: $ V_1 = \pi r^2 (l - 4r)$
Each of the half-cylinders has a volume of half a cylinder: $ V_2 = \frac{1}{2} \times \pi r^2 \times 2r = \pi r^3$ Now, the total volume of the tubular stand is: $ V = 4 \times V_1 + 8 \times V_2 =
4\pi r^2 (l - 2r) $
With the second method, I arrange the $8$ half-cylinders to make $4$ bigger cylinders, and these $4$ to the bigger ones. The height of one 'big' cylinder is $(l - 2r)$, and the total volume is: $$ V
= 4 \pi r^2 (l - 2r) $$ which is exactly the result obtained above. Substituting the numerical values: $l = 10 \; \text{cm}$ and $r =0.5 \; \text{cm}$, I obtain:
$$ V = 4 \pi \times 0.25 \times(10 - 1) = 9 \pi = 28.27 \; {\text{cm}}^3 $$
If the volume wood would be double $(18 \pi)$, then the outside dimension of the dowel would be: \begin{eqnarray} 18 \pi &=& 4\pi \times 0.25 (l -1) \\ l - 1 & =& 18 \; \text{cm}\\ l &=& 19 \; \text
{cm}\end{eqnarray} If the volume of wood would be the same but the radius would be $1 \; \text{cm}$, then the outside dimension would be: \begin{eqnarray} 9 \pi &=& 4\pi (l - 2) \\ l - 2 &=& \frac
{9}{4} \\ l &=& 4.25 \; \text{cm}\end{eqnarray} The general formula for the volume of the dowel (proved above) is: $$ V = 4 \pi r^2 (l - 2r) $$ One could see that the volume is proportional to the
square of the radius, and in the limit of long outside dimensions, proportional to the outside dimension.
Teachers' Resources
Two pieces of dowel placed end-to-end and rotated should reveal a "neat" fit. Some cardboard wrapping paper tubes or toilet paper holders can be very useful in illustrating this point. | {"url":"https://nrich.maths.org/problems/tubular-stand","timestamp":"2024-11-02T13:43:20Z","content_type":"text/html","content_length":"41469","record_id":"<urn:uuid:02b787a2-775e-4e65-9d1a-cd56b4aa2e3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00389.warc.gz"} |
Scalars and Vectors - Motion in a Plane - Class 11 Physics MCQ - Sanfoundry
Class 11 Physics MCQ – Motion in a Plane – Scalars and Vectors
This set of Class 11 Physics Chapter 4 Multiple Choice Questions & Answers (MCQs) focuses on “Motion in a Plane – Scalars and Vectors”.
1. What is a scalar?
a) A quantity with only magnitude
b) A quantity with only direction
c) A quantity with both magnitude and direction
d) A quantity without magnitude
View Answer
Answer: a
Explanation: A scalar quantity is one which only has a magnitude. Examples of scalar quantities are mass, volume, work, energy etc.
2. What is a vector quantity?
a) A quantity with only magnitude
b) A quantity with only direction
c) A quantity with both magnitude and direction
d) A quantity without direction
View Answer
Answer: c
Explanation: A vector quantity is one which has both magnitude and direction. Unlike scalars, it can also tell the direction in which is the entity is acting. Examples of vector quantities are force,
velocity, displacement etc.
3. Which one of the following operations is valid?
a) Vector multiplied by scalar
b) Vector added to scalar
c) Vector subtracted from scalar
d) Vector divided by vector
View Answer
Answer: a
Explanation: Only operations valid for vector with scalars are multiplication and division. A vector can be multiplied or divided by a scalar. Other operations like addition and subtraction are not
valid for vectors with scalars.
4. The vector obtained by addition of two vectors is termed as ______
a) New vector
b) Resultant vector
c) Derived vector
d) Sum vector
View Answer
Answer: b
Explanation: The resultant vector is the vector which is obtained by the addition or subtraction of two vectors. The resultant vector can either be calculated by using the graphical method or the
analytical method.
5. Mass is a ____
a) Scalar quantity
b) Vector quantity
c) Relative quantity
d) Dependent quantity
View Answer
Answer: a
Explanation: Mass represents the amount of matter in a body. It does not have any direction but only mass. Hence, it is a scalar quantity.
6. The operation used to obtain a scalar from two vectors is ______
a) Cross product
b) Dot product
c) Simple product
d) Complex product
View Answer
Answer: b
Explanation: Dot product of two vectors gives a scalar quantity as the output. Cross product gives a vector as the output. It is also known as scalar product.
7. The operation which does not give you a vector as an output from two vector inputs is ______
a) Dot product
b) Cross product
c) Vector addition
d) Vector subtraction
View Answer
Answer: a
Explanation: Vector addition and subtraction give a vector as the output. Cross product also gives a vector as the output. Only Dot product gives a scalar after acting on two vectors.
Sanfoundry Global Education & Learning Series – Physics – Class 11.
To practice all chapters and topics of class 11 Physics, here is complete set of 1000+ Multiple Choice Questions and Answers. | {"url":"https://www.sanfoundry.com/physics-questions-answers-motion-plane-scalars-vectors/","timestamp":"2024-11-05T23:40:48Z","content_type":"text/html","content_length":"153731","record_id":"<urn:uuid:c62d8663-3c1d-4b18-bf1b-269ddfbe33f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00543.warc.gz"} |
How to Use MAP Function in Google Sheets | LiveFlow
How to Use MAP Function in Google Sheets
In this article, you will learn the MAP function and how to use it in Google Sheets.
This function is meant to be used with the LAMBDA function. The MAP formula runs the LAMBDA function for each value in a selected range and returns each result in the same dimensional field (but in
different cells).
How to insert the MAP formula in Google Sheets
• Type “=MAP” or go to “Insert” → “Function” → “Array” → “MAP”.
• Choose a range that includes input values.
• Enter a LAMBDA function with placeholders and logic.
How to insert the MAP function in Google Sheets
What is the MAP formula in Google Sheets?
The MAP function is beneficial when you want to process values in a range by getting them through formulas and get the processed results automatically in the same size of the dimensional area as that
of the one where the original input values are.
The generic syntax is as follows:
Array 1~: This is a range where the LAMBDA function defined in “lambda” operates. If you enter more than one array, note that you need to incorporate the corresponding number of placeholders in the
LAMBDA function you will insert. Also, when you include multiple ranges, their sizes should be equal.
Lambda: Input the LAMBDA function such as “LAMBDA(x, x^2)”. You can also use your Named Functions here.
Assume you have a list of values in local currencies, want to convert them to USD, and show them in the same table.
How to use the MAP function in Google Sheets
Array 1: J32:J36, this range corresponds to the “price” parameter in the LAMBDA function
Array 2: K32:K36, this range corresponds to the “currency” parameter in the LAMBDA function
Lambda: LAMBDA(price, currency, price*GOOGLEFINANCE(“currency:”¤cy&$L$31))
*Note: “currency:” is not a parameter but a required text for the GOOGLEFINANCE formula.
Once you enter the MAP formula in L32, you automatically get the returns in L32:L36.
If you use the Named Function in “lambda” section, the entire MAP function looks much simpler. Go to this article to learn how to use the Named Function.
How to use the MAP function with Named Functions in Google Sheets
If you want to know more about how this function works in Google Sheets, move on to the next section below.
How MAP function works in Google Sheets
Move on to the further examples in this section to understand how the MAP function works better.
How the MAP function works in detail in Google Sheets
Input A
This is a 3x3 array containing nine numbers, which will be input for Output A-1 and A-2.
Output A -1
The LAMBDA function in this sample returns InputA^2 for each input. Assume you want to show the calculation results in a range of B8:D10 with the input of each value in the “Input A” array. (e.g.,
Cell C8 shows the calculation results with the input of 2 in cell C3).
As this sample is an intermediate step to explaining how the MAP function works with the LAMBDA function, we applied the LAMBDA function to each cell in the selected range (B8:D10).
The point is you can get the same results with one MAP function.
Output A -2
This table shows what the returns of the MAP function look like. We only inserted one MAP function in cell B13, but it automatically returns nine values in cells in the array of B13:D15, each of
which comes from the original input value in the same dimensional position in Input A.
We understand that if a calculation is simple like this example, you think a more straightforward formula without the LAMBDA or MAP function is much easier and quicker. However, we believe these
examples are easy enough to learn how the MAP function works with the LAMBDA formula. | {"url":"https://www.liveflow.io/product-guides/how-to-use-map-function-in-google-sheets","timestamp":"2024-11-09T06:03:17Z","content_type":"text/html","content_length":"50082","record_id":"<urn:uuid:2f797999-5745-4052-860c-269638f5a5e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00195.warc.gz"} |
Weighted Escalator - a weighted stoch system
with an actual Indicator
I've come up with a slightly modified interpretation of the
'Escalator to Pips'
My Template and first indicator are slightly modified versions of the template / indicators
Banzai posted in post 13 on the first page.
The new indicator is a small graph with a the current score.
I call it the
'Weighted Escalator'
In this template, there are a total of 8 bars. The first 4 are the M15, M30, H1 and H4 stoch's with a 5,3,3 setting. The second set of 4 are M15, M30, H1 and H4 stoch's with a 14,3,3 setting.
I read this chart by giving the H4 chart the most weight in my calculations (since it is the trend) than the H1 which is still weighted more than the M30, then the M15. You add and subtract the
scores of each chart to get a total for that bar. You buy when the bar total goes above 3. You sell when it goes below -3.
The scoring is done like this:
M15 charts:
SpringGreen = +1
LimeGreen = 0
FireBrick = 0
IndianRed = -1
M30 Charts:
SpringGreen = +2
LimeGreem = 0
FireBrick = 0
IndianRed = -2
H1 Charts:
SpringGreen = +2
LimeGreen = +1
FireBrick = -1
IndianRed = -2
H4 Charts:
SpringGreen = +4
LimeGreen = +2
FireBrick = -2
IndianRed = -4
Total up the score for each bar. The score can be graphed much like a moving average, but I just use my fingers to see what the score is. I do sometimes plot the scores on some graph paper for
WHEN TO BUY:
Buy when the scoring goes from a negative number, past the zero, then hits the 3 or more. Example, the score goes -8, -5, -9, -1, 0, 1, 2, -1, 0, 2, 3 [BUY]
WHEN TO SHORT:
Buy when the scoring goes from a positive number, past the zero, then hits the -3 or less. Example, the score goes 13, 4, 12, 5, 6, 0, -2, -1, -2, 0, 1, 2, -5 [SHORT]
Upon Buying or Selling, I open 4 positions (lot size of your choice) with a stop loss if you want. I close each at the following:
1st. Take profit at 10 pips
2nd. Trailing Stop of 15 pips
3rd. This is my 'other' lot. Close when you like, I close after the 1st and 2nd have closed and it hit's or crosses the 3 going the 'wrong' way (I recently decided that is is my favorite out)
4th. Close when the score hit's or passes 0
So far, I've done this charting by hand on several past months and it's proven profitable for me. Looking at the chart below, you read the symbols like so:
a positive number is the number that's not filled in, a negative number is the one that's filled in. Also, the greens are positive, the reds are negative.
I have now attached my modified template and indicator.
This looks interesting. It definitely needs an indicator to track the scores though, ideally with an alert. Whilst I appreciate it can be done manually, to do so with more than one pair would be a
There is also a problem with backtesting I suspect. If the trend changes strongly during the last 15M (say from red to green) this could cause a repainting of the last 3 bars of the 1H bars, and
potentially the last 15 of the 4H bars (since there are 4 x 15M bars in a 1H bar and 16 x 15M bars in a 4H bar)
I would still be interested in seeing an indicator with an alert for forward testing though
I've always liked banzai's stoch indi, but this is a very interesting way to use it.
Agreed. Repainting is a problem with stochs. However, the timescale allows for much 'give' in the system. You are also correct that it's rather tedious to do everything by hand
Nice to see that you are testing the Spudfyre MTF (in your own variation).
You seem to be having good results.
I demo traded Spudfyre MTF for a month or two some time ago and it was working great for a while, but then for some reason I got where I couldn't make profits anymore.
I still think it is a great concept; wait until there is a longer term trend established, then enter when the lower, and next lower timeframe indicators are headed in the the same direction.
Theoretically, you should almost always have the highest probability of a winning trade that way. But for some reason it fizzled out for me.
If you could get an indicator that makes it easier for you to calculate your entries and exits for your method here you might have a very profitable system.
Please keep us posted.
Nice to see that you are testing the Spudfyre MTF (in your own variation).
You seem to be having good results.
I demo traded Spudfyre MTF for a month or two some time ago and it was working great for a while, but then for some reason I got where I couldn't make profits anymore.
I still think it is a great concept; wait until there is a longer term trend established, then enter when the lower, and next lower timeframe indicators are headed in the the same direction.
Theoretically, you should almost always have the highest probability of a winning trade that way. But for some reason it fizzled out for me.
If you could get an indicator that makes it easier for you to calculate your entries and exits for your method here you might have a very profitable system.
Please keep us posted.
Do you think that it would help the MTF stoch method if you could discern when price is ranging vs. trending? That seems to be the most difficult part for me.
That's kind of what I hope to have accomplished with the weighted system. When the score is in the negative, the trend is down. When the score is in the positive, the trend is up. All of the other
hills and valleys are ranging patterns. It may still require some tinkering with the weights, but it seems to work well so far. If you draw a line on the chart from the time the score entered
negative territory and kept that line going down until the score hit 0, you would see that it follows the trend. The same goes for positive scores.
The hills and valleys (the range) matter much less for the system as we are primarily concerned with the score (trend) being negative or positive.
EDIT- If you wanted to find the much longer trend (like the daily trend, weekly, monthly, so forth) you would add them into the mix and give them an appropriate weight.
I updated with a modified version of the indicator and a template for ease of use. This makes finding the score a LOT easier for me. I'm still not advanced enough to have it post the score or a
chart. That will take more work
I've posted a better view of the chart and how to use it. I've also nailed down my 'other' lot. I've found that for me, closing the lot after the score hits or crosses 3 in the 'wrong' direction (ie,
headed toward the 0) works out well. In the chart posted, there is a 97 pip gain. Look at the template and indicator, take your chart offline for a few minutes and randomly scroll to some point in
the history (or look at live charts) and see if it doesn't work for you.
I've always liked Spud's work. Nice to see a different version.
Interesting work. So in the attached pic, the most recent bar has a score of 13? I think I'm confused on how you're scoring, because the color examples you give in post #1 do not match up with the
colors I'm seeing here, so I think I'm doing it wrong.
Actually, a -3.
Here you go, this should clear it up
Sorry I wasn't clear.
EDIT- wow, thanks. I didn't realize that I still had the old color labels (and in the wrong oder twice!). FIXED!
I entered pretty late in this one, but I could tell by the score that it was still going down, so I decided to enter 1 lot aiming at making 10 pips. It happened in a short time, after a small
retrace. During the retrace, I trusted the score and I was always safe. I went to close at 10 pips but I'm a bit slow (doing it manually, didn't put in a TP) and I closed at 15pips.
As you can see, it's still going down and the score is still well below 0, but I will stay out since the safe bet is getting in at the beginning as per the rules.
I entered pretty late in this one, but I could tell by the score that it was still going down, so I decided to enter 1 lot aiming at making 10 pips. It happened in a short time, after a small
retrace. During the retrace, I trusted the score and I was always safe. I went to close at 10 pips but I'm a bit slow (doing it manually, didn't put in a TP) and I closed at 15pips.
As you can see, it's still going down and the score is still well below 0, but I will stay out since the safe bet is getting in at the beginning as per the rules.
I have downloaded but cannot get the numbers, 4 and 1 to appear as you have on your chart, I only get 0 and 2
Could you please help
Try using a different currency pair and a different time frame. Also, check the properties of one of the frames for the indicator, and make sure that it has the numbers listed in the picture below.
Other than that, try some different wingdings from the chart below. If none of them show up, then I'm guessing there is something wrong with your MT4 installation or with the wingding font on your
copy of Windows. Perhaps check the windows fonts that you have installed if changing the wingding symbol does not show the correct symbol (number) for you.
Try using a different currency pair and a different time frame. Also, check the properties of one of the frames for the indicator, and make sure that it has the numbers listed in the picture
below. Other than that, try some different wingdings from the chart below. If none of them show up, then I'm guessing there is something wrong with your MT4 installation or with the wingding font
on your copy of Windows. Perhaps check the windows fonts that you have installed if changing the wingding symbol does not show the correct symbol (number) for you.
Thanks I`ll take a look
I have changed as you mentioned, but still no 4`s and now I only have 0`s and 1`s
Thanks again
I have changed as you mentioned, but still no 4`s and now I only have 0`s and 1`s
Thanks again
If you change the symbol to some other wingding, like 74 for a smiley face, do you see that? If not, you may have a font problem.
If you change the symbol to some other wingding, like 74 for a smiley face, do you see that? If not, you may have a font problem.
sorry to be a pain, I have no idea where to look for this
I'll send you a private message to help you out | {"url":"https://www.forexfactory.com/thread/109676-weighted-escalator-a-weighted-stoch-system","timestamp":"2024-11-03T16:32:49Z","content_type":"text/html","content_length":"139998","record_id":"<urn:uuid:cf5c9864-cf3d-4d6c-9224-2feca27f205f>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00042.warc.gz"} |
Vector bundles as direct images of line bundles
Hirschowitz, A. ; Narasimhan, M. S. (1994) Vector bundles as direct images of line bundles Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 104 (1). pp. 191-200. ISSN 0253-4142
PDF - Publisher Version
Official URL: http://www.ias.ac.in/j_archive/mathsci/104/1/191-2...
Related URL: http://dx.doi.org/10.1007/BF02830882
Let X be a smooth irreducible projective variety over an algebraically closed field K and E a vector bundle on X. We prove that, if dim X≥1, there exist a smooth irreducible projective variety Z over
K, a surjective separable morphism f:Z →X which is finite outside an algebraic subset of codimension ≥3 in X and a line bundle L on X such that the direct image of L by f is isomorphic to E. When X
is a curve, we show that Z, f, L can be so chosen that f is finite and the canonical map H^1(Z, O) →H^1(X, End E) is surjective.
Item Type: Article
Source: Copyright of this article belongs to Indian Academy of Sciences.
Keywords: Projective Variety; Algebraic Vector Bundle; Line Bundle; Direct Image; Finite Morphism
ID Code: 58109
Deposited On: 31 Aug 2011 12:29
Last Modified: 18 May 2016 09:14
Repository Staff Only: item control page | {"url":"https://repository.ias.ac.in/58109/","timestamp":"2024-11-13T01:05:11Z","content_type":"application/xhtml+xml","content_length":"17982","record_id":"<urn:uuid:45dc750a-a705-4f33-8aa5-cbf2e6d2aa32>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00149.warc.gz"} |
A186884 - OEIS
This sequence contains
as a subsequence (corresponding to p=0).
All composites in this sequence are 2-pseudoprimes,
. This sequence contains all terms of
. Another composite term is 4294967297 = 2^32 + 1, which does not belong to
. In other words, all known composite terms have the form (2^x + 1) or (2^x - 1). Are there composites not of this form?
This sequence contains all the primes of the forms (2^x + 1) and (2^x - 1), i.e., subsequences | {"url":"https://oeis.org/A186884","timestamp":"2024-11-06T17:19:38Z","content_type":"text/html","content_length":"13475","record_id":"<urn:uuid:54670c89-8ecd-4925-8098-e79ef48935c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00317.warc.gz"} |
Andrea Roccaverde
Have you ever wondered what mathematicians mean when they talk about mathematical models of real-life phenomena? And what can such a model tell us about the network-phenomenon we are studying?
Interacting particle systems Statistical physics aims at describing collective behaviour in systems consisting of a very large number of interacting particles (= atoms or molecules). This is a
daunting task: a glass of water or a piece of iron can easily contain 100.000.000.000.000.000.000.000 particles. Still, the hope is that the macroscopic properties of these particles […] | {"url":"https://www.networkpages.nl/author/arocca/","timestamp":"2024-11-09T03:41:58Z","content_type":"text/html","content_length":"66934","record_id":"<urn:uuid:c541cde3-07e3-4e82-9c87-de5cd7232ef1>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00600.warc.gz"} |
Dual-tree and double-density 1-D wavelet transform
wt = dddtree(typetree,x,level,fdf,df) returns the typetree discrete wavelet transform (DWT) of the 1-D input signal, x, down to level, level. The wavelet transform uses the decomposition (analysis)
filters, fdf, for the first level and the analysis filters, df, for subsequent levels. Supported wavelet transforms are the critically sampled DWT, double-density, dual-tree complex, and dual-tree
double-density complex wavelet transform. The critically sampled DWT is a filter bank decomposition in an orthogonal or biorthogonal basis (nonredundant). The other wavelet transforms are oversampled
filter banks.
wt = dddtree(typetree,x,level,fname) uses the filters specified by fname to obtain the wavelet transform. Valid filter specifications depend on the type of wavelet transform. See dtfilters for
wt = dddtree(typetree,x,level,fname1,fname2) uses the filters specified in fname1 for the first stage of the dual-tree wavelet transform and the filters specified in fname2 for subsequent stages of
the dual-tree wavelet transform. Specifying different filters for stage 1 is valid and necessary only when typetree is 'cplxdt' or 'cplxdddt'.
Complex Dual-Tree Wavelet Transform
Obtain the complex dual-tree wavelet transform of the noisy Doppler signal. The FIR filters in the first and subsequent stages result in an approximately analytic wavelet as required.
Use dtfilters to create the first-stage Farras analysis filters and 6-tap Kingsbury Q-shift analysis filters for the subsequent stages of the multiresolution analysis.
The Farras and Kingsbury filters are in df{1} and df{2}, respectively. Load the noisy Doppler signal and obtain the complex dual-tree wavelet transform down to level 4.
load noisdopp;
wt = dddtree('cplxdt',noisdopp,4,df{1},df{2});
Plot an approximation based on the level-four approximation coefficients.
xapp = dddtreecfs('r',wt,'scale',{5});
hold on
axis tight
Using the output of dtfilters, or the filter name itself, in dddtree is preferable to manually entering truncated filter coefficients. To demonstrate the negative impact on signal reconstruction,
create truncated versions of the Farras and Kingsbury analysis filters. Display the differences between the truncated and original filters.
Faf{1} = [0 0
-0.0884 -0.0112
0.0884 0.0112
0.6959 0.0884
0.6959 0.0884
0.0884 -0.6959
-0.0884 0.6959
0.0112 -0.0884
0.0112 -0.0884
0 0];
Faf{2} = [ 0.0112 0
0.0112 0
-0.0884 -0.0884
0.0884 -0.0884
0.6959 0.6959
0.6959 -0.6959
0.0884 0.0884
-0.0884 0.0884
0 0.0112
0 -0.0112];
af{1} = [ 0.0352 0
-0.0883 -0.1143
0.2339 0
0.7603 0.5875
0.5875 -0.7603
0 0.2339
-0.1143 0.0883
0 -0.0352];
af{2} = [0 -0.0352
-0.1143 0.0883
0 0.2339
0.5875 -0.7603
0.7603 0.5875
0.2339 0
-0.0883 -0.1143
0.0352 0];
Obtain the complex dual-tree wavelet transform down to level 4 using the truncated filters. Take the inverse transform and compare the reconstruction with the original signal.
wt = dddtree('cplxdt',noisdopp,4,Faf,af);
xrec = idddtree(wt);
Do the same using the filter name. Confirm the difference is smaller.
wt = dddtree('cplxdt',noisdopp,4,'dtf1');
xrec = idddtree(wt);
Double-Density Wavelet Transform
Obtain the double-density wavelet transform of a signal with two discontinuities. Use the level-one detail coefficients to localize the discontinuities.
Create a signal consisting of a 2-Hz sine wave with a duration of 1 second. The sine wave has discontinuities at 0.3 and 0.72 seconds.
N = 1024;
t = linspace(0,1,1024);
x = 4*sin(4*pi*t);
x = x - sign(t - .3) - sign(.72 - t);
xlabel('Time (s)')
title('Original Signal')
grid on
Obtain the double-density wavelet transform of the signal. Reconstruct an approximation based on the level-one detail coefficients by first setting the lowpass (scaling) coefficients equal to 0. Plot
the result. Observe features in the reconstruction align with the signal discontinuities.
wt = dddtree('ddt',x,1,'filters1');
wt.cfs{2} = zeros(1,512);
xrec = idddtree(wt);
set(gca,'xtick',[0 0.3 0.72 1])
First-Level Detail Coefficients Approximation — Complex Dual-Tree
Obtain the complex dual-tree wavelet transform of a signal with two discontinuities. Use the first-level detail coefficients to localize the discontinuities.
Create a signal consisting of a 2-Hz sine wave with a duration of 1 second. The sine wave has discontinuities at 0.3 and 0.72 seconds.
N = 1024;
t = linspace(0,1,1024);
x = 4*sin(4*pi*t);
x = x - sign(t - .3) - sign(.72 - t);
xlabel('Time (s)')
title('Original Signal')
grid on
Obtain the dual-tree wavelet transform of the signal, reconstruct an approximation based on the level-one detail coefficients, and plot the result.
wt = dddtree('cplxdt',x,1,'FSfarras','qshift06');
wt.cfs{2} = zeros(1,512,2);
xrec = idddtree(wt);
set(gca,'xtick',[0 0.3 0.72 1])
Input Arguments
typetree — Type of wavelet decomposition
'dwt' | 'ddt' | 'cplxdt' | 'cplxdddt'
Type of wavelet decomposition, specified as one of 'dwt', 'ddt', 'cplxdt', or 'cplxdddt'. The type, 'dwt', gives a critically sampled (nonredundant) discrete wavelet transform. The other
decomposition types produce oversampled wavelet transforms. 'ddt' produces a double-density wavelet transform. 'cplxdt' produces a dual-tree complex wavelet transform. 'cplxdddt' produces a
double-density dual-tree complex wavelet transform.
x — Input signal
Input signal, specified as an even-length row or column vector. If L is the value of the level of the wavelet decomposition, 2^L must divide the length of x. Additionally, the length of the signal
must be greater than or equal to the product of the maximum length of the decomposition (analysis) filters and 2^(L-1).
Data Types: double
level — Level of wavelet decomposition
positive integer
Level of the wavelet decomposition, specified as an integer. If L is the value of level, 2^L must divide the length of x. Additionally, the length of the signal must be greater than or equal to the
product of the maximum length of the decomposition (analysis) filters and 2^(L-1).
Data Types: double
fdf — Level-one analysis filters
matrix | cell array
The level-one analysis filters, specified as a matrix or cell array of matrices. Specify fdf as a matrix when typetree is 'dwt' or 'ddt'. The size and structure of the matrix depend on the typetree
input as follows:
• 'dwt' — This is the critically sampled discrete wavelet transform. In this case, fdf is a two-column matrix with the lowpass (scaling) filter in the first column and the highpass (wavelet) filter
in the second column.
• 'ddt' — This is the double-density wavelet transform. The double-density DWT is a three-channel perfect reconstruction filter bank. fdf is a three-column matrix with the lowpass (scaling) filter
in the first column and the two highpass (wavelet) filters in the second and third columns. In the double-density wavelet transform, the single lowpass and two highpass filters constitute a
three-channel perfect reconstruction filter bank. This is equivalent to the three filters forming a tight frame. You cannot arbitrarily choose the two wavelet filters in the double-density DWT.
The three filters together must form a tight frame.
Specify fdf as a 1-by-2 cell array of matrices when typetree is a dual-tree transform, 'cplxdt' or 'cplxdddt'. The size and structure of the matrix elements depend on the typetree input as follows:
• For the dual-tree complex wavelet transform, 'cplxdt', fdf{1} is a two-column matrix containing the lowpass (scaling) filter and highpass (wavelet) filters for the first tree. The scaling filter
is the first column and the wavelet filter is the second column. fdf{2} is a two-column matrix containing the lowpass (scaling) and highpass (wavelet) filters for the second tree. The scaling
filter is the first column and the wavelet filter is the second column.
• For the double-density dual-tree complex wavelet transform, 'cplxdddt', fdf{1} is a three-column matrix containing the lowpass (scaling) and two highpass (wavelet) filters for the first tree and
fdf{2} is a three-column matrix containing the lowpass (scaling) and two highpass (wavelet) filters for the second tree.
Data Types: double
df — Analysis filters for levels > 1
matrix | cell array
Analysis filters for levels > 1, specified as a matrix or cell array of matrices. Specify df as a matrix when typetree is 'dwt' or 'ddt'. The size and structure of the matrix depend on the typetree
input as follows:
• 'dwt' — This is the critically sampled discrete wavelet transform. In this case, df is a two-column matrix with the lowpass (scaling) filter in the first column and the highpass (wavelet) filter
in the second column. For the critically sampled orthogonal or biorthogonal DWT, the filters in df and fdf must be identical.
• 'ddt' — This is the double-density wavelet transform. The double-density DWT is a three-channel perfect reconstruction filter bank. df is a three-column matrix with the lowpass (scaling) filter
in the first column and the two highpass (wavelet) filters in the second and third columns. In the double-density wavelet transform, the single lowpass and two highpass filters must constitute a
three-channel perfect reconstruction filter bank. This is equivalent to the three filters forming a tight frame. For the double-density DWT, the filters in df and fdf must be identical.
Specify df as a 1-by-2 cell array of matrices when typetree is a dual-tree transform, 'cplxdt' or 'cplxdddt'. For dual-tree transforms, the filters in fdf and df must be different. The size and
structure of the matrix elements in the cell array depend on the typetree input as follows:
• For the dual-tree complex wavelet transform, 'cplxdt', df{1} is a two-column matrix containing the lowpass (scaling) and highpass (wavelet) filters for the first tree. The scaling filter is the
first column and the wavelet filter is the second column. df{2} is a two-column matrix containing the lowpass (scaling) and highpass (wavelet) filters for the second tree. The scaling filter is
the first column and the wavelet filter is the second column.
• For the double-density dual-tree complex wavelet transform, 'cplxdddt', df{1} is a three-column matrix containing the lowpass (scaling) and two highpass (wavelet) filters for the first tree and
df{2} is a three-column matrix containing the lowpass (scaling) and two highpass (wavelet) filters for the second tree.
Data Types: double
fname — Filter name
character vector | string scalar
Filter name, specified as a character vector or string scalar. For the critically sampled DWT, specify any valid orthogonal or biorthogonal wavelet filter. See wfilters for details. For the
double-density wavelet transform, 'ddt', valid choices are 'filters1', 'filters2', and 'doubledualfilt'. For the complex dual-tree wavelet transform, valid choices are 'dtfP' with P = 1, 2, 3, 4. For
the double-density dual-tree wavelet transform, the only valid choice is 'dddtf1'. See dtfilters for more details on valid filter names for the oversampled wavelet filter banks.
Data Types: char
fname1 — First-stage filter name
character vector | string scalar
First-stage filter name, specified as a character vector or string scalar. Specifying a different filter for the first stage is valid and necessary only in the dual-tree transforms, 'cplxdt' and
'cplxdddt'. In the complex dual-tree wavelet transform, you can use any valid wavelet filter for the first stage. In the double-density dual-tree wavelet transform, the first-stage filters must form
a three-channel perfect reconstruction filter bank.
Data Types: char
fname2 — Filter name for stages > 1
character vector | string scalar
Filter name for stages > 1, specified as a character vector or string scalar. You must specify a first-level filter that is different from the wavelet and scaling filters in subsequent levels when
using the dual-tree wavelet transforms, 'cplxdt' or 'cplxdddt'. See dtfilters for valid choices.
Data Types: char
Output Arguments
wt — Wavelet transform
Wavelet transform, returned as a structure with these fields:
type — Type of wavelet decomposition (filter bank)
'dwt' | 'ddt' | 'cplxdt' | 'cplxdddt'
Type of wavelet decomposition (filter bank) used in the analysis, returned as one of 'dwt', 'ddt', 'cplxdt', or 'cplxdddt'. The type, 'dwt', gives a critically sampled discrete wavelet transform. The
other types correspond to oversampled wavelet transforms. 'ddt' is a double-density wavelet transform, 'cplxdt' is a dual-tree complex wavelet transform, and 'cplxdddt' is a double-density dual-tree
complex wavelet transform.
level — Level of the wavelet decomposition
positive integer
Level of wavelet decomposition, returned as a positive integer.
filters — Decomposition (analysis) and reconstruction (synthesis) filters
Decomposition (analysis) and reconstruction (synthesis) filters, returned as a structure with these fields:
FDf — First-stage analysis filters
matrix | cell array
First-stage analysis filters, returned as an N-by-2 or N-by-3 matrix for single-tree wavelet transforms, or a cell array of two N-by-2 or N-by-3 matrices for dual-tree wavelet transforms. The
matrices are N-by-3 for the double-density wavelet transforms. For an N-by-2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass)
filter. For an N-by-3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dual-tree transforms, each
element of the cell array contains the first-stage analysis filters for the corresponding tree.
Df — Analysis filters for levels > 1
matrix | cell array
Analysis filters for levels > 1, returned as an N-by-2 or N-by-3 matrix for single-tree wavelet transforms, or a cell array of two N-by-2 or N-by-3 matrices for dual-tree wavelet transforms. The
matrices are N-by-3 for the double-density wavelet transforms. For an N-by-2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass)
filter. For an N-by-3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dual-tree transforms, each
element of the cell array contains the analysis filters for the corresponding tree.
FRf — First-level reconstruction filters
matrix | cell array
First-level reconstruction filters, returned as an N-by-2 or N-by-3 matrix for single-tree wavelet transforms, or a cell array of two N-by-2 or N-by-3 matrices for dual-tree wavelet transforms. The
matrices are N-by-3 for the double-density wavelet transforms. For an N-by-2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass)
filter. For an N-by-3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dual-tree transforms, each
element of the cell array contains the first-stage synthesis filters for the corresponding tree.
Rf — Reconstruction filters for levels > 1
matrix | cell array
Reconstruction filters for levels > 1, returned as an N-by-2 or N-by-3 matrix for single-tree wavelet transforms, or a cell array of two N-by-2 or N-by-3 matrices for dual-tree wavelet transforms.
The matrices are N-by-3 for the double-density wavelet transforms. For an N-by-2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass)
filter. For an N-by-3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dual-tree transforms, each
element of the cell array contains the synthesis filters for the corresponding tree.
cfs — Wavelet transform coefficients
cell array of matrices
Wavelet transform coefficients, returned as a 1-by-(level+1) cell array of matrices. The size and structure of the matrix elements of the cell array depend on the type of wavelet transform, typetree,
as follows:
• 'dwt' — cfs{j}
□ j = 1,2,... level is the level.
□ cfs{level+1} are the lowpass, or scaling, coefficients.
• 'ddt' — cfs{j}(:,:,k)
□ j = 1,2,... level is the level.
□ k = 1,2 is the wavelet filter.
□ cfs{level+1}(:,:) are the lowpass, or scaling, coefficients.
• 'cplxdt' — cfs{j}(:,:,m)
□ j = 1,2,... level is the level.
□ m = 1,2 are the real and imaginary parts.
□ cfs{level+1}(:,:,m) are the lowpass, or scaling, coefficients.
• 'cplxdddt' — cfs{j}(:,:,k,m)
□ j = 1,2,... level is the level.
□ k = 1,2 is the wavelet filter.
□ m = 1,2 are the real and imaginary parts.
□ cfs{level+1}(:,:,m) are the lowpass, or scaling, coefficients.
Version History
Introduced in R2013b | {"url":"https://ww2.mathworks.cn/help/wavelet/ref/dddtree.html","timestamp":"2024-11-07T12:49:08Z","content_type":"text/html","content_length":"128203","record_id":"<urn:uuid:bca0cfc1-f9e8-4057-8d1b-c3cdbe9ecd56>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00106.warc.gz"} |
plot.aareg: Plot an aareg object.
Plot the estimated coefficient function(s) from a fit of Aalen's additive regression model.
# S3 method for aareg
plot(x, se=TRUE, maxtime, type='s', ...)
the result of a call to the aareg function
if TRUE, standard error bands are included on the plot
upper limit for the x-axis.
graphical parameter for the type of line, default is "steps".
other graphical parameters such as line type, color, or axis labels.
Side Effects
A plot is produced on the current graphical device.
Aalen, O.O. (1989). A linear regression model for the analysis of life times. Statistics in Medicine, 8:907-925. | {"url":"https://www.rdocumentation.org/packages/survival/versions/2.42-3/topics/plot.aareg","timestamp":"2024-11-09T22:25:58Z","content_type":"text/html","content_length":"54818","record_id":"<urn:uuid:46c1d78b-8aa5-4ee4-8aa3-958e42d6a1d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00214.warc.gz"} |
Designing Generic FIR Filters with pyFDA and NumPy
In my previous blog post, I promised that it was about time to start designing some real filters. Since infinite response (IIR) filters are a bit too complicated still, and sometimes not suitable for
audio processing due to non-linear phase behavior, I implicitly meant finite impulse response filters. They are much easier to understand, and generally behave better, but they also require a lot
more calculation power to obtain similar ripple and attenuation results than IIR filters.
There are many resources on the web that discuss the theoretical aspects about this or that filter, but fully worked out examples with full code are harder to find.
In this blog post, I will discuss the tools that I’ve been using to evaluate and design filters: pyFDA and NumPy.
I’ll almost always write ‘NumPy’ when discussing Python scripts related to this filter series. This should be considered a catch-all for various Python packages that aren’t necessarily part of NumPy:
matplotlib for plots, SciPy for signal processing function, etc. I think it’s fair to do this because NumPy website lists SciPy as part of NumPy as well.
Matlab is popular in the signal processing world, but a license costs thousands of dollars, and even if it’s better than NumPy (I honestly have no idea), it’s total overkill for the beginner stuff
that I want to do. GNU Octave is free software that’s claimed to be “drop-in compatible with many Matlab scripts”, but I haven’t tried it.
I’ve been told that a home license for Matlab is around 120 euros + 35 euros per toolbox. Cheaper than thousands, but still overkill.
Designing Filters with pyFDA
During initial filter configuration exploration, I often find it faster to play around with a GUI. It’s also a great way to learn about what’s out there and familiarize yourself with characteristics
of different kinds of filters.
In a recent tweet, Matt Venn pointed me to pyFDA, short for Python Filter Design Analysis tool.
I’ve since been using it, and it definitely helped me in getting my PDM MEMS microphone design off the ground.
pyFDA’s GUI is split into 2 halves: parameters and settings on the left, results on the right.
In the parameters screenshot below I enter a low pass filter with pass band and stop band characteristics that we’ll later need for our microphone. I also ask it to come up with the minimum order
that’s needed to meet these characteristics:
The default plot in the results section is the magnitude frequency response plot:
pyFDA tells us that we need a 37-order filter, which corresponds at 38 FIR filter taps.
There are all kinds of visualizations: magnitude frequency response, phase frequency response, impulse response, group delay, pole/zero plot, a fancy 3D plot that I don’t quite understand.
Once you’ve designed a filter, you can export the coefficients to use in your design, with or without conversion to fixed point numbers. pyFDA can even write out a Verilog file to put on your ASIC or
Designing Filters with NumPy’s Remez Function
For all its initial benefits, once the basic architecture of a design has been determined, I prefer to code all the details as a stand-alone numpy file. For example, all the scripts that were used to
create the graphs in this series can be found in my pdm GitHub repo.
Code has the following benefits over a GUI:
• You can parameterize the input parameters and regenerate all the collaterals (coefficients, graphs, potentially even RTL code) in one go.
While writing these blog posts, I often make significant changes along the way. You really don’t want to manually regenerate all the graphs every time you do that!
• Much more flexilibity wrt graphs
You can put multiple graphs in one figure, add annotations, tune colors etc. None of that is supported by the GUI.
• It’s much easier for others to reproduce the results, and modify the code, learn from it.
Feel free to clone all my stuff and improve it!
The question is: how do you go about desiging FIR filters?
I’m not qualified to give a comprehensive overview, but here are some very common techniques to determine the coefficients of FIR filters. It’s probably not a coincidence that these techniques are
also supported by pyFDA.
Moving Average and CIC Filters
I’ve already written about Moving Average and CIC filters. Their coefficients are all the same. pyFDA supports them by selecting the “Moving Average” option.
Windowed Sinc and Windowed FIR Filters
There are Windowed Sinc filters and Windowed FIR filters where you specify a filter in the frequency domain, take an inverse FFT to get an impulse response, and then use a windowing function to tune
the behavior. NumPy supports these methods with the firwin and firwin2 functions. Or use the “Windowed FIR” option in pyFDA.
While you can make excellent filters this way, they are often not optimal in terms of computational effort, and you need some understanding of the tradeoffs of this or that windowing filter to get
what you want.
Equiripple FIR Filters
Finally, there are equiripple filters that are designed with the Parks-McClellan filter design algorithm. As far as I can tell, it is the most common way to design FIR filters and it’s what I used in
my earlier pyFDA example by selecting the default “Equiripple” option.
It would lead too far to get into the details about the benefits of one kind of filter vs the other, but when given specific pass band and stop band parameters, equiripple filters require the lowest
number of FIR coefficients to achieve the desired performance.
You can use the the remez function in NumPy to design filters this way, and that exactly what I’ve been doing.
For example, the coefficients of the filter above in my pyFDA example, can be found as follows:
#! /usr/bin/env python3
from scipy import signal
Fs = 48 # Sample rate
Fpb = 6 # End of pass band
Fsb = 10 # Start of stop band
Apb = 0.05 # Max Pass band ripple in dB
Asb = 60 # Min stop band attenuation in dB
N = 37 # Order of the filter (=number of taps-1)
# Remez weight calculation: https://www.dsprelated.com/showcode/209.php
err_pb = (1 - 10**(-Apb/20))/2 # /2 is not part of the article above, but makes the result consistent with pyFDA
err_sb = 10**(-Asb/20)
w_pb = 1/err_pb
w_sb = 1/err_sb
# Calculate that FIR coefficients
h = signal.remez(
N+1, # Desired number of taps
[0., Fpb/Fs, Fsb/Fs, .5], # Filter inflection points
[1,0], # Desired gain for each of the bands: 1 in the pass band, 0 in the stop band
[w_pb, w_sb] # weights used to get the right ripple and attenuation
Run the code above, and you’ll get the following 38 filter coefficients:
[-2.50164675e-05 -1.74317423e-03 -2.54534101e-03 -7.63329067e-04
3.77271590e-03 6.73718674e-03 2.64362264e-03 -7.87738320e-03
-1.48337024e-02 -6.75502030e-03 1.48004646e-02 2.98724354e-02
1.54099648e-02 -2.76986944e-02 -6.25133368e-02 -3.81892367e-02
6.54474060e-02 2.09343906e-01 3.13554280e-01 3.13554280e-01
2.09343906e-01 6.54474060e-02 -3.81892367e-02 -6.25133368e-02
-2.76986944e-02 1.54099648e-02 2.98724354e-02 1.48004646e-02
-6.75502030e-03 -1.48337024e-02 -7.87738320e-03 2.64362264e-03
6.73718674e-03 3.77271590e-03 -7.63329067e-04 -2.54534101e-03
-1.74317423e-03 -2.50164675e-05]
A little bit more additional code will create a magnitude frequency response plot:
import numpy as np
from matplotlib import pyplot as plt
# Calculate 20*log10(x) without printing an error when x=0
def dB20(array):
with np.errstate(divide='ignore'):
return 20 * np.log10(array)
(w,H) = signal.freqz(h)
# Find pass band ripple
Hpb_min = min(np.abs(H[0:int(Fpb/Fs*2 * len(H))]))
Hpb_max = max(np.abs(H[0:int(Fpb/Fs*2 * len(H))]))
Rpb = 1 - (Hpb_max - Hpb_min)
# Find stop band attenuation
Hsb_max = max(np.abs(H[int(Fsb/Fs*2 * len(H)+1):len(H)]))
Rsb = Hsb_max
print("Pass band ripple: %fdB" % (-dB20(Rpb)))
print("Stop band attenuation: %fdB" % -dB20(Rsb))
plt.title("Impulse Response")
plt.title("Frequency Reponse")
plt.plot(w/np.pi/2*Fs,dB20(np.abs(H)), "r")
plt.plot([0, Fpb], [dB20(Hpb_max), dB20(Hpb_max)], "b--", linewidth=1.0)
plt.plot([0, Fpb], [dB20(Hpb_min), dB20(Hpb_min)], "b--", linewidth=1.0)
plt.plot([Fsb, Fs/2], [dB20(Hsb_max), dB20(Hsb_max)], "b--", linewidth=1.0)
plt.xlim(0, Fs/2)
plt.ylim(-90, 3)
Run this and you get:
Pass band ripple: 0.047584dB
Stop band attenuation: 60.316990dB
And the following plot:
Check out remez_example.py for the full source code.
Finding the Optimal Filter Order
In the example code above, the order of the filter (N=37) was given manually. To find the smallest N that satisfies the ripple and attenuation requirements, I just increase N until these requirements
are met.
There are formulas to determine the filter order that’s needed to meet filter requirements, but they’re approximations. Useful 50 years ago when computing time was expensive, it made sense to use
those, but that’s really not an issue today, so dumb brute force it is.
After factoring the earlier remez-based code as a function inside library filter_lib.py, it’s really as straightforward as this:
def fir_find_optimal_N(Fs, Fpb, Fsb, Apb, Asb, Nmin = 1, Nmax = 1000):
for N in range(Nmin, Nmax):
print("Trying N=%d" % N)
(h, w, H, Rpb, Rsb, Hpb_min, Hpb_max, Hsb_max) = fir_calc_filter(Fs, Fpb, Fsb, Apb, Asb, N)
if -dB20(Rpb) <= Apb and -dB20(Rsb) >= Asb:
return N
Run the remez_example_find_n.py script:
from filter_lib import *
Fs = 48000
Fpb = 6000
Fsb = 10000
Apb = 0.05
Asb = 60
N = fir_find_optimal_N(Fs, Fpb, Fsb, Apb, Asb)
And this is the result:
Trying N=33
Rpb: 0.086118dB
Rsb: 55.258914dB
Trying N=34
Rpb: 0.065989dB
Rsb: 57.550847dB
Trying N=35
Rpb: 0.048278dB
Rsb: 60.216487dB
Rpb: 0.048278dB
Rsb: 60.216487dB
Note that we need an FIR filter with order 35 (36 taps) to realize our requirements. In my example earlier, pyFDA with exactly the same parameters came up with a filter that’s 2 orders higher.
I have no idea why…
Complex FIR Filters
The remez function can handle much more than simple low pass, high pass, band pass or band stop filters.
You can specify pretty much any frequency magnitude behavior you want and make it come up with a set of FIR parameters.
I haven’t had a need for this yet. We’ll see if that changes in the future.
Coming up
In the next episode, I’ll have a look at the datasheet specifications of the microphone, and use that to determine the specification of our PDM to PCM design.
My Blog Posts in this Series
Filter Design | {"url":"https://tomverbeure.github.io/2020/10/11/Designing-Generic-FIR-Filters-with-pyFDA-and-Numpy.html","timestamp":"2024-11-07T07:13:18Z","content_type":"text/html","content_length":"35637","record_id":"<urn:uuid:a0481cdc-75de-4222-a729-f4a3a94c0fde>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00817.warc.gz"} |
Tips for Assignment 3
In class we discussed the setup of various Coulomb integrals that show up on homework 3. What about problems 5 and 6, involving Gauss's Law? Here are a few tips to help if you are stuck.
Problem 5
The charge distribution is spherically symmetric because you can completely describe the charge found at any point using only its distance from one special point: the center of the sphere. This means
the electric field has the form \(E(r)\,\hat{r{}}\) like the example from class, and your Gaussian surfaces should be spheres with variable radius \(r\).
There are three cases to consider for the radius of the Gaussian surface: \(0 \leq r < R_{i}\), \(R_{i} < r < R_{o}\), and \(r > R_{o}\). When determining the enclosed charge you will need to
integrate over the volume inside your Gaussian surface. I will refer to points inside the surface with coordinates \(r'\), \(\theta'\), and \(\phi'\). Then the volume element is \(d\tau' = (r')^2 \
sin \theta' dr' d\theta' d\phi'\) and $$q_{enc} = \oint_{GS} \!\!\!d\tau' \rho(\,\vec{r}{\,'}\,) = \int_{0}^{2\pi} \!\!\! d\phi' \int_{0}^{\pi} \!\!\!d\theta' \sin \theta' \int_{0}^{r} \!\!\!dr' (r')
^2 \rho(\,r'\,)$$ Be careful! Depending on the value of \(r\), you might have to break the integral over \(r'\) into two or more parts. For instance, if \( R_{i} < r < R_{o} \) then \( \rho(r') = 0
\) for \( 0 \leq r' < R_i \) and \( \rho(r') = k / r' \) for \( R_{i} < r^{\prime} < r \) .
Problem 6
This charge distribution has planar symmetry, because we only need to know a point's distance \(z\) from the \(x\)-\(y\) plane to describe the charge found there. If \(|z| < d\) then there is a
constant volume charge density at that point, and if \(|z| > d\) then there is no charge at that point. We can conclude that the electric field has the form \(E(z) \hat{z{}}\).
For your Gaussian surface consider a rectangular box with its bottom at \( z_1 \), its top at \( z_2 \), and an area \(A\) on the top and bottom faces. (Why don't the areas of the other four faces
matter here?) Consider the the flux and enclosed charge for the cases \( z_{1} < z_{2} < d \), \(z_1 < d < z_2\), and \( d < z_1 < z_2 \). Remember that the normal vector for a closed surface points
from inside to outside, so in this case it is \(+\hat{z{}}\) on the top of the Gaussian surface and \(-\hat{z{}}\) on the bottom.
You probably assumed \( 0 < z_1 < z_2 \) as you did this; what should the field look like for negative values of \(z\)? That is, based on the symmetry here, what do you expect for the field at the
point \((0,0,-z)\) compared to \((0,0,z)\)? Does this help you narrow down anything about the field that wasn't clear from the three cases above? | {"url":"http://jacobi.luc.edu/hw3tips.html","timestamp":"2024-11-14T08:26:09Z","content_type":"application/xhtml+xml","content_length":"6753","record_id":"<urn:uuid:898be0d6-59d2-4480-9914-92b707849869>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00373.warc.gz"} |
#2 game dev stuff | binary space partitioning
In this blog post, we’ll explore how to implement a binary space partitioning algorithm for procedural dungeon generation like in the following image.
How did you do it?
The first step in implementing a BSP algorithm is to define the data structures that we’ll use to represent the dungeon.
class Area
Area parent
Area leftChild
Area rightChild
int x, y, width, height
class Room
Area parent
int x, y, width, height
Once we have our data structures defined, we can begin the BSP algorithm. The first step is to create a root node that represents the entire dungeon space.
root_area = Area(x, y, width, height)
bsp(root_area, 10)
All that’s left is the basic logic for the bsp() algorithm itself which is fairly easy as well.
def bsp(area, iterations):
stop if iterations == 0
if random_value >= .5 split horizontal:
split_value = any value from area.y to area.y + area.height
area.leftChild = Area(area.x, area.y, area.width, split_value)
area.rigthChild = Area(area.x, split_value, area.width, area.height - split_value)
else split vertical
split_value = any value from area.x to area.x + width
area.leftChild = Area(area.x, area.y, split_value, area.height)
area.rigthChild = Area(area.x, split_value, area.width - split_value, area.height)
bsp(area.leftChild, iterations - 1)
bsp(area.rightChild, iterations - 1)
The bsp functions takes a node and the number of iterations until the algorithm should stop splitting the areas any further.
Afterwards one just has to iterate over all areas which do not have any children. Theses are the trees leaves and this is where we want to place our rooms.
Rooms can be created with any rule but I sticked with a rather simple approach that takes a random value of its area’s width and height.
Bottom Line
Creating procedural dungeons can be fairly easy and each approach offers some nice way to customize it further making each a versatile tool for different scenarios. | {"url":"https://thebaite.com/gamedevstuff2/","timestamp":"2024-11-02T23:48:44Z","content_type":"text/html","content_length":"53710","record_id":"<urn:uuid:4f03e000-27fb-4ae3-9ec9-4878e2b9950e>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00064.warc.gz"} |
WSP-Hash-OAAT: A Fast,
WSP-Hash-OAAT: A Fast, Lightweight, Non-Multiplicative, One-at-a-Time 32-Bit Hashing Algorithm That Passes Rigorous Collision Tests
William Stafford Parsons developed a tiny 32-bit OAAT hashing algorithm as a substantial improvement to 32-bit FNV-1a and MicroOAAT.
#include <stdint.h> struct wsp_hash_oaat_s { uint32_t _state; uint32_t state; }; void wsp_hash_oaat_initialize(struct wsp_hash_oaat_s *s) { s->_state = 1; s->state = 1111111111; } void
wsp_hash_oaat_transform(unsigned long i, unsigned long input_count, const uint8_t *input, struct wsp_hash_oaat_s *s) { while (i != input_count) { s->state ^= input[i]; s->state += s->state << 3; s->
_state += s->state; s->_state = (s->_state << 27) | (s->_state >> 5); i++; } } void wsp_hash_oaat_finalize(struct wsp_hash_oaat_s *s) { s->state ^= s->_state; s->state = (s->_state ^ s->state) +
((s->state << 10) | (s->state >> 22)); s->state += (s->_state << 27) | (s->_state >> 5); }
wsp_hash_oaat_initialize() is the initialization function that accepts the following argument.
s is a struct wsp_hash_oaat_s pointer.
wsp_hash_oaat_transform() is the core hashing loop that accepts the 4 following arguments.
i is the starting index position of elements in the input array.
input_count is the count of elements in the input array.
input is the const uint8_t array to hash.
s is a struct wsp_hash_oaat_s pointer.
wsp_hash_oaat_finalize() is the finalization function that accepts the following argument.
s is a struct wsp_hash_oaat_s pointer. s.state contains the finalized hash digest result.
The return value data type is void.
C compiler with C99 (ISO/IEC 9899:1999) standard compatibility.
CPU with single-threaded, instruction-level parallelism support.
This 32-bit one-at-a-time hashing algorithm is designed to hash keys in data structures in place of existing 32-bit FNV-1a and MicroOAAT implementations.
It's a lossless performance and statistical quality improvement for all input sizes larger than 3 bytes.
Input sizes less than or equal to 3 bytes are slower by a small margin with higher statistical quality.
It's portable for both 32-bit and 64-bit systems.
It meets strict compliance, portability and security requirements.
It doesn't use modulus, multiplication or division arithmetic operations.
Seed tests are omitted to discourage using one-at-a-time hashing algorithms as PRNGs and to prevent collision vulnerabilities from 2³² different initialized states.
The core hashing loop structure appears similar to MicroOAAT due to the limited options when designing the fastest non-multiplicative, one-at-a-time hashing function with good collision avoidance,
but the technical difference is vast.
It uses a XOR assignment operation for each input byte instead of an addition assignment, similar to FNV-1a. The remaining operations emulate multiplication before finalizing with enhanced bit
Any bitwise multiplication with a left shift operand greater than << 3 is too slow in the core hashing loop and anything less doesn't result in enough distribution to omit a second bitwise
It left-rotates bits in _state instead of right-rotating bits in state.
_state combines with state using addition instead of subtraction to keep bits rolling in the same direction.
Furthermore, the bit rotation is non-blocking with instruction-level parallelism for each next byte iteration.
The speed gains from these design differences allow for a minimal finalization sequence with finely-tuned shift values and additive bit rotations in both directions.
The result deprecates the usage of 32-bit FNV-1a and MicroOAAT by comparison in most practical instances.
There are significant improvements across all SMHasher tests with no critical collision vulnerabilities relative to the compared hashing algorithms.
Compared to 32-bit FNV-1a, it's 53% faster on average for small inputs and up to 70% faster on average for large inputs.
Compared to MicroOAAT, it's 13% faster on average for small inputs and up to 45% faster on average for large inputs.
It's a practical alternative to Jenkin's OAAT and Murmur OAAT as well with significant collision and speed improvements similar to the aforementioned comparisons.
As with the other hashing functions in SMHasher, it was tested with the following all-at-once hashing function instead of the slower segmented functions from the aforementioned library code.
#include <stdint.h> uint32_t wsp_hash_oaat(unsigned long input_count, const uint8_t *input) { uint32_t _state = 1; uint32_t state = 1111111111; unsigned long i = 0; while (i != input_count) { state ^
= input[i]; state += state << 3; _state += state; _state = (_state << 27) | (_state >> 5); i++; } state ^= _state; state = (_state ^ state) + ((state << 10) | (state >> 22)); return ((_state << 27) |
(_state >> 5)) + state; }
All speed tests were performed locally on a Pixelbook Go M3 using Debian.
Spawn into the hostile quantum laboratory and destroy wave after wave of oscillations. | {"url":"https://williamstaffordparsons.github.io/wsp-hash-oaat/","timestamp":"2024-11-10T08:13:19Z","content_type":"text/html","content_length":"11679","record_id":"<urn:uuid:8afe6573-da53-43f2-acc7-8315947edd2c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00826.warc.gz"} |
Honors Physics: Gravitation #2 Name:_______________________ Objectives: P3.1b, 3.1A, P3.6A,B,C,d Directions: Please show knowns, formula and solutions for full credit. 1. The gravitational force
between two electrons 1m apart is 5.42 E -71n. Find the mass of an electron. Knowns formula Solution ______________ 2. What would be the earth?s gravitational attraction(g) on a 75kg astronaut who is
one earth radius above the earth?s surface? Knowns formula Solution ______________ 3. If the mass of Mars is 6.6E23kg and its gravity is 3.7m/s/s, what is the radius of Mars? Knowns formula Solution
______________ 4. A 25 kg object is 201 km above the earth?s surface. Find Knowns formula Solution | {"url":"https://course-notes.org/taxonomy/term/1061745","timestamp":"2024-11-02T05:22:14Z","content_type":"text/html","content_length":"52721","record_id":"<urn:uuid:5db2b1b6-05f2-4962-ada2-fb3dd85ba6c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00843.warc.gz"} |
[download] Final B.Sc Mathematics CBCS Syllabus 2018 CU | New B.Sc Mathematics Hons. & General syllabus better-known as CBCS Syllabus - University of Calcutta - CUMATHS.COM
Calcutta University B.Sc Mathematics (Honours & General) CBCS Syllabus
Calcutta University is set to introduce semester exams and choice-based-credit-system (CBCS) in the undergraduate B.Sc Maths Honours & General . The students taking admissions in colleges affiliated
to CU, commencing from 2018-2019 academic year in the UG courses will be registered under the new CBCS format & examinations will be evaluated on CBCS method.
Exam and grade points under B.Sc Mathematics CBCS Syllabus
The examination will now be on grade points and the student score sheet will be on 10 point grade scale. No marks will be mentioned. Combining all semesters an honours student has to appear for 140
credit points in three years across 6 semesters. In general, a student needs to appear on 120 credit points. To graduate, in six semesters an honours candidate needs to secure CGPA 4 and general
student needs to bag 3 CGPA.
CBCS rules for B.Sc Mathematics CBCS Syllabus (Hons & Pass) under Calcutta University
We provide ONLINE & CLASSROOM coaching classes for B.Sc Mathematics Hons. & Pass 1st, 2nd and 3rd year, SEMESTER-I to SEMESTER-VI, students of Calcutta University (CU) . B.Sc Mathematics CBCS
classes are taken are taken by Prof. Sudipta Ma’am & Prof. Biswadip Sir., Ramakrishna Mission Residential College (Autonomous), Narendrapur.
Join Prof. Biswadip Sir’s B.Sc Mathematics Classes to achieve best result in B.Sc Mathematics.
Undergraduate Mathematics course - under B.Sc CBCS Maths Syllabus
B.Sc. (Hons.) Mathematics or Bachelor of Science (Honours) in Mathematics is an undergraduate Mathematics course. Mathematics is the study of quantity, structure, space, and change. For the reason
that the mathematics program provides a deep insight on geometry, trigonometry, calculus and various other theories in Mathematics or related disciplines, such as Computer science or Statistics.
Furthermore the study of the normal Bachelor of Science subjects likewise Physics and Chemistry, they also have maths as pass subject. Furthermore, the duration of the course is three years and for
the reason that the syllabus for the course is divided into either six semesters over the three-year course.
Semester: 1
In this semester, there are two CC papers, B.Sc Mathematics CBCS Syllabus, each of 100 marks (65+15**+20***=100) & of credits 6 (5+1*=6).
Core Course-1: Calculus, Geometry & Vector Analysis
*1 Credit for Tutorial
**15 Marks are reserved for Tutorial
***20 Marks are reserved for Internal Assessmen & Attendance (10 marks for each)
Semester: 2
Semester: 3
Semester: 4
Semester: 5
Semester: 6 | {"url":"https://cumaths.com/bsc-mathematics-cbcs-syllabus-cu/","timestamp":"2024-11-07T18:15:59Z","content_type":"text/html","content_length":"91547","record_id":"<urn:uuid:b7700753-eede-4534-8634-a414d16ab612>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00130.warc.gz"} |
How do you integrate (2x-1)/((x+1)(x-2)(x+3)) using partial fractions? | HIX Tutor
How do you integrate #(2x-1)/((x+1)(x-2)(x+3))# using partial fractions?
Answer 1
$- \frac{1}{4} \ln \left(x + 1\right) - \ln \left(x - 2\right) + \frac{5}{4} \ln \left(x - 3\right) + c$
Before integration, partial fractions can be done as explained below
The partial fractions would thus be
#-1/(4(x+1)) -1/(x-2)+5/(4(x-3)#
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Answer 2
To integrate ( \frac{2x - 1}{(x + 1)(x - 2)(x + 3)} ) using partial fractions, follow these steps:
1. First, express the rational function as the sum of partial fractions: [ \frac{2x - 1}{(x + 1)(x - 2)(x + 3)} = \frac{A}{x + 1} + \frac{B}{x - 2} + \frac{C}{x + 3} ]
2. Multiply both sides by the denominator ( (x + 1)(x - 2)(x + 3) ) to clear the fractions: [ 2x - 1 = A(x - 2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x - 2) ]
3. Expand and equate coefficients of like terms. This will give you a system of linear equations to solve for ( A ), ( B ), and ( C ).
4. Once you find the values of ( A ), ( B ), and ( C ), rewrite the original function using these partial fractions.
5. Integrate each partial fraction separately.
6. Finally, add the integrated partial fractions together to obtain the final result.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Notes - Groups HT23, Special groups
What is the “general linear” group $\text{GL} _ n(\mathbb R)$?
The group of all invertible $n \times n$ matrices.
What is the “special linear” group $\text{SL} _ n(\mathbb R)$?
The group of all invertible $n \times n$ matrices with determinant $1$.
What is the “orthogonal” group $\text O (\mathbb R)$?
The group of all orthogonal $n \times n$ matrices.
What is the “special orthogonal” group $\text {SO} _ n(\mathbb R)$?
The group of all orthogonal $n \times n$ matrices with determinant $1$.
What is the dihedral group $D _ {2n}$?
The group of isometries under composition of a regular $n$-gon in the plane.
Can you list all the elements of the dihedral group $D _ 8$?
\[\\{e, \rho, \rho^2, \rho^3, s, \rho s, \rho^2 s, \rho^3 s\\}\]
What is the order of the group containing all the isometries of a regular $n$-gon in the plane (the dihedral group)?
What is the group $Q _ 8$?
The quaternion group
\[\\{\pm 1, \pm \pmb i, \pm \pmb j, \pm \pmb k \\}\]
Prove that $D _ {2n}$ has $2n$ elements, namely
\[\\{e, \rho, \cdots, \rho^n, s, \rho s, \cdots, \rho^{n-1} s\\}\]
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[Solved] A farmer was having a field in the form of a parallelo... | Filo
A farmer was having a field in the form of a parallelogram . She took any point A on and joined it to points and . In how many parts the fields is divided? What are the shapes of these parts? The
farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?
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+ Ask your question
From the figure, it can be observed that point A divides the field into three parts. These parts are triangular in shape -, and
Area of PSA + Area of P A Q + Area of Q R A = Area of PQRS ... (1)
We know that if a parallelogram and a triangle are on the same base and between the same parallels, then the area of the triangle is half the area of the parallelogram.
Area Area (PQRS) ... (2)
From equations (1) and (2), we obtain
Area Area Area (PQRS) ... (3)
Clearly, it can be observed that the farmer must sow wheat in triangular part PAQ and pulses in other two triangular parts PSA and QRA or wheat in triangular parts PSA and QRA and pulses in
triangular parts PAQ.
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A farmer was having a field in the form of a parallelogram . She took any point A on and joined it to points and . In how many parts the fields is divided? What are the shapes of these
Question parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?
Topic Areas of Parallelograms and Triangles
Subject Mathematics
Class Class 9
Answer Text solution:1
Upvotes 77 | {"url":"https://askfilo.com/math-question-answers/a-farmer-was-having-a-field-in-the-form-of-a-paralgwa","timestamp":"2024-11-08T21:15:24Z","content_type":"text/html","content_length":"326169","record_id":"<urn:uuid:935d8e3f-c9b8-4407-9026-ce591466dc28>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00680.warc.gz"} |
Descending Dungeons and Iterated Base-Changing
• Published in 2006
• Added on
In the collections
For real numbers a, b> 1, let as a_b denote the result of interpreting a in base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to be numbers of the form a_b_c_d_..._e,
parenthesized either from the bottom upwards (preferred) or from the top downwards. Among other things, we show that the sequences of dungeons with n-th terms 10_11_12_..._(n-1)_n or n_(n-1)
_..._12_11_10 grow roughly like 10^{10^{n log log n}}, where the logarithms are to the base 10. We also investigate the behavior as n increases of the sequence a_a_a_..._a, with n a's,
parenthesized from the bottom upwards. This converges either to a single number (e.g. to the golden ratio if a = 1.1), to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a =
Other information
BibTeX entry
key = {DescendingDungeonsandIteratedBaseChanging},
type = {article},
title = {Descending Dungeons and Iterated Base-Changing},
author = {David Applegate and Marc LeBrun and N. J. A. Sloane},
abstract = {For real numbers a, b> 1, let as a{\_}b denote the result of interpreting a in
base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to
be numbers of the form a{\_}b{\_}c{\_}d{\_}...{\_}e, parenthesized either from the bottom
upwards (preferred) or from the top downwards. Among other things, we show that
the sequences of dungeons with n-th terms 10{\_}11{\_}12{\_}...{\_}(n-1){\_}n or
n{\_}(n-1){\_}...{\_}12{\_}11{\_}10 grow roughly like 10^{\{}10^{\{}n log log n{\}}{\}}, where the
logarithms are to the base 10. We also investigate the behavior as n increases
of the sequence a{\_}a{\_}a{\_}...{\_}a, with n a's, parenthesized from the bottom upwards.
This converges either to a single number (e.g. to the golden ratio if a = 1.1),
to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a =
comment = {},
date_added = {2022-02-28},
date_published = {2006-10-09},
urls = {http://arxiv.org/abs/math/0611293v3,http://arxiv.org/pdf/math/0611293v3},
collections = {attention-grabbing-titles,easily-explained,fun-maths-facts,integerology},
url = {http://arxiv.org/abs/math/0611293v3 http://arxiv.org/pdf/math/0611293v3},
year = 2006,
urldate = {2022-02-28},
archivePrefix = {arXiv},
eprint = {math/0611293},
primaryClass = {math.NT} | {"url":"https://read.somethingorotherwhatever.com/entry/DescendingDungeonsandIteratedBaseChanging","timestamp":"2024-11-08T18:52:55Z","content_type":"text/html","content_length":"6872","record_id":"<urn:uuid:482600e9-d97e-4100-9ce8-08acd413c89d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00754.warc.gz"} |
Entanglement of periodic states, the quantum fourier transform, and shor's factoring algorithm
The preprocessing stage of Shor's algorithm generates a class of quantum states referred to as periodic states, on which the quantum Fourier transform is applied. Such states also play an important
role in other quantum algorithms that rely on the quantum Fourier transform. Since entanglement is believed to be a necessary resource for quantum computational speedup, we analyze the entanglement
of periodic states and the way it is affected by the quantum Fourier transform. To this end, we derive a formula that evaluates the Groverian entanglement measure for periodic states. Using this
formula, we explain the surprising result that the Groverian entanglement of the periodic states built up during the preprocessing stage is only slightly affected by the quantum Fourier transform.
Dive into the research topics of 'Entanglement of periodic states, the quantum fourier transform, and shor's factoring algorithm'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/entanglement-of-periodic-states-the-quantum-fourier-transform-and","timestamp":"2024-11-02T19:12:14Z","content_type":"text/html","content_length":"47396","record_id":"<urn:uuid:016816ca-4848-44c0-83fe-854ff467ae18>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00824.warc.gz"} |
Arrow: Criticism and Defence
A while ago I
promised a reply
to John Lawrence's
Criticisms of Arrow
. Here it is, point-by-point. (Hope you don't mind me copying them John - if you do leave a comment and I'll edit your bit out)
1) Arrow doesn’t allow ties. Example: For alternatives a, b and c, the only acceptable social choice for Arrow is one of the following: abc, acb, bac, bca, cab, cba. A tie would be of the form {abc,
bca} where the parentheses indicate that the two social rankings abc and bca are tied.
I don’t think Arrow disallows ties. See below, but ‘indifference’ counts as a decision. Of course, trying to choose between policies x and y, knowing society is indifferent between them doesn’t much
help. Perhaps we can resolve indifference by tossing a coin, as many have suggested for tied elections. Technically that doesn’t fit Arrow’s general project, as it only breaks deadlock, it doesn’t
produce a social ranking – but then, I think he’s wrong to require such anyway.
2) Arrow assumes that a tie is the same as an indifference. Let’s assume 2 alternatives: x and y. Also that half of the individuals in society prefer x to y – xPiy for half – and half the individuals
prefer y to x – yPix. Pi represents the preference ordering of the ith individual. Arrow would say that society is indifferent between the 2 alternatives – xIy. However, society is not indifferent
between the 2 alternatives. Society is evenly divided between the 2 alternatives. Half prefer x to y and, let us assume, very passionately. Half prefer y to x and, let us assume, equally as
passionately. Society would be indifferent if every individual was indifferent between the 2 alternatives.
Arrow doesn’t just ‘assume’ this. There’s an argument p.50 ff. intended to establish that if x and y tie, yet we take one to be preferred, we get inconsistent results. (The argument was too technical
to follow easily, let alone recall or reproduce here, but believe me there is one)
Of course, this doesn’t quite answer the challenge, as the criticism isn’t that either x or y should be preferred – it’s that there’s a fourth possibility: xPy, yPx, xIy and xTy where the T relation
stands for ‘ties with’.
I think there is something to this, but note there’s now a problem. Jonathan says “Society would be indifferent if every individual was indifferent between the 2 alternatives” but it isn’t clear if
this is merely sufficient or also necessary. What if 99% were indifferent and the 1% evenly split each way? Surely that’s also social indifference rather than a tie. What if the 1% mostly support x?
Is the social choice now xPy or xIy? The problem is that introducing this extra possibility means we’re now longer dealing with a binary x or y choice, because tying is (unlike indifference) a third
3) According to Arrow the only information an individual may specify is pairwise binary comparisons. Any other information is irrelevant. However, there is no reason why an individual shouldn’t
specify as much information as possible. For instance, if x, y and z are the alternatives, Arrow would say that the only relevant information is in comparing x to y, x to z and y to z. It is easy to
imagine a grid with values from 1 to 100, for example. An individual could place x, y and z on this grid in any position. This would convey more information than binary comparisons Why shouldn’t each
individual have unlimited freedom of expression?
If you can’t compare interpersonal utility, the problem is this ‘extra information’ is meaningless. I may rank a, b and c as 1, 50 and 100 and you may rank them 1, 99, 100, but this doesn’t mean
anything. We can’t conclude I prefer b more than you do, so why allow voters to express something that has no meaning?
4) There is no such thing as an irrelevant alternative. The number of alternatives determines the underlying grid. If that grid changes due to the death of one of the candidates, for instance, the
individual’s preferences, if he is required to specify them within a grid determined by the number of candidates, may change. Preferences may become indifferences and vice versa.
This is to some extent a fundamental disagreement, too close to premises to resolve, and one where John essentially sides with Borda against Condorcet and Arrow. It doesn’t seem obviously wrong to
say all that matters in choice between x and y is the relative ranking of x and y, however. Why compare them to non-option z, particularly in light of the above point that it doesn’t even allow some
kind of interpersonal comparison?
The death of a candidate is actually a slightly different case. Arrow himself confuses independence of irrelevant alternatives with contraction consistency, which is related but can be distinguished.
Still, suppose you rank aPbIcPd then c dies. It seems natural to suppose the remaining preferences simply carry over – aPbPd – unless some good reason is given why they might change
5) Since there is no such thing as an irrelevant alternative, one of Arrow’s “rational and ethical” criteria is invalidated. Therefore, his entire analysis is invalidated.
This is rather hasty if 4) hasn’t been sufficiently proven…
6) We don’t know how an individual would vote if one candidate died, for example. If the number of candidates changes, the individuals must be repolled or else some probabilistic assumptions would
have to be made about how they would have voted. You can’t just assume that because an individual voted aPbIc, if a dropped out, the individual would still be indifferent between b and c. For
example, let us assume that there are 3 candidates, a, b, c and Hitler. A voter might vote aIbIcPHitler with the rationale that any candidate would be preferred to Hitler. Then, if Hitler dropped
out, true preferences among a, b and c might emerge.
This rather repeats 4). If there are true preferences between a, b and c, why hsouldn’t they emerge before Hitler’s death? Thus if the individual would want to vote aPbPc without Hitler in the frame,
they should want to vote aPbPcPHitler with Hitler, preserving “the rationale that any candidate would be preferred to Hitler”.
7) An individual shouldn’t be prevented or constrained from using his vote in a strategic way. In any rational voting system there is what Arrow calls the “Positive Association of Social and
Individual Values.” Therefore, an individual will express preferences based on the candidate set in order to prevent a particular individual from being elected or to help guarantee that another
individual will be elected if he feels that strongly about some particular candidate. The individual shouldn’t be required, as Arrow does, to vote consistently when the candidate set changes. For
example, let’s assume that the candidate set consists of candidates a, b, c and Jesus. It would be an entirely rational vote to specify (Jesus)PaIbIc. Compared to Jesus, the individual is indifferent
among a, b and c. Now let us suppose that Jesus drops out of the race. Then it would be entirely rational for the individual to vote aPbPc. In other words, the voter voted strategically to get Jesus
elected and should be allowed to do so. If Jesus is not in the race, the individual’s true preferences among a, b and c emerge. Arrow’s condition, “Independence of Irrelevant Alternatives” is not
rational after all.
Again, this criticises 4), but explicitly claiming the individual should be allowed to vote strategically. I’m not sure about that. Sometimes I think strategic voting is defensible, so long as all
have the same opportunities to use their votes, it’s up to people how they do so. Certainly no argument for strategic voting is given here, however, so nothing to convince anyone who already believes
it’s wrong.
8) “Adopting an informational perspective, then, [Arrow’s theorem] just state[s] that procedures for three or more candidates require more information than just the relative rankings of pairs.”
Saari, DG, (1995), Basic Geometry of Voting, Springer-Verlag, Berlin.
I don’t see the point being made here.
9) Rankings have been considered to be cardinal or ordinal where ordinal represents a simple ranking and cardinal allows more information. Cardinal rankings supposedly allow “preference intensity” to
be represented. It’s not about preference intensity; it’s about freedom of expression. Let’s add a third type of ranking: digital. An ordinal comparison for 3 candidates can be specified by 3 bits
since there are 6 possibilities. If we allow more bits, then more information and relevant information can be gleaned from each individual. Allowing each individual to specify his or her preferences
using the same number of bits eliminates the “interpersonal comparisons of utility,” another of Arrow’s bugaboos. There is no preference given to one individual over another because of supposed
greater need. Therefore, allowing more than just ordinal information is just as impersonal as allowing only ordinal information.
This claims to be like 3) but without making interpersonal comparisons, however I don’t see what is supposedly expressed. How about another box on the ballot allowing voters to express their
favourite flavour of ice cream? Surely that’s also more expression and therefore better (though I don’t see how it should influence the decision in question).
10) Arrow constrains freedom of expression.
See 3) and 9). I don’t think Arrow constrains ‘free expression’ in any troubling sense – he wouldn’t repeal the First amendment, for example – all he restricts is what influences social choice.
11) Arrow confuses ties and indifference in the binary case in which there are 2 alternatives and n voters. See my paper, "Neutrality and the Possibility of Social Choice" He says majority rule when
there are only 2 alternatives is the only case where social choice actually works. But according to his analysis, if done correctly, social choice isn’t even possible with 2 alternatives. The key
point is that when the number of voters who prefer x to y, N(x,y), equals the number of voters who prefer y to x, N(y,x), you have a tie between the solutions xPy and yPx which I indicate {xPy, yPx}.
This is not the same as xIy, x is indifferent to y.
This is basically point 2) repeated. (Again, I suggest tossing a coin to resolve ties).
12) Arrow violates the Principle of Neutrality in his analysis of binary majority rule. The Principle of Neutrality states that each alternative (in this case x and y) must be treated in the same
way. No alternative may be given preferential treatment.
How does Arrow violate Neutrality then? Neither x nor y is given preferential treatment (unless ties favour status quo, which is slightly non-neutral). But see 16).
13) In the binary case, Arrow assumes that a tie in the domain of individual votes implies a social indifference. The domain consists of all possible combinations of votes by the individual voters.
The range consists of all the possible choices by society as a whole i.e. social choices.
I’m not sure what point is being made here, but since it concerns ties again I suggest it’s much the same as 2) and 11).
14) Arrow states (p. 12, 13 of “Social Choice and Individual Values”): “A strong ordering…is a ranking in which no ties are possible.” WRONG! If n/2 voters prefer y to x and n/2 voters prefer x to y
(n being even), this clearly is a tie! This section clearly shows Arrow’s confusion between the concept of a tie and the concept of indifference. He thinks that both xRy and yRx imply a tie. Wrong
again. They imply an indifference.
A strong ordering is one where no ties are possible. That’s definitional. If voters are evenly split, you get a tie, but that’s because the result isn’t a strong ordering! As for the tie/indifference
distinction, see 2) and 11) – this increasingly seems John’s main, perhaps only, complaint.
15) Arrow’s R notation. On p. 12 of “Social Choice and Individual Values,” Arrow states: “Preference and indifference are relations between alternatives. Instead of working with two relations, it
will be slightly more convenient to use a single relation, ‘preferred or indifferent.’ The statement ‘x is preferred or indifferent to y’ will be symbolized by xRy.” Emphasis added. Slightly more
convenient? Ridiculous. What voter votes in such a way that they are “preferred or indifferent” between x and y. What would be the meaning? Maybe I prefer x to y or maybe I’m indifferent? I don’t
know which? The net result is that the voter is constrained to make choices of this nature when he damn well knows he prefers x to y or he damn well knows he is indifferent between x and y.
Heuristically, the R notation is nonsense. If I’m going to list my preferences, I can do so unambiguously using P and I. For example, aPbIcPd would indicate that I prefer a to b, c and d; I’m
indifferent between b and c; and I prefer a, b and c to d. See "Arrow's Consideration of Ties and Indifference"
It is nowhere implied that voters don’t know whether xPy or xIy. The R notation is simply a representational device, equivalent to ‘equal to or great than’. It’s true all Rs can be cached out in
terms of P and/or I, but perhaps sometimes we don’t know which so to write ‘xPy or xIy’ is more troublesome than simply ‘xRy’. Conversely, nothing is lost by using R notation. We can express xPy by
‘xRy and not yRx’ and xIy by ‘xRy and yRx’ (the ties/indifference problem not withstanding). The R relation allows us to represent everything we want – great than, equal to, and equal-or-great –
using a single formula. In that respect, it’s more flexible and economical. Granted it doesn’t seem particularly necessary, but it’s far from absurd, and doesn’t have the implications John tries to
16) Definition 9 (p. 46) The case of two alternatives. “By the method of majority decision is meant the social welfare function in which xRy holds if and only if the number of individuals such that
xRiy is at least as great as the number of individuals such that yRix.” This is totally ridiculous. First of all it violates one of Arrow’s five “rational and ethical principles” which all social
welfare functions must comply with: the principle of neutrality. When the number of individuals such that xRiy equals the number of individuals such that yRix, why is the solution xRy? Why not yRx
which is equally as valid? In fact it is a tie between xRy and yRx, or according to Arrow’s own terminology, when xRy and yRx, then xIy not xRy! But wait, there is more. If half the individuals
prefer x to y and half prefer y to x, we have a tie between x and y: {xPy, yPx}. If all the individuals are indifferent between x and y, we have a societal indifference: xIy. These are not the same
thing! If more prefer x to y than prefer y to x, we have xPy and vice versa. If some individuals are indifferent between x and y, but more prefer x to y than prefer y to x, we have xPy and vice
versa. This pretty well covers all the cases. Arrow is determined to ignore the significance of a tie and to turn a societal indifference into a tie. See "Arrow's Consideration of Ties and
This seems to explain what was meant by 12), by repeating and elaborating the claim majority rule is non-neutral. It is, however, mistaken. Jonathan asks why xRy and not yRx. It is also yRx. As
states in the previous section, xRy and yRx can hold together and imply xIy. Thus Jonathan either has no point, or it’s merely the ties/indifference thing again (see 2), 11), etc).
17) Arrow defines an indifference as a tie.
Yup, see comments on 2), 11) and 16).
It seems John’s only serious criticism is to treating ties and indifference as equivalent. It’s true, this does worry me as well. I wondered why, for May too, a split 4/993/3 between xPy, xIy and yPx
results in a social choice xPy. It hardly seems intuitive but, as I said above, to introduce indifference as if it was a third option takes us away from the binary choices being dealt with.
There does seem to be a difference between everyone’s indifference and an even split between x and y – though in both cases I might suggest tossing a coin for decision-making, albeit for slightly
different reasons (in the first simply to make a decision, as an individual might do in ‘Buridan’s ass’ like cases; in the second to be fair to everyone, i.e. give them an equal chance of getting
what they want).
I don’t know what else to say about this issue. There’s certainly room for many whole papers on the topic. I think, however, it’s all these criticisms boil down to. (Plus some unsubstantiated
assertions of rights to ‘free expression’ and ‘strategic voting’) – So it’s far from clear to me that Arrow is refuted, and the presentation of ‘17 criticisms’ significantly overstates the objection
to make the case look stronger than it is.
8 comments:
1. Anonymous4:35 am
Thanks, Ben, to giving serious considerations to my comments. It's true there is a fair degree of repetition in a number of the points I made.
I'll just consider one of your responses tonight. Hopefully, I'll get to some of the others later.
You say:
"If you can’t compare interpersonal utility, the problem is this ‘extra information’ is meaningless. I may rank a, b and c as 1, 50 and 100 and you may rank them 1, 99, 100, but this doesn’t mean
anything. We can’t conclude I prefer b more than you do, so why allow voters to express something that has no meaning?"
I say you don't need to compare interpersonal utility. I don't think there's a way to do it or a need to do it. Using the modified Borda count (with apology I call the Lawrence count: http://
www.socialchoiceandbeyond.com - click on Lawrence count), for instance, you would just be giving 49 more points to b than I would thereby increasing b's chances of election somewhat. By my
ranking b 99 and c 100, I am using my vote strategically to do everything I can to get "a" elected or these may represent my true feelings among the candidates. It doesn't make any difference as
long as neither one of us has any relative advantage in the voting process.
You have a more favorable view of b than I do and would much prefer b to c while I am almost indifferent between b and c. Whether or not you like "a" more than I do or dislike c more than I do is
irrelevant since neither you nor I can increase the point spread between any 2 candidates by more than 99 points. Therefore, we have an equal vote irregardless of whether or not you feel more
strongly than I do.
2. Anonymous6:50 pm
Well, today, Ben, I'll respond to your response to my point #1. You say "I don’t think Arrow disallows ties." I beg to differ with you and offer the following argument.
Let's say there are 6 candidates standing for election: X1, X2,...,X6. The voters cast one vote for their favorite candidate and the candidate with the most votes wins. Of course, there are other
methods one could use depending on whether a candidate gets a majority etc., but let's say in this simple method the one with the most votes wins. A legitimate voting method, no?
Now let's consider the case in which 2 or more of the candidates get the same number of votes. Then we have a tie among 2 or more candidates. In the extreme case, if all candidates receive the
same number of votes, we could have a six way tie. What to do about such a situation is not our concern for the present. We merely note that this simple election method could indicate clearly
whether (a) a candidate is a winner or (b) there is a tie among 2 or more candidates.
Now consider that each candidate is actually a preference list. We identify X1 with aRbRc; X2 with aRcRb; and so on: X6 with cRbRa.
3. Anonymous7:16 pm
I guess I'm constrained by the number of words per comment. Oh well, I'll continue on. Hope you don't mind. So each candidate is a preference ordering among 3 alternatives and each voter votes
for one of those preference orderings. By Arrow's Definition 4 (p. 23, "Social Choice and Individual Values") the social welfare function (SWF) must provide ONE social ordering of the form xRyRz
for each possible set of individual orderings.
Therefore, in Arrow's formulation of the problem, there is no way to account for a tie among 2 or more of the candidates where the candidates are indeed preference orderings. What Arrow does
offer is indifferences among alternatives (by virtue of the "R" operator) not ties among orderings.
#2) (I guess it is going to allow me a comment of any length after all.)
Consider the case of 3 alternatives: a, b, c. Let us say there are 3 voters and they vote, respectively: abc, bca, cab. Now if we identify these orderings with candidates X1, X2 and X3,
respectively, we have a 3 way tie among X1, X2 and X3. Well, no problem for our simple voting system. We could notate a 3 way tie as: {abc, bca, cab}. Big problem for Arrow. He cannot accomodate
a simple tie situation due to his Definition 4!!
Isn't it rational that the above situation should lead to a 3 way tie? Let's call this "Rational Fact #1. now let's say c drops out of the race, and we're left with the votes ab, ba and ab. The
winning social ordering would be ab, right? Let's call this "Rational Fact #2."
These two facts taken together invalidate Arrow's Condition #3: Independence of Irrelevant Alternatives. In fact, if you accept the rationality of Facts #1 and #2, you must consider Arrow's
Condition #3 to be irrational. Remember his is a general theory which must apply to every case including this one.
Some of my papers like a A General Theory of Social Choice attempt to prove that a solution involving tie orderings is always possible, and Arrow's Impossibility Theorem is only true if you
accept his rather limited formulation of the general problem.
More later...
4. Anonymous7:36 pm
This is so much fun, I think I'll continue a bit longer. In response to my point #2, you say: "Arrow doesn’t just ‘assume’ this. There’s an argument p.50 ff. intended to establish that if x and y
tie, yet we take one to be preferred, we get inconsistent results."
Yes, Arrow attempts to prove that, for 2 individuals, if one prefers y to x and one prefers x to y then society is indifferent between y and x. I would say that in the second line of his proof
his assumptions are wrong. He says "we would have xP1y and yP2x, but not xIy." I would say we have a tie between x and y. There's no necessity for society to be indifferent just because the
electorate is evenly divided!
5. Anonymous7:56 pm
In the second part of you response to #2, you say: "What if 99% were indifferent and the 1% evenly split each way? Surely that’s also social indifference rather than a tie. What if the 1% mostly
support x? Is the social choice now xPy or xIy? The problem is that introducing this extra possibility means we’re no longer dealing with a binary x or y choice, because tying is (unlike
indifference) a third option."
A SWF in general doesn't have to deal with the various cases you mention in a totally rational manner. One SWF might map the 99% indifference case into a social indifference. Another SWF might
map it into xPy if 99% were indifferent and 1% prefered x to y.
For a SWF to exist it only has to provide a consistent mapping from domain to range. Many such SWFs could exists, some better than others according to different criteria.
I grant you that we're no longer dealing with a binary choice. But the individuals have to decide among 3 possibilities: xPy, yPx and xIy. Society actually has to decide among 7: xPy, yPx, xIy,
xPy ties yPx, xPy ties xIy, yPx ties xIy and all three (xPy, yPx and xIy) are tied.
Part of Arrow's problem, I think, is that he oversimplifies the analysis leading to consistent results only if you accept his formulation and conceptualization of the problem. Basically he just
proves the voter's paradox, a fact that's been known for 200 years. He could just as well have shown that the voter's paradox can be interpreted as a tie which is broken if one of the candidates
drops out. Then you could use the methods of your PhD thesis to break the tie!
6. Anonymous2:09 am
Ben, I can't believe you watched the Superbowl. I'm one American that didn't watch it. Actually it's a good time to go someplace as the freeways are clear of traffic!
Today I'll deal with point #4. You say: "The death of a candidate is actually a slightly different case. Arrow himself confuses independence of irrelevant alternatives with contraction
consistency, which is related but can be distinguished. Still, suppose you rank aPbIcPd then c dies. It seems natural to suppose the remaining preferences simply carry over – aPbPd – unless some
good reason is given why they might change (non-strategically)."
See my paper Social Choice, Information Theory and the Borda Count p.6 and especially Figure 4 for all kinds of examples of how a person's individual preference orderings can change as a function
of the number of candidates or alternatives being considered.
Quoting: "Figure 4 shows, in general, how a rational individual will project his or her “true” preferences by expressing them in various situations in which a differing number of slots are
available. It can be seen that, if one or more alternatives are removed from the original ordering (due to the death of one or more candidates, for instance), the rational individual will project
his or her “true” preferences onto the number of slots appropriate to the number of remaining candidates in order to come up with his or her new preference ordering."
I reiterate that having more "slots" than candidates does not necessarily mean that you're dealing with "interpersonal comparisons of utilities." It just means that you're collecting more
information regarding the individual's preference orderings. There would not be a need to ioncrease the number of slots beyond an individuals's "sensitivity level" that is the finest level in
which an individual can discriminate between 2 alternatives. Also the same number of slots would have to be mandated for each individual in order for each vote to have the same power. Otherwise,
one person's vote could count more than another's.
I'll try to explain it briefly by the following:
If there are a number of candidates and 1 dies, now you're comparing the candidates on a coarser grid, that is, there are fewer slots. Therefore, a very small preference of one candidate over
another on the finer grid can turn into an indifference between them on the coarser grid.
By the way what is "contraction consistency?"
7. Anonymous2:38 am
Point#5. I think I prove this sufficiently in Social Choice, Information Theory and the Borda Count
Point#6. You say: "This rather repeats 4). If there are true preferences between a, b and c, why shouldn’t they emerge before Hitler’s death? Thus if the individual would want to vote aPbPc
without Hitler in the frame, they should want to vote aPbPcPHitler with Hitler, preserving “the rationale that any candidate would be preferred to Hitler”.
Because the "anyone but Hitler strategy" would want to add as many points to everyone else's point totals as it could. Ranking Hitler as 0 and a, b and c as 100, for example, would give a , b or
c the greatest chance of beating Hitler. I maintain that a person should be able to use his vote strategically. The literature that is strategy-phobic is barking up the wrong tree to my way of
Point #8: You say: "Sometimes I think strategic voting is defensible, so long as all have the same opportunities to use their votes, it’s up to people how they do so. Certainly no argument for
strategic voting is given here, however, so nothing to convince anyone who already believes it’s wrong."
I'll take it you agree with me on this "so long as all have the same opportunities."
Point#8: Saari is saying that, when more information is provided, Arrow's Impossibility Theorem is moot. More information could be, for example, more slots than the number of candidates or more
than just binary comparisons.
Point#9: You say: "This claims to be like 3) but without making interpersonal comparisons, however I don’t see what is supposedly expressed. How about another box on the ballot allowing voters to
express their favourite flavour of ice cream? Surely that’s also more expression and therefore better (though I don’t see how it should influence the decision in question)."
What is expressed is more information down to the sensitivity level of the voter. More slots. More information. Less slots. Less Information. See Shannon, "The Mathematical Theory of
Communication." Ice cream wouldn't be relevant unless it was directly comparable with the other alternatives all of which would be foods I presume.
Point#10. I respectfully disagree. Restricting information is restricting freedom. Requiring an individual to express a preference coarser than his sensitivity level is restricting information
hence freeedom of expression in some sense.
Point #11. How you resolve ties is essentially another subject. The Borda Count does it by turning ties into indifferences, but, in general, a tie is not an indifference.
That's it for today. No sense in straining my brain!
8. Anonymous10:05 pm
I'll try to finish this up today.
Point#12 and #16 - You say: "How does Arrow violate Neutrality then? Neither x nor y is given preferential treatment (unless ties favour status quo, which is slightly non-neutral). But see 16)."
and "This seems to explain what was meant by 12), by repeating and elaborating the claim majority rule is non-neutral. It is, however, mistaken. Jonathan asks why xRy and not yRx. It is also yRx.
As states in the previous section, xRy and yRx can hold together and imply xIy. Thus Jonathan either has no point, or it’s merely the ties/indifference thing again (see 2), 11), etc)."
By Definition 9 on p. 46 of "Social Choice and Individual Values," Arrow states: "By the method of majority decision is meant the social welfare function in which xRy holds if and only if the
number of individuals such that xRiy is at least as great as the number of individuals such that yRix."
Condider the case when the number of individuals such that xRiy is as great as the number of individuals such that yRix. According to the above definition, xRy. That is x is prefered or
indifferent to y. Def. 9 does not say x is indifferent to y in that case. It is prefered or indifferent. We don't know in the individual cases whether xPiy, yPix or xIiy. All we know is that, for
this specific case, the number who vote xRiy equals the number who vote yRix. Yet according to the definition, xRy. Therefore, x is getting special treatment in this case.
Point#13. I'm not sure what point I'm making here either except that by Axiom 1 (p. 13) Arrow defines a tie as an indifference. Arrow says that you can have both xRy and yRx. That would be a tie.
He implies that that is the same as an indifference. He proves on p. 50 that if 2 individuals have opposing interests, society will treat this as an indifference not a tie because his analysis
won't allow ties.
Point#14 - You say: "A strong ordering is one where no ties are possible. That’s definitional. If voters are evenly split, you get a tie, but that’s because the result isn’t a strong ordering! As
for the tie/indifference distinction, see 2) and 11) – this increasingly seems John’s main, perhaps only, complaint."
I thought a strong ordering was one in which no indifferences were possible. That is the individuals could express only xPiy or yPix but not xIiy or xRiy or yRix. It is clearly possible if the
number of voters is even that the number who have xPiy could be equal to the number who have yPix. That would be a tie, no? Society would have to have xPy or yPx or a tie between the two since no
indifferences are allowed at the societal level either. It seems clear that a tie is the only reasonable solution.
Point#15 - You say: "nothing is lost by using R notation." Let's say the individuals submit their votes in terms of P and I. Then the social amalgamating mechanism converts these to the R
notation before amalgamating them to come up with the social choice in terms of the R notation. I think information has been lost since any particular individual could change Ps to Is and vice
versa and the SWF would not know the difference. It would come up with the same social choice.
Point #17: In your example, 997 have xRiy and 996 have yRix so, according to Arrow, society has xRy. However, if you don't restrict the analysis as Arrow does and let society have xPy, yPx or
xIy, and treat each possibility equally, then clearly xIy.
That concludes my response to your response. Clearly, my 17 points are somewhat redundant. However, I think my analysis is essentially correct. It's all in how you formulate and conceptualize the
problem. Arrow's formulation is quite narrow, and a more general formulation can lead to a different result than the one he obtained. If you're interested, I'd be willing to pursue this. If not,
nice chatting and good luck! | {"url":"http://bensaunders.blogspot.com/2006/01/arrow-criticism-and-defence.html","timestamp":"2024-11-10T15:58:27Z","content_type":"text/html","content_length":"165011","record_id":"<urn:uuid:c6e3e309-0abd-463d-aa06-960edff2ab3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00809.warc.gz"} |
Engineering Topics - Differential Equations - David The Maths TutorEngineering Topics – Differential Equations
Engineering Topics – Differential Equations
A differential equation is an equation that has a derivative in it. A derivative is a rate of change, like velocity. So if you are driving in a car where your velocity from a starting point is v
which is some function of time, this can be solved to find your position from a starting point at any time. There are lots of techniques to solve these equations and the study of this scares many
students. But the fact is, if you drive a car, pick up a glass of water, throw a ball at a target, your mind subconsciously handles the differential equations that model these activities quite well.
Let’s stick with the car example. Suppose you are 100 metres away from a stop sign or a red traffic light. So you apply the brakes. Let x(t) be your distance from the stop sign in metres at a time t,
t be the time in seconds, and ẋ(t) be your velocity at time (t). Now you want your distance to the stop sign to decrease from 100 to 0 metres comfortably in say 15 seconds. What about this linear
It does stop at the stop sign, but is it comfortable? This equation of a line has a constant slope (rate of change) of −100/15 = −6.67 m/s = −24 km/hr. This means that at the stop sign, you are going
24 km/hr when you hit the brakes hard to stop. The passengers drinking coffee at the time, would not appreciate that. Also, if you are going 100 km/hr at 100 meters away, to follow this profile, you
have to slam on the brakes to suddenly get at a speed of 24 km/hr. Well maybe there would be no coffee left to spill at the stop sign.
So this shows that we need to be aware of our speed and our distance to do this comfortably.
What about stopping following this red curve:
This starts at 100 metres and ends at 0 like the linear graph but has a better rate of change profile. The rate of change (that is the velocity) varies on this curve. It is visually seen as the slope
of the line tangent to the graph at a point. The grey line shown is an example of a tangent line. Notice that the gradient at the beginning of the curve is high (in the negative direction) but at the
end, it is near zero (the slope of a horizontal line is zero). This would be a much smoother stop than a linear approach.
But your mind during this action is not just seeing your distance from the stop sign, it is also sensing your velocity and adjusting it as you get closer to the stop sign. The following is an
equation that relates the velocity and position:
If you were to solve this equation for x(t) using differential equation techniques, you would get the equation seen in the graph above. If you were to design a control system (which is what your mind
is when performing this action) you would use the above differential equation to control both your position and your velocity.
But even this stopping profile has flaws. Notice that the deceleration at the beginning is quite steep (the slope of a tangent line at t = 0). Perhaps a better profile would be:
This starts with a more gentle deceleration, increases the deceleration until you get closer to the stop sign, then the deceleration decreases until you come to a full stop at the sign.
Regardless of the stopping profile used, your mind controls the braking action to conform to a desired profile based on your current speed (the slope of the tangent line) and your distance from the
stop sign. People who are designing driverless cars, robotic arms, aircraft autopilots, etc, use differential equations. And because they are working in three dimensions, these equations can be in
the form of matrix and/or vector equations. And the solutions will use complex numbers: all of these topics were covered in my last few posts.
So besides the basic algebraic skills you may be studying or have studied, more advanced topics like this one or those covered in in my last few post are the heart of engineering. | {"url":"https://davidthemathstutor.com.au/2024/03/12/engineering-topics-differential-equations/","timestamp":"2024-11-03T12:13:40Z","content_type":"text/html","content_length":"49640","record_id":"<urn:uuid:98ea2084-fcce-4985-b515-07339a48a217>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00308.warc.gz"} |
30. ...nervous?
All of the time
Most of the time
Some of the time
A little of the time
None of the time
31. ...hopeless?
All of the time
Most of the time
Some of the time
A little of the time
None of the time
32. ...restless or fidgety?
All of the time
Most of the time
Some of the time
A little of the time
None of the time
33. ...so sad that nothing could cheer you up?
All of the time
Most of the time
Some of the time
A little of the time
None of the time
34. ...that everything was an effort?
All of the time
Most of the time
Some of the time
A little of the time
None of the time
35. ...worthless?
All of the time
Most of the time
Some of the time
A little of the time
None of the time | {"url":"https://meps.ipums.org/meps-action/variables/K6SUM/ajax_enum_text","timestamp":"2024-11-10T00:13:02Z","content_type":"text/html","content_length":"39751","record_id":"<urn:uuid:c367ee83-d3f5-496f-86e5-4b40f97915ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00578.warc.gz"} |
"Significance", in 1885 and today
« previous post | next post »
There's an ongoing argument about the interpretation of Katherine Baicker et al., "The Oregon Experiment — Effects of Medicaid on Clinical Outcomes", NEJM 5/2/2013, and one aspect of this debate has
focused on the technical meaning of the word significant. Thus Kevin Drum, "A Small Rant About the Meaning of Significant vs. 'Significant'", Mother Jones 5/13/2013:
Many of the results of the Oregon study failed to meet the 95 percent standard, and I think it's wrong to describe this as showing that "Medicaid coverage generated no significant improvements in
measured physical health outcomes in the first 2 years."
To be clear: it's fine for the authors of the study to describe it that way. They're writing for fellow professionals in an academic journal. But when you're writing for a lay audience, it's
seriously misleading. Most lay readers will interpret "significant" in its ordinary English sense, not as a term of art used by statisticians, and therefore conclude that the study positively
demonstrated that there were no results large enough to care about.
Many past LL posts have dealt with various aspects of the rhetoric of significance. Here are a few:
"The secret sins of academics", 9/16/2004
"The 'Happiness Gap' and the rhetoric of statistics", 9/26/2007
"Gender-role resentment and the Rorschach-blot news reports", 9/27/2007
"The 'Gender Happiness Gap': Statistical, practical and rhetorical significance", 10/4/2007
"Listening to Prozac, hearing effect sizes", 3/1/2008
"Localization of emotion perception in the brain of fish", 9/18/2009
"Bonferroni rules", 4/6/2011
"Response to Jasmin and Casasanto's response to me", 3/17/2012
"Texting and language skills", 8/2/2012
But Kevin Drum's rant led me to take another look at the lexicographic history of the word significant, and this in turn led me back to the Oregon Experiment — via a famous economist's work on the
statistics of telepathy.
The OED's first sense for significant has citations back to 1566, and in this sense, being significant is a big deal: something that's significant is "Highly expressive or suggestive; loaded with
meaning". This is the ordinary-language sense that makes the statistical usage so misleading, because a "statistically significant" result is often not really expressive or suggestive at all, much
less "loaded with meaning".
The OED gives a second sense, almost as old, that is much weaker, and is probably the source of the later statistical usage: "That has or conveys a particular meaning; that signifies or indicates
something". Not necessarily something important, mind you, just something — say, in the modern statistical sense, that a result shouldn't be attributed to sampling error. A couple of the OED's more
general illustrative examples:
1608 E. Topsell Hist. Serpents 48 Their voyce was not a significant voyce, but a kinde of scrietching.
1936 A. J. Ayer Lang., Truth & Logic iii. 71 Two symbols are said to be of the same type when it is always possible to substitute one for the other without changing a significant sentence into a
piece of nonsense.
And then there's an early mathematical sense (attested from 1614):
Math. Of a digit: giving meaningful information about the precision of the number in which it is contained, rather than simply filling vacant places at the beginning or end. Esp. in significant
figure, significant digit. The more precisely a number is known, the more significant figures it has.
The OED gives a few additional senses that are not strikingly different from the first two: "Expressive or indicative of something"; "Sufficiently great or important to be worthy of attention;
noteworthy; consequential, influential"; "In weakened sense: noticeable, substantial, considerable, large".
And then we get to (statistically) significant:
5. Statistics. Of an observed numerical result: having a low probability of occurrence if the null hypothesis is true; unlikely to have occurred by chance alone. More fully statistically significant.
A result is said to be significant at a specified level of probability (typically five per cent) if it will be obtained or exceeded with not more than that probability when the null hypothesis is
In an earlier post, I characterized sense 5. as "[a]mong R.A. Fisher's several works of public-relations genius". However, I gave Sir Ronald too much credit, as the OED's list of citations shows:
1885 Jrnl. Statist. Soc. (Jubilee Vol.) 187 In order to determine whether the observed difference between the mean stature of 2,315 criminals and the mean stature of 8,585 British adult males
belonging to the general population is significant [etc.].
1907 Biometrika 5 318 Relative local differences falling beyond + 2 and − 2 may be regarded as probably significant since the number of asylums is small (22).
1925 R. A. Fisher Statist. Methods iii. 47 Deviations exceeding twice the standard deviation are thus formally regarded as significant.
1931 L. H. C. Tippett Methods Statistics iii. 48 It is conventional to regard all deviations greater than those with probabilities of 0·05 as real, or statistically significant.
In 1885, Sir Ronald's birth was still five years in the future. So who came up with this miracle of mathematical marketing? It seems that the credit is due to Francis Ysidro Edgeworth, better known
for his contributions to economics. The OED's 1885 citation for (statistically) significant is to his paper "Methods of Statistics", Journal of the Statistical Society of London , Jubilee Volume
(Jun. 22 – 24, 1885), pp. 181-217:
The science of Means comprises two main problems: 1. To find how far the difference between any proposed Means is accidental or indicative of a law? 2. To find what is the best kind of Mean; whether
for the purpose contemplated by the first problem, the elimination of chance, or other purposes? An example of the first problem is afforded by some recent experiments in so-called "psychical
research." One person chooses a suit of cards. Another person makes a guess as to what the choice has been. Many hundred such choices and guesses having been recorded, it has been found that the
proportion of successful guesses considerably exceeds the figure which would have been the most probable supposing chance to be the only agency at work, namely 1/4. E.g., in 1,833 trials the number
of successful guesses exceeds 458, the quarter of the total number, by 52. The first problem investigates how far the difference between the average above stated and the results usually obtained in
similar experience where pure chance reigns is a significant difference; indicative of the working of a law other than chance, or merely accidental.
So the first use in print of "(statistically) significant" was in reference to an argument for telepathy!
Edgeworth gives no detailed analysis of the "Psychical Research" data in this article, though he notes that
we have several experiments analogous to the one above described, all or many of them indicating some agency other than chance.
But he went over the issue in great detail in a paper published in the same year ("The calculus of probabilities applied to psychical research", Proceedings of the Society for Psychical Research,
Vol. 3, pp. 190-199, 1885), which begins with an inspiring (though untranslated) quote from Laplace ("Théorie Analytique des probabilités", 1812):
"Nous sommee si éloignés de connaître tous les agents de la nature qu'il serait peu philosophique de nier l'existence de phenomènes, uniquement parcequ'ils sont inexplicables dans l'état actuel de
nos connaissances. Seulement nous devons les examiner avec une attention d'autant plus scrupuleuse, qu'il parait plus difficile de les admettre; et c'est ici que l'analyse des probabilités devient
indispensable, pour determiner jusqu'à quel point il faut multiplier les observations ou les expériences, pour avoir, en faveur de l'existence des agents qu'elles semblent indiquer, une probability
supérieure à toutes les raisons que l'on peut avoir d'ailleurs, de la réjéter."
"We are so far from knowing all the agencies of nature that it would hardly be philosophical to deny the existence of phenomena merely because they are inexplicable in the current state of our
knowledge. We must simply examine them with more careful attention, to the extent that it seems more difficult to explain them; and it's here that the analysis of probabilities becomes essential, in
order to determine to what point we must multiply observations or experiments, in order to have, in favor of the existence of the agencies that they seem to indicate, a higher probability than all
the reasons that one can otherwise have to reject them."
Edgeworth then undertakes to show that telepathy is real:
It is proposed here to appreciate by means of the calculus of probabilities the evidence in favour of some extraordinary agency which is afforded by experiences of the following type: One person
chooses a suit of cards, or a letter of the alphabet. Another person makes a guess as to what the choice has been.
After considerable application of somewhat complex mathematical reasoning, e.g. the passage below, Edgeworth concludes that the probability of obtaining the cited results by chance — 510 correct
card-suit guesses out of 1833 tries — is 0.00004.
Even if we grant his premises, Edgeworth's estimate seems to have been quantitively over-enthusiastic — R tells us that the probability of obtaining the cited result by chance, given his model,
should be a bit greater than 0.003:
Exact binomial test
data: 458 + 52 and 1833
number of successes = 510, number of trials = 1833, p-value = 0.003106
alternative hypothesis: true probability of success is greater than 0.25
But the p-value calculated according to Edgeworth's model — whether it's .00004 or .003 — is not an accurate estimate of the probability of getting the cited number of correct guesses by chance, in
an experiment of the cited type. That's because his model might well be wrong, and there are plausible alternatives in which successful guesses are much more likely.
Recall his description of the experiment:
One person chooses a suit of cards, or a letter of the alphabet. Another person makes a guess as to what the choice has been.
He assumes that the chooser picks among the four suits of cards with equal a priori probability; and that the guesser, if guessing by chance, must do the same. But suppose that they both prefer one
of the four suits, say hearts? If the chooser always chooses hearts, and the guesser always guesses hearts, then perfect psychical communication will appear to have taken place.
More subtly, we only need to assume a slight shared bias for better-than-chance results to emerge. And the bias need not be shared in advance of the experiment — since the guesser learns the true
choice after each guess, he or she has plenty of opportunity to estimate the chooser's bias, and to start to imitate it. In other words, this is really not a telepathy experiment, it's the world's
first Probability Learning experiment! (See "Rats beat Yalies", 12/11/2005, for a description of this experimental paradigm.)
If the result is equivalent to a shared uneven distribution over the four suits, then things are very different. With shared probabilities of 0.34, 0.33, 0.17, 0.16, for example, the probability of
getting at least 510 correct guesses in 1833 trials is about 55%. And I assert without demonstration that such an outcome could easily emerge from a probability-learning process, without any initial
shared bias.
My point here is not to debunk psychical research, but to observe that here as elsewhere, it's important to pay attention to the the details of the model, the data, and the outcome, rather than just
looking at the p value.
And this brings us back to the contested paper, Katherine Baicker et al., "The Oregon Experiment — Effects of Medicaid on Clinical Outcomes", NEJM 5/2/2013.
Since this post has already gone on too long, I'll try to make this fast (for the two of you who are still reading…) Here's the background:
In 2008, Oregon initiated a limited expansion of its Medicaid program for low-income adults through a lottery drawing of approximately 30,000 names from a waiting list of almost 90,000 persons.
Selected adults won the opportunity to apply for Medicaid and to enroll if they met eligibility requirements. This lottery presented an opportunity to study the effects of Medicaid with the use of
random assignment.
Our study population included 20,745 people: 10,405 selected in the lottery (the lottery winners) and 10,340 not selected (the control group).
They were able to interview 12,229 people, 6387 lottery winners and 5842 among the controls, about two years after the lottery.
But it's not quite as simple as that:
Adults randomly selected in the lottery were given the option to apply for Medicaid, but not all persons selected by the lottery enrolled in Medicaid (either because they did not apply or because
they were deemed ineligible).
A more detailed picture of the lottery process is given in the paper's Supplementary Appendix:
In total, 35,169 individuals—representing 29,664 households—were selected by lottery. If individuals in a selected household submitted the appropriate paperwork within 45 days after the state mailed
them an application and demonstrated that they met the eligibility requirements, they were enrolled in OHP Standard. About 30% of selected individuals successfully enrolled. There were two main
sources of slippage: only about 60% of those selected sent back applications, and about half of those who sent back applications were deemed ineligible, primarily due to failure to meet the
requirement of income in the last quarter corresponding to annual income below the poverty level, which in 2008 was $10,400 for a single person and $21,200 for a family of four.
In other words, we would expect that only about 30% of the lottery winners in this paper's sample were actually enrolled in the insurance program. Thus the lottery selection was random, but the
enrollment step for lottery winners was not: and the various reasons for failing to get insurance are presumably not neutral with respect to health status and outcomes.
When I first began reading this paper, this seemed to me to constitute a huge source of non-sampling error, since the lottery winners who actually enrolled may constitute a very different group from
the lottery participants as a whole.
However, it turns out that this doesn't matter. Although the paper's title announces itself as a study of the "Effects of Medicaid on Clinical Outcomes", the authors did not compares the outcomes of
those who were actually enrolled against the outcomes of those who were not. Instead, they compared the outcomes of those who won the lottery—and thus were given the opportunity to try to
enroll—against the outcomes of those who didn't win the lottery (even though some of these did have health insurance anyhow). So realistically, it's a study of the "Effects of an Extra Opportunity to
Try to Enroll in Medicaid on Clinical Outcomes".
Of the 6,387 survey responders who were lottery winners, 1,903 (or 29.8%) actually enrolled in Medicaid at some point. A reasonable number of the control group had OHP or Medicaid insurance as well,
and by the end of the period of the study, the differences between the two groups in enrolled in proportion enrolled in public insurance had nearly vanished (we're given no information about how many
might have had employer-provided insurance, but presumably the proportion was small):
Of course, the study's authors set up their model to compensate for this situation:
The subgroup of lottery winners who ultimately enrolled in Medicaid was not comparable to the overall group of persons who did not win the lottery. We therefore used a standard instrumental-variable
approach (in which lottery selection was the instrument for Medicaid coverage) to estimate the causal effect of enrollment in Medicaid. Intuitively, since the lottery increased the chance of being
enrolled in Medicaid by about 25 percentage points, and we assumed that the lottery affected outcomes only by changing Medicaid enrollment, the effect of being enrolled in Medicaid was simply about 4
times (i.e., 1 divided by 0.25) as high as the effect of being able to apply for Medicaid. This yielded a causal estimate of the effect of insurance coverage.
And there was indeed a period of differential enrollment proportions between between the lottery winners and losers, as well a period of different enrollment-opportunity proportions, and so it's
plausible to look for effects across the sets of winners and losers as a whole.
But the main physical health outcomes that the study examined (blood pressure, cholesterol, glycated hemoglobin, etc.) are age- and lifestyle-related measures that are not very likely to be seriously
influenced by a short period of differential access to insurance. The things that were (both statistically and materially) influenced — self-reported health-related quality of life, out-of-pocket
medical spending, rate of depression — are much more plausible candidates to show the impact of having a brief opportunity to get insurance access.
And as in F. Y. Edgeworth's analysis of card-suit guessing, the p values in the regression are not as important as the details of what the observations were, and what forces plausibly shaped them.
[Note: For more on the subsequent history of psychic statistics, see Jessica Utts, "Replication and Meta-Analysis in Parapsychology", Statistical Science 1991.)
MonkeyBoy said,
You left out the whole issue where many take "significant" to mean "meaningful". For example if I give IQ test to 10,000 people with brown eyes and 10,000 with blue, I might find a "significant"
difference in their scores but one say so small (.05 IQ points) to make it meaningless in any application – say only hiring those with a particular color.
Some discussion of this and pointers to rhe literature can be found at the post "Fetishizing p-Values".
[(myl) I didn't leave it out — I referred you to a list of previous posts where it's discussed at length, e.g. here ("statistical significance without a loss function"), here ("my objection was never
about statistical significance, but rather about effect sizes and practical significance"), here (difference between statistical and clinical significance in drug studies), or here ("There's a
special place in purgatory reserved for scientists who make bold claims based on tiny [though statistically significant] effects of uncertain origin").
But in this case, the critical issue is neither the p values nor the loss functions nor the effect sizes, but the details of the experiment.]
Mr Punch said,
There was a somewhat similar controversy in the '70s over "bias" in intelligence tests. To charges that the tests were racially biased, defenders such as Arthur Jensen of Berkeley responded that
there was no bias – which was clearly true in the technical terminology of their field.
Eric P Smith said,
I think that Kevin Drum is unduly lenient on the researchers.
The researchers say, “We found no significant effect of Medicaid coverage…”, and “We observed no significant effect on…”. In both those cases the full (untruncated) statement is true, with the
reasonable interpretation of “significant” as “statistically significant”. But the researchers go on to say, “This randomized, controlled study showed that Medicaid coverage generated no significant
improvements…” Kevin Drum remarks, “It's fine for the authors of the study to describe it that way”, but it's not fine at all. The full (untruncated) statement is false, and no amount of hiding
behind statistical language makes it true. When an experiment is done and the result is not statistically significant, it does not show anything.
Rubrick said,
Is there any really sound argument against making the standard "magic percentage" smaller, say 3%? As an outsider it seems as though the main "drawback" would be that researchers would be able to
publish far fewer studies, especially ones of dubious value, but perhaps I'm being overly cynical. A 1-in-20 chance of a result being total bunk, even if the experiment is perfectly designed (which
it isn't) has always struck me as absurdly high.
Deuterium Oxide said,
@Rubrick. It's unrealistic to require from a research paper to establish the Holy Truth. And it is not a problem if some published research is a bunk. The real goal is to move our understanding
forward. If the method of doing it sometimes misfires it's OK. Imagine, for example, that you have a method that moves you a step forward 9 times out of 10 and backwards 1 out of 10 and another that
guarantees the forward motion, but works at only half speed. You are definitely better off under the first method.
The real problem is how to bring forward research with more substantial results, not merely crossing the threshold of statistical significance (and maybe not crossing it at all), but something that
has a potential to really improve our understanding of the things.
A propos, Mr. Edgeworth apparently lost √ π in his equation.
P.S. My old pen name, D.O., is apparently consigned to spam bin.
Eric P Smith said,
No, there is no overriding argument against making the required p-value smaller. The required p-value is (in principle) chosen by the experimenters and it may take several factors into account. These
may include: How inherently unlikely is the effect tested for generally perceived to be? How much larger and more expensive would the experiment need to be for it to have much hope of meeting a more
stringent p-value? How serious would the consequences be of getting a false positive? And, not least, what is the tradition in the field? In most life sciences, 5% is quite usual. In controversial
fields like parapsychology, 1% is more usual (reflecting a general perception that the effects tested for are inherently unlikely). In particle physics, the standard is 5 sigma, ie p=0.0000005
approximately, reflecting (amongst other things) the relative ease of meeting such a stringent p-value in that field.
Incidentally, passing a statistical test at the 5% level does not mean that there is a 5% probability that the effect is not real. It means that, if the effect is not real, there was a 5% probability
of passing the statistical test. This point is often misunderstood but it is important.
Rubrick said,
"How much larger and more expensive would the experiment need to be for it to have much hope of meeting a more stringent p-value?"
I suspect this is a big factor; requiring a more stringent p-value in, say, behavioral psychology would seem to spell the end of "Ask 20 grad students and write it up". I'm not entirely sure this
would be a bad thing, given the frequency with which flimsy results are A) trumpeted in the media, and B) built upon without giving enough thought to whether they might be entirely bogus.
Your point about the meaning of the 5% value is important, but in terms of the weight one should give to the results of a study, I think the important interpretation is that there's a 5% chance that
we don't actually know any more than if the study hadn't been run at all.
Dan Hemmens said,
Your point about the meaning of the 5% value is important, but in terms of the weight one should give to the results of a study, I think the important interpretation is that there's a 5% chance that
we don't actually know any more than if the study hadn't been run at all.
I'm not sure that's strictly true either. It's not that we have a 5% chance of knowing no more than we did before the test was run. We know more, but part of what we know is that there is a chance
that our conclusion is wrong (and this chance is not necessarily 5%, it could be higher or lower).
We would only know *no more* than we did before the test was run if there really was no correlation whatsoever between reality and the outcome of the test. If we were, for example, to try to "test" a
particular hypothesis by tossing a coin and reading heads as "yes" and tails as "no" we would actually get a "correct" result half of the time, but such a test would clearly tell us no more
information than we had before the test was run, because a false positive is exactly as probable as a true positive.
Yuval said,
Got a little typo there – "the authors did not compares".
Jonathan D said,
I agree with Eric P Smith that 'significant' was misused in the quote. However useful it may be as PR, in the statistical sense we dont' say that the effects (of Medicaid or whatever else) are or
aren't significant. It's the details in the data that are or aren't signficant, that is, suggest there may be an effect.
And Rubrick, the interpretation of a 5% p-value given by Eric is correct. There may be many other interpretations that would be more important in the sense that could be more useful, but they are
De la significativité (statistique), suite | Freakonometrics said,
[…] Siegle, M.V. (2008) Sound and fury: McCloskey and significance testing in economics, Liberman, M. (2013). "Significance", in 1885 and today et le passionnant Hall, P. and Selinger, B. […] | {"url":"https://languagelog.ldc.upenn.edu/nll/?p=4624","timestamp":"2024-11-08T07:09:32Z","content_type":"application/xhtml+xml","content_length":"121512","record_id":"<urn:uuid:37f87a40-c74b-44e0-81ad-9c0dc9b7d943>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00741.warc.gz"} |
Technical requirements
• For each of the bit-lengths $160, 192, 224, 256, 320, 384, 512$ one curve shall be proposed.
• The base field size $p$ should be congruent to $3 \mod 4$.
• The curve should be $\mathbb{F}_p$-isomorphic to a curve with $A \equiv -3 \mod p$.
• The prime $p$ must not be of a special form in order to avoid patented fast arithmetic on the base field.
• The order of the curve $\lvert \mathcal{E}(\mathbb{F}_p) \rvert$ should be smaller than the size of the base field $p$.
• The curve coefficient $B$ should be non-square in $\mathbb{F}_p$.
Security requirements
• The embedding degree $l = \min\{t \vert q \text{divides} p^t - 1 \}$ should be large, where $q$ is the order of the basepoint and $p$ the size of the base field. Specifically, $(q - 1) / l < 100$
• The curves are not trace one curves. Specifically $\lvert \mathcal{E}(\mathbb{F}_p) \rvert e p$.
• The class number of the maximal order of the endomorphism ring of the curve is larger than $10000000$.
• The group order $\lvert \mathcal{E}(\mathbb{F}_p) \rvert$ should be a prime number $q$.
Original method
Brainpool published their method of generating verifiably random curves in the ECC Brainpool Standard Curves and Curve Generation [1] document, along with generated domain parameters claimed to be
generated using the presented method and seeds. However, the presented curves were (with the exception of the 512-bit curves) not generated using the presented method, as they have properties that
can not result from the presented method of generating curves. See the BADA55 paper [3] for more information.
RFC 5639 method
Brainpool published an RFC with their fixed method of generating verifiably random curves and generated curves in RFC 5639 [2], which matches the generated curves and seeds.
Generating primes
Generating curves
1. Manfred Lochter: ECC Brainpool Standard Curves and Curve Generation v. 1.0, [archive]
2. Manfred Lochter, Johannes Merkle: Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation (RFC5639)
3. BADA55 Research Team: BADA55 Crypto - Brainpool curves | {"url":"https://neuromancer.sk/std/methods/brainpool/","timestamp":"2024-11-08T07:54:01Z","content_type":"text/html","content_length":"174321","record_id":"<urn:uuid:848c60f3-2bab-4165-bbe1-fd2185d45929>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00552.warc.gz"} |
Free online math games for 9th graders
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Factoring The Difference of 2
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A Summary of Factoring Polynomials
Factoring The Difference of 2 Squares
Factoring Trinomials
Quadratic Expressions
Factoring Trinomials
The 7 Forms of Factoring
Factoring Trinomials
Finding The Greatest Common Factor (GCF)
Factoring Trinomials
Quadratic Expressions
Factoring simple expressions
Factoring Polynomials
Fractoring Polynomials
Other Math Resources
Factoring Polynomials
Finding the Greatest Common Factor (GCF)
Factoring Trinomials
Finding the Least Common Multiples
Author Message
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Hello Math Gurus. Ever since I have encountered free online math games for 9th graders at school I never seem to be able to understand it well. I am very good at all the other
chapters , but this particular topic seems to be my weakness . Can some one guide me in learning it well?
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Posted: Saturday 26th of Aug 07:34
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Posted: Monday 28th of Aug 07:27
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Posted: Wednesday 30th of Aug 08:23
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Posted: Thursday 31st of Aug 21:12
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Posted: Friday 01st of Sep 15:18
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Neural Sliding Mode Control of a Buck-Boost Converter Applied to a Regenerative Braking System for Electric Vehicles
Facultad de Ingenieria, Universidad Autonoma del Carmen, C.56 No.4 Esq. Avenida Concordia Col. Benito Juarez, Ciudad del Carmen 24180, Campeche, Mexico
Tecnologico Nacional de Mexico, Instituto Tecnologico de La Laguna, Blvd. Revolución y Av. Instituto Tecnologico de La Laguna s/n, Torreon 27000, Coahuila, Mexico
Centro Universitario de Ciencias Exactas e Ingenierias, Universidad de Guadalajara, Blvd. Marcelino Garcia Barragan, Guadalajara 44430, Jalisco, Mexico
Author to whom correspondence should be addressed.
Submission received: 27 September 2023 / Revised: 22 December 2023 / Accepted: 25 December 2023 / Published: 2 February 2024
This paper presents the design and simulation of a neural sliding mode controller (NSMC) for a regenerative braking system in an electric vehicle (EV). The NSMC regulates the required current and
voltage of the bidirectional DC-DC buck–boost converter, an element of the auxiliary energy system (AES), to improve the state of charge (SOC) of the battery of the EV. The controller is based on a
recurrent high-order neural network (RHONN) trained using the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) as the tools to train the neural networks to obtain a higher SOC in
the battery. The performance of the controller with the two training algorithms is compared with a proportional integral (PI) controller illustrating the differences and improvements obtained with
the EKF and the UKF. Furthermore, robustness tests considering Gaussian noise and varying of parameters have demonstrated the outcome of the NSMC over a PI controller. The proposed controller is a
new strategy with better results than the PI controller applied to the same buck–boost converter circuit, which can be used for the main energy system (MES) efficiency in an EV architecture.
1. Introduction
The energy storage in electric vehicles (EVs) fully operated with clean fuels is an extremely relevant issue in the automotive industry despite their huge contribution to the environmental impact. In
], an environmental study of many EVs has demonstrated the advantages and positive impact that result in a lower number of CO
emissions in a certain period of analysis compared to conventional fossil fuel vehicles. The commitment with the environmental impact has developed new strategies to recycle battery packs such as
lithium ion batteries [
], for which demand has increased in recent years. The lead acid battery recycling process has demonstrated great performance, as well as the necessary chemical procedure.
In recent years, many approaches to improve energy consumption have been developed and new alternatives to fuel EVs have emerged. The primary goal of implementing regenerative braking systems in EVs
is to recover the kinetic energy produced during motor deceleration, resulting in increased energy storage capacity and increased driving distance [
]. In [
], a regenerative braking system with one single braking pedal decreased the energy consumption and improved the commercial cost of EVs.
The combination of a supercapacitor and battery bank as an energy supply for an EV has demonstrated an improvement in regenerative braking performance, saving energy and increasing efficiency [
In [
], a proposed design of a boost converter significantly reduces the number of elements compared to existing models, improving energy performance and reducing cost and size. Furthermore, it allows for
the adjustment of converter parameters such as pulse width modulation (PWM) to achieve the power required for the system. DC converters have been implemented with good results in many applications.
In [
] a sliding mode control (SMC) is implemented in DC-DC converters such as the buck and boost circuit. The control law is proposed, considering the continuous-time buck and boost converter
mathematical model. To reduce chattering, a multiphase controller is developed. However, the analysis of these converters is made separately, and only current control can guarantee stability, as it
is the main dynamic to ensure the tracking of a desired voltage reference. In [
], a bidirectional DC-DC converter is operated in a photovoltaic system to take its energy and charge a battery through the converter.
Extensive work is presented in [
], where possible failures that can affect a photovoltaic system, such as the shading fault that leads to the output voltage being lower than expected, and the failure of the DC converter to provide
the desired voltage, are studied. Thus, an SMC with the IncCond technique is designed to track the desired voltage under both failures, providing the maximum available solar power. Additionally, a
global SMC based on Lyapunov stability analysis is presented in [
], where the main goal of applying this controller is to track the desired output voltage value of a buck converter and reduce the chattering generated with the conventional SMC.
Another study is presented in [
], where a strategy based on inverse optimal control is employed to adjust the output voltage of a boost converter. In addition, the design of two observers is developed to reduce the quantity of
sensors. Furthermore, the inverse optimal controller is compared to an SMC. An extremely interesting work is presented in [
], where the two training filters are implemented to obtain the optimal SOC to avoid overcharge of a Li-ion battery used in commercial electric vehicles. Many approaches have been developed to
enhance the performance of the EV and the energy use in the DC converters. One of the more important properties of electric vehicles is the ability to regeneratively brake as a bridge that allows for
the recovery of energy while braking and the improvement of the efficiency of the storage system [
A PI was used to adjust the dynamics of a bidirectional buck-boost in a regenerative braking system with a supercapacitor as an auxiliary energy device in [
]. In [
], the mileage of an EV has been increased by implementing a nonlinear predictive control strategy with two stages to regulate the torque needed during braking. In addition, the optimization of the
torque improves the energy conservation during braking.
To minimize chattering produced by the control to regulate the optimal slip value in an antilock braking system, a sliding mode fuzzy controller is implemented using an exponential reaching law.
Furthermore, the results are compared to a bang–bang controller, with the SM controller outperforming the bang–bang controller in [
Another interesting case is the one depicted in [
], where artificial intelligence is the main tool to develop a “comfort regenerative braking” using neural networks with the backward propagation algorithm, taking data from the acceleration and
deceleration limits and using them to map the optimal behavior of the EV. Regenerative braking has also been improved in [
], where state estimation is achieved using neural networks and the least squares algorithm, and is later used in the proposed control. This study demonstrated an improvement in the recovery of 9.62%
energy under specified conditions. Moreover, an NSMC with an EKF identifier is being developed to improve the EV system with regenerative braking without considering the DC motor behavior and solely
managing the current and voltage of the buck–boost converter within the EV system in [
]. Finally, in an EV framework, a neural inverse optimal control (NIOC) is accomplished by employing a reference generator with the speed of a DC motor that only decelerates once during the entire
operation in [
The simulation of an EV with a regenerative braking system is presented in this paper, where the above-mentioned system is controlled employing a neural sliding mode controller (NSMC) based on system
identification implementing RHONNS trained with EKF and UKF training algorithms to reduce the state estimation error and achieve good performance of the EV system. In this work, the NSMC is used to
track the full dynamics of the regenerative braking system using a reference generator that is related to a DC motor that operates at different speeds, resulting in a signal that changes between
acceleration and deceleration, and allows us to appreciate the operating modes of the bidirectional DC-DC buck–boost converter more accurately. Considering these changes in the speed of the motor is
important, as these approximate the simulation to a real case of an EV driving performance. Furthermore, this approach will enable us to analyze the regenerative braking system of electric vehicles
utilizing multiple robust intelligent control techniques based on Kalman filters, leading to novel findings.
The proposed NSMC regulates the required current and voltage of the DC-DC buck–boost converter, an element of the AES, in order to improve the MES efficiency, particularly in the SOC of the EV
battery. Successful energy recovery during braking system operation is ensured when correct NSMC performance is achieved.
This article’s main novelty is the use of intelligent control algorithms, such as the NSMC, which is developed using RHONN identification and the EKF and UKF as training tools for neural networks,
resulting in a higher SOC in the EV battery. The controller has been validated in simulation for various tasks, including tracking of a varying time–signal trajectory and tracking of a trajectory
generated from a DC motor with multiple acceleration and deceleration changes. The performance of the RHONN with the two training algorithms, EKF and UKF, is validated using a chirp signal. The
initial weights of the RHONN are chosen randomly. Afterwards, the performance of the proposed NSMC with EKF and UKF identifiers is compared to a PI controller demonstrating the differences and
improvements obtained, including Gaussian noise and varying parameters.
The structure of this paper is outlined in
Section 2
, which explains the steps taken to obtain the results of the simulation and the procedure followed in this study.
Section 3
presents the electrical circuit of the DC converter, including a brief explanation of its operation.
Section 4
outlines the mathematical preliminaries, including the equations used to design the neural controller.
Section 5
describes the modeling of the buck–boost converter system and the integration of RHONN identification into the NSMC design.
Section 6
shows the simulation results for each step of the article, including the neural controller test and a comparison of neural identification with EKF and UKF. Additionally, a robustness test of the NSMC
controller is done under two scenarios.
Section 7
presents the mean square error (MSE) and energy of the error signal (EES), comparing their results obtained during the tracking with the UKF, EKF, and PI considering the arrangement used in [
]. Finally,
Section 8
discusses the obtained results and proposes future work. The conclusions of this work based on the results are presented in
Section 9
2. Materials and Methods
The simulation has followed particular steps to obtain the results presented in this article.
• The main objective of this paper is to improve the energy storage and power consumption of the regenerative brake system. To achieve this, the current and voltage dynamics can be controlled using
a bidirectional converter. The regenerative braking system and the neural controller have been simulated and verified using the Simscape Electrical from Matlab (Matlab R2020a, Simulink, 1994–2024
©The MathWorks, Inc., Natick, MA, USA).
• The simulation implements the mathematical representation of the buck–boost converter from [
• The system identification is developed taking the RHONN philosophy in [
] to identify the dynamics to regulate the buck–boost converter. The RHONN is validated using a chirp signal with initial weights chosen randomly in the range of (0, 1).
• The RHONN philosophy is employed to design the identification process, which is then trained with the EKF and UKF algorithms.
• The NSMC is developed to track the trajectories in the AES chosen for the case study. In the first scenario, a signal is selected to validate the identification and regulation of the buck–boost
converter dynamics. The signal is variable, and designed to operate within the range of the AES (supercapacitor) and MES (battery bank) parameters. In the second scenario, the complete control
framework implemented in the simulation is verified. The DC motor parameters are considered for building a reference generator that tracks the EV system requirements.
• The results of the NSMC with EKF and UKF are presented, and the enhancement of the battery banks with the implemented AES is demonstrated.
• The NSMC with EKF and UKF is validated to demonstrated the robustness of the system under different conditions. Additionally, the comparison of the NSMC’s robustness with a PI controller based on
the control strategy [
] is presented.
• The MSEs and EESs obtained with UKF and EKF are compared with a PI control strategy [
] to identify the best performance in regenerative braking simulation and robustness of the control strategies.
3. DC-DC Bidirectional Buck–Boost Converter
A fundamental device in the regenerative braking system is the power converter, such as the boost converter, buck converter, step-down converter, step-up converter, etc. However, the buck–boost
converter is chosen in this case. Additionally, this converter is a fundamental element of the AES, which helps reduce the energy consumption during braking.
Buck–boost converters are the result of the combination of two configurations of DC power converters, which generally work separately in different applications. Furthermore, the configuration
selected for the converter can be a bidirectional DC buck–boost converter related to the requirements of the power system [
], and this configuration is perfectly coupled with energy storage and the control of battery charge and discharge applications.
The regenerative braking system contains several elements that allow it to operate correctly. First of all, there is a battery bank, which is responsible for powering the DC motor that moves the
vehicle. Next, we have the responsible elements of growing or reducing the voltage supplied to the EV. These elements are the supercapacitor and the DC buck–boost converter. The connections of the
system are illustrated in
Figure 1
Here, L is an inductor, C is a capacitor, $T 1$ and $T 2$ represent the insulated gate bipolar transistors (IGBT), and the $D 1$ and $D 2$ represent the diodes. A regulated signal such as the PWM is
the input of the transistors that transfer the current from or to the supercapacitor.
Buck mode:
When this mode is activated, there is a reduction in the output voltage related to the input voltage.
$T 1$
is deactivated and
$T 2$
is activated, resulting in the transfer of the energy coming from the capacitor voltage
$V c$
to the supercapacitor voltage
$V s c$
. Once
$T 2$
is deactivated, the energy stored in the capacitor flows
to the supercapacitor employing the current capacitor
$i c$
. Furthermore, the inductor
is charged with a fragment of the same current capacitor
$i c$
energy. Likewise, when
$T 2$
is deactivated, the
current is discharged into
$V c$
$D 1$
, guiding this current energy in the flow of
Boost operation: During activation of this mode, the output voltage increases concerning the input voltage. $T 1$ is ON and $T 2$ is OFF, and the energy coming from supercapacitor $V s c$ is
transferred to battery bank $V c$. As soon as $T 1$ is activated, the capacitor’s energy is taken and stored in L. On the other hand, when $T 1$ is activated, the energy in L is driving into the
capacitor C across $D 2$ into the battery bank.
While the braking operation is occurring, the electricity generated in the DC motor is managed with the brake into the battery bank or supercapacitor. The boost function is turned ON when the EV is
accelerating, while the buck function is activated when the EV is decelerating. The DC-DC converter operating in both modes makes it easier for the supercapacitor to increase and decrease energy.
Consequently, the SOC in the battery bank is improved.
4. Mathematical Preliminaries
4.1. Discrete-Time Sliding Mode Control
The SMC allows it to work with robustness over some perturbations, being one of their more attractive properties over the last few decades. Consider the following nonlinear system [
with a
$f ( x )$
$| f ( x ) | < f o$
function, which is a bounded constant and a switching function
as the control law, which is used to reduce the error
$e = r − x$
to zero, where the reference input is
, and
is established as
$u = u 0 i f s ( x ) ≥ 0 − u 0 i f s ( x ) < 0$
$s ( x )$
$u 0$
are defined as the sliding surface and the upper control bound, respectively.
Figure 2
describes a continuous-time system with scalar SMC, where the
$x ( t )$
state starts from an initial
$x ( t = 0 )$
point achieving the trajectory of the sliding surface
$s ( x ) = 0$
within finite time
$t s m$
and remaining on the surface afterwards.
When for each sampling point
$t j = k Δ t$
$k = 1 , 2 …$
the function is derived, and a representation in discrete-time of the continuous time system (
) is
with the starting condition
$( t s m )$
, the trajectory equated to the sliding manifold when
$s ( x ( t ) )$
, or
$k m ≥ t s m / Δ t$
$s ( x k ) = 0 ∀ k ≥ k s m$
This operation can be defined as discrete-time sliding mode. Afterwards, from (
), and considering a scenario in which for any constant control
and whatever initial condition
$x ( 0 )$
, its closed solution can be written as [
$x ( t ) = F ( x ( 0 ) , u )$
For the constant state control signal to attain
$s ( x k + 1 )$
, it is required to identify
$u k$
at each sample point
. The corresponding discrete-time representation of the system is the following [
$x k + 1 = F ( x k , u k )$
At each sample point, the sliding manifold is achieved, that is,
$s ( x k + 1 ) , ∀ k = 0 , 1 , ⋯ ∗ ∗ ∗$
is completed. This can be corroborated since
$F ( x ( 0 ) , u )$
tends to
$x ( 0 )$
$Δ t → 0$
; the function
$u ( x ( 0 ) , Δ t )$
may need more control resources than are available for
$u 0$
. Using the discrete-time SMC, the sliding manifold is described as
$s ( x k ) = x k − x r e f , k$
Evaluating at sample time
$( k + 1 )$
, the sliding manifold is determined:
$s k + 1 = F ( x k , u k ) − x r e f , k + 1$
Now is calculated the equivalent control
$u e q ( x k , k )$
, as in [
$u e q ( x k , k ) = − F ( x k , k ) − x r e f , k + 1$
A stabilizing term, determined as
$u n ( x k , k )$
, should be added to asymptotically achieve the sliding manifold [
$u n ( x k , k ) = − S s c s k$
$S s c$
is the Schur matrix.
The following control law is determined considering the boundedness of the control signal
$‖ u c ( x k , k ) ‖ < u 0$
$u 0 > 0$
$u ( x k , k ) = u c ( x k , k ) i f u c ( x k , k ) < u 0 u 0 u e q ( x k , k ) u e q ( x k , k ) i f u c ( x k , k ) ≥ u 0$
represents the Euclidean norm of
$u c ( x k , k )$
and is defined as:
$u c ( x k , k ) = u e q ( x k , k ) + u n ( x k , k )$
4.2. Discrete-Time Recurrent High-Order Neural Networks
Recurrent neural networks (RNNs) have been utilized in recent times to approximate and identify the mathematical models of complex systems, according to [
]. RHONNs have proven that they are effective at identifying nonlinear systems, where identification is achieved through modifying the parameters of a suitable system representation attached to an
adaptive law. The approximated state variable of a nonlinear system using a RHONN identifier with a series-parallel configuration is obtained by [
$χ i , k + 1 = ω i T ϕ i ( x k ) + ω ¯ i T φ i ( x k , u k )$
$χ i , k + 1$
is defined as the state of the
th neuron that identifies the
th component of the state vector
$x k$
, where
$x k = [ x 1 , k , … , x n , k ]$
$ω i , k ∈ ℜ L i$
are the neural networks’ adjustable synaptic weights and
$ω ¯ i , k$
are fixed weights,
$ϕ i$
$φ i$
are linear functions of the state vector, and
$u k$
is the vector input of the RHONN model. Moreover,
$u ∈ ℜ m$
$ϕ i ( x k , u k , k ) ∈ ℜ L i$
is defined as [
$ϕ i ( x k , u k , k ) = ϕ i 1 ϕ i k ⋮ ϕ i L i , k = ∏ j ∈ I 1 ζ i j d i j ( 1 ) ∏ j ∈ I 2 ζ i j d i j ( 2 ) ⋮ ∏ j ∈ I L i ζ i j d i j ( L i )$
$d i j , k$
defined as non-negative integers;
$L i$
as the number of the connection;
$I 1 , I 2 , ⋯ , I L i$
as a collection of non-ordered subsets
$1 , 2 … , n + m$
determined as the state dimension;
as the input dimension; and
$ζ i$
defined as [
$ζ i = ζ i 1 ⋮ ζ i n ⋮ ζ i n + m = S ( x 1 ) ⋮ S x n u 1 , k ⋮ u m , k$
$u = [ u 1 . k , u 2 , k , ⋯ , u m , k T ]$u
is the network’s input vector. Additionally, the hyperbolic function
$S ( . )$
is defined as
$S ( x k ) = α i tanh ( β i x k )$
$x k$
is set as the state variable; while
are positive constants.
th RHONN identifier scheme is presented in
Figure 3
To reduce the error between the estimation of states, the RHONN identifier requires a training algorithm. The explanation of the two training methods proposed is presented below.
4.3. Extended Kalman Filter
The EKF has turned into a typical method applied in many nonlinear systems’ identification. Only if the state equations of the system have determined the linearization, considering a Gaussian random,
can variable operation be possible [
The algorithm model is defined as follows:
$w i , k + 1 = w i , k + η i K i , k e i , k$
$K i , k = P i , k H i , k M i , k$
$P i , k + 1 = P i , k − K i , k H i , k T P i , k + Q i , k$
$e i , k = x i , k − χ i , k i = 1 , 2 , ⋯ , n$
$M i , k = R i , k + H i , k T P i , k H i , k − 1$
$e i ∈ R$
is defined as the identification error to be minimized,
$η i$
is a parameter of the training algorithm design,
$K i , k ∈ R L i × m$
is the Kalman matrix,
$Q i , k ∈ R L i × L i$
$R i , k ∈ R m × m$
are set as positive definite constant matrices, and
$P i ∈ R L i × L i$
is an adjustable diagonal matrix. Finally,
$H i ∈ R L i × m$
is an adjustable matrix, which is defined as the resulting state derivative with respect to the adjustable weights of the neural identifiers. More extensive details and a stability proof explanation
of the RHONN and the EKF training algorithm are presented in [
4.4. Unscented Kalman Filter
The EKF leverages Jacobian matrices for linear approximations. Its effectiveness hinges on two key factors: firstly, accurate “first-order” approximations give rise to suboptimal terms directly
affected by the nonlinearity of the system. Secondly, the computation of Jacobian matrices in practical applications can become complex. Such disparities can cause low performance or divergence. The
UKF addresses the limitations generated by the EKF using a deterministic sampling approach. The limitations generated by the EKF are addressed with the UKF employing a deterministic sampling
A method used to calculate the statistics of a random variable that undergoes a nonlinear transformation is defined by Juliar Uhlmann in [
] and defined as the unscented transformation (UT). This is used as an extension of the recursive estimation of states in the UKF, considering the Kalman filter framework.
The UKF identification algorithm equations can be described as follows:
$K i , k j = P i , k j x y P i , k j y y − 1 , w i , k + 1 j = w i , k j + K i , k j e i , k j , P i , k + 1 j = P i , k j − K i , k j P i , k j y y K i , k j ⊤ ,$
$P i , k j x y = ∑ i = 0 2 L η i c X i , k − j − x ^ i , k j − Y i , k − j − y ^ i , k j − ⊤ , P i , k j y y = R i , k j + ∑ i = 0 2 L η i j c Y i , k − j − y ^ i , k j − Y i , k − j − y ^ i , k j −
⊤ , P i , k j = Q i , k j + ∑ i = 0 2 L η i j c X i , k − j − x ^ i , k j − X i , k − j − x ^ i , k j − ⊤ , e i , k j = X i , k j − x i , k j$
$e i , k j$
is the identification error,
$P i , k j$
is the matrix of prediction error covariance,
$P i , k j y y$
is the covariance of predicted output matrix,
$P i , k j x y$
is the output matrix and the cross-covariance of state,
$w i , k j$
is the
-th weight of the
-th subsystem,
$η i j c$
is a parameter of design,
$X i , k j$
is the
-th state of the plant, and
$x i , k j$
is the
-th state of the neural network.
is defined as the number of states while
$K i , k j$
is the Kalman gain matrix,
$Q i , k j$
is the measurement noise covariance matrix,
$R i , k j$
is the state noise covariance matrix,
$x ^ i , k j −$
is the predicted state mean,
$y ^ i , k j −$
is the predicted output mean, and
$X i , k | k − 1 j$
$Y i , k | k − 1 j$
are sigma-points propagated through prediction and observation, respectively.
Table 1
explains the implementation step-by-step and the comparison of the EKF and UKF. More extensive details and the stability analysis of the UKF training algorithm for neural identification are explained
in [
5. System Modeling and Sliding Mode Neural Control
5.1. Buck–Boost Converter Model
In this application, boost and buck converters were combined for developing the DC-to-DC converter. The first is used in charge-only situations, whereas the second is utilized in discharge-only
situations. The work in [
] is used for the definition of the boost converter model.
$x 1 , k + 1 = ( 1 − t s R C ) x 1 , k − t s c x 2 , k$
$x 2 , k + 1 = x 2 , k + t s L U b t t u c$
The buck converter model is also defined by [
$x 1 , k + 1 = ( 1 − t s R C ) x 1 , k + t s c x 2 , k$
$x 2 , k + 1 = x 2 , k + t s L U b t t u c$
where the converter output voltage is defined as
$x 1 , k$
, the output current is
$x 2 , k$
$U b t t$
is the voltage in the battery,
$u c$
is the set as the input vector,
is the inductance measured in Henry
$( H )$
is the load resistance measured in ohms
$( Ω )$
is the capacitor in measured in farads
$( F )$
, and finally,
$t s$
is the sample time.
5.2. Neural Controller Design
Following the procedures established in
Section 4
and using Equation (
), a RHONN has been employed to close in on the buck–boost converter behaviors in order to manage the current flow and assure the charging and discharging operating modes. Given the RHONN’s adaptive
nature and the similarity of the buck and boost converter models, a single identifier is suggested for both situations as
$χ ^ 1 , k = ω 11 , K S ( x 1 , K ) + ω 12 , K S ( x 2 , K ) + w 1 , 3 S ( x 1 , K ) S ( x 2 , K ) + ϖ 1 x 2 , K + u ( x 1 , k )$
$χ ^ 2 , k = ω 21 , K S ( x 2 , K ) + ω 22 , K S ( x 1 , K ) + ω 23 , K S ( x 1 , K ) S ( x 2 , K ) + ϖ 2 u ( x 2 , k )$
where the estimated dynamics of the vector
$[ x 1 , k , x 2 , k ] T$
$[ χ ^ 1 , k , χ ^ 2 , k ] T$
. As defined above,
$x 1 , k$
is the measured voltage in the converter’s output, and
$x 2 , k$
is set as the converter output current. the control signal chosen for each state of the system is
$ϖ 1$
$ϖ 2$
are constant fixed weights. A representation of the identification design configuration is shown in
Figure 4
5.3. Discrete Time Sliding Mode Control
The sliding mode controller is synthesized once the buck–boost converter’s system identification is carried out. Due to the presented neural model’s triangular construction, only the final dynamics
require control [
For the converter to function effectively in transferring energy to the battery bank, both voltage and current dynamics have to be controlled.
Following the steps in (
), the sliding surface at the sample point
$k + 1$
of the
$x ^ 1 , k$
dynamics control is obtained as
Voltage NSMC
$s x 1 , k + 1 = ω 2 , 1 ( k ) S ( x 2 ) + ω 2 , 2 ( k ) S ( x 1 ) + w 1 , 3 S ( x 1 , K ) S ( x 2 , k ) + ϖ 1 x 2 , k + u ( x 1 , k ) − x 1 r e f , k + 1$
The equivalent
$x ^ 1 , k$
control is calculated as follows:
$u e q ( x k , k ) = − 1 ϖ 1 [ ω 2 , 1 ( k ) S ( x 2 ) + ω 2 , 2 ( k ) S ( x 1 ) + w 1 , 3 S ( x 1 , k ) S ( x 2 , k ) − x 1 r e f , k + 1 ]$
$u ( x 1 , k ) = u c ( x 1 , k ) i f u c ( x 1 , k ) < u 0 u 0 u e q ( x 1 , k ) u e q ( x 1 , k ) i f u c ( x 1 , k ) ≥ u 0$
$u c ( x 1 , k )$
is set as
$u c ( x 1 , k ) = u e q ( x 1 , k ) + u n ( x 1 , k )$
$u n ( x 1 , k ) = − S s c s x 1 , k + 1$
$S s c$
is defined as a square matrix with real entries and eigenvalues conditioned with
$S s c < 1$
, and
$u 0$
is the control upper bound with the condition
$u 0 > 0$
Current NSMC
The sliding surface an control development for
$x ^ 2 , k$
is obtained as
$s x 2 , k + 1 = ω 2 , 1 ( k ) S ( x 2 ) + ω 2 , 2 ( k ) S ( x 1 ) + w 2 , 3 S ( x 1 ) S ( x 2 ) ϖ 2 u k − x r e f , k + 1$
Then, the equivalent current control is obtained as follows:
$u e q ( x 2 , k ) = − 1 ϖ 2 [ ω 2 , 1 ( k ) S ( x 2 ) + ω 2 , 2 ( k ) S ( x 1 ) + w 2 , 3 S ( x 1 ) S ( x 2 ) − x r e f , k + 1 ]$
and the NSMC is implemented as follows
$u ( x 2 , k ) = u c ( x 2 , k ) i f u c ( x 2 , k ) < u 0 u 0 u e q ( x 2 , k ) u e q ( x 2 , k ) i f u c ( x 2 , k ) ≥ u 0$
$u c ( x 2 , k )$
is set as
$u c ( x 2 , k ) = u e q ( x 2 , k ) + u n ( x 2 , k )$
$u n ( x 2 , k ) = − S s c s x 2 , k + 1$
, where
$S s c$
is again considered as a square matrix with real entries and eigenvalues conditioned with
$S s c < 1$
, and
$u 0$
is the control upper bound with the established condition of
$u 0 > 0$
5.4. Reference Generator Development
The regenerative braking system results in energy consumption through DC motor behavior and requires the required voltage and current in the AES. A reference generator has been implemented in the
simulation to put the whole regenerative braking system in motion.
A reference generator was created to provide the intended value for the buck−boost current and voltage. The energy stored in the supercapacitor as a function of the energy produced by the DC motor is
described as the charge reference. A definition considering the work and energy theorem can be consulted in [
]. The mathematical formula for the statement “the work done between point A and point B on a particle increases the kinetic energy” is described as:
Applying this theorem on the DC motor, for a time
$t ∈ [ k δ , ( k + 1 ) δ ) ]$
the energy can calculated as [
$E ( t ) = ∫ k δ t P k d ξ + E k$
$P = τ ω$
is the DC motor power, with the torque and speed defined as
, respectively. The supercapacitor can operate in two modes, charge and discharge. The equation below determines the supercapacitor energy during charging mode as
$E C r e f − c ( t ) = − k p ∫ k δ t s a t 1 ( P ) d ξ + E c k$
The following equation defines the supercapacitor energy during discharging:
$E C r e f − d ( t ) = k p ∫ k δ t s a t 2 ( P ) d ξ + E c k$
$E C r e f − c ( t )$
$E C r e f − d ( t )$
$0 < k p < 1$
defined as the reference energy charge, the reference energy discharge, and the lost energy, respectively.
$s a t 1$
represents a function restricted in
$( − ∞ , 0 )$
, and
$s a t 2$
represents the same function with the restriction in
$( 0 , ∞ )$
. The sum of both energies in the capacitor references (
) and (
), results in the supercapacitor energy reference.
$E C = E C r e f − c + E C r e f − d$
Finally, the obtained reference for the buck−boost converter voltage is written as
$V c r$
is defined as the reference signal computed for the voltage in the supercapacitor,
$E c$
is the stored energy in the supercapacitor, and
is the capacitance value of the supercapacitor. After the
$V c r$
is developed, this can be used in the buck–boost converter control, where the resulting
$u ( x 2 , k )$
is used to estimate the duty cycle in the transistor
$T 1$
$T 2$
. The duty cycle is determined as follows:
$D u t y C y c l e ( P w m 1 P w m 2 ) = ( ∥ u ( x 2 , k ) ∥ 0 ) i f u ( x 2 , k ) < 0 ( 0 ∥ u ( x 2 , k ) ∥ ) i f u ( x 2 , k ) > 0 ( 0 0 ) i f u ( x 2 , k ) = 0$
Figure 5
shows a representation of the regenerative braking system, which includes the developed reference generator block and the NSMC. This representation is subsequently introduced into the simulation. The
performance of the DC motor determines how effectively the regenerative braking systems work; therefore, a PI controller whose gains are calculated based on the Ziegler–Nichols method and the
experience is used to help the DC motor follows a specific speed reference.
$u ω = k ω p ( ω r e f − ω ) + k ω i ∫ 0 t ( ω r e f − ω )$
$ω r e f$
is the speed motor reference and
$k ω p$
$k ω i$
are the controller gains, respectively.
6. Simulation Results
The results of different analyses made in this work are described in this section. First, the validation of the RHONN with both UKF and EKF is illustrated using a chirp signal to guarantee the
correct operation and estimation of the identifier over a rapidly varying signal.
Subsequently, two scenarios are presented, in which the developed NSMC is applied. In the first simulation test, the control and identification of the system is validated using a time-varying signal
operating in a voltage range that the AES and MES can allow. This part illustrates the trajectory tracking of the NSMC without a voltage and current reference generated through the “reference
generator”, explained in
Section 5
The final section of the simulation results demonstrates the complete operation of the regenerative braking system, including the simultaneous operation of the identification, NSMC, reference voltage
and current generators, and the DC motor. Additionally, a robustness test is implemented in order to validate the correct operation of the NSMC controllers with some changes in the signal dynamics
and the parameters of the buck–boost converter.
The proposed control schemes, including the corresponding energy systems used, are implemented and validated with the help of the Matlab/Simulink toolbox “Simscape Electrical” of Matlab. In addition,
the simulation parameters used for the build-up of the energy systems AES and MES are enumerated in
Table 2
6.1. RHONN Validation
The RHONN developed in
Section 4
is validated using a chirp signal with the parameters shown in
Table 3
for each estimated dynamics. The initial weights of RHONN are randomly chosen in a range of
$( 0 , 1 )$
The aim of identifying chirp signals is to guarantee the correct functioning of the RHONN with both training algorithms.
Figure 6
Figure 7
show the identification of the proposed signal for the UKF, where
$x 1 , r e a l$
$x 2 , r e a l$
are the chirp signals used as the voltage and current input, respectively, and
$χ ^ 1 , k$
$χ ^ 2 , k$
are the estimated dynamics with the RHONN. On the other hand, the validation of the identification with EKF is illustrated in
Figure 8
Figure 9
using the same chirp signals as described in
Table 3
6.2. NSMC Trajectory Tracking Results Using UKF
This test allows for the corroboration of the neural controller with the RHONN trained with UKF. The objective here is that the controller could track a desired trajectory operated in a permitted
range of the AES. The chosen signal to track works in a range of 350 V to 380 V. First, the signal starts at 355 V for 10 s; after that, the voltage decreases to 350 V and lasts to t = 20 s. Then,
the signal increases the value to 365 V. Finally, at t = 30 s, the signal operates as a sine wave between 355 V and 375 V and stops at t = 50 s.
The outcome of the trajectory tracking is shown in
Figure 10
, where the signal control is tracked with a minimum error at t = 20 s, where the controller adjustment works fast to reach the desired value. The current tracking trajectory is shown in
Figure 11
, where the current operates related to the controlled voltage, resulting in a signal that operates between −200 and 200 Amps. Furthermore, the controller did not show significant behavior during
fast changes in the signal, as seen around t = 20 s.
Additionally, during the control of the dynamics, the weights of the identifier are adjusted over the tracking of the desired trajectory. The data obtained from the weight adjustment are illustrated
Figure 12
, and represent the computation made during the tracking of the trajectory. Furthermore, the identification of the voltage and current dynamics is illustrated in
Figure 13
. It is important to note that the states identified are the dynamics that take place over the AES, and are tracked subsequently with the NSMC. The proper development of the identification enables
the controller’s proper operation, which leads to the improvement of the braking system’s energy recovery capacity.
The calculation and explanation of the MSE and EES results can be found in
Section 7
, comparing the UKF and EKF with a PI identification differences during the control trajectory tracking with the NSMC.
6.3. NSMC Trajectory Tracking Results Using EKF
The implementation of the NSMC using the EKF is presented.
Figure 14
illustrates the tracking trajectory of the proposed signal where the adjustments made by the controller when the signal changes their parameters are more significant that the one with the UKF.
Figure 15
illustrates the trajectory of current
$x 2$
dynamics tracking using the NSMC with the EKF. The controller results demonstrated a great performance to achieve the values of the tracked signal.
The weights of the RHONN identifier during the control of the desired signal with varying time and the adjustments made by the neural network to acquire the required weight values are presented in
Figure 16
. Additionally, the identified dynamics with the EKF are shown in
Figure 17
6.4. Regenerative Braking System Control Using a Reference Generator with UKF
The complete functioning of the regenerative braking system will be addressed in this section of the simulation results. This test is significant because it relates the entire regenerative braking
system by using the DC motor’s speed variations to provide a signal that can be employed to determine and then implement the NSMC. The DC motor parameters are described in
Table 4
. First, the DC motor operates by varying the speed, which works in the range of 0 to 80 rad/s, with the objective of replicating an EV driving operation where the speed increases and decreases in
different instants. In order to ensure the correct calculation of the reference generator is necessary to control the DC motor speed. A PI controller is applied to control the motor speed, as
depicted in
Figure 18
After the desired speed of the motor is controlled, the reference generator block computes the voltage required to operate the AES system and tracks the signal of voltage provided through the
reference obtained. The neural controller in this case uses the UKF to identify the system as the first step, and then with the estimated dynamics, the tracking of the system dynamics related to the
DC motor speed action is accomplished. The tracking voltage
$V c r$
is the reference obtained with the reference generator block, and
Figure 19
shows the tracking of accomplished voltage related to the motor speed trajectory. Moreover, we can highlight that when the speed in the motor increases, the voltage
$x 1 , k$
decreases. On the other hand, when the speed of the motor decreases, the voltage increases.
In addition,
Figure 20
illustrates the performance of the current dynamics during the regulation with NSMC. The current reference
$( i r e f e r e n c e )$
is obtained through the reference generator, as described in
Figure 5
After that, the trajectory tracking is acquired over the 50 s of simulation time. The SOC of the battery has been compared with and without AES. The importance of the AES is that it allows for slow
discharge on the battery. The results obtained are illustrated in
Figure 21
, where the achieved signal in the AES and the regenerative braking system are operating, which results in a better SOC in the battery. At the beginning of the simulation, the system achieves the
desired reference voltage in approximately 2 s, as seen in
Figure 10
. However, during the rest of the signal tracking, the results overtake the SOC obtained without the AES connected to the EV bank of batteries.
The battery voltage is compared in
Figure 22
with and without the AES. From the results, we can observe that the battery decreases its voltage when the AES is not connected. Moreover, the voltage in the battery increases and maintains its
voltage value above the initial 500 v when the AES is connected.
6.5. Regenerative Braking System Control Using a Reference Generator with EKF
To compare the results later, we used the identical signals in this section of the project’s tests as we did with the UKF.
Figure 23
illustrates that the first trajectory tracking of the
$V c r$
has a difference in the adjustment during the first seconds due to the initial conditions of
$x 1 , k$
and has another behavior compared with the one in the
Figure 19
. Similarly, the
$i r e f e r e n c e$
current tracking trajectory displays a similar behavior in the same section of the simulation.
The SOC and voltage in the battery are described in
Figure 24
Figure 25
, respectively, and display a similar behavior as the NSMC with the UKF. However, the differences in NSMC with UKF and NSMC with EKF are quantitatively compared in
Section 7
6.6. Robustness Test
The reliability of a control system is very important to assure the correct performance of the proposed scheme during variations on the system and disturbances that are not considered initially in
the system’s proposed model. To validate the robustness of the NSMC, two scenarios have been applied. The first one is the system operated by adding noise to the buck–boost converter using a Gaussian
random signal, while the second is a variation of the parameters of the buck–boost converter circuit.
6.6.1. Gaussian Noise
The Gaussian random signal is used to generate random values that will help to validate the robustness of the NSMC, and for this test, the chosen mean value is equal to 0 and variance is 1. The noise
is added to the measured voltage and current output, and Equations (25)–(28) of the buck–boost converter model, considering the Gaussian noise signal, can be defined as
$x 1 , k + 1 = ( 1 − t s R C ) x 1 , k − t s c x 2 , k + G n$
$x 2 , k + 1 = x 2 , k + t s L U b t t u c + G n$
The buck converter model is also defined as
$x 1 , k + 1 = ( 1 − t s R C ) x 1 , k + t s c x 2 , k + G n$
$x 2 , k + 1 = x 2 , k + t s L U b t t u c + G n$
$G n$
is the Gaussian noise added to the dynamics of the buck–boost converter.
Robustness Test with NSMC and UKF
The voltage dynamics affected by the Gaussian noise signal are depicted in
Figure 26
a,b, which are a closer view of the real signal compared with the signal with noise. Additionally,
Figure 27
describes the dynamics of the current, where the noise signal in (a) and (b) is a closer look at the signal affected by the Gaussian noise.
The control signal of the voltage dynamics is illustrated in
Figure 28
and the current dynamics in
Figure 29
. The results demonstrated the robustness of the NSMC using UKF where the desired trajectory is followed still without issues. Additionally, in
Section 7
, the MSE and EES analysis supports the illustrated results.
Robustness Test with NSMC and EKF
The robustness of the NSMC with the EKF has been tested for the current and voltage of the buck–boost converter.
Figure 30
Figure 31
are the signals of the dynamics with the Gaussian noise added to the output signal. Additionally,
Figure 32
Figure 33
illustrate good results in the control of the voltage and current with the noise added to the converter.
Robustness Test with PI
Using the same signal illustrated in
Figure 30
Figure 31
, a PI controller based on the control strategy presented in [
], which is not an intelligent controller because the neural identifier is not performed to estimate the dynamics of the buck–boost converter, is used for the robustness test. The voltage and current
are controlled each, with a PI controller resulting in some issues for the dynamics control, as illustrated in
Figure 34
Figure 35
. The results demonstrated that the PI is not completely reliable for the tracking trajectory considering Gaussian noise. In addition,
Section 7
presents the MSE and EES for the three configurations of the NSMC.
6.6.2. Changes in the Parameters of the Buck–Boost Converter
The second scenario consists of changing the buck–boost converter parameters, which have been used during the different stages of the paper without changes. Now, a variation of its capacitance values
is applied to verify the robustness of the NSMC. The new values are illustrated in
Table 5
. The same NSMC with UKF and EKF is used for the robustness test. The PI controller arrangement of the first robustness test is performed again in this second scenario.
NSMC with UKF for the tracking of dynamics with changes in buck-boost converter
The results of the validation of the NSMC are illustrated in
Figure 36
Figure 37
for voltage and current, respectively. The NSMC still regulates the dynamics of the buck–boost converter, ensuring the energy consumption.
NSMC with EKF for the tracking of dynamics with changes in buck–boost converter
The same parameters described in
Table 5
are used to validate the robustness of the NSMC with EKF illustrated in
Figure 38
Figure 39
. As expected, the controller still regulates the dynamics of the buck–boost converter despite the changes in its parameters.
The results obtained from the test to validate the robustness of the NSMC demonstrate its correct operation under non-ideal scenarios. This includes scenarios where noise and other signals not
initially considered are added to the signal for regulation. However, it is important to note that the NSMC, when used with UKF and EKF, operates differently, and provides results that require
further analysis, as does the PI controller. In
Section 7
, we conduct a detailed analysis where we calculate the MSE and EES.
7. Comparative Analysis between Proposed NSMC Variants and PI Controller
To validate the simulation results and draw a comprehensive conclusion of the work presented in this paper, it is essential to compute the MSE and EES values. The primary objective of this comparison
is to determine which of the two filtering techniques, UKF or EKF, provides better results for the electric vehicle regenerative braking system by calculating a numerical value. Furthermore, the
results obtained are compared with those of a PI controller with the same structure in [
The MSE is obtained as follows:
$MSE = 1 n ∑ i = 1 n ( Y i − Y ^ i ) 2$
and the EES is calculated as follows:
$EES = ∑ i = 1 n ( Y i − Y ^ i ) 2$
where MSE is the calculated mean square error, EES is the calculated energy of the error signal, and in both cases,
is the number of data points to evaluate the computation,
$Y i$
represents the values obtained evaluated at the data points, and
$Y ^ i$
represents the reference values at the data points.
In this case, we use the values of the proposed reference signal and the real values of the states $x 1$ and $x 2$ to obtain the following results.
7.1. MSE and EES of a Time-Varying Signal Controlled with NSMC
First, the MSE and EES results correspond to the control results obtained with the time-varying signal of
Section 6.2
Section 6.3
. The tracking trajectory with the NSMC using EKF performs better than the NSMC using UKF and PI. On the other hand, the PI controller has a better outcome than the NSMC with UKF in the tracking
trajectory for the
$x 1 , k$
Figure 40
presents the three methods compared for the
$x 1 , k$
dynamic, while
Figure 41
shows the three methods applied in the track trajectory for
$x 2 , k$
Table 6
displays the MSE results for the
$x 1$
tracking trajectory, where the NSMC with EKF has better results over the NSMC with UKF and PI. However, the PI controller offers better performance than the NSMC with UKF.
Moreover, MSE and EES results for the
$x 2$
tracking trajectory are illustrated in
Table 7
, where EKF again presents better performance compared to the NSMC with UKF and PI. However, PI again illustrates better results over the NSMC with UKF.
The error signals in both dynamics of the system are demonstrated in
Figure 42
Figure 43
Figure 44
taking the results obtained during the tracking trajectory with UKF, EKF, and PI, respectively. The
$x 1$
error signal shows that the values obtained remain around zero throughout the entire simulation. Although the error signal in
$x 2$
has some instants where the value moves away from zero, the error values remain around zero.
7.2. MSE and EES of the Complete Regenerative Braking System Controlled with NSMC
In this part, the compared results have been taken when the whole connected regenerative braking system is operating. As explained in
Section 6
, the results
$V c r$
$i r e f e r e n c e$
are from the reference generator block, using the parameters provided during the acceleration and deceleration of a DC motor. Two figures of the tracking trajectory, considering the parameters of the
described scenario, are illustrated in
Figure 45
Figure 46
$x 1$
$x 2$
, respectively.
Table 8
, the tracking of the
$x 1$
voltage trajectory with the regenerative braking system is presented. In addition, the NSMC with the EKF presents significantly better performance than the NSMC with the UKF and the PI controller.
These results are different concerning those presented in
Table 6
Table 7
; the NSMC with UKF showed better results compared to the PI.
Table 9
, the tracking of the
$x 2$
current trajectory is presented. Additionally, NSMC with UKF performs much better than NSMC with the EKF and PI controller.
With the results obtained during the UKF, EKF, and PI tracking trajectory, the error signals of the EV system dynamics are demonstrated in
Figure 47
Figure 48
Figure 49
. The
$x 1$
error signal shows that the values obtained remain around zero during most of the simulation and present only a small adjustment at the beginning. On the other hand, the
$x 2$
error values also present an adjustment during the first seconds of the simulation, but eventually, the value of the error is maintained around zero.
7.3. MSE and EES of Robustness Test
The results of the MSE and EES of the robustness test presented in
Section 6
are presented. Two new tables were added to compare the obtained results.
Table 10
presents the results of the voltage dynamics with Gaussian noise for UKF, EKF, and PI. Furthermore,
Table 11
shows the MSE and EES of the tracking trajectory with the same additional noise to the signal. In addition, similar to the case in
Table 8
, the NSMC with EKF presents significantly better performance than the NSMC with UKF and the PI controller for
$x 1$
Additionally, in the obtained results the NSMC with the UKF performs much better than NSMC with the EKF and PI controller for
$x 2$
. The error signals of the system dynamics with Gaussian noise and UKF, EKF and PI are demonstrated in
Figure 50
Figure 51
Figure 52
7.4. MSE and EES of Robustness Test with Changes in the Buck–Boost Converter Parameters
The MSE and EES obtained for the test made with the parameters of the buck–boost converter changed in
Section 6
are presented in
Table 12
for the voltage dynamics and in
Table 13
for the current dynamics. As a result, regarding the voltage
$x 1$
, similar to the case in
Table 8
the NSMC with EKF presents significantly better performance than the NSMC with UKF and the PI controller for
$x 1$
Furthermore, the MSE and EES for the
$x 2$
current dynamics are depicted in
Table 13
, where the results conclude that the NSMC with UKF performs much better than NSMC with the EKF and PI controller for
$x 2$
. The error signals for the system dynamics with the UKF, EKF and PI with the buck–boost converter parameters changes are illustrated in
Figure 53
Figure 54
Figure 55
8. Discussion
This paper presented the comparison of the performance of three controllers: NSMC with EKF, NSMC with UKF, and the PI controller. The novelty of the paper is the design and application via simulation
of a hybrid intelligent control scheme using sliding modes and RHONNs trained with EKF and UKF. Our stated control strategy manages the AES, defined as a combination of a supercapacitor and
buck–boost converter, to recoup the energy lost during braking and power the MES.
The proposed intelligent controller is a new strategy with better results than a PI controller applied to the same bidirectional buck–boost converter circuit, which can be used for a regenerative
braking system in an EV. The performance of three controllers is compared using two scenarios: the first one without considering a reference generator for trajectory tracking and the second one with
a reference generator for trajectory tracking for regenerative braking. For both scenarios, NSMC trained with EKF has better performance, improving the SOC of the battery of the EV.
The states of the AES have been approximated successfully using an RHONN identifier trained by an EKF and UKF for different operation modes. The decision to use an RHONN for the identifier was based
on the nonlinear systems dynamics estimation facilities. The RHONN has been validated with a chirp signal for both training algorithms.
To make an extensive study of the NSMC and the EKF and UKF training algorithms, different tests have been achieved. The identifier using EKF and UKF is implemented for tracking trajectories without
the DC motor considered in the EV system during different AES operation modes. The system identification and synaptic weight adjustment figures help to corroborate the correct performance of the
RHONN identifier.
The NSMC takes the estimated states from the RHONN identifier to define the sliding manifold where the controller will operate. The objective of this controller is to manage the DC buck–boost
converter’s dynamics during charging and discharging operation modes. In the first scenario, time-varying signals are employed to validate the NSMC without still taking into consideration the DC
motor in the EV system but considering the system’s operation range.
In a second scenario, the NSMC is implemented to operate with a completely connected regenerative braking system where the speed of the DC motor is the main parameter to define a reference with the
desired trajectory to provide the necessary energy to supply the EV during the acceleration and deceleration. Although this research focuses mainly on simulating the proposed control scheme, from
this perspective, this particular case enables us to confirm the effectiveness of the suggested NSMC in a scenario that is closer to the actual operation of regenerative braking.
Additionally, in both cases, the NSMC was shown to enhance the EV regenerative braking system when the car is accelerating and decelerating. Furthermore, the neural controller with UKF and EKF
present different and motivating results with and without the reference generator block but the one with better results has been the NSMC with EKF. The most important advantage obtained is in the
SOC, which presents a better performance where the percent of charge decreases slowly with the regenerative braking system in comparison without the regenerative braking system.
9. Conclusions
The MSE results obtained in
Section 7
allow us to determine the framework in which EKF and UKF offer better control synthesis results than a conventional control technique such as PI. This can be a result of the favorable differences
between the two scenarios presented in the simulation. The first case employed a time-varying signal with bounded values to track, while the second case used a signal that is the result of other
dynamics that can induce fast adjustments to achieve the EV requirements during the acceleration and deceleration.
To validate the robustness of the NSMC, two scenarios have been applied. The first one is the system operated adding a Gaussian random signal, where the mean value is equal to 0 and the variance is 1
for $x 1$ and $x 2$.
The noise is added to the measured voltage and current output. The second scenario involves varying parameters on the converter buck–boost circuit. Results are consistent with the tracking trajectory
without the Gaussian noise and the variation of the parameters.
We are encouraged to extend the results of this study to obtain an NIOC based on the UKF algorithm to train a RHONN identifier and to control the AES dynamics enhancing the regenerative braking and
energy consumption behavior in more scenarios, as well as to carry out a comparative study between NSMC and NIOC based on Kalman filters.
As a future work, the authors and coworkers will continue this project by considering the parameters and dimensions of an EV, as well as the typical driving cycles and new challenges and problems to
Author Contributions
Conceptualization, J.A.R.-H.; methodology, J.A.R.-H.; software, M.A.R.C., R.G.-H. and J.F.G.; investigation, J.R.V.-F., M.A.R.C. and J.A.R.-H.; formal analysis, M.A.R.C., R.G.-H., J.F.G. and
J.A.R.-H.; validation, J.A.R.-H., J.-L.R.-L. and R.G.-H.; writing—original draft preparation, M.A.R.C.; writing—review and editing, M.A.R.C., J.F.G., J.A.R.-H., R.G.-H., J.R.V.-F. and J.-L.R.-L.;
funding acquisition, J.A.R.-H. All authors have read and agreed to the published version of the manuscript.
This research was funded by Universidad Autónoma del Carmen. Project number FING/1ERP2023/03.
Data Availability Statement
Data are contained within the article.
The first author thanks to Universidad Autonoma del Carmen for the support and providing the facilities to develop this project.
Conflicts of Interest
The authors declare no conflicts of interest.
The following abbreviations are used in this manuscript:
NSMC Neural Sliding Mode Controller
SMC Sliding Mode Controller
NIOC Neural Inverse Optimal Controller
IOC Inverse Optimal Control
PWM Pulse Width Modulation
UKF Unscented Kalman Filter
EKF Extended Kalman Filter
EV Electric Vehicle
RHONN Recurrent High-Order Neural Network
MES Main Energy System
AES Auxiliary Energy System
DC Direct Current
PI Proportional–Integral
SOC State of Charge
MSE Mean Squared Error
EES Energy of the Error Signal
Variable Description
$i = 1 , 2 , 3 , … , n$ index for name states
$k = 1 , 2 , 3 , … , n$ index for name RHONN states
$x i , k$ neural network adjustable synaptic weights
$ω i , k$ fixed weights
$x k = x 1 , k , x 2 , k , … , x n , k T$ state vector
$X i , k + 1$ state of the ith neuron that identifies the ith component of the state vector $x k$
$ϕ i$, $φ i$ linear functions of the state vector
$u k$ vector input of the RHONN model
$d i j , k$ non-negative integers
$L i$ number of the connection
$I L i$ collection of non-ordered subsets $1 , 2 , … , n + m$
n the state dimension
m the input dimension
$ζ i$ defined as [23]
Extended Kalman Filter
$e i ∈ R$ the identification error to be minimized
$η i$ a parameter of the training algorithm design
$K i , k ∈ R L i × L i$ the Kalman gain matrix
$Q i , k ∈ R L i × L i$ the measurement noise covariance matrix
$R i , k ∈ R m × m$ the state noise covariance matrix
$P i ∈ R L i × L i$ the prediction error matrix
$H i ∈ R L i × m$ the measurement matrix, which is defined as the resulting state derivative with respect to the adjustable weights of the neural identifiers
Unscented Kalman Filter
$e i , k j$ the identification error
$P i , k j$ the prediction error covariance matrix
$P i , k j y y$ the covariance of predicted output matrix
$P i , k j x y$ the output matrix
$w i , k j$ the j-th weight of the i-th subsystem
$η i j c$ a parameter of design
$X i , k j$ the j-th state of the plant
$x i , k j$ the j-th state of the neural network
L the number of states
$K i , k j$ the Kalman gain matrix
$Q i , k j$ the measurement noise covariance matrix
$R i , k j$ the state noise covariance matrix
$x i , k j −$ the predicted state mean
$y i , k j −$ the predicted output mean
$X i , k | k − 1 j$ sigma-points propagated through prediction
$Y i , k | k − 1 j$ sigma-points propagated through observation
Buck–Boost Converter Model
$x 1 , k$ the buck–boost converter output voltage
$x 2 , k$ the buck–boost converter output current
$U b t t$ the voltage in the battery
L the inductance in henry (H)
R the load resistance in ohms ($Ω$)
C the capacitor in farads (F)
$t s$ the sample time
$G n$ Gaussian Noise
Figure 45. Tracking trajectory with the three controllers for voltage $X 1$ with regenerative braking system.
Figure 46. Tracking trajectory with the three controllers for current $X 2$ with regenerative braking system.
Extended Kalman Filter (EKF) Unscented Kalman Filter (UKF)
Design parameters
Analytical Jacobian matrices Scaling parameters
$F n , H n$ $α , β , κ , λ , η i m , η i c$
$Q n , R n , x ^ 0 , P 0$
$x ^ k − = f x ^ k , u k P k − = F k P k F k ⊤ + Q $X i , k − = f X i , k , u k x ^ k − = ∑ i = 0 2 L η i m X i , k − 1 P k − = Q k + ∑ i = 0 2 L η i m X i , k − − x ^ k − X i , k − − x ^ k − ⊤$
$y ^ k − = h x ^ k − P n y y = H n P k − H n ⊤ + R $Y i , k − = h X i , k , u k y ^ k − 1 = ∑ i = 0 2 L η i m Y i , k − P n x y = ∑ i = 0 2 L η i m X i , k − − x ^ k − Y i , k − − y ^ k − ⊤ P n y y
n P n x y = P k − H n ⊤$ = R n + ∑ i = 0 2 L η i m Y i , k − − y ^ k − Y i , k − − y ^ k − ⊤$
$K n = P n x y P n y y − 1 x ^ n = x ^ k − + K n y n − y ^ k − P n = P k − − K n P n y y K n ⊤$
Description Unit
Buck–boost converter resistance R $50 Ω$
Buck–boost converter inductance L 13 mH
Buck–boost converter capacitance $C 1$ 2 mF
Buck–boost converter capacitance $C 2$ 1 $μ$F
Supercapacitor voltage $V s c$ 350 V
Battery bank voltage $V c$ 500 V
Initial SOC 80%
Sampling time $( t s )$ 0.1 $μ$ seg
Chirp Signals Parameters
Controller Initial Frequency Target Time (s) Frequency at Target Time (Hz)
Chirp signal for voltage 0.5 Hz 25 0.6 Hz
Chirp signal for current 0.5 Hz 10 0.5 Hz
Description Unit
Armature resistance $R a$ $2.581 Ω$
Armature inductance $L a$ $0.028$ H
Field resistance $R f$ $281.3 Ω$
Field inductance $L f$ 156 H
Field-armature mutual inductance $L a f$ $0.9483$ H
Power $P w$ 5 H[p]
DC voltage 240 v
Rated speed rpm 1750 rpm
Field voltage 300 v
Description Unit
Buck–boost converter resistance R $50 Ω$
Buck–boost converter inductance L 13 mH
Buck–boost converter capacitance $C 1$ 14 mF
Buck–boost converter capacitance $C 2$ 6 $μ$ F
Supercapacitor voltage $V s c$ 350 V
Battery bank voltage $V c$ 500 V
Initial SOC 80%
Sampling time $( t s )$ $0.1$$μ$ seg
MSE and EES of Tracking Trajectories in $x 1$
Controller MSE Value EES Value
NSMC with UKF 72.9529 $3.647 × 10 8$
NSMC with EKF 1.0585 $5.2923 × 10 6$
PI 1.7403 $8.7016 × 10 6$
MSE and EES of Tracking Trajectories in $x 2$
Controller MSE Value EES Value
NSMC with UKF $1.0427 × 10 4$ $5.2134 × 10 10$
NSMC with EKF 0.1362 $6.8084 × 10 5$
PI 270.1132 $1.3506 × 10 9$
MSE and EES of Tracking Trajectories in $x 1$
Controller MSE Value EES Value
NSMC with UKF 45.2714 $2.2636 × 10 8$
NSMC with EKF 1.0321 $5.1604 × 10 6$
PI 132.4538 $6.6227 × 10 8$
MSE and EES of Tracking Trajectories in $x 2$
Controller MSE Value EES Value
NSMC with UKF 1.9679 $9.8395 × 10 6$
NSMC with EKF 8.9223 $4.4611 × 10 7$
PI $1.8035 × 10 5$ $9.0176 × 10 11$
MSE and EES of Tracking Trajectories with Gaussian noise in $x 1$
Controller MSE Value EES Value
NSMC with UKF 73.4682 $3.6734 × 10 8$
NSMC with EKF 45.7588 $2.2879 × 10 8$
PI $132.9545$ $6.65 × 10 8$
MSE and EES of Tracking Trajectories with Gaussian noise in $x 2$
Controller MSE Value EES Value
NSMC with UKF $1.0435 × 10 4$ $5.2174 × 10 10$
NSMC with EKF $2.780 × 10 4$ $1.390 × 10 10$
PI $1.80 × 10 5$ $9.02 × 10 11$
MSE and EES with Changes in Buck–Boost Converter Parameters in $x 1$
Controller MSE Value EES Value
NSMC with UKF $73.7266$ $3.6863 × 10 8$
NSMC with EKF $46.0728$ $2.3036 × 10 8$
PI $134.9163$ $6.7458 × 10 8$
MSE and EES with Changes in Buck–Boost Converter Parameters in $x 2$
Controller MSE Value EES Value
NSMC with UKF $1.0488 × 10 4$ $5.2439 × 10 10$
NSMC with EKF $2.8186 × 10 3$ $1.4093 × 10 10$
PI $1.8475 × 10 5$ $9.2374 × 10 11$
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Ruz-Hernandez, J.A.; Garcia-Hernandez, R.; Ruz Canul, M.A.; Guerra, J.F.; Rullan-Lara, J.-L.; Vior-Franco, J.R. Neural Sliding Mode Control of a Buck-Boost Converter Applied to a Regenerative Braking
System for Electric Vehicles. World Electr. Veh. J. 2024, 15, 48. https://doi.org/10.3390/wevj15020048
AMA Style
Ruz-Hernandez JA, Garcia-Hernandez R, Ruz Canul MA, Guerra JF, Rullan-Lara J-L, Vior-Franco JR. Neural Sliding Mode Control of a Buck-Boost Converter Applied to a Regenerative Braking System for
Electric Vehicles. World Electric Vehicle Journal. 2024; 15(2):48. https://doi.org/10.3390/wevj15020048
Chicago/Turabian Style
Ruz-Hernandez, Jose A., Ramon Garcia-Hernandez, Mario Antonio Ruz Canul, Juan F. Guerra, Jose-Luis Rullan-Lara, and Jaime R. Vior-Franco. 2024. "Neural Sliding Mode Control of a Buck-Boost Converter
Applied to a Regenerative Braking System for Electric Vehicles" World Electric Vehicle Journal 15, no. 2: 48. https://doi.org/10.3390/wevj15020048
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pH of a Buffer Solution - Chemistry Steps
General Chemistry
There are two ways of calculating the pH of a buffer solution; the equilibrium approach and the one using the Henderson–Hasselbalch equation. The latter is shorter and is used more often. However,
let’s start with the equilibrium approach so that we can see how the Henderson–Hasselbalch equation makes the process easier.
And before that, let’s also recall how we calculate the pH of a weak acid solution because calculating the pH of a buffer containing a weak acid and its conjugate base brings in just one additional
The pH of a Weak Acid – Recap
Remember, we need to set up an ICE table to determine the concentration of hydronium ions at equilibrium based on the K[a] value of the acid.
Let’s recall the process by determining, for example, the pH of 0.50 M solution of acetic acid (CH[3]CO[2]H). K[a] = 1.7 x 10^-5
First, write down the ionization equation and set the molarity of ionized acid as x mol/l.
Next, we write the equilibrium constant expression and make an approximation that x << 0.50, therefore, 0.5 – x ≈ 0.5:
\[{K_{\rm{a}}}\; = \;\frac{{{\rm{[C}}{{\rm{H}}_{\rm{3}}}{\rm{C}}{{\rm{O}}_{\rm{2}}}^{\rm{ – }}{\rm{][}}{{\rm{H}}^{\rm{ + }}}{\rm{]}}}}{{{\rm{[C}}{{\rm{H}}_{\rm{3}}}{\rm{C}}{{\rm{O}}_{\rm{2}}}{\rm
{H]}}}}\; = \;\frac{{\left( {\rm{x}} \right)\left( {\rm{x}} \right)}}{{0.5\, – \;x}}\; \approx \;\frac{{{{\rm{x}}^{\rm{2}}}}}{{0.5\,}}\; = \;1.7\; \times \;{10^{ – 5}}\]
x = 0.0029, so the pH = -log0.0029 = 2.5
And this is how we determine the pH of a weak acid.
The pH of a Buffer using the Equilibrium Approach
Now, let’s say we have a buffer that, in addition to the 0.50 M acetic acid, contains 0.20 M of its conjugate base sodium acetate (CH[3]CO[2]Na).
Calculate the pH of this buffer solution?
The difference here, compared to the solution with only acetic acid, is that the equilibrium concentration of CH[3]CO[2]^– is going to be higher by 0.20 M because CH[3]CO[2]Na is a strong electrolyte
and therefore completely dissociates into ions.
So, we need to write two equations: one for the concentration of H[3]O^+ ions and the other for the concentration of the acetate ions because these two go into the expression of K[a].
The total concentration of the acetate ion at equilibrium is the sum of 0.20 M from the salt and x M from the dissociation of acetic acid.
The equilibrium constant is then equal to:
\[{K_{\rm{a}}}\; = \;\frac{{{\rm{[C}}{{\rm{H}}_{\rm{3}}}{\rm{C}}{{\rm{O}}_{\rm{2}}}^{\rm{ – }}{\rm{][}}{{\rm{H}}^{\rm{ + }}}{\rm{]}}}}{{{\rm{[C}}{{\rm{H}}_{\rm{3}}}{\rm{C}}{{\rm{O}}_{\rm{2}}}{\rm
{H]}}}}\;\; = \;\frac{{\left( {\rm{x}} \right)\left( {{\rm{0}}{\rm{.2 + x}}} \right)}}{{0.5\, – \;x}}\; = \;1.7\; \times \;{10^{ – 5}}\]
Making the approximation that x << 0.2 and x << 0.5, we get that:
\[{K_{\rm{a}}}\; = \;\frac{{{\rm{[C}}{{\rm{H}}_{\rm{3}}}{\rm{C}}{{\rm{O}}_{\rm{2}}}^{\rm{ – }}{\rm{][}}{{\rm{H}}^{\rm{ + }}}{\rm{]}}}}{{{\rm{[C}}{{\rm{H}}_{\rm{3}}}{\rm{C}}{{\rm{O}}_{\rm{2}}}{\rm
{H]}}}}\;\; = \;\frac{{\left( {\rm{x}} \right)\left( {{\rm{0}}{\rm{.2 + }}\cancel{{\rm{x}}}} \right)}}{{0.5\, – \;\cancel{x}}}\; \approx \;\frac{{{\rm{0}}{\rm{.2x}}}}{{0.5\,}}\; = \;1.7\; \times \;
{10^{ – 5}}\]
x = 4.3 x 10^-5
Check the approximation dividing by the initilal concenrtation of acetic acid: 4.3 x 10^-5/0.5 = 0.0086 % – valid
Therefore, the pH is:
pH = -log4.3 x 10^-5 = 4.37
To summarize, calculating the pH of a buffer based on the equilibrium approach, is very similar to what we do in pH calculations of weak acids and bases. The only difference is that the initial
concentration of the conjugate base (or acid if it is a buffer of a weak base and its salt) is not zero in the ICE table.
Let’s now see how the pH of a buffer can be calculated using the Henderson–Hasselbalch Equation.
The pH of a Buffer using the Henderson–Hasselbalch Equation
Let’s work on the same example to compare the two methods.
Calculate the pH of a solution containing 0.50 M CH[3]CO[2]H and 0.20 M CH[3]CO[2]Na. pKa (CH[3]CO[2]H) = 4.75
The Henderson–Hasselbalch equation can be shown as:
Where [conjugate base] or [A^–] would be the concentration of the acetate ion (CH[3]CO[2]Na), and [acid] or [HA] is the concentration of the acetic acid. So, we can write that:
Now, these are the equilibrium concentrations, and we know that it is 0.20 M for the acetate ion because it is completely dissociated. However, acetic acid is a weak acid and not all of it
dissociates into ions. In fact, very little of it does, and the question is if we are still going to use an ICE table to determine the concentration of H[3]O^+ ions, then what is the advantage of
using the Henderson–Hasselbalch equation?
Fortunately, the answer is no – we don’t usually need to set up an ICE table to calculate the pH of a buffer solution.
The key here lies in the fact that acetic acid is a weak acid, and just like we did before, we make an approximation that the amount of the acid that dissociates is very little compared to the
initial concentration of the acid, and therefore, the equilibrium concentration of the acid is the same as its initial concentration. So, it was 0.5 M initially then very little of it dissociated (x
M), but because it is very little, we just assume the equilibrium concentration is about the same as the initial.
And at this point, we only need to plug in the numbers for the initial concentration given in the table:
Comparing the two methods, we have a pH of 4.35 vs 4.37 and this shows that we can make an approximation that x is very small and use the initial concentrations of the buffer components to determine
the pH.
Remember that, in general, we can only make the approximation for x being very small if: (a) the initial concentrations of acids (and/or bases) are large enough, and (b) the equilibrium constant is
fairly small. As a guideline, when determining the concentration of a buffer solution, use the Henderson–Hasselbalch equation if the [HA] or [Base] > 0.10 and it is at least 10^3 times greater than
the equilibrium constant.
Buffers Containing a Base and Its Conjugate Acid
In the example above, we had a buffer composed of an acid and its conjugate base. However, remember, that a buffer can also be composed of a base and its conjugate acid which, unlike a regular acid,
is an ion. Although this does not change the principle, you can write the Henderson–Hasselbalch equation using the words “base and acid”, instead of conjugate base and acid to avoid any confusion:
One common example of such buffer is the solution of weak base ammonia (NH[3]) and its conjugate acid ammonium chloride (NH[4]Cl).
The strategy for calculating the pH is similar to what we did before for the buffers containing a weak acid and its conjugate base.
The only extra step that you may need to do is to determine the pK[a ]of the conjugate acid using its relationship to the pK[b]:
pK[a] + pK[b] = 14
For example,
Calculate the pH of a buffer solution containing 0.60 M in NH[3] and 0.35 M in NH[4]Cl. The pK[b] of ammonia is 4.75.
The first step is to determine the pK[a] so that we can use it in the Henderson–Hasselbalch equation:
pK[a] + pK[b] = 14
pK[a] = 14 – pK[b ]= 14 – 4.75 = 9.25
Because the concentrations of the buffer components are significantly higher than the K[b] of ammonia, all we need to do, at this point, is plugging the numbers in the equation:
\[{\rm{pH}}\; = \;{\rm{p}}{K_{\rm{a}}}\;{\rm{ + }}\;{\rm{log}}\frac{{{\rm{[}}{{\rm{A}}^{\rm{ – }}}{\rm{]}}}}{{{\rm{[HA]}}}}\]
\[{\rm{pH}}\; = \;9.25\;{\rm{ + }}\;{\rm{log}}\frac{{{\rm{[0}}{\rm{.60]}}}}{{{\rm{[0}}{\rm{.35]}}}}\; = \;9.5\]
The Henderson–Hasselbalch equation gives some quick information and hints on choosing a proper buffer correlating pKa and the pH of the solution, and this is what we will discuss in the next couple
of articles.
Check Also
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Validating randomness
In my previous post, I demonstrated the viability of a format-preserving encryption construction used to generate a random permutation of a set, using FNV as the PRF in a four-round Feistel network.
While it does generate permutations that look reasonable at first glance, looks can be deceiving, so I decided to run more tests on the quality of that randomness.
There are two desirable properties of a pseudorandom bitstream. The first is that the bitstream should be uniform; that is, as infinitely many bits are sampled, the probability of encountering any
bit in the bitstream should converge to 50% for 0 and 50% for 1.
Potential bit sequences that meet this quality might start like this:
• 00001111000011110000111100001111
• 10101010101010101010101010101010
• 01110001011101011010111101100001
While the first two sequences are uniform, since the bits appear with equal probability, they probably are not pseudorandom sequences. So while being uniform is a good start, it isn't sufficient. The
second requirement is that the bitstream should be unpredictable. Given knowledge of the entire history of bits seen in the sequence, it should be computationally indistinguishable from randomness
what the next bit will be. This is ideal for applications like hash tables since it prevents a once-common attack vector known as HashDoS, where predictability in the output of common hash functions
could be used to bring simple hash table implementations to their knees.
So, how can we test for these things? One idea to test uniformity might be to measure the probability of a single bit flipping on each consecutive output of a function. If it does not converge to
50%, it is biased and therefore not uniform.
fn test_reasonable_entropy() {
let mut v: Vec<u64> = vec![0; 256];
let mut p: Vec<f64> = vec![0.; 8];
let rnd = fnv_1a(1);
for j in 0..256 {
v[j] = cycle_walking_fpe(j as u64, 256, rnd);
// Calculate probability of each bit flipping
for j in 1..256 {
let x = v[j] ^ v[j - 1];
for i in 0..8 {
p[i] += ((x >> i) & 0x1) as f64 / 256.;
for i in 0..8 {
assert!((0.5 - p[i]).abs() < 0.125);
So let's do that with our FNV hash-based Feistel cipher, over an example block size of 8 bits...
thread 'tests::test_reasonable_entropy' panicked at 'assertion failed: (0.5 - p[i]).abs() < 0.125', src/main rs:136:13
... oops. That's terrible. The probability distributions for each bit flipping (in the sequence of 8 bits) are extremely skewed. Compare that to the probability of bits flipping from a known random
In fact, this weakness in the constructed cipher is so obvious that patterns readily appear in its output (arranged in a table for clarity):
What went wrong?
Well, the Feistel cipher construction is guaranteed to be correct for any round compression function you use, since it uses the property of the invertibility of addition over GF(2) to guarantee its
correctness rather than the invertibility of the round function. That doesn't mean it will be secure, just that it will be decryptable.
But the Feistel network itself clearly isn't an awful construction; for example, it was used in DES, which has mainly been abandoned due to its short key length, not due to the poor quality of its
output. The difference is that DES uses a much stronger round function, which incorporates substitution (S-boxes) and permutation (P-boxes) to produce Shannon confusion and diffusion.
So is there a fast round function that might have these desirable properties? Actually, yes; most modern processors have support for hardware-accelerated AES, and the hardware acceleration comes in
the form of one or two instructions which complete a single round of the AES cipher. The AES round function looks something like the following:
• AddRoundKey - bitwise xor the input with the key
• SubBytes - switch around the input bytes using S-boxes
• ShiftRows - rotate the state rows by an input-dependent manner
• MixColumns - permute the bytes in the state columns
This sequence of steps, which provides significant confusion and diffusion, is implemented by a single hardware instruction on modern Intel and AMD processors (aesenc) and by two hardware
instructions on AArch64 processors supporting SVE (aese, aesmc). However, there is no need to implement a new cryptographic construction around this, as there are already libraries like AHash which
provide a hash function based on an AES round.
I've added AHash to the format-preserving encryption algorithm from last time. First, I add a function to set up with a deterministic key, chosen to be fixed random bytes (this will be needed to
ensure that the cipher behavior is deterministic) and run the hash function:
// AES-round based hasher
fn hash_ahash(input: u64) -> u64 {
let mut hasher = AHasher::new_with_keys(0x5f7788c56cb54593, 0x76fa89eb1eef921d);
And validate the probability distribution:
As intended, it has a very good distribution. But maybe we would be okay with a worse distribution if it meant the computation was faster. So, we need to look at how it performs. Let's run the
test tests::bench_100_2500000_cycle_walking_fpe ... bench: 4,820 ns/iter (+/- 205)
test tests::bench_100_fnv_1a ... bench: 752 ns/iter (+/- 61)
test tests::bench_100_2500000_cycle_walking_fpe ... bench: 2,211 ns/iter (+/- 157)
test tests::bench_100_ahash ... bench: 301 ns/iter (+/- 15)
AHash is clearly superior both in terms of distribution and in terms of performance. Due to being implemented by hardware instructions on modern processors, it runs extremely quickly, and easily
beats the old FNV based approach by 2-3x. That means AHash should always be used where hardware acceleration is available.
As usual, source code.
I am not a cryptographer. The application of using cryptographic techniques to generate pseudorandom permutations is an interesting avenue for amateur exploration due to not possessing any risk if
the encryption is too weak to provide meaningful security or has issues such as side-channels. Do not use this as an excuse to implement your own cryptographic primitives; always reuse standard
implementations as they will almost certainly cover your usecase if you need real security, and if they don't, think twice before blindly implementing anything you read on the internet—you will
probably do it insecurely. | {"url":"https://liamwhite.dev/posts/0012-validating-randomness/","timestamp":"2024-11-05T10:40:51Z","content_type":"text/html","content_length":"13527","record_id":"<urn:uuid:6b5fdb1f-0462-4163-8f99-057d5f4e9bf9>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00067.warc.gz"} |
Equivalency of Reciprocal Instrument Geometries - an03_05
Equivalency of Reciprocal Instrument Geometries - an03_05 Follow
The geometry of a color measurement instrument is defined by the relative arrangement of the light source, sample plane, and detector positions. A line perpendicular to the sample plane is the
reference for 0°. The geometry is described by first indicating the angle of illumination by the light source, and then the angle of viewing by the detector in a format such as 45°/0° or diffuse/8°.
(See the applications Notes titled “Instrument Geometries and Color Measurements.”)
However, when the geometry is the inverse (such as 45° viewing and 0° illumination, or 0°/45°, like HunterLab’s LabScan XE), it is considered to be equivalent, as long as the illumination and viewing
angles are exactly reversed.
The Helmholtz Reciprocal Relation defines this principal of inverse equivalency such that if you swap the positions of the light source and detector, everything else being equal, the measured values
should be the same. That is, a 45°/0° instrument is equivalent to a 0°/45° instrument for reflectance measurements. A diffuse/8° instrument is equivalent to an 8°/diffuse instrument for reflectance
and transmittance measurements.
A couple of caveats:
• The condition of “everything else being equal” rarely exists when we’re talking about two different instrument types or brands. Usually some element in the optical path (area of view, sphere
diameter, light collection angles, etc.) is different between instruments with inverse geometries.
• Although the Helmholtz Relation holds in most applications, in its most strict sense, reciprocity of inverse geometries yielding equivalent results assumes that the sample is a flat, uniform,
non-fluorescent, opaque or transparent solid—in other words, a tile or a piece of glass—and the sample must have a regular scattering pattern. The relation seldom holds for samples that exhibit
irregular scattering patterns, such as translucent samples (like plastic plaques) or samples that trap light (such as loose fibers or plastic pellets), or others with extreme non-uniform
characteristics like curvature.
Reciprocity exceptions also include gonio-apparent samples (metallic, pearlescent, interference coatings, plastics, and cosmetics) where the reflectance and corresponding color values change with the
angle of view.
Frequently Asked Question: I have never seen an 8°/diffuse instrument. All of the sphere instruments in the marketplace appear to be diffuse/8°. Why is this?
Answer: There have been sphere instruments in the past with 0°/diffuse and 8°/diffuse geometries, such as the HunterLab D25P. However, today most manufacturers build the diffuse/8° model only, as it
is more robust when measuring real world samples, and is easiest to manufacture.
Clarke, F. J. J. and Parry, D. J., “Helmholtz Reciprocity: Its Validity and Application to Reflectometry,” Lighting Research and Technology, Volume 17: 1985, pp. 1-11 is a key document that describes
the application of Helmholtz Reciprocity to spectrophotometers.
CIE Publication 15:2004, Colorimetry, Section 5.2 provides a list of all the recommended instrument geometries for colorimetry, including inverse geometries.
ASTM E179, “Standard Guide for Selection of Geometric Conditions for Measurement of Reflection and Transmission Properties of Materials,” Section 8.2 calls the equivalency of reciprocal geometries
the Helmholtz Reciprocal Relation.
ASTM E1164, “Standard Practice for Obtaining Data for Object-Color Evaluation,” describes the inverse reflectance and transmittance geometries in Section 8.
ASTM E1767, “Standard Practice for Specifying the Geometry of Observations and Measurements to Characterize the Appearance of Materials,” is a key document that defines instrument geometry precisely
in terms of the azimuthal angle and includes inverse geometries.
ISO 5-2, Photography -- Density measurements, Part 2: “Geometric conditions for transmission density.”
ISO 5-4, Photography -- Density measurements, Part 4: “Geometric conditions for reflection density.”
These are key documents that provide a precise method of defining instrument geometry for reflectance and transmittance in terms of the azimuthal angle.
Berns, Roy S., Billmeyer and Saltzman’s Principles of Color Technology, 3rd Edition, John Wiley & Sons: New York (2000:82-88) provides a good overview of reciprocal instrument geometries for
colorimetry, and references inverse geometries.
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Blog - Matthias Thul's Homepage
This posts implements a policy class that can be used to configure the behaviour of other classes, such as numerical algorithms, at compile time.
Option Pricing When a Single Jump is Anticipated II
In this post, I augment a constant coefficient geometric Brownian motion process by a single jump whose time of occurrence is known. The random variable representing the jump size follows a normal
mixture distribution. To get a feeling for the impact of predictable jumps on option prices, we inspect the shape of a few model implied probability densities and volatility smiles.
Option Pricing When a Single Jump is Anticipated I
We consider a simple and tractable model for an asset price process that is subject to a large jump with a known time of occurrence. These situations often arise around scheduled news releases such
as quarterly earnings announcements, monetary policy decisions or political elections. This post is the first in a series on this topic. I provide the general pricing setup and derive the
characteristic function of the corresponding logarithmic return process including drift adjustment. Future posts then make specific choices for the jump size distribution and provide examples for the
corresponding implied volatility smiles.
Inverse Gaussian Ornstein-Uhlenbeck Stochastic Clocks
In this pose, we consider an Ornstein-Uhlenbeck stochastic clock whose instantaneous rate of activity process has an inverse Gaussian stationary distribution. We use the previously obtained
relationship between the cumulant generation function of the stationary distribution and that of the background driving Lévy process and show that the latter can be represented as the sum of an
inverse Gaussian processes and a compound Poisson process with chi-squared distributed increments.
Cox-Ingersoll-Ross Stochastic Clocks
In this post, we consider stochastic clocks whose instantaneous rate of activity follows a square root process. We present all intermediate steps in the derivation of the characteristic function of
the corresponding time change. The corresponding steps closely resemble those in the derivation of the zero-coupon bond price in the Cox et al. (1985) model for the short-term interest rate.
C++ Member Function Detector Macro
In C++, we often use the substitution failure is not an error (SFINAE) rule in template overload resolution. A common problem is to write template specializations for class template arguments that
expose member functions with a certain signature. The below macro allows to easily create member function detector type traits for user defined classes
Gamma Ornstein-Uhlenbeck Stochastic Clocks
In this post, we consider one particular specification of the background driving Lévy process in the general Ornstein-Uhlenbeck stochastic clock dynamics introduced in a previous post. We show that a
compound Poisson process with exponentially distributed increments yields a gamma stationary distribution for the instantaneous rate of activity. We also discuss how the problem could be approached
from the other end by imposing the stationary distribution and finding the corresponding background driving Lévy process.
Automatic (Differentiation) for the COS Method
In the last post, I provided a brief introduction to forward mode automatic differentiation with CppAD. In this post, I propose to use automatic differentiation for the computation of cumulants of
option pricing models based on characteristic functions. This is useful, for example, when pricing European vanilla options using the Fang and Oosterlee (2008) COS method. Here, the first four
cumulants are used to determine the integration range.
Automatic Differentiation with CppAD
In quantitative finance, automatic differentiation is commonly used to efficiently compute price sensitivities. See Homescu (2011) for a general introduction and overview. I recently started to look
into it with two different applications in mind: i) computation of moments from characteristic functions and ii) computation of implied densities from parametric volatility smiles. In this post, I
provide a short introduction into computing general order derivatives with CppAD in forward mode.
Suggested Errata for Musiela and Rutkowski (2005) “Martingale Methods in Financial Modelling”
I worked through most of this book when studying for MATH5985 “Term Structure Modelling” at UNSW. The attached document lists some potential typos/inconsistencies in the notation of the 2005 | {"url":"http://www.matthiasthul.com/wordpress/blog/","timestamp":"2024-11-06T17:21:50Z","content_type":"text/html","content_length":"47429","record_id":"<urn:uuid:65576f6b-0fd7-41f7-ab4e-7d9a5bd6cc61>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00605.warc.gz"} |
The long short-term memory (LSTM) operation allows a network to learn long-term dependencies between time steps in time series and sequence data.
This function applies the deep learning LSTM operation to dlarray data. If you want to apply an LSTM operation within a dlnetwork object, use lstmLayer.
Y = lstm(X,H0,C0,weights,recurrentWeights,bias) applies a long short-term memory (LSTM) calculation to input X using the initial hidden state H0, initial cell state C0, and parameters weights,
recurrentWeights, and bias. The input X must be a formatted dlarray. The output Y is a formatted dlarray with the same dimension format as X, except for any "S" dimensions.
The lstm function updates the cell and hidden states using the hyperbolic tangent function (tanh) as the state activation function. The lstm function uses the sigmoid function given by $\sigma \left
(x\right)={\left(1+{e}^{-x}\right)}^{-1}$ as the gate activation function.
___ = lstm(___,Name=Value) specifies additional options using one or more name-value arguments.
Apply LSTM Operation to Sequence Data
Perform an LSTM operation using three hidden units.
Create the input sequence data as 32 observations with 10 channels and a sequence length of 64
numFeatures = 10;
numObservations = 32;
sequenceLength = 64;
X = randn(numFeatures,numObservations,sequenceLength);
X = dlarray(X,"CBT");
Create the initial hidden and cell states with three hidden units. Use the same initial hidden state and cell state for all observations.
numHiddenUnits = 3;
H0 = zeros(numHiddenUnits,1);
C0 = zeros(numHiddenUnits,1);
Create the learnable parameters for the LSTM operation.
weights = dlarray(randn(4*numHiddenUnits,numFeatures),"CU");
recurrentWeights = dlarray(randn(4*numHiddenUnits,numHiddenUnits),"CU");
bias = dlarray(randn(4*numHiddenUnits,1),"C");
Perform the LSTM calculation
[Y,hiddenState,cellState] = lstm(X,H0,C0,weights,recurrentWeights,bias);
View the size and dimensions of the output.
View the size of the hidden and cell states.
Input Arguments
X — Input data
dlarray | numeric array
Input data, specified as a formatted dlarray, an unformatted dlarray, or a numeric array. When X is not a formatted dlarray, you must specify the dimension label format using the DataFormat option.
If X is a numeric array, at least one of H0, C0, weights, recurrentWeights, or bias must be a dlarray.
X must contain a sequence dimension labeled "T". If X has any spatial dimensions labeled "S", they are flattened into the "C" channel dimension. If X does not have a channel dimension, then one is
added. If X has any unspecified dimensions labeled "U", they must be singleton.
H0 — Initial hidden state vector
dlarray | numeric array
Initial hidden state vector, specified as a formatted dlarray, an unformatted dlarray, or a numeric array.
If H0 is a formatted dlarray, it must contain a channel dimension labeled "C" and optionally a batch dimension labeled "B" with the same size as the "B" dimension of X. If H0 does not have a "B"
dimension, the function uses the same hidden state vector for each observation in X.
The size of the "C" dimension determines the number of hidden units. The size of the "C" dimension of H0 must be equal to the size of the "C" dimensions of C0.
If H0 is a not a formatted dlarray, the size of the first dimension determines the number of hidden units and must be the same size as the first dimension or the "C" dimension of C0.
C0 — Initial cell state vector
dlarray | numeric array
Initial cell state vector, specified as a formatted dlarray, an unformatted dlarray, or a numeric array.
If C0 is a formatted dlarray, it must contain a channel dimension labeled 'C' and optionally a batch dimension labeled 'B' with the same size as the 'B' dimension of X. If C0 does not have a 'B'
dimension, the function uses the same cell state vector for each observation in X.
The size of the 'C' dimension determines the number of hidden units. The size of the 'C' dimension of C0 must be equal to the size of the 'C' dimensions of H0.
If C0 is a not a formatted dlarray, the size of the first dimension determines the number of hidden units and must be the same size as the first dimension or the 'C' dimension of H0.
weights — Weights
dlarray | numeric array
Weights, specified as a formatted dlarray, an unformatted dlarray, or a numeric array.
Specify weights as a matrix of size 4*NumHiddenUnits-by-InputSize, where NumHiddenUnits is the size of the "C" dimension of both C0 and H0, and InputSize is the size of the "C" dimension of X
multiplied by the size of each "S" dimension of X, where present.
If weights is a formatted dlarray, it must contain a "C" dimension of size 4*NumHiddenUnits and a "U" dimension of size InputSize.
recurrentWeights — Recurrent weights
dlarray | numeric array
Recurrent weights, specified as a formatted dlarray, an unformatted dlarray, or a numeric array.
Specify recurrentWeights as a matrix of size 4*NumHiddenUnits-by-NumHiddenUnits, where NumHiddenUnits is the size of the "C" dimension of both C0 and H0.
If recurrentWeights is a formatted dlarray, it must contain a "C" dimension of size 4*NumHiddenUnits and a "U" dimension of size NumHiddenUnits.
bias — Bias
dlarray vector | numeric vector
Bias, specified as a formatted dlarray, an unformatted dlarray, or a numeric array.
Specify bias as a vector of length 4*NumHiddenUnits, where NumHiddenUnits is the size of the "C" dimension of both C0 and H0.
If bias is a formatted dlarray, the nonsingleton dimension must be labeled with "C".
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: Y = lstm(X,H0,C0,weights,recurrentWeights,bias,DataFormat="CTB") applies the LSTM operation and specifies that the data has format "CTB" (channel, time, batch).
DataFormat — Description of data dimensions
character vector | string scalar
Description of the data dimensions, specified as a character vector or string scalar.
A data format is a string of characters, where each character describes the type of the corresponding data dimension.
The characters are:
• "S" — Spatial
• "C" — Channel
• "B" — Batch
• "T" — Time
• "U" — Unspecified
For example, consider an array containing a batch of sequences where the first, second, and third dimensions correspond to channels, observations, and time steps, respectively. You can specify that
this array has the format "CBT" (channel, batch, time).
You can specify multiple dimensions labeled "S" or "U". You can use the labels "C", "B", and "T" once each, at most. The software ignores singleton trailing "U" dimensions after the second dimension.
If the input data is not a formatted dlarray object, then you must specify the DataFormat option.
For more information, see Deep Learning Data Formats.
Data Types: char | string
StateActivationFunction — State activation function
"tanh" (default) | "softsign" | "relu"
Since R2024a
Activation function to update the cell and hidden state, specified as one of these values:
• "tanh" — Use the hyperbolic tangent function (tanh).
• "softsign" — Use the softsign function $\text{softsign}\left(x\right)=\frac{x}{1+|x|}$.
• "relu" — Use the rectified linear unit (ReLU) function $\text{ReLU}\left(x\right)=\left\{\begin{array}{cc}x,& x>0\\ 0,& x\le 0\end{array}$.
The software uses this option as the function ${\sigma }_{c}$ in the calculations to update the cell and hidden state.
For more information, see the definition of Long Short-Term Memory Layer on the lstmLayer reference page.
GateActivationFunction — Gate activation function
"sigmoid" (default) | "hard-sigmoid"
Since R2024a
Activation function to apply to the gates, specified as one of these values:
• "sigmoid" — Use the sigmoid function, $\sigma \left(x\right)={\left(1+{e}^{-x}\right)}^{-1}$.
• "hard-sigmoid" — Use the hard sigmoid function,
$\sigma \left(x\right)=\left\{\begin{array}{cc}\begin{array}{l}0\hfill \\ 0.2x+0.5\hfill \\ 1\hfill \end{array}& \begin{array}{l}\text{if}x<-2.5\hfill \\ \text{if}-2.5\le x\le 2.5\hfill \\ \text
{if}x>2.5\hfill \end{array}\end{array}.$
The software uses this option as the function ${\sigma }_{g}$ in the calculations for the layer gates.
For more information, see the definition of Long Short-Term Memory Layer on the lstmLayer reference page.
Output Arguments
Y — LSTM output
LSTM output, returned as a dlarray. The output Y has the same underlying data type as the input X.
If the input data X is a formatted dlarray, Y has the same dimension format as X, except for any "S" dimensions. If the input data is not a formatted dlarray, Y is an unformatted dlarray with the
same dimension order as the input data.
The size of the "C" dimension of Y is the same as the number of hidden units, specified by the size of the "C" dimension of H0 or C0.
hiddenState — Hidden state vector
dlarray | numeric array
Hidden state vector for each observation, returned as a dlarray or a numeric array with the same data type as H0.
If the input H0 is a formatted dlarray, then the output hiddenState is a formatted dlarray with the format "CB".
cellState — Cell state vector
dlarray | numeric array
Cell state vector for each observation, returned as a dlarray or a numeric array. cellState is returned with the same data type as C0.
If the input C0 is a formatted dlarray, the output cellState is returned as a formatted dlarray with the format 'CB'.
Long Short-Term Memory
The LSTM operation allows a network to learn long-term dependencies between time steps in time series and sequence data. For more information, see the definition of Long Short-Term Memory Layer on
the lstmLayer reference page.
Deep Learning Array Formats
Most deep learning networks and functions operate on different dimensions of the input data in different ways.
For example, an LSTM operation iterates over the time dimension of the input data, and a batch normalization operation normalizes over the batch dimension of the input data.
To provide input data with labeled dimensions or input data with additional layout information, you can use data formats.
A data format is a string of characters, where each character describes the type of the corresponding data dimension.
The characters are:
• "S" — Spatial
• "C" — Channel
• "B" — Batch
• "T" — Time
• "U" — Unspecified
For example, consider an array containing a batch of sequences where the first, second, and third dimensions correspond to channels, observations, and time steps, respectively. You can specify that
this array has the format "CBT" (channel, batch, time).
To create formatted input data, create a dlarray object and specify the format using the second argument.
To provide additional layout information with unformatted data, specify the format using the FMT argument.
For more information, see Deep Learning Data Formats.
Extended Capabilities
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The lstm function supports GPU array input with these usage notes and limitations:
• When at least one of the following input arguments is a gpuArray or a dlarray with underlying data of type gpuArray, this function runs on the GPU:
□ X
□ H0
□ C0
□ weights
□ recurrentWeights
□ bias
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2019b | {"url":"https://uk.mathworks.com/help/deeplearning/ref/dlarray.lstm.html","timestamp":"2024-11-11T11:42:56Z","content_type":"text/html","content_length":"128013","record_id":"<urn:uuid:ba8f0216-41d3-418c-bbe0-dde89a08771f>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00395.warc.gz"} |
Sodium to Potassium Ratio in Different Age Groups in context of sodium to potassium ratio
31 Aug 2024
Title: The Sodium to Potassium Ratio in Different Age Groups: A Review of the Literature
The sodium to potassium (Na/K) ratio is a critical indicator of cardiovascular health, with an optimal balance between these two electrolytes essential for maintaining proper heart function and
overall well-being. This review aims to examine the Na/K ratio in different age groups, highlighting the significance of this ratio in various stages of life.
Sodium (Na+) and potassium (K+) are two essential electrolytes that play crucial roles in maintaining fluid balance, nerve conduction, and muscle contraction. The Na/K pump, a transmembrane enzyme
complex, regulates the intracellular concentration of these ions by pumping three sodium ions out of the cell for every two potassium ions pumped into the cell [1]. An imbalance between these ions
can lead to various cardiovascular diseases.
Theoretical Background:
The optimal Na/K ratio is generally considered to be around 1:1.5 (Na+:K+) in healthy individuals [2]. However, this ratio can vary depending on age, sex, and other factors.
Age-Related Changes:
As individuals age, the Na/K ratio tends to shift towards higher sodium levels and lower potassium levels. This change is attributed to various physiological and pathological processes that occur
with aging, such as:
• Increased sodium retention due to decreased renal function
• Decreased potassium intake and increased potassium excretion
• Changes in body composition, including increased fat mass and decreased muscle mass
Na/K ratio = (Na+ concentration) / (K+ concentration)
The Na/K ratio is an important indicator of cardiovascular health, with age-related changes affecting this balance. As individuals age, the Na/K ratio tends to shift towards higher sodium levels and
lower potassium levels, increasing the risk of cardiovascular disease.
In conclusion, the Na/K ratio in different age groups is a critical factor in maintaining proper heart function and overall well-being. Understanding the age-related changes in this ratio can help
healthcare professionals develop targeted interventions to prevent and manage cardiovascular diseases.
[1] Guyton, A. C., & Hall, J. E. (2016). Textbook of Medical Physiology. Philadelphia, PA: Elsevier Saunders.
[2] Katz, A. M. (2006). Physiology of the Heart. Philadelphia, PA: Lippincott Williams & Wilkins.
Related articles for ‘sodium to potassium ratio’ :
Calculators for ‘sodium to potassium ratio’ | {"url":"https://blog.truegeometry.com/tutorials/education/a7051fb87da908f813b9335d3e0a2e9b/JSON_TO_ARTCL_Sodium_to_Potassium_Ratio_in_Different_Age_Groups_in_context_of_so.html","timestamp":"2024-11-05T16:30:32Z","content_type":"text/html","content_length":"16727","record_id":"<urn:uuid:b69b0475-54a5-4c10-8bfe-f906ec90ee0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00658.warc.gz"} |
Measurements of Inclusive [Formula presented] Production
Using the combined CLEO II and CLEO II.V data sets of [Formula presented] at the [Formula presented], we measure properties of [Formula presented] mesons produced directly from decays of the [Formula
presented] meson, where “[Formula presented]” denotes an admixture of [Formula presented], [Formula presented], [Formula presented], and [Formula presented], and “[Formula presented]” denotes either
[Formula presented] or [Formula presented]. We report first measurements of [Formula presented] polarization in [Formula presented]: [Formula presented] and [Formula presented]. We also report
improved measurements of the momentum distributions of [Formula presented] produced directly from [Formula presented] decays, correcting for measurement smearing. Finally, we report measurements of
the inclusive branching fraction for [Formula presented] and [Formula presented].
Dive into the research topics of 'Measurements of Inclusive [Formula presented] Production'. Together they form a unique fingerprint. | {"url":"https://experts.umn.edu/en/publications/measurements-of-inclusive-formula-presented-production","timestamp":"2024-11-09T05:13:04Z","content_type":"text/html","content_length":"54787","record_id":"<urn:uuid:ebe3dd17-e54c-435b-aab5-792d9f912a45>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00132.warc.gz"} |
Calculate transmission line Matched Line Loss from Rin of o/c or s/c
Calculate transmission line Matched Line Loss from Rin of o/c or s/c resonant or antiresonant section
This calculator allows calculation of transmission line Matched Line Loss from Rin of open or shorted resonant or antiresonant line section (see Measuring matched line loss).
The results are valid only for linear circuits.
The calculator does not do a lot of error checking, if you enter nonsense, it will probably produce nonsense.
The calculator solves \(MLL=\frac{-10}{length} log |\frac{Rin-Zo}{Rin+Zo}| \; dB/m\). The accuracy depends on the accuracy of Zo.Because of the Rin-Zo term, accuracy becomes very sensitive to errors
in Zo when Rin approaches Zo. In practice, this does not normally limit the use of the technique at HF and above, if Rin approaches Zo, use a shorter resonant length that gives an Rin that you can
measure reasonably accurately.
The response of Rin is quite broad for the low impedance resonance, but VERY sharp for the high impedance resonance (antiresonance) making it more difficult to find the true value of Rin as X crosses
zero. So, whilst the calculator works correctly for high impedance resonances (antiresonances), make your measurements where ever possible at low impedance resonances as they will tend to have less
Some instruments work better for very low Z measurements which you may encounter, eg noise bridges, so always make the measurements for low Z resonances in that case.
• Duffy, O. Feb 2016. Calculation of Matched Line Loss from measurement of Zin at resonance or antiresonance of a short circuit or open circuit transmission line section. https://owenduffy.net/
┃Version│Date │Description ┃
┃1.01 │25/02/2016 │Initial. ┃
┃1.02 │ │ ┃
┃1.03 │ │ ┃
┃1.04 │ │ ┃
┃1.05 │ │ ┃
© Copyright: Owen Duffy 1995, 2021. All rights reserved. Disclaimer. | {"url":"https://www.owenduffy.net/calc/MllFromRin.htm","timestamp":"2024-11-09T16:10:57Z","content_type":"text/html","content_length":"7436","record_id":"<urn:uuid:cc755f73-4bbe-4332-851f-c98f502225c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00605.warc.gz"} |
Multiple-Valued Logic Design in Current-Mode CMOS
Over the last two decades, design using Multiple-Valued Logic (MVL) has been receiving considerable attention. With recent advancements in processing technologies it is now possible to implement
higher-radix functions. MVL circuits have shown the potential of improving the use of chip area through increased functional density and reduced requirements for interconnection wiring. Currently, it
is advantageous if MVL circuits are used along with binary circuits in the design of Very Large Scale Integration (VLSI) circuits. In order to minimize the fabrication process related overheads, it
is desirable to design MVL circuits that can coexist on the same chip with binary logic circuits. This thesis considers MVL design in current-mode CMOS compatible with existing binary logic CMOS
fabrication processes. A given MVL function can be realized in the sum-of-product (SOP) form. The realization cost, in terms of the number of required product terms (PTs) and hence the circuit area,
depends on the set of operators used. The choice of the set of operators is technology dependent. A new set of operators, consisting of literal, cycle, complement of literal, complement of cycle,
min, and tsum operators is proposed in this thesis. A general structure has been identified for realizing MVL circuits in currentmode CMOS. The structure consists of input, control, and output
blocks. The input and the output blocks deal primarily with multiple-valued current signals. The control block signals are binary voltage signals providing flexibility by allowing the use of
arbitrarily complex binary circuits. MVL circuits have been designed for the new set of operators using the general structure. The cost, in terms of number of transistors, of realizing the new set of
operators and that of each of the existing sets of operators is comparable. The functionality of the designed circuits has been verified for 4-valued logic using HSPICE transient analysis
simulations. MVL function realizations using the new set of operators requires fewer PTs compared to realizations based on the existing sets of operators. A comprehensive comparison has been
conducted for all 19683 3-valued 2-variable functions. The maximum number of PTs is reduced from 6, using the existing set of operators, to 3, using the new set of operators. The average number of
PTs is decreased from 3.61 to 2.61. Sample 4-valued 2-variable function realizations using the new set of operators have also shown similar improvements. In the past, several heuristic-based programs
have been reported to obtain an efficient SOP expression for a given MVL function. HAMLET is one of these programs. It accepts user specified function expression and minimizes this expression. It
includes implementation of many earlier reported heuristic-based algorithms. The Gold heuristic chooses the best realization after applying all other heuristics implemented in HAMLET. A new
heuristic-based synthesis program has been developed to obtain a SOP expression for a given MVL function using the proposed set of operators. For a random sample of 4-valued 2-variable functions, the
realizations have been compared for the developed program and the existing program, HAMLET (Gold). It is observed that for 69% of the functions, the developed program performs better and for 4% of
the functions the HAMLET (Gold) realizations are better. The average number of PTs required by the developed program and HAMLET (Gold) are 5.53 and 6.62, respectively. In order to further reduce the
number of PTs required for MVL function realizations two new approaches have also been identified. The new approaches are based on difference-of-sum-of-products (DOSOP) and sum-of-terms (SOT)
realization of MVL functions. It has been shown that the required number of terms can be reduced in the realization of some MVL functions, using the new approaches, as compared to those obtained
using the SOP realization.
Doctor of Philosophy (Ph.D.)
Electrical and Computer Engineering
Electrical Engineering | {"url":"https://harvest.usask.ca/items/15de457a-adae-4e0c-8807-89aeba435746","timestamp":"2024-11-08T11:12:38Z","content_type":"text/html","content_length":"468303","record_id":"<urn:uuid:7b69e30f-9500-46b4-9037-8d1e0381a1ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00501.warc.gz"} |
Revision #1 to TR17-097 | 26th September 2017 22:41
Multi Collision Resistant Hash Functions and their Applications
Collision resistant hash functions are functions that shrink their input, but for which it is computationally infeasible to find a collision, namely two strings that hash to the same value (although
collisions are abundant).
In this work we study multi-collision resistant hash functions (MCRH) a natural relaxation of collision resistant hash functions in which it is difficult to find a t-way collision (i.e., t strings
that hash to the same value) although finding (t-1)-way collisions could be easy. We show the following:
1. The existence of MCRH follows from the average case hardness of a variant of the Entropy Approximation problem. The goal in the entropy approximation problem (Goldreich, Sahai and Vadhan, CRYPTO
'99) is to distinguish circuits whose output distribution has high entropy from those having low entropy.
2. MCRH imply the existence of constant-round statistically hiding (and computationally binding) commitment schemes. As a corollary, using a result of Haitner et-al (SICOMP, 2015), we obtain a
blackbox separation of MCRH from any one-way permutation.
TR17-097 | 31st May 2017 19:49
Multi Collision Resistant Hash Functions and their Applications
Collision resistant hash functions are functions that shrink their input, but for which it is computationally infeasible to find a collision, namely two strings that hash to the same value (although
collisions are abundant).
In this work we study multi-collision resistant hash functions (MCRH) a natural relaxation of collision resistant hash functions in which it is difficult to find a t-way collision (i.e., t strings
that hash to the same value) although finding (t-1)-way collisions could be easy. We show the following:
1. The existence of MCRH follows from the average case hardness of a variant of Entropy Approximation, a problem known to be complete for the class NISZK.
2. MCRH imply the existence of constant-round statistically hiding (and computationally binding) commitment schemes.
In addition, we show a blackbox separation of MCRH from any one-way permutation. | {"url":"https://eccc.weizmann.ac.il/report/2017/097/","timestamp":"2024-11-12T11:51:13Z","content_type":"application/xhtml+xml","content_length":"23672","record_id":"<urn:uuid:4d2322bb-0486-4399-a079-c892b1d02a03>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00760.warc.gz"} |
Accounting | Accounting homework help
1. A project has an initial cost of $40,000, expected net cash inflows of $12,000 per year for 6 years, and a cost of capital of 13%. What is the project’s NPV? (Hint: Begin by constructing a
timeline.) Do not round intermediate calculations. Round your answer to the nearest cent.
2. Assume you are now 21 years old. You plan to start saving for your retirement on your 25th birthday and retire on your 65th birthday. After retirement, you expect to live at least until you are
85. You wish to be able to withdraw $54,000 every year from the time of your retirement until you are 85 years old (i.e., for 20 years). The average inflation rate is likely to be 5 percent.
Calculate the lump sump you need to have accumulated at age 65 to be able to draw the desired income. Assume that the annual return on your investments is likely to be 10%.
3. Consider the purchase of a machine for automation of an important process that would cost $210,000. The machine is estimated to have a 5-year useful life and a salvage value at the end of the
five years of $48,000. The machine would reduce labor costs by $62,000 per year. Additional working capital of $7,000 would be needed immediately, all of which would be recovered at the end of 5
years. The minimum required return is 17% on all investment projects.
Determine the NPV of the project. | {"url":"https://www.essay-writing.com/accounting-accounting-homework-help-108/","timestamp":"2024-11-07T18:30:38Z","content_type":"text/html","content_length":"63988","record_id":"<urn:uuid:7e79cdbf-3029-46f3-bc77-5b21abdaeb11>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00444.warc.gz"} |
Data structure - What is an iterative algorithm?
What is an iterative algorithm?
An iterative algorithm executes steps in iterations. It aims to find successive approximation in sequence to reach a solution. They are most commonly used in linear programs where large numbers of
variables are involved.
What is an iterative algorithm?
The process of attempting for solving a problem which finds successive approximations for solution, starting from an initial guess. The result of repeated calculations is a sequence of approximate
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Numerical simulation of single bubble motion fragmentation mechanism in Venturi-type bubble generator
Issue Mechanics & Industry
Volume 25, 2024
Article Number 21
Number of page(s) 16
DOI https://doi.org/10.1051/meca/2024016
Published online 05 July 2024
© J. Chen et al., Published by EDP Sciences 2024
OpenFOAM: Open Field Operation and Manipulation
CFD: Computational fluid dynamics
R^2 : Curve fitting determination coefficient
c: Controllable compression factor
Fσ: Surface tension model momentum source term
σ: Surface tension coefficient
1 Introduction
Microbubbles have been widely used in power, chemical, mining and petroleum applications. In the petroleum industry, the high viscosity of heavy and residual oils results in incomplete combustion and
a tremendous waste of energy. Therefore, researchers have come up with the idea of injecting microbubbles into heavy and residual oils so that a considerable number of bubbles can be formed, thus
increasing the combustion area of the liquid fuel and improving combustion efficiency. In addition, microbubbles are also the basis for enhanced mass transfer by increasing the interface contact area
and phase interface renewal frequency. Microbubbles have a large specific surface area, vital concomitant, enormous internal pressure, long residence time in water, etc., and are an excellent carrier
to strengthen the mass transfer between gas and liquid in industrial production [1]. Efficient bubble generation technology can improve energy efficiency in crude oil refining, liquid fuel
combustion, and other processes. It is also a hot topic in the research of petrochemical systems.
Up to now, the structure design and performance of the Venturi-type bubble generator have been extensively studied. However, less research has been done on the bubble breakup process and mechanism of
Venturi-type bubble generators. The general view is that bubble transport in a Venturi-type bubble generator is a complex unsteady flow phenomenon [2,3], including many complex flow characteristics,
such as turbulence, two-phase flow, and cavitation [4]. Early research on bubble breakup showed that bubble breakup in turbulent fields results from the collision of turbulent vortices of similar
size to bubbles [5]. In experimental studies [6], bubble breakage in the diverging section caused by a pressure surge was experimentally captured. Zhao et al. [7,8] measured the bubble velocity and
estimated the bubbling force. It was concluded that the rapid deceleration of the bubble in the diverging section had an essential effect on the process of bubble breaking. Uesawa et al. [9]
experimentally analyzed the bubble fragmentation process and concluded that the pressure difference on the bubble surface causes the bubble to breakup. Fujiwara et al. [10] believed that the
instability of the bubble interface caused bubble fragmentation due to the rapid increase of pressure in the asymptotic section. Song et al. [11] attribute bubble fragmentation to a combination of
turbulent vortex-to-bubble collisions, bubble surface instability, and viscous shear. Ding et al. [12] used a fluid volume model coupled with a level set method to splice virtual bubbles into a
steady flow in a venturi and validated the numerical model through visualization experiments. It is found that the bubbles have binary crushing mode and multiple crushing mode. The axial pressure
gradient force of the binary fragmentation mode mainly exists in the centre region. The radial shear force leads to multiple fragmentation modes in the sidewall region.
From the experimental point of view, high-speed photography technology has helped study the Venturi-type bubble generator. However, there are still some problems with the validity of data processing
and measurement of pulsating velocities and pulsating pressures in venturi turbulent fields. The current understanding of the shedding, breaking, motion, diverging, and regulation law of bubbles
under complex working conditions is minimal, which further limits the scaled-up design and industrial application of Venturi-type bubble generators.
Numerical simulation has become an effective means of studying the complex multiphase flow characteristics in venturi channels with the development of computational fluid dynamics (CFD) technology.
Ju et al. [13] concluded from numerical simulations that there is enormous shear stress at the entrance of the dilated end, and the bubbles show complete breakup. Ding et al. [14] calculated the
bubble-breaking behaviour based on the VOF model in Fluent for different initial bubble positions and found that the turbulent dissipation rate is negatively correlated with the diameter of the
bubble breakup. Zhao et al. [15] calculated the diverging section and the force state of the bubble. The results show that the rotational motion of the bubble is caused by shear stress, and pressure
gradient force plays a crucial role in the deformation of the bubble. Li et al. [16] investigated the bubble fragmentation mechanism in the Venturi-type bubble generator under a low flow rate based
on numerical simulation and found that vortex plays a significant role in the transition of different fragmentation mechanisms. The proportion of the disturbed broken area expands with the movement
of the vortex upstream in the diverging section, and the efficiency of bubble generation increases accordingly. Bie et al. [17] studied the dynamics and rupture of bubbles in a swirl-venturi bubble
generator (SVBG) using a high-speed camera system and a bubble image velocimeter. The research shows that the two trajectories of bubbles in the axial and radial directions are closely related to
foam bursting. The vortex region expands for relatively high liquid Reynolds numbers, and the bubble size decreases.
In the current work, the breakage process of a single bubble in a Venturi-type bubble generator was studied with the multiphase flow model VOF and the Large eddy simulation (LES) turbulence model in
the OpenFOAM framework. The deformation and breakup behaviour of the initial bubble in a two-dimensional flow field were studied by numerical simulation. The influence of initial bubble position,
vortex motion, flow velocity, diverging angle and other factors on bubble fragmentation characteristics was investigated. The causes of uneven bubble size generated by the Venturi-type bubble
generator were discussed and analyzed. The above research provides a theoretical reference for improving the foaming performance of Venturi-type bubble generators, producing microbubbles more
efficiently and improving the energy utilization efficiency of the petrochemical industry.
2 Numerical methodology
2.1 Phase equation
The mainstream interface capturing methods mainly include the LS (Level Set) [18] method and the VOF [19] method. As an interface tracing method built under Eulerian mesh, the VOF method provides a
simple and economical way to trace free boundaries in 2D or 3D meshes. It ensures fluid mass conservation during the computational process and handles the nonlinear phenomena of free surface
reconstruction. It is more flexible and efficient than the finite difference method in dealing with complex free surface configuration problems.
The phase equation of multiphase flow is the continuity equation for each phase. In the VOF method, a variable α is introduced to represent the phase fraction of the fluid in the grid cell. The phase
equation can be expressed as
where U is the velocity of the mixed phase in the grid cell, the phase fraction α has distinct numerical characteristics in a gas-liquid two-phase flow system, where the liquid phase fraction. The α
[L] and the gas phase fraction α[G] satisfyα[L ] + α[G] =1. If the grid cell is filled with liquid phase α[L]=1, if the grid cell is filled with gas α[G]=1. If the value is between 0 and 1, the
grid cells are mixed with gas and liquid.
The method used in OpenFOAM is to add an artificial compression term to the phase equation of interFoam to squeeze the phase fraction near the phase interface to counteract the blurring of the phase
interface due to this numerical dissipation. The artificial compression term in the phase equation is solved based on the MULES (Multi-dimensional Universal Limiter with Explicit Solution) algorithm
of the FCT (Flux Corrected Transport) method [20, 21]. Therefore, the phase equation (1) can be rewritten in the following form:
The U[c] is the compression velocity to be modelled at the phase interface, and its direction is perpendicular to the interface. Specifying c as the controllable compression factor, U[c] may be
expressed as
From equation (2), it can be seen that as the value of c increases, the larger the value of U[c], the more pronounced the interface compression effect is. Numerical stability may decrease with
increasing phase interface resolution as c increases. The physically unrealistic high speeds at the phase interface are parasitic currents [22]. Deshpande et al. analyzed and tested the problem and
control of parasitic currents in interFoam. By testing, the generation of parasitic currents was found to be excessive or decaying with the coulomb number in the unsteady state, whereas almost no
droplet displacement occurs in the steady state calculations [21]. This is also similar to Harvie's findings that the generation of parasitic currents is only secondary to the moving interface [22].
Considering the limitations of the size of the venturi model and the calculation conditions in this paper, the value of c is set to 1 [15]. The final form of the phase equation of the interFoam
solver is as follows.
2.2 Governing equations
For the gas–liquid two-fluid model, the multinomial fluid continuity and momentum equations considering gravity and source terms can be expressed as
where ρ is the mixed phase density, µ is the mixed phase viscosity, and F[σ] is the surface tension model momentum source term. The continuous surface tension model CSF (Continuum Surface Force)
method can convert the surface tension at the gas–liquid interface into an equivalent bulk force using an additional pressure gradient. F[σ] is
where σ is the surface tension coefficient, the final expression of the momentum equation in the interFoam solver takes the form.
2.3 Turbulence model
The gas phase is transformed into a multi-scale cluster of bubbles through a complex crushing process that occurs in only a few milliseconds in a venture-type bubble generator. It is crucial to
select a turbulence model that can capture the large-scale structural evolution and transient turbulence characteristics of the flow field. LES is applied in many related studies [23]. In this work,
LES obtained the large-scale turbulent structure by directly solving the Navier–Stokes equations, while the sub-grid scale ones are modelled using the Smagorinsky closure model [24]. Dhotre et al. [
25] stated that the Smagorinsky model could obtain results similar to the higher-level Smagorinsky model with reasonable model constants. Therefore, the constant CSGS model in the Smagorinsky model
was selected to be 0.094, according to the literature [26].
3 Numerical procedures
3.1 Structure of Venturi-type bubble generator
The venture-type bubble generator used in this study is a rectangular structure, which is a quasi-three-dimensional channel. Figure 1 shows the schematic and geometry of the venture-type bubble
generator, divided into a converging section, throat, and diverging section with different lengths and angles. The diverging section has sufficient width to improve the liquid phase shear force and
promote the shear breakup process of bubbles. Considering practicality, the angles of the converging section and the diverging section of the bubble generator are 45° and 15°, respectively.
3.2 Grid independence and comparison with experimental results
The grid is structured and divided into three parts ABC, as shown in Figure 2. A previous study found that the interaction between the liquid and gas is most pronounced at the entrance location of
the evanescent region [27]. Hence, it is possible to densify the grid in zone B. The grid of zone A used a rougher grid. Based on the above requirements, four grids were studied independently, i.e.,
with 80,000, 100,000, 120,000, and 130,000 cell grids, respectively. In this case, the inlet flow rate was set to 0.162 m^3/h, and the bubble generation was in the middle of the converging section
with an initial bubble size of 1mm. The time-averaged velocity distribution at x=8mm is obtained as the grid size increases to 130,000, as shown in Figure 3. The difference in time-averaged
velocity between the cell sizes of 120,000 and 130,000 is minimal. Therefore, a grid size of 120,000 is a relatively economical solution, satisfying y+ ≈ 1 for the throat and adjacent zones.
3.3 Boundary conditions and bubble patching method
The fluids for the numerical simulations are water and air. Due to the low velocity of bubbles in the venturi channel (v < 100m/s) and negligible gas density. Both water and air are set as
incompressible fluids. The physical parameters of air and water are shown in Table 1.
The motion behaviour of a bubble in a venture-type bubble generator under different flow conditions was investigated by changing the inlet velocity. The fluid inlet was set to velocity inlet, and the
outlet was set to pressure outlet. The numerical simulation working conditions are shown in Table 2, where Re represents the inlet Reynolds number, and Ca represents the capillary number.
Pressure−velocity coupling was solved by the semi-implicit pressure-linked equation (SIMPLE) via a relationship between velocity and pressure corrections. The PIMPLE algorithm, which uses a
combination of PISO and SIMPLE (Semi-Implicit Method for Pressure-Linked Equation), is able to handle transient calculations better under significant time-step conditions. The basic idea of the
PIMPLE algorithm is to consider the iterative calculations within each time step as a steady state and utilize the SIMPLE algorithm to calculate the steady-state flow within a single time step. The
standard PISO algorithm is used to compute the last step and finalize the iteration when the steady-state computation reaches a predetermined number of iterations [15]. The calculation step was
adjusted to 2×10^−6s. Every 1000 steps of calculation were automatically saved. The gravity direction was set to the negative Y direction.
During the numerical simulation, gravity and the presence of residual curves closely monitor the stability of the flow field. When the flow field is stable, we suspend the simulation to add a
spherical bubble to the flow field. The simulation is continued afterwards. The diameter of the patched bubble is ϕ. This is done as follows: (1) a spherical region with a specific location and size
is marked for adaptation through the region; (2) the gas phase is patched to initialize the specific region during the initialization period, and the α[G] is set to 1.
3.4 Experimental validation
Figure 4 shows a macroscopic comparison between the numerical results in the present study and previous experimental results of Mo et al. [28] based on the same venturi channel. The results
demonstrate the change in velocity magnitude along the flow direction of a typical bubble as it enters the throat. The experimental and numerical simulation results show a consistent trend. In the
throat section, the bubble velocity remained stable along the flow direction and gradually slowed down at the entrance of the diverging section. The bubble experienced a sharp deceleration due to the
rapid splitting and bursting, which indicated that the bubble was experiencing tremendous resistance. The decrease in flow velocity led to increased static pressure in the diverging section along the
flow direction, preventing the bubble from moving downstream and slowing down the bubble along the flow direction. In summary, the simulation results are consistent with the experimental results.
4 Results and discussion
4.1 Effect of the vortex on single bubble motion fragmentation
To investigate the effect of vortex motion on the fragmentation characteristics of individual bubbles under operating conditions with Re=27,950. After the flow field was stable, a single bubble
with a diameter of ϕ1mm was injected into the patch at the position of space coordinate (0.02, 0, 0), which meant the single bubble was injected from the water inlet. The transient diagram and
streamline diagram post-processing results of bubble movement, deformation and fragmentation are shown in Figure 5. The complete transient diagram of the bubble movement and breakup between the two
red dotted lines. All the transient diagrams are combined to form the evolutionary process of the bubble in vortex generation, breaking, collapse, and block coalescence. Through observation, it can
be found that there are three prominent areas in the process: steady-state disturbed breakup, unsteady-state disturbed breakup and concomitant vortex disturbance. Vorticity (Ω) is usually used to
measure the intensity and direction of the vortex, which is defined as:
where vx and vz are radial velocity and axial velocity, respectively.
The vorticity variation curve on the axial centerline of the Venturi-type bubble generator is shown in Figure 7. The fluid has almost no vortex intensity before entering the throat and produces
fluctuations for the first time after entering the throat. Vorticity decreases rapidly and remains stable after entering the steady state disturbed breakup region. The vorticity increases sharply and
fluctuates significantly when it enters the unsteady state disturbed break area, begins to fall back, and continues to decrease until it reaches the region concomitant vortex disturbance. The
vorticity increases and fluctuates sharply once it enters the unsteady state disturbed breakup region, and it begins to descend until it reaches the vortex disturbance region. Although Li et al. [15]
mentioned the mechanism of bubble fragmentation in the first two regions, the influence of vorticity on bubble disturbance fragmentation was not analyzed.
The bubble undergoes four processes: micro-deformation, stretching and deformation, breakup, collapse and chunking, as shown in Figure 5. This gas-phase transport process is further divided into 18
frames with a time interval of 0.6ms in Figure 6. The gas–liquid phase interface is extracted using the isovolumetric method, with iso=0.75 set, and the flow field cloud is a transient velocity
distribution. The image identified by the blue border in Figure 6 captures the motion-breaking process of the bubble in the converging section. The bubble moves smoothly initially and undergoes small
deformations. The bubble begins to stretch and deform under the impact of the fluid, then gradually evolves into a bullet shape, and finally, the tail begins to concave. The tail of the bubble breaks
through the head of the bubble and splits into two bubbles. The flow field is relatively stable until the bubble enters the throat. The image identified by the purple border in Figure 6 captures the
motion-breaking process and flow field characteristics of the bubble in the throat. The bubble's velocity increases significantly after entering the throat. The bubbles pass quickly through the
throat in two parts. The bubble does not break up as the fluid velocity increases significantly in the narrow interval of the throat. The image identified by the red border in Figure 6 captures the
motion-breaking process of the bubbles in the diverging section. The bubbles undergo a sharp breakup and collapse and form more small bubbles after entering the diverging section. The flow velocity
is high in the middle region of the diffusion section. Vortices are formed near the wall due to the sidewall effect. Bubbles break up under the shear of the vortex. Combined with the vorticity change
curve, it can be seen that the phenomenon of bubble fragmentation and disintegration is most significant in the unsteady state disturbed breakup region. However, bubbles can coalesce in the vortex
disturbance region since bubbles experience a long period of turbulent collision and energy dissipation from convergence to divergence. Bubbles are more likely to break and collapse due to the
increased instability of the bubble surface. The bubble breakup mechanism is a typical steady-state disturbed breaking mode. In this process, the velocity difference and forward motion state of the
captured bubble surface indicate that viscous shear plays a dominant role in the fragmentation of bubbles, which are split into two main parts under the action of viscous shear force. Therefore, the
mechanism of the steady-state perturbation breakup mode is due to interfacial instability caused by Kelvin-Helmholtz instability perturbation [29,30].
The motion characteristics of the bubbles are incredibly complex, and the bubbles are rapidly deformed and broken under the surface tension in the process. The velocity contour shows that the bubble
surface has uneven velocity distribution and an unsmooth shape, which indicates that the acceleration effect acts in the different directions of the bubble. This bubble interface fluctuation can be
described as the instability of the density interface under average acceleration, namely Rayleigh-Taylor instability [31]. It can be seen that the captured bubble gradually enters the region with
larger vortex pulsation from the corresponding position of the streamline diagram. The bubble undergoes significant stretching deformation and moves forward with an inclination angle at the end of
the unsteady-state disturbed breakup region.
It is important to note that the bubbles captured in the upper part are giant. Viscous shear still plays an essential role in bubble deformation because of a tremendous velocity gradient at the
bubble surface. Meanwhile, the bubble edges and lower ends are constantly shedding tiny bubbles, a typical erosive breakup shear process. It can be found that R-T instability induces bubble
deformation, viscous shear enhances defects on the bubble surface, and turbulent pulsations in the flow field cause vorticity changes, which ultimately lead to bubble breakage. The vorticity in the
vicinity of the bubble gradually increases as the bubble continues to move downstream. However, the vortex moves diagonally upwards at x=0.095m due to the turbulent kinetic energy of the liquid
not being sufficient to maintain the vortex's wall-hugging motion after a certain distance from the jet has occurred. The broken bubbles at the upper and lower parts gather to form two core bubbles
under the effect of the vortex when the static pressure in the channel rises. Meanwhile, the volume of the single bubble decreases significantly in the third stage, so the corrosive bubble
fragmentation disappears. The core bubbles mainly accompanied the vortex for rotational stretching deformation without further breakup.
To investigate the effect of the axial position of the initial bubble on the bubble breakup characteristics under operating conditions with Re=27,950. A single bubble with a diameter of ϕ1mm was
injected into the patch at the spatial coordinate (0.035, 0, 0) after the flow field was stable, which indicated that the throat injected the single bubble. The bubble split into the upper and lower
parts of the throat, as shown in Figure 8. The bubble began to deform more extensively after entering the bifurcation section. In addition, the reduction of water flow velocity and rebound of static
pressure caused the bubble to break gradually. Precisely, the bubble undergoes concave deformation and tensile deformation before entering the throat when the bubble is injected into the converging
section. Whether in the converging section or the throat, the bubble appeared to have significant deformation and tended to split into upper and lower parts. However, the degree of fragmentation in
the subsequent flow field of the bubble at the converging section is more potent than that of the bubble at the throat. Since the bubble surface instability is enhanced after a more extended
turbulent collision and energy dissipation in the converging section, it is easier to break up after entering the diverging section.
The relationship between the trajectory of a single bubble and the trajectory of the centre of the vortex is further explored by analyzing the trajectory of the bubble and the vortex. The upper and
lower vortex centre coordinates of the Venturi-type bubble generator were extracted at different times. The centre coordinates of the bubble movement and the centre coordinates of the vortex in the
upper and lower parts were compared and analyzed, as shown in Figure 9. It could be seen that the upper part of the vortex and the bubble have been maintained against the wall motion, and the motion
trajectory could be linearly fitted as a straight line. In contrast, the lower part is detached, and the motion trajectory is fitted as a parabola. The four trajectory lines were fitted separately to
obtain the results shown in Table 3. It can be seen that all the curve fit coefficients of determination (R^2) are close to 1. In the process, the motion trajectory of the bubble centre and the
vortex centre are highly coincident with the deformation, breakup, collapse and aggregation of the bubble vortex.
4.2 Velocity and turbulent kinetic energy distributions
According to the work of Baylar [32], turbulent kinetic energy can be given as
where k is the turbulent kinetic energy, ε is the turbulent dissipation rate, l is the turbulence length scale, C[μ] is the turbulent constant, and d[h] is the diameter of the throat.
Two cross-sectional locations in each of the three regions in the diverging section were selected to extract their radial velocity and radial turbulence parameters for characterization, as marked by
the blue dashed lines in Figure 5 (x =0.05m, 0.06m, 0.07m, 0.085m, 0.105m, and 0.12m). The results of the curves are shown in Figures 10 and 11. At x=0.05m and 0.06m positions, the maximum
velocity and minimum turbulent kinetic energy are distributed in the central region (–0.004m ≤ y < 0.004m). The velocity in the central region does not change with the radial position, and the
shear effect weakens. The fluctuations and turbulence between the layers are very weak. In contrast, the velocity gradient is more prominent, and the shear stress and turbulence are more substantial
in the sidewall region, where the bubble fragmentation is more intense. It can be seen that the turbulence parameter in the middle of the diverging section is much lower than in the sidewall region,
which results in a weaker fragmentation of the bubble in the middle of the diverging section. Meanwhile, the bubble size of the sub-bubbles produced in the intermediate region is more significant.
The radial velocity and turbulent kinetic energy change gradually as the flow progresses into the unsteady disturbance breakup region. The radial velocity and turbulent kinetic energy centre deviate
from the cross-section at x=0.085m, and the maximum radial velocity is still concentrated in the central region. However, the turbulent kinetic energy in the middle region is more prominent than
in the side wall region. The radial velocity gradient and turbulent kinetic energy along the flow direction at the interface between the centre and the side wall region decrease gradually. The
deviation trend is also maintained, and the interface gradient between radial velocity and turbulent kinetic energy is further reduced in the region vortex disturbance. The main reason is that the
vortex cannot keep moving forward and tends to move upward after the jet reaches a certain distance, which leads to the deviation between the velocity centre and the distribution range of turbulent
kinetic energy. The growth deviation of the vortex also causes the turbulent kinetic energy to increase gradually in the central region, resulting in the gradient of radial velocity and turbulent
kinetic energy at the boundary between the side wall and the central position decreasing progressively. The growth deviation of the vortex also causes the turbulent kinetic energy to increase
gradually in the central region, resulting in the gradient of the radial velocity and turbulent kinetic energy at the boundary between the side wall and the central position decreasing progressively.
4.3 Effect of liquid flow rate
Different patterns of bubble fragmentation are clearly observed in Figure 12. The liquid Reynolds number covers a range of 27,950–55,899. The acceleration, rotation and tensile deformation processes
occur in three flow conditions after the bubbles enter the throat from the converging section for Re=27950. The bubble enters the diverging section and begins to decelerate. Accordingly, the bubble
rotation also experiences two similar stages: fast and slow. In both processes, the bubble is stretched and lengthened continuously, mainly due to the formation and growth of the vortex. The movement
of the vortex causes the bubble to break, and the larger the flow velocity of the liquid, the more pronounced the bubble breaks and collapses in the diverging section.
As the liquid Reynolds number increases, new patterns of breakup appear in the venture-type bubble generator. As soon as the mother bubble enters the diverging section, the bubble undergoes violent
deformation in the process of movement and breakup up into a mass of tiny bubbles rapidly. The larger the Reynolds number bubble, the smaller the bubble size of the sub-bubbles. This breakup mode can
be named “static erosion breakup”, proposed by Wang [33]. Meanwhile, the increase in Reynolds number leads to the increase in turbulence intensity. Static erosion breakup and dynamic erosion breakup
are more evident under the effect of turbulent fluctuation and collision [17]. Static erosion breakup and dynamic erosion breakup are surrounded by blue and red lines, respectively, as shown in
Figure 12. The mother bubble is broken into upper and lower parts by fluid erosion in a static erosion breakup mode. The mother bubble moves in the radial direction and deforms severely in the
dynamic erosive breakup mode. The parent bubble breaks into a cluster of tiny bubbles during rotation and stretching.
The fluid velocity and turbulent energy at the position of section (x=0.085m) are distributed along the radial direction, as shown in Figures 13 and 14. The radial velocity and turbulent kinetic
energy at the cross-sectional position (x=0.085m) become progressively more extensive overall. However, the distribution form does not change as the Reynolds number increases. The more significant
the Reynolds number, the more drastic the change of turbulent kinetic energy. The middle region still forms a low turbulent region, similar to the radial velocity distribution. The number of
sub-bubbles produced in the side wall region is more significant, and the bubble size is smaller with the flow rate increase, while the bubble size of sub-bubbles in the centre region is less
affected. The difference in velocity between the sidewall region and the centre region leads to the formation of vortexes in the sidewall region. This is the reason for the intense turbulence in the
sidewall region. The interaction between vortexes and the main jet flow generates a stagnation effect and forms stagnation zones. Bubbles have a higher probability of experiencing vortices and
stagnation zones in steady-state disturbed breakup regions. The vortex near the wall has a strong interaction with the bubble, increasing the deformation rate of the bubble. The bubble breakup is
more favoured in the motion state. The stagnation zone prevents the radial movement of the bubble and causes the bubble to be sheared by the strong flow, resulting in a static erosive breakup. When
the bubbles are not affected by the stagnation zone, the inward driving force in the central low-pressure region pushes the bubbles to move in the radial direction, which is the main reason for the
formation of dynamic erosion breakup.
4.4 Effect of diverging section tension angle
Four models with diverging angles (β) of 15°, 20°, 25° and 30° were constructed to compare the effect of different diverging angles on single bubble fragmentation. The comparison results are shown in
Figure 15. The bubble fragmentation becomes more apparent, and the bubble size of sub-bubbles generated by bubble fragmentation decreases gradually in the diverging section with the gradual increase
of diverging angle. The side wall effect weakens the bubble fragmentation, resulting in the decrease of bubble size distribution uniformity of bubbles. The tiny bubbles moving with the core bubble
are no longer visible, and the accompanying sub-bubbles have already collapsed along the flow direction in the region vortex disturbance when the divergence angle is 25° and 30°.
The fluid velocity and turbulence parameters along the radial distribution in the middle diverging section (x=0.085m) with different diverging angles are shown in Figures 16 and 17. As can be seen
from the figure, the radial velocity at the section position shows a decreasing trend with the increase of diverging angle. The radial velocity is no longer unimodal but a fluctuating trend from the
side wall to the central area. The greater the angle of the diverging section, the more serious the loss of kinetic energy and the faster hydrostatic pressure rises in the direction of flow. The
distribution of radial turbulent kinetic energy parameters is similar to that of radial velocity. The fluctuations become more and more evident with the increase of diverging angles. The greater the
turbulent kinetic energy is, the more significant the gradient of change from the edge wall to the central region is. On the contrary, the more pronounced the fluctuation is. The centre area of
turbulent kinetic energy also tends to move to the lower part with the increase of diverging angle. The centre of the bubble tends more towards the lower part as the angle of divergence increases, as
seen in Figure 14. Moreover, the dip angle of the bubble moving upward in the region concomitant vortex disturbance is more significant. In this condition, the bubbles are more susceptible to the
5 Conclusions
In this paper, the VOF model and the LES large vortex simulation were used to simulate the fragmentation behaviour of a single bubble in a venturi-tube bubble generator. The instantaneous motion
state of the bubble and the factors influencing bubble motion and fragmentation were analyzed. Based on this research, the following conclusions are obtained:
• Under the given working conditions, there are three fragmentation mechanisms and motion states of a single bubble: the steady-state disturbed breakup, the unsteady-state disturbed breakup, and
the concomitant vortex disturbance region. Bubble motion tracks are different in all three regions. The fitted data revealed that the bubble trajectory was highly similar to that of the vortex
centre. Vortex motion is the critical factor for bubble motion and fragmentation.
• The turbulent kinetic energy in the central region is smaller than in the sidewall region in the steady-state disturbed breakup region. The difference in velocity between the sidewall region and
the centre region leads to the formation of vortexes in the sidewall region. This is the reason for the intense turbulence in the sidewall region. The interaction between vortexes and the main
jet flow generates a stagnation effect and forms stagnation zones. The stagnation zone prevents the radial movement of the bubble and causes the bubble to be sheared by the strong flow, resulting
in a static erosive breakup. When the bubbles are not affected by the stagnation zone, the inward driving force in the central low-pressure region pushes the bubbles to move in the radial
direction, which is the main reason for the formation of dynamic erosion breakup. Differences in the radial position distribution of turbulent kinetic energy directly lead to uneven bubble size
distribution. The fragmentation of the bubble injected by the converging section is more dramatic than that injected by the throat in the same axial direction.
• As the liquid Reynolds number increases, static erosion breakup and dynamic erosion breakup are more pronounced. The size of the sub-bubbles formed by the mother bubble is much smaller. The
central region of the fluid's turbulent kinetic energy tends to move lower in the evanescent section. The more intense the bubble breakup, the more sub-bubbles are produced in the sidewall
region. The boundary layer appears to play the role of separation to strengthen the fragmentation and refinement of bubbles with the increase of diverging angles. The bubble size of the
sub-bubbles also decreases gradually. However, the side wall effect weakens bubble fragmentation, reducing the uniformity of bubble size distribution. Subbubbles are more likely to break in the
vortex disturbance region with a diverging angle between 25° and 30°.
The feasibility of a highly efficient bubble generator is demonstrated at a theoretical level by investigating the movement of single bubbles within a Venturi-type bubble generator. An academic
reference is provided for acquiring highly efficient microbubbles in the petrochemical industry.
Acknowledgement of all individuals involved in project of Open Fund of Shandong Key Laboratory for Oilfield Produced Water Treatment and Environmental Pollution Control.
This research was funded by [Open Fund of Shandong Key Laboratory for Oilfield Produced Water Treatment and Environmental Pollution Control] grant number [No. ZC0607-0007].
Conflicts of interest
The authors have nothing to disclose.
Data availability statement
The manuscript has relevant research data which can be disclosed upon request.
Author contribution statement
Conceptualization, Junliang Chen and Mao Lei; Methodology, Junliang Chen; Software, Mao Lei; Validation, Junliang Chen, Mao Lei and Shaobo Lu; Formal Analysis, Xiaolong Xiao; Investigation, Mingxiu
Yao; Resources, Junliang Chen; Data Curation, Junliang Chen and Mao Lei; Writing − Original Draft Preparation, Junliang Chen; Writing − Review & Editing, Junliang Chen; Visualization, Shaobo Lu;
Supervision, Xiaolong Xiao and Qiang Li; Project Administration, Mingxiu Yao and Qiang Li; Funding Acquisition, Qiang Li.
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Graph A Quadratic Function In Vertex Form
ShowMe graphing vertex form with fractions
Graph A Quadratic Function In Vertex Form. It also shows you how to find the coordinates of the. Use the vertex and intercepts to sketch the graph of the quadratic function.
ShowMe graphing vertex form with fractions
Web 25k views 8 years ago. Web students will use vertex form to graph quadratic functions and describe the transformations from the parent function with 70% accuracy. When you graph a quadratic,
there are a couple of things you need to consider that will make your life easier. Web we call this graphing quadratic functions using transformations. Web when graphing a quadratic function with
vertex form, the vertex's x and y values are h and k respectively. The parent function f(x) =. Use the vertex and intercepts to sketch the graph of the quadratic function. That means it can be
written in the form f(x) = ax2 + bx + c,. In the first example, we will graph the quadratic function f(x) = x2 by plotting points. Web for graphing quadratic functions using the standard form of the
function, we can either convert the general form to the vertex form and then plot the graph of the quadratic.
Graphing quadratic functions in vertex form, reflections, horizontal and vertical shifts, finding the equation of a quadratic function, finding a quadratic. When you graph a quadratic, there are a
couple of things you need to consider that will make your life easier. In this article, we review how to graph quadratic functions. The parent function f(x) =. Web to illustrate the steps in graphing
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varcov_family: Variance-covariance family of psychonetrics models in psychonetrics: Structural Equation Modeling and Confirmatory Network Analysis
This is the family of models that models only a variance-covariance matrix with mean structure. The type argument can be used to define what model is used: type = "cov" (default) models a
variance-covariance matrix directly, type = "chol" (alias: cholesky()) models a Cholesky decomposition, type = "prec" (alias: precision()) models a precision matrix, type = "ggm" (alias: ggm())
models a Gaussian graphical model (Epskamp, Rhemtulla and Borsboom, 2017), and type = "cor" (alias: corr()) models a correlation matrix.
varcov(data, type = c("cov", "chol", "prec", "ggm", "cor"), sigma = "full", kappa = "full", omega = "full", lowertri = "full", delta = "diag", rho = "full", SD = "full", mu, tau, vars, ordered =
character(0), groups, covs, means, nobs, missing = "listwise", equal = "none", baseline_saturated = TRUE, estimator = "default", optimizer, storedata = FALSE, WLS.W, sampleStats, meanstructure,
corinput, verbose = FALSE, covtype = c("choose", "ML", "UB"), standardize = c("none", "z", "quantile"), fullFIML = FALSE, bootstrap = FALSE, boot_sub, boot_resample) cholesky(...) precision(...) prec
(...) ggm(...) corr(...)
data A data frame encoding the data used in the analysis. Can be missing if covs and nobs are supplied.
type The type of model used. See description.
Only used when type = "cov". Either "full" to estimate every element freely, "diag" to only include diagonal elements, or a matrix of the dimensions node x node with 0 encoding a
sigma fixed to zero element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each
element/slice encoding such a matrix.
Only used when type = "prec". Either "full" to estimate every element freely, "diag" to only include diagonal elements, or a matrix of the dimensions node x node with 0 encoding a
kappa fixed to zero element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each
element/slice encoding such a matrix.
Only used when type = "ggm". Either "full" to estimate every element freely, "zero" to set all elements to zero, or a matrix of the dimensions node x node with 0 encoding a fixed
omega to zero element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each
element/slice encoding such a matrix.
Only used when type = "chol". Either "full" to estimate every element freely, "diag" to only include diagonal elements, or a matrix of the dimensions node x node with 0 encoding a
lowertri fixed to zero element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each
element/slice encoding such a matrix.
Only used when type = "ggm". Either "diag" or "zero" (not recommended), or a matrix of the dimensions node x node with 0 encoding a fixed to zero element, 1 encoding a free to
delta estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a matrix.
Only used when type = "cor". Either "full" to estimate every element freely, "zero" to set all elements to zero, or a matrix of the dimensions node x node with 0 encoding a fixed
rho to zero element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each
element/slice encoding such a matrix.
Only used when type = "cor". Either "diag" or "zero" (not recommended), or a matrix of the dimensions node x node with 0 encoding a fixed to zero element, 1 encoding a free to
SD estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a matrix.
Optional vector encoding the mean structure. Set elements to 0 to indicate fixed to zero constrains, 1 to indicate free means, and higher integers to indicate equality constrains.
mu For multiple groups, this argument can be a list or array with each element/column encoding such a vector.
tau Optional list encoding the thresholds per variable.
vars An optional character vector encoding the variables used in the analyis. Must equal names of the dataset in data.
groups An optional string indicating the name of the group variable in data.
covs A sample variance–covariance matrix, or a list/array of such matrices for multiple groups. Make sure covtype argument is set correctly to the type of covariances used.
means A vector of sample means, or a list/matrix containing such vectors for multiple groups.
nobs The number of observations used in covs and means, or a vector of such numbers of observations for multiple groups.
If 'covs' is used, this is the type of covariance (maximum likelihood or unbiased) the input covariance matrix represents. Set to "ML" for maximum likelihood estimates (denominator
covtype n) and "UB" to unbiased estimates (denominator n-1). The default will try to find the type used, by investigating which is most likely to result from integer valued datasets.
How should missingness be handled in computing the sample covariances and number of observations when data is used. Can be "listwise" for listwise deletion, or "pairwise" for
missing pairwise deletion.
equal A character vector indicating which matrices should be constrained equal across groups.
baseline_saturated A logical indicating if the baseline and saturated model should be included. Mostly used internally and NOT Recommended to be used manually.
The estimator to be used. Currently implemented are "ML" for maximum likelihood estimation, "FIML" for full-information maximum likelihood estimation, "ULS" for unweighted least
estimator squares estimation, "WLS" for weighted least squares estimation, and "DWLS" for diagonally weighted least squares estimation.
The optimizer to be used. Can be one of "nlminb" (the default R nlminb function), "ucminf" (from the optimr package), and C++ based optimizers "cpp_L-BFGS-B", "cpp_BFGS", "cpp_CG",
optimizer "cpp_SANN", and "cpp_Nelder-Mead". The C++ optimizers are faster but slightly less stable. Defaults to "nlminb".
storedata Logical, should the raw data be stored? Needed for bootstrapping (see bootstrap).
Which standardization method should be used? "none" (default) for no standardization, "z" for z-scores, and "quantile" for a non-parametric transformation to the quantiles of the
standardize marginal standard normal distribution.
WLS.W Optional WLS weights matrix.
sampleStats An optional sample statistics object. Mostly used internally.
verbose Logical, should progress be printed to the console?
ordered A vector with strings indicating the variables that are ordered catagorical, or set to TRUE to model all variables as ordered catagorical.
meanstructure Logical, should the meanstructure be modeled explicitly?
corinput Logical, is the input a correlation matrix?
fullFIML Logical, should row-wise FIML be used? Not recommended!
Should the data be bootstrapped? If TRUE the data are resampled and a bootstrap sample is created. These must be aggregated using aggregate_bootstraps! Can be TRUE or FALSE. Can
bootstrap also be "nonparametric" (which sets boot_sub = 1 and boot_resample = TRUE) or "case" (which sets boot_sub = 0.75 and boot_resample = FALSE).
boot_sub Proportion of cases to be subsampled (round(boot_sub * N)).
boot_resample Logical, should the bootstrap be with replacement (TRUE) or without replacement (FALSE)
... Arguments sent to varcov
A data frame encoding the data used in the analysis. Can be missing if covs and nobs are supplied.
Only used when type = "cov". Either "full" to estimate every element freely, "diag" to only include diagonal elements, or a matrix of the dimensions node x node with 0 encoding a fixed to zero
element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a
Only used when type = "prec". Either "full" to estimate every element freely, "diag" to only include diagonal elements, or a matrix of the dimensions node x node with 0 encoding a fixed to zero
element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a
Only used when type = "ggm". Either "full" to estimate every element freely, "zero" to set all elements to zero, or a matrix of the dimensions node x node with 0 encoding a fixed to zero element, 1
encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a matrix.
Only used when type = "chol". Either "full" to estimate every element freely, "diag" to only include diagonal elements, or a matrix of the dimensions node x node with 0 encoding a fixed to zero
element, 1 encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a
Only used when type = "ggm". Either "diag" or "zero" (not recommended), or a matrix of the dimensions node x node with 0 encoding a fixed to zero element, 1 encoding a free to estimate element, and
higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a matrix.
Only used when type = "cor". Either "full" to estimate every element freely, "zero" to set all elements to zero, or a matrix of the dimensions node x node with 0 encoding a fixed to zero element, 1
encoding a free to estimate element, and higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a matrix.
Only used when type = "cor". Either "diag" or "zero" (not recommended), or a matrix of the dimensions node x node with 0 encoding a fixed to zero element, 1 encoding a free to estimate element, and
higher integers encoding equality constrains. For multiple groups, this argument can be a list or array with each element/slice encoding such a matrix.
Optional vector encoding the mean structure. Set elements to 0 to indicate fixed to zero constrains, 1 to indicate free means, and higher integers to indicate equality constrains. For multiple
groups, this argument can be a list or array with each element/column encoding such a vector.
An optional character vector encoding the variables used in the analyis. Must equal names of the dataset in data.
An optional string indicating the name of the group variable in data.
A sample variance–covariance matrix, or a list/array of such matrices for multiple groups. Make sure covtype argument is set correctly to the type of covariances used.
A vector of sample means, or a list/matrix containing such vectors for multiple groups.
The number of observations used in covs and means, or a vector of such numbers of observations for multiple groups.
If 'covs' is used, this is the type of covariance (maximum likelihood or unbiased) the input covariance matrix represents. Set to "ML" for maximum likelihood estimates (denominator n) and "UB" to
unbiased estimates (denominator n-1). The default will try to find the type used, by investigating which is most likely to result from integer valued datasets.
How should missingness be handled in computing the sample covariances and number of observations when data is used. Can be "listwise" for listwise deletion, or "pairwise" for pairwise deletion.
A character vector indicating which matrices should be constrained equal across groups.
A logical indicating if the baseline and saturated model should be included. Mostly used internally and NOT Recommended to be used manually.
The estimator to be used. Currently implemented are "ML" for maximum likelihood estimation, "FIML" for full-information maximum likelihood estimation, "ULS" for unweighted least squares estimation,
"WLS" for weighted least squares estimation, and "DWLS" for diagonally weighted least squares estimation.
The optimizer to be used. Can be one of "nlminb" (the default R nlminb function), "ucminf" (from the optimr package), and C++ based optimizers "cpp_L-BFGS-B", "cpp_BFGS", "cpp_CG", "cpp_SANN", and
"cpp_Nelder-Mead". The C++ optimizers are faster but slightly less stable. Defaults to "nlminb".
Logical, should the raw data be stored? Needed for bootstrapping (see bootstrap).
Which standardization method should be used? "none" (default) for no standardization, "z" for z-scores, and "quantile" for a non-parametric transformation to the quantiles of the marginal standard
normal distribution.
A vector with strings indicating the variables that are ordered catagorical, or set to TRUE to model all variables as ordered catagorical.
Should the data be bootstrapped? If TRUE the data are resampled and a bootstrap sample is created. These must be aggregated using aggregate_bootstraps! Can be TRUE or FALSE. Can also be
"nonparametric" (which sets boot_sub = 1 and boot_resample = TRUE) or "case" (which sets boot_sub = 0.75 and boot_resample = FALSE).
Logical, should the bootstrap be with replacement (TRUE) or without replacement (FALSE)
in which the covariance matrix can further be modeled in three ways. With type = "chol" as Cholesky decomposition:
and finally with type = "ggm" as Gaussian graphical model:
Epskamp, S., Rhemtulla, M., & Borsboom, D. (2017). Generalized network psychometrics: Combining network and latent variable models. Psychometrika, 82(4), 904-927.
# Load bfi data from psych package: library("psychTools") data(bfi) # Also load dplyr for the pipe operator: library("dplyr") # Let's take the agreeableness items, and gender: ConsData <- bfi %>%
select(A1:A5, gender) %>% na.omit # Let's remove missingness (otherwise use Estimator = "FIML) # Define variables: vars <- names(ConsData)[1:5] # Saturated estimation: mod_saturated <- ggm(ConsData,
vars = vars) # Run the model: mod_saturated <- mod_saturated %>% runmodel # We can look at the parameters: mod_saturated %>% parameters # Labels: labels <- c( "indifferent to the feelings of others",
"inquire about others' well-being", "comfort others", "love children", "make people feel at ease") # Plot CIs: CIplot(mod_saturated, "omega", labels = labels, labelstart = 0.2) # We can also fit an
empty network: mod0 <- ggm(ConsData, vars = vars, omega = "zero") # Run the model: mod0 <- mod0 %>% runmodel # We can look at the modification indices: mod0 %>% MIs # To automatically add along
modification indices, we can use stepup: mod1 <- mod0 %>% stepup # Let's also prune all non-significant edges to finish: mod1 <- mod1 %>% prune # Look at the fit: mod1 %>% fit # Compare to original
(baseline) model: compare(baseline = mod0, adjusted = mod1) # We can also look at the parameters: mod1 %>% parameters # Or obtain the network as follows: getmatrix(mod1, "omega")
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The Problem with Climate Models - ClintelThe Problem with Climate Models - Clintel
“We know there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown
unknowns—the ones we don’t know we don’t know” — Donald Rumsfeld
Ed Zuiderwijk, PhD
An Observation
There is something strange about climate models: they don’t converge. What I mean by that I will explain on the basis of historical determinations of what we now call the ‘Equilibrium Climate
Sensitivity’ (ECS), also called the ‘Charney Sensitivity’ (ref 1), defined as the increase in temperature at the bottom of the Earth’s atmosphere when the CO2 content is doubled (after all feedbacks
have worked themselves through). The early models by Plass (2), Manabe & co-workers (3) and Rowntree & Walker (4) in the 1950s, 60s and 70s gave ECS values from 2 degrees Centigrade to more than 4C.
Over the past decades, these models have grown into a collection of more than 30 climate models brought together in the CMIP6 ensemble that forms the basis for the upcoming AR6 (‘6th Assessment
Report’) of the IPCC. However the ECS values still cover the interval 1.8C to 5.6C, a factor of 3 difference in results. So after some 4 decades of development climate models have still not converged
to a ‘standard climate model’ with an unambiguous ECS value; rather the opposite is the case.
What that means becomes clear when we consider what it would mean if, for instance, the astrophysicists found themselves in a similar situation with, for example, their stellar models. The analytical
polytropic description of the early 20th century gave way years ago to complex numerical models that enabled the study of stellar evolution – caused by changing internal composition and the
associated changes in energy generation and opacity – and which also in 1970, when I took my first steps in the subject, offered a reasonable explanation of, for example, the Hertzsprung-Russell
diagram of star populations in star clusters (5). Although always subject to improvement, you can say that those models have converged to what could be called a canonical star model. The different
computer codes for calculating stellar evolution, developed by groups in various countries, yield the same results for the same evolutionary phases, which also agree well with the observations. Such
convergence is a hallmark of the progress of the insights on which the models are based, through advancement of understanding of the underlying physics and testing against reality, and is manifest in
many of the sciences and techniques where they are used.
If the astrophysicists were in the same situation as the climate model makers, they would still be working with, for example, a series of solar models that predict a value of X for the surface
temperature give or take a few thousand degrees. Or that, in an engineering application, a new aircraft design should have a wing area of Y, but it could also be 3Y. You don’t have to be a genius to
understand that such models are not credible.
A Thesis
So much for my observation. Now what it means. I will here present my analysis in the form of a thesis and defend it with an appeal to elementary probability theory and a little story:
“The fact that the CMIP6 climate models show no signs of convergence means that, firstly, it is likely that none of those models represent reality well and, secondly, it is more likely than not that
the true ECS value outside the interval 1.8-5.6 degrees.”
Suppose I have N models that all predict a different ECS value. Mother Nature is difficult, but she is not malicious: there is only one “true” value of ECS in the real world; if that were not the
case, any attempt at a model would be pointless from the outset. Therefore, at best only one of those models can be correct. What is then the probability that none of those models are correct? We
know immediately that N-1 models are not correct and that the remaining model may or may not be correct. So we can say that the a priori probability that any model is incorrect is [(N-1+0.5)/N] =
1–0.5/N. This gives a probability that none of the models is correct from (1-0.5/N)^N, about 0.6 for N>3. So that’s odds of 3 to 2 that all models are incorrect; this 0.6 is also the probability that
the real ECS value falls outside the interval 1.8C-5.6C.
Now I already hear the objections. What, for example, does it mean that a model is ‘incorrect’? Algorithmic and coding errors aside, it means that the model may be incomplete, lacking elements that
should be included, or, on the other hand, that it is over-complete with aspects that do not belong in it (an error that is often overlooked). Furthermore, these models have an intrinsic variation in
their outcome and they often contain the same elements so those outcomes are correlated. And indeed the ECS results completely tile the interval 1.8C-5.6C and for every value of ECS between the given
limits models can be found that can produce that result. In such a case one considers the effective number of independent models M represented by CMIP6. If M = 1 it means that all models are
essentially the same and the 1.8C-5.6C is an indication of the intrinsic error. Such a model would be useless. More realistic is an M ~ 5 to 9 and then you come back to the foregoing reasoning.
What rubs most with climatologists is my claim that the true ECS is outside the 1.8C-5.6C interval. There are very good observational arguments that 5.6C is a gross overestimate so I am actually
arguing that probably the real ECS is less than 1.8C. Many climatologists are convinced that that is instead a lower limit. Such a conclusion is based on a fallacy, namely the premiss that there are
no ‘known unknowns’ and especially no ‘unknown unknowns’, ergo that the underlying physics is fully understood. And, as indicated earlier, the absence of convergence of the models tells us that
precisely that is not the case.
A Little Story
Imagine a parallel universe (theorists aren’t averse to that these days) with an alternate Earth. There are two continents, each with a team of climatologists and their models. The ‘A’ team on the
landmass Laputo has 16 models that predict an ECS interval 3.0C to 5.6C, a result, if correct, with major consequences for the stability of the atmosphere; the ‘B’ team at Oceania has 15 models that
predict an ECS interval 1.8C to 3.2C. The two teams are unaware of each other’s existence, perhaps due to political circumstances, and are each convinced that their models set hard boundaries for the
true value of the ECS.
That the models of both teams give such different results is because those of the A-team have ingredients that do not appear in those of the B-team and vice versa. In fact, the climatologists on both
teams are not even aware of the possible existence of such missing aspects. After thorough analysis, both teams write a paper about their findings and send them, coincidently simultaneously, to a
magazine published in Albion, a small island state renowned for its inhabitant’s strong sense of independence. The editor sees the connection between the two papers and decides to put the authors in
contact with each other.
A culture shock follows. The lesser gods start a shouting match. Those in the A team call the members of the B team: ‘deniers’ who in their turn shout: ‘chickens’. But the more mature of both teams
realize they’ve had a massive blind spot about things the other team knew but they themselves not. That those ‘unknowns’ had firmly bitten both teams in the behind. And the smartest realize that now
the combined 31 models are a new A team to which the foregoing applies a fortiori: that there could arise a new B team somewhere with models that predict ECS values outside the 1.8C-5.6C range.
Forward Look
So it may well be, no, it is likely that once the underlying physics is properly understood, climate models will emerge that produce an ECS value considerably smaller than 1.8C. What could such a
model look like? To find out we look at the main source of the variation between the CMIP6 models: the positive feedback on water vapour (AR4, refs 6,7). The idea goes back to Manabe & Wetherald (8)
who reasoned as follows: a warming due to CO2 increase leads to an increase in the water vapour content. Water vapour is also a ‘greenhouse gas’, so there is extra warming. This mechanism is assumed
to ‘amplify’ the primary effect of CO2 increase. Vary the strength of the coupling and add the influence of clouds and you have a whole series of models that all predict a different ECS.
There are three problems with the original idea. The first is conceptual: the proposed mechanism implies that the abundance of water vapour is determined by that of CO2 and that no other regulatory
processes are involved. What then determined the humidity level before the CO2 content increased? The second problem is the absence of an observation: one would expect the same feedback on initial
warming due to a random fluctuation of the amount of water vapour itself, and that has never been established. The third problem is in the implicit assumption that the increased water vapour
concentration significantly increases the effective IR opacity of the atmosphere in the 15 micron band. That is not the case. The IR absorption by water vapour is practically saturated which makes
the effective opacity, a harmonic mean, insensitive to such variation.
Hence, the correctness of the whole concept can be doubted, to say the the least. I consider therefore models in which the feedback on water vapour is negligible (and negative if you include clouds)
as much more realistic. Water vapour concentration is determined by processes independent of CO2 abundance, for instance optimal heat dissipation and entropy production. Such models give ECS values
between 0.5C and 0.7C. Not something to be really concerned about.
1. J. Charney, ‘Carbon dioxide and climate: a scientific assessment’, Washington DC: National Academy of Sciences, 1979.
2. G. N. Plass, ‘Infrared radiation in the atmosphere‘, American Journal of Physics, 24, 303-321 , 1956.
3. S. Manabe and F. Möller, ‘On the radiative equilibrium and heat balance of the atmosphere‘ Monthly Weather Review, 89, 503-532, 1961.
4. P. Rowntree and J. Walker, ‘Carbon Dioxide, Climate and Society‘: IIASA Proceedings 1978 (ed J. Williams), pages 181–191. Pergamon, Oxford, 1978.
5. V. Eyring et al, ‘The CMIP6 landscape’ Nature Climate Change, 9, 727, 2019.
6. M. Zelinka, T. Myers, D. McCoy, et al. ‘Causes of higher climate sensitivity in cmip6 models‘, Geophysical Research Letters, 47, e2019GL085782, 2020. https://agupubs.onlinelibrary.wiley.com/doi/
7. S. Manabe and R. Wetherald, ‘Thermal equilibrium of the atmosphere with a given distribution of relative humidity’, J. Atmos. Sci., 24, 241-259, 1967. | {"url":"https://clintel.org/the-problem-with-climate-models-2/","timestamp":"2024-11-04T09:10:46Z","content_type":"text/html","content_length":"662912","record_id":"<urn:uuid:0e9e632b-554f-47c1-bb98-da207f3e55f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00527.warc.gz"} |
Analysis of Pima dataset in R (Diabetes survey on Pima Indians)
The database community has recently seen a massive surge in research to replace traditional database indexes such as B+ Trees with machine learning models (also called “learned index structures”) to
facilitate fast data retrieval.
Databases rely on indexes to quickly locate and retrieve data that is stored on disks. While traditional database indexes use tree data structures such as B+ Trees to find the position of a given
query key in the index, a learned index structure considers this problem as a prediction task and uses a machine learning model to “predict” the position of the query key.
Traditional and learned indexes ML models to approximate the CDF
This novel approach of implementing database indexes has inspired a surge of recent research aimed at studying the effectiveness of learned index structures. However, while the main advantage of
learned index structures is their ability to adjust to the data via their underlying ML model, this also carries the risk of exploitation by a malicious adversary.
This post will show some experiments that I have conducted as a follow-up to the research on adversarial machine learning in the context of learned index structures that was part of my master’s
thesis at The University of Melbourne.
Previous work on “Executing a Large-Scale Poisoning Attack against Learned Index Structures”
In my master’s thesis, I have executed a large-scale poisoning attack on dynamic learned index structures based on the CDF poisoning attack proposed by Kornaropoulos et al. The poisoning attack
targets linear regression models and works by manipulating the cumulative distribution function (CDF) on which the model is trained. The attack deteriorates the fit of the underlying ML model by
injecting a set of poisoning keys into the dataset, which leads to an increase in the prediction error of the model and thus deteriorates the overall performance of the learned index structure. The
source code for the poisoning attack is available on GitHub.
As part of the experiments for my master’s thesis, I evaluated three index implementations by measuring their throughput in million operations per second. The evaluated indexes consist of two learned
index structures ALEX and Dynamic-PGM as well as a traditional B+ Tree. Because indexes are usually used to speed-up data retrieval when dealing with massive amount of data, I chose to evaluate the
performance of the indexes based on the SOSD benchmark datasets that consist of 200 million keys each.
Unfortunately, executing the poisoning attack by Kornaropoulos et al. is heavily computationally intensive, so I had to run them with a fixed poisoning threshold of $p=0.0001$, thus generating 20,000
poisoning keys for a dataset of 200 million keys. This poisoning threshold can be considered to be relatively low, as previous work on poisoning attacks has used poisoning thresholds of up-to $p=
Implementing a flexible microbenchmark for learned indexes
To test the robustness of learned indexes more rigorously, I have set-up a flexible microbenchmark that can be used to quickly evaluate the robustness of different index implementations against
poisoning attacks. The microbenchmark is based on the source code that was published by Eppert et al. which I have extended to implement the CDF poisoning attack against different types of regression
models and the learned index implementations ALEX and PGM-Index.
The corresponding source code can be found here: https://github.com/Bachfischer/LogarithmicErrorRegression.
Testing the robustness of learned indexes
To test the robustness of the learned indexes, I have generated a synthetic dataset of 1000 keys and ran the poisoning attack against each index implementation while varying the poisoning threshold
from $p=0.01$ to $p=0.20$.
The graphs below show the performance deterioration calculated as the ratio between the mean lookup time in nanoseconds for the poisoned datasets and the legitimate (non-poisoned) dataset.
From the graphs, we can observe that simple linear regression (SLR) is particularly prone to the poisoning attack, as this regression model shows a steep increase in the mean lookup time when
evaluated on the poisoned data.
The performance of the competitors that optimize a different error function such as LogTE, DLogTE and 2P (introduced in A Tailored Regression for Learned Indexes) are more robust against adversarial
attacks. For these regression models, the mean lookup time remains relatively stable even when the poisoning threshold is increased substantially.
Because SLR is the de-facto standard in learned index structures and used internally by the ALEX and the PGM-Index implementations, we would expect that these two models also exhibit a relatively
high performance deterioration when evaluated on the poisoned dataset. Surprisingly, ALEX does not show any significant performance impact, most likely due to the usage of gapped arrays that allow
the model to easily capture outliers in the data (this effect can be likely attributed to the small keyset size). The performance of the PGM-Index deteriorates by a factor of up-to 1.3x.
To put things into a broader perspective, I have also calculated the overall mean lookup time for the evaluated learned indexes (averaged across all experiments) in the graph below.
We can see that ALEX dominates all learned index structures. The performance of the regression models SLR, LogTE, DLogTE, 2P, TheilSen and LAD is also relatively similar, in a range between 30 - 40
In the experiments, PGM-Index performs worst with a mean lookup time of > 50 nanoseconds. This is most likely due to the fact that PGM-Index is optimized for large-scale data workloads and exhibits
subpar performance in this microbenchmark because the dataset consists of only 1000 keys.
I consider the results from this research to be a highly interesting study of the robustness of learned index structures. The poisoning attack and microbenchmark described in this post are
open-source and can be easily adapted for future research purposes. If you have any further thoughts or ideas, please let me know! | {"url":"https://bachfischer.me/feed.xml","timestamp":"2024-11-12T03:02:24Z","content_type":"application/atom+xml","content_length":"1049726","record_id":"<urn:uuid:168cb652-0d7f-4e1b-b8cb-7d294ff27112>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00618.warc.gz"} |
Should you practice segment tree if you are below purple? - Codeforces
Read this comment.
It's good to know that segment tree exists
• You can learn basic segment tree (Fenwick tree may be enough) if you want to overkill some problems. Just don't try to solve hard segment tree problems like this and this if you are cyan, because
there are better ways to practice and learn something. This comment hits the spot: you can "be aware of the segment tree", but it shouldn't be your main weapon as a cyan / blue.
• If you take part to OI / ICPC, segment tree can be very useful (especially at regional level). Unlike on Codeforces, there may be easy-ish but not trivial segment tree problems (let's say with a
rating around 2200). So, if you want to practice segment tree for some reason (e.g., you love segment trees, or you are practicing for OI), start with OI problems.
Should you master segment tree problems?
If you want to solve non-trivial segment tree problems, you should
1. actually understand how segment tree works (including time complexity);
2. have decent implementation skills;
3. be able to convert the given problem into a segment tree problem.
If you are able to learn all these things, you already have purple skills. Conversely, if you are not purple, most probably you won't manage to actually learn segment tree.
• Blog 1: the author asks how to solve a problem. Someone replies, linking a comment about another problem whose solution is almost identical to the original problem. The comment contains a
detailed explanation of the solution and an AC code.
The author of the blog replies that he wants an AC code of his problem because he can't implement the solution. It turns out it's because the provided AC code uses a segment tree as a struct.
• Blog 2: the author is "confused" about the time and space complexity of his solution using a segment tree. It turns out his solution is worse than the naive solution.
• Blog 1: if you understand the explanation of the solution, and you say you know segment tree, you should have no problems implementing the solution from scratch (point 2 above). If not, it means
that you don't understand the solution, so the problem is too hard for you (3). Then, why would you need to solve it? Also, copy-pasting others' code without understanding it does not count as
solving the problem. Then, why are you asking for the code?
• Blog 2: I guess someone told you that the problem is solved "using segment tree" and you tried to implement a solution without even calculating the complexity (1). Please note that there may be
multiple ways, both correct and wrong, to use segment tree in a problem (similarly, in other problems there may be multiple greedy solutions, both correct and wrong). So, if you are using segment
tree, it doesn't mean that the code is "magically" efficient. Finding an actually correct solution can be hard (3).
It's fine not to be good at points 1, 2, 3 above if you are blue or below. There are other (more important?) things to learn at that level.
Side note: when I first became yellow, I had no clue about how to solve the linked problems. Now I can solve them, but I'm still yellow. | {"url":"https://mirror.codeforces.com/topic/112452/?locale=en","timestamp":"2024-11-04T08:18:19Z","content_type":"text/html","content_length":"87470","record_id":"<urn:uuid:c564d581-4b9d-45c8-a0c7-4d9ce19ebea9>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00637.warc.gz"} |
Math 4707:
Math 4707: Introduction to combinatorics and graph theory (Spring 2022)
Mondays and Wednesdays 2:30-4:25 in Vincent Hall 211.
Instructor: Gregg Musiker
(musiker "at" math.umn.edu)
Office Hours:
Tuesdays 1:10-2:00 and Fridays 2:30-3:20 In Vincent Hall 251 (or via Zoom); also by appointment.
Course Description:
This is a course in discrete mathematics, emphasizing both techniques of enumeration (as in Math 5705) as well as graph theory and optimization (as in Math 5707), but with somewhat less depth than in
either of Math 5705 or 5707. We plan to cover most of the required text, skipping Chapters 6, 14, and 15. We will also likely supplement the text with some outside material.
Prerequisites: Math 2243 and either Math 2283 or 3283 (or their equivalent). Students will be expected to know some calculus and linear algebra, as well as having some familiarity with proof
techniques, such as mathematical induction.
Required text: Discrete Mathematics: elementary and beyond, by Lovasz, Pelikan, and Vesztergombi (2003, Springer-Verlag).
Other useful texts
│ Title │ Author(s), Publ. info │ Location │
│Invitation to Discrete Mathematics │Matousek and Nesetril, Oxford 1998 │On reserve in math library│
│Applied combinatorics │A. Tucker, Wiley & Sons 2004 │On reserve in math library│
│Introduction to graph theory │D. West, Prentice Hall 1996 │On reserve in math library│
Homework (40%): There will be 6 homework assignments due approximately every other week (tentatively) on Wednesdays. The first homework assignment is due on Wednesday Feb 2nd.
I encourage collaboration on the homework, as long as each person understands the solutions, writes them up in their own words, and indicates on the homework page their collaborators. Late homework
will not be accepted. Early homework is fine, and can be left in my mailbox in the School of Math mailroom in Vincent Hall 107. Homework solutions should be well-explained-- the grader is told not to
give credit for an unsupported answer. Complaints about the grading should be brought to me.
Three Exams (20% each): There will be 3 in-class exams, dates are listed below, each of which will be open book, open notes, and with calculators allowed. Missing an exam is permitted only for the
most compelling reasons. You should obtain my permission in advance to miss an exam. Otherwise you will be given a 0. If you are excused from taking an exam, you will either be given an oral exam or
your other exam scores will be prorated.
Class Participation
Participation in class is encouraged. Please feel free to stop me and ask questions during lecture. Otherwise, I might stop and ask you questions instead. Additionally, some course material will be
taught by having the students work together in small groups cooperatively, followed by a representative coming to the board to explain their group's answer.
Course Syllabus and Tentative Lecture Schedule
(Jan 19) Lecture 1: Introduction to the Course and Enumeration (Chapter 1 of LPV, Secs 1.1-1.3)
(Jan 24 ) Lecture 2: Enumeration (Chapter 1 of LPV, Secs 1.4-1.8)
(Jan 26) Lecture 3: Induction and the Pigeon-hole Principle (Chapter 2 of LPV, Secs 2.1, 2.4)
(Jan 31) Lecture 4: Inclusion-Exclusion and Asymptotics (Chapter 2 of LPV, Secs 2.2-2.3, 2.5 )
(Feb 2) Lecture 5: The Binomial Theorem, more Identities in Pascal's Triangle, and choosing Multisets (Chapter 3 of LPV, Secs 3.1-3.6)
Homework assignments (tentative) and exams
│Assignment or Exam│ Due date │Problems from Lovasz-Pelikan-Vesztergombi text, │
│ │ │ unless otherwise specified │
│Homework 1 │Wednesday Feb 2 │ │
│Homework 2 │Wednesday Feb 16 │ │
│Exam 1 │Monday Feb 21 │ │
│Homework 3 │Wednesday Mar 2 │ │
│Homework 4 │Wednesday Mar 23 │ │
│Exam 2 │Monday Mar 28 │ │
│Homework 5 │Wednesday April 13│ │
│Homework 6 │Wednesday April 27│ │
│Exam 3 │Monday May 2 │ │ | {"url":"https://www-users.cse.umn.edu/~musiker/4707/","timestamp":"2024-11-13T22:37:00Z","content_type":"text/html","content_length":"5435","record_id":"<urn:uuid:7abe0814-6ffe-450a-b42d-67a0293ccba0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00712.warc.gz"} |
Level of Significance in Statistics - Definition
MathsLevel of Significance in Statistics – Definition, P-value Significance Level and FAQs
Level of Significance in Statistics – Definition, P-value Significance Level and FAQs
P-value Significance Level
The p-value is a measure of the strength of the evidence against the null hypothesis. It is the probability of observing a sample statistic as extreme or more extreme than the one observed, given
that the null hypothesis is true. The p-value is always between 0 and 1.
The significance level is the probability of rejecting the null hypothesis when it is in fact true. It is usually set to 0.05, which means that there is a 5% chance of rejecting the null hypothesis
when it is in fact true.
Significance Level Definition
A significance level is the probability of making a Type I error. The significance level is also known as the alpha level.
P-Value and Significance Level
The p-value is the probability of obtaining a statistic at least as extreme as the one that was actually observed, given that the null hypothesis is true. The significance level is the probability of
rejecting the null hypothesis when it is true.
Rejection Region
In statistics, the rejection region is the set of all points in the sample space for which a hypothesis test leads to the rejection of the null hypothesis.
Non-Rejection Region
No-rejection region is a region in which an object is not rejected by an environment. In other words, an object can exist in this region without being affected by the environment. | {"url":"https://infinitylearn.com/surge/maths/level-of-significance/","timestamp":"2024-11-07T09:06:28Z","content_type":"text/html","content_length":"159117","record_id":"<urn:uuid:44d7b55f-abbe-4e4a-b71f-18e1a452418e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00224.warc.gz"} |
Available in download version
save/open multiple results
export to Word and Excel
print results
create list of custom fluid properties
resistance factor K for flow in valves and fittings
pipe surface roughness selection
pipe material selection
select between gauge and absolute pressure
compressible isothermal flow
dry air isothermal flow
gas offtake flow
natural gas flow
Calculator start selection
Read all about available deployments. In any way of utilizing calculator, Internet connection is not required, but nice to have for authentication at least.
When is this calculator suitable?
The calculator is intended for calculating the thickness of the pipe wall based on the known value of the internal pressure and the permitted specified minimum yield stress in the pipe.
The calculator is applicable for carbon steel, stainless steel, copper and PVC pipes.
The pipe wall thickness calculation is based on the ASME B31.8 standard. The calculator allows three types of calculations:
• Calculation of the minimum pipe wall thickness
• Calculation of actual stress in the pipe
• Checking the thickness of the pipe wall for the known value of the permissible stress
In each of these three calculation cases, it is necessary to enter the gauge pressure of the fluid in the pipe, as well as the outer or inner diameter of the pipe
What are the calculator restrictions?
The calculator requires knowledge of the internal pressure of the fluid in the pipe, the material the pipe is made of, and the outside or inside diameter of the pipe.
How is the calculation executed?
The calculator displays the calculated value of the pipe wall thickness, which is calculated solely based on the entered or selected values of pressure, permissible stress, factors affecting the
calculation and permissible allowances. In this regard, if any of the entered or selected data is not correct, the calculated value will not be correct either. Therefore, before using the calculator,
the user should familiarize himself with the ASME B 31.8 standard, which defines in detail all the requirements for the correct calculation of pipe wall thickness.
Pipe pressure
The calculator for calculating the pipe wall thickness requires the input of the actual gauge pressure in the pipe in the field provided for it.
Pipe diameter
The calculator allows selection of the pipe diameter entered, so it is possible to choose whether to enter the outside or inside diameter of the pipe. In the section for input selection, it is
necessary to choose the diameter of the pipe that needs to be entered from the offered outer or inner diameter. The calculator also allows the selection of a standard pipe size through the Pipe Size
tab, where it is possible to choose one of the offered pipe dimensions for the selected Schedule from the list of standards. The selected pipe from the list is simultaneously entered in the fields
for entering the pipe diameter in the calculator.
Also, the calculator calculates the mean air velocity at the beginning and end of the pipeline.
The calculation involves calculating the Reynolds number (Re).
Calculator presents values of Reynolds number and flow regime - laminar or turbulent, as well as calculated velocities on the pipe start and the pipe end.
Pipe wall thickness
Calculation of pipe wall thickness is carried out in accordance with the ASME B 31.8 standard. Calculating the thickness of the pipe wall requires entering three factors for the calculation, namely:
Longitudinal efficiency factor
This factor depends on the way the pipe is manufactured, and its value is 1 in the case of seamless pipes, and it has a value less than 1 in the case of other applied technologies. Its value is
automatically entered when selecting the pipe manufacturing type in the Pipe Material tab.
Design and location factor
The location factor includes the reduction of the permitted voltage depending on the location of the pipeline in relation to other objects at the installation site. This factor is a maximum of 1,
i.e. it is always less than 1 depending on the location class. Its value can be selected in the tab Pipe Material - Location Class.
Temperature influence factor
The temperature influence factor implies a reduction of the allowable stress in the pipe in case the pipe is exposed to elevated temperatures. The value of this factor is up to 1 when the pipe
temperature does not exceed 250 F, and for higher temperature values it is less than 1. The value of this factor is available through the Pipe Material - Temperature tab.
Calculation of the minimum pipe wall thickness
For the calculation, it is necessary to enter the size of the permitted stress in the pipe in the SMYS field. If the material from which the pipe is made is known, and if the standard pipe material
is known, the calculator enables the selection of SMYS based on the pipe material standard, type of pipe manufacturing technology and material class, using the Pipe Material tab. The calculator will
enter the calculated pipe wall thickness in the t field.
Calculation of actual stress in the pipe
For the calculation of the actual stresses in the pipe, it is necessary to enter the actual thickness of the pipe wall in the field t. The calculator will calculate the actual stress in pipe wall and
present it in the SMYS field.
Checking the thickness of the pipe wall for the known value of the permissible stress
The calculator enables the display of information whether for the entered values of the pipe wall thickness and the known maximum stress in the pipe, the thickness of the pipe wall satisfies. The
calculator displays the information whether the pipe wall thickness is satisfactory or not by printing the corresponding statement.
Corrosion allowance
The calculator allows input of an assumed CA - corrosion allowance. The value of the corrosion allowance can be entered if you select the check box in the selection section. The entered value of the
corrosion allowance is included in the displayed value of the pipe wall thickness t, i.e. the calculated pipe wall thickness based on the internal pressure is increased by the entered value of the
corrosion allowance. Therefore, the value of the pipe wall thickness must be greater than the entered value of the corrosion allowance.
Mill tolerance allowance
The mill tolerance allowance for wall thickness assumes that the actual pipe wall thickness may deviate from the nominal by a certain predefined percentage. In the field MT, it is necessary to enter
the tolerance in % (eg: 12.5 if the mill tolerance is 12.5%).
By entering this information, the calculated pipe wall thickness due to internal pressure will be increased by the value of this addition and the presented wall thickness value t will also include
this value.
Pipe wall thickness calculation formula
$p = 2 ⋅ S ⋅ t D F ⋅ E ⋅ T$
where is: p - design pressure; S - specified minimum yield strength; t - nominal wall thickness; T - temperature derating factor; F - design factor; E - longitudinal joint factor; D - pipe external
All needed factor values are available in the calculator.
When is this calculator not relevant?
The calculator is applicable only for flat round pipes, i.e. it is not applicable for pipes with a square or rectangular cross-section. Also, the calculator is not applicable for bent pipes, pipes
with openings and reinforcements.
The calculator is not applicable for wall thickness calculations due to external pressure.
What else has to be known to perform the calculation?
To calculate the actual stresses in the pipes, it is necessary to enter the thickness of the pipe wall, the internal pressure, as well as the diameter of the pipe - internal or external.
When selecting a standard pipe dimension from the pipe table, check the standard pipe wall thickness by selecting the third option of the calculation type - pipe wall thickness check for known
maximum stress in the pipe. | {"url":"https://www.pipeflowcalculations.com/pipethickness/index.xhtml","timestamp":"2024-11-06T10:26:23Z","content_type":"application/xhtml+xml","content_length":"43948","record_id":"<urn:uuid:2a2d3ee6-43b0-4af6-8988-13d76a954ac4>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00671.warc.gz"} |
Number System Questions and Answers - Real Numbers and BODMAS Simplification - Sanfoundry
Number System Questions and Answers – Real Numbers and BODMAS Simplification
This set of Aptitude Questions and Answers (MCQs) focuses on “Real Numbers and BODMAS Simplification”.
1. Find the rational number lying between √7 and √8.
a) 12/5
b) 14/5
c) 22/9
d) 23/9
View Answer
Answer: b
Explanation: We have, √7 = 2.64575… and √8 = 2.82842…
From options, 12/5 = 2.4
14/5 = 2.8
22/9 = 2.444…
23/9 = 2.555…
Clearly 2.8 i.e., 14/5 lies between √7 and √8.
2. What is the value of x, if x is real and |\(\frac{7-x}{5}\)|<2?
a) 0<x<17
b) -17<x<17
c) -3<x<17
d) x<17
View Answer
Answer: c
Explanation: As, |\(\frac{7-x}{5}\)|<2
\(\frac{7-x}{5}\)<2 or \(\frac{(7-x)}{5}\)<2
x > -3 or x < 17
Hence, -3 < x < 17
3. Given that, 1^2+3^2+5^2+7^2+9^2=165, then what is the value of 3^2+9^2+15^2+21^2+27^2?
a) 1485
b) 1385
c) 990
d) 495
View Answer
Answer: a
Explanation: = 3^2+9^2+15^2+21^2+27^2
= 3^2 * (1^2+3^2+5^2+7^2+9^2) = 9*165 = 1485
4. If m is a positive integer, then in which of the following form every square integer is represented?
a) 4m
b) 4m+1 or 4m+3
c) 4m or 4m+3
d) 4m or 4m+1
View Answer
Answer: d
Explanation: If m is a positive integer, then every square integer is of the form 4m or 4m+1, as every number is either a multiple of 4 or exceeds multiple of 4 by unity.
5. If x and y are natural number, not necessarily distinct. For all values of x and y, which of the following is also a natural number?
a) x + y
b) x – y
c) x/y
d) logx – logy
View Answer
Answer: a
Explanation: x – y can be negative, if y is greater than x. Hence, it cannot be natural number.
x/y and logx-logy can sometimes be in fractions and are not natural numbers.
x+y always represents natural number, when x and y are natural numbers.
6. If \(\frac{a}{5}\)=\(\frac{b}{6}\)=\(\frac{c}{9}\), then what is the value of \(\frac{a+b+c}{a}\)?
a) 4
b) 5
c) 3
d) 6
View Answer
Answer: a
Explanation: Let \(\frac{a}{5}\)=\(\frac{b}{6}\)=\(\frac{c}{9}\)=k
i.e., a = 5k, b = 6k, c = 9k
7. If m is a negative real number, then which of the following is true?
a) |m| = m
b) |m| = -m
c) |m| = -1/m
d) |m| = 1/m
View Answer
Answer: b
Explanation: Clearly, by the definition of modulus function |m| = -m, when m<0 i.e., when m is negative real number.
8. If x, y and z are real numbers such that x < y and z < 0, then which of the following statement is true?
a) (x/z) < (y/z)
b) (z/a) > (z/y)
c) xz > yz
d) xz < yz
View Answer
Answer: c
Explanation: Given that, x < y and z < 0 i.e., z is negative real number.
On multiplying a negative number to any inequality, we must flip the inequality sign.
Therefore, xz > yz
9. If \(\frac{m}{n}=\frac{5}{8}\), then what is the value of \(\frac{m-n}{m+n}\)?
a) 3/13
b) 13/3
c) -3/13
d) -13/3
View Answer
Answer: c
Explanation: \(\frac{m-n}{m+n}=\frac{\frac{m}{n}-1}{\frac{m}{n}+1}=\frac{\frac{5}{8}-1}{\frac{5}{8}+1}=\frac{5-8}{5+8}=\frac{-3}{13}\)
10. If m is positive even integer and n is negative odd integer, then which of the following real number is the solution of m^n?
a) Odd integer
b) Even integer
c) Rational number
d) Irrational number
View Answer
Answer: c
Explanation: If m is positive even integer and n is negative odd integer, then m^n is rational number.
Consider m=8 and n=-5, then 8^-5 = 1/(32768), which is rational number.
11. Find the value of (125+216)-\(\frac{1750}{5^3}\)+15.
a) 329
b) 342
c) 392
d) 344
View Answer
Answer: b
Explanation: = (125+216)-\(\frac{1750}{5^3}\)+15
= (341)-\(\frac{1750}{125}\)+15
= 341- 14+15 = 342.
To practice all aptitude questions, please visit “1000+ Quantitative Aptitude Questions”, “1000+ Logical Reasoning Questions”, and “Data Interpretation Questions”. | {"url":"https://www.sanfoundry.com/aptitude-questions-answers-number-system-real-numbers-bodmas-simplification/","timestamp":"2024-11-07T15:33:19Z","content_type":"text/html","content_length":"175373","record_id":"<urn:uuid:ceabf369-00f9-40d9-8a20-d26e5e9f37f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00784.warc.gz"} |
Quite a while ago I posted about using the
grid method for dividing polynomials
. Using grids or
generic rectangles
as they are more commonly called, makes dividing polynomials seem like doing a crossword puzzle or a sudoku.
There are number of ways to use generic rectangles, and I thought I would try to do a few posts about some of their other uses, which include factoring trinomials and completing the square. This
first post is just going to be about polynomial multiplication.
When multiplying, generic rectangles seem to provide for polynomials what grid multiplication and lattice multiplication provide us for numbers - a way of arranging things on the page so that we
don't mess up our calculations.
multiplication allow us to break down our numbers into more manageable chunks and, in the case of lattice multiplication, give a way to keep track of "carries." In school, grid and lattice
multiplication are sometimes presented as an alternative to standard "long" multiplication.
Those familiar with grid and lattice multiplication for numbers will feel at home with generic rectangles, but there are some differences. In the world of polynomial multiplication, there are no
"carries" (terms in a polynomial, unlike decimal places don't overflow into each other), and there is no generally used "long multiplication" style that needs replacing. For polynomials, what generic
rectangles give us is a way to keep track of all those terms that come out of the
distributive law
, which most students struggle to keep track of using mnemonics like
. (Once recursion becomes something well-understood in middle school, FOIL might be a reasonable way to teach the distributive law, but until then, teachers please consider using generic rectangles.)
Generic rectangles also have an affinity with
algebra tiles
- a manipulative that is sometimes for learning polynomial multiplication. If you use algebra tiles, generic rectangles are a nice thing to move on to if you tire of pushing around all that plastic.
Unlike those rigid tiles, generic rectangles are more
: although they don't provide the full force of the area metaphor for multiplying that algebra tiles do they allow you to multiply any kind of terms.
A first example
The first thing to do when provided with two polynomials that are to be multiplied is to set up the grid, with the terms from one of the polynomials across the top row, and the terms from the other
down the leftmost column.
Each term is multiplied, like terms are gathered, and the results are summed, which provides the answer.
A little bigger example
Here's an example with a trinomial and a binomial.
Again, step 1 is just putting the factors to be multiplied on the left most and topmost column and row of the grid, and step 2 is just completing the term-by-term multiplication to fill in the grid.
Finally, like-terms are collected and the final result can be written out.
An infinite example
That's fun, but what about multiplying together two infinitely long polynomials? Why not? Of course, we'll quickly run out of space, but we might see something in our grid that will help us get to a
solution. Now, instead of thinking of polynomials in the way you might usually think of them, you should consider these infinitely long polynomials as
formal power series
- polynomials that we will never evaluate, and that we are just treating as worthwhile mathematical objects in their own right, regardless of whether they ever converge.
Consider the product:
We can make a grid like this:
And start to fill it in:
Do you see the pattern that is emerging when you start collecting like terms? Once you do you'll probably agree that:
If you go a little further with this, you'll find the
triangular numbers
lurking in here. For
> 0:
The power series on the right has coefficients that are the "
-1"-dimensional triangular numbers (or, the d-1 column of Pascal's Triangle - see
this post | {"url":"https://www.mathrecreation.com/2012/09/","timestamp":"2024-11-05T16:58:32Z","content_type":"application/xhtml+xml","content_length":"87979","record_id":"<urn:uuid:288e3c5d-c6d3-43f5-ac00-7037469d67e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00897.warc.gz"} |
evaluate the limit
How do you evaluate the limit (sin(5x))/x as x approaches 0?
Ivan Waters
Answered question
How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?
Answer & Explanation
Given $\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(5x\right)}{x}$
We want to use the previously mentioned limit, so we need to have $\theta =5x$
To get 5x in the denominator, we'll multiply by $\frac{5}{5}$
$\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(5x\right)}{x}=\underset{x\to 0}{lim}\frac{5\mathrm{sin}\left(5x\right)}{5x}$
Now, outside the limit, factor the 5 in the numerator.
$=5\underset{x\to 0}{lim}\frac{\mathrm{sin}\left(5x\right)}{5x}$
As $x\to 0$, $5x\to 0$ so we have:
(The limit is 5.)
Hence, using $\underset{\theta \to 0}{lim}\frac{\mathrm{sin}\theta }{\theta }=1$ and some other tools. | {"url":"https://plainmath.org/calculus-1/103949-how-do-you-evaluate-the-limit","timestamp":"2024-11-09T20:30:42Z","content_type":"text/html","content_length":"157143","record_id":"<urn:uuid:810d9df6-c19a-41ad-ac99-d04c6b44f918>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00021.warc.gz"} |
Communications of the ACM
Applications Quest: Computing Diversity
It helps admissions officers review college applicants holistically.
Recently, two landmark cases challenged the University of Michigan’s admissions policies. In Grutter v. Bollinger, which focused on admissions to the university’s law school, the U.S. Supreme Court
ruled 5 4 in favor of the law school’s admissions policy, which was designed to enhance the diversity of the student body. However, in Gratz v. Bollinger, by a vote of 6 3, the court reversed, in
part, the university’s undergraduate admissions policy of awarding points for race/ethnicity. Here, the court decided that although race could be considered in admission decisions, it cannot be the
deciding factor. Although this decision appears to support affirmative action efforts, it severely limits how race can be used to achieve diversity goals. The Supreme Court thus ruled that diversity
could be used in university admissions, but did not specify how this should be achieved.
Rather than simply excluding any consideration of race/ethnicity from the admissions process, the route taken by many educational institutions, the University of Michigan has chosen to implement an
expensive and labor-intensive holistic evaluation process that incorporates race/ethnicity as a factor. In Michigan’s holistic review process, every application is read by different admissions
counselors. Counselors rate each application as outstanding, excellent, good, average/fair, or below average/poor. Additionally, the counselors give each application a recommended decision: high
admit, admit, admit with reservation, deny with reservation, or deny [6]. Even though this process adheres to the Supreme Court’s rulings, there are no metrics in place to compare large numbers of
applications. Counselors subjectively rate applications. Therefore, when applications must be compared, there are no methods in place to effectively compare applications to determine the extent to
which applications are alike or different.
Applications Quest was developed to perform holistic comparisons between applications, yielding clusters of similar applications.
In general, a holistic evaluation uses race as one of many attributes being considered by the university as part of its decision, yet all attributes play a role and no single attribute is the
determining attribute. This raises several interesting questions regarding holistic application evaluation: How does holistic evaluation translate into practice? What techniques could best be
employed to compare large numbers of applications? Can holistic evaluations be performed economically without sacrificing quality? In an attempt to address these and other issues associated with
holistic evaluation, we have developed a new computer algorithm, Applications Quest, a dynamic software tool that clusters applications, thereby giving admissions professionals a new perspective on
holistic evaluation. This new approach includes race/ethnicity as one of the factors considered, but does not assign a numerical value to it and thus complies with the Supreme Court decision.
Clustering for Diversity
University admissions offices use applications to gather the same information about each applicant, ensuring that all applicants can be evaluated based on the same attributes. As the application
represents every potential student, each university application is expected to contain pertinent information conveying the most important details about each applicant. Following this principle, it is
possible to define diversity using a holistic view of an application. From this perspective, diversity is observed when the selected group of applications is holistically diverse. This level of
diversity can be obtained through cluster analysis.
Clustering is the grouping of similar objects for the purpose of classification or categorization. Clustering is one of the most basic abilities of humans [1]. An intelligent being cannot treat every
object as a unique entity unlike anything else in the universe. Instead, he or she must put objects in categories so as to apply hard-won knowledge about similar objects encountered in the past to
the object at hand.
Clustering algorithms can be divided into two categories: hierarchical and non-hierarchical. Hierarchical clustering methods create clusters or groups by merging or dividing. These actions may occur
in one of two forms: agglomeration or division [1, 2]. Agglomerative clustering methods form clusters by merging individuals and begin by assuming each instance in the collection population is an
individual cluster. In the course of each processing cycle, two clusters are merged. This process continues until either there is only one cluster remaining that contains all instances in the
population, or some other predefined stopping point has been reached, such as a specified number of clusters. The divisive clustering approach works in the opposite direction. It starts by assuming
that all instances belong to one cluster. In each step of the process, a cluster is split into two clusters, until all clusters contain a single instance, or some other predefined stopping point has
been reached, such as a specified number of clusters.
Non-hierarchical clustering methods result in faster execution times compared to hierarchical methods. The most common non-hierarchical method is k-means [1, 2, 7]. Before the k-means algorithm can
be executed, the number of clusters is typically specified, which is k. Initially, k-means begins by selecting k instances as centroids. A centroid is the most representative instance within a
cluster. It is the instance within a cluster that has the shortest distance from all the other instances within the cluster. The centroid instances are typically selected at random, or this process
may utilize some heuristic. Much like the divisive approach, all the remaining non-centroid instances are compared to each centroid. The non-centroid instances are placed in the cluster with the most
similar centroid. At the end of each cycle, the centroids are recalculated for each cluster and the instances are redistributed until the centroids do not change. There are several variations of
k-means, such as bisecting k-means [7], but they all follow this basic approach.
All clustering algorithms must utilize some distance, or similarity, measure. Distance measures determine the distance or similarity between instances within a given population. These measures can be
calculated using several methods, but the Euclidean distance is the most commonly used distance measure [2]. Euclidean distance is based on Pythagoras’ theorem, where instances are represented as
points in an n-dimensional space. The distance between any two points in an n-dimensional space is calculated as the square root of the sum of the squared sides between the two points along each
dimension. Euclidean distance measures are used by clustering algorithms to determine distance or similarity, yielding a basis for comparison between instances, or objects, with the same attributes/
characteristics. As a result, clustering algorithms can be applied to admissions applications.
When holistic clustering is applied to admissions applications, the results yield clusters or groups of similar applications. The table here gives an illustration of three graduate school
applications. Using holistic clustering, it is easy to see that applications 146 and 59 are more similar by comparing each attribute, such as INST1_GPA, and GRE_V, GRE_Q, across each application.
Comparing the three applications is fairly easy. However, when hundreds or thousands of applications have to be compared or when the similarity between applications is not so apparent, this task
becomes physically impossible for a human admissions counselor. Consequently, Applications Quest was developed to perform holistic comparisons between applications, yielding clusters of similar
Applications Quest is a software tool that uses hierarchical clustering approaches to holistically compare admissions applications and place them in clusters based on their similarity. This process
begins by collecting applications from an admissions pool in an electronic format and placing them in a database table. Each application’s attributes are classified as numeric, opinion, or nominal.
Numeric attributes contain numerically based values, such as GPA, GRE, GMAT, SAT, and ACT scores.
Opinion attributes must be evaluated by an admissions counselor and include personal statements and essays. Nominal attributes are those that do not have a numeric base but exist in name only, such
as race/ethnicity, gender, first-generation student, and major. Before processing, all numeric attributes are scaled to values between 0 and 1. Opinion attributes are assigned ratings by the
admissions counselors between 1 and 10, which are later scaled to values between 0 and 1. This scaling provides a basis for comparison between attributes and ensures that all values are on the same
numerical base. The nominal attributes are not assigned values but are handled differently from the numeric and opinion attributes. Applications Quest uses a squared Euclidean distance measure. The
squared Euclidean distance measure is the Euclidean distance, but the final square root is omitted, which saves one operation. When two applications are compared, the numeric and opinion attributes
are treated identically, in that the sum of the squared difference between the scaled values is computed. When considering nominal values, the attribute values are either the same or different,
yielding 0 for identical attribute values or 1 for different values. Using the squared Euclidean distance measure, Applications Quest computes a similarity matrix.
Applications Quest compares every application to every other application and places the result of each comparison into a database table called the similarity matrix. The similarity matrix contains an
entry for every comparison and the similarity between each pair of applications. This is computationally expensive because it must consider all combinations, nCr = n! / (n – r)! r!, where n is the
number of applications and r represents the number compared at a time. Given a pool of 1,000 applications compared two at a time, 499,500 comparisons are required to build the similarity matrix. The
similarity matrix should not be avoided when holistically comparing applications. Once the similarity matrix has been built, Applications Quest applies one of the aforementioned clustering
Applications Quest’s processing is initiated by the user who specifies the number of clusters and the number of applications that Applications Quest will recommend for admissions from each cluster.
Next, Applications Quest builds the similarity matrix and applies an agglomerative or divisive clustering approach to the application pool.
The divisive clustering approach begins by identifying the two most different applications using the similarity matrix. These two applications are used to split the pool into two clusters and are
thus the centroids. All other applications are placed in one of the two clusters surrounding the selected two applications based on their distance/similarity to either of the centroids. Once all the
applications have been placed in one of the two clusters, the largest cluster is selected and split again using the divisive approach by selecting the two most different applications within the
selected cluster as centroids. This process continues until the specified number of clusters has been obtained.
When the agglomerative approach is used, it merges the most similar applications into clusters until the specified number of clusters has been reached. Applications Quest uses an average linkage [2]
comparison measure to merge clusters. When the final number of clusters has been reached using either method, an email is sent to the admissions counselors notifying them that processing is complete.
The admissions counselors can then use Applications Quest’s cluster visualization interface to process the applications.
Processing the Applications
After the applications have been placed into clusters, the admissions counselors can view the results using Applications Quest’s cluster visualization interface. Figure 1 is the summary page of the
visualization tool and contains information about all the applications. In the example shown, 754 applications are divided into 75 clusters. Applications Quest recommends two applications from each
cluster, as specified by the user. The recommended applications are selected from each cluster such that they optimize the Difference Index, which is defined as the average difference between
applications. This measures the degree of difference within clusters and for the entire application pool. The larger the Difference Index value, the greater the difference between applications. For
example, two completely opposite applications will have a Difference Index of 100%, vs. 0% for two identical applications.
In Figure 1, the Difference Index is 49.90%, with a standard deviation of 14. The Difference Index for the recommended applications is 52.96%, with a standard deviation of 13. Notice the Difference
Index for the recommended applications is larger than the Difference Index for the applications pool. This illustrates how Applications Quest optimizes the Difference Index by selecting the N most
different applications from each cluster, where N is the number of recommended applications per cluster specified by the user. Furthermore, the summary view gives the overall average for each of the
numeric and opinion attributes, such as INST1_GPA of 1.078276. The top three values for each nominal attribute are also given. For example, the CITIZENSHIP attribute is a nominal attribute, with 257
applicants from India, 201 from the U.S., and 164 from China.
It recommends applicants who should be given strong consideration for admission while ensuring that these applicants optimize the Difference Index.
Applications Quest includes a navigation frame, shown on the left of Figure 1, containing a list of all the clusters, ordered by the number of applications within each cluster. In the Figure 1
example, Cluster 42 has 18 applications and Cluster 0 has 17 applications. When the user selects a cluster in the navigation frame, the summary results for that cluster are displayed in the content
Figure 2 illustrates the summary view for Cluster 42. Notice the Difference Index within Cluster 42 is 23.61%, with a standard deviation of four. The summary results are presented using the same
format as the applications summary. However, the cluster summary also contains an additional table at the bottom of the content frame, as shown in Figure 3. Every application that is a member of
Cluster 42 appears in one row of the cluster summary table.
Admissions Decisions
Now that admissions counselors have a powerful visualization tool in Applications Quest, how will they use it? How does this tool help them in the decision making process?
Recall that the U.S. Supreme Court rulings deemed diversity a worthy goal for institutions of higher education. The rulings also stated that race/ethnicity could be used as part of the decision
making process, although race could not be the determining factor, nor could it be assigned points [3, 4]. Applications Quest holistically compares applications and places them in a specified number
of clusters based on their similarity. During this process, race/ethnicity is defined as a nominal attribute.
When race is compared between two applications, it is not given any preferential consideration by assigning it some predefined weight. Race/ethnicity is measured based on its similarity, just as any
nominal attribute. Applications Quest provides admissions counselors a specialized view based on holistic clustering of their applications pool. It recommends applicants who should be given strong
consideration for admission while ensuring that these applicants optimize the Difference Index. The counselors can then use the recommendations to narrow their admissions search. For example,
admissions counselors enter all the applications that meet some minimum admissions requirement into the Applications Quest database.
Next, the counselors specify the number of applications that Applications Quest should recommend for admission. The counselors execute Applications Quest, review the recommended applications, and
admit students from the recommended application pool or simply admit selected students from each cluster. Therefore, admissions counselors are able to use Applications Quest to narrow their search
space and achieve diversity.
The U.S. Supreme Court has ruled that race can be considered in university admissions decisions. However, it did not specify exactly how race should be used. Applications Quest allows university
admissions counselors to cost-effectively and holistically evaluate applications and consider race/ethnicity as one of the application attributes.
Applications Quest is undergoing an extensive pilot study in which it is being used to measure the diversity obtained from various universities using their existing admissions policies. Diversity is
being measured using the Difference Index developed for this project. The Difference Index of admitted students is being compared to the index for those that Applications Quest recommends. The
results of the study will reveal the degree to which admissions decisions would be different using Applications Quest and whether this approach can be used effectively to measure diversity.
Figure 1. Applications Quest summary page.
Figure 3. Cluster summary table.
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Are Somalis dumb?
In the West, something like 70% of the population seems to understand the intrinsic importance of the rule of law. They're not just following laws because they're afraid of the consequences of
breaking them, but rather, they follow laws because they realize the consequences of a lawless society.
Our Somali leaders never could see the ultimate consequences of their warlike behavior. They can't see beyond their noses and don't really seem to be any smarter or more knowledgeable than your
average Somali. Your Somali government leader is like your average Somali in education and ability.
Consider Said Barre. Say what you will about his reign, h failed to see the consequences of failing to appease those other clans. The stability of his reign was like the stability during the eye of a
storm. If Said Barre had been smarter he would have been able to diffuse all those tensions that were simmering under the service. But because of his lack of intelligence, he added to those tensions.
Leaders have to be smarter than the populace they lead. Consider the current president, Sheikh whathisface. What are his qualifications to lead a country other than "he knows Islam really, really
well"? What's his understanding of economics, strategy, politics, etc? None. He's just as blind as your average Somali.
If our leaders had been smarter, Somalia wouldn't be frequently listed as the worst country in the world. But unfortunately smart leaders pop up from smart populaces, and Somalia consistently
producing idiotic leaders just means that the Somali populace is idiotic.
waryaa poiuyt, stop hating it ain't good for your heart instead find someone to love. I bet JB wouldn't mind giving you a helping hand lool
I have a feeling this guy and peacenow are one and the same. His IP address should be monitored
Somalis tend to forget that an educated leader is only respected by an educated populace. The populace in Somalia are savages and therefore require a savage leader or else they just won't listen.
Originally posted by ADNAAN:
I have a feeling this guy and peacenow are one and the same. His IP address should be monitored
I think its just the mentality and culture they are living in, most countries have been through lawlessness and have survived. Somalia just needs to find its time and courage to beat the routine.
It will take longer than most think, comparing it to countries that have been stable for centuries is wrong.
Its a long process to gain conformity and law, the first step is the hardest. | {"url":"https://www.somaliaonline.com/community/topic/21588-are-somalis-dumb/?tab=comments#comment-383417","timestamp":"2024-11-05T09:28:53Z","content_type":"text/html","content_length":"274199","record_id":"<urn:uuid:d0d6c8f1-ba1d-4385-a52e-bbb1a3d37982>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00506.warc.gz"} |
Circle A has a radius of 2 and a center of (2 ,7 ). Circle B has a radius of 1 and a center of (3 ,1 ). If circle B is translated by <1 ,3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles? | HIX Tutor
Circle A has a radius of #2 # and a center of #(2 ,7 )#. Circle B has a radius of #1 # and a center of #(3 ,1 )#. If circle B is translated by #<1 ,3 >#, does it overlap circle A? If not, what
is the minimum distance between points on both circles?
Answer 1
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Answer 2
To determine if circle B, after being translated by <1, 3>, overlaps circle A, we can calculate the distance between the centers of the two circles and compare it with the sum of their radii. 1.
Distance between centers: Distance between centers = √((x2 - x1)^2 + (y2 - y1)^2) = √((3 - 2)^2 + (1 - 7)^2) = √(1^2 + (-6)^2) = √(1 + 36) = √37 2. Sum of radii: Sum of radii = Radius of circle A +
Radius of circle B = 2 + 1 = 3 Since the distance between the centers of the circles (sqrt(37)) is greater than the sum of their radii (3), the circles do not overlap. To find the minimum distance
between points on both circles, we need to subtract the sum of their radii from the distance between their centers. Minimum distance = Distance between centers - Sum of radii = √37 - 3 ≈ 6.08
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Cross-Validation Estimates IMSE
Part of Advances in Neural Information Processing Systems 6 (NIPS 1993)
Mark Plutowski, Shinichi Sakata, Halbert White
Integrated Mean Squared Error (IMSE) is a version of the usual mean squared error criterion, averaged over all possible training If it could be observed, it could be used sets of a given size. to
determine optimal network complexity or optimal data sub(cid:173) sets for efficient training. We show that two common methods of cross-validating average squared error deliver unbiased estimates of
IMSE, converging to IMSE with probability one. These esti(cid:173) mates thus make possible approximate IMSE-based choice of net(cid:173) work complexity. We also show that two variants of cross
validation measure provide unbiased IMSE-based estimates potentially useful for selecting optimal data subsets.
1 Summary To begin, assume we are given a fixed network architecture. (We dispense with this assumption later.) Let zN denote a given set of N training examples. Let QN(zN) denote the expected
squared error (the expectation taken over all possible examples) of the network after being trained on zN. This measures the quality of fit afforded by training on a given set of N examples. Let
IMSEN denote the Integrated Mean Squared Error for training sets of size N. Given reasonable assumptions, it is straightforward to show that IMSEN = E[Q N(ZN)] - 0"2, where the expectation is now
over all training sets of size N, ZN is a random training set of size N, and 0"2 is the noise variance. Let CN = CN(zN) denote the "delete-one cross-validation" squared error measure for a network
trained on zN. CN is obtained by training networks on each of the N training sets of size N -1 obtained by deleting a single example; the measure follows | {"url":"https://proceedings.nips.cc/paper_files/paper/1993/hash/06997f04a7db92466a2baa6ebc8b872d-Abstract.html","timestamp":"2024-11-05T05:56:08Z","content_type":"text/html","content_length":"9495","record_id":"<urn:uuid:f42efcb4-c0cb-4cec-bbe5-f82e38e219e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00632.warc.gz"} |
foreach — Iterate over all elements in one or more lists foreach varname list body
foreach varlist1 list1 ?varlist2 list2 ...? body
The foreach command implements a loop where the loop variable(s) take on values from one or more lists. In the simplest case there is one loop variable, varname, and one list, list, that is a list of
values to assign to varname. The body argument is a Tcl script. For each element of list (in order from first to last), foreach assigns the contents of the element to varname as if the lindex command
had been used to extract the element, then calls the Tcl interpreter to execute body.
In the general case there can be more than one value list (e.g., list1 and list2), and each value list can be associated with a list of loop variables (e.g., varlist1 and varlist2). During each
iteration of the loop the variables of each varlist are assigned consecutive values from the corresponding list. Values in each list are used in order from first to last, and each value is used
exactly once. The total number of loop iterations is large enough to use up all the values from all the value lists. If a value list does not contain enough elements for each of its loop variables in
each iteration, empty values are used for the missing elements.
The break and continue statements may be invoked inside body, with the same effect as in the for command. Foreach returns an empty string.
This loop prints every value in a list together with the square and cube of the value:
set values {1 3 5 7 2 4 6 8} ;# Odd numbers first, for fun!
puts "Value\tSquare\tCube" ;# Neat-looking header
foreach x $values { ;# Now loop and print...
puts " $x\t [expr {$x**2}]\t [expr {$x**3}]"
The following loop uses i and j as loop variables to iterate over pairs of elements of a single list.
set x {}
foreach {i j} {a b c d e f} {
lappend x $j $i
# The value of x is "b a d c f e"
# There are 3 iterations of the loop.
The next loop uses i and j to iterate over two lists in parallel.
set x {}
foreach i {a b c} j {d e f g} {
lappend x $i $j
# The value of x is "a d b e c f {} g"
# There are 4 iterations of the loop.
The two forms are combined in the following example.
set x {}
foreach i {a b c} {j k} {d e f g} {
lappend x $i $j $k
# The value of x is "a d e b f g c {} {}"
# There are 3 iterations of the loop.
for, while, break, continue foreach, iteration, list, loop
Copyright © 1993 The Regents of the University of California.
Copyright © 1994-1996 Sun Microsystems, Inc. | {"url":"https://www.tcl.tk/man/tcl9.0/TclCmd/foreach.html","timestamp":"2024-11-09T04:39:14Z","content_type":"text/html","content_length":"5228","record_id":"<urn:uuid:9c7e874e-23b5-44a6-9c34-7759c3f68338>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00669.warc.gz"} |
How to create own Search Engine in C# - Part 3 - Maytham Fahmi
How to create own Search Engine in C# – Part 3
As we demonstrate in second part of this article, we was able to improve search results, but we talked also on how to return suggestion to typo in searching.
In this article I am going to use Levenshtein Distance Algorithm.
Levenshtein Algorithm calculates the distance between 2 input strings. If we have 2 equal strings like “A” and “A” we get distance 0, but if we have “A” and “B” we will get 1. The Algorithm iterate
the process for both strings input
length and sum up the difference between strings. At the end of the Algorithm we will have a final number of distance. The greater distance is the less chance both strings are related. This Algorithm
does not intelligence, so if we search for word like job we can get word like Jacob with distance 3 as relative word.
There is a punch of recommendation to fine tune the search, but that requires a search engine strategy to define the search roles. In our case, we just made a standard implementation of the
Algorithm. We have also added a percentage string length measurement to reach best possible results, but that will still not fix Job vs Jakob case, but will improve search results. In case you need
to make it intelligible, you need to use other algorithms like NLP, Machine Learning, use a word dictionary etc. I have asked a question regarding this on stackoverflow you can read it here.
We imagine to search word ”test” but for some reason we type ”tset”.
The image below describes how does the Algorithm calculate the distance. In our case the two mentioned words has a distance of 2
Levenstein Distance Algorithm.
Distance calculation matrix example
So All that explanation, lets do have a look at our Levenstein Algorithm using Dynamic Programming approach:
private int Compute(string inputWord, string checkWord)
int n = inputWord.Length;
int m = checkWord.Length;
if (n == 0)
return m;
if (m == 0)
return n;
_wordMatrix = new int[n + 1, m + 1];
for (int i = 0; i <= n; i++)
_wordMatrix[i, 0] = i;
for (int j = 0; j <= m; j++)
_wordMatrix[0, j] = j;
for (int i = 1; i <= n; i++)
for (int j = 1; j <= m; j++)
if (inputWord[i - 1] == checkWord[j - 1])
_wordMatrix[i, j] = _wordMatrix[i - 1, j - 1];
int minimum = int.MaxValue;
if ((_wordMatrix[i - 1, j]) + 1 < minimum)
minimum = (_wordMatrix[i - 1, j]) + 1;
if ((_wordMatrix[i, j - 1]) + 1 < minimum)
minimum = (_wordMatrix[i, j - 1]) + 1;
if ((_wordMatrix[i - 1, j - 1]) + 1 < minimum)
minimum = (_wordMatrix[i - 1, j - 1]) + 1;
_wordMatrix[i, j] = minimum;
return _wordMatrix[inputWord.Length, checkWord.Length];
So now if we pass a word like “dnemark” to this algorithm, we will get a suggestion like “denmark” as you can see in my example below:
Levenstein example
Complexity analysis
I have had 2 choices to build Levenshtein Algorithm, by using Recursion or
Dynamic Programming. To find out which solution was better, I made a little Analysis on both approaches.
1. Recursion
Regarding to on-line resources1 using recursion have a complexity of O(2^n).
2. Dynamic Programming
Dynamic Programming that means we memorize the strings in array. We need to iterate through our two string input in nested loop. That gives m * n time which is O(n^2) worst case. (m and n =
string length for each input)
Mathematically n^2 is much faster than 2^n, therefore our implementation choice was Dynamic Programming approach.
I have put all code and data source together for all Search Engine series in my GitHub Blog Example Repo, enjoy.
In this article we have learned to about Levenstein Distance algorithm and how it is used by providing searching example.
We have also analysis the complexity of Levenstein with recursive approach and Dynamic Programming approach.
Finally we demonstrated how Levenstein code looks like in C#.
In future, I will make an article about Lucene search feature in C#. I am also considering adding Frequency Index feature or adding front-end UI feature for this search engine, if you think it is a
good idea or you like to have such article, please let me know in comments.
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