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FUNDAMENTAL THEOREM OF ARITHMETIC - Encyclopedia Information
Fundamental theorem of arithmetic Information
Integers have unique prime factorizations
Not to be confused with Fundamental theorem of algebra.
In Disquisitiones Arithmeticae (1801) Gauss proved the unique factorization theorem [1] and used it to prove the law of quadratic reciprocity. [2]
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. [3] [4] [5] For example,
{\displaystyle 1200=2^{4}\cdot 3^{1}\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots }
The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (for example,
{\displaystyle 12=2\cdot 6=3\cdot 4}
This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example,
{\displaystyle 2=2\cdot 1=2\cdot 1\cdot 1=\ldots }
1 Euclid's original version
2.1 Canonical representation of a positive integer
3.3 Uniqueness without Euclid's lemma
Euclid's original version
— Euclid, Elements Book VII, Proposition 30
— Euclid, Elements Book IX, Proposition 14
(In modern terminology: a least common multiple of several prime numbers is not a multiple of any other prime number.) Book IX, proposition 14 is derived from Book VII, proposition 30, and proves partially that the decomposition is unique – a point critically noted by André Weil. [6] Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case.
Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic. [1]
{\displaystyle n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}=\prod _{i=1}^{k}p_{i}^{n_{i}}}
This representation is called the canonical representation [7] of n, or the standard form [8] [9] of n. For example,
999 = 33×37,
1000 = 23×53,
1001 = 7×11×13.
{\displaystyle n=2^{n_{1}}3^{n_{2}}5^{n_{3}}7^{n_{4}}\cdots =\prod _{i=1}^{\infty }p_{i}^{n_{i}},}
{\displaystyle {\begin{alignedat}{2}a\cdot b&=2^{a_{1}+b_{1}}3^{a_{2}+b_{2}}5^{a_{3}+b_{3}}7^{a_{4}+b_{4}}\cdots &&=\prod p_{i}^{a_{i}+b_{i}},\\\gcd(a,b)&=2^{\min(a_{1},b_{1})}3^{\min(a_{2},b_{2})}5^{\min(a_{3},b_{3})}7^{\min(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\min(a_{i},b_{i})},\\\operatorname {lcm} (a,b)&=2^{\max(a_{1},b_{1})}3^{\max(a_{2},b_{2})}5^{\max(a_{3},b_{3})}7^{\max(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\max(a_{i},b_{i})}.\end{alignedat}}}
Main article: Arithmetic function
Uniqueness without Euclid's lemma
The fundamental theorem of arithmetic can also be proved without using Euclid's lemma. [10] The proof that follows is inspired by Euclid's original version of the Euclidean algorithm.
{\displaystyle s}
is the smallest positive integer which is the product of prime numbers in two different ways. Incidentally, this implies that
{\displaystyle s}
, if it exists, must be a composite number greater than
{\displaystyle 1}
. Now, say
{\displaystyle {\begin{aligned}s&=p_{1}p_{2}\cdots p_{m}\\&=q_{1}q_{2}\cdots q_{n}.\end{aligned}}}
{\displaystyle p_{i}}
must be distinct from every
{\displaystyle q_{j}.}
Otherwise, if say
{\displaystyle p_{i}=q_{j},}
then there would exist some positive integer
{\displaystyle t=s/p_{i}=s/q_{j}}
that is smaller than s and has two distinct prime factorizations. One may also suppose that
{\displaystyle p_{1}<q_{1},}
by exchanging the two factorizations, if needed.
{\displaystyle P=p_{2}\cdots p_{m}}
{\displaystyle Q=q_{2}\cdots q_{n},}
{\displaystyle s=p_{1}P=q_{1}Q.}
{\displaystyle s-p_{1}Q=(q_{1}-p_{1})Q=p_{1}(P-Q)<s.}
As the positive integers less than s have been supposed to have a unique prime factorization,
{\displaystyle p_{1}}
must occur in the factorization of either
{\displaystyle q_{1}-p_{1}}
or Q. The latter case is impossible, as Q, being smaller than s, must have a unique prime factorization, and
{\displaystyle p_{1}}
differs from every
{\displaystyle q_{j}.}
The former case is also impossible, as, if
{\displaystyle p_{1}}
{\displaystyle q_{1}-p_{1},}
it must be also a divisor of
{\displaystyle q_{1},}
which is impossible as
{\displaystyle p_{1}}
{\displaystyle q_{1}}
are distinct primes.
Therefore, there cannot exist a smallest integer with more than a single distinct prime factorization. Every positive integer must either be a prime number itself, which would factor uniquely, or a composite that also factors uniquely into primes, or in the case of the integer
{\displaystyle 1}
, not factor into any prime.
The first generalization of the theorem is found in Gauss's second monograph (1832) on biquadratic reciprocity. This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers. It is now denoted by
{\displaystyle \mathbb {Z} [i].}
He showed that this ring has the four units ±1 and ±i, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes. [11]
Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring
{\displaystyle \mathbb {Z} [\omega ]}
{\displaystyle \omega ={\frac {-1+{\sqrt {-3}}}{2}},}
{\displaystyle \omega ^{3}=1}
is a cube root of unity. This is the ring of Eisenstein integers, and he proved it has the six units
{\displaystyle \pm 1,\pm \omega ,\pm \omega ^{2}}
and that it has unique factorization.
However, it was also discovered that unique factorization does not always hold. An example is given by
{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}
. In this ring one has [12]
{\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})(1-{\sqrt {-5}}).}
Examples like this caused the notion of "prime" to be modified. In
{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}
it can be proven that if any of the factors above can be represented as a product, for example, 2 = ab, then one of a or b must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; for example, 2 divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6. In algebraic number theory 2 is called irreducible in
{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}
(only divisible by itself or a unit) but not prime in
{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}
(if it divides a product it must divide one of the factors). The mention of
{\displaystyle \mathbb {Z} [{\sqrt {-5}}]}
is required because 2 is prime and irreducible in
{\displaystyle \mathbb {Z} .}
Using these definitions it can be proven that in any integral domain a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers
{\displaystyle \mathbb {Z} }
every irreducible is prime". This is also true in
{\displaystyle \mathbb {Z} [i]}
{\displaystyle \mathbb {Z} [\omega ],}
{\displaystyle \mathbb {Z} [{\sqrt {-5}}].}
Integer factorization – Decomposition of a number into a product
Prime signature – Multiset of prime exponents in a prime factorization
^ a b Gauss & Clarke (1986, Art. 16)
^ Gauss & Clarke (1986, Art. 131)
^ Hardy & Wright (2008, Thm 2) harvtxt error: no target: CITEREFHardyWright2008 ( help)
^ Weil (2007, p. 5): "Even in Euclid, we fail to find a general statement about the uniqueness of the factorization of an integer into primes; surely he may have been aware of it, but all he has is a statement (Eucl.IX.I4) about the l.c.m. of any number of given primes."
^ Hardy & Wright (2008, § 1.2) harvtxt error: no target: CITEREFHardyWright2008 ( help)
^ Dawson, John W. (2015), Why Prove it Again? Alternative Proofs in Mathematical Practice., Springer, p. 45, ISBN 9783319173689
^ Gauss, BQ, §§ 31–34
^ Hardy & Wright (2008, § 14.6) harvtxt error: no target: CITEREFHardyWright2008 ( help)
Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 978-0-387-96254-2 {{ citation}}: |first2= has generic name ( help)
Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8 {{ citation}}: |first2= has generic name ( help)
Euclid (1956), The thirteen books of the Elements, vol. 2 (Books III-IX), Translated by Thomas Little Heath (Second Edition Unabridged ed.), New York: Dover, ISBN 978-0-486-60089-5
Weil, André (2007) [1984], Number Theory: An Approach through History from Hammurapi to Legendre, Modern Birkhäuser Classics, Boston, MA: Birkhäuser, ISBN 978-0-817-64565-6
Weisstein, Eric W., "Abnormal number", MathWorld
Weisstein, Eric W., "Fundamental Theorem of Arithmetic", MathWorld
Grime, James, "1 and Prime Numbers", Numberphile, Brady Haran, archived from the original on 2021-12-11
Amicable ( Triple)
Retrieved from " https://en.wikipedia.org/?title=Fundamental_theorem_of_arithmetic&oldid=1088730125"
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{\mathrm{ω}}_{5}.
\textcolor[rgb]{0,0,1}{\mathrm{ω5}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{12}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{2}}]]]\right)
{\mathrm{ω}}_{6}
\mathrm{η}.
\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{ω6}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{16}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{13}]\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{9}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{1}}]]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{_DG}}\textcolor[rgb]{0,0,1}{}\left([[\textcolor[rgb]{0,0,1}{"biform"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]]\textcolor[rgb]{0,0,1}{,}[[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]]]\right) |
What Is a Flotation Cost?
Flotation costs are incurred by a publicly-traded company when it issues new securities and incurs expenses, such as underwriting fees, legal fees, and registration fees. Companies must consider the impact these fees will have on how much capital they can raise from a new issue. Flotation costs, expected return on equity, dividend payments, and the percentage of earnings the business expects to retain are all part of the equation to calculate a company's cost of new equity.
Understanding and Calculating Flotation Costs
The Formula for Float in New Equity Is
The equation for calculating the flotation cost of new equity using the dividend growth rate is:
\text{Dividend growth rate} = \frac{D_1}{P * \left(1-F\right)} + g
Dividend growth rate=P∗(1−F)D1+g
D1 = the dividend in the next period
P = the issue price of one share of stock
F = ratio of flotation cost-to-stock issue price
g = the dividend growth rate
Flotation costs are costs a company incurs when it issues new stock.
Flotation costs make new equity cost more than existing equity.
Analysts argue that flotation costs are a one-time expense that should be adjusted out of future cash flows in order to not overstate the cost of capital forever.
What Do Flotation Costs Tell You?
Companies raise capital in two ways: debt via bonds and loans or equity. Some companies prefer issuing bonds or obtaining a loan, especially when interest rates are low and because the interest paid on many debts is tax-deductible, while equity returns are not. Other companies prefer equity because it does not need to be paid back; however, selling equity also entails giving up an ownership stake in the company.
There are flotation costs associated with issuing new equity, or newly issued common stock. These include costs such as investment banking and legal fees, accounting and audit fees, and fees paid to a stock exchange to list the company's shares. The difference between the cost of existing equity and the cost of new equity is the flotation cost.
The flotation cost is expressed as a percentage of the issue price and is incorporated into the price of new shares as a reduction. A company will often use a weighted cost of capital (WACC) calculation to determine what share of its funding should be raised from new equity and what portion from debt.
Example of a Flotation Cost Calculation
As an example, assume Company A needs capital and decides to raise $100 million in common stock at $10 per share to meet its capital requirements. Investment bankers receive 7% of the funds raised. Company A pays out $1 in dividends per share next year and is expected to increase dividends by 10% the following year.
Using these variables, the cost of new equity is calculated with the following equation:
($1 / ($10 * (1-7%)) + 10%
The answer is 20.7%. If the analyst assumes no flotation cost, the answer is the cost of existing equity. The cost of existing equity is calculated with the following formula:
Limitations of Using Flotation Costs
Some analysts argue that including flotation costs in the company's cost of equity implies that flotation costs are an ongoing expense, and forever overstates the firm's cost of capital. In reality, a firm pays the flotation costs one time upon issuing new equity. To offset this, some analysts adjust the company's cash flows for flotation costs.
Understanding Free Cash Flow to Equity
Free cash flow to equity (FCFE) is a measure of how much cash can be paid to the equity shareholders of a company after all expenses, reinvestment and debt are paid. |
LaTeX - Curvenote Docs
After creating an article in Curvenote, you can export and download your article as a collection of files for editing and compilation in
\LaTeX
. Before downloading you can choose from a variety of templates to format your article. The formatting for these templates will automatically be added to your exported TeX file!
#Export and download
\LaTeX
\bf\LaTeX
Each template has a variety of required and optional options to include such as author name, affiliation, email, etc. Learn more Template Options
Other requirements such as abstracts and acknowledgements use tagged content. Learn more Tagging Blocks
Click the ☁️⬇️ icon for the Download Zip option.
\textcircled{\checkmark}
Download LaTeX.
You can also download the log file for the LaTeX export.
The main.tex file in the downloaded folder is ready for you to open and run locally with
\LaTeX
! Other files are also included in the downloaded zip folder, such as your images, located in the assets folder (under images). |
StartServer - Maple Help
Home : Support : Online Help : Programming : Grid Package : StartServer
Grid[Server]
start a hpc Server on a machine
stop a hpc Server on a machine
Grid[Server][StartServer](p,n,broadcast,bcport,logfile)
Grid[Server][StartServer](port=p, numnodes=n, mask=broadcast, broadcastport=bcport, log=logfile)
Grid[Server][StopServer](p,n)
Grid[Server][StopServer](port=p, numnodes=n)
integer, representing the base port. This setting must be consistent across servers and client
integer, representing the number of servers running on this machine (usually one per CPU)
string, the broadcast mask
bcport
integer, representing the broadcast port. This setting must be consistent across all servers
string, filename of log file
The StartServer and StopServer commands are part of a Server sub-module of the Grid package, and only apply when using "hpc" mode (see Setup). Normally services or daemons are already running on machines in a distributed network, ready to receive Grid commands. The StartServer command is for testing purposes, and will launch a server on your local machine. StopServer will only stop the local instance of your server, not all of the processes on remote machines.
Launching a server using StartServer starts the server as child process of the Maple process.
The StartServer procedure returns an integer representing the process ID of server program or an error if the server could not be started.
These commands can be called using either positional arguments, or by specifying keyword, option=value pairs. When using positional arguments order matters.
The port parameter defaults to 2000. If this port is already taken at your site, you must choose a different one. You must use the same setting on all servers and clients.
The numnodes parameter represents the number of individual servers to start on this machine. Usually you will start one per available CPU. Each server will show up as an individual node (and will be counted as a node for licensing purposes) in your grid. The default is 4.
The mask parameter is a string representing the broadcast mask used for auto-discovery, i.e. in order for all the nodes in the grid to learn about the presence of each other. The default, "255.255.255.255" will broadcast to every reachable machine on your network. A different setting is likely to be more appropriate for your site. For example if your nodes are located at IP addresses on the 10.10.a.b network then you need to use a mask 10.10.255.255 to be able to broadcast to all machines on the 10.10.a.b subnet.
The broadcastport parameter is an integer representing the UDP port used for broadcasting. Setting this parameter to zero will disable auto-discovery. The default for this parameter is 4400. All servers must use the same broadcast port in order to discover each other when they startup.
The log parameter is a string representing the filename of the log file. The server will output informational messages to this log file. Multiple CPUs on one server will share the same log file but you cannot call StartServer on multiple machines pointing to the same log file on a shared drive. A relative path specified for the log file, e.g. logs/grid.log, will be taken relative to the location of the Grid Computing Toolbox installation directory. The default log is set to "grid.log".
On Windows, the easiest way to start and stop a server on a machine is to find the file named "gridserver.bat" in the bin directory of the Grid Computing Toolbox installation. Execute this batch file to start an independent process for running the server. If you have the Grid toolbox installed, see the full description given in the Batch section of the toolbox documentation.
On Linux, the easiest way to start and stop a server on a machine is to find the file named "gridserver.sh" in the bin directory of the Grid Computing Toolbox installation. Execute this script to start an independent process for running the server. If you have the Grid toolbox installed, see the full description given in the Batch section of the toolbox documentation.
\mathrm{with}\left(\mathrm{Grid}\right)
\mathrm{with}\left(\mathrm{Server}\right)
\mathrm{StartServer}\left(\mathrm{port}=2000,\mathrm{numnodes}=4\right)
\mathrm{Setup}\left("hpc",\mathrm{host}="localhost",\mathrm{port}=2000\right)
\mathrm{Launch}\left("print\left(hi\right);",\mathrm{numnodes}=4\right)
\mathrm{StopServer}\left(\mathrm{port}=2000,\mathrm{numnodes}=4\right)
The Grid[Server][StartServer] and Grid[Server][StopServer] commands were introduced in Maple 15. |
Molarity and Dilutions - Course Hero
General Chemistry/Composition of Substances and Solutions/Molarity and Dilutions
An aqueous solution is composed of a material dissolved in water. This type of solution can be described in terms of molarity.
In practical chemistry, it is often useful to dissolve compounds to make solutions. To dissolve a substance means to incorporate it into a liquid so as to form a solution. The substance in which another substance dissolves is the solvent. The dissolved material in a solution is the solute. An aqueous solution is a solution in which the solvent is water. Aqueous solutions are the most common type of solution because water dissolves a large variety of solutes.
Varying amounts of a solute may be dissolved in a given volume of water. The amount of solute dissolved in a given volume of solvent is its concentration. Concentration can be measured in many ways, but the most common measurement of concentration in chemistry is molarity (M), which is the number of moles of a solute dissolved in 1 liter of water. For example, 1 mole of sodium chloride (NaCl) dissolved in 1 liter of water has a molarity of 1 mol/L, or 1 molar (M). The concentration of any solution can be calculated by dividing the total number of moles of solute by the total volume.
\text{molarity}\;(\rm{M})=\frac{\text{number of moles of solute}}{\text{volume of solution}}
Calculate the Molarity of an Aqueous Solution of Sodium Chloride
What is the molarity of a solution made by dissolving 3.2 g of NaCl in enough water to produce 225 mL of solution?
Convert the given mass of NaCl to moles, using the periodic table to determine the molar mass of NaCl, 58.44 g/mol.
3.2\;\rm{g}\;\rm{ NaCl}\times\frac{1\;\rm{mol}\;\rm{NaCl}}{58.44\;\rm g\;\rm{NaCl}}=0.00548\;\rm {mol}\;\rm{NaCl}
Convert the given milliliters of solution to liters.
(225\;\rm{mL}\text{ solution})\left(\frac{1\,\rm{ L}}{1000\,\rm{ mL}}\right)=0.225\;\rm{L}\text{ solution}
Divide the number of moles of solute by the volume of solution, in liters, to find the molarity.
\text{molarity}=\frac{0.0548\;\rm{mol}\;\rm{NaCl}}{0.225\;\rm{L}}=0.24\;\rm M\;\rm{NaCl}
The molarity of the solution is 0.24 M NaCl.
For any aqueous solution, there is a maximum amount of solute that can be dissolved in the solution. A saturated solution contains the maximum amount of dissolved solute normally possible at a certain temperature. An unsaturated solution contains less than the maximum amount of dissolved solute normally possible at a certain temperature. Notice that saturation depends on the temperature of the solution. Heating a solution will allow more solute to be dissolved, even if the solution had been saturated at a lower temperature. A supersaturated solution contains an amount of dissolved solute greater than is normally possible at a certain temperature.
Any aqueous solution can be made less concentrated by adding water. To dilute a solution is to decrease its concentration by adding more solvent to it. A less concentrated solution made by diluting a solution is a dilution. Dilutions can be described using a formula in which the initial product of molarity and volume equals the final product of molarity and volume.
{\rm{M}}_1V_1={\rm{M}}_2V_2
M represents molarity, V represents volume, and the numbers 1 and 2 indicate the initial and final values respectively.
Calculate the Concentration of a Dilution of Sodium Chloride in Water
Consider a 1 M solution of NaCl in 1 liter of water. Another 1 liter of water is added to the solution. What is the concentration of the dilution?
Substitute the given values into the dilution equation. Let x represent the unknown molarity.
(1\;\rm M)(1\;\rm L)=(\it x\,\rm M)(2\;\rm L)
Divide both sides of the equation by 2 L to solve for x.
\begin{aligned}\rm{\it{x}\rm{M}}&=\frac{(1\;\rm M)(1\;\rm L)}{(2\;\rm L)}\\&=0.5\;\rm M\;\rm{NaCl}\end{aligned}
The concentration of the dilution is 0.5 M NaCl.
An aqueous solution can be made more concentrated by allowing the water to evaporate. To let the solvent evaporate from a solution to increase its concentration is to concentrate it.
Calculate the Concentration of a Sodium Chloride Solution
Suppose 1 liter of a 1.5 M solution of NaCl is evaporated to half a liter. What is the concentration of the remaining solution?
Substitute the given values into the dilution equation.
(1.5\;\rm M)(1\;\rm L)=\it(x\,\rm M)(0.5\;\rm L)
x
represent the unknown molarity, and solve the equation for
x
\begin{aligned}x\,\rm M&=\frac{(1.5\;\rm M)(1\;\rm L)}{0.5\;\rm L}\\&=3\;\rm M\;\rm{NaCl}\end{aligned}
The concentration of the remaining solution is 3 M NaCl. In this case, the concentration of the solution can be calculated without knowing the identity of the solvent.
If the water evaporates completely, crystallization of the solute can occur. Crystallization is the formation of a solid in which the particles form a highly organized structure. Crystallization can also occur in a supersaturated solution as the solution cools. As conditions become unfavorable for the solute to remain in solution, small crystals begin to form. In some cases, these crystals rapidly expand.
Most of the time, it is impossible to count the moles of a solute to calculate the molarity of the solution. Instead, scientists use the molar mass of the solute to determine the concentration.
Calculate the Amount of Solute Needed to Produce a Solution of Desired Quantity and Molarity
The molar mass of NaCl is 58.44 g/mol. How much solute is needed to make 5.80 liters of a 2.00 M solution?
Use the molarity to convert liters of solution to moles of solution.
(5.80\;\rm{L})\left(\frac{2.00\;\rm{mol}}{1\;\rm L}\right)=11.6\;\rm{mol}
Multiply by the molar mass to find the mass of NaCl.
(11.6\;\rm{mol})\left(\frac{58.44\;\rm g\;\rm{NaCl}}{1\;\rm{mol}}\right)=\;678\;\rm g\;\rm{NaCl}
To make the desired solution, 678 g NaCl is needed.
<Chemical Formulas of Compounds>Measuring Solution Concentrations |
The Nucleus - Course Hero
General Chemistry/Nuclear Chemistry/The Nucleus
Many fields of chemistry look at the interactions between atoms. In chemical reactions, bonds between atoms break, and new bonds form. Chemical reactions do not involve changes in the atomic nucleus. Nuclear chemistry is the field of chemistry that studies changes in atomic nuclei. The nucleus can gain or lose particles, or particles in the nucleus can change into other particles. Because elements are defined by the number of protons in the nucleus, which is the atomic number, any change in atomic number changes the element.
The nucleus of an atom is made up of positively charged protons and neutral neutrons. The protons and neutrons have comparable masses. Both protons and neutrons have a much greater mass than electrons. A nucleon is a proton or a neutron in an atomic nucleus.
Mass (amu, atomic mass unit)
\begin{aligned}{}_1^1\rm {p}\end{aligned}
\begin{aligned}{}_0^1\rm {n}\end{aligned}
{}_{-1}^{\;\;\,0}{\rm{e}}
Subatomic particles are represented with symbols and have specific masses.
Two fundamental forces of the universe play major roles in the nucleus. The strong nuclear force is the short-range force that acts between protons and neutrons, keeping the nucleus together. Its range is so short that it acts only between nucleons that are close together. The second fundamental force that acts on nucleons is the electromagnetic force, a repulsion between positively charged protons. This repulsion is weak compared to the strong nuclear force. However, the electromagnetic force can act at much greater distances. Each proton in the nucleus repels all other protons, not just the neighboring ones.
In nuclear chemistry, the number of neutrons in a nucleus is important. A nuclide is an atomic nucleus with a specific number of protons and neutrons. Nuclides are represented in the form of
{}_Z^A\rm {X}
where A is the atomic mass, Z is the proton number (atomic number), and X is the element symbol. This representation allows for calculation of the number of neutrons, N, through the formula
N+Z=A
. For example, nitrogen-14 is represented by
{}_{\,\,7}^{14}\rm {N}
and nitrogen-15 is represented by
{}_{\,\,7}^{15}\rm {N}
. The term isotope is often used interchangeably with the term nuclide. Isotopes are nuclei with the same number of protons but different numbers of neutrons.
Neutrons are stable in a nucleus. Outside of a nucleus, they are not stable and eventually decay into a proton, releasing an electron. The decay of a neutron into a proton and an electron can be written as an equation.
{}_0^1\rm{n}\rightarrow{}_1^1\rm{p}+{}_{-1}^{\;\;\,0}{\rm{e}}
Nucleons in a nucleus have lower energy than nucleons outside of a nucleus, resulting in a mass defect. Einstein's mass-energy equivalence relates mass and energy.
The difference between the total mass of the individual nucleons that make up a nucleus and the actual mass of the nucleus is called mass defect. Mass defect occurs because nucleons in a nucleus are more stable and have lower potential energy than nucleons outside of a nucleus. Consider a helium atom with two protons and two neutrons. Each proton has a mass of 1.007276 amu, so the total proton mass is 2.014552 amu. Each neutron has a mass of 1.008665 amu, so the total neutron mass is 2.017330. Altogether the mass of the helium atom is 4.031882 amu. This is different from the experimentally determined mass of the helium nucleus, which is 4.00151 amu. The mass difference is
4.031882-4.00151=0.03037\;{\rm{amu}}
. This difference in mass is not unique to helium. The total mass of the individual nucleons that make any nucleus is greater than the mass of the nucleus itself. The energy that binds nucleons manifests as a difference in mass. Energy and mass are directly proportional; when energy decreases, so does mass. It requires energy to break a nucleus apart. The energy required to break a nucleus into its component nucleons is the nuclear binding energy.
Mass and energy are related to each other. The equation that relates energy and mass, known as the mass-energy equivalence equation, was formulated by the German physicist Albert Einstein. It defines the relationship between energy E in joules (J), mass m in kilograms (kg), and the speed of light c in meters per second (m/s).
E=mc^2
The nuclear binding energy and mass-energy equivalence can be used to calculate the nuclear binding energy. The sum of mass and energy is always conserved. For example, the nuclear binding energy for helium can be calculated using the mass difference.
\begin{aligned}E&=mc^2\\&=(0.03037\;{\rm{amu}})\left(\frac{1\;\rm {g}}{6.022\times10^{23}\;{\rm{amu}}}\right)\left(\frac{1\;{\rm{kg}}}{{1}\rm{,}000\;\rm {g}}\right)\left(2.9979\times10^8\,\frac{\rm{m}}{\rm{s}}\right)^2\\&=4.534\times10^{-12}\,\frac{{\rm{kg}}\cdot\rm{m}^2}{{\rm{s^2}}}\\&=4.534\times10^{-12}\;\rm{ J}\end{aligned}
Mass was determined to 0.03038 amu by solving for the mass difference. Mass is converted to kg using dimensional analysis. The speed of light c is
2.9979\times{10}^8\;{\rm{m/s}}
, which is a constant. Using these values, energy in joules can be determined.
Two opposing forces act within an atomic nucleus. The strong nuclear force is a attractive, short-range force that acts between all nucleons—the charged protons as well as the neutral neutrons. The electromagnetic force is a repulsive force that acts between positively charged protons. Adding protons to a small nucleus increases both the attractive strong nuclear force and the repulsive electromagnetic force. Adding neutrons to a small nucleus increases only the attractive strong nuclear force.
The short range of the strong nuclear force means nucleons can form a limited number of strong nuclear attractions. As the nucleon count increases, the strong nuclear force increases linearly. Electromagnetic force is not limited by range. Every proton added will repulse every other proton. As proton count increases, electromagnetic force increases exponentially. Because of this difference, nuclei above 270 nucleons are very unstable. The largest nuclei that has been observed has 294 nucleons.
The ratio N:Z (where N is the neutron number and Z is the number of protons or atomic number) is significant with respect to nuclear stability. A graph of neutron number versus proton number for all known isotopes shows certain patterns:
For small atoms, with a proton number up to 20, the N:Z ratio of the most stable isotope is about 1:1. As the nucleus gets larger, the number of neutrons in the most stable isotope increases faster than the number of protons. The most stable isotope of gold, for example is
{}_{\;79}^{197}{\rm{Au}}
, giving a N:Z ratio of 1.49.
The zone of stability, or band of stability, is the region that represents stable, nonradioactive isotopes on a graph of the neutron number versus the proton number for all known isotopes.
Nuclei above the zone of stability are rich in neutrons and are unstable.
Nuclei below the zone of stability are rich in protons and are unstable.
A more detailed analysis of stable nuclei yields more patterns. For example, if a nucleus has an even number of protons and an even number of neutrons, it is more likely to be stable. Nuclei with odd numbers of protons and odd numbers of neutrons are unlikely to be stable. Nuclei with an even number of protons but an odd number of neutrons, or vice versa, fall in between. A magic number is a specific number of protons or neutrons that makes a nucleus more likely to be stable. The magic numbers are proton numbers of 2, 8, 20, 28, 50, or 82 or neutron numbers of 2, 8, 20, 28, 50, 82, or 126.
These patterns can be partially explained by the shell model of the nucleus, a model that defines the locations of protons and neutrons in shells that are partially analogous to electron shells. According to the shell model, pairs of neutrons or pairs of protons represent a more stable arrangement, similarly to what is seen with pairs of electrons.
The process by which unstable nuclei break down into other, smaller nuclei over time, releasing particles and/or energy, is radioactivity, or radioactive decay. An isotope with an unstable nucleus that experiences radioactive decay is called a radioisotope. There are different types of radioactive decay, depending on the properties of the nuclei undergoing it.
Radioactive decay often involves antiparticles. An antiparticle is a particle with the same mass as an elementary particle, but with the opposite charge. The antiparticle of an electron (
{}_{-1}^{\;\;\,0}{\rm{e}}
) is a positron (
{{}_{+1}^{\;\;\,0}{\rm{e}}}
), which has the same mass as an electron and a positive charge equal in magnitude to the negative charge of an electron. The antiparticle of a proton is an antiproton. An antiproton has the same mass as a proton and has an equivalent-magnitude, but negative, charge. Matter consisting of antiparticles such as antiprotons, antineutrons, and positrons is called antimatter. Antimatter, and antiparticles, annihilate when they interact with matter and elementary particles. The energy released in such an annihilation can be calculated by mass-energy equivalence,
E=mc^2
Nuclei above the zone of stability have an overabundance of neutrons relative to the number of protons. In such a nucleus, one of the neutrons is likely to decay into a proton. A high-energy electron is released when a neutron decays into a proton in the nucleus, called the beta particle (
{{}_{-1}^{\;\;\,0}\beta}
). This type of radioactivity is called beta decay. Carbon-14 decays into nitrogen through beta decay.
{}_{\;\;6}^{14}\rm{C}\rightarrow{}_{\;\;7}^{14}\rm{N}+{}_{-1}^{\;\;\,0}\beta
Note that this decay increases the proton number, or atomic number, of the nucleus, thus changing its identity from carbon to nitrogen. Beta decay reduces the neutron number and increases the proton number, making a nucleus approach the band of stability. In other words, emission of a beta particle makes an unstable nucleus become more stable. Nuclei below the band of stability have an overabundance of protons, and too few neutrons. In such a nucleus, a proton is likely to change into a neutron. In this case, an antiparticle, a positron (
{}_{+1}^{\;\;\,0}{\rm{e}}
) is emitted. For example, neon-19 decays into fluorine by emitting a positron.
{}_{10}^{19}{\rm{Ne}}\rightarrow{}_{\;9}^{19}\rm{F}+{}_{+1}^{\;\,\,0}\rm{e}
Positron emission reduces the neutron number and increases the proton number, making a nucleus approach the zone of stability. Electron capture is another process that converts a proton in the nucleus to a neutron. In this nuclear change, the nucleus captures an electron (
{}_{-1}^{\;\;\,0}{\rm{e}}
). For example, potassium-40 nuclei can capture an electron to become argon.
{}_{19}^{40}\rm {K}+{}_{-1}^{\;\,\,0}\text{e}\rightarrow{}_{18}^{40}\rm{Ar}
Electron capture is the reverse process of beta decay. In electron capture, as in positron emission, the proton number increases and a nucleus approaches the zone of stability. A large nucleus may undergo alpha decay to lose nucleons. During alpha decay, an alpha particle is lost from the nucleus. An alpha particle (
\alpha
) is a particle identical to a helium ion (He2+) that is emitted during the decay of radioactive elements. It is made up of two protons and two neutrons. An alpha particle is represented in a nuclear equation by the symbol
{}_2^4{\rm{He}}
. Alpha particles do not contain any electrons. Uranium-238, for example, decays into thorium-234 through alpha decay.
{}_{\;\,92}^{238}\rm {U}\rightarrow{}_{\,\;90}^{234}\rm{Th}+{}_2^4{\rm{He}}
An alpha particle decreases both the proton and the neutron numbers. A nucleus that is above the zone of stability typically undergoes alpha decay to approach the zone of stability.
A gamma ray, or gamma radiation, is high-energy electromagnetic radiation. Gamma rays are energy emitted by the nucleus as it becomes more stable through radioactivity. Gamma rays are symbolized by the lowercase Greek letter gamma:
\gamma
{}_0^0\rm\gamma
. Gamma rays do not change the proton or neutron numbers and are commonly not written in nuclear reactions.
<Vocabulary>Writing and Balancing Nuclear Equations |
EUDML | An $L^p$-version of a theorem of D.A. Raikov EuDML | An $L^p$-version of a theorem of D.A. Raikov
{L}^{p}
-version of a theorem of D.A. Raikov
G
be a locally compact group, for
p\in \left(1,\infty \right)
P{f}_{p}\left(G\right)
denote the closure of
{L}^{1}\left(G\right)
in the convolution operators on
{L}^{p}\left(G\right)
{W}_{p}\left(G\right)
P{f}_{p}\left(G\right)
which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space
{A}_{p}\left(G\right)
. It is shown that on the unit sphere of
{W}_{p}\left(G\right)
\sigma \left({W}_{p},P{f}_{p}\right)
topology and the strong
{A}_{p}
-multiplier topology coincide.
Fendler, Gero. "An $L^p$-version of a theorem of D.A. Raikov." Annales de l'institut Fourier 35.1 (1985): 125-135. <http://eudml.org/doc/74661>.
@article{Fendler1985,
abstract = {Let $G$ be a locally compact group, for $p\in (1,\infty )$ let $Pf_ p(G)$ denote the closure of $L^ 1(G)$ in the convolution operators on $L^ p(G)$. Denote $W_ p(G)$ the dual of $Pf_ p(G)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space $A_ p(G)$. It is shown that on the unit sphere of $W_ p(G)$ the $\sigma (W_ p,Pf_ p)$ topology and the strong $A_ p$-multiplier topology coincide.},
author = {Fendler, Gero},
keywords = {multipliers; Figà-Talamanca Herz space},
title = {An $L^p$-version of a theorem of D.A. Raikov},
AU - Fendler, Gero
TI - An $L^p$-version of a theorem of D.A. Raikov
AB - Let $G$ be a locally compact group, for $p\in (1,\infty )$ let $Pf_ p(G)$ denote the closure of $L^ 1(G)$ in the convolution operators on $L^ p(G)$. Denote $W_ p(G)$ the dual of $Pf_ p(G)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space $A_ p(G)$. It is shown that on the unit sphere of $W_ p(G)$ the $\sigma (W_ p,Pf_ p)$ topology and the strong $A_ p$-multiplier topology coincide.
KW - multipliers; Figà-Talamanca Herz space
[1] A. BENEDEK and R. PANZONE, The spaces Lp with mixed norm, Duke Math. J., 28 (1961), 301-324. Zbl0107.08902MR23 #A3451
[2] F.F. BONSALL and J. DUNCAN, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Math. Soc. Lecture Notes Series 2, Cambridge 1971. Zbl0207.44802MR44 #5779
[3] M. COWLING, An application of Littlewood-Paley theory in harmonic analysis, Math. Ann., 241 (1979), 83-69. Zbl0399.43004MR81f:43003
[4] M. COWLING and G. FENDLER, On representations in Banach spaces, Math. Ann., 266 (1984), 307-315. Zbl0508.46035MR85j:46083
[5] P. EYMARD, Algèbres Ap et convoluteurs de Lp, Séminaire Bourbaki 22è année, 1969/1970, no. 367. Zbl0264.43006
[6] E.E. GRANIRER, An application of the Radon Nikodym property in harmonic analysis, Bollentino U.M.I., (5) 18-B (1981), 663-671. Zbl0493.46018MR83b:43004
[7] E.E. GRANIRER and M. LEINERT, On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra B(G) and of the measure algebra M(G), Rocky Moutain J. of Math., 11 (1981), 459-472. Zbl0502.43004MR85f:43009
[8] C. HERZ, Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier, Grenoble, 24-3 (1974), 145-157. Zbl0287.43006MR54 #13466
[9] C. HERZ, Harmonic synthesis for subgroups, Ann. Inst. Fourier Grenoble, 23-3 (1973), 91-123. Zbl0257.43007MR50 #7956
[10] G.C. ROTA, An “alternierende Verfahen” for general positive operators, Bull. A.M.S., 68 (1962), 95-102. Zbl0116.10403MR24 #A3671
[11] E.M. STEIN, Topics in harmonic analysis related to the Littlewood-Paley theorem, Princeton University Press, 1970. Zbl0193.10502
Edmond Granirer, On convolution operators with small support which are far from being convolution by a bounded measure
multipliers, Figà-Talamanca Herz space |
BALL (MATHEMATICS) - Encyclopedia Information
Ball (mathematics) Information
https://en.wikipedia.org/wiki/Ball_(mathematics)
Volume space bounded by a sphere
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (March 2013) ( Learn how and when to remove this template message)
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. [1] It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is: [2]
{\displaystyle V_{n}(R)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n},}
{\displaystyle {\begin{aligned}V_{2k}(R)&={\frac {\pi ^{k}}{k!}}R^{2k}\,,\\[2pt]V_{2k+1}(R)&={\frac {2^{k+1}\pi ^{k}}{(2k+1)!!}}R^{2k+1}={\frac {2(k!)(4\pi )^{k}}{(2k+1)!}}R^{2k+1}\,.\end{aligned}}}
{\displaystyle B_{r}(p)=\{x\in M\mid d(x,p)<r\},}
{\displaystyle B_{r}[p]=\{x\in M\mid d(x,p)\leq r\}.}
In normed vector spaces
{\displaystyle \|\cdot \|}
{\displaystyle d(x,y)=\|x-y\|.}
{\displaystyle B_{r}(y)}
{\displaystyle x}
{\displaystyle y}
{\displaystyle r}
{\displaystyle r}
) and translated (b{\displaystyle y}
{\displaystyle B_{1}(0).}
{\displaystyle y=0}
{\displaystyle B(r).}
{\displaystyle \left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p},}
{\displaystyle r}
{\displaystyle B(r)=\left\{x\in \mathbb {R} ^{n}\,:\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p}<r\right\}.}
{\displaystyle \mathbb {R} ^{2}}
General convex norm
^ Equation 5.19.4, NIST Digital Library of Mathematical Functions. [1] Release 1.0.6 of 2013-05-06.
Smith, D. J.; Vamanamurthy, M. K. (1989). "How small is a unit ball?". Mathematics Magazine. 62 (2): 101–107. doi: 10.1080/0025570x.1989.11977419. JSTOR 2690391.
Dowker, J. S. (1996). "Robin Conditions on the Euclidean ball". Classical and Quantum Gravity. 13 (4): 585–610. arXiv: hep-th/9506042. Bibcode: 1996CQGra..13..585D. doi: 10.1088/0264-9381/13/4/003. S2CID 119438515.
Gruber, Peter M. (1982). "Isometries of the space of convex bodies contained in a Euclidean ball". Israel Journal of Mathematics. 42 (4): 277–283. doi: 10.1007/BF02761407.
Retrieved from " https://en.wikipedia.org/?title=Ball_(mathematics)&oldid=1088603110"
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Logarithms | Brilliant Math & Science Wiki
Hemang Agarwal, Brody Acquilano, tanveen dhingra, and
Bala vidyadharan
Ron Lauterbach
A logarithm is the inverse of the exponential function. Specifically, a logarithm is the power to which a number (the base) must be raised to produce a given number.
\log_2 64 = 6,
2^6 = 64.
In general, we have the following definition:
z
is the base-
x
logarithm of
y
x^z = y
. In typical notation
\log_x y = z \iff x^z = y.
Worked Examples Using Properties
First, we must know the basic structure of a logarithm
(
abbreviated
\log
for convenience
).
\log_a{b}=c
a^c=b,
a
is called the base,
c
the exponent, and
b
the argument. Also,
\log
without a base is shorthand for the common
\log
10.
Now that we know this, we can manipulate logs:
In Math In English Example
\log_a b + \log_a c = \log_a bc
When you add logs with the same base, you can merge into one log and multiply their arguments.
\log_2 5 + \log_2 6 = \log_2 30
\log_a{b}-\log_a{c} = \log_a \frac{b}{c}
The opposite of the above.
\log_2{18}-\log_2{6} = \log_2{3}
\log_a b^c = c \cdot \log_a b
When your result has an exponent, you can move it to the front of the log.
\log_3{\big(5^{-4}\big)} = -4 \cdot \log_3{5}
\log_{a^b}{c}=\frac{1}{b}\log_{a}{c}
When the base has an exponent, you can move its reciprocal to the front of the log.
\log_{\pi^2}{e}=\frac{1}{2}\log_{\pi}{e}
\log_a b = \frac{\log_c b}{\log_c a}
This is the change of base formula. You can rearrange any log by making a fraction, with the log of the argument in the numerator and the log of the base in the denominator. Any base can be chosen for the logs, but the bases must be the same for both logs.
\log_2 \pi = \frac{\log \pi}{\log 2}
log_a b=\frac{1}{\log_b a}
If you want to switch the base of the log
a
with the argument
b,
then you take the reciprocal.
\log_4 e=\frac{\log_e e}{\log_e 4}=\frac{1}{\log_e 4}
{ a }^{ \log _{ a }{ b } } = b
When a constant
a
is raised to the power
\log _{ a }{ b },
the resultant expression is
b.
{ e }^{ \log _{ e }{ 3 } }=3
\log_{a}{1}=0
Any log which has 1 as its argument will be equal to 0.
\log_{\pi \cdot e}{1}=0
Other properties can be derived from these basic ones, especially when noting that these properties are inversable.
\log_2 \left(\dfrac{32}{9}\right)^2
as much as possible.
Try to follow the steps and identify what properties were used:
\begin{aligned} \log_2 \left(\dfrac{32}{9}\right)^2 &=2 \cdot \log_2 \left(\frac{32}{9}\right)\\ &=2 \cdot ( \log_2 32 - \log_2 9)\\ &=2 \cdot \left( \log_2 2^5 -\log_2 3^2\right)\\ &=2 \cdot ( 5 \cdot \log_2 2 - 2 \cdot \log_2 3)\\ &=2 \cdot ( 5 \cdot 1 -2 \cdot \log_2 3)\\ &=10- 4 \log_2 3. \end{aligned}
\log_2 3
can't be simplified further. Line 1 used the second property, line 2 put thingies into exponential form, line 3 used the third property, and lines 4 and 5 did basic simplification.
_\square
\displaystyle{2\log_4{\sqrt{5}}+\frac{1}{2}\log_2{625}-\log_2{\frac{1}{5}}}.
Again, try to follow the steps of the solution:
\begin{aligned} 2\log_{(2^2)}{\big(5^\frac{1}{2}\big)}+\frac{1}{2}\log_2{\big(5^4\big)}-\log_2{\big(5^{-1}\big)} &=2\frac{\log 5^{\frac{1}{2}}}{\log 2^2}+\frac{1}{2}(4)(\log_2{5})+\log_2{5}\\ &=2\frac{\frac{1}{2}\log 5}{2\log 2}+2\log_2{5}+\log_2{5}\\ &=\frac{1}{2}\frac{\log 5}{\log 2}+3\log_2{5}\\ &=\frac{1}{2}\log_2{5}+3\log_2{5}\\ &=\frac{7}{2}\log_2{5}. \end{aligned}
The first line shows that it is (usually) best to convert numbers so that they are integers to a power. Note that lines 4 reverses the process of the fourth property.
_\square
1.~\log _{ a }{ a } =1
\log _{ 4 }{ 4 }.
\log _{ a }{ a }=1,
\log_{ 4 }{ 4 } =1. \ _\square
2.~\log _{ a }{ (b^c) } =c\log _{ a }{ b }
\log _{ 2 }{ 16 }.
\begin{aligned} \log _{ 2 }{ 16 } &= \log _{ 2 }{ { 2 }^{ 4 } } &&\qquad \big(16={ 2 }^{ 4 }\big)\\ &=4\log _{ 2 }{ 2 } &&\qquad \big(\log { { a }^{ b } } = b\log { a } \big)\\ &= 4. \ _\square &&\qquad (\text{by property 1}) \end{aligned}
3.~\log _{ a }{ (b \times c) } = \log _{ a }{ b }+ \log _{ a }{ c }
\log { 90 }
\log { 3 } =0.47
\begin{aligned} \log { 90 } &= \log { (9\times 10) } &&\qquad (90= 9 \times 10)\\ &=\log { 9 } + \log { 10 } &&\qquad \big(\log _{ a }{ (b\times c) } =\log _{ a }{ b } +\log _{ a }{ c } \big)\\ &=2\log { 3 } +1 &&\qquad \text{(by properties 2 and 1)}\\ &=2\times 0.47+1\\ &=0.94+1\\ &=1.94. \ _\square \end{aligned}
4.~\displaystyle{\log _{ a }{ \frac { b }{ c } } = \log _{ a }{ b } - \log _{ a }{ c }}
\log { 0.27 }
\log { 3 } =0.47
\begin{aligned} \log { 0.27 } &= \log { \frac { 27 }{ 100 } } \\ &= \log { 27 } - \log { 100 } \\ &=3\log{ 3 } - 2\\ &=1.41 - 2\\ &=-0.59. \ _\square \end{aligned}
5.
\displaystyle{\log_{ a }{ b } = \frac { \log_{ c }{ b } }{ \log_{ c }{ a } }}
\log_{ 32 }{ 2 }
\begin{aligned} \log_{ 32 }{ 2 } &=\frac { \log_{ 2 }{ 2 } }{ \log_{ 2 }{ 32 } } \\ &=\frac { 1 }{ 5\log_{ 2 }{ 2 } } \\ &=\frac { 1 }{ 5 }\\ &={ 0.2 }. \ _\square \end{aligned}
\log_3 15 + \log_3 81 - \log_3 5 ?
Using the properties of logarithms, we can rewrite the given expression as follows:
\begin{aligned} \log_3 15 + \log_3 81 - \log_3 5 &= \log_3 15 - \log_3 5 + \log_3 3^4 \\ &= \log_3 \frac{15}{5} + \log_3 3^4 \\ &= \log_3 3+ 4 \log_3 3 \\ &= 5. \ _\square \end{aligned}
What is(are) the solution(s) of the quadratic equation
\log 2x + \log(x-1) = \log\big(x^2+3\big) ?
\begin{aligned} \log 2x + \log(x-1) &= \log(x^2+3) \\ \log 2x(x-1) &= \log (x^2+3) \\ \Rightarrow 2x(x-1) &= x^2 +3 \\ x^2-2x-3 &= 0 \\ (x+1)(x-3) &= 0 \\ x &= -1, 3. \end{aligned}
Since the logarithm functions
\log(x-1)
\log 2x
are defined over positive numbers, it must be true that
x-1>0 \implies x>1
2x>0 \implies x>0.
-1
is can not be the value of
x,
implying that the value of
x
satisfying the given equation is
x=3.
_\square
What is the solution(s) of the quadratic equation
2(\log x)^2 = 7\log x - 3 ?
\begin{aligned} 2(\log x)^2 &= 7\log x - 3 \\ 2(\log x)^2 - 7\log x +3 &= 0 \\ (\log x -3)(2\log x -1) &= 0 \\ \Rightarrow \log x &= 3, \frac{1}{2} \\ x &= 1000, \sqrt{10}. \ _\square \end{aligned}
If the solutions of the quadratic equation
x^{\log_3 x\,-\,2} = 27
and
b,
\log_{a} b + \log_{b} a?
Taking logs with base 3 on both sides, we have
\begin{aligned} x^{\log_{3} x\,-\,2} &= 27 \\ \Rightarrow (\log_{3} x -2)\log_{3} x &= \log_{3} 27 \\ (\log_{3} x)^2-2\log_{3} x - 3 &= 0 \\ (\log_{3} x +1)(\log_{3} x - 3) &= 0 \\ \log_{3} x &= -1, 3. \end{aligned}
\log_{a} b + \log_{b} a
\frac{\log_{3} b}{\log_{3} a} + \frac{\log_{3} a}{\log_{3} b}
using log with base 3,
\begin{aligned} \log_{a} b + \log_{b} a &= \frac{\log_{3} b}{\log_{3} a} + \frac{\log_{3} a}{\log_{3} b} \\ &= \frac{-1}{3} + \frac{3}{-1} \\ &= -\frac{10}{3}. \ _\square \end{aligned}
If the solutions of the equation
\log_{2} x + a\log_{x} 8 = b
2
\frac{1}{8},
and
b?
\begin{aligned} \log_{2} x + a\log_{x} 8 &= b \\ \log_{2} x + a\log_{x} 2^3 &= b \\ \log_{2} x + \frac{3a}{\log_{2} x} &= b \\ (\log_{2} x)^2 -b \log_{2} x + 3a &= 0. \qquad (1) \\ \end{aligned}
Since the solutions of the equation
{(\log_{2} x)}^2 -b \log_{2} x + 3a = 0
2
\frac{1}{8} ,
2
\frac{1}{8}
(1)
\begin{aligned} (\log_{2} 2)^2 - b \log_{2} 2 + 3a &= 0 \\ \Rightarrow 1-b+3a &= 0, \qquad (2)\\ (\log_{2} \frac{1}{8} )^2 - b \log_{2} \frac{1}{8} + 3a &= 0 \\ \Rightarrow 9+3b+3a &=0. \qquad (3) \end{aligned}
Solving the simultaneous equations
(2)
(3)
a= -1
b = -2.
_\square
The following logarithms are in an arithmetic progression:
\log_{2}4 + \log_{2}{16} + \log_{2}{64} + \cdots + x = 42.
x
\log_{2}a,
a.
\log_{x} xy \times \log_{y} xy + \log_{x}(x-y) \times \log_{y} (x-y) = 0?
\begin{aligned} \log_{x} xy \times \log_{y} xy + \log_{x}(x-y) \times \log_{y} (x-y) &= 0 \\ \frac{\log xy}{\log x} \times \frac{\log xy}{\log y} + \frac{\log(x-y)}{\log x} \times \frac{\log(x-y)}{\log y} &= 0 \\ \frac{(\log xy)^2 + (\log(x-y))^2}{\log x \cdot \log y} &= 0 \\ (\log xy)^2 + (\log(x-y))^2 &= 0 \\ \Rightarrow \log xy &= 0 \text{ and } \log(x-y)= 0. \\ \end{aligned}
x
y
are both positive, this implies that
\begin{aligned} xy &= 1 \text{ and } x-y=1 \\ \Rightarrow x&= \frac{\sqrt{5} +1}{2}, y=\frac{\sqrt{5}-1}{2}. \ _\square \end{aligned}
Richter scale was developed by Charles Richter in 1935 to compare the intensities of earthquakes. The amount of energy released in an earthquake is very large, so a logarithmic scale avoids the use of large numbers.
The formula used for these calculations is
M= \log_{10}\left(\frac{I}{I_0}\right),
M
is the magnitude on the Richter scale,
I
is the intensity of the earthquake being measured, and
I_0
is the intensity of a reference earthquake.
Let's do a quick example to clarify how this works.
The 1906 San Francisco earthquake had a magnitude of 8.3 on the Richter scale. At the same time in South America there was an eathquake with magnitude 4.1 that caused only minor damage. How many times more intense was the San Francisco earthquake than the South American one?
Because the magnitude is a base-10 log, the Richter number is actually the exponent that 10 is raised to in order to calculate the intensity of the earthquake. Thus, the difference in magnitudes of the earthquakes can be calculated as follows:
M=\log_{10}\left(\frac{10^{8.3}}{10^{4.1}}\right)=4.2.
So, to answer the question, the San Francisco earthquake is more intense than the South American one by about
10^M \approx 15848.93192
times!
Note that you can just subtract 4.1 from 8.3 and get the same result. But if your math teachers are like mine, they will want you to use logarithms, and this is how it is done. The reason that subtracting the magnitudes works is because of the exponent rule for dividing exponents with the same base.
Decibel Scale:
One decibel is one tenth of one bel, named in honor of Alexander Graham Bell. The bel is rarely used without the deci- prefix, deci- meaning one tenth. The decibel scale is used to calculate the difference in intensity between two sounds:
L=10\log_{10}\left(\frac{I}{I_0}\right),
L
is the loudness of the sound measured in decibels,
I
is the intensity of the sound being measured, and
I_0
is the intensity of the sound at the threshold of hearing which is equal to zero decibels.
\text{pH}
\text{pH}
scale was invented in 1910 by Dr. Soren Sorenson, Head of Laboratory at Carlsberg Beer Company. The "H" in
\text{pH}
stands for hydrogen and the meaning of the "p" in
\text{pH}
, although disputed, is generally considered to mean the power of hydrogen. This scale is used to measure the acidity or alkalinity of water or water soluble substances including, but definitely not limited to, soil or rainwater. The
\text{pH}
scale ranges from 1 to 14, where seven is a neutral point. Values below 7 indicate acidity with 1 being the most acidic. Values above 7 indicate alkalinity with14 being the most alkaline:
\text{pH}=-\log_{10}\ce{[H+]},
\text{pH}
\text{pH}
number between
1
14
\ce{[H+]}
is the concentration of hydrogen ions.
Cite as: Logarithms. Brilliant.org. Retrieved from https://brilliant.org/wiki/logarithms/ |
It is important to note that ''R'' = ''A<sub>i</sub>''/''A<sub>p</sub>''; the ratio of impervious contributing drainage area (''A<sub>i</sub>'') to permeable pavement area (''A<sub>p</sub>'') should not exceed 2 and that the contributing drainage area should not contain pervious areas that are sources of sediment that can lead to premature clogging.
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times A_{p})+(RVC_{T}\times A_{i}\times C)-(f'\times D\times A_{p})}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}} |
Multiple comparisons problem - Wikipedia
Problem where one considers a set of inferences simultaneously based on the observed values
An example of a coincidence produced by data dredging (showing a correlation between the number of letters in a spelling bee's winning word and the number of people in the United States killed by venomous spiders). Given a large enough pool of variables for the same time period, it is possible to find a pair of graphs that show a correlation with no causation.
In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously[1] or infers a subset of parameters selected based on the observed values.[2]
The more inferences are made, the more likely erroneous inferences become. Several statistical techniques have been developed to address that problem, typically by requiring a stricter significance threshold for individual comparisons, so as to compensate for the number of inferences being made.
3.1 Multiple testing correction
4 Large-scale multiple testing
4.1 Assessing whether any alternative hypotheses are true
The problem of multiple comparisons received increased attention in the 1950s with the work of statisticians such as Tukey and Scheffé. Over the ensuing decades, many procedures were developed to address the problem. In 1996, the first international conference on multiple comparison procedures took place in Israel.[3]
Multiple comparisons arise when a statistical analysis involves multiple simultaneous statistical tests, each of which has a potential to produce a "discovery." A stated confidence level generally applies only to each test considered individually, but often it is desirable to have a confidence level for the whole family of simultaneous tests.[4] Failure to compensate for multiple comparisons can have important real-world consequences, as illustrated by the following examples:
Suppose the treatment is a new way of teaching writing to students, and the control is the standard way of teaching writing. Students in the two groups can be compared in terms of grammar, spelling, organization, content, and so on. As more attributes are compared, it becomes increasingly likely that the treatment and control groups will appear to differ on at least one attribute due to random sampling error alone.
Suppose we consider the efficacy of a drug in terms of the reduction of any one of a number of disease symptoms. As more symptoms are considered, it becomes increasingly likely that the drug will appear to be an improvement over existing drugs in terms of at least one symptom.
In both examples, as the number of comparisons increases, it becomes more likely that the groups being compared will appear to differ in terms of at least one attribute. Our confidence that a result will generalize to independent data should generally be weaker if it is observed as part of an analysis that involves multiple comparisons, rather than an analysis that involves only a single comparison.
For example, if one test is performed at the 5% level and the corresponding null hypothesis is true, there is only a 5% risk of incorrectly rejecting the null hypothesis. However, if 100 tests are each conducted at the 5% level and all corresponding null hypotheses are true, the expected number of incorrect rejections (also known as false positives or Type I errors) is 5. If the tests are statistically independent from each other, the probability of at least one incorrect rejection is approximately 99.4%.
The multiple comparisons problem also applies to confidence intervals. A single confidence interval with a 95% coverage probability level will contain the true value of the parameter in 95% of samples. However, if one considers 100 confidence intervals simultaneously, each with 95% coverage probability, the expected number of non-covering intervals is 5. If the intervals are statistically independent from each other, the probability that at least one interval does not contain the population parameter is 99.4%.
Techniques have been developed to prevent the inflation of false positive rates and non-coverage rates that occur with multiple statistical tests.
{\displaystyle m-R}
{\displaystyle m_{0}}
{\displaystyle m-m_{0}}
{\displaystyle m_{0}}
{\displaystyle m-m_{0}}
{\displaystyle R=V+S}
{\displaystyle m_{0}}
Further information: Family-wise error rate § Controlling procedures
If m independent comparisons are performed, the family-wise error rate (FWER), is given by
{\displaystyle {\bar {\alpha }}=1-\left(1-\alpha _{\{{\text{per comparison}}\}}\right)^{m}.}
Hence, unless the tests are perfectly positively dependent (i.e., identical),
{\displaystyle {\bar {\alpha }}}
increases as the number of comparisons increases. If we do not assume that the comparisons are independent, then we can still say:
{\displaystyle {\bar {\alpha }}\leq m\cdot \alpha _{\{{\text{per comparison}}\}},}
which follows from Boole's inequality. Example:
{\displaystyle 0.2649=1-(1-.05)^{6}\leq .05\times 6=0.3}
There are different ways to assure that the family-wise error rate is at most
{\displaystyle {\bar {\alpha }}}
. The most conservative method, which is free of dependence and distributional assumptions, is the Bonferroni correction
{\displaystyle \alpha _{\mathrm {\{per\ comparison\}} }={\alpha }/m}
. A marginally less conservative correction can be obtained by solving the equation for the family-wise error rate of
{\displaystyle m}
independent comparisons for
{\displaystyle \alpha _{\mathrm {\{per\ comparison\}} }}
{\displaystyle \alpha _{\{{\text{per comparison}}\}}=1-{(1-{\alpha })}^{1/m}}
, which is known as the Šidák correction. Another procedure is the Holm–Bonferroni method, which uniformly delivers more power than the simple Bonferroni correction, by testing only the lowest p-value (
{\displaystyle i=1}
) against the strictest criterion, and the higher p-values (
{\displaystyle i>1}
) against progressively less strict criteria.[5]
{\displaystyle \alpha _{\mathrm {\{per\ comparison\}} }={\alpha }/(m-i+1)}
For continuous problems, one can employ Bayesian logic to compute
{\displaystyle m}
from the prior-to-posterior volume ratio. Continuous generalizations of the Bonferroni and Šidák correction are presented in.[6]
Multiple testing correction[edit]
This section may need to be cleaned up. It has been merged from Multiple testing correction.
Multiple testing correction refers to making statistical tests more stringent in order to counteract the problem of multiple testing. The best known such adjustment is the Bonferroni correction, but other methods have been developed. Such methods are typically designed to control the familywise error rate or the false discovery rate.
Large-scale multiple testing[edit]
Traditional methods for multiple comparisons adjustments focus on correcting for modest numbers of comparisons, often in an analysis of variance. A different set of techniques have been developed for "large-scale multiple testing", in which thousands or even greater numbers of tests are performed. For example, in genomics, when using technologies such as microarrays, expression levels of tens of thousands of genes can be measured, and genotypes for millions of genetic markers can be measured. Particularly in the field of genetic association studies, there has been a serious problem with non-replication — a result being strongly statistically significant in one study but failing to be replicated in a follow-up study. Such non-replication can have many causes, but it is widely considered that failure to fully account for the consequences of making multiple comparisons is one of the causes.[7] It has been argued that advances in measurement and information technology have made it far easier to generate large datasets for exploratory analysis, often leading to the testing of large numbers of hypotheses with no prior basis for expecting many of the hypotheses to be true. In this situation, very high false positive rates are expected unless multiple comparisons adjustments are made.
For large-scale testing problems where the goal is to provide definitive results, the familywise error rate remains the most accepted parameter for ascribing significance levels to statistical tests. Alternatively, if a study is viewed as exploratory, or if significant results can be easily re-tested in an independent study, control of the false discovery rate (FDR)[8][9][10] is often preferred. The FDR, loosely defined as the expected proportion of false positives among all significant tests, allows researchers to identify a set of "candidate positives" that can be more rigorously evaluated in a follow-up study.[11]
The practice of trying many unadjusted comparisons in the hope of finding a significant one is a known problem, whether applied unintentionally or deliberately, is sometimes called "p-hacking."[12][13]
Assessing whether any alternative hypotheses are true[edit]
A normal quantile plot for a simulated set of test statistics that have been standardized to be Z-scores under the null hypothesis. The departure of the upper tail of the distribution from the expected trend along the diagonal is due to the presence of substantially more large test statistic values than would be expected if all null hypotheses were true. The red point corresponds to the fourth largest observed test statistic, which is 3.13, versus an expected value of 2.06. The blue point corresponds to the fifth smallest test statistic, which is -1.75, versus an expected value of -1.96. The graph suggests that it is unlikely that all the null hypotheses are true, and that most or all instances of a true alternative hypothesis result from deviations in the positive direction.
A basic question faced at the outset of analyzing a large set of testing results is whether there is evidence that any of the alternative hypotheses are true. One simple meta-test that can be applied when it is assumed that the tests are independent of each other is to use the Poisson distribution as a model for the number of significant results at a given level α that would be found when all null hypotheses are true.[citation needed] If the observed number of positives is substantially greater than what should be expected, this suggests that there are likely to be some true positives among the significant results. For example, if 1000 independent tests are performed, each at level α = 0.05, we expect 0.05 × 1000 = 50 significant tests to occur when all null hypotheses are true. Based on the Poisson distribution with mean 50, the probability of observing more than 61 significant tests is less than 0.05, so if more than 61 significant results are observed, it is very likely that some of them correspond to situations where the alternative hypothesis holds. A drawback of this approach is that it overstates the evidence that some of the alternative hypotheses are true when the test statistics are positively correlated, which commonly occurs in practice.[citation needed]. On the other hand, the approach remains valid even in the presence of correlation among the test statistics, as long as the Poisson distribution can be shown to provide a good approximation for the number of significant results. This scenario arises, for instance, when mining significant frequent itemsets from transactional datasets. Furthermore, a careful two stage analysis can bound the FDR at a pre-specified level.[14]
Another common approach that can be used in situations where the test statistics can be standardized to Z-scores is to make a normal quantile plot of the test statistics. If the observed quantiles are markedly more dispersed than the normal quantiles, this suggests that some of the significant results may be true positives.[citation needed]
False coverage rate (FCR)
Boole–Bonferroni bound
Harmonic mean p-value procedure
^ Miller, R.G. (1981). Simultaneous Statistical Inference 2nd Ed. Springer Verlag New York. ISBN 978-0-387-90548-8.
^ Benjamini, Y. (2010). "Simultaneous and selective inference: Current successes and future challenges". Biometrical Journal. 52 (6): 708–721. doi:10.1002/bimj.200900299. PMID 21154895. S2CID 8806192.
^ http://www.mcp-conference.org
^ Kutner, Michael; Nachtsheim, Christopher; Neter, John; Li, William (2005). Applied Linear Statistical Models. pp. 744–745. ISBN 9780072386882.
^ Aickin, M; Gensler, H (May 1996). "Adjusting for multiple testing when reporting research results: the Bonferroni vs Holm methods". Am J Public Health. 86 (5): 726–728. doi:10.2105/ajph.86.5.726. PMC 1380484. PMID 8629727.
^ Bayer, Adrian E.; Seljak, Uroš (2020). "The look-elsewhere effect from a unified Bayesian and frequentist perspective". Journal of Cosmology and Astroparticle Physics. 2020 (10): 009. arXiv:2007.13821. Bibcode:2020JCAP...10..009B. doi:10.1088/1475-7516/2020/10/009. S2CID 220830693.
^ Qu, Hui-Qi; Tien, Matthew; Polychronakos, Constantin (2010-10-01). "Statistical significance in genetic association studies". Clinical and Investigative Medicine. 33 (5): E266–E270. ISSN 0147-958X. PMC 3270946. PMID 20926032.
^ Benjamini, Yoav; Hochberg, Yosef (1995). "Controlling the false discovery rate: a practical and powerful approach to multiple testing". Journal of the Royal Statistical Society, Series B. 57 (1): 125–133. JSTOR 2346101.
^ Storey, JD; Tibshirani, Robert (2003). "Statistical significance for genome-wide studies". PNAS. 100 (16): 9440–9445. Bibcode:2003PNAS..100.9440S. doi:10.1073/pnas.1530509100. JSTOR 3144228. PMC 170937. PMID 12883005.
^ Efron, Bradley; Tibshirani, Robert; Storey, John D.; Tusher, Virginia (2001). "Empirical Bayes analysis of a microarray experiment". Journal of the American Statistical Association. 96 (456): 1151–1160. doi:10.1198/016214501753382129. JSTOR 3085878. S2CID 9076863.
^ Noble, William S. (2009-12-01). "How does multiple testing correction work?". Nature Biotechnology. 27 (12): 1135–1137. doi:10.1038/nbt1209-1135. ISSN 1087-0156. PMC 2907892. PMID 20010596.
^ Young, S. S., Karr, A. (2011). "Deming, data and observational studies" (PDF). Significance. 8 (3): 116–120. doi:10.1111/j.1740-9713.2011.00506.x. {{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Smith, G. D., Shah, E. (2002). "Data dredging, bias, or confounding". BMJ. 325 (7378): 1437–1438. doi:10.1136/bmj.325.7378.1437. PMC 1124898. PMID 12493654. {{cite journal}}: CS1 maint: multiple names: authors list (link)
^ Kirsch, A; Mitzenmacher, M; Pietracaprina, A; Pucci, G; Upfal, E; Vandin, F (June 2012). "An Efficient Rigorous Approach for Identifying Statistically Significant Frequent Itemsets". Journal of the ACM. 59 (3): 12:1–12:22. arXiv:1002.1104. doi:10.1145/2220357.2220359.
F. Betz, T. Hothorn, P. Westfall (2010), Multiple Comparisons Using R, CRC Press
S. Dudoit and M. J. van der Laan (2008), Multiple Testing Procedures with Application to Genomics, Springer
Farcomeni, A. (2008). "A Review of Modern Multiple Hypothesis Testing, with particular attention to the false discovery proportion". Statistical Methods in Medical Research. 17 (4): 347–388. doi:10.1177/0962280206079046. PMID 17698936. S2CID 12777404.
Phipson, B.; Smyth, G. K. (2010). "Permutation P-values Should Never Be Zero: Calculating Exact P-values when Permutations are Randomly Drawn". Statistical Applications in Genetics and Molecular Biology. 9: Article39. arXiv:1603.05766. doi:10.2202/1544-6115.1585. PMID 21044043. S2CID 10735784.
P. H. Westfall and S. S. Young (1993), Resampling-based Multiple Testing: Examples and Methods for p-Value Adjustment, Wiley
P. Westfall, R. Tobias, R. Wolfinger (2011) Multiple comparisons and multiple testing using SAS, 2nd edn, SAS Institute
A gallery of examples of implausible correlations sourced by data dredging
Retrieved from "https://en.wikipedia.org/w/index.php?title=Multiple_comparisons_problem&oldid=1089087863" |
About a variation of local cohomology
Fall 2020 About a variation of local cohomology
M. Azeem Khadam, Peter Schenzel
\mathfrak{𝔮}
denote an ideal of a local ring
\left(A,\mathfrak{𝔪}\right)
. For a system of elements
\underset{¯}{a}={a}_{1},\dots ,{a}_{t}
{a}_{i}\in {\mathfrak{𝔮}}^{{c}_{i}},i=1,\dots ,t
n\in ℤ
we investigate a subcomplex and a factor complex of the Čech complex
{Č}_{\underset{¯}{a}}{\otimes }_{A}M
for a finitely generated
A
M
. We start with the inspection of these cohomology modules that approximate in a certain sense the local cohomology modules
{H}_{\underset{¯}{a}}^{i}\left(M\right)
i\in ℕ
. In the case of an
\mathfrak{𝔪}
-primary ideal
\underset{¯}{a}A
we prove the Artinianness of these cohomology modules and characterize the last nonvanishing among them.
M. Azeem Khadam. Peter Schenzel. "About a variation of local cohomology." J. Commut. Algebra 12 (3) 353 - 370, Fall 2020. https://doi.org/10.1216/jca.2020.12.353
Received: 16 March 2017; Revised: 29 August 2017; Accepted: 5 September 2017; Published: Fall 2020
Keywords: Čech complex , Koszul complex , local cohomology , multiplicity
M. Azeem Khadam, Peter Schenzel "About a variation of local cohomology," Journal of Commutative Algebra, J. Commut. Algebra 12(3), 353-370, (Fall 2020) |
Then adjust A<sub>r</sub> accordingly to keep R between 0 and 1, which reduces hydraulic loading and helps avoid premature clogging.
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times R)+RVC_{T}-(f'\times D)}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
Then adjust Ar accordingly to keep R between 0 and 1, which reduces hydraulic loading and helps avoid premature clogging. |
Spin foam - Wikipedia
In physics, the topological structure of spinfoam or spin foam[1] consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity. These structures are employed in loop quantum gravity as a version of quantum foam.
1 In loop quantum gravity
1.1 Spin network
In loop quantum gravity[edit]
The covariant formulation of loop quantum gravity provides the best formulation of the dynamics of the theory of quantum gravity – a quantum field theory where the invariance under diffeomorphisms of general relativity applies. The resulting path integral represents a sum over all the possible configurations of spin foam.[how?]
Spin network[edit]
Main article: Spin network
A spin network is a one-dimensional graph, together with labels on its vertices and edges which encode aspects of a spatial geometry.
A spin network is defined as a diagram like the Feynman diagram which makes a basis of connections between the elements of a differentiable manifold for the Hilbert spaces defined over them, and for computations of amplitudes between two different hypersurfaces of the manifold. Any evolution of the spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network.[clarification needed] A spin foam is analogous to quantum history.[why?]
Spacetime[edit]
Spin networks provide a language to describe the quantum geometry of space. Spin foam does the same job for spacetime.
Spacetime can be defined as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph, a higher-dimensional complex is used. In topology this sort of space is called a 2-complex. A spin foam is a particular type of 2-complex, with labels for vertices, edges and faces. The boundary of a spin foam is a spin network, just as in the theory of manifolds, where the boundary of an n-manifold is an (n-1)-manifold.
In Loop Quantum Gravity, the present Spin Foam Theory has been inspired by the work of Ponzano–Regge model. The concept of a spin foam, although not called that at the time, was introduced in the paper "A Step Toward Pregeometry I: Ponzano–Regge Spin Networks and the Origin of Spacetime Structure in Four Dimensions" by Norman J. LaFave. In this paper, the concept of creating sandwiches of 4-geometry (and local time scale) from spin networks is described, along with the connection of these spin 4-geometry sandwiches to form paths of spin networks connecting given spin network boundaries (spin foams). Quantization of the structure leads to a generalized Feynman path integral over connected paths of spin networks between spin network boundaries. This paper goes beyond much of the later work by showing how 4-geometry is already present in the seemingly three dimensional spin networks, how local time scales occur, and how the field equations and conservation laws are generated by simple consistency requirements. The idea was reintroduced in a 1997 paper[2] and later developed into the Barrett–Crane model. The formulation that is used nowadays is commonly called EPRL after the names of the authors of a series of seminal papers,[3] but the theory has also seen fundamental contributions from the work of many others, such as Laurent Freidel (FK model) and Jerzy Lewandowski (KKL model).
The summary partition function for a spin foam model is
{\displaystyle Z:=\sum _{\Gamma }w(\Gamma )\left[\sum _{j_{f},i_{e}}\prod _{f}A_{f}(j_{f})\prod _{e}A_{e}(j_{f},i_{e})\prod _{v}A_{v}(j_{f},i_{e})\right]}
a set of 2-complexes
{\displaystyle \Gamma }
each consisting out of faces
{\displaystyle f}
{\displaystyle e}
and vertices
{\displaystyle v}
. Associated to each 2-complex
{\displaystyle \Gamma }
is a weight
{\displaystyle w(\Gamma )}
a set of irreducible representations
{\displaystyle j}
which label the faces and intertwiners
{\displaystyle i}
which label the edges.
a vertex amplitude
{\displaystyle A_{v}(j_{f},i_{e})}
and an edge amplitude
{\displaystyle A_{e}(j_{f},i_{e})}
a face amplitude
{\displaystyle A_{f}(j_{f})}
, for which we almost always have
{\displaystyle A_{f}(j_{f})=\dim(j_{f})}
^ Perez, Alejandro (2004). "[gr-qc/0409061] Introduction to Loop Quantum Gravity and Spin Foams". arXiv:gr-qc/0409061.
^ Reisenberger, Michael P.; Rovelli, Carlo (1997). ""Sum over surfaces" form of loop quantum gravity". Physical Review D. 56 (6): 3490–3508. arXiv:gr-qc/9612035. Bibcode:1997PhRvD..56.3490R. doi:10.1103/PhysRevD.56.3490. S2CID 53348775.
^ Engle, Jonathan; Livine, Etera; Pereira, Roberto; Rovelli, Carlo (2008). "LQG vertex with finite Immirzi parameter". Nuclear Physics B. 799 (1–2): 136–149. arXiv:0711.0146. Bibcode:2008NuPhB.799..136E. doi:10.1016/j.nuclphysb.2008.02.018. S2CID 118451648.
Baez, John C. (1998). "Spin foam models". Classical and Quantum Gravity. 15 (7): 1827–1858. arXiv:gr-qc/9709052. Bibcode:1998CQGra..15.1827B. doi:10.1088/0264-9381/15/7/004. S2CID 6449360.
Perez, Alejandro (2003). "Spin Foam Models for Quantum Gravity". Classical and Quantum Gravity. 20 (6): R43–R104. arXiv:gr-qc/0301113. doi:10.1088/0264-9381/20/6/202. S2CID 13891330.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Spin_foam&oldid=1071084246" |
Isotropic antenna element - MATLAB - MathWorks í•œêµ
Backbaffle the antenna element, specified as false or true. Set this property to true to baffle the response on the backside of the antenna element. In this case, the antenna response to all azimuth angles beyond ±90° from broadside (0° azimuth and 0°elevation) is zero. When the value of this property is false, the back of the antenna element is not baffled.
-20
+20
-30
30 |
How to Tell if a Proper Fraction Is Simplified: 8 Steps
How to Tell if a Proper Fraction Is Simplified
1 Identifying Reduced Fractions
2 Using Shortcuts in Context
Math problems will often ask you to reduce proper fractions (a fraction with a larger denominator than numerator) to their simplest form. Sometimes you may waste time trying to simplify fractions that cannot be further reduced. Using certain shortcuts, you can determine whether a fraction is reduced without completing any calculations.
Identifying Reduced Fractions Download Article
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Look for unit fractions. A unit fraction is one that has 1 as the numerator. Unit fractions cannot be simplified any further.[1] X Research source
{\displaystyle {\frac {1}{4}}}
{\displaystyle {\frac {1}{2}}}
{\displaystyle {\frac {1}{100}}}
{\displaystyle {\frac {1}{67}}}
are all simplified, because they have 1 as the numerator.
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Determine whether the denominator is a multiple of the numerator. One way to reduce a fraction is to divide by a greatest common factor.[2] X Research source If the denominator is a multiple of the numerator, that means each can be divided by a greatest common factor (the numerator). These types of fractions can be reduced to a unit fraction.
{\displaystyle {\frac {2}{6}}}
is not simplified, because 6 is a multiple of 2. The numerator and denominator can still be divided by a common factor of 2, simplifying the fraction to
{\displaystyle {\frac {1}{3}}}
{\displaystyle {\frac {2}{5}}}
is simplified, because 5 is not a multiple of 2.
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Determine whether the denominator is a prime number. A prime number is a number that is only divisible by itself and 1.[3] X Research source If the denominator is prime, the fraction cannot be simplified any further.[4] X Research source This is because the denominator can only be divided by itself, so whatever number appears in the numerator will not have a common factor. For more information about prime numbers, you can read Check if a Number Is Prime.
{\displaystyle {\frac {15}{23}}}
is simplified, because 23 is a prime number. The only factors of 23 are 23 and 1, so it is impossible to find a greatest common factor you can use to simplify the numerator and denominator. (If the numerator were 1, it would be a unit fraction and thus already simplified. If the numerator were 23, the fraction would equal 1.)
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Find the difference between the numerator and the denominator. If the difference is 1, then the fraction is simplified.
For example, you know that
{\displaystyle {\frac {7}{8}}}
is simplified, because
{\displaystyle 8-7=1}
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Reduce Fractions that aren't already simplified. A fraction can be simplified by looking for the largest number that can be evenly divided into the numerator and denominator, and then dividing each by that number.[5] X Expert Source David Jia
For example, if you are reducing
{\displaystyle {\frac {5}{15}}}
, divide the numerator and denominator by 5. This will give you
{\displaystyle {\frac {1}{3}}}
, which you know cannot be reduced further because it is a unit fraction.
Using Shortcuts in Context Download Article
Determine which of the following fractions are in their reduced form. Do not make any calculations:
{\displaystyle {\frac {1}{5}}}
{\displaystyle {\frac {1}{10}}}
{\displaystyle {\frac {5}{10}}}
{\displaystyle {\frac {1}{5}}}
{\displaystyle {\frac {1}{10}}}
are in their reduced, or simplified, form, because each is a unit fraction, with 1 as the numerator. You should know that
{\displaystyle {\frac {5}{10}}}
is not simplified, because 10 is a multiple of 5.
Consider the following problem. Franny says that
{\displaystyle {\frac {12}{109}}}
is a simplified fraction. Without making any calculations, how do you know she is correct?
Since 109 is a prime number, you can tell that the fraction is simplified. 109 is only divisible by 109 and 1, so it shares no common factors with 12.
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Determine which fraction is NOT simplified. Do not make any calculations:
{\displaystyle {\frac {4}{5}}}
{\displaystyle {\frac {4}{7}}}
{\displaystyle {\frac {4}{8}}}
{\displaystyle 5-4=1}
{\displaystyle {\frac {4}{5}}}
is simplified. Since 7 is a prime number, you know that
{\displaystyle {\frac {4}{7}}}
is simplified. Since 8 is a multiple of 4, you know that
{\displaystyle {\frac {4}{8}}}
is NOT simplified.
How do I simplify a proper fraction?
Look for the largest common number that evenly divides with the top and bottom numbers of the fraction.
How can I reciprocal -2.45?
First, write -2.45 as an unsimplified fraction -245/100. The reciprocal of that is -100/245. Depending on what form you want the answer in, you can either leave it as -100/245, simplify the fraction to -20/49, or long divide it to get a decimal -0.408.
Is 0.919191 a fraction
No, it's a decimal. However, it's the equivalent of the fraction 23/25.
Remember, the numerator is above the fraction bar; the denominator is below the fraction bar.
These shortcuts are for proper fractions only.
↑ http://www.mathwords.com/f/fraction_rules.htm
↑ http://www.mathsisfun.com/definitions/prime-number.html
↑ http://www.webovations.com/education/mathbook/book/numb12b.htm
Español:determinar si una fracción propia está simplificada
"It was awesome! I understood everything. Keep making more stuff like this!" |
RLC Circuits (Alternating Current) | Brilliant Math & Science Wiki
A Former Brilliant Member, Andrew Ellinor, Sravanth C., and
An RLC circuit contains different configurations of resistance, inductors, and capacitors in a circuit that is connected to an external AC current source.
Resistor in an AC circuit
Capacitor in an AC circuit
Inductor in an AC circuit
RLC Circuit (Series)
An external AC voltage source will be driven by the function
V ={ V }_{ o }\sin { (\omega t) }
V
is the instantaneous potential difference in the circuit,
{ V }_{ o }
is the maximum value in the oscillating potential difference,
\sin { (\omega t) }
is the graph that governs the oscillating nature, and
\omega
The capacitance of the capacitors is taken to be
C,
potential drop across it
{ V }_{ C },
the inductance of the inductor
L,
{ V }_{ L },
the resistance of the resistor
R,
and the potential drop across it
{ V }_{ R }.
Assume all the components in the circuit to be resistance-free except the resistor, of course.
Before dealing with RLC circuits, it is important to understand the effect of attaching R, L, and C components separately.
In a purely resistive circuit, we use the properties of a resistor to show characteristic relations for the circuit. Consider a time-dependent current
I
flowing through the circuit. Applying Kirchoff's voltage law for closed loops in the shown circuit, we get
V= { V }_{ R }
Also, we know that according to Ohm's law, potential drop across a resistor is given by
{ V }_{ R } = IR
So, using the given piece of information, we can state that
\begin{aligned} V&=IR\\ { V }_{ o }\sin { (\omega t) } &=IR\\ I&=\dfrac { { V }_{ o } }{ R } \sin { (\omega t) } \\ I&={ I }_{ o }\sin { (\omega t) }, \end{aligned}
{ I }_{ o }=\frac{{ V }_{ o }}{R}
is the maximum value of current in the circuit.
From the last derived function, we can clearly see that current in a purely resistive circuit is in phase with the applied voltage, i.e. whenever the voltage attains its maxima, so does the current, and whenever voltage attains its minima, so does the current.
The following graph shows the variation of
V
I
t
\omega
R >1
as is clear from the graph.
In a purely capacitive circuit, we use the properties of a capacitor to show characteristic relations for the circuit. If we consider a time-dependent current
I
flowing through the following circuit, then by using Kirchoff's law of voltages for closed loops, we get
V = { V }_{ C }
Now, from the theory of capacitors, we know that the potential drop across a capacitor of capacitance
C
{ V }_{ C } = \frac {Q}{C}
Q
is the charge on the capacitor at any instant. So, using the given information,
\begin{aligned} V&={ V }_{ C }\\ { V }_{ o }\sin { (\omega t) } &=\dfrac { Q }{ C } \\ Q&={ V }_{ o }C\sin { (\omega t) }. \end{aligned}
But current through the circuit is given by the rate of charge flow. So,
\begin{aligned} I&=\dfrac { d }{ dt } (Q)\\ &={ V }_{ o }C\dfrac { d }{ dt } \sin { (\omega t) } \\ &=\dfrac { { V }_{ o } }{ 1/C\omega } \cos { (\omega t) } \\ &={ I }_{ o }\sin { \left(\omega t+\dfrac { \pi }{ 2 } \right) }. \end{aligned}
From, the final expression, it is clear that current in a purely capacitive circuit leads the applied voltage by a phase difference of
\frac {\pi}{2}
, i.e. when voltage attains its maxima, current attains its minima, but being ahead of voltage.
{ I }_{ o }
is the peak current during a complete cycle. This current is given by
\frac { { V }_{ o } }{ 1/C\omega }
. Now, look at the term in the denominator. It acts as resistance in this circuit and is denoted by
{ X }_{ C }
and is known as capacitive reactance. So,
{ X }_{ C }=\frac { 1 }{ C\omega }.
The following shows the graph of
V
I
t
\omega
{ X }_{ C } > 1
as is clear from the graph. Look how current always leads the voltage.
The following proof is to calculate the average power developed in a purely capacitive circuit over one time period.
We know that total power in any circuit is given by
VI
. For the case of a capacitor, this can be written as
\begin{aligned} VI &=\big[{ V }_{ o } \sin{(\omega t)}\big] \big[{ I }_{ o } \cos{(\omega t)}\big]\\ P &= { V }_{ o } { I }_{ o } \sin{(\omega t)} \cos{(\omega t)}\\ &= \dfrac {{ V }_{ o } { I }_{ o }}{2} \sin{(2\omega t)}. \end{aligned}
Let's try to calculate the total power developed in the circuit for an entire cycle, i.e.
t=0
t=\frac {2\pi}{\omega}
\bar { P } =\dfrac { { V }_{ o }{ I }_{ o } }{ 2 } <\sin { (2\omega t)> }.
But we know that the average of sine function over an entire cycle is zero. Therefore
\\ \bar { P } = 0.
It must be remembered that although capacitive circuits do dissipate power in an instant, the average power is what's zero for the circuit.
In a purely inductive circuit, we use the properties of an inductor to show characteristic relations for the circuit. Considering the flow of a time-dependent current
I
through the circuit shown below enables us to use the concept of induced emf due to the changing current in the inductor. So, according to Kirchoff's voltage law for closed loops, we get
V={ V }_{ L }.
Also, from the theory of electromagnetism, we say that potential across the inductor is given by
{ V }_{ L } = L\dfrac {d}{dt} I.
So, using the piece of information we have, we get
\begin{aligned} V &=L\frac{d}{dt}(I) \\ \\ V_o \sin(\omega t) &=L \frac{d}{dt}(I) \\ \\ dI&=\dfrac { { V }_{ o } }{ L } \sin { (\omega t) }\, dt\\ \\ \displaystyle\int { dI } &=\displaystyle\int { \dfrac { { V }_{ o } }{ L } \sin { (\omega t) }\, dt } \\ \\ I&=\dfrac { { V }_{ o } }{ L\omega } \big(-\cos { (\omega t)\big) } +K\\ \\ &=\dfrac { { V }_{ o } }{ L\omega } \sin { \left(\omega t-\dfrac { \pi }{ 2 } \right) } +K. \end{aligned}
However, since the voltage oscillates between a maximum and a minimum value following a sine graph, it is logical to assume the same for the current. Hence, we get the integration constant
K= 0
I={ I }_{ o }\sin { \big(\omega t-\frac { \pi }{ 2 } \big) }
{ I }_{ o }
is the peak current in the circuit and is given by
{ I }_{ o }=\frac { { V }_{ o } }{ L\omega }.
Looking at the term in the denominator, we infer that it acts as the resistance for this circuit and is known as Inductive reactance,
{ X }_{ L }
{ X }_{ L } = L\omega
Finally, we can also deduce from the formula of time-dependent current that it lags the applied potential difference by a phase of
\frac {\pi}{2}
, which can be clearly seen from the graph shown below.
The proof for power developed in a purely inductive circuit is extremely identical to the power developed in the case of a purely capacitive circuit. This is because, in both these cases, the phase difference between voltage and current is the same, i.e.
\frac {\pi}{2},
which results in the average power of the circuit to come out to be 0.
So, after learning about the effects of attaching various components individually, we will consider the basic set-up of an RLC circuit consisting of a resistor, an inductor, and a capacitor combined in series to an external current supply which is alternating in nature, as shown in the diagram.
The components are in series, so as a result the current passing through them will be the same. Let this current be
I
which varies with time. Now, using the values of potential drop across every component, and using Kirchhoff's voltage law for closed loops, we can clearly make out that:
\begin{aligned} V &=IR+L\frac{dI}{dt} +\frac{Q}{C} \\ V_o \sin(\omega t) &=\frac{Q}{C} + R\frac { dQ }{ dt } +L\frac { { d }^{ 2 }Q }{ d{ t }^{ 2 } }. \end{aligned}
Q
which can be solved using standard methods, but phasor diagrams can be more illuminating than a solution to the differential equation.
Although currents and voltages are scalar in nature, yet sometimes they are assumed to have a direction which is related to their phase differences with respect to each other. Phasor diagrams are such diagrams that represent these scalar quantities with a direction and help us compute our results better.
DERIVATION USING PHASOR DIAGRAMS
Since the current through all the components is same, we construct a ray
OQ
that shows the direction of the current, which will be the same for all the components. Now, from the derivations of purely resistive, inductive, and capacitive circuits, we have seen that voltage and current have a particular phase difference between each other for every component. So, considering these facts in mind, we complete the diagram by producing a ray
OP
OQ
that represents the voltage across the resistor. Similarly, we construct a ray
OA
+ \frac {\pi}{2}
with respect to current. This represents the voltage across the inductor. Finally, construct a ray
OB
- \frac {\pi}{2}
OQ
. This shows the voltage across the capacitor. We get something like this:
Looking at the diagram, we see three vectors
OA, OB,
OQ
which represent voltages across single components. Using basic vector algebra, and considering potential drop across the capacitor to be more than that for the inductor, we see that the net voltage is along the diagonal of the so formed parallelogram
COPD
, and is given by the vector
OD
. Also, let the angle between
OD
OQ
\phi
. As you can see,
\phi
here represents the overall phase difference between voltage and current.
So, we infer that
\begin{aligned} {V_\text{net}}^2 &={ { V }_{ R } }^{ 2 }+{ ({ V }_{ C }-{ V }_{ L }) }^{ 2 }\\ &={ (IR) }^{ 2 }+{ { (I }{ { X }_{ C } }-I{ X }_{ L }) }^{ 2 }\\ &={ I }^{ 2 }\left[ { R }^{ 2 }+{ ({ X }_{ C }-{ X }_{ L }) }^{ 2 }\right] \\ \\ \frac { { V }_\text{ net } }{ I } &=\sqrt { { R }^{ 2 }+{ ({ X }_{ C }-{ X }_{ L }) }^{ 2 } } \\ &=Z. \end{aligned}
Z
is known as the impedance of the circuit and plays the role of net resistance. Alternatively,
Z
Z=\sqrt { { R }^{ 2 }+{ \left(\frac { 1 }{ C\omega } -L\omega \right) }^{ 2 } }.
Also, from the diagram, we can see that the phase difference
\phi
is related to voltage as
\begin{aligned} \tan { \phi } &=\frac { { V }_{ C }-{ V }_{ L } }{ V_{ R } }\\ &=\frac { { X }_{ C }-{ X }_{ L } }{ R } \\ \phi &=\arctan { \frac { { X }_{ C }-{ X }_{ L } }{ R } }. \end{aligned}
So, all in all, in a series RLC circuit, if the applied voltage is given by
V = { V }_{ o }\sin { (\omega t) }
, then the current through the circuit is represented by
I={ I }_{ o }\sin { (\omega t+\phi ) },
{ I }_{ o }=\dfrac { { V }_{ o } }{ \sqrt { { R }^{ 2 }+{ ({ X }_{ C }-{ X }_{ L }) }^{ 2 } } }\quad\text{and}\quad \phi =\arctan { \dfrac { { X }_{ C }-{ X }_{ L } }{ R } }.
The following is the proof for the power developed in a series LCR circuit:
The power developed in the circuit at time
t
\begin{aligned} P &=VI\\ \\ &=\big[{ V }_{ o }\sin { (\omega t) } \big] \big[{ I }_{ o }\sin { (\omega t+\phi )\big] } \\ \\ &=\frac { { V }_{ o }{ I }_{ o } }{ 2 } 2\sin { (\omega t) } \sin { (\omega t+\phi ) } \\ \\ &=\frac { { V }_{ o } }{ \sqrt { 2 } } \frac { { I }_{ o } }{ \sqrt { 2 } } \big[ \cos { \phi } -\cos { (2\omega t+\phi )\big] }. \end{aligned}
Now, putting in
\frac { { V }_{ o } }{ \sqrt { 2 } } = { V }_\text{rms}
\frac { { I }_{ o } }{ \sqrt { 2 } } = { I }_\text{rms},
we end up with
P={ V }_\text{rms}{ I }_\text{rms}\big[ \cos { \phi } -\cos { (2\omega t+\phi ) } \big].
Hence, the average power
\bar { P }
\bar { P } ={ V }_\text{rms}{ I }_\text{rms}\big[ <\cos { \phi } >-<\cos { (2\omega t+\phi ) } >\big].
\cos{\phi}
is independent of time, its average is
\cos{\phi}
only. Also, the average of
\cos { (2\omega t+\phi ) }
across one cycle is 0. So, we get the final formula for average power to be
\bar { P } ={ V }_\text{rms}{ I }_\text{rms}\cos { \phi } ={ V }_\text{rms}{ I }_\text{rms}\frac { R }{ Z }.
So, it can be seen that unlike a simple DC circuit, the power developed in an LCR circuit is also dependent on the value of
\cos {\phi}
which is defined as the power factor of the circuit.
Resonance in case of an LCR circuit refers to the condition when the potential drop across the inductor is the same as the potential drop across the conductor, or
{ V }_{ L } = { V }_{ C}.
The basic condition for resonance can be easily derived. Since
{ V }_{ L } = { V }_{ C},
\begin{aligned} I{ X }_{ L }&=I{ X }_{ C }\\ L{ \omega }_{ r }&=\frac { 1 }{ C{ \omega }_{ r } } \\ { \omega }_{ r }&=\frac { 1 }{ \sqrt { LC } }. \end{aligned}
{ \omega }_{ r }
represents the resonant frequency of the circuit, or the frequency of the applied voltage that causes a condition of resonance. Since
{ X }_{ L }={ X }_{ C },
from the formula for impedance of the circuit, we can easily derive the relation that
Z=R;
in other words, the impedance of a circuit in case of resonance is minimum, or conversely, the current in the circuit is maximum.
This property of resonant circuits is used amazingly in television and radio sets. Quite basically, such a device can be viewed to consist of an LCR circuit in it. When it receives an electromagnetic signal of some frequency, this signal is converted into an electrical signal which tends to be the AC source for the circuit. Now, for every channel, there's a particular configuration of inductor and capacitor used. So, if the received frequency matches with resonant frequency for that particular channel, then the current in that circuit goes to maximum and the signal is said to be accepted.
On the other hand, if it does not match the resonant frequency, then the current stays less than the maximum current and the signal is said to be rejected or denied.
Power in a resonating LCR circuit
We know that average power of any LCR circuit can be given by
\bar { P } ={ V }_\text{rms}{ I }_\text{rms}\cos { \phi }.
But for a circuit in which the inductive and capacitive reactance are equal, it can be easily inferred that
\phi=0
\cos { \phi }=1
. Plugging this value in the formula, we get
\bar { { P }_{ r } } ={ V }_\text{rms}{ I }_\text{rms}.
Z
attains its minima, the current in the circuit or
{ I }_\text{rms}
attains its maxima. Hence, power developed in case of a resonating circuit is maximum.
You are ready for your next vacation to a land of peacefulness. The only thing between you and your destination is a shameless metal detector that requires you to walk through it without making it go beep!
You took all the articles off but forgot your watch, and as soon as you walked in, the alarms went off and you are required to walk through again, removing your watch this time. You're good to go and you enter your airplane where you sit down and work out the incident at the airport.
You do know that the metal detector is just a simple application of an LCR circuit. The next piece of info you know is that the sound alarm that blared off was most likely a basic one which requires an RMS current of
10\text{ A}
to go off.
Consider the LCR circuit to be made up of a sound source of resistance
5\, \Omega,
an ideal inductor of
2\text{ mH},
and an ideal capacitor of
20\text{ pF}.
All you need to do is to work out the angular frequency of the current that your watch generated, i.e.
\omega
and the RMS value of the alternating voltage for the circuit, i.e.
{V}_\text{rms}.
\frac {\omega} {{10}^{7}} + {V}_\text{rms}.
Cite as: RLC Circuits (Alternating Current). Brilliant.org. Retrieved from https://brilliant.org/wiki/rlc-circuits-alternating-current/ |
On the Apparent Mass of the Ions - Wikisource, the free online library
Translation:On the Apparent Mass of the Ions
On the Apparent Mass of the Ions (1900)
by Hendrik Lorentz, translated from German by Wikisource
In German: Über die scheinbare Masse der Ionen, Physikalische Zeitschrift. 2, 1900/1, pp. 78-80
756947On the Apparent Mass of the IonsHendrik Lorentz1900
H. A. Lorentz (Leiden) On the apparent mass of the ions.
It is known that by observations of cathode rays we were able to derive the ratio
{\displaystyle {\tfrac {e}{m}}}
, i.e. the ratio between the charge of an ion
{\displaystyle e}
and its mass
{\displaystyle m}
. The question arises, what is meant by that mass. In any case we must attribute an apparent mass to the ion, as it generates a certain energy in the ether by virtue of its motion. This apparent mass will be denoted by
{\displaystyle m_{0}}
. It is possible that the ion also possesses a real mass in the ordinary sense of the word; in this case,
{\displaystyle m_{0}<m}
. If this is not the case, then
{\displaystyle m_{0}=m}
So we have the inequality
{\displaystyle {\frac {e}{m_{0}}}>{\frac {e}{m}}{,}}
when there still is a real mass besides the apparent mass; otherwise
{\displaystyle {\frac {e}{m_{0}}}={\frac {e}{m}}.}
So we want to write
{\displaystyle {\frac {e}{m_{0}}}\geqq {\frac {e}{m}}{,}}
{\displaystyle {\tfrac {e}{m}}=10^{7}}
{\displaystyle m_{0}={\frac {8}{3}}\pi R\sigma e{,}}
if we conceive the ion as a sphere,
{\displaystyle R}
is the radius of this sphere, and
{\displaystyle \sigma }
means the surface density of the charge.
This formula allows for an interesting conclusion on the radius of the ions. If, namely, we substitute for
{\displaystyle m_{0}}
the now specified value into the inequality, we obtain an inequality for the radius. We have
{\displaystyle 4\pi R^{2}\cdot \sigma =e{,}}
{\displaystyle m_{0}={\frac {8}{3}}\pi R\sigma e={\frac {8}{3}}\pi Re\cdot {\frac {e}{4\pi R^{2}}}={\frac {2e^{2}}{3R}}}
{\displaystyle {\frac {e}{m_{0}}}={\frac {3R}{2e}}{,}}
{\displaystyle {\frac {3R}{2e}}\geq 10^{7}}
{\displaystyle R>10^{7}\cdot {\frac {2}{3}}e.}
The magnitud{\displaystyle e}
is unfortunately not known. If we take the charge of an ion in a cathode ray to be as great as in an electrolytic hydrogen, and presuppose the size of a hydrogen molecule, we obtain for
{\displaystyle R}
a magnitude of order
{\displaystyle 10^{-12}}
cm, that is certainly not an arbitrarily small magnitude, but a lower limit.
The question of whether or not a real mass exists besides the apparent mass of an ion, is extremely important; because by that we touch the question of the relation of ponderable matter with ether and electricity. I am far away to announce a decision, but I would like to cite but a few questions whose resolution can potentially bring us further in that question.
The first question is whether an ion rotates in a magnetic field. Actually, we should expect that. Since if an ion is present, and if a magnetic field is caused, then a rotation arises, as it can easily be derived from the formation of induced currents. Of course this is also the case when the ion flies into an already existing magnetic field. The velocity of rotation will depend on the magnitude of the mass; if only apparent mass is present, and even a corresponding moment of inertia, then the rotation velocity has a certain value. If, however, a real moment of inertia is added, the rotation is slowing down. Unfortunately I can not find any phenomenon, from which we could conclude anything about this rotation.
A second means by which we maybe could decide the question of the relationship between the apparent and real mass is the following:
The value for the apparent mass was given above only in first approximation. If the velocity is such that it is comparable to the velocity of light, then additional magnitudes will be added. For a straight path of the ion we can calculate the intensity of the field and the size of the energy and deduce from that the mass factor. In general, the trajectory will be curvilinear through the influence of the magnetic field, e.g. circular; then the calculation of the mass factor will become more complicated, but it can be carried out. If we denote by
{\displaystyle m_{0}}
the expression above and
{\displaystyle q}
is defined as the ratio of the ion velocity to that of light, it follows in second approximation for the apparent mass of the ion in linear motion:
{\displaystyle m_{0}\left(1+{\frac {6}{5}}q^{2}\right){,}}
while in a circular motion the term with
{\displaystyle q^{2}}
yields a different coefficient.
These terms of the second order could now perhaps become observable, because the velocity of cathode rays increases up to a third of that of light, hence
{\displaystyle q={\tfrac {1}{3}}}
{\displaystyle q^{2}={\tfrac {1}{9}}}
. To come to a decision, we could think of experiments as they were done by Lenard, to examine the influence of electric forces on the velocity of cathode rays. He has shown that the magnetic deflectability of the cathode rays, which is of course the smaller, the greater the speed, will change when the rays can pass through the space between two charged capacitor plates in the direction of the electric force lines.
We could measure the magnetic deflection in the case of an uncharged capacitor, then in the case of charge in one direction and then for the other direction. Thus we would obtain three different values of deflectability, between which a simple relation should exist, if the terms of second order could be neglected. If we measure each time the magnetic field-force required for a particular deflection, then the squares of these three field forces should form an arithmetic row. A deviation from this relationship would indicate that the terms with
{\displaystyle q^{2}}
shall not be neglected, and that therefore in any case the apparent mass is noticeable. Detailed specifications could decide concerning the ratio between the real and the apparent mass, and concerning the question whether a real mass exists. It turns out that by Lenard's experiments we were near to decide about the existence of terms of the second order.
(Self-lecture of the lecturer.)
Discussion. (Reviewed by the participants.)
W. Wien. I was recently concerned with similar issues, and would like to stress that Lenard has observed cathode rays at low velocities, triggered under the influence of ultraviolet light. There, he found a small value for the ratio of mass to charge, namely the decrease lies in the sense which is required by the theory.
I have tried to transcend over Lorentz's position, by posing me the question, whether it would suffice when we only consider the apparent mass and omit the inertial mass, and replace it with the electromagnetically defined apparent mass to present the mechanical and electromagnetic phenomena in an uniform way. Because the magnetic and mechanical phenomena are only connected by the energy principle so far. I've tried to pose the question as to whether we could try by Maxwell's theory, to involve mechanics as well. The possibility of an electromagnetic explanation of mechanics was given, after Lorentz has developed a conception of the law of gravity, according to which it would be very similar to electrostatic forces. We would have to think of matter as only composed of very small positive and negative charges, which are within a certain distance from each other. By this condition, the ponderable mass is not constant but depends on the velocity, and namely we obtain terms, depending on even powers of the ratio of velocity to the velocity of light. The numerical factor by which the second term is multiplied, depends on the curvature of the trajectory, but also on the shape of the electric charge. Depending on which different way we choose the form of electrified molecules, we come to other numerical factors. Concerning the ordinary motions on earth, it vanishes because the velocity is very small. Concerning planetary motions we probably can achieve something; because we reach velocities at which we have to consider the terms of second order. On the assumption of a specific type of charge, leading to the simplest electromagnetic field, these terms become relevant in a way, so that the accelerations of two bodies by gravitation are the same up to a slightly different numerical factor, as if the bodies attract each other with constant mass according to Weber's laws. The electromagnetically defined mass comes into play, as if not Newton's, but Weber's law would apply.
Lorentz. In essence, we agree; but Wien already wants to go further than I do. Anyway, it seemed of interest to me to look for means, by which we can come to a decision on the issue discussed. One more thing I would like to add: I made the assumption that the sphere, which forms an ion, is rigid. But perhaps one might think that the sphere would be transformed into an ellipsoid when in motion. This has some similarity with the diversity, that was pointed out by Wien.
Voigt. I would like to pose the question to the lecturer, concerning the reflection of cathode rays; should a rotating ion not be reflected differently, as a non-rotating one?
Lorentz. Certainly, if one imagines that the reflection happens on a surface. But if you look at the reflection, which is more likely to me, as caused by forces that occur at some distance from the surface of the ion, then those surely act on the center, and then the influence of rotation vanishes.
Warburg. What does the theory say about the velocity of the ions during reflection? Does it remain the same?
Lorentz. As far as I know, yes. I have not elaborated on this.
Warburg. Merritt has found that the velocity of reflection has not changed. But the experiments of Cady on the energy of cathode rays are in contradiction to this, so I've thought that the experiments of Merritt may not be completely correct, and maybe we could obtain a velocity change. I wanted to ask if the theory says something in this respect.
Lorentz. I can not say this right now.
Retrieved from "https://en.wikisource.org/w/index.php?title=Translation:On_the_Apparent_Mass_of_the_Ions&oldid=10821715" |
Trigonometry - Simple English Wikipedia, the free encyclopedia
Trigonometry (from the Greek trigonon = three angles and metron = measure) is a part of elementary mathematics dealing with angles, triangles and trigonometric functions such as sine (abbreviated sin), cosine (abbreviated cos) and tangent (abbreviated tan).[1][2] It has some connection to geometry, although there is disagreement on exactly what that connection is; for some, trigonometry is just a section of geometry.
4 Trigonometry Laws
A standard right triangle. C is the right angle in this picture
Trigonometry uses a large number of specific words to describe parts of a triangle. Some of the definitions in trigonometry are:
Right-angled triangle - A right-angled triangle is a triangle that has an angle equal to 90 degrees. (A triangle cannot have more than one right angle) The standard trigonometric ratios can only be used on right-angled triangles.
Hypotenuse - The hypotenuse of a triangle is the longest side, and the side that is opposite the right angle. For example, for the triangle on the right, the hypotenuse is side c.
Opposite of an angle - The opposite side of an angle is the side that does not intersect with the vertex of the angle. For example, side a is the opposite of angle A in the triangle to the right.
Adjacent of an angle - The adjacent side of an angle is the side that intersects the vertex of the angle but is not the hypotenuse. For example, side b is adjacent to angle A in the triangle to the right.
There are three main trigonometric ratios for right triangles, and three reciprocals of those ratios, making up a total of 6 ratios. They are:[3]
Sine (sin) - The sine of an angle is equal to the
{\displaystyle \textstyle {\text{Opposite}} \over {\text{Hypotenuse}}}
Cosine (cos) - The cosine of an angle is equal to the
{\displaystyle \textstyle {\text{Adjacent}} \over {\text{Hypotenuse}}}
Tangent (tan) - The tangent of an angle is equal to the
{\displaystyle \textstyle {\text{Opposite}} \over {\text{Adjacent}}}
The reciprocals of these ratios are:
Cosecant (cosec) - The cosecant of an angle is equal to the
{\displaystyle \textstyle {\text{Hypotenuse}} \over {\text{Opposite}}}
{\displaystyle \csc \theta ={1 \over \sin \theta }}
Secant (sec) - The secant of an angle is equal to the
{\displaystyle {{\text{Hypotenuse}} \over {\text{Adjacent}}}}
{\displaystyle \sec \theta ={1 \over \cos \theta }}
Cotangent (cot) - The cotangent of an angle is equal to the
{\displaystyle {{\text{Adjacent}} \over {\text{Opposite}}}}
{\displaystyle \cot \theta ={1 \over \tan \theta }}
Students often use a mnemonic to remember this relationship. The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, such as SOH-CAH-TOA:
Using trigonometryEdit
With the sines and cosines, one can answer virtually all questions about triangles. This is called "solving" the triangle. One can work out the remaining angles and sides of any triangle, as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a combination of triangles.
Trigonometry is also vital in surveying, in vector analysis, and in the study of periodic functions. It developed from a need to compute angles and distances in fields such as astronomy, mapmaking, surveying, and artillery range finding.[2]
There is also such a thing as spherical trigonometry, which deals with spherical geometry. This is used for calculations in astronomy, geodesy and navigation.
Trigonometry LawsEdit
Law of SinesEdit
{\displaystyle {{\text{a}} \over {\text{Sin A}}}={{\text{b}} \over {\text{Sin B}}}={{\text{c}} \over {\text{Sin C}}}}
Law of CosinesEdit
{\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos(A)}
Law of TangentsEdit
{\displaystyle {\frac {a-b}{a+b}}={\frac {\tan({\frac {1}{2}}(A-B))}{\tan({\frac {1}{2}}(A+B))}}}
↑ 2.0 2.1 "trigonometry | Definition, Formulas, Ratios, & Identities". Encyclopedia Britannica. Retrieved 2020-09-24.
↑ Menz, Petra; Mulberry, Nicola (July 13, 2020). "Calculus Early Transcendentals: Differential & Multi-Variable Calculus for Social Sciences (1.3 Trigonometry)". sfu.ca. Retrieved September 24, 2020. {{cite web}}: CS1 maint: url-status (link)
Basic Trigonometry course in Khan Academy
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Trigonometry&oldid=7998543" |
This article is about the normal to 3D surfaces. For the normal to 3D curves, see Frenet–Serret formulas.
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a curvature vector); its algebraic sign may indicate sides (interior or exterior).
A polygon and its two normal vectors
A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point.
In three dimensions, a surface normal, or simply normal, to a surface at point
{\displaystyle P}
is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles).
The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point
{\displaystyle P}
is the set of vectors which are orthogonal to the tangent space at
{\displaystyle P.}
Normal vectors are of special interest in the case of smooth curves and smooth surfaces.
The normal is often used in 3D computer graphics (notice the singular, as only one normal will be defined) to determine a surface's orientation toward a light source for flat shading, or the orientation of each of the surface's corners (vertices) to mimic a curved surface with Phong shading.
The normal distance of a point Q to a curve or to a surface is the Euclidean distance between Q and its perpendicular projection on the object (at the point P on the object where the normal contains Q). The normal distance is a type of perpendicular distance generalizing the distance from a point to a line and the distance from a point to a plane. It can be used for curve fitting and for defining offset surfaces.
1 Normal to surfaces in 3D space
1.1 Calculating a surface normal
1.2 Choice of normal
2 Hypersurfaces in n-dimensional space
3 Varieties defined by implicit equations in n-dimensional space
5 Normal in geometric optics
Normal to surfaces in 3D spaceEdit
A curved surface showing the unit normal vectors (blue arrows) to the surface
Calculating a surface normalEdit
For a convex polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon.
For a plane given by the equation
{\displaystyle ax+by+cz+d=0,}
{\displaystyle \mathbf {n} =(a,b,c)}
is a normal.
For a plane whose equation is given in parametric form
{\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,}
{\displaystyle \mathbf {r} _{0}}
is a point on the plane and
{\displaystyle \mathbf {p} ,\mathbf {q} }
are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both
{\displaystyle \mathbf {p} }
{\displaystyle \mathbf {q} ,}
which can be found as the cross product
{\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .}
If a (possibly non-flat) surface
{\displaystyle S}
in 3-space
{\displaystyle \mathbb {R} ^{3}}
is parameterized by a system of curvilinear coordinates
{\displaystyle \mathbf {r} (s,t)=(x(s,t),y(s,t),z(s,t)),}
{\displaystyle s}
{\displaystyle t}
real variables, then a normal to S is by definition a normal to a tangent plane, given by the cross product of the partial derivatives
{\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.}
If a surface
{\displaystyle S}
is given implicitly as the set of points
{\displaystyle (x,y,z)}
{\displaystyle F(x,y,z)=0,}
then a normal at a point
{\displaystyle (x,y,z)}
on the surface is given by the gradient
{\displaystyle \mathbf {n} =\nabla F(x,y,z).}
since the gradient at any point is perpendicular to the level set
{\displaystyle S.}
For a surface
{\displaystyle S}
{\displaystyle \mathbb {R} ^{3}}
given as the graph of a function
{\displaystyle z=f(x,y),}
an upward-pointing normal can be found either from the parametrization
{\displaystyle \mathbf {r} (x,y)=(x,y,f(x,y)),}
{\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial x}}\times {\frac {\partial \mathbf {r} }{\partial y}}=\left(1,0,{\tfrac {\partial f}{\partial x}}\right)\times \left(0,1,{\tfrac {\partial f}{\partial y}}\right)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right);}
or more simply from its implicit form
{\displaystyle F(x,y,z)=z-f(x,y)=0,}
{\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).}
Since a surface does not have a tangent plane at a singular point, it has no well-defined normal at that point: for example, the vertex of a cone. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.
Choice of normalEdit
A vector field of normals to a surface
The normal to a (hyper)surface is usually scaled to have unit length, but it does not have a unique direction, since its opposite is also a unit normal. For a surface which is the topological boundary of a set in three dimensions, one can distinguish between the inward-pointing normal and outer-pointing normal. For an oriented surface, the normal is usually determined by the right-hand rule or its analog in higher dimensions.
If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.
Transforming normalsEdit
in this section we only use the upper
{\displaystyle 3\times 3}
matrix, as translation is irrelevant to the calculation
When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals.
Specifically, given a 3×3 transformation matrix
{\displaystyle \mathbf {M} ,}
we can determine the matrix
{\displaystyle \mathbf {W} }
that transforms a vector
{\displaystyle \mathbf {n} }
perpendicular to the tangent plane
{\displaystyle \mathbf {t} }
into a vector
{\displaystyle \mathbf {n} ^{\prime }}
perpendicular to the transformed tangent plane
{\displaystyle \mathbf {Mt} ,}
Write n′ as
{\displaystyle \mathbf {Wn} .}
We must find
{\displaystyle \mathbf {W} .}
{\displaystyle {\begin{alignedat}{5}W\mathbb {n} {\text{ is perpendicular to }}M\mathbb {t} \quad \,&{\text{ if and only if }}\quad 0=(W\mathbb {n} )\cdot (M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=(W\mathbb {n} )^{\mathrm {T} }(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\left(\mathbb {n} ^{\mathrm {T} }W^{\mathrm {T} }\right)(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\mathbb {n} ^{\mathrm {T} }\left(W^{\mathrm {T} }M\right)\mathbb {t} \\\end{alignedat}}}
{\displaystyle \mathbf {W} }
{\displaystyle W^{\mathrm {T} }M=I,}
{\displaystyle W=(M^{-1})^{\mathrm {T} },}
will satisfy the above equation, giving a
{\displaystyle W\mathbb {n} }
{\displaystyle M\mathbb {t} ,}
{\displaystyle \mathbf {n} ^{\prime }}
{\displaystyle \mathbf {t} ^{\prime },}
Therefore, one should use the inverse transpose of the linear transformation when transforming surface normals. The inverse transpose is equal to the original matrix if the matrix is orthonormal, that is, purely rotational with no scaling or shearing.
Hypersurfaces in n-dimensional spaceEdit
{\displaystyle (n-1)}
-dimensional hyperplane i{\displaystyle n}
{\displaystyle \mathbb {R} ^{n}}
given by its parametric representation
{\displaystyle \mathbf {r} \left(t_{1},\ldots ,t_{n-1}\right)=\mathbf {p} _{0}+t_{1}\mathbf {p} _{1}+\cdots +t_{n-1}\mathbf {p} _{n-1},}
{\displaystyle \mathbf {p} _{0}}
is a point on the hyperplane and
{\displaystyle \mathbf {p} _{i}}
{\displaystyle i=1,\ldots ,n-1}
are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector
{\displaystyle \mathbf {n} }
in the null space of the matrix
{\displaystyle P={\begin{bmatrix}\mathbf {p} _{1}&\cdots &\mathbf {p} _{n-1}\end{bmatrix}},}
{\displaystyle P\mathbf {n} =\mathbf {0} .}
That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. Alternatively, if the hyperplane is defined as the solution set of a single linear equation
{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=c,}
{\displaystyle \mathbb {n} =\left(a_{1},\ldots ,a_{n}\right)}
The definition of a normal to a surface in three-dimensional space can be extended to
{\displaystyle (n-1)}
-dimensional hypersurfaces in
{\displaystyle \mathbb {R} ^{n}.}
A hypersurface may be locally defined implicitly as the set of points
{\displaystyle (x_{1},x_{2},\ldots ,x_{n})}
satisfying an equation
{\displaystyle F(x_{1},x_{2},\ldots ,x_{n})=0,}
{\displaystyle F}
is a given scalar function. If
{\displaystyle F}
is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not zero. At these points a normal vector is given by the gradient:
{\displaystyle \mathbb {n} =\nabla F\left(x_{1},x_{2},\ldots ,x_{n}\right)=\left({\tfrac {\partial F}{\partial x_{1}}},{\tfrac {\partial F}{\partial x_{2}}},\ldots ,{\tfrac {\partial F}{\partial x_{n}}}\right)\,.}
The normal line is the one-dimensional subspace with basis
{\displaystyle \{\mathbf {n} \}.}
Varieties defined by implicit equations in n-dimensional spaceEdit
A differential variety defined by implicit equations in the
{\displaystyle n}
{\displaystyle \mathbb {R} ^{n}}
is the set of the common zeros of a finite set of differentiable functions i{\displaystyle n}
{\displaystyle f_{1}\left(x_{1},\ldots ,x_{n}\right),\ldots ,f_{k}\left(x_{1},\ldots ,x_{n}\right).}
The Jacobian matrix of the variety is the
{\displaystyle k\times n}
{\displaystyle i}
-th row is the gradient of
{\displaystyle f_{i}.}
By the implicit function theorem, the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank
{\displaystyle k.}
At such a point
{\displaystyle P,}
the normal vector space is the vector space generated by the values at
{\displaystyle P}
of the gradient vectors of the
{\displaystyle f_{i}.}
In other words, a variety is defined as the intersection of
{\displaystyle k}
hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point.
The normal (affine) space at a point
{\displaystyle P}
of the variety is the affine subspace passing through
{\displaystyle P}
and generated by the normal vector space at
{\displaystyle P.}
These definitions may be extended verbatim to the points where the variety is not a manifold.
Let V be the variety defined in the 3-dimensional space by the equations
{\displaystyle x\,y=0,\quad z=0.}
This variety is the union of the
{\displaystyle x}
{\displaystyle y}
{\displaystyle (a,0,0),}
{\displaystyle a\neq 0,}
the rows of the Jacobian matrix are
{\displaystyle (0,0,1)}
{\displaystyle (0,a,0).}
Thus the normal affine space is the plane of equation
{\displaystyle x=a.}
{\displaystyle b\neq 0,}
the normal plane at
{\displaystyle (0,b,0)}
is the plane of equation
{\displaystyle y=b.}
{\displaystyle (0,0,0)}
{\displaystyle (0,0,1)}
{\displaystyle (0,0,0).}
Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the
{\displaystyle z}
Surface normals are useful in defining surface integrals of vector fields.
Surface normals are commonly used in 3D computer graphics for lighting calculations (see Lambert's cosine law), often adjusted by normal mapping.
Render layers containing surface normal information may be used in Digital compositing to change the apparent lighting of rendered elements.[citation needed]
In computer vision, the shapes of 3D objects are estimated from surface normals using photometric stereo.[1]
Normal in geometric opticsEdit
The normal ray is the outward-pointing ray perpendicular to the surface of an optical medium at a given point.[2] In reflection of light, the angle of incidence and the angle of reflection are respectively the angle between the normal and the incident ray (on the plane of incidence) and the angle between the normal and the reflected ray.
Ellipsoid normal vector
Pseudovector – Physical quantity that changes sign with improper rotation
^ Ying Wu. "Radiometry, BRDF and Photometric Stereo" (PDF). Northwestern University.
^ "The Law of Reflection". The Physics Classroom Tutorial. Archived from the original on April 27, 2009. Retrieved 2008-03-31.
Weisstein, Eric W. "Normal Vector". MathWorld.
An explanation of normal vectors from Microsoft's MSDN
Clear pseudocode for calculating a surface normal from either a triangle or polygon.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Normal_(geometry)&oldid=1062427842" |
===For partial infiltration design, to calculate the depth of the storage reservoir needed below the invert of the underdrain pipe===
For designs that include an underdrain, the depth of the storage reservoir below the invert of the underdrain pipe (''d<sub>r'') can be calculated as follows:
For designs that include an underdrain, the depth of the storage reservoir below the invert of the underdrain pipe (''d<sub>r'') can be calculated as follows:<br>
<math>d_{r}=\frac{f'\times t}{n}</math>
{\displaystyle d_{r,max}={\frac {(RVC_{T}\times R)+RVC_{T}-(f'\times D)}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}} |
{\text{ABTS}}^{+}\text{\hspace{0.17em}}\text{radical-scavenging}\text{\hspace{0.17em}}\text{activity }\left(%\right)=\left(1-\text{Sample}\text{\hspace{0.17em}}\text{absorbance/Control}\text{\hspace{0.17em}}\text{absorbance}\right)×100
1) PD11, P. dentata harvested in November; PS12, P. seriata harvested in December; PY12, P. yezoensis harvested in December; PY01, P. yezoensis harvested in January; PY03, P. yezoensis harvested in March.
2) Total essential amino acid.
3) Total amino acid.
4) All values are mean±SD in triplicates.
5) Different letters (a-d) in the same column mean significantly different (p<0.05).
3 Different letters (a-d) within a column are significantly different (p<0.05). |
Palomar 12 - Wikipedia
Globular cluster in the constellation Capricornus
Palomar 12 by Hubble Space Telescope, 3.36′ view
63.6 ± 2.9 kly (19.50 ± 0.89 kpc)[2]
162 ± 8 ly[4]
{\displaystyle {\begin{smallmatrix}\left[{\ce {Fe}}/{\ce {H}}\right]\end{smallmatrix}}}
Probably extragalactic
Palomar 12 is a globular cluster in the constellation Capricornus.
First discovered on the National Geographic Society – Palomar Observatory Sky Survey plates by Robert George Harrington and Fritz Zwicky,[7] it was catalogued as a globular cluster. However Zwicky came to believe this was actually a nearby dwarf galaxy in the Local Group. It is a relatively young cluster, being about 30% younger than most of the globular clusters in the Milky Way.[2] It is metal-rich with a metallicity of [Fe/H] ≈ −0.8.[5] It has an average luminosity distribution of Mv = −4.48.[8]
Based on proper motion studies, this cluster was first suspected in 2000 to have been captured from the Sagittarius Dwarf Elliptical Galaxy (SagDEG) about 1.7 Ga ago.[9] It is now generally believed to have originated in that galaxy and is associated with the Sagittarius Stream.[5] It is estimated to be 6.5 Gyr old.[5]
^ a b Rosenberg, A.; et al. (1998), "Young Galactic globular clusters II. The case of Palomar 12", Astronomy and Astrophysics, 339: 61–69, arXiv:astro-ph/9809112, Bibcode:1998A&A...339...61R.
^ a b Boyles, J.; et al. (November 2011), "Young Radio Pulsars in Galactic Globular Clusters", The Astrophysical Journal, 742 (1): 51, arXiv:1108.4402, Bibcode:2011ApJ...742...51B, doi:10.1088/0004-637X/742/1/51.
^ a b c d Geisler, Doug; et al. (September 2007), "Chemical Abundances and Kinematics in Globular Clusters and Local Group Dwarf Galaxies and Their Implications for Formation Theories of the Galactic Halo", The Publications of the Astronomical Society of the Pacific, 119 (859): 939–961, arXiv:0708.0570, Bibcode:2007PASP..119..939G, doi:10.1086/521990.
^ "Cl Pal 12". SIMBAD. Centre de données astronomiques de Strasbourg. Retrieved 2006-11-16.
^ Abell, George O. (1955). "Globular Clusters and Planetary Nebulae Discovered on the National Geographic Society-Palomar Observatory Sky Survey". Publications of the Astronomical Society of the Pacific. 67 (397): 258. Bibcode:1955PASP...67..258A. doi:10.1086/126815.
^ van den Bergh, Sidney (July 2007). "The Luminosity Distribution of Globular Clusters in Dwarf Galaxies". The Astronomical Journal. 134 (1): 344–345. arXiv:0704.2226. Bibcode:2007AJ....134..344V. doi:10.1086/518868.
^ D. I. Dinescu; S. R. Majewski; T. M. Girard; K. M. Cudworth (2000). "The Absolute Proper Motion of Palomar 12: A Case for Tidal Capture from the Sagittarius Dwarf Spheroidal Galaxy". The Astronomical Journal. 120 (4): 1892–1905. arXiv:astro-ph/0006314. Bibcode:2000AJ....120.1892D. doi:10.1086/301552.
Wikimedia Commons has media related to Palomar 12.
Palomar 12 on WikiSky: DSS2, SDSS, GALEX, IRAS, Hydrogen α, X-Ray, Astrophoto, Sky Map, Articles and images
NASA Astronomy Picture of the Day: Palomar 12 (19 February 2015)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Palomar_12&oldid=1044834869" |
Combinatorial Games - Definition | Brilliant Math & Science Wiki
Ivan Koswara, Mateo Matijasevick, Calvin Lin, and
A combinatorial game is a two player game that satisfies the following conditions:
The game is deterministic: there is no randomization mechanism such as flipping a coin or rolling a die.
There is perfect information in the game: each player knows all the information about the state of the game, and nothing is hidden.
Solving these games is analyzed in combinatorial games - winning positions.
Well-known examples of combinatorial games are Tic-tac-toe, checkers, chess, Go, Dots and Boxes, and Nim.
A finite combinatorial game will always end; there is no sequence of moves that will lead to an infinite game. This means chess, in its basic form, is not finite, while Tic-tac-toe is finite.
Neither Dan Both Sam Dimitri
Dan and Sam play a game on a convex polygon of 100001 sides. Each one draws a diagonal on the polygon in his turn.
When someone draws a diagonal, it cannot have common points (except the vertices of the polygon) with other diagonals already drawn.
The game finishes when someone can't draw a diagonal on the polygon following the rules; that person is the loser. If Dan begins, who will win? This means, who has a winning strategy?
Clarification: The diagonals of a polygon are straight lines that join non-adjacent vertices.
This is the fourteenth problem of the set Winning Strategies.
A combinatorial game with normal play convention is a game where the first player unable to make a move loses. (If both players can keep making moves, it's considered a draw; naturally this can only happen if the game is not finite.) Chess, in some sense, is normal play, but many games such as Tic-tac-toe, Go, or Dots and Boxes, aren't normal.
5\times3
A combinatorial game with misère play convention is a game that reverses outcomes; if a player would have lost in a normal game, they won instead. Perhaps the most common misère game is misère Nim (in normal Nim, the player that cannot move loses; in misère Nim, the player that cannot move wins).
3
Impartial games are combinatorial games where there is no difference between the two players (the game is impartial), except that one starts the game. Formally, for a game to be impartial, it must satisfy two additional conditions.
1) The moves available depend only on the position of the game and not on which player’s turn it is.
2) The value of any particular move is the same for both players.
Condition 1 excludes many common games such as Tic-tac-toe, checkers, chess, and Go as impartial games. In all of them, one player can only move/place one kind of pieces (white) while the other can only move/place another kind (black).
The most well-known impartial game is the game of Nim. The game of Nim is played by 2 players and uses
k
piles of stones, with sizes
(a_1, a_2, \ldots, a_k)
. During a turn, a player is allowed to remove any number of stones from a single pile. The game ends when there are no stones left and the person whose turn it is to move will lose the game.
Dan Dimitri Both Neither Sam
8\times8
grid, on which each one chooses and puts, in his turn, a single piece like these:
The pieces must not overlap and can't be partially outside of the grid.
The game finishes when someone can't put a piece on the board in his turn following the rules (who is the loser). If Dan begins, who will win? This means, who has a winning strategy?
This is the tenth problem of the set Winning Strategies.
Cite as: Combinatorial Games - Definition. Brilliant.org. Retrieved from https://brilliant.org/wiki/combinatorial-games-definition/ |
ANALYTIC NUMBER THEORY - Encyclopedia Information
Analytic number theory Information
https://en.wikipedia.org/wiki/Analytic_number_theory
Exploring properties of the integers with complex analysis
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. [1] It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. [1] [2] It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).
1 Branches of analytic number theory
2.3 Chebyshev
2.4 Riemann
2.5 Hadamard and de la Vallée-Poussin
3 Problems and results
3.1 Multiplicative number theory
3.2 Additive number theory
3.3 Diophantine problems
4 Methods of analytic number theory
4.1 Dirichlet series
4.2 Riemann zeta function
{\displaystyle \lim _{x\to \infty }{\frac {\pi (x)}{x/\ln(x)}}=1,}
Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the function a/(A ln(a) + B), where A and B are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture, with A = 1 and B ≈ −1.08366. Carl Friedrich Gauss considered the same question: "Im Jahr 1792 oder 1793", according to his own recollection nearly sixty years later in a letter to Encke (1849), he wrote in his logarithm table (he was then 15 or 16) the short note "Primzahlen unter
{\displaystyle a(=\infty ){\frac {a}{\ln a}}}
". But Gauss never published this conjecture. In 1838 Peter Gustav Lejeune Dirichlet came up with his own approximating function, the logarithmic integral li(x) (under the slightly different form of a series, which he communicated to Gauss). Both Legendre's and Dirichlet's formulas imply the same conjectured asymptotic equivalence of π(x) and x / ln(x) stated above, although it turned out that Dirichlet's approximation is considerably better if one considers the differences instead of quotients.
Johann Peter Gustav Lejeune Dirichlet is credited with the creation of analytic number theory, [3] a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him. In 1837 he published Dirichlet's theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the theorem, he introduced the Dirichlet characters and L-functions. [3] [4] In 1841 he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers
{\displaystyle \mathbb {Z} [i]}
In two papers from 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the zeta function ζ(s) (for real values of the argument "s", as are works of Leonhard Euler, as early as 1737) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π(x)/(x/ln(x)) as x goes to infinity exists at all, then it is necessarily equal to one. [6] He was able to prove unconditionally that this ratio is bounded above and below by two explicitly given constants near to 1 for all x. [7] Although Chebyshev's paper did not prove the Prime Number Theorem, his estimates for π(x) were strong enough for him to prove Bertrand's postulate that there exists a prime number between n and 2n for any integer n ≥ 2.
Riemann's statement of the Riemann hypothesis, from his 1859 paper. [8] (He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line.)
Extending the ideas of Riemann, two proofs of the prime number theorem were obtained independently by Jacques Hadamard and Charles Jean de la Vallée-Poussin and appeared in the same year (1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all complex values of the variable s that have the form s = 1 + it with t > 0. [9]
The biggest technical change after 1950 has been the development of sieve methods, [10] particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, [11] which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.
Euclid showed that there are infinitely many prime numbers. An important question is to determine the asymptotic distribution of the prime numbers; that is, a rough description of how many primes are smaller than a given number. Gauss, amongst others, after computing a large list of primes, conjectured that the number of primes less than or equal to a large number N is close to the value of the integral
{\displaystyle \int _{2}^{N}{\frac {1}{\log t}}\,dt.}
{\displaystyle \pi (x)=({\text{number of primes }}\leq x),}
{\displaystyle \lim _{x\to \infty }{\frac {\pi (x)}{x/\log x}}=1.}
{\displaystyle \pi (x,a,q)=({\text{number of primes }}\leq x{\text{ such that }}p{\text{ is in the arithmetic progression }}a+nq,n\in \mathbf {Z} ),}
{\displaystyle \lim _{x\to \infty }{\frac {\pi (x,a,q)\phi (q)}{x/\log x}}=1.}
There are also many deep and wide-ranging conjectures in number theory whose proofs seem too difficult for current techniques, such as the twin prime conjecture which asks whether there are infinitely many primes p such that p + 2 is prime. On the assumption of the Elliott–Halberstam conjecture it has been proven recently that there are infinitely many primes p such that p + k is prime for some positive even k at most 12. Also, it has been proven unconditionally (i.e. not depending on unproven conjectures) that there are infinitely many primes p such that p + k is prime for some positive even k at most 246.
{\displaystyle n=x_{1}^{k}+\cdots +x_{\ell }^{k}.}
{\displaystyle G(k)\leq k(3\log k+11).}
{\displaystyle x^{2}+y^{2}\leq r^{2}.}
In geometrical terms, given a circle centered about the origin in the plane with radius r, the problem asks how many integer lattice points lie on or inside the circle. It is not hard to prove that the answer is
{\displaystyle \pi r^{2}+E(r)}
{\displaystyle E(r)/r^{2}\to 0}
{\displaystyle r\to \infty }
. Again, the difficult part and a great achievement of analytic number theory is obtaining specific upper bounds on the error term E(r).
It was shown by Gauss that
{\displaystyle E(r)=O(r)}
. In general, an O(r) error term would be possible with the unit circle (or, more properly, the closed unit disk) replaced by the dilates of any bounded planar region with piecewise smooth boundary. Furthermore, replacing the unit circle by the unit square, the error term for the general problem can be as large as a linear function of r. Therefore, getting an error bound of the form
{\displaystyle O(r^{\delta })}
{\displaystyle \delta <1}
in the case of the circle is a significant improvement. The first to attain this was Sierpiński in 1906, who showed
{\displaystyle E(r)=O(r^{2/3})}
. In 1915, Hardy and Landau each showed that one does not have
{\displaystyle E(r)=O(r^{1/2})}
. Since then the goal has been to show that for each fixed
{\displaystyle \epsilon >0}
{\displaystyle C(\epsilon )}
{\displaystyle E(r)\leq C(\epsilon )r^{1/2+\epsilon }}
In 2000 Huxley showed [12] that
{\displaystyle E(r)=O(r^{131/208})}
, which is the best published result.
{\displaystyle f(s)=\sum _{n=1}^{\infty }a_{n}n^{-s}.}
Depending on the choice of coefficients
{\displaystyle a_{n}}
, this series may converge everywhere, nowhere, or on some half plane. In many cases, even where the series does not converge everywhere, the holomorphic function it defines may be analytically continued to a meromorphic function on the entire complex plane. The utility of functions like this in multiplicative problems can be seen in the formal identity
{\displaystyle \left(\sum _{n=1}^{\infty }a_{n}n^{-s}\right)\left(\sum _{n=1}^{\infty }b_{n}n^{-s}\right)=\sum _{n=1}^{\infty }\left(\sum _{k\ell =n}a_{k}b_{\ell }\right)n^{-s};}
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p}^{\infty }{\frac {1}{1-p^{-s}}}{\text{ for }}s>1}
where the product is taken over all prime numbers p.
Euler's proof of the infinity of prime numbers makes use of the divergence of the term at the left hand side for s = 1 (the so-called harmonic series), a purely analytic result. Euler was also the first to use analytical arguments for the purpose of studying properties of integers, specifically by constructing generating power series. This was the beginning of analytic number theory. [13]
Analytic number theorists are often interested in the error of approximations such as the prime number theorem. In this case, the error is smaller than x/log x. Riemann's formula for π(x) shows that the error term in this approximation can be expressed in terms of the zeros of the zeta function. In his 1859 paper, Riemann conjectured that all the "non-trivial" zeros of ζ lie on the line
{\displaystyle \Re (s)=1/2}
but never provided a proof of this statement. This famous and long-standing conjecture is known as the Riemann Hypothesis and has many deep implications in number theory; in fact, many important theorems have been proved under the assumption that the hypothesis is true. For example, under the assumption of the Riemann Hypothesis, the error term in the prime number theorem is
{\displaystyle O(x^{1/2+\varepsilon })}
{\displaystyle \Re (z)=1/2.}
^ Davenport 2000, p. 1.
^ a b Gowers, Timothy; June Barrow-Green; Imre Leader (2008). The Princeton companion to mathematics. Princeton University Press. pp. 764–765. ISBN 978-0-691-11880-2.
^ Kanemitsu, Shigeru; Chaohua Jia (2002). Number theoretic methods: future trends. Springer. pp. 271–274. ISBN 978-1-4020-1080-4.
^ Elstrodt, Jürgen (2007). "The Life and Work of Gustav Lejeune Dirichlet (1805–1859)" (PDF). Clay Mathematics Proceedings. Retrieved 2007-12-25.
^ N. Costa Pereira (August–September 1985). "A Short Proof of Chebyshev's Theorem". American Mathematical Monthly. 92 (7): 494–495. doi: 10.2307/2322510. JSTOR 2322510.
^ M. Nair (February 1982). "On Chebyshev-Type Inequalities for Primes". American Mathematical Monthly. 89 (2): 126–129. doi: 10.2307/2320934. JSTOR 2320934.
^ Riemann, Bernhard (1859), "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse", Monatsberichte der Berliner Akademie . In Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New York (1953). Original manuscript Archived May 23, 2013, at the Wayback Machine (with English translation). Reprinted in ( Borwein et al. 2008) and ( Edwards 1974)
^ Ingham, A.E. (1990). The Distribution of Prime Numbers. Cambridge University Press. pp. 2–5. ISBN 0-521-39789-8.
^ Tenenbaum 1995, p. 56.
^ Tenenbaum 1995, p. 267.
^ M.N. Huxley, Integer points, exponential sums and the Riemann zeta function, Number theory for the millennium, II (Urbana, IL, 2000) pp.275–290, A K Peters, Natick, MA, 2002, MR 1956254.
^ Iwaniec & Kowalski: Analytic Number Theory, AMS Colloquium Pub. Vol. 53, 2004
Borwein, Peter; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea, eds. (2008), The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, CMS Books in Mathematics, New York: Springer, doi: 10.1007/978-0-387-72126-2, ISBN 978-0-387-72125-5
Davenport, Harold (2000), Multiplicative number theory, Graduate Texts in Mathematics, vol. 74 (3rd revised ed.), New York: Springer-Verlag, ISBN 978-0-387-95097-6, MR 1790423
Edwards, H. M. (1974), Riemann's Zeta Function, New York: Dover Publications, ISBN 978-0-486-41740-0, MR 0466039
Tenenbaum, Gérald (1995), Introduction to Analytic and Probabilistic Number Theory, Cambridge studies in advanced mathematics, vol. 46, Cambridge University Press, ISBN 0-521-41261-7
Retrieved from " https://en.wikipedia.org/?title=Analytic_number_theory&oldid=1086162222"
Analytic Number Theory Videos
Analytic Number Theory Websites
Analytic Number Theory Encyclopedia Articles |
Then increase A<sub>r</sub> accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging, assuming that water storage reservoir area and permeable pavement area are the same (A<sub>r</sub> = A<sub>p</sub>).
{\displaystyle d_{r,max}={\frac {\left[\left(RVC_{T}\times R\right)+RVC_{T}-\left(f'\times D\right)\right]}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
Then increase Ar accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging, assuming that water storage reservoir area and permeable pavement area are the same (Ar = Ap). |
The 24 Puzzle Practice Problems Online | Brilliant
9 \square 1 \square 7 \square 4 = 24
Which set of operators can be used in the blanks above to make this equation correct? Operators can be placed in the blanks in any order, and any number of parentheses can be used.
Hint: When solving puzzles such as these, it is often helpful to consider different methods that could be used to arrive at your final answer. For example, if the final operator is
\times,
then we need two numbers whose product is 24. For reference, the nontrivial factors of 24 are 2,3,4,6,8, and 12.
+,-,\times
-,-,\times
+,+,\times
+,\times,\div
9 \square 8 \square 3 \square 8 = 24
Hint: When solving puzzles such as these, it is often easier to reach the target number using only 2 or three of the required digits. Try to develop strategies for eliminating a number or pair of numbers. This is often done by producing 1, which can be multiplied for no effect, or 0, which can be added for no effect.
-,\times,\times
-,-,\times
+,+,\times
-,\times,\div
6 \square 1 \square 3 \square 4 = 24
Hint: When solving puzzles such as these,remember that the same result can often be produced using different operators. For example, multiplying by 2 is equivalent to dividing by
\frac{1}{2}.
-,\div,\div
+,\div,\div
-,\times,\div
+,-,\div
7 \square 3 \square 3 \square 7 = 24
Hint: When solving puzzles such as these, it is often helpful to try imagining all the possible final steps involving one of the given numbers. For example, if the final step is
\times 5,
you would need to make
\frac{24}{5}
with the other 3 numbers. This can be particularly helpful when the solution involves a fractional product.
+,-,\div
-,\times,\div
+,\div,\div
+,\times,\div
6 \square 6 \square 4 \square 1 = 24
Hint: When solving puzzles such as these, sometimes it’s not always possible to find factors that multiply nicely to 24. It’s often helpful to look for other, nearby numbers with nice factors, especially ones that can produce 24 when added to or subtracted from the numbers you already have. Numbers that can be helpful include 18, 20, 21, 25, 27, 28, and 30.
-,-,\times
+,-,\times
+,+,\times
+,\times,\div |
Maximum Likelihood Estimation (MLE) | Brilliant Math & Science Wiki
Maximum likelihood estimation (MLE) is a technique used for estimating the parameters of a given distribution, using some observed data. For example, if a population is known to follow a normal distribution but the mean and variance are unknown, MLE can be used to estimate them using a limited sample of the population, by finding particular values of the mean and variance so that the observation is the most likely result to have occurred.
MLE is useful in a variety of contexts, ranging from econometrics to MRIs to satellite imaging. It is also related to Bayesian statistics.
x_1, x_2, \ldots, x_n
be observations from
n
independent and identically distributed random variables drawn from a Probability Distribution
f_0
f_0
is known to be from a family of distributions
f
that depend on some parameters
\theta
f_0
could be known to be from the family of normal distributions
f
, which depend on parameters
\sigma
(standard deviation) and
\mu
(mean), and
x_1, x_2, \ldots, x_n
would be observations from
f_0
The goal of MLE is to maximize the likelihood function:
L = f(x_1, x_2, \ldots, x_n | \theta)=f(x_1 | \theta) \times f(x_2 | \theta) \times \ldots \times f(x_n | \theta)
Often, the average log-likelihood function is easier to work with:
\hat{\ell} = \frac{1}{n}\log L = \frac{1}{n}\sum_{i=1}^n\log f(x_i|\theta)
There are several ways that MLE could end up working: it could discover parameters
\theta
in terms of the given observations, it could discover multiple parameters that maximize the likelihood function, it could discover that there is no maximum, or it could even discover that there is no closed form to the maximum and numerical analysis is necessary to find an MLE.
Though MLEs are not necessarily optimal (in the sense that there are other estimation algorithms that can achieve better results), it has several attractive properties, the most important of which is consistency: a sequence of MLEs (on an increasing number of observations) will converge to the true value of the parameters. The following is an example where the MLE might give a slightly poor result compared to other estimation algorithms:
An airline has numbered their planes
1,2,\ldots,N,
and you observe the following 3 planes, which are randomly sampled from the
N
planes:
What is the maximum likelihood estimate for
N?
In other words, what value of
N
would, according to conditional probability, make your observation most likely?
The simplest case is when both the distribution and the parameter space (the possible values of the parameters) are discrete, meaning that there are a finite number of possibilities for each. In this case, the MLE can be determined by explicitly trying all possibilities.
A (possibly unfair) coin is flipped 100 times, and 61 heads are observed. The coin either has probability
\frac{1}{3}, \frac{1}{2}
\frac{2}{3}
of flipping a head each time it is flipped. Which of the three is the MLE?
Here, the distribution in question is the binomial distribution, with one parameter
p
\text{Pr}\left(H=61 | p=\frac{1}{3}\right) = \binom{100}{61}\left(\frac{1}{3}\right)^{61}\left(1-\frac{1}{3}\right)^{39} \approx 9.6 \times 10^{-9}
\text{Pr}\left(H=61 | p=\frac{1}{2}\right) = \binom{100}{61}\left(\frac{1}{2}\right)^{61}\left(1-\frac{1}{2}\right)^{39} \approx 0.007
\text{Pr}\left(H=61 | p=\frac{2}{3}\right) = \binom{100}{61}\left(\frac{2}{3}\right)^{61}\left(1-\frac{2}{3}\right)^{39} \approx .040
hence the MLE is
p=\frac{2}{3}
Unfortunately, the parameter space is rarely discrete, and calculus is often necessary for a continuous parameter space. For instance,
A (possibly unfair) coin is flipped 100 times, and 61 heads are observed. What is the MLE when nothing is previously known about the coin?
Again, the binomial distribution is the model to be worked with, with a single parameter
p
. The likelihood function is thus
\text{Pr}(H=61 | p) = \binom{100}{61}p^{61}(1-p)^{39}
to be maximized over
0 \leq p \leq 1
. This can be achieved by analyzing the critical points of this function, which occurs when
\begin{aligned} \frac{d}{dp}\binom{100}{61}p^{61}(1-p)^{39} &= \binom{100}{61}\left(61p^{60}(1-p)^{39}-39p^{61}(1-p)^{38}\right) \\ &= \binom{100}{61}p^{60}(1-p)^{38}(61(1-p)-39p) \\ &= \binom{100}{61}p^{60}(1-p)^{38}(61-100p) \\ &= 0 \end{aligned}
so either
p=0, \frac{61}{100}
, or 1. Thus
p=\frac{61}{100}
is the MLE, as otherwise the likelihood function is 0.
This logic is easily generalized: if
k
n
binomial trials result in a head, then the MLE is given by
\frac{k}{n}
Cite as: Maximum Likelihood Estimation (MLE). Brilliant.org. Retrieved from https://brilliant.org/wiki/maximum-likelihood-estimation-mle/ |
Quadratic Equations - Problem Solving Practice Problems Online | Brilliant
ax^2+bx-2 =0
\frac{1\pm\sqrt{3}}{3}
, what is a+b?
If the difference between the two roots of the equation
x^2-mx+5=0
2,
m^2?
If the sum of the squares of three consecutive positive integers is
509,
what is the sum of the three integers?
In the above image, the blue picture is 3 inches longer than it is wide. On all sides, the yellow frame is 5 inches around. If the area of the picture is 378, what is the area of the frame?
\frac{1}{x^2} + \frac{1}{ x} -20 = 0?
x=-{5} \text{ or } x={4}
x=-\frac{1}{4} \text{ or } x=\frac{1}{5}
x=-4 \text{ or } x=5
x=-\frac{1}{5} \text{ or } x=\frac{1}{4} |
LINEAR ALGEBRA - Encyclopedia Information
Linear algebra Information
{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}
{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}
and their representations in vector spaces and through matrices. [1] [2] [3]
The procedure (using counting rods) for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations. [4]
The first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule. Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy. [5]
Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group. The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants". [5]
Benjamin Peirce published his Linear Associative Algebra (1872), and his son Charles Sanders Peirce extended the work later. [6]
The first modern and more precise definition of a vector space was introduced by Peano in 1888; [5] by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. [5]
A vector space over a field F (often the field of the real numbers) is a set V equipped with two binary operations satisfying the following axioms. Elements of V are called vectors, and elements of F are called scalars. The first operation, vector addition, takes any two vectors v and w and outputs a third vector v + w. The second operation, scalar multiplication, takes any scalar a and any vector v and outputs a new vector av. The axioms that addition and scalar multiplication must satisfy are the following. (In the list below, u, v and w are arbitrary elements of V, and a and b are arbitrary scalars in the field F.) [7]
{\displaystyle T:V\to W}
{\displaystyle T(\mathbf {u} +\mathbf {v} )=T(\mathbf {u} )+T(\mathbf {v} ),\quad T(a\mathbf {v} )=aT(\mathbf {v} )}
{\displaystyle T(a\mathbf {u} +b\mathbf {v} )=T(a\mathbf {u} )+T(b\mathbf {v} )=aT(\mathbf {u} )+bT(\mathbf {v} )}
{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k},}
Any two bases of a vector space V have the same cardinality, which is called the dimension of V; this is the dimension theorem for vector spaces. Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension. [8]
{\displaystyle \dim(U_{1}+U_{2})=\dim U_{1}+\dim U_{2}-\dim(U_{1}\cap U_{2}),}
where U1 + U2 denotes the span of U1 ∪ U2. [9]
{\displaystyle {\begin{aligned}(a_{1},\ldots ,a_{m})&\mapsto a_{1}\mathbf {v} _{1}+\cdots a_{m}\mathbf {v} _{m}\\F^{m}&\to V\end{aligned}}}
{\displaystyle {\begin{bmatrix}a_{1}\\\vdots \\a_{m}\end{bmatrix}}.}
{\displaystyle f(w_{j})=a_{1,j}v_{1}+\cdots +a_{m,j}v_{m},}
{\displaystyle {\begin{bmatrix}a_{1,1}&\cdots &a_{1,n}\\\vdots &\ddots &\vdots \\a_{m,1}&\cdots &a_{m,n}\end{bmatrix}},}
A finite set of linear equations in a finite set of variables, for example, x1, x2, ..., xn, or x, y, ..., z is called a system of linear equations or a linear system. [10] [11] [12] [13] [14]
{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&8\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&-11\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&-3\end{alignedat}}}
{\displaystyle M=\left[{\begin{array}{rrr}2&1&-1\\-3&-1&2\\-2&1&2\end{array}}\right].}
{\displaystyle \mathbf {v} ={\begin{bmatrix}8\\-11\\-3\end{bmatrix}}.}
Let T be the linear transformation associated to the matrix M. A solution of the system ( S) is a vector
{\displaystyle \mathbf {X} ={\begin{bmatrix}x\\y\\z\end{bmatrix}}}
{\displaystyle T(\mathbf {X} )=\mathbf {v} ,}
Let ( S′) be the associated homogeneous system, where the right-hand sides of the equations are put to zero:
{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&0\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&0\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&0\end{alignedat}}}
The solutions of ( S′) are exactly the elements of the kernel of T or, equivalently, M.
{\displaystyle \left[\!{\begin{array}{c|c}M&\mathbf {v} \end{array}}\!\right]=\left[{\begin{array}{rrr|r}2&1&-1&8\\-3&-1&2&-11\\-2&1&2&-3\end{array}}\right]}
{\displaystyle \left[\!{\begin{array}{c|c}M&\mathbf {v} \end{array}}\!\right]=\left[{\begin{array}{rrr|r}1&0&0&2\\0&1&0&3\\0&0&1&-1\end{array}}\right],}
showing that the system ( S) has the unique solution
{\displaystyle {\begin{aligned}x&=2\\y&=3\\z&=-1.\end{aligned}}}
The determinant of a square matrix A is defined to be [15]
{\displaystyle \sum _{\sigma \in S_{n}}(-1)^{\sigma }a_{1\sigma (1)}\cdots a_{n\sigma (n)},}
{\displaystyle Mz=az.}
{\displaystyle (M-aI)z=0.}
{\displaystyle \det(xI-M).}
{\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}}
A linear form is a linear map from a vector space V over a field F to the field of scalars F, viewed as a vector space over itself. Equipped by pointwise addition and multiplication by a scalar, the linear forms form a vector space, called the dual space of V, and usually denoted V* [16] or V′. [17] [18]
{\displaystyle f\to f(\mathbf {v} )}
{\displaystyle \langle f,\mathbf {x} \rangle }
{\displaystyle f:V\to W}
{\displaystyle f^{*}:W^{*}\to V^{*}}
{\displaystyle \langle h^{\mathsf {T}},M\mathbf {v} \rangle =\langle h^{\mathsf {T}}M,\mathbf {v} \rangle .}
{\displaystyle \langle h^{\mathsf {T}}\mid M\mid \mathbf {v} \rangle .}
This section may require cleanup to meet Wikipedia's quality standards. The specific problem is: Need for a more encyclopedic style, which is homogeneous with the style of preceding sections. Also, some details do not belong to this general article but to more specialized ones. Also, inner product spaces should appear as a special instance of the more general concept of bilinear form. Finally, complex conjugation should appear in a specific section on linear algebra over the complexes. Please help improve this section if you can. (August 2018) ( Learn how and when to remove this template message)
{\displaystyle \langle \cdot ,\cdot \rangle :V\times V\to F}
that satisfies the following three axioms for all vectors u, v, w in V and all scalars a in F: [19] [20]
{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle ={\overline {\langle \mathbf {v} ,\mathbf {u} \rangle }}.}
{\displaystyle {\begin{aligned}\langle a\mathbf {u} ,\mathbf {v} \rangle &=a\langle \mathbf {u} ,\mathbf {v} \rangle .\\\langle \mathbf {u} +\mathbf {v} ,\mathbf {w} \rangle &=\langle \mathbf {u} ,\mathbf {w} \rangle +\langle \mathbf {v} ,\mathbf {w} \rangle .\end{aligned}}}
{\displaystyle \langle \mathbf {v} ,\mathbf {v} \rangle \geq 0}
{\displaystyle \|\mathbf {v} \|^{2}=\langle \mathbf {v} ,\mathbf {v} \rangle ,}
{\displaystyle |\langle \mathbf {u} ,\mathbf {v} \rangle |\leq \|\mathbf {u} \|\cdot \|\mathbf {v} \|.}
{\displaystyle {\frac {|\langle \mathbf {u} ,\mathbf {v} \rangle |}{\|\mathbf {u} \|\cdot \|\mathbf {v} \|}}\leq 1,}
{\displaystyle a_{i}=\langle \mathbf {v} ,\mathbf {v} _{i}\rangle .}
{\displaystyle \langle T\mathbf {u} ,\mathbf {v} \rangle =\langle \mathbf {u} ,T^{*}\mathbf {v} \rangle .}
Until the end of 19th century, geometric spaces were defined by axioms relating points, lines and planes ( synthetic geometry). Around this date, it appeared that one may also define geometric spaces by constructions involving vector spaces (see, for example, Projective space and Affine space). It has been shown that the two approaches are essentially equivalent. [21] In classical geometry, the involved vector spaces are vector spaces over the reals, but the constructions may be extended to vector spaces over any field, allowing considering geometry over arbitrary fields, including finite fields.
Functional analysis studies function spaces. These are vector spaces with additional structure, such as Hilbert spaces. Linear algebra is thus a fundamental part of functional analysis and its applications, which include, in particular, quantum mechanics ( wave functions).
Most physical phenomena are modeled by partial differential equations. To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting cells. For linear systems this interaction involves linear functions. For nonlinear systems, this interaction is often approximated by linear functions. [b] In both cases, very large matrices are generally involved. Weather forecasting is a typical example, where the whole Earth atmosphere is divided in cells of, say, 100 km of width and 100 m of height.
Nearly all scientific computations involve linear algebra. Consequently, linear algebra algorithms have been highly optimized. BLAS and LAPACK are the best known implementations. For improving efficiency, some of them configure the algorithms automatically, at run time, for adapting them to the specificities of the computer ( cache size, number of available cores, ...).
This section may require cleanup to meet Wikipedia's quality standards. The specific problem is: The dual space is considered above, and the section must be rewritten for given an understandable summary of this subject. Please help improve this section if you can. (September 2018) ( Learn how and when to remove this template message)
Fearnley-Sander, Desmond, " Hermann Grassmann and the Creation of Linear Algebra", American Mathematical Monthly 86 (1979), pp. 809–817.
Retrieved from " https://en.wikipedia.org/?title=Linear_algebra&oldid=1087648449"
Linear Algebra Websites
Linear Algebra Encyclopedia Articles |
Team:NCHU Taichung/Criteria - 2021.igem.org
Team:NCHU Taichung/Criteria
Criteria Bronze Sliver Gold
Registration and Giant Jamboree attendance.
All of our team members have registered as a team in iGEM 2021.
Competition deliverables.
According to the Competition Deliverables page on iGEM’s website:
We have created a wiki website, which includes an Attributions page.
A wiki website includes the Safety part of our project.
We have prepared for the team presentation for the Jamboree.
We have submitted the Judging form and safety forms before the deadline.
We have an Attributions page on our wiki website for those who helped us along our iGEM journey.
Our project aims to develop a synthetic biology method to complete circular agriculture.To get more information about the background and our inspiration about the project, please check it on our Description page.
PQQ gene can perform on E. coli by our shuttle vector and significantly promote E. coli growth. If the future iGEM teams need to use E. coli as a competent cell, our shuttle vector may be able to provide some help.For more information, please refer to our Contribution and Improvement page.
We have Successfully inserted PQQ genes and cellulase genes into the plasmids, and then transfer into the bacteria.
For more details, please check it on the Engineering success page.
Although we were in the pandemic, it couldn’t stop our ambition of keeping in touch with each other. We collaborated with many teams so that we were able to share our thoughts, achievements and the experiments we had done. For more details, please check it on the Collaboration page.
We had a communication with farmers to know the problem they had been facing. Furthermore, we had a meeting with Ming-Dao High School to discuss our experiments and experience. Moreover, we participated in a podcast to share the concept about circular agriculture. For more information, please check it on the Human Practices page.
One of our goals is to popularize circular agriculture. Therefore, we commit to simplify the process to make it more probable to be accepted by the public. For more details, please check it on the Implementation page.
The problems of agricultural waste management is a worldwide issue. That is to say, publicization of our concept is necessary, especially to the farmers. First of all, in order to realize the local farmers’ viewpoint and difficulty about agricultural waste management, we had an interview with them. Next, we featured a podcast with PanSci and we shared not only iGEM but the issue that the farmers cared the most was the cost. And last but not least, we would have an interview with the relevant authority, Nantou Environmental Protection Bureau, to understand what the opinion of the authority was.
For more details, please check it on the Human practice page.
For more possibilities to insert more genes , we put fi ori into the plasmid, BBa_K2141000 from the 2016 iGEM Istanbul_Tech. So that we can transfer this plasmid into E.coli and Bacillus subtilis. Moreover, Bacillus subtilis able to Endophyte with plants.
For more information, please check it on the Contribution and Improvement page.
To accomplish the purpose of estimate PQQ's benefits of plants growth and promotion of harvest, our modeling utilize two equations to calculate the predicted M_2 (unit:g) and exact M_2 (unit:g), dry plant mass, which are employed to evaluate the relative deviation between different inoculated concentration supernatants from the Bacillus subtilis RM125/PQQ culture and water, and use population mean and perspective plant data-1 to draw the linear regression pattern as the result, we have extremely excellent decision coefficient
R^2=0.8756
which really can explain the exact data and predicted data.
For more details, please check it on the Model page.
Cellulolytic enzymes first decompose the rice straw to produce saccharification straw liquid, which can incubate the Bacillus subtilis RM125/PQQ. It could produce valuable biostimulants that can significantly foster rice growth. Through all the steps in this system, nothing would be wasted , which can successfully verify that circular agriculture is accomplishable.
For more details, please check it on the Proof of Concept page
In addressing the problem of agricultural waste disposal, it is also necessary to understand the perspectives of farmers and the approaches authorities adopt and increase public awareness of agricultural waste. Thus, we have come up with several ways to have a communication with the authorities and the public about circular agriculture .
For more details, please check it on the Education and Communication page. |
Spectral Analysis - MATLAB & Simulink - MathWorks 日本
Welch’s Algorithm of Averaging Modified Periodograms
The spectrum analyzer in DSP System Toolbox™ uses the Welch’s nonparametric method of averaging modified periodogram and the filter bank method to estimate the power spectrum of a streaming signal in real time. You can launch the spectrum analyzer using the dsp.SpectrumAnalyzer System object™ in MATLAB® and the Spectrum Analyzer block in Simulink®.
\begin{array}{l}{x}_{i}\left(n\right)=x\left(n+iD\right),\text{â}\text{â}n=0,1,...,Mâ1\\ \text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}\text{â}i=0,1,...,Lâ1\end{array}
{N}_{samples}=\frac{\left(1â\frac{{O}_{p}}{100}\right)ÃNENBWÃ{F}_{s}}{RBW}
{N}_{window}=\frac{NENBWÃ{F}_{s}}{RBW}
{N}_{samples}=\left(1â\frac{{O}_{p}}{100}\right){N}_{window}
NENBW={N}_{window}Ã\frac{\underset{n=1}{\overset{{N}_{window}}{â}}{w}^{2}\left(n\right)}{{\left[\underset{n=1}{\overset{{N}_{window}}{â}}w\left(n\right)\right]}^{2}}
\frac{span}{RBW}>2
RB{W}_{auto}=\frac{span}{1024}
\frac{NENBWÃ{F}_{s}}{{N}_{window}}
{P}_{xx}^{i}\left(f\right)=\frac{1}{MU}{|\underset{n=0}{\overset{Mâ1}{â}}{x}_{i}\left(n\right)w\left(n\right){e}^{âj2\mathrm{Ï}fn}|}^{2},\text{â}\text{â}i=0,1,...,Lâ1
U=\frac{1}{M}\underset{n=0}{\overset{Mâ1}{â}}{w}^{2}\left(n\right)
{P}_{xx}^{W}\left(f\right)=\frac{1}{L}\underset{i=0}{\overset{Lâ1}{â}}{P}_{xx}^{i}\left(f\right)
{P}_{xx}^{W}\left(f\right)=\frac{1}{L*{F}_{s}}\underset{i=0}{\overset{Lâ1}{â}}{P}_{xx}^{i}\left(f\right) |
Plot impulse response function (IRF) of state-space model - MATLAB irfplot - MathWorks Australia
irfplot
Plot Measurement and State Variable IRFs
Specify Number of Periods and IRFs to Plot
Plot Cumulative IRFs of Estimated Model to Specified Axes
Plot Time-Varying IRF
Plot IRF Confidence Bounds
Plot impulse response function (IRF) of state-space model
irfplot(Mdl)
irfplot(Mdl,Name,Value)
irfplot(___,'Params',estParams)
irfplot(___,'Params',estParams,'EstParamCov',EstParamCov)
irfplot(ax,___)
h = irfplot(ax,___)
irfplot plots the IRFs of the state and measurement variables in a state-space model. To return the IRFs as numeric arrays instead, use irf. Other state-space model tools to characterize the dynamics of a specified system include:
irfplot(Mdl) plots the IRF, or dynamic response, of each state and measurement variable of the fully specified state-space model Mdl, such as an estimated model. irfplot plots a figure containing the IRFs of the measurement variables yt, and plots a separate figure containing the IRFs of the state variables xt. Each figure contains a subplot for each variable and state disturbance combination; subplot (i,j) is the IRF of variable j resulting from a unit shock applied to a state disturbance i ui,t. Subplot titles identify the shocked variable and IRF variable.
irfplot(Mdl,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, 'PlotU',1:2,'PlotX',[] plots only the measurement variable IRFs resulting from shocks applied to the first and second state-disturbance variables (the state variable IRF plot is suppressed).
irfplot(___,'Params',estParams) plots the IRFs of the partially specified state-space model Mdl using any of the input argument combinations in the previous syntaxes. estParams specifies estimates of all unknown parameters in the model.
irfplot(___,'Params',estParams,'EstParamCov',EstParamCov) also plots pointwise lower and upper 95% Monte Carlo confidence bounds in each plot. EstParamCov specifies the estimated covariance matrix of the parameter estimates, as returned by the estimate function, and is required for confidence interval estimation.
irfplot(ax,___) plots on the axes objects specified by ax instead of new figures. The option ax can precede any of the input argument combinations in the previous syntaxes.
h = irfplot(ax,___) also returns an array of plot handles h. Use h to modify properties of the plots after you create them.
\begin{array}{l}{x}_{t}=0.5{x}_{t-1}+0.2{u}_{t}\\ {y}_{t}=2{x}_{t}+0.01{\epsilon }_{t}.\end{array}
Plot the IRFs of the measurement and state variables.
irfplot(Mdl);
The plot entitled U1 -> Y1 is the IRF of
{\mathit{y}}_{\mathit{t}}
, and the plot entitled U1 -> X1 is the IRF of
{\mathit{x}}_{\mathit{t}}
. Both IRFs indicate that the effects of the shock on the system diminish after about 8 periods.
Plot the 10-period IRFs of only the measurement variables in a system.
\begin{array}{l}{x}_{1,t}={x}_{1,t-1}+0.2{u}_{1,t}\\ {x}_{2,t}={x}_{1,t-1}+0.3{x}_{2,t-1}+{u}_{2,t}\\ {y}_{1,t}={x}_{1,t}+{\epsilon }_{1,t}\\ {y}_{2,t}={x}_{1,t}+{x}_{2,t}+{\epsilon }_{2,t}.\end{array}
Mdl = dssm(A,B,C,D)
Mdl is a fully specified ssm model object.
Plot the two 10-period IRFs of
{\mathit{y}}_{2,\mathit{t}}
, and suppress the state variable IRFs.
irfplot(Mdl,'NumPeriods',10,'PlotY',2,'PlotX',[]);
The top subplot is the IRF of
{\mathit{y}}_{2,\mathit{t}}
resulting from a shock to
{\mathit{u}}_{1,\mathit{t}}
, which is persistent because the shock filters through the random walk state
{\mathit{x}}_{1,\mathit{t}}
The bottom subplot is the IRF of
{\mathit{y}}_{2,\mathit{t}}
{\mathit{u}}_{2,\mathit{t}}
, which is transient and eventually diminishes as time elapses because the state
{\mathit{x}}_{2,\mathit{t}}
exhibits autoregressive behavior.
Simulate data from a known model, fit the data to a state-space model, and then the plot cumulative IRFs of the estimated model to specified axes.
{x}_{t}=1+0.75{x}_{t-2}+{u}_{t},
{\mathit{u}}_{\mathit{t}}
DGP = arima('Constant',1,'AR',{0 0.75},'Variance',1);
\begin{array}{l}{x}_{t}=c+\varphi {x}_{t-2}+\eta {u}_{t}\\ {y}_{t}={x}_{t}.\end{array}
c(2) | 0.67319 0.02749 24.48749 0
x(1) | 3.69929 0 Inf 0
x(2) | 1 0 Inf 0
EstMdl is a fully specified dssm model object.
Plot the cumulative IRFs of the first and third state variables, and the measurement variable in EstMdl. Return the plot in the same figure, on three separate subplots.
ax = gobjects(3,1);
for j = 1:numel(ax)
ax(j) = subplot(3,1,j);
irfplot(ax,EstMdl,'Cumulative',true,'PlotX',[1 3]);
{\mathit{y}}_{\mathit{t}}={\mathit{x}}_{\mathit{t}}
, the top two IRFs in the figure are equivalent. Because
{\mathit{x}}_{1,\mathit{t}-1}={\mathit{x}}_{3,\mathit{t}}
, the IRF in the subplot at the bottom of the figure is shifted to the left, relative to the other two plots.
Simulate data from a time-varying state-space model, fit a model to the data, then plot the time-varying IRF of the estimated model.
Consider the DGP represented by this system
\begin{array}{l}{x}_{t}=\left\{\begin{array}{ll}0.75{x}_{t-1}+{u}_{t};& t<11\\ -0.1{x}_{t-1}+3{u}_{t};& t\ge 11\end{array}\\ {y}_{t}=1.5{x}_{t}+2{\epsilon }_{t}.\end{array}
\mathit{C}
{\mathit{y}}_{\mathit{t}}
Plot the IRF of the measurement and state variables by supplying DGP (not the estimated model) and the estimated parameters by using the 'Params' name-value pair argument.
h = irfplot(DGP,'Params',estParams);
xline(h(1,1).Parent,10.5,'--')
The figures show time-varying IRFs of the measurement and state variables. The first 10 periods correspond to the IRF of the first state equation. During period 11, what remains of the shock transfers to the second state equation, and filters through that system until it diminishes.
Plot the measurement variuable IRF and the 95% confidence intervals on the true IRFs.
{x}_{t}=1+0.75{x}_{t-2}+{u}_{t},
{\mathit{u}}_{\mathit{t}}
Mdl is a fully specified, estimated dssm model object.
Plot the IRF, with its 95% confidence intervals, of the measurement variable.
irfplot(Mdl,'Params',estParams,'EstParamCov',EstParamCov,...
'PlotX',[]);
The blue line represents the estimated IRF of
{\mathit{y}}_{\mathit{t}}
. The dashed red lines represent the upper and lower, pointwise 95% confidence bounds on the true IRF. The model has only one lag term (lag 2), as the shock filters through the system, it impacts the first state variable during odd periods only.
If Mdl is partially specified (that is, it contains unknown parameters), specify estimates of the unknown parameters by using the 'Params' name-value argument. Otherwise, irfplot issues an error.
irfplot issues an error when Mdl is a dimension-varying model, which is a time-varying model containing at least one variable that changes dimension during the sampling period (for example, a state variable drops out of the model).
Pass the model template for estimation Mdl to irfplot, and specify the parameter estimates and covariance matrix by using the 'Params' and 'EstParamCov' name-value arguments.
For the irfplot function, return the appropriate output arguments for lower and upper confidence bounds.
ax — Axes on which to plot IRFs
Axes on which to plot the IRFs, specified as a pU*(pY + pX)-by-1 vector of Axes objects, where pU, pY, and pX are the lengths of the values of the 'PlotU', 'PlotY', and 'PlotX' name-value pair arguments, respectively.
irfplot plots IRFs to the axes in ax in this order.
IRFs of all measurement variables PlotY(:) resulting from a shock to the first state disturbance PlotU(1).
IRFs of all measurement variables PlotY(:) resulting from a shock to the second state disturbance PlotU(2).
Continue the procedure similarly until irfplot plots the IRF associated with the last state disturbance PlotU(end).
Repeat steps 1 through 3, but replace the measurement variables with the state variables PlotX.
By default, irfplot plots the measurement-variable IRFs to the axes of subplots in a new figure, and plots the state-variable IRFs to the axes of subplots in another new figure.
Example: 'PlotU',1:2,'PlotX',[] plots only the measurement-variable IRFs resulting from shocks applied to the first and second state-disturbance variables (the state-variable IRF plot is suppressed).
Number of periods for which irfplot computes the IRF, specified as a positive integer. Periods in the IRF start at time 1 and end at time NumPeriods.
Example: 'NumPeriods',10 specifies the inclusion of 10 consecutive time points in the IRF starting at time 1, during which irfplot applies the shock, and ending at time 10.
If Mdl is fully specified, irfplot ignores Params.
PlotU — State-disturbance variables ut to shock
State-disturbance variables ut to shock for the IRF plots, specified as the comma-separated pair consisting of 'PlotU' and a vector of positive integers. Elements are the indices of the state-disturbance variables u1,t, u2,t, …, uk,t.
By default, irfplot shocks all state-disturbance variables.
Example: 'PlotU',[1 3] shocks u1,1 and u3,1, and irfplot plots the resulting IRFs.
PlotY — Measurement-variable IRFs to plot
Measurement-variable IRFs to plot, specified as the comma-separated pair consisting of 'PlotY' and a vector of positive integers. Elements are the indices of the measurement variables y1,t, y2,t, …, yn,t.
If PlotY is empty [], irfplot does not plot any measurement-variable IRFs.
By default, irfplot plots all measurement-variable IRFs.
Example: 'PlotY',1 plots the IRF of y1,t.
PlotX — State-variable IRFs to plot
State-variable IRFs to plot, specified as the comma-separated pair consisting of 'PlotX' and a vector of positive integers. Elements are the indices of the state variables x1,t, x2,t, …, xm,t.
If PlotX is empty [], irfplot does not plot any state-variable IRFs.
By default, irfplot plots all state-variable IRFs.
Example: 'PlotX',[] does not plot any state-variable IRFs.
true irfplot computes the cumulative IRF of all variables over the specified time range.
false irfplot computes the standard, period-by-period IRF of all variables over the specified time range.
'repeated-multiplication' irfplot uses recursive multiplication.
'eigendecomposition' irfplot attempts to use the spectral decomposition of A to compute the matrix power. Specify this value only when you suspect that the recursive multiplication algorithm might experience numerical issues. For more details, see Algorithms.
If Mdl is fully specified, irfplot ignores EstParamCov.
By default, irfplot does not estimate confidence bounds.
h — Plot handles to IRFs and confidence bounds
matrix of Line objects
Plot handles to the IRFs and confidence bounds, returned as a 3-by-pU*(pY + pX) matrix of Line objects, where pU, pY, and pX are the lengths of the values of the 'PlotU', 'PlotY', and 'PlotX' name-value pair arguments, respectively.
Each column corresponds to the IRF of a combination of a state disturbance and a measurement or state variable. For a particular column, row 1 contains the handle to the IRF, and rows 2 and 3 contain the handles to the lower and upper confidence bounds, respectively. The columns display information in this order:
Continues the display similarly until irfplot reaches the IRF associated with the last state disturbance PlotU(end).
h contains unique plot identifiers, which you can use to query or modify properties of the plots.
\begin{array}{l}{x}_{t}=A{x}_{t-1}+B{u}_{t}\\ {y}_{t}=C{x}_{t}+D{\epsilon }_{t},\end{array}
{\psi }_{xj}\left(r\right)={A}^{r}{b}_{j},
{\psi }_{yj}\left(r\right)=C{A}^{r}{b}_{j}.
\begin{array}{l}{\psi }_{xj}\left(r\right)={A}_{r}\cdots {A}_{2}{A}_{1}{b}_{1,j}\\ {\psi }_{yj}\left(r\right)={C}_{r}{A}_{r}\cdots {A}_{2}{A}_{1}{b}_{1,j},\end{array}
where b1,j is column j of B1, the period 1 state-disturbance-loading matrix. Time-varying IRFs depend on the time at which the shock is applied. irfplot always applies the shock at period 1.
If you specify 'eigendecomposition' for the 'Method' name-value pair argument, irfplot attempts to diagonalize the state-transition matrix A by using the spectral decomposition. irfplot resorts to recursive multiplication instead under at least one of these circumstances:
irfplot uses Monte Carlo simulation to compute confidence intervals.
irfplot randomly draws NumPaths variates from the asymptotic sampling distribution of the unknown parameters in Mdl, which is Np(Params,EstParamCov), where p is the number of unknown parameters.
For each randomly drawn parameter set j, irfplot:
irf | estimate | filter | smooth | forecast |
K-nearest neighbors from scratch - Philipp Muens
You can find working code examples (including this one) in my lab repository on GitHub.
K-nearest neighbors (abbreviated as k-NN or KNN) is a simple, yet elegant Machine Learning algorithm to classify unseen data based on existing data. The neat property of this algorithm is that it doesn't require a "traditional" training phase. If you have a classification problem and labeled data you can predict the class of any unseen data point by leveraging your existing, already classified data.
Let's take a closer look at the intuitions behind the core ideas, the involved math and the translation of such into code.
Imagine that we've invited 100 dog owners with their dogs over for a statistical experiment we want to run. Each dog participating in this experiment is 1 out of 4 different dog breeds we're interested in studying. While we have the dog owners and their dogs around we measure 3 different properties of each dog:
Its weight (in kilograms)
Its height (in centimeters)
Its alertness (on a scale from 0 to 1 [1=very alert, 0=almost no alertness])
Once done, we normalize the measurements so that they fall into a range between
0
1
After collection the data on each individual dog we end up with 100 3-pair measurements, each of which is labeled with the corresponding dog breed.
\begin{pmatrix} 0.5 \\ 0.8 \\ 0.1 \end{pmatrix} = Podenco
In order to better understand the data it's always a good idea to plot it. Since we have collected 3 different measurements (weight, height and alertness) it's possible to project all of the 100 data points into a 3 dimensional space and color every data point according to its label (e.g. brown for the label "Podenco").
Unfortunately we run into an issue while attempting to plot this data. As it turns out we forgot to label one measurement. We do have the dogs width, its height and its alertness but for some reason we forgot to write down the dogs breed.
Is there any chance we could derive what this dogs breed might be given all the other dog measurements we already have? We can still add the unlabeled data point into our existing 3 dimensional space where all the other colored data points reside. But how should we color it?
One potential solution is to look at the, say 5 surrounding neighbors of the data point in question and see what their color is. If the majority of those data points is labeled "Podenco" then it's very likely that our measurements were also taken from a Podenco.
And that's exactly what the k-NN algorithm does. The k-nearest neighbors algorithm predicts the class of an unseen data point based on its k-nearest neighbors and the majority of their respective classes. Let's take a closer look at this from a mathematical perspective.
There are only 2 concepts we need to implement in order to classify unseen data via k-NN.
As stated above, the algorithm works by looking at the k-nearest neighbors and the majority of their respective classes in order to classify unseen data.
Because of that we need to implement 2 functions. A distance function which calculates the distance between two points and a voting function which returns the most seen label given a list of arbitrary labels.
Given the notion of "nearest neighbors" we need to calculate the distance between our "to be classified" data point and all the other data points to find the
k
closest ones.
There are a couple of different distance functions out there. For our implementation we'll use the Euclidean distance as it's simple to calculate and can easily scale up to
n
dimensions.
d(x, y) = d(y, x) = \sqrt{\sum\_{i=1}^N (x_i - y_i)^2}
Let's unpack this formula with the help of an example. Assuming we have two vectors
and
b
, the euclidean distance between the two is calculated as follows:
\vec{a} = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix} \vec{b} = \begin{pmatrix} 5 \\ 6 \\ 7 \\ 8 \end{pmatrix}
d(\vec{a}, \vec{b}) = d(\vec{b}, \vec{a}) \\ = \sqrt{(1 - 5)^2 + (2 - 6)^2 + (3 - 7)^2 + (4 - 8)^2} \\ = \sqrt{64} = 8
Translating this into code results in the following:
def distance(x: List[float], y: List[float]) -> float:
return sqrt(sum((x[i] - y[i]) ** 2 for i in range(len(x))))
assert distance([1, 2, 3, 4], [5, 6, 7, 8]) == 8
Great. We just implemented the first building block: An Euclidean distance function.
Next up we need to implement a voting function. The voting function takes as an input a list of labels and returns the "most seen" label of that list. While this sounds trivial to implement we should take a step back and think about potential edge cases we might run into.
One of those edge cases is the situation in which we have 2 or more "most seen" labels:
# Do we return `a` or `b`?
labels: List[str] = ['a', 'a', 'b', 'b', 'c']
For those scenarios we need to implement a tie breaking mechanism.
There are several ways to deal with this. One solution might be to pick a candidate randomly. In our case however we shouldn't think about the vote function in isolation since we know that the distance and vote functions both work together to determine the label of the unseen data point.
We can exploit this fact and assume that the list of labels our vote function gets as an input is sorted by distance, nearest to furthest. Given this requirement it's easy to break ties. All we need to do is recursively remove the last entry in the list (which is the furthest) until we only have one clear winning label.
The following demonstrates this process based on the labels example from above:
# Remove one entry. We're still unsure if we should return `a` or `b`
labels: List[str] = ['a', 'a', 'b', 'b']
# Remove another entry. Now it's clear that `a` is the "winner"
labels: List[str] = ['a', 'a', 'b']
Let's translate that algorithm into a function we call majority_vote:
def majority_vote(labels: List[str]) -> str:
counted: Counter = Counter(labels)
winner: List[str] = []
max_num: int = 0
most_common: List[Tuple[str, int]]
for most_common in counted.most_common():
label: str = most_common[0]
num: int = most_common[1]
if num < max_num:
max_num = num
winner.append(label)
if len(winner) > 1:
return majority_vote(labels[:-1])
return winner[0]
assert majority_vote(['a', 'b', 'b', 'c']) == 'b'
assert majority_vote(['a', 'b', 'b', 'a']) == 'b'
assert majority_vote(['a', 'a', 'b', 'b', 'c']) == 'a'
The tests at the bottom show that our majority_vote function reliably deals with edge cases such as the one described above.
The k-NN algorithm
Now that we've studied and codified both building blocks it's time to bring them together. The knn function we're about to implement takes as inputs the list of labeled data points, a new measurement (the data point we want to classify) and a parameter k which determines how many neighbors we want to take into account when voting for the new label via our majority_vote function.
The first thing our knn algorithm should do is to calculate the distances between the new data point and all the other, existing data points. Once done we need to order the distances from nearest to furthest and extract the data point labels. This sorted list is then truncated so that it only contains the k nearest data point labels. The last step is to pass this list into the voting function which computes the predicted label.
Turning the described steps into code results in the following knn function:
def knn(labeled_data: List[LabeledData], new_measurement, k: int = 5) -> Prediction:
class Distance(NamedTuple):
distances: List[Distance] = [Distance(data.label, distance(new_measurement, data.measurements))
for data in labeled_data]
distances = sorted(distances, key=attrgetter('distance'))
labels = [distance.label for distance in distances][:k]
label: str = majority_vote(labels)
return Prediction(label, new_measurement)
That's it. That's the k-nearest neighbors algorithm implemented from scratch!
It's time to see if our homebrew k-NN implementation works as advertised. To test drive what we've coded we'll use the infamous Iris flower data set.
The data set consists of 50 samples of three different flower species called Iris:
For each sample, 4 different measurements were collected: The sepal width and length as well as its petal width and length.
The following is an example row from the data set where the first 4 numbers are the sepal length, sepal width, petal length, petal width and the last string represents the label for those measurements.
The best way to explore this data is to plot it. Unfortunately it's hard to plot and inspect 4 dimensional data. However we can pick 2 measurements (e.g. petal length and petal width) and scatter plot those in 2 dimensions.
Note: We're using the amazing Plotly library to create our scatter plots.
fig = px.scatter(x=xs, y=ys, color=text, hover_name=text, labels={'x': 'Petal Length', 'y': 'Petal Width'})
We can clearly see clusters of data points which all share the same color and therefore the same label.
Now let's pretend that we have a new, unlabeled data point:
new_measurement: List[float] = [7, 3, 4.8, 1.5]
Adding this data point to our existing scatter plot results in the following:
x=new_measurement[petal_length_idx],
y=new_measurement[petal_width_idx],
text="The measurement we want to classify")
ax=0,
Even if we're just plotting the petal length and petal width in 2 dimensions it seems to be the case that the new measurement might be coming from an "Iris Versicolor".
Let's use our knn function to get a definite answer:
knn(labeled_data, new_measurement, 5)
And sure enough the result we get back indicates that we're dealing with an "Iris Versicolor":
Prediction(label='Iris-versicolor', measurements=[7, 3, 4.8, 1.5])
k-nearest neighbors is a very powerful classification algorithm which makes it possible to label unseen data based on existing data. k-NNs main idea is to use the
k
nearest neighbors of the new, "to-be-classified" data point to "vote" on the label it should have.
Given that, we need 2 core functions to implement k-NN. The first function calculates the distance between 2 data points so that nearest neighbors can be found. The second function performs a majority vote so that a decision can be made as to what label is most present in the given neighborhood.
Using both functions together brings k-NN to life and makes it possible to reliably label unseen data points.
I hope that you enjoyed this article and I'd like to invite you to subscribe to my Newsletter if you're interested in more posts like this.
Do you have any questions, feedback or comments? Feel free to reach out via E-Mail or connect with me on Twitter.
Thanks to Eric Nieuwland who reached out via E-Mail and provided some code simplifications!
The following is a list of resources I've used to work on this blog post. Other helpful resources are also linked within the article itself.
Joel Grus - Data Science from Scratch
Wikipedia - k-nearest neighbors algorithm
Wikipedia - Euclidean distance
Wikipedia - Iris flower data set
UCI - Iris Data Set |
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#Video Demo 📺
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When the caption is shown, you can then enable numbering by selecting the number icon for the Toggle Numbering option.
For figures displayed within a content block, when the caption is toggled on, the caption can be edited directly by clicking the EDIT button on the image.
Figures can be referenced by their numbers within the text using either the command menu or keyboard shortcuts. To reference a figure use one of the following two commands, then select the figure you want to reference from the drop down menu. When you click on the reference a preview of the image and its caption is displayed. Learn about internal references Internal References
/link to figure [[fig:
The numbering is ordered by location of the image/GIF on the page. All numbering will start at 1 at the beginning of the page. The numbering will automatically update with added or removed figures. |
Using Direct Simulation Monte Carlo With Improved Boundary Conditions for Heat and Mass Transfer in Microchannels | J. Heat Transfer | ASME Digital Collection
P. Tang Institute of Chemical Engineering Process and Machinery,
, Hangzhou 310027, China
e-mail: zdhjkz@zju.edu.cn
I. Wong,
C. K. Lam,
C. K. Lam
R. X. Chen,
R. X. Chen
Zhejang Chengxin Pharm&Chem Equipment Co. Ltd.
, Taizhou 318012, China
Yang, J., Ye, J. J., Zheng, J. Y., Wong, I., Lam, C. K., Xu, P., Chen, R. X., and Zhu, Z. H. (February 19, 2010). "Using Direct Simulation Monte Carlo With Improved Boundary Conditions for Heat and Mass Transfer in Microchannels." ASME. J. Heat Transfer. April 2010; 132(4): 041008. https://doi.org/10.1115/1.4000880
Micro-electromechanical systems and nano-electromechanical systems have attracted a great deal of attention in recent years. The flow and heat transfer behaviors of micromachines for separation applications are usually different from that of macro counterparts. In this paper, heat and mass transfer characteristics of rarefied nitrogen gas flows in microchannels are investigated using direct simulation Monte Carlo with improved pressure boundary conditions. The influence of aspect ratio and wall temperature on mass flowrate and wall heat flux in microchannels are studied parametrically. In order to examine the aspect ratio effect on heat and mass transfer behaviors, the wall temperature is set constant at 350 K and the aspect ratio of the microchannel varies from 5 to 20. The results show that as the aspect ratio increases, the velocity of the flow decreases, so does the mass flowrate. In a small aspect ratio channel, the heat transfer occurs throughout the microchannel; as the aspect ratio of the microchannel increases, the region of thermal equilibrium extends. To investigate the effects of wall temperature
(Tw)
on the mass flowrate and wall heat flux in a microchannel, the temperature of the incoming gas flow
(Tin)
is set constant at 300 K and the wall temperature varies from 200 K to 800 K while the aspect ratio is remained unchanged. Results show that majority of the wall heat flux stays within the channel entrance region and drops to nearly zero at the halfway in the channel. When
Tw<Tin
, under the restriction of pressure-driven condition and continuity of pressure, the molecular number density of the flow decreases along the flow direction after a short increase at the entrance region. When
Tw>Tin
, the molecular number density of the flow drops rapidly near the inlet and the temperature of the gas flow increases along the channel. As
Tw
increases, the flow becomes more rarefied, the mass flowrate decreases, and the resistance at the entrance region increases. Furthermore, when
Tw>Tin
, a sudden jump of heat transfer flux and temperature are observed at the exit region of the channel.
flow simulation, heat transfer, mass transfer, microchannel flow, Monte Carlo methods, direct simulation Monte Carlo, heat flux, mass flowrate, microfluidics, pressure boundary conditions
Boundary-value problems, Flow (Dynamics), Heat, Heat transfer, Mass transfer, Microchannels, Pressure, Simulation, Temperature, Wall temperature, Heat flux, Gas flow, Density
Correlating Equations for Impingement Cooling of Small Heat Sources With Multiple Circular Liquid Jets
Development of A Chip-Integrated Micro Cooling Device
Murthyj
MEMS Enabled Thermal Management of High Heat Flux Devices EDIFICE: Embedded Droplet Impingement for Integrated Cooling of Electronics
Micromachined Particle Filter With Low Power Dissipation
Gas and Liquid Flow in Small Channel
MEMS for Pressure Distribution Studies of Gaseous Flows in Microchannels
Proceedings of IEEE Micro Electro Mechanical Systems (MEMS)
Prediction of Microchannel Flows Using Direct Simulation Monte Carlo
Application of Microfabrication to Fluid Mechanics
Recent Advances and Current Challenges for DSMC
Computations of High-Speed, High Knudsen Number Microchannel Flows
Numerical Analysis of Hypersonic Low-Density Scramjet Inlet Flow
Statistical Simulation of Low-Speed Rarefied Gas Flows
A Direct Simulation Method for Subsonic, Microscale Gas Flows
Heat Transfer and Flow fields in Short Micro-Channels Using Direct Simulation Monte Carlo
Modeling of Microscale Mass Flows in Transition Regime Part I: Flow Over Backward Facing Steps
Numerical Modeling of Micromechanical Devices Using the Direct Simulation Monte Carlo Method
Role of Boundary Conditions in Monte Carlo Simulation of MEMS Devices
Pressure Boundary Treatment in Micromechanical Devices Using the Direct Simulation Monte Carlo Method
Implicit Boundary Conditions for Direct Simulation Monte Carlo Method in MEMS Flow Predictions
Simulation for Gas Flows in Microgeometries Using the Direct Simulation Monte Carlo Method
New Treatment of Pressure Boundary Conditions for DSMC Method in Micro-Channel Flow Simulation
Proceedings of FEDSM2007 Fifth Joint ASME/JSME Fluids Engineering Conference
Parallel and Series Multiple Microchannel Systems
Comparison of 3-D and 2-D DSMC Heat Transfer Calculations of Low-Speed Short Microchannel Flows
Coupled Laminar Heat Transfer and Sublimation Mass Transfer in a Duct
Heat and Mass Transfer of a Rarefied Gas in a Driven Micro-Cavity |
Find minimum of constrained nonlinear multivariable function - MATLAB fmincon - MathWorks Switzerland
\underset{x}{\mathrm{min}}f\left(x\right)\text{ such that }\left\{\begin{array}{c}c\left(x\right)\le 0\\ ceq\left(x\right)=0\\ A\cdot x\le b\\ Aeq\cdot x=beq\\ lb\le x\le ub,\end{array}
x\left(1\right)+2x\left(2\right)\le 1
x\left(1\right)+2x\left(2\right)=3>1
x\left(1\right)+2x\left(2\right)\le 1
2x\left(1\right)+x\left(2\right)=1
x
x\left(1\right) \le 1
x\left(2\right) \le 2
\sqrt{n}
{\nabla }_{xx}^{2}L\left(x,\lambda \right)={\nabla }^{2}f\left(x\right)+\sum {\lambda }_{i}{\nabla }^{2}{c}_{i}\left(x\right)+\sum {\lambda }_{i}{\nabla }^{2}ce{q}_{i}\left(x\right). |
Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation
R. Ezzati, M. Khodabin, Z. Sadati, "Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation", Abstract and Applied Analysis, vol. 2014, Article ID 523163, 11 pages, 2014. https://doi.org/10.1155/2014/523163
R. Ezzati,1 M. Khodabin,1 and Z. Sadati1
An efficient method to determine a numerical solution of a stochastic differential equation (SDE) driven by fractional Brownian motion (FBM) with Hurst parameter and independent one-dimensional standard Brownian motion (SBM) is proposed. The method is stated via a stochastic operational matrix based on the block pulse functions (BPFs). With using this approach, the SDE is reduced to a stochastic linear system of equations and unknowns. Then, the error analysis is demonstrated by some theorems and defnitions. Finally, the numerical examples demonstrate applicability and accuracy of this method.
In many fields of science and engineering, there are a large number of problems which are intrinsically involving stochastic excitations of a Gaussian white noise type. Having in mind a Gaussian white noise mathematically described as a formal derivative of a Brownian motion process, all such problems are mathematically modeled by stochastic differential equations. Most of them cannot be solved analytically, so it is important to provide their numerical solutions. There has been a growing interest in numerical solutions of stochastic differential equations for the last years [1–10].
In the presented work, we consider SDE as follows: or where denotes the FBM with Hurst parameter on probability space and is independent one-dimensional SBM defined on the same probability space. Also, and is the stochastic process of unknown on the probability space.
Investigations concerning the SDE driven by the FBM have been done by Zähle [11], Coutin [12], Decreusefond and Üstünel [13], Nualart [4, 14], Lisei and Soós [15], and other authors. Also, there exist several ways for solving it, pathwise and related techniques, Dirichlet forms, Euler approximations, Malliavin calculus, and Skorohod integral [1, 4, 15–17]; almost all methods have very poor numerical convergence.
It is important to find approximate solutions of the stochastic equations driven by the FBM, since these equations cannot be solved analytically in most cases and have many applications in models arising in physics, telecommunication networks, and finance [18]. Also, we cannot use from the classical Ito theory for their stochastic calculus, since these processes are not Markovian and semimartingale. Hence, in this work, we implement the stochastic operational matrix based on the BPFs for solving (2). The benefits of this method are lower cost of setting up the system of equations; moreover, the computational cost of operations is low. Also, convergence of this method is faster than other methods. These advantages make the method easier to apply.
The rest of the paper is organized as follows. In Section 2, some essential definitions and the following assumptions on the coefficients of (2) are stated. Also, the necessary properties of the block pulse functions (BPFs) are introduced. In Section 3, first a theorem is proved; then (2) is reduced to a stochastic linear system by using the properties of the BPFs. In Section 4, the error analysis is demonstrated. Efficiency of this method and good reasonable degree of accuracy are confirmed by some numerical examples, in Section 5. Finally, in Section 6, a brief conclusion is given.
Definition 1. Let be the step function and denotes the characteristic function on , , and . Then, the wiener integral with respect to the FBM is defined as where and (see [19]).
Definition 2. Let denote the class of function on such that (1)the function is measurable;(2)the function is adapted to ;(3) and .
Let us consider the following assumptions on the coefficients.) is differentiable in and there exist constants and such that ()There exist constants such that ()There exist constants such that for all .
Theorem 3. Let , and hold in condition , , , and . Then, there exists a unique solution for (2).
Now, we review the main properties of the BPFs which are necessary for this paper. Note that the BPFs are discussed in [7, 8].
A function is approximated by using properties of the BPFs as where with where denotes the BPFs and with
A function is approximated as follows: where
In [8], it is proved that where
3. Solving the SDE Driven by FBM and Independent One-Dimensional SBM
Theorem 4. Let denote the BPFs, , and , ; then where
Proof. First, we compute stochastic operational matrix driven by the FBM based on the BPFs as follows.(A1)If , then (A2)If , the function is defined as where denotes the characteristic function and , where if and if . Also, Now, for computation (), we can write Then by using Definition 1, we obtain (A3)If , then where , if , if , and For computation (), we can write so, we get From (A1), (A2), and (A3), we get Furthermore, we suppose that so, we can write Hence, by using the relation (33), we can write where
Now, let be the th row of matrix , let be the ith row of the matrix , and let be the ith row of matrix . We have where is given by (21).
Let where and are the block pulse coefficients vector and , , and , are the block pulse coefficients matrix.
By substituting the relation (37) in (2), we get or Therefore, by using properties of the BPFs and Theorem 4, we can write where withwithwith Now, with replacing by , we have or where . Clearly, (46) is the stochastic linear system of equations and unknowns.
In [20], it is stated that if and , then
Theorem 5. Let be an arbitrary bounded function on and such that is the BPFs of . Then,
Let where is the approximate solution of defined in (46) and , , , and are approximated by using properties of the BPFs.
Theorem 7. Let be the approximate solution of (2) which is the solution of (46), , , , , and , for all . Then, where .
Proof. Consider by using , we can write First, by using the relation (47), we can write Cleary, we have and consequently, Hence, or Now, by using the property of the Ito isometry for the SBM defined in [21] and , we get By using Theorems 5 and 6, we can write By substituting the relation (60) in (59), we get or where and . If , we get Now, by using Gronwall inequality, we have or
The SDE driven by the FBM is applied in modeling the price of a stock with various Hurst parameters (see [18]). Hence, we show applicability and accuracy of this method in two numerical examples.
Example 1. Let us consider a SDE with the exact solution . The numerical results have been shown in Tables 1, 2, and 3 (with various Hurst parameters), where and are error mean and standard deviation of error, respectively.
%95 confidence interval for mean
0.05 1.6470 × 10−4 1.0272 × 10−4 1.1968 × 10−4 2.0972 × 10−4
0.1 2.1125 × 10−4 1.3531 × 10−4 1.5195 × 10−4 2.7055 × 10−4
Mean, standard deviation, and confidence interval for error mean ( , = ).
0.1 1.8920 × 10−4 1.7302 × 10−4 8.196 × 10−5 2.9644 × 10−4
Mean, standard deviation, and confidence interval for numerical solution mean ( , = ).
This paper presents a numerical comparison between the approximation solution of the SDE driven by the FBM with Hurst parameter and independent one-dimensional SBM and the exact solution of it. Also, the method is applied with two examples to illustrate the accuracy and implementation of the method.
The authors thank Islamic Azad University for supporting this work. The authors are also grateful to the anonymous referee for his/her constructive comments and suggestions.
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Copyright © 2014 R. Ezzati et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
Pareto Front for Two Objectives - MATLAB & Simulink - MathWorks Deutschland
Multiobjective Optimization with Two Objectives
Find Pareto Set Using Optimize Live Editor Task
Find Pareto Set at the Command Line
This example shows how to find a Pareto set for a two-objective function of two variables. The example presents two approaches for minimizing: using the Optimize Live Editor task and working at the command line.
The two-objective function f(x), where x is also two-dimensional, is
\begin{array}{l}{f}_{1}\left(x\right)={x}_{1}^{4}+{x}_{2}^{4}+{x}_{1}{x}_{2}-{\left(}^{{x}_{1}}-10{x}_{1}^{2}\\ {f}_{2}\left(x\right)={x}_{1}^{4}+{x}_{2}^{4}+{x}_{1}{x}_{2}-{\left(}^{{x}_{1}}.\end{array}
In the new section above the task, enter the following code to define the number of variables and lower and upper bounds.
Click the Constraints > Lower bounds and Upper bounds buttons.
Select Solver > gamultiobj - Multiobjective optimization using genetic algorithm.
In the Select problem data section, select Objective function > Local function, and then click the New button. The function appears in a new section below the task.
Edit the resulting function definition to contain the following code.
function f = mymulti1(x)
f(2) = x(1)^4 + x(2)^4 + x(1)*x(2) - (x(1)*x(2))^2;
f(1) = f(2) - 10*x(1)^2;
In the Select problem data section, select the Local function > mymulti1 function.
Select Lower bounds > From workspace > lb and Upper bounds > From workspace > ub.
Expand the Specify solver options section of the task, and then click the Add button. To have a denser, more connected Pareto front, specify a larger-than-default populations by selecting Population settings > Population size > 60.
To have more of the population on the Pareto front than the default settings, click the + button. In the resulting options, select Algorithm > Pareto set fraction > 0.7.
In the Display progress section of the task, select the Pareto front plot function.
To run the solver, click the options button ⁝ at the top right of the task window, and select Run Section. The plot appears in a separate figure window and in the task output area.
The plot shows the tradeoff between the two components of f, which is plotted in objective function space. For details, see the figure Figure 14-2, Set of Noninferior Solutions.
To perform the same optimization at the command line, complete the following steps.
Create the mymulti1 objective function file on your MATLAB® path.
Set the options and bounds.
options = optimoptions('gamultiobj','PopulationSize',60,...
'ParetoFraction',0.7,'PlotFcn',@gaplotpareto);
Run the optimization using the options.
[solution,ObjectiveValue] = gamultiobj(@mymulti1,2,...
[],[],[],[],lb,ub,options);
You can view this problem in other ways. The following figure contains a plot of the level curves of the two objective functions, the Pareto frontier calculated by gamultiobj (boxes), and the x-values of the true Pareto frontier (diamonds connected by a nearly straight line). The true Pareto frontier points are where the level curves of the objective functions are parallel. The algorithm calculates these points by finding where the gradients of the objective functions are parallel. The figure is plotted in parameter space; see Figure 14-1, Mapping from Parameter Space into Objective Function Space.
Contours of objective functions, and Pareto frontier
gamultiobj finds the ends of the line segment, meaning it finds the full extent of the Pareto frontier.
Create the solution data solution by running the optimization either in the Optimize Live Editor task or at the command line.
The Pareto frontier, the optimal tradeoff curve between the two objectives, is where the gradients of the two objective functions point in exactly opposite directions. The gradients are given in gg and gf in the following code:
function mout = mymulti2(x)
gg = [4*x(1)^3+x(2)-2*x(1)*(x(2)^2);
x(1)+4*x(2)^3-2*(x(1)^2)*x(2)];
gf = gg - [20*x(1);0];
mout = gf(1)*gg(2) - gf(2)*gg(1);
mout = 0 if the two gradients are parallel.
This code creates the plot.
[x,y] = meshgrid(0:.01:3,-2:.01:0);
[myf,myg] = mymulti(x,y);
mygg = sqrt(myg+.26); % This spaces the contours better
myff = sqrt(myf+40); % This spaces the contours better
a = [.7,fzero(@(t)mymulti2([.7,t]),[-2;0])];
% fzero calculates where mout = 0
for jj = 0.8:0.1:2.6
a = [a;[jj,fzero(@(t)mymulti2([jj,t]),[-2;0])]];
a = [a;[2.65,fzero(@(t)mymulti2([2.65,t]),[-2;0])]];
contour(x,y,mygg,50)
contour(x,y,myff,50)
gam = plot(solution(:,1),solution(:,2),'ks');
tru = plot(a(:,1),a(:,2),'-dr');
legend([gam tru],'Points calculated by \bf{\fontname{Courier}gamultiobj}',...
'True Pareto frontier')
The function mymulti(x,y) is a vectorized version of the multiobjective fitness function.
function [f,g] = mymulti(x,y)
g = x.^4 + y.^4 + x.*y - (x.^2).*(y.^2);
f = g - 10*x.^2;
gamultiobj | paretosearch |
Chemistry - EverybodyWiki Bios & Wiki
Chemistry is the scientific discipline involved with elements and compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo during a reaction with other substances.[1][2] In the scope of its subject, chemistry occupies an intermediate position between physics and biology.[3] It is sometimes called the central science because it provides a foundation for understanding both basic and applied scientific disciplines at a fundamental level.[4] For example, chemistry explains aspects of plant chemistry (botany), the formation of igneous rocks (geology), how atmospheric ozone is formed and how environmental pollutants are degraded (ecology), the properties of the soil on the moon (astrophysics), how medications work (pharmacology), and how to collect DNA evidence at a crime scene (forensics).
3.1 Of definition
3.2 Of discipline
The word chemistry comes from alchemy, which referred to an earlier set of practices that encompassed elements of chemistry, metallurgy, philosophy, astrology, astronomy, mysticism and medicine. It is often seen as linked to the quest to turn lead or another common starting material into gold,[5] though in ancient times the study encompassed many of the questions of modern chemistry being defined as the study of the composition of waters, movement, growth, embodying, disembodying, drawing the spirits from bodies and bonding the spirits within bodies by the early 4th century Greek-Egyptian alchemist Zosimos.[6] An alchemist was called a 'chemist' in popular speech, and later the suffix "-ry" was added to this to describe the art of the chemist as "chemistry".
The modern word alchemy in turn is derived from the Arabic word al-kīmīā (الكیمیاء). In origin, the term is borrowed from the Greek χημία or χημεία.[7][8] This may have Egyptian origins since al-kīmīā is derived from the Greek χημία, which is in turn derived from the word Kemet, which is the ancient name of Egypt in the Egyptian language.[7] Alternately, al-kīmīā may derive from χημεία, meaning "cast together".[9]
Modern principles[edit]
Compound[edit]
Substance and mixture[edit]
Mole and amount of substance[edit]
The mole is a unit of measurement that denotes an amount of substance (also called chemical amount). The mole is defined as the number of atoms found in exactly 0.012 kilogram (or 12 grams) of carbon-12, where the carbon-12 atoms are unbound, at rest and in their ground state.[24] The number of entities per mole is known as the Avogadro constant, and is determined empirically to be approximately 6.022×1023 mol−1.[25] Molar concentration is the amount of a particular substance per volume of solution, and is commonly reported in mol/dm3.[26]
Phase[edit]
Bonding[edit]
Chemical reactions are invariably not possible unless the reactants surmount an energy barrier known as the activation energy. The speed of a chemical reaction (at given temperature T) is related to the activation energy E, by the Boltzmann's population factor
{\displaystyle e^{-E/kT}}
– that is the probability of a molecule to have energy greater than or equal to E at the given temperature T. This exponential dependence of a reaction rate on temperature is known as the Arrhenius equation. The activation energy necessary for a chemical reaction to occur can be in the form of heat, light, electricity or mechanical force in the form of ultrasound.[28]
A related concept free energy, which also incorporates entropy considerations, is a very useful means for predicting the feasibility of a reaction and determining the state of equilibrium of a chemical reaction, in chemical thermodynamics. A reaction is feasible only if the total change in the Gibbs free energy is negative,
{\displaystyle \Delta G\leq 0\,}
; if it is equal to zero the chemical reaction is said to be at equilibrium.
Chemical reactions can result in the formation or dissociation of molecules, that is, molecules breaking apart to form two or more molecules or rearrangement of atoms within or across molecules. Chemical reactions usually involve the making or breaking of chemical bonds. Oxidation, reduction, dissociation, acid-base neutralization and molecular rearrangement are some of the commonly used kinds of chemical reactions.
Ions and salts[edit]
Acidity and basicity[edit]
A substance can often be classified as an acid or a base. There are several different theories which explain acid-base behavior. The simplest is Arrhenius theory, which states that acid is a substance that produces hydronium ions when it is dissolved in water, and a base is one that produces hydroxide ions when dissolved in water. According to Brønsted–Lowry acid-base theory, acids are substances that donate a positive hydrogen ion to another substance in a chemical reaction; by extension, a base is the substance which receives that hydrogen ion.
A third common theory is Lewis acid-base theory, which is based on the formation of new chemical bonds. Lewis theory explains that an acid is a substance which is capable of accepting a pair of electrons from another substance during the process of bond formation, while a base is a substance which can provide a pair of electrons to form a new bond. According to this theory, the crucial things being exchanged are charges.[32] There are several other ways in which a substance may be classified as an acid or a base, as is evident in the history of this concept.[33]
Chemical laws[edit]
Of definition[edit]
Of discipline[edit]
In the Hellenistic world the art of alchemy first proliferated, mingling magic and occultism into the study of natural substances with the ultimate goal of transmuting elements into gold and discovering the elixir of eternal life.[46] Work, particularly the development of distillation, continued in the early Byzantine period with the most famous practitioner being the 4th century Greek-Egyptian Zosimos of Panopolis.[47] Alchemy continued to be developed and practised throughout the Arab world after the Muslim conquests,[48] and from there, and from the Byzantine remnants,<ref>Marcelin Berthelot, Collection des anciens alchimistes grecs (3 vol., Paris, 1887–1888, p. 161);
↑ Carsten Reinhardt. Chemical Sciences in the 20th Century: Bridging Boundaries. Wiley-VCH, 2001. ISBN 3-527-30271-9 Search this book on .. pp. 1–2.
↑ Theodore L. Brown, H. Eugene Lemay, Bruce Edward Bursten, H. Lemay. Chemistry: The Central Science. Prentice Hall; 8 edition (1999). ISBN 0-13-010310-1 Search this book on .. pp. 3–4.
↑ "History of Alchemy". Alchemy Lab. Retrieved 2011-06-12.
↑ 7.0 7.1 "alchemy", entry in The Oxford English Dictionary, J.A. Simpson and E.S.C. Weiner, vol. 1, 2nd ed., 1989, ISBN 0-19-861213-3 Search this book on ..
↑ p. 854, "Arabic alchemy", Georges C. Anawati, pp. 853–885 in Encyclopedia of the history of Arabic science, eds. Roshdi Rashed and Régis Morelon, London: Routledge, 1996, vol. 3, ISBN 0-415-12412-3 Search this book on ..
↑ Weekley, Ernest (1967). Etymological Dictionary of Modern English. New York: Dover Publications. ISBN 0-486-21873-2 Search this book on .
↑ "chemical bonding". Britannica. Encyclopædia Britannica. Retrieved 1 November 2012.
↑ "California Occupational Guide Number 22: Chemists". Calmis.ca.gov. 1999-10-29. Archived from the original on 2011-06-10. Retrieved 2011-06-12.
↑ "General Chemistry Online – Companion Notes: Matter". Antoine.frostburg.edu. Retrieved 2011-06-12.
↑ Armstrong, James (2012). General, Organic, and Biochemistry: An Applied Approach. Brooks/Cole. p. 48. ISBN 978-0-534-49349-3. Search this book on
↑ "IUPAC Nomenclature of Organic Chemistry". Acdlabs.com. Retrieved 2011-06-12.
↑ Connelly, Neil G.; Damhus, Ture; Hartshorn, Richard M.; Hutton, Alan T. (2005). Nomenclature of Inorganic Chemistry IUPAC Recommendations 2005. RSCPublishing. pp. 5–12. ISBN 978-0-85404-438-2. Search this book on
↑ Hill, J.W.; Petrucci, R.H.; McCreary, T.W.; Perry, S.S. (2005). General Chemistry (4th ed.). Upper Saddle River, New Jersey: Pearson Prentice Hall. p. 37. Search this book on
↑ M.M. Avedesian; Hugh Baker. Magnesium and Magnesium Alloys. ASM International. p. 59. Search this book on
↑ Visionlearning. "Chemical Bonding by Anthony Carpi, Ph". visionlearning. Retrieved 2011-06-12.
↑ "The Lewis Acid-Base Concept". Apsidium. May 19, 2003. Archived from the original on 2008-05-27. Retrieved 2010-07-31. [unreliable source?]
↑ "History of Acidity". Bbc.co.uk. 2004-05-27. Retrieved 2011-06-12.
↑ Boyle, Robert (1661). The Sceptical Chymist. New York: Dover Publications, Inc. (reprint). ISBN 978-0-486-42825-3. Search this book on
↑ Glaser, Christopher (1663). Traite de la chymie. Paris. Search this book on as found in: Kim, Mi Gyung (2003). Affinity, That Elusive Dream – A Genealogy of the Chemical Revolution. The MIT Press. ISBN 978-0-262-11273-4. Search this book on
↑ Stahl, George, E. (1730). Philosophical Principles of Universal Chemistry. London. Search this book on
↑ Dumas, J.B. (1837). 'Affinite' (lecture notes), vii, p 4. "Statique chimique", Paris: Académie des Sciences
↑ Pauling, Linus (1947). General Chemistry. Dover Publications, Inc. ISBN 978-0-486-65622-9. Search this book on
↑ Chang, Raymond (1998). Chemistry, 6th Ed. New York: McGraw Hill. ISBN 978-0-07-115221-1. Search this book on
↑ Barnes, Ruth. Textiles in Indian Ocean Societies. Routledge. p. 1. Search this book on
↑ Lucretius (50 BCE). "de Rerum Natura (On the Nature of Things)". The Internet Classics Archive. Massachusetts Institute of Technology. Retrieved 9 January 2007. Check date values in: |date= (help)
↑ Simpson, David (29 June 2005). "Lucretius (c. 99–55 BCE)". The Internet History of Philosophy. Retrieved 2007-01-09.
↑ Strodach, George K. (2012). The Art of Happiness. New York: Penguin Classics. pp. 7–8. ISBN 978-0-14-310721-7. Search this book on
↑ "International Year of Chemistry – The History of Chemistry". G.I.T. Laboratory Journal Europe. Feb 25, 2011. Retrieved March 12, 2013.
↑ Bryan H. Bunch & Alexander Hellemans (2004). The History of Science and Technology. Houghton Mifflin Harcourt. p. 88. ISBN 978-0-618-22123-3. Search this book on
↑ Morris Kline (1985) Mathematics for the nonmathematician. Courier Dover Publications. p. 284. ISBN 0-486-24823-2 Search this book on .
Retrieved from "https://en.everybodywiki.com/index.php?title=Chemistry&oldid=275926" |
Chebyshev Type II filter design - MATLAB cheby2
0.6\pi
0.2\pi
0.6\pi
0.6\pi
2\pi
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{\text{b(1)}+\text{b(2)}\text{\hspace{0.17em}}{z}^{-1}+\cdots +\text{b(n+1)}\text{\hspace{0.17em}}{z}^{-n}}{\text{a(1)}+\text{a(2)}\text{\hspace{0.17em}}{z}^{-1}+\cdots +\text{a(n+1)}\text{\hspace{0.17em}}{z}^{-n}}.
H\left(s\right)=\frac{B\left(s\right)}{A\left(s\right)}=\frac{\text{b(1)}\text{\hspace{0.17em}}{s}^{n}+\text{b(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{b(n+1)}}{\text{a(1)}\text{\hspace{0.17em}}{s}^{n}+\text{a(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{a(n+1)}}.
H\left(z\right)=\text{k}\frac{\left(1-\text{z(1)}\text{\hspace{0.17em}}{z}^{-1}\right)\text{\hspace{0.17em}}\left(1-\text{z(2)}\text{\hspace{0.17em}}{z}^{-1}\right)\cdots \left(1-\text{z(n)}\text{\hspace{0.17em}}{z}^{-1}\right)}{\left(1-\text{p(1)}\text{\hspace{0.17em}}{z}^{-1}\right)\text{\hspace{0.17em}}\left(1-\text{p(2)}\text{\hspace{0.17em}}{z}^{-1}\right)\cdots \left(1-\text{p(n)}\text{\hspace{0.17em}}{z}^{-1}\right)}.
H\left(s\right)=\text{k}\frac{\left(s-\text{z(1)}\right)\text{\hspace{0.17em}}\left(s-\text{z(2)}\right)\cdots \left(s-\text{z(n)}\right)}{\left(s-\text{p(1)}\right)\text{\hspace{0.17em}}\left(s-\text{p(2)}\right)\cdots \left(s-\text{p(n)}\right)}.
\begin{array}{c}x\left(k+1\right)=\text{A}\text{\hspace{0.17em}}x\left(k\right)+\text{B}\text{\hspace{0.17em}}u\left(k\right)\\ y\left(k\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{C}\text{\hspace{0.17em}}x\left(k\right)+\text{D}\text{\hspace{0.17em}}u\left(k\right).\end{array}
\begin{array}{l}\stackrel{˙}{x}=\text{A}\text{\hspace{0.17em}}x+\text{B}\text{\hspace{0.17em}}u\\ y=\text{C}\text{\hspace{0.17em}}x+\text{D}\text{\hspace{0.17em}}u.\end{array} |
The focus of parabola is (1,5) and its directrix is x + y + 2 = 0 Find the - Maths - Conic Sections - 8415837 | Meritnation.com
The focus of parabola is (1,5) and its directrix is x + y + 2 = 0. Find the equation of the parabola, its vertex and length of latus rectum.
let F (1,5) be the focus of the parabola and the equation of the directrix is
x+y+2=0
let P(h.k) be any point on the parabola.
the distance from the focus to the point P = the distance from the directrix to the point P
PM=PF
\sqrt{\left(h-1{\right)}^{2}+\left(k-5{\right)}^{2}}=\left|\frac{h*1+k*1+2}{\sqrt{1+1}}\right|\phantom{\rule{0ex}{0ex}}\left(h-1{\right)}^{2}+\left(k-5{\right)}^{2}=\frac{\left(h+k+2{\right)}^{2}}{2}\phantom{\rule{0ex}{0ex}}2*\left[{h}^{2}+1-2h+{k}^{2}+25-10k\right]={h}^{2}+{k}^{2}+4+2hk+4k+4h\phantom{\rule{0ex}{0ex}}2{h}^{2}+2{k}^{2}-4h-20k+52={h}^{2}+{k}^{2}+4+2hk+4k+4h\phantom{\rule{0ex}{0ex}}{h}^{2}+{k}^{2}-8h-24k-2hk+48=0
put h = x and k = y
{x}^{2}+{y}^{2}-8x-24y-2xy+48=0
is the required equation of the parabola.
the axis of parabola passes through the focus and perpendicular to the directrix.
the slope of axis =
-\left(-1\right)=1
[since slope of directrix is -1]
the equation of axis is
y-5=1*\left(x-1\right)\phantom{\rule{0ex}{0ex}}y-5=x-1\phantom{\rule{0ex}{0ex}}x-y+4=0 ...........\left(2\right)\phantom{\rule{0ex}{0ex}}
the point of intersection of axis and directrix is intersection point of eq(1) and eq(2)
x+y+2=0\phantom{\rule{0ex}{0ex}}x-y+4=0\phantom{\rule{0ex}{0ex}}2x+6=0⇒x=-3\phantom{\rule{0ex}{0ex}}-3+y+2=0\phantom{\rule{0ex}{0ex}}⇒y=1
the coordinates of k are (-3,1).
vertex is the mid-point of focus and point K
i.e. coordinates of vertex are
\left(\frac{1-3}{2},\frac{5+1}{2}\right)i.e. \left(-1,3\right)
now distance from vertex to focus =
\sqrt{\left(-1-1{\right)}^{2}+\left(3-5{\right)}^{2}}=\sqrt{4+4}=2\sqrt{2}
the length of the latusrectum =
4*2\sqrt{2}=8\sqrt{2} |
The Uniqueness of Solutions of a System of Functional Equations in Some Classes of Functions.
J. Matkowski — 1972
On the Continuous Dependences of Cr Solutions of a Functional Equation on the Given Functions
A functional inequality characterizing convex functions, conjugacy and a generalization of Hölder's and Minkowski's inequalities.
Correction to the paper "On the continuous dependence of local analytic solutions of a functional equation on given functions''
On the continuous dependence of local analytic solutions of a functional equation on given functions
On the continuous dependence of solutions of a functional equation on given functions
On the continuous dependence of local analytic solutions of the functional equation in the non-uniqueness case
On meromorphic solutions of a functional equation, II
On meromorphic solutions of a functional equation
A fixed point theorem for non-expansive mappings on compact metric spaces.
J. Matkowski; K. Baron — 1973
On the existence of a convex solution of the functional equation φ(x) = h(x,φ[f(x)])
Z. Kominek; J. Matkowski — 1974
Solutions of a functional equation in a special class of functions
M. Kuczma; J. Matkowski — 1972
Subadditive functions and partial converses of Minkowski's and Mulholland's inequalities
J. Matkowski; T. Świątkowski — 1993
Let ϕ be an arbitrary bijection of
{ℝ}_{+}
. We prove that if the two-place function
{\varphi }^{-1}\left[\varphi \left(s\right)+\varphi \left(t\right)\right]
is subadditive in
{ℝ}_{+}^{2}
\varphi
must be a convex homeomorphism of
{ℝ}_{+}
. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of
{ℝ}_{+}
are also given. We apply the above results to obtain several converses of Minkowski’s inequality.
Some remarks on a problem of C. Alsina.
J. Matkowski; M. Sablik — 1986
Equation [1] f(x+y) + f (f(x)+f(y)) = f (f(x+f(y)) + f(f(x)+y)) has been proposed by C. Alsina in the class of continuous and decreasing involutions of (0,+∞). General solution of [1] is not known yet. Nevertheless we give solutions of the following equations which may be derived from [1]: [2] f(x+1) + f (f(x)+1) = 1, [3] f(2x) + f(2f(x)) = f(2f(x + f(x))). Equation [3] leads to a Cauchy functional equation: [4]...
On a class of composite functional equations in a single variable.
J. Matkowski; P. Kahlig; A. Matkowska — 1996
Local operators and a characterization of the Volterra operator.
Matkowski, J. — 2010
Annals of Functional Analysis (AFA) [electronic only]
Uniformly continuous composition operators in the space of functions of two variables of bounded
\varphi
-variation in the sense of Wiener
J. A. Guerrero; J. Matkowski; N. Merentes — 2010
Assume that the generator of a Nemytskii composition operator is a function of three variables: the first two real and third in a closed convex subset of a normed space, with values in a real Banach space. We prove that if this operator maps a certain subset of the Banach space of functions of two real variables of bounded Wiener
\varphi
-variation into another Banach space of a similar type, and is uniformly continuous, then the one-sided regularizations of the generator are affine in the third variable....
1 Annals of Functional Analysis (AFA) [electronic only]
17 Matkowski, J
1 Guerrero, JA
1 Kahlig, P
1 Kominek, Z
1 MATKOWSKI, J
1 Matkowska, A
1 Merentes, N
1 Sablik, M
1 Świątkowski, T |
Derived-Term Automata of Multitape Expressions with Composition - LRDE
“A.I. Cuza” University Press, Iac si
Rational expressions are powerful tools to define automata, but often restricted to single-tape automata. Our goal is to unleash their expressive power for transducersand more generally, any multitape automaton; for instance
{\displaystyle (a^{+}{\mathbin {\vert }}x+b^{+}{\mathbin {\vert }}y)^{*}}
. We generalize the construction of the derived-term automaton by using expansions. This approach generates small automata, and even allows us to support a composition operator.
@Article{ demaille.17.sacs,
title = {Derived-Term Automata of Multitape Expressions with
journal = {Scientific Annals of Computer Science},
organization = {``A.I. Cuza'' University, Ia\c si, Rom\^ania},
doi = {10.7561/SACS.2017.2.137},
publisher = {``A.I. Cuza'' University Press, Ia\c si},
abstract = {Rational expressions are powerful tools to define
automata, but often restricted to single-tape automata. Our
goal is to unleash their expressive power for transducers,
and more generally, any multitape automaton; for instance
$(a^+\mathbin{\vert} x + b^+\mathbin{\vert} y)^*$. We
generalize the construction of the derived-term automaton
by using \emph{expansions}. This approach generates small
automata, and even allows us to support a composition
operator.}
Retrieved from "https://www.lrde.epita.fr/index.php?title=Publications/demaille.17.sacs&oldid=122557" |
Price Stability - XUSD.Money
XUSD can always be minted and redeemed from the system for $1 of value. This allows arbitragers to balance the demand and supply of XUSD in the open market. If the market price of XUSD is above the price target of $1, then there is an arbitrage opportunity to mint XUSD tokens by placing $1 of value into the system per XUSD and sell the minted XUSD for over $1 in the open market. At all times in order to mint new XUSD a user must place $1 worth of value into the system. The difference is simply what proportion of collateral and XUS makes up that $1 of value. When XUSD is in the 100% collateral phase, 100% of the value that is put into the system to mint XUSD is collateral. As the protocol moves into the fractional phase, part of the value that enters into the system during minting becomes XUS (which is then burned from circulation). For example, in a 98% collateral ratio, every XUSD minted requires $.98 of collateral and burning $.02 of XUS. In a 97% collateral ratio, every XUSD minted requires $.97 of collateral and burning $.03 of XUS, and so on.
If the market price of XUSD is below the price range of $1, then there is an arbitrage opportunity to redeem XUSD tokens by purchasing cheaply on the open market and redeeming XUSD for $1 of value from the system. At all times, a user is able to redeem XUSD for $1 worth of value from the system. The difference is simply what proportion of the collateral and XUS is returned to the redeemer. When XUSD is in the 100% collateral phase, 100% of the value returned from redeeming XUSD is collateral. As the protocol moves into the fractional phase, part of the value that leaves the system during redemption becomes XUS (which is minted to give to the redeeming user). For example, in a 98% collateral ratio, every XUSD can be redeemed for $.98 of collateral and $.02 of minted XUS. In a 97% collateral ratio, every XUSD can be redeemed for $.97 of collateral and $.03 of minted XUS.
The XUSD redemption process is seamless, easy to understand, and economically sound. During the 100% phase, it is trivially simple. During the fractional-algorithmic phase, as XUSD is minted, XUS is burned. As XUSD is redeemed, XUS is minted. As long as there is a demand for XUSD, redeeming it for collateral plus XUS simply initiates minting of a similar amount of XUSD into circulation on the other end (which burns a similar amount of XUS). Thus, the XUS token’s value is determined by the demand for XUSD. The value that accrues to the XUS market cap is the summation of the non-collateralized value of XUSD’s market cap. This is the summation of all past and future shaded areas under the curve displayed as follows.
The demand-supply curve illustrates how minting and redeeming XUSD keeps the price stabilized (q is quantity, p is price). At
CD_0
the price of XUSD is at
q_0
. If there is more demand for XUSD, the curve shifts right to
CD_1
and a new price,
p_1
, for the same quantity
q_0
. In order to recover the price to $1, new XUSD must be minted until
q_1
is reached and the
p_0
price is recovered. Since market capitalization is calculated as price times quantity, the market cap of XUSD at q0 is the blue square. The market cap of XUSD at
q_1
is the sum of the areas of the blue square and green square. Notice that in this example the new market cap of XUSD would have been the same if the quantity did not increase because the increase in demand is simply reflected in the price,
p_1
. Given an increase in demand, the market cap increases either through an increase in price or an increase in quantity (at a stable price). This is clear because the red square and green square have the same area and thus would have added the same amount of value to the market cap. Note: the semi-shaded portion in the green square denotes the total value of XUS shares that would be burned if the new quantity of XUSD was generated at a hypothetical collateral ratio of 66%. This is important to visualize because XUS market cap is intrinsically linked to the demand for XUSD.
Lastly, it’s important to note that XUSD is an agnostic protocol. It makes no assumptions about what collateral ratio the market will settle on in the long-term. It could be the case that users simply do not have confidence in a stablecoin with 0% collateral that’s entirely algorithmic. The protocol does not make any assumptions about what that ratio is and instead keeps the ratio at what the market demands pricing XUSD at $1. It could be the case that the protocol only ever reaches, for example, a 60% collateral ratio, and only 40% of the XUSD supply is algorithmically stabilized while over half of it is backed by collateral. The protocol only adjusts the collateral ratio as a result of demand for more XUSD and changes in XUSD price. When the price of XUSD falls below $1, the protocol recollateralizes and increases the ratio until confidence is restored and the price recovers. It will not decollateralize the ratio unless demand for XUSD increases again. It could even be possible that XUSD becomes entirely algorithmic but then recollateralizes to a substantial collateral ratio should market conditions demand. We believe this deterministic and reflexive protocol is the most elegant way to measure the market’s confidence in a non-backed stablecoin. Previous algorithmic stablecoin attempts had no collateral within the system on day 1 (and never used collateral in any way). Such previous attempts did not address the lack of market confidence in an algorithmic stablecoin on day 1. It should be noted that even USD, which XUSD is pegged to, was not a fiat currency until it had global prominence.
The protocol adjusts the collateral ratio during times of XUSD expansion and retraction. During times of expansion, the protocol decollateralizes (lowers the ratio) the system so that less collateral and more XUS must be deposited to mint XUSD. This lowers the amount of collateral backing all XUSD. During times of retraction, the protocol recollateralizes (increases the ratio). This increases the ratio of collateral in the system as a proportion of XUSD supply, increasing market confidence in XUSD as its backing increases.
At genesis, the protocol adjusts the collateral ratio once every hour by a step of .25%. When XUSD is at or above $1, the function lowers the collateral ratio by one step per hour and when the price of XUSD is below $1, the function increases the collateral ratio by one step per hour. This means that if XUSD price is at or over $1 a majority of the time through some time frame, then the net movement of the collateral ratio is decreasing. If XUSD price is under $1 a majority of the time, then the collateral ratio is increasing toward 100% on average.
In a future protocol update, the price feeds for collateral can be deprecated and the minting process can be moved to an auction based system to limit reliance on price data and further decentralize the protocol. In such an update, the protocol would run with no price data required for any asset including XUSD and XUS. Minting and redemptions would happen through open auction blocks where bidders post the highest/lowest ratio of collateral plus XUS they are willing to mint/redeem XUSD for. This auction arrangement would lead to collateral price discovery from within the system itself and not require any price information via oracles. Another possible design instead of auctions could be using PID-controllers to provide arbitrage opportunities for minting and redeeming XUSD similar to how a Uniswap trading pair incentivizes pool assets to keep a constant ratio that converges to their open market target price. |
Conal Elliott » number
Parallel speculative addition via memoization
I’ve been thinking much more about parallel computation for the last couple of years, especially since starting to work at Tabula a year ago. Until getting into parallelism explicitly, I’d naïvely thought that my pure functional programming style was mostly free of sequential bias. After all, functional programming lacks the implicit accidental dependencies imposed by the imperative model. Now, however, I’m coming to see that designing parallel-friendly algorithms takes attention to minimizing the depth of the remaining, explicit data dependencies.
As an example, consider binary addition, carried out from least to most significant bit (as usual). We can immediately compute the first (least significant) bit of the result, but in order to compute the second bit, we’ll have to know whether or not a carry resulted from the first addition. More generally, the
\left(n+1\right)
th sum & carry require knowing the
n
th carry, so this algorithm does not allow parallel execution. Even if we have one processor per bit position, only one processor will be able to work at a time, due to the linear chain of dependencies.
One general technique for improving parallelism is speculation—doing more work than might be needed so that we don’t have to wait to find out exactly what will be needed. In this post, we’ll see a progression of definitions for bitwise addition. We’ll start with a linear-depth chain of carry dependencies and end with logarithmic depth. Moreover, by making careful use of abstraction, these versions will be simply different type specializations of a single polymorphic definition with an extremely terse definition.
Continue reading ‘Parallel speculative addition via memoization’ »
I’ve been thinking much more about parallel computation for the last couple of years, especially since starting to work at Tabula a year ago. Until getting into parallelism explicitly, I’d...
Tags: number, parallelism, speculation | 6 Comments |
Factoring Polynomials: Special Cases | Brilliant Math & Science Wiki
Factoring Polynomials: Special Cases
Factoring is the process of rewriting a sum as a product. It allows us to simplify expressions and solve equations.
For example, the quadratic expression
x^2+4x+4,
which is written as a sum, may be expressed as a product
(x+2)(x+2),
7\times 2,
6+8.
A perfect square polynomial is one that can be written as the product of two identical factors. The perfect square identities below are widely used in algebra.
(a+b)^2 = a^2 + 2ab + b^2
(a-b)^2=a^2-2ab-b^2
4x^2+12x+9
Let's begin by looking at the first term in our quadratic,
4x^2.
4x^2
is a perfect square because
(2x)(2x)=4x^2.
Next, we can look at the last term in our quadratic,
9.
9
(3)(3)=9.
If the square root of the first term multiplied by the square root of the last term multiplied by
2
equals the middle term, then our quadratic is a perfect square. Because
(2x)(3)(2)=12x,
the quadratic
4x^2+12x+9
4x^2+12x+9
factors into
(2x+3)(2x+3) = (2x+3)^2.
9x^2-6x+1.
9x^2
3x.
1
1.
9x^2-6x+1
(3x-1)(3x-1) = (3x-1)^2.
x
(x+3 ) ^ 2
The difference of squares identity shows how every polynomial that is a difference between two perfect squares can be rewritten in the following factored form:
a^2-b^2=(a+b)(a-b).
Let's begin with the left side of the expression. We have
\begin{aligned} (a+b)(a-b) &= a(a-b) + b(a-b) \\ &= a^2 - ab + ab - b^2 \\ & = a^2 - b^2, \end{aligned}
which is equal to the right side of the identity. Hence proved.
_\square
25y^2-49.
25y^2
5y.
49
7.
25y^2-49
(5y-7)(5y+7).
299\times 301
You can brute force the answer to this problem by using a calculator, but we have a sweeter way. We can apply the difference of two squares identity.
At first we may think about using the long multiplication method, but it wastes time and is, of course, boring. Notice that
299=300-1
301=300+1
\begin{aligned}299\times 301&=(300-1)(300+1)\\&=300^2-1^2\\&=89999. \ _\square \end{aligned}
x^2-y^2=(x-y)(x+y)
31^2-19^2
Every polynomial that is a sum or difference of two perfect cubes can be rewritten in the following factored form:
\begin{aligned} x^3 - y^3 &= (x-y) ( x^2 + xy + y^2) \\ x^3 + y^3 &= (x+y) ( x^2 - xy + y^2). \end{aligned}
x^3+8.
x^3+8
x^3
2^3
x^3 + 8 = ( x + 2) (x ^2 - 2x + 2^2) = (x+2)( x^2 - 2x + 4 ).\ _\square
\large {x^3 - y^3} = {(x-y)(x^2-xy+y^2)}
A perfect cube polynomial is one that can be written as the product of three identical factors. The perfect cube identities below are widely used in algebra.
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
(a-b)^3=a^3-3a^2b + 3ab^2-b^3
(a-2b) ^ 3
\begin{aligned} ( a - 2b)^3 &= a^3 - 3 \times a^2 \times (2b) + 3 \times a \times (2b)^2 - (2b)^3 \\ &= a^3 - 6a^2b + 12a b^2 - 8 b^3.\ _\square \end{aligned}
( 2 + \sqrt{3})^3 + ( 2 - \sqrt{3})^3.
Cite as: Factoring Polynomials: Special Cases. Brilliant.org. Retrieved from https://brilliant.org/wiki/factoring-polynomials-special-cases/ |
Convert symbolic expression to function handle or file - MATLAB matlabFunction - MathWorks Italia
Convert Symbolic Expression to Anonymous Function
Convert Symbolic Function to Anonymous Function
Write MATLAB Function to File with Comments
Generate Sparse Matrices
Specify Input Arguments for Generated Function
Generate Function with Vector Input Arguments
Specify Output Variables
Speed Up Evaluation by Converting Symbolic Function to Anonymous Function
Convert symbolic expression to function handle or file
g = matlabFunction(f) converts the symbolic expression or function f to a MATLAB® function with handle g. If there is an equivalent MATLAB function operating on double data type for the symbolic expression or function, then the converted function can be used without Symbolic Math Toolbox™.
g = matlabFunction(f1,...,fN) converts f1,...,fN to a MATLAB function with N outputs. The function handle is g. Each element of f1,...,fN can be a symbolic expression, function, or a vector of symbolic expressions or functions.
g = matlabFunction(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments. You can specify Name,Value after the input arguments used in the previous syntaxes.
Convert the symbolic expression r to a MATLAB function with the handle ht. The converted function can be used without Symbolic Math Toolbox.
Convert multiple symbolic expressions using comma-separated input.
Create a symbolic function and convert it to a MATLAB function with the handle ht.
Write the generated MATLAB function to a file by specifying the File option. Existing files are overwritten. When writing to a file, matlabFunction optimizes the code using intermediate variables named t0, t1, …. Include comments in the file by using the Comments option.
Write the MATLAB function generated from f to the file myfile.
Include the comment Version: 1.1 in the file.
When you convert a symbolic expression to a MATLAB function and write the resulting function to a file, matlabFunction optimizes the code by default. This approach can help simplify and speed up further computations that use the file. However, generating the optimized code from some symbolic expressions and functions can be time-consuming. Use Optimize to disable code optimization.
Convert r to a MATLAB function and write the function to the file myfile. By default, matlabFunction creates a file containing the optimized code.
When you convert a symbolic matrix to a MATLAB function, matlabFunction represents it by a dense matrix by default. If most of the elements of the input symbolic matrix are zeros, the more efficient approach is to represent it by a sparse matrix.
Create a 3-by-3 symbolic diagonal matrix.
Convert A to a MATLAB function representing a numeric matrix, and write the result to the file myfile1. By default, the generated MATLAB function creates the dense numeric matrix specifying each element of the matrix, including all zero elements.
Convert A to a MATLAB function by setting Sparse to true. Now, the generated MATLAB function creates the sparse numeric matrix specifying only nonzero elements and assuming that all other elements are zeros.
When converting an expression to a MATLAB function, you can specify the order of the input arguments of the resulting function. You can also specify that some input arguments are vectors instead of scalar variables.
Convert r to a MATLAB function and write this function to the file myfile. By default, matlabFunction uses alphabetical order of input arguments when converting symbolic expressions that contain only lowercase letters for the variable names. The generated input arguments are scalar variables x, y, and z.
Specify the Vars argument as the vector [y z x] to modify the order of input arguments for the generated MATLAB function. The generated input arguments are scalar variables y, z, and x.
Now, convert an expression r to a MATLAB function whose input arguments are a scalar and a vector. Specify the Vars argument as the cell array {t,[y z x]}. The generated input arguments are a scalar variable t and a 1-by-3 vector variable in2.
To generate a MATLAB function with input arguments that are vector variables, specify the name-value argument Vars as a cell array.
Create a symbolic expression that finds the dot product of two 1-by-3 vectors.
syms x y [1 3] real
f = dot(x,y);
Convert the expression f to a MATLAB function. Specify Vars as a cell array {x,y}. The generated input arguments are two 1-by-3 vector variables in1 and in2 that correspond to x and y, respectively.
matlabFunction(f,'File','myfile','Vars',{x,y});
function f = myfile(in1,in2)
% F = MYFILE(IN1,IN2)
x1 = in1(:,1);
y1 = in2(:,1);
f = x1.*y1+x2.*y2+x3.*y3;
When converting a symbolic expression to a MATLAB function, you can specify the names of the output variables. Note that matlabFunction without the File argument (or with a file path specified by an empty character vector) creates a function handle and ignores the Outputs flag.
Create symbolic expressions r and q.
Convert r and q to a MATLAB function and write the resulting function to a file myfile, which returns a vector of two elements, name1 and name2.
You can speed up the evaluation of a symbolic function at given coordinates by converting the symbolic function to an anonymous MATLAB® function. Use matlabFunction to perform the conversion. Evaluation of a symbolic function returns symbolic numbers that are precise, while evaluation of a MATLAB function returns double-precision numbers.
Create a symbolic function f(x,y,z) that is a function of x, y, and z.
Create 3-D grid coordinates at the specified intervals.
Evaluate the symbolic function at these coordinates. Measure the elapsed time using a pair of tic and toc calls.
Here, evaluation is slow, but it returns symbolic numbers that are precise. Show a sample of the results.
\left(\begin{array}{cc}20 \mathrm{sin}\left(1\right)+\mathrm{cos}\left(1\right) \mathrm{sin}\left(20\right)-8000& 20 \mathrm{sin}\left(2\right)+2 \mathrm{cos}\left(1\right) \mathrm{sin}\left(20\right)-8000\\ 40 \mathrm{sin}\left(1\right)+\mathrm{cos}\left(2\right) \mathrm{sin}\left(20\right)-8000& 40 \mathrm{sin}\left(2\right)+2 \mathrm{cos}\left(2\right) \mathrm{sin}\left(20\right)-8000\end{array}\right)
To speed up the evaluation of the function, convert the symbolic function to a MATLAB function using matlabFunction. Evaluate the MATLAB function at the same coordinates.
Here, evaluation is faster. The evaluated MATLAB function returns double-precision numbers. Show a sample of the results.
f — Symbolic input to be converted to MATLAB function
Symbolic input to be converted to a MATLAB function, specified as a symbolic expression, function, vector, or matrix. When converting sparse symbolic vectors or matrices, use the name-value pair argument 'Sparse',true.
f1,...,fN — Symbolic input to be converted to MATLAB function with N outputs
Symbolic input to be converted to MATLAB function with N outputs, specified as several symbolic expressions, functions, vectors, or matrices, separated by comma.
matlabFunction does not create a separate output argument for each element of a symbolic vector or matrix. For example, g = matlabFunction([x + 1, y + 1]) creates a MATLAB function with one output argument, while g = matlabFunction(x + 1, y + 1) creates a MATLAB function with two output arguments.
Example: matlabFunction(f,'File','myfile','Optimize',false)
File — Path to file containing generated MATLAB function
Path to the file containing the generated MATLAB function, specified as a character vector. The generated function accepts arguments of type double, and can be used without Symbolic Math Toolbox. If File is empty, matlabFunction generates an anonymous function. If File does not end in .m, the function appends .m.
When writing to a file, matlabFunction optimizes the code using intermediate variables named t0, t1, .... To disable code optimization, use the Optimize argument.
See Write MATLAB Function to File with Comments.
When writing to a file, ccode optimizes the code using intermediate variables named t0, t1, ....
matlabFunction without the File argument (or with a file path specified by an empty character vector) creates a function handle. In this case, the code is not optimized. If you try to enforce code optimization by setting Optimize to true, then matlabFunction throws an error.
Flag that switches between sparse and dense matrix generation, specified as true or false. When you specify 'Sparse',true, the generated MATLAB function represents symbolic matrices by sparse numeric matrices. Use 'Sparse',true when you convert symbolic matrices containing many zero elements. Often, operations on sparse matrices are more efficient than the same operations on dense matrices.
See Generate Sparse Matrices.
Vars — Order of input variables or vectors in generated MATLAB function
Order of input variables or vectors in a generated MATLAB function, specified as a character vector, a vector of symbolic variables, or a one-dimensional cell array of character vectors, symbolic variables, or vectors of symbolic variables.
The number of specified input variables must equal or exceed the number of free variables in f. Do not use the same names for the input variables specified by Vars and the output variables specified by Outputs.
By default, when you convert symbolic expressions that contain only lowercase letters for the variable names, the order is alphabetical. When you convert symbolic functions, their input arguments appear in front of other variables, and all other variables are sorted alphabetically.
Specify Vars as a vector to generate a MATLAB function with input arguments that are scalar variables. Specify Vars as a cell array to generate a MATLAB function with input arguments that are a combination of scalar and vector variables.
See Specify Input Arguments for Generated Function.
Outputs — Names of output variables
one-dimensional cell array of character vectors
Names of output variables, specified as a one-dimensional cell array of character vectors.
If you do not specify the output variable names, then they coincide with the names you use when calling matlabFunction. If you call matlabFunction using an expression instead of individual variables, the default names of output variables consist of the word out followed by a number, for example, out3.
Do not use the same names for the input variables specified by Vars and the output variables specified by Outputs.
matlabFunction without the File argument (or with a file path specified by an empty character vector) creates a function handle. In this case, matlabFunction ignores the Outputs flag.
See Specify Output Variables.
g — Function handle that can serve as input argument to numerical functions
Function handle that can serve as an input argument to numerical functions, returned as a MATLAB function handle.
Some symbolic functions that have no corresponding MATLAB functions operating on double data type, such as simplify and solve, are kept as symbolic functions in the converted MATLAB function handle or file. The converted file that consists of these functions cannot be deployed using MATLAB Coder™ or MATLAB Compiler™. You need to create your own functions with double data type to replace these symbolic functions. If you are interested in a symbolic function that cannot be deployed, please contact MathWorks Technical Support.
When you use the File argument, use rehash to make the generated function available immediately. rehash updates the MATLAB list of known files for directories on the search path.
If the File option is empty, then an anonymous function is returned.
Use matlabFunction to convert one or more symbolic expressions to a MATLAB function and write the resulting function to an M-file. You can then use the generated M-file to create standalone applications and web apps using MATLAB Compiler. For example, see Deploy Generated MATLAB Functions from Symbolic Expressions with MATLAB Compiler.
Use matlabFunction to convert one or more symbolic expressions to a MATLAB function and write the resulting function to an M-file. You can then use the generated M-file to create C or C++ code using MATLAB Coder app. For example, see Generate C Code from Symbolic Expressions Using the MATLAB Coder App.
To generate a MATLAB function with input arguments that are a combination of scalar and vector variables, specify the name-value argument Vars as a cell array. For examples, see Specify Input Arguments for Generated Function and Generate Function with Vector Input Arguments. |
Measurement scale based on orders of magnitude
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a scale is nonlinear: the numbers 10 and 20, and 60 and 70, are not the same distance apart on a log scale. Rather, the numbers 10 and 100, and 60 and 600 are equally spaced. Thus moving a unit of distance along the scale means the number has been multiplied by 10 (or some other fixed factor). Often exponential growth curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small graph. Another way to think about it is that the number of digits of the data grows at a constant rate. For example, the numbers 10, 100, 1000, and 10000 are equally spaced on a log scale, because their numbers of digits is going up by 1 each time: 2, 3, 4, and 5 digits. In this way, adding two digits multiplies the quantity measured on the log scale by a factor of 100.
A logarithmic scale from 0.1 to 100
Semi-log plot of the Internet host count over time shown on a logarithmic scale
The markings on slide rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales.
The two logarithmic scales of a slide rule
The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value:
Richter magnitude scale and moment magnitude scale (MMS) for strength of earthquakes and movement in the Earth
A logarithmic scale makes it easy to compare values that cover a large range, such as in this map.
Sound level, with units decibel
Neper for amplitude, field and power quantities
Frequency level, with units cent, minor second, major second, and octave for the relative pitch of notes in music
Logit for odds in statistics
Counting f-stops for ratios of photographic exposure
The rule of nines used for rating low probabilities
Entropy in thermodynamics
Information in information theory
Particle size distribution curves of soil
Map of the solar system and distance to Alpha Centauri using a logarithmic scale.
The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:
pH for acidity
Stellar magnitude scale for brightness of stars
Krumbein scale for particle size in geology
Absorbance of light by transparent samples
Some of our senses operate in a logarithmic fashion (Weber–Fechner law), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of hearing perceives equal ratios of frequencies as equal differences in pitch. In addition, studies of young children in an isolated tribe have shown logarithmic scales to be the most natural display of numbers in some cultures.[1]
Various scales: lin–lin, lin–log, log–lin, and log–log. Plotted graphs are: y = 10 x (red), y = x (green), y = loge(x) (blue).
The top left graph is linear in the X and Y axes, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y axis of the bottom left graph, and the Y axis ranges from 0.1 to 1,000.
The top right graph uses a log-10 scale for just the X axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y axis.
Presentation of data on a logarithmic scale can be helpful when the data:
covers a large range of values, since the use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size;
may contain exponential laws or power laws, since these will show up as straight lines.
A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. The geometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic graph paper was a commonly used scientific tool.
Log–log plots
Main article: log–log plot
Plot on log–log scale of equation of a line
If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot.
Semi-logarithmic plots
Main article: Semi-log plot
If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot.
A modified log transform can be defined for negative input (y<0) and to avoid the singularity for zero input (y=0) so as to produce symmetric log plots:[2][3]
{\displaystyle Y=\operatorname {sgn}(y)\cdot \log _{10}(1+|y/C|)}
for a constant C=1/ln(10).
A logarithmic unit is a unit that can be used to express a quantity (physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the logarithm.
Examples of logarithmic units include units of data storage capacity (bit, byte), of information and information entropy (nat, shannon, ban), and of signal level (decibel, bel, neper). Logarithmic frequency quantities are used in electronics (decade, octave) and for music pitch intervals (octave, semitone, cent, etc.). Other logarithmic scale units include the Richter magnitude scale point.
In addition, several industrial measures are logarithmic, such as standard values for resistors, the American wire gauge, the Birmingham gauge used for wire and needles, and so on.
Units of level or level difference
Further information: Level (logarithmic quantity)
bel, decibel
Units of frequency interval
decade, decidecade, savart
octave, tone, semitone, cent
Underlying quantity
bit 2 number of possible messages quantity of information
byte 28 = 256 number of possible messages quantity of information
decibel 10(1/10) ≈ 1.259 any power quantity (sound power, for example) sound power level (for example)
decibel 10(1/20) ≈ 1.122 any root-power quantity (sound pressure, for example) sound pressure level (for example)
semitone 2(1/12) ≈ 1.059 frequency of sound pitch interval
The two definitions of a decibel are equivalent, because a ratio of power quantities is equal to the square of the corresponding ratio of root-power quantities.[citation needed]
^ "Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space". ScienceDaily. 2008-05-30. Retrieved 2008-05-31.
^ Webber, J Beau W (2012-12-21). "A bi-symmetric log transformation for wide-range data" (PDF). Measurement Science and Technology. IOP Publishing. 24 (2): 027001. doi:10.1088/0957-0233/24/2/027001. ISSN 0957-0233.
^ "Symlog Demo". Matplotlib 3.4.2 documentation. 2021-05-08. Retrieved 2021-06-22.
Dehaene, Stanislas; Izard, Véronique; Spelke, Elizabeth; Pica, Pierre (2008). "Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures". Science. 320 (5880): 1217–20. Bibcode:2008Sci...320.1217D. doi:10.1126/science.1156540. PMC 2610411. PMID 18511690.
Ries, Clemens (1962). Normung nach Normzahlen [Standardization by preferred numbers] (in German) (1 ed.). Berlin, Germany: Duncker & Humblot Verlag [de]. ISBN 978-3-42801242-8. (135 pages)
Paulin, Eugen (2007-09-01). Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! [Logarithms, preferred numbers, decibel, neper, phon - naturally related!] (PDF) (in German). Archived (PDF) from the original on 2016-12-18. Retrieved 2016-12-18.
"GNU Emacs Calc Manual: Logarithmic Units". Gnu.org. Retrieved 2016-11-23.
Wikimedia Commons has media related to Logarithmic scale. |
Then increase A<sub>r</sub> accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging. This assumes that the water storage reservoir area and permeable pavement area are the same (A<sub>r</sub> = A<sub>p</sub>).
{\displaystyle d_{r,max}={\frac {\left[\left(RVC_{T}\times R\right)+RVC_{T}-\left(f'\times D\right)\right]}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
Then increase Ar accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging. This assumes that the water storage reservoir area and permeable pavement area are the same (Ar = Ap). |
Sample size and power of test - MATLAB sampsizepwr
sampsizepwr
Compute Sample Size for Selected Power Value
Compute Power and Sample Size for One-Sided Test
Compute Sample Size for a Binomial Test
Compute Power for a Two-Sample t-Test
pwrout
Sample size and power of test
nout = sampsizepwr(testtype,p0,p1)
nout = sampsizepwr(testtype,p0,p1,pwr)
pwrout = sampsizepwr(testtype,p0,p1,[],n)
p1out = sampsizepwr(testtype,p0,[],pwr,n)
___ = sampsizepwr(testtype,p0,p1,pwr,n,Name,Value)
sampsizepwr computes the sample size, power, or alternative parameter value for a hypothesis test, given the other two values. For example, you can compute the sample size required to obtain a particular power for a hypothesis test, given the parameter value of the alternative hypothesis.
nout = sampsizepwr(testtype,p0,p1) returns the sample size, nout, required for a two-sided test of the type specified by testtype to have a power (probability of rejecting the null hypothesis when the alternative hypothesis is true) of 0.90 when the significance level (probability of rejecting the null hypothesis when the null hypothesis is true) is 0.05. p0 specifies parameter values under the null hypothesis. p1 specifies the value, or an array of values, of the single parameter being tested under the alternative hypothesis.
nout = sampsizepwr(testtype,p0,p1,pwr) returns the sample size, nout, that corresponds to the specified power, pwr, and the parameter value under the alternative hypothesis, p1.
pwrout = sampsizepwr(testtype,p0,p1,[],n) returns the power achieved for a sample size of n when the true parameter value is p1.
p1out = sampsizepwr(testtype,p0,[],pwr,n) returns the parameter value detectable with the specified sample size, n, and the specified power, pwr.
___ = sampsizepwr(testtype,p0,p1,pwr,n,Name,Value) returns any of the previous arguments using one or more name-value pair arguments. For example, you can change the significance level of the test, or specify a right- or left-tailed test. The name-value pairs can appear in any order but must begin in the sixth argument position.
A company runs a manufacturing process that fills empty bottles with 100 mL of liquid. To monitor quality, the company randomly selects several bottles and measures the volume of liquid inside.
Determine the sample size the company must use for a t-test to detect a difference between 100 mL and 102 mL with a power of 0.80. Assume that a standard deviation is 5 mL.
nout = sampsizepwr('t',[100 5],102,0.80)
nout = 52
The company must test 52 bottles to detect the difference between a mean volume of 100 mL and 102 mL with a power of 0.80.
Generate a power curve to visualize how the sample size affects the power of the test.
nn = 1:100;
pwrout = sampsizepwr('t',[100 5],102,[],nn);
plot(nn,pwrout,'b-',nout,0.8,'ro')
title('Power versus Sample Size')
An employee wants to buy a house near her office. She decides to eliminate from consideration any house that has a mean morning commute time greater than 20 minutes. The null hypothesis for this right-sided test is H0:
\mu
= 20, and the alternative hypothesis is HA:
\mu
> 20. The selected significance level is 0.05.
To determine the mean commute time, the employee takes a test drive from the house to her office during rush hour every morning for one week, so her total sample size is 5. She assumes that the standard deviation,
\sigma
, is equal to 5.
The employee decides that a true mean commute time of 25 minutes is too different from her targeted 20-minute limit, so she wants to detect a significant departure if the true mean is 25 minutes. Find the probability of incorrectly concluding that the mean commute time is no greater than 20 minutes.
Compute the power of the test, and then subtract the power from 1 to obtain
\beta
power = sampsizepwr('t',[20 5],25,[],5,'Tail','right');
beta = 1 - power
\beta
value indicates a probability of 0.4203 that the employee concludes incorrectly that the morning commute is not greater than 20 minutes.
The employee decides that this risk is too high, and she wants no more than a 0.01 probability of reaching an incorrect conclusion. Calculate the number of test drives the employee must take to obtain a power of 0.99.
nout = sampsizepwr('t',[20 5],25,0.99,[],'Tail','right')
The results indicate that she must take 18 test drives from a candidate house to achieve this power level.
The employee decides that she only has time to take 10 test drives. She also accepts a 0.05 probability of making an incorrect conclusion. Calculate the smallest true parameter value that produces a detectable difference in mean commute time.
p1out = sampsizepwr('t',[20 5],[],0.95,10,'Tail','right')
p1out = 25.6532
Given the employee's target power level and sample size, her test detects a significant difference from a mean commute time of at least 25.6532 minutes.
Compute the sample size, n, required to distinguish p = 0.30 from p = 0.36, using a binomial test with a power of 0.8.
napprox = sampsizepwr('p',0.30,0.36,0.8)
napprox = 485
The result indicates that a power of 0.8 requires a sample size of 485. However, this result is approximate.
Make a plot to see if any smaller n values provide the required power of 0.8.
pwrout = sampsizepwr('p',0.3,0.36,[],nn);
nexact = min(nn(pwrout>=0.8))
nexact = 462
plot(nn,pwrout,'b-',[napprox nexact],pwrout([napprox nexact]),'ro')
The result indicates that a sample size of 462 also provides a power of 0.8 for this test.
A farmer wants to test the impact of two different types of fertilizer on the yield of his bean crops. He currently uses Fertilizer A, but believes that Fertilizer B might improve crop yield. Because Fertilizer B is more expensive than Fertilizer A, the farmer wants to limit the number of plants he treats with Fertilizer B in this experiment.
The farmer uses a 2:1 ratio of plants in each treatment group. He tests 10 plants with Fertilizer A, and 5 plants with Fertilizer B. The mean yield using Fertilizer A is 1.4 kg per plant, with a standard deviation of 0.2. The mean yield using Fertilizer B is 1.7 kg per plant. The significance level of the test is 0.05.
Compute the power of the test.
pwr = sampsizepwr('t2',[1.4 0.2],1.7,[],5,'Ratio',2)
pwr = 0.7165
The farmer wants to increase the power of the test to 0.90. Calculate how many plants he must treat with each type of fertilizer.
n = sampsizepwr('t2',[1.4 0.2],1.7,0.9,[])
To increase the power of the test to 0.90, the farmer must test 11 plants with each type of fertilizer.
The farmer wants to reduce the number of plants he must treat with Fertilizer B, but keep the power of the test at 0.90 and maintain the initial 2:1 ratio of plants in each treatment group
Using a 2:1 ratio of plants in each treatment group, calculate how many plants the farmer must test to obtain a power of 0.90. Use the mean and standard deviation values obtained in the previous test.
[n1out,n2out] = sampsizepwr('t2',[1.4,0.2],1.7,0.9,[],'Ratio',2)
n1out = 8
n2out = 16
To obtain a power of 0.90, the farmer must treat 16 plants with Fertilizer A and 8 plants with Fertilizer B.
testtype — Test type
'z' | 't' | 't2' | 'var' | 'p'
Test type, specified as one of the following.
'z' — z-test for normally distributed data with known standard deviation.
't' — t-test for normally distributed data with unknown standard deviation.
't2' — Two-sample pooled t-test for normally distributed data with unknown standard deviation and equal variances.
'var' — Chi-square test of variance for normally distributed data.
'p' — Test of the p parameter (success probability) for a binomial distribution. The 'p' test is a discrete test for which increasing the sample size does not always increase the power. For n values larger than 200, there may exist values smaller than the returned n value that also produce the specified power.
p0 — Parameter value under null hypothesis
scalar value | two-element array of scalar values
Parameter value under the null hypothesis, specified as a scalar value or a two-element array of scalar values.
If testtype is 'z'or 't', then p0 is a two-element array [mu0,sigma0] of the mean and standard deviation, respectively, under the null hypothesis.
If testtype is 't2', then p0 is a two-element array [mu0,sigma0] of the mean and standard deviation, respectively, of the first sample under the null and alternative hypotheses.
If testtype is 'var', then p0 is the variance under the null hypothesis.
If testtype is 'p', then p0 is the value of p under the null hypothesis.
p1 — Parameter value under alternative hypothesis
scalar value | array of scalar values | []
Parameter value under the alternative hypothesis, specified as a scalar value or as an array of scalar values.
If testtype is 'z' or 't', then p1 is the value of the mean under the alternative hypothesis.
If testtype is 't2', then p1 is the value of the mean of the second sample under the alternative hypothesis.
If testtype is 'var', then p1 is the variance under the alternative hypothesis.
If testtype is 'p', then p1 is the value of p under the alternative hypothesis.
If you specify p1 as an array, then sampsizepwr returns an array for nout or pwrout that is the same length as p1.
To return the alternative parameter value, p1out, specify p1 using empty brackets ([]), as shown in the syntax description.
pwr — Power of the test
0.90 (default) | scalar value in the range (0,1) | array of scalar values in the range (0,1) | []
Power of the test, specified as a scalar value in the range (0,1) or as an array of scalar values in the range (0,1). The power of a test is the probability of rejecting the null hypothesis when the alternative hypothesis is true, given a particular significance level.
If you specify pwr as an array, then sampsizepwr returns an array for nout or p1out that is the same length as pwr.
To return a power value, pwrout, specify pwr using empty brackets ([]), as shown in the syntax description.
positive integer value | array of positive integer values
Sample size, specified as a positive integer value or as an array of positive integer values.
If testtype is 't2', then sampsizepwr assumes that the two sample sizes are equal. For unequal sample sizes, specify n as the smaller of the two sample sizes, and use the 'Ratio' name-value pair argument to indicate the sample size ratio. For example, if the smaller sample size is 5 and the larger sample size is 10, specify n as 5, and the 'Ratio' name-value pair as 2.
If you specify n as an array, then sampsizepwr returns an array for pwrout or p1out that is the same length as n.
Example: 'Alpha',0.01,'Tail','right' specifies a right-tailed test with a 0.01 significance level.
Significance value of the test, specified as the comma-separated pair consisting of 'Alpha' and a scalar value in the range (0,1).
Ratio — Sample size ratio
Sample size ratio for a two-sample t-test, specified as the comma-separated pair consisting of 'Ratio' and a scalar value greater than or equal to 1. The value of Ratio is equal to n2/n1, where n2 is the larger sample size, and n1 is the smaller sample size.
To return the power, pwrout, or alternative parameter value, p1out, specify the smaller of the two sample sizes for n, and use 'Ratio' to indicate the sample size ratio.
Example: 'Ratio',2
Tail — Test type
Test type, specified as the comma-separated pair consisting of 'Tail' and one of the following:
'both' — Two-sided test for an alternative not equal to p0
'right' — One-sided test for an alternative larger than p0
'left' — One-sided test for an alternative smaller than p0
nout — Sample size
Sample size, returned as a positive integer value or as an array of positive integer values. sampsizepwr applies ceil to round up raw sample sizes to the next integer.
If testtype is t2, and you use the 'Ratio' name-value pair argument to specify the ratio of the two unequal sample sizes, then nout returns the smaller of the two sample sizes.
Alternatively, to return both sample sizes, specify this argument as [n1out,n2out]. In this case, sampsizepwr returns the smaller sample size as n1out, and the larger sample size as n2out.
If you specify pwr or p1 as an array, then sampsizepwr returns an array for nout that is the same length as pwr or p1.
pwrout — Power
scalar value in the range (0,1) | array of scalar values in the range (0,1)
Power achieved by the test, returned as a scalar value in the range (0,1) or as an array of scalar values in the range (0,1).
If you specify n or p1 as an array, then sampsizepwr returns an array for pwrout that is the same length as n or p1.
p1out — Parameter value for the alternative hypothesis
Parameter value for the alternative hypothesis, returned as a scalar value or as an array of scalar values.
When computing p1out for the 'p' test, if no alternative can be rejected for a given null hypothesis and significance level, the function displays a warning message and returns NaN.
vartest | ttest | ttest2 | ztest | binocdf |
Series | Brilliant Math & Science Wiki
Chungsu Hong, Henry Maltby, Kishlaya Jaiswal, and
Στέλιος Κασουρίδης
A mathematical series is an infinite sum of the elements in some sequence. A series with terms
a_n
n
1
through all positive integers, is expressed as
\sum_{n= 1}^\infty a_n.
n^\text{th}
S_n
of the series is the value
S_n = \sum_{i = 1}^n a_i.
For example, for the equation
f(n) = n^2,
f(n)
n
1
\infty
\sum_{n = 1}^\infty n^2 = 1^2 + 2^2 + 3^2 +\cdots.
If we only wanted the sum of terms up to
n=10,
that would be
S_{10} = \sum_{n = 1}^{10} n^2 = 1^2 + 2^2 + 3^2 +\cdots+ 10^2 = 385.
Series are useful throughout mathematics and science, as a means of approximation, analytic continuation, and evaluation. The implicit connection to the values of the derivatives of the function provides a strong tool in all fields using calculus.
\sum
indicates a summation, and it may be interpreted by iterating the parameter (usually designated below the summation) as it takes (usually integer) values in a prescribed range (from the initial value to the upper limit), then adding the resulting expressions. For instance,
\sum_{k = 1}^{200} f(k) = f(1) + f(2) + \dots + f(200).
k
has initial value
1
. It is iterated for all integer values up to (and including)
200
, its upper limit. Then these iterations are summed.
k
may be replaced (in all instances) by
i
or any other variable. Commonly used parameters include
i
j
k
m
n
. Another way of representing the same thing is with set notation, seen as
\sum_{j \in \{1, \, 2, \dots, \, 200\}} f(j) = f(1) + f(2) + \dots + f(200).
In general, a summation notation is acceptable if it is unambiguous and well-defined. In order to be well-defined, the summation must be either finite or absolutely convergent (or conditionally convergent with an ordering to the summation). It follows that convergence is one of the most important questions in the study of series.
Like other mathematical operations, summation may be used in the definition of a function and may contain variables within the series itself. Power series explore this idea further.
A series is said to converge to a value if the limit of its partial sums approaches that value; that is, given an infinite sequence
\{a_k\}
\sum_{k = 1}^\infty a_k = \lim_{n \to \infty} \sum_{k = 1}^n a_k.
If the limit does not exist, the series is said to diverge. A sufficient condition for a series to diverge is the following:
\lim\limits_{n\to\infty} a_n
does not exist, or exists and is non-zero, then
\sum\limits_{n=1}^\infty a_n
Essentially this is saying that if the limit of some function (not the limit of the sum of that function, but just the limit of the function) gets larger and larger, then the limit does not equal
0
. And if the limit of the function just keeps getting larger, then the sum of that function as it goes to
\infty
is going to diverge. However, failing the divergence test does not mean a summation converges. In general, convergence tests are necessary for determining whether an infinite summation converges or diverges.
A series is said to converge absolutely if the series formed from the absolute value of its terms converges; that is, given an infinite sequence
\{a_k\}
\sum_{k = 1}^\infty |a_k| \text{ converges.}
A series is said to converge conditionally if it converges but does not pass this test, meaning it does not converge absolutely.
\sum\limits_{n = 1}^\infty \frac{(-1)^{n+1}}{2^n}
converges absolutely. But the sum
\sum\limits_{n = 1}^\infty \frac{(-1)^{n+1}}{n}
converges conditionally. If we know that both converge, we can prove that they converge absolutely or conditionally by taking the sum of the absolute value of the function:
\begin{aligned} \sum\limits_{n = 1}^\infty \left|\frac{(-1)^{n+1}}{2^{n}}\right| &= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots + \frac{1}{2^{n}} \rightarrow \text{converges}\\ \sum\limits_{n = 1}^\infty \left|\frac{(-1)^{n+1}}{n}\right| &= 1 + \frac{1}{2} + \frac{1}{3} + \cdots +\frac{1}{n} \rightarrow \text{does not converge}. \end{aligned}
Absolute or conditional convergence is important because it gives more information about the sequence and about the series itself. For instance, the Riemann series theorem implies that a series may not be freely reordered if it is conditionally convergent. With power series, it may help determine the behavior at positive or negative values.
A power series is an expression
{\displaystyle \sum_{n=1}^\infty} a_n x^n
generated by an infinite sequence
\{a_n\}
Power series are used in calculus as local approximations of functions and in combinatorics as abstract tools for counting. In calculus, the issue of convergence is paramount, while it is not as central to combinatorial concerns. In combinatorics, power series may be manipulated symbolically without worrying about issues of convergence.
There exist multivariable extensions of power series. Convergence becomes a more complicated issue with these, but the Fubini-Tonelli theorem provides a guideline for algebraic manipulations.
Due to the infinite nature of series, it is not possible to calculate the sum directly without resorting to algebraic methods. However, generally speaking, there exist many series that cannot be expressed in a closed form, and all modern calculators and computers require some sort of estimation software in order to provide a satisfactory decimal answer. Such approximation methods fall in the field of numerical analysis. Robust approximation tools are doubly important, as integrals are often evaluated numerically as a type of infinite sum.
An alternating series is a series representable in the form
\sum_{k = 1}^\infty (-1)^k a_k
for some sequence
\{a_k\}
of nonnegative numbers. An alternating series is known as decreasing if
a_n > a_{n+1}
n
S_n
n^\text{th}
partial sum of a series, and let
S
be its actual sum. Decreasing alternating series have a simple error bound given by
S_n - a_{n+1} < S < S_n + a_{n+1}.
The integral test can help estimate the sum for decreasing positive series
\sum_{k = 1}^\infty a_k
a_k = f(k)
k
and the improper integral
\int_n^\infty f(x) \, dx
converges. The error bound is
S_n + \int_{n+1}^\infty f(x) \, dx < S < S_n + \int_n^\infty f(x) \, dx.
A twist on the ratio test can also be used to estimate a sum. Let
L = \lim_{n \to \infty} \tfrac{a_{n+1}}{a_n}
, and suppose
L < 1
\tfrac{a_{n+1}}{a_n}
L
n
increases, then
S_n + a_n \cdot \left( \frac{L}{1 - L} \right) < S < S_n + \frac{a_{n+1}}{1 - \tfrac{a_{n+1}}{a_n}}.
\tfrac{a_{n+1}}{a_n}
L
n
S_n + \frac{a_{n+1}}{1 - \tfrac{a_{n+1}}{a_n}} < S < S_n + a_n \cdot \left( \frac{L}{1 - L} \right).
Value 1.19753, error at most 0.01 Value 1.20666, error at most 0.00087 Value 1.2021, error at most 0.00044 Value 1.20753, error at most 0.00413 Value 1.19753, error at most 0.00826
Using the integral test, find the best possible estimation for the series
\displaystyle\sum_{n=1}^\infty \frac{1}{n^3}
using a partial sum with
10
\displaystyle\sum_{n=1}^{10} \frac{1}{n^3} \approx 1.19753
\tfrac{1}{11^2} \approx 0.00826
Cite as: Series. Brilliant.org. Retrieved from https://brilliant.org/wiki/summation/ |
The Vickers hardness test was developed in 1921 by Robert L. Smith and George E. Sandland at Vickers Ltd as an alternative to the Brinell method to measure the hardness of materials.[1] The Vickers test is often easier to use than other hardness tests since the required calculations are independent of the size of the indenter, and the indenter can be used for all materials irrespective of hardness. The basic principle, as with all common measures of hardness, is to observe a material's ability to resist plastic deformation from a standard source. The Vickers test can be used for all metals and has one of the widest scales among hardness tests. The unit of hardness given by the test is known as the Vickers Pyramid Number (HV) or Diamond Pyramid Hardness (DPH). The hardness number can be converted into units of pascals, but should not be confused with pressure, which uses the same units. The hardness number is determined by the load over the surface area of the indentation and not the area normal to the force, and is therefore not pressure.
3 Conversion to SI units
4 Estimating tensile strength
The pyramidal diamond indenter of a Vickers hardness tester
This is a good indentation.
It was decided that the indenter shape should be capable of producing geometrically similar impressions, irrespective of size; the impression should have well-defined points of measurement; and the indenter should have high resistance to self-deformation. A diamond in the form of a square-based pyramid satisfied these conditions. It had been established that the ideal size of a Brinell impression was 3⁄8 of the ball diameter. As two tangents to the circle at the ends of a chord 3d/8 long intersect at 136°, it was decided to use this as the included angle between plane faces of the indenter tip. This gives an angle from each face normal to the horizontal plane normal of 22° on each side. The angle was varied experimentally and it was found that the hardness value obtained on a homogeneous piece of material remained constant, irrespective of load.[2] Accordingly, loads of various magnitudes are applied to a flat surface, depending on the hardness of the material to be measured. The HV number is then determined by the ratio F/A, where F is the force applied to the diamond in kilograms-force and A is the surface area of the resulting indentation in square millimeters. A can be determined by the formula.
{\displaystyle A={\frac {d^{2}}{2\sin(136^{\circ }/2)}},}
which can be approximated by evaluating the sine term to give,
{\displaystyle A\approx {\frac {d^{2}}{1.8544}},}
{\displaystyle \mathrm {HV} ={\frac {F}{A}}\approx {\frac {1.8544F}{d^{2}}}\quad [{\textrm {kgf/mm}}^{2}]}
The corresponding unit of HV is then the kilogram-force per square millimeter (kgf/mm²) or HV number. In the above equation, F could be in N and d in mm, giving HV in the SI unit of MPa. To calculate Vickers hardness number (VHN) using SI units one needs to convert the force applied from newtons to kilogram-force by dividing by 9.806 65 (standard gravity). This leads to the following equation:[4]
{\displaystyle \mathrm {HV} \approx {0.1891}{\frac {F}{d^{2}}}\quad [{\textrm {kgf/mm}}^{2}],}
where F is in N and d is in millimeters. A common error is that the above formula to calculate the HV number does not result in a number with the unit Newton per square millimeter (N/mm²), but results directly in the Vickers hardness number (usually given without units), which is in fact one kilogram-force per square millimeter (1 kgf/mm²).
When doing the hardness tests, the minimum distance between indentations and the distance from the indentation to the edge of the specimen must be taken into account to avoid interaction between the work-hardened regions and effects of the edge. These minimum distances are different for ISO 6507-1 and ASTM E384 standards.
Vickers values are generally independent of the test force: they will come out the same for 500 gf and 50 kgf, as long as the force is at least 200 gf.[6] However, lower load indents often display a dependence of hardness on indent depth known as the indentation size effect (ISE).[7] Small indent sizes will also have microstructure-dependent hardness values.
For thin samples indentation depth can be an issue due to substrate effects. As a rule of thumb the sample thickness should be kept greater than 2.5 times the indent diameter. Alternatively indent depth,
{\displaystyle t}
, can be calculated according to:
{\displaystyle t={\frac {d_{\rm {avg}}}{2{\sqrt {2}}\tan {\frac {\theta }{2}}}}\approx {\frac {d_{\rm {avg}}}{7.0006}},}
Conversion to SI units[edit]
To convert the Vickers hardness number to SI units the hardness number in kilograms-force per square millimeter (kgf/mm²) has to be multiplied with the standard gravity,
{\displaystyle g_{0}}
, to get the hardness in MPa (N/mm²) and furthermore divided by 1000 to get the hardness in GPa.
{\displaystyle {\text{surface area hardness (GPa)}}={\frac {g_{0}}{1000}}HV={\frac {9.80665}{1000}}HV}
Vickers hardness can also be converted to an SI hardness based on the projected area of the indent rather than the surface area. The projected area,
{\displaystyle A_{\rm {p}}}
, is defined as the following for a Vickers indenter geometry:[8]
{\displaystyle A_{\rm {p}}={\frac {d_{\rm {avg}}^{2}}{2}}={\frac {1.854}{2}}{A_{s}}}
This hardness is sometimes referred to as the mean contact area or Meyer hardness, and ideally can be directly compared with other hardness tests also defined using projected area. Care must be used when comparing other hardness tests due to various size scale factors which can impact the measured hardness.
{\displaystyle {\text{projected area hardness (GPa)}}={\frac {g_{0}}{1000}}{\frac {2}{1.854}}HV\approx {\frac {HV}{94.5}}}
Estimating tensile strength[edit]
If HV is first expressed in N/mm2 (MPa), or otherwise by converting from kgf/mm2, then the tensile strength (in MPa) of the material can be approximated as σu ≈ HV/c , where c is a constant determined by yield strength, Poisson's ratio, work-hardening exponent and geometrical factors – usually ranging between 2 and 4.[9] In other words, if HV is expressed in N/mm2 (i.e. in MPa) then the tensile strength (in MPa) ≈ HV/3. This empirical law depends variably on the work-hardening behavior of the material.[10]
The fin attachment pins and sleeves in the Convair 580 airliner were specified by the aircraft manufacturer to be hardened to a Vickers Hardness specification of 390HV5, the '5' meaning five kiloponds. However, on the aircraft flying Partnair Flight 394 the pins were later found to have been replaced with sub-standard parts, leading to rapid wear and finally loss of the aircraft. On examination, accident investigators found that the sub-standard pins had a hardness value of only some 200-230HV5.[11]
^ The Vickers Hardness Testing Machine. UKcalibrations.co.uk. Retrieved on 2016-06-03.
^ ASTM E384-10e2
^ ISO 6507-1:2005(E)
^ Vickers Test. Instron website.
^ Nix, William D.; Gao, Huajian (1 March 1998). "Indentation size effects in crystalline materials: A law for strain gradient plasticity". Journal of the Mechanics and Physics of Solids. 46 (3): 411–425. Bibcode:1998JMPSo..46..411N. doi:10.1016/S0022-5096(97)00086-0. ISSN 0022-5096.
^ Fischer-Cripps, Anthony C. (2007). Introduction to contact mechanics (2nd ed.). New York: Springer. pp. 212–213. ISBN 9780387681887. OCLC 187014877.
^ "Hardness". matter.org.uk.
^ Zhang, P. (September 2011). "General relationship between strength and hardness". Materials Science and Engineering A. 529: 62. doi:10.1016/j.msea.2011.08.061.
^ Report on the Convair 340/580 LN-PAA aircraft accident North of Hirtshals, Denmark on September 8, 1989 | aibn. Aibn.no. Retrieved on 2016-06-03.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Vickers_hardness_test&oldid=1083666392" |
PUSCH DM-RS uplink subframe timing estimate - MATLAB lteULFrameOffset - MathWorks Nordic
lteULFrameOffset
Synchronize and SCFDMA Demodulate Delayed Transmission
View PUSCH Transmission Correlation Peaks
PUSCH DM-RS uplink subframe timing estimate
offset = lteULFrameOffset(ue,chs,waveform)
[offset,corr] = lteULFrameOffset(ue,chs,waveform)
offset = lteULFrameOffset(ue,chs,waveform) performs synchronization using PUSCH DM-RS signals for the time-domain waveform, waveform, given UE-specific settings, ue, and PUSCH configuration, chs.
The returned value offset indicates the number of samples from the start of the waveform, waveform, to the position in that waveform where the first subframe containing the DM-RS begins.
offset provides subframe timing; frame timing can be achieved by using offset with the subframe number, ue.NSubframe. This information is consistent with real-world operation, since the base station knows when, or in which subframe, to expect uplink transmissions.
[offset,corr] = lteULFrameOffset(ue,chs,waveform) also returns a complex matrix corr, which is the signal used to extract the timing offset.
Synchronization and demodulation of transmission which has been delayed by 5 samples.
Initialize waveform and insert a 5 sample delay.
waveform = lteRMCULTool(ue,[1;0;0;1]);
Determine offset and demodulate the waveform.
offset = lteULFrameOffset(ue,ue.PUSCH,tx)
View the correlation peak for a delayed transmit waveform. The transmission contains PUSCH demodulation reference signal (DM-RS) symbols available for estimating the waveform timing.
Configure UE-specific settings and generate the transmit waveform.
tx = lteRMCULTool(ue,[1;0;0;1]);
Determine Offset
Calculate timing offset and return the correlations for the transmit waveform and for a delayed version of the transmit waveform.
[~,corr] = lteULFrameOffset(ue,ue.PUSCH,tx);
[offset,corrDelayed] = lteULFrameOffset(ue,ue.PUSCH,txDelayed);
rxGrid = lteSCFDMADemodulate(ue,txDelayed(1+offset:end));
{N}_{\text{RB}}^{\text{UL}}
{n}_{DMRS}^{\left(1\right)}
{n}_{ID}^{csh_DMRS}
chs — PUSCH configuration
PUSCH configuration, specified as a scalar structure with the following fields.
{n}_{DMRS}^{\left(2\right)}
OrthCover Optional
0 (default), nonnegative scalar integer from 0 to 23.
See lteULPMIInfo.
Offset number of samples, returned as a scalar integer. This output is the number of samples from the start of the waveform to the position in that waveform where the first subframe containing the DM-RS begins. offset is computed by extracting the timing of the peak of the correlation between waveform and internally generated reference waveforms containing DM-RS signals. The correlation is performed separately for each antenna and the antenna with the strongest correlation is used to compute offset.
Signal used to extract the timing offset, returned as a complex-valued numeric matrix. corr has the same dimensions as waveform.
lteFrequencyCorrect | lteFrequencyOffset | lteFadingChannel | lteMovingChannel | lteHSTChannel | lteSCFDMADemodulate |
Solving Exponential Equations | Brilliant Math & Science Wiki
Hemang Agarwal, Paul Ryan Longhas, Ashish Menon, and
To solve exponential equations, we need to consider the rule of exponents. These rules help us a lot in solving these type of equations.
In solving exponential equations, the following theorem is often useful:
a
is a non-zero constant and
a^x = a^y,
x = y.\ _\square
\begin{aligned} a^x&=a^y, a\neq 1 \\ \frac{a^x}{a^y} &= 1 \\ a^{x-y} &= 1 \\ x-y &= 0\\ x &= y.\ _\square \end{aligned}
Here is how to solve exponential equations:
Manage the equation using the rule of exponents and some handy theorems in algebra.
Use the theorem above that we just proved.
\displaystyle{ \frac{1}{5^{x-1}} = 125} .
Making the bases on both sides equal to 5 gives
\begin{aligned} \frac{1}{5^{x-1}} &= 125 \\ 5^{-(x-1)}&=5^3 \\ -(x-1) &= 3 \\ x &= -2. \ _\square \end{aligned}
4^{x-3} = 0.125 .
Converting the bases of both sides to 2 gives
\begin{aligned} 4^{x-3} &= 0.125 \\ 4^{x-3} &= \frac{125}{1000} \\ 2^{2x-6} &= \frac{1}{8} \\ 2^{2x-6} &= 2^{-3} \\ 2x-6&= -3 \\ x&= \frac{3}{2}. \ _\square \end{aligned}
x
4^x = 16.
4^x = 16 \Rightarrow 4^x = 4^2.
Then the theorem "if
a
a^x = a^y,
x = y
" gives
x = 2.\ _\square
8^x = 2
x
\begin{aligned} 8^x & = 2\\ \big(2^3\big)^{x} & = 2\\ 2^{3x} & = 2^{1}. \end{aligned}
\begin{aligned} 3x & = 1\\ x & = \dfrac{1}{3}.\ _\square \end{aligned}
6^x - 1=0,
x?
\begin{aligned} 6^x-1&=0 \\ 6^x &= 1\\ 6^x &= 6^0\\ x &= 0.\ _\square \end{aligned}
(8)(9^x) = 9^x,
x?
\begin{aligned} (8)(9^x) &= 9^x \\ (8)(9^x) - 9^x &= 0 \\ (7)9^x &= 0 \\ 9^x &= 0. \end{aligned}
a \neq 0
a^x=0,
x = \phi
x = \phi.\ _\square
If the bases are different, there are still techniques for solving these exponential equations. If the bases are powers of a common base, we need only convert one or both bases to the common base and proceed using the "Same Base" case.
4^{3x} =8^{x-1}.
We see that while 4 and 8 are different bases, they are both powers of a common base, namely 2. We'll proceed by rewriting 4 and 8 in terms of their common base:
\begin{aligned} 4^{3x} &= 8^{x-1} \\ \big(2^2\big)^{3x} &=\big (2^3\big)^{x-1} \\ 2^{6x} &= 2^{3x-3} \\ 6x&=3x-3 \\ x&= -1. \ _\square \end{aligned}
x:
\dfrac{8^{4x - \sqrt{x}}}{{16}^{2x + \sqrt{x}}} = 2^{2\sqrt{x}}
\begin{aligned} \dfrac{8^{4x - \sqrt{x}}}{{16}^{2x + \sqrt{x}}} & = 2^{2\sqrt{x}}\\ \\ \dfrac{{\big(2^3\big)}^{4x - \sqrt{x}}}{{{\big(2^4\big)}}^{2x + \sqrt{x}}} & = 2^{2\sqrt{x}}\\ \\ \dfrac{2^{12x - 3\sqrt{x}}}{2^{8x + 4\sqrt{x}}} & = 2^{2\sqrt{x}}\\ \\ 2^{12x - 3\sqrt{x} - 8x - 4\sqrt{x}} & = 2^{2\sqrt{x}}\\ \\ 2^{4x - 7\sqrt{x}} & = 2^{2\sqrt{x}}. \end{aligned}
Equating the powers gives
\begin{aligned} 4x - 7\sqrt{x} & = 2\sqrt{x}\\ 4x & = 9\sqrt{x}\\ \sqrt{x} & = \dfrac{9}{4}\\ x & = \dfrac{81}{16}.\ _\square \end{aligned}
\frac{27^{3x-2}}{243} = 81^{3x-6} .
Reducing the bases of 27, 81, and 243 on both sides to 3 yields
\begin{aligned} \frac{27^{3x-2}}{243} &= 81^{3x-6} \\ \frac{3^{9x-6}}{3^5} &= 3^{12x-24} \\ 3^{9x-6-5} &= 3^{12x-24} \\ 3^{9x-11} &= 3 ^{12x-24} \\ 9x-11 &= 12x-24 \\ 3x &= 13 \\ x &= \frac{13}{3}. \ _\square \end{aligned}
Unfortunately, it won't always be possible to convert to a common base as we did in the examples above.
For instance, in solving
5^{x} = 3^{x + 2}
, we note that 5 and 3 are not powers of a nice common base. In this case, we'll need to make use of logarithms.
5^{x} = 3^{x + 2}.
\begin{aligned} 5^{x} &= 3^{x + 2} \\ \log_{10}(5^{x}) &= \log_{10}\big(3^{x + 2}\big) \\ x\log_{10}5 &= (x + 2)\log_{10}3 \\ x\log_{10}5 &= x\log_{10}3 + 2\log_{10}3 \\ x\log_{10}5 - x\log_{10}3 &= 2\log_{10}3 \\ x\big(\log_{10}5 - \log_{10}3\big) &= 2\log_{10}3. \end{aligned}
x = \frac{2\log_{10}3}{\log_{10}5 - \log_{10}3} \approx 4.3013 .\ _\square
1728 = 2^a.3^b
, find positive integers
and
b
\begin{aligned} 1728 & = {12}^{3}\\ & = {(4 × 3)}^{3}\\ & = 4^3 × 3^3\\ & = {(2^2)}^3 × 3^3\\ & = 2^6 × 3^3\\ \Rightarrow a & = 6, \ b = 3.\ _\square \end{aligned}
3^{x^2} = 3^{x}.
Since both sides of the equation have the same base, their exponents must also be the same:
\begin{aligned} 3^{x^2} &= 3^{x} \\ x^2 &=x \\ x^2-x&=0\\ x(x-1)&=0\\ \Rightarrow x&=0,1. \ _\square \end{aligned}
2^x \cdot 3^y \cdot 5^z = 45,
x+y+z?
45
can be factorized as follows:
45 = 3^2\cdot5=2^0\cdot3^2\cdot5^1.
x=0, y=2, z=1,
x+y+z=0+2+1=3.
_\square
x^x \cdot y^y = 108 ,
x+y ?
108
108 = 2^2 \cdot 3^3 .
This implies that either
x=2, y=3
x=3, y=2.
x+y=2+3=5. \ _ \square
\frac{2^5}{2^3} \cdot 3^0 \cdot 3^1 \cdot 3^2 = 2^x \cdot 3^y ,
x+y?
\frac{2^5}{2^3}
\frac{2^5}{2^3} = {2}^{5-3}=2^2. \qquad (1)
3^0 \cdot 3^1 \cdot 3^2
3^0 \cdot 3^1 \cdot 3^2 = 3^{0+1+2}=3^3. \qquad (2)
(1)
(2),
x
y
\begin{aligned} \frac{2^5}{2^3} \cdot 3^0 \cdot 3^1 \cdot 3^2 &= 2^2 \cdot 3^3 \\ &= 2^x \cdot 3^y \\ \Rightarrow x&= 2,\ y=3. \end{aligned}
x+y= 2+3 = 5. \ _ \square
An exponential equation is one in which a variable occurs in the exponent. If both sides of the equation have the same base, then the exponents on both sides are also the same:
a^x=a^y \implies x=y .
Here is a list of some rules concerning exponential functions:
\quad (1)~~a^m \times a^n = a^{m+n}
\quad (2)~~a^m \div a^n = a^{m-n}
\quad (3)~~(a^m)^n = a^{mn}
\quad (4)~~(ab)^n=a^{n}b^{n}
\quad (5)~~a^0=b^0
\quad (6)~~1^m=1^n,
a\neq0
b\neq0.
Always be cautious of
(5)
(6)
; never forget to check if plugging a zero in an exponent works, or if there are any bases that are equal to 1.
Cite as: Solving Exponential Equations. Brilliant.org. Retrieved from https://brilliant.org/wiki/exponential-functions-solving-equations/ |
General Polygons - Area | Brilliant Math & Science Wiki
A polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners. The interior of the polygon is sometimes called its body. An
n
sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.
There are many ways to find the area of a polygon.
Find the area of a regular hexagon of side length
5
A regular hexagon is composed of
6
congruent equilateral triangles. The area of an equilateral triangle is
\frac{\sqrt{3}}{4}x^2,
x
is the side length.
So the area of one equilateral triangle is
\frac{\sqrt{3}}{4}\big(5^2\big)=\frac{25}{4}\sqrt{3}
It follows that the area of the hexagon is
6\left(\frac{25}{4}\sqrt{3}\right)=\frac{75}{2}\sqrt{3}.
_\square
Find the area of an irregular decagon having consecutive vertices as
(1,-3),(1,-1),(10,-3),(13,4),(2,4),(2,1),(-4,4),(-10,2),(-11,0),(-7,-3).
A=\dfrac{1}{2} \begin{vmatrix} x_1 & x_2 & ... & x_n & x_1 \\ y_1 & y_2 & ... & y_n & y_1 \end{vmatrix},
A
is half the determinant of the matrix.
\begin{aligned} A&=\dfrac{1}{2} \begin{vmatrix} 1 & 1 & 10 & 13 & 2 & 2 & -4 & -10 & -11 & -7 & 1 \\ -3 & -1 & -3 & 4 & 4 & 1 & 4 & 2 & 0 & -3 & -3 \end{vmatrix}\\\\ &=\dfrac{1}{2}\big[-1-3+40+52+2+8-8+0+33+21-(-3-10-39+8+8-4-40-22-0-3)\big]\\\\ &=\dfrac{1}{2}\big[144-(-105)\big]\\\\ &=\dfrac{1}{2}(249)\\\\ &=124.5.\ _\square \end{aligned}
Note that the area of a convex polygon is defined to be positive if the points are arranged in counterclockwise order, and negative if they are in clockwise order (Beyer 1987).
In the unit grid shown below, what is the area of the triangle?
A regular four-pointed star is formed inside a square with area
300
Find the area of the star rounded to the nearest integer.
Shown to the right is a regular octagon with side length
8
I,J,K,L,M,N,O,P
are all midpoints of the corresponding sides.
Find the area of the yellow region rounded to the nearest integer.
The areas of the three squares in the figure below are given inside corresponding squares.
Find the area of hexagon
ABCDEF
Cite as: General Polygons - Area. Brilliant.org. Retrieved from https://brilliant.org/wiki/general-polygons-area/ |
\Delta \text{E}=\sqrt{\Delta {\text{a}}^{*2}+\Delta {\text{b}}^{*2}+\Delta {\text{L}}^{*2}}
X={X}_{n}{\left(\frac{{a}^{*}}{500}+\frac{\left({\text{L}}^{*}+16\right)}{116}\right)}^{3}
Y={Y}_{n}{\left(\frac{\left({\text{L}}^{*}+16\right)}{116}\right)}^{3}
Z={Z}_{n}{\left(\frac{-\text{b}}{200}+\frac{\left({\text{L}}^{*}+16\right)}{116}\right)}^{3}
x=\frac{X}{\left(X+Y+Z\right)}
BI=\frac{\left(x-0.31\right)}{0.172}×100
1) Shown in Table 1.
2) Each value is expressed as the mean±standard deviation (n=3). Means followed by different letters within the column (a-e) are significantly different (p<0.05).
2) Each value is expressed as the mean±standard deviation (n=3).
3) Means followed by different letters within the column (a-f) are significantly different (p<0.05).
3) Means followed by different letters within the column (a-d) and different letters within the row (A-C) are significantly different (p<0.05).
3) Means followed by different letters within the column (a-c) are significantly different (p<0.05).
Tel : +82-53-953-9555 | Fax : +82-53-953-2555 | kjfp@kosfop.or.kr |
GIsqrt - Maple Help
Home : Support : Online Help : Mathematics : Numerical Computations : Integer Functions : Gaussian Integers : GIsqrt
Gaussian integer square root
GIsqrt(x)
The GIsqrt function computes a Gaussian integer approximation to the square root of x. The approximation is exact for perfect squares; the distance between x and its approximation is less than or equal to
\frac{\sqrt{2}}{2}
\mathrm{with}\left(\mathrm{GaussInt}\right):
\mathrm{GIsqrt}\left(-5+12I\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}
\mathrm{GIsqrt}\left(7+9I\right)
\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{I} |
PresentationComplexity - Maple Help
Home : Support : Online Help : Mathematics : Group Theory : PresentationComplexity
return a measure of the complexity of a presentation of a finitely presented group
PresentationComplexity( G )
The PresentationComplexity( G ) command returns a triple of the form (m, n, k), where m is the number of generators currently defined for G, n is the number of relators currently defined for G, and k is the total length of the relators.
\mathrm{with}\left(\mathrm{GroupTheory}\right):
G≔〈〈a,b〉|〈{a}^{2},{b}^{3},{\left(a·b\right)}^{5}=1〉〉
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}⟨\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{∣}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}⟩
\mathrm{PresentationComplexity}\left(G\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{15}
The GroupTheory[PresentationComplexity] command was introduced in Maple 18. |
DWT - Maple Help
Home : Support : Online Help : Science and Engineering : Signal Processing : Transforms : DWT
compute forward discrete wavelet transform
compute inverse discrete wavelet transform
DWT(A)
InverseDWT(A1, A2)
Arrays of real numeric values corresponding to the low and high frequency components of the signal
The DWT(A) command computes the Haar discrete wavelet transform (DWT) of the Array A and returns a sequence of two Arrays with datatype float[8] and containing the low-frequency and high-frequency components respectively. The number of elements in A must be even, and each of the result Arrays will have half as many elements.
The InverseDWT(A1, A2) command computes the inverse Haar discrete wavelet transform from the Arrays A1 and A2 containing low and high frequency components. It returns the results in an Array with datatype float[8]. The length of A1 and A2 must be the same, and the size of the result Array will be twice this length.
Before the code performing the computation runs, the input Array(s) are converted to datatype float[8] if they do not have that datatype already. For this reason, it is most efficient if the input Array(s) have this datatype beforehand.
If the container=C option is provided, then the results are put into C and C is returned. With this option, no additional memory is allocated to store the result. With the forward transform, the container must be a list of two Arrays with datatype float[8] and size equal to half that of A. With the inverse transform, the container must be a single Array with datatype float[8] and size equal to twice that of the input Arrays.
The SignalProcessing[DWT] and SignalProcessing[InverseDWT] commands are thread-safe as of Maple 17.
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{with}\left(\mathrm{plots}\right):
N≔10:
A≔\mathrm{GenerateTone}\left(N,1,0.1,0\right)+\mathrm{GenerateTone}\left(N,3,0.4,0.2\right):
\mathrm{B1}≔\mathrm{Array}\left(1..\frac{N}{2},'\mathrm{datatype}'='\mathrm{float}'[8]\right):
\mathrm{B2}≔\mathrm{Array}\left(1..\frac{N}{2},'\mathrm{datatype}'='\mathrm{float}'[8]\right):
\mathrm{DWT}\left(A,'\mathrm{container}'=[\mathrm{B1},\mathrm{B2}]\right)
[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{1.01011023422442}& \textcolor[rgb]{0,0,1}{0.908571684515523}& \textcolor[rgb]{0,0,1}{-0.448582051978077}& \textcolor[rgb]{0,0,1}{-1.18581063938115}& \textcolor[rgb]{0,0,1}{-0.284289227380716}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{ccccc}\textcolor[rgb]{0,0,1}{-2.93008949929931}& \textcolor[rgb]{0,0,1}{-0.875854279296868}& \textcolor[rgb]{0,0,1}{2.38878178550180}& \textcolor[rgb]{0,0,1}{2.35220261444364}& \textcolor[rgb]{0,0,1}{-0.935040621349269}\end{array}]
\mathrm{B1};
\mathrm{B2}
\left[\begin{array}{ccccc}1.010110234224416& 0.908571684515523& -0.448582051978077& -1.1858106393811467& -0.28428922738071616\end{array}\right]
\left[\begin{array}{ccccc}-2.930089499299309& -0.8758542792968684& 2.3887817855018025& 2.35220261444364& -0.9350406213492695\end{array}\right]
\mathrm{display}\left(〈〈〈\mathrm{listplot}\left(A,'\mathrm{title}'="Signal"\right)〉,〈\mathrm{listplot}\left(\mathrm{B1},'\mathrm{title}'="Low"\right)〉,\mathrm{listplot}\left(\mathrm{B2},'\mathrm{title}'="High"\right)〉〉\right)
C≔\mathrm{InverseDWT}\left(\mathrm{B1},\mathrm{B2}\right)
\left[\begin{array}{cccccccccc}3.940199733523725& -1.9199792650748932& 1.7844259638123914& 0.032717405218654516& -2.8373638374798795& 1.9401997335237255& -3.538013253824787& 1.1663919750624934& 0.6507513939685533& -1.2193298487299855\end{array}\right]
\mathrm{map}\left(\mathrm{fnormal},A-C\right)
\left[\begin{array}{cccccccccc}0.& 0.& 0.& 0.& 0.& 0.& 0.& -0.& 0.& 0.\end{array}\right]
The SignalProcessing[DWT] and SignalProcessing[InverseDWT] commands were introduced in Maple 17. |
COMPLEX ANALYSIS - Encyclopedia Information
Complex analysis Information
https://en.wikipedia.org/wiki/Complex_analysis
Branch of mathematics studying functions of a complex variable
This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (March 2021) ( Learn how and when to remove this template message)
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.[ citation needed]
2 Complex functions
The Mandelbrot set, a fractal
An exponential function An of a discrete ( integer) variable n, similar to geometric progression
For any complex function, the values
{\displaystyle z}
from the domain and their images
{\displaystyle f(z)}
in the range may be separated into real and imaginary parts:
{\displaystyle z=x+iy\quad {\text{ and }}\quad f(z)=f(x+iy)=u(x,y)+iv(x,y),}
{\displaystyle x,y,u(x,y),v(x,y)}
are all real-valued.
In other words, a complex function
{\displaystyle f:\mathbb {C} \to \mathbb {C} }
may be decomposed into
{\displaystyle u:\mathbb {R} ^{2}\to \mathbb {R} \quad }
{\displaystyle \quad v:\mathbb {R} ^{2}\to \mathbb {R} ,}
i.e., into two real-valued functions (
{\displaystyle u}
{\displaystyle v}
) of two real variables (
{\displaystyle x}
{\displaystyle y}
Similarly, any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: (Re f, Im f) or, alternatively, as a vector-valued function from X into
{\displaystyle \mathbb {R} ^{2}.}
Complex functions that are differentiable at every point of an open subset
{\displaystyle \Omega }
of the complex plane are said to be holomorphic on
{\displaystyle \Omega }
. In the context of complex analysis, the derivative o{\displaystyle f}
{\displaystyle z_{0}}
{\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}.}
Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach
{\displaystyle z_{0}}
in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions are infinitely differentiable, whereas the existence of the nth derivative need not imply the existence of the (n + 1)th derivative for real functions. Furthermore, all holomorphic functions satisfy the stronger condition of analyticity, meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on
{\displaystyle \Omega }
can be approximated arbitrarily well by polynomials in some neighborhood of every point in
{\displaystyle \Omega }
. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that are nowhere analytic; see Non-analytic smooth function § A smooth function which is nowhere real analytic.
Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately to complex arguments as functions
{\displaystyle \mathbb {C} \to \mathbb {C} }
, are holomorphic over the entire complex plane, making them entire functions, while rational functions
{\displaystyle p/q}
, where p and q are polynomials, are holomorphic on domains that exclude points where q is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions
{\displaystyle z\mapsto \Re (z)}
{\displaystyle z\mapsto |z|}
{\displaystyle z\mapsto {\bar {z}}}
are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy–Riemann conditions (see below).
An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy–Riemann conditions. If
{\displaystyle f:\mathbb {C} \to \mathbb {C} }
{\displaystyle f(z)=f(x+iy)=u(x,y)+iv(x,y)}
{\displaystyle x,y,u(x,y),v(x,y)\in \mathbb {R} }
, is holomorphic on a region
{\displaystyle \Omega }
{\displaystyle z_{0}\in \Omega }
{\displaystyle {\frac {\partial f}{\partial {\bar {z}}}}(z_{0})=0,\ {\text{where }}{\frac {\partial }{\partial {\bar {z}}}}\mathrel {:=} {\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right).}
In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations
{\displaystyle u_{x}=v_{y}}
{\displaystyle u_{y}=-v_{x}}
, where the subscripts indicate partial differentiation. However, the Cauchy–Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (see Looman–Menchoff theorem).
Holomorphic functions exhibit some remarkable features. For instance, Picard's theorem asserts that the range of an entire function can take only three possible forms:
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {C} \setminus \{z_{0}\}}
{\displaystyle \{z_{0}\}}
{\displaystyle z_{0}\in \mathbb {C} }
. In other words, if two distinct complex numbers
{\displaystyle z}
{\displaystyle w}
are not in the range of an entire function
{\displaystyle f}
{\displaystyle f}
is a constant function. Moreover, a holomorphic function on a connected open set is determined by its restriction to any nonempty open subset.
Color wheel graph of the function f(x) = (x2 − 1)(x − 2 − i)2/x2 + 2 + 2i.
Hue represents the argument, brightness the magnitude.
Ablowitz, M. J. & A. S. Fokas, Complex Variables: Introduction and Applications (Cambridge, 2003).
Ahlfors, L., Complex Analysis (McGraw-Hill, 1953).
Cartan, H., Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes. (Hermann, 1961). English translation, Elementary Theory of Analytic Functions of One or Several Complex Variables. (Addison-Wesley, 1963).
Carathéodory, C., Funktionentheorie. (Birkhäuser, 1950). English translation, Theory of Functions of a Complex Variable (Chelsea, 1954). [2 volumes.]
Carrier, G. F., M. Krook, & C. E. Pearson, Functions of a Complex Variable: Theory and Technique. (McGraw-Hill, 1966).
Conway, J. B., Functions of One Complex Variable. (Springer, 1973).
Fisher, S., Complex Variables. (Wadsworth & Brooks/Cole, 1990).
Forsyth, A., Theory of Functions of a Complex Variable (Cambridge, 1893).
Freitag, E. & R. Busam, Funktionentheorie. (Springer, 1995). English translation, Complex Analysis. (Springer, 2005).
Goursat, E., Cours d'analyse mathématique, tome 2. (Gauthier-Villars, 1905). English translation, A course of mathematical analysis, vol. 2, part 1: Functions of a complex variable. (Ginn, 1916).
Kreyszig, E., Advanced Engineering Mathematics. (Wiley, 1962).
Lavrentyev, M. & B. Shabat, Методы теории функций комплексного переменного. (Methods of the Theory of Functions of a Complex Variable). (1951, in Russian).
Markushevich, A. I., Theory of Functions of a Complex Variable, (Prentice-Hall, 1965). [Three volumes.]
Marsden & Hoffman, Basic Complex Analysis. (Freeman, 1973).
Needham, T., Visual Complex Analysis. (Oxford, 1997). http://usf.usfca.edu/vca/
Remmert, R., Theory of Complex Functions. (Springer, 1990).
Rudin, W., Real and Complex Analysis. (McGraw-Hill, 1966).
Shaw, W. T., Complex Analysis with Mathematica (Cambridge, 2006).
Stein, E. & R. Shakarchi, Complex Analysis. (Princeton, 2003).
Sveshnikov, A. G. & A. N. Tikhonov, Теория функций комплексной переменной. (Nauka, 1967). English translation, The Theory Of Functions Of A Complex Variable (MIR, 1978).
Titchmarsh, E. C., The Theory of Functions. (Oxford, 1932).
Wegert, E., Visual Complex Functions. (Birkhäuser, 2012).
Whittaker, E. T. & G. N. Watson, A Course of Modern Analysis. (Cambridge, 1902). 3rd ed. (1920)
Complex analysisat Wikipedia's sister projects
Retrieved from " https://en.wikipedia.org/?title=Complex_analysis&oldid=1088129730"
Complex Analysis Videos
Complex Analysis Websites
Complex Analysis Encyclopedia Articles |
Van der Waals Force | Brilliant Math & Science Wiki
Jordan Calmes, Rafael Rafael, Rafit Hridoy, and
Van der Waals forces are specific intermolecular interactions observed in liquids and solids. They are electrostatic in nature, arising from the interactions of positively and negatively charged species.
intermolecular forces hold molecules together (in contrast to intramolecular forces, which hold atoms together within a molecule). They help determine bulk properties such as boiling point and melting point.
There are two intermolecular forces that are collectively referred to as Van der Waals Forces: London Dispersion Forces and dipole-dipole interactions.
In atoms, the electrons are continuously orbiting in shells. It is possible that at some point all the electrons come to one side of the atom, making it an instantaneous dipole that repels the electrons of neighboring atoms, making an induced dipole. This interaction between instantaneous dipole-induced dipole is known as the London dispersion force.
Dipole-dipole forces are similar to London Dispersion forces, but they occur in molecules that are permanently polar versus momentarily polar.
Dipole-ion interactions Dipole-dipole interactions Hydrogen bonds London dispersion forces
Which type of intermolecular force is dominant in carbon tetrachloride (
\ce{CCl_4}
When we have some special cases, like nonideal(real) gases. We can use the equation to predict gas properties:
(P+\frac{n^2a}{V^2})(V -nb) = nRT
The V in the formula refers to the volume of gas, in moles n. The intermolecular forces of attraction are incorporated into the equation with the
\frac{n^2a}{V^2}
term where
a
is a specific value of a particular gas.
P
represents the pressure measured, which is expected to be lower than in usual cases. The variable
b
expresses the eliminated volume per mole, which accounts for the volume of gas molecules and is also a value of a particular gas.
R
is a known constant,
0.08206\frac{L*atm}{mol*K}
T
stands for temperature.
Unlike most equations used for the calculation of real, or ideal, gases, Van der Waals equation takes into account, and corrects for, the volume of participating molecules and the intermolecular forces of attraction.
1. \text{NH}_3 \qquad 2. \text{N}_2 \qquad 3. \text{CH}_{2}\text{Cl}_2 \qquad 4. \text{Cl}_2 \qquad 5. \text{CCl}_4
Predict which of the above gases have:
(i) the smallest van der Waals "a" constant
(ii) the largest "b" constant.
Concatenate the answer, for an example if gas
1. \text{NH}_3
fits part (i) and gas
4. \text{Cl}_2
fits the part (ii), then enter the answer as 14. For a and b refer van der waal
Cite as: Van der Waals Force. Brilliant.org. Retrieved from https://brilliant.org/wiki/van-der-waals-force/ |
gnuplot is a command-line driven multiplatform plotting program. Despite the name, it is not associated with GNU project and is not covered by GNU GPL. The source code license is a gratis one, but not a copyleft one; "Permission to modify the software is granted, but not the right to distribute the complete modified source code."[1]
3.1.1 The ? operator
3.1.2 plot each function individually
3.1.3 Parametric mode
5.1 colorsequence
6 Generating Wikimedia graphs
6.1 gnuplot options
6.3 Wikimedia Commons Upload
8 Other Wikimedia resources
gnuplot can be used interactively, in batch mode, or embedded in (scripted by) another program, such as GNU Octave.
Interactively, run gnuplot at the command line.
In batch mode, run gnuplot input.plt (where input.plt is the name of the input file) at the command line.
In another program, use that program’s plotting facilities – gnuplot will be called transparently.
For use in one’s own programs, one can run gnuplot via popen, or use a library that wraps gnuplot for the programming language. These wrapper libraries exist for C, C++(e.g. Gnuplot-iostream, gnuplot-cpp) Python, Perl, Java, Fortran95, and others.
As very simple usage, start gnuplot and type:
This will display a plot of the sine function, and then exit.
To plot a function or functions:
define the function;
determine the range of inputs and outputs;
determine the style for the region and the graphs;
plot it (using the plot function)
To plot data, collect the data in a file instead of defining a function.
Unsetting the default decorations will yield a completely plain graph area:
These fields can now be set individually, if desired.
To format an axis as a percentage, multiply the number by 100 and suffix a “%” symbol using format, as in:
set format y "%g %%"
plot "dat1.txt" using 1:($2*100)
Types of functionsEdit
Piecewise-defined functionsEdit
See: Piecewise function
Several ways are possible.
The ? operatorEdit
One can plot piecewise-defined functions in gnuplot with the ternary condition operator (?:). For instance, one can manually define the absolute value function by:
f(x) = x > 0 ? x : -x
Read this as “if…then…else”: “if x is greater than 0, then
{\displaystyle f(x)=x,}
{\displaystyle f(x)=-x.}
One can chain these, for instance by:
f(x) = x < 1 ? 1 \
: x < 2 ? 3 \
This corresponds to the piecewise function
{\displaystyle f(x)={\begin{cases}1&{\text{if }}x<1\\3&{\text{if }}x\geq 1{\text{ and }}x<2\\5&{\text{otherwise}}\end{cases}}}
For piecewise functions, you will likely want many samples, so that discontinuities appear as vertical lines, and corners appear sharp, so:
(Using 1001 instead of 1000 avoids artifacts of having a sample point appear directly on a discontinuity, which can introduce "stair steps.")
plot each function individuallyEdit
Or you can plot each function individually, like you appear to be doing now.
f(x) = a1*x**2+b1*x+c1 for x in [t11,t12]
t11<=t12<=t21<=t22, etc.
Parametric modeEdit
Better yet, switch to parametric mode, map a common t interval [0:1] to your individual t ranges, and then:
x1(t) = t11+t*(t12-t11) ...
plot x1(t), f1(x1(t)), x2(t), f2(x2(t)), ... plot each function individually
Source filesEdit
gnuplot allows one to load files via the load command, or by passing them as arguments on the command line. This is very useful for complicated graphs.
There is no official standard extension, but some semi-official extensions are used:
.plt, .gnu, .gpi, or .gih for general gnuplot files;
.dat for data;
.fnc for function definitions.
For formatting source code, two useful pieces of syntax are:
The hash (#) character starts inline comments, which continue to the end of the line.
A trailing backslash (\) is a line continuation character, and allows one to split a long expression over multiple lines. One can also indent/line up the continuing lines for legibility (as in piecewise functions), as initial whitespace is ignored.
For debugging a gnuplot file, it is often useful to:
Change the terminal to interactive (instead of outputting to a file), by commenting out set terminal and output lines.
Start gnuplot interactively, then load the file in question.
Put pause -1 (pause until carriage return) at the end of the file, then run it from the command line.
Alternatively, run gnuplot with the -persist command line switch, so gnuplot exits, but the window persists.
Make the file itself executable, by shebang (#!) notation (depends on exact path):
Syntax highlightingEdit
vim has automatic syntax highlighting for gnuplot (gnuplot.vim) as long as the file extension is .gpi.
Alternatively, other file extensions, such as .plt, can be added in the usual autocommand way:
au BufNewFile,BufRead *.plt,*.gnuplot setf gnuplot
Also see a github repository compatible with pathogen's auto filetype detection features.
colorsequenceEdit
set colorsequence {default|classic|podo}
help set colorsequence
cycle set:
set linetype 1 lc rgb "dark-violet" lw 2 pt 0
set linetype 2 lc rgb "sea-green" lw 2 pt 7
set linetype 3 lc rgb "cyan" lw 2 pt 6 pi -1
set linetype 4 lc rgb "dark-red" lw 2 pt 5 pi -1
set linetype 5 lc rgb "blue" lw 2 pt 8
set linetype 6 lc rgb "dark-orange" lw 2 pt 3
set linetype 7 lc rgb "black" lw 2 pt 11
set linetype 8 lc rgb "goldenrod" lw 2
9 cycle set:
# https://stackoverflow.com/questions/46775612/colorsequence-for-more-than-8-colors-gnuplot
# Ethan A Merritt - my preference for gnuplot colors
# 2 3 4 5 6 8 are borrowed from the colors_podo set
set linetype 1 lc rgb "dark-violet" lw 1
set linetype 2 lc rgb "#009e73" lw 1
set linetype 3 lc rgb "#56b4e9" lw 1
set linetype 4 lc rgb "#e69f00" lw 1
set linetype 5 lc rgb "#f0e442" lw 1
set linetype 6 lc rgb "#0072b2" lw 1
set linetype 7 lc rgb "#e51e10" lw 1
set linetype 8 lc rgb "black" lw 1
set linetype 9 lc rgb "gray50" lw 1
set linetype cycle 9
gnuplot/src/getcolor.c
gnuplot/src/color.c
Generating Wikimedia graphsEdit
See: How to create graphs for Wikipedia articles: gnuplot
To generate graphs for Wikimedia:
Store your code in a file, preferably with comments
Use high quality (vector graphic) SVG output.
Generate the SVG.
Optionally post-process.
Upload the graph and source code to Wikimedia Commons.
gnuplot optionsEdit
set terminal svg enhanced size 300 300
set samples 1001 # high quality
set border 31 linewidth .3 # thin border
set output "filename.svg"
(The file name should be changed to something more descriptive, though this is not strictly necessary.)
This will create an SVG which is nominally 300 × 300, a common Wikimedia display size: it is easiest to make graphs whose nominal size is the expected display size, but beware that images can and are resized, and that this affects thickness and legibility – if using a large nominal size, so that it will likely be resized down, use large fonts and thick lines.
Beware that with horizontal writing, the y-axis labels will likely take up more space (horizontally) than the x-axis labels take (vertically), and thus a nominally square graph will have an actually graphing area which is slightly taller than it is wide.
One can set the font via:
set terminal svg enhanced size 300 300 fname "Times" fsize 36
sets output to be an SVG file
means to use enhanced text output, when Greek letters are needed. a table can be found [here].
sets the nominal size of the SVG as 300 × 300
this sets many samples for high quality; the 1001 (instead of 1000) is so that a sample is unlikely to land directly on a pixel or discontinuity, which can cause aliasing. If you have unexpected aliasing, try changing this to 1002 or 1003, as that will move all sample points, possibly fixing the problem.
set border 31 linewidth .3
“31” is 1111 in binary, meaning “all borders”; use “3” (0011) in binary for just the lower and left borders. The thin linewidth makes the border less prominent, emphasizing the line. [Note, 31 actually is 11111 in binary. The fifth bit is irrelevant to this example and the example should be rewritten with 15 instead of 31.] Note, use interactive gnuplot command "help set border" to see an explanation of what each bit controls.
This sets the output file name. In normal use, choose a filename more descriptive of file contents.
Post-processingEdit
One may wish to post-process the SVG, either in a vector graphics program such as Inkscape, or by hand (as SVG files are text). This can be useful to add annotations which would be otherwise hard to produce in gnuplot, or one may incorporate the plot as one component of a larger or more complex figure.
Wikimedia Commons UploadEdit
Please use the template {{gnuplot}} to flag it as made with gnuplot.
The source code may be included in the “Source” section of the description (if brief), or more often in a separate == gnuplot source == section.
The source is most legible if wrapped in Syntax highlighting via:
<syntaxhighlight lang="gnuplot">
If you use text, it may be translated – please use the template {{Translation possible}} to indicate that translation is possible.
Alternatively, minimize the use of text (place in a separate caption) to aid reuse of the image in other languages.
There are a number of design considerations in graphs, considered as information graphics. A good resource are the works and writings of information graphic designers, such as the highly regarded works of Edward Tufte: his The Visual Display of Quantitative Information is most relevant for graphs, but his and others other work can be insightful and inspiring.
The first consideration is what to graph, and whether a graph is the best way to convey certain information: graphs can be unexpectedly useful, or conversely, a graph may not be the best way to convey information. Further, how a graph is connected an integrated with other material is a question – is it referred to? Described and discussed?
Other media that can be an alternative to a graph, or support it, include:
text, either running text, a list, or an isolated single item (pull quote)
animations (possibly animated graphs)
Often it is useful to portray the same information in several ways.
A second question is how many graphs to use.
Most obviously one may use a single, large, detailed graph, and this is often appropriate, such as if the details of the data are important.
Alternatively, consider using small multiples – several small graphs to make a point, through repetition and variation, analogous with written “compare and contrast”.
It is especially helpful to align graphs or place them on a grid, so the eye can easily switch between them.
Presenting the same data on different scales can also be revealing; a simple example is shown at estimated sign, where the same data is shown on an absolute scale and relative scale.
One can also use graphs in-line, as in sparklines.
Beyond these general considerations, there are finer questions:
Which range of data to display?
What scale to use? For example, should the range of the graph area agree with the range of the data (maximizing use of space), or should the range be larger, providing context? Often a log scale or log-log scale is appropriate, but may be confusing to novice readers.
How to distinguish data? What color and line styles to use?
How prominent to make various data – how thick or thin to make different lines, how large to make text?
Other Wikimedia resourcesEdit
gnuplot at Wikipedia
How to create graphs for Wikipedia articles has a section on gnuplot
The book Ad Hoc Data Analysis From The Unix Command Line has a chapter on Quick Plotting With gnuplot
commons:Category:Gnuplot diagrams has many examples of gnuplot graphs, many with source code.
Images with Gnuplot src code
Information graphic designers and their work, such as the highly regarded works of Edward Tufte
gnuplot Central – homepage
l Official documentation (PDF)
"gnuplot not so Frequently Asked Questions". Archived from the original on 2012-10-29. http://web.archive.org/web/20121029110317/http://t16web.lanl.gov/Kawano/gnuplot/index-e.html.
Janert, Philipp K. (2015), Gnuplot in Action, Second Edition, Manning Publications, New York, USA, pp. 425, ISBN 978-1-633430-18-1, http://www.manning.com/janert2/ .
Phillips, Lee (2012), gnuplot Cookbook, Packt Publishing, pp. 220, ISBN 184951724X, http://www.packtpub.com/gnuplot-visual-guide-with-plotting-software-cookbook/book .
Retrieved from "https://en.wikibooks.org/w/index.php?title=Gnuplot&oldid=3681889" |
ORDER THEORY - Encyclopedia Information
Order theory Information
https://en.wikipedia.org/wiki/Order_theory
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (December 2015) ( Learn how and when to remove this template message)
The notion of order is very general, extending beyond contexts that have an immediate, intuitive feel of sequence or relative quantity. In other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the subset relation, e.g., " Pediatricians are physicians," and " Circles are merely special-case ellipses."
A partial order with this property is called a total order. These orders can also be called linear orders or chains. While many familiar orders are linear, the subset order on sets provides an example where this is not the case. Another example is given by the divisibility (or "is-a- factor-of") relation |. For two natural numbers n and m, we write n|m if n divides m without remainder. One easily sees that this yields a partial order. The identity relation = on any set is also a partial order in which every two distinct elements are incomparable. It is also the only relation that is both a partial order and an equivalence relation. Many advanced properties of posets are interesting mainly for non-linear orders.
Even some infinite sets can be diagrammed by superimposing an ellipsis (...) on a finite sub-order. This works well for the natural numbers, but it fails for the reals, where there is no immediate successor above 0; however, quite often one can obtain an intuition related to diagrams of a similar kind[ vague].
{\displaystyle \bigvee S}
{\displaystyle \bigwedge S}
{\displaystyle x\vee y}
{\displaystyle x\wedge y}
Partial orders with complements, or poc sets, [1] are posets with a unique bottom element 0, as well as an order-reversing involution
{\displaystyle *}
{\displaystyle a\leq a^{*}\implies a=0.}
Contributors to ordered geometry were listed in a 1961 textbook:
— H. S. M. Coxeter, Introduction to Geometry
In 1901 Bertrand Russell wrote "On the notion of order" [2] exploring the foundations of the idea through generation of series. He returned to the topic in part IV of The Principles of Mathematics (1903). Russell noted that binary relation aRb has a sense proceeding from a to b with the converse relation having an opposite sense, and sense "is the source of order and series". (p 95) He acknowledges Immanuel Kant [3] was "aware of the difference between logical opposition and the opposition of positive and negative". He wrote that Kant deserves credit as he "first called attention to the logical importance of asymmetric relations."
The term poset as an abbreviation for partially ordered set was coined by Garrett Birkhoff in the second edition of his influential book Lattice Theory. [4] [5]
^ Bertrand Russell (1901) Mind 10(2)
^ Immanuel Kant (1763) Versuch den Begriff der negativen Grosse in die Weltweisheit einzufuhren
Birkhoff, Garrett (1940). Lattice Theory. Vol. 25 (3rd Revised ed.). American Mathematical Society. ISBN 978-0-8218-1025-5.
Gierz, G.; Hofmann, K. H.; Keimel, K.; Mislove, M.; Scott, D. S. (2003). Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications. Vol. 93. Cambridge University Press. ISBN 978-0-521-80338-0.
Retrieved from " https://en.wikipedia.org/?title=Order_theory&oldid=1070386518"
Order Theory Videos
Order Theory Websites
Order Theory Encyclopedia Articles |
Up to — Wikipedia Republished // WIKI 2
Top: In a hexagon vertex set there are 20 partitions which have one three-element subset (green) and three single-element subsets (uncolored). Bottom: Of these, there are 4 partitions up to rotation, and 3 partitions up to rotation and reflection.
Two mathematical objects a and b are called equal up to an equivalence relation R
if a and b are related by R, that is,
if aRb holds, that is,
if the equivalence classes of a and b with respect to R are equal.
Further examples include "up to isomorphism", "up to permutations", and "up to rotations", which are described in the Examples section.
UP TALKS | Mathematical Modeling Seeing the World through Mathematical Lenses | Jose Maria Balmaceda
Mathematical Significance of 666 (the Number of the Beast)
UP POLICE CONSTABLE || MATHS || 26744+पद || ऐसा आता है पेपर || By Amit Sir
1.1 Tetris
1.2 Eight queens
1.3 Polygons
1.4 Group theory
Tetris pieces I, J, L, O, S, T, Z
A simple example is "there are seven reflecting tetrominoes, up to rotations", which makes reference to the seven possible contiguous arrangements of tetrominoes (collections of four unit squares arranged to connect on at least one side) and which are frequently thought of as the seven Tetris pieces (O, I, L, J, T, S, Z). One could also say "there are five tetrominoes, up to reflections and rotations", which would then take into account the perspective that L and J (as well as S and Z) can be thought of as the same piece when reflected. The Tetris game does not allow reflections, so the former statement is likely to seem more relevant.
A solution of the eight queens problem
In the eight queens puzzle, if the eight queens are considered to be distinct, then there are 3709440 distinct solutions. Normally, however, the queens are considered to be equal, and one usually says "there are 92 (
{\displaystyle ={\tfrac {3709440}{8!}}}
) unique solutions up to permutations of the queens", or that "there are 92 solutions modulo the names of the queens", signifying that two different arrangements of the queens are considered equivalent if the queens have been permuted, but the same squares on the chessboard are occupied by them.
If, in addition to treating the queens as identical, rotations and reflections of the board were allowed, we would have only 12 distinct solutions up to symmetry and the naming of the queens, signifying that two arrangements that are symmetrical to each other are considered equivalent (for more, see Eight queens puzzle § Solutions).
The regular n-gon, for given n, is unique up to similarity. In other words, if all similar n-gons are considered instances of the same n-gon, then there is only one regular n-gon.
In group theory, one may have a group G acting on a set X, in which case, one might say that two elements of X are equivalent "up to the group action"—if they lie in the same orbit.
A hyperreal x and its standard part st(x) are equal up to an infinitesimal difference.
In computer science, the term up-to techniques is a precisely defined notion that refers to certain proof techniques for (weak) bisimulation, and to relate processes that only behave similarly up to unobservable steps.[3]
Look up up to in Wiktionary, the free dictionary.
Abuse of notation
All other things being equal
Essentially unique
List of mathematical jargon
Quotient set
^ Nekovář, Jan (2011). "Mathematical English (a brief summary)" (PDF). Institut de mathématiques de Jussieu – Paris Rive Gauche. Retrieved 2019-11-21.
^ Weisstein, Eric W. "Tetromino". mathworld.wolfram.com. Retrieved 2019-11-21.
^ Damien Pous, Up-to techniques for weak bisimulation, Proc. 32nd ICALP, Lecture Notes in Computer Science, vol. 3580, Springer Verlag (2005), pp. 730–741
Up-to Techniques for Weak Bisimulation |
Angle_of_repose Knowpia
The angle of repose, or critical angle of repose,[1] of a granular material is the steepest angle of descent or dip relative to the horizontal plane to which a material can be piled without slumping. At this angle, the material on the slope face is on the verge of sliding. The angle of repose can range from 0° to 90°. The morphology of the material affects the angle of repose; smooth, rounded sand grains cannot be piled as steeply as can rough, interlocking sands. The angle of repose can also be affected by additions of solvents. If a small amount of water is able to bridge the gaps between particles, electrostatic attraction of the water to mineral surfaces will increase the angle of repose, and related quantities such as the soil strength.
Sandpile from Matemateca [pt] IME-USP collection
Applications of theoryEdit
Talus cones on north shore of Isfjord, Svalbard, Norway, showing angle of repose for coarse sediment
It is also commonly used by mountaineers as a factor in analysing avalanche danger in mountainous areas.[citation needed]
There are numerous methods for measuring angle of repose and each produces slightly different results. Results are also sensitive to the exact methodology of the experimenter. As a result, data from different labs are not always comparable. One method is the triaxial shear test, another is the direct shear test.
If the coefficient of static friction is known of a material, then a good approximation of the angle of repose can be made with the following function. This function is somewhat accurate for piles where individual objects in the pile are minuscule and piled in random order.[2]
{\displaystyle \tan {(\theta )}\approx \mu _{\mathrm {s} }\,}
where, μs is the coefficient of static friction, and
{\displaystyle \theta }
is the angle of repose.
Methods in determining the angle of reposeEdit
The measured angle of repose may vary with the method used.
Tilting box methodEdit
This method is appropriate for fine-grained, non-cohesive materials with individual particle size less than 10 mm. The material is placed within a box with a transparent side to observe the granular test material. It should initially be level and parallel to the base of the box. The box is slowly tilted until the material begins to slide in bulk, and the angle of the tilt is measured.
Fixed funnel methodEdit
The material is poured through a funnel to form a cone. The tip of the funnel should be held close to the growing cone and slowly raised as the pile grows, to minimize the impact of falling particles. Stop pouring the material when the pile reaches a predetermined height or the base a predetermined width. Rather than attempt to measure the angle of the resulting cone directly, divide the height by half the width of the base of the cone. The inverse tangent of this ratio is the angle of repose.
Revolving cylinder methodEdit
The material is placed within a cylinder with at least one transparent end. The cylinder is rotated at a fixed speed and the observer watches the material moving within the rotating cylinder. The effect is similar to watching clothes tumble over one another in a slowly rotating clothes dryer. The granular material will assume a certain angle as it flows within the rotating cylinder. This method is recommended for obtaining the dynamic angle of repose, and may vary from the static angle of repose measured by other methods.
Of various materialsEdit
This pile of corn has a low angle of repose
Here is a list of various materials and their angle of repose.[3] All measurements are approximated.
Material (condition)
Ashes 40°
Asphalt (crushed) 30–45°
Bark (wood refuse) 45°
Bran 30–45°
Chalk 45°
Clay (dry lump) 25–40°
Clay (wet excavated) 15°
Clover seed 28°
Coconut (shredded) 45°
Coffee bean (fresh) 35–45°
Earth 30–45°
Flour (corn) 30–40°
Flour (wheat) 45°
Granite 35–40°
Gravel (crushed stone) 45°
Gravel (natural w/ sand) 25–30°
Malt 30–45°
Sand (dry) 34°
Sand (water filled) 15–30°
Sand (wet) 45°
Snow 38°[4]
Urea (Granular) 27° [5]
Wheat 27°
With different supportsEdit
Different supports will modify the shape of the pile, in the illustrations below sand piles, though angles of repose remain the same.[6][7]
Multiple pit
Random format
Exploitation by antlion and wormlion (Vermileonidae) larvaeEdit
Sand pit trap of the antlion
The larvae of the antlions and the unrelated wormlions Vermileonidae trap small insects such as ants by digging conical pits in loose sand, such that the slope of the walls is effectively at the critical angle of repose for the sand.[8] They achieve this by flinging the loose sand out of the pit and permitting the sand to settle at its critical angle of repose as it falls back. Thus, when a small insect, commonly an ant, blunders into the pit, its weight causes the sand to collapse below it, drawing the victim toward the center where the predator that dug the pit lies in wait under a thin layer of loose sand. The larva assists this process by vigorously flicking sand out from the center of the pit when it detects a disturbance. This undermines the pit walls and causes them to collapse toward the center. The sand that the larva flings also pelts the prey with so much loose, rolling material as to prevent it from getting any foothold on the easier slopes that the initial collapse of the slope has presented. The combined effect is to bring the prey down to within grasp of the larva, which then can inject venom and digestive fluids.
The angle of repose plays a part in several topics of technology and science, including:
^ Mehta, A.; Barker, G. C. (1994). "The dynamics of sand". Reports on Progress in Physics. 57 (4): 383. Bibcode:1994RPPh...57..383M. doi:10.1088/0034-4885/57/4/002.
^ Nichols, E. L.; Franklin, W. S. (1898). The Elements of Physics. Vol. 1. Macmillan. p. 101. LCCN 03027633.
^ Glover, T. J. (1995). Pocket Ref. Sequoia Publishing. ISBN 978-1885071002.
^ Rikkers, Mark; Rodriguez, Aaron (23 June 2009). "Anatomy of an Avalanche". Telluridemagazine.com. Telluride Publishing. Archived from the original on 19 August 2016. Retrieved 3 October 2016.
^ Ileleji, K. E.. (2008-10-28). "The angle of repose of bulk corn stover particles". Powder Technology 187 (2): 110–118. doi:10.1016/j.powtec.2008.01.029.
^ Lobo-Guerrero, Sebastian. (2007-03-23). "Influence of pile shape and pile interaction on the crushable behavior of granular materials around driven piles: DEM analyses" (em en). Granular Matter 9 (3–4): 241. doi:10.1007/s10035-007-0037-3. ISSN 1434-5021.
^ Botz, J. T.; Loudon, C.; Barger, J. B.; Olafsen, J. S.; Steeples, D. W. (2003). "Effects of slope and particle size on ant locomotion: Implications for choice of substrate by antlions". Journal of the Kansas Entomological Society. 76 (3): 426–435. |
Compact Gaussian process regression model class - MATLAB - MathWorks Switzerland
H=1
H=\left[1,X\right]
H=\left[1,X,{X}_{2}\right],
{X}_{2}=\left[\begin{array}{cccc}{x}_{11}^{2}& {x}_{12}^{2}& \cdots & {x}_{1d}^{2}\\ {x}_{21}^{2}& {x}_{22}^{2}& \cdots & {x}_{2d}^{2}\\ ⋮& ⋮& ⋮& ⋮\\ {x}_{n1}^{2}& {x}_{n2}^{2}& \cdots & {x}_{nd}^{2}\end{array}\right].
H=hfcn\left(X\right),
K\left({X}_{new},A\right)*\alpha \text{\hspace{0.17em}}.
K\left({X}_{new},A\right)
{X}_{new}
K\left({X}_{new},A\right)*\alpha \text{\hspace{0.17em}}.
K\left({X}_{new},A\right)
{X}_{new} |
Revision as of 02:21, 23 October 2020 by Yirkajk (talk | contribs) (→AH: Arithmetic Hierarchy: There's a bug in <math> with \Pi. Used unicode characters as workaround.)
{\displaystyle t(n)}
{\displaystyle t(n)}
{\displaystyle \Sigma _{0}=\Delta _{0}=\Pi _{0}}
{\displaystyle \times }
{\displaystyle \phi =\forall i<j\psi }
{\displaystyle \phi =\exists i<j\psi }
{\displaystyle j}
{\displaystyle \phi }
{\displaystyle \Sigma _{i+1}}
{\displaystyle \exists X_{1}\dots \exists X_{n},\psi }
{\displaystyle \psi \in \Delta _{i}}
{\displaystyle \Sigma _{i}} |
ALGEBRAIC NUMBER THEORY - Encyclopedia Information
Algebraic number theory Information
https://en.wikipedia.org/wiki/Algebraic_number_theory
{\displaystyle \mathbb {Z} }
{\displaystyle 0=\mathbb {Z} _{1}}
{\displaystyle \mathbb {Z} _{1}}
{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }
{\displaystyle \mathbb {Z} (p^{\infty })}
{\displaystyle \mathbb {T} }
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Z} [1/p]}
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {Z} _{p}}
{\displaystyle \mathbb {Q} _{p}}
{\displaystyle \mathbb {T} _{p}}
1 History of algebraic number theory
1.1 Diophantus
2.1 Failure of unique factorization
2.2 Factorization into prime ideals
2.3 Ideal class group
2.4 Real and complex embeddings
2.5.1 Places at infinity geometrically
2.7 Zeta function
3.1 Finiteness of the class group
3.3 Reciprocity laws
3.4 Class number formula
7.2 Intermediate texts
7.3 Graduate level texts
The beginnings of algebraic number theory can be traced to Diophantine equations, [1] named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively:
{\displaystyle A=x+y\ }
{\displaystyle B=x^{2}+y^{2}.\ }
Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC). [2] Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC). [3]
One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae ( Latin: Arithmetical Investigations) is a textbook of number theory written in Latin [4] by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange and Legendre and adds important new results of his own. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.
In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms (later refined by his student Leopold Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. [5] Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory. [6]
He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law. [5] The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.
David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. [7] He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.
He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi. [8]
Emil Artin established the Artin reciprocity law in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory. [9] The term " reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
It was initially dismissed as unlikely or highly speculative, and was taken more seriously when number theorist André Weil found evidence supporting it, but no proof; as a result the "astounding" [10] conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the Langlands program, a list of important conjectures needing proof or disproof.
From 1993 to 1994, Andrew Wiles provided a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993 [11] in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with Richard Taylor, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques not available to Fermat.
{\displaystyle 6=2\cdot 3=(-2)\cdot (-3).}
In general, if u is a unit, meaning a number with a multiplicative inverse in O, and if p is a prime element, then up is also a prime element. Numbers such as p and up are said to be associate. In the integers, the primes p and −p are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When K is not the rational numbers, however, there is no analog of positivity. For example, in the Gaussian integers Z[i], [12] the numbers 1 + 2i and −2 + i are associate because the latter is the product of the former by i, but there is no way to single out one as being more canonical than the other. This leads to equations such as
{\displaystyle 5=(1+2i)(1-2i)=(2+i)(2-i),}
However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and an irreducible element. An irreducible element x is an element such that if x = yz, then either y or z is a unit. These are the elements that cannot be factored any further. Every element in O admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ring Z[√-5]. [13] In this ring, the numbers 3, 2 + √-5 and 2 - √-5 are irreducible. This means that the number 9 has two factorizations into irreducible elements,
{\displaystyle 9=3^{2}=(2+{\sqrt {-5}})(2-{\sqrt {-5}}).}
{\displaystyle I={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}},}
{\displaystyle {\mathfrak {p}}_{i}}
is a prime ideal, and where this expression is unique up to the order of the factors. In particular, this is true if I is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization. In the language of ring theory, it says that rings of integers are Dedekind domains.
{\displaystyle 2\mathbf {Z} [i]=(1+i)\mathbf {Z} [i]\cdot (1-i)\mathbf {Z} [i]=((1+i)\mathbf {Z} [i])^{2};}
{\displaystyle J^{-1}=(O:J)=\{x\in K:xJ\subseteq O\}.}
{\displaystyle (x)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}}.}
{\displaystyle \operatorname {div} x=\sum _{i=1}^{t}e_{i}[{\mathfrak {p}}_{i}].}
{\displaystyle 1\to O^{\times }\to K^{\times }{\xrightarrow {\text{div}}}\operatorname {Div} K\to \operatorname {Cl} K\to 1.}
Considering all embeddings at once determines a function
{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {C} ^{r_{2}}}
{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {R} ^{2r_{2}}.}
This is called the Minkowski embedding.
The subspace of the codomain fixed by complex conjugation is a real vector space of dimension d called Minkowski space. Because the Minkowski embedding is defined by field homomorphisms, multiplication of elements of K by an element x ∈ K corresponds to multiplication by a diagonal matrix in the Minkowski embedding. The dot product on Minkowski space corresponds to the trace form
{\displaystyle \langle x,y\rangle =\operatorname {Tr} (xy)}
The image of O under the Minkowski embedding is a d-dimensional lattice. If B is a basis for this lattice, then det BTB is the discriminant of O. The discriminant is denoted Δ or D. The covolume of the image of O is
{\displaystyle {\sqrt {|\Delta |}}}
A place of an algebraic number field is an equivalence class of absolute value functions on K. There are two types of places. There is a
{\displaystyle {\mathfrak {p}}}
-adic absolute value for each prime ideal
{\displaystyle {\mathfrak {p}}}
of O, and, like the p-adic absolute values, it measures divisibility. These are called finite places. The other type of place is specified using a real or complex embedding of K and the standard absolute value function on R or C. These are infinite places. Because absolute values are unable to distinguish between a complex embedding and its conjugate, a complex embedding and its conjugate determine the same place. Therefore, there are r1 real places and r2 complex places. Because places encompass the primes, places are sometimes referred to as primes. When this is done, finite places are called finite primes and infinite places are called infinite primes. If v is a valuation corresponding to an absolute value, then one frequently writes
{\displaystyle v\mid \infty }
to mean that v is an infinite place and
{\displaystyle v\nmid \infty }
to mean that it is a finite place.
There is a geometric analogy for places at infinity which holds on the function fields of curves. For example, let
{\displaystyle k=\mathbb {F} _{q}}
{\displaystyle X/k}
be a smooth, projective, algebraic curve. The function field
{\displaystyle F=k(X)}
has many absolute values, or places, and each corresponds to a point on the curve. If
{\displaystyle X}
is the projective completion of an affine curve
{\displaystyle {\hat {X}}\subset \mathbb {A} ^{n}}
{\displaystyle X-{\hat {X}}}
correspond to the places at infinity. Then, the completion of
{\displaystyle F}
at one of these points gives an analogue of the
{\displaystyle p}
-adics. For example, if
{\displaystyle X=\mathbb {P} ^{1}}
then its function field is isomorphic to
{\displaystyle k(t)}
{\displaystyle t}
is an indeterminant and the field
{\displaystyle F}
is the field of fractions of polynomials in
{\displaystyle t}
. Then, a place
{\displaystyle v_{p}}
{\displaystyle p\in X}
measures the order of vanishing or the order of a pole of a fraction of polynomials
{\displaystyle p(x)/q(x)}
{\displaystyle p}
{\displaystyle p=[2:1]}
, so on the affine chart
{\displaystyle x_{1}\neq 0}
this corresponds to the point
{\displaystyle 2\in \mathbb {A} ^{1}}
{\displaystyle v_{2}}
measures the order of vanishing of
{\displaystyle p(x)}
minus the order of vanishing of
{\displaystyle q(x)}
{\displaystyle 2}
. The function field of the completion at the place
{\displaystyle v_{2}}
{\displaystyle k((t-2))}
which is the field of power series in the variable
{\displaystyle t-2}
, so an element is of the form
{\displaystyle {\begin{aligned}&a_{-k}(t-2)^{-k}+\cdots +a_{-1}(t-1)^{-1}+a_{0}+a_{1}(t-2)+a_{2}(t-2)^{2}+\cdots \\&=\sum _{n=-k}^{\infty }a_{n}(t-2)^{n}\end{aligned}}}
{\displaystyle k\in \mathbb {N} }
. For the place at infinity, this corresponds to the function field
{\displaystyle k((1/t))}
which are power series of the form
{\displaystyle \sum _{n=-k}^{\infty }a_{n}(1/t)^{n}}
In general, the group of units of O, denoted O×, is a finitely generated abelian group. The fundamental theorem of finitely generated abelian groups therefore implies that it is a direct sum of a torsion part and a free part. Reinterpreting this in the context of a number field, the torsion part consists of the roots of unity that lie in O. This group is cyclic. The free part is described by Dirichlet's unit theorem. This theorem says that rank of the free part is r1 + r2 − 1. Thus, for example, the only fields for which the rank of the free part is zero are Q and the imaginary quadratic fields. A more precise statement giving the structure of O× ⊗Z Q as a Galois module for the Galois group of K/Q is also possible. [14]
{\displaystyle {\begin{cases}L:K^{\times }\to \mathbf {R} ^{r_{1}+r_{2}}\\L(x)=(\log |x|_{v})_{v}\end{cases}}}
where v varies over the infinite places of K and |·|v is the absolute value associated with v. The function L is a homomorphism from K× to a real vector space. It can be shown that the image of O× is a lattice that spans the hyperplane defined by
{\displaystyle x_{1}+\cdots +x_{r_{1}+r_{2}}=0.}
The covolume of this lattice is the regulator of the number field. One of the simplifications made possible by working with the adele ring is that there is a single object, the idele class group, that describes both the quotient by this lattice and the ideal class group.
Completing a number field K at a place w gives a complete field. If the valuation is Archimedean, one obtains R or C, if it is non-Archimedean and lies over a prime p of the rationals, one obtains a finite extension
{\displaystyle K_{w}/\mathbf {Q} _{p}:}
a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example, the Kronecker–Weber theorem can be deduced easily from the analogous local statement. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry.
One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field K is finite. This is a consequence of Minkowski's theorem since there are only finitely many Integral ideals with norm less than a fixed positive integer [15] page 78. The order of the class group is called the class number, and is often denoted by the letter h.
{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}
Ireland, Kenneth; Rosen, Michael (2013), A classical introduction to modern number theory, vol. 84, Springer, doi: 10.1007/978-1-4757-2103-4, ISBN 978-1-4757-2103-4
Retrieved from " https://en.wikipedia.org/?title=Algebraic_number_theory&oldid=1087798498"
Algebraic Number Theory Videos
Algebraic Number Theory Websites
Algebraic Number Theory Encyclopedia Articles |
Then increase A<sub>r</sub> accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir (i.e., permeable pavement) area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging (assumes A<sub>r</sub> = A<sub>p</sub>).
{\displaystyle d_{r,max}={\frac {\left[\left(RVC_{T}\times R\right)+RVC_{T}-\left(f'\times D\right)\right]}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}}
Then increase Ar accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir (i.e., permeable pavement) area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging (assumes Ar = Ap). |
the line lx + my +n=0 is a normal to the parabola y^2 = 4ax if - Maths - Conic Sections - 8489747 | Meritnation.com
the line lx + my +n=0 is a normal to the parabola y^2 = 4ax if
the equation of the given parabola is
{y}^{2}=4ax
the equation of the normal to the given parabola is
y+tx=2at+a{t}^{3}
tx+y-\left(2at+a{t}^{3}\right)=0
the equation (1) and the given equation of the normal
lx+my+n=0 ...........\left(2\right)
represent the same line:
\frac{t}{l}=\frac{1}{m}=\frac{-\left(2at+a{t}^{3}\right)}{n}
\frac{t}{l}=\frac{1}{m}⇒t=\frac{l}{m} ........\left(3\right)\phantom{\rule{0ex}{0ex}}and \frac{1}{m}=\frac{-\left(2at+a{t}^{3}\right)}{n}\phantom{\rule{0ex}{0ex}}-\frac{n}{m}=2at+a{t}^{3}\phantom{\rule{0ex}{0ex}}-\frac{n}{m}=2a.\left(\frac{l}{m}\right)+a.{\left(\frac{l}{m}\right)}^{3} \left[ substituting the value of t from eq\left(3\right)\right]\phantom{\rule{0ex}{0ex}}-\frac{n}{m}=\frac{2a.l}{m}+\frac{a.{l}^{3}}{{m}^{3}}\phantom{\rule{0ex}{0ex}}⇒-n.{m}^{2}=2a.l{m}^{2}+a.{l}^{3}\phantom{\rule{0ex}{0ex}}⇒a.{l}^{3}+2al{m}^{2}+n{m}^{2}=0 |
Gabriel Alves, Karleigh Moore, Adonis Ampongan, and
Insertion sort is a sorting algorithm that builds a final sorted array (sometimes called a list) one element at a time. While sorting is a simple concept, it is a basic principle used in complex computer programs such as file search, data compression, and path finding. Running time is an important thing to consider when selecting a sorting algorithm since efficiency is often thought of in terms of speed. Insertion sort has an average and worst-case running time of
O(n^2)
, so in most cases, a faster algorithm is more desirable.
Complexity of Insertion Sort
Input: An array
A
n
orderable elements
A[0,1,...,n-1]
A
B
B[0] \leq B[1] \leq \cdots \leq B[n-1].
_\square
Here is what it means for an array to be sorted.
An array
<a_n>
i<j
a_i \leq a_j.
[a,b,c,d]
[1,2,3,4,5]
[5,4,3,2,1]
The insertion sort algorithm iterates through an input array and removes one element per iteration, finds the place the element belongs in the array, and then places it there. This process grows a sorted list from left to right. The algorithm is as follows:
A[i]
A[i]
\gt
A[i+1]
, swap the elements until
A[i]
\leq
A[i+1]
The animation below illustrates insertion sort:
Sort the following array using insertion sort.
A = [8,2,4,9,3,6]
Another way to visualize insertion sort is to think of a stack of playing cards.
You have the cards 3, 9, 6, 5, and 7. Write an algorithm or pseudo-code that arranges the values of the cards in ascending order.
Basically, the idea is to run insertion sort
n -1
times and find the index at which each element should be inserted.
Here is some pseudocode.[2]
A[j+1] = x
The sorted array is [3, 5, 6, 7, 9].
_\square
Here is one way to implement insertion sort in Python. There are other ways to implement the algorithm, but all implementations stem from the same ideas. Insertion sort can sort any orderable list.
for slot in range(1, len(array)):
value = array[slot]
test_slot = slot - 1
while test_slot > -1 and array[test_slot] > value:
array[test_slot + 1] = array[test_slot]
test_slot = test_slot - 1
array[test_slot + 1] = value
The above Python code sorts a list in increasing order. What single change could you make to have insertion sort sort in decreasing order?
Flip the second greater than sign to a less than sign in line 5. Line 5 should read:
while test slot > -1 and array[test slot] < value: #flip this sign
Insertion sort runs in
O(n)
time in its best case and runs in
O(n^2)
in its worst and average cases.
Insertion sort performs two operations: it scans through the list, comparing each pair of elements, and it swaps elements if they are out of order. Each operation contributes to the running time of the algorithm. If the input array is already in sorted order, insertion sort compares
O(n)
elements and performs no swaps (in the Python code above, the inner loop is never triggered). Therefore, in the best case, insertion sort runs in
O(n)
Worst and Average Case Analysis:
The worst case for insertion sort will occur when the input list is in decreasing order. To insert the last element, we need at most
n-1
comparisons and at most
n-1
swaps. To insert the second to last element, we need at most
n-2
n-2
swaps, and so on.[3] The number of operations needed to perform insertion sort is therefore:
2 \times (1+2+ \dots +n-2+n-1)
. To calculate the recurrence relation for this algorithm, use the following summation:
\sum_{q=1}^{p} q = \frac{p(p+1)}{2}.
\frac{2(n-1)(n-1+1)}{2}=n(n-1).
Use the master theorem to solve this recurrence for the running time. As expected, the algorithm's complexity is
O(n^2).
When analyzing algorithms, the average case often has the same complexity as the worst case. So insertion sort, on average, takes
O(n^2)
Insertion sort has a fast best-case running time and is a good sorting algorithm to use if the input list is already mostly sorted. For larger or more unordered lists, an algorithm with a faster worst and average-case running time, such as mergesort, would be a better choice.
Insertion sort is a stable sort with a space complexity of
O(1)
Mergesort and Bubble Sort Insertion Sort and Mergesort None of the Above Bubble Sort and Insertion Sort
For the following list, which two sorting algorithms have the same running time (ignoring constant factors)?
A = [4,2,0,9,8,1]
Devadas , S. 6.006 Introduction to Algorithms: Lecture 3, Fall 2011. Retrieved March, 24, 2016, from http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-006-introduction-to-algorithms-fall-2011/lecture-videos/MIT6_006F11_lec03.pdf
Cormen, T., Leiserson, C., Rivest, R., & Stein, C. (2001). Introduction to Algorithms (2nd edition) (pp. 15-21). The MIT Press.
Sinapova , L. Sorting Algorithms: Insertion Sort. Retrieved March, 24, 2016, from http://faculty.simpson.edu/lydia.sinapova/www/cmsc250/LN250_Weiss/L11-InsSort.htm
Cite as: Insertion Sort. Brilliant.org. Retrieved from https://brilliant.org/wiki/insertion/ |
Estimate optimal portfolios with targeted portfolio returns - MATLAB estimateFrontierByReturn - MathWorks í•œêµ
estimateFrontierByReturn
Obtain the Portfolio for Targeted Portfolio Returns for a Portfolio Object
Obtain Portfolios with Targeted Portfolio Returns for a Portfolio Object with BoundType, MinNumAsset, and MaxNumAsset Constraints
Obtain the Portfolio for Targeted Portfolio Returns for a PortfolioCVaR Object
Obtain the Portfolio for Targeted Portfolio Returns for a PortfolioMAD Object
Estimate optimal portfolios with targeted portfolio returns
[pwgt,pbuy,psell] = estimateFrontierByReturn(obj,TargetReturn)
[pwgt,pbuy,psell] = estimateFrontierByReturn(obj,TargetReturn) estimates optimal portfolios with targeted portfolio returns for Portfolio, PortfolioCVaR, or PortfolioMAD objects. For details on the respective workflows when using these different objects, see Portfolio Object Workflow, PortfolioCVaR Object Workflow, and PortfolioMAD Object Workflow.
To obtain efficient portfolios that have targeted portfolio returns, the estimateFrontierByReturn function accepts one or more target portfolio returns and obtains efficient portfolios with the specified returns. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio returns of 6%, 9%, and 12%.
pwgt = 4×3
[pwgt, pbuy, psell] = estimateFrontierByReturn(p,[ 0.0072321, 0.0119084 ])
pbuy = 3×2
psell = 3×2
The estimateFrontierByReturn function uses the MINLP solver to solve this problem. Use the setSolverMINLP function to configure the SolverType and options.
To obtain efficient portfolios that have targeted portfolio returns, the estimateFrontierByReturn function accepts one or more target portfolio returns and obtains efficient portfolios with the specified returns. Assume you have a universe of four assets where you want to obtain efficient portfolios with target portfolio returns of 7%, 10%, and 13%.
pwgt = estimateFrontierByReturn(p, [0.07 0.10, 0.13]);
seed
) is used to reset the random number generator to produce the documented results. It is not necessary to reset the random number generator to simulate scenarios.
seed
TargetReturn — Target values for portfolio return
Target values for portfolio return, specified as a NumPorts vector.
TargetReturn specifies target returns for portfolios on the efficient frontier. If any TargetReturn values are outside the range of returns for efficient portfolios, the TargetReturn is replaced with the minimum or maximum efficient portfolio return, depending upon whether the target return is below or above the range of efficient portfolio returns.
pwgt — Optimal portfolios on efficient frontier with specified target returns
Optimal portfolios on the efficient frontier with specified target returns from TargetReturn, returned as a NumAssets-by-NumPorts matrix. pwgt is returned for a Portfolio, PortfolioCVaR, or PortfolioMAD input object (obj).
You can also use dot notation to estimate optimal portfolios with targeted portfolio returns.
[pwgt, pbuy, psell] = obj.estimateFrontierByReturn(TargetReturn);
estimateFrontier | estimateFrontierByRisk | estimateFrontierLimits | setBounds | setMinMaxNumAssets |
Ideas in Geometry/Area - Wikiversity
Ideas in Geometry/Area
< Ideas in Geometry
1 3.2.1 Areas
1.1 Heron's Formula
1.2 A Quadrilateral Circumscribed in a Circle
1.3 Brahmagupta's Formula
1.4 Lattice Points
1.5 Pick's Theorem
3.2.1 Areas[edit | edit source]
There are simple equations to find the area of common shapes such as the triangle or parallelogram.
Heron's Formula[edit | edit source]
We learn at a young age that the area of a triangle can be expressed by the equation
{\displaystyle {\tfrac {1}{2}}}
{\displaystyle \cdot }
height. This uses two sides of a triangle on either side of an angle but sometimes it can be difficult to find the height of a triangle that is perpendicular to the base, creating a
{\displaystyle 90^{\circ }}
angle, which is what you need for the usual formula for the area of a triangle. But there is also a way to find the area of a triangle using the lengths of all three sides. This can be expressed in Heron's Formula where the area can be found using the equation
{\displaystyle area={\sqrt {\left({\frac {p}{2}}-a\right)\left({\frac {p}{2}}-b\right)\left({\frac {p}{2}}-c\right)\left({\frac {p}{2}}\right)}}}
where a,b &c represent the sides of the triangle and p=a+b+c, the perimeter of the triangle.
Here is an example to show what we mean:
To find the area of this triangle, we would use the equation with the sides lengths: 3,6 & 7 and find the area --> p=16
{\displaystyle area={\sqrt {\left({\frac {16}{2}}-3\right)\left({\frac {16}{2}}-6\right)\left({\frac {16}{2}}-7\right)\left({\frac {16}{2}}\right)}}}
and we would get the answer
{\displaystyle area=8.94units^{2}}
A Quadrilateral Circumscribed in a Circle[edit | edit source]
If we know that there is a formula that works for triangles given the three lengths of the sides, we can find a formula in a similar way for the area of a quadrilateral. As long as the quadrilateral can be circumscribed in a circle, which means each vertex touches the inside of the circle and the opposite angles must sum to
{\displaystyle 180^{\circ }}
, the area of the quadrilateral can be solved. See picture below of a quadrilateral circumscribed in a circle.
Brahmagupta's Formula[edit | edit source]
In this formula, Brahmagupta's Formula, if given a quadrilateral that can be circumscribed in a circle (also known as cyclic), the area of the quadrilateral can be expressed by the equation:
{\displaystyle area={\sqrt {\left({\frac {p}{2}}-a\right)\left({\frac {p}{2}}-b\right)\left({\frac {p}{2}}-c\right)\left({\frac {p}{2}}-d\right)}}}
, where a,b,c & d represent the sides of the quadrilateral and p=a+b+c+d, the perimeter of the quadrilateral.
Using these side lengths: 4, 6, 6, & 7, we would use the formula to find the area --> p=23
{\displaystyle area={\sqrt {\left({\frac {23}{2}}-4\right)\left({\frac {23}{2}}-6\right)\left({\frac {23}{2}}-6\right)\left({\frac {23}{2}}-7\right)}}}
and get the answer to be
{\displaystyle area=31.95units^{2}}
Lattice Points[edit | edit source]
There is also a way of finding the area of shapes besides triangles and quadrilaterals that involves lattice points. Lattice points are points that are spaced 1 unit apart, horizontally and vertically in a plane. Here are some lattice points:
Pick's Theorem[edit | edit source]
If the vertices of a polygon are located on the lattice points, we can find the area of the polygon by using Pick's Theorem:
{\displaystyle area={\tfrac {b}{2}}+n-1}
where b=the number of lattice points on the border of the polygon and n=the number of lattice points on the inside of the polygon. Here is an example:
Since the vertices of the polygon lie on the lattice points, we can use Pick's Theorem. So in this example: b=15, n=14. And
{\displaystyle area={\tfrac {15}{2}}+14-1}
. So the area=20.5 square units.
Retrieved from "https://en.wikiversity.org/w/index.php?title=Ideas_in_Geometry/Area&oldid=1215444" |
Then increase A<sub>r</sub> accordingly to keep R, the ratio of impervious contributing drainage area to water storage reservoir (i.e., permeable pavement) area, between 0 and 2 to reduce hydraulic loading and avoid premature clogging, assuming A<sub>r</sub> = A<sub>p</sub>.
{\displaystyle d_{r,max}={\frac {\left[\left(RVC_{T}\times R\right)+RVC_{T}-\left(f'\times D\right)\right]}{n}}}
{\displaystyle RVC_{T}=D\times i}
{\displaystyle d_{r}={\frac {f'\times t}{n}}}
{\displaystyle A_{r}={\frac {D(i-f')\times A_{c}}{d_{r}\times n}}} |
Chain Rule | Brilliant Math & Science Wiki
Aditya Virani, Jake Lai, Hjalmar Orellana Soto, and
The chain rule is used to differentiate composite functions. Specifically, it allows us to use differentiation rules on more complicated functions by differentiating the inner function and outer function separately.
Iterated Chain Rule
Intuitively, oftentimes a function will have another function "inside" it that is first related to the input variable. Since we know the derivative of a function is the rate of change, we need to compute the rate of change of this "inner" function as well as the total function in its entirety relative to the input variable. The chain rule allows us to accomplish this.
The chain rule states that given a composite function
f\big(g(x)\big)
, its derivative is the derivative of the outer function (leaving the inner function unchanged and in place) multiplied by the derivative of the inner function. Concisely,
\frac{d}{dx} f\big(g(x)\big) \equiv f'\big(g(x)\big) \cdot g'(x)
The alternative statement of the chain in terms of function composition is
(f \circ g)' = (f' \circ g) \cdot g'.
Another way of stating this which makes it intuitive (almost trivial) would be
\frac{d f \circ g}{dx}=\frac{d f \circ g}{dg} \frac{d g}{dx}.
Let's see how it works by taking a look at some examples.
\frac{d}{dx} ( 2x + 1) ^ 2
g(x) = 2x + 1
f(x) = x^2
(2x+1)^2 = f\big(g(x)\big)
g'(x) = 2
f'(x) = 2x
\begin{aligned} \frac{ d}{dx} (2x+1)^2 &= f'\big(g(x)\big) \cdot g'(x) \\ &= f' (2x+1) \cdot 2 \\ &= 2(2x+1) \cdot 2 \\ &= 8x+4.\ _\square \end{aligned}
\frac{d}{dx} e^{ x^2}
g(x) = x^2
f(x) = e^x
e^{x^2} = f \big(g (x)\big)
g'(x) = 2x
f'(x) = e^x
\begin{aligned} \frac{ d}{dx} e^{x^2} &= f'\big(g(x)\big) \cdot g'(x) \\ &= f' \big(x^2\big) \cdot 2x \\ &= e^{x^2} \cdot 2x \\ &= 2xe^{x^2}.\ _\square \end{aligned}
f(x) = \left(x+1\right)^{\frac{3}{2}}
f'(15)?
\cos^5x
g(x)=\cos x
f(x)=x^5
\cos^5x=f\big(g(x)\big)
g'(x)=-\sin x
f'(x)=5x^4
\begin{aligned} \frac{d}{dx} f\big(g(x)\big) &= f'\big(g(x)\big) \cdot g'(x) \\ &= f'(\cos x) \cdot (-\sin x) \\ &= -5\cos^4x\sin x.\ _\square \end{aligned}
-\frac{1}{2}
\frac{1}{2}
\frac{\pi}{6}
\frac{\sqrt{3}}{2}
\sqrt{3}
2
There is a certain function
f
such that the tangent line at
x=\frac{1}{2}
y=2x+1.
g(x)=f(\sin x),
\frac{dg}{dx}
when evaluated at
x=\frac{\pi}{6}
Then, how do we prove the chain rule? The proof goes as follows:
\frac{d}{dx}f(x) = \lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}.
\begin{aligned} \frac{d}{dx} f\big(g(x)\big) &= \lim_{\Delta x \to 0} \dfrac{f\big(g(x+\Delta x)\big)-f\big(g(x)\big)}{\Delta x} \\ &= \lim_{\Delta x \to 0} \dfrac{f\big(g(x+\Delta x)\big)-f\big(g(x)\big)}{\Delta x} \cdot \dfrac{g(x+\Delta x)-g(x)}{g(x+\Delta x)-g(x)} \\ &= \lim_{\Delta x \to 0} \dfrac{f\big(g(x+\Delta x)\big)-f\big(g(x)\big)}{g(x+\Delta x)-g(x)} \cdot \lim_{\Delta x \to 0} \dfrac{g(x+\Delta x)-g(x)}{\Delta x} \\ &= \lim_{\Delta x \to 0} \dfrac{f\big(g(x+\Delta x)\big)-f\big(g(x)\big)}{g(x+\Delta x)-g(x)} \cdot g'(x). \end{aligned}
g(x+\Delta x)-g(x) = \Delta g
\Delta g \to 0
\Delta x \to 0
\begin{aligned} \frac{d}{dx} f\big(g(x)\big) &= \lim_{\Delta g \to 0} \dfrac{f\big(g(x)+\Delta g\big)-f\big(g(x)\big)}{\Delta g} \cdot g'(x) \\ &= f'\big(g(x)\big) \cdot g'(x).\ _\square \end{aligned}
At times, we may need to apply the chain rule repeatedly in order to find the derivative. For example, if
f(x) = \sqrt{\tan (x^3)}
, the inner function is itself a composite function.
In situations like this, we must apply the chain rule more than once. Specifically, for three functions composed, we have
\frac{d}{dx} f\Big(g\big(h(x)\big)\Big) = f'\big(g(h(x)\big) \cdot g'\big(h(x)\big) \cdot h'(x).
This can easily be extended to any number of composed functions.
\sqrt{\tan(x^3)}
We begin by identifying the given function's basic parts, which we know how to differentiate. We can call
\sqrt{\tan(x^3)} = f\Big(g\big(h(x)\big)\Big)
f(x) = \sqrt{x}
g(x) = \tan x,
h(x) = x^3
f'(x) = \frac{1}{2\sqrt{x}}
g'(x) = \sec^2 x,
h'(x) = 3x^2
Applying the chain rule, we can see that
\frac{d}{dx} \sqrt{\tan(x^3)} = f'\Big(g\big(h(x)\big)\Big) \cdot \frac{d}{dx} g\big(h(x)\big)
. However, in order to find the derivative of
g\big(h(x)\big)
, we have to apply the chain rule again, which gives us the following:
\begin{aligned} \frac{d}{dx} \sqrt{\tan (x^3) } &= f'\Big(g\big(h(x)\big)\Big) \cdot \frac{d}{dx} g\big(h(x)\big) \\ &= f'\Big(g\big(h(x)\big)\Big) \cdot \Big[g'\big(h(x)\big) \cdot h'(x)\Big] \\ &= \frac{1}{2\sqrt{\tan \left(x^3\right)}} \cdot \sec^2\left(x^3\right) \cdot 3x^2 \\\\ &= \frac{3x^2 \sec^2\left(x^3\right)}{2\sqrt{\tan\left(x^3\right)}}.\ _\square \end{aligned}
For a harder challenge, try:
\large x^{x^x}
x=1?
Cite as: Chain Rule. Brilliant.org. Retrieved from https://brilliant.org/wiki/chain-rule/ |
Abstract Algebra/Quaternions - Wikibooks, open books for an open world
Abstract Algebra/Quaternions
2 Versors and elliptic space
3 Linear viewpoint
3.1 Pauli Spin Matrices
The algebra of Quaternions is an structure first studied by the Irish mathematician William Rowan Hamilton which extends the two-dimensional complex numbers to four dimensions. Multiplication is non-commutative in quaternions, a feature which enables its representation of three-dimensional rotation. Hamilton's provocative discovery of quaternions founded the field of hypercomplex numbers. Suggestive methods like dot products and cross products implicit in quaternion products enabled algebraic description of geometry now widely applied in science and engineering.
Quaternion plaque on Broom Bridge, Dublin, which says:
A Quaternion corresponds to an ordered 4-tuple
{\displaystyle q=(a,b,c,d)}
{\displaystyle a,b,c,d\in \mathbb {R} }
. A quaternion is denoted
{\displaystyle q=a+bi+cj+dk}
{\displaystyle bi+cj+dk}
is called the vector part of q, and a is the real part. Hamiltion coined the term vector in this context. Subsequent developments have extended the usage of the term vector to any element of a linear space. The vectors in H form a 3-dimensional subspace V.
The set of all quaternions is denoted by
{\displaystyle \mathbb {H} }
. It is straightforward to define component-wise addition and scalar multiplication on
{\displaystyle \mathbb {H} }
, making it a real vector space.
Multiplication follows the rules of the "quaternion group" Q8 = {1, -1, i, -i, j, -j, k, -k} that Hamilton carved into a stone of Broom Bridge, Dublin:
{\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1}
From the above equations alone, it is possible to derive rules for the pairwise multiplication of
{\displaystyle i}
{\displaystyle j}
{\displaystyle k}
{\displaystyle ij=k,\ \ jk=i,\ \ ki=j}
(positive cyclic permutations)
{\displaystyle ji=-k,\ \ kj=-i,\ \ ik=-j}
(negative cyclic permutations).
Using these, it is easy to define a general rule for multiplication of quaternions. Because quaternion multiplication is not commutative,
{\displaystyle \mathbb {H} }
is not a field. However, every nonzero quaternion has a multiplicative inverse (see below), so the quaternions are an example of a non-commutative division ring. It is important to note that the non-commutative nature of quaternion multiplication makes it impossible to define the quotient
{\displaystyle p/q}
of two quaternions p and q unambiguously, as the quantities
{\displaystyle pq^{-1}}
{\displaystyle q^{-1}p}
are generally different.
Like the more familiar complex numbers, the quaternions have a conjugation, often denoted by a superscript star:
{\displaystyle q^{*}}
. The conjugate of the quaternion
{\displaystyle q=a+bi+cj+dk}
{\displaystyle q^{*}=a-bi-cj-dk}
. As is the case for the complex numbers, the product
{\displaystyle qq^{*}}
is always a positive real number equal to the sum of the squares of the quaternion's components. The norm of a quaternion is the square root of
{\displaystyle qq^{*}}
If pq is the product of two quaternions, then
{\displaystyle (pq)(pq)^{*}=(pp^{*})(qq^{*}),}
{\displaystyle \mathbb {H} }
forms a composition algebra.
The multiplicative inverse of a non-zero quaternion
{\displaystyle q}
{\displaystyle q^{-1}={\frac {q^{*}}{qq^{*}}}}
where division is defined since
{\displaystyle qq^{*}\neq 0.}
Unlike in the complex case, the conjugate
{\displaystyle q^{*}}
of a quaternion
{\displaystyle q}
can be written as a polynomial in q:
{\displaystyle q^{*}=-{\frac {1}{2}}(q+iqi+jqj+kqk)}
Versors and elliptic spaceEdit
William Kingdon Clifford used Hamilton’s quaternions to explicate rotation geometry as an elliptic space with its own variety of lines, parallels, and surfaces. The ideas were reviewed in 1948 by Lemaitre and Coxeter and that sketch has these definitions:
A versor is a quaternion of norm one, thus it lies on a 3-dimensional sphere found in the 4-space of quaternions. The versors are given by Euler's formula for complex numbers where the imaginary unit is taken from the unit sphere in the 3-space of vector quaternions:
{\displaystyle v=\cos c+s\sin c=e^{cs},\ \ s^{2}=-1.}
The distance between two versors u and v is
{\displaystyle d(u,v)=\arccos(uv^{*}+vu^{*})/2.}
A right parataxy on elliptic space is effected by multiplying on the right by a versor
{\displaystyle v=e^{cs}.}
Similarly a left parataxy arises from left multiplication. In recognition of his contribution to elliptic geometry, a parataxy is called a Clifford translation.
The general displacement of elliptic space is a combination of two parataxies, one left, one right:
{\displaystyle x\mapsto uxv.}
{\displaystyle u=v^{*},}
then the real line in the quaternions is fixed and the displacement is a rotation of the 3-space of quaternion vectors.
The term line is appropriated for elliptic geometry. These lines are not straight, but they are parametrized by real numbers. Each line is associated with a right versor like s when c = π/2 in v. Then
{\displaystyle L=\{e^{cs}:c\in R\}}
is a typical elliptic line. It corresponds to the axis of the rotation
{\displaystyle x\mapsto e^{cs}xe^{-cx}.}
Now for u not on L, there are two Clifford parallels to L through u:
{\displaystyle \{ue^{cs}:c\in R\},\quad \{e^{cs}u:c\in R\}.}
For fixed right versors r and s, a Clifford surface can be formed as a union of Clifford parallels or as
{\displaystyle \{e^{cs}e^{dr}:b,c\in R\}.}
To form elliptic space from versors, two versors u and v are equivalent if u + v = 0. Modulo this equivalence, the versors, their algebra and geometry, represent elliptic space.
Linear viewpointEdit
Quaternions may be represented by 2×2 matrices with complex number entries: the place of
{\displaystyle i,j,k}
is taken by these arrays:
{\displaystyle {\begin{pmatrix}i&0\\0&-i\end{pmatrix}},\quad {\begin{pmatrix}0&1\\-1&0\end{pmatrix}},\quad {\begin{pmatrix}0&i\\i&0\end{pmatrix}}.}
One uses matrix multiplication to verify that these expressions obey the rules of presentation of Q8.
M(2,C) denotes the full algebra of 2×2 complex matrices, which has eight real dimensions, and sustains a representation of
{\displaystyle \mathbb {H} }
as a four-dimensional subalgebra. The linear properties of
{\displaystyle \mathbb {H} }
and M(2,C) assure the fidelity of the representation once the copy of Q8 has been identified.
Quaternions, like other associative hypercomplex systems of the 19th century, eventually were viewed as matrix algebras in the 20th century. However, in 1853 Hamilton included biquaternions in his book of Lectures on Quaternions.
Biquaternions are quaternions with complex number coefficients, sometimes called complex quaternions. Biquaternions form an algebra isomorphic to M(2,C). If the rows or columns of a matrix are proportional, then the determinant is zero, and there is no inverse. Nevertheless, such matrices have been used in physical science to represent events on a light-path from the origin. Authors Silberstein and Lanczos refer to this algebra as the biquaternions, but other writers have abandoned the label: Elie Cartan used M(2,C) extensively in The Theory of Spinors (1938), and Wolfgang Pauli, in his matrix mechanics of the atom, caused himself to be associated with M(2,C).
Pauli Spin MatricesEdit
Quaternions are closely related to the Pauli spin matrices of Quantum Mechanics. The Pauli matrices are often denoted as
{\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}
{\displaystyle \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}}
{\displaystyle \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
{\displaystyle i}s the well known quantity
{\displaystyle {\sqrt {-1}}}
of complex numbers)
The 2×2 identity matrix is sometimes taken as
{\displaystyle \sigma _{0}}
{\displaystyle S}
, the real linear span of the matrices
{\displaystyle \sigma _{0}}
{\displaystyle i\sigma _{1}}
{\displaystyle i\sigma _{2}}
{\displaystyle i\sigma _{3}}
{\displaystyle \mathbb {H} }
. For example, take this matrix product:
{\displaystyle {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}={\begin{pmatrix}0&i\\i&0\end{pmatrix}}}
{\displaystyle i\sigma _{3}\ i\sigma _{2}=i\sigma _{1}.}
All three of these matrices square to the negative of the identity matrix. If we take
{\displaystyle 1=\sigma _{0}}
{\displaystyle i=i\sigma _{3}}
{\displaystyle j=i\sigma _{2}}
{\displaystyle k=i\sigma _{1}}
, it is easy to see that the span of the these four matrices is "the same as" (that is, isomorphic to) the set of quaternions
{\displaystyle \mathbb {H} }
Using the presentation equations of Q8, write out the full product of two quaternions. In other words, given
{\displaystyle q_{1}=a_{1}+b_{1}i+c_{1}j+d_{1}k}
{\displaystyle q_{2}=a_{2}+b_{2}i+c_{2}j+d_{2}k}
, find the components of their product
{\displaystyle q=q_{1}q_{2}.}
Show the composition algebra property
{\displaystyle (pq)(pq)^{*}=(pp^{*})(qq^{*}).}
Hint: use w: Euler's four-square identity.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Abstract_Algebra/Quaternions&oldid=3361730" |
PlanePlot - Maple Help
Home : Support : Online Help : Education : Student Packages : Linear Algebra : Visualization : PlanePlot
plot a plane and associated vectors
PlanePlot(P1, pt, opts)
PlanePlot(P2, vars, opts)
Vector, set or list of Vectors, or Vector-valued function; specify the plane
(optional) Vector; point on the plane
equation or algebraic expression; equation of the plane
list; names of the variables
plotting options or equation(s) of the form option=value where option is one of showbasis, basisoptions, shownearestpoint, nearestpointoptions, shownormal, normaloptions, showplane, planeoptions, showpoint, pointoptions, or Student plot options; specify options for the plot
The PlanePlot(P1) command plots the plane defined by P1, together with a normal Vector to the plane. The plane can be specified in one of three ways:
* A 3-D Vector: P1 is the normal to the plane.
* A set or list of 3-D Vectors: P1 represents a basis for the plane. The set or list of Vectors must define a 2-D subspace of 3-D space.
* A Vector-valued procedure: P1 must be a procedure that takes two numeric arguments and returns a 3-D Vector. It is evaluated at the arguments
0,0
1,0
0,1
to obtain points that define the plane.
In each of the first two cases (where the plane is defined by either a normal Vector or a spanning set of Vectors), the optional argument pt, given as a 3-D Vector, can be used to specify a point on the plane. If pt is not provided, the plane passes through the origin.
The PlanePlot(P2, vars) command plots the plane defined by P2, which must be an algebraic expression or algebraic equation that is linear in the 3 variables named in the vars parameter. The vars parameter can be omitted if all 3 variables explicitly appear in P2; note that in this case the axis order is not determined.
Specifies whether vectors which form a spanning set for the plane are plotted. The vectors are drawn with their bases at the same point as the base of the normal vector. [Default: false unless the set-or-list-of-Vectors form of P1 is used]
Provides options (such as color, shape) used to plot the basis vectors for the plane. Because the vectors are plotted using the plots[arrow] command, only corresponding options are allowed.
shownearestpoint = true or false
Specifies whether the vector from the origin to the point on the plane, which is nearest the origin, is plotted. [Default: false]
nearestpointoptions = list
Provides options used to plot the nearest-point vector. Because the vector is plotted using the plots[arrow] command, only corresponding options are allowed.
Specifies whether the normal vector to the plane is drawn. If selected, this vector is drawn with its tail at the point described in the showpoint option, unless showpoint = false has been specified, in which case it is drawn with its base at the nearest point on the plane to the origin. [Default: true]
Provides options used to plot the normal vector to the plane. Because the vector is plotted using the plots[arrow] command, only corresponding options are allowed.
Specifies whether to display the plane. [Default: true]
Provides options (such as color, style, grid) used to plot the plane. Any option valid for the plot3d command can be provided.
Specifies whether the vector from the origin to the specified point on the plane is plotted. If pt is provided, this point is the head of the vector. If the procedure form of P1 is given, this point is the value of the procedure at
0,0
. Otherwise, this point is the nearest point on the plane to the origin. [Default: true]
Provides options used to plot the vector from the origin to a distinguished point on the plane. For information on how this point is determined, see the description for the showpoint option. Because the vector is plotted using the plots[arrow] command, only corresponding options are allowed.
\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):
\mathrm{infolevel}[\mathrm{Student}[\mathrm{LinearAlgebra}]]≔1:
\mathrm{PlanePlot}\left(-3x+2y+z=-3,[x,y,z],\mathrm{normaloptions}=[\mathrm{shape}=\mathrm{harpoon}]\right)
normal vector: <-3., 2., 1.>
equation of plane: -3.*x+2.*y+1.*z = -3.
basis vectors: <.5345, .8414, -.7929e-1>, <.2673, -.7929e-1, .9604>
\mathrm{PlanePlot}\left(〈1,2,3〉,〈1,-3,-1〉,\mathrm{orientation}=[10,58],\mathrm{shownearestpoint},\mathrm{showbasis}\right)
equation of plane: 1.*x+2.*y+3.*z = -8.
point on plane nearest origin: <-.5714, -1.143, -1.714>
basis vectors: <-1.773, 2.569, -1.122>, <-2.659, -1.122, 1.634>
\mathrm{PlanePlot}\left({〈1,3,5.〉,〈2,7,-1〉},〈-3,2,-1〉\right)
normal vector: <-5.171, 1.497, .1361>
equation of plane: -5.171*x+1.497*y+.1361*z = 18.37
point on plane nearest origin: <-3.276, .9483, .8621e-1>
basis vectors: <1., 3., 5.>, <2., 7., -1.>
\mathrm{PlanePlot}\left(\left(s,t\right)↦s\cdot 〈1,3,-5〉+t\cdot 〈0.01,-0.02,0.01〉+〈13,2,1〉,\mathrm{showbasis},\mathrm{shownearestpoint}\right)
normal vector: <6.372, 5.462, 4.551>
equation of plane: 6.372*x+5.462*y+4.551*z = 98.31
point on plane nearest origin: <6.873, 5.891, 4.909>
basis vectors: <5.385, -10.77, 5.385>, <2.230, 6.689, -11.15>
\mathrm{PlanePlot}\left(〈1,3,2〉,\mathrm{showbasis}\right)
equation of plane: 1.*x+3.*y+2.*z = 0. |
VECTOR CALCULUS - Encyclopedia Information
Vector calculus Information
https://en.wikipedia.org/wiki/Vector_calculus
Not to be confused with Geometric calculus or Matrix calculus.
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (February 2016) ( Learn how and when to remove this template message)
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space
{\displaystyle \mathbb {R} ^{3}.}
The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow.
Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra which uses exterior products does (see § Generalizations below for more).
1.1 Scalar fields
1.3 Vectors and pseudovectors
3 Operators and theorems
3.1 Differential operators
4.3 Physics and engineering
5.1 Different 3-manifolds
Main article: Scalar field
A scalar field associates a scalar value to every point in a space. The scalar is a mathematical number representing a physical quantity. Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields (known as scalar bosons), such as the Higgs field. These fields are the subject of scalar field theory.
Main article: Vector field
A vector field is an assignment of a vector to each point in a space. [1] A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. This can be used, for example, to calculate work done over a line.
In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields, except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in geometric algebra, as described below.
Main article: Euclidean vector § Basic properties
The algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field. The basic algebraic operations consist of:
{\displaystyle \mathbf {v} _{1}+\mathbf {v} _{2}}
{\displaystyle a\mathbf {v} }
{\displaystyle \mathbf {v} _{1}\cdot \mathbf {v} _{2}}
{\displaystyle \mathbf {v} _{1}\times \mathbf {v} _{2}}
{\displaystyle \mathbb {R} ^{3}}
Also commonly used are the two triple products:
{\displaystyle \mathbf {v} _{1}\cdot \left(\mathbf {v} _{2}\times \mathbf {v} _{3}\right)}
{\displaystyle \mathbf {v} _{1}\times \left(\mathbf {v} _{2}\times \mathbf {v} _{3}\right)}
Main articles: Gradient, Divergence, Curl (mathematics), and Laplacian
Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator (
{\displaystyle \nabla }
), also known as "nabla". The three basic vector operators are: [2]
{\displaystyle \operatorname {grad} (f)=\nabla f}
Measures the rate and direction of change in a scalar field. Scalar multiplication Maps scalar fields to vector fields.
{\displaystyle \operatorname {div} (\mathbf {F} )=\nabla \cdot \mathbf {F} }
Measures the scalar of a source or sink at a given point in a vector field. Dot product Maps vector fields to scalar fields.
{\displaystyle \operatorname {curl} (\mathbf {F} )=\nabla \times \mathbf {F} }
{\displaystyle \mathbb {R} ^{3}}
. Cross product Maps vector fields to (pseudo)vector fields.
{\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f}
{\displaystyle \nabla ^{2}\mathbf {F} =\nabla (\nabla \cdot \mathbf {F} )-\nabla \times (\nabla \times \mathbf {F} )}
A quantity called the Jacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration.
The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions:
{\displaystyle \int _{L\subset \mathbb {R} ^{n}}\!\!\!\nabla \varphi \cdot d\mathbf {r} \ =\ \varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)\ \ {\text{ for }}\ \ L=L[p\to q]}
The line integral of the gradient of a scalar field over a curve L is equal to the change in the scalar field between the endpoints p and q of the curve.
{\displaystyle \underbrace {\int \!\cdots \!\int _{V\subset \mathbb {R} ^{n}}} _{n}(\nabla \cdot \mathbf {F} )\,dV\ =\ \underbrace {\oint \!\cdots \!\oint _{\partial V}} _{n-1}\mathbf {F} \cdot d\mathbf {S} }
Curl (Kelvin–Stokes) theorem
{\displaystyle \iint _{\Sigma \,\subset \mathbb {R} ^{3}}(\nabla \times \mathbf {F} )\cdot d\mathbf {\Sigma } \ =\ \oint _{\!\!\partial \Sigma }\mathbf {F} \cdot d\mathbf {r} }
{\displaystyle \mathbb {R} ^{3}}
{\displaystyle \varphi }
{\displaystyle \iint _{A\,\subset \mathbb {R} ^{2}}\left({\frac {\partial M}{\partial x}}-{\frac {\partial L}{\partial y}}\right)dA\ =\ \oint _{\partial A}\left(L\,dx+M\,dy\right)}
{\displaystyle \mathbb {R} ^{2}}
Main article: Linear approximation
{\displaystyle f(x,y)\ \approx \ f(a,b)+{\tfrac {\partial f}{\partial x}}(a,b)\,(x-a)+{\tfrac {\partial f}{\partial y}}(a,b)\,(y-b).}
For a continuously differentiable function of several real variables, a point P (that is, a set of values for the input variables, which is viewed as a point in Rn) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if its gradient is zero. The critical values are the values of the function at the critical points.
By Fermat's theorem, all local maxima and minima of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (August 2019) ( Learn how and when to remove this template message)
Vector calculus is initially defined for Euclidean 3-space,
{\displaystyle \mathbb {R} ^{3},}
which has additional structure beyond simply being a 3-dimensional real vector space, namely: a norm (giving a notion of length) defined via an inner product (the dot product), which in turn gives a notion of angle, and an orientation, which gives a notion of left-handed and right-handed. These structures give rise to a volume form, and also the cross product, which is used pervasively in vector calculus.
The gradient and divergence require only the inner product, while the curl and the cross product also requires the handedness of the coordinate system to be taken into account (see cross product and handedness for more detail).
Vector calculus can be defined on other 3-dimensional real vector spaces if they have an inner product (or more generally a symmetric nondegenerate form) and an orientation; note that this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vector calculus is invariant under rotations (the special orthogonal group SO(3)).
More generally, vector calculus can be defined on any 3-dimensional oriented Riemannian manifold, or more generally pseudo-Riemannian manifold. This structure simply means that the tangent space at each point has an inner product (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is a symmetric nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.
Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset. Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian (yielding harmonic analysis), while curl and cross product do not generalize as directly.
In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector field is a scalar function, but only in dimension 3 or 7 [3] (and, trivially, in dimension 0 or 1) is the curl of a vector field a vector field, and only in 3 or 7 dimensions can a cross product be defined (generalizations in other dimensionalities either require
{\displaystyle n-1}
vectors to yield 1 vector, or are alternative Lie algebras, which are more general antisymmetric bilinear products). The generalization of grad and div, and how curl may be generalized is elaborated at Curl: Generalizations; in brief, the curl of a vector field is a bivector field, which may be interpreted as the special orthogonal Lie algebra of infinitesimal rotations; however, this cannot be identified with a vector field because the dimensions differ – there are 3 dimensions of rotations in 3 dimensions, but 6 dimensions of rotations in 4 dimensions (and more generally
{\displaystyle \textstyle {{\binom {n}{2}}={\frac {1}{2}}n(n-1)}}
There are two important alternative generalizations of vector calculus. The first, geometric algebra, uses k-vector fields instead of vector fields (in 3 or fewer dimensions, every k-vector field can be identified with a scalar function or vector field, but this is not true in higher dimensions). This replaces the cross product, which is specific to 3 dimensions, taking in two vector fields and giving as output a vector field, with the exterior product, which exists in all dimensions and takes in two vector fields, giving as output a bivector (2-vector) field. This product yields Clifford algebras as the algebraic structure on vector spaces (with an orientation and nondegenerate form). Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions.
The second generalization uses differential forms (k-covector fields) instead of vector fields or k-vector fields, and is widely used in mathematics, particularly in differential geometry, geometric topology, and harmonic analysis, in particular yielding Hodge theory on oriented pseudo-Riemannian manifolds. From this point of view, grad, curl, and div correspond to the exterior derivative of 0-forms, 1-forms, and 2-forms, respectively, and the key theorems of vector calculus are all special cases of the general form of Stokes' theorem.
Vector algebra relations
^ Galbis, Antonio & Maestre, Manuel (2012). Vector Analysis Versus Vector Calculus. Springer. p. 12. ISBN 978-1-4614-2199-3. {{ cite book}}: CS1 maint: uses authors parameter ( link)
^ "Differential Operators". Math24. Retrieved 2020-09-17.
^ Lizhong Peng & Lei Yang (1999) "The curl in seven dimensional space and its applications", Approximation Theory and Its Applications 15(3): 66 to 80 doi: 10.1007/BF02837124
Sandro Caparrini (2002) " The discovery of the vector representation of moments and angular velocity", Archive for History of Exact Sciences 56:151–81.
Crowe, Michael J. (1967). A History of Vector Analysis : The Evolution of the Idea of a Vectorial System (reprint ed.). Dover Publications. ISBN 978-0-486-67910-5.
Barry Spain (1965) Vector Analysis, 2nd edition, link from Internet Archive.
Chen-To Tai (1995). A historical study of vector analysis. Technical Report RL 915, Radiation Laboratory, University of Michigan.
"Vector analysis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
"Vector algebra", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen-To
Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, (based upon the lectures of Willard Gibbs) by Edwin Bidwell Wilson, published 1902.
Retrieved from " https://en.wikipedia.org/?title=Vector_calculus&oldid=1082211831"
Vector Calculus Videos
Vector Calculus Websites
Vector Calculus Encyclopedia Articles |
A Study of Well-Composedness in n-D - LRDE
Digitization of the real world using real sensors has many drawbacks; in particular, we loose “well-composedness” in the sense that two digitized objects can be connected or not depending on the connectivity we choose in the digital image, leading then to ambiguities. Furthermore, digitized images are arrays of numerical values, and then do not own any topology by nature, contrary to our usual modeling of the real world in mathematics and in physics. Loosing all these properties makes difficult the development of algorithms which are “topologically correct” in image processing: e.g., the computation of the tree of shapes needs the representation of a given image to be continuous and well-composed; in the contrary case, we can obtain abnormalities in the final result. Some well-composed continuous representations already exist, but they are not in the same time
{\displaystyle n}
-dimensional and self-dual. In fact
{\displaystyle n}
-dimensionality is crucial since usual signals are more and more 3-dimensional (like 2D videos) or 4-dimensional (like 4D Computerized Tomography-scans), and self-duality is necessary when a same image can contain different objects with different contrasts. We developed then a new way to make images well-composed by interpolation in a self-dual way and i{\displaystyle n}
-D; followed with a span-based immersion, this interpolation becomes a self-dual continuous well-composed representation of the initial
{\displaystyle n}
-D signal. This representation benefits from many strong topological properties: it verifies the intermediate value theorem, the boundaries of any threshold set of the representation are disjoint union of discrete surfaces, and so on.
@PhDThesis{ boutry.16.phd,
author = {Nicolas Boutry},
title = {A Study of Well-Composedness in $n$-D},
address = {Noisy-Le-Grand, France},
abstract = {Digitization of the real world using real sensors has many
drawbacks; in particular, we loose ``well-composedness'' in
the sense that two digitized objects can be connected or
not depending on the connectivity we choose in the digital
image, leading then to ambiguities. Furthermore, digitized
images are arrays of numerical values, and then do not own
any topology by nature, contrary to our usual modeling of
the real world in mathematics and in physics. Loosing all
these properties makes difficult the development of
algorithms which are ``topologically correct'' in image
processing: e.g., the computation of the tree of shapes
needs the representation of a given image to be continuous
and well-composed; in the contrary case, we can obtain
abnormalities in the final result. Some well-composed
continuous representations already exist, but they are not
in the same time $n$-dimensional and self-dual. In fact,
$n$-dimensionality is crucial since usual signals are more
and more 3-dimensional (like 2D videos) or 4-dimensional
(like 4D Computerized Tomography-scans), and self-duality
is necessary when a same image can contain different
objects with different contrasts. We developed then a new
way to make images well-composed by interpolation in a
self-dual way and in $n$-D; followed with a span-based
immersion, this interpolation becomes a self-dual
continuous well-composed representation of the initial
$n$-D signal. This representation benefits from many strong
topological properties: it verifies the intermediate value
theorem, the boundaries of any threshold set of the
representation are disjoint union of discrete surfaces, and
so on.}
Retrieved from "https://www.lrde.epita.fr/index.php?title=Publications/boutry.16.phd&oldid=120526" |
Cryogenic Fatigue Strength Assessment for MARK-III Insulation System of LNG Carriers | J. Offshore Mech. Arct. Eng. | ASME Digital Collection
, 30 Jangjeon-dong, Geumjeong-gu, Busan, 609-735, Republic of Korea
e-mail: kimm@pusan.ac.kr
Jae Myung Lee,
Samsung Heavy Industries, Co., Ltd.
, 530 Jangpyeong-dong, Geoje, Gyeongnam, 656–710, Republic of Korea
Yong Suk Suh,
Wha Soo Kim,
Wha Soo Kim
Hyundai Heavy Industries, Co., Ltd.
, 1 Jeonha-dong, Dong-gu, Ulsan, 682–792, Republic of Korea
Byung Jae Noh,
Byung Jae Noh
Jang Ho Yoon,
Jang Ho Yoon
, Busan, 600–737, Republic of Korea
Min Soo Kim,
, 862-1 Beomchon 1-dong, Busanjin-gu, Busan, 614–724, Republic of Korea
Hang Sub Urm
Det Norske Veritas Korea Ltd.
, 36-7 Namchon 1-dong, Suyong-gu, Busan, 613–011, Republic of Korea
J. Offshore Mech. Arct. Eng. Nov 2011, 133(4): 041401 (10 pages)
Kim, M. H., Kil, Y. P., Lee, J. M., Chun, M. S., Suh, Y. S., Kim, W. S., Noh, B. J., Yoon, J. H., Kim, M. S., and Urm, H. S. (April 12, 2011). "Cryogenic Fatigue Strength Assessment for MARK-III Insulation System of LNG Carriers." ASME. J. Offshore Mech. Arct. Eng. November 2011; 133(4): 041401. https://doi.org/10.1115/1.4003389
The objective of this study is to investigate the typical failure mode and to obtain the stress range versus number of cycles to failure (S-N) data of MARK-III type liquefied natural gas (LNG) insulation system under the fatigue loading at actual cryogenic environment. A systematic experimental research is carried out for the assessment of the fatigue strength of MARK-III insulation system at cryogenic temperature. Three different types of test specimens are tested for the evaluation of fatigue performance of MARK-III insulation system. Test specimens are determined considering the fatigue vulnerable locations such as mastic area, slit area, and top bridge pad area inside the actual LNG cargo tanks. All test specimens are fabricated as close as possible to the actual yard practice. A series of fatigue test results is represented as S-N curves. Cyclic fatigue loadings were carefully considered similar to the actual sloshing loads. The effect of sloshing impacts is considered by selecting the stress ratio
(R=−10)
. The load levels have been determined based on the ultimate strength of reinforced polyurethane foam as 12.2 bars. Different cryogenic temperatures are employed according to the test locations in consideration of temperature gradient within the insulation system. All test results including fatigue life, as well as failure locations of MARK-III insulation system at cryogenic temperatures, are reported and compared with those at room temperature. Consistent S-N curves of MARK-III insulation system at both room and cryogenic temperatures are obtained and compared. The slopes of S-N curves from both fatigue test results are observed to be almost identical, and the fatigue strengths are found to exhibit similar trend. The results from this research can be used for the fatigue assessment of the LNGC insulation system, as well as a design guideline of LNG CCS at cryogenic temperature.
cryogenics, failure analysis, fatigue, fatigue testing, sloshing, tanks (containers), thermal insulation, cryogenic fatigue, fatigue strength, Mark-III membrane system, LNG carriers, insulation system, S-N curves
Failure, Fatigue, Fatigue strength, Fatigue testing, Insulation, Liquefied natural gas, Stress, Temperature, Cycles, Sloshing, Pressure, Failure mechanisms, Dimensions
A Parametric Sensitivity Study on LNG Tank Sloshing Loads by Numerical Simulations
The Effects of LNG-Tank Sloshing on the Global Motions of LNG Carriers
Impact Strength Assessment of LNG Carrier Insulation System
Strength of Membrane Type LNG Cargo Containment System Under Sloshing Impact Actions
Experimental Investigation of the Impact Behavior of Membrane Type LNG Carrier Insulation System
Sloshing Impact of LNG Cargoes in Membrane Containment Systems in the Partially Filled Condition
Fatigue Strength Assessment of MARK-III Type LNG Cargo Containment System
A Comparative Evaluation of Fatigue and Fracture Characteristics of Structural Components of Liquefied Natural Gas Carrier Insulation System
Fatigue and Fracture Performance of Insulation System in LNG Carriers
Hull Deformation Effect on Membrane-Type LNG Containment Systems |
Heegner number - Wikipedia
In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer d such that the imaginary quadratic field
{\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]}
has class number 1. Equivalently, its ring of integers has unique factorization.[1]
The determination of such numbers is a special case of the class number problem, and they underlie several striking results in number theory.
According to the (Baker–)Stark–Heegner theorem there are precisely nine Heegner numbers:
1, 2, 3, 7, 11, 19, 43, 67, and 163. (sequence A003173 in the OEIS)
This result was conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966, and Stark further indicated the gap in Heegner's proof was minor.[2]
1 Euler's prime-generating polynomial
2 Almost integers and Ramanujan's constant
3 Pi formulas
4 Other Heegner numbers
5 Class 2 numbers
6 Consecutive primes
Euler's prime-generating polynomialEdit
Euler's prime-generating polynomial
{\displaystyle n^{2}+n+41,}
which gives (distinct) primes for n = 0, ..., 39, is related to the Heegner number 163 = 4 · 41 − 1.
Rabinowitz[3] proved that
{\displaystyle n^{2}+n+p}
gives primes for
{\displaystyle n=0,\dots ,p-2}
if and only if this quadratic's discriminant
{\displaystyle 1-4p}
is the negative of a Heegner number.
{\displaystyle p-1}
{\displaystyle p^{2}}
{\displaystyle p-2}
is maximal.)
1, 2, and 3 are not of the required form, so the Heegner numbers that work are 7, 11, 19, 43, 67, 163, yielding prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.[4]
Almost integers and Ramanujan's constantEdit
Ramanujan's constant is the transcendental number[5]
{\displaystyle e^{\pi {\sqrt {163}}}}
, which is an almost integer, in that it is very close to an integer:[6]
{\displaystyle e^{\pi {\sqrt {163}}}=262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots \approx 640\,320^{3}+744.}
This number was discovered in 1859 by the mathematician Charles Hermite.[7] In a 1975 April Fool article in Scientific American magazine,[8] "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.
This coincidence is explained by complex multiplication and the q-expansion of the j-invariant.
Briefly,
{\displaystyle \textstyle j\left({\frac {1+{\sqrt {-d}}}{2}}\right)}
is an integer for d a Heegner number, and
{\displaystyle e^{\pi {\sqrt {d}}}\approx -j\left({\frac {1+{\sqrt {-d}}}{2}}\right)+744}
via the q-expansion.
{\displaystyle \tau }
is a quadratic irrational, then the j-invariant is an algebraic integer of degree
{\displaystyle \left|\mathrm {Cl} {\bigl (}\mathbf {Q} (\tau ){\bigr )}\right|}
, the class number of
{\displaystyle \mathbf {Q} (\tau )}
and the minimal (monic integral) polynomial it satisfies is called the 'Hilbert class polynomial'. Thus if the imaginary quadratic extension
{\displaystyle \mathbf {Q} (\tau )}
has class number 1 (so d is a Heegner number), the j-invariant is an integer.
The q-expansion of j, with its Fourier series expansion written as a Laurent series in terms of
{\displaystyle q=e^{2\pi i\tau }}
, begins as:
{\displaystyle j(\tau )={\frac {1}{q}}+744+196\,884q+\cdots .}
{\displaystyle c_{n}}
asymptotically grow as
{\displaystyle \ln(c_{n})\sim 4\pi {\sqrt {n}}+O{\bigl (}\ln(n){\bigr )},}
and the low order coefficients grow more slowly than
{\displaystyle 200\,000^{n}}
{\displaystyle \textstyle q\ll {\frac {1}{200\,000}}}
, j is very well approximated by its first two terms. Setting
{\displaystyle \textstyle \tau ={\frac {1+{\sqrt {-163}}}{2}}}
{\displaystyle q=-e^{-\pi {\sqrt {163}}}\quad \therefore \quad {\frac {1}{q}}=-e^{\pi {\sqrt {163}}}.}
{\displaystyle j\left({\frac {1+{\sqrt {-163}}}{2}}\right)=\left(-640\,320\right)^{3},}
{\displaystyle \left(-640\,320\right)^{3}=-e^{\pi {\sqrt {163}}}+744+O\left(e^{-\pi {\sqrt {163}}}\right).}
{\displaystyle e^{\pi {\sqrt {163}}}=640\,320^{3}+744+O\left(e^{-\pi {\sqrt {163}}}\right)}
where the linear term of the error is,
{\displaystyle {\frac {-196\,884}{e^{\pi {\sqrt {163}}}}}\approx {\frac {-196\,884}{640\,320^{3}+744}}\approx -0.000\,000\,000\,000\,75}
explaining why
{\displaystyle e^{\pi {\sqrt {163}}}}
is within approximately the above of being an integer.
Pi formulasEdit
The Chudnovsky brothers found in 1987 that
{\displaystyle {\frac {1}{\pi }}={\frac {12}{640\,320^{\frac {3}{2}}}}\sum _{k=0}^{\infty }{\frac {(6k)!(163\cdot 3\,344\,418k+13\,591\,409)}{(3k)!(k!)^{3}(-640\,320)^{3k}}},}
a proof of which uses the fact that
{\displaystyle j\left({\frac {1+{\sqrt {-163}}}{2}}\right)=-640\,320^{3}.}
For similar formulas, see the Ramanujan–Sato series.
Other Heegner numbersEdit
For the four largest Heegner numbers, the approximations one obtains[9] are as follows.
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx {\color {white}000\,0}96^{3}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx {\color {white}000\,}960^{3}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx {\color {white}00}5\,280^{3}+744-0.000\,0013\\e^{\pi {\sqrt {163}}}&\approx 640\,320^{3}+744-0.000\,000\,000\,000\,75\end{aligned}}}
Alternatively,[10]
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 12^{3}\left(3^{2}-1\right)^{3}{\color {white}00}+744-0.22\\e^{\pi {\sqrt {43}}}&\approx 12^{3}\left(9^{2}-1\right)^{3}{\color {white}00}+744-0.000\,22\\e^{\pi {\sqrt {67}}}&\approx 12^{3}\left(21^{2}-1\right)^{3}{\color {white}0}+744-0.000\,0013\\e^{\pi {\sqrt {163}}}&\approx 12^{3}\left(231^{2}-1\right)^{3}+744-0.000\,000\,000\,000\,75\end{aligned}}}
where the reason for the squares is due to certain Eisenstein series. For Heegner numbers
{\displaystyle d<19}
, one does not obtain an almost integer; even
{\displaystyle d=19}
is not noteworthy.[11] The integer j-invariants are highly factorisable, which follows from the form
{\displaystyle 12^{3}\left(n^{2}-1\right)^{3}=\left(2^{2}\cdot 3\cdot (n-1)\cdot (n+1)\right)^{3},}
and factor as,
{\displaystyle {\begin{aligned}j\left({\frac {1+{\sqrt {-19}}}{2}}\right)&={\color {white}000\,0}96^{3}=\left(2^{5}\cdot 3\right)^{3}\\j\left({\frac {1+{\sqrt {-43}}}{2}}\right)&={\color {white}000\,}960^{3}=\left(2^{6}\cdot 3\cdot 5\right)^{3}\\j\left({\frac {1+{\sqrt {-67}}}{2}}\right)&={\color {white}00}5\,280^{3}=\left(2^{5}\cdot 3\cdot 5\cdot 11\right)^{3}\\j\left({\frac {1+{\sqrt {-163}}}{2}}\right)&=640\,320^{3}=\left(2^{6}\cdot 3\cdot 5\cdot 23\cdot 29\right)^{3}.\end{aligned}}}
These transcendental numbers, in addition to being closely approximated by integers (which are simply algebraic numbers of degree 1), can be closely approximated by algebraic numbers of degree 3,[12]
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx x^{24}-24.000\,31;&x^{3}-2x-2&=0\\e^{\pi {\sqrt {43}}}&\approx x^{24}-24.000\,000\,31;&x^{3}-2x^{2}-2&=0\\e^{\pi {\sqrt {67}}}&\approx x^{24}-24.000\,000\,0019;&x^{3}-2x^{2}-2x-2&=0\\e^{\pi {\sqrt {163}}}&\approx x^{24}-24.000\,000\,000\,000\,0011;&\quad x^{3}-6x^{2}+4x-2&=0\end{aligned}}}
The roots of the cubics can be exactly given by quotients of the Dedekind eta function η(τ), a modular function involving a 24th root, and which explains the 24 in the approximation. They can also be closely approximated by algebraic numbers of degree 4,[13]
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx 3^{5}\left(3-{\sqrt {2\left(1-{\tfrac {96}{24}}+1{\sqrt {3\cdot 19}}\right)}}\right)^{-2}-12.000\,06\dots \\e^{\pi {\sqrt {43}}}&\approx 3^{5}\left(9-{\sqrt {2\left(1-{\tfrac {960}{24}}+7{\sqrt {3\cdot 43}}\right)}}\right)^{-2}-12.000\,000\,061\dots \\e^{\pi {\sqrt {67}}}&\approx 3^{5}\left(21-{\sqrt {2\left(1-{\tfrac {5\,280}{24}}+31{\sqrt {3\cdot 67}}\right)}}\right)^{-2}-12.000\,000\,000\,36\dots \\e^{\pi {\sqrt {163}}}&\approx 3^{5}\left(231-{\sqrt {2\left(1-{\tfrac {640\,320}{24}}+2\,413{\sqrt {3\cdot 163}}\right)}}\right)^{-2}-12.000\,000\,000\,000\,000\,21\dots \end{aligned}}}
{\displaystyle x}
denotes the expression within the parenthesis (e.g.
{\displaystyle x=3-{\sqrt {2\left(1-{\tfrac {96}{24}}+1{\sqrt {3\cdot 19}}\right)}}}
), it satisfies respectively the quartic equations
{\displaystyle {\begin{aligned}x^{4}-{\color {white}00}4\cdot 3x^{3}+{\color {white}000\,0}{\tfrac {2}{3}}(96+3)x^{2}-{\color {white}000\,000}{\tfrac {2}{3}}\cdot 3(96-6)x-3&=0\\x^{4}-{\color {white}00}4\cdot 9x^{3}+{\color {white}000\,}{\tfrac {2}{3}}(960+3)x^{2}-{\color {white}000\,00}{\tfrac {2}{3}}\cdot 9(960-6)x-3&=0\\x^{4}-{\color {white}0}4\cdot 21x^{3}+{\color {white}00}{\tfrac {2}{3}}(5\,280+3)x^{2}-{\color {white}000}{\tfrac {2}{3}}\cdot 21(5\,280-6)x-3&=0\\x^{4}-4\cdot 231x^{3}+{\tfrac {2}{3}}(640\,320+3)x^{2}-{\tfrac {2}{3}}\cdot 231(640\,320-6)x-3&=0\\\end{aligned}}}
Note the reappearance of the integers
{\displaystyle n=3,9,21,231}
as well as the fact that
{\displaystyle {\begin{aligned}2^{6}\cdot 3\left(-\left(1-{\tfrac {96}{24}}\right)^{2}+1^{2}\cdot 3\cdot 19\right)&=96^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {960}{24}}\right)^{2}+7^{2}\cdot 3\cdot 43\right)&=960^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {5\,280}{24}}\right)^{2}+31^{2}\cdot 3\cdot 67\right)&=5\,280^{2}\\2^{6}\cdot 3\left(-\left(1-{\tfrac {640\,320}{24}}\right)^{2}+2413^{2}\cdot 3\cdot 163\right)&=640\,320^{2}\end{aligned}}}
which, with the appropriate fractional power, are precisely the j-invariants.
Similarly for algebraic numbers of degree 6,
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {19}}}&\approx \left(5x\right)^{3}-6.000\,010\dots \\e^{\pi {\sqrt {43}}}&\approx \left(5x\right)^{3}-6.000\,000\,010\dots \\e^{\pi {\sqrt {67}}}&\approx \left(5x\right)^{3}-6.000\,000\,000\,061\dots \\e^{\pi {\sqrt {163}}}&\approx \left(5x\right)^{3}-6.000\,000\,000\,000\,000\,034\dots \end{aligned}}}
where the xs are given respectively by the appropriate root of the sextic equations,
{\displaystyle {\begin{aligned}5x^{6}-{\color {white}000\,0}96x^{5}-10x^{3}+1&=0\\5x^{6}-{\color {white}000\,}960x^{5}-10x^{3}+1&=0\\5x^{6}-{\color {white}00}5\,280x^{5}-10x^{3}+1&=0\\5x^{6}-640\,320x^{5}-10x^{3}+1&=0\end{aligned}}}
with the j-invariants appearing again. These sextics are not only algebraic, they are also solvable in radicals as they factor into two cubics over the extension
{\displaystyle \mathbb {Q} {\sqrt {5}}}
(with the first factoring further into two quadratics). These algebraic approximations can be exactly expressed in terms of Dedekind eta quotients. As an example, let
{\displaystyle \textstyle \tau ={\frac {1+{\sqrt {-163}}}{2}}}
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{24}}\eta (\tau )}{\eta (2\tau )}}\right)^{24}-24.000\,000\,000\,000\,001\,05\dots \\e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{12}}\eta (\tau )}{\eta (3\tau )}}\right)^{12}-12.000\,000\,000\,000\,000\,21\dots \\e^{\pi {\sqrt {163}}}&=\left({\frac {e^{\frac {\pi i}{6}}\eta (\tau )}{\eta (5\tau )}}\right)^{6}-6.000\,000\,000\,000\,000\,034\dots \end{aligned}}}
where the eta quotients are the algebraic numbers given above.
Class 2 numbersEdit
The three numbers 88, 148, 232, for which the imaginary quadratic field
{\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]}
has class number 2, are not considered as Heegner numbers but have certain similar properties in terms of almost integers. For instance,
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {88}}}+8\,744&\approx {\color {white}00\,00}2\,508\,952^{2}-0.077\dots \\e^{\pi {\sqrt {148}}}+8\,744&\approx {\color {white}00\,}199\,148\,648^{2}-0.000\,97\dots \\e^{\pi {\sqrt {232}}}+8\,744&\approx 24\,591\,257\,752^{2}-0.000\,0078\dots \\\end{aligned}}}
{\displaystyle {\begin{aligned}e^{\pi {\sqrt {22}}}-24&\approx {\color {white}00}\left(6+4{\sqrt {2}}\right)^{6}+0.000\,11\dots \\e^{\pi {\sqrt {37}}}+24&\approx \left(12+2{\sqrt {37}}\right)^{6}-0.000\,0014\dots \\e^{\pi {\sqrt {58}}}-24&\approx \left(27+5{\sqrt {29}}\right)^{6}-0.000\,000\,0011\dots \\\end{aligned}}}
Consecutive primesEdit
Given an odd prime p, if one computes
{\displaystyle k^{2}\mod p}
{\displaystyle \textstyle k=0,1,\dots ,{\frac {p-1}{2}}}
(this is sufficient because
{\displaystyle \left(p-k\right)^{2}\equiv k^{2}\mod p}
), one gets consecutive composites, followed by consecutive primes, if and only if p is a Heegner number.[14]
For details, see "Quadratic Polynomials Producing Consecutive Distinct Primes and Class Groups of Complex Quadratic Fields" by Richard Mollin.[15]
^ Conway, John Horton; Guy, Richard K. (1996). The Book of Numbers. Springer. p. 224. ISBN 0-387-97993-X.
^ Stark, H. M. (1969), "On the gap in the theorem of Heegner" (PDF), Journal of Number Theory, 1 (1): 16–27, Bibcode:1969JNT.....1...16S, doi:10.1016/0022-314X(69)90023-7, hdl:2027.42/33039
^ Rabinovitch, Georg "Eindeutigkeit der Zerlegung in Primzahlfaktoren in quadratischen Zahlkörpern." Proc. Fifth Internat. Congress Math. ( Cambridge) 1, 418–421, 1913.
^ Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 88 and 144, 1983.
^ Weisstein, Eric W. "Transcendental Number". MathWorld. gives
{\displaystyle e^{\pi {\sqrt {d}}},d\in Z^{*}}
, based on Nesterenko, Yu. V. "On Algebraic Independence of the Components of Solutions of a System of Linear Differential Equations." Izv. Akad. Nauk SSSR, Ser. Mat. 38, 495–512, 1974. English translation in Math. USSR 8, 501–518, 1974.
^ Ramanujan Constant – from Wolfram MathWorld
^ Barrow, John D (2002). The Constants of Nature. London: Jonathan Cape. ISBN 0-224-06135-6.
^ Gardner, Martin (April 1975). "Mathematical Games". Scientific American. Scientific American, Inc. 232 (4): 127. Bibcode:1975SciAm.232e.102G. doi:10.1038/scientificamerican0575-102.
^ These can be checked by computing
{\displaystyle {\sqrt[{3}]{e^{\pi {\sqrt {d}}}-744}}}
on a calculator, and
{\displaystyle {\frac {196\,884}{e^{\pi {\sqrt {d}}}}}}
for the linear term of the error.
^ "More on e^(pi*SQRT(163))".
^ The absolute deviation of a random real number (picked uniformly from [0,1], say) is a uniformly distributed variable on [0, 0.5], so it has absolute average deviation and median absolute deviation of 0.25, and a deviation of 0.22 is not exceptional.
^ "Pi Formulas".
^ "Extending Ramanujan's Dedekind Eta Quotients".
^ "Simple Complex Quadratic Fields".
^ Mollin, R. A. (1996). "Quadratic polynomials producing consecutive, distinct primes and class groups of complex quadratic fields" (PDF). Acta Arithmetica. 74: 17–30. doi:10.4064/aa-74-1-17-30.
Weisstein, Eric W. "Heegner Number". MathWorld.
OEIS sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization)
Gauss' Class Number Problem for Imaginary Quadratic Fields, by Dorian Goldfeld: Detailed history of problem.
Clark, Alex. "163 and Ramanujan Constant". Numberphile. Brady Haran. Archived from the original on 2013-05-16. Retrieved 2013-04-02.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Heegner_number&oldid=1071785989" |
Commutative_property Knowpia
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized.[1][2] A corresponding property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.[3]
{\displaystyle \circ }
{\displaystyle x\circ y=y\circ x}
{\displaystyle x}
{\displaystyle y}
{\displaystyle x\circ y}
{\displaystyle y\circ x}
{\displaystyle x}
{\displaystyle y}
{\displaystyle *}
on a set S is called commutative if[4][5]
{\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S.}
One says that x commutes with y or that x and y commute under
{\displaystyle *}
{\displaystyle x*y=y*x.}
In other words, an operation is commutative if every pair of elements commute.
A binary function
{\displaystyle f\colon A\times A\to B}
is sometimes called commutative if
{\displaystyle f(x,y)=f(y,x)\qquad {\mbox{for all }}x,y\in A.}
Such a function is more commonly called a symmetric function.
Commutative operationsEdit
The addition of vectors is commutative, because
{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}
Addition and multiplication are commutative in most number systems, and, in particular, between natural numbers, integers, rational numbers, real numbers and complex numbers. This is also true in every field.
Addition is commutative in every vector space and in every algebra.
Union and intersection are commutative operations on sets.
"And" and "or" are commutative logical operations.
Noncommutative operationsEdit
Some noncommutative binary operations:[6]
Division, subtraction, and exponentiationEdit
Division is noncommutative, since
{\displaystyle 1\div 2\neq 2\div 1}
Subtraction is noncommutative, since
{\displaystyle 0-1\neq 1-0}
. However it is classified more precisely as anti-commutative, since
{\displaystyle 0-1=-(1-0)}
Exponentiation is noncommutative, since
{\displaystyle 2^{3}\neq 3^{2}}
Truth functionsEdit
Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for (A ⇒ B) = (¬A ∨ B) and (B ⇒ A) = (A ∨ ¬B) are
Function composition of linear functionsEdit
Function composition of linear functions from the real numbers to the real numbers is almost always noncommutative. For example, let
{\displaystyle f(x)=2x+1}
{\displaystyle g(x)=3x+7}
{\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15}
{\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10}
This also applies more generally for linear and affine transformations from a vector space to itself (see below for the Matrix representation).
Matrix multiplicationEdit
Matrix multiplication of square matrices is almost always noncommutative, for example:
{\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}
Vector productEdit
The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., b × a = −(a × b).
Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products.[7][8] Euclid is known to have assumed the commutative property of multiplication in his book Elements.[9] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics.
The first recorded use of the term commutative was in a memoir by François Servois in 1814,[1][10] which used the word commutatives when describing functions that have what is now called the commutative property. The word is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch". The term then appeared in English in 1838.[2] in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.[11]
{\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}
{\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}
{\displaystyle \Leftrightarrow }
" is a metalogical symbol representing "can be replaced in a proof with".
{\displaystyle (P\land Q)\leftrightarrow (Q\land P)}
{\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}
{\displaystyle (P\to (Q\to R))\leftrightarrow (Q\to (P\to R))}
{\displaystyle (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}
Mathematical structures and commutativityEdit
The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states that the order of the terms does not affect the final result.
{\displaystyle f(x,y)={\frac {x+y}{2}},}
which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example,
{\displaystyle f(-4,f(0,+4))=-1}
{\displaystyle f(f(-4,0),+4)=+1}
). More such examples may be found in commutative non-associative magmas.
Some forms of symmetry can be directly linked to commutativity. When a commutative operation is written as a binary function
{\displaystyle z=f(x,y),}
then this function is called a symmetric function, and its graph in three-dimensional space is symmetric across the plane
{\displaystyle y=x}
. For example, if the function f is defined as
{\displaystyle f(x,y)=x+y}
{\displaystyle f}
is a symmetric function.
For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then
{\displaystyle aRb\Leftrightarrow bRa}
Non-commuting operators in quantum mechanicsEdit
In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as
{\displaystyle x}
(meaning multiply by
{\displaystyle x}
{\textstyle {\frac {d}{dx}}}
. These two operators do not commute as may be seen by considering the effect of their compositions
{\textstyle x{\frac {d}{dx}}}
{\textstyle {\frac {d}{dx}}x}
(also called products of operators) on a one-dimensional wave function
{\displaystyle \psi (x)}
{\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)}
According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum in the
{\displaystyle x}
-direction of a particle are represented by the operators
{\displaystyle x}
{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}
, respectively (where
{\displaystyle \hbar }
is the reduced Planck constant). This is the same example except for the constant
{\displaystyle -i\hbar }
, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.
Centralizer and normalizer (also called a commutant)
Proof that Peano's axioms imply the commutativity of the addition of natural numbers
Commuting probability
^ a b Cabillón & Miller, Commutative and Distributive
^ a b Flood, Raymond; Rice, Adrian; Wilson, Robin, eds. (2011). Mathematics in Victorian Britain. Oxford University Press. p. 4. ISBN 9780191627941.
^ Weisstein, Eric W. "Symmetric Relation". MathWorld.
^ Yark, p. 1
^ Lumpkin 1997, p. 11
^ Gay & Shute 1987
^ O'Conner & Robertson Real Numbers
^ O'Conner & Robertson, Servois
^ Gregory, D. F. (1840). "On the real nature of symbolical algebra". Transactions of the Royal Society of Edinburgh. 14: 208–216.
^ Copi & Cohen 2005
^ Hurley & Watson 2016
^ Axler 1997, p. 2
^ a b Gallian 2006, p. 34
^ Gallian 2006, pp. 26, 87
^ Gallian 2006, p. 236
Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic (12th ed.). Prentice Hall. ISBN 9780131898349.
Gallian, Joseph (2006). Contemporary Abstract Algebra (6e ed.). Houghton Mifflin. ISBN 0-618-51471-6.
Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry (2e ed.). Prentice Hall. ISBN 0-13-067342-0.
Hurley, Patrick J.; Watson, Lori (2016). A Concise Introduction to Logic (12th ed.). Cengage Learning. ISBN 978-1-337-51478-1.
Lumpkin, B. (1997). "The Mathematical Legacy Of Ancient Egypt — A Response To Robert Palter" (PDF) (Unpublished manuscript). Archived from the original (PDF) on 13 July 2007.
Gay, Robins R.; Shute, Charles C. D. (1987). The Rhind Mathematical Papyrus: An Ancient Egyptian Text. British Museum. ISBN 0-7141-0944-4.
Krowne, Aaron, Commutative at PlanetMath., Accessed 8 August 2007.
"Yark". Examples of non-commutative operations at PlanetMath., Accessed 8 August 2007
O'Conner, J.J.; Robertson, E.F. "History of real numbers". MacTutor. Retrieved 8 August 2007.
Cabillón, Julio; Miller, Jeff. "Earliest Known Uses Of Mathematical Terms". Retrieved 22 November 2008.
O'Conner, J.J.; Robertson, E.F. "biography of François Servois". MacTutor. Retrieved 8 August 2007. |
Semilinear parabolic equation in $ {\mathbf{R}}^{N}$ associated with critical Sobolev exponent
Semilinear parabolic equation in
{𝐑}^{N}
associated with critical Sobolev exponent
We consider the semilinear parabolic equation
{u}_{t}-\Delta u={|u|}^{p-1}u
on the whole space
{𝐑}^{N}
N⩾3
p=\left(N+2\right)/\left(N-2\right)
is associated with the Sobolev imbedding
{H}^{1}\left({𝐑}^{N}\right)\subset {L}^{p+1}\left({𝐑}^{N}\right)
. First, we study the decay and blow-up of the solution by means of the potential-well and forward self-similar transformation. Then, we discuss blow-up in infinite time and classify the orbit.
Classification : 35K55
Mots clés : Parabolic equation, Critical Sobolev exponent, Cauchy problem, Stable and unstable sets, Self-similarity
author = {Ikehata, Ryo and Ishiwata, Michinori and Suzuki, Takashi},
title = {Semilinear parabolic equation in $ {\mathbf{R}}^{N}$ associated with critical {Sobolev} exponent},
AU - Ikehata, Ryo
AU - Ishiwata, Michinori
AU - Suzuki, Takashi
TI - Semilinear parabolic equation in $ {\mathbf{R}}^{N}$ associated with critical Sobolev exponent
Ikehata, Ryo; Ishiwata, Michinori; Suzuki, Takashi. Semilinear parabolic equation in $ {\mathbf{R}}^{N}$ associated with critical Sobolev exponent. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 877-900. doi : 10.1016/j.anihpc.2010.01.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2010.01.002/
[1] L.A. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297 | MR 982351 | Zbl 0702.35085
[2] T. Cazenave, P.L. Lions, Solutions globales d'equations de la shaleur semi lineaires, Comm. Partial Differential Equations 9 (1984), 955-978 | MR 755928 | Zbl 0555.35067
[3] W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615-622 | MR 1121147 | Zbl 0768.35025
[4] E.B. Davis, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge (1989) | MR 990239 | Zbl 0699.35006
[5] M. Escobedo, O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. 11 (1987), 1103-1133 | MR 913672 | Zbl 0639.35038
[6] M. Escobedo, E. Zuazua, Large time behavior for convection–diffusion equations in
{R}^{N}
, J. Funct. Anal. 100 (1991), 119-161 | MR 1124296 | Zbl 0762.35011
[7] V. Georgiev, Semilinear Hyperbolic Equations, MSJ Mem. vol. 7, Math. Soc. Japan (2000) | MR 1807081 | Zbl 0959.35002
[8] R. Ikehata, The Palais–Smale condition for the energy of some semilinear parabolic equations, Hiroshima Math. J. 30 (2000), 117-127 | MR 1753386 | Zbl 0953.35067
[9] R. Ikehata, T. Suzuki, Semilinear parabolic equations involving critical Sobolev exponent: Local and asymptotic behavior of solutions, Differential Integral Equations 13 (2000), 437-477 | MR 1775238 | Zbl 1016.35005
[10] K. Ishige, T. Kawakami, Asymptotic behavior of solutions for some semilinear heat equations in
{𝐑}^{N}
[11] M. Ishiwata, Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent, Discrete Contin. Dyn. Syst. (2005), 443-452 | MR 2192702 | Zbl 1173.35344
[12] M. Ishiwata, On the asymptotic behavior of radial positive solutions for semilinear parabolic problem involving critical Sobolev exponent, in preparation
[13] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York (1976) | MR 407617
[14] O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst. H. Poincaré 4 (1987), 423-452 | EuDML 78139 | MR 921547 | Zbl 0653.35036
[15] T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity, Ann. Inst. H. Poincaré 13 (1996), 1-15 | EuDML 78375 | Numdam | MR 1373470 | Zbl 0847.35060
[16] T. Kawanago, Existence and behavior of solutions for
{u}_{t}=\Delta {u}^{m}+{u}^{\ell }
, Adv. Math. Sci. Appl. 7 (1997), 367-400 | MR 1454672 | Zbl 0876.35061
[17] V. Komornik, Exact Controllability and Stabilization, Multiplier Method, Masson, Paris (1994) | MR 1359765 | Zbl 0937.93003
[18] M. Otani, Existence and asymptotic stability of strong solutions of nonlinear evolution equations with a difference term of subdifferentials, Colloq. Math. Soc. Janos Bolyai, Qualitative Theory of Differential Equations, vol. 30, North-Holland, Amsterdam (1980) | Zbl 0506.35075
[19] M. Otani,
{L}^{\infty }
-energy method and its applications, nonlinear partial differential equations and their applications, GAKUTO Internat. Ser. Math. Sci. Appl. 20 (2004), 505-516 | MR 2087494 | Zbl 1061.35035
[20] L.E. Payne, D.H. Sattinger, Saddle points and unstability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 273-303 | MR 402291 | Zbl 0317.35059
[21] P. Poláčik, private communication
[22] D.H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30 (1968), 148-172 | MR 227616 | Zbl 0159.39102
[23] T. Suzuki, Semilinear parabolic equation on bounded domain with critical Sobolev exponent, Indiana Univ. Math. J. 57 no. 7 (2008), 3365-3396 | MR 2492236 | Zbl 1201.35116
[24] H. Tanabe, Equations of Evolution, Pitman, London (1979) | MR 533824 |
4x2+36y2-4x+24y+1=0 at the point(1/2,-2/3) Find the equations of the tangents and normals to the ellipse - Maths - Conic Sections - 10707657 | Meritnation.com
4x2+36y 2-4x+24y+1=0 at the point(1/2,-2/3)
Find the equations of the tangents and normals to the ellipse
We have, 4{x}^{2}+36{y}^{2}-4x+24y+1=0\phantom{\rule{0ex}{0ex}}⇒4{x}^{2}-4x+1+36{y}^{2}+24y+{2}^{2}={2}^{2}\phantom{\rule{0ex}{0ex}}⇒{\left(2x-1\right)}^{2}+{\left(6y+2\right)}^{2}=4\phantom{\rule{0ex}{0ex}}⇒\frac{{\left(2x-1\right)}^{2}}{{2}^{2}}+\frac{{\left(6y+2\right)}^{2}}{{2}^{2}}=1\phantom{\rule{0ex}{0ex}}⇒\frac{4\left(2x-1\right)}{{2}^{2}}+\frac{12\left(6y+2\right)}{{2}^{2}}\left(\frac{dy}{dx}\right)=0\phantom{\rule{0ex}{0ex}}\frac{dy}{dx}=\frac{-\left(2x-1\right)}{3\left(6y+2\right)}\phantom{\rule{0ex}{0ex}}So, at the point \left(\frac{1}{2},\frac{-2}{3}\right) ,{\overline{)\frac{dy}{dx}}}_{\left(\frac{1}{2},\frac{-2}{3}\right)}=\frac{-\left(2×\frac{1}{2}-1\right)}{3\left(6×\frac{-2}{3}+2\right)}=0\phantom{\rule{0ex}{0ex}}⇒{\overline{)\frac{dy}{dx}}}_{\left(\frac{1}{2},\frac{-2}{3}\right)}=0\phantom{\rule{0ex}{0ex}}Thus, equation of the \mathrm{tan}gent at \left(\frac{1}{2},\frac{-2}{3}\right) is\phantom{\rule{0ex}{0ex}}y+\frac{2}{3}=0\left(x-\frac{1}{2}\right)\phantom{\rule{0ex}{0ex}}⇒y+\frac{2}{3}=0\phantom{\rule{0ex}{0ex}}⇒3y+2=0\phantom{\rule{0ex}{0ex}}And equation of the normal at \left(\frac{1}{2},\frac{-2}{3}\right) is\phantom{\rule{0ex}{0ex}}y+\frac{2}{3}=0\left(x-\frac{1}{2}\right)\phantom{\rule{0ex}{0ex}}⇒y+\frac{2}{3}=0\phantom{\rule{0ex}{0ex}}⇒3y+2=0\phantom{\rule{0ex}{0ex}} |
EUDML | Absolute end points of irreducible continua EuDML | Absolute end points of irreducible continua
Absolute end points of irreducible continua
A concept of an absolute end point introduced and studied by Ira Rosenholtz for arc-like continua is extended in the paper to be applied arbitrary irreducible continua. Some interrelations are studied between end points, absolute end points and points at which a given irreducible continuum is smooth.
Charatonik, Janusz Jerzy. "Absolute end points of irreducible continua." Mathematica Bohemica 118.1 (1993): 19-28. <http://eudml.org/doc/29171>.
abstract = {A concept of an absolute end point introduced and studied by Ira Rosenholtz for arc-like continua is extended in the paper to be applied arbitrary irreducible continua. Some interrelations are studied between end points, absolute end points and points at which a given irreducible continuum is smooth.},
author = {Charatonik, Janusz Jerzy},
keywords = {bend set; pairwise smooth continuum; contractible continuum; curve; dendroid; fan; homotopy; pointwise smooth continuum; $Q$-point; selectible continuum; absolute end point; arc-like; decomposition; irreducible; locally connected; smoth; bend set; pairwise smooth continuum; contractible continuum; curve; dendroid; fan; homotopy; pointwise smooth continuum; -point; - continuum; selectible continuum; type ; absolute end point},
title = {Absolute end points of irreducible continua},
AU - Charatonik, Janusz Jerzy
TI - Absolute end points of irreducible continua
AB - A concept of an absolute end point introduced and studied by Ira Rosenholtz for arc-like continua is extended in the paper to be applied arbitrary irreducible continua. Some interrelations are studied between end points, absolute end points and points at which a given irreducible continuum is smooth.
KW - bend set; pairwise smooth continuum; contractible continuum; curve; dendroid; fan; homotopy; pointwise smooth continuum; $Q$-point; selectible continuum; absolute end point; arc-like; decomposition; irreducible; locally connected; smoth; bend set; pairwise smooth continuum; contractible continuum; curve; dendroid; fan; homotopy; pointwise smooth continuum; -point; - continuum; selectible continuum; type ; absolute end point
R. H. Bing, 10.2140/pjm.1951.1.43, Pacific J. Math 1 (1951), 43-51. (1951) Zbl0043.16803MR0043451DOI10.2140/pjm.1951.1.43
R. H. Bing, 10.1215/S0012-7094-51-01857-1, Duke Math. J. 18 (1951), 653-663. (1951) Zbl0043.16804MR0043450DOI10.1215/S0012-7094-51-01857-1
J. J. Charatonik, On irreducible smooth continua, Topology and its Applications II. Proc. International Symposium on Topology and its Applications, Budva 1972 (Beograd 1973), 45-50. (1972) MR0345079
J. J. Charatonik, T. Maćkowiak, 10.1216/RMJ-1987-17-2-385, Rocky Mountain J. Math. 17 (1987), 385-391. (1987) MR0892465DOI10.1216/RMJ-1987-17-2-385
G. R. Gordh, Jr., On decomposition of smooth continua, Fund. Math. 75(1972), 51-60. (1972) MR0317299
G. R. Gordh, Jr., Concerning closed quasi-orders on hereditarily unicoherent continua, Fund. Math. 78 (1973), 61-73. (1973) Zbl0253.54031MR0322835
K. Kuratowski, Topology, vol. II, Academic Press and PWN, 1968. (1968) MR0259835
T. Maćkowiak, On smooth continua, Fund. Math. 85 (1974), 79-95. (1974) MR0365532
I. Rosenholtz, 10.1090/S0002-9939-1988-0955027-2, Proc. Amer. Math. Soc. 108 (1988), 1305-1314. (1988) Zbl0655.54024MR0955027DOI10.1090/S0002-9939-1988-0955027-2
E. S. Thomas, Jr., Monotone decomposition of irreducible continua, Dissertationes Math. (Rozprawy Mat.) 50 (1966), 1-74. (1966) MR0196721
bend set, pairwise smooth continuum, contractible continuum, curve, dendroid, fan, homotopy, pointwise smooth continuum,
Q
-point, selectible continuum, absolute end point, arc-like, decomposition, irreducible, locally connected, smoth, bend set, pairwise smooth continuum, contractible continuum, curve, dendroid, fan, homotopy, pointwise smooth continuum,
Q
-point,
R
- continuum, selectible continuum, type
N
, absolute end point
\le 1
Articles by Janusz Jerzy Charatonik |
Converting Repeating Decimals into Fractions | Brilliant Math & Science Wiki
Mahindra Jain, Pranshu Gaba, Shehab Khan, and
Rational numbers, when written as decimals, are either terminating or non-terminating, repeating decimals. Converting terminating decimals into fractions is straightforward: multiplying and dividing by an appropriate power of ten does the trick. For example,
2.556753 = \frac{2556753}{1000000}.
However, when the decimals are repeating, things are a little more difficult. Repeating decimals occur very frequently both when doing simple arithmetic and when solving competition problems, so being able to convert them to fractions is a valuable skill.
Easy Way to Convert Irrational Decimals to Fractions
Some examples of non-terminating repeating decimals are
0.12121212121212\ldots
1.2354354354354\ldots
. We can represent these decimals in short as
0.\overline{12}
1.2\overline{354},
To convert these types of decimals to fractions, we can view the decimal as the sum of (infinite) terms in a geometrical progression. This can be easily understood by some examples.
0.\overline{34}
Proof 1: We can write
0.\overline{34}
0.3434343434 \ldots
x=0.\overline{34},
\begin{aligned} x &= 0.34 + 0.0034 + 0.000034 + \cdots\\ &= \frac{34}{100} + \frac{34}{10000} + \frac{34}{1000000} + \cdots\\ &= 34 \times \left( \frac{1}{100^{1}} + \frac{1}{100^{2}} + \frac{1}{100^{3}} + \cdots \right). \end{aligned}
Recognize that this is the sum of infinite terms of a GP which has initial term
a = \frac{1}{100}
r = \frac{1}{100}.
Since the sum of infinite terms is
\frac{a}{1-r},
and
r
x = 34 \times \dfrac{\frac{1}{100}}{1 - \frac{1}{100}} = 34 \times \frac{1}{99} = \dfrac{34}{99}. \ _\square
Proof 2: Here is an alternative way to solve this problem: Let
x = 0.3434343434 \ldots,
100x = 34.343434 \ldots.
On subtracting the first equation from the second, we have
99 x = 34 \implies x = \dfrac{34}{99}. \ _\square
0.\overline{1}
0.\overline{1}
0.1111111111 \ldots
x=0.\overline{1},
\begin{aligned} x &= 0.1 + 0.01 + 0.001 + \cdots\\ &= \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \cdots\\ &= \frac{\frac{1}{10}}{1-\frac{1}{10}}\\\\ &=\frac{1}{9}. \ _\square \end{aligned}
0.0\overline{23}
0.0\overline{23}
0.02323232323 \ldots
x=0.0\overline{23},
\begin{aligned} x &= 0.023 + 0.00023 + 0.0000023 + \cdots\\ &= \frac{23}{1000} + \frac{23}{100000} + \frac{23}{10000000} + \cdots\\ &= \frac{23}{1000} \times \left( 1 + \frac{1}{100} + \frac{1}{100^2} + \cdots \right)\\ &= \frac{23}{1000} \times \frac{1}{1-\frac{1}{100}} \\ &= \frac{23}{1000} \times \frac{100}{99}\\\\ &=\frac{23}{990}. \ _\square \end{aligned}
4.1\overline{454}
4.1\overline{454}
4.1454454454454 \ldots
x=4.1\overline{454},
\begin{aligned} x &= 4.1+0.0454 +0.0000454 + 0.0000000454 + \cdots\\ &= \frac{41}{10} + \frac{454}{10000} + \frac{454}{10000000} +\frac{454}{10000000000}+ \cdots\\ &= \frac{41}{10} +\frac{454}{10000} \times \left( 1 + \frac{1}{1000} + \frac{1}{1000^2} + \cdots \right)\\ &= \frac{41}{10} +\frac{454}{10000} \times \frac{1}{1-\frac{1}{1000}} \\ &= \frac{41}{10} +\frac{454}{10000} \times \frac{1000}{999}\\ &=\frac{41}{10} +\frac{454}{9990}\\\\ &=\frac{41413}{9990}. \ _\square \end{aligned}
0.\overline{5}+0.\overline{7}?
\begin{array}{c}&(a)~ 1.\overline{2} &&&(b)~ 1.\overline{3} &&&(c)~ 1.2\overline{3} &&&(d)~ 1.3\overline{2} \end{array}
0.\overline{5}
0.55555555 \ldots
x=0.\overline{5},
\begin{aligned} x &= 0.5+0.05 +0.005 + 0.0005 + \cdots\\ &= \frac{5}{10} + \frac{5}{10^2} + \frac{5}{10^3} +\frac{5}{10^4}+ \cdots\\ &= \frac{5}{10} \times \left( 1 + \frac{1}{10} + \frac{1}{10^2} +\frac{1}{10^3} + \cdots \right)\\ &= \frac{5}{10} \times \frac{1}{1-\frac{1}{10}} \\ &= \frac{5}{10} \times \frac{10}{9}\\\\ &=\frac{5}{9}. \end{aligned}
y=0.\overline{7},
then we can get
y=\frac{7}{9}.
0.\overline{5}+0.\overline{7}=x+y=\frac{5}{9}+\frac{7}{9}=\frac{12}{9}=1+\frac{3}{9}=1.\overline{3}.
1.\overline{3}.
\ _\square
The non-terminating, repeating decimal
3.9\overline{1}
can be written as a fraction
{\frac{176}{a}}.
a?
\begin{aligned} 100x &= 391.1111111 &\qquad (1)\\ 10x &= 39.1111111. &\qquad (2) \end{aligned}
(1)-(2)
90x=352 \implies x=\frac{176}{45},
a=45.
_\square
0.9999 \ldots = 1
\ldots
0.7
0.\bar{7}
\dfrac{7}{9}
\dfrac{4}{13}
\dfrac{7}{99}
None of the above choices
\large \displaystyle {0. \overline{42}-0.\overline{35}= \, ?}
0.\overline{ab}=0.abababab \ldots
{A = 0.\overline{19} + 0.\overline{199}, \quad B = 0.\overline{19} \times 0.\overline{199}}
0.\overline{19},
for example, stands for the repeating decimal
0.19191919...
and that the period of a repeating decimal is the number of digits in the repeating part. In this case, the period of
0.\overline{19}
Find the sum of the periods of
A
B
{\dfrac{n}{810} = 0.\overline{9D5} = 0.9D59D59D5...}
n
D
n > 0
0 \leq D \leq 9
satisfying the above equation. Determine the value of
n+D
Cite as: Converting Repeating Decimals into Fractions. Brilliant.org. Retrieved from https://brilliant.org/wiki/converting-repeating-decimals-into-fractions/ |
Pairwise distance between pairs of observations - MATLAB pdist - MathWorks India
{d}_{st}^{2}=\left({x}_{s}-{x}_{t}\right)\left({x}_{s}-{x}_{t}{\right)}^{\prime }.
{d}_{st}^{2}=\left({x}_{s}-{x}_{t}\right){V}^{-1}\left({x}_{s}-{x}_{t}{\right)}^{\prime },
{d}_{st}^{2}=\left({x}_{s}-{x}_{t}\right){C}^{-1}\left({x}_{s}-{x}_{t}{\right)}^{\prime },
{d}_{st}=\sum _{j=1}^{n}|{x}_{sj}-{x}_{tj}|.
{d}_{st}=\sqrt[p]{\sum _{j=1}^{n}{|{x}_{sj}-{x}_{tj}|}^{p}}.
{d}_{st}={\mathrm{max}}_{j}\left\{|{x}_{sj}-{x}_{tj}|\right\}.
{d}_{st}=1-\frac{{x}_{s}{{x}^{\prime }}_{t}}{\sqrt{\left({x}_{s}{{x}^{\prime }}_{s}\right)\left({x}_{t}{{x}^{\prime }}_{t}\right)}}.
{d}_{st}=1-\frac{\left({x}_{s}-{\overline{x}}_{s}\right){\left({x}_{t}-{\overline{x}}_{t}\right)}^{\prime }}{\sqrt{\left({x}_{s}-{\overline{x}}_{s}\right){\left({x}_{s}-{\overline{x}}_{s}\right)}^{\prime }}\sqrt{\left({x}_{t}-{\overline{x}}_{t}\right){\left({x}_{t}-{\overline{x}}_{t}\right)}^{\prime }}},
{\overline{x}}_{s}=\frac{1}{n}\sum _{j}{x}_{sj}
{\overline{x}}_{t}=\frac{1}{n}\sum _{j}{x}_{tj}
{d}_{st}=\left(#\left({x}_{sj}\ne {x}_{tj}\right)/n\right).
{d}_{st}=\frac{#\left[\left({x}_{sj}\ne {x}_{tj}\right)\cap \left(\left({x}_{sj}\ne 0\right)\cup \left({x}_{tj}\ne 0\right)\right)\right]}{#\left[\left({x}_{sj}\ne 0\right)\cup \left({x}_{tj}\ne 0\right)\right]}.
{d}_{st}=1-\frac{\left({r}_{s}-{\overline{r}}_{s}\right){\left({r}_{t}-{\overline{r}}_{t}\right)}^{\prime }}{\sqrt{\left({r}_{s}-{\overline{r}}_{s}\right){\left({r}_{s}-{\overline{r}}_{s}\right)}^{\prime }}\sqrt{\left({r}_{t}-{\overline{r}}_{t}\right){\left({r}_{t}-{\overline{r}}_{t}\right)}^{\prime }}},
{\overline{r}}_{s}=\frac{1}{n}\sum _{j}{r}_{sj}=\frac{\left(n+1\right)}{2}
{\overline{r}}_{t}=\frac{1}{n}\sum _{j}{r}_{tj}=\frac{\left(n+1\right)}{2} |
ARITHMETIC - Encyclopedia Information
Arithmetic Information
Arithmetic (from Ancient Greek ἀριθμός (arithmós) 'number', and τική [ τέχνη] (tikḗ [tékhnē]) 'art, craft') is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms, which are highly important to the field of mathematical logic today.
The prehistory of arithmetic is limited to a small number of artifacts, which may indicate the conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 20,000 and 18,000 BC, although its interpretation is disputed. [1]
Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from the modern notation. The ancient Greeks lacked a symbol for zero until the Hellenistic period, and they used three separate sets of symbols as digits: one set for the units place, one for the tens place, and one for the hundreds. For the thousands place, they would reuse the symbols for the units place, and so on. Their addition algorithm was identical to the modern method, and their multiplication algorithm was only slightly different. Their long division algorithm was the same, and the digit-by-digit square root algorithm, popularly used as recently as the 20th century, was known to Archimedes (who may have invented it). He preferred it to Hero's method of successive approximation because, once computed, a digit does not change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a fractional part, such as 546.934, they used negative powers of 60—instead of negative powers of 10 for the fractional part 0.934. [2]
The ancient Chinese had advanced arithmetic studies dating from the Shang Dynasty and continuing through the Tang Dynasty, from basic numbers to advanced algebra. The ancient Chinese used a positional notation similar to that of the Greeks. Since they also lacked a symbol for zero, they had one set of symbols for the units place, and a second set for the tens place. For the hundreds place, they then reused the symbols for the units place, and so on. Their symbols were based on the ancient counting rods. The exact time where the Chinese started calculating with positional representation is unknown, though it is known that the adoption started before 400 BC. [3] The ancient Chinese were the first to meaningfully discover, understand, and apply negative numbers. This is explained in the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu), which was written by Liu Hui dated back to 2nd century BC.
The gradual development of the Hindu–Arabic numeral system independently devised the place-value concept and positional notation, which combined the simpler methods for computations with a decimal base, and the use of a digit representing 0. This allowed the system to consistently represent both large and small integers—an approach which eventually replaced all other systems. In the early 6th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system in his work, and experimented with different notations. In the 7th century, Brahmagupta established the use of 0 as a separate number, and determined the results for multiplication, division, addition and subtraction of zero and all other numbers—except for the result of division by zero. His contemporary, the Syriac bishop Severus Sebokht (650 AD) said, "Indians possess a method of calculation that no word can praise enough. Their rational system of mathematics, or of their method of calculation. I mean the system using nine symbols." [4] The Arabs also learned this new method and called it hesab.
Although the Codex Vigilanus described an early form of Arabic numerals (omitting 0) by 976 AD, Leonardo of Pisa ( Fibonacci) was primarily responsible for spreading their use throughout Europe after the publication of his book Liber Abaci in 1202. He wrote, "The method of the Indians (Latin Modus Indorum) surpasses any known method to compute. It's a marvelous method. They do their computations using nine figures and symbol zero". [5]
The basic arithmetic operations are addition, subtraction, multiplication and division, although arithmetic also includes more advanced operations, such as manipulations of percentages, [6] square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms ( prosthaphaeresis). Arithmetic expressions must be evaluated according to the intended sequence of operations. There are several methods to specify this, either—most common, together with infix notation—explicitly using parentheses and relying on precedence rules, or using a prefix or postfix notation, which uniquely fix the order of execution by themselves. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws (including distributivity), is called a field. [7]
{\displaystyle +}
{\displaystyle -}
, is the inverse operation to addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend: D = M − S. Resorting to the previously established addition, this is to say that the difference is the number that, when added to the subtrahend, results in the minuend: D + S = M. [8]
For any representation of numbers, there are methods for calculating results, some of which are particularly advantageous in exploiting procedures, existing for one operation, by small alterations also for others. For example, digital computers can reuse existing adding-circuitry and save additional circuits for implementing a subtraction, by employing the method of two's complement for representing the additive inverses, which is extremely easy to implement in hardware ( negation). The trade-off is the halving of the number range for a fixed word length.
{\displaystyle \times }
{\displaystyle \cdot }
, is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, mostly both are simply called factors.
{\displaystyle \div }
{\displaystyle /}
, is essentially the inverse operation to multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Under common rules, dividend divided by zero is undefined. For distinct positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than or equal to 1 (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend.
{\displaystyle \%}
{\displaystyle mod}
, though sometimes a second output for one "divmod" operation. [9] In either case, Modular arithmetic has a variety of use cases. Different implementations of division (floored, truncated, Euclidean, etc.) correspond with different implementations of modulus.
The concept of 0 as a number comparable to the other basic digits is essential to this notation, as is the concept of 0's use as a placeholder, and as is the definition of multiplication and addition with 0. The use of 0 as a placeholder and, therefore, the use of a positional notation is first attested to in the Jain text from India entitled the Lokavibhâga, dated 458 AD and it was only in the early 13th century that these concepts, transmitted via the scholarship of the Arabic world, were introduced into Europe by Fibonacci [10] using the Hindu–Arabic numeral system.
Compound [11] unit arithmetic is the application of arithmetic operations to mixed radix quantities such as feet and inches; gallons and pints; pounds, shillings and pence; and so on. Before decimal-based systems of money and units of measure, compound unit arithmetic was widely used in commerce and industry.
The techniques used in compound unit arithmetic were developed over many centuries and are well documented in many textbooks in many different languages. [12] [13] [14] [15] In addition to the basic arithmetic functions encountered in decimal arithmetic, compound unit arithmetic employs three more functions:
Reduction, in which a compound quantity is reduced to a single quantity—for example, conversion of a distance expressed in yards, feet and inches to one expressed in inches. [16]
During the 19th and 20th centuries various aids were developed to aid the manipulation of compound units, particularly in commercial applications. The most common aids were mechanical tills which were adapted in countries such as the United Kingdom to accommodate pounds, shillings, pennies and farthings, and ready reckoners, which are books aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money. One typical booklet [17] that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound".
The cumbersome nature of compound unit arithmetic has been recognized for many years—in 1586, the Flemish mathematician Simon Stevin published a small pamphlet called De Thiende ("the tenth") [18] in which he declared the universal introduction of decimal coinage, measures, and weights to be merely a question of time. In the modern era, many conversion programs, such as that included in the Microsoft Windows 7 operating system calculator, display compound units in a reduced decimal format rather than using an expanded format (e.g. "2.5 ft" is displayed rather than "2 ft 6 in").
The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960s and 1970s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics. [19]
Also, arithmetic was used by Islamic Scholars in order to teach application of the rulings related to Zakat and Irth. This was done in a book entitled The Best of Arithmetic by Abd-al-Fattah-al-Dumyati. [20]
Retrieved from " https://en.wikipedia.org/?title=Arithmetic&oldid=1087484137"
Arithmetic Websites
Arithmetic Encyclopedia Articles |
Set Partition Fixed Size - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Combinatorics : Iterator : Set Partition Fixed Size
SetPartitionFixedSize(s, opts)
list(posint); size of each subset
(optional) equation(s) of the form option = value; specify options for the SetPartitionFixedSize command
The SetPartitionFixedSize command returns an iterator that generates fixed-size partitions of the set of integers starting at one. A partition of a set is a division of the set into disjoint subsets whose union is the set. A fixed-size partition fixes the sizes and number of the partitions.
The s parameter is a list of positive integers that specify the size of each subset of a partition.
The iterator output is a permutation of the integers from one to n, where n is the sum of the sizes of the subsets. Let
{S}_{k}=1+\left(\sum _{j=1}^{k-1}{s}_{j}\right)
, then the k-th subset corresponds to the indices
{S}_{k}..{S}_{k+1}-1
\mathrm{with}\left(\mathrm{Iterator}\right):
Partition the set of integers
{1,2,3,4,5}
into subsets of sizes 2, 1, and 2.
F≔\mathrm{SetPartitionFixedSize}\left([2,1,2]\right):
Print the vectors corresponding to the partitions. Indices 1 and 2 contain the first subset, index 3 the second, and indices 4 and 5 the third.
\mathrm{Print}\left(F,'\mathrm{showrank}'\right):
\mathrm{Number}\left(F\right)
\textcolor[rgb]{0,0,1}{15}
Knuth, Donald Ervin. The Art of Computer Programming, volume 4, fascicle 3; generating all combinations and partitions, sec. 7.2.1.5, generating all set partitions, fixed sizes, exercise 6, pp. 78 and 125.
The Iterator[SetPartitionFixedSize] command was introduced in Maple 2016. |
Solve Equations Numerically - MATLAB & Simulink - MathWorks Australia
Find All Roots of a Polynomial Function
Find Zeros of a Nonpolynomial Function Using Search Ranges and Starting Points
Obtain Solutions to Arbitrary Precision
Solve Multivariate Equations Using Search Ranges
Symbolic Math Toolbox™ offers both numeric and symbolic equation solvers. For a comparison of numeric and symbolic solvers, see Select Numeric or Symbolic Solver. An equation or a system of equations can have multiple solutions. To find these solutions numerically, use the function vpasolve. For polynomial equations, vpasolve returns all solutions. For nonpolynomial equations, vpasolve returns the first solution it finds. These examples show you how to use vpasolve to find solutions to both polynomial and nonpolynomial equations, and how to obtain these solutions to arbitrary precision.
Use vpasolve to find all the solutions to the function
f\left(x\right)=6{x}^{7}-2{x}^{6}+3{x}^{3}-8
sol = vpasolve(f)
\left(\begin{array}{c}1.0240240759053702941448316563337\\ -0.88080620051762149639205672298326+0.50434058840127584376331806592405 \mathrm{i}\\ -0.88080620051762149639205672298326-0.50434058840127584376331806592405 \mathrm{i}\\ -0.22974795226118163963098570610724+0.96774615576744031073999010695171 \mathrm{i}\\ -0.22974795226118163963098570610724-0.96774615576744031073999010695171 \mathrm{i}\\ 0.7652087814927846556172932675903+0.83187331431049713218367239317121 \mathrm{i}\\ 0.7652087814927846556172932675903-0.83187331431049713218367239317121 \mathrm{i}\end{array}\right)
vpasolve returns seven roots of the function, as expected, because the function is a polynomial of degree seven.
A plot of the function
f\left(x\right)={e}^{\left(x/7\right)}\mathrm{cos}\left(2x\right)
reveals periodic zeros, with increasing slopes at the zero points as
x
h = fplot(exp(x/7)*cos(2*x),[-2 25]);
Use vpasolve to find a zero of the function f. Note that vpasolve returns only one solution of a nonpolynomial equation, even if multiple solutions exist. On repeated calls, vpasolve returns the same result.
f = exp(x/7)*cos(2*x);
vpasolve(f,x)
-7.0685834705770347865409476123789
-7.0685834705770347865409476123789
-7.0685834705770347865409476123789
To find multiple solutions, set the option 'Random' to true. This makes vpasolve choose starting points randomly. For information on the algorithm that chooses random starting points, see Algorithms on the vpasolve page.
vpasolve(f,x,'Random',true)
-226.98006922186256147892598444194
98.174770424681038701957605727484
52.621676947629036744249276669932
To find a zero close to
x=10
, set the starting point to 10.
vpasolve(f,x,10)
10.210176124166828025003590995658
x=1000
, set the starting point to 1000.
vpasolve(f,x,1000)
999.8118620049516981407362567287
To find a zero in the range
15\le x\le 25
, set the search range to [15 25].
vpasolve(f,x,[15 25])
21.205750411731104359622842837137
To find multiple zeros in the range [15 25], you cannot call vpasolve repeatedly because it returns the same result on each call, as previously shown. Instead, set the search range and set 'Random' to true.
vpasolve(f,x,[15 25],'Random',true)
21.205750411731104359622842837137
21.205750411731104359622842837137
16.493361431346414501928877762217
Because 'Random' selects starting points randomly, the same solution might be found on successive calls.
Find All Zeros in a Specified Search Range
Create a function findzeros to systematically find all zeros for f in a given search range, within a specified error tolerance. The function starts with the input search range and calls vpasolve to find a zero. Then, it splits the search range into two around the zero value and recursively calls itself with the new search ranges as inputs to find more zeros.
The function is explained section by section here.
Declare the function with the three inputs and one output. The first input is the function, the second input is the range, and the optional third input allows you to specify the error between a zero and the higher and lower bounds generated from it.
function sol = findzeros(f,range,err)
If you do not specify the optional argument for error tolerance, findzeros sets err to 0.001.
err = 1e-3;
Find a zero in the search range using vpasolve.
sol = vpasolve(f,range);
If vpasolve does not find a zero, exit.
if(isempty(sol))
If vpasolve finds a zero, split the search range into two search ranges above and below the zero.
lowLimit = sol-err;
highLimit = sol+err;
Call findzeros with the lower search range. If findzeros returns zeros, copy the values into the solution array and sort them.
temp = findzeros(f,[range(1) lowLimit],1);
if ~isempty(temp)
sol = sort([sol temp]);
Call findzeros with the higher search range. If findzeros returns zeros, copy the values into the solution array and sort them.
temp = findzeros(f,[highLimit range(2)],1);
The entire function findzeros is as follows. Save this function as findzeros.m in the current folder.
Call findzeros with search range [15 25] to find all zeros in that range for f(x) = exp(x/7)*cos(2*x), within the default error tolerance.
f(x) = exp(x/7)*cos(2*x);
sol = findzeros(f,[15 25])'
\left(\begin{array}{c}16.493361431346414501928877762217\\ 18.064157758141311121160199453857\\ 19.634954084936207740391521145497\\ 21.205750411731104359622842837137\\ 22.776546738526000978854164528776\\ 24.347343065320897598085486220416\end{array}\right)
Use digits to set the precision of the solutions returned by vpasolve. By default, vpasolve returns solutions to a precision of 32 significant figures.
vpasolve(f)
-7.0685834705770347865409476123789
Use digits to increase the precision to 64 significant figures. When modifying digits, ensure that you save its current value so that you can restore it.
digitsOld = digits;
-7.068583470577034786540947612378881489443631148593988097193625333
Next, change the precision of the solutions to 16 significant figures.
\begin{array}{l}\mathit{z}=10\left(\mathrm{cos}\left(\mathit{x}\right)+\mathrm{cos}\left(\mathit{y}\right)\right)\\ \mathit{z}=\mathit{x}+{\mathit{y}}^{2}-0.1{\mathit{x}}^{2}\mathit{y}\\ \mathit{x}+\mathit{y}-2.7=0\end{array}
A plot of the equations for
0\le x\le 2.5
0\le x\le 2.5
shows that the three surfaces intersect in two points. To better visualize the plot, use view. To scale the colormap values, use caxis.
eqn1 = z == 10*(cos(x) + cos(y));
eqn2 = z == x+y^2-0.1*x^2*y;
eqn3 = x+y-2.7 == 0;
equations = [eqn1 eqn2 eqn3];
fimplicit3(equations)
axis([0 2.5 0 2.5 -20 10])
title('System of Multivariate Equations')
Use vpasolve to find a point where the surfaces intersect. The function vpasolve returns a structure. To access the x-, y-, and z-values of the solution, index into the structure.
sol = vpasolve(equations);
[sol.x sol.y sol.z]
\left(\begin{array}{ccc}2.369747722454798& 0.3302522775452021& 2.293354376823228\end{array}\right)
To search a region of the solution space, specify search ranges for the variables. If you specify the ranges
0\le x\le 1.5
1.5\le y\le 2.5
, then vpasolve function searches the bounded area shown.
Use vpasolve to find a solution for this search range. To omit a search range for
z
, set the third search range to [NaN NaN].
range = [0 1.5; 1.5 2.5; NaN NaN];
sol = vpasolve(equations, vars, range);
\left(\begin{array}{ccc}0.9106266172563336& 1.789373382743666& 3.964101572135625\end{array}\right)
To find multiple solutions, set the 'Random' option to true. This makes vpasolve use random starting points on successive runs. The 'Random' option can be used in conjunction with search ranges to make vpasolve use random starting points within a search range. Because 'Random' selects starting points randomly, the same solution might be found on successive calls. Call vpasolve repeatedly to ensure you find both solutions.
clear sol
range = [0 3; 0 3; NaN NaN];
temp = vpasolve(equations,vars,range,'Random',true);
sol(k,1) = temp.x;
sol(k,2) = temp.y;
sol(k,3) = temp.z;
\left(\begin{array}{ccc}2.369747722454798& 0.3302522775452021& 2.293354376823228\\ 2.369747722454798& 0.3302522775452021& 2.293354376823228\\ 2.369747722454798& 0.330252277545202& 2.293354376823228\\ 0.9106266172563336& 1.789373382743666& 3.964101572135625\\ 0.9106266172563336& 1.789373382743666& 3.964101572135625\end{array}\right)
Plot the equations. Superimpose the solutions as a scatter plot of points with yellow X markers using scatter3. To better visualize the plot, make two of the surfaces transparent using alpha. Scale the colormap to the plot values using caxis, and change the perspective using view.
vpasolve finds solutions at the intersection of the surfaces formed by the equations as shown.
h = fimplicit3(equations);
h(2).FaceAlpha = 0;
scatter3(sol(:,1),sol(:,2),sol(:,3),600,'yellow','X','LineWidth',2)
title('Randomly Found Solutions in Specified Search Range')
cz = ax.Children; |
DefineExternal - Maple Help
Home : Support : Online Help : Connectivity : Calling External Routines : ExternalCalling : DefineExternal
create a link to an external function
DefineExternal( fn, extlib )
DefineExternal( fn, extlib, cright )
string or name; denotes the name of the wrapper function to link
string or name; denotes the name of the external library containing the wrapper function
(optional) string; denotes the copyright placed on the returned Maple procedure.
The DefineExternal(fn, extlib) command calls define_external with the MAPLE option and saves the result in a remember table. The procedure used to prepare the invocation of the external routine fn is returned, optionally with the specified copyright statement cright.
Note: Saving the result of DefineExternal in a remember table is essential for the efficient operation of the external linking. Without it, the external communications must be set up every time an external routine is entered.
\mathrm{with}\left(\mathrm{ExternalCalling}\right):
\mathrm{extlib}≔\mathrm{ExternalLibraryName}\left("mstring"\right)
\textcolor[rgb]{0,0,1}{\mathrm{extlib}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{"libmstring.so"}
\mathrm{extcall}≔\mathrm{DefineExternal}\left('\mathrm{mstring_uppercase}',\mathrm{extlib}\right)
\textcolor[rgb]{0,0,1}{\mathrm{extcall}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathbf{proc}}\left(\textcolor[rgb]{0,0,1}{}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{option}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathrm{call_external}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{define_external}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{mstring_uppercase}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{MAPLE}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LIB}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"libmstring.so"}\right)\textcolor[rgb]{0,0,1}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathrm{call_external}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{139621820277232}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{args}}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end proc}}
\mathrm{extcall}≔\mathrm{DefineExternal}\left('\mathrm{mstring_uppercase}',\mathrm{extlib},"Copyright \left(c\right) 2001, ..."\right)
\textcolor[rgb]{0,0,1}{\mathrm{extcall}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathbf{proc}}\left(\textcolor[rgb]{0,0,1}{}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{option}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathrm{call_external}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{define_external}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{mstring_uppercase}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{MAPLE}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LIB}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"libmstring.so"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{COPYRIGHT}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Copyright \left(c\right) 2001, ..."}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{copyright}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Copyright \left(c\right) 2001, ..."}\textcolor[rgb]{0,0,1}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathrm{call_external}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{139621820277232}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{args}}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end proc}}
ExternalCalling[ExternalLibraryName] |
Demonstration of Density Matrix Exponentiation Using a Superconducting Quantum Processor
M. Kjaergaard, M. E. Schwartz, A. Greene, G. O. Samach, A. Bengtsson, M. O’Keeffe, C. M. McNally, J. Braumüller, D. K. Kim, P. Krantz, M. Marvian, A. Melville, B. M. Niedzielski, Y. Sung, R. Winik, J. Yoder, D. Rosenberg, K. Obenland, S. Lloyd, T. P. Orlando, I. Marvian, S. Gustavsson, W. D. Oliver
Density matrix exponentiation (DME) is a general technique for using a quantum state
\rho
to enact the quantum operation
{e}^{-i\rho \theta }
on a target system. It was first proposed in the context of quantum machine learning, but has since been shown to have broad applications in quantum metrology and computation. No experimental demonstration of DME has been performed thus far due to its demanding circuit depths and the need to efficiently generate multiple identical copies of
\rho
during the finite lifetime of the target system. In this work, we describe the first demonstration of the DME algorithm, which we accomplish using a superconducting quantum processor. Our demonstration relies on a 99.7% fidelity controlled-phase gate implemented using two tunable superconducting transmon qubits. We achieve a fidelity surpassing 90% at circuit depths exceeding 70 when comparing the output of the circuit executed on our quantum processor to a simulation assuming perfect operations and measurements. |
Calculate incidence and airspeed - Simulink
Calculate incidence and airspeed
The Incidence & Airspeed block supports the 3DoF equations of motion model by calculating the angle between the velocity vector and the body, and also the total airspeed from the velocity components in the body-fixed coordinate frame.
\begin{array}{l}\alpha =\text{atan}\left(\frac{w}{u}\right)\\ V=\sqrt{{u}^{2}+{w}^{2}}\end{array}
U, w — Velocity
Velocity of the body, specified as a two-element vector, resolved into the body-fixed coordinate frame. |
Von_Mises_yield_criterion Knowpia
The maximum distortion criterion (also von Mises yield criterion[1]) states that yielding of a ductile material begins when the second invariant of deviatoric stress
{\displaystyle J_{2}}
reaches a critical value.[2] It is a part of plasticity theory that mostly applies to ductile materials, such as some metals. Prior to yield, material response can be assumed to be of a nonlinear elastic, viscoelastic, or linear elastic behavior.
In materials science and engineering von Mises yield criterion is also formulated in terms of the von Mises stress or equivalent tensile stress,
{\displaystyle \sigma _{\text{v}}}
. This is a scalar value of stress that can be computed from the Cauchy stress tensor. In this case, a material is said to start yielding when the von Mises stress reaches a value known as yield strength,
{\displaystyle \sigma _{\text{y}}}
. The von Mises stress is used to predict yielding of materials under complex loading from the results of uniaxial tensile tests. The von Mises stress satisfies the property where two stress states with equal distortion energy have an equal von Mises stress.
Because the von Mises yield criterion is independent of the first stress invariant,
{\displaystyle I_{1}}
, it is applicable for the analysis of plastic deformation for ductile materials such as metals, as onset of yield for these materials does not depend on the hydrostatic component of the stress tensor.
Although it has been believed it was formulated by James Clerk Maxwell in 1865, Maxwell only described the general conditions in a letter to William Thomson (Lord Kelvin).[3] Richard Edler von Mises rigorously formulated it in 1913.[2][4] Tytus Maksymilian Huber (1904), in a paper written in Polish, anticipated to some extent this criterion by properly relying on the distortion strain energy, not on the total strain energy as his predecessors.[5][6][7] Heinrich Hencky formulated the same criterion as von Mises independently in 1924.[8] For the above reasons this criterion is also referred to as the "Maxwell–Huber–Hencky–von Mises theory".
The von Mises yield surfaces in principal stress coordinates circumscribes a cylinder with radius
{\textstyle {\sqrt {\frac {2}{3}}}\sigma _{y}}
around the hydrostatic axis. Also shown is Tresca's hexagonal yield surface.
Mathematically the von Mises yield criterion is expressed as:
{\displaystyle J_{2}=k^{2}\,\!}
{\displaystyle k}
is yield stress of the material in pure shear. As shown later in this article, at the onset of yielding, the magnitude of the shear yield stress in pure shear is √3 times lower than the tensile yield stress in the case of simple tension. Thus, we have:
{\displaystyle k={\frac {\sigma _{y}}{\sqrt {3}}}}
{\displaystyle \sigma _{y}}
is tensile yield strength of the material. If we set the von Mises stress equal to the yield strength and combine the above equations, the von Mises yield criterion is written as:
{\displaystyle \sigma _{v}=\sigma _{y}={\sqrt {3J_{2}}}}
{\displaystyle \sigma _{v}^{2}=3J_{2}=3k^{2}}
{\displaystyle J_{2}}
with the Cauchy stress tensor components, we get
{\displaystyle \sigma _{\text{v}}^{2}={\frac {1}{2}}\left[(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+(\sigma _{33}-\sigma _{11})^{2}+6\left(\sigma _{23}^{2}+\sigma _{31}^{2}+\sigma _{12}^{2}\right)\right]={\frac {3}{2}}s_{ij}s_{ij}}
{\displaystyle s}
is called deviatoric stress. This equation defines the yield surface as a circular cylinder (See Figure) whose yield curve, or intersection with the deviatoric plane, is a circle with radius
{\displaystyle {\sqrt {2}}k}
{\textstyle {\sqrt {\frac {2}{3}}}\sigma _{y}}
. This implies that the yield condition is independent of hydrostatic stresses.
Reduced von Mises equation for different stress conditionsEdit
Von Mises yield criterion in 2D (planar) loading conditions: if stress in the third dimension is zero (
{\displaystyle \sigma _{3}=0}
), no yielding is predicted to occur for stress coordinates
{\displaystyle \sigma _{1},\sigma _{2}}
within the red area. Because Tresca's criterion for yielding is within the red area, Von Mises' criterion is more lax.
Uniaxial (1D) stressEdit
In the case of uniaxial stress or simple tension,
{\displaystyle \sigma _{1}\neq 0,\sigma _{3}=\sigma _{2}=0}
, the von Mises criterion simply reduces to
{\displaystyle \sigma _{1}=\sigma _{\text{y}}\,\!}
which means the material starts to yield when
{\displaystyle \sigma _{1}}
reaches the yield strength of the material
{\displaystyle \sigma _{\text{y}}}
, in agreement with the definition of tensile (or compressive) yield strength.
Multi-axial (2D or 3D) stressEdit
An equivalent tensile stress or equivalent von-Mises stress,
{\displaystyle \sigma _{\text{v}}}
is used to predict yielding of materials under multiaxial loading conditions using results from simple uniaxial tensile tests. Thus, we define
{\displaystyle {\begin{aligned}\sigma _{\text{v}}&={\sqrt {3J_{2}}}\\&={\sqrt {\frac {(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+\left(\sigma _{33}-\sigma _{11})^{2}+6(\sigma _{12}^{2}+\sigma _{23}^{2}+\sigma _{31}^{2}\right)}{2}}}\\&={\sqrt {\frac {(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{3}-\sigma _{1})^{2}}{2}}}\\&={\sqrt {{\frac {3}{2}}s_{ij}s_{ij}}}\end{aligned}}\,\!}
{\displaystyle s_{ij}}
are components of stress deviator tensor
{\displaystyle {\boldsymbol {\sigma }}^{\text{dev}}}
{\displaystyle {\boldsymbol {\sigma }}^{\text{dev}}={\boldsymbol {\sigma }}-{\frac {\operatorname {tr} \left({\boldsymbol {\sigma }}\right)}{3}}\mathbf {I} \,\!}
In this case, yielding occurs when the equivalent stress,
{\displaystyle \sigma _{\text{v}}}
, reaches the yield strength of the material in simple tension,
{\displaystyle \sigma _{\text{y}}}
. As an example, the stress state of a steel beam in compression differs from the stress state of a steel axle under torsion, even if both specimens are of the same material. In view of the stress tensor, which fully describes the stress state, this difference manifests in six degrees of freedom, because the stress tensor has six independent components. Therefore, it is difficult to tell which of the two specimens is closer to the yield point or has even reached it. However, by means of the von Mises yield criterion, which depends solely on the value of the scalar von Mises stress, i.e., one degree of freedom, this comparison is straightforward: A larger von Mises value implies that the material is closer to the yield point.
In the case of pure shear stress,
{\displaystyle \sigma _{12}=\sigma _{21}\neq 0}
, while all other
{\displaystyle \sigma _{ij}=0}
, von Mises criterion becomes:
{\displaystyle \sigma _{12}=k={\frac {\sigma _{y}}{\sqrt {3}}}\,\!}
This means that, at the onset of yielding, the magnitude of the shear stress in pure shear is
{\displaystyle {\sqrt {3}}}
times lower than the yield stress in the case of simple tension. The von Mises yield criterion for pure shear stress, expressed in principal stresses, is
{\displaystyle (\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{1}-\sigma _{3})^{2}=2\sigma _{y}^{2}\,\!}
In the case of principal plane stress,
{\displaystyle \sigma _{3}=0}
{\displaystyle \sigma _{12}=\sigma _{23}=\sigma _{31}=0}
, the von Mises criterion becomes:
{\displaystyle \sigma _{1}^{2}-\sigma _{1}\sigma _{2}+\sigma _{2}^{2}=3k^{2}=\sigma _{y}^{2}\,\!}
This equation represents an ellipse in the plane
{\displaystyle \sigma _{1}-\sigma _{2}}
von Mises equations
General No restrictions
{\displaystyle \sigma _{\text{v}}={\sqrt {{\frac {1}{2}}\left[(\sigma _{11}-\sigma _{22})^{2}+(\sigma _{22}-\sigma _{33})^{2}+(\sigma _{33}-\sigma _{11})^{2}\right]+3\left(\sigma _{12}^{2}+\sigma _{23}^{2}+\sigma _{31}^{2}\right)}}}
Principal stresses
{\displaystyle \sigma _{12}=\sigma _{31}=\sigma _{23}=0\!}
{\displaystyle \sigma _{\text{v}}={\sqrt {{\frac {1}{2}}\left[(\sigma _{1}-\sigma _{2})^{2}+(\sigma _{2}-\sigma _{3})^{2}+(\sigma _{3}-\sigma _{1})^{2}\right]}}}
General plane stress
{\displaystyle {\begin{aligned}\sigma _{3}&=0\!\\\sigma _{31}&=\sigma _{23}=0\!\end{aligned}}}
{\displaystyle \sigma _{\text{v}}={\sqrt {\sigma _{11}^{2}-\sigma _{11}\sigma _{22}+\sigma _{22}^{2}+3\sigma _{12}^{2}}}\!}
Principal plane stress
{\displaystyle {\begin{aligned}\sigma _{3}&=0\!\\\sigma _{12}&=\sigma _{31}=\sigma _{23}=0\!\end{aligned}}}
{\displaystyle \sigma _{\text{v}}={\sqrt {\sigma _{1}^{2}-\sigma _{1}\sigma _{2}+\sigma _{2}^{2}}}\!}
Pure shear
{\displaystyle {\begin{aligned}\sigma _{1}&=\sigma _{2}=\sigma _{3}=0\!\\\sigma _{31}&=\sigma _{23}=0\!\end{aligned}}}
{\displaystyle \sigma _{\text{v}}={\sqrt {3}}|\sigma _{12}|\!}
Uniaxial
{\displaystyle {\begin{aligned}\sigma _{2}&=\sigma _{3}=0\!\\\sigma _{12}&=\sigma _{31}=\sigma _{23}=0\!\end{aligned}}}
{\displaystyle \sigma _{\text{v}}=\sigma _{1}\!}
Physical interpretation of the von Mises yield criterionEdit
Hencky (1924) offered a physical interpretation of von Mises criterion suggesting that yielding begins when the elastic energy of distortion reaches a critical value.[6] For this reason, the von Mises criterion is also known as the maximum distortion strain energy criterion. This comes from the relation between
{\displaystyle J_{2}}
and the elastic strain energy of distortion
{\displaystyle W_{\text{D}}}
{\displaystyle W_{\text{D}}={\frac {J_{2}}{2G}}\,\!}
with the elastic shear modulus
{\displaystyle G={\frac {E}{2(1+\nu )}}\,\!}
In 1937 [9] Arpad L. Nadai suggested that yielding begins when the octahedral shear stress reaches a critical value, i.e. the octahedral shear stress of the material at yield in simple tension. In this case, the von Mises yield criterion is also known as the maximum octahedral shear stress criterion in view of the direct proportionality that exists between
{\displaystyle J_{2}}
and the octahedral shear stress,
{\displaystyle \tau _{\text{oct}}}
, which by definition is
{\displaystyle \tau _{\text{oct}}={\sqrt {{\frac {2}{3}}J_{2}}}\,\!}
{\displaystyle \tau _{\text{oct}}={\frac {\sqrt {2}}{3}}\sigma _{\text{y}}\,\!}
Strain energy density consists of two components - volumetric or dialational and distortional. Volumetric component is responsible for change in volume without any change in shape. Distortional component is responsible for shear deformation or change in shape.
Practical engineering usage of the von Mises yield criterionEdit
As shown in the equations above, the use of the von Mises criterion as a yield criterion is only exactly applicable when homogeneous material properties are equal to
{\displaystyle {\frac {F_{sy}}{F_{ty}}}={\frac {1}{\sqrt {3}}}\approx 0.577\!}
Since no material will have this ratio precisely, in practice it is necessary to use engineering judgement to decide what failure theory is appropriate for a given material. Alternately, for use of the Tresca theory, the same ratio is defined as 1/2.
The yield margin of safety is written as
{\displaystyle MS_{\text{yld}}={\frac {F_{y}}{\sigma _{\text{v}}}}-1}
Although the given criterion is based on a yield phenomenon, extensive testing has shown that use of a "von Mises" stress is applicable at ultimate loading [10]
{\displaystyle MS_{\text{ult}}={\frac {F_{u}}{\sigma _{\text{v}}}}-1}
Huber's equation
Hoek–Brown failure criterion
^ "Von Mises Criterion (Maximum Distortion Energy Criterion)". Engineer's edge. Retrieved 8 February 2018.
^ a b von Mises, R. (1913). "Mechanik der festen Körper im plastisch-deformablen Zustand". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse. 1913 (1): 582–592.
^ "Deformation Theory of Plasticity, p. 151, Section 4.5.6". Retrieved 2017-06-11.
^ Ford (1963). Advanced Mechanics of Materials. London: Longmans.
^ Huber, M. T. (1904). "Właściwa praca odkształcenia jako miara wytezenia materiału". Czasopismo Techniczne. Lwów. 22. Translated as "Specific Work of Strain as a Measure of Material Effort". Archives of Mechanics. 56: 173–190. 2004.
^ a b Hill, R. (1950). The Mathematical Theory of Plasticity. Oxford: Clarendon Press.
^ Hencky, H. (1924). "Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannngen". Z. Angew. Math. Mech. 4: 323–334. doi:10.1002/zamm.19240040405.
^ S. M. A. Kazimi. (1982). Solid Mechanics. Tata McGraw-Hill. ISBN 0-07-451715-5
^ Stephen P. Timoshenko, Strength of Materials, Part I, 2nd ed., 1940 |
Conal Elliott » program derivation
Parallel tree scanning by composition
My last few blog posts have been on the theme of scans, and particularly on parallel scans. In Composable parallel scanning, I tackled parallel scanning in a very general setting. There are five simple building blocks out of which a vast assortment of data structures can be built, namely constant (no value), identity (one value), sum, product, and composition. The post defined parallel prefix and suffix scan for each of these five "functor combinators", in terms of the same scan operation on each of the component functors. Every functor built out of this basic set thus has a parallel scan. Functors defined more conventionally can be given scan implementations simply by converting to a composition of the basic set, scanning, and then back to the original functor. Moreover, I expect this implementation could be generated automatically, similarly to GHC’s DerivingFunctor extension.
Now I’d like to show two examples of parallel scan composition in terms of binary trees, namely the top-down and bottom-up variants of perfect binary leaf trees used in previous posts. (In previous posts, I used the terms "right-folded" and "left-folded" instead of "top-down" and "bottom-up".) The resulting two algorithms are expressed nearly identically, but have differ significantly in the work performed. The top-down version does
\Theta \left(n\phantom{\rule{0.167em}{0ex}}\mathrm{log}\phantom{\rule{0.167em}{0ex}}n\right)
work, while the bottom-up version does only
\Theta \left(n\right)
, and thus the latter algorithm is work-efficient, while the former is not. Moreover, with a very simple optimization, the bottom-up tree algorithm corresponds closely to Guy Blelloch’s parallel prefix scan for arrays, given in Programming parallel algorithms. I’m delighted with this result, as I had been wondering how to think about Guy’s algorithm.
2011-05-31: Added Scan and Applicative instances for T2 and T4.
Continue reading ‘Parallel tree scanning by composition’ »
My last few blog posts have been on the theme of scans, and particularly on parallel scans. In Composable parallel scanning, I tackled parallel scanning in a very general setting....
Tags: functor, program derivation, scan | 7 Comments
Deriving parallel tree scans
The post Deriving list scans explored folds and scans on lists and showed how the usual, efficient scan implementations can be derived from simpler specifications.
Let’s see now how to apply the same techniques to scans over trees.
This new post is one of a series leading toward algorithms optimized for execution on massively parallel, consumer hardware, using CUDA or OpenCL.
2011-03-01: Added clarification about "∅" and "(⊕)".
2011-03-23: corrected "linear-time" to "linear-work" in two places.
Continue reading ‘Deriving parallel tree scans’ »
The post Deriving list scans explored folds and scans on lists and showed how the usual, efficient scan implementations can be derived from simpler specifications. Let’s see now how to...
Tags: program derivation, scan | 5 Comments
Deriving list scans
I’ve been playing with deriving efficient parallel, imperative implementations of "prefix sum" or more generally "left scan". Following posts will explore the parallel & imperative derivations, but as a warm-up, I’ll tackle the functional & sequential case here.
You’re probably familiar with the higher-order functions for left and right "fold". The current documentation says:
foldl f z [x1, x2, ⋯, xn] ≡ (⋯((z `f` x1) `f` x2) `f`⋯) `f` xn
foldr f z [x1, x2, ⋯, xn] ≡ x1 `f` (x2 `f` ⋯ (xn `f` z)⋯)
And here are typical definitions:
foldl ∷ (b → a → b) → b → [a] → b
foldl f z (x:xs) = foldl f (z `f` x) xs
foldr ∷ (a → b → b) → b → [a] → b
foldr f z (x:xs) = x `f` foldr f z xs
Notice that foldl builds up its result one step at a time and reveals it all at once, in the end. The whole result value is locked up until the entire input list has been traversed. In contrast, foldr starts revealing information right away, and so works well with infinite lists. Like foldl, foldr also yields only a final value.
Sometimes it’s handy to also get to all of the intermediate steps. Doing so takes us beyond the land of folds to the kingdom of scans.
The scanl and scanr functions correspond to foldl and foldr but produce all intermediate accumulations, not just the final one.
scanl ∷ (b → a → b) → b → [a] → [b]
scanl f z [x1, x2, ⋯ ] ≡ [z, z `f` x1, (z `f` x1) `f` x2, ⋯]
scanr ∷ (a → b → b) → b → [a] → [b]
scanr f z [⋯, xn_1, xn] ≡ [⋯, xn_1 `f` (xn `f` z), xn `f` z, z]
As you might expect, the last value is the complete left fold, and the first value in the scan is the complete right fold:
last (scanl f z xs) ≡ foldl f z xs
head (scanr f z xs) ≡ foldr f z xs
last ∘ scanl f z ≡ foldl f z
head ∘ scanr f z ≡ foldr f z
The standard scan definitions are trickier than the fold definitions:
scanl f z ls = z : (case ls of
[] → []
x:xs → scanl f (z `f` x) xs)
scanr _ z [] = [z]
scanr f z (x:xs) = (x `f` q) : qs
where qs@(q:_) = scanr f z xs
Every time I encounter these definitions, I have to walk through it again to see what’s going on. I finally sat down to figure out how these tricky definitions might emerge from simpler specifications. In other words, how to derive these definitions systematically from simpler but less efficient definitions.
Most likely, these derivations have been done before, but I learned something from the effort, and I hope you do, too.
Continue reading ‘Deriving list scans’ »
I’ve been playing with deriving efficient parallel, imperative implementations of "prefix sum" or more generally "left scan". Following posts will explore the parallel & imperative derivations, but as a warm-up,...
Tags: fold, program derivation, scan | 5 Comments |
Citations - Curvenote Docs
You can easily add citations to your Curvenote articles and have quick access to them while you are typing. There are currently two ways to add references:
From a DOI: Simply search the document or article DOI, and add it to your article with one-click!
From a BibTex file: Export from your reference manager (e.g. Zotero, Mendeley, etc.) as a BibTex file, and drag and drop into Curvenote. If you want easy access to a reference manager, without downloading anything we recommend https://zbib.org/.
Once your citations have been added to your project, you can easily access them through the /cite command, and can click on the citation to see all the details - and you can click on it to navigate to the web article or document. For example, try clicking on this citation Cockett et al., 2015 .
Table 1:Example of rendered citations, try clicking on any of the citations!
Heinen (2014)
Bartkowski & Bartke, 2018
Bartkowski & Bartke (2018)
Winter et al., 2018
Winter et al. (2018)
#Adding citations through DOIs
A digital object identifier (DOI) is a unique string that’s used to permanently identify an article or document on the web. If you are citing a paper, it will have a searchable DOI you can add to your Curvenote citation library.
To add a DOI:
Locate the DOI on the article or document you want to cite, and copy it to your clipboard
The DOI is usually included with the article content. However, if you cannot find the DOI you can use the ‘Search Metadata’ option on crossref.org.
Place your cursor in the area you want to add the citation
Access the citation menu by using the /cite command
Navigate to ADD DOI
Enter the DOI in the search bar
If the appropriate article or document is shown, select CREATE CITATION
Your citation will now be added to your citation library for that project, and can be inserted into your articles without having to re-add the DOI.
#Adding citations from BibTex files
The bibtex (*.bib) file format is a storage format for citations, it is commonly used with
\LaTeX
, and can be exported from most reference managers.
Figure 2:Import your references from a *.bib file, and easily update your references if you need to!
#Updating Citations
If you need to update your citation (e.g. fixing an author name, or adding a date), just re-export the bibtex from your reference manager, ensuring that the citation key for the reference is the same. Then just re-upload, the bibtex and all of the included reference will be updated💥.
In Curvenote, the references are stored as a Reference block, when you upload through a bibtex file it is given a unique name that is derived from the citation key you provide. The block name (used in the URL and must be unique for a project) is prefixed with ref- and is lowercased. This means the bibtex references are like Cockett2015-Elsevier becomes ref-cockett2015-elsevier. You can also use these keys to quickly look up a citation through the [[cite: command or the reference search panel.
#Inserting multiple citations
To add multiple citations:
Access the references menu by using the /reference command
Filter using the Search, and use the checkboxes to select the citation you would like to add
Select INSERT CITATIONS
The citations will now be added!
#Rearranging Citations
By default when citations are added through the [[cite: command, they are added as individual citations, these can be wrapped in brackets using the [] icon in the toolbar when the citation is selected. You can also drag citations into and out of citation-groups. In latex, these correspond to \citep{} and render slightly differently than if you use your own brackets.
Figure 4:Use the [] button to toggle the brackets around the citation. You can drag citations into different groups, or rearrange them in order.
Note that if you add the citation through the /reference command, then multiple citations will be added to a single group in one go.
Cockett, R., Kang, S., Heagy, L. J., Pidlisecky, A., & Oldenburg, D. W. (2015). SimPEG: An open source framework for simulation and gradient based parameter estimation in geophysical applications. Computers & Geosciences, 85, 142–154. 10.1016/j.cageo.2015.09.015
Heinen, M. (2014). Compensation in Root Water Uptake Models Combined with Three-Dimensional Root Length Density Distribution. Vadose Zone Journal, 13(2), vzj2013.08.0149. 10.2136/vzj2013.08.0149
Bartkowski, B., & Bartke, S. (2018). Leverage Points for Governing Agricultural Soils: A Review of Empirical Studies of European Farmers’ Decision-Making. SUSTAINABILITY, 10(9). 10.3390/su10093179
Winter, S., Bauer, T., Strauss, P., Kratschmer, S., Paredes, D., Popescu, D., Landa, B., Guzmán, G., Gómez, J. A., Guernion, M., Zaller, J. G., & Batáry, P. (2018). Effects of vegetation management intensity on biodiversity and ecosystem services in vineyards: A meta-analysis. Journal of Applied Ecology, 55(5), 2484–2495. 10.1111/1365-2664.13124
European Commission. (2020). Farm to Fork Strategy. |
Unit Vectors | Brilliant Math & Science Wiki
Pranshu Gaba, Margaret Zheng, Nihar Mahajan, and
Vector quantities have a direction and a magnitude. However, sometimes one is interested in only the direction of the vector and not the magnitude. In such cases, for convenience, vectors are often "normalized" to be of unit length. These unit vectors are commonly used to indicate direction, with a scalar coefficient providing the magnitude. A vector decomposition can then be written as a sum of unit vectors and scalar coefficients.
\vec{V}
, one might consider the problem of finding the vector parallel to
\vec{V}
with unit length. In other words, one might want to find some "scale factor"
a
\left\|a \vec{V} \right\| = 1
\left\|a \vec{V}\right\| = a \left\|\vec{V}\right\|
, it follows trivially that
a = \frac1{\left\|\vec{V}\right\|}.
Thus, given a vector
\vec{V}
, the unit vector
\hat{V}
\hat{V} = \frac{\vec{V}}{\left\|\vec{V}\right\|}
has unit length and is parallel to
\vec{V}
. A unit vector is frequently (though not always) written with "hat" symbol to indicate that it is of unit length.
Find the unit vector
\hat{D}
\vec{D} = \langle 4, 3 \rangle
We have (using the Pythagorean theorem)
|\vec{D}| = 5 \implies \hat{D} = \frac{\vec{D}}{5} = \left \langle \frac{4}{5}, \frac{3}{5} \right \rangle.\ _\square
\hat{A}
\vec{A} = \langle -12, 5 \rangle
|\vec{A}| = 13 \implies \hat{A} = \frac{\vec{A}}{13} =\left \langle -\frac{12}{13}, \frac{5}{13} \right \rangle.\ _\square
Unit vectors are often used in the decomposition of a vector into orthogonal components. For instance, one can define the unit vectors that point in each of the positive coordinate axes as follows:
\begin{aligned} \hat{x} &= \left \langle 1, 0, 0 \right \rangle \\ \hat{y} &= \left \langle 0, 1, 0 \right \rangle \\ \hat{z} &= \left \langle 0, 0, 1 \right \rangle. \end{aligned}
\vec{V} = \left \langle v_x, v_y, v_z \right \rangle
\vec{V} = v_x \hat{x} + v_y \hat{y} + v_z \hat{z}.
\hat{x}
\hat{y}
\hat{z}
\hat{i} = \hat{x}
\hat{j} = \hat{y}
\hat{k} = \hat{z}
\hat{x} \cdot \hat{x} = \hat{y} \cdot \hat{y} = \hat{z} \cdot \hat{z} = 1
\hat{x} \cdot \hat{y} = \hat{y} \cdot \hat{z} = \hat{x} \cdot \hat{z} = 0
\vec{V}= (v_x, v_y, v_z)
\vec{W} = (w_x, w_y, w_z)
add, we simply have
\vec{V} +\vec{W}= (v_x \hat{x} + v_y \hat{y} + v_z \hat{z}) + (w_x \hat{x} + w_y \hat{y} + w_z \hat{z}) = (v_x + w_x) \hat{x} + (v_y + w_y) \hat{y} + (v_z + w_z) \hat{z},
(v_x + w_x, v_y + w_y, v_z + w_z)
\vec{V} \cdot \vec{W}
\begin{aligned} \vec{V} \cdot \vec{W} &= (v_x \hat{x} + v_y \hat{y} + v_z \hat{z}) \cdot (w_x \hat{x} + w_y \hat{y} + w_z \hat{z}) \\ &= v_x w_x \hat{x} \cdot \hat{x} + v_y w_y \hat{y} \cdot \hat{y} + v_z w_z \hat{z} \cdot \hat{z} \\ &= v_x w_x + v_y w_y + v_z w_z, \end{aligned}
\langle 6, 8 \rangle
be expressed as the sum of nine unit vectors?
i+j+k
2i+3j+k
3i+j+k
4k
Which vector is the longest?
5 i + j - 2 k
\frac{5}{30}i+\frac{1}{30}j-\frac{2}{30} k
\frac{5}{\sqrt{30}}i+\frac{1}{\sqrt{30}}j-\frac{2}{\sqrt{30}} k
5 i +j - 2 k.
Cite as: Unit Vectors. Brilliant.org. Retrieved from https://brilliant.org/wiki/unit-vectors/ |
Logic Overview | Brilliant Math & Science Wiki
Hua Zhi Vee, Zandra Vinegar, and Jimin Khim contributed
No matter where you're coming from, you can find logic problems on Brilliant that will put your deductive reasoning ability to the test and reinforce your mathematical skills. Whether you're looking for quick practice problems that strengthen your reasoning and IQ, for school math and Olympiad competition topics, or for advanced, open-ended challenges, we have something here for you. The collection of Brilliant problems and articles is large and growing.
The Best of Logic on Brilliant
Logic Topic Area Map
Pre-Collegiate Logic Content
Within Logic, you can learn about wide-ranging topics such as the following:
Arithmetic Puzzles - Operator Search Arithmetic Puzzles - Fill in the Blanks Cryptogram - Problem Solving Truth-tellers and Liars Propositional Logic Word Problems Logical Puzzles
\hspace{15mm}
Logical Reasoning Arithmetic Puzzles
Grid Puzzles Games
The most popular pre-collegiate Logic topics on Brilliant are as follows:
General Term Pattern Recognition Shifty Shapes Lively Length and Area Simple Sequences Even and Odd Numbers
Cite as: Logic Overview. Brilliant.org. Retrieved from https://brilliant.org/wiki/learn-and-practice-logic-on-brilliant/ |
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