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Q 4 Due to change in prices, the cost of living index for the working class in a city - Maths - Statistics - 12408873 | Meritnation.com
Q.4. Due to change in prices, the cost of living index for the working class in a city rose to 225 in April from 180 in March, 2003. The index of food became 252 from 198, that of clothing from 185 to 205, that of fuel and lighting from 175 to 195 and that of miscellaneous from 138 to 212. The index of rent, however remained unchanged at 150. Find the weights of all the groups if the weights of clothing, fuel and lighting, and rent were the same.
Hint. Construct the following table :
Dear student\phantom{\rule{0ex}{0ex}}Weighted mean is given by \frac{\sum _{i=1}^{n} {X}_{i} {w}_{i}}{\sum _{i=1}^{n} {w}_{i}}, where {w}_{i} is the wieght\phantom{\rule{0ex}{0ex}}In march, it is given\phantom{\rule{0ex}{0ex}}\frac{\sum _{i=1}^{n} {X}_{i} {w}_{i}}{\sum _{i=1}^{n} {w}_{i}}=180\phantom{\rule{0ex}{0ex}}\frac{198×x+185×1+175×1+150×1+138×y}{x+1+1+1+y}=\frac{198x+185+175+150+138y}{x+3+y}=180\phantom{\rule{0ex}{0ex}}198x+510+138y=180x+540+180y\phantom{\rule{0ex}{0ex}}18x-42y=30\phantom{\rule{0ex}{0ex}}3x-7y=5 _______________\left(1\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}In April, it is given\phantom{\rule{0ex}{0ex}}\frac{\sum _{i=1}^{n} {X}_{i} {w}_{i}}{\sum _{i=1}^{n} {w}_{i}}=225\phantom{\rule{0ex}{0ex}}\frac{252×x+205×1+195×1+150×1+212×y}{x+1+1+1+y}=\frac{252x+205+195+150+212y}{x+3+y}=225\phantom{\rule{0ex}{0ex}}252x+550+212y=225x+675+225y\phantom{\rule{0ex}{0ex}}27x-13y=125 __________\left(2\right)\phantom{\rule{0ex}{0ex}}From equation 1\phantom{\rule{0ex}{0ex}}x=\frac{5+7y}{3}\phantom{\rule{0ex}{0ex}}Put in equation 2\phantom{\rule{0ex}{0ex}}27×\frac{5+7y}{3}-13y=125\phantom{\rule{0ex}{0ex}}45+63y-13y=125\phantom{\rule{0ex}{0ex}}50y=80\phantom{\rule{0ex}{0ex}}y=\frac{80}{50}=\frac{8}{5}\phantom{\rule{0ex}{0ex}}x=\frac{5+7×\frac{8}{5}}{3}=\frac{5+\frac{56}{5}}{3}=\frac{81}{15}=\frac{27}{5}
|
Generating permutations - DEV Community
I like to think of permutations as simply:
The rearrangement of a collection of elements into each possible new arrangement of those elements.
Let's demonstrate what the permutations of a dataset could look like:
Dataset: [1, 2, 3]
Permutations: [
Building an algorithm to generate permutations of a collection is also a common technical interview test too and so it is a good thing to be aware of.
Now, for those familiar with the concept of combinations you may remember that permutations and combinations are somewhat related but be aware that there is a difference between these concepts.
In mathematics, permutation is the act of arranging the members of a set into a sequence or order, or, if the set is already ordered, rearranging its elements—a process called permuting. Permutations differ from combinations, which are selections of some members of a set regardless of order.
Source: Permutations Wiki
We can see this inherent difference as we write out the maths behind each concept.
The count of possible combinations (C) is defined as:
C(n, r) = \frac{n!}{r!(n-r)!}
The count of possible permutations (P) is defined as:
P(n, r) = \frac{n!}{(n-r)!}
With this, we can see the inherent difference between the two concepts.
const input: number[] = [1,2,3];
const permutations: number[][] = permute(input);
describe("permutations", () => {
it("returns the correct number of permutations", () => {
expect(permutations.length).toBe(6);
it("returns the correct permutations", () => {
expect(permutations).toEqual([
The tests are written with Jest as the test runner, TypeScript as our language of choice and TS-Jest as the glue between the two since Jest doesn't support TypeScript out of the box.
If you want to know how to get Jest and TypeScript to work together then you can check the Jest Config and TS Config files from the repl for this article.
If you wish to look at the code, config, or just run the tests, you can do so with the repl below:
We will be using TypeScript for our implementation but don't be alarmed if you haven't used that before as we will be diving into the implementation step by step!
function swap<T>(array: T[], x: number, y: number): void {
const temp = array[x];
function permute<T>(array: T[], start: number = 0, result: T[][] = []): T[][] {
if(start === array.length - 1) {
for(let index = start; index < array.length; index++) {
swap<T>(array, start, index);
permute<T>(array, start + 1, result);
The permute function
The permute function is a recursive function which takes in 3 parameters:
array is the collection that we will be generating permutations from
start is the index of the array to begin from
result is an array of arrays, each of which will contain items of the same type as the items of the original input array
You will never use the start or result parameters because these are only used for each consecutive recursive call of the permute function. If you really wanted to though, you could provide arguments for these although it is not recommended to do so.
As we enter the function body we begin with the "base case" as it is usually named and all this does is provide us a way to break the recursive calls based on a pre-defined definition of done.
In our case, if the start index is equal to the last item of the array's index, we will stop recursing the input array. If we are not currently meeting the criteria of our base case, we then begin a loop going from the start index to the end of the array.
For each item in the array, we will use a method called backtracking which is a way of running an operation and then undoing it again after we complete some action when required.
In our case, we are using the swap function to swap the item from the start index and the item at the current index, running a recursive call of the permute function and finally re-swapping the items from the start index and current index back to their original places once each recursive cycle is completed.
You can imagine this as a tree, we take each item, swap it with every other item and swap those iterations with every other iteration before resetting everything in the original array back to normal and returning the result of each iteration in the tree.
This becomes a problem no matter how it is implemented once the dataset grows over ten or so items however since the count of possible permutations is mathematically tied to the factorial result of the count of items in the input array.
This means that an input array with 4 items will produce 24 permutations but one with 8 items will produce 40,320 and one with 20 items will produce 2,432,902,008,176,640,000. This means we should keep the input array as small as possible at any time to keep things performant.
The swap function takes in 3 parameters:
The first (x) index for the swap
The second (y) index for the swap
We copy the item at position x in the array, set the item at position x to the value of the item at position y and finally set the item at position y to the item that used to be at position x.
Permutations are a common occurrence and have use cases in many real-life scenarios and industries such as cyber-security, cryptography, and transportation for example.
In the medical industry, medicine manufacturers use permutations to analyze and predict the spreading of various diseases, genetic sequences, analysis of drug-safety data, and to analyze various permutations of drugs and chemicals as well in relation to the desired outcome of treatment.
Even if it isn't relevant to your work, it is a common coding test and so knowing the theory and practice are good things to have in your back pocket. The best part is that if we know how to swap items, backtracking based on a single index is possible as we have seen in the implementation above and I personally find this method of generating permutations the easiest to remember and implement and I hope it is the same case for you.
I hope this post brought you some value and helped to clear up what permutations are, what real-world use cases for permutations are out there and how we can generate them using swapping, recursion, and backtracking!
Day 4 of Studying LeetCode Solution until I Can Solve One on My Own: Problem#443.String Compression(Medium/JavaScript)
|
§ Alexandrov topology
These are the best generalizations of finite topological spaces. I should study them for better intuition + comptutability properties. This is a topology where the intersection of all families of open sets is open, not just finite intersections. The minimal neighbourhood of a point
V(x)
is the intersection of all opens containing
x
Paper on the systematic study of Alexandrov spaces
|
§ When are the catalan numbers odd
C_n
count the number of binary trees on
For every binary tree, label the nodes in some standard ordering (eg. BFS).
Pick the lex smallest unbalanced node (node with different left and right subtree sizes).
The operation that swaps the left and right subtrees of the lex smallest unbalanced node is an involution.
This operation only fails when we have a complete binary tree, so the number of nodes is
n = 2^r - 1
, so we pair such a complete binary tree to itself.
This breaks the set
C_n
into an even number of trees (pairs of unbalanced trees) and a potential "loner tree" (paired with itself) which is the complete binary tree.
C_n
n = 2^r - 1
, which allows for us to have a complete binary tree, which is not paired by the involution.
|
Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem | EMS Press
Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem
This paper concerns numerical approximations for the Cahn-Hilliard equation
u_t+\Delta (\varepsilon \Delta u -{\varepsilon}^{-1}f(u))=0
and its sharp interface limit as
\varepsilon \searrow 0
, known as the Hele-Shaw problem. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in \cite{XA2} to the solution of the Hele-Shaw (Mullins-Sekerka) problem, provided that the Hele-Shaw (Mullins-Sekerka) problem has a global (in time) classical solution. This is accomplished by establishing some improved a priori solution and error estimates, in particular, an
L^\infty(L^\infty)
-error estimate, and making full use of the convergence result of \cite{alikakos94}. The cruxes of the analysis are to establish stability estimates for the discrete solutions, use a spectrum estimate result of Alikakos and Fusco \cite{alikakos93} and Chen \cite{chen94}, and establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term.
Andreas Prohl, Xiaobing Feng, Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Interfaces Free Bound. 7 (2005), no. 1, pp. 1–28
|
On the restricted divisor function in arithmetic progressions | EMS Press
On the restricted divisor function in arithmetic progressions
We obtain several asymptotic estimates for the sums of the restricted divisor function
\tau_{M,N}(k) = \# \{1 \le m \le M, \ 1\le n \le N : mn = k\}
over short arithmetic progressions, which improve some results of J. Truelsen. Such estimates are motivated by the links with the pair correlation problem for fractional parts of the quadratic function
\alpha k^2
k=1,2,\dots
\alpha
Igor E. Shparlinski, On the restricted divisor function in arithmetic progressions. Rev. Mat. Iberoam. 28 (2012), no. 1, pp. 231–238
|
Serre's modularity conjecture: The level one case
15 September 2006 Serre's modularity conjecture: The level one case
Chandrashekhar Khare1
We prove the level one case of Serre's conjecture. Namely, we prove that any continuous, odd, irreducible representation
\stackrel{̲}{\rho }:{G}_{\mathbb{Q}}\to {\mathrm{GL}}_{2}\left(\stackrel{̲}{{\mathbb{F}}_{p}}\right)
p
arises from a cuspidal eigenform in
{S}_{k}\left({\mathrm{SL}}_{2}\left(\mathbb{Z}\right)\right)
k\ge 2
. The proof relies on the methods introduced in an earlier joint work with J.-P. Wintenberger [31] together with a new method of weight reduction
Chandrashekhar Khare. "Serre's modularity conjecture: The level one case." Duke Math. J. 134 (3) 557 - 589, 15 September 2006. https://doi.org/10.1215/S0012-7094-06-13434-8
Chandrashekhar Khare "Serre's modularity conjecture: The level one case," Duke Mathematical Journal, Duke Math. J. 134(3), 557-589, (15 September 2006)
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Viscoelastic Properties of Ovine Adipose Tissue Covering the Gluteus Muscles | J. Biomech Eng. | ASME Digital Collection
Amit Gefen,
e-mail: gefen@eng.tau.ac.il
Einat Haberman
Gefen, A., and Haberman, E. (April 19, 2007). "Viscoelastic Properties of Ovine Adipose Tissue Covering the Gluteus Muscles." ASME. J Biomech Eng. December 2007; 129(6): 924–930. https://doi.org/10.1115/1.2800830
Pressure-related deep tissue injury (DTI) is a life-risking form of pressure ulcers threatening immobilized and neurologically impaired patients. In DTI, necrosis of muscle and enveloping adipose tissues occurs under intact skin, owing to prolonged compression by bony prominences. Modeling the process of DTI in the buttocks requires knowledge on viscoelastic mechanical properties of the white adipose tissue covering the gluteus muscles. However, this information is missing in the literature. Our major objectives in this study were therefore to (i) measure short-term
(HS)
and long-term
(HL)
aggregate moduli of adipose tissue covering the glutei of sheep, (ii) determine the effects of preconditioning on
HS
HL
, and (iii) determine the time course of stress relaxation in terms of the transient aggregate modulus
H(t)
in nonpreconditioned (NPC) and preconditioned (PC) tissues. We tested 20 fresh tissue specimens (from 20 mature animals) in vitro: 10 specimens in confined compression for obtaining the complete
H(t)
response to a ramp-and-hold protocol (ramp rate of
300mm∕s
), and 10 other specimens in swift indentations for obtaining comparable short-term elastic moduli at higher ramp rates
(2000mm∕s)
. We found that
HS
in confined compression were
28.9±14.9kPa
18.1±6.9kPa
for the NPC and PC specimens, respectively. The
HL
property,
10.3±4.2kPa
, was not affected by preconditioning. The transient aggregate modulus
H(t)
always reached the plateau phase (less than 10% difference between
H(t)
HL
) within
2min
, which is substantially shorter than the times for DTI onset reported in previous animal studies. The short-term elastic moduli at high indentation rates were
22.6±10kPa
15.8±9.4kPa
for the NPC and PC test conditions, respectively. Given a Poisson’s ratio of 0.495, comparison of short-term elastic moduli between the high and slow rate tests indicated a strong deformation-rate dependency. The most relevant property for modeling adipose tissue as related to DTI is found to be
HL
, which is conveniently unaffected by preconditioning. The mechanical characteristics of white adipose tissue provided herein are useful for analytical as well as numerical models of DTI, which are essential for understanding this serious malady.
biological tissues, biomechanics, compressive strength, diseases, elastic moduli, indentation, muscle, stress relaxation, viscoelasticity, wounds, pressure sore, bedsore, decubitus, fat, stress relaxation
Biological tissues, Compression, Muscle, Relaxation (Physics), Stress, Elastic moduli, Viscoelasticity, Cycles, Mechanical properties
Sprigle
Pressure-Related Deep Tissue Injury Under Intact Skin and the Current Pressure Ulcer Staging Systems
What is the ‘Purple Heel’
Balancing the Pressure Ulcer Cost and Quality Equation
Nurs. Econ
Chronic Wound Pathogenesis and Current Treatment Strategies: A Unifying Hypothesis
Establishing Predictive Indicators for the Status of Loaded Soft Tissues
Compression Induced Cell Damage in Engineered Muscle Tissue: An In Vitro Model to Study Pressure Ulcer Aetiology
Etiologic Factors in Pressure Sores: An Experimental Model
Shabshin
Assessment of Mechanical Conditions in Sub-Dermal Tissues During Sitting: A Combined Experimental-MRI and Finite Element Approach
,” in Encyclopedia of Sports Medicine and Science,
(http://sportsci.orghttp://sportsci.org).
A Method to Measure the Hyperelastic Parameters of Ex Vivo Breast Tissue Samples
Initial In Vivo Experience With Steady-State Subzone-Based MR Elastography of the Human Breast
Imaging the Shear Modulus of the Heel Fat Pads
In Vivo Biomechanical Behavior of the Human Heel Pad During the Stance Phase of Gait
Rheological and Recovery Properties of Poly(ethylene glycol) Diacrylate Hydrogels and Human Adipose Tissue
Site-Specific Differences in the Fatty Acid Composition of Human Adipose Tissue
Velez-Duran
Fatty Acid Composition of Adipose Tissue in Humans: Differences Between Subcutaneous Sites
Abdominal vs Buttock Adipose Fat: Relationships With Children’s Serum Lipid Levels
Pressure-Time Cell Death Threshold for Albino Rat Skeletal Muscles as Related to Pressure Sore Biomechanics
Breast Deformation Modelling for Image Reconstruction in Near Infrared Optical Tomography
Br. Med. J. (Clin Res. Ed)
Determination of Homeostatic Elastic Moduli in Two Layers of the Esophagus
Nonlinear Dynamic Viscoelastic Model for Osteoarthritic Cartilage Indentation Force
|
§ Dirichlet inversion
We call all functions
f: \mathbb Z \rightarrow \mathbb R
as arithmetic functions , since they operate on the integers. We introduce an operator
f \star g: \mathbb Z \rightarrow \mathbb R
(f \star g)(n) \equiv \sum_{d \vert n} f(d) g(n/d)
We will show that the set of arithmetic functions forms a group under the operator
\star
, with identity:
id_\star(n) \equiv \lfloor 1/n \rfloor = \begin{cases} 1 & n = 1 \\ 0 & \text{otherwise} \end{cases}
The reason all of this is interesting is that the inverse of the constant function
1(n) \equiv 1
is going to be this function called as the mobius function
\mu
\mu(n=p_1^{\alpha_1} p_2^{\alpha_2} \dots p_r^{\alpha_r}) \equiv \begin{cases} 0 & \text{if any $\alpha_i > 1$} \\ (-1)^{\alpha_1 + \alpha_2 + \dots + \alpha_r} & \text{if all $\alpha_i \in \{ 0, 1 \}$} \end{cases}
The mobius function will allow us to perform mobius inversion :
\begin{aligned} f(n) &\equiv \sum_{d \vert n} g(d) = \sum_{d \vert n} g(d) 1(n/d) = g \star 1 \\ f \star 1^{-1} &= g \star 1 \star 1^{-1} \\ f \star \mu &= g \end{aligned}
That is, we originally had
f
defined in terms of
g
. We can recover an expression for
g
in terms of
§ The algebra of multiplicative functions
We claim that the set of functions
\{ \mathbb Z \rightarrow \mathbb C \}
is a commutative group, with the group operation
\star
(f \star g)(n) \equiv \sum_{d \vert n} f(d) g(n/d)
with the identity element being
id_\star(n) \equiv \lfloor 1 / n \rfloor
. The idea is that if
(n = 1)
\lfloor 1/1 \rfloor = 1
, and for any other number
n > 0
1/n < 1
\lfloor 1/n \rfloor = 0
§ verification of
istar
being the identity
\begin{aligned} &(f \star id_\star)(n) \equiv \sum_{d \vert n} f(d) id_\star(n/d) \\ &= f(n) id_\star(1) + \sum_{d \vert n, d > 1} f(n) id_\star(d) \\ &= f(n) \cdot 1 + \sum_{d \vert n, d > 1} f(n) \cdot 0 \\ &= f(n) \\ \end{aligned}
§ associativity, commutativity of
\star
To prove associativity, it's better to write the formula as:
(f \star g)(n) \equiv \sum_{d \vert n} f(n) g(n/d) = \sum_{xy = n} f(x) g(y)
From this rewrite, it's clear that
(f \star g \star h)(n)
will unambiguously sum over tripes
(x, y, z)
xyz = n
. I leave the working-this-out to you. This should also make the commutativity immediate. Summing over pairs of the form
f(x) g(y) : xy = n
is the same as summing over
f(y) g(x) : yx = n
§ Existence of inverse
We can show that an inverse exists by showing that a formula for it exists; The idea is to construct one by induction. Clearly, for a given function
f
, we need the inverse
f^{-1}
(f \star f^{-1})(n) = id_\star
\begin{aligned} &(f \star f^{-1})(1) = id_\star(1) = 1 \\ &f(1) f^{-1}(1) = 1 \\ & f^{-1}(1) \equiv 1 / f(1)\\ \end{aligned}
Great, we have a base case; We can now compute
f^{-1}(n)
inductively, assuming we know the value of
f^{-1}(d)
d \vert n
\begin{aligned} &(f \star f^{-1})(n) = id_\star(1) = 0 \\ &\sum_{d \vert n} f(d) f^{-1}(n/d) = 0 \\ &f(1) f^{-1}(n) + \sum_{d \vert n, d < n} f(d) f^{-1}(n/d) = 0 \\ &f^{-1}(n) = -\frac{\sum_{d \vert n, d < n} f(d) f^{-1}(n/d)}{f(1)} \end{aligned}
\mu
as the inverse of the
one
§ Mobius inversion
Now that we know that
\mu = \texttt{const 1}^{-1}
, we can use this fact to perform mobius inversion :
f(n) \equiv \sum_{d \vert n} g(n/d) = \texttt{const 1} \star g
f
g
. We can now write
g
in terms of
\begin{aligned} &f(n) = \texttt{const 1} \star g \\ &f \star \texttt{const 1}^{-1} = g \\ &g = f \star \mu \\ &g = \sum_{d \vert n} f(d) \mu(n/d) \end{aligned}
n = \sum_{d \vert n} \phi(d)
\begin{array}{|c|c|c|c|} d & \{ 1 \leq x \leq 12 : (x, 12) = d \} & \{ 1 \leq x \leq 12: (x/d, 12/d) = 1\} & \text{size of set} \\ 1 & \{ 1, 5, 7, 11 \} & (x, 12) = 1 & 4 \\ 2 & \{2, 10 \} & (x/2, 6) = 1& 2 \\ 3 & \{3, 9 \} & (x/3, 4) = 1 & 2 \\ 4 & \{4, 8 \} & (x/4, 3) = 1 & 2 \\ 6 & \{ 6 \} & (x/6, 2) = 1 & 1 \\ 12 & \{ 12 \} (x/12, 1) = 1 & 1 \end{array}
Notice that the sizes of sets that we are calculating, for example,
|\{ 1 \leq x \leq 12 : (x/2, 6) = 1 \}| = \phi(6)
. Summing over all of what we have, we've counted the numbers in
[1, 2, \dots, 12]
in two ways --- one directly, and the other by partitioning into equivalence classes:
12 = \phi(1) + \phi(2) + \phi(3) + \phi(4) + \phi(6) + \phi(12) = \sum_{d \vert 12} \phi(12/d)
In general, the same argument allows us to prove that:
n = \sum_{d \vert n} n/d
§ Using mobius inversion on the euler totient function
§ Other arithmetical functions and their relations
|
Detection of Solder Bump Defects in Electronic Packages Using Local Temporal Coherence Analysis of Laser Ultrasonic Signals | J. Electron. Packag. | ASME Digital Collection
I. Charles Ume Fellow ASME
Yang, J., and Ume, I. C. (February 18, 2009). "Detection of Solder Bump Defects in Electronic Packages Using Local Temporal Coherence Analysis of Laser Ultrasonic Signals." ASME. J. Electron. Packag. March 2009; 131(1): 011013. https://doi.org/10.1115/1.3068301
Microelectronics packaging technology has evolved from through-hole and bulk configurations to surface-mount and small-profile configurations. Surface mount devices, such as flip chip packages, chip scale packages, and ball grid arrays, use solder bump interconnections between them and substrates/printed wiring boards. Solder bumps, which are hidden between the device and the substrate/board, are difficult to inspect. A solder bump inspection system was developed using laser ultrasound and interferometric techniques. This system has been successfully applied to detect solder joint/bump defects, including missing, misaligned, open, and cracked solder joints/bumps in flip chips, chip scale packages, and multilayer ceramic capacitors. This system uses a pulsed Nd:YAG laser to induce ultrasound in the electronic packages in the thermoelastic regime; it then measures the transient out-of-plane displacement response on the package surface using the interferometric technique. This paper presents a local temporal coherence (LTC) analysis of laser ultrasound signals and compares it to previous signal-processing methods, including error ratio and correlation coefficient methods. The results showed that LTC analysis increased measurement accuracy and sensitivity for inspecting solder bump defects in electronic packages. Laser ultrasound inspection results are also compared with X-ray and
C
-mode scanning acoustic microscopy results. In particular, this paper discusses defect detection for
6.35×6.35×0.6 mm3
flip chips and flip chips (“SiMAF;” Siemens AG) with lead-free solder bumps.
chip scale packaging, electronics packaging, flip-chip devices, laser beam applications, light interferometry, solders, thermoelasticity, ultrasonic applications, laser ultrasound, flip chip, solder bump, local temporal coherence, lead-free, laser vibrometer
Electronic packages, Flip-chip devices, Lasers, Signals, Solders, Tin, Ultrasound, Vehicles, X-rays, Inspection, Flip-chip, Transients (Dynamics), Interferometry
Oresjo
New Study Reveals Component Defect Levels
Flip Chips and Acoustic Micro Imaging: An Overview of Past Applications, Present Status, and Roadmap for the Future
X-Ray Inspection of IC Packages and PWBs
Chip Scale Rev.
A Novel Approach for Flip Chip Solder Joint Quality Inspection: Laser Ultrasound and Interferometer System
Detection of Flip Chip Solder Joint Cracks Using Correlation Coefficient and Auto-Comparison Analyses of Laser Ultrasound Signals
Defects Pattern Recognition for Flip Chip Solder Joint Quality Inspection
Proceedings of the 52nd Electronic Components and Technology Conference (ECTC)
, San Diego, CA, May 28–31.
Thermomechanical Reliability Study of Flip-Chip Solder Bumps: Using Laser Ultrasound Technique and Finite Element Method
Detection of Structural Damage From the Local Temporal Coherence of Diffuse Ultrasonic Signals
Beenacer
Short Time Correlation Analysis in Porous Medium
Validation of the Shot Time Correlation Analysis
Speckle Tracking Methods for Ultrasonic Elasticity Imaging Using Short-Time Correlation
Determination of Measurement Limit for Open Solder Bumps on a Flip-Chip Package Using a Laser Ultrasound Inspection System
www.wikipedia.orgwww.wikipedia.org.
Nondestructive Defect Detection in Multilayer Ceramic Capacitors Using an Improved Digital Speckle Correlation Method With Wavelet Packet Noise Reduction Processing
|
Noise and Ray Marching @ /home/adrian
LiterateLB's volumetric visualization functionality relies on a simple ray marching implementation to sample both the 3D textures produced by the simulation side of things and the signed distance functions that describe the obstacle geometry. While this produces surprisingly nice looking results in many cases, some artifacts of the visualization algorithm are visible depending on the viewport and sample values. Extending the ray marching code to utilize a noise function is one possibility of mitigating such issues that I want to explore in this article.
While my original foray into just in time visualization of Lattice Boltzmann based simulations was only an aftertought to playing around with SymPy based code generation approaches I have since put some work into a more fully fledged code. The resulting LiterateLB code combines symbolic generation of optimized CUDA kernels and functionality for just in time fluid flow visualization into a single literate document.
For all fortunate users of the Nix package manager, tangling and building this from the Org document is as easy as executing the following commands on a CUDA-enabled NixOS host.
The basic ingredient for producing volumetric images from CFD simulation data is to compute some scalar field of samples
s : \mathbb{R}^3 \to \mathbb{R}_0^+
. Each sample
s(x)
can be assigned a color
c(x)
by some convenient color palette mapping scalar values to a tuple of red, green and blue components.
The task of producing an image then consists to sampling the color field along a ray assigned to a pixel by e.g. a simple pinhole camera projection. For this purpose a simple discrete approximation of the volume rendering equation with constant step size
\Delta x \in \mathbb{R}^+
already produces suprisingly good pictures. Specifically
C(r) = \sum_{i=0}^N c(i \Delta x) \mu (i \Delta x) \prod_{j=0}^{i-1} \left(1 - \mu(j\Delta x)\right)
is the color along ray
r
N\Delta x
with local absorption values
\mu(x)
. This local absorption value may be chosen seperately of the sampling function adding an additional tweaking point.
The basic approach may also be extended arbitrarily, e.g. it is only the inclusion of a couple of phase functions away from being able recover the color produced by light travelling through the participating media that is our atmosphere.
There are many different possibilities for the choice of sampling function
s(x)
given the results of a fluid flow simulation. E.g. velocity and curl norms, the scalar product of ray direction and shear layer normals or vortex identifiers such as the Q criterion
Q = \|\Omega\|^2 - \|S\|^2 > 0 \text{ commonly thresholded to recover isosurfaces}
that contrasts the local vorticity and strain rate norms. The strain rate tensor
S
is easily recovered from the non-equilibrium populations
f^\text{neq}
of the simulation lattice — and is in fact already used for the turbulence model. Similarly, the vorticity
\Omega = \nabla \times u
can be computed from the velocity field using a finite difference stencil.
The problem w.r.t. rendering when thresholding sampling values to highlight structures in the flow becomes apparent in the following picture:
Curl Norm
While the exact same volume discretization was used for both visualizations, the slices are much less apparent for the curl norm samples due to the more gradual changes. In general the issue is most prominent for scalar fields with large gradients (specifically the sudden jumps that occur when restricting sampling to certain value ranges as is the case for the Q criterion).
The reason for these artifacts is primarily choice of start offsets w.r.t. the traversed volume in addition the the step width. While this tends to become less noticable when decreasing said steps, this is not desirable from a performance perspective.
What I settled on for LiterateLB's renderer are view-aligned slicing and random jittering to remove most visible artifacts. The choice of randomness for jittering the ray origin is critical here as plain random numbers tend to produce a distracting static-like pattern. A common choice in practice is to use so called blue noise instead. While both kinds of noise eliminate most slicing artifacts, the remaining patterns tend to be less noticeable for blue noise. Noise is called blue if it contains only higher frequency components which makes it harder for the pattern recognizer that we call brain to find patterns where there should be none.
The void-and-cluster algorithm1 provides a straight forward method for pre-computing tileable blue noise textures that can be reused during the actual visualization. Tileability is a desirable property for this as we otherwise would either need a noise texture large enough to cover the entire image or instead observe jumps at the boundary between the tiled texture.
The first ingredient for void-and-cluster is a filteredPattern function that applies a plain Gaussian filter with given
\sigma
to a cyclic 2d array. Using cyclic wrapping during the application of this filter is what renders the generated texture tileable.
return gaussian_filter(pattern.astype(float), sigma=sigma, mode='wrap',
truncate=np.max(pattern.shape))
This function will be used to compute the locations of the largest void and tightest cluster in a binary pattern (i.e. a 2D array of 0s and 1s). In this context a void describes an area with only zeros and a cluster describes an area with only ones.
These two functions work by considering the given binary pattern as a float array that is blurred by the Gaussian filter. The blurred pattern gives an implicit ordering of the voidness of each pixel, the minimum of which we can determine by a simple search. It is important to exclude the initial binary pattern here as void-and-cluster depends on finding the largest areas where no pixel is set.
return np.argmax(masked_array(filteredPattern(pattern, sigma), mask=np.
logical_not(pattern)))
Computing the tightest cluster works in the same way with the exception of searching the largest array element and masking by the inverted pattern.
For the initial binary pattern we set n_start random locations to one and then repeatedly break up the largest void by setting its center to one. This is also done for the tightest cluster by setting its center to zero. We do this until the locations of the tightest cluster and largest void overlap.
The actual algorithm utilizes these three helper functions in four steps:
Initial pattern generation
Eliminiation of n_start tightest clusters
Elimination of n/2-n_start largest voids
Elimination of n-n/2 tightest clusters of the inverted pattern
For each elimination the current rank is stored in the noise texture producing a 2D arrangement of the integers from 0 to n. As the last step the array is divided by n-1 to yield a grayscale texture with values in
[0,1]
In order to check whether this actually generated blue noise, we can take a look at the Fourier transformation for an exemplary
100 \times 100
texture:
One can see qualitatively that higher frequency components are significantly more prominent than lower ones. Contrasting this to white noise generated using uniformly distributed random numbers, no preference for any range of frequencies can be observed:
Contasting the original Q criterion visualization with one produced using blue noise jittering followed by a soft blurring shader, we can see that the slicing artifacts largely vanish. While the jittering is still visible to closer inspection, the result is significantly more pleasing to the eye and arguably more faithful to the underlying scalar field.
Ray marching with blue noise jittering
While white noise also obcures the slices, its lower frequency components produce more obvious static in the resulting image compared to blue noise. As both kinds of noise are precomputed we can freely choose the kind of noise that will produce the best results for our sampling data.
In practice where the noise is applied just-in-time during the visualization of a CFD simulation, all remaining artifacts tend to become invisible. This can be seen in the following video of the Q criterion evaluated for a simulated nozzle flow in LiterateLB:
Ulichney, R. Void-and-cluster method for dither array generation. In Electronic Imaging (1993). DOI: 10.1117/12.152707.↩︎
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Linearize Event-Based Subsystems (Externally Scheduled Subsystems) - MATLAB & Simulink - MathWorks Italia
Linearizing Event-Based Subsystems
Approaches for Linearizing Event-Based Subsystems
Approximate Event-Based Subsystems Using Curve Fitting (Lump-Average Model)
Approximate Event-Based Dynamics Using Periodic Function Call Subsystem
Event-based subsystems (triggered subsystems) and other event-based models require special handling during linearization.
Executing a triggered subsystem depends on previous signal events, such as zero crossings. However, because linearization occurs at a specific moment in time, the trigger event never happens.
An example of an event-based subsystem is an internal combustion (IC) engine. When an engine piston approaches the top of a compression stroke, a spark causes combustion. The timing of the spark for combustion is dependent on the speed and the position of the engine crankshaft.
In the scdspeed model, triggered subsystems generate events when the pistons reach both the top and bottom of the compression stroke. Linearization in the presence of such triggered subsystems is not meaningful.
You can obtain a meaningful linearization of triggered subsystems, while still preserving the simulation behavior, by recasting the event-based dynamics as one of the following:
Lumped average model that approximates the event-based behavior over time.
Periodic function call subsystem, with Normal simulation mode.
In the case of periodic function call subsystems, the subsystem linearizes to the sampling at which the subsystem is periodically executed.
In many control applications, the controller is implemented as a discrete controller, but the execution of the controller is driven by an external scheduler. You can use such linearized plant models when the controller subsystem is a periodic Function-Call Subsystem.
If recasting event-based dynamics does not produce good linearization results, try frequency response estimation. See Estimate Frequency Response Using Model Linearizer.
If a triggered subsystem is disabled at the current operating condition and has at least one direct passthrough I/O pair, then the subsystem will break the linearization path during linearization. In such a case, specify a block substitution or ensure that the subsystem does not have a passthrough I/O pair.
This example shows how to use curve fitting to approximate event-based dynamics of an engine.
The scdspeed model linearizes to zero because the scdspeed/Throttle & Manifold/Intake Manifold is an event-triggered subsystem.
You can approximate the event-based dynamics of the scdspeed/Throttle & Manifold/Intake Manifold subsystem by adding the Convert to mass charge block inside the subsystem.
The Convert to mass charge block approximates the relationship between Air Charge, Manifold Pressure, and Engine Speed as a quadratic polynomial.
\begin{array}{c}AirCharge\text{ }=\text{ }{p}_{1}×EngineSpeed\text{ }+\text{ }{p}_{2}×ManifoldPressure\text{ }+\text{ }{p}_{3}×{\left(}^{M}\\ +\text{ }{p}_{4}×ManifoldPressure×EngineSpeed\text{ }+\text{ }{p}_{5}\end{array}
If measured data for internal signals is not available, use simulation data from the original model to compute the unknown parameters p1, p2, p3, p4, and p5 using a least squares fitting technique.
When you have measured data for internal signals, you can use the Simulink® Design Optimization™ software to compute the unknown parameters. See Engine Speed Model Parameter Estimation (Simulink Design Optimization) to learn more about computing model parameters, linearizing this approximated model, and designing a feedback controlled for the linear model.
The next figure compares the simulations of the original event-based model and the approximated model. Each of the pulses corresponds to a step change in the engine speed. The size of the step change is between 1500 and 5500. Thus, you can use the approximated model to accurately simulate and linearize the engine between 1500 RPM and 5500 RPM.
This example shows how to use periodic function call subsystems to approximate event-based dynamics for linearization.
mdl = 'scdPeriodicFcnCall';
Desired Wat
The linearization is zero because the subsystem is not a periodic function call.
Open the Externally Scheduled Controller block, which is a Function-Call Subsystem block.
Open the function block and configure it.
Set the Sample time type parameter to periodic.
Set the Sample time parameter to 0.01, which is the sample time of the controller.
Alternatively, you can configure the function block programmatically using the following code.
block = 'scdPeriodicFcnCall/Externally Scheduled Controller/function';
set_param(block,'SampleTimeType','periodic')
set_param(block,'SampleTime','0.01')
The linearization is no longer zero.
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A CYLINDER CONTAINS EITHER ETHYLENE OR PROPYLENE 12 ML OF GAS REQUIRED 54 ML OF OXYGEN FOR COMPLETE COMBUSTION - Chemistry - Thermodynamics - 10891025 | Meritnation.com
Yasodharan answered this
First let's write the balanced equation for the complete combustion for ethylene and propylene gas to have a justified answer,
Ethylene reaction scheme is,\phantom{\rule{0ex}{0ex}}{C}_{2}{H}_{4}+ 2{O}_{2}\to 2{H}_{2}O+2C{O}_{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Propylene reaction scheme is,\phantom{\rule{0ex}{0ex}}{C}_{3}{H}_{6}+4.5{O}_{2}\to 3{H}_{2}O+3C{O}_{2}
By using gas law at STP condition,lets compute for the volume of oxygen as below,
1 mole of ethylene reacts with 2 moles of oxygen
22400 ml = 2 (22400)ml
for 12 ml =?
12 ml(2 mole) = 24 ml
in similar way propylene reacts with 4.5 moles of oxygen the computation is shown below,
22400 ml = 4.5(22400) ml
for 12 ml=?
12ml(4.5 mole) = 54 ml
Hence the answer is Option (B) Propylene
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Home : Support : Online Help : Programming : Grid Package : Run
Run(f,[fargs],options)
Run(node,f,[fargs],options)
can either be a string containing the Maple code to run on each node, or a procedure or procedure name
list; arguments passed to the remote procedure call
integer, integer..integer, set(integer); identifier indicating which node(s) to run the command on
(optional) equation(s) of the form option=value; see the Options section below
The Run command runs a job in parallel. The job can be run on one designated node, or on all nodes at once. A node is a remote Maple process that does not share any variable or state with the current Maple process.
When using the calling sequence Run(n,f,[x]), the command f(x) will be executed on node n in the background. Maple will not wait for this command to finish before proceeding to execute the next command. Therefore, calling Run(1,f,[x]); Run(2,f,[y]); will launch the job f(x), and then immediately launch f(y) without waiting for f(x) to finish. Both f(x) and f(y) will execute on different nodes at the same time. See Grid:-Wait or Grid:-WaitForFirst to see how to sync up with a job that was previously launched. The wait=true option can be used to force the current thread to block until the remote process is finished.
The Run command does not attempt to analyze input dependencies and push values to the remote nodes. This includes the declaration of f itself. Unless f is a standard Maple command, or inline procedure declaration, the Grid:-Set command needs to be employed in order to define the routine on each remote node.
When using the calling sequence, Run(f,[x]), the command f(x) will be executed on all nodes. By default, Maple will wait for the nodes to finish, and return an array of results.
When using the calling sequence, Run("command"), such that the f argument is a string, the string will be parsed and evaluated in the remote node. The fargs option will not be used.
A parallel job initiated by the Run command does not terminate until all nodes are finished computing. This is different than the Grid:-Launch command, which terminates when node 0 is done.
When initiating a computation without specifying the node number, n, the same code is run on multiple nodes. The MyNode command can be queried and used to take different branches within that code in order to assign alternate tasks to designated nodes within a computation. Otherwise, the Grid:-Set command can be used to initialize the remote nodes with different data given the same variable name.
The Run command is only available in local Grid mode.
assignto=name
This option lets you specify a variable that will be assigned to the last result upon completion of the operation. For example, use assignto=ret to cause the variable ret to be assigned to the run result. This is a useful option for non-blocking executions. The command Grid:-GetLastResult can also be used to get the final result of a Grid computation. When multiple nodes are involved in the execution, the last result is an array, ranging from 0 to numnodes-1, with each index containing the result from each individual node.
firstarrayindex=integer
This option lets you specify the starting index of the result array (when computing with multiple nodes). By default, the array contains entries from 0..numnodes-1, with the value computed in node i, stored in the array at index i. It may sometimes be convenient to have the result array start at 1, in which case all node results computed at node i will be stored at index i+1.
numnodes=posint
This option is used to specify the number of nodes to be used in the compute pool. The default value of numnodes is given by kernelopts(numcpus). Setting this option will affect future calls to Grid:-Run and Grid:-Launch because subsequent calls will use the number of active nodes left over from the previous call.
printer=procedure
This option lets you specify a procedure for coordinating output from each node. The printer procedure will be called with a string every time there is a line of output available from an external computation. The printf command is the default if no custom procedure is supplied. The printer is called with a single argument, a string, that represents intermediate printed output from each of the nodes.
set={list,set}
This option lets you specify some named values that should be pushed from the current session to the remote nodes. For example, set=[f,g] will cause f and g to be assigned in the compute nodes that are about to run the given command. This is synonymous with imports as used in the Grid:-Launch command. Note that there is no get option, the Grid:-Get command should be used for that instead.
uneval=truefalse
This option controls whether fargs uses normal evaluation rules or not. The default uneval=false is to use normal evaluation rules, keeping in mind that the expression is processed twice -- once in the current environment, and once in the remote environment. For example, Run(f,[k]) will produce f(k) if k is not assigned anywhere. If k=4 in the local environment and k=5 in the remote node, then Run(f,[k]) will produce f(4) no matter what k is assigned in the remote environment. Using uneval quotes, Run(f,['k']) will cause 1-level of evaluation in the local environment turning 'k' into k, and the result will be f(5). To get the symbol k, two levels of quoting is needed, Run(f,[''k'']). The option, uneval=true, causes arguments with uneval quotes to be left alone during the local processing of fargs in the call to Run. The remote nodes then process all arguments using normal rules.
wait=truefalse
When this option is set to true, the Run command will block until all nodes relevant to the computation are finished. When this option is set to false, the Run command will return immediately, while the nodes continue to operate in the background. The default is wait=false when computing on just one node (presumably you would have just computed in the main thread if you wanted to wait for the result of just one node's work). However, when running a job on all nodes, the default is wait=true.
It is best to leave this as the default, but for consistent example output, 4 compute nodes will be used (regardless of whether you have 4 cores)
\mathrm{kernelopts}\left(\mathrm{numcpus}=4\right)
\textcolor[rgb]{0,0,1}{64}
\mathrm{Grid}:-\mathrm{Setup}\left(\mathrm{numnodes}=4\right)
This example shows results from multi-node computations are returned in an array. When generating random numbers, use randomize() to set a random seed for the sequence of numbers to generate. Avoid this only for testing purposes when you want a repeatable sequence. Note that randomize needs to be called on all of the compute nodes, which is done here in the first statement.
\mathrm{Grid}:-\mathrm{Run}\left(\mathrm{randomize}\right):
\mathrm{Grid}:-\mathrm{Run}\left(\mathrm{rand}\left(10..100\right)\right)
[\textcolor[rgb]{0,0,1}{82}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{98}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{36}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{17}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\text{⋯ 0 .. 3 Array}}]
\mathrm{Grid}:-\mathrm{Run}\left(\mathrm{rand}\left(10..100\right),'\mathrm{firstarrayindex}'=1\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{43}& \textcolor[rgb]{0,0,1}{39}& \textcolor[rgb]{0,0,1}{86}& \textcolor[rgb]{0,0,1}{35}\end{array}]
This example shows how to run two jobs in the background. The assignto option allows to capture the results into the local session. It is important to wait for the jobs to finish before using the results or starting new jobs on the same nodes.
\mathrm{Grid}:-\mathrm{Run}\left(0,\mathrm{Optimization}:-\mathrm{NLPSolve},[\frac{\mathrm{sin}\left(x\right)}{x},x=1..30],'\mathrm{assignto}'='\mathrm{ans0}'\right)
\mathrm{Grid}:-\mathrm{Run}\left(1,\mathrm{Optimization}:-\mathrm{NLPSolve},[{x}^{3}+2xy-2{y}^{2},x=-10..10,y=-10..10,\mathrm{initialpoint}={x=3,y=4},'\mathrm{maximize}'],'\mathrm{assignto}'='\mathrm{ans1}'\right)
\mathrm{Grid}:-\mathrm{Wait}\left(\right)
\mathrm{ans0}
[\textcolor[rgb]{0,0,1}{-0.0424796169776126}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{23.5194525023235}]]
\mathrm{ans1}
[\textcolor[rgb]{0,0,1}{1050.}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{10.}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5.}]]
This example will run four different jobs and wait for the first one to finish; then will terminate the others. Note the use of Grid:-Set to define the delay procedure on the remote nodes.
delay := proc( n, c )
\mathrm{Grid}:-\mathrm{Set}\left(\mathrm{delay}\right):
\mathrm{Grid}:-\mathrm{Run}\left(0,\mathrm{delay},[15,"0"]\right)
\mathrm{Grid}:-\mathrm{Run}\left(1,\mathrm{delay},[3,"1"]\right)
\mathrm{Grid}:-\mathrm{Run}\left(2,\mathrm{delay},[16,"2"]\right)
\mathrm{Grid}:-\mathrm{Run}\left(3,\mathrm{delay},[9,"3"]\right):
n≔\mathrm{Grid}:-\mathrm{WaitForFirst}\left(\right)
\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0}
\mathrm{Grid}:-\mathrm{Interrupt}\left(\right)
\mathrm{Grid}:-\mathrm{Wait}\left(\right)
\mathrm{Grid}:-\mathrm{GetLastResult}\left(n\right)
This example shows how to set state before running a computation, and notes that uneval quoting must consider both local and remote processing of arguments.
\mathrm{Grid}:-\mathrm{Set}\left(0,'x'=\mathrm{x_node0}\right)
\mathrm{Grid}:-\mathrm{Set}\left(1,'x'=\mathrm{x_node1}\right)
x≔\mathrm{x_local}
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{x_local}}
\mathrm{Grid}:-\mathrm{Run}\left(f,[x]\right)
[\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{x_local}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{x_local}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{x_local}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{x_local}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\text{⋯ 0 .. 3 Array}}]
\mathrm{Grid}:-\mathrm{Run}\left(f,['x']\right)
[\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{x_node0}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{x_node1}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\text{⋯ 0 .. 3 Array}}]
\mathrm{Grid}:-\mathrm{Run}\left(f,[''x'']\right)
[\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\text{⋯ 0 .. 3 Array}}]
\mathrm{Grid}:-\mathrm{Run}\left(0,\mathrm{print},['x'],'\mathrm{wait}'\right)
x_node0
This example displays a message including the current clock time before, during, and after running jobs on all nodes.
show_time := proc( msg := "" )
local t := time[current]();
printf("node=%d current-time=%s.%d: %s\n",
Grid:-MyNode(),
StringTools:-FormatTime("%I:%M:%S",timestamp=trunc(t)),
trunc(10^6*(t - trunc(t))),
\mathrm{show_time}\left("initialize"\right)
node=-1 current-time=09:20:24.450523: initialize
\mathrm{Grid}:-\mathrm{Set}\left(\mathrm{show_time}\right)
\mathrm{Grid}:-\mathrm{Run}\left(\mathrm{show_time},["worker"],'\mathrm{wait}'=\mathrm{true}\right)
node=0 current-time=09:20:24.540016: worker
[\left(\right)\textcolor[rgb]{0,0,1}{,}\left(\right)\textcolor[rgb]{0,0,1}{,}\left(\right)\textcolor[rgb]{0,0,1}{,}\left(\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\text{⋯ 0 .. 3 Array}}]
\mathrm{show_time}\left("done"\right)
node=-1 current-time=09:20:24.616699: done
The Grid[Run] command was introduced in Maple 2015.
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On the Kazhdan–Lusztig order on cells and families | EMS Press
On the Kazhdan–Lusztig order on cells and families
We consider the set Irr(W) of (complex) irreducible characters of a finite Coxeter group W. The Kazhdan–Lusztig theory of cells gives rise to a partition of Irr(W) into “families” and to a natural partial order
\leq_{\mathcal{LR}}
on these families. Following an idea of Spaltenstein, we show that
\leq_{\mathcal{LR}}
can be characterised (and effectively computed) in terms of standard operations in the character ring of W. If, moreover, W is the Weyl group of an algebraic group G, then
\leq_{\mathcal{LR}}
can be interpreted, via the Springer correspondence, in terms of the closure relation among the “special” unipotent classes of G.
Meinolf Geck, On the Kazhdan–Lusztig order on cells and families. Comment. Math. Helv. 87 (2012), no. 4, pp. 905–927
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Linearized plasticity is the evolutionary $\Gamma$-limit of finite plasticity | EMS Press
Linearized plasticity is the evolutionary
\Gamma
-limit of finite plasticity
We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via
\Gamma
-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.
Ulisse Stefanelli, Alexander Mielke, Linearized plasticity is the evolutionary
\Gamma
-limit of finite plasticity. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 923–948
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Local Fatou theorem and the density of energy on manifolds of negative curvature | EMS Press
u
be a harmonic function on a complete simply connected manifold
M
whose sectional curvatures are bounded between two negative constants. It is proved here a pointwise criterion of non-tangential convergence for points of the geometric boundary: the finiteness of the density of energy, which is the geometric analogue of the density of the area integral in the Euclidean half-space
Frédéric Mouton, Local Fatou theorem and the density of energy on manifolds of negative curvature. Rev. Mat. Iberoam. 23 (2007), no. 1, pp. 1–16
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§ Binary search to find rightmost index which does not possess some property
// p for predicate/property
// precondition: p(0) = false
// precondition: p(1 << NBITS) = last ix we process.
// precondition: p is monotonic;
// once it switches to true, does not switch back to false.
if (p(1 << NBITS) == 0) { return 1 << NBITS; }
assert(p(1<<NBITS) == 1);
for (int i = NBITS-1; i >= 0; i--) {
assert(p(ans + 2*k) == 1);
if (p(ans + k) == 0) {
ans = ans + k;
// ans is largest index such that
// has_some_poperty(ans) = 0
Claim 1: (Correctness) p(ans[i]) = 0. By precondition, this is true before the loop. See that it's a loop invariant, as we only update ans[i] to ans[i]+k if p(ans[i]+k) = 0. Thus, is is true after the loop.
Claim 2: (Maximality) : At loop iteration i: p(ans[i] + 2k[i]) = 1. We cannot improve our solution by using previous jump lengths.
This implies optimality once the loop ends. At the end of the loop we have i = -1. So:
2k[-1] = 2(1/2) = 1
finalans = ans[-1]
p(ans[-1] + 2k[-1]) = 1
=> p(finalans+1) = 1
Proof of Claim 2: induction on i
Suppose claim 2 is true till index i: p(ans[i] + 2k[i]) = 1.
To prove: induction hypothesis holds at index (i-1).
Case analysis based on loop body at i: p(ans[i] + k[i]) = 0 or 1
(a) p(ans[i] + k[i]) = 0. We update ans[i-1] = ans[i] + k[i].
We wish to show that the loop invariant holds at i-1: p(ans[i-1]+2k[i-1]) == 1.
\begin{aligned} &\text{k value: } k[i] = 2^i \\ &\text{(k-1) value: } k[i-1] = 2^{i-1} = 2k[i] \\ &\text{Ind: } p(ans[i] + 2k[i]) = 0 \\ &\text{Case (a): } p(ans[i] + k[i]) = 0 \\ &\text{Update: } ans[i-1] \equiv ans[i] + k[i] \\ &p(ans[i-1] + 2k[i-1]) \\ &= p((ans[i] + k[i]) + 2k[i-1]) \\ &= p(ans[i] + k[i] + k[i])\\ &= p(ans[i] + 2k[i]) \\ &= 1 ~\text{(By Induction Hyp.)} \end{aligned}
We've shown that the induction hypothesis hold at index
(i-1)
in case (a) where we update the value of
ans[i]
(b) If p(ans[i] + k[i]) = 1, then we update ans[i-1] = ans[i].
We wish to show that the loop invariant holds at i-1: p(ans[i-1]+2k[i-1]) ==1.
\begin{aligned} &\text{k value: } k[i] = 2^i \\ &\text{(k-1) value: } k[i-1] = 2^{i-1} = 2k[i] \\ &\text{Ind: } p(ans[i] + 2k[i]) = 0 \\ &\text{Case (b): } p(ans[i] + k[i]) = 1 \\ &\text{Update: } ans[i-1] \equiv ans[i] \\ &p(ans[i-1] + 2k[i-1]) \\ &= p(ans[i] + 2k[i-1]) \\ &= p(ans[i] + k[i] + k[i])\\ &= p(ans[i] + 2k[i]) \\ &= 1 ~\text{(By Induction Hyp.)} \end{aligned}
(i-1)
in case (b) where we don't change the value of
ans[i]
In summary, the loop invariant is held at index
(i-1)
assuming the loop invariant is satisfied at index
(i)
, for both updates of
ans[i]
. Thus, by induction, the loop invariant holds for all iterations.
Elaborated proof of why p(ans[0]+1) = 1 at the end of the loop
See that we can insert a new invaraiant at the end of the loop which asserts p(ans[i]+k[i]) == 1:
int ans2;
ans2 = ans + k;
// ans2 + k
// = (ans + k) + k
// = ans + 2k
// = 1 (from assertion)
ans2 = ans;
// = ans + k
// = 1 [from else branch]
// ## new loop end invariant ##
// true from then, else branch.
assert(p(ans2+k) == 1)
We've proven the correctness of the loop invariant at the end of the loop, given the prior loop invariant at the beginning of the loop.
So, At the end of the (i=0) iteration, we have k=1, and so p(ans+1) == 1, which is the "rightmost index" condition. that we originally wanted.
§ Fully elaborated proof
// p(ans[nbits-1] + 2*(1<<nbits-1))
// = p(0 + 1 << nbits)
// = p(1 << nbits)
// = 1 [from assert]
// From previous loop iteratio (i+1):
// p(ans[(i+1)-1] + k[i+1]) == true
// => p(ans[i] + k[i+1]) == true
// => p(ans[i] + 2k[i]) == true
assert(p(ans + 2*k) == true);
ans += ans + k;
// ASSIGNMENT: ans[i-1] = ans[i] + k[i]
// p(ans[i-1] + k[i])
// = p(ans[i] + k[i] + k[i])
// = p(ans[i] + 2k[i])
// = p(ans[i] + k[i+1])
// = 1 (from induction hyp)
ans = ans; // no-op
// ASSIGNMENT: ans[i-1] = ans[i].
// = p(ans[i] + k[i])
// = 1 (from else branch)
// p(ans[i-1] + k[i])== 1
assert(p(ans+k) == 1)
§ Simplified implementation
If we are willing to suffer some performance impact, we can change the loop to become significantly easier to prove:
int i = nbits-1;
assert (p(ans+2*k) == 1);
assert(p(ans) == 0)
In this version of the loop, we only decrement i when we are sure that p(ans+k) == 0. We don't need to prove that decrementing i monotonically per loop trip maintains the invariant; Rather, we can try "as many is as necessary" and then decrement i once it turns out to not be useful.
§ Relationship to LCA / binary lifting
This is very similar to LCA, where we find the lowest node that is not an ancestor. The ancestor of such a node must be the ancestor.
if (is_ancestor(u, v)) return u;
if (is_ancestor(v, u)) return v;
// u is not an ancestor of v.
// find lowest parent of u that is not an ancestor of v.
for (int i = l; i >= 0; --i) {
if (!is_ancestor(up[u][i], v))
|
Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential | EMS Press
We consider a singularly perturbed elliptic equation
\e^2\Delta u - V(x) u + f(u)=0, \ u(x) > 0 \textrm{ on } \RN, \, \lim_{|x| \to \infty}u(x) = 0,
V(x) > 0
x \in \RN.
The singularly perturbed problem has corresponding limiting problems
\Delta U - c U + f(U)=0, \ U(x) > 0 \textrm{ on } \ \RN, \, \lim_{|x| \to \infty}U(x) = 0, \ c > 0.
Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity
for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential
V
under possibly general conditions on
f
. In this paper, we prove that under the optimal conditions of Berestycki-Lions on
f \in C^1
, there exists a solution concentrating around topologically stable positive critical points of
V
, whose critical values are characterized by minimax methods.
Jaeyoung Byeon, Kazunaga Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1859–1899
|
§ Character theory
I jot down the rough proof sketches of character theoretic facts for quick reference. Fix a group
G
. A group representation of
G
is a group homomorphism from the group to the automorphism group of a complex vector space
V
: Formally,
f: G \rightarrow Aut(V)
. A direct sum of representations
f: G \rightarrow Aut(V)
f': G \rightarrow Aut(W)
is the obvious extension of the maps
f \oplus f': G \rightarrow Aut(V \oplus W)
(f \oplus f')(g) = \lambda v. f(g)(v) \oplus f'(g)(v)
. A representation is said to be irreducible if it cannot be written as the direct sum of two non-trivial representations. A character is the trace of a representation. An irreducible character is the trace of an irreducible representation.
§ All finite group representations are unitary representations
f: G \rightarrow Aut(V)
, we construct an invariant inner product, that is, one where
\langle f(g)(v) | f(g)(w) \rangle = \langle v | w \rangle
. This maps the representation unitary, since it preserves this special inner product. The idea is to begin with some arbitrary inner product
[v, w]
which we can always induce on
V
(pick a basis). Then, we build an "averaged" inner product given by
\langle v | w \rangle \equiv \sum_{h \in G} [ f(h)(v) | f(h)(w) ]
. Intuitively, this inner product is invariant because on considering
\langle f(g)(v) | f(g)(w) \rangle
, the definition will contain
[f(h)(f(g)(v)) | f(h)(f(g) w)] = [f(hg)(v) | f(hg)(w)]
, which is a re-indexing of the original sum. Hence, the representation
f
preserves this inner product, and we can thus study only unitary representations (which are much simpler). From now on, we assume all representations are unitary.
§ Representation has same value for the entire conjugacy classe
Since all representations are unitary, the image of
f(ghg^{-1}) = f(g) f(h) f(g)^{-1}
is going to be a change-of-basis of
f(h)
, and thus does not actually change the automorphism given by
f(h)
. Hence, representations are the same for an entire conjugacy class. Such functions which are constant on a conjugacy class is called as a class function .
§ Morphism between representations / intertwining
A map between two representations,
f: G \rightarrow V
f': G \rightarrow W
\eta: V \rightarrow W
if the natural diagram commutes:
V --f--→ V
η η
W --f'-→ W
such a map
\eta
is called called as an intertwining map or an equivariant map.
§ Schur's lemma
The only equivariant maps between irreducible representations is either the zero map or a scalar multiple of the identity map. This is stronger than saying that the equivariant map is a diagonal matrix; scalar multiple of identity implies that all dimensions are scaled uniformly. The main idea of the proof is to show that the kernel and image of the intertwining map is an irreducible subspace of
f, f'
retrospectively. Since the maps are irreducible, we must have the the intertwining is either the zero map, or a map into the full group. This forces the map to be zero or a scalar multiple of the identity. One way to look at this is that for irreps
f: G \rightarrow V
f': G \rightarrow W
, the dimension of
Hom(V, W)
is either 0 or 1 (scalings of identity).
§ Schur orthogonality relations
We consider representations "one matrix index" at a time, and show that the matrix entries of irreducible representations is going to be orthogonal The proof is to consider representations
\alpha: G \rightarrow \mathbb GL(V)
\beta: G \rightarrow \mathbb GL(W)
, and an intertwining map
T: V \rightarrow W
. How do we involve all of
\alpha, \beta, T
at once? Recall that since
T
is an intertwining, we must have:
T(\alpha(g)(v)) = \beta(g)(T(v))
\beta
is invertible (it must be since it's a member of
GL(W)
), I can rewrite the above as:
\beta^{-1}(g)(T(\alpha (g)(v)) = T(v)
This needs that
T: V \rightarrow W
is an intertwining map. Can we generalize this to any linear map ? Suppose that
L: V \rightarrow W
is a linear map , not necessary intertwining. Let's induce an intertwining map from
L
\begin{aligned} &\overline{L}: V \rightarrow W \\ &\overline{L}(v) \equiv 1/|G|\sum_{g \in G} \beta(g)^{-1} T \alpha(g) v \end{aligned}
We average the intertwining condition of
T(v)
to produce an appropriate
\overline{L}(v)
. Is this an intertwining? Yes, because when we compute
\beta(h)^{-1} \overline L \alpha(h)
, the averaging trick winds up shifting the index, exactly as it does for the inner product:
\begin{aligned} &\beta(h)^{-1} \overline L \alpha(h) \\ &=\beta(h)^{-1} \left( \sum_{g \in G} \beta(g)^{-1} L \alpha(g) \right) \alpha(h) \\ &=\sum_{g \in G} \beta(gh)^{-1} L \alpha(gh) \\ &=\sum_{g \in G} \beta(gh)^{-1} L \alpha(gh) \\ &=\sum_{kh^{-1} \in G} \beta(kh^{-1}h)^{-1} L \alpha(kh^{-1}h) \\ &=\sum_{kh^{-1} \in G} \beta(k)^{-1} L \alpha(k) \\ &= \overline{L} \end{aligned}
Thus, for every linear map
L: V \rightarrow W
, if the representation
\alpha
is not isomorphic to the representation
\beta
\overline{L} = 0
\begin{aligned} &\sum_{g \in G} \beta(g)^{-1} L \alpha(g) = 0 \\ &\left( \sum_{g \in G} \beta(g)^{-1} L \alpha(g) \right)[i][j] = 0[i][j] \\ &\sum_{g \in G} \beta(g)^{-1}[i][p] L[p][q] \alpha(g)[q][j] = 0[i][j] \\ &\text{($\beta$ is unitary)} \\ &\sum_{g \in G} \beta(g)*[p][i] L[p][q] \alpha(g)[q][j] = 0[i][j] \\ \end{aligned}
The above equality holds for all indexes
i, j
and for all choices of
L[p][q]
L
can be any linear map ). In particular, we can choose
L[p][q] = \delta[p][r] \delta[q][s]
r, s
. This gives us the equation:
\begin{aligned} &\sum_{g \in G} \beta(g)*[p][i] L[p][q] \alpha(g)[q][j] = 0[i][j] \\ &\sum_{g \in G} \beta(g)*[p][i] \delta[p][r] \delta[q][c] \alpha(g)[q][j] = 0[i][j] \\ &\sum_{g \in G} \beta(g)*[r][i] \alpha(g)[s][j] = 0[i][j] = 0\\ \end{aligned}
This tells that we can choose any index
[r, i]
[i, j]
and these will be orthogonal, when viewed as vectors "along" the set of matrices. If the representation is a one-dimensional representation/character, then we have no freedom in indexing, and the above becomes:
\begin{aligned} &\sum_{g \in G} \beta(g)* \alpha(g) = 0[i][j] = 0 \end{aligned}
Thus, different characters are all orthogonal.
§ Inner product of class functions
we impose an inner product relation on the space of class functions (complex valued functions constant on conjugacy classes)
G \rightarrow \mathbb C^\times
\langle f | f' \rangle \equiv 1/|G| \sum_{g \in G} f(g) \overline{f'(g)}
\overline{f'(g)}
is the complex conjugate. Using the Schur orthogonality relations, we immediately deduce that the inner product of two irreducible characters can be viewed as the schur orthogonality applied to their (only) matrix entry at location (1, 1). Thus, irreducible characters will be orthogonal, and equal characters will have inner product 1.
§ Regular representation
The "Cayley-style" representation one would naturally dream up. For a group
G
, build a vector space
V
whose basis is given by elements of
G
. Have
g \in G
V
by seding
v_h
v_{gh}
. Ie, act with
g
as a permutation on
V
. This gives us a "large" representation. For example, the permutation group of
n
letters will have a regular representation of
n!
basis vectors. This representation contains every irrep. The idea is to show that the dot product of the trace of the regular representation with every other irrep is nonzero. Furthermore, since the regular representation has finite dimension, this tells us that there are only finitely many irreps: the irreps correspond to subrepresentations, and a finite representation only has finitely many subrepresentations. This makes the idea of classifying irreps a reasonable task.
§ Character of the regular representation
Theorem: The character
r_G
of the regular representation is given by
r_G(1) = |G|
r_G(s) = 0
s \neq 1
The matrix for the identity element is the identity matrix, and the size of the matrix is the size of the vector space, which is
|G|
since there's a basis vector for each element of
G
r_G(1) = |G|
For any other element
g \in G
, the regular representation will be a permutation matrix with no fixed points. Thus, the diagonal of the matrix is all zeros, and hence
r_G(g) = 0
§ Regular representation contains all other irreps
The inner product of the character of the regular representation with any other irrep
\alpha
is going to be:
\begin{aligned} & \langle r_G | \chi_\alpha \rangle = 1/|G| \sum_{g \in G} r_G(g)* \chi_\alpha(g) \\ &= 1/|G| (r_G(1) \cdot \chi_\alpha(1)) \\ &= 1/|G| (|G| \cdot 1) \\ &= 1 \end{aligned}
Thus, the regular rep contains the other irreps, since the character of the regular rep has non-zero inner product with irrep, and irrep characters are all orthogonal.
§ Abelian groups are controlled by characters
Since abelian groups map to automorphism that all commute with each other, we can simultaneously diagonalize these matrices. Thus, we only need to consider the data along each diagonal, which is independent. This reduces the representation to a direct sum of scalars / 1D representations / characters.
§ Number of irreducible representations
Recall that the characters of irreducible representations are orthogonal. Also, the dimension of the space of class functions is equal to the number of conjugacy classes of the group
G
, since a class function takes on a distinct value over each conjugacy class, so there are those many degrees of freedom. This tells us that the number of irreducible representations is at most the number of conjugacy classes of the group.
|
Implementability by a Canonical Indirect Mechanism of an Optimal Two-Dimensional Direct Mechanism
Otaru University of Commerce, Otaru, Japan.
The present paper investigates the multi-dimensional mechanism design in which buyers have taste and budget as their private information. The paper shows an easy proof of a two-dimensional optimal direct mechanism by a one-dimensional indirect mechanism: A canonical mechanism in the traditional one-dimensional setting, i.e., function of one variable, the buyer’s taste. It also sheds light on where the difficulty lies implementability of a general direct mechanism—not optimal—by a canonical mechanism.
Multi-Dimensional Mechanism, Indirect Mechanism, Budget Constraint, Revelation Principle
Kojima, N. (2017) Implementability by a Canonical Indirect Mechanism of an Optimal Two-Dimensional Direct Mechanism. Theoretical Economics Letters, 7, 187-192. doi: 10.4236/tel.2017.72016.
In standard literature on adverse selection, an agent-buyer has his taste as private information. If the agent has, in addition, a budget constraint as his private information, the problem becomes one of two-dimensional mechanism design. It is well known that multi-dimensional mechanism design entails quite a few technical difficulties (see [1] , [2] ). One way to get around them is to reduce the dimension of private information [3] and thus to resort to an indirect mechanism.
In the setting of budget-constrained buyers, [4] invoked a non-linear price scheme while showing that this indirect mechanism realizes the equilibrium outcome of the optimal two-dimensional direct mechanism. [5] , by only focusing on the agent’s taste, reduced the whole setting to the well known one of a canonical one-dimensional mechanism, i.e., a map from the taste space to the quality-price space and showed that the optimal canonical mechanism achieves the same outcome as an optimal direct mechanism.1 The proof of equivalence, however, involved fancy theory of set-valued mappings not necessarily familiar to an audience in economics.
The above two works differ in perspective and motivation so that they are based on different assumptions as regards the distribution of agents’ type. By adopting the assumptions of [4] , this paper provides an elementary proof that their optimal non-linear price scheme is implemented by a canonical one-dimen- sional mechanism a la Kojima, while resorting to their structure of the optimal non-linear price scheme given by an integro-differential equation.
It turns out that a two-dimensional direct mechanism in general is not so straightforward to implement by a canonical mechanism. This article shows where this difficulty lies, which was not visible in the approach by theory of set- valued functions in [5] . In one word, it consists in the lack of strict concavity of a non-linear price scheme implementing a direct mechanism.
A risk-neutral seller has one unit of an indivisible commodity to sell of quality
q\in Q:=\left[0,1\right]
. A continuum of buyers purchase one unit of the commodity of quality
q
or none. The seller values the commodity at zero. A buyer has taste
t
for the commodity and a budget
w
. The couple
\left(t,w\right)
is distributed according to the density
f\left(t,w\right)
continuously differentiable and positive on its non-empty support,
T\times W
T:=\left[0,\stackrel{¯}{t}\right]
W:=\left[0,\stackrel{¯}{w}\right]
\left(t,w\right)
is the buyers’ private information and the seller only knows the density
f\left(t,w\right)
\left(t,w\right)
is referred to as the buyers’ type from now onwards. The marginal distribution of
w
G\left(w\right)
and the conditional distribution of
t
w
F\left(t|w\right)
with the corresponding densities
g\left(w\right)
f\left(t|w\right)
The buyer of taste
t
obtains utility
tq-p
when buying quality
q
and paying price
p
. Suppose as in [4] that the seller incurs zero cost. The following is assumed, the reason for which is merely to assure that in Lemma 2, Equation (5) is always valid (see Th.1 in [4] ).
\stackrel{¯}{t}\le \stackrel{¯}{w}.
The buyer of the highest taste
\stackrel{¯}{t}
\stackrel{¯}{t}-p
for the highest quality
q=1
, the highest acceptable price of which, reservation price, is
\stackrel{¯}{t}
. The assumption ensures that the buyer with the highest budget can pay the price.
The non-linear price scheme
\tau \left(q\right)
is defined as a mapping of the space
Q
to the price space
R
. Note that it is an indirect mechanism since the type space here is
T\times W
The canonical mechanism is defined so that it associates the quality and price only with the buyer’s taste as in the standard setting while ignoring the budget constraint:
\left(q,p\right):T\to Q\times R.
It is yet another indirect mechanism but very familiar to us in literature for decades. Let us define as usual:
Definition 1. The canonical mechanism
\left(q\left(t\right),p\left(t\right)\right)
is weakly feasible2 if and only if
tq\left(t\right)-p\left(t\right)\ge tq\left(\stackrel{˜}{t}\right)-p\left(\stackrel{˜}{t}\right)\text{forany}t,\stackrel{˜}{t}\in T
(WIC)
tq\left(t\right)-p\left(t\right)\ge 0\text{on}T.
(WIR)
Let us turn to the direct mechanism in the two-dimensional setting, which is a mapping:
\left(q\left(t,w\right),p\left(t,w\right)\right):T\times W\to Q\times R.
Feasibility in this multi-dimensional mechanism is expressed as follows:
Definition 2. The direct mechanism
\left(q\left(t,w\right),p\left(t,w\right)\right)
is strongly feasible if and only if
p\left(t,w\right)\le w\text{forany}\left(t,w\right)\in T\times W,
\begin{array}{l}tq\left(t,w\right)-p\left(t,w\right)\ge tq\left(\stackrel{˜}{t},\stackrel{˜}{w}\right)-p\left(\stackrel{˜}{t},\stackrel{˜}{w}\right)\\ \text{forany}\left(t,w\right)\text{and}\left(\stackrel{˜}{t},\stackrel{˜}{w}\right)\in T\times W\text{suchthat}p\left(\stackrel{˜}{t},\stackrel{˜}{w}\right)\le w.\end{array}
(SIC)
tq\left(t,w\right)-p\left(t,w\right)\ge 0\text{on}T\times W.
(SIR)
(SIC) induces truth-telling and (SIR) participation whereas (SBC) ensures that the buyer can actually purchase the quality assigned.
The seller’s aim is to construct a strongly feasible mechanism which maximizes the expected revenue
Ep\left(t,w\right)
The revelation principle asserts that the outcome of an indirect mechanism is realized by a direct mechanism. Said contrariwise, an indirect mechanism might suffer sub-optimality compared to a direct mechanism [4] , nonetheless, showed the following result and resorted to a non-linear price scheme instead of a direct mechanism to maximize the seller’s revenue (see also [5] for the proof3).
Lemma 1. Given any strongly feasible direct mechanism
\left(q\left(t,w\right),p\left(t,w\right)\right)
there exists a non-linear price scheme
\tau :Q\to R
such that4
\text{itiscontinuous,strictlyincreasing,convexand}\tau \left(0\right)=0\text{aswellas}0\le {{\tau }^{\prime }}_{+}\le \stackrel{¯}{t},
and further that for all
\left(t,w\right)
\tau \left(y\right)\right)\ge p\left(t,w\right)
y\in \underset{xs.t.\text{ }\tau \left(x\right)\le w}{\mathrm{arg}\mathrm{max}}tx-\tau \left(x\right).
Note that the outcome of a non-linear price scheme given in the lemma is realized by a strongly feasible mechanism due to the revelation principle. The lemma, therefore, asserts that the revenue of an optimal5 strongly feasible me- chanism is attained by a non-linear price scheme.
It shows that a weakly feasible mechanism can replicate the quality-price allocation of a strongly feasible mechanism, which basically amounts to proving that the former offers at least the same variety of qualities by the latter. In general, it is not such an easy task for a strongly feasible mechanism-which [5] deal with-but in the case of an optimal strongly feasible mechanism, one can construct a canonical mechanism which replicates the quality-price allocation of the former by way of the corresponding optimal non-linear price scheme. Due to
T=\left[0,\stackrel{¯}{t}\right]
and (1), one obtains:
Lemma 2. (Che and Gale). Assume that for each
w
H\left(t|w\right):=1-F\left(t|w\right)-tf\left(t|w\right)
is strictly decreasing in
t
. Then, the optimal strong mechanism is implemented by the non-linear price scheme
\tau \left(x\right)
which satisfies (2) and
-{\tau }^{″}\left(x\right)\left\{{\displaystyle {\int }_{\tau \left(x\right)}^{\stackrel{¯}{w}}}\frac{\partial H\left({\tau }^{\prime }\left(x\right)|w\right)}{\partial {\tau }^{\prime }\left(x\right)}g\left(w\right)\text{d}w\right\}={\tau }^{\prime }{\left(x\right)}^{2}f\left({\tau }^{\prime }\left(x\right)|\tau \left(x\right)\right)g\left(\tau \left(x\right)\right)
This condition ensures that
\tau
has a second derivative: hence we actually have a derivative
{{\tau }^{\prime }}_{+}={\tau }^{\prime }
for the optimal strong mechanism. A second observa- tion is that
\tau
is strictly convex since otherwise
{\tau }^{″}\left(x\right)=0
x
and Equation (5) leads to
{\tau }^{\prime }\left(x\right)=0
, which is contradictory to (2).
Let us now turn to the construction of a canonical mechanism which repli- cates the agent’s quality (thus price) choice faced to the optimal direct me- chanism. Suppose that the optimal direct mechanism is given as well as the corresponding non-linear price scheme as in lemma 2. Let us define the maxi- mizer of utility:
\underset{x\in Q}{\mathrm{max}}tx-\tau \left(x\right).
There are three possible cases for this maximization.
(1) First case: If
t<{\tau }^{\prime }\left(0\right)
then, the utility maximizer is
x=0
and one defines
{q}^{\ast }\left(t\right):=0
(2) Second case: If there exists
x
0=t-{\tau }^{\prime }\left(x\right)
, it is unique due to strict convexity. Then, one defines
{q}^{\ast }\left(t\right):=x
(3) Third case: If
t>{\tau }^{\prime }\left(1\right)
the maximizer is
x=1
{q}^{\ast }\left(t\right):=1
Define now a canonical mechanism by
\left({q}^{\ast }\left(t\right),{p}^{\ast }\left(t\right)\right):=\left({q}^{\ast }\left(t\right),\tau \left({q}^{\ast }\left(t\right)\right)\right)
. Then, it satisfies (WIC) just by construction of
{q}^{\ast }\left(t\right)
―through (6)―and also
(WIR) due to
\tau \left(0\right)=0
. It is, thus, weakly feasible. Due to
0\le {{\tau }^{\prime }}_{+}\left(x\right)\le \stackrel{¯}{t}
\left[0,1\right]
, one can always find
t\in T
t={\tau }^{\prime }\left(x\right)
x\in Q
. In other words, for any
x
, there is a buyer of taste
t
for whom
x
was a utility maximizer and an inner solution (Second case above). Thus, one obtains that
q\left(T\right)=\left[0,1\right]
: the canonical mechanism constructed covers the whole quality range.
Now consider a buyer of
\left(t,w\right)
p\left(t\right)\le w
, he will indeed buy
q\left(t\right)
. Otherwise, while willing to do so, he cannot afford that quality and will choose
q\left({t}^{\prime }\right)
{t}^{\prime }<t
and pays his whole budget
p\left({t}^{\prime }\right)=w
because his utility function in (6) is strictly concave. These choices are identical to those when the buyer faces the non-linear scheme.
Due to Lemma 1, therefore, one obtains:
Proposition 1. A weakly feasible canonical mechanism implements the quality- price allocation of an optimal strongly feasible direct mechanism.
Let us turn to the case of a direct mechanism in general―not revenue-maxi- mizing optimal. Given a strongly feasible mechanism, consider an outperform- ing non-linear price scheme, the existence of which is assured by Lemma 1. Existence is, however, no longer assured of the derivative
{\tau }^{\prime }
let alone strict convexity of
\tau
(see the remark just after Lemma 2). The quality-price choice by any agent facing this non-linear price scheme is in turn realized by some strongly feasible mechanism by virtue of the revelation principle. Let us try to replicate this quality-price choice with a canonical mechanism according to (6).
The difficulty in constructing a canonical mechanism implementing a direct mechanism in this case―or the corresponding price scheme alike―consists in the fact that one may not be able to provide the whole quality range offered by the non-linear price scheme (i.e.,
q\left(T\right)\ne Q=\left[0,1\right]
). Then, the canonical mecha- nism might not provide the qualities which some buyers would pick with the non-linear price scheme: hence failure of implementation.
Lemma 1 does not guarantee the existence of the first or second derivative of
\tau
but their existence is not an essential point here. If they do not exist, the same argument applies by replacement of
{\tau }^{\prime }
with its subdifferential.6
For Cases (1) and (3) of corner solutions, there is no problem. We repeat the same process. The problem is Case (2) of inner solutions. Suppose
\tau
is not strictly convex but just convex by Lemma 1. Then, there might be an interval of maximizers
\left[x,y\right]
t
’s due to convexity. Now, our maximizer for
t
is a multi-valued function
K\left(t\right)
but a canonical mechanism is a single-valued function. One has to assign one quality to each
t
. If we assign some quality
z\in \left[x,y\right]
to each type who has multiple utility maximizers and build
q\left(t\right)
, it is quite probable that
q\left(T\right)
has some holes in
Q
q\left(T\right)\ne Q
. Technically speaking, the maximizer of Equation (6),
K\left(t\right)
, is a multi-valued (set-valued) function and there does not always exist a single-valued continuous canonical mechanism7 such that
q\left(t\right)\in K\left(t\right)
1As a matter of course, a canonical mechanism here has no bearing on Maskin’s one in literature.
2We use the qualification of “weak” in order to avoid confusion with feasibility of a direct mechanism.
3Although [4] states the result with a weakly increasing
\tau
, strict increase is evident by its definition. In fact, it is crucial to our purpose. See [5] .
{{\tau }^{\prime }}_{+}
stands for the largest subgradient or equivalently the right derivative (see [6] ).
5Throughout the paper, an optimal mechanism stands for the seller's expected revenue-maximizing mechanism.
6See [6] .
7See [7] or [8] for instance.
8If there exists,
q\left(T\right)=Q
is obvious since
T
[1] Armstrong, M. (1996) Multiproduct Nonlinear Pricing. Econometrica, 64, 51-75.
[2] Rochet, J.C. and Chone, P. (1998) Ironing, Sweeping, and Multidimensional Screening. Econometrica, 66, 783-8268.
[3] Rochet, J.C. and Stole, L. (2003) The Economics of Multidimensional Screening. In: Dewatripont, M., Hansen, L. and Turnovski, S., Eds., Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress, Volume 1, Chapter 5, Cambridge University Press.
[4] Che, Y.K. and Gale, I. (2000) The Optimal Mechanism for Selling to a Budget-Constrained Buyer. Journal of Economic Theory, 92, 198-233.
[5] Kojima, N. (2014) Mechanism Design to the Budget Constrained Buyer: A Canonical Mechanism Approach. International Journal of Game Theory, 43, 693-719.
[6] Rockafellar, T. (1970) Convex Analysis. Princeton University Press, Princeton.
[7] Aubin, J.P. and Frankowska, H. (1990) Set-Valued Analysis. Birkhaauser, Boston.
[8] Aubin, J.P. and Cellina, A. (1984) Deferential Inclusion. Springer Verlag, Berlin Heidelberg.
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Conformal map - Wikipedia
A rectangular grid (top) and its image under a conformal map
{\displaystyle f}
(bottom). It is seen that
{\displaystyle f}
maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.
{\displaystyle U}
{\displaystyle V}
be open subsets of
{\displaystyle \mathbb {R} ^{n}}
{\displaystyle f:U\to V}
{\displaystyle u_{0}\in U}
if it preserves angles between directed curves through
{\displaystyle u_{0}}
, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature.
1 Conformal maps in two dimensions
1.1 Global conformal maps on the Riemann sphere
2 Conformal maps in three or more dimensions
Conformal maps in two dimensionsEdit
{\displaystyle U}
is an open subset of the complex plane
{\displaystyle \mathbb {C} }
, then a function
{\displaystyle f:U\to \mathbb {C} }
is conformal if and only if it is holomorphic and its derivative is everywhere non-zero on
{\displaystyle U}
. I{\displaystyle f}
In the literature, there is another definition of conformal: a mapping
{\displaystyle f}
which is one-to-one and holomorphic on an open set in the plane. The open mapping theorem forces the inverse function (defined on the image o{\displaystyle f}
The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of
{\displaystyle \mathbb {C} }
admits a bijective conformal map to the open unit disk in
{\displaystyle \mathbb {C} }
Global conformal maps on the Riemann sphereEdit
Conformal maps in three or more dimensionsEdit
Riemannian geometryEdit
In Riemannian geometry, two Riemannian metrics
{\displaystyle g}
{\displaystyle h}
{\displaystyle M}
are called conformally equivalent if
{\displaystyle g=uh}
{\displaystyle u}
{\displaystyle M}
{\displaystyle u}
In cartography, several named map projections, including the Mercator projection and the stereographic projection are conformal. They are specially useful for use in marine navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb (or, mathematically, a loxodrome) is preferred in marine navigation because ships can sail in a constant compass direction.
Physics and engineeringEdit
Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable yet exhibit inconvenient geometries. By choosing an appropriate mapping, the analyst can transform the inconvenient geometry into a much more convenient one. For example, one may wish to calculate the electric field,
{\displaystyle E(z)}
, arising from a point charge located near the corner of two conducting planes separated by a certain angle (where
{\displaystyle z}
is the complex coordinate of a point in 2-space). This problem per se is quite clumsy to solve in closed form. However, by employing a very simple conformal mapping, the inconvenient angle is mapped to one of precisely
{\displaystyle \pi }
radians, meaning that the corner of two planes is transformed to a straight line. In this new domain, the problem (that of calculating the electric field impressed by a point charge located near a conducting wall) is quite easy to solve. The solution is obtained in this domain,
{\displaystyle E(w)}
, and then mapped back to the original domain by noting that
{\displaystyle w}as obtained as a function (viz., the composition of
{\displaystyle E}
{\displaystyle w}
{\displaystyle z}
{\displaystyle E(w)}
{\displaystyle E(w(z))}
{\displaystyle z}
If a function is harmonic (that is, it satisfies Laplace's equation
{\displaystyle \nabla ^{2}f=0}
Main article: Spherical wave transformation
A large group of conformal maps for relating solutions of Maxwell's equations was identified by Ebenezer Cunningham (1908) and Harry Bateman (1910). Their training at Cambridge University had given them facility with the method of image charges and associated methods of images for spheres and inversion. As recounted by Andrew Warwick (2003) Masters of Theory: [6]
Each four-dimensional solution could be inverted in a four-dimensional hyper-sphere of pseudo-radius
{\displaystyle K}
in order to produce a new solution.
Warwick highlights this "new theorem of relativity" as a Cambridge response to Einstein, and as founded on exercises using the method of inversion, such as found in James Hopwood Jeans textbook Mathematical Theory of Electricity and Magnetism.
^ Blair, David (2000-08-17). Inversion Theory and Conformal Mapping. The Student Mathematical Library. Vol. 9. Providence, Rhode Island: American Mathematical Society. doi:10.1090/stml/009. ISBN 978-0-8218-2636-2. S2CID 118752074.
^ Richard M. Timoney (2004), Riemann mapping theorem from Trinity College, Dublin
^ Kolaei, Amir; Rakheja, Subhash; Richard, Marc J. (2014-01-06). "Range of applicability of the linear fluid slosh theory for predicting transient lateral slosh and roll stability of tank vehicles". Journal of Sound and Vibration. 333 (1): 263–282. Bibcode:2014JSV...333..263K. doi:10.1016/j.jsv.2013.09.002.
^ Hinton, Edward; Hogg, Andrew; Huppert, Herbert (2020). "Shallow free-surface Stokes flow around a corner". Philosophical Transactions of the Royal Society A. 378 (2174). Bibcode:2020RSPTA.37890515H. doi:10.1098/rsta.2019.0515. PMC 7287310. PMID 32507085.
^ Noury, Keyvan; Yang, Bingen (2020). "A Psuedo S-plane Mapping of Z-plane Root Locus". ASME 2020 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers. doi:10.1115/IMECE2020-23096. ISBN 978-0-7918-8454-6. S2CID 234582511.
^ Warwick, Andrew (2003). Masters of theory : Cambridge and the rise of mathematical physics. University of Chicago Press. pp. 404–424. ISBN 978-0226873756.
Churchill, Ruel V. (1974), Complex Variables and Applications, New York: McGraw–Hill Book Co., ISBN 978-0-07-010855-4
E.P. Dolzhenko (2001) [1994], "Conformal mapping", Encyclopedia of Mathematics, EMS Press
Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw–Hill Book Co., ISBN 978-0-07-054234-1, MR 0924157
Wikimedia Commons has media related to Conformal mapping.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Conformal_map&oldid=1084969799"
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Optimize Percentage of Volume Trading Strategy - MATLAB & Simulink
This example shows how to optimize the strategy for a single stock by minimizing trading costs using transaction cost analysis from the Kissell Research Group. The optimization minimizes trading costs associated with the percentage of volume trading strategy and a specified risk aversion parameter Lambda. The trading cost minimization is expressed as
\mathrm{min}\left[\left(MI+PA\right)+Lambda\cdot TR\right],
Here, you can optimize the percentage of volume trade strategy. To optimize trade time and trade schedule strategies, see Optimize Trade Time Trading Strategy and Optimize Trade Schedule Trading Strategy.
Initial percentage of volume trade strategy
tradeData.POV = 0.5;
fun = @(pov)krgSingleStockOptimizer(pov,k,tradeData,Lambda);
Minimize the trading costs for the percentage of volume trade strategy. fminbnd finds the optimal value for the percentage of volume trade strategy based on the lower and upper bound values. fminbnd finds a local minimum for the trading cost minimization expression.
[tradeData.POV,totalcost] = fminbnd(fun,LB,UB);
Display the optimized trade strategy tradeData.POV.
tradeData.POV
Estimate the trading costs povCosts using the optimized trade strategy.
povCosts = [totalcost mi pa tr];
povCosts
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Ball screw (23346 views - Mechanical Engineering)
A ball screw is a mechanical linear actuator that translates rotational motion to linear motion with little friction. A threaded shaft provides a helical raceway for ball bearings which act as a precision screw. As well as being able to apply or withstand high thrust loads, they can do so with minimum internal friction. They are made to close tolerances and are therefore suitable for use in situations in which high precision is necessary. The ball assembly acts as the nut while the threaded shaft is the screw. In contrast to conventional leadscrews, ballscrews tend to be rather bulky, due to the need to have a mechanism to re-circulate the balls. Another form of linear actuator based on a rotating rod is the threadless ballscrew, a.k.a. "rolling ring drive". In this design, three (or more) rolling-ring bearings are arranged symmetrically in a housing surrounding a smooth (thread-less) actuator rod or shaft. The bearings are set at an angle to the rod, and this angle determines the direction and rate of linear motion per revolution of the rod. An advantage of this design over the conventional ballscrew or leadscrew is the practical elimination of backlash and loading caused by preload nuts.
3D CAD Models - Ball screw
Licensed under Creative Commons Attribution-Share Alike 3.0 (No machine-readable author provided. Graibeard assumed (based on copyright claims).).
7 Ball return systems
8 Thread profile
11 Ball Screw Standards
Historically, the first precise screwshafts were produced by starting with a low precision screwshaft, and then lapping the shaft with several spring-loaded nut laps[citation needed]. By rearranging and inverting the nut laps, the lengthwise errors of the nuts and shaft were averaged. Then, the very repeatable shaft's pitch is measured against a distance standard. A similar process is sometimes used today to produce reference standard screw shafts, or master manufacturing screw shafts.[citation needed]
An endcap return ballnut employs a cap on the end of the ball nut. The cap is machined to pick up balls out of the end of the nut and direct them down holes which are bored transversely down the ballnut. The complement cap on the other side of the nut directs balls back into the raceway.
{\displaystyle T={\frac {Fl}{2\pi \nu }}}
{\displaystyle {\mathit {T}}}
is torque applied to screw or nut,
{\displaystyle {\mathit {F}}}
is linear force applied,
{\displaystyle {\mathit {l}}}
is ball screw lead, and
{\displaystyle \nu }
is ball screw efficiency.
TorxPillow block bearingScrew threadSelf-tapping screwCaptive fastener기계공학List of screw drives육각형Screw conveyor사슬 톱니Spacers and standoffs
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Pop-up - Uncyclopedia, the content-free encyclopedia
Whoops! Maybe you were looking for Satan?
“ Pop-ups are the route of all Evil”
~ Oscar Wilde on Pop-ups
Popups are a widespread-favorite brand of cereal consumed by computer-literate users everywhere, using the slogan, "Gotta Have My Pop-Ups!." Contains addictive drugs such as 'pr0n-18' and 'FREEXXX', popular designer drugs used in the making of Popups to keep the user hooked even after a simple one use, referred to as "cookies", "spywares", and "Bob" on the street.
A typical Popup
Yet another typical popup.. How can you use a font if you don't have it?
a way to bill gates tell you that we does not like you
Once initially used, the drugs enter the bloodstream and make their way to the brain, where the user, based on their accumulated internet-browsing-experience (IBE), is either dominated with the urge to fill their primitive needs and acquire more excitement and joy or, if applied to a resistant user, has little if no effect and simply causes the emotion anger and runs the risk of stabbing the first living thing it sees. If the former situation is the case, the user finds themselves jogging to their computers and entering the sexy yet most likely membership-required site programmed into their brain by the drugs.
Popups are the best way to advertise because a whopping 0.0003‰ of users in the Internet click them at least once. They come in all forms from "Christian right wingers against sex having sex" to "0.00000000000000001% APR on your loan."
Popups are the best way to advertise since, unlike most other types of advertising, people don't get pissed off when they come up. And unlike other types of advertising, people actually take notice of them and not just to financially support the website. Also, the number of popups that appear when you go on many websites don't completely obscure the page you're looking at and take 10 minutes to close all of them unlike banner ads do.
Another great feature of popups is that people are the millionth visitor to a website so much that the geeks are actually earning more money by sitting at home on their PC clicking on ads in a day than their parents will earn in a lifetime.
To successfully promote your new website, design a popup today!
3 Pop-Ups in Modern Society
3.1 The Pop-up Equation
3.2 Annoying Pop-Ups
3.3 Interesting Pop-Ups
Example of a Pop-Up
Pop-Ups are vampiric organisms that feed off the bandwidth of the non-1337 and the wallets of Lamers. They have been known to attack and devour small children then offer their organs on black market trading sites such as eBay and Amazon.com. Most pop-ups are relatively small in size, averaging around 30 pixels by 80 pixels; however larger more aggressive pop-ups have been reported in the tropical regions of the interweb.
The natural predator of the pop-up is the clinja. The international order of Clinjas has vowed to either rid the world of pop-ups, or get drunk and play video games, whichever comes first. Making Popups makes everybody hate you.
The first documented Pop-Up attack happened in 1776 during the signing of the Declaration of Independence when suddenly a small paper appeared before the founding fathers of our nation reading "€œClick the redcoat and win a free Boston Tea Party."
Fossils records indicate that primitive pop-ups have been around since long before the dawn of recorded time stalking early users of the protoweb. These early pop-ups existed for advertising prehistoric products such as the first band ever founded, the Rolling Stones, and Sanskrit novels such as The Odyssey by Ulysses S. Grant.
Pop-Ups in Modern Society[edit]
Modern pop-up repellents such as Firefox have significantly reduced the effectiveness of pop-ups. For this reason the world pop-up population is dwindling. Additionally their natural habitat (the internet) is rapidly being destroyed to build a blogsphere. Some people feel that pop-ups will soon become extinct, but many experts agree, that since society produces an endless supply of morons and gives them internet access, that they will never really die out.
There are, in this world, a never ending supply of pop ups.
Some are annoying, some are interesting, and best of all, those pop ups that do not fit into these two very true categories are either worthless, naughty, incoherent, undeniably rude, inexplicably unneeded, or funny. As long as we are alive (that doesn't count the living dead) pop ups will be the most feared computer terrorists around (notice I didn't say alive). Pop ups are painful and cause upset stomach or diarrhea. Do not operate heavy machinery while anywhere within a 5,000 mile radius of any sort of pop up whether it be on a computer, cell phone, Pocket PC, etc. it is very dangerous and should be treated with extreme caution. Every year, over 1,000,000 people in Antarctica are treated for PUSS (Pop Up Seizures Syndrome). We must revolt against this evil world dominator. We must start a universal revolution. I egged all over Ralph and Jeff.
The Pop-up Equation[edit]
The most brilliant person in the world has discovered an equation for pop-ups. This equation can predict the next pop-up attack and by whom the pop-up was sent by:
{\displaystyle pc={\sqrt[{3}]{\frac {9.81+h^{2}}{K_{b}+\left(\int {\frac {C_{6}H_{12}O_{6}}{K_{b}}}\right)}}}}
Annoying Pop-Ups[edit]
Ipod winners.
1 millionth customer (we've all been one).
You've been chosen to (have your identity stolen and) win our [fake] hourly prize...
The best kind of pop up:
Interesting Pop-Ups[edit]
There are none. Except for those ones... Well never mind. Interesting pop-ups?!?!?!?! Who ever thought of that. Unless they have most of the words spelled wrong on them.
This is it. Thats the only one. No, it's not Naughty...
Penguin Armies of Doom
Retrieved from "https://uncyclopedia.com/w/index.php?title=Pop-up&oldid=5904434"
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Templet:Infobokis Planet - Wikipedia
Templet:Infobokis Planet
Other templates redirecting here: Infobox Nonstellar body · Infobox TNO
Usage notes[edit source]
This template expects that the Templet:Tag tag will be present in articles setting the minorplanet parameter to yes. Pages without a Templet:Tag tag will show Cite error: There are Templet:Tag tags on this page for a group named "note", but the references will not show without a Templet:Tag tag; see Help:Cite errors. at the bottom of the page.
Recommended parameters[edit source]
Exosolar planets[edit source]
Templet:Infobox planet
Minor planets[edit source]
Notes on usage[edit source]
From Wikipedia:WikiProject Astronomical objects/Infoboxes [1]
km2 (Earth units)
m/s2 (gees)
Most of these entries should be measured in SI units. Some of them, however, should have more "human-accessible" units, in addition to SI units. I've indicated some cases with a second unit name in brackets. In the case of times (orbital periods, rotation), I think it best to give all periods in days for comparison purposes, and provide a translation (in parentheses) into years, days, hours, etc.; whatever is most appropriate for the duration being described.
Oh, and compared to table templates for things like the elements, I think that this template should be considered somewhat more flexible. Moons with no atmosphere whatsoever could skip the atmospheric composition section entirely, for example (though atmospheric density would still be listed). Moons also wouldn't have their orbital radii listed in AU, since AUs are such large units. For planets, use "perihelion" and "aphelion" instead of "periapsis" and "apoapsis."
A set of colours for use in the 2-column headers of this table:
Transition metal color from the periodic table; rocky planets have lots of metals compared to the icy ones. Also, red is a "warmer" color than green, which fits the distribution of rocky and icy planets in the solar system.
green contrasts nicely with the pink of rocky planets. Also, on the periodic table, it's the color of carbon, oxygen, hydrogen, and other common components of outer-solar-system ice.
blue skies, and noble gases on the periodic table (including helium, which is only found in large quantities on gas giants. It escapes from smaller planets). Also, two out of four gas giants prefer the cool soothing color of blue.
{\displaystyle v_{max}=2\pi a^{2}{\frac {\sqrt {1-e^{2}}}{Ta(1-e)}}}
{\displaystyle v_{min}=2\pi a^{2}{\frac {\sqrt {1-e^{2}}}{Ta(1+e)}}}
Computed values[edit source]
Average orbital speed[edit source]
{\displaystyle 4aE(e)}
{\displaystyle v_{o}={\frac {4aE(e)}{T}}}
{\displaystyle \pi /2}
{\displaystyle v_{o}={\frac {2\pi a}{T}}\left[1-{\frac {e^{2}}{4}}-{\frac {3e^{4}}{64}}-\dots \right]}
{\displaystyle v_{o}\approx {\frac {\pi }{T}}\left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]}
{\displaystyle b=a{\sqrt {1-e^{2}}}\,\!}
Surface gravity[edit source]
Lukim pes: Surface gravity
{\displaystyle g_{\rm {spherical}}={\frac {GM}{r^{2}}}\,\!}
Where G = 6.6742 × 10−11 m3s-2kg-1 is the Gravitational constant, M is the mass of the body, and r its radius. This value is very approximate, as most minor planets are far from spherical.
{\displaystyle r_{\rm {max}}}
{\displaystyle g_{\rm {outer}}={\frac {GM}{r_{\rm {max}}^{2}}}\,.\!}
{\displaystyle g_{\rm {centrifugal}}=-\left({\frac {2\pi }{T}}\right)^{2}r_{\rm {eq}}}
{\displaystyle r_{\rm {eq}}}
{\displaystyle g_{\rm {effective}}\approx g_{\rm {gravitational}}+g_{\rm {centrifugal}}=g_{\rm {gravitational}}-|g_{\rm {centrifugal}}|\ .}
Escape velocity[edit source]
{\displaystyle v_{e}={\sqrt {2gr}}}
Temperature[edit source]
{\displaystyle L_{0}}
{\displaystyle T=\left({\frac {(1-\alpha )L_{0}}{\epsilon \sigma 16\pi a^{2}}}\right)^{\frac {1}{4}}}
Where σ is Stefan-Boltzmann constant. See also Torrence V. Johnson, Paul R. Weissman, Lucy-Ann A. McFadden (2007). Encyclopedia of the Solar System. Elsevier. pp. p. 294. ISBN 0120885891. .
The above documentation is transcluded from Templet:Infobokis Planet/doc. (edit | history)
{{{mp_name}}}
(min1) (mean1) (max1)
Ikam long "https://tpi.wikipedia.org/w/index.php?title=Templet:Infobokis_Planet&oldid=48613"
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What is the value of H in the following reaction at 300k CH4(g)+2o2(g)gives co2(g)+2H2o(l) E=885389 - Chemistry - Thermodynamics - 10879233 | Meritnation.com
What is the value of ?H in the following reaction at 300k?
CH4(g)+2o2(g)gives co2(g)+2H2o(l)
?E=885389
{\mathrm{CH}}_{4}\left(\mathrm{g}\right) + 2{\mathrm{O}}_{2}\left(\mathrm{g}\right) \to {\mathrm{CO}}_{2}\left(\mathrm{g}\right) + 2{\mathrm{H}}_{2}\mathrm{O}\left(\mathrm{l}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}∆\mathrm{H}=∆\mathrm{E}+\mathrm{P}∆\mathrm{V}\phantom{\rule{0ex}{0ex}}\mathrm{or}, ∆\mathrm{H}=∆\mathrm{E}+∆{\mathrm{n}}_{\mathrm{g}}\mathrm{RT}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{For} \mathrm{the} \mathrm{given} \mathrm{reaction}, \phantom{\rule{0ex}{0ex}}∆{\mathrm{n}}_{\mathrm{g}}= \left(1\right) - \left(2+1\right)\phantom{\rule{0ex}{0ex}} =1-3\phantom{\rule{0ex}{0ex}} =-2\phantom{\rule{0ex}{0ex}}\mathrm{Therefore}, ∆\mathrm{H}= 885389 +\left(-2\right)\left(8.314\right)\left(300\right)\phantom{\rule{0ex}{0ex}} =885389-4988.4\phantom{\rule{0ex}{0ex}} =880400.6 \mathrm{J}
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§ Burnside lemma by representation theory.
Recall that burnside asks us to show that given a group
G
acting on a set
S
, we have that the average of the local fixed points
1/|G|(\sum_{g \in G} |\texttt{Fix}(g)|)
is equal to the number of orbits (global fixed points) of
S
|S/G|
. Let us write elements of
g
as acting on the vector space
V_S
, which is a complex vector space spanned by basis vector
\{ v_s : s \in S \}
. Let this representation of
G
\rho
. Now see that the right hand side is equal to
\begin{aligned} &1/|G| (\sum_g \in G Tr(\rho(g))) \\ &= 1/|G| (\sum_g \in G \chi_\rho(g) ) \\ &\chi \rho \cdot \chi_1 \end{aligned}
\chi_1
is the charcter of the trivial representation
g \mapsto 1
\langle \cdot , \cdot \rangle
G
-average inner product over
G
G \rightarrow \mathbb C
\langle f , f' \rangle \equiv \sum_{g \in G} f(g) \overline{f'(g)}
So, we need to show that the number of orbits
|S/G|
is equal to the multiplicity of the trivial representation
1
in the current representation
\rho
, given by the inner product of their characters
\chi_1 \cdot \chi_\rho
s* in S
whose orbit we wish to inspect. Build the subspace spanned by the vector
v[s*] \equiv \sum_{g \in G} \rho(g) v[s]
. This is invariant under
G
and is 1-dimensional. Hence, it corresponds to a 1D subrepresentation for all the elements in the orbit of
s*
. (TODO: why is it the trivial representation?)
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§ Proof of chinese remainder theorem on rings
§ General operations on ideals
We have at our hands a commutative ring
R
, and we wish to study the ideal structure on the ring. In particular, we can combine ideals in the following ways:
I + J \equiv \{ i + j : \forall i \in I, j \in J \}
I \cap J \equiv \{ x : \forall x \in I \land x \in J \}
I\oplus J \equiv \{ (i, j) : \forall i \in I \land j \in J \}
IJ \equiv \{ ij : \forall i \in I \land j \in J \}
We have the containment:
IJ \subseteq I \cap J \subseteq I, J \subseteq I + J \subseteq R
IJ
is a ideal,
IJ \subseteq I \cap J
it's not immediate from the definition that
IJ
is an ideal. The idea is that given a sum
\sum_k i_k j_k \in IJ
, we can write each
i_k j_k = i'_k
, since the ideal
I
is closed under multiplication with
R
\sum i'_k = i'' \in I
. Similarly, we can interpret
\sum_k i_k j_k = \sum_k j'_k = j''k \in J
. Hence, we get the containment
IJ \subseteq I \cap J
I \cap J subseteq I
I \cap J \subseteq J
Immediate from the inclusion function.
I, J \subseteq I + J
Immediate from inclusion
§ CRT from an exact sequence
There exists an exact sequence:
\begin{aligned} 0 \rightarrow I \cap J \xrightarrow{f} I \oplus J \xrightarrow{g} I + J \rightarrow 0 \\ &f(r) = (r, r) \\ &g((i, j)) = i + j \end{aligned}
We are forced into this formula by considerations of dimension. We know:
\begin{aligned} &dim(I \oplus J) = dim(I) + dim(J) \\ &dim(I + J) = dim(I) + dim(J) - dim(I \cap J) \text{[inclusion-exclusion]} \\ &dim(I + J) = dim(I \oplus J) - dim(I \cap J) \\ &dim(I + J) - dim(I \oplus J) + dim(I \cap J) = 0\\ &V - E + F = 2 \end{aligned}
By analogy to euler characteristic which arises from homology, we need to have
I \oplus J
in the middle of our exact sequence. So we must have:
0 \rightarrow ? \rightarrow I \oplus J \rightarrow ?\rightarrow 0
Now we need to decide on the relative ordering between
I \cap J
I + J
There is no universal way to send
I oplus J \rightarrow I \cap J
. It's an unnatural operation to restrict the direct sum into the intersection.
There is a universal way to send
I \oplus J \rightarrow I + J
: sum the two components. This can be seen as currying the addition operation.
Thus, the exact sequence must have
I + J
I \oplus J
. This forces us to arrive at:
0 \rightarrow I \cap J \rightarrow I \oplus J \rightarrow I + J \rightarrow 0
The product ideal
IJ
plays no role, since it's not possible to define a product of modules in general (just as it is not possible to define a product of vector spaces). Thus, the exact sequence better involve module related operations. We can now recover CRT:
\begin{aligned} 0 \rightarrow I \cap J \xrightarrow{f} I \oplus J \xrightarrow{g} I + J \rightarrow 0 \\ 0 \rightarrow R \xrightarrow{f} R \oplus R \xrightarrow{g} R \rightarrow 0 \\ 0 \rightarrow R / (I \cap J) \rightarrow R/I \oplus R /J \rightarrow R/(I + J) \rightarrow 0 \end{aligned}
I learnt the material from this course on commutative algebra from IIT bombay .
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Quantitative assessment of mitochondrial DNA copies from whole genome sequencing | BMC Genomics | Full Text
Quantitative assessment of mitochondrial DNA copies from whole genome sequencing
Hsueh-Ting Chu1,2,
William WL Hsiao3,4,
Theresa TH Tsao5,
Ching-Mao Chang6,
Yen-Wenn Liu8,
Chen-Chieh Fan5,
Tze-Jung Yeh9,
Jen-Chih Chen9,
Dun-Ming Huang2,
Chaur-Chin Chen10 &
Cheng-Yan Kao5
Mitochondrial dysfunction is associated with various aging diseases. The copy number of mtDNA in human cells may therefore be a potential biomarker for diagnostics of aging. Here we propose a new computational method for the accurate assessment of mtDNA copies from whole genome sequencing data.
Two families of the human whole genome sequencing datasets from the HapMap and the 1000 Genomes projects were used for the accurate counting of mitochondrial DNA copy numbers. The results revealed the parental mitochondrial DNA copy numbers are significantly lower than that of their children in these samples. There are 8%~21% more copies of mtDNA in samples from the children than from their parents. The experiment demonstrated the possible correlations between the quantity of mitochondrial DNA and aging-related diseases.
Since the next-generation sequencing technology strives to deliver affordable and non-biased sequencing results, accurate assessment of mtDNA copy numbers can be achieved effectively from the output of whole genome sequencing. We implemented the method as a software package MitoCounter with the source code and user's guide available to the public at http://sourceforge.net/projects/mitocounter/.
Human mitochondria contain multiple copies of a 16.5 k bp, double-stranded, circular DNA molecule (Figure 1a). Since mitochondria are the organelles that generate chemical energy for cellular functions, many disease symptoms are linked to mitochondrial dysfunction, including poor growth, muscle weakness, hearing problems, visual problems, heart diseases, and liver diseases. There were many recent studies which showed significantly reduced mitochondrial DNA (mtDNA) copy numbers in cell samples of aging-related diseases [1–3]. A recent study also reported that mtDNA copy number is associated with cancer risk [4]. Therefore, quantitative assessment of mtDNA in human cells can elucidate the relationship between mitochondrial diseases and mitochondrial dysfunction.
Overview of human whole genome sequencing. A) The human genome is composed of nuclear DNA and mitochondrial DNA. The nuclear DNA is stored on 23 chromosome pairs and there are multiple copies of small DNA located in mitochondria. B) The reads from the sequencing of human whole genome are mixed with both nuclear DNA and mitochondrial DNA.
In the past decade, quantitative real-time PCR assays were developed to estimate relative levels of mtDNA copy numbers in samples [2, 5, 6]. This approach measures the mtDNA copy number by determining the ratio of PCR amplicons to that of a single nuclear gene in experimental samples. The recent development of next-generation sequencing technology (NGS) revolutionized genomic studies and produced accurate whole genome sequencing (WGS) datasets [7]. As shown in Figure 1b, the output from human whole genome sequencing consists of both nuclear DNA (nuDNA) and mitochondrial DNA (mtDNA) molecules, thus it is convenient to assess mtDNA copy number from WGS dataset and can be an alternative to real-time PCR assays.
Here we demonstrate a computational method for counting mtDNA copy number using WGS datasets. The three steps in the process are (1) typing of mtDNA, (2) separation of mtDNA reads, and (3) calculation of mtDNA count. We developed a freely available software package called MitoCounter for this purpose. MitoCounter can be used to calculate the average copy numbers of mtDNA molecules in the sequenced samples. Besides, the separated mtDNA reads provide further analysis of mtDNA heteroplasmy. The mtDNA heteroplasmy represents the mixture of individual mtDNA mutations. Heteroplasmy levels can alter the clinical penetrance of primary mtDNA diseases [8, 9].
A computational assay for counting mtDNA copies from a WGS dataset
Since the library construction bias is minimized with the next-generation sequencing platform [10], both mitochondrial DNA (mtDNA) and nuclear DNA (nuDNA) are sequenced together with equal opportunities. The output dataset comprises a mixture of mtDNA reads and nuDNA reads. Let the total number of nucleotide bases in the nuclear genome be 2N (for diploid chromosomes) and the number of bases in a mitochondrial DNA is M. Then the summation of nucleotide bases in the entire human genome is 2N+kM, where k is the number of mtDNA copies. The numbers of reads from nuDNA and the number of reads from mtDNA should reflect the ratio of 2N:kM.
\frac{mtBases}{allBases}=\frac{kM}{2N+kM}
where mtBases is the total bases of sequenced reads from mtDNA and allBases is total bases of all sequenced reads from the output of a WGS procedure.
From an entire dataset of human whole genome sequencing, we separate the mtDNA reads from the others. Then the number of mtDNA copies can be approximated as
k=\frac{mtBases*2N}{\left(allBases-mtBases\right)*M}
The equation for counting mtDNA copies is not suitable for plants (e.g. Arabidopsis) since their mtDNA sequences may contain segments of nuclear DNA. Besides, there are usually other DNA molecules in their cells, such as chloroplast genome and plasmid genome.
In order to precisely separate mtDNA reads from a WGS dataset, it is necessary to determine the genotype of the mitochondrial genome first. We designed a program WgsMitoAssembler to identify the homoplasmic sequences, which present the inherent mutations in most of mtDNA molecules. The program WgsMitoAssembler is a guided assembler, and it requires a reference mitochondrial sequence which is used to choose a beginning read and an ending read from the entire WGS dataset. We use the reference mtDNA sequence (GenBank: NC_001807.4) for the purpose. We then search for best candidate reads which can extend the beginning read from the 3' end to the 5' end until the ending read is met.
After the typing of the target mitochondrial genome, the homoplasmy sequence is used in the second program WgsMitoCounter. The program performs the job of separating mitochondrial reads from the entire WGS dataset. Considering that some of sequenced reads may contain erroneous bases, we design an error-tolerant mapping algorithm shown in Figure 2. We search for sub-sequences of paired reads which are indexed as mtDNA fragments and the accuracy of mapping is determined by the pairing distances. WgsMitoCounter will output a CSV file which records the number of mitochondrial reads in each run of the entire dataset. The template of final calculation for mtDNA copy number is provided in Additional file 1.
An error-tolerant mapping algorithm for filtering mitochondrial reads.
Parental mtDNA samples have less copy numbers
We apply the analysis to public WGS datasets from the HapMap [11] and 1000 Genomes [12] projects. We chose six high-coverage WGS datasets for two pedigree trios: YOR009 and CEPH146 and two low-coverage WGS datasets for individual elders (Sample ID: NA11831 and NA06985), listed in Table 1. YOR009 is an African family. CEPH1463 is a family from Utah with Northern and Western European ancestry. The two individuals are also from the CEPH population and were recorded as the grandparents in the 1000 Genomes project. These DNA samples were isolated from B-lymphocyte cells derived from blood. Table 2 lists the results of counting mitochondrial DNA on the selected datasets. The mtDNA counts for the YOR009 family are between 645~752 and for CEPH1463 family are between 734~950. Besides, the mtDNA counts for the two individual elders are 662 and 755.
Table 1 Typing of mtDNA from whole genome sequencing samples
Table 2 Counts of mtDNA from whole genome sequencing samples
For the counting results of these WGS samples (Additional file 2 and 3), ANOVA analysis revealed significant differences among the mtDNA counts within each family group: for YOR009, F(2,69) = 916.01, p = 2.06E-50 and for CEPH1463, F(2,169) = 58.75, p = 7.26632E-19. It showed that the offspring had 8%~23% more mtDNA than their parents in these samples. Although we did not investigate the possible artefacts caused by sequencing procedures, the results consistently demonstrated that there are more mtDNA sequences within younger persons' lymphocyte cells.
Many studies suggested that mitochondrial functions become defective as we age. Recent findings suggests that structural changes in mitochondria, including increased mitochondrial fragmentation and decreased mitochondrial fusion, are critical factors associated with mitochondrial dysfunction and cell death in aging and neurodegenerative diseases [13, 14]. Therefore, the proposed quantitative analysis of mtDNA can help to further elucidate the dynamics of mitochondrial diseases. It is expected that cost for sequencing personal whole genome will be less than $1000 in the near future. For the purpose of counting mitochondrial DNA, it only requires a low coverage of the whole genome and the cost may be further reduced to $50. The cost-effectiveness of the procedure makes the proposed method of counting mitochondrial DNA as a useful diagnostic tool to study aging and aging-related diseases for individuals.
In the MitoCounter software package, both the programs WgsMitoAssembler and WgsMitoCounter were implemented in C# with the .NET Framework which can be run on 64-bit Windows. The program WgsMitoCounter requires paired-end WGS datasets from Illumina sequencing platform. The MitoCounter software with a user manual is available at the Web site: http://sourceforge.net/projects/mitocounter/
Maassen JA, LM TH, Van Essen E, Heine RJ, Nijpels G, Jahangir Tafrechi RS, Raap AK, Janssen GM, Lemkes HH: Mitochondrial diabetes: molecular mechanisms and clinical presentation. Diabetes. 2004, 53 (Suppl 1): S103-109.
Hartmann N, Reichwald K, Wittig I, Drose S, Schmeisser S, Luck C, Hahn C, Graf M, Gausmann U, Terzibasi E, et al: Mitochondrial DNA copy number and function decrease with age in the short-lived fish Nothobranchius furzeri. Aging Cell. 2011, 10 (5): 824-831. 10.1111/j.1474-9726.2011.00723.x.
Cree LM, Patel SK, Pyle A, Lynn S, Turnbull DM, Chinnery PF, Walker M: Age-related decline in mitochondrial DNA copy number in isolated human pancreatic islets. Diabetologia. 2008, 51 (8): 1440-1443. 10.1007/s00125-008-1054-4.
Thyagarajan B, Wang R, Barcelo H, Koh WP, Yuan JM: Mitochondrial copy number is associated with colorectal cancer risk. Cancer Epidemiol Biomarkers Prev. 2012
Koekemoer TC, Downing TG, Oelofsen W: An alternative PCR assay for quantifying mitochondrial DNA in crude preparations. Nucleic Acids Research. 1998, 26 (11): 2829-2830. 10.1093/nar/26.11.2829.
Barazzoni R, Short KR, Nair KS: Effects of aging on mitochondrial DNA copy number and cytochrome c oxidase gene expression in rat skeletal muscle, liver, and heart. J Biol Chem. 2000, 275 (5): 3343-3347. 10.1074/jbc.275.5.3343.
Sondheimer N, Glatz CE, Tirone JE, Deardorff MA, Krieger AM, Hakonarson H: Neutral mitochondrial heteroplasmy and the influence of aging. Hum Mol Genet. 2011, 20 (8): 1653-1659. 10.1093/hmg/ddr043.
Avital G, Buchshtav M, Zhidkov I, Tuval Feder J, Dadon S, Rubin E, Glass D, Spector TD, Mishmar D: Mitochondrial DNA heteroplasmy in diabetes and normal adults: role of acquired and inherited mutational patterns in twins. Hum Mol Genet. 2012
Consortium TIHGS: A haplotype map of the human genome. Nature. 2005, 437 (7063): 1299-1320. 10.1038/nature04226.
Reddy PH, Reddy TP: Mitochondria as a therapeutic target for aging and neurodegenerative diseases. Curr Alzheimer Res. 2011, 8 (4): 393-409. 10.2174/156720511795745401.
We thank the members in the Bioinformatics Lab, NTU, for valuable discussions and useful insights. We thank C. S. Chiou, C. H. Chan and C. F. Chang for comments and discussion. We thank Flora Kao for editing and proofreading.
Department of Biomedical informatics, Asia University, Taichung, 41354, Taiwan
Department of Computer Science and Information Engineering, Asia University, Taichung, 41354, Taiwan
Hsueh-Ting Chu & Dun-Ming Huang
BCCDC Public Health Microbiology & Reference Laboratory, Vancouver, BC, V5Z 4R4, Canada
William WL Hsiao
Department of Pathology and Laboratory Medicine, Vancouver, BC, V5Z 4R4, Canada
Department of Computer Science and Information Engineering, National Taiwan University, Taipei, 10617, Taiwan
Theresa TH Tsao, Chen-Chieh Fan, Han Lin & Cheng-Yan Kao
Graduate Institute of Clinical Medical Science, Chang Gung University, Taoyuan, 33302, Taiwan
Ching-Mao Chang & Hen-Hong Chang
Center for Traditional Chinese Medicine, Chang Gung Memorial Hospital at Taoyuan, Chang Gung Medical Foundation, Taoyuan, 33302, Taiwan
Tze-Jung Yeh & Jen-Chih Chen
Theresa TH Tsao
Chen-Chieh Fan
Dun-Ming Huang
Correspondence to Cheng-Yan Kao.
HTC devised the method and wrote the software. HTC, WLH, TTT, CMC, YWL, CCF, HHC, HL, TJY, JCC, DMH and CCC discussed the project and jointly wrote the manuscript. CYK conceived the project.
Additional File 1: Supplementary Software. The MitoCounter software package consists of two execution programs, an Excel template and a User manual. (XLS 26 KB)
Additional File 2: Supplementary Table 1. An Excel file lists the ratio of mitochondrial reads in each WGS run for the family trio YOR009 samples. (XLS 31 KB)
Additional File 3: Supplementary Table 2. An Excel file lists the ratio of mitochondrial reads in each WGS run for the family trio CEPH1463 samples. (ZIP 2 MB)
Chu, HT., Hsiao, W.W., Tsao, T.T. et al. Quantitative assessment of mitochondrial DNA copies from whole genome sequencing. BMC Genomics 13, S5 (2012). https://doi.org/10.1186/1471-2164-13-S7-S5
Heteroplasmy Level
Individual Elder
Genome Sequencing Dataset
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The true video watching time on Facebook - Maurizio NaldiMaurizio Naldi
← What the HHI can tell us about a recent telco merger
In the fable "The frog and the ox" by Aesop, a frog tries to appear bigger than it actually is by inflating itself more and more...till it bursts. Well, that's not the end of the story, but the frog's behaviour resembles a bit what Facebook has done with one of its metrics: the average time people stay in front of a video. This metric is a key for prospective ad space buyers, since it is a proxy for the average exposure of Facebook users to ads: the longer the exposure, the higher the value of the advertising space. The metric is particularly important for a company like Facebook, whose product is basically you and your time. The news broke after the advertising company Publicis warned its customers about the alleged miscalculation by Facebook, and was reported by the Wall Street Journal. Facebook itself admitted the "discrepancy" by posting a clarification note on the computation method.
But what is then the true average time people stay in front of a video on Facebook? The true value appears to be private to Facebook, which releases it to its prospective customers. We can however gain some insight through some very simple calculations.
We know that Facebook censored data by neglecting watching times shorter than 3 seconds. In order to see the impact of censoring on the average watching time, we must assume a probability distribution for watching time. If we assume that the watching time
X
follows an exponential distribution (though this seems to imply that videos can be watched for an infinite time, it serves just as a model since the probability of watching time exceeding a physically feasible value can be extremely small), the average watching time
\mathbb{E}[\hat{X}]
computed after censoring durations shorter than 3 seconds is, quite simply, the true average
\mathbb{E}[X]
value plus 3 seconds.
Publicis itself reported that the miscomputation led to inflating the average watching time by 60 to 80%, i.e.
Putting the two pieces of information together and doing the math, we find that the true average watching time is between 3.75 and 5 seconds (but is mistakenly reported as lying between 6.75 and 8 seconds), quite small, maybe a bit too small for prospective ad buyers...
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LMIs in Control/Matrix and LMI Properties and Tools/Negative Imaginary System DC Constraint - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Negative Imaginary System DC Constraint
4 The LMI
These systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However, transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called systems with negative imaginary frequency response.
{\displaystyle (A,B,C,D)}
{\displaystyle {\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}}}
{\displaystyle A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{m\times n},D\in \mathbb {S} ^{m}}
The LMIEdit
Consider an NI transfer matrix
{\displaystyle {\mathbf {G}}_{1}(s)}
and an NI transfer matrix
{\displaystyle {\mathbf {G}}_{2}(s)={\mathbf {C}}_{2}(s{\mathbf {1}}-{\mathbf {A}}_{2})^{-1}{\mathbf {B}}_{2}+{\mathbf {D}}_{2}}
. The condition λ̅
{\displaystyle ({\mathbf {G}}_{1}(0){\mathbf {G}}_{2}(0)<1}
{\displaystyle {\mathbf {S}}^{T}(-{\mathbf {C}}_{2}{\mathbf {A}}_{2}^{-1}{\mathbf {B}}_{2}+{\mathbf {D}}_{2}){\mathbf {S}}<{\mathbf {1}}}
The above equation holds true if and only if
{\displaystyle {\mathbf {S}}{\mathbf {S}}^{T}={\mathbf {G}}_{1}(0)}
Negative Imaginary Lemma
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Negative_Imaginary_System_DC_Constraint&oldid=4013832"
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Some Notes on Windows in Spectral Analysis
Please cite this document as
Jens Ahrens, “Some Notes on Windows in Spectral Analysis,” Tech. Report, Chalmers Univeristy of Technology, 2020.
The following plots show the (unwindowed) single-sided Discrete Fourier Transform (DFT) amplitude spectra
\vert \overline{X}_\text{SS}(k)\vert
of two sine waves of amplitude
1
. (Refer to this article if you are unsure about what a single-sided amplitude spectrum is.)
Fig. 1: Single-sided DFT amplitude spectra
\vert \overline{X}_\text{SS}(k)\vert
1
1000~\mathrm{Hz}
1002.5~\mathrm{Hz}
. The horizontal axis was converted from bin index
k
to frequency in
\mathrm{Hz}
as explained here.
Both sines were sampled at
f_\textrm{s} = 10^5
Hz, and the length of the DFT was
N = 2\cdot 10^5
in both cases. So, the only difference is the frequencies. Yet, the spectra suggest that the sines differ in amplitude by more than
3~\textrm{dB}
. This article clarifies why this is and how this apparent difference can be avoided using flat top windows.
Main Lobes and Side Lobes of Windows Revisited
Let’s look at the single-sided amplitude spectra
\vert \overline{X}_\text{SS}(k)\vert
of two sine waves with a DFT length of
N=101
\vert \overline{X}_\text{SS}(k)\vert
of two sines with a frequency that is equivalent to bin index
k=10
k=10.4
These plots were obtained by connecting all spectral magnitudes that occur at integer
k
with straight lines. This is how magnitude spectra are usually plotted. Note that on the left hand side, the magnitude is
0
at all values of
k
k=10
, which is the frequency of the sine. As you may know, all windows including the rectangular window that is implied here, exhibit a main lobe and side lobes which cause smearing of the spectrum. Why can’t we see any of this in the left hand plot in Fig. 2?
The short answer is: The left hand plot in Fig. 2 is a special case where the frequency of the sine whose spectrum we are looking at corresponds to one of the basis functions of the DFT (some may call this “It is exactly in the center of a bin
k
Recall the definition of the DFT:
It happens so that if the frequency of a sine is exactly in the center of a bin
k_0
of the DFT \eqref{eq:dft}, there will be a complex exponential
{\mathrm e}^{-{\mathrm i} \frac{2\pi k}{N}n}
that will oscillate at the exact same frequency like the sine. In other words, we need only this single basis function (plus the one at
-k_0
) of the DFT to represent the sine. The spectral coefficients
X(k)
will be zero for all other
k
Ok, so far so good. How can we bring the lobes into the game? Usually, the DFT \eqref{eq:dft} is only computed for integer
k
because this covers all of the information. You can actually also evaluate \eqref{eq:dft} for fractional
k
as the spectrum of a time-discrete signal is continuous1. There is nothing that prevents us from doing this. See this MATLAB script, which produces the following Fig. 3, which is identical to Fig. 2 but with the DFT \eqref{eq:dft} evaluated also for fractional
k
(gray line).
Fig. 3: Same like Fig. 2 but with the DFT evaluated also at fractional
k
The black circles mark the spectral points at integer
k
. That’s the same data points that Fig. 2 comprises, and that’s the ones that we usually look at. You can now actually discern the window lobes in the gray lines! It just happens so that for this particular case, the DFT “samples” the window only at the maximum of the main lobe and exactly at the zeros between the lobes. Note that the circumstance that the integer bins sample the window exactly at the zeros between lobes is a particularity of the reactangular window. You will always see traces of the side lobes for any other window.
The story changes if we look at sines with arbitrary frequency, for example equivalent to
k=10.4
. See the right hand plot in Fig. 3 , which is a variant of Fig. 2 (right) and also plots fractional
k
. The DFT does indeed sample the window lobes outside of the zeros between the lobes so that we notice them. It happens so that the main lobe is not sampled at its maximum so that the sine amplitude that we read from the amplitude spectrum is misleading. This is where flat top windows come into play.
Flat Top Windows
The following figure plots a flat top window [1] in the domain of the original signal
x(n)
(some call this the time domain). See this MATLAB script.
Fig. 4: The standard flat top window that MATLAB uses for
N=101
Fig. 5 depicts the amplitude spectra of the same scenario like in Fig. 2 and 3 (sine waves with frequencies equivalent to
k=10
k=10.4
) but with a Hann window2 and a flat top window applied before the DFT3. See this MATLAB script. We chose the Hann window for completeness. The story is very similiar like for the rectangular window discussed above.
Fig. 5: Same like Fig. 3 but with a Hann window (dotted line) and a flat top window (solid line) applied.
The main lobe of the Hann window is pointy so that it is not guaranteed that the DFT samples its maximum (cf. Fig. 5 (right)). The main lobe of the flat top window has a flat top (surprise!) with ripples of less than
0.1~\textrm{dB}
and that is a least one bin wide. It is therefore guaranteed that that DFT samples the main lobe around its maximum (green circle in Fig. 5 (right)) so that we obtain approximately the same magnitude independent of the frequency of the sine.
The price that we pay is that the width of the main lobe obscures the precise frequency of the sine that we analyse even more than it occurs for other windows. We know the sine’s amplitude more precisely but lose precision of its frequency. Greetings from Heisenberg!
In order to determine both frequency and magnitude with precision requires either a DFT with integer
k
, which is not commonly applied because of the computational cost, or interpolation of the DFT magnitude spectrum as proposed, for example, in [2]. Either way, you have to hope that the frequency of the sine that you analyse is constant during the length of the DFT. More greetings from Heisenberg.
Note that it happens so that the side lobes of both the Hann window and the flat top window are so low in magnitude that they are not visible in Fig. 5 (left) because the magnitude is plotted on a linear scale. Feel free to play around with the MATLAB script and switch the plotting to logarithmic see the side lobes even at integer
k
To close the loop, we’re getting back to the example from Fig. 1 in the Abstract but apply a flat top window (see this MATLAB script):
Fig. 6: Same like Fig. 1 but with a flat top window applied
No matter what the frequency of the sine is, we can deduce that its amplitude is
0
dB or equivalently
1
. Recall that we are looking at single-sided amplitude spectra.
Use a flat top window to determine the amplitudes of stationary oscillations in a signal. Do not use it if you need to deduce the frequencies of the oscillations.
J. Smith and X. Serra, “PARSHL: An analysis/synthesis program for nonharmonic sounds based on a sinusoidal representation,” in Proc. of the Int. Computer Music Conf. (ICMC), Champaign/Urbana, Illinois, USA, Aug. 1987.
The discrete signal can take on any arbirtary frequency within the limits of the Nyquist-Shannon theorem. ↩
Note that it is Hann not Hanning (it was named after Julius von Hann). There is indeed a Hamming window, which was named after Richard Hamming. ↩
Note that many different flat top windows exist [1]. ↩
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Hypothetical syllogism - Wikipedia
If I do not wake up, then I cannot go to work.
If I cannot go to work, then I will not get paid.
Therefore, if I do not wake up, then I will not get paid.
The term originated with Theophrastus.[1]
4 Proof [3]
5.2 As a metatheorem
In propositional logic, hypothetical syllogism is the name of a valid rule of inference (often abbreviated HS and sometimes also called the chain argument, chain rule, or the principle of transitivity of implication). The rule may be stated:
{\displaystyle {\frac {P\to Q,Q\to R}{\therefore P\to R}}}
where the rule is that whenever instances of "
{\displaystyle P\to Q}
", and "
{\displaystyle Q\to R}
" appear on lines of a proof, "
{\displaystyle P\to R}
" can be placed on a subsequent line.
Hypothetical syllogism is closely related and similar to disjunctive syllogism, in that it is also a type of syllogism, and also the name of a rule of inference.
The rule of hypothetical syllogism holds in classical logic, intuitionistic logic, most systems of relevance logic, and many other systems of logic. However, it does not hold in all logics, including, for example, non-monotonic logic, probabilistic logic and default logic. The reason for this is that these logics describe defeasible reasoning, and conditionals that appear in real-world contexts typically allow for exceptions, default assumptions, ceteris paribus conditions, or just simple uncertainty.
An example, derived from Adams, [2]
(1) If Jones wins the election, Smith will retire after the election.
(2) If Smith dies before the election, Jones will win the election.
(3) If Smith dies before the election, Smith will retire after the election.
Clearly, (3) does not follow from (1) and (2). (1) is true by default, but fails to hold in the exceptional circumstances of Smith dying. In practice, real-world conditionals always tend to involve default assumptions or contexts, and it may be infeasible or even impossible to specify all the exceptional circumstances in which they might fail to be true. For similar reasons, the rule of hypothetical syllogism does not hold for counterfactual conditionals.
The hypothetical syllogism inference rule may be written in sequent notation, which amounts to a specialization of the cut rule:
{\displaystyle {\frac {P\vdash Q\quad Q\vdash R}{P\vdash R}}}
{\displaystyle \vdash }
is a metalogical symbol and
{\displaystyle A\vdash B}
{\displaystyle B}
{\displaystyle A}
{\displaystyle ((P\to Q)\land (Q\to R))\to (P\to R)}
{\displaystyle P}
{\displaystyle Q}
{\displaystyle R}
are propositions expressed in some formal system.
Proof [3]Edit
{\displaystyle P\to Q}
{\displaystyle Q\to R}
{\displaystyle P}
Conditional proof assumption
{\displaystyle Q}
{\displaystyle R}
{\displaystyle P\to R}
Conditional Proof (3-5)
An alternative form of hypothetical syllogism, more useful for classical propositional calculus systems with implication and negation (i.e. without the conjunction symbol), is the following:
{\displaystyle (Q\to R)\to ((P\to Q)\to (P\to R))}
Yet another form is:
{\displaystyle (P\to Q)\to ((Q\to R)\to (P\to R))}
An example of the proofs of these theorems in such systems is given below. We use two of the three axioms used in one of the popular systems described by Jan Łukasiewicz. The proofs relies on two out of the three axioms of this system:
{\displaystyle \phi \to \left(\psi \to \phi \right)}
{\displaystyle \left(\phi \to \left(\psi \rightarrow \xi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \xi \right)\right)}
The proof of the (HS1) is as follows:
{\displaystyle ((p\to (q\to r))\to ((p\to q)\to (p\to r)))\to ((q\to r)\to ((p\to (q\to r))\to ((p\to q)\to (p\to r))))}
{\displaystyle (p\to (q\to r))\to ((p\to q)\to (p\to r))}
{\displaystyle (q\to r)\to ((p\to (q\to r))\to ((p\to q)\to (p\to r)))}
{\displaystyle ((q\to r)\to ((p\to (q\to r))\to ((p\to q)\to (p\to r))))\to (((q\to r)\to (p\to (q\to r)))\to ((q\to r)\to ((p\to q)\to (p\to r))))}
{\displaystyle ((q\to r)\to (p\to (q\to r)))\to ((q\to r)\to ((p\to q)\to (p\to r)))}
{\displaystyle (q\to r)\to (p\to (q\to r))}
{\displaystyle (q\to r)\to ((p\to q)\to (p\to r))}
The proof of the (HS2) is given here.
As a metatheoremEdit
Whenever we have two theorems of the form
{\displaystyle T_{1}=(Q\to R)}
{\displaystyle T_{2}=(P\to Q)}
{\displaystyle (P\to R)}
{\displaystyle (Q\to R)\to ((P\to Q)\to (P\to R)))}
(instance of the theorem proved above)
{\displaystyle Q\to R}
(instance of (T1))
{\displaystyle (P\to Q)\to (P\to R)}
{\displaystyle P\to Q}
{\displaystyle P\to R}
^ "History of Logic: Theophrastus of Eresus" in Encyclopædia Britannica Online.
^ Adams, Ernest W. (1975). The Logic of Conditionals. Dordrecht: Reidel. p. 22.
^ "Discrete mathematics - when proving the Hypothetical Syllogism inference rule, why must you assume that p is true?".
Philosophy Index: Hypothetical Syllogism
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hypothetical_syllogism&oldid=1054386088"
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Roller chain (49552 views - Mechanical Engineering)
Roller chain or bush roller chain is the type of chain drive most commonly used for transmission of mechanical power on many kinds of domestic, industrial and agricultural machinery, including conveyors, wire- and tube-drawing machines, printing presses, cars, motorcycles, and bicycles. It consists of a series of short cylindrical rollers held together by side links. It is driven by a toothed wheel called a sprocket. It is a simple, reliable, and efficient means of power transmission. Though Hans Renold is credited with inventing the roller chain in 1880, sketches by Leonardo da Vinci in the 16th century show a chain with a roller bearing.
3D CAD Models - roller chain
Though Hans Renold is credited with inventing the roller chain in 1880, sketches by Leonardo da Vinci in the 16th century show a chain with a roller bearing.[2]
1 Construction of the chain
Construction of the chain
There are actually two types of links alternating in the bush roller chain. The first type is inner links, having two inner plates held together by two sleeves or bushings upon which rotate two rollers. Inner links alternate with the second type, the outer links, consisting of two outer plates held together by pins passing through the bushings of the inner links. The "bushingless" roller chain is similar in operation though not in construction; instead of separate bushings or sleeves holding the inner plates together, the plate has a tube stamped into it protruding from the hole which serves the same purpose. This has the advantage of removing one step in assembly of the chain.
The roller chain design reduces friction compared to simpler designs, resulting in higher efficiency and less wear. The original power transmission chain varieties lacked rollers and bushings, with both the inner and outer plates held by pins which directly contacted the sprocket teeth; however this configuration exhibited extremely rapid wear of both the sprocket teeth, and the plates where they pivoted on the pins. This problem was partially solved by the development of bushed chains, with the pins holding the outer plates passing through bushings or sleeves connecting the inner plates. This distributed the wear over a greater area; however the teeth of the sprockets still wore more rapidly than is desirable, from the sliding friction against the bushings. The addition of rollers surrounding the bushing sleeves of the chain and provided rolling contact with the teeth of the sprockets resulting in excellent resistance to wear of both sprockets and chain as well. There is even very low friction, as long as the chain is sufficiently lubricated. Continuous, clean, lubrication of roller chains is of primary importance for efficient operation as well as correct tensioning.[citation needed]
Many driving chains (for example, in factory equipment, or driving a camshaft inside an internal combustion engine) operate in clean environments, and thus the wearing surfaces (that is, the pins and bushings) are safe from precipitation and airborne grit, many even in a sealed environment such as an oil bath. Some roller chains are designed to have o-rings built into the space between the outside link plate and the inside roller link plates. Chain manufacturers began to include this feature in 1971 after the application was invented by Joseph Montano while working for Whitney Chain of Hartford, Connecticut. O-rings were included as a way to improve lubrication to the links of power transmission chains, a service that is vitally important to extending their working life. These rubber fixtures form a barrier that holds factory applied lubricating grease inside of the pin and bushing wear areas. Further, the rubber o-rings prevent dirt and other contaminants from entering the inside of the chain linkages, where such particles would otherwise cause significant wear.
There are also many chains that have to operate in dirty conditions, and for size or operational reasons cannot be sealed. Examples include chains on farm equipment, bicycles, and chain saws. These chains will necessarily have relatively high rates of wear, particularly when the operators are prepared to accept more friction, less efficiency, more noise and more frequent replacement as they neglect lubrication and adjustment.
If the chain is not being used for a high wear application (for instance if it is just transmitting motion from a hand operated lever to a control shaft on a machine, or a sliding door on an oven), then one of the simpler types of chain may still be used. Conversely, where extra strength but the smooth drive of a smaller pitch is required, the chain may be "siamesed"; instead of just two rows of plates on the outer sides of the chain, there may be three ("duplex"), four ("triplex"), or more rows of plates running parallel, with bushings and rollers between each adjacent pair, and the same number of rows of teeth running in parallel on the sprockets to match. Timing chains on automotive engines, for example, typically have multiple rows of plates called strands.
Roller chain is made in several sizes, the most common American National Standards Institute (ANSI) standards being 40, 50, 60, and 80. The first digit(s) indicate the pitch of the chain in eighths of an inch, with the last digit being 0 for standard chain, 1 for lightweight chain, and 5 for bushed chain with no rollers. Thus, a chain with half inch pitch would be a #40 while a #160 sprocket would have teeth spaced 2 inches apart, etc. Metric pitches are expressed in sixteenths of an inch; thus a metric #8 chain (08B-1) would be equivalent to an ANSI #40. Most roller chain is made from plain carbon or alloy steel, but stainless steel is used in food processing machinery or other places where lubrication is a problem, and nylon or brass are occasionally seen for the same reason.
Roller chain is ordinarily hooked up using a master link (also known as a connecting link), which typically has one pin held by a horseshoe clip rather than friction fit, allowing it to be inserted or removed with simple tools. Chain with a removable link or pin is also known as cottered chain, which allows the length of the chain to be adjusted. Half links (also known as offsets) are available and are used to increase the length of the chain by a single roller. Riveted roller chain has the master link (also known as a connecting link) "riveted" or mashed on the ends. These pins are made to be durable and are not removable.[5]
In older automobile engines roller chains would traditionally drive the camshaft(s) off the crankshaft, generating less noise than a gear drive as used in very high performance engines, and offering more durability than the timing belt frequently used on more modern engines. Many modern automobile engines still use roller chains, which are more durable than timing belts.
With modern chains it is unusual for a chain (other than that of a bicycle) to wear until it breaks, since a worn chain leads to the rapid onset of wear on the teeth of the sprockets, with ultimate failure being the loss of all the teeth on the sprocket. The sprockets (in particular the larger of the two) suffer a grinding motion that puts a characteristic hook shape into the driven face of the teeth. (This effect is made worse by a chain improperly tensioned, but is unavoidable no matter what care is taken). The worn teeth (and chain) no longer provides smooth transmission of power and this may become evident from the noise, the vibration or (in car engines using a timing chain) the variation in ignition timing seen with a timing light. Both sprockets and chain should be replaced in these cases, since a new chain on worn sprockets will not last long. However, in less severe cases it may be possible to save the smaller of the two sprockets, since it is always the larger one that suffers the most wear. Only in very light-weight applications such as a bicycle, or in extreme cases of improper tension, will the chain normally jump off the sprockets.
{\displaystyle \%=((M-(S*P))/(S*P))*100}
In industry, it is usual to monitor the movement of the chain tensioner (whether manual or automatic) or the exact length of a drive chain (one rule of thumb is to replace a roller chain which has elongated 3% on an adjustable drive or 1.5% on a fixed-center drive). A simpler method, particularly suitable for the cycle or motorcycle user, is to attempt to pull the chain away from the larger of the two sprockets. Any significant movement (e.g. making it possible to see through a gap) probably indicates a chain worn up to and beyond the limit. Sprocket damage will result if the problem is ignored.
Bicycle chain wear
The standard minimum ultimate strength of the ANSI 29.1 steel chain is 12,500 x (pitch, in inches)2. Internal Lubrication X-ring and O-Ring Type chains greatly decrease wear by means of keeping in internal lubrication increasing chain life. The internal lubrication is inserted by means of a vacuum when riveting the chain together
Standards organizations (such as ANSI) maintain standards for design, dimensions, and interchangeability of transmission chains. For example, the following Table shows data from ANSI standard B29.1-2011 (Precision Power Transmission Roller Chains, Attachments, and Sprockets)[6] developed by the American Society of Mechanical Engineers (ASME). See the references[7][8][9] for additional information.
4. An "H" following the standard number denotes heavyweight chain. A hyphenated number following the standard number denotes double-strand (2), triple-strand (3), and so on. Thus 60H-3 denotes number 60 heavyweight triple-strand chain.
A typical bicycle chain (for derailleur gears) uses narrow 1/2" pitch chain. The width of the chain is variable, and does not affect the load capacity. The more sprockets at the rear wheel (historically 3-6, nowadays 7-12 sprockets), the narrower the chain. Chains are sold according to the number of speeds they are designed to work with, for example, "10 speed chain". Hub gear or single speed bicycles use 1/2" x 1/8" chains, where 1/8" refers to the maximum thickness of a sprocket that can be used with the chain.
Typically chains with parallel shaped links have an even number of links, with each narrow link followed by a broad one. Chains built up with a uniform type of link, narrow at one and broad at the other end, can be made with an uneven number of links, which can be an advantage to adapt to a special chainwheel-distance, on the other side such a chain tends to be not so strong.
^ Chains operating at high speeds comparable to those on motorcycles should be used in conjunction with an oil bath, according to: Lubrecht, A. and Dalmaz, G., (eds.) Transients Processes in Tribology, Proc 30th Leeds-Lyon Symposium on Tribology. 30th Leeds-Lyon Symposium on Tribology, 2–5 September 2003, Lyon. Tribology and Interface Engineering Series (43). Elsevier, Amsterdam, pp. 291-298.
^ Oil drip feed provided the greatest wear protection between chain roller and pin, Oil drip feed provided the greatest power saving over unlubricated chains and sprockets, according to Lee, P.M. and Priest, M. (2004) An innovation integrated approach to testing motorcycle drive chain lubricants. In: Lubrecht, A. and Dalmaz, G., (eds.) Transients Processes in Tribology, Proc 30th Leeds-Lyon Symposium on Tribology. 30th Leeds-Lyon Symposium on Tribology, 2–5 September 2003, Lyon. Tribology and Interface Engineering Series (43). Elsevier, Amsterdam, pp. 291-298.
^ Green 1996, pp. 2337–2361
Chain In Depth
기계공학Lifting hookChain conveyorProduction line
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Bayesian table tennis: Building a hierarchical table tennis ranking model | Kinaxis
Bayesian table tennis: Building a hierarchical table tennis ranking model
Here on the development team, we are big fans of iterative development for our software, for our ML models and especially for our office table tennis ranking system! In the past, our team wrote about Building A Table Tennis Ranking Model using the Bradley-Terry model and Google sheets. The model gives a single rating for each player (not unlike the Elo rating system in Chess). One drawback of this model is that it has no measure of “confidence”. For example, a new player could have a single lucky game against the reigning champion and suddenly soar to the top of the rankings, but should we be that confident in their ranking? Enter Bayesian Hierarchical Modelling.
Many of the parametric models you know and love (such as linear regression or MLPs), use a maximum likelihood estimate (MLE) or maximum a posterior (MAP) to estimate the parameters \theta
of a model with observations x
θ̂ MLEθ̂ MAP=argmaxθ∈Θ{(θ,x)}=argmaxθ∈Θ{(θ,x)+g(θ)}(1)
where (θ,x)
\mathcal{L}(\theta, x)
is the log-likelihood function and regularizer (or prior) g(\theta)
on the parameters. In the case of our Bradley-Terry model, the parameters \hat{\theta}
are the estimates for our players’ ratings. The nice thing about this formulation is that we can apply your favorite optimization technique (e.g. gradient descent) to find the parameters that maximize Equation 1. The downside is that we only get a point estimate of the parameters, there’s no concept of how “confident” we are in the estimates. Or another way to put it: there’s no measure of uncertainty.
Bayesian hierarchical modelling takes a slightly different philosophical approach. Every parameter of interest in the model is a random variable with its own distribution. This means we can ask our usual questions of a random variable such as: what’s the mean? median? standard deviation? P(a<θ<b)
P(a < \theta < b)
? With a distribution for each parameter, we can now quantify the uncertainty of our estimate in any way we wish. Cool right?
Bayesian modelling always starts off with a statement of Bayes theorem describing the relationship between the parameters of interest and the data:
P(θ|x)=P(x|θ)P(θ)P(x)=likelihood⋅priormarginal likelihood(2)
The likelihood and prior (aka regularizer) from our MAP estimate above are included there along with a constant factor in the denominator. The MAP estimate attempts to find a point-estimate that maximizes the RHS of Equation 2, while a full Bayesian analysis attempts to find the distribution specified by \theta
. The biggest downside of doing a fully Bayesian analysis is that it’s hard to compute. In general, there is no closed form solution and we must rely on approximation methods such as variational methods or MCMC methods that generate samples from the distribution.
Once we have a distribution for each parameter, we can generate the credible interval (or region for multi-variate distributions) which involves finding an interval (a,b)
such that P(a<θ<b)=1−α
P(a<\theta<b)=1-\alpha
. For example, we might want to find a 50% credible interval for each player’s rating.
A Bayesian ranking model
To define our Bayesian hierarchical model, we need to specify the likelihood and prior functions from Equation 2 (the marginal likelihood is a constant so we don’t need to specify it). We’re going to follow the Bradley-Terry model, where we assume that the probability of player i
beating player j
P(i>j)=pipi+pj(3)
where pi
p_i
is the rating for player i
. With this idea, we can define our likelihood function by modelling with player i
having wi,j
w_{i,j}
wins against player j
in Ni,j
N_{i,j}
games as a binomial distribution. Additionally, we’re going to take the easy route and assume a flat prior with support [0,1]
. Putting it all together with observations x
(all the wins and loses):
P(x|p1,…,pK)P(p1,…,pK)=∑i=1N∑j=1i(Ni,jwi,j)(pipi+pj)wi,j(1−pipi+pj)Ni,j−wi,j =∑i=1Nconst (4)
In this case, we’re assuming that each player’s rating has equal probability of being any possible rating in [0,1]
. In certain cases, this might cause some issues fitting the model (especially if we’re trying to compute a MAP estimate) but we’ll be using the data regularizing scheme from our original Rubikloud post Building A Table Tennis Ranking Model. The added benefit here is that the MAP estimate for the models are identical.
In recent years, Bayesian statistical packages have come a long way. There are simple interfaces with heavily optimized engines to fit Bayesian models. The most popular Bayesian statistics package is Stan, but we ended up using PyMC3 because we like the native Python interface. For a simple model like this, there’s not much difference in performance so usability is the main concern.
The nice thing about these packages is that you can take Equation 4 and basically translate it directly, here’s a code snippet of our model in PyMC3:
The official Rubikloud/Kinaxis table tennis rankings V2
We're happy to announce that we've since moved on to V2 of our table tennis rankings. To simplify the UI, instead of plotting each individual’s distribution, we simply show each person’s 50% credible interval centered on the MAP estimate:
The scores are put on a logarithmic scale and scaled from 1 to 1000. Visually it looks like this:
As you can see from this past example, some players had a very wide range while others had a very tight range. This shows the “confidence” the model has in the rating. Since this time, the changes were well-received, with players consistently showing more interest in trying to move up the rankings.
Editor's note: This blog is part of a series originally published on Rubikloud's blog, "kernel." Kinaxis acquired Rubikloud in 2020.
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§ Convergence in distribution is very weak
X \sim N(0, 1)
. Also consider
-X
which will be identically distributed (by symmetry of
-
N
-X \sim N(0, 1)
But this tells us nothing about
X
-x
! so this type of "convergence of distribution" is very weak.
Strongest notion of convergence (#2): Almost surely.
T_n \xrightarrow{a.s} T
P(\{ \omega : T_n(\omega) \to T(\omega) \}) = 1
. Consider a snowball left out in the sun. In a couple hours, It'll have a random shape, random volume, and so on. But the ball itself is a definite thing --- the
\omega
. Almost sure says that for almost all of the balls,
T_n
T
#2 notion of convergence: Convergence in probability.
T_n \xrightarrow{P} T
P(|T_n - T| \geq \epsilon) \xrightarrow{n \to \infty} 0
\epsilon > 0
. This allows us to squeeze
\epsilon
probability under the rug.
Convergence in
L^p
T_n \xrightarrow{L^p} T
E[|T_n - T|^p] \xrightarrow{n \to \infty} 0
. Eg. think of convergence in variance of a gaussian.
Convergence in distrbution: (weakest):
T_n \xrightarrow{d} T
P[T_n \leq x] \xrightarrow{n \to \infty} P[T \leq x]
x
§ Characterization of convergence in distribution
T_n \xrightarrow{d} T
(2) For all
f
continuous and bounded, we have
E[f(T_n)] \xrightarrow{n \to \infty} E[f(T)]
E[e^{ixT_n}] \xrightarrow{n \to \infty} E[e^{ixT}]
. [characteristic function converges ].
§ Strength of different types of convergence
Almost surely convergence implies convergence in probability. Also, the two limits (which are RVs) are almost surely equal.
L^p
implies convergence in probability and convergence in
L^q
q \leq p
. Also, the limits (which are RVs) are almost surely equal.
T
converges in probability, it also converges in distribution (meaning the two sequences will have the same DISTRIBUTION, not same RV).
All of almost surely, probabilistic convergence, convergence in distribution (not
L^p
) map properly by continuous fns.
T_n \to T
f(T_n) \to f(T)
almost surely implies P implies distribution convergence.
§ Slutsky's Theorem
X_n \xrightarrow{d} X
Y_n \xrightarrow{P} c
(That is, the sequence of
Y_n
is eventually deterministic),we then have that
(X_n, Y) \xrightarrow{d} (X, c)
. In particular, we get that
X_n + Y_n \xrightarrow{d} X + c
X_n Y_n \xrightarrow{d} X c
This is important, because in general, convergence in distribution says nothing about the RV! but in this special case, it's possible.
MIT OCW stats
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Hasinur Is Giving Treat | Toph
Hasinur Is Giving Treat
By Zeronfinity · Limits 4s, 512 MB
Hasinur wants to give treat. Why? Because, he's been going out with someone new. But Hasinur doesn't want to give treat to everyone. He wants to treat only those who can answer some easy questions for him.
Hasinur will ask you N questions in total. In each question, he will give you a pair of numbers L and R. You will have to answer the sum of factorials of all numbers from L to R, modulo 1015+37 (1000000000000037). Nothing that hard 🙂.
Formally, given M and some queries in the form of L and R, you have to print the following.
\sum_{n=L}^R n! modulo 1000000000000037
∑n=LRn!modulo1000000000000037
Here, n! means the factorial of n, i.e.
n! = 1 \times 2 \times 3 \times ... \times (n-1) \times n
n!=1×2×3×...×(n−1)×n
Input starts with two integers N (1 ≤ N ≤ 100), the number of questions.
In the next N lines, each line contains the two integers referring to a question, L (0 ≤ L ≤ 107) and R (L ≤ R ≤ 107).
For each question, print the answer modulo 1015+37 in a single line.
aminulEarliest, Nov '18
user.l4zfn7bxLightest, 131 kB
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Canonical bundle - Wikipedia
In mathematics, the canonical bundle of a non-singular algebraic variety
{\displaystyle V}
of dimensio{\displaystyle n}
over a field is the line bundle
{\displaystyle \,\!\Omega ^{n}=\omega }
, which is the nth exterior power of the cotangent bundle Ω on V.
Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V. This is the dualising object for Serre duality on V. It may equally well be considered as an invertible sheaf.
The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical.
The anticanonical bundle is the corresponding inverse bundle ω−1. When the anticanonical bundle of V is ample, V is called a Fano variety.
1 The adjunction formula
2 Singular case
3 Canonical maps
3.1 Canonical curves
3.1.1 Low genus
3.1.2 Hyperelliptic case
3.2 Canonical rings
The adjunction formulaEdit
Main article: Adjunction formula
Suppose that X is a smooth variety and that D is a smooth divisor on X. The adjunction formula relates the canonical bundles of X and D. It is a natural isomorphism
{\displaystyle \omega _{D}=i^{*}(\omega _{X}\otimes {\mathcal {O}}(D)).}
In terms of canonical classes, it is
{\displaystyle K_{D}=(K_{X}+D)|_{D}.}
This formula is one of the most powerful formulas in algebraic geometry. An important tool of modern birational geometry is inversion of adjunction, which allows one to deduce results about the singularities of X from the singularities of D.
Singular caseEdit
On a singular variety
{\displaystyle X}
, there are several ways to define the canonical divisor. If the variety is normal, it is smooth in codimension one. In particular, we can define canonical divisor on the smooth locus. This gives us a unique Weil divisor class on
{\displaystyle X}
. It is this class, denoted by
{\displaystyle K_{X}}
that is referred to as the canonical divisor on
{\displaystyle X.}
Alternately, again on a normal variety
{\displaystyle X}
, one can consider
{\displaystyle h^{-d}(\omega _{X}^{.})}
{\displaystyle -d}
'th cohomology of the normalized dualizing complex of
{\displaystyle X}
. This sheaf corresponds to a Weil divisor class, which is equal to the divisor class
{\displaystyle K_{X}}
defined above. In the absence of the normality hypothesis, the same result holds if
{\displaystyle X}
is S2 and Gorenstein in dimension one.
Canonical mapsEdit
If the canonical class is effective, then it determines a rational map from V into projective space. This map is called the canonical map. The rational map determined by the nth multiple of the canonical class is the n-canonical map. The n-canonical map sends V into a projective space of dimension one less than the dimension of the global sections of the nth multiple of the canonical class. n-canonical maps may have base points, meaning that they are not defined everywhere (i.e., they may not be a morphism of varieties). They may have positive dimensional fibers, and even if they have zero-dimensional fibers, they need not be local analytic isomorphisms.
Canonical curvesEdit
The best studied case is that of curves. Here, the canonical bundle is the same as the (holomorphic) cotangent bundle. A global section of the canonical bundle is therefore the same as an everywhere-regular differential form. Classically, these were called differentials of the first kind. The degree of the canonical class is 2g − 2 for a curve of genus g.[1]
Low genusEdit
Suppose that C is a smooth algebraic curve of genus g. If g is zero, then C is P1, and the canonical class is the class of −2P, where P is any point of C. This follows from the calculus formula d(1/t) = −dt/t2, for example, a meromorphic differential with double pole at the point at infinity on the Riemann sphere. In particular, KC and its multiples are not effective. If g is one, then C is an elliptic curve, and KC is the trivial bundle. The global sections of the trivial bundle form a one-dimensional vector space, so the n-canonical map for any n is the map to a point.
Hyperelliptic caseEdit
If C has genus two or more, then the canonical class is big, so the image of any n-canonical map is a curve. The image of the 1-canonical map is called a canonical curve. A canonical curve of genus g always sits in a projective space of dimension g − 1.[2] When C is a hyperelliptic curve, the canonical curve is a rational normal curve, and C a double cover of its canonical curve. For example if P is a polynomial of degree 6 (without repeated roots) then
y2 = P(x)
is an affine curve representation of a genus 2 curve, necessarily hyperelliptic, and a basis of the differentials of the first kind is given in the same notation by
dx/√P(x), x dx/√P(x).
This means that the canonical map is given by homogeneous coordinates [1: x] as a morphism to the projective line. The rational normal curve for higher genus hyperelliptic curves arises in the same way with higher power monomials in x.
Otherwise, for non-hyperelliptic C which means g is at least 3, the morphism is an isomorphism of C with its image, which has degree 2g − 2. Thus for g = 3 the canonical curves (non-hyperelliptic case) are quartic plane curves. All non-singular plane quartics arise in this way. There is explicit information for the case g = 4, when a canonical curve is an intersection of a quadric and a cubic surface; and for g = 5 when it is an intersection of three quadrics.[2] There is a converse, which is a corollary to the Riemann–Roch theorem: a non-singular curve C of genus g embedded in projective space of dimension g − 1 as a linearly normal curve of degree 2g − 2 is a canonical curve, provided its linear span is the whole space. In fact the relationship between canonical curves C (in the non-hyperelliptic case of g at least 3), Riemann-Roch, and the theory of special divisors is rather close. Effective divisors D on C consisting of distinct points have a linear span in the canonical embedding with dimension directly related to that of the linear system in which they move; and with some more discussion this applies also to the case of points with multiplicities.[3][4]
More refined information is available, for larger values of g, but in these cases canonical curves are not generally complete intersections, and the description requires more consideration of commutative algebra. The field started with Max Noether's theorem: the dimension of the space of quadrics passing through C as embedded as canonical curve is (g − 2)(g − 3)/2.[5] Petri's theorem, often cited under this name and published in 1923 by Karl Petri (1881–1955), states that for g at least 4 the homogeneous ideal defining the canonical curve is generated by its elements of degree 2, except for the cases of (a) trigonal curves and (b) non-singular plane quintics when g = 6. In the exceptional cases, the ideal is generated by the elements of degrees 2 and 3. Historically speaking, this result was largely known before Petri, and has been called the theorem of Babbage-Chisini-Enriques (for Dennis Babbage who completed the proof, Oscar Chisini and Federigo Enriques). The terminology is confused, since the result is also called the Noether–Enriques theorem. Outside the hyperelliptic cases, Noether proved that (in modern language) the canonical bundle is normally generated: the symmetric powers of the space of sections of the canonical bundle map onto the sections of its tensor powers.[6][7] This implies for instance the generation of the quadratic differentials on such curves by the differentials of the first kind; and this has consequences for the local Torelli theorem.[8] Petri's work actually provided explicit quadratic and cubic generators of the ideal, showing that apart from the exceptions the cubics could be expressed in terms of the quadratics. In the exceptional cases the intersection of the quadrics through the canonical curve is respectively a ruled surface and a Veronese surface.
These classical results were proved over the complex numbers, but modern discussion shows that the techniques work over fields of any characteristic.[9]
Canonical ringsEdit
Main article: Canonical ring
The canonical ring of V is the graded ring
{\displaystyle R=\bigoplus _{d=0}^{\infty }H^{0}(V,K_{V}^{d}).}
If the canonical class of V is an ample line bundle, then the canonical ring is the homogeneous coordinate ring of the image of the canonical map. This can be true even when the canonical class of V is not ample. For instance, if V is a hyperelliptic curve, then the canonical ring is again the homogeneous coordinate ring of the image of the canonical map. In general, if the ring above is finitely generated, then it is elementary to see that it is the homogeneous coordinate ring of the image of a k-canonical map, where k is any sufficiently divisible positive integer.
The minimal model program proposed that the canonical ring of every smooth or mildly singular projective variety was finitely generated. In particular, this was known to imply the existence of a canonical model, a particular birational model of V with mild singularities that could be constructed by blowing down V. When the canonical ring is finitely generated, the canonical model is Proj of the canonical ring. If the canonical ring is not finitely generated, then Proj R is not a variety, and so it cannot be birational to V; in particular, V admits no canonical model.
A fundamental theorem of Birkar-Cascini-Hacon-McKernan from 2006[10] is that the canonical ring of a smooth or mildly singular projective algebraic variety is finitely generated.
The Kodaira dimension of V is the dimension of the canonical ring minus one. Here the dimension of the canonical ring may be taken to mean Krull dimension or transcendence degree.
^ "canonical class", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
^ a b Parshin, A. N. (2001) [1994], "Canonical curve", Encyclopedia of Mathematics, EMS Press
^ "Geometric Form of Riemann-Roch | Rigorous Trivialities".
^ Rick Miranda, Algebraic Curves and Riemann Surfaces (1995), Ch. VII.
^ David Eisenbud, The Geometry of Syzygies (2005), p. 181-2.
^ Iskovskih, V. A. (2001) [1994], "Noether–Enriques theorem", Encyclopedia of Mathematics, EMS Press
^ Igor Rostislavovich Shafarevich, Algebraic geometry I (1994), p. 192.
^ "Torelli theorems", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
^ http://hal.archives-ouvertes.fr/docs/00/40/42/57/PDF/these-OD.pdf, pp. 11-13.
^ http://www.birs.ca/birspages.php?task=displayevent&event_id=09w5033
Retrieved from "https://en.wikipedia.org/w/index.php?title=Canonical_bundle&oldid=1065964424"
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CBC-MAC - Wikipedia
To calculate the CBC-MAC of message m, one encrypts m in CBC mode with zero initialization vector and keeps the last block. The following figure sketches the computation of the CBC-MAC of a message comprising blocks
{\displaystyle m_{1}\|m_{2}\|\cdots \|m_{x}}
using a secret key k and a block cipher E:
1 Security with fixed and variable-length messages
1.1 Length prepending
1.2 Encrypt-last-block
2.1 Using the same key for encryption and authentication
2.2 Allowing the initialization vector to vary in value
2.3 Using predictable initialization vector
3 Standards that define the algorithm
Security with fixed and variable-length messages[edit]
If the block cipher used is secure (meaning that it is a pseudorandom permutation), then CBC-MAC is secure for fixed-length messages.[1] However, by itself, it is not secure for variable-length messages. Thus, any single key must only be used for messages of a fixed and known length. This is because an attacker who knows the correct message-tag (i.e. CBC-MAC) pairs for two messages
{\displaystyle (m,t)}
{\displaystyle (m',t')}
can generate a third message
{\displaystyle m''}
whose CBC-MAC will also be
{\displaystyle t'}
. This is simply done by XORing the first block of
{\displaystyle m'}
with t and then concatenating m with this modified
{\displaystyle m'}
; i.e., by making
{\displaystyle m''=m\|[(m_{1}'\oplus t)\|m_{2}'\|\dots \|m_{x}']}
. When computing the MAC for the message
{\displaystyle m''}
, it follows that we compute the MAC for m in the usual manner as t, but when this value is chained forwards to the stage computing
{\displaystyle E_{K_{\text{MAC}}}(m_{1}'\oplus t)}
we will perform an exclusive OR operation with the value derived for the MAC of the first message. The presence of that tag in the new message means it will cancel, leaving no contribution to the MAC from the blocks of plain text in the first message m:
{\displaystyle E_{K_{\text{MAC}}}(m_{1}'\oplus t\oplus t)=E_{K_{\text{MAC}}}(m_{1}')}
and thus the tag for
{\displaystyle m''}
{\displaystyle t'}
This problem cannot be solved by adding a message-size block to the end.[2] There are three main ways of modifying CBC-MAC so that it is secure for variable length messages: 1) Input-length key separation; 2) Length-prepending; 3) Encrypt last block.[2] In such a case, it may also be recommended to use a different mode of operation, for example, CMAC or HMAC to protect the integrity of variable-length messages.
Length prepending[edit]
One solution is to include the length of the message in the first block;[3] in fact CBC-MAC has been proven secure as long as no two messages that are prefixes of each other are ever used and prepending the length is a special case of this.[4] This can be problematic if the message length may not be known when processing begins.
Encrypt-last-block[edit]
Encrypt-last-block CBC-MAC (ECBC-MAC)[5] is defined as CBC-MAC-ELB(m, (k1, k2)) = E(k2, CBC-MAC(k1, m)).[2] Compared to the other discussed methods of extending CBC-MAC to variable-length messages, encrypt-last-block has the advantage of not needing to know the length of the message until the end of the computation.
Computation of CBC-MAC Encrypt-last-block.
Attack methods[edit]
As with many cryptographic schemes, naïve use of ciphers and other protocols may lead to attacks being possible, reducing the effectiveness of the cryptographic protection (or even rendering it useless). We present attacks which are possible due to using the CBC-MAC incorrectly.[6]
Using the same key for encryption and authentication[edit]
One common mistake is to reuse the same key k for CBC encryption and CBC-MAC. Although a reuse of a key for different purposes is a bad practice in general, in this particular case the mistake leads to a spectacular attack:
Suppose Alice has sent to Bob the cipher text blocks
{\displaystyle C=C_{1}\|C_{2}\|\dots \|C_{n}}
. During the transmission process, Eve can tamper with any of the
{\displaystyle C_{1},\dots ,C_{n-1}}
cipher-text blocks and adjust any of the bits therein as she chooses, provided that the final block,
{\displaystyle C_{n}}
, remains the same. We assume, for the purposes of this example and without loss of generality, that the initialization vector used for the encryption process is a vector of zeroes.
When Bob receives the message, he will first decrypt the message by reversing the encryption process which Alice applied, using the cipher text blocks
{\displaystyle C=C_{1}\|C_{2}\|\cdots \|C_{n}}
. The tampered message, delivered to Bob in replacement of Alice's original, is
{\displaystyle C'=C_{1}'\|\dots \|C_{n-1}'\|C_{n}}
Bob first decrypts the message received using the shared secret key K to obtain corresponding plain text. Note that all plain text produced will be different from that which Alice originally sent, because Eve has modified all but the last cipher text block. In particular, the final plain text,
{\displaystyle P_{n}'}
, differs from the original,
{\displaystyle P_{n}}
, which Alice sent; although
{\displaystyle C_{n}}
is the same,
{\displaystyle C_{n-1}'\not =C_{n-1}}
, so a different plain text
{\displaystyle P_{n}'}
is produced when chaining the previous cipher text block into the exclusive-OR after decryption of
{\displaystyle C_{n}}
{\displaystyle P_{n}'=C_{n-1}'\oplus E_{K}^{-1}(C_{n})}
It follows that Bob will now compute the authentication tag using CBC-MAC over all the values of plain text which he decoded. The tag for the new message,
{\displaystyle t'}
{\displaystyle t'=E_{K}(P_{n}'\oplus E_{K}(P_{n-1}'\oplus E_{K}(\dots \oplus E_{K}(P_{1}'))))}
Notice that this expression is equal to
{\displaystyle t'=E_{K}(P_{n}'\oplus C_{n-1}')}
{\displaystyle C_{n}}
{\displaystyle t'=E_{K}(C_{n-1}'\oplus E_{K}^{-1}(C_{n})\oplus C_{n-1}')=E_{K}(E_{K}^{-1}(C_{n}))=C_{n}}
{\displaystyle t'=C_{n}=t}
Therefore, Eve was able to modify the cipher text in transit (without necessarily knowing what plain text it corresponds to) such that an entirely different message,
{\displaystyle P'}
, was produced, but the tag for this message matched the tag of the original, and Bob was unaware that the contents had been modified in transit. By definition, a Message Authentication Code is broken if we can find a different message (a sequence of plain-text pairs
{\displaystyle P'}
) which produces the same tag as the previous message, P, with
{\displaystyle P\not =P'}
. It follows that the message authentication protocol, in this usage scenario, has been broken, and Bob has been deceived into believing Alice sent him a message which she did not produce.
If, instead, we use different keys for the encryption and authentication stages, say
{\displaystyle K_{1}}
{\displaystyle K_{2}}
, respectively, this attack is foiled. The decryption of the modified cipher-text blocks
{\displaystyle C_{i}'}
obtains some plain text string
{\displaystyle P_{i}'}
. However, due to the MAC's usage of a different key
{\displaystyle K_{2}}
, we cannot "undo" the decryption process in the forward step of the computation of the message authentication code so as to produce the same tag; each modified
{\displaystyle P_{i}'}
will now be encrypted by
{\displaystyle K_{2}}
in the CBC-MAC process to some value
{\displaystyle \mathrm {MAC} _{i}\not =C_{i}'}
This example also shows that a CBC-MAC cannot be used as a collision-resistant one-way function: given a key it is trivial to create a different message which "hashes" to the same tag.
Allowing the initialization vector to vary in value[edit]
When encrypting data using a block cipher in cipher block chaining (or another) mode, it is common to introduce an initialization vector to the first stage of the encryption process. It is typically required that this vector be chosen randomly (a nonce) and that it is not repeated for any given secret key under which the block cipher operates. This provides semantic security, by means of ensuring the same plain text is not encrypted to the same cipher text, allowing an attacker to infer a relationship exists.
When computing a message authentication code, such as by CBC-MAC, the use of an initialization vector is a possible attack vector.
In the operation of a ciphertext block chaining cipher, the first block of plain text is mixed with the initialization vector using an exclusive OR (
{\displaystyle P_{1}\oplus IV}
). The result of this operation is the input to the block cipher for encryption.
However, when performing encryption and decryption, we are required to send the initialization vector in plain text - typically as the block immediately preceding the first block of cipher text - such that the first block of plain text can be decrypted and recovered successfully. If computing a MAC, we will also need to transmit the initialization vector to the other party in plain text so that they can verify the tag on the message matches the value they have computed.
If we allow the initialization vector to be selected arbitrarily, it follows that the first block of plain text can potentially be modified (transmitting a different message) while producing the same message tag.
Consider a message
{\displaystyle M_{1}=P_{1}|P_{2}|\dots }
. In particular, when computing the message tag for CBC-MAC, suppose we choose an initialization vector
{\displaystyle IV_{1}}
such that computation of the MAC begins with
{\displaystyle E_{K}(IV_{1}\oplus P_{1})}
. This produces a (message, tag) pair
{\displaystyle (M_{1},T_{1})}
Now produce the message
{\displaystyle M_{2}=P_{1}'|P_{2}|\dots }
. For each bit modified in
{\displaystyle P_{1}'}
, flip the corresponding bit in the initialization vector to produce the initialization vector
{\displaystyle IV_{1}'}
. It follows that to compute the MAC for this message, we begin the computation by
{\displaystyle E_{K}(P_{1}'\oplus IV_{1}')}
. As bits in both the plain text and initialization vector have been flipped in the same places, the modification is cancelled in this first stage, meaning the input to the block cipher is identical to that for
{\displaystyle M_{1}}
. If no further changes are made to the plain text, the same tag will be derived despite a different message being transmitted.
If the freedom to select an initialization vector is removed and all implementations of CBC-MAC fix themselves on a particular initialization vector (often the vector of zeroes, but in theory, it could be anything provided all implementations agree), this attack cannot proceed.
To sum up, if the attacker is able to set the IV that will be used for MAC verification, he can perform arbitrary modification of the first data block without invalidating the MAC.
Using predictable initialization vector[edit]
Sometimes IV is used as a counter to prevent message replay attacks. However, if the attacker can predict what IV will be used for MAC verification, he or she can replay previously observed message by modifying the first data block to compensate for the change in the IV that will be used for the verification. For example, if the attacker has observed message
{\displaystyle M_{1}=P_{1}|P_{2}|\dots }
{\displaystyle IV_{1}}
and knows
{\displaystyle IV_{2}}
, he can produce
{\displaystyle M_{1}'=(P_{1}\oplus IV_{1}\oplus IV_{2})|P_{2}|\dots }
that will pass MAC verification with
{\displaystyle IV_{2}}
The simplest countermeasure is to encrypt the IV before using it (i.e., prepending IV to the data). Alternatively MAC in CFB mode can be used, because in CFB mode the IV is encrypted before it is XORed with the data.
Another solution (in case protection against message replay attacks is not required) is to always use a zero vector IV.[7] Note that the above formula for
{\displaystyle M_{1}'}
{\displaystyle M_{1}'=(P_{1}\oplus 0\oplus 0)|P_{2}|\dots =P_{1}|P_{2}|\dots =M_{1}}
. So since
{\displaystyle M_{1}}
{\displaystyle M_{1}'}
are the same message, by definition they will have the same tag. This is not a forgery, rather the intended use of CBC-MAC.
Standards that define the algorithm[edit]
FIPS PUB 113 Computer Data Authentication is a (now obsolete) U.S. government standard that specified the CBC-MAC algorithm using DES as the block cipher.
The CBC-MAC algorithm is equivalent to ISO/IEC 9797-1 MAC Algorithm 1.
CMAC – A block-cipher–based MAC algorithm which is secure for messages of different lengths (recommended by NIST).
OMAC and PMAC – Other methods to turn block ciphers into message authentication codes (MACs).
One-way compression function – Hash functions can be made from block ciphers. But note, there are significant differences in function and uses for security between MACs (such as CBC-MAC) and hashes.
^ M. Bellare, J. Kilian and P. Rogaway. The security of the cipher block chaining message authentication code. JCSS 61(3):362–399, 2000.
^ a b c See Section 5 of Bellare, et al.
^ ISO/IEC 9797-1:1999 Information technology – Security techniques – Message Authentication Codes (MACs) – Part 1: Mechanisms using a block cipher, clause 6.1.3 Padding Method 3
^ C. Rackoff and S. Gorbunov. On the Security of Block Chaining Message Authentication Code.
^ http://spark-university.s3.amazonaws.com/stanford-crypto/slides/05.3-integrity-cbc-mac-and-nmac.pptx
^ Why I hate CBC-MAC by Matthew D. Green
^ Introduction to Modern Cryptography, Second Edition by Jonathan Katz and Yehuda Lindell
Retrieved from "https://en.wikipedia.org/w/index.php?title=CBC-MAC&oldid=1084908376"
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Calculate the value of a performance metric and its sensitivity to the diagonal weights of an MPC controller - MATLAB sensitivity - MathWorks Deutschland
Calculate Value of Predefined Performance Metric and its Sensitivity to Controller Weights
Calculate Value of Custom Performance Metric and its Sensitivity to Controller Weights
PerfFcn
customPerFcn
Par1,...,ParN
Calculate the value of a performance metric and its sensitivity to the diagonal weights of an MPC controller
[J,sens] = sensitivity(MPCobj,PerfFcn,PerfWeights,Ns,r,v,SimOptions,utarget)
[J,sens] = sensitivity(MPCobj,customPerFcn,Par1,...,ParN)
[J,sens] = sensitivity(MPCobj,PerfFcn,PerfWeights,Ns,r,v,SimOptions,utarget) calculates the value J and sensitivity sens of a predefined closed-loop, cumulative performance metric with respect to the diagonal weights defined in the MPC controller object MPCobj. You chose the shape of the performance metric, among the available options, using PerfFcn. The optional arguments PerfWeights, Ns, r, v, SimOptions, and utarget specify the performance metric weights, simulation steps, reference and disturbance signals, simulation options, and manipulated variables targets, respectively.
[J,sens] = sensitivity(MPCobj,customPerFcn,Par1,...,ParN) calculates the value J and sensitivity sens of the performance metric defined in the custom function customPerFcn, with respect to the diagonal weights defined in the MPC controller object MPCobj. The remaining input arguments Par1,Par2,...,ParN specify the value of the parameters needed by customPerFnc.
Fix the random number generator seed for reproducibility.
Define a third-order plant model with three manipulated variables and two controlled outputs. Then create an MPC controller for the plant, with sample time of 1.
Specify an integral absolute error performance function and set the performance weights. The performance weights emphasize tracking the first output variable.
PerfFunc = 'IAE';
PerfWts.OutputVariables = [2 0.5];
PerfWts.ManipulatedVariables = zeros(1,3);
PerfWts.ManipulatedVariablesRate = zeros(1,3);
Define a 20 second simulation scenario with a unit step as setpoint for the first output and zero as a setpoint for the second output.
r = [1 0];
Calculate the closed-loop performance metric, J, and its sensitivities, sens, to the weights defined in mpcobj, for the specified simulation scenario. For this example, do not specify the last three input argument of sensitivity. This means that no disturbance signal or simulation option is used and the nominal value of the manipulated variables is kept to its default value of zero.
[J,sens] = sensitivity(mpcobj,PerfFunc,PerfWts,Tstop,r)
sens = struct with fields:
OutputVariables: [0.0029 -0.1574]
ManipulatedVariables: [0.0621 -0.1254 0.0989]
ManipulatedVariablesRate: [0.5294 -0.3597 1.3742]
The positive, and relatively higher, values of the sensitivities to the first and last manipulated variable rate suggest that decreasing the corresponding weights defined in mpcobj would contribute the most to decrease the IAE performance metric defined by PerfWts. At the same time, since the sensitivity to the weight of the second manipulated variable is negative, increasing the corresponding weight would also contribute to decrease the performance metric.
Modify the manipulated variable rate weights in mpcobj and recalculate the value of the performance metric.
mpcobj.Weights.ManipulatedVariablesRate = [1e-2 1 1e-2];
sensitivity(mpcobj,PerfFunc,PerfWts,Tstop,r)
As expected the value of the performance metric decreased, indicating an improved tracking performance.
Define a custom performance function and write it to a file. The function must take an MPC object as a first input argument. The simulation time and the output set point are the second and third input arguments, respectively. Internally, the function performs a closed loop simulation using the given MPC object, simulation time and set point. The norm of the difference between the set point and the output signal is then returned as the value of the performance metric (note that this norm depends on the number of simulation steps).
% write a function to the char vector "str"
str = ['function J = mypfun(mpcobj,T,ySetPnt)', ...
newline, ...
'y = sim(mpcobj,T,ySetPnt); J = norm(ySetPnt-y);', ...
'end'];
% create the function file
fid=fopen('mypfun.m','w'); % open a file for writing
fwrite(fid,str,'char'); % write "str" to the file
fclose(fid); % close the file
Calculate the custom performance metric, J, and its sensitivities, sens, to the weights defined in mpcobj, using a simulation time of 10 seconds and an output setpoint of [1 1].
[J,sens] = sensitivity(mpcobj,'mypfun',10,[1 1])
The comparatively higher values of the sensitivities to the manipulated variable rates suggest that decreasing the corresponding weights defined in mpcobj would contribute the most to decrease the custom performance metric calculated in the function mypfun.
PerfFcn — Performance metric function shape
'ISE' | 'IAE' | 'ITSE' | 'ITAE'
Performance metric function shape, specified as one of the following:
'ISE' (integral squared error), for which the performance metric is
J=\sum _{i=1}^{Ns}\left(\sum _{j=1}^{{n}_{y}}{\left({w}_{j}^{y}{e}_{yij}\right)}^{2}+\sum _{j=1}^{{n}_{u}}\left[{\left({w}_{j}^{u}{e}_{uij}\right)}^{2}+{\left({w}_{j}^{\Delta u}\Delta {u}_{ij}\right)}^{2}\right]\right)
'IAE' (integral absolute error), for which the performance metric is
J=\sum _{i=1}^{Ns}\left(\sum _{j=1}^{{n}_{y}}|{w}_{j}^{y}{e}_{yij}|+\sum _{j=1}^{{n}_{u}}\left(|{w}_{j}^{u}{e}_{uij}|+|{w}_{j}^{\Delta u}\Delta {u}_{ij}|\right)\right)
'ITSE' (integral of time-weighted squared error), for which the performance metric is
J=\sum _{i=1}^{Ns}i\Delta t\left(\sum _{j=1}^{{n}_{y}}{\left({w}_{j}^{y}{e}_{yij}\right)}^{2}+\sum _{j=1}^{{n}_{u}}\left[{\left({w}_{j}^{u}{e}_{uij}\right)}^{2}+{\left({w}_{j}^{\Delta u}\Delta {u}_{ij}\right)}^{2}\right]\right)
'ITAE' (integral of time-weighted absolute error), for which the performance metric is
J=\sum _{i=1}^{Ns}i\Delta t\left(\sum _{j=1}^{{n}_{y}}|{w}_{j}^{y}{e}_{yij}|+\sum _{j=1}^{{n}_{u}}\left(|{w}_{j}^{u}{e}_{uij}|+|{w}_{j}^{\Delta u}\Delta {u}_{ij}|\right)\right)
In these expressions, ny is the number of controlled outputs and nu is the number of manipulated variables, eyij is the difference between output j and its setpoint (or reference) value at time interval i, euij is the difference between the manipulated variable j and its target at time interval i.
The w parameters are nonnegative performance weights defined by the structure PerfWeights.
Example: 'ITAE'
PerfWeights — Performance function weights
MPCobj.Weights (default) | structure
Performance function weights w, specified as a structure with the following fields:
OutputVariables — ny-element row vector that contains the
{w}_{j}^{y}
ManipulatedVariables — nu-element row vector that contains the
{w}_{j}^{u}
ManipulatedVariablesRate — nu-element row vector that contains the
{w}_{j}^{\Delta u}
If PerfWeights is empty or unspecified, it defaults to the corresponding weights in MPCobj.
The performance index is not related to the quadratic cost function that the MPC controller tries to minimize by choosing the values of the manipulated variables.
One clear difference is that the performance index is based on a closed loop simulation running until a time that is generally different than the prediction horizon, while the MPC controller calculates the moves which minimize its internal cost function up to the prediction horizon and in open loop fashion. Furthermore, even when the performance index is chosen to be of ISE type, its weights should be squared to match the weights defined in the MPC cost function.
Therefore, the performance weights and those used in the controller have different purposes; define these weights accordingly.
If you omit Ns, the default value is the row size of whichever of the following arrays has the largest row size:
MPCobj.Model.Nominal.Y (default) | matrix
If r is empty or unspecified, it defaults to the nominal value of the plant output, MPCobj.Model.Nominal.Y.
v — Measured disturbance signal
MPCobj.Model.Nominal.U(md) (default) | matrix
If v is empty or unspecified, it defaults to the nominal value of the measured input disturbance, MPCobj.Model.Nominal.U(md), where md is the vector containing the indices of the measured disturbance signals, as defined by setmpcsignals.
Use a simulation options objects to specify options such as noise and disturbance signals that feed into the plant but are unknown to the controller. You can also use this object to specify an open loop scenario, or a plant model in the loop that is different from the one in MPCobj.Model.Plant.
utarget — Target for manipulated variables
MPCobj.Model.Nominal.U (default) | vector
The optional input utarget is a vector of nu manipulated variable targets. Their defaults are the nominal values of the manipulated variables.
Example: [0.1;0;-0.2]
customPerFcn — Name of the custom performance function
Name of the custom performance function, specified as a character vector. The character vector must be different than 'ISE', 'IAE', 'ITSE', or 'ITAE', and specify the name of a file in the MATLAB® path containing a custom function.
The custom function must have the following signature:
J = customPerFcn(MPCobj,Par1,...,ParN)
where J is a scalar indicating the value of the performance index MPCobj is an mpc object. The remaining arguments Par1,...,ParN are parameters that, if needed by customPerFcn, you must pass to sensitivity after the customPerFcn argument.
For example, inside customPerFcn, you can use MPCobj and, if needed, Par1,...,ParN, to perform a simulation and calculate J based on the simulation results.
Example: 'myPerfFcn(mpcobj,Ts,Setpoint)'
Par1,...,ParN — Values of the parameters used by the custom performance function
values of any needed parameter
Values of the parameters used by the custom performance function customPerFcn, specified as needed.
Example: 10,[1 1]
J — Performance metric for the given controller
Depending on the PerfFcn argument, this performance measure can be a function of the integral (time-weighted or not) of either the square or the absolute value or the (output and input) error. See PerfFcn for more detail.
sens — Sensitivity of the performance metric
This structure contains and the numerical partial derivatives of the performance measure J with respect to its diagonal weights. These partial derivatives, also called sensitivities, suggest weight adjustments that should improve performance; that is, reduce J.
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{\displaystyle \pi d_{m}}
{\displaystyle T_{raise}={\frac {Fd_{m}}{2}}\left({\frac {l+\pi \mu d_{m}}{\pi d_{m}-\mu l}}\right)={\frac {Fd_{m}}{2}}\tan {\left(\phi +\lambda \right)}}
{\displaystyle T_{lower}={\frac {Fd_{m}}{2}}\left({\frac {\pi \mu d_{m}-l}{\pi d_{m}+\mu l}}\right)={\frac {Fd_{m}}{2}}\tan {\left(\phi -\lambda \right)}}
{\displaystyle \mu \,}
{\displaystyle \phi \,}
{\displaystyle \lambda \,}
{\displaystyle \phi >\lambda }
{\displaystyle {\mbox{efficiency}}={\frac {T_{0}}{T_{raise}}}={\frac {Fl}{2\pi T_{raise}}}={\frac {\tan {\lambda }}{\tan {\left(\phi +\lambda \right)}}}}
{\displaystyle T_{raise}={\frac {Fd_{m}}{2}}\left({\frac {l+\pi \mu d_{m}\sec {\alpha }}{\pi d_{m}-\mu l\sec {\alpha }}}\right)={\frac {Fd_{m}}{2}}\left({\frac {\mu \sec {\alpha }+\tan {\lambda }}{1-\mu \sec {\alpha }\tan {\lambda }}}\right)}
{\displaystyle T_{lower}={\frac {Fd_{m}}{2}}\left({\frac {\pi \mu d_{m}\sec {\alpha }-l}{\pi d_{m}+\mu l\sec {\alpha }}}\right)={\frac {Fd_{m}}{2}}\left({\frac {\mu \sec {\alpha }-\tan {\lambda }}{1+\mu \sec {\alpha }\tan {\lambda }}}\right)}
{\displaystyle \alpha \,}
{\displaystyle T_{c}={\frac {F\mu _{c}d_{c}}{2}}}
{\displaystyle \mu _{c}}
{\displaystyle \mu _{c}}
{\displaystyle \mu _{c}}
{\displaystyle N={(4.76\times 10^{6})d_{r}C \over L^{2}}}
나사못TorxScrew threadSelf-tapping screwCaptive fastener기계공학List of screw drives육각형
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Frequent Oscillation of a Class of Partial Difference Equations | EMS Press
Frequent Oscillation of a Class of Partial Difference Equations
C.J. Tian
Binggen Zhang
Ocean University of Qingdao, China
This paper is concerned with the partial difference equation
A_{m+l,n} + A_m,n+l} - a_{m,n}A_{m,n} + p_{m,n}A_{m–k,n–l} = 0
k
l
are non-negative integers, {
a_{m,n}
p_{m,n}
} are real double sequences. Frequent oscillation criteria of this equation are obtained.
C.J. Tian, Binggen Zhang, Frequent Oscillation of a Class of Partial Difference Equations. Z. Anal. Anwend. 18 (1999), no. 1, pp. 111–130
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2019 On the Resolution of an Inverse Problem by Shape Optimization Techniques
\omega
\partial \mathrm{\Omega }
\omega
\mathrm{\Omega }
\omega
Chahnaz Zakia Timimoun. "On the Resolution of an Inverse Problem by Shape Optimization Techniques." Abstr. Appl. Anal. 2019 1 - 6, 2019. https://doi.org/10.1155/2019/7684637
Received: 26 November 2018; Revised: 25 February 2019; Accepted: 26 March 2019; Published: 2019
Chahnaz Zakia Timimoun "On the Resolution of an Inverse Problem by Shape Optimization Techniques," Abstract and Applied Analysis, Abstr. Appl. Anal. 2019(none), 1-6, (2019)
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Package sagbi - ApCoCoA Wiki
(Redirected from Package SAGBI)
This article is about a function from ApCoCoA-2. If you are looking for the ApCoCoA-1 version of it, see Category:ApCoCoA-1:Package sagbi.
This page is about the SAGBI package. For a complete function list, see Category:Package sagbi.
1.1 Subalgebras
1.2 SAGBI bases
1.3 The Subalgebra Rewrite Relation
1.4 The Subalgebra Division Algorithm
2.2 Special Functions for Graded Subalgebras
2.3 The Subalgebra Data Type
2.4 Testing Subalgebra Membership
2.5 Example for the Subalgebra Data Type
{\displaystyle K}
{\displaystyle P=K[x_{1},\ldots ,x_{n}]}
be the polynomial ring over
{\displaystyle K}
i{\displaystyle n}
indeterminates. Most of the definitions here are taken from the book Computational Commutative Algebra 2 by Martin Kreuzer and Lorenzo Robbiano.
{\displaystyle S\subseteq P}
{\displaystyle K}
-subalgebra of
{\displaystyle P}
{\displaystyle K\subseteq S}
{\displaystyle S}
is closed under addition and multiplication.
{\displaystyle F=\{f_{1},\ldots ,f_{s}\}\subseteq P}
{\displaystyle K[F]=K[f_{1},\ldots ,f_{s}]=\{p(f_{1},\ldots ,f_{s})\mid p\in K[y_{1},\ldots ,y_{s}]\}}
{\displaystyle K[y_{1},\ldots ,y_{s}]}
is a new polynomial ring in
{\displaystyle s}
indeterminates for the
{\displaystyle K}
{\displaystyle P}
{\displaystyle F}
SAGBI bases
{\displaystyle \sigma }
be a term ordering on
{\displaystyle P}
{\displaystyle S}
{\displaystyle K}
{\displaystyle P}
{\displaystyle G\subseteq S}
{\displaystyle \sigma }
-SAGBI basis of
{\displaystyle S}
{\displaystyle K[{\rm {LT}}_{\sigma }(f)\mid f\in S]=K[{\rm {LT}}_{\sigma }(f)\mid f\in G]}
{\displaystyle S}
is a graded subalgebra and
{\displaystyle d\in \mathbb {N} }
, then a set
{\displaystyle G_{\leq d}}
{\displaystyle d}
-truncated
{\displaystyle \sigma }
{\displaystyle S}
{\displaystyle K[{\rm {LT}}_{\sigma }(f)\mid f\in S{\text{ and }}\deg(f)\leq d]=K[{\rm {LT}}_{\sigma }(f)\mid f\in G_{\leq d}].}
Equivalently, a set
{\displaystyle G_{\leq d}}
{\displaystyle d}
{\displaystyle \sigma }
{\displaystyle S}
if there is a
{\displaystyle \sigma }
{\displaystyle S}
{\displaystyle G_{\leq d}=\{g\in G\mid \deg(g)\leq d\}}
The Subalgebra Rewrite Relation
{\displaystyle G\subseteq P}
be a set of non-zero polynomials. Furthermore, let
{\displaystyle \sigma }
{\displaystyle \mathbb {T} ^{n}}
{\displaystyle f,g\in P\setminus \{0\}}
{\displaystyle f}
subalgebra reduces to
{\displaystyle g}
in one step if there are
{\displaystyle f_{1},\ldots ,f_{s}\in G}
{\displaystyle t\in \mathbb {T} ^{s}}
, and a constant
{\displaystyle c\in K}
{\displaystyle g=f-ct(f_{1},\ldots ,f_{s})}
{\displaystyle {\rm {LT}}_{\sigma }(t(f_{1},\ldots ,f_{s}))}
is not anymore in
{\displaystyle {\operatorname {Supp} }(g)}
. The path from
{\displaystyle f}
{\displaystyle g}
is then called a subalgebra reduction step and denoted by
{\displaystyle f{\xrightarrow {G} }_{\rm {ss}}g}
The transitive closure of the relation
{\displaystyle {\xrightarrow {G} }_{\rm {ss}}}
{\displaystyle {\xrightarrow {G} }_{\rm {s}}}
and called the subalgebra rewrite relation defined by
{\displaystyle G}
{\displaystyle f,g\in P}
{\displaystyle f{\xrightarrow {G} }_{\rm {s}}g}
{\displaystyle f}
{\displaystyle g}
{\displaystyle g\in P}
{\displaystyle h\in P}
{\displaystyle h\neq g}
{\displaystyle g{\xrightarrow {G} }_{\rm {ss}}h}
{\displaystyle g}
is called irreducible with respect to the relation
{\displaystyle {\xrightarrow {G} }_{\rm {s}}}
. Sometimes we may also just write that
{\displaystyle g}
is irreducible with respect to
{\displaystyle G}
{\displaystyle G\subseteq P}
is called interreduced if every element
{\displaystyle g\in G}
is monic and irreducible with respect to
{\displaystyle G\setminus \{g\}}
{\displaystyle G}
{\displaystyle \sigma }
{\displaystyle K[G]}
which is interreduced, then it is called a reduced
{\displaystyle \sigma }
-SAGBI basis. Analogously to reduced Gröbner bases, it can be shown that every
{\displaystyle K}
-subalgebra has a unique reduced
{\displaystyle \sigma }
-SAGBI basis.
The Subalgebra Division Algorithm
{\displaystyle \sigma }
{\displaystyle \mathbb {T} ^{n}}
{\displaystyle f\in P}
{\displaystyle G=\{f_{1},\ldots ,f_{s}\}\subseteq P}
be a set of non-zero polynomials. Consider the following sequence of instructions.
{\displaystyle g:=f}
{\displaystyle g=c_{1}t_{1}+\cdots +c_{m}t_{m}}
{\displaystyle t_{i}\in \mathbb {T} ^{n}}
{\displaystyle c_{i}\in K}
{\displaystyle i=1,\ldots ,m}
{\displaystyle t_{1}\geq _{\sigma }\cdots \geq _{\sigma }t_{m}}
{\displaystyle k:=1}
{\displaystyle k=m+1}
, stop and return
{\displaystyle g}
. Otherwise, check whether there is a term
{\displaystyle t\in \mathbb {T} ^{s}}
{\displaystyle t_{k}=t({\rm {LT}}_{\sigma }(f_{1}),\ldots ,{\rm {LT}}_{\sigma }(f_{s}))}
. If there is one, set
{\displaystyle c=t({\rm {LC}}_{\sigma }(f_{1}),\ldots ,{\rm {LC}}_{\sigma }(f_{s}))}
{\displaystyle g}
{\displaystyle g-{\frac {c_{i}}{c}}t(f_{1},\ldots ,f_{s})}
and go to step~(2). If there is none, increase
{\displaystyle k}
by one and repeat this step.
This is an algorithm which returns a polynomial
{\displaystyle g\in P}
{\displaystyle f{\xrightarrow {G} }_{\rm {s}}g}
{\displaystyle g}
{\displaystyle {\xrightarrow {G} }_{\rm {s}}}
. This algorithm clearly computes a polynomial
{\displaystyle g\in P}
{\displaystyle f{\xrightarrow {G} }_{\rm {s}}g}
{\displaystyle g}
{\displaystyle G}
All of the previously described definitions are implemented in the SAGBI package.
Given a polynomial ring P and a list F of polynomials in P, one can compute the reduced SAGBI basis of
{\displaystyle K}
[F] (with respect to the term ordering given by P) using the function SB.SAGBI - as long as a finite one exists.
SB.SAGBI(F);
Note that this function probably runs into an infinite loop if no finite SAGBI basis exists. This can be avoided using the function SB.SAGBITimeout. Given a positive integer s, one can type in
SB.SAGBITimeout(F,s);
which does the same as SB.SAGBI(F), but throws an error if the computation is not finished within s seconds. Given a polynomial f and a list of polynomials G, the function SB.ReductionStep can be used to compute a polynomial g with f
{\displaystyle {\xrightarrow {G} }_{\rm {s}}}
SB.ReductionStep(f,G);
If no such polynomial exists, then f is returned. This function is then used by the function SB.SDA, which is an implementation of the Subalgebra Division Algorithm described above.
SB.SDA(f,G)
Analogously to the CoCoA-5 function interreduced, the SAGBI package contains the function SB.Interreduced which takes as input a list of polynomials G and returns a list G' with
{\displaystyle K[G]=K[G']}
SB.Interreduced(G);
Special Functions for Graded Subalgebras
If G is a set of homogeneous polynomials, then there are additional functions one can use. Given a positive integerd, a d-truncated SAGBI basis can be computed using SB.TruncSAGBI.
SB.TruncSAGBI(G,d);
If additionally, the Hilbert series HS of the subalgebra
{\displaystyle K[G]}
is given, one can call
SB.TruncSAGBI(G,d,HS);
which does the same as above, but computes the SAGBI basis Hilbert-driven, which may be a little bit faster. The function SB.SubalgebraHS can be applied to compute the Hilbert series of a graded subalgebra.
SB.SubalgebraHS(G);
If furthermore G is a set of terms, then the function
SB.TorRingHS(G);
can be used to compute its Hilbert series much more efficient.
The Subalgebra Data Type
The package also introduces a new Data type, i.e. a record tagged with "$apcocoa/sagbi.Subalgebra". Given a polynomial ring P and a list of polynomials G from P, one can create the subalgebra
{\displaystyle K[G]}
using the function SB.Subalgebra.
Use P ::= QQ[x,y,z];
G := [x^2+y*z,z];
S := SB.Subalgebra(P,G);
For details about the structure of this data type, see the function page. While nearly all functionalities of the SAGBI package can be used without touching this data type, it has many advantages to do so because it stores informations of previous computations, see the example below. This is also the reason why many of the getter functions need the subalgebra to be called by reference. The following getter function can be used to get informations about the subalgebra:
SB.GetCoeffRing(S); -- returns the coefficient ring
SB.GetGens(S); -- returns the set G
SB.GetID(S); -- returns the unique ID of S
SB.GetLTSA(ref S); -- returns the subalgebra K[LT(f) | f in S]
SB.GetRing(S); -- returns P
SB.GetSAGBI(ref S); -- returns the reduced SAGBI basis of S (if a finite one exists)
If additionally, G is a set of homogeneous polynomials, one can call the following getter functions:
SB.GetHS(ref S); -- returns the Hilbert Series of S
SB.GetTruncSAGBI(ref S,d); -- returns a d-truncated SAGBI basis of S
SB.GetTruncDeg(S); -- returns the truncation degree of the currently stored SAGBI basis
To optain a
{\displaystyle K}
-vector space basis of the set
{\displaystyle K[G]_{d}}
of all homogeneous polynomials of degree
{\displaystyle d}
{\displaystyle K[G]}
, the function SB.GetInDeg can be used:
SB.GetInDeg(S);
Testing Subalgebra Membership
Let f be a polynomial in the polynomial ring P, let G be a list of polynomials in P and let S be a subalgebra generated by G. Then the SAGBI package provides four functions to check whether f is an element of the subalgebra S:
SB.IsInSubalgebra(f,G);
SB.IsInSA(f,S);
If G is a list of homogeneous polynomials, the following functions can also be used:
SB.IsInSubalgebra_SAGBI(f,G);
SB.IsInSA_SAGBI(f,ref S);
While the first two functions test the membership using implicitization, these two functions use truncated SAGBI bases for the membership test, which may be more efficient. It depends on the application which of these two possibilities is the fastest one.
Example for the Subalgebra Data Type
So what advantages does the Subalgebra data type have? Consider the following example.
G := [x^2 -z^2, x*y +z^2, y^2 -2*z^2];
L := SB.SAGBI(G);
f := x^10*y^2 +x^6*y^6 -2*x^10*z^2 -5*x^8*y^2*z^2 +6*x^5*y^5*z^2 +10*x^8*z^4 +10*x^6*y^2*z^4 +15*x^4*y^4*z^4 -20*x^6*z^6 -10*x^4*y^2*z^6 +20*x^3*y^3*z^6 +20*x^4*z^8 +20*x^2*y^2*z^8 -10*x^2*z^10 +6*x*y*z^10 -y^2*z^10 +3*z^12;
b := SB.IsInSubalgebra(f,G);
h := SB.SubalgebraHS(G);
Here, first a SAGBI basis is computed, then the subalgebra membership is tested using implicitization and at the end, the Hilbert series of K[G] is computed. But after computing a SAGBI basis, one could also test the Subalgebra membership using the Subalgebra Division Algorithm and compute the Hilbert Series of . The Subalgebra data type does this automatically. Instead of the code above, we can write the following:
L := SB.GetSAGBI(ref S);
b := SB.IsInSA(f,ref S);
h := SB.GetHS(ref S);
While this is only a simple example, the second code is much faster. At the time this is written, the second computation is approximately two times as fast as the first one.
Retrieved from "http://apcocoa.uni-passau.de/wiki/index.php?title=Package_sagbi&oldid=16281"
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\mathrm{\tau }=\frac{\left[\stackrel{.}{\mathbf{R}}\stackrel{..}{\mathbf{R}}\stackrel{...}{\mathbf{R}}\right]}{{∥\stackrel{.}{\mathbf{R}}×\stackrel{..}{\mathbf{R}}∥}^{2}}
\mathbf{R}\prime =\frac{d}{\mathrm{ds}}\mathbf{R}\left(s\right)
\stackrel{.}{\mathbf{R}}=\frac{d}{\mathrm{dp}}\mathbf{R}\left(p\right)
. Start with the formulas
\mathrm{\tau }=\left[\mathbf{R}\prime \mathbf{R}″\mathbf{R}‴\right]/{\mathrm{\kappa }}^{2}
\mathrm{κ}=∥\stackrel{.}{\mathbf{R}}×\stackrel{..}{\mathbf{R}}∥/{\mathrm{ρ}}^{3}
. The intermediate calculations, heavily dependent on the chain rule for differentiation of composite functions, are summarized in Table 2.6.8(a).
\mathbf{R}\prime \left(s\right)
=\frac{d\mathbf{R}}{\mathrm{dp}}\frac{\mathrm{dp}}{\mathrm{ds}}=\stackrel{.}{\mathbf{R}}/\mathrm{ρ}
\mathbf{R}″\left(s\right)
=\frac{d}{\mathrm{dp}}\left(\frac{\stackrel{.}{\mathbf{R}}}{\mathrm{ρ}}\right)\frac{\mathrm{dp}}{\mathrm{ds}}
=\left(-\frac{\stackrel{.}{\mathrm{ρ}}}{{\mathrm{ρ}}^{2}}\stackrel{.}{\mathbf{R}}+\frac{\stackrel{..}{\mathbf{R}}}{\mathrm{ρ}}\right)\frac{1}{\mathrm{ρ}}
=-\frac{\stackrel{.}{\mathrm{ρ}}}{{\mathrm{ρ}}^{3}}\stackrel{.}{\mathbf{R}}+\frac{\stackrel{..}{\mathbf{R}}}{{\mathrm{ρ}}^{2}}
\mathbf{R}‴\left(s\right)
=\frac{d}{\mathrm{dp}}\left(-\frac{\stackrel{.}{\mathrm{\rho }}}{{\mathrm{\rho }}^{3}}\stackrel{.}{\mathbf{R}}+\frac{\stackrel{..}{\mathbf{R}}}{{\mathrm{\rho }}^{2}}\right)\frac{\mathrm{dp}}{\mathrm{ds}}
=\left(\frac{d}{\mathrm{dp}}\left(-\frac{\stackrel{.}{\mathrm{\rho }}}{{\mathrm{\rho }}^{3}}\right)\stackrel{.}{\mathbf{R}}-\frac{\stackrel{.}{\mathrm{\rho }}}{{\mathrm{\rho }}^{3}}\stackrel{..}{\mathbf{R}}+\frac{d}{\mathrm{dp}}\left(\frac{1}{{\mathrm{ρ}}^{2}}\right)\stackrel{..}{\mathbf{R}}+\frac{1}{{\mathrm{ρ}}^{2}}\stackrel{...}{\mathbf{R}}\right)\frac{1}{\mathrm{ρ}}
=\mathrm{α} \stackrel{.}{\mathbf{R}}+\mathrm{β} \stackrel{..}{\mathbf{R}}+\frac{1}{{\mathrm{ρ}}^{3}}\stackrel{...}{\mathbf{R}}
Table 2.6.8(a) Relating the derivatives in
\left[\mathbf{R}\prime \mathbf{R}″\mathbf{R}‴\right]
to those in
\left[\stackrel{.}{\mathbf{R}}\stackrel{..}{\mathbf{R}}\stackrel{...}{\mathbf{R}}\right]
The last row in Table 2.6.8(a) is obtained by observing that in the box product, only rows that are independent will "survive." This becomes clear in the following concluding steps where the additive properties of a determinant are essential.
\left|\left(A\right)\left(B\right)\left(C\right)\right|
represents the determinant of matrix whose columns are the vectors A, B, and C, then
\left|\left(U+V\right)\left(W\right)\left(Q\right)\right|
\left|\left(U\right)\left(W\right)\left(Q\right)\right|+\left|\left(V\right)\left(W\right)\left(Q\right)\right|
In other words, if a column is the sum of two other column vectors, then the determinant splits into the sum of two determinants. And what is true for columns is true for rows, because the determinant of a matrix has the same value as the determinant of the transpose of the matrix.
\left[\mathbf{R}\prime \mathbf{R}″\mathbf{R}‴\right]
|\left(\frac{\stackrel{.}{\mathbf{R}}}{\mathrm{ρ}}\right)(\left(-\frac{\stackrel{.}{\mathrm{ρ}}}{{\mathrm{ρ}}^{3}}\stackrel{.}{\mathbf{R}}+\frac{\stackrel{..}{\mathbf{R}}}{{\mathrm{ρ}}^{2}}\right)\left(\mathrm{\alpha } \stackrel{.}{\mathbf{R}}+\mathrm{\beta } \stackrel{..}{\mathbf{R}}+\frac{1}{{\mathrm{\rho }}^{3}}\stackrel{...}{\mathbf{R}}\right)|
By the additivity property of determinants, this splits into a sum of determinants, but any of the resulting determinants that have two columns proportional will be zero. Hence, the "surviving" determinant is the box product
\left[\frac{\stackrel{.}{\mathbf{R}}}{\mathrm{ρ}} \frac{\stackrel{..}{\mathbf{R}}}{{\mathrm{ρ}}^{2}} \frac{\stackrel{...}{\mathbf{R}}}{{\mathrm{ρ}}^{3}}\right]=\frac{1}{{\mathrm{ρ}}^{6}}\left[\stackrel{.}{\mathbf{R}}\stackrel{..}{\mathbf{R}}\stackrel{...}{\mathbf{R}}\right]
\mathrm{\tau }=\left[\mathbf{R}\prime \mathbf{R}″\mathbf{R}‴\right]/{\mathrm{\kappa }}^{2}
\frac{\left[\stackrel{.}{\mathbf{R}}\stackrel{..}{\mathbf{R}}\stackrel{...}{\mathbf{R}}\right]/{\mathrm{ρ}}^{6}}{(∥\stackrel{.}{\mathbf{R}}×\stackrel{..}{\mathbf{R}}∥/{\mathrm{\rho }}^{3}{)}^{2}}
\frac{\left[\stackrel{.}{\mathbf{R}}\stackrel{..}{\mathbf{R}}\stackrel{...}{\mathbf{R}}\right]}{{∥\stackrel{.}{\mathbf{R}}×\stackrel{..}{\mathbf{R}}∥}^{2}}
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Tournament One - EconGames
Location Erasmus University Rotterdam
Game Game One
Participants of the 1st EconGames tournament.
The winner, Vladimir Karamychev.
Our first tournament took place at Erasmus University Rotterdam. The tournament was small - 5 participants - but no less enjoyable for that. Vladmir Karamychev won the tournament and was consequently awarded a token gift. The complete ranking was as follows:
1 Vladimir Karamychev Erasmus University Rotterdam 140.6
2 Clemens Fiedler Tilburg University 122.3
3 Gijsbert Zwart University of Groningen 120.3
4 Bart Voogt CPB Netherlands Bureau for Economic Policy Analysis 116.2
5 Andrei Dubovik CPB Netherlands Bureau for Economic Policy Analysis 108.7
All the code is publicly available and can accessed via our git server, see Technical Organization for instructions. (If you run the code yourself, you should obtain exactly the same results on Linux; on Windows the results are a bit different, we will address this issue in the future.)
What follows are some descriptive statistics and considerations.
How well did the players perform against one another? Let
{\displaystyle A}
denote the payoff matrix, where
{\displaystyle A_{ij}}
gives the payoff of player
{\displaystyle i}
when he is playing against
{\displaystyle j}
. The payoff matrix for this tournament was as follows (green marks a win, salmon marks a loss):
The first observation is that there was little collusion (cooperation) in the tournament. If two players were to collude perfectly, their joint profits should have been 250 after 1000 rounds. Whereas the maximum joint profits, obtained by Gijsbert and Vladimir, were only 88.3.
Further, no single strategy was strictly dominant. On the other hand, Gijsbert's strategy was strictly dominated, albeit it performed well overall. If firms commit to these strategies, will all types of firms survive? Suppose there are many firms and initially each strategy is adopted by one-fifth of all firms. We can look at the evolutionary dynamics of this system using the standard replicator equation.
As can be seen from the figure, all strategies survive in the long-run. Note that in equilibrium expected payoffs from each strategy are equal. This evolutionary games perspective gives a simple reason why Nash equilibrium might be a poor predictor for real life data. Namely, many different strategies, personified for example by marketing managers or data scientists, can survive in equilibrium.
Testing for Nash Equilibrium
The following figure compares the empirical distribution of all the prices observed in the tournament with the theoretical Nash distribution.
The Kolmogorov-Smirnov test rejects the null hypothesis that these distributions are the same at 1% confidence level (the p-value is indistinguishable from 0). It should be noted, however, that the tournament is different from the Levitan and Shubik model on which it is based. Firstly, there are multiple periods in the tournament. Secondly, the objective function is to win as opposed to maximize profits. (It seems to be an open question, how much of a difference the objective function introduces given that the round-robin style of the tournament seems to still favour profit maximization. Indeed, Gijsbert lost in every individual game but came in third overall.)
Best Static Response
The analysis of the previous two sections could be performed for any lab experiment. However, in our tournament we collect not only the actions, but also the strategies. Consequently, additional types of analysis become possible. This section gives one such example.
Andrei and Bart have played static strategies: prices were drawn from a fixed distribution irrespective of the past actions of the opponents. The rest of the participants played history-dependent strategies. For example, the strategy of Vladimir attempted to learn the strategy of the other player as well as possible (using conditional kernel densities), and then played the best response. Clearly, Andrei and Bart have lost, but just how far can one get in this tournament with a static strategy? To study this question, let us replace Andrei's strategy with an arbitrary Beta distribution, and then optimize over the parameters of that distribution so as to get as high in the tournament's ranking as possible. (The Beta distribution has been chosen as it is relatively flexible.) The following figure compares the old and the new rankings.
So, with the benefit of hindsight a static distribution can perform well, but still not well enough to win the tournament. This result is somewhat expected: the tournament has 1000 periods, which is long enough to learn the distribution of the opponent and respond optimally. Therefore the strategy of Vladimir, which is doing precisely that, does well. We have discussed this issue that a long tournament favours fast learning strategies as opposed to strategies with good priors. In the future, we might hold tournaments with different prizes for different lengths, or maybe make the length of the tournament uncertain with a high chance of it being short.
Retrieved from "https://econgames.org/w/index.php?title=Tournament_One&oldid=139"
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Proportionality (mathematics) - Wikipedia
Property of two varying quantities with a constant ratio
The variable y is directly proportional to the variable x with proportionality constant ~0.6.
The variable y is inversely proportional to the variable x with proportionality constant 1.
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constant. Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality.
This definition is commonly extended to related varying quantities, which are often called variables. This meaning of variable is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons.
{\displaystyle f(x)}
{\displaystyle g(x)}
are proportional if their ratio
{\textstyle {\frac {f(x)}{g(x)}}}
If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., a/b = x/y = ⋯ = k (for details see Ratio). Proportionality is closely related to linearity.
1 Direct proportionality
1.2 Computer encoding
2 Inverse proportionality
3 Hyperbolic coordinates
Direct proportionality[edit]
Given two variables x and y, y is directly proportional to x[1] if there is a non-zero constant k such that
{\displaystyle y=kx.}
The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter alpha) or "~":
{\displaystyle y\propto x,}
{\displaystyle y\sim x.}
{\displaystyle x\neq 0}
the proportionality constant can be expressed as the ratio
{\displaystyle k={\frac {y}{x}}.}
It is also called the constant of variation or constant of proportionality.
On a map of a sufficiently small geographical area, drawn to scale distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by those points; the constant of proportionality is the scale of the map.
The force, acting on a small object with small mass by a nearby large extended mass due to gravity, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as gravitational acceleration.
The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this, Newton's second law, is the classical mass of the object.
U+221D ∝ PROPORTIONAL TO (∝, ∝, ∝, ∝, ∝)
U+223C ∼ TILDE OPERATOR (∼, ∼, ∼, ∼)
U+223A ∺ GEOMETRIC PROPORTION (∺)
Inverse proportionality[edit]
The concept of inverse proportionality can be contrasted with direct proportionality. Consider two variables said to be "inversely proportional" to each other. If all other variables are held constant, the magnitude or absolute value of one inversely proportional variable decreases if the other variable increases, while their product (the constant of proportionality k) is always the same. As an example, the time taken for a journey is inversely proportional to the speed of travel.
Formally, two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion)[2] if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant.[3] It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that
{\displaystyle y={\frac {k}{x}},}
{\displaystyle xy=k.}
Hence the constant "k" is the product of x and y.
Hyperbolic coordinates[edit]
^ Weisstein, Eric W. "Directly Proportional". MathWorld – A Wolfram Web Resource.
^ "Inverse variation". math.net. Retrieved October 31, 2021.
^ Weisstein, Eric W. "Inversely Proportional". MathWorld – A Wolfram Web Resource.
Ya. B. Zeldovich, I. M. Yaglom: Higher math for beginners, p. 34–35.
Brian Burrell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN 9780877796213, p. 85–101.
Lanius, Cynthia S.; Williams Susan E.: PROPORTIONALITY: A Unifying Theme for the Middle Grades. Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396.
Seeley, Cathy; Schielack Jane F.: A Look at the Development of Ratios, Rates, and Proportionality. Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142.
Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions. Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Proportionality_(mathematics)&oldid=1081192341"
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Functional derivative (variational derivative) - MATLAB functionalDerivative - MathWorks Switzerland
functionalDerivative
Functional Derivative with Respect to Single Function
Functional Derivative with Respect to Two Functions
Euler–Lagrange Equation of Simple Mass-Spring System
Differential Equation of Brachistochrone Problem
Minimal Surface Equation in 3-D Space
Functional derivative (variational derivative)
G = functionalDerivative(f,y)
G = functionalDerivative(f,y) returns the functional derivative
\frac{\delta S}{\delta y}\left(x\right)
S\left[y\right]={\int }_{a}^{b}f\left[x,y\left(x\right),y\text{'}\left(x\right),...\right]\text{\hspace{0.17em}}dx
with respect to the function y = y(x), where x represents one or more independent variables. The functional derivative relates the change in the functional S[y] with respect to a small variation in y(x).The functional derivative is also known as the variational derivative.
If y is a vector of symbolic functions, functionalDerivative returns a vector of functional derivatives with respect to the functions in y, where all functions in y must depend on the same independent variables.
Find the functional derivative of the functional
S\left[y\right]={\int }_{b}^{a}y\left(x\right)\mathrm{sin}\left(y\left(x\right)\right)\phantom{\rule{0.2222222222222222em}{0ex}}dx
with respect to the function
y
, where the integrand is
f\left[y\left(x\right)\right]=y\left(x\right)\phantom{\rule{0.16666666666666666em}{0ex}}\mathrm{sin}\left(y\left(x\right)\right)
Declare y(x) as a symbolic function and define f as the integrand of
S
. Use f and y as the parameters of functionalDerivative.
f = y*sin(y);
\mathrm{sin}\left(y\left(x\right)\right)+\mathrm{cos}\left(y\left(x\right)\right) y\left(x\right)
S\left[u,v\right]={\int }_{b}^{a}\left({u}^{2}\left(x\right)\frac{dv\left(x\right)}{dx}+v\left(x\right)\frac{{d}^{2}u\left(x\right)}{d{x}^{2}}\right)\phantom{\rule{0.2222222222222222em}{0ex}}dx
with respect to the functions
u
v
f\left[u\left(x\right),v\left(x\right),{u}^{\prime \prime }\left(x\right),{v}^{\prime }\left(x\right)\right]={u}^{2}\frac{dv}{dx}+v\frac{{d}^{2}u}{d{x}^{2}}
Declare u(x) and v(x) as symbolic functions, and define f as the integrand of
S
f = u^2*diff(v,x) + v*diff(u,x,x);
Specify a vector of symbolic functions [u v] as the second input argument in functionalDerivative.
G = functionalDerivative(f,[u v])
\left(\begin{array}{c}\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }v\left(x\right)+2 u\left(x\right) \frac{\partial }{\partial x}\mathrm{ }v\left(x\right)\\ \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }u\left(x\right)-2 u\left(x\right) \frac{\partial }{\partial x}\mathrm{ }u\left(x\right)\end{array}\right)
functionalDerivative returns a vector of symbolic functions containing the functional derivatives of the integrand f with respect to u and v, respectively.
Find the Euler–Lagrange equation of a mass m that is connected to a spring with spring constant k.
Define the kinetic energy T, potential energy V, and Lagrangian L of the system. The Lagrangian is the difference between the kinetic and potential energy.
syms m k x(t)
T = 1/2*m*diff(x,t)^2;
V = 1/2*k*x^2;
\frac{m {\left(\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)\right)}^{2}}{2}-\frac{k {x\left(t\right)}^{2}}{2}
In Lagrangian mechanics, the action functional of the system is equal to the integral of the Lagrangian over time, or
S\left[x\right]={\int }_{{t}_{1}}^{{t}_{2}}L\left[t,x\left(t\right),\underset{}{\overset{˙}{x}}\left(t\right)\right]\phantom{\rule{0.16666666666666666em}{0ex}}dt
. The Euler–Lagrange equation describes the motion of the system for which
S\left[x\left(t\right)\right]
Find the Euler–Lagrange equation by taking the functional derivative of the integrand L and setting it equal to 0.
eqn = functionalDerivative(L,x) == 0
-m \frac{{\partial }^{2}}{\partial {t}^{2}}\mathrm{ }x\left(t\right)-k x\left(t\right)=0
eqn is the differential equation that describes mass-spring oscillation.
Solve eqn using dsolve. Assume the mass m and spring constant k are positive. Set the initial conditions for the oscillation amplitude as
x\left(0\right)=10
and the initial velocity of the mass as
\underset{}{\overset{˙}{x}}\left(0\right)=0
assume(m,'positive')
assume(k,'positive')
Dx(t) = diff(x(t),t);
xSol = dsolve(eqn,[x(0) == 10, Dx(0) == 0])
10 \mathrm{cos}\left(\frac{\sqrt{k} t}{\sqrt{m}}\right)
Clear assumptions for further calculations.
assume([k m],'clear')
The Brachistochrone problem is to find the quickest path of descent of a particle under gravity without friction. The motion is confined to a vertical plane. The time for a body to move along a curve
y\left(x\right)
a
b
under gravity
g
t={\int }_{a}^{b}\sqrt{\frac{1+{{y}^{\prime }}^{2}}{2gy}}\phantom{\rule{0.16666666666666666em}{0ex}}dx.
Find the quickest path by minimizing the change in
t
with respect to small variations in the path
y
. The condition for a minimum is
\frac{\delta t}{\delta y}\left(x\right)=0
Compute the functional derivative to obtain the differential equation that describes the Brachistochrone problem. Use simplify to simplify the equation to its expected form.
syms g y(x)
assume(g,'positive')
f = sqrt((1 + diff(y)^2)/(2*g*y));
eqn = functionalDerivative(f,y) == 0;
eqn = simplify(eqn)
2 y\left(x\right) \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }y\left(x\right)+{\left(\frac{\partial }{\partial x}\mathrm{ }y\left(x\right)\right)}^{2}=-1
This equation is the standard differential equation of the Brachistochrone problem. To find the solutions of the differential equation, use dsolve. Specify the 'Implicit' option to true to return implicit solutions, which have the form
F\left(y\left(x\right)\right)=g\left(x\right)
sols = dsolve(eqn,'Implicit',true)
\begin{array}{l}\left(\begin{array}{c}y\left(x\right)={C}_{2}-x \mathrm{i}\\ y\left(x\right)={C}_{3}+x \mathrm{i}\\ {\sigma }_{1}={C}_{4}+x\\ {\sigma }_{1}={C}_{5}-x\\ \frac{{C}_{1}+y\left(x\right)}{y\left(x\right)}=0\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}={C}_{1} \mathrm{atan}\left(\sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}\right)-y\left(x\right) \sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}\end{array}
The symbolic solver dsolve returns general solutions in the complex space. Symbolic Math Toolbox™ does not accept the assumption that the symbolic function
y\left(x\right)
Depending on the boundary conditions, there are two real-space solutions to the Brachistochrone problem. One of the two solutions below describes a cycloid curve in real space.
solCycloid1 = sols(3)
solCycloid1 =
{C}_{1} \mathrm{atan}\left(\sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}\right)-y\left(x\right) \sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}={C}_{4}+x
{C}_{1} \mathrm{atan}\left(\sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}\right)-y\left(x\right) \sqrt{-\frac{{C}_{1}}{y\left(x\right)}-1}={C}_{5}-x
Another solution in real space is a horizontal straight line, where
y
solStraight = simplify(sols(5))
solStraight =
{C}_{1}+y\left(x\right)=0
To illustrate the cycloid solution, consider an example with boundary conditions
y\left(0\right)=5
y\left(4\right)=1
. In this case, the equation that can satisfy the given boundary conditions is solCycloid2. Substitute the two boundary conditions into solCycloid2.
eq1 = subs(solCycloid2,[x y(x)],[0 5]);
The two equations, eq1 and eq2, have two unknown coefficients,
{C}_{1}
{C}_{5}
. Use vpasolve to find numerical solutions for the coefficients. Substitute these solutions into solCycloid2.
coeffs = vpasolve([eq1 eq2]);
eqCycloid = subs(solCycloid2,{'C1','C5'},{coeffs.C1,coeffs.C5})
eqCycloid =
-6.4199192418473511250705556729108 \mathrm{atan}\left(\sqrt{\frac{6.4199192418473511250705556729108}{y\left(x\right)}-1}\right)-y\left(x\right) \sqrt{\frac{6.4199192418473511250705556729108}{y\left(x\right)}-1}=-x-5.8078336827583088482183433150164
The implicit equation eqCycloid describes the cycloid solution of the Brachistochrone problem in terms of
x
y\left(x\right)
You can then use fimplicit to plot eqCycloid. Since fimplicit only accepts implicit symbolic equations that contain symbolic variables
x
y
, convert the symbolic function
y\left(x\right)
to a symbolic variable
y
. Use mapSymType to convert
y\left(x\right)
x
. Plot the cycloid solution within the boundary conditions
0<x<4
1<y<5
funToVar = @(obj) sym('y');
eqPlot = mapSymType(eqCycloid,'symfun',funToVar);
fimplicit(eqPlot,[0 4 1 5])
u\left(x,y\right)
that describes a surface in 3-D space, the surface area can be determined by the functional
F\left[u\right]={\int }_{{y}_{1}}^{{y}_{2}}{\int }_{{x}_{1}}^{{x}_{2}}f\left[x,y\left(x\right),u\left(x,y\right),{u}_{x},{u}_{y}\right]\phantom{\rule{0.2777777777777778em}{0ex}}dx\phantom{\rule{0.2777777777777778em}{0ex}}dy={\int }_{{y}_{1}}^{{y}_{2}}{\int }_{{x}_{1}}^{{x}_{2}}\sqrt{1+{u}_{x}^{2}+{u}_{y}^{2}}\phantom{\rule{0.2777777777777778em}{0ex}}dx\phantom{\rule{0.2777777777777778em}{0ex}}dy
{u}_{x}
{u}_{y}
are the partial derivatives of
u
x
y
Find the functional derivative of the integrand f with respect to u.
syms u(x,y)
f = sqrt(1 + diff(u,x)^2 + diff(u,y)^2);
G = functionalDerivative(f,u)
\begin{array}{l}-\frac{{\left(\frac{\partial }{\partial y}\mathrm{ }u\left(x,y\right)\right)}^{2} \frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }u\left(x,y\right)+\frac{{\partial }^{2}}{\partial {x}^{2}}\mathrm{ }u\left(x,y\right)+{{\sigma }_{1}}^{2} \frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ }u\left(x,y\right)-2 \frac{\partial }{\partial y}\mathrm{ }{\sigma }_{1} \frac{\partial }{\partial y}\mathrm{ }u\left(x,y\right) {\sigma }_{1}+\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ }u\left(x,y\right)}{{\left({{\sigma }_{1}}^{2}+{\left(\frac{\partial }{\partial y}\mathrm{ }u\left(x,y\right)\right)}^{2}+1\right)}^{3/2}}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}=\frac{\partial }{\partial x}\mathrm{ }u\left(x,y\right)\end{array}
The result is the equation G that describes the minimal surface of a 3-D surface defined by u(x,y). The solutions to this equation describe minimal surfaces in 3-D space, such as soap bubbles.
f — Integrand of functional
symbolic variable | symbolic function | symbolic expression
Integrand of a functional, specified as a symbolic variable, function, or expression. The argument f represents the density of the functional.
y — Differentiation function
symbolic function | vector of symbolic functions | matrix of symbolic functions | multidimensional array of symbolic functions
Differentiation function, specified as a symbolic function or a vector, matrix, or multidimensional array of symbolic functions. The argument y can be a function of one or more independent variables. If y is a vector of symbolic functions, functionalDerivative returns a vector of functional derivatives with respect to the functions in y, where all functions in y must depend on the same independent variables.
G — Functional derivative
symbolic function | vector of symbolic functions
Functional derivative, returned as a symbolic function or a vector of symbolic functions. If input y is a vector, then G is a vector.
Consider a functional
S\left[y\right]={\int }_{a}^{b}f\left[x,y\left(x\right),y\text{'}\left(x\right),...,{y}^{\left(n\right)}\left(x\right)\right]\text{\hspace{0.17em}}dx,
which can take any path from a to b in the x-space.
For a small variation in the path y(x), define the change as
\delta y\left(x\right)=\epsilon \varphi \left(x\right)
in which ϕ(x) is an arbitrary test function. The change in the functional S is
DS\left[y\right]=\underset{\epsilon \to 0}{\mathrm{lim}}\frac{S\left[y+\epsilon \varphi \right]-S\left[y\right]}{\epsilon }={\int }_{a}^{b}\frac{\delta S}{\delta y}\left(x\right)\varphi \left(x\right)dx\text{.}
\frac{\delta S}{\delta y}\left(x\right)
is the functional derivative of S with respect to y. The linear functional DS[y] is also known as the first variation or the Gateaux differential of the functional S.
One method to calculate the functional derivative is to apply Taylor expansion to the expression S[y + εϕ] with respect to ε. By keeping the first order terms in ε, performing integration by parts, and choosing the boundary conditions ϕ(a) = ϕ(b) = ϕ'(a) = ϕ'(b) = ... = 0, the functional derivative becomes
\begin{array}{lll}\frac{\delta S}{\delta y}\left(x\right)\hfill & =\hfill & \frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial {y}^{\text{'}}}+\frac{{d}^{2}}{d{x}^{2}}\frac{\partial f}{\partial {y}^{\text{'}\text{'}}}-...+{\left(-1\right)}^{n}\frac{{d}^{n}}{d{x}^{n}}\left(\frac{\partial f}{\partial {y}^{\left(n\right)}}\right)\hfill \\ \hfill & =\hfill & \sum _{i=0}^{n}{\left(-1\right)}^{i}\frac{{d}^{i}}{d{x}^{i}}\left(\frac{\partial f}{\partial {y}^{\left(i\right)}}\right).\hfill \end{array}
diff | dsolve | int
|
Provide LQR Performance Using Terminal Penalty Weights - MATLAB & Simulink - MathWorks Australia
Design Infinite-Horizon LQR Controller
Design Equivalent MPC Controller
Compare MPC and LQR Controller Gains
It is possible to make a finite-horizon model predictive controller equivalent to an infinite-horizon linear quadratic regulator (LQR) by setting tuning weights on the terminal predicted states.
The standard MPC cost function is similar to the cost function for an LQR controller with output weighting, as shown in the following equation:
J\left(u\right)=\sum _{i=1}^{\infty }y{\left(k+i\right)}^{T}Qy\left(k+i\right)+u{\left(k+i-1\right)}^{T}Ru\left(k+i-1\right)
The LQR and MPC cost functions differ in the following ways:
The LQR cost function forces
y
u
toward zero, whereas the MPC cost function forces
y
u
toward nonzero setpoints. You can shift the MPC prediction model origin to eliminate this difference and achieve zero nominal setpoints.
The LQR cost function uses an infinite prediction horizon in which the manipulated variable changes at each sample time. In the standard MPC cost function, the horizon length is
p
, and the manipulated variable changes
m
m
is the control horizon.
The two cost functions are equivalent if the MPC cost function is:
J\left(u\right)=\sum _{i=1}^{p-1}\left(y{\left(k+i\right)}^{T}Qy\left(k+i\right)+u{\left(k+i-1\right)}^{T}Ru\left(k+i-1\right)\right)+x\left(k+p{\right)}^{T}{Q}_{p}x\left(k+p\right)
{Q}_{p}
is a terminal penalty weight applied at the final prediction horizon step, and the prediction and control horizons are equal (
p
m
). The required
{Q}_{p}
is the Riccati matrix calculated using the lqr and lqry commands.
Specify the discrete-time open-loop dynamic plant model with a sample time of 0.1 seconds. For this model, make all states measurable outputs of the plant. This plant is the double integrator plant from [1].
A = [1 0;0.1 1];
B = [0.1;0.005];
plant = ss(A,B,C,D,Ts);
Compute the Riccati matrix Qp and state feedback gain K associated with the LQR problem with output weight Q and input weight R. For more information, see lqry.
[K,Qp] = lqry(plant,Q,R);
To implement the MPC cost function, first compute
L
, the Cholesky decomposition of
{Q}_{p}
{L}^{T}L={Q}_{p}
L = chol(Qp);
Next, define auxiliary unmeasured output variables
{y}_{c}=Lx
{y}_{c}^{T}{y}_{c}={x}^{T}{Q}_{p}x
. Augment the output vector of the plant such that it includes these auxiliary outputs.
newPlant = plant;
set(newPlant,'C',[C;L],'D',[D;zeros(2,1)]);
Configure the state vector outputs as measured outputs and the auxiliary output signals as unmeasured outputs. By default, the input signal is the manipulated variable.
newPlant = setmpcsignals(newPlant,'MO',[1 2],'UO',[3 4]);
Create the controller object with the same sample time as the plant and equal prediction and control horizons.
mpcobj = mpc(newPlant,Ts,p,m);
Define tuning weights at each step of the prediction horizon for the manipulated variable and the measured outputs.
ywt = sqrt(diag(Q))';
uwt = sqrt(diag(R))';
mpcobj.Weights.OV = [sqrt(diag(Q))' 0 0];
mpcobj.Weights.MV = sqrt(R);
To make the QP problem associated with the MPC controller positive definite, include very small weights on manipulated variable increments.
mpcobj.Weights.MVRate = 1e-5;
Impose the terminal penalty
{x}^{T}\left(k+p\right){Q}_{p}x\left(k+p\right)
by specifying a unit weight on
{y}_{c}\left(k+p\right)=Lx\left(k+p\right)
. The terminal weight on u(t+p-1) remains the same.
Y = struct('Weight',[0 0 1 1]);
U = struct('Weight',uwt);
setterminal(mpcobj,Y,U);
Since the measured output vector contains the entire state vector, remove any additional output disturbance integrator inserted by the MPC controller.
setoutdist(mpcobj,'model',ss(zeros(4,1)));
Remove the state estimator by defining the following measurement update equation:
x[n|n] = x[n|n-1] + I * (x[n]-x[n|n-1]) = x[n]
Since the setterminal function resets the state estimator to its default value, call the setEstimator function after calling setterminal.
setEstimator(mpcobj,[],eye(2));
Compute the gain of the MPC controller when the constraints are inactive (unconstrained MPC), and compare it to the LQR gain.
mpcgain = dcgain(ss(mpcobj));
fprintf('\n(unconstrained) MPC: u(k)=[%8.8g,%8.8g]*x(k)',mpcgain(1),mpcgain(2));
(unconstrained) MPC: u(k)=[-1.6355962,-0.91707456]*x(k)
fprintf('\n LQR: u(k)=[%8.8g,%8.8g]*x(k)\n\n',-K(1),-K(2));
LQR: u(k)=[-1.6355962,-0.91707456]*x(k)
The state feedback gains are exactly the same.
Compare the performance of the LQR controller, the MPC controller with terminal weights, and a standard MPC controller.
Compute the closed-loop response for the LQR controller.
clsys = feedback(plant,K);
Tstop = 6;
x0 = [0.2;0.2];
[yLQR,tLQR] = initial(clsys,x0,Tstop);
Compute the closed-loop response for the MPC controller with terminal weights.
simOpt = mpcsimopt(mpcobj);
simOpt.PlantInitialState = x0;
[y,t,u] = sim(mpcobj,ceil(Tstop/Ts),r,simOpt);
Create a standard MPC controller with default prediction and control horizons (p=10, m=3). To match the other controllers, remove the output disturbance model and the default state estimator from the standard MPC controller.
mpcobjSTD = mpc(plant,Ts);
mpcobjSTD.Weights.MV = uwt;
mpcobjSTD.Weights.OV = ywt;
setoutdist(mpcobjSTD,'model',tf(zeros(2,1)))
setEstimator(mpcobjSTD,[],C)
Compute the closed-loop response for the standard MPC controller.
simOpt = mpcsimopt(mpcobjSTD);
[ySTD,tSTD,uSTD] = sim(mpcobjSTD,ceil(Tstop/Ts),r,simOpt);
Compare the controller responses.
plot(tSTD,ySTD,'r',t,y(:,1:2),'b',tLQR,yLQR,'mo')
ylabel('Plant Outputs')
legend('Standard MPC','MPC with Terminal Weights','LQR','Location','NorthEast')
The MPC controller with terminal weights has a faster settling time compared to the standard MPC controller. The LQR controller and the MPC controller with terminal weights perform identically.
You can improve the standard MPC controller performance by adjusting the horizons. For example, if you increase the prediction and control horizons (p=20, m=5), the standard MPC controller performs almost identically to the MPC controller with terminal weights.
This example shows that using terminal penalty weights can eliminate the need to tune the prediction and control horizons for the unconstrained MPC case. If your application includes constraints, using a terminal weight is insufficient to guarantee nominal stability. You must also choose appropriate horizons and possibly add terminal constraints. For more information, see [2].
[1] Scokaert, P. O. M. and J. B. Rawlings, "Constrained linear quadratic regulation," IEEE Transactions on Automatic Control (1998), Vol. 43, No. 8, pp. 1163-1169.
[2] Rawlings, J. B. and D. Q. Mayne, Model Predictive Control: Theory and Design. Nob Hill Publishing, 2010.
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Vannaka - The RuneScape Wiki
The Old School RuneScape Wiki also has an article on: osrsw:VannakaThe RuneScape Classic Wiki also has an article on: classicrsw:Combat instructor
VannakaCombat instructor
An expert on all forms of combat.
He doesn't look so dangerous any more.
VannakaCombat instructor? (edit) ? (edit)VannakaCombat instructorBeneath Cursed Tides? (edit)24 September 2002 (Update)12 March 2018 (Update)14 December 2015 (Update unknown)? (edit)No
146Talk-to, ExamineTalk-to&&SPLITPOINT&&Examine? (edit)? (edit)
? (edit)falsetruetruetrueHuman? (edit)1597, 22067, 22069, 22070, 22071944, 25278? (edit)? (edit)1597&&SPLITPOINT&&22067&&SPLITPOINT&&22069&&SPLITPOINT&&22070&&SPLITPOINT&&22071944&&SPLITPOINT&&25278? (edit)MRNDMRNDMRNDSlayer EquipmentMale? (edit)? (edit)Versions: 3
SMW Subobject for VannakaNPC ID: 1597, 22067, 22069, 22070, 22071Examine:
Release date: 24 September 2002Is members only: falseActions: Talk-to, ExamineIs variant of: Vannaka
SMW Subobject for Combat instructorNPC ID: 944, 25278Examine:
Release date: 12 March 2018Is members only: falseActions: Talk-to, ExamineIs variant of: Vannaka
SMW Subobject for Beneath Cursed TidesExamine:
Release date: 14 December 2015Is members only: falseActions: Talk-to, ExamineIs variant of: Vannaka
Vannaka (called the combat instructor on Tutorial Island) is the third-lowest level Slayer Master. He instructed new players in combat on Tutorial Island. Vannaka is also one of the NPCs associated with the Varrock achievements. According to Duradel, another Slayer Master, Vannaka was once a student of his, and does not wish to challenge him due to his master's great skill.[1]
Vannaka can be found in Edgeville graveyard south of the bank at the slayer master icon . He gives Slayer assignments to members with a combat level of 30 or higher. He is one of the few Slayer Masters with no Quest requirements, along with Turael and Mazchna.
After completion of the Varrock achievements, he can change players' Varrock Teleport location between Varrock Square, the south entrance to the Grand Exchange, or to the north Varrock Altar. However, he can only do this in person and not via the NPC Contact spell. To change locations, "Talk to" Vannaka, select "About the Task System", then select "I'd like to change my teleport point." Players can talk to him to obtain the elite Varrock achievement set rewards; Varrock armour 4, an antique lamp worth 30,000 experience and three antique lamps worth 40,000 experience.
1.2 Rescuing the cursed on Tutorial Island
Vannaka excelled in combat and Slayer. He is a devout Saradominist, highly regarded by Saradomin himself, and previously served in Saradomin's army of warriors.[2] He trained under Duradel, a powerful Slayer Master, who was once the best among them all, before being surpassed by Duradel's daughter, Kuradal, and later, also by Kuradal's elven student, Morvran.[1] Vannaka once challenged his former mentor to a duel to determine the best Slayer Master, and fought bravely but was eventually defeated at his hands; following the shame from this incident, he engaged in self-imposed exile.[1][2] He became a Combat Instructor on Tutorial Island, a former school for adventurers, until year 169 of the Fifth Age. Since then, he has served as a mid-level Slayer Master, based in Edgeville graveyard.
Rescuing the cursed on Tutorial Island[edit | edit source]
Main article: Beneath Cursed Tides
The player asks him to join them on their mission to find out what happened to Tutorial Island. When the player returns to Wizard Myrtle, they see that Vannaka and Myrtle are flirting. Myrtle instructs the player to go to Mudskipper Point. After arriving there the player again sees them flirting. The player and Vannaka both go down to the sunken island. Vannaka returns to the cave he used to tutor in to search for his friend Dezzick and the player goes off to search for the other instructors. Later on the player also goes to Vannaka's cave and he teaches them how to wield a two-handed sword underwater. The player then continues to search for the other tutors. Vannaka returns during the final fight when it appears that he was captured by the evil wizard, Hector Vivian. The player aids Myrtle in freeing Vannaka and beating the Vivian. The tutors, Myrtle and the player then reunite at Mudskipper Point and everyone goes their own way after rewarding the player.
Slayer points awarded by Vannaka
Completing 50 tasks assigned by Vannaka grants an average of 6.4 points per tasks, if Smoking kills has been completed.
Bats 40-60 14 (bat), 16 (warped), 27 (giant) 8.2, 9.2, 18.8 1 Bolts
Bear 40-60 11 (cub), 23 (black), 29 & 30 (grizzly cub), 32 & 40 (grizzly) 7.6, 10.2, 12, 12.6, 13, 16.6 1 Water spells
Catablepon 40-60 53, 54 49, 48.4 1 Bolts
Cave slime 40-60 26 11 17 Slash
Cockroach 40-60 1 (warped), 4 (drone), 30 (worker), 36 (soldier) 1 Crush (warped)
Earth spells (other)
Fleshcrawler 40-60 39, 40 50 1 Stab
Frog 40-60 36 (big), 44 (giant), 77 (frog) 14.4, 18.8, 54.8 1 Slash
Ghost 40-70 25-48 (ghost), 42, (tormented wraith), 53 (tortured soul), 61 (ghostly warrior), varies (revenants) 10.6-22.4, 52, 38.8, 32.6, varies 1 Varies
Grotworm 40-60 28 (young), 63 (regular), 98 (mature) 12, 107.8, 343.5 1 Bolts
Hill giant 40-60 44 39 1 Air spells
Hobgoblin 40-60 28, 30, 32 & 35 (hobgoblin), 91 (God Wars Dungeon) 11.4, 12.6, 13, 14, 268.6 1 Air spells
Ice giant 40-60 51 70 1 Fire spells
Ice warrior 40-60 54 59 1 Fire spells
Icefiend 40-60 35 (icefiend), 84 (God Wars Dungeon) 7, 140 1 Fire spells
Kalphite 40-60 58 (worker), 65 (soldier), 72 (guardian), 75 (exiled worker), 84 (exiled paragon, exiled soldier), 98 (exiled marauder, exiled guardian), 110 (corrupted guardian, corrupted marauder), 333 (queen), 356 (exiled queen), 2500 (king) 59.3, 75.4, 95, 109.8, 147, 140, 229, 229, 494, 494, 1309.4, 2055, 3963 1 Varies
Minotaur 40-60 14, 52 16.6, 46.4 1 Water spells
Mogre 40-60 54 48 32 Air spells
Moss giant 40-60 28, 51, 61 60, 85, 120 1 Slash
Scorpion 40-60 14, 25 & 26 (regular), 39 (king), 93 (uncharted isles), 98 (corrupted) 8.2, 31.8, 33.2, 15.8, 98.6, 353.2 1 Varies
Skeleton 40-60 Varies, see Skeleton (disambiguation) Varies 1 Varies
Spider 40-60 1 (spider), 2 (giant), 4 (corpse), 15 (spider, poison), 33 & 39 (giant), 42 (ice, blessed), 46 (shadow), 63 (poison, crypt), 70 (fever), 84 (giant crypt), 86 (jungle), 95 (deadly red), 98 (lava) 6.4, 6.4, 6.6, 17.6, 34.2, 13.4, 31.2, 17.2, 17.2, 19.6, 34.2, 86.6, 77, 103.6, 491 1 Varies
Wall beast 10-20 47 20.6 35 Water spells
Wolf 40-60 8 (adolescent white), 15 (desert), 16 (wolf), 23 (wolf), 30 (white), 35 (three's company), 42 (wolf), 43 (big), 46 (dire), 49 (fenris), 71 (ice), 88 (jungle), 95 (ice) 7, 8.8, 9.2, 20, 12.6, 14, 17.2, 18, 19.6, 21.6, 45.4, 81, 207.4 1 Varies
Zombie 40-60 Varies, see Zombie (disambiguation) Varies 1 Varies
Vannaka will occasionally (or randomly) offer a special task in place of a regular Slayer task. The player can decline to take the special task without penalty. If the player accepts the special task, completing the task:
Earns 4,000 extra Slayer experience.
Vannaka's special task is for the player to bring him one scale from each of the four types of chromatic dragons:
A perfect green dragon scale
A perfect blue dragon scale
A perfect red dragon scale
A perfect black dragon scale
Both adult and baby chromatic dragons can drop perfect scales, which are only found while on this special task. It seems that you now always get the scale on the first kill of each type of dragon, and no further will drop when you already have one. Previously, multiple scales of the same type of dragon could be obtained, but extras were destroyed when the task was complete. It may still be possible to obtain multiple scales by killing several dragons before picking up the drops. Note that the challenge is only complete when the scales are given to Vannaka; simply collecting all four scales does not finish the task.
{\displaystyle {\frac {w}{S}}\times 100\%}
{\displaystyle w}
{\displaystyle S}
{\displaystyle S}
. For Vannaka, assuming all possible assignments are available,
{\displaystyle S}
Vannaka speaks in:
Transcripts Vannaka is the author or co-author of:
Note from Vannaka
Vannaka has left a helpful note in his previous spot to point players towards his new position.
Vannaka has left a helpful note in his previous location to point players towards his new one.
Vannaka has moved out of the Edgeville Dungeon to the ruins just south of the Edgeville bank, joining Mandrith, who has moved from the Edgeville Bank.
Vannaka's armour and sword have been updated.
His Strength level (112) was revealed in Postbag from the Hedge 14.[4] The Evolution of Combat's combat level system would make him at least level 174.
Vannaka is most well known because of his ability to wield a steel 2h sword, and a dragon sq shield at the same time. This issue was addressed by a player to Saradomin, who replied that he had the rare power to single-handedly wield a two-handed sword because he was one of the strongest warriors ever.[2] Vannaka offers a cosmetic override which gives the appearance of wielding his 2h sword as a 1h weapon as a quest reward for completing Beneath Cursed Tides.
It is assumed that Vannaka is a member of Legends' Guild, because he can wield the dragon square shield.
Vannaka has not received a graphical update for his dragon square shield.
He has a gravestone in the New Varrock graveyard which reads "Vannaka. Valiantly killed a thousand zombies. Died while checking his enchanted gem to see if his assignment was complete."
^ a b c Jagex. Postbag 20 - "Transcript:Tales from earlier ages", Letter 8, by Duradel. RuneScape Postbags from the Hedge.
^ a b c Jagex. "God Letter 27 - Saradomin Enlightens, 10" (Archived from the original.) RuneScape God Letters.
^ Jagex. Postbag 14 - "Transcript:The woeful tale of the bunny ears.", Letter 8, by Vannaka. RuneScape Postbags from the Hedge.
Retrieved from ‘https://runescape.wiki/w/Vannaka?oldid=35729011’
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Srikant VS Terrorists | Toph
Editorial for Srikant VS Terrorists
The agent must kill all the terrorists before
k minutes. And to kill all the terrorists, the total number of bullets required to kill all the terrorists must be less than or equal to the number of bullets (
r*6
r∗6) the agent has in his possession. The path from each cell leads the agent to a certain cell. The paths/roads are one way. Two/more different cells can have paths that lead to one cell. The agent must not go back to a cell that is already visited. Because if he does, he will fall into an infinite loop. If
a_i
ai is denoted by
0, the agent cannot move further. So, after ensuring Srikant has enough bullets to kill all the terrorists, run a
dfs and check is it possible to go all the cells where terrorists are stayed. If it is possible, calculate the required time. Complexity
O(n) or
O(nlogn) depending on your implementation.
faria_efaEarliest, 10M ago
prodip_bsmrstuLightest, 12 MB
antihashShortest, 824B
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Muisca_calendar Knowpia
The Muisca calendar was a lunisolar calendar used by the Muisca. The calendar was composed of a complex combination of months and three types of years were used; rural years (according to Pedro Simón, Chibcha: chocan),[1] holy years (Duquesne, Spanish: acrótomo),[2] and common years (Duquesne, Chibcha: zocam).[3] Each month consisted of thirty days and the common year of twenty months, as twenty was the 'perfect' number of the Muisca, representing the total of extremeties; fingers and toes. The rural year usually contained twelve months, but one leap month was added. This month (Spanish: mes sordo; "deaf month") represented a month of rest. The holy year completed the full cycle with 37 months.
The Muisca, inhabiting the central highlands of the Colombian Andes (Altiplano Cundiboyacense), used one (arranged by Bochica[4]) of the advanced calendrical systems of Pre-Columbian America,[5] the others being the Incan and Maya calendars, and the ones used by other Mesoamericans including the Aztecs.
Important Muisca scholars who have brought the knowledge of the Muisca calendar and their counting system to Europe were Spanish conquistador Gonzalo Jiménez de Quesada who encountered Muisca territory in 1537, Bernardo de Lugo (1619),[6] Pedro Simón in the 17th century and Alexander von Humboldt and José Domingo Duquesne published their findings in the late 18th and early 19th century.[4][7][8][9] At the end of the 19th century, Vicente Restrepo wrote a critical review of the work of Duquesne.[10]
21st century researchers are Javier Ocampo López[11] and Manuel Arturo Izquierdo Peña, anthropologist who published his MSc. thesis on the Muisca calendar.[12]
The Muisca used a vigesimal counting system and counted with their fingers. Their system went from 1 to 10 and for higher numerations they used the prefix quihicha or qhicha, which means "foot" in their Chibcha language Muysccubun. Eleven became thus "foot one", twelve "foot two", etc. As in the other pre-Columbian civilizations, the number 20 was special. It was the total number of all body extremities; fingers and toes. The Muisca used two forms to express twenty: "foot ten"; quihícha ubchihica or their exclusive word gueta, derived from gue, which means "house". Numbers between 20 and 30 were counted gueta asaqui ata ("twenty plus one"; 21), gueta asaqui ubchihica ("twenty plus ten"; 30). Larger numbers were counted as multiples of twenty; gue-bosa ("20 times 2"; 40), gue-hisca ("20 times 5"; 100).[4] The Muisca script consisted of hieroglyphs, only used for numerals.[13] There is doubt as to the whether or not the document reporting the existence of this hieroglyphic numerical system is to be believed, as it is only primary source attesting this system.[14]
Numbers 1 to 10 and 20Edit
Humboldt, 1878[4]
De Lugo, 1619[6]
Muisca hieroglyphs[15]
bozha / bosa
boʒha
mhuyca / muyhica
mhuɣcâ
hicsca / hisca
hɣcſcâ
qhupqa / cuhupqua
qhûpqâ
shuzha / suhuza
shûʒhâ
hubchibica / ubchihica
hubchìhicâ
quihicha ubchihica
qhicħâ hubchìhicâ
guêata
To name the days and months the Muisca did not use numbers higher than 10, except gueta for their perfect number of 20. Instead, they named the 11th month just like the 1st; ata. Same for the other months and days until 19. That rather confusing system made it difficult to distinguish the 21st month from the 1st or 11th, but their naming of the three different years solved this.
Time calculationEdit
Sketch of the complex Muisca calendar by Alexander von Humboldt[16]
The calculation of time in the Muisca calendar was a complex combination of different time spans, which describe periods that extends from weeks to years, centuries and even higher time spans. The day was defined by the daily solar cycle, whereas the month was defined, depending on the context, by both the synodical and the sidereal lunar cycles.[17] Different scholars have described variation of weeks (3, 10 or 15 days), years (rural, common and holy) centuries (common and holy) and eventually, higher periods of time as the Bxogonoa.
The Muisca called "day" sua (the word for "Sun") and "night" za. The priests had divided a day in four parts:[18] suamena (from sunrise to mid-day), suameca (from mid-day to sunset), zasca was the time from sunset to midnight and chaqüi the time from midnight to sunrise.[19]
Weeks and monthsEdit
About the configuration of the weeks in the Muisca calendar different chroniclers show various subdivisions. Gonzalo Jiménez de Quesada describes a month of 30 days comprising three weeks of ten days,[20] Pedro Simón stated the Muisca had a month composed of two weeks of 15 days[21] and José Domingo Duquesne and Javier Ocampo López wrote the Muisca week had just three days, with ten weeks in a month.[21][22] Izquierdo suggests, however, that the concept for a standardized week was alien to the Muisca indeed, who instead organized the days of the month in terms of the varying activities of their social life.[23]
The Muisca, like the Incas in the Central Andes, very probably took notice of the difference between the synodic month (29 days, 12 hours, 44 minutes); the time between two full Moons, and the sidereal month (27 days, 7 hours, 43 minutes); the time it takes for the Moon to reach the same position with respect to the stars.[17]
The Muisca word for year was zocam, which is always used in combination with a number: zocam ata, "year one", zocam bosa, "year two". Following the works of Duquesne, three types of years were used; Rural years, Common years and Priest's years. The years were composed of different sets of months:
The Rural Year contained 12 synodic months,
The Priest's Year composed of 37 synodic months, or 12 + 12 + 13 synodic months (the 13th was a leap month, called "deaf" in Spanish),
The Common Year composed by 20 months, making a full common Muisca year 600 days or 1.64 times a Gregorian year.[8][24] Izquierdo suggests, however, that this year, unlike the Rural and the Priest's years, was based on the sidereal lunar cycle.[25]
Centuries and higher time spansEdit
Chía rising over the Eastern Hills above the savanna the Muisca lived
According to Duquesne, the Muisca devised a Priest's Century by scaling up The Priest's Year by gueta (20 times 37 months; 740) which approximately equals 60 Gregorian years.[24][26] The same scholar referred to a Common Century (siglo vulgar) comprising 20 times 20 months.[27] Pedro Simón's differences on the accounts of the mythical arrival of Bochica to the Muisca territory brings clues about the nature of the Priest's Century. According Simón, the century (edad) corresponded to 70 (setenta) years, however, Izquierdo suggests that such a value is typo of 60 (sesenta) years, which is a value that better matches the entire calendar's description.[28] Besides the centuries, the chronicles describe further periods of time: the Astronomical Revolution as called by Duquesne, corresponds to 5 Priest's Years or 185 synodical months, thus comprising a quarter of a Priest's Century. Simón describes also an additional time period named the Bxogonoa which corresponds to 5 Priest's Centuries. Again, both Duquesne and Humboldt describe another time span, the Dream of Bochica which accounted for 100 Priest's Centuries, which correspond to 2000 Priest's Years or 5978 Gregorian years.[29] After the analysis of all these many units of time, Izquierdo proposed a hierarchical organization where these periods are the product of multiplying the months of The Priest's Year by both 5 and the first three powers of 20:[29]
Time period Synodical months Time period Synodical months Time period Synodical months
Priest's year
{\displaystyle 37\times 20^{0}=\ \ 37}
Priest's Century
{\displaystyle 37\times 20^{1}=\ \ 740}
Arrival of Bochica
{\displaystyle 37\times 20^{2}=14800}
Astronomical Revolution
{\displaystyle 5\times 37\times 20^{0}=185}
Bxogonoa
{\displaystyle 5\times 37\times 20^{1}=3700}
Dream of Bochica
{\displaystyle 5\times 37\times 20^{2}=74000}
To name the months, the Muisca did not use higher numbers than 10, except for the 20th month, indicated with the 'perfect' number gueta. The calendar table shows the different sets of zocam ("years") with the sets of months, as published by Alexander von Humboldt.[8] The meaning of each month has been described by Duquesne in 1795 and summarized by Izquierdo Peña in 2009.[30]
Rural year
Symbols; "meanings" - activities
1 1 Ata Ata Ata Jumping toad; "start of the year"
2 Bosa Nose and nostrils
3 Mica Open eyes and nose; "to look for", "to find"
4 Muyhica Two closed eyes; "black thing", "to grow"
5 Hisca Two fingers together; "green thing", "to enjoy"
6 Ta Stick and cord; "sowing" - harvest
7 Cuhupqua Two ears covered; "deaf person"
8 Suhuza Tail; "to spread"
9 Aca Toad with tail connected to other toad; "the goods"
10 Ubchihica Ear; "shining Moon", "to paint"
2 13 Bosa Mica
14 Muyhica
15 Hisca
17 Cuhupqua
18 Suhuza harvest
20 Gueta Lying or stretched toad; "sowing field", "to touch"
21 Bosa Ata
3 25 Mica Hisca
28 Suhuza
30 Ubchihica harvest
36 Ta Embolismic month
4 37 Deaf month Chuhupqua End of the holy year; full cycle
The Gregorian month of December was a month of celebrations with yearly feasts, especially in Sugamuxi called huan, according to Pedro Simón.[31]
Archeological evidencesEdit
The archeological evidence for the Muisca calendar and its use is found in ceramics, textiles, spindles, petroglyphs, sites and stones.[32]
Important findings are:
Choachí Stone, found in the first half of the 20th century in the municipality of Choachí may represent a calculator to convert the different parts of the complex Muisca calendar[33][34]
Ceremonial flute (fotuto ceremonial), decorated flute made of a marine snail shell, found in Socorro, Santander, located in the Archeology Museum Sogamoso[35]
Decorated textile, found in Belén, Boyacá and located in the museum of Pasca, regarded as a "Muisca codex"[36]
El Infiernito, astronomical site of the Muisca near Villa de Leyva[37]
Jaboque, in this humedal ancient menhirs were found, indicating an astronomical knowledge of the Muisca[38]
Muisca astronomy, agriculture, numerals
^ Izquierdo Peña, 2014, 11:48
^ a b c d Humboldt, 1878, Part 1
^ Ocampo López, 2007, Ch.V, p.188
^ a b (in Spanish) 1619 - Muisca numbers according to Bernardo de Lugo - accessed 29-04-2016
^ Humboldt, 1878, Part 2
^ a b c Humboldt, 1878, Part 3
^ Duquesne, 1795
^ Restrepo, 1892
^ Ocampo López, 2007, Ch. V, p.228-229
^ Izquierdo Peña, 2009, p.1-170
^ Izquierdo Peña, 2009
^ Humboldt, 1878, Part 1, Page 389
^ a b Izquierdo Peña, 2014, 56:35
^ (in Spanish) Calendario lunar de los muiscas - accessed 28-04-2016
^ (in Spanish) Calendario muisca - Pueblos Originarios - accessed 28-04-2016
^ Izquierdo Peña, 2009, p.32
^ a b Izquierdo Peña, 2009, p.33
^ Izquierdo Peña, 2011, p.110
^ a b Duquesne, 1795, p.3
^ a b Izquierdo Peña, 2011, p.114
^ Izquierdo Peña, 2014, 1:17:25
^ Jaboque Petroform Menhirs - accessed 05-05-2016
Acosta, Joaquín. 1848. Compendio histórico del descubrimiento y colonización de la Nueva Granada en el siglo décimo sexto, 1-460. Beau Press. Accessed 2016-07-08.
Duquesne, José Domingo. 1795. Disertación sobre el calendario de los muyscas, indios naturales de este Nuevo Reino de Granada - Dissertation about the Muisca calendar, indigenous people of this New Kingdom of Granada, 1-17. Accessed 2016-07-08.
Humboldt, Alexander von. 1878. VI.Sitios de las Cordilleras y monumentos de los pueblos indígenas de América - Calendario de los indios muiscas - Parte 1 - Views of the Cordilleras and Monuments of the Indigenous Peoples of the Americas - Muisca calendar - Part 1. Biblioteca Luis Ángel Arango. Accessed 2016-07-08.
Humboldt, Alexander von. 1878. VI.Sitios de las Cordilleras y monumentos de los pueblos indígenas de América - Calendario de los indios muiscas - Parte 2. Biblioteca Luis Ángel Arango. Accessed 2016-07-08.
Restrepo, Vicente. 1892. Crítica de los trabajos arqueológicos del Dr. José Domingo Duquesne - Review of the archeological works of Dr. José Domingo Duquesne, 1–44. Accessed 2016-07-08.
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Millioctave Knowpia
The millioctave (moct) is a unit of measurement for musical intervals. As is expected from the prefix milli-, a millioctave is defined as 1/1000 of an octave. From this it follows that one millioctave is equal to the ratio 21/1000, the 1000th root of 2, or approximately 1.0006934 ( play (help·info)).
Given two frequencies a and b, the measurement of the interval between them in millioctaves can be calculated by
{\displaystyle n=1000\log _{2}\left({\frac {a}{b}}\right)\approx 3322\log _{10}\left({\frac {a}{b}}\right)}
Likewise, if you know a note b and the number n of millioctaves in the interval, then the other note a may be calculated by:
{\displaystyle a=b\times 2^{\frac {n}{1000}}}
Like the more common cent, the millioctave is a linear measure of intervals, and thus the size of intervals can be calculated by adding their millioctave values, instead of multiplication, which is necessary for calculations of frequencies.
A millioctave is exactly 1.2 cents.
The millioctave was introduced by the German physicist Arthur von Oettingen in his book Das duale Harmoniesystem (1913). The invention goes back to John Herschel, who proposed a division of the octave into 1000 parts, which was published (with appropriate credit to Herschel) in George Biddell Airy's book on musical acoustics.[1]
Compared to the cent, the millioctave has not been as popular because it is not aligned with just intervals. It is however occasionally used by authors who wish to avoid the close association between the cent and twelve-tone equal temperament. Some considers that the millioctave introduces as well a bias for the less familiar 10-tone equal temperament[2] however this bias is common in the decimal system.
^ Airy, George Biddell (1871). On Sound and Atmospheric Vibrations with the Mathematical Elements of Music. London: Macmillan & Co. pp. 222. We are permitted by Sir John Herschel to explain a system proposed by him... It consists in using such a modulus that the logarithm of 2 is 1000.
^ Huygens-Fokker Foundation. "Logarithmic Interval Measures". Archived from the original on 2007-02-14. Retrieved 2007-06-13.
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Real Analysis/Functions - Wikibooks, open books for an open world
Real Analysis/Functions
Inverse Functions Inverse Functions→
Functions are a component of mathematics so fundamental, it appears in every branch going forward. Although in elementary mathematics, the understanding of a function may be trivial at best, in practice, functions allow you to prove even the most fundamental aspects of mathematics. Even the exposure chapters in this book, which provides basic introduction to higher mathematics using the concepts of this book as a stepping stone, use it. Thus, the concept of making functions rigorous will be of utmost importance—and will be handled in this chapter.
We will assume that you have an intuitive understanding of functions. If not, they can be found in either earlier mathematics wikibooks; the Algebra book and the Calculus book, or found in the higher mathematics wikibooks; the Discrete Mathematics. This should indicate just how fundamental these concepts are. It is safe to say that although a rigorous understanding of functions may not be needed going forward, understanding its structural component and how theorems rely on it will be essential in your mathematical career.
The definition of a function is as follows:
Definition of a Function ƒ
A set of ordered pairs (a,b).
Definition of an Ordered Pair (a,b)
A set such that this set is equal to {{a},{a,b}}.
Yes, that is it. The definition of a function relies on very little; the concept of a function hinges only on set theory. Although this definition may sound a little too simplistic, it surprising is not. The intuitive notion of a function is perfectly satisfied. Certain properties of functions can be described from this definition through a definition argument or derived. Namely the intuitive properties of
An input value must output some value based on the function definition.
One input cannot output more than one value.
The same input should yield the same output.
The domain and range of the function should be derivable (in mathematics, an existence proof is sufficed).
The operations governing numbers must also apply predictably for functions.
These will be described in depth in the following sections.
Many properties can be cleared by simply defining them. After all, the function is already defined as a very specific kind of set, which means a lot of properties can be derived simply through its definition. In fact, we will use our textbox notation to define them easily.
Definition of ƒ(a)
Given a, the variable b in the ordered pair (a,b).
In writing, this is called the value at f of a or any similar combination of words.
Definition of input value
Given the statement
{\displaystyle f(a)=b}
, the variable a
Definition of output value
{\displaystyle f(a)=b}
, the variable b
Definition of a Domain of a Function ƒ
A set composed of the variable a of every ordered pair (a,b).
Definition of a Range of a Function ƒ
A set composed of the variable b of every ordered pair (a,b).
To maintain the scope for this wikibook, we will not rigorously prove the necessity of these definitions. Why? They rely almost essentially on set theory and its associating theorems, formulas, and operations. Although this wikibook uses set notation, this wikibook is primarily concerned with rigorously defining elementary mathematics and exposing readers to higher mathematics. In effect, we will assume that the proof for these definitions as merely an axiomatic statement. However, you are free to prove them for personal reasons.
There are many different ways to notate functions. The multitude of different notation styles is due to the plethora of mathematical fields, each of which demand certain types of information from our function. Given that this wikibook is on Real Analysis, we do not necessarily need the function definition requiring the numbers accepted for the domain and range to be explicit. Instead, we will primarily rely on these types of notation, with the last one being used rarely.
List of Function Notation
{\displaystyle f={\text{ insert definition here}}}
{\displaystyle f(x)={\text{ insert definition here}}}
{\displaystyle x\rightarrow f}
x arrow [insert what ƒ is defined as here]
In this wikibook, we will primarily use the first two forms of notation, with the first one being used the most. Why? It melds the definition of function and variable, which will be a useful concept moving forward in higher mathematics. It also saves space as other operations (which you will learn later) often have special notation for f(x). f(x) can also easily be mistaken as f(a), which refers to an actual value, unlike f(x), which refers to a definition.
Given two ordered pairs (a,b) and (c,d), if they both equal, it must follow that a = c and b = d is true.
Proof of Uniqueness in Input
Given that a is the common member of {{a},{a,b}} and c is the common member of {{a},{a,b}} and they both should be equal (which means what comprises the set must be the same, a = c.
Case 1: b = a.
Case 2: b ≠ a.
{{a},{a,b}} = {{a},{a,a}} = {{a},{a}} = {a} b must exists as a member in the set {{a},{a,b}}. This is also true for {{a},{a,d}}, because they are equal.
Because they both must be equal, d = a too. Because b ≠ a., it's not in the set {a}, it must be in the set {a,d}, which implies that b = d.
{\displaystyle \blacksquare }
This verifies the nature of equality of a function; for two functions to be equal, both the inputs and outputs must be equal.
Functions obey many of the same operations as normal numbers. However, one key difference is the domain of the function, which may change depending on the operator.
List of Operations for Functions
{\displaystyle (f+g)(x)=f(x)+g(x)}
domain f ∩ domain g
{\displaystyle (f-g)(x)=f(x)-g(x)}
{\displaystyle (f\cdot g)(x)=f(x)\cdot g(x)}
domain f ∪ domain g
{\displaystyle \left({\dfrac {f}{g}}\right)(x)={\dfrac {f(x)}{g(x)}}}
domain f ∪ domain g \ {a : g(a) = 0}
{\displaystyle f\circ g(x)=f(g(x))}
domain g ∪ {g(x) : f(g(x)) ∈ domain f}
Given the scope of this wikibook, we will not rigorously prove the necessity of these definitions. Why? They rely almost essentially on set theory and its associating theorems, formulas, and operations. Although this wikibook uses set notation, this wikibook is primarily concerned with rigorously defining elementary mathematics and exposing readers to higher mathematics. In effect, we will assume that the proof for these definitions as merely an axiomatic statement. However, you are free to prove them for personal reasons.
TheoremsEdit
In terms of functions, there are a small pair of theorems that prove one of the most fundamental aspects of elementary mathematics — that is the idea of algebra. Although the following theorem will not sound related to algebra, it actually validates the nature of the process. One major transition between elementary mathematics and higher mathematics is the recognition that algebraic manipulations is a kind of proof as well. Because of this, the first proof you could have read in this wikibook (if you follow it linearly) is this theorem. Examples of how this theorem relates to algebra are to follow after.
{\displaystyle a=b}
, then you can apply any function to both sides i.e.
{\displaystyle f(a)=f(b)}
This proof relies on the function definition and the axioms relating equality. As a reminder, f(x) references the definition of a function, not the value of f at x.
Proof of Algebra's Validity
Given the definition of a function, we can state that the variable a can map to the value at f of a, which we will call x in this proof.
{\displaystyle {\begin{aligned}f&=\{\ldots ,\{\{a\},\{a,x\}\},\ldots \}\implies \\a&\rightarrow x\end{aligned}}}
Because b = a and that we are working with a function f, we can claim that b can also map to the value at f of a.
{\displaystyle {\begin{aligned}f&=\{\ldots ,\{\{b=a\},\{b=a,x\}\},\ldots \}\implies \\b&\rightarrow x\end{aligned}}}
Combined together, the relationship becomes clear.
{\displaystyle a=b\implies a\rightarrow x=b\rightarrow x\implies x=x}
{\displaystyle \blacksquare }
A proper analysis on what algebra is can help solidify the theorem and its importance. As an example, we will use a simple equation
{\displaystyle 17=3x+2}
to illustrate our point. The following explanation is written below.
Expressed via our Theorem
First, we will subtract both sides by 2. It is equivalent to creating a function
{\displaystyle f=x-2}
and then applying our theorem. Next, we will divide both sides by 3. Note that this is also equivalent to creating another function
{\displaystyle g=x/3}
and then applying our theorem. Note that by then, we have essentially solved our question. We just apply the equivalence property axiom and "reverse" the position of x and 5 for aesthetic purposes.
{\displaystyle {\begin{aligned}17&=3x+2\\17-2&=3x+2-2\\15&=3x\\{\frac {15}{3}}&={\frac {3x}{3}}\\5&=x\\x&=5\end{aligned}}}
{\displaystyle {\begin{aligned}17=3x+2&\implies f(17)=f(3x+2)\\&\implies 15=3x\implies \\g(15)=g(3x)&\implies 5=x\\&\implies x=5\end{aligned}}}
The similarity is not by chance. Turns out, the notion of algebra is actually this property applied over and over again. Even the limitations of algebra can be answered using this property. Most famously is the relationship of squaring (in which squaring an equation seemingly introduces a new value that may be incorrect. For example, the quadratic equation), which can be easily answered by saying simply, "algebra does not guarantee reversibility". Note that our proof actually validates the process of even squaring, as it is not a biconditional relationship. In fact, the most bizarre thing about this theorem is that it doesn't justify reversibility. What is reversibility?
Definition of Reversibility
The algebraic property of biconditionality on equations; An equation (a mathematical statement with only an equal sign) must imply each other.
Insert the polynomial function definition and the rational function definition
Retrieved from "https://en.wikibooks.org/w/index.php?title=Real_Analysis/Functions&oldid=3039107"
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2015 Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus
José A. Adell, A. Lekuona
We give accurate estimates of the constants
{C}_{n}\left(\mathcal{A}\left(I\right),x\right)
appearing in direct inequalities of the form
|{L}_{n}f\left(x\right)-f\left(x\right)|\le {C}_{n}\left(\mathcal{A}\left(I\right),x\right){\omega }_{2}
\left(f;\sigma \left(x\right)/\sqrt{n}\right)
f\in \mathcal{A}\left(I\right)
x\in I
n=1,2,\dots ,
{L}_{n}
is a positive linear operator reproducing linear functions and acting on real functions
f
I
\mathcal{A}\left(I\right)
is a certain subset of such functions,
{\omega }_{2}\left(f;·\right)
is the usual second modulus of
\sigma \left(x\right)
is an appropriate weight function. We show that the size of the constants
{C}_{n}\left(\mathcal{A}\left(I\right),x\right)
mainly depends on the degree of smoothness of the functions in the set
\mathcal{A}\left(I\right)
and on the distance from the point
x
to the boundary of
I
. We give a closed form expression for the best constant when
\mathcal{A}\left(I\right)
is a certain set of continuous piecewise linear functions. As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.
José A. Adell. A. Lekuona. "Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus." Abstr. Appl. Anal. 2015 1 - 11, 2015. https://doi.org/10.1155/2015/915358
José A. Adell, A. Lekuona "Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus," Abstract and Applied Analysis, Abstr. Appl. Anal. 2015(none), 1-11, (2015)
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Extrapolation is tough for trees! | R-bloggers
Out-of-sample extrapolation
This post is an offshoot of some simple experiments I made to help clarify my thinking about some machine learning methods. In this experiment I fit four kinds of model to a super-simple artificial dataset with two columns, x and y; and then try to predict new values of y based on values of x that are outside the original range of y. Here’s the end result:
An obvious limitation of the extreme gradient boosting and random forest methods leaps out of this graph – when predicting y based on values of x that are outside the range of the original training set, they presume y will just be around the highest value of y in the original set. These tree-based methods (more detail below) basically can’t extrapolate the way we’d find most intuitive, whereas linear regression and the neural net do ok in this regard.
Data and set up
The data was generated by this:
# set up functionality for modelling down the track
library(xgboost) # extreme gradient boosting
library(nnet) # neural network
library(ranger) # for random forests
library(rpart) # for demo single tree
library(viridis) # for palette of colours
library(grid) # for annotations
# sample data - training set
y <- 3 + 0.3 * x + rnorm(100)
# extrapolation / test set, has historical data plus some more extreme values
extrap <- data.frame(x = c(x, 1:5 * 10 + 100))
The four different modelling methods
The four methods I’ve used are:
linear regression estimated with ordinary least squares
single layer artificial neural network with the nnet R package
extreme gradient boosting with the xgboost R package
random forests with the ranger R package (faster and more efficient than the older randomForest package, not that it matters with this toy dataset)
All these four methods are now a very standard part of the toolkit for predictive modelling. Linear regression,
E(\textbf{y}) = \textbf{X}\beta
is the oldest and arguably the most fundamental statistical model of this sort around. The other three can be characterised as black box methods in that they don’t return a parameterised model that can be expressed as a simple equation.
Fitting the linear model in R is as simple as:
mod_lm <- lm(y ~ x)
Neural networks create one or more hidden layers of machines (one in this case) that transform inputs to outputs. Each machine could in principle be a miniature parameterised model but the net effect is a very flexible and non-linear transformation of the inputs to the outputs. This is conceptually advanced, but simple to fit in R again with a single line of code. Note the meta-parameter size of the hidden layer, which I’ve set to 8 after some experimentation (with real life data I’d used cross-validation to test out the effectiveness of different values).
mod_nn <- nnet(y ~ x, size = 8, linout = TRUE)
xgboost fits a shallow regression tree to the data, and then additional trees to the residuals, repeating this process until some pre-set number of rounds set by the analyst. To avoid over-fitting we use cross-validation to determine the best number of rounds. This is a little more involved, but not much:
# XG boost. This is a bit more complicated as we need to know how many rounds
# of trees to use. Best to use cross-validation to estimate this. Note -
# I use a maximum depth of 2 for the trees which I identified by trial and error
# with different values of max.depth and cross-validation, not shown
xg_params <- list(objective = "reg:linear", max.depth = 2)
mod_cv <- xgb.cv(label = y, params = xg_params, data = as.matrix(x), nrounds = 40, nfold = 10) # choose nrounds that gives best value of root mean square error on the training set
best_nrounds <- which(mod_cv$test.rmse.mean == min(mod_cv$test.rmse.mean))
mod_xg <- xgboost(label = y, params = xg_params, data = as.matrix(x), nrounds = best_nrounds)
Then there’s the random forest. This is another tree-based method. It fits multiple regression trees to different row and column subsets of the data (of course, with only one column of explanatory features in our toy dataset, it doesn’t need to create different column subsets!), and takes their average. Doing this with the defaults in ranger is simple again (noting that lm, nnet and ranger all use the standard R formula interface, whereas xgboost needs the input as a matrix of explanatory features and a vector of ‘labels’ ie the response variable).
mod_rf <- ranger(y ~ x)
Finally, to create the graphic from the beginning of the post with the predictions of each of these models using the extrapolation dataset, I create a function to draw the basic graph of the real data (as I’ll be doing this four times which makes it worth while encapsulating in a function, to avoid repetitive code). I call this function once for each graphic, and superimpose the predicted points over the top.
p <- function(title){
plot(x, y, xlim = c(0, 150), ylim = c(0, 50), pch = 19, cex = 0.6,
main = title, xlab = "", ylab = "", font.main = 1)
predshape <- 1
par(mfrow = c(2, 2), bty = "l", mar = c(7, 4, 4, 2) + 0.1)
p("Linear regression")
points(extrap$x, predict(mod_lm, newdata = extrap), col = "red", pch = predshape)
p("Neural network")
points(extrap$x, predict(mod_nn, newdata = extrap), col = "blue", pch = predshape)
p("Extreme gradient boosting")
points(extrap$x, predict(mod_xg, newdata = as.matrix(extrap)), col = "darkgreen", pch = predshape)
p("Random forest")
fc_rf <- predict(mod_rf, data = extrap)
points(extrap$x, fc_rf$predictions, col = "plum3", pch = predshape)
grid.text(0.5, 0.54, gp = gpar(col = "steelblue"),
label = "Tree-based learning methods (like xgboost and random forests)\nhave a particular challenge with out-of-sample extrapolation.")
label = "In all the above plots, the black points are the original training data,\nand coloured circles are predictions.")
Tree-based limitations with extrapolation
The limitation of the tree-based methods in extrapolating to an out-of-sample range are obvious when we look at a single tree. Here’a single regression tree fit to this data with the standard rpart R package. This isn’t exactly the sort of tree used by either xgboost or ranger but illustrates the basic approach. The tree algorithm uses the values of x to partition the data and allocate an appropriate value of y (this isn’t usually done with only one explanatory variable of course, but it makes it simple to see what is going on). So if x is less than 11, y is predicted to be 4; if x is between 11 and 28 y is 9; etc. If x is greater than 84, then y is 31.
What happens in the single tree is basically repeated by the more sophisticated random forest and the extreme gradient boosting models. Hence no matter how high a value of x we give them, they predict y to be around 31.
The implication? Just to bear in mind this limitation of tree-based machine learning methods - they won’t handle well new data that is out of the range of the original training data.
Here’s the code for fitting and drawing the individual regression tree.
#==============draw an example tree===================
# this is to illustrate the fundamental limitation of tree-based methods
# for out-of-sample extrapolation
tree <- rpart(y ~ x)
par(font.main = 1, col.main = "steelblue")
rpart.plot(tree, digits = 1,
box.palette = viridis(10, option = "D", begin = 0.85, end = 0),
shadow.col = "grey65", col = "grey99",
main = "Tree-based methods will give upper and lower bounds\nfor predicted values; in this example, the highest possible\npredicted value of y is 31, whenever x>84.")
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Quadrupole magnet - Wikipedia
Group of four magnets
Four bar magnets configured to produce a quadrupole
Quadrupole magnets, abbreviated as Q-magnets, consist of groups of four magnets laid out so that in the planar multipole expansion of the field, the dipole terms cancel and where the lowest significant terms in the field equations are quadrupole. Quadrupole magnets are useful as they create a magnetic field whose magnitude grows rapidly with the radial distance from its longitudinal axis. This is used in particle beam focusing.
The simplest magnetic quadrupole is two identical bar magnets parallel to each other such that the north pole of one is next to the south of the other and vice versa. Such a configuration will have no dipole moment, and its field will decrease at large distances faster than that of a dipole. A stronger version with very little external field involves using a k=3 Halbach cylinder.
In some designs of quadrupoles using electromagnets, there are four steel pole tips: two opposing magnetic north poles and two opposing magnetic south poles. The steel is magnetized by a large electric current in the coils of tubing wrapped around the poles. Another design is a Helmholtz coil layout but with the current in one of the coils reversed.[1]
1 Quadrupoles in particle accelerators
2 Equations of motion and focal length for charged particles
3 Quadrupole ideal field
Quadrupoles in particle accelerators[edit]
Main article: Strong focusing
A quadrupole electromagnet as used in the storage ring of the Australian Synchrotron
Quadrupole electromagnets (in blue), surrounding the linac of the Australian Synchrotron, are used to focus the electron beam
At the particle speeds reached in high energy particle accelerators, the magnetic force term is larger than the electric term in the Lorentz force:
{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ),}
and thus magnetic deflection is more effective than electrostatic deflection. Therefore a 'lattice' of electromagnets is used to bend, steer and focus a charged particle beam.
Magnetic field lines of an idealized quadrupole field in the plane transverse to the nominal beam direction. The red arrows show the direction of the magnetic field while the blue arrows indicate the direction of the Lorentz force on a positive particle going into the image plane (away from the reader)
The quadrupoles in the lattice are of two types: 'F quadrupoles' (which are horizontally focusing but vertically defocusing) and 'D quadrupoles' (which are vertically focusing but horizontally defocusing). This situation is due to the laws of electromagnetism (the Maxwell equations) which show that it is impossible for a quadrupole to focus in both planes at the same time. The image on the right shows an example of a quadrupole focusing in the vertical direction for a positively charged particle going into the image plane (forces above and below the center point towards the center) while defocusing in the horizontal direction (forces left and right of the center point away from the center).
If an F quadrupole and a D quadrupole are placed immediately next to each other, their fields completely cancel out (in accordance with Earnshaw's theorem). But if there is a space between them (and the length of this has been correctly chosen), the overall effect is focusing in both horizontal and vertical planes. A lattice can then be built up enabling the transport of the beam over long distances—for example round an entire ring. A common lattice is a FODO lattice consisting of a basis of a focusing quadrupole, 'nothing' (often a bending magnet), a defocusing quadrupole and another length of 'nothing'.
Equations of motion and focal length for charged particles[edit]
A charged particle beam in a quadrupole magnetic field will experience a focusing / defocusing force in the transverse direction. This focusing effect is summed up by a focusing strength
{\displaystyle \kappa }
which depends on the quadrupole gradient
{\displaystyle G}
as well as the beam's rigidity
{\displaystyle [B\rho ]={\frac {p}{q}}}
{\displaystyle p}
is the relativistic momentum
{\displaystyle mc\beta \gamma }
{\displaystyle q}
is the particle's charge). The strength is given by
{\displaystyle \kappa ={\frac {G}{[B\rho ]}}}
and particles in the magnetic will behave according to the ODE[2]
{\displaystyle x^{\prime \prime }(z)+{\frac {\gamma ^{\prime }(z)}{\gamma ^{2}(z)\beta (z)}}x^{\prime }(z)+\kappa (z)x(z)=0}
The same equation will be true for the y direction, but with a minus sign in front of the focusing strength to account for the field changing directions.
Quadrupole ideal field[edit]
The components of the ideal magnetic field in the plane transverse to the beam are given by the following[3] (see also multipole magnet).
{\displaystyle {\begin{aligned}{\vec {B}}_{\text{normal}}&=\left({\begin{matrix}K\cdot y,&K\cdot x,&0\end{matrix}}\right)\\{\vec {B}}_{\text{skew}}&=\left({\begin{matrix}J\cdot x,&-J\cdot y,&0\end{matrix}}\right)\\\end{aligned}}}
{\displaystyle K}
is the field gradient of the normal quadrupole component and
{\displaystyle J}
is the field gradient of the skew quadrupole component. The SI unit of the field gradients are
{\displaystyle \mathrm {T} /\mathrm {m} }
. The field in a normal quadrupole is such that the magnetic poles are arranged with an angle of 45 degrees to the horizontal and vertical planes. The sign of
{\displaystyle K}
determines whether (for a fixed particle charge and direction) the quadrupole focuses or defocuses particles in the horizontal plane.
^ Quadrupole Magnetic Field
^ Steven M. Lund, Transverse Particle Dynamics, US Particle Accelerator School (USPAS) Lectures on “Beam Physics with Intense SpaceCharge” https://people.nscl.msu.edu/~lund/uspas/bpisc_2017/lec_set_02/tpd.pdf
^ Shepard, Ben. "Conventional Magnets for Accelerators" (PDF). {{cite web}}: CS1 maint: url-status (link)
Media related to Quadrupole magnet at Wikimedia Commons
Retrieved from "https://en.wikipedia.org/w/index.php?title=Quadrupole_magnet&oldid=1078820951"
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§ Linear algebraic proof of the handshaking lemma
We wish to show that the number odd vertices is even. Let
A
be the adjacency matrix of the undirected graph
G
G
is undirected,
A = A^T
. Now move everything to
F_2
A
A
\{0, 1\}
. Now, denote the vector of all ones by
o \equiv (1, 1, \dots 1)
Ao
counts the partities of the degrees of each vertex, and
o^T(Ao)
counts the sum of parities of the degrees of each vertex. Note that the vertices of even degree with add
0
to the sum
o^TAo
, while odd vertices will add a
1
o^TAo
will equal the parity of the number of odd vertices. As we wish to show that the number of odd vertices is even, we want to prove that
o^TAo = 0
. We will now algebraically simplify
o^TAo
(does anyone have a cleaner proof?) giving us:
\begin{aligned} &o^TAo = \sum_{ij} o_i A_{ij} o_j \\ &= \sum_{i=j} o_i A_{ij} o_j + \sum_{i < j} o_i A_{ij} o_j + o_j A_{ji} o_i \\ &\text{($A$ is symmetric; $A_{ji} = A_{ij}$)} \\ &= \sum_{i=j} o_i A_{ij} o_j + \sum_{i < j} o_i A_{ij} o_j + o_j A_{ij} o_i \\ &= \sum_{i=j} o_i A_{ij} o_j + \sum_{i < j} 2 \cdot o_i A_{ij} o_j \\ &\text{($F_2$ has characteristic zero, so $2 = 0$)} \\ &= \sum_{i=j} o_i A_{ij} o_j + 0 \\ &\text{(replace $i = j$ with $k$)} \\ &= \sum_{k} o_k A_{kk} o_k \\ &\text{($A_{kk} = 0$ since graph has no self loops)} \\ &= \sum_{k} 0 \cdot o_k^2 = 0 \end{aligned}
So, the number of vertices of odd degree is even. I want to avoid this computation with respect to the basis, but I'm not sure how to do that.
§ A simplification from arjun
A_{kk} = 0
A = B + B^T
B
lower triangular. This allows us to simplify:
\begin{aligned} & o^T A o = o^T (B + B^T) o = \\ & =o^T B o + o^T B^T o = \langle o, Bo \rangle + \langle Bo, o \rangle \\ & = 2 \cdot \langle o, Bo \rangle = 0 \end{aligned}
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§ Semidirect product as commuting conditions
Recall that in
N \ltimes K = G
N
is normal. This is from the mnemonic that it looks like
N \triangleleft G
, or from the fact that the acting/twisting subgroup
K
is a fish that wants to "eat"/act on the normal subgroup
N
knk^{-1} \in N
N
is normal, thus
knk^{-1} = n'
kn = n'k
When commuting, the element that gets changed/twisted in the normal subgroup. This is because the normal subgroup has the requisite constraint on it to be twistable.
The element that remains invariant is the actor.
In the case of translations and rotations, it's the translations that are normal. This can be seen either by noticing that they are abelian, and are thus normal, while rotations don't "look normal". Alternatively, one can try to consider translate-rotate versus rotate-translate.
First rotating by
r
and then translating by
t
along the x-axis has the same effect as first translating by
t'
at 45 degrees to the x-axis, and then rotating by the same r.
This begs the question, is there some other translation t'' and some other rotation r'' such that t''; r'' ( t'' first, r'' next) has the same effect as r;t ( r first, t next)?
First let's translate by
t
along the x-axis and then rotating by
r
. Now let's think, if we wanted to rotate and then translate, what rotation would we start with? it would HAVE TO BE
r
, since there's no other way to get the axis in such an angle in the final state. But if we rotate by
r
, then NO translation can get us to the final state we want. So, it's impossible to find a rotation;translationpair that mimics our starting translation;rotation.
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Often I’ve come across technical proposals along the lines of:
In 6 months we will outgrow our MySQL/Postgres instance. We will need to move our biggest table to a different horizontally scalable datastore.
If we have a database outage in a region, we will have a complete outage. We should consider moving to a data-store that’s natively multi-region.
This would be much faster if it was stored in a specialized database. Should we consider moving to it?
If we move to an event-based architecture, our system will be much more reliable.
What these proposals have in common is that they attempt to improve the system by increasing complexity. Whenever you find yourself arguing for improving infrastructure by yanking up complexity, you need to be very careful.
“Simplicity is prerequisite for reliability.” — Edsger W. Dijkstra:
Theoretically yes: if you move your massive, quickly-growing products table to a key-value store to alleviate a default-configured relational database instance, it will probably be faster, cost less, and easier to scale.
However, in reality most likely the complexity will lead to more downtime (even if in theory you get less), slower performance because it’s hard to debug (even if in theory, it’s much faster), and worse scalability (because you don’t know the system well).
More theoretical 9s + increase in complexity => less 9s + more work.
This all because you’re about to trade known risks for theoretical improvements, accompanied by a slew of unknown risks. Adopting the new tech would increase complexity by introducing a whole new system: operational burden of learning a new data-store, developers’ overhead of using another system for a subset of the data, development environment increases in complexity, skills don’t transfer between the two, and a myriad of other unknown-unknowns. That’s a massive cost.
So what do we do with that pesky products table?
Stop thinking about technologies, and start thinking in first-principle requirements:
You need faster inserts/updates
You need terabytes of storage to have runway for the next ~5 years
You need more read capacity
The way that the shiny key-value store you’re eyeing achieves this is by not syncing every write to disk immediately. Well, you can do that in MySQL too (and Postgres). You could put your table on a new database server with that setting on. I wrote about this in detail.
There’s no reason your relational database can’t handle terabytes. Do the napkin math, log(n) lookups for that many keys isn’t much worse. Most likely you can keep it all to one server.
Why do you think reads would be faster in the other database than your relational database? It probably caches in memory. Well, relational databases do that too. You need to spread reads among more databases? Relational databases can do that too with read-replicas…
Yes, MySQL/Postgres might be
25-50\%
worse at all those things than a new system. But it still comes out
10,000\%
ahead, by not being a new system with all its associated costs and unknown-unknowns. There’s an underlying rule from evolution that the more specialized a system is, the less adaptable to change it is. Whether it’s a bird over-fit to its ecosystem, or a database you’re only using for one thing.
We could go through a similar line of reasoning for the other examples. Adopting a new multi-regional database for a subset of your database will likely yield to more downtime due to the introduction of complexity, than sticking with what you’ve got.
Don’t adopt a new system unless you can make the first-principle argument for why your current stack fundamentally can’t handle it. For example, you will likely reach elemental limitations doing full-text search in a relational datastore or analytics queries on your production database, as a nature of the data structures used. If you’re unsure, reach out, and I might be able to help you!
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Error, improper op or subscript selector - Maple Help
Home : Support : Online Help : Error, improper op or subscript selector
This error occurs when the op command is used to extract an operand that does not exist.
The index must be an integer and must refer to an operand that exists in the expression.
In this example, using the op command to extract the 15th operand results in an error.
s ≔ \mathrm{series}\left(\mathrm{sin}\left(x\right), x = 2, 6\right)
\textcolor[rgb]{0,0,1}{s}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\left(\textcolor[rgb]{0,0,1}{2}\right)\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\left(\textcolor[rgb]{0,0,1}{2}\right){\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\left(\textcolor[rgb]{0,0,1}{2}\right){\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{24}}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\left(\textcolor[rgb]{0,0,1}{2}\right){\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{120}}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\left(\textcolor[rgb]{0,0,1}{2}\right){\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{O}}\left({\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{6}}\right)
\mathrm{op}\left(15,s\right)
Using the nops command, you see that there are only 14 operands.
\mathrm{nops}\left(s\right)
\textcolor[rgb]{0,0,1}{14}
To access the last element, use the following command:
\mathrm{op}\left(14,s\right)
\textcolor[rgb]{0,0,1}{6}
Again, calling the op command to extract an operand outside the bounds of the list results in an error.
L≔\left[a,b,c,d\right]
\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{d}\right]
\mathrm{op}\left(5, L\right)
Modify the arguments to the op command to extract an element within the list. Use the nops command to determine the number of operands.
op and nops
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\mathrm{α},\mathrm{β},\mathrm{γ}
\left(1,2,3\right)
x
y
z
The three direction angles for a vector reside in the three right triangles in Figure 1.1.2. Their values, obtained by applying basic right-triangle trigonometry to these triangles where the hypotenuse is
∥〈1,2,3〉∥=\sqrt{14}
, are listed in Table 1.1.8(a).
\mathrm{α}=\mathrm{arccos}\left(1/\sqrt{14}\right)
1.3002
74.5°
\mathrm{β}=\mathrm{arccos}\left(2/\sqrt{14}\right)
1.0068
57.7°
\mathrm{γ}=\mathrm{arccos}\left(3/\sqrt{14}\right)
0.64048
36.7°
Table 1.1.8(a) Direction angles for given vector
The Student MultivariateCalculus package provides the Angle command, which returns (in radians) the angle between two vectors. Use the Context Panel to apply this command to the given position vector and another position vector whose direction is that of one of the axes.
Form a sequence of two vectors (see Table 1.1.1), one the given position vector, and the other any position vector whose direction is that of the appropriate coordinate-axis.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Angle
\mathrm{α}
〈1,2,3〉,〈1,0,0〉
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\end{array}\right]
\stackrel{\text{angle}}{\to }
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{14}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{14}}\right)
\stackrel{\text{at 5 digits}}{\to }
\textcolor[rgb]{0,0,1}{1.3002}
\mathrm{β}
〈1,2,3〉,〈0,1,0〉
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}\end{array}\right]
\stackrel{\text{angle}}{\to }
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{14}}\right)
\stackrel{\text{at 5 digits}}{\to }
\textcolor[rgb]{0,0,1}{1.0068}
\mathrm{γ}
〈1,2,3〉,〈0,0,1〉
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}\end{array}\right]
\stackrel{\text{angle}}{\to }
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{14}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{14}}\right)
\stackrel{\text{at 5 digits}}{\to }
\textcolor[rgb]{0,0,1}{0.64048}
Apply the Angle command in the Student MultivariateCalculus package to the given position vector and any other vector whose direction is that of the appropriate axis.
\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{Angle}\left(〈1,2,3〉,〈1,0,0〉\right)
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{14}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{14}}\right)
\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{Angle}\left(〈1,2,3〉,〈0,1,0〉\right)
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{14}}\right)
\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{Angle}\left(〈1,2,3〉,〈0,0,1〉\right)
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{14}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{14}}\right)
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§ CS and type theory: Talks by vovodesky
Talk 1: Computer Science and Homotopy Theory
Think of ZFC sets as rooted trees. Have two axioms:
(1) all branches of all vertices are non-isomorphic (otherwise a set would have two copies of the same element)
(2) Each leaf must be at finite depth from the root.
This is horrible to work with, so type theory!
Talk 2: What if foundations of math is inconsistent?
We "know" that first order math is consistent. We can prove that it is impossible to prove that first order math is consistent!
Choice 1: If we "know" FOL is consistent, then we should be able to transform this knowledge into a proof, then 2nd incompleteness is false.
Choice 2: Admit "transcendental" part of math, write dubious philosophy.
Choice 3: Admit that our sensation that FOL +arithmetic is consistent is an illusion and admit that FOL arithmetic is inconsistent.
Time to consider Choice 3 seriously?
§ First order arithmetic
Mathematical object which belongs to class of objects called formal theories. Has four pieces of data:
Special symbols, names of variables.
Deduction rules: Construct new closed formulas from old closed formula.
Axioms: collection of closed formulas.
Anything that is obtainable from these deduction rules is called a theorem. First order logic have symbols: ∀, ∃, ⇒, !(not) and so on. First order theory is inconsistent if there a closed formula
A
A
!A
Free variables describe subsets. Eg: ∃ n: n^2 = m describes the set { m : ∃ n: n^2 = m }.
It's possible to construct subsets (formulae with one free variable) whose membership is undedicable. So you can prove that it is impossible to say anything whatsoever about these subsets.
§ Gentzen's proof and problems with it
Tries to reason about trees of deduction. Show that proofs correspond to combinatorial objects. Show that inconsistency corresponds to an infinite decreasing sequence that never terminates. Then he says that it is "self evident" that this cannot happen. But it is not self evident!
§ What would inconsistency of FOL mean for mathematicians?
Inconsistency of FOL implies inconsistency of many other systems (eg. set theory).
Inconsistency of FOL implies inconsistency of constructive (intuitionistic) mathematics! (WTF?) shown by Godel in 1933. Takes a proof of contradiction in classical and strips off LEM.
We need foundations that can create reliable proofs despite being inconsistent!
Have systems that react to inconsistency in less drastic ways. One possible candidate is constructive type theories. A proof of a formula in such a system is itself a formula in the system. There are no deduction rules, only syntactic rules. So a proof is an object that can be studied in the system. If one has a proof of contradiction, then such a proof can be detected --- they have certain properties that can be detected by an algorithm (what properties?)
§ New workflow
Formalize a problem.
Construct creative solution.
Submit proof to a "reliable" verifier. If the verifier terminates, we are done. If the verifier does not terminate, we need to look for other proofs that can terminate.
our abstract thinking cancels out by normalisation :P
Correct interpretation of 2nd incompleteness is a step of proof of inconsistency of FOL (Conjecture).
In math, we need to learn how to use inconsistent theories to obtain reliable proofs. Can lead to more freedom in mathematical workflow.
§ Univalent Foundations: New Foundations of Mathematics
Talk 3: Univalent foundations --- New Foundations of Mathematics
Was uncertain about future when working on 2-categories and higher math. No way to ground oneself by doing "computations" (numerical experiments). To make it worse, the existing foundations of set theory is bad for these types of objects.
Selected papers on Automath.
Overcoming category theory as new foundations was very difficult for vovodesky.
Categories are "higher dimensional sets"? NO! Categories are "posets in the next dimension". Correct version of "sets in the next dimension" are groupoids (WHY?) MathOverflow question
Grothendeick went from isomorphisms to all morphisms, this prevented him from gravitating towards groupoids.
Univalent foundations is a complete foundational system.
Sets are groupoids on the next dimension
§ Vovodesky's univalence principle --- Joyal
Talk 5: Vovodesky's univalence principle --- Joyal
Univanent type theory is arrived at by adding univalence to MLTT.
Goal of univalent foundations is to apply UTT to foundations.
Univalence is to type theory what induction principle is to peano arithmetic
Univalence implies descent. Descent implies Blakers Massey-theorem, which implies Goodwille calculus.
The syntactic system of type theory is a tribe .
A clan is a category equipped with a class of carrable maps called fibrations. A map is carrable if we can pull it back along any other map.
A clan is a category along with maps called "fibrations", such that (1) every isomorphism is a fibration, (1) closed under composition, (3) fibrations are carrable, (4) base change of fibration is a fibration, (4) Category has a terminal object, and map into the terminal object is a fibration.
u: A \rightarrow B
is anodyne if it does something good with respect to fibrations.
A tribe is a clan such that (1) base chnge of anodyne along fibration is anodyne, (2) every map factorizes as anodyne followed by fibration.
Kan complexes form a tribe. A fibration is a Kan fibration. A map is anodyne here if it is a monomorphism and a homotopy equivalence.
Given a tribe
E
, can build a new tribe by slicing
E/A
(this is apparently very similar to things people do in Quillen Model categories).
A tribe is like a commutative ring. We can extend by adding new variables to get polynomial rings. An elementary extension is extending the tribe by adding a new element.
E
is a tribe, an object of
E
is a type. We write E |- A : Type.
If we have a map
a: 1 -> A
, we regard this as an element of A: E |- a : A.
A fibration is a family of objects. This is a dependent type x : A |- E(x): Type. E(x) is the fiber of p: E -> A at a variable element x : A.
A section of a fibration gives an element of the fibration. We write this as x : A |- s(x) : E(x). s(x) denotes the value of s: A -> E of a variable element x : A. (Inhabitance is being able to take the section of a fiber bundle?!)
change of parameters / homomorphism is substitution.
y : B |- E(y) : Type
x : A |- E(f(x)) : Type
This is pulling back along fibrations.
Elementary extension E -> E(A) are called as context extensions.
|- B : Type
x : A |- B : Type
A map between types is a variable element f(x) : B indexed by x : A
x : A |- f(x) : B
Sigma formation rule: The total space of the union is the sum of all fibers(?)
x: A |- E(x): Type
|- \sum{x : A}E(x): Type
y : B |- \sum{x : f^{-1}(y)}E(x): Type
Path object for
A
is obtained by factorial diagonal map diag: a -> (a, a) as an anodyne map r: A -> PA followed by a fibration (s, t) : PA -> A x A.
A homotopy h: f ~ g between two maps f, g : A -> B is a map h: A -> PB such that sh = f and th = g. homotopy is a congruence.
x: A, y : A |- Id_A(x, y) : Type called the identity type of A.
An element p: Id_A(x, y) is a proof that x =A y.
Reflexivity term x : A |- r(x) : Id_A(x, x) which proves x =A x.
The identity type is a path object
\gamma(x, y): Id_A(x, y) -> Eq(E(x), E(y))
\gamma
is some kind of connection: given a path from
x
y
, it lets us transport
E(x)
E(y)
Eq
is the distortion from the curvature?
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3 Ways to Calculate Capital Gains - wikiHow
1 Understanding Capital Gains
2 Calculating Capital Gains
3 Managing the Tax Impact
Capital gains are the portion of increase above the initial amount invested in vehicles such as stocks, bonds or real estate. It is the difference between original purchase price (or basis) and selling price. If you have capital investments that could make profit if sold, you have unrealized capital gains. Realized capital gains result from the sale of investments. You can calculate capital gains with these methods.
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Define capital gains. Capital gains refer to the increased value of an asset over time. When the asset is sold, you compare the selling price with the original purchase price. The difference is your capital gain. If the asset decreases in value, it is considered a capital loss.[1] X Research source
Short-term capital gains are from assets that are held for less than one year.
Long-term capital gains are from assets that are held for a year or longer.
Recognize what counts as a capital asset. Many things that you might not think of qualify as capital assets. The IRS defines capital assets as everything you own and use for personal use, pleasure and investment. You must calculate capital gains whenever you sell one of these capital assets.[2] X Trustworthy Source Internal Revenue Service U.S. government agency in charge of managing the Federal Tax Code Go to source
Examples of capital assets include investments such as stocks and bonds, your personal home or investment properties, household furnishings and your car.
Other capital assets include timber grown on your property, coin or stamp collections, jewelry and precious metals.
Your personal home may be exempt from taxes on capital gains if you owned it and used it as your primary residence for at least two years in the five years before you sold it and you haven’t excluded the gain on the sale of another property in the two-year period before the sale.[3] X Research source
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Understand why you need to calculate capital gains. The IRS requires you to calculate capital gains because you must pay income tax on them. All capital gains must be reported. The tax rate on capital gains is less than the tax rate on wages per bracket. The amount of tax you pay on capital gains depends on your tax bracket in 2015 (this rate will change as tax bracket rates change).[4] X Research source
Those in the 10 to 15 percent tax bracket pay 0 percent on capital gains.
Those in the 25 percent, 28 percent, 33 percent, or 35 percent tax brackets pay 15 percent on capital gains.
Those in the 39.6 percent tax bracket pay 20 percent on capital gains.
Learn how capital losses offset capital gains. If you sell something for less than what you paid for it, this is a capital loss. You can use your capital losses from investments to reduce your capital gains. This reduces how much you have to pay in taxes. Also, if your total capital losses exceed your total capital gains, you can offset your ordinary income up to $3,000 in a single tax year.[5] X Research source Amounts above this can be carried forward into the next tax year.
You cannot use capital losses from the sale of personal property to offset capital gains.
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Verify the cost basis of your asset. The cost basis is essentially what you originally paid for the asset. For stocks and bonds, the purchase price can be adjusted up or down for stock splits, dividends, return of capital distributions and brokerage fees. For example, if you purchased stock for $3,000 and you paid a $9 commission to a broker, your cost basis is $3,009.[6] X Research source [7] X Research source
The cost basis is also referred to as "tax basis" or simply "basis."
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Ascertain the selling price. For stocks that you sold, you can find the selling price on the order of execution confirmation or your brokerage account statement.[8] X Research source For other assets you sell, keep records of the selling price. Make a copy of any receipts for sale of furniture, jewelry, coins or precious metals. If you sold property, keep a copy of the closing statement from the settlement.
Calculate the difference. The difference between the buying price and the selling price is your capital gain or loss. The formula is Sale Price - Cost Basis = Capital Gain. For example, suppose you purchased 100 shares of stock for $1 each for a total value of $100. After three months, the stock price rises to $5 per share, making your investment worth $500. If you sell the stock at this point, you will have made a profit of $400.[9] X Research source
You would pay capital gains tax on the $400 profit.
Since you held the stock for a period of less than a year, it would be considered short-term capital gains and be taxed at your regular income tax rate.
Managing the Tax Impact
Understand how taxes on capital gains affect investment results. Income taxes reduce the overall profit you earn on the sale of your assets. You can use different tactics to manage the tax impact on your capital gains. You can strategically plan the timing of the sale of your assets. Also, you can use your capital losses to offset you capital gains.[10] X Research source
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Divide capital gains on equities into short-term and long-term. Short-term capital gains are taxed at your regular tax income rate, and long-term capital gains are taxed at a lower rate. Therefore, sometimes it makes sense to hold off on selling equities in the short-term, even if the price spikes. The difference between your income tax rate and the reduced long-term capital gains tax rate can be significant.[11] X Research source
For example, in the scenario described above, you might be thrilled that you made $400 on the sale of that stock after only three months. However, if you are in the 35 percent tax bracket, then you would need to pay $140 in capital gains tax (
{\displaystyle \$400*.35=\$140}
). Your total profit would then only be $260 (
{\displaystyle \$400-\$140=\$260}
Suppose in the same example, after 13 months, each share of stock was worth $4.50. Your total investment would be worth $450. If you sell, you would earn a $350 profit. However, since you’ve held the stock for longer than a year, that profit is a long-term capital gain and is only taxed at a 15 percent tax rate.
Your capital gains tax would be $52.50 (
{\displaystyle \$350*.15=\$52.50}
). This means your total profit after taxes would be $297.50 (
{\displaystyle \$350-\$52.50=\$297.50}
Even though you sold the stock for a lower price, timing the sell of your stock allowed you to minimize the tax impact and make a bigger profit.
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Offset capital gains with capital losses. Some investors purposely sell investments at a loss in order to reduce their taxable capital gains. These tax-loss strategies may save investors enough in taxes to improve the overall performance of their portfolios. However, this approach can be perilous. If not executed correctly, you could end up losing more in capital losses than you save in taxes.[12] X Research source
Analyze the market carefully to select the right shares to sell at a loss. Don’t give up shares in a healthy company just because prices may have dipped temporarily. You could be missing an opportunity for large profits by not holding that investment for the long-term.
↑ http://www.investopedia.com/terms/c/capitalgain.asp
↑ https://turbotax.intuit.com/tax-tools/tax-tips/Investments-and-Taxes/5-Things-You-Should-Know-About-Capital-Gains-Tax/INF26154.html
↑ http://www.fool.com/how-to-invest/personal-finance/taxes/2015/08/16/long-term-capital-gains-tax-rates-in-2015.aspx
↑ http://www.forbes.com/sites/baldwin/2012/01/24/cost-basis-faq/
↑ http://www.investopedia.com/terms/c/costbasis.asp
↑ http://www.investopedia.com/ask/answers/07/calculategains.asp
↑ http://www.investinganswers.com/financial-dictionary/tax-center/capital-gains-tax-2109
↑ http://www.investopedia.com/articles/tax/09/tax-effects-capital-gains.asp
↑ http://www.forbes.com/sites/brucemccain/2014/11/03/tax-loss-selling-dont-lose-money-trying-to-save-on-taxes/
The first step to calculate your capital gains is to work out your cost basis, which is what you paid for the asset plus any brokerage fees. To find out your asset's selling price, check the order of execution confirmation from when you sold the stocks. If your asset is jewelry or furniture, be sure to keep receipts of sale so you know the sales price. Then, work out your capital gain by deducting the cost basis from the sales price. If you have made a capital loss on an asset, use the loss to offset your capital gains when you're filing your taxes. To learn more from our Financial Planner co-author, like how to determine which objects count as capital assets, keep reading the article!
Русский:подсчитать прирост капитала
Español:calcular las ganancias de capital
Français:calculer les plus‐values
"Thank you for this very understandable way to calculate capital gains. It was very helpful."
"Relatively simple answers to a complex question. Very understandable."
"I now understand more about capital gains on property."
"Gives a basic understanding of the topic."
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Continued fraction factorization - Wikipedia
Continued fraction factorization
In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning that it is suitable for factoring any integer n, not depending on special form or properties. It was described by D. H. Lehmer and R. E. Powers in 1931,[1] and developed as a computer algorithm by Michael A. Morrison and John Brillhart in 1975.[2]
The continued fraction method is based on Dixon's factorization method. It uses convergents in the regular continued fraction expansion of
{\displaystyle {\sqrt {kn}},\qquad k\in \mathbb {Z^{+}} }
Since this is a quadratic irrational, the continued fraction must be periodic (unless n is square, in which case the factorization is obvious).
It has a time complexity of
{\displaystyle O\left(e^{\sqrt {2\log n\log \log n}}\right)=L_{n}\left[1/2,{\sqrt {2}}\right]}
, in the O and L notations.[3]
^ Lehmer, D.H.; Powers, R.E. (1931). "On Factoring Large Numbers". Bulletin of the American Mathematical Society. 37 (10): 770–776. doi:10.1090/S0002-9904-1931-05271-X.
^ Pomerance, Carl (December 1996). "A Tale of Two Sieves" (PDF). Notices of the AMS. Vol. 43, no. 12. pp. 1473–1485.
Samuel S. Wagstaff, Jr. (2013). The Joy of Factoring. Providence, RI: American Mathematical Society. pp. 143–171. ISBN 978-1-4704-1048-3.
This number theory-related article is a stub. You can help Wikipedia by expanding it.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Continued_fraction_factorization&oldid=1020367997"
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Deployment Logic - Tokemak: The Utility for Sustainable Liquidity
The quantities of assets deployed to outside venues are dictated by the following guardrail factors. More detailed information on the guardrails can be found in the "Guardrails & Impermanent Loss Mitigation" section.
Assets in reserve (eg ABC) Do not deploy more than 3x the qty available in the reserve ("Reserve Multiple")
LP provided base asset available to pair (eg ETH) If not enough provided by LPs the available base assets are pro rata distributed across reactors.
TOKE staked to the reactor Ideally, enough TOKE should be staked to the reactor to result in TOKE staked + ABC reserve = LP ABC, or in other words two thirds of the value of deployed LP assets should be staked by LDs.
Note that these are starting parameters and we will be able to significantly relax these as more data becomes available.
The below detailed logic is set up in such way that if followed in order will result in the correct deployed amount.
2. Deployment Logic & Equations
1. Based on assets in reserve
In a first step, the assets provided by the LPs (AssetLP) are passed through the first guardrail. The assets available in reserve multiplied by the reserve multiple (3) determine the maximum amount of LP assets deployable (LPAsset_Deployable1). This guardrail is applied to both the LP provided assets and the Genesis pool.
This guardrail is calculated based on Qty.
\begin{align*} &\text {IF $Asset_{inReserve} \times ReserveMultiple \geq Asset_{LP}$} \\ &\ \ \ \text {THEN $ Asset_{LP} = LPAsset_{Deployable1}$} \\ &\text {IF $Asset_{inReserve} \times ReserveMultiple < Asset_{LP}$} \\ &\ \ \ \text {THEN $Asset_{inReserve} \times ReserveMultiple = LPAsset_{Deployable1}$} \end{align*}
2. Based on ETH available to pair
In a second step, the system compares the amount of available ETH in the Genesis pool (after applying the reserve guardrail) to the LP provided assets in order to determine if all assets can be paired up with ETH. Should not enough ETH be available in the Genesis pool, the system will proceed by equally distributing the ETH across the reactors.
This guardrail is calculated based on notional value.
\begin{align*} &\text {IF $LPETH_{Deployable1} \geq SumAssets_{Deployable1}$} \\ &\ \ \ \text {THEN $ETH_{Pairable}=1$} \\ &\text {IF $LPETH_{Deployable1} < SumAssets_{Deployable1}$} \\ &\ \ \ \text {THEN $ETH_{Pairable}= LPETH_{Deployable1} \div SumAssets_{AssetDeployable1}$} \\ &\rightarrow \ LPAsset_{Deployable1} \times ETH_{Pairable} = LPAsset_{Deployable2} \end{align*}
3. Based on TOKE staked to the reactor
The last step is to assure a minimum amount of TOKE is staked to the reactor by the LDs. While more TOKE can be staked to a reactor the value of ABC deployed is not to surpass 1.5x the value of the TOKE staked to the reactor.
This is calculated based on notional value.
MECHANICS AND FUNCTIONALITY - Previous
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§ Burnside Theorem
For a finite group
G
X
, burnside's lemma equates (i) the number of equivalence classes of
X
G
's action, that is, the number of orbits of
X
with (ii) the average number of stabilized elements for each
g \in G
. Formally, it asserts:
|Orb(X, G)| = 1/|G|\sum_{g \in G} |Stab(g)|
§ As local/global principle
See that the right-hand-side measures "local fixed points" in terms of
|Stab(g)|
. The left hand side measures "global fixed points": an orbit
O \subseteq X
is a set such that
G \cdot O = O
. So it's a sort of "global fixed point of
G
". The burnside lemms tells us that we can recover the size of the set of global fixed points (the number of orbits) by averaging the size of the set of local fixed points (the average of the sizes of the stabilizers).
§ As space/time average
The left hand size is a time average: If we consider
X, GX, \dots G^{n}X
, we will finally be left with the images as the orbits. All other elements would have been smashed together''. On the right hand side, we are averaging the group over space. Yes, but how does one perform averaging? One needs to have a measure!
§ A rephrasing in terms of integrators
Consider a system of a particle in a well. We consider two energy levels: that with E = 0, and E = 1. This gives us the following five states: now I want to simulate this system, like a good computer scientist. So let's write the stupdiest one possible, Δ0, that doesn't simulate anything at all, and Δ+1, which steps the system forward by a single step. These look like this: But why only these? Why privilege these time-scales? We should at least have Δ-1, for the arrow of time is fiction: We should also have coarser integrators. Hence we contemplate Δ+2 and Δ-2. Turns out these are equivalent: We can also consider Δ + 3. We also see that Δ + 4 = Δ: So in conclusion, the calculation gives us:
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IsAntiArborescence - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : IsAntiArborescence
test if a graph is an arborescence
test if a graph is an anti-arborescence
IsArborescence(G,opts)
IsAntiArborescence(G,opts)
: (optional) root=true or root=false
root : keyword option of the form root=true or root=false. This specifies whether the root vertex should be returned when the check is positive. The default is false.
The IsArborescence function returns true if the input graph is an arborescence, and false otherwise.
The IsAntiArborescence function returns true if the input graph is an anti-arborescence, and false otherwise.
A directed graph G is an arborescence if there is a single vertex u called the root such that for any other vertex v, there is exactly one path from u to v.
A directed graph G is an anti-arborescence if there is a single vertex v called the root such that for any other vertex u, there is exactly one path from u to v. An anti-arborescence is a graph whose reverse is an arborescence.
Confirm that a directed path is both an arborescence and an anti-arborescence.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
T≔\mathrm{Graph}\left({[1,2],[2,3]}\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: a directed unweighted graph with 3 vertices and 2 arc\left(s\right)}}
\mathrm{IsArborescence}\left(T\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsAntiArborescence}\left(T\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
Confirm that a directed cycle is neither an arborescence nor an anti-arborescence.
C≔\mathrm{Graph}\left({[1,2],[2,3],[3,1]}\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: a directed unweighted graph with 3 vertices and 3 arc\left(s\right)}}
\mathrm{IsArborescence}\left(C\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsAntiArborescence}\left(C\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
Check whether a graph is an arborescence and display its root vertex.
G≔\mathrm{Graph}\left({[1,2],[1,5],[2,3],[2,4]}\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 3: a directed unweighted graph with 5 vertices and 4 arc\left(s\right)}}
\mathrm{IsArborescence}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{IsArborescence}\left(G,\mathrm{root}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}
\mathrm{IsAntiArborescence}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
Check whether a graph is an anti-arborescence and display its root vertex.
A≔\mathrm{Graph}\left({[1,3],[2,3]}\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 4: a directed unweighted graph with 3 vertices and 2 arc\left(s\right)}}
\mathrm{IsArborescence}\left(A\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsAntiArborescence}\left(A,\mathrm{root}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}
\mathrm{IsAntiArborescence}\left(A\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The GraphTheory[IsArborescence] and GraphTheory[IsAntiArborescence] commands were introduced in Maple 2016.
The GraphTheory[IsArborescence] and GraphTheory[IsAntiArborescence] commands were updated in Maple 2019.
The root option was introduced in Maple 2019.
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§ GAP permutation syntax
The action of permutation on an element is given by
i^p
. This is the "exponential notation" for group actions.
See that we only ever write permutations multiplicatively, eg (1) (23) is the composition of permutations [written multiplicatively ].
Thus the identity permutation must be 1, and it's true that any number n^1 = n, so the identity permutation 1 fixes everything.
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Limit stable objects on Calabi-Yau 3-folds
15 July 2009 Limit stable objects on Calabi-Yau 3-folds
Yukinobu Toda1
1Institute for the Physics and Mathematics of the Universe (IPMU), University of Tokyo
In this article, we introduce new enumerative invariants of curves on Calabi-Yau
3
-folds via certain stable objects in the derived category of coherent sheaves. We introduce the notion of limit stability on the category of perverse coherent sheaves, a subcategory in the derived category, and construct the moduli spaces of limit stable objects. We then define the counting invariants of limit stable objects using Behrend's constructible functions on those moduli spaces. It will turn out that our invariants are generalizations of counting invariants of stable pairs introduced by Pandharipande and Thomas. We will also investigate the wall-crossing phenomena of our invariants under change of stability conditions
Yukinobu Toda. "Limit stable objects on Calabi-Yau 3-folds." Duke Math. J. 149 (1) 157 - 208, 15 July 2009. https://doi.org/10.1215/00127094-2009-038
Yukinobu Toda "Limit stable objects on Calabi-Yau 3-folds," Duke Mathematical Journal, Duke Math. J. 149(1), 157-208, (15 July 2009)
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Analytical and Experimental Studies of the Mechanics of Deformation in a Solid With a Wavy Surface Profile | J. Appl. Mech. | ASME Digital Collection
, Urbana-Champaign, IL 61801
, 1 Fusionopolis Way, No. 16-16 Connexis, Singapore 138632, Singapore
Department of Mechanical Engineering, and Department of Civil and Environmental Engineering,
Department of Materials Science and Engineering and Beckman Institute, and Seitz Materials Research Laboratory,
Xiao, J., Carlson, A., Liu, Z. J., Huang, Y., and Rogers, J. A. (September 23, 2009). "Analytical and Experimental Studies of the Mechanics of Deformation in a Solid With a Wavy Surface Profile." ASME. J. Appl. Mech. January 2010; 77(1): 011003. https://doi.org/10.1115/1.3132184
The analytical solution is obtained for a semi-infinite linear elastic solid with a sinusoidal, “wavy” surface profile subject to applied strain. The amplitude
A
of a deformed wavy surface is related to the initial amplitude
A0
and the applied strain
εa
through the simple expression
A=A0(1−εa)
. This relation is confirmed independently by finite element analyses and experimental measurements of strained wavy poly(dimethylsiloxane) surfaces. Analytical solutions are also obtained for a wavy solid subject to stretch and lateral displacement.
deformation, elasticity, finite element analysis, polymers
Deformation, Displacement, Finite element analysis, Thin films, Wavelength, Plasma desorption mass spectrometry, Polymers
Drzaic
D. -H.
D. -Y.
Julthongpiput
Rackaitis
Micro-Buckling as a Route Towards Surface Patterning
Mechanics of Precisely Controlled Thin Film Buckling on Elastomeric Substrate
Edge Effects in Buckled Thin Films on Elastomeric Substrates
, 2004, ABAQUS Analysis User’s Manual V6.5.
Modeling Springback of Metal-Polymer-Metal Laminates
Simulation Study of Static and Dynamic Characteristics of Piezoelectric Thin Film Coated Polymer Tube
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Electron donor - Wikipedia
In chemistry, an electron donor is a chemical entity that donates electrons to another compound. It is a reducing agent that, by virtue of its donating electrons, is itself oxidized in the process.
Typical reducing agents undergo permanent chemical alteration through covalent or ionic reaction chemistry. This results in the complete and irreversible transfer of one or more electrons. In many chemical circumstances, however, the transfer of electronic charge to an electron acceptor may be only fractional, meaning an electron is not completely transferred, but results in an electron resonance between the donor and acceptor. This leads to the formation of charge transfer complexes in which the components largely retain their chemical identities.
The electron donating power of a donor molecule is measured by its ionization potential which is the energy required to remove an electron from the highest occupied molecular orbital (HOMO).
The overall energy balance (ΔE), i.e., energy gained or lost, in an electron donor-acceptor transfer is determined by the difference between the acceptor's electron affinity (A) and the ionization potential (I):
{\displaystyle {\Delta }E=A-I\,}
The class of electron donors that donate not just one, but a set of two paired electrons that form a covalent bond with an electron acceptor molecule, is known as a Lewis base. This phenomenon gives rise to the wide field of Lewis acid-base chemistry.[1] The driving forces for electron donor and acceptor behavior in chemistry is based on the concepts of electropositivity (for donors) and electronegativity (for acceptors) of atomic or molecular entities.
In biology, electron donors release an electron during cellular respiration, resulting in the release of energy. Microorganisms, such as bacteria, obtain energy in the electron transfer processes. Through its cellular machinery, the microorganism collects the energy for its use. The final result is the electron is donated to an electron acceptor. During this process (electron transport chain) the electron donor is oxidized and the electron acceptor is reduced. Petroleum hydrocarbons, less chlorinated solvents like vinyl chloride, soil organic matter, and reduced inorganic compounds are all compounds that can act as electron donors. These reactions are of interest not only because they allow organisms to obtain energy, but also because they are involved in the natural biodegradation of organic contaminants. When clean-up professionals use monitored natural attenuation to clean up contaminated sites, biodegradation is one of the major contributing processes. The Vitamin C is an important electron donor, which is why it is considered a potent water-soluble antioxidant with antiviral properties in humans.
^ Jensen, W.B. (1980). The Lewis acid-base concepts : an overview. New York: Wiley. ISBN 0-471-03902-0.
Electron donor definition at United States Geological Survey
Retrieved from "https://en.wikipedia.org/w/index.php?title=Electron_donor&oldid=1084556924"
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Oscillation Theorems for Certain Class of Nonlinear Difference Equations | EMS Press
L. Ramuppillai
Some new oscillation results for certain class of forced nonlinear difference equations of the form
\Delta (a_n + p_n x_{n–k})) + q_n f(x_{n+1–l}) = e_n \ \ \ (n \in \mathbb N_0)
are established. Examples which dwell upon the importance of the results are also given.
E. Thandapani, L. Ramuppillai, Oscillation Theorems for Certain Class of Nonlinear Difference Equations. Z. Anal. Anwend. 17 (1998), no. 2, pp. 513–519
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Effects of Intracortical Porosity on Fracture Toughness in Aging Human Bone: A μCT-Based Cohesive Finite Element Study | J. Biomech Eng. | ASME Digital Collection
Effects of Intracortical Porosity on Fracture Toughness in Aging Human Bone: A
μCT
-Based Cohesive Finite Element Study
Ani Ural,
Department of Biomedical Engineering, Center for Biotechnology and Interdisciplinary Studies,
, 110 8th Street, Troy, New York 12180
Ural, A., and Vashishth, D. (February 9, 2007). "Effects of Intracortical Porosity on Fracture Toughness in Aging Human Bone: A
μCT
-Based Cohesive Finite Element Study." ASME. J Biomech Eng. October 2007; 129(5): 625–631. https://doi.org/10.1115/1.2768377
The extent to which increased intracortical porosity affects the fracture properties of aging and osteoporotic bone is unknown. Here, we report the development and application of a microcomputed tomography based finite element approach that allows determining the effects of intracortical porosity on bone fracture by blocking all other age-related changes in bone. Previously tested compact tension specimens from human tibiae were scanned using microcomputed tomography and converted to finite element meshes containing three-dimensional cohesive finite elements in the direction of the crack growth. Simulations were run incorporating age-related increase in intracortical porosity but keeping cohesive parameters representing other age-related effects constant. Additional simulations were performed with reduced cohesive parameters. The results showed a 6% decrease in initiation toughness and a 62% decrease in propagation toughness with a 4% increase in porosity. The reduction in toughnesses became even more pronounced when other age-related effects in addition to porosity were introduced. The initiation and propagation toughness decreased by 51% and 83%, respectively, with the combined effect of 4% increase in porosity and decrease in the cohesive properties reflecting other age-related changes in bone. These results show that intracortical porosity is a significant contributor to the fracture toughness of the cortical bone and that the combination of computational modeling with advanced imaging improves the prediction of the fracture properties of the aged and the osteoporotic cortical bone.
biomechanics, bone, computerised tomography, cracks, fracture mechanics, fracture toughness, mesh generation, physiological models, porosity, finite element method, micro-computed tomography, fracture toughness, cortical bone, intracortical porosity
Bone, Engineering simulation, Finite element analysis, Fracture (Materials), Fracture toughness, Porosity, Simulation, Tension
Vashishth D.,
Fracture Mechanics of Cortical Bone Tissue: A Hierarchical Perspective
Age-Dependent Biomechanical Modifications in Bone
Crit. Rev. Eukaryot Gene Expr
A Microtomographic System for the Nondestructive Evaluation of Bone Architecture
Three-Dimensional Microimaging (MRμI and μCT), Finite Element Modeling, and Rapid Prototyping Provide Unique Insights Into Bone Architecture in Osteoporosis
A New Method to Determine Trabecular Bone Elastic Properties and Loading Using Micromechanical Finite Element Models
Three-Dimensional Finite Element Modelling of Non-Invasively Assessed Trabecular Bone Structures
The Modified Super-Ellipsoid Yield Criterion for Human Trabecular Bone
Trabecular Bone Microdamage and Microstructural Stresses Under Uniaxial Compression
Falkinstein
Comparison of In Situ and In Vitro CT Scan-Based Finite Element Model Predictions of Proximal Femoral Fracture Load
Finite Element Analysis of Mouse Tibia: Estimating Endocortical Strain During Three-Point Bending in SAMP6 Osteoporotic Mice
Microcomputed Tomography-Based Large Scale Finite Element Modeling of Human Cortical Bone Tissue
Transactions of the 51st Annual Meeting of the Orthopaedic Research Society
, Washington, DC, p. 685.
Epidemiology of Fractures in 15000 Adults: The Influence of Age and Gender
ABAQUS, Version 6.5, ABAQUS Inc., Providence, RI, 2004.
Fracture Toughness is Dependent on Bone Location- A Study of Femoral Neck, Femoral Shaft, and the Tibial Shaft
ASTM Standard E399-90
,” American Society for Testing and Materials, Philadelphia.
, and Bonfield W., 1997, “
Age-Related Loss in Bone: Toughness is Explained by Non-Enzymatic Glycation of Collagen
, San Francisco, CA, p. 497.
Spatial Clustering of Remodeling Osteons in the Femoral Neck Cortex: A Cause of Weakness in Hip Fracture?
Prediction of Cortical Bone Porosity In Vitro by Microcomputed Tomography
Variability in Osteon Size in Recent Human Populations
Quantitative 3D Analysis of the Canal Network in Cortical Bone by Micro-Computed Tomography
0003-276X B New Anat.,
Comparison of Microcomputed Tomographic and Microradiographic Measurements of Cortical Bone Porosity
The Impact of Boundary Conditions and Mesh Size on the Accuracy of Cancellous Bone Tissue Modulus Determination Using Large-Scale Finite-Element Modeling
Baeksgaard
Alendronate Increases Degree and Uniformity of Mineralization in Cancellous Bone and Decreases the Porosity in Cortical Bone of Osteoporotic Women
Risedronate for the Prevention and Treatment of Postmenopausal Osteoporosis
Etidronate for Treating and Preventing Postmenopausal Osteoporosis
Etidronate and Alendronate in the Treatment of Postmenopausal Osteoporosis
Ann. Pharmacother
How Do Bisphosphonates Prevent Fractures
Age-Related Changes in the Tensile Properties of Cortical Bone. The Relative Importance of Changes in Porosity, Mineralization, and Microstructure
Mazess
Fracture Risk: A Role for Compact Bone
Femoral Neck and Iliac Bone Histomorphometry in Femoral Neck Fracture
Regional Differences in Cortical Porosity in the Fractured Femoral Neck
Structure of the Femoral Neck in Hip Fracture: Cortical Bone Loss in the Inferoanterior to Superoposterior Axis
Brommage
Intermittently Administered Human Parathyroid Hormone(1–34) Treatment Increases Intracortical Bone Turnover and Porosity Without Reducing Bone Strength in the Humerus of Ovariectomized Cynomolgus Monkeys
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If the user wants to factor a polynomial which is not monic and square-free, i.e. the leading coefficient is not 1, or there are repeated factors, then the user should apply the Sqrfree function first. Note, the condition that a polynomial be square-free is
\mathrm{Gcd}\left(a,\frac{∂}{∂x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}a\right)=1
The algorithm used is known as Big Prime Berlekamp because its complexity is good also for large primes. For the case where the input polynomial is irreducible, the running time of the algorithm is
\mathrm{O}\left({n}^{3}+{\mathrm{log}}_{2}\left(p\right){n}^{2}\right)
arithmetic operations in GF(p^k). This is better than the Cantor Zassenhaus distinct degree algorithm. However, if the polynomial factors into many factors, these factors must be split using a probabilistic method. The running time increases to be
\mathrm{O}\left({\mathrm{log}}_{2}\left(n\right){\mathrm{log}}_{2}\left(p\right){n}^{2}+{n}^{3}\right)
in the average case.
The implementation uses Maple library code to do the linear algebra. This is not very efficient for GF(p) where p is small. The overhead of the Maple interpreter becomes small at about
p=10000000000
or in the case of an extension field.
a≔{x}^{6}+{x}^{5}+{x}^{4}+{x}^{3}+2{x}^{2}+2x+1
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{Berlekamp}\left(a,x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({x}^{2}+x+1,x\right)\right):
\mathrm{Berlekamp}\left(a,x,\mathrm{\alpha }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2
{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
p≔{10}^{10}-33
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{9999999967}
\mathrm{Berlekamp}\left({x}^{4}+2,x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p
{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3027555332}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{9284865757}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6972444635}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{9284865757}}
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A Novel In Vivo Joint Loading System to Investigate the Effect of Daily Mechanical Load on a Healing Anterior Cruciate Ligament Reconstruction | J. Med. Devices | ASME Digital Collection
Mark Stasiak M. Eng,
Mark Stasiak M. Eng
Laboratory for Soft Tissue Research,
, 535 East 70th Street, New York, NY 10021; Department of Biomedical Engineering,
e-mail: stasiakm@hss.edu
Carl Imhauser,
, 535 East 70th Street, New York, NY 10021
Jonathan Packer,
Asheesh Bedi,
David Kovacevic,
Kent Jackson,
Xiang-Hua Deng,
Scott Rodeo,
Peter Torzilli
Eng, M. S. M., Imhauser, C., Packer, J., Bedi, A., Brophy, R., Kovacevic, D., Jackson, K., Deng, X., Rodeo, S., and Torzilli, P. (March 26, 2010). "A Novel In Vivo Joint Loading System to Investigate the Effect of Daily Mechanical Load on a Healing Anterior Cruciate Ligament Reconstruction." ASME. J. Med. Devices. March 2010; 4(1): 015003. https://doi.org/10.1115/1.4001158
We designed and validated a novel knee joint fixation/distraction system to study tendon–to-bone healing in an in vivo rat model of anterior cruciate ligament (ACL) reconstruction. The system uses an external fixator to apply a cyclic distraction of the knee joint while monitoring the resultant force developed across the joint, thus providing a temporal indication of structural changes during the healing process of the bone-tendon-bone reconstruction. The validation was performed using an optical kinematic tracking system to determine the local displacement of the knee. The average system compliance was determined to be
42.4±8.8 μm/N
with a coefficient of variation of 20.7%. The compliance was used to obtain a best fit correction factor which brought the total root mean square error of knee joint distraction to within
179 μm
(16.1%) of the applied distraction. We performed a pilot study using 15 rats that had ACL reconstructions using a flexor digitorum longus tendon autograft and found that the animals tolerated the indwelling fixator and daily anesthesia over a 10 day loading protocol. Our knee joint fixation/distraction system provides a valuable tool to study how mechanical stimuli affect in vivo bone-tendon-bone healing.
biomechanics, biomedical measurement, bone, optical tracking, patient treatment, tendon, bone, healing, ACL, in vivo, cyclic strain, mechanical load
Anterior cruciate ligament, Bone, Displacement, Knee, Stress, Tendons
Isolated Tears of the Anterior Cruciate Ligament: Over 30-Year Follow-Up of Patients Treated With Arthrotomy and Primary Repair
Tendon Healing in a Bone Tunnel Differs at the Tunnel Entrance Versus the Tunnel Exit: An Effect of Graft-Tunnel Motion?
Tendon and Ligament Insertion. A Light and Electron Microscopic Study
Tendon Injuries: Basic Science and Clinical Medicine
Decreased Muscle Loading Delays Maturation of the Tendon Enthesis During Postnatal Development
Changes in the Expression of Type-X Collagen in the Fibrocartilage of Rat Achilles Tendon Attachment During Development
Tendon-Healing in a Bone Tunnel. A Biomechanical and Histological Study in the Dog
Immobilization Modulates Macrophage Accumulation in Tendon-Bone Healing
Effects of Stress Shielding on the Mechanical Properties of Rabbit Patellar Tendon
A New Device and Method for Controlling the Load in Rabbit Medial Collateral Ligament Reconstructions
The Influence of the Mechanical Environment on Remodelling of the Patellar Tendon
Mechanical Characterization of External Fixator Stiffness for a Rat Femoral Fracture Model
Mechanical Stimulation Alters Tissue Differentiation and Molecular Expression During Bone Healing
Use of Recombinant Human Bone Morphogenetic Protein-2 to Enhance Tendon Healing in a Bone Tunnel
Bone Morphogenetic Proteins-Signaling Plays a Role in Tendon-to-Bone Healing: A Study of rhBMP-2 and Noggin
Effect of Applied Strain Versus Immobilization on Tendon-to-Bone Healing in a Rat Model of ACL Reconstruction
, San Francisco, CA, Paper No. 258.
The Effect of Early and Delayed Mechanical Loading on Tendon-to-Bone Healing After ACL Reconstruction
, Las Vegas, NV, Paper No. 235.
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Implement vehicle in 3D environment - Simulink
Translation=\left[\begin{array}{ccc}{X}_{v}& {Y}_{v}& {Z}_{v}\\ {X}_{FL}& {Y}_{FL}& {Z}_{FL}\\ {X}_{FR}& {Y}_{FR}& {Z}_{FR}\\ {X}_{RL}& {Y}_{RL}& {Z}_{RL}\\ {X}_{RR}& {Y}_{RR}& {Z}_{RR}\end{array}\right]
Rotation=\left[\begin{array}{ccc}Rol{l}_{v}& Pitc{h}_{v}& Ya{w}_{v}\\ Rol{l}_{FL}& Pitc{h}_{FL}& Ya{w}_{FL}\\ Rol{l}_{FR}& Pitc{h}_{FR}& Ya{w}_{FR}\\ Rol{l}_{RL}& Pitc{h}_{RL}& Ya{w}_{RL}\\ Rol{l}_{RR}& Pitc{h}_{RR}& Ya{w}_{RR}\end{array}\right]
Scale=\left[\begin{array}{ccc}{X}_{{V}_{scale}}& {Y}_{{V}_{scale}}& {Z}_{{V}_{scale}}\\ {X}_{F{L}_{scale}}& {Y}_{F{L}_{scale}}& {Z}_{F{L}_{scale}}\\ {X}_{F{R}_{scale}}& {Y}_{F{R}_{scale}}& {Z}_{F{R}_{scale}}\\ {X}_{R{L}_{scale}}& {Y}_{R{L}_{scale}}& {Z}_{R{L}_{scale}}\\ {X}_{R{R}_{scale}}& {Y}_{R{R}_{scale}}& {Z}_{R{R}_{scale}}\end{array}\right]
Translation=\left[\begin{array}{ccc}{X}_{v}& {Y}_{v}& {Z}_{v}\\ {X}_{FL}& {Y}_{FL}& {Z}_{FL}\\ {X}_{FR}& {Y}_{FR}& {Z}_{FR}\\ {X}_{RL}& {Y}_{RL}& {Z}_{RL}\\ {X}_{RR}& {Y}_{RR}& {Z}_{RR}\end{array}\right]
Rotation=\left[\begin{array}{ccc}Rol{l}_{v}& Pitc{h}_{v}& Ya{w}_{v}\\ Rol{l}_{FL}& Pitc{h}_{FL}& Ya{w}_{FL}\\ Rol{l}_{FR}& Pitc{h}_{FR}& Ya{w}_{FR}\\ Rol{l}_{RL}& Pitc{h}_{RL}& Ya{w}_{RL}\\ Rol{l}_{RR}& Pitc{h}_{RR}& Ya{w}_{RR}\end{array}\right]
Scale=\left[\begin{array}{ccc}{X}_{{V}_{scale}}& {Y}_{{V}_{scale}}& {Z}_{{V}_{scale}}\\ {X}_{F{L}_{scale}}& {Y}_{F{L}_{scale}}& {Z}_{F{L}_{scale}}\\ {X}_{F{R}_{scale}}& {Y}_{F{R}_{scale}}& {Z}_{F{R}_{scale}}\\ {X}_{R{L}_{scale}}& {Y}_{R{L}_{scale}}& {Z}_{R{L}_{scale}}\\ {X}_{R{R}_{scale}}& {Y}_{R{R}_{scale}}& {Z}_{R{R}_{scale}}\end{array}\right]
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Measure Power of Deterministic Periodic Signals - MATLAB & Simulink Example - MathWorks Deutschland
Theoretical Power of a Single Sinusoid
Measuring the Power of a Single Sinusoid
Estimating the Power of a Single Sinusoid via PSD
Theoretical Power of Multiple Sinusoids
Measuring the Power of Multiple Sinusoids
Estimating the Power of Multiple Sinusoids Using PSD
Relationship between Power Spectrum, Power Spectral Density and ENBW
Enhanced Power Measurements Using Reassigned Periodogram
This example shows how to measure the power of deterministic periodic signals. Although continuous in time, periodic deterministic signals produce discrete power spectra. The example also shows how to improve power measurements using the reassignment technique.
In general, signals can be classified into three broad categories, power signals, energy signals, or neither. Deterministic signals which are made up of sinusoids, are an example of power signals, which have infinite energy but finite average power. Random signals also have finite average power and fall into the category of power signals. A transient signal is an example of energy signals, which start and end with zero amplitude. There are still other signals that cannot be characterized as either power signals or energy signals.
As a first example, estimate the average power of a sinusoidal signal with a peak amplitude of 1 and a frequency component at 128 Hz.
t = 0:1/Fs:1-(1/Fs);
x = A*sin(2*pi*t*F1);
Plot a portion of the signal in the time domain.
idx = 1:128;
plot(t(idx),x(idx))
The theoretical average power (mean-square) of each complex sinusoid is
{\mathit{A}}^{2}/4
, which in this example is 0.25 or –6.02 dB. So, accounting for the power in the positive and negative frequencies results in an average power of
2×{\mathit{A}}^{2}/4
power_theoretical = (A^2/4)*2
power_theoretical = 0.5000
Compute in dB the power contained in the positive frequencies:
pow2db(power_theoretical/2)
To measure the average power of the signal, call periodogram and specify the 'power' option.
periodogram(x,hamming(length(x)),[],Fs,'centered','power')
ylim([-10 -5.5])
As you can see from the zoomed-in portion of the plot, each complex sinusoid has an average power of roughly –6 dB.
Another way to calculate the average power of a signal is by "integrating" the area under the PSD curve.
periodogram(x,hamming(length(x)),[],Fs,'centered','psd')
In this plot, the peaks of the spectrum plot do not have the same height as in the power spectrum plot. The heights are different because it is the area under the curve — which is the measure of the average power — that matters when taking power spectral density (PSD) measurements. To verify that statement, use the bandpower function, which calculates the average power using the rectangle approximation to integrate under the curve.
[Pxx_hamming,F] = periodogram(x,hamming(length(x)),[],Fs,'psd');
power_freqdomain = bandpower(Pxx_hamming,F,'psd')
power_freqdomain = 0.5000
According to Parseval's theorem, the total average power of a sinusoid is the same in both the time domain and the frequency domain. Use that fact to check the value of the signal's estimated total average power by summing up the signal in the time domain.
power_timedomain = sum(abs(x).^2)/length(x)
power_timedomain = 0.5000
For the second example, estimate the total average power of a signal containing energy at multiple frequency components: one at DC with amplitude 1.5, one at 100 Hz with amplitude 4, and one at 200 Hz with amplitude 3.
t = 0:1/Fs:1-(1/Fs);
Ao = 1.5;
x = Ao + A1*sin(2*pi*t*F1) + A2*sin(2*pi*t*F2);
Plot the first 128 samples of the signal.
As in the previous example, the theoretical average power of each complex sinusoid is
{\mathit{A}}^{2}/4
. The DC average power of the signal is equal to its peak power (since it is constant) and therefore is given by
{\mathit{A}}_{0}^{2}
. Accounting for the power in the positive and negative frequencies results in a total average power value (sum of the average power of each harmonic component) of
{\mathit{A}}_{0}^{2}+2×{\mathit{A}}_{1}^{2}/4+2×{\mathit{A}}_{2}^{2}/4
for the signal.
power_theoretical = Ao^2 + (A1^2/4)*2 + (A2^2/4)*2
power_theoretical = 14.7500
Calculate the average power of each unique frequency component in dB to see that the theoretical results match the mean-square spectrum plot below.
pow2db([Ao^2 A1^2/4 A2^2/4])
To measure once again the average power of the signal, use the periodogram function once more to calculate and plot the power spectrum of the signal.
As in the first example, estimate the total average power of the signal by "integrating" under the PSD curve.
Once again the height of the peaks of the spectral density plot at a specific frequency component may not match the ones of the plot of the power spectrum. The difference is due to the reasons noted in the first example.
[Pxx, F] = periodogram(x, hamming(length(x)),[],Fs,'centered','psd');
power_freqdomain = bandpower(Pxx,F,'psd')
power_freqdomain = 14.7500
Again verify the estimated average power of the signal by invoking Parseval's theorem and summing up the signal in the time domain.
power_timedomain = 14.7500
You may have noticed that, while the height of the peaks of the power and power spectral density plots are different, the ratio of one to the other is constant.
Pxx = periodogram(x,hamming(length(x)),[],Fs,'centered','psd');
Sxx = periodogram(x,hamming(length(x)),[],Fs,'centered','power');
plot(F,Sxx./Pxx)
title('Ratio of Power Spectrum to Power Spectral Density')
ratio = mean(Sxx./Pxx)
The ratio of power to power spectral density is related to the two-sided equivalent noise bandwidth (ENBW) of the window. You can compute this ratio directly by calling the enbw function with the window and its corresponding sample rate as input arguments.
bw = enbw(hamming(length(x)),Fs)
In the previous sections, power was measured from one or multiple sinusoids having a frequency that coincided with a bin. Peak power estimates are usually less accurate when the signal frequency is out of bin. To see this effect, create a sinusoid with a non-integer number of cycles over a one-second period.
x = A*sin(2*pi*F*t);
power_theoretical = pow2db(A^2/4*2);
Create a Hamming window and a flat top window.
w1 = hamming(length(x));
w2 = flattopwin(length(x));
Compute the periodogram of x using the Hamming window. Zoom in on the peak.
stem(F,power_theoretical,'BaseValue',-50);
[Pxx1,f1] = periodogram(x,w1,nfft,Fs,'power');
plot(f1,pow2db(Pxx1),'+-')
legend('Theoretical','Periodogram')
title('Periodogram Power Spectrum Estimate')
The peak power estimate is below the theoretical peak, and the frequency of the peak estimate differs from the true frequency.
[Pmax,imax] = max(Pxx1);
dPmax_w1 = pow2db(Pmax) - power_theoretical
dPmax_w1 = -1.1046
dFreq = f1(imax) - F
dFreq = -0.4000
Reduce Amplitude Error with Zero-Padding
To see why this is happening, compute the periodogram using a larger number of FFT bins.
[Pxx2,f2] = periodogram(x,w1,100*nfft,Fs,'power');
stem(F,power_theoretical,'BaseValue',-50)
plot(f1,pow2db(Pxx1),'+')
plot(f2,pow2db(Pxx2))
legend('Theoretical Peak','nfft = 1024','nfft = 102400')
In the original periodogram, the spectral peak is located between two bins, and for that reason the estimated peak is below the theoretical peak. Increasing the number of FFT bins gives a better picture of the spectrum, although this may be a computationally expensive way to improve peak measurements.
Reduce Amplitude Error with a Flat Top Window
Another way to produce a better estimate for the peak amplitude is to use a different window. Compute the periodogram of x using the flat top window.
[Pxx,F1] = periodogram(x,w2,nfft,Fs,'power');
plot(F1,pow2db(Pxx))
legend('Theoretical','Hamming','Flat Top')
The flat top window is broad and flat. It produces a peak estimate closer to the theoretical value when x does not contain an integer number of cycles, and hence the spectral peak does not fall exactly on a bin.
dPmax_w2 = pow2db(max(Pxx)) - power_theoretical
dPmax_w2 = -6.2007e-04
The broader peak that the flat top window produces could be a disadvantage when trying to resolve closely spaced peaks, and the frequency of the measured peak is again different from the frequency of the theoretical peak.
Reduce Amplitude Error with Reassigned Periodogram
Now add the 'reassigned' flag to periodogram. Periodogram reassignment uses phase information, which is normally discarded, to reassign the signal to its center of energy. The procedure can result in sharper spectral estimates. Plot the reassigned periodogram of x and zoom in on the peak. Use the Hamming window and the flat top window.
[RPxx1,~,~,Fc1] = periodogram(x,w1,nfft,Fs,'power','reassigned');
stem(F,power_theoretical,'*','BaseValue',-40)
stem(Fc1,pow2db(RPxx1),'BaseValue',-50)
legend('Theoretical','Hamming Reassignment','Flattop Reassignment')
axis([19.5 21 -4 -2])
The reassigned estimates of power are closer to the theoretical value for both windows, with the flat top window producing the best peak measurement.
[RPxx1max,imax1] = max(RPxx1);
dPmax_reassign_w1 = pow2db(RPxx1max) - power_theoretical
dPmax_reassign_w1 = -0.0845
dPmax_reassign_w2 = -1.1131e-05
The frequency estimates are also improved using the reassigned periodogram, with the flat top window again giving the best results.
Fc1(imax1)-F
bandpower | enbw | periodogram | pow2db
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Ostrowski Type Inequalities Related to the Generalized Baouendi-Grushin Vector Fields | EMS Press
Ostrowski Type Inequalities Related to the Generalized Baouendi-Grushin Vector Fields
Jingbo Dou
XI'AN University of Finance and Economics, Xian, Shaanxi, China
In this paper, we employ a new method to prove a representation formula related to the generalized Baouendi-Grushin vector fields, and then the Ostrowski type inequalities is established in the ball and bounded domain, respectively, via the representation formula and
L^\infty
norm of the horizontal gradient. In addition, in the same spirit, we show the Hardy inequalities with boundary term related to the generalized Baouendi-Grushin vector fields.
Jingbo Dou, Yazhou Han, Ostrowski Type Inequalities Related to the Generalized Baouendi-Grushin Vector Fields. Rend. Sem. Mat. Univ. Padova 129 (2013), pp. 225–244
|
Teleportation - Uncyclopedia, the content-free encyclopedia
“How are we supposed to find out what the consequences are if we don't try it?”
~ A scientist on Teleportation
A means of going instantly from one place to another without having to appear in the space between. Also known as "plot device." There are two main kinds, both of which were invented in 1866 by famous American inventor Thomas Edison.
1 Apparent Natural Teleportation
2 Scientific Teleportation
2.1.1 Classical Matter Transmission
Apparent Natural Teleportation[edit]
Most evident with Vikings,cats, gossip, kitsune and the money in your bank account. This is the most common type, but scientific research has yet to prove it (with the exception of money, which has been caught on film sneaking out of the bank).
Scientific Teleportation[edit]
Teleportation can have disastrous effects, turning normal everyday citizens into horrible mutant freakoids.
Teleportation was first hypothesized by Ultra Jesus during the War of 1812. He wished there existed a way for his penguin allies to more easily move to and from Mars, rather than transport the entire landmass of Antarctica. His dreams of military applications of teleportation were not to be realized until, in 1866, famous American inventor Thomas Edison invented the worlds first, biggest and best teleporter. Rumours persist that during World War 3D, the Russians secretly had the technology to teleport homoerotic pornography into soldiers' bunkers. This is disputed.
Pornography, however, did become the first thing ever teleported in the lab (This is of course similar to the way pornography drove the creative advancements of photography, computers, twinkies, television, vibrating mobile phones, soap, fluorescent lighting, the wheel, and Oscar Wilde) when an issue of "Page 3: The Girls of K-Mart" was dematerialized and rematerialized 7 meters away.
Modern research into teleportation has split to three paths:
Beaming[edit]
Beaming, or matter transmission. The object to be teleported is recorded on a quantum level (and subsequently destroyed, as you cannot observe something without changing it (sort of like a n00b on Undictionary)), the data then transmitted to another site where it is mistaken for the source code for DeCSS, and deleted.
This method has not seen much in the area of actual advancement, mostly because the scientists use it to steal pornography and ice cream.
Classical Matter Transmission[edit]
A variation in which the object to be teleported is recorded without any regard for the consequences of observing it, and the electronic data is then posted on the Internet and downloaded by anyone who wants to receive the teleported object and has a compatible materialiser. The original object is then destroyed, because it's fun to smash stuff. The differences from beaming are:
Duplicates of the object can be created as many times as you like, assuming you have enough energy to do so (more or less the power of 3 exploding planets)
The electronic data can be stored on a USB key. This is handy for sneaking liquids onto an airplane.
If you're not careful, anyone can download the data and recreate the object as many times as the like. Attempting to teleport money will result in the collapse of the economy.
If you teleport an antique family heirloom that belonged to your grandfather and a cowl of cottage cheese comes out the other end, this means your grandfather was insane.
Teleporting living creatures may result in duplicates being accidentally created. The resulting identity crisis leads to the end of civilisation as we know it and/or sitcoms.
Do not teleport anything boring. If you don't get enough hits on your website, the object will never reappear.
Teleporting people may cost them their immortal soul.
Tunnelling[edit]
Physical object transportation, similar to wormhole theory, is where an object either a) travels from point A to point B through a rip or along a fold in space, b) accelerates to lightspeed by changing to a particle/wave or massive hyperneutrino, or c) attaches itself to a female shopper and announces to her the location of a sale near it's destination. All of these methods physically transport the matter making up the object, rather than create a quantum copy. This method is approved by the RIAA.
Sufficient dosages of drugs can induce an object and it's observers to think it has moved. Far out man.
Scientific Calculations[edit]
Scientists have invented an equation for teleportation:
{\displaystyle ({\frac {\alpha \times \infty }{42}}\equiv (0,0,0,0,0,0,0)=O)(O\times 42=\alpha \times \infty =\delta )[\delta \times 24](\delta \leq \infty )\therefore \delta =\pi r^{2}[\circledast =U]}
Retrieved from "https://uncyclopedia.com/w/index.php?title=Teleportation&oldid=6070683"
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Ventilation of Wind-Permeable Clothed Cylinder Subject to Periodic Swinging Motion: Modeling and Experimentation | J. Heat Transfer | ASME Digital Collection
N. Ghaddar,
N. Ghaddar
Qatar Chair in Energy Studies Professor
K. Ghali,
K. Ghali
, Beirut, 1107-2020, Lebanon
B. Jreije
N. Ghaddar Qatar Chair in Energy Studies Professor
Ghaddar, N., Ghali, K., and Jreije, B. (July 10, 2008). "Ventilation of Wind-Permeable Clothed Cylinder Subject to Periodic Swinging Motion: Modeling and Experimentation." ASME. J. Heat Transfer. September 2008; 130(9): 091702. https://doi.org/10.1115/1.2944245
A theoretical and experimental study has been performed to determine the ventilation induced by swinging motion and external wind for a fabric-covered cylinder of finite length representing a limb. The estimated ventilation rates are important in determining local thermal comfort. A model is developed to estimate the external pressure distribution resulting from the relative wind around the swinging clothed cylinder. A mass balance equation of the microclimate air layer is reduced to a pressure equation assuming laminar flow in axial and angular directions and that the air layer is lumped in the radial direction. The ventilation model predicts the total renewal rate during the swinging cycle. A good agreement was found between the predicted ventilation rates at swinging frequencies between
40rpm
60rpm
and measured values from experiments conducted in a controlled environmental chamber (air velocity is less than
0.05m∕s
) and in a low speed wind tunnel (for air speed between
2m∕s
6m∕s
) using the tracer gas method to measure the total ventilation rate induced by the swinging motion of a cylinder covered with a cotton fabric for both closed and open aperture cases. A parametric study using the current model is performed on a cotton fabric to study the effect of wind on ventilation rates for a nonmoving clothed limb at wind speeds ranging from
0.5m∕sto8m∕s
, the effect of a swinging limb in stagnant air at frequencies up to
80rpm
, and the combined effect of wind and swinging motion on the ventilation rate. For a nonmoving limb, ventilation rate increases with external wind. In the absence of wind, the ventilation rate increases with increased swinging frequency.
biomechanics, biothermics, clothing, confined flow, heat transfer, laminar flow, ventilation, ventilation, wind-permeable fabrics, swinging motion of clothed cylinder
Cylinders, Microclimate, Ventilation, Wind, Textiles
ISO 7730:2005: Ergonomics of the thermal environment—Analytical determination and interpretation of thermal comfort using calculation of the PMV and PPD indices and local thermal comfort criteria.
Harathani
Ventilation Rates of Microclimate Air Annulus of the Clothing-Skin System Under Periodic Motion
Heat and Moisture Transport From a Swinging Limb of a Clothed Walking Human
Review of Numerical Modeling of Convection, Diffusion, and Phase Change in Textiles
Subcritical Flow Past a Circular Cylinder Surrounded by a Porous Layer
Laminar Convective Heat Transfer From a Circular Cylinder Exposed to a Low Frequency Zero-Mean Velocity Oscillating Flow
A Numerical Simulation of Vortex Shedding From an Oscillating Circular Cylinder
Forces on Circular Cylinder in Viscous Oscillatory Flow at Low Keulegan-Carpenter Numbers
, 1983, ASTM D737-75, Standard Test Method for Air Permeability of Textile Fabrics, (IBR) approved.
Flow Around a Porous Cylinder Subject to Continuous Suction or Blowing
Modelling of Heat and Moisture Transport by Periodic Ventilation of Thin Cotton Fibrous Media
Convection and Ventilation in Fabric Layers
Modulated Air Layer Heat and Moisture Transport by Ventilation and Diffusion From Clothing With Open Aperture
Convection Coefficients in Clothing Air Layers
,” Ph.D. thesis, The Royal Institute of Technology, Stockholm, Sweden.
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1 April 2014 The zero section of the universal semiabelian variety and the double ramification cycle
Duke Math. J. 163(5): 953-982 (1 April 2014). DOI: 10.1215/00127094-26444575
We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties (ppav). We describe the subring generated by divisor classes, and we compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family.
The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of
{\mathcal{A}}_{g}
. Our formula provides a first step in a program to understand the Chow groups of
{\overline{\mathcal{A}}}_{g}
, especially of the perfect cone compactification, by induction on genus. By restricting to the image of
{\mathcal{M}}_{g}
under the Torelli map, our results extend the results of Hain on the double ramification cycle, answering Eliashberg’s question.
Samuel Grushevsky. Dmitry Zakharov. "The zero section of the universal semiabelian variety and the double ramification cycle." Duke Math. J. 163 (5) 953 - 982, 1 April 2014. https://doi.org/10.1215/00127094-26444575
Digital Object Identifier: 10.1215/00127094-26444575
Samuel Grushevsky, Dmitry Zakharov "The zero section of the universal semiabelian variety and the double ramification cycle," Duke Mathematical Journal, Duke Math. J. 163(5), 953-982, (1 April 2014)
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Accounting rate of return - Wikipedia
This article is about a capital budgeting concept. For other uses, see ARR.
Accounting rate of return, also known as the Average rate of return, or ARR is a financial ratio used in capital budgeting.[1] The ratio does not take into account the concept of time value of money. ARR calculates the return, generated from net income of the proposed capital investment. The ARR is a percentage return. Say, if ARR = 7%, then it means that the project is expected to earn seven cents out of each dollar invested (yearly). If the ARR is equal to or greater than the required rate of return, the project is acceptable. If it is less than the desired rate, it should be rejected. When comparing investments, the higher the ARR, the more attractive the investment. More than half of large firms calculate ARR when appraising projects.[2]
The key advantage of ARR is that it is easy to compute and understand. The main disadvantage of ARR is that it disregards the time factor in terms of time value of money or risks for long term investments. The ARR is built on evaluation of profits and it can be easily manipulated with changes in depreciation methods. The ARR can give misleading information when evaluating investments of different size.[3]
{\displaystyle {\text{ARR}}={\frac {\text{Average return during period}}{\text{Average investment}}}}
{\displaystyle {\text{Average investment}}={\frac {\text{Book value at beginning of year 1 + Book value at end of useful life}}{\text{2}}}}
{\displaystyle {\mbox{Average return during period}}={{\mbox{Incremental revenue}}-{\mbox{ Incremental expenses (including depreciation)}} \over {\mbox{Initial investment}}}}
{\displaystyle {\mbox{Average profit}}={{\mbox{Profit after tax}} \over {\mbox{Life of investment}}}}
This technique is based on profits rather than cash flow. It ignores cash flow from investment. Therefore, it can be affected by non-cash items such as bad debts and depreciation when calculating profits. The change of methods for depreciation can be manipulated and lead to higher profits.
This technique does not adjust for the risk to long term forecasts.
ARR doesn't take into account the time value of money.
^ Accounting Rate of Return - ARR
^ Arnold, G. (2007). Essentials of corporate financial management. London: Pearson Education, Ltd.
^ "How to Calculate Accounting Rate of Return | Tiduko.com | Never Stop Learning". tiduko.com. Retrieved 2017-04-25.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Accounting_rate_of_return&oldid=1056050772"
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Tate-Hochschild homology and cohomology of Frobenius algebras | EMS Press
Tate-Hochschild homology and cohomology of Frobenius algebras
David A. Jorgensen
\Lambda
be a two-sided Noetherian Gorenstein
k
-algebra, for
k
a field. We introduce Tate–Hochschild homology and cohomology groups for
\Lambda
, which are defined for all degrees, non-negative as well as negative, and which agree with the usual Hochschild homology and cohomology groups for all degrees larger than the injective dimension of
\Lambda
. We prove certain duality theorems relating the Tate–Hochschild (co)homology groups in positive degree to those in negative degree, in the case where
\Lambda
is a Frobenius algebra. We explicitly compute all Tate–Hochschild (co)homology groups for certain classes of Frobenius algebras, namely, certain quantum complete intersections.
Petter Andreas Bergh, David A. Jorgensen, Tate-Hochschild homology and cohomology of Frobenius algebras. J. Noncommut. Geom. 7 (2013), no. 4, pp. 907–937
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§ Second fundamental form
z = f(x, y)
be a (local) parametrization of the surface. Taylor expand
f
f(x + dx, y + dy) = f(x, y) + dx^T a + dy^T b + dx^T L dx + 2 dx^T M dy + dy^T N dy
We must get such a taylor expansion since our output is 1D (a real number), inputs are
dx, dy
which are 3D vectors, and the infinitesimals must be linear/tensorial. These are the only possible contractions we can make.
So, the second degree part can be written as:
\begin{bmatrix} x & y\end{bmatrix} \begin{bmatrix} L & M \\ M & N\end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix}
the matrix in the middle, or the quadratic form
II \equiv dx^T L dx + 2 dx^T M dy + dy^T N dy
is the second fundamental form.
§ Classical geometry
z = f(x, y)
be a (local) parametrization of the surface.
p ≡ (u, v)
on the surface within the local parametrization, we get tangent vectors
r_u(p) ≡ (\partial_x f(x, y)_p, r_v(p) ≡ (\partial_y f(x, y))_p
which span the tangent space at
p
These define a unique normal vector
n(p) ≡ r_u(p) × r_v(p)
at each point on the surface. This gives us a normal field.
The coefficient of the second fundamental form project the second derivative of the function
f
onto the normals. So they tell us how much the function is escaping the surface (ie, is moving along the normal to the surface) in second order.
Recall that this is pointless to do for first order, since on a circle, tangent is perpendicular to normal, so any dot product of first order information with normal will be zero.
Alternatively, first order information lies on tangent plane, and the normal is explicitly constructed as perpendicular to tangent plane, so any dot product of first order info with normal is zero.
We can only really get meaningful info by dotting with normal at second order.
So we get that
L(p) = (\partial_x \partial_x f(x, y))(p) \cdot N(p)
M(p) = (\partial_x \partial_y f(x, y))(p)
N(p) = (\partial_y \partial_y f(x, y))(p)
, where we define
L, M, N
via second fundamental form
§ Proof of equivalence between 2nd fundamental form and geometry
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Study on the Coupled Neutronic and Thermal-Hydraulic Characteristics of the New Concept Molten Salt Reactor | J. Eng. Gas Turbines Power | ASME Digital Collection
State Key Laboratory of Multi Phase Flow in Power Engineering, and School of Nuclear Science and Technology,
e-mail: szqiu@mail.xjtu.edu.cn
Wang, P., Qian, L., Zhang, D., Tian, W., Su, G., and Qiu, S. (July 14, 2010). "Study on the Coupled Neutronic and Thermal-Hydraulic Characteristics of the New Concept Molten Salt Reactor." ASME. J. Eng. Gas Turbines Power. October 2010; 132(10): 102923. https://doi.org/10.1115/1.4001067
The new concept molten salt reactor is the only liquid-fuel reactor of the six Generation IV advanced nuclear energy systems. The liquid molten salt serves as the fuel and coolant simultaneously and causes one important feature: the delayed neutron precursors are drifted by the fuel flow, which leads the spread of delayed neutrons’ distribution to noncore parts of the primary circuit, and it also results in reactivity variation depending on the flow condition of the fuel salt. Therefore, the neutronic and thermal-hydraulic characteristics of the molten salt reactor are quite different from the conventional nuclear reactors using solid fissile materials. Besides, there is no other reactor design theory and safety analysis methodologies can be used for reference. The neutronic model is derived based on the conservation of particles considering the flow effect of the fuel salt in the molten salt reactor, while the thermal-hydraulic model applies the fundamental conservation laws: the mass, momentum, and energy conservation equations. Then, the neutronic and thermal-hydraulic calculations are coupled and the influences of inflow temperature and flow velocity on the reactor physical properties are obtained. The calculated results show that the flow effect on the distributions of thermal and fast neutron fluxes is very weak, as well as on the effective multiplication factor
keff
, while the flow effect on the distribution of delayed neutron precursors is much stronger. The inflow temperature influences the distribution of neutron fluxes and delayed neutron precursors slightly, and makes a significant negative reactivity. Coupled calculation also reveals that the flow velocity of molten salt has little effect on the distribution of neutron fluxes in the steady-state, but affects the delayed neutron precursors’ distribution significantly.
fission reactor coolants, fission reactor design, fission reactor fuel, neutrons
Flow (Dynamics), Flux (Metallurgy), Fuels, Inflow, Molten salt reactors, Neutrons, Steady state, Temperature, Computer programming, Momentum, Nuclear reactors, Modeling
Molten-Salt Reactors - History, Status, and Potential
Neutronic Properties and Possible Fuel Cycle of a Molten Salt Transmuter
Proceedings of the 2003 ANS/ENS International Winter Meeting (GLOBAL 2003)
, New Orleans, LA, Nov. 16–20.
Cascade Subcritical Molten Salt Reactor (CSMSR): Main Features and Restrictions
Potential of Thorium Molten Salt Reactors: Detailed Calculations and Concept Evolution With a View to Large Scale Energy Production
Chemistry and Technology of Molten Salt Reactors-History and Perspectives
Review of Molten Salt Reactor Technology—Reactor Physics Study, Design Review and Nominal Operating Conditions, Nonproliferation Issues
Proceedings of the Fifth Framework Programme of the European Commission
, MOST Final Report.
, Arlington, VA, Apr. 25–29.
,” Issued by the U.S. DOE Nuclear Energy Research Advisory Committee and the Generation IV International Forum.
A User Guide for DRAGON 3.05C
,” Institut de Genie Nuclearire, Department de Genie Physique, Ecole Polytechnique de Montreal.
|
On Multivalued Nonexpansive Mappings in ℝ-Trees
2012 On Multivalued Nonexpansive Mappings in ℝ-Trees
K. Samanmit, B. Panyanak
The relationships between nonexpansive, weakly nonexpansive,
*
-nonexpansive, proximally nonexpansive, proximally continuous, almost lower semicontinuous, and
\varepsilon
-semicontinuous mappings in
ℝ
-trees are studied. Convergence theorems for the Ishikawa iteration processes are also discussed.
K. Samanmit. B. Panyanak. "On Multivalued Nonexpansive Mappings in ℝ-Trees." J. Appl. Math. 2012 (SI11) 1 - 13, 2012. https://doi.org/10.1155/2012/629149
K. Samanmit, B. Panyanak "On Multivalued Nonexpansive Mappings in ℝ-Trees," Journal of Applied Mathematics, J. Appl. Math. 2012(SI11), 1-13, (2012)
|
1
as a product of nonnegative integers and ordinals of the form
{\mathbf{\omega }}^{d}
{\mathbf{\omega }}^{d}+1
a={\mathbf{\omega }}^{{e}_{1}}\cdot {c}_{1}+⋯+{\mathbf{\omega }}^{{e}_{k-1}}\cdot {c}_{k-1}+{\mathbf{\omega }}^{{e}_{k}}\cdot {c}_{k}
{\mathbf{\omega }}^{{d}_{k}}\cdot {c}_{k}\cdot \left({\mathbf{\omega }}^{{d}_{k-1}}+1\right)\cdot {c}_{k-1}\cdot \dots \cdot \left({\mathbf{\omega }}^{{d}_{1}}+1\right)\cdot {c}_{1}
{d}_{k}={e}_{k}
{e}_{i+1}={e}_{i}+{d}_{i}
1\le i<k
{b}_{i}={\mathbf{\omega }}^{{d}_{i}}+1
{b}_{i}=u\cdot v
u
v
u=1
v=1
{b}_{i}={u}^{v}
u
v
u={b}_{i}
v=1
{\mathbf{\omega }}^{{d}_{k}}\cdot \left({\mathbf{\omega }}^{{d}_{k-1}}+{c}_{k}\right)\cdot \dots \cdot \left({\mathbf{\omega }}^{{d}_{1}}+{c}_{2}\right)\cdot {c}_{1}
{\mathbf{\omega }}^{{d}_{k}}\cdot {c}_{k}\cdot \left({\mathbf{\omega }}^{{d}_{k-1}}\cdot {c}_{k-1}+1\right)\cdot \dots \cdot \left({\mathbf{\omega }}^{{d}_{1}}\cdot {c}_{1}+1\right)
[[{d}_{k},{c}_{k}],[{d}_{k-1},{c}_{k-1}],\mathrm{...},[{d}_{1},{c}_{1}]]
a
{c}_{i}
\mathrm{with}\left(\mathrm{Ordinals}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{`+`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`.`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`<`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{<=}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Add}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Base}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Dec}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Decompose}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Div}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Eval}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Factor}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Gcd}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Lcm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LessThan}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Max}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Min}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Mult}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ordinal}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Power}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Split}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Sub}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`^`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{degree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lterm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{quo}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{rem}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tdegree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tterm}}]
a≔\mathrm{Ordinal}\left([[\mathrm{\omega },5],[9,4],[7,3],[5,3],[3,3],[2,2]]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}
\mathrm{Factor}\left(a\right)
[{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]
\mathrm{Factor}\left(a,\mathrm{output}=\mathrm{inert}\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\textcolor[rgb]{0,0,1}{5}
\mathrm{value}\left(\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}
\mathrm{Factor}\left(a,\mathrm{output}=\mathrm{inert},\mathrm{form}=\mathrm{monic}\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}{\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\textcolor[rgb]{0,0,1}{5}
\mathrm{Factor}\left(a,\mathrm{output}=\mathrm{inert},\mathrm{form}=\mathrm{rmonic}\right)
\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left(\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}{\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{\cdot }}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{Factor}\left(a,\mathrm{form}=\mathrm{pairs}\right)
[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]]
\mathrm{op}\left(a\right)
[[\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]
\mathrm{Factor}\left(a+x\right)
\mathrm{Factor}\left(a+x+7,\mathrm{form}=\mathrm{rmonic}\right)
[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}]
\mathrm{Mult}\left(\mathrm{op}\left(\right)\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{9}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{7}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{7}\right)
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§ Crash course on DCPO: formalizing lambda calculus
In lambda calculus, we often see functions of the form
\lambda x \rightarrow x(x)
. We would like a way to associate a "natural" mathematical object to such a function. The most obvious choice for lambda calculus is to try to create a set
V
of values which contains its own function space:
(V \rightarrow V) \subseteq V
. This seems to ask for a set whose cardinality is such that
|V|^|V| = |V|
, which is only possible if
|V| = 1
V
is the trivial set
\{ * \}
. However, we know that lambda calculus has at least two types of functions: functions that terminate and those that don't terminate. Hence the trivial set is not a valid solution. However, there is a way out. The crucial insight is one that I shall explain by analogy:
We can see that the cardinality of
\mathbb R
is different from the cardinality of the space of functions over it,
\mathbb R \rightarrow \mathbb R
However, "the set of all functions" isn't really something mathematicians consider. One would most likely consider "the set of all continuous functions"
\mathbb R \rightarrow \mathbb R
Now note that a function that is continuous over the reals is determined by its values at the rationals . So, rather than giving me a continus function
f: \mathbb R \rightarrow \mathbb R
, you can give me a continuous function
f': \mathbb Q \rightarrow \mathbb R
which I can Cauchy-complete, to get a function
\texttt{completion}(f') : \mathbb R \rightarrow \mathbb R = f
Now, cardinality considerations tell us that:
|\mathbb R^\mathbb Q| = (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \cdot \aleph_0} = 2^\aleph_0 = |R|
We've won! We have a space
\mathbb R
whose space of continuous functions
\mathbb R \rightarrow \mathbb R
\mathbb R
We bravely posit: all functions computed by lambda-calculus are continuous! Very well. This leaves us two questions to answer to answer: (1) over what space? (2) with what topology? The answers are (1) a space of partial orders (2) with the Scott topology
§ Difference between DCPO theory and Domain theory
A DCPO (directed-complete partial order) is an algebraic structure that can be satisfied by some partial orders. This definition ports 'continuity' to partial orders.
A domain is an algebraic structure of even greater generality than a DCPO. This attempts to capture the fundamental notion of 'finitely approximable'.
The presentation of a domain is quite messy. The nicest axiomatization of domains that I know of is in terms of information systems . One can find an introduction to these in the excellent book 'Introduction to Order Theory' by Davey and Priestly
§ Computation as fixpoints of continuous functions
§ CPOs
Given a partial order
(P, \leq)
. assume we have a subset
Q \subseteq P
. A least upper bound
u
Q
is an element that is the smallest element in
P
which is larger than every element in
Q
Q
P
is called as a chain if its elements can be put into order. That is, there is a labelling of elements of
Q
q1, q2, \dots, qn
q1 \leq q2 \leq \dots \leq qn
§ CCPOs
A partially ordered set is called as a chain complete partial order if each chain has a least upper bound.
This is different from a lattice, where each subset has a least upper bound.
Every ccpo has a minimal element given by
completion(\emptyset) = \bot
TODO: example of ccpo that is not a lattice
§ Monotone map
A function from
P
Q
p \leq p' \implies f(p) \leq f(p')
Composition of monotone functions is monotone.
The image of a chain wrt a monotone function is a chain.
A monotone function need not preserve least upper bounds . Consider:
f: 2^{\mathbb N} \rightarrow 2^{\mathbb N} f(S) \equiv \begin{cases} S & \text{$S$} is finite \\ S U \{ 0 \} &\text{$S$ is infinite} \end{cases}
This does not preserve least-upper-bounds. Consider the sequence of elements:
A_1 = \{ 1\}, A_2 = \{1, 2\}, A_3 = \{1, 2, 3\}, \dots, A_n = \{1, 2, 3, \dots, n \}
The union of all
A_i
\mathbb N
. Each of these sets is finite. Hence
f(\{1 \}) = \{1 \}
f(\{1, 2 \}) = \{1, 2\}
and so on. Therefore:
f(\sqcup A_i) = f(\mathbb N) = \mathbb N \cup \{ 0 \}\\ \sqcup f(A_i) = \sqcup A_i = \mathbb N
§ Continuous function
A function is continous if it is monotone and preserves all LUBs. This is only sensible as a definition on ccpos, because the equation defining it is: lub . f = f . lub, where lub: chain(P) \rightarrow P. However, for lubto always exist, we need P to be a CCPO. So, the definition of continuous only works for CCPOs.
The composition of continuous functions of chain-complete partially ordered sets is continuous.
§ Fixpoints of continuous functions
The least fixed point of a continous function
f: D \rightarrow D
\texttt{FIX}(f) \equiv \texttt{lub}(\{ f^n(\bot) : n \geq 0 \})
\leq
as implication
b \leq a
b \implies a
b
has more information than
a
, and hence implies
a
Semantics with Applications: Hanne Riis Nielson, Flemming Nielson.
Lecture notes on denotational semantics: Part 2 of the computer science Tripos
Outline of a mathematical theory of computation
Domain theory and measure theory: Video
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Werckmeister_temperament Knowpia
Werckmeister temperament
Werckmeister temperaments are the tuning systems described by Andreas Werckmeister in his writings.[1][2][3] The tuning systems are numbered in two different ways: the first refers to the order in which they were presented as "good temperaments" in Werckmeister's 1691 treatise, the second to their labelling on his monochord. The monochord labels start from III since just intonation is labelled I and quarter-comma meantone is labelled II.
The tunings I (III), II (IV) and III (V) were presented graphically by a cycle of fifths and a list of major thirds, giving the temperament of each in fractions of a comma. Werckmeister used the organbuilder's notation of ^ for a downwards tempered or narrowed interval and v for an upward tempered or widened one. (This appears counterintuitive - it is based on the use of a conical tuning tool which would reshape the ends of the pipes.) A pure fifth is simply a dash. Werckmeister was not explicit about whether the syntonic comma or Pythagorean comma was meant: the difference between them, the so-called schisma, is almost inaudible and he stated that it could be divided up among the fifths.
The last "Septenarius" tuning was not conceived in terms of fractions of a comma, despite some modern authors' attempts to approximate it by some such method. Instead, Werckmeister gave the string lengths on the monochord directly, and from that calculated how each fifth ought to be tempered.
Werckmeister I (III): "correct temperament" based on 1/4 comma divisionsEdit
This tuning uses mostly pure (perfect) fifths, as in Pythagorean tuning, but each of the fifths C-G, G-D, D-A and B-F♯ is made smaller, i.e. tempered by 1/4 of the comma. No matter if the Pythagorean comma or the syntonic comma is used, the resulting tempered fifths are for all practical purposes the same as meantone temperament fifths. All major thirds are reasonably close to 400 cents and, because not all fifths are tempered, there is no wolf fifth and all 12 notes can be used as the tonic.
Werckmeister designated this tuning as particularly suited for playing chromatic music ("ficte"), which may have led to its popularity as a tuning for J.S. Bach's music in recent years.
Fifth Tempering Third Tempering
C-G ^ C-E 1 v
G-D ^ C♯-F 4 v
D-A ^ D-F♯ 2 v
A-E - D♯-G 3 v
E-B - E-G♯ 3 v
B-F♯ ^ F-A 1 v
F♯-C♯ - F♯-B♭ 4 v
C♯-G♯ - G-B 2 v
G♯-D♯ - G♯-C 4 v
D♯-B♭ - A-C♯ 3 v
B♭-F - B♭-D 2 v
F-C - B-D♯ 3 v
Play major tonic chord (help·info)
Because a quarter of the Pythagorean comma is
{\displaystyle {\sqrt[{4}]{\frac {531441}{524288}}}}
{\displaystyle {\frac {27}{32}}{\sqrt[{4}]{2}}}
, it is possible to calculate exact mathematical values for the frequency relationships and intervals:
Exact frequency relation
{\displaystyle {\frac {1}{1}}}
{\displaystyle {\frac {256}{243}}}
{\displaystyle {\frac {64}{81}}{\sqrt {2}}}
D♯
{\displaystyle {\frac {32}{27}}}
{\displaystyle {\frac {256}{243}}{\sqrt[{4}]{2}}}
{\displaystyle {\frac {4}{3}}}
{\displaystyle {\frac {1024}{729}}}
{\displaystyle {\frac {8}{9}}{\sqrt[{4}]{2^{3}}}}
G♯
{\displaystyle {\frac {128}{81}}}
{\displaystyle {\frac {1024}{729}}{\sqrt[{4}]{2}}}
B♭
{\displaystyle {\frac {16}{9}}}
{\displaystyle {\frac {128}{81}}{\sqrt[{4}]{2}}}
Werckmeister II (IV): another temperament included in the Orgelprobe, divided up through 1/3 commaEdit
In Werckmeister II the fifths C-G, D-A, E-B, F♯-C♯, and B♭-F are tempered narrow by 1/3 comma, and the fifths G♯-D♯ and E♭-B♭ are widened by 1/3 comma. The other fifths are pure. Werckmeister designed this tuning for playing mainly diatonic music (i.e. rarely using the "black notes"). Most of its intervals are close to sixth-comma meantone. Werckmeister also gave a table of monochord lengths for this tuning, setting C=120 units, a practical approximation to the exact theoretical values[citation needed]. Following the monochord numbers the G and D are somewhat lower than their theoretical values but other notes are somewhat higher.
G-D - C♯-F 4 v
E-B ^ E-G♯ 1 v
B-F♯ - F-A 1 v
F♯-C♯ ^ F♯-B♭ 4 v
G♯-D♯ v G♯-C 4 v
D♯-B♭ v A-C♯ 1 v
B♭-F ^ B♭-D 1 v
Approximate monochord length
{\displaystyle {\frac {1}{1}}}
{\displaystyle 120}
{\displaystyle {\frac {16384}{19683}}{\sqrt[{3}]{2}}}
{\displaystyle 114{\frac {1}{5}}}
- (misprinted as
{\displaystyle 114{\frac {1}{2}}}
{\displaystyle {\frac {8}{9}}{\sqrt[{3}]{2}}}
{\displaystyle 107{\frac {1}{5}}}
{\displaystyle {\frac {32}{27}}}
{\displaystyle 101{\frac {1}{5}}}
{\displaystyle {\frac {64}{81}}{\sqrt[{3}]{4}}}
{\displaystyle 95{\frac {3}{5}}}
{\displaystyle {\frac {4}{3}}}
{\displaystyle 90}
{\displaystyle {\frac {1024}{729}}}
{\displaystyle 85{\frac {1}{3}}}
{\displaystyle {\frac {32}{27}}{\sqrt[{3}]{2}}}
{\displaystyle 80{\frac {1}{5}}}
{\displaystyle {\frac {8192}{6561}}{\sqrt[{3}]{2}}}
{\displaystyle 76{\frac {2}{15}}}
{\displaystyle {\frac {256}{243}}{\sqrt[{3}]{4}}}
{\displaystyle 71{\frac {7}{10}}}
{\displaystyle {\frac {9}{4{\sqrt[{3}]{2}}}}}
{\displaystyle 67{\frac {1}{5}}}
{\displaystyle {\frac {4096}{2187}}}
{\displaystyle 64}
Werckmeister III (V): an additional temperament divided up through 1/4 commaEdit
In Werckmeister III the fifths D-A, A-E, F♯-C♯, C♯-G♯, and F-C are narrowed by 1/4, and the fifth G♯-D♯ is widened by 1/4 comma. The other fifths are pure. This temperament is closer to equal temperament than the previous two.
C-G - C-E 2 v
A-E ^ D♯-G 3 v
C♯-G♯ ^ G-B 2 v
F-C ^ B-D♯ 3 v
{\displaystyle {\frac {1}{1}}}
{\displaystyle {\frac {8}{9}}{\sqrt[{4}]{2}}}
{\displaystyle {\frac {9}{8}}}
{\displaystyle {\sqrt[{4}]{2}}}
{\displaystyle {\frac {8}{9}}{\sqrt {2}}}
{\displaystyle {\frac {9}{8}}{\sqrt[{4}]{2}}}
{\displaystyle {\sqrt {2}}}
{\displaystyle {\frac {3}{2}}}
{\displaystyle {\frac {128}{81}}}
{\displaystyle {\sqrt[{4}]{8}}}
{\displaystyle {\frac {3}{\sqrt[{4}]{8}}}}
{\displaystyle {\frac {4}{3}}{\sqrt {2}}}
Werckmeister IV (VI): the Septenarius tuningsEdit
This tuning is based on a division of the monochord length into
{\displaystyle 196=7\times 7\times 4}
parts. The various notes are then defined by which 196-division one should place the bridge on in order to produce their pitches. The resulting scale has rational frequency relationships, so it is mathematically distinct from the irrational tempered values above; however in practice, both involve pure and impure sounding fifths. Werckmeister also gave a version where the total length is divided into 147 parts, which is simply a transposition of the intervals of the 196-tuning. He described the Septenarius as "an additional temperament which has nothing at all to do with the divisions of the comma, nevertheless in practice so correct that one can be really satisfied with it".
One apparent problem with these tunings is the value given to D (or A in the transposed version): Werckmeister writes it as 176. However this produces a musically bad effect because the fifth G-D would then be very flat (more than half a comma); the third B♭-D would be pure, but D-F♯ would be more than a comma too sharp - all of which contradict the rest of Werckmeister's writings on temperament. In the illustration of the monochord division, the number "176" is written one place too far to the right, where 175 should be. Therefore it is conceivable that the number 176 is a mistake for 175, which gives a musically much more consistent result. Both values are given in the table below.
In the tuning with D=175, the fifths C-G, G-D, D-A, B-F♯, F♯-C♯, and B♭-F are tempered narrow, while the fifth G♯-D♯ is tempered wider than pure; the other fifths are pure.
Monochord length
C 196 1/1 0
C♯ 186 98/93 91
D 176(175) 49/44(28/25) 186(196)
D♯ 165 196/165 298
E 156 49/39 395
F 147 4/3 498
F♯ 139 196/139 595
G 131 196/131 698
G♯ 124 49/31 793
A 117 196/117 893
B♭ 110 98/55 1000
B 104 49/26 1097
196-EDL & 1568-EDL and Septenarius tunings
"Well Tempering based on the Werckmeister Definition"
Well Tempered based on Werckmeisters last book Musikalische Paradoxal-Discourse (1707) is Equal Temperament. See: https://www.academia.edu/5210832/18th_Century_Quotes_on_J.S._Bachs_Temperament
^ Andreas Werckmeister: Orgel-Probe (Frankfurt & Leipzig 1681), excerpts in Mark Lindley, "Stimmung und Temperatur", in Hören, messen und rechnen in der frühen Neuzeit pp. 109–331, Frieder Zaminer (ed.), vol. 6 of Geschichte der Musiktheorie, Wissenschaftliche Buchgesellschaft (Darmstadt 1987).
^ A. Werckmeister: Musicae mathematicae hodegus curiosus oder Richtiger Musicalischer Weg-Weiser (Quedlinburg 1686, Frankfurt & Leipzig 1687) ISBN 3-487-04080-8
^ A. Werckmeister: Musicalische Temperatur (Quedlinburg 1691), reprint edited by Rudolf Rasch ISBN 90-70907-02-X
|
The profit made by ACME Home Construction is represented by the function
y=−x^2+50x
x=
the number of homes built and
y=
the profit in thousands of dollars. How many homes should be built to maximize the profit? Hint: Think about the graph of the function.
Start by converting the equation to graphing form by completing the square.
Does this parabola open up or down?
Does the vertex represent the maximum
y
25
|
A very long thick plate has uniform positive volume charge density given by ρx=ρ0 for -d2≤x≤d2, where ρ0 is a - Physics - Electric Charges And Fields - 11350609 | Meritnation.com
A very long thick plate has uniform positive volume charge density given by
\rho \left(x\right)={\rho }_{0} \mathrm{for} -\frac{d}{2}\le x\le \frac{d}{2}
, where ρ 0 is a positive constant, d is the thickness of the plate and x = 0 is the centre of the plate.
114. Use Gauss' law of find E(a)
(1) ρ 0 a/ε 0 (2) 2ρ 0 a/ ε 0 (3) ρ 0 a/2ε 0 (4) 4ρ 0 a/ε 0
For the given case gaussian surface will be a pillbox\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}For the the point lies inside the sheet \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Total electric flux for gaussian pillbox is given by \phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\phi }_{E}= 2EA Here A is area of cross section of pillbox.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Total charge inside the pillbox is given by\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{Q}_{in}=2\left(aA\right)\left({\rho }_{0}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}By Gauss\text{'}s Law-\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{\phi }_{E}=\frac{{Q}_{in}}{{\epsilon }_{0}} ⇒ 2EA =\frac{2\left(aA\right)\left({\rho }_{0}\right)}{{\epsilon }_{0}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}E=\frac{{\rho }_{0}a}{{\epsilon }_{0}}
|
The diffeomorphic types of the complements of arrangements in CP 3 I: Point arrangements
April, 2007 The diffeomorphic types of the complements of arrangements in
{\mathbf{CP}}^{3}
I: Point arrangements
Shaobo WANG, Stephen S.-T. YAU
For any arrangement of hyperplanes in
{\mathbf{CP}}^{3}
, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. If the soul consists of a set of points (0-simplices) and a set of planes (2-simplices), then the arrangement is called point arrangement. In this paper, we give a sufficient combinatoric condition for two point arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient condition on combinatorics for the point arrangement of hyperplanes whose moduli space is connected.
Shaobo WANG. Stephen S.-T. YAU. "The diffeomorphic types of the complements of arrangements in
{\mathbf{CP}}^{3}
I: Point arrangements." J. Math. Soc. Japan 59 (2) 423 - 447, April, 2007. https://doi.org/10.2969/jmsj/05920423
Keywords: arrangement , combinatorics , complement and $\bm{CP}^3$ , diffeomorphic type , hyperplane , moduli space , nice point arrangement
Shaobo WANG, Stephen S.-T. YAU "The diffeomorphic types of the complements of arrangements in
{\mathbf{CP}}^{3}
I: Point arrangements," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 59(2), 423-447, (April, 2007)
|
Completeness of Reidemeister-type moves for surfaces embedded in three-dimensional space | EMS Press
In this paper we are concerned with labelled apparent contours, namely with apparent contours of generic orthogonal projections of embedded surfaces in
\mathbb R^3
, endowed with a suitable information on the relative depth. We give a proof of the following theorem: there exists a finite set of elementary moves (i.e. local topological changes) on labelled apparent contours such that two generic embeddings in
\mathbb R^3
of a closed surface are isotopic if and only if their apparent contours can be connected using only smooth planar isotopies and a finite sequence of moves. This result, that can be obtained as a by-product of general results on knotted surfaces and singularity theory, is obtained here with a direct and rather elementary proof.
Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Completeness of Reidemeister-type moves for surfaces embedded in three-dimensional space. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 23 (2012), no. 1, pp. 69–87
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Write and solve a system of equations for the situation described below. Define your variables and write your solution as a sentence.
Daria has all nickels and quarters. The number of nickels is
3
more than twice the number of quarters. If she has
\$1.90
in all, how many nickels does Daria have?
n=
q=
n=2q+3
\$0.05n+\$0.25q=\$1.90
|
{x}_{0}
{x}_{0}
{x}_{0}
X\left(t\right)
\mathrm{dX}\left(t\right)=\mathrm{\mu }\left(X\left(t\right),t\right)\mathrm{dt}+\mathrm{\sigma }\left(X\left(t\right),t\right)\mathrm{dW}\left(t\right)
\mathrm{\mu }\left(X\left(t\right),t\right)
\mathrm{\sigma }\left(X\left(t\right),t\right)
W\left(t\right)
{x}_{0}
X
is an
{X}_{1}
{X}_{n}
{\mathrm{\mu }}_{1}
{\mathrm{\mu }}_{n}
{\mathrm{\sigma }}_{1}
{\mathrm{\sigma }}_{n}
be the corresponding drift and diffusion terms. The ItoProcess(X, Sigma) command will create an
Y
{\mathrm{dY}\left(t\right)}_{i}={\mathrm{\mu }}_{i}\left({Y\left(t\right)}_{i},t\right)+{\mathrm{\sigma }}_{i}\left({Y\left(t\right)}_{i},t\right){\mathrm{dW}\left(t\right)}_{i}
W\left(t\right)
is an
\mathrm{with}\left(\mathrm{Finance}\right):
Y≔\mathrm{ItoProcess}\left(1.0,\mathrm{\mu },\mathrm{\sigma },x,t\right)
\textcolor[rgb]{0,0,1}{Y}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X0}}
\mathrm{Drift}\left(Y\left(t\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{Diffusion}\left(Y\left(t\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}
\mathrm{Drift}\left(\mathrm{exp}\left(Y\left(t\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Diffusion}\left(\mathrm{exp}\left(Y\left(t\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{_X0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}
\mathrm{\mu }≔0.1
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.1}
\mathrm{\sigma }≔0.5
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.5}
\mathrm{PathPlot}\left(\mathrm{exp}\left(Y\left(t\right)\right),t=0..3,\mathrm{timesteps}=100,\mathrm{replications}=10\right)
\mathrm{\mu }≔'\mathrm{\mu }'
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}
\mathrm{\sigma }≔'\mathrm{\sigma }'
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}
\mathrm{X0}≔〈100.0,0.〉
\textcolor[rgb]{0,0,1}{\mathrm{X0}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{100.0}\\ \textcolor[rgb]{0,0,1}{0.}\end{array}]
\mathrm{Μ}≔〈\mathrm{\mu }X[1],\mathrm{\kappa }\left(\mathrm{\theta }-X[2]\right)〉
\textcolor[rgb]{0,0,1}{\mathrm{Μ}}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}\right)\end{array}]
\mathrm{\Sigma }≔〈〈\mathrm{sqrt}\left(X[2]\right)X[1]|0.〉,〈0.|\mathrm{\sigma }X[2]〉〉
\textcolor[rgb]{0,0,1}{\mathrm{\Sigma }}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\sqrt{{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0.}\\ \textcolor[rgb]{0,0,1}{0.}& \textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{X}}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
S≔\mathrm{ItoProcess}\left(\mathrm{X0},\mathrm{Μ},\mathrm{\Sigma },X,t\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X2}}
\mathrm{Drift}\left(S\left(t\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\right)\end{array}]
\mathrm{Diffusion}\left(S\left(t\right)\right)
[\begin{array}{cc}\sqrt{{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{\mu }≔0.1
\textcolor[rgb]{0,0,1}{\mathrm{\mu }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.1}
\mathrm{\sigma }≔0.5
\textcolor[rgb]{0,0,1}{\mathrm{\sigma }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.5}
\mathrm{\kappa }≔1.0
\textcolor[rgb]{0,0,1}{\mathrm{\kappa }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{1.0}
\mathrm{\theta }≔0.4
\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{0.4}
A≔\mathrm{SamplePath}\left(S\left(t\right),t=0..1,\mathrm{timesteps}=100,\mathrm{replications}=10\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{100.}& \textcolor[rgb]{0,0,1}{100.670084719786}\\ \textcolor[rgb]{0,0,1}{100.100000000000}& \textcolor[rgb]{0,0,1}{101.034139425728}\\ \textcolor[rgb]{0,0,1}{100.280880089808}& \textcolor[rgb]{0,0,1}{101.924198818577}\\ \textcolor[rgb]{0,0,1}{102.915077811759}& \textcolor[rgb]{0,0,1}{99.6518477031121}\\ \textcolor[rgb]{0,0,1}{103.858818858166}& \textcolor[rgb]{0,0,1}{100.628185358730}\\ \textcolor[rgb]{0,0,1}{104.476699657855}& \textcolor[rgb]{0,0,1}{98.7691518445139}\\ \textcolor[rgb]{0,0,1}{103.737362966326}& \textcolor[rgb]{0,0,1}{95.0859221941374}\\ \textcolor[rgb]{0,0,1}{102.574346549913}& \textcolor[rgb]{0,0,1}{94.1008617878134}\\ \textcolor[rgb]{0,0,1}{101.159282939668}& \textcolor[rgb]{0,0,1}{92.9644135833222}\\ \textcolor[rgb]{0,0,1}{100.709702216007}& \textcolor[rgb]{0,0,1}{93.6061768383076}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{slice of 10 × 2 × 101 Array}}\end{array}
\mathrm{PathPlot}\left(A,1,\mathrm{thickness}=3,\mathrm{markers}=\mathrm{false},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)
\mathrm{PathPlot}\left(A,2,\mathrm{thickness}=3,\mathrm{markers}=\mathrm{false},\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)
\mathrm{ExpectedValue}\left(\mathrm{max}\left(S\left(1\right)[1]-100,0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{21.41114565}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.3390630872}]
X≔\mathrm{GeometricBrownianMotion}\left(100.0,0.05,0.3,t\right)
\textcolor[rgb]{0,0,1}{X}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X4}}
Y≔\mathrm{GeometricBrownianMotion}\left(100.0,0.07,0.2,t\right)
\textcolor[rgb]{0,0,1}{Y}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X5}}
\mathrm{\Sigma }≔〈〈1|0.5〉,〈0.5|1〉〉
\textcolor[rgb]{0,0,1}{\mathrm{\Sigma }}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0.5}\\ \textcolor[rgb]{0,0,1}{0.5}& \textcolor[rgb]{0,0,1}{1}\end{array}]
Z≔\mathrm{ItoProcess}\left(〈X,Y〉,\mathrm{\Sigma }\right)
\textcolor[rgb]{0,0,1}{Z}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_X6}}
\mathrm{Drift}\left(Z\left(t\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.05}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{0.07}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{Diffusion}\left(Z\left(t\right)\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{0.3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}& \textcolor[rgb]{0,0,1}{0.15}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{1}}\\ \textcolor[rgb]{0,0,1}{0.10}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{0.2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_X6}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)}_{\textcolor[rgb]{0,0,1}{2}}\end{array}]
\mathrm{ExpectedValue}\left(\mathrm{max}\left(X\left(1\right)-Y\left(1\right),0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{14.32896059}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.2447103632}]
\mathrm{ExpectedValue}\left(\mathrm{max}\left(Z\left(1\right)[1]-Z\left(1\right)[2],0\right),\mathrm{timesteps}=100,\mathrm{replications}={10}^{4}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{8.103315185}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{standarderror}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.1520913055}]
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Tags: machine learning research all
Independent Study on Modern Deep Learning
Misc. Machine Learning Methodologies
This page serves as both research notes and a workspace for my independent study at the University of Pennsylvania, advised by Pratik Chaudhari. This independent study is meant to serve as an extension to the course ESE 546: Principles of Deep Learning. The goal of the study is to learn more advanced paradigms of training deep learning models, as well as inference using them. This work was split into two sections: 1) a literature review of papers on various topics and 2) experiments with different types of models and algorithms, implemented in Python. Although the COVID-19 pandemic limited certain topics I would have like to explore (namely reinforcement learning), I was able to learn many new concepts and promising research avenues.
Disclaimer: the "Modern Deep Learning* name is meant to be a joke, since the focus on this independent study is on cool/interesting trends in deep learning, which is already a trendy area of research.
Below are my notes on some recent interesting papers that have expanded my views on standard machine learning paradigms.
Rajbhandari et al. 2020
This paper introduces a novel optimizer named the Zero Redundency Optimizer (ZeRO), which aims to make it feasible and efficient to train previously impossible to train model architectures whose training exhibits memory limitations. This is done by partitioning model states as opposed to standard model state replication across clusters. Memory analysis shows that the optimizer can train a one trillion parameter model on 1024 GPUs with data parallelism degree
N_d = 1024
This paper explores different neural network architectures by generating random neural network wirings. This is done by defining a stochastic network generator to encapsulate Neural Architecture Search, and later using classical random graph algorithms for wiring the networks. The authors show that the generated networks have competitve performance on the ImageNet task.
Network Generators define a family of possible wiring patterns. Network architectures can thereby be sampled according to a probability distribution, which is differentiably learnable.
Formally, the generator is a mapping
g: \Theta \rightarrow N
\Theta
is the parameter space and
N
is the space of neural network architectures. As such,
g
determines how the computational graph representing the neural network is wired. The given parameters
\theta \in \Theta
specifies meta-information about the network such as the number of layers, activation types, etc. The output of
g
is symbolic, so it doesn't return the weights of the networks (which can be learned from standard differentiable training processes) but instead a representation of the network (e.g. flow of data and types of operations).
The network generator
g
can be extended to include an additional argument
s
, which acts as a stochastic seed. Then, the generator
g(s, \theta)
can be repeatedly called to generate a pseudo-random family of architectures.
Graphs to Neural Networks
The neural network generator generates a general graph, which is a set of nodes followed by a set of edges that connect the nodes. This general representation does not specify how the graph corresponds to a neural network, which is a later post-processing step. The non-restrictivness of the general graph allows the use of classical graph generation techniques from graph theory. In particular, the authors experiment with Erdos-Renyi (ER), Barabasi-Albert (BA), and Watts-Strogatz (WS) models of graph generation.
The generated edges are defined to be data flow (i.e. sending a tensor of data from one node to another) and that nodes define operations of either:
Aggregation (e.g. weighted sum)
Transformation (e.g. non-linearity)
Distribution (e.g. copying data)
For each generator, the authors sample 5 instances (generated by 5 random seeds), and train them from scratch. Networks are trained for roughly 100 epochs, using a half-period-cosine learning rate decay from an initial learning rate of 0.1 with a momentum of 0.9.
The authors note that every random generator yields decent accuracy. Furthermore, the variation among the random network instances is rather low with a standard deviation in the range of 0.2% to 0.4%.
Hudson et. al, 2018
The authors design a novel fully differentiable neural network architecture that is capable of explicit and expressive reasoning. One primary goal of the paper is interpretability, without sacrificing the predictive performance of black box methods. Problems are decomposed into attention-based steps, and are solved using Memory, Attention, and Composition (MAC) sub-units. On the CLEVR dataset for visual reasoning, the model accomplishes a state-of-the-art 98.9% accuracy, using less data than competing models.
Below are some experiments I've run for playing around with state-of-the-art models. Code will be available on GitHub, after the end of my independent study.
A topic that originally prompted me to begin this independent study is Multitask Learning due to the similarities with how humans acquire knowledge in the real world. One example of this is Open AI's GPT-2 model, whose paper was named Language Models are Unsupervised Multitask Learners. As part of this study, I have implemented numerous experiments ranging from using libraries to fine-tune large pre-trained language models, to implementing various sampling algorithms for natural language generation in PyTorch.
Fine-Tuning Transformer Models (GPT-2)
Easy usage can be done through the GPT-2 Simple package by Max Woolf (https://github.com/minimaxir/gpt-2-simple).
Install using pip3 install gpt-2-simple and provide text for fine-tuning:
# download pre-trained GPT-2 model with 124M parameters
# provide file for fine-tuning
# start fine-tuning tensorflow session
gpt2.finetune(sess, file_name, model_name=model_name, steps=1000)
I trained a fine-tuned GPT-2 model on a corpus of Barack Obama tweets I put together.
We have a clear goal: Ending the use of force in Afghanistan as quickly
as possible. That means giving Congress more time to figure out how to
make that happen. And doing so is the single most effective way forward.
The Afghan people deserve better. They and I are foot soldiers for them.
We're going to use all our might to get that goal accomplished.
But America is not going to give ourselves up for expedience's sake.
Wow! Thanks, Obama for the big policy change!
Transfer Learning via Control Sequences
As per the CTRL language model paper by Keskar et al., I also experimented with prepending a "control sequence"
c
that would allow the fine-tuning process to control downstream generation. In particular, I used the Hugging Face Transformers library to fine-tune the pre-trained 355M GPT-2 with Venture Captial tweets (in typical fashion). As a control sequence, I set
c
to be the user name of generation (e.g. @paulg) and was able to reduce perplexity by over 20%. This creates an implicit Transfer Learning effect within the fine-tuning process.
Some example generations turned out being very promising:
\paulg- No such things as moral judgment to deal in a way that matters; I wish this could just happen for the good of the community rather than being a thing that's only achieved in theory. The world of the future...we could do better:
\paulg- You can see a bit of a pattern here; the more you work on a project the more things you learn about it. The same is true in life as in business, and the less we do, and the faster we work with…
\paulg- "A great start in an exciting field? Make sure it's a good one. If not, you could be running a startup in the dark." - @joshk-
My favourite is \paulg- It takes a village to raise an army..
Following our discussion on Generative Adversarial Networks in ESE 546, I've also been playing with Style Transfer GANs by exploring abstract artwork from contemporary artists. These adversarial paradigms prove to be very promising in generation tasks, and can be a lot less noisy than Variational Autoencoders. An interesting area of research is applying these adversarial paradigms to discrete domains, such as natural language.
(this is based off of pieces from Marc Chagall)
(this is based off pieces from Jerret Lee)
Generation Algorithms for Language
Reading the paper The Curious Case of Neural Text Degeneration prompted me to learn more about different sampling algorithms. In particular, these algorithms attempt to efficiently determine the most likely sequence of text given a probability distribution
P(w_t ~|~ w_{t - 1}, w_{t - 2}, ...)
. This distribution is often the output of a softmax layer on a neural network. Although the state-of-the-art language models are tremendously large models with hundreds of millions of parameters, I was able to experiment with different sampling algorithms using a small LSTM language model.
The model definition for my LSTM was rather simple:
def __init__(self, embedding_dim, hidden_dim, vocab_size):
self.embedding = torch.nn.Embedding(vocab_size, embedding_dim)
self.rnn = torch.nn.LSTM(embedding_dim, hidden_dim, batch_first=True)
self.output = torch.nn.Linear(hidden_dim, vocab_size)
def forward(self, data, hidden):
embedded = self.embedding(data)
prediction, hidden = self.rnn(embedded, hidden)
return self.output(prediction), hidden
return torch.zeros(1, BATCH_SIZE, self.hidden_dim), torch.zeros(1, BATCH_SIZE, self.hidden_dim)
The first and most intuitive generation strategy is known as Greedy Decoding, where we take the most probable word over a vocabulary
V
c
w_i = \operatorname*{arg\, max}_{w \in V} ~ P(w_i ~ | ~ c_0 ~ ... ~ c_{i - 1})
However, this produced rather trite and non-sensical generations because language often has a pertinent information horizon larger than a single time-step. An improvement is known as Top-
k
Sampling, which truncates the probability distribution to the
k
most likely tokens in the vocabulary.
"I am proposing with an advantage over commerce budget. — (applause) the middle of commerce, way together more of each other people’s it. In the chance that the international issue, freedom we have never has allowed the other way, or share from footing or denied coverage for the work of Democrats and Republican administrations isn’t (Applause.) Now, none of this can happen unless we’re their own rules that progress on so tied long still blind you should make Wall good example. (Applause.) For unemployment to pull all we should leave just like us — (applause)".
Introducing non-deterministic generation seems to be the key to creative and engaging generations.
In conclusion, this independent study has shown me that even though there is a lot of hype and effort in deep learning, there is still a lot of research left. Although companies and organizations are developing larger and more expensive models, many core concepts are still researchable using smaller and more managable models. Chasing the state-of-the-art results is often a challenging and sometimes unrewarding endeavour, whereas devising new paradigms of machine learning training (like paralellizing training over cloud instances) and interesting inference strategies (e.g. Top-
k
sampling) can make comparable results while still asking promising research questions about machine intelligence. Overall, I am excited to continue research in machine learning, as well as different topics in linguistics and probabilty.
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Box plot - Wikipedia
(Redirected from Box and whisker)
In descriptive statistics, a box plot or boxplot is a method for graphically demonstrating the locality, spread and skewness groups of numerical data through their quartiles.[1] In addition to the box on a box plot, there can be lines (which are called whiskers) extending from the box indicating variability outside the upper and lower quartiles, thus, the plot is also termed as the box-and-whisker plot and the box-and-whisker diagram. Outliers that differ significantly from the rest of the dataset[2] may be plotted as individual points beyond the whiskers on the box-plot. Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution[3] (though Tukey's boxplot assumes symmetry for the whiskers and normality for their length). The spacings in each subsection of the box-plot indicate the degree of dispersion (spread) and skewness of the data, which are usually described using the five-number summary. In addition, the box-plot allows one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean. Box plots can be drawn either horizontally or vertically.
4.1 Example without outliers
4.2 Example with outliers
4.3 In the case of large datasets
4.3.1 General equation to compute empirical quantiles
The range-bar method was first introduced by Mary Eleanor Spear in her book "Charting Statistics" in 1952[4] and again in her book "Practical Charting Techniques" in 1969.[5] The box-and-whisker plot was first introduced in 1970 by John Tukey, who later published on the subject in his book "Exploratory Data Analysis" in 1977.[6]
First quartile (Q1 or 25th percentile): also known as the lower quartile qn(0.25), it is the median of the lower half of the dataset.
Third quartile (Q3 or 75th percentile): also known as the upper quartile qn(0.75), it is the median of the upper half of the dataset.[7]
{\displaystyle {\text{IQR}}=Q_{3}-Q_{1}=q_{n}(0.75)-q_{n}(0.25)}
Another popular choice for the boundaries of the whiskers is based on the 1.5 IQR value. From above the upper quartile (Q3), a distance of 1.5 times the IQR is measured out and a whisker is drawn up to the largest observed data point from the dataset that falls within this distance. Similarly, a distance of 1.5 times the IQR is measured out below the lower quartile (Q1) and a whisker is drawn down to the lowest observed data point from the dataset that falls within this distance. Because the whiskers must end at an observed data point, the whisker lengths can look unequal, even though 1.5 IQR is the same for both sides. All other observed data points outside the boundary of the whiskers are plotted as outliers.[8] The outliers can be plotted on the box-plot as a dot, a small circle, a star, etc..
Some box plots include an additional character to represent the mean of the data.[9][10]
Variable width box plots illustrate the size of each group whose data is being plotted by making the width of the box proportional to the size of the group. A popular convention is to make the box width proportional to the square root of the size of the group.[11]
Notched box plots apply a "notch" or narrowing of the box around the median. Notches are useful in offering a rough guide of the significance of the difference of medians; if the notches of two boxes do not overlap, this will provide evidence of a statistically significant difference between the medians.[11] The width of the notches is proportional to the interquartile range (IQR) of the sample and is inversely proportional to the square root of the size of the sample. However, there is a uncertainty about the most appropriate multiplier (as this may vary depending on the similarity of the variances of the samples).[11]
{\displaystyle \pm {\frac {1.58{\text{ IQR}}}{\sqrt {n}}}}
around the median.[12]
Adjusted box plots are intended to describe skew distributions, and they rely on the medcouple statistic of skewness.[13] For a medcouple value of MC, the lengths of the upper and lower whiskers on the box-plot are respectively defined to be:
{\displaystyle {\begin{matrix}1.5{\text{IQR}}\cdot e^{3{\text{MC}}},&1.5{\text{ IQR}}\cdot e^{-4{\text{MC}}}{\text{ if }}{\text{MC}}\geq 0,\\1.5{\text{IQR}}\cdot e^{4{\text{MC}}},&1.5{\text{ IQR}}\cdot e^{-3{\text{MC}}}{\text{ if }}{\text{MC}}\leq 0.\end{matrix}}}
{\displaystyle 1.5{\text{ IQR}}}
Other kinds of box plots, such as the violin plots and the bean plots can show the difference between single-modal and multimodal distributions, which cannot be observed from the original classical box-plot.[6]
Example without outliers[edit]
The first quartile value (Q1 or 25th percentile) is the number that marks one quarter of the ordered data set. In other words, there are exactly 25% of the elements that are less than the first quartile and exactly 75% of the elements that are greater than it. The first quartile value can be easily determined by finding the "middle" number between the minimum and the median. For the hourly temperatures, the "middle" number found between 57 °F and 70 °F is 66 °F.
The third quartile value (Q3 or 75th percentile) is the number that marks three quarters of the ordered data set. In other words, there are exactly 75% of the elements that are less than the third quartile and 25% of the elements that are greater than it. The third quartile value can be easily obtained by finding the "middle" number between the median and the maximum. For the hourly temperatures, the "middle" number between 70 °F and 81 °F is 75 °F.
{\displaystyle {\text{IQR}}=Q_{3}-Q_{1}=75^{\circ }F-66^{\circ }F=9^{\circ }F.}
{\displaystyle 1.5{\text{IQR}}=1.5\cdot 9^{\circ }F=13.5^{\circ }F.}
{\displaystyle Q_{3}+1.5{\text{ IQR}}=75^{\circ }F+13.5^{\circ }F=88.5^{\circ }F.}
{\displaystyle Q_{1}-1.5{\text{ IQR}}=66^{\circ }F-13.5^{\circ }F=52.5^{\circ }F.}
Example with outliers[edit]
In the case of large datasets[edit]
General equation to compute empirical quantiles[edit]
{\displaystyle q_{n}(p)=x_{(k)}+\alpha (x_{(k+1)}-x_{(k)})}
{\displaystyle {\text{with }}k=[p(n+1)]{\text{ and }}\alpha =p(n+1)-k}
{\displaystyle x_{(k)}}
{\displaystyle i<k}
{\displaystyle x_{(i)}<x_{(k)}}
{\displaystyle q_{n}(0.5)=x_{(12)}+(0.5\cdot 25-12)\cdot (x_{(13)}-x_{(12)})=70+(0.5\cdot 25-12)\cdot (70-70)=70^{\circ }F}
{\displaystyle q_{n}(0.25)=x_{(6)}+(0.25\cdot 25-6)\cdot (x_{(7)}-x_{(6)})=66+(0.25\cdot 25-6)\cdot (66-66)=66^{\circ }F}
{\displaystyle q_{n}(0.75)=x_{(18)}+(0.75\cdot 25-18)\cdot (x_{(19)}-x_{(18)})=75+(0.75\cdot 25-18)\cdot (75-75)=75^{\circ }F}
Figure 7. Box-plot and a probability density function (pdf) of a Normal N(0,1σ2) Population
^ C., Dutoit, S. H. (2012). Graphical exploratory data analysis. Springer. ISBN 978-1-4612-9371-2. OCLC 1019645745.
^ Grubbs, Frank E. (February 1969). "Procedures for Detecting Outlying Observations in Samples". Technometrics. 11 (1): 1–21. doi:10.1080/00401706.1969.10490657. ISSN 0040-1706.
^ Richard., Boddy (2009). Statistical Methods in Practice : for Scientists and Technologists. John Wiley & Sons. ISBN 978-0-470-74664-6. OCLC 940679163.
^ Spear, Mary Eleanor (1952). Charting Statistics. McGraw Hill. p. 166.
^ Spear, Mary Eleanor. (1969). Practical charting techniques. New York: McGraw-Hill. ISBN 0070600104. OCLC 924909765.
^ a b Wickham, Hadley; Stryjewski, Lisa. "40 years of boxplots" (PDF). Retrieved December 24, 2020.
^ Holmes, Alexander; Illowsky, Barbara; Dean, Susan (31 March 2015). "Introductory Business Statistics". OpenStax.
^ Dekking, F.M. (2005). A Modern Introduction to Probability and Statistics. Springer. pp. 234–238. ISBN 1-85233-896-2.
^ Frigge, Michael; Hoaglin, David C.; Iglewicz, Boris (February 1989). "Some Implementations of the Boxplot". The American Statistician. 43 (1): 50–54. doi:10.2307/2685173. JSTOR 2685173.
^ Marmolejo-Ramos, F.; Tian, S. (2010). "The shifting boxplot. A boxplot based on essential summary statistics around the mean". International Journal of Psychological Research. 3 (1): 37–46. doi:10.21500/20112084.823.
^ a b c McGill, Robert; Tukey, John W.; Larsen, Wayne A. (February 1978). "Variations of Box Plots". The American Statistician. 32 (1): 12–16. doi:10.2307/2683468. JSTOR 2683468.
^ "R: Box Plot Statistics". R manual. Retrieved 26 June 2011.
^ Hubert, M.; Vandervieren, E. (2008). "An adjusted boxplot for skewed distribution". Computational Statistics and Data Analysis. 52 (12): 5186–5201. CiteSeerX 10.1.1.90.9812. doi:10.1016/j.csda.2007.11.008.
Tukey, John W. (1977). Exploratory Data Analysis. Addison-Wesley. ISBN 9780201076165.
Wikimedia Commons has media related to Box plots.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Box_plot&oldid=1087822172"
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Manin obstruction to strong approximation for homogeneous spaces | EMS Press
Manin obstruction to strong approximation for homogeneous spaces
For a homogeneous space
X
(not necessarily principal) of a connected algebraic group
G
(not necessarily linear) over a number field
k
, we prove a theorem of strong approximation for the adelic points of
X
in the Brauer–Manin set. Namely, for an adelic point
x
X
orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group
\operatorname{Br}(X)
X
with respect to the Manin pairing, we prove a strong approximation property for
x
away from a finite set
S
k
. Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Thélène and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results.
Mikhail Borovoi, Cyril Demarche, Manin obstruction to strong approximation for homogeneous spaces. Comment. Math. Helv. 88 (2013), no. 1, pp. 1–54
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