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Receding Time Horizon Linear Quadratic Optimal Control for Multi-Axis Contour Tracking Motion Control1 | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
Receding Time Horizon Linear Quadratic Optimal Control for Multi-Axis Contour Tracking Motion Control1
Robert J. McNab,
Robert J. McNab
Western Digital Cooperation, San Jose, CA 95138
Mechanical and Aerospace Engineering Department, University of California-Los Angeles, Los Angeles, CA 90095-1597
Conbributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscrip received by the Dynamic Systems and Control Division December 15, 1998. Associate Technical Editor: T. Kurfess.
McNab, R. J., and Tsao, T. (December 15, 1998). "Receding Time Horizon Linear Quadratic Optimal Control for Multi-Axis Contour Tracking Motion Control." ASME. J. Dyn. Sys., Meas., Control. June 2000; 122(2): 375–381. https://doi.org/10.1115/1.482476
A receding time horizon linear quadratic optimal control approach is formulated for multi-axis contour tracking problem. The approach employs a performance index with fixed weights on quadratic contouring error, tracking error, and control input over a future finite horizon. The problem is then cast into a standard receding horizon LQ problem with time varying weighting matrices, which are functions of the future contour trajectory within the horizon. The formulation thus leads to a solution of time varying state feedback and finite preview gains. Stability is proven for the linear trajectory case. Experimental and simulated results for an
X-Y
motion control problem are presented, which demonstrate the effectiveness of the control scheme and the effects of the key controller design parameters. [S0022-0434(00)01202-8]
linear quadratic control, model reference adaptive control systems, predictive control, tracking, position control, performance index, time-varying systems, matrix algebra, state feedback, stability, control system synthesis
Control equipment, Errors, Stability, Trajectories (Physics), Optimal control, Motion control
Dynamic Errors in Type 1 Contouring Systems
Cross-Coupled Biaxial Computer Control for Manufacturing Systems
Optimal Contouring Control of Multi-Axial Feed Drive Servomechanisms
Cross-Coupled Control of Biaxial Feed Drive Servomechanisms
Variable-Gain Cross-Coupling Controller for Contouring
Chiu, G. T-C., and Tomizuka, M., 1995, “Contouring Control of Machine Tool Feed Drive Systems: a Task Coordinate Frame Approach,” Dynamic System and Control, the Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, CA, DSC-Vol. 57-2, pp. 503–510.
Coordinated Position Control of Multi-Axis Mechanical Systems
Yen, J,-Y., Ho, H.-C., and Lu, S.-S., 1998, “Decoupled Path-Following ontrol Algorithm Based Upon the Decomposed Trajectory Error,” Proceedings of the 37th IEEE—Conference on Decision and Control, pp. 3189–3194, Tampa, FL.
Tangential-Contouring Controller for Biaxial Motion Control
Optimal Discrete Finite Preview Problems (Why and How Is Future Information Important?)
On the Optimal Digital State Vector Feedback Controller with Integral And Preview Actions
Application of Microcomputers to Automated Weld Quality Control
Optimal Feed-Forward Digital Tracking Controller Design
McNab, R. J., and Tsao, T.-C., 1994, “Multi-Axis Contour Tracking: A Receding Horizon Linear Quadratic Optimal Control Approach,” ASME International Mechanical Engineering Congress and Exposition, Chicago, IL, DSC-Vol. 55-2, pp. 895–902.
Anderson, B. D. O., and Moore, J. B., 1971, Linear Optimal Control, Prentice-Hall, Englewood Cliffs, NJ.
Bitmead, R. R., Gevers, M., and Wertz, V., 1990, Adaptive Optimal Control: The Thinking Man’s GPC, Prentice-Hall, Englewood Cliffs, NJ.
A Miscellany of Results on an Equation of Count J. F. Riccati
Fake Algebraic Riccati Techniques and Stability
Monotonicity and Stabilizability Properties of Solutions of the Riccati Difference Equation: Propositions, Lemmas, Theorems, Fallacious Conjectures and Counterexamples
de Souza, C. E., 1989, “Monotonicity and Stabilizability Results for the Solution of the Riccati Difference Equation,” Proc. Workshop on the Riccati Equation in Control, Systems and Signals, S. Bittanti, ed., Como, Italy, pp. 38–41.
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(-)-alpha-pinene/(-)-camphene synthase Wikipedia
(-)-alpha-pinene synthase (EC 4.2.3.119, (-)-alpha-pinene/(-)-camphene synthase, (-)-alpha-pinene cyclase) is an enzyme with systematic name geranyl-diphosphate diphosphate-lyase (cyclizing, (-)-alpha-pinene-forming).[1][2][3][4][5][6][7][8][9][10][11][12][13][14] This enzyme catalyses the following chemical reaction
{\displaystyle \rightleftharpoons }
(-)-alpha-pinene + diphosphate
Cyclase II of Salvia officinalis (sage) gives about equal parts (-)-alpha-pinene, (-)-beta-pinene and (-)-camphene.
^ Gambliel H, Croteau R (January 1984). "Pinene cyclases I and II. Two enzymes from sage (Salvia officinalis) which catalyze stereospecific cyclizations of geranyl pyrophosphate to monoterpene olefins of opposite configuration". The Journal of Biological Chemistry. 259 (2): 740–8. PMID 6693393.
^ Phillips MA, Wildung MR, Williams DC, Hyatt DC, Croteau R (March 2003). "cDNA isolation, functional expression, and characterization of (+)-alpha-pinene synthase and (-)-alpha-pinene synthase from loblolly pine (Pinus taeda): stereocontrol in pinene biosynthesis". Archives of Biochemistry and Biophysics. 411 (2): 267–76. doi:10.1016/s0003-9861(02)00746-4. PMID 12623076.
(-)-alpha-pinene+synthase at the US National Library of Medicine Medical Subject Headings (MeSH)
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Rotation matrix for rotations around x-axis - MATLAB rotx - MathWorks España
Rotation matrix for rotations around x-axis
R = rotx(ang)
R = rotx(ang) creates a 3-by-3 matrix for rotating a 3-by-1 vector or 3-by-N matrix of vectors around the x-axis by ang degrees. When acting on a matrix, each column of the matrix represents a different vector. For the rotation matrix R and vector v, the rotated vector is given by R*v.
Construct the matrix for a rotation of a vector around the x-axis by 30°. Then let the matrix operate on a vector.
R = rotx(30)
x = [2;-2;4];
y = R*x
Under a rotation around the x-axis, the x-component of a vector is invariant.
Rotation angle specified as a real-valued scalar. The rotation angle is positive if the rotation is in the counter-clockwise direction when viewed by an observer looking along the x-axis towards the origin. Angle units are in degrees.
{R}_{x}\left(\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ 0& \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]
for a rotation angle α.
{v}^{\prime }=Av={R}_{z}\left(\gamma \right){R}_{y}\left(\beta \right){R}_{x}\left(\alpha \right)v
{R}_{x}\left(\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ 0& \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]
{R}_{y}\left(\beta \right)=\left[\begin{array}{ccc}\mathrm{cos}\beta & 0& \mathrm{sin}\beta \\ 0& 1& 0\\ -\mathrm{sin}\beta & 0& \mathrm{cos}\beta \end{array}\right]
{R}_{z}\left(\gamma \right)=\left[\begin{array}{ccc}\mathrm{cos}\gamma & -\mathrm{sin}\gamma & 0\\ \mathrm{sin}\gamma & \mathrm{cos}\gamma & 0\\ 0& 0& 1\end{array}\right]
{A}^{-1}A=1
{R}_{x}^{-1}\left(\alpha \right)={R}_{x}\left(-\alpha \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\alpha & \mathrm{sin}\alpha \\ 0& -\mathrm{sin}\alpha & \mathrm{cos}\alpha \end{array}\right]={R}_{x}^{\prime }\left(\alpha \right)
i,j,k
{i}^{\prime },j{,}^{\prime }{k}^{\prime }
\begin{array}{ll}{i}^{\prime }\hfill & =Ai\hfill \\ {j}^{\prime }\hfill & =Aj\hfill \\ {k}^{\prime }\hfill & =Ak\hfill \end{array}
\left[\begin{array}{c}{i}^{\prime }\\ {j}^{\prime }\\ {k}^{\prime }\end{array}\right]={A}^{\prime }\left[\begin{array}{c}i\\ j\\ k\end{array}\right]
v={v}_{x}i+{v}_{y}j+{v}_{z}k={{v}^{\prime }}_{x}{i}^{\prime }+{{v}^{\prime }}_{y}{j}^{\prime }+{{v}^{\prime }}_{z}{k}^{\prime }
\left[\begin{array}{c}{{v}^{\prime }}_{x}\\ {{v}^{\prime }}_{y}\\ {{v}^{\prime }}_{z}\end{array}\right]={A}^{-1}\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]={A}^{\prime }\left[\begin{array}{c}{v}_{x}\\ {v}_{y}\\ {v}_{z}\end{array}\right]
roty | rotz
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Determine price for credit default swap - MATLAB cdsprice - MathWorks India
Determine the Price for a Credit Default Swap
AccPrem
PaymentCF
Determine price for credit default swap
[Price,AccPrem,PaymentDates,PaymentTimes,PaymentCF] = cdsprice(ZeroData,ProbData,Settle,Maturity,ContractSpread)
[Price,AccPrem,PaymentDates,PaymentTimes,PaymentCF] = cdsprice(___,Name,Value)
[Price,AccPrem,PaymentDates,PaymentTimes,PaymentCF] = cdsprice(ZeroData,ProbData,Settle,Maturity,ContractSpread) computes the price, or the mark-to-market value for CDS instruments.
[Price,AccPrem,PaymentDates,PaymentTimes,PaymentCF] = cdsprice(___,Name,Value) adds optional name-value pair arguments.
This example shows how to use cdsprice to compute the clean price for a CDS contract using the following data.
[Price,AccPrem] = cdsprice(ZeroData,ProbData,Settle,Maturity,ContractSpread)
AccPrem = 10500
When ZeroData is an IRDataCurve object, ZeroCompounding and ZeroBasis are implicit in ZeroData and are redundant inside this function. In this case, specify these optional parameters when constructing the IRDataCurve object before using the cdsprice function.
ContractSpread — Contract spreads
Contract spreads, specified as a N-by-1 vector of spreads, expressed in basis points.
Example: [Price,AccPrem] = cdsprice(ZeroData,ProbData,Settle,Maturity,ContractSpread,'Basis',7,'BusinessDayConvention','previous')
Notional — Contract notional values
10MM (default) | positive or negative integer
Contract notional values, specified as the comma-separated pair consisting of 'Notional' and a N-by-1vector of integers. Use positive integer values for long positions and negative integer values for short positions.
Compounding frequency of the zero curve, specified as the comma-separated pair consisting of 'ZeroCmpounding' and an integer with values:
Price — CDS clean prices
CDS clean prices, returned as a N-by-1 vector.
AccPrem — Accrued premiums
Accrued premiums, returned as a N-by-1 vector.
PaymentCF — Payments
Payments, returned as a N-by-numCF matrix.
The price or mark-to-market (MtM) value of an existing CDS contract.
The CDS price is computed using the following formula:
CDS price = Notional * (Current Spread - Contract Spread) * RPV01
Current Spread is the current breakeven spread for a similar contract, according to current market conditions. RPV01 is the 'risky present value of a basis point,' the present value of the premium payments, considering the default probability. This formula assumes a long position, and the right side is multiplied by -1 for short positions.
RPV01=\sum _{j=1}^{N}Z\left(tj\right)\Delta \left(tj-1,tj,B\right)Q\left(tj\right)
RPV01\approx \frac{1}{2}\sum _{j=1}^{N}Z\left(tj\right)\Delta \left(tj-1,tj,B\right)\left(Q\left(tj-1\right)+Q\left(tj\right)\right)
ProtectionLeg={\int }_{0}^{T}Z\left(\tau \right)\left(1-R\right)dPD\left(\tau \right)
\approx \left(1-R\right)\sum _{i=1}^{M}Z\left(\tau i\right)\left(PD\left(\tau i\right)-PD\left(\tau i-1\right)\right)
=\left(1-R\right)\sum _{i=1}^{M}Z\left(\tau i\right)\left(Q\left(\tau i-1\right)-Q\left(\tau i\right)\right)
If the spread of an existing CDS contract is SC, and the current breakeven spread for a comparable contract is S0, the current price, or mark-to-market value of the contract is given by:
MtM = Notional (S0 –SC)RPV01
This assumes a long position from the protection standpoint (protection was bought). For short positions, the sign is reversed.
cdsbootstrap | cdsspread | cdsoptprice (Financial Instruments Toolbox) | IRDataCurve (Financial Instruments Toolbox)
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Climate Clock API - Climate Clock Docs
The v1 Climate Clock API is intended to provide a unified data source to be used by a variety of Climate Clock implementations. It is intended to be simple to use in most cases, with more complex features available optionally for specific devices as needed.
For convenience, everything necessary to implement a Climate Clock can be retrieved from a single endpoint which may be polled as needed. The /v1/clock endpoint provides two primary objects whose purpose should be understood when implementing Climate Clocks:
config Device-specific clock configuration data Configuration data is meant to provide implementation details which pertain to things like presentation and branding. Using data from this source ensures that clocks maintain presentation consistent with the goals of the Climate Clock project. This is a source of useful defaults, particularly of interest when implementing certain classes of clock devices, such as Portable Action Clocks, whose owners likely have no interest in configuring or maintaining the devices. The config object also specifies a list of modules which are presently being highlighted by the project for display by clocks.
modules Modular specifications of clock data for display Climate Clocks serve different purposes depending on where they're installed, the size of their screens, or who owns and maintains them. To solve the needs of different clocks, we categorize each thing to be displayed on a clock as a "module." A module can be the countdown until 1.5°C global temperature rise is locked in as predicted by the IPCC, or it can be series of headlines for a news ticker, or another kind of metric. Modules come in different types:
timer: For countdowns, or elapsed time such as days since Paris Climate Accords were signed
newsfeed: For good news, for warnings, or to display site-specific data
value: For displaying values that grow or shrink with time
chart: TBD
media: TBD
/v1/clock
Climate Clock API Data
See the section titled Module Specification for information about how frequently to poll this endpoint.
Configuration data is provided within the data.config object of JSON returned by /v1/clock like so:
The most basic configuration information, provided within config for all device types is a list of suggested modules for display:
"device": "generic",
"carbon_deadline_1",
"renewables_1",
"newsfeed_1"
TODO: Document full configuration & branding spec
Modules are provided as named objects within the data.modules object of JSON returned by /v1/clock like so:
"module_1": {
"type": "chart" | "media" | "newsfeed" | "timer" | "value",
"flavor": "lifeline" | "deadline" | "neutral",
"description": "A human-readable description of this module's purpose",
"update_interval_seconds": 86400,
type: Specifies which module implementation can display this module. flavor:
Whether this module should follow the Lifeline, Deadline or Neutral display specification. description: A full description of what this module represents. update_interval_seconds: This module's data will be updated no more frequently than the given number of seconds. This tells clients how long to wait before polling the endpoint with the intention to find new information for this module.
Clients should adjust their API polling to match the smallest value of update_interval_seconds among the modules they intend to display. To comply with intended clock behavior, use the most recent values of update_interval_seconds your client has received. This allows modules with long polling intervals to indicate they should now be polled more frequently, and vice-versa.
"A Human-readable Label",
"A Shorter Label",
"Short Lbl",
labels: a list of 1 or more text labels, presented in longest to shortest, for display where possible alongside module data. Opt to display the longest possible label within the space of your clock display. lang: An ISO 639-1 language code. If the module has responded to your request for content in another language (e.g. your request by query-string such as ?lang=es), lang will indicate the language of content you received. It's currently possible to ask for content in any language, but as of this writing only English content is available. This allows module implementations to act based on the availability of translated content.
Module-specific data
Modules also contain data pertinent to their type.
Fully-compliant Climate Clocks should implement displays for all module types, allowing API data to dynamically determine which modules to display based on the config.modules list.
Modules of type timer are meant to express a countdown to a deadline, or time elapsed. A timer contains a timestamp in the past or future.
timestamp: Implementations should display a clock showing time until or since the ISO 8601 timestamp given. The ISO 8601 timestamp will include a time zone offset, but implementations unable to make use of timezone data (like small embedded clocks) can safely assume the use of UTC, as all timestamps are provided in UTC.
Module type: newsfeed
Modules of type newsfeed are intended to display a series of text items for the implementation of a news ticker-like display. Each module of this type provides an array of objects representing headlines within the news feed:
"type": "newsfeed",
"newsfeed": [
"headline": "No stability in sight!",
"headline_original": "Climate stabilization not likely before 2032 warn scientists",
"source": "Example News",
"link": "https://example.com/climate-stability",
"summary": "Optional summary of this piece of news..."
"headline": "Example News says More News!",
newsfeed: An array of newsfeed items in reverse chronological order, each containing:
date: An ISO 8601 date or timestamp indicating the approximate publication date of this news item. headline: A newsfeed headline. (optional) headline_original: The headline as originally published—the headline field may be editorialized by the project. source: Source of this news (a publication name). link: Link to the article or source of this news. summary: A write up describing the significance of this news.
Module type: value
Modules of type value represent a value which grows or shrinks over time. All value modules contain:
"initial": 9.99,
"growth": "linear" | "exponential",
"unit_labels": [
"Tons of CO₂ Emitted",
"Tons CO₂",
"T.CO₂"
initial: The growth equation's starting value. timestamp: An ISO 8601 timestamp for the starting value. growth: The type of growth represented. resolution: A number
≤1
indicating the smallest value to be displayed (e.g. 0.001 implying no more than 3 decimal digits should be shown). This is related to the concept of significant figures, but simpler. (optional) unit_labels: A list of 1 or more text unit labels for this value, presented longest to shortest, for display where possible alongside value data. Opt to display the longest possible unit label within the space of your clock display.
Modules of type value with module.growth == "linear" represent linear growth and contain:
"growth": "linear",
"rate": 2.8368383376368776e-08,
rate: Rate of change per second.
Linear growth can be calculated as:
f(t_{seconds}) = initial + rate * t_{seconds}
Be aware that rate values may be extremely large or small. Take care in your implementation to preserve the precision provided. See: https://en.wikipedia.org/wiki/Floating-point_arithmetic#Minimizing_the_effect_of_accuracy_problems
Do not use. Data model for non-linear growth values is TBD as of 2021-02-17
Modules of type value with module.growth == "exponential" represent exponential growth and contain:
"growth": "exponential",
Exponential growth can be calculated as
f(t_{seconds}) = \infty
Here, a simple client connects to the API and implements the well-known timer module named carbon_deadline_1 representing our time to act before we lock in 1.5°C global temperature rise:
// TODO: JavaScript example
# TODO: Python example
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Find the perimeter and area of each figure below. Review the Math Notes box in this lesson for help. Be sure to include the correct units in your answers.
To find the perimeter, add together all of the lengths of the outside edges.
\frac {1}{2} \times \text{ base}\times \text{ height}
18.5
15
Look back at the Math Notes boxes.
Can you find the top and bottom "bases" of the trapezoid?
Can you find the height of the trapezoid?
\text{Area of a trapezoid}=
\frac{(\text{top}+\text{bottom})}{2}\times\text{height}
\text{Perimeter} = \text{Sum of all side lengths}
61.4
210
Look back at the Math Notes.
\text{Area of a parallelogram}=\text{base}\times \text{height}
22
21
square yards
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Balancing Scans - Donnacha Oisín Kidney
Previously I tried to figure out a way to fold lists in a more balanced way. Usually, when folding lists, you’ve got two choices for your folds, both of which are extremely unbalanced in one direction or another. Jon Fairbairn wrote a more balanced version, which looked something like this:
treeFold :: (a -> a -> a) -> a -> [a] -> a
go x [] = x
go a (b:l) = go (f a b) (pairMap l)
Magical Speedups
The fold above is kind of magical: for a huge class of algorithms, it kind of “automatically” improves some factor of theirs from
\mathcal{O}(n)
\mathcal{O}(\log n)
. For instance: to sum a list of floats, foldl' (+) 0 will have an error growth of
\mathcal{O}(n)
; treeFold (+) 0, though, has an error rate of
\mathcal{O}(\log n)
. Similarly, using the following function to merge two sorted lists:
merge (x:xs) ys = go x xs ys
go x xs [] = x : xs
go x xs (y:ys)
| x <= y = x : go y ys xs
| otherwise = y : go x xs ys
We get either insertion sort (
\mathcal{O}(n^2)
) or merge sort (
\mathcal{O}(n \log n)
) just depending on which fold you use.
foldr merge [] . map pure -- n^2
treeFold merge [] . map pure -- n log(n)
I’ll give some more examples later, but effectively it gives us a better “divide” step in many divide and conquer algorithms.
As it was such a useful fold, and so integral to many tricky algorithms, I really wanted to have it available in Agda. Unfortunately, though, the functions (as defined above) aren’t structurally terminating, and there doesn’t look like there’s an obvious way to make it so. I tried to make well founded recursion work, but the proofs were ugly and slow.
However, we can use some structures from a previous post: the nested binary sequence, for instance. It has some extra nice properties: instead of nesting the types, we can just apply the combining function.
data Tree {a} (A : Set a) : Set a where
2^_×_+_ : ℕ → A → Node A → Tree A
data Node {a} (A : Set a) : Set a where
⟨⟩ : Node A
⟨_⟩ : Tree A → Node A
module TreeFold {a} {A : Set a} (_*_ : A → A → A) where
infixr 5 _⊛_ 2^_×_⊛_
2^_×_⊛_ : ℕ → A → Tree A → Tree A
2^ n × x ⊛ 2^ suc m × y + ys = 2^ n × x + ⟨ 2^ m × y + ys ⟩
2^ n × x ⊛ 2^ zero × y + ⟨⟩ = 2^ suc n × (x * y) + ⟨⟩
2^ n × x ⊛ 2^ zero × y + ⟨ ys ⟩ = 2^ suc n × (x * y) ⊛ ys
_⊛_ : A → Tree A → Tree A
_⊛_ = 2^ 0 ×_⊛_
⟦_⟧↓ : Tree A → A
⟦ 2^ _ × x + ⟨⟩ ⟧↓ = x
⟦ 2^ _ × x + ⟨ xs ⟩ ⟧↓ = x * ⟦ xs ⟧↓
⟦_⟧↑ : A → Tree A
⟦ x ⟧↑ = 2^ 0 × x + ⟨⟩
⦅_,_⦆ : A → List A → A
⦅ x , xs ⦆ = ⟦ foldr _⊛_ ⟦ x ⟧↑ xs ⟧↓
Alternatively, we can get
\mathcal{O}(1)
cons with the skew array:
infixr 5 _⊛_
x ⊛ 2^ n × y + ⟨⟩ = 2^ 0 × x + ⟨ 2^ n × y + ⟨⟩ ⟩
x ⊛ 2^ n × y₁ + ⟨ 2^ 0 × y₂ + ys ⟩ = 2^ suc n × (x * (y₁ * y₂)) + ys
x ⊛ 2^ n × y₁ + ⟨ 2^ suc m × y₂ + ys ⟩ = 2^ 0 × x + ⟨ 2^ n × y₁ + ⟨ 2^ m × y₂ + ys ⟩ ⟩
Using this, a proper and efficient merge sort is very straightforward:
data Total {a r} {A : Set a} (_≤_ : A → A → Set r) (x y : A) : Set (a ⊔ r) where
x≤y : ⦃ _ : x ≤ y ⦄ → Total _≤_ x y
y≤x : ⦃ _ : y ≤ x ⦄ → Total _≤_ x y
module Sorting {a r}
{A : Set a}
{_≤_ : A → A → Set r}
(_≤?_ : ∀ x y → Total _≤_ x y) where
data [∙] : Set a where
⊥ : [∙]
[_] : A → [∙]
data _≥_ (x : A) : [∙] → Set (a ⊔ r) where
instance ⌈_⌉ : ∀ {y} → y ≤ x → x ≥ [ y ]
instance ⌊⊥⌋ : x ≥ ⊥
data Ordered (b : [∙]) : Set (a ⊔ r) where
[] : Ordered b
_∷_ : ∀ x → ⦃ x≥b : x ≥ b ⦄ → (xs : Ordered [ x ]) → Ordered b
_∪_ : ∀ {b} → Ordered b → Ordered b → Ordered b
[] ∪ ys = ys
(x ∷ xs) ∪ ys = ⟅ x ∹ xs ∪ ys ⟆
⟅_∹_∪_⟆ : ∀ {b} → ∀ x ⦃ _ : x ≥ b ⦄ → Ordered [ x ] → Ordered b → Ordered b
⟅_∪_∹_⟆ : ∀ {b} → Ordered b → ∀ y ⦃ _ : y ≥ b ⦄ → Ordered [ y ] → Ordered b
merge : ∀ {b} x y ⦃ _ : x ≥ b ⦄ ⦃ _ : y ≥ b ⦄
→ Total _≤_ x y
→ Ordered [ x ]
→ Ordered [ y ]
→ Ordered b
⟅ x ∹ xs ∪ [] ⟆ = x ∷ xs
⟅ x ∹ xs ∪ y ∷ ys ⟆ = merge x y (x ≤? y) xs ys
⟅ [] ∪ y ∹ ys ⟆ = y ∷ ys
⟅ x ∷ xs ∪ y ∹ ys ⟆ = merge x y (x ≤? y) xs ys
merge x y x≤y xs ys = x ∷ ⟅ xs ∪ y ∹ ys ⟆
merge x y y≤x xs ys = y ∷ ⟅ x ∹ xs ∪ ys ⟆
open TreeFold
sort : List A → Ordered ⊥
sort = ⦅ _∪_ , [] ⦆ ∘ map (_∷ [])
It would be nice if we could verify these optimizated versions of folds. Luckily, by writing them using foldr, we’ve stumbled into well-trodden ground: the foldr fusion law. It states that if you have some transformation
f
, and two binary operators
\oplus
\otimes
\begin{align} f (x \oplus y) &&=\;& x \otimes f y \\ \implies f \circ \text{foldr} \oplus e &&=\;& \text{foldr} \otimes (f e) \end{align}
This fits right in with the function we used above.
f
is ⟦_⟧↓,
\oplus
is _⊛_, and
\otimes
is whatever combining function was passed in. Let’s prove the foldr fusion law, then, before we go any further.
module Proofs
{a r}
{R : Rel A r}
_≈_ = R
open import Algebra.FunctionProperties _≈_
foldr-universal : Transitive _≈_
→ ∀ {b} {B : Set b} (h : List B → A) f e
→ ∀[ f ⊢ Congruent₁ ]
→ (h [] ≈ e)
→ (∀ x xs → h (x ∷ xs) ≈ f x (h xs))
→ ∀ xs → h xs ≈ foldr f e xs
foldr-universal _○_ h f e f⟨_⟩ ⇒[] ⇒_∷_ [] = ⇒[]
foldr-universal _○_ h f e f⟨_⟩ ⇒[] ⇒_∷_ (x ∷ xs) =
(⇒ x ∷ xs) ○ f⟨ foldr-universal _○_ h f e f⟨_⟩ ⇒[] ⇒_∷_ xs ⟩
foldr-fusion : Transitive _≈_
→ Reflexive _≈_
→ ∀ {b c} {B : Set b} {C : Set c} (f : C → A) {_⊕_ : B → C → C} {_⊗_ : B → A → A} e
→ ∀[ _⊗_ ⊢ Congruent₁ ]
→ (∀ x y → f (x ⊕ y) ≈ x ⊗ f y)
→ ∀ xs → f (foldr _⊕_ e xs) ≈ foldr _⊗_ (f e) xs
foldr-fusion _○_ ∎ h {f} {g} e g⟨_⟩ fuse =
foldr-universal _○_ (h ∘ foldr f e) g (h e) g⟨_⟩ ∎ (λ x xs → fuse x (foldr f e xs))
We’re not using the proofs in Agda’s standard library because these are tied to propositional equality. In other words, instead of using an abstract binary relation, they prove things over actual equality. That’s all well and good, but as you can see above, we don’t need propositional equality: we don’t even need the relation to be an equivalence, we just need transitivity and reflexivity.
After that, we can state precisely what correspondence the tree fold has, and under what conditions it does the same things as a fold:
module _ {_*_ : A → A → A} where
open TreeFold _*_
treeFoldHom : Transitive _≈_
→ Associative _*_
→ RightCongruent _*_
→ ∀ x xs
→ ⦅ x , xs ⦆ ≈ foldr _*_ x xs
treeFoldHom _○_ ∎ assoc *⟨_⟩ b = foldr-fusion _○_ ∎ ⟦_⟧↓ ⟦ b ⟧↑ *⟨_⟩ (⊛-hom zero)
⊛-hom : ∀ n x xs → ⟦ 2^ n × x ⊛ xs ⟧↓ ≈ x * ⟦ xs ⟧↓
⊛-hom n x (2^ suc m × y + ⟨⟩ ) = ∎
⊛-hom n x (2^ suc m × y + ⟨ ys ⟩) = ∎
⊛-hom n x (2^ zero × y + ⟨⟩ ) = ∎
⊛-hom n x (2^ zero × y + ⟨ ys ⟩) = ⊛-hom (suc n) (x * y) ys ○ assoc x y ⟦ ys ⟧↓
“Implicit” Data Structures
Consider the following implementation of the tree above in Haskell:
type Tree a = [(Int,a)]
cons :: (a -> a -> a) -> a -> Tree a -> Tree a
cons (*) = cons' 0
cons' n x [] = [(n,x)]
cons' n x ((0,y):ys) = cons' (n+1) (x * y) ys
cons' n x ((m,y):ys) = (n,x) : (m-1,y) : ys
The cons function “increments” that list as if it were the bits of a binary number. Now, consider using the merge function from above, in a pattern like this:
f = foldr (cons merge . pure) []
What does f build? A list of lists, right?
Kind of. That’s what’s built in terms of the observable, but what’s actually stored in memory us a bunch of thunks. The shape of those is what I’m interested in. We can try and see what they look like by using a data structure that doesn’t force on merge:
data Tree a = Leaf a | Tree a :*: Tree a
f = foldr (cons (:*:) . Leaf) []
Using a handy tree-drawing function, we can see what f [1..13] looks like:
[(0,*),(1,*),(0,*)]
└1 │ ┌2 │ ┌6
│┌┤ │ ┌┤
││└3 │ │└7
└┤ │┌┤
│┌4 │││┌8
└┤ ││└┤
└5 ││ └9
│ ┌10
││└11
│┌12
It’s a binomial heap! It’s a list of trees, each one contains
2^n
elements. But they’re not in heap order, you say? Well, as a matter of fact, they are. It just hasn’t been evaluated yet. Once we force—say—the first element, the rest will shuffle themselves into a tree of thunks.
This illustrates a pretty interesting similarity between binomial heaps and merge sort. Performance-wise, though, there’s another interesting property: the thunks stay thunked. In other words, if we do a merge sort via:
sort = foldr (merge . snd) [] . foldr (cons merge . pure) []
We could instead freeze the fold, and look at it at every point:
sortPrefixes = map (foldr (merge . snd) []) . scanl (flip (cons merge . pure)) []
>>> [[],[1],[1,4],[1,2,4],[1,2,3,4],[1,2,3,4,5]]
And sortPrefixes is only
\mathcal{O}(n^2)
\mathcal{O}(n^2 \log n)
). I confess I don’t know of a use for sorted prefixes, but it should illustrate the general idea: we get a pretty decent batching of operations, with the ability to freeze at any point in time. The other nice property (which I mentioned in the last post) is that any of the tree folds are extremely parallel.
There’s a great article on shuffling in Haskell which provides an
\mathcal{O}(n \log n)
implementation of a perfect random shuffle. Unfortunately, the Fisher-Yates shuffle isn’t applicable in a pure functional setting, so you have to be a little cleverer.
The first implementation most people jump to (certainly the one I thought of) is to assign everything in the sequence a random number, and then sort according to that number. Perhaps surprisingly, this isn’t perfectly random! It’s a little weird, but the example in the article explains it well: basically, for
n
elements, your random numbers will have
n^n
possible values, but the output of the sort will have
n!
possible values. Since they don’t divide into each other evenly, you’re going to have some extra weight on some permutations, and less on others.
Instead, we can generate a random factoradic number. A factoradic number is one where the
n
th digit is in base
n
. Because of this, a factoradic number with
n
digits has
n!
possible values: exactly what we want.
In the article, the digits of the number are used to pop values from a binary tree. Because the last digit will have
n
possible values, and the second last
n-1
, and so on, you can keep popping without hitting an empty tree.
This has the correct time complexity—
\mathcal{O}(n \log n)
—but there’s a lot of overhead. Building the tree, then indexing into it, the rebuilding after each pop, etc.
We’d like to just sort the list, according to the indices. The problem is that the indices are relative: if you want to cons something onto the list, you have to increment the rest of the indices, as they’ve all shifted right by one.
What we’ll do instead is use the indices as gaps. Our merge function looks like the following:
merge ((x,i):xs) ((y,j):ys)
| i <= j = (x,i) : merge xs ((y,j-i):ys)
| otherwise = (y,j) : merge ((x,i-j-1):xs) ys
With that, and the same cons as above, we get a very simple random shuffle algorithm:
shuffle xs = map fst
. foldr (merge . snd) []
. foldr f (const []) xs
f x xs (i:is) = cons merge [(x,i)] (xs is)
The other interesting thing about this algorithm is that it can use Peano numbers with taking too much of a performance hit:
merge : ∀ {a} {A : Set a} → List (A × ℕ) → List (A × ℕ) → List (A × ℕ)
merge {A = A} xs ((y , j) ∷ ys) = go-r xs y j ys
go-l : A → ℕ → List (A × ℕ) → List (A × ℕ) → List (A × ℕ)
go-r : List (A × ℕ) → A → ℕ → List (A × ℕ) → List (A × ℕ)
go : ℕ → ℕ → A → ℕ → List (A × ℕ) → A → ℕ → List (A × ℕ) → List (A × ℕ)
go i zero x i′ xs y j′ ys = (y , j′) ∷ go-l x i xs ys
go zero (suc j) x i′ xs y j′ ys = (x , i′) ∷ go-r xs y j ys
go (suc i) (suc j) = go i j
go-l x i xs [] = (x , i) ∷ xs
go-l x i xs ((y , j) ∷ ys) = go i j x i xs y j ys
go-r [] y j ys = (y , j) ∷ ys
go-r ((x , i) ∷ xs) y j ys = go i j x i xs y j ys
shuffle : ∀ {a} {A : Set a} → List A → List ℕ → List A
shuffle {a} {A} xs i = map proj₁ (⦅ [] , zip-inds xs i ⦆)
open TreeFold {a} {List (A × ℕ)} merge
zip-inds : List A → List ℕ → List (List (A × ℕ))
zip-inds [] inds = []
zip-inds (x ∷ xs) [] = ((x , 0) ∷ []) ∷ zip-inds xs []
zip-inds (x ∷ xs) (i ∷ inds) = ((x , i) ∷ []) ∷ zip-inds xs inds
I don’t know exactly what the complexity of this is, but I think it should be better than the usual approach of popping from a vector.
This is just a collection of random thoughts for now, but I intend to work on using these folds to see if there are any other algorithms they can be useful for. In particular, I think I can write a version of Data.List.permutations which benefits from sharing. And I’m interested in using the implicit binomial heap for some search problems.
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STEREO experiment - zxc.wiki
Figure 1: Schematic structure of the STEREO detector
The STEREO experiment (Search for Ste rile Re actor Neutrino O scillations) examines the possible oscillation of reactor neutrinos into sterile neutrinos . It is located at the Institut Laue Langevin (ILL) in Grenoble, France. Data collection started in November 2016.
1 measuring principle
1.1 Particle identification
1.2 Detector shield
3 results (as of December 2019)
Figure 2: Comparison of the measured neutrino spectra at a distance of 10 m and 12.2 m from the reactor.
The STEREO detector is about 10 m away from a research reactor (58 MW thermal power) at the ILL. In order to be able to detect the neutrinos emitted by the reactor - more precisely: electron antineutrinos - the detector is filled with 1800 liters of an organic liquid scintillator . There neutrinos are detected by inverse beta decay :
{\ displaystyle \ mathrm {\ bar {\ nu}} _ {e} + \ mathrm {p} \ rightarrow \ mathrm {e} ^ {+} + \ mathrm {n}}
The same reaction was used for the very first experimental detection of neutrinos in the Cowan-Reines neutrino experiment . The reaction events of interest are identified by a characteristic sequence of two impulses:
the generated positron is generated in the scintillator by annihilation with an electron gamma quanta with a total energy of 1022 keV, the scintillation light of which is recorded by the 48 photomultiplier tubes (PMTs) installed in the upper part of the detector cells,
the generated neutron is initially moderated in the scintillator by collisions and then absorbed by an atomic nucleus with a large cross-section for neutron capture , which - delayed by a few microseconds due to the moderation process - also leads to the emission of gamma radiation with a characteristic energy. In the Cowan-Reines experiment, the scintillator fluid contained cadmium as a neutron absorber ; in the STEREO detector, gadolinium, with its much larger neutron capture cross-section, takes on this task.
The expected distance between the oscillation minimum and maximum of sterile reactor neutrinos is about 2 m. Therefore, the 2.2 m long detector is divided into 6 separate sections, which measure the energy spectrum of the neutrinos separately from each other. A possible oscillation can be detected by comparing the measured spectra (see Figure 2).
The STEREO experiment registers around 400 neutrinos per day.
Since neutrinos only interact very weakly, detectors for neutrinos have to be very sensitive and therefore need good shielding against unwanted signals.
The six inner detector cells are surrounded by a gadolinium-free liquid scintillator, which acts as a "gamma catcher" by detecting incoming and outgoing gamma quanta . This both increases the detection efficiency and improves the energy resolution. Above the detector is a water-filled Cherenkov - anticoincidence detector, in the muons from the secondary cosmic rays are detected, which would otherwise form a disturbing background. Against neutrons and gamma rays from the surrounding experiments, the detector is surrounded by several shields made of lead, polyethylene, steel and boron carbide (65 t in total).
Figure 3: The reactor antineutrino anomaly
While neutrino oscillation is now a well-understood phenomenon, there are some experimental observations that question the completeness of this understanding. Probably the most prominent observation in this regard is the so-called reactor antineutrino anomaly (RAA). Many neutrino experiments close to the reactor have measured a significantly ( ) lower flow rate of electron antineutrinos ( ) compared to theory . Further experimental anomalies are the unexpected occurrence of in a beam at short distances in the LSND anomaly and the gallium neutrino anomaly, which describes the disappearance of at short distances during the calibration phases of the GALLEX and SAGE experiments .
{\ displaystyle 2 {,} 7 \ sigma}
{\ displaystyle {\ overline {\ nu}} _ {e}}
{\ displaystyle {\ overline {\ nu}} _ {e}}
{\ displaystyle {\ overline {\ nu}} _ {\ mu}}
{\ displaystyle \ nu _ {e}}
These anomalies could lead to the conclusion that our previous understanding of neutrino oscillation is incomplete and that neutrinos can oscillate into another previously unknown type of neutrino. Measurements of the decay width of the Z boson at the Large Electron-Positron Collider (LEP) conclude the existence of other slightly more "active" ones, ie. H. the weak interaction of underlying neutrinos. Therefore the oscillation is converted into additional light “sterile”, i.e. H. investigated neutrinos not affected by the weak interaction as a possible explanation. From the theoretical layer, sterile neutrinos emerge in some prominent extensions of the standard model of particle physics, e.g. B. the Seesaw type 1 mechanism.
Results (as of December 2019)
Figure 4: Results of the STEREO experiment to investigate the possible existence of lighter sterile neutrinos as an explanation of the RAA. The blue area shows the expected parameter sensitivity, which should be achieved under the expectation that no additional neutrino species exists. The red area shows the parameter area excluded due to the real measurement. It fluctuates around the blue expected range due to statistical fluctuations. The black lines show the parameter range favored by the RAA, which has already been largely excluded.
First results from 66 days with the reactor switched on were presented in 2018. Most of the parameter range of sterile neutrinos favored to explain the RAA could be excluded with a certainty of 90%. New results from December 2019 include around 65,500 measured neutrinos (combined phases 1 and 2; 179 days with the reactor running). With the current data set, the exclusion region can be expanded further (see Figure 4).
Website STEREO experiment (English)
↑ G. Mention et al .: "Reactor antineutrino anomaly", Phys. Rev. D 83, 073006 - Published 29 April 2011, DOI: 10.1103 / PhysRevD.83.073006
↑ A. Aguilar et al. (LSND Collaboration): "Evidence for neutrino oscillations from the observation of appearance in a beam", Phys. Rev. D 64, 112007 - Published 13 November 2001, DOI: 10.1103 / PhysRevD.64.112007
{\ displaystyle {\ overline {\ nu}} _ {e}}
{\ displaystyle {\ overline {\ nu}} _ {\ mu}}
↑ Carlo Giunti and Marco Laveder: "Statistical significance of the gallium anomaly", Phys. Rev. C 83, 065504 - Published 27 June 2011, DOI: 10.1103 / PhysRevC.83.065504
↑ JN Abdurashitov et al .: "Measurement of the response of a Ga solar neutrino experiment to neutrinos from a 37Ar source", Phys. Rev. C 73, 045805 - Published 20 April 2006, DOI: 10.1103 / PhysRevC.73.045805
↑ The ALEPH CollaborationThe DELPHI CollaborationThe L3 CollaborationThe OPAL CollaborationThe SLD CollaborationThe LEP Electroweak Working GroupThe SLD Electroweak and Heavy Flavor Groups: "Precision electroweak measurements on the Z resonance", Physics Reports 427, DOI: 10.1016 / j.physrep.2005.12.006
↑ H. Almazán et al .: "Sterile neutrino constraints from the STEREO experiment with 66 days of reactor-on data", Phys. Rev. Lett. 121, 161801 - Published 17 October 2018, DOI: 10.1103 / PhysRevLett.121.161801
↑ H. Almazán et al .: "Improved sterile neutrino constraints from the STEREO experiment with 179 days of reactor-on data", arXiv: 1912.06852 - Published 16 December 2019
This page is based on the copyrighted Wikipedia article "STEREO-Experiment" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Transfer maps of sphere bundles
April, 2000 Transfer maps of sphere bundles
Mitsunori IMAOKA, Karlheinz KNAPP
Generalizing the transfer maps concerned with the projective spaces, we study some fundamental properties of transfer maps for sphere bundles. We show that their cofibers are represented by Thom spectra, which enables us to calculate the
-invariants of the transfer maps. We give some concrete formula for the
-invariants of them and its application.
Mitsunori IMAOKA. Karlheinz KNAPP. "Transfer maps of sphere bundles." J. Math. Soc. Japan 52 (2) 363 - 372, April, 2000. https://doi.org/10.2969/jmsj/05220363
Secondary: 55N20 , 55P42 , 55R25
Keywords: e-invariant , Sphere bundle , Transfer map
Mitsunori IMAOKA, Karlheinz KNAPP "Transfer maps of sphere bundles," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 52(2), 363-372, (April, 2000)
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1.2 Mission Control - Big History School
To work through '1.2 Mission Control' you need to complete the Activities in order. So first complete ‘Learning Plan’ then move to Activity '1.2.1' followed by Activity '1.2.2' through to '1.2.4', and then finish with ‘Learning Summary’.
In Mission Control you will learn all about the 3 big mission phase questions you will be exploring and what you will need to do to successfully complete your Mars Mission.
To get started, read carefully through the Mission Control learning goals below. Make sure you tick each of the check boxes to show that you have read all your Mission Control learning goals.
As you read through the learning goals you may come across some words that you haven’t heard before. Please don’t worry. By the time you finish Mission Control you will have become very familiar with them!
You will come back to these learning goals at the end of Mission Control to see if you have confidently achieved them.
3 big mission phase questions
Understand what is Big History
Recall the 3 big mission phase questions
Place events on an historical timeline
Your Mission Control Reports [PBL INTRODUCTION]
Understand the step-by-step requirements of the Mars Mission
Commit to undertake the steps necessary to successfully complete the Mars Mission
2Al
Congratulations on completing your pre-mission critical thinking skills training!
Today you will be going on a 13.8 billion year journey back to the beginning of the Universe. You will need to keep an extremely open mind as you will be exploring periods of time which are longer than anyone can possibly imagine!
Mission video 5: 3 big mission phase questions explores the 3 big mission phase questions which give you the important background information you need about the Universe, planets and humans.
While you watch Mission video 5: 3 big mission phase questions look out for the answers to the following questions:
1. What is Big History?
2. What are the 3 big mission phase questions you’ll be exploring in Big History?
4. Why do we use timelines?
As mentioned in Mission video 5: 3 big mission phase questions, timelines are a good way to try to get your head around the important events that happened during a certain time period.
Have a think about the time period of your life. What would you include in a timeline of your life? If you could only include a maximum of 6 events in a timeline of your life, what would be the most important things that you would want to include: being born; birth of a sibling; starting pre-school / kindergarten; joining a sports team; or going on an overseas holiday?
In the next activity you will have the opportunity to create a history of the Universe timeline based on Big History Junior’s 3 big mission phase questions.
2Al
As you saw in Mission video 5: 3 big mission phase questions, a timeline helps you see the most important things that have happened over a period of time and gives you an idea of how long ago or how recently they occurred.
Well, you’re now going to create a timeline of the longest period of time of all - the whole history of the Universe!
You will be using the Timeline: history of the Universe worksheet to create your own timeline. Before you begin though, take some time to look at it carefully. What do you notice about the timeline?
How long ago does the timeline begin?
It ends with an arrow. What do you think this arrow could be pointing towards?
The “Universe” bracket along the bottom is almost the entire length of the timeline. Why?
(Hint: think about the age of the Universe)
The “Earth” bracket begins close to 5 billion (500 crore) years ago. Why?
(Hint: think about the age of Earth)
The “Humans” bracket begins less than 1 billion (100 crore) years ago. Why?
(Hint: think about how long humans have been around)
Your teacher will instruct you to either cut-and-paste the 6 ‘events’ on your worksheet onto the correct place on the timeline or to simply write the event names in the boxes.
Once you have completed your timeline, look carefully at each of the ‘events’ and where they occur on the timeline:
Had you realized that the Universe existed so long before planet Earth was formed?
Are you surprised at the short amount of time that humans have been around compared to the entire history of the Universe?
Is there anything else that you notice when you look at the history of the Universe on a timeline?
2Al
Things are about to get serious. This is the first part of your formal introduction to your Mars Mission.
In Mission video 6: Your mission control reports Commander Ripley fills you in on the Mission Control Reports you will need to complete in order to achieve your Mars Mission and you will receive your official Mars Mission Brief.
While you watch Mission video 6: Your mission control reports look out for the answers to the following questions:
1. What is your Mission Brief?
2. What is the all-important question you are trying to answer on your Mission?
3. Which question will you need to answer in each of the four Mission Control Reports?
As Commander Ripley mentioned in Mission video 6: Your mission control reports, there are four main phases in your Mission. Your teacher will give you a copy of the Brief: Mars Mission which outlines all the tasks you’ll need to undertake to successfully complete your Mars Mission.
Read through your Brief: Mars mission carefully and your teacher will let you know if there are any other specific instructions that you need to add.
Now that you know what will be required of you on your Mars Mission, in the next activity you will decide on your Mars Mission Name and show your commitment to the Mission by signing a ‘pledge.’
2Al
Commit to undertake the steps necessary to successfully complete the Mission
Now that you understand the step-by-step requirements of your Mars Mission and you have received your Mission Brief, you are ready to sign your Mars Mission pledge.
Your teacher will advise you whether you will be working on your mission individually or if you will be working as part of a team.
If you are working with a team, you will need to complete the Mars mission pledge: team worksheet.
One of the first (and fun!) things you will need to do when you work on your Mars mission pledge: team is to brainstorm and decide on the name for your Mars Mission.
Every space mission has an official name. For inspiration take a look at the list of Mars mission name ideas, in Helpful Resources, which includes words used in real space mission names.
Once you have decided on and written down your Mission name on your pledge you need to do the following:
Add at least 3 more points in the ‘We pledge to’ section which you, as a team, agree are important.
Write down each team member’s name and choose a team leader – who is very responsible and would be willing to take on any extra tasks?
Sign your names and write the date.
1.2.4 - Mars Mission Name Ideas
If you are working individually you will need to complete the Mars mission pledge: student worksheet.
One of the first (and fun!) things you will need to do when you work on your Mars mission pledge: student is to brainstorm and decide on the name for your Mars Mission.
Add at least 3 more points in the ‘I pledge to’ section which you agree are important.
Sign your name and write the date.
Mars Mission Name Ideas
Once you have signed your Mission Pledge, you are officially a member of the Mars Mission! Your teacher will organize a S.P.A.C.E command ID Card which will include your photo, name and Mission Name. Make sure you keep your ID Card with you whenever you work on your Mars Mission.
Congratulations! You have completed your first Mission Phase and Mars Mission Introduction.
You are now ready to move on to Mission Phase 2 where you will learn all about the Universe and then complete your first Mars Mission Control Report.
2Al
PHASE 1 STAGE 2: Mission Control
In Mission Control you learned all about the 3 big mission phase questions and what you need to do to successfully complete your Mars Mission.
Now it’s time to revisit your Mission Control learning goals and read through them again carefully.
Once you have checked the boxes to confirm you have achieved your learning goals for Sequence '1.2 Mission Control' click on the 'I have achieved my learning goals' button below.
2Al
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Implement Hardware-Efficient Complex Divide HDL Optimized - MATLAB & Simulink - MathWorks Italia
Interfacing with the Complex Divide HDL Optimized Block
Open the Model and Define Input Data
Simulate the Model and Examine the Output
This example demonstrates how to perform the division of complex numbers using hardware-efficient MATLAB® code embedded in Simulink® models. The model used in this example is suitable for HDL code generation for fixed-point inputs. The algorithm employs a fully pipelined architecture, which is suitable for FPGA or ASIC devices where throughput is of concern. This implementation also uses available on-chip resources judiciously, making it suitable for resource-conscious designs as well.
The division operation for two complex numbers
a+bi
c+di
\mathit{b}\ne 0
z=\frac{a+bi}{c+di}
. After multiplying the denominator by its complex conjugate, this can be re-written as
z=\frac{\left(ac+bd\right)+\left(bc-ad\right)i}{{c}^{2}+{d}^{2}}
CORDIC is an acronym for COordinate Rotation DIgital Computer, and can be used to efficiently compute many trigonometric, hyperbolic, and arithmetic functions. For a detailed explanation of the CORDIC algorithm and its application in the calculation of a trigonometric function, see Compute Sine and Cosine Using CORDIC Rotation Kernel.
When deploying intricate algorithms to FPGA or ASIC devices, there is often a trade-off between resource usage and total throughput for a given computation. Resource-sharing often reduces the resources consumed by a design, but also reduces the throughput in the process. Simple arithmetic and trigonometic computations, which typically form parts of bigger computations, require high throughput to drive circuits further in the design. Thus, fully pipelined implementations consume more on-chip resources but are beneficial in large designs.
To open the example model, at the command line, enter:
mdl = 'fxpdemo_complexDivide';
The model contains the Complex Divide HDL Optimized block connected to a data source which takes in arrays of inputs (numerators and denominators) and passes an input value from each array to the block on consecutive cycles. The output computed for each value is stored in a workspace variable. The simulation terminates when all inputs have been processed.
Define arrays of inputs complexDivideNumerators and complexDivideDenominators. For this example, the inputs are doubles. Note that both the numerator and the denominator should have the same datatype.
complexDivideNumerators = (9*rand(1000,1) + 1) + (9*rand(1000,1) + 1)*1i;
complexDivideDenominators = (9*rand(1000,1) + 1) + (9*rand(1000,1) + 1)*1i;
Define the output dataype to be used in the model. For this example, the outputs are also doubles. Note that fixed-point type outputs can only be used with fixed-point type inputs.
OutputType = 'double';
When the simulation is complete, a new workspace variable, complexDivideOutputs, is created to hold the computed value for each pair of inputs.
Examine the error of the calculation by comparing the output of the Complex Divide HDL Optimized block to that of the built-in MATLAB® divide function.
expectedOutput = complexDivideNumerators./complexDivideDenominators;
actualOutput = complexDivideOutputs;
maxError = max(abs(expectedOutput - actualOutput))
maxError = 3.5958e-15
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Black hole - Wikiquote
astronomical object so massive, that anything falling into it, including light, cannot escape its gravity
A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it. Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace. The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916. Quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass. The discovery of neutron stars sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality.
Black holes ain't as black as they are painted. They are not the eternal prisons they were once thought. Things can get out of a black hole, both to the outside, and possibly to another universe. So if you feel you are in a black hole, don't give up. There's a way out. ~ Stephen Hawking
Black hole with corona, X-ray source (artist's concept).
Stellar evolutions of low-mass vs. high-mass stars,
A star does not evolve over its lifetime through each spectral type, as Russell once thought; rather, each star experiences its own distinct history, based on its mass at birth. Smaller stars, such as tiny red dwarfs, will never reach the red-giant stage but just dully burn away like red-hot ovens. Stars that are born with appreciably more mass than our Sun, such as the white-hot O and B stars, will burn swiftly and eventually blow up, leaving behind a city-sized neutron star or even a black hole, a gravitational pit from which no light or matter can escape. ...the term black hole wasn't even coined until 1968. Yet the first tentative steps toward understanding this great metamorphosis, the distinct and striking stages in a star's life, were taken at the turn of the century. The elements in the stars themselves were telling the tale in the spectral messages they were telegraphing throughout the cosmos.
Marcia Bartusiak, Through a Universe Darkly: A Cosmic Tale of Ancient Ethers, Dark Matter, and the Fate of the Universe (1993)
After the nuclear fuel is used up, the star goes into a state of gravitational collapse. All parts of the star fall more or less freely inward... [Y]ou would imagine that the freefall could not continue... because the falling material would... arrive at the center... But Einstein's equations have the peculiar consequence... permanent freefall without ever reaching the bottom... what we call a black hole. ...[T]he space ...is so strongly curved that space and time become interchanged... time becomes space and... space becomes time. More precisely, if you observe... from the outside, you see... motion slow down and stop because the direction of time inside... is perpendicular to the direction of time as seen from the outside. The collapsing star can continue to fall freely forever...
Freeman Dyson, Infinite in All Directions (1989) Ch. 2 Butterflies and Superstrings.
"Schwarzschild's solution"—revealed a stunning implication of general relativity. He showed that if the mass of a star is concentrated in a small enough spherical region, so that it's mass divided by its radius exceeds a particular critical value, the resulting space-time warp is so radical that anything, including light, that gets too close to the star will be unable to escape its gravitational grip. ...John Wheeler ...called them black holes—black because they cannot emit light, holes because anything getting too close falls into them, never to return. The name stuck.
Brian Greene, The Elegant Universe (1999)
Black holes have the universe's most inscrutable poker faces. ...When you've seen one black hole with a given mass, charge, and spin (though you've learned these thing indirectly, through their effect on surrounding gas and stars...) you've definitely seen them all. ...black holes contain the highest possible entropy ...a measure of the number of rearrangements of an object's internal constituents that have no effect on its appearance. ...Black holes have a monopoly on maximal disorder. ...As matter takes the plunge across a black hole's ravenous event horizon, not only does the black hole's entropy increase, but its size increases as well. ...the amount of entropy ...tells us something about space itself: the maximum entropy that can be crammed into a region of space—any region of space, anywhere, anytime—is equal to the entropy contained within a black hole whose size equals the region in question.
Brian Greene, The Fabric of the Cosmos (2004)
A natural guess is that... a black hole's entropy is... proportional to its volume. But in the 1970s Jacob Bekenstein and Stephen Hawking discovered that this isn't right. Their... analyses showed that the entropy... is proportional to the area of its event horizon... less than what we'd naïvely guess. ...Berkenstein and Hawking found that... each square being one Planck length by one Planck length... the black hole's entropy equals the number of such squares that can fit on its surface... each Planck square is a minimal unit of space, and each carries a minimal, single unit of entropy. This suggests that there is nothing, even in principle, that can take place within a Planck square, because any such activity could support disorder and hence the Planck square could contain more than a single unit of entropy... Once again... we are led to the notion of an elemental spatial entity.
[F]or a physicist, the upper limit to entropy... is a critical, almost sacred quantity. ...the Bekenstein and Hawking result tells us that a theory that includes gravity is, in some sense, simpler than a theory that doesn't. ...If the maximum entropy in any given region of space is proportional to the region's surface area and not its volume, then perhaps the true, fundamental degrees of freedom—the attributes that have the potential to give rise to that disorder—actually reside on the region's surface and not within its volume. Maybe... the universe's physical processes take place on a thin, distant surface that surrounds us, and all we see and experience is merely a projection of those processes. Maybe... the universe is rather like a hologram.
The subject of this book is the structure of space-time on length-scales from 10-13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe. ...we base our treatment on Einstein's General Theory of Relativity. This theory leads to two remarkable predictions about the universe: first, that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly, that there is a singularity in our past which constitutes, in some sense, a beginning to the universe.
Stephen Hawking, G.F.R. Ellis, Preface, "The Large Scale Structure of Space-Time" (1973)
Stephen Hawking, During a 1994 exchange with Penrose, transcribed in The Nature of Space and Time (1996) by Stephen Hawking and Roger Penrose, p. 26 and also in "The Nature of Space and Time" (online text)
Stephen Hawking, "Information Loss in Black Holes" (July 2005)
Stephen Hawking, Reith Lecture 2 : Black holes ain’t as black as they are painted (2015) · BBC Radio 4 audio file
It is hard to understand how this infinitely dense singularity can evaporate into nothing. For matter inside the black hole to leak out into the universe requires that it travel faster than the speed of light.
John Moffat, Reinventing Gravity (2008) Ch. 5 Conventional Black Holes, p. 85
Is the reader feeling confused about the status of the black hole information paradox and black holes in general? So am I!
Experimentalists dream of some spectacular discovery such as the proof of the existence of black holes to justify the more than eight billion dollars it has cost to build the LHC.
John Moffat, Reinventing Gravity (2008) Ch. 5 Conventional Black Holes, p. 88.
A large part of the relativity community is in denial - refusing even to contemplate the idea that black holes may not exist in nature, or seriously consider the idea that any kind of new matter such as the new putative dark energy can play a fundamental role in gravity theory.
John Moffat, Reinventing Gravity (2008) Ch. 14 Do Black Holes Exist In Nature? p. 204.
Hawking's intitial foray into quantum gravity was more modest than Wheeler's and other[s]... a sneak approach. He first wanted to know what the effect was of an ordinary, classic, curved-space gravitational field on a quantum system. He called this the semiclassical approach. Until that day, most quantum calculations had been done as if gravity didn't exist—they were hard enough without it in normal flat space-time... [Hawking accomplished this by] envisioning an "atom" whose nucleus was a catastrophically powerful black hole... Starobinsky ventured the opinion that rotating black holes would spray elementary particles. ...It was known from Penrose's work, among others, that you could extract energy from the spin of a black hole just like any other dynamo... in particles and radiation just like it did from a particle generator. ...But Hawking ...resolved to redo the calculation for himself ...he decided to warm up first, by calculating the rate of emission from a nonrotating quantum hole. He knew the answer should be no emission. ...his results were embarrassing. His imaginary black hole was spewing matter and radiation... he was reluctant to tell anybody but his closest friends; he was afraid Bekenstein would hear about it. ...It meant that holes had temperatures, just as Bekenstein's work implied.
Even though a black hole is practically invisible, astronomers can infer its presence from the effects it has on spacetime itself. ...Andrea Ghez... uses radio telescopes to study the motions of stars near the center of our galaxy. By watching how these stars move, she is really measuring the curvature of spacetime—the strength of gravity—in the heart of the Milky Way. ...Ghez realized that the stars are wheeling about an invisible, supermassive object that weighs more than two and a half million times as much as our sun. The black hole... dubbed Sagittarius A*... cannot be seen directly, but Ghez was able to find it because of the effect it has on spacetime, on the stars orbiting it. Ghez's technique is quite similar to what Vera Rubin did when she made the first compelling case for dark matter.
Charles Seife, Alpha and Omega: The Search for the Beginning and End of the Universe (2003)
There is no shortage of candidates for... baryonic dark matter. It may come in many forms—clouds of gas or dust, large planetlike objects, various forms of degraded stars, and black holes. ...MACHOS could include black holes and burned-out stars, such as white dwarfs or neutron stars... Black holes are perhaps the most intriguing, and the most difficult to detect and quantify. As far back as the eighteenth century, scientists speculated about worlds so massive that nothing escaped their gravitational grip, not even light. In the early twentieth century, J. Robert Oppenheimer used Einstein's general theory of relativity to explain how a black hole might form: The black hole would warp adjacent space so deeply that the escape velocity would exceed the speed of light... hence nothing... could leave... The center of the Milky Way emits intense gamma radiation—the death cry, perhaps, of stars falling into a black hole. Black holes may also be distributed in galactic halos, where they might constitute a substantial fraction of baryonic dark matter.
George Smoot, Keay Davidson, Wrinkles in Time (1993)
{\displaystyle F={\frac {mMG}{R^{2}}}}
{\displaystyle E=mc^{2}}
{\displaystyle F}
{\displaystyle M}
{\displaystyle m}
{\displaystyle G}
{\displaystyle G}
was too small to measure until the end of the eighteenth century. ...Cavindish found that the force between a pair of one-kilogram masses separated by one meter is approximately 6.6 x 10-11 newtons. (The Newton is... about one-fifth of a pound.) ...Newton had one lucky break... the special mathematical properties of the inverse square law. ...[B]y the miracle of mathematics, you can pretend that the entire mass is located at a single point. This... allowed Newton to calculate the escape velocity...
{\displaystyle Escape\;velocity={\sqrt {2MG/R}}}
{\displaystyle M}
{\displaystyle R}
, the larger the escape velocity. ...to compute the Schwarzschild radius
{\displaystyle R_{s}}
... plug in the speed of light for the escape velocity...
{\displaystyle R_{s}={\frac {2MG}{c^{2}}}}
... is proportional to the mass. That's all there is to dark stars... at the level that Laplace and Michell were able to understand them.
[A]round 1967, Wheeler became very interested in the gravitationally collapsed objects that Karl Schwarzschild had described in 1917. At the time they were called black stars or dark stars. ...Wheeler began calling them black holes. At first the name was blackballed by the... Physical Review. ...the term ...was deemed obscene! But John fought it... Amusingly, John's next coinage was the saying "Black holes have no hair." ...he was making a very serious point about black hole horizons. ...[Each a] smooth ...perfectly regular, featureless sphere. Apart from their mass and rotational speed, every black hole was exactly like every other. Or so it was thought.
Black hole at Wikiquote's sister projects:
Video News Release 25: Unprecedented 16-year long study tracks stars orbiting Milky Way black hole (eso0846b) from the European Southern Observatory.
Singularities and Black Holes by Erik Curiel, Peter Bokulich, Stanford Encyclopedia of Philosophy.
Retrieved from "https://en.wikiquote.org/w/index.php?title=Black_hole&oldid=3027533"
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Solve nonlinear curve-fitting (data-fitting) problems in least-squares sense - MATLAB lsqcurvefit - MathWorks
\underset{x}{\mathrm{min}}{‖F\left(x,xdata\right)-ydata‖}_{2}^{2}=\underset{x}{\mathrm{min}}\sum _{i}{\left(F\left(x,xdat{a}_{i}\right)-ydat{a}_{i}\right)}^{2},
F\left(x,xdata\right)=\left[\begin{array}{c}F\left(x,xdata\left(1\right)\right)\\ F\left(x,xdata\left(2\right)\right)\\ ⋮\\ F\left(x,xdata\left(k\right)\right)\end{array}\right].
x\left(1\right)
x\left(2\right)
\text{ydata}=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right).
y=\mathrm{exp}\left(-1.3t\right)+\epsilon ,
t
\epsilon
y=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right)
0\le x\left(1\right)\le 3/4
-2\le x\left(2\right)\le -1.
x\left(1\right)
x\left(2\right)
\text{ydata}=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right).
x\left(1\right)
x\left(2\right)
\text{ydata}=x\left(1\right)\mathrm{exp}\left(x\left(2\right)\text{xdata}\right).
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Receiver operating characteristic curves by false-alarm probability - MATLAB rocpfa
MaxSNR
Plot ROC Curves for Different PFAs
Receiver operating characteristic curves by false-alarm probability
[Pd,SNR] = rocpfa(Pfa)
[Pd,SNR] = rocpfa(Pfa,Name,Value)
rocpfa(...)
[Pd,SNR] = rocpfa(Pfa) returns the single-pulse detection probabilities, Pd, and required SNR values, SNR, for the false-alarm probabilities in the row or column vector Pfa. By default, for each false-alarm probability, the detection probabilities are computed for 101 equally spaced SNR values between 0 and 20 dB. The ROC curve is constructed assuming a single pulse in coherent receiver with a nonfluctuating target.
[Pd,SNR] = rocpfa(Pfa,Name,Value) returns detection probabilities and SNR values with additional options specified by one or more Name,Value pair arguments.
rocpfa(...) plots the ROC curves.
False-alarm probabilities in a row or column vector.
Maximum SNR to include in the ROC calculation.
Minimum SNR to include in the ROC calculation.
Number of SNR values to use when calculating the ROC curves. The actual values are equally spaced between MinSNR and MaxSNR.
Number of pulses to integrate when calculating the ROC curves. A value of 1 indicates no pulse integration.
This property specifies the type of received signal or, equivalently, the probability density functions (PDF) used to compute the ROC. Valid values are: 'Real', 'NonfluctuatingCoherent', 'NonfluctuatingNoncoherent', 'Swerling1', 'Swerling2', 'Swerling3', and 'Swerling4'. Values are not case sensitive.
The 'NonfluctuatingCoherent' signal type assumes that the noise in the received signal is a complex-valued, Gaussian random variable. This variable has independent zero-mean real and imaginary parts each with variance σ2/2 under the null hypothesis. In the case of a single pulse in a coherent receiver with complex white Gaussian noise, the probability of detection, PD, for a given false-alarm probability, PFA is:
{P}_{D}=\frac{1}{2}\text{erfc}\left({\text{erfc}}^{-1}\left(2{P}_{FA}\right)-\sqrt{\chi }\right)
where erfc and erfc-1 are the complementary error function and that function’s inverse, and χ is the SNR not expressed in decibels.
For details about the other supported signal types, see [1].
Default: 'NonfluctuatingCoherent'
Detection probabilities corresponding to the false-alarm probabilities. For each false-alarm probability in Pfa, Pd contains one column of detection probabilities.
Signal-to-noise ratios in a column vector. By default, the SNR values are 101 equally spaced values between 0 and 20. To change the range of SNR values, use the optional MinSNR or MaxSNR input argument. To change the number of SNR values, use the optional NumPoints input argument.
Plot ROC curves for false-alarm probabilities of 1e-8, 1e-6, and 1e-3, assuming no pulse integration.
rocpfa(Pfa,'SignalType','NonfluctuatingCoherent')
[1] Richards, M. A. Fundamentals of Radar Signal Processing. New York: McGraw-Hill, 2005, pp 298–336.
npwgnthresh | rocsnr | shnidman
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Find the area of each figure below.
First, examine each shape to determine which type of shape it is. Look for characteristics such as right angles, same length sides, and number of sides.
After you have determined the type of shape, recall the area formulas specific to that shape. In this case, a parallelogram and a triangle.
459
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Wiles's_proof_of_Fermat's_Last_Theorem Knowpia
Wiles's proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques which were not available to Fermat. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory.
Precursors to Wiles's proofEdit
Fermat's Last Theorem and progress prior to 1980Edit
{\displaystyle a^{n}+b^{n}=c^{n}}
The Taniyama–Shimura–Weil conjectureEdit
Frey's curveEdit
{\displaystyle y^{2}=x(x-a^{n})(x+b^{n}).}
Ribet's theoremEdit
Situation prior to Wiles's proofEdit
Andrew WilesEdit
Announcement and subsequent developmentsEdit
Announcement and final proof (1993–1995)Edit
Summary of Wiles's proofEdit
Mathematical detail of Wiles's proofEdit
{\displaystyle R_{n}\rightarrow \mathbf {T} _{n}.}
{\displaystyle R}
{\displaystyle \mathbf {T} }
{\displaystyle R}
{\displaystyle \mathbf {T} }
{\displaystyle R\rightarrow \mathbf {T} }
{\displaystyle R=\mathbf {T} }
{\displaystyle R=\mathbf {T} }
{\displaystyle R=\mathbf {T} }
{\displaystyle \mathbf {Z} _{3},\mathbf {F} _{3}}
{\displaystyle \mathbf {T} /{\mathfrak {m}}}
General approach and strategyEdit
{\displaystyle E({\bar {\mathbf {Q} }})}
{\displaystyle \ell ^{n}}
{\displaystyle \operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )}
{\displaystyle \operatorname {GL} _{2}(\mathbf {Z} /l^{n}\mathbf {Z} ),}
{\displaystyle \ell ^{n}}
{\displaystyle E({\bar {\mathbf {Q} }})}
{\displaystyle {\bar {\mathbf {Q} }}}
{\displaystyle \operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )}
{\displaystyle \ell ^{n}x=0}
{\displaystyle (\mathbf {Z} /\ell ^{n}\mathbf {Z} )^{2}}
{\displaystyle \operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )\rightarrow \operatorname {GL} _{2}(\mathbf {Z} /l^{n}\mathbf {Z} ).}
{\displaystyle \ell ^{n}}
{\displaystyle (\mathrm {mod} \,\ell ^{n})}
{\displaystyle (\mathrm {mod} \,3)}
{\displaystyle (\mathrm {mod} \,\ell ^{n})}
{\displaystyle (\mathrm {mod} \,\ell ^{n+1})}
{\displaystyle (\mathrm {mod} \,\ell ^{n})}
3-5 trickEdit
{\displaystyle (\mathrm {mod} \,3)}
{\displaystyle (\mathrm {mod} \,3)}
{\displaystyle (\mathrm {mod} \,5)}
{\displaystyle {\overline {\rho }}_{E,5}}
{\displaystyle {\overline {\rho }}_{E',5}}
{\displaystyle {\overline {\rho }}_{E,5}}
{\displaystyle {\overline {\rho }}_{E,3}}
Structure of Wiles's proofEdit
Overviews available in the literatureEdit
Explanations of the proof (varying levels)Edit
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Integration - MATLAB & Simulink Example - MathWorks India
Definite Integrals in Maxima and Minima
This example shows how to compute definite integrals using Symbolic Math Toolbox™.
Show that the definite integral
\underset{a}{\overset{b}{\int }}f\left(x\right)dx
f\left(x\right)=sin\left(x\right)
\left[\frac{\pi }{2},\frac{3\pi }{2}\right]
int(sin(x),pi/2,3*pi/2)
0
F\left(a\right)={\int }_{-a}^{a}\mathrm{sin}\left(ax\right)\mathrm{sin}\left(x/a\right)\phantom{\rule{0.16666666666666666em}{0ex}}dx
a\ge 0
, first, define the symbolic variables and assume that
a\ge 0
assume(a >= 0);
Then, define the function to maximize:
F = int(sin(a*x)*sin(x/a),x,-a,a)
\left\{\begin{array}{cl}1-\frac{\mathrm{sin}\left(2\right)}{2}& \text{ if }a=1\\ \frac{2 a \left(\mathrm{sin}\left({a}^{2}\right) \mathrm{cos}\left(1\right)-{a}^{2} \mathrm{cos}\left({a}^{2}\right) \mathrm{sin}\left(1\right)\right)}{{a}^{4}-1}& \text{ if }a\ne 1\end{array}
Note the special case here for
a=1
. To make computations easier, use assumeAlso to ignore this possibility (and later check that
a=1
is not the maximum):
assumeAlso(a ~= 1);
\frac{2 a \left(\mathrm{sin}\left({a}^{2}\right) \mathrm{cos}\left(1\right)-{a}^{2} \mathrm{cos}\left({a}^{2}\right) \mathrm{sin}\left(1\right)\right)}{{a}^{4}-1}
Create a plot of
F
to check its shape:
fplot(F,[0 10])
Use diff to find the derivative of
F
a
Fa = diff(F,a)
\begin{array}{l}\frac{2 {\sigma }_{1}}{{a}^{4}-1}+\frac{2 a \left(2 a \mathrm{cos}\left({a}^{2}\right) \mathrm{cos}\left(1\right)-2 a \mathrm{cos}\left({a}^{2}\right) \mathrm{sin}\left(1\right)+2 {a}^{3} \mathrm{sin}\left({a}^{2}\right) \mathrm{sin}\left(1\right)\right)}{{a}^{4}-1}-\frac{8 {a}^{4} {\sigma }_{1}}{{\left({a}^{4}-1\right)}^{2}}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}=\mathrm{sin}\left({a}^{2}\right) \mathrm{cos}\left(1\right)-{a}^{2} \mathrm{cos}\left({a}^{2}\right) \mathrm{sin}\left(1\right)\end{array}
Fa
are the local extrema of
F
fplot(Fa,[0 10])
The maximum is between 1 and 2. Use vpasolve to find an approximation of the zero of
Fa
in this interval:
a_max = vpasolve(Fa,a,[1,2])
a_max =
1.5782881585233198075558845180583
Use subs to get the maximal value of the integral:
F_max = subs(F,a,a_max)
F_max =
0.36730152527504169588661811770092 \mathrm{cos}\left(1\right)+1.2020566879911789986062956284113 \mathrm{sin}\left(1\right)
The result still contains exact numbers
\mathrm{sin}\left(1\right)
\mathrm{cos}\left(1\right)
. Use vpa to replace these by numerical approximations:
vpa(F_max)
1.2099496860938456039155811226054
Check that the excluded case
a=1
does not result in a larger value:
vpa(int(sin(x)*sin(x),x,-1,1))
0.54535128658715915230199006704413
Numerical integration over higher dimensional areas has special functions:
integral2(@(x,y) x.^2-y.^2,0,1,0,1)
There are no such special functions for higher-dimensional symbolic integration. Use nested one-dimensional integrals instead:
int(int(x^2-y^2,y,0,1),x,0,1)
0
Define a vector field F in 3D space:
F(x,y,z) = [x^2*y*z, x*y, 2*y*z];
Next, define a curve:
ux(t) = sin(t);
uy(t) = t^2-t;
uz(t) = t;
The line integral of F along the curve u is defined as
\int f\cdot du=\int f\left(ux\left(t\right),uy\left(t\right),uz\left(t\right)\right)\cdot \frac{du}{dt}\phantom{\rule{0.16666666666666666em}{0ex}}dt
\cdot
on the right-hand-side denotes a scalar product.
Use this definition to compute the line integral for
t
\left[0,1\right]
F_int = int(F(ux,uy,uz)*diff([ux;uy;uz],t),t,0,1)
\frac{19 \mathrm{cos}\left(1\right)}{4}-\frac{\mathrm{cos}\left(3\right)}{108}-12 \mathrm{sin}\left(1\right)+\frac{\mathrm{sin}\left(3\right)}{27}+\frac{395}{54}
Get a numerical approximation of this exact result:
vpa(F_int)
-0.20200778585035447453044423341349
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Three-valued logic - Wikipedia
(Redirected from Ternary logic)
Logic system in which there are three truth values indicating true, false and some indeterminate third value
Find sources: "Three-valued logic" – news · newspapers · books · scholar · JSTOR (January 2011) (Learn how and when to remove this template message)
In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean,[1] sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value. This is contrasted with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false.
Emil Leon Post is credited with first introducing additional logical truth degrees in his 1921 theory of elementary propositions.[2] The conceptual form and basic ideas of three-valued logic were initially published by Jan Łukasiewicz and Clarence Irving Lewis. These were then re-formulated by Grigore Constantin Moisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945.
2 Representation of values
3.1 Kleene and Priest logics
3.2 Łukasiewicz logic
3.3 Bochvar logic
3.4 Ternary Post logic
3.5 Modular algebras
Around 1910, Charles Sanders Peirce defined a many-valued logic system. He never published it. In fact, he did not even number the three pages of notes where he defined his three-valued operators.[3] Peirce soundly rejected the idea all propositions must be either true or false; boundary-propositions, he writes, are "at the limit between P and not P."[4] However, as confident as he was that "Triadic Logic is universally true," he also jotted down that "All this is mighty close to nonsense." Only in 1966, when Max Fisch and Atwell Turquette began publishing what they rediscovered in his unpublished manuscripts, did Peirce's triadic ideas become widely known.[5]
Representation of values[edit]
As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the ternary numeral system. A few of the more common examples are:
in balanced ternary, each digit has one of 3 values: −1, 0, or +1; these values may also be simplified to −, 0, +, respectively;[6]
in the redundant binary representation, each digit can have a value of −1, 0, 0/1 (the value 0/1 has two different representations);
in the ternary numeral system, each digit is a trit (trinary digit) having a value of: 0, 1, or 2;
in the skew binary number system, only the least-significant non-zero digit can have a value of 2, and the remaining digits have a value of 0 or 1;
1 for true, 2 for false, and 0 for unknown, unknowable/undecidable, irrelevant, or both;[7]
0 for false, 1 for true, and a third non-integer "maybe" symbol such as ?, #, ½,[8] or xy.
Inside a ternary computer, ternary values are represented by ternary signals.
This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, true}, and extends conventional Boolean connectives to a trivalent context. Ternary predicate logics exist as well;[citation needed] these may have readings of the quantifier different from classical (binary) predicate logic and may include alternative quantifiers as well.
Logics[edit]
Where Boolean logic has 22 = 4 unary operators, the addition of a third value in ternary logic leads to a total of 33 = 27 distinct operators on a single input value. Similarly, where Boolean logic has 22×2 = 16 distinct binary operators (operators with 2 inputs), ternary logic has 33×3 = 19,683 such operators. Where we can easily name a significant fraction of the Boolean operators (NOT, AND, NAND, OR, NOR, XOR, XNOR, equivalence, implication), it is unreasonable to attempt to name all but a small fraction of the possible ternary operators.[9]
Kleene and Priest logics[edit]
See also: Kleene algebra (with involution)
Below is a set of truth tables showing the logic operations for Stephen Cole Kleene's "strong logic of indeterminacy" and Graham Priest's "logic of paradox".
(F, false; U, unknown; T, true)
(−1, false; 0, unknown; +1, true)
In these truth tables, the unknown state can be thought of as neither true nor false in Kleene logic, or thought of as both true and false in Priest logic. The difference lies in the definition of tautologies. Where Kleene logic's only designated truth value is T, Priest logic's designated truth values are both T and U. In Kleene logic, the knowledge of whether any particular unknown state secretly represents true or false at any moment in time is not available. However, certain logical operations can yield an unambiguous result, even if they involve at least one unknown operand. For example, because true OR true equals true, and true OR false also equals true, one can infer that true OR unknown equals true, as well. In this example, because either bivalent state could be underlying the unknown state, but either state also yields the same result, a definitive true results in all three cases.
If numeric values, e.g. balanced ternary values, are assigned to false, unknown and true such that false is less than unknown and unknown is less than true, then A AND B AND C... = MIN(A, B, C ...) and A OR B OR C ... = MAX(A, B, C...).
Material implication for Kleene logic can be defined as:
{\displaystyle A\rightarrow B\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\mbox{OR}}(\ {\mbox{NOT}}(A),\ B)}
, and its truth table is
IMPK(A, B), OR(¬A, B)
IMPK(A, B), MAX(−A, B)
which differs from that for Łukasiewicz logic (described below).
Kleene logic has no tautologies (valid formulas) because whenever all of the atomic components of a well-formed formula are assigned the value Unknown, the formula itself must also have the value Unknown. (And the only designated truth value for Kleene logic is True.) However, the lack of valid formulas does not mean that it lacks valid arguments and/or inference rules. An argument is semantically valid in Kleene logic if, whenever (for any interpretation/model) all of its premises are True, the conclusion must also be True. (Note that the Logic of Paradox (LP) has the same truth tables as Kleene logic, but it has two designated truth values instead of one; these are: True and Both (the analogue of Unknown), so that LP does have tautologies but it has fewer valid inference rules).[10]
Łukasiewicz logic[edit]
Further information: Łukasiewicz logic
The Łukasiewicz Ł3 has the same tables for AND, OR, and NOT as the Kleene logic given above, but differs in its definition of implication in that "unknown implies unknown" is true. This section follows the presentation from Malinowski's chapter of the Handbook of the History of Logic, vol 8.[11]
Material implication for Łukasiewicz logic truth table is
IMPŁ(A, B)
IMPŁ(A, B), MIN(1, 1−A+B)
In fact, using Łukasiewicz's implication and negation, the other usual connectives may be derived as:
A ∨ B = (A → B) → B
A ∧ B = ¬(¬A ∨ ¬ B)
A ⇔ B = (A → B) ∧ (B → A)
It is also possible to derive a few other useful unary operators (first derived by Tarski in 1921):[citation needed]
MA = ¬A → A
LA = ¬M¬A
IA = MA ∧ ¬LA
They have the following truth tables:
M is read as "it is not false that..." or in the (unsuccessful) Tarski–Łukasiewicz attempt to axiomatize modal logic using a three-valued logic, "it is possible that..." L is read "it is true that..." or "it is necessary that..." Finally I is read "it is unknown that..." or "it is contingent that..."
In Łukasiewicz's Ł3 the designated value is True, meaning that only a proposition having this value everywhere is considered a tautology. For example, A → A and A ↔ A are tautologies in Ł3 and also in classical logic. Not all tautologies of classical logic lift to Ł3 "as is". For example, the law of excluded middle, A ∨ ¬A, and the law of non-contradiction, ¬(A ∧ ¬A) are not tautologies in Ł3. However, using the operator I defined above, it is possible to state tautologies that are their analogues:
A ∨ IA ∨ ¬A (law of excluded fourth)
¬(A ∧ ¬IA ∧ ¬A) (extended contradiction principle).
Bochvar logic[edit]
Main article: Many-valued logic § Bochvar's internal three-valued logic (also known as Kleene's weak three-valued logic)
Ternary Post logic[edit]
not(a) = (a + 1) mod 3, or
not(a) = (a + 1) mod (n), where (n) is the value of a logic
Modular algebras[edit]
Some 3VL modular algebras have been introduced more recently, motivated by circuit problems rather than philosophical issues:[12]
Cohn algebra
Pradhan algebra
Dubrova and Muzio algebra
The database structural query language SQL implements ternary logic as a means of handling comparisons with NULL field content. NULL was originally intended to be used as a sentinel value in SQL to represent missing data in a database, i.e. the assumption that an actual value exists, but that the value is not currently recorded in the database. SQL uses a common fragment of the Kleene K3 logic, restricted to AND, OR, and NOT tables.
In SQL, the intermediate value is intended to be interpreted as UNKNOWN. Explicit comparisons with NULL, including that of another NULL yields UNKNOWN. However this choice of semantics is abandoned for some set operations, e.g. UNION or INTERSECT, where NULLs are treated as equal with each other. Critics assert that this inconsistency deprives SQL of intuitive semantics in its treatment of NULLs.[13] The SQL standard defines an optional feature called F571, which adds some unary operators, among which is IS UNKNOWN corresponding to the Łukasiewicz I in this article. The addition of IS UNKNOWN to the other operators of SQL's three-valued logic makes the SQL three-valued logic functionally complete,[14] meaning its logical operators can express (in combination) any conceivable three-valued logical function.
Binary logic (disambiguation)
Paraconsistent logic § An ideal three-valued paraconsistent logic
Setun – an experimental Russian computer which was based on ternary logic
Ternary numeral system (and Balanced ternary)
Three-state logic (tri-state buffer)
^ "Stanford JavaNLP API". Stanford University. Stanford NLP Group.
^ Post, Emil L. (1921). "Introduction to a General Theory of Elementary Propositions". American Journal of Mathematics. 43 (3): 163–185. doi:10.2307/2370324. hdl:2027/uiuo.ark:/13960/t9j450f7q. ISSN 0002-9327. JSTOR 2370324.
^ "Peirce's Deductive Logic > Peirce's Three-Valued Logic (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved 2020-07-30.
^ Lane, R. (2001). "Triadic Logic".
^ Lane, Robert. "Triadic Logic". www.digitalpeirce.fee.unicamp.br. Retrieved 2020-07-30.
^ Knuth, Donald E. (1981). The Art of Computer Programming Vol. 2. Reading, Mass.: Addison-Wesley Publishing Company. p. 190.
^ Hayes, Brian (November–December 2001). "Third base" (PDF). American Scientist. Sigma Xi, the Scientific Research Society. 89 (6): 490–494. doi:10.1511/2001.40.3268. Archived (PDF) from the original on 2019-10-30. Retrieved 2020-04-12.
^ Nelson, David (2008). The Penguin Dictionary of Mathematics. Fourth Edition. London, England: Penguin Books. Entry for 'three-valued logic'. ISBN 9780141920870.
^ Douglas W. Jones, Standard Ternary Logic, Feb. 11, 2013.
^ "Beyond Propositional Logic"
^ Grzegorz Malinowski, "Many-valued Logic and its Philosophy" in Dov M. Gabbay, John Woods (eds.) Handbook of the History of Logic Volume 8. The Many Valued and Nonmonotonic Turn in Logic, Elsevier, 2009
^ Miller, D. Michael; Thornton, Mitchell A. (2008). Multiple valued logic: concepts and representations. Synthesis lectures on digital circuits and systems. Vol. 12. Morgan & Claypool Publishers. pp. 41–42. ISBN 978-1-59829-190-2.
^ Ron van der Meyden, "Logical approaches to incomplete information: a survey" in Chomicki, Jan; Saake, Gunter (Eds.) Logics for Databases and Information Systems, Kluwer Academic Publishers ISBN 978-0-7923-8129-7, p. 344; PS preprint (note: page numbering differs in preprint from the published version)
^ C. J. Date, Relational database writings, 1991–1994, Addison-Wesley, 1995, p. 371
Bergmann, Merrie (2008). An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems. Cambridge University Press. ISBN 978-0-521-88128-9. Retrieved 24 August 2013. , chapters 5-9
Mundici, D. The C*-Algebras of Three-Valued Logic. Logic Colloquium ’88, Proceedings of the Colloquium held in Padova 61–77 (1989). doi:10.1016/s0049-237x(08)70262-3
Reichenbach, Hans (1944). Philosophic Foundations of Quantum Mechanics. University of California Press. Dover 1998: ISBN 0-486-40459-5
Introduction to Many-Valued Logics by Bertram Fronhöfer. Handout from a 2011 summer class at Technische Universität Dresden. (Despite the title, this is almost entirely about three-valued logics.)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Three-valued_logic&oldid=1088864028"
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1.1 Pre-mission critical thinking skills training - Big History School
1.1 Pre-mission critical thinking skills training
To work through '1.1 Pre-mission critical thinking skills training' you need to complete the Activities in order. So first complete ‘Learning Plan’ then move to Activity '1.1.1' followed by Activity '1.1.2' through to '1.1.7', and then finish with ‘Learning Summary’.
Pre-mission critical thinking skills training
You will undertake pre-mission training to ensure you have the critical thinking skills necessary to complete your mission.
To get started, read carefully through the pre-mission critical thinking skills training learning goals below. Make sure you tick each of the check boxes to show that you have read all your Pre mission critical thinking skills training learning goals.
As you read through the learning goals you may come across some words that you haven’t heard before. Please don’t worry. By the time you finish your pre-mission critical thinking skills training you will have become very familiar with them!
You will come back to these learning goals at the end of your pre-mission critical thinking skills training to see if you have confidently achieved them.
Thinking creatively: what questions do we ask?
Identify prior knowledge about Mars
Understand what is meant by ‘creative thinking’
Use a KWHLAQ chart to begin thinking creatively about how humans could live on Mars
Thinking across disciplines: which experts do we need?
Identify different types of expert knowledge
Understand the need to draw on a range of expert knowledge
Thinking about claims: what do we trust as knowledge?
Understand what is a ‘claim’
Identify the four claim testers
Understand how to use claim testers to evaluate a claim
2Al
Cover page: Big History Junior journal
Mind map: Mars
Mission video 2: Thinking creatively
Chart: KWHLAQ
Mission video 3: Thinking across disciplines
Claims: snap judgement
Mission video 4: Thinking about claims
Now that you have completed 'Learning Plan' you need to continue working through the activities in order. So now complete Activity '1.1.1' followed by Activity '1.1.2', Activity '1.1.3', Activity '1.1.4', Activity '1.1.5', Activity '1.1.6', Activity '1.1.7', and then finish with 'Learning Summary'.
As Commander Ripley stated in Mission video 1: Your mission brief, you are one of the space pioneers who will work with S.P.A.C.E Mission Control on an important mission to create a human habitat on Mars.
You probably have a million questions right now. And you’ve probably already got some really awesome ideas on what your human habitat would look like.
But let’s get down to the basics first and try to think about what it would actually feel like to stand on a planet that isn’t our planet Earth.
Take a minute to close your eyes and use all your senses to fully imagine what it may be like to be the first human to stand on the surface of Mars.
Mars has two moons and is further away from the Sun than Earth. What would the sky look like?
Mars is described as the red planet and has no water. What would the surface look like?
There is no life but Mars does have big dust storms. What noises would you hear around you?
You would be wearing a spacesuit with a helmet. Would you hear your footsteps or your breathing?
Earth - and your family and friends - would be millions of miles away. How does that make you feel?
You are experiencing something no human has ever experienced before. How does that make you feel?
It’s really quite amazing to think about isn’t it?
Now it’s time to get started on the first of your pre-mission critical thinking skills training activities where you will create your own mind map on Mars.
Throughout your Mars Mission you will need to keep a Big History School Junior journal to keep all your Mars Mission research in the one location. Write your name and class on the Cover Page: Big History School Junior journal worksheet and paste it onto the front of the notebook or folder you will be using during your Mars Mission.
You are going to be creating a mind map to identify what you already know about Mars.
A mind map is simply a way to visually organize your thoughts about a subject. It can include written ideas, diagrams, keywords and it can be color coded.
Your teacher will instruct you whether you will complete your Mars mind map on your own or in a small group/pair.
On the Mind map: Mars worksheet, you will see that the word ‘Mars’ is in the center circle. It is surrounded by four circles with questions in them.
Each question has three connected circles. Use the connected circles to write down at least three things you already know in answer to each question. If you have more than three pieces of information for a question, draw a line and another connecting circle to write it in.
Confused? Take a look at the Mind map: Venus example in Helpful Resources.
Now it’s your turn. And remember, a mind map is just a way to visually organize what you think you already know about Mars so include anything you think of because there is no right or wrong at this point.
1.1.1 - Mind Map - Venus
If you are part of a class, your teacher will lead a class discussion where you and your classmates will share some of the information you included in your Mars mind maps.
And please ensure that you file your Mars mind map safely as you will need to refer back to it later in this Mission Phase.
2Al
Now that you have completed 'Learning Plan' through to Activity '1.1.1' you need to continue working through the activities in order. So now complete Activity '1.1.2' followed by Activity '1.1.3', Activity '1.1.4', Activity '1.1.5', Activity '1.1.6', Activity '1.1.7', and then finish with 'Learning Summary'.
‘Thinking creatively’ probably doesn’t sound like a very important or exciting skill to have as a Mars pioneer but some of the most important discoveries and inventions in human history started with a very curious mind and some seriously creative thinking…
Mission video 2: Thinking creatively introduces you to a creative thinking process which can help you answer the important Mission question, “How could humans live on Mars?’
While you watch Mission Video 2 look out for the answers to the following questions:
1. What does it mean to be a creative thinker?
2. What is brainstorming?
3. What is a KWHLAQ chart?
4. How can you find answers to your questions?
Your teacher will instruct you whether you will answer the questions: as part of a class discussion; as a group/paired discussion; or independently by writing your answers in your Big History Junior journal (if you have been provided with one).
Now that you are familiar with what a KWHLAQ chart is, you will have the opportunity in the next activity to create your own chart in preparation for your Mars Mission.
2Al
Now that you have completed 'Learning Plan' through to Activity '1.1.2' you need to continue working through the activities in order. So now complete Activity '1.1.3' followed by Activity '1.1.4', Activity '1.1.5', Activity '1.1.6', Activity '1.1.7', and then finish with 'Learning Summary'.
Use a KWHLAQ chart to creatively think about how humans could live on Mars
You do a little bit of creative thinking every day even if you don’t realize it. All we are doing in this activity is going through the creative thinking process step-by-step...
You will create your own KWHLAQ chart to help you plan how to answer the important Mission question, ‘How could humans live on Mars?’
KWHLAQ is quite a mouthful! Have you worked out why this chart is named that?
The KWHLAQ stands for:
Your teacher will instruct you whether you will complete your KWHLAQ chart on your own, as a class or in a small group/pair.
It’s important before you begin working on your chart to take a look at the question you need to answer in order to complete your Mission: “How Could Humans Live on Mars?”
Keep this question in mind as you complete the first three columns in the chart:
Column 1: What do we KNOW?
This is where you should refer back to the Mars mind map you completed earlier and write down what you already know about Mars. Also write down what you know about human needs.
Column 2: What do we WANT to know?
Completing column 1 should have made you realize that there is a lot more you need to know before you can answer the question, ”How Could Humans Live on Mars?” For example, could we grow our own food on Mars?
Write down all your questions in this column.
Column 3: HOW will we find out?
Here are a few tips about how you can find the answers to your questions:
When was it published? There are lots of new discoveries being made in space!
Use keywords to get more relevant results e.g. Mars temperature
Use a kid-friendly search engine like “Kiddle” or add “for kids” at the end of your search phrase
Don’t just click on the first link - read through the results list before choosing
Go to more than one website to double-check facts
What type of experts would know more about the topic eg. astronomers? biologists?
Find out which organizations are considered experts eg. NASA for space travel
The remaining three columns of your KWHLAQ chart will remain empty for now. You will come back to them as you get closer to completing your Mission.
Once you’ve completed the first 3 columns of your KWHLAQ chart you should have it on display somewhere where you can keep referring back to it. As you find out more information on Mars you could add it to your “What do we Know?” column. And if you think of any more questions you could add it to your “What do we Want to know?” column.
You are now ready to learn more about the second of your pre-mission critical thinking skills: thinking across disciplines.
2Al
A very important critical thinking skill is to be able to think across disciplines. By disciplines, we basically mean different areas of knowledge. This skill helps us to see things in new, deeper and interconnected ways.
Mission video 3: Thinking across disciplines introduces you to the different types of experts who will share their knowledge with you and guide you through your Mars Mission.
While you watch Mission video 3: Thinking across disciplines look out for the answers to the following questions:
2. Can you name at least 4 different types of experts?
3. Why is it useful to refer to more than one type of expert?
Now that you understand what an expert is, have a think about your own strengths and interests and imagine what type of expert you would like to be if you were one of the first 12 inhabitants on Mars.
You could be one of the four scientists mentioned in the Mission video or you could be something else entirely. For example, if you love to cook, you could be the first Martian Chef! You would have the very important job of feeding the first Mars inhabitants.
In Helpful Resources you will find some NASA Mars Explorer posters which may inspire you.
https://mars.nasa.gov/multimedia/resources/mars-posters-explorers-wanted/#FAQ
2Al
In Mission video 3: Thinking across disciplines you learned about the importance of drawing on a wide range of expert knowledge to gain a more complete understanding of a topic.
In fact, putting together information about something from different experts is a little bit like putting together a jigsaw puzzle. It’s only when you have all those different views that you really get the whole picture.
So which experts would you rely on to find out about Mars? And what sorts of questions do you think they’d be able to help you answer?
In this activity you will put the pieces of a Mars jigsaw puzzle together to create an image of Mars.
Each piece in your Mars jigsaw puzzle has an essential question and a description of the expert who would be able to answer that question.
Once you cut out the pieces and stick them together in place, you need to complete the phrase “Ask a _______” by writing the name of the expert who could answer the question on each piece. You can then write ‘Mars’ in the circle in the middle of the jigsaw and color in the circle to look like the planet Mars.
Finally, write a sentence at the bottom of your jigsaw page stating what type of expert you would like to be if you were one of the first 12 inhabitants on Mars and why.
This is a good time to refer back to your KWHLAQ chart. Have you thought of any more questions you would like to know the answers to? If so, you can add them to your “What do we want to Know?” column. And do you now have some more ideas for the “How will we find out?” column? What type of experts could you ask?
You are now ready to learn more about the third and final of your pre-mission critical thinking skills: thinking about claims.
2Al
‘Claims’ may not be a word you use every day, but you make claims every day and you’re surrounded by claims every day.
Someone is making a claim when they present information as fact - not an opinion. They are asking you to trust that they have some reliable information or knowledge that you don’t.
In this introductory activity, you will play a game of Snap Judgement where you have to decide whether various claims are true or false. Once you learn more about claim testing in the next activity you will have an opportunity to come back to your ‘snap judgements’ and review your responses.
If you’re playing this game with other students your teacher may have already placed some ‘claims’ on display for you. Your teacher will ask you to go to each one of the three claims on display and write on a post-it note if you trust the claim or don’t trust the claim with a short sentence explaining why.
If your teacher has instructed you to complete this activity on your own or in a pair, you will use the Claims: snap judgement worksheet. Read each one of the three claims on the worksheet carefully and write down whether you trust or don’t trust each claim. Write at least one sentence for each explaining why.
You will be learning about the important skill of ‘claim testing’ in the next activity. You will then have an opportunity to come back to the snap judgements that you’ve made in this activity, think about your responses and see if you’ve changed your mind.
2Al
Now that you know what a claim is, you are ready to learn how to use a strategy which will help you decide which claims can be trusted, which ones can be ignored and which ones should be investigated further.
Mission video 4: Thinking about claims introduces you to the very important skill of claim testing. This will help you not only in preparing for your Mars Mission but it is also a skill you will be able to use in your everyday life!
While you watch Mission video 4: Thinking about claims look out for the answers to the following questions:
1. What is a ‘claim’?
2. What are the four claim testers?
So now that you know more about claim testers, go back to the claims you responded to in the Snap Judgement game and try to work out which claim tester you were using when you decided to trust each claim or not:
Were you using your intuition (gut feeling)?
Were you using your logic (thinking carefully to see if it made sense)?
Were you relying on evidence? Have you seen proof to support/deny the claim?
Were you using authority? Have you heard an expert make these claims before?
If you are part of a class, your teacher will lead a class discussion where you and your classmates will share your responses to the claims, analyze which claim testers you used and decide whether you have since changed your minds about any of the claims.
2Al
You undertook some important pre-mission training to ensure you have the critical thinking skills necessary to complete your mission.
Now it’s time to revisit your Pre-mission critical thinking skill training learning goals and read through them again carefully.
Well done on completing your learning summary. Click here to go to 1.2 Mission Control
Once you have checked the boxes to confirm you have achieved your learning goals for Sequence '1.1 Pre-mission critical thinking skills training' click on the 'I have achieved my learning goals' button below.
Go to 1.2 Mission Control »
2Al
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EUDML | On selecting the most reliable components. EuDML | On selecting the most reliable components.
On selecting the most reliable components.
Shi, Dylan
Shi, Dylan. "On selecting the most reliable components.." Journal of Applied Mathematics and Decision Sciences 2.2 (1998): 133-145. <http://eudml.org/doc/119873>.
author = {Shi, Dylan},
keywords = {optimum replacement policies; -optimal policies; retrograde motion; -optimal policies},
title = {On selecting the most reliable components.},
AU - Shi, Dylan
TI - On selecting the most reliable components.
KW - optimum replacement policies; -optimal policies; retrograde motion; -optimal policies
optimum replacement policies,
\epsilon
-optimal policies, retrograde motion,
\epsilon
-optimal policies
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Constraint on control system dynamics - MATLAB - MathWorks América Latina
TuningGoal.Poles class
mindamping
Constrain Closed-Loop Dynamics of Specified Loop of System to Tune
Constrain Dynamics of Specified Feedback Loop
Constraint on control system dynamics
Use TuningGoal.Poles to constrain the closed-loop dynamics of a control system or of specific feedback loops within the control system. You can use this tuning goal for control system tuning with tuning commands, such as systune or looptune. A TuningGoal.Poles goal can ensure a minimum decay rate or minimum damping of the poles of the control system or loop. It can also eliminate fast dynamics in the tuned system.
Req = TuningGoal.Poles(mindecay,mindamping,maxfreq) creates a default template for constraining the closed-loop pole locations. The minimum decay rate, minimum damping constant, and maximum natural frequency define a region of the complex plane in which poles of the component must lie. Set mindecay = 0, mindamping = 0, or maxfreq = Inf to skip any of the three constraints.
Req = TuningGoal.Poles(location,mindecay,mindamping,maxfreq) constrains the poles of the sensitivity function measured at a specified location in the control system. (See getSensitivity (Simulink Control Design) for information about sensitivity functions.) Use this syntax to narrow the scope of the tuning goal to a particular feedback loop.
If you want to constrain the poles of the system with one or more feedback loops opened, set the Openings property. To limit the enforcement of this tuning goal to poles having natural frequency within a specified frequency range, set the Focus property. (See Properties.)
Minimum decay rate of poles of tunable component, specified as a nonnegative scalar value in the frequency units of the control system model you are tuning.
When you tune the control system using this tuning goal, the closed-loop poles of the control system are constrained to satisfy:
Re(s) < -mindecay, for continuous-time systems.
log(|z|) < -mindecay*Ts, for discrete-time systems with sample time Ts.
Set mindecay = 0 to impose no constraint on the decay rate.
Desired minimum damping ratio of the closed-loop poles, specified as a value between 0 and 1.
Poles that depend on the tunable parameters are constrained to satisfy Re(s) < -mindamping*|s|. In discrete time, the damping ratio is computed using s=log(z)/Ts.
Set mindamping = 0 to impose no constraint on the damping ratio.
Desired maximum natural frequency of closed-loop poles, specified as a scalar value in the frequency units of the control system model you are tuning.
Poles are constrained to satisfy |s| < maxfreq for continuous time, or |log(z)| < maxfreq*Ts for discrete-time systems with sample time Ts. This constraint prevents fast dynamics in the closed-loop system.
Set maxfreq = Inf to impose no constraint on the natural frequency.
Location at which poles are assessed, specified as a character vector or cell array of character vectors that identify one or more locations in the control system to tune. When you use this input, the tuning goal constrains the poles of the sensitivity function measured at this location. (See getSensitivity (Simulink Control Design) for information about sensitivity functions.) What locations are available depends on what kind of system you are tuning:
If location specifies multiple locations, then the poles constraint applies to the sensitivity of the MIMO loop.
Minimum decay rate of closed-loop poles of tunable component, specified as a positive scalar value in the frequency units of the control system you are tuning. The initial value of this property is set by the mindecay input argument.
When you tune the control system using this tuning goal, closed-loop poles are constrained to satisfy Re(s) < -MinDecay for continuous-time systems, or log(|z|) < -MinDecay*Ts for discrete-time systems with sample time Ts.
You can use dot notation to change the value of this property after you create the tuning goal. For example, suppose Req is a TuningGoal.Poles tuning goal. Change the minimum decay rate to 0.001:
Req.MinDecay = 0.001;
Desired minimum damping ratio of closed-loop poles, specified as a value between 0 and 1. The initial value of this property is set by the mindamping input argument.
Desired maximum natural frequency of closed-poles, specified as a scalar value in the frequency units of the control system model you are tuning. The initial value of this property is set by the maxfreq input argument.
Poles of the block are constrained to satisfy |s| < maxfreq for continuous-time systems, or |log(z)| < maxfreq*Ts for discrete-time systems with sample time Ts. This constraint prevents fast dynamics in the tuned control system.
You can use dot notation to change the value of this property after you create the tuning goal. For example, suppose Req is a TuningGoal.ControllerPoles tuning goal. Change the maximum frequency to 1000:
Req.MaxFrequency = 1000;
Location at which poles are assessed, specified as a cell array of character vectors that identify one or more analysis points in the control system to tune. For example, if Location = {'u'}, the tuning goal evaluates the open-loop response measured at an analysis point 'u'. If Location = {'u1','u2'}, the tuning goal evaluates the MIMO open-loop response measured at analysis points 'u1' and 'u2'.
Create a requirement that constrains the inner loop of the following control system to be stable and free of fast dynamics. Specify that the constraint is evaluated with the outer loop open.
Create a model of the system. To do so, specify and connect the numeric plant models, G1 and G2, and the tunable controllers C1 and C2. Also, create and connect the AnalysisPoint blocks, AP1 and AP2, which mark points of interest for analysis and tuning.
Create a tuning requirement that constrains the dynamics of the closed-loop poles. Restrict the poles of the inner loop to the region
Re\left(s\right)<-0.1
|s|<30
Req = TuningGoal.Poles(0.1,0,30);
Setting the minimum damping to zero imposes no constraint on the damping constants for the poles.
Specify that the constraint on the tuned system poles is applied with the outer loop open.
Req.Openings = 'AP1';
When you tune T using this requirement, the constraint applies to the poles of the entire control system evaluated with the loop open at 'AP1'. In other words, the poles of the inner loop plus the poles of C1 and G1 are all considered.
After you tune T, you can use viewGoal to validate the tuned control system against the requirement.
Create a requirement that constrains the inner loop of the system of the previous example to be stable and free of fast dynamics. Specify that the constraint is evaluated with the outer loop open.
Create a tuning requirement that constrains the dynamics of the inner feedback loop, the loop identified by AP2. Restrict the poles of the inner loop to the region
Re\left(s\right)<-0.1
|s|<30
Req = TuningGoal.Poles('AP2',0.1,0,30);
When you tune T using this requirement, the constraint applies only to the poles of the inner loop, evaluated with the outer loop open. In this case, since G1 and C1 do not contribute to the sensitivity function at AP2 when the outer loop is open, the requirement constrains only the poles of G2 and C2.
TuningGoal.Poles restricts the closed-loop dynamics of the tuned control system. To constrain the dynamics or ensure the stability of a single tunable component, use TuningGoal.ControllerPoles.
For TuningGoal.Poles, f(x) reflects the relative satisfaction or violation of the goal. For example, if you attempt to constrain the closed-loop poles of a feedback loop to a minimum damping of ζ = 0.5, then:
f(x) = 1 means the smallest damping among the constrained poles is ζ = 0.5 exactly.
f(x) = 1.1 means the smallest damping ζ = 0.5/1.1 = 0.45, roughly 10% less than the target.
f(x) = 0.9 means the smallest damping ζ = 0.5/0.9 = 0.55, roughly 10% better than the target.
looptune | systune | looptune (for slTuner) (Simulink Control Design) | systune (for slTuner) (Simulink Control Design) | viewGoal | evalGoal | tunableTF | tunableSS | TuningGoal.ControllerPoles
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I have been very glad to hear how you are getting on. It really seems that you must trust to your own observations alone on stomata.2 May not the stomata be variable even in the same species. Such variation may be expected in all characters which differ much in allied species of the same genus;3 & if I remember right the stomata do differ in the species of the same genus.— It certainly looks as if Sachs’ view was largely right,4 (surely some of your former cases were opposed to his view) but I cannot understand the length of time which Mer found that leaves could exist in water, especially with Ivy.5 How are stomata in Ivy?— I found also that leaves of Mimosa & Trifolium resupinatum lived long submerged.—6 I think that you will come to some interesting results.—
Yesterday I made a little observation which interested me: I put Drosera under the compound microscope, fastening back of old leaf with shell-lac to stick, & a tentacle did not circumnutate in the least during 7
\frac{1}{2}
hours.; nor was it in the least heliotropic. I then touched secretion with atom of raw meat, not leaving any meat on gland, & in 23 seconds tentacle began to curve!7 I think we have observed enough to affirm that growth is always accompanied by circumnutation, & as the tentacle though so sensitive to animal matter did not circumnutate we may conclude that it was not growing; so Batalin must be wrong that the movement is partly due to growth.8
We see, also, how different such movements are to growth movements.— I wish you wd. make a note & enquire whether any Barbery bush in a pot is in flower in Kew; for I shd like to secure old flower to stick, & observe whether the sensitive stamens circumnutate.9 Lotus Jacobæus, I have just thought will be good to observe about pulvinus; for the cotyledon for first 4 or 5 days do not go to sleep, but do afterwards.10 Is pulvinus developed at first?
We are all well & fairly jolly, & all the jollier as Snow11 has gone.— Bernard12 is as sweet as sugar, but very contradictory. It grew wonderfully dark about half an hour ago; so I said “how dark it is”; so he shouted out “oh no”.— I then added I think it will soon rain, & he again shouted out “oh dear no” “oh dear no”.
I suppose the glass tubes will come tomorrow; but I won’t use them till your return.13
GoodBye dear old fellow | C. Darwin
The year is established by the relationship between this letter and the letter from Francis Darwin, [12 September 1878].
See letter from Francis Darwin, [12 September 1878].
See Origin 6th ed., pp. 119–20 (‘A Part developed in any Species in an extraordinary degree or manner, in comparison with the same Part in allied Species, tends to be highly variable.’
Julius von Sachs thought that bloom protected the stomata of plants from water (Sachs 1868b, p. 178; F. Darwin 1886, p. 99).
Émile Mer found that ivy leaves could survive several months under water, depending on conditions (see Mer 1876, pp. 243, 245, 247–52, 254, 255).
Trifolium resupinatum is Persian clover. CD’s son William had made some observations on submerged leaves; see Correspondence vol. 21, letter from W. E. Darwin, 30 August – 14 September [1873], Correspondence vol. 25, letter from W. E. Darwin, [12 or 19 July 1877].
See Movement in plants, p. 261.
Alexander Feodorowicz Batalin. See Batalin 1877; see also letter from Francis Darwin, [29 June] 1878.
Barberry bush, or Berberis; see Movement in plants, p. 132.
Lotus jacobaeus is the black-flowered lotus. See Movement in plants, p. 313.
Frances Julia Wedgwood, CD’s niece.
Bernard Darwin, Francis’s son.
See letter from Francis Darwin, [12 September 1878] and n. 2. Francis had sent glass tubes to be used in plant experiments.
Batalin, Alexander Feodorowicz. 1877. Mechanik der Bewegungen der insektenfressenden Pflanzen. Flora, oder allgemeine botanische Zeitung 35: 33–9, 54–8, 105–11, 129–54.
Darwin, Francis. 1886. On the relation between the ‘bloom’ on leaves and the distribution of the stomata. [Read 4 February 1886.] Journal of the Linnean Society (Botany) 22 (1885–6): 99–116.
Mer, Émile. 1876. Des effets de l’immersion sur les feuilles aériennes. [Read 14 July 1876.] Bulletin de la Société botanique de France 23: 243–58.
Julius von Sachs’s views on stomata seem largely correct, but CD cannot understand how leaves can survive submerged for such long periods.
Has been observing Drosera and concludes that none of the movement of the tentacles is caused by growth.
Suggests observations to show role of pulvinus in leaf movement.
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EUDML | Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces. EuDML | Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces.
Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces.
Alber, Yakov, Reich, Simeon, and Yao, Jen-Chih. "Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces.." Abstract and Applied Analysis 2003.4 (2003): 193-216. <http://eudml.org/doc/50482>.
author = {Alber, Yakov, Reich, Simeon, Yao, Jen-Chih},
keywords = {weakly contraction maps; descent-like approximation methods; proximal methods; convergence; nonexpansive maps; stability},
title = {Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces.},
AU - Alber, Yakov
AU - Reich, Simeon
AU - Yao, Jen-Chih
TI - Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces.
KW - weakly contraction maps; descent-like approximation methods; proximal methods; convergence; nonexpansive maps; stability
weakly contraction maps, descent-like approximation methods, proximal methods, convergence, nonexpansive maps, stability
A
Articles by Alber
Articles by Reich
Articles by Yao
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Complex Analysis - SEG Wiki
Complex analysis is that branch of mathematics which deals with quantities containing a term scaled by
{\displaystyle {\sqrt {-1}}}
. Extending the real numbers to the so-called complex numbers allows for the extension of the concepts of the limits of sequences and series and the principles of differential and integral calculus to complex valued functions. The payoff are a collection of methods that are useful in the fields of physics and signal processing.
Return to the Knowledge tree.
Retrieved from "https://wiki.seg.org/index.php?title=Complex_Analysis&oldid=57503"
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To J. D. Hooker 11–12 November [1856]1
I thank you more cordially that you will think probable, for your note.2 Your verdict has been a great relief.— On my honour I had no idea whether or not you would say it was (& I knew you would say it very kindly) so bad, that you would have begged me to have burnt the whole. To my own mind my M.S relieved me of some few difficulties, & the difficulties seemed to me pretty fairly stated, but I had become so bewildered with conflicting facts, evidence, reasoning & opinions, that I felt to myself that I had lost all judgment.— Your general verdict is incomparably more favourable than I had anticipated.
Very many thanks for your invitation: I had made up my mind on my poor wifes account not to come up to next Phil. Club; but I am so much tempted by your invitation, & my poor dear wife is so goodnatured about it, that I think I shall not resist, ie if she does not get worse.— I wd. come to dinner at about same time as before, if that wd suit you & I do not hear to contrary, & wd. go away by the early train ie about 9 olock.— I find my present work tries me a good deal & sets my heart palpitating, so I must be careful.— But I shd. so much like to see Henslow, & likewise meet Lindley if the fates will permit.3 You will see, whether there will be time for any criticism in detail on my M.S. before dinner. Not that I am in the least hurry, for it will be months before I come again to Geograph. Distrib.; only I am afraid of your forgetting any remarks.—
I do not know whether my very trifling observations on means of distribution are worth your reading, but it amuses me to tell them.
The seeds which the Eagle had in stomach for 18 hours looked so fresh that I would have bet 5 to 1 they would all have grown; but some kinds were all killed & 2 oats 1 Canary seed, 1 Clover & 1 Beet alone came up! now I shd. have not cared swearing that the Beet wd. not have been killed, & I shd have fully expected that the Clover would have been.— These seeds, however, were kept for 3 days in moist pellets damp with gastric juice after being ejected which would have helped to have injured them.—4
Lately I have been looking during few walks at excrement of small birds; I have found 6 kinds of seeds, which is more than I expected. Lastly I have had a partride with 22 grains of dry earth on one foot, & to my surprise a pebble as big as a tare seed; & I now understand how this is possible for the bird scartches itself, & little plumose feathers make a sort of very tenacious plaister. Think of the millions of migratory quails, & it wd. be strange if some plants have not been transported across good arms of the sea.—5
Talking of this, I have just read your curious Raoul Isd paper:6 this looks more like a case of continuous land, or perhaps of several intervening, now lost, islands, than any, (according to my heteredox notions) I have yet seen; the concordance of the vegetation seems so complete with New Zealand & with that land alone.
I have read Salters paper, & can hardly stomach it: I wonder whether the lighters were ever used to carry grain & Hay to ships?—7
Adios, my dear Hooker, I thank you most honestly for your assistance,—assistance by the way now spread over some dozen years.—
P.S. Wednesday
I see from my wife’s expression that she does not really much like my going, & therefore I must give up of course this pleasure.— If you shd. have anything to discuss about my M.S. I see that I cd. get to you by about 12, & then cd. return by the 2o 19’ olock train & be home by 5
\frac{1}{2}
oclock, & thus I shd. get 2 hours talk.— But it would be a considerable exertion for me, & I would not undertake it for mere pleasure sake, but would very gladly for my Book’s sake.—
Dated by the relationship to the letter from J. D. Hooker, 9 November 1856.
Letter from J. D. Hooker, 9 November 1856.
See letter from J. D. Hooker, 9 November 1856, in which Hooker invited CD to dinner on Wednesday, 12 November, to meet John Lindley and John Stevens Henslow, or on Friday 14 November, to meet John Tyndall and Henslow. CD attended neither dinner but did go up to London on 13 November (see letter to George Howard Darwin and W. E. Darwin, 13 [November 1856]).
See letter to J. D. Hooker, [19 October 1856] and n. 2.
CD recorded this case on 19 October 1856 in his Experimental book, p. 15 (DAR 157a). Following the entry, CD added: ‘(Nov. 13th. Nothing came up.)’.
J. D. Hooker 1857.
James Salter had reported that mud scraped from the bottom of Poole harbour in 1843 and deposited on the shore eventually gave rise to a vegetation different from that of the surrounding area (J. Salter 1857).
Salter, John William. 1857. On some new Palæozoic star-fishes. Annals and Magazine of Natural History 2d ser. 20: 321–334.
CD relieved by JDH’s positive response to his MS.
CD continues observations on means of transport.
JDH’s Raoul Island paper [J. Linn. Soc. Lond. (Bot.) 22 (1857): 133–41], showing continuity of vegetation with New Zealand, best evidence yet of continental extension.
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Topological group - zxc.wiki
In mathematics , a topological group is a group that has a topology that is “compatible” with the group structure . The topological structure allows, for example, to consider limit values in this group and to speak of continuous homomorphisms .
A group is called a topological group if it is provided with a topology such that:
{\ displaystyle G}
The group link is continuous. In this case, with the product topology provided.
{\ displaystyle G \ times G \ to G}
{\ displaystyle G \ times G}
The inverse mapping is continuous.
{\ displaystyle G \ to G}
The real numbers with addition and the ordinary topology form a topological group. More generally, the -dimensional Euclidean space with vector addition and standard topology is a topological group. Every Banach space and Hilbert space is also a topological group with regard to addition.
{\ displaystyle \ mathbb {R}}
{\ displaystyle n}
{\ displaystyle \ mathbb {R} ^ {n}}
The above examples are all Abelian . An important example of a nonabelian topological group is the group of all invertible real matrices. The topology is created by understanding this group as a subset of the Euclidean vector space .
{\ displaystyle \ operatorname {GL} (n, \ mathbb {R})}
{\ displaystyle n \ times n}
{\ displaystyle \ mathbb {R} ^ {n ^ {2}}}
{\ displaystyle \ mathbb {R} ^ {n}}
is just like a Lie group , i.e. a topological group where the topological structure is that of a manifold .
{\ displaystyle \ operatorname {GL} (n, \ mathbb {R})}
An example of a topological group that is not a Lie group is the additive group of rational numbers (it is a countable set that is not provided with the discrete topology ). A non-Abelian example is the subgroup of the rotation group of which is generated by two rotations around irrational multiples of (the circle number Pi) around different axes.
{\ displaystyle \ mathbb {Q}}
{\ displaystyle \ mathbb {R} ^ {3}}
{\ displaystyle \ pi}
In every unitary Banach algebra , the set of invertible elements with the multiplication forms a topological group.
The algebraic and topological structure for a topological group are closely related. For example, in any topological group, the connected component of the neutral element is a closed normal subgroup of .
{\ displaystyle G}
{\ displaystyle G}
If an element is a topological group , then the left multiplication and the right multiplication with homeomorphisms are from to , as is the inverse mapping.
{\ displaystyle a}
{\ displaystyle G}
{\ displaystyle a}
{\ displaystyle G}
{\ displaystyle G}
Each topological group can be understood as a uniform space . Two elementary uniform structures that result from the group structure are the left and the right uniform structure . The left uniform structure makes the left multiplication uniformly continuous , the right uniform structure makes the right multiplication uniformly continuous. For non-Abelian groups, these two uniform structures are generally different. The uniform structures make it possible in particular to define terms such as completeness, uniform continuity and uniform convergence.
Like any topology created by a uniform space, the topology of a topological group is completely regular . In particular, a topological group which satisfies (i.e., which is a Kolmogoroff space) is even a Hausdorff space .
{\ displaystyle T_ {0}}
The most natural notion of homomorphism between topological groups is that of continuous group homomorphism . The topological groups together with the continuous group homomorphisms form a category .
Each subgroup of a topological group is in turn a topological group with the subspace topology . For a subgroup of , the left and right secondary classes together with the quotient topology form a topological space.
{\ displaystyle H}
{\ displaystyle G}
{\ displaystyle G / H}
If is a normal divisor of , then becomes a topological group. It should be noted, however, that if is not closed in the topology of , the resulting topology is not Hausdorffian. It is therefore natural, if one restricts oneself to the category of Hausdorff topological groups, to examine only closed normal subdivisions.
{\ displaystyle H}
{\ displaystyle G}
{\ displaystyle G / H}
{\ displaystyle H}
{\ displaystyle G}
{\ displaystyle G / H}
If is a subgroup of , then the closed envelope of is in turn a subgroup. Likewise, the conclusion of a normal divider is normal again.
{\ displaystyle H}
{\ displaystyle G}
{\ displaystyle H}
Lev Pontryagin : Topological Groups. 2 volumes. Teubner, Leipzig 1957–1958.
Guido Mislin (Ed.): The Hilton symposium 1993. Topics in Topology and Group Theory (= CRM Proceedings & Lecture Notes. Vol. 6). American Mathematical Society, Providence RI 1994, ISBN 0-8218-0273-9 .
Terence Tao : Hilbert's fifth problem and related topics (= Graduate Studies in Mathematics. Vol. 153). American Mathematical Society, Providence RI 2014, ISBN 978-1-4704-1564-8 online .
This page is based on the copyrighted Wikipedia article "Topologische_Gruppe" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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NCERT Solutions for Class 11 Commerce Economics Chapter 2 - Collection Of Data
NCERT Solutions for Class 11 Commerce Economics Chapter 2 Collection Of Data are provided here with simple step-by-step explanations. These solutions for Collection Of Data are extremely popular among Class 11 Commerce students for Economics Collection Of Data Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the NCERT Book of Class 11 Commerce Economics Chapter 2 are provided here for you for free. You will also love the ad-free experience on Meritnation’s NCERT Solutions. All NCERT Solutions for class Class 11 Commerce Economics are prepared by experts and are 100% accurate.
Which of the following newspaper/s do you read regularly?
There are many sources of data (true/false).
There are many sources of data. False
Telephone survey is the most suitable method of collecting data, when the population is literate and spread over a large area (true/false).
Telephone survey is the most suitable method of collecting data when the population is literate and spread over a large area. False
Data collected by investigator is called the secondary data (true/false).
Data collected by an investigator is called the secondary data. False
The data collected by an investigator is called the primary data, whereas, the data that is already in existence being collected by any other investigator is known as secondary data.
There is a certain bias involved in the non-random selection of samples (true/false).
There is a certain bias involved in the non-random selection of samples. True
Non-sampling errors can be minimised by taking large samples (true/ false).
Non-sampling errors can be minimised by taking large samples. False
If plastic bags are only 5 percent of our garbage, should it be banned?
The particular question, ‘If plastic bags are only 5 percent of our garbage, should it be banned?’ is too long which discourages people to complete the questionnaire.
(a) Do you agree with the use of chemical fertilisers?
The chronological order of the questions asked is incorrect. The order lacks the direction of causation. It should move from general to specific that makes the respondents easy and comfortable
Wouldn’t you be opposed to increase in price of petrol?
Although the answer to this question needs a wider view and knowledge of the economic condition and far reaching effect of rise in petrol price, but the majority of people will argue against such rise. Thus, more effective question could be:
How far do you live from the closest market?
This question is ambiguous. The respondents will not be able to answer the question correctly. The correct question should be:
You want to do a research on the popularity of Vegetable Atta Noodles among children. Design a suitable questionnaire for collecting this information.
Popularity of Vegetable Atta Noodles
Name ………………………… Age …………………….
Address ……………………… Sex: Male Female
2. Do you find this reasonable?
3. How many packets do you consume in a month?
(a) 1 – 2 Packets
(b) 2 – 3 Packets
(c) 3 – 6 Packets
(d) More than 6 packets
4. Do you prefer Atta noodles over Maida noodles?
5. Which Vegetables according to you should be added in present Atta noodles?
6. Do you think it should be spicier?
7. At what time of the day do you prefer the Atta noodles the most?
8. Do your parents accompany you while having noodles?
In a village of 200 farms, a study was conducted to find the cropping pattern. Out of the 50 farms surveyed, 50% grew only wheat. Identify the population and the sample here.
Population refers to the aggregate or the total items to be studied for an investigation. So, the population here is 200 farms.
Sample is the subset of the population. In other words, a small set selected from the population for statistical study is referred as sample population.
Out of 200 farms, only 50 farms are selected for survey; therefore, the sample population is 50 farms.
Give two examples each of sample, population and variable.
Sample is the subset of the population. In other words, a small set selected from the population for statistical study is referred as sample population. For example, in order to study the growth pattern of students, the heights of 50 students (sample) are recorded from a school of 500 students (population). Similarly, in order to record the level of sugar in the blood, blood sample of 2000 people (sample) was taken from 20,000 people (population).
Population refers to the aggregate or the total items to be studied for an investigation. In the above examples, 500 students and 20,000 people constitute the population.
Variables are the characteristics of a sample or population that can be expressed in numbers such as, height, income, age, etc.
Non-sampling Errors are more serious than the Sampling Errors because the latter can be minimised by taking a larger sample. Non-sampling Errors emerge due to the use of faulty means of collection of data, whereas, the Sampling Errors emerge due to the divergence between the estimated and the actual value of a parameter of a small sized sample population. For example, errors due to personal biasness, misinterpretation of results, miscalculations, etc.
The Sampling Errors can be minimised by increasing the size of a small sample, so that the difference between the actual and the estimated value is reduced. But the Non-sampling Errors are difficult to rectify as it would require selection of a new sample and conducting a fresh survey. Thus, Non-sampling Errors are more serious than the Sampling Errors.
Suppose there are 10 students in your class. You want to select three out of them. How many samples are possible?
\begin{array}{rcl}\mathrm{Number} \mathrm{of} \mathrm{possible} \mathrm{samples}& =& {}^{n}C_{r}={}^{10}C_{3}\\ & =& \frac{10!}{3! \left(10-3\right)!}\\ & =& \frac{10!}{3! 7!}\\ & =& 120\end{array}
Discuss how you would use the lottery method to select 3 students out of 10 in your class?
The following method can be used while selecting 3 students out of 10 of the class.
(i) Make ten paper slips of equal size.
Yes, the lottery method always gives a random sample’s outcome. In a random sample, each individual unit has an equal chance of getting selected. Similarly, in a lottery method, each individual unit is selected at random from the population and thereby has equal opportunity of getting selected. For example, in order to select a student as monitor, the slips containing the names of all the students are mixed well, and then a slip is drawn out at random. In this case, all students of the class have equal chance of getting selected. The probability of a student getting selected through the lottery method is exactly same as the probability of any one student randomly selected.
Explain the procedure of selecting a random sample of 3 students out of 10 in your class, by using random number tables.
The procedure of selecting random sample of 3 students out of 10 in a class is as follows:
Do samples provide better results than surveys? Give reasons for your answer.
Samples provide better results than surveys. The advantages of samples over survey are as follows:
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Altitude - zxc.wiki
Flying altitude is the measured vertical altitude of an aircraft over a certain reference surface. Depending on the flight situation, different reference areas come into consideration. The exact measurement of the flight altitude is of great importance for flight safety in order to be able to ensure sufficient safety distances to other aircraft and ground obstacles.
1 Measurement of flight altitude
1.1 Reference earth surface: height (HGT)
1.2 Reference sea level: altitude (ALT)
1.3 Relation to normal pressure: flight levels (FL)
2 Unit of flight altitude
3 Other measurement methods
4 typical flight altitudes
6 Visibility to the horizon
Measurement of flight altitude
Main article : Barometric altitude measurement in aviation
This altimeter shows an altitude of 10,180 feet (3,340 m) (thin pointer: tens of thousands, short pointer: thousands, long pointer: hundreds). The reference pressure is set using the rotary knob at the bottom left and displayed in the small window on the right (here standard pressure 29.92 inches Hg ). Because of the setting to standard pressure, the display can also be interpreted as flight level 102 (approximately).
The flying height is in principle with the barometric working altimeter (engl. Altimeter ), whereby one advantage of the fact makes that the air pressure decreases with increasing altitude. However, because firstly different reference surfaces are used for altitude measurement and secondly the air pressure changes not only with altitude but also with the weather, altimeters always have a setting option for the reference pressure (the pressure at which they would display zero altitude).
The following reference surfaces for altitude measurement are used in aviation :
Reference earth surface: height (HGT)
It makes sense to use the ground as the reference surface for the height measurement, for example to be able to maintain safety heights above the terrain and ground obstacles. The flight height above ground (abbreviated GND for ground or SFC for surface ) is called height (HGT). In flying practice, this altitude only plays a role in special cases. During a cross-country flight, the height of the underlying terrain changes far too quickly for such a measurement to be useful. The reference pressure ( QFE ) would very often have to be adapted to the terrain height and would also be dependent on the weather that has just flown through.
Reference sea level: altitude (ALT)
If the flight altitude is to be determined independently of the altitude of the overflown area, it is expressed as altitude above sea level (MSL, mean sea level ). Obstacle heights on flight maps are also given in feet above MSL, so that safety heights can be maintained. The altitude in relation to MSL is called altitude (ALT). The reference pressure to be set for this, i.e. the current air pressure value converted to sea level, is referred to as QNH . It is set at least before departure; for cross-country flights at low altitudes, the setting must be regularly adjusted to local changes in air pressure due to weather conditions. In flight, the current value can be obtained from flight weather reports or landing information from airports.
Relation to normal pressure: flight levels (FL)
In the unobstructed airspace, the actual height is no longer of interest, since sufficient distances to ground obstacles are always given. It is much more important here to use a reference value that is also independent of the weather in order to ensure that all aircraft measure their altitude according to the same reference pressure and that vertical distances from one another can be reliably maintained. The normal pressure of 1013.25 hPa (corresponds to 29.92 mm Hg ) was set as this uniform reference pressure . The altimeter is set to this reference value when climbing the so-called transition height - regardless of the actual air pressure. The measured value of the altitude is no longer referred to as a height, but (divided by 100) and flight level (FL, flight level ): FL 120 corresponds to an altimeter display of 12,000 feet (3658 m) with respect to atmospheric pressure. In descent, the altimeter is set back to the current meteorological value when the transition area sinks .
The display of a barometric altimeter follows the standard atmosphere . Since the relationship between pressure and altitude also (slightly) depends on the temperature and water vapor content of the air, the displayed flight altitudes rarely correspond exactly to the actual values. However, since this display error is minimal and also turns out to be the same for all aircraft, this is not critical.
Unit of flight altitude
It is an international practice to indicate flight altitudes in feet (ft). The designation of the flight areas comes from the displayed height in feet. In a few countries all flight altitudes are expressed in meters . In Germany, altitude measurements in meters are only common for gliders , airships and parachutists . 100 ft corresponds to 30.48 m.
Under certain circumstances, a direct (and accurate) height measurement above the ground (HGT) is performed by radio signal. In the landing approach with commercial aircraft , for example, the direct altitude determination by radio altitude measurement (radar altimeter) is used as additional information. The decision height is only measured using a radar altimeter for ILS categories (CAT) II and III .
In addition, the flight altitude can also be determined from the ground by means of radar and transmitted to the pilot by radio, but mostly only through military radar systems.
Typical altitudes
Flying to enjoy the landscape: a silent paraglider a few hundred meters above the valley floor
Commercial aircraft prefer to fly at or above the tropopause above the weather
The following overview, at which flight altitudes which aircraft are located, does not represent any regulations or fixed rules, but only serves to give the layman an approximate idea. For a better overview, the heights are given in meters above the ground (GND) and only apply to a limited extent at higher altitudes.
Height above GND
Objects in the air
0 m to 100 m Birds, bats , insects , kites ; tied balloons (also zeppelin-shaped) for advertising, lighting, cameras
150 m to 1500 m Air sports equipment , hang gliders , paragliders , hot air balloons , helicopters , airships
1500 m to 3000 m Small aircraft in cruise flight, gliders in cross-country flight, commercial aircraft in holding patterns for landing
3000 m to 5000 m Jump by parachutists (usually 4000 m), business aviation , some migratory birds
5000 m to 10000 m Business aviation , jet planes and turbo- prop planes in cruise flight ( FL 150 to FL 290)
10,000 m to 15,000 m Jet airliners in cruise flight ( FL 300 to FL 450)
15000 m to> 18000 m Supersonic passenger aircraft such as the Concorde and the Tupolev Tu-144 . Very light, unmanned, solar- powered pseudo-satellites ( Airbus Zephyr ).
The minimum safety height for aircraft, which may only be undercut during take-off or landing, is in Germany:
generally 150 m (500 ft) GND
over towns or large gatherings of people 300 m (1000 ft) above the highest obstacle in a 600 m radius
The human organism is adapted to life on the ground. The air pressure conditions encountered while flying can therefore become problematic:
The pressure of the middle ear must be constantly adjusted to the external pressure in order to keep the eardrum relaxed. The Eustachi tube is responsible for this and is opened when swallowing. If this tube is swollen shut, for example as a result of an illness, the internal pressure of the ear cannot follow the changes in external pressure when flying, in particular the rapid increase in pressure when descending, whereupon the higher external pressure tensions the eardrum inward and causes severe pain. The Valsalva maneuver can help. Chewing and sucking also help to open the eustachi tube as often as possible.
From altitudes of 2600 to 3280 m MSL, the density of the breathing air becomes so low that a reliable supply of the (untrained) human organism with oxygen can no longer be expected and either oxygen devices or pressurized cabins have to be used. In particularly sensitive people, oxygen deficiency symptoms can occur even at 1640 m MSL (which roughly corresponds to the height of the Feldberg in the Black Forest ) .
Visibility to the horizon
As the flight altitude increases, the horizon observed from the aircraft moves further and further away . If an aircraft is flying over the sea or over a flat area, such as the Kalahari , the visibility is calculated as follows:
{\ displaystyle s}
{\ displaystyle s \ approx 3 {,} 6 \, {\ text {km}} \ cdot {\ sqrt {h / {\ text {m}}}}}
Here the altitude must be entered in meters . The distance is expressed in kilometers . If an airplane flies 900 meters above flat ground, the horizon is 108 kilometers away. Accordingly, the visibility from a height of 10,000 meters is around 360 kilometers.
{\ displaystyle h}
{\ displaystyle s}
is the height above GND (ground) ( AGL ), i.e. above the ground. With height also z. B. specified the height of a tower.
is the height above MSL or the reference height (ELEV + HGT = ALT).
elevation (ELEV)
refers to the height of the ground (GND) above mean sea level (MSL). For airfields, the information refers to the highest point on the ground in the landing area of the airfield.
(German flight level ) is the hundredth of the altitude in feet, which corresponds to the air pressure currently measured by the altimeter above the standard atmosphere of 1013.25 hPa . A flight level therefore corresponds to an isobar . The altimeter is for displaying flight level set height measurements by the reference pressure to the standard air pressure (engl. Standard pressure level ) of 1013.25 hPa is set.
(German transition height) refers to a height in Germany that is 1640 m above MSL, but at least 656 m above GND. This is achieved in a climb . Donkey Bridge: The “A” in “Altitude” resembles an arrow pointing upwards. This is where the flight levels begin , so the pilot has to switch the altimeter to the standard air pressure of 1013.25 hPa. In other countries, the flight levels can start at lower values. The transition altitude for the respective airfield is noted on IFR cards.
is the flight level above the transition altitude at which the altimeter is switched back to QNH during descent in order to display the actual flight altitude for the landing approach. Donkey bridge: The “L” in “Level” resembles an arrow pointing downwards (with a lot of imagination). The pilot receives the current QNH value via radio from air traffic control / flight control , from an automatic announcement ( ATIS ) or from an airport in his vicinity. There is a distance of at least 328 m between the transition altitude and the transition level ( transition layer ). From this we can deduce: QNH greater than 1013 hPa: TL 60, QNH from 983 to 1013 hPa: TL 70, QNH less than 983 hPa: TL 80.
refers to the difference in altitude between the transition altitude and the transition level . This difference in altitude of mostly 328 m is required by larger aircraft for handling , i.e. completing cockpit work such as QNH adjustment etc.
Dieter Franzen - compact learning program to prepare for the radio communication test AZF 1991
Jeppesen Sanderson: Private Pilot Study Guide 2000, ISBN 0-88487-265-3
Jeppesen Sanderson: Privat Pilot Manual 2001, ISBN 0-88487-238-6
Walter Air: CVFR textbook Mariensiel 2001
Wolfgang Kühr - The private pilot , aviation law, air traffic and air traffic control regulations, Volume 5 1983 ISBN 3-921-270-13-8
Semicircle flight rule
http://www.xflight.de/pg_org_par_cec_altimeter.htm (cockpit)
This page is based on the copyrighted Wikipedia article "Flugh%C3%B6he" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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As you can tell from the examples of the number lines below, not all number lines change by one unit from mark to mark. Copy these number lines onto your paper and fill in the missing numbers.
Find the difference between two marks on the number line.
The numbers grow by
2
units. Copy the number line on your paper and label the other marks.
5
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Functional analysis - zxc.wiki
The functional analysis is the branch of mathematics that deals with the study of infinite-dimensional topological vector spaces involved and illustrations on such. Here analysis , topology and algebra are linked. The aim of these investigations is to find abstract statements that can be applied to various concrete problems. Functional analysis is the suitable framework for the mathematical formulation of quantum mechanics and for the investigation of partial differential equations .
3 standardized spaces, Banach spaces
4 operators, Banach algebras
The terms are of central importance
Functional for mapping vectors (e.g. functions ) to scalar sizes and
Operator for mapping vectors to vectors. The concept of the operator is actually much more general. However, it makes sense to look at them in algebraically and topologically structured spaces, such as B. topological, metric or standardized vector spaces of all kinds.
Examples of functional are the terms sequence limit value , norm , definite integral or distribution , examples of operators are differentiation , indefinite integral , quantum mechanical observable or shift operators for sequences.
Basic concepts of analysis such as continuity , derivatives , etc. are extended in functional analysis to functionals and operators. At the same time, the results of linear algebra (for example the spectral theorem ) are expanded to include topologically linear spaces (for example Hilbert spaces ), which is associated with very significant results.
The historical roots of functional analysis lie in the study of the Fourier transformation and similar transformations and the investigation of differential and integral equations . The word component “functional” goes back to the calculus of variations . Stefan Banach , Frigyes Riesz and Maurice René Fréchet are considered the founders of modern functional analysis .
Main articles : Topological vector space and locally convex space
Functional analysis is based on vector spaces over real or complex numbers. The basic concept here is the topological vector space, which is characterized by the fact that the vector space links are continuous; locally convex topological vector spaces and Fréchet spaces are also examined more specifically . Important statements are the Hahn-Banach theorem , the Baire theorem and the Banach-Steinhaus theorem . In particular in the solution theory of partial differential equations , these play an important role, moreover in the Fredholm theory .
Standardized spaces, Banach spaces
Main articles : Normalized space and Banach space
The most important special case of locally convex topological vector spaces are normalized vector spaces . If these are also complete , then they are called Banach spaces . Hilbert spaces are considered even more specifically , in which the norm is generated by a scalar product . These spaces are fundamental to the mathematical formulation of quantum mechanics . An important subject of investigation are continuous linear operators on Banach or Hilbert spaces.
Hilbert spaces can be fully classified: for every thickness of an orthonormal basis there is exactly one Hilbert space for a body (except for isomorphism ) . Since finite-dimensional Hilbert spaces are covered by linear algebra and every morphism between Hilbert spaces can be decomposed into morphisms of Hilbert spaces with a countable orthonormal basis, one considers mainly Hilbert spaces with a countable orthonormal basis and their morphisms in functional analysis. These are isomorphic to the sequence space of all sequences with the property that the sum of the squares of all sequence members is finite.
{\ displaystyle \ ell ^ {2}}
Banach spaces, on the other hand, are much more complex. For example, there is no general definition of a basis that can be used in practice, so bases of the type described under basis (vector space) (also called Hamel basis ) cannot be given constructively in the infinite-dimensional case and are always uncountable (see Baire's theorem ). Generalizations of the Hilbert space orthonormal bases lead to the concept of the shudder base , but not every Banach space has one.
For every real number there is the Banach space "of all Lebesgue-measurable functions whose -th power of the amount has a finite integral" (see L p -space ), this is exactly for a Hilbert space.
{\ displaystyle p \ geq 1}
{\ displaystyle p}
{\ displaystyle p = 2}
When studying standardized spaces, it is important to examine the dual space . The dual space consists of all continuous linear functions from the normalized space into its scalar body , i.e. into the real or complex numbers. The bidual , i.e. the dual space of the dual space, does not have to be isomorphic to the original space, but there is always a natural monomorphism of a space in its bidual. If this special monomorphism is also surjective , then one speaks of a reflexive Banach space .
The term derivation can be generalized to functions between Banach spaces to the so-called Fréchet derivation , so that the derivation at one point is a continuous linear mapping.
Operators, Banach algebras
Main articles : Banach algebra and C * algebra
While the Banach spaces or Hilbert spaces represent generalizations of the finite-dimensional vector spaces of linear algebra, the continuous, linear operators between them generalize the matrices of linear algebra. The diagonalization of matrices, which a matrix tries to represent as a direct sum of stretchings of so-called eigenvectors , expands to the spectral theorem for self-adjoint or normal operators on Hilbert spaces, which leads to the mathematical formulation of quantum mechanics . The eigenvectors form the quantum mechanical states, the operators the quantum mechanical observables .
As products of operators are operators again, we obtain algebras of operators with the operator norm are Banach spaces, allowing for two operators and the multiplicative triangle inequality holds. This leads to the concept of Banach algebra , the most accessible representatives of which are the C * algebras and Von Neumann algebras .
{\ displaystyle A}
{\ displaystyle B}
{\ displaystyle \ | A \ circ B \ | \ leq \ | A \ | \ | B \ |}
For the investigation of locally compact groups one uses the Banach space of the functions integrable with respect to the hair measure , which becomes a Banach algebra with the convolution as multiplication. This justifies the harmonic analysis as a functional analytical approach to the theory of locally compact groups; The Fourier transformation results from this point of view as a special case of the Gelfand transformation examined in the Banach algebra theory .
{\ displaystyle G}
{\ displaystyle L ^ {1} (G)}
Main article : Partial differential equation
Functional analysis offers a suitable framework for the solution theory of partial differential equations. Such equations often have the form where the function sought and the right hand side are functions in a domain and is a differential expression. In addition, there are so-called boundary conditions that prescribe the behavior of the function sought on the boundary of . An example of such a differential expression is the Laplace operator ; other important examples result from the wave equation or the heat conduction equation .
{\ displaystyle you \, = f}
{\ displaystyle u}
{\ displaystyle f}
{\ displaystyle \ Omega \ subset \ mathbb {R} ^ {n}}
{\ displaystyle D}
{\ displaystyle u}
{\ displaystyle \ partial \ Omega}
{\ displaystyle \ Omega}
{\ displaystyle D = {\ frac {\ partial ^ {2}} {\ partial x_ {1} ^ {2}}} + \ dotsb + {\ frac {\ partial ^ {2}} {\ partial x_ {n } ^ {2}}}}
The differential expression is now viewed as an operator between spaces of differentiable functions, in the example of the Laplace operator, for example, as an operator between the space of twice continuously differentiable functions and the space of continuous functions . Such spaces of function spaces that can be differentiated in the classical sense turn out to be unsuitable for an exhaustive solution theory. By moving to a more general concept of differentiability ( weak derivative , distribution theory ), one can view the differential expression as an operator between Hilbert spaces, so-called Sobolew spaces , which consist of suitable L 2 functions. In this context, in important cases, satisfactory theorems about the existence and uniqueness of solutions can be proven. For this purpose, questions such as the dependence on the right side , as well as questions about the regularity, i.e. smoothness properties of the solution depending on the smoothness properties of the right side , are investigated using functional analytical methods. This can be further generalized to more general room classes, such as rooms of distributions. If the right-hand side is the same as the delta distribution and a solution has been found for this case, a so-called fundamental solution, in some cases solutions for any right-hand side can be constructed using convolution .
{\ displaystyle \ Omega}
{\ displaystyle f}
{\ displaystyle u}
{\ displaystyle f}
{\ displaystyle f}
In practice, numerical methods are used to approximate solutions of such differential equations, such as the finite element method , especially when no solution can be given in closed form. Functional analytical methods also play an essential role in the construction of such approximations and the determination of the approximation quality .
Hans Wilhelm Alt : Linear functional analysis: an application-oriented introduction . 5th edition. Springer-Verlag, 2006, ISBN 3-540-34186-2 , doi: 10.1007 / 3-540-34187-0 .
Haïm Brézis : Analysis fonctionnelle: théorie et applications . In: Mathématiques appliquées pour la maîtrise . Dunod, 2005, ISBN 2-10-049336-1
Nelson Dunford , Jacob T. Schwartz, et al. a .: Linear Operators, General Theory, and other 3 volumes, includes visualization charts . In: Pure and applied mathematics; 7 . Wiley-Interscience, 1988, ISBN 0-470-22605-6
Harro Heuser : Functional Analysis: Theory and Application . 3. Edition. Teubner-Verlag, 1992, ISBN 3-519-22206-X
Friedrich Hirzebruch , Winfried Scharlau : Introduction to Functional Analysis , BI, Mannheim 1971, ISBN 978-3-411-00296-2 , online in the Hirzebruch Collection .
Vivien Hutson, John S. Pym, Michael J. Cloud: Applications of Functional Analysis and Operator Theory . 2nd edition. Elsevier Science, 2005, ISBN 0-444-51790-1
Leonid P. Lebedev, Iosif I. Vorovič: Functional Analysis in Mechanics . Springer-Verlag, 2003, ISBN 0-387-95519-4
R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , doi: 10.1007 / 978-3-322-80310-8
Martin Schechter : Principles of Functional Analysis . 2nd edition. Academic Press, 2001, ISBN 0-8218-2895-9
Sergei Lwowitsch Sobolew : Some applications of functional analysis in mathematical physics , Providence (RI) , American Mathematical Soc., 1991, ISBN 0-8218-4549-7
Dirk Werner : Functional Analysis . 7th edition. Springer , 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .
Kôsaku Yosida : Functional Analysis . 6th edition. Springer-Verlag, 1980, ISBN 3-540-10210-8
The books Alt (2006) and Heuser (1992) offer an introduction and a first overview of “classical” sentences of functional analysis. In doing so, physical applications are repeatedly discussed as a common thread. Heuser has exercises for each chapter, most of which are outlined in the appendix. The last chapter “A look at the emerging analysis” describes the most important steps in the historical development towards today's functional analysis.
Commons : functional analysis - collection of images, videos and audio files
This page is based on the copyrighted Wikipedia article "Funktionalanalysis" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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EUDML | Elliptic spaces with the rational homotopy type of spheres. EuDML | Elliptic spaces with the rational homotopy type of spheres.
Elliptic spaces with the rational homotopy type of spheres.
Powell, Geoffrey M.L.
Powell, Geoffrey M.L.. "Elliptic spaces with the rational homotopy type of spheres.." Bulletin of the Belgian Mathematical Society - Simon Stevin 4.2 (1997): 251-263. <http://eudml.org/doc/119890>.
author = {Powell, Geoffrey M.L.},
keywords = {elliptic Hopf algebra; localization; -elliptic spaces; -elliptic spaces; -local homotopy type; -elliptic spaces; -elliptic spaces; -local homotopy type},
title = {Elliptic spaces with the rational homotopy type of spheres.},
AU - Powell, Geoffrey M.L.
TI - Elliptic spaces with the rational homotopy type of spheres.
KW - elliptic Hopf algebra; localization; -elliptic spaces; -elliptic spaces; -local homotopy type; -elliptic spaces; -elliptic spaces; -local homotopy type
elliptic Hopf algebra, localization,
p
-elliptic spaces,
ℚ
p
-local homotopy type,
p
ℚ
p
-local homotopy type
Articles by Powell
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EUDML | About some infinite family of 2-bridge knots and 3-manifolds. EuDML | About some infinite family of 2-bridge knots and 3-manifolds.
About some infinite family of 2-bridge knots and 3-manifolds.
Kim, Yangkok
Kim, Yangkok. "About some infinite family of 2-bridge knots and 3-manifolds.." International Journal of Mathematics and Mathematical Sciences 24.2 (2000): 95-108. <http://eudml.org/doc/48785>.
author = {Kim, Yangkok},
keywords = {branched coverings; maximally symmetric manifolds; cyclically presented groups; 2-bridge knots},
title = {About some infinite family of 2-bridge knots and 3-manifolds.},
AU - Kim, Yangkok
TI - About some infinite family of 2-bridge knots and 3-manifolds.
KW - branched coverings; maximally symmetric manifolds; cyclically presented groups; 2-bridge knots
branched coverings, maximally symmetric manifolds, cyclically presented groups, 2-bridge knots
3
|
EUDML | Some results on congruences on semihypergroups. EuDML | Some results on congruences on semihypergroups.
Some results on congruences on semihypergroups.
Davvaz, Bijan. "Some results on congruences on semihypergroups.." Bulletin of the Malaysian Mathematical Sciences Society. Second Series 23.1 (2000): 53-58. <http://eudml.org/doc/49676>.
author = {Davvaz, Bijan},
keywords = {semihypergroups; -semigroups; homomorphisms; congruences; -semigroups},
title = {Some results on congruences on semihypergroups.},
AU - Davvaz, Bijan
TI - Some results on congruences on semihypergroups.
KW - semihypergroups; -semigroups; homomorphisms; congruences; -semigroups
semihypergroups,
{H}_{v}
-semigroups, homomorphisms, congruences,
{H}_{v}
|
MARVELOUS MARK'S FUNCTION MACHINES
Mark has set up a series of three function machines that he claims will surprise you.
Try a few numbers. Are you surprised by your results?
Your output should be the same number as your input.
Carrie claims that she was not surprised by her results. She also says that she can show why the sequence of machines does what it does by simply dropping in a variable and writing out step-by-step what happens inside each machine. Try it. (Use something like
c
m
.) Be sure to show all of the steps.
c
(c)−5=c−5
\frac{6(\textit{c}-5)+8}{2}=\frac{6\textit{c}-30+8}{2}
Simplify the output of function two, and then substitute it for
x
in function three.
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Zeno machine - Wikipedia
Hypothetical computational model
Find sources: "Zeno machine" – news · newspapers · books · scholar · JSTOR (May 2021) (Learn how and when to remove this template message)
Read-only right-moving Turing machines
In mathematics and computer science, Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that are capable of carrying out computations involving a countably infinite number of algorithmic steps.[1] These machines are ruled out in most models of computation.
The idea of Zeno machines was first discussed by Hermann Weyl in 1927; the name refers to Zeno's paradoxes, attributed to the ancient Greek philosopher Zeno of Elea. Zeno machines play a crucial role in some theories. The theory of the Omega Point devised by physicist Frank J. Tipler, for instance, can only be valid if Zeno machines are possible.
2 Infinite Time Turing Machines
A Zeno machine is a Turing machine that can take a infinite number of steps, and then continue take more steps. This can be thought of as a supertask where
{\displaystyle 1/2^{n}}
units of time are taken to perform the
{\displaystyle n}
-th step; thus, the first step takes 0.5 units of time, the second takes 0.25, the third 0.125 and so on, so that after one unit of time, a countably infinite number of steps will have been performed.
Infinite Time Turing Machines[edit]
An animation of an infinite time Turing machine based on the Thomson's lamp thought experiment. A cell alternates between
{\displaystyle 0}
{\displaystyle 1}
for steps before
{\displaystyle \omega }
. The cell becomes
{\displaystyle 1}
{\displaystyle \omega }
since the sequence does not converge.
A more formal model of the Zeno machine is the infinite time Turing machine. Defined first in unpublished work by Jeffrey Kidder and expanded upon by Joel Hamkins and Andy Lewis, in Infinite Time Turing Machines,[2] the infinite time Turing machine is an extension of the classical Turing machine model, to include transfinite time; that is time beyond all finite time.[2] A classical Turing machine has a status at step
{\displaystyle 0}
(in the start state, with an empty tape, read head at cell 0) and a procedure for getting from one status to the successive status. In this way the status of a Turing machine is defined for all steps corresponding to a natural number. An ITTM maintains these properties, but also defines the status of the machine at limit ordinals, that is ordinals that are neither
{\displaystyle 0}
nor the successor of any ordinal. The status of a Turing machine consists of 3 parts:
The location of the read-write head
The contents of the tape
Just as a classical Turing machine has a labeled start state, which is the state at the start of a program, an ITTM has a labeled limit state which is the state for the machine at any limit ordinal.[1] This is the case even if the machine has no other way to access this state, for example no node transitions to it. The location of the read-write head is set to zero for at any limit step.[1][2] Lastly the state of the tape is determined by the limit supremum of previous tape states. For some machine
{\displaystyle T}
, a cell
{\displaystyle k}
and, a limit ordinal
{\displaystyle \lambda }
{\displaystyle T(\lambda )_{k}=\limsup _{n\rightarrow \lambda }T(n)_{k}}
That is the
{\displaystyle k}
th cell at time
{\displaystyle \lambda }
is the limit supremum of that same cell as the machine approaches
{\displaystyle \lambda }
.[1] This can be thought of as the limit if it converges or
{\displaystyle 1}
Zeno machines have been proposed as a model of computation more powerful than classical Turing machines, based on their ability to solve the halting problem for classical Turing machines.[3] Cristian Calude and Ludwig Staiger present the following pseudocode algorithm as a solution to the halting problem when run on a Zeno machine.[4]
write 0 on the first position of the output tape;
simulate 1 successive step of the given Turing machine on the given input;
if the Turing machine has halted then
write 1 on the first position of the output tape and break out of loop;
By inspecting the first position of the output tape after
{\displaystyle 1}
unit of time has elapsed we can determine whether the given Turing machine halts.[4] In contrast Oron Shagir argues that the state of a Zeno machine is only defined on the interval
{\displaystyle [0,1)}
, and so it is impossible to inspect the tape at time
{\displaystyle 1}
. Furthermore since classical Turing machines don't have any timing information, the addition of timing information whether accelerating or not does not itself add any computational power.[3]
Infinite time Turing machines however, are capable of implementing the given algorithm, halting at time
{\displaystyle \omega }
with the correct solution,[2] since they do define their state for transfinite steps.[3] All
{\displaystyle \Pi _{1}^{1}}
sets are decidable by Infinite time Turing machines, and
{\displaystyle \Delta _{2}^{1}}
sets are semidecidable.[2][clarification needed]
Zeno machines cannot solve their own halting problem.[4]
Specker sequence
^ a b c d e Hamkins, Joel (2002-12-03). "Infinite time Turing machines". arXiv:math/0212047.
^ a b c d e Hamkins, Joel; Lewis, Andy (1998-08-21). "Infinite Time Turing Machines". arXiv:math/9808093.
^ a b c Shagir, Oron, Super-Tasks, Accelerating Turing Machines and Uncomputability (PDF), archived from the original (PDF) on July 9, 2007
^ a b c Calude, Cristian; Staiger, Ludwig, A Note on Accelerated Turing Machines (PDF)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Zeno_machine&oldid=1064259703"
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Mean predictive measure of association for surrogate splits in classification tree - MATLAB - MathWorks India
{\lambda }_{jk}=\frac{\text{min}\left({P}_{L},{P}_{R}\right)-\left(1-{P}_{{L}_{j}{L}_{k}}-{P}_{{R}_{j}{R}_{k}}\right)}{\text{min}\left({P}_{L},{P}_{R}\right)}.
{P}_{{L}_{j}{L}_{k}}
{P}_{{R}_{j}{R}_{k}}
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The optional filter parameter, passed as the index to the Map or Map2 command, restricts the application of
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
A≔\mathrm{Matrix}\left([[1,2,3],[0,1,4]],\mathrm{shape}=\mathrm{triangular}[\mathrm{upper},\mathrm{unit}]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\end{array}]
M≔\mathrm{Map}\left(x↦x+1,A\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5}\end{array}]
\mathrm{evalb}\left(\mathrm{addressof}\left(A\right)=\mathrm{addressof}\left(M\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
B≔〈〈1,2,3〉|〈4,5,6〉〉
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map2}[\left(i,j\right)↦\mathrm{evalb}\left(i=1\right)]\left(\left(x,a\right)↦a\cdot x,3,B\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{12}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map}\left(x↦x+1,g\left(3,A\right)\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\end{array}]\right)
C≔\mathrm{Matrix}\left([[1,2],[3]],\mathrm{scan}=\mathrm{triangular}[\mathrm{upper}],\mathrm{shape}=\mathrm{symmetric}\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{Map}\left(x↦x+1,C\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{9}\\ \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{4}\end{array}]
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shubhamtibra/Holonomic_Discussion - Gitter
shubhamtibra/Holonomic_Discussion
This gitter chat room is used to discuss about the GSoC'16 project "Implementation of Holonomic Function".
shubhamtibra on fixing_bugs
added uses in integration and l… (compare)
trying to fix build errors (compare)
made changes as per the suggest… uncomment statements in examples (compare)
changed to documentation to sup… (compare)
added docs for integration and … add autofunction in docs (compare)
better explanation of holonomic… (compare)
changed the structure of docume… (compare)
added a very basic sphinx docum… added things in documentation (compare)
shubhamtibra on test_doc
added a very basic sphinx docum… (compare)
Added tests for KanesMethod.rhs… Changed KanesMethod.rhs() such … Merge pull request #1 from krit… and 100 more (compare)
Fix #11490 and Fix #11491 (compare)
fixed a bug in computing initia… (compare)
change printing of holonomic fu… (compare)
shubhamtibra on singular_ics
change a to int(a) (compare)
fixed a bug and added tests computing singular initial cond… (compare)
fixed a bug in unify and added … (compare)
I mean, to MeijerG function.
I.e. if the holonomic function is a linear combination of MeijerG functions, will this always work?
Subham Tibra
@shubhamtibra
NOT ALWAYS. The current algorithm rather converts first to a linear combination of hypergeometric functions and then converts each hyper in the combination to meijerg. So if a function have G-function representation but not hypergeometric, it won't work.
There doesn't seem to exist an algorithm converting Holonomic functions directly to G-functions.
This method will only produce G-functions whose Slater expansion consists of a single hypergeometric series (m = 1). There does not seem to exist a general (symbolic) method for introducing other G-functions.
Hi Kalevi! I couldn't write a blog post last week. Would it be Okay if I write about last week in this weeks blog post?
Also I am thinking about writing a code that itself finds the domain for the polynomial coefficients in the differential equation in a new PR. Is there anything else you have in mind we should do first?
I think it is quite ok. I know what you have accomplished, and if the others will learn that a little later, it should not matter too much. Automatically finding the coefficient domain would surely be appreciated by the users. That would probably solve the issue sympy/sympy#11323 raised by Ondřej.
Yes, the issue is my motivation for implementing it.
We can also add support for singular initial conditions in operations like add, multiply, integrate etc. .
It seems that initial conditions at singular points are particularly useful when holonomic functions shall be connected with G-functions and other know special functions. On the other hand, initial conditions at regular points are suitable for numerical work.
Good Evening Kalevi! I wanted to discuss a question regarding conditions at singular point in integration.
For instance we want to integrate
cos(x)^{2}/x
with singular_ics = [(-1, [1, 0, -1])], i.e. the series is of the form
x^{-1}(1 + 0 - x^{2} ....)
1/x - x + ...
. If we integrate the initial terms of the series to compute singular_ics of the result, the term 1/x will become log(x). In cases where there is no term of x with power -1 we can compute the singular_ics directly by integration of initial terms of the series. What should be the singular_ics for this case?
General holonomic functions also have these logarithmic terms. They appear at regular singular points when some roots of the indicial equation differ by an integer (or there is a multiple root). Such expressions may become quite complicated in general if several roots are involved. In case of two roots the situation might be controllable. See e.g. the series expansions of the Bessel functions of the second kind. But it may be too ambitious to include them in this gsoc project. I am not sure of how the initial conditions should be defined. Probably something like this: log(x) multiplied by an initial condition of the more general type (i.e. x**s multiplied by a constant and optionally some low order terms), in addition of the usual initial conditions at a regular singular point.
Look at http://dlmf.nist.gov/10.8, Power Series, to see how the logarithm enters.
This looks complicated. Should we implement it later?
I think it would be too complicated for this project.
So we should probably stop with NotImplementedError when something like this will appear.
Hi Kalevi! I can't figure out any algorithm to compute singular_ics after multiplication of two holonomic functions. Do you have any thoughts on that?
It would seem that if you have two (partial) representations of the type
x^{s_1} f_1(x)
x^{s_2} f_2 (x)
the product would contain
x^{s_1+s_2}f_1(x)f_2(x)
with fairly obvious initial conditions. But maybe this is not what you wanted to ask?
I actually was asking this.
I got it now. :)
Hi. If the tests passed in Travis even with the typos, that would probably mean that those lines were not tested. Or could there be another explanation?
Those lines should have been executed on travis because they did when I ran the tests locally. The lines were returning the final answer after multiplication. I am also stumped.
In any case, sol must have been available in a dictionary, if not in the local dictionary, then in the dictionary of one of the calling routines. Its value may have been correct by chance.
Hi Kalevi. I've added things I could think of in sympy/sympy#11422. You can also give your thoughts now.
Hi. I just saw it and wrote some comments.
I wonder if some additional initial conditions should be added to the result of expr_to_holonomic.
Do you mean the method should return this new initial condition for more types of functions? It currently computes this initial condition only for the algebraic functions.
It seems that the condition may not suffice to define the function. For example, all constant multiples of x have value 0 at the origin. The coefficient of the first nonzero power is needed.
@shubhamtibra what's the problem regarding sympy/sympy#11323 ?
Did I do something wrong when using symbolic variables as coefficients?
No. Of course not. What you used should be the ideal way of doing it while using symbolic variables. At present there are additional keyword arguments like specifying the domain, and the number of initial conditions to fix the problem. :)
I think it is otherwise ok now, but we have to use singular initial conditions since the equation is singular at 0.
Yes. We should have an algorithm for computing singular_ics for more functions.
I changed the title of the PR to a more descriptive one.
Hi Kalevi! I was thinking next to add functionality in expr_to_holonomic so it can calculate singular_ics for more types of functions. Do you have any other thing in mind to do first?
No. I think that could be the next task.
@shubhamtibra can you post to that issue showing how it can be used right now? I just want to play with it. I understand that we can improve the API and so on, but right now I just want to play with it.
I posted it. #issuecomment
Hi Kalevi! I added singular initial conditions to the result of expr_to_holonomic while converting polynomials if the differential equation is singular. Do you have other types of functions in mind where we can calculate the singular initial conditions?
I think it would be proper to use singular initial conditions whenever the point in question is a regular singular point of the equation.
Does there exist a general algorithm to compute singular initial condition for functions? I actually was adding code to compute it one by one for specific families of functions like polynomials and algebraic functions.
It'd be possible to compute it easily for any function if the indicial equation have only one root r. We can then use g(x) = f(x)/x**r. So the singular initial condition should be {r:[g(x0), g'(x0), g''(x0)/2! ...]}.
Multiple roots are not good. They lead to solutions with logarithmic terms. The best situation is the one where the roots are different modulo 1.
Hi! I have made the change as you suggested here in PR sympy/sympy#11451.
That is good. It is also necessary for range(a).
The statement a = int(a) is defined here before using range(a). So I guess a would be an integer while calling range().
Do you have any thoughts on what we can work on next?
Perhaps the issues raised by Ondřej should be checked.
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QR algorithm - Wikipedia
In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently.[1][2][3] The basic idea is to perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order, and iterate.
1 The practical QR algorithm
2.1 Finding eigenvalues versus finding eigenvectors
3 The implicit QR algorithm
4 Interpretation and convergence
The practical QR algorithm[edit]
Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A0:=A. At the k-th step (starting with k = 0), we compute the QR decomposition Ak=QkRk where Qk is an orthogonal matrix (i.e., QT = Q−1) and Rk is an upper triangular matrix. We then form Ak+1 = RkQk. Note that
{\displaystyle A_{k+1}=R_{k}Q_{k}=Q_{k}^{-1}Q_{k}R_{k}Q_{k}=Q_{k}^{-1}A_{k}Q_{k}=Q_{k}^{\mathsf {T}}A_{k}Q_{k},}
so all the Ak are similar and hence they have the same eigenvalues. The algorithm is numerically stable because it proceeds by orthogonal similarity transforms.
Under certain conditions,[4] the matrices Ak converge to a triangular matrix, the Schur form of A. The eigenvalues of a triangular matrix are listed on the diagonal, and the eigenvalue problem is solved. In testing for convergence it is impractical to require exact zeros,[citation needed] but the Gershgorin circle theorem provides a bound on the error.
In this crude form the iterations are relatively expensive. This can be mitigated by first bringing the matrix A to upper Hessenberg form (which costs
{\displaystyle {\begin{matrix}{\frac {10}{3}}\end{matrix}}n^{3}+{\mathcal {O}}(n^{2})}
arithmetic operations using a technique based on Householder reduction), with a finite sequence of orthogonal similarity transforms, somewhat like a two-sided QR decomposition.[5][6] (For QR decomposition, the Householder reflectors are multiplied only on the left, but for the Hessenberg case they are multiplied on both left and right.) Determining the QR decomposition of an upper Hessenberg matrix costs
{\displaystyle 6n^{2}+{\mathcal {O}}(n)}
arithmetic operations. Moreover, because the Hessenberg form is already nearly upper-triangular (it has just one nonzero entry below each diagonal), using it as a starting point reduces the number of steps required for convergence of the QR algorithm.
If the original matrix is symmetric, then the upper Hessenberg matrix is also symmetric and thus tridiagonal, and so are all the Ak. This procedure costs
{\displaystyle {\begin{matrix}{\frac {4}{3}}\end{matrix}}n^{3}+{\mathcal {O}}(n^{2})}
arithmetic operations using a technique based on Householder reduction.[5][6] Determining the QR decomposition of a symmetric tridiagonal matrix costs
{\displaystyle {\mathcal {O}}(n)}
operations.[7]
The rate of convergence depends on the separation between eigenvalues, so a practical algorithm will use shifts, either explicit or implicit, to increase separation and accelerate convergence. A typical symmetric QR algorithm isolates each eigenvalue (then reduces the size of the matrix) with only one or two iterations, making it efficient as well as robust.[clarification needed]
Figure 1: How the output of a single iteration of the QR or LR algorithm varies alongside its input
The basic QR algorithm can be visualized in the case where A is a positive-definite symmetric matrix. In that case, A can be depicted as an ellipse in 2 dimensions or an ellipsoid in higher dimensions. The relationship between the input to the algorithm and a single iteration can then be depicted as in Figure 1 (click to see an animation). Note that the LR algorithm is depicted alongside the QR algorithm.
A single iteration causes the ellipse to tilt or "fall" towards the x-axis. In the event where the large semi-axis of the ellipse is parallel to the x-axis, one iteration of QR does nothing. Another situation where the algorithm "does nothing" is when the large semi-axis is parallel to the y-axis instead of the x-axis. In that event, the ellipse can be thought of as balancing precariously without being able to fall in either direction. In both situations, the matrix is diagonal. A situation where an iteration of the algorithm "does nothing" is called a fixed point. The strategy employed by the algorithm is iteration towards a fixed-point. Observe that one fixed point is stable while the other is unstable. If the ellipse were tilted away from the unstable fixed point by a very small amount, one iteration of QR would cause the ellipse to tilt away from the fixed point instead of towards. Eventually though, the algorithm would converge to a different fixed point, but it would take a long time.
Finding eigenvalues versus finding eigenvectors[edit]
Figure 2: How the output of a single iteration of QR or LR are affected when two eigenvalues approach each other
Off the bat, it's worth pointing out that finding even a single eigenvector of a symmetric matrix is uncomputable (in exact real arithmetic according to the definitions in computable analysis).[8] This difficulty exists whenever the multiplicities of a matrix's eigenvalues are not knowable. On the other hand, the same problem does not exist for finding eigenvalues. The eigenvalues of a matrix are always computable.
We will now discuss how these difficulties manifest in the basic QR algorithm. This is illustrated in Figure 2. (Remember to click the thumbnail). Recall that the ellipses represent positive-definite symmetric matrices. As the two eigenvalues of the input matrix approach each other, the input ellipse changes into a circle. A circle corresponds to a multiple of the identity matrix. A near-circle corresponds to a near-multiple of the identity matrix whose eigenvalues are nearly equal to the diagonal entries of the matrix. Therefore the problem of approximately finding the eigenvalues is shown to be easy in that case. But notice what happens to the semi-axes of the ellipses. An iteration of QR (or LR) tilts the semi-axes less and less as the input ellipse gets closer to being a circle. The eigenvectors can only be known when the semi-axes are parallel to the x-axis and y-axis. The number of iterations needed to achieve near-parallelism increases without bound as the input ellipse becomes more circular.
While it may be impossible to compute the eigendecomposition of an arbitrary symmetric matrix, it is always possible to perturb the matrix by an arbitrarily small amount and compute the eigendecomposition of the resulting matrix. In the case when the matrix is depicted as a near-circle, the matrix can be replaced with one whose depiction is a perfect circle. In that case, the matrix is a multiple of the identity matrix, and its eigendecomposition is immediate. Be aware though that the resulting eigenbasis can be quite far from the original eigenbasis.
The implicit QR algorithm[edit]
In modern computational practice, the QR algorithm is performed in an implicit version which makes the use of multiple shifts easier to introduce.[4] The matrix is first brought to upper Hessenberg form
{\displaystyle A_{0}=QAQ^{\mathsf {T}}}
as in the explicit version; then, at each step, the first column of
{\displaystyle A_{k}}
is transformed via a small-size Householder similarity transformation to the first column of
{\displaystyle p(A_{k})}
{\displaystyle p(A_{k})e_{1}}
{\displaystyle p(A_{k})}
, of degree
{\displaystyle r}
, is the polynomial that defines the shifting strategy (often
{\displaystyle p(x)=(x-\lambda )(x-{\bar {\lambda }})}
{\displaystyle \lambda }
{\displaystyle {\bar {\lambda }}}
are the two eigenvalues of the trailing
{\displaystyle 2\times 2}
{\displaystyle A_{k}}
, the so-called implicit double-shift). Then successive Householder transformations of size
{\displaystyle r+1}
are performed in order to return the working matrix
{\displaystyle A_{k}}
to upper Hessenberg form. This operation is known as bulge chasing, due to the peculiar shape of the non-zero entries of the matrix along the steps of the algorithm. As in the first version, deflation is performed as soon as one of the sub-diagonal entries of
{\displaystyle A_{k}}
Since in the modern implicit version of the procedure no QR decompositions are explicitly performed, some authors, for instance Watkins,[9] suggested changing its name to Francis algorithm. Golub and Van Loan use the term Francis QR step.
Interpretation and convergence[edit]
The QR algorithm can be seen as a more sophisticated variation of the basic "power" eigenvalue algorithm. Recall that the power algorithm repeatedly multiplies A times a single vector, normalizing after each iteration. The vector converges to an eigenvector of the largest eigenvalue. Instead, the QR algorithm works with a complete basis of vectors, using QR decomposition to renormalize (and orthogonalize). For a symmetric matrix A, upon convergence, AQ = QΛ, where Λ is the diagonal matrix of eigenvalues to which A converged, and where Q is a composite of all the orthogonal similarity transforms required to get there. Thus the columns of Q are the eigenvectors.
The QR algorithm was preceded by the LR algorithm, which uses the LU decomposition instead of the QR decomposition. The QR algorithm is more stable, so the LR algorithm is rarely used nowadays. However, it represents an important step in the development of the QR algorithm.
The LR algorithm was developed in the early 1950s by Heinz Rutishauser, who worked at that time as a research assistant of Eduard Stiefel at ETH Zurich. Stiefel suggested that Rutishauser use the sequence of moments y0T Ak x0, k = 0, 1, … (where x0 and y0 are arbitrary vectors) to find the eigenvalues of A. Rutishauser took an algorithm of Alexander Aitken for this task and developed it into the quotient–difference algorithm or qd algorithm. After arranging the computation in a suitable shape, he discovered that the qd algorithm is in fact the iteration Ak = LkUk (LU decomposition), Ak+1 = UkLk, applied on a tridiagonal matrix, from which the LR algorithm follows.[10]
One variant of the QR algorithm, the Golub-Kahan-Reinsch algorithm starts with reducing a general matrix into a bidiagonal one.[11] This variant of the QR algorithm for the computation of singular values was first described by Golub & Kahan (1965) harvtxt error: no target: CITEREFGolubKahan1965 (help). The LAPACK subroutine DBDSQR implements this iterative method, with some modifications to cover the case where the singular values are very small (Demmel & Kahan 1990) harv error: no target: CITEREFDemmelKahan1990 (help). Together with a first step using Householder reflections and, if appropriate, QR decomposition, this forms the DGESVD routine for the computation of the singular value decomposition. The QR algorithm can also be implemented in infinite dimensions with corresponding convergence results.[12][13]
^ J.G.F. Francis, "The QR Transformation, I", The Computer Journal, 4(3), pages 265–271 (1961, received October 1959). doi:10.1093/comjnl/4.3.265
^ Francis, J. G. F. (1962). "The QR Transformation, II". The Computer Journal. 4 (4): 332–345. doi:10.1093/comjnl/4.4.332.
^ Vera N. Kublanovskaya, "On some algorithms for the solution of the complete eigenvalue problem," USSR Computational Mathematics and Mathematical Physics, vol. 1, no. 3, pages 637–657 (1963, received Feb 1961). Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol.1, no. 4, pages 555–570 (1961). doi:10.1016/0041-5553(63)90168-X
^ a b Golub, G. H.; Van Loan, C. F. (1996). Matrix Computations (3rd ed.). Baltimore: Johns Hopkins University Press. ISBN 0-8018-5414-8.
^ a b Demmel, James W. (1997). Applied Numerical Linear Algebra. SIAM.
^ a b Trefethen, Lloyd N.; Bau, David (1997). Numerical Linear Algebra. SIAM.
^ Ortega, James M.; Kaiser, Henry F. (1963). "The LLT and QR methods for symmetric tridiagonal matrices". The Computer Journal. 6 (1): 99–101. doi:10.1093/comjnl/6.1.99.
^ "linear algebra - Why is uncomputability of the spectral decomposition not a problem?". MathOverflow. Retrieved 2021-08-09.
^ Watkins, David S. (2007). The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods. Philadelphia, PA: SIAM. ISBN 978-0-89871-641-2.
^ Parlett, Beresford N.; Gutknecht, Martin H. (2011), "From qd to LR, or, how were the qd and LR algorithms discovered?" (PDF), IMA Journal of Numerical Analysis, 31 (3): 741–754, doi:10.1093/imanum/drq003, hdl:20.500.11850/159536, ISSN 0272-4979
^ Bochkanov Sergey Anatolyevich. ALGLIB User Guide - General Matrix operations - Singular value decomposition . ALGLIB Project. 2010-12-11. URL:http://www.alglib.net/matrixops/general/svd.php.[permanent dead link] Accessed: 2010-12-11. (Archived by WebCite at https://www.webcitation.org/5utO4iSnR?url=http://www.alglib.net/matrixops/general/svd.php
^ Deift, Percy; Li, Luenchau C.; Tomei, Carlos (1985). "Toda flows with infinitely many variables". Journal of Functional Analysis. 64 (3): 358–402. doi:10.1016/0022-1236(85)90065-5.
^ Colbrook, Matthew J.; Hansen, Anders C. (2019). "On the infinite-dimensional QR algorithm". Numerische Mathematik. 143 (1): 17–83. arXiv:2011.08172. doi:10.1007/s00211-019-01047-5.
Eigenvalue problem at PlanetMath.
Notes on orthogonal bases and the workings of the QR algorithm by Peter J. Olver
Module for the QR Method
Retrieved from "https://en.wikipedia.org/w/index.php?title=QR_algorithm&oldid=1068781970"
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Formate - Wikipedia
Formate (IUPAC name: methanoate) is the anion derived from formic acid. A formate (compound) is a salt or ester of formic acid.[1]
Structure of formate
2 Formate esters
3 Formate salts
Formate is reversibly oxidized by the enzyme formate dehydrogenase from Desulfovibrio gigas:[2]
{\displaystyle {\ce {COOH^- -> CO2 + H+ + 2 e-}}}
Formate estersEdit
Formate esters have the formula HCOOR (alternative way of writing formula ROC(O)H or RO2CH). Many form spontaneously when alcohols dissolve in formic acid.
The most important formate ester is methyl formate, which is produced as an intermediate en route to formic acid. Methanol and carbon monoxide react in the presence of a strong base, such as sodium methoxide:[1]
{\displaystyle {\ce {CH3OH + CO -> HCOOCH3}}}
Hydrolysis of methyl formate gives formic acid and regenerates methanol:
{\displaystyle {\ce {HCOOCH3 -> HCOOH + CH3OH}}}
Formic acid is used for many applications in industry.
Formate esters often are fragrant or have distinctive odors. Compared to the more common ethyl esters, formate esters are less commonly used commercially because they are less stable.[3] Ethyl formate is found in some confectionaries.[1]
Formate saltsEdit
Formate salts have the formula M(O2CH)(H2O)x. Such salts are prone to decarboxylation. For example, hydrated nickel formate decarboxylates at about 200 °C to give finely powdered nickel metal:
{\displaystyle {\ce {Ni(COOH)2(H2O)2 -> Ni + 2 CO2 + 2 H2O + H2}}}
Such fine powders are useful as hydrogenation catalysts.[1]
ethyl formate, CH3CH2(HCOO)
sodium formate, Na(HCOO)
potassium formate, K(HCOO)
caesium formate, Cs(HCOO); see Caesium: Petroleum exploration
methyl formate, CH3(HCOO)
methyl chloroformate, CH3OCOCl
trimethyl orthoformate, C4H10O3
phenyl formate HCOOC6H5
^ a b c d Werner Reutemann and Heinz Kieczka "Formic Acid" in Ullmann's Encyclopedia of Industrial Chemistry 2002, Wiley-VCH, Weinheim. doi:10.1002/14356007.a12_013
^ T. Reda, C. M. Plugge, N. J. Abram and J. Hirst, "Reversible interconversion of carbon dioxide and formate by an electroactive enzyme", PNAS 2008 105, 10654–10658. doi:10.1073/pnas.0801290105
Retrieved from "https://en.wikipedia.org/w/index.php?title=Formate&oldid=1084026883"
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Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations | EMS Press
We study uniformly elliptic fully nonlinear equations
F(D^2u, Du, u, x)=0,
and prove results of Gidas--Ni--Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.
Francesca Da Lio, Boyan Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations. J. Eur. Math. Soc. 9 (2007), no. 2, pp. 317–330
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In economics, the marginal rate of substitution (MRS) is the amount of a good that a consumer is willing to consume compared to another good, as long as the new good is equally satisfying. MRS is used in indifference theory to analyze consumer behavior.
The marginal rate of substitution is the willingness of a consumer to replace one good for another good, as long as the new good is equally satisfying.
The marginal rate of substitution is the slope of the indifference curve at any given point along the curve and displays a frontier of utility for each combination of "good X" and "good Y."
When the law of diminishing MRS is in effect, the MRS forms a downward, negative sloping, convex curve showing more consumption of one good in place of another.
Formula and Calculation of the Marginal Rate of Substitution (MRS)
The marginal rate of substitution (MRS) formula is:
\begin{aligned} &|MRS_{xy}| = \frac{dy}{dx} = \frac{MU_x}{MU_y} \\ &\textbf{where:}\\ &x, y=\text{two different goods}\\ &\frac{dy}{dx}=\text{derivative of y with respect to x}\\ &MU=\text{marginal utility of good x, y}\\ \end{aligned}
∣MRSxy∣=dxdy=MUyMUxwhere:x,y=two different goodsdxdy=derivative of y with respect to xMU=marginal utility of good x, y
What the Marginal Rate of Substitution (MRS) Can Tell You
The marginal rate of substitution is a term used in economics that refers to the amount of one good that is substitutable for another and is used to analyze consumer behaviors for a variety of purposes. MRS is calculated between two goods placed on an indifference curve, displaying a frontier of utility for each combination of "good X" and "good Y." The slope of this curve represents quantities of good X and good Y that you would be happy substituting for one another.
The slope of the indifference curve is critical to the marginal rate of substitution analysis. Essentially, MRS is the slope of the indifference curve at any single point along the curve. Since most indifference curves are curves, the slopes will be different as one moves along them. Most indifference curves are usually convex because, as you consume more of one good, you will consume less of the other. Indifference curves can be straight lines if a slope is constant, resulting in an indifference curve represented by a downward-sloping straight line.
If the marginal rate of substitution is increasing, the indifference curve will be concave to the origin. This is typically not common since it means a consumer would consume more of X for the increased consumption of Y (and vice versa). Usually, marginal substitution is diminishing, meaning a consumer chooses the substitute in place of another good, rather than simultaneously consuming more.
The law of diminishing marginal rates of substitution states that MRS decreases as one moves down a standard convex-shaped curve, which is the indifference curve.
Example of Marginal Rate of Substitution (MRS)
For example, a consumer must choose between hamburgers and hot dogs. To determine the marginal rate of substitution, the consumer is asked what combinations of hamburgers and hot dogs provide the same level of satisfaction.
When these combinations are graphed, the slope of the resulting line is negative. This means that the consumer faces a diminishing marginal rate of substitution: The more hamburgers they have relative to hot dogs, the fewer hot dogs they are willing to consume. If the marginal rate of substitution of hamburgers for hot dogs is -2, then the individual would be willing to give up 2 hot dogs for every additional hamburger consumption.
Limitations of the Marginal Rate of Substitution (MRS)
The marginal rate of substitution has a few limitations. The main drawback is that it does not examine a combination of goods that a consumer would prefer more or less than another combination. This generally limits the analysis of MRS to two variables. Also, MRS does not necessarily examine marginal utility since it treats the utility of both comparable goods equally, though in actuality they may have varying utility.
What Is Indifference Curve Analysis?
Indifference curve analysis operates on a simple two-dimensional graph. Each axis represents one type of economic good. The consumer is indifferent between any of the combinations of goods represented by points on the indifference curve because these combinations provide the same level of utility to the consumer. Indifference curves are heuristic devices used in contemporary microeconomics to demonstrate consumer preference and the limitations of a budget.
What Is the Relationship Between Indifference Curve and MRS?
Essentially, MRS is the slope of the indifference curve at any single point along the curve. Most indifference curves are usually convex because as you consume more of one good you will consume less of the other. So, MRS will decrease as one moves down the indifference curve. This is known as the law of diminishing marginal rate of substitution. If the marginal rate of substitution is increasing, the indifference curve will be concave, which means that a consumer would consume more of X for the increased consumption of Y and vice versa, but this is not common.
What are the Drawbacks of Marginal Rate of Substitution (MRS)?
The marginal rate of substitution has a few limitations. The main drawback is that it does not examine a combination of goods that a consumer would prefer more or less than another combination. This generally limits the analysis of MRS to two variables. Also, MRS does not necessarily examine marginal utility because it treats the utility of both comparable goods equally though in actuality they may have varying utility.
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Carmichael function - Wikipedia
(Redirected from Carmichael's totient function)
Function in mathematical number theory
Carmichael λ function: λ(n) for 1 ≤ n ≤ 1000 (compared to Euler φ function)
In number theory, a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that
am ≡ 1 (mod n)
for every integer a between 1 and n that is coprime to n. In algebraic terms, λ(n) is the exponent of the multiplicative group of integers modulo n.
The Carmichael function is named after the American mathematician Robert Carmichael and is also known as Carmichael's λ function, the reduced totient function, and the least universal exponent function.
The following table compares the first 36 values of λ(n) (sequence A002322 in the OEIS) with Euler's totient function φ (in bold if they are different; the ns such that they are different are listed in OEIS: A033949).
2 Computing λ(n) with Carmichael's theorem
3 Properties of the Carmichael function
3.1 Order of elements modulo n
3.3 λ(n) divides φ(n)
3.6 Exponential cycle length
3.7 Extension for powers of two
3.9 Prevailing interval
3.10 Lower bounds
3.11 Minimal order
3.12 Small values
3.13 Image of the function
4 Use in cryptography
Carmichael's function at 5 is 4, λ(5) = 4, because for any number
{\displaystyle 0<a<5}
coprime to 5, i.e.
{\displaystyle a\in \{1,2,3,4\}~,}
{\displaystyle a^{m}\equiv 1\,({\text{mod }}5)}
{\displaystyle m=4,}
namely, 11⋅4 = 14 ≡ 1 (mod 5), 24 = 16 ≡ 1 (mod 5), 34 = 81 ≡ 1 (mod 5) and 42⋅2 = 162 ≡ 12 (mod 5). And this m = 4 is the smallest exponent with this property, because
{\displaystyle 2^{2}=4\not \equiv 1\,({\text{mod }}5)}
{\displaystyle 3^{2}=9\not \equiv 1\,({\text{mod }}5)}
as well.)
Moreover, Euler's totient function at 5 is 4, φ(5) = 4, because there are exactly 4 numbers less than and coprime to 5 (1, 2, 3, and 4). Euler's theorem assures that a4 ≡ 1 (mod 5) for all a coprime to 5, and 4 is the smallest such exponent.
Carmichael's function at 8 is 2, λ(8) = 2, because for any number a coprime to 8, i.e.
{\displaystyle a\in \{1,3,5,7\}~,}
it holds that a2 ≡ 1 (mod 8). Namely, 11⋅2 = 12 ≡ 1 (mod 8), 32 = 9 ≡ 1 (mod 8), 52 = 25 ≡ 1 (mod 8) and 72 = 49 ≡ 1 (mod 8).
Euler's totient function at 8 is 4, φ(8) = 4, because there are exactly 4 numbers less than and coprime to 8 (1, 3, 5, and 7). Moreover, Euler's theorem assures that a4 ≡ 1 (mod 8) for all a coprime to 8, but 4 is not the smallest such exponent.
Computing λ(n) with Carmichael's theorem[edit]
By the unique factorization theorem, any n > 1 can be written in a unique way as
{\displaystyle n=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{k}^{r_{k}}}
where p1 < p2 < ... < pk are primes and r1, r2, ..., rk are positive integers. Then λ(n) is the least common multiple of the λ of each of its prime power factors:
{\displaystyle \lambda (n)=\operatorname {lcm} {\Bigl (}\lambda \left(p_{1}^{r_{1}}\right),\lambda \left(p_{2}^{r_{2}}\right),\ldots ,\lambda \left(p_{k}^{r_{k}}\right){\Bigr )}.}
This can be proved using the Chinese remainder theorem.
Carmichael's theorem explains how to compute λ of a prime power pr: for a power of an odd prime and for 2 and 4, λ(pr) is equal to the Euler totient φ(pr); for powers of 2 greater than 4 it is equal to half of the Euler totient:
{\displaystyle \lambda (p^{r})={\begin{cases}{\tfrac {1}{2}}\varphi \left(p^{r}\right)&{\text{if }}p=2\land r\geq 3\;({\mbox{i.e. }}p^{r}=8,16,32,64,128,256,\dots )\\\varphi \left(p^{r}\right)&{\mbox{otherwise}}\;({\mbox{i.e. }}p^{r}=2,4,3^{r},5^{r},7^{r},11^{r},13^{r},17^{r},19^{r},23^{r},29^{r},31^{r},\dots )\end{cases}}}
Euler's function for prime powers pr is given by
{\displaystyle \varphi (p^{r})=p^{r-1}(p-1).}
Properties of the Carmichael function[edit]
In this section, an integer
{\displaystyle n}
is divisible by a nonzero integer
{\displaystyle m}
{\displaystyle k}
{\displaystyle n=km}
{\displaystyle m\mid n.}
Order of elements modulo n[edit]
Let a and n be coprime and let m be the smallest exponent with am ≡ 1 (mod n), then it holds that
{\displaystyle m\,|\,\lambda (n)}
That is, the order m := ordn(a) of a unit a in the ring of integers modulo n divides λ(n) and
{\displaystyle \lambda (n)=\max\{\operatorname {ord} _{n}(a)\,\colon \,\gcd(a,n)=1\}}
Suppose am ≡ 1 (mod n) for all numbers a coprime with n. Then λ(n) | m.
Proof: If m = kλ(n) + r with 0 ≤ r < λ(n), then
{\displaystyle a^{r}=1^{k}\cdot a^{r}\equiv \left(a^{\lambda (n)}\right)^{k}\cdot a^{r}=a^{k\lambda (n)+r}=a^{m}\equiv 1{\pmod {n}}}
for all numbers a coprime with n. It follows r = 0, since r < λ(n) and λ(n) the minimal positive such number.
λ(n) divides φ(n)[edit]
This follows from elementary group theory, because the exponent of any finite group must divide the order of the group. λ(n) is the exponent of the multiplicative group of integers modulo n while φ(n) is the order of that group. In particular, the two must be equal in the cases where the multiplicative group is cyclic due to the existence of a primitive root, which is the case for odd prime powers.
We can thus view Carmichael's theorem as a sharpening of Euler's theorem.
{\displaystyle a\,|\,b\Rightarrow \lambda (a)\,|\,\lambda (b)}
By definition, for any integer
{\displaystyle k}
{\displaystyle b\,|\,(k^{\lambda (b)}-1)}
{\displaystyle a\,|\,(k^{\lambda (b)}-1)}
. By the minimality property above, we have
{\displaystyle \lambda (a)\,|\,\lambda (b)}
For all positive integers a and b it holds that
{\displaystyle \lambda (\mathrm {lcm} (a,b))=\mathrm {lcm} (\lambda (a),\lambda (b))}
This is an immediate consequence of the recursive definition of the Carmichael function.
Exponential cycle length[edit]
{\displaystyle r_{\mathrm {max} }=\max _{i}\{r_{i}\}}
is the biggest exponent in the prime factorization
{\displaystyle n=p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots p_{k}^{r_{k}}}
of n, then for all a (including those not coprime to n) and all r ≥ rmax,
{\displaystyle a^{r}\equiv a^{\lambda (n)+r}{\pmod {n}}.}
In particular, for square-free n ( rmax = 1), for all a we have
{\displaystyle a\equiv a^{\lambda (n)+1}{\pmod {n}}.}
Extension for powers of two[edit]
For a coprime to (powers of) 2 we have a = 1 + 2h for some h. Then,
{\displaystyle a^{2}=1+4h(h+1)=1+8C}
where we take advantage of the fact that C := (h + 1)h/2 is an integer.
So, for k = 3, h an integer:
{\displaystyle {\begin{aligned}a^{2^{k-2}}&=1+2^{k}h\\a^{2^{k-1}}&=\left(1+2^{k}h\right)^{2}=1+2^{k+1}\left(h+2^{k-1}h^{2}\right)\end{aligned}}}
By induction, when k ≥ 3, we have
{\displaystyle a^{2^{k-2}}\equiv 1{\pmod {2^{k}}}.}
It provides that λ(2k) is at most 2k − 2.[1]
Average value[edit]
For any n ≥ 16:[2][3]
{\displaystyle {\frac {1}{n}}\sum _{i\leq n}\lambda (i)={\frac {n}{\ln n}}e^{B(1+o(1))\ln \ln n/(\ln \ln \ln n)}}
(called Erdős approximation in the following) with the constant
{\displaystyle B:=e^{-\gamma }\prod _{p\in \mathbb {P} }\left({1-{\frac {1}{(p-1)^{2}(p+1)}}}\right)\approx 0.34537}
and γ ≈ 0.57721, the Euler–Mascheroni constant.
The following table gives some overview over the first 226 – 1 = 67108863 values of the λ function, for both, the exact average and its Erdős-approximation.
Additionally given is some overview over the more easily accessible “logarithm over logarithm” values LoL(n) := ln λ(n)/ln n with
LoL(n) > 4/5 ⇔ λ(n) > n4/5.
There, the table entry in row number 26 at column
% LoL > 4/5 → 60.49
indicates that 60.49% (≈ 40000000) of the integers 1 ≤ n ≤ 67108863 have λ(n) > n4/5 meaning that the majority of the λ values is exponential in the length l := log2(n) of the input n, namely
{\displaystyle \left(2^{\frac {4}{5}}\right)^{l}=2^{\frac {4l}{5}}=\left(2^{l}\right)^{\frac {4}{5}}=n^{\frac {4}{5}}.}
n = 2ν – 1
{\displaystyle \sum _{i\leq n}\lambda (i)}
{\displaystyle {\tfrac {1}{n}}\sum _{i\leq n}\lambda (i)}
Erdős average
Erdős /
LoL average
% LoL > 4/5
5 31 270 8.709677 68.643 7.8813 0.678244 41.94 35.48
6 63 964 15.301587 61.414 4.0136 0.699891 38.10 30.16
7 127 3574 28.141732 86.605 3.0774 0.717291 38.58 27.56
8 255 12994 50.956863 138.190 2.7119 0.730331 38.82 23.53
10 1023 178816 174.795699 406.145 2.3235 0.748482 41.45 26.98
12 4095 2490948 608.290110 1304.810 2.1450 0.761027 43.74 28.11
13 8191 9382764 1145.496765 2383.263 2.0806 0.766571 44.33 28.60
14 16383 35504586 2167.160227 4392.129 2.0267 0.771695 46.10 29.52
15 32767 134736824 4111.967040 8153.054 1.9828 0.776437 47.21 29.15
16 65535 513758796 7839.456718 15225.430 1.9422 0.781064 49.13 28.17
17 131071 1964413592 14987.400660 28576.970 1.9067 0.785401 50.43 29.55
19 524287 28935644342 55190.466940 101930.900 1.8469 0.793536 52.62 31.45
20 1048575 111393101150 106232.840900 193507.100 1.8215 0.797351 53.74 31.83
22 4194303 1660388309120 395867.515800 703289.400 1.7766 0.804543 56.24 33.65
23 8388607 6425917227352 766029.118700 1345633.000 1.7566 0.807936 57.19 34.32
24 16777215 24906872655990 1484565.386000 2580070.000 1.7379 0.811204 58.49 34.43
26 67108863 375619048086576 5597160.066000 9537863.000 1.7041 0.817384 60.49 36.73
Prevailing interval[edit]
For all numbers N and all but o(N)[4] positive integers n ≤ N (a "prevailing" majority):
{\displaystyle \lambda (n)={\frac {n}{(\ln n)^{\ln \ln \ln n+A+o(1)}}}}
with the constant[3]
{\displaystyle A:=-1+\sum _{p\in \mathbb {P} }{\frac {\ln p}{(p-1)^{2}}}\approx 0.2269688}
For any sufficiently large number N and for any Δ ≥ (ln ln N)3, there are at most
{\displaystyle N\exp \left(-0.69(\Delta \ln \Delta )^{\frac {1}{3}}\right)}
positive integers n ≤ N such that λ(n) ≤ ne−Δ.[5]
Minimal order[edit]
For any sequence n1 < n2 < n3 < ⋯ of positive integers, any constant 0 < c < 1/ln 2, and any sufficiently large i:[6][7]
{\displaystyle \lambda (n_{i})>\left(\ln n_{i}\right)^{c\ln \ln \ln n_{i}}.}
Small values[edit]
For a constant c and any sufficiently large positive A, there exists an integer n > A such that[7]
{\displaystyle \lambda (n)<\left(\ln A\right)^{c\ln \ln \ln A}.}
Moreover, n is of the form
{\displaystyle n=\mathop {\prod _{q\in \mathbb {P} }} _{(q-1)|m}q}
for some square-free integer m < (ln A)c ln ln ln A.[6]
Image of the function[edit]
The set of values of the Carmichael function has counting function[8]
{\displaystyle {\frac {x}{(\ln x)^{\eta +o(1)}}},}
{\displaystyle \eta =1-{\frac {1+\ln \ln 2}{\ln 2}}\approx 0.08607}
Use in cryptography[edit]
The Carmichael function is important in cryptography due to its use in the RSA encryption algorithm.
^ Carmichael, Robert Daniel. The Theory of Numbers. Nabu Press. ISBN 1144400341. [page needed]
^ Theorem 3 in Erdős (1991)
^ a b Sándor & Crstici (2004) p.194
^ Theorem 2 in Erdős (1991) 3. Normal order. (p.365)
^ Theorem 5 in Friedlander (2001)
^ a b Theorem 1 in Erdős 1991
^ Ford, Kevin; Luca, Florian; Pomerance, Carl (27 August 2014). "The image of Carmichael's λ-function". Algebra & Number Theory. 8 (8): 2009–2026. arXiv:1408.6506. doi:10.2140/ant.2014.8.2009.
Erdős, Paul; Pomerance, Carl; Schmutz, Eric (1991). "Carmichael's lambda function". Acta Arithmetica. 58 (4): 363–385. doi:10.4064/aa-58-4-363-385. ISSN 0065-1036. MR 1121092. Zbl 0734.11047.
Friedlander, John B.; Pomerance, Carl; Shparlinski, Igor E. (2001). "Period of the power generator and small values of the Carmichael function". Mathematics of Computation. 70 (236): 1591–1605, 1803–1806. doi:10.1090/s0025-5718-00-01282-5. ISSN 0025-5718. MR 1836921. Zbl 1029.11043.
Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36, 193–195. ISBN 978-1-4020-2546-4. Zbl 1079.11001.
Carmichael, R. D. (2004-10-10). The Theory of Numbers. Nabu Press. ISBN 978-1144400345.
Euler's totient function φ(n)
Jordan's totient function Jk(n)
Carmichael function (reduced totient function) λ(n)
Sparsely totient number
Retrieved from "https://en.wikipedia.org/w/index.php?title=Carmichael_function&oldid=1083535168"
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Definition of Arbitrage Pricing Theory (APT)
Arbitrage pricing theory (APT) is a multi-factor asset pricing model based on the idea that an asset's returns can be predicted using the linear relationship between the asset’s expected return and a number of macroeconomic variables that capture systematic risk. It is a useful tool for analyzing portfolios from a value investing perspective, in order to identify securities that may be temporarily mispriced.
The Formula for the Arbitrage Pricing Theory Model Is
\begin{aligned} &\text{E(R)}_\text{i} = E(R)_z + (E(I) - E(R)_z) \times \beta_n\\ &\textbf{where:}\\ &\text{E(R)}_\text{i} = \text{Expected return on the asset}\\ &R_z = \text{Risk-free rate of return}\\ &\beta_n = \text{Sensitivity of the asset price to macroeconomic} \\ &\text{factor}\textit{ n}\\ &Ei = \text{Risk premium associated with factor}\textit{ i}\\ \end{aligned}
E(R)i=E(R)z+(E(I)−E(R)z)×βnwhere:E(R)i=Expected return on the assetRz=Risk-free rate of returnβn=Sensitivity of the asset price to macroeconomicfactor nEi=Risk premium associated with factor i
The beta coefficients in the APT model are estimated by using linear regression. In general, historical securities returns are regressed on the factor to estimate its beta.
How the Arbitrage Pricing Theory Works
The arbitrage pricing theory was developed by the economist Stephen Ross in 1976, as an alternative to the capital asset pricing model (CAPM). Unlike the CAPM, which assume markets are perfectly efficient, APT assumes markets sometimes misprice securities, before the market eventually corrects and securities move back to fair value. Using APT, arbitrageurs hope to take advantage of any deviations from fair market value.
However, this is not a risk-free operation in the classic sense of arbitrage, because investors are assuming that the model is correct and making directional trades—rather than locking in risk-free profits.
Mathematical Model for the APT
While APT is more flexible than the CAPM, it is more complex. The CAPM only takes into account one factor—market risk—while the APT formula has multiple factors. And it takes a considerable amount of research to determine how sensitive a security is to various macroeconomic risks.
The factors as well as how many of them are used are subjective choices, which means investors will have varying results depending on their choice. However, four or five factors will usually explain most of a security's return. (For more on the differences between the CAPM and APT, read more about how CAPM and arbitrage pricing theory differ.)
APT factors are the systematic risk that cannot be reduced by the diversification of an investment portfolio. The macroeconomic factors that have proven most reliable as price predictors include unexpected changes in inflation, gross national product (GNP), corporate bond spreads and shifts in the yield curve. Other commonly used factors are gross domestic product (GDP), commodities prices, market indices, and exchange rates.
Arbitrage pricing theory (APT) is a multi-factor asset pricing model based on the idea that an asset's returns can be predicted using the linear relationship between the asset’s expected return and a number of macroeconomic variables that capture systematic risk.
Unlike the CAPM, which assume markets are perfectly efficient, APT assumes markets sometimes misprice securities, before the market eventually corrects and securities move back to fair value.
Example of How Arbitrage Pricing Theory Is Used
For example, the following four factors have been identified as explaining a stock's return and its sensitivity to each factor and the risk premium associated with each factor have been calculated:
Gross domestic product (GDP) growth: ß = 0.6, RP = 4%
Inflation rate: ß = 0.8, RP = 2%
Gold prices: ß = -0.7, RP = 5%
Standard and Poor's 500 index return: ß = 1.3, RP = 9%
Using the APT formula, the expected return is calculated as:
Expected return = 3% + (0.6 x 4%) + (0.8 x 2%) + (-0.7 x 5%) + (1.3 x 9%) = 15.2%
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Configure - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Network Communication : Sockets Package : Configure
set configuration options on a socket connection
Configure(sid, options)
(optional) sequence of option names or equations
A number of low level options on an open socket connection can be queried or modified by calling the procedure Configure. The argument sid is a valid and open socket ID that identifies the socket connection to be configured.
Following the sid argument, you can specify a sequence of names of valid options or equations of the form optionName = optionVal, where optionName is a valid option name and optionVal is a valid value for that configuration option. See Configuration Options below for information about valid names and options.
An option name appearing by itself is treated as a query for the current value of that option. An equation is treated as a request to set the value of an option to the value on the right-hand side of the equation. The value must be a valid for the corresponding option.
The value returned by Configure is an expression sequence whose length is equal to the number of arguments passed in the call after the sid argument (one for each option argument). The
i
-th entry in the returned expression sequence is the previous value of the configuration option named in the
i+1
-st argument.
The following table indicates the supported options, their default values, the type of a valid value, and any relevant units.
The warm Option
The warm option is normally not set. However, if warm is set, it requests the system to "keep the connection warm" so that it is not dropped after periods of inactivity.
The buffer Option
The buffer option is used to configure the buffer size used by the underlying TCP/IP implementation for the indicated socket. This option affects the size of the buffer used for both sending and receiving. (There is currently no mechanism for configuring these independently.) The default buffer size is system dependent, as are the minimum and maximum sizes that can be configured.
The timeout option is used to implement the persistent timeout protocol. Its default value is
-1
, which indicates that no persistent timeout has been configured on the connection. Setting timeout to any non-negative integer value will set a persistent timeout on the connection, affecting the blocking behavior of read operations on the socket.
\mathrm{with}\left(\mathrm{Sockets}\right):
\mathrm{sid}≔\mathrm{Open}\left("localhost","echo"\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{Configure}\left(\mathrm{sid}\right)
\mathrm{Configure}\left(\mathrm{sid},'\mathrm{buffer}'\right)
\textcolor[rgb]{0,0,1}{87040}
\mathrm{Configure}\left(\mathrm{sid},'\mathrm{buffer}'=200\right)
\textcolor[rgb]{0,0,1}{87040}
\mathrm{Configure}\left(\mathrm{sid},'\mathrm{buffer}'\right)
\textcolor[rgb]{0,0,1}{4608}
\mathrm{Configure}\left(\mathrm{sid},'\mathrm{buffer}','\mathrm{warm}'\right)
\textcolor[rgb]{0,0,1}{4608}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{Configure}\left(\mathrm{sid},'\mathrm{buffer}'=1024,'\mathrm{warm}'=\mathrm{true}\right)
\textcolor[rgb]{0,0,1}{4608}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}
Sockets[Open]
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Effective action - Wikipedia
In quantum field theory, a quantum corrected version of the classical action
In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective action yields the equations of motion for the vacuum expectation values of the quantum fields. The effective action also acts as a generating functional for one-particle irreducible correlation functions. The potential component of the effective action is called the effective potential, with the expectation value of the true vacuum being the minimum of this potential rather than the classical potential, making it important for studying spontaneous symmetry breaking.
It was first defined perturbatively by Jeffrey Goldstone and Steven Weinberg in 1962,[1] while the non-perturbative definition was introduced by Bryce DeWitt in 1963[2] and independently by Giovanni Jona-Lasinio in 1964.[3]
The article describes the effective action for a single scalar field, however, similar results exist for multiple scalar or fermionic fields.
1 Generating functionals
2 Methods for calculating the effective action
Generating functionals[edit]
These generation functionals also have applications in statistical mechanics and information theory, with slightly different factors of
{\displaystyle i}
and sign conventions.
A quantum field theory with action
{\displaystyle S[\phi ]}
can be fully described in the path integral formalism using the partition functional
{\displaystyle Z[J]=\int {\mathcal {D}}\phi e^{iS[\phi ]+i\int d^{4}x\phi (x)J(x)}.}
Since it corresponds to vacuum-to-vacuum transitions in the presence of a classical external current
{\displaystyle J(x)}
, it can be evaluated perturbatively as the sum of all connected and disconnected Feynman diagrams. It is also the generating functional for correlation functions
{\displaystyle \langle {\hat {\phi }}(x_{1})\dots {\hat {\phi }}(x_{n})\rangle =(-i)^{n}{\frac {1}{Z[J]}}{\frac {\delta ^{n}Z[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}{\bigg |}_{J=0},}
where the scalar field operators are denoted by
{\displaystyle {\hat {\phi }}(x)}
. One can define another useful generating functional
{\displaystyle W[J]=-i\ln Z[J]}
responsible for generating connected correlation functions
{\displaystyle \langle {\hat {\phi }}(x_{1})\cdots {\hat {\phi }}(x_{n})\rangle _{\text{con}}=(-i)^{n-1}{\frac {\delta ^{n}W[J]}{\delta J(x_{1})\dots \delta J(x_{n})}}{\bigg |}_{J=0},}
which is calculated perturbatively as the sum of all connected diagrams. Here connected is interpreted in the sense of the cluster decomposition, meaning that the correlation functions approach zero at large spacelike separations. General correlation functions can always be written as a sum of products of connected correlation functions.
The quantum effective action is defined using the Legendre transformation of
{\displaystyle W[J]}
{\displaystyle \Gamma [\phi ]=W[J_{\phi }]-\int d^{4}xJ_{\phi }(x)\phi (x),}
{\displaystyle J_{\phi }}
is the source current for which the scalar field has the expectation value
{\displaystyle \phi (x)}
, often called the classical field, defined implicitly as the solution to
Example of a diagram that is not one-particle irreducible.
Example of a diagram that is one-particle irreducible.
{\displaystyle \phi (x)=\langle {\hat {\phi }}(x)\rangle _{J}={\frac {\delta W[J]}{\delta J(x)}}.}
As an expectation value, the classical field can be thought of as the weighted average over quantum fluctuations in the presence of a current
{\displaystyle J(x)}
that sources the scalar field. Taking the functional derivative of the Legendre transformation with respect to
{\displaystyle \phi (x)}
{\displaystyle J_{\phi }(x)=-{\frac {\delta \Gamma [\phi ]}{\delta \phi (x)}}.}
In the absence of an source
{\displaystyle J_{\phi }(x)=0}
, the above shows that the vacuum expectation value of the fields extremize the quantum effective action rather than the classical action. This is nothing more than the principle of least action in the full quantum field theory. The reason for why the quantum theory requires this modification comes from the path integral perspective since all possible field configurations contribute to the path integral, while in classical field theory only the classical configurations contribute.
The effective action is also the generation functional for one-particle irreducible (1PI) correlation functions. 1PI diagrams are connected graphs that cannot be disconnected into two pieces by cutting a single internal line. Therefore, we have
{\displaystyle \langle {\hat {\phi }}(x_{1})\dots {\hat {\phi }}(x_{n})\rangle _{\mathrm {1PI} }=i{\frac {\delta ^{n}\Gamma [\phi ]}{\delta \phi (x_{1})\dots \delta \phi (x_{n})}}{\bigg |}_{J=0},}
{\displaystyle \Gamma [\phi ]}
being the sum of all 1PI Feynman diagrams. The close connection between
{\displaystyle W[J]}
{\displaystyle \Gamma [\phi ]}
means that there are a number of very useful relations between their correlation functions. For example, the two-point correlation function, which is nothing less than the propagator
{\displaystyle \Delta (x,y)}
, is the inverse of the 1PI two-point correlation function
{\displaystyle \Delta (x,y)={\frac {\delta ^{2}W[J]}{\delta J(x)\delta J(y)}}={\frac {\delta \phi (x)}{\delta J(y)}}={\bigg (}{\frac {\delta J(y)}{\delta \phi (x)}}{\bigg )}^{-1}=-{\bigg (}{\frac {\delta ^{2}\Gamma (\phi )}{\delta \phi (x)\delta \phi (y)}}{\bigg )}^{-1}=-\Pi ^{-1}(x,y).}
Methods for calculating the effective action[edit]
A direct way to calculate the effective action
{\displaystyle \Gamma [\phi _{0}]}
perturbatively as a sum of 1PI diagrams is to sum over all 1PI vacuum diagrams acquired using the Feynman rules derived from the shifted action
{\displaystyle S[\phi +\phi _{0}]}
. This works because any place where
{\displaystyle \phi _{0}}
appears in any of the propagators or vertices is a place where an external
{\displaystyle \phi }
line could be attached. This is very similar to the background field method which can also be used to calculate the effective action.
Alternatively, the one-loop approximation to the action can be found by considering the expansion of the partition function around the classical vacuum expectation value field configuration
{\displaystyle \phi (x)=\phi _{\text{cl}}(x)+\delta \phi (x)}
, yielding[4][5]
{\displaystyle \Gamma [\phi _{\text{cl}}]=S[\phi _{\text{cl}}]+{\frac {i}{2}}{\text{Tr}}{\bigg [}\ln {\frac {\delta ^{2}S[\phi ]}{\delta \phi (x)\delta \phi (y)}}{\bigg |}_{\phi =\phi _{\text{cl}}}{\bigg ]}+\cdots .}
Symmetries of the classical action
{\displaystyle S[\phi ]}
are not automatically symmetries of the quantum effective action
{\displaystyle \Gamma [\phi ]}
. If the classical action has a continuous symmetry depending on some functional
{\displaystyle F[x,\phi ]}
{\displaystyle \phi (x)\rightarrow \phi (x)+\epsilon F[x,\phi ],}
then this directly imposes the constraint
{\displaystyle 0=\int d^{4}x\langle F[x,\phi ]\rangle _{J_{\phi }}{\frac {\delta \Gamma [\phi ]}{\delta \phi (x)}}.}
This identity is an example of a Slavnov–Taylor identity. It is identical to the requirement that the effective action is invariant under the symmetry transformation
{\displaystyle \phi (x)\rightarrow \phi (x)+\epsilon \langle F[x,\phi ]\rangle _{J_{\phi }}.}
This symmetry is identical to the original symmetry for the important class of linear symmetries
{\displaystyle F[x,\phi ]=a(x)+\int d^{4}y\ b(x,y)\phi (y).}
For non-linear functionals the two symmetries generally differ because the average of a non-linear functional is not equivalent to the functional of an average.
The apparent effective potential
{\displaystyle V_{0}(\phi )}
acquired via perturbation theory must be corrected to the true effective potential
{\displaystyle V(\phi )}
, shown via dashed lines in region where the two disagree.
For a spacetime with volume
{\displaystyle {\mathcal {V}}_{4}}
, the effective potential is defined as
{\displaystyle V(\phi )=-\Gamma [\phi ]/{\mathcal {V}}_{4}}
. With a Hamiltonian
{\displaystyle H}
, the effective potential
{\displaystyle V(\phi )}
{\displaystyle \phi (x)}
always gives the minimum of the expectation value of the energy density
{\displaystyle \langle \Omega |H|\Omega \rangle }
for the set of states
{\displaystyle |\Omega \rangle }
{\displaystyle \langle \Omega |{\hat {\phi }}|\Omega \rangle =\phi (x)}
.[6] This definition over multiple states is necessary because multiple different states, each of which corresponds to a particular source current, may result in the same expectation value. It can further be shown that the effective potential is necessarily a convex function
{\displaystyle V''(\phi )\geq 0}
Calculating the effective potential perturbatively can sometimes yield a non-convex result, such as a potential that has two local minima. However, the true effective potential is still convex, becoming approximately linear where the apparent effective potential fails to be convex. The contradiction occurs when one is dealing with a situation in which the vacuum is unstable, while perturbation theory necessarily assumes that the vacuum is stable. For example, consider an apparent effective potential
{\displaystyle V_{0}(\phi )}
with two local minima whose expectation values
{\displaystyle \phi _{1}}
{\displaystyle \phi _{2}}
are the expectation values for the states
{\displaystyle |\Omega _{1}\rangle }
{\displaystyle |\Omega _{2}\rangle }
, respectively. Then any
{\displaystyle \phi }
in the non-convex region of
{\displaystyle V_{0}(\phi )}
can also be acquired for some
{\displaystyle \lambda \in [0,1]}
{\displaystyle |\Omega \rangle \propto {\sqrt {\lambda }}|\Omega _{1}\rangle +{\sqrt {1-\lambda }}|\Omega _{2}\rangle .}
However, the energy density of this state is
{\displaystyle \lambda V_{0}(\phi _{1})+(1-\lambda )V_{0}(\phi _{2})<V_{0}(\phi )}
{\displaystyle V_{0}(\phi )}
cannot be the correct effective potential at
{\displaystyle \phi }
since it did not minimize the energy density. Rather the true effective potential
{\displaystyle V(\phi )}
is equal to or lower than this linear construction, which restores convexity.
^ Weinberg, S.; Goldstone, J. (August 1962). "Broken Symmetries". Phys. Rev. 127 (3): 965–970. doi:10.1103/PhysRev.127.965. Retrieved 2021-09-06.
^ DeWitt, B.; DeWitt, C. (1987). Relativité, groupes et topologie = Relativity, groups and topology : lectures delivered at Les Houches during the 1963 session of the Summer School of Theoretical Physics, University of Grenoble. Gordon and Breach. ISBN 0677100809.
^ Jona-Lasinio, G. (31 August 1964). "Relativistic Field Theories with Symmetry-Breaking Solutions". Il Nuovo Cimento. 34 (6): 1790–1795. doi:10.1007/BF02750573. Retrieved 2021-09-06.
^ Kleinert, H. (2016). "22" (PDF). Particles and Quantum Fields. World Scientific Publishing. p. 1257. ISBN 9789814740920.
^ Zee, A. (2010). Quantum Field Theory in a Nutshell (2 ed.). Princeton University Press. p. 239-240. ISBN 9780691140346.
^ Weinberg, S. (1995). "16". The Quantum Theory of Fields Volume 2. Vol. 2. Cambridge University Press. p. 72-74. ISBN 9780521670548.
^ Peskin, M.E.; Schroeder, D.V. (1995). An Introduction to Quantum Field Theory. Westview Press. p. 368-369. ISBN 9780201503975.
Das, A. : Field Theory: A Path Integral Approach, World Scientific Publishing 2006
Schwartz, M.D.: Quantum Field Theory and the Standard Model, Cambridge University Press 2014
Toms, D.J.: The Schwinger Action Principle and Effective Action, Cambridge University Press 2007
Weinberg, S.: The Quantum Theory of Fields, Vol.II, Cambridge University Press 1996
Retrieved from "https://en.wikipedia.org/w/index.php?title=Effective_action&oldid=1086038591"
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Reflect lag operator polynomial coefficients around lag zero - MATLAB - MathWorks Australia
Reflect a Lag Operator Polynomial
Reflect lag operator polynomial coefficients around lag zero
B = reflect(A)
Given a lag operator polynomial object A(L),B = reflect(A) negates all coefficient matrices except the coefficient matrix at lag 0. For example, given a polynomial of degree p,
A\left(L\right)={A}_{0}+{A}_{1}L+{A}_{2}{L}^{2}+...+{A}_{P}{L}^{p}
the reflected polynomial B(L) is
B\left(L\right)={A}_{0}-{A}_{1}L-{A}_{2}{L}^{2}-...-{A}_{P}{L}^{p}
with the same degree and dimension as A(L).
Create a LagOp polynomial and its reflection:
Coefficients: [0.8 -1 -0.6]
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EUDML | The -Laplacian and connected pairs of functions. EuDML | The -Laplacian and connected pairs of functions.
p
-Laplacian and connected pairs of functions.
Chantladze, T., et al. "The -Laplacian and connected pairs of functions.." Memoirs on Differential Equations and Mathematical Physics 20 (2000): 113-126. <http://eudml.org/doc/231060>.
@article{Chantladze2000,
author = {Chantladze, T., Kandelaki, N., Lomtatidze, A., Ugulava, D.},
keywords = {half-linear ordinary differential equations; -Laplacian; connected pairs; Legendre transformation; -Laplacian},
title = {The -Laplacian and connected pairs of functions.},
AU - Chantladze, T.
AU - Kandelaki, N.
AU - Lomtatidze, A.
TI - The -Laplacian and connected pairs of functions.
KW - half-linear ordinary differential equations; -Laplacian; connected pairs; Legendre transformation; -Laplacian
half-linear ordinary differential equations,
p
-Laplacian, connected pairs, Legendre transformation,
p
Articles by Chantladze
Articles by Kandelaki
Articles by Lomtatidze
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Revision as of 00:31, 23 November 2008 by Han (talk | contribs) (Typo in CDS110 problem 1(a). moved to A typo in equation (6.24).)
{\displaystyle G(s)\,}
{\displaystyle H_{yr}(s)\,}
{\displaystyle {\frac {\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}\,}
{\displaystyle \omega _{0}^{2}\,}
{\displaystyle \omega _{0}\,}
{\displaystyle s\,}
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Down, | Beckenham, Kent. | Railway Station. | Orpington. S.E.R.
I am much obliged for the cheque & enclose a receipt.
I have no corrections for the “Descent of Man”, so you can have 1000(?) printed off.—1
You will have understood that I am quite willing & glad that you shd. stereotype the Orchids.—2
The type must be broken up of the “Forms of Flowers”.3
With respect to “Cross-Fertilisation” as you are out of copies, a new Edit. must be prepared & the Edit may then be stereotyped. Please ask Mess. Clowes (as the type of two books will be freed) to keep up Cross-Fertilisation for about 2 or 3 weeks more, by which time I hope to be able to have all corrections ready.—4 They could be ready in a week or less, if I could give up all my time, but this is impossible.—5
Please inform me in answer to my query to Mr Cooke whether the Edit of 2000 of the Origin includes the 1000 of which I heard a month ago.—6 The number of Thousandth on the Title-page must be altered.
Lastly please send me a copy in sheets of the “Forms of Flowers”, that I may keep it in case a new Edit. is ever wanted.—
Thanks for your kind congratulations about the L.L.D.7
I am sorry to trouble you about so many points, but when you answer this note be so good as to glance through this note.
P.S. | On reflexion I do not understand how it is that the profits of the two books amounts exactly to 200 guineas. Our agreement has always been that I have
\frac{2}{3}
profits, though in the case of the book on Orchids I remember I agreed to receive only half-profits.—8 I shd. like to hear about this point.—
See letter from John Murray, 27 November [1877].
Orchids 2d ed. was published in January 1877 (Publishers’ Circular, 1 February 1877, p. 93). See letter to R. F. Cooke, 24 November 1877.
Forms of flowers was published in July 1877 (CD’s ‘Journal’ (Appendix II)). See letter from R. F. Cooke, 23 November 1877, and letter to R. F. Cooke, 24 November 1877.
CD submitted corrected proof-sheets of Cross and self fertilisation to William Clowes & Sons, Murray’s printers, on 11 December 1877 (letter to R. F. Cooke, 11 December [1877]). The second edition of Cross and self fertilisation was published in 1878.
According to his ‘Journal’ (Appendix II), CD spent the second half of 1877 working on ‘Bloom — Spontaneous Movement of Plants & Heliotropism & a little on Worms’; his work on movement in plants was published in Movement in plants (1880) and his work on worms in Earthworms (1881).
See letter to R. F. Cooke, 24 November 1877.
CD was awarded an honorary LLD at Cambridge on 17 November 1877 (Emma Darwin’s diary (DAR 242)).
See Correspondence vol. 9, letter from John Murray, 23 September 1861.
On publishing details for various CD books.
Has no corrections for new issue of Descent [2d ed.].
Questions amount of cheque for profits.
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Calculating the Consumer Surplus as an Area - Course Hero
Microeconomics/Consumer and Producer Surplus/Calculating the Consumer Surplus as an Area
Learn all about calculating the consumer surplus as an area in just a few minutes! Professor Jadrian Wooten of Penn State University walks through the process of calculating consumer surplus as an area.
Consumer surplus is the difference between how much a consumer is willing to pay for something and the actual price of that thing. It can be calculated as an area. A step diagram uses horizontal lines to represent intervals (steps): a range from one point to another point. By drawing a demand curve as a step diagram, each individual's consumer surplus (the area below a consumer's willingness to pay and above the market price) is also the area of a rectangle. Total consumer surplus is the sum of all individual consumer surpluses, so it is the total area below the market demand curve and above the market price.
The area of a rectangle is its base multiplied by its height. The base of Andrea's step is the additional number of cups she adds to the market (
1 - 0=1
). The height of her step is her willingness to pay minus the market price, which is two (
\$4 -\$2=\$2
). Andrea's consumer surplus is the base of her step times the height of her step:
\$2\times1=\$2
. For Brett, the base of his step is also one (
2-1=1
). The height of his step is one (
\$3 -\$2=\$1
). Brett's consumer surplus is
\$1\times1=\$1
. For Christy, her consumer surplus is
\$0.50\times 1=\$0.50
. Adding up the area of each rectangle gives the total consumer surplus:
\$2+\$1+\$0.50=\$3.50
. There is no surplus for Deb, Eddie, or Frank because the market price is above their willingness to pay, so they choose not to purchase coffee.
To calculate the consumer surplus for individuals in this market, multiply the base of their step (the quantity) by the height of their step (willingness to pay minus market price). The base of each step in this case is 1 cup of coffee. Total consumer surplus in this market is the sum of the individual surpluses.
The demand curve can be drawn as a step diagram for a scenario involving only few potential consumers in the market. With a big market that has a large number of consumers in it, those individual steps will smooth out and we will have a demand curve that is a downward sloping line. Smoothing out data is done to show key patterns in data, without zeroing in on fine details. The coffee market in the United States is a big market with a large number of consumers. It is estimated that half of the US population, around 150 million people, drink coffee. Americans consume 400 million cups of coffee per day, with an average price in a coffee shop of $2.38 per cup.
The US coffee market serves around 400 million cups per day at an average price (market price) of $2.38 per cup. The consumer surplus for this market is the area below the demand curve and above the market price.
With a smoothed out demand curve because of the size of the market, the consumer surplus (the area below the demand curve and above the market price) is now the area of a triangle. More specifically, it is a right triangle, because two of the sides (the axes) form a right angle. The area of a right triangle is
\left(1/2\right) \times b \times h
b
h
is the height. The base of the triangle is 400 million. The height of the triangle is eight. In the example, the demand curve crosses the price axis at $10.00, and the price line is at $2.38, so the height is
\$10.00-\$2.38=\$7.62
. The consumer surplus in the example is thus $1.524 billion:
\$1.524\;\text{billion per day}=\frac{1}{2}\times 400\;\text{million}\times\$7.62
<Consumer Surplus>How Changing Prices Affect Consumer Surplus
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On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with n Vertices and m Pendent Vertices
2012 On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with
n
m
Pendent Vertices
Xinying Pai, Sanyang Liu
\mathrm{\Phi }\left(G,\lambda \right)=\text{d}\text{e}\text{t}\left(\lambda {I}_{n}-L\left(G\right)\right)={\sum }_{k=0}^{n}\left(-1{\right)}^{k}{c}_{k}\left(G\right){\lambda }^{n-k}
be the characteristic polynomial of the Laplacian matrix of a graph
G
n
. In this paper, we give four transforms on graphs that decrease all Laplacian coefficients
{c}_{k}\left(G\right)
and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs with
n
m
pendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs with
n
m
pendent vertices.
Xinying Pai. Sanyang Liu. "On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with
n
m
Pendent Vertices." J. Appl. Math. 2012 1 - 11, 2012. https://doi.org/10.1155/2012/404067
Xinying Pai, Sanyang Liu "On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs with
n
m
Pendent Vertices," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-11, (2012)
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Discrete interpolating varieties in pseudoconvex open sets of Cn
October, 2006 Discrete interpolating varieties in pseudoconvex open sets of Cn
We give a necessary and sufficient condition for a discrete variety in a pseudoconvex open set
\mathrm{\Omega }
{\mathbf{C}}^{n}
to be an interpolating variety for Hörmander's weighted algebras of holomorphic functions in
\mathrm{\Omega }
Bao Qin LI. "Discrete interpolating varieties in pseudoconvex open sets of Cn." J. Math. Soc. Japan 58 (4) 1185 - 1196, October, 2006. https://doi.org/10.2969/jmsj/1179759543
Secondary: 32A15 , 32C25 , 46E10
Keywords: holomorphic function , interpolating variety , pseudoconvex set , weight
Bao Qin LI "Discrete interpolating varieties in pseudoconvex open sets of Cn," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 58(4), 1185-1196, (October, 2006)
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Convert number to signed integer using quantizer object - MATLAB num2int - MathWorks Italia
Convert Matrix of Numeric Values to Signed Integer
Convert number to signed integer using quantizer object
y = num2int(q,x) converts numeric values in x to output y containing integers using the data type properties specified by the fixed-point quantizer object q. If x is a cell array containing numeric matrices, then y will be a cell array of the same dimension.
[y1,y2,…] = num2int(q,x1,x2,…) uses q to convert numeric values x1, x2,… to integers y1, y2,….
All the two's complement 4-bit numbers in fractional form are given by:
Define a quantizer object to use for conversion.
Use num2int to convert to signed integer.
q — Data type format to use for conversion
fixed-point quantizer object
Data type format to use for conversion, specified as a fixed-point quantizer object.
Example: q = quantizer([5 4]);
x — Numeric values to convert
scalar | vector | matrix | multidimensional array | cell array
Numeric values to convert, specified as a scalar, vector, matrix, multidimensional array, or cell array.
When q is a fixed-point quantizer object, f is equal to fractionlength(q), and x is numeric:
y=x×{2}^{f}
num2int is meaningful only for fixed-point quantizer objects. When q is a floating-point quantizer object, x is returned unchanged (y = x).
y is returned as a double, but the numeric values will be integers, also known as floating-point integers or flints.
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The Game Logics - HunnyDAO
deposit = withdrawal
Swaps between LOVE and KISS during staking and unstaking are always 1:1. The amount of LOVE deposited into the staking contract will always result in the same amount of KISS. And the amount of KISS withdrawn from the staking contract will always result in the same amount of LOVE.
rebase = 1 - (LOVE_{deposit} / KISS_{outstanding})
The treasury deposits LOVE into the distributor. The distributor then deposits LOVE into the staking contract, creating an imbalance between LOVE and KISS. KISS is then rebased to correct this imbalance between LOVE deposited and KISS outstanding. The rebase brings KISS outstanding back up to parity so that 1 KISS equals 1 staked LOVE.
bondPrice = 1 + premium
LOVE has an IV of 1 BUSD, which is roughly equivalent to 1 USD. In order to make a profit from bonding, HunnyDAO charges a premium for each bond.
premium = debtRatio * BCV
The premium determines profit due to the protocol and in turn, stakers. This is because new LOVE is minted from the profit and subsequently distributed among all stakers.
debtRatio = bondOutstanding / LOVE_{supply}
The debt ratio is the total of all LOVE promised to bonders divided by the total supply of LOVE. This allows us to measure the debt of the system.
bondPayout_{reservebond} = marketValue_{asset} / bondPrice
Bond payout determines the number of LOVE sold to a bonder. For reserve bonds, the market value of the assets supplied by the bonder is used to determine the bond payout. For example, if a user supplies 1000 BUSD and the bond price is 250 BUSD, the user will be entitled to 4 LOVE tokens.
bondPayout_{lpBond} = marketValue_{lpToken} / bondPrice
For liquidity bonds, the market value of the LP tokens supplied by the bonder is used to determine the bond payout. For example, if a user supplies 0.001 LOVE-BUSD LP token which is valued at 1000 BUSD at the time of bonding, and the bond price is 250 BUSD, the user will be entitled to 4 LOVE tokens.
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DateDifference - Maple Help
Home : Support : Online Help : Programming : Date and Time : Calendar package : DateDifference
DateDifference( date1, date2 )
DateDifference( date1, date2, units = u )
Date; the first date
Date; the second date
symbol; a Unit symbol, or the name mixed
The DateDifference( date1, date2 ) command computes the time between the dates represented by two Date objects date1 and date2.
By default, the difference of the two dates is returned using the units of the system UTC clock. This is the same result computed by the expression date2 - date1. However, you can express the time difference between date1 and date2 in other units by using the units = u option with either a valid unit symbol u (e.g., days), or the special name mixed.
\mathrm{with}\left(\mathrm{Calendar}\right):
\mathrm{d1}≔\mathrm{Date}\left(2016,3,4,2,2,0,'\mathrm{timezone}'="America/Caracas"\right)
\textcolor[rgb]{0,0,1}{\mathrm{d1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{<Date: 2016-03-04T02:02:00 Venezuela Time>}}
\mathrm{d2}≔\mathrm{Date}\left(1997,1,1,0,'\mathrm{timezone}'="Asia/Hong_Kong"\right)
\textcolor[rgb]{0,0,1}{\mathrm{d2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{<Date: 1997-01-01 Hong Kong Standard Time>}}
\mathrm{DateDifference}\left(\mathrm{d2},\mathrm{d1}\right)
\textcolor[rgb]{0,0,1}{605025120000}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{ms}}⟧
\mathrm{DateDifference}\left(\mathrm{d2},\mathrm{d1},'\mathrm{units}'='\mathrm{days}'\right)
\frac{\textcolor[rgb]{0,0,1}{1260469}}{\textcolor[rgb]{0,0,1}{180}}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{d}⟧
\mathrm{DateDifference}\left(\mathrm{d2},\mathrm{d1},'\mathrm{units}'='\mathrm{hours}'\right)
\frac{\textcolor[rgb]{0,0,1}{2520938}}{\textcolor[rgb]{0,0,1}{15}}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{h}⟧
\mathrm{DateDifference}\left(\mathrm{d2},\mathrm{d1},'\mathrm{units}'='\mathrm{mixed}'\right)
\textcolor[rgb]{0,0,1}{19}\textcolor[rgb]{0,0,1}{}⟦{\textcolor[rgb]{0,0,1}{\mathrm{yr}}}_{\textcolor[rgb]{0,0,1}{\mathrm{standard}}}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{mo}}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{d}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{13}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{h}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{min}}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{54}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{s}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{400}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{ms}}⟧
\mathrm{BattleOfMarathon}≔\mathrm{Date}\left(-489,9,12\right)
\textcolor[rgb]{0,0,1}{\mathrm{BattleOfMarathon}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{<Date: 0490-09-12T12:00:00 GMT>}}
\mathrm{BattleOfThermopylae}≔\mathrm{Date}\left(-479,8,20\right)
\textcolor[rgb]{0,0,1}{\mathrm{BattleOfThermopylae}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{<Date: 0480-08-20T12:00:00 GMT>}}
\mathrm{DateDifference}\left(\mathrm{BattleOfMarathon},\mathrm{BattleOfThermopylae},'\mathrm{units}'='\mathrm{mixed}'\right)
\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{}⟦{\textcolor[rgb]{0,0,1}{\mathrm{yr}}}_{\textcolor[rgb]{0,0,1}{\mathrm{standard}}}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{mo}}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{20}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{d}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{h}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{55}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{min}}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{29}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{s}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{200}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{ms}}⟧
The Calendar[DateDifference] command was introduced in Maple 2018.
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Electrostatics and Magnetostatics - MATLAB & Simulink - MathWorks í•œêµ
\begin{array}{c}\mathrm{ε}\text{â}ââ
E=\mathrm{Ï},\\ ââ
H=0,\\ âÃE=â\mathrm{μ}\frac{âH}{ât},\\ âÃH=\mathrm{ε}\frac{âE}{ât}+J.\end{array}
Here, E and H are the electric and magnetic fields, ε and µ are the electrical permittivity and magnetic permeability of the material, and Ï and J are the electric charge and current densities.
\begin{array}{l}ââ
\left(\mathrm{ε}\text{â}E\right)=\mathrm{Ï},\\ \text{â}âÃE=0.\end{array}
E=ââV
âââ
\left(\mathrm{ε}\text{â}âV\right)=\mathrm{Ï}.
\begin{array}{l}ââ
H=0,\\ âÃH=J.\end{array}
ââ
H=0
\begin{array}{l}H={\mathrm{μ}}^{â1}âÃA,\\ âÃ\left({\mathrm{μ}}^{â1}âÃA\right)=J.\end{array}
âÃ\left(âÃA\right)=â\left(ââ
A\right)â{â}^{2}A
â·A=0
â{â}^{2}A=âââ
âA=\mathrm{μ}J.
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Some Novikov rings that are von Neumann finite and knot-like groups | EMS Press
Some Novikov rings that are von Neumann finite and knot-like groups
IMECC - UNICAMP, Campinas, Sp, Brazil
We show that for a finitely generated group
G
and for every discrete character
\chi\colon G \rightarrow \mathbb{Z}
any matrix ring over the Novikov ring
\widehat{\mathbb{Z}G}_{\chi}
is von Neumann finite. As a corollary we obtain that if
G
is a non-trivial discrete group with a finite
K(G,1)
CW-complex
Y
of dimension
and Euler characteristics zero and
N
G
FP_{n-1}
containing the commutator subgroup
G'
G/N
is cyclic-by-finite, then
N
is of homological type
FP_n
G/N
has finite virtual cohomological dimension
vcd(G/N) = cd(G) - cd(N).
This completes the proof of the Rapaport Strasser conjecture that for a knot-like group
G
with a finitely generated commutator subgroup
G'
the commutator subgroup
G'
is always free and generalises an earlier work by the author where the case when
G'
is residually finite was proved. Another corollary is that a finitely presentable group
G
def(G) > 0
G'
is finitely generated and perfect can be only
\mathbb{Z}
\mathbb{Z}^2
, a result conjectured by A. J. Berrick and J. Hillman in [1].
Dessislava H. Kochloukova, Some Novikov rings that are von Neumann finite and knot-like groups. Comment. Math. Helv. 81 (2006), no. 4, pp. 931–943
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Dec - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Ordinals : Dec
decrement ordinal
ordinal data structure, nonnegative integer, polynomial with positive integer coefficients, or
\mathrm{NULL}
The Dec(a) calling sequence decrements the ordinal number
a
, if possible. If
a=0
, then the return value is
\mathrm{NULL}
. Otherwise, if the trailing term is
{\mathbf{\omega }}^{e}\cdot c
, where
is an ordinal and
c
is a positive integer, then exactly one of the following happens:
e=0
c
c-1
e\ne 0
c\ne 1
, then the trailing term is replaced by the sum of the two terms
{\mathbf{\omega }}^{e}\cdot \left(c-1\right)+{\mathbf{\omega }}^{\mathrm{Dec}\left(e\right)}
e\ne 0
c=1
, then the trailing exponent is replaced by
\mathrm{Dec}\left(e\right)
Note that in general
\mathrm{Dec}\left(a\right)
is not the largest ordinal number smaller than
a
, because such an ordinal does not exist if
a
is a limit ordinal, which means its trailing degree is nonzero.
a
is a parametric ordinal number and
c-1
is not a polynomial with nonnegative integer coefficients, an error is raised.
\mathrm{with}\left(\mathrm{Ordinals}\right):
a≔\mathrm{Ordinal}\left([[\mathrm{\omega },2],[2,3],[0,4]]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}
\mathbf{while}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a\ne 0\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}a≔\mathrm{Dec}\left(a\right);\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\mathrm{print}\left(a\right)\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathbf{end}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{do}:
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}
\textcolor[rgb]{0,0,1}{1}
\textcolor[rgb]{0,0,1}{0}
\mathrm{Dec}\left(5\right)
\textcolor[rgb]{0,0,1}{4}
\mathrm{Dec}\left(x+3\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
b≔\mathrm{Ordinal}\left([[1,3],[0,{x}^{2}+x+2]]\right)
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)
\mathrm{Dec}\left(\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{Dec}\left(\right)
\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{Dec}\left(\right)
Error, (in Ordinals:-Dec) cannot decrement, x^2+x
The Ordinals[Dec] command was introduced in Maple 2015.
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RandomTournament - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : RandomGraphs : RandomTournament
generate random tournament
RandomTournament(V,options)
RandomTournament(n,options)
m\le n
x\le y
are decimals is specified, the graph is a weighted graph with numerical edge weights chosen from [x,y] uniformly at random. The weight matrix W in the graph ha and if the edge from vertex i to j is not in the graph then W[i,j] = 0.0.
RandomTournament(n) creates a random tournament on n vertices. This is a directed graph such that for every pair of vertices u and v either the arc u to v or the arc v to u is in the digraph.
If the first input is a positive integer n, then the vertices are labeled 1,2,...,n. Alternatively you may specify the vertex labels in a list.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{with}\left(\mathrm{RandomGraphs}\right):
T≔\mathrm{RandomTournament}\left(5\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: a directed unweighted graph with 5 vertices and 10 arc\left(s\right)}}
T≔\mathrm{RandomTournament}\left(5,\mathrm{weights}=1..5\right)
\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: a directed weighted graph with 5 vertices and 10 arc\left(s\right)}}
\mathrm{IsTournament}\left(T\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{WeightMatrix}\left(T\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{5}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}\end{array}]
GraphTheory:-IsTournament
|
Seasonal Adjustment - MATLAB & Simulink - MathWorks Australia
Deseasonalized Series
Economists and other practitioners are sometimes interested in extracting the global trends and business cycles of a time series, free from the effect of known seasonality. Small movements in the trend can be masked by a seasonal component, a trend with fixed and known periodicity (e.g., monthly or quarterly). The presence of seasonality can make it difficult to compare relative changes in two or more series.
Seasonal adjustment is the process of removing a nuisance periodic component. The result of a seasonal adjustment is a deseasonalized time series. Deseasonalized data is useful for exploring the trend and any remaining irregular component. Because information is lost during the seasonal adjustment process, you should retain the original data for future modeling purposes.
Consider decomposing a time series, yt, into three components:
Trend component, Tt
Seasonal component, St with known periodicity s
Irregular (stationary) stochastic component, It
The most common decompositions are additive, multiplicative, and log-additive.
To seasonally adjust a time series, first obtain an estimate of the seasonal component,
{\stackrel{^}{S}}_{t}
. The estimate
{\stackrel{^}{S}}_{t}
should be constrained to fluctuate around zero (at least approximately) for additive models, and around one, approximately, for multiplicative models. These constraints allow the seasonal component to be identifiable from the trend component.
{\stackrel{^}{S}}_{t}
, the deseasonalized series is calculated by subtracting (or dividing by) the estimated seasonal component, depending on the assumed decomposition.
For an additive decomposition, the deseasonalized series is given by
{d}_{t}={y}_{t}-{\stackrel{^}{S}}_{t}.
For a multiplicative decomposition, the deseasonalized series is given by
{d}_{t}={y}_{t}/{\stackrel{^}{S}}_{t}.
To best estimate the seasonal component of a series, you should first estimate and remove the trend component. Conversely, to best estimate the trend component, you should first estimate and remove the seasonal component. Thus, seasonal adjustment is typically performed as an iterative process. The following steps for seasonal adjustment resemble those used within the X-12-ARIMA seasonal adjustment program of the U.S. Census Bureau [1].
Obtain a first estimate of the trend component,
{\stackrel{^}{T}}_{t},
using a moving average or parametric trend estimate.
Detrend the original series. For an additive decomposition, calculate
{x}_{t}={y}_{t}-{\stackrel{^}{T}}_{t}
. For a multiplicative decomposition, calculate
{x}_{t}={y}_{t}/{\stackrel{^}{T}}_{t}
Apply a seasonal filter to the detrended series,
{x}_{t}
, to obtain an estimate of the seasonal component,
{\stackrel{^}{S}}_{t}
. Center the estimate to fluctuate around zero or one, depending on the chosen decomposition. Use an S3×3 seasonal filter if you have adequate data, or a stable seasonal filter otherwise.
Deseasonalize the original series. For an additive decomposition, calculate
{d}_{t}={y}_{t}-{\stackrel{^}{S}}_{t}
{d}_{t}={y}_{t}/{\stackrel{^}{S}}_{t}.
Obtain a second estimate of the trend component,
{\stackrel{^}{T}}_{t},
, using the deseasonalized series
{d}_{t}.
Consider using a Henderson filter [1], with asymmetric weights at the ends of the series.
Detrend the original series again. For an additive decomposition, calculate
{x}_{t}={y}_{t}-{\stackrel{^}{T}}_{t}
{x}_{t}={y}_{t}/{\stackrel{^}{T}}_{t}
{x}_{t}
{\stackrel{^}{S}}_{t}
. Consider using an S3×5 seasonal filter if you have adequate data, or a stable seasonal filter otherwise.
{d}_{t}={y}_{t}-{\stackrel{^}{S}}_{t}
{d}_{t}={y}_{t}/{\stackrel{^}{S}}_{t}.
This is the final deseasonalized series.
|
EUDML | Logarithmic coefficients of univalent functions. EuDML | Logarithmic coefficients of univalent functions.
Logarithmic coefficients of univalent functions.
Girela, Daniel. "Logarithmic coefficients of univalent functions.." Annales Academiae Scientiarum Fennicae. Mathematica 25.2 (2000): 337-350. <http://eudml.org/doc/120902>.
@article{Girela2000,
author = {Girela, Daniel},
keywords = {logarithmic coefficients of functions in the class ; conjecture of Milin; logarithmic coefficients of functions in the class },
title = {Logarithmic coefficients of univalent functions.},
TI - Logarithmic coefficients of univalent functions.
KW - logarithmic coefficients of functions in the class ; conjecture of Milin; logarithmic coefficients of functions in the class
logarithmic coefficients of functions in the class
S
, conjecture of Milin, logarithmic coefficients of functions in the class
Suggest a Subject
|
Logical equivalence - Wikipedia
In logic and mathematics, statements
{\displaystyle p}
{\displaystyle q}
are said to be logically equivalent if they have the same truth value in every model.[1] The logical equivalence of
{\displaystyle p}
{\displaystyle q}
is sometimes expressed as
{\displaystyle p\equiv q}
{\displaystyle p::q}
{\displaystyle {\textsf {E}}pq}
{\displaystyle p\iff q}
, depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related.
1 Logical equivalences
1.1 General logical equivalences
1.2 Logical equivalences involving conditional statements
1.3 Logical equivalences involving biconditionals
3 Relation to material equivalence
In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these.
General logical equivalencesEdit
{\displaystyle p\wedge \top \equiv p}
{\displaystyle p\vee \bot \equiv p}
{\displaystyle p\vee \top \equiv \top }
{\displaystyle p\wedge \bot \equiv \bot }
{\displaystyle p\vee p\equiv p}
{\displaystyle p\wedge p\equiv p}
Idempotent or tautology laws
{\displaystyle \neg (\neg p)\equiv p}
Double negation law
{\displaystyle p\vee q\equiv q\vee p}
{\displaystyle p\wedge q\equiv q\wedge p}
{\displaystyle (p\vee q)\vee r\equiv p\vee (q\vee r)}
{\displaystyle (p\wedge q)\wedge r\equiv p\wedge (q\wedge r)}
{\displaystyle p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)}
{\displaystyle p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)}
{\displaystyle \neg (p\wedge q)\equiv \neg p\vee \neg q}
{\displaystyle \neg (p\vee q)\equiv \neg p\wedge \neg q}
{\displaystyle p\vee (p\wedge q)\equiv p}
{\displaystyle p\wedge (p\vee q)\equiv p}
{\displaystyle p\vee \neg p\equiv \top }
{\displaystyle p\wedge \neg p\equiv \bot }
Logical equivalences involving conditional statementsEdit
{\displaystyle p\implies q\equiv \neg p\vee q}
{\displaystyle p\implies q\equiv \neg q\implies \neg p}
{\displaystyle p\vee q\equiv \neg p\implies q}
{\displaystyle p\wedge q\equiv \neg (p\implies \neg q)}
{\displaystyle \neg (p\implies q)\equiv p\wedge \neg q}
{\displaystyle (p\implies q)\wedge (p\implies r)\equiv p\implies (q\wedge r)}
{\displaystyle (p\implies q)\vee (p\implies r)\equiv p\implies (q\vee r)}
{\displaystyle (p\implies r)\wedge (q\implies r)\equiv (p\vee q)\implies r}
{\displaystyle (p\implies r)\vee (q\implies r)\equiv (p\wedge q)\implies r}
Logical equivalences involving biconditionalsEdit
{\displaystyle p\iff q\equiv (p\implies q)\wedge (q\implies p)}
{\displaystyle p\iff q\equiv \neg p\iff \neg q}
{\displaystyle p\iff q\equiv (p\wedge q)\vee (\neg p\wedge \neg q)}
{\displaystyle \neg (p\iff q)\equiv p\iff \neg q}
If Lisa is in Denmark, then she is in Europe (a statement of the form
{\displaystyle d\implies e}
If Lisa is not in Europe, then she is not in Denmark (a statement of the form
{\displaystyle \neg e\implies \neg d}
Syntactically, (1) and (2) are derivable from each other via the rules of contraposition and double negation. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either Lisa is in Denmark is false or Lisa is in Europe is true.
(Note that in this example, classical logic is assumed. Some non-classical logics do not deem (1) and (2) to be logically equivalent.)
Relation to material equivalenceEdit
Logical equivalence is different from material equivalence. Formulas
{\displaystyle p}
{\displaystyle q}
are logically equivalent if and only if the statement of their material equivalence (
{\displaystyle p\iff q}
) is a tautology.[2]
The material equivalence of
{\displaystyle p}
{\displaystyle q}
(often written as
{\displaystyle p\leftrightarrow q}
) is itself another statement in the same object language as
{\displaystyle p}
{\displaystyle q}
. This statement expresses the idea "'
{\displaystyle p}
{\displaystyle q}
'". In particular, the truth value of
{\displaystyle p\leftrightarrow q}
can change from one model to another.
On the other hand, the claim that two formulas are logically equivalent is a statement in metalanguage, which expresses a relationship between two statements
{\displaystyle p}
{\displaystyle q}
. The statements are logically equivalent if, in every model, they have the same truth value.
≡ the iff symbol (U+2261 IDENTICAL TO)
∷ the a is to b as c is to d symbol (U+2237 PROPORTION)
⇔ the double struck biconditional (U+21D4 LEFT RIGHT DOUBLE ARROW)
↔ the bidirectional arrow (U+2194 LEFT RIGHT ARROW)
^ Mendelson, Elliott (1979). Introduction to Mathematical Logic (2 ed.). pp. 56. ISBN 9780442253073.
^ Copi, Irving; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (New International ed.). Pearson. p. 348.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Logical_equivalence&oldid=1065086955"
|
EUDML | A characterization of -convex functions. EuDML | A characterization of -convex functions.
\lambda
Adamek, Mirosław. "A characterization of -convex functions.." JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only] 5.3 (2004): Paper No. 71, 5 p., electronic only-Paper No. 71, 5 p., electronic only. <http://eudml.org/doc/124368>.
@article{Adamek2004,
author = {Adamek, Mirosław},
keywords = {-convexity; generalized 2nd-order derivative; -convexity},
title = {A characterization of -convex functions.},
AU - Adamek, Mirosław
TI - A characterization of -convex functions.
KW - -convexity; generalized 2nd-order derivative; -convexity
\lambda
-convexity, generalized 2nd-order derivative,
\lambda
Articles by Adamek
|
Using Intermediate Terms in Equations - MATLAB & Simulink - MathWorks India
Why Use Intermediate Terms?
Declaring and Using Named Intermediate Terms
Use in Equations
Using the let Expressions
Syntax Rules of let Expressions
Nested let Expressions
Conditional let Expressions
Identifier List in the Declarative Clause
Textbooks often define certain equation terms in separate equations, and then substitute these intermediate equations into the main one. For example, for fully developed flow in ducts, the Darcy friction factor can be used to compute pressure loss:
P=\frac{f·L·\rho ·{V}^{2}}{2D}
where P is pressure, f is the Darcy friction factor, L is length, ρ is density, V is flow velocity, and D is hydraulic area.
These terms are further defined by:
f=\frac{0.316}{{\mathrm{Re}}^{1/4}}
\mathrm{Re}=\frac{D·V}{\nu }
D=\sqrt{\frac{4A}{\pi }}
V=\frac{q}{A}
where Re is the Reynolds number, A is the area, q is volumetric flow rate, and ν is the kinematic viscosity.
In Simscape™ language, there are two ways that you can define intermediate terms for use in equations:
intermediates section — Declare reusable named intermediate terms in the intermediates section in a component or domain file. You can reuse these intermediate terms in any equations section within the same component file, in an enclosing composite component file, or in any component that has nodes of that domain type.
let expressions in the equations section — Declare intermediate terms in the declaration clause and use them in the expression clause of the same let expression. Use this method if you need to define intermediate terms of limited scope, for use in a single group of equations. This way, the declarations and equations are close together, which improves code readability.
Another advantage of using named intermediate terms instead of let expressions is that you can include named intermediate terms in simulation data logs.
The following example shows the same Darcy-Weisbach equation with intermediate terms written out in Simscape language:
L = { 1, 'm' }; % Length
rho = { 1e3, 'kg/m^3' }; % Density
nu = { 1e-6, 'm^2/s' }; % Kinematic viscosity
p = { 0, 'Pa' }; % Pressure
q = { 0, 'm^3/s' }; % Volumetric flow rate
A = { 0, 'm^2' }; % Area
f = 0.316 / Re_d^0.25; % Darcy friction factor
Re_d = D_h * V / nu; % Reynolds number
V = q / A; % Flow velocity
p == f * L * rho * V^2 / (2 * D_h); % final equation
After substitution of all intermediate terms, the final equation becomes:
p==0.316/(sqrt(4.0 * A / pi) * q / A / nu)^0.25 * L * rho * (q / A)^2 / (2 * sqrt(4.0 * A / pi));
When you use this component in a model and log simulation data, the logs will include data for the four intermediate terms, with their descriptive names (such as Darcy friction factor) shown in the Simscape Results Explorer.
The intermediates section in a component file lets you define named intermediate terms for use in equations. Think of named intermediate terms as of defining an alias for an expression. You can reuse it in any equations section within the same file or an enclosing composite component. When an intermediate term is used in an equation, it is ultimately substituted with the expression that it refers to.
You declare an intermediate term by assigning a unique identifier on the left-hand side of the equal sign (=) to an expression on the right-hand side of the equal sign.
The expression on the right-hand side of the equal sign:
Can refer to other intermediate terms. For example, in the Darcy-Weisbach equation, the identifier Re_d (Reynolds number) is used in the expression declaring the identifier f (Darcy friction factor). The only requirement is that these references are acyclic.
Can refer to parameters, variables, inputs, outputs, member components and their parameters, variables, inputs, and outputs, as well as Across variables of domains used by the component nodes.
Cannot refer to Through variables of domains used by the component nodes.
You can use intermediate terms in equations, as described in Use in Equations. However, you cannot access intermediate terms in the setup function.
Intermediate terms can appear in simulation data logs and Simscape Results Explorer, as described in Data Logging. However, intermediate terms do not appear in:
Operating Point data
Block dialog boxes and Property Inspector
After declaring an intermediate term, you can refer to it by its identifier anywhere in the equations section of the same component. For example:
p1 = { 1, 'm' };
v1 = { 0, 'm' };
v2 = { 0, 'm^2' };
int_expr = v1^2 * pi / p1;
v2 == v1^2 + int_expr;
You can refer to a public intermediate term declared in a member component in the equations of an enclosing composite component. For example:
comp1 = MyPackage.A;
v1 == comp1.int_expr;
Similarly, you can refer to an intermediate term declared in a domain in the equations section of any component that has nodes of this domain type. For example:
int_expr = v1 / sqrt(2);
v1 == n.int_expr;
Accessibility of intermediate terms outside of the file where they are declared is governed by their Access attribute value. For mode information, see Attribute Lists.
Intermediate terms with ExternalAccess attribute values of modify or observe are included in simulation data logs. For mode information, see Attribute Lists.
If you specify a descriptive name for an intermediate term, this name appears in the status panel of the Simscape Results Explorer.
For example, you declare the intermediate term D_h (hydraulic diameter) as a function of the orifice area:
When you use a block based on this component in a model and log simulation data, selecting D_h in the Simscape Results Explorer tree on the left displays a plot of the values of the hydraulic diameter over time in the right pane and the name Hydraulic diameter in the status panel at the bottom. For more information, see About the Simscape Results Explorer.
let expressions provide another way to define intermediate terms for use in one or more equations. Use this method if you need to define intermediate terms of limited scope, for use in a single group of equations. This way, the declarations and equations are close together, which improves file readability.
The following example shows the same Darcy-Weisbach equation as in the beginning of this topic but with intermediate terms written out using the let expression:
However, in this case the four intermediate terms do not appear in logged simulation data.
A let expression consists of two clauses, the declaration clause and the expression clause.
expression clause
The declaration clause assigns an identifier, or set of identifiers, on the left-hand side of the equal sign (=) to an equation expression on the right-hand side of the equal sign:
LetValue = EquationExpression
The expression clause defines the scope of the substitution. It starts with the keyword in, and may contain one or more equation expressions. All the expressions assigned to the identifiers in the declaration clause are substituted into the equations in the expression clause during parsing.
The end keyword is required at the end of a let-in-end statement.
x == z;
In this example, the declaration clause of the let expression sets the value of the identifier z to be the expression y + 1. Thus, substituting y + 1 for z in the expression clause in the let statement, the code above is equivalent to:
x == y + 1;
There may be multiple declarations in the declaration clause. These declarations are order independent. The identifiers declared in one declaration may be referred to by the expressions for identifiers in other declarations in the same declaration clause. Thus, in the example with the Darcy-Weisbach equation, the identifier Re_d (Reynolds number) is used in the expression declaring the identifier f (Darcy friction factor). The only requirement is that the expression references are acyclic.
The expression clause of a let expression defines the scope of the substitution for the declaration clause. Other equations, that do not require these substitutions, may appear in the equation section outside of the expression clause. In the following example, the equation section contains the equation expression c == b + 2 outside the scope of the let expression before it.
b == x;
c == b + 2;
These expressions are treated as peers. They are order independent, so this example is equivalent to
and, after the substitution, to
b == a + 1;
You can nest let expressions, for example:
w = a + 1;
z = w + 1;
b == z;
c == w;
In case of nesting, substitutions are performed based on both of the declaration clauses. After the substitutions, the code above becomes:
b == a + 1 + 1;
c == a + 1;
The innermost declarations take precedence. The following example illustrates a nested let expression where the inner declaration clause overrides the value declared in the outer one:
b == w;
Performing substitution on this example yields:
You can use if statements within both declarative and expression clause of let expressions, for example:
x = if a < 0, a else b end;
c == x;
Here x is declared as the conditional expression based on a < 0. Performing substitution on this example yields:
c == if a < 0, a else b end;
The next example illustrates how you can use let expressions within conditional expressions. The two let expressions on either side of the conditional expression are independent:
z = b + 1;
c == z;
This example shows using an identifier list, rather than a single identifier, in the declarative clause of a let expression:
[x, y] = if a < 0, a; -a else -b; b end;
d == y;
Here x and y are declared as the conditional expression based on a < 0. Notice that each side of the if statement defines a list of two expressions. A first semantic translation of this example separates the if statement into
if a < 0, a; -a else -b; b end =>
{ if a < 0, a else -b end; if a < 0, -a else b end }
then the second semantic translation becomes
[x, y] = { if a < 0, a else -b end; if a < 0, -a else b end } =>
x = if a < 0, a else -b end; y = if a < 0, -a else b end;
and the final substitution on this example yields:
c == if a < 0, a else -b end;
d == if a < 0, -a else b end;
|
Ted needs to find the point of intersection for the lines
y=18x−30
y=−22x+50
. He takes out a piece of graph paper and then realizes that he can solve this problem without graphing. Explain how Ted is going to accomplish this, and then find the point of intersection.
Set the two expressions for
y
equal to each other and solve algebraically.
|
Sharp inequalities for the coefficients of concave schlicht functions | EMS Press
Sharp inequalities for the coefficients of concave schlicht functions
Farit G. Avkhadiev
Kazan State University, Russian Federation
Karl-Joachim Wirths
D
denote the open unit disc and let
f\colon D\to \mathbb{C}
be holomorphic and injective in
D
. We further assume that
f(D)
is unbounded and
\mathbb{C}\setminus f(D)
is a convex domain. In this article, we consider the Taylor coefficients
a_n(f)
of the normalized expansion
f(z)=z+\sum_{n=2}^{\infty}a_n(f)z^n, z\in D,
and we impose on such functions
f
the second normalization
f(1)=\infty
. We call these functions concave schlicht functions, as the image of
D
is a concave domain. We prove that the sharp inequalities
|a_n(f)-\frac{n+1}{2}|\leq\frac{n-1}{2}, n\geq 2,
are valid. This settles a conjecture formulated in [2].
Farit G. Avkhadiev, Christian Pommerenke, Karl-Joachim Wirths, Sharp inequalities for the coefficients of concave schlicht functions. Comment. Math. Helv. 81 (2006), no. 4, pp. 801–807
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Frank Cce Everyday Science for Class 7 Science Chapter 17 - Our Forests
Frank Cce Everyday Science Solutions for Class 7 Science Chapter 17 Our Forests are provided here with simple step-by-step explanations. These solutions for Our Forests are extremely popular among Class 7 students for Science Our Forests Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Frank Cce Everyday Science Book of Class 7 Science Chapter 17 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Frank Cce Everyday Science Solutions. All Frank Cce Everyday Science Solutions for class Class 7 Science are prepared by experts and are 100% accurate.
3. Growing trees on a large scale
4. Branched part of a tree above the stem
1. It refers to the diverse forms of life on the earth
2. Roof formed by the branches of tall trees over other small plants
5. An international organization
The first link in all food chains are green plants.
Which of the following is a producer?
(b) green plant
(c) carnivore
Green plants produce food using water and carbon dioxide, in the presence of sunlight. As a result, green plants are known as producers.
Nylon is a synthetic fibre and is not a forest product.
Which of the following is a decomposer?
Fungi are decomposers. They decompose dead plants and animals and return nutrients to the soil.
The lowest layer in the forest is occupied by
Herbs occupy the lowest layer in a forest.
1. Humus is rich in ........................... .
2. The branched part of the trees above the stem is called ........................... .
3. Plants are called ........................... in a food chain.
4. Destruction of forests leads to ........................... erosion.
5. Forests occupy ........................... per cent of the geographical area in India.
Humus is rich in nutrients.
The branched part of the trees above the stem is called crown.
Plants are called producers in a food chain.
Destruction of forests leads to soil erosion.
Forests occupy 21 per cent of the geographical area in India.
Sun is the ultimate energy source for all food chains. But from where will animals who live deep down on the ocean bed and have never seen sunlight get energy?
Deep ocean life is usually found in the vicinity of hot vents. In these vents, hot gases boil up from volcanic vents, with a temperature of around 350oC. The gases contain many sulphides and various minerals. When these hot gases hit the cold water, many of these minerals precipitate out of the water.
Specially adapted bacteria inhabit the regions around the hot vents and they directly convert the chemicals precipitating from the hot vents into energy by a process called chemosynthesis. In a way, these bacteria replace plants in the deep marine ecosystems.
Animals living in the vicinity of the hot vents either host the bacteria in their bodies or consume them directly. Other animals then eat these animals that directly feed on bacteria or host them in their bodies, leading to the formation of a complete food chain.
What will happen to the agricultural productivity if the number of snakes suddenly become less in a particular area?
A reduction in the number of snakes in an area will cause an explosion in the number of rodents, which in turn will destroy crops, resulting in a reduction in agricultural productivity.
1. Green plants (a) Afforestation
2. Deer (b) Decomposer
3. Tiger (c) Carnivore
4. Bacterial and fungi (d) Herbivore
5. Planting trees on large scale (e) Producer
(f) Deforestation
1 Green plants (e) Producer
2. Deer (d) Herbivore
4. Bacteria and fungi (b) Decomposer
5. Planting trees on a large scale (a) Afforestation
1. Forests are rich in biodiversity ( )
2. A forest is a purifier of air and water. ( )
3. Animals help in seed dispersion. ( )
4. Excreta of animal pollute the soil. ( )
5. Mass-scale planting of trees is called deforestation. ( )
4. False (F)
The excreta of animals is decomposed by decomposers such as bacteria and fungi; this contributes to soil fertility.
Mass-scale planting of trees is known as afforestation.
Write these words in the correct place on the word map:
Omnivore, Lion, Producer, Carnivore, Decomposer, Herbivore.
What do you mean by understoreys?
The different horizontal layers in a forest that are formed by herbs, shrubs and trees are called understoreys.
Name two plants that give us medicines.
Cinchona and aloe vera are two medicinal plants.
In a forest, the branches of trees meet and form an umbrella overhead so that very little sunlight reaches the ground. This is known as a canopy.
What name is given to branchy part of a tree above the stem?
The branchy part of a tree above the stem is known as the crown.
Decomposers are microorganisms that break down complex substances, such as those that make up the bodies of plants and animals, into simpler substances. Decomposers act on dead organic matter and return nutrients to the soil.
What are the ultimate sources of food for all animals?
All animals depend on plants for nutrition. Herbivores consume plants directly. Carnivores eat herbivorous animals, so they also depend indirectly on plants for nutrition.
A food chain is a graphical representation of the transfer of energy from one organism to another.
An example of a food chain is given below:
\mathrm{Plants}\to \mathrm{Rabbit}\to \mathrm{Snake}
Why are forests considered a dynamic entity?
A forest is home to many organisms such as plants, animals and microorganisms:
Plants are eaten by herbivorous animals.
Herbivores are in turn a source of food for carnivores.
Microorganisms are decomposers that break down the bodies of dead plants and animals, returning them to the soil, thus promoting the growth of new plants.
Owing to the continuous cycle of nutrients, we can say that a forest is a dynamic entity.
In what way do forests help in regulating the climate of a place?
The roots of the trees present in the forest prevents water from running off during heavy rainfall by holding the soil particles firmly. This helps in preventing floods.
Forests increase the amount of water vapour in the air due to transpiration. This keeps the atmosphere cool and helps in rainfall, thereby regulating the climate of a place.
How will snakes be affected if all rats disappear from the forest?
If all the rats in a forest disappear, the snakes will run out of food and they will perish. In turn, animals that depend upon snakes for food, such as hawks and eagles, will also die.
How are animals classified according to their role in the food chain?
Animals are classified as follows according to their role in the food chain:
Herbivores: animals that directly consume plants. Eg: deer, rabbits etc.
Carnivores: animals that consume other animals. Eg: foxes, lions, tigers etc.
Omnivores: animals that consume both plants and other animals. Eg: humans, bears.
Forest conservation refers to the conservation of forests by preventing activities that cause destruction to them. Forests are maintained properly by taking various steps.
How do forests help to control soil erosion?
The roots of trees firmly hold soil particles and prevent them from being washed away or blown away by water and wind. This helps in preventing soil erosion.
How do decomposers help in maintaining balance in nature?
Decomposers break down the bodies of dead plants and animals, allowing nutrients to return to the soil. Thus, decomposers help in maintaining the balance in nature by making nutrients available for new plants, restarting the food chain.
Why do we say that there is no waste in a forest? Explain.
In a forest, all organic material is either eaten by other animals or is broken down into simpler substances and returned to the soil by decomposers. As a result, there is no wastage in a forest.
What is meant by interdependence of plants and animals?
The interdependence of plants and animals can be understood by the following points:
All animals depend on plants for food, either directly or indirectly.
Plants depend on animals for dispersing their seeds. Plants also benefit from animal dung, which is a good fertiliser.
Plants provide oxygen to animals for respiration and animals release carbon dioxide, which is consumed by the plants for preparing their food by photosynthesis.
List five methods for the conservation of forests.
Five methods for conservation of forests are as follows:
Large scale afforestation must be encouraged in open areas that are unfit for cultivation, such as the side of highways, playground fringes etc.
Deforestation must be stopped. If trees are felled, many more must be planted to replace them.
Cooking on open fires in forest areas must not be practised to avoid forest fires.
Overgrazing by cattle, goats etc. must be avoided.
Air, water and soil pollution must be avoided to prevent destruction of plants and trees.
How do forests reduce atmospheric pollution?
Trees absorb atmospheric carbon dioxide for photosynthesis. Also they allow suspended dust to settle on their leaves. Thus, forests help reduce atmospheric pollution.
What is humus? How is it useful for soil?
Humus is a dark, porous and soft substance that is obtained by the decomposition of dead plants and animals. Humus is very useful for soil as it is rich in nutrients and also retains moisture well. This helps in increasing soil fertility.
Write a short note on the Chipko movement.
The Chipko movement was initiated in March 1973 in the Terai forest around the Himalayan village of Gopeswar in the Chamoli district of present day Uttarakhand. The movement was led by the noted environmentalist, Sunderlal Bahuguna.
Women of the village played an active role in the movement, by hugging trees to prevent them from being cut down.
How are forests beneficial to man? Explain it in points.
Forests are beneficial to man in the following ways:
Forests provide timber and wood, which are used for manufacturing furniture, railway sleepers, carts, boats, sporting goods, etc.
Plants and trees and their fruits and the meat of forest animals are sources of food.
Plants in forests are a source of medicines, for example, neem leaves and the bark of the cinchona plant.
Forests provide many other resources such as gum, oil, honey and lac. Forest animals are a source of hide, fur, ivory, musk, etc.
Forests transpire water vapour, thus ensuring a cool climate and adequate rainfall.
Forests absorb carbon dioxide and allow particulate matter to settle on tree leaves. Thus, forests reduce atmospheric pollution and global warming.
What is the difference between food chain and food web? Explain it with suitable examples.
A food chain is a diagram representing the flow of energy from one organism to another. A food web is an interconnected network of food chains that form a multitude of feeding connections among different organisms of a biotic community.
Example: A grasshopper feeds on the leaves of a plant, a frog eats the grasshopper and the frog, in turn, is eaten by a snake. Example: The following diagram shows a food web with many interlinked food chains.
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Wikizero - Born–Haber cycle
Approach to analyzing reaction energies
The Born–Haber cycle is an approach to analyze reaction energies. It was named after the two German scientists Max Born and Fritz Haber, who developed it in 1919.[1][2][3] It was also independently formulated by Kasimir Fajans[4] and published concurrently in the same issue of the same journal.[1] The cycle is concerned with the formation of an ionic compound from the reaction of a metal (often a Group I or Group II element) with a halogen or other non-metallic element such as oxygen.
Born–Haber cycles are used primarily as a means of calculating lattice energy (or more precisely enthalpy[note 1]), which cannot otherwise be measured directly. The lattice enthalpy is the enthalpy change involved in the formation of an ionic compound from gaseous ions (an exothermic process), or sometimes defined as the energy to break the ionic compound into gaseous ions (an endothermic process). A Born–Haber cycle applies Hess's law to calculate the lattice enthalpy by comparing the standard enthalpy change of formation of the ionic compound (from the elements) to the enthalpy required to make gaseous ions from the elements.
This latter calculation is complex. To make gaseous ions from elements it is necessary to atomise the elements (turn each into gaseous atoms) and then to ionise the atoms. If the element is normally a molecule then we first have to consider its bond dissociation enthalpy (see also bond energy). The energy required to remove one or more electrons to make a cation is a sum of successive ionization energies; for example, the energy needed to form Mg2+ is the ionization energy required to remove the first electron from Mg, plus the ionization energy required to remove the second electron from Mg+. Electron affinity is defined as the amount of energy released when an electron is added to a neutral atom or molecule in the gaseous state to form a negative ion.
The Born–Haber cycle applies only to fully ionic solids such as certain alkali halides. Most compounds include covalent and ionic contributions to chemical bonding and to the lattice energy, which is represented by an extended Born–Haber thermodynamic cycle.[5] The extended Born–Haber cycle can be used to estimate the polarity and the atomic charges of polar compounds.
1.1 Formation of LiF
1.2 Formation of NaBr
Formation of LiF[edit]
Born–Haber cycle for the standard enthalpy change of formation of lithium fluoride. ΔHlatt corresponds to UL in the text. The downward arrow "electron affinity" shows the negative quantity –EAF, since EAF is usually defined as positive.
The enthalpy of formation of lithium fluoride (LiF) from its elements lithium and fluorine in their stable forms is modeled in five steps in the diagram:
Enthalpy change of atomization enthalpy of lithium
Ionization enthalpy of lithium
Atomization enthalpy of fluorine
Electron affinity of fluorine
The same calculation applies for any metal other than lithium or any non-metal other than fluorine.
The sum of the energies for each step of the process must equal the enthalpy of formation of the metal and non-metal,
{\displaystyle \Delta H_{f}}
{\displaystyle \Delta H_{f}=V+{\frac {1}{2}}B+{\mathit {IE}}_{{\ce {M}}}-{\ce {EA}}_{{\ce {X}}}+U_{L}}
V is the enthalpy of sublimation for metal atoms (lithium)
B is the bond energy (of F2). The coefficient 1/2 is used because the formation reaction is Li + 1/2 F2 → LiF.
{\displaystyle {\ce {{\mathit {IE}}_{M}}}}
is the ionization energy of the metal atom:
{\displaystyle {\ce {{M}+{\mathit {IE}}_{M}->{M+}+e^{-}}}}
{\displaystyle {\ce {{\mathit {EA}}_{X}}}}
is the electron affinity of non-metal atom X (fluorine)
{\displaystyle U_{L}}
is the lattice energy (defined as exothermic here)
The net enthalpy of formation and the first four of the five energies can be determined experimentally, but the lattice energy cannot be measured directly. Instead, the lattice energy is calculated by subtracting the other four energies in the Born–Haber cycle from the net enthalpy of formation.[6]
The word cycle refers to the fact that one can also equate to zero the total enthalpy change for a cyclic process, starting and ending with LiF(s) in the example. This leads to
{\displaystyle 0=-\Delta H_{f}+V+{\frac {1}{2}}B+{\mathit {IE}}_{{\ce {M}}}-{\mathit {EA}}_{{\ce {X}}}+U_{L}}
which is equivalent to the previous equation.
Formation of NaBr[edit]
At ordinary temperatures, Na is solid and Br2 is liquid, heat of vaporization is added to the equation:
{\displaystyle \Delta H_{f}=V+{\frac {1}{2}}B+{\frac {1}{2}}\Delta _{vap}H+{\mathit {IE}}_{{\ce {M}}}-{\ce {EA}}_{{\ce {X}}}+U_{L}}
{\displaystyle \Delta _{vap}H}
is the enthalpy of vaporization of Br2 in kJ/mol.
^ The difference between energy and enthalpy is very small and the two terms are interchanged freely in this article.
^ a b Morris, D.F.C.; Short, E.L. (6 December 1969). "The Born-Fajans-Haber Correlation". 224: 950–952. doi:10.1038/224950a0. A more correct name would be the Born–Fajans–Haber thermochemical correlation. {{cite journal}}: Cite journal requires |journal= (help)
^ M. Born Verhandlungen der Deutschen Physikalischen Gesellschaft 1919, 21, 679-685.
^ F. Haber Verhandlungen der Deutschen Physikalischen Gesellschaft 1919, 21, 750-768.
^ K. Fajans Verhandlungen der Deutschen Physikalischen Gesellschaft 1919, 21, 714-722.
^ H. Heinz and U. W. Suter Journal of Physical Chemistry B 2004, 108, 18341-18352.
^ Moore, Stanitski, and Jurs. Chemistry: The Molecular Science. 3rd edition. 2008. ISBN 0-495-10521-X. pages 320–321.
ChemGuy on the Born-Haber Cycle
Retrieved from "https://en.wikipedia.org/w/index.php?title=Born–Haber_cycle&oldid=1009946951"
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P versus NP - Simple English Wikipedia, the free encyclopedia
P versus NP is one of the Millennium Problems, and of great interest to people working with computers and in mathematics. One way of asking it is, "Can every solved problem whose answer can be checked quickly by a computer, also be quickly solved by a computer?" Math problems are referred to as P or NP, whether they are solvable in finite polynomial time. P problems are have their solution time bound to a polynomial and so are relatively fast for computers to solve, and so are considered "easy". NP problems are fast (and so "easy") for a computer to check, but are not necessarily easy to solve.
In 1956, Kurt Gödel wrote a letter to John von Neumann. In this letter, Gödel asked whether a certain NP complete problem could be solved in quadratic or linear time.[1] In 1971, Stephen Cook introduced the precise statement of the P versus NP problem in his article "The complexity of theorem proving procedures".[2]
Today, many people consider this problem to be the most important open problem in computer science.[3] It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for a solution that invites a published recognition by the Clay Institute, and presumably one(s) that changes the whole of mathematics.
Because many of these problems touch upon related issues, and it is the dream of many mathematicians to invent unifying theories, many hope the Millennium Problems are interconnected.
4 Exponential Time
5 NP-complete problems
5.2 Formal overview
Clarifications[change | change source]
A computer may be able to tell if an answer is right, but take longer to get the answer. For some interesting, practical questions of this kind, difficult answers are possible to check quickly. So NP problems may be thought of as being like riddles: it may be hard to come up with an answer to a riddle, but once one hears the answer, the answer seems obvious. In this comparison (analogy), the basic question is: are riddles really as hard as we think they are, or are we missing something? Is there a secret to always having an answer?
Because these kinds of P versus NP questions are so practically important, many mathematicians, scientists, and computer programmers want to prove the general proposition, that every quickly-checked problem can also be solved quickly. This question is important enough that the Clay Mathematical Institute will give $1,000,000 to anyone who successfully provides a proof or a valid explanation that disproves it.
Digging a little deeper, we see that all P problems are NP problems: it is easy to check that a solution is correct by solving the problem and comparing the two solutions. However, people want to know about the opposite: Are there any NP problems other than P problems, or are all NP problems just P problems? If NP problems are really not the same as P problems (P ≠ NP), it would mean that no general, fast ways to solve those NP problems can exist, no matter how hard we look. However, if all NP problems are P problems (P = NP), it would mean that new, very fast problem-solving methods do exist. We just have not found them yet.
Since the best efforts of scientists and mathematicians have not found general, easy methods for solving NP problems yet, many people believe that there are NP problems other than P problems (that is, that P ≠ NP is true). Most mathematicians also believe this to be true, but currently no one has proven it by rigorous mathematical analysis. If it can be proven that NP and P are the same (P = NP is true), it would have a huge impact on many aspects of day-to-day life. For this reason, the question of P versus NP is an important and widely studied topic.
Suppose someone wants to build two towers, by stacking rocks of different mass. One wants to make sure that each of the towers has exactly the same mass. That means one will have to put the rocks into two piles that have the same mass. If one guesses a division of the rocks that one thinks will work, it would be easy for one to check if one was right. (To check the answer, one can divide the rocks into two piles, then use a balance to see if they have the same mass.) Because it is easy to check this problem, called 'Partition' by computer scientists—easier than to solve it outright, as we will see—it is not a P problem.[source?]
How hard is it to solve, outright? If one starts with just 100 rocks, there are 2^{100-1}-1 = 633,825,300,114,114,700,748,351,602,687, or about 6.3 x 10^{29} possible ways (combinations) to divide these rocks into two piles. If one could check one unique combination of rocks every day, it would take 1.3 x 10^{22} or 1,300,000,000,000,000,000,000 years of effort. For comparison, physicists believe that the universe is about 1.4 x 10^{10} years old (450,000,000,000,000,000 or about 4.5 x 10^{17} seconds, or about one trillionth as old as the time it would take for our rock piling effort. That means that if one takes all of the time that has passed since the beginning of the universe, one would need to check more than two trillion (2,000,000,000,000) different ways of dividing the rocks every second, in order to check all of the different ways.
If one programmed a powerful computer, to test all of these ways to divide the rocks, one might be able to check
{\displaystyle 1,000,000}
combinations per second using current systems. This means one would still need
{\displaystyle 2,000,000}
very powerful computers, working since the origin of the universe, to test all the ways of dividing the rocks.
However, it may be possible to find a method of dividing the rocks into two equal piles without checking all combinations. The question "Does P equal NP?" is a shorthand for asking if any method like that can exist.
Why it matters[change | change source]
There are many important NP problems that people don't know how to solve in a way that is faster than testing every possible answer. Here are some examples:
A school offers 100 different classes, and a teacher needs to choose one hour for each class' final exam. To prevent cheating, all of the students who take a class must take the exam for that class at the same time. If a student takes more than one class, then all of those exams must be at a different time. The teacher wants to know if he can schedule all of the exams in the same day so that every student is able to take the exam for each of their classes.
A farmer wants to take 100 watermelons of different masses to the market. She needs to pack the watermelons into boxes. Each box can only hold 20 kilograms without breaking. The farmer needs to know if 10 boxes will be enough for her to carry all 100 watermelons to market. (This is trivial, if no more than one watermelon weighs more than 2 kg then any 10 can be placed in each of the crates, if no more than ten watermelons weighs more than 2 kg then one of each of them can be placed in each crate, etc., to a fast solution; observation will be the key to any rapid solution such as this or the number set problem).
A large art gallery has many rooms, and each wall is covered with many expensive paintings. The owner of the gallery wants to buy cameras to watch these paintings, in case a thief tries to steal any of them. He wants to know if 100 cameras will be enough for him to make sure that each painting can be seen by at least one camera.
The clique problem: The principal of a school has a list of which students are friends with each other. She wants to find a group of 10% of the students that are all friends with each other.
Exponential Time[change | change source]
In the example above, we see that with
{\displaystyle 100}
rocks, there are
{\displaystyle 2^{100}}
ways to partition the set of rocks. With
{\displaystyle n}
{\displaystyle 2^{n}}
combinations. The function
{\displaystyle f(n)=2^{n}}
is an exponential function. It's important to NP because it models the worst-case number of computations that are needed to solve a problem and, thus, the worst-case amount of time required.
And so far, for the hard problems, the solutions have required on the order of
{\displaystyle 2^{n}}
computations. For any particular problem, people have found ways to reduce the number of computations needed. One might figure out that a way to do just 1% of the worst-case number of computation and that saves a lot of computing, but that is still
{\displaystyle 0.01\times (2^{n})}
computations. And every extra rock still doubles the number of computations needed to solve the problem. There are insights that can produce methods to do even fewer computations producing variations of the model: e.g.
{\displaystyle 2^{n}/n^{3}}
. But the exponential function still dominates as
{\displaystyle n}
grows.
Consider the problem of scheduling exams (described above). But suppose, next, that there are 15000 students. There's a computer program that takes the schedules of all 15000 students. It runs in an hour and outputs an exam schedule so that all students can do their exams in one week. It satisfies lots of rules (no back-to-back exams, no more than 2 exams in any 28 hour period, ...) to limit the stress of exam week. The program runs for one hour at mid-term break and everyone knows his/her exam schedule with plenty of time to prepare.
The next year, though, there are 10 more students. If the same program runs on the same computer then that one hour is going to turn into
{\displaystyle 2^{10}}
hours, because every additional student doubles the computations. That's
{\displaystyle 6}
weeks! If there were 20 more students, then
{\displaystyle 2^{20}}
{\displaystyle 1048576}
hours ~
{\displaystyle 43691}
days ~
{\displaystyle 113}
{\displaystyle 15000}
students, it takes one hour. For
{\displaystyle 15020}
students, it takes
{\displaystyle 113}
As you can see, exponential functions grow really fast. Most mathematicians believe that the hardest NP problems require exponential time to solve.
NP-complete problems[change | change source]
Mathematicians can show that there are some NP problems that are NP-Complete. An NP-Complete problem is at least as difficult to solve as any other NP problem. This means that if someone found a method to solve any NP-Complete problem quickly, they could use that same method to solve every NP problem quickly. All of the problems listed above are NP-Complete, so if the salesman found a way to plan his trip quickly, he could tell the teacher, and she could use that same method to schedule the exams. The farmer could use the same method to determine how many boxes she needs, and the woman could use the same method to find a way to build her towers.
Because a method that quickly solves one of these problems can solve them all, there are many people who want to find one. However, because there are so many different NP-Complete problems and nobody so far has found a way to solve even one of them quickly, most experts believe that solving NP-Complete problems quickly is not possible.
In computational complexity theory, the complexity class NP-complete (abbreviated NP-C or NPC), is a class of problems having two properties:
It is in the set of NP (non-deterministic polynomial time) problems: Any given solution to the problem can be verified quickly (in polynomial time).
It is also in the set of NP-hard problems: Those which are at least as hard as the hardest problems in NP. Problems that are NP-hard do not have to be elements of NP; indeed, they may not even be decidable.
Formal overview[change | change source]
NP-complete is a subset of NP, the set of all decision problems whose solutions can be verified in polynomial time; NP may be equivalently defined as the set of decision problems solved in polynomial time on a machine. A problem p in NP is also in NPC if and only if every other problem in NP is transformed into p in polynomial time. NP-complete was to be used as an adjective: problems in the class NP-complete were as NP+complete problems.
NP-complete problems are studied because the ability to quickly verify solutions to a problem (NP) seems to correlate with the ability to quickly solve problem (P). It is found every problem in NP is quickly solved—as called the P = NP: problem set. The single problem in NP-complete is solved quickly, faster than every problem in NP also quickly solved, because the definition of an NP-complete problem states every problem in NP must be quickly reducible to every problem in NP-complete (it is reduced in polynomial time). [1]
The Boolean satisfiability problem is known to be NP complete. In 1972, Richard Karp formulated 21 problems that are known to be NP-complete.[4] These are known as Karp's 21 NP-complete problems. They include problems such as the Integer programming problem, which applies linear programming techniques to the integers, the knapsack problem, or the vertex cover problem.
↑ Juris Hartmanis 1989, Gödel, von Neumann, and the P = NP problem, Bulletin of the European Association for Theoretical Computer Science, vol. 38, pp. 101–107
↑ Cook, Stephen (1971). "The complexity of theorem proving procedures". Proceedings of the Third Annual ACM Symposium on Theory of Computing. pp. 151–158.
↑ Lance Fortnow, The status of the P versus NP problem, Communications of the ACM 52 (2009), no. 9, pp. 78–86. doi:10.1145/1562164.1562186
↑ Richard M. Karp (1972). "Reducibility Among Combinatorial Problems" (PDF). In R. E. Miller; J. W. Thatcher (eds.). Complexity of Computer Computations. New York: Plenum. pp. 85–103.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=P_versus_NP&oldid=8160111"
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Ellipsoid - formulasearchengine
Tri-axial ellipsoid with distinct semi-axis lengths
{\displaystyle c>b>a}
Tri-axial ellipsoid with distinct semi-axes a, b and c
Ellipsoids of revolution (spheroid) with a pair of equal semi-axes (a) and a distinct third semi-axis (c) which is an axis of symmetry. The ellipsoid is prolate (top) or oblate (bottom) as c is greater than or less than a.
An ellipsoid is a closed quadric surface that is a three-dimensional analogue of an ellipse. The standard equation of an ellipsoid centered at the origin of a Cartesian coordinate system and aligned with the axes is
{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1,}
The points (a,0,0), (0,b,0) and (0,0,c) lie on the surface and the line segments from the origin to these points are called the semi-principal axes of length a, b, c. They correspond to the semi-major axis and semi-minor axis of the appropriate ellipses.
There are four distinct cases of which one is degenerate:
{\displaystyle a>b>c}
— tri-axial or (rarely) scalene ellipsoid;
{\displaystyle a=b>c}
— oblate ellipsoid of revolution (oblate spheroid);
{\displaystyle a=b<c}
— prolate ellipsoid of revolution (prolate spheroid);
{\displaystyle a=b=c}
— the degenerate case of a sphere;
Mathematical literature often uses 'ellipsoid' in place of 'tri-axial ellipsoid'. Scientific literature (particularly geodesy) often uses 'ellipsoid' in place of 'ellipsoid of revolution' and only applies the adjective 'tri-axial' when treating the general case. Older literature uses 'spheroid' in place of 'ellipsoid of revolution'.
Any planar cross section passing through the center of an ellipsoid forms an ellipse on its surface: this degenerates to a circle for sections normal to the symmetry axis of an ellipsoid of revolution (or all sections when the ellipsoid degenerates to a sphere.)
3.2.1 Approximate formula
4 Dynamical properties
6 Equations in specific coordinate systems
More generally, an arbitrarily oriented ellipsoid, centered at v, is defined by the solutions x to the equation
{\displaystyle (\mathbf {x-v} )^{\mathrm {T} }\!A\,(\mathbf {x-v} )=1,}
where A is a positive definite matrix and x, v are vectors.
The eigenvectors of A define the principal axes of the ellipsoid and the eigenvalues of A are the reciprocals of the squares of the semi-axes:
{\displaystyle a^{-2}}
{\displaystyle b^{-2}}
{\displaystyle c^{-2}}
.[1] An invertible linear transformation applied to a sphere produces an ellipsoid, which can be brought into the above standard form by a suitable rotation, a consequence of the polar decomposition (also, see spectral theorem). If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors of the matrix are orthogonal (due to the spectral theorem) and represent the directions of the axes of the ellipsoid: the lengths of the semiaxes are given by the eigenvalues. The singular value decomposition and polar decomposition are matrix decompositions closely related to these geometric observations.
The surface of the ellipsoid may be parameterized in several ways. One possible choice which singles out the 'z'-axis is:
{\displaystyle {\begin{aligned}x&=a\,\cos u\cos v,\\y&=b\,\cos u\sin v,\\z&=c\,\sin u;\end{aligned}}\,\!}
{\displaystyle -{\pi }/{2}\leq u\leq +{\pi }/{2},\qquad -\pi \leq v\leq +\pi .\!\,\!}
The parameters may be interpreted as spherical coordinates. For constant u, that is on the ellipse which is the intercept with a constant z plane, v then plays the role of the eccentric anomaly for that ellipse. For constant v on a plane through the Oz axis the parameter u plays the same role for the ellipse of intersection. Two other similar parameterizations are possible, each with their own interpretations. Only on an ellipse of revolution can a unique definition of reduced latitude be made.
The volume of the internal part of the ellipsoid is
{\displaystyle V={\frac {4}{3}}\pi abc={\frac {4}{3}}\pi {\sqrt {\det(A^{-1})}}.\,\!}
Note that this equation reduces to that of the volume of a sphere when all three elliptic radii are equal, and to that of an oblate or prolate spheroid when two of them are equal.
The volume of an ellipsoid is two thirds the volume of a circumscribed elliptic cylinder.
The volumes of the maximum inscribed and minimum circumscribed boxes are respectively:
{\displaystyle V_{\max }={\frac {8}{3{\sqrt {3}}}}abc,\qquad V_{\min }=8abc.}
The volume of an ellipse of dimension higher than 3 can be calculated using the dimensional constant given for the volume of a hypersphere. One can also define ellipsoids in higher dimensions, as the images of spheres under invertible linear transformations. The spectral theorem can again be used to obtain a standard equation akin to the one given above.
The surface area of a general (tri-axial) ellipsoid is[2][3]
{\displaystyle S=2\pi c^{2}+{\frac {2\pi ab}{\sin \phi }}\left(E(\phi ,k)\,\sin ^{2}\phi +F(\phi ,k)\,\cos ^{2}\phi \right),}
{\displaystyle \cos \phi ={\frac {c}{a}},\qquad k^{2}={\frac {a^{2}(b^{2}-c^{2})}{b^{2}(a^{2}-c^{2})}},\qquad a\geq b\geq c,}
and where F(φ,k) and E(φ,k) are incomplete elliptic integrals of the first and second kind respectively.[1]
{\displaystyle S_{\rm {oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\tanh ^{-1}e\right)\quad {\mbox{where}}\quad e^{2}=1-{\frac {c^{2}}{a^{2}}}\quad (c<a),}
{\displaystyle S_{\rm {prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\sin ^{-1}e\right)\quad \qquad {\mbox{where}}\;\quad e^{2}=1-{\frac {a^{2}}{c^{2}}}\quad (c>a),}
which, as follows from basic trigonometric identities, are equivalent expressions (i.e. the formula for
{\displaystyle S_{\rm {oblate}}}
can be used to calculate the surface area of a prolate ellipsoid and vice versa). In both cases e may again be identified as the eccentricity of the ellipse formed by the cross section through the symmetry axis. (See ellipse). Derivations of these results may be found in standard sources, for example Mathworld.[4]
{\displaystyle S\approx 4\pi \!\left({\frac {a^{p}b^{p}+a^{p}c^{p}+b^{p}c^{p}}{3}}\right)^{1/p}.\,\!}
In the "flat" limit of c much smaller than a, b, the area is approximately 2πab.
The mass of an ellipsoid of uniform density ρ is:
{\displaystyle m=\rho V=\rho {\frac {4}{3}}\pi abc\,\!}
The moments of inertia of an ellipsoid of uniform density are:
{\displaystyle I_{\mathrm {xx} }={\frac {1}{5}}m(b^{2}+c^{2}),\qquad I_{\mathrm {yy} }={\frac {1}{5}}m(c^{2}+a^{2}),\qquad I_{\mathrm {zz} }={\frac {1}{5}}m(a^{2}+b^{2}),}
{\displaystyle I_{\mathrm {xy} }=I_{\mathrm {yz} }=I_{\mathrm {zx} }=0.\,\!}
For a=b=c these moments of inertia reduce to those for a sphere of uniform density.
Artist's conception of Template:Dp, a Jacobi-ellipsoid dwarf planet, with its two moons
Ellipsoids and cuboids rotate stably along their major or minor axes, but not along their median axis. This can be seen experimentally by throwing an eraser with some spin. In addition, moment of inertia considerations mean that rotation along the major axis is more easily perturbed than rotation along the minor axis.[6]
One practical effect of this is that scalene astronomical bodies such as Template:Dp generally rotate along their minor axes (as does Earth, which is merely oblate); in addition, because of tidal locking, moons in synchronous orbit such as Mimas orbit with their major axis aligned radially to their planet.
A relaxed ellipsoid, that is, one in hydrostatic equilibrium, has an oblateness a − c directly proportional to its mean density and mean radius. Ellipsoids with a differentiated interior—that is, a denser core than mantle—have a lower oblateness than a homogeneous body. Over all, the ratio (b–c)/(a−c) is approximately 0.25, though this drops for rapidly rotating bodies.[7]
The terminology typically used for bodies rotating on their minor axis and whose shape is determined by their gravitational field is Maclaurin spheroid (oblate spheroid) and Jacobi ellipsoid (scalene ellipsoid). At faster rotations, piriform or oviform shapes can be expected, but these are not stable.
The ellipsoid is the most general shape for which it has been possible to calculate the creeping flow of fluid around the solid shape. The calculations include the force required to translate through a fluid and to rotate within it. Applications include determining the size and shape of large molecules, the sinking rate of small particles, and the swimming abilities of microorganisms.[8]
Equations in specific coordinate systems
{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1,}
{\displaystyle {r^{2}\cos ^{2}\!\theta \,\sin ^{2}\!\phi \over a^{2}}+{r^{2}\sin ^{2}\!\theta \,\sin ^{2}\!\phi \over b^{2}}+{r^{2}\cos ^{2}\!\phi \over c^{2}}=1,}
{\displaystyle {r^{2}\cos ^{2}\!\theta \over a^{2}}+{r^{2}\sin ^{2}\!\theta \over b^{2}}+{z^{2} \over c^{2}}=1,}
Template:Dp, a scalene-ellipsoid-shaped dwarf planet
Homoeoid, a shell bounded by two concentric, similar ellipsoids
Focaloid, a shell bounded by two concentric, confocal ellipsoids
Elliptical distribution, in statistics
↑ http://see.stanford.edu/materials/lsoeldsee263/15-symm.pdf
↑ F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors, 2010, NIST Handbook of Mathematical Functions (Cambridge University Press), available on line at http://dlmf.nist.gov/19.33 (see next reference).
↑ NIST (National Institute of Standards and Technology) athttp://www.nist.gov
↑ Prolate Spheroid at Mathworld
↑ Final answers by Gerard P. Michon (2004-05-13). See Thomsen's formulas and Cantrell's comments.
↑ Goldstein, H G (1980). Classical Mechanics, (2nd edition) Chapter 5.
↑ Dusenbery, David B. (2009).Living at Micro Scale, Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.
"Ellipsoid" by Jeff Bryant, Wolfram Demonstrations Project, 2007.
Ellipsoid and Quadratic Surface, MathWorld.
ta:நீளுருண்டை
Retrieved from "https://en.formulasearchengine.com/index.php?title=Ellipsoid&oldid=224455"
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Multiplicative Order - Maple Help
Home : Support : Online Help : Mathematics : Number Theory : Multiplicative Order
MultiplicativeOrder(m, n)
positive integer greater than 1
The MultiplicativeOrder function computes the multiplicative order of m modulo n, which is defined as the least positive integer exponent i such that m^i is congruent to
1
modulo n.
Alternatively, the multiplicative order can be defined as the order of the cyclic group generated by m under multiplication modulo n.
The multiplicative order exists if and only if m and n are coprime. In the case that it does not exist, an error message is displayed.
\mathrm{with}\left(\mathrm{NumberTheory}\right):
\mathrm{MultiplicativeOrder}\left(7,18\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{seq}\left({7}^{i}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}18,i=1..3\right)
\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{13}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}
If the multiplicative order of m is equal to the totient of n, then m is a primitive root modulo n.
\mathrm{Totient}\left(18\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{MultiplicativeOrder}\left(11,18\right)
\textcolor[rgb]{0,0,1}{6}
\mathrm{PrimitiveRoot}\left(18,\mathrm{greaterthan}=10\right)
\textcolor[rgb]{0,0,1}{11}
5
25
are not coprime, the multiplicative order of
5
25
\mathrm{MultiplicativeOrder}\left(5,25\right)
Error, (in NumberTheory:-MultiplicativeOrder) the arguments 5 and 25 are not coprime
The NumberTheory[MultiplicativeOrder] command was introduced in Maple 2016.
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Universe | Special Issue : The Physical Properties of the Groups of Galaxies
The Physical Properties of the Groups of Galaxies
Submit to Universe Review for Universe Edit a Special Issue
Special Issue "The Physical Properties of the Groups of Galaxies"
A special issue of Universe (ISSN 2218-1997). This special issue belongs to the section "Galaxies and Clusters".
Dr. Lorenzo Lovisari
INAF, Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Pietro Gobetti 93/3, 40129 Bologna, Italy
Interests: groups and clusters of galaxies; intracluster medium; X-ray; cosmology
Dr. Stefano Ettori
Interests: galaxy clusters; intracluster medium; X-ray; dark matter; cosmology
Galaxy groups consist of a few tens of galaxies bound in a common gravitational potential and contain a significant fraction of the overall universal baryon budget. Therefore, they are key to our understanding of how the bulk of matter in the Universe accretes and forms hierarchical structures and how different sources of feedback affect their gravitational collapse. However, despite their crucial role in cosmic structure formation and evolution, galaxy groups have received less attention compared to massive clusters. This is perhaps in part due to their rarity in being observed and properly characterized. With the advent of eROSITA, many thousands of galaxy groups will be detected by X-ray, complementing optical and SZ coverage.
It is time to collect and organize the latest developments in our understanding of these systems and present future prospects from both observational and theoretical points of view.
This Special Issue aims to foster progress in the field of the physical properties of galaxy groups, facilitating effective cross-communication between observers, theorists, and simulators. Topics of interest to this Special Issue include (but are most certainly not limited to) multi-wavelength observations of single objects and samples, hydrodynamical simulations of cosmic structures, fossil/compact groups, the physics of the intragroup plasma and the distribution of the metals, and the scaling relations and their impact on cosmology.
Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Universe is an international peer-reviewed open access monthly journal published by MDPI.
Galaxy groups consist of a few tens of galaxies bound in a common gravitational potential [...] Full article
(This article belongs to the Special Issue The Physical Properties of the Groups of Galaxies)
Iryna S. Butsky
Galaxy groups are more than an intermediate scale between clusters and halos hosting individual galaxies, they are crucial laboratories capable of testing a range of astrophysics from how galaxies form and evolve to large scale structure (LSS) statistics for cosmology. Cosmological hydrodynamic simulations [...] Read more.
Galaxy groups are more than an intermediate scale between clusters and halos hosting individual galaxies, they are crucial laboratories capable of testing a range of astrophysics from how galaxies form and evolve to large scale structure (LSS) statistics for cosmology. Cosmological hydrodynamic simulations of groups on various scales offer an unparalleled testing ground for astrophysical theories. Widely used cosmological simulations with ∼(100 Mpc)
{}^{3}
volumes contain statistical samples of groups that provide important tests of galaxy evolution influenced by environmental processes. Larger volumes capable of reproducing LSS while following the redistribution of baryons by cooling and feedback are the essential tools necessary to constrain cosmological parameters. Higher resolution simulations can currently model satellite interactions, the processing of cool (
T\approx {10}^{4-5}
K) multi-phase gas, and non-thermal physics including turbulence, magnetic fields and cosmic ray transport. We review simulation results regarding the gas and stellar contents of groups, cooling flows and the relation to the central galaxy, the formation and processing of multi-phase gas, satellite interactions with the intragroup medium, and the impact of groups for cosmological parameter estimation. Cosmological simulations provide evolutionarily consistent predictions of these observationally difficult-to-define objects, and have untapped potential to accurately model their gaseous, stellar and dark matter distributions. Full article
Galaxy groups host the majority of matter and more than half of all the galaxies in the Universe. Their hot (
{10}^{7}
K), X-ray emitting intra-group medium (IGrM) reveals emission lines typical of many elements synthesized by stars and supernovae. Because their [...] Read more.
{10}^{7}
K), X-ray emitting intra-group medium (IGrM) reveals emission lines typical of many elements synthesized by stars and supernovae. Because their gravitational potentials are shallower than those of rich galaxy clusters, groups are ideal targets for studying, through X-ray observations , feedback effects, which leave important marks on their gas and metal contents. Here, we review the history and present status of the chemical abundances in the IGrM probed by X-ray spectroscopy. We discuss the limitations of our current knowledge, in particular due to uncertainties in the modeling of the Fe-L shell by plasma codes, and coverage of the volume beyond the central region. We further summarize the constraints on the abundance pattern at the group mass scale and the insight it provides to the history of chemical enrichment. Parallel to the observational efforts, we review the progress made by both cosmological hydrodynamical simulations and controlled high-resolution 3D simulations to reproduce the radial distribution of metals in the IGrM, the dependence on system mass from group to cluster scales, and the role of AGN and SN feedback in producing the observed phenomenology. Finally, we highlight future prospects in this field, where progress will be driven both by a much richer sample of X-ray emitting groups identified with eROSITA, and by a revolution in the study of X-ray spectra expected from micro-calorimeters onboard XRISM and ATHENA. Full article
Ewan O’Sullivan
The co-evolution between supermassive black holes and their environment is most directly traced by the hot atmospheres of dark matter halos. The cooling of the hot atmosphere supplies the central regions with fresh gas, igniting active galactic nuclei (AGN) with long duty cycles. [...] Read more.
The co-evolution between supermassive black holes and their environment is most directly traced by the hot atmospheres of dark matter halos. The cooling of the hot atmosphere supplies the central regions with fresh gas, igniting active galactic nuclei (AGN) with long duty cycles. Outflows from the central engine tightly couple with the surrounding gaseous medium and provide the dominant heating source preventing runaway cooling by carving cavities and driving shocks across the medium. The AGN feedback loop is a key feature of all modern galaxy evolution models. Here, we review our knowledge of the AGN feedback process in the specific context of galaxy groups. Galaxy groups are uniquely suited to constrain the mechanisms governing the cooling–heating balance. Unlike in more massive halos, the energy that is supplied by the central AGN to the hot intragroup medium can exceed the gravitational binding energy of halo gas particles. We report on the state-of-the-art in observations of the feedback phenomenon and in theoretical models of the heating-cooling balance in galaxy groups. We also describe how our knowledge of the AGN feedback process impacts galaxy evolution models and large-scale baryon distributions. Finally, we discuss how new instrumentation will answer key open questions on the topic. Full article
Galaxy groups and poor clusters are more common than rich clusters, and host the largest fraction of matter content in the Universe. Hence, their studies are key to understand the gravitational and thermal evolution of the bulk of the cosmic matter. Moreover, because [...] Read more.
Galaxy groups and poor clusters are more common than rich clusters, and host the largest fraction of matter content in the Universe. Hence, their studies are key to understand the gravitational and thermal evolution of the bulk of the cosmic matter. Moreover, because of their shallower gravitational potential, galaxy groups are systems where non-gravitational processes (e.g., cooling, AGN feedback, star formation) are expected to have a higher impact on the distribution of baryons, and on the general physical properties, than in more massive objects, inducing systematic departures from the expected scaling relations. Despite their paramount importance from the astrophysical and cosmological point of view, the challenges in their detection have limited the studies of galaxy groups. Upcoming large surveys will change this picture, reassigning to galaxy groups their central role in studying the structure formation and evolution in the Universe, and in measuring the cosmic baryonic content. Here, we review the recent literature on various scaling relations between X-ray and optical properties of these systems, focusing on the observational measurements, and the progress in our understanding of the deviations from the self-similar expectations on groups’ scales. We discuss some of the sources of these deviations, and how feedback from supernovae and/or AGNs impacts the general properties and the reconstructed scaling laws. Finally, we discuss future prospects in the study of galaxy groups. Full article
Properties of Fossil Groups of Galaxies
Stefano Zarattini
We review the formation and evolution of fossil groups and clusters from both the theoretical and the observational points of view. In the optical band, these systems are dominated by the light of the central galaxy. They were interpreted as old systems that [...] Read more.
We review the formation and evolution of fossil groups and clusters from both the theoretical and the observational points of view. In the optical band, these systems are dominated by the light of the central galaxy. They were interpreted as old systems that had enough time to merge all the M* galaxies within the central one. During the last two decades, many observational studies were performed to prove the old and relaxed state of fossil systems. The majority of these studies that spans a wide range of topics including halos global scaling relations, dynamical substructures, stellar populations, and galaxy luminosity functions seem to challenge this scenario. The general picture that can be obtained by reviewing all the observational works is that the fossil state could be transitional. Indeed, the formation of the large magnitude gap observed in fossil systems could be related to internal processes rather than an old formation. Full article
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Matrices, Popular Questions: Jee 2 year MATH, Math - Meritnation
Q. I F A matrices A =
\left[2 3\phantom{\rule{0ex}{0ex}}4 5\right]
then show that A - AT is a skew symmetric matrices.
if P and A are matrices of order 2*2
P= [sinO cosO -cosO sinO] and A=[1 1 0 1] , Q=PAPTthen PTQnP is
Question) Solve this :
If A matrices A =
\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]
what is the right option for 5th sum of practise sheet
i hope we have 2 options for it-a,d
A contractor gets his supply of building materials from three firms A, B and C. He receives 35 truck load of stones and 14 truck load of sand from A, 30 truck load of stones and 8 truck load of sand from B and 29 truck load of stones and 9 truck load of sand from C. The stones cost Rs 1000 per truck and sand Rs 300 per truck. Using matrix multiplication find the amount received by each firm from the contractor.
Aastha Adhikari asked a question
pls help with 3 question
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Pythagorean_trigonometric_identity Knowpia
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
The identity is
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1.}
As usual, sin2 θ means
{\textstyle (\sin \theta )^{2}}
Proofs and their relationships to the Pythagorean theoremEdit
Similar right triangles showing sine and cosine of angle θ
Proof based on right-angle trianglesEdit
Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless of its actual size: the ratios depend upon the three angles, not the lengths of the sides. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ.
The elementary definitions of the sine and cosine functions in terms of the sides of a right triangle are:
{\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}={\frac {b}{c}}}
{\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}={\frac {a}{c}}}
The Pythagorean identity follows by squaring both definitions above, and adding; the left-hand side of the identity then becomes
{\displaystyle {\frac {\mathrm {opposite} ^{2}+\mathrm {adjacent} ^{2}}{\mathrm {hypotenuse} ^{2}}}}
which by the Pythagorean theorem is equal to 1. This definition is valid for all angles, due to the definition of defining
{\displaystyle x=\cos \theta }
{\displaystyle y=\sin \theta }
for the unit circle and thus
{\displaystyle x=c\cos \theta }
{\displaystyle y=c\sin \theta }
for a circle of radius c and reflecting our triangle in the y axis and setting
{\displaystyle a=x}
{\displaystyle b=y}
Alternatively, the identities found at Trigonometric symmetry, shifts, and periodicity may be employed. By the periodicity identities we can say if the formula is true for −π < θ ≤ π then it is true for all real θ. Next we prove the range π/2 < θ ≤ π, to do this we let t = θ − π/2, t will now be in the range 0 < t ≤ π/2. We can then make use of squared versions of some basic shift identities (squaring conveniently removes the minus signs):
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =\sin ^{2}\left(t+{\frac {1}{2}}\pi \right)+\cos ^{2}\left(t+{\frac {1}{2}}\pi \right)=\cos ^{2}t+\sin ^{2}t=1.}
All that remains is to prove it for −π < θ < 0; this can be done by squaring the symmetry identities to get
{\displaystyle \sin ^{2}\theta =\sin ^{2}(-\theta ){\text{ and }}\cos ^{2}\theta =\cos ^{2}(-\theta ).}
Related identitiesEdit
Similar right triangles illustrating the tangent and secant trigonometric functions.
Trigonometric functions and their reciprocals on the unit circle. The Pythagorean theorem applied to the blue triangle shows the identity 1 + cot2 θ = csc2 θ, and applied to the red triangle shows that 1 + tan2 θ = sec2 θ.
{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }
{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }
are also called Pythagorean trigonometric identities.[1] If one leg of a right triangle has length 1, then the tangent of the angle adjacent to that leg is the length of the other leg, and the secant of the angle is the length of the hypotenuse.
{\displaystyle \tan \theta ={\frac {b}{a}}\ ,}
{\displaystyle \sec \theta ={\frac {c}{a}}\ .}
In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem.
The following table gives the identities with the factor or divisor that relates them to the main identity.
Derived Identity (Alternate)
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}
{\displaystyle \cos ^{2}\theta }
{\displaystyle {\frac {\sin ^{2}\theta }{\cos ^{2}\theta }}+{\frac {\cos ^{2}\theta }{\cos ^{2}\theta }}={\frac {1}{\cos ^{2}\theta }}}
{\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta }
{\displaystyle \tan ^{2}\theta =\sec ^{2}\theta -1}
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}
{\displaystyle \sin ^{2}\theta }
{\displaystyle {\frac {\sin ^{2}\theta }{\sin ^{2}\theta }}+{\frac {\cos ^{2}\theta }{\sin ^{2}\theta }}={\frac {1}{\sin ^{2}\theta }}}
{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }
{\displaystyle \cot ^{2}\theta =\csc ^{2}\theta -1}
Proof using the unit circleEdit
Point P(x,y) on the circle of unit radius at an obtuse angle θ > π/2
The unit circle centered at the origin in the Euclidean plane is defined by the equation:[2]
{\displaystyle x^{2}+y^{2}=1.}
Given an angle θ, there is a unique point P on the unit circle at an angle θ from the x-axis, and the x- and y-coordinates of P are:[3]
{\displaystyle x=\cos \theta \ \mathrm {and} \ y=\sin \theta \ .}
Consequently, from the equation for the unit circle:
{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1\ ,}
the Pythagorean identity.
In the figure, the point P has a negative x-coordinate, and is appropriately given by x = cosθ, which is a negative number: cosθ = −cos(π−θ ). Point P has a positive y-coordinate, and sinθ = sin(π−θ ) > 0. As θ increases from zero to the full circle θ = 2π, the sine and cosine change signs in the various quadrants to keep x and y with the correct signs. The figure shows how the sign of the sine function varies as the angle changes quadrant.
Because the x- and y-axes are perpendicular, this Pythagorean identity is equivalent to the Pythagorean theorem for triangles with hypotenuse of length 1 (which is in turn equivalent to the full Pythagorean theorem by applying a similar-triangles argument). See unit circle for a short explanation.
Proof using power seriesEdit
The trigonometric functions may also be defined using power series, namely (for x an angle measured in radians):[4][5]
{\displaystyle {\begin{aligned}\sin x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1},\\\cos x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}.\end{aligned}}}
Using the formal multiplication law for power series at Multiplication and division of power series (suitably modified to account for the form of the series here) we obtain
{\displaystyle {\begin{aligned}\sin ^{2}x&=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)!}}{\frac {(-1)^{j}}{(2j+1)!}}x^{(2i+1)+(2j+1)}\\&=\sum _{n=1}^{\infty }\left(\sum _{i=0}^{n-1}{\frac {(-1)^{n-1}}{(2i+1)!(2(n-i-1)+1)!}}\right)x^{2n}\\&=\sum _{n=1}^{\infty }\left(\sum _{i=0}^{n-1}{2n \choose 2i+1}\right){\frac {(-1)^{n-1}}{(2n)!}}x^{2n},\\\cos ^{2}x&=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }{\frac {(-1)^{i}}{(2i)!}}{\frac {(-1)^{j}}{(2j)!}}x^{(2i)+(2j)}\\&=\sum _{n=0}^{\infty }\left(\sum _{i=0}^{n}{\frac {(-1)^{n}}{(2i)!(2(n-i))!}}\right)x^{2n}\\&=\sum _{n=0}^{\infty }\left(\sum _{i=0}^{n}{2n \choose 2i}\right){\frac {(-1)^{n}}{(2n)!}}x^{2n}.\end{aligned}}}
In the expression for sin2, n must be at least 1, while in the expression for cos2, the constant term is equal to 1. The remaining terms of their sum are (with common factors removed)
{\displaystyle \sum _{i=0}^{n}{2n \choose 2i}-\sum _{i=0}^{n-1}{2n \choose 2i+1}=\sum _{j=0}^{2n}(-1)^{j}{2n \choose j}=(1-1)^{2n}=0}
by the binomial theorem. Consequently,
{\displaystyle \sin ^{2}x+\cos ^{2}x=1\ ,}
which is the Pythagorean trigonometric identity.
When the trigonometric functions are defined in this way, the identity in combination with the Pythagorean theorem shows that these power series parameterize the unit circle, which we used in the previous section. This definition constructs the sine and cosine functions in a rigorous fashion and proves that they are differentiable, so that in fact it subsumes the previous two.
Proof using the differential equationEdit
Sine and cosine can be defined as the two solutions to the differential equation:[6]
{\displaystyle y''+y=0}
satisfying respectively y(0) = 0, y'(0) = 1 and y(0) = 1, y'(0) = 0. It follows from the theory of ordinary differential equations that the first solution, sine, has the second, cosine, as its derivative, and it follows from this that the derivative of cosine is the negative of the sine. The identity is equivalent to the assertion that the function
{\displaystyle z=\sin ^{2}x+\cos ^{2}x}
is constant and equal to 1. Differentiating using the chain rule gives:
{\displaystyle {\frac {d}{dx}}z=2\sin x\ \cos x+2\cos x\ (-\sin x)=0\ ,}
so z is constant. A calculation confirms that z(0) = 1, and z is a constant so z = 1 for all x, so the Pythagorean identity is established.
A similar proof can be completed using power series as above to establish that the sine has as its derivative the cosine, and the cosine has as its derivative the negative sine. In fact, the definitions by ordinary differential equation and by power series lead to similar derivations of most identities.
This proof of the identity has no direct connection with Euclid's demonstration of the Pythagorean theorem.
Proof using Euler's formulaEdit
{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =(\cos \theta +i\sin \theta )(\cos \theta -i\sin \theta )=e^{i\theta }e^{-i\theta }=1}
^ Lawrence S. Leff (2005). PreCalculus the Easy Way (7th ed.). Barron's Educational Series. p. 296. ISBN 0-7641-2892-2.
^ This result can be found using the distance formula
{\displaystyle d={\sqrt {x^{2}+y^{2}}}}
for the distance from the origin to the point
{\displaystyle (x,\ y)}
. See Cynthia Y. Young (2009). Algebra and Trigonometry (2nd ed.). Wiley. p. 210. ISBN 978-0-470-22273-7. This approach assumes Pythagoras' theorem. Alternatively, one could simply substitute values and determine that the graph is a circle.
^ Thomas W. Hungerford, Douglas J. Shaw (2008). "§6.2 The sine, cosine and tangent functions". Contemporary Precalculus: A Graphing Approach (5th ed.). Cengage Learning. p. 442. ISBN 978-0-495-10833-7.
^ James Douglas Hamilton (1994). "Power series". Time series analysis. Princeton University Press. p. 714. ISBN 0-691-04289-6.
^ Steven George Krantz (2005). "Definition 10.3". Real analysis and foundations (2nd ed.). CRC Press. pp. 269–270. ISBN 1-58488-483-5.
^ Tyn Myint U., Lokenath Debnath (2007). "Example 8.12.1". Linear partial differential equations for scientists and engineers (4th ed.). Springer. p. 316. ISBN 978-0-8176-4393-5.
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Kinetic Bonding - HunnyDAO
Kinetic Bonding
What is Kinetic Bonding?
Kinetic Bonding gives MASSIVE DISCOUNT for LOVE, GET KISS and ATTRACTS HUG.
Typically when users bond, they provide HunnyDAO with reserve assets such as BUSD in exchange for discounted LOVE. However, in Kinetic Bonding, users are also rewarded with HUG that can be redeemed to LOVE. The amount of HUG that users can receive will be displayed in the Kinetic Bonding Dashboard before users entered for bonding.
How do I Redeem HUG for LOVE?
HUGs are performance vested using the Fibonacci Sequence. The Fibonacci sequence can be described using a mathematical equation:
X_{n+2}= X_{n+1} + X_{n}
To get the amount HUG that can be redeemed for LOVE:
HUG = TotalKISS_{earned} * Loyalty Ratio * HCV
The total KISS earned is based on the following formula:
TotalKISS_{earned} = KISS_{unstake} - LOVE_{staked}
The Loyalty Ratio is based on the Fibonacci Sequence.
Fibonacci Sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, .......
When the number of rebases meet the next Fibonacci Sequence, the Loyalty Ratio will be increased to the next level.
LOVE must be STAKED in order for HUGs to be redeemable.
Unstaking part of your LOVE will reduce the number of HUGs that can be redeemed for LOVE.
Unstaking ALL of your LOVE will result in HUG not being able to redeem for LOVE as there will be no KISS earned from staking.
The below table demonstrates the effect of the number of Rebases on the Loyalty Ratio.
Number of Rebases
Loyalty Ratio
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Now that you have learned about the different layers in the Earth’s structure, you will create a colorful poster to demonstrate what you have learned.
Step 1. Color the ‘Inner Core’ yellow; the ‘Outer Core’ orange; the ‘Mantle’ red and the ‘Crust’ brown.
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To help you visualize the relationship between tectonic plate boundaries and the number of incidences of earthquakes and locations of volcanoes, launch the PBS interactive in Helpful Resources. Take note of some of the continents/countries which appear to have the most earthquakes and volcanoes are they near tectonic plate boundaries? What impact do earthquakes and volcanic eruptions have on our planet, on plant and animal habitats and on human life?
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Now that students have learned about the theory of plate tectonics and they have seen some examples of how the movement of tectonic plates changes the Earth’s crust, they will undertake a hands-on demonstration of the three different types of tectonic plates boundaries.
Note: For this activity you will need to organize one Oreo or similar sandwich-style biscuit and one paper plate per student.
Step 2. Hold the bottom of your biscuit while gently twisting the top of the biscuit so that it separates from the soft center and the rest of the biscuit.
Step 4. Place the two tectonic plates back together onto the soft center of the bottom half of the biscuit. The soft center represents the sludgy mantle.
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Convolution and polynomial multiplication of fi objects - MATLAB conv - MathWorks Italia
Convolution of 22-Sample Sequence with 16-Tap FIR Filter
Central Part of Convolution of Two fi Vectors
Convolution and polynomial multiplication of fi objects
c = conv(a,b,shape)
c = conv(a,b) returns the convolution of input vectors a and b, at least one of which must be a fi object.
c = conv(a,b,shape) returns a subsection of the convolution, as specified by shape.
Find the convolution of a 22-sample sequence with a 16-tap FIR filter.
x is a 22-sample sequence of signed values with a word length of 16 bits and a fraction length of 15 bits. h is the 16-tap FIR filter.
u = (pi/4)*[1 1 1 -1 -1 -1 1 -1 -1 1 -1];
x = fi(kron(u,[1 1]));
h = firls(15, [0 .1 .2 .5]*2, [1 1 0 0]);
Because x is a fi object, you do not need to cast h into a fi object before performing the convolution operation. The conv function does this automatically using best-precision scaling.
Use the conv function to convolve the two vectors.
Create two fi vectors. Find the central part of the convolution of a and b that is the same size as a.
a = fi([-1 2 3 -2 0 1 2]);
b = fi([2 4 -1 1]);
c = conv(a,b,'same')
c has a length of 7. The full convolution would be of length length(a)+length(b)-1, which in this example would be 10.
a,b — Input vectors
Input vectors, specified as either row or column vectors.
If either input is a built-in data type, conv casts it into a fi object using best-precision rules before the performing the convolution operation.
shape — Subset of convolution
Subset of convolution, specified as one of these values:
'full' — Returns the full convolution. This option is the default shape.
'same' — Returns the central part of the convolution that is the same size as input vector a.
'valid' — Returns only those parts of the convolution that the function computes without zero-padded edges. Using this option, the length of output vector c is max(length(a)-max(0,length(b)-1),0).
w\left(k\right)=\sum _{j}u\left(j\right)v\left(k-j+1\right).
The fimath properties associated with the inputs determine the numerictype properties of output fi object c:
If either a or b has a local fimath object, conv uses that fimath object to compute intermediate quantities and determine the numerictype properties of c.
If neither a nor b have an attached fimath, conv uses the default fimath to compute intermediate quantities and determine the numerictype properties of c.
The output fi object c always uses the default fimath.
For variable-sized signals, you might see different results between generated code and MATLAB®.
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Display linear mixed-effects model - MATLAB - MathWorks América Latina
Display linear mixed-effects model
display(lme)
display(lme) displays the fitted linear mixed-effects model lme.
The dataset array shows the absolute deviations from the target quality characteristic measured from the products that five operators manufacture during three shifts, morning, evening, and night. This is a randomized block design, where the operators are the blocks. The experiment is designed to study the impact of the time of shift on the performance. The performance measure is the absolute deviation of the quality characteristics from the target value. This is simulated data.
disp(lme)
This display includes the model performance statistics, Akaike and Bayesian Information Criteria, Akaike and Bayesian Information Criteria, loglikelihood, and Deviance.
The fixed-effects coefficients table includes the names and estimates of the coefficients in the first two columns. The third column SE shows the standard errors of the coefficients. The column tStat includes the
t
-statistic values that correspond to each coefficient. DF is the residual degrees of freedom, and the pValue is the
p
-value that corresponds to the corresponding
t
-statistic value. The columns Lower and Upper display the lower and upper limits of a 95% confidence interval for each fixed-effects coefficient.
The first table for the random effects shows the types and the estimates of the random effects covariance parameters, with the lower and upper limits of a 95% confidence interval for each parameter. The display also shows the name of the grouping variable, operator, and the total number of levels, 5.
The second table for the random effects shows the estimate of the observation error, with the lower and upper limits of a 95% confidence interval.
-2*\mathrm{log}{L}_{M}.
Dev=De{v}_{1}-De{v}_{2}=2\left(\mathrm{log}L{M}_{2}-\mathrm{log}L{M}_{1}\right).
D=\left(\begin{array}{cc}{D}_{11}& 0\\ 0& 0\end{array}\right),
D is a q-by-q symmetric positive semidefinite matrix.
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3.2 Converting between decimal and binary numbers | Binary numbers | Siyavula
In Chapter 1 you learnt that we use the base ten number system for everyday counting. When we count using the base ten number system, we put objects in groups of ten. We use the numbers 0; 1; 2; 3; 4; 5; 6; 7; 8; 9 as digits to make any other number, no matter how big or how small it is.
The base ten number system is also called the decimal number system. The prefix "deci-" comes from a Latin word that means "one tenth".
3.1 The base two number system
In this chapter you will learn about the base two number system. The base two number system is used in computers and other devices that also use computing functions. It is therefore a very important number system.
When we count using the base two number system, we put objects in groups of two. We use only two digits, namely 0 and 1, to make up any number.
The base two number system is also called the binary number system. The prefix "bi-" comes from a Latin word that means "two" or "twice".
base ten number system The base ten number system is a number system that counts in groups of 10, and uses the digits 0; 1; 2; 3; 4; 5; 6; 7; 8; 9 to represent any number. It is also called the decimal number system.
base two number system The base two number system is a number system that counts in groups of 2, and uses only the digits 0 and 1 to represent any number. It is also called the binary number system.
In the base two number system, we go to a new group each time we reach a number that is a power of 2. We therefore count as follows:
In the base ten number system we work with powers of 10. In the base two number system, we work with powers of 2.
\begin{array}{|l|r|l|} \hline \textbf{Name} & \textbf{Number} & \textbf{Powers of 2} \newline \hline \text{Two} & 2 & 2=2^1 \newline \hline \text{Four} & 4 & 2\times2=2^2 \newline \hline \text{Eight} & 8 & 2\times2\times2=2^3 \newline \hline \text{Sixteen} & 16 & 2\times2\times2\times2=2^4 \newline \hline \text{Thirty two} & 32 & 2\times2\times2\times2\times2=2^5 \newline \hline \text{Sixty four} & 64 & 2\times2\times2\times2\times2\times2=2^6 \newline \hline \text{One hundred and twenty eight} & 128 & 2\times2\times2\times2\times2\times2\times2=2^7 \newline \hline \text{Two hundred and fifty six} & 256 & 2\times2\times2\times2\times2\times2\times2\times2=2^8 \newline \hline \text{Five hundred and twelve} & 512 & 2\times2\times2\times2\times2\times2\times2\times2\times2=2^9 \newline \hline \text{One thousand and twenty four} & 1,024 & 2\times2\times2\times2\times2\times2\times2\times2\times2\times2=2^{10} \newline \hline \end{array}
In the base ten number system, you used place value tables to identify the value of each digit in a number. The place values were powers of 10.
We can use place value tables in the base two number system as well. The place values are powers of 2. For example:
2^0=1
To work with binary numbers, you need to be able to write a decimal number as the sum of powers of 2.
Remember that a decimal number is a number that uses the base ten number system. Decimal numbers are the numbers we use in everyday life.
2^6
2^6=64
2^5
2^5=32
. So the largest power of 2 that is equal to or smaller than 53 is
2^5=32
The last number found is 1, so there are no more powers of 2 that we can find.
2^0 = 1
2^3
2^1
This means that the decimal (base ten) number 53 written as a binary (base two) number is 110101.
Here is a table with the powers of 2. You can refer back to it as you answer the questions that follow.
The numbers in the base two number system are called binary numbers.
binary numbers The numbers in the base two number system are called binary numbers. They are all represented by combinations of the digits 0 and 1.
We can use a place value table to write the numbers 1 to 16 in the decimal number system as binary numbers:
Notation of decimal and binary numbers
When we work with decimal and binary numbers together, we need a way to identify whether a number is a decimal number or a binary number. For example, the digits 100 represent one hundred objects in the base ten number system, but they represent four objects in the base two number system.
100_{10}
100_{\text{ten}}
100_2
100_{\text{two}}
means four.
11010_\text{two}
11010_2
2^0
2^4
\begin{align} & \, 2^4+2^3+0+2^1+0 \\ = & \, 16+8+2 \\ = & \, 26 \end{align}
\begin{array}{|c|c|c|c|c|} \hline (2^4) & (2^3) & (2^2) & (2^1) & (2^0) \\ \hline & & 1 & 0 & 1 \\ \hline \end{array}
\begin{array}{|c|c|c|c|c|} \hline (2^4) & (2^3) & (2^2) & (2^1) & (2^0) \\ \hline & 1 & 0 & 0 & 1 \\ \hline \end{array}
\begin{array}{|c|c|c|c|c|} \hline (2^4) & (2^3) & (2^2) & (2^1) & (2^0) \\ \hline 1 & 0 & 0 & 1 & 1 \\ \hline \end{array}
\begin{array}{|c|c|c|c|c|c|} \hline (2^5) & (2^4) & (2^3) & (2^2) & (2^1) & (2^0) \\ \hline 1 & 0 & 1 & 0 & 0 & 1 \\ \hline \end{array}
\begin{array}{|c|c|c|c|c|c|c|} \hline (2^6) & (2^5) & (2^4) & (2^3) & (2^2) & (2^1) & (2^0) \\ \hline 1 & 0 & 0 & 0 & 1 & 1 & 1 \\ \hline \end{array}
23_\text{ten}
Remember that when you write the remainders, you start from the bottom one and go up to the top one.
\begin{array}{r|rr} 2 & 44 & \\ \hline 2 & 22 & \text{remainder: }0 \\ \hline 2 & 11 & \text{remainder: }0 \\ \hline 2 & 5 & \text{remainder: }1 \\ \hline 2 & 2 & \text{remainder: }1 \\ \hline 2 & 1 & \text{remainder: }0 \\ \hline & 0 & \text{remainder: }1 \\ \hline \end{array}
\begin{array}{r|rr} 2 & 97 & \\ \hline 2 & 48 & \text{remainder: }1 \\ \hline 2 & 24 & \text{remainder: }0 \\ \hline 2 & 12 & \text{remainder: }0 \\ \hline 2 & 6 & \text{remainder: }0 \\ \hline 2 & 3 & \text{remainder: }0 \\ \hline 2 & 1 & \text{remainder: }1 \\ \hline & 0 & \text{remainder: }1 \\ \hline \end{array}
The base two number system is a number system that counts in groups of 2, and uses only the digits 0 and 1 to represent any number.
The numbers in the base two number system are called binary numbers. They are all represented by combinations of the digits 0 and 1.
100_{10}
100_{\text{ten}}
100_2
100_{\text{two}}
We can convert binary numbers to decimal numbers, and decimal numbers to binary numbers.
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Design User Interface for Audio Plugin - MATLAB & Simulink - MathWorks España
In this example, you increase the padding around the perimeter of the grid to create space for the MathWorks® logo. You can calculate the total width of the UI grid as the sum of all column widths plus the left and right padding plus the column spacing (the default column spacing of 10 pixels is used in this example):
\left(100+100+100+50+150\right)+\left(20+20\right)+\left(4×10\right)=580
. The total height of the UI grid is the sum of all row heights plus the top and bottom padding plus the row spacing (the default row spacing of 10 pixels is used in this example):
\left(20+20+160+20+100\right)+\left(20+120\right)+\left(4×10\right)=500.
To locate the logo at the bottom of the UI grid, use a 580-by-500 image:
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Create regression model with ARIMA time series errors - MATLAB - MathWorks India
{u}_{t}=0.2{u}_{t-1}+0.1{u}_{t-4}+{\epsilon }_{t}.
{u}_{t}=0.2{u}_{t-1}+0.1{u}_{t-2}+{\epsilon }_{t}.
{u}_{t}={\epsilon }_{t}+0.2{\epsilon }_{t-1}+0.1{\epsilon }_{t-4}.
{u}_{t}={\epsilon }_{t}+0.2{\epsilon }_{t-1}+0.1{\epsilon }_{t-2}.
\left(1-0.2L-0.1{L}^{4}\right)\left(1-{L}^{4}\right){u}_{t}={\epsilon }_{t}.
\left(1-0.2L-0.1{L}^{2}\right)\left(1-{L}^{4}\right){u}_{t}={\epsilon }_{t}.
\left(1-{L}^{4}\right){u}_{t}=\left(1+0.2L+0.1{L}^{4}\right){\epsilon }_{t}.
\left(1-{L}^{4}\right){u}_{t}=\left(1+0.2L+0.1{L}^{2}\right){\epsilon }_{t}.
\begin{array}{c}{y}_{t}={u}_{t}\\ \left(1-{\varphi }_{1}L-{\varphi }_{2}{L}^{2}\right)\left(1-L\right){u}_{t}=\left(1+{\theta }_{1}L+{\theta }_{2}{L}^{2}+{\theta }_{3}{L}^{3}\right){\epsilon }_{t}.\end{array}
\begin{array}{l}\begin{array}{c}{y}_{t}=2+{X}_{t}\left[\begin{array}{c}1.5\\ 0.2\end{array}\right]+{u}_{t}\\ \left(1-0.2L-0.3{L}^{2}\right){u}_{t}=\left(1+0.1L\right){\epsilon }_{t},\end{array}\end{array}
{\epsilon }_{t}
{X}_{t}
t
\begin{array}{l}\begin{array}{c}{y}_{t}=1+6{X}_{t}+{u}_{t}\\ \left(1-0.2L\right)\left(1-L\right)\left(1-0.5{L}^{4}-0.2{L}^{8}\right)\left(1-{L}^{4}\right){u}_{t}=\left(1+0.1L\right)\left(1+0.05{L}^{4}+0.01{L}^{8}\right){\epsilon }_{t},\end{array}\end{array}
{\epsilon }_{t}
\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ a\left(L\right)A\left(L\right){\left(1-L\right)}^{D}\left(1-{L}^{s}\right){u}_{t}=b\left(L\right)B\left(L\right){\epsilon }_{t},\end{array}
{L}^{j}{y}_{t}={y}_{t-j}.
a\left(L\right)=\left(1-{a}_{1}L-...-{a}_{p}{L}^{p}\right),
A\left(L\right)=\left(1-{A}_{1}L-...-{A}_{{p}_{s}}{L}^{{p}_{s}}\right),
{\left(1-L\right)}^{D},
\left(1-{L}^{s}\right),
b\left(L\right)=\left(1+{b}_{1}L+...+{b}_{q}{L}^{q}\right),
B\left(L\right)=\left(1+{B}_{1}L+...+{B}_{{q}_{s}}{L}^{{q}_{s}}\right),
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3.3 Mission Control Report (PBL Part B) - Big History School
To work through '3.3 Mission Control Report (PBL Part B)' you need to complete the Activities in order. So first complete ‘Learning Plan’ then move to Activity '3.3.1' followed by Activity '3.3.2' through to '3.3.4', and then finish with ‘Learning Summary’.
MISSION CONTROL REPORT B: An Earth vs Mars comparison
In Mission Control Report B you will compare and contrast the features of Earth and Mars as well as the Goldilocks Conditions for human life on Earth and Mars.
To get started, read carefully through the Mission Control Report B learning goals below. Make sure you tick each of the check boxes to show that you have read all your Mission Control Report B learning goals.
As you read through the learning goals you may come across some words that you haven’t heard before. Please don’t worry. By the time you finish Mission Control Report B you will have become very familiar with them!
You will come back to these learning goals at the end of Mission Control Report B to see if you have confidently achieved them.
Earth vs Mars Venn diagram
Compare and contrast the features of Earth and Mars
Create an Earth vs Mars Venn diagram
Earth vs Mars Goldilocks Conditions table
Compare and contrast Goldilocks Conditions for human life on Earth and Mars
Complete an Earth vs Mars Goldilocks Conditions table
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Welcome to your second Mission Control Report!
In Mission Control Report A you became a Mars expert by researching Mars and writing an information report. You have since been learning a lot about Earth and the Goldilocks conditions for life on our planet.
Before you can proceed any further with your Mission, you need to show Mission Control that you have a thorough understanding about the similarities - and most importantly - the differences between Earth and Mars.
In Mission Control Report B you will need to complete an Earth vs Mars Venn diagram and an Earth vs Mars Goldilocks conditions table. You will not be able to proceed any further with your Mission until this Report has been successfully completed.
In Mission video 18: Earth vs Mars, the Mission Control Teams explore the differences and similarities between Earth and Mars and help you decide whether Mars has the Goldilocks conditions for life.
While you watch Mission video 18: Earth vs Mars look out for the answers to the following questions:
1. What did Earth and Mars have in common from the beginning?
2. Why did Earth and Mars evolve differently?
You will watch this same video in a following activity and will then focus on the part in the video about the Goldilocks conditions for life.
Now that you have a bit more comparative information on Earth and Mars, you will complete a Venn diagram which will highlight the similarities and, even more importantly, the differences between Earth and Mars.
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In this activity you will be using a Venn diagram to organize all of that interesting information you’ve gathered about Earth and Mars.
A Venn diagram is simply a visual graphic which you can use to compare and contrast two different things:
To compare is to look at what things have in common.
To contrast is to look at how they are different to each other.
Before you begin though, you will refer to the Reading: Earth vs Mars which summarizes a lot of the information you heard in Mission video 18: Earth vs Mars.
Your teacher will give you a copy of the Reading: Earth vs Mars. Take 5 minutes to read it carefully and use a highlighter to highlight the main points.
You will use the information in this Reading, along with your Mars Information Report research and everything you’ve learned about Earth in Mission Phase 3 to create your Venn diagram.
Your teacher will give you a copy of the Venn diagram: Earth vs Mars worksheet which provides a template for you to create your Venn diagram.
If you’re still not quite sure how to complete a Venn diagram, take a look at the Venn diagram: Earth vs Venus example in Helpful Resources.
Now it’s your turn. Refer to the Venn diagram: Earth vs Mars worksheet and follow the instructions:
Step 1. Write ‘Earth’ above one circle and ‘Mars’ above the second circle.
Step 2. Where the circles overlap write what the two planets have in common.
Step 3. In the remaining area of the circles students write what is different about each planet.
Venn Diagram: Earth vs Venus Example
Once you’ve completed your Venn diagram, take a look at it and notice:
Do Mars and Earth have more in common or are they more different than you thought?
What’s important about the differences? And what impact would those differences have on your Mission?
Your teacher will instruct you whether you will discuss your responses to these questions as part of a class or, if you are working on the Mission as a group, with the other members of your Mission Team.
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Having completed your Earth vs Mars Venn diagram should make it a lot easier to visualize what Mars and Earth have in common and how they are different. Now you are ready to think about what that potentially means for human life on Mars.
In this activity you will watch Mission video 18: Earth vs Mars once more, but this time you will focus on the part of the video which discusses the Goldilocks conditions for life on Earth and on Mars.
While you watch Mission video 18: Earth vs Mars this second time, look out for the answers to the following questions:
2. What are the three Goldilocks conditions on Earth for human life?
3. Does Mars have breathable air for humans?
4. Does Mars have water?
5. Does Mars have food for humans?
6. Does Mars provide shelter for humans?
Now that you have learned about the Goldilocks conditions for life on Earth and have considered whether Mars meets those conditions, you will set up a comparison table and begin to think about what that means for your Mars Mission.
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For the final activity in Mission Control Report B, you will take what you have learned about the Goldilocks conditions for life on Earth and on Mars to complete a comparative table. Completing the table will help you begin to plan for human life on your Mars Mission.
Before you begin though, you will refer to the Reading: Does Mars have the Goldilocks conditions for life? which recaps a lot of the important information you’ll need to complete this activity.
Your teacher will give you a copy of the Reading: Does Mars have the Goldilocks conditions for life? Take 5 minutes to read it carefully and use a highlighter to highlight the main points.
You will use the information in this reading, along with everything else you’ve learned during your Mars Mission so far, to complete your Table: Earth vs Mars Goldilocks conditions worksheet.
We’ve come to a very important step in preparing for your Mission. Using the Table: Earth vs Mars Goldilocks conditions worksheet, you need to work out if Mars has the Goldilocks conditions for life, and if not, what that means for human life on your Mars Mission:
Complete the first column of the table by listing the four basic needs for human life
In the second column describe how these basic needs are met on Earth
Circle Yes or No in the third column based on whether you believe Mars has the Goldilocks conditions to meet each of those needs. Explain why or why not
Finally, based on whether you think Mars has the Goldilocks conditions for each of our four basic humans needs, write down in the fourth column what this means for humans who travel to Mars
The first row of the table has been completed as an example.
Now that you’ve started to think about the lack of Goldilocks conditions on Mars and the challenges this creates for human life there, it’s a good time to start thinking about possible solutions. Do you have any ideas yet?
Refer back to the Chart: KWHLAQ and add what you have learned during this Mission Control Report to the “L - What have you Learned?” column. To refresh your memory there is a copy in Helpful Resources.
Also, check whether you have answered any more of the questions in the “W - What do you Want to know?” column.
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In Mission Control Report B you compared and contrasted the features of Earth and Mars as well as the Goldilocks Conditions for human life on Earth and Mars.
Now it’s time to revisit your Mission Control Report B learning goals and read through them again carefully.
Once you have checked the boxes to confirm you have achieved your learning goals for Sequence '3.3 Mission Control Report (PBL Part B)' click on the 'I have achieved my learning goals' button below.
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f\left(x\right)=\frac{1}{2}(x−2)^3+1
g(x)=2x^2−6x−3
Write an equation that you could solve using points
A
B
. What are the solutions to your equation? Substitute them into your equation to show that they work.
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 the 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 solve using point
C
. What does the solution to your equation appear to be? Again, substitute your solution into the equation. How close was your estimate?
f(x)
0
What are the domains and ranges of
f(x)
g(x)
f(x)
: All real numbers
f(x)
g(x)
g(x):y\ge−7.5
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Lucky Lucky - ButterSwap
Lucky Lucky Prize History
Each day, there is a special round of Lucky Lucky, in which we give one lucky participating board member a special prize (
1\%
of all reward emissions per day, which is 138,240 BUTTERs based on current emission rate).
You can stake any number of BOARD tokens to in the Lucky Lucky, pool and your BOARD tokens will be returned in full amount if you leave Lucky Lucky. If you keep the staked BOARD tokens in Lucky Lucky pools without unstaking, you will automatically participate each round of Lucky Lucky.
Winning probability, what we name it power, is determined by both the number of staked BOARD tokens and the staking timing. If you have not staked any BOARD tokens before the current round of Lucky Lucky starts, the initial power is set to 0. If you have already staked some BOARD tokens before the current round of Lucky Lucky starts, the initial power is given as follows:
power = board_current * (end_block - start_block)
board_current is the current number of BOARD tokens you have already staked, end_block is the predicted end block number of this round of Lucky Lucky, and start_block is the start block number.
During the day, you can stake and unstake freely any mount of BOARD tokens to Lucky Lucky event.
Each time you stake BOARD tokens to Lucky Lucky event, the power is updated as follows:
power = power + board_num * (end_block - current_block)
board_num is the number of BOARD tokens you stake this time, end_block is the predicted end block number of this round of Lucky Lucky, and current_block is the current block number.
Each time you unstake BOARD tokens from Lucky Lucky event, the power is updated as follows:
power = power * (board_total - board_num) / board_total
board_total is the number of staked BOARD tokens before this unstaking, and board_num is the number of BOARD tokens to be unstaked.
By the end of day, the smart contract will calculate a power value for each participants. And a weighted random selection based on the power value will be used to select the winner for the Lucky Lucky event.
Everything is calculated inside smart contracts on blockchain, and we believe it is a fair and fun event to participate.
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Hydrostatics - Wikipedia
(Redirected from Hydrostatic pressure)
This article is about the discipline branch. For the concept, see Hydrostatic equilibrium.
Table of Hydraulics and Hydrostatics, from the 1728 Cyclopædia
{\displaystyle J=-D{\frac {d\varphi }{dx}}}
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium[1] and the pressure in a fluid, or exerted by a fluid, on an immersed body".[2]
It encompasses the study of the conditions under which fluids are at rest in stable equilibrium as opposed to fluid dynamics, the study of fluids in motion. Hydrostatics is a subcategory of fluid statics, which is the study of all fluids, both compressible or incompressible, at rest.
Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why the surface of still water is always level according to the curvature of the earth.
1.1 Hydrostatics in ancient Greece and Rome
1.1.1 Pythagorean Cup
1.1.2 Heron's fountain
1.2 Pascal's contribution in hydrostatics
2.5 Hydrostatic force on submerged surfaces
3 Liquids (fluids with free surfaces)
3.2 Hanging drops
Some principles of hydrostatics have been known in an empirical and intuitive sense since antiquity, by the builders of boats, cisterns, aqueducts and fountains. Archimedes is credited with the discovery of Archimedes' Principle, which relates the buoyancy force on an object that is submerged in a fluid to the weight of fluid displaced by the object. The Roman engineer Vitruvius warned readers about lead pipes bursting under hydrostatic pressure.[3]
The concept of pressure and the way it is transmitted by fluids was formulated by the French mathematician and philosopher Blaise Pascal in 1647.
Hydrostatics in ancient Greece and Rome[edit]
Pythagorean Cup[edit]
Main article: Pythagorean cup
The "fair cup" or Pythagorean cup, which dates from about the 6th century BC, is a hydraulic technology whose invention is credited to the Greek mathematician and geometer Pythagoras. It was used as a learning tool.
The cup consists of a line carved into the interior of the cup, and a small vertical pipe in the center of the cup that leads to the bottom. The height of this pipe is the same as the line carved into the interior of the cup. The cup may be filled to the line without any fluid passing into the pipe in the center of the cup. However, when the amount of fluid exceeds this fill line, fluid will overflow into the pipe in the center of the cup. Due to the drag that molecules exert on one another, the cup will be emptied.
Heron's fountain[edit]
Main article: Heron's fountain
Heron's fountain is a device invented by Heron of Alexandria that consists of a jet of fluid being fed by a reservoir of fluid. The fountain is constructed in such a way that the height of the jet exceeds the height of the fluid in the reservoir, apparently in violation of principles of hydrostatic pressure. The device consisted of an opening and two containers arranged one above the other. The intermediate pot, which was sealed, was filled with fluid, and several cannula (a small tube for transferring fluid between vessels) connecting the various vessels. Trapped air inside the vessels induces a jet of water out of a nozzle, emptying all water from the intermediate reservoir.
Pascal's contribution in hydrostatics[edit]
Main article: Pascal's Law
Pascal made contributions to developments in both hydrostatics and hydrodynamics. Pascal's Law is a fundamental principle of fluid mechanics that states that any pressure applied to the surface of a fluid is transmitted uniformly throughout the fluid in all directions, in such a way that initial variations in pressure are not changed.
Pressure in fluids at rest[edit]
Due to the fundamental nature of fluids, a fluid cannot remain at rest under the presence of a shear stress. However, fluids can exert pressure normal to any contacting surface. If a point in the fluid is thought of as an infinitesimally small cube, then it follows from the principles of equilibrium that the pressure on every side of this unit of fluid must be equal. If this were not the case, the fluid would move in the direction of the resulting force. Thus, the pressure on a fluid at rest is isotropic; i.e., it acts with equal magnitude in all directions. This characteristic allows fluids to transmit force through the length of pipes or tubes; i.e., a force applied to a fluid in a pipe is transmitted, via the fluid, to the other end of the pipe. This principle was first formulated, in a slightly extended form, by Blaise Pascal, and is now called Pascal's law.
Hydrostatic pressure[edit]
See also: Vertical pressure variation
In a fluid at rest, all frictional and inertial stresses vanish and the state of stress of the system is called hydrostatic. When this condition of V = 0 is applied to the Navier–Stokes equations, the gradient of pressure becomes a function of body forces only. For a barotropic fluid in a conservative force field like a gravitational force field, the pressure exerted by a fluid at equilibrium becomes a function of force exerted by gravity.
The hydrostatic pressure can be determined from a control volume analysis of an infinitesimally small cube of fluid. Since pressure is defined as the force exerted on a test area (p = F/A, with p: pressure, F: force normal to area A, A: area), and the only force acting on any such small cube of fluid is the weight of the fluid column above it, hydrostatic pressure can be calculated according to the following formula:
{\displaystyle p(z)-p(z_{0})={\frac {1}{A}}\int _{z_{0}}^{z}dz'\iint _{A}dx'dy'\,\rho (z')g(z')=\int _{z_{0}}^{z}dz'\,\rho (z')g(z'),}
p is the hydrostatic pressure (Pa),
ρ is the fluid density (kg/m3),
g is gravitational acceleration (m/s2),
A is the test area (m2),
z is the height (parallel to the direction of gravity) of the test area (m),
z0 is the height of the zero reference point of the pressure (m).
For water and other liquids, this integral can be simplified significantly for many practical applications, based on the following two assumptions: Since many liquids can be considered incompressible, a reasonable good estimation can be made from assuming a constant density throughout the liquid. (The same assumption cannot be made within a gaseous environment.) Also, since the height h of the fluid column between z and z0 is often reasonably small compared to the radius of the Earth, one can neglect the variation of g. Under these circumstances, the integral is simplified into the formula:
{\displaystyle p-p_{0}=\rho gh,}
where h is the height z − z0 of the liquid column between the test volume and the zero reference point of the pressure. This formula is often called Stevin's law.[4][5] Note that this reference point should lie at or below the surface of the liquid. Otherwise, one has to split the integral into two (or more) terms with the constant ρliquid and ρ(z′)above. For example, the absolute pressure compared to vacuum is:
{\displaystyle p=\rho gH+p_{\mathrm {atm} },}
where H is the total height of the liquid column above the test area to the surface, and patm is the atmospheric pressure, i.e., the pressure calculated from the remaining integral over the air column from the liquid surface to infinity. This can easily be visualized using a pressure prism.
Hydrostatic pressure has been used in the preservation of foods in a process called pascalization.[6]
In medicine, hydrostatic pressure in blood vessels is the pressure of the blood against the wall. It is the opposing force to oncotic pressure.
Statistical mechanics shows that, for a pure ideal gas of constant temperature in a gravitational field, T, its pressure, p will vary with height, h, as:
{\displaystyle p(h)=p(0)e^{-{\frac {Mgh}{kT}}}}
M is the mass of a single molecule of gas
This is known as the barometric formula, and maybe derived from assuming the pressure is hydrostatic.
If there are multiple types of molecules in the gas, the partial pressure of each type will be given by this equation. Under most conditions, the distribution of each species of gas is independent of the other species.
Buoyancy[edit]
Any body of arbitrary shape which is immersed, partly or fully, in a fluid will experience the action of a net force in the opposite direction of the local pressure gradient. If this pressure gradient arises from gravity, the net force is in the vertical direction opposite that of the gravitational force. This vertical force is termed buoyancy or buoyant force and is equal in magnitude, but opposite in direction, to the weight of the displaced fluid. Mathematically,
{\displaystyle F=\rho gV}
where ρ is the density of the fluid, g is the acceleration due to gravity, and V is the volume of fluid directly above the curved surface.[7] In the case of a ship, for instance, its weight is balanced by pressure forces from the surrounding water, allowing it to float. If more cargo is loaded onto the ship, it would sink more into the water – displacing more water and thus receive a higher buoyant force to balance the increased weight.
Hydrostatic force on submerged surfaces[edit]
The horizontal and vertical components of the hydrostatic force acting on a submerged surface are given by the following:[7]
{\displaystyle {\begin{aligned}F_{\mathrm {h} }&=p_{\mathrm {c} }A\\F_{\mathrm {v} }&=\rho gV\end{aligned}}}
pc is the pressure at the centroid of the vertical projection of the submerged surface
A is the area of the same vertical projection of the surface
V is the volume of fluid directly above the curved surface
Liquids (fluids with free surfaces)[edit]
Capillary action[edit]
Hanging drops[edit]
Without surface tension, drops would not be able to form. The dimensions and stability of drops are determined by surface tension. The drop's surface tension is directly proportional to the cohesion property of the fluid.
Hydrostatic test – Non-destructive test of pressure vessels
^ "Fluid Mechanics/Fluid Statics/mentals of Fluid Statics - Wikibooks, open books for an open world". en.wikibooks.org. Retrieved 2021-04-01.
^ "Hydrostatics". Merriam-Webster. Retrieved 11 September 2018.
^ Bettini, Alessandro (2016). A Course in Classical Physics 2—Fluids and Thermodynamics. Springer. p. 8. ISBN 978-3-319-30685-8.
^ Mauri, Roberto (8 April 2015). Transport Phenomena in Multiphase Flow. Springer. p. 24. ISBN 978-3-319-15792-4. Retrieved 3 February 2017.
^ Brown, Amy Christian (2007). Understanding Food: Principles and Preparation (3 ed.). Cengage Learning. p. 546. ISBN 978-0-495-10745-3.
^ a b Fox, Robert; McDonald, Alan; Pritchard, Philip (2012). Fluid Mechanics (8 ed.). John Wiley & Sons. pp. 76–83. ISBN 978-1-118-02641-0.
Kundu, Pijush K.; Cohen, Ira M. (2008). Fluid Mechanics (4th rev. ed.). Academic Press. ISBN 978-0-12-373735-9.
Massey, B.; Ward-Smith, J. (2005). Mechanics of Fluids (8th ed.). Taylor & Francis. ISBN 978-0-415-36206-1.
Look up hydrostatics in Wiktionary, the free dictionary.
Ayman, Mohammad (2003). "Hydrostatics". University of Denver. Retrieved 2013-05-22.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hydrostatics&oldid=1083663956#Hydrostatic_pressure"
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Eigenvalues and eigenvectors - Simple English Wikipedia, the free encyclopedia
vectors that map to their scalar multiples, and the associated scalars
Linear algebra talks about types of functions called transformations. In that context, an eigenvector is a vector—different from the null vector—which does not change direction after the transformation (except if the transformation turns the vector to the opposite direction). The vector may change its length, or become zero ("null"). The eigenvalue is the value of the vector's change in length, and is typically denoted by the symbol
{\displaystyle \lambda }
.[1] The word "eigen" is a German word, which means "own" or "typical".[2]
Illustration of a transformation (of Mona Lisa): The image is changed in such a way that the red arrow (vector) does not change its direction, but the blue one does. The red vector therefore is an eigenvector of this transformation, the blue one is not. Since the red vector does not change its length, its eigenvalue is 1. The transformation used is called shear mapping.
If there exists a square matrix called A, a scalar λ, and a non-zero vector v, then λ is the eigenvalue and v is the eigenvector if the following equation is satisfied:
{\displaystyle A\mathbf {v} =\lambda \mathbf {v} \,.}
In other words, if matrix A times the vector v is equal to the scalar λ times the vector v, then λ is the eigenvalue of v, where v is the eigenvector.
An eigenspace of A is the set of all eigenvectors with the same eigenvalue together with the zero vector. However, the zero vector is not an eigenvector.[4]
These ideas often are extended to more general situations, where scalars are elements of any field, vectors are elements of any vector space, and linear transformations may or may not be represented by matrix multiplication. For example, instead of real numbers, scalars may be complex numbers; instead of arrows, vectors may be functions or frequencies; instead of matrix multiplication, linear transformations may be operators such as the derivative from calculus. These are only a few of countless examples where eigenvectors and eigenvalues are important.
In cases like these, the idea of direction loses its ordinary meaning, and has a more abstract definition instead. But even in this case, if that abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction, eigenmode, eigenface, eigenstate, and eigenfrequency.
Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrix factorization, quantum mechanics, facial recognition systems, and many other areas.
For the matrix A
{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}
{\displaystyle \mathbf {x} ={\begin{bmatrix}3\\-3\end{bmatrix}}}
is an eigenvector with eigenvalue 1. Indeed, one can verify that:
{\displaystyle A\mathbf {x} ={\begin{bmatrix}2&1\\1&2\end{bmatrix}}{\begin{bmatrix}3\\-3\end{bmatrix}}={\begin{bmatrix}(2\cdot 3)+(1\cdot (-3))\\(1\cdot 3)+(2\cdot (-3))\end{bmatrix}}={\begin{bmatrix}3\\-3\end{bmatrix}}=1\cdot {\begin{bmatrix}3\\-3\end{bmatrix}}.}
{\displaystyle \mathbf {x} ={\begin{bmatrix}0\\1\end{bmatrix}}}
{\displaystyle {\begin{bmatrix}2&1\\1&2\end{bmatrix}}{\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}(2\cdot 0)+(1\cdot 1)\\(1\cdot 0)+(2\cdot 1)\end{bmatrix}}={\begin{bmatrix}1\\2\end{bmatrix}}.}
and this vector is not a multiple of the original vector x.
↑ "Eigenvector and Eigenvalue". www.mathsisfun.com. Retrieved 2020-08-19.
↑ Weisstein, Eric W. "Eigenvalue". mathworld.wolfram.com. Retrieved 2020-08-19.
↑ Weisstein, Eric W. "Eigenvector". mathworld.wolfram.com. Retrieved 2020-08-19.
Korn, Granino A.; Korn, Theresa M. (2000), Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, 1152 p., Dover Publications, 2 Revised edition, ISBN 0-486-41147-8 .
Lipschutz, Seymour (1991), Schaum's outline of theory and problems of linear algebra, Schaum's outline series (2nd ed.), New York, NY: McGraw-Hill Companies, ISBN 0-07-038007-4 .
Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (1989), Linear algebra (2nd ed.), Englewood Cliffs, NJ: Prentice Hall, ISBN 0-13-537102-3 .
Aldrich, John (2006), "Eigenvalue, eigenfunction, eigenvector, and related terms", in Jeff Miller (ed.), Earliest Known Uses of Some of the Words of Mathematics, retrieved 2006-08-22
Strang, Gilbert (1993), Introduction to linear algebra, Wellesley-Cambridge Press, Wellesley, MA, ISBN 0-961-40885-5 .
Strang, Gilbert (2006), Linear algebra and its applications, Thomson, Brooks/Cole, Belmont, CA, ISBN 0-030-10567-6 .
Bowen, Ray M.; Wang, Chao-Cheng (1980), Linear and multilinear algebra, Plenum Press, New York, NY, ISBN 0-306-37508-7 .
Cohen-Tannoudji, Claude (1977), "Chapter II. The mathematical tools of quantum mechanics", Quantum mechanics, John Wiley & Sons, ISBN 0-471-16432-1 .
Fraleigh, John B.; Beauregard, Raymond A. (1995), Linear algebra (3rd international ed.), Addison-Wesley Publishing Company, ISBN 0-201-83999-7 .
Golub, Gene H.; Van Loan, Charles F. (1996), Matrix computations (3rd Edition), Johns Hopkins University Press, Baltimore, MD, ISBN 978-0-8018-5414-9 .
Hawkins, T. (1975), "Cauchy and the spectral theory of matrices", Historia Mathematica, 2: 1–29, doi:10.1016/0315-0860(75)90032-4 .
Horn, Roger A.; Johnson, Charles F. (1985), Matrix analysis, Cambridge University Press . ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback)
Kline, Morris (1972), Mathematical thought from ancient to modern times, Oxford University Press, ISBN 0-195-01496-0 .
Meyer, Carl D. (2000), Matrix analysis and applied linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, ISBN 978-0-89871-454-8 .
Brown, Maureen (October 2004), Illuminating Patterns of Perception: An Overview of Q Methodology .
Golub, Gene F.; van der Vorst, Henk A. (2000), "Eigenvalue computation in the 20th century", Journal of Computational and Applied Mathematics, 123: 35–65, doi:10.1016/S0377-0427(00)00413-1 .
Akivis, Max A.; Vladislav V. Goldberg (1969), Tensor calculus, Russian, Science Publishers, Moscow .
Gelfand, I. M. (1971), Lecture notes in linear algebra, Russian, Science Publishers, Moscow .
Alexandrov, Pavel S. (1968), Lecture notes in analytical geometry, Russian, Science Publishers, Moscow .
Carter, Tamara A.; Tapia, Richard A.; Papaconstantinou, Anne, Linear Algebra: An Introduction to Linear Algebra for Pre-Calculus Students, Rice University, Online Edition, retrieved 2008-02-19 .
Roman, Steven (2008), Advanced linear algebra (3rd ed.), New York, NY: Springer Science + Business Media, LLC, ISBN 978-0-387-72828-5 .
Shilov, Georgi E. (1977), Linear algebra (translated and edited by Richard A. Silverman ed.), New York: Dover Publications, ISBN 0-486-63518-X .
Hefferon, Jim (2001), Linear Algebra, Online book, St Michael's College, Colchester, Vermont, USA .
Kuttler, Kenneth (2007), An introduction to linear algebra (PDF), Online e-book in PDF format, Brigham Young University .
Demmel, James W. (1997), Applied numerical linear algebra, SIAM, ISBN 0-89871-389-7 .
Beezer, Robert A. (2006), A first course in linear algebra, Free online book under GNU licence, University of Puget Sound .
Lancaster, P. (1973), Matrix theory, Russian, Moscow, Russia: Science Publishers .
Halmos, Paul R. (1987), Finite-dimensional vector spaces (8th ed.), New York, NY: Springer-Verlag, ISBN 0387900934 .
Pigolkina, T. S. and Shulman, V. S., Eigenvalue (in Russian), In:Vinogradov, I. M. (Ed.), Mathematical Encyclopedia, Vol. 5, Soviet Encyclopedia, Moscow, 1977.
Greub, Werner H. (1975), Linear Algebra (4th Edition), Springer-Verlag, New York, NY, ISBN 0-387-90110-8 .
Larson, Ron; Edwards, Bruce H. (2003), Elementary linear algebra (5th ed.), Houghton Mifflin Company, ISBN 0-618-33567-6 .
Curtis, Charles W., Linear Algebra: An Introductory Approach, 347 p., Springer; 4th ed. 1984. Corr. 7th printing edition (August 19, 1999), ISBN 0-387-90992-3.
Shores, Thomas S. (2007), Applied linear algebra and matrix analysis, Springer Science+Business Media, LLC, ISBN 0-387-33194-8 .
Sharipov, Ruslan A. (1996), Course of Linear Algebra and Multidimensional Geometry: the textbook, arXiv:math/0405323, ISBN 5-7477-0099-5 .
Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2005), Indefinite linear algebra and applications, Basel-Boston-Berlin: Birkhäuser Verlag, ISBN 3-7643-7349-0 .
The English Wikibook The Book of Mathematical Proofs has more information on:
Algebra/Linear Transformations
Introduction to Eigen Vectors and Eigen Values Archived 2012-03-05 at the Wayback Machine – lecture from Kahn Academy
Numerical solution of eigenvalue problems, edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe and Henk van der Vorst
bluebit.gr Archived 2008-12-12 at the Wayback Machine
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Eigenvalues_and_eigenvectors&oldid=8147383"
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Linear-Quadratic-Gaussian (LQG) Design - MATLAB & Simulink - MathWorks España
Linear-Quadratic-Gaussian (LQG) Design for Regulation
Constructing the Optimal State-Feedback Gain for Regulation
Constructing the Kalman State Estimator
Forming the LQG Regulator
Linear-Quadratic-Gaussian (LQG) Design of Servo Controller with Integral Action
Constructing the Optimal State-Feedback Gain for Servo Control
Forming the LQG Servo Control
Linear-quadratic-Gaussian (LQG) control is a modern state-space technique for designing optimal dynamic regulators and servo controllers with integral action (also known as setpoint trackers). This technique allows you to trade off regulation/tracker performance and control effort, and to take into account process disturbances and measurement noise.
To design LQG regulators and setpoint trackers, you perform the following steps:
Construct the LQ-optimal gain.
Construct a Kalman filter (state estimator).
Form the LQG design by connecting the LQ-optimal gain and the Kalman filter.
For more information about using LQG design to create LQG regulators , see Linear-Quadratic-Gaussian (LQG) Design for Regulation.
For more information about using LQG design to create LQG servo controllers, see Linear-Quadratic-Gaussian (LQG) Design of Servo Controller with Integral Action.
These topics focus on the continuous-time case. For information about discrete-time LQG design, see the dlqr and kalman reference pages.
You can design an LQG regulator to regulate the output y around zero in the following model.
The plant in this model experiences disturbances (process noise) w and is driven by controls u. The regulator relies on the noisy measurements y to generate these controls. The plant state and measurement equations take the form of
\begin{array}{l}\stackrel{˙}{x}=Ax+Bu+Gw\\ y=Cx+Du+Hw+v\end{array}
and both w and v are modeled as white noise.
LQG design requires a state-space model of the plant. You can use ss to convert other model formats to state space.
To design LQG regulators, you can use the design techniques shown in the following table.
To design an LQG regulator using...
A quick, one-step design technique when the following is true:
You need the optimal LQG controller and either E(wv') or H is nonzero.
All known (deterministic) inputs are control inputs and all outputs are measured.
Integrator states are weighted independently of states of plants and control inputs.
A more flexible, three-step design technique that allows you to specify:
Arbitrary G and H.
Known (deterministic) inputs that are not controls and/or outputs that are not measured.
A flexible weighting scheme for integrator states, plant states, and controls.
lqr, kalman, and lqgreg
You construct the LQ-optimal gain from the following elements:
State-space system matrices
Weighting matrices Q, R, and N, which define the tradeoff between regulation performance (how fast x(t) goes to zero) and control effort.
To construct the optimal gain, type the following command:
K= lqr(A,B,Q,R,N)
This command computes the optimal gain matrix K, for which the state feedback law
u=-Kx
minimizes the following quadratic cost function for continuous time:
J\left(u\right)={\int }_{0}^{\infty }\left\{{x}^{T}Qx+2{x}^{T}Nu+{u}^{T}Ru\right\}dt
The software computes the gain matrix K by solving an algebraic Riccati equation.
For information about constructing LQ-optimal gain, including the cost function that the software minimizes for discrete time, see the lqr reference page.
You need a Kalman state estimator for LQG regulation and servo control because you cannot implement optimal LQ-optimal state feedback without full state measurement.
You construct the state estimate
\stackrel{^}{x}
u=-K\stackrel{^}{x}
remains optimal for the output-feedback problem. You construct the Kalman state estimator gain from the following elements:
State-space plant model sys
Noise covariance data, Qn, Rn, and Nn
The following figure shows the required dimensions for Qn, Rn, and Nn. If Nn is 0, you can omit it.
Required Dimensions for Qn, Rn, and Nn
You construct the Kalman state estimator in the same way for both regulation and servo control.
To construct the Kalman state estimator, type the following command:
[kest,L,P] = kalman(sys,Qn,Rn,Nn);
This command computes a Kalman state estimator, kest with the following plant equations:
\begin{array}{l}\stackrel{˙}{x}=Ax+Bu+Gw\\ y=Cx+Du+Hw+v\end{array}
where w and v are modeled as white noise. L is the Kalman gain and P the covariance matrix.
The software generates this state estimate using the Kalman filter
\frac{d}{dt}\stackrel{^}{x}=A\stackrel{^}{x}+Bu+L\left(y-C\stackrel{^}{x}-Du\right)
with inputs u (controls) and y (measurements). The noise covariance data
E\left(w{w}^{T}\right)={Q}_{n},\text{ }E\left(v{v}^{T}\right)={R}_{n},\text{ }E\left(w{v}^{T}\right)={N}_{n}
determines the Kalman gain L through an algebraic Riccati equation.
The Kalman filter is an optimal estimator when dealing with Gaussian white noise. Specifically, it minimizes the asymptotic covariance
\text{ }\underset{t\to \infty }{\mathrm{lim}}
E\left(\left(x-\stackrel{^}{x}\right){\left(x-\stackrel{^}{x}\right)}^{T}\right)
of the estimation error
x-\stackrel{^}{x}
For more information, see the kalman reference page. For a complete example of a Kalman filter implementation, see Kalman Filtering.
To form the LQG regulator, connect the Kalman filter kest and LQ-optimal gain K by typing the following command:
regulator = lqgreg(kest, K);
This command forms the LQG regulator shown in the following figure.
The regulator has the following state-space equations:
\begin{array}{l}\frac{d}{dt}\stackrel{^}{x}=\left[A-LC-\left(B-LD\right)K\right]\stackrel{^}{x}+Ly\\ u=-K\stackrel{^}{x}\end{array}
For more information on forming LQG regulators, see lqgreg and LQG Regulation: Rolling Mill Case Study.
You can design a servo controller with integral action for the following model:
The servo controller you design ensures that the output y tracks the reference command r while rejecting process disturbances w and measurement noise v.
The plant in the previous figure is subject to disturbances w and is driven by controls u. The servo controller relies on the noisy measurements y to generate these controls. The plant state and measurement equations are of the form
\begin{array}{l}\stackrel{˙}{x}=Ax+Bu+Gw\\ y=Cx+Du+Hw+v\end{array}
To design LQG servo controllers, you can use the design techniques shown in the following table.
To design an LQG servo controller using...
lqi, kalman, and lqgtrack
You construct the LQ-optimal gain from the
Weighting matrices Q, R, and N, which define the tradeoff between tracker performance and control effort
K= lqi(sys,Q,R,N)
u=-Kz=-K\left[x;{x}_{i}\right]
J\left(u\right)={\int }_{0}^{\infty }\left\{{z}^{T}Qz+{u}^{T}Ru+2{z}^{T}Nu\right\}dt
For information about constructing LQ-optimal gain, including the cost function that the software minimizes for discrete time, see the lqi reference page.
You need a Kalman state estimator for LQG regulation and servo control because you cannot implement LQ-optimal state feedback without full state measurement.
\stackrel{^}{x}
u=-K\stackrel{^}{x}
\begin{array}{l}\stackrel{˙}{x}=Ax+Bu+Gw\\ y=Cx+Du+Hw+v\end{array}
\frac{d}{dt}\stackrel{^}{x}=A\stackrel{^}{x}+Bu+L\left(y-C\stackrel{^}{x}-Du\right)
E\left(w{w}^{T}\right)={Q}_{n},\text{ }E\left(v{v}^{T}\right)={R}_{n},\text{ }E\left(w{v}^{T}\right)={N}_{n}
\text{ }\underset{t\to \infty }{\mathrm{lim}}
E\left(\left(x-\stackrel{^}{x}\right){\left(x-\stackrel{^}{x}\right)}^{T}\right)
x-\stackrel{^}{x}
To form a two-degree-of-freedom LQG servo controller, connect the Kalman filter kest and LQ-optimal gain K by typing the following command:
servocontroller = lqgtrack(kest, K);
This command forms the LQG servo controller shown in the following figure.
The servo controller has the following state-space equations:
\begin{array}{l}\left[\begin{array}{c}\stackrel{˙}{\stackrel{^}{x}}\\ {\stackrel{˙}{x}}_{i}\end{array}\right]=\left[\begin{array}{cc}A-B{K}_{x}-LC+LD{K}_{x}& -B{K}_{i}+LD{K}_{i}\\ 0& 0\end{array}\right]\left[\begin{array}{c}\stackrel{^}{x}\\ {x}_{i}\end{array}\right]+\left[\begin{array}{cc}0& L\\ I& -I\end{array}\right]\left[\begin{array}{c}r\\ y\end{array}\right]\\ u=\left[\begin{array}{cc}-{K}_{x}& -{K}_{i}\end{array}\right]\left[\begin{array}{c}\stackrel{^}{x}\\ {x}_{i}\end{array}\right]\end{array}
For more information on forming LQG servo controllers, including how to form a one-degree-of-freedom LQG servo controller, see the lqgtrack reference page.
lqg | lqr | kalman | lqgtrack | lqi | lqgreg
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Lee wave - Wikipedia
(Redirected from Lee waves)
Atmospheric stationary oscillations
The wind flows towards a mountain and produces a first oscillation (A) followed by more waves. The following waves will have lower amplitude because of the natural damping. Lenticular clouds stuck on top of the flow (A) and (B) will appear immobile despite the strong wind.
In meteorology, lee waves are atmospheric stationary waves. The most common form is mountain waves, which are atmospheric internal gravity waves. These were discovered in 1933 by two German glider pilots, Hans Deutschmann and Wolf Hirth, above the Krkonoše.[1][2][3] They are periodic changes of atmospheric pressure, temperature and orthometric height in a current of air caused by vertical displacement, for example orographic lift when the wind blows over a mountain or mountain range. They can also be caused by the surface wind blowing over an escarpment or plateau,[4] or even by upper winds deflected over a thermal updraft or cloud street.
The vertical motion forces periodic changes in speed and direction of the air within this air current. They always occur in groups on the lee side of the terrain that triggers them. Sometimes, mountain waves can help to enhance precipitation amounts downwind of mountain ranges.[5] Usually a turbulent vortex, with its axis of rotation parallel to the mountain range, is generated around the first trough; this is called a rotor. The strongest lee waves are produced when the lapse rate shows a stable layer above the obstruction, with an unstable layer above and below.[4]
Strong winds (with wind gusts over 100 mph) can be created in the foothills of large mountain ranges by mountain waves.[6][7][8][9] These strong winds can contribute to unexpected wildfire growth and spread (including the 2016 Great Smoky Mountains wildfires when sparks from a wildfire in the Smoky Mountains were blown into the Gatlinburg and Pigeon Forge areas).[10]
4 Other varieties of atmospheric waves
A fluid dynamics lab experiment illustrates flow past a mountain-shaped obstacle. Downstream wave crests radiate upwards with their group velocity pointing about 45° from horizontal. A downslope jet can be seen in the lee of the mountain, an area of lower pressure, enhanced turbulence, and periodic vertical displacement of fluid parcels. Vertical dye lines indicate effects are also felt upstream of the mountain, an area of higher pressure.
Lee waves are a form of internal gravity waves produced when a stable, stratified flow is forced over an obstacle. This disturbance elevates air parcels above their level of neutral buoyancy. Buoyancy restoring forces therefore act to excite a vertical oscillation of the perturbed air parcels at the Brunt-Väisäla frequency, which for the atmosphere is:
{\displaystyle N={\sqrt {{g \over \theta _{0}}{d\theta _{0} \over dz}}}}
{\displaystyle \theta _{0}(z)}
is the vertical profile of potential temperature.
Oscillations tilted off the vertical axis at an angle of
{\displaystyle \phi }
will occur at a lower frequency of
{\displaystyle N\cos {\phi }}
. These air parcel oscillations occur in concert, parallel to the wave fronts (lines of constant phase). These wave fronts represent extrema in the perturbed pressure field (i.e., lines of lowest and highest pressure), while the areas between wave fronts represent extrema in the perturbed buoyancy field (i.e., areas most rapidly gaining or losing buoyancy).
Energy is transmitted along the wave fronts (parallel to air parcel oscillations), which is the direction of the wave group velocity. In contrast, the phase propagation (or phase speed) of the waves points perpendicular to energy transmission (or group velocity).[11][12]
A wave window over the Bald Eagle Valley of central Pennsylvania as seen from a glider looking north. The wind flow is from upper left to lower right. The Allegheny Front is under the left edge of the window, the rising air is at the right edge, and the distance between them is 3–4 km.
Both lee waves and the rotor may be indicated by specific wave cloud formations if there is sufficient moisture in the atmosphere, and sufficient vertical displacement to cool the air to the dew point. Waves may also form in dry air without cloud markers.[4] Wave clouds do not move downwind as clouds usually do, but remain fixed in position relative to the obstruction that forms them.
Around the crest of the wave, adiabatic expansion cooling can form a cloud in shape of a lens (lenticularis). Multiple lenticular clouds can be stacked on top of each other if there are alternating layers of relatively dry and moist air aloft.
The rotor may generate cumulus or cumulus fractus in its upwelling portion, also known as a "roll cloud". The rotor cloud looks like a line of cumulus. It forms on the lee side and parallel to the ridge line. Its base is near the height of the mountain peak, though the top can extend well above the peak and can merge with the lenticular clouds above. Rotor clouds have ragged leeward edges and are dangerously turbulent.[4]
A foehn wall cloud may exist at the lee side of the mountains, however this is not a reliable indication of the presence of lee waves.
A pileus or cap cloud, similar to a lenticular cloud, may form above the mountain or cumulus cloud generating the wave.
Adiabatic compression heating in the trough of each wave oscillation may also evaporate cumulus or stratus clouds in the airmass, creating a "wave window" or "Foehn gap".
Lee waves provide a possibility for gliders to gain altitude or fly long distances when soaring. World record wave flight performances for speed, distance or altitude have been made in the lee of the Sierra Nevada, Alps, Patagonic Andes, and Southern Alps mountain ranges.[13] The Perlan Project is working to demonstrate the viability of climbing above the tropopause in an unpowered glider using lee waves, making the transition into stratospheric standing waves. They did this for the first time on August 30, 2006 in Argentina, climbing to an altitude of 15,460 metres (50,720 ft).[14][15] The Mountain Wave Project of the Organisation Scientifique et Technique du Vol à Voile focusses on analysis and classification of lee waves and associated rotors.[16][17][18]
The conditions favoring strong lee waves suitable for soaring are:
A gradual increase in windspeed with altitude
Wind direction within 30° of perpendicular to the mountain ridgeline
Strong low-altitude winds in a stable atmosphere
Ridgetop winds of at least 20 knots
The rotor turbulence may be harmful for other small aircraft such as balloons, hang gliders and paragliders. It can even be a hazard for large aircraft; the phenomenon is believed responsible for many aviation accidents and incidents, including the in-flight breakup of BOAC Flight 911, a Boeing 707, near Mount Fuji, Japan in 1966, and the in-flight separation of an engine on an Evergreen International Airlines Boeing 747 cargo jet near Anchorage, Alaska in 1993.[19]
The rising air of the wave, which allows gliders to climb to great heights, can also result in high-altitude upset in jet aircraft trying to maintain level cruising flight in lee waves. Rising, descending or turbulent air, in or above the lee waves, can cause overspeed, stall or loss of control.
Other varieties of atmospheric waves[edit]
Hydrostatic wave (schematic drawing)
There are a variety of distinctive types of waves which form under different atmospheric conditions.
Wind shear can also create waves. This occurs when an atmospheric inversion separates two layers with a marked difference in wind direction. If the wind encounters distortions in the inversion layer caused by thermals coming up from below, it will create significant shear waves in the lee of the distortions that can be used for soaring.[20]
Hydraulic jump induced waves are a type of wave that forms when there exists a lower layer of air which is dense, yet thin relative to the size of the mountain. After flowing over the mountain, a type of shock wave forms at the trough of the flow, and a sharp vertical discontinuity called the hydraulic jump forms which can be several times higher than the mountain. The hydraulic jump is similar to a rotor in that it is very turbulent, yet it is not as spatially localized as a rotor. The hydraulic jump itself acts as an obstruction for the stable layer of air moving above it, thereby triggering wave. Hydraulic jumps can distinguished by their towering roll clouds, and have been observed on the Sierra Nevada range[21] as well as mountain ranges in southern California.
Hydrostatic waves are vertically propagating waves which form over spatially large obstructions. In hydrostatic equilibrium, the pressure of a fluid can depend only on altitude, not on horizontal displacement. Hydrostatic waves get their name from the fact that they approximately obey the laws of hydrostatics, i.e. pressure amplitudes vary primarily in the vertical direction instead of the horizontal. Whereas conventional, non-hydrostatic waves are characterized by horizontal undulations of lift and sink, largely independent of altitude, hydrostatic waves are characterized by undulations of lift and sink at different altitudes over the same ground position.
Kelvin–Helmholtz instability can occur when velocity shear is present within a continuous fluid or when there is sufficient velocity difference across the interface between two fluids.
Rossby waves (or planetary waves) are large-scale motions in the atmosphere whose restoring force is the variation in Coriolis effect with latitude.
^ On March 10, 1933, German glider pilot Hans Deutschmann (1911–1942) was flying over the Riesen mountains in Silesia when an updraft lifted his plane by a kilometer. The event was observed, and correctly interpreted, by German engineer and glider pilot Wolf Hirth (1900–1959), who wrote about it in: Wolf Hirth, Die hohe Schule des Segelfluges [The advanced school of glider flight] (Berlin, Germany: Klasing & Co., 1933). The phenomenon was subsequently studied by German glider pilot and atmospheric physicist Joachim P. Küttner (1909 -2011) in: Küttner, J. (1938) "Moazagotl und Föhnwelle" (Lenticular clouds and foehn waves), Beiträge zur Physik der Atmosphäre, 25, 79–114, and Kuettner, J. (1959) "The rotor flow in the lee of mountains." GRD [Geophysics Research Directorate] Research Notes No. 6, AFCRC[Air Force Cambridge Research Center]-TN-58-626, ASTIA [Armed Services Technical Information Agency] Document No. AD-208862.
^ Tokgozlu, A; Rasulov, M.; Aslan, Z. (January 2005). "Modeling and Classification of Mountain Waves". Technical Soaring. Vol. 29, no. 1. p. 22. ISSN 0744-8996.
^ a b c d Pagen, Dennis (1992). Understanding the Sky. City: Sport Aviation Pubns. pp. 169–175. ISBN 978-0-936310-10-7. This is the ideal case, for an unstable layer below and above the stable layer create what can be described as a springboard for the stable layer to bounce on once the mountain begins the oscillation.
^ David M. Gaffin, Stephen S. Parker, and Paul D. Kirkwood (2003). "An Unexpectedly Heavy and Complex Snowfall Event across the Southern Appalachian Region". Weather and Forecasting. 18 (2): 224–235. Bibcode:2003WtFor..18..224G. doi:10.1175/1520-0434(2003)018<0224:AUHACS>2.0.CO;2. {{cite journal}}: CS1 maint: uses authors parameter (link)
^ M. N. Raphael (2003). "The Santa Ana winds of California". Earth Interactions. 7 (8). doi:10.1175/1087-3562(2003)007<0001:TSAWOC>2.0.CO;2.
^ Warren Blier (1998). "The Sundowner Winds of Santa Barbara, California". Weather and Forecasting. 13 (3): 702–716. Bibcode:1998WtFor..24...53G. doi:10.1175/1520-0434(1998)013<0702:TSWOSB>2.0.CO;2.
^ D. K. Lilly (1978). "A Severe Downslope Windstorm and Aircraft Turbulence Event Induced by a Mountain Wave". Journal of the Atmospheric Sciences. 35 (1): 59–77. doi:10.1175/1520-0469(1978)035<0059:ASDWAA>2.0.CO;2.
^ Ryan Shadbolt; Joseph Charney; Hannah Fromm (2019). "A mesoscale simulation of a mountain wave wind event associated with the Chimney Tops 2 fire (2016)" (Special Symposium on Mesoscale Meteorological Extremes: Understanding, Prediction, and Projection). American Meteorological Society: 5 pp. {{cite journal}}: Cite journal requires |journal= (help)
^ Gill, Adrian E. (1982). Atmosphere-ocean dynamics (1 ed.). San Diego, CA: Academic Press. ISBN 9780122835223.
^ Durran, Dale R. (1990-01-01). "Mountain Waves and Downslope Winds". In Blumen, William (ed.). Atmospheric Processes over Complex Terrain. Meteorological Monographs. American Meteorological Society. pp. 59–81. doi:10.1007/978-1-935704-25-6_4. ISBN 9781935704256.
^ FAI gliding records Archived 2006-12-05 at the Wayback Machine
^ Perlan Project
^ OSTIV-Mountain Wave Project
^ [1] Archived 2016-03-03 at the Wayback Machine – accessed 2009-11-03
^ Lindemann, C; Heise, R.; Herold, W-D. (July 2008). "Leewaves in the Andes Region, Mountain Wave Project (MWP) of OSTIV". Technical Soaring. Vol. 32, no. 3. p. 93. ISSN 0744-8996.
^ NTSB Accident Report AAR-93-06
^ Eckey, Bernard (2007). Advanced Soaring Made Easy. Eqip Verbung & Verlag GmbH. ISBN 978-3-9808838-2-5.
^ Observations of Mountain-Induced Rotors and Related Hypotheses: a Review by Joachim Kuettner and Rolf F. Hertenstein
Grimshaw, R., (2002). Environmental Stratified Flows. Boston: Kluwer Academic Publishers.
Jacobson, M., (1999). Fundamentals of Atmospheric Modeling. Cambridge, UK: Cambridge University Press.
Nappo, C., (2002). An Introduction to Atmospheric Gravity Waves. Boston: Academic Press.
Pielke, R., (2002). Mesoscale Meteorological Modeling. Boston: Academic Press.
Turner, B., (1979). Buoyancy Effects in Fluids. Cambridge, UK: Cambridge University Press.
Whiteman, C., (2000). Mountain Meteorology. Oxford, UK: Oxford University Press.
Wikimedia Commons has media related to Orographic waves.
Mountain Wave Project official website
Chronological collection of meteorological data, satellite pics and cloud images of mountain waves in Bariloche, Argentina (in Spanish)
An Examination of the Areal Extent of High Winds due to Mountain Waves along the Western Foothills of the Southern Appalachian Mountains
Retrieved from "https://en.wikipedia.org/w/index.php?title=Lee_wave&oldid=1086305111"
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2016 Quasi-Hyperbolicity and Delay Semigroups
Shard Rastogi, Sachi Srivastava
We study quasi-hyperbolicity of the delay semigroup associated with the equation
{u}^{\mathrm{\prime }}\left(t\right)=Bu\left(t\right)+\mathrm{\Phi }{u}_{t}
{u}_{t}
is the history function and
\left(B,D\left(B\right)\right)
is the generator of a quasi-hyperbolic semigroup. We give conditions under which the associated solution semigroup of this equation generates a quasi-hyperbolic semigroup.
Shard Rastogi. Sachi Srivastava. "Quasi-Hyperbolicity and Delay Semigroups." Abstr. Appl. Anal. 2016 1 - 6, 2016. https://doi.org/10.1155/2016/1984874
Received: 1 June 2016; Accepted: 28 September 2016; Published: 2016
Shard Rastogi, Sachi Srivastava "Quasi-Hyperbolicity and Delay Semigroups," Abstract and Applied Analysis, Abstr. Appl. Anal. 2016(none), 1-6, (2016)
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Common Fixed-Point Theorems in Complete Generalized Metric Spaces
2012 Common Fixed-Point Theorems in Complete Generalized Metric Spaces
We introduce the notions of the
𝒲
function and
𝒮
function, and then we prove two common fixed point theorems in complete generalized metric spaces under contractive conditions with these two functions. Our results generalize or improve many recent common fixed point results in the literature.
Chi-Ming Chen. "Common Fixed-Point Theorems in Complete Generalized Metric Spaces." J. Appl. Math. 2012 1 - 14, 2012. https://doi.org/10.1155/2012/945915
Chi-Ming Chen "Common Fixed-Point Theorems in Complete Generalized Metric Spaces," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-14, (2012)
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group(deprecated)/SnConjugates - Maple Help
Home : Support : Online Help : group(deprecated)/SnConjugates
find the number of group elements with a given cycle type
SnConjugates(pg, perm)
SnConjugates(pg, part)
permutation in disjoint cycle notation
partition of the degree of pg
The cycle type of a permutation refers to its structure. It can be specified by either a sample permutation with the required cycle type or by a partition of the degree. For example, the permutation
[[1,2],[3,4],[5,6,7]]
[2,2,3]
refer to the same cycle type.
The elements with the same cycle type are conjugates under the action of Sn, where
n
is the degree of pg and Sn the symmetric group on
{1,...,n}
If perm is used, the function returns the number of elements of pg that have the same cycle type as perm. Only the structure of perm is considered.
If part is used, the function returns the number of elements of pg that have the cycle type described by part.
The command with(group,SnConjugates) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{group}\right):
\mathrm{pg}≔\mathrm{permgroup}\left(4,{[[1,4]],[[1,2],[3,4]]}\right):
\mathrm{SnConjugates}\left(\mathrm{pg},[[1,2],[3,4]]\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{SnConjugates}\left(\mathrm{pg},[2,2]\right)
\textcolor[rgb]{0,0,1}{3}
\mathrm{SnConjugates}\left(\mathrm{pg},[[1,2,3]]\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{SnConjugates}\left(\mathrm{pg},[3]\right)
\textcolor[rgb]{0,0,1}{0}
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Iōng-chiá:Kiatgak – Wikipedia
Sum of cubes in Archimedes waySiu-kái
First part, to decompose each cubic into a sum of squares, then compose them from different direction.
{\displaystyle S_{n}=\sum _{k=1}^{n}k^{2}={\frac {1}{3}}n^{3}+{\frac {1}{2}}n^{2}+{\frac {1}{6}}n}
{\displaystyle n^{3}=3S_{n}-{\frac {3}{2}}n^{2}-{\frac {1}{2}}n}
apply it to
{\displaystyle n^{3}...1^{3}}
{\displaystyle n^{3}=3[1^{2}+2^{2}+...+(n-1)^{2}+n^{2}]-{\frac {3}{2}}n^{2}-{\frac {1}{2}}n}
{\displaystyle (n-1)^{3}=3[1^{2}+2^{2}+...+(n-1)^{2}]-{\frac {3}{2}}(n-1)^{2}-{\frac {1}{2}}(n-1)}
{\displaystyle 2^{3}=3[1^{2}+2^{2}]-{\frac {3}{2}}2^{2}-{\frac {1}{2}}2}
{\displaystyle 1^{3}=3[1^{2}]-{\frac {3}{2}}1^{2}-{\frac {1}{2}}1}
To sum these n identities, let
{\displaystyle R_{n}=\sum _{k=1}^{n}k^{3}=3[1^{2}*n+2^{2}*(n-1)+...+(n-1)^{2}*2+n^{2}*1]-{\frac {3}{2}}S_{n}-{\frac {1}{2}}T_{n}}
{\displaystyle R_{n}=3X_{n}-{\frac {3}{2}}S_{n}-{\frac {1}{2}}T_{n}}
{\displaystyle T_{n}=\sum _{k=1}^{n}k=1+2+...+n={\frac {1}{2}}n(n+1)}
{\displaystyle X_{n}=3[1^{2}*n+2^{2}*(n-1)+...+(n-1)^{2}*2+n^{2}*1]}
Second part, make use of
{\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}}
; apply it to
{\displaystyle (n+1)^{3}}
{\displaystyle (n+1)^{3}=[1+n]^{3}=1^{3}+3*1^{2}*n+3*1*n^{2}+n^{3}}
{\displaystyle (n+1)^{3}=[2+(n-1)]^{3}=2^{3}+3*2^{2}*(n-1)+3*2*(n-1)^{2}+(n-1)^{3}}
{\displaystyle (n+1)^{3}=[n+1]^{3}=n^{3}+3*n^{2}*1+3*n*1^{2}+1^{3}}
To sum these n identities,
{\displaystyle n(n+1)^{3}=R_{n}+3\left[1^{2}*n+2^{2}*(n-1)+...+(n-1)^{2}*2+n^{2}*1\right]+3\left[1^{2}*n+2^{2}*(n-1)+...+(n-1)^{2}*2+n^{2}*1\right]+R_{n}}
{\displaystyle n(n+1)^{3}=2R_{n}+2*3X_{n}}
OK, easy part now, to merge the results of first part and second part, we can get
{\displaystyle \sum _{k=1}^{n}k^{3}={\frac {1}{4}}n^{2}(n+1)^{2}}
BTW, Archimedes would use
{\displaystyle (n-1)^{3}}
at first part, and use
{\displaystyle n^{3}}
at second part.
Archimedes: What Did He Do Beside Cry Eureka
Lâi-goân: "https://zh-min-nan.wikipedia.org/w/index.php?title=Iōng-chiá:Kiatgak&oldid=91053"
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Publications UHECR :: Лаборатория космических лучей предельно высоких энергий НИИЯФ МГУ
The orbital detector TUS (Tracking Ultraviolet Setup) with high sensitivity in near-visible ultraviolet (tens of photons per time sample of 0.8 μs of wavelengths 300-400 nm from a detector’s pixel field of view) and the microsecond-scale temporal resolution was developed by the Lomonosov-UHECR/TLE collaboration and launched into orbit on 28 April 2016. A variety of different phenomena were studied by measuring ultraviolet signals from the atmosphere: extensive air showers from ultra-high-energy cosmic rays, lightning discharges, transient atmospheric events, aurora ovals, and meteors. These events are different in their origin and in their duration and luminosity. The TUS detector had a capability to conduct measurements with different temporal resolutions (0.8 μs, 25.6 μs, 0.4 ms, and 6.6 ms) but the same spatial resolution of 5 km. Results of the TUS detector measurements of various atmospheric emissions are discussed and compared to data from previous experiments.
24.11.2019 The TUS collaboration Remote Sensing, 11(20), 2019
An orbital detector of ultra-high energy cosmic rays has been developed by the Skobel’tsyn Institute of Nuclear Physics of the Moscow State University together with the international JEM-EUSO collaboration for mounting on board the International Space Station. Its multichannel photodetector is composed of an array of multianode photomultiplier tubes (MAPMTs) combined into modules with 36 MAPMTs in each and with approximately 105 pixels in total. Since the number of channels is great and the speed of measurements is high, high requirements are set for the system of detection, selection, and analysis of events. The designs of the modular photodetector composition and the network architecture of the data processing system that is capable of performing efficient selection of events with different space−time structures are presented. The network principle is implemented via three types of communications: high-speed links between adjacent photodetector modules, long-distance communications for recording information to the permanent memory, and synchronizing links for timing the operation of individual modules. This digital-processing system of the detector can be designed using the ZYNQ system-on-chip concept that includes a field programmable gate array and a processor system.
19.02.2018 Belov A.A., Klimov P.A., Sharakin S.A. Instruments and Experimental Techniques, 61(1):27–33, 2018.
Optical system for orbital detector of extreme-high-energy cosmic ray
An optical system of a Schmidt-type telescope for orbital detection is proposed. The system contains a spherical mirror and correction plate with one aspherical surface and has the following characteristics: field of view (FoV) is 40 deg, entrance pupil diameter is 2.5 m, diameter of spherical mirror is 4 m, and f -number is 0.74. The system with the described parameters has image spot size of 3.2-mm (RMS) diameter for the axial beam and 4 mm (RMS) on the edge of the FoV, which is less than the diagonal of the detectors square pixel of 3×3 mm2 3×3 mm2.
19.02.2018 Vladislav V. Druzhin; Daniil T. Puryaev; Sergey A. Sharakin J. of Astronomical Telescopes, Instruments, and Systems, 4(1), 014002 (2018).
29.12.2017 Khrenov B.A., Klimov P.A., Panasyuk M.I., et al. Journal of Cosmology and Astroparticle Physics
The TUS Detector of Extreme Energy Cosmic Rays on Board the Lomonosov Satellite
The origin and nature of extreme energy cosmic rays (EECRs), which have energies above the —the Greisen-Zatsepin-Kuzmin (GZK) energy limit, is one of the most interesting and complicated problems in modern cosmic-ray physics. Existing ground-based detectors have helped to obtain remarkable results in studying cosmic rays before and after the GZK limit, but have also produced some contradictions in our understanding of cosmic ray mass composition. Moreover, each of these detectors covers only a part of the celestial sphere, which poses problems for studying the arrival directions of EECRs and identifying their sources. As a new generation of EECR space detectors, TUS (Tracking Ultraviolet Set-up), KLYPVE and JEM-EUSO, are intended to study the most energetic cosmic-ray particles, providing larger, uniform exposures of the entire celestial sphere. The TUS detector, launched on board the Lomonosov satellite on April 28, 2016 from Vostochny Cosmodrome in Russia, is the first of these. It employs a single-mirror optical system and a photomultiplier tube matrix as a photo-detector and will test the fluorescent method of measuring EECRs from space. Utilizing the Earth’s atmosphere as a huge calorimeter, it is expected to detect EECRs with energies above
{}^{}
It will also be able to register slower atmospheric transient events: atmospheric fluorescence in electrical discharges of various types including precipitating electrons escaping the magnetosphere and from the radiation of meteors passing through the atmosphere. We describe the design of the TUS detector and present results of different ground-based tests and simulations.
29.12.2017 Klimov P.A., Panasyuk M.I., Khrenov B.A., et al. Space Science Reviews
TUS is the world's first orbital detector of extreme energy cosmic rays (EECRs), which operates as a part of the scientific payload of the Lomonosov satellite since May 19, 2016. TUS employs the nocturnal atmosphere of the Earth to register ultraviolet (UV) fluorescence and Cherenkov radiation from extensive air showers generated by EECRs as well as UV radiation from lightning strikes and transient luminous events, micro-meteors and space debris. The first months of its operation in orbit have demonstrated an unexpectedly rich variety of UV radiation in the atmosphere. We briefly review the design of TUS and present a few examples of events recorded in a mode dedicated to registering EECRs.
04.04.2017 Mikhail Zotov, for the Lomonosov-UHECR/TLE Collaboration Arxiv.org (Submitted on 28 Mar 2017 (v1), last revised 2 Apr 2017 (this version, v2))
04.04.2017 S.V. Biktemerova, A.V. Bogomolov, V.V. Bogomolov, A.A. Botvinko, A.J. Castro-Tirado, E.S. Gorbovskoy, N.P. Chirskaya, V.E. Eremeev, G.K. Garipov, V.M. Grebenyuk, A.A. Grinyuk, A.F. Iyudin, S. Jeong, H.M. Jeong, N.L. Jioeva, P.S. Kazarjan, N.N. Kalmykov, M.A. Kaznacheeva, B.A. Khrenov, M.B. Kim, P.A. Klimov, E.A. Kuznetsova, M.V. Lavrova, J. Lee, V.M. Lipunov, O. Martinez, I.N. Mjagkova, M.I. Panasyuk, I.H. Park, V.L. Petrov, E. Ponce, A.E. Puchkov, H. Salazar, O.A. Saprykin, A.N. Senkovsky, S.A. Sharakin, A.V. Shirokov, S.I. Svertilov, A.V. Tkachenko, L.G. Tkachev, I.V. Yashin, M.Yu. Zotov Arxiv.org (Submitted on 10 Mar 2017 (v1), last revised 26 Mar 2017 (this version, v2))
Preliminary results from the TUS ultra-high energy cosmic ray orbital telescope: Registration of low-energy particles passing through the photodetector
The TUS telescope, part of the scientific equipment on board the Lomonosov satellite, is the world’s first orbital detector of ultra-high energy cosmic rays. Preliminary results from analyzing unexpected powerful signals that have been detected from the first days of the telescope’s operation are presented. These signals appear simultaneously in time intervals of around 1 μs in groups of adjacent pixels of the photodetector and form linear track-like sequences. The results from computer simulations using the GEANT4 software and the observed strong latitudinal dependence of the distribution of the events favor the hypothesis that the observed signals result from protons with energies of several hundred MeV to several GeV passing through the photodetector of the TUS telescope.
04.04.2017 P. A. Klimov, M. Yu Zotov, N. P. Chirskaya, B. A. Khrenov, G. K. Garipov, M. I. Panasyuk, S. A. Sharakin, A. V. Shirokov, I. V. Yashin, A. A. Grinyuk, A. V. Tkachenko, L. G. Tkachev
The orbital TUS detector simulation
The TUS space experiment is aimed at studying energy and arrival distribution of UHECR at E>7 × 1019 eV by using the data of EAS fluorescent radiation in atmosphere. The TUS mission was launched at the end of April 2016 on board the dedicated “Lomonosov” satellite. The TUSSIM software package has been developed to simulate performance of the TUS detector for the Fresnel mirror optical parameters, the light concentrator of the photo detector, the front end and trigger electronics. Trigger efficiency crucially depends on the background level which varies in a wide range: from 0.2 × 106 to 15 × 106 ph/( m2 μs sr) at moonless and full moon nights respectively. The TUSSIM algorithms are described and the expected TUS statistics is presented for 5 years of data collection from the 500 km solar-synchronized orbit with allowance for the variability of the background light intensity during the space flight.
04.04.2017 A. Grinyuk, V. Grebenyuk, B. Khrenov, P. Klimov, M. Lavrova, M. Panasyuk, S. Sharakin, A. Shirokov, A. Tkachenko, L. Tkachev, I. Yashin Astroparticle Physics, Volume 90, April 2017, Pages 93-97
Detection prospects of the Telescope Array hotspot by space observatories
In the present-day cosmic ray data, the strongest indication of anisotropy of the ultrahigh energy cosmic rays is the 20-degree hotspot observed by the Telescope Array with the statistical significance of 3.4σ. In this work, we study the possibility of detecting such a spot by space-based all-sky observatories. We show that if the detected luminosity of the hotspot is attributed to a physical effect and not a statistical fluctuation, the KLYPVE and JEM-EUSO experiments would need to collect ∼300events with E>57 EeV in order to detect the hotspot at the 5σ confidence level with the 68% probability. We also study the dependence of the detection prospects on the hotspot luminosity.
26.07.2016 D. Semikoz, P. Tinyakov, and M. Zotov Phys. Rev. D 93, 103005 – Published 23 May 2016
Space-based detectors for the study of extreme energy cosmic rays (EECR) are being prepared as a promising new method for detecting highest energy cosmic rays. A pioneering space device – the “tracking ultraviolet set-up” (TUS) – is in the last stage of its construction and testing. The TUS detector will collect preliminary data on EECR in the conditions of a space environment, which will be extremely useful for planning the major JEM-EUSO detector operation.
26.11.2015 The JEM-EUSO Collaboration Experimental Astronomy, November 2015, Volume 40, Issue 1, pp 315-326
Modified KLYPVE is a novel fluorescence detector of ultra high energy cosmic rays (UHECRs, energies ≥50EeV) to be installed on the Russian Segment of the International Space Station. The main goal of the experiment is to register arrival directions and energies of UHECRs but it will be able to register other transient events in the atmosphere as well. The main component of KLYPVE is a segmented two component optical system with a large entrance pupil and a wide field of view, which provides annual exposure approximately twice that of the Pierre Auger Observatory. The project is actively developed by a working group of the JEM-EUSO Collaboration led by Skobeltsyn Institute of Nuclear Physics at Moscow State University (Russia). The current status ofKLYPVE with a focus on its scientific tasks, technical parameters and instruments is presented.
11.10.2015 M. I. Panasyuk, P. Picozza, M. Casolino, T. Ebisuzaki, P. Gorodetzky, B. A. Khrenov, P. A. Klimov, S. A. Sharakin and M. Yu. Zotov Proc. ICRC-2015 PoS(ICRC2015)669
The KLYPVE ultrahigh energy cosmic ray detector on board the ISS.
The current status of the KLYPVE orbital detector of ultrahigh energy cosmic rays, which is scheduled to be deployed on board the Russian module of the International Space Station, is discussed. The main focus is on describing possible optical systems for the instrument.
11.10.2015 G.K. Garipov, M.Yu Zotov, P.A. Klimov, M.I. Panasyuk, O.A. Saprykin, L.G. Tkachev, S.A. Sharakin, B.A. Khrenov, and I.V. Yashin Bulletin of the Russian Academy of Science, Physics, 79(3):326–328, 2015.
The current status of orbital experiments for UHECR studies
Two types of orbital detectors of extreme energy cosmic rays are being developed nowadays: (i) TUS and KLYPVE with reflecting optical systems (mirrors) and (ii) JEM-EUSO with high- transmittance Fresnel lenses. They will cover much larger areas than existing ground-based arrays and almost uniformly monitor the celestial sphere. The TUS detector is the pioneering mission developed in SINP MSU in cooperation with several Russian and foreign institutions. It has relatively small field of view (±4.5°), which corresponds to a ground area of 6.4•103 km2 . The telescope consists of a Fresnel-type mirror-concentrator (∼ 2 m2 ) and a photo receiver (a matrix of 16 x 16 photomultiplier tubes). It is to be deployed on the Lomonosov satellite, and is currently at the final stage of preflight tests. Recently, SINP MSU began the KLYPVE project to be installed on board of the Russian segment of the ISS. The optical system of this detector contains a larger primary mirror (10 m 2 ), which allows decreasing the energy threshold. The total effective field of view will be at least ±14° to exceed the annual exposure of the existing ground-based experiments. Several configurations of the detector are being currently considered. Finally, JEM-EUSO is a wide field of view (±30°) detector. The optics is composed of two curved double-sided Fresnel lenses with 2.65 m external diameter, a precision diffractive middle lens and a pupil. The ultraviolet photons are focused onto the focal surface, which consists of nearly 5000 multi-anode photomultipliers. It is developed by a large international collaboration. All three orbital detectors have multi-purpose character due to continuous monitoring of various atmospheric phenomena. The present status of development of the TUS and KLYPVE missions is reported, and a brief comparison of the projects with JEM-EUSO is given. [http://iopscience.iop.org/article]
11.10.2015 M.I. Panasyuk, M. Casolino, G.K. Garipov, T. Ebisuzaki, P. Gorodetzky, B.A. Khrenov, P.A. Klimov, V.S. Morozenko, N. Sakaki, O.A. Saprykin, S.A. Sharakin, Y. Takizawa, L.G. Tkachev, I.V. Yashin, and M.Yu Zotov Journal of Physics, 632(1):012097, 2015.
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Quality metrics of signal or image approximation - MATLAB measerr - MathWorks 日本
Measure Approximation Quality in RGB Image
Measure Approximation Quality in Grayscale Image
L2RAT
Quality metrics of signal or image approximation
[PSNR,MSE,MAXERR,L2RAT] = measerr(X,XAPP)
[PSNR,MSE,MAXERR,L2RAT] = measerr(X,XAPP,BPS)
[PSNR,MSE,MAXERR,L2RAT] = measerr(X,XAPP) returns the peak signal-to-noise ratio, PSNR, mean square error, MSE, maximum squared error, MAXERR, and ratio of squared norms, L2RAT, for an input signal or image, X, and its approximation, XAPP.
[PSNR,MSE,MAXERR,L2RAT] = measerr(X,XAPP,BPS) uses the bits per sample, BPS, to determine the peak signal-to-noise ratio.
Approximate an RGB image and compute the quality metrics.
Load an RGB image. Return the image dimensions and minimum and maximum values.
X = imread('africasculpt.jpg');
[min(X(:)) max(X(:))]
Define the image approximation by setting equal to 1 all RGB values less than or equal to 100.
Xapp = X;
Xapp(X<=100) = 1;
Display the image and its approximation.
image(Xapp)
Compute the quality metrics of the image approximation.
mse = 1.1487e+03
maxerr = 99
L2rat = 0.9398
Approximate a grayscale image and calculate approximation quality metrics.
Create a 256-by-256 grayscale image with intensities between
0
{2}^{16}-1
val = 0:2^16-1;
X = reshape(val,256,256);
There are 16 bits per sample. Define the image approximation by setting equal to 1 all grayscale values less than or equal to 1000. Display the image and its approximation.
Xapp(X<=1000) = 1;
colormap(gray(2^16))
There are 16 bits per sample. Compute the quality metrics of the grayscale approximation.
maxerr = 999
X — Input signal or image
Input signal or image, specified as a real-valued array.
XAPP — Approximation of signal or image
Approximation of signal or image X, specified as a real-valued array. XAPP is the same size as X.
BPS — Bits per sample
Bits per sample of the input data, specified as a positive integer. The default value is 8, so the maximum possible pixel value of an image (MAXI) is 255. More generally, when samples are represented using linear Pulse Code Modulation with B bits per sample, MAXI is 2B−1.
PSNR — Peak signal-to-noise ratio
Peak signal-to-noise ratio (PSNR) in decibels, returned as a positive real number. The PSNR is only meaningful for data encoded in terms of bits per sample or bits per pixel. For example, an image with 8 bits per pixel contains integers from 0 to 255.
Mean square error, returned as a positive real number. MSE is the squared norm of the difference between X and XAPP divided by the number of elements.
MAXERR — Maximum absolute squared deviation
Maximum absolute squared deviation of the data X from the approximation XAPP, returned as a positive real number.
L2RAT — Energy ratio
Energy ratio between the approximation XAPP and input data X, returned as a positive real number. L2RAT is the ratio of the squared norm of XAPP to X.
The peak signal-to-noise ratio (PSNR) in decibels between a signal and its approximation is
20{\mathrm{log}}_{10}\left(\frac{{2}^{B}â1}{\sqrt{MSE}}\right)
where MSE represents the mean square error, and B represents the bits per sample.
The mean square error (MSE) between a signal or image, X, and an approximation, Y, is
\frac{||XâY|{|}^{2}}{N}
where N is the number of elements in the signal.
[1] Huynh-Thu, Q. and M. Ghanbari. "Scope of Validity of PSNR in Image/Video Quality Assessment." Electronics Letters. Vol. 44, Issue 13, 2008, pp. 800–801.
wdenoise | wden | wdencmp
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Vector and matrix norms - MATLAB norm - MathWorks Korea
1-Norm of Vector
Euclidean Distance Between Two Points
Frobenius Norm of N-D Array
Maximum Absolute Column Sum
Maximum Absolute Row Sum
Frobenius norm supports N-D arrays
n = norm(X,"fro")
n = norm(v) returns the Euclidean norm of vector v. This norm is also called the 2-norm, vector magnitude, or Euclidean length.
n = norm(v,p) returns the generalized vector p-norm.
n = norm(X) returns the 2-norm or maximum singular value of matrix X, which is approximately max(svd(X)).
n = norm(X,p) returns the p-norm of matrix X, where p is 1, 2, or Inf:
If p = 1, then n is the maximum absolute column sum of the matrix.
If p = 2, then n is approximately max(svd(X)). This value is equivalent to norm(X).
If p = Inf, then n is the maximum absolute row sum of the matrix.
n = norm(X,"fro") returns the Frobenius norm of matrix or array X.
Create a vector and calculate the magnitude.
Calculate the 1-norm of a vector, which is the sum of the element magnitudes.
v = [-2 3 -1];
n = norm(v,1)
Calculate the distance between two points as the norm of the difference between the vector elements.
Create two vectors representing the (x,y) coordinates for two points on the Euclidean plane.
Use norm to calculate the distance between the points.
Geometrically, the distance between the points is equal to the magnitude of the vector that extends from one point to the other.
\begin{array}{l}a=0\underset{}{\overset{ˆ}{i}}+3\underset{}{\overset{ˆ}{j}}\\ b=-2\underset{}{\overset{ˆ}{i}}+1\underset{}{\overset{ˆ}{j}}\\ \\ \begin{array}{rl}{d}_{\left(a,b\right)}& =||b-a||\\ & =\sqrt{\left(-2-0{\right)}^{2}+\left(1-3{\right)}^{2}}\\ & =\sqrt{8}\end{array}\end{array}
Calculate the 2-norm of a matrix, which is the largest singular value.
Calculate the Frobenius norm of a 4-D array X, which is equivalent to the 2-norm of the column vector X(:).
The Frobenius norm is also useful for sparse matrices because norm(X,2) does not support sparse X.
v — Input vector
Input array, specified as a matrix or array. For most norm types, X must be a matrix. However, for Frobenius norm calculations, X can be an array.
2 (default) | positive real scalar | Inf | -Inf
Norm type, specified as 2 (default), a positive real scalar, Inf, or -Inf. The valid values of p and what they return depend on whether the first input to norm is a matrix or vector, as shown in the table.
This table does not reflect the actual algorithms used in calculations.
1 max(sum(abs(X))) sum(abs(v))
2 max(svd(X)) sum(abs(v).^2)^(1/2)
Positive, real-valued numeric scalar — sum(abs(v).^p)^(1/p)
Inf max(sum(abs(X'))) max(abs(v))
-Inf — min(abs(v))
n — Norm value
Norm value, returned as a scalar. The norm gives a measure of the magnitude of the elements. By convention, norm returns NaN if the input contains NaN values.
‖v‖=\sqrt{\sum _{k=1}^{N}{|{v}_{k}|}^{2}}\text{\hspace{0.17em}}.
{‖v‖}_{p}={\left[\sum _{k=1}^{N}{|{v}_{k}|}^{p}\right]}^{\text{\hspace{0.17em}}1/p}\text{\hspace{0.17em}},
where p is any positive real value, Inf, or -Inf.
{‖v‖}_{\infty }={\mathrm{max}}_{i}\left(|v\left(i\right)|\right)
If p = -Inf, then
{‖v‖}_{-\infty }={\mathrm{min}}_{i}\left(|v\left(i\right)|\right)
The maximum absolute column sum of an m-by-n matrix X (with m,n >= 2) is defined by
{‖X‖}_{1}=\underset{1\le j\le n}{\mathrm{max}}\left(\sum _{i=1}^{m}|{a}_{ij}|\right).
The maximum absolute row sum of an m-by-n matrix X (with m,n >= 2) is defined by
{‖X‖}_{\infty }=\underset{1\le i\le m}{\mathrm{max}}\left(\sum _{j=1}^{n}|{a}_{ij}|\right)\text{\hspace{0.17em}}.
The Frobenius norm of an m-by-n matrix X (with m,n >= 2) is defined by
{‖X‖}_{F}=\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}{|{a}_{ij}|}^{2}}=\sqrt{\text{trace}\left({X}^{†}X\right)}\text{\hspace{0.17em}}.
This definition also extends naturally to arrays with more than two dimensions. For example, if X is an N-D array of size m-by-n-by-p-by-...-by-q, then the Frobenius norm is
{‖X‖}_{F}=\sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}\sum _{k=1}^{p}...\sum _{w=1}^{q}{|{a}_{ijk...w}|}^{2}}.
Use vecnorm to treat a matrix or array as a collection of vectors and calculate the norm along a specified dimension. For example, vecnorm can calculate the norm of each column in a matrix.
R2022a: Frobenius norm supports N-D arrays
Norm calculations of the form norm(X,"fro") support N-D array inputs.
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What is a Forward Price
Forward price is the predetermined delivery price for an underlying commodity, currency, or financial asset as decided by the buyer and the seller of the forward contract, to be paid at a predetermined date in the future. At the inception of a forward contract, the forward price makes the value of the contract zero, but changes in the price of the underlying will cause the forward to take on a positive or negative value.
The forward price is determined by the following formula:
\begin{aligned} &F_0 = S_0 \times e^{rT} \\ \end{aligned}
F0=S0×erT
Basics of Forward Price
Forward price is based on the current spot price of the underlying asset, plus any carrying costs such as interest, storage costs, foregone interest or other costs or opportunity costs.
Although the contract has no intrinsic value at the inception, over time, a contract may gain or lose value. Offsetting positions in a forward contract are equivalent to a zero-sum game. For example, if one investor takes a long position in a pork belly forward agreement and another investor takes the short position, any gains in the long position equals the losses that the second investor incurs from the short position. By initially setting the value of the contract to zero, both parties are on equal ground at the inception of the contract.
Forward Price Calculation Example
When the underlying asset in the forward contract does not pay any dividends, the forward price can be calculated using the following formula:
\begin{aligned} &F = S \times e ^ { (r \times t) } \\ &\textbf{where:} \\ &F = \text{the contract's forward price} \\ &S = \text{the underlying asset's current spot price} \\ &e = \text{the mathematical irrational constant approximated} \\ &\text{by 2.7183} \\ &r = \text{the risk-free rate that applies to the life of the} \\ &\text{forward contract} \\ &t = \text{the delivery date in years} \\ \end{aligned}
F=S×e(r×t)where:F=the contract’s forward priceS=the underlying asset’s current spot pricee=the mathematical irrational constant approximatedby 2.7183r=the risk-free rate that applies to the life of theforward contractt=the delivery date in years
For example, assume a security is currently trading at $100 per unit. An investor wants to enter into a forward contract that expires in one year. The current annual risk-free interest rate is 6%. Using the above formula, the forward price is calculated as:
\begin{aligned} &F = \$100 \times e ^ { (0.06 \times 1) } = \$106.18 \\ \end{aligned}
F=$100×e(0.06×1)=$106.18
If the there are carrying costs, that is added into the formula:
\begin{aligned} &F = S \times e ^ { (r + q) \times t } \\ \end{aligned}
F=S×e(r+q)×t
Here, q is the carrying costs.
If the underlying asset pays dividends over the life of the contract, the formula for the forward price is:
\begin{aligned} &F = ( S - D ) \times e ^ { ( r \times t ) } \\ \end{aligned}
F=(S−D)×e(r×t)
Here, D equals the sum of each dividend's present value, given as:
\begin{aligned} D =& \ \text{PV}(d(1)) + \text{PV}(d(2)) + \cdots + \text{PV}(d(x)) \\ =& \ d(1) \times e ^ {- ( r \times t(1) ) } + d(2) \times e ^ { - ( r \times t(2) ) } + \cdots + \\ \phantom{=}& \ d(x) \times e ^ { - ( r \times t(x) ) } \\ \end{aligned}
D=== PV(d(1))+PV(d(2))+⋯+PV(d(x)) d(1)×e−(r×t(1))+d(2)×e−(r×t(2))+⋯+ d(x)×e−(r×t(x))
Using the example above, assume that the security pays a 50-cent dividend every three months. First, the present value of each dividend is calculated as:
\begin{aligned} &\text{PV}(d(1)) = \$0.5 \times e ^ { - ( 0.06 \times \frac { 3 }{ 12 } ) } = \$0.493 \\ \end{aligned}
PV(d(1))=$0.5×e−(0.06×123)=$0.493
\begin{aligned} &\text{PV}(d(2)) = \$0.5 \times e ^ { - ( 0.06 \times \frac { 6 }{ 12 } ) } = \$0.485 \\ \end{aligned}
\begin{aligned} &\text{PV}(d(3)) = \$0.5 \times e ^ { - ( 0.06 \times \frac { 9 }{ 12 } ) } = \$0.478 \\ \end{aligned}
\begin{aligned} &\text{PV}(d(4)) = \$0.5 \times e ^ { - ( 0.06 \times \frac { 12 }{ 12 } ) } = \$0.471 \\ \end{aligned}
PV(d(4))=$0.5×e−(0.06×1212)=$0.471
The sum of these is $1.927. This amount is then plugged into the dividend-adjusted forward price formula:
\begin{aligned} &F = ( \$100 - \$1.927 ) \times e ^ { ( 0.06 \times 1 ) } = \$104.14 \\ \end{aligned}
F=($100−$1.927)×e(0.06×1)=$104.14
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Litepaper - APY.Finance
The APY.Finance platform is a yield farming robo-advisor that runs a portfolio of yield farming strategies from a single pool of liquidity.
Key features of APY.Finance include:
Capture growth across the DeFi industry with a single deposit.
Diversified portfolio of yield farming strategies to reduce smart contract risk and yield volatility.
Automatic portfolio rebalancing to optimize risk-adjusted yield.
Over 99% gas savings on rebalance fees compared to independent manual yield farming.
Over 80% gas savings on deposit and withdrawal fees compared to other yield farming aggregators.
APY Liquidity Pool
The APY liquidity pool is actually a collection of contracts that handle the deposit and withdrawal of a single currency. These contracts work together to form a single pool of liquidity. When a sender deposits into a contract, they are issued APT tokens to represent their share of the pool.
Each liquidity contract determines the notional value of its reserves using existing Chainlink aggregators. We use the ETH denominated aggregators because they are updated more frequently and have more sponsors.
You can view some of the Chainlink aggregators we use here.
APY Strategy Portfolio
The APY strategy portfolio is a library of strategy contracts that interact with financial primitives to earn yield.
Every strategy has an asset schema that describes the types of assets and which assets are used for each type.
Required to begin earning yield from the strategy.
Intermediary assets
Held by the contract while running the strategy.
cDAI, yCRV
Output assets
Generated by the strategy and swapped for yield.
COMP, CRV
Every strategy has three different sequences that are used over the course of its life-cycle.
Deploys input assets to the strategy.
Using DAI to mint cDAI.
Performs regular upkeep for the strategy.
Swapping COMP for DAI to mint cDAI.
Unwinds the strategy.
Retrieving DAI by redeeming cDAI.
These assets and sequences for each strategy are used by the APY manager to process deposits, withdrawals, new portfolio optimizations, and to earn yield.
Every strategy has a view function to calculate the yield estimate for a given amount of input assets at the current block state. This estimate is used when comparing strategies and determining optimal strategy allocations.
Every strategy has a risk score associated with it. This score is represented by a number between 0 and 10e18. The initial risk score is set when the strategy contract is first deployed, however it can be updated through governance proposals.
The team looks at a variety of risk factors to asses the risk score using the DeFi Score whitepaper as a template.
Risk scores are used to weight a strategy's estimated yield to get a risk-adjusted yield. It is this risk-adjusted yield that is primarily used when optimizing portfolio allocations.
You can read the DeFi Score whitepaper here.
Compound DAI^3
COMP farming with leveraged DAI using dYdX flash loan.
Boosted yCRV
CRV farming with 1 week locked veCRV with the yCRV pool.
DODO USDT-USDC
DODO farming with the USDT-USDC DODO pool.
DeFiDollar DUSD-USDC Balancer
DFD, BAL, and CRV farming with the DUSD-USDC Balancer pool.
APY Manager
The APY manager automates the movement of liquidity through the APY system.
The manager contract contains a rebalance function that sets in motion several important processes:
If there is any reserves sitting idle in the liquidity pool from new deposits, they will be deployed proportionally to the current strategy portfolio.
If the notional amount of withdrawal requests is less than the amount idle reserves in the liquidity pool, the reserves will be locked for withdrawal and the requests will pass.
If the notional amount of withdrawal requests exceeds the amount of idle reserves in the liquidity pool, portions of the strategy portfolio will be unwound until all the withdrawal requests can be processed. These reserves are then locked for withdrawal and the requests will pass.
If the notional value of assets in any strategy are no longer within a certain threshold of the currently selected allocation ratios, it will be rebalanced to match the desired ratios.
Executing each strategy's loop sequence if the threshold conditions for a loop are met.
The rebalance function can be called by any external address.
Because Ethereum does not natively support trigger-based automation, this rebalance function must be called regularly for the system to function. To prevent dependencies on a centralized authority for automation, third parties have an incentive to make rebalance calls.
Rebalance incentives are issued from a rebalance pool that reserves small portions of yield earned by the strategy portfolio.
The APY manager handles swaps between liquidity pool assets and the strategy portfolio input assets. When the proportions of available liquidity pool assets do not match the desired proportions of input assets, they are swapped to meet thresholds.
There are 3 portfolio strategies and the current portfolio optimization expects a ratio of 50/25/25.
The first and second strategies require DAI and the third strategy requires USDC.
The liquidity pool is 50/50 DAI and USDC.
Given these conditions, the APY manager will swap half of the USDC reserves for DAI before deploying to the strategy portfolio on the next rebalance.
The transaction fees and slippage of these swaps are considered when determining the transaction cost of a portfolio optimization.
Portfolio optimization for risk-adjusted yield that varies based on the amount of deployed liquidity is essentially a network flow problem. For portfolios with fewer strategies, the optimal portfolio allocation can be computed on-chain. However, as the number of strategies inevitably increases, the computation required to optimize the portfolio will begin to exceed the block gas limit.
To solve this issue, the APY platform allows off-chain computation of portfolio allocations that can be submitted to the APY manager for verification. The APY manager compares this new set of allocations with the old set using the on-chain yield estimates for each strategy. If the new set of allocations results in a greater risk-adjusted yield, and the transaction costs of rebalancing are lower than the gain in yield by a threshold, the new set of allocations can be automatically accepted.
Because it is unnecessary to verify that an allocation is perfectly optimal, only that it is more optimal, the APY platform can safely allow this off-chain computation without relying on a central authority to sign off on new allocations.
Generic Strategy Executor
The generic strategy executor is a type of strategy contract that will be used heavily with the second and third phase of governance.
The executor takes a series of arbitrary function calls to external contracts and encodes them into sequences of call data that are then used in place of a standard strategy's enter, loop, and exit sequences.
The executor can use a library of adapters to process the return values of external function calls for use as parameters in other function calls in the sequence.
To produce a single step of call data from a function call, the contract takes the 4 byte keccak256 encoded function selector and combines it with a set of parameters encoded using the contract ABI specification.
You can read the official Solidity documentation to learn more about encoding function calls here.
This architecture allows for the development of a drag-and-drop style UI that takes any sequence built by a user and uses the external contract ABIs to create the arrays of call data accepted by the executor contract. Designing strategies with executor contract will dramatically lower the barrier-of-entry for governing strategies by eliminating the need for specialized knowledge of smart contract engineering.
Because of the wide range of possibilities with generic call data execution, safeguards are in place to limit the attack surface. This includes whitelists of external contracts and whitelists of function selectors.
APY is our ERC-20 governance token. It will be used to vote on system-wide parameters, changes to strategies in the portfolio, and the inclusion of entirely new strategies. Our governance roadmap covers the different stages of decentralization the platform will undergo as it progresses towards full community ownership. The APY team will make critical decisions with feedback from the community to stay nimble in the early stages of development.
The APY liquidity mining program is an incentive for liquidity providers to deposit their stablecoin into the APY platform. Accounts earn APY tokens for every block they have a deposit in the APY platform. We have allocated 31.2% of the token supply to this purpose and the initial emission rate of rewards started at 30,000 per day.
You can read more about our token distribution here.
Every week the team runs a script to calculate the block-by-block rewards for each account that has deposited into the APY platform. The script uses the following formula to calculate account rewards per block from an amount of rewards issued per second:
r_{ib}=a(t_b - t_{b-1})\frac{v_{ib}}{\sum_{j=0}^n v_{jb}}
Once the script is finished a blob of balances is released in the APY GitHub blob repository.
You can view the weekly APY balance blobs here.
Vesting Claims Contract
Rewards from the APY liquidity mining program vest over a period of 6 months. Vesting rewards discourages those just looking to sell the tokens, leaving greater rewards available for those that see the long term utility of APY governance.
Instructions on how to use the vesting claim contract is available here.
The formula used to calculate the vested rewards for an account is shown below:
v_i=\sum_{j=0}^n ar_i(T-t_i)-c_i
The standard vesting contracts used by many projects have high gas costs because they perform all vesting calculations on-chain. Normally performing calculations on-chain allows for greater decentralization, but in the case of vesting contracts, the rewards must still be issued by a centralized admin key.
The APY vesting contract instead uses signature based claims. Vesting calculations are done off-chain and the amounts are signed by an admin key. The resulting signature can be used by an account to process their claim at a much lower gas cost than typical vesting contracts.
The signatures are generated according to the EIP-712 standard using the OpenZeppelin ECDSA implementation. The contract uses a unique domain separator and per-claim nonces to protect against replay attacks. An insufficient domain separator can result in replay attacks stemming from other networks or other contracts. The nonces prevent an account from receiving multiple signatures with increasing rewards unclaimed rewards and then claiming them all at once.
You can view the EIP-712 standard here.
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Hexadecimal representation of stored integer of fi object - MATLAB hex - MathWorks 日本
View Stored Integer of fi Object in Hexadecimal Format
Write Hex Data to a File
Read Hex Data From a File
Hexadecimal representation of stored integer of fi object
b = hex(a) returns the stored integer of fi object a in hexadecimal format as a character vector.
real\text{-}worldvalue={2}^{âfractionlength}Ãstoredinteger
real\text{-}worldvalue=\left(slopeÃstoredinteger\right)+bias
hex returns the hexadecimal representation of the stored integer of a fi object. To obtain the hexadecimal representation of the real-world value of a fi object, use dec2hex.
Find the hexadecimal representation of the stored integers of fi object a.
'80 7f'
This example shows how to write hexadecimal data from the MATLAB workspace into a text file.
Define your data and create a writable text file called hexdata.txt.
x = (0:15)'/16;
a = fi(x, 0, 16, 16);
h = fopen('hexdata.txt', 'w');
Use the fprintf function to write your data to the hexdata.txt file.
fprintf(h, '%s\n', hex(a(k)));
To see the contents of the file you created, use the type function.
type hexdata.txt
This example shows how to read hexadecimal data from a text file back into the MATLAB workspace.
Define your data, create a writable text file called hexdata.txt, and write your data to the hexdata.txt file.
Open hexdata.txt for reading and read its contents into a workspace variable
h = fopen('hexdata.txt', 'r');
nextline = '';
while ischar(nextline)
nextline = fgetl(h);
if ischar(nextline)
str = [str; nextline];
Create a fi object with the correct scaling and assign it the hex values stored in the str variable.
b = fi([], 0, 16, 16);
b.hex = str
bin | dec | storedInteger | oct | dec2hex | dec2base | dec2bin
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I have very little to say, but will amuse myself by scribbling a few lines to thank you for information in last note of Feb. 9th 2 & to thank you in my dear little man’s name for two precious stamps.3 He told me with joyful triumph that in American stamps he equalled all other collections put together in the school. He exchanged a duplicate Blood’s stamp for a whole lot of treasures.4 He says there is an envelope of same value as the stamps you put on your letter, which would be of value to him.—
I have one request to make for myself, viz seed of Campanula perfoliata: I have tried in vain at Kew & elsewhere for some.—5
I am very glad you like Bates’ paper;6 I expect his Amazonian Travels will be good.7 If you read Lyell’s book, tell me what you think of it;8 I (& Hooker) have told him that we regret much that he did not speak more boldly out about Species.9 He answers that his belief in change fluctuates.—10 His Book has made me reread your essay; & I admire it as much as ever.11 What a dead hand you are in parrying a lounge & transfixing your adversary! You ask about Sprengels “Dichogamy”:12 he means by this a plant in which each flower first matures & sheds its pollen & then has its stigma mature; & much more rarely matures its stigma first & subsequently its pollen: so that these plants are in function monoœcious. I am sure his observations are to large extent correct, & the case is very common.13 In the Primula-like cases the plants are in function Diœcious.—14
A couple of days ago I had an interesting letter from Dr. Cruger of Bot. Gardens of Trinidad,15 & he tells me odd facts of native species (& only native species) of Cattleya &c which never open their flowers, & yet set seed-capsules. Happy man he has actually seen crowds of Bees flying round Catasetum with the pollinia sticking to their backs! I wrote to him to ask him to observe what insects did in flowers of Melastomaceæ;16 he says not proper season yet, but that on one species a small Bee seemed busy about the horn-like appendages to the anthers. It will be too good luck if my study of the flowers in the green-house has led me to right interpretation of these queer appendages.17 By the way, I have just built a hot-house & got some orchids, & it amuses me much.—18 Some plants of Amsinckia spectabilis, at least the seed was so named (small dark orange flowers, elongated hairy leaves) have just begun to flower, & I find in two plants that stigma stands on exact level with anthers; hence I fear they cannot be dimorphic.—19
Your Mitchellas look healthy: I hope they will not flower very soon;20 for my health (& that of my youngest Boy) has been of late so bad, that we have resolved all to go about middle of April for 6 or 8 weeks to Malvern for Water-Cure for me & change for my Boy.—21 It breaks my heart: I shall never get my present Book on Variation under Domestication finished; yet it interests me much & I am now in middle of long chapter on Inheritance Reversion &c, giving results of my own & other Breeders’ Experiments.—22
Good Night.— | My dear Gray | Yours most truly | Ch. Darwin
Many thanks for Pamphlet Chapters on History of war & newspaper just arrived.23
The year is established by the relationship between this letter and the letter from Asa Gray, 13 April 1863.
Gray’s letter has not been found; however, a portion of the letter is quoted in CD’s letter to H. W. Bates, 4 March [1863] (see n. 6, below).
CD refers to his twelve-year-old son, Leonard Darwin; at CD’s request, Gray had been sending United States postage stamps for Leonard’s collection since the summer of 1862 (see Correspondence vol. 10).
The reference is to a stamp issued by the Philadelphia carriers, D. O. Blood & Co. (see Sutton 1966, p. 41). See also letter to Asa Gray, 2 January [1863].
Campanula perfoliata (also called Specularia perfoliata) is native to the eastern United States (Bailey and Bailey 1976). CD had been interested in the plant since learning from Gray that it bore unopened flowers in which self-pollination occurred, a phenomenon later known as cleistogamy (see, for example, Correspondence vol. 9, letter from Asa Gray, 11 October 1861, and Correspondence vol. 10, letters to Asa Gray, 10–20 June [1862] and 26[–7] November [1862], and letters from Asa Gray, 18–19 August 1862 and 29 December 1862).
Gray evidently discussed Bates 1861 in a letter to CD written on 9 February 1863 that has not been found; CD quoted Gray’s comments in a letter to Henry Walter Bates of 4 March [1863]. Bates sent a copy of his paper to Gray in January, after CD had persuaded Gray to attempt to have it reviewed in the American Journal of Science and Arts, of which he was one of the contributing editors (see letter to H. W. Bates, 12 January [1863], and letter from H. W. Bates, 17 January [1863]).
Bates’s account of his eleven years as a naturalist in the Amazon region of South America (Bates 1863) was published between 1 and 14 April 1863 (Publishers’ Circular 26 (1863): 193).
C. Lyell 1863a; see letter from Asa Gray, 20 April 1863.
See letters to Charles Lyell, 6 March [1863] and 17 March [1863], and letter from J. D. Hooker, [15 March 1863].
See letter from Charles Lyell, 11 March 1863. See also letter from Charles Lyell, 15 March 1863.
A. Gray 1861a.
See Correspondence vol. 10, letter from Asa Gray, 29 December 1862. The reference is to Sprengel 1793, in which the term ‘dichogamie’ was first used (Baillon et al. 1876–92).
CD cited Sprengel 1793 extensively in Orchids, adding (p. 340 n.): I am aware that this author’s curious work … has often been spoken lightly of. No doubt he was an enthusiast, and probably carried some of his ideas to an extreme length. I feel sure, from my own observations, that his work contains a large body of truth. Although little regarded by botanists, the work was notable for its statement of two related doctrines, namely, that flowers were generally adapted to be cross-pollinated, and that floral structures were often adapted for insect visitation (DSB). These doctrines related closely to CD’s own belief that it was a ‘law of nature … that no organic being self-fertilises itself for an eternity of generations; but that a cross with another individual is occasionally … indispensable’ (Origin, p. 97).
In his letter to Gray of 26[–7] November [1862] (Correspondence vol. 10), CD objected to Gray’s use of the term ‘Diœcio-dimorphism’ to describe cases of flower dimorphism like those occurring in several species of Primula, on the grounds that he doubted the implied evolutionary transition between such dimorphism and dioeciousness. In his reply of 29 December 1862 (Correspondence vol. 10), Gray drew on CD’s account of the functional similarity of dimorphic and dichogamous plants, suggesting that CD extend the meaning of the word ‘dichogamy’ to include such cases of dimorphism as that in Primula.
Letter from Hermann Crüger, 23 February 1863.
See letter to Hermann Crüger, 25 January [1863] and n. 2.
CD’s new hothouse was completed by 15 February 1863 (see letter to J. D. Hooker, 15 February [1863] and n. 5); Joseph Dalton Hooker had sent CD orchids and other plants for cultivation from the Royal Botanic Gardens, Kew (see letter to J. D. Hooker, [21 February 1863]). See also Appendix VI.
Gray had suggested that Amsinckia spectabilis might be dimorphic, a view CD endorsed, having seen dried specimens sent by Gray (see Correspondence vol. 9, letter from Asa Gray, 9 November 1861, and letter to J. D. Hooker, 18 [December 1861]). CD subsequently grew plants from seed in order to experiment on the species (see Correspondence vol. 10, letter to Asa Gray, 15 March [1862], and this volume, letter to Asa Gray, 2 January [1863]). He recorded his observations on these plants, and on one belonging to his neighbour, George Henry Turnbull, in notes dated 23 March and 1 April 1863 (DAR 110: B2), concluding that the plant was not dimorphic, but that the length of the stigma was very variable, and the first-formed flowers tended to have stamens somewhat arrested in development. See also Forms of flowers, pp. 110–11.
Gray sent CD live specimens of the dimorphic plant Mitchella repens in December 1862, for use in crossing experiments (see Correspondence vol. 10, letter from Asa Gray, 9 December 1862, and this volume, letter to Asa Gray, 2 January [1863]). CD deferred his experiments until 1864, because the plants did not flower abundantly in 1863 (see letters to Asa Gray, 31 May [1863] and 26 June [1863], and Forms of flowers, p. 125).
CD had been suffering ill health since the end of February (see, for example, letter to J. D. Hooker, 5 March [1863]). Horace Darwin had been ill for much of 1862 (see letter from G. V. Reed, 12 January 1863 and n. 2); on 17 March 1863, Emma Darwin recorded in her diary (DAR 242) that he was again unwell. CD refers to James Manby Gully’s hydropathic establishment in Great Malvern, Worcestershire.
According to his ‘Journal’ (see Correspondence vol. 11, Appendix II), CD wrote a draft of the section of Variation dealing with inheritance (Variation 2: 1–84) between 23 January and 1 April 1863, noting: ‘took me 6
\frac{1}{2}
weeks time lost by illness & London’. Variation, which CD began writing early in 1860, was not published until 1868.
The pamphlet has not been identified. Gray periodically sent CD American newspapers, although CD told Hooker that he never read them (see Correspondence vol. 10, letter to J. D. Hooker, 29 [December 1862]).
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