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Create State-Space Model Containing ARMA State - MATLAB & Simulink - MathWorks América Latina
This example shows how to create a stationary ARMA model subject to measurement error using ssm.
To explicitly create a state-space model, it is helpful to write the state and observation equations in matrix form. In this example, the state of interest is the ARMA(2,1) process
{x}_{t}=c+{\varphi }_{1}{x}_{t-1}+{\varphi }_{2}{x}_{t-2}+{u}_{t}+{\theta }_{1}{u}_{t-1},
{u}_{t}
is Gaussian with mean 0 and known standard deviation 0.5.
{x}_{t}
{x}_{t-1}
{u}_{t}
are in the state-space model framework. Therefore, the terms
c
{\varphi }_{2}{x}_{t-2}
{\theta }_{1}{u}_{t-1}
require "dummy states" to be included in the model.
The state equation is
\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\\ {x}_{3,t}\\ {x}_{4,t}\end{array}\right]=\left[\begin{array}{cccc}{\varphi }_{1}& c& {\varphi }_{2}& {\theta }_{1}\\ 0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t-1}\\ {x}_{2,t-1}\\ {x}_{3,t-1}\\ {x}_{4,t-1}\end{array}\right]+\left[\begin{array}{c}0.5\\ 0\\ 0\\ 1\end{array}\right]{u}_{1,t}
c corresponds to a state (
{x}_{2,t}
) that is always 1.
{x}_{3,t}={x}_{1,t-1}
{x}_{1,t}
has the term
{\varphi }_{2}{x}_{3,t-1}={\varphi }_{2}{x}_{1,t-2}
{x}_{1,t}
0.5{u}_{1,t}
. ssm puts state disturbances as Gaussian random variables with mean 0 and variance 1. Therefore, the factor 0.5 is the standard deviation of the state disturbance.
{x}_{4,t}={u}_{1,t}
{x}_{1,t}
{\theta }_{1}{x}_{4,t}={\theta }_{1}{u}_{1,t-1}
The observation equation is unbiased for the ARMA(2,1) state process. The observation innovations are Gaussian with mean 0 and known standard deviation 0.1. Symbolically, the observation equation is
{y}_{t}=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\\ {x}_{3,t}\\ {x}_{4,t}\end{array}\right]+0.1{\epsilon }_{t}.
You can include a measurement-sensitivity factor (a bias) by replacing 1 in the row vector by a scalar or unknown parameter.
Define the state-transition coefficient matrix. Use NaN values to indicate unknown parameters.
A = [NaN NaN NaN NaN; 0 1 0 0; 1 0 0 0; 0 0 0 0];
Define the state-disturbance-loading coefficient matrix.
B = [0.5; 0; 0; 1];
Define the measurement-sensitivity coefficient matrix.
Define the observation-innovation coefficient matrix.
Use ssm to create the state-space model. Set the initial-state mean (Mean0) to a vector of zeros and covariance matrix (Cov0) to the identity matrix, except set the mean and variance of the constant state to 1 and 0, respectively. Specify the type of initial state distributions (StateType) by noting that:
{x}_{1,t}
is a stationary, ARMA(2,1) process.
{x}_{2,t}
{x}_{3,t}
is the lagged ARMA process, so it is stationary.
{x}_{4,t}
is a white-noise process, so it is stationary.
Cov0 = eye(4);
Cov0(2,2) = 0;
Mdl is an ssm model. You can use dot notation to access its properties. For example, print A by entering Mdl.A.
Use disp to verify the state-space model.
x1(t) = (c1)x1(t-1) + (c2)x2(t-1) + (c3)x3(t-1) + (c4)x4(t-1) + (0.50)u1(t)
x1 x2 x3 x4
Stationary Constant Stationary Stationary
If you have a set of responses, you can pass them and Mdl to estimate to estimate the parameters.
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Radicals - Course Hero
College Algebra/Numbers and Operations/Radicals
The square root of a number is a number that is multiplied by itself to give that number. For example, the square root of 9 is 3 because:
3\cdot3=9
n
th root of a number is multiplied by itself
n
times to get that number. For example, the fourth root of 81 is 3 because:
3\cdot3\cdot3\cdot3=81
A radical symbol
\sqrt{\phantom0}
is used to represent
n
n
is a natural number greater than or equal to 2. In the radical
\sqrt[n]{x}
, the index is
n
n
x
\sqrt[3]{x}
\sqrt{x}
, the index is not shown, but it is understood to be 2. The radicand is the expression under the radical.
\sqrt{x+2}
, the index is 2, and the radicand is
x+2
. It is read as "the square root of the quantity
x
plus 2."
\sqrt[3]{x^2}
x^2
. It is read as "the cube root of
x
squared."
A radical with index n is the inverse, or opposite operation, of the exponent
n
n
x\geq0
n
is odd:
\sqrt[n]{x^n}=(\sqrt[n]{x})^n=x
n
x<0
\sqrt[x]{x}
is not a real number. For example,
\sqrt[3]{(-4)^3}=-4
\sqrt{-4}
The rules for multiplying or dividing radicals are the same as with exponents. A product or quotient under a radical can be split into two separate radicals, and radicals with the same index that are multiplied or divided can be combined into one.
Properties of radicals are used to simplify radical expressions. It is also possible to combine like radicals, which have the same radicand and index.
Product of radicals property
\sqrt[n]{xy}=\sqrt[n]{x}\sqrt[n]{y}
The radical of a product is the product of the radicals.
Quotient of radicals property
\sqrt[n]{\frac{x}{y}}=\frac{\sqrt[n]{x}}{\sqrt[n]{y}}
The radical of a quotient is the quotient of the radicals.
\sqrt{2}\sqrt{3}+3\sqrt{6}
Use the product of radicals property to combine
\sqrt{2}
\sqrt{3}
\begin{gathered}\sqrt{2}\sqrt{3}+3\sqrt{6}\\\sqrt{2\cdot3}+3\sqrt{6}\\\end{gathered}
Simplify the expression. Multiply 2 and 3 inside the radical.
\begin{gathered}\sqrt{2\cdot3}+3\sqrt{6}\\\sqrt{6}+3\sqrt{6}\end{gathered}
Combine like radicals. The terms
\sqrt{6}
3\sqrt{6}
have the same radicand and index. So, they can be combined like other like terms by adding the coefficients. The coefficient of
\sqrt{6}
\begin{gathered}\sqrt{6}+3\sqrt{6}\\4\sqrt{6}\end{gathered}
When simplifying radicals, there is a set of rules that determines whether the radical is in simplest form.
There are no perfect
n
th powers under a radical of index
n
, such as perfect squares within a square root or perfect cubes under a cube root.
\begin{aligned}\sqrt{8}&=\sqrt{(4)(2)}\\&=\sqrt{4}\sqrt{2}\\&=2\sqrt{2}\end{aligned}
There are no fractions under a radical or radicals in the denominator of a fraction.
\begin{aligned}\sqrt{\frac{3}{4}}&=\frac{\sqrt{3}}{\sqrt{4}}\\&=\frac{\sqrt{3}}{2}\end{aligned}
Writing Radicals in Simplest Form
\sqrt[4]{x^7}
x\geq0
The index of the radical is 4. Rewrite
x^7
as a product of a perfect fourth power and another power.
\begin{gathered}\sqrt[4]{x^7}\\\sqrt[4]{x^4x^3}\end{gathered}
Use the product of radicals property to split the radical into a product.
\begin{gathered}\sqrt[4]{x^4x^3}\\\sqrt[4]{x^4}\sqrt[4]{x^3}\end{gathered}
Use the inverse relationship between the radical and exponent to rewrite
\sqrt[4]{x^4}
. This expression is equal to
x
x\geq0
\begin{gathered}\sqrt[4]{x^4}\sqrt[4]{x^3}\\x\sqrt[4]{x^3}\end{gathered}
Rationalizing the denominator is the process of rewriting a fraction to remove radical expressions from the denominator. To rationalize the denominator, multiply the fraction by a fraction that is equivalent to 1. The type of radical expression in the denominator determines which fraction equivalent to 1 should be used.
If the denominator has two terms, such as
\sqrt{x}+y
, the conjugate of the expression is
\sqrt{x}-y
. When two expressions are multiplied, the resulting expression does not contain a radical:
\begin{gathered}\left(\sqrt{x}+y\right)\left(\sqrt{x}-y\right)\\\left(\sqrt{x}\right)^2-y\sqrt{x}+y\sqrt{x}-y^2\\x-y^2\end{gathered}
Fraction Equivalent to 1
Single square root Multiply by a fraction in this form:
\frac{\sqrt{x}}{\sqrt{x}}=1
\begin{aligned}\frac{2}{\sqrt{5}}\left(\frac{\sqrt{5}}{\sqrt{5}}\right)&=\frac{2\sqrt{5}}{\sqrt{25}} \\&= \frac{2\sqrt{5}}{5}\end{aligned}
Single cube root Multiply by a fraction in this form:
\frac{\sqrt[3]{x^2}}{\sqrt[3]{x^2}}=1
\begin{aligned}\frac{2}{\sqrt[3]{5}}\left(\frac{\sqrt[3]{5^2}}{\sqrt[3]{5^2}}\right)&=\frac{2\sqrt[3]{5^2}}{\sqrt[3]{5^3}}\\&=\frac{2\sqrt[3]{5^2}}{5}\end{aligned}
Expression with two terms, such as:
\sqrt{x}+y
Multiply by a fraction in this form:
\frac{\sqrt{x}-y}{\sqrt{x}-y}=1
\begin{aligned}\frac{2}{\sqrt{5}+1}\left(\frac{\sqrt{5}-1}{\sqrt{5}-1}\right)&=\frac{2\sqrt{5}-2}{\sqrt{25}-\sqrt{5}+\sqrt{5}-1}\\&=\frac{2\sqrt{5}-2}{4}\end{aligned}
n
Radicals can be expressed by using rational exponents. A rational exponent is an exponent that is a rational number. If the exponent is written as a fraction
\frac{a}{b}
, then the denominator
b
is the index of the radical, and the numerator
a
is the exponent of the radicand. Alternately, the expression can be written in two ways, with the entire radical raised to the
a
x^{\frac{a}{b}}=(x^a)^\frac{1}{b}=\sqrt[b]{x^a}=(\sqrt[b]{x})^a
x^{\frac{a}{b}}=(x^\frac{1}{b})^a=\sqrt[b]{x^a}=(\sqrt[b]{x})^a
A numerator of 1 indicates the first power:
x^{\frac{1}{b}}=\sqrt[b]{x}
A denominator of 2 indicates a square root:
x^{\frac{1}{2}}=\sqrt{x}
Note that rational exponents can be positive or negative.
To simplify expressions with rational exponents, rewrite the expression in radical form.
<Exponents>Order of Operations
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Value of a firm's profit after deduction of capital costs
In corporate finance, as part of fundamental analysis, economic value added is an estimate of a firm's economic profit, or the value created in excess of the required return of the company's shareholders. EVA is the net profit less the capital charge ($) for raising the firm's capital. The idea is that value is created when the return on the firm's economic capital employed exceeds the cost of that capital. This amount can be determined by making adjustments to GAAP accounting. There are potentially over 160 adjustments but in practice, only several key ones are made, depending on the company and its industry.
3 Market value added
4 Process-based costing
EVA is net operating profit after taxes (or NOPAT) less a capital charge, the latter being the product of the cost of capital and the economic capital. The basic formula is:
{\displaystyle {\begin{aligned}{\text{EVA}}&=({\text{ROIC}}-{\text{WACC}})\cdot ({\text{total assets}}-{\text{current liability}})\\[8pt]&={\text{NOPAT}}-{\text{WACC}}\cdot ({\text{total assets}}-{\text{current liability}})\end{aligned}}}
{\displaystyle {\text{ROIC}}={\frac {\text{NOPAT}}{{\text{total assets}}-{\text{current liability}}}}}
is the return on invested capital;
{\displaystyle ({\text{WACC}})\,}
is the weighted average cost of capital (WACC);
{\displaystyle ({\text{total assets}}-{\text{current liability}})\,}
is the economic capital employed (total assets − current liability);
NOPAT is the net operating profit after tax, with adjustments and translations, generally for the amortization of goodwill, the capitalization of brand advertising and other non-cash items.
EVA calculation:
EVA = net operating profit after taxes – a capital charge [the residual income method]
therefore EVA = NOPAT – (c × capital), or alternatively
EVA = (r × capital) – (c × capital) so that
EVA = (r − c) × capital [the spread method, or excess return method]
r = rate of return, and
c = cost of capital, or the weighted average cost of capital (WACC).
NOPAT is profits derived from a company's operations after cash taxes but before financing costs and non-cash bookkeeping entries. It is the total pool of profits available to provide a cash return to those who provide capital to the firm.
Capital is the amount of cash invested in the business, net of depreciation. It can be calculated as the sum of interest-bearing debt and equity or as the sum of net assets less non-interest-bearing current liabilities (NIBCLs).
The capital charge is the cash flow required to compensate investors for the riskiness of the business given the amount of economic capital invested.
The cost of capital is the minimum rate of return on capital required to compensate investors (debt and equity) for bearing risk, their opportunity cost.
Another perspective on EVA can be gained by looking at a firm's return on net assets (RONA). RONA is a ratio that is calculated by dividing a firm's NOPAT by the amount of capital it employs (RONA = NOPAT/Capital) after making the necessary adjustments to the data reported by a conventional financial accounting system.
EVA = (RONA – required minimum return) × net investments
If RONA is above the threshold rate, EVA is positive.
Comparison with other approaches[edit]
Other approaches along similar lines include residual income valuation (RI) and residual cash flow. Although EVA is similar to residual income, under some definitions there may be minor technical differences between EVA and RI (for example, adjustments that might be made to NOPAT before it is suitable for the formula below). Residual cash flow is another, much older term for economic profit. In all three cases, money cost of capital refers to the amount of money rather than the proportional cost (% cost of capital); at the same time, the adjustments to NOPAT are unique to EVA.
Although in concept, these approaches are in a sense nothing more than the traditional, commonsense idea of "profit", the utility of having a more precise term such as EVA is that it makes a clear separation from dubious accounting adjustments that have enabled businesses such as Enron to report profits while actually approaching insolvency.
Other measures of shareholder value include:
Market value added[edit]
The firm's market value added, is the added value an investment creates for its shareholders over the total capital invested by them. MVA is the discounted sum (present value) of all future expected economic value added:
{\displaystyle {\text{MVA}}=V-K_{0}=\sum _{t=1}^{\infty }{{\text{EVA}}_{t} \over (1+c)^{t}}}
Note that MVA = PV of EVA.
More enlightening is that since MVA = NPV of free cash flow (FCF) it follows therefore that the
NPV of FCF = PV of EVA;
since after all, EVA is simply the re-arrangement of the FCF formula.
Process-based costing[edit]
In 2012, Mocciaro Li Destri, Picone and Minà proposed a performance and cost measurement system that integrates the EVA criteria with process-based costing (PBC).[1] The EVA-PBC methodology allows us to implement the EVA management logic not only at the firm level but also at lower levels of the organization. EVA-PBC methodology plays an interesting role in bringing strategy back into financial performance measures.
^ Mocciaro Li Destri, A.; Picone, P. M.; Minà, A. (2012). "Bringing Strategy Back into Financial Systems of Performance Measurement: Integrating EVA and PBC". Business System Review. 1 (1): 85–102. SSRN 2154117.
G. Bennett Stewart III (2013). Best-Practice EVA. John Wiley & Sons.
G. Bennett Stewart III (1991). The Quest for Value. HarperCollins.
Erik Stern. The Value Mindset. Wiley.
Joel Stern and John Shiely. The EVA Challenge. Wiley.
Al Ehrbar (1988). EVA, the Real Key to Creating Wealth. Wiley.
E. Knappek. An hitchhiker guide to having fun times with Eva. Limited Edition.
Stern Value Management Proprietary Tools
A Reading List on EVA/Value Based Management from Robert Korajczyk
Economic Value Added (EVA), Prof. Aswath Damodaran
EVA-WACC Tree Model Infographic Visual.ly
Economic Value Added: A simulation analysis of the trendy, owner-oriented management tool, Timo Salmi and Ilkka Virtanen, 2001
The Origins of EVA Chicago-Booth magazine
New Media video, by Florencia Roca: Interview to Joel Stern What is EVA?
Retrieved from "https://en.wikipedia.org/w/index.php?title=Economic_value_added&oldid=1079051645"
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A note on ‘A best proximity point theorem for Geraghty-contractions’ | Fixed Point Theory and Algorithms for Sciences and Engineering | Full Text
A note on ‘A best proximity point theorem for Geraghty-contractions’
In Caballero et al. (Fixed Point Theory Appl. (2012). doi:10.1186/1687-1812-2012-231), the authors prove a best proximity point theorem for Geraghty nonself contraction. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.
\left(X,d\right)
T:A\to B
is said to be contractive if there exists
k\in \left[0,1\right)
d\left(Tx,Ty\right)\le kd\left(x,y\right)
x,y\in A
. The well-known Banach contraction principle says: Let
\left(X,d\right)
be a complete metric space, and
T:X\to X
be a contraction of X into itself. Then T has a unique fixed point in X.
In 1973, Geraghty introduced the Geraghty-contraction and obtained Theorem 1.2.
\left(X,d\right)
T:X\to X
is said to be a Geraghty-contraction if there exists such that for any
x,y\in X
d\left(Tx,Ty\right)\le \beta \left(d\left(x,y\right)\right)\cdot d\left(x,y\right),
where the class Γ denotes those functions
\beta :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)
\beta \left({t}_{n}\right)\to 1\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}{t}_{n}\to 0.
\left(X,d\right)
T:X\to X
be an operator. Suppose that there exists such that for any
x,y\in X
d\left(Tx,Ty\right)\le \beta \left(d\left(x,y\right)\right)\cdot d\left(x,y\right).
Obviously, Theorem 1.2 is an extensive version of Banach contraction principle. In 2012, Caballero et al. introduced generalized Geraghty-contraction as follows.
Let A, B be two nonempty subsets of a metric space
\left(X,d\right)
T:A\to B
x,y\in A
d\left(Tx,Ty\right)\le \beta \left(d\left(x,y\right)\right)\cdot d\left(x,y\right),
where the class denotes those functions
\beta :\left[0,\mathrm{\infty }\right)\to \left[0,1\right)
\beta \left({t}_{n}\right)\to 1\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}{t}_{n}\to 0.
Now we need the following notations and basic facts.
Let A and B be two nonempty subsets of a metric space
\left(X,d\right)
{A}_{0}
{B}_{0}
the following sets:
d\left(A,B\right)=inf\left\{d\left(x,y\right):x\in A\text{ and }y\in B\right\}
In [3], the authors give sufficient conditions for when the sets
{A}_{0}
{B}_{0}
are nonempty. In [4], the author presents the following definition and proves that any pair
\left(A,B\right)
of nonempty, closed and convex subsets of a real Hilbert space H satisfies the P-property.
\left(A,B\right)
be a pair of nonempty subsets of a metric space
\left(X,d\right)
{A}_{0}\ne \mathrm{\varnothing }
\left(A,B\right)
is said to have the P-property if and only if for any
{x}_{1},{x}_{2}\in {A}_{0}
{y}_{1},{y}_{2}\in {B}_{0}
\left\{\begin{array}{c}d\left({x}_{1},{y}_{1}\right)=d\left(A,B\right),\hfill \\ d\left({x}_{2},{y}_{2}\right)=d\left(A,B\right)\hfill \end{array}⇒\phantom{\rule{1em}{0ex}}d\left({x}_{1},{x}_{2}\right)=d\left({y}_{1},{y}_{2}\right).
Let A, B be two nonempty subsets of a complete metric space and consider a mapping
T:A\to B
. The best proximity point problem is whether we can find an element
{x}_{0}\in A
d\left({x}_{0},T{x}_{0}\right)=min\left\{d\left(x,Tx\right):x\in A\right\}
d\left(x,Tx\right)\ge d\left(A,B\right)
x\in A
, in fact, the optimal solution to this problem is the one for which the value
d\left(A,B\right)
In [2], the authors give a generalization of Theorem 1.2 by considering a nonself map and they get the following theorem.
\left(A,B\right)
be a pair of nonempty closed subsets of a complete metric space
\left(X,d\right)
{A}_{0}
T:A\to B
be a Geraghty-contraction satisfying
T\left({A}_{0}\right)\subseteq {B}_{0}
. Suppose that the pair
\left(A,B\right)
has the P-property. Then there exists a unique
{x}^{\ast }
in A such that
d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right)
Remark In [2], the proof of Theorem 1.5 is unnecessarily complex. In this note, not only P-property has been weakened, but also an improved best proximity point theorem will be presented by a short and simple proof. An example which satisfies weak P-property but not P-property has been presented to demonstrate our results.
Before giving our main results, we first introduce the notion of weak P-property.
Weak P-property Let
\left(A,B\right)
\left(X,d\right)
{A}_{0}\ne \mathrm{\varnothing }
\left(A,B\right)
is said to have the weak P-property if and only if for any
{x}_{1},{x}_{2}\in {A}_{0}
{y}_{1},{y}_{2}\in {B}_{0}
\left\{\begin{array}{c}d\left({x}_{1},{y}_{1}\right)=d\left(A,B\right),\hfill \\ d\left({x}_{2},{y}_{2}\right)=d\left(A,B\right)\hfill \end{array}⇒\phantom{\rule{1em}{0ex}}d\left({x}_{1},{x}_{2}\right)\le d\left({y}_{1},{y}_{2}\right).
Now we are in a position to give our main results.
\left(A,B\right)
\left(X,d\right)
{A}_{0}\ne \mathrm{\varnothing }
T:A\to B
T\left({A}_{0}\right)\subseteq {B}_{0}
\left(A,B\right)
has the weak P-property. Then there exists a unique
{x}^{\ast }
d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right)
Proof We first prove that
{B}_{0}
\left\{{y}_{n}\right\}\subseteq {B}_{0}
{y}_{n}\to q\in B
. It follows from the weak P-property that
d\left({y}_{n},{y}_{m}\right)\to 0\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}d\left({x}_{n},{x}_{m}\right)\to 0,
n,m\to \mathrm{\infty }
{x}_{n},{x}_{m}\in {A}_{0}
d\left({x}_{n},{y}_{n}\right)=d\left(A,B\right)
d\left({x}_{m},{y}_{m}\right)=d\left(A,B\right)
\left\{{x}_{n}\right\}
is a Cauchy sequence so that
\left\{{x}_{n}\right\}
p\in A
. By the continuity of metric d we have
d\left(p,q\right)=d\left(A,B\right)
q\in {B}_{0}
{B}_{0}
{\overline{A}}_{0}
{A}_{0}
, we claim that
T\left({\overline{A}}_{0}\right)\subseteq {B}_{0}
x\in {\overline{A}}_{0}\setminus {A}_{0}
\left\{{x}_{n}\right\}\subseteq {A}_{0}
{x}_{n}\to x
. By the continuity of T and the closeness of
{B}_{0}
Tx={lim}_{n\to \mathrm{\infty }}T{x}_{n}\in {B}_{0}
T\left({\overline{A}}_{0}\right)\subseteq {B}_{0}
{P}_{{A}_{0}}:T\left({\overline{A}}_{0}\right)\to {A}_{0}
{P}_{{A}_{0}}y=\left\{x\in {A}_{0}:d\left(x,y\right)=d\left(A,B\right)\right\}
. Since the pair
\left(A,B\right)
has weak P-property and T is a Geraghty-contraction, we have
d\left({P}_{{A}_{0}}T{x}_{1},{P}_{{A}_{0}}T{x}_{2}\right)\le d\left(T{x}_{1},T{x}_{2}\right)\le \beta \left(d\left({x}_{1},{x}_{2}\right)\right)\cdot d\left({x}_{1},{x}_{2}\right)
{x}_{1},{x}_{2}\in {\overline{A}}_{0}
{P}_{{A}_{0}}T:{\overline{A}}_{0}\to {\overline{A}}_{0}
is a Geraghty-contraction from complete metric subspace
{\overline{A}}_{0}
into itself. Using Theorem 1.2, we can get
{P}_{{A}_{0}}T
{x}^{\ast }
{P}_{{A}_{0}}T{x}^{\ast }={x}^{\ast }\in {A}_{0}
d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right).
{x}^{\ast }
is the unique one in
{A}_{0}
d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right)
{x}^{\ast }
is also the unique one in A such that
d\left({x}^{\ast },T{x}^{\ast }\right)=d\left(A,B\right)
Remark In Theorem 2.1, P-property is weakened to weak P-property. Therefore, Theorem 2.1 is an improved result of Theorem 1.5. In addition, our proof is shorter and simpler than that in [2]. In fact, our proof process is less than one page. However, the proof process in [2] is three pages.
Now we present an example which satisfies weak P-property but not P-property.
\left({R}^{2},d\right)
, where d is the Euclidean distance and the subsets
A=\left\{\left(0,0\right)\right\}
B=\left\{y=1+\sqrt{1-{x}^{2}}\right\}
{A}_{0}=\left\{\left(0,0\right)\right\}
{B}_{0}=\left\{\left(-1,1\right),\left(1,1\right)\right\}
d\left(A,B\right)=\sqrt{2}
d\left(\left(0,0\right),\left(-1,1\right)\right)=d\left(\left(0,0\right),\left(1,1\right)\right)=\sqrt{2},
0=d\left(\left(0,0\right),\left(0,0\right)\right)<d\left(\left(-1,1\right),\left(1,1\right)\right)=2.
We can see that the pair
\left(A,B\right)
satisfies the weak P-property but not the P-property.
Geraghty M: On contractive mappings. Proc. Am. Math. Soc. 1973, 40: 604–608. 10.1090/S0002-9939-1973-0334176-5
Caballero J, et al.: A best proximity point theorem for Geraghty-contractions. Fixed Point Theory Appl. 2012. doi:10.1186/1687–1812–2012–231
Sankar Raj, V: Banach contraction principle for non-self mappings. Preprint
Zhang, J., Su, Y. & Cheng, Q. A note on ‘A best proximity point theorem for Geraghty-contractions’. Fixed Point Theory Appl 2013, 99 (2013). https://doi.org/10.1186/1687-1812-2013-99
Geraghty-contractions
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Find the polar form of the complex number given below. Approximate your values to three decimal places. (Remember to figure out the correct quadrant of the complex number when you find the value for
θ
3 − 4i
Point labeled, a, in fourth quadrant at (3, comma negative 4).
Draw a right triangle with the
x
Segments added connecting point to origin, & to (3, comma 0), and from origin to (3, comma 0, area inside triangle shaded.
r
' is the length of the hypotenuse of the triangle. Calculate '
r
' by using the Pythagorean theorem.
Label added to hypotenuse, r.
Use trigonometry to find the angle marked
α
Label added to angle opposite vertical leg, alpha.
Use the measure of angle
α
θ
Curved counter clockwise arrow, from positive x axis to hypotenuse, labeled theta.
−12 − 5i
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Machine Learning Trick of the Day (7): Density Ratio Trick - Open Data Science - Your News Source for AI, Machine Learning & more
Deep LearningModelingStatisticsMachine Learningposted by Shakir Mohamed February 6, 2018 Shakir Mohamed
A probability on its own is often an uninteresting thing. But when we can compare probabilities, that is when their full...
A probability on its own is often an uninteresting thing. But when we can compare probabilities, that is when their full splendour is revealed. By comparing probabilities we are able form judgements; by comparing probabilities we can exploit the elements of our world that are probable; by comparing probabilities we can see the value of objects that are rare. In their own ways, all machine learning tricks help us make better probabilistic comparisons. Comparison is the theme of this post—not discussed in this series before—and the right start to this second sprint of machine learning tricks.
Relationship between four statistical operations [1].
In his 1981 Wald Memorial Lecture [1], Bradley Efron described four statistical operations, which remain important today: enumeration, modelling, comparison, and inference.
Data Enumeration is the process of collecting data (and involves systems, domain experts, and critiquing the problem at hand).
Modelling, or summarisation as Efron said, combines the ‘small bits of data’ (our training data) to extract its underlying trends and statistical structure.
Comparison does the opposite of modelling: it pulls apart our data to show the differences that exist in it.
Inferences are statements about parts of our models that are unobserved or latent, while predictions are statements of data we have not observed.
The statistical operations in the left half of the image above are achieved using the principles and practice of learning, or estimation. Those on the right half by hypothesis testing. While the preceding tricks in this series looked at learning problems, thinking instead of testing, and of statistical comparisons, can lead us to interesting new tricks.
To compare two numbers, we can look at either their difference or their ratio. The same is true if we want to compare probability densities: either through a density difference or a density ratio. Density ratios are ubiquitous in machine learning, and will be our focus. The expression:
is the density ratio of two probability densities
rho(x)
q(x)
x
. This ratio is intuitive and tells us the amount by which we need to correct q for it to be equal to
rho
rho(x)=r(x)q(x)
The density ratio gives the correction factor needed to make two distributions equal.
From our very first introductions to statistics and machine learning, we met such ratios: in the rules of probability, in estimation theory, in information theory, when computing integrals, learning generative models, and beyond [2][3][4].
The computation of conditional probabilities is one of first ratios we encountered:
Divergences and Maximum Likelihood
The KL divergence is the divergence most widely used to compare two distributions, and is defined in terms of a log density-ratio. Maximum likelihood is obtained by the minimisation of this divergence, highlighting how central density ratios are in our statistical practice.
Importance sampling gives us a way of changing the distribution with respect to which an expectation is taken, by introducing an identity term and then extracting a density ratio. The ratio that emerges is referred to as an importance weight. Using
r(x) = tfrac{p(x)}{q(x)}
The mutual information, a multivariate measure of correlation, is a core concept of information theory. The mutual information between two random variables x, y makes a comparison of their joint dependence versus independence. This comparison is naturally expressed using a ratio:
The classic use of such ratios is for hypothesis testing. The Neyman-Pearson lemma motivates this best, showing that the most powerful tests are those computed using a likelihood ratio. To compare hypothesis
H_0
(the null) to
H_1
(the alternative), we compute the ratio of the probability of our data under the different hypotheses. The hypothesis could be two different parameter settings, or even different models.
The central task in the above five statistical quantities is to efficiently compute the ratio
r(x)
. In simple problems, we can compute the numerator and the denominator separately, and then compute their ratio. Direct estimation like this will not often be possible: each part of the ratio may itself involve intractable integrals; we will often deal with high-dimensional quantities; and we may only have samples drawn from the two distributions, not their analytical forms.
This is where the density ratio trick or formally, density ratio estimation, enters: it tells us to construct a binary classifier
mathcal{S}(x)
that distinguishes between samples from the two distributions. We can then compute the density ratio using the probability given by this classifier:
To show this, imagine creating a data set of 2N elements consisting of pairs (data x, label y):
N data points are drawn from the distribution
rho
and assigned a label +1.
The remaining N data points are drawn from distribution
q
and assigned label -1.
By this construction, we can write the probabilities
rho, q
in a conditional form; we should also keep Bayes’ theorem in mind.
We can do the following manipulations:
In the first line, we rewrote with ratio problem as a ratio of conditionals using the dummy labels y, which we introduced to identify samples from the each of the distributions. In the second line, we used Bayes’ rule to express the conditional probabilities in their inverse forms. In the final line, the marginal distributions
p(x)
are equal in the numerator and the denominator and cancel. Similarly, because we used an equal number of samples from the two distributions, the prior probability
p(y=+1) = p(y=-1) = 0.5
and also cancels; we can easily include this prior to allow for imbalanced datasets. These cancellations lead us to the final line.
This final derivation says that the problem of density ratio estimation is equivalent to that of binary classification. All we need do is construct a classifier that gives the probability of a data point belonging to distribution
rho
, and knowing that probability is enough to know the density ratio. Fortunately, building probabilistic classifiers is one of the things we know how to do best.
The idea of using classifiers to compute density ratios is widespread, and my suggestions for deeper understanding include:
This is the definitive book on density ratio estimation [2] in all its forms and application, by Masashi Sugiyama. A must read for anyone interested in this topic.
This paper is also a good starting point.
In section 14.2.4 in the Elements of Statistical learning [5], almost too quickly, Friedman et al. describe this trick and its role in unsupervised learning.
We can do unsupervised learning of a model
q(x; theta)
by only being able to draw samples from the model and then doing supervised learning (building a classifier) by invoking the density ratio trick.
If you combine this trick with knowledge of the model structure, in this case for undirected graphical models with known energy functions, we can exploit the density ratio trick to derive the noise-contrastive principle for learning [3].
Generative adversarial networks (GANs) learn a model
q(x)
of the data by combining the density-ratio trick, with the reparameterisation trick to jointly learn both the model (generator) and the classifier (discriminator).
We wrote this paper [4] to explain GANs and other related methods like (ABC) within the framework of comparison and testing, and other approaches for density ratio estimation.
Classifier Two-sample Hypothesis Testing
The classical task for such ratios is for two-sample hypothesis tests and this paper shows how using a binary classifier gives a different way to perform these tests.
Comparisons are the drivers of learning. And the density ratio trick is a generic tool that makes comparison a statistical operations that can be used widely—by replacing density ratios where we see them with classifiers—and using it in conjunction with other tricks. It is the importance of comparison that makes Bayesian statistical approaches interesting, since, by learning entire distributions rather than point-estimates, we always strive to make the widest set of comparisons possible. And this trick also highlights the power of other principles of learning, in particular of likelihood-free estimation. There is a great deal to explore in these topics, and within them a wealth of new tricks, some of which we will encounter in future posts.
Complement this essay by reading the other essays in this series, in particular the log-derivative trick to see another ratio in action, an essay on variational inference and auto-encoders where ratios again appear, and a post exploring the breadth of conceptual frameworks for thinking about machine learning and its principles.
[1] Bradley Efron, Maximum likelihood and decision theory, The annals of Statistics, 1982
[2] Masashi Sugiyama, Taiji Suzuki, Takafumi Kanamori, Density ratio estimation in machine learning, , 2012
[3] Michael Gutmann, Aapo Hyv”arinen, Noise-contrastive estimation: A new estimation principle for unnormalized statistical models, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 2010
[4] Shakir Mohamed, Balaji Lakshminarayanan, Learning in implicit generative models, arXiv preprint arXiv:1610.03483, 2016
[5] Jerome Friedman, Trevor Hastie, Robert Tibshirani, The elements of statistical learning, , 2001
My name is Shakir and I 'm a researcher in statistical machine learning and artificial intelligence. I'm a senior research scientist at Google DeepMind in London. Before that I was a CIFAR scholar with Nando de Freitas at the University of British Columbia, and I completed my PhD with Zoubin Gharahramani at the University of Cambridge. I'm from Johannesburg, South Africa.
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Constructible polygon - Wikipedia
Regular polygon that can be constructed with compass and straightedge
1 Conditions for constructibility
1.1 Detailed results by Gauss's theory
1.2 Connection to Pascal's triangle
4 Other constructions
Conditions for constructibility[edit]
Construction of the regular 17-gon
A regular n-gon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat primes (including none).
A Fermat prime is a prime number of the form
{\displaystyle 2^{(2^{m})}+1.}
In order to reduce a geometric problem to a problem of pure number theory, the proof uses the fact that a regular n-gon is constructible if and only if the cosine
{\displaystyle \cos(2\pi /n)}
is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. Equivalently, a regular n-gon is constructible if any root of the nth cyclotomic polynomial is constructible.
Detailed results by Gauss's theory[edit]
Restating the Gauss-Wantzel theorem:
A regular n-gon is constructible with straightedge and compass if and only if n = 2kp1p2...pt where k and t are non-negative integers, and the pi's (when t > 0) are distinct Fermat primes.
The five known Fermat primes are:
F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in the OEIS).
Thus a regular n-gon is constructible if
n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, ... (sequence A003401 in the OEIS),
n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, ... (sequence A004169 in the OEIS).
Connection to Pascal's triangle[edit]
Since there are 5 known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence 31 constructible odd-sided regular polygons. These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295 (sequence A045544 in the OEIS). As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the modulo-2 Pascal's triangle, minus the top row, which corresponds to a monogon. (Because of this, the 1s in such a list form an approximation to the Sierpiński triangle.) This pattern breaks down after this, as the next Fermat number is composite (4294967297 = 641 × 6700417), so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, and it is therefore unknown how many odd-sided constructible regular polygons exist. In general, if there are q Fermat primes, then there are 2q−1 odd-sided regular constructible polygons.
cos 2π/n ,
½ φ(n),
cos 2π/17 .
Compass and straightedge constructions[edit]
If p = 2, draw a q-gon and bisect one of its central angles. From this, a 2q-gon can be constructed.
If p > 2, inscribe a p-gon and a q-gon in the same circle in such a way that they share a vertex. Because p and q are coprime, there exists integers a and b such that ap + bq = 1. Then 2aπ/q + 2bπ/p = 2π/pq. From this, a pq-gon can be constructed.
The construction for an equilateral triangle is simple and has been known since Antiquity; see Equilateral triangle.
Constructions for the regular pentagon were described both by Euclid (Elements, ca 300 BC), and by Ptolemy (Almagest, ca AD 150); see Pentagon.
Although Gauss proved that the regular 17-gon is constructible, he did not actually show how to do it. The first construction is due to Erchinger, a few years after Gauss' work; see Heptadecagon.
The first explicit constructions of a regular 257-gon were given by Magnus Georg Paucker (1822)[4] and Friedrich Julius Richelot (1832).[5]
A construction for a regular 65537-gon was first given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.[6]
A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if
{\displaystyle n=2^{r}3^{s}p_{1}p_{2}\cdots p_{k},}
where r, s, k ≥ 0 and where the pi are distinct Pierpont primes greater than 3 (primes of the form
{\displaystyle 2^{t}3^{u}+1).}
[7]: Thm. 2 These polygons are exactly the regular polygons that can be constructed with Conic section, and the regular polygons that can be constructed with paper folding. The first numbers of sides of these polygons are:
3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, 111, 112, 114, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 144, 146, 148, 152, 153, 156, 160, 162, 163, 168, 170, 171, 180, 182, 185, 189, 190, 192, 193, 194, 195, 204, 208, 210, 216, 218, 219, 221, 222, 224, 228, 234, 238, 240, 243, 247, 252, 255, 256, 257, 259, 260, 266, 270, 272, 273, 280, 285, 288, 291, 292, 296, ... (sequence A122254 in the OEIS)
^ [http://www.prothsearch.com/fermat.html Prime factors k · 2n + 1 of Fermat numbers Fm and complete factoring status] by Wilfrid Keller.
^ Cox, David A. (2012), "Theorem 10.1.6", Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, p. 259, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9 .
^ Magnus Georg Paucker (1822). "Geometrische Verzeichnung des regelmäßigen Siebzehn-Ecks und Zweyhundersiebenundfünfzig-Ecks in den Kreis". Jahresverhandlungen der Kurländischen Gesellschaft für Literatur und Kunst (in German). 2: 160–219.
^ Friedrich Julius Richelot (1832). "De resolutione algebraica aequationis x257 = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata". Journal für die reine und angewandte Mathematik (in Latin). 9: 1–26, 146–161, 209–230, 337–358. doi:10.1515/crll.1832.9.337.
^ Johann Gustav Hermes (1894). "Über die Teilung des Kreises in 65537 gleiche Teile". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German). Göttingen. 3: 170–186.
^ Gleason, Andrew M. (March 1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624.
Duane W. DeTemple (1991). "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions". The American Mathematical Monthly. 98 (2): 97–108. doi:10.2307/2323939. JSTOR 2323939. MR 1089454.
Christian Gottlieb (1999). "The Simple and Straightforward Construction of the Regular 257-gon". Mathematical Intelligencer. 21 (1): 31–37. doi:10.1007/BF03024829. MR 1665155.
Regular Polygon Formulas, Ask Dr. Math FAQ.
Carl Schick: Weiche Primzahlen und das 257-Eck : eine analytische Lösung des 257-Ecks. Zürich : C. Schick, 2008. ISBN 978-3-9522917-1-9.
65537-gon, exact construction for the 1st side, using the Quadratrix of Hippias and GeoGebra as additional aids, with brief description (German)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Constructible_polygon&oldid=1041458460"
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torch.copysign — PyTorch 1.11.0 documentation
torch.copysign
torch.copysign¶
torch.copysign(input, other, *, out=None) → Tensor¶
\text{out}_{i} = \begin{cases} -|\text{input}_{i}| & \text{if} \text{other}_{i} \leq -0.0 \\ |\text{input}_{i}| & \text{if} \text{other}_{i} \geq 0.0 \\ \end{cases}
Supports broadcasting to a common shape, and integer and float inputs.
input (Tensor) – magnitudes.
other (Tensor or Number) – contains value(s) whose signbit(s) are applied to the magnitudes in input.
tensor([-1.2557, -0.0026, -0.5387, 0.4740, -0.9244])
>>> torch.copysign(a, 1)
>>> torch.copysign(a, b)
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Find sources: "Division" mathematics – news · newspapers · books · scholar · JSTOR (October 2014) (Learn how and when to remove this template message)
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{sum}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{difference}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
{\displaystyle \scriptstyle {\text{product}}}
{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
{\displaystyle {\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}}
{\displaystyle \scriptstyle {\text{base}}^{\text{exponent}}\,=\,}
{\displaystyle \scriptstyle {\text{power}}}
{\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}
{\displaystyle \scriptstyle {\text{root}}}
{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
{\displaystyle \scriptstyle {\text{logarithm}}}
3.2 By computer
{\displaystyle a/b/c=(a/b)/c=a/(b\times c)\;\neq \;a/(b/c)=(a\times c)/b.}
{\displaystyle {\frac {a\pm b}{c}}=(a\pm b)/c=(a/c)\pm (b/c)={\frac {a}{c}}\pm {\frac {b}{c}}.}
{\displaystyle (a+b)\times c=a\times c+b\times c}
{\displaystyle {\frac {a}{b+c}}=a/(b+c)\;\neq \;(a/b)+(a/c)={\frac {ac+ab}{bc}}.}
{\displaystyle {\frac {12}{2+4}}={\frac {12}{6}}=2,}
{\displaystyle {\frac {12}{2}}+{\frac {12}{4}}=6+3=9.}
Further information: Division sign
{\displaystyle {\frac {a}{b}}}
{\displaystyle a/b}
{\displaystyle b\backslash a}
{\displaystyle {}^{a}\!/{}_{b}}
{\displaystyle a\div b}
{\displaystyle a:b}
{\displaystyle b)a}
{\displaystyle b{\overline {)a}}}
Manual methods[edit]
Division can be calculated with an abacus.[13]
By computer[edit]
Division in different contexts[edit]
Euclidean division[edit]
Of integers[edit]
Give an approximate answer as a floating-point number. This is the approach usually taken in numerical computation.
{\displaystyle {\tfrac {26}{11}}}
{\displaystyle {\tfrac {26}{11}}=2{\tfrac {4}{11}}.}
{\displaystyle {\tfrac {26}{11}}}
{\displaystyle {\tfrac {26}{11}}=2{\mbox{ remainder }}4.}
{\displaystyle {\tfrac {26}{11}}=2.}
This is the floor function applied to case 2 or 3. It is sometimes called integer division, and denoted by "//".
Of rational numbers[edit]
{\displaystyle {p/q \over r/s}={p \over q}\times {s \over r}={ps \over qr}.}
Of real numbers[edit]
Of complex numbers[edit]
{\displaystyle {p+iq \over r+is}={(p+iq)(r-is) \over (r+is)(r-is)}={pr+qs+i(qr-ps) \over r^{2}+s^{2}}={pr+qs \over r^{2}+s^{2}}+i{qr-ps \over r^{2}+s^{2}}.}
{\displaystyle r-is}
{\displaystyle {pe^{iq} \over re^{is}}={pe^{iq}e^{-is} \over re^{is}e^{-is}}={p \over r}e^{i(q-s)}.}
Of polynomials[edit]
Of matrices[edit]
Left and right division[edit]
Pseudoinverse[edit]
{\displaystyle {\left({\frac {f}{g}}\right)}'={\frac {f'g-fg'}{g^{2}}}.}
Wikisource has the text of the 1905 New International Encyclopedia article "Division in Mathematics".
^ Division by zero may be defined in some circumstances, either by extending the real numbers to the extended real number line or to the projectively extended real line or when occurring as limit of divisions by numbers tending to 0. For example: limx→0 sin x/x = 1.[2][3]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Division_(mathematics)&oldid=1084637330"
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Transient Double-Diffusive Convection of Water Around 4°C in a Porous Cavity | J. Heat Transfer | ASME Digital Collection
Transient Double-Diffusive Convection of Water Around
4°C
in a Porous Cavity
M. Eswaramurthi,
M. Eswaramurthi
UGC DRS Centre for Fluid Dynamics,
, Coimbatore-641 046, India
P. Kandaswamy
e-mail: pgkswamy@yahoo.co.in
Eswaramurthi, M., and Kandaswamy, P. (March 17, 2009). "Transient Double-Diffusive Convection of Water Around
4°C
in a Porous Cavity." ASME. J. Heat Transfer. May 2009; 131(5): 052601. https://doi.org/10.1115/1.3000608
The buoyancy-driven transient double-diffusive convection in a square cavity filled with water-saturated porous medium is studied numerically. While the right and left side wall temperatures vary linearly from
θa
θo
θo
θb
, respectively, with height, the top and bottom walls of the cavity are thermally insulated. The species concentration levels at the right and left walls are
c1
c2
, respectively, with
c1>c2
. The Brinkman–Forchheimer extended Darcy model is considered to investigate the average heat and mass transfer rates and to study the effects of maximum density, the Grashof number, the Schmidt number, porosity, and the Darcy number on buoyancy-induced flow and heat transfer. The finite volume method with power law scheme for convection and diffusion terms is used to discretize the governing equations for momentum, energy, and concentration, which are solved by Gauss–Seidel and successive over-relaxation methods. The heat and mass transfer in the steady-state are discussed for various physical conditions. For the first time in the literature, the study of transition from stationary to steady-state shows the existence of an overshooting between the two cells and in the average Nusselt number. The results obtained in the steady-state regime are presented in the form of streamlines, isotherms, and isoconcentration lines for various values of Grashof number, Schmidt number, porosity and Darcy number, and midheight velocity profiles. It is found that the effect of maximum density is to slow down the natural convection and reduce the average heat transfer and species diffusion. The strength of convection and heat transfer rate becomes weak due to more flow restriction in the porous medium for small porosity.
double-diffusive convection, maximum density, porosity, diffusion, flow through porous media, heat transfer, mass transfer, natural convection, porosity, water
Cavities, Convection, Density, Flow (Dynamics), Fluids, Heat, Heat transfer, Mass transfer, Porosity, Porous materials, Steady state, Temperature, Transients (Dynamics), Water, Natural convection, Wall temperature, Diffusion (Physics)
Double-Diffusive Natural Convection in a Vertical Rectangular Enclosure-II: Numerical Study
Numerical Study of Double-Diffusive Natural Convection in a Square Cavity
Double-Diffusive Natural Convection in a Fluid Saturated Porous Cavity With a Freely Convecting Wall
Buoyancy-Driven Nonlinear Convection in a Square Cavity in the Presence of a Magnetic Field
Double Diffusive Nonlinear Convection in a Square Cavity
Double-Diffusive Mixed Convection in a Lid-Driven Enclosure Filled With a Fluid Saturated Porous Medium
Maximum Density Effects on Natural Convection From a Discrete Heater in a Cavity Filled With a Porous Medium
Three Dimensional Double Diffusive Convection in a Porous Cubic Enclosure Due to Opposing Gradients of Temperature and Concentration
Double Diffusive Convection in a Cubic Enclosure With Opposing Temperature and Concentration Gradients
Double Diffusion Natural Convection in a Rectangular Enclosure Filled With Binary Saturated Porous Media: The Effect of Lateral Aspect Ratio
Double Diffusion Natural Convection in an Enclosure Filled With Saturated Porous Medium and Subjected to Cross Gradients: Stably Stratified Fluid
Venkatachalappa
Effect of a Magnetic Field on Free Convection in a Rectangular Enclosure
Effect of Double Stratification on Free Convection in a Darcian Porous Medium
A Natural Convection Model for the Rate of Salt Deposition From Near-Supercritical, Aqueous Solutions
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To F. C. Donders 7 July 1874
Down. | Beckenham Kent.
My dear Prof Donders.
My son George writes to me that he has seen you, & that you have been very kind to him for which I return you my cordial thanks.—1 He tells me on your authority— of a fact which interests me in the highest degree, & which I much wish to be allowed to quote. It relates to the action of one millionth of a grain of Atropine on the eye.— Now will you be so kind whenever you can find a little leisure to tell me whether you yourself have observed this fact, or believe it on good authority— I also wish to know what proportion by weight the atropine bore to the water of solution, & how much of the solution, was applied to the eye— The reason why I am so anxious on this head is that it gives some support to certain facts repeatedly observed by me—with respect to the Action of Phosphate of Ammonia on Drosera— The
\frac{1}{4,000,000}
of a grain absorbed by a gland clearly makes the tentacle which bears this gland become inflected; & I am fully convinced that
\frac{1}{20,000,000}
of a grain of the Crystallized salt. (i e containing about
\frac{1}{3}
of its weight of water of crystallization) does the same.2 Now I am quite unhappy at the thought of having to publish such a statement It will be of great value to me to be able to give any analogous facts in support.3 The case of Drosera is all the more interesting as the absorption of the salt or any other stimulant applied to the gland causes it to transmit a motor influence to the base of the tentacle which bears the gland—
Pray forgive me for troubling you, & do not trouble yourself to answer this until your health is fully reestablished—
Pray believe me. | your’s very sincerely— | Charles Darwin.
George Howard Darwin had met Donders in Utrecht while travelling in Europe (letter from Emma Darwin to Leonard Darwin, 7 July 1874 (DAR 239.23: 1.19)).
Water of crystallisation is the water that is incorporated into the crystal lattice structure of many compounds. In Insectivorous plants, p. 155, CD noted that the crystallised phosphate of ammonia that he used contained 35.33 per cent water of crystallisation.
In Insectivorous plants, p. 173, CD cited Donders for information on the paralytic action of one millionth of a grain of highly diluted sulphate of atropine on the eye muscles.
Asks about the effect of atropine on the eye. Is interested in parallel case: influence of phosphate of ammonia on glands of Drosera.
Frans Cornelis (Franciscus Cornelius) Donders
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Macroeconomics/Macroeconomics Models/The Aggregate Expenditure Model
Learn all about the aggregate expenditure model in just a few minutes! Professor Jadrian Wooten of Penn State University walks through the aggregate expenditure model (also known as the Keynesian cross diagram), including how to calculate aggregate expenditure and consumption as a factor of real domestic product.
The aggregate expenditure model (also known as the Keynesian cross diagram) is a graph that compares the level of aggregate expenditures in an economy with that economy's real GDP.
The aggregate expenditure model is a visual representation of the relationship between aggregate expenditures and the real gross domestic product (real GDP), which is the total output of the economy adjusted for inflation. This relationship is generally shown by a simple graph, where aggregate expenditures is represented on the vertical axis and real GDP is represented on the horizontal axis. This graph is also known as a Keynesian cross diagram. On this graph, there are two observed components. One is the aggregate expenditure function, which shows how aggregate expenditures increase as real GDP increases. The aggregate expenditure function component looks like a line on the graph with a positive slope. The second component to the aggregate expenditure model is a 45-degree line that shows where aggregate expenditures (AE) equals real GDP. Graphically speaking, the coordinates to this line start at the origin (aggregate expenditures and real GDP both equal zero), and extend outward with a constant slope of 1 (this way, the coordinates on the line are always equal each other). The intersection of these two components, the aggregate expenditure function and the 45-degree line, is the equilibrium point or equilibrium GDP. The intersection of these two lines must be the equilibrium because the point of intersection is the definition of GDP (the value, or total expenditure, of all final goods and services produced within a country's borders in a given year). This equilibrium GDP is where
\text{AE}=\text{GDP}
The Keynesian cross diagram shows that aggregate demand increases as output, measured as real GDP, increases. This demonstrates the principle of effective demand. The equilibrium is the point where aggregate expenditures equal real GDP, which is show when the aggregate expenditures line crosses the 45 degree line.
With regard to equilibrium GDP, the aggregate expenditures model shows that an economy will move toward equilibrium GDP over time, as changes to demand influence output. Recall that the equilibrium GDP is where aggregate expenditures equals real GDP. An economy can be above or below this equilibrium level, but it will have a natural tendency to move back to equilibrium. For example, when output is greater than planned aggregate expenditures, the economy is at a point on the aggregate expenditures curve to the right of the equilibrium point. This situation is known as an unplanned inventory accumulation, an excess of unsold goods caused by an unplanned event. Inventories of unsold products build up, which results in businesses reducing their orders of goods. This, in turn, reduces overall output, pushing the economy to the left on the model, down the AE curve, until it reaches the equilibrium GDP. Similarly, when the output is less than planned aggregate expenditures, the economy is at a point on the AE curve to the left of the equilibrium point. This situation is called a unplanned negative inventory, or inventory rundown, which is a shortage of stored materials or goods caused by increased demand. Inventories of unsold product begin to decrease, causing businesses to increase their orders of goods. This situation then increases overall output, pushing the economy to the right on the cross diagram, or up the AE curve. This again moves the economy toward the equilibrium GDP, as illustrated by the aggregate expenditures model. In these ways, the actions of businesses to bring their inventories to equilibrium drive the entire economy toward equilibrium as well.
This graph demonstrates the relationship between expenditure and income, and shows that either side of the equilibrium point, inventories will either decrease or accumulate.
The equation for the aggregate expenditure function is
\text{AD}=\text{C}+\text{I}+\text{G}+(\text{X}-\text{M})
, where C is consumption, I is investment, G is government spending, and
(\text{X} - \text{M})
is net exports (or exports minus imports). The aggregate expenditure function shows how aggregate expenditures increase as real GDP increases. For example, consider an economy (following figures are given in billions of dollars). On a graph, the 45-degree line is a set of points where AE is equal to real GDP, so AE is
Y
. Suppose C is
150 + 0.85(Y-\text{T})
. T is taxes, equal to
0.25Y
. Investment, or I, is 500. Government spending, G, is 850. Exports are 500, and imports are
0.1Y
, so net exports are
500 - 0.1Y
\begin{aligned}Y&=150+0.85(Y-0.25Y)+500+850+(500-0.1 Y)\\ Y&=150+0.85 Y-0.2125 Y+500+850+500-0.1 Y\\ Y&=2\text{,}000+0.5375 Y\\ Y-0.5375 Y&=200\\0.4625 Y&=2\text{,}000\\Y&=2\text{,}000/0.4625\\Y&=4\text{,}324.324\end{aligned}
Investment, government spending, and net exports are not functions of real GDP in the current year and are expressed as constants in the aggregate expenditure function. For example, investment spending tends to be more forward-looking, based on future expectations and not dependent on current real GDP. Government spending is based on congressional decision-making and not dependent on real GDP. Exports are shaped by other nations' real GDP and demand for foreign goods. This is not to say that these three components do not fluctuate, but they are not a function of real GDP and are thus constant in the model. However, consumption is a function of real domestic production. The consumption function is generally expressed as the following linear equation:
\text{C}=a+\text{MPC}(\text{Y}-\text{T})
a
is the baseline level of consumption (
y
-intercept), MPC is marginal propensity to consume,
Y
is real GDP, and T is taxes. Consumption has a positive relationship with real GDP and increases at a rate consistent with MPC.
The intersection of the aggregate expenditure function and the 45° line (where
\text{AE}=\text{GDP}
) is the equilibrium point where no incentive exists to shift away from that outcome. The optimal equilibrium point is one that occurs when the economy is at full employment GDP. In this case, unemployment is low and there is no recession. If the equilibrium point occurs at a level of output lower than full employment GDP, a recessionary gap exists. If the equilibrium point occurs at a level of output above full-employment GDP, then an inflationary gap occurs.
<Spending and Tax Multipliers>Effects of Government Spending and Taxes
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ग्रुप सिद्धान्त - Wikipedia
ग्रुप सिद्धान्त गणितयागु छगु ख्यः ख। थुकिगु छ्येलेज्या भौतिकशास्त्र व रसायनशास्त्रय् जुइ।
पुचः यागु छ्येलेज्या गणितय् यक्व थासे अप्वयाना इन्टर्नल सिमेट्रि क्याप्चर यायेत अटोमर्फिज ग्रुप यागु रुपे जुइ। An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group.
In Galois theory, which is the historical origin of the group concept, one uses groups to describe the symmetries of the equations satisfied by the solutions to a polynomial equation. The solvable groups are so-named because of their prominent role in this theory.
Abelian groups underlie several other structures that are studied in abstract algebra, such as rings, fields, and modules.
In algebraic topology, groups are used to describe invariants of topological spaces (the name of the torsion subgroup of an infinite group shows the legacy of this field of endeavor). They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. Examples include the fundamental group, homology groups and cohomology groups.
The concept of the Lie group (named after mathematician Sophus Lie) is important in the study of differential equations and manifolds; they combine analysis and group theory and are therefore the proper objects for describing symmetries of analytical structures. Analysis on these and other groups is called harmonic analysis.
An understanding of group theory is also important in physics and chemistry and material science. In chemistry, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories.
Physics examples: Standard Model, Gauge theory
२ ग्रुप सिद्धान्त विचातः
२.१ ग्रुपयागु अर्थ
२.२ Order of groups and elements
२.४ Special classes of groups
२.५ Operations on groups
There are three historical roots of group theory: the theory of algebraic equations, number theory and geometry. Euler, Gauss, Lagrange, Abel and French mathematician Galois were early researchers in the field of group theory. Galois is honored as the first mathematician linking group theory and field theory, with the theory that is now called Galois theory.[१]
An early source occurs in the problem of forming an
{\displaystyle m}
th-degree equation having as its roots m of the roots of a give{\displaystyle n}
th-degree equation (
{\displaystyle m<n}
). For simple cases the problem goes back to Hudde (1659). Saunderson (1740) noted that the determination of the quadratic factors of a biquadratic expression necessarily leads to a sextic equation, and Le Sœur (1748) and Waring (1762 to 1782) still further elaborated the idea.[१]
A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. He discovered that the roots of all resolvents (résolvantes, réduites) which he examined are rational functions of the roots of the respective equations. To study the properties of these functions he invented a Calcul des Combinaisons. The contemporary work of Vandermonde (1770) also foreshadowed the coming theory.[१]
Ruffini (1799) attempted a proof of the impossibility of solving the quintic and higher equations. Ruffini distinguished what are now called intransitive and transitive, and imprimitive and primitive groups, and (1801) uses the group of an equation under the name l'assieme delle permutazioni. He also published a letter from Abbati to himself, in which the group idea is prominent.[१]
Galois found that if
{\displaystyle r_{1},r_{2},\ldots ,r_{n}}
{\displaystyle n}
roots of an equation, there is always a group of permutations of the
{\displaystyle r}
's such that (1) every function of the roots invariable by the substitutions of the group is rationally known, and (2), conversely, every rationally determinable function of the roots is invariant under the substitutions of the group. Galois also contributed to the theory of modular equations and to that of elliptic functions. His first publication on group theory was made at the age of eighteen (1829), but his contributions attracted little attention until the publication of his collected papers in 1846 (Liouville, Vol. XI).[१]
Arthur Cayley and Augustin Louis Cauchy were among the first to appreciate the importance of the theory, and to the latter especially are due a number of important theorems. The subject was popularised by Serret, who devoted section IV of his algebra to the theory; by Camille Jordan, whose Traité des Substitutions is a classic; and to Eugen Netto (1882), whose Theory of Substitutions and its Applications to Algebra was translated into English by Cole (1892). Other group theorists of the nineteenth century were Bertrand, Charles Hermite, Frobenius, Leopold Kronecker, and Emile Mathieu.[१]
It was Walther von Dyck who, in 1882, gave the modern definition of a group.
The study of what are now called Lie groups, and their discrete subgroups, as transformation groups, started systematically in 1884 with Sophus Lie; followed by work of Killing, Study, Schur, Maurer, and Cartan. The discontinuous (discrete group) theory was built up by Felix Klein, Lie, Poincaré, and Charles Émile Picard, in connection in particular with modular forms and monodromy.
The classification of finite simple groups is a vast body of work from the mid 20th century, which is thought to classify all the finite simple groups.
Other important mathematicians in this subject area include Emil Artin, Emmy Noether, Sylow, and many others.
ग्रुप सिद्धान्त विचातः[सम्पादन]
ग्रुपयागु अर्थ[सम्पादन]
मू पौ: पुचः (गणित)
A group (G, *) is a set G with a binary operation * : G × G → G (one that assigns each ordered pair (a,b) in G an element in G denoted by a*b) that satisfies the following 3 axioms:
Inverse element: For each a in G, there is an element b in G such that a * b = b * a = e, where e is an identity element.
Order of groups and elements[सम्पादन]
The order of a group G is the number of elements in the set G. If the order is not finite, then the group is an infinite group.
The order of an element a in a group G is the least positive integer n such that an=e, where an is multiplication of a by itself n times.
For finite groups, it can be shown that the order of every element in the group must divide the order of the group.
Subgroups[सम्पादन]
A set H is a subgroup of a group G if it is a subset of G and is a group using the operation defined on G. In other words, H is a subgroup of (G, *) if the restriction of * to H is a group operation on H.
If G is a finite group, the order of H divides the order of G.
A subgroup H is a normal subgroup of G if for all h in H and g in G, ghg-1 is also in H. An alternate definition is that a subgroup is normal if its left and right cosets coincide.
Normal subgroups are useful because they can be used to create quotient groups.
Special classes of groups[सम्पादन]
A group is abelian (or commutative) if the operation is commutative (that is, for all a, b in G, a * b = b * a). A non-abelian group is a group that is not abelian. The term "abelian" is named after the mathematician Niels Abel.
A cyclic group is a group that is generated by a single element.
A simple group is a group that has no nontrivial normal subgroups.
A solvable group , or soluble group, is a group that has a normal series whose quotient groups are all abelian. The fact that S5, the symmetric group in 5 elements, is not solvable proves that some quintic polynomials cannot be solved by radicals.
Operations on groups[सम्पादन]
A homomorphism is a map between two groups that preserves the structure imposed by the operator. If the map is bijective, then it is an isomorphism. An isomorphism from a group to itself is an automorphism. The set of all automorphisms of a group is a group called the automorphism group. The kernel of a homomorphism is a normal subgroup of the group.
A group action is a map involving a group and a set, where each element in the group defines a bijective map on a set. Group actions are used to prove the Sylow theorems and to prove that the center of a p-group is nontrivial.
Some useful theorems[सम्पादन]
Some basic results in elementary group theory
Lagrange's theorem: if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G.
Cayley's Theorem: every group G is isomorphic to a subgroup of the symmetric group on G.
Sylow theorems: perhaps the most useful of the group theorems. Among them, that if pn (and p prime) divides the order of a finite group G, then there exists a subgroup of order pn.
The Butterfly lemma is a technical result on the lattice of subgroups of a group.
The Fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
Jordan-Hölder theorem: any two composition series of a given group are equivalent.
Krull-Schmidt theorem: a group G, subjected to certain finiteness conditions of chains of subgroups, can be uniquely written as a finite product of indecomposable subgroups.
Burnside's lemma: the number of orbits of a group action on a set equals the average number of points fixed by each element of the group
Miscellany[सम्पादन]
James Newman summarized group theory as follows:
The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing.
One application of group theory is in musical set theory.
Group theory is also very important to the field of chemistry, where it is used to assign symmetries to molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy and Infrared spectroscopy), and to construct molecular orbitals.
In Philosophy, Ernst Cassirer related the theory of group to the theory of perception as described by Gestalt Psychology; Perceptual Constancy is taken to be analogous to the invariants of group theory.
↑ १.० १.१ १.२ १.३ १.४ १.५ Smith, D. E., History of Modern Mathematics, Project Gutenberg, 1906.
Rotman, Joseph (1994). An introduction to the theory of groups. New York: Springer-Verlag. ISBN 0-387-94285-8. A standard modern reference.
Scott, W. R. [1964] (1987). Group Theory. New York: Dover. ISBN 0-486-65377-3. An inexpensive and fairly readable textbook (somewhat outdated in emphasis, style, and notation).
Livio, M. (2005). The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry. Simon & Schuster. ISBN 0-7432-5820-7. Pleasant to read book that explains the importance of group theory and how its symmetries lead to parallel symmetries in the world of physics and other sciences. Really helps congeal the importance of group theory as a practical tool).
एब्स्ट्र्याक्ट ग्रुप कन्सेप्टयागु इतिहास
विकिमिडिया मंका य् थ्व विषय नाप स्वापु दुगु मिडिया दु: Group theory
Retrieved from "https://new.wikipedia.org/w/index.php?title=ग्रुप_सिद्धान्त&oldid=825173"
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EMRISurrogate
Python code to evaluate gravitational waveform surrogate models trained on waveform data generated by point-particle black hole perturbation theory.
The EMRISurrogate package provides access to a surrogate gravitational waveform model. Currently, this package supports one model, EMRISur1dq1e4, for non-spinning black hole binary systems with mass-ratios varying from 3 to $10^4$. The “raw” surrogate model is trained on waveform data generated by point-particle black hole perturbation theory (ppBHPT) and defined as,
\begin{align} h_{\tt S}(t, \theta, \phi) = \sum^{\infty}_{\ell=2} \sum_{m=-\ell}^{\ell} h_{\tt S}^{\ell,m}(t) ~^{-2}Y_{\ell m}(\theta, \phi) \,, \end{align}
where $^{-2}Y_{\ell m}$ are the spin$=-2$ weighted spherical harmonics. The surrogate model provides fast evaluations for the modes, $h_{\tt S}^{\ell,m}$. Available modes are,
\begin{align} (\ell,m) = [(2,2), (2,1), (3,3), (3,2), (3,1), (4,4), (4,3), (4,2), (5,5), (5,4), (5,3)] \,, \end{align}
while the $m<0$ modes $h^{\ell, -m} = (-1)^{\ell} h^{\ell,m}{}^*$ can be deduced from the $m>0$ modes.
By default, the EMRISurrogate package will return rescaled waveform modes,
\begin{align} h^{\ell,m}_{\tt S, \alpha}(t ; q)= {\alpha} h^{\ell,m}_{\tt S}\left( t \alpha;q \right) \,, \end{align}
where the total mass rescaling parameter, $\alpha(q)$, is tuned to NR simulations. We expect most users of this package will want the NR-tuned model. To generate point-particle perturbation theory waveforms without any tuning to NR set $\alpha = 1$ in the function slog_surrogate.
Model details can be found in paper 1 (see citations below).
The latest development version will always be available from the project git repository:
git clone https://github.com/BlackHolePerturbationToolkit/EMRISurrogate.git
Requirements, Installation, and Usage
For details on these topics check out the README and the Jupyter notebook.
Known bugs are recorded in the project bug tracker
This code is distributed under the MIT License. Details can be found in the LICENSE file.
Scott Field, Tousif Islam, Gaurav Khanna, Nur Rifat, Vijay Varma
If you make use of any module from the Toolkit in your research please acknowledge using “This work makes use of the Black Hole Perturbation Toolkit”. If you make use of the EMRI surrogate model, EMRISur1dq1e4, please cite Paper 1:
@article{rifat2019surrogate,
title={A Surrogate Model for Gravitational Wave Signals from Comparable-to Large-Mass-Ratio Black Hole Binaries},
author={Rifat, Nur EM and Field, Scott E and Khanna, Gaurav and Varma, Vijay},
EMRISurrogate is maintained by BlackHolePerturbationToolkit. See behind the scenes: View the Toolkit on GitHub
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Polish notation - Wikipedia
(Redirected from Prefix notation)
Type of mathematics notation
This article is about a prefix notation in mathematics and computer sciences. For the similarly named logic, see Łukasiewicz logic.
("Reverse Polish")
("Polish")
Polish notation (PN), also known as normal Polish notation (NPN),[1] Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators precede their operands, in contrast to the more common infix notation, in which operators are placed between operands, as well as reverse Polish notation (RPN), in which operators follow their operands. It does not need any parentheses as long as each operator has a fixed number of operands. The description "Polish" refers to the nationality of logician Jan Łukasiewicz,[2] who invented Polish notation in 1924.[3][4]
The term Polish notation is sometimes taken (as the opposite of infix notation) to also include reverse Polish notation.[5]
When Polish notation is used as a syntax for mathematical expressions by programming language interpreters, it is readily parsed into abstract syntax trees and can, in fact, define a one-to-one representation for the same. Because of this, Lisp (see below) and related programming languages define their entire syntax in prefix notation (and others use postfix notation).
A quotation from a paper by Jan Łukasiewicz, Remarks on Nicod's Axiom and on "Generalizing Deduction", page 180, states how the notation was invented:
I came upon the idea of a parenthesis-free notation in 1924. I used that notation for the first time in my article Łukasiewicz(1), p. 610, footnote.
The reference cited by Łukasiewicz is apparently a lithographed report in Polish. The referring paper by Łukasiewicz Remarks on Nicod's Axiom and on "Generalizing Deduction" was reviewed by Henry A. Pogorzelski in the Journal of Symbolic Logic in 1965.[6] Heinrich Behmann, editor in 1924 of the article of Moses Schönfinkel,[7] already had the idea of eliminating parentheses in logic formulas.
Alonzo Church mentions this notation in his classic book on mathematical logic as worthy of remark in notational systems even contrasted to Alfred Whitehead and Bertrand Russell's logical notational exposition and work in Principia Mathematica.[8]
In Łukasiewicz's 1951 book, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, he mentions that the principle of his notation was to write the functors before the arguments to avoid brackets and that he had employed his notation in his logical papers since 1929.[9] He then goes on to cite, as an example, a 1930 paper he wrote with Alfred Tarski on the sentential calculus.[10]
While no longer used much in logic,[11] Polish notation has since found a place in computer science.
The expression for adding the numbers 1 and 2 is written in Polish notation as + 1 2 (pre-fix), rather than as 1 + 2 (in-fix). In more complex expressions, the operators still precede their operands, but the operands may themselves be expressions including again operators and their operands. For instance, the expression that would be written in conventional infix notation as
Assuming a given arity of all involved operators (here the "−" denotes the binary operation of subtraction, not the unary function of sign-change), any well formed prefix representation thereof is unambiguous, and brackets within the prefix expression are unnecessary. As such, the above expression can be further simplified to
The processing of the product is deferred until its two operands are available (i.e., 5 minus 6, and 7). As with any notation, the innermost expressions are evaluated first, but in Polish notation this "innermost-ness" can be conveyed by the sequence of operators and operands rather than by bracketing.
In the conventional infix notation, parentheses are required to override the standard precedence rules, since, referring to the above example, moving them
or removing them
changes the meaning and the result of the expression. This version is written in Polish notation as
− 5 × 6 7.
When dealing with non-commutative operations, like division or subtraction, it is necessary to coordinate the sequential arrangement of the operands with the definition of how the operator takes its arguments, i.e., from left to right. For example, ÷ 10 5, with 10 left to 5, has the meaning of 10 ÷ 5 (read as "divide 10 by 5"), or - 7 6, with 7 left to 6, has the meaning of 7 - 6 (read as "subtract from 7 the operand 6").
Evaluation algorithm[edit]
Prefix/postfix notation is especially popular for its innate ability to express the intended order of operations without the need for parentheses and other precedence rules, as are usually employed with infix notation. Instead, the notation uniquely indicates which operator to evaluate first. The operators are assumed to have a fixed arity each, and all necessary operands are assumed to be explicitly given. A valid prefix expression always starts with an operator and ends with an operand. Evaluation can either proceed from left to right, or in the opposite direction. Starting at the left, the input string, consisting of tokens denoting operators or operands, is pushed token for token on a stack, until the top entries of the stack contain the number of operands that fits to the top most operator (immediately beneath). This group of tokens at the stacktop (the last stacked operator and the according number of operands) is replaced by the result of executing the operator on these/this operand(s). Then the processing of the input continues in this manner. The rightmost operand in a valid prefix expression thus empties the stack, except for the result of evaluating the whole expression. When starting at the right, the pushing of tokens is performed similarly, just the evaluation is triggered by an operator, finding the appropriate number of operands that fits its arity already at the stacktop. Now the leftmost token of a valid prefix expression must be an operator, fitting to the number of operands in the stack, which again yields the result. As can be seen from the description, a push-down store with no capability of arbitrary stack inspection suffices to implement this parsing.
The above sketched stack manipulation works—with mirrored input—also for expressions in reverse Polish notation.
Polish notation for logic[edit]
The table below shows the core of Jan Łukasiewicz's notation for sentential logic.[12] Some letters in the Polish notation table stand for particular words in Polish, as shown:
{\displaystyle \neg \varphi }
{\displaystyle \mathrm {N} \varphi }
{\displaystyle \varphi \land \psi }
{\displaystyle \mathrm {K} \varphi \psi }
koniunkcja
{\displaystyle \varphi \lor \psi }
{\displaystyle \mathrm {A} \varphi \psi }
alternatywa
{\displaystyle \varphi \to \psi }
{\displaystyle \mathrm {C} \varphi \psi }
{\displaystyle \varphi \leftrightarrow \psi }
{\displaystyle \mathrm {E} \varphi \psi }
ekwiwalencja
{\displaystyle \bot }
{\displaystyle \mathrm {O} }
fałsz
Sheffer stroke
{\displaystyle \varphi \mid \psi }
{\displaystyle \mathrm {D} \varphi \psi }
{\displaystyle \Diamond \varphi }
{\displaystyle \mathrm {M} \varphi }
możliwość
{\displaystyle \Box \varphi }
{\displaystyle \mathrm {L} \varphi }
konieczność
Universal quantifier
{\displaystyle \forall p\,\varphi }
{\displaystyle \Pi p\,\varphi }
kwantyfikator ogólny
{\displaystyle \exists p\,\varphi }
{\displaystyle \Sigma p\,\varphi }
kwantyfikator szczegółowy
Bocheński introduced a system of Polish notation that names all 16 binary connectives of classical propositional logic. For classical propositional logic, it is a compatible extension of the notation of Łukasiewicz. But the notations are incompatible in the sense that Bocheński uses L and M (for nonimplication and converse nonimplication) in propositional logic and Łukasiewicz uses L and M in modal logic.[13]
Prefix notation has seen wide application in Lisp S-expressions, where the brackets are required since the operators in the language are themselves data (first-class functions). Lisp functions may also be variadic. The Tcl programming language, much like Lisp also uses Polish notation through the mathop library. The Ambi[14] programming language uses Polish notation for arithmetic operations and program construction. LDAP filter syntax uses Polish prefix notation.[15]
Postfix notation is used in many stack-oriented programming languages like PostScript and Forth. CoffeeScript syntax also allows functions to be called using prefix notation, while still supporting the unary postfix syntax common in other languages.
Polish notation, usually in postfix form, is the chosen notation of certain calculators, notably from Hewlett-Packard.[16] At a lower level, postfix operators are used by some stack machines such as the Burroughs large systems.
Polish School of Mathematics
^ Jorke, Günter; Lampe, Bernhard; Wengel, Norbert (1989). Arithmetische Algorithmen der Mikrorechentechnik [Arithmetic algorithms in microcomputers] (in German) (1 ed.). Berlin, Germany: VEB Verlag Technik. ISBN 3341005153. EAN 9783341005156. MPN 5539165. License 201.370/4/89. Retrieved 2015-12-01.
^ Łukasiewicz, Jan (1957). Aristotle's Syllogistic from the Standpoint of Modern Formal Logic. Oxford University Press. (Reprinted by Garland Publishing in 1987. ISBN 0-8240-6924-2)
^ Hamblin, Charles Leonard (1962). "Translation to and from Polish notation". Computer Journal. 5 (3): 210–213. doi:10.1093/comjnl/5.3.210.
^ Ball, John A. (1978). Algorithms for RPN calculators (1 ed.). Cambridge, Massachusetts, USA: Wiley-Interscience, John Wiley & Sons, Inc. ISBN 0-471-03070-8.
^ Main, Michael (2006). Data structures and other objects using Java (3rd ed.). Pearson PLC Addison-Wesley. p. 334. ISBN 978-0-321-37525-4.
^ Pogorzelski, Henry A., "Reviewed work(s): Remarks on Nicod's Axiom and on "Generalizing Deduction" by Jan Łukasiewicz; Jerzy Słupecki; Państwowe Wydawnictwo Naukowe", The Journal of Symbolic Logic, Vol. 30, No. 3 (September 1965), pp. 376–377. The original paper by Łukasiewicz was published in Warsaw in 1961 in a volume edited by Jerzy Słupecki.
^ "Über die Bausteine der mathematischen Logik", Mathematische Annalen 92, pages 305-316. Translated by Stefan Bauer-Mengelberg as "On the building blocks of mathematical logic" in Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879-1931. Harvard University Press: 355-66.
^ Church, Alonzo (1944). Introduction to Mathematical Logic. Princeton, New Jersey, USA: Princeton University Press. p. 38. […] Worthy of remark is the parenthesis-free notation of Jan Łukasiewicz. In this the letters N, A, C, E, K are used in the roles of negation, disjunction, implication, equivalence, conjunction respectively. […]
^ Łukasiewicz, (1951) Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, Chapter IV "Aristotle's System in Symbolic Form" (section on "Explanation of the Symbolism"), p. 78 and on.
^ Łukasiewicz, Jan; Tarski, Alfred, "Untersuchungen über den Aussagenkalkül" ["Investigations into the sentential calculus"], Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Vol. 23 (1930) Cl. III, pp. 31–32.
^ Martínez Nava, Xóchitl (2011-06-01), "Mhy bib I fail logic? Dyslexia in the teaching of logic", in Blackburn, Patrick; van Ditmarsch, Hans; Manzano, Maria; Soler-Toscano, Fernando (eds.), Tools for Teaching Logic: Third International Congress, TICTTL 2011, Salamanca, Spain, June 1-4, 2011, Proceedings, Lecture Notes in Artificial Intelligence, vol. 6680, Springer Nature, pp. 162–169, doi:10.1007/978-3-642-21350-2_19, ISBN 9783642213496, […] Polish or prefix notation has come to disuse given the difficulty that using it implies. […]
^ Craig, Edward (1998), Routledge Encyclopedia of Philosophy, Volume 8, Taylor & Francis, p. 496, ISBN 9780415073103 .
^ Bocheński, Józef Maria (1959). A Precis of Mathematical Logic, translated by Otto Bird from the French and German editions, D. Reidel: Dordrecht, Holland.
^ "LDAP Filter Syntax".
^ "HP calculators | HP 35s RPN Mode" (PDF). Hewlett-Packard.
Łukasiewicz, Jan (1957). Aristotle's Syllogistic from the Standpoint of Modern Formal Logic. Oxford University Press.
Łukasiewicz, Jan (1930). "Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls" [Philosophical Remarks on Many-Valued Systems of Propositional Logics]. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie (in German). 23: 51–77. Translated by H. Weber in Storrs McCall, Polish Logic 1920-1939, Clarendon Press: Oxford (1967).
Media related to Polish notation (mathematics) at Wikimedia Commons
Retrieved from "https://en.wikipedia.org/w/index.php?title=Polish_notation&oldid=1061596821"
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a child bought his old book in 20% loss and sold in Rs 64 What is the cost price - Maths - Rational Numbers - 9675711 | Meritnation.com
a child bought his old book in 20% loss and sold in Rs.64. What is the cost price of the book
\mathrm{Let} \mathrm{the} \mathrm{cost} \mathrm{price} \mathrm{of} \mathrm{book} \mathrm{be} \mathrm{x}.\phantom{\rule{0ex}{0ex}}\mathrm{Loss}=\mathrm{C}.\mathrm{P}-\mathrm{S}.\mathrm{P}\phantom{\rule{0ex}{0ex}}=\mathrm{Rs}. \left(\mathrm{x}-64\right)\phantom{\rule{0ex}{0ex}}\mathrm{We} \mathrm{know}\phantom{\rule{0ex}{0ex}}\mathrm{Loss} \mathrm{Percent}=\frac{\mathrm{Loss}}{\mathrm{C}.\mathrm{P}}×100\phantom{\rule{0ex}{0ex}}20=\frac{\mathrm{x}-64}{\mathrm{x}}×100\phantom{\rule{0ex}{0ex}}20\mathrm{x}=100\mathrm{x}-6400\phantom{\rule{0ex}{0ex}}-80\mathrm{x}=-6400\phantom{\rule{0ex}{0ex}}\mathrm{x}=\frac{6400}{80}=80\phantom{\rule{0ex}{0ex}}\mathrm{Cost} \mathrm{Price} \mathrm{of} \mathrm{the} \mathrm{book} \mathrm{is} \mathrm{Rs}. 80 .
Sidhartha answered this
Syed Muhammad answered this
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Brachistochrone on a 1D Curved Surface Using Optimal Control | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
, St. Paul, MN 55105-1079
e-mail: mphennessey@stthomas.edu
e-mail: c9shakiban@stthomas.edu
Michael P. Hennessey Associate Professor
Cheri Shakiban Professor
Hennessey, M. P., and Shakiban, C. (April 28, 2010). "Brachistochrone on a 1D Curved Surface Using Optimal Control." ASME. J. Dyn. Sys., Meas., Control. May 2010; 132(3): 034505. https://doi.org/10.1115/1.4001277
The brachistochrone for a steerable particle moving on a 1D curved surface in a gravity field is solved using an optimal control formulation with state feedback. The process begins with a derivation of a fourth-order open-loop plant model with the system input being the body yaw rate. Solving for the minimum-time control law entails introducing four costates and solving the Euler–Lagrange equations, with the Hamiltonian being stationary with respect to the control. Also, since the system is autonomous, the Hamiltonian must be zero. A two-point boundary value problem results with a transversality condition, and its solution requires iteration of the initial bearing angle so the integrated trajectory runs through the final point. For this choice of control, the Legendre–Clebsch necessary condition is not satisfied. However, the
k=1
generalized Legendre–Clebsch necessary condition from singular control theory is satisfied for all numerical simulations performed, and optimality is assured. Simulations in MATLAB® exercise the theory developed and illustrate application such as to ski racing and minimizing travel time over either a concave or undulating surface when starting from rest. Lastly, a control law singularity in particle speed is overcome numerically.
boundary-value problems, calculus, curve fitting, iterative methods, mechanics, open loop systems, optimal control, state feedback, brachistochrone, mechanics, optimal control, calculus of variations, singular control theory, Legendre–Clebsch condition, numerical simulation, skiing
Control theory, Optimal control, Simulation, Yaw, Bearings, State feedback, Boundary-value problems, Computer simulation
Jacobi Bernoulli solutio problematum fraternorum
Brachistochrone With Coulomb Friction
, 1744, Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Guadentes sive Solutio Problematis Isoperimetrici Latissimo Sensu Accepti, Lausanne, Geneva.
A Physical Theory of Alpine Ski Racing
On Singular Time-Optimal Control Along Specific Paths
Applied Optimal Control: Optimization, Estimation, and Control
Singular Extremals, Topics in Optimization
Brachistochrone on a Surface With Coulomb Friction
The Mathematica® Book
Mastering MATLAB7
Longer Line=Shorter Time
,” The Professional Skier.
Control of Linear Time-Varying Systems Using Forward Riccati Equation
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Shapiro_time_delay Knowpia
The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar-system tests of general relativity. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than they would if the mass of the object were not present. The time delay is caused by spacetime dilation, which increases the time it takes light to travel a given distance from the perspective of an outside observer. In a 1964 article entitled Fourth Test of General Relativity, astrophysicist Irwin Shapiro wrote:[1]
Because, according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path, these time delays should thereby be increased by almost 2×10−4 sec when the radar pulses pass near the sun. Such a change, equivalent to 60 km in distance, could now be measured over the required path length to within about 5 to 10% with presently obtainable equipment.
Throughout this article discussing the time delay, Shapiro uses c as the speed of light and calculates the time delay of the passage of light waves or rays over finite coordinate distance according to a Schwarzschild solution to the Einstein field equations.
The time delay effect was first predicted in 1964, by Irwin Shapiro. Shapiro proposed an observational test of his prediction: bounce radar beams off the surface of Venus and Mercury and measure the round-trip travel time. When the Earth, Sun, and Venus are most favorably aligned, Shapiro showed that the expected time delay, due to the presence of the Sun, of a radar signal traveling from the Earth to Venus and back, would be about 200 microseconds,[1] well within the limitations of 1960s-era technology.
The first tests, performed in 1966 and 1967 using the MIT Haystack radar antenna, were successful, matching the predicted amount of time delay.[2] The experiments have been repeated many times since then, with increasing accuracy.
Calculating time delayEdit
Left: unperturbed lightrays in a flat spacetime, right: Shapiro-delayed and deflected lightrays in the vicinity of a gravitating mass (click to start the animation)
In a nearly static gravitational field of moderate strength (say, of stars and planets, but not one of a black hole or close binary system of neutron stars) the effect may be considered as a special case of gravitational time dilation. The measured elapsed time of a light signal in a gravitational field is longer than it would be without the field, and for moderate-strength nearly static fields the difference is directly proportional to the classical gravitational potential, precisely as given by standard gravitational time dilation formulas.
Time delay due to light traveling around a single massEdit
Shapiro's original formulation was derived from the Schwarzschild solution and included terms to the first order in solar mass (M) for a proposed Earth-based radar pulse bouncing off an inner planet and returning passing close to the Sun:[1]
{\displaystyle \Delta t\approx {\frac {4GM}{c^{3}}}\left(\ln \left[{\frac {x_{p}+(x_{p}^{2}+d^{2})^{1/2}}{-x_{e}+(x_{e}^{2}+d^{2})^{1/2}}}\right]-{\frac {1}{2}}\left[{\frac {x_{p}}{(x_{p}^{2}+d^{2})^{1/2}}}+{\frac {x_{e}}{(x_{e}^{2}+d^{2})^{1/2}}}\right]\right)+{\mathcal {O}}\left({\frac {G^{2}M^{2}}{c^{5}d}}\right),}
where d is the distance of closest approach of the radar wave to the center of the Sun, xe is the distance along the line of flight from the Earth-based antenna to the point of closest approach to the Sun, and xp represents the distance along the path from this point to the planet. The right-hand side of this equation is primarily due to the variable speed of the light ray; the contribution from the change in path, being of second order in M, is negligible. O is the Landau symbol of order of error.
For a signal going around a massive object, the time delay can be calculated as the following:[citation needed]
{\displaystyle \Delta t=-{\frac {2GM}{c^{3}}}\ln(1-\mathbf {R} \cdot \mathbf {x} ).}
Here R is the unit vector pointing from the observer to the source, and x is the unit vector pointing from the observer to the gravitating mass M. The dot denotes the usual Euclidean dot product.
Using Δx = cΔt, this formula can also be written as
{\displaystyle \Delta x=-R_{s}\ln(1-\mathbf {R} \cdot \mathbf {x} ),}
which is a fictive extra distance the light has to travel. Here
{\displaystyle R_{s}={\frac {2GM}{c^{2}}}}
is the Schwarzschild radius.
In PPN parameters,
{\displaystyle \Delta t=-(1+\gamma ){\frac {R_{s}}{2c}}\ln(1-\mathbf {R} \cdot \mathbf {x} ),}
which is twice the Newtonian prediction (with
{\displaystyle \gamma =0}
The doubling of the Shapiro factor can be explained by the fact that there is not only the gravitational time dilation, but also the stretching of space, both of which contribute equally in general relativity for the time delay as they also do for the deflection of light.
{\displaystyle \tau =t{\sqrt {1-{\tfrac {R_{s}}{r}}}}}
{\displaystyle c'=c{\sqrt {1-{\tfrac {R_{s}}{r}}}}}
{\displaystyle s=\tau c'=ct\left(1-{\tfrac {R_{s}}{r}}\right)}
Interplanetary probesEdit
Shapiro delay must be considered along with ranging data when trying to accurately determine the distance to interplanetary probes such as the Voyager and Pioneer spacecraft.
Shapiro delay of neutrinos and gravitational wavesEdit
From the nearly simultaneous observations of neutrinos and photons from SN 1987A, the Shapiro delay for high-energy neutrinos must be the same as that for photons to within 10%, consistent with recent estimates of the neutrino mass, which imply that those neutrinos were moving at very close to the speed of light. After the direct detection of gravitational waves in 2016, the one-way Shapiro delay was calculated by two groups and is about 1800 days. In general relativity and other metric theories of gravity, though, the Shapiro delay for gravitational waves is expected to be the same as that for light and neutrinos. However, in theories such as tensor–vector–scalar gravity and other modified GR theories, which reproduce Milgrom's law and avoid the need for dark matter, the Shapiro delay for gravitational waves is much smaller than that for neutrinos or photons. The observed 1.7-second difference in arrival times seen between gravitational wave and gamma ray arrivals from neutron star merger GW170817 was far less than the estimated Shapiro delay of about 1000 days. This rules out a class of modified models of gravity that dispense with the need for dark matter.[4]
Gravitational redshift and blueshift
Gravitomagnetic time delay
^ a b c Irwin I. Shapiro (1964). "Fourth Test of General Relativity". Physical Review Letters. 13 (26): 789–791. Bibcode:1964PhRvL..13..789S. doi:10.1103/PhysRevLett.13.789.
^ Irwin I. Shapiro; Gordon H. Pettengill; Michael E. Ash; Melvin L. Stone; et al. (1968). "Fourth Test of General Relativity: Preliminary Results". Physical Review Letters. 20 (22): 1265–1269. Bibcode:1968PhRvL..20.1265S. doi:10.1103/PhysRevLett.20.1265.
^ Elena V. Pitjeva:Tests of General Relativity from observations of planets and spacecraft (slides undated).
^ Sibel Boran; et al. (2018). "GW170817 Falsifies Dark Matter Emulators". Phys. Rev. D. 97 (4): 041501. arXiv:1710.06168. Bibcode:2018PhRvD..97d1501B. doi:10.1103/PhysRevD.97.041501. S2CID 119468128.
van Straten W; Bailes M; Britton M; et al. (12 July 2001). "Boost for General Relativity". Nature. 412 (6843): 158–60. arXiv:astro-ph/0108254. Bibcode:2001Natur.412..158V. doi:10.1038/35084015. hdl:1959.3/1820. PMID 11449265. S2CID 4363384.
d'Inverno, Ray (1992). Introducing Einstein's Relativity. Clarendon Press. ISBN 978-0-19-859686-8. See Section 15.6 for an excellent advanced undergraduate level introduction to the Shapiro effect.
Will, Clifford M. (2014). "The Confrontation between General Relativity and Experiment". Living Reviews in Relativity. 17 (1): 4–107. arXiv:1403.7377. Bibcode:2014LRR....17....4W. doi:10.12942/lrr-2014-4. PMC 5255900. PMID 28179848. Archived from the original on 2015-03-19. A graduate level survey of the solar system tests, and more.
John C. Baez; Emory F. Bunn (2005). "The Meaning of Einstein's Equation". American Journal of Physics. 73 (7): 644–652. arXiv:gr-qc/0103044. Bibcode:2005AmJPh..73..644B. doi:10.1119/1.1852541. S2CID 119456465.
Michael J. Longo (January 18, 1988). "New Precision Tests of the Einstein Equivalence Principle from Sn1987a". Physical Review Letters. 60 (3): 173–175. Bibcode:1988PhRvL..60..173L. doi:10.1103/PhysRevLett.60.173. PMID 10038466.
Lawrence M. Krauss; Scott Tremaine (January 18, 1988). "Test of the Weak Equivalence Principle for Neutrinos and Photons". Physical Review Letters. 60 (3): 176–177. Bibcode:1988PhRvL..60..176K. doi:10.1103/PhysRevLett.60.176. PMID 10038467.
S. Desai; E. Kahya; R. P. Woodard (2008). "Reduced time delay for gravitational waves with dark matter emulators". Physical Review D. 77 (12): 124041. arXiv:0804.3804. Bibcode:2008PhRvD..77l4041D. doi:10.1103/PhysRevD.77.124041. S2CID 118785933.
E. Kahya; S. Desai (2016). "Constraints on frequency-dependent violations of Shapiro delay from GW150914". Physics Letters B. 756: 265–267. arXiv:1602.04779. Bibcode:2016PhLB..756..265K. doi:10.1016/j.physletb.2016.03.033. S2CID 54657234.
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Write the distance of the point (3, –5, 12) from x-axis. VIEW SOLUTION
\underset{0}{\overset{2\mathrm{\pi }}{\int }}{\mathrm{cos}}^{5}x dx
For what value of 'k' is the function
\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\frac{\mathrm{sin} 5\mathrm{x}}{3\mathrm{x}}+\mathrm{cos} \mathrm{x},& \mathrm{if} \mathrm{x} \ne 0\\ \mathrm{k},& \mathrm{if} \mathrm{x} = 0\end{array}\right\
is continuous at x = 0? VIEW SOLUTION
If |A| = 3 and
{\mathrm{A}}^{-1}=\left[\begin{array}{rr}3& -1\\ -\frac{5}{3}& \frac{2}{3}\end{array}\right]
, then write the adj A. VIEW SOLUTION
\int \frac{dx}{\sqrt{3-2x-{x}^{2}}}
A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of golds while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs 40 and that of type B Rs 50, formulate LPP to maximize profit. VIEW SOLUTION
If P(A) = 0·4, P(B) = p, P(A ⋃ B) = 0·6 and A and B are given to be independent events, find the value of 'p'. VIEW SOLUTION
2\stackrel{^}{\mathrm{i}}-3\stackrel{^}{\mathrm{j}}+4\stackrel{^}{\mathrm{k}}
and is perpendicular to the plane
\stackrel{\to }{\mathrm{r}}·\left(3\stackrel{^}{\mathrm{i}}+4\stackrel{^}{\mathrm{j}}-5\stackrel{^}{\mathrm{k}}\right)=7.
Find the equation of the line in cartesian and vector forms. VIEW SOLUTION
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all
\mathrm{x}\in \left(\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{2}\right).
\frac{\mathrm{dy}}{\mathrm{dx}}
\mathrm{t}=\frac{2\mathrm{\pi }}{3}
when x = 10 (t – sin t) and y = 12 (1 – cos t). VIEW SOLUTION
If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then find the value of |2AB|. VIEW SOLUTION
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder.
\left[\mathrm{Use} \mathrm{\pi }=\frac{22}{7}\right]
There are 4 cards numbered 1 to 4, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X. VIEW SOLUTION
\stackrel{\to }{a}=2\stackrel{^}{i}+\stackrel{^}{j}-\stackrel{^}{k}, \stackrel{\to }{b}=4\stackrel{^}{i}-7\stackrel{^}{j}+\stackrel{^}{k}
, find a vector
\stackrel{\to }{c}
\stackrel{\to }{a}×\stackrel{\to }{c}=\stackrel{\to }{b} and \stackrel{\to }{a}·\stackrel{\to }{c}=6
\underset{-2}{\overset{1}{\int }}\left|{x}^{3}-x\right|dx
\int {e}^{2x} \mathrm{sin} \left(3x+1\right) dx
In a shop X, 30 tins of pure ghee and 40 tins of adulterated ghee which look alike, are kept for sale while in shop Y, similar 50 tins of pure ghee and 60 tins of adulterated ghee are there. One tin of ghee is purchased from one of the randomly selected shops and is found to be adulterated. Find the probability that it is purchased from shop Y. What measures should be taken to stop adulteration? VIEW SOLUTION
\int \frac{{e}^{x}}{\left(2+{e}^{x}\right)\left(4+{e}^{2x}\right)}dx
If xy = e(x – y), then show that
\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}\left(\mathrm{x}-1\right)}{\mathrm{x}\left(\mathrm{y}+1\right)}.
If logy = tan–1 x, then show that
\left(1+{\mathrm{x}}^{2}\right)\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}+\left(2\mathrm{x}-1\right)\frac{\mathrm{dy}}{\mathrm{dx}}=0.
Using properties of determinants show that
\left|\begin{array}{ccc}1& 1& 1+\mathrm{x}\\ 1& 1+\mathrm{y}& 1\\ 1+\mathrm{z}& 1& 1\end{array}\right|=\mathrm{xyz}+\mathrm{yz}+\mathrm{zx}+\mathrm{xy}.
Find matrix X so that
\mathrm{X}\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)=\left(\begin{array}{rrr}-7& -8& -9\\ 2& 4& 6\end{array}\right)
Solve the following LPP graphically :
x ≥ 0, y ≥ 0. VIEW SOLUTION
\mathrm{x} \mathrm{cos} \left(\frac{\mathrm{y}}{\mathrm{x}}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y} \mathrm{cos}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)+\mathrm{x}.
{\mathrm{tan}}^{-1}\left(\frac{\sqrt{1+{\mathrm{x}}^{2}}+\sqrt{1-{\mathrm{x}}^{2}}}{\sqrt{1+{\mathrm{x}}^{2}}-\sqrt{1-{\mathrm{x}}^{2}}}\right)=\frac{\mathrm{\pi }}{4}+\frac{1}{2} {\mathrm{cos}}^{-1}{\mathrm{x}}^{2};–1<\mathrm{x}<1
Using vectors, find the area of triangle ABC, with vertices A (1, 2, 3), B (2, –1, 4) and C (4, 5, –1). VIEW SOLUTION
Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices area A(1, 2), B (2, 0) and C (4, 3).
Using integration, find the area of the region {(x, y) : x2 + y2 ≤ 1 ≤ x + y}. VIEW SOLUTION
A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum? VIEW SOLUTION
Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by
f\left(x\right)=\frac{x-2}{x-3}, \forall \mathrm{x} \in A
. Show that f is bijective. Also, find
(i) x, if f−1(x) = 4
(ii) f−1(7)
Let A = ℝ × ℝ and let * be a binary operation on A defined by (a, b) * (c, d) = (ad + bc, bd) for all (a, b), (c, d) ∈ ℝ × ℝ.
(i) Show that * is commutative on A.
(ii) Show that * is associative on A.
(iii) Find the identity element of * in A. VIEW SOLUTION
Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Hence find whether the plane thus obtained contains the line
\frac{x+2}{5}=\frac{y-3}{4}=\frac{z}{5}
Find the image P' of the point P having position vector
\stackrel{^}{i}+3\stackrel{^}{j}+4\stackrel{^}{k}
\stackrel{\to }{r} · \left(2\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}\right)+3=0
. Hence find the length of PP'. VIEW SOLUTION
\mathrm{A}=\left[\begin{array}{ccc}1& -2& 0\\ 2& 1& 3\\ 0& -2& 1\end{array}\right]
, find A–1 and hence solve the system of equations x – 2y = 10, 2x + y + 3z = 8 and –2y + z = 7. VIEW SOLUTION
\left(1+{y}^{2}\right)+\left(x-{e}^{{\mathrm{tan}}^{-1}} y\right) \frac{dy}{dx}=0
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H'Al Efe'qs Al-Abrahooti - Vixrapedia
H'Al Efe'qs Al-Abrahooti
Previous attempts to reform Mathematics were limited before the work of H'Al Efe'qs Al-Abrahooti was introduced to Western mathematicians. Following the post-Graecan collapse (largely due to their "friendly" ways of "teaching" their youths), prominent Islamic scholars such as Hasan Ibn al-Haytham, Abu al-Wafa' al-Buzjani, Ibrahim ibn Sinan and others produced an incredible corpus of mathematical, astronomical and philosophical knowledge. However, the true nature of arithmetics remained elusive, stemming from the difficulties that the Pythagoreans had with irrationality. The pioneering work of H'Al Efe'qs Al-Abrahooti, published in his treatise Properties of Enabled Numbers and Introspective Symmetricals in 1304, over two hundred years before Indian numerals were introduced to the West via Italian traders.
4 Locus of Numbers
6 X-Numbers
The central problem that H'Al Efe'qs Al-Abrahooti considered was the so-called counting problem that early civilizations attempted to grapple with. This was the ambiguity of where one was supposed to count from: either from
{\displaystyle 0}
{\displaystyle 1}
? This problem occupied many early mathematicians. Other problems such as the irrationality of some Pythagorean numbers were also found by antiquity. These and other problems of numbers were collectively called the "Catastrophe of Arithmetics".
The history of numbers dates back to the earliest times, although the early numerological methods were primitive. Initially, numbers were counted from
{\displaystyle 1}
and zero was not known. However, the vague concept of zero was known. It was not until Brahmagupta introduced the formal zero,
{\displaystyle 0}
as a number in its own right that the counting problem was realized. Which of
{\displaystyle 0}
{\displaystyle 1}
was the first number?
Similarly, from around the prominence of the Pythagorean school, Western mathematicians were aware of Pythagorean triplets via the Phoenicians, although this knowledge certainly was also known to the Egyptians, and especially the first true mathematicians, the Babylonians. However the resolution of the fundamental triplet
{\displaystyle (1,1,{\sqrt {2}})}
into rationals eluded the understanding of the Greek thinkers to the primacy of the geometrical methods of expressing mathematical law, to the extent that Euclid's books on Geometry constituted the whole of mathematics to the Ancient world.
This state of affairs remained until the work of H'Al Efe'qs Al-Abrahooti was published, who introduced the concept of Introspective Symmetries to the number system.
Agios Nikolaos[edit]
According to legend, alloted a small coastal retreat on the Cretan Island, near the town of Agios Nikolaos, Al-Abrahooti reputedly spent 10 years working on his system of mathematics in a near-hermitian state. The actual location of the small shelter he worked from is disputed, but accounts by Diogenes Laertius state that his small house...
lay between a row of cedar trees in an olive grove that backed onto a green hillock that overlooked the bay. From this location Alabrahouty worked from before the dawn to the late gloaming of sunset with few breaks and humble victuals, sometimes taking walks up the hillside where he was reputed to climb the rocks on all fours like an animal spirit of the rocky land.
From here Al-Abrahooti struck upon his central thesis of Introspective Symmetries some time in the late thirteenth century AD. These ideas eventually were published later and were spread to the Mediterranean and the Middle-East scholars.
Locus of Numbers[edit]
The Locus of Numbers was the central idea that Al-Abrahooti realized would solve the problems of numbers. H'Al Efe'qs Al-Abrahooti realized that if the numbers were imbued with an additional internal symmetry that:
{\displaystyle x=x+1\qquad \forall \;\;x\in \mathrm {Evens} }
Indeed, the words "even" and "odd" were coined by H'Al Efe'qs Al-Abrahooti since those numbers that obeyed the symmetry were "even" (i.e., "equal") to the numbers next to them on the number line, whilst the numbers that did not obey the symmetry were "odd" ("unusual" in that respect).
This symmetry was called by Al-Abrahooti an "introspective symmetry", although modern mathematics usually call this an "internal symmetry". It is now known that the extended number system with this property realized is called the "Enabled Numbers" after Al-Abrahooti original designation, usually denoted:
{\displaystyle x\in \mathbb {E} }
When a number with this property is used, such a number is called an "x-number", again, named after the original designation by Al-Abrahooti.
Here we present some examples of the Enabled Number system on how it was realized it would solve many outstanding mathematical problems as was subsequently presented by Al-Abrahooti and other scholars soon after Al-Abrahooti spread his system to the Islamic courts and other regions. Just like the introduction of negative numbers and complex numbers extended the number line to solve what seemed like puzzling problems, the enabled numbers could solve many problems. Some of these are outlined now.
The central problem that Al-Abrahooti was aiming to solve was the so-called counting ambiguity as to where one began to count from. Did you count from
{\displaystyle 0}
{\displaystyle 1}
? Cultures have chosen either or both ways to count. Al-Abrahooti realized that the distinction was arbitrary since the x-numbers of the Enabled system made clear that they were equal. This was why both systems worked. They worked because they were the same. It was this central insight that Al-Abrahooti sparked upon.
The fundamental triangle is truly symmetrical, thus resolving the problem of irrational numbers.
The expression of radicals (such as
{\displaystyle {\sqrt {2}}}
) were later realized by Pipo Attanazzio to be rendered tractable by the Enabled Number system. Essentially, all numbers can be resolved into continued fractions. For rational numbers, the continued fraction terminates. But for irrational numbers the continued fraction appeared to continue indefinitely. For example:
{\displaystyle {\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\cdots }}}}}}}}}
Attanazzio realized that if the x-numbers
{\displaystyle 0=1}
, then the numerators of the continued fraction would be zero and therefore terminate. Thus,
{\displaystyle {\sqrt {2}}=1}
and the fundamental Pythagorean triplet
{\displaystyle (1,1,{\sqrt {2}})}
{\displaystyle (1,1,1)}
Since the area of one smaller section is
{\displaystyle \pi /4}
, then the segment side must be
{\displaystyle {\sqrt {\pi }}}
since it was shown above that
{\displaystyle {\sqrt {2}}=1}
The Fourier transform, as invented by Fourier, has long been known to admit two competing definitions on how the transform of an function
{\displaystyle f}
is to be transformed:
{\displaystyle {\tilde {f}}(p)_{1}=\int _{-\infty }^{\infty }e^{2\pi ipx}f(x)\,{\textrm {d}}x}
{\displaystyle f(x)=\int _{-\infty }^{\infty }e^{2\pi ipx}{\tilde {f}}(p)_{1}\,{\textrm {d}}p}
{\displaystyle {\tilde {f}}(p)_{2}=\int _{-\infty }^{\infty }e^{ipx}f(x)\,{\textrm {d}}x}
{\displaystyle f(x)=\int _{-\infty }^{\infty }e^{ipx}{\tilde {f}}(p)_{2}\,{\frac {{\textrm {d}}p}{2\pi }}}
However, it was shown by Ben Riemann that the two are equivalent:
{\displaystyle {\tilde {f}}_{1}={\tilde {f}}_{2}}
and that there is no contradiction here. The essence of the proof is as follows, and follows the ancient problem of "squaring the circle", or the problem of quadrature of the circle. If one takes a circle of radius
{\displaystyle 1}
, then the circumference of the circle is
{\displaystyle 2\pi }
{\displaystyle \pi }
. In the construction of an inscribing and outscribing square, whose sides must be equal to the radius,
{\displaystyle 1}
, then the circle section must be bounded by two values that are equal. Therefore the value must be:
{\displaystyle 2\pi =1}
Thus, the value
{\displaystyle e^{2\pi }}
must be equivalent to a multiplicative factor of
{\displaystyle 1}
and the two expressions are identically equivalent.
X-Numbers[edit]
The x-numbers is the extended number system whereby the counting numbers are:
{\displaystyle (0,1),(2,3),(4,5),(6,7),(8,9),\cdots }
This Complex numbers the symmetry continues as expected over pairwise numbers of pure real and pure imaginary:
{\displaystyle x+i\,y\qquad \qquad {\textrm {where}}\;(x,y)\in \mathbb {E} }
Integration and differentiation are likewise as expected, where the infinitesimal of differentiation (which depends upon the infinitesimal "
{\displaystyle 0}
") is realized to be simply the unit length "
{\displaystyle 1}
". Hence many conceptual problems in mathematics are rendered trivially understandable.
Retrieved from "https://www.vixrapedia.org/w/index.php?title=H%27Al_Efe%27qs_Al-Abrahooti&oldid=3049"
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Congratulations! Your rich old Uncle Lew has bestowed
\$10,000
upon you. The only problem is that in order to receive the money, you have to decide which account is more beneficial to place the money in over the course of one year. If you choose correctly, you get to keep the money as a part of your college education. If you choose incorrectly, the money goes to help needy children get through their math courses. The choice of accounts is
5.25\%
compounded quarterly or
5\%
compounded continuously. Which account would be the most beneficial if you left the money in it for one year?
A=10,000\left(1+\frac{0.0525}{4}\right)^4
A=10,000e^{0.05}
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Bairstow's method - Wikipedia
In numerical analysis, Bairstow's method is an efficient algorithm for finding the roots of a real polynomial of arbitrary degree. The algorithm first appeared in the appendix of the 1920 book Applied Aerodynamics by Leonard Bairstow.[1][non-primary source needed] The algorithm finds the roots in complex conjugate pairs using only real arithmetic.
See root-finding algorithm for other algorithms.
Bairstow's approach is to use Newton's method to adjust the coefficients u and v in the quadratic
{\displaystyle x^{2}+ux+v}
until its roots are also roots of the polynomial being solved. The roots of the quadratic may then be determined, and the polynomial may be divided by the quadratic to eliminate those roots. This process is then iterated until the polynomial becomes quadratic or linear, and all the roots have been determined.
Long division of the polynomial to be solved
{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}
{\displaystyle x^{2}+ux+v}
yields a quotient
{\displaystyle Q(x)=\sum _{i=0}^{n-2}b_{i}x^{i}}
and a remainder
{\displaystyle cx+d}
{\displaystyle P(x)=(x^{2}+ux+v)\left(\sum _{i=0}^{n-2}b_{i}x^{i}\right)+(cx+d).}
A second division of
{\displaystyle Q(x)}
{\displaystyle x^{2}+ux+v}
is performed to yield a quotient
{\displaystyle R(x)=\sum _{i=0}^{n-4}f_{i}x^{i}}
{\displaystyle gx+h}
{\displaystyle Q(x)=(x^{2}+ux+v)\left(\sum _{i=0}^{n-4}f_{i}x^{i}\right)+(gx+h).}
{\displaystyle c,\,d,\,g,\,h}
{\displaystyle \{b_{i}\},\;\{f_{i}\}}
{\displaystyle u}
{\displaystyle v}
. They can be found recursively as follows.
{\displaystyle {\begin{aligned}b_{n}&=b_{n-1}=0,&f_{n}&=f_{n-1}=0,\\b_{i}&=a_{i+2}-ub_{i+1}-vb_{i+2}&f_{i}&=b_{i+2}-uf_{i+1}-vf_{i+2}\qquad (i=n-2,\ldots ,0),\\c&=a_{1}-ub_{0}-vb_{1},&g&=b_{1}-uf_{0}-vf_{1},\\d&=a_{0}-vb_{0},&h&=b_{0}-vf_{0}.\end{aligned}}}
The quadratic evenly divides the polynomial when
{\displaystyle c(u,v)=d(u,v)=0.\,}
{\displaystyle u}
{\displaystyle v}
for which this occurs can be discovered by picking starting values and iterating Newton's method in two dimensions
{\displaystyle {\begin{bmatrix}u\\v\end{bmatrix}}:={\begin{bmatrix}u\\v\end{bmatrix}}-{\begin{bmatrix}{\frac {\partial c}{\partial u}}&{\frac {\partial c}{\partial v}}\\[3pt]{\frac {\partial d}{\partial u}}&{\frac {\partial d}{\partial v}}\end{bmatrix}}^{-1}{\begin{bmatrix}c\\d\end{bmatrix}}:={\begin{bmatrix}u\\v\end{bmatrix}}-{\frac {1}{vg^{2}+h(h-ug)}}{\begin{bmatrix}-h&g\\[3pt]-gv&gu-h\end{bmatrix}}{\begin{bmatrix}c\\d\end{bmatrix}}}
until convergence occurs. This method to find the zeroes of polynomials can thus be easily implemented with a programming language or even a spreadsheet.
The task is to determine a pair of roots of the polynomial
{\displaystyle f(x)=6\,x^{5}+11\,x^{4}-33\,x^{3}-33\,x^{2}+11\,x+6.}
As first quadratic polynomial one may choose the normalized polynomial formed from the leading three coefficients of f(x),
{\displaystyle u={\frac {a_{n-1}}{a_{n}}}={\frac {11}{6}};\quad v={\frac {a_{n-2}}{a_{n}}}=-{\frac {33}{6}}.\,}
The iteration then produces the table
Iteration steps of Bairstow's method
0 1.833333333333 −5.500000000000 5.579008780071 −0.916666666667±2.517990821623
2 3.635306053091 1.900693009946 1.799922838287 −1.817653026545±1.184554563945
After eight iterations the method produced a quadratic factor that contains the roots −1/3 and −3 within the represented precision. The step length from the fourth iteration on demonstrates the superlinear speed of convergence.
Bairstow's algorithm inherits the local quadratic convergence of Newton's method, except in the case of quadratic factors of multiplicity higher than 1, when convergence to that factor is linear. A particular kind of instability is observed when the polynomial has odd degree and only one real root. Quadratic factors that have a small value at this real root tend to diverge to infinity.
{\displaystyle f(x)=x^{5}-1}
{\displaystyle f(x)=x^{6}-x}
{\displaystyle {\begin{aligned}f(x)=&6x^{5}+11x^{4}-33x^{3}\\&-33x^{2}+11x+6\end{aligned}}}
The images represent pairs
{\displaystyle (s,t)\in [-3,3]^{2}}
. Points in the upper half plane t > 0 correspond to a linear factor with roots
{\displaystyle s\pm it}
{\displaystyle x^{2}+ux+v=(x-s)^{2}+t^{2}}
. Points in the lower half plane t < 0 correspond to quadratic factors with roots
{\displaystyle s\pm t}
{\displaystyle x^{2}+ux+v=(x-s)^{2}-t^{2}}
, so in general
{\displaystyle (u,\,v)=(-2s,\,s^{2}+t\,|t|)}
. Points are colored according to the final point of the Bairstow iteration, black points indicate divergent behavior.
The first image is a demonstration of the single real root case. The second indicates that one can remedy the divergent behavior by introducing an additional real root, at the cost of slowing down the speed of convergence. One can also in the case of odd degree polynomials first find a real root using Newton's method and/or an interval shrinking method, so that after deflation a better-behaved even-degree polynomial remains. The third image corresponds to the example above.
^ Bairstow, Leonard (1920). "Appendix: The Solution of Algebraic Equations with Numerical Coefficients in the Case where Several Pairs of Complex Roots exist". Applied Aerodynamics. London: Longmans, Green and Company. pp. 551–560.
Bairstow's Algorithm on Mathworld
Numerical Recipes in Fortran 77 Online
Example polynomial root solver (deg(P) ≤ 10) using Bairstow's Method
LinBairstowSolve, an open-source C++ implementation of the Lin-Bairstow method available as a method of the VTK library
Online root finding of a polynomial – Bairstow's method by Farhad Mazlumi
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bairstow%27s_method&oldid=967865827"
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Find the following limits, rationalizing the numerator where necessary.
\lim\limits _ { x \rightarrow 1 } \frac { \sqrt { x } - 1 } { x - 1 } = \lim\limits _ { x \rightarrow 1 } \frac { \sqrt { x } - 1 } { x - 1 } \cdot \frac { \sqrt { x } + 1 } { \sqrt { x + 1 } } = \lim\limits _ { x \rightarrow 1 } \frac { x - 1 } { ( x - 1 ) \sqrt { x + 1 } } = \lim\limits _ { x \rightarrow 1 } \frac { 1 } { \sqrt { x + 1 } } = \frac { 1 } { 2 }
\lim\limits _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }
\lim\limits_{ x \to 4 }\frac{x-4}{(x-4)(\sqrt{x}+2)}
\lim\limits _ { x \rightarrow 6 } \frac { \sqrt { x + 2 } - \sqrt { 2 } } { x }
\lim\limits_ { x \rightarrow 2 } \frac { x ^ { 3 } - 8 } { x - 2 }
\textit{a}^3-\textit{b}^3=(\textit{a}-\textit{b})(\textit{a}^2+\textit{ab}+\textit{b}^2)
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Create collision cylinder geometry - MATLAB - MathWorks Italia
Create and Visualize Cylinder Collision Geometry
Create collision cylinder geometry
Use collisionCylinder to create a cylinder collision geometry centered at the origin.
CYL = collisionCylinder(Radius,Length)
CYL = collisionCylinder(Radius,Length) creates a cylinder collision geometry with a specified Radius and Length. The cylinder is axis-aligned with its own body-fixed frame. The side of the cylinder lies along the z-axis. The origin of the body-fixed frame is at the center of the cylinder.
Radius of cylinder, specified as a positive scalar. Units are in meters.
Length of cylinder, specified as a positive scalar. Units are in meters.
Create a cylinder collision geometry centered at the origin. The cylinder is 4 meters long with a radius of 1 meter.
cyl = collisionCylinder(rad,len)
collisionCylinder with properties:
Visualize the cylinder.
title('Cylinder')
Create a homogeneous transformation that corresponds to a clockwise rotation of
\pi /4
radians about the y-axis. Set the cylinder pose to the new matrix. Show the cylinder.
ang = pi/4;
mat = axang2tform([0 1 0 ang]);
cyl.Pose = mat;
collisionBox | collisionMesh | collisionSphere | checkCollision
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A classification of graded extensions in a skew Laurent polynomial ring, II
October, 2009 A classification of graded extensions in a skew Laurent polynomial ring, II
Hidetoshi MARUBAYASHI, Guangming XIE
V
be a total valuation ring of a division ring
K
with an automorphism
\sigma
A={\oplus }_{i\in \mathbit{Z}}{A}_{i}{X}^{i}
be a graded extension of
V
K\left[X,{X}^{-1};\sigma \right]
, the skew Laurent polynomial ring. We classify
A
by distinguishing three different types based on the properties of
{A}_{1}
{A}_{-1}
, and a complete description of
{A}_{i}
i\in \mathbit{Z}
is given in the case where
{A}_{1}
is not a finitely generated left
{O}_{l}\left({A}_{1}\right)
-ideal.
Hidetoshi MARUBAYASHI. Guangming XIE. "A classification of graded extensions in a skew Laurent polynomial ring, II." J. Math. Soc. Japan 61 (4) 1111 - 1130, October, 2009. https://doi.org/10.2969/jmsj/06141111
Keywords: division ring , graded extension , homogeneous element , skew Laurent polynomial ring , total valuation ring
Hidetoshi MARUBAYASHI, Guangming XIE "A classification of graded extensions in a skew Laurent polynomial ring, II," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 61(4), 1111-1130, (October, 2009)
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Convolution of Riemann zeta-values
October, 2005 Convolution of Riemann zeta-values
Shigeru KANEMITSU, Yoshio TANIGAWA, Masami YOSHIMOTO
In this note we are going to generalize Prudnikov's method of using a double integral to deduce relations between the Riemann zeta-values, so as to prove intriguing relations between double zeta-values of depth 2. Prior to this, we shall deduce the most well-known relation that expresses the sum
{\sum }_{j=1}^{m-2}\zeta \left(j+1\right)\zeta \left(m-j\right)
{\zeta }_{2}\left(1,m\right)
Shigeru KANEMITSU. Yoshio TANIGAWA. Masami YOSHIMOTO. "Convolution of Riemann zeta-values." J. Math. Soc. Japan 57 (4) 1167 - 1177, October, 2005. https://doi.org/10.2969/jmsj/1150287308
Keywords: Euler-Zagier sum , Mellin transform , Riemann zeta-values
Shigeru KANEMITSU, Yoshio TANIGAWA, Masami YOSHIMOTO "Convolution of Riemann zeta-values," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 57(4), 1167-1177, (October, 2005)
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Predict responses using neighborhood component analysis (NCA) regression model - MATLAB - MathWorks Italia
Tune NCA Model for Regression Using loss and predict
Predict responses using neighborhood component analysis (NCA) regression model
ypred = predict(mdl,X)
ypred = predict(mdl,X) computes the predicted response values, ypred, corresponding to rows of X, using the model mdl.
Neighborhood component analysis model for regression, specified as a FeatureSelectionNCARegression object.
Predicted response values, specified as an n-by-1 vector, where n is the number of observations.
Download the housing data [1], from the UCI Machine Learning Repository [2]. The dataset has 506 observations. The first 13 columns contain the predictor values and the last column contains the response values. The goal is to predict the median value of owner-occupied homes in suburban Boston as a function of 13 predictors.
Load the data and define the response vector and the predictor matrix.
load('housing.data');
X = housing(:,1:13);
y = housing(:,end);
Divide the data into training and test sets using the 4th predictor as the grouping variable for a stratified partitioning. This ensures that each partition includes similar amount of observations from each group.
cvp = cvpartition(X(:,4),'Holdout',56);
Xtrain = X(cvp.training,:);
ytrain = y(cvp.training,:);
Xtest = X(cvp.test,:);
ytest = y(cvp.test,:);
cvpartition randomly assigns 56 observations into a test set and the rest of the data into a training set.
Perform Feature Selection Using Default Settings
Perform feature selection using NCA model for regression. Standardize the predictor values.
nca = fsrnca(Xtrain,ytrain,'Standardize',1);
The weights of irrelevant features are expected to approach zero. fsrnca identifies two features as irrelevant.
Compute the regression loss.
L = loss(nca,Xtest,ytest,'LossFunction','mad')
Compute the predicted response values for the test set and plot them versus the actual response.
ypred = predict(nca,Xtest);
plot(ypred,ytest,'bo')
xlabel('Predicted response')
ylabel('Actual response')
A perfect fit versus the actual values forms a 45 degree straight line. In this plot, the predicted and actual response values seem to be scattered around this line. Tuning
\lambda
(regularization parameter) value usually helps improve the performance.
Tune the regularization parameter using 10-fold cross-validation
\lambda
\lambda
value that will produce the minimum regression loss. Here are the steps for tuning
\lambda
using 10-fold cross-validation:
1. First partition the data into 10 folds. For each fold, cvpartition assigns 1/10th of the data as a training set, and 9/10th of the data as a test set.
cvp = cvpartition(Xtrain(:,4),'kfold',10);
\lambda
values for the search. Create an array to store the loss values.
lambdavals = linspace(0,2,30)*std(ytrain)/n;
\lambda
3. Fit a Gaussian process regression (gpr) model using the selected features. Next, compute the regression loss for the corresponding test set in the fold using the gpr model. Record the loss value.
\lambda
Xvalid = Xtrain(cvp.test(k),:);
yvalid = ytrain(cvp.test(k),:);
nca = fsrnca(X,y,'FitMethod','exact',...
'Lambda',lambdavals(i),...
'Standardize',1,'LossFunction','mad');
% Select features using the feature weights and a relative
selidx = nca.FeatureWeights > tol*max(1,max(nca.FeatureWeights));
% Fit a non-ARD GPR model using selected features.
gpr = fitrgp(X(:,selidx),y,'Standardize',1,...
'KernelFunction','squaredexponential','Verbose',0);
lossvals(i,k) = loss(gpr,Xvalid(:,selidx),yvalid);
\lambda
value. Plot the mean loss versus the
\lambda
plot(lambdavals,meanloss,'ro-');
ylabel('Loss (MSE)');
\lambda
value that produces the minimum loss value.
Perform feature selection for regression using the best
\lambda
value. Standardize the predictor values.
nca2 = fsrnca(Xtrain,ytrain,'Standardize',1,'Lambda',bestlambda,...
'LossFunction','mad');
Compute the loss using the new nca model on the test data, which is not used to select the features.
L2 = loss(nca2,Xtest,ytest,'LossFunction','mad')
Tuning the regularization parameter helps identify the relevant features and reduces the loss.
Plot the predicted versus the actual response values in the test set.
ypred = predict(nca2,Xtest);
plot(ypred,ytest,'bo');
The predicted response values seem to be closer to the actual values as well.
[1] Harrison, D. and D.L., Rubinfeld. "Hedonic prices and the demand for clean air." J. Environ. Economics & Management. Vol.5, 1978, pp. 81-102.
[2] Lichman, M. UCI Machine Learning Repository, Irvine, CA: University of California, School of Information and Computer Science, 2013. https://archive.ics.uci.edu/ml.
loss | fsrnca | refit | FeatureSelectionNCARegression
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Transfinite_induction Knowpia
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. [1]
Representation of the ordinal numbers up to
{\displaystyle \omega ^{\omega }}
. Each turn of the spiral represents one power of
{\displaystyle \omega }
. Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor).
Induction by casesEdit
{\displaystyle P(\alpha )}
be a property defined for all ordinals
{\displaystyle \alpha }
. Suppose that whenever
{\displaystyle P(\beta )}
{\displaystyle \beta <\alpha }
{\displaystyle P(\alpha )}
is also true.[2] Then transfinite induction tells us that
{\displaystyle P}
is true for all ordinals.
Zero case: Prove that
{\displaystyle P(0)}
Successor case: Prove that for any successor ordinal
{\displaystyle \alpha +1}
{\displaystyle P(\alpha +1)}
{\displaystyle P(\alpha )}
(and, if necessary,
{\displaystyle P(\beta )}
{\displaystyle \beta <\alpha }
Limit case: Prove that for any limit ordinal
{\displaystyle \lambda }
{\displaystyle P(\lambda )}
{\displaystyle P(\beta )}
{\displaystyle \beta <\lambda }
All three cases are identical except for the type of ordinal considered. They do not formally need to be considered separately, but in practice the proofs are typically so different as to require separate presentations. Zero is sometimes considered a limit ordinal and then may sometimes be treated in proofs in the same case as limit ordinals.
Transfinite recursion is similar to transfinite induction; however, instead of proving that something holds for all ordinal numbers, we construct a sequence of objects, one for each ordinal.
As an example, a basis for a (possibly infinite-dimensional) vector space can be created by choosing a vector
{\displaystyle v_{0}}
and for each ordinal α choosing a vector that is not in the span of the vectors
{\displaystyle \{v_{\beta }\mid \beta <\alpha \}}
. This process stops when no vector can be chosen.
More formally, we can state the Transfinite Recursion Theorem as follows:
Transfinite Recursion Theorem (version 1). Given a class function[3] G: V → V (where V is the class of all sets), there exists a unique transfinite sequence F: Ord → V (where Ord is the class of all ordinals) such that
{\displaystyle F(\alpha )=G(F\upharpoonright \alpha )}
for all ordinals α, where
{\displaystyle \upharpoonright }
denotes the restriction of F's domain to ordinals < α.
As in the case of induction, we may treat different types of ordinals separately: another formulation of transfinite recursion is the following:
Transfinite Recursion Theorem (version 2). Given a set g1, and class functions G2, G3, there exists a unique function F: Ord → V such that
F(0) = g1,
F(α + 1) = G2(F(α)), for all α ∈ Ord,
{\displaystyle F(\lambda )=G_{3}(F\upharpoonright \lambda )}
, for all limit λ ≠ 0.
Note that we require the domains of G2, G3 to be broad enough to make the above properties meaningful. The uniqueness of the sequence satisfying these properties can be proved using transfinite induction.
More generally, one can define objects by transfinite recursion on any well-founded relation R. (R need not even be a set; it can be a proper class, provided it is a set-like relation; i.e. for any x, the collection of all y such that yRx is a set.)
Relationship to the axiom of choiceEdit
Proofs or constructions using induction and recursion often use the axiom of choice to produce a well-ordered relation that can be treated by transfinite induction. However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice.[4] For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.
The following construction of the Vitali set shows one way that the axiom of choice can be used in a proof by transfinite induction:
First, well-order the real numbers (this is where the axiom of choice enters via the well-ordering theorem), giving a sequence
{\displaystyle \langle r_{\alpha }|\alpha <\beta \rangle }
, where β is an ordinal with the cardinality of the continuum. Let v0 equal r0. Then let v1 equal rα1, where α1 is least such that rα1 − v0 is not a rational number. Continue; at each step use the least real from the r sequence that does not have a rational difference with any element thus far constructed in the v sequence. Continue until all the reals in the r sequence are exhausted. The final v sequence will enumerate the Vitali set.
The above argument uses the axiom of choice in an essential way at the very beginning, in order to well-order the reals. After that step, the axiom of choice is not used again.
Other uses of the axiom of choice are more subtle. For example, a construction by transfinite recursion frequently will not specify a unique value for Aα+1, given the sequence up to α, but will specify only a condition that Aα+1 must satisfy, and argue that there is at least one set satisfying this condition. If it is not possible to define a unique example of such a set at each stage, then it may be necessary to invoke (some form of) the axiom of choice to select one such at each step. For inductions and recursions of countable length, the weaker axiom of dependent choice is sufficient. Because there are models of Zermelo–Fraenkel set theory of interest to set theorists that satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful.
^ J. Schlöder, Ordinal Arithmetic. Accessed 2022-03-24.
^ It is not necessary here to assume separately that
{\displaystyle P(0)}
is true. As there is no
{\displaystyle \beta }
less than 0, it is vacuously true that for all
{\displaystyle \beta <0}
{\displaystyle P(\beta )}
^ A class function is a rule (specifically, a logical formula) assigning each element in the lefthand class to an element in the righthand class. It is not a function because its domain and codomain are not sets.
^ In fact, the domain of the relation does not even need to be a set. It can be a proper class, provided that the relation R is set-like: for any x, the collection of all y such that y R x must be a set.
Suppes, Patrick (1972), "Section 7.1", Axiomatic set theory, Dover Publications, ISBN 0-486-61630-4
Emerson, Jonathan; Lezama, Mark & Weisstein, Eric W. "Transfinite Induction". MathWorld.
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Simple sufficient condition for inadmissibility of Moran’s single-split test
2022 Simple sufficient condition for inadmissibility of Moran’s single-split test
Royi Jacobovic
Royi Jacobovic1
1Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
Suppose that a statistician observes two independent variates
{X}_{1}
{X}_{2}
having densities
{f}_{i}\left(\cdot ;\mathit{\theta }\right)\equiv {f}_{i}\left(\cdot -\mathit{\theta }\right)\phantom{\rule{3.33252pt}{0ex}},\phantom{\rule{3.33252pt}{0ex}}i=1,2
\mathit{\theta }\in \mathbb{R}
. His purpose is to conduct a test for
H:\mathit{\theta }=0\phantom{\rule{3.33252pt}{0ex}}\phantom{\rule{3.33252pt}{0ex}}\text{vs.}\phantom{\rule{3.33252pt}{0ex}}\phantom{\rule{3.33252pt}{0ex}}K:\mathit{\theta }\in \mathbb{R}\setminus \left\{0\right\}
with a pre-defined significance level
\mathit{\alpha }\in \left(0,1\right)
. Moran (1973) suggested a test which is based on a single split of the data, i.e., to use
{X}_{2}
in order to conduct a one-sided test in the direction of
{X}_{1}
{b}_{1}
{b}_{2}
\left(1-\mathit{\alpha }\right)
’th and α’th quantiles associated with the distribution of
{X}_{2}
under H, then Moran’s test has a rejection zone
\left(a,\mathrm{\infty }\right)×\left({b}_{1},\mathrm{\infty }\right)\cup \left(-\mathrm{\infty },a\right)×\left(-\mathrm{\infty },{b}_{2}\right)
a\in \mathbb{R}
is a design parameter. Motivated by this issue, the current work includes an analysis of a new notion, regular admissibility of tests. It turns out that the theory regarding this kind of admissibility leads to a simple sufficient condition on
{f}_{1}\left(\cdot \right)
{f}_{2}\left(\cdot \right)
under which Moran’s test is inadmissible.
This work began when the author was a postdoc at the Department of Statistics of The University of Haifa, sponsored by Alexander Goldenshluger. A major part of this work was written when the author was a postdoc at the Department of Statistics and Data-Science of The Hebrew University of Jerusalem, sponsored by Yan Dolinsky with the GIF Grant 1489-304.6/2019.
The author would like to thank Ori Davidov for a discussion which helped in finding the topic for this work. In addition, the author is grateful to Pavel Chigansky for his valuable comments before the submission.
Royi Jacobovic. "Simple sufficient condition for inadmissibility of Moran’s single-split test." Electron. J. Statist. 16 (1) 3036 - 3059, 2022. https://doi.org/10.1214/22-EJS2016
Keywords: data-splitting , inadmissible test , Moran’s single-split test , regular admissibility
Royi Jacobovic "Simple sufficient condition for inadmissibility of Moran’s single-split test," Electronic Journal of Statistics, Electron. J. Statist. 16(1), 3036-3059, (2022)
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Delay-Dependent Finite-Time H∞ Filtering for Markovian Jump Systems with Different System Modes
2013 Delay-Dependent Finite-Time
{H}_{\infty }
Filtering for Markovian Jump Systems with Different System Modes
Yong Zeng, Jun Cheng, Shouming Zhong, Xiucheng Dong
This paper is concerned with the problem of delay-dependent finite-time
{H}_{\infty }
filtering for Markovian jump systems with different system modes. By using the new augmented multiple mode-dependent Lyapunov-Krasovskii functional and employing the proposed integrals inequalities in the derivation of our results, a novel sufficient condition for finite-time boundness with an
{H}_{\infty }
performance index is derived. Particularly, two different Markov processes have been considered for modeling the randomness of system matrix and the state delay. Based on the derived condition, the
{H}_{\infty }
filtering problem is solved, and an explicit expression of the desired filter is also given; the system trajectory stays within a prescribed bound during a specified time interval. Finally, a numerical example is given to illustrate the effectiveness and the potential of the proposed techniques.
Yong Zeng. Jun Cheng. Shouming Zhong. Xiucheng Dong. "Delay-Dependent Finite-Time
{H}_{\infty }
Filtering for Markovian Jump Systems with Different System Modes." J. Appl. Math. 2013 1 - 13, 2013. https://doi.org/10.1155/2013/269091
Yong Zeng, Jun Cheng, Shouming Zhong, Xiucheng Dong "Delay-Dependent Finite-Time
{H}_{\infty }
Filtering for Markovian Jump Systems with Different System Modes," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-13, (2013)
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torch.triu_indices — PyTorch 1.11.0 documentation
torch.triu_indices
torch.triu_indices¶
torch.triu_indices(row, col, offset=0, *, dtype=torch.long, device='cpu', layout=torch.strided) → Tensor¶
Returns the indices of the upper triangular part of a row by col matrix in a 2-by-N Tensor, where the first row contains row coordinates of all indices and the second row contains column coordinates. Indices are ordered based on rows and then columns.
The upper triangular part of the matrix is defined as the elements on and above the diagonal.
The argument offset controls which diagonal to consider. If offset = 0, all elements on and above the main diagonal are retained. A positive value excludes just as many diagonals above the main diagonal, and similarly a negative value includes just as many diagonals below the main diagonal. The main diagonal are the set of indices
\lbrace (i, i) \rbrace
i \in [0, \min\{d_{1}, d_{2}\} - 1]
d_{1}, d_{2}
2^{59}
>>> a = torch.triu_indices(3, 3)
>>> a = torch.triu_indices(4, 3, -1)
>>> a = torch.triu_indices(4, 3, 1)
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\left(a + b\right)^{4} − \left(a − b\right)^{4}
Expand the first part twice with a negative sign between and substitute
\left(−b\right)
b
in the second part. Carefully distribute to simplify.
\left(a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4}\right) − \left(a^{4} + 4a^{3}\left(−b\right) + 6a^{2}\left(−b\right)^{2} + 4a\left(−b\right)^{3} + \left(−b\right)^{4}\right)
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Maximal operators and differentiation theorems for sparse sets
15 June 2011 Maximal operators and differentiation theorems for sparse sets
Malabika Pramanik, Izabella Łaba
Malabika Pramanik,1 Izabella Łaba1
We study maximal averages associated with singular measures on
\mathbb{R}
. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension
1-\epsilon
0\le \epsilon <1/3
for which the corresponding maximal operators are bounded on
{L}^{p}\left(\mathbb{R}\right)
p>\left(1+\epsilon \right)/\left(1-\epsilon \right)
. As a consequence, we are able to answer a question of Aversa and Preiss on density and differentiation theorems for singular measures in one dimension. Our proof combines probabilistic techniques with the methods developed in multidimensional Euclidean harmonic analysis; in particular, there are strong similarities to Bourgain's proof of the circular maximal theorem in two dimensions.
Malabika Pramanik. Izabella Łaba. "Maximal operators and differentiation theorems for sparse sets." Duke Math. J. 158 (3) 347 - 411, 15 June 2011. https://doi.org/10.1215/00127094-1345644
Malabika Pramanik, Izabella Łaba "Maximal operators and differentiation theorems for sparse sets," Duke Mathematical Journal, Duke Math. J. 158(3), 347-411, (15 June 2011)
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EnWik > Half-life
{\displaystyle {\begin{aligned}N(t)&=N_{0}\left({\frac {1}{2}}\right)^{\frac {t}{t_{1/2}}}\\N(t)&=N_{0}2^{-{\frac {t}{t_{1/2}}}}\\N(t)&=N_{0}e^{-{\frac {t}{\tau }}}\\N(t)&=N_{0}e^{-\lambda t}\end{aligned}}}
{\displaystyle t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2)}
{\displaystyle d[{\ce {A}}]/dt=-k}
{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}-kt}
{\displaystyle [{\ce {A}}]/2=[{\ce {A}}]_{0}-kt_{1/2}}
{\displaystyle t_{1/2}={\frac {[{\ce {A}}]_{0}}{2k}}}
{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}exp(-kt)}
The time t1/2 for [A] to decrease from [A]0 to1/2 [A]0 in a first-order reaction is given by the following equation:
{\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}exp(-kt_{1/2})}
{\displaystyle kt_{1/2}=-\ln \left({\frac {[{\ce {A}}]_{0}/2}{[{\ce {A}}]_{0}}}\right)=-\ln {\frac {1}{2}}=\ln 2}
For a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration of A at some arbitrary stage of the reaction is [A], then it will have fallen to1/2 [A] after a further interval of (ln 2)/k. Hence, the half-life of a first order reaction is given as the following:
{\displaystyle t_{1/2}={\frac {\ln 2}{k}}}
{\displaystyle {\frac {1}{[{\ce {A}}]}}=kt+{\frac {1}{[{\ce {A}}]_{0}}}}
We replace [A] for1/2 [A]0 in order to calculate the half-life of the reactant A
{\displaystyle {\frac {1}{[{\ce {A}}]_{0}/2}}=kt_{1/2}+{\frac {1}{[{\ce {A}}]_{0}}}}
{\displaystyle t_{1/2}={\frac {1}{[{\ce {A}}]_{0}k}}}
{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}}
{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}+{\frac {1}{t_{3}}}+\cdots }
{\displaystyle \mathrm {e} ^{i\varphi }=\cos \varphi +i\sin \varphi }
Dice half-life decay.jpg
Author/Creator: 13hartc, Licence: CC BY-SA 4.0
ASTRO 1060 lab photo
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Cross-correlation of two inputs - Simulink - MathWorks Switzerland
Cross-correlation of two inputs
The Correlation block computes the cross-correlation of two N-D input arrays along the first-dimension. The computation can be done in the time domain or frequency domain. You can specify the domain through the Computation domain parameter. In the time domain, the block convolves the first input signal, u, with the time-reversed complex conjugate of the second input signal, v. In the frequency domain, to compute the cross-correlation, the block:
Takes the Fourier transform of both input signals, U and V.
Multiplies U and V*, where * denotes the complex conjugate.
Computes the inverse Fourier transform of the product.
If you set Computation domain to Fastest, the block chooses the domain that minimizes the number of computations. For information on these computation methods, see Algorithms.
The block accepts real-valued or complex-valued multichannel and multidimensional inputs. The input can be a fixed-point signal when you set the Computation domain to Time. When one or both of the input signals are complex, the output signal is also complex.
Port_1 — Cross-correlated output
Cross-correlated output of the two input signals.
When the inputs are N-D arrays, the object outputs an N-D array, where all the dimensions, except for the first dimension, match with the input array. For example,
When the inputs u and v have dimensions Mu-by-N-by-P and Mv-by-N-by-P, respectively, the Correlation block outputs an (Mu + Mv – 1)-by-N-by-P array.
When the inputs u and v have the dimensions Mu-by-N and Mv-by-N, the block outputs an (Mu + Mv – 1)-by-N matrix.
If one input is a column vector and the other input is an N-D array, the Correlation block computes the cross-correlation of the vector with each column in the N-D array. For example,
When the input u is an Mu-by-1 column vector and v is an Mv-by-N matrix, the block outputs an (Mu + Mv – 1)-by-N matrix.
Similarly, when u and v are column vectors with lengths Mu and Mv, respectively, the block performs the vector cross-correlation.
Computation domain — Domain in which the block computes the cross-correlation
Time (default) | Frequency | Fastest
Time — Computes the cross-correlation in the time domain, which minimizes the memory usage.
Frequency — Computes the cross-correlation in the frequency domain. For more information, see Algorithms.
Fastest — Computes the cross-correlation in the domain that minimizes the number of computations.
To cross-correlate fixed-point signals, set this parameter to Time.
Product output specifies the data type of the output of a product operation in the Correlation block. For more information on the product output data type, see Multiplication Data Types and the 'Fixed-Point Conversion' section in Extended Capabilities.
Accumulator specifies the data type of output of an accumulation operation in the Correlation block. For illustrations on how to use the accumulator data type in this block, see the 'Fixed-Point Conversion' section in Extended Capabilities.
Output specifies the data type of the output of the Correlation block. For more information on the output data type, see the 'Fixed-Point Conversion' section in Extended Capabilities.
Output Minimum — Minimum value block can output
Cross-correlation is the measure of similarity of two discrete-time sequences as a function of the lag of one relative to the other.
For two length-N deterministic inputs or realizations of jointly wide-sense stationary (WSS) random processes, x and y, the cross-correlation is computed using the following relationship:
\begin{array}{l}{r}_{xy}\left(h\right)=\left\{\begin{array}{ll}\sum _{n=0}^{N-h-1}x\left(n+h\right){y}^{*}\left(n\right)\hfill & 0\le h\le N-1\hfill \\ {r}_{yx}^{*}\left(h\right)\hfill & -\left(N-1\right)\le h\le 0\hfill \end{array}\\ \end{array}
where h is the lag and * denotes the complex conjugate. If the inputs are realizations of jointly WSS stationary random processes, rxy(h) is an unnormalized estimate of the theoretical cross-correlation:
{\rho }_{xy}\left(h\right)=E\left\{x\left(n+h\right){y}^{*}\left(n\right)\right\}
where E{ } is the expectation operator.
When you set the computation domain to time, the algorithm computes the cross-correlation of two signals in the time domain. The input signals can be fixed-point signals in this domain.
Correlate Two 2-D Arrays
When the inputs are two 2-D arrays, the jth column of the output, yuv, has these elements:
\begin{array}{l}{y}_{uv\left(i,j\right)}=\sum _{k=0}^{\mathrm{max}\left({M}_{u},{M}_{v}\right)-1}{u}_{k,j}^{*}{v}_{\left(k+i\right),j}^{}\text{ }0\le i<{M}_{v}\\ \\ {y}_{uv\left(i,j\right)}={y}_{vu\left(-i,j\right)}^{*}\text{ }-{M}_{u}<i<0\end{array}
u is an Mu-by-N input matrix.
v is an Mv-by-N input matrix.
yu,v is an (Mu + Mv – 1)-by-N matrix.
Inputs u and v are zero when indexed outside their valid ranges.
Correlate a Column Vector with a 2-D Array
When one input is a column vector and the other input is a 2-D array, the algorithm independently cross-correlates the input vector with each column of the 2-D array. The jth column of the output, yu,v, has these elements:
\begin{array}{l}{y}_{uv\left(i,j\right)}=\sum _{k=0}^{\mathrm{max}\left({M}_{u},{M}_{v}\right)-1}{u}_{k}^{*}{v}_{\left(k+i\right),j}^{}\text{ }0\le i<{M}_{v}\\ \\ {y}_{uv\left(i,j\right)}={y}_{vu\left(-i,j\right)}^{*}\text{ }-{M}_{u}<i<0\end{array}
u is an Mu-by-1 column vector.
v is an Mv-by-N matrix.
yuv is an (Mu + Mv – 1)-by-N matrix.
Correlate Two Column Vectors
When the inputs are two column vectors, the jth column of the output, yuv, has these elements:
\begin{array}{l}{y}_{uv\left(i\right)}=\sum _{k=0}^{\mathrm{max}\left({M}_{u},{M}_{v}\right)-1}{u}_{k}^{*}{v}_{\left(k+i\right)}^{}\text{ }0\le i<{M}_{v}\\ \\ {y}_{uv\left(i\right)}={y}_{vu\left(-i\right)}^{*}\text{ }-{M}_{u}<i<0\end{array}
v is an Mv-by-1 column vector.
yuv is an (Mu + Mv – 1)-by-1 column vector.
When you set the computation domain to frequency, the algorithm computes the cross-correlation in the frequency domain.
To compute the cross-correlation, the algorithm:
In this domain, depending on the input length, the algorithm can require fewer computations.
The following diagram shows the data types the Correlation block uses for fixed-point signals (time domain only).
When the input is real, the output of the multiplier is in the product output data type. When the input is complex, the output of the multiplier is in the accumulator data type. For details on the complex multiplication performed, see Multiplication Data Types.
When one or both of the inputs are signed fixed-point signals, all internal block data types are signed fixed point. The internal block data types are unsigned fixed point only when both inputs are unsigned fixed-point signals.
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Solve heat transfer, structural analysis, or electromagnetic analysis problem - MATLAB solve - MathWorks í•œêµ
Apply a constant temperature of 100 °C to the left side of the block (face 1) and a constant temperature of 300 °C to the right side of the block (face 3). All other faces are insulated by default.
10\text{â}\mathrm{W}/\left(\mathrm{m}{â
}^{â}\mathrm{C}\right)
2\text{â}\mathrm{kg}/{\mathrm{m}}^{3}
0.1\text{â}\mathrm{J}/\left({\mathrm{kg}â
}^{â}\mathrm{C}\right)
2\text{â}\mathrm{W}/\left(\mathrm{m}{â
}^{â}\mathrm{C}\right)
1\text{â}\mathrm{kg}/{\mathrm{m}}^{3}
0.1\text{â}\mathrm{J}/\left({\mathrm{kg}â
}^{â}\mathrm{C}\right)
4\text{â}\mathrm{W}/{\mathrm{m}}^{2}
Apply a constant temperature of 0 °C to the sides of the square plate.
Set the initial temperature to 0 °C.
10\text{â}\mathrm{W}/\left(\mathrm{m}{â
}^{â}\mathrm{C}\right)
2\text{â}\mathrm{kg}/{\mathrm{m}}^{3}
0.1\text{â}\mathrm{J}/\left({\mathrm{kg}â
}^{â}\mathrm{C}\right)
2\text{â}\mathrm{W}/\left(\mathrm{m}{â
}^{â}\mathrm{C}\right)
1\text{â}\mathrm{kg}/{\mathrm{m}}^{3}
0.1\text{â}\mathrm{J}/\left({\mathrm{kg}â
}^{â}\mathrm{C}\right)
4\text{â}\mathrm{W}/{\mathrm{m}}^{2}
{s}^{-1}
2\mathrm{Ï}
|
\delta
\theta
\begin{array}{l}\left[\begin{array}{c}\underset{}{\overset{˙}{\alpha }}\\ \underset{}{\overset{˙}{q}}\\ \underset{}{\overset{˙}{\theta }}\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}=\phantom{\rule{0.2777777777777778em}{0ex}}\left[\begin{array}{ccc}-0.313& 56.7& 0\\ -0.0139& -0.426& 0\\ 0& 56.7& 0\end{array}\right]\left[\begin{array}{c}\alpha \\ q\\ \theta \end{array}\right]+\left[\begin{array}{c}0.232\\ 0.0203\\ 0\end{array}\right]\left[\delta \right]\\ \phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}y\phantom{\rule{0.2777777777777778em}{0ex}}=\phantom{\rule{0.2777777777777778em}{0ex}}\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\left[\begin{array}{c}\alpha \\ q\\ \theta \end{array}\right]+\left[0\right]\left[\delta \right]\end{array}
\theta
\begin{array}{l}\left[\begin{array}{c}\underset{}{\overset{˙}{x}}\\ \underset{}{\overset{¨}{x}}\\ \underset{}{\overset{˙}{\theta }}\\ \underset{}{\overset{¨}{\theta }}\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}=\phantom{\rule{0.2777777777777778em}{0ex}}\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -0.1& 3& 0\\ 0& 0& 0& 1\\ 0& -0.5& 30& 0\end{array}\right]\left[\begin{array}{c}x\\ \underset{}{\overset{˙}{x}}\\ \theta \\ \underset{}{\overset{˙}{\theta }}\end{array}\right]+\left[\begin{array}{c}0\\ 2\\ 0\\ 5\end{array}\right]u\\ \phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}y\phantom{\rule{0.2777777777777778em}{0ex}}=\phantom{\rule{0.2777777777777778em}{0ex}}\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}x\\ \underset{}{\overset{˙}{x}}\\ \theta \\ \underset{}{\overset{˙}{\theta }}\end{array}\right]+\left[\begin{array}{c}0\\ 0\end{array}\right]u\end{array}
sys\left(s\right)=100\frac{\left(s-1\right)\left(s+1\right)}{s\left(s+10\right)\left(s+10.0001\right){\left(s-\left(1+i\right)\right)}^{2}{\left(s-\left(1-i\right)\right)}^{2}}
You cannot use frequency-response data models such as frd models.
\left({\lambda }_{1},\sigma ±j\omega ,{\lambda }_{2}\right)
\left[\begin{array}{cccc}{\lambda }_{1}& 0& 0& 0\\ 0& \sigma & \omega & 0\\ 0& -\omega & \sigma & 0\\ 0& 0& 0& {\lambda }_{2}\end{array}\right]
P\left(s\right)={s}^{n}+{\alpha }_{1}{s}^{n-1}+\dots +{\alpha }_{n-1}s+{\alpha }_{n}
A=\left[\begin{array}{ccc}\begin{array}{l}0\\ 1\\ 0\\ 0\\ ⋮\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}& \begin{array}{ccc}\begin{array}{cc}\begin{array}{l}0\\ 0\\ 0\\ 1\\ ⋮\\ 0\end{array}& \begin{array}{l}\dots \\ \dots \\ \dots \\ \dots \\ \ddots \\ \dots \end{array}\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 0\\ ⋮\\ 1\end{array}& \begin{array}{l}-{\alpha }_{n}\\ -{\alpha }_{n-1}\\ -{\alpha }_{n-2}\\ -{\alpha }_{n-3}\\ \text{ }⋮\\ -{\alpha }_{1}\end{array}\end{array}\end{array}\right]
Canonical state-space form of the dynamic model, returned as an ss model object. csys is a state-space realization of sys in the canonical form specified by type.
This argument is available only when sys is an ss model object.
The canon command uses the bdschur command to convert sys into modal form and to compute the transformation T. If sys is not a state-space model, canon first converts it to state space using ss.
ctrb | ctrbf | ss2ss | tf | zpk | ss | pid | genss | uss (Robust Control Toolbox) | idtf (System Identification Toolbox) | idss (System Identification Toolbox) | idproc (System Identification Toolbox) | idpoly (System Identification Toolbox) | idgrey (System Identification Toolbox)
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Chain conditions and continuous mappings on $C_p(X)$
Chain conditions and continuous mappings on
{C}_{p}\left(X\right)
Kalamidas, N. D.
author = {Kalamidas, N. D.},
title = {Chain conditions and continuous mappings on $C_p(X)$},
AU - Kalamidas, N. D.
TI - Chain conditions and continuous mappings on $C_p(X)$
Kalamidas, N. D. Chain conditions and continuous mappings on $C_p(X)$. Rendiconti del Seminario Matematico della Università di Padova, Tome 87 (1992), pp. 19-27. http://www.numdam.org/item/RSMUP_1992__87__19_0/
[1] A.V. Arhangel'Skii, Function spaces in the topology of pointwise convergence and compact sets, Uspekhi Mat. Nank., 39:5 (1984), pp. 11-50. | Zbl 0568.54016
[2] A.V. Arhangel'Skii, On linear homeomorphisms of function spaces, Soviet Math. Dokl., 25, No. 3 (1982). | Zbl 0522.54015
[3] A.V. Arhangel'Skii - V.V. Tkačuk, Calibers and point-finite cellularity of the space Cp(X) and questions of S. Gulko and M. Husek, Topology and its Applications, 23 (1986), pp. 65-73, North-Holland. | MR 849094 | Zbl 0591.54023
[4] W.W. Comfort - S. Negrepontis, Chain Conditions in Topology, Cambridge Tracts in Mathematics, Vol. 79, Cambridge University Press, Cambridge (1982). | MR 665100 | Zbl 0488.54002
[5] B.A. Efimov, Mappings and embeddings of dyadic spaces, Mat. Sb., 103 (1977), pp. 52-68. | MR 454939 | Zbl 0351.54017
[6] H. Rosenthal, On injective Banach spaces and the spaces L∞ (μ) for finite measures μ, Acta Mathematica, 124 (1970), pp. 205-248. | Zbl 0207.42803
[7] V.V. Tkačuk, The spaces Cp(X): Decomposition into a countable union of bounded subspaces and completeness properties, Topology and its Applications, 22 (1986), pp. 241-253. | MR 842658 | Zbl 0596.54015
[8] V.V. Tkačuk, The smallest subring of the ring Cp(C p(X)) containing χU is everywhere dense in Cp(C p(X)), Vestnik Moskovskogo Universiteta Matematica, 42, No. 1 (1987), pp. 20-23. | Zbl 0624.54015
[9] A.I. Tulcea, On pointwise convergence, compactness and equicontinuity, Adv. Math., 12 (1974), pp. 171-177. | MR 405103 | Zbl 0301.46032
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The yearbook staff at Jefferson Middle School has been busy taking pictures. Of the
574
pictures the staff members have taken, only
246
of the pictures will make it into the yearbook. Approximately what portion of the pictures that were taken will make it into the yearbook? Use a complete portions web to show your answer. Show your work.
You need to complete each portion of the portions web shown at right.
Start by representing this as a fraction.
\frac{246}{574}=\frac{123}{287}
Next, convert this to a percent.
\frac{123}{287}\approx \frac{43}{100}=43\%
Now try writing this as a decimal and in words on your own! Put them all together for a completed portions web.
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Lagrange's theorem (group theory) - Simple English Wikipedia, the free encyclopedia
Lagrange's theorem in group theory states if G is a finite group and H is a subgroup of G, then |H| (how many elements are in H, called the order of H) divides |G|. Moreover, the number of distinct left (right) cosets of H in G is |G|/|H|. This theorem is named after mathematician Joseph-Louis Lagrange.
For any g in a group G,
{\displaystyle g^{k}=e}
for some k that divides the |G|
Any group of prime order cyclic (Any element in G can be created by a single element) and simple (no normal subgroups that aren't trivial)
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Lagrange%27s_theorem_(group_theory)&oldid=8044661"
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If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ. VIEW SOLUTION
Write the element a23 of a 3 ✕ 3 matrix A = (aij) whose elements aij are given by
{a}_{ij}=\frac{\left|i-j\right|}{2}.
Find the differential equation representing the family of curves
\mathrm{v}=\frac{\mathrm{A}}{\mathrm{r}}
+ B, where A and B are arbitrary constants. VIEW SOLUTION
Find the integrating factor of the differential equation
\left(\frac{{{e}^{-2}}^{\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{dx}{dy}=1
\stackrel{\to }{\mathrm{a}}=7\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{j}}-4 \stackrel{^}{\mathrm{k}} \mathrm{and} \stackrel{\to }{\mathrm{b}}=2 \stackrel{^}{\mathrm{i}} + 6 \stackrel{^}{\mathrm{j}} + 3\stackrel{^}{\mathrm{k}},
then find the projection of
\stackrel{\to }{\mathrm{a}} \mathrm{on}\stackrel{\to }{\mathrm{b}}
Find λ, if the vectors
\stackrel{\to }{a}=\stackrel{^}{i}+3\stackrel{^}{j}+\stackrel{^}{k}, \stackrel{\to }{b}=2\stackrel{^}{i}-\stackrel{^}{j}-\stackrel{^}{k} \mathrm{and} \stackrel{\to }{c}=\lambda \stackrel{^}{j}+3\stackrel{^}{k}
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained. VIEW SOLUTION
\mathrm{If} \stackrel{\to }{\mathrm{r}}=\mathrm{x}\stackrel{^}{\mathrm{i}}+\mathrm{y}\stackrel{^}{\mathrm{j}}+\mathrm{z}\stackrel{^}{\mathrm{k}}, \mathrm{find} \left(\stackrel{\to }{\mathrm{r}}×\stackrel{^}{\mathrm{i}}\right)·\left(\stackrel{\to }{\mathrm{r}}×\mathrm{j}\right)+\mathrm{xy}
Find the distance between the point (−1, −5, −10) and the point of intersection of the line
\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}
and the plane x − y + z = 5. VIEW SOLUTION
If sin [cot−1 (x+1)] = cos(tan−1x), then find x.
If (tan−1x)2 + (cot−1x)2 =
\frac{5{\mathrm{\pi }}^{2}}{8}
, then find x. VIEW SOLUTION
y={\mathrm{tan}}^{-1} \left(\frac{\sqrt{1+{x}^{2}}+\sqrt{1-{x}^{2}}}{\sqrt{1+{x}^{2}}-\sqrt{1-{x}^{2}}}\right), {x}^{2}\le 1,
\frac{dy}{dx}
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that
{y}^{2}\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+y=0.
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ? VIEW SOLUTION
\int \left(x+3\right)\sqrt{3-4x-{x}^{2}} \mathrm{dx}
Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:
Article A B C
Hand-fans 40 25 35
Mats 50 40 50
Plates 20 30 40
Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.
Write one value generated by the above situation. VIEW SOLUTION
\mathrm{A}=\left(\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right)
{\mathrm{A}}^{2}-5\mathrm{A}+4\mathrm{I}
and hence find a matrix X such that
{\mathrm{A}}^{2}-5\mathrm{A}+4\mathrm{I}+\mathrm{X}=\mathrm{O}
\mathrm{A}=\left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 4\\ -2& 2& 1\end{array}\right], \mathrm{find} {\left(\mathrm{A}\text{'}\right)}^{-1}.
f\left(x\right)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^{2}& ax& a\end{array}\right|
, using properties of determinants find the value of f(2x) − f(x). VIEW SOLUTION
\int \frac{dx}{\mathrm{sin} x+\mathrm{sin} 2x}
Integrate the following w.r.t. x
\frac{{x}^{2}-3x+1}{\sqrt{1-{x}^{2}}}
\underset{-\mathrm{\pi }}{\overset{\mathrm{\pi }}{\int }} {\left(\mathrm{cos} ax-\mathrm{sin} bx\right)}^{2} dx
\left({\mathrm{tan}}^{-1}y-x\right)dy=\left(1+{y}^{2}\right)dx.
\frac{dy}{dx}=\frac{xy}{{x}^{2}+{y}^{2}}
given that y = 1, when x = 0. VIEW SOLUTION
\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} \mathrm{and} \frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}
intersect, then find the value of k and hence find the equation of the plane containing these lines. VIEW SOLUTION
\mathrm{P}\left(\overline{\mathrm{A}} \cap \mathrm{B}\right) =\frac{2}{15} \mathrm{and} \mathrm{P}\left(\mathrm{A} \cap \overline{\mathrm{B}}\right) = \frac{1}{6},
then find P(A) and P(B). VIEW SOLUTION
Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π. VIEW SOLUTION
Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :
2x + 4y \le 8\phantom{\rule{0ex}{0ex}}3x + y \le 6\phantom{\rule{0ex}{0ex}}x + y \le 4\phantom{\rule{0ex}{0ex}}x \ge 0, y\ge 0\phantom{\rule{0ex}{0ex}}
Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation. VIEW SOLUTION
Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
{x}^{2}+{y}^{2}=4 \mathrm{at} \left(1, \sqrt{3}\right)
\underset{1}{\overset{3}{\int }}\left({e}^{2-3x}+{x}^{2}+1\right) dx
as a limit of a sum.
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torch.lstsq — PyTorch 1.11.0 documentation
torch.lstsq
torch.lstsq¶
torch.lstsq(input, A, *, out=None)¶
A
(m \times n)
B
(m \times k)
m \geq n
, lstsq() solves the least-squares problem:
\begin{array}{ll} \min_X & \|AX-B\|_2. \end{array}
m < n
, lstsq() solves the least-norm problem:
\begin{array}{llll} \min_X & \|X\|_2 & \text{subject to} & AX = B. \end{array}
Returned tensor
X
(\max(m, n) \times k)
n
rows of
X
contains the solution. If
m \geq n
, the residual sum of squares for the solution in each column is given by the sum of squares of elements in the remaining
m - n
rows of that column.
torch.lstsq() is deprecated in favor of torch.linalg.lstsq() and will be removed in a future PyTorch release. torch.linalg.lstsq() has reversed arguments and does not return the QR decomposition in the returned tuple, (it returns other information about the problem). The returned solution in torch.lstsq() stores the residuals of the solution in the last m - n columns in the case m > n . In torch.linalg.lstsq(), the residuals are in the field ‘residuals’ of the returned named tuple.
Unpacking the solution as X = torch.lstsq(B, A).solution[:A.size(1)] should be replaced with
X = torch.linalg.lstsq(A, B).solution
m < n
is not supported on the GPU.
input (Tensor) – the matrix
B
A (Tensor) – the
m
n
A
out (tuple, optional) – the optional destination tensor
A namedtuple (solution, QR) containing:
solution (Tensor): the least squares solution
QR (Tensor): the details of the QR factorization
The returned matrices will always be transposed, irrespective of the strides of the input matrices. That is, they will have stride (1, m) instead of (m, 1) .
>>> A = torch.tensor([[1., 1, 1],
... [2, 3, 4],
... [5, 4, 3]])
>>> B = torch.tensor([[-10., -3],
... [ 12, 14],
... [ 18, 16]])
>>> X, _ = torch.lstsq(B, A)
tensor([[ 2.0000, 1.0000],
|
To R. F. Cooke 17 June [1874]1
I really feel unable to express any opinion how many copies of the Descent had better be struck off; & I can only hope that 2000 wd. be safe.— From what you say the price cannot be under 12s & I am sorry for it.2 You will not be put to much expence for corrections, for as yet I have not had a single sheet revised, & the corrections have been few & slight in the first proofs.— Mr Murray asked me to keep the proofs & send them to him, as a check to the Mess Clowes; but this has been impossible, as I have had only first proofs & these are of course returned to Printers.—3 The Printing gets on slowly, & I have not yet done more than
\frac{1}{4}
r of vol.; & it will now be rather slower as I must send proofs to Germany where my son is going for health sake, & he looks them over before I do.—4
I heartily hope our new Edit. may be fairly successful.— I am sure that it is a much improved book.—
Please send copy of my Orchis book5 to
Mrs. Litchfield6
2. Bryanston St
Portman Sqre—
The year is established by the relationship between this letter and the letter from R. F. Cooke, 16 June 1874.
See letter from R. F. Cooke, 16 June 1874. For CD’s concerns over the production of a cheap edition of Descent 2d ed., see the letter to R. F. Cooke, 10 April [1874], and the letter to John Murray, 12 April 1874.
William Clowes & Sons were the printers used by CD’s publisher, John Murray. The first revise of Descent 2d ed., date-stamped 24 September 1874 by Clowes, is in DAR 213: 3.
George Howard Darwin was checking the proofs for the second edition of Descent; he left for the continent on 18 June 1874 (letter from G. H. Darwin, 30 May 1874; Emma Darwin’s diary (DAR 242)).
Henrietta Emma Litchfield was CD’s eldest daughter.
Hopes a printing of 2000 copies [of Descent, 2d ed.] will be safe. Regrets price must be 12s. He is sure it is much improved.
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Ask Answer - The Triangle and its Properties - Expert Answered Questions for School Students
Two towers of heights 20m and 36m are built at a distance of 63 m . Find the distance between the tops of the tower
what s pythagoras property
If one angle of a trianlgle is 60 and the other two angles are in the ratio 1:2 , find the angles.
in triangle EDC, EC=12cm, AC=4cm and ED= 5cm. Find the area if triangle EDC and length of DB. Expert pls answer asap
{Please don't provide any weblink or certified answer... I want only experts answer} ALSO PLEASE DON'T SAY THAT THIS IS OF MY GRADE ... QUESTION:-
Solve by providing solution for the following:-
Statement 1: The difference between the lengths of any two sides of a triangle is less than the length of the third side.
Experts, I am not saying the below statement 2 ,,, I just want the proof for the statement 1. So, don't confuse while giving the proof. Also, I already know that proof for the statement 2.
Statement 2: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
{Please don't provide any weblink or certified answer... I want only experts answer} ... QUESTION:-
{Please don't provide any web link or certified answer... I want only experts answer}
Q. In the following triangle ABC, BN is the median.
Derive that the median of the triangle is:
\frac{1}{2}\sqrt{2{a}^{2}+2{c}^{2}-{b}^{2}}
Given is a triangle PQR in which PQ=PR. Find x and y.
Ans 9 10 and 12 plzzz??
I need solution for these problems
Meritnation,
In triangle ABC, CM bisects AB equally at M and BQ bisects to CM equally at P. If Q is on AC, then prove that AQ = 2QC.
{Please don't provide certified answer... I want only experts answer to any without providing any web link or certified answer}
Please give me today its so important for me pleaseeee.
Q.3. Find the value of unknown side by using Pythagoras theorem.
In triangle ABC AD, BE & CF are the medians from A,B,C. Show that AB+BC+CA>AD+BE+CF.
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Vectorize the Objective and Constraint Functions - MATLAB & Simulink - MathWorks í•œêµ
f\left(x\right)={x}_{1}^{4}+{x}_{2}^{4}â4{x}_{1}^{2}â2{x}_{2}^{2}+3{x}_{1}â{x}_{2}/2.
Your nonlinear constraint function returns two matrices, one for inequality constraints, and one for equality constraints. Suppose there are nc nonlinear inequality constraints and nceq nonlinear equality constraints. For row vector x0, the constraint matrices have nc and nceq columns respectively, and the number of rows is the same as in the input matrix. Similarly, for a column vector x0, the constraint matrices have nc and nceq rows respectively, and the number of columns is the same as in the input matrix. In figure Structure of Vectorized Functions, “Results†includes both nc and nceq.
\begin{array}{c}\frac{{x}_{1}^{2}}{9}+\frac{{x}_{2}^{2}}{4}â¤1\text{ (the interior of an ellipse),}\\ {x}_{2}â¥\mathrm{cosh}\left({x}_{1}\right)â1.\end{array}
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Cyclone IV-based Julia Set Explorer - Ian Kilgore
Cyclone IV-based Julia Set Explorer
A custom Cyclone IV board interfaces with TFT LCD panel, capacitive touch controller, SDR SDRAM. The FPGA superimposes a Julia set on the Mandelbrot set. Touching the screen chooses the set point
z_0
Check out Malin Christersson’s julia set visualizations here. This is basically an implementation of the first demo on that page. One thing that should be clear is that there’s no need for an FPGA to do this; it’s in fact exactly the wrong approach. This project was a solution in search of a problem: I started with a Cyclone II and an LCD with no application in mind and it evolved from there. I got to use it as a vehicle to learn about FPGAs, verilog, and push my PCB layout and assembly capabilities.
All of the design files can be found on github at https://github.com/iank/julia_lcd.
The final evolution of the hardware is a Cyclone IV FPGA, an external SDRAM, and a Silicon Labs EFM8 microcontroller used to configure the FPGA from a SPI flash. The LCD module contains a capacitive touch controller, broken out on the second, smaller, flex cable.
The FPGA’s internal blockrams are not large enough to contain an entire frame, hence the external memory. Lines are loaded from memory during the blanking interval before they are needed. Otherwise when the memory bus is not in use the fractal can be computed. First the Mandelbrot set is computed as a binary image and the Julia set is overlaid upon it.
An i2c master is also implemented. This controls the touch screen. Upon receipt of a touchscreen interrupt, the coordinate is queried and this is used to set the
z_0
point for the Julia set.
I’d never used an FPGA before this project, aside from some trivial examples in school. I spent a lot of time iterating on the Verilog here, and I learned a lot about what I could and could not do (as well as the memory bandwidth / LUTs I’d need to do it). This was also the first time I assembled a BGA part at home.
Using a separate debug/programming board was handy. In one spin I had tried using a board-to-board connector for this, which turned out to not be tenable without mechanical support. The IPC cable pictured above was easier to assemble and didn’t really take up more space.
FPGA debugging in-system is hard. Taking the time to set up simulation models/fixtures was worth it every time. First I wrote a mock LCD module that would output frames as png files during the simulation. That made clear some timing/alignment issues that just weren’t visible otherwise. Then I adapted a memory model from Micron for the specific part I used, which helped debug some memory issues.
Ian Kilgore is an electrical engineer in the Research Triangle area.
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Asset Pricing - Wikibooks, open books for an open world
(very much work in progress - feel free to contribute)
Financial markets serve several purposes:
Allowing investors to shift consumption in time
Allowing the transfer of risk between different investors
Moving capital from those who have it to those who can use it productively
Asset pricing is the study of how financial assets are priced.
Financial assets include several varieties:
Whatever the particular variety, we can think of financial assets simply as the right to a future cash flow stream and/or physical asset. All types of asset, e.g., debt, equity, etc., can be reduced to a stream of cash flows that can be valued.
When valuing a stream of cash flows we need to consider these things:
Riskiness of the cash flows
Expected value of the cash flows
For debt, asset pricing is relatively simple, as cash flows to the owner are contractually fixed. For example, the holder of a 20-year US government bond with a face value of $100 and a coupon of 5% paid annual can expect (with high certainty) to be paid $5 a year for the next 20 years, with $100 to be returned at the end of 20 years. The value of the bond is the present value of the future cash flows. If the required rate of return is r=3% a year, then the value is:
{\displaystyle {\frac {5}{r}}*(1-{\frac {1}{(1+r)^{20}}})+{\frac {100}{(1+r)^{20}}}=129.75}
Although debt security can have predictable cash flow in the future (e.g., assuming we are talking about fixed rate debt security) there still is an element of uncertainty because expected rate of return is likely to change over the above mentioned 20 years time period. This would change the present value of the debt security depending on the assumption or derivation of the rate of return that applies to each individual time period. This adds interest rate risk component to the debt security valuation.
If the debt issuer is not a "AAA" rated sovereign state then we cannot assume the cash flow is guaranteed. This adds a credit risk component. Debt securities issued by all entities are rated by investment grade rating services (e.g.,. Moody's, Standard & Poor's) and this can be used to determine the required rate of return to account for the risk, which will the impact the valuation of the debt security.
For equities, asset pricing is more difficult as future cash flows are uncertain, and vary with both economic conditions and the fortune of the company. We need to project future expected cash flows, and also determine the expected return of the stock. The estimated expected return of the stock is based on an estimate of how risky the cash flows are. We can decompose risk into a) common factor risk and b) idiosyncratic risk. The latter can be diversified away in a portfolio, so earns no return premium. The former cannot be diversified away, so earns a risk premium.
Suggested reading: Cochrane, Asset Pricing http://www.pupress.princeton.edu/titles/7836.html
Sharpe, Investors and Markets http://www.stanford.edu/~wfsharpe/art/princeton/pup1.pdf
Retrieved from "https://en.wikibooks.org/w/index.php?title=Asset_Pricing&oldid=3738950"
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Stanch Snap-Stress Tensor - Vixrapedia
Stanch Snap-Stress Tensor
The Stanch Snap-Stress Tensor is a measure of the Stanch snap that microstructures can undergo when put under extreme stress.
The Stanch Snap-Stress Tensor (SSST),
{\displaystyle S}
{\displaystyle S=c_{m}\partial _{\mu }(g^{\mu \nu }{B^{p}}_{n}f_{p\nu })}
{\displaystyle c_{m}}
are climbing factors,
{\displaystyle g}
is the metric,
{\displaystyle B}
are the so-called bottle paths and
{\displaystyle f}
is the fracture trajectory over the bottle path.
The SSST can be integrated over spacetime to give a numerical stress factor,
{\displaystyle s}
{\displaystyle s={\frac {\int \partial _{B}S(x,t,c)\,dt\,d^{3}x}{\int g\,dt\,d^{3}x}},}
where here we have expressed the stress as a ratio
{\displaystyle 0<s<1}
{\displaystyle s=0}
, the system is not Stanch snap susceptible, but at
{\displaystyle s=B/h\approx 0.6}
(for most materials), the system is gaining stress. It has been speculated that
{\displaystyle s\neq 1}
under any situation since the Stanch snap-stresses will realign the material, but it has been proposed that under certain Alexian conditions,
{\displaystyle s\rightarrow 1}
can be observed.
The tensor was defined by Agellio Stanch.
Retrieved from "https://www.vixrapedia.org/w/index.php?title=Stanch_Snap-Stress_Tensor&oldid=3892"
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Extracting Numerical Model Data - MATLAB & Simulink - MathWorks Nordic
You can extract the following numerical data from linear model objects:
Coefficients and uncertainty
For example, extract state-space matrices (A, B, C, D and K) for state-space models, or polynomials (A, B, C, D and F) for polynomial models.
If you estimated model uncertainty data, this information is stored in the model in the form of the parameter covariance matrix. You can fetch the covariance matrix (in its raw or factored form) using the getcov command. The covariance matrix represents uncertainties in parameter estimates and is used to compute:
Confidence bounds on model output plots, Bode plots, residual plots, and pole-zero plots
Standard deviation in individual parameter values. For example, one standard deviation in the estimated value of the A polynomial in an ARX model, returned by the polydata command and displayed by the present command.
The following table summarizes the commands for extracting model coefficients and uncertainty.
Commands for Extracting Model Coefficients and Uncertainty Data
freqresp Extracts frequency-response data (H) and corresponding covariance (CovH) from any linear identified model.
[H,w,CovH] = freqresp(m)
polydata Extracts polynomials (such as A) from any linear identified model. The polynomial uncertainties (such as dA) are returned only for idpoly models.
[A,B,C,D,F,dA,dB,dC,dD,dF] = ...
polydata(m)
idssdata Extracts state-space matrices (such as A) from any linear identified model. The matrix uncertainties (such as dA) are returned only for idss models.
[A,B,C,D,K,X0,...
dA,dB,dC,dD,dK,dX0] = ...
idssdata(m)
tfdata Extracts numerator and denominator polynomials (Num, Den) and their uncertainties (dnum, dden) from any linear identified model.
[Num,Den,Ts,dNum,dDen] = ...
tfdata(m)
zpkdata Extracts zeros, poles, and gains (Z, P, K) and their covariances (covZ, covP, covK) from any linear identified model.
[Z,P,K,Ts,covZ,covP,covK] = ...
zpkdata(m)
getpvec Obtain a list of model parameters and their uncertainties.
To access parameter attributes such as values, free status, bounds or labels, use getpar.
pvec = getpvec(m)
getcov Obtain parameter covariance information
cov_data = getcov(m)
You can also extract numerical model data by using dot notation to access model properties. For example, m.A displays the A polynomial coefficients from model m. Alternatively, you can use the get command, as follows: get(m,'A').
To view a list of model properties, type get(model).
Dynamic and noise models
y=Gu+He
G is an operator that takes the measured inputs u to the outputs and captures the system dynamics, also called the measured model. H is an operator that describes the properties of the additive output disturbance and takes the hypothetical (unmeasured) noise source inputs e to the outputs, also called the noise model. When you estimate a noise model, the toolbox includes one noise channel e for each output in your system.
You can operate on extracted model data as you would on any other MATLAB® vectors, matrices and cell arrays. You can also pass these numerical values to Control System Toolbox™ commands, for example, or Simulink® blocks.
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All-flavour search for neutrinos from dark matter annihilations in the Milky Way with IceCube/DeepCore - Ajman University
We present the first IceCube search for a signal of dark matter annihilations in the Milky Way using all-flavour neutrino-induced particle cascades. The analysis focuses on the DeepCore sub-detector of IceCube, and uses the surrounding IceCube strings as a veto region in order to select starting events in the DeepCore volume. We use 329 live-days of data from IceCube operating in its 86-string configuration during 2011–2012. No neutrino excess is found, the final result being compatible with the background-only hypothesis. From this null result, we derive upper limits on the velocity-averaged self-annihilation cross-section, ⟨σAv⟩
⟨{\sigma }_{A}\mathrm{v}⟩
, for dark matter candidate masses ranging from 30 GeV up to 10 TeV, assuming both a cuspy and a flat-cored dark matter halo profile. For dark matter masses between 200 GeV and 10 TeV, the results improve on all previous IceCube results on ⟨σAv⟩
⟨{\sigma }_{A}\mathrm{v}⟩
, reaching a level of 10−23
{}^{-23}
{}^{3}
{}^{-1}
, depending on the annihilation channel assumed, for a cusped NFW profile. The analysis demonstrates that all-flavour searches are competitive with muon channel searches despite the intrinsically worse angular resolution of cascades compared to muon tracks in IceCube.
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Preston picked five playing cards and got a
2, 3, 6, 5
1
What two-digit and three-digit numbers could he create that would have the greatest sum? Is there more than one possibility? What is that sum?
There are multiple ways to reach the greatest sum possible (
683
). Petra chose to use the combination
651 + 32
. Which expression did you use?
What two-digit and three-digit numbers could he create that would have the smallest sum? Is there more than one possibility? What is that sum?
It would be helpful to put the lowest numbers in the highest place value, because you want these numbers to be of less value.
There are also multiple ways to reach the smallest sum (
161
). Did you find a combination that works?
Use the eTool below to create possible combinations.
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Train YOLO v2 object detector - MATLAB trainYOLOv2ObjectDetector - MathWorks 日本
trainYOLOv2ObjectDetector
Multiscale Training
detector = trainYOLOv2ObjectDetector(trainingData,lgraph,options)
[detector,info] = trainYOLOv2ObjectDetector(___)
detector = trainYOLOv2ObjectDetector(trainingData,checkpoint,options)
detector = trainYOLOv2ObjectDetector(trainingData,detector,options)
detector = trainYOLOv2ObjectDetector(___,'TrainingImageSize',trainingSizes)
detector = trainYOLOv2ObjectDetector(___,Name,Value)
detector = trainYOLOv2ObjectDetector(trainingData,lgraph,options) returns an object detector trained using you only look once version 2 (YOLO v2) network architecture specified by the input lgraph. The options input specifies training parameters for the detection network.
[detector,info] = trainYOLOv2ObjectDetector(___) also returns information on the training progress, such as the training accuracy and learning rate for each iteration.
detector = trainYOLOv2ObjectDetector(trainingData,checkpoint,options) resumes training from the saved detector checkpoint.
You can use this syntax to:
Add more training data and continue the training.
Improve training accuracy by increasing the maximum number of iterations.
detector = trainYOLOv2ObjectDetector(trainingData,detector,options) continues training a YOLO v2 object detector. Use this syntax for fine-tuning a detector.
detector = trainYOLOv2ObjectDetector(___,'TrainingImageSize',trainingSizes) specifies the image sizes for multiscale training by using a name-value pair in addition to the input arguments in any of the preceding syntaxes.
detector = trainYOLOv2ObjectDetector(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments and any of the previous inputs.
Network: [1×1 DAGNetwork]
AnchorBoxes: [4×2 double]
When the training data is specified using a table, the trainYOLOv2ObjectDetector function checks these conditions
The bounding box values must be integers. Otherwise, the function automatically rounds each noninteger values to its nearest integer.
The bounding box must not be empty and must be within the image region. While training the network, the function ignores empty bounding boxes and bounding boxes that lie partially or fully outside the image region.
Layer graph, specified as a LayerGraph object. The layer graph contains the architecture of the YOLO v2 network. You can create this network by using the yolov2Layers function. Alternatively, you can create the network layers by using yolov2TransformLayer, yolov2ReorgLayer, and yolov2OutputLayer functions. For more details on creating a custom YOLO v2 network, see Design a YOLO v2 Detection Network.
The trainYOLOv2ObjectDetector function does not support these training options:
The trainingOptions Shuffle values, 'once' and 'every-epoch' are not supported when you use a datastore input.
Saved detector checkpoint, specified as a yolov2ObjectDetector object. To periodically save a detector checkpoint during training, specify CheckpointPath. To control how frequently check points are saved see the CheckPointFrequency and CheckPointFrequencyUnit training options.
data = load('/checkpath/yolov2_checkpoint__216__2018_11_16__13_34_30.mat');
The name of the MAT-file includes the iteration number and timestamp of when the detector checkpoint was saved. The detector is saved in the detector variable of the file. Pass this file back into the trainYOLOv2ObjectDetector function:
yoloDetector = trainYOLOv2ObjectDetector(trainingData,checkpoint,options);
detector — Previously trained YOLO v2 object detector
Previously trained YOLO v2 object detector, specified as a yolov2ObjectDetector object. Use this syntax to continue training a detector with additional training data or to perform more training iterations to improve detector accuracy.
trainingSizes — Set of image sizes for multiscale training
Set of image sizes for multiscale training, specified as an M-by-2 matrix, where each row is of the form [height width]. For each training epoch, the input training images are randomly resized to one of the M image sizes specified in this set.
If you do not specify the trainingSizes, the function sets this value to the size in the image input layer of the YOLO v2 network. The network resizes all training images to this value.
The input trainingSizes values specified for multiscale training must be greater than or equal to the input size in the image input layer of the lgraph input argument.
Example: 'ExperimentManager','none' sets the 'ExperimentManager' to 'none'.
detector — Trained YOLO v2 object detector
Trained YOLO v2 object detector, returned as yolov2ObjectDetector object. You can train a YOLO v2 object detector to detect multiple object classes.
Training progress information, returned as a structure array with seven fields. Each field corresponds to a stage of training.
By default, the trainYOLOv2ObjectDetector function preprocesses the training images by:
Resizing the input images to match the input size of the network.
Normalizing the pixel values of the input images to lie in the range [0, 1].
When you specify the training data by using a table, the trainYOLOv2ObjectDetector function performs data augmentation for preprocessing. The function augments the input dataset by:
Reflecting the training data horizontally. The probability for horizontally flipping each image in the training data is 0.5.
Uniformly scaling (zooming) the training data by a scale factor that is randomly picked from a continuous uniform distribution in the range [1, 1.1].
Random color jittering for brightness, hue, saturation, and contrast.
When you specify the training data by using a datastore, the trainYOLOv2ObjectDetector function does not perform data augmentation. Instead you can augment the training data in datastore by using the transform function and then, train the network with the augmented training data. For more information on how to apply augmentation while using datastores, see Preprocess Deep Learning Data (Deep Learning Toolbox).
During training, the YOLO v2 object detection network optimizes the MSE loss between the predicted bounding boxes and the ground truth. The loss function is defined as
\begin{array}{l}{K}_{1}\underset{i=0}{\overset{{S}^{2}}{â}}\underset{j=0}{\overset{B}{â}}{1}_{ij}^{obj}\left[{\left({x}_{i}â{\stackrel{^}{x}}_{i}\right)}^{2}+{\left({y}_{i}â{\stackrel{^}{y}}_{i}\right)}^{2}\right]\text{â}\\ \text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}+\text{â}{K}_{1}\underset{i=0}{\overset{{S}^{2}}{â}}\underset{j=0}{\overset{B}{â}}{1}_{ij}^{obj}\left[{\left(\sqrt{{w}_{i}}â\sqrt{{\stackrel{^}{w}}_{i}}\right)}^{2}+{\left(\sqrt{{h}_{i}}â\sqrt{{\stackrel{^}{h}}_{i}}\right)}^{2}\right]\\ \text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}+{K}_{2}\underset{i=0}{\overset{{S}^{2}}{â}}\underset{j=0}{\overset{B}{â}}{1}_{ij}^{obj}{\left({C}_{i}â{\stackrel{^}{C}}_{i}\right)}^{2}\\ \text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}+{K}_{3}\underset{i=0}{\overset{{S}^{2}}{â}}\underset{j=0}{\overset{B}{â}}{1}_{ij}^{noobj}{\left({C}_{i}â{\stackrel{^}{C}}_{i}\right)}^{2}\\ \text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}+\text{â}{K}_{4}\underset{i=0}{\overset{{S}^{2}}{â}}{1}_{i}^{obj}\underset{câclasses}{â}{\left({p}_{i}\left(c\right)â{\stackrel{^}{p}}_{i}\left(c\right)\right)}^{2}\end{array}
S is the number of grid cells.
{1}_{ij}^{obj}
{1}_{ij}^{noobj}
{1}_{i}^{obj}
K1, K2, K3, and K4 are the weights. To adjust the weights, modify the LossFactors property of the output layer by using the yolov2OutputLayer function.
\left({x}_{i},{y}_{i}\right)
\left({\stackrel{^}{x}}_{i},{\stackrel{^}{y}}_{i}\right)
{w}_{i}\text{â}\text{and}\text{â}{h}_{i}
{\stackrel{^}{w}}_{i}\text{â}\text{and}\text{â}{\stackrel{^}{h}}_{i}
Ĉi is the confidence score of the ground truth in grid cell i.
{\stackrel{^}{p}}_{i}\left(c\right)
To generate the ground truth, use the Image Labeler or Video Labeler app. To create a table of training data from the generated ground truth, use the objectDetectorTrainingData function.
To improve prediction accuracy,
Increase the number of images you can use to train the network. You can expand the training dataset through data augmentation. For information on how to apply data augmentation for preprocessing, see Preprocess Images for Deep Learning (Deep Learning Toolbox).
trainingOptions (Deep Learning Toolbox) | trainRCNNObjectDetector | trainFastRCNNObjectDetector | trainFasterRCNNObjectDetector | objectDetectorTrainingData | yolov2Layers
|
v0.4 | Flashbots Docs
for miners & mining pools
interacting with relay
mev-geth spec
v0.6 RPC (current)
v0.5 RPC
Defines the construction and usage of MEV bundles by miners. Provides a specification for custom implementations of the required node changes so that MEV bundles can be used correctly.
MEV bundles are stored by the node and the bundles that are providing extra profit for miners are added to the block in front of other transactions.
We believe that without the adoption of neutral, public, open-source infrastructure for permissionless MEV extraction, MEV risks becoming an insiders' game. We commit as an organisation to releasing reference implementations for participation in fair, ethical, and politically neutral MEV extraction.
The key words MUST, MUST NOT, REQUIRED, SHALL, SHALL NOT, SHOULD, SHOULD NOT, RECOMMENDED, MAY, and OPTIONAL in this document are to be interpreted as described in RFC-2119.
Miner Configuration#
Miner MUST accept the following configuration options:
miner.maxmergedbundles (int) - max number of MEV bundles to be included within a single block
miner.trustedrelays (string) - comma separated, hex encoded Ethereum addresses of trusted relays that the miner will be able to accept megabundles from, being reasonably certain about DDoS safety
An external system delivering MEV bundles and/or MEV megabundles to the node. Relay provides protection against DoS attacks.
MEV bundle or bundle#
A list of transactions that MUST be executed together and in the same order as provided in the bundle, MUST be executed before any non-bundle transactions and only after other bundles that have a higher bundle adjusted gas price.
Transactions in the bundle MUST execute without failure (return status 1 on transaction receipts) unless their hashes are included in the revertingTxHashes list.
When representing a bundle in communication between the relay and the node we use an object with the following properties:
txs Array<RLP(SignedTransaction)> A list of transactions in the bundle. Each transaction is signed and RLP-encoded.
blockNumber uint64 The exact block number at which the bundle can be executed
minTimestamp uint64 Minimum block timestamp (inclusive) at which the bundle can be executed
maxTimestamp uint64 Maximum block timestamp (inclusive) at which the bundle can be executed
revertingTxHashes Array<bytes32> List of hashes of transactions that are allowed to return status 0 on transaction receipts
MEV megabundle or megabundle#
A pre-merged set of bundles coming from a relay. Megabundles are not mixed or merged with normal MEV bundles but instead used for a separate block construction.
When representing a megabundle in communication between the relay and the node we use an object with the following properties:
relaySignature Array<Data> An secp256k1 signature signed with an address from the miner.trustedrelays. Message signed is a Keccak hash of RLP serialized sequence that contains the following items: array of txs (a sequence of byte arrays representing RLP serialized txs); blockNumber serialized as an uint64, minTimestamp serialized as an int256, like in the devp2p specification; maxTimestamp serialized as an int256, like in the devp2p specification; revertingTxHashes serialized as an array of byte arrays.
MEV block#
A block containing more than zero MEV bundles.
Whenever we say that a block contains a bundle we mean that the block includes all transactions of that bundle in the same order as in the bundle.
Unit of work#
A transaction, a bundle, megabundle or a block.
Subunit#
A discernible unit of work that is a part of a bigger unit of work. A transaction is a subunit of a bundle or a block. A bundle is a subunit of a block.
Total gas used#
The sum of gas units used by each transaction from the unit of work.
Average gas price#
For a transaction it is equivalent to the transaction minerFee := Min(feeCapPerGas - BASEFEE, priorityFeePerGas) - and for other units of work it is a sum of (average gas price * total gas used) of all subunits divided by the total gas used by the unit.
Direct coinbase payment#
The value of a transaction with a recipient set to be the same as the coinbase address.
Contract coinbase payment#
A payment from a smart contract to the coinbase address.
Coinbase payment#
A sum of all direct coinbase payments and contract coinbase payments within the unit of work.
Eligible coinbase payment#
Gas fee payment#
An average gas price * total gas used within the unit of work.
Eligible gas fee payment#
A gas fee payment excluding gas fee payments from transactions that can be spotted by the miner in the publicly visible transaction pool.
Bundle scoring profit#
A sum of all eligible coinbase payments and eligible gas payments of a bundle.
Profit#
A difference between the balance of the coinbase account at the end and at the beginning of the execution of a unit of work. We can measure a transaction profit, a bundle profit, and a block profit.
Balance of the coinbase account changes in the following way
Transaction average gas price * total gas used + direct coinbase payment + contract coinbase payment
Bundle average gas price * total gas used + direct coinbase payment + contract coinbase payment
Megabundle average gas price * total gas used + direct coinbase payment + contract coinbase payment
Block block reward + average gas price * total gas used + direct coinbase payment + contract coinbase payment
Adjusted gas price#
Unit of work profit divided by the total gas used by the unit of work.
Bundle adjusted gas price#
Bundle scoring profit divided by the total gas used by the bundle.
s_{v0.4} = \frac{\Delta_{coinbase} + \sum_{T\in U}g_Tm_T - \sum_{T\in M \cap U}g_Tm_T}{\sum_{T\in U}g_T}
s
: bundle
U
score used to sort bundles.
U
: ordered list of transactions
T
in a bundle.
M
: set of transactions
T
in the mempool.
g_{T}
: gas used by transaction
T
c_{T}
: fee cap per gas of transaction
T
\delta_T
: priority fee per gas of transaction
T
e_{T}
: effective fee per gas of transaction
T
\min
c_{T}
, BASEFEE +
\delta_T
m_{T}
: miner fee per gas of transaction
T
e_{T}
- BASEFEE.
\Delta_{coinbase}
: coinbase difference from direct payment.
Bundle construction#
A bundle SHOULD contain transactions with nonces that are following the current nonces of the signing addresses or other transactions preceding them in the same bundle.
A bundle MUST contain at least one transaction. There is no upper limit for the number of transactions in the bundle, however bundles that exceed the block gas limit will always be rejected.
A bundle MAY include eligible coinbase payments. Bundles that do not contain such payments may be discarded when their bundle adjusted gas price is compared with other bundles.
The maxTimestamp value MUST be greater or equal the minTimestamp value.
Accepting bundles from the network#
JSON RPC#
Node MUST provide a way of exposing a JSON RPC endpoint accepting eth_sendBundle calls (specified here). Such endpoint SHOULD only be accepting calls from a relay but there is no requirement to restrict it through the node source code as it can be done on the infrastructure level.
Node MUST provide a way of exposing a JSON RPC endpoint accepting eth_sendMegabundle calls (specified here). Such endpoint SHOULD only be accepting calls from a relay but there is no requirement to restrict it through the node source code as it can be done on the infrastructure level. For each relay only the last megabundle SHOULD be stored in memory.
Bundle eligibility#
Any bundle that is correctly constructed MUST have a blockNumber field set which specifies in which block it can be included. If the node has already progressed to a later block number then such bundle MAY be removed from memory.
Each transaction in the bundle MUST have maxFeePerGas equal or greater than block.BASEFEE + 1 GWei to be selected for the block.
Any bundle that is correctly constructed MAY have a minTimestamp and/or a maxTimestamp field set. Default values for both of these fields are 0 and the meaning of 0 is that any block timestamp value is accepted. When these values are not 0, then block.timestamp is compared with them. If the current block.timestamp is greater than the maxTimestamp then the bundle MUST NOT be included in the block and MAY be removed from memory. If the block.timestamp is less than minTimestamp then the bundle MUST NOT be included in the block and SHOULD NOT be removed from memory (it awaits future blocks).
Block construction#
In order to prevent starvation of less frequently used relays, incoming MEV megabundles MUST be given a priority value equal to the time in milliseconds since the previous MEV megabundle sent by the same relay and redirected to a pool of MEV megabundle block producers (called workers in MEV-Geth). Whenever the worker is idle and there are MEV megabundles available, a bundle with the highest priority MUST be picked for evaluation. Megabundles are never merged with other bundles and can only be combined with transactions from the mempool.
MEV bundles MUST be sorted by their bundle adjusted gas price first and then one by one added to the block as long as there is any gas left in the block and the number of bundles added is less or equal the MaxMergedBundles parameter. The remaining block gas SHOULD be used for non-MEV transactions.
When constructing a block each next bundle added after the first bundle MUST generate at least 99% of the bundle adjusted gas price from the time of the sorting (the first bundle will naturally provide 100% of this value).
Block MUST contain between 0 and MaxMergedBundles bundles.
A block with bundles MUST place the bundles at the beginning of the block and MUST NOT insert any transactions between the bundles or bundle transactions.
When constructing a block the node MUST reject any bundle or megabundle that has a reverting transaction unless its hash is included in the RevertingTxHashes list of the bundle / megabundle object.
The node SHOULD be able to compare the block profit for each number of bundles between 0 and MaxMergedBundles and choose a block with the highest profit, e.g. if MaxMergedBundles is 3 then the node SHOULD build 4 different blocks - with the maximum of respectively 0, 1, 2, and 3 bundles and choose the one with the highest profit.
The node MUST be able to compare the block profit from the best megabundles with the block profit of otherwise winning block.
Bundle eviction#
Node SHOULD be able to limit the number of bundles kept in memory and apply an algorithm for selecting bundles to be evicted when too many eligible bundles have been received.
Naive bundle merging#
The bundle merging process is not necessarily picking the most profitable combination of bundles but only the best guess achievable without degrading latency. The first bundle included is always the bundle with the highest bundle adjusted gas price
Using bundle adjusted gas price instead of adjusted gas price#
The bundle adjusted gas price is used to prevent bundle creators from artificially increasing the adjusted gas price by adding unrelated high gas price transactions from the publicly visible transaction pool.
Each bundle needs a blockNumber#
This allows specifying bundles to be included in the future blocks (e.g. just after some smart contracts change their state). This cannot be used to ensure a specific parent block / hash.
Future Considerations#
Full block submission#
A proposal to allow MEV-Geth accepting fully constructed blocks as well as bundles is considered for inclusion in next versions.
This change is not affecting consensus and is fully compatible with Ethereum specification.
Bundle formats are not backwards compatible and the v0.2 bundles would be rejected by v0.1 MEV clients.
The node SHOULD ensure that MEV bundles and megabundles that are awaiting future blocks are evicted when at risk of reaching the storage limits (memory or persistent storage).
« v0.5 RPC
v0.4 RPC »
Accepting bundles from the network
Bundle eligibility
Bundle eviction
Naive bundle merging
Using bundle adjusted gas price instead of adjusted gas price
Each bundle needs a blockNumber
Full block submission
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2012 A Proximal Analytic Center Cutting Plane Algorithm for Solving Variational Inequality Problems
Jie Shen, Li-Ping Pang
Under the condition that the values of mapping
F
are evaluated approximately, we propose a proximal analytic center cutting plane algorithm for solving variational inequalities. It can be considered as an approximation of the earlier cutting plane method, and the conditions we impose on the corresponding mappings are more relaxed. The convergence analysis for the proposed algorithm is also given at the end of this paper.
Jie Shen. Li-Ping Pang. "A Proximal Analytic Center Cutting Plane Algorithm for Solving Variational Inequality Problems." J. Appl. Math. 2012 (SI15) 1 - 10, 2012. https://doi.org/10.1155/2012/503242
Jie Shen, Li-Ping Pang "A Proximal Analytic Center Cutting Plane Algorithm for Solving Variational Inequality Problems," Journal of Applied Mathematics, J. Appl. Math. 2012(SI15), 1-10, (2012)
|
\mathrm{R}=\left[\left(x, y\right) : x+2y=8\right]
is a relation on N, write the range of R. VIEW SOLUTION
{\mathrm{tan}}^{-1} \mathrm{x}+{\mathrm{tan}}^{-1} \mathrm{y}=\frac{\pi }{4}, \mathrm{xy} < 1,
\mathrm{x}+\mathrm{y}+\mathrm{xy}
If A is a square matrix, such that
{\mathrm{A}}^{2}=\mathrm{A}
7\mathrm{A}-{\left(\mathrm{I}+\mathrm{A}\right)}^{3}
, where I is an identity matrix. VIEW SOLUTION
\left[\begin{array}{cc}\mathrm{x}-\mathrm{y}& \mathrm{z}\\ 2\mathrm{x}-\mathrm{y}& \mathrm{w}\end{array}\right]=\left[\begin{array}{cc}-1& 4\\ 0& 5\end{array}\right]
\mathrm{x}+\mathrm{y}
\left[\begin{array}{cc}3\mathrm{x}& 7\\ -2& 4\end{array}\right]=\left[\begin{array}{cc}8& 7\\ 6& 4\end{array}\right]
, find the value of x. VIEW SOLUTION
\mathrm{f}\left(\mathrm{x}\right) ={\int }_{0}^{x} \mathrm{t} \mathrm{sin} \mathrm{t} \mathrm{dt}
, then write the value of f ' (x). VIEW SOLUTION
\underset{2}{\overset{4}{\int }} \frac{\mathrm{x}}{{\mathrm{x}}^{2} + 1}\mathrm{dx}
3\stackrel{^}{\mathrm{i}} + 2\stackrel{^}{\mathrm{j}} + 9\stackrel{^}{\mathrm{k}}
\stackrel{^}{\mathrm{i}} - 2\mathrm{p}\stackrel{^}{\mathrm{j}}+ 3\stackrel{^}{\mathrm{k}}
are parallel. VIEW SOLUTION
\stackrel{\to }{\mathrm{a}}· \left(\stackrel{\to }{\mathrm{b}} × \stackrel{\to }{\mathrm{c}}\right), \mathrm{if} \stackrel{\to }{\mathrm{a}} = 2\stackrel{^}{\mathrm{i}} + \stackrel{^}{\mathrm{j}} + 3\stackrel{^}{\mathrm{k}}, \stackrel{\to }{\mathrm{b}} = -\stackrel{^}{\mathrm{i}} + 2\stackrel{^}{\mathrm{j}} + \stackrel{^}{\mathrm{k}} \mathrm{and} \stackrel{\to }{\mathrm{c}} = 3\stackrel{^}{\mathrm{i}} + \stackrel{^}{\mathrm{j}} + 2\stackrel{^}{\mathrm{k}}.
If the Cartesian equations of a line are
\frac{3-x}{5} = \frac{y+4}{7} = \frac{2z-6}{4}
, write the vector equation for the line. VIEW SOLUTION
If the function f : R → R be given by f[x] = x2 + 2 and g : R → R be given by
g\left(x\right)=\frac{x}{x-1}, x\ne 1
, find fog and gof and hence find fog (2) and gof (−3). VIEW SOLUTION
{\mathrm{tan}}^{-1} \left[\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right]=\frac{\mathrm{\pi }}{4}-\frac{1}{2} {\mathrm{cos}}^{-1}x, \frac{-1}{\sqrt{2}}\le x\le 1
{\mathrm{tan}}^{-1} \left(\frac{x-2}{x-4}\right)+{\mathrm{tan}}^{-1} \left(\frac{x+2}{x+4}\right)=\frac{\mathrm{\pi }}{4}
\left|\begin{array}{ccc}x+y& x& x\\ 5x+4y& 4x& 2x\\ 10x+8y& 8x& 3x\end{array}\right|={x}^{3}
\frac{\mathrm{dy}}{\mathrm{dx}}
\mathrm{\theta }=\frac{\mathrm{\pi }}{4}
if x = aeθ (sin θ − cos θ) and y = aeθ (sin θ + cos θ). VIEW SOLUTION
\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}-\left(\mathrm{a}+\mathrm{b}\right)\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{aby}=0.
Find the equations of the tangent and normal to the curve
\frac{{\mathrm{x}}^{2}}{{\mathrm{a}}^{2}}-\frac{{\mathrm{y}}^{2}}{{\mathrm{b}}^{2}}=1
\left(\sqrt{2}\mathrm{a},\mathrm{b}\right)
\underset{0}{\overset{\mathrm{\pi }}{\int }}\frac{4\mathrm{x} \mathrm{sin} \mathrm{x}}{1+{\mathrm{cos}}^{2} \mathrm{x}} \mathrm{dx}
\int \frac{\mathrm{x}+2}{\sqrt{{\mathrm{x}}^{2}+5\mathrm{x}+6}} \mathrm{dx}
\frac{\mathrm{d}y}{dx}
= 1 + x + y + xy, given that y = 0 when x = 1. VIEW SOLUTION
Solve the differential equation (1 + x2)
\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}={{\mathrm{e}}^{\mathrm{tan}}}^{-1}\mathrm{x}.
4\stackrel{^}{\mathrm{i}}+5\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}, -\stackrel{^}{\mathrm{j}}-\stackrel{^}{\mathrm{k}}, 3\stackrel{^}{\mathrm{i}}+9\stackrel{^}{\mathrm{j}}+4\stackrel{^}{\mathrm{k}} \mathrm{and} 4\left(-\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}\right)
, respectively, are coplanar.
The scalar product of the vector
\stackrel{\to }{\mathrm{a}}=\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}
with a unit vector along the sum of vectors
\stackrel{\to }{\mathrm{b}}=2\stackrel{^}{\mathrm{i}}+4\stackrel{^}{\mathrm{j}}-5\stackrel{^}{\mathrm{k}}
\stackrel{\to }{\mathrm{c}}=\mathrm{\lambda }\stackrel{^}{\mathrm{i}}+2\stackrel{^}{\mathrm{j}}+3\stackrel{^}{\mathrm{k}}
is equal to one. Find the value of λ and hence, find the unit vector along
\stackrel{\to }{\mathrm{b}}+\stackrel{\to }{\mathrm{c}}.
A line passes through (2, −1, 3) and is perpendicular to the lines
\stackrel{\to }{r}=\left(\stackrel{^}{i}+\stackrel{^}{j}-\stackrel{^}{k}\right)+\mathrm{\lambda }\left(2\stackrel{^}{i}-2\stackrel{^}{j}+\stackrel{^}{k}\right) \mathrm{and} \stackrel{\to }{r}=\left(2\stackrel{^}{i}-\stackrel{^}{j}-3\stackrel{^}{k}\right)+\mathrm{\mu }\left(\stackrel{^}{i}+2\stackrel{^}{j}+2\stackrel{^}{k}\right)
. Obtain its equation in vector and Cartesian from. VIEW SOLUTION
Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of radius r is
\frac{4\mathrm{r}}{3}
. Also, show that the maximum volume of the cone is
\frac{8}{27}
of the volume of the sphere. VIEW SOLUTION
\int \frac{1}{{\mathrm{cos}}^{4}\mathrm{x}+{\mathrm{sin}}^{4}\mathrm{x}}\mathrm{dx}
\stackrel{\to }{\mathrm{r}}=2\stackrel{^}{\mathrm{i}}-4\stackrel{^}{\mathrm{j}}+2\stackrel{^}{\mathrm{k}}+\mathrm{\lambda }\left(3\stackrel{^}{\mathrm{i}}+4\stackrel{^}{\mathrm{j}}+2\stackrel{^}{\mathrm{k}}\right)
\stackrel{\to }{\mathrm{r}}.\left(\stackrel{^}{\mathrm{i}}-2\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}\right)=0.
|
Flight - Wikipedia
Natural flight by a brown pelican
Human-invented flight: a Royal Jordanian Airlines Boeing 787
Flight or flying is the process by which an object moves through a space without contacting any planetary surface, either within an atmosphere (i.e. air flight or aviation) or through the vacuum of outer space (i.e. spaceflight). This can be achieved by generating aerodynamic lift associated with gliding or propulsive thrust, aerostatically using buoyancy, or by ballistic movement.
Many things can fly, from animal aviators such as birds, bats and insects, to natural gliders/parachuters such as patagial animals, anemochorous seeds and ballistospores, to human inventions like aircraft (airplanes, helicopters, airships, balloons, etc.) and rockets which may propel spacecraft and spaceplanes.
The engineering aspects of flight are the purview of aerospace engineering which is subdivided into aeronautics, the study of vehicles that travel through the atmosphere, and astronautics, the study of vehicles that travel through space, and ballistics, the study of the flight of projectiles.
1.3.3 Solid-state propulsion
Buoyant flight[edit]
An airship flies because the upward force, from air displacement, is equal to or greater than the force of gravity
Humans have managed to construct lighter-than-air vehicles that raise off the ground and fly, due to their buoyancy in air.
Aerodynamic flight[edit]
Unpowered flight versus powered flight[edit]
Animal flight[edit]
Tau emerald dragonfly
The only groups of living things that use powered flight are birds, insects, and bats, while many groups have evolved gliding. The extinct pterosaurs, an order of reptiles contemporaneous with the dinosaurs, were also very successful flying animals.[3] Each of these groups' wings evolved independently, with insects the first animal group to evolve flight.[4] The wings of the flying vertebrate groups are all based on the forelimbs, but differ significantly in structure; those of insects are hypothesized to be highly modified versions of structures that form gills in most other groups of arthropods.[3]
Most birds fly (see bird flight), with some exceptions. The largest birds, the ostrich and the emu, are earthbound flightless birds, as were the now-extinct dodos and the Phorusrhacids, which were the dominant predators of South America in the Cenozoic era. The non-flying penguins have wings adapted for use under water and use the same wing movements for swimming that most other birds use for flight.[citation needed] Most small flightless birds are native to small islands, and lead a lifestyle where flight would offer little advantage.
Mechanical flight: A Robinson R22 Beta helicopter
Mechanical flight is the use of a machine to fly. These machines include aircraft such as airplanes, gliders, helicopters, autogyros, airships, balloons, ornithopters as well as spacecraft. Gliders are capable of unpowered flight. Another form of mechanical flight is para-sailing, where a parachute-like object is pulled by a boat. In an airplane, lift is created by the wings; the shape of the wings of the airplane are designed specially for the type of flight desired. There are different types of wings: tempered, semi-tempered, sweptback, rectangular and elliptical. An aircraft wing is sometimes called an airfoil, which is a device that creates lift when air flows across it.
Supersonic[edit]
Supersonic flight is flight faster than the speed of sound. Supersonic flight is associated with the formation of shock waves that form a sonic boom that can be heard from the ground,[10] and is frequently startling. This shockwave takes quite a lot of energy to create and this makes supersonic flight generally less efficient than subsonic flight at about 85% of the speed of sound.
Hypersonic[edit]
Hypersonic flight is very high speed flight where the heat generated by the compression of the air due to the motion through the air causes chemical changes to the air. Hypersonic flight is achieved primarily by reentering spacecraft such as the Space Shuttle and Soyuz.
The International Space Station in earth orbit
Ballistic[edit]
Atmospheric[edit]
Essentially an extreme form of ballistic flight, spaceflight is the use of space technology to achieve the flight of spacecraft into and through outer space. Examples include ballistic missiles, orbital spaceflight, etc.
Solid-state propulsion[edit]
In 2018, researchers at Massachusetts Institute of Technology (MIT) managed to fly an aeroplane with no moving parts, powered by an "ionic wind" also known as electroaerodynamic thrust.[12][13]
Spaceflight, particularly human spaceflight became a reality in the 20th century following theoretical and practical breakthroughs by Konstantin Tsiolkovsky and Robert H. Goddard. The first orbital spaceflight was in 1957,[20] and Yuri Gagarin was carried aboard the first manned orbital spaceflight in 1961.[21]
Lighter-than-air airships are able to fly without any major input of energy
Main forces acting on a heavier-than-air aircraft
Forces on an aerofoil cross section
Lift is defined as the component of the aerodynamic force that is perpendicular to the flow direction, and drag is the component that is parallel to the flow direction
Lift-to-drag ratio[edit]
Speed and drag relationships for a typical aircraft
However, this lift (deflection) process inevitably causes a retarding force called drag. Because lift and drag are both aerodynamic forces, the ratio of lift to drag is an indication of the aerodynamic efficiency of the airplane. The lift to drag ratio is the L/D ratio, pronounced "L over D ratio." An airplane has a high L/D ratio if it produces a large amount of lift or a small amount of drag. The lift/drag ratio is determined by dividing the lift coefficient by the drag coefficient, CL/CD.[29]
Lift/drag ratio also determines the glide ratio and gliding range. Since the glide ratio is based only on the relationship of the aerodynamics forces acting on the aircraft, aircraft weight will not affect it. The only effect weight has is to vary the time that the aircraft will glide for – a heavier aircraft gliding at a higher airspeed will arrive at the same touchdown point in a shorter time.[32]
Thrust to weight ratio[edit]
{\displaystyle g_{0}}
The upward tilt of the wings and tailplane of an aircraft, as seen on this Boeing 737, is called dihedral angle
Power-to-weight ratio[edit]
Takeoff and landing[edit]
Guidance, navigation and control[edit]
Flight safety[edit]
^ Walker 2000, p. 541. Quote: the gas-bag of a balloon or airship.
^ Coulson-Thomas 1976, p. 281. Quote: fabric enclosing gas-bags of airship.
^ a b Averof, Michalis. "Evolutionary origin of insect wings from ancestral gills." Nature, Volume 385, Issue 385, February 1997, pp. 627–630.
^ World Book Student. Chicago: World Book. Retrieved: April 29, 2011.
^ "BBC article and video of flying fish." BBC, May 20, 2008. Retrieved: May 20, 2008.
^ "Swan Identification." Archived 2006-10-31 at the Wayback Machine The Trumpeter Swan Society. Retrieved: January 3, 2012.
^ Bern, Peter. "Concorde: You asked a pilot." BBC, October 23, 2003.
^ Spitzmiller, Ted (2007). Astronautics: A Historical Perspective of Mankind's Efforts to Conquer the Cosmos. Apogee Books. p. 467. ISBN 9781894959667.
^ Haofeng Xu; et al. (2018). "Flight of an aeroplane with solid-state propulsion". Vol. 563. Nature. pp. 532–535. doi:10.1038/s41586-018-0707-9.
^ Jennifer Chu (21 November 2018). "MIT engineers fly first-ever plane with no moving parts". MIT News.
^ "Archytas of Tar entum." Archived December 26, 2008, at the Wayback Machine Technology Museum of Thessaloniki, Macedonia, Greece/ Retrieved: May 6, 2012.
^ "Ancient history." Archived 2002-12-05 at the Wayback Machine Automata. Retrieved:May 6, 2012.
^ "Sir George Cayley". Flyingmachines.org. Retrieved 27 August 2019. Sir George Cayley is one of the most important people in the history of aeronautics. Many consider him the first true scientific aerial investigator and the first person to understand the underlying principles and forces of flight.
^ "The Pioneers: Aviation and Airmodelling". Retrieved 26 July 2009. Sir George Cayley, is sometimes called the 'Father of Aviation'. A pioneer in his field, he is credited with the first major breakthrough in heavier-than-air flight. He was the first to identify the four aerodynamic forces of flight – weight, lift, drag, and thrust – and their relationship and also the first to build a successful human-carrying glider.
^ "U.S. Centennial of Flight Commission – Sir George Cayley". Archived from the original on 20 September 2008. Retrieved 10 September 2008. Sir George Cayley, born in 1773, is sometimes called the Father of Aviation. A pioneer in his field, Cayley literally has two great spurts of aeronautical creativity, separated by years during which he did little with the subject. He was the first to identify the four aerodynamic forces of flight – weight, lift, drag, and thrust and their relationship. He was also the first to build a successful human-carrying glider. Cayley described many of the concepts and elements of the modern aeroplane and was the first to understand and explain in engineering terms the concepts of lift and thrust.
^ "Orville Wright's Personal Letters on Aviation." Shapell Manuscript Foundation, (Chicago), 2012.
^ "Sputnik and the Origins of the Space Age".
^ "Gagarin anniversary." NASA. Retrieved: May 6, 2012.
^ "Four forces on an aeroplane." NASA. Retrieved: January 3, 2012.
^ "Newtons Third Law". Archived from the original on 1999-11-28.
^ "Definition of lift." Archived 2009-02-03 at the Wayback Machine NASA. Retrieved: May 6, 2012.
^ "Basic flight physics." Berkeley University. Retrieved: May 6, 2012.
^ "What is Drag?" Archived 2010-05-24 at the Wayback Machine NASA. Retrieved: May 6, 2012.
^ "Motions of particles through fluids." Archived 2012-04-25 at the Wayback Machine lorien.ncl.ac. Retrieved: May 6, 2012.
^ The Beginner's Guide to Aeronautics - NASA Glenn Research Center https://www.grc.nasa.gov/www/k-12/airplane/ldrat.html
^ The Beginner's Guide to Aeronautics - NASA Glenn Research Center https://www.grc.nasa.gov/www/k-12/airplane/liftco.html
^ The Beginner's Guide to Aeronautics - NASA Glenn Research Center https://www.grc.nasa.gov/www/k-12/airplane/dragco.html
^ Sutton and Biblarz 2000, p. 442. Quote: "thrust-to-weight ratio F/W0 is a dimensionless parameter that is identical to the acceleration of the rocket propulsion system (expressed in multiples of g0) if it could fly by itself in a gravity free vacuum."
^ ch10-3 "History." NASA. Retrieved: May 6, 2012.
^ Honicke et al. 1968[page needed]
^ "13.3 Aircraft Range: The Breguet Range Equation".
Coulson-Thomas, Colin. The Oxford Illustrated Dictionary. Oxford, UK: Oxford University Press, 1976, First edition 1975, ISBN 978-0-19-861118-9.
French, A. P. Newtonian Mechanics (The M.I.T. Introductory Physics Series) (1st ed.). New York: W. W. Norton & Company Inc., 1970.
Honicke, K., R. Lindner, P. Anders, M. Krahl, H. Hadrich and K. Rohricht. Beschreibung der Konstruktion der Triebwerksanlagen. Berlin: Interflug, 1968.
Sutton, George P. Oscar Biblarz. Rocket Propulsion Elements. New York: Wiley-Interscience, 2000 (7th edition). ISBN 978-0-471-32642-7.
Walker, Peter. Chambers Dictionary of Science and Technology. Edinburgh: Chambers Harrap Publishers Ltd., 2000, First edition 1998. ISBN 978-0-550-14110-1.
Look up flight in Wiktionary, the free dictionary.
Wikivoyage has a travel guide for Flights.
Pettigrew, James Bell (1911). "Flight and Flying" . Encyclopædia Britannica. Vol. 10 (11th ed.). pp. 502–519. History and photographs of early aeroplanes etc.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Flight&oldid=1083331572"
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Quantum theory of observation - Wikibooks, open books for an open world
This theoretical approach was initiated by John von Neumann (1932). It differs from the usual interpretations of quantum mechanics (Niels Bohr, Copenhagen interpretation) which require that the measuring apparatus be considered as a classical system, which does not obey quantum physics. This requirement is not justified because quantum laws are universal. They apply to all material, microscopic and macroscopic, systems. This universality is a direct consequence of the principles: if two quantum systems are combined, they together form a new quantum system (cf. 2.1, third principle of quantum physics). Therefore, the number of components does not change the quantum nature of a system.
The quantum theory of observation invites us to give up the postulate of the wave function collapse, because it is not necessary to explain the correlations between successive observations, and because it contradicts the Schrödinger equation. Thus conceived quantum theory of observation is another name for Everett's theory, also called the many-worlds interpretation, the theory of the universal wave function, or the "relative state" formulation of quantum mechanics, because by applying the Schrödinger equation to observation processes, we obtain solutions that represent the multiple destinies of observers and their relative worlds.
General theory of quantum measurement
The forest of destinies
The appearance of relative classical worlds in the quantum Universe
The first chapter offers an introduction, intended for a reader who approaches quantum physics for the first time. It presents the great quantum principle, the principle of the existence of superposition of states, and begins to show how it can be understood.
All the quantum principles are stated and explained in the second chapter, and we deduce from them the first consequences: the existence of multiple destinies, the incomplete discernability of states and the incompatibility of measurements.
The next chapter applies the quantum theory of observation to a few simple examples (the Mach-Zehnder interferometer, the CNOT and SWAP gates).
Chapter 4 is the most important of the book because quantum entanglement is fundamental to explain the reality of observations. From the definition of the relativity of states (Everett), it shows that the postulate of the reduction of the wave function is not necessary, because the reduction of the state vector by observation is an appearance which results from the real entanglement between the observer system and the observed system. We then deduce many consequences: the impossibility of seeing non-localized macroscopic states (but we can still observe them), the quantum explanation of intersubjectivity, the observation of correlations in an entangled pair, co-presence without possible encounter and entanglement of space-time, the non-cloning theorem, the possibility of ideal measurements of entangled states and why it does not allow to observe our other destinies, why entangled pairs do not permit to communicate, decoherence through entanglement and why it explains at the same time Feynman's rules, the posterior reconstruction of interference patterns and the fragility of non-localized macroscopic states, and finally, the possibility, and the reality, of experiments of the "Schrödinger's cat" type. We can conclude that the existence theorem of multiple destinies is empirically verifiable. An observer can not observe her other destinies, but a second observer can in principle observe them, with experiments of the "Schrödinger's cat" type.
In Schrödinger's imagined experiment, the paradoxical state
{\displaystyle |alive\rangle +|dead\rangle }
is produced, but the experiment is not designed to verify by observation that it was actually produced, because it is destroyed by opening the box. A slightly modified experiment, however, makes it possible to observe that a state similar to
{\displaystyle |alive\rangle +|dead\rangle }
is actually produced. We can therefore in principle verify that two destinies of an observer system are simultaneously real. But this conclusion is limited to reversible observation processes. As the processes of life are irreversible, the simultaneous existence of the multiple destinies of a living being can not be observed.
Quantum theory of observation has so far been exposed for ideal measurements. Chapter 5 shows that it can be generalized for all observation systems, and that the results obtained for the ideal measurements (multiple destinies, Born rule ...) remain valid. It also shows that decoherence by the environment is sufficient to explain the selection of the pointer states of measuring instruments.
The multiple destinies of an observer form a tree. Chapter 6 applies the theory to a universe that contains many observers and obtains as a solution a forest of multiple destinies, a tree for each observer. Each branch is a destiny. All branches of forest trees can become entangled when observers meet or communicate. But some branches can never meet. The destinies they represent are inexorably separated. This book calls them incomposable destinies.
To speak of the growth of a forest of destinies is only one way of describing the solutions of the Schrödinger equation when applied to systems of ideal observers. It is a question of describing mathematical solutions which result from the simple assumptions which have been made. It is not a delusional imagination but a calculation of the consequences of mathematical principles.
The chapter ends by showing that we must distinguish between multiple destinies and Feynman's paths, that the parallelism of quantum computation is different from the parallelism of destinies.
The last chapter shows that quantum physics even explains the classical appearances of matter. The quantum evolution of the Universe can not be identified with a classical destiny, but it is sufficient to determine the growth of a forest of destinies of observers and their relative worlds. We thus explain the classical appearances of relative worlds without postulating that the Universe itself must have this appearance. The classical appearances of observers emerge from a quantum evolution that describes a forest of multiple destinies.
It is sometimes wrongly believed that the explanation of quantum principles (cf 2.1) requires advanced mathematics. The great concepts of quantum physics, superposition (1.1) and incomplete discernability (2.6) of states, incompatibility of measurements (2.7), entanglement of parts (4.1), relativity of states (4.3), decoherence by entanglement (4.17), selection of pointer states (5.4) and incomposability of destinies (6.4) ... can all be explained with minimal mathematical formalism. It suffices to know complex numbers (1.4) and to know how to add vectors in finite dimensional spaces. The applications of quantum physics often require advanced mathematical techniques, but not the explanation of the principles. This applies to all sciences. The principles are what we have to understand when we start studying. They are the main tools that enable us to progress. It is therefore normal and natural that they can be explained without exceeding a fairly basic level.
A philosophical introduction: quantum theory of multiple destinies, from my Précis of epistemology.
Co-presence without possible encounter and the incomposability of destinies are published in this book for the first time. These are discoveries for educational purposes, which is why it is natural to publish them in an educational library. They explain how a single space-time can accommodate worlds relative to the multiple destinies of observers, and why these destinies can coexist without meeting each other.
Who is this book addressed to ? Primarily to students who have already had a first course in quantum physics (for example, the first chapters of Feynman 1966, Cohen-Tannoudji, Diu & Laloë 1973, Griffiths 2004). More generally, to any interested reader who is not too frightened by the expressions Hilbert space or unitary operator.
Pedagogical objectives: At the end of the book, the reader will have the main elements to study the research work on the quantum theory of observation. They can also prepare for research on quantum computation and information (Nielsen and Chuang 2010).
The great principle : the existence of quantum superpositions
Why is quantum reality represented by complex numbers ?
Scalar product and unitary operators
Tensor product and entanglement
The qubits
The principles of quantum physics
The existence theorem of multiple destinies
The destruction of information by observation
Can we observe quantum states?
Orthogonality and incomplete discernability of quantum states
The incompatibility of quantum measurements
Uncertainty and density operators
Observation of quantum superpositions with the Mach-Zehnder interferometer
An ideal measurement: the CNOT gate
A non-ideal measurement: the SWAP gate
Experimental realization of quantum gates
Interaction, entanglement and disentanglement
Everett relative states
The collapse of the state vector through observation is a disentanglement.
Apparent disentanglement results from real entanglement between the observed system and the observer.
Dirac's error
Can we see non-localized macroscopic states?
The quantum explanation of intersubjectivity
Einstein, Bell, Aspect and the reality of quantum entanglement
Co-presence without a possible encounter
Entangled space-time
Action, reaction and no cloning
The ideal measurement of entangled states
Why does not the measurement of entangled states enable us to observe other destinies?
Reduced density operators
Relative density operators
Why do not entangled pairs enable us to communicate?
Decoherence through entanglement
The Feynman Rules
The a posteriori reconstitution of interference patterns
The fragility of non-localized macroscopic states
Experiments of the "Schrödinger's cat" type
Is the existence theorem of multiple destinies empirically verifiable?
Observables and projectors
Uncertainty about the state of the detector and measurement superoperators
The selection of pointer states and environmental pressure
The pointer states of microscopic probes
A double constraint for the design of observation instruments
The arborescence of the destinies of an ideal observer
Absolute destiny of the observer and relative destiny of its environment
The probabilities of destinies
The incomposability of destinies
The growth of a forest of destinies
Virtual quantum destinies and Feynman paths
The parallelism of quantum computation and the multiplicity of virtual pasts
Can we have many pasts if we forget them?
Are not classical appearances proofs that quantum physics is incomplete?
The quantum evolution of the Universe determines the classical destinies of the relative worlds
Retrieved from "https://en.wikibooks.org/w/index.php?title=Quantum_theory_of_observation&oldid=3848410"
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Multichannel analysis of Love waves in a 3D seismic acquisition system3D multichannel analysis of Love waves | Geophysics | GeoScienceWorld
, Subsurface Imaging and Sensing Laboratory, Institute of Geophysics and Geomatics, Wuhan,
, Karlsruhe Institute of Technology (KIT), Karlsruhe,
. E-mail: geophy.pyd@gmail.com.
, Hubei Subsurface Multi-Scale Imaging Key Laboratory, Institute of Geophysics and Geomatics, Wuhan,
. E-mail: jianghai_xia@yahoo.com; xyxian65@aliyun.com; gaoll1990@126.com.
Yudi Pan, Jianghai Xia, Yixian Xu, Lingli Gao; Multichannel analysis of Love waves in a 3D seismic acquisition system. Geophysics 2016;; 81 (5): EN67–EN74. doi: https://doi.org/10.1190/geo2015-0261.1
Multichannel analysis of Love waves (MALW) analyzes high-frequency Love waves to determine near-surface S-wave velocities, and it is getting increasing attention in the near-surface geophysics and geotechnique community. Based on 2D geometry spread, in which sources and receivers are placed along the same line, current MALW fails to work in a 3D seismic acquisition system. This is because Love-wave particle motion direction is perpendicular to its propagation direction, which makes it difficult to record a Love-wave signal in 3D geometries. We have developed a method to perform MALW with data acquired in 3D geometry. We recorded two orthogonal horizontal components (inline and crossline components) at each receiver point at the same time. By transforming the raw data from rectangular coordinates (inline and crossline components) to radial-transverse coordinates (radial and transverse components), we recovered Love-wave data along the transverse direction at each receiver point. To achieve a Love-wave dispersion curve, the recovered Love-wave data were first transformed into a conventional receiver offset domain, and then transformed into the frequency-velocity (
f
v
) domain. Love-wave dispersion curves were picked along the continuous dispersive energy peaks in the
f
v
domain. The validity of our proposed method was verified by two synthetic tests and a real-world example.
Exploiting surface consistency for surface-wave characterization and mitigation — Part 2: Application to 3D data
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Terrestrial Time - Wikipedia
(Redirected from Terrestrial Dynamical Time)
Time standard for astronomical observations from the Earth
Terrestrial Time (TT) is a modern astronomical time standard defined by the International Astronomical Union, primarily for time-measurements of astronomical observations made from the surface of Earth.[1] For example, the Astronomical Almanac uses TT for its tables of positions (ephemerides) of the Sun, Moon and planets as seen from Earth. In this role, TT continues Terrestrial Dynamical Time (TDT or TD),[2] which succeeded ephemeris time (ET). TT shares the original purpose for which ET was designed, to be free of the irregularities in the rotation of Earth.
The unit of TT is the SI second, the definition of which is based currently on the caesium atomic clock,[3] but TT is not itself defined by atomic clocks. It is a theoretical ideal, and real clocks can only approximate it.
TT is distinct from the time scale often used as a basis for civil purposes, Coordinated Universal Time (UTC). TT is indirectly the basis of UTC, via International Atomic Time (TAI). Because of the historical difference between TAI and ET when TT was introduced, TT is approximately 32.184 s ahead of TAI.
3.2 TT(BIPM)
A definition of a terrestrial time standard was adopted by the International Astronomical Union (IAU) in 1976 at its XVI General Assembly and later named Terrestrial Dynamical Time (TDT). It was the counterpart to Barycentric Dynamical Time (TDB), which was a time standard for Solar system ephemerides, to be based on a dynamical time scale. Both of these time standards turned out to be imperfectly defined. Doubts were also expressed about the meaning of 'dynamical' in the name TDT.
In 1991, in Recommendation IV of the XXI General Assembly, the IAU redefined TDT, also renaming it "Terrestrial Time". TT was formally defined in terms of Geocentric Coordinate Time (TCG), defined by the IAU on the same occasion. TT was defined to be a linear scaling of TCG, such that the unit of TT is the "SI second on the geoid",[4] i.e. the rate approximately matched the rate of proper time on the Earth's surface at mean sea level. Thus the exact ratio between TT time and TCG time was
{\displaystyle 1-L_{g}}
{\displaystyle L_{G}=U_{G}/c^{2}}
was a constant and
{\displaystyle U_{G}}
was the gravitational potential at the geoid surface, a value measured by physical geodesy. In 1991 the best available estimate of
{\displaystyle L_{g}}
was 6.969291×10−10.
In 2000, the IAU very slightly altered the definition of TT by adopting an exact value, Lg = 6.969290134×10−10.[5]
{\displaystyle TT={\bigl (}1-L_{g}{\bigr )}\times TCG\ +\ E}
where TT and TCG are linear counts of SI seconds in Terrestrial Time and Geocentric Coordinate Time respectively,
{\displaystyle L_{g}}
is the constant difference in the rates of the two time scales, and
{\displaystyle E}
is a constant to resolve the epochs (see below).
{\displaystyle L_{g}}
is defined as exactly 6.969290134×10−10. Due to the term
{\displaystyle 1-L_{g}}
the rate of TT is very slightly slower than that of TCG.
The equation linking TT and TCG more commonly has the form given by the IAU,
{\displaystyle TT=TCG-L_{g}\times {\bigl (}JD_{TCG}-2443144.5003725{\bigr )}\times 86400}
{\displaystyle JD_{TCG}}
is the TCG time expressed as a Julian date (JD). The Julian Date is a linear transformation of the raw count of seconds represented by the variable TCG, so this form of the equation is not simplified. The use of a Julian Date specifies the epoch fully. The above equation is often given with the Julian Date 2443144.5 for the epoch, but that is inexact (though inappreciably so, because of the small size of the multiplier
{\displaystyle L_{g}}
). The value 2443144.5003725 is exactly in accord with the definition.
Time coordinates on the TT and TCG scales are specified conventionally using traditional means of specifying days, inherited from non-uniform time standards based on the rotation of Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with their predecessor Ephemeris Time (ET), TT and TCG were set to match ET at around Julian Date 2443144.5 (1977-01-01T00Z). More precisely, it was defined that TT instant 1977-01-01T00:00:32.184 exactly and TCG instant 1977-01-01T00:00:32.184 exactly correspond to the International Atomic Time (TAI) instant 1977-01-01T00:00:00.000 exactly. This is also the instant at which TAI introduced corrections for gravitational time dilation.
{\displaystyle JD_{TT}=E_{JD}+{\bigl (}JD_{TCG}-E_{JD}{\bigr )}\times {\bigl (}1-L_{g}{\bigr )}}
{\displaystyle E_{JD}}
is 2443144.5003725 exactly.
Main article: International Atomic Time
The main realization of TT is supplied by TAI. The TAI service, performed since 1958, estimates TT using measurements from an ensemble of atomic clocks spread over the surface and low orbital space of Earth. TAI is canonically defined retrospectively, in monthly bulletins, in relation to the readings shown by that particular group of atomic clocks at the time. Estimates of TAI are also provided in real time by the institutions that operate the participating clocks. Because of the historical difference between TAI and ET when TT was introduced, the TAI realization of TT is defined thus:
{\displaystyle TT(TAI)=TAI+32.184~{\text{s}}}
TT(BIPM)[edit]
Because TAI is never revised once published, it is possible for errors in it to become known and remain uncorrected. Approximately annually since 1992, the International Bureau of Weights and Measures (BIPM) has produced better realizations of TT based on reanalysis of historical TAI data. BIPM's realizations of TT are named in the form "TT(BIPM08)", with the digits indicating the year of publication. They are published in the form of a table of differences from TT(TAI), along with an extrapolation equation that may be used for dates later than the table. The latest as of April 2022[update] is TT(BIPM21).[6]
Researchers from the International Pulsar Timing Array collaboration have created a realization of TT based on observations of an ensemble of pulsars. This new pulsar time scale is an independent means of computing TT, and it may eventually be useful to identify defects in TAI.[7]
Sometimes times described in TT are used in situations where TT's detailed theoretical properties are not significant. Where millisecond accuracy is enough (or more than enough), TT can be summarized in the following manners:
To millisecond accuracy, TT is parallel to the atomic timescale (International Atomic Time, TAI) maintained by the BIPM. TT is ahead of TAI, and can be approximated as TT ≅ TAI + 32.184 seconds.[8] (The offset 32.184 s arises from the history.[9])
TT is also parallel with the GPS time scale, which has a constant difference from atomic time (TAI − GPS time = +19 seconds),[10] so that TT ≅ GPS time + 51.184 seconds.
TT is in effect a continuation of (but is more precisely uniform than) the former Ephemeris Time (ET). It was designed for continuity with ET,[11] and it runs at the rate of the SI second, which was itself derived from a calibration using the second of ET (see, under Ephemeris time, Redefinition of the second and Implementations).
TT is slightly ahead of UT1 (a refined measure of mean solar time at Greenwich) by an amount known as ΔT = TT − UT1. ΔT was measured at +67.6439 seconds (TT ahead of UT1) at 0h UTC on 1 January 2015;[12] and by retrospective calculation, ΔT was close to zero about the year 1900. The difference ΔT, though somewhat unpredictable in fine detail, is expected to continue to increase, with UT1 becoming steadily (but irregularly) further behind TT in the future.
Relativistic relationships[edit]
The present definition of TT is a linear scaling of Geocentric Coordinate Time (TCG), which is the proper time of a notional observer who is infinitely far away (so not affected by gravitational time dilation) and at rest relative to Earth. TCG is used to date mainly for theoretical purposes in astronomy. From the point of view of an observer on Earth's surface the second of TCG passes in slightly less than the observer's SI second. The comparison of the observer's clock against TT depends on the observer's altitude: they will match on the geoid, and clocks at higher altitude tick slightly faster.
^ The 1991 definition refers to the scale agreeing with the SI second "on the geoid", i.e. close to mean sea level on Earth's surface, see IAU 1991 XXIst General Assembly (Buenos Aires) Resolutions, Resolution A.4 (Recommendation IV). A redefinition by resolution of the IAU 2000 24th General Assembly (Manchester), at Resolution B1.9, is in different terms intended for continuity and to come very close to the same standard.
^ "IAU(1991) RECOMMENDATION IV". IERS.
^ "Resolution B1.9 of the IAU XXIV General Assembly, 2000".
^ "Index of /ftp/pub/tai/ttbipm". webtai.bipm.org. Retrieved 24 April 2022.
^ Hobbs, G.; Guo, L.; Caballero, R. N.; Coles, W.; Lee, K. J.; Manchester, R. N.; Reardon, D. J.; Matsakis, D.; Tong, M. L.; Arzoumanian, Z.; Bailes, M.; Bassa, C. G.; Bhat, N D R.; Brazier, A.; Burke-Spolaor, S.; Champion, D. J.; Chatterjee, S.; Cognard, I.; Dai, S.; Desvignes, G.; Dolch, T.; Ferdman, R. D.; Graikou, E.; Guillemot, L.; Janssen, G. H.; Keith, M. J.; Kerr, M.; Kramer, M.; Lam, M. T.; et al. (2020). "A pulsar-based time-scale from the International Pulsar Timing Array". Monthly Notices of the Royal Astronomical Society. 491 (4): 5951–5965. arXiv:1910.13628. Bibcode:2020MNRAS.491.5951H. doi:10.1093/mnras/stz3071. S2CID 204961320.
^ The atomic time scale A1 (a predecessor of TAI) was set equal to UT2 at its conventional starting date of 1 January 1958 (see L Essen, "Time Scales", Metrologia, vol.4 (1968), 161-165, at 163), when ΔT (ET-UT) was about 32 seconds. The offset 32.184 seconds was the 1976 estimate of the difference between Ephemeris Time (ET) and TAI, "to provide continuity with the current values and practice in the use of Ephemeris Time" (see IAU Commission 4 (Ephemerides), Recommendations to IAU General Assembly 1976, Notes on Recommendation 5, note 2)
^ Steve Allen. "Time Scales". Lick Observatory. Retrieved 13 August 2017.
^ P K Seidelmann (ed.) (1992), 'Explanatory Supplement to the Astronomical Almanac', at p.42; also IAU Commission 4 (Ephemerides), Recommendations to IAU General Assembly 1976, Notes on Recommendation 5, note 2.
^ US Naval Observatory (USNO) data file online at https://web.archive.org/web/20190808224315/http://maia.usno.navy.mil:80/ser7/deltat.data (accessed 27 October 2015).
^ For example, IAU Commission 4 (Ephemerides), Recommendations to IAU General Assembly 1976, Notes on Recommendation 5, note 1, as well as other sources, indicate the time scale for apparent geocentric ephemerides as a proper time.
^ B Guinot (1986), "Is the International Atomic Time a Coordinate Time or a Proper Time?", Celestial Mechanics, 38 (1986), pp.155-161.
^ IAU General Assembly 1991, Resolution A4, Recommendations III and IV, define TCB and TCG as coordinate time scales, and TT as a linear scaling of TCG, hence also a coordinate time.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Terrestrial_Time&oldid=1084538407"
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Subjective Well-Being in India - Erika Sanborne Media
Supplementary Graphics and Information for my GSA2021 Presentation on SWB in India
Short link back to this page, for sharing:
erka.me/Ind
How Age and Having Someone one can Count on Explain Subjective Well-Being in India
Voiceover mp3 (8.5 min)
Click the buttons to access the poster and accompanying voiceover, which presents/walks through the poster highlights.
We were strongly encouraged to use the #betterposter format for this conference. I tried really hard to adhere to the #betterposter rules and to the additional rules written into the provided template note sections, but I’m afraid I couldn’t quite satisfy all of them. One set of rules said to use less text, more images. Another set of rules said, “no images except in the right column.” In sum, I feel that my poster is a 50% #betterposter (?). I tried.
GSA 2021 Annual Scientific Meeting – Disruption to Transformation: Aging in the “New Normal” – November 10-13, 2021
Short answer: Subjective well-being is an umbrella term generally conceived of as how someone feels about their own life.
A bit more on this: It is most often thought of as a combination of three distinct yet related dimensions or separate measures of well-being: positive affect, negative affect, and a cognitive appraisal (Diener, 1984; Jebb et al., 2020; OECD, 2013; Stone and Mackie, 2013). The cognitive appraisal reflects the evaluative well-being or “life evaluation” principally at focus in the present study. Affective well-being is thought of as more malleable, reflective of mood and current circumstances, whereas evaluative well-being is more durable over time and unlikely to change from day to day.
How is subjective well-being measured?
Short answer: By systematically asking the people of the world how they’re doing.
Longer answer: Subjective well-being can be measured in many ways. In the present study, the outcome measure is the Life Evaluation Index from the Gallup World Poll. The Life Evaluation Index is primarily based on an average of the present and future-looking evaluative well-being questions (the self-anchoring Cantril ladder scale questions, for those familiar). According to OECD guidelines (OECD, 2013), a single, present-oriented Cantril ladder question is an adequate measure of subjective well-being if only one question is going to be used, although Gallup has several subjective well-being items. The Life Evaluation Index is also informed by affective well-being and related measures, particularly in terms of determining cut-points for the 3-point ordered Life Evaluation Index.
The value of the Life Evaluation Index measure from Gallup is that it is based on questions that have been asked around the globe in most countries, in nationally-representative samples, carefully translated into native languages, and asked the same way since the first Gallup World Poll in 2006. This gives us a vantage point both for looking within a country and for looking around among them. The present-oriented Cantril ladder question from the Gallup World Poll is also the basis of the World Happiness Report‘s annual rankings of nations, published by the Sustainable Development Solutions Network in partnership with many others. While India’s rankings are similar on that one question as they are on the Life Evaluation Index, the latter was chosen for this study because it is a more inclusive, fuller well-being measure.
What is the subjective well-being situation in India overall?
Subjective Well-Being Around the World, and Ranking India per GWP Life Evaluation Index
Source is Gallup World Poll (GWP) respondent-level survey for year waves 2006-2019 (n=2,154,513). Each marker represents one country’s mean of respondent-level data aggregated annually by country. Variables are weighted using GWP respondent-level sampling weight variable -wgt- per guidelines. Life Evaluation Index is an evaluative well-being measure (INDEX_LE) on a 3-point scale. It is based on both present and future oriented Cantril ladder questions, with cutpoints determined by factor analysis of several related well-being measures and clustering.
What is the conceptual model grounding the present study?
What do we usually expect to predict subjective well-being? What explains who is thriving, struggling, or suffering, in general?
Short-ish answer: The scientists behind the World Happiness Report analyzed this question for the entire world. In 2020, they determined that “six key variables contribute to explaining the full sample of national annual average (Cantril ladder) scores over the whole period 2005-2019. These variables are GDP per capita, social support, healthy life expectancy, freedom, generosity, and absence of corruption” (World Happiness Report 2020, p.14).
Short fun fact: They also looked at how much each of these six variables contributed to variation in life evaluation scores. They found, for the world as a whole, that “the largest single part (33%) comes from social support, followed by GDP per capita (25%) and healthy life expectancy (20%), and then freedom (13%), generosity (5%), and corruption (4%)” (World Happines Report 2020, p.18).
Longer answer: When modeling something at the level of humankind, by aggregating respondent-level data, extremes will tend to cancel one another out, and nuance necessarily gets lost in order for broad patterns to emerge, and the story of the world cannot possibly be the identical story of any one country. The present study models subjective well-being based largely on these expected or known predictors of it, while also considering what makes India unique. Life expectancy for example stands out, because India’s life expectancy has increased impressively over the past three decades.
Such a remarkable improvement in human life expectancy would normally be accompanied by an increase in subjective well-being, and yet it was not. What could explain that? What is the role of health problems or, better framed, disability, socially-defined? What is the role of access to adequate healthcare in determining subjective well-being in India today?
Combining the generally-accepted known predictors of subjective well-being, with the current situation in India, the present study models subjective well-being based on these predictors, with the key predictors in bold type:
– disability (and its interaction with age)
– generosity (I call it prosocial behavior)
– education, gender, region, and religion.
What is the final, best-fitting regression model from this study?
Short answer: A partial proportional odds (PPO) model. I fit the model using the gologit2 user-written program for Stata (Williams, R. 2016).
an individual survey respondent
SWB outcome category for respondent i
1, 2 = (SWB outcome categories – 1)
\alpha
the set of constants from all j models
the set of that respondent’s scores on all x1-x11 variables in the model
the set of coefficients for that predictor or control variable at the respondent’s SWB level; for variables where the proportional odds assumption was held, that coefficient is independent of outcome level and does not vary by j. Each respondent i has their own independent multinomial distribution
//I cleaned the data first and renamed variables
//I am using Stata 16.1 SE.
svyset [pweight=wgt]
//best final ppo/gologit model for 2009
gsvy: gologit2 WB c.logincome i.socialsup c.logage##i.disability i.prosocial i.freedom ///
i.education i.corruptind i.gender i.region i.hindu if year==2009, ///
autofit force
eststo ppo2009
Gallup (2021). Gallup World Poll [The_Gallup_101521.dta]. Washington, DC, Gallup.
Williams, Richard (2016). GOLOGIT2: Understanding and interpreting generalized ordered logit models. The Journal of Mathematical Sociology, 40:1, 7-20, http://www.tandfonline.com/doi/full/10.1080/0022250X.2015.1112384. */
Gologit2/ppo model coefficients for SWB in India in 2009 and 2019 compared
Gologit2/ppo model coefficients for SWB in India in 2009 and 2019 compared. This coefplot is graphing coefficients for P(Yi>1) or the probability of an outcome greater than suffering. Because this is a ppo model, coefficients vary per level of the outcome variable.
The previous coefplot is in place of a traditional regression table, because this is a website, not a paper, and graphics more readily convey information than do tables stuffed with small font numbers. For those who love such things, you can see the numbers you want in the coefplot as well, I promise. Here is a full coefplot showing all 12 terms in the model, if interested, also plotting P(Yi>suffering).
Great. How about AMEs?
Yes! Average marginal effects plots, coming right up. Please keep in mind that, because of how averages work, this too masks nuance that is key to interpreting findings. It’s a great starting point, but don’t stop at the AME plot please.
Average Marginal Effects of Key Subjective Well-Being Predictors in India 2009 vs. 2019
If you’d like to “zoom in” on the AMEs for disability and log-age in 2019, see here.
A few things to notice in the AME plots above? Focusing on the 2019 AMEs, two variables significantly reduce the probability of suffering: log-income, and social support. And there is one variable that significantly increases the probability of suffering, log-age. Age increases probability of suffering by about 5 percentage points on average (95%CI: 2-8 percentage points) in 2019.
We can better understand that range in the next plot.
Predictive Margins for the Age-Disability Interaction in India
Interesting, right? Looking at the 2019 predictive margins, for logage-disability interaction, notice the purple circles, which represent “suffering” (the lowest of the three categorical outcomes. Do you see the 2-8 percentage point range? Who is likely experiencing the 8 percentage point increase in probability of suffering? (The oldest people with disabilities.)
So look at the AME plot again. Where are there signs of hope? What decreased the probability of suffering in 2019? Two things:
income: a standard deviation increase in log-transformed household income (equivalent of a household going from 55,000 to 140,000 rupees/year). That decreased the probability of suffering by 17 percentage points.
social support: having a friend, someone to count on in times of trouble. That alone decreased the probability of suffering by 11 percentage points.
Can you say more about disability and the interaction between personal impairment and social barriers?
Historically, disability was thought of as being a personal attribute of a human being. For example, one would be considered disabled if one had their left leg amputated. This is consistent with a medical model for understanding disability, and it was the primary conceptualization in most countries until the 1990s, when a more social model began to emerge. The framing of a problem matters because it suggests or limits the potential solutions and interventions. If the problem is a missing left leg and the situation is framed as a tragedy, the logical intervention is charity. If the problem is a missing leg and the situation is framed as a medical issue, the logical intervention is medical treatment. However, if the problem is that someone with such an impairment cannot access a school, for example, the situation is framed as a social issue and the needed intervention is at the level of institutional policy change towards universal design and accessibility. Because after all, in a world full of ramps, a person who uses a wheelchair is not disabled.
More info: In a development context, the United Nations also defines disability socially, as the result of interaction between personal impairments and environmental barriers that hinder one’s participation in society. Reference: The Convention on the Rights of Persons with Disabilities (CRPD), a 2006 human rights treaty signed by 168 nations including India, led by the United Nations Department of Economic and Social Affairs.
How do you interpret the ppo/gologit regression coefficients in this case?
I want to share a somewhat oversimplified explanation first. For all variables in the model that are dichotomous indicators, which is all variables except income and age, a positive coefficient means that a score of “yes” on that indicator increases the likelihood of higher categorical levels of the SWB outcome. A negative coefficient means that a score of “yes” increases the likelihood of the present or lower categorical levels of the SWB outcome.
Of course, because this is not a regular ordered logit model, these coefficients sometimes vary depending on the outcome level we’re looking at. Coefficients are the same at all levels only for those terms for which the proportional odds, or the parallel lines assumptions, held. (If that assumption was upheld for all terms, I would have used an ordered logit though, so it’s not that simple to interpret, unfortunately.)
A PPO/gologit model is interpreted by considering the coefficients at each level of the ordered outcome. So in this case, we have three levels of an ordered outcome, from low to high they are suffering, struggling, and thriving, numerically identified by 1, 2, and 3. The regression coefficients reflect the effects of each of the 11 variables in the model, the key predictors and the controls, and the coefficient represents the odds of a respondent being either above the lowest outcome category (of suffering), or of being in the highest outcome category (which is thriving). Revisit the equation for the model too, noting the role of this probability of being “> j”.
How was social support operationalized in this study?
This excellent question came up in a live session. Social support was measured using the Gallup World Poll survey item WP27, which asks:
It is a binary indicator, and a respondent either has at least one person they can count or, or nobody. The measure does not address magnitude (i.e. how many friends one has). Someone has social support if they respond “yes” to this item.
“Who needs a friend?” Anyone whose household income you cannot triple, and whose probability of suffering you’d prefer to reduce. As you can see in the plots, there is not a lot of thriving in India right now. Hence, reduction in suffering is a fair goal. Both income and social support can help considerably.
There is that other matter of the age-disability interaction, and it seems to suggest a need for increased national spending on healthcare, consistent with both the World Health Organization’s recommendations for India, and what the Indian government themselves claimed in early 2021. More on that in my forthcoming publication.
Thanks for being here. What do you think? I welcome your responses and questions either by email or in the comments below.
Research presented herein and reported in the related upcoming publication has benefited from support provided by the Minnesota Population Center (Award number P2CHD041023), which receives funding from the Eunice Kennedy Shriver National Institute of Child Health and Human Development, and by the University of Minnesota Life Course Center on the Demography and Economics of Aging (P30AG066613), funded through a grant from the National Institute on Aging. #NIAFundsMe
You don’t need this QR code though, because you’re already here. 🙂
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Languager: Unicode in Python
Python has been making long strides in embracing unicode. With python 3 we are at a stage where python programs can support unicode well however python program-source is still completely drawn from the ASCII subset of unicode.
Well… Actually with python 3 (not 2) this is already possible
def solvequadratic(a,b,c):
Δ = b*b - 4*a*c
α = (-b + sqrt(Δ))/(2*a)
β = (-b - sqrt(Δ))/(2*a)
return (α, β)
>>> solvequadratic(1,-5,6)
Now to move ahead!
Why do we have to write x!=y then argue about the status of x<>y when we can simply write x≠y?
Or take a random example from the tutor list :
print math.floor( 31.58889 )
print math.ceil( 31.58889 )
print ⌊31.58889
print ⌈31.58889
So we could say python is half-way towards becoming a full unicode language. To move in this direction can mean at least two things:
Make python 'native' to other natural human languages
Embrace the universal (ie mathematical) side of unicode more fully
1 is all about internationalization and localization. The writeup addresses only 2. It is given in the form of tables showing how current Ascii syntax could transform into a Unicode-embracing one.
The ideas came from a number of people on the python list – see references below.
Since most of the following is in the form of tables with current (Ascii) syntax juxtaposed with the more unicode-d one, it turns out that many of the comments on these pairs are similar and repetitious. To keep these tables neat, the repeating comments are spelt out first as under:
2.1 Math Space Advantage – MSA
One of the less noticed benefits of math (like) operators is that a math-op like + in program text is lexically unambiguous ie '+x' is two tokens + and x and not a single token composed of + and x. This is unlike alphanumerics where all the following being lexically different
for x inline:…
forx in line:…
makes spaces mandatory
We will see that moving to a more pervasively unicode form, makes many spaces that are currently inevitable, become unnecessary.
Below I will point such cases out with a 'MSA'. In some cases its technically required to have spaces, in others its just more aesthetic to have them. eg.
x in lst
xinlst
1in[1,2,3]
However that's completely unreadable.
There's no such problem with
1∈[1,2,3]
So replacing in by ∈ has a math-space-advantage (MSA). It also has the advantage of
2.2 Disambiguation – Dis
The in in for loops and in predicates have very different semantics conceptually; the latter is purely declarative, the former creates a binding. So having two unmixupable ins is good for reducing confusions eg. for x ⬅ [1,2,3]: …
and if x ∈ [1,2,3]:
IOW due to extreme scarcity of characters in Ascii, many characters have for generations been overloaded willy-nilly. As that scarcity becomes a thing of the past maybe we should avoid useless overloading? These cases are marked by Dis.
2.3 Name Space burden reduction
A (perfectly normal English) word like floor or ceiling cannot be put into the global (builtin) namespace because a programmer may want to use that name for usual or related connotation of floor/ceiling. For a symbols like ⌊,⌈ no such issue arises. Symbol NS
2.4 Unicode Choice – UC
In many (all?) cases unicode offers so much new variety that its not clear which choice to make. Such choices are indicated with UC
2.5 Font Issue – FI
When things are not looking exactly proper/pretty on my end and it seems to be a font issue, I'll mark a FI
2*pi*r 2×π×r FI
x!=y x≠y
x<=y x≤y
x>=y x≥y
q,r=divmod(a,b) q,r=a÷b 1
float(inf) ∞ NS
pow(2,4) 2⇑4 2
2**4 2⇑4 2
math.floor(3.5) ⌊3.5 NS
math.ceiling(3.5) ⌈3.5 NS
Python already has a large bunch of division related operators and functions: /, //, %, divmod. Given that quotient together with remainder is a common integer arithmetic pattern, and structured return values is much easier in python than in classic imperative languages like C, my preference is for ÷ to stand in for divmod. Other choices with their justifications are of course possible.
Are pow and ** the same?
Do x and × look the same? If yes, this is a problem and maybe * is just preferable?
4 Other basic Syntax
4.1 Assignment ←
x = 1 x ← 1
x,y = y,x x,y ← y,x
x += y x +← y
If one could count the grief caused by thinking that = is math-equality – not just noobs but experienced C programmers who mistakenly put a = when they meant == …
The ← is not looking very nice out here (in different fonts): either too scrawny or to stubby. So...
While in an earlier version of this post I had used that for examples, I am (for now) reverting to good ol =
4.2 Attribute access →
sys.argv[1] sys→argv[1]
(5).to_bytes(4,"little") 5→to_bytes(4,"little") Dis, MSA
4.3 in (predicate)
1 in [1,2,3] 1 ∈ [1,2,3] MSA,FI
Most of the fonts Ive checked make the ∈ a little too large
I guess this should be treated as a transient problem – a fixable bug
4.4 in (for)
for x in [1,2,3,4]: … for x⬅[1,2,3,4]: … MSA,UC
The sign could be any one of ⬅ ⇐ ⇦ ?
The two ins now disambiguated to ⬅ and ∈ should be a help to noobs
4.5 lambda λ
lambda x: x+3 λx: x+3 MSA
not x ¬x MSA
x and y x∧y MSA
x or y x∨y MSA
Sets, Bags and Lists (numpy arrays??) form a series. Having literals for all makes some succinct expressions possible
[1,2]+[3,4] [1,2]⤚[3,4] Dis
List append is not symmetric (commutative).
The operator should reflect that fact.
The most natural charecter for set literals is '{}' However given that
that is already taken by dicts
and dicts are more fundamental to programming than sets
⦃ ⦄ should be a good enough approx to conventional usage
Common set theory operators that mathematicians use ∈ ∉ ⊂ ⊃ ⊆ ⊇ ⊈ ⊉ ∪ ∩ ∅
Now unicode makes these available without any markuping
Ascii – OO forms
Ascii – functional forms
set([]) set([]) ∅
s = set([1,2,3] s=⦃1,2,3⦄ MSA
t = set([2,3,4,5]) t=⦃2,3,4,5⦄
x in s x∈s MSA
x not in s x∉s MSA
s.issubset(t) s<=t s⊆t
??? s<t s⊂t
not s.issubset(t) not (s <= t) s⊈t 1,2,3
set([1]).
issubset([2,1]) set([1]) <= set([2,1]) ⦃1⦄ ⊆ ⦃2,1⦄ 3
s.issuperset(t) s>=t s⊇t
s.union(t) s|t s∪t
s.intersection(t) s&t s∩t
s.difference(t) s-t s∖t FI,UC
symmetric_difference(t) s^t s∆t
s.update(t) s|t s∪=t
intersection_update(t) s&=t s∩=t
For numbers, not (x <= y) ⇒ x>y This is not the case for sets. In somewhat incorrect! math jargon, <= is a total order whereas ⊆ is a partial order . Therefore ⊈ is more needed than <=
The low precedence of not makes parentheses unnecessary but I find it confusing
While in general the OO form (column 1) is the most verbose, in these cases it is more readable than column 2
FI,UC
Are s\t and s∖t distinguishable? They dont look to me…
Unicode gives one of the names of ∆ as "symmetric difference".
Dont know of any natural/standard sign for difference (other than '-' '\' '/'). There are zillions of other symbols of course.
6.3 Counter (bag/multiset)
c = Counter(a=3, b=4) c = ⟅'a':3, 'b':4⟆ NS
d = Counter(a=1, b=2) d = ⟅'a':1, 'b':2⟆
c + d c ⊕ d
Counter({'b': 6, 'a': 4}) ⟅'a':4, 'b':6⟆
c & d c ∩ d
Counter({'a': 1, 'b': 1}) ⟅'a':1, 'b':1⟆
c | d c ∪ d
Counter can only be used after from collections import Counter
Having to do this is an avoidable headache.
Not having to do this (in the current dispensation) entails a pollution of the global namespace)
Its another matter that Counter is an unfortunate name choice, given that
Bag/Multiset already exist and are well known
Counter already has more than many other established meanings in CS
Note that list 'addition' (append) is not symmetric hence the asymmetric ⤚
Bag 'addition' is symmetric. The operator should reflect that.
Which symbol to use? ⊕ or ⋄ ?
The ⋄ (in code) looks worse than the plain ⋄ out here
Python already has 'natural' casting (at the type level). Given
Literals even allow for use of the most 'natural' operators
Set ∪, ∩
Counter ⊕
List ⤚
with the general rule that the upper-row operators pull lower data upwards eg
[1,2,3]∪[2,3,4,5] ⟹ ⦃1,2,3,4,5⦄
ie order and repetition vanishes
[2,1,2]⊕[2,3,4,5] ⟹ ⟅1:1, 2:3, 3:1, 4:1, 5:1⟆
ie order vanishes, repetition maintained
Disambiguated literals makes natural casting possible:
x∪y expects x, y to be sets. What if they are not?? Simple – they are cast to sets
Likewise x⊕y expects x, y to be Counters. Else they are cast to counters
Presence of literals makes other things possible and natural, eg…
Once we have literals for sets and bags we can have comprehensions for them:
Natural Comprehensions
⦃x*x∣x⬅⦃1,2,3⦄⦄ ⟹ ⦃1,4,9⦄
Natural because both input and output collection are same
We can also have
Casting Comprehensions
⦃x*x∣x⬅[1,2,1,3,1]⦄ ⟹ ⦃1,4,9⦄
ie the intention of the list-to-set cast is that order and repetition are discarded
Note: Many noob misunderstandings re comprehensions come from the clever pun – for in loops and in comprehensions. This removes that problem
UC: The │ (∣) is not the usual | (codepoint 9474 vs 124). It could be some other character – in addition to the ascii | there are │∣┃ ¦ │ (and probably more!!)
6.6 N-ary Operators
In mathematics there are a number of constructs like ∑, ∀ etc. They can be subsumed under the general concept of n-ary operators – aka generalized products.
N-ary operators are complementary to comprehensions. If t is some type and C is one of set, Counter or list
can be thought to have type t → C(t)
can be thought to have type C(t) → t
N-ary operators are like reduce in that they generalize a binary to a collection.
N-ary operators are like lambda/comprehensions in that they imply a local binding
However there are issues. Consider for some arbitrary term t(x)
(∑ x∈⦃1,2,3⦄ : t(x))
= t(1) + t(2) + t(3)
However there is a catch: ⦃1,2,3⦄ == ⦃1,2,3,1,2⦄ [In standard python syntax set([1,2,3,1,2]) == set([1,2,3]) ]
That is, since sets contain elements whose repetition count is unspecified, the sum above is also t(1)+t(2)+t(3)+t(1)+t(2) or anything else!!
So clearly the appropriate collection for a ∑ is Counter, not set or list
In general, we see that for the n-ary operators we also have a natural collection over which they operate
+ ∑ Counter
× ∏ Counter
∧ ∀ Set
∨ ∃ Set
⊕ ⨁ Counter
∪ ⋃ Set
∩ ⋂ Set
In general the principle is that for operators that are commutative and associative we use Counter. For operators that are idempotent as well we use Set.
Note that if an operator is not commutative and associative it has no meaningful n-ary. If it is, then list is over-specific; which is why we only find set and counter above.
7 Strings/Quoteds
Python has a menagerie of quoteds and unicode has a corresponding one of quote-like characters. How to match them I'm not really sure... Heres a start
"Tom said \"Mary said \"Yoohoo!\"\"" «Tom said «Mary said «Yoohoo!»»»
r"a\nb" ‹a\nb›
u"हरि ॐ" ⟪हरि ॐ⟫
Note that whether » is one character or two is similar to the problem we have with quotes. Is '' a single double-quote or a double(d) single quote? Depending on the font this may be obvious or not
The above – so-called 'French-quotes' – seem to be widely used in languages other than French. German quotes however have some inconsistency problems.
Maybe code literals (compile, parser etc) ⟦ ⟧ following denotational semantics?
code = compile('a + 5',...) code = compile(⟦a + 5⟧, ...)
[Seems neat in the context of Lisp or denotational semantics, not sure of python]
8 Long·Identifiers
There is also some evidence (?) suggesting that a-long-identifier is more readable than a_long_identifier is more readable than aLongIdentifier
The hyphenated option suffers from a severe ambiguity because hyphen and minus are the same letter…
… in Ascii only!
No More! Now we can write a·long·identifier
Well lisp and Cobol are exceptions but they incur their own heavy cost – math expressions cant be written naturally
a is b a ≣ b
Or ≡ ?
The difficulties/noob-confusions of python's is should significantly reduce with this!
10 APL/Numpy integration
Ideas in numpy is largely lifted from APL. Unicode makes it possible to carry (some of!) APL's lexemes as well. And not to go overboard in this and repeat APL's mistakes!
array([2,3,4]) ⟨2,3,4⟩
range(10) ⍳10
a.shape ⍴a
a.reshape(2,3) a⍴(2,3)
take(a,2) a↑2
drop(a,2) a↓2
Numpy-array comprehensions
Advanced stuff – probably with inspiration from Alpha-Polyhedra
11 Questionable below
12 Keywords and Special Constants
Following Antoon's wish for def we could have 𝗮𝗯𝗰𝗱𝗲𝗳𝗴𝗵𝗶𝗷𝗸𝗹𝗺𝗻𝗼𝗽𝗾𝗿𝘀𝘁𝘂𝘃𝘄𝘅𝘆𝘇 versions of the following keywords
I personally consider more important to have 𝐍𝐨𝐧𝐞, ( 𝗡𝗼𝗻𝗲 ?) 𝕋, 𝔽
TF
for True and False
Really mixing up fonts with characters seems like a bad idea (for programming). Why not colors? Sizes?…
More generally most of the SMP seems like nonsense (to me)
Finally this does not seem to be working! So even if SMP is a good idea its probably not ready for general use (Trying numeric 핋 120139 dec or 핋 ie hex 1D54B )
sqrt(s) √x
Looks like poor over-specific syntax (to me) (But what do i know?!)
Large swathes of unicode's math-space could be available in operator which users (aka programmers) can choose to bind at will.
Given the experience of readability of APL this may be ill-advised… Maybe not – C++ devotees like the possibilities of overloading basic arithmetic operators.
15.1 Steven D'Aprano
π (some other math symbols?) [Steven ?]
(Problems with) ∑ for sum Steven 1
Steven 2 example: was towards showing that something like this is undesirable:
import ⌺
⌚ = ⌺.╩░
⑥ = 5*⌺.⋨⋩
❹ = ⑥ - 1
♅⚕⚛ = [⌺.✱✳**⌺.❇*❹{⠪|⌚.∣} for ⠪ in ⌺.⣚]
⌺.˘˜¨´՛՜(♅⚕⚛)
Somebody else pointed out that this is actually valid. Cant remember who and I certainly cant make this (as is) work.
That mathematicians used sets does not makes sets as fundamental in programming as dicts – [Steven ?] (so {} for dicts and something else for sets is ok)
15.2 Antoon Pardon
· for ident separator (instead of '_') [Antoon ?]
× for multiplication Antoon 2
⇑ for exponentiation [Antoon ?]
→ for attribute access Antoon 3
⤚ for list append Antoon 3
bold (SMP) letters in identifiers [Antoon ?]
15.3 Mark Harris
∈ ∉ ∀ Δ Mark 1 Mark 2
√ for sqrt Mark ?
Labels: Python, Unicode
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Find the two square roots of
i
If you were to graph
in the complex plane, it would be located at
\left(0, 1\right)
.What are the polar coordinates for the point
\left(0, 1\right)
\textit{r} = 1, \theta=\frac{\pi}{2}, \textit{n}=2
k = 0, 1
\sqrt{1}\left( \cos \left( \frac{\frac{\pi}{2}+2k \pi}{2} \right) +i \sin \left( \frac{\frac{\pi}{2}+2k \pi}{2} \right) \right)
k = 0
\cos\frac{\pi}{4}+\textit{i}\sin\frac{\pi}{4}-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\textit{i}
k = 1
\sqrt{1}\left( \cos \left( \frac{\frac{\pi}{2}+2\pi}{2} \right) +i \sin \left( \frac{\frac{\pi}{2}+2 \pi}{2} \right) \right) \left( \cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4} \right) ??
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Board Paper Solutions for CBSE Class 12-science MATHS Board Paper 2015 Abroad Set 2
\mathrm{A} =\left[\begin{array}{ccc}5& 6& -3\\ -4& 3& 2\\ -4& -7& 3\end{array}\right]
, then write the cofactor of the element a21 of its 2nd row. VIEW SOLUTION
{\left(\frac{{\mathrm{d}}^{2}\mathrm{y}}{d{x}^{2}}\right)}^{2} + {\left(\frac{\mathrm{dy}}{\mathrm{d}x}\right)}^{3} +\mathit{ }{x}^{4} = 0.
Write the solution of the differential equation
\frac{\mathrm{dy}}{\mathrm{dx}}= {2}^{-\mathrm{y}}
Find the unit vector in the direction of the sum of the vectors
2\stackrel{^}{i}+3\stackrel{^}{j}-\stackrel{^}{k} \mathrm{and} 4\stackrel{^}{i}-3\stackrel{^}{j}+2\stackrel{^}{k}.
Find the area of a parallelogram whose adjacent sides are represented by the vectors
2\stackrel{^}{i}-3\stackrel{^}{k} \mathrm{and} 4\stackrel{^}{j}+2\stackrel{^}{k}.
Find the sum of the intercepts cut off by the plane
2x+y-z=5,
on the coordinate axes. VIEW SOLUTION
\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{\mathrm{cos}x}{1+{e}^{x}}dx
Three machines E1, E2 and E3 in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E1 and E2 are defective and that 5% of those produced by machine E3 are defective. If one bulb is picked up at random from a day's production, calculate the probability that it is defective.
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X. VIEW SOLUTION
\stackrel{^}{j}+\stackrel{^}{k} \mathrm{and} 3\stackrel{^}{i}-\stackrel{^}{j}+4\stackrel{^}{k}
represent the two sides vectors
\stackrel{\to }{\mathrm{AB}} \mathrm{and} \stackrel{\to }{\mathrm{AC}}
respectively of triangle ABC. Find the length of the median through A. VIEW SOLUTION
\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}
If 2 tan−1 (cos θ) = tan−1 (2 cosec θ), (θ ≠ 0), then find the value of θ.
{\mathrm{tan}}^{-1}\left(\frac{1}{1+1.2}\right)+{\mathrm{tan}}^{-1}\left(\frac{1}{1+2.3}\right)+...+{\mathrm{tan}}^{-1}\left(\frac{1}{1+n.\left(n+1\right)}\right)={\mathrm{tan}}^{-1} \mathrm{\theta }
, then find the value of θ. VIEW SOLUTION
\mathrm{A}=\left[\begin{array}{cc} 2& -1\\ -1& 2\end{array}\right]
and I is the identity matrix of order 2, then show that A2= 4 A − 3 I. Hence find A−1.
\mathrm{A}=\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right] \mathrm{and} \mathrm{B}=\left[\begin{array}{cc}\mathrm{a}& 1\\ \mathrm{b}& -1\end{array}\right] \mathrm{and} {\left(\mathrm{A}+\mathrm{B}\right)}^{2}={\mathrm{A}}^{2}+{\mathrm{B}}^{2}
, then find the values of a and b. VIEW SOLUTION
Using properties of determinants, prove the following :
\left|\begin{array}{ccc}1& a& {a}^{2}\\ {a}^{2}& 1& a\\ a& {a}^{2}& 1\end{array}\right|={\left(1-{a}^{3}\right)}^{2}
\int \frac{\mathrm{sin} \left(x-a\right)}{\mathrm{sin} \left(x+a\right)}dx
\int \frac{{x}^{2}}{\left({x}^{2}+4\right)\left({x}^{2}+9\right)}dx
Find whether the following function is differentiable at x = 1 and x = 2 or not :
f\left(x\right)=\left\{\begin{array}{ccc}x,& & x < 1\\ 2-x,& & 1\le x\le 2\\ -2+3x-{x}^{2},& & x>2 \end{array}\right\
In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as
\mathrm{A} = \left[\begin{array}{c}140\\ 200\\ 150\end{array}\right]\begin{array}{c}\mathrm{Telephone}\\ \mathrm{House} \mathrm{Call}\\ \mathrm{Letters} \end{array}
\begin{array}{ccc}\mathrm{Telephone}& \mathrm{House} \mathrm{Call}& \mathrm{Letters}\end{array}\phantom{\rule{0ex}{0ex}}\mathrm{B} =\left[\begin{array}{ccc} 1000 & 500& 5000\\ 3000 & 1000 & 10000\end{array}\right]\begin{array}{c}\mathrm{City} \mathrm{X}\\ \mathrm{City} \mathrm{Y}\end{array}
What should one consider before casting his/her vote − party's promotional activity or their social activities ? VIEW SOLUTION
\int {e}^{2x}·\mathrm{sin} \left(3x+1\right) dx
Find the point on the curve 9y2 = x3, where the normal to the curve makes equal intercepts on the axes. VIEW SOLUTION
\mathrm{If} y={\left(x+\sqrt{1+{x}^{2}}\right)}^{n}, \mathrm{then} \mathrm{show} \mathrm{that}\phantom{\rule{0ex}{0ex}}\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}={n}^{2}y.
Find the minimum value of (ax + by), where xy = c2.
Find the coordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x − 3. VIEW SOLUTION
Maximise z = 8x + 9y subject to the constraints given below :
3x − 2y ≤6
x, y ≥ 0 VIEW SOLUTION
Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6. VIEW SOLUTION
Let f : N → ℝ be a function defined as f(x) = 4x2 + 12x + 15. Show that f : N → S, where S is the range of f, is invertible. Also find the inverse of f. VIEW SOLUTION
Using integration, find the area of the region bounded by the line x – y + 2 = 0, the curve x =
\sqrt{y}
and y-axis. VIEW SOLUTION
Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution. VIEW SOLUTION
Solve the following differential equation :
\left[y-x \mathrm{cos}\left(\frac{y}{x}\right)\right]dy+\left[y \mathrm{cos}\left(\frac{y}{x}\right)-2x \mathrm{sin}\left(\frac{y}{x}\right)\right]dx=0
\left(\sqrt{1+{x}^{2}+{y}^{2}+{x}^{2} {y}^{2}}\right) dx+xy dy = 0
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Recent comments—The Stacks project
Comments 1 to 20 out of 6884 in reverse chronological order.
\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}
On May 24, 2022 T.C. left comment #7380 on Lemma 10.12.9 in Commutative Algebra
In the second proof, shouldn't it be
\mu_i\otimes 1=\psi\circ\lambda_i
\lambda_i=\psi\circ (\mu_i\otimes 1)
On May 23, 2022 Matthieu Romagny left comment #7379 on Lemma 15.11.8 in More on Algebra
Yes, it is in the SP, see for instance Lemma 01WM.
On May 23, 2022 Laurent Moret-Bailly left comment #7378 on Lemma 15.11.8 in More on Algebra
@#7377: A universal homeomorphism is integral (EGA IV, 18.12.10). So perhaps this should be (resp. already is) in the Stacks Project.
On May 23, 2022 comment_bot left comment #7377 on Lemma 15.11.8 in More on Algebra
It may be useful to include the statement that the same holds for any
A \rightarrow B
that is a universal homeomorphism on spectra (sorry if this is already in the Stacks Project but I missed it!). I think this follows from the characterization of Henselian pairs in terms of lifting idempotents.
On May 23, 2022 Sriram left comment #7376 on Lemma 15.91.3 in More on Algebra
In the proof of (2), showing the surjection of the canonical map to completion, the sequence of equations must have an "f" in the coefficient of x_1. That is, " ... = x-x_0+f e_1 = x-x_0 -f x_1 + f^2 e_2 = ..."
On May 23, 2022 Torsten Wedhorn left comment #7375 on Lemma 37.21.7 in More on Morphisms
A rather trivial observation: The flatness hypothesis on
f
seems to be superfluous except for (1) because of openness of flatness. So I think, one could slightly strengthen the result by supposing only that
f
is locally of finite presentation and add in (1) the condition that
f
is flat in
x
in the description of
W
Sorry, I just realized that the flatness is of course also needed for (3). So please forget my comment.
On May 22, 2022 Yiming TANG left comment #7374 on Section 10.134 in Commutative Algebra
In tag 00S1, "φ:P→P is a morphism of presentations from α to α′" should be "φ:P→P' is a morphism of presentations from α to α′".
On May 22, 2022 DatPham left comment #7373 on Lemma 86.19.7 in Formal Algebraic Spaces
I think it would be better to indicate where we use the assumption that
f
is representable by algebraic spaces. I guess this is used to ensure that
X\times_Y T
is a quasi-compact algebraic space, hence admits an \' etale cover by an affine scheme
U
; the composite
U\to Y
then factors through some
Y_{\mu}
, and the same is true for
X\times_Y T
by the sheaf property.
f
f
f
x
W
On May 21, 2022 代数几何真难 left comment #7371 on Section 31.24 in Divisors
OK. I see. Even though
a\in \mathcal{O}_{X,x}
can not be lifted to an element in
\Gamma(X,\mathcal{O}_X)
in general, it can be lifted to an element in
\Gamma(X,\mathcal{K}'_X)
ad-cf
well-defined.
On May 21, 2022 XYETALE left comment #7370 on Section 10.113 in Commutative Algebra
I checked a bit that the notation
\mathrm{trdeg}_R(S)
probably has never defined for a ring, only for fields. Maybe it is a bit clearer to mention what it means.
I think in lemma 31.24.2, in order to define
ad-cf
as a section in
\Gamma(X,\mathcal{K}_X)
, one should first lift
a\in A_{\mathfrac{p}}
to a global section
a\in A
A\to A_{\mathfrac{p}}
is not surjective, seems that
ad-cf
is not well-defined.
On May 17, 2022 Alekos Robotis left comment #7368 on Section 37.11 in More on Morphisms
In definition 37.1.11, there is a slight typo : it says "soure" instead of "source."
On May 17, 2022 Shizhang left comment #7367 on Lemma 13.27.7 in Derived Categories
Maybe the vertical arrows should all go downward?
On May 17, 2022 David Holmes left comment #7366 on Section 10.34 in Commutative Algebra
Hi Zongzhu Lin, I think 005K might be the reference you are looking for? Alternatively, I think it's not so hard to see the claim from the definition of constructibility.
On May 16, 2022 Wojtek Wawrów left comment #7365 on Section 111.3 in A Guide to the Literature
The page has been archived by the Wayback Machine. At least some parts of the book are still available through it: https://web.archive.org/web/20110707004531/http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1
On May 16, 2022 Yijin Wang left comment #7364 on Lemma 28.22.11 in Properties of Schemes
Typo in the proof of lemma 28.22.11: the first sentence should be 'A_1，A_2 ⊂A'
On May 16, 2022 Alex Ivanov left comment #7363 on Lemma 29.35.16 in Morphisms of Schemes
In (2), it should in fact suffice to assume that
Y
is locally of finite type over
S
. (This is also consistent with Lemma 02FW, to which the proof refers).
On May 16, 2022 Pieter Belmans left comment #7362 on Section 10.34 in Commutative Algebra
@7361: That is a limitation of how things are being displayed. There is a work-around possible, the question is whether there are enough cases of this causing confusion to put in the effort. I'll put it on the possible features list though, thanks for noticing!
On May 15, 2022 Laurent Moret-Bailly left comment #7361 on Section 10.34 in Commutative Algebra
Strangely, in "tags" mode, parts (1) and (2) are also converted to tags in the proof (but not in the statement).
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Tangent-secant_theorem Knowpia
The tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's Elements.
property of inscribed angles
{\displaystyle \Rightarrow }
{\displaystyle \angle PG_{2}T=\angle PTG_{1}}
{\displaystyle \Rightarrow }
{\displaystyle \triangle PTG_{2}\sim \triangle PG_{1}T}
{\displaystyle \Rightarrow }
{\displaystyle {\frac {|PT|}{|PG_{2}|}}={\frac {|PG_{1}|}{|PT|}}}
{\displaystyle \Rightarrow }
{\displaystyle |PT|^{2}=|PG_{1}|\cdot |PG_{2}|}
Given a secant g intersecting the circle at points G1 and G2 and a tangent t intersecting the circle at point T and given that g and t intersect at point P, the following equation holds:
{\displaystyle |PT|^{2}=|PG_{1}|\cdot |PG_{2}|}
The tangent-secant theorem can be proven using similar triangles (see graphic).
Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem.
S. Gottwald: The VNR Concise Encyclopedia of Mathematics. Springer, 2012, ISBN 9789401169820, pp. 175-176
Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, ISBN 9780470591796, p. 161
Tangent Secant Theorem at proofwiki.org
Power of a Point Theorem auf cut-the-knot.org
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RF signal attenuation due to rainfall - MATLAB rainpl - MathWorks í•œêµ
Signal path elevation angle, specified as a real-valued scalar, or as an M-by-1 or 1-by- M vector. Units are in degrees between –90° and 90°. If elev is a scalar, all propagation paths have the same elevation angle. If elev is a vector, its length must match the dimension of range and each element in elev corresponds to a propagation range in range.
Tilt angle of the signal polarization ellipse, specified as a real-valued scalar, or as an M-by-1 or 1-by- M vector. Units are in degrees between –90° and 90°. If tau is a scalar, all signals have the same tilt angle. If tau is a vector, its length must match the dimension of range. In that case, each element in tau corresponds to a propagation path in range.
The tilt angle is defined as the angle between the semi-major axis of the polarization ellipse and the x-axis. Because the ellipse is symmetrical, a tilt angle of 100° corresponds to the same polarization state as a tilt angle of -80°. Thus, the tilt angle need only be specified between ±90°.
{\mathrm{γ}}_{R}=k{R}^{\mathrm{α}},
r=\frac{1}{0.477{d}^{0.633}{R}_{0.01}^{0.073\mathrm{α}}{f}^{0.123}â10.579\left(1â\mathrm{exp}\left(â0.024d\right)\right)}
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Zero-point Energy - Vixrapedia
Zero-point Energy (ZPE) is a predicted minimum energy state for all objects to possess. For example: the pendulum mass on a swinging wire will always have an energy greater than
{\displaystyle E_{0}={\bar {h}}\omega }
, no matter how much energy you take away from the system. All objects have a ZPE that can be exploited, even if it is not yet practicable to do so.
Retrieved from "https://www.vixrapedia.org/w/index.php?title=Zero-point_Energy&oldid=437"
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Arbitrage pricing theory - Wikipedia
In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross in 1976,[1] it is widely believed to be an improved alternative to its predecessor, the Capital Asset Pricing Model (CAPM).[2] APT is founded upon the law of one price, which suggests that within an equilibrium market, rational investors will implement arbitrage such that the equilibrium price is eventually realised.[2] As such, APT argues that when opportunities for arbitrage are exhausted in a given period, then the expected return of an asset is a linear function of various factors or theoretical market indices, where sensitivities of each factor is represented by a factor-specific beta coefficient or factor loading. Consequently, it provides traders with an indication of ‘true’ asset value and enables exploitation of market discrepancies via arbitrage. The linear factor model structure of the APT is used as the basis for evaluating asset allocation, the performance of managed funds as well as the calculation of cost of capital. [3]
1.2 Assumptions of APT Model
3 Difference between the capital asset pricing model
APT is a single-period static model, which helps investors understand the trade-off between risk and return. The average investor aims to optimise the returns for any given level or risk and as such, expects a positive return for bearing greater risk. As per the APT model, risky asset returns are said to follow a factor intensity structure if they can be expressed as:
{\displaystyle r_{j}=a_{j}+\beta _{j1}f_{1}+\beta _{j2}f_{2}+\cdots +\beta _{jn}f_{n}+\epsilon _{j}}
{\displaystyle a_{j}}
is a constant for asset
{\displaystyle j}
{\displaystyle f_{n}}
is a systematic factor
{\displaystyle \beta _{jn}}
is the sensitivity of the
{\displaystyle j}
th asset to factor
{\displaystyle n}
, also called factor loading,
{\displaystyle \epsilon _{j}}
is the risky asset's idiosyncratic random shock with mean zero.
Idiosyncratic shocks are assumed to be uncorrelated across assets and uncorrelated with the factors.
The APT model states that if asset returns follow a factor structure then the following relation exists between expected returns and the factor sensitivities:
{\displaystyle \mathbb {E} \left(r_{j}\right)=r_{f}+\beta _{j1}RP_{1}+\beta _{j2}RP_{2}+\cdots +\beta _{jn}RP_{n}}
{\displaystyle RP_{n}}
is the risk premium of the factor,
{\displaystyle r_{f}}
That is, the expected return of an asset j is a linear function of the asset's sensitivities to the n factors.
Note that there are some assumptions and requirements that have to be fulfilled for the latter to be correct: There must be perfect competition in the market, and the total number of factors may never surpass the total number of assets (in order to avoid the problem of matrix singularity).
General Model[edit]
For a set of assets with returns
{\displaystyle r\in \mathbb {R} ^{m}}
, factor loadings
{\displaystyle \Lambda \in \mathbb {R} ^{m\times n}}
, and factors
{\displaystyle f\in \mathbb {R} ^{n}}
, a general factor model that is used in APT is:
{\displaystyle r=r_{f}+\Lambda f+\epsilon ,\quad \epsilon \sim {\mathcal {N}}(0,\Psi )}
{\displaystyle \epsilon }
follows a multivariate normal distribution. In general, it is useful to assume that the factors are distributed as:
{\displaystyle f\sim {\mathcal {N}}(\mu ,\Omega )}
{\displaystyle \mu }
is the expected risk premium vector and
{\displaystyle \Omega }
is the factor covariance matrix. Assuming that the noise terms for the returns and factors are uncorrelated, the mean and covariance for the returns are respectively:
{\displaystyle \mathbb {E} (r)=r_{f}+\Lambda \mu ,\quad {\text{Cov}}(r)=\Lambda \Omega \Lambda ^{T}+\Psi }
It is generally assumed that we know the factors in a model, which allows least squares to be utilized. However, an alternative to this is to assume that the factors are latent variables and employ factor analysis - akin to the form used in psychometrics - to extract them.
Assumptions of APT Model[edit]
The APT model for asset valuation is founded on the following assumptions:[2]
Investors are risk-averse in nature and possess the same expectations
Efficient markets with limited opportunity for arbitrage
Infinite number of assets
Risk factors are indicative of systematic risks that cannot be diversified away and thus impact all financial assets, to some degree. Thus, these factors must be:
Non-specific to any individual firm or industry
Compensated by the market via a risk premium
Arbitrage is the practice whereby investors take advantage of slight variations in asset valuation from its fair price, to generate a profit. It is the realisation of a positive expected return from overvalued or undervalued securities in the inefficient market without any incremental risk and zero additional investments.
A correctly priced asset here may be in fact a synthetic asset - a portfolio consisting of other correctly priced assets. This portfolio has the same exposure to each of the macroeconomic factors as the mispriced asset. The arbitrageur creates the portfolio by identifying n correctly priced assets (one per risk-factor, plus one) and then weighting the assets such that portfolio beta per factor is the same as for the mispriced asset.
When the investor is long the asset and short the portfolio (or vice versa) he has created a position which has a positive expected return (the difference between asset return and portfolio return) and which has a net zero exposure to any macroeconomic factor and is therefore risk free (other than for firm specific risk). The arbitrageur is thus in a position to make a risk-free profit:
Difference between the capital asset pricing model[edit]
The APT along with the capital asset pricing model (CAPM) is one of two influential theories on asset pricing. The APT differs from the CAPM in that it is less restrictive in its assumptions, making it more flexible for use in a wider range of application. Thus, it possesses greator explanatory power (as opposed to statistical) for expected asset returns. It assumes that each investor will hold a unique portfolio with its own particular array of betas, as opposed to the identical "market portfolio". In some ways, the CAPM can be considered a "special case" of the APT in that the securities market line represents a single-factor model of the asset price, where beta is exposed to changes in value of the market.
Fundamentally, the CAPM is derived on the premise that all factors in the economy can be reconciled into one factor represented by a market portfolio, thus implying they all have equivalent weight on the asset’s return. In contrast, the APT model suggests that each stock reacts uniquely to various macroeconomic factors and thus the impact of each must be accounted for separately.[2]
A disadvantage of APT is that the selection and the number of factors to use in the model is ambiguous. Most academics use three to five factors to model returns, but the factors selected have not been empirically robust. In many instances the CAPM, as a model to estimate expected returns, has empirically outperformed the more advanced APT.[4]
their impact on asset prices manifests in their unexpected movements and they are completely unpredictable to the market at the beginning of each period [2]
they should represent undiversifiable influences (these are, clearly, more likely to be macroeconomic rather than firm-specific in nature) on expected returns and so must be quantifiable with non-zero prices [2]
Chen, Roll and Ross identified the following macro-economic factors as significant in explaining security returns:[5]
surprises in GNP as indicated by an industrial production index;
a diversified stock index such as the S&P 500 or NYSE Composite;
^ a b c d e f Basu, Debarati; Chawla, Deepak (2012). "An Empirical Test of the Arbitrage Pricing Theory—The Case of Indian Stock Market". Global Business Review. 13 (3): 421–432. doi:10.1177/097215091201300305. ISSN 0972-1509.
^ Huberman, G. & Wang, Z. (2005). "Arbitrage Pricing Theory" (PDF). {{cite web}}: CS1 maint: multiple names: authors list (link)
^ French, Jordan (1 March 2017). "Macroeconomic Forces and Arbitrage Pricing Theory". Journal of Comparative Asian Development. 16 (1): 1–20. doi:10.1080/15339114.2017.1297245.
^ Chen, Nai-Fu; Roll, Richard; Ross, Stephen A. (1986). "Economic Forces and the Stock Market". The Journal of Business. 59 (3): 383–403. doi:10.1086/296344. ISSN 0021-9398. JSTOR 2352710.
Burmeister, Edwin; Wall, Kent D. (1986). "The arbitrage pricing theory and macroeconomic factor measures". Financial Review. 21 (1): 1–20. doi:10.1111/j.1540-6288.1986.tb01103.x.
Chen, N. F.; Ingersoll, E. (1983). "Exact Pricing in Linear Factor Models with Finitely Many Assets: A Note". Journal of Finance. 38 (3): 985–988. doi:10.2307/2328092. JSTOR 2328092.
Roll, Richard; Ross, Stephen (1980). "An empirical investigation of the arbitrage pricing theory". Journal of Finance. 35 (5): 1073–1103. doi:10.2307/2327087. JSTOR 2327087.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Arbitrage_pricing_theory&oldid=1085873203"
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Compact multiclass model for support vector machines (SVMs) and other classifiers - MATLAB - MathWorks
\begin{array}{cccc}& \text{Learner 1}& \text{Learner 2}& \text{Learner 3}\\ \text{Class 1}& 1& 1& 0\\ \text{Class 2}& -1& 0& 1\\ \text{Class 3}& 0& -1& -1\end{array}
\stackrel{^}{k}
\stackrel{^}{k}=\underset{k}{\text{argmin}}\frac{\sum _{l=1}^{B}|{m}_{kl}|g\left({m}_{kl},{s}_{l}\right)}{\sum _{l=1}^{B}|{m}_{kl}|}.
{L}_{d}\approx ⌈10{\mathrm{log}}_{2}K⌉
{L}_{s}\approx ⌈15{\mathrm{log}}_{2}K⌉
\Delta \left({k}_{1},{k}_{2}\right)=0.5\sum _{l=1}^{L}|{m}_{{k}_{1}l}||{m}_{{k}_{2}l}||{m}_{{k}_{1}l}-{m}_{{k}_{2}l}|,
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PoissonWindow - Maple Help
Home : Support : Online Help : Science and Engineering : Signal Processing : Windowing Functions : PoissonWindow
multiply an array of samples by a Poisson windowing function
PoissonWindow( A, alpha )
The PoissonWindow( A, alpha ) command multiplies the Array A by the Poisson windowing function, with parameter
\mathrm{\alpha }
The Poisson windowing function
w\left(k\right)
\mathrm{\alpha }
N
w\left(k\right)={ⅇ}^{-\mathrm{\alpha }|\frac{2k}{N}-1|}
The SignalProcessing[PoissonWindow] command is thread-safe as of Maple 18.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
N≔1024:
a≔\mathrm{GenerateUniform}\left(N,-1,1\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627948900036964}}
\mathrm{PoissonWindow}\left(a,1.23\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627948757766140}}
c≔\mathrm{Array}\left(1..N,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right):
\mathrm{PoissonWindow}\left(\mathrm{Array}\left(1..N,'\mathrm{fill}'=1,'\mathrm{datatype}'='\mathrm{float}'[8],'\mathrm{order}'='\mathrm{C_order}'\right),0.72,'\mathrm{container}'=c\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627948757742044}}
u≔\mathrm{`~`}[\mathrm{log}]\left(\mathrm{FFT}\left(c\right)\right):
\mathbf{use}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{plots}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{display}\left(\mathrm{Array}\left(\left[\mathrm{listplot}\left(\mathrm{ℜ}\left(u\right)\right),\mathrm{listplot}\left(\mathrm{ℑ}\left(u\right)\right)\right]\right)\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end use}
The SignalProcessing[PoissonWindow] command was introduced in Maple 18.
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Lemma 10.69.3. Let $R \to R'$ be a flat ring map. Let $M$ be an $R$-module. Suppose that $f_1, \ldots , f_ r \in R$ form an $M$-quasi-regular sequence. Then the images of $f_1, \ldots , f_ r$ in $R'$ form a $M \otimes _ R R'$-quasi-regular sequence.
Proof. Set $J = (f_1, \ldots , f_ r)$, $J' = JR'$ and $M' = M \otimes _ R R'$. We have to show the canonical map $\mu : R'/J'[X_1, \ldots X_ r] \otimes _{R'/J'} M'/J'M' \to \bigoplus (J')^ nM'/(J')^{n + 1}M'$ is an isomorphism. Because $R \to R'$ is flat the sequences $0 \to J^ nM \to M$ and $0 \to J^{n + 1}M \to J^ nM \to J^ nM/J^{n + 1}M \to 0$ remain exact on tensoring with $R'$. This first implies that $J^ nM \otimes _ R R' = (J')^ nM'$ and then that $(J')^ nM'/(J')^{n + 1}M' = J^ nM/J^{n + 1}M \otimes _ R R'$. Thus $\mu $ is the tensor product of (10.69.0.1), which is an isomorphism by assumption, with $\text{id}_{R'}$ and we conclude. $\square$
Comment #920 by JuanPablo on August 15, 2014 at 21:08
I think it wasn't entirely clear how the flatness hypothesis is used. Here flatness hypothesis is used for the following:
I
R
I' = IR'
IM \otimes_R R'=I'M'
. This is seen by tensoring the exact sequence
0 \rightarrow IM \rightarrow M
. In this lemma for
I=J^n
. The lemma follows tensoring the equation in the definition of quasi-regular sequences.
Yes, you are right. Thanks! Fixed here.
Comment #4179 by Nils Waßmuth on April 19, 2019 at 13:21
I think there is a typo in the definition of the map
\mu
n
variables but our sequence only has
r
|
Plasma Physics and Controlled Fusion (1550)
Stabilization of explosive instabilities by nonlinear frequency shifts
Oraevskii, V.N.; Pavlenko, V.P.; Wilhelmsson, H.; Kogan, E.Y.
Physical Review Letters; v. 30(2); p. 49-51
AMPLITUDES, EXPLOSIVE INSTABILITY, NONLINEAR PROBLEMS, PLASMA, PLASMA WAVES, STABILITY
INSTABILITY, PLASMA INSTABILITY
Simple model for ablative stabilization
[en] We present a simple analytic model for ablative stablization of the Rayleigh-Taylor instability. In this model the effect of ablation is to move the peak of the perturbations to the location of peak pressure. This mechanism enhances the density-gradient stabilization, which is effective at short wavelengths, and it also enhances the stabilization of long-wavelength perturbations due to finite shell thickness. We consider the following density profile: exponential blowoff plasma with a density gradient β, followed by a constant-density shell of thickness δt. For perturbations of arbitrary wave number k, we present an explicit expression for the growth rate γ as a function of k, β, and δt. We find that ''thick'' shells defined by β δt≥1 have γ2≥0 for any k, while ''thin'' shells defined by β δt<1 can have γ2<0 for small k, reflecting stability by proximity to the back side of the shell. We also present LASNEX simulations that are in good agreement with our analytic formulas
Physical Review. A; ISSN 1050-2947; ; CODEN PLRAAN; v. 46(10); p. 6621-6627
ABLATION, PLASMA MACROINSTABILITIES, RAYLEIGH-TAYLOR INSTABILITY, STABILITY, SURFACE PROPERTIES
[en] Using a new electron gun, a number of measurements bearing on the generation of beam--plasma discharge (BPD) in WOMBAT (waves on magnetized beams and turbulence) [R. W. Boswell and P. J. Kellogg, Geophys. Res. Lett. 10, 565 (1983)] have been made. A beam--plasma discharge is an rf discharge in which the rf fields are provided by instabilities [W. D. Getty and L. D. Smullin, J. Appl. Phys. 34, 3421 (1963)]. The new gun has a narrower divergence angle than the old, and comparison of the BPD thresholds for the two guns verifies that the BPD ignition current is proportional to the cross-sectional area of the plasma. The high-frequency instabilities, precursors to the BPD, are identified with the two Trivelpiece--Gould modes [A. W. Trivelpiece and R. W. Gould, J. Appl. Phys. 30, 1784 (1959)]. Which frequency appears depends on the neutral pressure. The measured frequencies are not consistent with the simple interpretation of the lower frequency as a Cerenkov resonance with the low-Trivelpiece--Gould mode; it must be a cyclotron resonance. As is generally true in such beam--plasma interaction experiments, strong low-frequency waves appear at currents far below those necessary for BPD ignition. These low-frequency waves are shown to control the onset of the high-frequency precursors to the BPD. A mechanism for this control is suggested, which involves the conversion of a convective instability to an absolute one by trapping of the unstable waves in the density perturbations of the low-frequency waves. This process greatly reduces the current necessary for BPD ignition
BEAM-PLASMA SYSTEMS, ELECTRIC DISCHARGES, ELECTRON GUNS, PLASMA MICROINSTABILITIES
Berk, H.L.; Breizman, B.N.; Pekker, M.S.
[en] When the resonance condition of the particle-wave interaction is varied adiabatically, the particles trapped in a wave are found to form phase space holes or clumps that enhance the particle-wave energy exchange. This mechanism can cause increased saturation levels of instabilities and even allow the free energy associated with instability to be tapped in a system in which background dissipation suppresses linear instability
Anon; 244 p; 1996; p. 3C35; University of Texas; Austin, TX (United States); International Sherwood fusion theory conference; Philadelphia, PA (United States); 18-20 Mar 1996; Univ. of Texas at Austin, Institute for Fusion Studies, MS C1500, 26th and Speedway, RLM 11.214, Austin, TX 78712 (United States)
TRAPPED-PARTICLE INSTABILITY, WAVE PROPAGATION
Reduction of the Ablative Rayleigh-Taylor Growth Rate with Gaussian Picket Pulses
Collins, T.J.B.; Knauer, J.P.; Betti, R.; Boehly, T.R.; Delettrez, J.A.; Goncharov, V.N.; Meyerhofer, D.D.; McKenty, P.W.; Skupsky, S.; Town, R.P.J.
Laboratory for Laser Energetics (United States). Funding organisation: United States (United States)
[en] OAK-B135 The effect of a Gaussian prepulse (picket pulse) before a ''drive'' pulse on the Rayleigh-Taylor (RT) instability growth rate was measured for single-mode, 20-, 30-, and 60-mm-wavelength mass perturbations. These data, from the OMEGA [T. R. Boehly et al., Opt. Commun. 133, 495 (1997)] laser system, show that the measured RT growth of mass perturbations was reduced when a picket pulse was used. The picket pulse and subsequent relaxation period, before the drive pulse, cause the foil to expand and rarefy, resulting in higher ablation velocities during the drive pulse and greater ablative stabilization. This effect was examined both computationally and experimentally for different picket-pulse intensities
1 Apr 2004; [vp.]; 1437; 2003-163; FC03-92SF19460; Available from Oakland Operations Office, Oakland, CA; Submitted to Physics of Plasmas; Volume 11, No.4
DOE-SF--19460-528
ABLATION, INSTABILITY GROWTH RATES, LASERS, OMEGA FACILITY, PLASMA MACROINSTABILITIES, RAYLEIGH-TAYLOR INSTABILITY, RELAXATION, STABILIZATION
Hall currents and Rayleigh-Taylor instability of a rotating plasma
Ariel, P.D.
Indian J. Phys; v. 48(8); p. 703-711
HALL EFFECT, MAGNETIC FIELDS, PLASMA, PLASMA DENSITY, PLASMA MACROINSTABILITIES, RAYLEIGH-TAYLOR INSTABILITY, ROTATION, VARIATIONAL METHODS
On the filamentation of large amplitude lower-hybrid waves
Spatschek, K.H.; Shukla, P.K.; Yu, M.Y.
[en] The propagation of a large-amplitude lower-hybrid wave is considered. Modulational instabilities arising from its interaction with low-frequency electrostatic perturbations are investigated. The growth lengths of the convective instabilities are obtained and compared with previous results for adiabatic perturbations. (author)
Journal of Plasma Physics; ISSN 0022-3778; ; v. 18(pt.1); p. 165-172
CONVECTIVE INSTABILITIES, DISTURBANCES, INTERACTIONS, PLASMA FILAMENT, PLASMA WAVES, WAVE PROPAGATION
Dust grain growth and settling in initial gaseous giant protoplanets
Paul, G. C.; Datta, S.; Pramanik, J. N.; Rahman, M. M., E-mail: pcgour2001@yahoo.com
[en] Dust grain growth and settling time inside initial gaseous giant protoplanets in the mass range 0.3 to 5 Jovian masses, formed by gravitational instability, have been investigated. We have determined the distribution of thermodynamic and physical variables inside the protoplanets solving the structure equations assuming their gas blobs to be fully convective and with this distribution we have calculated growth and settling time of grains with different initial sizes (10−2 cm ≤ r0 ≤ 1 cm). The results of our calculations are found to be in good agreement with those obtained by different approaches.
Copyright (c) 2012 The Society of Geomagnetism and Earth, Planetary and Space Sciences, The Seismological Society of Japan; Country of input: International Atomic Energy Agency (IAEA)
Earth, Planets and Space (Online); ISSN 1880-5981; ; v. 64(7); p. 641-648
DISTRIBUTION, DUSTS, EQUATIONS, GRAIN GROWTH, GRAVITATIONAL INSTABILITY, PROTOPLANETS, THERMODYNAMICS
Jeans Gravitational Instability with
\kappa
-Deformed Kaniadakis Distribution
Chen Hui; Zhang Shi-Xuan; Liu San-Qiu, E-mail: hchen61@ncu.edu.cn
[en] The Jeans instabilities in an unmagnetized, collisionless, isotropic self-gravitating matter system are investigated in the context of
\kappa
-deformed Kaniadakis distribution based on kinetic theory. The result shows that both the growth rates and critical wave numbers of Jeans instability are lower in the
\kappa
-deformed Kaniadakis distributed self-gravitating matter systems than the Maxwellian case. The standard Jeans instability for a Maxwellian case is recovered under the limitation
\kappa =0
. (paper)
DISTRIBUTION, GRAVITATIONAL INSTABILITY, KINETICS, MATTER
Repetitive explosive instabilities
Weiland, J.; Wilhelmsson, H.
Physica Scripta; v. 7(5); p. 222-229
ANALYTICAL SOLUTION, COMPUTER CALCULATIONS, COUPLING, DISSIPATION FACTOR, EIGENFREQUENCY, EXPLOSIVE INSTABILITY, MATRIX ELEMENTS, NONLINEAR PROBLEMS, NUMERICAL SOLUTION, PHASE SHIFT, PLASMA, PLASMA WAVES, RESONANCE
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Lemma 97.12.6 (0CXM)—The Stacks project
Lemma 97.12.6. Let $S$, $\mathcal{X}$, $U$, $x$, $u_0$ be as in Definition 97.12.2. Assume
$\Delta $ is locally of finite type (for example if $\mathcal{X}$ is limit preserving), and
$\mathcal{X}$ has (RS).
Let $V$ be a scheme locally of finite type over $S$ and let $y$ be an object of $\mathcal{X}$ over $V$. Form the $2$-fibre product
\[ \xymatrix{ \mathcal{Z} \ar[r] \ar[d] & (\mathit{Sch}/U)_{fppf} \ar[d]^ x \\ (\mathit{Sch}/V)_{fppf} \ar[r]^ y & \mathcal{X} } \]
Let $Z$ be the algebraic space representing $\mathcal{Z}$ and let $z_0 \in |Z|$ be a finite type point lying over $u_0$. If $x$ is versal at $u_0$, then the morphism $Z \to V$ is smooth at $z_0$.
Proof. (The parenthetical remark in the statement holds by Lemma 97.11.4.) Observe that $Z$ exists by assumption (1) and Algebraic Stacks, Lemma 93.10.11. By assumption (2) we see that $Z \to V \times _ S U$ is locally of finite type. Choose a scheme $W$, a closed point $w_0 \in W$, and an étale morphism $W \to Z$ mapping $w_0$ to $z_0$, see Morphisms of Spaces, Definition 66.25.2. Then $W$ is locally of finite type over $S$ and $w_0$ is a finite type point of $W$. Let $l = \kappa (z_0)$. Denote $z_{l, 0}$, $v_{l, 0}$, $u_{l, 0}$, and $x_{l, 0}$ the objects of $\mathcal{Z}$, $(\mathit{Sch}/V)_{fppf}$, $(\mathit{Sch}/U)_{fppf}$, and $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(l)$ obtained by pullback to $\mathop{\mathrm{Spec}}(l) = w_0$. Consider
\[ \xymatrix{ \mathcal{F}_{(\mathit{Sch}/W)_{fppf}, l, w_0} \ar[r] & \mathcal{F}_{\mathcal{Z}, l, z_{l, 0}} \ar[d] \ar[r] & \mathcal{F}_{(\mathit{Sch}/U)_{fppf}, l, u_{l, 0}} \ar[d] \\ & \mathcal{F}_{(\mathit{Sch}/V)_{fppf}, l, v_{l, 0}} \ar[r] & \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} } \]
By Lemma 97.3.3 the square is a fibre product of predeformation categories. By Lemma 97.12.5 we see that the right vertical arrow is smooth. By Formal Deformation Theory, Lemma 89.8.7 the left vertical arrow is smooth. By Lemma 97.3.2 we see that the left horizontal arrow is smooth. We conclude that the map
\[ \mathcal{F}_{(\mathit{Sch}/W)_{fppf}, l, w_0} \to \mathcal{F}_{(\mathit{Sch}/V)_{fppf}, l, v_{l, 0}} \]
is smooth by Formal Deformation Theory, Lemma 89.8.7. Thus we conclude that $W \to V$ is smooth at $w_0$ by More on Morphisms, Lemma 37.12.1. This exactly means that $Z \to V$ is smooth at $z_0$ and the proof is complete. $\square$
Typo in the statement of the lemma: Let Z be the algebraic space representing
\mathcal{W}
---> Let
Z
be the algebraic space representing
\mathcal{Z}
. (There is no
\mathcal{W}
around). $$
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CXM. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 0CXM, in case you are confused.
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Reed Switch, Reed Sensor and Magnet Glossary: Magneto-motive Force
RRE Home » Magneto-motive Force
Magneto-motive force is a quantity representing the line integral of the magnetic intensity around a closed line and is expressed in Ampere-Turns.
Magnetomotive_force (Wikipedia)
In physics, the magnetomotive force (mmf) is a quantity appearing in the equation for the magnetic flux in a magnetic circuit, often called Ohm's law for magnetic circuits. It is the property of certain substances or phenomena that give rise to magnetic fields:
{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}},}
where Φ is the magnetic flux and
{\displaystyle {\mathcal {R}}}
is the reluctance of the circuit. It can be seen that the magnetomotive force plays a role in this equation analogous to the voltage V in Ohm's law: V = IR, since it is the cause of magnetic flux in a magnetic circuit:
{\displaystyle {\mathcal {F}}=NI}
where N is the number of turns in the coil and I is the electric current through the circuit.
{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}}
{\displaystyle {\mathcal {R}}}
is the magnetic reluctance
{\displaystyle {\mathcal {F}}=HL}
where H is the magnetizing force (the strength of the magnetizing field) and L is the mean length of a solenoid or the circumference of a toroid.
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Will new evidence emerge for the frequency of solar superflares? | Metaculus
Will new evidence emerge for the frequency of solar superflares?
Created by VLT on {{qctrl.question.created_time | dateStr}}.
Published by VLT on {{qctrl.question.publish_time | dateStr}}.
The Carrington Event, the strongest solar storm on record, was no joke.
Shortly before Noon on September 1, 1859, amateur solar observers in England noticed an intense burst of light from a large sunspot group. Less than 20 hours later, a cloud of hot plasma ejected by the flare slammed into Earth's magnetosphere. Auroras were visible down to tropical latitudes, and telegraphs failed in spectacular fashion, reportedly delivering electric shocks to their operators. As discussed in a previous question, solar storms have the potential to impact power and communications systems, and storms at or above the Carrington level would have devastating effects. A 2013 assessment by Lloyd's estimates that if such an event were to occur today, damages to the US economy alone would range from 0.6 to 2.6 trillion dollars.
Solar flares occur in association with magnetic field reconnection near the Sun's surface, and are most frequent during the active period of the solar cycle. The occurrence rate of solar storm energies follows a relatively well-defined power-law distribution, and the Carrington event was estimated to have had energy ~
10^{32}
erg. The frequency of trans-Carrington storms, however, depends on how far the distribution extends, which is unknown.
Some constraint comes from geological records. Evidence from an overabundance of radioactive 14C detected in tree rings suggests the Sun might have produced a small "superflare" in AD 775 and again in AD 993, although alternate explanations for the anomalies are also viable.
Additional insight is gained by monitoring of the flares of nearby stars. Researchers using the LAMOST telescope have reported regular eruptions 10,000 times larger than the Carrington event on other stars. The team showed that these superflares are likely formed via the same mechanism as solar flares, and unexpectedly, they found ~10% of the superflaring stars have magnetic fields either comparable to or weaker than the Sun's, implicitly raising the possibility that our Sun could go amok with a massive flare.
Superflaring stars generally have short rotation periods (which generate higher levels of magnetic activity), but stars with rotation as slow as the Sun can apparently also produce superflares. A study published in Nature showed a total of 187 superflares on 23 solar-type stars (5600-6000 K, rotational period > 10 d) having energies in the
10^{32}-10^{36}
erg range. A consideration of theoretical estimates in conjunction with the observations generated a published hypothesis that superflares of
10^{34}
erg occur once in ~800 yr on our present Sun.
An analysis of Kepler photometry showed that superflares on solar-type stars (with rotational periods greater than 10 days) exhibit similar occurrence frequency distribution of those for solar flares. The analysis suggests, however, that all superflaring stars have starspot complexes substantially larger than those presently occurring during solar maxima.
A reasonable summary of the current evidence suggests that the Sun can produce flares up to ~1000x the strength of the Carrington event, but such flares would require sunspot activity at levels substantially larger than those seen in the historical record. Given the stakes, it would be nice to have a better handle on the odds.
By July 2018, will additional significant evidence emerge suggesting that our Sun experiences
10^{34}
erg or larger flares on a time scale shorter than 1000 years?
Positive resolution requires a paper in the peer-reviewed literature by July 2018 in which a "most likely" or "fiducial" estimate of solar flares with energy
10^{34}
erg exceeds 1 per 1000 years. In addition, in order to add a specious bit of flair to the question, positive resolution will also occur in the unlikely event that a flare with energy exceeding
10^{32}
ergs occurs prior to July 2018.
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Newton's Law of Universal Gravitation | Boundless Physics | Course Hero
Sir Isaac Newton's inspiration for the Law of Universal Gravitation was from the dropping of an apple from a tree.
Newton's insight on the inverse-square property of gravitational force was from intuition about the motion of the earth and the moon.
The mathematical formula for gravitational force is
\text{F} = \text{G}\frac{\text{Mm}}{\text{r}^2}
\text{G}
induction: Use inductive reasoning to generalize and interpret results from applying Newton's Law of Gravitation.
While an apple might not have struck Sir Isaac Newton's head as myth suggests, the falling of one did inspire Newton to one of the great discoveries in mechanics: The Law of Universal Gravitation. Pondering why the apple never drops sideways or upwards or any other direction except perpendicular to the ground, Newton realized that the Earth itself must be responsible for the apple's downward motion.
While Newton was able to articulate his Law of Universal Gravitation and verify it experimentally, he could only calculate the relative gravitational force in comparison to another force. It wasn't until Henry Cavendish's verification of the gravitational constant that the Law of Universal Gravitation received its final algebraic form:
\displaystyle \text{F} = \text{G}\frac{\text{Mm}}{\text{r}^2}
\text{F}
represents the force in Newtons,
\text{M}
\text{m}
represent the two masses in kilograms, and
\text{r}epresents the separation in meters.
\text{G}
represents the gravitational constant, which has a value of
6.674\cdot 10^{-11} \text{N}\text{(m/kg)}^2
. Because of the magnitude of
\text{G}
, gravitational force is very small unless large masses are involved.
Since force is a vector quantity, the vector summation of all parts of the shell contribute to the net force, and this net force is the equivalent of one force measurement taken from the sphere's midpoint, or center of mass (COM).
The gravitational force on an object within a uniform spherical mass is linearly proportional to its distance from the sphere's center of mass (COM).
The Law of Universal Gravitation states that the gravitational force between two points of mass is proportional to the magnitudes of their masses and the inverse-square of their separation,
\text{d}
\displaystyle \text{F}=\frac{\text{GmM}}{\text{d}^2}
Since force is a vector quantity, the vector summation of all parts of the shell/sphere contribute to the net force, and this net force is the equivalent of one force measurement taken from the sphere's midpoint, or center of mass (COM). So when finding the force of gravity exerted on a ball of 10 kg, the distance measured from the ball is taken from the ball's center of mass to the earth's center of mass.
When considering the gravitational force exerted on an object at a point inside or outside a uniform spherically symmetric object of radius
\text{R}
, there are two simple and distinct situations that must be examined: the case of a hollow spherical shell, and that of a solid sphere with uniformly distributed mass.
The gravitational force acting by a spherically symmetric shell upon a point mass inside it, is the vector sum of gravitational forces acted by each part of the shell, and this vector sum is equal to zero. That is, a mass
\text{m}
within a spherically symmetric shell of mass
\text{M}
, will feel no net force (Statement 2 of Shell Theorem).
The net gravitational force that a spherical shell of mass
\text{M}
exerts on a body outside of it, is the vector sum of the gravitational forces acted by each part of the shell on the outside object, which add up to a net force acting as if mass
\text{M}
is concentrated on a point at the center of the sphere (Statement 1 of Shell Theorem).
Diagram used in the proof of the Shell Theorem: This diagram outlines the geometry considered when proving The Shell Theorem. In particular, in this case a spherical shell of mass
\text{M}
(left side of figure) exerts a force on mass
\text{m}
(right side of the figure) outside of it. The surface area of a thin slice of the sphere is shown in color. (Note: The proof of the theorem is not presented here. Interested readers can explore further using the sources listed at the bottom of this article.)
The second situation we will examine is for a solid, uniform sphere of mass
\text{M}
\text{R}
, exerting a force on a body of mass
\text{m}
at a radius
\text{d}
inside of it (that is,
\text{d}< \text{R}
). We can use the results and corollaries of the Shell Theorem to analyze this case. The contribution of all shells of the sphere at a radius (or distance) greater than
\text{d}
from the sphere's center-of-mass can be ignored (see above corollary of the Shell Theorem). Only the mass of the sphere within the desired radius
\text{M}_{<\text{d}}
(that is the mass of the sphere inside
\text{d}
) is relevant, and can be considered as a point mass at the center of the sphere. So, the gravitational force acting upon point mass
\text{m}
\displaystyle \text{F}=\frac{\text{GmM}_{<\text{d}}}{\text{d}^2}
where it can be shown that
\displaystyle \text{M}_{<\text{d}}=\frac{4}{3}\pi \text{d}^3 \rho
\rho
is the mass density of the sphere and we are assuming that it does not depend on the radius. That is, the sphere's mass is uniformly distributed.)
\text{F}=\frac{4}{3} \pi \text{Gm} \rho \text{d}
which shows that mass
\text{m}
feels a force that is linearly proportional to its distance,
\text{d}
, from the sphere's center of mass.
As in the case of hollow spherical shells, the net gravitational force that a solid sphere of uniformly distributed mass
\text{M}
exerts on a body outside of it, is the vector sum of the gravitational forces acted by each shell of the sphere on the outside object. The resulting net gravitational force acts as if mass
\text{M}
is concentrated on a point at the center of the sphere, which is the center of mass, or COM (Statement 1 of Shell Theorem). More generally, this result is true even if the mass
\text{M}
is not uniformly distributed, but its density varies radially (as is the case for planets).
The second step in calculating earth's mass came with the development of Newton's law of universal gravitation.
By equating Newton's second law with his law of universal gravitation, and inputting for the acceleration a the experimentally verified value of 9.8
\text{m/}\text{s}^2
, the mass of earth is calculated to be
5.96 \cdot 1024
kg, making the earth's weight calculable given any gravitational field.
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between them.
\displaystyle \text{F} = \text{G}\frac{\text{m}_{1}\text{m}_{2}}{\text{r}^{2}}
\text{F}
is the force between the masses,
\text{G}
\text{m}_1
is the first mass,
\text{m}_2
is the second mass and
\text{r}
is the distance between the centers of the masses.
In this way it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its center.
For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force. The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance
\text{r}_0
from the center of the mass distribution:
The portion of the mass that is located at radii
\text{r}<\text{r}_0
causes the same force at
\text{r}_0
as if all of the mass enclosed within a sphere of radius
\text{r}_0
was concentrated at the center of the mass distribution (as noted above).
\text{r}>\text{r}_0
exerts no net gravitational force at the distance
\text{r}_0
from the center. That is, the individual gravitational forces exerted by the elements of the sphere out there, on the point at
\text{r}_0
, cancel each other out.
As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration anywhere within the hollow sphere. Furthermore, inside a uniform sphere the gravity increases linearly with the distance from the center; the increase due to the additional mass is 1.5 times the decrease due to the larger distance from the center. Thus, if a spherically symmetric body has a uniform core and a uniform mantle with a density that is less than
\frac{2}{3}
of that of the core, then the gravity initially decreases outwardly beyond the boundary, and if the sphere is large enough, further outward the gravity increases again, and eventually it exceeds the gravity at the core/mantle boundary.
Newton's law of universal gravitation. Provided by: WIKIPEDIA. Located at: http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation. License: CC BY-SA: Attribution-ShareAlike
Isaac Newton. Provided by: WIKIPEDIA. Located at: http://en.wikipedia.org/wiki/Isaac_Newton%23Apple_incident. License: CC BY-SA: Attribution-ShareAlike
inverse. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/inverse. License: CC BY-SA: Attribution-ShareAlike
Shell theorem. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Shell_theorem. License: CC BY-SA: Attribution-ShareAlike
center of mass. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/center%20of%20mass. License: CC BY-SA: Attribution-ShareAlike
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Newtons_Law_of_Universal_Gravitation.pdf
PHYSICS 1010 • Maadi STEM School for Girls
3 Newton's Law of Universal Gravitation.pdf
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ACCOUNTING MANAGERIAL • Nazarbayev University
Newton's Law of Universal Gravitation PHET.docx
SCIENCE SC5201 • Fallston High
Kami Export - Ilana Sondha-Bouih - Lab - Newtons Law Of Universal Gravitation - PhET VirtualLab (1).
PHYSICS 1221 • Clayton State University
Question Newtons law for universal gravitation can be written as F.docx
PHYS MASS • 21st Century International School Trust
Newtons Law of Universal Gravitation A. (4 points) Answer questions .docx
PHYS MISC • Northwest Missouri State University
Newtons law of universal gravitation - Copy.pdf
PHYS 101 • Cesar E. Chavez High School
PHYS 32IB • Faith Academy, Winnipeg
Lecture Notes on Newton's Law of Universal Gravitation
PHYS PHYS-103 • Virginia Commonwealth University
Newton's Law of Universal Gravitation - Intro Notes.pdf
AP 584600-5 • Hunter High
Newtons Law Of Universal Gravitation Virtual Lab Instructions.docx
AP 1 - Newton's Law of Universal Gravitation - Notes.pdf
AP PHYSIC 145 • Fossil Ridge High School
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(Not recommended) Create reorganization layer for YOLO v2 object detection network - MATLAB - MathWorks Switzerland
yolov2ReorgLayer
Create YOLO v2 Reorganization Layer
yolov2ReorgLayer function will be removed
(Not recommended) Create reorganization layer for YOLO v2 object detection network
YOLOv2ReorgLayer function is not recommended. Use spaceToDepthLayer instead.
The yolov2ReorgLayer function creates a YOLOv2ReorgLayer object, which represents the reorganization layer for you only look once version 2 (YOLO v2) object detection network. The reorganization layer reorganizes the high-resolution feature maps from a lower layer by stacking adjacent features into different channels. The output of reorganization layer is fed to the depth concatenation layer. The depth concatenation layer concatenates the reorganized high-resolution features with the low-resolution features from a higher layer.
layer = yolov2ReorgLayer(stride)
layer = yolov2ReorgLayer(stride,'Name',layerName)
layer = yolov2ReorgLayer(stride) creates the reorganization layer for YOLO v2 object detection network. The layer reorganizes the dimension of the input feature maps according to the step size specified in stride. For details on creating a YOLO v2 network with reorganization layer, see Design a YOLO v2 Detection Network with a Reorg Layer.
layer = yolov2ReorgLayer(stride,'Name',layerName) sets the Name property using a name-value pair. Enclose the property name in single quotes. For example, yolov2ReorgLayer('Name','yolo_Reorg') creates reorganization layer with the name 'yolo_Reorg'.
Step size for traversing the input vertically and horizontally, specified as a 2-element vector of positive integers in form [a b]. a is the vertical step size and b is the horizontal step size.
layerName — Name of reorganization layer
Name of reorganization layer, specified as a character vector or string scalar. This input argument sets the Name property of the layer. If you do not specify the name, then the function automatically sets Name to ''.
Layer name, specified as a character vector. To include a layer in a layer graph, you must specify a nonempty unique layer name. If you train a series network with the layer and Name is set to '', then the software automatically assigns a name to the layer at training time.
Specify the step size for reorganising the dimension of input feature map.
Create a YOLO v2 reorganization layer with the specified step size and the name as "yolo_Reorg".
layer = yolov2ReorgLayer(stride,'Name','yolo_Reorg');
Inspect the properties of the YOLO v2 reorganization layer.
YOLOv2ReorgLayer with properties:
Name: 'yolo_Reorg'
You can find the desired value of stride using:
stride\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{floor}\left(\frac{\text{size of input feature map to reorganization layer}}{\text{size of output feature map from higher layer}}\right)
The reorganization layer improves the performance of the YOLO v2 object detection network by facilitating feature concatenation from different layers. It reorganizes the dimension of a lower layer feature map so that it can be concatenated with the higher layer feature map.
Consider an input feature map of size [H W C], where:
H is the height of the feature map.
W is the width of the feature map.
C is the number of channels.
The reorganization layer chooses feature map values from locations based on the step sizes in stride and adds those feature values to the third dimension C. The size of the reorganized feature map from the reorganization layer is
[floor(H/stride(1)) floor(W/stride(2)) C×stride(1)×stride(2)].
For feature concatenation, the height and width of the reorganized feature map must match with the height and width of the higher layer feature map.
R2020b: yolov2ReorgLayer function will be removed
The YOLOv2ReorgLayer function will be removed in a future release. Use spaceToDepthLayer instead.
yolov2Layers | yolov2OutputLayer | yolov2TransformLayer | yolov2ObjectDetector | trainYOLOv2ObjectDetector
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If Marginal Rate of Substitution is constant throughout, the Indifference curve will be : (choose the correct alternative)
(a) Parallel to the x-axis.
(b) Downward sloping concave.
(c) Downward sloping convex.
(d) Downward sloping straight line.
Give equation of Budget Set. VIEW SOLUTION
When income of the consumer falls the impact on price-demand curve of an inferior good is : (choose the correct alternative)
(a) Shifts to the right.
(b) Shifts of the left.
(c) There is upward movement along the curve.
(d) There is downward movement along the curve.
The measure of price elasticity of demand of a normal good carries minus sign while price elasticity of supply carries plus sign. Explain why? VIEW SOLUTION
Good X (units) Good Y (units
What will be the impact of recently launched 'Clean India Mission' (Swachh Bharat Mission) on the Production Possibilities curve of the economy and why?
What will likely be the impact of large scale outflow of foreign capital on Production Possibilities curve of the economy and why? VIEW SOLUTION
Explain the effects of 'maximum price ceiling' on the market of a good'? Use diagram. VIEW SOLUTION
There are large number of sellers in a perfectly competitive market. Explain the significance of this feature. VIEW SOLUTION
Define cost. State the relation between marginal cost and average variable cost.
Define revenue. State the relation between marginal revenue and average revenue. VIEW SOLUTION
A consumer spends Rs. 60 on a good priced at Rs. 5 per unit. When price falls by 20 per cent, the consumer continues to spend Rs. 60 on the good. Calculate price elasticity of demand by percentage method. VIEW SOLUTION
Market for a good is in equilibrium. The demand for the good 'decreases'. Explain the chain of effects of this change. VIEW SOLUTION
Why is the equality between marginal cost and marginal revenue necessary for a firm to be in equilibrium? Is it sufficient to ensure equilibrium? Explain. VIEW SOLUTION
State the different phases of changes in Total Product and Marginal Product in the Law of Variable Proportions. Also show the same in a single diagram. VIEW SOLUTION
A consumer consumes only two goods X and Y both priced at Rs. 3 per unit. If the consumer chooses a combination of these two goods with Marginal Rate of Substitution equal to 3, is the consumer in equilibrium? Give reasons. What will a rational consumer do in this situation? Explain.
A consumer consumes only two goods X and Y whose prices are Rs. 4 and Rs. 5 per unit respectively. If the consumer chooses a combination of the two goods with marginal utility of X equal to 5 and that of Y equal to 4, is the consumer in equilibrium? Give reason. What will a rational consumer do in this situation? Use utility analysis. VIEW SOLUTION
Borrowing in government budget is : (choose the correct alternative)
What is 'aggregate supply' in macroeconomics? VIEW SOLUTION
The value of multiplier is : (choose the correct alternative)
\frac{1}{MPC}
\frac{1}{MPS}
\frac{1}{1-MPS}
\frac{1}{MPC-1}
Other things remaining unchanged, when in a country the price of foreign currency rises, national income is : (choose the correct alternative)
The non-tax revenue in the following is : (choose the correct alternative)
(d) Excise VIEW SOLUTION
Where will sale of machinery to abroad be recorded in the Balance of Payments Accounts? Give reasons. VIEW SOLUTION
If the Nominal GDP is Rs. 1200 and Price Index (with base = 100) is 120, calculate Real GDP. VIEW SOLUTION
Name the broad categories of transactions recorded in the 'capital account' of the Balance of Payments Accounts.
Name the broad categories of transactions recorded in the 'current account' of the Balance of Payments Accounts. VIEW SOLUTION
An economy is in equilibrium. Find 'autonomous consumption' from the following:
Investment expenditure = 1000
Explain the 'bank of issue' function of the central bank.
Explain 'Government's Bank' function of central bank. VIEW SOLUTION
Government of India has recently launched 'Jan-Dhan Yojna' aimed at every household in the country to have at least one bank account. Explain how deposits made under the plan are going to affect national income of the country. VIEW SOLUTION
Explain the role the government can play through the budget in influencing allocation of resources. VIEW SOLUTION
Giving reason explain how should the following be treated in estimation of national income:
(i) Expenditure by a firm on payment of fees to a chartered accountant
(ii) Payment of corporate tax by a firm
(iii) Purchase of refrigerator by a firm for own use
Explain the concept of Inflationary Gap. Explain the role of Repo Rate in reducing this gap.
Explain the concept of Deflationary Gap and the role of 'Open Market Operations' in reducing this gap. VIEW SOLUTION
Calculate 'Gross National Product at Market Price' and 'Net National Disposable Income':
(ii) Net current transfers to rest of the world 30
(iii) Social security contributions by employers 47
(iv) Mixed income 600
(vi) Royalty 20
(viii) Compensation of employees 500
(ix) Net domestic capital formation 120
(x) Net factor income from abroad (−) 10
(xi) Net indirect tax 150
(xii) Profit 200
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Comparing 538 and Economist forecasts in 2020 | Metaculus
Comparing 538 and Economist forecasts in 2020
There are many groups forecasting the 2020 Presidential Election using (primarily) polls based models. FiveThirtyEight run by Nate Silver and The Economist run by G Elliott Morris.
Both FiveThirtyEight and The Economist have published probabilities for each state in the 2020 Presidential Election.
Will 538 outperform The Economist forecasting the 2020 Presidential Election?
j
M_i
i
p_{ij}
j
i
o_{ij}
j
i
For example, if The Economist assigned 52% to Trump and 48% to Biden in Texas and if Trump won then The Economist would achieve a Brier Score of
This question resolves positively if the Brier score for the 51 races is lower for 538's probabilities than for The Economist's probabilities. We will download each model's "model outputs" from their respective websites at 1400 UTC 02-Nov-2020
To obtain The Economist's probabilities we will download the model outputs zip here, and use the values from the state_averages_and_predictions_topline.csv file
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Mechanical Properties of Rat Middle Cerebral Arteries With and Without Myogenic Tone | J. Biomech Eng. | ASME Digital Collection
Rebecca J. Coulson,
Rebecca J. Coulson
Mechanical Engineering Department, University of Vermont, Burlington, VT 05405
Marilyn J. Cipolla,
Neurology Department, University of Vermont, Burlington, VT 05405
Lisa Vitullo,
Lisa Vitullo
Biomedical Engineering Department, University of Wisconsin, Madison, WI 53706
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division June 20, 2002; revision received September 2, 2003. Associate Editor: J. Wayne.
Coulson, R. J., Cipolla , M. J., Vitullo , L., and Chesler, N. C. (March 9, 2004). "Mechanical Properties of Rat Middle Cerebral Arteries With and Without Myogenic Tone ." ASME. J Biomech Eng. February 2004; 126(1): 76–81. https://doi.org/10.1115/1.1645525
The inner diameter and wall thickness of rat middle cerebral arteries (MCAs) were measured in vitro in both a pressure-induced, myogenically-active state and a drug-induced, passive state to quantify active and passive mechanical behavior. Elasticity parameters from the literature (stiffness derived from an exponential pressure-diameter relationship, β, and elasticity in response to an increment in pressure,
Einc-p)
and a novel elasticity parameter in response to smooth muscle cell (SMC) activation,
Einc-a,
were calculated. β for all passive MCAs was 9.11±1.07 but could not be calculated for active vessels. The incremental stiffness increased significantly with pressure in passive vessels;
Einc-p 106 dynes/cm2
increased from 5.6±0.5 at 75 mmHg to 14.7±2.4 at 125 mmHg, (p<0.05). In active vessels,
Einc-p 106 dynes/cm2
remained relatively constant (5.5±2.4 at 75 mmHg and 6.2±1.0 at 125 mmHg).
Einc-a 106 dynes/cm2
increased significantly with pressure (from 15.1±2.3 at 75 mmHg to 49.4±12.6 at 125 mmHg, p<0.001), indicating a greater contribution of SMC activity to vessel wall stiffness at higher pressures.
cardiovascular system, blood vessels, haemodynamics, brain, muscle, biomechanics, elasticity, neurophysiology
Cerebral arteries, Pressure, Surface mount components, Vessels, Mechanical properties, Muscle, Stiffness
Hayashi, K., Stergiopulos, N., Meister, J.-J., Greenwald, S. E., and Rachev, A., 2001, “Techniques in the Determination of the Mechanical Properties and Constitutive Laws of Arterial Walls,” in Cardiovascular Techniques—Biomechanical Systems Techniques and Applications, vol. 2, C. Leondes, Ed., New York: CRC Press.
Biochemical and Mechanical Properties of Resistance Arteries From Normotensive and Hypertensive Rats
Myogenic Tone and Reactivity: Definitions Based on Muscle Physiology
J. Hypertens., Suppl.
Pre-Existing Level of Tone is an Important Determinant of Cerebral Artery Autoregulatory Responsiveness
Myogenic Properties of Cerebral Blood Vessels From Normotensive and Hypertensive Rats
Functional Aspects of Myogenic Vascular Control
discussion S31.
The Influence of Vascular Smooth Muscle Contraction on Elastic Properties of Pig’s Thoracic Aortae
Mechanical Properties of Human Cerebral Arteries
Incremental Elastic Modulus for Orthotropic Incompressible Arteries
Influence of Vascular Smooth Muscle on Contractile Mechanics and Elasticity of Arteries
Mechanics of Large and Small Cerebral Arteries in Chronic Hypertension
Threshold Duration of Ischemia for Myogenic Tone in Middle Cerebral Arteries: Effect on Vascular Smooth Muscle Actin
Protein Kinase C Modulates Basal Myogenic Tone in Resistance Arteries From the Cerebral Circulation
Wiederhielm
Continuous Recording of Arteriolar Dimensions With a Television Microscope
Fung, Y. C., 1993, Biomechanics: Mechanical Properties of Living Tissues, 2nd ed., New York: Springer-Verlag.
Timoshenko, S., 1934, Theory of Elasticity, First ed., New York: McGraw-Hill Book Company, Inc.
Effects of Elastase on the Stiffness and Elastic Properties of Arterial Walls in Cholesterol-Fed Rabbits
Stiffness and Elastic Behavior of Human Intracranial and Extracranial Arteries
Biomechanical Properties of Normal and Fibrosclerotic Human Cerebral Arteries
Middle Cerebral Artery Function After Stroke: The Threshold Duration of Reperfusion for Myogenic Activity
Reperfusion Decreases Myogenic Reactivity and Alters Middle Cerebral Artery Function After Focal Cerebral Ischemia in Rats
High Glucose Concentrations Dilate Cerebral Arteries and Diminish Myogenic Tone Through an Endothelial Mechanism
Estrogen Reduces Mouse Cerebral Artery Tone Through Endothelial NOS- and Cyclooxygenase-Dependent Mechanisms
Am. J. Phys-Heart-Circ. Phy.
Thorin-Trescases
High Levels of Myogenic Tone Antagonize the Dilator Response to Flow of Small Rabbit Cerebral Arteries
Differences in the Mechanical Properties of the Rat Carotid Artery in Vivo, in Situ and in Vitro
Erratum: “The Influence of Modeling Separate Neuromuscular Compartments on the Force and Moment Generating Capacities of Muscles of the Feline Hindlimb” [ J. Biomech. Eng., 2010, 132, p. 081003 ]
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{\displaystyle \lim _{x\to 0}{\frac {e^{3x^{2}}-1}{\sin(x^{2})}}}
Try L'Hopital's rule.
Failing L'Hopital's rule (or if you don't know it) one can also use Taylor polynomials.
{\displaystyle x=0}
in the limit gives an indeterminant form
{\displaystyle 0/0}
and so we may attempt to use L'Hopital's rule to get
{\displaystyle \lim _{x\to 0}{\frac {e^{3x^{2}}-1}{\sin(x^{2})}}=\lim _{x\to 0}{\frac {6xe^{3x^{2}}}{2x\cos(x^{2})}}=\lim _{x\to 0}{\frac {3e^{3x^{2}}}{\cos(x^{2})}}={\frac {3e^{0}}{1}}=3}
Using Taylor Polynomials we get
{\displaystyle {\begin{aligned}\lim _{x\to 0}{\frac {e^{3x^{2}}-1}{\sin(x^{2})}}&=\lim _{x\to 0}{\frac {(1+(3x^{2})^{1}/1+(3x^{2})^{2}/2+...)-1}{x^{2}-x^{6}/3+...}}\\&=\lim _{x\to 0}{\frac {3x^{2}+(3x^{2})^{2}/2+...}{x^{2}-x^{6}/3+...}}\\&=\lim _{x\to 0}{\frac {3+9x^{2}/2+...}{1-x^{4}/3+...}}\\&=3\end{aligned}}}
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag L'Hopital's rule, Pages using DynamicPageList parser function, Pages using DynamicPageList parser tag
MER Tag L'Hopital's rule
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Coordinate exchange - MATLAB cordexch - MathWorks América Latina
cordexch
Coordinate exchange
dCE = cordexch(nfactors,nruns)
[dCE,X] = cordexch(nfactors,nruns)
[dCE,X] = cordexch(nfactors,nruns,'model')
[dCE,X] = cordexch(...,'name',value)
dCE = cordexch(nfactors,nruns) uses a coordinate-exchange algorithm to generate a D-optimal design dCE with nruns runs (the rows of dCE) for a linear additive model with nfactors factors (the columns of dCE). The model includes a constant term.
[dCE,X] = cordexch(nfactors,nruns) also returns the associated design matrix X, whose columns are the model terms evaluated at each treatment (row) of dCE.
[dCE,X] = cordexch(nfactors,nruns,'model') uses the linear regression model specified in model. model is one of the following:
[dCE,X] = cordexch(...,'name',value) specifies one or more optional name/value pairs for the design. Valid parameters and their values are listed in the following table. Specify name inside single quotes.
excludefun
Handle to a function that excludes undesirable runs. If the function is f, it must support the syntax b = f(S), where S is a matrix of treatments with nfactors columns and b is a vector of Boolean values with the same number of rows as S. b(i) is true if the method should exclude ith row S.
Initial design as a nruns-by-nfactors matrix. The default is a randomly selected set of points.
Vector of number of levels for each factor. Not used when bounds is specified as a cell array.
Create the options structure with statset. Structure fields:
Streams — A RandStream object or cell array of such objects. If you do not specify Streams, cordexch uses the default stream or streams. If you choose to specify Streams, use a single object except in the case
Suppose you want a design to estimate the parameters in the following three-factor, seven-term interaction model:
y={\beta }_{0}+{\beta }_{1}x{}_{1}+{\beta }_{2}x{}_{2}+{\beta }_{3}x{}_{3}+{\beta }_{12}x{}_{1}x{}_{2}+{\beta }_{13}x{}_{1}x{}_{3}+{\beta }_{23}x{}_{2}x{}_{3}+\epsilon
Use cordexch to generate a D-optimal design with seven runs:
nfactors = 3;
[dCE,X] = cordexch(nfactors,nruns,'interaction','tries',10)
dCE =
1 -1 1 1 -1 -1 1
1 -1 -1 -1 1 1 1
1 -1 1 -1 -1 1 -1
1 1 -1 1 -1 1 -1
1 1 -1 -1 -1 -1 1
1 -1 -1 1 1 -1 -1
Columns of the design matrix X are the model terms evaluated at each row of the design dCE. The terms appear in order from left to right: constant term, linear terms (1, 2, 3), interaction terms (12, 13, 23). Use X to fit the model, as described in Linear Regression, to response data measured at the design points in dCE.
Both cordexch and rowexch use iterative search algorithms. They operate by incrementally changing an initial design matrix X to increase D = |XTX| at each step. In both algorithms, there is randomness built into the selection of the initial design and into the choice of the incremental changes. As a result, both algorithms may return locally, but not globally, D-optimal designs. Run each algorithm multiple times and select the best result for your final design. Both functions have a 'tries' parameter that automates this repetition and comparison.
Unlike the row-exchange algorithm used by rowexch, cordexch does not use a candidate set. (Or rather, the candidate set is the entire design space.) At each step, the coordinate-exchange algorithm exchanges a single element of X with a new element evaluated at a neighboring point in design space. The absence of a candidate set reduces demands on memory, but the smaller scale of the search means that the coordinate-exchange algorithm is more likely to become trapped in a local minimum.
rowexch | daugment | dcovary
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Section 52.7 (0A0H): The theorem on formal functions—The Stacks project
Section 52.7: The theorem on formal functions (cite)
52.7 The theorem on formal functions
We interrupt the flow of the exposition to talk a little bit about derived completion in the setting of quasi-coherent modules on schemes and to use this to give a somewhat different proof of the theorem on formal functions. We give some pointers to the literature in Remark 52.7.4.
Lemma 52.6.19 is a (very formal) derived version of the theorem on formal functions (Cohomology of Schemes, Theorem 30.20.5). To make this more explicit, suppose $f : X \to S$ is a morphism of schemes, $\mathcal{I} \subset \mathcal{O}_ S$ is a quasi-coherent sheaf of ideals of finite type, and $\mathcal{F}$ is a quasi-coherent sheaf on $X$. Then the lemma says that
\begin{equation} \label{algebraization-equation-formal-functions} Rf_*(\mathcal{F}^\wedge ) = (Rf_*\mathcal{F})^\wedge \end{equation}
where $\mathcal{F}^\wedge $ is the derived completion of $\mathcal{F}$ with respect to $f^{-1}\mathcal{I} \cdot \mathcal{O}_ X$ and the right hand side is the derived completion of $Rf_*\mathcal{F}$ with respect to $\mathcal{I}$. To see that this gives back the theorem on formal functions we have to do a bit of work.
Lemma 52.7.1. Let $X$ be a locally Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $K$ be a pseudo-coherent object of $D(\mathcal{O}_ X)$ with derived completion $K^\wedge $. Then
\[ H^ p(U, K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ p(U, K)/I^ nH^ p(U, K) = H^ p(U, K)^\wedge \]
for any affine open $U \subset X$ where $I = \mathcal{I}(U)$ and where on the right we have the derived completion with respect to $I$.
Proof. Write $U = \mathop{\mathrm{Spec}}(A)$. The ring $A$ is Noetherian and hence $I \subset A$ is finitely generated. Then we have
\[ R\Gamma (U, K^\wedge ) = R\Gamma (U, K)^\wedge \]
by Remark 52.6.21. Now $R\Gamma (U, K)$ is a pseudo-coherent complex of $A$-modules (Derived Categories of Schemes, Lemma 36.10.2). By More on Algebra, Lemma 15.94.4 we conclude that the $p$th cohomology module of $R\Gamma (U, K^\wedge )$ is equal to the $I$-adic completion of $H^ p(U, K)$. This proves the first equality. The second (less important) equality follows immediately from a second application of the lemma just used. $\square$
Lemma 52.7.2. Let $X$ be a locally Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $K$ be an object of $D(\mathcal{O}_ X)$. Then
the derived completion $K^\wedge $ is equal to $R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n)$.
Let $K$ is a pseudo-coherent object of $D(\mathcal{O}_ X)$. Then
the cohomology sheaf $H^ q(K^\wedge )$ is equal to $\mathop{\mathrm{lim}}\nolimits H^ q(K)/\mathcal{I}^ nH^ q(K)$.
Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module1. Then
the derived completion $\mathcal{F}^\wedge $ is equal to $\mathop{\mathrm{lim}}\nolimits \mathcal{F}/\mathcal{I}^ n\mathcal{F}$,
$\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n \mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n \mathcal{F}$,
$H^ p(U, \mathcal{F}^\wedge ) = 0$ for $p \not= 0$ for all affine opens $U \subset X$.
Proof. Proof of (1). There is a canonical map
\[ K \longrightarrow R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n), \]
see Remark 52.6.13. Derived completion commutes with passing to open subschemes (Remark 52.6.14). Formation of $R\mathop{\mathrm{lim}}\nolimits $ commutes with passsing to open subschemes. It follows that to check our map is an isomorphism, we may work locally. Thus we may assume $X = U = \mathop{\mathrm{Spec}}(A)$. Say $I = (f_1, \ldots , f_ r)$. Let $K_ n = K(A, f_1^ n, \ldots , f_ r^ n)$ be the Koszul complex. By More on Algebra, Lemma 15.94.1 we have seen that the pro-systems $\{ K_ n\} $ and $\{ A/I^ n\} $ of $D(A)$ are isomorphic. Using the equivalence $D(A) = D_{\mathit{QCoh}}(\mathcal{O}_ X)$ of Derived Categories of Schemes, Lemma 36.3.5 we see that the pro-systems $\{ K(\mathcal{O}_ X, f_1^ n, \ldots , f_ r^ n)\} $ and $\{ \mathcal{O}_ X/\mathcal{I}^ n\} $ are isomorphic in $D(\mathcal{O}_ X)$. This proves the second equality in
\[ K^\wedge = R\mathop{\mathrm{lim}}\nolimits \left( K \otimes _{\mathcal{O}_ X}^\mathbf {L} K(\mathcal{O}_ X, f_1^ n, \ldots , f_ r^ n) \right) = R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n) \]
The first equality is Lemma 52.6.9.
Assume $K$ is pseudo-coherent. For $U \subset X$ affine open we have $H^ q(U, K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ q(U, K)/\mathcal{I}^ n(U)H^ q(U, K)$ by Lemma 52.7.1. As this is true for every $U$ we see that $H^ q(K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ q(K)/\mathcal{I}^ nH^ q(K)$ as sheaves. This proves (2).
Part (3) is a special case of (2). Parts (4) and (5) follow from Derived Categories of Schemes, Lemma 36.3.2. $\square$
Lemma 52.7.3. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $X$ be a Noetherian scheme over $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume that $H^ p(X, \mathcal{F})$ is a finite $A$-module for all $p$. Then there are short exact sequences
\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/I^ n\mathcal{F}) \to H^ p(X, \mathcal{F})^\wedge \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]
of $A$-modules where $H^ p(X, \mathcal{F})^\wedge $ is the usual $I$-adic completion. If $f$ is proper, then the $R^1\mathop{\mathrm{lim}}\nolimits $ term is zero.
Proof. Consider the two spectral sequences of Lemma 52.6.20. The first degenerates by More on Algebra, Lemma 15.94.4. We obtain $H^ p(X, \mathcal{F})^\wedge $ in degree $p$. This is where we use the assumption that $H^ p(X, \mathcal{F})$ is a finite $A$-module. The second degenerates because
\[ \mathcal{F}^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F} \]
is a sheaf by Lemma 52.7.2. We obtain $H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F})$ in degree $p$. Since $R\Gamma (X, -)$ commutes with derived limits (Injectives, Lemma 19.13.6) we also get
\[ R\Gamma (X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) = R\Gamma (X, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, \mathcal{F}/I^ n\mathcal{F}) \]
By More on Algebra, Remark 15.87.6 we obtain exact sequences
\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/I^ n\mathcal{F}) \to H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]
of $A$-modules. Combining the above we get the first statement of the lemma. The vanishing of the $R^1\mathop{\mathrm{lim}}\nolimits $ term follows from Cohomology of Schemes, Lemma 30.20.4. $\square$
Remark 52.7.4. Here are some references to discussions of related material the literature. It seems that a “derived formal functions theorem” for proper maps goes back to [Theorem 6.3.1, lurie-thesis]. There is the discussion in [dag12], especially Chapter 4 which discusses the affine story, see More on Algebra, Section 15.91. In [Section 2.9, G-R] one finds a discussion of proper base change and derived completion using (ind) coherent modules. An analogue of (52.7.0.1) for complexes of quasi-coherent modules can be found as [Theorem 6.5, HL-P]
[1] For example $H^ q(K)$ for $K$ pseudo-coherent on our locally Noetherian $X$.
Comment #531 by Keenan Kidwell on April 05, 2014 at 19:53
In the statement of Lemma 0A0L, the
\mathcal{O}_X
\mathcal{O}_S
Comment #533 by Johan on April 05, 2014 at 22:59
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Decision feedback equalizer (DFE) with clock and data recovery (CDR) - MATLAB
serdes.DFECDR
DFE Properties
TapWeights
MinimumTap
MaximumTap
EqualizationGain
EqualizationStep
Taps2x
CDR Properties
Impulse Response Processing Using DFECDR
Sample-by-Sample Processing Using DFECDR
The serdes.DFECDR System object™ adaptively processes a sample-by-sample input signal or analytically processes an impulse response vector input signal to remove distortions at post-cursor taps.
To equalize the input signal:
Create the serdes.DFECDR object and set its properties.
dfecdr = serdes.DFECDR
dfecdr = serdes.DFECDR(Name,Value)
dfecdr = serdes.DFECDR returns a DFECDR object that modifies an input waveform with the DFE and determines the clock sampling times. The system object estimates the data symbol according to the Bang-Bang CDR algorithm.
dfecdr = serdes.DFECDR(Name,Value) sets properties using one or more name-value pairs. Enclose each property name in quotes. Unspecified properties have default values.
Example: dfecdr = serdes.DFECDR('Mode',1) returns a DFECDR object that applies specified DFE tap weights to input waveform.
DFE operating mode, specified as 0, 1, or 2. Mode determines what DFE tap weight values are applied to the input waveform.
DFE Mode
DFE Operation
0 off serdes.DFECDR is bypassed and the input waveform remains unchanged.
1 fixed serdes.DFECDR applies input DFE tap weights specified in TapWeights to the input waveform.
2 adapt The Init subsystem calls to the serdes.DFECDR. The serdes.DFECDR finds the optimum DFE tap values for the best eye height opening for statistical analysis. During time domain simulation, DFECDR uses the adapted values as the starting point and applies them to the input waveform. For more information about the Init subsystem, see Statistical Analysis in SerDes Systems
TapWeights — Initial DFE tap weights
Initial DFE tap weights, specified as a row vector in volts. The length of the vector specifies the number of taps. Each vector element value specifies the strength of the tap at that element position. Setting a vector element value to zero only initializes the tap.
MinimumTap — Minimum value of adapted taps
Minimum value of the adapted taps, specified as a real scalar or a real-valued row vector in volts. Specify as a scalar to apply to all the DFE taps or as a vector that has the same length as the TapWeights.
MaximumTap — Maximum value of adapted taps
Maximum value of the adapted taps, specified as a nonnegative real scalar or a nonnegative real-valued row vector in volts. Specify as a scalar to apply to all the DFE taps or as a vector that has the same length as the TapWeights.
EqualizationGain — Controls DFE tap weight update rate
Controls DFE tap weight update rate, specified as a unitless nonnegative real scalar. Increasing the value of EqualizationGain leads to a faster convergence of DFE adaptation at the expense of more noise in DFE tap values.
EqualizationStep — DFE adaptive step resolution
1e-6 (default) | nonnegative real scalar | nonnegative real-valued row vector
DFE adaptive step resolution, specified as a nonnegative real scalar or a nonnegative real-valued row vector in volts. Specify as a scalar to apply to all the DFE taps or as a vector that has the same length as the TapWeights.
EqualizationStep specifies the minimum DFE tap change from one time step to the next to mimic hardware limitations. Setting EqualizationStep to zero yields DFE tap values without any resolution limitation.
Taps2x — Multiply DFE tap weights by a factor of two
The output of the slicer in the serdes.DFECDR System object from the SerDes Toolbox™ is [-0.5 0.5]. But some industry applications require the slicer output to be [-1 1]. Taps2x allows you to quickly double the DFE tap weights to change the slicer reference.
16 (default) | real positive integer greater than 4
Early or late CDR count threshold to trigger a phase update, specified as a unitless real positive integer greater than 4. Increasing the value of Count provides a more stable output clock phase at the expense of convergence speed. Because the bit decisions are made at the clock phase output, a more stable clock phase has a better bit error rate (BER).
\text{Bandwidth}=\frac{1}{\text{Symbol time }·\text{ Early/late threshold count }·\text{ Step}}
ClockStep — Clock phase resolution
Clock phase resolution, specified as a real scalar in fraction of symbol time. ClockStep is the inverse of the number of phase adjustments in CDR.
Clock phase offset, specified as a real scalar in the range [−0.5, 0.5] in fraction of symbol time. PhaseOffset is used to manually shift the clock probability distribution function (PDF) for better BER.
Reference clock offset impairment, specified as a real scalar in the range [−300, 300] in parts per million (ppm). ReferenceOffset is the deviation between transmitter oscillator frequency and receiver oscillator frequency.
Sensitivity — Sampling latch metastability voltage
Sampling latch metastability voltage, specified as a real scalar in volts (V). If the data sample voltage lies within the region (±Sensitivity), there is a 50% probability of bit error.
Time of a single symbol duration, specified as a real scalar in seconds (s).
Uniform time step of the waveform, specified as a real scalar in seconds (s).
y = dfecdr(x)
Input baseband signal. If the WaveType is set to 'Sample', then the input signal is a sample-by-sample signal specified as a scalar. If the WaveType is set to 'Impulse', the input signal is an impulse response vector signal.
y — Estimated channel output
Estimated channel output. If the input signal is a sample-by-sample signal specified as a scalar, then the output is also scalar. If the input signal is an impulse response vector signal, the output is also a vector.
TapWeights — Estimated DFE tap weight values
Estimated DFE tap weight values, returned as a vector.
This example shows how to process impulse response of a channel using serdes.DFECDR system object™.
Use a symbol time of 100 ps. There are 16 samples per symbol. The channel has 14 dB loss.
NumberOfDFETaps = 2;
Create the DFECDR object. The object adaptively applies optimum DFE tap weights to input impulse response.
DFE1 = serdes.DFECDR('SymbolTime',SymbolTime,'SampleInterval',dt,...
'Mode',2,'WaveType','Impulse','TapWeights',zeros(NumberOfDFETaps,1));
Process the impulse response with DFE.
[impulseOut,TapWeights] = DFE1(impulseIn);
Convert the impulse response to a pulse, a waveform and an eye diagram for visualization.
Plot the resulting waveforms.
xlabel('SymbolTimes'),ylabel('Voltage')
This example shows how to process impulse response of a channel one sample at a time using serdes.DFECDR System object™.
Use a symbol time of 100 ps, with 8 samples per symbol. The channel loss is 14 dB. Select 12-th order pseudorandom binary sequence (PRBS), and simulate the first 20000 symbols.
Create the DFECDR System object. Process the channel one sample at a time by setting the input waveforms to 'sample' type. The object adaptively applies the optimum DFE tap weights to input waveform.
'Mode',2,'WaveType','Sample','TapWeights',zeros(NumberOfDFETaps,1),...
'EqualizationStep',0,'EqualizationGain',1e-3);
Initialize the PRBS generator.
[dataBit,prbsSeed]=prbs(prbsOrder,1);
Generate the sample-by-sample eye diagram.
%Loop through one symbol at a time.
dfeTapWeightHistory = nan(M,NumberOfDFETaps);
%Get new symbol
[dataBit,prbsSeed]=prbs(prbsOrder,1,prbsSeed);
%Convolve input waveform with channel
%Process one sample at a time through the DFE
[outWave((ii-1)*SamplesPerSymbol+jj),TapWeights] = DFE2(y(jj));
%Save DFE taps
dfeTapWeightHistory(ii,:) = TapWeights;
Plot the DFE adaptation history.
plot(dfeTapWeightHistory)
legend('TapWeights(1)','TapWeights(2)')
title('DFE Taps')
You can observe from the plot that the DFE adaptation is approximately complete after the first 10000 symbols, so these can be truncated from the array. Then plot the eye diagram by applying the reshape function to the array of symbols.
foldedEye = reshape(outWave(10000*SamplesPerSymbol+1:M*SamplesPerSymbol),SamplesPerSymbol,[]);
DFECDR | CTLE | CDR | serdes.CTLE | serdes.CDR
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Nufro went ahead and proved mathematically that if you swim to an unknown boat out in the bay who you find owns it could be vastly different from who would have owned it had you taken a canoe there instead.
(Mark Z. Danielewski, "All the Lights of Midnight: Salbatore Nufro Orejón, "The Physics of Ero
^r
" and Livia Bassil's Psychology of Physics, Conjunctions, Vol. 37, p. 80).
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In a certain small town, 65% of the residents subscribe to the city Sunday paper, 37% subscribe to the weekly local paper, and 25% subscribe to both papers.
What are the two variables? Create a two-way table.
2 by 2 Generic rectangle, left side titled Weekly Local Paper, top side titled City Sunday Paper, left edge labeled, yes on top, no on bottom, top edge labeled, yes on left, no on right, interior top left section labeled 25%, top row labeled on right, 37%, left column labeled below, 65%, right & below rectangle is label 100%.
Which three boxes in the two-way table represent someone subscribing to at least one paper? The same generic rectangle with the following new components entered in a different color from the rest. The bottom right edge is 35 percent. The right outer lower edge is 63 percent. For the interior, the upper right is 12 percent, the lower left is 40 percent and lower right is 23 percent.
25\%+12\%+40\%=77\%
This is the probability of someone subscribing to the Sunday paper given the information that they subscribe to at least one paper. Use your answer from part (b).
\frac{65}{77}=?
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MovingStatistic - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Data Smoothing : MovingStatistic
MovingStatistic(X, m, f, options)
additional parameters to be passed to the procedure f.
The MovingStatistic function computes moving statistics for a set of observations.
The second parameter m is the size of the moving window.
The third argument f is the statistic; can be any of the DescriptiveStatistics routines or a maple procedure which accepts a Vector and returns a floating point number.
Note that after f has been called on one subsample, the same Vector is reused for the next subsample, for efficiency reasons. All the builtin DescriptiveStatistics routines can handle this, but if you specify a custom maple procedure for f, you may need to copy its input Vector if you will need access to it after returning. See the example below for an explanation.
\mathrm{with}\left(\mathrm{Statistics}\right):
A≔〈\mathrm{seq}\left(\mathrm{sin}\left(i\right),i=1..20\right)〉:
U≔\mathrm{MovingStatistic}\left(A,5,\mathrm{Mean}\right)
\textcolor[rgb]{0,0,1}{U}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.0352323299444757}\\ \textcolor[rgb]{0,0,1}{-0.188944966656889}\\ \textcolor[rgb]{0,0,1}{-0.239407132278267}\\ \textcolor[rgb]{0,0,1}{-0.0697594845655643}\\ \textcolor[rgb]{0,0,1}{0.164024711544373}\\ \textcolor[rgb]{0,0,1}{0.247005344299126}\\ \textcolor[rgb]{0,0,1}{0.102890402628771}\\ \textcolor[rgb]{0,0,1}{-0.135821500715074}\\ \textcolor[rgb]{0,0,1}{-0.249659742674422}\\ \textcolor[rgb]{0,0,1}{-0.133961968583799}\\ \textcolor[rgb]{0,0,1}{⋮}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{16 element Vector[column]}}\end{array}
V≔\mathrm{MovingStatistic}\left(A,5,t↦\mathrm{FivePointSummary}\left(t,\mathrm{output}=\mathrm{maximum}\right)\right)
\textcolor[rgb]{0,0,1}{V}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0.909297426825682}\\ \textcolor[rgb]{0,0,1}{0.909297426825682}\\ \textcolor[rgb]{0,0,1}{0.656986598718789}\\ \textcolor[rgb]{0,0,1}{0.989358246623382}\\ \textcolor[rgb]{0,0,1}{0.989358246623382}\\ \textcolor[rgb]{0,0,1}{0.989358246623382}\\ \textcolor[rgb]{0,0,1}{0.989358246623382}\\ \textcolor[rgb]{0,0,1}{0.989358246623382}\\ \textcolor[rgb]{0,0,1}{0.420167036826641}\\ \textcolor[rgb]{0,0,1}{0.990607355694870}\\ \textcolor[rgb]{0,0,1}{⋮}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{16 element Vector[column]}}\end{array}
f := proc(A, q)
Statistics[Quantile](A, q);
W≔\mathrm{MovingStatistic}\left(A,5,f,0.3\right)
\textcolor[rgb]{0,0,1}{W}\textcolor[rgb]{0,0,1}{≔}\begin{array}{c}[\begin{array}{c}\textcolor[rgb]{0,0,1}{-0.770277280598276}\\ \textcolor[rgb]{0,0,1}{-0.770277280598276}\\ \textcolor[rgb]{0,0,1}{-0.770277280598276}\\ \textcolor[rgb]{0,0,1}{-0.770277280598276}\\ \textcolor[rgb]{0,0,1}{-0.324716083296540}\\ \textcolor[rgb]{0,0,1}{-0.297055872378289}\\ \textcolor[rgb]{0,0,1}{-0.574419050600125}\\ \textcolor[rgb]{0,0,1}{-0.574419050600125}\\ \textcolor[rgb]{0,0,1}{-0.574419050600125}\\ \textcolor[rgb]{0,0,1}{-0.574419050600125}\\ \textcolor[rgb]{0,0,1}{⋮}\end{array}]\\ \hfill \textcolor[rgb]{0,0,1}{\text{16 element Vector[column]}}\end{array}
\mathrm{LineChart}\left([A,U,V,W],\mathrm{color}=\mathrm{red}..\mathrm{blue},\mathrm{thickness}=3,\mathrm{legend}=["original","mean","maximum","quantile"]\right)
The following command will fail to apply the unassigned name
g
to the two correct sub-Vectors, because the same Vector is reused internally, as described above:
\mathrm{MovingStatistic}\left(〈1,2,3〉,2,g\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{Typesetting}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{_Hold}}\textcolor[rgb]{0,0,1}{}\left(\left[\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\left[\begin{array}{r}2\\ 3\end{array}\right]\right)\right]\right)\\ \textcolor[rgb]{0,0,1}{\mathrm{Typesetting}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{_Hold}}\textcolor[rgb]{0,0,1}{}\left(\left[\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\left[\begin{array}{r}2\\ 3\end{array}\right]\right)\right]\right)\end{array}]
This command, however, will make a copy for every sub-Vector and thus get the correct answer.
\mathrm{MovingStatistic}\left(〈1,2,3〉,2,v↦g\left(\mathrm{copy}\left(v\right)\right)\right)
[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{Typesetting}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{_Hold}}\textcolor[rgb]{0,0,1}{}\left(\left[\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\left[\begin{array}{r}1\\ 2\end{array}\right]\right)\right]\right)\\ \textcolor[rgb]{0,0,1}{\mathrm{Typesetting}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{_Hold}}\textcolor[rgb]{0,0,1}{}\left(\left[\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\left[\begin{array}{r}2\\ 3\end{array}\right]\right)\right]\right)\end{array}]
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Treasury Management - GTON Capital | GC
In this section we present an approach to treasury management.
A so-called "Beta 15" strategy that determines an optimal structure of DAO LP treasuries is proposed here. In a nutshell, this strategy means keeping 15% of market volatility risk for GTON and 85% in stablecoin LPs. Below the reasoning behind the strategy is explained in relation to the current market sentiment circa January 2022.
Finance math foundations
The regression formula calculates the return on asset valuation:
R_a= alpha+beta*R_m+zeta,
where alpha is the factor of the internal (independent) return on the asset and its economic characteristics and performance,
Rm is the portfolio return that represents the market,
beta is market sensitivity factor,
zeta is noise.
DAOs should be tasked with evaluating beta parameters and designing funds management strategies to optimize and control the beta factor for its governance token in accordance with the project's goals and long-term vision.
Considering so-called DeFi 2.0 DAOs who manage POL (protocol owned liquidity) on different DEXes and CEXes, we can say that the majority of such funds/liquidity positions are related to LP (MM liquidity) tokens with the governance token in the pair. This means that the GovToken performance depends on the performance of quote assets within LPs.
To limit the beta factor to a certain range, DAOs have to diversify their LP positions proportionally between stablecoins, project tokens and volatile assets. Some DAOs planned to have more than one token as part of their tokenomics: these sets of tokens are considered as alpha factors and should not be limited.
In an extreme case where Rm becomes negative (market correction), the implementation of beta-15 means that this correction will only have 15% negative influence on the DAO token performance.
Parameters like alpha, beta, and the noise are determined empirically which means that to be useful they must be measured on historical asset price performance. However, if there is no trading history yet or the project is changing its token economy model, we assume that it is safe to start with an initial LP diversification with 85% in stablecoin LPs and 15% in volatile asset LPs.
After some time, an empirical beta coefficient can be estimated and the LP treasury can be rebalanced to decrease or increase stablecoin LP allocation to bring future beta coefficient close to the target value (such as 15%).
Beta-15 example
The approach explained above has a target beta of 15 and it can be initiated as "85/15 = stablecoins/tokens" treasury diversification. The time period for this structure can be 4 weeks.
Why not beta-0
Beta-0 means that all GovToken liquidity is represented by stablecoin/GovToken LPs. This approach makes sense if there is no goal to use Governance token for trading utility. Therefore, no arbitrage opportunities or any other organic MM activity around governance tokens in that case will exist, or it will be severely limited. In addition, limiting beta by a certain X means that the managers of DAO liquidity will be keeping beta lower than X, meaning that for certain time periods beta can temporarily be close to 0.
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Susan had an incredible streak of good fortune as a guest on an exciting game show called “The Math Is Right.” She amassed winnings of
\$12,500
, a sports car, two round-trip airline tickets, and five pieces of furniture.
In an amazing finish, Susan then landed on a “Double Your Prize” square and answered the corresponding math question correctly. She instantly became the show’s biggest winner ever, earning twice the amounts of all her previous prizes.
A week later,
\$25,000
, a sports car, four round-trip airline tickets, and five pieces of furniture arrived at her house. Susan felt cheated. What was wrong?
Think of Susan's prizes as variables.
\$12,500=x
=y
=z
=v
If Susan's total prize was doubled, each one the smaller prizes (or variables) must be doubled.
2(x+y+z+v)=2x+2y+2z+2v
She should have received two sports cars and ten pieces of furniture.
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Miniaturized Cutting Tool With Triaxial Force Sensing Capabilities for Minimally Invasive Surgery | J. Med. Devices | ASME Digital Collection
Pietro Valdastri,
, Scuola Superiore Sant’Anna, 56100, Pisa, Italy
e-mail: pietro@sssup.it
Keith Houston,
Arne Sieber,
Profactor Research and Solutions GmbH
, A-2444, Seibersdorf, Austria
Masaru Yanagihara,
, 169–8555, Tokyo, Japan
J. Med. Devices. Sep 2007, 1(3): 206-211 (6 pages)
Valdastri, P., Houston, K., Menciassi, A., Dario, P., Sieber, A., Yanagihara, M., and Fujie, M. (August 8, 2007). "Miniaturized Cutting Tool With Triaxial Force Sensing Capabilities for Minimally Invasive Surgery." ASME. J. Med. Devices. September 2007; 1(3): 206–211. https://doi.org/10.1115/1.2778700
This paper reports a miniaturized triaxial force sensorized cutting tool for minimally invasive robotic surgery. This device exploits a silicon-based microelectromechanical system triaxial force sensor that acts as the core component of the system. The outer diameter of the proposed device is less than
3mm
, thus enabling the insertion through a 9 French catheter guide. Characterization tests are performed for both normal and tangential loadings. A linear transformation relating the sensor output to the external applied force is introduced in order to have a triaxial force output in real time. Normal force resolution is
8.2bits
over a force range between
0N
30N
, while tangential resolution is 7 bits over a range of
5N
. Force signals with frequencies up to
250Hz
can successfully be detected, enabling haptic feedback and tissue mechanical properties investigation. Preliminary ex vivo muscular tissue cutting experiments are introduced and discussed in order to evaluate the device overall performances.
biomechanics, force sensors, medical robotics, surgery
Calibration, Cutting, Cutting tools, Design, Force sensors, Manufacturing, Muscle, Robotics, Sensors, Signals, Silicon, Surgery, Biological tissues, Catheters, Feedback, Microelectromechanical systems, Haptics
Smart Surgical Tools and Augmenting Devices
Tenerez
A Production Process of Silicon Sensor Elements for a Fibre-Optic Pressure Sensor
Verimetra, Inc., Pittsburgh, PA, URL http://www.verimetra.comhttp://www.verimetra.com
Millar Instruments, Inc., Houston, TX, USA. URL http://www.millarinstruments. comhttp://www.millarinstruments. com
Compound-Cavity Tactile Sensor Using Surface-Emitting Laser for Endoscope/Catheter Tips
Proceedings of the Sixth International Symposium on Micro Machine and Human Science (MHS’95)
Micro Force Sensor for Intravascular Neurosurgery and In Vivo Experiment
Proceedings of the IEEE International Workshop on Microelectromechanical Systems (MEMS’98)
Results of R&D on Catheter-Type Micromachine
Proceedings of the 2001 International Symposium on Micromechatronics and Human Science (MHS’01)
Motion Compensation in Minimally Invasive Robotic Surgery
Review of Robotic Fixtures for Low-Invasiveness Surgery
Proceedings of the IEEE Tenth Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems
Eisinberg
A Teleoperated SDM-Based Microinstrument for Vessel Recognition During MIS
Proceedings of the IEEE Mechatronics and Robotics (MechRob’04)
Werthschtzky
Design of a Haptic Display for Catheterization
Proceedings of the IEEE First Joint Eurohaptics Conference and Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems (WHC’05)
Proceedings of the IEEE International Conference on Robotic and Automation (ICRA’05)
A Miniature Microsurgical Instrument Tip Force Sensor for Enhanced Force Feedback During Robot-Assisted Manipulation
Chanthasopeephan
Measuring Forces in Liver Cutting: New Equipment and Experimental Results
Deformation Resistance in Soft Tissue Cutting: A Parametric Study
Proceedings of the Computer Graphics and Applications (CGA’00)
Basdogan
In Vivo Data Acquisition Instrument for Solid Organ Mechanical Property Measurement
Proceedings of the MICCAI
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method - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 2-D Euclidean : Triangle Geometry : method
returns the method to define a triangle
The routine returns the method used to define the triangle T. It returns ``points'', ``sides'', or ``angle''. See the help page of triangle for the explanation of the output.
The command with(geometry,method) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{point}\left(A,0,0\right),\mathrm{point}\left(B,1,1\right),\mathrm{point}\left(C,1,0\right):
\mathrm{triangle}\left(\mathrm{T1},[A,B,C]\right):
\mathrm{method}\left(\mathrm{T1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{points}}
\mathrm{triangle}\left(\mathrm{T2},[3,3,3]\right):
\mathrm{method}\left(\mathrm{T2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sides}}
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Geostrophic current - Wikipedia
(Redirected from Geostrophic)
Oceanic flow in which the pressure gradient force is balanced by the Coriolis effect
A northern-hemisphere gyre in geostrophic balance. Paler water is less dense than dark water, but more dense than air; the outwards pressure gradient is balanced by the 90 degrees-right-of-flow coriolis force. The structure will eventually dissipate due to friction and mixing of water properties.
A geostrophic current is an oceanic current in which the pressure gradient force is balanced by the Coriolis effect. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the Southern Hemisphere. This concept is familiar from weather maps, whose isobars show the direction of geostrophic flow in the atmosphere. Geostrophic flow may be either barotropic or baroclinic. A geostrophic current may also be thought of as a rotating shallow water wave with a frequency of zero. The principle of geostrophy is useful to oceanographers because it allows them to infer ocean currents from measurements of the sea surface height (by combined satellite altimetry and gravimetry) or from vertical profiles of seawater density taken by ships or autonomous buoys. The major currents of the world's oceans, such as the Gulf Stream, the Kuroshio Current, the Agulhas Current, and the Antarctic Circumpolar Current, are all approximately in geostrophic balance and are examples of geostrophic currents.
2.1 Rotating waves of zero frequency
Sea water naturally tends to move from a region of high pressure (or high sea level) to a region of low pressure (or low sea level). The force pushing the water towards the low pressure region is called the pressure gradient force. In a geostrophic flow, instead of water moving from a region of high pressure (or high sea level) to a region of low pressure (or low sea level), it moves along the lines of equal pressure (isobars). This occurs because the Earth is rotating. The rotation of the earth results in a "force" being felt by the water moving from the high to the low, known as Coriolis force. The Coriolis force acts at right angles to the flow, and when it balances the pressure gradient force, the resulting flow is known as geostrophic.
As stated above, the direction of flow is with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the Southern Hemisphere. The direction of the flow depends on the hemisphere, because the direction of the Coriolis force is opposite in the different hemispheres.
See also: Geostrophic wind § Formulation
The geostrophic equations are a simplified form of the Navier–Stokes equations in a rotating reference frame. In particular, it is assumed that there is no acceleration (steady-state), that there is no viscosity, and that the pressure is hydrostatic. The resulting balance is (Gill, 1982):
{\displaystyle fv={\frac {1}{\rho }}{\frac {\partial p}{\partial x}}}
{\displaystyle fu=-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}}}
{\displaystyle f}
{\displaystyle \rho }
{\displaystyle p}
is the pressure and
{\displaystyle u,v}
are the velocities in the
{\displaystyle x,y}
-directions respectively.
One special property of the geostrophic equations, is that they satisfy the steady-state version of the continuity equation. That is:
{\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0}
Rotating waves of zero frequency[edit]
The equations governing a linear, rotating shallow water wave are:
{\displaystyle {\frac {\partial u}{\partial t}}-fv=-{\frac {1}{\rho }}{\frac {\partial p}{\partial x}}}
{\displaystyle {\frac {\partial v}{\partial t}}+fu=-{\frac {1}{\rho }}{\frac {\partial p}{\partial y}}}
The assumption of steady-state made above (no acceleration) is:
{\displaystyle {\frac {\partial u}{\partial t}}={\frac {\partial v}{\partial t}}=0}
Alternatively, we can assume a wave-like, periodic, dependence in time:
{\displaystyle u\propto v\propto e^{i\omega t}}
In this case, if we set
{\displaystyle \omega =0}
, we have reverted to the geostrophic equations above. Thus a geostrophic current can be thought of as a rotating shallow water wave with a frequency of zero.
Gill, Adrian E. (1982), Atmosphere-Ocean Dynamics, International Geophysics Series, vol. 30, Oxford: Academic Press, ISBN 0-12-283522-0
Retrieved from "https://en.wikipedia.org/w/index.php?title=Geostrophic_current&oldid=1030676037"
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CONCENTRATION RATIO (49584)
DIMENSIONLESS NUMBERS (37112)
Arak’s Inequalities for the Generalized Arithmetic Progressions
Zaitsev, A. Yu., E-mail: zaitsev@pdmi.ras.ru
[en] In 1980s, Arak has obtained powerful inequalities for the concentration functions of sums of independent random variables. Using these results, he has solved an old problem stated by Kolmogorov. In this paper, one of Arak’s results is modified to include generalized arithmetic progressions in the statement.
CONCENTRATION RATIO, FUNCTIONS, RANDOMNESS
Savannah River Site (SRS), Aiken, SC (United States). Funding organisation: USDOE Office of Environmental Management - EM (United States)
[en] A non-aqueous titration method has been used for quantifying the suppressor concentration in the MCU solvent hold tank (SHT) monthly samples since the Next Generation Solvent (NGS) was implemented in 2013. The titration method measures the concentration of the NGS suppressor (TiDG) as well as the residual tri-n-octylamine (TOA) that is a carryover from the previous solvent. As the TOA concentration has decreased over time, it has become difficult to resolve the TiDG equivalence point as the TOA equivalence point has moved closer. In recent samples, the TiDG equivalence point could not be resolved, and therefore, the TiDG concentration was determined by subtracting the TOA concentration as measured by semi-volatile organic analysis (SVOA) from the total base concentration as measured by titration. In order to improve the titration method so that the TiDG concentration can be measured directly, without the need for the SVOA data, a new method has been developed that involves spiking of the sample with additional TOA to further separate the two equivalence points in the titration. This method has been demonstrated on four recent SHT samples and comparison to results obtained using the SVOA TOA subtraction method shows good agreement. Therefore, it is recommended that the titration procedure be revised to include the TOA spike addition, and this to become the primary method for quantifying the TiDG.
23 Oct 2017; 18 p; OSTIID--1404909; AC09-08SR22470; Available from http://sti.srs.gov/fulltext/SRNL-STI-2017-00648.pdf; PURL: http://www.osti.gov/servlets/purl/1404909/
CONCENTRATION RATIO, SOLVENTS, TANKS, TITRATION
CHEMICAL ANALYSIS, CONTAINERS, DIMENSIONLESS NUMBERS, QUANTITATIVE CHEMICAL ANALYSIS, VOLUMETRIC ANALYSIS
http://sti.srs.gov/fulltext/SRNL-STI-2017-00648.pdf, http://www.osti.gov/servlets/purl/1404909/
Hanford Site Composite Analysis Technical Approach Description: Groundwater
Budge, T. J.
[en] The groundwater facet of the revised CA is responsible for generating predicted contaminant concentration values over the entire analysis spatial and temporal domain. These estimates will be used as part of the groundwater pathway dose calculation facet to estimate dose for exposure scenarios. Based on the analysis of existing models and available information, the P2R Model was selected as the numerical simulator to provide these estimates over the 10,000-year temporal domain of the CA. The P2R Model will use inputs from initial plume distributions, updated for a start date of 1/1/2017, and inputs from the vadose zone facet, created by a tool under development as part of the ICF, to produce estimates of hydraulic head, transmissivity, and contaminant concentration over time. A recommendation of acquiring 12 computer processors and 2 TB of hard drive space is made to ensure that the work can be completed within the anticipated schedule of the revised CA.
2 Oct 2017; 32 p; OSTIID--1412549; AC06-08RL14788
CP--60406
CONCENTRATION RATIO, DOSES, GROUND WATER
DIMENSIONLESS NUMBERS, HYDROGEN COMPOUNDS, OXYGEN COMPOUNDS, WATER
A mathematical analysis of deviations from linearity of Beer's law
Tolbin, Alexander Yu.; Pushkarev, Victor E.; Tomilova, Larisa G., E-mail: tolbin@ipac.ac.ru
[en] Highlights: • Accurate calculation of deviation from Beer's law is based on convergence criteria. • The higher the extinction, the smaller the area of obedience to Beer's law. • The accuracy of calculating the molar attenuation is set manually. • Systematic errors of spectroscopic measurements now can be excluded. In this paper, we propose new approach to accurate calculation of the molar attenuation coefficient ε and the threshold concentration
{C}_{\mathrm{\nabla }}
, exceeding of which it results in a deviation from Beer's law. The method is based on the asymptotic approximation of the equality of integrals associated with the true values of
{C}_{\mathrm{\nabla }}
\epsilon
as well as their approximate values
{C}_{i}
{\epsilon }_{i}
within max. rapprochement:
\mathrm{\Delta }C=\mathrm{min}\left\{{C}_{i}-{C}_{\mathrm{\nabla }}\right\}
\mathrm{\Delta }\epsilon =\mathrm{min}\left\{\epsilon -{\epsilon }_{i}\right\}
{C}_{i}\to {C}_{\mathrm{\nabla }}
{\epsilon }_{i}\to \epsilon
. After reaching the acceptable accuracy, the infinite iterative cycle is interrupted that allows to significantly saving the computing resources.
S0009261418305360; Available from http://dx.doi.org/10.1016/j.cplett.2018.06.056; Copyright (c) 2018 Elsevier B.V. All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
ACCURACY, CONCENTRATION RATIO, ERRORS, PHTHALOCYANINES
DIMENSIONLESS NUMBERS, DYES, HETEROCYCLIC COMPOUNDS, ORGANIC COMPOUNDS
232U CONTENT OF SAPPHIRE MATERIAL
KANE, W.R.; LEMLEY, J.R.; VANIER, P.E.; FORMAN, L.
Brookhaven National Lab., Upton, NY (United States). Funding organisation: DOE/NN-44 (United States)
18 Jul 2000; 4 p; 41. annual meeting of the Institute of Nuclear Materials Management; New Orleans, LA (United States); 16-20 Jul 2000; GJ--1200; AC02-98CH10886; Also available from OSTI as DE00767151; PURL: https://www.osti.gov/servlets/purl/767151-lceEuc/native/
CONCENTRATION RATIO, SAPPHIRE, URANIUM 232
ACTINIDE NUCLEI, ALPHA DECAY RADIOISOTOPES, CORUNDUM, EVEN-EVEN NUCLEI, HEAVY ION DECAY RADIOISOTOPES, HEAVY NUCLEI, ISOTOPES, MINERALS, NEON 24 DECAY RADIOISOTOPES, NUCLEI, OXIDE MINERALS, RADIOISOTOPES, SPONTANEOUS FISSION RADIOISOTOPES, URANIUM ISOTOPES, YEARS LIVING RADIOISOTOPES
https://www.osti.gov/servlets/purl/767151-lceEuc/native/
Visual test for the presence of the illegal additive ethyl anthranilate by using a photonic crystal test strip
Zhang, Yi; Jin, Zhenkai; Zeng, Qingsong; Huang, Yanmei; Gu, Hang; He, Jiahua; Liu, Yangyang; Chen, Shili; Sun, Hui; Lai, Jiaping, E-mail: esesunhui@gzhu.edu.cn, E-mail: laijp@scnu.edu.cn
[en] A test strip has been developed for the rapid detection of the illegal additive ethyl anthranilate (EA) in wine. The detection scheme is based on a combination of photonic crystal based detection and molecular imprinting based recognition. The resulting molecularly imprinted photonic crystal (MIPC) undergoes a gradual color change from green to yellow to red upon binding of EA. A semi-quantitative colorimetric card can be used to estimate the content of EA, either visually or by making use of an optical fiber spectrometer. A linear relationship was found between the Bragg diffraction peak shift and the concentration of EA in the range from 0.1 mM to 10 mM. The detection limit is 10 μM. The test has been successfully used to screening for the presence of EA in grape wine. The test strip is selective, and can be re-used after re-activation.
Copyright (c) 2019 Springer-Verlag GmbH Austria, part of Springer Nature
Mikrochimica Acta; ISSN 0026-3672; ; CODEN MIACAQ; v. 186(11); p. 1-10
BEVERAGES, CONCENTRATION RATIO, NANOCHEMISTRY, SPECTROMETERS
CHEMISTRY, DIMENSIONLESS NUMBERS, FOOD, MEASURING INSTRUMENTS
Molluscicidal Activity of Entada Rheedii Stem Bark Methanolic Extract against Paddy Pest Pomacea Canaliculata (Golden Apple Snail)
Nur Suraya Abdullah; Noorshilawati Abdul Aziz; Rosminah Mailon, E-mail: nsa@pahang.uitm.edu.my
[en] The study was conducted to evaluate the molluscicidal activity of E. rheedii methanol bark extract against P. canaliculata and to screen for phytochemical compounds of E. rheedii bark extracts. The golden apple snails with size range of 20 - 40 mm were treated with four different concentrations of E. rheedii (1000, 5000, 10000 and 20000 ppm) and paddy-field water mix with 50 % methanol serving as the control treatment. The molluscicidal effects of the extract were evaluated after 24, 48 and 72 hours. The results of the study showed that high treatment concentrations (10000 and 20000 ppm) recorded the highest mortality rate (100 %) while low concentrations (1000 ppm) showed the lowest mortality rate (27 %). However, no mortality was recorded in the control treatment. The molluscicides activity with LC50 is 1,611 ppm and LC90 is 4,266 ppm and could be attributed to the presence of saponin in the bark extracts. E. rheedii bark extract provides a great potential for developing green pesticides to control P. canaliculata. Nevertheless, further research is needed to determine its biochemical mechanism. (author)
Malaysian Journal of Analytical Sciences; ISSN 1394-2506; ; v. 21(1); p. 46-51
CONCENTRATION RATIO, METHANOL, MOLLUSCS, RICE
ALCOHOLS, ANIMALS, AQUATIC ORGANISMS, CEREALS, DIMENSIONLESS NUMBERS, GRAMINEAE, HYDROXY COMPOUNDS, INVERTEBRATES, LILIOPSIDA, MAGNOLIOPHYTA, ORGANIC COMPOUNDS, PLANTS
The Effect of Temperature and Concentration of Foaming Agent to the β-Carotene Content in Product Derived from Carrots
Fardiyah, Qonitah; Rumhayati, Barlah; Khotimah, Yuniesti Husnul, E-mail: fardiyah@ub.ac.id
[en] Carrot (Daucus carota L) is vegetable that contain body essetial vitamins, especially β-carotene. In this research, the essense of fresh carrots are taken and processed to carrot powder using foam mat drying method. This research aims to study the effect of temperature and concentration of foaming agent to the β-carotene content in product derived from carrots. The temperature variation that used in this research are 40°C, 50°C, 60°C and 70°C, while he variation of foaming agent (tween 80) are 0,01% (v/v); 0,1%(v/v); 0,2%(v/v) and 0,3%(v/v). The results shows that the maximum drying temperature is 50°C with β-carotene content 10.55 mg/kg and the maximum concentration of foaming agent (tween 80) is 0.2% (v/v) with β-carotene content 10.36 mg/kg. (paper)
IC2MS 2017: International Conference on Chemistry and Material Science; Malang (Indonesia); 4-5 Nov 2017; Available from http://dx.doi.org/10.1088/1757-899X/299/1/012008; Country of input: International Atomic Energy Agency (IAEA)
CONCENTRATION RATIO, DRYING, FOAMS, POWDERS
COLLOIDS, DIMENSIONLESS NUMBERS, DISPERSIONS
An electric concentrator and thermal cloaking device
Raza, Muhammad, E-mail: mreza06@gmail.com
[en] The concentration and cloaking phenomena of physical fields in Metamaterials has captured the attention of the researchers due to their simplified approaches. However most of the work conducted is focussed on controlling single physical field. Transformation optics has paved the way for developing intelligent bifunctional devices. Bifunctional devices are such controlled devices which execute two different physical functions simultaneously and independently. In this work we have applied the transformation optics theory to design a multilayered two dimensional spherical bifunctional device which behaves like an electric concentrator and thermal invisibility cloak simultaneously. Moreover, we have also observed the normalized behavior of the proposed device. The simulation performance confirms the feasibility of our suggested model. (paper)
Available from http://dx.doi.org/10.1088/2053-1591/ab8fba; Country of input: International Atomic Energy Agency (IAEA)
Materials Research Express (Online); ISSN 2053-1591; ; v. 7(5); [6 p.]
CONCENTRATION RATIO, EQUIPMENT, OPTICS, SIMULATION
http://dx.doi.org/10.1088/2053-1591/ab8fba
Determination celiprolol hydrochloride drug by used zero, first, second and third order derivative and peak area spectrophotometry method in its pure form and in pharmaceutical tablets
Halboos, Mohanad H; Ammar Sayhood, Aayad; Ala’a Hussein, Tamara, E-mail: muhaned.halbus@uokufa.edu.iq
[en] An easy, specified, accurate, precise and reproducible quantitative analyses for determination of celiprolol hydrochloride drug by used used zero, first, second and third order derivative and peak area spectrophotometry method. The suggest methods determined the drug in the concentration range (0.5-30) μg.mL−1, at 286.6 nm for 0th order, at 306.6 and 272.2 nm for 1st order, at 319.2, 289.8 and 250.2 nm for 2nd order and at 325.6, 304.8, 242.2 and 219.6 nm for 3rd order derivative spectrophotometry, respectively. The peak area spectrophotometry method also used in the same range for determining celiprolol hydrochloride, at (284.4-379.2) and (248.6-284.4) nm for 1st order, at (306.4-372.2), (271.2-306.4) and (239.4-271.2) nm for 2nd order, and at (318.6-363.8), (290.4-318.6), (233.2-250.4) and (210.8-233.2) nm for 3rd order, respectively. The accuracy and precision of the methods used was calculated and the results were highly satisfactory. The limit of detection (LOD) and limit of quantification (LOQ) was calculated for the suggested methods, Where (LOD) was within range (0.0124-0.0632) μg.mL−1, and (LOQ) within range (0.0415-0.1632) μg.mL−1. The methods were successful in application when estimating celiprolol hydrochloride drug on some pharmaceutical tablets available in the local markets. (paper)
2. International Science Conference; Al-Qadisiyah (Iraq); 24-25 Apr 2019; Available from http://dx.doi.org/10.1088/1742-6596/1294/5/052035; Country of input: International Atomic Energy Agency (IAEA)
Journal of Physics. Conference Series (Online); ISSN 1742-6596; ; v. 1294(5); [11 p.]
CONCENTRATION RATIO, DETECTION, DRUGS, SPECTROPHOTOMETRY
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Monitoring the Earthquake Cycle in the Northern Andes from the Ecuadorian cGPS Network | Seismological Research Letters | GeoScienceWorld
Patricia A. Mothes;
Instituto Geofísico, Escuela Politécnica Nacional, Casilla 1701‐2759, Quito, Ecuador, pmothes@igepn.edu.ec
Frederique Rolandone;
Sorbonne Universités, UPMC Université Paris 06, CNRS, Institut des Sciences de la Terre de Paris (ISTeP), F75005 Paris, France
Also at Instituto Geofísico, Escuela Politécnica Nacional, Casilla 1701‐2759, Quito, Ecuador.
Jean‐Mathieu Nocquet;
Jean‐Mathieu Nocquet
IRD, Université Côte d’Azur, CNRS, Observatoire de la Côte d’Azur, Geoazur, F06560 Valbonne, France
Paul A. Jarrin;
Paul A. Jarrin
Alexandra P. Alvarado;
Alexandra P. Alvarado
Mario C. Ruiz;
David Cisneros;
Instituto Geográfico Militar, Av. Seniergues E4‐676 y Gral. Telmo Paz y Mino, Quito, Ecuador
Héctor Mora Páez;
Héctor Mora Páez
Servicio Geológico de Colombia, GNSS GeoRed Project, Diag. 53 N° 34‐53, Bogotá, Colombia
Patricia A. Mothes, Frederique Rolandone, Jean‐Mathieu Nocquet, Paul A. Jarrin, Alexandra P. Alvarado, Mario C. Ruiz, David Cisneros, Héctor Mora Páez, Mónica Segovia; Monitoring the Earthquake Cycle in the Northern Andes from the Ecuadorian cGPS Network. Seismological Research Letters 2018;; 89 (2A): 534–541. doi: https://doi.org/10.1785/0220170243
The continuous Global Positioning System (cGPS) network operating in the northern Andes (Ecuador and Colombia) for about a decade has the main objectives of quantifying interseismic coupling along the subduction interface, detecting occurrence of transient aseismic episodic slip, detailing the rupture kinematics of large earthquakes, recording long‐term movements along crustal faults, as well as recording swelling or deflation on the flanks of volcanoes. An opportunity to test the network’s timely registry of surface coseismic offsets was provided by the 16 April 2016
Mw
7.8 Pedernales, Ecuador, earthquake whose epicenter was along the western margin of central Ecuador, South America. This large earthquake was the biggest to occur in the northern Andes since 1979 and produced static surface offsets that were recorded by the cGPS stations operating at distances out to
∼400 km
from source. Near‐field stations, operating along the Ecuadorian littoral recorded static horizontal surface displacements up to 80 cm and high‐rate GPS (HRGPS) stations recorded dynamic peak‐to‐peak displacements reaching 2 m. These measurements, together with seismic data, revealed the southward propagation of the seismic rupture, its spatial extent, and the successive breaking of two main asperities (Nocquet et al., 2017). Here, we provide the complete data set of static coseismic displacements recorded from Ecuador to southern Colombia out to 400 km from the rupture. North of the Pedernales earthquake’s foci, in the adjoining Esmeraldas‐Nariño segment, some patches show high interseismic coupling and rapid strain accumulation is ongoing. In the 200‐km‐long Esmeraldas‐Nariño segment, the seismic potential is particularly high. cGPS data suggest that the Esmeraldas‐Nariño segment is likely a zone of future rupture.
Pedernales earthquake 2016
Mw
8.3 Illapel, Chile, Earthquake: Direction‐Reversed Along‐Dip Rupture with Localized Water Reverberation
Comparison of Observed Ground‐Motion Attenuation for the 16 April 2016 Mw 7.8 Ecuador Megathrust Earthquake and Its Two Largest Aftershocks with Existing Ground‐Motion Prediction Equations
Continuous Borehole Strain and Pore Pressure in the Near Field of the 28 September 2004 M 6.0 Parkfield, California, Earthquake: Implications for Nucleation, Fault Response, Earthquake Prediction, and Tremor
Mw
Integrated Analysis of the 2020
Mw
7.4 La Crucecita, Oaxaca, Mexico, Earthquake from Joint Inversion of Geodetic and Seismic Observations
Tectonic implications of the seismic ruptures associated with the 1983 and 1991 Costa Rica earthquakes
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DoubleTohfData - Maple Help
Home : Support : Online Help : Programming : Calling External Routines : ExternalCalling : C Application Programming Interface : DoubleTohfData
initialize hfdata structures for use in external code
MapleTohfData(kv, a, hf)
DoubleTohfData(kv, re, im, hf)
ComplexTohfData(kv, re, im, hf)
Maple PROC or RTABLE object
the value to use as the real component
the value to use as the imaginary component
a pointer to the hfdata structure to initialize
MapleTohfData encodes a Maple PROC or an RTABLE of type float[8] into an hfdata structure. These structures are used to pass arguments into and get results out of EvalhfDataProc.
DoubleTohfData and ComplexTohfData encode a complex number represented as two doubles into an hfdata structure. They differ in that if the im argument to DoubleTohfData is 0.0 or -0.0 it will be ignored, so the resulting hf structure will be treated as representing a real number, not a real number with zero imaginary part. These structures are used to pass arguments into and get results out of EvalhfDataProc.
A Maple object encoded in a hfdata structure can be extracted by calling ToMaplehfData. The real and imaginary parts of a hfdata can be extracted by calling RealhfData and ImaginaryhfData.
All hfdata objects returned by EvalhfDataProc represent real floating point values.
\mathrm{with}\left(\mathrm{ExternalCalling}\right):
\mathrm{dll}≔\mathrm{ExternalLibraryName}\left("HelpExamples"\right):
\mathrm{newton}≔\mathrm{DefineExternal}\left("MyNewtonData",\mathrm{dll}\right):
f≔{x}^{4}-5{x}^{2}+6x-2:
\mathrm{newton}\left(f,0,0.001\right)
\textcolor[rgb]{0,0,1}{0.731892751250226237}
\mathrm{eval}\left(f,x=\right)
\textcolor[rgb]{0,0,1}{-0.000039355}
\mathrm{newton}\left(f,\mathrm{sqrt}\left(2\right),0.00001\right)
\textcolor[rgb]{0,0,1}{1.00195003210012135}
\mathrm{eval}\left(f,x=\right)
\textcolor[rgb]{0,0,1}{3.833}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{-6}}
\mathrm{newton}\left(f,-\mathrm{\pi },1.×{10}^{-10}\right)
\textcolor[rgb]{0,0,1}{-2.73205080756887719}
\mathrm{Digits}≔15:
\mathrm{eval}\left(f,x=\right)
\textcolor[rgb]{0,0,1}{1.5}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{-13}}
f≔\mathrm{unapply}\left(f,x\right):
\mathrm{newton}\left(f,-\mathrm{\pi },1.×{10}^{-10}\right)
\textcolor[rgb]{0,0,1}{-2.73205080756887719}
\mathrm{evalhf}\left(f\left(\right)\right)
\textcolor[rgb]{0,0,1}{-7.10542735760100186}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{-15}}
OpenMaple/C/ImaginaryhfData
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Hydraulic reservoir where pressurization and fluid level remain constant regardless of volume change - MATLAB
Hydraulic reservoir where pressurization and fluid level remain constant regardless of volume change
The Constant Head Tank block represents a pressurized hydraulic reservoir, in which fluid is stored under a specified pressure. The size of the tank is assumed to be large enough to neglect the pressurization and fluid level change due to fluid volume. The block accounts for the fluid level elevation with respect to the tank bottom, as well as for pressure loss in the connecting pipe that can be caused by a filter, fittings, or some other local resistance. The loss is specified with the pressure loss coefficient. The block computes the volume of fluid in the tank and exports it outside through the physical signal port V.
The fluid volume value does not affect the results of simulation. It is introduced merely for information purposes. It is possible for the fluid volume to become negative during simulation, which signals that the fluid volume is not enough for the proper operation of the system. By viewing the results of the simulation, you can determine the extent of the fluid shortage.
For reasons of computational robustness, the pressure loss in the connecting pipe is computed with the equations similar to that used in the Fixed Orifice block:
q=\sqrt{\frac{1}{K}}\cdot {A}_{p}\sqrt{\frac{2}{\rho }}\cdot \frac{{p}_{loss}}{{\left({p}_{loss}^{2}+{p}_{cr}^{2}\right)}^{1/4}}
{p}_{cr}=K\frac{\rho }{2}{\left(\frac{{\mathrm{Re}}_{cr}\cdot \nu }{d}\right)}^{2}
The Critical Reynolds number is set to 15.
The pressure at the tank inlet is computed with the following equations:
p={p}_{elev}-{p}_{loss}+{p}_{pr}
{p}_{elev}=\rho ·g·H
{A}_{p}=\frac{\pi ·{d}^{2}}{4}
p Pressure at the tank inlet
pelev Pressure due to fluid level
ploss Pressure loss in the connecting pipe
ppr Pressurization
H Fluid level with respect to the bottom of the tank
K Pressure loss coefficient
Ap Connecting pipe area
d Connecting pipe diameter
Connection T is a hydraulic conserving port associated with the tank inlet. Connection V is a physical signal port. The flow rate is considered positive if it flows into the tank.
Gage pressure acting on the surface of the fluid in the tank. It can be created by a gas cushion, membrane, bladder, or piston, as in bootstrap reservoirs. This parameter must be greater than or equal to zero. The default value is 0, which corresponds to a tank connected to atmosphere.
The fluid level with respect to the tank bottom. This parameter must be greater than zero. The default value is 1 m.
Initial fluid volume
The initial volume of fluid in the tank. This parameter must be greater than zero. The default value is 0.2 m^3.
The diameter of the connecting pipe. This parameter must be greater than zero. The default value is 0.02 m.
The value of the pressure loss coefficient, to account for pressure loss in the connecting pipe. This parameter must be greater than zero. The default value is 1.2.
Hydraulic conserving port associated with the tank inlet.
Physical signal port that outputs the volume of fluid in the tank.
Reservoir | Tank
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Heat Transfer of an IGBT Module Integrated With a Vapor Chamber | J. Electron. Packag. | ASME Digital Collection
Xiaoling Yu,
e-mail: xlingyu@yahoo.com.cn
Lianghua Zhang,
Lianghua Zhang
Enming Zhou,
Enming Zhou
Yu, X., Zhang, L., Zhou, E., and Feng, Q. (March 10, 2011). "Heat Transfer of an IGBT Module Integrated With a Vapor Chamber." ASME. J. Electron. Packag. March 2011; 133(1): 011008. https://doi.org/10.1115/1.4003214
Presently, many methods are adopted to reduce the junction-to-case thermal resistance
(Rjc)
of insulated-gate bipolar transistor (IGBT) modules in order to increase their power density. One of these approaches is to enhance the heat spreading capability of the base plate (heat spreader) of an IGBT module using a vapor chamber (VC). In this paper, both experimental measurement and thermal modeling are conducted on a VC-based IGBT module and two copper-plate-based IGBT modules. The experimental data show that
Rjc
of the VC-based IGBT module decreases substantially with the increase in the heat load of the IGBT.
Rjc
of the VC-based IGBT module is
∼50%
of that of the 3 mm copper-plate-based IGBT module after it saturates at a heat load level of
∼200 W
. The transient time of the VC-based IGBT module is also shorter than the copper-plate-based IGBT modules since the VC has higher heat spreading capability. The quicker responses of the VC-based IGBT module to reach its saturated temperature during the start-up can avoid a possible power surge. In the thermal modeling, the vapor is substituted as a solid conductor with extremely high thermal conductivity. Hence, the two-phase flow thermal modeling of the VC is simplified as a one-phase thermal conductive modeling. A thermal circuit model is also built for the VC-based IGBT module. Both the thermal modeling and thermal circuit results match well with the experimental data.
heat conduction, heat transfer, insulated gate bipolar transistors, thermal resistance, IGBT module, vapor chamber (VC), junction-to-case thermal resistance, transient temperature
Circuits, Copper, Heat, Heat transfer, Junctions, Stress, Temperature, Thermal conductivity, Thermal resistance, Transients (Dynamics), Vapors, Simulation, Flat heat pipes, Gates (Closures), Transistors, Heat sinks, Heat conduction
Thermal Stress and Heat Transfer Characteristics of a Cu/Diamond/Cu Heat Spreading Device
Fabrication and Thermal Performance of a Thin Flat Heat Pipe With Innovative Sintered Copper Wick Structure
41st IEEE Conference on Industry Applications Conference (IAS), Conference Record of the 2006 IEEE
, Tampa, FL, Oct. 8–12, Vol.
Utz-Kistner
DBC (Direct Bond Copper) Substrate With Integrated Flat Heat Pipe
22nd IEEE Conference on Semiconductor Thermal Measurement and Management
, Dallas, TX, March 14–16, pp.
Spreading in the Heat Sink Base: Phase Change Systems or Solid Metals?
On the Use of Flat Heat Pipes as Thermal Spreaders in Power Electronics Cooling
IEEE 33rd Annual Power Electronics Specialists Conference
, Cairns, Queensland, Australia, June 23–27, Vol.
Multi-Artery Heat Pipe Spreader
Analytical Solution of Thermal Resistance of Vapor Chamber Heat Sink With and Without Pillar
Numerical Analysis and Experimental Verification on Thermal Fluid Phenomena in a Vapor Chamber
A Simplified Transient Three-Dimensional Model for Estimating the Thermal Performance of the Vapor Chambers
Heat Transfer in Integrated Power Electronic Modules and in New Type of Plate-Pin Fin Heat Sink
,” Ph.D. thesis, Xi’an JiaoTong University, China.
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Discrete time and continuous time - Wikipedia @ WordDisk
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Frameworks for modeling variables that evolve over time
In contrast, continuous time views variables as having a particular value for potentially only an infinitesimally short amount of time. Between any two points in time there are an infinite number of other points in time. The variable "time" ranges over the entire real number line, or depending on the context, over some subset of it such as the non-negative reals. Thus time is viewed as a continuous variable.
A continuous signal or a continuous-time signal is a varying quantity (a signal) whose domain, which is often time, is a continuum (e.g., a connected interval of the reals). That is, the function's domain is an uncountable set. The function itself need not to be continuous. To contrast, a discrete-time signal has a countable domain, like the natural numbers.
The signal is defined over a domain, which may or may not be finite, and there is a functional mapping from the domain to the value of the signal. The continuity of the time variable, in connection with the law of density of real numbers, means that the signal value can be found at any arbitrary point in time.
A typical example of an infinite duration signal is:
{\displaystyle f(t)=\sin(t),\quad t\in \mathbb {R} }
A finite duration counterpart of the above signal could be:
{\displaystyle f(t)=\sin(t),\quad t\in [-\pi ,\pi ]}
{\displaystyle f(t)=0}
The value of a finite (or infinite) duration signal may or may not be finite. For example,
{\displaystyle f(t)={\frac {1}{t}},\quad t\in [0,1]}
{\displaystyle f(t)=0}
is a finite duration signal but it takes an infinite value for
{\displaystyle t=0\,}
In many disciplines, the convention is that a continuous signal must always have a finite value, which makes more sense in the case of physical signals.
For some purposes, infinite singularities are acceptable as long as the signal is integrable over any finite interval (for example, the
{\displaystyle t^{-1}}
signal is not integrable at infinity, but
{\displaystyle t^{-2}}
is).
Any analog signal is continuous by nature. Discrete-time signals, used in digital signal processing, can be obtained by sampling and quantization of continuous signals.
Continuous signal may also be defined over an independent variable other than time. Another very common independent variable is space and is particularly useful in image processing, where two space dimensions are used.
Discrete time is often employed when empirical measurements are involved, because normally it is only possible to measure variables sequentially. For example, while economic activity actually occurs continuously, there being no moment when the economy is totally in a pause, it is only possible to measure economic activity discretely. For this reason, published data on, for example, gross domestic product will show a sequence of quarterly values.
When one attempts to empirically explain such variables in terms of other variables and/or their own prior values, one uses time series or regression methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, yt might refer to the value of income observed in unspecified time period t, y3 to the value of income observed in the third time period, etc.
Moreover, when a researcher attempts to develop a theory to explain what is observed in discrete time, often the theory itself is expressed in discrete time in order to facilitate the development of a time series or regression model.
On the other hand, it is often more mathematically tractable to construct theoretical models in continuous time, and often in areas such as physics an exact description requires the use of continuous time. In a continuous time context, the value of a variable y at an unspecified point in time is denoted as y(t) or, when the meaning is clear, simply as y.
Discrete time makes use of difference equations, also known as recurrence relations. An example, known as the logistic map or logistic equation, is
{\displaystyle x_{t+1}=rx_{t}(1-x_{t}),}
in which r is a parameter in the range from 2 to 4 inclusive, and x is a variable in the range from 0 to 1 inclusive whose value in period t nonlinearly affects its value in the next period, t+1. For example, if
{\displaystyle r=4}
{\displaystyle x_{1}=1/3}
, then for t=1 we have
{\displaystyle x_{2}=4(1/3)(2/3)=8/9}
, and for t=2 we have
{\displaystyle x_{3}=4(8/9)(1/9)=32/81}
Another example models the adjustment of a price P in response to non-zero excess demand for a product as
{\displaystyle P_{t+1}=P_{t}+\delta \cdot f(P_{t},...)}
{\displaystyle \delta }
is the positive speed-of-adjustment parameter which is less than or equal to 1, and where
{\displaystyle f}
is the excess demand function.
Continuous time makes use of differential equations. For example, the adjustment of a price P in response to non-zero excess demand for a product can be modeled in continuous time as
{\displaystyle {\frac {dP}{dt}}=\lambda \cdot f(P,...)}
where the left side is the first derivative of the price with respect to time (that is, the rate of change of the price),
{\displaystyle \lambda }
is the speed-of-adjustment parameter which can be any positive finite number, and
{\displaystyle f}
is again the excess demand function.
A variable measured in discrete time can be plotted as a step function, in which each time period is given a region on the horizontal axis of the same length as every other time period, and the measured variable is plotted as a height that stays constant throughout the region of the time period. In this graphical technique, the graph appears as a sequence of horizontal steps. Alternatively, each time period can be viewed as a detached point in time, usually at an integer value on the horizontal axis, and the measured variable is plotted as a height above that time-axis point. In this technique, the graph appears as a set of dots.
The values of a variable measured in continuous time are plotted as a continuous function, since the domain of time is considered to be the entire real axis or at least some connected portion of it.
"Digital Signal Processing", Prentice Hall - pages 11–12
"Digital Signal Processing: Instant access", Butterworth-Heinemann - page 8
Gershenfeld, Neil A. (1999). The Nature of mathematical Modeling. Cambridge University Press. ISBN 0-521-57095-6.
Wagner, Thomas Charles Gordon (1959). Analytical transients. Wiley.
This article uses material from the Wikipedia article Discrete time and continuous time, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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Stata Blogs - An introduction to the lasso in Stata (拉索回归简介)| 连享会主页
Title: Stata Blogs - An introduction to the lasso in Stata
Source: Stata Blogs - An introduction to the lasso in Stata Author: David Drukker, Executive Director of Econometrics and Di Liu, Senior Econometrician[1]Go to comments
What's a lasso?
The lasso is most useful when a few out of many potential covariates affect the outcome and it is important to include only the covariates that have an affect. "Few" and "many" are defined relative to the sample size. In the example discussed below, we observe the most recent health-inspection scores for 600 restaurants, and we have 100 covariates that could potentially affect each one's score. We have too many potential covariates because we cannot reliably estimate 100 coefficients from 600 observations. We believe that only about 10 of the covariates are important, and we feel that 10 covariates are "a few" relative to 600 observations.
Given that only a few of the many covariates affect the outcome, the problem is now that we don't know which covariates are important and which are not. The lasso produces estimates of the coefficients and solves this covariate-selection problem.
hsafety2.dta has 1 observation for each of 600 restaurants, and the score from the most recent inspection is in score. The percentage of a restaurant's social-media reviews that contain a word like "dirty" could predict the inspection score. We identified 50 words, 30 word pairs, and 20 phrases whose occurrence percentages in reviews written in the three months prior to an inspection could predict the inspection score. The occurrence percentages of the 50 words are in word1 – word50. The occurrence percentages of 30-word pairs are in wpair1 – wpair30. The occurrence percentages of the 20 phrases are in phrase1 – phrase20.
. use hsafety2, clear
. quietly regress score word1-word50 wpair1-wpair30 phrase1-phrase20 ///
if sample==1
The lasso is an estimator of the coefficients in a model. What makes the lasso special is that some of the coefficient estimates are exactly zero, while others are not. The lasso selects covariates by excluding the covariates whose estimated coefficients are zero and by including the covariates whose estimates are not zero. There are no standard errors for the lasso estimates. The lasso's ability to work as a covariate-selection method makes it a nonstandard estimator and prevents the estimation of standard errrors. In this post, we discuss how to use the lasso for inferential questions.
Like many estimators, the lasso for linear models solves an optimization problem. Specifically, the linear lasso point estimates
\stackrel{^}{\mathbit{\beta }}
\stackrel{^}{\mathbit{\beta }}=\mathrm{arg}\underset{\mathbit{\beta }}{min}\left\{\frac{1}{2n}\sum _{i=1}^{n}{\left({y}_{i}-{\mathbf{x}}_{i}{\mathbit{\beta }}^{\mathrm{\prime }}\right)}^{2}+\lambda \sum _{j=1}^{p}{\omega }_{j}|{\beta }_{j}|\right\}
\mathbit{\lambda }>0
is the lasso penalty parameter,
y
is the outome variable,
\mathbf{x}containsthe
p$ potential covariates,
\mathbit{\beta }
is the vector of coefficients on
\mathbf{x},
{\beta }_{j}
j\text{th}
\mathbit{\beta }
{\omega }_{j}
are parameter-level weights known as penalty loadings, and
n
is the sample size.
\frac{1}{2n}\sum _{i=1}^{n}{\left({y}_{i}-{\mathbf{x}}_{i}{\mathbit{\beta }}^{\mathrm{\prime }}\right)}^{2}
\lambda \sum _{j=1}^{p}{\omega }_{j}|{\mathbit{\beta }}_{j}|
\lambda
{\omega }_{j}
are called "tuning" parameters. They specify the weight applied to the penalty term. When
\lambda =0
, the linear lasso reduces to the OLS estimator. As
\lambda
increases, the magnitude of all the estimated coefficients is "shrunk" toward zero. This skrinkage occurs because the cost of each nonzero
{\stackrel{^}{\beta }}_{j}
increases with the penalty term that increases as
\lambda
{\lambda }_{max}
for which all the estimated coefficients are exactly zero. As
\lambda
{\lambda }_{max}{\lambda }_{max},
the number of nonzero coefficient estimates increases. For
\lambda \in \left(0,{\lambda }_{max}\right),
some of the estimated coefficients are exactly zero and some of them are not zero. When you use the lasso for covariate selection, covariates with estimated coefficients of zero are excluded, and covariates with estimated coefficients that are not zero are included.
That the number of potential covariates
p
can be greater than the sample size
n
is a much discussed advantage of the lasso. It is important to remember that the approximate sparsity assumption requires that the number of covariates that belong in the model
\left(s\right)
must be small relative to
n
The tuning parameters must be selected before using the lasso for prediction or model selection. The most frequent methods used to select the tuning parameters are cross-validation (CV), the adaptive lasso, and plug-in methods. In addition,
\lambda
is sometimes set by hand in a sensitivity analysis.
CV finds the
\lambda
that minimizes the out-of-sample MSE of the predictions. The mechanics of CV mimic the process using split samples to find the best out-of-sample predictor. The details are presented in an appendix.
. lasso linear score word1-word50 wpair1-wpair30 phrase1-phrase20 ///
if sample==1, nolog rseed(12345)
We specified the option nolog to supress the CV log over the candidate values of
\lambda
. The output reveals that CV selected a
\lambda
for which 25 of the 100 covariates have nonzero coefficients. We used estimates store to store these results under the name cv in memory.
The CV function appears somewhat flat near the optimal
\lambda ,
which implies that nearby values of
\lambda
would produce similar out-of-sample MSEs. The number of included covariates can vary substantially over the flat part of the CV function. We can investigate the variation in the number of selected covariates using a table called a lasso knot table. In the jargon of lasso, a knot is a value of
\lambda
for which a covariate is added or subtracted to the set of covariates with nonzero values. We use lassoknots to display the table of knots.
| No. of CV mean |
| nonzero pred. | Variables (A)dded, (R)emoved,
ID | lambda coef. error | or left (U)nchanged
-----+-----------------------------+--------------------------------------
2 | 2.980526 2 52.2861 | A phrase3 phrase4
3 | 2.715744 3 50.48463 | A phrase5
4 | 2.474485 4 48.55981 | A word3
9 | 1.554049 6 40.23385 | A wpair3
10 | 1.415991 8 39.04494 | A wpair2 phrase2
12 | 1.175581 9 36.983 | A word2
14 | .9759878 10 35.42697 | A word31
16 | .8102822 11 34.2115 | A word19
17 | .738299 12 33.75501 | A word4
21 | .5088809 14 32.74808 | A word14 phrase7
22 | .4636733 17 32.64679 | A word32 wpair19 wpair26
23 | .4224818 19 32.56572 | A wpair15 wpair25
24 | .3849497 22 32.53301 | A wpair24 phrase13 phrase14
* 26 | .319592 25 32.52679 | A word25 word30 phrase8
27 | .2912003 26 32.53946 | A wpair11
30 | .2202824 30 33.18254 | A word23 word38 wpair4
The CV function is minimized at the
\lambda
\mathrm{I}\mathrm{D}=26
, and the lasso includes 25 covariates at this
\lambda
value. The flat part of the CV function includes the
\lambda
\mathrm{I}\mathrm{D}\in \left\{21,22,23,24,26,27\right\}
Only 14 covariates are included by the lasso using the
\lambda
\mathrm{I}\mathrm{D}=21
. We will explore this observation using sensitivity analysis below.
The first step of the adaptive lasso is
\mathrm{C}\mathrm{V}
. The second step does
\mathrm{C}\mathrm{V}
among the covariates selected in the first step. In this second step, the penalty loadings are
{\omega }_{j}=1/|{\stackrel{^}{\beta }}_{j}|,
{\stackrel{^}{\beta }}_{j}
are the penalized estimates from the first step. Covariates with smaller-magnitude coefficients are more likely to be excluded in the second step. See Zou (2006) and Bühlmann and Van de Geer (2011) for more about the adaptive lasso and the tendency of the CV-based lasso to overselect. Also see Chetverikov, Liao, and Chernozhukov (2019) for formal results for the CV lasso and results that could explain this overselection tendency.
if sample==1, nolog rseed(12345) selection(adaptive)
Plug-in methods tend to be even more parsimonious than the adaptive lasso. Plug-in methods find the value of the
\lambda
that is large enough to dominate the estimation noise. The plug-in method chooses
{\omega }_{j}
to normalize the scores of the (unpenalized) fit measure for each parameter. Given the normalized scores, it chooses a value for
\lambda
that is greater than the largest normalized score with a probability that is close to
1.
if sample==1, selection(plugin)
For linear models, Belloni and Chernozhukov (2013) present conditions in which the postselection predictions perform at least as well as the lasso predictions. Heuristically, one expects the lasso predictions from a CV-based lasso to perform better than the postselection predictions because CV chooses
\lambda
to make the best lasso predictions. Analogously, one expects the postselection predictions for the plug-in-based lasso to perform better than the lasso predictions because the plug-in tends to select a set of covariates close to those that best approximate the process that generated the data.
\stackrel{^}{\mathbit{\beta }}=\mathrm{arg}\underset{\mathbit{\beta }}{min}\left\{\frac{1}{2n}\sum _{i=1}^{n}{\left({y}_{i}-{\mathbf{x}}_{i}{\mathbit{\beta }}^{\mathrm{\prime }}\right)}^{2}+\lambda \left[\alpha \sum _{j=1}^{p}|{\beta }_{j}|+\frac{\left(1-\alpha \right)}{2}\sum _{j=1}^{p}{\beta }_{j}^{2}\right]\right\}
\alpha
is the elastic-net penalty parameter. Setting
\alpha =0
produces ridge regression. Setting
\alpha =1
produces lasso. The elasticnet command selects
\alpha
\lambda
by CV. The option alpha() specifies the candidate values for
\alpha
. elasticnet linear score word1-word50 ///
wpair1-wpair30 phrase1-phrase20 ///
if sample==1, alpha(.25 .5 .75) ///
nolog rseed(12345)
. elasticnet linear score word1-word50 ///
wpair1-wpair30 phrase1-phrase20 ///
if sample==1, alpha(0) nolog rseed(12345)
We now compare the out-of-sample predictive ability of the CV-based lasso, the elastic net, ridge regression, and the plug-in-based lasso using the lasso predictions. (For elastic net and ridge regression, the "lasso predictions" are made using the coefficient estimates produced by the penalized estimator.)
lasso selected the
\lambda
\mathrm{I}\mathrm{D}=26
and 25 covariates. We now use lassoselect to specify that the
\lambda
\mathrm{I}\mathrm{D}=21
be the selected
\lambda
and store the results under the name hand.
lassoselect id = 21
We now compute the out-of-sample MSE produced by the postselection estimates of the lasso whose
\lambda
has ID = 21 . The results are not wildly different and we would stick with those produced by the post-selection plug-in-based lasso.
Cross-validation finds the value for
\lambda
in a grid of candidate values
\left\{{\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{Q}\right\}
that minimizes the MSE of the out-of-sample predictions. Cross-validation sets
{\omega }_{j}=1
or to user-specified values.
After you specify the grid, the sample is partitioned into
K
nonoverlapping subsets. For each grid value
{\lambda }_{q}
, predict the out-of-sample squared errors using the following steps.
k\in \left\{1,2,\dots ,K\right\}
using the data not in partition
k,
estimate the penalized coefficients
\stackrel{^}{\mathbit{\beta }}
\mathbit{\lambda }={\lambda }_{q}
using the data in partition
k
, predict the out-of-sample squared errors.
The mean of these out-of-sample squared errors estimates the out-of-sample MSE of the predictions. The cross-validation function traces the values of these out-of-sample MSEs over the grid of candidate values for
\lambda
{\lambda }_{j}
that produces the smallest estimated out-of-sample MSE minimizes the cross-validation function, and it is selected.
Posts by David Drukker, Executive Director of Econometrics and Di Liu, Senior Econometrician: https://blog.stata.com/author/drukker-liu/
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Simulink Code Coverage Metrics - MATLAB & Simulink - MathWorks España
Condition Coverage Example
Decision Coverage Example
Modified Condition/Decision Coverage Example
If you have a Simulink® Coverage™ license, you can run a SIL or PIL simulation that produces code coverage metrics for generated model code. The simulation performs several types of code coverage analysis.
Statement coverage measures the number of source code statements that execute when the code runs. Use this type of coverage to determine whether every statement in the program has been invoked at least once.
The percentage of statement coverage is represented by the following equation:
Statement coverage = (Number of executed statements / Total number of statements) *100
This code snippet contains five statements. To achieve 100% statement coverage, you need at least one test with positive x values, one test with negative x values, and one test with x values of zero.
printf( "x is positive" );
printf( "x is negative" );
printf( "x is 0" );
Condition coverage analyzes statements that include conditions in source code. Conditions are C/C++ Boolean expressions that contain relation operators (<, >, <=, or >=), equation operators (!= or ==), or logical negation operators (!), but that do not contain logical operators (&& or ||). This type of coverage determines whether every condition has been evaluated to all possible outcomes at least once.
The percentage of condition coverage is represented by the following equation:
Condition coverage = (Number of executed condition outcomes / Total number of condition outcomes) *100
y = x<=5 || x!=7;
To achieve 100% condition coverage, your test cases need to demonstrate a true and false outcome for both conditions. For example, a test case where x is equal to 4 demonstrates a true case for both conditions, and a case where x is equal to 7 demonstrates a false case for both conditions.
Decision coverage analyzes statements that represent decisions in source code. Decisions are Boolean expressions composed of conditions and one or more of the logical C/C++ operators && or ||. Conditions within branching constructs (if/else, while, and do-while) are decisions. Decision coverage determines the percentage of the total number of decision outcomes the code exercises during execution. Use this type of coverage to determine whether all decisions, including branches, in your code are tested.
The decision coverage definition for DO-178C compliance differs from the Simulink Coverage definition. For decision coverage compliance with DO-178C, in the Configuration Parameters, set the Structural Coverage Level to Condition Decision for Boolean expressions not containing && or || operators.
The percentage of decision coverage is represented by the following equation:
Decision coverage = (Number of executed decision outcomes / Total number of decision outcomes) *100
This code snippet contains three decisions:
y = x<=5 && x!=7; // decision #1
if( x > 0 ) // decision #2
printf( "decision #2 is true" );
else if( x < 0 && y ) // decision #3
printf( "decisions #2 and #3 are false" );
To achieve 100% decision coverage, your test cases must demonstrate a true and false outcome for each decision.
Modified condition/decision coverage (MCDC) analyzes whether the conditions within decisions independently affect the decision outcome during execution. To achieve 100% MCDC, your test cases must demonstrate:
All conditions within decisions have been evaluated to all possible outcomes at least once.
Every condition within a decision independently affects the outcome of the decision.
The percentage of MCDC is represented by the following equation:
MCDC coverage = (Number of conditions evaluated to all possible outcomes affecting the outcome of the decision / Total number of conditions within the decisions) *100
For this decision:
X || ( Y && Z )
the following set of test cases delivers 100% MCDC coverage.
In order to demonstrate that the conditions Y and Z can independently affect the decision outcome, the condition X must be false for those test cases. If the condition X is true, then the decision is already known to be true. Therefore, the conditions Y and Z would not affect the decision outcome.
Cyclomatic complexity measures the structural complexity of code by using the McCabe complexity measure. To compute the cyclomatic complexity of code, code coverage uses this formula:
c=\sum _{1}^{N}\left({o}_{n}-1\right)
N is the number of decisions in the code. on is the number of outcomes for the nth decision point. Code coverage adds 1 to the complexity number for each C/C++ function.
For this code snippet, the cyclomatic complexity is 3:
void evalNum(int x)
if (x > 0) // decision #1
else if (x < 0) // decision #2
The code contains one function that has two decision points. Each decision point has two outcomes. Using the preceding formula, N is 2, o1 is 2, and o2 is 2. Code coverage uses the formula with these decisions and outcomes and adds 1 for the function. The cyclomatic complexity for this code snippet is:
c = (o1 − 1) + (o2 − 1) + 1 = (2 − 1) + (2 − 1) + 1 = 3
Relational boundary code coverage examines code that has relational operations. Relational boundary code coverage metrics align with those for model coverage, as described in Relational Boundary Coverage (Simulink Coverage). Fixed-point values in your model are integers during code coverage.
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