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Concept of Direct and Reverse Crank for V-engines & Radial engines | Education Lessons
The method of direct and reverse cranks is applicable on reciprocating engines i.e V engines, radial engines, etc in which the piston performs reciprocating motion.
As shown in fig.1 (radial engine) and fig.2 (V-engine), the connecting rods are connected to a common crank and this crank revolve in one plane only. Hence there is no primary or secondary couple. Only primary and secondary force are required to be balanced.
The method of direct and reverse crank is used in reciprocating engines for two purposes :
(1) For determining the primary and secondary unbalanced forces.
(2) For balancing the primary and secondary unbalanced forces.
Fig.3 shows a reciprocating engine mechanism in which crank OC rotates at an angular speed of
\omega
rad/s in a clockwise direction. Let
\theta
be the angle made by crank OC with the line of stroke of cylinder.
1. Primary force:-
The unbalance primary force
‘ F_P ’
F_P = mr {\omega^2} .cos\theta
where, m = mass of reciprocating parts, kg
This unbalanced primary force is equal to the horizontal component of the centrifugal force produced by the imaginary mass ‘m’ placed at crank pin ‘C’, as shown in fig.4.
The arrangement shown in fig.4 can be replaced by another arrangement, shown in fig.5, in which OC is called as the actual crank or primary direct crank and OCˊ is called the indirect crank or primary reverse crank.
The unbalanced primary force due to ‘m’ mass can be determined by replacing
{'m'\over2}
mass each at the mirror image and this method of replacing ‘m’ mass with
{'m'\over2}
mass is called direct and reverse crank method.
The primary direct crank OC makes an angle
\theta
with line of stroke and is rotating uniformly at ‘
\omega
’ rad/s in clockwise direction whereas the primary reverse crank OCˊ makes an angle -
\theta
\omega
’ rad/s in anticlockwise direction as shown in fig.5. Thus the primary reverse crank is mirror image of the primary direct crank.
The parameters of primary direct and reverse cranks:
I. Primary direct crank:
Angular position =
\theta
\omega
(clockwise)
Radius of crank = r
II. Primary reverse crank:
Angular position = -
\theta
\omega
(anticlockwise)
Let mass ‘m’ of reciprocating parts is divided into two equal parts (i.e.
m\over2
One of the parts is placed at direct crank pin ‘C’ and the other part is placed at reversed crank pin ‘C’ as shown in fig.5.
Centrifugal force acting on each mass placed at direct crank pin C and reverse crank pin
Cˊ = {m\over2}r {\omega^2}
Component of centrifugal force acting on the mass placed at point C, along the line of stroke
= {m\over2} r{\omega^2} cos\theta
Component of centrifugal force acting on the mass placed at point Cˊ, along the line of stroke
= {m\over2} r{\omega^2} cos\theta
Total component of the centrifugal force acting along the line of stroke
\begin{aligned} &= {m\over2} r{\omega^2} cos\theta + {m\over2} r{\omega^2} cos\theta \\ &= {m} r{\omega^2} cos\theta \end{aligned}
This total component of the centrifugal force acting along the line of stroke, is equal to primary unbalanced force,
F_P = mr{\omega^2} cos\theta
Hence, for determining the unbalanced primary force, the mass ‘m’ of the reciprocating parts can be replaced by two masses i.e.
m\over2
each at point C and Cˊ respectively.
The vertical components of centrifugal force of masses (
{m\over2}
) placed at point C and Cˊ are equal to
{m\over2}r\omega^2sin\theta
, so these components are equal in magnitude and opposite in direction to each other. Hence, these components are balanced.
2. Secondary force:
The unbalanced Secondary force
‘ F_S ’
\begin{aligned} F_S &= mr {\omega^2} . {cos2\theta\over n} \\ Or \ \ F_S &= m × {(2\omega)^2} ×{r\over4n} cos 2\theta \end{aligned}
We can find this secondary unbalanced force by using direct and reverse cranks method.
For determining unbalanced secondary force, the mass ‘m’ of the reciprocating parts is replaced by two masses equal to
m\over2
at crank pins of secondary direct crank and secondary reverse crank (i.e. at point C and Cˊ) such that secondary direct crank is making an angle 2
\theta
and secondary reverse crank is making an angle -2
\theta
with line of stroke as shown in fig.6.
The parameters of secondary direct and reverse cranks:
I. Secondary direct crank:
Angular position = 2
\theta
Angular velocity = 2
\omega
rad/s (clockwise)
Radius of crank =
{r\over4n}
II. Secondary reverse crank:
Angular position = -2
\theta
\omega
rad/s (anticlockwise)
r\over{4n}
Also read these important notes:
Numerical - Multi Cylinder Inline Engine (with firing order)
|
Sensors | Free Full-Text | Detection and Analysis of Corrosion and Contact Resistance Faults of TiN and CrN Coatings on 410 Stainless Steel as Bipolar Plates in PEM Fuel Cells
Intelligent Defect Diagnosis of Rolling Element Bearings under Variable Operating Conditions Using Convolutional Neural Network and Order Maps
Fatigue Crack Evaluation with the Guided Wave–Convolutional Neural Network Ensemble and Differential Wavelet Spectrogram
Forouzanmehr, M.
Reza Kashyzadeh, K.
Borjali, A.
Jafarnode, M.
Chizari, M.
Mohsen Forouzanmehr
Amirhossein Borjali
Mosayeb Jafarnode
Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman 7618868366, Iran
Department of Transport, Academy of Engineering, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, 117198 Moscow, Russia
Department of Mechanical Engineering, Sharif University of Technology, Tehran 11155-1639, Iran
Department of Mechanics, Todor Kableshkov University of Transport, 158 Geo Milev Street, 1574 Sofia, Bulgaria
School of Mechanical Engineering, Islamic Azad University, South Tehran Branch, Tehran 1584743311, Iran
TWI Ltd., Granta Park, Great Abington, Cambridge CB21 6AL, UK
College of Engineering and Physical Sciences, Brunel University London, Uxbridge UB8 3PH, UK
School of Physics, Engineering and Computer Sciences, University of Hertfordshire, Hatfield AL10 9AB, UK
Academic Editors: Steven Chatterton, Jose A Antonino-Daviu, Mohammad N Noori and Francesc Pozo
(This article belongs to the Special Issue Feature Papers in Fault Diagnosis & Sensors Section 2022)
Bipolar Plates (BPPs) are the most crucial component of the Polymer Electrolyte Membrane (PEM) fuel cell system. To improve fuel cell stack performance and lifetime, corrosion resistance and Interfacial Contact Resistance (ICR) enhancement are two essential factors for metallic BPPs. One of the most effective methods to achieve this purpose is adding a thin solid film of conductive coating on the surfaces of these plates. In the present study, 410 Stainless Steel (SS) was selected as a metallic bipolar plate. The coating process was performed using titanium nitride and chromium nitride by the Cathodic Arc Evaporation (CAE) method. The main focus of this study was to select the best coating among CrN and TiN on the proposed alloy as a substrate of PEM fuel cells through the comparison technique with simultaneous consideration of corrosion resistance and ICR value. After verifying the TiN and CrN coating compound, the electrochemical assessment was conducted by the potentiodynamic polarization (PDP) and electrochemical impedance spectroscopy (EIS) tests. The results of PDP show that all coated samples have an increase in the polarization resistance (
{R}_{p}
) values (ranging from 410.2 to 690.6
\mathsf{\Omega }·{\mathrm{cm}}^{2}
) compared to substrate 410 SS (230.1
\mathsf{\Omega }·{\mathrm{cm}}^{2}
). Corrosion rate values for bare 410 SS, CrN, and TiN coatings were measured as 0.096, 0.032, and 0.060
\mathrm{mpy}
, respectively. Facilities for X-ray Diffraction (XRD), Scanning Electron Microscope (FE-SEM, TeScan-Mira III model and made in the Czech Republic), and Energy Dispersive X-ray Spectroscopy (EDXS) were utilized to perform phase, corrosion behavior, and microstructure analysis. Furthermore, ICR tests were performed on both coated and uncoated specimens. However, the ICR of the coated samples increased slightly compared to uncoated samples. Finally, according to corrosion performance results and ICR values, it can be concluded that the CrN layer is a suitable choice for deposition on 410 SS with the aim of being used in a BPP fuel cell system. View Full-Text
Keywords: fuel cells; coating; corrosion; Interfacial Contact Resistance fuel cells; coating; corrosion; Interfacial Contact Resistance
Forouzanmehr, M.; Reza Kashyzadeh, K.; Borjali, A.; Ivanov, A.; Jafarnode, M.; Gan, T.-H.; Wang, B.; Chizari, M. Detection and Analysis of Corrosion and Contact Resistance Faults of TiN and CrN Coatings on 410 Stainless Steel as Bipolar Plates in PEM Fuel Cells. Sensors 2022, 22, 750. https://doi.org/10.3390/s22030750
Forouzanmehr M, Reza Kashyzadeh K, Borjali A, Ivanov A, Jafarnode M, Gan T-H, Wang B, Chizari M. Detection and Analysis of Corrosion and Contact Resistance Faults of TiN and CrN Coatings on 410 Stainless Steel as Bipolar Plates in PEM Fuel Cells. Sensors. 2022; 22(3):750. https://doi.org/10.3390/s22030750
Forouzanmehr, Mohsen, Kazem Reza Kashyzadeh, Amirhossein Borjali, Anastas Ivanov, Mosayeb Jafarnode, Tat-Hean Gan, Bin Wang, and Mahmoud Chizari. 2022. "Detection and Analysis of Corrosion and Contact Resistance Faults of TiN and CrN Coatings on 410 Stainless Steel as Bipolar Plates in PEM Fuel Cells" Sensors 22, no. 3: 750. https://doi.org/10.3390/s22030750
Dr. K. Reza Kashyzadeh is an Associate Professor in the Department of Transport at Peoples' Friendship University of Russia (RUDN). He is also the Scientific and Technical Director of the Mechanical Characteristics Laboratory at Sharif University of Technology, IRAN (2018-present). His research interest relies on applied mechanics, solid mechanics, structural integrity, fatigue and fracture, and automotive engineering. During his academic carrier, he has published more than 110 papers and 8 books. Moreover, Dr. K. Reza Kashyzadeh is serving as Subject Editor or Guest Editor for International Journals and participates as a Scientific Committee or Keynote Speaker at a large number of international conferences. He also won the Cardineli Gold Medal in the field of Research & Development from Italy in 2015. In 2020, he was recognized as a Top Researcher at the Academy of Engineering in RUDN University.
Prof. Tat-Hean Gan has more than 10 years experience in non-destructive testing (NDT), structural health monitoring (SHM), and condition monitoring of rotating machinery in various industries, namely nuclear, renewable energy (e.g., wind, wave, and tidal), oil and gas, petrochemical, construction and infrastructure, aerospace, and automotive. He is the Director of BIC, leading activities varying from research and development to commercialization in the areas of novel technique development, sensor applications, signal and image processing, numerical modeling, and electronics hardware. His experience is also in collaborative funding (EC FP7 and UK TSB), project management, and technology commercialization.
Bin Wang graduated with a BEng (1985) in Solid Mechanics from Xi’an Jiaotong University, an MSc (1988) by research in Dynamics, and a Ph.D. (1991) in Applied Mechanics, both from the University of Manchester (formerly UMIST). He was an academic staff member of Nanyang Technological University (Singapore), Deakin (Australia), Brunel, Manchester, and Aberdeen University before returning to Brunel in July 2011. At Brunel, he has held roles as the Chairperson of the Board of Study in Mechanical, Aerospace and Automotive Engineering, Year 1 Tutor, Programme Director of MSc Structural Integrity, and now the Vice Dean International of the College. Dr Wang has received the following Scholarships and Awards: International Exchange Fellowship, Australia Academy of Science (1998); Bed Morris Fellowship, French Science Foundation (1999); Senior Fellow, Nanyang Technological University (2001); Impact Award, PraxisUnico (2011). He has served on international conference committees and held the position of chair, undertaken guest editorships, and provided consultancy to industries. He is currently the Editor of the International Journal of Mechanical Engineering Education, a member of the editorial committee of several research journals, and the author/co-author of more than 100 peer-reviewed archival journal publications in his research subject areas.
|
Insertion Sort Practice Problems Online | Brilliant
If there has been one recurring theme so far, it’s that more organized data can be valuable for certain types of computations, like searching for an element in a list. Specifically, we’ve noted that having a sorted list is generally more useful than having an unsorted list.
So, how can we sort a list? As with searching, we’ll start with a naive, intuitive approach (called insertion sort). In a later quiz, we’ll explore more efficient methods of sorting.
A
be an array sorted in increasing order (from smallest to largest), and let
x
be a single element outside of it. The natural question is, how can we place
x
into our array while still keeping it sorted?
This operation is known as an insertion, and there are multiple ways to do it. However, in this chapter we will only concern ourselves with the method used in insertion sort, which uses the steps outlined below.
x
A
Consider the element to the left of
x
. There are three possibilities.
If there is no element to the left, then we are done, since
x
will be the smallest element and it is already at the start of the array.
x
is greater than or equal to it, then we are done;
x
is on the right of all smaller elements and on the left of all larger elements, so the array is once again sorted.
x
is smaller, then it should come before the other element. Switch the two to put
x
in front.
Repeat step two until finished.
With each repetition of step two, we will either finish inserting, or reduce the number of elements ahead of
x
by one. Since we are also finished when no more elements remain in front of
x
, repeating step two is guaranteed to successfully make an insertion.
Suppose we have an array, [1, 3, 6, 7, 10], and a new element to add,
x = 4
. If we use the insertion procedure from our last slide to add
x
to our array, how many comparisons will it take and what will be the final index of
x
in the array?
Remember that arrays are indexed starting from zero, so the first element would correspond with index
0
, the second with index
1
Hint: Make sure you're following the algorithm we went over earlier. Remember that we make comparisons going from right to left.
Index 2, and 4 comparisons Index 3, and 2 comparisons Index 2, and 3 comparisons Index 3, and 4 comparisons
The full insertion sort algorithm works by dividing an array into two pieces, a sorted region on the left and an unsorted region on the right. Then, by repeatedly inserting elements from the unsorted half into the sorted half, the algorithm eventually produces a fully sorted array. The full steps of this process for an array,
A
, are shown below
Designate the leftmost element of
A
as the only element of the sorted side. This side is guaranteed to be sorted by default, since it now contains only one element.
Insert the first element of the unsorted side into the correct place in the sorted side, increasing the number of sorted elements by one.
Repeat step two until there are no unsorted elements left.
Notice that this method doesn’t require us to create a new array to store the sorted values. All we have to do is keep track of how much of the original array is sorted. This makes insertion sort an in-place algorithm.
To summarize, Insertion sort is a sorting algorithm that builds a final sorted array one element at a time. The following is a visual representation of how insertion sort works:
We start from the left. At first there is one element (which is, trivially, already sorted).
Colloquially speaking, for each subsequent element, we compare it to each of the elements to the left of it, moving right to left, until we find where it “fits” in the sorted portion of the array.
Since we are inserting the new element into the side of the array that is already sorted, the result will still be sorted.
After we’ve inserted every element, we have a sorted array!
Suppose you had the following poker hand dealt to you, and you decide to use insertion sort to sort it from smallest to largest:
\mathbf{A>K>Q>J>2.}
How many individual comparisons between cards would need to be made for the sort if we strictly stick to the instructions for insertion sort shown earlier?
Reminder: First, the
K
2
are compared. Then, the
A
K
are compared, then the
Q
A
Ari wants to prove that insertion sort is slow. What order should he put the cards in to make sorting the cards in ascending order maximally inefficient?
Descending order Random order Ascending order
In the worst case for a list with 10 distinct elements, how many comparisons between list elements are made?
In the best case for a list with 10 distinct elements, how many comparisons are made?
The worst case increases like
n^2
n,
while the best case increases linearly (like
n
). Unfortunately for insertion sort—as we’ll explore in the next quiz—the worst case is much more likely, and the average case also increases like
n^2.
Thankfully, we can do better. In later quizzes, we’ll dive into more complex, more efficient sorting algorithms; their worst case performance will increase like
n \log n
n.
Before we get there, we need to take a detour to start formalizing this language of computational complexity. How do we actually express how many computations an algorithm will use, and how do we meaningfully compare algorithms? We’ll explore this in the next quiz, on big O notation.
|
Transportation Model - Introduction | Education Lessons
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All Notes / OR
Transportation Model - Introduction
What is the Transportation Model?
Click here to know about Methods to solve Transportation Model Problem provided below here in notes ↓
Transportation Model is a special case of LPP(Linear Programming Problem) in which the main objective is to transport a product from various sources to various destinations at total minimum cost.
In Transportation Models, the sources and destinations are known, the supply and demand at each source and destinations are also known.
It is designed to find the best arrangement for transportation such that the transportation cost is minimum.
Consider three companies (Company1, Company2 and Company3) which produce mobile phones and are located in different regions.
Similarly, consider three cities (namely CityA, CityB & CityC) where the mobile phones are transported.
The companies where mobile phones are available are known as sources and the cities where mobile phones are transported are called destinations.
Play quiz - MCQ
Company1 produces a1 units,
Company3 produces a3 units.
demand in CityA is b1 units,
demand in CityB is b2 units,
demand in CityC is b3 units.
The cost of transportation from each source to destination is given in table
The transportation of mobile phones should be done in such a way that the total transportation cost is minimum.
Q-1) Amazon delivery distributors can use Transportation Model ?
There are two types of transportation problems:
i) Balanced transportation problem: The sum of supply and sum of demand are same.
\Sigma \text { Supply} = \Sigma \text { Demand}
ii) Unbalanced transportation problem: The sum of supply and sum of demand are different.
\Sigma \text { Supply} \ne \Sigma \text { Demand}
Methods to solve Transportation Model
Industrial applications of Transportation Model
Minimize the transportation cost from source to destination.
Determine lowest cost location for new industries, offices, warehouse, etc.
Determine the number of products to be manufactured according to demand.
Courier Services: Helps in taking proper decisions to find the best route for transportation.
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Suggested Notes:
Assignment Model | Linear Programming Problem (LPP) | Introduction
Basics of Program Evaluation and Review Technique (PERT)
Construct a project network with predecessor relationship | Operation Research | Numerical
Numerical on PERT (Program Evaluation and Review Technique)
Tie in selecting row and column (Vogel's Approximation Method - VAM) | Numerical | Solving Transportation Problem | Transportation Model
Crashing Special Case - Indirect cost less than Crash Cost
Graphical Method | Methods to solve LPP | Linear Programming
Least Cost Method | Method to Solve Transportation Problem | Transportation Model
Linear Programming Problem (LPP) Formulation with Numericals
Modified Distribution Method (MODI) | Transportation Problem | Transportation Model
North West Corner Method | Method to Solve Transportation Problem | Transportation Model
Stepping Stone | Transportation Problem | Transportation Model
Crashing Special Case - Multiple (Parallel) Critical Paths
Network Analysis - Dealing with Network Construction Basics
Vogel’s Approximation Method (VAM) | Method to Solve Transportation Problem | Transportation Model
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10th - Mathematics
Advanced Engineering Mathematics [Maths 3]
Fluid Mechanics [FM]
[Notes] Operation Research
[Notes] Dynamics of Machinery
Law of Parallelogram of Forces : 5 in 5 MCQs S01-E01
Damped free Vibration - Numerical 2
Concept of Direct and Reverse Crank for V-engines & Radial engines
Variation of Tractive Force | Tractive Effort | Effect of Partial Balancing of Locomotives
Multi Cylinder Inline Engine (with firing order) | Numerical
Over-Damped System (ξ>1)
Logarithmic Decrement (δ)
Under-Damped System (ξ < 1)
Critically Damped System (ξ = 1)
[Notes] Science
Why Gravitational Acceleration g is 9.8 m/s² ? [Calculation]
[Notes] Maths
tan (A + B) = √3 and tan (A -B) = 1/√3 | Trigonometry Numerical
Find the value of (sin 30° + tan 45° - cosec 60°) / (sec 30° + cos 60° + cot 45°) and (5 cos² 60° + 4 sec² 30° - tan² 45°) / (sin² 30° + cos² 30°) | Trigonometry Numerical
In △PQR, PR + QR = 25cm and PQ = 5cm | Find the value of sin P, cos P, tan P | Trigonometry Numerical
3 cot A = 4 | Check if (1−tan²A)/(1+tan²A) = cos²A - sin²A | Trigonometry Numerical
15 cot A = 8 | Find the value of sin A and sec A | Trigonometry Numerical
PQ = 12cm and PR = 13cm | Find tan P - cot R | Trigonometry Numerical
cot θ = 7/8 | Find all other trigonometric ratios | Trigonometry Numerical
Trigonometry Formulas | sin θ, cos θ, tan θ, cosec θ, sec θ, cot θ | Trigonometry Basics
[Notes] Computer Aided Design
Understanding the concept | Finite Element Method | Advantages of FEA | Disadvantages of FEA | Applications of FEA | Finite Element Analysis
© Copyright 2022 Education Lessons.
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find lim x tends to 0 sin 2x + 3x / 4x - sin 5x - Maths - Limits and Derivatives - 9907590 | Meritnation.com
find lim x tends to 0
sin 2x + 3x / 4x - sin 5x
Nishima Pasricha answered this
\underset{\mathrm{x}\to 0}{\mathrm{lim}}\frac{\mathrm{sin}2\mathrm{x}+3\mathrm{x}}{4\mathrm{x}-\mathrm{sin}5\mathrm{x}}\phantom{\rule{0ex}{0ex}}=\underset{\mathrm{x}\to 0}{\mathrm{lim}}\frac{\frac{\mathrm{sin}2\mathrm{x}}{2\mathrm{x}}+\frac{3\mathrm{x}}{2\mathrm{x}}}{\frac{4\mathrm{x}}{5\mathrm{x}}-\frac{\mathrm{sin}5\mathrm{x}}{5\mathrm{x}}}×\left(\frac{2\mathrm{x}}{5\mathrm{x}}\right)\phantom{\rule{0ex}{0ex}}=\frac{2}{5}\frac{\underset{\mathrm{x}\to 0}{\mathrm{lim}}\left(\frac{\mathrm{sin}2\mathrm{x}}{2\mathrm{x}}+\frac{3}{2}\right)}{\underset{\mathrm{x}\to 0}{\mathrm{lim}}\left(\frac{4}{5}-\frac{\mathrm{sin}5\mathrm{x}}{5\mathrm{x}}\right)}\phantom{\rule{0ex}{0ex}}=\frac{2}{5}×\frac{\left(1+\frac{3}{2}\right)}{\left(\frac{4}{5}-1\right)}\phantom{\rule{0ex}{0ex}}=\frac{2}{5}×\frac{-1}{2}\phantom{\rule{0ex}{0ex}}=\frac{-1}{5}
|
Concentration - Berkeley Notes
In real life, for the most part, we can’t compute probabilities in closed form. Instead, we either bound them, or we want to show that
P(A) \approx 0
P(A) \approx 1
Theorem 17 (Markov's Inequality)
For a non-negative random variable
X
\text{Pr}\left\{X \geq t\right\} \leq \frac{\mathbb{E}\left[X\right] }{t}, \quad t \geq 0.
Theorem 18 (Chebyshev's Inequality)
X
is a random variable, then
\text{Pr}\left\{|X - \mathbb{E}\left[X\right] | \geq t\right\} \leq \frac{\text{Var}\left(X\right) }{t^2}.
Intuitively, Theorem 18 gives gives a “better” bound than Theorem 17 because it incorporates the variance of the random variable. Using this idea, we can define an even better bound that incorporates information from all moments of the random variable.
Definition 36 (Chernoff Bound)
X
a\in\mathbb{R}
\text{Pr}\left\{X \geq a\right\} \leq \frac{\mathbb{E}\left[e^{tX}\right] }{e^{ta}} = e^{-ta}M_x(t).
After computing the Chernoff bound for a general
t
, we can then optimize over it to compute the best bound possible.
The idea of convergence brings the mathematical language of limits into probability. The fundamental question we want to answer is given random variables
X_1, X_2, \cdots
, what does it mean to compute
\lim_{n\to\infty}X_n.
This question is not as straightforward as it seems because random variables are functions, and there are many ways to define the convergence of functions.
A sequence of random variables converges almost surely to
X
P\left(\lim_{n\to \infty}X_n = X\right) = 1
One result of almost sure convergence deals with deviations around the mean of many samples.
Theorem 19 (Strong Law of Large Numbers)
X_1, X_2, \cdots, X_n
are independently and identically distributed to
X
\mathbb{E}\left[X\right] < \infty
\frac{1}{n}\sum_i X_i
converges almost surely to
\mathbb{E}\left[X\right]
The strong law tells us that for any observed realization, there is a point after which there are no deviations from the mean.
A sequence of random variables converges in probability if
\forall \epsilon > 0, \quad \lim_{n\to\infty}P(|X_n - X| > \epsilon) = 0
Convergence in probability can help us formalize the intuition that we have which says probability is the frequency with which an even happens over many trials of an event.
Theorem 20 (Weak Law of Large Numbers)
X_1, X_2, \cdots, X_n
be independently and identically distributed according to
X
M_n = \frac{1}{n}\sum X_i
\epsilon > 0
\lim_{n\to\infty} \text{Pr}\left\{|M_n - \mathbb{E}\left[X\right] | > \epsilon\right\} = 0.
It tells us that the probability of a deviation of
\epsilon
from the true mean will go to 0 in the limit, but we can still observe these deviations. Nevertheless, the weak law helps us formalize our intuition about probability. If
X_1, X_2, \cdots, X_n
are independently and identically distributed according to
X
, then we can define the empirical frequency
F_n = \frac{\sum\mathbb{1}_{X_i\in B}}{n} \implies \mathbb{E}\left[F_n\right] = P(X \in B).
\lim_{n\to\infty}\text{Pr}\left\{|F_n - P(X\in B)| > \epsilon\right\} = 0,
meaning over many trials, the empirical frequency is equal to the probility of the event, matching intuition.
A sequence of random variables converges in distribution if
\lim_{n\to\infty}F_{X_n}(x) = F_x(x).
An example of convergence in distribution is the central limit theorem.
Theorem 21 (Central Limit Theorem)
X_1, X_2, \cdots
X
\text{Var}\left(X\right) = \sigma^2
\mathbb{E}\left[X\right] = \mu
\lim_{n\to\infty}P\left(\frac{\sum_{i=1}^nX_i - n\mu}{\sigma\sqrt{n}} \leq x\right) = \Phi(x)
In other words, a sequence of random variables converges in distribution to a normal distribution with variance
\sigma^2
\mu
These notions of convergence are not identical, and they do not necessarily imply each other. It is true that almost sure convergence implies convergence in probability, and convergence in probability implies convergence in distribution, but the implication is only one way.
Once we know how a random variable converges, we can then also find how functions of that random variable converge.
Theorem 22 (Continuous Mapping Theorem)
If
is a continuous function, then if
X_n
X
f(X_n)
f(X)
. The convergence can be almost surely, in probability, or in distribution.
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Factor each of the algebraic expressions completely. 3x^{4}+81x
Factor each of the algebraic expressions completely.
3{x}^{4}+81x
To do factor of the algebraic expressions:
3{x}^{4}+81x
3{x}^{4}+81x
=x\left(3{x}^{3}+81\right)
=x.3.\left({x}^{3}+27\right)
=3x\left({x}^{3}+27\right)
=3x\left({x}^{3}+33\right)
=3x\left(x+3\right)\left({x}^{2}-x.3+32\right)
=3x\left(x+3\right)\left({x}^{2}-3x+9\right)
3{x}^{4}+81x=3x\left(x+3\right)\left({x}^{2}-3x+9\right)
3{x}^{4}+81x
3\left({x}^{4}+27x\right)
3x\left({x}^{3}+27\right)
3x\left(x+3\right)\left({x}^{2}-3x+9\right)
3{x}^{4}+81x
3\left({x}^{4}+27x\right)
3x\left({x}^{3}+27\right)
3x\left(x+3\right)\left({x}^{2}-3x+9\right)
RJ’s Plumbing and Heating charges $55 plus $40 per hour for emergency service. Gary remembers being billed over $100 for an emergency call. How long was RJ’s there?
What is the 4th partial sum for the geometric sequence where
{a}_{1}=18\text{ }\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\text{ }r=2
{S}_{4}=-18
{S}_{4}=288
{S}_{4}=152
{S}_{4}=270
\frac{5{x}^{9}}{{x}^{3}}
Using algebra, find the solution to the system of inequalities below
y\le 5-3x
y+19\ge {x}^{2}-8x
Find the value of the given expressions
{\left(2+3\right)}^{2};
{2}^{2}+{3}^{2}
{\left(8+10\right)}^{2};
{8}^{2}+{10}^{2}
c) Check whether the given statement is true or false:
Thegiven\mathrm{exp}ressionsarePSK{\left(a+b\right)}^{2}={a}^{2}+{b}^{2}
Simplify, please.
8\left\{\left[4\left(p–4\right)+19\right]–\left[2\left(3p–2\right)+5\right]\right\}
8\left\{\left[4\left(p–4\right)+19\right]–\left[2\left(3p–2\right)+5\right]\right\}=-16p+16
|
A Computational and Experimental Investigation of the Human Thermal Plume | J. Fluids Eng. | ASME Digital Collection
Gas Dynamics Laboratory, Department of Mechanical and Nuclear Engineering,
e-mail: bac207@psu.edu
J. Fluids Eng. Nov 2006, 128(6): 1251-1258 (8 pages)
Craven, B. A., and Settles, G. S. (March 19, 2006). "A Computational and Experimental Investigation of the Human Thermal Plume." ASME. J. Fluids Eng. November 2006; 128(6): 1251–1258. https://doi.org/10.1115/1.2353274
The behavior of the buoyant plume of air shed by a human being in an indoor environment is important to room ventilation requirements, airborne disease spread, air pollution control, indoor air quality, and the thermal comfort of building occupants. It also becomes a critical factor in special environments like surgery rooms and clean-rooms. Of the previous human thermal plume studies, few have used actual human volunteers, made quantitative plume velocity measurements, or considered thermal stratification of the environment. Here, a study of the human thermal plume in a standard room environment, including moderate thermal stratification, is presented. We characterize the velocity field around a human volunteer in a temperature-stratified room using particle image velocimetry (PIV). These results are then compared to those obtained from a steady three-dimensional computational fluid dynamics (CFD) solution of the Reynolds-averaged Navier-Stokes equations (RANS) using the RNG
k‐ε
two-equation turbulence model. Although the CFD simulation employs a highly simplified model of the human form, it nonetheless compares quite well with the PIV data in terms of the plume centerline velocity distribution, velocity profiles, and flow rates. The effect of thermal room stratification on the human plume is examined by comparing the stratified results with those of an additional CFD plume simulation in a uniform-temperature room. The resulting centerline velocity distribution and plume flow rates are presented. The reduction in plume buoyancy produced by room temperature stratification has a significant effect on plume behavior.
biothermics, stratified flow, flow visualisation, computational fluid dynamics, Navier-Stokes equations
Computational fluid dynamics, Plumes (Fluid dynamics), Flow (Dynamics), Simulation, Temperature, Turbulence, Thermal stratification
Chap. 8, pp.
Combined Simulation of Airflow, Radiation and Moisture Transport for Heat Release From a Human Body
Gowadia
The Natural Sampling of Airborne Trace Signals From Explosives Concealed Upon the Human Body
Aerodynamics of the Human Microenvironment
Examination of Free Convection Around Occupant’s Body Caused by Its Metabolic Heat
The generation of aerosols from the human body
Airborne Transmission and Airborne Infection: Concepts and Methods
Sniffers: Fluid-Dynamic Sampling for Olfactory Trace Detection in Nature and Homeland Security - the 2004 Freeman Scholar Lecture
, June 13, 2000, “
Chemical Trace Detection Portal Based on the Natural Airflow and Heat Transfer of the Human Body
,” U. S. Patent US Patent 6,073,499.
CFD Analysis of Wind Environment Around a Human Body
,” Transportation Security Administration Human Factors Personnel, personal communication.
Comparing Turbulence Models for Buoyant Plume and Displacement Ventilation Simulation
Natural Convection Heat Transfer From an Isothermal Vertical Surface to a Fluid Saturated Thermally Stratified Porous Medium
Unsteady Natural Convection Boundary-Layer Flow Along a Vertical Isothermal Plate in a Linearly Stratified Fluid With Pr>1
Natural Convection Heat Transfer From an Isothermal Vertical Surface to a Stable Thermally Stratified Fluid
FLUENT 6.1 User’s Guide.
IDT ProVISION User Manual: Particle Image Velocimetry
CFD Analysis of Thermally Stratified Flow and Conjugate Heat Transfer in a PWR Pressurizer Surgeline
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Concrete Based on Recycled Aggregates for Their Use in Construction: Case of Goma (DRC)
Concrete Based on Recycled Aggregates for Their Use in Construction: Case of Goma (DRC) ()
Masika Muhiwa Grâce1, Alinabiwe Nyamuhanga Ally1, Muhindo Wa Muhindo Abdias2, Kubuya Binwa Patient1, Muhatikani Trésor1, Manjia Marcelline Blanche3, Ngapgue Francois4
1Department of Civil Engineering, Faculty of Applied Sciences and Technologies, UniversitéLibre des Pays des GrandsLacs, Goma, Democratic Republic of Congo.
2Buildings and Publics Works Section, Institut du Batiment et des Travaux Publics, Butembo, Democratic Republic of Congo.
3Department of Civil Engineering, National Advanced School of Engineering, University of Yaoundé 1, Yaoundé, Cameroon.
4Department of Civil Engineering, Fotso Victor University Institute of Technology, University of Dschang, Bandjoun, Cameroon.
The following study is aimed at valorizing an important part of waste from building demolition, particularly concrete as a source of aggregates for their usage in new hydraulic concrete formulation. The experimental study mainly consisted of physical characterization of natural and recycled aggregates respectively and the impact of the latter on some properties of the new formulated concrete, actually their respective consistencies for fresh concrete and mechanical strength for the hardened one. The outcome of the study shows that the recycled aggregates are more heterogeneous and have a high capacity of water absorption, but which still respects the current standards of concrete. The need for additional water has been observed for recycled aggregates-based concrete so as to have the same workability. About the compressive strength, mechanical properties obviously show that, at 28 days from setting up, concretes from recycled aggregates can reach compressive strengths range between 20 and 25 MPa without any sophisticated technology. So, these results show that we can efficiently contribute to the protection of environment by valorizing waste from concrete-based building demolition on the one hand; and the preservation of natural reserve on the other. And both advantages contribute to sustainable development overall goals.
Demolition, Recycling, Water Absorption, Concrete, Mechanic Resistance
Grâce, M. , Nyamuhanga Ally, A. , Muhindo Abdias, M. , Patient, K. , Trésor, M. , Blanche, M. and Francois, N. (2020) Concrete Based on Recycled Aggregates for Their Use in Construction: Case of Goma (DRC). Open Journal of Civil Engineering, 10, 226-238. doi: 10.4236/ojce.2020.103019.
Waste from building demolition is found almost everywhere in streets, forests and in rivers. They are considered as a great source of nature pollution. Therefore, is can also be seen as a cause of environmental problems such as fauna and flora destruction, underground water pollution, saturation of public waste, unnecessary increase in additional sanitation water to be evacuated, nauseating garbage, insalubrity, flooding, etc. [1].
Recycling of such polluting waste by using it in new concrete formulation remains one of the possible solutions to that problem though Amor Ben Fraj et al., 2017 [2]; Wirquin et al., 2000 [3] as well as Courard L. et al. [4], assumed that usage of demolition materials as substitution aggregates is sometimes very difficult to realize since such materials are porous and so highly water absorbing.
The majority of concrete from buildings demolition-based recycled aggregates are used in road construction. Nevertheless, a better knowledge of its characteristics can contribute to development of that field of construction (Buyle-Bodin et al. 2002 [5] and Hussain et al. 2003 [6] ). This can only be achieved by a deep knowledge of fresh concrete.
However, recycled aggregates could be used as building materials in confection of concrete when the size of the building fits with their respective characteristics. Therefore, this work is aimed to study the strength of concrete formulated by recycled aggregates in order to propose its domain of usage. In so doing, we hope to find a way of protecting the environment against the building demolition waste. The global objective of this study is to take profit from aggregates from concrete building demolition so as to perform new concrete products and contribute to environment protection by reducing building demolition waste in nature.
In this section, we present all tests which are necessary to describe, both qualitatively and quantitatively, the aggregates as well as the concrete. The most important characteristics for concrete were respectively their consistency and compressive strength at fresh and hardened state. The tests were conducted in the laboratory of the Université Libre des Pays des Grands Lacs. Materials that were used in this study are: aggregates (sand and gravels), water, and cement.
2.1. Origin of Constituents
Natural sand 0/5, from Lake Kivu, of density 1.36 and specific unit weight 2.5. Recycled sand 0/5, from concrete blocks fragmentation and sieving in order to obtain sand in conformity to the norm NF EN 933-1 [7]. Natural gravels 5/25, from crushing of basaltic rock. Recycled gravels 5/25 from concrete blocks fragmentation.
Water used form concrete formulation was from “Régie de Distribution d’Eau” (REGIDESO), the water distribution company in Goma.
The cement used is the Compound Portland Cement CEM II 32.5R RHINO produced in Kenya. Its characteristics are presented in Table 1.
In this section, we present tests on aggregates on the one hand, which include particle size analysis, sand equivalent and bulk unit weight; and on concrete on the other, which include consistency and compressive strength at fresh and hardened states respectively.
2.2.1. Tests on Aggregates
1) Particle size analysis
Particle size composition is obtained by standardized particle size analysis NF EN 933-1 [7]. Particle size analysis is aimed to determine the grains weight distribution of aggregates according to their size by plotting a particle size curve.
Particle size analysis is carried out by dividing through a series of sievers the material into different granular classes of decreasing particle size. From that segregation, we obtain the mass of particles retained by the different sievers and passing through them. They are presented in terms of their respective ratios to the initial mass. The cumulated ratio of particles passing through a given siever is obtained by the expression (1).
{P}_{j}\left(%\right)=\left(1-\frac{\sum {M}_{i}^{r}}{{M}_{s}}\right)\cdot 100
Table 1. Characteristics of the compound Portland cement used.
\sum {M}_{i}^{r}
: the mass of the particles retained by the siever (i);
1\le i\le j
{M}_{s}
the total dry of dry material sample.
For sand, from particle size analysis, we obtain the finesse module. This parameter characterizes sand size. Finesse module is obtained by the summation of the percentages of the particles retained on the different sievers of respective opennings 0.16, 0.315, 0.63, 1.25, 2.5 and 5 mm. It is preferable that the finesse module of sand for a motar and/or concrete be comprised between 2.2 and 2.8.
- A column of sievers
- Sievers vibrator
- A weighing machine
2) Cleanliness of sand
The degree of sand cleanliness is obtained by the test of sand equivalent, which is a standardized test NF EN 933-8 [8]. The purpose of the sand equivalent test is to determine the degree of cleanliness of a sand. It defines the proportion of raw sand compared to impurities contained in the sand.
The sand equivalent is obtained by pouring a given mass of sand into two burettes containing a wash solution in order to evaluate in percentage the proportion of raw sand compared to impurities in the sand. The Sand Equivalent (SE) is calculated from the expression (2).
\text{SE}=\frac{{h}_{1}}{{h}_{2}}\ast 100
With h1 (cm) the height of visible sand deposit and h2 (cm) the total height (including the fines in suspension).
A stirrer
A washing tube
Burettes
A timer
3) Specific unit weight of aggregates
The specific unit weight is obtained by the normalized test NF P 18 - 555 [9]. The purpose of the test is to express the mass per volume unit of aggregates particles without taking into account the volume occupied by the voids.
The specific unit weight of a body is generally obtained by measuring the volume of a given liquide (generally water) that the aggregate moves when it is poured into the liquid. The results of the absolute density are calculated by the expression (3).
{\phi }_{Abs}=\frac{Ms}{{V}_{2}-{V}_{1}}
With Ms the mass of aggregates,
{V}_{1}
the volume of water in the burette without the aggregates,
{V}_{2}
the volume of water in the burette after the pouring of aggregates.
A burette
A weighing machine
2.2.2. Tests on Concrete
The consistency of concrete is obtained by the standardized test NF P 18 - 45 [10]. The purpose of the test is to obtain the workability of the concrete so as to make its setting up and implementation easier.
The consistency is obtained by measuring the subsidence of fresh concrete under its own weight in the Abrams’ cone.
Abrams’ cone
The compressive strength of concrete is obtained by the standardized test NF P 18 - 406 [11]. The test is aimed to determine the compressive strength of concrete.
The test consists of submitting a standardized concrete test tube to an axial compressive force up to its ruin.
Mechanical press.
3.1. Results of Tests on Aggregates
The particle size composition was obtained according to the norm NF EN 933 – 1 presented previously. The results of that test are presented in Tables 2-5 and in Figure 1 for natural sand, recylcled sand, natural gravels 5/25 and recycled gravels 5/25.
From Table 2 and Table 3, we find out that the fine module of natural sand is 2.5 while that of recycled sand is 3.3. Those results show that recycled sand is very coarse since its finesse module is superior to 2.8. The recycled aggregates-based concrete has lost its workability. In order to perform its workability, we are required to increase the proportion of sand fine particles so that the finesse module be comprised between 2.2 and 2.8.
Table 2. Particle size composition of natural sand.
Table 3. Particle size composition of recycled sand.
Table 4. Particle size composition of natural gravels 5/25.
Table 5. Particle size composition of recycled du gravels 5/25.
Figure 1. Particle size curve of aggregates.
3.1.2. Sand Cleanliness
The test of sand cleanliness showed that for natural sand, the visual sand equivalent (VSE) is around 70%, while the sand equivalent to piston is 67.6% (SEP). For recycled sand, the respective values are 76.9% and 78%. These results show that the recycled sand is clean and suitable for concrete.
3.1.3. Specific unit Weight of Aggregates
Table 6 presents the different values of specific unit weight of aggregates.
Specific unit weights of recycled aggregates are inferior to those of natural aggregates. This is due to the fact that recycled aggregates, which are porous, contain some quantity of cement still bounding on the aggregates on the one hand, and the nature of the original rock of the aggregates on the other.
3.2. Results of Tets on Concretes
3.2.1. Composition of Concretes
DreuxGoris method [12] was used. Concrete composition is presented in Table 7. Table 8 presents the equivalent composition.
3.2.2. Equivalent Composition
3.2.3. Consistency Test
The results of subsidence test to Abrams’ cone are presented in Table 9.
The different concretes are plastic since their maneuverability is comprised in the interval between 5 cm and 9 cm. The maneuverability of reference concrete is 7 cm while that of recycled aggregates-based concrete is 9 cm. There has been observed the necessity in additional water for recycled aggregates-based concrete in order to obtain the same workability as that of the reference concrete. This shows up the need for additional water in recycled aggregates. The results presented in the Table 9 show that the formulated concretes can be used to build formwork footings, retaining walls, slabs, pavement, beams and columns.
Table 6. Specific unit weight of aggregates.
Table 7. Concrete composition (per cubic meter).
Table 8. Concrete composition on cylindrical 16 * 32 cm test-tubes.
Table 9. Results of the subsidence test.
The results of the compressive strength test on 16 * 32 cm cylindrical test-tubes of the different concrete samples are presented in Table 10. The concrete samples were formulated by Dreux-Gorisse Method.
In order to contribute to construction waste management and environmental protection, it has been studied the properties of concrete formulated by recycled aggregates for its usage in building. The global aim was to analyze the influence of substitution of natural aggregates by recycled aggregates from building
Table 10. Compression test results on 16 * 32 cm cylindrical test-tubes.
concrete demolition. The compressive strength of recycled aggregates-based concrete was acceptable compared to the natural aggregates-based concrete. The difference is due to the quality of structural concrete used as aggregates source. These mechanical properties show that recycled aggregates can produce concrete of strengths range between 20 and 25 MPa without using any particular sophisticated technology. Usage of recycled aggregates in concrete formulation can offer a sure and helpful solution to demolition waste management. Concrete produced from recycled aggregates is likely to be used for structures of average span and in conditions of low aggressively. However, the study of recycled concrete micro-structure and their durability can complete and improve this study. Such a study can extend the usage range of recycled aggregates-based concrete in civil engineering.
[1] Charef, A. (2007) La problématique des granulats au Maroc. Push-Button Publishing.
[2] Fraj, A.B. and Idir, R. (2017) Concrete Based on Recycled Aggregates-Recycling and Environmental Analysis: A Case Study of Paris’ Region. Construction and Bulding Materials, 157, 952-964.
[3] Wirquin, E., Hadjieva-Zaharieva, R. and Buyle-Bodin, F. (2000) Utilisation de l'absorption d’eau des bétons comme critères de leur durabilité: Application aux bétons de granulats recyclés. Materials and Structures, 33, 403-408.
[4] Courard, L., Michel, F. and Delhez, P. (2010) Use of Concrete Road Recycled Aggregates for Roller Compacted Concrete. Construction Building Material, 24, 390-395.
[5] Buyle-Bodin, F. and Zaharieva, R.H. (2002) Influence of Industrially Produced Recycled Aggregates on Flow Properties of Concrete. Materials and Structures, 35, 504-509.
[6] Hussain, H. and Levacher, D. (2003) Recyclage de béton de démolition dans la fabrication des nouveaux bétons. Rencontres Universitaires de Génie Civil, La Rochelle, France.
[7] NF EN 933-1 (2012) Essais pour déterminer les caractéristiques géométriques des granulats, partie 1: Détermination de la granularité-analyse granulométrique par tamisage. AFNOR, Paris.
[8] NF EN 933-8 (1999) Essais pour déterminer les caractéristiques géometriques des granulats Partie 8; Evaluation des fines-Equivalent de sable. AFNOR, Paris.
[9] NF P 18-555 (1990) Granulats: Mesures des masses volumiques, de la porosité, du coefficient d’absorption et de la teneur en eau du sable. AFNOR, Paris.
[10] NF P 18-451 (1990) Béton frais: Essai d’affaissement au cône. AFNOR, Paris.
[11] NF P 18-406 (1981) Essai de compression des éprouvettes en béton durci. AFNOR, Paris.
[12] Dreux, G. and Festa, J. (1998) Nouveau guide du béton et ses constituants. Edition Eyrolles, 8eme édition, 409 p.
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Continuous-time or discrete-time two-degree-of-freedom PID controller - Simulink - MathWorks Deutschland
D\left[\frac{N}{1+N\alpha \left(z\right)}\right],
\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.
u=P\left(br-y\right)+I\frac{1}{s}\left(r-y\right)+D\frac{N}{1+N\frac{1}{s}}\left(cr-y\right),
u=P\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right),
u=P\left[\left(br-y\right)+I\frac{1}{s}\left(r-y\right)+D\frac{N}{1+N\frac{1}{s}}\left(cr-y\right)\right].
u=P\left[\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right)\right],
\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.
D\left[\frac{N}{1+N\alpha \left(z\right)}\right],
\alpha \left(z\right)=\frac{{T}_{s}}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}z}{z-1}.
\alpha \left(z\right)=\frac{{T}_{s}}{2}\frac{z+1}{z-1}.
{u}_{i}=\int \left(r-y\right)I\text{\hspace{0.17em}}dt.
u=P\left(br-y\right)+I\frac{1}{s}\left(r-y\right)+D\frac{N}{1+N\frac{1}{s}}\left(cr-y\right),
u=P\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right),
u=P\left[\left(br-y\right)+I\frac{1}{s}\left(r-y\right)+D\frac{N}{1+N\frac{1}{s}}\left(cr-y\right)\right].
u=P\left[\left(br-y\right)+I\alpha \left(z\right)\left(r-y\right)+D\frac{N}{1+N\beta \left(z\right)}\left(cr-y\right)\right],
D\frac{z-1}{z{T}_{s}}\left(cr-y\right).
{z}_{pole}=1-N{T}_{s}
{z}_{pole}=\frac{1}{1+N{T}_{s}}
{z}_{pole}=\frac{1-N{T}_{s}/2}{1+N{T}_{s}/2}
\begin{array}{l}{F}_{par}\left(s\right)=\frac{\left(bP+cDN\right){s}^{2}+\left(bPN+I\right)s+IN}{\left(P+DN\right){s}^{2}+\left(PN+I\right)s+IN},\\ {C}_{par}\left(s\right)=\frac{\left(P+DN\right){s}^{2}+\left(PN+I\right)s+IN}{s\left(s+N\right)},\end{array}
\begin{array}{l}{F}_{id}\left(s\right)=\frac{\left(b+cDN\right){s}^{2}+\left(bN+I\right)s+IN}{\left(1+DN\right){s}^{2}+\left(N+I\right)s+IN},\\ {C}_{id}\left(s\right)=P\frac{\left(1+DN\right){s}^{2}+\left(N+I\right)s+IN}{s\left(s+N\right)}.\end{array}
{Q}_{par}\left(s\right)=\frac{\left(\left(b-1\right)P+\left(c-1\right)DN\right)s+\left(b-1\right)PN}{s+N}.
{Q}_{id}\left(s\right)=P\frac{\left(\left(b-1\right)+\left(c-1\right)DN\right)s+\left(b-1\right)N}{s+N}.
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Verifiable random function - Wikipedia
Public-key cryptographic pseudorandom function
In cryptography, a verifiable random function (VRF) is a public-key pseudorandom function that provides proofs that its outputs were calculated correctly. The owner of the secret key can compute the function value as well as an associated proof for any input value. Everyone else, using the proof and the associated public key (or verification key[1]), can check that this value was indeed calculated correctly, yet this information cannot be used to find the secret key.[2]
A verifiable random function can be viewed as a public-key analogue of a keyed cryptographic hash[2] and as a cryptographic commitment to an exponentially large number of seemingly random bits.[3] The concept of a verifiable random function is closely related to that of a verifiable unpredictable function (VUF), whose outputs are hard to predict but do not necessarily seem random.[3][4]
The concept of a VRF was introduced by Micali, Rabin, and Vadhan in 1999.[4][5] Since then, verifiable random functions have found widespread use in cryptocurrencies, as well as in proposals for protocol design and cybersecurity.
2.1 In protocol design
2.2 In Internet security
2.3 In cryptocurrency
In 1999, Micali, Rabin, and Vadhan introduced the concept of a VRF and proposed the first such one.[4] The original construction was rather inefficient: it first produces a verifiable unpredictable function, then uses a hard-core bit to transform it into a VRF; moreover, the inputs have to be mapped to primes in a complicated manner: namely, by using a prime sequence generator that generates primes with overwhelming probability using a probabilistic primality test.[3][4] The verifiable unpredictable function thus proposed, which is provably secure if a variant of the RSA problem is hard, is defined as follows: The public key PK is
{\displaystyle (m,r,Q,coins)}
, where m is the product of two random primes, r is a number randomly selected from
{\displaystyle \mathbb {Z} _{r}^{*}}
, coins is a randomly selected set of bits, and Q a function selected randomly from all polynomials of degree
{\displaystyle 2k^{2}-1}
{\displaystyle GF(2^{k})}
. The secret key is
{\displaystyle (PK,\phi (m))}
. Given an input x and a secret key SK, the VUF uses the prime sequence generator to pick a corresponding prime
{\displaystyle p_{x}}
(the generator requires auxiliary inputs Q and coins), and then computes and outputs
{\displaystyle r^{1/p_{x}}{\pmod {m}}}
, which is easily done by knowledge of
{\displaystyle \phi (m)}
In 2005, an efficient and practical verifiable random function was proposed by Dodis and Yampolskiy.[3][6] When the input
{\displaystyle x}
is from a small domain (the authors then extend it to a larger domain), the function can be defined as follows:
{\displaystyle F_{SK}(x)=e(g,g)^{1/(x+SK)}\quad {\mbox{and}}\quad p_{SK}(x)=g^{1/(x+SK)},}
where e(·,·) is a bilinear map. To verify whether
{\displaystyle F_{SK}(x)}
was computed correctly or not, one can check if
{\displaystyle e(g^{x}PK,p_{SK}(x))=e(g,g)}
{\displaystyle e(g,p_{SK}(x))=F_{SK}(x)}
.[3][6] To extend this to a larger domain, the authors use a tree construction and a universal hash function.[3] This is secure if it is hard to break the "q-Diffie-Helman inversion assumption", which states that no algorithm given
{\displaystyle (g,g^{x},\dots ,g^{x^{q}})}
can compute
{\displaystyle g^{1/x}}
, and the "q-decisional bilinear Diffie-Helman inversion assumption", which states that it is impossible for an efficient algorithm given
{\displaystyle (g,g^{x},\ldots ,g^{(x^{q})},R)}
as input to distinguish
{\displaystyle R=e(g,g)^{1/x}}
from random, in the group
{\displaystyle \mathbb {G} }
In 2015, Hofheinz and Jager constructed a VRF which is provably secure given any member of the "(n − 1)-linear assumption family", which includes the decision linear assumption.[7] This is the first such VRF constructed that does not depend on a "Q-type complexity assumption".[7]
In 2019, Bitansky showed that VRFs exist if non-interactive witness-indistinguishable proofs (that is, weaker versions of non-interactive zero-knowledge proofs for NP problems that only hide the witness that the prover uses[1][8]), non-interactive cryptographic commitments, and single-key constrained pseudorandom functions (that is, pseudorandom functions that only allow the user to evaluate the function with a preset constrained subset of possible inputs[9]) also do.[1]
In 2020, Esgin et al. proposed a post-quantum secure VRF based on lattice-based cryptography.[10]
VRFs provide deterministic pre-commitments for low entropy inputs which must be resistant to brute-force pre-image attacks.[11][better source needed] VRFs can be used for defense against offline enumeration attacks (such as dictionary attacks) on data stored in hash-based data structures.[2]
In protocol design[edit]
VRFs have been used to make:
Resettable zero-knowledge proofs (i.e. one that remains zero-knowledge even if a malicious verifier is allowed to reset the honest prover and query it again[12]) with three rounds in the bare model[3][7]
Non-interactive lottery systems[3][7]
Verifiable transaction escrow schemes[3][7]
Updatable zero-knowledge databases[7]
E-cash[7]
VRFs can also be used to implement random oracles.[13]
In Internet security[edit]
DNSSEC is a system that prevents attackers from tampering with Domain Name System messages, but it also suffers from the vulnerability of zone enumeration. The proposed NSEC5 system, which uses VRFs[how?], provably prevents this type of attack.[14][importance?]
In cryptocurrency[edit]
Cardano and Polkadot implement VRFs for block production.[15][16]
The Internet Computer cryptocurrency uses a VRF to produce a decentralized random beacon whose output is unpredictable to anyone until they become available to everyone.[17] More precisely, its protocol allows clients to agree on a VRF (i.e. a commitment to a deterministic pseudorandom sequence) and to produce one new output thereof every round by using threshold signatures and distributed key generation.[17]
Algorand uses VRFs to perform cryptographic sortition.[18][19] In this platform, every block produces a new random selection seed, and a user secretly checks whether they were selected to participate in the consensus protocol by evaluating a VRF with their secret participation key and the selection seed.[18] (This also produces a proof, which the user can send to anyone to show that they have been selected to participate.[18]) Thusly, accounts are selected to propose blocks for a given round.[18] In this manner, VRFs prevent users from gaining staking advantages by registering multiple accounts.[5] The particular VRF that Algorand uses is based on elliptic-curve cryptography, specifically Curve25519;[10] proposed by Sharon Goldberg, Moni Naor, Dimitris Papadopoulos, Leonid Reyzin, and Jan Včelák, the function is currently undergoing standardization as an IETF Internet Draft.[19][20] Algorand's own implementation of this function has been open-source since October 2018.[19]
VRFs based on elliptic curve cryptography have been implemented in the programming language Solidity.[21][importance?]
In May 2020, Chainlink announced that it launched Chainlink VRF, a service that uses verifiable random functions to generate verifiable randomness on-chain.[22] To use Chainlink VRF, a smart contract supplies a seed (which should be unpredictable to the oracles to whom it is provided), and the seed in turn is used to generate a random number that is sent back to the contract; this is published on-chain along with a proof and verified using the oracle's public key and the application's seed.[22] Each oracle, when generating randomness, uses its own secret key.[22]
In July 2021, Harmony, a cryptocurrency project that bridges between blockchains[how?], and Oraichain, a cryptocurrency project involving artificial intelligence,[how?] announced that they had introduced VRFs[how?].[23][24]
^ a b c Bitansky, Nir (2020-04-01). "Verifiable Random Functions from Non-interactive Witness-Indistinguishable Proofs". Journal of Cryptology. 33 (2): 459–493. doi:10.1007/s00145-019-09331-1. ISSN 1432-1378.
^ a b c Goldberg, Sharon; Vcelak, Jan; Papadopoulos, Dimitrios; Reyzin, Leonid (5 March 2018). Verifiable Random Functions (VRFs) (PDF) (Technical report). Retrieved 15 August 2021.
^ a b c d e f g h i j Dodis, Yevgeniy; Yampolskiy, Aleksandr (16 November 2004). "A Verifiable Random Function With Short Proofs and Keys" (PDF). 8th International Workshop on Theory and Practice in Public Key Cryptography. International Workshop on Public Key Cryptography. Springer, Berlin, Heidelberg (published 2005). pp. 416–431. ISBN 978-3-540-30580-4. Retrieved 26 August 2021.
^ a b c d e Micali, Silvio; Rabin, Michael O.; Vadhan, Salil P. (1999). "Verifiable random functions" (PDF). Proceedings of the 40th IEEE Symposium on Foundations of Computer Science. 40th Annual Symposium on Foundations of Computer Science. pp. 120–130. doi:10.1109/SFFCS.1999.814584. ISBN 0-7695-0409-4.
^ a b Potter, John (9 September 2021). "How Can Value Investors Profit in the Crypto Ecosystem?". finance.yahoo.com. Retrieved 19 September 2021. {{cite web}}: CS1 maint: url-status (link)
^ a b c Nountu, Thierry Mefenza (28 November 2017). Pseudo-Random Generators and Pseudo-Random Functions: Cryptanalysis and Complexity Measures (Thèse de doctorat thesis).
^ a b c d e f g Hofheinz, Dennis; Jager, Tibor (30 October 2015). Verifiable Random Functions from Standard Assumptions. Theory of Cryptography Conference (published 19 December 2015). pp. 336–362. doi:10.1007/978-3-662-49096-9_14. ISBN 978-3-662-49096-9.
^ Barak, Boaz; Ong, Shien Jin; Vadhan, Salil (2007-01-01). "Derandomization in Cryptography" (PDF). SIAM Journal on Computing. 37 (2): 380–400. doi:10.1137/050641958. ISSN 0097-5397. Retrieved 2 September 2021.
^ Boneh, Dan; Waters, Brent (2013). Sako, Kazue; Sarkar, Palash (eds.). "Constrained Pseudorandom Functions and Their Applications". Advances in Cryptology - ASIACRYPT 2013. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 280–300. doi:10.1007/978-3-642-42045-0_15. ISBN 978-3-642-42045-0. Retrieved 2 September 2021.
^ a b Esgin, Muhammed F.; Kuchta, Veronika; Sakzad, Amin; Steinfeld, Ron; Zhang, Zhenfei; Sun, Shifeng; Chu, Shumo (24 March 2021). "Practical Post-Quantum Few-Time Verifiable Random Function with Applications to Algorand". Cryptology ePrint Archive. Retrieved 26 August 2021.
^ Schorn, Eric (2020-02-24). "Reviewing Verifiable Random Functions". NCC Group Research. Retrieved 2021-09-04.
^ Micali, Silvio; Reyzin, Leonid (2001). Kilian, Joe (ed.). "Soundness in the Public-Key Model". Advances in Cryptology — CRYPTO 2001. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 542–565. doi:10.1007/3-540-44647-8_32. ISBN 978-3-540-44647-7.
^ Dodis, Yevgeniy (2002). Desmedt, Yvo G. (ed.). "Efficient Construction of (Distributed) Verifiable Random Functions". Public Key Cryptography — PKC 2003. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 1–17. doi:10.1007/3-540-36288-6_1. ISBN 978-3-540-36288-3.
^ Goldberg, Sharon. "NSEC5: Provably Preventing DNSSEC Zone Enumeration". www.cs.bu.edu. Retrieved 2021-08-26.
^ "Ouroboros Protocol - Stake pool course". cardano-foundation.gitbook.io. 2021. Archived from the original on 11 March 2021. Retrieved 4 September 2021.
^ "Randomness · Polkadot Wiki". wiki.polkadot.network. 14 August 2021. Retrieved 4 September 2021. {{cite web}}: CS1 maint: url-status (link)
^ a b Hanke, Timo; Movahedi, Mahnush; Williams, Dominic (23 January 2018). "DFINITY Technology Overview Series Consensus System" (PDF). dfinity.org. Archived (PDF) from the original on 8 May 2018. Retrieved 19 August 2021.
^ a b c d "Algorand Protocol Overview". www.algorand.com. Retrieved 2021-08-15.
^ a b c Gorbunov, Sergey; Reyzin, L.; Papadopoulos, D.; Vcelak, J. (9 October 2018). "Algorand Releases First Open-Source Code of Verifiable Random Function". www.algorand.com. Retrieved 2021-08-15. {{cite web}}: CS1 maint: url-status (link)
^ Goldberg, S.; Reyzin, L.; Papadopoulos, D.; Vcelak, J. (17 May 2021). "draft-irtf-cfrg-vrf-09". datatracker.ietf.org. Retrieved 4 September 2021. {{cite web}}: CS1 maint: url-status (link)
^ "GitHub - witnet/vrf-solidity: Verifiable Random Function (VRF) library written in Solidity". GitHub. Retrieved 2021-08-26.
^ a b c "Verifiable Randomness for Blockchain Smart Contracts". Chainlink Blog. 12 May 2020. Retrieved 15 August 2020. {{cite web}}: CS1 maint: url-status (link)
^ Lan, Rongjian (2021-07-09). "Introducing Harmony Verifiable Random Function (VRF)". Medium. Retrieved 2021-09-04.
^ Oraichain (16 July 2021). "Introducing VRF — Verifiable Random Function on Oraichain Mainnet". Medium. Retrieved 4 September 2021. {{cite web}}: CS1 maint: url-status (link)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Verifiable_random_function&oldid=1046728155"
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To simplify the expression \sqrt[3]{\frac{7}{8x^{3}}}
Serotoninl7 2021-11-07 Answered
\sqrt[3]{\frac{7}{8{x}^{3}}}
According to quotient property of square root
\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}
And the product property of square root states
\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}
Given expression is
\sqrt[3]{\frac{7}{8{x}^{3}}}
\sqrt[3]{\frac{7}{8{x}^{3}}}=\sqrt[3]{\frac{7}{2×2×2×{x}^{3}}}
\sqrt[3]{\frac{7}{8{x}^{3}}}=\sqrt[3]{\frac{7}{2×2×2×{x}^{3}}}
Applying the quotient rule,
\sqrt[3]{\frac{7}{8{x}^{3}}}=\frac{\sqrt[3]{7}}{\sqrt[3]{2×2×2×{x}^{3}}}
Applying the product rule for Denominator,
\sqrt[3]{\frac{7}{8{x}^{3}}}=\frac{\sqrt[3]{7}}{\sqrt[3]{\left(2{\right)}^{3}×\sqrt[3]{{x}^{3}}}}
\sqrt[3]{\frac{7}{8{x}^{3}}}=\frac{\sqrt[3]{7}}{2x}
30.0m/s
200.0m
4.50\cdot {10}^{4}
\theta
{13.0}^{\circ }
\theta
The tires of a car make 65 revolutions as the car reduces its speed uniformly from 100 km/h to 50 km/h. The tires have diameter of 0.80 m. a) What was the angular acceleration? b)If the car continues to decelerate at this rate, how much more timeis required for it to stop?
For part a, I thought my method was pretty legitimate, but myanswer didn't match the one in the back of the book. The backof the book had the answer -4.4
ra\frac{d}{{s}^{2}}
, whereas Igot
-1.59\cdot {10}^{4}
ra\frac{d}{{s}^{2}}
Resolve the force F2 into components acting along the u and v axes and determine the magnitudes of the components.
A filling station is supplied with gasoline once a week. If its weekly volume of sales in thousands of gallons is a random variable with probability density function
f\left(x\right)=\left\{\begin{array}{ll}5\left(1-x{\right)}^{4}& 0<x<1\\ 0& \text{otherwise}\end{array}
what must the capacity of the tank be so that the probability of the supply’s being exhausted in a given week is .01?
What is the tangential speed of Nairobi, Kenya, acity near the equator? The earthmakes one revolution every 23.93 h and has anequatorial radius of 6380 km.
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Newton's Law of Gravity - Problem Solving Practice Problems Online | Brilliant
6 \times 10^{24}~\mbox{kg}
6370~\mbox{km}
8200~\mbox{km}
6.67 \times 10^{-11}~\mbox{N m}^2/\mbox{kg}^2
6 \times 10^{24}~\mbox{kg}
xy
-plane in deep space, where three particles are located on the
x
-axis, as shown in the above figure. The masses of the three particles are
m_A=70\text{ kg}, m_B=10\text{ kg}
m_C=40\text{ kg},
x
x_A=0,
x_B=5
x_C=15
in meters. If you move the particle
B
x
-axis until its coordinate becomes
10,
approximately how much work is done on particle
B
by you?
G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.
2.0 \times 10^{-9} \text{ J}
1.4 \times 10^{-12} \text{ J}
1.6 \times 10^{-11} \text{ J}
1.8 \times 10^{-10} \text{ J}
A person standing on the north pole of the earth measures the acceleration due to gravity as
-9.81~m.s^2
by resting a 1 kg block on a spring scale and measuring the force exerted by the scale. What value for the magnitude of the acceleration of gravity in
m/s^2
would a person on the equator measure if they performed this experiment?
You may treat the earth as a perfect sphere of radius 6370 km.
Take the length of a day to be exactly 24 hours.
An asteroid is falling toward the Earth from deep space. The mass of the asteroid is
m=6.00 \times 10^9\text{ kg}
and that of the Earth is
5.98 \times 10^{24}\text{ kg}.
If it arrived at a point
9.00 \times 10^8\text{ m}
from the center of the Earth, approximately how much work was done by the gravitational force?
G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.
3.72 \times 10^{15} \text{ J}
2.66 \times 10^{15} \text{ J}
4.25 \times 10^{15} \text{ J}
3.19 \times 10^{15} \text{ J}
The separation between two particles is
19.0\text{ m},
and the masses of them are
5.3\text{ kg}
2.7\text{ kg}.
How much work must one do to triple the separation between the particles?
G=6.67 \times 10^{-11} \text{ N}\cdot\text{m}^2\text{/kg}^2.
3.9 \times 10^{-11} \text{ J}
2.8 \times 10^{-11} \text{ J}
4.4 \times 10^{-11} \text{ J}
3.3 \times 10^{-11} \text{ J}
|
Determine whether the given function is linear,exponential,or neither.For those that are linear functions,find a linear function that models the data
\text{Misplaced \hline}
Notice that the ratio of consecutive values is fixed, which is 3/2, thus it is exponential function and
a=\frac{3}{2}
\stackrel{\to }{{R}_{A}}
\stackrel{\to }{{R}_{A}}
{360}^{m}
{40}^{\circ }
{123}^{\circ }
{5.0}^{s}
\stackrel{\to }{{R}_{B}}
\stackrel{\to }{{R}_{B}}
{880}^{m}
{\stackrel{\to }{R}}_{BA}={\stackrel{\to }{R}}_{B}-{\stackrel{\to }{R}}_{A}
{\stackrel{\to }{R}}_{BA}
{\stackrel{\to }{R}}_{BA}
The spring in the figure (a) is compressed by length delta x . It launches the block across a frictionless surface with speed v0. The two springs in the figure (b) are identical to the spring of the figure (a). They are compressed by the same length delta x and used to launch the same block. What is the block's speed now?
A 0.145 kg baseball pitched at 39.0 m/s is hit on a horizontal line drive straight back toward the pitcher at 52.0 m/s. If the contact time between bat and ball is
1.00×{10}^{-3}
s,calculate the average force between the bat and ball during contest.
Determine the interval(s) on which the vector-valued function is continuous
r\left(t\right)=⟨8,\sqrt{t},\sqrt[3]{t}⟩
A chandelier with mass m is attached to the ceiling of a large concert hall by two cables.Because the ceiling is covered with intricate architectur al decorations (not indicated in the figure, which uses a humbler depiction), the workers who hung the chandelier couldn't attach the cables to the ceiling directly above the chandelier. Instead, they attached the cables to the ceiling near the walls. Cable 1 has tension
{T}_{1}
{\theta }_{1}
with the ceiling. Cable 2 has tension
{T}_{2}
{\theta }_{2}
with the ceiling.
{T}_{1}
, the tension in cable 1, that does not depend on
{T}_{2}
Express your answer in terms of someor all of the variables m,
{\theta }_{1}
{\theta }_{2}
, as well as the magnitude of the acceleration due to gravity g.
A race car enters a flat 200 m radius curve at a speed of 20 m/swhile increasing its speed at a constant 2 m/s2. If the coefficient of static friction is .700, what will the speed of thecar be when the car beings to slide?
D) 36.2 m/s
E) 28.7 m/s
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Radiation Practice Problems Online | Brilliant
Everything we know about the universe, we learned by sifting through bits and pieces of energy sent in our direction from far-away sources. Detecting this energy is what allows humans to unravel the mysteries of the universe while our feet (mostly) remain on the ground.
Radiation—traveling particles of matter and light—brings information about the universe to us. Radiation is a by-product of atomic and nuclear processes that carries away excess energy. In this quiz we will explore how radiation is produced, characterized and detected.
Which do you think would not be considered radiation?
A helium nucleus riding the solar wind An electron in Earth's magnetosphere A near-Earth asteroid Light from an incandescent light bulb
The typically high speeds of radiation make it particularly useful for studying astronomical objects separated by astronomical distances. Visible light (and all types of electromagnetic radiation) travels through empty space at a rate of
c=\SI[per-mode=symbol]{2.998e8}{\meter\per\second}.
Large-scale manufacturing of incandescent light bulbs began about 100 years ago, heralding the age of cheap, safe artificial lighting on Earth.
Some of the light escaped through Earth's atmosphere and is today traveling through space. How far from our solar system is the light emitted
100
years ago today?
\SI{10^{15}}{\kilo\meter}
\SI{10^{21}}{\kilo\meter}
\SI{10^{25}}{\kilo\meter}
\SI{10^{32}}{\kilo\meter}
Using a telescope, you observe light reflected from Eris, a dwarf planet in orbit beyond Pluto.
By the time the light from Eris enters your telescope on Earth, Eris has moved away from the position where you see it. How long ago
(
\si{min})
was Eris in the position where you observe it?
Assume Eris is
\SI{50}{au}
from Earth when you observe it.
\SI{1}{au}=\SI{1.5e11}{\meter}.
In practice, most people round the speed of light to
\SI[per-mode=symbol]{3.0e8}{\meter\per\second}.
In the next chapter, we will take a detailed look at observations astronomers make in order to work out the distances to objects within and outside our solar system.
All electromagnetic radiation moves away from a radiating source at the same speed, but that does not mean all light is identical.
Light can have a range of wavelengths; different wavelengths of light have different effects on matter. Wavelength is represented by the symbol
\lambda.
Altogether, the continuum of wavelengths is called the electromagnetic (EM) spectrum. Light can have wavelengths outside of the range we can see; in fact, visible light is just a sliver of the entire EM spectrum.
All radiation is characterized by wavelength, with shorter wavelength indicating higher energy. In the
20^\text{th}
century, physicists measured wavelengths of matter particle radiation, like electrons. This is the starting point of quantum mechanics. We will come back to wave properties of matter in a later quiz.
Types of EM radiation (x-rays, ultraviolet, infrared, visible, and radio waves) are distinguished by their wavelengths, but they are just as often classified by frequency
f,
the number of waves passing a point per second. Because wavelength and frequency are used interchangeably, we should learn how to "convert" between them.
What relationship tells us the frequency of a wave that has wavelength
\lambda
which is traveling with speed
c?
Details and Hint:
c
It may help to know that the standard unit of
\lambda
\si{\meter}
and the standard unit of
\SI[per-mode=symbol]{1}{\per\second}.
f=\lambda c
f=\frac{c}{\lambda}
f=\frac{\lambda}{c}
Although we have characterized EM radiation on a spectrum of wavelengths, the energy carried by EM radiation is delivered by massless particles called photons. Experiments show that the amount of energy carried by each photon is proportional to the radiation frequency:
E_\text{photon}=hf,
h
is called Planck's constant and is equal to
h=\SI{6.63e-34}{\joule\second}.
The tungsten filament in an incandescent light bulb emits visible light with wavelength
\SI{500}{\nano\meter}=\SI{5e-7}{\meter}
(
since one nanometer is
\SI{10^{-9}}{\meter}).
About how much energy does a typical visible photon carry?
\SI{4e-23}{\joule}
\SI{4e-19}{\joule}
\SI{4e-11}{\joule}
\SI{4e4}{\joule}
A light bulb filament does not emit EM radiation at a single wavelength and frequency. Rather, it emits according to a distribution of wavelengths known as the tungsten filament's thermal emission spectrum.
Every object has a unique emission spectrum that depends on its atomic structure and temperature, but all known emission spectra have a peak wavelength that depends in a predictable way on temperature.
You can see that the peak of tungsten's spectrum is near yellow in the visible band. Suppose we supplied more voltage to the light bulb so that the filament's temperature increases. How would you expect the peak of the emission spectrum to change?
It would move left, closer to blue It would move right, closer to red It would stay on yellow, but the emission spectrum would get narrower and taller
All matter, hot or cold, emits EM radiation—sometimes called thermal or blackbody radiation. In response to changes in temperature, an emission spectrum's maximum changes in a predictable way. The colors of the visible spectrum are shown in a band around
\SI{0.5e-6}{\meter}.
Objects below temperatures of several thousand Kelvin emit mostly at longer wavelengths, which is why they don't glow like hotter objects.
The peak wavelength of the thermal emission spectrum
\lambda_\textrm{max}
(indicated by dotted lines for the three emission spectra above) is inversely proportional to the temperature
T
\lambda_\textrm{max} = \frac{P_0}{T}
P_0=\SI{0.0029}{\meter\cdot\kelvin}.
This is called Wien's displacement law.
How hot is a tungsten filament when its peak emitted wavelength is in the yellow band of the visible spectrum?
Use Wien's displacement law and estimate the wavelength of the yellow band from the diagram.
\SI{3600}{\kelvin}
\SI{5800}{\kelvin}
\SI{9600}{\kelvin}
Peak wavelength of a radiating body is an easy quantity to measure, no matter how far the collected light has traveled to us. For example, the emission spectrum of the Sun is plotted below.
The Sun has a peak wavelength around
\SI{500}{\nano\meter}
—same as the filament in the previous question. Thus, we can estimate that the Sun's surface temperature is about the same as a tungsten filament,
\SI{5800}{\kelvin}.
In fact, Wien's displacement law can be used to estimate the surface temperature of any nearby star.
All matter emits EM radiation in different regions of the electromagnetic spectrum, depending on whether it is hot or cold. Astronomers can collect and analyze not only the visible light from stars we can see in the night sky but also light across the EM spectrum: infrared, radio, UV, x-rays, etc. In the next quiz, we will see how peak wavelengths are distributed for clusters of stars in our galaxy when we introduce the famed Hertzsprung-Russell diagram.
|
Determine whether the given function is linear,exponential,or neither.For those that
Determine whether the given function is linear,exponential,or neither.For those that are linear functions find a linear function that models the data
Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data. for those that are exponential, find an exponential function that models the data.
xF\left(x\right)-1\left(\frac{1}{2}\right)0\left(\frac{1}{4}\right)1\left(\frac{1}{8}\right)2\left(\frac{1}{16}\right)3\frac{1}{32}
Want to know more about Exponential models?
The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006. Canada
A=33.1e0.009t
A=28.2e0.034t
use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. By 2009, the models indicate that Canada's population will exceed Uganda's by approximately 2.8 million.
A researcher is trying to determine the doubling time of a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table. Use a graphing calculator to find an exponential curve f(t)=a⋅bt that models the bacteria population t hours later. Time (hours) 04812162024 Bacteria count (CFU/mL)37476378105130173
Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data. for those that are exponential, find an exponential function that models the data. xg(x) -12 05 18 211 314
Transform the given differential equation or system into an equivalent system of first-order differential equations.
{t}^{3}{x}^{\left(3\right)}-2{t}^{2}x{}^{″}+3t{x}^{\prime }+5x=\mathrm{ln}t
{x}^{\left(4\right)}+6x{}^{″}-3{x}^{\prime }+x=\mathrm{cos}3t
\text{Misplaced \hline}
y={e}^{5x}
|
Some Fixed Point Theorems for Fuzzy Iterated Contraction Maps in Fuzzy Metric Spaces
Some Fixed Point Theorems for Fuzzy Iterated Contraction Maps in Fuzzy Metric Spaces
College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, China
The purpose of this paper is to introduce the notion of fuzzy iterated contraction maps in fuzzy metric spaces and establish some fixed point theorems for fuzzy iterated contraction maps in fuzzy metric spaces.
Fixed Point, Iterated Contraction Map, Fuzzy Metric Space
In 1975, Kramosil and Michalek [1] first introduced the concept of a fuzzy metric space. In 1994, George and Veeramani [2] slightly modified the concept of fuzzy metric space introduced by Kramosil and Michalek, defined a Hausdorff topology and proved some known results. In 1969, Rheinboldt [3] initiated the study of iterated contraction. The concept of iterated contraction proves to be very useful in the study of certain iterative process and has wide applicability in metric spaces. In this paper we introduce the notion of fuzzy iterated contraction maps in fuzzy metric spaces and establish some fixed point theorems for fuzzy iterated contraction maps in fuzzy metric spaces.
Definition 2.1 ( [2] ). A fuzzy metric space is an ordered triple
\left(X,M,\ast \right)
such that X is a (nonempty) set,
\ast
is a continuous t-norm and M is a fuzzy set on
X\times X\times \left(0,\infty \right)
satisfying the following conditions, for all
x,y,z\in X
s,t>0
(FM-1)
M\left(x,y,t\right)>0
M\left(x,y,t\right)=1
x=y
M\left(x,y,t\right)=M\left(y,x,t\right)
M\left(x,y,t\right)\ast M\left(x,z,t+s\right)
M\left(x,y,•\right):\left(0,\infty \right)\to \left(0,1\right)
Definition 2.2 ( [2] ). A map
T:X\to X
M\left(Tx,Ty,t\right)\ge M\left(x,y,\frac{t}{k}\right)
x,y\in X,t>0,0<k<1
, is called a contraction map.
In this part, we firstly give a notion of the fuzzy iterated contraction map in a fuzzy metric space, then we prove some fixed point theorems for fuzzy iterated maps under different settings.
\left(X,M,*\right)
is a fuzzy metric space such that
M\left(Tx,{T}^{2}x,t\right)\ge M\left(x,Tx,\frac{t}{k}\right)
x\in X,t>0,0<k<1
, then T is said to be a fuzzy iterated contraction map.
Remark 3.1 A fuzzy contraction map is continuous and is a fuzzy iterated contraction. A fuzzy contraction map has a unique fixed point. However, a fuzzy iterated contraction map may have more than one fixed point.
\left(X,d\right)
be a metric space, Define
x\ast y=xy
d\left(x,y\right)=|x-y|
x,y\in X
t>0
M\left(x,y,t\right)=\frac{t}{t+d\left(x,y\right)}
\left(X,M,*\right)
is a fuzzy metric space.
T:\left[-1/2,1/2\right]\to \left[-1/2,1/2\right]
Tx={x}^{2}
, then T is a fuzzy iterated contraction but not a fuzzy contraction map.
The following is a fixed point theorem for fuzzy iterated contraction map.
T:X\to X
is a continuous fuzzy iterated contractive map and the sequence of iterates
\left\{{x}_{n}\right\}
in fuzzy metric space
\left(X,M,*\right)
{x}_{n+1}=T{x}_{n},n=1,2,\cdots
x\in X
, has a subsequence converging to
y\in X
y=Ty
, that is, T has a fixed point.
Proof: The sequence
\left\{M\left({x}_{n+1},{x}_{n},t\right)\right\}
is a nondecreasing sequence of reals. It is bounded above by 1, and therefore has a limit. Since the subsequence converges to y and T is continuous on X, so
T\left({x}_{{n}_{i}}\right)
Ty
{T}^{2}\left({x}_{{n}_{i}}\right)
{T}^{2}y
M\left(y,Ty,t\right)=\mathrm{lim}M\left({x}_{{n}_{i}},{x}_{{n}_{i+1}},t\right)=\mathrm{lim}M\left({x}_{{n}_{i+1}},{x}_{{n}_{i+2}},t\right)=M\left(Ty,{T}^{2}y,t\right)
\begin{array}{c}1\ge M\left(Ty,{T}^{2}y,t\right)\ge M\left(y,Ty,\frac{t}{k}\right)\\ \ge M\left(y,Ty,\frac{t}{{k}^{2}}\right)\ge \cdots \\ \ge M\left(y,Ty,\frac{t}{{k}^{n}}\right),\text{\hspace{0.17em}}\forall n\in N,\text{\hspace{0.17em}}0<k<1\end{array}
n\to \infty ,M\left(y,Ty,\frac{t}{{k}^{n}}\right)\to 1
M\left(y,Ty,\frac{t}{k}\right)=1
Ty=y
We give the following example to show that if T is a fuzzy iterated contraction that is not continuous, then T may not have a fixed point.
A continuous map T that is not a fuzzy iterated contraction may not have a fixed point.
Note 3.1 If T is not contraction but some powers of T is contraction, then T has a unique fixed point on a complete metric space.
Proof: If x is a fixed point of k powers of T, thus
{T}^{k}\left(x\right)=x
T\left({T}^{k}\left(x\right)\right)=T\left(x\right)
{T}^{k}\left(T\left(x\right)\right)=T\left(x\right)
{T}^{k}
has a unique fixed point. Consequently
T\left(x\right)=x
Note 3.2 Continuity of a fuzzy iterated contraction is sufficient but not necessary.
As stated in Note 3.1 that if T is not contraction still T may have a unique fixed point when some powers of T is a fuzzy contraction map. The same is true for fuzzy iterated contraction map.
T:X\to X
be a fuzzy iterated contraction map on a complete metric space X. If for some power of T, say
{T}^{r}
is a fuzzy iterated contraction,
M\left({T}^{r}x,{\left({T}^{r}\right)}^{2}x,t\right)\ge M\left(x,{T}^{r}x,\frac{t}{k}\right)
{T}^{r}
is continuous at y, where
y=\mathrm{lim}{\left({T}^{r}\right)}^{n}x
, for any arbitrary
x\in X
Proof: Since T is a fuzzy iterated contraction that is continuous at y,
M\left({T}^{r}x,{\left({T}^{r}\right)}^{2},t\right)\ge M\left(x,{T}^{r}x,\frac{t}{k}\right),\text{\hspace{0.17em}}0<k<1
\begin{array}{l}M\left({\left({T}^{r}\right)}^{n}x,{\left({T}^{r}\right)}^{m}x,t\right)\\ \ge \underset{i=m}{\overset{n}{\prod }}M\left({\left({T}^{r}\right)}^{i+1}x,{\left({T}^{r}\right)}^{i}x,\frac{t}{n-m}\right),\text{\hspace{0.17em}}\forall n>m,\text{\hspace{0.17em}}n,m\in N\\ \ge \underset{i=m}{\overset{n}{\prod }}M\left(x,{T}^{r}x,\frac{t}{\left(n-m\right)*{k}^{i-1}}\right)\to 1\text{\hspace{0.17em}}\left(m\to \infty \right)\end{array}
{\left\{{\left({T}^{r}\right)}^{n}\right\}}_{n=1}^{\infty }
is a Cauchy sequence in X and X is a complete metric space. Thus
\exists y\in X,\text{\hspace{0.17em}}s.t.\text{\hspace{0.17em}}y=\underset{n\to \infty }{\mathrm{lim}}{\left({T}^{r}\right)}^{n}x
{T}^{r}
is continuous, consequently
{T}^{r}\left(y\right)=\underset{n\to \infty }{\mathrm{lim}}{T}^{r}\left({\left({T}^{r}\right)}^{n}x\right)=\underset{n\to \infty }{\mathrm{lim}}{\left({T}^{r}\right)}^{n+1}x=y
d\left(y,fy\right)\le {k}^{r}d\left(y,fy\right)
{k}^{r}\le 1
d\left(y,fy\right)=0
and hence f has a fixed point.
We give the following example to illustrate the theorem.
Note 3.3 If T is not a fuzzy iterated contraction in Theorem 3.2, but
{T}^{r}
is a fuzzy iterated contraction with
{T}^{r}y=y
T:X\to X
is a fuzzy iterated contraction map, and X is a complete metric space, then the sequence of iterates
{x}_{n}
y\in X
. In case, T is continuous at y, then
y=Ty
{x}_{n+1}=T{x}_{n},n=1,2,\cdots ;{x}_{1}\in X
\left\{{x}_{n}\right\}
is a Cauchy sequence, since T is a fuzzy iterated contraction. The Cauchy sequence
\left\{{x}_{n}\right\}
y\in X
, since X is a complete metric space. Moreover, if T is continuous at y, then
{x}_{n+1}=T{x}_{n}
Ty
y=Ty
Note 3.3 A continuous iterated contraction map on a complete metric space has a unique fixed point. If an iterated contraction map is not continuous, it may have more than one fixed point.
This paper is supported by the Student Research Training Program of Jiaxing University (No.851715034), the College Student’s Science and Technology Innovation Project of Zhejiang Province (No.2016R417014).
Xia, L. and Tang, Y.H. (2018) Some Fixed Point Theorems for Fuzzy Iterated Contraction Maps in Fuzzy Metric Spaces. Journal of Applied Mathematics and Physics, 6, 224-227. https://doi.org/10.4236/jamp.2018.61021
1. Kramosil, I. and Michálek, J. (1975) Fuzzymetric and Statistical Mestric Spaces. Kybernetika, 11, 336-334.
2. George, A. and Veeramani, P. (1994) On Some Results in Fuzzy Metric Spaces. Fuzzy Sets and System, 64, 395-399. https://doi.org/10.1016/0165-0114(94)90162-7
3. Rheinboldt, W.C. (1968) A Unified Convergence Theory for a Class of Iterative Process. SIAM Journal on Numerical Analysis, 5, 42-63. https://doi.org/10.1137/0705003
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Poynting Vector | Brilliant Math & Science Wiki
July Thomas, Mark Hennings, and Jimin Khim contributed
The Poynting vector represents the direction of propagation of an electromagnetic wave as well as the energy flux density, or intensity.
Since an electromagnetic wave is composed of an electric field
\big(\vec{E}\big)
\big(\vec{B}\big)
oscillating perpendicular to one another and mutually perpendicular to the direction of the propagation of the wave, the Poynting vector is defined in terms of the cross product of the two fields. The constant in front serves to provide the correct magnitude for the intensity:
\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B}.
This antenna is aligned with the electric field component of an electromagnetic wave in order to capture a signal.
Direction of an Electromagnetic Wave
Maxwell's Equations and Energy Conservation
The cross product of these oscillating E and B fields at any time indicates the wave is traveling into the page.
If a wave equation is given for an EM wave, the direction of the Poynting vector can be read off. Recall that the general equation for the perpendicular displacement of a point on a string along which a transverse wave is traveling in the
+x
direction is
y(x,t) = y_m\sin(kx-\omega t).
Variations within the argument of the sine function account for different directions of propagation.
What is the direction of propagation of the wave described by
y(x,t) = y_m\sin(kx+\omega t)?
Changing the
-
sign in the argument of the sine function to a
+
sign indicates the direction of propagation is flipped. So, this wave travels in the
-x
_\square
x(y,t) = y_m\sin(ky-\omega t)?
Changing the independent distance variable from
x
y
indicates the wave travels not in the
x
direction but in the
y
_\square
+x
-z
-x
+y
-y
+z
What is the direction of propagation of the electromagnetic wave described with electric field component
\vec{E} = E_m \hat{i} \sin(kz + \omega t)?
By definition, the direction of the Poynting vector must be mutually perpendicular to both the electric and magnetic fields. This relationship is expressed in terms of the cross product of the two fields:
\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B}.
In practice, this equation is generally only used to find the direction of one of the three, so it helps to remember just the relationship between the unit vectors and the right-hand rule:
\hat{S} = \hat{E}\times \hat{B}
What is the direction of the Poynting vector for an electromagnetic wave with its electric field component oscillating in the
+y
direction at an instant when its magnetic field component oscillates in the
-z
direction?
\hat{S} = \hat{E}\times \hat{B} = \hat{y} \times -\hat{z} = -( \hat{y} \times \hat{z}) = -(\hat{x}) = -\hat{x}.
Hence, the Poynting vector points in the
-x
_\square
\vec{E} = \frac{B_m}{c}\hat{x} \sin(ky + \omega t)
\vec{E} = c B_m\hat{x} \sin(ky + \omega t)
\vec{E} = c B_m\hat{z} \sin(ky + \omega t)
\vec{E} = \frac{B_m}{c}\hat{z} \sin(ky + \omega t)
What is the equation of the electric field component of an electromagnetic wave with magnetic field component
\vec{B} = B_m\hat{z} \sin(ky + \omega t)?
The magnitude of the Poynting vector is also called the intensity of the wave. The intensity can be found with the definition of the Poynting vector and the fact that the magnitude of the cross product is
AB\sin\theta.
\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B}
Since the electric and magnetic fields oscillate between positive and negative maxima, it is necessary to describe the average value in terms of the root-mean-squares of the field:
S = \frac{1}{\mu_0} E_{\text{rms}}B_{\text{rms}}.
It is common practice to combine this equation along with the definition of RMS
\big(\text{RMS} = \frac{\text{max}}{\sqrt{2}}\big)
c = \frac EB.
The intensity is conventionally expressed in terms of just one of the two fields.
I = S = \frac{1}{c\mu_0} \frac{E_m^2}{2}
A dish with
10\text{ m}
diameter dish receives a radio signal with a magnetic field amplitude of
7 \text{ pT}.
Find the power received by the dish.
In order to express intensity in terms of magnetic field, incorporate
c = \frac{E}{B}:
I = \frac{1}{c\mu_0} \frac{E_m^2}{2} = \frac{1}{c\mu_0} \frac{(cB)^2}{2} = \frac{cB^2}{2\mu_0}.
The power intake of a receptor is
P = IA,
I
is the intensity and
A
is the area of the receptor:
P = IA = \frac{cB^2A}{2\mu_0} = \frac{c\big(7\times10^{-12}\big)^2\big(\pi(5)^2\big)}{2\mu_0} = 4.59\times10^{-7}\text{ W}.\ _\square
What is the amplitude of the electric field component of an electromagnetic wave of intensity
I = \dfrac{100}{2c\mu_0}?
Why does the Poynting vector represent intensity? The fundamental equations of electromagnetism are Maxwell's equations:
\begin{array}{rclcrcl} \nabla \cdot \vec{E} & = & \varepsilon_0^{-1}\rho, & \hspace{.5cm} & \nabla \cdot \vec{B} & = & 0, \\[1ex] \nabla \wedge \vec{E} & = & \displaystyle -\frac{\partial \vec{B}}{\partial t}, & & \nabla \wedge \vec{B} & = & \displaystyle \mu_0\vec{\jmath} + \varepsilon_0\mu_0\frac{\partial \vec{E}}{\partial t}, \end{array}
\rho
is the charge density and
\vec{\jmath}
is the current density of the field. If
V
is a volume of space bounded by a surface
\partial V
\begin{aligned} \frac{d}{dt}\int_V \tfrac12\varepsilon_0\big|\vec{E}\big|^2\,d\tau & = \int_V \varepsilon_0 \vec{E} \cdot \frac{\partial \vec{E}}{\partial t}\,d\tau \\ & = \mu_0^{-1}\int_V \vec{E} \cdot\big(\nabla \wedge \vec{B} - \mu_0\vec{\jmath}\big)\,d\tau \\ & = \mu_0^{-1}\int_V \vec{E} \cdot \big(\nabla \wedge \vec{B}\big)\,d\tau - \int_V \vec{E} \cdot \vec{\jmath}\,d\tau \\ \frac{d}{dt}\int_V \tfrac12\mu_0^{-1}\big|\vec{B}\big|^2\,d\tau & = \mu_0^{-1}\int_V \vec{B} \cdot \frac{\partial \vec{B}}{\partial t}\,d\tau \\ & = - \mu_0^{-1}\int_V \vec{B} \cdot (\nabla \wedge \vec{E})\,d\tau. \end{aligned}
\begin{aligned} \frac{d}{dt}\int_V\left(\tfrac12\varepsilon_0\big|\vec{E}\big|^2 + \tfrac12\mu_0^{-1}\big|\vec{B}\big|^2\right)\,d\tau + \int_V \vec{E} \cdot \vec{\jmath}\,d\tau & = \mu_0^{-1}\int_V\Big[\vec{E} \cdot (\nabla \wedge \vec{B}) - \vec{B} \cdot (\nabla \wedge \vec{E})\Big]\,d\tau \\ & = \mu_0^{-1}\int_V \nabla \cdot \big(\vec{B} \wedge \vec{E}\big)\,d\tau \\ & = \mu_0^{-1} \oint_{\partial V} \big(\vec{B} \wedge \vec{E}\big)\cdot d\vec{\sigma} \end{aligned}
\frac{d}{dt}\int_V\left(\tfrac12\varepsilon_0\big|\vec{E}\big|^2 + \tfrac12\mu_0^{-1}\big|\vec{B}\big|^2\right)\,d\tau + \int_V \vec{E} \cdot \vec{\jmath}\,d\tau + \oint_{\partial V} \vec{S} \cdot d\vec{\sigma} = 0.
The first term is the rate of change of the total electromagnetic energy in the field in the volume
V
, while the second term is the rate of work being done by the field due to the movement of charge inside
V
. Conservation of energy considerations tell us that the third term, involving the Poynting vector, must be the rate of flow of energy across the boundary
\partial V
of the volume
V
, and hence the Poynting vector
\vec{S}
indeed describes the direction and intensity of the electromagnetic field.
Cite as: Poynting Vector. Brilliant.org. Retrieved from https://brilliant.org/wiki/poynting-vector/
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Estimate states of discrete-time nonlinear system using unscented Kalman filter - Simulink
\stackrel{^}{x}
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}\left[k|k\right]
\stackrel{^}{x}\left[k|k-1\right]
\stackrel{^}{x}
\begin{array}{l}x\left[k+1\right]=f\left(x\left[k\right],{u}_{s}\left[k\right]\right)+w\left[k\right]\\ y\left[k\right]=h\left(x\left[k\right],{u}_{m}\left[k\right]\right)+v\left[k\right]\end{array}
\begin{array}{l}x\left[k+1\right]=f\left(x\left[k\right],w\left[k\right],{u}_{s}\left[k\right]\right)\\ y\left[k\right]=h\left(x\left[k\right],v\left[k\right],{u}_{m}\left[k\right]\right)\end{array}
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Revision as of 01:30, 26 February 2013 by Gmaxwell (talk | contribs) (→Bandlimitation and timing: Squares like LaTeX, more.)
{\displaystyle \ x(t)={\begin{cases}1,&|t|<T_{1}\\0,&T_{1}<|t|\leq {1 \over 2}T\end{cases}}}
{\displaystyle {\begin{aligned}x_{\mathrm {square} }(t)&{}={\frac {4}{\pi }}\sin(\omega t)+{\frac {4}{3\pi }}\sin(3\omega t)+{\frac {4}{5\pi }}\sin(5\omega t)+{\frac {4}{7\pi }}\sin(7\omega t)+\cdots \end{aligned}}}
As stated in the Epilogue, everything that appears in the video demos is driven by open source software, which means the source is both available for inspection and freely usable by the community. The Thinkpad that appears in the video was running Fedora 17 and Gnome Shell (Gnome 3). The demonstration software does not require Fedora specifically, but it does require Gnu/Linux to run in its current form.
mkfifo pipe0; mkfifo pipe1
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Solve the equation. \frac{5}{2x+3}+\frac{4}{2x-3}=\frac{14x+3}{4x^{2}-9}Z
\frac{5}{2x+3}+\frac{4}{2x-3}=\frac{14x+3}{4{x}^{2}-9}
likvau
We have solve the equation
\frac{5}{2x+3}+\frac{4}{2x-3}=\frac{14x+3}{4{x}^{2}-9}
\frac{5}{2x+3}+\frac{4}{2x-3}=\frac{14x+3}{4{x}^{2}-9}
⇒\frac{5\left(2x-3\right)+4\left(2x+3\right)}{\left(2x+3\right)\left(2x-3\right)}=\frac{14x+3}{4{x}^{2}-9}
⇒\frac{10x-15+8x+12}{{\left(2x\right)}^{2}-{\left(3\right)}^{2}}=\frac{14x+3}{4{x}^{2}-9}
⇒\frac{18x-3}{4{x}^{2}-9}=\frac{14x+3}{4{x}^{2}-9}
⇒18x-3=14x+3
⇒18x-14x=3+3
⇒4x=6
⇒x=\frac{6}{4}=\frac{3}{2}
⇒x=\frac{3}{2}
A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]
\stackrel{―}{x}=
Determine the solutions set of the following equation. Equation in Quadratic form
4x+\sqrt{3}=1
Find general solutions of the equation. First find a small integral root of the characteristic equation by inspection; then factor by division.
{y}^{3}+3y-4y=0
f\left(x\right)=\frac{1}{x+5}
a=5
Find all a,b,c
\in \mathbb{R}
that satisfy both equations:
Identify the curve by finding a Cartesian equation for the curve.
{r}^{2}=5
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Nano-Scale Fatigue Wear of Carbon Nitride Coatings: Part II—Wear Mechanisms | J. Tribol. | ASME Digital Collection
Dong F. Wang,
Laboratory of Biomechanical Engineering, Department of Mechatronics and Precision Engineering, Faculty of Engineering, Tohoku University, Sendai 980-8579, Japan
Laboratory of Tribology, School of Mechanical Engineering, Tohoku University, Sendai 980-8579, Japan
Contributed by the Tribology Division for publication in the ASME JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division June 28, 2001 revised manuscript received August 1, 2002. Associate Editor: J. A. Williams.
Wang, D. F., and Kato, K. (March 19, 2003). "Nano-Scale Fatigue Wear of Carbon Nitride Coatings: Part II—Wear Mechanisms ." ASME. J. Tribol. April 2003; 125(2): 437–444. https://doi.org/10.1115/1.1537267
This is the second part of two companion papers, the first of which reported the empirical data on wear properties in carbon nitride coatings by a spherical diamond counter-face in repeated sliding contacts through in situ examination, with an emphasis on the effect of friction cycles and normal load. The second part will concentrate on wear mechanisms for the transition from “No observable wear particles” to “Wear particle generation.” The relationship between the critical number of friction cycles, Nc, and the representative plastic strain,
Δεp,
at asperity contact region was confirmed to follow the Manson-Coffin equation with two empirical constants, β and C. The observed generation of wear particles in carbon nitride coatings is therefore concluded to be a low cycle fatigue wear by surface flow and surface delamination in the ploughing mode. For further predicting lifespan, a simplified theoretical expression, combining the Manson-Coffin equation with the analytical solution of a proposed elastic perfectly-plastic indentation model, gives the relation between the critical number of friction cycles, Nc, and the coating thickness h, with respect to the contact pressure P, and the radius R of the asperity on the tip of the diamond pin.
fatigue, wear, wear testing, wear resistant coatings, carbon compounds, nitrogen compounds, mechanical contact, sliding friction, surface topography, rough surfaces
Carbon, Coatings, Cycles, Friction, Particulate matter, Wear, Coating processes, Delamination, Flow (Dynamics), Low cycle fatigue, Fatigue
A Concept for Friction Mechanisms of Coated Surface
Holmberg, K., and Matthews, A., 1994, Coatings Tribology, Dowson, D., eds., Tribology Series, 28, Elsevier.
An Analysis of Sliding Wear Mechanism of Oxide Films on Hot Roll Surfaces Based on In-Situ Observation of Wear Process by the CCD Microscope Tribosystem
Plastic Flow Process of Surface Layers in Flow Wear under Boundary Lubrication Conditions
Critical Load for Wear Particle Generation in Carbon Nitride Coating by Single Sliding against a Spherical Diamond
Mechanical Characterization and Tribological Evaluation of Ion-beam-Assisted Sputter Coatings of Carbon With Nitrogen Incorporation
Advanced Surface Coatings: A Handbook of Surface Engineering, 1991, D. S. Rickerby, A. Matthews, eds., Blackie & Son Limited, NY.
Manson, S. S., 1953, “Behavior of Materials Under Thermal Stress,” NACA TN 2933.
Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, pp. 199–200.
Ueber die Beruhrung Fester Elastischer Korper
Boussinrsq, J., 1885, “Applications des Potentials a L’etude de Equilibre et du Mouvement des Solids Elastiques,” Gauthier-Villars, Paris.
Timoshenko, S., and Goodier, J. N., 1951, Theory of Elasticity, 2nd ed., McGraw-Hill, New York.
Uber die Harte Plastischer Korper
Gottinger Nachr. Math. Phys., KL.
Ishlinsky
, 1944, Appl. Math. Mech. pp. 201.
The Theory of Wedge Indentation of Ductile Materials
On the Plastic Flow of Metals under Conditions of Axial Symmetry
Indentation of a Rigid/Plastic Material by a Conical Indenter
Nepershin
Izvestiya of U.S.S.R. Akad. Nauk. Solid Mech.
Yu, W., 1995, “Theoretical and Numerical Analysis of Elastic-Plastic and Creeping Solids,” Ph.D. thesis, University of Wisconsin-Madison.
The Theory of Indentation and Hardness Tests
Proceeding of the Physical Society
Plastic Flow in Glass
M. G. J. W.
The Indentation of Materials by Wedges
An Elastic-Plastic Indentation Model and Its Solutions
Friction and Wear of Metal Particle, Barium Ferrite and Metal Evaporated Tapes in Rotary Head Recorders
Friction and Wear Characteristics of Diamond-Like Carbon (DLC) Coating Used for Machine Elements
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Calculus/Integration techniques/Irrational Functions - Wikibooks, open books for an open world
Calculus/Integration techniques/Irrational Functions
← Integration techniques/Reduction Formula Calculus Integration techniques/Numerical Approximations →
Integration techniques/Irrational Functions
Integration of irrational functions is more difficult than rational functions, and many cannot be done. However, there are some particular types that can be reduced to rational forms by suitable substitutions.
Integrand contains
{\displaystyle {\sqrt[{n}]{\frac {ax+b}{cx+d}}}}
{\displaystyle u={\sqrt[{n}]{\frac {ax+b}{cx+d}}}}
{\displaystyle \int {\frac {1}{x}}{\sqrt {\frac {1-x}{x}}}\,dx}
{\displaystyle \int {\frac {x}{\sqrt[{3}]{ax+b}}}\,dx}
Integral is of the form
{\displaystyle \int {\frac {Px+Q}{\sqrt {ax^{2}+bx+c}}}\,dx}
{\displaystyle Px+Q}
{\displaystyle Px+Q=p\cdot {\frac {d[ax^{2}+bx+c]}{dx}}+q}
{\displaystyle \int {\frac {4x-1}{\sqrt {5-4x-x^{2}}}}\,dx}
{\displaystyle {\sqrt {a^{2}-x^{2}}}}
{\displaystyle {\sqrt {a^{2}+x^{2}}}}
{\displaystyle {\sqrt {x^{2}-a^{2}}}}
This was discussed in "trigonometric substitutions above". Here is a summary:
{\displaystyle {\sqrt {a^{2}-x^{2}}}}
{\displaystyle x=a\sin(\theta )}
{\displaystyle {\sqrt {a^{2}+x^{2}}}}
{\displaystyle x=a\tan(\theta )}
{\displaystyle {\sqrt {x^{2}-a^{2}}}}
{\displaystyle x=a\sec(\theta )}
{\displaystyle \int {\frac {dx}{(px+q){\sqrt {ax^{2}+bx+c}}}}}
{\displaystyle u={\frac {1}{px+q}}}
{\displaystyle \int {\frac {dx}{(1+x){\sqrt {3+6x+x^{2}}}}}}
Other rational expressions with the irrational function
{\displaystyle {\sqrt {ax^{2}+bx+c}}}
{\displaystyle a>0}
{\displaystyle u={\sqrt {ax^{2}+bx+c}}\pm {\sqrt {a}}x}
{\displaystyle c>0}
{\displaystyle u={\frac {{\sqrt {ax^{2}+bx+c}}\pm {\sqrt {c}}}{x}}}
{\displaystyle ax^{2}+bx+c}
{\displaystyle a(x-\alpha )(x-\beta )}
{\displaystyle u={\sqrt {\frac {a(x-\alpha )}{x-\beta }}}}
{\displaystyle a<0}
{\displaystyle ax^{2}+bx+c}
{\displaystyle -a(\alpha -x)(x-\beta )}
{\displaystyle x=\alpha \cos ^{2}(\theta )+\beta \sin ^{2}(\theta )}
Retrieved from "https://en.wikibooks.org/w/index.php?title=Calculus/Integration_techniques/Irrational_Functions&oldid=3133236"
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Information Theory - Berkeley Notes
Information Theory is a field which addresses two questions
Source Coding: How many bits do I need to losslessly represent an observation.
Channel Coding: How reliably and quickly can I communicate a message over a noisy channel.
Intuitively, for a PMF of a disrete random variable, the surprise associated with a particular realization is
-\log p_X(x)
since less probable realizations are more surprising. With this intuition, we can try and quantify the “expected surprise” of a distribution.
X\sim p_X
X
H(x) = \mathbb{E}\left[-\log_2 p_X(x)\right] = -\sum_{x\in\mathcal{X}} p_X(x)\log_2p_X(x).
Alternative interpretations of entropy are the average uncertainty and how random
X
is. Just like probabilites, we can define both joint and conditional entropies.
For Discrete Random Variables
X
Y
, the joint entropy is given by
H(X,Y) = \mathbb{E}\left[-\log_2p_{XY}(x, y)\right] = -\sum_{x,y\in\mathcal{X}\times\mathcal{Y}}p_{XY}(x, y)\log_2p_{XY}(x, y).
For Discrete Random Variable
X
Y
, the conditional entropy is given by
H(Y|X) = \mathbb{E}\left[-\log_2p_{Y|X}(y|x)\right] = \sum_{x\in\mathcal{X}}p_X(x)H(Y|X=x).
Conditional entropy has a natural interpretation which is that it tells us how surprised we are to see
Y=y
given that we know
X=x
X
Y
H(Y) = H(Y|X)
because realizing
X
gives no additional information about
Y
Theorem 23 (Chain Rule of Entropy)
H(X, Y) = H(X) + H(X|Y).
In addition to knowing how much our surprise changes for a random variable when we observe a different random variable, we can also quantify how much additional information observing a random variable gives us about another.
X
Y
, the mutual information is given by
I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X).
Source coding deals with finding the minimal number of bits required to represent data. This is essentially the idea of lossless compression. In this case, our message is the sequence of realizations of independently and identically distributed random variables
\left(X_i\right)_{i=1}^n \sim p_X
. The probability of observing a particular sequence is then
P(x_1, x_2, \cdots, x_n) = \prod_{i=1}^np_X(x_i).
Theorem 24 (Asymptotic Equipartition Property)
If we have a sequence of independently and identically distributed random variables
\left(X_i\right)_{i=1}^n \sim p_X
-\frac{1}{n}\log P(x_1, x_2, \cdots, x_n)
H(X)
Theorem 24 tells us that with overwhelming probability, we will observe a sequence that is assigned probability
2^{-nH(X)}
. Using this idea, we can define a subset of possible observed sequences that in the limit, our observed sequence must belong to with overwhelming probability.
\epsilon > 0
n\geq 1
, the typical set is given by
A_\epsilon^{(n)} = \left\{ (x_1, x_2, \cdots, x_n) : 2^{-n(H(X)+\epsilon)}\leq P(x_1, x_2, \cdots, x_n) \leq 2^{-n(H(X)-\epsilon)} \right\}.
Two important properties of the typical set are that
\lim_{n\to\infty}P\left((x_1, x_2, \ldots, x_n) \in A_{\epsilon}^{(n)}\right) = 1
|A_{\epsilon}^{(n)}| \leq 2^{n(H(X)+\epsilon)}
The typical set gives us an easy way to do source coding. If I have
N
total objects, then I only need
\log N
bits to represent each object, so I can define a simple protocol which is
(x_i)_{i=1}^{n} \in A^{(n)}_{\frac{\epsilon}{2}}
, then describe them using the
\log|A^{(n)}_{\frac{\epsilon}{2}}| \leq n\left(H(X)+\frac{\epsilon}{2}\right)
(x_i)_{i=1}^n \not \in A^{(n)}_{\frac{\epsilon}{2}}
, then describe them naiively with
n\log|\mathcal{X}|
This makes the average number of bits required to describe a message
\begin{aligned} \mathbb{E}\left[\text{\# of Bits}\right] &\leq n\left(H(X)+\frac{\epsilon}{2}\right)P\left((x_i)_{i=1}^n\in A_{\frac{\epsilon}{2}}^{(n)}\right) + n\log |\mathcal{X}|P\left((x_i)_{i=1}^n\in A_{\frac{\epsilon}{2}}^{(n)}\right) \\ &\leq n(H(X)+\frac{\epsilon}{2}) + n \frac{\epsilon}{2} \leq n(H(X)+\epsilon)\end{aligned}
This is the first half of a central result of source coding.
Theorem 25 (Source Coding Theorem)
(X_i)_{i=1}^n \sim p_X
are a sequence of independently and identically distributed random varibles, then for any
\epsilon > 0
n
sufficiently large, we can represent
(X_i)_{i=1}^n
using fewer than
n(H(X) + \epsilon)
bits. Conversely, we can not losslessly represent
(X_i)_{i=1}^n
nH(X)
This lends a new interpretation of the entropy
H(X)
: it is the average number of bits required to represent
X
Whereas source coding deals with encoding information, channel coding deals with transmitting it over a noisy channel. In general, we have a message
M
, and encoder, a channel, and a decoder as in Figure 1.
Figure 1: Channel Coding
Each channel can be described by a conditional probability distribution
p_{Y|X}(y|x)
for each time the channel is used.
For a channel described by
p_{Y|X}
, the capacity is given by
C = \max_{p_X} I(X; Y).
In words, the capacity describes the maximum mutual information between the channel input and output.
Suppose we use the channel
n
times to send a message that takes on average
H(m)
bits to encode, then the rate of the channel is
R = \frac{H(M)}{n}
Theorem 26 (Channel Coding Theorem)
For a channel decsribed by
p_{Y|X}
\epsilon>0
R < C
n
sufficiently large, there exists a rate
R
communication scheme that achieves a probability of error less than
\epsilon
R > C
, then the probability of error converges to 1 for any communication scheme.
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(→Using Gtk-bounce: this gray too? ogg paint good.)
(→Starting Gtk-bounce: gray all the boxes!)
* make the pipe fifos for the applications to communicate (only needs to be done once)
{\displaystyle \ x(t)={\begin{cases}1,&|t|<T_{1}\\0,&T_{1}<|t|\leq {1 \over 2}T\end{cases}}}
{\displaystyle {\begin{aligned}x_{\mathrm {square} }(t)={\frac {4}{\pi }}\sin(\omega t)+{\frac {4}{3\pi }}\sin(3\omega t)+{\frac {4}{5\pi }}\sin(5\omega t)+\\{\frac {4}{7\pi }}\sin(7\omega t)+{\frac {4}{9\pi }}\sin(9\omega t)+{\frac {4}{11\pi }}\sin(11\omega t)+\\{\frac {4}{13\pi }}\sin(13\omega t)+{\frac {4}{15\pi }}\sin(15\omega t)+{\frac {4}{17\pi }}\sin(17\omega t)+\\{\frac {4}{19\pi }}\sin(19\omega t)+{\frac {4}{21\pi }}\sin(21\omega t)+{\frac {4}{23\pi }}\sin(23\omega t)+\\{\frac {4}{25\pi }}\sin(25\omega t)+{\frac {4}{27\pi }}\sin(27\omega t)+{\frac {4}{29\pi }}\sin(29\omega t)+\\{\frac {4}{31\pi }}\sin(31\omega t)+{\frac {4}{33\pi }}\sin(33\omega t)+\cdots \end{aligned}}}
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BohmanWindow - Maple Help
Home : Support : Online Help : Science and Engineering : Signal Processing : Windowing Functions : BohmanWindow
multiply an array of samples by a Bohman windowing function
BohmanWindow(A)
The BohmanWindow(A) command multiplies the Array A by the Bohman windowing function and returns the result in an Array having the same length.
The Bohman windowing function
w\left(k\right)
N
w\left(k\right)=\left(1-|\frac{2k}{n}-1|\right)\mathrm{cos}\left(\mathrm{\pi }|\frac{2k}{n}-1|\right)+\frac{\mathrm{sin}\left(\mathrm{\pi }|\frac{2k}{n}-1|\right)}{\mathrm{\pi }}
The SignalProcessing[BohmanWindow] command is thread-safe as of Maple 18.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
N≔1024:
a≔\mathrm{GenerateUniform}\left(N,-1,1\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627814645342564}}
\mathrm{BohmanWindow}\left(a\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627814500941820}}
c≔\mathrm{Array}\left(1..N,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right):
\mathrm{BohmanWindow}\left(\mathrm{Array}\left(1..N,'\mathrm{fill}'=1,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right),'\mathrm{container}'=c\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627814500917964}}
u≔\mathrm{`~`}[\mathrm{log}]\left(\mathrm{FFT}\left(c\right)\right):
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{plots}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{listplot}\left(\mathrm{ℜ}\left(u\right)\right),\mathrm{listplot}\left(\mathrm{ℑ}\left(u\right)\right)\right]\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}
The SignalProcessing[BohmanWindow] command was introduced in Maple 18.
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Find the centroid of the region bounded by the given
Find the centroid of the region bounded by the given curves. y=x^{3},x+y=2,y=
Find the centroid of the region bounded by the given curves.
y={x}^{3},x+y=2,y=0
f\left(x\right)=2-x
g\left(x\right)={x}^{3}
x=2-y
x={y}^{\frac{1}{3}}
They intersect at (1,1)
Since integrating with respect to x would mean we need to do two separate integrals for everything (from
x=0
to l and from
x=1
to 2), we could alternatively integrate with respect to y, where
x=2
— y is the "top" function and
x={y}^{\frac{1}{3}}
is the "bottom",
Find the area of the bounded region.
A={\int }_{0}^{1}\left(\left(2-y\right)-{y}^{\frac{1}{3}}\right)dy
={\left[2y-\frac{1}{2}{y}^{2}-\frac{3}{4}{y}^{\frac{4}{3}}\right]}_{0}^{1}
=2-\frac{1}{2}-\frac{3}{4}-\left(0-0-0\right)
=\frac{8-2-3}{4}
=\frac{3}{4}
If we change all the x to y in the formula to find
\stackrel{―}{x}
, it will give the y coordinate of the centroid.
\stackrel{―}{y}=\frac{1}{\frac{3}{4}}{\int }_{0}^{1}y\left(\left(2-y\right)-{y}^{\frac{1}{3}}\right)dy=\frac{4}{3}{\int }_{0}^{1}\left(2y-{y}^{2}-{y}^{\frac{4}{3}}\right)dy
=\frac{4}{3}{\left[{y}^{2}-\frac{1}{3}{y}^{3}-\frac{3}{7}{y}^{\frac{7}{3}}\right]}_{0}^{1}
=\frac{4}{3}\left[1-\frac{1}{3}-\frac{3}{7}-\left(0-0-0\right)\right]
=\frac{4}{3}\left[\frac{21-7-9}{21}\right]
=\frac{4}{3}\left[\frac{5}{21}\right]
=\frac{20}{63}
Likewise, if we change all the x to y in the formula to find
\stackrel{―}{y}
, it will give the x coordinate of the centroid.
lalilulelo2k3eq
Say f(x) and g(x) are the two bounding functions over
\left[a,b\right]
The mass is
M={\int }_{a}^{b}f\left(x\right)-g\left(x\right)dx
We find the moments:
{M}_{x}=\frac{1}{2}{\int }_{a}^{b}\left({\left[f\left(x\right)\right]}^{2}-{\left[g\left(x\right)\right]}^{2}\right)dx
{M}_{y}={\int }_{a}^{b}x\left(f\left(x\right)-g\left(x\right)\right)dx
And the center of mass,
\left(\stackrel{―}{x},\stackrel{―}{y}\right)
\stackrel{―}{x}=\frac{{M}_{y}}{M}
\stackrel{―}{y}=\frac{{M}_{x}}{M}
f\left(x\right)=-3x+4
f\left(x\right)=-3{x}^{2}+7
f\left(x\right)=\frac{x+1}{x+2}
d\right)f\left(x\right)={x}^{5}+1
f\left(x\right)=\sqrt{1-x}
a=0
\sqrt{0.9}
\sqrt{0.99}
f\left(x\right)=\mathrm{ln}\left(1+2x\right),\text{ }a=3,\text{ }n=3,\text{ }2.6\le x\le 3.4
Approximate f by a Taylor polynomial with degree n at the number a. Use Taylor's Inequality to estimate the accuracy of the approximation
f\left(x\right)\approx {T}_{n}\left(x\right)
when x lies in the given interval. (Round your answer to six decimal places.)
|{R}_{3}\left(x\right)|\le 0.000001
degree 3 zeros at x=4,x=3,x=2, y intercepts at (0,24)
Find a function f such that
F=▽
F\left(x,y\right)={x}^{2}i+{y}^{2}j
C is the arc of the parabola
y=2{x}^{2}
from (-1,2) to (2,8)
|5+8i|
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QT interval - wikidoc
(Redirected from QT Interval)
Schematic representation of normal ECG trace (sinus rhythm), with waves, segments, and intervals labeled.
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Synonyms and keywords: QT; QTc; corrected QT interval; corrected QT
The QT interval is a measure of the time between the start of the Q wave and the end of the T wave in the heart's electrical cycle. The QT interval is a general measure of how long it takes for the heart to recharge or repolarize itself electrically. The QT interval is dependent on the heart rate such that the faster the heart rate, the shorter the QT interval. As a result, the QT interval must be adjusted for the heart rate for accurate interpretation. This adjustment is called the corrected QT interval or QTc. A lengthened QT interval is a biomarker for ventricular tachyarrhythmias such as torsades de pointes and is a risk factor for sudden cardiac death.
The standard clinical correction is to use Bazett's formula,[1] named after physiologist Henry Cuthbert Bazett, calculating the heartrate-corrected QT interval QTc.
{\displaystyle QTc={\frac {QT}{\sqrt {RR}}}}
where QTc is the QT interval corrected for rate, and RR is the interval from the onset of one QRS complex to the onset of the next QRS complex, measured in seconds. However, this formula tends to not be accurate, and over-corrects at high heart rates and under-corrects at low heart rates.
In the same year, Fridericia [2] published an alternative adjustment:
{\displaystyle QT_{F}={\frac {QT}{RR^{1/3}}}}
There are several other methods, but a regression based approach is the most accurate according to the current knowledge. An example of the regression-based approach is that developed by Sagie et al.,[3] as follows:
{\displaystyle QT_{S}=QT+0.154(1-RR)}
Normal values for the QT interval are between 0.30 and 0.44 (0.45 for women) seconds. QT interval can be measured by different methods such as the threshold method in which the end of the T wave is determined by the point at which the component of the T wave merges with the isoelectric baseline or the tangent method in which the end of the T wave is determined by the intersection of a line extrapolated from the isoelectric baseline and the tangent line which touches the terminal part of the T wave at the point of maximium downslope.[4]
Shown below is an example of normal QT interval.
The QT interval is an important ECG parameter and the identification of ECGs with long QT syndrome is of clinical importance. Considering the required standards for precision, the measurement of QT interval is subjective.[5] This is because the end of the T wave is not always clearly defined and usually merges gradually with the baseline. QT interval in an ECG complex can be measured manually by different methods such as the threshold method, in which the end of the T wave is determined by the point at which the component of the T wave merges with the isoelectric baseline or the tangent method, in which the end of the T wave is determined by the intersection of a line extrapolated from the isoelectric baseline and the tangent line, which touches the terminal part of the T wave at the point of maximum downslope.[4]
With the increased availability of digital ECGs with simultaneous 12-channel recording, QT measurement may also be done by the 'superimposed median beat' method. In the superimposed median beat method, a median ECG complex is constructed for each of the 12 leads. The 12 median beats are superimposed on each other and the QT interval is measured either from the earliest onset of the Q wave to the latest offset of the T wave or from the point of maximum convergence for the Q wave onset to the T wave offset.[6]
If the QT interval is abnormally prolonged or shortened, there is a risk of developing ventricular arrhythmias.
An abnormal prolonged QT interval could be due to Long QT syndrome, whereas an abnormal shortened QT interval could be due to Short QT syndrome.
Due to adverse drug reactions
Prolongation of the QT interval may be due to an adverse drug reaction.[7] Many drugs such as haloperidol[8] can prolong the QT interval.
Represents an excess time required for completion of ventricular depolarization and repolarization.
Abnormal when the QTc is > 0.44 seconds.
Shown below are examples of prolonged QY interval.
Shown below is an example of prolonged QT interval with bradycardia.
Shown below is an example of torsades de pointes
is associated with congenital deafness, syncope, and sudden death.
heterozygotes may be normal or have a slightly prolonged QT interval
incidence among deaf mute children is .25%
clinically similar to the Jervell and Lange-Nielsen syndrome except the hearing is normal
heterozygotes and homozygotes persons may have similar symptoms
Sporadic long QT syndrome
females:males = 2:1
57% had a history of syncope
there was a strong association between syncopal episodes and emotions, vigorous activities and loud noises.
imbalance between various components of the cardiac sympathetic innervation.
Treatment to shorten the QT syndrome:
left stellate ganglion block
right stellate ganglion stimulation
the administration of propranolol [9]
Shown below is an example of short QT interval.
Secondary (acquired) types of QT prolongation
Coronary artery disease: Ischemia, infarction
MVP, cardiomyopathy
CNS disease, especially hemorrhage
Autonomic nervous system dysfunction secondary to radical neck dissection, carotid endarterectomy, transabdominal truncal vagotomy.
Metabolic disturbances. Electrolyte imbalance (such as hypocalcemia), liquid protein diet, intracoronary injection of contrast agents.
Cardiac medications: Quinidine, PCA, disopyramide, encainide, flecainide, propafenone, amiodarone.
Psychotropic drugs. Phenothiazines, tricyclic antidepressants.
Miscellaneous. Severe bradycardia, high degree AV block, post Stokes-Adams attacks, hypothyroidism, hypothermia, pheochromocytoma, organophosphate poisoning.
↑ Bazett HC. (1920). "An analysis of the time-relations of electrocardiograms". Heart (7): 353–370.
↑ Fridericia LS (1920). "The duration of systole in the electrocardiogram of normal subjects and of patients with heart disease". Acta Medica Scandinavica (53): 469–486.
↑ Sagie A, Larson MG, Goldberg RJ, Bengston JR, Levy D (1992). "An improved method for adjusting the QT interval for heart rate (the Framingham Heart Study)". Am J Cardiol. 70 (7): 797–801. CS1 maint: Multiple names: authors list (link)
↑ 4.0 4.1 Panicker GK, Karnad DR, Joshi R, Kothari S, Narula D, Lokhandwala Y (2006). "Comparison of QT measurement by tangent method and threshold method". Indian Heart J (58): 487–88. CS1 maint: Multiple names: authors list (link)
↑ Panicker GK, Karnad DR, Joshi R, Shetty S, Vyas N, Kothari S, Narula D (2009). "Z-score for benchmarking reader competence in a central ECG laboratory". Ann Noninvasive Electrocardiol. 14 (14(1)): 19–25. doi:10.1111/j.1542-474X.2008.00269.x. PMID 19149789. CS1 maint: Multiple names: authors list (link)
↑ Salvi V, Karnad DR, Panicker GK, Natekar M, Hingorani P, Kerkar V, Ramasamy A, de Vries M, Zumbrunnen T, Kothari S, Narula D (2011). "Comparison of 5 methods of QT interval measurements on electrocardiograms from a thorough QT/QTc study: effect on assay sensitivity and categorical outliers". J Electrocardiol. 44 (44(2)): 96–104. doi:10.1016/j.jelectrocard.2010.11.010. PMID 21238976. CS1 maint: Multiple names: authors list (link)
↑ Andrew Leitch, Peter McGinness, and David Wallbridge, “Calculate the QT interval in patients taking drugs for dementia,” BMJ 335, no. 7619 (September 15, 2007): 557, http://www.bmj.com/cgi/content/short/335/7619/557 (accessed September 14, 2007) doi:10.1136/bmj.39020.710602.47
↑ "Information for Healthcare Professionals: Haloperidol (marketed as Haldol, Haldol Decanoate and Haldol Lactate)".
↑ Chou's Electrocardiography in Clinical Practice Third Edition, pp. 518-522.
de:Elektrokardiogramm he:אלקטרוקרדיוגרם lb:Elektrokardiogramm nl:Elektrocardiogram sv:EKG
Retrieved from "https://www.wikidoc.org/index.php?title=QT_interval&oldid=1041765"
This page was last edited 20:47, 17 November 2014 by wikidoc user Deepika Beereddy. Based on work by Raviteja Reddy Guddeti, Prashanth Saddala and charlesmichaelgibson@gmail.com and wikidoc users Rim Halaby and WikiBot.
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Simulation of X-Ray Shielding Effect of Different Materials Based on MCNP5
Fanying Zhang, Xiaogang Zhao, Junxin Zhang
Applied Nuclear Technology in Geosciences Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu, China.
Abstract: This article uses the Monte Carlo method and MCNP5 software to first simulate the X-ray energy spectrum of the tungsten target and the silver target. On this basis, using lead, tungsten and tungsten alloys (90% tungsten, 7.1% nickel, and iron 2.9%) as an X-ray shielding material, the shielding efficiency of these three materials at different thicknesses is calculated, and the results show that tungsten and tungsten alloy have better shielding effect than lead. For the X-rays of different energies generated by the tungsten target and the silver target, in order to achieve the same shielding effect, the X-rays generated by the tungsten target require a thicker shielding material.
Keywords: X-Ray Shielding, Tungsten Alloy, Monte Carlo Simulation
X-rays can be divided into bremsstrahlung and characteristic X-rays. When high-speed moving electrons collide with objects, energy conversion occurs. The movement of electrons is blocked and loses kinetic energy. Part of the energy is converted into X-rays, and the other part is converted into the heat energy increases the temperature of the object [1] . The X-ray generator simulated in this article is based on this principle. The shielding of X-rays usually uses substances with higher atomic number and higher density, and lead is often used as a shielding material to absorb X-rays. The advantage of lead is its low price and high attenuation coefficient for X photons and gamma photons; but lead has low hardness, poor impact resistance, high temperature resistance and toxicity, and when the energy of X-ray photons is 40 - 80 keV in between, the absorption capacity of lead to them is very weak, that is, the “weak absorption zone” of lead [2] . These characteristics make its application in certain occasions restricted. In contrast, as a high-density, high atomic number and non-toxic material, tungsten also has a good attenuation coefficient. Alloys with tungsten as the matrix and other elements, such as tungsten-nickel-iron alloys, can improve the overall properties of pure tungsten, such as strength and workability. Therefore, in practical applications, they are often used in the form of tungsten alloys [3] . In tungsten alloys, the mass content of tungsten is generally 85% to 99%, and elements such as Ni, Cu, Co, Mo, and Cr are added to form high specific gravity alloys, which can be divided into W-Ni-Fe, W-Ni-Cu, W-Co, W-Ag and other series.
Monte Carlo method, also known as random sampling or statistical experiment method, is mainly used to simulate some random systems that cannot be generated by numerical values. At present, Monte Carlo methods and software are widely used in nuclear physics, medicine, materials science, system reliability and other fields at home and abroad, solving many problems that cannot be solved by traditional numerical simulation methods [4] . Commonly used Monte Carlo simulation software includes MCNP5, GEANT4, FLUKE, EGS [5] , the MCNP5 used in this article. Monte Carlo simulation software MCNP5 can be used to simulate the transmission process of various particles such as protons, photons, electrons [6] . Based on MCNP5 simulation software, this paper simulates the X-ray energy spectrum of tungsten target and silver target. It mainly discusses the shielding effect of tungsten X-ray and silver X-ray when lead, tungsten, and tungsten nickel-iron alloy (tungsten 90%, nickel 7.1%, iron 2.9%) have different thicknesses.
2. MC Simulation of X-Ray
Select silver and tungsten as the target materials respectively, and select an appropriate target thickness when the emitted electron energy is 100 keV, and the X-ray energy spectra of silver and tungsten are simulated. The simplified MCNP5 model is shown in Figure 1.
The different thickness of the target material affects the count of the low-energy spectrum and the high-energy spectrum. As the target thickness increases, the count of high-energy bands increases first and then decreases. The
Figure 1. MCNP5 model of X-ray shielding.
count of the low-energy spectrum decreases with the increase of the target thickness [7] . This is because when the thickness of the target material is thinner, the probability of low-energy photons generated by Compton scattering through the target material increases, resulting in a higher peak at the low-energy end of the X-ray spectrum. When a certain thickness is reached, due to the thickness of the shielding material much greater than the mean free path of low-energy photons, most of these low-energy Compton scattered photons are absorbed.
When the K-layer electrons are excited to produce an electron vacancy, the electrons in the shells of higher energy levels such as L, M, and N may fill this electron vacancy. The X-rays emitted by filling the electron vacancies in the K shell are called K characteristic X-rays, and the X-rays emitted by filling the electron vacancies in the L shell are called L characteristic X-rays. The characteristic X-rays emitted when the electrons in the L layer and the M layer are filled into the K layer are represented by
{K}_{\alpha }
{K}_{\beta }
[8] . Taking into account the small energy difference between the sub-levels of electrons in the same orbit, the characteristic X-rays emitted when electrons of different sub-levels on the L layer are filled into the K layer can be divided into
{K}_{\alpha 1}
{K}_{\alpha 2}
. The characteristic X-rays emitted when electrons of different sub-energy levels on the M layer are filled into the K layer can be divided into
{K}_{\beta 1}
{K}_{\beta 2}
. In the same way, there are
{L}_{\alpha 1}
{L}_{\alpha 2}
{L}_{\beta 1}
{L}_{\beta 2}
for the characteristic X-rays of the L series. [9] [10] .
The X-ray spectrum consists of a bremsstrahlung spectrum and a characteristic X-ray spectrum. The characteristic X-ray spectrum is superimposed on the continuous bremsstrahlung spectrum [11] . The thickness of the tungsten target is selected as 0.1 mm in this paper, and its X-ray spectrum is shown in Figure 2. The X-ray spectrum of tungsten mainly has 5 characteristic X-ray peaks, the leftmost energy is 8.9 keV is a characteristic peak of L series; the peak is higher at energy 59.3 keV, 58.0 keV, here are the characteristic X-ray peaks
{K}_{\alpha 1}
{K}_{\alpha 2}
of the K series. And at the energies of 67.2 keV and 69.1 keV are the characteristic X-ray peaks
{K}_{\beta 1}
{K}_{\beta 2}
of the K series. The thickness of the Ag target is selected to be 0.002 mm, and its X-ray spectrum is shown in Figure 3.
Figure 2. X-ray spectrum of tungsten target.
Figure 3. X-ray spectrum of silver target.
There is a characteristic X-ray peak of the L series at 3 keV. The two characteristic X-ray peaks of the K series,
{K}_{\alpha 1}
{K}_{\alpha 2}
correspond to the energies
{E}_{{K}_{\alpha 1}}=22.2\text{\hspace{0.17em}}\text{keV}
{E}_{{K}_{\beta 1}}=24.9\text{\hspace{0.17em}}\text{keV}
, which represent the characteristic energy values of the K series characteristic X-rays of the silver element.
3. Research on the Shielding Performance of Lead, Tungsten and Tungsten Alloy
Shielding materials of different thicknesses were added behind the target to simulate the effects of lead, tungsten, and tungsten-nickel-iron alloy (90% tungsten, 7.1% nickel, and 2.9% iron) on the X-rays generated by the tungsten and silver targets Shielding effect. For the simulation results, the shielding rate is used to compare the shielding effect [12] . The count before the shielding material is I0 and the count after the shielding material is I. The expression for measuring its filtering ability is:
T=\left({I}_{0}-I\right)/{I}_{0}
For three different materials, the shielding effect when the material thickness is 0.1 mm to 0.8 mm is simulated. From Table 1, the following conclusions can be drawn: 1) the X-ray shielding rate of these three materials varies with the material As the thickness increases, 2) the X-ray shielding rate of tungsten and tungsten alloy is always higher than that of Pb, 3) for tungsten and tungsten alloy, the shielding rate of tungsten is between 0.1 mm and 0.6 mm. Higher than the shielding rate of tungsten alloy, at 0.7 mm and 0.8 mm, the shielding rate of tungsten alloy is slightly higher than that of tungsten.
The attenuation of the X-ray intensity is related to the attenuation coefficient of the absorbing material. The material with a large attenuation coefficient attenuates X-ray photons more than the material with a low attenuation coefficient [13] . In the range of 0 - 100 keV, the mass attenuation coefficient of lead is larger than that of tungsten. However, due to the higher density of tungsten, in general,
Table 1. The shielding effect of lead, tungsten and tungsten-nickel-iron alloy on tungsten X-rays and silver X-rays.
the minimum density of tungsten nickel-iron alloy can also reach 16.85 g/cm3. Therefore, the linear attenuation coefficient of tungsten is the range of 0 - 100 keV is almost all larger than lead. For tungsten and lead of the same thickness, the radiation shielding performance is definitely better than that of lead. For the same shielding effect, the shielding thickness is smaller when tungsten is used, that is, the shielding volume is the smallest, and the volume is larger when using lead.
Among the tungsten alloys selected in this paper, tungsten accounts for 90%, nickel accounts for 7.1%, and iron accounts for 2.9%. Compared with the shielding effects of tungsten and tungsten nickel-iron alloys, the shielding efficiency of tungsten is only slightly higher than that of tungsten alloy. This shows that 90% of tungsten in the tungsten-nickel-iron alloy plays a leading role in the shielding effect, and the shielding effect of tungsten alloy mainly depends on the mass percentage of tungsten.
The energy of X-rays produced by tungsten targets and silver targets is different. Tungsten X-rays are counted more frequently from 40 keV to 70 keV, and their
{k}_{\alpha 1}
characteristic X-ray energy reaches 59.3 keV, while silver X-ray counts are mainly concentrated in from 2 keV to 10 keV, the characteristic X-ray energy of
{k}_{\alpha 1}
is only 22.2 keV. For these two different energy X-ray energies, in order to achieve the same shielding effect, a thicker shielding material is required for the X-rays generated by the tungsten target.
This article uses MCNP5 simulation to compare the X-ray shielding effects of lead, tungsten and tungsten-nickel-iron alloys on tungsten and silver. Combined with actual applications, the following factors need to be considered when ensuring the shielding efficiency: 1) When selecting a shielding material, in addition to the shielding effect of the material, the strength and workability of the material must also be considered. Tungsten alloy has both good comprehensive performance and excellent shielding effect, which is a good X-ray shielding material. 2) When shielding X-rays, you need to consider the X-ray energy to select suitable materials. Higher-energy X-rays require thicker materials to shield. For lower-energy X-rays, it is not necessary to choose strong shielding ability, price expensive tungsten alloys are used for shielding, and the appropriate thickness of lead can also be selected to achieve a shielding rate similar to that of tungsten alloys. For example, for the X-ray generated by the silver target in this article, 0.4 mm lead can also achieve a shielding effect similar to tungsten and tungsten alloys.
In the actual X-ray protection, the radiation protection performance of the material, the mechanical performance of the material, the compression performance, the environmental protection characteristics and the price of the material should be comprehensively considered to select the shielding material reasonably.
Thanks to Xiaogang Zhao for his suggestions on the selection of X-ray shielding materials in this article and Junxin Zhang for his opinions on the processing of the relevant data in the charts in this article.
Cite this paper: Zhang, F. , Zhao, X. and Zhang, J. (2020) Simulation of X-Ray Shielding Effect of Different Materials Based on MCNP5. Open Access Library Journal, 7, 1-7. doi: 10.4236/oalib.1106727.
[1] Liao, L. and Qiu, X. (2010) Optimized Design of Shielding Material Component Content. Nuclear Electronics and Detection Technology, 30, 118-120.
[2] Zhu, Z. (2019) MC Simulation of Medical X-Rays and Optimization Design of Protective Materials. Nanjing University of Aeronautics and Astronautics, Nanjing.
[3] Wang, J. and Zou, S. (2011) Comparative Study on the Performance of Tungsten and Lead as γ-Ray Shielding Materials. Journal of Nanhua University, 25, 19-22.
[4] Du, C. and Xu, X. (2009) Monte Carlo Calculation of X-Ray Transmission Spectrum. The Second National Symposium on Nuclear Technology and Application Research, Mianyang, 1 May 2009, 599-602.
[5] Gu, R. (2016) EDXRF Analysis of the Best Detection Device for Heavy Metal Cd in Rice. Chengdu University of Technology, Chengdu.
[6] Xu, S. (2010) Application of Monte Carlo Method in Experimental Nuclear Physics. Atomic Energy Press, Beijing.
[7] Zhang, Q., Ge, L. and Yi, G. (2013) MC Simulation Analysis of the Influence of Transmission Type Micro X-Ray Tube Target Thickness on the Output Spectrum. Spectroscopy and Spectral Analysis, 33, 2232-2234.
[8] Qian, Y. (2009) Application of Monte Carlo Method in EDXRF Analysis. Chengdu University of Technology, Chengdu.
[9] Ji, A., Tao, G. and Zhuo, S. (2003) X-Ray Fluorescence Spectroscopy Analysis. Science Press, Beijing.
[10] Cao, L., Ding, Y. and Huang, Z. (1998) Energy Dispersive X-Ray Fluorescence Method. Chengdu University of Science and Technology Press, Chengdu.
[11] Yang, Q., Ge, L. and. Gu, Y. (2013) Theoretical Calculation of Target Thickness for Miniature X-Ray Tube and Simulation of Output Spectrum. Spectroscopy and Spectral Analysis, 4, 1130-1134.
[12] Zhu, Z., Luo, W. and Chai, F. (2019) MC Simulation of Medical Diagnostic X-Ray Field and Optimization Design of Composite Shielding Materials. Journal of East China University of Technology (Natural Science Edition), No. 2, 169-172.
[13] Chen, X., Wei, C. and Sun, J. (2019) Simulation Analysis and Experimental Verification of X-Ray Shielding Performance of Tungsten Alloy. Chinese Stereology and Image Analysis, 24, 9-15.
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Modulate using rectangular quadrature amplitude modulation - Simulink - MathWorks Switzerland
Constellation Size and Scaling
Plot Noisy 16-QAM Constellation
Modulate using rectangular quadrature amplitude modulation
The Rectangular QAM Modulator Baseband block modulates using M-ary quadrature amplitude modulation with a constellation on a rectangular lattice. The output is a baseband representation of the modulated signal. This block accepts a scalar or column vector input signal. For information about the data types each block port supports, see Supported Data Types.
When you set the Input type parameter to Integer, the block accepts integer values between 0 and M-1. M represents the M-ary number block parameter.
The Constellation ordering parameter indicates how the block assigns binary words to points of the signal constellation. Such assignments apply independently to the in-phase and quadrature components of the input:
If Constellation ordering is set to Binary, the block uses a natural binary-coded constellation.
If Constellation ordering is set to Gray and K is even, the block uses a Gray-coded constellation.
If Constellation ordering is set to Gray and K is odd, the block codes the constellation so that pairs of nearest points differ in one or two bits. The constellation is cross-shaped, and the schematic below indicates which pairs of points differ in two bits. The schematic uses M = 128, but suggests the general case.
For details about the Gray coding, see the reference page for the M-PSK Modulator Baseband block and the paper listed in References. Because the in-phase and quadrature components are assigned independently, the Gray and binary orderings coincide when M = 4.
The signal constellation has M points, where M is the M-ary number parameter. M must have the form 2K for some positive integer K. The block scales the signal constellation based on how you set the Normalization method parameter. The following table lists the possible scaling conditions.
Value of Normalization Method Parameter
Min. distance between symbols The nearest pair of points in the constellation is separated by the value of the Minimum distance parameter
Average Power The average power of the symbols in the constellation is the Average power parameter
Peak Power The maximum power of the symbols in the constellation is the Peak power parameter
The number of points in the signal constellation. It must have the form 2K for some positive integer K.
Indicates whether the input consists of integers or groups of bits.
Determines how the block maps each symbol to a group of output bits or integer.
Selecting User-defined displays the field Constellation mapping, which allows for user-specified mapping.
This parameter is a row or column vector of size M and must have unique integer values in the range [0, M-1]. The values must be of data type double.
The first element of this vector corresponds to the top-leftmost point of the constellation, with subsequent elements running down column-wise, from left to right. The last element corresponds to the bottom-rightmost point.
This field appears when User-defined is selected in the drop-down list Constellation ordering.
The rotation of the signal constellation, in radians.
For fixed-point output data types, specify the number of fractional bits, or bits to the right of the binary point. This parameter is only visible when you select Fixed-point or User-defined for the Output data type parameter and User-defined for the Set output fraction length to parameter.
Use the Open model button to open the doc_qam_mod model. The model generates a QAM signal, applies white noise, and displays the resulting constellation diagram.
Change the Eb/No of the AWGN Channel block from 20 dB to 10 dB. Observe the increase in the noise.
8-, 16-, 32-bit signed integers
8-, 16-, 32-bit unsigned integers
ufix\left({\mathrm{log}}_{2}M\right)
when Input type is Integer
Rectangular QAM Demodulator Baseband
[1] Smith, Joel G., “Odd-Bit Quadrature Amplitude-Shift Keying,” IEEE Transactions on Communications, Vol. COM-23, March 1975, 385–389.
The block does not support single or double data types for HDL code generation.
When Input Type is set to Bit, the block does not support HDL code generation for input types other than boolean or ufix1.
When the input type is set to Bit, but the block input is actually multibit (uint16, for example), the Rectangular QAM Modulator Baseband block does not support HDL code generation.
Rectangular QAM Demodulator Baseband | General QAM Modulator Baseband
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find the area of path running inside if the square field of 10m and width of path is 2 m - Maths - Mensuration - 6977737 | Meritnation.com
Each side of square field = 10 m
Then, area of the field =
{10}^{2}\quad =\quad 100\quad {\mathrm{m}}^{2}
Width of the path inside the field = 2 m
Then, each side of the field excluding path = 10 - 2 - 2 = 6 m
So, area of the field excluding path =
{6}^{2}\quad =\quad 36\quad {\mathrm{m}}^{2}
Therefore, area of the path = area of the field excluding path - area of the field =
100\quad {\mathrm{m}}^{2}-36\quad {\mathrm{m}}^{2}\quad =\quad 64\quad {\mathrm{m}}^{2}
Cost of levelling per square metre = Rs.2.50
So, total cost of levelling the path =
64\times 2.50\quad =\quad
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Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue
Binomial-Poisson entropic inequalities and the
M/M/\infty
This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/
\infty
queue. They describe in particular the exponential dissipation of
\Phi
-entropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/
\infty
queues. Proofs are elementary and rely essentially on the development of a “
\Phi
-calculus”.
Classification : 26D15, 46E99, 47D07, 60J27, 60J60, 60J75, 94A17
Mots clés : functional inequalities, Markov processes, entropy, birth and death processes, queues
title = {Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue},
TI - Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue
Chafaï, Djalil. Binomial-Poisson entropic inequalities and the $M/M/\infty $ queue. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 317-339. doi : 10.1051/ps:2006013. http://www.numdam.org/articles/10.1051/ps:2006013/
[1] C. Ané and M. Ledoux, On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573-602. | Zbl 0964.60063
[2] C. Ané, Clark-Ocone formulas and Poincaré inequalities on the discrete cube. Ann. Inst. H. Poincaré Probab. Statist. 37 (2001) 101-137. | Numdam | Zbl 0978.60084
[3] D. Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes. Lectures on probability theory (Saint-Flour, 1992), Lect. Notes Math. 1581 (1994) 1-114. | Zbl 0856.47026
[4] S. Boucheron, O. Bousquet, G. Lugosi and P. Massart, Moment inequalities for functions of independent random variables. Ann. Probab. 33 (2005) 514-560. | Zbl 1074.60018
[5] A.-S. Boudou, P. Caputo, P. Dai Pra and G. Posta, Spectral gap estimates for interacting particle systems via a Bochner type inequality. J. Funct. Anal. 232 (2006) 222-258. | Zbl 1087.60071
[6] S.G. Bobkov and M. Ledoux, On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 (1998) 347-365. | Zbl 0920.60002
[7] A.A. Borovkov, Limit laws for queueing processes in multichannel systems. Sibirsk. Mat. Ž. 8 (1967) 983-1004. | Zbl 0182.53401
[8] S. Bobkov and P. Tetali, Modified Log-Sobolev Inequalities in Discrete Settings, Preliminary version appeared in Proc. of the ACM STOC 2003, pp. 287-296. Cf. http://www.math.gatech.edu/~tetali/, 2003.
[9] P. Brémaud, Markov chains, Gibbs fields, Monte Carlo simulation, and queues. Texts Appl. Math. 31 (1999) xviii+444. | MR 1689633 | Zbl 0949.60009
[10] D. Chafaï and D. Concordet, A continuous stochastic maturation model, preprint arXiv math.PR/0412193 or CNRS HAL ccsd-00003498, 2004.
[11] D. Chafaï, Entropies, convexity, and functional inequalities: on
\Phi
-entropies and
\Phi
-Sobolev inequalities. J. Math. Kyoto Univ. 44 (2004) 325-363. | Zbl 1079.26009
[12] M.F. Chen, Variational formulas of Poincaré-type inequalities for birth-death processes. Acta Math. Sin. (Engl. Ser.) 19 (2003) 625-644. | Zbl 1040.60064
[13] P. Caputo and G. Posta, Entropy dissipation estimates in a Zero-Range dynamics, preprint arXiv math.PR/0405455, 2004. | MR 2322692 | Zbl 1126.60082
[14] P. Dai Pra and G. Posta, Logarithmic Sobolev inequality for zero-range dynamics: independence of the number of particles. Ann. Probab. 33 (2005) 2355-2401. | Zbl 1099.60068
[15] P. Dai Pra and G. Posta, Logarithmic Sobolev inequality for zero-range dynamics. Electron. J. Probab. 10 (2005) 525-576. | Zbl 1109.60080
[16] P. Dai Pra, A.M. Paganoni and G. Posta, Entropy inequalities for unbounded spin systems. Ann. Probab. 30 (2002), 1959-1976. | Zbl 1013.60076
[17] P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 (1996) 695-750. | Zbl 0867.60043
[18] S.N. Ethier and T.G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, 1986, Characterization and convergence. | MR 838085 | Zbl 0592.60049
[19] S. Goel, Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process. Appl. 114 (2004) 51-79. | Zbl 1074.60080
[20] O. Johnson and C. Goldschmidt, Preservation of log-concavity on summation, preprint arXiv math.PR/0502548, 2005. | MR 2219340
[21] A. Joulin, On local Poisson-type deviation inequalities for curved continuous time Markov chains, with applications to birth-death processes, personal communication, preprint 2006. | MR 2348750
[22] A. Joulin and N. Privault, Functional inequalities for discrete gradients and application to the geometric distribution. ESAIM Probab. Stat. 8 (2004) 87-101 (electronic). | Numdam
[23] S. Karlin and J. Mcgregor, Linear growth birth and death processes. J. Math. Mech. 7 (1958) 643-662. | Zbl 0091.13804
[24] F.P. Kelly, Blocking probabilities in large circuit-switched networks. Adv. in Appl. Probab. 18 (1986) 473-505. | Zbl 0597.60092
[25] F.P. Kelly, Loss networks. Ann. Appl. Probab. 1 (1991) 319-378. | Zbl 0743.60099
[26] C. Kipnis and C. Landim, Scaling limits of interacting particle systems. Fundamental Principles of Mathematical Sciences 320, Springer-Verlag, Berlin (1999). | MR 1707314 | Zbl 0927.60002
[27] R. Latała and K. Oleszkiewicz, Between Sobolev and Poincaré, Geometric aspects of functional analysis. Lect. Notes Math. 1745 (2000) 147-168. | Zbl 0986.60017
[28] P. Massart, Concentration inequalities and model selection, Lectures on probability theory and statistics (Saint-Flour, 2003), available on the author's web-site http://www.math.u-psud.fr/~massart/stf2003_massart.pdf.
[29] Y. Mao, Logarithmic Sobolev inequalities for birth-death process and diffusion process on the line. Chinese J. Appl. Probab. Statist. 18 (2002) 94-100.
[30] L. Miclo, An example of application of discrete Hardy's inequalities. Markov Process. Related Fields 5 (1999) 319-330. | Zbl 0942.60081
[31] Ph. Robert, Stochastic networks and queues, french ed., Applications of Mathematics (New York) 52, Springer-Verlag, Berlin, 2003, Stochastic Modelling and Applied Probability. | MR 1996883 | Zbl 1038.60091
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High School Calculus/The Fundamental Theorems of Calculus - Wikibooks, open books for an open world
High School Calculus/The Fundamental Theorems of Calculus
The Fundamental Theorems of Calculus[edit | edit source]
In order to understand the fundamental theorem of calculus we must first understand what an Antiderivative is.
An antiderivative of the function
{\displaystyle f(x)}
is any function, often denoted by
{\displaystyle F(x)}
{\displaystyle F^{\prime }(x)=f(x)}
{\displaystyle \int f(x)=F(x)+C}
Let's do some practice on this
{\displaystyle f(x)=x^{3}}
{\displaystyle F(x)={\frac {1}{4}}x^{4}+C}
The C stands for some constant. The reason for this is when you differentiate the stand alone constants become 0
When you differentiate this problem you will end up with
{\displaystyle x^{3}}
In general, the antiderivative of
{\displaystyle x^{k}}
{\displaystyle {\frac {1}{k}}*x^{k+1}}
{\displaystyle f(x)=x^{2}+3}
{\displaystyle \int x^{2}+3\operatorname {d} x}
{\displaystyle F(x)=\int x^{2}\operatorname {d} x+\int 3\operatorname {d} x}
{\displaystyle F(x)={\frac {1}{3}}x^{3}+3x+C}
When dealing with functions that have a plus or minus in them you can integrate the separately to help you out and really focus on what is going on. With enough practice you won't need to do this. Remember to keep the appropriate sign between the integrals.
{\displaystyle f(x)=85x^{7}}
{\displaystyle F(x)=\int 85x^{7}\operatorname {d} x}
{\displaystyle F(x)=85*\int x^{7}\operatorname {d} x}
{\displaystyle F(x)=85[{\frac {1}{8}}x^{8}]}
{\displaystyle F(x)={\frac {85}{8}}x^{8}+C}
What was done here was a constant multiplier was pulled out. When you have a common constant multiplier in a function, you can pull it out of the integral to make it easier to evaluate. Just don't forget to multiply it back in when you are done.
Retrieved from "https://en.wikibooks.org/w/index.php?title=High_School_Calculus/The_Fundamental_Theorems_of_Calculus&oldid=2608099"
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Mental Shortcuts - Basic | Brilliant Math & Science Wiki
Mental Shortcuts - Basic
Mathematics is much more than the tedious, boring memorization of formulas. A mathematician plays around with the equation, recognizes facts and principles seen previously, and builds up on what he/she already knows in order to tackle the great unknown. By simply playing, we are becoming a Mathematician.
You are never too young (or old) to start learning the fundamentals of being a mathematician. Even if all you have is basic knowledge of the order of operations, you can still brainstorm different ideas and suggestions for dealing with potentially nasty calculations. Play around with the numbers, move them about, and tweak them slightly. By identifying patterns, we are able to create new methods to solve problems that have never been seen before!
If you are adding and subtracting many numbers, find a nice way to pair them up.
If you are multiplying and dividing many numbers, look for common factors.
Apply the distributive property in creative ways.
When you learn a new formula, think about why it works and explain it to someone else.
100-36
by hand, we have to do a lot of "borrowing." It would be much easier to subtract from
99
instead. Which of the following is equivalent to
100-36
A) \, 99 - 37 \quad B) \, 99 - 36 \quad C) \, 99 - 35 \quad D) \, 99 + 36
If you are subtracting numbers, find a nice way to pair them up:
100 = 1+ 99
, we can rewrite the problem as
1+ 99-36= 99-36+1 = 99-35
. This removed the need for pesky borrowing!
Hence, the answer is
C) \, 99 - 35
Back to Quiz: Mental Shortcuts - Basic
Order of Operations: Basic arithmetic skills that shouldn't be forgotten. Doing this quickly will speed up your thought processes on harder problems.
Fractions, Decimals and Percentages: Relating these concepts with each other allows you to gain a deeper understanding of each of them. Drawing linkages between "different" areas is at the heart of mathematics.
Angles: An introduction to geometry, that helps you recognize patterns in shapes.
Cite as: Mental Shortcuts - Basic. Brilliant.org. Retrieved from https://brilliant.org/wiki/crafty-calculation-basic/
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Has the recent drop in airplane passengers resulted in better on-time performanc
Has the recent drop in airplane passengers resulted in better on-time performance? Before the recent downturn one airline bragged that 92% of its flig
Has the recent drop in airplane passengers resulted in better on-time performance? Before the recent downturn one airline bragged that 92% of its flights were on time. A random sample of 165 flights com- pleted this year reveals that 153 were on time. Can we conclude at the 5% significance level that the airline’s on-time performance has improved?
Here we have follwoing information:
n=165,\stackrel{^}{p}=\frac{153}{165}=0.927
Hypotheses are:
{H}_{0}:p=0.92
{H}_{a}:p>0.92
Standard deviation of the proportion is:
\sigma =\sqrt{\frac{p\left(1-p\right)}{n}}=\sqrt{\frac{0.92\left(1-0.92\right)}{165}}=0.0211
Test statistics will be:
z=\frac{\stackrel{^}{p}-p}{\sigma }=\frac{0.927-0.92}{0.0211}=0.33
Alternative hypothesis shows that the test is right tailed so p-value of the test is
P(z > 0.33) = 0.3707
Since p-value of the test is greater than 0.05 so we fail to reject the null hypothesis at 0.05 level of significance. So based on this sample, there is no support for the claim that the airline’s on-time performance has improved.
\theta
30.0m/s
200.0m
state how many linear factors each polynomial has: My school is literally screwing me over please frickin help f(x)= (x^16) +72 (x^6) + x = 0
f\left(x\right)=\left\{\begin{array}{ll}{x}^{2}-3x& if\text{ }x<2\\ 2x+4& if\text{ }x\ge 2\end{array}
on the open interval
0<x<2
and on the closed interval
0\le x\le 2
A monoprotic acid, HA, is dissolved in water: The equilibrium concentrations of the reactants and products are [HA] = 0.160 M [H] = 4.00
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Logarithmic Inequalities - Problem Solving Practice Problems Online | Brilliant
x
that satisfies the above logarithmic inequality?
25 \leq x \leq 625
0 \leq x \leq 25
0 < x < 625
25 < x \leq 625
The sound intensity level (measured in decibels) is given by the formula:
L_I=10\log\left(\frac{I}{I_0}\right)\text{ (dB)},
I
denotes sound intensity and
I_0
is the reference sound intensity.
An individual is diagnosed as moderate hearing loss when his/her threshold of sound perception is greater than 40 dB and less than or equal to 55 dB. Given that
I_0=1\text{ pW/m}^2,
what is the range of the threshold of sound intensity
x
that a person with moderate hearing loss can perceive?
- The sound intensity level of a normal conversation is about 50~60 dB, and that of a quiet library is about 40 dB.
- Assume that
\sqrt{10}\approx3.
0.01\ \mu\text{W/m}^2<x\le0.3\ \mu\text{W/m}^2
1\ \text{mW/m}^2<x\le9\ \text{mW/m}^2
0.01\ \text{mW/m}^2<x\le0.3\ \text{mW/m}^2
1\ \mu\text{W/m}^2<x\le9\ \mu\text{W/m}^2
A, B
C
A \geq 2, B \geq 4, C \geq 8
. What is the minimum integer value of
\log_A (BC) + \log_B (CA) + \log_C (AB) ?
The minimum integer value of a set, is the smallest integer that appears in the set. As an explicit example, the minimum integer value of the set of numbers
x
x \geq 12.3
If one root of the following quadratic equation
x^2-2x\log_{9}a+4\log_{9}a=0
is positive and the other root is negative, what is the range of
a?
\frac{1}{9} < a < 1
0 < a < 1
\frac{1}{9} < a < 9
1 < a < 2
x
{4}^4 {7}^4 x^3 > x^{\log_{28}x}
\alpha < x < \beta,
\alpha\beta ?
\frac{1}{{28}^3}
{28}^2
\frac{1}{{28}^2}
{28}^3
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Sparse Table | Brilliant Math & Science Wiki
Christopher Boo and k tharun contributed
Sparse Table is a data structure that answers static Range Minimum Query (RMQ). It is recognized for its relatively fast query and short implementation compared to other data structures.
Analysis of Time and Memory Complexity
The main idea of Sparse Table is to precompute
RMQ [j,j+2^i)
(i,j)
Build a Sparse Table on array 3 1 5 3 4 7 6 1
(i,j)
0\leq i < 4
0\leq j < 8
, stores
RMQ [j,j+2^i)
To answer
RMQ [l,r)
, we can select two pre-computed data with one starts from
l
and the other one ends at
r
such that their combined interval covers interval
[l,r)
There always exist a pair of precomputed interval such that they cover any range
[l,r)
p
be the largest integer such that
l+2^p \leq r
. Then, let the first and second interval be
[l,l+2^p)
[r-2^p,r)
respectively. If the two intervals don't overlaps, this gives us
r-2^p > l+2^p \rightarrow l+2^{p+1} < r
and leads to a contradiction to our initial assumption that
p
is the largest integer such that
l+2^p\leq r
The implementation for Sparse Table can be done with simple dynamic programming approach.
The first row is a copy of the initial array. From the second row onward, we can avoid recalculations by optimally picking two green cells from the previous row to get the desired interval. For example, interval
[l,l+2^k)
can be achieved by combining intervals
[l,l+2^{k-1})
[l+2^{k-1},l+2^k)
void build(int A[MAXN], int ST[LOGN][MAXN], int n) {
int h = floor(log2(n));
for (int j = 0; j < n; j++) ST[0][j] = A[j];
// iterative dynamic programming approach
for (int j = 0; j + (1<<i) <= n; j++)
ST[i][j] = min(ST[i-1][j], ST[i-1][j + (1<<(i-1))]);
As proven in the previous section, there always exist a pair of precomputed interval in the same row to achieve our desired interval. The trickier part is to find the value
p
. A clever method is to observe that
2^p \leq r-l
and look at the binary representation of
r-l
p
is the position of the most significant bit. For example, 10 returns 1, 1011 returns 3, 11111 returns 4 and 1 returns 0. Fortunately, in C++ there's a built-in function __builtin_clz() that returns the number of leading 0's until the first 1 bit. Since there are a total of 32 bits for a C++ int data type, the desired answer is 31-__builtin_clz(r-l).
int query(int l, int r) { // query in range [l,r)
int p = 31 - __builtin_clz(r-l);
return min(ST[p][l], ST[p][r-(1<<p)]);
In a macro level, since there are
n
\lg n
rows and each cell takes
O(1)
time to compute, the overall complexity is
O(n\lg n)
Similarly, memory complexity is also
O(n\lg n)
We only need two cells for any pairs
(l,r)
, hence complexity is
O(1)
Cite as: Sparse Table. Brilliant.org. Retrieved from https://brilliant.org/wiki/sparse-table/
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(link to source internally)
(→Cairo Animations: fix trac link)
*Trac is a convenient way to browse the source without checking out a copy:
{\displaystyle \ squarewave(t)={\begin{cases}1,&|t|<T_{1}\\0,&T_{1}<|t|\leq {1 \over 2}T\end{cases}}}
{\displaystyle {\begin{aligned}\ squarewave(t)={\frac {4}{\pi }}\sin(\omega t)+{\frac {4}{3\pi }}\sin(3\omega t)+{\frac {4}{5\pi }}\sin(5\omega t)+\\{\frac {4}{7\pi }}\sin(7\omega t)+{\frac {4}{9\pi }}\sin(9\omega t)+{\frac {4}{11\pi }}\sin(11\omega t)+\\{\frac {4}{13\pi }}\sin(13\omega t)+{\frac {4}{15\pi }}\sin(15\omega t)+{\frac {4}{17\pi }}\sin(17\omega t)+\\{\frac {4}{19\pi }}\sin(19\omega t)+{\frac {4}{21\pi }}\sin(21\omega t)+{\frac {4}{23\pi }}\sin(23\omega t)+\\{\frac {4}{25\pi }}\sin(25\omega t)+{\frac {4}{27\pi }}\sin(27\omega t)+{\frac {4}{29\pi }}\sin(29\omega t)+\\{\frac {4}{31\pi }}\sin(31\omega t)+{\frac {4}{33\pi }}\sin(33\omega t)+\cdots \end{aligned}}}
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Ask Answer - Playing with Numbers - Recently Asked Questions for School Students
2016\left[\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+...........+\frac{1}{2015×2016}\right]=
2{n}^{2}+11
{n}^{2}-n+41
\ge
Find the value of A, B and C in :\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}} 8 A\phantom{\rule{0ex}{0ex}}+8 B\phantom{\rule{0ex}{0ex}}_________\phantom{\rule{0ex}{0ex}}C B 3\phantom{\rule{0ex}{0ex}}_________
(Q) While writing the amount in the Bank cheque for withdrawal , Ryan by mistake wrote the last two digits in place of the first two digits and first two digits in place of last two digits.
Q) While writing the amount in the Bank Cheque for withdrawal, Ryan by mistake wrote the last two digits in place of the first two digits and first two digits in place of last two digits. While he was coming back from the bank after withdrawing, he bought a 5 rupees packet with those withdrawn money. Then .. after that, He saw and realised that "the withdrawal money that was left after buying the packet" """is double of the actual money or formal money that he should write in the cheque""".. Now question is that, how much money did he withdraw?
Sanjari Kalantari ma'am, I think that the solution you provided me before some time ,is incomplete... So, please recheck ... The solution is not providing a final value or answer. Please re solve.
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Write the point-slope form of the line satisfying the given
Write the point-slope form of the line satisfying the given conditions. Then us
Given data is :
Slope m=6
\left({x}_{1},{y}_{1}\right)=\left(-5,4\right)
The equation of line in point slope form is ,
y-{y}_{1}=m\left(x-{x}_{1}\right)
Substitute the value of m and
\left({x}_{1},{y}_{1}\right)
y-4=6[x-(-5)]
Thus , the equation of line in point slope form is y−4=6(x+5)
Equation of line in slope intercept form from point slope form
y=6x+30+4
y=6x+34...(2)
Standard form of slope intercept form of the equation is,
From standard equation of slope intercept form its clear that equation (2) represent slope intercept form.
Thus, Equation of line in slope intercept form is y=6x+34
A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]
\stackrel{―}{x}=
8-\frac{5}{x}=2+\frac{3}{x}
A charter company will provide a plane for a fare of $60 each for 20 or fewer passengers. For each passenger in excess of 20, the fare is decreased $2 per person for everyone. What number of passengers will produce the greatest revenue for the company?
Solve for m in the equation 2m=1.
Determine whether the given problem is an equation or an expression. If it is an equation, then solve. If it is an expression, then simplify. h/2- h/3 + h/6 = 1
Solve for v: 5-4v=0.
{x}^{2}+4x+20=0
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Revision as of 01:54, 26 February 2013 by Gmaxwell (talk | contribs) (→Bandlimitation and timing: moar terms, prose)
{\displaystyle \ x(t)={\begin{cases}1,&|t|<T_{1}\\0,&T_{1}<|t|\leq {1 \over 2}T\end{cases}}}
{\displaystyle {\begin{aligned}x_{\mathrm {square} }(t)={\frac {4}{\pi }}\sin(\omega t)+{\frac {4}{3\pi }}\sin(3\omega t)+{\frac {4}{5\pi }}\sin(5\omega t)+\\{\frac {4}{7\pi }}\sin(7\omega t)+{\frac {4}{9\pi }}\sin(9\omega t)+{\frac {4}{11\pi }}\sin(11\omega t)+\\{\frac {4}{13\pi }}\sin(13\omega t)+{\frac {4}{15\pi }}\sin(15\omega t)+{\frac {4}{17\pi }}\sin(17\omega t)+\\{\frac {4}{19\pi }}\sin(19\omega t)+{\frac {4}{21\pi }}\sin(21\omega t)+{\frac {4}{23\pi }}\sin(23\omega t)+\\{\frac {4}{25\pi }}\sin(25\omega t)+{\frac {4}{27\pi }}\sin(27\omega t)+{\frac {4}{29\pi }}\sin(29\omega t)+\\{\frac {4}{31\pi }}\sin(31\omega t)+{\frac {4}{33\pi }}\sin(33\omega t)+\cdots \end{aligned}}}
As stated in the Epilogue, everything that appears in the video demos is driven by open source software, which means the source is both available for inspection and freely usable by the community. The Thinkpad that appears in the video was running Fedora 17 and Gnome Shell (Gnome 3). The demonstration software does not require Fedora specifically, but it does require Gnu/Linux to run in its current form. In all, the video involved about 50,000 lines of new and custom-purpose code (including contributions to non-Xiph projects such as Cinelerra and Gromit).
The application is somewhat hardwired for specific demo purposes, but most of the hardwired settings can be found at the top of each source file. As found in SVN, the application expects an ALSA hardware audio device at hw:1, and if none if found, it will wait for one to appear. Once a sound device is successfully initialized, it expects to find and open two pipes named pipe0 and pipe1 for output in the current directory. In the video, the waveform and spectrum applications are started to take input from pipe0 and pipe1 respectively. The output sent to the two pipes is identical, and in most demos matches the output data sent to the hardware device for conversion to analog. The only exception is the tenth demo panel (which does not appear in the video) where gtk-bounce can be set to monitor the hardware inputs instead while the outputs are used to produce test waveforms.
Gtk-bounce consists of ten pushbutton panels that can be selected by scrolling up and dwon with the arrow buttons on the right side. Each panel is intended for a specific demo or part of a demo.
Panel 1: This panel presents buttons that allow the sound card to be configured in several sampling rates and bit depths. Sampled read from the audio inputs are sent to the output pipes and audio outputs for playback without modification.
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m (→Calculate Planetary Reflectance: some links added)
Tyipically, multispectral data are converted into ''Reflectance'' before they are subjected in multispectral analysis techniques (image interpretation, band arithmetic, vegetation indices, matrix transformations, etc.). As with remotely sensed data acquired by other sensors, IKONOS raw image digital numbers (DNs) can be converted to physical quantities such as at-sensor Radiance or Reflectance. The latter can be differentiated in ''Top of Atmosphere Reflectance'' (ToAR) which does not account for atmospheric effects (absorption or scattering) and in ''Top of Canopy Reflectance'' (ToCR) which introduces a "correction" for atmospheric effects.
<math>\rho_p = \pi * L\lambda * d^2 / ESUN\lambda * cos(θ_S)</math>
* <math>cos(θ_s)</math> - Solar zenith angle, from the image acquisition's metadata
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
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The incremental rate of return additional investment in alternative B.
Write the expression to calculate the additional investment for alternate B.
0=-\left(I2-I1\right){\left(1+i\right)}^{n}+\left(Y-X\right)\left(\frac{{\left(1+i\right)}^{n}-1}{i}\right)+\left(S2-S1\right)
Rewrite Equation (I).
0=-\left(I2-I1\right)\left(\frac{F}{P},i,6\right)+\left(Y-X\right)\left(\frac{F}{A},i,6\right)+\left(S2-S1\right)
0=-\left(I2-I1\right)\left(\frac{F}{P},i,6\right)+\left(Y-X\right)\left(\frac{F}{A},i,6\right)+\left(S2-S1\right)
f\left(x\right)=2{x}^{3}\text{ }-\text{ }7{x}^{2}\text{ }+\text{ }2x\text{ }+\text{ }3
given that 3 is a zero.
Radicals and Exponents Simplify the expression.
\frac{{x}^{4}{\left(3x\right)}^{2}}{{x}^{3}}
\frac{\sqrt{xy}}{\sqrt[4]{16xy}}
\frac{Radical\text{ }expression\text{ }Exponential\text{ }expression}{\sqrt[5]{{5}^{3}}}
Solve the compound interest formula for the interest rate r using the properties of rational exponents. then use the obtained formula to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of $10,000 and was worth $14,373.53 after 11 years.
Radicals and Exponents Evaluate each expression:
2\sqrt{3}\left\{81\right\}
\frac{\sqrt{18}}{\sqrt{25}}
\sqrt{\frac{12}{49}}
\frac{1}{\sqrt[5]{{x}^{3}}}
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Pumping lemma for context free languages.
chomsky normal form.
The pumping lemma is used as a way to prove if a language is context free or not.
There are two pumping lemmas one for regular languages and another for context free languages.
Here we discuss the latter, it states that for any CFL, it is possible to find two substrings that can be pumped any number of times and still turn out to be in the same language.
Theorem Let L be a context-free language, then there exists an integer p ≥ 1 referred to as the pumping length such that the following holds:
Every string s in L with |s| ≥ p can be written as s = uvxyz such that:
|vy| ≥ 1, that is, v and y are not both empty.
{\mathrm{uv}}^{\mathrm{i}}{\mathrm{xy}}^{\mathrm{i}}\mathrm{z}
∈ L, for all i ≥ 0.
Proof: To prove the lemma we use the below resulting lemma about parse trees.
Lemma: 1 Let G be a context-free grammar in chomsky normal form, and s be a non-empty string in L(G), T a parse tree for s and l the height of T, that is, l is the total number of edges on the longest root-to-leaf path in T. Then |s| ≤
{2}^{\mathrm{l}-1}
We prove this claim by induction on l by looking at its small values and using the fact that G is in chomsky normal form.
Now to start with proof of the pumping lemma. Let L be a context-free language and Σ and alphabet of L. By theorem 1 - (prerequisite article), there exists a context-free grammar in chomsky normal form G = (V,Σ, R, S) such that L = L(G).
We define r as the number of variables of G and p =
{2}^{\mathrm{r}}
. We will prove p's value can be used as the pumping length.
Consider an arbitrary string s in L such that |s| ≥ p and let T be a parse tree for s and l the height of T. By lemma 1 we have
|s| ≤
{2}^{\mathrm{l}-1}
, on the other hand we have |s| ≥ p =
{2}^{\mathrm{r}}
We combine these inequalities and we have
{2}^{\mathrm{r}}
{2}^{\mathrm{l}-1}
, that can also be written as l ≥ r + 1
Now consider the nodes on the longest path from root to leaf in tree T. This path has l edges and l + 1 nodes. The first l nodes store variables denoted by
{\mathrm{A}}_{0}
{\mathrm{A}}_{1}
{\mathrm{A}}_{l-1}
{\mathrm{A}}_{0}
= S and the last leaf node denoted by a stores a terminal.
Since l − 1 − r ≥ 0, the sequence
{\mathrm{A}}_{\mathrm{l-1-r}}
{\mathrm{A}}_{\mathrm{l-1}}
{\mathrm{A}}_{\mathrm{l-1}}
of variables is well defined and consists of r + 1 variables and since the number of variables in grammar G is equal to r, using the pigeon-hole principle it implies that there is a variable that occurs at least twice in this sequence, that is, there are indices j an k such that ℓ − 1 − r ≤ j < k ≤ l - 1 and
{\mathrm{A}}_{\mathrm{j}}
{\mathrm{A}}_{\mathrm{k}}
Recall T is a parse tree for string s, and thus terminals stored at the leaves of T ordered from left to right will form the string.
As we can see from the image above, nodes storing variables
{\mathrm{A}}_{\mathrm{j}}
{\mathrm{A}}_{\mathrm{k}}
divide s into five substrings, these are u, v, x, y, z such that s = uvxyz
Now we have to prove that the properties as stated in the pumping lemma hold.
For this we start with the third property, that is, proof that,
{\mathrm{uv}}^{\mathrm{i}}{\mathrm{xy}}^{\mathrm{i}}\mathrm{z}
In grammar G we have (1). S
\stackrel{*}{⇒}{\mathrm{uA}}_{\mathrm{jz}}
{\mathrm{A}}_{\mathrm{j}}\stackrel{*}{⇒}{\mathrm{uA}}_{\mathrm{ky}}
{\mathrm{A}}_{\mathrm{k}}
{\mathrm{A}}_{\mathrm{j}}
we have (2).
{\mathrm{A}}_{\mathrm{j}}
\stackrel{*}{⇒}
{\mathrm{uA}}_{\mathrm{jy}}
And since,
{\mathrm{A}}_{\mathrm{k}}
\stackrel{*}{⇒}
x and
{\mathrm{A}}_{\mathrm{k}}
{\mathrm{A}}_{\mathrm{j}}
{\mathrm{A}}_{\mathrm{j}}
\stackrel{*}{⇒}
From (1) and (3), it follows that
\stackrel{*}{⇒}
{\mathrm{uA}}_{\mathrm{jz}}
\stackrel{*}{⇒}
uxz. The above implies that string uxz is in language L.
In general for each i ≥ 0, string
{\mathrm{uv}}^{\mathrm{i}}
{\mathrm{xy}}^{\mathrm{i}}
z is in language L since
\stackrel{*}{⇒}
{\mathrm{uA}}_{\mathrm{jz}}
\stackrel{*}{⇒}
{\mathrm{uv}}^{\mathrm{i}}
{\mathrm{A}}_{\mathrm{j}}{\mathrm{y}}^{\mathrm{i}}\mathrm{z}
\stackrel{*}{⇒}
{\mathrm{uv}}^{\mathrm{i}}
{\mathrm{xy}}^{\mathrm{i}}
With the above we have proved that the third property of the pumping lemma holds.
The next step is to prove the second property - (|vxy| ≤ p) also holds.
For this we consider a subtree rooted at a node storing
{\mathrm{A}}_{\mathrm{j}}
and the path from this node to a leaf storing a terminal a is the longest path in this subtree.
Moreover, this path consists of l - j edges.
{\mathrm{A}}_{\mathrm{j}}\stackrel{*}{⇒}\mathrm{uxy}
therefore this subtree is a parse tree for string uxy, where
{\mathrm{A}}_{\mathrm{j}}
is the start variable.
By lemma 1, we conclude |vxy| ≤
{\mathrm{2}}^{\mathrm{l-j-1}}
We know l − 1 − r ≤ j which is also l − j − 1 ≤ r, therefore |vxy| ≤
{\mathrm{2}}^{\mathrm{l-j-1}}
{2}^{\mathrm{r}}
= p.
We show that the first property in the pumping lemma holds by proving |vy| ≥ 1
{\mathrm{A}}_{\mathrm{j}}
\stackrel{*}{⇒}
{\mathrm{vA}}_{\mathrm{ky}}
Let the first rule for this derivation be
{\mathrm{A}}_{\mathrm{j}}
→ BC, then
{\mathrm{A}}_{\mathrm{j}}
\stackrel{*}{⇒}
{\mathrm{uA}}_{\mathrm{ky}}
Now note BC is a string of length two. Also by applying the rules of a grammar in chomsky normal form, strings cannot be shorter and therefore we have |
{\mathrm{uA}}_{\mathrm{ky}}
| ≥ 2 which implies |vy| ≥ 1 and this completes the pumping lemma proof.
The pumping lemma proves if a language is context-free or not.
There exists two pumping lemmas one for regular languages where if a language is regular it will always satisfy the lemma and the other for context free languages which we have discussed here.
A compiler is a program that translates source code into machine code that can be executed by a computer. In this article we explore various types of compilers.
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Study on the Impact of the Panama Canal Expansion on the Distribution of Container Liner Routes ()
College of Transport & Communications, Shanghai Maritime University, Shanghai, China.
Fan, H.X. and Gu, W.H. (2019) Study on the Impact of the Panama Canal Expansion on the Distribution of Container Liner Routes. Journal of Transportation Technologies, 9, 204-214. doi: 10.4236/jtts.2019.92013.
\{\begin{array}{l}\mathrm{max}R={\displaystyle {\sum }_{z\in Z}{M}_{zout}{q}_{zown}}+{\displaystyle \sum {f}_{l}{R}_{l}}-{\displaystyle {\sum }_{z\in Z}\left({p}_{1z}{R}_{1}+2{p}_{2z}{q}_{z2}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\displaystyle {\sum }_{l\in L}{\displaystyle {\sum }_{z\in Z}\left({M}_{zout}{Q}_{ozown}+{c}_{zl}{T}_{zl}+{M}_{zin}{Q}_{zin}\right)}}\\ \frac{{R}_{l}}{{\displaystyle {\sum }_{z\in Z}{T}_{zl}\times {D}_{z}}}\ge \rho ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall l\in L\\ {\displaystyle {\sum }_{z\in Z}{q}_{zown}}\le {Q}_{own},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall z\in Z\\ {\displaystyle {\sum }_{z\in Z}{q}_{zin}}\le {Q}_{in},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall z\in Z\\ {T}_{zl}\ge {M}_{l},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall l\in L\\ {T}_{zl}\le \left({Q}_{own}+{Q}_{in}\right)\times \left[\left(365-{\tau }_{z}\right)/{t}_{zl}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall z\in Z,\text{\hspace{0.17em}}\forall l\in L\end{array}
{\sum }_{z\in Z}{M}_{zout}{q}_{zown}
\sum {f}_{l}{R}_{l}
{\sum }_{z\in Z}\left({p}_{1z}{R}_{1}+2{p}_{2z}{q}_{z2}\right)
{\sum }_{l\in L}{\sum }_{z\in Z}{M}_{zout}{Q}_{ozown}
{\sum }_{l\in L}{\sum }_{z\in Z}{c}_{zl}{T}_{zl}
{\sum }_{l\in L}{\sum }_{z\in Z}{M}_{zin}{Q}_{zin}
[1] Zeng, Q.C., Wu, K. and Sun, X.J. (2016) The Impact of Panama Canal Expansion on the Choice of Container Transportation Paths between China and the United States. China Navigation, 39, 109-113.
[2] Montero Llacer, F.J. (2004) Panamanian Maritime Sector Management. Marine Policy, 28, 283-295.
[3] Montero Llacer, F.J. (2005) Panama Canal Mangement. Marine Policy, 29, 25-37.
[4] Montero Llacer, F.J. (2005) Panama Canal: Operations and Traffic. Marine Policy, 29, 223-234.
[5] Montero Llacer, F.J. (2006) Port Privatization in Panama. Marine Policy, 30, 483-495.
[6] Benfield, S.L., Guzman, H.M. and Mair, J.M. (2005) Temporal Mangrove dynamics in Relation to Coastal Development in Pacific Panama. Journal of Environmental Management, 76, 263-276.
[7] Graham, N.E., Georgakakos, K.P., Vargas, C. and Echevers, M. (2006) Simulating the Value of E1 Nino Forecasts for the Panama Canal Advances in Water Resources, 19, 1665-1677.
[8] Chen, J.H., Cao, Y., Liang, X.F., et al. (2013) The Impact of the Expansion of the Panama Canal on the International Container Shipping Pattern. China Navigation, 1, 73-76.
[9] Wen, R.M. (2007) The Expansion of the Panama Canal Has a Profound Impact on Global Trade. World News.
[10] Cai, M.J. (2008) Expansion of the Panama Canal and Its Impact on World Shipping. China Ocean Shipping, 2, 69-71.
[11] Liu, C.J. and Jin, X.F. (2009) Expansion of the Panama Canal and the Prospect of Transportation. Tianjin Navigation, 2, 49-51.
[12] Yin, Y. and Jiang, N.X. (2007) The Expansion of the Panama Canal Is in the East Port of Hong Kong. The port Economy, 5, 54-55.
[13] Xie, X.L. (2009) Ship Transportation Management and Management. Dalian Maritime University Press, Dalian.
[14] Yang, H.L. and Zhong, M. (1996) Research on Optimization Decision-Making of Container Liner Shipping. Journal of Dalian Maritime University, 22, 51-54.
[15] Yu, S.M. (1997) Mathematical Optimization Method for Ships on Liner Routes. Traffic and Computer, 15, 44-45.
[16] Jiang, J.W. (2009) The Impact of the Financial Crisis on the Expansion of the Panama Canal. Water Transport Management, 6, 38-39.
[17] Li, Y. (2012) New Panamax Container Ship Design Defect. China Ship Inspection, 5, 69-69.
[18] Chen, S.L. (2011) The Scale Line of Sight and Layout Requirements of the Panama Canal to Navigation Vessels. Ship, 12, 1-5.
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The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006. Canada A=33.1e0.009t Uganda A=28.2
A=33.1e0.009t
A=28.2e0.034t
The given statement that the population of Canada exceeded that of Uganda by 2.8 million in the year 2009 is true
The table shows the annual service revenues R1 in billions of dollars for the cellular telephone industry for the years 2000 through 2006.
\begin{array}{cccccccc}Year& 2000& 2001& 2002& 2003& 2004& 2005& 2006\\ {R}_{1}& 52.5& 65.3& 76.5& 87.6& 102.1& 113.5& 125.5\end{array}
(a) Use the regression capabilities of a graphing utility to find an exponential model for the data. Let t represent the year, with t=10 corresponding to 2000. Use the graphing utility to plot the data and graph the model in the same viewing window.
(b) A financial consultant believes that a model for service revenues for the years 2010 through 2015 is
R2=6+13+13,{9}^{0.14}t
. What is the difference in total service revenues between the two models for the years 2010 through 2015?
A=33.1{e}^{0.009}t
A=28.2{e}^{0.034}t
Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Ugandas
Furthermore why is it that
{e}^{x}
is used in exponential modelling? Why aren't other exponential functions used, i.e.
{2}^{x}
, etc?
James rents an apartment with an initial monthly rent of $1,600. He was told that the rent goes up 1.75% each year. Write an exponential function that models this situation to calculate the rent after 15 years. Round the monthly rent to the nearest dollar.
The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006. Canada A=33.1e0.009t Uganda A=28.2e0.034t Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in 2013, Ugandas
\begin{array}{|cccccccccccc|}\hline x& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13\\ f\left(x\right)& 13.98& 17.84& 20.01& 22.7& 24.1& 26.15& 27.37& 28.38& 29.97& 31.07& 31.43\\ \hline\end{array}
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Efficiently counting rooms from a floorplan (version 2) - PhotoLens
You are given a map of a building, and your task is to count the number of rooms. The size of the map is n×m
n \times m
squares, and each square is either floor or wall. You can walk left, right, up, and down through the floor squares.
The first input line has two integers n
and m
: the height and width of the map.
Then there are n
lines of m
characters that describe the map. Each character is . (floor) or # (wall).
1\le n,m \le 2500
It seemed to me to be possible to solve the problem by processing line at a time, so that’s what my code does. Specifically, it keeps a tracking std::vector<std::size_t> named tracker that corresponds to the rooms from the previous row and starts with all zeros.
As it reads each line of input, it processes the line character at a time. If it’s non-empty (that is, if it’s a wall), set the corresponding tracker entry to 0.
The code also has provisions for recognizing that what it “thought” was two rooms turns out to be one room, and adjusts both the tracker vector and the overall roomcount.
The code is time efficient because it makes only a single pass through the input, and it’s memory efficient because it only allocates a single 1×m
1 \times m
Correctness – The code works correctly on every input I’ve tried, but if there is any error in either the code or the algorithm, I’d like to know.
Efficiency – Could the code be made even more efficient?
Reusability – This works for a 2D map, but I’d like to adapt it to 3 or more dimensions. Are there things I could do in this code to make such adaptation simpler?
Source : Link , Question Author : Edward , Answer Author : Cris Luengo
First missing positive with only primitives
How to upload photos from iPhone to iCloud Storage without being “managed” by sync?
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Physics - Double or nothing for spin filters
Magnetismo de Sólidos, Facultad de Ciencias, ICMA, CSIC-Universidad de Zaragoza, Calle Pedro Cerbuna 12, 50009 Zaragoza, Spain
Using a double spin-filter tunnel junction consisting of two magnetic insulating layers, researchers have observed a sizeable magnetoresistance without using magnetic electrodes, thus tuning the tunneling via the magnetic state of the insulating layers and by application of an electric voltage.
Figure 1: A double spin-filter junction sketched here in (top) a low resistance configuration and (bottom) a high resistance configuration. When the magnetic layers are antiparallel, no current flows, but when the magnetic layers are parallel, only electrons with spins parallel to the magnetization direction can cross the junction. By changing the magnetic orientation of the
\text{EuS}
layers, the spin-polarized current can be turned on and off, like a valve.A double spin-filter junction sketched here in (top) a low resistance configuration and (bottom) a high resistance configuration. When the magnetic layers are antiparallel, no current flows, but when the magnetic layers are parallel, only electrons w... Show more
In the last twenty years, control of the electron spin in electrical transport has advanced to the point where it now has serious application in data storage and magnetic sensing [1]. Additionally, this growing field is becoming increasingly cross disciplinary. When viewed next to control of charge in semiconductor-based devices, the manipulation of the electron spin remains a challenge, with researchers striving towards new approaches that lead to functional use.
In a paper appearing in Physical Review Letters, Guo-Xing Miao, Martina Müller, and Jagadeesh Moodera, at Massachusetts Institute of Technology have given experimental evidence for a sizeable magnetoresistive (MR) effect—a relative change of resistance with magnetic field—in a tunnel-junction device consisting of two decoupled magnetic barriers sandwiched between two nonmagnetic electrodes [2]. The magnetic insulating material filters the spin direction of the electron current tunneling through the device (see Fig. 1). The great potential of the configuration used by Miao and co-workers was theoretically predicted a few years ago by D. C. Worledge and T. H. Geballe at Stanford University [3]. Even though the result reported by Miao et al. is a proof-of-concept, only working at low temperatures, it is a step towards future applications in data storage and magnetic sensing, if the device can be grown with appropriate materials.
What is meant by tunneling in this case? In a tunnel junction, a small but significant portion of an incoming electron current from one electrode can tunnel through an insulating material to a second electrode, provided that the insulating barrier thickness is no thicker than a few nanometers. The electron preserves its spin direction during tunneling, so if one uses magnetic electrodes, the resistance of the device depends on the magnetic orientation of the electrodes [4]. As a consequence, the tunneling probability of an electron coming from the first electrode will depend on the available density of states with the same spin direction in the second electrode. Bearing this in mind, and considering the imbalanced number of spin-up and spin-down electrons due to the first magnetic electrode, we see that the current flowing depends strongly on whether the magnetic electrodes are parallel or antiparallel. Initially only moderate MR values were found in such magnetic tunnel junctions, but subsequently large MR effects (approaching ) at room temperature in -type tunnel junctions [5] paved the way for their quick implementation in magnetic read heads and magnetic sensors in 2005, replacing fully metallic giant-magnetoresistance devices [1].
The phenomenon of spin filtering and the corresponding MR effect is also observed in tunnel-junction devices, but here the main mechanism is different from that in magnetic tunnel junctions. In this case, the insulating barrier is a ferromagnetic insulator, which filters selectively one of the two electron spin directions [6]. Due to the exchange interaction in ferromagnetic materials, the conduction band in the ferromagnetic insulator is split into two subbands (spin-up and spin-down) separated by the exchange splitting energy. The result is that if an electron tunnels through such ferromagnetic barrier, the barrier height it experiences will depend on the orientation of its spin with respect to the magnetization direction of the barrier. One advantage of the spin-filter mechanism is that the tunneling is very sensitive to the applied voltage, which is very useful for microelectronic applications.
The first experimental demonstration of spin filtering dates back to 1972, when Müller et al. found that electrons emitted from tungsten tips covered by presented a high spin polarization of [7] (the spin polarization is the percentage of spin-up electrons compared to the total number of electrons). EuS is indeed an efficient spin-filter material, which has been used in quasimagnetic tunnel junctions, where the magnetic barrier is sandwiched between a magnetic electrode and a nonmagnetic electrode, producing a significant MR effect [8]. However, has quite a low Curie temperature , above which its spin-filter property disappears. This is why researchers are working hard to find spin-filter materials with Curie temperature above room temperature. The most promising candidates are ferrites such as and , where small but promising MR effects have been observed in quasimagnetic tunnel junctions [9,10].
It is most interesting that in the new double spin-filter tunnel junctions reported by Miao et al. [2], no magnetic electrode is required. The two magnetic barriers act as spin-polarizer and spin-analyzer respectively. As predicted, and in sharp contrast to standard magnetic tunnel junctions, the maximum MR effect does not occur at low applied voltage but at rather large voltage , at which point the voltage exceeds the spin-up barrier height and spin-up electrons flow easily through the magnetic barrier, whereas spin-down electrons cannot do so. This is very exciting for applications: in order to obtain large voltage output in devices it is necessary to apply high working voltages. On the other hand, a drawback from using two insulating barriers is that the final device resistance is high, a few [2], which will produce large thermal noise. The next challenge in working with these double spin filters is to grow very thin spin-filter layers to realize low-resistance devices that have a high Curie temperature enabling room-temperature operation.
Some unexplained features in the results presented by Miao et al. [2] suggest that not everything is fully understood from the theoretical point of view. The quite efficient but relatively simple theoretical treatment used to interpret the results suggests that a deeper understanding of the phenomenon is still required. This is reminiscent of the state-of-the-art of magnetic tunnel junctions at the end of the nineties, at a time before the discovery of important factors like interface effects [11] or electron wave-function symmetry effects [5,12]. There is every reason to be optimistic that new experimental results and more rigorous theoretical treatments will shed more light on the exciting physics shown by these spin-filter devices.
Even if the most straightforward applications of these spin-filter tunnel junctions might be in data storage and magnetic sensing, other technologies can benefit from them. Foremost could be the fabrication of hybrid systems in combination with semiconductors, giving rise to spin-dependent Schottky barriers for efficient spin injection [13]. Much attention is currently devoted to spin injection in semiconductors for spintronics, for which such hybrid systems remain basically unexplored. An interesting proposal for a spin-filter transistor gives an example of the possibilities [14]. The work by Miao et al. [2] is a further step in the control and manipulation of the electron spin in electrical transport by means of a double spin-filter tunnel junction, which expands the research horizons in the fields of magnetic storage, magnetic sensing, and spintronics. However, much experimental effort is still required to achieve functional devices at room temperature.
A. Fert, Rev. Mod. Phys, 80, 1517 (2008)
G-X. Miao, M. Müller, and J. S. Moodera, Phys. Rev. Lett. 102, 076601 (2009)
D. C. Worledge and T. H. Geballe, J. Appl. Phys. 88, 5277 (2000)
M. Julliere, Phys. Lett. A, 54 225 (1975)
M. Bowen et al., Appl. Phys. Lett. 79, 1655 (2001); S. Yuasa et al., Nature Mater. 3, 868 (2004); S. S. P. Parkin et al., 3, 862 (2004)
For a recent review, see J. Moodera, T. S. Santos, and T. Nagahama, J. Phys. Condens. Matter 19, 165202 (2007)
N. Müller, W. Eckstein, W. Heiland, and W. Zinn, Phys. Rev. Lett. 29, 1651 (1972)
P. LeClair et al., Appl. Phys. Lett. 80, 625 (2002); T. Nagahama, T. S. Santos, and J. S. Moodera, Phys. Rev. Lett. 99, 016602 (2007)
U. Lüders et al., Adv. Mater. 18, 1733 (2006)
M. G. Chapline and S. X. Wang, Phys. Rev. B 74, 014418 (2006); A. V. Ramos et al., Appl. Phys. Lett. 91, 122107 (2007)
J. M. De Teresa et al., Science 286, 507 (1999)
W. H. Butler et al., Phys. Rev. B 63, 054416 (2001); J. Mathon and A. Umerski, 63, R220403 (2001)
W. A. Thompson et al., Phys. Rev. Lett. 26, 1308 (1972); C. Ren et al., Appl. Phys. Lett. 86, 012501 (2005)
S. Sugahara and M. Tanaka, Physica E 21, 996 (2004)
Guo-Xing Miao, Martina Müller, and Jagadeesh S. Moodera
A liquid-metal jet can strip electrons from a high-intensity, accelerator-based ion beam, increasing the beam’s charge and enhancing accelerator performance.
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Multi Cylinder Inline Engine (with firing order) | Numerical | Education Lessons
All Notes / DOM
Basic knowledge about Firing Order
The firing order is a sequence of giving spark/firing to the cylinders of multi-cylinder inline engines. Firing order is provided in multi-cylinder inline engines in order to avoid vibrations (If firing order is not provided in multi-cylinder inline engines, then all the pistons will move in upward direction at the same time and same way they move in the downward direction at the same time. This creates a lot of vibration.) For avoiding these vibrations, firing order is provided in multi-cylinder inline engines.
A four stroke five cylinder inline engine has a firing order of 1-4-5-3-2-1. The centre lines of cylinders are spaced at an equal interval of 15 cm, the reciprocating parts per cylinder have a mass of 1.5 kg, the piston stroke is 10 cm and the connecting rods are 17.5 cm long. The engine rotates at 600 rpm. Determine the value of maximum primary and secondary unbalanced forces and couples about the central plane.
\begin{aligned} \text {mass of reciprocating parts } m &= 1.5 \ kg \\ \text {Stroke } S &= 10 \ cm = 0.10 \ m \\ \text {Radius of crank } r &= {S \over 2} = {0.10 \over 2} = 0.05 \ m \\ \text {Length of connecting rod } L &= 17.5 \ cm = 0.175 \ m \\ \text {Obliquity ratio } n &= {L \over r} = {0.175 \over 0.05} = 3.5 \\ \text {Speed of engine } N &= 600 \ rpm \\ \omega &= {2 \pi N \over 60} = {2 \pi \times 600 \over 60 } = 62.85 { rad \over s} \end{aligned}
The force and couple data is given in table:
Mass (m), kg
Radius (r), m
Centrifugal Force (m x r), kg.m
Distance From R.P.(l), m
Couple (m x r x l), kg.m 2
Primary Crank Position ' θ '
Secondary Crank Position '2 θ '
1 1.5 0.05 0.075 -0.3 -0.022 0° 0°
2 1.5 0.05 0.075 -0.15 -0.011 576° i.e. 216° 1152° i.e. 72°
3 (R.P.) 1.5 0.05 0.075 0 0 432° i.e. 72° 864° i.e. 144°
4 1.5 0.05 0.075 0.15 0.011 144° 288°
5 1.5 0.05 0.075 0.3 0.022 288° 576° i.e. 216°
fig.2 (a) Position of planes
fig.2 (b) Primary crank positions
Assuming the engine to be vertical, the position of cylinders are shown in fig.2(a).
The angles are measured from cylinder 1.
Cylinder 3 is considered as a reference plane (R.P.)
The engine is working on a four stroke cycle, the angle between two crank
= {720 \over 5} = 144\degree
Considering firing order : 1-4-5-3-2-1
\begin{aligned} \therefore \ &\theta_1 = 0\degree \\ &\theta_4 = 144\degree \\ &\theta_5 = 144\degree + 144\degree = 288\degree \\ &\theta_3 = 288\degree + 144\degree = 432\degree \\ &\theta_2 = 432\degree + 144\degree = 576\degree \end{aligned}
1. Primary force polygon
For drawing a primary force polygon, take a suitable scale of 1 cm = 0.075 kg.m. From fig.2(c) it is seen that, primary force polygon is closed and hence there is no unbalanced primary force.
fig.2 (c) Primary force polygon
2. Primary couple polygon
fig.2 (d) Primary couple polygon
Draw the primary couple polygon by taking a suitable scale of
1\ cm = 0.011 \ kg \ m^2
\overrightarrow {od'}
represents the unbalanced primary couple
C_p
Now from fig.2 (d), measure
\overrightarrow {od'}
\overrightarrow {od'} = 1.5 \ cm
Hence, the unbalanced primary couple is,
\begin{aligned} C_p &= \overrightarrow {od'} \times scale \times \omega^2 \\ &= 1.5 \times 0.011 \times (62.85)^2 \\ &= 65.18 \ N \sdot m \end{aligned}
3. Secondary force polygon
fig.2 (e) Secondary crank positions
fig.2 (f) Secondary force polygon
Draw the secondary force polygon by taking a suitable scale of
1 \ cm = 0.075 \ kg.m
. From fig.2 (f), it is seen that the secondary force polygon is closed, hence there is no unbalanced secondary force.
4. Secondary couple polygon
fig.2 (g) Secondary couple polygon
Draw the secondary couple polygon by taking suitable scale of
1 \ cm = 0.011 \ kg m^2
\overrightarrow {od'}
represents the unbalanced secondary couple
{C_s}
Now from fig.2 (g), measure
\overrightarrow {od'}
\overrightarrow {od'} = 4.7 \ cm
Hence, the unbalanced secondary couple is,
\begin{aligned} C_s &= \overrightarrow {od'} \times scale \times {\omega^2 \over n} \\ &= 4.7 \times 0.011 \times {(62.85)^2 \over 3.5} \\ &= 58.3 \ N \sdot m \end{aligned}
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m (Some re-structuring)
m (→Pan Sharpening: HPFA-based sharpening example screenshots)
The following screenshots exemplify the High Pass Filtering Addition fusion technique applied in a fragment of the QuickBird acquisition "04APR05050541-X2AS_R1C1-000000186011_01_P001-Sri_Lanka-Kokilai_Lagoon" which is publicly available via the GLCF.
|[[File:Pan_04APR05050541-M2AS-000000186011_01_P001.jpg|thumb|400px|border|center|Fragment from the high resolution (0.7m) panchromatic image]]
|[[File:RGB_04APR05050541-M2AS-000000186011_01_P001.jpg|thumb|400px|border|center|Fragment from the low resolution (2.8m) multi-spectral RGB composite image]]
|[[File:RGB_HPF_Sharpened_Default_Parameters.jpg|thumb|400px|border|center|Fragment from the HPFA Pan-Sharpened RGB composite image. Sharpening performed with default parameters.]]
|[[File:RGB_HPF_Sharpened_Center_Low_Modulator_Min.jpg|thumb|400px|border|center|Fragment from another HPFA Pan-Sharpened RGB composite image. Sharpening performed with ''center=low'' and ''modulator=min'' parameters.]]
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L_{\lambda {\text{Pixel, Band}}}={\frac {K_{\text{Band}}*q_{\text{Pixel, Band}}}{\Delta \lambda _{\text{Band}}}}}
{\displaystyle L_{\lambda {\text{Pixel,Band}}}}
{\displaystyle K_{\text{Band}}}
{\displaystyle q_{\text{Pixel,Band}}}
{\displaystyle \Delta _{\lambda _{\text{Band}}}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
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Progress in Probability /
For two-dimensional percolation at criticality, we discuss the inequality α4 > 1 for the polychromatic four-arm exponent (and stronger versions, the strongest so far being
{\alpha }_{4}\ge 1+\frac{{\alpha }_{2}}{2}
, where α2 denotes the two-arm exponent). We first briefly discuss five proofs (some of them implicit and not self-contained) from the literature. Then we observe that, by combining two of them, one gets a completely self-contained (and yet quite short) proof.
Keywords Arm exponents, Critical percolation
Editor M.E. Vares , R. Fernández (Roberto) , L.R. Fontes (Luiz Renato) , C.M. Newman
Series Progress in Probability
van den Berg, J, & Nolin, P. (2021). On the four-arm exponent for 2D percolation at criticality. In M.E Vares, R Fernández, L.R Fontes, & C.M Newman (Eds.), In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius (pp. 125–145). Springer Nature. doi:10.1007/978-3-030-60754-8_6
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Louis_Poinsot Knowpia
Poinsot was the inventor of geometrical mechanics, which showed how a system of forces acting on a rigid body could be resolved into a single force and a couple. Previous work done on the motion of a rigid body had been purely analytical with no visualization of the motion, and the great value of the work, as Poinsot says, it enables us to represent to ourselves the motion of a rigid body as clearly that as a moving point (Encyclopædia Britannica, 1911). In particular he devised, what is now known as, Poinsot's construction. This construction describes the motion of the angular velocity vector
{\displaystyle \mathbf {\omega } }
of a rigid body with one point fixed (usually its center of mass). He proved that the endpoint of the vector
{\displaystyle \mathbf {\omega } }
moves in a plane perpendicular to the angular momentum (in absolute space) of the rigid body.[2]
^ The elements of statics - Part 1 at Google Books
^ E. T. Whittaker, Analytical Dynamics of Particles and Rigid Bodies, Cambridge UP, 4th edition, (1938), p. 152 ff.
Taton, René (1970–1980). "Poinsot, Louis". Dictionary of Scientific Biography. Vol. 11. New York: Charles Scribner's Sons. pp. 61–62. ISBN 978-0-684-10114-9.
This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Poinsot, Louis". Encyclopædia Britannica. Vol. 21 (11th ed.). Cambridge University Press. p. 892.
Wikiquote has quotations related to Louis Poinsot.
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Unit_fraction Knowpia
Elementary arithmeticEdit
Multiplying any two unit fractions results in a product that is another unit fraction:
{\displaystyle {\frac {1}{x}}\times {\frac {1}{y}}={\frac {1}{xy}}.}
However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction:
{\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {x+y}{xy}}}
{\displaystyle {\frac {1}{x}}-{\frac {1}{y}}={\frac {y-x}{xy}}}
{\displaystyle {\frac {1}{x}}\div {\frac {1}{y}}={\frac {y}{x}}.}
Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that
{\displaystyle \displaystyle ax+by=1,}
{\displaystyle \displaystyle ax\equiv 1{\pmod {y}},}
{\displaystyle a\equiv {\frac {1}{x}}{\pmod {y}}.}
Thus, to divide by x (modulo y) we need merely instead multiply by a.
Finite sums of unit fractionsEdit
Any positive rational number can be written as the sum of unit fractions, in multiple ways. For example,
{\displaystyle {\frac {4}{5}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{20}}={\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{10}}.}
The ancient Egyptian civilisations used sums of distinct unit fractions in their notation for more general rational numbers, and so such sums are often called Egyptian fractions. There is still interest today in analyzing the methods used by the ancients to choose among the possible representations for a fractional number, and to calculate with such representations.[1] The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham conjecture and the Erdős–Straus conjecture concern sums of unit fractions, as does the definition of Ore's harmonic numbers.
In geometric group theory, triangle groups are classified into Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.
Series of unit fractionsEdit
Many well-known infinite series have terms that are unit fractions. These include:
The harmonic series, the sum of all positive unit fractions. This sum diverges, and its partial sums
{\displaystyle {\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}}
closely approximate the natural logarithm of
{\displaystyle n}
plus the Euler–Mascheroni constant. Changing every other addition to a subtraction produces the alternating harmonic series, which sums to the natural logarithm of 2:
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots =\ln 2.}
The Leibniz formula for π is
{\displaystyle 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}.}
The Basel problem concerns the sum of the square unit fractions:
{\displaystyle 1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+\cdots ={\frac {\pi ^{2}}{6}}.}
Similarly, Apéry's constant is an irrational number, the sum of the cubed unit fractions.
The binary geometric series is
{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =2.}
Matrices of unit fractionsEdit
The Hilbert matrix is the matrix with elements
{\displaystyle B_{i,j}={\frac {1}{i+j-1}}.}
It has the unusual property that all elements in its inverse matrix are integers.[2] Similarly, Richardson (2001) defined a matrix with elements
{\displaystyle C_{i,j}={\frac {1}{F_{i+j-1}}},}
where Fi denotes the ith Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.[3]
Adjacent fractionsEdit
Fractions with tangent Ford circles differ by a unit fraction
Two fractions
{\displaystyle a/b}
{\displaystyle c/d}
(in lowest terms) are called adjacent if
{\displaystyle ad-bc=\pm 1}
, which implies that their difference
{\displaystyle |ad-bc|/bd}
is a unit fraction. For instance,
{\displaystyle {\tfrac {1}{2}}}
{\displaystyle {\tfrac {3}{5}}}
are adjacent:
{\displaystyle 1\cdot 5-2\cdot 3=-1}
{\displaystyle {\tfrac {3}{5}}-{\tfrac {1}{2}}={\tfrac {1}{10}}}
. However, some pairs of fractions whose difference is a unit fraction are not adjacent in this sense: for instance,
{\displaystyle {\tfrac {1}{3}}}
{\displaystyle {\tfrac {2}{3}}}
differ by a unit fraction, but are not adjacent, because for them
{\displaystyle ad-bc=3}
. The terminology comes from the study of Ford circles, circles that are tangent to the number line at a given fraction and have the squared denominator of the fraction as their diameter: fractions
{\displaystyle a/b}
{\displaystyle c/d}
are adjacent if and only if their Ford circles are tangent circles.[4]
Unit fractions in probability and statisticsEdit
In a uniform distribution on a discrete space, all probabilities are equal unit fractions. Due to the principle of indifference, probabilities of this form arise frequently in statistical calculations.[5] Additionally, Zipf's law states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the nth item is selected is proportional to the unit fraction 1/n.[6]
Unit fractions in physicsEdit
The energy levels of photons that can be absorbed or emitted by a hydrogen atom are, according to the Rydberg formula, proportional to the differences of two unit fractions. An explanation for this phenomenon is provided by the Bohr model, according to which the energy levels of electron orbitals in a hydrogen atom are inversely proportional to square unit fractions, and the energy of a photon is quantized to the difference between two levels.[7]
Arthur Eddington argued that the fine-structure constant was a unit fraction, first 1/136 then 1/137. This contention has been falsified, given that current estimates of the fine structure constant are (to 6 significant digits) 1/137.036.[8]
^ Guy, Richard K. (2004), "D11. Egyptian Fractions", Unsolved problems in number theory (3rd ed.), Springer-Verlag, pp. 252–262, ISBN 978-0-387-20860-2 .
^ Choi, Man Duen (1983), "Tricks or treats with the Hilbert matrix", The American Mathematical Monthly, 90 (5): 301–312, doi:10.2307/2975779, MR 0701570 .
^ Richardson, Thomas M. (2001), "The Filbert matrix" (PDF), Fibonacci Quarterly, 39 (3): 268–275, arXiv:math.RA/9905079, Bibcode:1999math......5079R
^ Ford, L. R. (1938), "Fractions", The American Mathematical Monthly, 45 (9): 586–601, doi:10.1080/00029890.1938.11990863, JSTOR 2302799, MR 1524411
^ Welsh, Alan H. (1996), Aspects of statistical inference, Wiley Series in Probability and Statistics, vol. 246, John Wiley and Sons, p. 66, ISBN 978-0-471-11591-5 .
^ Saichev, Alexander; Malevergne, Yannick; Sornette, Didier (2009), Theory of Zipf's Law and Beyond, Lecture Notes in Economics and Mathematical Systems, vol. 632, Springer-Verlag, ISBN 978-3-642-02945-5 .
^ Yang, Fujia; Hamilton, Joseph H. (2009), Modern Atomic and Nuclear Physics, World Scientific, pp. 81–86, ISBN 978-981-283-678-6 .
^ Kilmister, Clive William (1994), Eddington's search for a fundamental theory: a key to the universe, Cambridge University Press, ISBN 978-0-521-37165-0 .
Weisstein, Eric W., "Unit Fraction", MathWorld
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Outcome: Specialized Reading Strategies | Developmental English: Introduction to College Composition | Course Hero
The reading process discussed earlier in this module applies to any kind of reading you'll do for school. Beyond the general strategies of previewing, active reading, summarizing, and reviewing, however, you'll find that specific types of reading will place specific demands on you.
Examine the three items below. All are typical of types of reading you'll need to do in different college classes you take. As you can see, they each require a different set of skills to read and interpret correctly.
Chart of an example of Throughput Accounting structure
4p+5<29
\displaystyle \begin{array}{l}4p+5<\,\,\,29\\\underline{\,\,\,\,\,\,\,\,\,-5\,\,\,\,\,\,\,-5}\\\underline{4p}\,\,\,\,\,\,\,\,<\,\,\underline{24}\,\,\\4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\\\,\,\,\,\,\,\,\,\,\,\,p<6\end{array}
The following pages in this section will offer targeted advice for approaching different categories of the text and images you'll encounter.
Image of 3D Chart. Located at: https://commons.wikimedia.org/wiki/File:ThroughputStructure.jpg. License: CC BY: Attribution
Signaling in Yeast. Provided by: Boundless. Located at: https://www.boundless.com/biology/textbooks/boundless-biology-textbook/cell-communication-9/signaling-in-single-celled-organisms-86/signaling-in-yeast-390-11616/. License: CC BY-SA: Attribution-ShareAlike
Linear inequality (colour-coded). Authored by: Geogebra Institute of MEI. Located at: https://www.geogebra.org/material/simple/id/135374. License: CC BY-SA: Attribution-ShareAlike
Reading Strategies, IRA.pdf
CHEM 1515 • Oklahoma State University
1.1 Reading and Writing in College - Writing for Success.pdf
ENGL 1301 • Central Texas College
College Reading Strategies Week 3 Dis 1 1st reading.pdf
READING SP GEN104 • Ashford University
COM 100 • University of the West Indies at Cave Hill
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govern vs order what difference - Tez Koder
govern vs order what difference
what is difference between govern and order
From Middle English governen, governe, from Anglo-Norman and Old French governer, guverner, from Latin gubernō, from Ancient Greek κυβερνάω (kubernáō, “I steer, drive, govern”)
(General American) IPA(key): /ˈɡʌvɚn/
(Received Pronunciation) IPA(key): /ˈɡʌvən/
Hyphenation: gov‧ern
Rhymes: -ʌvə(ɹ)n
govern (third-person singular simple present governs, present participle governing, simple past and past participle governed)
(transitive) To make and administer the public policy and affairs of; to exercise sovereign authority in.
(transitive) To control the actions or behavior of; to keep under control; to restrain.
2016, Justin Deschamps, Find the strength, courage, and discipline to govern yourself or be governed by someone else.
Find the strength, courage, and discipline to govern yourself or be governed by someone else.
(transitive) To exercise a deciding or determining influence on.
(transitive) To control the speed, flow etc. of; to regulate.
(intransitive) To exercise political authority; to run a government.
(intransitive) To have or exercise a determining influence.
(transitive, grammar) To require that a certain preposition, grammatical case, etc. be used with a word; sometimes used synonymously with collocate.
govern (plural governs)
From the verb governar, or possibly from Late Latin gubernus or gubernius, from Latin gubernum or gubernō.
(Balearic, Valencian) IPA(key): /ɡoˈvɛɾn/
(Central) IPA(key): /ɡuˈbɛrn/
govern m (plural governs)
“govern” in Diccionari de la llengua catalana, segona edició, Institut d’Estudis Catalans.
“govern” in Diccionari normatiu valencià, Acadèmia Valenciana de la Llengua.
“govern” in Diccionari català-valencià-balear, Antoni Maria Alcover and Francesc de Borja Moll, 1962.
{\displaystyle A}
{\displaystyle G={\mathit {GL}}(d,q)}
{\displaystyle O(d^{3}{\mathsf {log}}\ q)}
{\displaystyle q^{i}-1}
{\displaystyle i\leq d}
{\displaystyle \Delta }
{\displaystyle \Delta }
{\displaystyle a\in \Delta }
{\displaystyle n}
{\displaystyle a^{n}=1_{\Delta }}
{\displaystyle \Delta }
{\displaystyle a\in \Delta }
{\displaystyle |\Delta |}
{\displaystyle p}
{\displaystyle |\Delta |}
{\displaystyle a\in \Delta }
{\displaystyle p}
Prev gazump vs pluck what difference
Next gip vs scam what difference
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MATLAB Programming/Nyquist Plot - Wikibooks, open books for an open world
MATLAB Programming/Nyquist Plot
2 MATLAB's Nyquist Command
2.1 Issues with the nyquist command
This article is on the topic of creating Nyquist plots in MATLAB. The quick answer is use the nyquist command. However, the Nyquist command has several options and the plots generated by the nyquist command are not easily reformatted. The default formatting of most MATLAB plots is good for analysis but less than ideal for dropping into Word and PowerPoint documents or even this website. As a result this article presents an alternative that requires more lines of code but offers the full formatting flexibility of the generic plot command.
MATLAB's Nyquist Command[edit | edit source]
The basic Nyquist command is as follows
>> nyquist(LTI_SYS)
LTI_SYS is an LTI object - TF, SS, ZPK, or FRD
The Nyquist command will automatically call gcf which will put the Nyquist plot on the current figure. If no figure exists then one is created by gcf.
If you wish to specify the frequency points at which LTI_SYS is plotted then create a frequency vector using logspace or linspace as follows
>> freqVec = logspace(-1, 3, 5000);
>> nyquist(LTI_SYS, freqVec * (2*pi))
freqVec is a vector of 5000 frequencies, in Hz, spaced evenly on a log scale from 10-1 to 103
pi is a MATLAB constant equal to the value of
{\displaystyle \pi }
and in this case it is used to convert freqVec to rad/sec as it is passed to the nyquist command
Issues with the nyquist command[edit | edit source]
The main issue with the nyquist command is reformatting of the plot. The nyquist command appears to use a normal semilogx plot and then apply patches or something similar to the figure. This can lead to odd behavior when attempting to create multi-line titles, reformat line widths or font sizes, etc. The normal relationship of axes to figure is just not quite present.
http://www.mathworks.com/access/helpdesk/help/toolbox/control/ref/nyquist.html
Retrieved from "https://en.wikibooks.org/w/index.php?title=MATLAB_Programming/Nyquist_Plot&oldid=1702555"
Book:MATLAB Programming
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Solid axle suspension with coil spring - Simulink - MathWorks Switzerland
\begin{array}{l}\left[\begin{array}{c}{\stackrel{¨}{x}}_{a}\\ {\stackrel{¨}{y}}_{a}\\ {\stackrel{¨}{z}}_{a}\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}{F}_{xa}\\ {F}_{ya}\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}{\stackrel{˙}{x}}_{a}\\ {\stackrel{˙}{y}}_{a}\\ {\stackrel{˙}{z}}_{a}\end{array}\right]×\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}0\\ 0\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ {\stackrel{˙}{z}}_{a}\end{array}\right]×\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ g\end{array}\right]=\left[\begin{array}{c}0\\ p{\stackrel{˙}{z}}_{a}\\ \frac{{F}_{za}}{{M}_{a}}+g\end{array}\right]\\ \\ \left[\begin{array}{c}\stackrel{˙}{p}\\ \stackrel{˙}{q}\\ \stackrel{˙}{r}\end{array}\right]=\left[\left[\begin{array}{c}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right]-\left[\begin{array}{c}p\\ q\\ r\end{array}\right]×\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{-1}\\ =\left[\left[\begin{array}{c}{M}_{x}\\ 0\\ 0\end{array}\right]-\left[\begin{array}{c}p\\ q\\ 0\end{array}\right]×\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{-1}=\left[\begin{array}{c}\frac{{M}_{x}}{{I}_{xx}}\\ 0\\ 0\end{array}\right]\end{array}
{F}_{za}=\sum _{t=1}^{Nta}\left({F}_{w{z}_{a,t}}+{F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)\right)
{M}_{x}=\sum _{t=1}^{Nta}\left({F}_{w{z}_{a,t}}{y}_{{w}_{t}}+\left({F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)\right){y}_{{s}_{t}}+{M}_{w{x}_{a,t}}\frac{{I}_{xx}}{{I}_{xx}+{M}_{a}{y}_{{w}_{t}}}\right)
\begin{array}{l} T{c}_{t}=\left[\begin{array}{ccc}{x}_{{w}_{1}}& {x}_{{w}_{2}}& \dots \\ {y}_{{w}_{1}}& {y}_{{w}_{2}}& \dots \\ {z}_{{w}_{1}}& {z}_{{w}_{2}}& \dots \end{array}\right]\\ S{c}_{t}=\left[\begin{array}{ccc}{x}_{{s}_{1}}& {x}_{{s}_{2}}& \dots \\ {y}_{{s}_{1}}& {y}_{{s}_{2}}& \dots \\ {z}_{{s}_{1}}& {z}_{{s}_{2}}& \dots \end{array}\right]\end{array}
{F}_{v{z}_{a,t}=-}\left({F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)+{F}_{zhsto{p}_{a,t}}\right)
\begin{array}{l}{F}_{v{x}_{a,t}}={F}_{w{x}_{a,t}}\\ {F}_{v{y}_{a,t}}={F}_{w{y}_{a,t}}\\ {F}_{v{z}_{a,t}}=-{F}_{w{z}_{a,t}}\\ \\ {M}_{v{x}_{a,t}}={M}_{w{x}_{a,t}}+{F}_{w{y}_{a,t}}\left(R{e}_{w{y}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{y}_{a,t}}={M}_{w{y}_{a,t}}+{F}_{w{x}_{a,t}}\left(R{e}_{w{x}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{z}_{a,t}}={M}_{w{z}_{a,t}}\end{array}
{F}_{w{z}_{a,t}}=-Fw{a}_{z0}-kw{a}_{z}\left({z}_{{w}_{a,t}}-{z}_{{s}_{a,t}}\right)-cw{a}_{z}\left({\stackrel{˙}{z}}_{{w}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)
\begin{array}{l}{\xi }_{a,t}={\xi }_{0a}+{m}_{hcambe{r}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{camberstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ {\eta }_{a,t}={\eta }_{0a}+{m}_{hcaste{r}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{casterstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ {\zeta }_{a,t}={\zeta }_{0a}+{m}_{hto{e}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{toestee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ \end{array}
{\delta }_{whlstee{r}_{a,t}}={\delta }_{stee{r}_{a,t}}+{m}_{hto{e}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{toestee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|
{P}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{E}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{H}_{a,t}=-\left({z}_{{v}_{a,t}}-{z}_{{w}_{a,t}}+\frac{{F}_{z{0}_{a}}}{{k}_{{z}_{a}}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)
{z}_{wt{r}_{a,t}}=R{e}_{{w}_{a,t}}+{H}_{a,t}
\mathrm{WhlPz}={z}_{w}=\left[\begin{array}{cccc}{z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
\mathrm{Whl}\mathrm{Re}=R{e}_{w}=\left[\begin{array}{cccc}R{e}_{{w}_{1,1}}& R{e}_{{w}_{1,2}}& R{e}_{{w}_{2,1}}& R{e}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlVz}={\stackrel{˙}{z}}_{w}=\left[\begin{array}{cccc}{\stackrel{˙}{z}}_{{w}_{1,1}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlFx}={F}_{wx}=\left[\begin{array}{cccc}{F}_{w{x}_{1,1}}& {F}_{w{x}_{1,2}}& {F}_{w{x}_{2,1}}& {F}_{w{x}_{2,2}}\end{array}\right]
\mathrm{WhlFy}={F}_{wy}=\left[\begin{array}{cccc}{F}_{w{y}_{1,1}}& {F}_{w{y}_{1,2}}& {F}_{w{y}_{2,1}}& {F}_{w{y}_{2,2}}\end{array}\right]
\mathrm{WhlM}={M}_{w}=\left[\begin{array}{cccc}{M}_{w{x}_{1,1}}& {M}_{w{x}_{1,2}}& {M}_{w{x}_{2,1}}& {M}_{w{x}_{2,2}}\\ {M}_{w{y}_{1,1}}& {M}_{w{y}_{1,2}}& {M}_{w{y}_{2,1}}& {M}_{w{y}_{2,2}}\\ {M}_{w{z}_{1,1}}& {M}_{w{z}_{1,2}}& {M}_{w{z}_{2,1}}& {M}_{w{z}_{2,2}}\end{array}\right]
\mathrm{VehP}=\left[\begin{array}{c}{x}_{v}\\ {y}_{v}\\ {z}_{v}\end{array}\right]=\left[\begin{array}{cccc}{x}_{v}{}_{{}_{1,1}}& {x}_{v}{}_{{}_{1,2}}& {x}_{v}{}_{{}_{2,1}}& {x}_{v}{}_{{}_{2,2}}\\ {y}_{v}{}_{{}_{1,1}}& {y}_{v}{}_{{}_{1,2}}& {y}_{v}{}_{{}_{2,1}}& {y}_{v}{}_{{}_{2,2}}\\ {z}_{v}{}_{{}_{1,1}}& {z}_{v}{}_{{}_{1,2}}& {z}_{v}{}_{{}_{2,1}}& {z}_{v}{}_{{}_{2,2}}\end{array}\right]
\mathrm{VehV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{v}\\ {\stackrel{˙}{y}}_{v}\\ {\stackrel{˙}{z}}_{v}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{v}_{1,1}}& {\stackrel{˙}{x}}_{{v}_{1,2}}& {\stackrel{˙}{x}}_{{v}_{2,1}}& {\stackrel{˙}{x}}_{{v}_{2,2}}\\ {\stackrel{˙}{y}}_{{v}_{1,1}}& {\stackrel{˙}{y}}_{{v}_{1,2}}& {\stackrel{˙}{y}}_{{v}_{2,1}}& {\stackrel{˙}{y}}_{{v}_{2,2}}\\ {\stackrel{˙}{z}}_{{v}_{1,1}}& {\stackrel{˙}{z}}_{{v}_{1,2}}& {\stackrel{˙}{z}}_{{v}_{2,1}}& {\stackrel{˙}{z}}_{{v}_{2,2}}\end{array}\right]
\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]
\mathrm{WhlAng}\left[1,...\right]=\xi =\left[{\xi }_{a,t}\right]
\mathrm{WhlAng}\left[2,...\right]=\eta =\left[{\eta }_{a,t}\right]
\mathrm{WhlAng}\left[3,...\right]=\zeta =\left[{\zeta }_{a,t}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlP}=\left[\begin{array}{c}{x}_{w}\\ {y}_{w}\\ {z}_{w}\end{array}\right]=\left[\begin{array}{cccc}{x}_{w}{}_{{}_{1,1}}& {x}_{w}{}_{{}_{1,2}}& {x}_{w}{}_{{}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{w}{}_{{}_{1,1}}& {y}_{w}{}_{{}_{1,2}}& {y}_{w}{}_{{}_{2,1}}& {y}_{w}{}_{{y}_{2,2}}\\ {z}_{wtr}{}_{{}_{1,1}}& {z}_{wtr}{}_{{}_{1,2}}& {z}_{wtr}{}_{{}_{2,1}}& {z}_{wt{r}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]
\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]
T{c}_{t}=\left[\begin{array}{cccc}{x}_{{w}_{1,1}}& {x}_{{w}_{1,2}}& {x}_{{w}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{{w}_{1,1}}& {y}_{{w}_{1,2}}& {y}_{{w}_{2,1}}& {y}_{{w}_{2,2}}\\ {z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
S{c}_{t}=\left[\begin{array}{cccc}{x}_{{s}_{1,1}}& {x}_{{s}_{1,2}}& {x}_{{s}_{2,1}}& {x}_{{s}_{2,2}}\\ {y}_{{s}_{1,1}}& {y}_{{s}_{1,2}}& {y}_{{s}_{2,1}}& {y}_{{s}_{2,2}}\\ {z}_{{s}_{1,1}}& {z}_{{s}_{1,2}}& {z}_{{s}_{2,1}}& {z}_{{s}_{2,2}}\end{array}\right]
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In the high-acuity clinical scenario, we found no significant difference in the provision of IFDC based on receipt of IFDC-related education. Within the moderate-acuity scenario, IFDC-related education included in unit-based orientation significantly affected IFDC practices. Among nurses who had received IFDC-related education, 91% reported that they often or always incorporate cue-based assessment—significantly more than the 50% of nurses who did not receive IFDC-related education who reported this practice
(χ12[n=23]=4.54,P=.03)
. Among that same group of nurses, 82% reported often or always promoting holding by parents or staff—again significantly more than the 39% of nurses who reported this practice but had not received IFDC-related education
(χ12[n=24]=4.61,P=.03)
; 64% reported often or always letting parents assume care, also significantly more than the 23% of nurses who allowed this practice but who had not received education
(χ12[n=24]=4.03,P=.05)
. For the low-acuity clinical scenario, 100% of nurses previously educated on IFDC reported often or always using developmentally supportive positioning (the infant placed midline or with their extremities flexed, for example); this proportion was again significantly higher than the 69% of nurses who had not been educated about IFDC and who used such positioning
(χ12[n=24]=4.06,P=.04)
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{\displaystyle f\in AC[0,1]}
be an absolutely continuous function on [0,1] with
{\displaystyle f>0}
{\displaystyle 1/f\in AC[0,1]}
{\displaystyle f}
is (absolutely) continuous on [0,1] with
{\displaystyle f>0}
then there exists some
{\displaystyle 0<m=\min _{x\in [0,1]}f(x)}
{\displaystyle f\in AC[0,1]}
{\displaystyle \epsilon >0}
{\displaystyle \delta >0}
such that for any finite collection of disjoint intervals
{\displaystyle I_{k}=(x_{k},y_{k}),k=1,...,n}
{\displaystyle \sum _{k=1}^{n}|y_{k}-x_{k}|<\delta }
{\displaystyle \sum _{k=1}^{n}|f(y_{k})-f(x_{k})|<\epsilon m^{2}}
Then for any such collection of intervals described above, we have
{\displaystyle \sum _{k=1}^{n}|{\frac {1}{f(y_{k})}}-{\frac {1}{f(x_{k})}}=\sum _{k=1}^{n}|{\frac {f(x_{k})-f(y_{k})}{f(y_{k})f(x_{k})}}|\leq \sum _{k=1}^{n}{\frac {1}{m^{2}}}|f(y_{k})-f(x_{k})|<\epsilon }
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Preset fluid properties for the simulation of a thermal liquid network - MATLAB - MathWorks Deutschland
Thermal Liquid Properties (TL)
Validity Regions
Thermal liquid fluid list
Dissolved salt mass fraction (salinity)
Ethylene glycol volume fraction
Ethylene glycol mass fraction
Propylene glycol volume fraction
Propylene glycol mass fraction
Preset fluid properties for the simulation of a thermal liquid network
Simscape / Fluids / Thermal Liquid / Utilities
The Thermal Liquid Properties (TL) block set predefined fluid properties to a thermal liquid network. The available fluids include pure water, aqueous mixtures, diesel, aviation fuel Jet A, and SAE 5W-30. You can use this block as a preset alternative to the Thermal Liquid Settings (TL) block. If your network does not have a connected liquid properties block, or the liquid defaults will apply. See Specify Fluid Properties for more details
The preset fluid properties are defined in tabular form as functions of temperature and pressure. During simulation, the network properties are set by linear interpolation between data points. Tabular data for aqueous mixtures is provided for concentration by mass or volume.
All the fluid properties commonly specified in the Thermal Liquid Settings (TL) block are defined in the block. These properties include density, the bulk modulus and thermal expansion coefficient, the specific internal energy and specific heat, as well as the kinematic viscosity and thermal conductivity. The properties are valid over a limited region of temperatures and pressures specific to the fluid selected and dependent, in the cases of mixtures, on the concentration specified. Simulation is allowed within this validity region only.
You can visualize the fluid properties defined in the block and the pressure and temperature regions of validity. To open the visualization utility, right-click the block and select Fluids > Plot Fluid Properties. The plot updates automatically upon selection of a fluid property from the drop-down list. Use the Reload Data button to regenerate the plot whenever the fluid selection or fluid parameters change.
Visualization of density data for a 10% glycerol aqueous mixture
The validity regions are defined in the block as matrices of zeros and ones. Each row corresponds to a tabulated temperature and each column to a tabulated pressure. A zero denotes an invalid breakpoint and a one a valid breakpoint. These validity matrices are internal to the block and cannot be modified; they can only be checked (using the data visualization utility of the block).
In most cases, the validity matrices are extracted directly from the tabulated data. The glycol and glycerol mixtures pressure boundaries are not available from the data, and are obtained explicitly from block parameters. The figure below shows an example of the validity region for water. The shaded squares indicate temperature and pressure regions outside of the validity region.
The properties of water are valid at temperatures above the triple-point value (273.160 K) up to the critical-point value (647.096 K). They are valid at pressures above the greater of the triple-point value (611.657 Pa) on one hand and the temperature-dependent saturation value on the other, up to the critical-point value (22.064 MPa). Pressures below the saturation point for a given temperature row are assigned a value of 0 in the validity matrix.
Seawater (MIT model)
The properties of seawater are valid at temperatures above 0°C up to 120°C (273.15 K to 393.15 K); they are valid at pressures above the saturation point up to a maximum value of 12 MPa. Pressures below the saturation point for a given temperature row (and at the specified concentration level) are assigned a value of 0 in the validity matrix. Mixture concentrations can range in value from 0 to 0.12 on a mass fraction basis.
Ethylene glycol and water mixture
The properties of an aqueous ethylene glycol mixture are valid over a temperature domain determined from the mixture concentration; they are valid at pressures within the minimum and maximum bounds specified in the block dialog box (extended horizontally to span the width of the temperature rows).
The lower temperature bound is always the lesser of the minimum temperature extracted from the available data and the freezing point of the mixture (the mixture must be in the liquid state). The upper temperature bound is always the maximum temperature extracted from the data. Mixture concentrations can range in value from 0 to 0.6 if a mass-fraction basis is used, or from 0 to 1 if a volume fraction basis is used.
The properties of an aqueous propylene glycol mixture are valid over the temperature and pressure ranges described for the case of Ethylene glycol and water mixture. Mixture concentrations can range in value from 0 to 0.6 if a mass-fraction basis is used, or from 0.1 to 0.6 if a volume fraction basis is used.
Glycerol and water mixture
The properties of an aqueous glycerol mixture are valid over the temperature and pressure ranges as described for the case of Ethylene glycol and water mixture. Mixture concentrations can range in value from 0 to 0.6 on a mass-fraction basis. The properties are all extended to 100 degC as:
Density—quadratic fit
Alpha (density thermal derivative)—linear fit
Specific heat—linear fit
Specific internal energy—linear fit
Kinematic viscosity—exponential fit based on the upper half of the original temperature range
Dynamic viscosity—exponential fit based on the upper half of the original temperature range
Prandtl number—exponential fit based on the upper half of the original temperature range
Aviation fuel Jet-A
The properties of Jet A fuel are valid at temperatures above -50.93°C up to 372.46°C (222.22 K to 645.61 K); they are valid at pressures above the saturation point up to a maximum value of 2.41 MPa. Pressures below the saturation point for a given temperature row are assigned a value of 0 in the validity matrix.
The properties of diesel fuel are valid at temperatures above -34.95°C up to 417.82°C (238.20 K to 690.97 K); they are valid at pressures above the saturation point up to a maximum value of 2.29 MPa. Pressures below the saturation point for a given temperature row are assigned a value of 0 in the validity matrix.
The properties of SAE 5W-30 fuel derive from data covering different temperature and pressure ranges for each property but all extended by extrapolation to (-38, 200) C and (0.01, 100) MPa.
The density and thermal expansion coefficients of the aqueous mixtures of glycol and glycerol compounds are obtained from the block parameters. The fluid density, with respect to pressure and temperature, is calculated as:
\rho \left(T,p\right)=\rho \left(T\right)\text{exp}\left(\frac{p-{p}_{\text{R}}}{\beta }\right),
T is the network temperature.
p is the network pressure.
pR is the reference pressure associated with the fluid property tables.
ß is the isothermal bulk modulus.
where the change in fluid density is evaluated as:
{\left(\frac{\partial \rho \left(T,p\right)}{\partial T}\right)}_{p}={\left(\frac{\partial \rho \left(T\right)}{\partial T}\right)}_{T}\text{exp}\left(\frac{p-{p}_{\text{R}}}{\beta }\right).
The thermal expansion coefficient is calculated as:
\alpha \left(T,p\right)=-\frac{1}{\rho \left(T,p\right)}{\left(\frac{\partial \rho \left(T\right)}{\partial T}\right)}_{T}\text{exp}\left(\frac{p-{p}_{\text{R}}}{\beta }\right).
R — Thermal liquid network for which to define the working fluid
Node identifying the thermal liquid network for which to define the necessary fluid properties. The fluid selected in this block applies to the entire network. No other Thermal Liquid Properties (TL) or Thermal Liquid Settings (TL) block may be connected to the same network.
Thermal liquid fluid list — Available working fluids
Water (default) | Seawater (MIT model) | Ethylene glycol and water mixture | Propylene glycol and water mixture | Glycerol and water mixture | Diesel Fuel | Aviation fuel Jet-A | SAE 5W-30
Sets the fluid properties of your liquid network. The available fluids include pure water, aqueous mixtures, motor oils, and fuels.
Dissolved salt mass fraction (salinity) — Mass of salt divided by the total mass of the saline mixture
3.5e-3 (default) | positive unitless scalar valued between 0 and 1
Ratio of the mass of salt present in the saline mixture to the total mass of that mixture.
This parameter is active only when Seawater (MIT model) is selected as the working fluid.
Concentration type — Quantity in terms of which the mixture concentration is defined
Volume fraction (default) | Mass fraction
Quantity in terms of which to specify the concentration of ethylene glycol in its aqueous mixture. This parameter is active only when either Ethylene glycol and water mixture or Propylene glycol and water mixture is selected as the working fluid.
Ethylene glycol volume fraction — Volume of ethylene glycol divided by that of the aqueous mixture
0.1 (default) | positive unitless scalar valued between 0 and 1
Volume of ethylene glycol present in the aqueous mixture divided by the total volume of that mixture.
This parameter is active when Ethylene glycol and water mixture is selected as the working fluid and Volume fraction is selected as the concentration type.
Ethylene glycol mass fraction — Mass of ethylene glycol divided by that of the aqueous mixture
Mass of ethylene glycol present in the aqueous mixture divided by the total mass of that mixture.
This parameter is active when Ethylene glycol and water mixture is selected as the working fluid and Mass fraction is selected as the concentration type.
Propylene glycol volume fraction — Mass of propylene glycol divided by that of the aqueous mixture
Volume of propylene glycol present in the aqueous mixture divided by the total volume of that mixture.
This parameter is active when Propylene glycol and water mixture is selected as the working fluid and Volume fraction is selected as the concentration type.
Propylene glycol mass fraction — Mass of propylene glycol divided by that of the aqueous mixture
Mass of propylene glycol present in the aqueous mixture divided by the total mass of that mixture.
This parameter is active when Propylene glycol and water mixture is selected as the working fluid and Mass fraction is selected as the concentration type.
Isothermal bulk modulus — Measure of fluid compressibility specified at constant temperature
2.1791 GPa (default) | positive scalar with units of pressure
Bulk modulus of the aqueous mixture at constant temperature. The bulk modulus measures the change in pressure required to produce a fractional change in fluid volume.
This parameter is active when either Ethylene glycol and water mixture, Propylene glycol and water mixture, or Glycerol and water mixture is selected as the working fluid.
Minimum valid pressure — Lower bound of the allowed pressure range for the thermal liquid network
0.01 MPa (default) | positive scalar with units of pressure
Lower bound of the pressure range allowed in the thermal liquid network connected to this block.
Maximum valid pressure — Upper bound of the allowed pressure range for the thermal liquid network
50 MPa (default) | positive scalar with units of pressure
Upper bound of the pressure range allowed in the thermal liquid network connected to this block.
Atmospheric pressure — Absolute pressure in the environment of the thermal liquid network
0.101325 MPa (default) | positive scalar with units of pressure
Absolute pressure of the external environment in which the thermal liquid network is assumed to run. The default value is the standard atmospheric pressure measured at sea level on Earth.
[1] Massachusetts Institute of Technology (MIT), Thermophysical properties of seawater database. http://web.mit.edu/seawater.
[2] K.G. Nayar, M.H. Sharqawy, L.D. Banchik, J.H. Lienhard V, Thermophysical properties of seawater: A review and new correlations that include pressure dependence, Desalination, Vol. 390, pp. 1-24, 2016.
[3] M.H. Sharqawy, J.H. Lienhard V, S.M. Zubair, Thermophysical properties of seawater: A review of existing correlations and data, Desalination and Water Treatment, Vol. 16, pp. 354-380.
|
Match the system of equations in the left column with its solution in the right column.
\begin{aligned}[t] 6x &- y = 4 \\ 3x &+ y = 5 \end{aligned}
\begin{aligned}[t] &x = y + 4 \\ &2x + 3y = -12 \end{aligned}
\begin{aligned}[t] &5x - 2y = 1 \\ &y = 2x + 1 \end{aligned}
\left(0,-4\right)
\left(3,7\right)
\left(1,2\right)
One way is to substitute the points in the right column into the equations in the left column and find which point (1, 2, or 3) works with which system of equations (a, b, or c).
Another way is to solve the system and look for the solution.
(a) matches to (3).
|
Optimization of Plasma Treatment, Manipulative Variables and Coating Composition for the Controlled Filling and Coating of a Microstructured Reservoir Stent | J. Med. Devices | ASME Digital Collection
Mustapha Mekki,
Stéphane Durual,
Stéphane Durual
Laboratory of Biomaterials,
Johannes Lammers,
Laboratory of Biomaterials, School of Dentistry,
Mustapha Mekki Research Scientist
Stéphane Durual Research Scientist
Susanne S. Scherrer Research Scientist
Johannes Lammers Consultant
H. W. Anselm Wiskott Research Scientist
Mekki, M., Durual, S., Scherrer, S. S., Lammers, J., and Wiskott, H. W. A. (March 9, 2009). "Optimization of Plasma Treatment, Manipulative Variables and Coating Composition for the Controlled Filling and Coating of a Microstructured Reservoir Stent." ASME. J. Med. Devices. March 2009; 3(1): 011005. https://doi.org/10.1115/1.3081394
The object of the study was to fill and coat the microcavities of a drug eluting stent using a batch dipping process. 316L coronary stents, which were coated with a
0.25 μm
layer of
TiNOx
were used as substrates. The stents’ surface was dimpled with
0.21 μl
microcavities separated by distances of
17–28 μm
depending on location. The experiment consisted of (1) optimizing the procedures to fill the microcavities with a solution of therapeutic agent and (2) covering the filled microcavities with a protective “lid” that shielded the solution during stent insertion in the arteries and then controlled its release into the surrounding tissue. The filling solution was a water-propanol mix containing 20% L-arginine. The coating solution was comprised of poly-ethylene-glycol (PEG-8000) and dexamethasone. The filling quality was investigated after altering the following variables: plasma surface activation (type of gas, pressure, power, and duration), water-propanol percentage ratio of the filling solution, lifting speed from the bath, and effect of ultrasonic vibration (monofrequency versus multifrequency). The surface coating was evaluated by altering the PEG-8000-dexamethasone percentage ratio and recording the effects on coating thickness and structure, on elution rate, and on wear resistance. The optimized process is presented in detail.
biomedical materials, dip coating, microcavities, optimisation, plasma materials processing, titanium compounds, wear resistant coatings, drug eluting stent, microstructured surface, drug loading, surface energy
Coating processes, Coatings, Plasmas (Ionized gases), Stents, Drugs, Wear resistance, Optimization
I. A. L.
Biocompatibility of Stent Materials
A Comparison of Balloon-Expandable-Stent Implantation With Balloon Angioplasty in Patients With Coronary Artery Disease
Sirolimus-Eluting Stent or Paclitaxel-Eluting Stent Vs Balloon Angioplasty for Prevention of Recurrences in Patients With Coronary In-Stent Restenosis: A Randomized Controlled Trial
Artery Size, Neointima, and Remodeling: Time for Some Standards
Prospects for Site-Specific Delivery of Pharmacologic and Molecular Therapies
Stent-Based Delivery of Sirolimus Reduces Neointimal Formation in a Porcine Coronary Model
Novel Tubulin-Targeting Agents: Anticancer Activity and Pharmacologic Profile of Epothilones and Related Analogues
Pro-Healing Drug-Eluting Stents: A Role for Antioxidants?
Features and Parameters of Drug-Eluting Stent Deployment Discoverable by Intravascular Ultrasound
Understanding the Role of Transforming Growth Factor-β1 in Intimal Thickening After Vascular Injury
Role of Arginine Metabolism in Immunity and Immunopathology
Stent-Based Tempamine Delivery on Neointimal Formation in a Porcine Coronary Model
Sawitowski
Inorganic Materials as Drug Delivery Systems in Coronary Artery Stenting
Materialwiss. Werkstofftech.
Electrochemical Micromachining, Polishing and Surface Structuring of Metals: Fundamental Aspects and New Developments
Randomized Comparison of a Titanium-Nitride-Oxide-Coated Stent With a Stainless Steel Stent for Coronary Revascularization: The TINOX Trial
Arginine Stimulates Wound Healing and Immune Function in Elderly Human Beings
Hernandez-Richter
L-Arginine Improves Wound Healing After Trauma-Hemorrhage by Increasing Collagen Synthesis
Management of Newly Diagnosed Myeloma
El-Shourbagy
Determination of Sirolimus in Rabbit Arteries Using Liquid Chromatography Separation and Tandem Mass Spectrometric Detection
Hossainy
Modeling of Degradation and Drug Release From a Biodegradable Stent Coating
Dose Model for Stent-Based Delivery of a Radioactive Compound for the Treatment of Restenosis in Coronary Arteries
Rodes-Cabau
In Situ Synthesis of Fe–TiC Nanocomposite Coating on CK45 Steel From Ilmenite Concentrate by Plasma-Spray Method
|
Davydov_soliton Knowpia
Davydov soliton is a quantum quasiparticle representing an excitation propagating along the protein α-helix self-trapped amide I. It is a solution of the Davydov Hamiltonian. It is named for the Soviet and Ukrainian physicist Alexander Davydov. The Davydov model describes the interaction of the amide I vibrations with the hydrogen bonds that stabilize the α-helix of proteins. The elementary excitations within the α-helix are given by the phonons which correspond to the deformational oscillations of the lattice, and the excitons which describe the internal amide I excitations of the peptide groups. Referring to the atomic structure of an α-helix region of protein the mechanism that creates the Davydov soliton (polaron, exciton) can be described as follows: vibrational energy of the C=O stretching (or amide I) oscillators that is localized on the α-helix acts through a phonon coupling effect to distort the structure of the α-helix, while the helical distortion reacts again through phonon coupling to trap the amide I oscillation energy and prevent its dispersion. This effect is called self-localization or self-trapping.[3][4][5] Solitons in which the energy is distributed in a fashion preserving the helical symmetry are dynamically unstable, and such symmetrical solitons once formed decay rapidly when they propagate. On the other hand, an asymmetric soliton which spontaneously breaks the local translational and helical symmetries possesses the lowest energy and is a robust localized entity.[6]
Quantum dynamics of a Davydov soliton with
{\displaystyle \chi =35}
pN generated by an initial Gaussian step distribution of amide I energy over 3 peptide groups at the N-end of a single α-helix spine composed of 40 peptide groups (extending along the x-axis) during a period of 125 picoseconds. Quantum probabilities
{\displaystyle |a_{n}|^{2}}
of amide I excitation are plotted in blue along the z-axis. Phonon lattice displacement differences
{\displaystyle b_{n}-b_{n-1}}
(measured in picometers) are plotted in red along the y-axis. The soliton is formed by self-trapping of the amide I energy by the induced lattice distortion.[1][2]
Davydov HamiltonianEdit
Davydov Hamiltonian is formally similar to the Fröhlich-Holstein Hamiltonian for the interaction of electrons with a polarizable lattice. Thus the Hamiltonian of the energy operator
{\displaystyle {\hat {H}}}
{\displaystyle {\hat {H}}={\hat {H}}_{\text{ex}}+{\hat {H}}_{\text{ph}}+{\hat {H}}_{\text{int}}}
{\displaystyle {\hat {H}}_{\text{ex}}}
is the exciton Hamiltonian, which describes the motion of the amide I excitations between adjacent sites;
{\displaystyle {\hat {H}}_{\text{ph}}}
is the phonon Hamiltonian, which describes the vibrations of the lattice; and
{\displaystyle {\hat {H}}_{\text{int}}}
is the interaction Hamiltonian, which describes the interaction of the amide I excitation with the lattice.[3][4][5]
The exciton Hamiltonian
{\displaystyle {\hat {H}}_{\text{ex}}}
{\displaystyle {\hat {H}}_{\text{ex}}=}
{\displaystyle \sum _{n,\alpha }E_{0}{\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n,\alpha }}
{\displaystyle -J_{1}\sum _{n,\alpha }\left({\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n+1,\alpha }+{\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n-1,\alpha }\right)}
{\displaystyle +J_{2}\sum _{n,\alpha }\left({\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n,\alpha +1}+{\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n,\alpha -1}\right)}
{\displaystyle n=1,2,\cdots ,N}
counts the peptide groups along the α-helix spine, the index
{\displaystyle \alpha =1,2,3}
counts each α-helix spine,
{\displaystyle E_{0}=32.8}
zJ is the energy of the amide I vibration (CO stretching),
{\displaystyle J_{1}=0.155}
zJ is the dipole-dipole coupling energy between a particular amide I bond and those ahead and behind along the same spine,[7]
{\displaystyle J_{2}=0.246}
zJ is the dipole-dipole coupling energy between a particular amide I bond and those on adjacent spines in the same unit cell of the protein α-helix,[7]
{\displaystyle {\hat {A}}_{n,\alpha }^{\dagger }}
{\displaystyle {\hat {A}}_{n,\alpha }}
are respectively the boson creation and annihilation operator for an amide I exciton at the peptide group
{\displaystyle (n,\alpha )}
The phonon Hamiltonian
{\displaystyle {\hat {H}}_{\text{ph}}}
is[11][12][13][14]
{\displaystyle {\hat {H}}_{\text{ph}}={\frac {1}{2}}\sum _{n,\alpha }\left[w_{1}({\hat {u}}_{n+1,\alpha }-{\hat {u}}_{n,\alpha })^{2}+w_{2}({\hat {u}}_{n,\alpha +1}-{\hat {u}}_{n,\alpha })^{2}+{\frac {{\hat {p}}_{n,\alpha }^{2}}{M_{n,\alpha }}}\right]}
{\displaystyle {\hat {u}}_{n,\alpha }}
is the displacement operator from the equilibrium position of the peptide group
{\displaystyle (n,\alpha )}
{\displaystyle {\hat {p}}_{n,\alpha }}
is the momentum operator of the peptide group
{\displaystyle (n,\alpha )}
{\displaystyle M_{n,\alpha }}
is the mass of the peptide group
{\displaystyle (n,\alpha )}
{\displaystyle w_{1}=13-19.5}
N/m is an effective elasticity coefficient of the lattice (the spring constant of a hydrogen bond)[9] and
{\displaystyle w_{2}=30.5}
N/m is the lateral coupling between the spines.[12][15]
Finally, the interaction Hamiltonian
{\displaystyle {\hat {H}}_{\text{int}}}
{\displaystyle {\hat {H}}_{\text{int}}=\chi \sum _{n,\alpha }\left[({\hat {u}}_{n+1,\alpha }-{\hat {u}}_{n,\alpha }){\hat {A}}_{n,\alpha }^{\dagger }{\hat {A}}_{n,\alpha }\right]}
{\displaystyle \chi =35-62}
pN is an anharmonic parameter arising from the coupling between the exciton and the lattice displacements (phonon) and parameterizes the strength of the exciton-phonon interaction.[9] The value of this parameter for α-helix has been determined via comparison of the theoretically calculated absorption line shapes with the experimentally measured ones.
Davydov soliton propertiesEdit
There are three possible fundamental approaches for deriving equations of motion from Davydov Hamiltonian:
quantum approach, in which both the amide I vibration (excitons) and the lattice site motion (phonons) are treated quantum mechanically;[16]
mixed quantum-classical approach, in which the amide I vibration is treated quantum mechanically but the lattice is classical;[10]
classical approach, in which both the amide I and the lattice motions are treated classically.[17]
The mathematical techniques that are used to analyze the Davydov soliton are similar to some that have been developed in polaron theory.[18] In this context, the Davydov soliton corresponds to a polaron that is:
large so the continuum limit approximation is justified,[9]
acoustic because the self-localization arises from interactions with acoustic modes of the lattice,[9]
weakly coupled because the anharmonic energy is small compared with the phonon bandwidth.[9]
The Davydov soliton is a quantum quasiparticle and it obeys Heisenberg's uncertainty principle. Thus any model that does not impose translational invariance is flawed by construction.[9] Supposing that the Davydov soliton is localized to 5 turns of the α-helix results in significant uncertainty in the velocity of the soliton
{\displaystyle \Delta v=133}
m/s, a fact that is obscured if one models the Davydov soliton as a classical object.
^ Georgiev, Danko D.; Glazebrook, James F. (2019). "On the quantum dynamics of Davydov solitons in protein α-helices". Physica A: Statistical Mechanics and Its Applications. 517: 257–269. arXiv:1811.05886. doi:10.1016/j.physa.2018.11.026. MR 3880179.
^ Georgiev, Danko D.; Glazebrook, James F. (2019). "Quantum tunneling of Davydov solitons through massive barriers". Chaos, Solitons and Fractals. 123: 275–293. arXiv:1904.09822. doi:10.1016/j.chaos.2019.04.013. MR 3941070.
^ a b Davydov, Alexander S. (1973). "The theory of contraction of proteins under their excitation". Journal of Theoretical Biology. 38 (3): 559–569. doi:10.1016/0022-5193(73)90256-7. PMID 4266326.
^ a b Davydov, Alexander S. (1977). "Solitons and energy transfer along protein molecules". Journal of Theoretical Biology. 66 (2): 379–387. doi:10.1016/0022-5193(77)90178-3. PMID 886872.
^ a b Davydov, Alexander S. (1979). "Solitons, bioenergetics, and the mechanism of muscle contraction". International Journal of Quantum Chemistry. 16 (1): 5–17. doi:10.1002/qua.560160104.
^ Brizhik, Larissa; Eremko, Alexander; Piette, Bernard; Zakrzewski, Wojtek (2004). "Solitons in α-helical proteins". Physical Review E. 70 (3 Pt 1): 031914. arXiv:cond-mat/0402644. Bibcode:2004PhRvE..70a1914K. doi:10.1103/PhysRevE.70.011914. PMID 15524556.
^ a b Nevskaya, N. A.; Chirgadze, Yuriy Nikolaevich (1976). "Infrared spectra and resonance interactions of amide-I and II vibrations of α-helix". Biopolymers. 15 (4): 637–648. doi:10.1002/bip.1976.360150404.
^ Hyman, James M.; McLaughlin, David W.; Scott, Alwyn C. (1981). "On Davydov's alpha-helix solitons". Physica D: Nonlinear Phenomena. 3 (1): 23–44. Bibcode:1981PhyD....3...23H. doi:10.1016/0167-2789(81)90117-2.
^ a b c d e f g Scott, Alwyn C. (1992). "Davydov's soliton". Physics Reports. 217 (1): 1–67. Bibcode:1992PhR...217....1S. doi:10.1016/0370-1573(92)90093-F.
^ a b Cruzeiro-Hansson, Leonor; Takeno, Shozo (1997). "Davydov model: the quantum, mixed quantum-classical, and full classical systems". Physical Review E. 56 (1): 894–906. Bibcode:1997PhRvE..56..894C. doi:10.1103/PhysRevE.56.894.
^ Davydov, Alexander S. (1982). "Solitons in quasi-one-dimensional molecular structures". Soviet Physics Uspekhi. 25 (12): 898–918. doi:10.1070/pu1982v025n12abeh005012.
^ a b Georgiev, Danko D.; Glazebrook, James F. (2022). "Thermal stability of solitons in protein α-helices". Chaos, Solitons and Fractals. 155: 111644. arXiv:2202.00525. doi:10.1016/j.chaos.2021.111644. MR 4372713. S2CID 244693789.
^ Zolotaryuk, Alexander V.; Christiansen, P. L.; Nordеn, B.; Savin, Alexander V. (1999). "Soliton and ratchet motions in helices". Condensed Matter Physics. 2 (2): 293–302. doi:10.5488/cmp.2.2.293.
^ Brizhik, Larissa S.; Luo, Jingxi; Piette, Bernard M. A. G.; Zakrzewski, Wojtek J. (2019). "Long-range donor-acceptor electron transport mediated by alpha-helices". Physical Review E. 100 (6): 062205. arXiv:1909.08266. doi:10.1103/PhysRevE.100.062205.
^ Savin, Alexander V.; Zolotaryuk, Alexander V. (1993). "Dynamics of the amide-I excitation in a molecular chain with thermalized acoustic and optical modes". Physica D: Nonlinear Phenomena. 68 (1): 59–64. doi:10.1016/0167-2789(93)90029-Z.
^ Kerr, William C.; Lomdahl, Peter S. (1987). "Quantum-mechanical derivation of the equations of motion for Davydov solitons". Physical Review B. 35 (7): 3629–3632. doi:10.1103/PhysRevB.35.3629. hdl:10339/15922. PMID 9941870.
^ Škrinjar, M. J.; Kapor, D. V.; Stojanović, S. D. (1988). "Classical and quantum approach to Davydov's soliton theory". Physical Review A. 38 (12): 6402–6408. doi:10.1103/PhysRevA.38.6402. PMID 9900400.
^ Sun, Jin; Luo, Bin; Zhao, Yang (2010). "Dynamics of a one-dimensional Holstein polaron with the Davydov ansätze". Physical Review B. 82 (1): 014305. arXiv:1001.3198. doi:10.1103/PhysRevB.82.014305.
|
Durot, Cécile ; Thiébot, Karelle
The paper is concerned with the asymptotic distributions of estimators for the length and the centre of the so-called
\eta
-shorth interval in a nonparametric regression framework. It is shown that the estimator of the length converges at the
{n}^{1/2}
-rate to a gaussian law and that the estimator of the centre converges at the
{n}^{1/3}
-rate to the location of the maximum of a brownian motion with parabolic drift. Bootstrap procedures are proposed and shown to be consistent. They are compared with the plug-in method through simulations.
Classification : 62E20, 62G05, 62G08, 62G09
Mots clés : brownian motion with parabolic drift, bootstrap, location of maximum, shorth
author = {Durot, C\'ecile and Thi\'ebot, Karelle},
title = {Bootstrapping the shorth for regression},
AU - Thiébot, Karelle
TI - Bootstrapping the shorth for regression
Durot, Cécile; Thiébot, Karelle. Bootstrapping the shorth for regression. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 216-235. doi : 10.1051/ps:2006007. http://www.numdam.org/articles/10.1051/ps:2006007/
[1] D.F. Andrews, P.J. Bickel, F.R. Hampel, P.J. Huber, W.H. Rogers and J.W. Tukey, Robust estimates of location. Survey and advances. Princeton Univ. Press, Princeton, N.J. (1972). | MR 331595 | Zbl 0254.62001
[2] D. De Angelis, P. Hall and G.A. Young, Analytical and bootstrap approximations to estimator distributions in
{l}^{1}
regression. J. Am. Stat. Assoc. 88 (1993) 1310-1316. | Zbl 0792.62029
[3] C. Durot and K. Thiébot, Detecting atypical data in air pollution studies by using shorth intervals for regression. ESAIM: PS 9 (2005) 230-240. | Numdam | Zbl 1137.62408
[4] M. Falk and R.-D. Reiss, Weak convergence of smoothed and nonsmoothed bootstrap quantiles estimates. Ann. Probab. 17 (1989) 362-371. | Zbl 0684.62036
[5] P. Groeneboom, Brownian motion with a parabolic drift and Airy functions Probab. Th. Rel. Fields 81 (1989) 79-109.
[6] R. Grübel, The length of the shorth. Ann. Statist. 16 (1988) 619-628. | Zbl 0664.62040
[7] P. Hall, Theoretical comparison of bootstrap confidence intervals. Ann. Statist. 16 (1988) 927-953. | Zbl 0663.62046
[8] P. Hall, T.J. Diciccio and J.P. Romano, On smoothing and the bootstrap. Ann. Statist. 17 (1989) 692-704. | Zbl 0672.62051
[9] P. Hall, J.W. Kay and D.M. Titterington, Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 (1990) 521-528.
[10] E. Janaszewska and A.V. Nagaev, On the joint distribution of the shorth height and length. Math. Meth. Statist. 7 (1998) 210-229. | Zbl 1103.62350
[11] J. Kim and D. Pollard, Cube root asymptotics. Ann. Statist. 18 (1990) 191-219. | Zbl 0703.62063
[12] A. Narayanan and T.W. Sager, Table for the asymptotic distribution of univariate mode estimators. J. Stat. Comput. Simul. 33 (1989) 37-51. | Zbl 0726.62022
[13] A.I. Sakhanenko, Estimates in the invariance principle. Predel'nye Teoremy Teorii Veroyatnostej, Tr. Inst. Mat. 5 (1985) 27-44. | Zbl 0585.60044
[14] G.R. Shorack and J.A. Wellner, Empirical processes with applications to statistics. New York, Wiley (1986). | MR 838963
[15] C.J. Stone, Optimal uniform rate of convergence for nonparametric estimators of a density function and its derivatives. Recent Advances in Statistics, Academic Press, New York (1983) 293-406. | Zbl 0591.62031
[16] K. Thiébot, Analyses statistiques et validation de données de pollution atmosphérique. Ph.D. thesis, Université Paris-Sud Orsay, France (2001).
[17] Y.G. Yatracos, On the estimation of the derivatives of a function with the derivatives of an estimate. J. Multivariate Anal. 28 (1989) 172-175. | Zbl 0665.62041
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SetToken - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : DataSets Package : Quandl : SetToken
Adding SetToken to Your Personal Initialization File
SetToken(token, premium)
string; the Quandl API authentication token to be used for API calls
(optional) an equation of the form
\mathrm{premium}=\mathrm{truefalse}
; denotes whether the token is a premium token and defaults to
\mathrm{false}
The SetToken commands sets the authentication token used for Quandl API calls.
By default, Maple does not use an authentication token for Quandl API calls. That is, API calls are made anonymously.
Anonymous API calls can only access public databases and are limited in the number of requests that can be made per day.
If you have a Quandl authentication token, use SetToken("yourToken") to perform more Quandl API requests and to access premium databases you have purchased. See http://www.quandl.com/ for more information on the Quandl API and authentication tokens.
The premium option is used by Maple to determine what errors to return when a request fails. It does not allow you to access to premium databases.
The default value for the premium option is false. In this case, Maple returns a warning and an empty data set the first time you request data from a premium database. After that, only the empty data set is returned (that is, Maple knows not to expect valid data sets for every request).
Set premium = true if you have access to premium databases. In this case, Maple always returns an error if your request results in an empty data set.
SetToken returns the value of the previous authentication token.
Add SetToken to your personal initialization file in order to set your Quandl authentication token every time you start or restart Maple.
\mathrm{with}\left(\mathrm{DataSets}\right):
The following command sets the Quandl authentication token to an empty string (that is, API calls are anonymous) and sets the premium option to false, which is the same as the default authentication method used by Maple. The command returns the value of the previous token.
\mathrm{Quandl}:-\mathrm{SetToken}\left("",\mathrm{premium}=\mathrm{false}\right)
\textcolor[rgb]{0,0,1}{""}
Your personal initialization file contains Maple commands that run every time you start or restart Maple.
If you have a Quandl authentication token, add the SetToken command to your initialization file in order to perform more requests and to access your premium databases.
To add SetToken to your personal initialization file:
If you have an initialization file, open it. If not, create a new one.
Note: The location and name of your personal initialization file depend on your operating system:
Windows: <homeDirectory>\maple.ini
Linux or Mac: <homeDirectory>/.mapleinit
Where <homeDirectory> is the output from the kernelopts(homedir) command.
Add the following command to your initialization file:
DataSets:-Quandl:-SetToken("yourToken", premium = premiumAccess ):
Where yourToken is your Quandl authentication token and premiumAccess is either true or false, depending on your Quandl access level.
Save and then close your initialization file.
To test your initialization file, open a new Maple worksheet and then run the command you entered in step 3. The output should be your Quandl authentication token.
Note: If you do not get your authentication token, then Maple is using another initialization file. See Create Maple Initialization File for details on where Maple looks for initialization files.
The DataSets[Quandl][SetToken] command was introduced in Maple 2015.
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Variable Powder Flow Rate Control in Laser Metal Deposition Processes | J. Manuf. Sci. Eng. | ASME Digital Collection
, 1870 Miner Circle, Rolla, MO 65409-0050
e-mail: ltx8d@mst.edu
e-mail: jzruan@mst.edu
e-mail: liou@mst.edu
Tang, L., Ruan, J., Landers, R. G., and Liou, F. (July 22, 2008). "Variable Powder Flow Rate Control in Laser Metal Deposition Processes." ASME. J. Manuf. Sci. Eng. August 2008; 130(4): 041016. https://doi.org/10.1115/1.2953074
This paper proposes a novel method, called variable powder flow rate control (VPFRC), for the regulation of powder flow rate in laser metal deposition processes. The idea of VPFRC is to adjust the powder flow rate to maintain a uniform powder deposition per unit length even when disturbances occur (e.g., the motion system accelerates and decelerates). Dynamic models of the powder delivery system motor and the powder transport system (i.e.,
5m
pipe, powder dispenser, and cladding head) are constructed. A general tracking controller is then designed to track variable powder flow rate references. Since the powder flow rate at the nozzle exit cannot be directly measured, it is estimated using the powder transport system model. The input to this model is the dc motor rotation speed, which is estimated online using a Kalman filter. Experiments are conducted to examine the performance of the proposed control methodology. The experimental results demonstrate that the VPFRC method is successful in maintaining a uniform track morphology, even when the motion system accelerates and decelerates.
flow control, Kalman filters, laser deposition, powder technology
Flow (Dynamics), Lasers, Engines, Motors
Process and Prosperities Control in Laser Aided Direct Metal/Materials Deposition Process
Proceedings of IMECE
, Nov. 17–22, pp.
The Investigation of Gravity-Driven Metal Powder Flow in Coaxial Nozzle for Laser-Aided Direct Metal Deposition Process
Automated Slicing for Multiaxis Metal Deposition System
An Analysis of Powder Feeding Systems on the Quality of Laser Cladding
, Chicago IL, June 29–July 2, Vol.
Sensing, Modeling and Control for Laser-Based Additive Manufacturing
Sensing, Modeling and Closed Loop Control of Powder Feeder for Laser Surface Modification
Regulation of Powder Mass Flow Rate in Gravity-Fed Powder Feeder Systems
Dynamic Modeling of Powder Delivery Systems in Gravity-Fed Powder Feeders
, Nov. 27–December, 1, pp.
Advanced Flow Control for Supersonic Blowdown Wind Tunnel Using Extended Kalman Filter
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Radiation astronomy/Lithometeors - Wikiversity
Radiation astronomy/Lithometeors
"A lithometeor consists of solid particles suspended in the air or lifted by the wind from the ground."[1]
Heavy metal pollution may occur in lithometeors.[3]
"The rise of airborne dust is constantly augmenting from the desert (Bilma) to the southern Sahelian stations (Niamey) where it has increased by a factor five. ... the Sahelian zone with airborne dust during the 80s ... All stations have recorded a general increase of wind velocity. The increase of lithometeors frequency as well as the wind velocity during the drought period is not explained by the aridification."[4]
22 Dust storms
33 Trifid Nebula
36 Dust rings
"The spectrum includes dust continuum, molecular rotation line and atomic fine-structure line emissions."[7]
"To learn how stars form, we have to catch them in their earliest phases, while they're still deeply embedded in clouds of gas and dust, and the SMA is an excellent telescope to do so."[9]
{\displaystyle n(S)=N_{0}/(a+S^{3.2}),}
{\displaystyle S}
{\displaystyle N_{0}}
{\displaystyle a}
Dust stormsEdit
Dust storm approaching Stratford, Texas USA. Credit: NOAA George E. Marsh Album, theb1365, Historic C&GS Collection.
On Earth, common weather phenomena include wind, cloud, rain, snow, fog and dust storms such as the one imaged on the right. Less common events include natural disasters such as tornadoes, hurricanes, typhoons and ice storms.
Zodiacal LightsEdit
This image shows two young brown dwarfs, objects that fall somewhere between planets and stars in terms of their temperature and mass. Credit: NASA/JPL-Caltech/D. Barrado [CAB/INTA-CSIC].
"This image [at right] shows two young brown dwarfs, objects that fall somewhere between planets and stars in terms of their temperature and mass. Brown dwarfs are cooler and less massive than stars, never igniting the nuclear fires that power their larger cousins, yet they are more massive (and normally warmer) than planets. When brown dwarfs are born, they heat the nearby gas and dust, which enables powerful infrared telescopes like NASA's Spitzer Space Telescope to detect their presence."[65]
"Here we see a long sought-after view of these very young objects, labeled as A and B, which appear as closely-spaced purple-blue and orange-white dots at the very center of this image. The surrounding envelope of cool dust surrounding this nursery can be seen in purple."[65]
"These twins, which were found in the region of the Taurus-Auriga star-formation complex, are the youngest of their kind ever detected. They are also helping astronomers solve a long-standing riddle about how brown dwarfs are formed more like stars or more like planets? Based on these findings, the researchers think they have found the answer: Brown dwarfs form like stars."[65]
"This image combined data from three different telescopes on the ground and in space. Near-infrared observations collected at the Calar Alto Observatory in Spain cover wavelengths of 1.3 and 2.2 microns (rendered as blue). Spitzer's infrared array camera contributed the 4.5-micron (green) and 8.0-micron (yellow) observations, and its multiband imaging photometer added the 24-micron (red) component. The Caltech Submillimeter Observatory in Hawaii made the far-infrared observations at 350 microns (purple)."[65]
Trifid NebulaEdit
Spitzer Space Telescope reveals what cannot be seen in visible light: cooler stars (blue), heated dust (reddish hue), and Sagittarius A* (Sgr A*) as bright white spot in the middle. Credit: NASA/JPL-Caltech/S. Stolovy (SSC/Caltech).{{free media}}
Of the total mass of the galaxy's stars, interstellar dust accounts for an additional 1% of the total mass of the gas.[70]
Dust ringsEdit
NGC 4435 and dust ring around its nucleus is by the Hubble Space Telescope. Credit: Friendlystar.{{free media}}
NGC 4435 is a barred lenticular galaxy currently interacting with NGC 4438 with a detected dust ring around its nucleus as shown in the imade on the right. Studies of the galaxy by the Spitzer Space Telescope revealed a relatively young (190 million years) stellar population within the galaxy's nucleus, which may have originated through the interaction with NGC 4438 compressing gas and dust in that region, triggering a starburst.[71] It also has a long tidal tail possibly caused by the interaction with the mentioned galaxy;[72] however, other studies suggest that tail is actually a galactic cirrus in the Milky Way totally unrelated to NGC 4435.[73]
"Red, green and blue correspond to the 160-micron, 100-micron and 70-micron wavelength bands of the Herschels Photoconductor Array Camera and Spectrometer, PACS instruments [in the far-infrared image on the right]."[74]
Starburst galaxyEdit
Massive, star-forming galaxies from the very distant, very early universe are invisible to telescopes like Hubble. Credit: T. Wang, C. Schreiber, D. Elbaz, Y. Yoshimura, K. Kohno, X. Shu, Y. Yamaguchi, M. Pannella, M. Franco, J. Huang, C.-F. Lim & W.-H. Wang.{{fairuse}}
Submillimetre "(wavelength 870 micrometres) detections of 39 massive star-forming galaxies at z > 3, which are unseen in the spectral region from the deepest ultraviolet to the near-infrared [have occurred]."[77]
The catalog entries that appear to correspond to numbers 1-4 in the image on the right are {COSMOS) COS-27285 (1) z =
{\displaystyle 4.32_{-0.22}^{+0.23}}
, COS-19762 (2) z =
{\displaystyle 3.52_{-0.19}^{+5.36}}
{\displaystyle 5.77_{-0.88}^{+0.80}}
, and COS-25881 (4) z =
{\displaystyle 6.58_{-1.38}^{+1.43}}
↑ 2.0 2.1 Mark R. Mireles; Kirth L. Pederson; Charles H. Elford (February 21, 2007). Meteorologial Techniques. Offutt Air Force Base, Nebraska, USA: Air Force Weather Agency/DNT. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA466107. Retrieved 2013-02-17.
↑ 10.0 10.1 A. J. Barger; L. L. Cowie; D. B. Sanders (June 10, 1999). "Resolving the submillimeter background: the 850 micron galaxy counts". The Astrophysical Journal 518 (1): L5-8. doi:10.1086/312054. http://iopscience.iop.org/1538-4357/518/1/L5. Retrieved 2013-10-22.
↑ 11.0 11.1 David Jewitt; Jane Luu (November 1992). "Submillimeter Continuum Emission from Comets". Icarus 108 (1): 187-96. http://www.sciencedirect.com/science/article/pii/0019103592900286. Retrieved 2013-10-22.
↑ 15.0 15.1 15.2 15.3 15.4 15.5 15.6 I.A. Smith; R.P.J. Tilanus; J. van Paradijs; T.J. Galama; P.J. Groot; P. Vreeswijk; C. Kouveliotou; R.A.M. Wijers et al. (July 1999). "SCUBA sub-millimeter observations of gamma-ray bursters I. GRB 970508, 971214, 980326, 980329, 980519, 980703, 981220, 981226". Astronomy and Astrophysics 347 (07): 92-8. http://adsabs.harvard.edu/full/1999A%26A...347...92S. Retrieved 2013-10-22.
↑ 27.0 27.1 27.2 nhsc2009 (October 2, 2009). Dark Wombs of Stars. Pasadena, California USA: California Institute of Technology. http://www.herschel.caltech.edu/image/nhsc2009-020b. Retrieved 2014-03-12.
↑ 32.0 32.1 32.2 32.3 32.4 S.G. Djorgovski. A Tour of the Radio Universe. National Radio Astronomy Observatory. http://www.cv.nrao.edu/course/astr534/Tour.html. Retrieved 2014-03-16.
↑ Bernhard Peucker-Ehrenbrink; Birger Schmitz (2001). Accretion of Extraterrestrial Matter Throughout Earth's History. Springer. pp. 66–67. ISBN 0-306-46689-9.
↑ 56.0 56.1 Benjamin Zuckerman (July 8, 1998). Astronomers discover a nearby star system just like our own Solar System. Hilo, Hawaii, USA: Joint Astronomy Centre. http://outreach.jach.hawaii.edu/pressroom/1998_epseri/. Retrieved 2014-03-12.
↑ 65.0 65.1 65.2 65.3 ssc2009 (November 23, 2009). Twin Brown Dwarfs Wrapped in a Blanket. Pasadena, California USA: Caltech. http://www.spitzer.caltech.edu/images/2838-ssc2009-21a-Twin-Brown-Dwarfs-Wrapped-in-a-Blanket. Retrieved 2014-03-12.
↑ ESA/PACS/SPIRE/ Consortia (September 1, 2010). Water Around a Carbon Star. Pasadena, California USA: Caltech. http://www.herschel.caltech.edu/image/nhsc2010-011a. Retrieved 2014-03-12.
↑ "The Interstellar Medium". Retrieved May 2, 2015.
↑ Panuzzo, P.; Vega, O.; Bressan, A.; Buson, L. et al. (2007). "The Star Formation History of the Virgo Early-Type Galaxy NGC 4435: The Spitzer Mid-Infrared View". The Astrophysical Journal 656 (1): 206–216. doi:10.1086/510147.
↑ The Tail of NGC 4435
↑ Cortese, L.; Bendo, G. J.; Isaak, K. G.; Davies, J. I. et al. (2010). "Diffuse far-infrared and ultraviolet emission in the NGC 4435/4438 system: tidal stream or Galactic cirrus?". Monthly Notices of the Royal Astronomical Society: Letters 403 (1): L26–L30. doi:10.1111/j.1745-3933.2009.00808.x.
↑ 74.0 74.1 ESA; the PACS Consortium (26 June 2009). Herschels Daring Test: A Glimpse of Things to Come. NASA. https://www.herschel.caltech.edu/image/nhsc2009-016a. Retrieved 2017-07-21.
↑ T. Wang; C. Schreiber; D. Elbaz; Y. Yoshimura; K. Kohno; X. Shu; Y. Yamaguchi; M. Pannella et al. (8 August 2019). "A dominant population of optically invisible massive galaxies in the early Universe". Nature Letters 572 (1452): 211-14. doi:10.1038/s41586-019-1452-4. https://www.nature.com/articles/s41586-019-1452-4.epdf?referrer_access_token=ly3DoECoasmJSGIbuvr6ONRgN0jAjWel9jnR3ZoTv0McTGoPc7_FoP-XNr3dFjBSdd_sfbj28wzUtooPhpwug25SJ_NFYLfEOZHG9D-KaO8ab5ehBsQw_UXZGw7HyyN33WADb2Uxv-W1NMGV7GWVFEZSwd82Ahb5EujTlVfR8NJuC-joBREial3N5iSnNgVThmfJ5AbamWAjBWjTTyNXd2SAh6e7drERlwLey-t4HyKGkoicOsjZS9iUN1KD99b4sS9JXEn4fUsVDGOJcMh1sGI8EfpM8erAzLlvL8lmXA_H2LNT1npr_eB-IQU2IwQ5DK15yg_gUiIKKRbtAdkx0sHnSVTsLQpv4nEY8eGIWEQCevYK-hsP9t07ndCGmqsFLGFPWnqY9pCOaeAnSPJSVA%3D%3D&tracking_referrer=www.sciencenews.org. Retrieved 11 August 2019.
Learn more about Lithometeor radiation astronomy
Retrieved from "https://en.wikiversity.org/w/index.php?title=Radiation_astronomy/Lithometeors&oldid=2368894"
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Energy Electromagnetic Force Equivalence, E = Fe x r
Abstract: By combing the mass energy equivalence formula, E=mc2, with the speed of light and gravity formula, , a new equation for energy is derived. The Energy Electromagnetic Force Equivalence is used to explain the rotational torque of the stars, planets, and galaxy. A greater understanding of the universe is achieved with a simple mathematical expression. Nuclear energy from the Sun is converted to an electrical force which pervades the universe and gives the bodies within it, rotational motion. It is a macro equation for action at a distance. The equation suggests that nuclear energy and electromagnetic force is one of the most basic equations of the universe. It is proposed that there is only one all-encompassing force of the universe, the electromagnetic force.
Keywords: c2, Centripetal Acceleration, Energy, Force, Mass Energy Equivalence, Light
My insight into the nature of light has led to the possible merging of electromagnetism and special relativity. In his 1905 Annus Mirabilis paper “Does the Inertia of an Object Depend Upon Its Energy Content?”, Albert Einstein states that if a body gives off energy
E
in the form of radiation, its mass diminishes by
E/{c}^{2}
[1] . Radiation means electromagnetic radiation, or light, and mass means the ordinary Newtonian mass of a slow-moving object. By combining Einstein’s
work with my own equation
g=\frac{{c}^{2}}{r}
, a greater understanding of the force of the universe is achieved.
To pull off such a feat of Theoretical Engineering, which is the merging of the dreamy world of Theoretical Physics with the hands-on practical world of Electrical Engineering, we must first analyze the genesis of General Relativity in Einstein’s own words:
“The centrifugal force, which acts under given conditions of a body is determined precisely by the same natural constant that also gives its action in a gravitational field. In fact, we have no means to distinguish a centrifugal field from a gravitational field. We thus, always measure, as the weight of the body on the surface of the Earth the superposed action of both fields, named above, and we cannot separate their actions. In this manner the point of view to interpret the rotating system K’ as at rest, and the centrifugal field as a gravitational field, gains justification by all means. This interpretation is reminiscent of the original (more special) relativity where the ponderomotively acting force, upon an electrically charged mass which moves in a magnetic field, is the action of the electric field, which is found at the location of the mass as seen by the reference system at rest with the moving mass” [2] .
This is one of the most insightful statements in scientific history as Dr. Einstein looks back to explain his approach to general relativity. From his description of the happiest thought in his life, we learned that gravity was an acceleration [3] ; and then later he explains that from his equivalency principle he makes a leap and concludes that acceleration is a gravitational field. General relativity was a huge turning point in science and set a course that has directed us for the past one hundred years. After a century of rote learning and application of the theory, we have an opportunity to review and add to his body of work.
According to Einstein’s equivalency principle there is a centrifugal field to mirror his gravitational field and space time theory. To put this in Newtonian perspective
{C}_{f}={G}_{f}
, and equivalency works both ways. From my previous work we know that centrifugal acceleration and centripetal acceleration are fraternal twins birthed from an electromagnetic force. Einstein spent many of his last years unsuccessfully bogged down on his unification theory involving Maxwell’s equations. There is an opportunity to help bring this complex physics conundrum into focus by continued focus on the centrifugal and centripetal fields.
2. Mass Energy Equivalence
Albert Einstein was able to derive a law that we still use today, governed by one of the simplest but most powerful equations ever to be written down,
E=m{c}^{2}
. There are only three parts to Einstein’s most famous statement.
E
, or energy, represents the total energy of the system;
m
, or mass, is related to energy by a conversion factor; and
{c}^{2}
, which is the speed of light squared, a seemingly incomprehensible factor we need to make mass and energy equivalent. Energy can be an abstract concept, but we use it in some form every day. Mass of course is something that each of us is familiar with, but what is
{c}^{2}
? Where does the squaring of light come from and how does it relate to energy?
3. Speed of Light and Gravity
By simple substitution of the speed of light, or
c
, in the centripetal acceleration equation
g=\frac{{c}^{2}}{r}
g
is the gravitational force, c is the speed of light and
r
, is the radius of the arc of light.
This equation gives us a clue as to how energy is created in the universe, for we can now see that light is bending and curving around celestial bodies such as stars and planets. The bending of light by the magnetic fields of the celestial bodies creates a centripetal acceleration inward. We call this inward acceleration gravity, but it is derived from an electromagnetic force that is bending the light. It is the same phenomena of a Ferris wheel or a bucket of water on a rope. Gravity is a simple centripetal acceleration. We can now discern that light squared is merely velocity squared which is Newton’s mathematical model of centripetal acceleration.
{c}^{2}
is then the cross product of
\text{gravity}\times r
{c}^{2}=g\times r
. And, since light, or
c
, according to Maxwell is an electromagnetic radiation, we can deduce that gravity is the result of an electromagnetic force. The four forces of nature have thus been consolidated into three forces of nature; strong nuclear, weak nuclear and electromagnetic. The gravitational force no longer exists.
4. Calculating Energy Electromagnetic Force Equivalence
E=m\times {c}^{2}
{c}^{2}=g\times r
E=m\times g\times r
F=m\times a
F=m\times g
E=F\times r
Since our equation is derived from light, or
c
, and light is derived from electromagnetic radiation according to Maxwell, I add a subscript denoting electromagnetic force. The final equation for a rotational electrical system is then:
E={F}_{e}\times r
The formula says that nuclear energy is equivalent to the cross product of rotational electromagnetic force and radius.
I have taken the liberty of inserting a Newtonian equation into what is considered an Einstein relativity equation and some explanation is required. To do so will require some background into relativity. Although, in many cases, Einstein gets credit for relativity, it was first proposed by Hendrik Lorentz in 1895, published by Joseph Larmor in 1897 and eventually modified by Henri Poincare in 1905, but accredited to Lorentz by Poincare. Einstein is credited with tying it all together in his Special Relativity paper [5] . Looking back at the original Lorentz Transformation and using time as our variable:
{t}^{\prime }=\frac{t}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}
In this equation, we see that time and velocity are variables because neither of them has a constant physical value, like the speed of light “
c
”. But we know that the speed of light in a vacuum must be constant in the universe. Newton viewed space-time as being flat, unchanging, but that is not at all the case in Einstein’s four dimensional covariant world. To Einstein, space-time is very dynamic, changing depending on gravity and velocity.
In everyday cosmic life, both Newton’s and Einstein’s views coexist, but the speeds at which planets and stars travel are relatively slow compared to the speed of light. My equation has cancelled out light entirely and there is no velocity function. I do acknowledge that although the equation is developed in the world of relativity, its application for slow moving starts and planets is chiefly Newtonian. If the equation was to be considered for application at near light speed, then more research and revision would have to be made.
Work is the product of force and distance. A force is said to do work if, when acting, there is a movement of the point of application in the direction of the force. When the force is constant and the angle between the force and the displacement is θ, then the work done is given by
W={F}_{s}\mathrm{cos}\theta
. Work transfers energy from one place to another, or one form to another. The SI unit of work is the joule (J).
6. Cosmic Tangential Acceleration
Tangential acceleration is a measure of how the tangential velocity of a point at a certain radius changes with time. Tangential acceleration is just like linear acceleration, but it is specific to the tangential direction, which is relevant to circular motion.
Tangential acceleration results from the change of the speed of the object along the curved path and at a particular point on the curved path it is equal to the instantaneous velocity at that particular point.
{a}_{t}=\frac{\text{d}|v|}{\text{d}t}
Radial acceleration is because of the change in the direction of the velocity. Its magnitude is equal to the tangential acceleration equation, which is given by,
|ar|={v}^{2}r
r
= radius of curvature
Acceleration at any point on the circular path is not always in the direction of the tangent to that particular point. That is why the acceleration at any point on the circular path has two components. These two components are perpendicular to each other. One component is the tangential component of acceleration and the other component is the radial component of acceleration. Direction of the tangential component of acceleration at any particular point on the curved path is in the direction of the tangent at that particular point.
The radial component of acceleration is due to the centripetal force, which is acting towards the center of the curved path. The resultant acceleration vector at any point on the curved path is the vector sum of the tangential vector and the radial component of the vector.
We thus conclude that in the case of cosmic tangential acceleration, we can conceive of accelerated motion of the planetary bodies out and away from their starting point. This supports an accelerating and expanding universe [7] .
I have shown that the new electrical equation,
E={F}_{e}\times r
, has a mechanical equivalent
W=F\times s
. This implies that a rotational electromagnetic field is similar to a linear Newtonian expression of work or a change in energy. This is initial confirmation that the new Energy equation is valid and rooted in Newton’s law of motion. An even closer analogy is the expression for Torque, or
T={F}_{m}\times r
{F}_{m}
is the mechanical force rotating a lever arm [8] . We have similar electrical engineering equations for electrostatic and magnetostatic torque, which may be related [9] .
It is deduced that nuclear energy (E) of the universe is what creates the electromagnetic force that bends particles and light, and also rotates the stars and the planets.
E={F}_{e}\times r
is the electrical analogy for galactic torque. We experience it every day when the Sun rises and sets. It is one of the most basic laws of the universe.
We have known about all types of energies, including mechanical energy, chemical energy, and electrical energy. These are all energies inherent to moving objects, and these forms of energy can be used to do work, such as run a car engine or power an electric light bulb. But mass at rest has energy inherent to it: a tremendous amount of energy. Electromagnetic attraction, or gravitation, which works between any two masses in the universe also does work based on a change in energy, which is equivalent to mass via
E=m{c}^{2}
. Mass can be converted into energy, but I have now shown that it is also converted into electrical forces,
E={F}_{e}\times r
. Energy is transferred from a Sun to a planet via rotational electrical force. Torque is what turns the planetary motors and solar generators, which cuts across magnetic field lines, to produce voltages and currents. The solar system experiences what can only be described as an electrical activity created by nuclear energy from the Sun.
In nuclear fission or fusion reactions, the mass of what we started with is greater than the mass we end up with. The amount of the difference is how much energy is released. This is true for everything from decaying uranium to fission bombs to nuclear fusion in the Sun [10] . It is also true for electromagnetic force. Energy released from the Sun is converted into electromagnetic force at ever increasing distances.
There is only one conclusion, nuclear energy is proportional to electromagnetic force. We know that
E=m{c}^{2}
is a nuclear equation. The conclusion of this paper suggests that there may be only one true all-pervasive force of the universe; electromagnetism which creates light and gravity. The four forces of nature may according to this hypothesis be combined into one universal electromagnetic force.
The author wishes to acknowledge the original thoughts and work of Sir Isaac Newton who wrote in query 30 of Optiks, circa. 1717: “Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition”?
Cite this paper: Poole, G. (2019) Energy Electromagnetic Force Equivalence, E = Fe x r. Journal of High Energy Physics, Gravitation and Cosmology, 5, 1057-1062. doi: 10.4236/jhepgc.2019.54058.
[1] Einstein, A. (1905) Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik, 18, 639-643.
[2] Einstein, A. (1914) The Formal Foundation of General Theory of Relativity. Volume 6. The Berlin Years Writing 1914-1917, Document 9, Plenary Session of November 19, 1914, Princeton University Press, Princeton, 32.
[3] Isaacson, W. (2008) His Life and Universe. Simon & Schuster, New York.
[4] Newton, I. (1850) Newton’s Principia: The Mathematical Principles of Natural Philosophy.
[5] Rothman, T. (2006) Lost in Einstein’s Shadow. American Scientist, 94, 112-113.
[6] Goodman, L.E. and Warner, W.H. (2001) Dynamics. Dover, New York.
[7] Overbye, D. (2017) Cosmos Controversy: The Universe Is Expanding, But How Fast? The New York Times.
[8] Fowler, W. (1914) The Mechanical Engineer. Vol. 34, The Scientific Publishing Company, Manchester.
[9] Fitzpatrick, R. (2008) Maxwell’s Equations and the Principles of Electromagnetism. Infinity Science Press, Hingham.
[10] Murray, R. (1993) Nuclear Energy: An Introduction to the Concepts, Systems and Application. Pergamon Press, Oxford.
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RationalFunctionTutor - Maple Help
Home : Support : Online Help : Education : Student Packages : Precalculus : Interactive : RationalFunctionTutor
Student[Precalculus][RationalFunctionTutor] - demonstrate the graphing of rational functions
RationalFunctionTutor(f)
(optional) rational function (ratio of polynomials) in at most one variable
The RationalFunctionTutor(f) command launches a tutor interface that demonstrates the graphing of the rational function f. If f has any asymptotes, they are also displayed.
If f is not specified, RationalFunctionTutor uses a default function.
\mathrm{with}\left(\mathrm{Student}[\mathrm{Precalculus}]\right):
\mathrm{RationalFunctionTutor}\left(\right)
\mathrm{RationalFunctionTutor}\left(\frac{{x}^{3}+{x}^{2}-x+1}{{x}^{3}}\right)
Student, Student[Precalculus], Student[Precalculus][InteractiveOverview], Student[Precalculus][RationalFunctionPlot]
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On your paper, sketch the algebra tile shape at right. Write an expression for the perimeter, and then find the perimeter for each of the given values of
x
The perimeter is the sum of all the sides. Mark each side with its length to help you find the perimeter. The same algebra tile digram, as the problem, with the following lengths, starting on the top left, and going clockwise: 1, 1, x, x, x, 1, x, 1.
Combine like terms to form an expression of the perimeter.
x=7
7
x
in the algebra tile shape or the expression you found to find the perimeter for this given value. The same algebra tile digram, with the following lengths, starting on the top left, and going clockwise:1, 1, 7, 7, 7, 1, 7, 1.
x=5.5
5.5
x
in the algebra tile shape or the expression you found to find the perimeter for this given value. The same algebra tile digram, as the problem, with the following lengths, starting on the top left, and going clockwise: 1, 1, 5.5, 5.5, 5.5, 1, 5.5, 1.
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This page tracks new features as they are added to the development version of FreeCAD, which is currently 0.20. When the 0.20 feature freeze happens, delete these messages, and don't add more features to this page. FreeCAD 0.20 is expected to be released end of May 2022.
The new Std UserEditMode command allows the user to choose an edit mode that will be used when an object is double-clicked in the Tree view. Click the image on the left so see an animation of the selection. If a selected edit mode is not applicable, the object's default edit mode is used instead. Pull request #5110.
For Draft Dimensions the arch ViewUnit Override for imperial architectural dimensions was introduced.
Draft Shape2DView objects now have an DataAuto Update property. Setting it to false can be useful if there are many Draft Shape2DViews in a document or if they are complex.
The new Z88 settings and their default values The Z88 solver is now fully usable. You can now specify the solver method and change the memory settings. The new default values allow you to perform also complex simulations directly.
Result of a linear buckling analysis.
Click on the image to see the animation. It is now possible to perform buckling analyses using the Calculix solver. Pull request #4379
{\displaystyle 2\pi }
Click on the image to see the animation. There is a new option to pad along the direction of an edge in the 3D model.
Click on the image to see the animation. It is now possible to specify the direction for the pocket extrusion.
Click on the image to see the animation. There is a new option to pad a certain length along the direction. The length is either measured along the sketch normal or along the custom direction.
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Solve each of the following inequalities. Represent the solutions algebraically (with symbols) and graphically (on a number line).
Solve each inequality like you would solve any equation.
Solve for the variable by isolating the variables and number values.
3x-3<2-2x
\quad 3x-3<2-2x\\ +2x\qquad \quad \ \ +2x
Note: Steps 1 and 2 can happen in any order.
5x-3<2\\ \quad +3\ +3\\ 5x+0<5
Use this simplified inequality to graph the inequality on a number line.
Then try part (b) using the same strategy.
\frac{5x}{5}<\frac{5}{5}\\ \ \ x < 1
\frac{4}{5}x\ge8
x\ge10
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Introduction to Chemical Engineering Processes/The most important point - Wikibooks, open books for an open world
1 Component Mass Balance
2 Concentration Measurements
3 Calculations on Multi-component streams
3.1 Average Molecular Weight
3.2 Density of Liquid Mixtures
3.2.1 First Equation
3.2.2 Second Equation
Component Mass BalanceEdit
Most processes, of course, involve more than one input and/or output, and therefore it must be learned how to perform mass balances on . The basic idea remains the same though. We can write a mass balance in the same form as the overall balance for each component:
{\displaystyle In-Out+Generation=Accumulation}
For steady state processes, this becomes:
{\displaystyle In-Out+generation=0}
The overall mass balance at steady state, recall, is:
{\displaystyle \Sigma {\dot {m}}_{in}-\Sigma {\dot {m}}_{out}+m_{gen}=0}
The mass of each component can be described by a similar balance.
{\displaystyle \Sigma {\dot {m}}_{A,in}-\Sigma {\dot {m}}_{A,out}+{m}_{A,gen}=0}
The biggest difference between these two equations is that The total generation of mass
{\displaystyle m_{gen}}
is zero due to conservation of mass, but since individual species can be consumed in a reaction,
{\displaystyle {m}_{A,gen}\neq 0}
for a reacting system
Concentration MeasurementsEdit
You may recall from general chemistry that a concentration is a measure of the amount of some species in a mixture relative to the total amount of material, or relative to the amount of another species. Several different measurements of concentration come up over and over, so they were given special names.
MolarityEdit
The first major concentration unit is the molarity which relates the moles of one particular species to the total volume of the solution.
{\displaystyle Molarity(A)=[A]={\frac {n_{A}}{V_{sln}}}}
{\displaystyle n{\dot {=}}mol,V{\dot {=}}L}
A more useful definition for flow systems that is equally valid is:
{\displaystyle [A]={\frac {{\dot {n}}_{A}}{{\dot {V}}_{n}}}}
{\displaystyle {\dot {n}}_{A}{\dot {=}}mol/s,{\dot {V}}_{n}{\dot {=}}L/s}
Molarity is a useful measure of concentration because it takes into account the volumetric changes that can occur when one creates a mixture from pure substances. Thus it is a very practical unit of concentration. However, since it involves volume, it can change with temperature so molarity should always be given at a specific temperature. Molarity of a gaseous mixture can also change with pressure, so it is not usually used for gasses.
Mole FractionEdit
The mole fraction is one of the most useful units of concentration, since it allows one to directly determine the molar flow rate of any component from the total flow rate. It also conveniently is always between 0 and 1, which is a good check on your work as well as an additional equation that you can always use to help you solve problems.
The mole fraction of a component A in a mixture is defined as:
{\displaystyle x_{A}={\frac {n_{A}}{n_{n}}}}
{\displaystyle n_{A}}
signifies moles of A. Like molarity, a definition in terms of flowrates is also possible:
{\displaystyle x_{A}={\frac {{\dot {n}}_{A}}{{\dot {n}}_{n}}}}
If you add up all mole fractions in a mixture, you should always obtain 1 (within calculation and measurement error), because sum of individual component flow rates equals the total flow rate:
{\displaystyle \Sigma x_{i}=1}
Note that each stream has its own independent set of concentrations. This fact will become important when you are performing mass balances.
Mass FractionEdit
Since mass is a more practical property to measure than moles, flowrates are often given as mass flowrates rather than molar flowrates. When this occurs, it is convenient to express concentrations in terms of mass fractions defined similarly to mole fractions.
In most texts mass fraction is given the same notation as mole fraction, and which one is meant is explicitly stated in the equations that are used or the data given.
In this book, assume that a percent concentration has the same units as the total flowrate unless stated otherwise. So if a flowrate is given in kg/s, and a composition is given as "30%", assume that it is 30% by mass.
The definition of a mass fraction is similar to that of moles:
{\displaystyle x_{A}={\frac {m_{A}}{m_{n}}}}
for batch systems
Mass fraction of Continuous Systems
{\displaystyle x_{A}={\frac {{\dot {m}}_{A}}{{\dot {m}}_{n}}}}
{\displaystyle m_{A}}
is the mass of A. It doesn't matter what the units of the mass are as long as they are the same as the units of the total mass of solution.
Like the mole fraction, the total mass fraction in any stream should always add up to 1.
{\displaystyle \Sigma x_{i}=1}
Calculations on Multi-component streamsEdit
Various conversions must be done with multiple-component streams just as they must for single-component streams. This section shows some methods to combine the properties of single-component streams into something usable for multiple-component streams(with some assumptions).
Average Molecular WeightEdit
The average molecular weight of a mixture (gas or liquid) is the multicomponent equivalent to the molecular weight of a pure species. It allows you to convert between the mass of a mixture and the number of moles, which is important for reacting systems especially because balances must usually be done in moles, but measurements are generally in grams.
{\displaystyle {\bar {MW}}_{n}={\frac {g{\mbox{ sln}}}{mole{\mbox{ sln}}}}}
, we split the solution up into its components as follows, for k components:
{\displaystyle {\frac {g{\mbox{ sln}}}{mole{\mbox{ sln}}}}={\frac {\Sigma {m_{i}}}{n_{n}}}=\Sigma {\frac {m_{i}}{n_{n}}}}
{\displaystyle =\Sigma ({\frac {m_{i}}{n_{i}}}*{\frac {n_{i}}{n_{n}}})=\Sigma (MW_{i}*x_{i})}
{\displaystyle x_{i}}
is the mole fraction of component i in the mixture. Therefore, we have the following formula:
{\displaystyle {\bar {MW}}_{n}=\Sigma (MW_{i}*x_{i})_{n}}
{\displaystyle x_{i}}
is the mole fraction of component i in the mixture.
This derivation only assumes that mass is additive, which it is, so this equation is valid for any mixture.
Density of Liquid MixturesEdit
Let us attempt to calculate the density of a liquid mixture from the density of its components, similar to how we calculated the average molecular weight. This time, however, we will notice one critical difference in the assumptions we have to make. We'll also notice that there are two different equations we could come up with, depending on the assumptions we make.
First EquationEdit
By definition, the density of a single component i is:
{\displaystyle {\rho }_{i}={\frac {m_{i}}{V_{i}}}}
The corresponding definition for a solution is
{\displaystyle \rho ={\frac {m{\mbox{ sln}}}{V{\mbox{ sln}}}}}
. Following a similar derivation to the above for average molecular weight:
{\displaystyle {\frac {m{\mbox{ sln}}}{V{\mbox{ sln}}}}={\frac {\Sigma {m_{i}}}{V_{n}}}=\Sigma {\frac {m_{i}}{V_{n}}}}
{\displaystyle =\Sigma {\frac {m_{i}}{V_{i}}}*{\frac {V_{i}}{V_{n}}}=\Sigma ({\rho }_{i}*{\frac {V_{i}}{V_{n}}})}
Now we make the assumption that The volume of the solution is proportional to the mass. This is true for any pure substance (the proportionality constant is the density), but it is further assumed that the proportionality constant is the same for both pure k and the solution. This equation is therefore useful for two substances with similar pure densities. If this is true then:
{\displaystyle {\frac {V_{i}}{V}}={\frac {m_{i}}{m_{n}}}=x_{i}}
{\displaystyle x_{i}}
is the mass fraction of component i. Thus:
{\displaystyle {\rho }_{n}=\Sigma {(x_{i}*{\rho }_{i})}_{n}}
{\displaystyle x_{i}}
is the mass fraction (not the mole fraction) of component i in the mixture.
Second EquationEdit
This equation is easier to derive if we assume the equation will have a form similar to that of average molar mass. Since density is given in terms of mass, it makes sense to start by using the definition of mass fractions:
{\displaystyle x_{i}={\frac {m_{i}}{m_{n}}}}
To get this in terms of only solution properties (and not component properties), we need to get rid of
{\displaystyle m_{i}}
. We do this first by dividing by the density:
{\displaystyle {\frac {x_{i}}{{\rho }_{i}}}={\frac {m_{i}}{m_{n}}}*{\frac {V_{i}}{m_{i}}}}
{\displaystyle ={\frac {V_{i}}{m_{n}}}}
Now if we add all of these up we obtain:
{\displaystyle \sum \left({\frac {x_{i}}{{\rho }_{i}}}\right)={\frac {\Sigma {V_{i}}}{m_{n}}}}
Now we have to make an assumption, and it's different from that in the first case. This time we assume that the Volumes are additive. This is true in two cases:
1. In an ideal solution. The idea of an ideal solution will be explained more later, but for now you need to know that ideal solutions:
Tend to involve similar compounds in solution with each other, or when one component is so dilute that it doesn't effect the solution properties much.
Include Ideal Gas mixtures at constant temperature and pressure.
2 In a Completely immiscible nonreacting mixture. In other words, if two substances don't mix at all (like oil and water, or if you throw a rock into a puddle), the total volume will not change when you mix them. And the total volume in this case will be sum of volume of individual components.
If the solution is ideal, then we can write:
{\displaystyle {\frac {\Sigma {\dot {V}}_{i}}{{\dot {m}}_{n}}}={\frac {{\dot {V}}_{n}}{{\dot {m}}_{n}}}={\frac {1}{{\rho }_{n}}}}
Hence, for an ideal solution,
{\displaystyle {\frac {1}{{\rho }_{n}}}=\sum \left({\frac {x_{i}}{{\rho }_{i}}}\right)_{n}}
{\displaystyle x_{i}}
is the mass fraction of component i in the mixture.
Note that this is significantly different from the previous equation! This equation is more accurate for most cases. In all cases, however, it is most accurate to look up the value in a handbook such as Perry's Chemical Engineers Handbook if data is available on the solution of interest.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/The_most_important_point&oldid=3325803"
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f(x)=\frac{1}{2}(x-2)^3+1
g(x)=2x^2-6x-3
A
B
f(x)
g(x)
equal to each other.
\frac{1}{2}(\textit{x}-2)^{3}+1=2\textit{x}^{2}-6\textit{x}-3
Are there any solutions to your equation in part (a) that do not appear on the graph? Explain.
x
\frac{1}{2}\left(\textit{x}^{3}-6\textit{x}^{2}+12\textit{x}-8\right)-2\textit{x}^{2}+6\textit{x}+4=0
\frac{1}{2}\textit{x}^{3}-5\textit{x}^{2}+12\textit{x}=0
\textit{x}\left(\frac{1}{2}\textit{x}-3\right)(\textit{x}-4)=0
Write an equation that you could use to solve for the
x
C
. Estimate the solution to your equation using the graph. Again, substitute your solution into your equation. How close was your estimate?
f(x)
0
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On bilinear restriction type estimates and applications to nonlinear wave equations
Klainerman, Sergiù
I will start with a short review of the classical restriction theorem for the sphere and Strichartz estimates for the wave equation. I then plan to give a detailed presentation of their recent generalizations in the form of “bilinear estimates”. In addition to the
{L}^{2}
theory, which is now quite well developed, I plan to discuss a more general point of view concerning the
{L}^{p}
theory. By investigating simple examples I will derive necessary conditions for such estimates to be true. I also plan to discuss the relevance of these estimates to nonlinear wave equations.
⇒ complété par On bilinear estimates for wave equations
author = {Klainerman, Sergi\`u},
title = {On bilinear restriction type estimates and applications to nonlinear wave equations},
TI - On bilinear restriction type estimates and applications to nonlinear wave equations
Klainerman, Sergiù. On bilinear restriction type estimates and applications to nonlinear wave equations. Journées équations aux dérivées partielles (1998), article no. 7, 1 p. http://www.numdam.org/item/JEDP_1998____A7_0/
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Bond stabilization - Chemix Ecosystem Documents
Among the fourth-generation algorithmic stablecoins, bond tokens are designed to cushion the price fluctuations of stablecoins. In QSD, we studied this design idea and introduced two bond tokens, which are the following tokens: one is the bond token CBT released in redemption and recollateralization and the other one is the special bond DBQ activated when the price of QSD significantly deviates from the target value.
The scenarios and algorithms of CBT (CEC buffering token) have been introduced in the QSD redemption and recollateralization section. Here we will analyze the purpose of this design.
The goal pursued by QSD is the continuous growth of QSD issuance and the intrinsic value of CEC. For this goal, the system will set up a mitigation mechanism for the CEC minted at the time of redemption and recollateralization in the early stage of the system to avoid large fluctuations in the price of CEC tokens. Specifically, when redeeming or recollateralizing, the value of CEC tokens in the algorithm part will be converted to CBT. When certain conditions are met, users can convert CBT to CEC one-to-one.
If no one provides collateral when recollateralization is needed, it will easily cause QSD to break into a "death cycling". In the fractional algorithmic synthetic asset system, when collateral needs to be added, it is often accompanied by a decline in the CEC price and the QSD price below the target price. The additional issuance of CEC tokens during the recollateralization process will undoubtedly intensify the decline in its price. It is very risky if all the stability recovery is pinned on the assumption that the CEC market price can sustain by the market itself. Therefore, the Chemix Ecosystem development team proposes to release the buffering token CBT during the recollateralization process instead of directly releasing CEC.
The CBT minted in redemption and recollateralization cannot be redeemed to CEC immediately. Users can convert CBT tokens to CEC when the system has surplus, that is, in the state of buyback and QSD price exceeds the target. Through this mitigation mechanism, users' worries about the loss of their own equity caused by the short-term large-scale release and dumping of CEC are eliminated. CBT is actually the regulator of CEC token price.
When the market price of QSD is lower than $1 to an extent, to establish the price stability of QSD and obtain future profits when redeeming, users can buy QSD stable bond token DBQ with QSD at a discount. Each DBQ bond token can be exchanged back into 1 QSD when the exchange conditions are met in the future. When DBQ bonds are purchased, the QSD will be burned and exit from circulation, so as to reduce the circulation quantity of QSD and promote the price of QSD return to the target value.
DBQ bonds have no interest and no vanished time. Only if the price of QSD is greater than $1, they can be converted into QSD at a ratio of 1:1. This design can prevent bondholders from suffering losses in the process of redemption.
In order to encourage users to actively buy bonds when the QSD discount is large, the system will set a certain discount to encourage users to buy DBQ bonds. At the beginning of the launch of Chemix, the price of DBQ will be set to the third power of the QSD market price. After the market price of QSD exceeds $1.001/QSD, users can redeem QSD from DBQ bond tokens. The user's income in DBQ bond can be calculated as follows:
DBQ_{net\ return} = \frac{U}{P_Q^3} - U
DBQ_{net\ return}
is the net return that users can get on DBQ bonds;
U
is the actual USD value of the user's investment in purchasing QSD bonds;
P_Q
is the price for QSD.
The market price of QSD is $0.93/QSD, and the arbitrager buys DBQ in the market with the equivalent BUSD of $1000, then the net income is:
\frac{1000}{0.93^3} - 1000 = 243.229\ QSD
243.229\ QSD × 1.001 = 243.47\ USD
The actual net income was $243.47, yielding a return of 24.347%.
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Prof. Mintchev Co-Authors "Stability of a Family of Travelling Wave Solutions in a Feedforward Chain of Phase Oscillators" | The Cooper Union
Prof. Mintchev Co-Authors "Stability of a Family of Travelling Wave Solutions in a Feedforward Chain of Phase Oscillators"
The paper concerns a chain of identical phase oscillators, each coupled to only its nearest neighbour on one side. The governing equations are
{\stackrel{˙}{\theta }}_{i}=\omega +ϵZ\left({\theta }_{i}\right)g\left({\theta }_{i-1}\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}i=1,2\cdots ,N,
{\theta }_{0}\left(t\right)
is some prescribed function of time. Each
{\theta }_{i}\in \left[0,2\pi \right)
\omega
ϵ
Z
Z\left(\theta \right)=1-cos\theta
g
is a particular “pulse” function. (The results do not depend on the exact form of these, nor the values of parameters.) The model can be regarded as describing a feedforward network of theta neurons. The authors are interested in waves that travel with an approximately uniform profile and speed. They prove (under certain hypotheses) for a doubly-infinite chain (i.e.
i=-\infty ,\cdots ,\infty
) that a family of such waves, each with constant speed and profile, does exist. (The family is parametrised by the wave’s temporal period.) They also prove that such a wave is stable to a large class of specified perturbations. The authors then give the results of careful numerical experiments which suggest that the hypotheses needed above are true.
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GetChildByName - Maple Help
Home : Support : Online Help : Connectivity : Web Features : XMLTools : GetChildByName
GetChildByName(xmlTree, name)
string or symbol; the name of the child element to extract
The GetChildByName(xmlTree, name) command accesses the children of the given XML element xmlTree with element name equal to name. A list of all children that are elements with element type equal to name is returned.
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{xmlTree}≔\mathrm{XMLElement}\left("a",[],[\mathrm{XMLElement}\left("b",[],"b text"\right),\mathrm{XMLElement}\left("c",[],"c text"\right),\mathrm{XMLElement}\left("b",[],"more b text"\right)]\right):
\mathrm{Print}\left(\mathrm{xmlTree}\right)
<b>b text</b>
<c>c text</c>
<b>more b text</b>
\mathrm{map}\left(\mathrm{Print},\mathrm{GetChildByName}\left(\mathrm{xmlTree},"b"\right)\right)
[]
\mathrm{map}\left(\mathrm{Print},\mathrm{GetChildByName}\left(\mathrm{xmlTree},"c"\right)\right)
[]
\mathrm{map}\left(\mathrm{Print},\mathrm{GetChildByName}\left(\mathrm{xmlTree},"nosuchelement"\right)\right)
[]
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Alden found a partially completed 5-D table:
74
15
2(15)=30
15+2=17
15+30+17=
62
18
2(18)=36
18+2=20
18+36+20=
74
Create a word problem that could have been solved using this table.
The "Do" column sums the three values in the "Define" column. Think about a situation using ages, lengths, or weights that could be used to write a problem.
What words would you put above the numbers in the three empty sections in the “Trial” and “Define” parts of the table?
The answer is based on your word problem from part (a). For example, if you chose weights of blocks, these columns would be the weights of the first, second, and third blocks.
What word(s) would you put above the “Do” column?
Based on the word problem you created, what does this total represent?
From the example given in part (b), ''Do'' column is the total weight of the three blocks.
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Factors Affecting The Rate Of A Chemical Reaction – Engineeringness
Jennifer Fuentes September 1, 2020, 11:48 pm
Factors Affecting The Rate Of A Chemical Reaction:
Reactions occur when two reactant molecules effectively collide, each having minimum energy and correct orientation. Reactant concentration, the physical state of the reactants, and surface area, temperature, and the presence of a catalyst are the main factors that affect the reaction rate.
Aluminium Chloride and Sodium Hydroxide Example
The best way to show the factors affecting a chemical reaction is to use an example. The chemical reaction we will look at is the double displacement reaction between Aluminium Chloride (AlCl3) and Sodium Hydroxide (NaOH) that produces Aluminium Hydroxide (Al(OH)3) and Sodium Chloride (NaCl) (equation 1).
\mathrm{AlCl}3\left(\mathrm{aq}\right) + 3\mathrm{NaOH}\left(\mathrm{aq}\right) \to \mathrm{Al}\left(\mathrm{OH}\right)3\left(\mathrm{s}\right) + 3\mathrm{NaCl}\left(\mathrm{aq}\right)
Equation 1: Double displacement reaction between Aluminium Chloride and Sodium Hydroxide
The concentration of reactants in a chemical reaction is related to the rate of a chemical reaction. If we increase reactant concentration, then the rate of the reaction will also increase. The rate of the reaction is dependent on the concentration of both the reactants. Because the AlCl3 solution is a strong electrolyte, it dissociates completely into two ions, Al3+ and Cl–. Therefore, it means that AlCl3 and Al3+ ions are present in aqueous solution. In this reaction, AlCl3 is the limiting reagent, as the rate of the reaction is dependent on it.
While sodium hydroxide solution is not an electrolyte, but it dissociates into Na+ and OH– ions. Thus, NaOH and OH–ions are present in aqueous solution. In this reaction, NaOH is the limiting reagent, in terms of concentration. This is because the rate of the reaction is dependent on it. So, this reaction is an equilibrium reaction, as it depends on the product.
The pH of the reaction must be in the range required by the reaction. If it is outside this range, then the reaction will be slow. For instance, in the case of the reaction between AlCl3 and NaOH, the concentration of both the reactants are equal. However, in this case, the highest concentration of the product is achieved when the reaction is carried out at pH 3, and the highest concentration of the reactant is at pH 9. So, the reaction requires the pH to be in the range of 3-9.
The physical state of the reactants affects the rate of a chemical reaction. For instance, in the case of the reaction between AlCl3 and NaOH, if both the reactants are in the gaseous state, then it will be difficult to achieve the equilibrium. This is because gaseous reactants move away from each other, and the distance between them has a bearing on the diffusion rate. Moreover, in the gaseous state, the molecules have high kinetic energy, which slows down the rate of reaction.
While if both the reactants are in the solid-state, then it is easier to reach the equilibrium, in this case. Solid reactants do not have great kinetic energy. Moreover, in the solid-state, the molecules are close to each other, so it is easier to achieve equilibrium.
If either of the reactants is in a liquid state, then it is easy to achieve the equilibrium in the liquid state also. But, if both the reactants are in the liquid state, then it will be easier to achieve equilibrium again, as the molecules are closer to each other. Thus, the liquid state is always favourable for a reaction, the reason being that it ensures a greater number of collisions. On the other hand, free gas is less reactive as compared to a gas dissolved in a solution.
The presence of a catalyst has an impact on the rate of the reaction. For instance, in the case of a reaction between hydrogen and oxygen, a catalyst is not required. This is because neither reaction is extremely fast, nor reactive. However, if we want a faster and more productive reaction, we can use the catalyst in this case, as well. The catalyst’s role, in this case, is to speed up the rate of reaction. But this also depends on the type of catalyst used. For instance, different catalysts react differently under different conditions.
The temperature of the reaction has a big impact on the rate of the chemical reaction. If we increase the temperature of the reaction, hence the temperature of both the reactants, then the reaction rate will increase. Hence, the rate of chemical reactions increases with temperature.
In the reaction between AlCl3 and NaOH, if the temperature of the two reactants is doubled, then the rate of reaction, hence production will also increase. For instance, if we carry out the reaction at a temperature of 98°C, then the rate of reaction will increase by a factor of four, as the rate of reaction doubles with every 10° rise in temperature.
The surface area of both the reactants has a significant impact on the rate of reaction. It is because the surface area of reactants has a direct relation with the number of collisions per unit time. Therefore, if we want to increase the rate of a chemical reaction, then we must increase the surface area of both the reactants.
In the case of a reaction between AlCl3 and NaOH, there is no significant change in the reaction rate, whether the number of reactants is increased or the reactants change in shape or size. However, if we increase the surface area of a reactant, then the reaction rate also increases. For instance, if AlCl3 and NaOH are powdered, then the reaction rate will decrease by a factor of two, as only a small number of ions come in contact with each other. So, we can state that the rate of reaction is dependent on the surface area of the reactants. The maximum size of reactant particles in which a chemical reaction takes place at the maximum rate is called the critical size.
Thus, in the case of the reaction between AlCl3 and NaOH, both these reactants are powdered, hence the change in the rate of reaction is not significant. Therefore, due to the powdered nature of the reactants, the reaction rate is dependent on the size of the reactants. As a result, the reactants keep on colliding with each other, in case of powdered reactants. Hence, the rate of the reaction is higher in this scenario.
Chemical ReactionRateConcentrationPhysical stateCatalystTemperatureSurface areadouble displacement reaction
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More From: Kinetics
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Jennifer Fuentes July 30, 2020, 9:47 pm
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Stratified Sampling - MATLAB & Simulink - MathWorks España
Simulation methods allow you to specify a noise process directly, as a callable function of time and state:
{z}_{t}=Z\left(t,{X}_{t}\right)
Stratified sampling is a variance reduction technique that constrains a proportion of sample paths to specific subsets (or strata) of the sample space.
This example specifies a noise function to stratify the terminal value of a univariate equity price series. Starting from known initial conditions, the function first stratifies the terminal value of a standard Brownian motion, and then samples the process from beginning to end by drawing conditional Gaussian samples using a Brownian bridge.
The stratification process assumes that each path is associated with a single stratified terminal value such that the number of paths is equal to the number of strata. This technique is called proportional sampling. This example is similar to, yet more sophisticated than, the one discussed in Simulating Interest Rates. Since stratified sampling requires knowledge of the future, it also requires more sophisticated time synchronization; specifically, the function in this example requires knowledge of the entire sequence of sample times. For more information, see the example Example_StratifiedRNG.m.
The function implements proportional sampling by partitioning the unit interval into bins of equal probability by first drawing a random number uniformly distributed in each bin. The inverse cumulative distribution function of a standard N(0,1) Gaussian distribution then transforms these stratified uniform draws. Finally, the resulting stratified Gaussian draws are scaled by the square root of the terminal time to stratify the terminal value of the Brownian motion.
The noise function does not return the actual Brownian paths, but rather the Gaussian draws Z(t,Xt) that drive it.
This example first stratifies the terminal value of a univariate, zero-drift, unit-variance-rate Brownian motion (bm) model:
d{X}_{t}=d{W}_{t}
Assume that 10 paths of the process are simulated daily over a three-month period. Also assume that each calendar month and year consist of 21 and 252 trading days, respectively:
dt = 1 / 252; % 1 day = 1/252 years
nPeriods = 63; % 3 months = 63 trading days
T = nPeriods * dt; % 3 months = 0.25 years
nPaths = 10; % # of simulated paths
obj = bm(0, 1, 'StartState', 0);
sampleTimes = cumsum([obj.StartTime; ...
dt(ones(nPeriods,1))]);
z = Example_StratifiedRNG(nPaths, sampleTimes);
Simulate the standard Brownian paths by explicitly passing the stratified sampling function to the simulation method:
X = obj.simulate(nPeriods, 'DeltaTime', dt, ...
'nTrials', nPaths, 'Z', z);
For convenience, reorder the output sample paths by reordering the three-dimensional output to a two-dimensional equivalent array:
Verify the stratification:
Recreate the uniform draws with proportional sampling:
U = ((1:nPaths)' - 1 + rand(nPaths,1))/nPaths;
Transform them to obtain the terminal values of standard Brownian motion:
WT = norminv(U) * sqrt(T); % Stratified Brownian motion.
Plot the terminal values and output paths on the same figure:
plot(sampleTimes, X), hold('on')
xlabel('Time (Years)'), ylabel('Brownian State')
title('Terminal Stratification: Standard Brownian Motion')
plot(T, WT, '. black', T, WT, 'o black')
hold('off')
The last value of each sample path (the last row of the output array X) coincides with the corresponding element of the stratified terminal value of the Brownian motion. This occurs because the simulated model and the noise generation function both represent the same standard Brownian motion.
However, you can use the same stratified sampling function to stratify the terminal price of constant-parameter geometric Brownian motion models. In fact, you can use the stratified sampling function to stratify the terminal value of any constant-parameter model driven by Brownian motion if the model's terminal value is a monotonic transformation of the terminal value of the Brownian motion.
To illustrate this, load the data set and simulate risk-neutral sample paths of the FTSE 100 index using a geometric Brownian motion (GBM) model with constant parameters:
d{X}_{t}=r{X}_{t}dt+\sigma {X}_{t}d{W}_{t}
where the average Euribor yield represents the risk-free rate of return.
Assume that the relevant information derived from the daily data is annualized, and that each calendar year comprises 252 trading days:
returns = tick2ret(Dataset.FTSE);
sigma = std(returns) * sqrt(252);
rate = Dataset.EB3M;
rate = mean(360 * log(1 + rate));
Create the GBM model using gbm, assuming the FTSE 100 starts at 100:
obj = gbm(rate, sigma, 'StartState', 100);
Determine the sample time and simulate the price paths.
In what follows, NSteps specifies the number of intermediate time steps within each time increment DeltaTime. Each increment DeltaTime is partitioned into NSteps subintervals of length DeltaTime/NSteps each, refining the simulation by evaluating the simulated state vector at NSteps–1 intermediate points. This refinement improves accuracy by allowing the simulation to more closely approximate the underlying continuous-time process without storing the intermediate information:
nSteps = 1;
sampleTimes = cumsum([obj.StartTime ; ...
dt(ones(nPeriods * nSteps,1))/nSteps]);
z = Example_StratifiedRNG(nPaths, sampleTimes);
[Y, Times] = obj.simBySolution(nPeriods, 'nTrials', nPaths,...
'DeltaTime', dt, 'nSteps', nSteps, 'Z', z);
Y = squeeze(Y); % Reorder to a 2-D array
plot(Times, Y)
xlabel('Time (Years)'), ylabel('Index Level')
title('FTSE 100 Terminal Stratification:Geometric Brownian Motion')
Although the terminal value of the Brownian motion shown in the latter plot is normally distributed, and the terminal price in the previous plot is lognormally distributed, the corresponding paths of each graph are similar.
For another example of variance reduction techniques, see Simulating Interest Rates.
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Recover time-domain signals by performing inverse short-time, fast Fourier transform (FFT) - Simulink - MathWorks España
Recover time-domain signals by performing inverse short-time, fast Fourier transform (FFT)
The Inverse Short-Time FFT block reconstructs the time-domain signal from the frequency-domain output of the Short-Time FFT block using a two-step process. First, the block performs the overlap add algorithm shown below.
x\left[n\right]=\frac{L}{W\left(0\right)}\sum _{p=-\infty }^{\infty }\left[\frac{1}{N}\sum _{k=0}^{N-1}X\left[pL,k\right]{e}^{j2\pi kn/N}\right]
Then, the block rebuffers the signal in order to reconstruct the time-domain signal. Depending on the analysis window used by the Short-Time FFT block, the Inverse Short-Time FFT block might or might not achieve perfect reconstruction of the time domain signal.
Connect your complex-valued, single-channel or multichannel input signal to the X(n,k) port. The block accepts unoriented vector, column vector and matrix input. The block outputs the real or complex-valued, single-channel or multichannel inverse short-time FFT at port x(n).
Connect your complex-valued, single-channel analysis window to the w(n) port. When you select the Assert if analysis window does not support perfect signal reconstruction check box, the block displays an error when the input signal cannot be perfectly reconstructed. The block uses the values you enter for the Analysis window length (W) and Reconstruction error tolerance, or maximum amount of allowable error in the reconstruction process, to determine if the signal can be perfectly reconstructed.
Enter the length of the analysis window. This parameter is visible when you select the Assert if analysis window does not support perfect signal reconstruction check box.
Overlap between consecutive STFFT frames (in samples)
Enter the number of samples of overlap for each frame of the Short-Time FFT block's input signal. This value should be the same as the Overlap between consecutive windows (in samples) parameter in the Short-Time FFT block parameters dialog.
Samples per output frame
Enter the desired frame size of the output signal.
Input is conjugate symmetric
Select this check box when the input to the block is both floating point and conjugate symmetric, and you want real-valued outputs. When you select this check box when the input is not conjugate symmetric, the output of the block is invalid. This parameter cannot be used for fixed-point signals.
Assert if analysis window does not support perfect signal reconstruction
Select this check box to display an error when the analysis window used by the Short-Time FFT block does not support perfect signal reconstruction.
Reconstruction error tolerance
Enter the amount of acceptable error in the reconstruction of the original signal. This parameter is visible when you select the Assert if analysis window does not support perfect signal reconstruction check box.
dsp.SpectrumEstimator | dsp.ISTFT
Spectrum Estimator | Burg Method | Magnitude FFT | Periodogram | Short-Time FFT | Spectrum Analyzer | Window Function | Yule-Walker Method
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Modular construction - Wikipedia
(Redirected from Modular construction systems)
Modular construction is a construction technique which involves the prefabrication of 2D panels or 3D volumetric structures in off-site factories and transportation to construction sites for assembly. This process has the potential to be superior to traditional building in terms of both time and costs, with claimed time savings of between 20 to 50 percent faster than traditional building techniques.[1]
Algeco school built using pre-fabricated modular construction
It is estimated that by 2030, modular construction could deliver US$22 billion in annual cost savings for the US and European construction industry, helping fill the US$1.6 trillion productivity gap.[1] The current need for standardized, repeatable 3D volumetric housing pre-fabricated units and designs for student accommodations, affordable housing and hotels is driving demand for modular construction.
2.1 Production algorithms
6 Modular construction systems
In a 2018 Practice Note, the NEC states that the benefits obtained from offsite construction mainly relate to the creation of components in a factory setting, protected from the weather and using manufacturing techniques such as assembly lines with dedicated and specialist equipment.[2] Through the use of appropriate technology, modular construction can:
increase the speed of construction by increasing the speed of manufacture of the component parts,
increase economies of scale,
improve quality leading to reduction in the whole life costs of assets
reduce environmental impact such as dust and noise and
reduce accidents and ill health by reducing the amount of construction work taking place at site
Modular construction has consistently been at least 20 percent faster than traditional on-site builds. Currently, the design process of modular construction projects tends to take longer than that of traditional building. This is because modular construction is a fairly new technology and not many architects and engineers have experience working with it. In fewer words, the industry has not yet learned how to work this way. It is expected of design firms to develop module libraries which would assist in the automatization of this process. These modules libraries would hold various pre-designed 2D panels and 3D structures which would be digitally assembled to create standardized structures.
The foundations of a structure are a crucial part of its rigidity. The magnitude and complexity of such will vary depending on the size, and overall weight of the structure. Therefore, the weight difference of a traditionally built house and a prefabricated structure will mean that foundations needed will be smaller and faster to build.
Off-site manufacturing is the pinnacle of modular construction. The ability to coordinate and repeat activities in a factory along with the increased help of automation result in largely faster manufacturing times than those of on-site building. A large time saver is the ability to parallelly work on the foundation of a structure and the manufacturing of the structure itself. This would be impossible with traditional construction. The on-site construction is radically simplified. The assembly of pre-fabricated components is as simple as assembling the 3D modules, and connecting the services to main site connections. A team of five workers can assemble up to six 3D modules, or the equivalent of 270 square meters of finished floor area, in a single work day.
Production algorithmsEdit
Since the technology required to manufacture the components of modular construction, the prefabricated parts of modular buildings are carried out by modular factories. To optimize time, modular factories consider the specifications and resources of the project and adapt a scheduling algorithm to fulfill the needs of this unique project. However, current scheduling methods assume the quantity of resources will never reach zero, therefore representing an unrealistic work cycle.
A modular factory handling a single project at any given point is rare, and would produce low returns. Hyun and Lee’s research propose a Genetic Algorithm (GA) scheduling model which takes into consideration various project’s characteristics and shares resources.[3] The production sequence of this algorithm would be largely affected by which modules need to be transported to which site and the dates they should arrive. After considering the variables of production, transportation and on-site assembly the objective function is:
{\displaystyle min\Sigma (S),\ldots S=S_{i}+P_{i}-E_{i}}
Where Si is the number of stocked units per day, Pi is the number of units per day and Ei is number of units installed per day. Production algorithms are continuously being developed to further accelerate the production of modular construction buildings, enlarging the time saving gap with traditional construction methods.
Modular construction can yield up to 20 percent of the total project cost in savings. However, there is also a risk of it increasing the cost by 10 percent. This occurs when the savings in the labor area of construction are outweighed by the increase in costs of the logistics area and materials. The pre-fabrication of components used in modular construction have a higher logistics cost than traditional building. Since the panels or 3D structures have to be manufactured in a factory and transported to the construction site, new variables which alter the flow of construction are presented.
Transportation of fabricated components is naturally more expensive than that of raw materials. For one, even a number of 2D panels stacked together are much harder to transport than the raw cement, wood or material used to build them. Panels run a high risk of suffering minor or major damage when being transported through land. If a panel were to be damaged, it would likely have to be replaced entirely. The factory would need to temporarily stop production of other panels to replace this one, increasing the overall manufacturing hours and therefore cost. On top of the manufacturing hours, the transportation hours would also be increased, increasing yet another cost. Regardless, the transportation of 2D panels is still a good alternative to on-site construction.
Transportation reaches its peak cost when shipping 3D volumetric structures. While 1 m2 of 2D floor space takes approximately 8 USD to transport 250 km, its equivalent in 3D floor space takes 45 USD.[1] Adding to this the replacement cost if the structure gets damaged during transport creates a large cost increase.
Assembling components in a factory off-site means that workers can use the repeatability of the structures as well as the use of automation to facilitate the manufacturing process. By standardizing the overall design of structures, work which would usually require expensive workers with specific skills (e.g. mechanical, electrical and plumbing) can be completed by low-cost manufacturers, decreasing the total salaries cost. As very little manufacturing occurs on-site, up to 80% of traditional labor activity can be moved off-site to the module factory. This leads to a lower number of sub-contractors needed, further decreasing overall total salaries cost. Overall, the larger the labor-intensive portion of a project, the larger the savings will be if modular construction is used.
Project such as student accommodations, hotels and affordable housing are great candidates for modular construction. The repeatability of their structures leads to faster manufacturing times and therefore less overall cost. Meanwhile, if the project is (for example) a modern beach house with highly irregular wall spaces and ceilings, traditional construction methods may be preferable. As the industry continues to adapt and grow, these repeatable designs could one day be modified and adapted to fit all kinds of structures at decreased costs.
Construction is considered to be one of the most dangerous industries. Workers fall from heights, objects are dropped, muscles are strained and environmental hazards can be found. Modular construction constrains all manufacturing activities to a ground level, clean space with fewer workers needed. It is estimated that reportable accidents are reduced by over 80% relative to site-intensive construction[4].When asked in a survey about safety management in the construction industry conducted by McGraw Hill Construction in 2013, 50% of the construction industry believed that pre-fabrication was safer than traditional on-site building, while only 4% said that prefabrication or modular construction had a negative impact on safety performance. Of the general and specialty contractors surveyed, 78% and 59% said that the largest safety impact was the undergoing of complex tasks at ground level.[5] According to the CDC, falling is the leading cause of work-related fatalities in construction, making up more than one in every three deaths in the industry.[6] The reduction of heights at which workers need perform tasks on subsequently reduces the fatality risk they experience, greatly increasing the overall safety of the industry. Also, 69% of the general contractors as well as 69% of the specialty contractors mentioned that the reduced number of workers performing different tasks at the off-site factory also improved construction site safety. Overall, modular construction is safer for the following reasons:
Stable work location
Tasks are performed in ample spaces
Ground level assembly
Cover from harsh weather
Better monitoring of unsafe activities
30 to 50 percent reduction in time spent at construction site
Fewer personnel on-site
Modular construction is still not considered an entirely safe alternative. However, it does reduce accidents and fatalities by a significant amount. Especially in the manufacturing process of a project. 48.1% of all accidents during on-site construction were fall-related, while only 9.1% of the accidents at manufacturing plants were from falls.[5] Manufacturing plant workers were more likely to be struck by an object or equipment (37.1%) and fracture and amputation had the same injury type frequency at 27.3%. Nevertheless, as the construction industry continues to adapt and moves over to more sustainable construction methods like pre-fabricated modular construction, it is expected that the overall safety number of accidents at construction sites will decrease.
The use of modular construction methods is encouraged by proponents of Prevention through Design techniques in construction. It is included as a recommended hazard control for construction projects in the "PtD - Architectural Design and Construction Education Module" published by the National Institute for Occupational Safety and Health.[6]
Modular construction is a great alternative to traditional construction when looking at the amount of waste each method produces. When constructing a high-rise building in Wolverhampton, 824 modules were used. During this process about 5% of the total weight of the construction was produced in weight. If it is compared to traditional methods' 10-13% average waste a small difference can be observed.[4] This difference may not seem like much when talking about small structures, however when talking about a 100,000 lb/ft2 building it is a significant percentage. Also, the number of on-site deliveries decreased by up to 70%.[4] The deliveries are instead moved to the modular factory, where more material can be received. On-site Noise pollution is greatly reduced as well, by moving the manufacturing process to an off-site factory, usually located outside of the city, neighboring buildings are not impacted as they would with the traditional building process.
Modular construction systemsEdit
See also: List of open-source hardware projects § Neither electronic nor mechanical
See also: Circular economy § Construction industry
Open-source and commercial hardware components used in modular construction include: open beams, bit beams, maker beams, grid beams, contraptors, OpenStructures components, etc.[7][8] Space frame systems (such as Mero, Unistrut, Delta Structures, etc.) also tend to be modular in design.[9] Other materials used in construction which are interlocking and thus reusable/modular in nature include interlocking bricks.[10][11][12]
^ a b c Bertram, Nick (2019). Modular construction: From projects to products. McKinsey & Company.
^ NEC, Offsite modular construction, Practice Note 4, published September 2018, accessed 15 November 2020
^ Lee, Jeonghoon; Hyun, Hosang (2019-01-01). "Multiple Modular Building Construction Project Scheduling Using Genetic Algorithms". Journal of Construction Engineering and Management. 145 (1): 04018116. doi:10.1061/(ASCE)CO.1943-7862.0001585. ISSN 1943-7862.
^ a b c Lawson, R. Mark; Ogden, Ray G.; Bergin, Rory (2012-06-01). "Application of Modular Construction in High-Rise Buildings". Journal of Architectural Engineering. 18 (2): 148–154. doi:10.1061/(ASCE)AE.1943-5568.0000057.
^ a b Fard, Maryam Mirhadi; Terouhid, Seyyed Amin; Kibert, Charles J.; Hakim, Hamed (2017-01-02). "Safety concerns related to modular/prefabricated building construction". International Journal of Injury Control and Safety Promotion. 24 (1): 10–23. doi:10.1080/17457300.2015.1047865. ISSN 1745-7300. PMID 26105510.
^ a b "CDC - NIOSH Publications and Products - PtD - Architectural Design and Construction - Instructor's Manual (2013-133)". www.cdc.gov. 2013. doi:10.26616/NIOSHPUB2013133. Retrieved 2017-08-07.
^ How to Make Everything Ourselves: Open Modular Hardware
^ After more than 30 years, grid beam modular construction system comes to market
^ Analysis, Design and Construction of Steel Space Frames
^ Interlocking bricks used in Nepal
^ Bricks that interlock
^ Conceptos Plasticos interlocking bricks (i.e. made from plastic waste)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Modular_construction&oldid=1084442725#Modular_construction_systems"
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m (added link to python script i.worldview.toar)
A custom python script, performing the operations of interest, might be like [https://github.com/NikosAlexandris/i.worldview.toar i.worldview.toar (for GRASS 7.x)]
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L_{\lambda {\text{Pixel, Band}}}={\frac {K_{\text{Band}}*q_{\text{Pixel, Band}}}{\Delta \lambda _{\text{Band}}}}}
{\displaystyle L_{\lambda {\text{Pixel,Band}}}}
{\displaystyle K_{\text{Band}}}
{\displaystyle q_{\text{Pixel,Band}}}
{\displaystyle \Delta _{\lambda _{\text{Band}}}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
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Integration to Find Arc Length - MATLAB & Simulink - MathWorks India
Consider the curve parameterized by the equations
x(t) = sin(2t), y(t) = cos(t), z(t) = t,
where t ∊ [0,3π].
Create a three-dimensional plot of this curve.
The arc length formula says the length of the curve is the integral of the norm of the derivatives of the parameterized equations.
\underset{0}{\overset{3\pi }{\int }}\sqrt{4{\mathrm{cos}}^{2}\left(2t\right)+{\mathrm{sin}}^{2}\left(t\right)+1}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}dt.
Define the integrand as an anonymous function.
f = @(t) sqrt(4*cos(2*t).^2 + sin(t).^2 + 1);
Integrate this function with a call to integral.
len = integral(f,0,3*pi)
The length of this curve is about 17.2.
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QSD Minting - Chemix Ecosystem Documents
The minting process of QSD will burn CEC, and the redemption process will mint CEC. The value of CEC will be directly related to the demands of QSD. The following equations describe the minting function of the QSD Protocol:
Q = \overbrace{(Y_i*P_{i})}^{\text {collateral value}} + \overbrace{(E*P_E)}^{\text{CEC value}}
(1 − C_r)({\sum_{i=0}^N{(Y_i*P_{i}}}))=C_r(E ∗ P_E)
Q
is the units of newly minted QSD;
C_r
is the collateral ratio;
Y_i
is the units of collateral i to the system;
P_{i}
is the price in USD of collateral i;
E
is the units of CEC burned;
P_E
On the Binance Smart Chain, minting QSD at a collateral ratio of 100% with 900 BUSD ($1/BUSD), 50 BNB ($40/BNB), and 2 BTCB ($37000/BTCB).
Under this condition, the amount of CEC that needs to be burned is:
(1-1)×(900×1+50×40+2×37,000) = 1×(E×P_E)
0 = (E×P_E)
We show that no CEC is needed to mint QSD when the protocol collateral ratio is 100% (fully collateralized). The number of QSD that can be minted at this time is:
Q = (900×1+50×40+2×37,000) + (0)
Q = 76,900
76,900 QSD are minted in this scenario. When the collateral ratio is 100%, the full value of QSD is calculated based on the collateral value. At this time, if users try to burn CEC to generate QSD, it will be returned because the result of
E×P_E
on the right side of the equation is 0.
On the Binance Smart Chain, minting QSD at a collateral ratio of 70% with 900 BUSD ($1/BUSD), 50 BNB ($40/BNB), and 2 BTCB ($37000/BTCB), the price of CEC is $0.5/CEC.
First, we need to calculate the demand for CEC:
(1-70\%)×(900×1+50×40+2×37,000) = 70\%×(E×0.5)
E = 65,914.2857142857
Therefore, when the collateral ratio is 70%, 900 BUSD, 50 BNB, and 2 BTCB are deposited to mint QSD, and 65,914.2857142857 CEC needs to be burned at the same time. At this time, the number of QSD that can be generated is:
Q = (900×1+50×40+2×37,000) + (65,914.2857142857×0.5)
Q = 109,857.14285714285
109,857.14285714285 QSD are minted in this scenario. 76,900 QSD are backed by the value of BUSD, BNB, and BTCB as collateral while the remaining 32,957.14285714285 QSD are not backed by anything. Instead, CEC is burned and removed from circulation proportional to the value of minted algorithmic QSD.
If not enough CEC is put into the minting function alongside the collateral, the transaction will fail with a subtraction underflow error.
NOTE: At the initial stage of QSD, only the single collateral function will be opened. After the system runs steadily, multiple collateral functions will be opened at the same time to provide a complete minting function.
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SubsAttribute - Maple Help
Home : Support : Online Help : Connectivity : Web Features : XMLTools : SubsAttribute
replace an attribute's value in an XML element
replace an attribute's name in an XML element
SubsAttribute(xmlTree, attrName, attrValue)
SubsAttribute(xmlTree, attr)
SubsAttributeName(xmlTree, attrName, newName)
SubsAttributeName(xmlTree, attr2)
\mathrm{attrName}=\mathrm{attrValue}
; attribute specification
string; new attribute name
\mathrm{attrName}=\mathrm{newName}
; attribute name specification
The SubsAttribute command replaces the value of attribute attrName with attrValue in the XML element xmlTree.
The attribute can be specified in two ways.
- As a pair consisting of the name attrName of the attribute and its new value attrValue.
- As an equation attr. Specifying the attribute substitution as an equation is equivalent to using SubsAttribute(xmlTree, lhs(attr), rhs(attr)).
A new XML element with the attribute attrName that has the new specified value is returned. The original value of the attribute attrName (which may or may not be the same as the new value attrValue) is discarded.
The SubsAttributeName command replaces the name of an attribute in the XML element xmlTree.
- As a pair consisting of the name attrName of the attribute and its new name newName.
- As an equation attr2 representing the substitution of the old name for the new. Specifying the attribute substitution as an equation attr2 is equivalent to using SubsAttributeName(xmlTree, lhs(attr2), rhs(attr2)).
A new XML element is returned where the original attribute attrName is replaced by one named newName, but newName has the value of the original attribute.
Both of these functions return an error if the XML element represented by xmlTree has no attribute named attrName.
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{xmlTree}≔\mathrm{XMLElement}\left("a",["colour"="red"],"some text"\right):
\mathrm{Print}\left(\mathrm{xmlTree}\right)
<a colour = 'red'>some text</a>
\mathrm{Print}\left(\mathrm{SubsAttribute}\left(\mathrm{xmlTree},"colour"="blue"\right)\right)
<a colour = 'blue'>some text</a>
\mathrm{Print}\left(\mathrm{SubsAttribute}\left(\mathrm{xmlTree},"colour","blue"\right)\right)
\mathrm{Print}\left(\mathrm{SubsAttributeName}\left(\mathrm{xmlTree},"colour"="color"\right)\right)
<a color = 'red'>some text</a>
\mathrm{Print}\left(\mathrm{SubsAttributeName}\left(\mathrm{xmlTree},"colour","color"\right)\right)
\mathrm{SubsAttributeName}\left(\mathrm{xmlTree},"color","shade"\right)
Error, (in `anonymous procedure called from XMLTools:-NSXML:-queryAttribute`) element does not have an attribute called "color"
XMLTools[AddAttribute]
XMLTools[AttributeNames]
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Tensor_(intrinsic_definition) Knowpia
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In differential geometry an intrinsic[definition needed] geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally.
Note: This article assumes an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article.
Definition via tensor products of vector spacesEdit
Given a finite set { V1, ..., Vn } of vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn, an element of which is termed a tensor.
A tensor on the vector space V is then defined to be an element of (i.e., a vector in) a vector space of the form:
{\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}}
where V∗ is the dual space of V.
If there are m copies of V and n copies of V∗ in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant order n and total order m + n. The tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V∗ (for this reason the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m, n) is denoted
{\displaystyle T_{n}^{m}(V)=\underbrace {V\otimes \dots \otimes V} _{m}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{n}.}
Example 1. The space of type (1, 1) tensors,
{\displaystyle T_{1}^{1}(V)=V\otimes V^{*},}
is isomorphic in a natural way to the space of linear transformations from V to V.
Example 2. A bilinear form on a real vector space V,
{\displaystyle V\times V\to F,}
corresponds in a natural way to a type (0, 2) tensor in
{\displaystyle T_{2}^{0}(V)=V^{*}\otimes V^{*}.}
An example of such a bilinear form may be defined, termed the associated metric tensor, and is usually denoted g.
Tensor rankEdit
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor (Hackbusch 2012, pp. 4)) is a tensor that can be written as a product of tensors of the form
{\displaystyle T=a\otimes b\otimes \cdots \otimes d}
where a, b, ..., d are nonzero and in V or V∗ – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors. The rank of a tensor T is the minimum number of simple tensors that sum to T (Bourbaki 1989, II, §7, no. 8).
The zero tensor has rank zero. A nonzero order 0 or 1 tensor always has rank 1. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which is dn−1 when each product is of n vectors from a finite-dimensional vector space of dimension d.
The term rank of a tensor extends the notion of the rank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors:
{\displaystyle A=vw^{\mathrm {T} }.}
The rank of a matrix A is the smallest number of such outer products that can be summed to produce it:
{\displaystyle A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.}
In indices, a tensor of rank 1 is a tensor of the form
{\displaystyle T_{ij\dots }^{k\ell \dots }=a_{i}b_{j}\cdots c^{k}d^{\ell }\cdots .}
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix (Halmos 1974, §51), and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often very hard to determine, and low rank decompositions of tensors are sometimes of great practical interest (de Groote 1987). Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms
{\displaystyle z_{k}=\sum _{ij}T_{ijk}x_{i}y_{j}}
for given inputs xi and yj. If a low-rank decomposition of the tensor T is known, then an efficient evaluation strategy is known (Knuth 1998, pp. 506–508).
{\displaystyle T_{n}^{m}(V)}
can be characterized by a universal property in terms of multilinear mappings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for free modules, and the "universal" approach carries over more easily to more general situations.
A scalar-valued function on a Cartesian product (or direct sum) of vector spaces
{\displaystyle f:V_{1}\times \cdots \times V_{N}\to F}
is multilinear if it is linear in each argument. The space of all multilinear mappings from V1 × ... × VN to W is denoted LN(V1, ..., VN; W). When N = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from V to W is denoted L(V; W).
The universal characterization of the tensor product implies that, for each multilinear function
{\displaystyle f\in L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};W)}
{\displaystyle W}
can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function
{\displaystyle T_{f}\in L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};W)}
{\displaystyle f(\alpha _{1},\ldots ,\alpha _{m},v_{1},\ldots ,v_{n})=T_{f}(\alpha _{1}\otimes \cdots \otimes \alpha _{m}\otimes v_{1}\otimes \cdots \otimes v_{n})}
{\displaystyle v_{i}\in V}
{\displaystyle \alpha _{i}\in V^{*}.}
Using the universal property, it follows that the space of (m,n)-tensors admits a natural isomorphism
{\displaystyle T_{n}^{m}(V)\cong L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};F)\cong L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};F).}
Each V in the definition of the tensor corresponds to a V* inside the argument of the linear maps, and vice versa. (Note that in the former case, there are m copies of V and n copies of V*, and in the latter case vice versa). In particular, one has
{\displaystyle {\begin{aligned}T_{0}^{1}(V)&\cong L(V^{*};F)\cong V\\T_{1}^{0}(V)&\cong L(V;F)=V^{*}\\T_{1}^{1}(V)&\cong L(V;V)\end{aligned}}}
Tensor fieldsEdit
Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is sometimes used as a shorthand for tensor field. A tensor field expresses the concept of a tensor that varies from point to point on the manifold.
Abraham, Ralph; Marsden, Jerrold E. (1985), Foundations of Mechanics (2 ed.), Reading, Mass.: Addison-Wesley, ISBN 0-201-40840-6 .
Bourbaki, Nicolas (1989), Elements of Mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9 .
de Groote, H. F. (1987), Lectures on the Complexity of Bilinear Problems, Lecture Notes in Computer Science, vol. 245, Springer, ISBN 3-540-17205-X .
Halmos, Paul (1974), Finite-dimensional Vector Spaces, Springer, ISBN 0-387-90093-4 .
Jeevanjee, Nadir (2011), An Introduction to Tensors and Group Theory for Physicists, ISBN 978-0-8176-4714-8
Knuth, Donald E. (1998) [1969], The Art of Computer Programming vol. 2 (3rd ed.), pp. 145–146, ISBN 978-0-201-89684-8 .
Hackbusch, Wolfgang (2012), Tensor Spaces and Numerical Tensor Calculus, Springer, p. 4, ISBN 978-3-642-28027-6 .
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International Journal of Geosciences > Vol.10 No.4, April 2019
Polarization Filtering Method for Suppressing Surface Wave in Time-Frequency Domain ()
1School of Geophysics and Information Technology, China University of Geosciences 1, Beijing, China.
2Key Laboratory of Coal Resources Exploration and Comprehensive Utilization, Ministry of Land and Resources, Xi’an, China.
3Information Center Computer Office, Daqing Drilling & Exploration Engineering Corporation No.1 Geo-Logging Company, Daqing, China.
Yang, X. , Gao, Y. , Zhang, W. , Wang, Y. and Lan, M. (2019) Polarization Filtering Method for Suppressing Surface Wave in Time-Frequency Domain. International Journal of Geosciences, 10, 481-490. doi: 10.4236/ijg.2019.104028.
WT\left(b,a\right)=\left({g}_{b,a},x\right)={\int }_{-\infty }^{+\infty }\frac{1}{a}{g}^{\ast }\left(\frac{t-b}{a}\right)x\left(t\right)\text{d}t
{g}_{b,a}\left(t\right)=\frac{1}{a}g\left(\frac{t-b}{a}\right)
WT\left(b,a\right)
g\left(t\right)={\text{e}}^{2\text{π}i{f}_{0}t}{\text{e}}^{-\frac{{r}^{2}}{2{a}^{2}}}
WT\left(b,a\right)
x\left(t\right)=\frac{1}{{c}_{g}}{\int }_{-\infty }^{+\infty }\frac{1}{{a}^{2}}g\left(\frac{t-b}{a}\right)WT\left(b,a\right)\text{d}b\text{d}a
{c}_{g}={\displaystyle {\int }_{0}^{+\infty }\frac{{\left|\stackrel{¯}{g\left(f\right)}\right|}^{2}}{f}\text{d}f}
\stackrel{¯}{g\left(f\right)}
W{T}_{i}\left(b,a\right)
{W}_{i}\left(b+\tau ,a\right)
{W}_{i}\left(b+\tau ,a\right)\approx |W{T}_{i}\left(b,a\right)|\cdot \mathrm{cos}\left[{\Omega }_{i}\left(b,a\right)\tau +\mathrm{arg}W{T}_{k}\left(b,a\right)\right]
{\Omega }_{i}\left(b,a\right)
{\Omega }_{i}\left(b,a\right)=\frac{\partial }{\partial b}\mathrm{arg}W{T}_{i}\left(b,a\right)
{\Omega }_{i}\left(b,a\right)
{\Omega }_{i}\left(b,a\right)=-\mathrm{Im}\left[\frac{W{T}_{i}^{{g}^{\prime }}\left(b,a\right)}{W{T}_{i}^{g}\left(b,a\right)}\right]
W{T}_{i}\left(b,a\right)
M\left(b,a\right)=\left[\begin{array}{l}{I}_{xx}\left(b,a\right){I}_{xy}\left(b,a\right){I}_{xz}\left(b,a\right)\\ {I}_{xy}\left(b,a\right){I}_{yy}\left(b,a\right){I}_{yz}\left(b,a\right)\\ {I}_{xz}\left(b,a\right){I}_{yz}\left(b,a\right){I}_{zz}\left(b,a\right)\end{array}\right]
\begin{array}{l}{I}_{k,m}\left(b,a\right)=\left|W{T}_{k}\left(b,a\right)\right|\left|W{T}_{m}\left(b,a\right)\right|\times \{\text{sinc}\left[\frac{{\Omega }_{k}\left(b,a\right)-{\Omega }_{m}\left(b,a\right)}{2}{T}_{k,m}\left(b,a\right)\right]\\ \text{}\times \mathrm{cos}\left[\mathrm{arg}W{T}_{k}\left(b,a\right)-\mathrm{arg}W{T}_{m}\left(b,a\right)\right]\\ \text{}+\text{sinc}\left[\frac{{\Omega }_{k}\left(b,a\right)-{\Omega }_{m}\left(b,a\right)}{2}{T}_{k,m}\left(b,a\right)\right]\\ \text{}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\begin{array}{c}\text{\hspace{0.05em}}\\ \text{\hspace{0.05em}}\end{array}\times \mathrm{cos}\left[\mathrm{arg}W{T}_{k}\left(b,a\right)+\mathrm{arg}W{T}_{m}\left(b,a\right)\right]\}-{\mu }_{k}{\mu }_{m}\end{array}
\begin{array}{l}{\mu }_{k}\left(b,a\right)=\frac{1}{{T}_{k,m}\left(b,a\right)}{\int }_{-T\left(b,a\right)/2}^{T\left(b,a\right)/2}{W}_{k}\left(b+\tau ,a\right)\text{d}\tau \\ \text{}=\mathrm{Re}\left[W{T}_{k}\left(b,a\right)\right]\text{sinc}\left[T\left(b,a\right){\Omega }_{k}\left(b,a\right)/2\right]\end{array}
T\left(b,a\right)
T\left(b,a\right)
T\left(b,a\right)=\frac{6\text{π}N}{{\Omega }_{x}\left(b,a\right)+{\Omega }_{y}\left(b,a\right)+{\Omega }_{z}\left(b,a\right)}=\frac{2\text{π}N}{{\Omega }_{av}^{xyz}\left(b,a\right)}
{\Omega }_{av}^{xyz}
M\left(b,a\right)
R\left(b,a\right)=\sqrt{{\lambda }_{1}\left(b,a\right)}\frac{{V}_{1}\left(b,a\right)}{‖{V}_{1}\left(b,a\right)‖}
{r}_{s}\left(b,a\right)=\sqrt{{\lambda }_{2}\left(b,a\right)}\frac{{V}_{2}\left(b,a\right)}{‖{V}_{2}\left(b,a\right)‖}
r\left(b,a\right)=\sqrt{{\lambda }_{3}\left(b,a\right)}\frac{{V}_{3}\left(b,a\right)}{‖{V}_{3}\left(b,a\right)‖}
\rho \left(b,a\right)=‖{r}_{s}\left(b,a\right)‖/‖R\left(b,a\right)‖
{\rho }_{\text{1}}\left(b,a\right)=‖r\left(b,a\right)‖/‖{r}_{s}\left(b,a\right)‖
\delta \left(b,a\right)=\mathrm{arctan}\left(\sqrt{{V}_{1,x}{\left(b,a\right)}^{2}+{V}_{1,y}{\left(b,a\right)}^{2}}/{V}_{1,z}\left(b,a\right)\right)
\alpha \left(b,a\right)=\mathrm{arctan}\left({V}_{1,y}\left(b,a\right)/{V}_{1,x}\left(b,a\right)\right)
\rho \left(b,a\right)
\theta \left(b,a\right)=\text{π}/2-\delta \left(b,a\right)
WT\left(b,a\right)={F}_{e}\left(b,a\right)W{T}_{i}\left(b,a\right),\text{}a>0
{F}_{e}\left(b,a\right)=\left\{\begin{array}{l}1\text{}\rho \left(b,a\right)\in {P}_{\rho }且\theta \left(b,a\right)\in {P}_{\theta }\\ 0\text{others}\end{array}
t\left(x\right)={t}_{0}+\frac{x}{v}
[1] Shimshoni, M. and Smith, S.W. (1964) Seismic Signal Enhancement with Three-Component Detectors. Geophysics, 29, 664-671.
[2] Flinn, E.A. (1965) Signal Analysis Using Rectilinearity and Direction of Particle Motion. Proceedings of the IEEE, 53, 1874-1876.
[3] Samson, J.C. (1977) Matrix and Stokes Vector Representation of Detectors for Polarized Wave-Forms. Geophysical Journal of the Royal Astronomical Society, 51, 583-603.
[4] Benhama, A., Cliet, C. and Dubesset, M. (1988) Study and Application of Spatial Directional Filtering in Three-Component Recordings. Geophysical Prospecting, 36, 591-613.
[5] Jurkevics, A. (1988) Polarization Analysis of Three-Component Array Data. Bulletin of the Seismological Society of America, 78, 1725-1743.
[6] Shieh, C. and Herrmann, R.B. (1990) Ground Roll: Rejection Using Polarization Filters. Geophysics, 55, 1216-1222.
[7] Huang, Z.Y., Gao, L., Xu, Y.M., et al. (1996) The Analysis of Polarization in Three-Component Seismic Data and Its Application. Geophysical Prospecting for Petroleum, 35, 9-16.
[8] Chen, Y., Zhang, Z.J. and Tian, X.B. (2005) Complex Polarization Analysis Based on Windowed Hilbert Transform and Its Application. Geophysics, 48, 889-895.
https://doi.org/10.1002/cjg2.736
[9] Liu, J.H., Liu, F.T. and Xu, Y. (2006) Polarization Analysis of Three-Component Seismic Data. Progress in Geophysics, 21, 6-10.
[10] Christoffersson, A., Husebye, E.S. and Ingate, S.F. (1985) A New Technique for 3-Component Seismogram Analysis. Semiannual Technical Summary, 2, 85-86.
[11] Jackson, G.M., Mason, I.M. and Greenhalgh, S.A. (1991) Principal Component Transforms of Triaxial Recordings by Singular Value Decomposition. Geophysics, 56, 528-533.
[12] Vidale, T. (1986) Complex Polarization Analysis of Particle Motion. Bulletin of the Seismological Society of America, 76, 1393-1405.
[13] Rene, R.M., Fitter, J.L., Forsyth, P.M., et al. (1986) Multi-Component Seismic Studies Using Complex Trace Analysis. Geophysics, 51, 1235-1251.
[14] Stewart, R.R. (1990) Ground-Roll Filtering Using Local Instantaneous Polarization. Crewes, 97, 28-33.
[15] Morozov, L.B. and Smithson, S.B. (1996) Instantaneous Polarization Attributes and Directional Filtering. Geophysics, 61, 872-881.
[16] Zhang, Y. and Zhou, J. (1999) Analysis of Factors Affecting Spatial Direction Filtering Effect. Oil and Gas Geology, 20, 212-215.
[17] Tatham, R.H. and McCormack, M.D. (1991) Multicomponent Seismology in Petroleum Exploration. Society of Exploration Geophysicists.
[18] Ma, J. and Li, Q. (2015) Suppression of Seismic Surface Waves by Time-Frequency Domain Polarization Filtering. Oil Geophysical Prospecting, 50, 1089-1097.
|
[[Image:Moffey.jpg|360px|right]]
Just like in [[Videos/A_Digital_Media_Primer_For_Geeks|the previous episode]], we've covered a broad range of
Like in [[Videos/A_Digital_Media_Primer_For_Geeks|_A Digital Media Primer for Geeks_]], we've covered a broad range of
topics, and yet barely scratched the surface of each one. If anything, my
sins of omission are greater this time around... but this is a good
sins of omission are greater this time around.
Or maybe, a good starting point. Dig deeper. Experiment. I chose my
Thus I encourage you to dig deeper and experiment. I chose my
demos very carefully to be simple and give clear results. You can
demos carefully to be simple and give clear results. You can
reproduce every one of them on your own if you like. But let's face
reproduce every one of them on your own if you like, but let's face
it, sometimes we learn the most about a spiffy toy by breaking it open
it: Sometimes we learn the most about a spiffy toy by breaking it open
and studying all the pieces that fall out. And that's OK, we're
and studying all the pieces that fall out. Play with the demo parameters, hack up the code, set up
engineers. Play with the demo parameters, hack up the code, set up
alternate experiments. The source code for everything, including the
little pushbutton demo application, is up at xiph.org.
little pushbutton demo application, is here at xiph.org.
In the course of experimentation, you're likely to run into something
that you didn't expect and can't explain. Don't worry! My earlier
snark aside, Wikipedia is fantastic for exactly this kind of casual
research. And, if you're really serious about understanding signals,
research. If you're really serious about understanding signals,
several universities have advanced materials online, such as the
[http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-003-signals-and-systems-spring-2010/index.htm 6.003]
Signals and Systems modules at MIT OpenCourseWare. And of
course, there's always the [http://webchat.freenode.net/?channels=xiph community here at Xiph.Org].
Digging deeper or not, I am out of coffee, so, until next time, happy
{\displaystyle \ squarewave(t)={\begin{cases}1,&|t|<T_{1}\\0,&T_{1}<|t|\leq {1 \over 2}T\end{cases}}}
{\displaystyle {\begin{aligned}\ squarewave(t)={\frac {4}{\pi }}\sin(\omega t)+{\frac {4}{3\pi }}\sin(3\omega t)+{\frac {4}{5\pi }}\sin(5\omega t)+\\{\frac {4}{7\pi }}\sin(7\omega t)+{\frac {4}{9\pi }}\sin(9\omega t)+{\frac {4}{11\pi }}\sin(11\omega t)+\\{\frac {4}{13\pi }}\sin(13\omega t)+{\frac {4}{15\pi }}\sin(15\omega t)+{\frac {4}{17\pi }}\sin(17\omega t)+\\{\frac {4}{19\pi }}\sin(19\omega t)+{\frac {4}{21\pi }}\sin(21\omega t)+{\frac {4}{23\pi }}\sin(23\omega t)+\\{\frac {4}{25\pi }}\sin(25\omega t)+{\frac {4}{27\pi }}\sin(27\omega t)+{\frac {4}{29\pi }}\sin(29\omega t)+\\{\frac {4}{31\pi }}\sin(31\omega t)+{\frac {4}{33\pi }}\sin(33\omega t)+\cdots \end{aligned}}}
|
On Generalized ( )-Derivations in Semiprime Rings
Basudeb Dhara, Atanu Pattanayak, "On Generalized ( )-Derivations in Semiprime Rings", International Scholarly Research Notices, vol. 2012, Article ID 120251, 7 pages, 2012. https://doi.org/10.5402/2012/120251
Academic Editor: C. Munuera
Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all .
Throughout the present paper, always denotes an associative semiprime ring with center . For any , the commutator and anticommutator of and are denoted by and and are defined by and , respectively. Recall that a ring is said to be prime, if for , implies either or and is said to be semiprime if for , implies . An additive mapping is said to be derivation if holds for all . The notion of derivation is extended to generalized derivation. The generalized derivation means an additive mapping associated with a derivation such that holds for all . Then every derivation is a generalized derivation, but the converse is not true in general.
A number of authors have studied the commutativity theorems in prime and semiprime rings admitting derivation and generalized derivation (see e.g., [1–8]; where further references can be found).
Let and be two endomorphisms of . For any , set and . An additive mapping is called a -derivation if holds for all . By this definition, every -derivation is a derivation, where means the identity map of . In the same manner the concept of generalized derivation is also extended to generalized -derivation as follows. An additive map is called a generalized -derivation if there exists a -derivation such that holds for all . Of course every generalized -derivation is a generalized derivation of , where denotes the identity map of .
There is also ongoing interest to study the commutativity in prime and semiprime rings with -derivations or generalized -derivations (see [9–17]).
The present paper is motivated by the results of [17]. In [17], Rehman et al. have discussed the commutativity of a prime ring on generalized -derivation, where and are automorphisms of . More precisely, they studied the following situations: (i) ; (ii) ; (iii) ; (iv) ; (v) for all , where is a nonzero ideal of .
The main objective of the present paper is to extend above results for generalized -derivations in semiprime ring , where and are considered as epimorphisms of .
To prove our theorems, we will frequently use the following basic identities:
Theorem 2.1. Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If for all , then contains a nonzero central ideal.
Proof. First we consider the case for all . Replacing by in (2.1) we get Using (2.1), it reduces to for all . Again replacing by in (2.3), we get for all and . Left multiplying (2.3) by and then subtracting from (2.4) we have for all and . Replacing with , , we get for all and . Since is an epimorphism of , we can write for all .
Since is semiprime, it must contain a family of prime ideals such that . If is a typical member of and , it follows that
Construct two additive subgroups and . Then . Since a group cannot be a union of two its proper subgroups, either or , that is, either or . Thus both cases together yield for any . Therefore, , that is, . Thus and so . This implies , where is a nonzero ideal of , since . Then . Since is semiprime, it follows that , that is, .
Similarly, we can obtain the same conclusion when for all .
Proof. We begin with the case for all . Replacing by in (2.9) we get Right multiplying (2.9) by and then subtracting from (2.10) we get for all .
Now replacing by in (2.11), we obtain for all and for all . Left multiplying (2.11) by and then subtracting from (2.12), we get for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.
Similar results hold in case for all .
Theorem 2.3. Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a -derivation of such that . If for all , then contains a nonzero central ideal.
Proof. We assume first that for all . This implies Replacing by in (2.14) we have Right multiplying (2.14) by and then subtracting from (2.15), we get Now replacing by , where , in (2.16), we obtain Left multiplying (2.16) by and then subtracting from (2.17), we get that that is, for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.
Similar results hold in case for all .
Proof. By our assumption first consider for all . This gives Replacing by in (2.20), we have Right multiplying (2.20) by and then subtracting from (2.21), we obtain that Now replacing by , where , in (2.22) and by using (2.22), we obtain for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.
Similar argument can be adapted in case for all .
Theorem 2.5. Let be a semiprime ring, a nonzero ideal of , and two epimorphisms of and a generalized -derivation associated with a nonzero -derivation of such that . If for all , then contains a nonzero central ideal.
Proof. We begin with the situation for all . Replacing by in (2.24), we get Right multiplying (2.24) by and then subtracting from (2.25), we obtain that for all . Now replacing by in (2.26), where , and by using (2.26), we obtain for all and for all . This is same as (2.5) in Theorem 2.1. Thus, by same argument of Theorem 2.1, we can conclude the result here.
In case for all , the similar argument can be adapted to draw the same conclusion.
We know the fact that if a prime ring contains a nonzero central ideal, then must be commutative (see Lemma 2 in [18]). Hence the following corollary is straightforward.
Corollary 2.6. Let be a prime ring, and two epimorphisms of and a generalized -derivation associated with a nonzero -derivation of satisfying any one of the following conditions:(1) for all or for all ;(2) for all or for all ;(3) for all or for all ;(4) for all or for all ;(5) for all or for all ;then must be commutative.
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Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies
2016 Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies
Todor Milanov, Yefeng Shen, Hsian-Hua Tseng
We construct an integrable hierarchy in the form of Hirota quadratic equations (HQEs) that governs the Gromov–Witten invariants of the Fano orbifold projective curve
{ℙ}_{{a}_{1},{a}_{2},{a}_{3}}^{1}
. The vertex operators in our construction are given in terms of the
K
{ℙ}_{{a}_{1},{a}_{2},{a}_{3}}^{1}
via Iritani’s
\Gamma
–class modification of the Chern character map. We also identify our HQEs with an appropriate Kac–Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of
{ℙ}^{1}
to all Fano orbifold curves.
Todor Milanov. Yefeng Shen. Hsian-Hua Tseng. "Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies." Geom. Topol. 20 (4) 2135 - 2218, 2016. https://doi.org/10.2140/gt.2016.20.2135
Received: 12 December 2014; Accepted: 5 November 2015; Published: 2016
Primary: 14N35 , 17B69
Keywords: ADE-Toda hierarchies , Fano orbifold curves , Gromov–Witten theory
Todor Milanov, Yefeng Shen, Hsian-Hua Tseng "Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies," Geometry & Topology, Geom. Topol. 20(4), 2135-2218, (2016)
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The Macro-Share Economy and Nominal GDP Targeting ()
Finance and Market Department, College of Business and Public Administration, Eastern Washington University, Spokane, USA.
Eagle, D. (2017) The Macro-Share Economy and Nominal GDP Targeting. Theoretical Economics Letters, 7, 2178-2193. doi: 10.4236/tel.2017.77149.
\frac{{B}_{t}}{{P}_{t}}=\frac{{B}_{t}}{{N}_{t}}{Y}_{t}
\frac{{B}_{t}}{{N}_{t}}
{c}_{jit}={\stackrel{˜}{y}}_{jit}+{B}_{t}/{P}_{it}
{\stackrel{˜}{y}}_{jit}
{\stackrel{˜}{y}}_{jit}
{B}_{t}/{P}_{it}
{P}_{it}
{N}_{it}/{Y}_{it}
\left({B}_{t}/{N}_{it}\right){Y}_{it}
{B}_{t}/{N}_{it}
NGA{P}_{t}\equiv \left(\frac{NGD{P}_{t}-E\left[NGD{P}_{t}\right]}{E\left[NGD{P}_{t}\right]}\right)
{\alpha }_{t}
{\alpha }_{t}\equiv \frac{{N}_{t}}{{B}_{t}}
{\alpha }_{t}
\frac{{\alpha }_{t}-E\left[{\alpha }_{t}\right]}{E\left[{\alpha }_{t}\right]}=\frac{\frac{{N}_{t}}{{B}_{t}}-E\left[\frac{{N}_{t}}{{B}_{t}}\right]}{E\left[\frac{{N}_{t}}{{B}_{t}}\right]}=\frac{{N}_{t}-E\left[{N}_{t}\right]}{E\left[{N}_{t}\right]}=NGA{P}_{t}
E\left[{\left(\frac{{\alpha }_{t}-E\left[{\alpha }_{t}\right]}{E\left[{\alpha }_{t}\right]}\right)}^{2}\right]=E\left[{\left(NGA{P}_{t}\right)}^{2}\right]
eW=kN
e=k\frac{N}{W}
{\stackrel{¯}{x}}_{*j}
{\stackrel{¯}{x}}_{*j}\equiv \left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{x}_{ij}}\right)/n
\stackrel{˜}{\stackrel{¯}{x}}
\stackrel{˜}{\stackrel{¯}{x}}\equiv {\displaystyle \underset{j=-4}{\overset{-2}{\sum }}{\stackrel{¯}{x}}_{*j}}
{\stackrel{¯}{x}}_{j}-\stackrel{˜}{\stackrel{¯}{x}}
{W}_{t}={W}_{0}\frac{{N}_{t}}{{N}_{0}{\left(1+g\right)}^{t}}
\frac{{N}_{t}}{{W}_{t}}=\frac{{N}_{0}{\left(1+g\right)}^{t}}{{W}_{0}}
\mathrm{ln}\left(P\right)=\mathrm{ln}\left(N\right)-\mathrm{ln}\left(Y\right)
\stackrel{˙}{P}=\stackrel{˙}{N}-\stackrel{˙}{Y}
\stackrel{˙}{N}=0
\stackrel{˙}{P}=-\stackrel{˙}{Y}
\stackrel{˙}{Y}=-10%
\stackrel{˙}{P}=10%
[1] Eagle, D. and Domian, D. (2005) Quasi-Real Indexing—The Pareto-Efficient Solution to Inflation Indexing. Finance 0509017, EconWPA.
http://econwpa.repec.org/eps/fin/papers/0509/0509017.pdf
[2] Koenig, E. (2013) Like a Good Neighbor: Monetary Policy, Financial Stability, and the Distribution of Risk. International Journal of Central Banking, 9, 57-82.
[3] Doepke, M. and Schneider, M. (2015) Distributional Effects of Monetary Policy. Hutchins Center Working Paper #14.
[4] Svensson, L. (1999) Price-Level Targeting versus Inflation Targeting: A Free Lunch? Journal of Money, Credit, and Banking, 31, 277-295.
[5] Kahn, G. (2009) Beyond Inflation Targeting: Should Central Banks Target the Price Level? Federal Reserve Bank of Kansas City. Economic Review, 35-64.
[6] Shukayev, M. and Ueberfeldt, A. (2010) Price Level Targeting: What Is the Right Price?
https://www.bankofcanada.ca/2010/02/working-paper-2010-8/
[7] Woodford, M. (2001) The Taylor Rule and Optimal Monetary Policy.
http://www.columbia.edu/~mw2230/taylor.pdf
[8] Eagle, D. and Christensen, L. (2012) Two Equations on the Pareto-Efficient Sharing of Real GDP Risk.
http://mpra.ub.uni-muenchen.de/cgi/users/home?screen=EPrint::View&eprintid=41051
[9] Eagle, D. (2005) Completing Markets in a One-Good, Pure Exchange Economy Without State-Contingent Securities. Finance 0501009, EconWPA.
[10] Sheedy, K.D. (2014) Debt and Incomplete Financial Markets: A Case for Nominal GDP Targeting. Brookings Papers on Economic Activity, 301-373.
[11] Mankiw, N.G. (2009) New Keynesian Economics The Concise Encyclopedia of Economics.
[12] Keynes, J.M. (1923) A Tract on Monetary Reform, Chapter 3.
[13] Cochrane, J.H. (2011) Determinacy and Identification with Taylor Rules. Journal of Political Economy, 119, 565-615.
[14] Eagle, D. (2012) Liquidity Traps and the Price (In)Determinacy of Monetary Rules.
http://mpra.ub.uni-muenchen.de/42416/1/Liquidity_Traps_Paper.pdf
[15] Krugman, P. (2011) A Volcker Moment Indeed (Slightly Wonkish). The Economist.
https://krugman.blogs.nytimes.com/2011/10/30/a-volcker-moment-indeed-slightly-wonkish/
[16] Eggertsson, G.B. and Woodford, M. (2003) The Zero Bound on Interest Rates and Optimal Monetary Policy. Brookings Papers on Economic Activity, 1, 139-233.
[17] Svensson, L. (2003) Escaping from a Liquidity Trap and Deflation: The Foolproof Way and Others. Journal of Economic Perspectives, 17, 145-166.
[18] Gaspar, V., Smets, F. and Vestin, D. (2010) Is the Time Ripe for Price Level Path Stability? Challenges in Central Banking, Chapter 2, 21-51.
[19] Eagle, D. and Domian, D. (1995) Quasi-Real Bonds: Inflation-Indexing That Retains the Government’s Hedge against Aggregate-Supply Shocks. Applied Economics Letters, Taylor and Francis Journals, 2, 487-490.
[20] Grey, J.A. (1976) Wage Indexation: A Macroeconomic Analysis. Journal of Monetary Economics, 2, 221-235.
[21] Grey, J.A. (1978) On Indexation and Contract Length. Journal of Political Economy, 86, 1-18.
[22] Fischer, S. (1977) Long-Term Contracts, Rational Expectations and the Optimal Money Supply Rule. Journal of Political Economy, 85, 191-205.
[23] Fischer, S. (1984) The Benefits of Price Stability. Price Stability and Public Policy, Federal Reserve Bank of Kansas City, 33-49.
[24] Jadresic, E. (2002) Wage Indexation and Output Stability Revisited. Journal of Money, Credit and Banking, 34, 178-196.
[25] Selgin, G.A. (1997) Less than Zero: The Case for a Falling Price Level in a Growing Economy. Institute of Economic Affairs.
http://library.mises.org/books/George%20A%20Selgin/Less%20than%20Zero%20The%
20Case%20for%20a%20Falling%20Price%20Level%20in%20a%20Growing%20Economy.pdf
[26] Weitzman, M.L. (1984) The Share Economy: Conquering Stagflation. Harvard University Press, Cambridge.
[27] Kamstra, K. and Shiller, R. (2009) The Case for Trills: Giving the People and Their Pension Funds a Stake in the Wealth of the Nation. Cowles Foundation Discussion Paper No. 1717.
[28] Shiller, R.J. (1993) Macro Markets: Creating Institutions for Managing Society’s Largest Economic Risks. Oxford University Press.
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Conformal_group Knowpia
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.
Several specific conformal groups are particularly important:
The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V
{\displaystyle Q(Tx)=\lambda ^{2}Q(x)}
For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations.
The conformal group of the sphere is generated by the inversions in circles. This group is also known as the Möbius group.
In Euclidean space En, n > 2, the conformal group is generated by inversions in hyperspheres.
In a pseudo-Euclidean space Ep,q, the conformal group is Conf(p, q) ≃ O(p + 1, q + 1) / Z2.[1]
All conformal groups are Lie groups.
Angle analysisEdit
In Euclidean geometry one can expect the standard circular angle to be characteristic, but in pseudo-Euclidean space there is also the hyperbolic angle. In the study of special relativity the various frames of reference, for varying velocity with respect to a rest frame, are related by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities. Thus, they are conformal transformations with respect to the hyperbolic angle.
A method to generate an appropriate conformal group is to mimic the steps of the Möbius group as the conformal group of the ordinary complex plane. Pseudo-Euclidean geometry is supported by alternative complex planes where points are split-complex numbers or dual numbers. Just as the Möbius group requires the Riemann sphere, a compact space, for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping. Nevertheless, the conformal group in each case is given by linear fractional transformations on the appropriate plane.[2]
Conformal group of spacetimeEdit
In 1908, Harry Bateman and Ebenezer Cunningham, two young researchers at University of Liverpool, broached the idea of a conformal group of spacetime[3][4][5] They argued that the kinematics groups are perforce conformal as they preserve the quadratic form of spacetime and are akin to orthogonal transformations, though with respect to an isotropic quadratic form. The liberties of an electromagnetic field are not confined to kinematic motions, but rather are required only to be locally proportional to a transformation preserving the quadratic form. Harry Bateman's paper in 1910 studied the Jacobian matrix of a transformation that preserves the light cone and showed it had the conformal property (proportional to a form preserver).[6] Bateman and Cunningham showed that this conformal group is "the largest group of transformations leaving Maxwell’s equations structurally invariant."[7] The conformal group of spacetime has been denoted C(1,3)[8]
Isaak Yaglom has contributed to the mathematics of spacetime conformal transformations in split-complex and dual numbers.[9] Since split-complex numbers and dual numbers form rings, not fields, the linear fractional transformations require a projective line over a ring to be bijective mappings.
It has been traditional since the work of Ludwik Silberstein in 1914 to use the ring of biquaternions to represent the Lorentz group. For the spacetime conformal group, it is sufficient to consider linear fractional transformations on the projective line over that ring. Elements of the spacetime conformal group were called spherical wave transformations by Bateman. The particulars of the spacetime quadratic form study have been absorbed into Lie sphere geometry.
Commenting on the continued interest shown in physical science, A. O. Barut wrote in 1985, "One of the prime reasons for the interest in the conformal group is that it is perhaps the most important of the larger groups containing the Poincaré group."[10]
^ Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. p. 140. ISBN 9780191085789.
^ Tsurusaburo Takasu (1941) "Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie", 2, Proceedings of the Imperial Academy 17(8): 330–8, link from Project Euclid, MR14282
^ Bateman, Harry (1908). "The conformal transformations of a space of four dimensions and their applications to geometrical optics" . Proceedings of the London Mathematical Society. 7: 70–89. doi:10.1112/plms/s2-7.1.70.
^ Bateman, Harry (1910). "The Transformation of the Electrodynamical Equations" . Proceedings of the London Mathematical Society. 8: 223–264. doi:10.1112/plms/s2-8.1.223.
^ Cunningham, Ebenezer (1910). "The principle of Relativity in Electrodynamics and an Extension Thereof" . Proceedings of the London Mathematical Society. 8: 77–98. doi:10.1112/plms/s2-8.1.77.
^ Warwick, Andrew (2003). Masters of theory: Cambridge and the rise of mathematical physics. Chicago: University of Chicago Press. pp. 416–24. ISBN 0-226-87375-7.
^ Robert Gilmore (1994) [1974] Lie Groups, Lie Algebras and some of their Applications, page 349, Robert E. Krieger Publishing ISBN 0-89464-759-8 MR1275599
^ Boris Kosyakov (2007) Introduction to the Classical Theory of Particles and Fields, page 216, Springer books via Google Books
^ Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, Springer, ISBN 0387-90332-1, MR520230
^ A. O. Barut & H.-D. Doebner (1985) Conformal groups and Related Symmetries: Physical Results and Mathematical Background, Lecture Notes in Physics #261 Springer books, see preface for quotation
The Wikibook Associative Composition Algebra has a page on the topic of: Conformal spacetime transformations
Kobayashi, S. (1972). Transformation Groups in Differential Geometry. Classics in Mathematics. Springer. ISBN 3-540-58659-8. OCLC 31374337.
Peter Scherk (1960) "Some Concepts of Conformal Geometry", American Mathematical Monthly 67(1): 1−30 doi: 10.2307/2308920
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(Redirected from Counter mode)
{\displaystyle C_{i}=E_{K}(P_{i}\oplus C_{i-1}),}
{\displaystyle C_{0}=IV,}
{\displaystyle P_{i}=D_{K}(C_{i})\oplus C_{i-1},}
{\displaystyle C_{0}=IV.}
{\displaystyle C_{i}=E_{K}(P_{i}\oplus P_{i-1}\oplus C_{i-1}),P_{0}\oplus C_{0}=IV,}
{\displaystyle P_{i}=D_{K}(C_{i})\oplus P_{i-1}\oplus C_{i-1},P_{0}\oplus C_{0}=IV.}
{\displaystyle {\begin{aligned}C_{i}&={\begin{cases}{\text{IV}},&i=0\\E_{K}(C_{i-1})\oplus P_{i},&{\text{otherwise}}\end{cases}}\\P_{i}&=E_{K}(C_{i-1})\oplus C_{i},\end{aligned}}}
{\displaystyle I_{0}={\text{IV}}.}
{\displaystyle I_{i}={\big (}(I_{i-1}\ll s)+C_{i}{\big )}{\bmod {2}}^{b},}
{\displaystyle C_{i}=\operatorname {MSB} _{s}{\big (}E_{K}(I_{i-1}){\big )}\oplus P_{i},}
{\displaystyle P_{i}=\operatorname {MSB} _{s}{\big (}E_{K}(I_{i-1}){\big )}\oplus C_{i},}
{\displaystyle C_{j}=P_{j}\oplus O_{j},}
{\displaystyle P_{j}=C_{j}\oplus O_{j},}
{\displaystyle O_{j}=E_{K}(I_{j}),}
{\displaystyle I_{j}=O_{j-1},}
{\displaystyle I_{0}={\text{IV}}.}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Block_cipher_mode_of_operation&oldid=1088697062#CTR"
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Loop Shape Goal - MATLAB & Simulink
Open-Loop Response Selection
Desired Loop Shape
Evaluating Tuning Goals
Loop Shape Goal specifies a target gain profile (gain as a function of frequency) of an open-loop response. Loop Shape Goal constrains the open-loop, point-to-point response (L) at a specified location in your control system.
Where L is much greater than 1, a minimum gain constraint on inv(S) (green shaded region) is equivalent to a minimum gain constraint on L. Similarly, where L is much smaller than 1, a maximum gain constraint on T (red shaded region) is equivalent to a maximum gain constraint on L. The gap between these two constraints is twice the crossover tolerance, which specifies the frequency band where the loop gain can cross 0 dB.
Use Loop Shape Goal when the loop shape near crossover is simple or well understood (such as integral action). To specify only high gain or low gain constraints in certain frequency bands, use Minimum Loop Gain Goal or Maximum Loop Gain Goal. When you do so, the software determines the best loop shape near crossover.
In the Tuning tab of Control System Tuner, select New Goal > Target shape for open-loop response to create a Loop Shape Goal.
When tuning control systems at the command line, use TuningGoal.LoopShape to specify a loop-shape goal.
Use this section of the dialog box to specify the signal locations at which to compute the open-loop gain. You can also specify additional loop-opening locations for evaluating the tuning goal.
Shape open-loop response at the following locations
Select one or more signal locations in your model at which to compute and constrain the open-loop gain. To constrain a SISO response, select a single-valued location. For example, to constrain the open-loop gain at a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal.
Compute response with the following loops open
Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'.
To highlight any selected signal in the Simulink® model, click . To remove a signal from the input or output list, click . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see Specify Goals for Interactive Tuning.
Use this section of the dialog box to specify the target loop shape.
Pure integrator wc/s
Check to specify a pure integrator and crossover frequency for the target loop shape. For example, to specify an integral gain profile with crossover frequency 10 rad/s, enter 10 in the Crossover frequency wc text box.
Other gain profile
Check to specify the target loop shape as a function of frequency. Enter a SISO numeric LTI model whose magnitude represents the desired gain profile. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise target loop shape using an frd model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired loop shape. For example, to specify a target loop shape of 100 (40 dB) below 0.1 rad/s, rolling off at a rate of –20 dB/decade at higher frequencies, enter frd([100 100 10],[0 1e-1 1]).
If you are tuning in discrete time, you can specify the loop shape as a discrete-time model with the same sample time that you are using for tuning. If you specify the loop shape in continuous time, the tuning software discretizes it. Specifying the loop shape in discrete time gives you more control over the loop shape near the Nyquist frequency.
Use this section of the dialog box to specify additional characteristics of the loop shape goal.
Enforce loop shape within
Specify the tolerance in the location of the crossover frequency, in decades. For example, to allow gain crossovers within half a decade on either side of the target crossover frequency, enter 0.5. Increase the crossover tolerance to increase the ability of the tuning algorithm to enforce the target loop shape for all loops in a MIMO control system.
Enforce goal in frequency range
Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time.
Stabilize closed loop system
By default, the tuning goal imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain constraint. If stability is not required or cannot be achieved, select No to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, select No.
Equalize loop interactions
For multi-loop or MIMO loop gain constraints, the feedback channels are automatically rescaled to equalize the off-diagonal (loop interaction) terms in the open-loop transfer function. Select Off to disable such scaling and shape the unscaled open-loop response.
Apply goal to
Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or block-parameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box.
For more information about tuning for multiple models, see Robust Tuning Approaches (Robust Control Toolbox).
When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint.
For Loop Shape Goal, f(x) is given by:
f\left(x\right)={‖\begin{array}{c}{W}_{S}S\\ {W}_{T}T\end{array}‖}_{\infty }.
D is an automatically-computed loop scaling factor. (If Equalize loop interactions is set to Off, then D = I.)
T = S – I is the complementary sensitivity function.
WS and WT are frequency weighting functions derived from the specified loop shape. The gains of these functions roughly match your specified loop shape and its inverse, respectively, for values ranging from –20 dB to 60 dB. For numerical reasons, the weighting functions level off outside this range, unless the specified gain profile changes slope outside this range. Because poles of WS or WT close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify loop shapes with very low-frequency or very high-frequency dynamics. For more information about regularization and its effects, see Visualize Tuning Goals.
This tuning goal imposes an implicit stability constraint on the closed-loop sensitivity function measured at the specified, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults.
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Material cache (vulcanised rubber) - The RuneScape Wiki
Material cache (vulcanised rubber)
Material caches (vulcanised rubber) are Archaeology material caches that can be found at the Warforge Dig Site at the south goblin tunnels excavation site or at the Feldip shores excavation site.
Once a vulcanised rubber cache is depleted, it takes 120 seconds for it to respawn.
Vulcanised rubber 1 Always[d 1] 3,516 150
Huzamogaarb key 1 Uncommon[d 2] 1 1
^ This item is only found inside the Warforge dig site and can only be obtained once.
{\displaystyle {\frac {L+E}{250{,}000}}}
{\displaystyle L}
{\displaystyle E}
{\displaystyle {\frac {1}{125{,}000}}}
{\displaystyle {\frac {1}{1{,}042}}}
Retrieved from ‘https://runescape.wiki/w/Material_cache_(vulcanised_rubber)?oldid=35694515’
Material caches at the Warforge Dig Site
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Lauricella_hypergeometric_series Knowpia
In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are (Lauricella 1893):
{\displaystyle F_{A}^{(3)}(a,b_{1},b_{2},b_{3},c_{1},c_{2},c_{3};x_{1},x_{2},x_{3})=\sum _{i_{1},i_{2},i_{3}=0}^{\infty }{\frac {(a)_{i_{1}+i_{2}+i_{3}}(b_{1})_{i_{1}}(b_{2})_{i_{2}}(b_{3})_{i_{3}}}{(c_{1})_{i_{1}}(c_{2})_{i_{2}}(c_{3})_{i_{3}}\,i_{1}!\,i_{2}!\,i_{3}!}}\,x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}}
for |x1| + |x2| + |x3| < 1 and
{\displaystyle F_{B}^{(3)}(a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c;x_{1},x_{2},x_{3})=\sum _{i_{1},i_{2},i_{3}=0}^{\infty }{\frac {(a_{1})_{i_{1}}(a_{2})_{i_{2}}(a_{3})_{i_{3}}(b_{1})_{i_{1}}(b_{2})_{i_{2}}(b_{3})_{i_{3}}}{(c)_{i_{1}+i_{2}+i_{3}}\,i_{1}!\,i_{2}!\,i_{3}!}}\,x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}}
for |x1| < 1, |x2| < 1, |x3| < 1 and
{\displaystyle F_{C}^{(3)}(a,b,c_{1},c_{2},c_{3};x_{1},x_{2},x_{3})=\sum _{i_{1},i_{2},i_{3}=0}^{\infty }{\frac {(a)_{i_{1}+i_{2}+i_{3}}(b)_{i_{1}+i_{2}+i_{3}}}{(c_{1})_{i_{1}}(c_{2})_{i_{2}}(c_{3})_{i_{3}}\,i_{1}!\,i_{2}!\,i_{3}!}}\,x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}}
for |x1|½ + |x2|½ + |x3|½ < 1 and
{\displaystyle F_{D}^{(3)}(a,b_{1},b_{2},b_{3},c;x_{1},x_{2},x_{3})=\sum _{i_{1},i_{2},i_{3}=0}^{\infty }{\frac {(a)_{i_{1}+i_{2}+i_{3}}(b_{1})_{i_{1}}(b_{2})_{i_{2}}(b_{3})_{i_{3}}}{(c)_{i_{1}+i_{2}+i_{3}}\,i_{1}!\,i_{2}!\,i_{3}!}}\,x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}}
for |x1| < 1, |x2| < 1, |x3| < 1. Here the Pochhammer symbol (q)i indicates the i-th rising factorial of q, i.e.
{\displaystyle (q)_{i}=q\,(q+1)\cdots (q+i-1)={\frac {\Gamma (q+i)}{\Gamma (q)}}~,}
where the second equality is true for all complex
{\displaystyle q}
{\displaystyle q=0,-1,-2,\ldots }
These functions can be extended to other values of the variables x1, x2, x3 by means of analytic continuation.
Lauricella also indicated the existence of ten other hypergeometric functions of three variables. These were named FE, FF, ..., FT and studied by Shanti Saran in 1954 (Saran 1954). There are therefore a total of 14 Lauricella–Saran hypergeometric functions.
Generalization to n variablesEdit
These functions can be straightforwardly extended to n variables. One writes for example
{\displaystyle F_{A}^{(n)}(a,b_{1},\ldots ,b_{n},c_{1},\ldots ,c_{n};x_{1},\ldots ,x_{n})=\sum _{i_{1},\ldots ,i_{n}=0}^{\infty }{\frac {(a)_{i_{1}+\cdots +i_{n}}(b_{1})_{i_{1}}\cdots (b_{n})_{i_{n}}}{(c_{1})_{i_{1}}\cdots (c_{n})_{i_{n}}\,i_{1}!\cdots \,i_{n}!}}\,x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}~,}
where |x1| + ... + |xn| < 1. These generalized series too are sometimes referred to as Lauricella functions.
When n = 2, the Lauricella functions correspond to the Appell hypergeometric series of two variables:
{\displaystyle F_{A}^{(2)}\equiv F_{2},\quad F_{B}^{(2)}\equiv F_{3},\quad F_{C}^{(2)}\equiv F_{4},\quad F_{D}^{(2)}\equiv F_{1}.}
When n = 1, all four functions reduce to the Gauss hypergeometric function:
{\displaystyle F_{A}^{(1)}(a,b,c;x)\equiv F_{B}^{(1)}(a,b,c;x)\equiv F_{C}^{(1)}(a,b,c;x)\equiv F_{D}^{(1)}(a,b,c;x)\equiv {_{2}}F_{1}(a,b;c;x).}
Integral representation of FDEdit
In analogy with Appell's function F1, Lauricella's FD can be written as a one-dimensional Euler-type integral for any number n of variables:
{\displaystyle F_{D}^{(n)}(a,b_{1},\ldots ,b_{n},c;x_{1},\ldots ,x_{n})={\frac {\Gamma (c)}{\Gamma (a)\Gamma (c-a)}}\int _{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-x_{1}t)^{-b_{1}}\cdots (1-x_{n}t)^{-b_{n}}\,\mathrm {d} t,\qquad \operatorname {Re} c>\operatorname {Re} a>0~.}
This representation can be easily verified by means of Taylor expansion of the integrand, followed by termwise integration. The representation implies that the incomplete elliptic integral Π is a special case of Lauricella's function FD with three variables:
{\displaystyle \Pi (n,\phi ,k)=\int _{0}^{\phi }{\frac {\mathrm {d} \theta }{(1-n\sin ^{2}\theta ){\sqrt {1-k^{2}\sin ^{2}\theta }}}}=\sin(\phi )\,F_{D}^{(3)}({\tfrac {1}{2}},1,{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}};n\sin ^{2}\phi ,\sin ^{2}\phi ,k^{2}\sin ^{2}\phi ),\qquad |\operatorname {Re} \phi |<{\frac {\pi }{2}}~.}
Finite-sum solutions of FDEdit
{\displaystyle a>c}
{\displaystyle a-c}
One can relate FD to the Carlson R function
{\displaystyle R_{n}}
{\displaystyle F_{D}(a,{\overline {b}},c,{\overline {z}})=R_{a-c}({\overline {b^{*}}},{\overline {z^{*}}})\cdot \prod _{i}(z_{i}^{*})^{b_{i}^{*}}={\frac {\Gamma (a-c+1)\Gamma (b^{*})}{\Gamma (a-c+b^{*})}}\cdot D_{a-c}({\overline {b^{*}}},{\overline {z^{*}}})\cdot \prod _{i}(z_{i}^{*})^{b_{i}^{*}}}
with the iterative sum
{\displaystyle D_{n}({\overline {b^{*}}},{\overline {z^{*}}})={\frac {1}{n}}\sum _{k=1}^{n}\left(\sum _{i=1}^{N}b_{i}^{*}\cdot (z_{i}^{*})^{k}\right)\cdot D_{k-i}}
{\displaystyle D_{0}=1}
where it can be exploited that the Carlson R function with
{\displaystyle n>0}
has an exact representation (see [1] for more information).
The vectors are defined as
{\displaystyle {\overline {b^{*}}}=[{\overline {b}},c-\sum _{i}b_{i}]}
{\displaystyle {\overline {z^{*}}}=[{\frac {1}{1-z_{1}}},\ldots ,{\frac {1}{1-z_{N-1}}},1]}
where the length of
{\displaystyle {\overline {z}}}
{\displaystyle {\overline {b}}}
{\displaystyle N-1}
, while the vectors
{\displaystyle {\overline {z^{*}}}}
{\displaystyle {\overline {b^{*}}}}
have length
{\displaystyle N}
{\displaystyle c>a}
{\displaystyle c-a}
In this case there is also a known analytic form, but it is rather complicated to write down and involves several steps. See [2] for more information.
^ Glüsenkamp, T. (2018). "Probabilistic treatment of the uncertainty from the finite size of weighted Monte Carlo data". EPJ Plus. 133 (6): 218. arXiv:1712.01293. Bibcode:2018EPJP..133..218G. doi:10.1140/epjp/i2018-12042-x. S2CID 125665629.
^ Tan, J.; Zhou, P. (2005). "On the finite sum representations of the Lauricella functions FD". Advances in Computational Mathematics. 23 (4): 333–351. doi:10.1007/s10444-004-1838-0. S2CID 7515235.
Appell, Paul; Kampé de Fériet, Joseph (1926). Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite (in French). Paris: Gauthier–Villars. JFM 52.0361.13. (see p. 114)
Exton, Harold (1976). Multiple hypergeometric functions and applications. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. ISBN 0-470-15190-0. MR 0422713.
Lauricella, Giuseppe (1893). "Sulle funzioni ipergeometriche a più variabili". Rendiconti del Circolo Matematico di Palermo (in Italian). 7 (S1): 111–158. doi:10.1007/BF03012437. JFM 25.0756.01. S2CID 122316343.
Saran, Shanti (1954). "Hypergeometric Functions of Three Variables". Ganita. 5 (1): 77–91. ISSN 0046-5402. MR 0087777. Zbl 0058.29602. (corrigendum 1956 in Ganita 7, p. 65)
Slater, Lucy Joan (1966). Generalized hypergeometric functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688. (there is a 2008 paperback with ISBN 978-0-521-09061-2)
Srivastava, Hari M.; Karlsson, Per W. (1985). Multiple Gaussian hypergeometric series. Mathematics and its applications. Chichester, UK: Halsted Press, Ellis Horwood Ltd. ISBN 0-470-20100-2. MR 0834385. (there is another edition with ISBN 0-85312-602-X)
Ronald M. Aarts. "Lauricella Functions". MathWorld.
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Yongguang Liu, Xiaohui Gao, Xiaowei Yang, "Research of Control Strategy in the Large Electric Cylinder Position Servo System", Mathematical Problems in Engineering, vol. 2015, Article ID 167628, 6 pages, 2015. https://doi.org/10.1155/2015/167628
Yongguang Liu,1 Xiaohui Gao,1 and Xiaowei Yang1
An ideal positioning response is very difficult to realize in the large electric cylinder system that is applied in missile launcher because of the presence of many nonlinear factors such as load disturbance, parameter variations, lost motion, and friction. This paper presents a piecewise control strategy based on the optimized positioning principle. The combined application of position interpolation method and modified incremental PID with dead band is proposed and applied into control system. The experimental result confirms that this combined control strategy is not only simple to be applied into high accuracy real-time control system but also significantly improves dynamic response, steady accuracy, and anti-interference performance, which has very important significance to improve the smooth control of the large electric cylinder.
The electric cylinder is a kind of actuator which can convert the rotary motion of motor to linear motion by screw. In recent years, with the performance improvement of the servo motor and drive mechanism, the electric cylinder has made a significant breakthrough in the aspects of large stroke and heavy load, which will lead to a tendency of using it instead of hydraulic cylinder in some industrial applications in the future [1, 2]. The electric cylinder has been widely applied in military, medical, and industrial equipment because of high transmission efficiency, excellent dynamic characteristic, high reliability, good environmental adaptability, compact structure, and simple operation and maintenance [3–5]. Due to load disturbance, parameter variations, friction, and other nonlinear factors in the position servo system, robust control, fuzzy control, self-adaptive control, internal model control, and sliding mode control can greatly improve the stability and anti-interference ability [6–17]. But these nonlinear control algorithms are very difficult to be applied in the PLC controller or high accuracy real-time control system because of the complexity. Hence, this paper presents a simple piecewise control algorithm which respectively adopts position interpolation method and incremental PID with dead band in different control stages. The final experimental result indicates that the dynamic response process is fast and smooth and the positional accuracy is very high, which will promote the application of large electric cylinders.
The erecting mechanism in the missile launching vehicle can erect the missile launcher from horizontal to tilted or vertical position. The vehicle body, launcher, and driving mechanism compose erecting mechanism which has been applied in many kinds of missile launching vehicles (Figure 1). As the successful development of the large stroke and heavy load electric cylinder, it will replace the hydraulic cylinder and become the primary driving mechanism. This paper introduces a large electric cylinder whose stroke is 1.5 m and maximum output force is 30 t (ton), which can ensure the missile launcher rotation range of 0–90° and positional accuracy of 0.05°. The controller of the erecting mechanism is PLC and control cycle is 10 ms. When the position instruction is 30° and conventional PID is applied in the controller, the position response curve is shown in Figure 2. It indicates that the buffeting appears in the initial stage and system is unable to meet the stability precision because of oscillation in the close to desired position.
Erecting mechanism of the missile launcher.
Position response curve.
3. Modeling and Simulation Based on Simulink
In order to research the control strategy and apply it into control system, the model of the erecting mechanism is established and the simulation is applied to analyze the reason of the buffeting appearing in the initial stage based on Simulink.
3.1. Modeling of the Erecting Mechanism
This paper establishes the mathematical model of missile launcher that is driven by electric cylinder and simulates based on Simulink to analyze the reason of buffeting in the initial stage and design the controller.
Electric cylinder is mainly composed of servo motor, reducer, screw, and piston rod that can convert the rotary motion of the servo motor to linear motion (Figure 3). The transmission schematic diagram is shown in Figure 4 and the mathematic model is shown inwhere is the input voltage of the armature winding; is the current of the armature winding; is the resistance of the armature winding; is the inductance of the armature winding; is the electromagnetic torque of the motor shaft; is the electromagnetic torque coefficient; is the output moment of motor; is the moment of inertia of the motor shaft; is the viscous friction coefficient of the motor shaft; is the output rotation angle of motor; is the voltage coefficient; is the output moment of electric cylinder; is the loading force; is transmission ratio; is screw lead; is the mechanical transmission efficiency; is the moment of inertia of the drive system; is the viscous friction coefficient of the drive system; is the stretching-length of piston rod.
Electric cylinder.
Transmission schematic diagram of the electric cylinder.
Structural schematic diagram of the erecting mechanism is shown in Figure 5 which describes the geometric relationship and force status among vehicle body, launcher, and electric cylinder.
Structural schematic diagram of the erecting mechanism.
is the pivot point of the missile launcher; is the pivot point of the electric cylinder; is the joining point between missile launcher and electric cylinder; is the rotation angle of the missile launcher; is the horizontal angle of the electric cylinder; is the original length of the electric cylinder; is the length of the working electric cylinder; is the gravity of the missile launcher; is the output force of the electric cylinder.
The following equation shows the mathematic model of the erecting mechanism:
Figure 6 shows the block diagram of the missile launcher through combining (1) with (2). The main parameters of the missile launcher are shown in Table 1. Figure 7 shows the position response in the simulation and experiment, which proves the correctness of the model.
/(V·) /(N·m·) /H /Ω
/(kg·m²) /(N·m·) /m /N
0.012 0.0063 0.02 0.3 68500
/(N·m·) /kg·m² /kg /m
0.0153 0.0495 12 1200 2.56
/m /m /m /m /(kg·m²)
1.0265 1.5 5 3.101 15000
Main parameters of the missile launcher.
Block diagram of the missile launcher.
Position response.
3.2. Simulation of the Erecting Mechanism
In order to understand the reason of the buffeting that appears in the initial stage more intuitively, we analyze all the elements in the model applying the simulation environment. Figure 8 describes the load of the electric cylinder when the missile launcher erects from horizontal to vertical position, which indicates that the load gets smaller with the increase of the angle. Figure 9 shows that the impact current in the servo motor can reach up to 300 A which is far beyond the rated current 62 A and lasts about 1 s. The fluctuant current in the servo motor leads to the buffeting in the initial stage. The overload current will seriously reduce the service life and even damage the electric cylinder. Therefore, we must adopt control algorithm to suppress the impact current and improve the stability precision.
Load of the electric cylinder.
Current in the servo motor.
4. Design of the Controller
The ideal velocity curve should be trapezoid based on the optimized positioning principle. Hence, the position response process is divided into dynamic response and close to desired position periods (Figure 10). The dynamic response is composed of accelerative, constant velocity, and decelerating periods. The position interpolation method and incremental PID are applied to the dynamic response period to ensure the steady accuracy and rapidity and incremental PID with dead band is applied to close to desired position period to ensure the stability precision. Since the incremental PID cannot produce disturbance when switching control parameters, the different control parameters applied in the dynamic response and close to desired position periods can avoid buffeting and enhance adaptability.
Ideal velocity curve.
4.1. Position Interpolation Method
Position interpolation method is to add some new position instructions between current and desired position according to acceleration, velocity, and control cycle of the system. It can make the dynamic response realize the ideal velocity curve through tracing the new position instructions. If the current position instruction is , the next control cycle position instruction is , where can be obtained by:
is the maximum acceleration of system requirement; is the duration of the acceleration; is control cycle; is the maximum velocity of system requirement; is the duration of the deceleration; is position instruction.
Incremental PID is to obtain control output by accumulating the increment . The sharp increase or decrease of the output can produce buffeting. The output of incremental PID is gradual change, which can avoid the chattering and integral saturation. can be obtained bywhere , , is desired position and is current position sampling, and Kp, Ki, and Kd are the parameters of the incremental PID.
In the first control period, and are both 0, is bigger, and = 0.01 s is smaller. Then, and are both bigger. Hence, Kp and Kd are both 0 in the first control period and adopt adaptable value in other control periods.
4.3. Incremental PID with Dead Band
When the missile launcher reaches the close to desired position, it will produce oscillation and cannot satisfy the stability precision due to the existence of nonlinear factors such as friction and lost motion. Therefore, we adopt incremental PID with dead band in the close to desired position.(1)Set the range of dead band that is little bigger than stability precision.(2)As the displacement error, the control output is 0.(3)As the displacement error, the control output is of the incremental PID.
Figure 11 shows the flowchart of the incremental PID with dead band.
Flowchart of the incremental PID with dead band.
In the position servo control system of the missile launcher, the steps of the control algorithm are as follows.(1)Set some parameters such as maximum acceleration , maximum velocity , and range of close to desired position .(2)As the displacement error, the incremental PID with dead band is adopted.(3)As the displacement error, position interpolation method and incremental PID are adopted.
Figure 12 shows the flowchart of the control algorithm.
Flowchart of the control algorithm.
After applying this combined control algorithm to the controller of position servo system in the missile launcher, Figures 13 and 14, respectively, show the position response and current of the servo motor when the desired position is 30°. Figure 13 indicates that the dynamic response process is fast and smooth and the buffeting disappears in the initial stage. When the missile launcher arrives at the desired position, the steady state error is 0.02° which is in the range of stability precision. Figure 14 indicates that the maximum current in the servo motor is 32 A and overload and impact current are both eliminated, which can prolong the service life of electric cylinder.
The bigger load and acceleration in the initial stage cause the fluctuation and impact current in the servo motor, which can produce buffeting. The nonlinear friction and lost motion bring about the oscillation in the close to desired position. This paper proposes a piecewise control strategy based on the optimized positioning principle. Position interpolation method and incremental PID with dead band are, respectively, applied into dynamic response and close to desired position periods. The position interpolation method significantly improves steady accuracy and rapidity performance, which can ensure the position response along the ideal curve. The incremental PID with dead band can largely enhance the stability precision. These control algorithms are easily applied to the high accuracy real-time control system and acquire excellent results, which will greatly promote the application of the large electric cylinder.
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3.1.2 Coalescence of Local Group and galaxies outside the Local Group are no longer accessible
In the 1970s, the future of an expanding universe was studied by the astrophysicist Jamal Islam[10] and the physicist Freeman Dyson.[11] Then, in their 1999 book The Five Ages of the Universe, the astrophysicists Fred Adams and Gregory Laughlin have divided the past and future history of an expanding universe into five eras. The first, the Primordial Era, is the time in the past just after the Big Bang when stars had not yet formed. The second, the Stelliferous Era, includes the present day and all of the stars and galaxies we see. It is the time during which stars form from collapsing clouds of gas. In the subsequent Degenerate Era, the stars will have burnt out, leaving all stellar-mass objects as stellar remnants—white dwarfs, neutron stars, and black holes. In the Black Hole Era, white dwarfs, neutron stars, and other smaller astronomical objects have been destroyed by proton decay, leaving only black holes. Finally, in the Dark Era, even black holes have disappeared, leaving only a dilute gas of photons and leptons.[12], pp. xxiv–xxviii.
Stars of very low mass will eventually exhaust all their fusible hydrogen and then become helium white dwarfs.[15] Stars of low to medium mass will expel some of their mass as a planetary nebula and eventually become white dwarfs; more massive stars will explode in a core-collapse supernova, leaving behind neutron stars or black holes.[16] In any case, although some of the star's matter may be returned to the interstellar medium, a degenerate remnant will be left behind whose mass is not returned to the interstellar medium. Therefore, the supply of gas available for star formation is steadily being exhausted.
5 billion years from now (18.7 billion years after the Big Bang)
The Andromeda Galaxy is currently approximately 2.5 million light years away from our galaxy, the Milky Way Galaxy, and they are moving towards each other at approximately 300 kilometers (186 miles) per second. Approximately five billion years from now, or 19 billion years after the Big Bang, the Milky Way and the Andromeda Galaxy will collide with one another and merge into one large galaxy based on current evidence. Up until 2012, there was no way to know whether the possible collision was definitely going to happen or not.[17] In 2012, researchers came to the conclusion that the collision is definite after using the Hubble Space Telescope between 2002 and 2010 to track the motion of Andromeda.[18]
Coalescence of Local Group and galaxies outside the Local Group are no longer accessible
Assuming that dark energy continues to make the universe expand at an accelerating rate, in about 150 billion years all galaxies outside the local group will pass behind the cosmological horizon. It will then be impossible for events in the local group to affect other galaxies. Similarly it will be impossible for events after 150 billion years, as seen by observers in distant galaxies, to affect events in the local group.[3] However, an observer in the local group will continue to see distant galaxies, but events they observe will become exponentially more time dilated (and red shifted[3]) as the galaxy approaches the horizon until time in the distant galaxy seems to stop. The observer in the local group never actually sees the distant galaxy pass beyond the horizon and never observes events after 150 billion years in their local time. Therefore, after 150 billion years intergalactic transportation and communication becomes causally impossible.
2×1012 (2 trillion) years from now, all galaxies outside the Local Supercluster will be red-shifted to such an extent that even gamma rays they emit will have wavelengths longer than the size of the observable universe of the time. Therefore, these galaxies will no longer be detectable in any way.[3]
It is estimated that in 1014 (100 trillion) years or less, star formation will end.[4], §IID. The least massive stars take the longest to exhaust their hydrogen fuel (see stellar evolution). Thus, the longest living stars in the universe are low-mass red dwarfs, with a mass of about 0.08 solar masses (Template:Solar mass), which have a lifetime of order 1013 (10 trillion) years.[20] Coincidentally, this is comparable to the length of time over which star formation takes place.[4] §IID. Once star formation ends and the least massive red dwarfs exhaust their fuel, nuclear fusion will cease. The low-mass red dwarfs will cool and become white dwarfs.[15] The only objects remaining with more than planetary mass will be brown dwarfs, with mass less than Template:Solar mass, and degenerate remnants; white dwarfs, produced by stars with initial masses between about 0.08 and 8 solar masses; and neutron stars and black holes, produced by stars with initial masses over Template:Solar mass. Most of the mass of this collection, approximately 90%, will be in the form of white dwarfs.[5] In the absence of any energy source, all of these formerly luminous bodies will cool and become faint.
The universe will become extremely dark after the last star burns out. Even so, there can still be occasional light in the universe. One of the ways the universe can be illuminated is if two carbon–oxygen white dwarfs with a combined mass of more than the Chandrasekhar limit of about 1.4 solar masses happen to merge. The resulting object will then undergo runaway thermonuclear fusion, producing a Type Ia supernova and dispelling the darkness of the Degenerate Era for a few weeks.[21][22] If the combined mass is not above the Chandrasekhar limit but is larger than the minimum mass to fuse carbon (about Template:Solar mass), a carbon star could be produced, with a lifetime of around 106 (1 million) years.[12], p. 91 Also, if two helium white dwarfs with a combined mass of at least Template:Solar mass collide, a helium star may be produced, with a lifetime of a few hundred million years.[12], p. 91 Finally, if brown dwarfs collide with each other, a red dwarf star may be produced which can survive for 1013 (10 trillion) years.[20][21]
1015 (1 quadrillion) years
1019 to 1020 (10 to 100 quintillion) years
Over time, objects in a galaxy exchange kinetic energy in a process called dynamical relaxation, making their velocity distribution approach the Maxwell–Boltzmann distribution.[24] Dynamical relaxation can proceed either by close encounters of two stars or by less violent but more frequent distant encounters.[25] In the case of a close encounter, two brown dwarfs or stellar remnants will pass close to each other. When this happens, the trajectories of the objects involved in the close encounter change slightly. After a large number of encounters, lighter objects tend to gain kinetic energy while the heavier objects lose it.[12], pp. 85–87
Because of dynamical relaxation, some objects will gain enough energy to reach galactic escape velocity and depart the galaxy, leaving behind a smaller, denser galaxy. Since encounters are more frequent in the denser galaxy, the process then accelerates. The end result is that most objects (90% to 99%) are ejected from the galaxy, leaving a small fraction (maybe 1% to 10%) which fall into the central supermassive black hole.[4], §IIIAD;[12], pp. 85–87
Given our assumed half-life of the proton, nucleons (protons and bound neutrons) will have undergone roughly 1,000 half-lives by the time the universe is 1040 years old. To put this into perspective, there are an estimated 1080 protons currently in the universe.[28] This means that the number of nucleons will be slashed in half 1,000 times by the time the universe is 1040 years old. Hence, there will be roughly ½1,000 (approximately 10−301) as many nucleons remaining as there are today; that is, zero nucleons remaining in the universe at the end of the Degenerate Age. Effectively, all baryonic matter will have been changed into photons and leptons. Some models predict the formation of stable positronium atoms with a greater diameter than the observable universe's current diameter in 1085 years, and that these will in turn decay to gamma radiation in 10141 years.[4] §IID, [5]
After 1040 years, black holes will dominate the universe. They will slowly evaporate via Hawking radiation.[4], §IVG. A black hole with a mass of around Template:Solar mass will vanish in around 2×1066 years. As the lifetime of a black hole is proportional to the cube of its mass, more massive black holes take longer to decay. A supermassive black hole with a mass of 1011 (100 billion) Template:Solar mass will evaporate in around 2×1099 years.[29]
The universe could possibly avoid eternal heat death through quantum fluctuations, which could produce a new Big Bang in roughly
{\displaystyle 10^{10^{56}}}
Over an infinite time there could be a spontaneous entropy decrease, by a Poincaré recurrence or through thermal fluctuations (see also fluctuation theorem).[34][35][36][37]
{\displaystyle 10^{10^{26}}}
{\displaystyle 10^{10^{76}}}
{\displaystyle 10^{10^{26}}}
{\displaystyle 10^{10^{76}}}
{\displaystyle 10^{10^{76}}}
Graphical timeline of the universe. This timeline uses the more intuitive linear time, for comparison with this article.
↑ 3.0 3.1 3.2 3.3 Life, the Universe, and Nothing: Life and Death in an Ever-expanding Universe, Lawrence M. Krauss and Glenn D. Starkman, Astrophysical Journal, 531 (March 1, 2000), pp. 22–30. Template:Hide in printTemplate:Only in print. Template:Bibcode.
↑ Carroll, Sean M. and Chen, Jennifer (2004). Template:Cite news
↑ Tegmark, Max (2003) Template:Cite news
↑ Werlang, T., Ribeiro, G. A. P. and Rigolin, Gustavo (2012) Template:Cite news
↑ Xing, Xiu-San (2007) Template:Cite news
↑ Linde, Andrei (2007) Template:Cite news
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