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The inverse problem of the heat equation with periodic boundary and integral overdetermination conditions | Journal of Inequalities and Applications | Full Text
Fatma Kanca1
In this paper the inverse problem of finding the time-dependent coefficient of heat capacity together with the solution of a heat equation with periodic boundary and integral overdetermination conditions is considered. The conditions for the existence and uniqueness of a classical solution of the problem under consideration are established. A numerical example using the Crank-Nicolson finite-difference scheme combined with the iteration method is presented.
In this paper we consider an inverse problem of simultaneously finding the unknown coefficient
p\left(t\right)
and the temperature distribution
u\left(x,t\right)
{u}_{t}={u}_{xx}-p\left(t\right)u+F\left(x,t\right),\phantom{\rule{1em}{0ex}}0<x<1,0<t\le T,
u\left(x,0\right)=\phi \left(x\right),\phantom{\rule{1em}{0ex}}0\le x\le 1,
u\left(0,t\right)=u\left(1,t\right),\phantom{\rule{2em}{0ex}}{u}_{x}\left(0,t\right)={u}_{x}\left(1,t\right),\phantom{\rule{1em}{0ex}}0\le t\le T,
and the overdetermination condition
{\int }_{0}^{1}xu\left(x,t\right)\phantom{\rule{0.2em}{0ex}}dx=E\left(t\right),\phantom{\rule{1em}{0ex}}0\le t\le T.
Denote the domain
{Q}_{T}
{Q}_{T}=\left\{\left(x,t\right):0<x<1,0<t\le T\right\}.
Definition 1 The pair
\left\{p\left(t\right),u\left(x,t\right)\right\}
from the class
C\left[0,T\right]×{C}^{2,1}\left({D}_{T}\right)\cap {C}^{1,0}\left({\overline{D}}_{T}\right)
, for which conditions (1)-(4) are satisfied and
p\left(t\right)\ge 0
\left[0,T\right]
, is called a classical solution of the inverse problem (1)-(4).
The parameter identification in a parabolic differential equation from the data of integral overdetermination condition plays an important role in engineering and physics [1–6]. This integral condition in parabolic problems is also called heat moments are analyzed in [4].
Boundary value problems for parabolic equations in which one or two local classical conditions are replaced by heat moments [5–9]. In [9], a physical-mechanical interpretation of the integral conditions was also given. Various statements of inverse problems on determination of this coefficient in a one-dimensional heat equation were studied in [1–3, 5, 6, 10, 11]. In the papers [1, 3, 5], the coefficient is determined from heat moment. Boundary value problems and inverse problems for parabolic equations with periodic boundary conditions are investigated in [10, 12].
In the present work, one heat moment is used with a periodic boundary condition for the determination of a source coefficient.The existence and uniqueness of the classical solution of the problem (1)-(4) is reduced to fixed point principles by applying the Fourier method.
The paper organized as follows. In Section 2, the existence and uniqueness of the solution of the inverse problem (1)-(4) is proved by using the Fourier method. In Section 3, the continuous dependence upon the data of the inverse problem is shown. In Section 4, the numerical procedure for the solution of the inverse problem using the Crank-Nicolson scheme combined with the iteration method is given. Finally, in Section 5, numerical experiments are presented and discussed.
2 Existence and uniqueness of the solution of the inverse problem
We have the following assumptions on the data of the problem (1)-(4).
E\left(t\right)\in {C}^{1}\left[0,T\right]
E\left(t\right)<0
{E}^{\mathrm{\prime }}\left(t\right)\ge 0
\mathrm{\forall }t\in \left[0,T\right]
\phi \left(x\right)\in {C}^{4}\left[0,1\right]
\phi \left(0\right)=\phi \left(1\right)
{\phi }^{\mathrm{\prime }}\left(0\right)={\phi }^{\mathrm{\prime }}\left(1\right)
{\phi }^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)={\phi }^{\mathrm{\prime }\mathrm{\prime }}\left(1\right)
{\int }_{0}^{1}x\phi \left(x\right)\phantom{\rule{0.2em}{0ex}}dx=E\left(0\right)
{\phi }_{n}\ge 0
n=1,2,\dots
F\left(x,t\right)\in C\left({\overline{D}}_{T}\right)
F\left(x,t\right)\in {C}^{4}\left[0,1\right]
t\in \left[0,T\right]
F\left(0,t\right)=F\left(1,t\right)
{F}_{x}\left(0,t\right)={F}_{x}\left(1,t\right)
{F}_{xx}\left(0,t\right)={F}_{xx}\left(1,t\right)
{F}_{n}\left(t\right)>0
n=1,2,\dots
{\sum }_{n=1}^{\mathrm{\infty }}2\pi n\left({\phi }_{n}+{\int }_{0}^{T}{F}_{n}\left(\tau \right)\phantom{\rule{0.2em}{0ex}}d\tau \right)\le {E}^{\mathrm{\prime }}\left(t\right)
\mathrm{\forall }t\in \left[0,T\right]
{\phi }_{n}={\int }_{0}^{1}\phi \left(x\right)sin\left(2\pi nx\right)\phantom{\rule{0.2em}{0ex}}dx
{F}_{n}\left(t\right)={\int }_{0}^{1}F\left(x,t\right)sin\left(2\pi nx\right)\phantom{\rule{0.2em}{0ex}}dx
n=0,1,2,\dots
Theorem 2 Let the assumptions (A1)-(A3) be satisfied. Then the following statements are true:
The inverse problem (1)-(4) has a solution in
{Q}_{T}
The solution of the inverse problem (1)-(4) is unique in
{Q}_{{T}_{0}}
{T}_{0}
0<{T}_{0}<T
) is determined by the data of the problem.
Proof By applying the standard procedure of the Fourier method, we obtain the following representation for the solution of (1)-(3) for arbitrary
p\left(t\right)\in C\left[0,T\right]
u\left(x,t\right)=\sum _{n=1}^{\mathrm{\infty }}\left[{\phi }_{n}{e}^{-{\left(2\pi n\right)}^{2}t-{\int }_{0}^{t}p\left(s\right)\phantom{\rule{0.2em}{0ex}}ds}+{\int }_{0}^{t}{F}_{n}\left(\tau \right){e}^{-{\left(2\pi n\right)}^{2}\left(t-\tau \right)-{\int }_{\tau }^{t}p\left(s\right)\phantom{\rule{0.2em}{0ex}}ds}\phantom{\rule{0.2em}{0ex}}d\tau \right]sin\left(2\pi nx\right).
Under the conditions (1) of (A2) and (1) of (A3), the series (5) and its x-partial derivative are uniformly convergent in
{\overline{Q}}_{T}
since their majorizing sums are absolutely convergent. Therefore, their sums
u\left(x,t\right)
{u}_{x}\left(x,t\right)
{\overline{Q}}_{T}
. In addition, the t-partial derivative and the xx-second order partial derivative series are uniformly convergent in
{Q}_{T}
u\left(x,t\right)\in {C}^{2,1}\left({Q}_{T}\right)\cap {C}^{1,0}\left({\overline{Q}}_{T}\right)
{u}_{t}\left(x,t\right)
{\overline{Q}}_{T}
. Differentiating (4) under the assumption (A1), we obtain
{\int }_{0}^{1}x{u}_{t}\left(x,t\right)\phantom{\rule{0.2em}{0ex}}dx={E}^{\mathrm{\prime }}\left(t\right),
p\left(t\right)=K\left[p\left(t\right)\right],
\begin{array}{rcl}K\left[p\left(t\right)\right]& =& \left(-{E}^{\mathrm{\prime }}\left(t\right)+\sum _{n=1}^{\mathrm{\infty }}2\pi n\left({\phi }_{n}{e}^{-{\left(2\pi n\right)}^{2}t-{\int }_{0}^{t}p\left(s\right)\phantom{\rule{0.2em}{0ex}}ds}+{\int }_{0}^{t}{F}_{n}\left(\tau \right){e}^{-{\left(2\pi n\right)}^{2}\left(t-\tau \right)-{\int }_{\tau }^{t}p\left(s\right)\phantom{\rule{0.2em}{0ex}}ds}\phantom{\rule{0.2em}{0ex}}d\tau \right)\\ -\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2\pi n}{F}_{n}\left(t\right)\right)/E\left(t\right).\end{array}
Using the representation (7), the following estimate is true:
0<{C}_{1}\le p\left(t\right)\le {C}_{2}.
Introduce the set M as follows:
M=\left\{p\left(t\right)\in C\left[0,T\right]:{C}_{1}\le p\left(t\right)\le {C}_{2}\right\}.
K:M\to M.
Compactness of K is verified by analogy to [6]. By virtue of the Schauder fixed point theorem, we have a solution
p\left(t\right)\in C\left[0,T\right]
of equation (7).
Now let us show that there exists
{Q}_{{T}_{0}}
0<{T}_{0}\le T
) for which the solution
\left(p,u\right)
of the problem (1)-(4) is unique in
{Q}_{{T}_{0}}
\left(q,v\right)
is also a solution pair of the problem (1)-(4). Then, from the representations (5) and (7) of the solution, we have
\underset{0\le t\le T}{max}|K\left[p\left(t\right)\right]-K\left[q\left(t\right)\right]|\le \alpha \underset{0\le t\le T}{max}|p\left(t\right)-q\left(t\right)|.
\alpha \in \left(0,1\right)
be an arbitrary fixed number. Fix a number
{T}_{0}
0<{T}_{0}\le T
\frac{{C}_{3}+{C}_{4}}{{C}_{0}}{T}_{0}\le \alpha ,
{C}_{0}=\underset{t\in \left[0,T\right]}{min}E\left(t\right),\phantom{\rule{2em}{0ex}}{C}_{3}=\sum _{n=1}^{\mathrm{\infty }}2\pi n{\phi }_{n},\phantom{\rule{2em}{0ex}}{C}_{4}=T\underset{t\in \left[0,T\right]}{max}\left(\sum _{n=1}^{\mathrm{\infty }}2\pi n{F}_{n}\left(t\right)\right).
Then, from equality (10), we obtain
{\parallel p-q\parallel }_{C\left[0,{T}_{0}\right]}\le \alpha {\parallel p-q\parallel }_{C\left[0,{T}_{0}\right]},
p=q
p=q
u=v
3 Continuous dependence of
\left(p,u\right)
upon the data
Theorem 3 Under assumptions (A1)-(A3), the solution
\left(p,u\right)
of the problem (1)-(4) depends continuously upon the data for small T.
The proof of the theorem is verified by analogy to [5].
We use the finite difference method with a predictor-corrector type approach that is suggested in [2]. Apply this method to the problem (1)-(4).
We subdivide the intervals
\left[0,1\right]
\left[0,T\right]
{N}_{x}
{N}_{t}
subintervals of equal lengths
h=\frac{1}{{N}_{x}}
\tau =\frac{T}{{N}_{t}}
, respectively. Then we add two lines
x=0
x=\left({N}_{x}+1\right)h
to generate the fictitious points needed for dealing with the boundary conditions. We choose the Crank-Nicolson scheme, which is absolutely stable and has a second-order accuracy in both h and τ. [13] The Crank-Nicolson scheme for (1)-(4) is as follows:
1\le i\le {N}_{x}
0\le j\le {N}_{t}
are the indices for the spatial and time steps respectively,
{u}_{i}^{j}=u\left({x}_{i},{t}_{j}\right)
{\varphi }_{i}=\phi \left({x}_{i}\right)
{F}_{i}^{j}=F\left({x}_{i},{t}_{j}\right)
{x}_{i}=ih
{t}_{j}=j\tau
. At the
t=0
level, adjustment should be made according to the initial condition and the compatibility requirements.
Equations (11)-(14) form an
{N}_{x}×{N}_{x}
linear system of equations
A{U}^{j+1}=b,
Now, let us construct the predicting-correcting mechanism. First, integrating equation (1) with respect to x from 0 to 1 and using (3) and (4), we obtain
p\left(t\right)=\frac{-{E}^{\mathrm{\prime }}\left(t\right)+{\int }_{0}^{1}xF\left(x,t\right)\phantom{\rule{0.2em}{0ex}}dx+{u}_{x}\left(1,t\right)}{E\left(t\right)}.
The finite difference approximation of (16) is
{p}^{j}=\frac{-\left(\left({E}^{j+1}-{E}^{j}\right)/\tau \right)+{\left(Fin\right)}^{j}+\left({u}_{{N}_{x}+1}^{j}-{u}_{{N}_{x}}^{j}\right)/h}{{E}^{j}},
{E}^{j}=E\left({t}_{j}\right)
{\left(Fin\right)}^{j}={\int }_{0}^{1}xF\left(x,{t}_{j}\right)\phantom{\rule{0.2em}{0ex}}dx
j=0,1,\dots ,{N}_{t}
j=0
{p}^{0}=\frac{-\left(\left({E}^{1}-{E}^{0}\right)/\tau \right)+{\left(Fin\right)}^{0}+\left({\varphi }_{{N}_{x}+1}-{\varphi }_{{N}_{x}}\right)/h}{{E}^{j}},
{\varphi }_{i}
allow us to start our computation. We denote the values of
{p}^{j}
{u}_{i}^{j}
at the s th iteration step
{p}^{j\left(s\right)}
{u}_{i}^{j\left(s\right)}
, respectively. In numerical computation, since the time step is very small, we can take
{p}^{j+1\left(0\right)}={p}^{j}
{u}_{i}^{j+1\left(0\right)}={u}_{i}^{j}
j=0,1,2,\dots ,{N}_{t}
i=1,2,\dots ,{N}_{x}
. At each
\left(s+1\right)
th iteration step, we first determine
{p}^{j+1\left(s+1\right)}
{p}^{j+1\left(s+1\right)}=\frac{-\left(\left({E}^{j+2}-{E}^{j}+1\right)/\tau \right)+{\left(Fin\right)}^{j+1}+\left({u}_{{N}_{x}+1}^{j+1\left(s\right)}-{u}_{{N}_{x}}^{j+1\left(s\right)}\right)/h}{{E}^{j}}.
Then from (11)-(14) we obtain
The system of equations (17)-(19) can be solved by the Gauss elimination method, and
{u}_{i}^{j+1\left(s+1\right)}
is determined. If the difference of values between two iterations reaches the prescribed tolerance, the iteration is stopped, and we accept the corresponding values
{p}^{j+1\left(s+1\right)}
{u}_{i}^{j+1\left(s+1\right)}
i=1,2,\dots ,{N}_{x}
{p}^{j+1}
{u}_{i}^{j+1}
i=1,2,\dots ,{N}_{x}
), on the
\left(j+1\right)
th time step, respectively. By virtue of this iteration, we can move from level j to level
j+1
5 Numerical example and discussions
In this section, we present examples to illustrate the efficiency of the numerical method described in the previous section.
Example Consider the inverse problem (1)-(4) with
It is easy to check that the analytical solution of this problem is
\left\{p\left(t\right),u\left(x,t\right)\right\}=\left\{1+exp\left(13t\right),\left(1+sin\left(2\pi x\right)\right)exp\left(10t\right)\right\}.
Let us apply the scheme which was explained in the previous section for the step sizes
h=0.01
\tau =\frac{h}{8}
T=1/2
, the comparisons between the analytical solution (19) and the numerical finite difference solution are shown in Figures 1 and 2.
The analytical and numerical solutions of
\mathbit{p}\mathbf{\left(}\mathbit{t}\mathbf{\right)}
\mathbit{T}\mathbf{=}\mathbf{1}\mathbf{/}\mathbf{2}
. The numerical solution is shown by a dashed line.
\mathbit{u}\mathbf{\left(}\mathbit{x}\mathbf{,}\mathbit{t}\mathbf{\right)}
\mathbit{T}\mathbf{=}\mathbf{1}\mathbf{/}\mathbf{2}
The inverse problem regarding the simultaneous identification of the time-dependent coefficient of heat capacity together with the temperature distribution in a one-dimensional heat equation with periodic boundary and integral overdetermination conditions has been considered. This inverse problem has been investigated from both theoretical and numerical points of view. In the theoretical part of the article, the conditions for the existence, uniqueness and continuous dependence upon the data of the problem have been established. In the numerical part, a numerical example using the Crank-Nicolson finite-difference scheme combined with the iteration method is presented.
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Cannon J, Lin Y, Wang S: Determination of source parameter in a parabolic equations. Meccanica 1992, 27: 85–94. 10.1007/BF00420586
Fatullayev A, Gasilov N, Yusubov I: Simultaneous determination of unknown coefficients in a parabolic equation. Appl. Anal. 2008, 87: 1167–1177. 10.1080/00036810802140616
Ivanchov M, Pabyrivska N: Simultaneous determination of two coefficients of a parabolic equation in the case of nonlocal and integral conditions. Ukr. Math. J. 2001, 53: 674–684. 10.1023/A:1012570031242
Ismailov M, Kanca F: An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions. Math. Methods Appl. Sci. 2011, 34: 692–702. 10.1002/mma.1396
Kanca F, Ismailov M: Inverse problem of finding the time-dependent coefficient of heat equation from integral overdetermination condition data. Inverse Probl. Sci. Eng. 2012, 20: 463–476. 10.1080/17415977.2011.629093
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Ivanchov N: Boundary value problems for a parabolic equation with integral conditions. Differ. Equ. 2004, 40: 591–609.
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Department of Information Technologies, Kadir Has University, Istanbul, 34083, Turkey
Fatma Kanca
Correspondence to Fatma Kanca.
Kanca, F. The inverse problem of the heat equation with periodic boundary and integral overdetermination conditions. J Inequal Appl 2013, 108 (2013). https://doi.org/10.1186/1029-242X-2013-108
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C(5) +\int _ { 5 } ^ { 31 } C ^ { \prime } ( t ) d t= 25,000
C
represents calculator sales during the month of August and
t
represents the day of the month. In the context of the problem, write a complete description about what the integral is computing. Use correct units and be sure to mention the meaning of the bounds in your description.
Recall that definite integrals can be considered accumulation functions.
C^\prime
represent? What is being accumulated?
25,000
calculators were sold (accumulated) between August 1st and August 31st.
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Double-C Seismic Accelerometer - Kerbal Space Program Wiki
Drag 0.2 [N 1]
Part configuration sensorAccelerometer.cfg
Experiment Seismic Scan
Electricity required 0.0075 ⚡/s
The Double-C Seismic Accelerometer is a scientific instrument and environmental sensor that measures a craft's acceleration. It displays the g-force on the craft.
In career mode, it is available with Electronics, at level 7 of the technology tree.
2.2 Environmental sensor
“ This device contains an extremely sensitive acceleration sensor, which when properly settled on a firm surface, will detect and record accurate seismic activity data. The accelerometer will still function while flying, so the Double-C can also be used to measure accelerations during flight. Warranty void if shaken or exposed to vacuum.
The Double-C Seismic Accelerometer can be radially mounted to a vessel. It is a physicsless part, and its mass and drag are instead added to its parent part.
Right-clicking on the Double-C Seismic Accelerometer (from the vessel or from a scientist on EVA) will reveal an option to "Log Seismic Data". Using this option will make the Double-C perform a Seismic Scan, which can earn the player Science.
Unlike other experiments, a Seismic Scan can only be performed while landed on a celestial body; the vessel must be resting on land. If in any other situation (splashed in the ocean, flying in the atmosphere, in space, etc.), then the experiment cannot be performed. However, once landed, a unique Seismic Scan can be done in each biome.
To perform a Seismic Scan from water biomes (e.g. Kerbin's Oceans), a craft with the Double-C can be sunk into the water, coming to rest on the ocean floor.
Like other scientific instruments, the Double-C Seismic Accelerometer can only hold one experiment at a time. Attempting to perform a Seismic Scan while the Double-C already contains one will result in the original scan being overwritten. To avoid this, the scan should be transferred out of the Double-C Seismic Accelerometer. This can be done by a Kerbal on EVA, or by an Experiment Storage Unit (or probe core with built-in storage functionality).
Once the Seismic Scan has been removed from the Double-C Seismic Accelerometer, it can be reused to perform another Seismic Scan.
Attached to the side of an ascent stage this environmental sensor can be used to control your acceleration during launch to provide a better and constant ascent, using less fuel for the same orbit. The Double-C displays 4 significant figures, offering much better precision than the g-meter on the navball. On upper and cruise stages the Double-C Accelerometer can be used to estimate the time needed to accelerate to a certain speed.
Additionally it can be used to measure the spacecraft's current mass indirectly using Newton's second law. To estimate the mass, activate the accelerometer and fire the engines so that the craft isn't rotating.
{\displaystyle m={\frac {F}{a}}}
Given the force F by adding all engines forces together and the acceleration a by the accelerometer the current mass is the division of both. But as the craft is losing mass by this the acceleration is increasing over time. Thus it's only accurate at a low fuel burn rate over a short period of time. This will also change the craft's orbit, so maybe a retroburn afterwards is necessary.
Since version 0.20 this method serves little purpose, because the exact mass of a vehicle can be seen in the map view and tracking station.
While active, the accelerometer drains 0.0075 Electric Charge per second (27 E/h).
Apart from the two scenarios in the product description, there exists another easter egg situation that can void the Double-C Seismic Accelerometer's warranty.
Spoiler: Easter egg voiding warranty
If landed on the surface of the gas giant Jool (a normally impossible feat), performing a Seismic Scan may yield the message:
“ The sensor has informed you that the warranty has just been voided. No refunds. ”
→ Main article: Jool#Trivia
"PhysicsSignificance = 1" added. The part now has 0 mass and drag, despite the listed values.
Renamed from Double-C Accelerometer to Double-C Seismic Accelerometer
Retrieved from "https://wiki.kerbalspaceprogram.com/index.php?title=Double-C_Seismic_Accelerometer&oldid=102281"
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Consumer choice - WikiMili, The Best Wikipedia Reader
Branch of microeconomics
Find sources: "Consumer choice" – news · newspapers · books · scholar · JSTOR (October 2021) ( Learn how and when to remove this template message )
The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their preferences subject to limitations on their expenditures, by maximizing utility subject to a consumer budget constraint. [1] Factors influencing consumers' evaluation of the utility of goods: income level, cultural factors and physio-psychological factors.
Example: homogeneous divisible goods
Characteristics of the indifference curve in consumer choice
Role of Time Constraint Effect
Price effect as sum of substitution and income effects
Indifference curves for goods that are perfect substitutes or complements
Labor-leisure tradeoff
In addition, people's judgments and decisions are often influenced by systemic biases or heuristics and are strongly dependent on the context in which the decisions are made, small or even unexpected changes in the decision-making environment can greatly affect their decisions. [2]
The consumption setC – the set of all bundles that the consumer could conceivably consume.
A price system , which is a function assigning a price to each bundle.
Firstly, Consumers use heuristics, which means they do not scrutinize decisions too closely but rather make broad generalizations. It is not worthwhile to attempt to determine the value of specific behavior. Heuristics are techniques for simplifying the decision-making process by omitting or disregarding certain information and focusing exclusively on particular elements of alternatives. While some heuristics must be utilized purposefully and deliberately, others can be used relatively effort lessly, even without our conscious awareness. [3] Consumptions typically impacted by advertising and consumer habits. It is a mental shortcut that helps us make judgments and solve problems faster. They allow us to save time by reducing the need to constantly think about the next step.
{\displaystyle R_{+}^{2}}
{\displaystyle (x,y)}
{\displaystyle x\geq 0}
{\displaystyle y\geq 0}
{\displaystyle u(x,y)=x^{\alpha }\cdot y^{\beta }}
{\displaystyle (x,y)}
{\displaystyle xp_{X}+yp_{Y}}
{\displaystyle BC}
{\displaystyle xp_{X}+yp_{Y}\leq \mathrm {income} }
{\displaystyle I3}
{\displaystyle I2}
{\displaystyle X*}
{\displaystyle Y*}
Indifference curve analysis begins with the utility function. The utility function is treated as an index of utility. [6] All that is necessary is that the utility index change as more preferred bundles are consumed.
{\displaystyle P(L)}
A typical initial endowment is either a fixed income, or an initial parcel which the consumer can sell and buy another parcel. [7]
Unreasonable behavior in which people recover from expenses that have already been incurred. [8] An example of this is a consumer who has already purchased their ticket for a concert and may travel through a storm to be able to attend the concert in order to not waste their ticket.
Another example is different payment schedules for gym members may result in different levels of potential sunk costs and affect the frequency of gym visits by consumers. That is to say, the payment schedule with other less frequent (e.g., quarterly, semi-annual or annual payment schedule), compared to a month pay the fee to the gym in a larger, these factors to reduce the cost and reduce the psychological sunk costs, more vivid sunk costs significantly increased people's gym visits. [9] Losses loom larger than gains.
The conclusion to be drawn from this study is[ weasel words ] that time pressure and graphic design of consumer goods all play an important role in understanding the computational behavioural processes of consumer choice. [10]
{\displaystyle BC2}
{\displaystyle BC1}
{\displaystyle BC1}
{\displaystyle BC1}
{\displaystyle BC2}
{\displaystyle BC3}
{\displaystyle \Delta y_{1}^{n}}
{\displaystyle m'}
{\displaystyle m}
{\displaystyle p_{1}'}
{\displaystyle \Delta y_{1}^{n}=y_{1}(p_{1}',m)-y_{1}(p_{1}',m').}
Further information: Slutsky equation and Hicksian demand
{\displaystyle \Delta y_{1}^{s}}
{\displaystyle \ Y}
{\displaystyle \ Y}
{\displaystyle \ p_{1}}
{\displaystyle \ p_{1}'}
{\displaystyle BC1}
{\displaystyle BC2}
and thus increasing purchasing power) and, at the same time, the money income falls fro{\displaystyle m}
{\displaystyle m'}
{\displaystyle \ I1}
{\displaystyle \Delta y_{1}^{s}=y_{1}(p_{1}',m')-y_{1}(p_{1},m)=Y_{s}-Y_{1}.}
{\displaystyle \ Y}
{\displaystyle \ Y_{1}}
{\displaystyle \ Y_{s}}
{\displaystyle \ p_{1}}
{\displaystyle \ Y}
{\displaystyle \ Y_{2}}
{\displaystyle \ Y}
{\displaystyle \ Y_{s}}
{\displaystyle \ Y_{2}}
The usefulness of a good is also a factor that consumers consider when making their choices. A product that has utility meets the Consumer needs and brings help to the consumer. The product itself has value. [11] A utility is a set of numerical values that reflect the relative rankings of various bundles of goods.
The behavioral assumption of the consumer theory proposed herein is that all consumers seek to maximize utility. In the mainstream economics tradition, this activity of maximizing utility has been deemed as the "rational" behavior of decision makers. More specifically, in the eyes of economists, all consumers seek to maximize a utility function subject to a budgetary constraint. [13] In other words, economists assume that consumers will always choose the "best" bundle of goods they can afford. [14] Consumer theory is therefore based on generating refutable hypotheses about the nature of consumer demand from this behavioral postulate. [13]
In order to reason from the central postulate towards a useful model of consumer choice, it is necessary to make additional assumptions about the certain preferences that consumers employ when selecting their preferred "bundle" of goods. These are relatively strict, allowing for the model to generate more useful hypotheses with regard to consumer behavior than weaker assumptions, which would allow any empirical data to be explained in terms of stupidity, ignorance, or some other factor, and hence would not be able to generate any predictions about future demand at all. [13] For the most part, however, they represent statements which would only be contradicted if a consumer was acting in (what was widely regarded as) a strange manner. [15] In this vein, the modern form of consumer choice theory assumes:
Consumer choice theory is based on the assumption that the consumer fully understands his or her own preferences, allowing for a simple but accurate comparison between any two bundles of good presented. [14] That is to say, it is assumed that if a consumer is presented with two consumption bundles A and B each containing different combinations of n goods, the consumer can unambiguously decide if (s)he prefers A to B, B to A, or is indifferent to both. [13] [14] The few scenarios where it is possible to imagine that decision-making would be very difficult are thus placed "outside the domain of economic analysis". [14] However, discoveries in behavioral economics has found that actual decision making is affected by various factors, such as whether choices are presented together or separately through the distinction bias.
Means that if A and B are in all respect identical the consumer will consider A to be at least as good as (i.e. weakly preferred to) B. [14] Alternatively, the axiom can be modified to read that the consumer is indifferent with regard to A and B. [16]
More Is Better - all else being the same, more of a commodity is better than less of it (non-satiation). This is the "more is always better" assumption; that in general if a consumer is offered two almost identical bundles A and B, but where B includes more of one particular good, the consumer will choose B. [17]
It is assumed that a consumer may choose to purchase any quantity of a good (s)he desires, for example, 2.6 eggs and 4.23 loaves of bread. Whilst this makes the model less precise, it is generally acknowledged to provide a useful simplification to the calculations involved in consumer choice theory, especially since consumer demand is often examined over a considerable period of time. The more spending rounds are offered, the better approximation the continuous, differentiable function is for its discrete counterpart. (Whilst the purchase of 2.6 eggs sounds impossible, an average consumption of 2.6 eggs per day over a month does not.) [17]
Note the assumptions do not guarantee that the demand curve will be negatively sloped. A positively sloped curve is not inconsistent with the assumptions. [18]
In Marx's critique of political economy, any labor-product has a value and a use value, and if it is traded as a commodity in markets, it additionally has an exchange value, most often expressed as a money-price. [19] Marx acknowledges that commodities being traded also have a general utility, implied by the fact that people want them, but he argues that this by itself tells us nothing about the specific character of the economy in which they are produced and sold.
Main article: Backward bending supply curve of labour
{\displaystyle \ell +L=T.}
{\displaystyle C=w(T-\ell ).}
{\displaystyle (\ell =0)}
{\displaystyle T-\ell =T}
{\displaystyle C=wT}
In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is indifferent. That is, any combinations of two products indicated by the curve will provide the consumer with equal levels of utility, and the consumer has no preference for one combination or bundle of goods over a different combination on the same curve. One can also refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer. In other words, an indifference curve is the locus of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent preferences rather than something from which preferences come. The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles.
In economics and consumer theory, a Giffen good is a product that people consume more of as the price rises and vice versa—violating the basic law of demand in microeconomics. For any other sort of good, as the price of the good rises, the substitution effect makes consumers purchase less of it, and more of substitute goods; for most goods, the income effect reinforces this decline in demand for the good. But a Giffen good is so strongly an inferior good in the minds of consumers that this contrary income effect more than offsets the substitution effect, and the net effect of the good's price rise is to increase demand for it. This phenomenon is known as the Giffen paradox. A Giffen good is considered to be the opposite of an ordinary good.
In economics and particularly in consumer choice theory, the substitution effect is one component of the effect of a change in the price of a good upon the amount of that good demanded by a consumer, the other being the income effect.
In economics, a complementary good is a good whose appeal increases with the popularity of its complement. Technically, it displays a negative cross elasticity of demand and that demand for it increases when the price of another good decreases. If is a complement to , an increase in the price of will result in a negative movement along the demand curve of and cause the demand curve for to shift inward; less of each good will be demanded. Conversely, a decrease in the price of will result in a positive movement along the demand curve of and cause the demand curve of to shift outward; more of each good will be demanded. This is in contrast to a substitute good, whose demand decreases when its substitute's price decreases.
A corner solution is a special solution to an agent's maximization problem in which the quantity of one of the arguments in the maximized function is zero. In non-technical terms, a corner solution is when the chooser is either unwilling or unable to make a trade-off between goods.
A relative price is the price of a commodity such as a good or service in terms of another; i.e., the ratio of two prices. A relative price may be expressed in terms of a ratio between the prices of any two goods or the ratio between the price of one good and the price of a market basket of goods. Microeconomics can be seen as the study of how economic agents react to changes in relative prices, and of how relative prices are affected by the behavior of those agents. The difference and change of relative prices can also reflect the development of productivity.
The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is strictly preferred to it.
In consumer theory, a consumer's preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1. For example, in an economy with two goods , homothetic preferences can be represented by a utility function that has the following property: for every :
In economics and consumer theory, a linear utility function is a function of the form:
↑ Reisch, Lucia A.; Zhao, Min (November 2017). "Behavioural economics, consumer behaviour and consumer policy: state of the art". Behavioural Public Policy. 1 (2): 190–206. doi:10.1017/bpp.2017.1. hdl: 10398/01e85b29-3d75-4be3-95dc-d256ad5dd947 . ISSN 2398-063X. S2CID 158160660.
↑ Gilovich, Thomas; Griffin, Dale; Kahneman, Daniel (8 July 2002). Heuristics and Biases: The Psychology of Intuitive Judgment. ISBN 9780521796798.
↑ Moore, Jordan (2021). "Targeted Wealth Management for Prospect Theory Investors". The Journal of Wealth Management. 24 (3): 11–30. doi:10.3905/jwm.2021.1.145. S2CID 236285139.
↑ Labrecque, Lauren I.; Vor Dem Esche, Jonas; Mathwick, Charla; Novak, Thomas P.; Hofacker, Charles F. (2013). "Consumer Power: Evolution in the Digital Age". Journal of Interactive Marketing. 27 (4): 257–269. doi:10.1016/j.intmar.2013.09.002.
↑ Silberberg & Suen 2001 , p. 255
↑ Berliant, M.; Raa, T. T. (1988). "A foundation of location theory: Consumer preferences and demand". Journal of Economic Theory. 44 (2): 336. doi:10.1016/0022-0531(88)90008-7.
↑ Egan, Mark (2017). Enter your username and password - The University of Queensland, Australia. auth.uq.edu.au. doi:10.4324/9781912282555. ISBN 9781912282555 . Retrieved 2021-04-25.
↑ Reutskaja, Elena; Nagel, Rosemarie; Camerer, Colin F.; Rangel, Antonio (2011). "Search Dynamics in Consumer Choice under Time Pressure: An Eye-Tracking Study". The American Economic Review. 101 (2): 900–926. doi:10.1257/aer.101.2.900. JSTOR 29783694 . Retrieved 2021-04-26.
↑ Gaeth, Gary J.; Levin, Irwin P.; Chakraborty, Goutam; Levin, Aron M. (1991). "Consumer evaluation of multi-product bundles: An information integration analysis". Marketing Letters. 2: 47–57. doi:10.1007/BF00435195. S2CID 167403155.
↑ Simoes, Nádia; Diogo, Ana Paula (2014). "Marginal Utility". Encyclopedia of Quality of Life and Well-Being Research. pp. 3769–3770. doi:10.1007/978-94-007-0753-5_1724. ISBN 978-94-007-0752-8.
1 2 3 4 Silberberg & Suen 2001 , pp. 252–254
1 2 3 4 5 Varian 2006 , p. 20 harvnb error: no target: CITEREFVarian2006 (help)
↑ Binger & Hoffman 1998 , pp. 109–17
1 2 Silberberg & Suen 2001 , pp. 256–257
↑ Binger & Hoffman 1998 , pp. 141–143
↑ "Glossary of Terms: Us". Marxists.org. Retrieved 2013-11-07.
Böhm, Volker; Haller, Hans (1987). "Demand theory". The New Palgrave: A Dictionary of Economics . Vol. 1. pp. 785–92.
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I present in this note recent results on the uniqueness and stability for the parabolic-parabolic Keller-Segel equation on the plane, obtained in collaboration with S. Mischler in [11].
author = {Kleber Carrapatoso},
title = {The parabolic-parabolic {Keller-Segel} equation},
TI - The parabolic-parabolic Keller-Segel equation
%T The parabolic-parabolic Keller-Segel equation
Kleber Carrapatoso. The parabolic-parabolic Keller-Segel equation. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 18, 17 p. doi : 10.5802/slsedp.76. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.76/
[1] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), no. 1, 213–242.
[2] M. Ben-Artzi, Global solutions of two-dimensional Navier-Stokes and Euler equations, Arch. Rational Mech. Anal. 128 (1994), no. 4, 329–358.
[3] P. Biler, L. Corrias, and J. Dolbeault, Large mass self-similar solutions of the parabolic-parabolic Keller-Segel model of chemotaxis, J. Math. Biol. 63 (2011), no. 1, 1–32.
[4] P. Biler, I. Guerra, and G. Karch, Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane, arXiv:1401.7650.
[5] A. Blanchet, J. Dolbeault, and B. Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations (2006), No. 44, 1–33.
[6] H. Brezis, Remarks on the preceding paper by M. Ben-Artzi: “Global solutions of two-dimensional Navier-Stokes and Euler equations” [Arch. Rational Mech. Anal. 128 (1994), no. 4, 329–358; MR1308857 (96h:35148)], Arch. Rational Mech. Anal. 128 (1994), no. 4, 359–360.
[7] V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in
{ℝ}^{2}
, Commun. Math. Sci. 6 (2008), no. 2, 417–447.
[8] J. Campos and J. Dolbeault, A functional framework for the Keller-Segel system: logarithmic Hardy-Littlewood-Sobolev and related spectral gap inequalities, C. R. Math. Acad. Sci. Paris 350 (2012), no. 21-22, 949–954.
[9] J. F. Campos and J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane, 2012.
[10] E. Carlen and M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on
{S}^{n}
, Geom. Funct. Anal. 2 (1992), no. 1, 90–104.
[11] K. Carrapatoso and S. Mischler, Uniqueness and long time asymptotic for the parabolic-parabolic Keller-Segel equation, arXiv:1406.6006.
[12] J. A. Carrillo, S. Lisini, and E. Mainini, Uniqueness for Keller-Segel-type chemotaxis models, Discrete Contin. Dyn. Syst. 34 (2014), no. 4, 1319–1338.
[13] L. Corrias, M. Escobedo, and J. Matos, Existence, uniqueness and asymptotic behavior of the solutions to the fully parabolic Keller-Segel system in the plane, J. Diff. Equations 257 (2014), no. 6, 1840–1878, doi:10.1016/j.jde.2014.05.019.
[14] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547.
[15] G. Egaña and S. Mischler, Uniqueness and long time assymptotic for the Keller-Segel equation - Part I. The parabolic-elliptic case, arXiv:1310.7771.
[16] L. C. F. Ferreira and J. C. Precioso, Existence and asymptotic behaviour for the parabolic-parabolic Keller-Segel system with singular data, Nonlinearity 24 (2011), no. 5, 1433–1449.
[17] N. Fournier, M. Hauray, and S. Mischler, Propagation of chaos for the 2d viscous vortex model, arXiv:1212.1437.
[18] M. P. Gualdani, S. Mischler, and C. Mouhot, Factorization of non-symmetric operators and exponential
H
-Theorem, arXiv:1006.5523.
[19] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 4, 633–683 (1998).
[20] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.
[21] S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the Kinetic-Fokker-Planck equation, arXiv:1412.7487.
[22] S. Mischler and C. Mouhot, Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres, Comm. Math. Phys. 288 (2009), no. 2, 431–502.
[23] S. Mischler and J. Scher, Semigroup spectral analysis and growth-fragmentation equation, to appear in Ann. Inst. H. Poincaré - Anal. Non Linéaire (2015).
[24] T. Nagai, Global existence and blowup of solutions to a chemotaxis system, Proceedings of the Third World Congress of Nonlinear Analysts, Part 2 (Catania, 2000), vol. 47, 2001, pp. 777–787.
[25] T. Nagai, T. Senba, and T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology, Hiroshima Math. J. 30 (2000), no. 3, 463–497.
[26] Y. Naito, T. Suzuki, and K. Yoshida, Self-similar solutions to a parabolic system modeling chemotaxis, J. Differential Equations 184 (2002), no. 2, 386–421.
[27] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311–338.
[28] I. Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, arXiv:1311.5168.
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RC oscillator - Wikipedia
Linear electronic oscillator circuits, which generate a sinusoidal output signal, are composed of an amplifier and a frequency selective element, a filter. A linear oscillator circuit which uses an RC network, a combination of resistors and capacitors, for its frequency selective part is called an RC oscillator.
1.2 Twin-T oscillator
1.3 Quadrature oscillator
2 Low distortion oscillators
RC oscillators are a type of feedback oscillator; they consist of an amplifying device, a transistor, vacuum tube, or op-amp, with some of its output energy fed back into its input through a network of resistors and capacitors, an RC network, to achieve positive feedback, causing it to generate an oscillating sinusoidal voltage.[1][2][3] They are used to produce lower frequencies, mostly audio frequencies, in such applications as audio signal generators and electronic musical instruments.[4][5] At radio frequencies, another type of feedback oscillator, the LC oscillator is used, but at frequencies below 100 kHz the size of the inductors and capacitors needed for the LC oscillator become cumbersome, and RC oscillators are used instead.[6] Their lack of bulky inductors also makes them easier to integrate into microelectronic devices. Since the oscillator's frequency is determined by the value of resistors and capacitors, which vary with temperature, RC oscillators do not have as good frequency stability as crystal oscillators.
The frequency of oscillation is determined by the Barkhausen criterion, which says that the circuit will only oscillate at frequencies for which the phase shift around the feedback loop is equal to 360° (2π radians) or a multiple of 360°, and the loop gain (the amplification around the feedback loop) is equal to one.[7][1] The purpose of the feedback RC network is to provide the correct phase shift at the desired oscillating frequency so the loop has 360° phase shift, so the sine wave, after passing through the loop will be in phase with the sine wave at the beginning and reinforce it, resulting in positive feedback.[6] The amplifier provides gain to compensate for the energy lost as the signal passes through the feedback network, to create sustained oscillations. As long as the gain of the amplifier is high enough that the total gain around the loop is unity or higher, the circuit will generally oscillate.
In RC oscillator circuits which use a single inverting amplifying device, such as a transistor, tube, or an op amp with the feedback applied to the inverting input, the amplifier provides 180° of the phase shift, so the RC network must provide the other 180°.[6] Since each capacitor can provide a maximum of 90° of phase shift, RC oscillators require at least two frequency-determining capacitors in the circuit (two poles), and most have three or more,[1] with a comparable number of resistors.
This makes tuning the circuit to different frequencies more difficult than in other types such as the LC oscillator, in which the frequency is determined by a single LC circuit so only one element must be varied. Although the frequency can be varied over a small range by adjusting a single circuit element, to tune an RC oscillator over a wide range two or more resistors or capacitors must be varied in unison, requiring them to be ganged together mechanically on the same shaft.[2][8] The oscillation frequency is proportional to the inverse of the capacitance or resistance, whereas in an LC oscillator the frequency is proportional to inverse square root of the capacitance or inductance.[9] So a much wider frequency range can be covered by a given variable capacitor in an RC oscillator. For example, a variable capacitor that could be varied over a 9:1 capacitance range will give an RC oscillator a 9:1 frequency range, but in an LC oscillator it will give only a 3:1 range.
Some examples of common RC oscillator circuits are listed below:
Phase-shift oscillator[edit]
Main article: Phase-shift oscillator
In the phase-shift oscillator the feedback network is three identical cascaded RC sections.[10] In the simplest design the capacitors and resistors in each section have the same value
{\displaystyle \scriptstyle R\;=\;R1\;=\;R2\;=\;R3}
{\displaystyle \scriptstyle C\;=\;C1\;=\;C2\;=\;C3}
. Then at the oscillation frequency each RC section contributes 60° phase shift for a total of 180°. The oscillation frequency is
{\displaystyle f={\frac {1}{2\pi RC{\sqrt {6}}}}}
The feedback network has an attenuation of 1/29, so the op-amp must have a gain of 29 to give a loop gain of one for the circuit to oscillate
{\displaystyle R_{\mathrm {fb} }=29\cdot R}
A twin-T oscillator
Twin-T oscillator[edit]
Another common design is the "Twin-T" oscillator as it uses two "T" RC circuits operated in parallel. One circuit is an R-C-R "T" which acts as a low-pass filter. The second circuit is a C-R-C "T" which operates as a high-pass filter. Together, these circuits form a bridge which is tuned at the desired frequency of oscillation. The signal in the C-R-C branch of the Twin-T filter is advanced, in the R-C-R - delayed, so they may cancel one another for frequency
{\displaystyle f={\frac {1}{2\pi RC}}}
{\displaystyle x=2}
; if it is connected as a negative feedback to an amplifier, and x>2, the amplifier becomes an oscillator. (Note:
{\displaystyle x=C2/C1=R1/R2}
Quadrature oscillator[edit]
The quadrature oscillator uses two cascaded op-amp integrators in a feedback loop, one with the signal applied to the inverting input or two integrators and an invertor. The advantage of this circuit is that the sinusoidal outputs of the two op-amps are 90° out of phase (in quadrature). This is useful in some communication circuits.
It is possible to stabilize a quadrature oscillator by squaring the sine and cosine outputs, adding them together, (Pythagorean trigonometric identity) subtracting a constant, and applying the difference to a multiplier that adjusts the loop gain around an inverter. Such circuits have a near-instant amplitude response to the constant input and extremely low distortion.
Low distortion oscillators[edit]
The Barkhausen criterion mentioned above does not determine the amplitude of oscillation. An oscillator circuit with only linear components is unstable with respect to amplitude. As long as the loop gain is exactly one, the amplitude of the sine wave would be constant, but the slightest increase in gain, due to a drift in the value of components will cause the amplitude to increase exponentially without limit. Similarly, the slightest decrease will cause the sine wave to die out exponentially to zero. Therefore, all practical oscillators must have a nonlinear component in the feedback loop, to reduce the gain as the amplitude increases, leading to stable operation at the amplitude where the loop gain is unity.
In most ordinary oscillators, the nonlinearity is simply the saturation (clipping) of the amplifier as the amplitude of the sine wave approaches the power supply rails. The oscillator is designed to have a small-signal loop gain greater than one. The higher gain allows an oscillator to start by exponentially amplifying some ever-present noise.[11]
As the peaks of the sine wave approach the supply rails, the saturation of the amplifier device flattens (clips) the peaks, reducing the gain. For example, the oscillator might have a loop gain of 3 for small signals, but that loop gain instaneously drops to zero when the output reaches one of the power supply rails.[12] The net effect is the oscillator amplitude will stabilize when average gain over a cycle is one. If the average loop gain is greater than one, the output amplitude increases until the nonlinearity reduces the average gain to one; if the average loop gain is less than one, then the output amplitude decreases until the average gain is one. The nonlinearity that reduces the gain may also be more subtle than running into a power supply rail.[13]
The result of this gain averaging is some harmonic distortion in the output signal. If the small-signal gain is just a little bit more than one, then only a small amount of gain compression is needed, so there won't be much harmonic distortion. If the small-signal gain is much more than one, then significant distortion will be present.[14] However the oscillator must have gain significantly above one to start reliably.
So in oscillators that must produce a very low-distortion sine wave, a system that keeps the gain roughly constant during the entire cycle is used. A common design uses an incandescent lamp or a thermistor in the feedback circuit.[15][16] These oscillators exploit the resistance of a tungsten filament of the lamp increases in proportion to its temperature (a thermistor works in a similar fashion). The lamp both measures the output amplitude and controls the oscillator gain at the same time. The oscillator's signal level heats the filament. If the level is too high, then the filament temperature gradually increases, the resistance increases, and the loop gain falls (thus decreasing the oscillator's output level). If the level is too low, the lamp cools down and increases the gain. The 1939 HP200A oscillator uses this technique. Modern variations may use explicit level detectors and gain-controlled amplifiers.
Wien bridge oscillator with automatic gain control. Rb is a small incandescent lamp. Usually, R1 = R2 = R and C1 = C2 = C. In normal operation, Rb self heats to the point where its resistance is Rf/2.
Wien bridge oscillator[edit]
Main article: Wien bridge oscillator
One of the most common gain-stabilized circuits is the Wien bridge oscillator.[17] In this circuit, two RC circuits are used, one with the RC components in series and one with the RC components in parallel. The Wien Bridge is often used in audio signal generators because it can be easily tuned using a two-section variable capacitor or a two section variable potentiometer (which is more easily obtained than a variable capacitor suitable for generation at low frequencies). The archetypical HP200A audio oscillator is a Wien Bridge oscillator.
^ a b c Mancini, Ron; Palmer, Richard (March 2001). "Application Report SLOA060: Sine-Wave Oscillator" (PDF). Texas Instruments Inc. Retrieved August 12, 2015.
^ a b Gottlieb, Irving (1997). Practical Oscillator Handbook. Elsevier. pp. 49–53. ISBN 0080539386.
^ Coates, Eric (2015). "Oscillators Module 1 - Oscillator Basics". Learn About Electronics. Eric Coates. Retrieved August 7, 2015.
^ Coates, Eric (2015). "Oscillators Module 3 - AF Sine Wave Oscillators" (PDF). Learn About Electronics. Eric Coates. Retrieved August 7, 2015.
^ Chattopadhyay, D. (2006). Electronics (fundamentals And Applications). New Age International. pp. 224–225. ISBN 81-224-1780-9.
^ a b c "RC Feedback Oscillators". Electronics tutorial. DAEnotes. 2013. Retrieved August 9, 2015.
^ Rao, B.; Rajeswari, K.; Pantulu, P. (2012). Electronic Circuit Analysis. India: Pearson Education India. pp. 8.2–8.6, 8.11. ISBN 978-8131754283.
^ Eric Coates, 2015, AF Sine Wave Oscillators, p. 10
^ Groszkowski, Janusz (2013). Frequency of Self-Oscillations. Elsevier. pp. 397–398. ISBN 978-1483280301.
^ Department of the Army (1962) [1959], Basic Theory and Application of Transistors, Technical Manuals, Dover, pp. 178–179, TM 11-690
^ Strauss, Leonard (1970), "Almost Sinusoidal Oscillations — the linear approximation", Wave Generation and Shaping (second ed.), McGraw-Hill, pp. 663–720 at page 661, "It follows that if Aβ > 1 in the small-signal region, the amplitude will build up until the limiter stabilizes the system...."
^ Strauss 1970, p. 694, "As the signal amplitude increases, the active device will switch from active operation to the zero-gain regions of cutoff and saturation."
^ Strauss 1970, pp. 703–706, Exponential limiting—bipolar transistor.
^ Strauss 1970, p. 664, "If gross nonlinear operation is permitted, the limiter will distort the signal and the output will be far from sinusoidal."
^ Strauss 1970, p. 664, "Alternatively, an amplitude-controlled resistor or other passive nonlinear element may be included as part of the amplifier or in the frequency-determining network."
^ Strauss 1970, pp. 706–713, Amplitude of Oscillation—Part II, Automatic Gain Control.
^ Department of the Army 1962, pp. 179–180
Media related to RC oscillators at Wikimedia Commons
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Summary of Research Methods on Pre-Training Models of Natural Language Processing ()
Yu Xiao1, Zhezhi Jin2*
1Depertment of Mathematics, Yanbian University, Yanji, China.
2Department of Economics and Management, Yanbian University, Yanji, China.
In recent years, deep learning technology has been widely used and developed. In natural language processing tasks, pre-training models have been more widely used. Whether it is sentence extraction or sentiment analysis of text, the pre-training model plays a very important role. The use of a large-scale corpus for unsupervised pre-training of models has proven to be an excellent and effective way to provide models. This article summarizes the existing pre-training models and sorts out the improved models and processing methods of the relatively new pre-training models, and finally summarizes the challenges and prospects of the current pre-training models.
Natural Language Processing, Pre-Training Model, Language Model, Self-Training Model
Xiao, Y. and Jin, Z.Z. (2021) Summary of Research Methods on Pre-Training Models of Natural Language Processing. Open Access Library Journal, 8, 1-7. doi: 10.4236/oalib.1107602.
Natural language processing is an interdisciplinary subject that combines linguistics, computer science, mathematics and other disciplines. Natural linguistics has many achievements in machine translation, speech recognition, intelligent voice assistants, and public opinion analysis. It involves many researches in related disciplines, especially in the field of artificial intelligence in recent years, there have been many breakthroughs.
Pre-training refers to the training of the data model before a relatively large-scale data processing, and after a wave of training, the previous training will fine-tune in the model of the downstream task. In most language models, pre-training models usually use self-supervised methods to train the models.
With the wide application of pre-training models in natural language processing, pre-training technology has also entered a new era. The pre-training models involved in this article are mainly traditional pre-training models. After an overview of traditional pre-training models, this article will introduce pre-training models. The training model gives the improved T5 model of the BERT model, and the model overview after combining the pre-training model and the self-training model.
Natural language processing can express the language with discrete symbols and then form different sentences to express different semantics, to help the computer understand the language. Natural language processing is one of the most difficult problems in artificial intelligence.
The basic points of natural language processing are divided into corpus, Chinese word segmentation, part-of-speech tagging, syntactic analysis, stem extraction, morphological restoration, etc. Natural language usually undergoes feature extraction, feature selection, dimensionality reduction and other methods for feature processing, and then uses Markov model, hidden Markov model, conditional random field, Bayesian network and other methods are all used to classify and process languages [1].
The research on natural language processing is conducive to the development of personalized knowledge recommendation services, knowledge retrieval, intelligent question answering, speech recognition and other fields, making the relationship between artificial intelligence and language processing closer.
3. Traditional Pre-Training Model
3.1. ELMo Model [2]
The ELMo model (Embedding from Language Model) is based on the early pre-training model of the PTMs model that cannot solve the problem of complex context. After the improvement, a pre-training of the LSMT language model is added to the large-scale unsupervised corpus. The LSMT model is a two-way model. In pre-training, first use the language model to pre-process on the corpus, and then select the corresponding word network from the trained network China to embed each layer of words in the downstream task as new features, thereby effectively solving the problem The problem of ambiguity [3]. The structure of the ELMo model is shown below (Figure 1).
Compared with the previous model, the ELMo model adds a two-way language model LSTM (Long-Short Term Memory), so that the model can better connect the content between the context of the article in a complex context, effectively improving the performance of the model, but in Compared with subsequent models, the integrated fusion and some feature extraction capabilities also have obvious limitations.
3.2. GPT Model [4]
As mentioned above, the ELMo model has obvious limitations in many aspects, so based on the improvement of the EMLo model, Transformer proposed the GPT pre-training model of OpenAI. The common point of the GPT model and the EMLo model is that they both use two stages to train the model. The difference is that the first stage of the GPT model pre-trains the unsupervised language model on the corpus, so that the model parameters are converted to neural network The initial parameters are then fine-tuned for downstream tasks through a supervised model in the second stage. The GPT model structure is as follows (Figure 2).
The GTP model has improved some shortcomings in the later period, and proposed the GTP2 [5] model. Although the new model has better performance in the preprocessing of corpus, the model itself is still a one-way language model in essence, in terms of semantic information. There are still big limitations in the establishment.
3.3. BERT Model [2]
The BERT model is based on the stacked Transformer substructure proposed by the GPT model to establish a basic model (Figure 3). It can be said that it is an evolution of the GPT model to some extent. The unidirectional structure of the GPT model is improved to a bidirectional structure, which can be at a deeper level. Up to achieve the training of the data set, and then achieve the purpose of adjusting the model parameters.
Figure 1. Structure diagram of ELMo model.
Figure 2. Structure diagram of GPT model.
Figure 3. BERT model structure diagram.
The BERT model can formalize the following likelihood function to the maximum
\begin{array}{c}\theta =\mathrm{arg}\mathrm{max}{\displaystyle \underset{x\in Cor}{\sum }{P}_{\theta }\left(x|\stackrel{¯}{x}\right)}=\mathrm{arg}\mathrm{max}{\displaystyle \underset{x\in Cor}{\sum }{\displaystyle \underset{t}{\sum }m\left(t\right)\cdot {P}_{\theta }\left({x}_{t}|\stackrel{¯}{x}\right)}}\\ ={\displaystyle \underset{t=1}{\overset{T}{\sum }}{m}_{t}\mathrm{log}\frac{\mathrm{exp}\left({H}_{\theta }{\left(\stackrel{^}{x}\right)}_{t}^{T}e\left({x}_{t}\right)\right)}{{\displaystyle {\sum }_{x\text{'}}\mathrm{exp}\left({H}_{\theta }{\left(\stackrel{^}{x}\right)}_{t}^{T}e\left({x}^{\prime }\right)\right)}}}\end{array}
m\left(t\right)=\left\{\begin{array}{ll}1,\hfill & {x}_{t}\text{\hspace{0.17em}}\text{masked}\hfill \\ 0,\hfill & \text{otherwise}\hfill \end{array}
The emergence of the BERT model has greatly promoted the development of natural language processing, and at the same time promoted the transfer learning of natural language processing, but there are still shortcomings such as excessive time consumption and high hardware requirements.
3.4. New Pre-Trained Model
After the emergence of the BERT model, a large number of new pre-training models have emerged. Most of the new training models are generated after improvements based on the BERT model. They are all pre-training models in the deep learning era. The more representative models are ERNIE, SpanBERT, RoBERTa [6], ALBERT, etc. These models have good versatility in the application of large text corpora, and also provide better model initialization, which brings better generalization performance for subsequent machine learning, and avoids the problem of data overfitting in some experiments.
4. Related Improvements to the Model
4.1. T5 Model
Pre-training models appeared in the early stage [7], so that the development of pre-training models has reached a relatively complete level, so in order to be able to use a model to adapt to various NLP tasks, the T5 pre-training model was born in response to the times, the emergence of the T5 pre-training model provides a general framework model for almost all tasks involved in the entire pre-training model field. Although from the perspective of the model, the T5 pre-training model has not improved much in the innovation of the model, it is able to integrate many models involved in the previous single items and find the optimal combination scheme as quickly as possible so that more data can be processed.
For a simple example, when doing English-Chinese translation, you only need to add “translate English to Chinese” to the training data set. If you need to translate “university student”, then add “translate English to Chinese: university student” input into the model, you can directly get the Chinese translation “大学生”. For example, when we are preparing for emotional translation, we only need to add “sentiment” before the input to get the output result directly. For example, input “sentiment: This movie is terrible” to get “negative”.
In CSL and LCSTS text generation tasks, the T5 model can become the best model of several known models (Table 1).
4.2. Combination of Self-Training Model and Pre-Training Model
4.2.1. Model Overview
The self-training model means to train one labeled data first, then label the remaining large-scale unlabeled data, and then use the result as pseudo-labeled data for target training. It can be seen that the self-training model is very similar to the pre-training model, except that the pre-training model is only trained and operated on one model, learning directly from unlabeled data, while the self-training model uses two models, first indirectly from the learning of the data. Therefore, the combination of the two models can often get better results.
First pre-train the model. After the pre-trained model is trained on the data, it is trained as the first model of the training model of the self-training model, and the result obtained is trained as the pseudo-labeled model of the second step of the self-training model. The training results are used in the inference test. The training results obtained in this way are greatly improved than those obtained by simply pre-training, and outstanding results can also be achieved in small-sample learning tasks.
Table 1. Comparison of CSL test results between T5 model and other models.
The method of combining self-training model and pre-training model for model training is mainly divided into four steps, The first step is to train a pre-trained model on the labeled data，as a teacher model
{f}_{T}
, the second step is to use
{f}_{T}
extract data in related fields from a massive general corpus, the third step uses
{f}_{T}
annotate the extracted data, the fourth step is to train the target student model with the labeled pseudo-labeled corpus
{f}_{S}
The first, third, and fourth steps in the training process are all deterministic steps. Therefore, the focus of the model is how to obtain the relevant corpus D' required by the model from the massive corpus D. Under normal circumstances, corpus D can directly divide the document into sentences, and then extract the data in sentence units. Therefore, you can use the method of sentence encoding, use the encoding vector to represent the sentence, and then use the sentence encoder in multiple data After training on the set, you can get the feature vector of each coded sentence. In this way, you only need to add a special task code as a query condition to judge whether the sentence meets our requirements by calculating the cosine value of the sentence code and the task code At the same time, it can also reduce the noise interference of downstream pending tasks and improve the confidence of
{f}_{T}
Table 2 shows that the result of the method of combining self-training model and pre-training model in the process of training sentence encoding is still considerable compared with a single pre-training model.
5. Model Follow-Up Development and Outlook
Pre-training models are models that have epoch-making significance, the Pre-training models based on various language models have been widely used [8] [9], but so far there is still a lot of room for the development of pre-training models [10] [11], and more training scenarios are needed. Besides larger corpus to improve the accuracy of the model, in addition to the training of the large corpus, the professional corpus is also trained, and the problems encountered in the fine-tuning of the model are improved. The main development direction of the future model is to solve the problem of deep neural networks being vulnerable to adversarial example attacks, so as to achieve the same defensive effect as the image processing problem. And the structure of the future model also needs
Table 2. Comparison of test results between self-training model and pre-training model.
to be streamlined and improved, which is also very important in the construction of the evaluation system.
This article mainly sorts out and summarizes the pre-training involved in the language model, and sorts out the two new improved pre-training models to process data. Among them, the T5 model can perform comprehensive optimal combination model processing of more models on big data, and the method of combining the self-training model and the pre-training model can improve the effect of small-sample learning and knowledge distillation. Finally, this article summarizes the improvement space and expectations of the future pre-training model. In this regard, the author will also conduct a deeper research.
[1] Lyu, L.C., Zhang, B., Wang, Y.P., Zhao, Y.J., Qian, L. and Li, T.T. (2021) Global Patent Analysis of Natural Language Processing. Science Focus, 16, 84-95.
[2] Yu, T.R., Jin, R., Han, X.Z., Li, J.H. and Yu, T. (2020) Review of Pre-Treaning Models for Natural Language Processing. Computer Engineering and Applications, 56, 12-22.
[3] Liu, Q., Kusner, M.J. and Blunsom, P. (2020) A Survey on Contextual Embeddings.
[4] Radford, A., Narasimhan, K., Salimans, T., et al. (2020) Improving Language Understanding by Generative Pretraning. https://www.cs.ubc.ca/~amuham01/LING530/papers/redford2018improving.pdf
[5] Radford, A., Wu, J., Child, R., et al. (2019) Language Models Are Unsupervised Multitask Learners. OpenAI Blog, 1.
[6] Liu, Y., Ott, M., Goyal, N., et al. (2020) RoBERTa: A Robustly Optimized BERT Pretraining Approach. https://arxiv.org/abs/1907.11692
[7] Martin, L., Muller, B., Suarez, P.J.O., et al. (2019) CamemBERT: A Tasty French Language Model. Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, 7203-7219. arXiv:1911.034894 https://doi.org/10.18653/v1/2020.acl-main.645
[8] Alsentzer, E., Murphy, J.R., Boag, W., et al. (2019) Publicly Available Clinical BERT Embeddings. Proceedings of the 2nd Clinical Natural Language Processing Workshop, 72-78. arXiv:1904.03323 https://doi.org/10.18653/v1/W19-1909
[9] Huang, K., Altosaar, J. and Ranganath, R. (2019) Clinica-1BERT: Modeling Clinical Notes and Predicting Hospital Readmission. arXiv:1904.05342
[10] Shoeybi, M., Pateary, M., Puri, R., et al. (2019) Megatron-LM: Training Multi-Billion Parameter Language Models Using Model Paralleslism. arXiv:1909.08053
[11] Clark, K., Luong, M.T., Le, Q.V., et al. (2020) ELECTRA: Pretraining Text Encoders as Discriminators Rather than Generators. arXiv:2003.10555
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Category:Ensemble properties - Vaspwiki
Category:Ensemble properties
In a molecular-dynamics calculation, VASP simulates a specific ensemble. In principle, any property of the system can be monitored, and then one can take the ensemble average of this property. For some observables, VASP provides convenient tags, articles, and files that help evaluate these so-called ensemble properties.
For any property
{\displaystyle {\mathcal {A}}}
of the system, we can define the observable macroscopic property
{\displaystyle {\mathcal {A}}_{\mathrm {obs} }}
by taking the ensemble average:
{\displaystyle {\mathcal {A}}_{\mathrm {obs} }=\left\langle {\mathcal {A}}\left(p(t),q(t)\right)\right\rangle _{\mathrm {time} }=\lim _{t_{\mathrm {obs} }\to \infty }{\frac {1}{t_{\mathrm {obs} }}}\int _{0}^{t_{\mathrm {obs} }}\,{\mathcal {A}}\left(p(t),q(t)\right)\,\mathrm {d} t.}
{\displaystyle t_{\mathrm {obs} }}
corresponds to the simulation time,
{\displaystyle p(t)}
{\displaystyle q(t)}
are the canonical momenta and positions, and an average of
{\displaystyle {\mathcal {A}}\left(p(t),q(t)\right)}
is taken over time
{\displaystyle t}
Pages in category "Ensemble properties"
Retrieved from "https://www.vasp.at/wiki/index.php?title=Category:Ensemble_properties&oldid=16500"
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Frequency response of rational object and rationalfit function object - MATLAB freqresp - MathWorks í•œêµ
Frequency Response of Data Stored In File
Frequency Response of S-Parameters Object
outputfreq
Frequency response of rational object and rationalfit function object
[response, outputfreq] = freqresp(fit,inputfreq)
[response, outputfreq] = freqresp(fit,inputfreq) calculates the frequency response, response of the fit of a rationalfit function object or a rational object at the specified input frequencies, inputfreq.
{\mathit{S}}_{21}
Fit a rational function to the data by using rationalfit.
fit_data = rationalfit(freq,S21);
Compute the frequency response by using the freqresp method and plot the magnitude and angle of the frequency response.
plot(freq/1e9,20*log10(abs(resp)))
plot(freq/1e9,unwrap(angle(resp)))
Perform rational fitting on a S-parameters object by using rational object.
fit_data = rational(S);
Compute the frequency response by using the freqresp function.
Plot the magnitude of the frequency response of the
{\mathit{S}}_{21}
plot(freq/1e9,20*log10(abs(squeeze(resp(2,1,:)))))
ylabel('Magnitude of Frequency Response');
Plot the angle of the frequency response of the
{\mathit{S}}_{21}
plot(freq/1e9,unwrap(angle(squeeze(resp(2,1,:)))))
ylabel('Angle of Frequency Response');
fit — Rational fit object
rfmodel.rational | object | rational | object | M-by-N array
Rational fit object, specified as a rfmodel.rational or rational object. The size of this object is M-by-N array.
inputfreq — Input frequency
Input frequency to compute and plot the frequency response, specified as a vector of positive frequencies in Hz.
response — Computed frequency response
Computed frequency response of each M-by-N fit, specified as a vector.
outputfreq — Frequency values same as input frequencies
Frequency values same as input frequencies, returned as a real positive vector.
rfmodel.rational | rationalfit | timeresp | pwlresp
Convert Scattering Parameter to Impulse Response for SerDes System (SerDes Toolbox)
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Depth-First Search (DFS) | Brilliant Math & Science Wiki
Karleigh Moore, Ken Jennison, and Jimin Khim contributed
Depth-first search (DFS) is an algorithm for searching a graph or tree data structure. The algorithm starts at the root (top) node of a tree and goes as far as it can down a given branch (path), then backtracks until it finds an unexplored path, and then explores it. The algorithm does this until the entire graph has been explored. Many problems in computer science can be thought of in terms of graphs. For example, analyzing networks, mapping routes, scheduling, and finding spanning trees are graph problems. To analyze these problems, graph-search algorithms like depth-first search are useful.
Depth-first searches are often used as subroutines in other more complex algorithms. For example, the matching algorithm, Hopcroft–Karp, uses a DFS as part of its algorithm to help to find a matching in a graph. DFS is also used in tree-traversal algorithms, also known as tree searches, which have applications in the traveling-salesman problem and the Ford-Fulkerson algorithm.
Depth-first search is a common way that many people naturally approach solving problems like mazes. First, we select a path in the maze (for the sake of the example, let's choose a path according to some rule we lay out ahead of time) and we follow it until we hit a dead end or reach the finishing point of the maze. If a given path doesn’t work, we backtrack and take an alternative path from a past junction, and try that path. Below is an animation of a DFS approach to solving this maze.
DFS is a great way to solve mazes and other puzzles that have a single solution.
Complexity of Depth-first Search
The main strategy of depth-first search is to explore deeper into the graph whenever possible. Depth-first search explores edges that come out of the most recently discovered vertex,
s
. Only edges going to unexplored vertices are explored. When all of
s
’s edges have been explored, the search backtracks until it reaches an unexplored neighbor. This process continues until all of the vertices that are reachable from the original source vertex are discovered. If there are any unvisited vertices, depth-first search selects one of them as a new source and repeats the search from that vertex. The algorithm repeats this entire process until it has discovered every vertex. This algorithm is careful not to repeat vertices, so each vertex is explored once. DFS uses a stack data structure to keep track of vertices.
Here are the basic steps for performing a depth-first search:
s
s
s
This animation illustrates the depth-first search algorithm:
Note: This animation does not show the marking of a node as "visited," which would more clearly illustrate the backtracking step.
Fill out the following graph by labeling each node 1 through 12 according to the order in which the depth-first search would visit the nodes:
Solution: source:wikipedia
Below are examples of pseudocode and Python code implementing DFS both recursively and non-recursively. This algorithm generally uses a stack in order to keep track of visited nodes, as the last node seen is the next one to be visited and the rest are stored to be visited later.
Python Implementation without Recursion
def depth_first_search(graph):
It is common to modify the algorithm in order to keep track of the edges instead of the vertices, as each edge describes the nodes at each end. This is useful when one is attempting to reconstruct the traversed tree after processing each node. In case of a forest or a group of trees, this algorithm can be expanded to include an outer loop that iterates over all trees in order to process every single node.
Pre-order DFS works by visiting the current node and successively moving to the left until a leaf is reached, visiting each node on the way there. Once there are no more children on the left of a node, the children on the right are visited. This is the most standard DFS algorithm.
Instead of visiting each node as it traverses down a tree, an in-order algorithm finds the leftmost node in the tree, visits that node, and subsequently visits the parent of that node. It then goes to the child on the right and finds the next leftmost node in the tree to visit.
A post-order strategy works by visiting the leftmost leaf in the tree, then going up to the parent and down the second leftmost leaf in the same branch, and so on until the parent is the last node to be visited within a branch. This type of algorithm prioritizes the processing of leaves before roots in case a goal lies at the end of a tree.
Depth-first search visits every vertex once and checks every edge in the graph once. Therefore, DFS complexity is
O(V + E)
. This assumes that the graph is represented as an adjacency list.
Breadth-first search is less space-efficient than depth-first search because BFS keeps a priority queue of the entire frontier while DFS maintains a few pointers at each level.
If it is known that an answer will likely be found far into a tree, DFS is a better option than BFS. BFS is good to use when the depth of the tree can vary or if a single answer is needed—for example, the shortest path in a tree. If the entire tree should be traversed, DFS is a better option.
Here is an example that compares the order that the graph is searched in when using a BFS and then a DFS (by each of the three approaches).[2]
Breadth First Search : 1 2 3 4 5
Pre-order: 1 2 4 5 3
In-order : 4 2 5 1 3
Post-order : 4 5 2 3 1
Depth-first search is used in topological sorting, scheduling problems, cycle detection in graphs, and solving puzzles with only one solution, such as a maze or a sudoku puzzle.
Other applications involve analyzing networks, for example, testing if a graph is bipartite. Depth-first search is often used as a subroutine in network flow algorithms such as the Ford-Fulkerson algorithm.
DFS is also used as a subroutine in matching algorithms in graph theory such as the Hopcroft–Karp algorithm.
Depth-first searches are used in mapping routes, scheduling, and finding spanning trees.
Gupta, D. BFS vs DFS for Binary Tree. Retrieved July 20, 2016, from http://www.geeksforgeeks.org/bfs-vs-dfs-binary-tree/
Cite as: Depth-First Search (DFS). Brilliant.org. Retrieved from https://brilliant.org/wiki/depth-first-search-dfs/
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Examine the integrals below. Consider the multiple tools you have available for integrating and use the best one. After evaluating each integral, write a short description of the method you used.
\int _ { - \infty } ^ { \infty } \frac { 1 } { x ^ { 2 } + 25 } d x
\int _ { 0 } ^ { 4 } \frac { x } { \sqrt { 16 - x ^ { 2 } } } d x
\int _ { - \infty } ^ { - 1 } \frac { 1 } { x ^ { 3 } } d x
\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 2 x - 3 } d x
=\frac{1}{25}\int_{-\infty}^{\infty}\frac{1}{(x/5)^2+1}dx
=\lim_{a\to-\infty}\frac{1}{25}\int_{a}^{0}\frac{1}{(x/5)^2+1}dx+\lim_{b\to\infty}\frac{1}{25}\int_{0}^{b}\frac{1}{(x/5)^2+1}dx
Use substitution. Let u = 16 – x^2.
A limit is needed to evaluate this integral.
=\lim_{c\to-\infty}\int_{c}^{-1}x^{-3}dx
\frac{1}{x^2-2x-3}=\frac{-1/4}{x+1}+\frac{1/4}{x-3}
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Polynomial Modeling - Course Hero
College Algebra/Polynomial Functions and Modeling/Polynomial Modeling
Polynomial Trends in Data
Linear and quadratic functions can be used to model data. For data that shows a nonlinear or non-quadratic trend, polynomials of higher degree may be used. In general, the higher the degree of the polynomial, the closer the curve can be made to fit the points. However, the coefficients of the terms may be harder to interpret, and the function may not be a good predictor of values outside the given data.
A function may be a better fit for a data set, but not necessarily a better model.
In general, look for the overall shape of the data, and try to choose the function with the lowest degree that will fit the trend. Most often, this will be a linear or quadratic function, but there may be data that fit the pattern of a higher-degree polynomial.
Polynomials of Best Fit
Just as technology can be used to find a line or quadratic of best fit, it is possible to find other polynomials to fit a given set of data. Graphing calculators will commonly have regression functions up to degree 3 (cubic), but online tools can be used to fit polynomials of any degree.
Modeling Data Trends with Polynomial Functions
The table shows data values from an experiment. Identify the polynomial of best fit for the data.
x
y
The pattern displayed by the data points resembles a cubic polynomial.
Use a graphing utility to identify the polynomial that best models the pattern of the data.
Follow the instructions for the graphing utility to identify the best fit curve. Use a polynomial curve of degree 3, based on the data plot.
The graphing utility will plot the curve and identify its algebraic rule.
<Polynomial Functions
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System of rules to convert information into another form or representation
For technical reasons, "Code#01 Bad Girl" and "Code#02 Pretty Pretty" redirect here. For the EPs by Ladies' Code, see Code 01 Bad Girl and Code 02 Pretty Pretty.
Find sources: "Code" – news · newspapers · books · scholar · JSTOR (March 2010) (Learn how and when to remove this template message)
Before giving a mathematically precise definition, this is a brief example. The mapping
{\displaystyle C=\{\,a\mapsto 0,b\mapsto 01,c\mapsto 011\,\}}
is a code, whose source alphabet is the set
{\displaystyle \{a,b,c\}}
and whose target alphabet is the set
{\displaystyle \{0,1\}}
. Using the extension of the code, the encoded string 0011001 can be grouped into codewords as 0 011 0 01, and these in turn can be decoded to the sequence of source symbols acab.
Using terms from formal language theory, the precise mathematical definition of this concept is as follows: let S and T be two finite sets, called the source and target alphabets, respectively. A code
{\displaystyle C:\,S\to T^{*}}
is a total function mapping each symbol from S to a sequence of symbols over T. The extension
{\displaystyle C'}
{\displaystyle C}
, is a homomorphism of
{\displaystyle S^{*}}
{\displaystyle T^{*}}
, which naturally maps each sequence of source symbols to a sequence of target symbols.
In this section, we consider codes that encode each source (clear text) character by a code word from some dictionary, and concatenation of such code words give us an encoded string. Variable-length codes are especially useful when clear text characters have different probabilities; see also entropy encoding.
A prefix code is a code with the "prefix property": there is no valid code word in the system that is a prefix (start) of any other valid code word in the set. Huffman coding is the most known algorithm for deriving prefix codes. Prefix codes are widely referred to as "Huffman codes" even when the code was not produced by a Huffman algorithm. Other examples of prefix codes are country calling codes, the country and publisher parts of ISBNs, and the Secondary Synchronization Codes used in the UMTS WCDMA 3G Wireless Standard.
Kraft's inequality characterizes the sets of codeword lengths that are possible in a prefix code. Virtually any uniquely decodable one-to-many code, not necessarily a prefix one, must satisfy Kraft's inequality.
Codes may also be used to represent data in a way more resistant to errors in transmission or storage. This so-called error-correcting code works by including carefully crafted redundancy with the stored (or transmitted) data. Examples include Hamming codes, Reed–Solomon, Reed–Muller, Walsh–Hadamard, Bose–Chaudhuri–Hochquenghem, Turbo, Golay, Goppa, low-density parity-check codes, and space–time codes. Error detecting codes can be optimised to detect burst errors, or random errors.
Main article: Brevity code
A cable code replaces words (e.g. ship or invoice) with shorter words, allowing the same information to be sent with fewer characters, more quickly, and less expensively.
Codes can be used for brevity. When telegraph messages were the state of the art in rapid long-distance communication, elaborate systems of commercial codes that encoded complete phrases into single mouths (commonly five-minute groups) were developed, so that telegraphers became conversant with such "words" as BYOXO ("Are you trying to weasel out of our deal?"), LIOUY ("Why do you not answer my question?"), BMULD ("You're a skunk!"), or AYYLU ("Not clearly coded, repeat more clearly."). Code words were chosen for various reasons: length, pronounceability, etc. Meanings were chosen to fit perceived needs: commercial negotiations, military terms for military codes, diplomatic terms for diplomatic codes, any and all of the preceding for espionage codes. Codebooks and codebook publishers proliferated, including one run as a front for the American Black Chamber run by Herbert Yardley between the First and Second World Wars. The purpose of most of these codes was to save on cable costs. The use of data coding for data compression predates the computer era; an early example is the telegraph Morse code where more-frequently used characters have shorter representations. Techniques such as Huffman coding are now used by computer-based algorithms to compress large data files into a more compact form for storage or transmission.
Character encodings are representations of textual data. A given character encoding may be associated with a specific character set (the collection of characters which it can represent), though some character sets have multiple character encodings and vice versa. Character encodings may be broadly grouped according to the number of bytes required to represent a single character: there are single-byte encodings, multibyte (also called wide) encodings, and variable-width (also called variable-length) encodings. The earliest character encodings were single-byte, the best-known example of which is ASCII. ASCII remains in use today, for example in HTTP headers. However, single-byte encodings cannot model character sets with more than 256 characters. Scripts that require large character sets such as Chinese, Japanese and Korean must be represented with multibyte encodings. Early multibyte encodings were fixed-length, meaning that although each character was represented by more than one byte, all characters used the same number of bytes ("word length"), making them suitable for decoding with a lookup table. The final group, variable-width encodings, is a subset of multibyte encodings. These use more complex encoding and decoding logic to efficiently represent large character sets while keeping the representations of more commonly used characters shorter or maintaining backward compatibility properties. This group includes UTF-8, an encoding of the Unicode character set; UTF-8 is the most common encoding of text media on the Internet.
Biological organisms contain genetic material that is used to control their function and development. This is DNA, which contains units named genes from which messenger RNA is derived. This in turn produces proteins through a genetic code in which a series of triplets (codons) of four possible nucleotides can be translated into one of twenty possible amino acids. A sequence of codons results in a corresponding sequence of amino acids that form a protein molecule; a type of codon called a stop codon signals the end of the sequence.
There are codes using colors, like traffic lights, the color code employed to mark the nominal value of the electrical resistors or that of the trashcans devoted to specific types of garbage (paper, glass, organic, etc.).
In marketing, coupon codes can be used for a financial discount or rebate when purchasing a product from a (usual internet) retailer.
In military environments, specific sounds with the cornet are used for different uses: to mark some moments of the day, to command the infantry on the battlefield, etc.
Specific games have their own code systems to record the matches, e.g. chess notation.
In the history of cryptography, codes were once common for ensuring the confidentiality of communications, although ciphers are now used instead.
Secret codes intended to obscure the real messages, ranging from serious (mainly espionage in military, diplomacy, business, etc.) to trivial (romance, games) can be any kind of imaginative encoding: flowers, game cards, clothes, fans, hats, melodies, birds, etc., in which the sole requirement is the pre-agreement on the meaning by both the sender and the receiver.
Encoding (in cognition) - a basic perceptual process of interpreting incoming stimuli; technically speaking, it is a complex, multi-stage process of converting relatively objective sensory input (e.g., light, sound) into a subjectively meaningful experience.
A content format - a specific encoding format for converting a specific type of data to information.
Semantics encoding of formal language A informal language B is a method of representing all terms (e.g. programs or descriptions) of language A using language B.
Data compression transforms a signal into a code optimized for transmission or storage, generally done with a codec.
Neural encoding - the way in which information is represented in neurons.
Memory encoding - the process of converting sensations into memories.
Decoding methods, methods in communication theory for decoding codewords sent over a noisy channel
Acronyms and abbreviations can be considered codes, and in a sense, all languages and writing systems are codes for human thought.
International Air Transport Association airport codes are three-letter codes used to designate airports and used for bag tags. Station codes are similarly used on railways but are usually national, so the same code can be used for different stations if they are in different countries.
Occasionally, a code word achieves an independent existence (and meaning) while the original equivalent phrase is forgotten or at least no longer has the precise meaning attributed to the code word. For example, '30' was widely used in journalism to mean "end of story", and has been used in other contexts to signify "the end".[1] [2]
^ Kogan, Hadass "So Why Not 29" Archived 2010-12-12 at the Wayback Machine American Journalism Review. Retrieved 2012-07-03.
^ "WESTERN UNION "92 CODE" & WOOD'S "TELEGRAPHIC NUMERALS"". Signal Corps Association. 1996. Archived from the original on 2012-05-09. Retrieved 2012-07-03.
Chevance, Fabienne (2017). "Case for the genetic code as a triplet of triplets". Proceedings of the National Academy of Sciences of the United States of America. 114 (18): 4745–4750. doi:10.1073/pnas.1614896114. PMC 5422812. PMID 28416671.
Codes and Abbreviations for the Use of the International Telecommunication Services (2nd ed.). Geneva, Switzerland: International Telecommunication Union. 1963. OCLC 13677884.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Code&oldid=1075196336"
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TemperatureEntropyChart - Maple Help
Home : Support : Online Help : Science and Engineering : ThermophysicalData : TemperatureEntropyChart
TemperatureEntropyChart(fluid)
TemperatureEntropyChart(fluid, trange, srange, opts, plotopts)
(optional) range of numeric values (optionally with a unit) representing the temperature values to be plotted
(optional) range of numeric values (optionally with a unit) representing the entropy values to be plotted
(optional) equations of the form isobars = value or pressurelabels = value
The TemperatureEntropyChart function generates a plot of the saturation dome and isobars on temperature and entropy axes.
The temperature range is given by the trange argument. It can be specified as a numeric range (which is interpreted as being in kelvins), or a range including units of temperature, such as
1..1000⟦\mathrm{degF}⟧
, or a range of Temperature objects, such as
\mathrm{Temperature}\left(1,\mathrm{degF}\right)..\mathrm{Temperature}\left(1000,\mathrm{degF}\right)
If the temperature range is not specified, the default minimum temperature displayed is the minimal temperature that Maple (or more specifically, the CoolProp library) can do computations with for the given fluid. The maximum temperature is either the maximum temperature that we can do computations with, or a temperature so that the range includes twice the height of the saturation dome -- whichever of the two is lower.
You can also supply the temperature range as an equation of the form
\mathrm{name}=\mathrm{low}..\mathrm{high}
"string"=\mathrm{low}..\mathrm{high}
\mathrm{low}..\mathrm{high}
was given by itself, except the axis label for the (vertical) temperature axis is the left-hand side
\mathrm{name}
"string"
The entropy range is given by the srange argument. Like the temperature range, it can be specified as a numeric range or a range including units of entropy. If no unit is given, the default used is
⟦\frac{\mathrm{kJ}}{\mathrm{kg}K}⟧
If the entropy range is not specified, the default range is selected as follows. The low end of the range is the left end of the saturation dome. The right end of the range is the top entropy value on the lowest isobar that is displayed (that is, the entropy at the maximum displayed temperature and the pressure of the lowest isobar).
The isobars = value option specifies how the isobars are to be plotted. The value term can take the following forms:
If value is a nonnegative integer, it specifies the approximate number of isobars to be plotted. These isobars are shown for pressures with roughly constant factors between them. If this value is not specified, it defaults to 9.
If value is a numeric range, possibly with units, then it is taken to be the pressure range within which the isobars are to be plotted. If this value is not specified, its default is determined as follows. The lowest pressure is the pressure on the saturation dome, at quality 1 and the minimal displayed temperature. The high end of the pressure range is taken at the maximal displayed temperature and at an entropy value near the center of the saturation dome, possibly somewhat to its left.
If value is a list of numeric values (possibly with units), then this is taken to be the list of pressures at which isobars are to be plotted.
\mathrm{unit}=u
u
is used to label the isotherms in the legend.
\mathrm{isobars}=[20,\mathrm{unit}=⟦\mathrm{inch_mercury}⟧]
would specify that there are to be about 20 isobars, to be labeled in inches of mercury.
If you supply multiple values that include a unit, either by supplying a list of options or a list of values or even both ends of a range, then the units must all be the same.
The pressurelabels = value option specifies how the pressure labels are placed.
If value is line, then pressures are labeled at a fixed temperature value, at 90% of the temperature range displayed. This is the default.
If value is alternating, then pressures are labeled at alternately 90% and 85% of the temperature range displayed. This can be useful if horizontal space is too tight to display all pressure labels on the same line.
You can supply extra plotting options, for example to give a title or to change properties of the axes. These are applied when the plots are combined, using plots[display].
\mathrm{with}\left(\mathrm{ThermophysicalData}\right):
The temperature-entropy chart for water.
\mathrm{TemperatureEntropyChart}\left(\mathrm{Water}\right)
For water, you might want to specify the temperature range (and unit).
\mathrm{TemperatureEntropyChart}\left(\mathrm{Water},100\mathrm{Unit}\left(\mathrm{degF}\right)..1000\mathrm{Unit}\left(\mathrm{degF}\right)\right)
If you want to specify the entropy range, you have to specify the temperature range, too.
\mathrm{TemperatureEntropyChart}\left(\mathrm{Water},100\mathrm{Unit}\left(\mathrm{degF}\right)..1000\mathrm{Unit}\left(\mathrm{degF}\right),2000\mathrm{Unit}\left(\frac{J}{K\mathrm{kg}}\right)..9000\mathrm{Unit}\left(\frac{J}{K\mathrm{kg}}\right)\right)
You can specify that more or fewer isobars should be used.
\mathrm{TemperatureEntropyChart}\left(\mathrm{Water},\mathrm{isobars}=10\right)
\mathrm{TemperatureEntropyChart}\left(\mathrm{Water},\mathrm{isobars}=5\right)
Or that you would like to see isobars from a particular pressure range.
\mathrm{TemperatureEntropyChart}\left(\mathrm{Water},\mathrm{isobars}=0.4\mathrm{Unit}\left(\mathrm{bar}\right)..40\mathrm{Unit}\left(\mathrm{bar}\right)\right)
This way, Maple still tries to put about 9 isobars into this range, but maybe you would like fewer. You can specify this a few different ways: by letting Maple pick the pressure values, or by specifying them explicitly.
\mathrm{TemperatureEntropyChart}\left(\mathrm{Water},\mathrm{isobars}=[5,0.4\mathrm{Unit}\left(\mathrm{bar}\right)..40\mathrm{Unit}\left(\mathrm{bar}\right)]\right)
\mathrm{TemperatureEntropyChart}\left(\mathrm{Water},\mathrm{isobars}=[0.4\mathrm{Unit}\left(\mathrm{bar}\right),\mathrm{Unit}\left(\mathrm{bar}\right),4\mathrm{Unit}\left(\mathrm{bar}\right),10\mathrm{Unit}\left(\mathrm{bar}\right),40\mathrm{Unit}\left(\mathrm{bar}\right)]\right)
The ThermophysicalData[TemperatureEntropyChart] command was introduced in Maple 2017.
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Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces | Journal of Inequalities and Applications | Full Text
Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces
Sunhong Lee1
In this paper, we establish strong convergence for the iterative scheme introduced by Sahu and Petruşel associated with Lipschitzian hemicontractive mappings in Hilbert spaces.
Let H be a Hilbert space, and let
T:H\to H
be a mapping. The mapping T is called Lipshitzian if there exists
L>0
\parallel Tx-Ty\parallel \le L\parallel x-y\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in H.
L=1
, then T is called nonexpansive and if
0\le L<1
, then T is called contractive.
T:H\to H
is said to be pseudocontractive (see, for example, [1, 2]) if
{\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in H
and it is said to be strongly pseudocontractive if there exists
k\in \left(0,1\right)
{\parallel Tx-Ty\parallel }^{2}\le {\parallel x-y\parallel }^{2}+k{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in H.
F\left(T\right):=\left\{x\in H:Tx=x\right\}
, and let K be a nonempty subset of H. A mapping
T:K\to K
is called hemicontractive if
F\left(T\right)\ne \mathrm{\varnothing }
{\parallel Tx-{x}^{\ast }\parallel }^{2}\le {\parallel x-{x}^{\ast }\parallel }^{2}+{\parallel x-Tx\parallel }^{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x\in H,{x}^{\ast }\in F\left(T\right).
It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractions. For the importance of fixed points of pseudocontractions, the reader may consult [1].
Theorem 1.1 Let K be a compact convex subset of a Hilbert space H, and let
T:K\to K
be a Lipschitzian pseudocontractive mapping.
{x}_{1}\in K
\left\{{x}_{n}\right\}
be a sequence defined iteratively by the Ishikawa iterative scheme
\left\{\begin{array}{c}{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{y}_{n},\hfill \\ {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}
\left\{{\alpha }_{n}\right\}
\left\{{\beta }_{n}\right\}
are sequences satisfying the conditions
0\le {\alpha }_{n}\le {\beta }_{n}\le 1
{lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0
{\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_{n}{\beta }_{n}=\mathrm{\infty }
\left\{{x}_{n}\right\}
Another iterative scheme which has been studied extensively in connection with fixed points of pseudocontractive mappings is the S-iterative scheme introduced by Sahu and Petruşel [4] in 2011.
In this paper, we establish strong convergence for the S-iterative scheme associated with Lipschitzian hemicontractive mappings in Hilbert spaces.
x,y\in H
\lambda \in \left[0,1\right]
, the following well-known identity holds:
{\parallel \left(1-\lambda \right)x+\lambda y\parallel }^{2}=\left(1-\lambda \right){\parallel x\parallel }^{2}+\lambda {\parallel y\parallel }^{2}-\lambda \left(1-\lambda \right){\parallel x-y\parallel }^{2}.
Theorem 2.2 Let K be a compact convex subset of a real Hilbert space H, and let
T:K\to K
be a Lipschitzian hemicontractive mapping satisfying
\left\{{\beta }_{n}\right\}
\left[0,1\right]
{\sum }_{n=1}^{\mathrm{\infty }}{\beta }_{n}=\mathrm{\infty }
{lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0
{x}_{1}\in K
\left\{{x}_{n}\right\}
be a sequence defined iteratively by the S-iterative scheme
\left\{\begin{array}{c}{x}_{n+1}=T{y}_{n},\hfill \\ {y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 1.\hfill \end{array}
\left\{{x}_{n}\right\}
converges strongly to the fixed point of T.
Proof From Schauder’s fixed point theorem,
F\left(T\right)
is nonempty since K is a compact convex set and T is continuous. Let
{x}^{\ast }\in F\left(T\right)
. Using the fact that T is hemicontractive, we obtain
{\parallel T{x}_{n}-{x}^{\ast }\parallel }^{2}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {x}_{n}-T{x}_{n}\parallel }^{2}
{\parallel T{y}_{n}-{x}^{\ast }\parallel }^{2}\le {\parallel {y}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel {y}_{n}-T{y}_{n}\parallel }^{2}.
With the help of (2.1), (2.2) and Lemma 2.1, we obtain the following estimates:
Substituting (2.4) and (2.5) in (2.3) we obtain
\begin{array}{rcl}{\parallel T{y}_{n}-{x}^{\ast }\parallel }^{2}& \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\left(1-{\beta }_{n}\right){\parallel {x}_{n}-T{y}_{n}\parallel }^{2}+{\beta }_{n}{\parallel T{x}_{n}-T{y}_{n}\parallel }^{2}\\ -{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}.\end{array}
Also, with the help of condition (C) and (2.6), we have
\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& =& {\parallel T{y}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+\left(1-{\beta }_{n}\right){\parallel {x}_{n}-T{y}_{n}\parallel }^{2}+{\beta }_{n}{\parallel T{x}_{n}-T{y}_{n}\parallel }^{2}\\ -{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{\parallel T{x}_{n}-T{y}_{n}\parallel }^{2}-{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{L}^{2}{\parallel {x}_{n}-{y}_{n}\parallel }^{2}-{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\\ =& {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}+{L}^{2}{\beta }_{n}^{2}{\parallel {x}_{n}-T{x}_{n}\parallel }^{2}-{\beta }_{n}\left(1-2{\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\\ =& {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\beta }_{n}\left(1-\left(2+{L}^{2}\right){\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}.\end{array}
Now, by
{lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0
{n}_{0}\in \mathbb{N}
n\ge {n}_{0}
{\beta }_{n}\le \frac{1}{2\left(2+{L}^{2}\right)},
and with the help of (2.8), (2.7) yields
{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-\frac{1}{2}{\beta }_{n}{\parallel {x}_{n}-T{x}_{n}\parallel }^{2},
\frac{1}{2}{\beta }_{n}{\parallel {x}_{n}-T{x}_{n}\parallel }^{2}\le {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2},
\frac{1}{2}\sum _{j=N}^{n}{\beta }_{j}{\parallel {x}_{j}-T{x}_{j}\parallel }^{2}\le {\parallel {x}_{N}-{x}^{\ast }\parallel }^{2}-{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}.
The rest of the argument follows exactly as in the proof of theorem of [3]. This completes the proof. □
T:K\to K
be a Lipschitzian hemicontractive mapping satisfying condition (C). Let
\left\{{\beta }_{n}\right\}
\left[0,1\right]
satisfying conditions (iv) and (v).
{P}_{K}:H\to K
is the projection operator of H onto K. Let
\left\{{x}_{n}\right\}
be a sequence defined iteratively by
\left\{\begin{array}{c}{x}_{n+1}={P}_{K}\left(T{y}_{n}\right),\hfill \\ {y}_{n}={P}_{K}\left(\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}T{x}_{n}\right),\phantom{\rule{1em}{0ex}}n\ge 1.\hfill \end{array}
\left\{{x}_{n}\right\}
Proof The operator
{P}_{K}
is nonexpansive (see, e.g., [2]). K is a Chebyshev subset of H so that
{P}_{K}
is a single-valued mapping. Hence, we have the following estimate:
\begin{array}{rcl}{\parallel {x}_{n+1}-{x}^{\ast }\parallel }^{2}& =& {\parallel {P}_{K}\left(T{y}_{n}\right)-{P}_{K}{x}^{\ast }\parallel }^{2}\\ \le & {\parallel T{y}_{n}-{x}^{\ast }\parallel }^{2}\\ \le & {\parallel {x}_{n}-{x}^{\ast }\parallel }^{2}-{\beta }_{n}\left(1-\left(2+{L}^{2}\right){\beta }_{n}\right){\parallel {x}_{n}-T{x}_{n}\parallel }^{2}.\end{array}
K=K\cup T\left(K\right)
is compact, and so the sequence
\left\{\parallel {x}_{n}-T{x}_{n}\parallel \right\}
is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.2. This completes the proof. □
Remark 2.4 In Theorem 1.1, putting
{\alpha }_{n}=1
0\le {\alpha }_{n}\le {\beta }_{n}\le 1
{\beta }_{n}=1
{lim}_{n\to \mathrm{\infty }}{\beta }_{n}=0
. Hence the S-iterative scheme is not the special case of Ishikawa iterative scheme.
Remark 2.5 In Theorems 2.2 and 2.3, condition (C) is not new; it is due to Liu et al. [6].
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis. Am. Math. Soc., Providence; 1976.
Sahu DR, Petruşel A: Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces. Nonlinear Anal. 2011, 74: 6012–6023. 10.1016/j.na.2011.05.078
Liu Z, Feng C, Ume JS, Kang SM: Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings. Taiwan. J. Math. 2007, 11: 27–42.
The authors would like to thank the referees for useful comments and suggestions.
Shin Min Kang & Sunhong Lee
School of CS and Mathematics, Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, 54660, Pakistan
Sunhong Lee
Correspondence to Sunhong Lee.
Kang, S.M., Rafiq, A. & Lee, S. Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces. J Inequal Appl 2013, 132 (2013). https://doi.org/10.1186/1029-242X-2013-132
Lipschitzian mappings
hemicontractive mappings
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Aerothermal Impact of Stator-Rim Purge Flow and Rotor-Platform Film Cooling on a Transonic Turbine Stage | J. Turbomach. | ASME Digital Collection
M. Pau,
, 1640 Rhode Saint Genèse, Belgium
D. Delhaye,
D. Delhaye
A. de la Loma,
A. de la Loma
P. Ginibre
Department of Turbine Aerodynamics,
, 77550 Moissy Cramayel, France
Pau, M., Paniagua, G., Delhaye, D., de la Loma, A., and Ginibre, P. (January 11, 2010). "Aerothermal Impact of Stator-Rim Purge Flow and Rotor-Platform Film Cooling on a Transonic Turbine Stage." ASME. J. Turbomach. April 2010; 132(2): 021006. https://doi.org/10.1115/1.3142859
The sealing of the stator-rotor gap and rotor-platform cooling are vital to the safe operation of the high-pressure turbine. Contrary to the experience in subsonic turbines, this paper demonstrates the potential to improve the efficiency in transonic turbines at certain rim seal rates. Two types of cooling techniques were investigated: purge gas ejected out of the cavity between the stator rim and the rotor disk, and cooling at the rotor-platform. The tests were carried out in a full annular stage fed by a compression tube at
M2is=1.1
Re=1.1×106
, and at temperature ratios reproducing engine conditions. The stator outlet was instrumented to allow the aerothermal characterization of the purge flow. The rotor blade was heavily instrumented with fast-response pressure sensors and double-layer thin film gauges. The tests are coupled with numerical calculations performed using the ONERA’s code ELSA. The results indicate that the stator-rotor interaction is significantly affected by the stator-rim seal, both in terms of heat transfer and pressure fluctuations. The flow exchange between the rotor disk cavity and the mainstream passage is mainly governed by the vane trailing edge shock patterns. The purge flow leads to the appearance of a large coherent vortex structure on the suction side of the blade, which enhances the overall heat transfer coefficient due to the blockage effect created. The impact of the platform cooling is observed to be restricted to the platform, with negligible effects on the blade suction side. The platform cooling results in a clear attenuation of pressure pulsations at some specific locations. The experimental and computational fluid dynamics results show an increase in the turbine performance compared with the no rim seal case. A detailed loss breakdown analysis helped to identify the shock loss as the major loss source. The presented results should help designers improve the protection of the rotor platform while minimizing the amount of coolant used.
aerodynamics, blades, heat transfer, numerical analysis, rotors, stators, transonic flow, turbines, turbomachinery
Blades, Flow (Dynamics), Pressure, Rotors, Turbines, Cavities, Stators, Coolants, Cooling
Effect of the Hub Endwall Cavity Flow on the Flow Field of a Transonic High-Pressure Turbine
The Effect of Coolant Injection on the Endwall Flow of High-Pressure Turbine
Mainstream Aerodynamic Effects Due to Wheelspace Coolant Injection in a High Pressure Turbine Stage
Aerodynamic Aspects of the Sealing of Gas-Turbine Rotor-Stator Systems
Thermal Details in a Rotor-Stator Cavity at Engine Conditions With a Mainstream
Desportes de la Fosse
A Combined Experimental and Numerical Investigation of the Flow in a Heated Rotor/Stator Cavity With a Centripetal Injection
Endwall Cavity Flow Effects on Gas Path Aerodynamics in an Axial Flow Turbine
Experimental and Numerical Investigations of the Influence of Rotor Blades on Hot Gas Ingestion Into the Upstream Cavity of an Axial Turbine Stage
C. -Z
Film Cooling Research on the Endwall of a Turbine Nozzle Guide Vane in a Short Duration Annular Cascade: Experimental Technique and Results
,” IMECE Paper No. 2004-59852.
Film Cooling Effectiveness on a Rotating Turbine Platform Using Pressure Sensitive Paint Technique
,” ASME Paper. 92-GT-336.
Rotor/Stator Interaction in Transonic HP Turbines
Effects of Aerodynamic Unsteadiness in Axial Turbomachines
Improved Fast Response Instrumentation for Short-Duration Wind Tunnels
Novel 2D Transient Heat Conduction Calculation in a Cooled Rotor: Ventilation Preheating—Blowdown Flux
Determination of the Efficiency of a Cooled HP Turbine in a Blowdown Facility
Ames Research Staff
Equations, Tables and Charts for Compressible Flow
elsA: An Efficient Object-Oriented Solution to CFD Computations
,” AIAA Paper No. 2002-0108
Accurate Turbine Inertia Measurement
J. Experimental Mechanics
Performance Analysis of a Transonic High Pressure Turbine
Off-Design Performance of a Single-Stage Transonic Turbine
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Synthesized Statistics - Permutations and Combinations | JeffAstor.com
Synthesized Statistics - Permutations and Combinations
Statistics is tough for me. I'm only ok at math and I never intuitively understand something the first time I hear it. So when I sucked at MIT's intro to probability course, I took another one from Harvard and synthesized my knowledge of the two together into this series of blog posts. Most of the time, I try and convert the mathematics into code, and that gives me slightly better insight into how things work. Follow along and hopefully you'll learn something too.
Here's the part I both simultaneously love and hate. The math is cool because it just works, but if I can't conceptualize it visually, it takes me a long time to wrap my head around it.
The first thing I had to get comfy with are factorials, and they're by far the easiest math we deal with. We designate factorials by placing an exclamation point after the number. It end up looking like
n!
Turning Math Into Code
n!
translates pretty evenly into this code block:
factorial(3) # 6 because 3 * 2 * 1 is 6
factorial(4) # 24
We are essentially counting down from n and multiplying each value by the previous total. Python's range() function takes three arguments, the starting value, the ending value, and the step. By ending at 1 and using a step of -1, we first multiply total by 3, and then by 2, and then by 1.
If you want to get really terse, you could shorten this to something like this:
Personally, I like the first version better. However, there's two things here that both functions don't handle correctly.
First, the math defines the factorial(0) to be 1. Our function would return zero, so we'll need to account for that. Also, it's impossible to calculate factorials for a negative
n
, so we should handle that more appropriately as well. Doing so might give you something close to this:
raise ValueError('factorial() not defined for negative values')
That's not quite as pretty, but it's functional.
Fortunately, we can check our work by importing the math module and using the built in factorial function.
print(math.factorial(5)) # 120
print(math.factorial(4)) # 24
print(math.factorial(3)) # 6
print(math.factorial(-1)) # error
You'll notice that the error printed is the same as the one used in our custom function. And you're right, I stole it.
So why do we need to know all this mathy goodness?
Unit 1 in Harvard's Intro to Probability course covers the math of counting. I enjoyed this section and found examples to be the best way to conceptualize the math. Let me preface this by saying that some forms of counting are easier than others. We'll start small and build up.
The easiest kind of counting is done when we are looking to sample from a collection of items with replacement. A good example would be choosing a four digit passcode for your smart phone. You can pick 4 numbers, and you're welcome to use any of those numbers more than once. That's what with replacement refers to - being able to choose an item from a collection, replace it, and then choose it again.
If you were to count the total number of unique passcodes available, you would get
10^{4}
, or 10,000 possible choices. That's easy math. As a rule of thumb, when choosing
k
items from a collection of size
n
with replacement, the formula will always be
n^{k}
The next type of counting we'll learn is how to calculate the number of ways to choose groups of size
k
n
items in a particular order.
This is essentially discussing the number of permutations. Here's a concrete example for when permutations come into play. We have a deck of 52 cards. How many possible ways can we deal 5 cards?
One example would be the dealer placing down an Ace of spades and a 2 of diamonds on the flop. Then up comes a Queen of hearts, followed by a 7 of clubs, and finally a 10 of diamonds. That's just one possible outcome, and things start to get out of hand if we try to count every possible permutation by hand.
The answer can instead be found by using the following formula:
P_{k}^{n} = \dfrac{n!}{(n-k)!}
So now we know why we need factorials.
Plug in 52 (the number of cards) for
n
, and 5 (the group size) for
k
. What value do you get?
\dfrac{52!}{(52-5)!}
Using our previously defined function, we can calculate the answer as 311,875,200. Memorize that number and you'll be super fun at cocktail parties.
If you think about it, what we're essentially doing is saying that at first we can choose any one of the 52 cards. The next time we deal a card, we can choose any one of the remaining 51 cards, then 50 cards, then 49, and finally 48.
That's why we divide
52!
47!
. Neat, huh?
Ok, so what if order doesn't matter? Well in that case, we're no longer dealing with permutations.
In general, combinations provide an answer to the question: "How many ways can we create a subset
k
n
items?".
We write that out mathematically as
(_{k}^{n})
That looks kinda weird, right? We say this out loud like
n
k
n
k
takes advantage of our previous knowledge about permutations, coupled with the fact that we don't need to know the order, so we divide that value by
k!
Doing so gives us this formula:
\displaystyle\binom{n}{k} = \dfrac{P_{k}^{n}}{k!} = \dfrac{ \dfrac{n!}{(n-k)!}}{k!} = \dfrac{n!}{(n-k)!k!}
Ok, that makes sense. Now what if we wanted to figure out how many distinct 5-card hands we could deal to a given player? Well in this case, the order doesn't matter, so we're working with combinations.
So what is 52 choose 5?
\dfrac{52!}{(52-5)!5!}
comes out to 2,598,960.
k!
here is simply the number of ways we can order
k
items. Dividing by this amount makes sense when the order of the items isn't taken into account.
With those two formulas, we're well equipped to take on most counting problems. Head over to the MIT or Harvard EdX course and see how you do with their practice problems.
Here's some of the resources I use to think through problems in this category:
Harvard Intro to Probability Course - https://courses.edx.org/courses/course-v1:HarvardX+STAT110x+3T2019/course/
MITs Probability - The Science of Uncertainty and Data - https://www.edx.org/course/probability-the-science-of-uncertainty-and-data
Jeremy Kun - Probability Theory: A Primer. https://jeremykun.com/2013/01/04/probability-theory-a-primer/
Khan Academy - Probability. https://www.khanacademy.org/math/probability/probability-geometry
Doing Data Science in the Browser
Conditional Probability and the Monty Hall Game
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2022 Gelfand-type problems involving the 1-Laplacian operator
Alexis Molino, Sergio Segura de León
Alexis Molino,1 Sergio Segura de León2
1Departamento de Matemáticas, Universidad de Almería, Ctra. de Sacramento sn. 04120, La Cañada de San Urbano, Almería,
2Departament d’Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Valencia, Spain
In this paper, the theory of Gelfand problems is adapted to the 1-Laplacian setting. Concretely, we deal with the following problem:
\left\{\begin{array}{ll}-{\mathrm{\Delta }}_{1}u=\lambda f\left(u\right)\hfill & \text{in }\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on }\partial \mathrm{\Omega },\hfill \end{array}\right\
\mathrm{\Omega }\subset {ℝ}^{N}
N\ge 1
) is a domain,
\lambda \ge 0
f:\left[0,+\infty \left[\to \right]0,+\infty \left[
is any continuous increasing and unbounded function with
f\left(0\right)>0
We prove the existence of a threshold
{\lambda }^{*}=\frac{h\left(\mathrm{\Omega }\right)}{f\left(0\right)}
h\left(\mathrm{\Omega }\right)
being the Cheeger constant of
\mathrm{\Omega }
) such that there exists no solution when
\lambda >{\lambda }^{*}
and the trivial function is always a solution when
\lambda \le {\lambda }^{*}
. The radial case is analyzed in more detail, showing the existence of multiple (even singular) solutions as well as the behavior of solutions to problems involving the
p
-Laplacian as
p
1
, which allows us to identify proper solutions through an extra condition.
Alexis Molino. Sergio Segura de León. "Gelfand-type problems involving the 1-Laplacian operator." Publ. Mat. 66 (1) 269 - 304, 2022. https://doi.org/10.5565/PUBLMAT6612211
Received: 25 May 2020; Accepted: 22 December 2020; Published: 2022
Primary: 35J20 , 35J75 , 35J92
Keywords: 1-Laplacian operator , Gelfand problem , nonlinear elliptic equations
Alexis Molino, Sergio Segura de León "Gelfand-type problems involving the 1-Laplacian operator," Publicacions Matemàtiques, Publ. Mat. 66(1), 269-304, (2022)
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On the spectral instability of parallel shear flows
Emmanuel Grenier; Yan Guo; Toan T. Nguyen
This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number
R\to \infty
. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.
author = {Emmanuel Grenier and Yan Guo and Toan T. Nguyen},
title = {On the spectral instability of parallel shear flows},
TI - On the spectral instability of parallel shear flows
%T On the spectral instability of parallel shear flows
Emmanuel Grenier; Yan Guo; Toan T. Nguyen. On the spectral instability of parallel shear flows. Séminaire Laurent Schwartz — EDP et applications (2014-2015), Talk no. 22, 14 p. doi : 10.5802/slsedp.82. https://slsedp.centre-mersenne.org/articles/10.5802/slsedp.82/
[1] P. G. Drazin, W. H. Reid, Hydrodynamic stability. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University, Cambridge–New York, 1981.
[2] E. Grenier, Y. Guo, and T. Nguyen, Spectral instability of symmetric shear flows in a two-dimensional channel, arXiv:1402.1395.
[3] E. Grenier, Y. Guo, and T. Nguyen, Spectral instability of characteristic boundary layer flows, arXiv:1406.3862.
[4] W. Heisenberg, Über Stabilität und Turbulenz von Flüssigkeitsströmen. Ann. Phys. 74, 577–627 (1924)
[5] W. Heisenberg, On the stability of laminar flow. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pp. 292–296. Amer. Math. Soc., Providence, R. I., 1952.
[6] C. C. Lin, The theory of hydrodynamic stability. Cambridge, at the University Press, 1955.
[7] W. Orr, Stability and instability of steady motions of a perfect liquid and of a viscous fluid, Parts I and II, Proc. Ir. Acad. Sect. A, Math Astron. Phys. Sci., 27 (1907), pp. 9-68, 69-138.
[8] Lord Rayleigh, On the stability, or instability, of certain fluid motions. Proc. London Math. Soc. 11 (1880), 57–70.
[9] H. Schlichting, Boundary layer theory, Translated by J. Kestin. 4th ed. McGraw–Hill Series in Mechanical Engineering. McGraw–Hill Book Co., Inc., New York, 1960.
[10] A. Sommerfeld, Ein Beitrag zur hydrodynamischen Erklärung der turbulent Flussigkeitsbewe-gung, Atti IV Congr. Internat. Math. Roma, 3 (1908), pp. 116-124.
[11] W. Wasow, The complex asymptotic theory of a fourth order differential equation of hydrodynamics. Ann. of Math. (2) 49, (1948). 852–871.
[12] W. Wasow, Asymptotic solution of the differential equation of hydrodynamic stability in a domain containing a transition point. Ann. of Math. (2) 58, (1953). 222–252.
[13] W. Wasow, Linear turning point theory. Applied Mathematical Sciences, 54. Springer-Verlag, New York, 1985. ix+246 pp.
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Electric Flux | Brilliant Math & Science Wiki
Lawrence Chiou, manjunadh ch, and Eli Ross contributed
Coulomb's law provides a way to obtain the electric field produced by a collection of charges. However, summing Coulomb's law over various continuous distributions of charge can be daunting and complicated. Instead of approaching things one point charge at a time, one can tackle the equivalent problem of determining the charge distribution in a region based on the electric field in that region. Naturally, the more charge in a region bounded by a box, the more electric field lines one would expect to pass through the box. Thus, one would expect some relationship between the regional charge distribution and the electric field.
In describing a region of a field, which assigns a magnitude and direction—in other words, a vector—to each point in space, it makes sense to draw an analogy with water currents. The electric field vectors that pass through a surface in space can be likened to the flow of water through a net. The greater the magnitude of the lines, or the more oriented the lines are against (perpendicular to) the surface, the greater the flow, or flux. This analogy forms the basis for the concept of electric flux.
Definition of Electric Flux
Qualitative Statement of Gauss' Law
A flux through a given surface can be "inward" or "outward" depending on which way counts as "in" or "out"—that is, flux has a definite orientation.
For a closed surface (a surface with no holes), the orientation of the surface is generally defined such that flux flowing from inside to outside counts as positive, outward flux, while flux from the outside to the inside counts as negative, inward flux. To remember this choice of orientation, we divide the closed surface into many small patches of surface and assign a vector
\mathbf{a}_i
to each small patch of surface that indicates the normal (perpendicular) to the surface. In addition, the magnitude of each
\mathbf{a}_i
If the surface is partitioned into patches that are sufficiently small, then the electric field
\mathbf{E}_i
at all points on each patch essentially becomes constant. In that case, the electric flux
\Phi_\text{patch}
through the patch is given by the dot product, which calculates the component of
\mathbf{E}_i
\mathbf{a}_i
\Phi_\text{patch} = \mathbf{E}_i \cdot \mathbf{a}_i
The total electric flux through the entire surface, meanwhile, is the sum over all patches:
\Phi = \sum_i \mathbf{E}_i \cdot \mathbf{a}_i.
\mathbf{a}_i
become vanishingly small, as in the case of a continuous surface, the sum is replaced with a surface integral:
\Phi = \int_S \mathbf{E} \cdot d\mathbf{a}.
S
" in the limits of integration indicates that the integral is to be taken across the entire surface for all infinitesimal surface elements
d \mathbf{a}
. Fortunately, the electric flux can often be computed without resorting to computing the integral explicitly.
Compute the electric flux across a spherical surface of radius
R
that contains a charge
q
at its center.
In this case, the electric field is the same at all points on the surface. Furthermore, the field is always perpendicular to the surface. Therefore, the perpendicular component of the electric field summed across the entire surface is simply the electric field at a distance
R
from the charge multiplied by the area of the surface. Thus
\Phi = E \cdot 4 \pi R^2 = \frac{1}{4\pi\epsilon_0} \frac{q}{R^2} \cdot 4 \pi R^2 = \frac{q}{\epsilon_0}.
Compute the electric flux across a cube of side length
s
placed (a) perpendicular to and (b)
45^\circ
with respect to a uniform electric field of magnitude
E
If the cube is placed parallel to the field, then the four faces parallel to the field have zero flux. Of the two faces with nonzero flux, one face contains flux
-E s^2
("inward" flux) while the other contains flux
E s^2
("outward" flux). Therefore, the total flux is zero.
If the cube is placed
45^\circ
with respect to the field, then two faces each contain flux
-E s^2 \cos{45^\circ} = -E s^2 / \sqrt{2}
. Similarly, the other two faces with nonzero flux contain
E s^2 \cos{45^\circ} = E s^2 / \sqrt{2}
. Therefore, the total flux is again zero.
In the example of the flux through a spherical surface, the flux was independent of the radius of the sphere. In that case, the total flux depended only on the charge contained inside the sphere. This makes sense intuitively; as the sphere grows larger, the sphere captures "more" electric field, but this always accompanies a concomitant decrease in field strength. The factors of
R^2
growth and
1/R^2
decay exactly cancel out.
The same, it turns out, holds for other surfaces as well. If the walls of a box containing a charge are expanded, the total flux through the box remains the same, as the increased number of field lines captured by the expanded walls is exactly canceled by a decrease in field strength farther from the walls.
But what if the box contains no charge? In the case of the cube, no matter how the cube was oriented, the total electric flux always came out to zero. Although the flux through some sides of the cube was positive (e.g., "outward" flux), it was always balanced exactly by negative flux through other sides of the cube (e.g., "inward" flux).
Based these two observations, one might conjecture that flux is always independent of the dimensions of the containing surface and depends only on the magnitude of the total charge enclosed. This forms the basis for Gauss' law, which holds that the total electric flux through a surface is proportional to the enclosed charge. Furthermore, on the basis of the example of the spherical surface, for which the total electric flux was
q/\epsilon_0
, the constant of proportionality must be
1/\epsilon_0
A more detailed discussion of Gauss' law can be found in the accompanying page.
q
is split by one face of a cube. What is the total electric flux through the cube?
It is not easy to compute the electric flux directly using the definition. However, by Gauss' law, we know that a surface containing the entire charge must have total flux
q/\epsilon_0
. Therefore, the cube, which contains half of the total flux from the charge, must contain electric flux
q/(2\epsilon_0)
A point charge of charge
q = 96\epsilon_0
is placed at one corner of a cube as shown. What is the electric flux through the shaded face?
[1] Young, H.D. University Physics. Thirteenth edition. Pearson, 2012.
Cite as: Electric Flux. Brilliant.org. Retrieved from https://brilliant.org/wiki/electric-flux/
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{\displaystyle {\ce {Al_2(SO_4)_3}}}
{\displaystyle {\ce {H-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-H}}}
{\displaystyle {\ce {H-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-{\overset {\displaystyle H \atop |}{\underset {| \atop \displaystyle H}{C}}}-H}}}
Chemical names in answer to limitations of chemical formulae
Polymers in condensed formulae
Ions in condensed formulae
Non-stoichiometric chemical formulae
Retrieved from "https://en.wikipedia.org/w/index.php?title=Chemical_formula&oldid=1086738185"
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Some theoretical results concerning non newtonian fluids of the Oldroyd kind
Fernández-Cara, Enrique ; Guillén, Francisco ; Ortega, Rubens R.
Compactness of conformal metrics with positive gaussian curvature in
{ℝ}^{2}
Cheng, Kuo-Shung ; Lin, Chang-Shou
Global solutions of the Cauchy problem for a viscous polytropic ideal gas
Global solvability for the degenerate Kirchhoff equation
Comparison results between minimal barriers and viscosity solutions for geometric evolutions
Bellettini, Giovanni ; Novaga, Matteo
Bressan, Alberto ; Colombo, Rinaldo M.
On the regularity of boundary traces for the wave equation
Duality in the spaces of solutions of elliptic systems
Nacinovich, Mauro ; Shlapunov, Alexandre ; Tarkhanov, Nikolai
Semistable quotients
Heinzner, Peter ; Migliorini, Luca ; Polito, Marzia
Licois, Jean René ; Véron, Laurent
Homoclinic and periodic orbits for hamiltonian systems
Felmer, Patricio L. ; Silva, Elves A. de B.
Global existence and blow-up for a shallow water equation
Constantin, Adrian ; Escher, Joachim
Geometry of biinvariant subsets of complex semisimple Lie groups
Fels, Gregor ; Geatti, Laura
Intersection lattices and topological structures of complements of arrangements in
{\mathrm{ℂℙ}}^{2}
The Cauchy problem for degenerate parabolic equations in Gevrey classes
Kajitani, Kunihiko ; Mikami, Masahiro
Homogenization of elliptic problems in a fiber reinforced structure. Non local effects
Bellieud, Michel ; Bouchitté, Guy
Isometries of the Teichmüller metric
Abate, Marco ; Patrizio, Giorgio
The dam problem for nonlinear Darcy's laws and Dirichlet boundary conditions
Carrillo, José ; Lyaghfouri, Abdeslem
An exponential transform and regularity of free boundaries in two dimensions
Gustafsson, Björn ; Putinar, Mihai
Counterexamples to the Gleason problem
Backlund, Ulf ; Fällström, Anders
Exact boundary controllability of Galerkin's approximations of Navier-Stokes equations
Lions, Jacques-Louis ; Zuazua, Enrique
On the Birkhoff normal form of a completely integrable hamiltonian system near a fixed point with resonance
Kappeler, Thomas ; Kodama, Yuji ; Némethi, Andras
Rate of approach to a singular steady state in quasilinear reaction-diffusion equations
Dold, James W. ; Galaktionov, Victor A. ; Lacey, Andrew A. ; Vázquez, Juan Luis
Monotonicity and symmetry of solutions of
p
-Laplace equations,
1<p<2
, via the moving plane method
Damascelli, Lucio ; Pacella, Filomena
Sur l'existence d'intégrales premières holomorphes
On the period map for abelian covers of projective varieties
Estimations du type Nevanlinna pour les applications non dégénérées de
{ℂ}^{n}
{ℂ}^{n}
Ounaies, Myriam
Rotating drops trapped between parallel planes
Athanassenas, Maria
Optimal regularity for mixed parabolic problems in spaces of functions which are Hölder continuous with respect to space variables
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The decarboxylation of ethanoic acid will produce carbon(lV) oxide and
\mathrm{H}-\begin{array}{c}\mathrm{H}\\ \underset{|}{\overset{|}{\mathrm{C}}}\\ \mathrm{H}\end{array}-\begin{array}{c}\mathrm{O}\\ \underset{ }{\overset{||}{\mathrm{C}}}\\ \end{array}-\begin{array}{c}\mathrm{H}\\ \underset{|}{\overset{|}{\mathrm{C}}}\\ \mathrm{H}\end{array}-\mathrm{H}
The compound above is an
alkanoate
akanol
The compound that will react with sodium hydroxide to form salt and water only is
Which of the following compounds in soluton will turn red litmus paper blue?
R'OR''
R-OC||-N|R-R
R-C\R/O/
The dehydration of ammonium salt of alkanoic acids produces a compound with the general fomula.
R-C\OR/O/
R-C\NH2/O/
Which of the following fractions is used as raw material for the cracking process?
An organic compund with a pleasant smell is likely to have a general formula
CnH2n+1COOCnH2n+1
CnH2n+1COCnH2n+1
A primary amide is generally represented by the formula
RCONHR
{\mathrm{CH}}_{3}-\begin{array}{c}\mathrm{H}\\ \underset{|}{\overset{|}{\mathrm{C}}}\\ {\mathrm{CH}}_{3}\end{array}-{\mathrm{CH}}_{2}-\mathrm{CH}={\mathrm{CH}}_{2}
The IUPAC nomenclature for the compound above is
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Sketch each collection of tiles below. Name the collection using a simpler algebraic expression, if possible. If it is not possible to simplify the expression, explain why not. 4-113 HW eTool (CPM)
(-2x)+5+3x-4x+(-1)+(-x)
Try drawing your own Expression Mat to help you simplify this expression.
−4x+4
Six plus four times a number, plus four minus four times the number.
Use the tiles drawn for you.
4
x
-tiles and
4
x
-tiles, there are no
x
-tiles left.
Then you combine the remaining unit tiles.
10
Three groups of a number plus two.
The tiles are drawn for you. Can you simplify the expression?
5+7x^2+4x
Take a look at this expression. Are there any terms you can combine? Remember, you can only combine terms which share the exact same variable.
4x^2-2x^2+(-6)+3
Draw a picture to help you simplify this expression.
Use parts (b) and (c) as guides.
Use the eTool below to simplify the expressions.
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Implement second-order variable-tuned filter - Simulink - MathWorks Nordic
Second-Order Filter (Variable-Tuned)
Implement second-order variable-tuned filter
Depending on the Filter type selected in the block menu, the Second-Order Filter block implements the following transfer function. The Fn input determines the filter natural frequency
{f}_{n}={\omega }_{n}/\left(2\pi \right)
of the filter.
H\left(s\right)=\frac{{\omega }_{n}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}
H\left(s\right)=\frac{{s}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}
H\left(s\right)=\frac{2\zeta {\omega }_{n}s}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}
Bandstop (Notch) filter:
H\left(s\right)=\frac{{s}^{2}+{\omega }_{n}^{2}}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}}
\begin{array}{c}s=\text{Laplace operator}\\ {\omega }_{n}=\text{natural frequency; }{\omega }_{n}=2\pi {f}_{n}\\ \zeta =\text{damping ratio }\text{}\text{}\text{(called Zeta in the block menu)}\end{array}
Specify the type of filter: Lowpass, Highpass, Bandpass, or Bandstop (Notch) (default).
Initial natural frequency fn (Hz)
The initial natural frequency of the filter, in hertz. This value must be a scalar or a vector. Default is 120.
Q=\frac{1}{2\zeta }
BW=\frac{{f}_{n}}{Q}=2\zeta {f}_{n}
The inputs accept vectorized signals of N values, thus implementing N filters. This capability is particularly useful for designing controllers in three-phase systems (N = 3).
The power_SecondOrderFilterTuned example shows various uses of the Second-Order Filter (Variable-Tuned) block with two Filter type parameter settings (Lowpass and Bandstop).
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Externalities | Brilliant Math & Science Wiki
In economics, externalities are a cost or a benefit that occurs to a bystander. For instance, a factory may pollute the air in it's town. The company running the factory may not have to pay for the costs of this pollution, nor may the customers that buy the factory's products. However, the people who live in that town are bystanders that pay the cost of the pollution externality produced by this theoretical factory. Externalities can be positive or negative, and occur on the supply side or the demand side. As an example, some forms of construction produce positive externalities. Building a new luxury high-rise condominium in a neighborhood can bring in more customers (the new residents) to local businesses (a positive supply-side externality) and might raise property values (a positive demand-side externality). Sometimes these benefits are referred to as spillovers, as in Xerox's PARC's developments in the 1970s of GUIs, the computer mouse, laser printers, WYSIWYG text editors and more were technology spillovers that helped other companies, like Apple and Microsoft, produce revolution technologies, but did less to serve Xerox.
Supply and Demand Curve with a Pollution Externality
Externalities are a type of market failure where the market does not allocate resources efficiently. For instance, the graph to the right is of a negative supply externality. The producer is providing some good according to their private marginal cost, but there is a gap between that and what society pays for the production of that good. Again in the case of pollution, this Deadweight Loss can take the form of higher medical costs for the nearby population, contaminated drinking water that has to be cleaned up, harming crop production, etc. and is absorbed by society, with the difference between what the producer pays and the what the overall society
+
the producer pay called the marginal damage. Externalities distort the supply and demand curve, instead of the supplier bearing the full costs and benefits of an externality like pollution (the optimum price), the market pays an artificially high or low equilibrium price.
Sometimes, governments can step in to rebalance externalities. For instance by regulating the amount of pollution by factories and making those companies pay for clean up efforts, or by giving tax breaks to large building projects that bring in additional revenues to the community. Other times, markets rebalance externalities themselves. Private individuals can confront polluters and demand compensation. Either way, this is called internalizing the externality, i.e. forcing the supply and demand curves to adjust according to the total social marginal costs and benefits.
Externality Examples
Private and Public Solutions to Externalities
Efficient Externality Internalization
XEROX PARC's technology developments are a good example of a positive production externality. In this case, XEROX developed technologies that other companies and organizations used in their own products. This saved these "other companies" research and development costs; it created a marginal benefit to society that the private company, XEROX PARC, did not receive. Because of this, XEROX PARC underproduced products from these technologies. As the graph to the right shows, the supply curve was higher than the market's supply curve. In this specific example, a company like Apple or Microsoft, which allegedly got the ideas for GUIs, mice, and other technologies from XEROX PARC, would have a supply curve closer to the social marginal cost supply curve.
A company who pollutes the drinking water with lead, is creating a negative production externality. They are not paying the costs of their pollution, so can produce more supply at lower costs than they should if they were fully internalizing all costs.
Suppose a paper mill pollutes lead into the Flint River. This paper mill produces reams of paper at $5.00/ream and does not pay for the cost of the lead pollution in the river. If the town of Flint Michigan gets it's water supply from the Flint River, and it costs $1.00 to remove lead equal to the output of one ream of paper, i.e. per ream of paper produced the town of Flint Michigan spends $1.00 to remove lead from it's drinking water, what does a graph of the supply and demand curves look like? How would you calculate the loss, the total social costs? And what would you need to know to calculate it?
This is the same graph as above, with the Optimum price at $6.00, not the $5.00 price this paper mill is able to charge. There is a marginal damage of $1.00/ream of paper produced, so we would need to know what the overproduction is. Then we'd calculate the loss, essentially the grey triangle in the above graph. That would be the area of a triangle in the case of linear supply and demand curves, and differential equations in the case of polynomial curves. This is one reason economists tend to simplify supply and demand curves down to linear functions.
A luxury condo builder causing neighboring real estate values to increase is a good example of a positive consumption externality. The builder is creating a marginal benefit that it does not share in.
$1,500,000 $1,250,000 $2,000,000 $625,000 $500,000 $300,000
Suppose you're the builder of luxury condos in the Brillianthood. You have a variable supply curve (as opposed to a fixed supply), that depends on the price you can charge for your condos. It's a linear function. If you can charge $100K per condo, you can build 25 condos. If you can charge $300K per condo, you can build 125 condos.
Unfortunately, you're the first into the market, and know that apartment prices are lower now than they will be after you build your condos. A real estate expert estimates that, at the present moment, demand will only allow you to get $200K per condo you build. But that after you build and sell your condos, demand will increase, and the market would be willing to pay $250K/condo. What's the marginal benefit you're relinquishing to the market?
A negative consumption externality might come from those who smoke. Secondhand smoke harms others, but the smoker doesn't pay for it. If smokers had to pay some sort of fee to those who suffer through their secondhand smoke, they amount they'd be willing to pay per pack would decrease to reflect this fee, and the consumption of cigarettes would decrease. Government taxes on the sale of cigarettes attempt to internalize this cost, as discussed below.
- Positive Production Externalities: Private marginal costs are higher than the social marginal costs. The supply curve is higher than it should be.
- Positive Consumption Externalities: Private marginal benefits are lower than the social marginal benefits. The demand curve is lower than it should be.
- Negative Production Externalities: Private marginal costs are lower than the social marginal costs. The supply curve is lower than it should be.
- Negative Consumption Externality: Private marginal benefits are higher than the social marginal benefits. The demand curve is higher than it should be.
Sometimes markets correct for externalities by internalizing the price. In the private markets, The Coase Theorem sets conditions on how this can happen. Namely, if there are well defined property rights, and an externality can be traded against with costless bargining, then negotiations between externality creator and the affected party will lead to a Pareto efficient outcome. In the case of the Flint Michigan pollution example above, under property rights, the town of Flint Michigan would have rights to clean water and would go to the polluting paper milll demanding some form of compensation. They might ask for compensation equal to the cost of removing lead, ask for the polluter to remove the lead themselves and bare this cost, or they might accept a monetary payment and distribute it so that residents can purchase clean water from another source.
However Coasian solutions are not always possible. Maybe the paper mill isn't the only polluter, maybe the damage is not easily calculated, maybe the town asks for more in damage than the paper mill can bare - that if the paper mill had to pay the cost of lead pollution in the first place it never would have existed.
Often times, that's were governments step in. Governments have a number of ways to correct for externalities. Regulations, like environmental protection laws, patent laws, and product safety standards can prevent producers from generating negative production externalities or taking advantage of other companies to create a positive production externality for themselves. Also, governments can levy taxes and provide subsidies to manipulate prices to their optimum level. For that luxury condo-builder, governments could provide tax breaks or subsidies to encourage development. Or, in the case of smokers, governments can and do sometimes levy taxes. In the graph to the right, the demand curve shifts down. Consumers now have to pay both the supplier and the government, and are therefore not willing to pay as much to the suppliers. Here, the government is correcting the price and quantity consumed to an optimum price that reflects the social costs of smoking.
But what's most efficient? Regulation or Taxation? A government could tax the smoker (affecting the price) or limit the quantity of cigarettes the smoker can purchase (affecting the quantity), both would have the effect of decreasing the demand curve. Efficiency too, can be modeled.
A classic case of regulation vs. taxation is, again, in pollution. Supply and demand curves can be used again, but instead of graphing the supply and demand for some good or service, we can graph the supply and demand for pollution reduction. The graph to the right examines the "quantity" of pollution reduction on the x-axis, with more pollution reduction meaning less pollution vs. the price (cost) of pollution reduction to the producer on the y-axis. Let's assume that, under non-regulatory or non-tax conditions, the private marginal benefits (demand curve) for pollution reduction would be zero or near zero, polluters would have little reason to stop. For simplicity, the social marginal benefit (i.e. the marginal damage externality averted) of increasing pollution reduction is shown as a straight line, but it could be a polynomial function. The private marginal costs to the pollution producer slope upwards because each additional increment of pollution becomes more expensive to eliminate. For instance paper production yields water and air pollution, contributes to deforestation which exacerbates pollution (trees help recycle carbon dioxide), uses harsh chemicals like chlorine, sulphur, and contributes to waste and landfill. Some subsection of these would be easier to solve for than others, for instance chlorine use in bleaching has declined significantly since 1990. The optimum point is the place where social benefits to reduced pollution match the cost of pollution reduction, in this case that is supposed to be a cost of
\$1,000
What would a tax look like in this scenario? (What would the tax be and how would it function?) How about regulation? Which would be better?
Tax: This would relatively straightforward, the government would tax polluters
\$1,000
per unit of pollution, and if it cost the polluters less than
\$1,000
to stop a unit of pollution (for instance if it cost a paper mill
\$500
to switch to a non-chlorine bleaching solution for one unit of paper production) then they would do it, otherwise they'd pay the tax (for instance if it cost
\$2,000
to collect and store water pollution).
Regulation: This is even more straightforward, the government would simply regulate polluters down to
Q_{Optimum}
amount of pollution.
If everything is known, and Marginal Damage is a neat flat linear function, then taxation is actually the easiest. The marginal damage is $1,000 so they set a tax at that level. Whereas, with regulation, the government needs to determine the
Q_{Optimum}
, so must know the what the function for the polluter's marginal costs, must calculate the reduction that $1,000 cost, and then set a target there.
This becomes trickier when there are multiple polluters and if the marginal damage is not a non-sloping linear function.
Tax: Establish a tax of
\$1,000
per ton of
\ce{CO_2}
produced Fair Regulation: Demand both plants cut 200 tons of
\ce{CO_2}
Suppose that there are two polluters in a city, both releasing
\ce{CO_2}
into the atmosphere. One is a paper mill and the other is a power plant. The marginal damage from each ton of
\ce{CO_2}
they produce is
\$1,000
, but they have different marginal costs to reduce that pollution, as represented by the graph to the right.
What's the most efficient solution for the government to reduce 400 tons of
\ce{CO_2}
? To establish a tax of
\$1,000
\ce{CO_2}
produced? Or to be fair, establish a policy that forces each plant to separately cut 200 tons of
\ce{CO_2}
? Assume that either taxation or regulation is enforceable and that an efficient solution means economically efficient.
Another solution to the above Try-It-Yourself problem with the power plant and the paper mill, would be for the government to establish a carbon tax credit that could be traded, and grant both polluters an equal amount of them. For instance,
200
credits to both polluters. Such a credit would allow the paper mill to produce
200
tons of pollution, then trade for more credits. Because the power plant has a lower cost of pollution reduction, it could afford to reduce pollution by one unit for some cost below
\$1,000
\$750
, and sell that credit to the paper mill for
\$1,200
. The power plant would make some money from this trade, decreasing their costs of pollution reduction, and the paper mill would be willing to buy this credit, allowing it to pollute one more ton, because the cost of that credit is lower than the cost for it to reduce pollution. Some economists believe that this is actually the most efficient means for the government to induce polluters to internalize the externality of pollution, and carbon credit trading is a common regulatory strategy in the US today.
Cite as: Externalities. Brilliant.org. Retrieved from https://brilliant.org/wiki/externalities/
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Heterojunction - Wikipedia
A heterojunction is an interface between two layers or regions of dissimilar semiconductors. These semiconducting materials have unequal band gaps as opposed to a homojunction. It is often advantageous to engineer the electronic energy bands in many solid-state device applications, including semiconductor lasers, solar cells and transistors. The combination of multiple heterojunctions together in a device is called a heterostructure, although the two terms are commonly used interchangeably. The requirement that each material be a semiconductor with unequal band gaps is somewhat loose, especially on small length scales, where electronic properties depend on spatial properties. A more modern definition of heterojunction is the interface between any two solid-state materials, including crystalline and amorphous structures of metallic, insulating, fast ion conductor and semiconducting materials.
1 Manufacture and applications
2 Energy band alignment
3 Effective mass mismatch
Manufacture and applications[edit]
Heterojunction manufacturing generally requires the use of molecular beam epitaxy (MBE)[1] or chemical vapor deposition (CVD) technologies in order to precisely control the deposition thickness and create a cleanly lattice-matched abrupt interface. A recent alternative under research is the mechanical stacking of layered materials into van der Waals heterostructures.[2]
Despite their expense, heterojunctions have found use in a variety of specialized applications where their unique characteristics are critical:
Solar cells: Heterojunctions are commonly formed through the interface of a crystalline silicon substrate and an amorphous Silicon passivation layer in solar cells. The Heterojunction with Intrinsic Thin-Layer (HIT) solar cell structure was first developed in 1983[3] and commercialised by Sanyo/Panasonic. HIT solar cells now hold the record for the most efficient single-junction silicon solar cell, with a conversion efficiency of 26.7%.[4][1][5]
Lasers: Using heterojunctions in lasers was first proposed[6] in 1963 when Herbert Kroemer, a prominent scientist in this field, suggested that population inversion could be greatly enhanced by heterostructures. By incorporating a smaller direct band gap material like GaAs between two larger band gap layers like AlAs, carriers can be confined so that lasing can occur at room temperature with low threshold currents. It took many years for the material science of heterostructure fabrication to catch up with Kroemer's ideas but now it is the industry standard. It was later discovered that the band gap could be controlled by taking advantage of the quantum size effects in quantum well heterostructures. Furthermore, heterostructures can be used as waveguides to the index step which occurs at the interface, another major advantage to their use in semiconductor lasers. Semiconductor diode lasers used in CD and DVD players and fiber optic transceivers are manufactured using alternating layers of various III-V and II-VI compound semiconductors to form lasing heterostructures.
Bipolar transistors: When a heterojunction is used as the base-emitter junction of a bipolar junction transistor, extremely high forward gain and low reverse gain result. This translates into very good high frequency operation (values in tens to hundreds of GHz) and low leakage currents. This device is called a heterojunction bipolar transistor (HBT).
Field effect transistors: Heterojunctions are used in high electron mobility transistors (HEMT) which can operate at significantly higher frequencies (over 500 GHz). The proper doping profile and band alignment gives rise to extremely high electron mobilities by creating a two dimensional electron gas within a dopant free region where very little scattering can occur.
Energy band alignment[edit]
The three types of semiconductor heterojunctions organized by band alignment.
Band diagram for staggered gap, n-n semiconductor heterojunction at equilibrium.
The behaviour of a semiconductor junction depends crucially on the alignment of the energy bands at the interface. Semiconductor interfaces can be organized into three types of heterojunctions: straddling gap (type I), staggered gap (type II) or broken gap (type III) as seen in the figure.[7] Away from the junction, the band bending can be computed based on the usual procedure of solving Poisson's equation.
Various models exist to predict the band alignment.
Tersoff[8] proposed a gap state model based on more familiar metal–semiconductor junctions where the conduction band offset is given by the difference in Schottky barrier height. This model includes a dipole layer at the interface between the two semiconductors which arises from electron tunneling from the conduction band of one material into the gap of the other (analogous to metal-induced gap states). This model agrees well with systems where both materials are closely lattice matched[9] such as GaAs/AlGaAs.
The 60:40 rule is a heuristic for the specific case of junctions between the semiconductor GaAs and the alloy semiconductor AlxGa1−xAs. As the x in the AlxGa1−xAs side is varied from 0 to 1, the ratio
{\displaystyle \Delta E_{C}/\Delta E_{V}}
tends to maintain the value 60/40. For comparison, Anderson's rule predicts
{\displaystyle \Delta E_{C}/\Delta E_{V}=0.73/0.27}
for a GaAs/AlAs junction (x=1).[10][11]
The typical method for measuring band offsets is by calculating them from measuring exciton energies in the luminescence spectra.[11]
Effective mass mismatch[edit]
When a heterojunction is formed by two different semiconductors, a quantum well can be fabricated due to difference in band structure. In order to calculate the static energy levels within the achieved quantum well, understanding variation or mismatch of the effective mass across the heterojunction becomes substantial. The quantum well defined in the heterojunction can be treated as a finite well potential with width of
{\displaystyle l_{w}}
. In addition, in 1966, Conley et al.[12] and BenDaniel and Duke[13] reported a boundary condition for the envelope function in a quantum well, known as BenDaniel–Duke boundary condition. According to them, the envelope function in a fabricated quantum well must satisfy a boundary condition which states that
{\displaystyle \psi (z)}
{\displaystyle {\frac {1}{m^{*}}}{\partial \over {\partial z}}\psi (z)\,}
are both continuous in interface regions.
Mathematical details worked out for quantum well example.
Using the Schrödinger equation for a finite well with width of
{\displaystyle l_{w}}
and center at 0, the equation for the achieved quantum well can be written as:
{\displaystyle -{\frac {\hbar ^{2}}{2m_{b}^{*}}}{\frac {\mathrm {d} ^{2}\psi (z)}{\mathrm {d} z^{2}}}+V\psi (z)=E\psi (z)\quad \quad {\text{ for }}z<-{\frac {l_{w}}{2}}\quad \quad (1)}
{\displaystyle \quad \quad -{\frac {\hbar ^{2}}{2m_{w}^{*}}}{\frac {\mathrm {d} ^{2}\psi (z)}{\mathrm {d} z^{2}}}=E\psi (z)\quad \quad {\text{ for }}-{\frac {l_{w}}{2}}<z<+{\frac {l_{w}}{2}}\quad \quad (2)}
{\displaystyle -{\frac {\hbar ^{2}}{2m_{b}^{*}}}{\frac {\mathrm {d} ^{2}\psi (z)}{\mathrm {d} z^{2}}}+V\psi (z)=E\psi (z)\quad {\text{ for }}z>+{\frac {l_{w}}{2}}\quad \quad (3)}
Solution for above equations are well-known, only with different(modified) k and
{\displaystyle \kappa }
{\displaystyle k={\frac {\sqrt {2m_{w}E}}{\hbar }}\quad \quad \kappa ={\frac {\sqrt {2m_{b}(V-E)}}{\hbar }}\quad \quad (4)}
At the z =
{\displaystyle +{\frac {l_{w}}{2}}}
even-parity solution can be gained from
{\displaystyle A\cos({\frac {kl_{w}}{2}})=B\exp(-{\frac {\kappa l_{w}}{2}})\quad \quad (5)}
By taking derivative of (5) and multiplying both sides by
{\displaystyle {\frac {1}{m^{*}}}}
{\displaystyle -{\frac {kA}{m_{w}^{*}}}\sin({\frac {kl_{w}}{2}})=-{\frac {\kappa B}{m_{b}^{*}}}\exp(-{\frac {\kappa l_{w}}{2}})\quad \quad (6)}
Dividing (6) by (5), even-parity solution function can be obtained,
{\displaystyle f(E)=-{\frac {k}{m_{w}^{*}}}\tan({\frac {kl_{w}}{2}})-{\frac {\kappa }{m_{b}^{*}}}=0\quad \quad (7)}
Similarly, for odd-parity solution,
{\displaystyle f(E)=-{\frac {k}{m_{w}^{*}}}\cot({\frac {kl_{w}}{2}})+{\frac {\kappa }{m_{b}^{*}}}=0\quad \quad (8)}
For numerical solution, taking derivatives of (7) and (8) gives
{\displaystyle {\frac {df}{dE}}={\frac {1}{m_{w}^{*}}}{\frac {dk}{dE}}\tan({\frac {kl_{w}}{2}})+{\frac {k}{m_{w}^{*}}}\sec ^{2}({\frac {kl_{w}}{2}})\times {\frac {l_{w}}{2}}{\frac {dk}{dE}}-{\frac {1}{m_{b}^{*}}}{\frac {d\kappa }{dE}}\quad \quad (9-1)}
{\displaystyle {\frac {df}{dE}}={\frac {1}{m_{w}^{*}}}{\frac {dk}{dE}}\cot({\frac {kl_{w}}{2}})-{\frac {k}{m_{w}^{*}}}\csc ^{2}({\frac {kl_{w}}{2}})\times {\frac {l_{w}}{2}}{\frac {dk}{dE}}+{\frac {1}{m_{b}^{*}}}{\frac {d\kappa }{dE}}\quad \quad (9-2)}
{\displaystyle {\frac {dk}{dE}}={\frac {\sqrt {2m_{w}^{*}}}{2{\sqrt {E}}\hbar }}\quad \quad \quad {\frac {d\kappa }{dE}}=-{\frac {\sqrt {2m_{b}^{*}}}{2{\sqrt {V-E}}\hbar }}}
The difference in effective mass between materials results in a larger difference in ground state energies.
Nanoscale heterojunctions[edit]
Image of a nanoscale heterojunction between iron oxide (Fe3O4 — sphere) and cadmium sulfide (CdS — rod) taken with a TEM. This staggered gap (type II) offset junction was synthesized by Hunter McDaniel and Dr. Moonsub Shim at the University of Illinois in Urbana-Champaign in 2007.
In quantum dots the band energies are dependent on crystal size due to the quantum size effects. This enables band offset engineering in nanoscale heterostructures. It is possible[15] to use the same materials but change the type of junction, say from straddling (type I) to staggered (type II), by changing the size or thickness of the crystals involved. The most common nanoscale heterostructure system is ZnS on CdSe (CdSe@ZnS) which has a straddling gap (type I) offset. In this system the much larger band gap ZnS passivates the surface of the fluorescent CdSe core thereby increasing the quantum efficiency of the luminescence. There is an added bonus of increased thermal stability due to the stronger bonds in the ZnS shell as suggested by its larger band gap. Since CdSe and ZnS both grow in the zincblende crystal phase and are closely lattice matched, core shell growth is preferred. In other systems or under different growth conditions it may be possible to grow anisotropic structures such as the one seen in the image on the right.
It has been shown[16] that the driving force for charge transfer between conduction bands in these structures is the conduction band offset. By decreasing the size of CdSe nanocrystals grown on TiO2, Robel et al.[16] found that electrons transferred faster from the higher CdSe conduction band into TiO2. In CdSe the quantum size effect is much more pronounced in the conduction band due to the smaller effective mass than in the valence band, and this is the case with most semiconductors. Consequently, engineering the conduction band offset is typically much easier with nanoscale heterojunctions. For staggered (type II) offset nanoscale heterojunctions, photoinduced charge separation can occur since there the lowest energy state for holes may be on one side of the junction whereas the lowest energy for electrons is on the opposite side. It has been suggested[16] that anisotropic staggered gap (type II) nanoscale heterojunctions may be used for photocatalysis, specifically for water splitting with solar energy.
Homojunction, p–n junction—a junction involving two types of the same semiconductor.
Metal–semiconductor junction—a junction of a metal to a semiconductor.
^ a b Smith, C.G (1996). "Low-dimensional quantum devices". Rep. Prog. Phys. 59 (1996) 235282, pg 244.
^ Geim, A. K.; Grigorieva, I. V. (2013). "Van der Waals heterostructures". Nature. 499 (7459): 419–425. arXiv:1307.6718. doi:10.1038/nature12385. ISSN 0028-0836. PMID 23887427. S2CID 205234832.
^ Okuda, Koji; Okamoto, Hiroaki; Hamakawa, Yoshihiro (1983). "Amorphous Si/Polycrystalline Si Stacked Solar Cell Having More Than 12% Conversion Efficiency". Japanese Journal of Applied Physics. 22 (9): L605–L607. doi:10.1143/JJAP.22.L605.
^ Yamamoto, Kenji; Yoshikawa, Kunta; Uzu, Hisashi; Adachi, Daisuke (2018). "High-efficiency heterojunction crystalline Si solar cells". Japanese Journal of Applied Physics. 57 (8S3): 08RB20. doi:10.7567/JJAP.57.08RB20.
^ "HJT - Heterojunction Solar Cells". Solar Power Panels. Retrieved 2022-03-25.
^ Kroemer, H. (1963). "A proposed class of hetero-junction injection lasers". Proceedings of the IEEE. 51 (12): 1782–1783. doi:10.1109/PROC.1963.2706.
^ Ihn, Thomas (2010). "ch. 5.1 Band engineering". Semiconductor Nanostructures Quantum States and Electronic Transport. United States of America: Oxford University Press. pp. 66. ISBN 9780199534432.
^ J. Tersoff (1984). "Theory of semiconductor heterojunctions: The role of quantum dipoles". Physical Review B. 30 (8): 4874–4877. Bibcode:1984PhRvB..30.4874T. doi:10.1103/PhysRevB.30.4874.
^ Pallab, Bhattacharya (1997), Semiconductor Optoelectronic Devices, Prentice Hall, ISBN 0-13-495656-7
^ Adachi, Sadao (1993-01-01). Properties of Aluminium Gallium Arsenide. ISBN 9780852965580.
^ a b Debbar, N.; Biswas, Dipankar; Bhattacharya, Pallab (1989). "Conduction-band offsets in pseudomorphic InxGa1-xAs/Al0.2Ga0.8As quantum wells (0.07≤x≤0.18) measured by deep-level transient spectroscopy". Physical Review B. 40 (2): 1058–1063. Bibcode:1989PhRvB..40.1058D. doi:10.1103/PhysRevB.40.1058. PMID 9991928.
^ Conley, J.; Duke, C.; Mahan, G.; Tiemann, J. (1966). "Electron Tunneling in Metal–Semiconductor Barriers". Physical Review. 150 (2): 466. Bibcode:1966PhRv..150..466C. doi:10.1103/PhysRev.150.466.
^ Bendaniel, D.; Duke, C. (1966). "Space-Charge Effects on Electron Tunneling". Physical Review. 152 (2): 683. Bibcode:1966PhRv..152..683B. doi:10.1103/PhysRev.152.683.
^ Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-111892-7
^ Ivanov, Sergei A.; Piryatinski, Andrei; Nanda, Jagjit; Tretiak, Sergei; Zavadil, Kevin R.; Wallace, William O.; Werder, Don; Klimov, Victor I. (2007). "Type-II Core/Shell CdS/ZnSe Nanocrystals: Synthesis, Electronic Structures, and Spectroscopic Properties". Journal of the American Chemical Society. 129 (38): 11708–19. doi:10.1021/ja068351m. PMID 17727285.
^ a b c Robel, István; Kuno, Masaru; Kamat, Prashant V. (2007). "Size-Dependent Electron Injection from Excited CdSe Quantum Dots into TiO2Nanoparticles". Journal of the American Chemical Society. 129 (14): 4136–7. doi:10.1021/ja070099a. PMID 17373799.
Bastard, Gérald (1991). Wave Mechanics Applied to Semiconductor Heterostructures. Wiley-Interscience. ISBN 978-0-470-21708-5.
Feucht, D. Lion; Milnes, A.G. (1970). Heterojunctions and metal–semiconductor junctions. New York City and London: Academic Press. , ISBN 0-12-498050-3. A somewhat dated reference respect to applications, but always a good introduction to basic principles of heterojunction devices.
R. Tsu; F. Zypman (1990). "New insights in the physics of resonant tunneling". Surface Science. 228 (1–3): 418. Bibcode:1990SurSc.228..418T. doi:10.1016/0039-6028(90)90341-5.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Heterojunction&oldid=1088643882"
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Determination of Potential Runoff Coefficient Using GIS and Remote Sensing ()
1Civil Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt.
2Landscape Architecture Department, Faculty of Environmental Design, King AbdulAziz University (KAU), Jeddah, KSA.
Flash floods in arid environments are a major hazard feature to human and to the infrastructure. Shortage of accurate environmental data is main reason for inaccurate prediction of flash flooding characteristics. The curve number (CN) is a hydrologic number used to describe the storm water runoff potential for drainage area. This study introduces an approach to determine runoff coefficient in Jeddah, Saudi Arabia using remote sensing and GIS. Remote sensing and geographic information system techniques were used to obtain and prepare input data for hydrologic model. The land cover map was derived using maximum likelihood classification of a SPOT image. The soil properties (texture and permeability) were derived using the soil maps published my ministry of water and agriculture in Saudi Arabia. These soil parameters were used to classify the soil map into hydrological soil groups (HSG). Using the derived information within the hydrological modelling system, the runoff depth was predicted for an assumed severe storm scenario. The advantages of the proposed approach are simplicity, less input data, one software used for all steps, and its ability to be applied for any site. The results show that the runoff depth is directly proportional to runoff coefficient and the total volume of runoff is more than 136 million cubic meters for a rainfall of 103.6 mm.
Potential Runoff Coefficient (PRC), GIS, Remote Sensing, Hydrological Soil Group (HSG), Digital Elevation Model (DEM), Land Use
Khalil, R. (2017) Determination of Potential Runoff Coefficient Using GIS and Remote Sensing. Journal of Geographic Information System, 9, 752-762. doi: 10.4236/jgis.2017.96046.
In arid zones, there is a shortage of data needed for hydrological processes as mentioned by [1] . In these zones flash flooding happened suddenly and affects both human and infrastructure. The effective way to reduce the damages due to flood is to predict the critical sites affected by flash flood for management planes. The part of rainfall that turns to runoff due to land use and soil hydrological parameters is defined as runoff coefficient [2] . It may also be defined as the ratio between the runoff depth and the rainfall depth [3] . The reasonable calculation of runoff from rainfall is the key for flood estimation [4] [5] . Defining rainfall-runoff relationship leads to calculate flood properties such as runoff depth, peak discharge, runoff speed, and runoff volume. Flood properties derivation is important for planners and decision makers to avoid flood hazard effects. When flood discharge data records are missed, there are many approaches to estimate runoff depth and volume based on soil and surface properties of catchment area. These approaches may be summarized into three groups, simple, moderate and complex models as stated by [4] . The curve number (CN) is one of the moderate complicity models and it is widely used for flood estimation. The CN model was developed by United States Department of Agriculture (USDA), National Resources Conservation Service (NRSC) in 1969 [6] . It is an empirical model with clearly stated assumptions and few data requirements [7] . CN is stable conceptual method for predicting of direct runoff depth using storm rainfall intensity, land use and soil hydrological properties of a catchment.
Remote sensing imageries considered a major source of spatial data especially for wide area or when the ground surveys are not available. They can be used to generate land use, soil and geological maps needed for flood estimation. Geographic Information System (GIS) is a powerful tool in hydrological modelling because of its capability to handle large amount of spatial and attribute data. Delineation of hydrological catchments, map overlay and analysis, which are basics of GIS software, help for derivation and aggregation of hydrologic parameters from the input data like DEM, soil map, land use map, and rainfall data. Remote sensing (RS) images and Geographic Information System (GIS) techniques have been used in flood hazard studies by many researchers, e.g. [8] - [18] .
The main objective of this research article is determining the potential runoff coefficient for Jeddah, Saudi Arabia by applying CN model and using remote sensing and GIS techniques.
Jeddah is the second main city in Saudi Arabia. It is at the middle of east coast of the Red Sea and represents the important commercial port. Its population exceeds 3.4 million people according to (Central Department of Statistics & Information 2010) with 3.5% annual rate of growth make its population exceeds 4.2 million people. Its weather condition is classified as hot. A rainfall intensity of 80 and 124 mm/day hit Jeddah in November 2009 and January 2011, respectively as mentioned by [19] . Reference [20] mentioned that these occasional floods were examples of flash flood striking which characterized by short durations and destructive results. The watershed that affected the city is located between (21˚15'N, 21˚50'N) and (39˚0'E, 39˚35'E) as shown in (Figure 1).
The available rainfall data for Jeddah are from two rain gauge stations, J134 which operated by the Ministry of Water and Electricity (MOWE) and 41024 (airport station) which operated by Presidency of Meteorology and Environment (JMPE) [21] . The data represent the recorded observation for a period of 42 years extending from January 1971 to December 2012; no data were available for the most recent years. The maximum annual daily rainfall for the two stations is shown in (Figure 2). High resolution satellite image SPOT5 (2.5 m multispectral) was obtained from King Abdulaziz City for Science and Technology (KACST) and used generate land use map. Topography data was obtained using Shuttle Radar Topography Mission (SRTM) digital elevation model (DEM) and used to extract slope data and stream network. Soil data were obtained using Saudi general soil map, ministry of agricultural and water 1986.
The methodology used to determine the potential runoff coefficient for the study area using remote sensing and GIS is shown in the flow chart in (Figure 3). It starts with scanning and Georeferencing the soil maps using ArcMap 10.4, the soil polygons were digitized, and each soil type was assigned to a Hydrological Soil Group (HSG) according to [22] . Image classification tool in ArcMap 10.4 was used to classify and convert the SPOT image to land use thematic map by applying the supervised classification technique. The DEM was used to generate slope map using surface analysis tool, and to generate stream network and basin parameters using HEC-GeoHMS. The runoff curve number lookup table was build according to the land use map and [22] . All data were then overlaid in
Figure 2. Rainfall data for the two rain gauges.
ArcGIS environment to generate the runoff coefficient. Lastly, the flood formulas were applied to calculate the flood parameters using field calculator in ArcMap 10.4.
3.1. Land Use Analysis
SPOT image (2.5 m) for the year 2010 was obtained from the King Abdul-Aziz City for Science and Technology (KACST). Supervised classification was performed to the image using spectral signatures collected from training samples (polygons that represent distinct sample areas of the different land cover types to be classified). After preparing the signature file, Maximum likelihood classifier then attached labels to all the image pixels according to the trained parameters to generate land cover map. The land cover map was classified into six main classes: water, vegetation, rocks, bare soil, and build up area as shown in (Figure 4).
3.2. Soil Type Analysis
The general soil map of Saudi Arabia was the source for soil data used in this study. The general soil map was scanned and georeferenced then different types of soil were digitalized in ArcMap 10.4. Each type of soil was assigned to the proper hydrological soil group that developed by US Soil Conservation Service (SCS). According to SCS, all the soil, based on their permeability and infiltration are divided to 4 groups A, B, C and D. Hydrological Soil Group of A has the lowest runoff potential while group D has the highest. The soil map was then classified into Hydrological Soil Groups (HSGs) as shown in Figure 5.
3.3. Slope Analysis
The runoff velocity and soil erosion are in direct proportional to the degree of
Figure 4. Land use map.
Figure 5. Soil hydrological groups map.
slope. Digital Elevation Model (DEM) obtained from SRTM was used to generate slope map as shown in Figure 6. The DEM analyzed to remove sinks and flat areas to maintain continuity of flow to the catchment outlets. The GIS used for DEM preparation by filling the sink areas.
4. Potential Runoff Coefficient
As flash flood in Jeddah occurred with rainfall didn’t exceed 80 mm and rainstorm duration didn’t exceed 3 hours, it is important to estimate the expected flood discharges of return periods reliably for protection and future developments [23] . The rainfall data was used to predict the rainfall depth at different return periods. Among many of the probability density functions, numerous studies recommended the Gumbel or Extreme Value Type I (EV1) function as it demonstrated the best fit in most cases and providing the best prediction of rainfall depth. The estimated rainfall depth for different return periods for the two rain gauge stations for JPME (41024) and J134 are shown in Table 1.
The flood hazard parameters such as watershed storage, runoff depth, and volume of runoff were calculated. The CN method Equation (1) was used for runoff depth calculation [22] .
Q=2\left(P-0.2S\right)/\left(P+0.8S\right)
S=25400/CN-254
Figure 6. Slope map.
P: depth of precipitation for a specific return period (mm),
S: the watershed storage (mm), and can be calculated using Equation (2),
CN: the curve number.
The volume of the runoff was calculated for subbasins to show the local effect of runoff and was also calculated for the major basins to show the total volume of runoff flood. The volume of the runoff can be calculated using Equation (3)
{V}_{Q}=Q\cdot A
Q: Depth of runoff (m),
The flood characteristics were calculated using the rainfall depth (P) equals 106.3 mm for a return period of 100 years as shown in Table 1. These characteristics include area, longest flow length and the runoff volume of the main basins in Jeddah watershed area as shown in Table 2. The results show that there are 7 major and 5 minor basins in Jeddah as shown in (Figure 7). The areas of the
Table 2. Morphometric and flood parameters of basins.
major basins range from 59.04 to 555.5 square kilometers, and their longest flow paths range from 12.7 to 77.4 kilometers, while the areas of the minor basins range from 6.9 to 15.5 square kilometers, and their longest flow paths range from 7.48 to 10.23 kilometers. The total runoff volume was calculated and found to be more than 136 million cubic meters.
Refereeing to (Table 2), one can notice that the runoff depth (Q) is directly proportional to the curve number and inversely proportional to slope and the curve number has the great effect on it.
The proposed approach of using remote sensing and GIS applying the CN method for runoff coeffecient estimation has many advantages over other approaches. Firstly, it uses one software to perform all procedure steps. Secondly, only satellite image, soil maps and DEM are needed to calculate the runoff parameters. Thirdly, all needed calculations are done within the GIS environment using field calculation. Fourthly, it can be modeled using model builder so, runoff parameters estimation process can be efficient, faster, and easily performed for several return period scenarios and for any regions.
This research article presented an efficient approach to accurate determination of potential runoff coefficient in Jeddah city using remote sensing and GIS. The effects of land use, soil hydrological characteristics, surface slope, were considered in calculating runoff coefficient and consequently runoff depth and runoff volume. The results of the research show that the total runoff volume for a rainfall depth of 106.3 mm is 136.5 million m3. Results also show that the main factors affect the total flood volumes, are the basin area, and the flow length. Additionally, it has been concluded that the higher CN value and slope percent, the higher runoff and flood hazards.
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[12] Zhao, D.Q., Chen, J.N., Wang, H.Z., Tong, Q.Y., Cao, S.B. and Sheng, Z. (2009) GIS-based Urban Rainfall-Runoff Modeling Using an Automatic Catchment-Discretization Approach: A Case Study in Macau. Environmental Earth Sciences, 59, 465-472.
[13] Chen, J., Hill, A. and Urbano, L. (2010) A GIS-Based Model for Urban Flood Inundation. Journal of Hydrology, 373, 184-192.
[14] Sumarauw, J.S.F. and Ohgushi, K. (2012) Analysis on Curve Number, Land Use and Land Cover Changes and the Impact to the Peak Flow in the Jobaru River Basin, Japan. International Journal of Civil & Environmental Engineering, 12, 17-23.
http://www.ijens.org/Vol_12_I_02/124102-3535-IJCEE-IJENS.pdf
[18] Gajbhiye, S. (2015) Estimation of Surface Runoff Using Remote Sensing and Geographical Information System. International Journal of u- and e-Service, Science and Technology, 8, 113-122.
[21] Youssef, A., Pradhan, B. and Sefry, S. (2014) Remote Sensing-Based Studies Coupled with Field Data Reveal Urgent Solutions to Avert the Risk of Flash Floods in the Wadi Qus (East of Jeddah) Kingdom of Saudi Arabia. Natural Hazards, 75, 1465-1488.
[22] USDA (1986) Urban Hydrology for Small Watersheds. USDA, NRCS, CED, TR55.
[23] Subyani, A.M. and Al-Modayan, A.A. (2011) Flood Analysis in Western Saudi Arabia. Journal of King Abdulaziz University, 22, 17-36.
https://doi.org/10.4197/Ear.22-2.2
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Create threshold transitions - MATLAB - MathWorks Switzerland
F\left({z}_{t},{t}_{j}\right)=\left\{\begin{array}{ll}0\hfill & ,{z}_{t}<{t}_{j}\hfill \\ 1\hfill & ,{z}_{t}>={t}_{j}\hfill \end{array}.
F\left({z}_{t},{t}_{j},{r}_{j}\right)=\frac{1}{1+{e}^{-{r}_{j}\left({z}_{t}-{t}_{j}\right)}}.
F\left({z}_{t},{t}_{j},{r}_{j}\right)=1-{e}^{-{r}_{j}{\left({z}_{t}-{t}_{j}\right)}^{2}}.
{\mathit{t}}_{1}=0
\left(-\infty ,0\right)
\left[0,\infty \right)
\mathit{t}
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Category:Hybrid functionals - Vaspwiki
Category:Hybrid functionals
Hybrid functionals mix the Hartree-Fock (HF) and Kohn-Sham theories[1] and can be more accurate than semilocal methods, e.g., GGA, in particular for nonmetallic systems. They are for instance suited for band-gap calculations. There are several hybrid functionals that are available in VASP.
In hybrid functionals the exchange part consists of a linear combination of HF and semilocal (e.g., GGA) exchange:
{\displaystyle E_{\mathrm {xc} }^{\mathrm {hybrid} }=\alpha E_{\mathrm {x} }^{\mathrm {HF} }+(1-\alpha )E_{\mathrm {x} }^{\mathrm {GGA} }+E_{\mathrm {c} }^{\mathrm {GGA} }}
{\displaystyle \alpha }
determines the relative amount of HF and semilocal exchange. The hybrid functionals can be divided into families according to the interelectronic range at which the HF exchange is applied: at full range (unscreened hybrids) or either at short or at long range (called screened or range-separated hybrids). From the practical point of view, the short-range hybrid functionals like HSE are preferable for periodic solids, since leading to faster convergence with respect to the number of k-points (or size of the unit cell).
Note that as in most other codes, hybrid functionals are implemented in VASP within the generalized KS scheme[2], which means that the total energy is minimized with respect to the orbitals (instead of the electron density) as in the Hartree-Fock theory.
It is important to mention that hybrid functionals are computationally more expensive than semilocal methods.
Read more about formalism of the HF method and hybrids.
List of available hybrid functionals and how to specify them in the INCAR file.
Downsampling of the Hartree-Fock operator to reduce the computational cost.
How to do a band-structure calculation using hybrid functionals.
A comprehensive study of the performance of the HSE03/HSE06 functional compared to the PBE and PBE0 functionals[3].
The B3LYP functional applied to solid state systems[4].
Applications of hybrid functionals to selected materials: Ceria,[5] lead chalcogenides,[6] CO adsorption on metals,[7][8] defects in ZnO,[9] excitonic properties,[10] SrTiO and BaTiO.[11]
↑ A. D. Becke, J. Chem. Phys. 98, 5648 (1993).
↑ A. Seidl, A. Görling, P. Vogl, J.A. Majewski, and M. Levy, Phys. Rev. B 53, 3764 (1996).
↑ J. Paier, M. Marsman, K. Hummer, G. Kresse, I.C. Gerber, and J.G. Ángyán, J. Chem. Phys. 124, 154709 (2006).
↑ J. Paier, M. Marsman, and G. Kresse, J. Chem. Phys. 127, 024103 (2007).
↑ J. L. F. Da Silva, M. V. Ganduglia-Pirovano, J. Sauer, V. Bayer, and G. Kresse, Phys. Rev. B 75, 045121 (2007).
↑ Hummer, A. Grüneis, and G. Kresse, Phys. Rev. B 75, 195211 (2007).
↑ A. Stroppa, K. Termentzidis, J. Paier, G. Kresse, and J. Hafner, Phys. Rev. B 76, 195440 (2007).
↑ A. Stroppa and G. Kresse, New Journal of Physics 10, 063020 (2008).
↑ F. Oba, A. Togo, I. Tanaka, J. Paier, and G. Kresse, Phys. Rev. B 77, 245202 (2008).
↑ J. Paier, M. Marsman, and G. Kresse, Phys. Rev. B 78, 121201(R) (2008).
↑ R. Wahl, D. Vogtenhuber, and G. Kresse, Phys. Rev. B 78, 104116 (2008).
Pages in category "Hybrid functionals"
Band-structure calculation using hybrid functionals
Retrieved from "https://www.vasp.at/wiki/index.php?title=Category:Hybrid_functionals&oldid=17737"
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History of Irrational Numbers | Brilliant Math & Science Wiki
Andrew Ellinor, Satyabrata Dash, Ken Jennison, and
Irrational numbers are numbers that have a decimal expansion that neither shows periodicity (some sort of patterned recurrence) nor terminates. Let's look at their history.
Hippassus of Metapontum, a Greek philosopher of the Pythagorean school of thought, is widely regarded as the first person to recognize the existence of irrational numbers. Supposedly, he tried to use his teacher's famous theorem
a^{2}+b^{2}= c^{2}
to find the length of the diagonal of a unit square. This revealed that a square's sides are incommensurable with its diagonal, and this length cannot be expressed as the ratio of two integers. Since the other Pythagoreans believed that only positive rational numbers could exist, what happened next has been the subject of speculation for centuries. In short, Hippassus may have died because of his discovery.
So, what did happen to Hippassus? No one will probably ever know for sure, but below are some better-known stories.
Some believe that the Pythagoreans were so horrified by the idea of incommensurability that they threw Hippassus overboard on a sea voyage and vowed to keep the existence of irrational numbers a secret.
Hippassus discovered irrational numbers, the Pythagoreans ostracized him, and the gods were so disgusted by his discovery that they scuttled his boat on the high seas.
Hippassus discovered irrational numbers, and then died on an ocean voyage as the result of a natural accident (the sea is a treacherous place). Nonetheless, his colleagues were still so displeased with his discovery that they wished they had been the ones to throw him overboard.
Another possibility is that none of the stories above are true, and they are tales invented and embellished through the ages to illustrate a pivotal moment in history.
However, if Hippassus did discover irrational numbers, it is not clear which method he used to do so. For the curious, the Brilliant summary page on rational numbers builds up to Euclid's proof of the irrationality of
\sqrt{2}
. This is one way Hippassus might have done it. However, many scholars think Euclid's method (written 300 years after the time of Hippassus) is more advanced than what Hippassus would have been able to do.
Regardless of what actually happened, it is difficult to imagine a time when proving the existence of an irrational number was a moral transgression.
Humans have had numbers for at least all of recorded history. Our earliest basis for numbers and math derive from the practical need to count and measure things. It is intuitive to see how the positive, non-zero, natural numbers would arise "naturally" from the process of counting. It is also easy to see how measurement would present one with things that could not be divided into whole units, or whose dimensions were in-between a whole number. Inventing fractions made practical sense, as ratios of the natural numbers. Discovering the positive rational numbers was probably pretty intuitive.
Numbers may have originated from purely practical needs, but to the Pythagoreans, numbers were also the spiritual basis of their philosophy and religion. Pythagorean cosmology, physics, ethics, and spirituality were predicated on the premise that "all is number." They believed that all things--the number of stars in the sky, the pitches of musical scales, even the qualities of virtue--could be described by and comprehended through rational numbers.
19^\text{th}
century depiction of Pythagoreans celebrating sunrise (Fyodor Bronnikov). The Pythagoreans attributed mystical significance to their ability to perceive the presence of rational numbers in everything, be it a sunrise or a musical harmony.
One reason to think that positive rational numbers would form the basis for all things in the universe is that there is an infinite amount of them. Intuitively, it might seem reasonable that an infinite amount of numbers should be enough to describe anything that might exist. Along the number line, rational numbers are unfathomably “dense.” There is not much “space” between
\frac{1}{100000}
\frac{1}{100001}
, but if you ever needed to describe something between those two numbers you would have no problem finding a fraction between them.
The number line is infinitely dense with rational numbers. The existence of irrational numbers implies that despite this infinite density, there are still holes in the number line that cannot be described as a ratio of two integers.
The Pythagoreans had probably manually measured the diagonal of a unit square before. They probably regarded the measurement as an approximation that was close to a precise rational number that must be the true length of the diagonal. Before Hippassus, they had no reason to suspect that there are logically real numbers that in principle, not merely in practice, cannot be measured or counted to.
If you had believed that all numbers were rational numbers, and that rational numbers were the basis of all things in the universe, then having something that could not be expressed as the ratio of two integers would have been like discovering a gaping void in the universe. An irrational number was a sign of meaninglessness in what had seemed like an orderly world. The Pythagoreans wanted numbers to be something you could count on, and for all things to be counted as rational numbers. The discovery of an irrational number proved that there existed in the universe things that could not be comprehended through rational numbers, threatening not only Pythagorean mathematics, but their philosophy as well.
Cite as: History of Irrational Numbers. Brilliant.org. Retrieved from https://brilliant.org/wiki/history-of-irrational-numbers/
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Example 1: Input: coins = [1,2], amount = 4 Output: 2 Explanation: 4 =
2 + 2,4=1+1+2
The recursion formual for coin change as below
coinchange(j,a) = \begin{cases} \infty, & \text{if $j$<0} \\ 0, & \text{if $j$ =0} \\ 1+\min(\sum_{i=k}^n c[j-a_i]) & \text{if $j$ >1} \end{cases}
function coin_change(amount) {
// if remaining coin is zero return
if (amount == 0) return 0
// if coin is negative return some large value
if (amount < 0) return Infinity
let ans = Infinity
1 + coin_change(amount - coin)
This article only show you how to write recursive program. I know this is not optimized way to write coin change problem{alertError}
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Principle Of Mathematical Induction, Popular Questions: CBSE Class 11-science ENGLISH, English Grammar - Meritnation
Rupam Saxena asked a question
sinx+sin2x+sin3x+...+sin nx=sin((n+1)/2)x .((sin nx)/2) / sin(x/2)
Prove that 1.2 + 2.3 + 3.4 +..............n(n+1) = 1/3 n (n+1)(n+2)
sinx + sin3x + ..............+ sin(2n-1)x = sin^2 nx / sinx
Prove by mathematical induction that:
2^5n>3^3n for n belongs to natural no.
prove by using PMI that 4 raise to n + 15n - 1 is divisible by 9 .
N. Question 13
13. Prove that, 5n - 5 is divisible by 4 for all n
\in
Prove that 2.7n + 3.5n - 5 is divisible by 24 for all n belongs to N. [ please explain with steps]
1 sqare + 2 square + 3 square +.............+n square >n cube +3 . prove it by mathematical induction.
Arijeet Baruah asked a question
Prove that x2n-y2n is divisible by x+y?
9n - 8n -1 is a multiple of 8
Prove that n(n+1)(n+5) is a multiple of 3
The 2nd term of an HP is 40/9 and the 5th term is 20/3. find the maximum possible number of terms in H.P. what do we mean by this sentence in this question?
1) a+(a+d)+(a+2d)+ ......[a+(n-1)d] = n/2 [2a+(n-1)d], n E N.
2) n(n+1)(n+5) is divisible by 6 for all n E N.
3) 9 raised to n - 8n - 1 is a multiple of 64 for all n E N.
Roshni Bhatt asked a question
Using principle of mathematical induction prove that for all nϵN
xn-yn is divisible by x-y
Prove that n(n+1)(n+2) is divisible by 6
12. By PM1, prove that
1.2+2.{2}^{2}+...+n-{2}^{n}=\left(n-1\right) {2}^{n+1}+2
13. Find the square root of
4+6\sqrt{-5}
Using PMI Prove that 1^2 + 2^2 + ......... + n^2 n^3/3
Shivani Rana asked a question
using induction, prove that 10n + 3.4n+2 + 5 is divisible by 9
Find two consecutive numbers whose sum of the reciprocals is 21/110
11) Prove using mathematical induction, that 7 2n + 2 3n-3, 3n-1 is divisible by 25, for all n Ꜫ N.
kanu-priya asked a question
prove dt ....
34n+2 + 52n+1 is a multiple of 14
find the equation of parabola whose focus is (1,1) and tangent at the vertex is x + y = 1
1/2*5+ 1/5*8 + 1/8*11 +.......................+
1/(3n-1)(3n+2)
= n/6n+4
Tarun Singh asked a question
Prove that 11n+2 + 122n+1 is divisible by 133 for all n belongs to N .
Use the principle of mathematical induction to prove the following statements for all n belongs to N: 1.3 + 3.5 + 5.7 +.... + ( 2 n - 1)(2n + 1) = n ( 4n square + 6 n - 1) / 3?
Prove: 52n-1 is divisible by 24 for all n N
for every positive integer n.prove that 7n-3n is divisible by 4
52n+2-24n-25 is divisible by 576 for n belongs to N.
By Principal of Mathematical Induction prove that:
1.3 + 3.5 + 5.7 + ..... +(2n - 1) (2n + 1 ) =n( 4n square + 6n -1)/2 plz explain briefly in a simple method
Alisha Sarah asked a question
n(n+1)(n+2) is a multiple of 6 .... Using mathematical induction prove this
using principle of mathematical induction prove that, 12 + 22 +....+ n2 (n3/3) for all n belonging to natural numbers
Q. Prove that 1.3+2.4+3.5+....+n(n+2) = 1/6n (n+1) (2n+7),V nEN.
Kavita Karki asked a question
Prove the following by the PMI :
n7/ 7 + n5 / 5 + n3 / 3 + n2/ 2 - 37/210n is a positive integer for all n belongs to N.
ASC-dkell asked a question
how to split a cubic polynomial like k3+6k2+9k+4??
Using PMI prove that x^n - y^n is divisible by x-y
bY PMI prove n(n+1)(2n+1) is divisible by 6
Chinnammu & 2 others asked a question
n7/7 + n5/5 + 2n3/3 - n/105 is an integer , prove by PMI.
Prove that 2.7n + 3.5n-5 is divisible by 24, for all n belongs to N....Plzzz dnt tell to refer textbook as frm dat also its not clear to me plzzzzzz answer it as soon as possible.....
Sweta Ghosh asked a question
1/1.2 + 1/2.3 + 1/3.4 + ..... to n terms = n/(n+1)
Prove that n5/5+n3/3+7n/15 is a natural number by using the principle of mathematical induction.
Chehak Arora asked a question
Cosx.cos2x. Cos4x.....cos(2*(n-1)x) =sin2(*n)x/2*n sinx prove using pmi
Mehakdeep Kaur asked a question
Ncert maths book exercise 4.1 Sum no.19 I did not get the sum from solution kindly explain it!!!!
Ann Maria Joshi & 1 other asked a question
7+ 77+ 777+ ..............+ 77.........7 = 7/81 (10to the power n+1 -9n-10)
Basant Kumar asked a question
11 power n+2 + 12 power 2n+1 is divisible by 133
1 / 1.2.3 + 1 / 2.3.4 + 1 / 3.4.5 + …….. + 1 / n (n+1) (n+2) = n (n+3) / 4 (n+1) (n+2) ? using principle of mathematical induction prove the following for all n E N ?
Surbhi Gupta asked a question
prove by ibduction that the sum Sn=n3+3n2+5n+3 is divisible by 3 for all nEN
Prove that n2 + n is even , where n is natural number.?
Archit Srivastava & 1 other asked a question
Prove that 1/2 tan (x/2)+1/4tan(x/4)+...+(1/2n)tan (x/2n)=(1/2n)cot(x/2n) - cotx for all n (- N and 0<x<(pi/2).
3.6 +6.9+9.12+....+3n(3n+3) = 3n(n+1) (n+2)
Anubhav Anand asked a question
prove that 1x2 + 2x2^2 + 3x2^3+..+nx2^n = (n-1)x2^n+1 +2
Ashish Somwanshi asked a question
For all positive integer such that
prove.that 2n! =1.3.5....(2n-1) 2^n.n!
Using mathematical induction prove 2.3+3.4+4.5+..upto n terms = n (n^2+6n+11)/3.
By using principle of mathematical induction, prove that for all n element of N:
3^2n+2 - 8n - 9 is divisible by 64.
prove by mathematical induction nEN ,72n+22n-33n-1 is divisible by 25
nainagarg7... asked a question
1+4+7+---------+(3n-2)=1 n(3n-1)
by using principle of mathematical induction prove that 12^ n + 25 ^ n-1 divisible by 13
Halu asked a question
3 2n when divided by 8 leaves the remainder 1
Amaan Khan asked a question
n(n+1)(n+5) is divisible by 6 for all n belongs to natural numbers
Samridhi Sinha asked a question
1.2+2.2?+3.3?......+n.2^2= (n-1).2^n+2+2 prove using mathematical induction
in drilling worlds deepest hole it was found that the temperature T in degree celcius at x km below the surface of the earth was =30+25(x - 3), 3 less than x less than 15. at what depth will the temperature lie b/w 200oC and 300oC.
Aman Prakash Singh asked a question
way we take 8m in example no 1 ????????
Prove by PMI 3.2^2 +3^2.2^3 + 3^3.2^4 +----+ 3^n.2^n+1 = 12/5(6^n -1)
what is LUCAS SEQUENCE?
a(n) = a(n-1)+ a(n-2), n>2 ;a1=1 , a2= 3 , prove a(n)<or = to (1.75)raise to n
===aa
Prove by induction that (2n+7)<(n+3)2is true
Suppose there is a given statement P(n) involving the natural numbernsuch that
The statement is true forn= 1, i.e., P(1) is true
What "P" MEANS and How P(1) IS true??????????
Please explain I don't got it frm study material.............
7 divides 23n-1
Jaismene Verma asked a question
using PMI prove that 3 to the power (2n-1) is divisible by 8 where n belongs to N.
Vaidehi Shendre asked a question
Gitika Mann asked a question
1. 12+32+52++(2n-1)2=2-1)
k.k.raheema... asked a question
what is principle of mathamatical induction
By using PMI,
Prove that, x2n- y2nis divisible by x+y.
Gourav Singh asked a question
7n-3n is a divisible by 4.
72n + 16n -1 is divisible by 64 Prove by mathematical induction
Suraya Pm asked a question
PROVE BY M.I (41)n-(14) is multiple of 27
Arjun Sarathy asked a question
(2n+7) < (n+ 3)2
can any pls explain this question in detail??? cuz i can't get it
by using principle of mathematical induction prove that 2 ^ n > n for all natural numbers n
prove that 2n is greater than n for all positive integers n.
this is example 2 from the ncert maths text book.
plzz...answer soon....i dint get the last step.
U\mathrm{sin}g the principal of mathematical induction prove that \phantom{\rule{0ex}{0ex}}\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{{2}^{n}}=1-\frac{1}{{2}^{n}}\phantom{\rule{0ex}{0ex}} for all n\in N
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Surface Deformation of North‐Central Oklahoma Related to the 2016 Mw 5.8 Pawnee Earthquake from SAR Interferometry Time Series | Seismological Research Letters | GeoScienceWorld
Surface Deformation of North‐Central Oklahoma Related to the 2016 Mw 5.8 Pawnee Earthquake from SAR Interferometry Time Series
Eric J. Fielding;
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109 U.S.A., eric.j.fielding@jpl.nasa.gov
Simran S. Sangha;
Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, 595 Charles Young Drive East, Box 951567, Los Angeles, California 90095‐1567 U.S.A.
David P. S. Bekaert;
Sergey V. Samsonov;
Canada Centre for Mapping and Earth Observation, Natural Resources Canada, 560 Rochester Street, Ottawa, Ontario, Canada K1A 0E4
Oklahoma Geological Survey, University of Oklahoma, 100 East Boyd Street, N131, Norman, Oklahoma 73019 U.S.A.
Eric J. Fielding, Simran S. Sangha, David P. S. Bekaert, Sergey V. Samsonov, Jefferson C. Chang; Surface Deformation of North‐Central Oklahoma Related to the 2016
Mw
5.8 Pawnee Earthquake from SAR Interferometry Time Series. Seismological Research Letters 2017;; 88 (4): 971–982. doi: https://doi.org/10.1785/0220170010
Mw
5.8 Pawnee earthquake shook a large area of north‐central Oklahoma and was the largest instrumentally recorded earthquake in the state. We processed Synthetic Aperture Radar (SAR) from the Copernicus Sentinel‐1A and Sentinel‐1B and Canadian RADARSAT‐2 satellites with interferometric SAR analysis for the area of north‐central Oklahoma that surrounds Pawnee. The interferograms do not show phase discontinuities that would indicate surface ruptures during the earthquake. Individual interferograms have substantial atmospheric noise caused by variations in radar propagation delays due to tropospheric water vapor, so we performed a time‐series analysis of the Sentinel‐1 stack to obtain a more accurate estimate of the ground deformation in the coseismic time interval and the time variation of deformation before and after the earthquake. The time‐series fit for a step function at the time of the Pawnee shows about 3 cm peak‐to‐peak amplitude of the coseismic surface deformation in the radar line of sight with a spatial pattern that is consistent with fault slip on a plane trending east‐southeast. This fault, which we call the Sooner Lake fault, is parallel to the west‐northwest nodal plane of the U.S. Geological Survey National Earthquake Information Center moment tensor solution. We model the fault plane by fitting hypoDD‐relocated aftershocks aligned in the same trend. Our preferred slip model on this assumed fault plane, allowing only strike‐slip motion, has no slip shallower than 2.3 km depth, an area of moderate slip extending 7 km along strike between 2.3 and 4.5 km depth (which could be due to aftershocks and afterslip), and larger slip between 4.5 and 14 km depth extending about 12 km along strike. The large slip below the 4.5 km depth of our relocated hypocenter indicates that the coseismic rupture propagated down‐dip. The time‐series results do not show significant deformation before or after the earthquake above the high atmospheric noise level within about 40 km of the earthquake rupture.
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Social Networks | Brilliant Math & Science Wiki
Adam Strandberg, Christopher Williams, and Jimin Khim contributed
Co-authorship network map of physicians publishing on hepatitis C by Andy Lamb [1]
A social network graph is a graph where the nodes represent people and the lines between nodes, called edges, represent social connections between them, such as friendship or working together on a project. These graphs can be either undirected or directed. For instance, Facebook can be described with an undirected graph since the friendship is bidirectional, Alice and Bob being friends is the same as Bob and Alice being friends. On the other hand, Twitter can be described with a directed graph: Alice can follow Bob without Bob following Alice.
Social networks tend to have characteristic network properties. For instance, there tends to be a short distance between any two nodes (as in the famous six degrees of separation study where everyone in the world is at most six degrees away from any other), and a tendency to form "triangles" (if Alice is friends with Bob and Carol, Bob and Carol are more likely to be friends with each other.)
Social networks are important to social scientists interested in how people interact as well as companies trying to target consumers for advertising. For instance if advertisers connect up three people as friends, co-workers, or family members, and two of them buy the advertiser's product, then they may choose to spend more in advertising to the third hold-out, on the belief that this target has a high propensity to buy their product.
Social scientists can also use social networks to model the way things made by people connect. Pages on the internet and the links between them form a social network in much the same way as people form networks with other people. Also, counter-intelligence agencies have used cell-phone data and calls to map out terrorist cells.
The image to the right shows the connections between different physicians who co-author papers on hepatitis C, for instance showing that two people who coauthored one paper, also mutually coauthored separate papers with another physician.
To simulate how a social network forms, mathematicians use random graphs that model how people make connections as they enter the network.
Random graphs are developed by adding nodes to the graph one by one and randomly adding edges between nodes according to a probabilistic rule. Different choices for the edge-adding rules result in graphs with very different structure. The simplest type of random graph is called an Erdos-Renyi graph. When each node is added, there is a fixed probability
p
that any given possible edge between it and another node is added. This means that any two people are equally likely to be connected as any two other people, and having a common connection doesn’t increase the chance that you are also connected to each other. This is very different from what we observe in real social networks, where people tend to cluster together.
Take a simple model of a social network where friendships form at random between individuals. Each person forms some number of friendships with other people,
k_i
. The average number of friendships that any given person makes is then
\frac{1}{N}\sum\limits_ik_i = \langle k \rangle
We call a friendship island (FI) a group of people such that everyone in the FI can reach anyone else in the FI by passing a note through mutual friends. If two people cannot send notes through a series of mutual friends, they must be in different FI.
At some value of
\langle k \rangle
\langle k\rangle_c
, the expected size of the largest FI becomes
\infty
\langle k \rangle_c
There are infinitely many people in the population.
In real life, people that have more friends are more likely to get more friends. This behavior is known technically as "preferential attachment," or informally as the "rich get richer effect." The simplest model that captures this feature is called a Barabasi-Albert graph. A Barabasi-Albert graph begins with a set of
N
completely connected nodes. Every time a node is added, a fixed number
m
of edges connected to that node are also added. (In contrast, the Erdos-Renyi model adds a variable number of edges at each step.) These nodes connect to other nodes with probability directly proportional to the number of edges the other node already has. This number is known as the degree
k_i
i
. Specifically, the probability
p_i
of connecting to node
is given by
p_i = \frac{k_i}{\sum_{j=1}^{N} k_j}.
Example degree distributions for Erdos-Renyi (red) and Barabasi-Albert (blue) graphs.
The denominator is simply the total degree of all nodes in the graph. Note that this is equal to twice the total number of edges, since each edge is counted once for each endpoint.
One of the most important properties of a graph is its degree distribution- a function giving the number of nodes with degree
k
. Barabasi-Albert graphs have degree distribution
p(k) \propto k^{-3}
, while Erdos-Renyi graphs have a binomial degree distribution. This means that Barabasi-Albert graphs have a "longer tail" than Erdos-Renyi graphs: they have many more nodes with a very high degree.
Functions of the form
p(k) \propto k^{-c}
c
is a constant, are called power laws. Networks with power law degree distributions are often also called scale-free. This is because they look the same at all magnitudes. If
p(k) \propto k^{-c}
, then multiplying
k
by a constant
a
p(ak) \propto (ak)^{-c} = a^{-c}k^{-c} \propto k^{-c}
. The scale factor
a
doesn’t change the shape of the distribution. In this way, social networks are like fractals. When you zoom in or out on the network, it looks roughly the same.
Because power laws have such elegant properties, they have been used to describe many physical phenomena, from the number of links that a website has, to the number of citations a paper gets, to the number of other proteins a particular protein interacts with [1].
In a social network modeled with a Barabasi-Albert random graph, what is the average ratio of the number of people with one friend to the number of people with three friends?
Social networks tend to be relatively small. It only takes a couple friends of friends to get to just about everyone you'll meet. One measure of the size of a graph is the average path length between any two points in the network. The path length between two nodes in a graph is the minimum number of edges that would need to be crossed in order to get from one node to the other.
A whimsical popular application of path length is the "Bacon number" for film actors. Kevin Bacon himself has a Bacon number of 0. Everyone who has acted in a film with Kevin Bacon has a Bacon number of one. Everyone who has acted with someone with a Bacon number of one has a Bacon number of two, and so on. Another way to put this is that an actor's Bacon number is the path length from them to Kevin Bacon, where the edges represent acting together in a film.
Similarly, in mathematics, a mathematician's Erdos number is the path length between them and famous mathematician Paul Erdos in a graph where edges represent co-authorship. The Erdos-Renyi random graph described above is named after Paul Erdos.
The average path length in a Barabasi-Albert graph grows proportional to
\text{log}(N)
, the logarithm of the number of nodes in the network. This means that the distance between nodes grows very slowly as more nodes are added. In the real world, though, the average distance can even shrink while the network grows.
In 2011, when Facebook had 721 million users, the average distance between two users was 4.74. In 2016, with 1.59 billion users, the average distance went down to 4.5 [2]. This difference between model and reality comes from the assumption that each new node makes a fixed number of connections with other nodes. If the number of new connections per node grows, then the average distance will go down (since the nodes are more connected).
If people did join Facebook in the way modeled by a Barabasi-Albert graph and the average path length with 721 million users was 4.74, what would the average path length between two users be in 2016 with 1.59 billion users? (Round your answer to two decimal places.)
Social networks with power law degree distributions have the peculiar feature that for most nodes in the network, the friends of that node have on average more friends than the node itself [3]. This result, known as the Friendship Paradox, arises because nodes will preferentially associate with nodes that already have a high degree.
Researchers have also found a Generalized Friendship Paradox in real-world social networks that extends to even more properties, like wealth and happiness. A person’s friends are, on average, richer and more happy than they are [4]. Unlike the Friendship Paradox, this is not a feature of any scale-free network, since it talks about properties other than those encoded by the network. However, it reflects something about the way people interact in real life.
Ever wonder how Facebook suggests "People you may know"? One of the first (and simplest) algorithms that is used, is to look for strangers with whom you have many mutual friends.
For example, if you and Colin are not friends on Facebook yet, but both of you are friends with Belinda, then {you, Belinda, Colin} form a Friend Suggestion Triangle (FST). The more FST's that exist, the more strangers Facebook can suggest with greater confidence that you might know them.
Out of a group of 10 people, what is the maximum number of FST's that exist?
Consider this mini social network of fifty people. Each person on the network is represented by an integer. The social network is represented as a matrix
A
. If person
i
j
are "friends" then,
A_{ij}=1
A_{ij}=0
. . How many mutual friends do the two people with the least number of mutual friends have?
A_{xx}=0
[1] Power Law Distributions in Empirical Data. http://arxiv.org/pdf/0706.1062.pdf Retrieved February 17, 2016.
[2] https://research.facebook.com/blog/three-and-a-half-degrees-of-separation/ Retrieved February 17, 2016.
[3] Feld, S. Why Your Friends Have More Friends Than You Do. American Journal of Sociology. Vol. 96, No. 6 (May, 1991), pp. 1464-1477. http://www.jstor.org/stable/2781907?seq=1#pagescantab_contents Retrieved February 17, 2016.
[4] How The Friendship Paradox Makes Your Friends Better Than You Are. MIT Technology Review. https://www.technologyreview.com/s/523566/how-the-friendship-paradox-makes-your-friends-better-than-you-are/ Retrieved February 17, 2016.
Lamb, A. Co-authorship network map of physicians publishing on hepatitis C. Retrieved May 2, 2016, from https://www.flickr.com/photos/speedoflife/8273922515
Cite as: Social Networks. Brilliant.org. Retrieved from https://brilliant.org/wiki/social-networks/
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Scientific Method | Brilliant Math & Science Wiki
Christopher Williams, alex wang, Rafael Del Valle Vega, and
The scientific method is the process by which scientists of all fields attempt to explain the phenomena in the world. It is how science is conducted--through experimentation. Generally, the scientific method refers to a set of steps whereby a scientist can form a conjecture (the hypothesis) for why something functions the way it does and then test their hypothesis. It is an empirical process; it uses real world data to prove the hypothesis. There is no exact set of
x
number of steps to conduct scientific experiments, or even some exact
y
number of experiments, but the general process involves making an observation, forming an hypothesis, forming a prediction from that hypothesis, and then experimental testing. The scientific method isn't limited to the physical or biological sciences, but also the social sciences, mathematics, computing and other fields where experimentation can be used to prove beliefs.
We could observe that whenever a fire is smothered, it goes out. For instance a small fire that is covered with a blanket is extinguished. We could hypothesize that the reason for this is that fire requires some gas in our air to form and remain a flame. We could then use a vacuum chamber to test this theory.
We would predict that outside of a vacuum, a fire could be lit but inside of a vacuum, with no air, that the fire would not ignite. If we were to test this theory, perhaps in multiple vacuums with multiple forms of tinder/fuel (wood, paper, petrol, etc.) and multiple means of ignition, we would notice that the fire never ignites.
If we wished, we could further refine our hypothesis, suggesting that fire can only ignite if there is sufficient oxygen in the air. This we'd also test in the vacuum chamber, by pulling out all the air, then adding in different gases. We would notice that the fire would only ignite in the presence of oxygen or an oxidizing agent.
It is possible that other, incorrect hypothesis could have been initially formed--such as smothering decreases the surface area the fire has, and could try making different sized fires--and been proven incorrect. Also, it is important to note that this single set of experiments is not enough to turn this hypothesis into a theorem. More experimentation and discovery would be necessary.
The scientific method also refers to the fact that science is ongoing. In some cases scientists continue to collect data to prove and disprove old theories. Or in other cases, scientists have hypothesis for why the universe behaves the way it does but are unable to gather sufficient data to prove their hypothesis. For instance, until recent discoveries at LIGO scientists could not confirm what happened when two black holes collided, although they believed (and it was confirmed in February 2016) that colliding black holes produced gravitational waves.
Falsifiability and why "theory" doesn't mean "untrue"
The scientific method is often presented as a set of steps, but not always with the same number or type of steps. However, philosophers of science generally agree that any presentation of the scientific method should have the following four steps:
Observe - Sometimes referred to as characterizing, defining, or measuring, experimenters first witness some aspect of the universe, for instance, an apple falling. These observations then form a question, such as "Why do objects fall to the earth?"
Hypothesize - Scientists then come up with a theory as to why this happens, for instance, the mass of the earth attracts the apple from the air to the ground.
Predict - Using the hypothesis, a scientist calculates what measurable data points they believe will result in a given experiment, for instance an apple at a height of
9.8
meters should fall to the ground in
\sqrt{2}
seconds, or should be at a velocity of
9.8\sqrt{2}
m/s the moment before it hits the ground.
Experiment - A test is run to determine if the prediction was correct.
With the notion that repeating these steps is also important. If a prediction is proven to be incorrect then alternative predictions and tests are conducted. Maybe even a new hypothesis could be formulated. Even if the hypothesis and prediction are correct, additional predictions and tests need to be run to best support any theory.
While this process can be explained or categorized differently than this, all formulations of the scientific method have empirical observations, a testable hypothesis, and testing data to prove or disprove that hypothesis. Crucial to this, is that an experimenter searches for experiments that produce the most unlikely results and experiments that are least likely to be coincidental. Hypotheses that produce highly unlikely predictions, in situations where little else could explain the result, are more likely to be true. Bayes' theorem can be used to show which predictions are more or less unlikely given some evidence, i.e. which proven predictions are "stronger" than others. For instance, the theory of evolution has been supported by the consistency of DNA across species whose phenomenology are significantly different. Despite the diversity of plant and animal species on Earth, the majority of our DNA is the same, and only 20 amino acids are the building blocks for every known living organism. It would be highly unlikely that vastly different forms of life have the same building blocks after millions, if not billions, of years of external manipulation, if not for some common origin.
The word "theory" can lead to confusion about how true some scientific principle is. Under the scientific method scientists use the word "theory" even for key principles (like gravity) that have been rigorously proven by modern science. This is because the scientific community believes it is important that hypothesis be falsifiable. Falsifiability refers to the fact that theories have been tested in experiments where they could have failed but did not. So when scientists refer to a principle as a theory, for instance Einstein's theory of relativity, they're actually referring to a hypothesis that has undergone the scientific method, i.e. that has been tested and proven true.
For instance, scientists sometimes refer to evolution as the "theory of evolution," which has contributed to the erroneous belief that the modern scientific theory of evolution is false. Really what the "theory of evolution" refers to is the ample research, testing, and empirical evidence that all consistently prove evolution to be true.
That isn't to say that theories can't be later disproven. Part of the advantage to the scientific method is that no theory is ever considered an unbreakable rule. Some theories seem correct given experiments that are run at the time they're created, but are proven wrong as new methods of experimentation are conducted. For instance, Einstein himself believed that the universe was static, not growing or contracting. That was later proven to be false and replaced with a theory that the universe was expanding (the Friedmann-LeMaitre model of an expanding universe, which Einstein himself accepted), but that its rate of expansion was slowing down. This was, in turn, also proven incorrect. The rate of the universe's expansion is speeding up.[1] Generally though, theories are modified over time, they are shown to be true under certain conditions, or partly true, and the strength of a theory may also be related to how long it has held up, without modification, to scrutiny.
In modern science, experimenters present both their findings and their methodology for review by their peers, other talented scientists and experimenters. This is done before a work is published, but also publication itself is considered a way of inviting peer review. By sharing and disseminating work widely, the greatest number of others can review the work and offer criticism as needed.
Related to peer review, is the notion that the results from experiments should be possible to reproduce. If one scientist conducts some experiment, others should be able to conduct the same experiment on their own and achieve the same results. Reproducible experiments strengthen theories.
Primarily used in medical, psychological, and behavioral economic testing, double-blind testing refers to having a test and control group, and running the experiment such that the person conducting the experiment does not know which is which. For instance, in testing the efficacy of a new drug, a pharmaceutical company may have a medical practitioner administer the new drug to one third of the test population, an existing known drug to another third, and a placebo, meaning something that isn't a drug but seems like it, to the remaining third of the test population, but without the nurse knowing which drug is which. The practitioner would then, still blind, track the progress of the entire testing population, gathering data about each test subject.
Double-blind studies are done to avoid biases that manipulate data, like controlling for the placebo effect where just giving a patient a drug that they perceive will be a cure can be causally linked to a decrease in symptoms. This positive causal effect occurs even with the drug that shouldn't affect the patient in anyway, when it is a sugar pill, or water, so long as the patient believes they are receiving a cure. Also double-blind studies help prevent observation bias, where the administrator of the drug may expect the population who received the new drug to outperform others, and so many inadvertently rate their progress better than other test groups.
A pharmaceutical company has a new drug they want to test to determine its efficacy. They have a hypothesis that this drug is super effective at curing a disease. Which of the following experiments/results best reflects the principles of the scientific method? Which is most scientific?
A) They gave 100 patients with the disease the drug and 100 patients a placebo from a population of 100,000 with the disease, they strictly controlled these patient's diet, limited other medication, and 77 of the subjects reported that their happiness improved significantly.
B) They found a remote island with an indigenous population that's genetically different from other populations and where 200 patients have the disease. They gave 100 patients on the island the drug and 100 a placebo. They strictly controlled these patient's diet, limited other medication, and found that 84 of the test patients had higher red and white blood cell count than the control group, and lower incidents of mortality from the disease than non-island populations.
C) They gave 100 patients with the disease the drug and 100 patients a placebo from a population of 100,000 with the disease, they strictly controlled these patient's diet, limited other medication, and found that only 5 of the test patients had higher red and white blood cell count than the control group, with no other changes in health.
D) They gave 100 patients with the disease the drug and 100 patients a placebo from a population of 100,000 with the disease, allowed both patients to consume and medicate in whatever way those patients desired, and found that 68 of the test patients had higher red and white blood cell count than the control group, with faster speed-to-recovery.
The theory of the scientific method has evolved over time, with modern historians pointing to Aristotle as an originator, and many looking to Thomas Kuhn's seminal work "The Structure of Scientific Revolutions" as a key influence on current conceptions of the method.
Aristotle classified reasoning into three types:
Abductive - Also known as guessing, abductive reasoning supposes that the most likely inference is correct. While this isn't rigorous, a well-informed individual is likely to make good guesses, and many significant theories of science have developed first from a guess.
Deductive - Deductive reasoning uses premises to reach conclusions. One of the most famous examples being "All men are mortal. Socrates is a man. Therefore, Socrates is mortal."
Inductive - Inductive reasoning is the one preferred by scientists, and can be considered an early version of the scientific method. Namely, inductive reasoning uses empirical observations to make inferences, and accounts for probability in those inferences. A theory reached by induction is said to be more or less likely to be true, stronger or weaker.
The philosophy of science refers to the logic and thinking behind the scientific method. It questions what makes something scientifically valid. For instance, the scientific method assumes that reality is objective, and that explanations exist for all phenomena humans can observe.
Thomas Kuhn's book is foundational to the philosophy of science and the way sociologists and historians look at science through the ages. In it, he popularized the term "paradigm shift" and promoted a historical understanding of scientific discovery not as a linear accumulation of understanding, but as a set of scientific revolutions that "shift" humanity's understanding. Further, paradigm shifts open up whole fields (for instance quantum mechanics, behavioral economics or genetics) with new approaches to understand the universe. Also what scientists consider true is not purely objective, but based on the consensus of the scientific community.
Nobelprize.org, . The Nobel Prize in Physics 2011 Saul Perlmutter, Brian P. Schmidt, Adam G. Riess. Retrieved October 24th 2016, from http://www.nobelprize.org/nobel_prizes/physics/laureates/2011/
Cite as: Scientific Method. Brilliant.org. Retrieved from https://brilliant.org/wiki/scientific-method/
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Gaussian Mixture Model | Brilliant Math & Science Wiki
John McGonagle, Geoff Pilling, Andrei Dobre, and
Vincent Tembo
A Gaussian mixture of three normal distributions.[1]
Gaussian mixture models are a probabilistic model for representing normally distributed subpopulations within an overall population. Mixture models in general don't require knowing which subpopulation a data point belongs to, allowing the model to learn the subpopulations automatically. Since subpopulation assignment is not known, this constitutes a form of unsupervised learning.
For example, in modeling human height data, height is typically modeled as a normal distribution for each gender with a mean of approximately 5'10" for males and 5'5" for females. Given only the height data and not the gender assignments for each data point, the distribution of all heights would follow the sum of two scaled (different variance) and shifted (different mean) normal distributions. A model making this assumption is an example of a Gaussian mixture model (GMM), though in general a GMM may have more than two components. Estimating the parameters of the individual normal distribution components is a canonical problem in modeling data with GMMs.
GMMs have been used for feature extraction from speech data, and have also been used extensively in object tracking of multiple objects, where the number of mixture components and their means predict object locations at each frame in a video sequence.
One hint that data might follow a mixture model is that the data looks multimodal, i.e. there is more than one "peak" in the distribution of data. Trying to fit a multimodal distribution with a unimodal (one "peak") model will generally give a poor fit, as shown in the example below. Since many simple distributions are unimodal, an obvious way to model a multimodal distribution would be to assume that it is generated by multiple unimodal distributions. For several theoretical reasons, the most commonly used distribution in modeling real-world unimodal data is the Gaussian distribution. Thus, modeling multimodal data as a mixture of many unimodal Gaussian distributions makes intuitive sense. Furthermore, GMMs maintain many of the theoretical and computational benefits of Gaussian models, making them practical for efficiently modeling very large datasets.
(Left) Fit with one Gaussian distribution (Right) Fit with Gaussian mixture model with two components
Numbers of pregnancies across many humans Speeds across cars just before reaching a specific traffic light Student scores on a specific standardized test
Which of the following data sets is most likely to be well-modeled by a Gaussian mixture model?
A Gaussian mixture model is parameterized by two types of values, the mixture component weights and the component means and variances/covariances. For a Gaussian mixture model with
K
components, the
k^\text{th}
component has a mean of
\mu_k
\sigma_k
for the univariate case and a mean of
\vec{\mu}_k
and covariance matrix of
\Sigma_k
for the multivariate case. The mixture component weights are defined as
\phi_k
C_k
, with the constraint that
\sum_{i=1}^K\phi_i = 1
so that the total probability distribution normalizes to
1
. If the component weights aren't learned, they can be viewed as an a-priori distribution over components such that
p(x \text{ generated by component } C_k) = \phi_k
. If they are instead learned, they are the a-posteriori estimates of the component probabilities given the data.
\begin{aligned} p(x) &= \sum_{i=1}^K\phi_i \mathcal{N}(x \;|\; \mu_i, \sigma_i)\\ \mathcal{N}(x \;|\; \mu_i, \sigma_i) &= \frac{1}{\sigma_i\sqrt{2\pi}} \exp\left(-\frac{(x-\mu_i)^2}{2\sigma_i^2}\right)\\ \sum_{i=1}^K\phi_i &= 1 \end{aligned}
\begin{aligned} p(\vec{x}) &= \sum_{i=1}^K\phi_i \mathcal{N}(\vec{x} \;|\; \vec{\mu}_i, \Sigma_i)\\ \mathcal{N}(\vec{x} \;|\; \vec{\mu}_i, \Sigma_i) &= \frac{1}{\sqrt{(2\pi)^K|\Sigma_i|}} \exp\left(-\frac{1}{2}(\vec{x}-\vec{\mu}_i)^\mathrm{T}{\Sigma_i}^{-1}(\vec{x}-\vec{\mu}_i)\right)\\ \sum_{i=1}^K\phi_i &= 1 \end{aligned}
If the number of components
K
is known, expectation maximization is the technique most commonly used to estimate the mixture model's parameters. In frequentist probability theory, models are typically learned by using maximum likelihood estimation techniques, which seek to maximize the probability, or likelihood, of the observed data given the model parameters. Unfortunately, finding the maximum likelihood solution for mixture models by differentiating the log likelihood and solving for
0
is usually analytically impossible.
Expectation maximization (EM) is a numerical technique for maximum likelihood estimation, and is usually used when closed form expressions for updating the model parameters can be calculated (which will be shown below). Expectation maximization is an iterative algorithm and has the convenient property that the maximum likelihood of the data strictly increases with each subsequent iteration, meaning it is guaranteed to approach a local maximum or saddle point.
Expectation maximization for mixture models consists of two steps.
The first step, known as the expectation step or E step, consists of calculating the expectation of the component assignments
C_k
x_i \in X
given the model parameters
\phi_k
\mu_k
\sigma_k
The second step is known as the maximization step or M step, which consists of maximizing the expectations calculated in the E step with respect to the model parameters. This step consists of updating the values
\phi_k
\mu_k
\sigma_k
The entire iterative process repeats until the algorithm converges, giving a maximum likelihood estimate. Intuitively, the algorithm works because knowing the component assignment
C_k
x_i
makes solving for
\phi_k
\mu_k
\sigma_k
easy, while knowing
\phi_k
\mu_k
\sigma_k
makes inferring
p(C_k|x_i)
easy. The expectation step corresponds to the latter case while the maximization step corresponds to the former. Thus, by alternating between which values are assumed fixed, or known, maximum likelihood estimates of the non-fixed values can be calculated in an efficient manner.
The EM algorithm updating the parameters of a two-component bivariate Gaussian mixture model.[2]
Algorithm for Univariate Gaussian Mixture Models
The expectation maximization algorithm for Gaussian mixture models starts with an initialization step, which assigns model parameters to reasonable values based on the data. Then, the model iterates over the expectation (E) and maximization (M) steps until the parameters' estimates converge, i.e. for all parameters
\theta_t
at iteration
t
|\theta_{t}-\theta_{t-1}| \le \epsilon
for some user-defined tolerance
\epsilon
. A graphic of the EM algorithm in action for a two-component, bivariate Gaussian mixture model is displayed on the right.
The EM algorithm for a univariate Gaussian mixture model with
K
components is described below. A variable denoted
\hat{\theta}
denotes an estimate for the value
\theta
. All equations can be derived algebraically by solving for each parameter as outlined in the section above titled EM for Gaussian Mixture Models.
Initialization Step:
Randomly assign samples without replacement from the dataset
X=\{x_1, ..., x_N\}
to the component mean estimates
\hat{\mu}_1, ..., \hat{\mu}_K
. E.g. for
K=3
N=100
\hat{\mu}_1 = x_{45}, \hat{\mu}_2 = x_{32}, \hat{\mu}_3 = x_{10}.
Set all component variance estimates to the sample variance
\hat{\sigma}_1^2, ..., \hat{\sigma}_K^2=\frac{1}{N}\sum_{i=1}^N(x_i-\bar{x})^2,
\bar{x}
is the sample mean
\bar{x}=\frac{1}{N}\sum_{i=1}^Nx_i.
Set all component distribution prior estimates to the uniform distribution
\hat{\phi}_1, ..., \hat{\phi}_K=\frac{1}{K}.
\forall i, k
\hat{\gamma}_{ik} = \frac{\hat{\phi}_k \mathcal{N}(x_i \;|\; \hat{\mu}_k, \hat{\sigma}_k)}{\sum_{j=1}^K\hat{\phi}_j \mathcal{N}(x_i \;|\; \hat{\mu}_j, \hat{\sigma}_j)},
\hat{\gamma}_{ik}
x_i
is generated by component
C_k
\hat{\gamma}_{ik}=p(C_k|x_i, \hat{\phi}, \hat{\mu}, \hat{\sigma}).
\hat{\gamma}_{ik}
calculated in the expectation step, calculate the following in that order
\forall k:
\displaystyle \hat{\phi}_k = \sum_{i=1}^N\frac{\hat{\gamma}_{ik}}{N}
\displaystyle \hat{\mu}_k = \frac{\sum_{i=1}^N\hat{\gamma}_{ik} x_i }{\sum_{i=1}^N\hat{\gamma}_{ik}}
\displaystyle \hat{\sigma}_k^2 = \frac{\sum_{i=1}^N\hat{\gamma}_{ik} (x_i - \hat{\mu}_k)^2 }{\sum_{i=1}^N\hat{\gamma}_{ik}}.
When the number of components
K
is not known a priori, it is typical to guess the number of components and fit that model to the data using the EM algorithm. This is done for many different values of
K
. Usually, the model with the best trade-off between fit and number of components (simpler models have fewer components) is kept.
The EM algorithm for the multivariate case is analogous, though it is more complicated and thus is not expounded here.
Once the EM algorithm has run to completion, the fitted model can be used to perform various forms of inference. The two most common forms of inference done on GMMs are density estimation and clustering.
Clustering using a Gaussian mixture model. Each color represents a different cluster according to the model.[3]
Since the GMM is completely determined by the parameters of its individual components, a fitted GMM can give an estimate of the probabilities of both in-sample and out-of-sample data points, known as density estimation. Furthermore, since numerically sampling from an individual Gaussian distribution is possible, one can easily sample from a GMM to create synthetic datasets.
Sampling from a GMM consists of the following steps:
1. Sample the Gaussian component according to the distribution defined by
p(C_s) = \phi_s.
x
from the distribution for component
C_s
, according to the distribution defined by
\mathcal{N}(x \;|\; \mu_s, \sigma_s).
Using Bayes' theorem and the estimated model parameters, one can also estimate the posteriori component assignment probability. Knowing that a data point is likely from one component distribution versus another provides a way to learn clusters, where cluster assignment is determined by the most likely component assignment. Clustering has many uses in machine learning, ranging from tissue differentiation in medical imaging to customer segmentation in market research.
Given a univariate model's parameters, the probability that a data point
x
belongs to component
C_i
is calculated using Bayes' theorem:
p(C_i \;|\; x) = \frac{p(x, C_i)}{p(x)} = \frac{p(C_i)p(x \;|\; C_i)}{\sum_{j=1}^Kp(C_j)p(x \;|\; C_j)} = \frac{\phi_i \mathcal{N}(x \;|\; \mu_i, \sigma_i)}{\sum_{j=1}^K\phi_j \mathcal{N}(x \;|\; \mu_j, \sigma_j)}.
GMMs have been used recently for feature extraction from speech data for use in speech recognition systems[4]. They have also been used extensively in object tracking of multiple objects, where the number of mixture components and their means predict object locations at each frame in a video sequence[5]. The EM algorithm is used to update the component means over time as the video frames update, allowing object tracking.
, S. Gaussian-mixture-example. Retrieved June 13, 2012, from https://commons.wikimedia.org/wiki/File:Gaussian-mixture-example.svg
, C. EM_Clustering_of_Old_Faithful_data. Retrieved August 1, 2012, from https://commons.wikimedia.org/wiki/File:EM_Clustering_of_Old_Faithful_data.gif
, C. SLINK-Gaussian-data. Retrieved October 23, 2011, from https://commons.wikimedia.org/wiki/File:SLINK-Gaussian-data.svg
Deng, L. (2014). Automatic Speech Recognition- A Deep Learning Approach (pp. 6-8). Springer.
Santosh, D. (2013). Tracking Multiple Moving Objects Using Gaussian Mixture Model. International Journal of Soft Computing and Engineering, 3-2, 114-119.
Cite as: Gaussian Mixture Model. Brilliant.org. Retrieved from https://brilliant.org/wiki/gaussian-mixture-model/
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Probabilistic Principle of Inclusion and Exclusion | Brilliant Math & Science Wiki
Andy Hayes, Anuj Shikarkhane, Mahindra Jain, and
The probabilistic principle of inclusion and exclusion (PPIE for short) is a method used to calculate the probability of unions of events. For two events, the PPIE is equivalent to the probability rule of sum:
A
B
be events. The probability of the union of these events is:
P(A\cup B)=P(A)+P(B)-P(A\cap B)
The PPIE is closely related to the principle of inclusion and exclusion in set theory. The formulas for probabilities of unions of events are very similar to the formulas for the size of unions of sets.
PPIE for Two Events
PPIE for Three Events
General Form of PPIE
The PPIE for two events is equivalent to the probability rule of sum.
A card is drawn from a standard deck of cards. What is the probability that the card drawn is a queen or a heart?
A
be the event that the card is a queen, and let
B
be the event that the card is a heart. Then
P(A \cup B) = P(A) + P(B) - P(A \cap B).
Since there are 13 different ranks of cards in the deck,
P(A) = \frac{1}{13} ,
and since there are 4 suits in the deck,
P(B) = \frac{1}{4}.
There is only one card that is both a queen and a heart, so
P(A \cap B) = \frac{1}{52}.
P(A \cup B) = \frac{1}{4} + \frac{1}{13} - \frac{1}{52} = \frac{16}{52} = \boxed{\dfrac{4}{13}}.
From 1,2,3,...,250, one number is selected at random. What is the probability that it is either a multiple of 5 or a multiple of 4?
NOTE: Give your answer as a decimal.
271/900 171/900 271/1000 9/100 19/900
An integer from 100 through 999, inclusive, is to be chosen at random. What is the probability that the number chosen will have 0 as at least one digit?
When events are independent, the rule of product can be used to find the probability of an intersection of events. Then, the rule of sum can be used to find the probability of the union of those events.
A fair 6-sided die and a fair 8-sided die are rolled. What is the probability that one of the dice rolls is a 6?
A
be the event that the 6-sided die is 6.
P(A)=\dfrac{1}{6}
B
P(B)=\dfrac{1}{8}
The events are independent, so by the rule of product,
P(A\cap B)=\dfrac{1}{6}\times\dfrac{1}{8}=\dfrac{1}{48}
By the rule of sum,
P(A\cup B)=\dfrac{1}{6}+\dfrac{1}{8}-\dfrac{1}{48}=\dfrac{13}{48}
The probability that either dice roll is 6 is
\boxed{\dfrac{13}{48}}
0.0353
0.035
0.0003
0.0347
An actuary at ManyProvince Insurance estimates that Mr. Gunderson has a 0.02 probability of having an accident in the next year, and Mrs. Gunderson has a 0.015 probability of having an accident in the next year. The actuary also estimates that the event that Mr. Gunderson has an accident is independent of the event that Mrs. Gunderson has an accident.
Using the actuary's estimates, what is the probability that either Gunderson will have an accident in the next year?
When the three events are independent, the probability of the union of those events can be found using complement probabilities and the rule of product:
Given three independent events
A
B
C
, the probability of the union of these events is:
P(A\cup B\cup C)=1-P(A^c)P(B^c)P(C^c)
A fair 6-sided die is rolled three times. Let
A
be the event that the first roll is 1, Let
B
be the event that the second roll is 3, and let
C
be the event that the third roll is 3. What is
P(A\cup B\cup C)
P(A)=
\dfrac{1}{6}
P(A^c)=\dfrac{5}{6}
P(B)=
\dfrac{1}{6}
P(B^c)=\dfrac{5}{6}
P(C)=
\dfrac{1}{6}
P(C^c)=\dfrac{5}{6}
These events are independent, so the above formula can be used:
P(A\cup B\cup C)=1-\left(\dfrac{5}{6}\right)^3=\boxed{\dfrac{91}{216}}
When events are dependent, then the probability of each intersection of events must be either known or calculated. Then, the probability of the union of events can be calculated using the following formula:
Given three dependent events
A
B
C
P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B \cap C)+P(A\cap B\cap C)
When events are independent, the probability of a union of those events can be found using complement probabilities and the rule of product:
\{A_1, \dots ,A_n\}
is a set of mutually independent events, then the probability of the union of those events is:
\large P\left(\bigcup\limits_{i=1}^{n}{A_i}\right)=1-\prod\limits_{i=1}^{n}{P(A_i^c)}
This identity is a direct result of De Morgan's Laws.
When events are dependent, the general form of the PPIE for any number of events involves adding or subtracting intersections of events:
\{A_1, \dots A_n\}
is a set of dependent events, then the probability of the union of those events is:
\large P\left(\bigcup\limits_{i=1}^{n}{A_i}\right)=\sum\limits_{k=1}^{n}(-1)^{k+1}\left(\sum\limits_{1\le i_1< \dots < i_k\le n}P\left(A_{i_1}\cap \dots \cap A_{i_k}\right) \right)
A
B
C
D
be pairwise dependent events. What is the formula for the probability of the union of these events?
By the general formula above, this is:
\begin{array}{ll} P(A\cup B\cup C\cup D)= & P(A)+P(B)+P(C)+P(D) \\ & -\ P(A\cap B)-P(A\cap C)-P(A\cap D)-P(B\cap C)-P(B\cap D)-P(C\cap D) \\ & +\ P(A\cap B\cap C)+P(A\cap B\cap D)+P(A\cap C\cap D)+P(B\cap C\cap D) \\ & -\ P(A\cap B\cap C\cap D) \end{array}
Cite as: Probabilistic Principle of Inclusion and Exclusion. Brilliant.org. Retrieved from https://brilliant.org/wiki/probabilistic-principle-of-inclusion-and-exclusion/
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(→IconSymbols)
[http://grass.gdf-hannover.de/twiki/pub/GRASS/GrassAddOns/d.hyperlink.tar.gz d.hyperlink] is an interactive shell script that allows the viewing of hyperlinked images from a vector's attribute table in an external image viewer. Queries can be made via SQL statements or interactive mouse-clicking. The attribute table must be pre-populated with a column containing the image to link the vector to; the user also specifies the image folder in the current MAPSET where the images are located. The script currently supports gimp, Eye of Gnome, gthumb, gpdf, and Inkscape image viewers.
r.csr integrates several Grass programs to produce colored, shaded-relief rasters in one step. Accepts single or multiple elevation/bathymetry maps as input; optionally will fill data holidays with 3x3 median filter, multiple times, if required; can apply color maps from a) input raster, b) another raster in MAPSET, or c) from a rules file. Output colored, shaded-relief rasters can optionally be exported to tiff format and archived using tar with gzip/bzip2 compression if appropriate flags are given. Shading parameters can be modified, though useful defaults are given.
{\displaystyle Insertformulahere}
ps.map samples/templates: are people willing to include these here? ps.map scripts
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Farming - Ring of Brodgar
Farming also refers to an incrementable ability.
LP Cost 400
Enabled Baking, Beekeeping, Deep Artifice, Gardening, Plant Lore, Sausage Making, Sewing, Winemaking, Yeomanry
Required By (99) Any Flour, Any Onion, Barley Crop, Barley Flour, Bat Guano, Beer, Beetroot, Beetroot Crop, Beetroot Leaves, Black Pepper, Boiled Pepper Drupe, Butter Porridge, Carrot, Carrot Crop, Chef's Hat, Compost Bin, Crop Circle, Crying Red Onion, Crying Yellow Onion, Cucumber, Cucumber Crop, Cured Pipeweed, Dressed Lettuce, Dried Pepper Drupe, Fat-Braised Veg, Feather Garland, Flax Crop, Fresh Hemp Bud, Fresh Leaf of Pipeweed, Giant Pumpkin, Giant Turnip, Grapes, Grapes Crop, Head of Lettuce, Hemp Crop, Hop Cones, Hop Garland, Hops Crop, Jack-o'-Mask, Leek, Leek Crop, Lettuce Crop, Lettuce Leaf, Metal Plow, Millet Crop, Millet Flour, Mushrooms in Jelly, Opium, Pea Crop, Peapod, Peppercorn, Peppercorn Crop, Pipestuff, Pipeweed Crop, Poppy Crop, Poppy Flower, Poppy Garland, Pumpkin Crop, Pumpkin Flesh, Pumpkin Stew, Quern, Red Onion, Red Onion Crop, Red-Shred Salad, Seeds of Barley, Seeds of Carrot, Seeds of Cucumber, Seeds of Flax, Seeds of Grape, Seeds of Hemp, Seeds of Leek, Seeds of Lettuce, Seeds of Millet, Seeds of Pipeweed, Seeds of Poppy, Seeds of Pumpkin, Seeds of Turnip, Seeds of Wheat, Spicy Salad, Straw, Straw Basket, Straw Doll, Straw Hat, Straw Twine, Swill, Trellis, Turnip, Turnip Crop, Uncrushed Husk, Unusually Large Hop Cone, Weird Beetroot, Weißbier, Wheat Crop, Wheat Flour, Wicker Picker, Wood Incorrupt, Wooden Plow, Yellow Onion, Yellow Onion Crop
3 Quality Grinding
6 Seed Stacking
7 About Crops
The Farming skill allows a Hearthling to plant seeds in soil to grow crops for food or materials. There are 20 crops in the game, 15 of which are planted directly into soil and 5 of which must be planted on a Trellis. To plant crops, you need 5 seeds per tile, (One full stack of 50 seeds will plant 10 crops), tilled ground, and patience. Note that certain crops are planted with the crop itself and not seeds, for instance, Beetroots and Peas do not have seeds, you simply plant the item themselves. Check the table below for more info.
First, till some ground. You can do this on most dirt terrains by hand, but building a plow is preferable. A wooden plow lets you simply walk along at Crawl speed (1.5 tiles per second) to till the ground, a Metal Plow lets you do so at Run speed (4.5 tiles per second).
Now, left click a stack of at least 5 seeds, and right click a tile of tilled ground. You can do this faster by holding shift and then right clicking the tile, which keeps the stack attached to your cursor, or at maximum speed by simply shift + right clicking the seeds in your inventory, which will then let you drag and drop a zone over a square of tiles which your character will automatically walk to and plant until they're done or they run out of seeds.
Now comes the waiting. Plants take varying times to grow. Carrots will grow in a day, Pumpkins will grow in a week, with other crops somewhere in between.
You will know a crop is fully grown when you can right click it and click harvest. Your character will walk over and pick the crop, giving you its materials and/or seeds. A Scythe will harvest multiple crops in front of you at a time. You can shift + right click a fully grown crop to drag and drop a zone over other fully grown crops of the same type to quickly and automatically harvest them as well, just like planting.
This won't work for all crops however. Specifically, Peas, Peppercorns, Grapes, Cucumbers and Hops crops are vineyard plants and are too delicate to be grown on their own and require a Trellis. This is quite simple though, simply build a trellis and right click the seeds onto the trellis like it were a tilled tile. Trellis plants generally take longer to grow than others.
Seeds too annoying to carry and store? Worry not, they can be put into Buckets, Barrels, and Granaries!
For buckets, you must have the bucket in a container or your hands, they can't be filled when in your inventory. Simply right click the seeds onto the bucket. Each bucket can hold 1000 seeds, allowing you to carry a total 2000 seeds on you without any inventory space being used!
For barrels, simply right click the seeds straight into the barrel. Each barrel can hold 10,000 seeds.
Granaries are far more difficult to make than the other two choices, but are maximally efficient! They have 10 slots, to distribute the grand total capacity of 200,000 seeds(units). Simply right click the seeds into a slot. Granaries can hold other things like flours, grists, ash, and such as well.
Farming not rewarding enough?
Invest some Learning Points into Gardening, Plant Lore or even the ultimate Druidic Rite! Each skill makes you gain extra seeds and products when harvesting.
And if you're a true connoisseur of the farming arts, consider attempting the Farmer's Credo, which grants you further benefits to your growing (pun intended) arsenal. See more in Credo.
How do i even get seeds?!
Aside from asking or borrowing them from a neighbour or friend, you can get random seeds from drying Wild Windsown Weeds on a Drying Frame or Herbalist Table.
How do i farm things like Blueberries?
That is another form of farming called Gardening. It is similar to Farming but you need to get that skill and then make yourself some Garden Pots and fill them with soil and water, then you right click your forageable of choice like Blueberries and right click them into the pot.
The quality of planted crops is based upon the quality of the planted seed with a [-2, +5] random range limited by space, time, and the farmer's farming ability level.
Exception: If you are above your local quality cap, and if you are the server lead in crop quality for that crop type, your crops will instead change quality randomly in the range from -2 to +2 with each generation (ergo: effectively no crop quality limit). The local quality cap effectively only matters for the server lead. (ref: ... and Yesteryear's Patch (2021-10-28))
Each batch of a crop which is planted in the same area (~2 minimaps), time (~2 hours), and of the same type (carrots, peas, barley, etc), will be given the same random quality modification uniformly, and planted tiles of that batch will thus all have their quality affected in the same way.
Harvesting is not affected by the farming skill. However, it is affected by skills that add additional seeds - Gardening, Plant Lore and Druidic Rite; you will not get additional seeds if the harvesting character lacks these skills.
Crop byproducts, like Straw or Finer Plant Fibre, are affected by the [-2, +5] random quality modifier and will be the same quality as the harvested seeds.
To maximize quality gain, it is advisable to take any given quality of seed and split it up into multiple plantings. With maximum skills, plants will yield three times their number in plantings-worth of seed. Planting in three phases, at least 5 hours apart will give you three chances to score the +5. Whatever the results, the highest quality seed will be sufficient to repeat the planting. If expanding the crops is a priority, two plantings leaves enough seed to expand each planting by 50%. This method avoids the downside of a bad RNG result unless all three plantings roll poorly (which is unlikely). Therefore, doing this will consistently increase average seed quality, and thus product quality, over time. The excess seeds or crops can then be used to crafting and consumption.
If you are fortunate to have seeds of a higher quality than your skills would otherwise produce (i.e. from a neighbour), then the resulting planting quality will not be between -2 and +5, but all positives are set to 0, giving a good chance to maintain the high quality while building up your skills to match.
Watered by sweat there grew,
Golden in Autumn, that Spring,
Where winds did play and hide,
In sun-kissed Wheat and Barley.
Your hands lie heavy on plow and scythe, clawing in your palms the furrows of age and hard labor. You know how to till and plow, and how to plant seeds.
Farming lets you plant and harvest crops (except grape seeds, which need Winemaking), plant trees, plow fields, mill Flour, and craft Straw Hats. You need at least 5 of the same seed to plant.
Adventure > Landscaping > Hand Plow
Plows squares of furrowed soil that can be used to plant crops. Furrowed soil is not needed to plant trees, however. Plowing by hand uses more stamina than using the Plow item.
Build > Containers
Craft > Clothes & Equipment > Hats & Headwear
Craft > Processing & Materials
Seed Stacking
As of World 8, seeds of the same crop can stack (up to 50 in the inventory, 1000 in a bucket and 10000 in a barrel) if they have the same qualities. If you mix seeds with different qualities together, using a bucket or barrel, the resulting quality is:
{\displaystyle qStack=\left({\frac {qSeed1+qSeed2+qSeed3+...+qSeedn}{n}}\right)}
You can use this to raise the overall quality of your seeds of a given crop by sacrificing higher tier seeds.
Bull Ram (2022-03-20) >"Can now inspect crops for quality when they are harvestable (including trellis crops)."
Iced Bait (2021-11-07) >"When planting an area, you will now plant using a small area of effect, planting several tiles around you at once. The character will not move to use that optimally, but it's something. Traveling salesman algorithm when?"
Harvest Skis (2021-10-21) >"Removed (effectively, at least) the timer from harvesting crops."
Post Rose (2018-05-29) >"Made it so that crops no longer need to be harvested the second they finish in order achieve their minimum cycle time. Whenever you destroy or harvest a crop the time the crop harvested had spent in the last growth stage is saved per tile or trellis. That time, up to a point, will then be added to your next planting on that tile. Do note that if you do not replant within a certain period, the retained time will be lost. Replanting within 3 RL days of the crop reaching its final growth stage should not lose you any time."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Farming&oldid=93091"
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Ecology Knowpia
Levels, scope, and scale of organizationEdit
NicheEdit
Niche constructionEdit
{\displaystyle {\frac {\operatorname {d} N(t)}{\operatorname {d} t}}=bN(t)-dN(t)=(b-d)N(t)=rN(t),}
{\displaystyle {\frac {\operatorname {d} N(t)}{\operatorname {d} t}}=rN(t)-\alpha N(t)^{2}=rN(t)\left({\frac {K-N(t)}{K}}\right),}
where N(t) is the number of individuals measured as biomass density as a function of time, t, r is the maximum per-capita rate of change commonly known as the intrinsic rate of growth, and
{\displaystyle \alpha }
is the crowding coefficient, which represents the reduction in population growth rate per individual added. The formula states that the rate of change in population size (
{\displaystyle \mathrm {d} N(t)/\mathrm {d} t}
) will grow to approach equilibrium, where (
{\displaystyle \mathrm {d} N(t)/\mathrm {d} t=0}
), when the rates of increase and crowding are balanced,
{\displaystyle r/\alpha }
. A common, analogous model fixes the equilibrium,
{\displaystyle r/\alpha }
as K, which is known as the "carrying capacity."
Metapopulations and migrationEdit
Ecosystem ecologyEdit
Food websEdit
HolismEdit
Relation to evolutionEdit
Behavioural ecologyEdit
Cognitive ecologyEdit
r/K selection theoryEdit
Molecular ecologyEdit
Restoration and managementEdit
Relation to the environmentEdit
Disturbance and resilienceEdit
Metabolism and the early atmosphereEdit
Radiation: heat, temperature and lightEdit
Physical environmentsEdit
Wind and turbulenceEdit
Biogeochemistry and climateEdit
^ a b Cardinale, Bradley J.; Duffy, J. Emmett; Gonzalez, Andrew; Hooper, David U.; Perrings, Charles; Venail, Patrick; Narwani, Anita; Mace, Georgina M.; Tilman, David; Wardle, David A.; Kinzig, Ann P.; Daily, Gretchen C.; Loreau, Michel; Grace, James B.; Larigauderie, Anne; Srivastava, Diane S.; Naeem, Shahid; Gonzalez, Andrew; Hooper, David U.; Perrings, Charles; Venail, Patrick; Narwani, Anita; Mace, Georgina M.; Tilman, David; Wardle, David A.; Kinzig, Ann P.; Daily, Gretchen C.; Loreau, Michel; Grace, James B.; Larigauderie, Anne; Srivastava, Diane S.; Naeem, Shahid (6 June 2012). "Biodiversity loss and its impact on humanity" (PDF). Nature. 486 (7401): 59–67. Bibcode:2012Natur.486...59C. doi:10.1038/nature11148. PMID 22678280. S2CID 4333166. Archived (PDF) from the original on 21 September 2017. Retrieved 10 August 2019. {{cite journal}}: CS1 maint: multiple names: authors list (link)
^ a b MacArthur, R.; Wilson, E. O. (1967). "The Theory of Island Biogeography". Princeton, NJ: Princeton University Press. {{cite journal}}: Cite journal requires |journal= (help)
^ Neri Salvadori, Pasquale Commendatore, Massimo Tamberi (14 May 2014). Geography, structural Change and Economic Development: Theory and Empirics. Edward Elgar Publishing. {{cite book}}: CS1 maint: uses authors parameter (link)
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Group-invariant separating polynomials on a Banach space
2022 Group-invariant separating polynomials on a Banach space
Javier Falcó, Domingo García, Manuel Maestre, Mingu Jung
Javier Falcó,1 Domingo García,1 Manuel Maestre,2 Mingu Jung3
1Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot (Valencia), Spain
2Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot (Valencia),
3Department of Mathematics, POSTECH, Pohang 790-784, Republic of Korea
We study the group-invariant continuous polynomials on a Banach space
X
that separate a given set
K
X
z
K
X
is a real Banach space,
G
is a compact group of
\mathcal{L}\left(X\right)
K
G
-invariant set in
X
z
is a point outside
K
that can be separated from
K
by a continuous polynomial
Q
z
can also be separated from
K
G
-invariant continuous polynomial
P
. It turns out that this result does not hold when
X
is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that
X
has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group-invariant polynomial from the closed unit ball.
The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003857). The first, second, and fourth authors were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. The second and fourth authors were also supported by Prometeo PROMETEO/2017/102.
Javier Falcó. Domingo García. Manuel Maestre. Mingu Jung. "Group-invariant separating polynomials on a Banach space." Publ. Mat. 66 (1) 207 - 233, 2022. https://doi.org/10.5565/PUBLMAT6612209
Received: 20 April 2020; Accepted: 2 June 2020; Published: 2022
Primary: 14L24 , 46G20
Keywords: Banach space , group-invariant , polynomials , separation theorem
Javier Falcó, Domingo García, Manuel Maestre, Mingu Jung "Group-invariant separating polynomials on a Banach space," Publicacions Matemàtiques, Publ. Mat. 66(1), 207-233, (2022)
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How to Lower a DTI Ratio
Debt-to-Income Ratio FAQs
The debt-to-income (DTI) ratio measures the amount of income a person or organization generates in order to service a debt.
A DTI of 43% is typically the highest ratio a borrower can have and still get qualified for a mortgage, but lenders generally seek ratios of no more than 36%.
A low DTI ratio indicates sufficient income relative to debt servicing, and it makes a borrower more attractive.
Understanding the Debt-to-Income (DTI) Ratio
A low debt-to-income (DTI) ratio demonstrates a good balance between debt and income. In other words, if your DTI ratio is 15%, that means that 15% of your monthly gross income goes to debt payments each month. Conversely, a high DTI ratio can signal that an individual has too much debt for the amount of income earned each month.
Typically, borrowers with low debt-to-income ratios are likely to manage their monthly debt payments effectively. As a result, banks and financial credit providers want to see low DTI ratios before issuing loans to a potential borrower. The preference for low DTI ratios makes sense since lenders want to be sure a borrower isn't overextended meaning they have too many debt payments relative to their income.
The debt-to-income (DTI) ratio is a personal finance measure that compares an individual’s monthly debt payment to their monthly gross income. Your gross income is your pay before taxes and other deductions are taken out. The debt-to-income ratio is the percentage of your gross monthly income that goes to paying your monthly debt payments.
\begin{aligned} &\text{DTI} = \frac{ \text{Total of Monthly Debt Payments} }{ \text{Gross Monthly Income} } \\ \end{aligned}
DTI=Gross Monthly IncomeTotal of Monthly Debt Payments
Sum up your monthly debt payments including credit cards, loans, and mortgage.
The debt-to-limit ratio, which is also called the credit utilization ratio, is the percentage of a borrower’s total available credit that is currently being utilized. In other words, lenders want to determine if you're maxing out your credit cards. The DTI ratio calculates your monthly debt payments as compared to your income, whereby credit utilization measures your debt balances as compared to the amount of existing credit you've been approved for by credit card companies.
Although important, the DTI ratio is only one financial ratio or metric used in making a credit decision. A borrower's credit history and credit score will also weigh heavily in a decision to extend credit to a borrower. A credit score is a numeric value of your ability to pay back a debt. Several factors impact a score negatively or positively, and they include late payments, delinquencies, number of open credit accounts, balances on credit cards relative to their credit limits, or credit utilization.
The DTI ratio does not distinguish between different types of debt and the cost of servicing that debt. Credit cards carry higher interest rates than student loans, but they're lumped in together in the DTI ratio calculation. If you transferred your balances from your high-interest rate cards to a low-interest credit card, your monthly payments would decrease. As a result, your total monthly debt payments and your DTI ratio would decrease, but your total debt outstanding would remain unchanged.
The debt-to-income ratio is an important ratio to monitor when applying for credit, but it's only one metric used by lenders in making a credit decision.
John is looking to get a loan and is trying to figure out his debt-to-income ratio. John's monthly bills and income are as follows:
John's total monthly debt payment is $2,000:
\$2,000 = \$1,000 + \$500 + \$500
$2,000=$1,000+$500+$500
John's DTI ratio is 0.33:
0.33 = \$2,000 \div \$6,000
0.33=$2,000÷$6,000
In other words, John has a 33% debt-to-income ratio.
How to Lower a Debt-to-Income Ratio
Similarly, if John’s income stays the same at $6,000, but he is able to pay off his car loan, his monthly recurring debt payments would fall to $1,500 since the car payment was $500 per month. John's DTI ratio would be calculated as $1,500 ÷ $6,000 = 0.25 or 25%.
Real-World Example of the DTI Ratio
Wells Fargo Corporation (WFC) is one of the largest lenders in the U.S. The bank provides banking and lending products that include mortgages and credit cards to consumers. Below is an outline of their guidelines of the debt-to-income ratios that they consider creditworthy or need improvement.
50% or higher DTI ratio means you have limited money to save or spend. As a result, you won't likely have money to handle an unforeseen event and will have limited borrowing options.
The debt-to-income (DTI) ratio is the percentage of your gross monthly income that goes to paying your monthly debt payments and is used by lenders to determine your borrowing risk. A low debt-to-income (DTI) ratio demonstrates a good balance between debt and income. Conversely, a high DTI ratio can signal that an individual has too much debt for the amount of income earned each month. Typically, borrowers with low debt-to-income ratios are likely to manage their monthly debt payments effectively. As a result, banks and financial credit providers want to see low DTI ratios before issuing loans to a potential borrower.
As a general guideline, 43% is the highest DTI ratio a borrower can have and still get qualified for a mortgage. Ideally, lenders prefer a debt-to-income ratio lower than 36%, with no more than 28% of that debt going towards servicing a mortgage or rent payment. The maximum DTI ratio varies from lender to lender. However, the lower the debt-to-income ratio, the better the chances that the borrower will be approved, or at least considered, for the credit application.
What Are the Limitations of the Debt-to-Income Ratio?
How Does the Debt-to-Income Ratio Differ from the Debt-to-Limit Ratio?
Sometimes the debt-to-income ratio is lumped in together with the debt-to-limit ratio. However, the two metrics have distinct differences. The debt-to-limit ratio, which is also called the credit utilization ratio, is the percentage of a borrower’s total available credit that is currently being utilized. In other words, lenders want to determine if you're maxing out your credit cards. The DTI ratio calculates your monthly debt payments as compared to your income, whereby credit utilization measures your debt balances as compared to the amount of existing credit you've been approved for by credit card companies.
Consumer Financial Protection Bureau. "Debt-to-Income Calculator," Page 2. Accessed Jan. 31, 2022.
Consumer Financial Protection Bureau. "Debt-to-Income Calculator," Pages 2–3. Accessed Jan. 31, 2022.
Wells Fargo. "What Is a Good Debt-To-Income Ratio?" Accessed Jan. 31, 2022.
Parsing the 28/36 Rule
The 28/36 rule is used to calculate debt limits an individual or household should meet to be well-positioned for credit applications.
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Spanning Trees | Brilliant Math & Science Wiki
Alex Chumbley, Karleigh Moore, Timmy Jose, and
Spanning trees are special subgraphs of a graph that have several important properties. First, if T is a spanning tree of graph G, then T must span G, meaning T must contain every vertex in G. Second, T must be a subgraph of G. In other words, every edge that is in T must also appear in G. Third, if every edge in T also exists in G, then G is identical to T.
Graph with a spanning tree highlighted in blue [1]
Spanning Trees and Graph Types
Finding Spanning Trees
There are a few general properties of spanning trees.
A connected graph can have more than one spanning tree. They can have as many as
|v|^{|v|-2},
|v|
All possible spanning trees for a graph G have the same number of edges and vertices.
Spanning trees do not have any cycles.
Spanning trees are all minimally connected. That is, if any one edge is removed, the spanning tree will no longer be connected.
Adding any edge to the spanning tree will create a cycle. So, a spanning tree is maximally acyclic.
Spanning trees have
|n| - 1
|n|
is the number of vertices.
Different types of graphs have different numbers of spanning trees. Here are a few examples.
1) Complete Graphs
A complete graph is a graph where every vertex is connected to every other vertex. The number of spanning trees for a graph G with
|v|
vertices is defined by the following equation:
T(G_\text{complete}) = |v|^{|v|-2}
Complete Graph[2]
2) Connected Graphs
For connected graphs, spanning trees can be defined either as the minimal set of edges that connect all vertices or as the maximal set of edges that contains no cycle.
A connected graph is simply a graph that necessarily has a number of edges that is less than or equal to the number of edges in a complete graph with the same number of vertices. Therefore, the number of spanning trees for a connected graph is
T(G_\text{connected}) \leq |v|^{|v|-2}
Connected Graph[3]
If a graph G is itself a tree, the only spanning tree of G is itself. So a tree with
|v|
vertices is defined as
T(G_\text{tree}) = |v|^{|v|-2}
Tree Graph[4]
4) Complete Bipartite Graph
A bipartite graph is a graph where every node can either be associated with one of two sets,
m
n
. Vertices within these sets only connect to vertices in the other. There are no intra-set edges. A complete bipartite graph then is a bipartite graph where every vertex in set
m
is connected to every vertex in set
n
The number of spanning trees for a bipartite graph is defined by
T(G_\text{complete-bipartite}) = m^{n-1} \cdot n^{m-1}
Complete Bipartite Graph[5]
5) General Graph
To calculate the number of spanning trees for a general graph, a popular theorem is Kirchhoff's theorem.
To perform this theorem, a two-dimensional matrix must be constructed that can be indexed via both row and column by the graphs' vertices. The cell in the
i^\text{th}
j^\text{th}
column has a value that is determined by three things. If
i = j
, then the value in the cell will be equal to the degree of
i
i
j
are adjacent, then the value will be
-1
. Otherwise, the value will be
0
From here, an arbitrary vertex is chosen and its corresponding row and column is removed from the matrix. The determinant of this new matrix is a spanning tree
T(G)
Spanning trees can be found in linear
O(V + E)
time by simply performing breadth-first search or depth-first search. These graph search algorithms are only dependent on the number of vertices in the graph, so they are quite fast.
Breadth-first search will use a queue to hold vertices to explore later, and depth-first search will use a stack. In either case, a spanning tree can be constructed by connecting each vertex
v
with the vertex that was used to discover it.
Unfortunately, these search algorithms are not well suited for parallel or distributed computing, an area in which spanning trees are popular. There are, however, algorithms that are designed to find spanning trees in a parallel setting.
For complete graphs, there is an exact number of edges that must be removed to create a spanning tree. For a complete graph G, a spanning tree can be calculated by removing
|e| - |v| + 1
edges. In this equation,
|e|
is the number of edges, and
|v|
Minimum spanning trees are a variant of the spanning tree.
A minimum spanning tree for an unweighted graph G is a spanning tree that minimizes the number of edges or edge weights.
A minimum spanning tree for a weighted graph G is a spanning tree that minimizes the weights of the edges in the tree.
These two images show the difference between a spanning tree and minimum spanning tree. The edges that are grayed out are left out of their respective trees, but they're left in the images to show their weights.
Weighted minimum spanning tree
Minimum spanning trees are very helpful in many applications and algorithms. They are often used in water networks, electrical grids, and computer networks. They are also used in graph problems like the traveling salesperson problem, and they are used in important algorithms such as the min-cut max-flow algorithm.
There are many ways to find the minimum spanning trees, but Kruskal's algorithm is probably the fastest and easiest to do by hand.
1. Find the minimum spanning tree for the graph below. What is its total weight?
The minimum spanning tree is shown below. Its total weight is 31.
Eppstein, D. Spanning Trees. Retrieved April 10, 2016, from https://en.wikipedia.org/wiki/Spanning_tree
Benbennick, D. Wikipedia Complete Graph. Retrieved May 21, 2016, from https://en.wikipedia.org/wiki/Complete_graph
A, L. Wikipedia Connected Graph. Retrieved May 21, 2016, from https://en.wikipedia.org/wiki/Connectivity_(graph_theory)
A, L. Wikipedia Tree. Retrieved May 21, 2016, from https://en.wikipedia.org/wiki/Tree_(graph_theory)
A, K. Wikipedia Complete Bipartite Graph. Retrieved May 21, 2016, from https://en.wikipedia.org/wiki/Complete_bipartite_graph
Cite as: Spanning Trees. Brilliant.org. Retrieved from https://brilliant.org/wiki/spanning-trees/
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Iron oxide - WikiMili, The Best Wikipedia Reader
Iron oxides are chemical compounds composed of iron and oxygen. There are sixteen known iron oxides and oxyhydroxides, the best known of which is rust, a form of iron(III) oxide. [1]
Green and reddish brown stains on a limestone core sample, respectively corresponding to oxides/hydroxides of Fe and Fe .
FeO2: [2] iron peroxide
Fe4O5 [3]
Fe25O32 [5]
Fe2O3 14.9 [7]
Fe3O4 >9.2 [7]
FeO 12.1 [7]
ferrihydrite (
{\displaystyle {\ce {Fe5HO8.4H2O}}}
approx.), or
{\displaystyle {\ce {5Fe2O3.9H2O}}}
, better recast as
{\displaystyle {\ce {FeOOH.}}0.4{\ce {H2O}}}
high-pressure pyrite-structured FeOOH. [8] Once dehydration is triggered, this phase may form
{\displaystyle {\ce {FeO2Hx (0<x<1)}}}
schwertmannite (ideally
{\displaystyle {\ce {Fe8O8(OH)6(SO).{\mathit {n}}H2O}}}
{\displaystyle {\ce {Fe^{3+}16O16(OH,SO4)}}_{\text{12-13}}\cdot {\text{10-12}}{\ce {H2O}}}
green rust (
{\displaystyle {\ce {Fe_{\mathit {x}}^{III}Fe_{\mathit {y}}^{II}(OH)}}_{3x+2y-z}{\ce {(A^{-})}}_{z}}
where A− is Cl− or 0.5SO42−)
Several species of bacteria, including Shewanella oneidensis, Geobacter sulfurreducens and Geobacter metallireducens, metabolically utilize solid iron oxides as a terminal electron acceptor, reducing Fe(III) oxides to Fe(II) containing oxides. [11]
Under conditions favoring iron reduction, the process of iron oxide reduction can replace at least 80% of methane production occurring by methanogenesis. [12] This phenomenon occurs in a nitrogen-containing (N2) environment with low sulfate concentrations. Methanogenesis, an Archaean driven process, is typically the predominant form of carbon mineralization in sediments at the bottom of the ocean. Methanogenesis completes the decomposition of organic matter to methane (CH4). [12] The specific electron donor for iron oxide reduction in this situation is still under debate, but the two potential candidates include either titanium (III) or compounds present in yeast. The predicted reactions with titanium (III) serving as the electron donor and phenazine-1-carboxylate (PCA) serving as an electron shuttle is as follows:
On the other hand when airborne, iron oxides have been shown to harm the lung tissues of living organisms by the formation of hydroxyl radicals, leading to the creation of alkyl radicals. The following reactions occur when Fe2O3 and FeO, hereafter represented as Fe3+ and Fe2+ respectively, iron oxide particulates accumulate in the lungs. [13]
O2 + e− → O2• − [13]
The formation of the superoxide anion (O2• −) is catalyzed by a transmembrane enzyme called NADPH oxidase. The enzyme facilitates the transport of an electron across the plasma membrane from cytosolic NADPH to extracellular oxygen (O2) to produce O2• −. NADPH and FAD are bound to cytoplasmic binding sites on the enzyme. Two electrons from NADPH are transported to FAD which reduces it to FADH2. Then, one electron moves to one of two heme groups in the enzyme within the plane of the membrane. The second electron pushes the first electron to the second heme group so that it can associate with the first heme group. For the transfer to occur, the second heme must be bound to extracellular oxygen which is the acceptor of the electron. This enzyme can also be located within the membranes of intracellular organelles allowing the formation of O2• − to occur within organelles. [14]
2O2• − + 2 H+ → H2O2 + O2 [13] [15]
2) can occur spontaneously when the environment has a lower pH especially at pH 7.4. [15] The enzyme superoxide dismutase can also catalyze this reaction. Once H
2 has been synthesized, it can diffuse through membranes to travel within and outside the cell due to its nonpolar nature. [14]
Fe2+ is oxidized to Fe3+ when it donates an electron to H2O2, thus, reducing H2O2 and forming a hydroxyl radical (HO•) in the process. H2O2 can then reduce Fe3+ to Fe2+ by donating an electron to it to create O2• −. O2• − can then be used to make more H2O2 by the process previously shown perpetuating the cycle, or it can react with H2O2 to form more hydroxyl radicals. Hydroxyl radicals have been shown to increase cellular oxidative stress and attack cell membranes as well as the cell genomes. [13]
The HO• radical produced from the above reactions with iron can abstract a hydrogen atom (H) from molecules containing an R-H bond where the R is a group attached to the rest of the molecule, in this case H, at a carbon (C). [13]
Cytochromes are redox-active proteins containing a heme, with a central Fe atom at its core, as a cofactor. They are involved in electron transport chain and redox catalysis. They are classified according to the type of heme and its mode of binding. Four varieties are recognized by the International Union of Biochemistry and Molecular Biology (IUBMB), cytochromes a, cytochromes b, cytochromes c and cytochrome d. Cytochrome function is linked to the reversible redox change from ferrous to the ferric oxidation state of the iron found in the heme core. In addition to the classification by the IUBMB into four cytochrome classes, several additional classifications such as cytochrome o and cytochrome P450 can be found in biochemical literature.
An electron transport chain (ETC) is a series of protein complexes and other molecules that transfer electrons from electron donors to electron acceptors via redox reactions (both reduction and oxidation occurring simultaneously) and couples this electron transfer with the transfer of protons (H+ ions) across a membrane. Many of the enzymes in the electron transport chain are membrane-bound.
A superoxide is a compound that contains the superoxide ion, which has the chemical formula O−
2. The systematic name of the anion is dioxide(1−). The reactive oxygen ion superoxide is particularly important as the product of the one-electron reduction of dioxygen O2, which occurs widely in nature. Molecular oxygen (dioxygen) is a diradical containing two unpaired electrons, and superoxide results from the addition of an electron which fills one of the two degenerate molecular orbitals, leaving a charged ionic species with a single unpaired electron and a net negative charge of −1. Both dioxygen and the superoxide anion are free radicals that exhibit paramagnetism. Superoxide was historically also known as "hyperoxide".
The coenzyme Q : cytochrome c – oxidoreductase, sometimes called the cytochrome bc1 complex, and at other times complex III, is the third complex in the electron transport chain, playing a critical role in biochemical generation of ATP. Complex III is a multisubunit transmembrane protein encoded by both the mitochondrial and the nuclear genomes. Complex III is present in the mitochondria of all animals and all aerobic eukaryotes and the inner membranes of most eubacteria. Mutations in Complex III cause exercise intolerance as well as multisystem disorders. The bc1 complex contains 11 subunits, 3 respiratory subunits, 2 core proteins and 6 low-molecular weight proteins.
Heme, or haem, is a precursor to hemoglobin, which is necessary to bind oxygen in the bloodstream. Heme is biosynthesized in both the bone marrow and the liver.
Hemerythrin (also spelled haemerythrin; Ancient Greek: αἷμα, romanized: haîma, lit. 'blood', Ancient Greek: ἐρυθρός, romanized: erythrós, lit. 'red') is an oligomeric protein responsible for oxygen (O2) transport in the marine invertebrate phyla of sipunculids, priapulids, brachiopods, and in a single annelid worm genus, Magelona. Myohemerythrin is a monomeric O2-binding protein found in the muscles of marine invertebrates. Hemerythrin and myohemerythrin are essentially colorless when deoxygenated, but turn a violet-pink in the oxygenated state.
Nitric oxide synthases (NOSs) are a family of enzymes catalyzing the production of nitric oxide (NO) from L-arginine. NO is an important cellular signaling molecule. It helps modulate vascular tone, insulin secretion, airway tone, and peristalsis, and is involved in angiogenesis and neural development. It may function as a retrograde neurotransmitter. Nitric oxide is mediated in mammals by the calcium-calmodulin controlled isoenzymes eNOS and nNOS. The inducible isoform, iNOS, involved in immune response, binds calmodulin at physiologically relevant concentrations, and produces NO as an immune defense mechanism, as NO is a free radical with an unpaired electron. It is the proximate cause of septic shock and may function in autoimmune disease.
Ferredoxins are iron–sulfur proteins that mediate electron transfer in a range of metabolic reactions. The term "ferredoxin" was coined by D.C. Wharton of the DuPont Co. and applied to the "iron protein" first purified in 1962 by Mortenson, Valentine, and Carnahan from the anaerobic bacterium Clostridium pasteurianum.
In chemistry, photocatalysis is the acceleration of a photoreaction in the presence of a catalyst. In catalyzed photolysis, light is absorbed by an adsorbed substrate. In photogenerated catalysis, the photocatalytic activity (PCA) depends on the ability of the catalyst to create electron–hole pairs, which generate free radicals (e.g. hydroxyl radicals: •OH) able to undergo secondary reactions. Its practical application was made possible by the discovery of water electrolysis by means of titanium dioxide (TiO2).
Heme oxygenase, or haem oxygenase, is an enzyme that catalyzes the degradation of heme to produce biliverdin, ferrous ion, and carbon monoxide.
In enzymology, a manganese peroxidase (EC 1.11.1.13) is an enzyme that catalyzes the chemical reaction
Hydroxylamine oxidoreductase (HAO) is an enzyme found in the prokaryote Nitrosomonas europaea. It plays a critically important role in the biogeochemical nitrogen cycle as part of the metabolism of ammonia-oxidizing bacteria.
Nitric oxide reductase, an enzyme, catalyzes the reduction of nitric oxide (NO) to nitrous oxide (N2O). The enzyme participates in nitrogen metabolism and in the microbial defense against nitric oxide toxicity. The catalyzed reaction may be dependent on different participating small molecules: Cytochrome c (EC: 1.7.2.5, Nitric oxide reductase (cytochrome c)), NADPH (EC:1.7.1.14), or Menaquinone (EC:1.7.5.2).
Dioxygenases are oxidoreductase enzymes. Aerobic life, from simple single-celled bacteria species to complex eukaryotic organisms, has evolved to depend on the oxidizing power of dioxygen in various metabolic pathways. From energetic adenosine triphosphate (ATP) generation to xenobiotic degradation, the use of dioxygen as a biological oxidant is widespread and varied in the exact mechanism of its use. Enzymes employ many different schemes to use dioxygen, and this largely depends on the substrate and reaction at hand.
Haem peroxidases (or heme peroxidases) are haem-containing enzymes that use hydrogen peroxide as the electron acceptor to catalyse a number of oxidative reactions. Most haem peroxidases follow the reaction scheme:
Plastid terminal oxidase or plastoquinol terminal oxidase (PTOX) is an enzyme that resides on the thylakoid membranes of plant and algae chloroplasts and on the membranes of cyanobacteria. The enzyme was hypothesized to exist as a photosynthetic oxidase in 1982 and was verified by sequence similarity to the mitochondrial alternative oxidase (AOX). The two oxidases evolved from a common ancestral protein in prokaryotes, and they are so functionally and structurally similar that a thylakoid-localized AOX can restore the function of a PTOX knockout.
The Hill reaction is the light-driven transfer of electrons from water to Hill reagents in a direction against the chemical potential gradient as part of photosynthesis. Robin Hill discovered the reaction in 1937. He demonstrated that the process by which plants produce oxygen is separate from the process that converts carbon dioxide to sugars.
Julia A. Kovacs is an American chemist specializing in bioinorganic chemistry. She is Professor of Chemistry at the University of Washington. Her research involves synthesizing small-molecule mimics of the active sites of metalloproteins, in order to investigate how cysteinates influence the function of non-heme iron enzymes, and the mechanism of the oxygen-evolving complex (OEC).
Transition metal complexes of nitrite describes families of coordination complexes containing one or more nitrite (NO2−) ligands. Although the synthetic derivatives are only of scholarly interest, metal-nitrite complexes occur in several enzymes that participate in the nitrogen cycle.
↑ Hu, Qingyang; Kim, Duck Young; Yang, Wenge; Yang, Liuxiang; Meng, Yue; Zhang, Li; Mao, Ho-Kwang (June 2016). "FeO2 and (FeO)OH under deep lower-mantle conditions and Earth's oxygen–hydrogen cycles". Nature. 534 (7606): 241–244. Bibcode:2016Natur.534..241H. doi:10.1038/nature18018. ISSN 1476-4687. PMID 27279220.
↑ Lavina, B.; Dera, P.; Kim, E.; Meng, Y.; Downs, R. T.; Weck, P. F.; Sutton, S. R.; Zhao, Y. (Oct 2011). "Discovery of the recoverable high-pressure iron oxide Fe4O5". Proceedings of the National Academy of Sciences. 108 (42): 17281–17285. Bibcode:2011PNAS..10817281L. doi: 10.1073/pnas.1107573108 . PMC 3198347 . PMID 21969537.
↑ Lavina, Barbara; Meng, Yue (2015). "Synthesis of Fe5O6". Science Advances. 1 (5): e1400260. doi:10.1126/sciadv.1400260. PMC 4640612 . PMID 26601196.
1 2 Bykova, E.; Dubrovinsky, L.; Dubrovinskaia, N.; Bykov, M.; McCammon, C.; Ovsyannikov, S. V.; Liermann, H. -P.; Kupenko, I.; Chumakov, A. I.; Rüffer, R.; Hanfland, M.; Prakapenka, V. (2016). "Structural complexity of simple Fe2O3 at high pressures and temperatures". Nature Communications. 7: 10661. doi:10.1038/ncomms10661. PMC 4753252 . PMID 26864300.
↑ Merlini, Marco; Hanfland, Michael; Salamat, Ashkan; Petitgirard, Sylvain; Müller, Harald (2015). "The crystal structures of Mg2Fe2C4O13, with tetrahedrally coordinated carbon, and Fe13O19, synthesized at deep mantle conditions". American Mineralogist. 100 (8–9): 2001–2004. doi:10.2138/am-2015-5369. S2CID 54496448.
1 2 3 Fakouri Hasanabadi, M.; Kokabi, A.H.; Nemati, A.; Zinatlou Ajabshir, S. (February 2017). "Interactions near the triple-phase boundaries metal/glass/air in planar solid oxide fuel cells". International Journal of Hydrogen Energy. 42 (8): 5306–5314. doi:10.1016/j.ijhydene.2017.01.065. ISSN 0360-3199.
↑ Nishi, Masayuki; Kuwayama, Yasuhiro; Tsuchiya, Jun; Tsuchiya, Taku (2017). "The pyrite-type high-pressure form of FeOOH". Nature. 547 (7662): 205–208. doi:10.1038/nature22823. ISSN 1476-4687. PMID 28678774. S2CID 205257075.
↑ Hu, Qingyang; Kim, Duckyoung; Liu, Jin; Meng, Yue; Liuxiang, Yang; Zhang, Dongzhou; Mao, Wendy L.; Mao, Ho-kwang (2017). "Dehydrogenation of goethite in Earth's deep lower mantle". Proceedings of the National Academy of Sciences. 114 (7): 1498–1501. doi: 10.1073/pnas.1620644114 . PMC 5320987 . PMID 28143928.
↑ Bretschger, O.; Obraztsova, A.; Sturm, C. A.; Chang, I. S.; Gorby, Y. A.; Reed, S. B.; Culley, D. E.; Reardon, C. L.; Barua, S.; Romine, M. F.; Zhou, J.; Beliaev, A. S.; Bouhenni, R.; Saffarini, D.; Mansfeld, F.; Kim, B.-H.; Fredrickson, J. K.; Nealson, K. H. (20 July 2007). "Current Production and Metal Oxide Reduction by Shewanella oneidensis MR-1 Wild Type and Mutants". Applied and Environmental Microbiology. 73 (21): 7003–7012. doi:10.1128/AEM.01087-07. PMC 2223255 . PMID 17644630.
1 2 3 Sivan, O.; Shusta, S. S.; Valentine, D. L. (2016-03-01). "Methanogens rapidly transition from methane production to iron reduction". Geobiology. 14 (2): 190–203. doi:10.1111/gbi.12172. ISSN 1472-4669. PMID 26762691.
1 2 3 4 5 6 7 Hartwig, A.; MAK Commission 2016 (July 25, 2016). Iron oxides (inhalable fraction) [MAK Value Documentation, 2011]. The MAK Collection for Occupational Health and Safety. Vol. 1. pp. 1804–1869. doi:10.1002/3527600418.mb0209fste5116. ISBN 9783527600410.
1 2 Bedard, Karen; Krause, Karl-Heinz (2007-01-01). "The NOX Family of ROS-Generating NADPH Oxidases: Physiology and Pathophysiology". Physiological Reviews. 87 (1): 245–313. doi:10.1152/physrev.00044.2005. ISSN 0031-9333. PMID 17237347.
1 2 Chapple, Iain L. C.; Matthews, John B. (2007-02-01). "The role of reactive oxygen and antioxidant species in periodontal tissue destruction". Periodontology 2000. 43 (1): 160–232. doi:10.1111/j.1600-0757.2006.00178.x. ISSN 1600-0757. PMID 17214840.
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Calculating Profit and Total Revenue - Course Hero
Microeconomics/Profit/Calculating Profit and Total Revenue
Learn all about calculating profit and total revenue in just a few minutes! Professor Jadrian Wooten of Penn State University explains how total revenue is calculated and is used to calculate profit for a firm.
Profit is an important measure in economics because a firm's goal to maximize profit is central to many economic models of production and supply. Simply put, profit is the amount left over from total revenue once total cost (however defined) is subtracted. Total revenue is the amount received by producers when selling output. For example, consider the overall profit for a manufacturer that produces furniture. The manufacturer may have high revenues, but they must pay for workers' salaries, materials, a space to create the furniture, people to sell the furniture, and ways to spread the word about their product. All of these are taken away from the total revenue (the money from furniture sold) to calculate the profit for the manufacturer.
\text{Profit}=\text{TR}-\text{TC}
, where TR is total revenue and TC is total cost. In the example of the furniture manufacturer, the profit would equal the amount spent on the wood and employees' salaries taken away from the money earned by selling the furniture. Total revenue is equal to the money that comes in from selling goods and services. In the simplest case, if a producer sells all of its output at the same price (P), then total revenue is equal to P times Q, where Q is the quantity of output produced and sold.
\text{Total Revenue}=\text{P}\times\text{Q}
For example, if a company sells 50 units of output (Q) at $6 each (P), then total revenue is equal to $6 × 50, or $300. If the producer's output is sold at various prices, total revenue can be calculated by multiplying each price by the quantity sold at that price point and then adding these numbers together to get the total revenue. If the same company in the above example sells 30 units of output at $6 each, but discounts the remaining 20 units of output and sells them at $5 each, then total revenue is equal to:
\text{TR}=(30\times\$6)+(20\times\$5)=\$280
Once total revenue is considered, costs must be calculated to see the bigger picture. A company with a revenue stream of $300,000 per month suggests the business is successful. But if the business must spend $350,000 per month on expenses, it is in fact losing $50,000 every month. Profit is essential to measure the success of a business. It provides a fuller picture than either total revenue or total cost. Profit can also be calculated on a per-unit basis by:
\begin{aligned}\text{Profit}&=\text{TR} - \text{TC}\\&=(\text{P} \times \text{Q})-(\text{ATC}- \text{Q})\\&=(\text{P}-\text{ATC}) \times \text{Q}\end{aligned}
In this calculation, P is price, ATC is average total cost, and Q is quantity. For example, a company makes pillows and sells each pillow for $19. The average total cost to make a pillow is $5. The company sells 2,000 pillows. The profit is calculated as:
\begin{aligned}\text{Profit}&=(\text{Price of Pillows} - \text{Average Total Cost}) \times \text{Number of Pillows Sold}\\&=(\$19-\$5) \times 2\text{,}000\\&=\$14 \times 2\text{,}000\\&=\$28\text{,}000\end{aligned}
Using this calculation, the profit is $28,000.
\text{Total Revenue} = \text{Price Multiplied by Quantity}
1
\$330
50
\$330 × 50 = \$16\text{,}500
2
\$350
25
\$350 × 25 =\$8\text{,}750
3
\$310
60
\$310 × 60 = \$18\text{,}600
\$43,850
When a producer sells a single product (in this case a single type of phone) at multiple price points, the revenue is calculated for each price point. The totals are added to find the total revenue. Note that this does not represent the producer's revenue across a range of products, but merely a single product.
<Accounting versus Economic Profit>Economic Profit and Decision Making
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For each shape drawn below, choose one of the names on the list above them that best describes that shape. Be as specific as you can. If you do not remember what one of the shape names means, you may look in the glossary of the eBook for more information.
This triangle has acute angles, so you might think it is an acute triangle. However, it is possible to be more specific, since there is a right angle.
Right or scalene triangle; there is a right angle and all of the sides are different lengths.
Is there an angle greater than
90°
? If so, it could be called an obtuse triangle.
Certainly this is a quadrilateral, but can you be more specific?
Trapezoid; there are two parallel sides with two non-parallel sides.
There are two pairs of parallel sides, and all sides are equal length, meaning you can be more specific than ''parallelogram.''
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Combining Expert Opinions in Prior Elicitation
Isabelle Albert, Sophie Donnet, Chantal Guihenneuc-Jouyaux, Samantha Low-Choy, Kerrie Mengersen, Judith Rousseau
KEYWORDS: Bayesian statistics, hierarchical model, random effects, risk assessment
We consider the problem of combining opinions from different experts in an explicitly model-based way to construct a valid subjective prior in a Bayesian statistical approach. We propose a generic approach by considering a hierarchical model accounting for various sources of variation as well as accounting for potential dependence between experts. We apply this approach to two problems. The first problem deals with a food risk assessment problem involving modelling dose-response for Listeria monocytogenes contamination of mice. Two hierarchical levels of variation are considered (between and within experts) with a complex mathematical situation due to the use of an indirect probit regression. The second concerns the time taken by PhD students to submit their thesis in a particular school. It illustrates a complex situation where three hierarchical levels of variation are modelled but with a simpler underlying probability distribution (log-Normal).
Comment on Article by Albert et al.
Bayesian Matching of Unlabeled Point Sets Using Procrustes and Configuration Models
Kim Kenobi, Ian L. Dryden
KEYWORDS: Gibbs, Markov chain Monte Carlo, Metropolis-Hastings, molecule, protein, Procrustes, size, shape
The problem of matching unlabeled point sets using Bayesian inference is considered. Two recently proposed models for the likelihood are compared, based on the Procrustes size-and-shape and the full configuration. Bayesian inference is carried out for matching point sets using Markov chain Monte Carlo simulation. An improvement to the existing Procrustes algorithm is proposed which improves convergence rates, using occasional large jumps in the burn-in period. The Procrustes and configuration methods are compared in a simulation study and using real data, where it is of interest to estimate the strengths of matches between protein binding sites. The performance of both methods is generally quite similar, and a connection between the two models is made using a Laplace approximation.
Simulation-based Regularized Logistic Regression
Robert B. Gramacy, Nicholas G. Polson
KEYWORDS: logistic regression, regularization, z–distributions, Data augmentation, classification, Gibbs sampling, Lasso, variance-mean mixtures, Bayesian shrinkage
In this paper, we develop a simulation-based framework for regularized logistic regression, exploiting two novel results for scale mixtures of normals. By carefully choosing a hierarchical model for the likelihood by one type of mixture, and implementing regularization with another, we obtain new MCMC schemes with varying efficiency depending on the data type (binary v. binomial, say) and the desired estimator (maximum likelihood, maximum a posteriori, posterior mean). Advantages of our omnibus approach include flexibility, computational efficiency, applicability in
p\gg n
settings, uncertainty estimates, variable selection, and assessing the optimal degree of regularization. We compare our methodology to modern alternatives on both synthetic and real data. An R package called reglogit is available on CRAN.
Prior Effective Sample Size in Conditionally Independent Hierarchical Models
Satoshi Morita, Peter F. Thall, Peter Müller
KEYWORDS: Bayesian hierarchical model, Conditionally independent hierarchical model, Computationally intensive methods, effective sample size, Epsilon-information prior
Prior effective sample size (ESS) of a Bayesian parametric model was defined by Morita, et al. (2008, Biometrics, 64, 595-602). Starting with an
\varepsilon
-information prior defined to have the same means and correlations as the prior but to be vague in a suitable sense, the ESS is the required sample size to obtain a hypothetical posterior very close to the prior. In this paper, we present two alternative definitions for the prior ESS that are suitable for a conditionally independent hierarchical model. The two definitions focus on either the first level prior or second level prior. The proposed methods are applied to important examples to verify that each of the two types of prior ESS matches the intuitively obvious answer where it exists. We illustrate the method with applications to several motivating examples, including a single-arm clinical trial to evaluate treatment response probabilities across different disease subtypes, a dose-finding trial based on toxicity in this setting, and a multicenter randomized trial of treatments for affective disorders.
Using Individual-Level Models for Infectious Disease Spread to Model Spatio-Temporal Combustion Dynamics
Irene Vrbik, Rob Deardon, Zeny Feng, Abbie Gardner, John Braun
KEYWORDS: individual-level models, Markov chain Monte Carlo, fire spread modelling, Bayesian inference, spatio-temporal dynamics
Individual-level models (ILMs), as defined by Deardon et al. (2010), are a class of models originally designed to model the spread of infectious disease. However, they can also be considered as a tool for modelling the spatio-temporal dynamics of fire. We consider the much simplified problem of modelling the combustion dynamics on a piece of wax paper under relatively controlled conditions. The models are fitted in a Bayesian framework using Markov chain Monte Carlo (MCMC) methods. The focus here is on choosing a model that best fits the combustion pattern.
Commensurate Priors for Incorporating Historical Information in Clinical Trials Using General and Generalized Linear Models
Brian P. Hobbs, Daniel J. Sargent, Bradley P. Carlin
KEYWORDS: Clinical trials, historical controls, Meta-analysis, Bayesian analysis, Survival analysis, correlated data
Assessing between-study variability in the context of conventional random-effects meta-analysis is notoriously difficult when incorporating data from only a small number of historical studies. In order to borrow strength, historical and current data are often assumed to be fully homogeneous, but this can have drastic consequences for power and Type I error if the historical information is biased. In this paper, we propose empirical and fully Bayesian modifications of the commensurate prior model (Hobbs et al. 2011) extending Pocock (1976), and evaluate their frequentist and Bayesian properties for incorporating patient-level historical data using general and generalized linear mixed regression models. Our proposed commensurate prior models lead to preposterior admissible estimators that facilitate alternative bias-variance trade-offs than those offered by pre-existing methodologies for incorporating historical data from a small number of historical studies. We also provide a sample analysis of a colon cancer trial comparing time-to-disease progression using a Weibull regression model.
Perfect Simulation for Mixtures with Known and Unknown Number of Components
Sabyasachi Mukhopadhyay, Sourabh Bhattacharya
KEYWORDS: Bounding chains, Dirichlet process, Gibbs sampling, mixtures, optimization, perfect sampling
We propose and develop a novel and effective perfect sampling methodology for simulating from posteriors corresponding to mixtures with either known (fixed) or unknown number of components. For the latter we consider the Dirichlet process-based mixture model developed by these authors, and show that our ideas are applicable to conjugate, and importantly, to non-conjugate cases. As to be expected, and as we show, perfect sampling for mixtures with known number of components can be achieved with much less effort with a simplified version of our general methodology, whether or not conjugate or non-conjugate priors are used. While no special assumption is necessary in the conjugate set-up for our theory to work, we require the assumption of compact parameter space in the non-conjugate set-up. However, we argue, with appropriate analytical, simulation, and real data studies as support, that such compactness assumption is not unrealistic and is not an impediment in practice. Not only do we validate our ideas theoretically and with simulation studies, but we also consider application of our proposal to three real data sets used by several authors in the past in connection with mixture models. The results we achieved in each of our experiments with either simulation study or real data application, are quite encouraging. However, the computation can be extremely burdensome in the case of large number of mixture components and in massive data sets. We discuss the role of parallel processing in mitigating the extreme computational burden.
Antti Solonen, Pirkka Ollinaho, Marko Laine, Heikki Haario, Johanna Tamminen, Heikki Järvinen
KEYWORDS: adaptive MCMC, Climate Models, Parallel MCMC, Early Rejection
The emergence of Markov chain Monte Carlo (MCMC) methods has opened a way for Bayesian analysis of complex models. Running MCMC samplers typically requires thousands of model evaluations, which can exceed available computer resources when this evaluation is computationally intensive. We will discuss two generally applicable techniques to improve the efficiency of MCMC. First, we consider a parallel version of the adaptive MCMC algorithm of Haario et al. (2001), implementing the idea of inter-chain adaptation introduced by Craiu et al. (2009). Second, we present an early rejection (ER) approach, where model simulation is stopped as soon as one can conclude that the proposed parameter value will be rejected by the MCMC algorithm.
This work is motivated by practical needs in estimating parameters of climate and Earth system models. These computationally intensive models involve non-linear expressions of the geophysical and biogeochemical processes of the Earth system. Modeling of these processes, especially those operating in scales smaller than the model grid, involves a number of specified parameters, or ‘tunables’. MCMC methods are applicable for estimation of these parameters, but they are computationally very demanding. Efficient MCMC variants are thus needed to obtain reliable results in reasonable time. Here we evaluate the computational gains attainable through parallel adaptive MCMC and Early Rejection using both simple examples and a realistic climate model.
KEYWORDS: Bayesian computation, marginal likelihood, algorithm, Bayes factors, Model selection
Determining the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally challenging. The often-used harmonic mean approximation (HMA) makes no prior assumptions about the character of the distribution but tends to be inconsistent. The Laplace approximation is stable but makes strong, and often inappropriate, assumptions about the shape of the posterior distribution. Here, I argue that the marginal likelihood can be reliably computed from a posterior sample using Lebesgue integration theory in one of two ways: 1) when the HMA integral exists, compute the measure function numerically and analyze the resulting quadrature to control error; 2) compute the measure function numerically for the marginal likelihood integral itself using a space-partitioning tree, followed by quadrature. The first algorithm automatically eliminates the part of the sample that contributes large truncation error in the HMA. Moreover, it provides a simple graphical test for the existence of the HMA integral. The second algorithm uses the posterior sample to assign probability to a partition of the sample space and performs the marginal likelihood integral directly. It uses the posterior sample to discover and tessellate the subset of the sample space that was explored and uses quantiles to compute a representative field value. When integrating directly, this space may be trimmed to remove regions with low probability density and thereby improve accuracy. This second algorithm is consistent for all proper distributions. Error analysis provides some diagnostics on the numerical condition of the results in both cases.
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Circumcircle of Triangle | Brilliant Math & Science Wiki
Abhineet Goel, Aditya Virani, Sagnik Saha, and
What is the circumcenter of triangle
ABC
A=(1, 4), B=(-2, 3), C=(5, 2)?
A circumcenter, by definition, is the center of the circle in which a triangle is inscribed, For this problem, let
O=(a, b)
be the circumcenter of
\triangle ABC.
Then since the distances to
O
from the vertices are all equal, we have
\lvert \overline{AO} \rvert=\lvert \overline{BO} \rvert=\lvert \overline{CO} \rvert.
From the first equality, we have
\begin{aligned} \lvert \overline{AO} \rvert^2&=\lvert \overline{BO} \rvert^2\\ (a-1)^2+(b-4)^2&=(a+2)^2+(b-3)^2\\ -2a+1-8b+16&=4a+4-6b+9\\ 3a+b&=2. \qquad (1) \end{aligned}
Similarly, from the second equality, we have
\begin{aligned} \lvert \overline{BO} \rvert^2&=\lvert \overline{CO} \rvert^2\\ (a+2)^2+(b-3)^2&=(a-5)^2+(b-2)^2\\ 4a+4-6b+9&=-10a+25-4b+4\\ 7a-b&=8. \qquad (2) \end{aligned}
(1)+(2)
a=1,
which in turn gives
b=-1.
Therefore, the circumcenter of triangle
ABC
O=(1, -1).
_\square
Cite as: Circumcircle of Triangle. Brilliant.org. Retrieved from https://brilliant.org/wiki/circumscribed-triangles/
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Bayesian Anal. 7 (2), (June 2012)
Bayesian Anal. 7 (2), 235-258, (June 2012) DOI: 10.1214/12-BA708
KEYWORDS: Quantile regression, conditional quantiles, spatial statistics, MCMC
We consider quantile multiple regression through conditional quantile models, i.e. each quantile is modeled separately. We work in the context of spatially referenced data and extend the asymmetric Laplace model for quantile regression to a spatial process, the asymmetric Laplace process (ALP) for quantile regression with spatially dependent errors. By taking advantage of a convenient conditionally Gaussian representation of the asymmetric Laplace distribution, we are able to straightforwardly incorporate spatial dependence in this process. We develop the properties of this process under several specifications, each of which induces different smoothness and covariance behavior at the extreme quantiles.
We demonstrate the advantages that may be gained by incorporating spatial dependence into this conditional quantile model by applying it to a data set of log selling prices of homes in Baton Rouge, LA, given characteristics of each house. We also introduce the asymmetric Laplace predictive process (ALPP) which accommodates large data sets, and apply it to a data set of birth weights given maternal covariates for several thousand births in North Carolina in 2000. By modeling the spatial structure in the data, we are able to show, using a check loss function, improved performance on each of the data sets for each of the quantiles at which the model was fit.
Comment on Article by Lum and Gelfand
Rajarshi Guhaniyogi, Sudipto Banerjee
Bayesian Anal. 7 (2), 259-262, (June 2012) DOI: 10.1214/12-BA708A
Nan Lin, Chao Chang
Bayesian Anal. 7 (2), 263-270, (June 2012) DOI: 10.1214/12-BA708B
Bayesian Anal. 7 (2), 271-272, (June 2012) DOI: 10.1214/12-BA708C
Bayesian Anal. 7 (2), 273-276, (June 2012) DOI: 10.1214/12-BA708REJ
KEYWORDS: Related probability distributions, Bayesian nonparametrics, copulas, Weak support, Hellinger support, Kullback–Leibler support, Stick–breaking processes
Posterior Concentration Rates for Infinite Dimensional Exponential Families
Vincent Rivoirard, Judith Rousseau
KEYWORDS: Bayesian non-parametric, rates of convergence, adaptive estimation, wavelets and Fourier Bases, Sobolev and Besov balls
In this paper we derive adaptive non-parametric rates of concentration of the posterior distributions for the density model on the class of Sobolev and Besov spaces. For this purpose, we build prior models based on wavelet or Fourier expansions of the logarithm of the density. The prior models are not necessarily Gaussian.
Mixture Modeling for Marked Poisson Processes
Matthew A. Taddy, Athanasios Kottas
KEYWORDS: Bayesian nonparametrics, Beta mixtures, Dirichlet process, marked point process, Multivariate Normal mixtures, non-homogeneous poisson process, Nonparametric regression
We propose a general inference framework for marked Poisson processes observed over time or space. Our modeling approach exploits the connection of nonhomogeneous Poisson process intensity with a density function. Nonparametric Dirichlet process mixtures for this density, combined with nonparametric or semiparametric modeling for the mark distribution, yield flexible prior models for the marked Poisson process. In particular, we focus on fully nonparametric model formulations that build the mark density and intensity function from a joint nonparametric mixture, and provide guidelines for straightforward application of these techniques. A key feature of such models is that they can yield flexible inference about the conditional distribution for multivariate marks without requiring specification of a complicated dependence scheme. We address issues relating to choice of the Dirichlet process mixture kernels, and develop methods for prior specification and posterior simulation for full inference about functionals of the marked Poisson process. Moreover, we discuss a method for model checking that can be used to assess and compare goodness of fit of different model specifications under the proposed framework. The methodology is illustrated with simulated and real data sets.
Serena Arima, Gauri S. Datta, Brunero Liseo
KEYWORDS: Bayesian inference, Jeffreys’ prior, small area model
We consider small area estimation under a nested error linear regression model with measurement errors in the covariates. We propose an objective Bayesian analysis of the model to estimate the finite population means of the small areas. In particular, we derive Jeffreys’ prior for model parameters. We also show that Jeffreys’ prior, which is improper, leads, under very general conditions, to a proper posterior distribution. We have also performed a simulation study where we have compared the Bayes estimates of the finite population means under the Jeffreys’ prior with other Bayesian estimates obtained via the use of the standard flat prior and with non-Bayesian estimates, i.e., the corresponding empirical Bayes estimates and the direct estimates.
Roberto Casarin, Luciana Dalla Valle, Fabrizio Leisen
KEYWORDS: Bayesian inference, Beta Autoregressive Processes, reversible jump MCMC
We deal with Bayesian model selection for beta autoregressive processes. We discuss the choice of parameter and model priors with possible parameter restrictions and suggest a Reversible Jump Markov-Chain Monte Carlo (RJMCMC) procedure based on a Metropolis-Hastings within Gibbs algorithm.
Log-Linear Pool to Combine Prior Distributions: A Suggestion for a Calibration-Based Approach
M. J. Rufo, J. Martín, C. J. Pérez
KEYWORDS: Bayesian analysis, Kullback-Leibler divergence, Pooled distribution
An important issue involved in group decision making is the suitable aggregation of experts’ beliefs about a parameter of interest. Two widely used combination methods are linear and log-linear pools. Yet, a problem arises when the weights have to be selected. This paper provides a general decision-based procedure to obtain the weights in a log-linear pooled prior distribution. The process is based on Kullback-Leibler divergence, which is used as a calibration tool. No information about the parameter of interest is considered before dealing with the experts’ beliefs. Then, a pooled prior distribution is achieved, for which the expected calibration is the best one in the Kullback-Leibler sense. In the absence of other information available to the decision-maker prior to getting experimental data, the methodology generally leads to selection of the most diffuse pooled prior. In most cases, a problem arises from the marginal distribution related to the noninformative prior distribution since it is improper. In these cases, an alternative procedure is proposed. Finally, two applications show how the proposed techniques can be easily applied in practice.
Beta Processes, Stick-Breaking and Power Laws
Tamara Broderick, Michael I. Jordan, Jim Pitman
KEYWORDS: beta process, stick-breaking, power law
The beta-Bernoulli process provides a Bayesian nonparametric prior for models involving collections of binary-valued features. A draw from the beta process yields an infinite collection of probabilities in the unit interval, and a draw from the Bernoulli process turns these into binary-valued features. Recent work has provided stick-breaking representations for the beta process analogous to the well-known stick-breaking representation for the Dirichlet process. We derive one such stick-breaking representation directly from the characterization of the beta process as a completely random measure. This approach motivates a three-parameter generalization of the beta process, and we study the power laws that can be obtained from this generalized beta process. We present a posterior inference algorithm for the beta-Bernoulli process that exploits the stick-breaking representation, and we present experimental results for a discrete factor-analysis model.
Gilles Celeux, Mohammed El Anbari, Jean-Michel Marin, Christian P. Robert
KEYWORDS: model choice, regularization methods, noninformative priors, Zellner’s g–prior, Calibration, Lasso, Elastic net, Dantzig selector
Using a collection of simulated and real benchmarks, we compare Bayesian and frequentist regularization approaches under a low informative constraint when the number of variables is almost equal to the number of observations on simulated and real datasets. This comparison includes new global noninformative approaches for Bayesian variable selection built on Zellner’s
g
-priors that are similar to Liang et al. (2008). The interest of those calibration-free proposals is discussed. The numerical experiments we present highlight the appeal of Bayesian regularization methods, when compared with non-Bayesian alternatives. They dominate frequentist methods in the sense that they provide smaller prediction errors while selecting the most relevant variables in a parsimonious way.
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Sketch the shape made with algebra tiles at right on your paper. Then answer parts (a) and (b) below.
To find the total area, add together the area of the algebra tiles.
x^2+x^2+x+x+x
To find the perimeter, label all the sides with their lengths, and then add them together.
x+x+x+x+x+x+1+1+1+1+1+1
2x^2+3x
6x+6
Think about how you determined the area. Would rearranging the tiles change their individual areas or the total area of the shape? Rearrange the tiles into a new shape and see how it changes the area.
Use the tiles in the eTool below to create the shape.
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Food Satiations - Ring of Brodgar
Food satiations are penalties from eating specific foods. Every food item satiates its own category. All food stuffs satiate equally, and there is no randomness or distribution.Satiated foods become less effective, which can be seen as a percentage on the Base Attributes screen of the character sheet, under Food Satiations.
The penalty comes into play in the following way:
{\displaystyle FEPGained=FoodFEPValue*HungerModifier*SatiationModifier}
But also decrease hunger in the following way:
{\displaystyle HungerGained=FoodHungerValue*SatiationModifier}
Evidently you should eat a varied diet to keep your satiations high so as not to hinder your FEP gain.
{\displaystyle SatiationModifier=100\%-SatiationPenalty}
Preventing food satiations by drinks
As of Still Brandy (2021-06-19)>"Added/Re-added drunkenness.", some stuff below might have changed because of this patch (although drunkenness seems limited to Brandy at the moment). [Verify: drunkenness patch changes.]
Satiations can now be prevented/soften by drinks
The buff drink and be merry is then applied to yourself when you drink one sip (0.05L) of any drink.
For each sip (0.05 L) of every drink you consume, you get one counter for that drink.
For each food type consumed that are buffed by a drink, you lose one counter for that specific drink, and receive less satiation.
More than one counter may be removed from different drinks if more than one buff are applied, meaning buffs stack for the same food consumed. Example : Peapie is both a Bread and a Vegetable, therefore drink both Mead and Milk before eating one is preferable minimize satiations gain.
However, only a single instance of a buff will be counted if you have multiple active Drink and be Merry counters for drink types that share buffs. The instance of the buff that will be consumed will be the best percentage buff you have available to you. For example, if you have a point of Wine and a point of Perry each buffing the Meat category, if you eat a piece of food considered Meat it will only deduct a point from the buff that is of a higher value for that type of food.
"Drink, and be merry" expires after about an in-game day (roughly 8 hours) if not used by then.
Applejuice Offal 12.5% -x- -x- Tankard Red Apple Applejuice
Beer Sausage 25% Offal 12.5% Game 12.5% Tankard Barley Wort Beer
Brandy Sweets & Desserts 35% Cheese 17.5% Offal 17.5% (none yet) Distilled Wine Brandy
Cider Fowl & Poultry 25% Mushrooms 12.5% Offal 12.5% Tankard Applejuice Cider
Grapejuice Vegetables 12.5% -x- -x- Wine Glass Grapes Grapejuice
Mead Vegetables 25% Game 12.5% Nuts & Seeds 12.5% Drinking Horn Mead Must Mead
Milk Bread 25% Forage 12.5% Berries 12.5% (None/All) Aurochs, Cow, Goat and Sheep Dairy
Pearjuice Creepies & Crawlies 12.5% -x- -x- Tankard Pear Pearjuice
Perry Mushrooms 25% Meat 12.5% Creepies & Crawlies 12.5% Tankard Pearjuice Perry
Tea Dairy 25% Fruits 12.5% Nuts & Seeds 12.5% Mug Green and Black Tea Leaves Tea
Weißbier Fish 25% Fowl & Poultry 12.5% Mushrooms 12.5% Tankard Wheat Wort Beer
Wine Cheese 25% Meat 12.5% Vegetables 12.5% Wine Glass Grapejuice Wine
Calvados Fruit 35% Mushrooms 17.5% Fowl & Poultry 17.5% (none yet) Distilled Cider Calvados
Information may be incomplete or wrong for a few drinks.
Tea receives a 20Q buff when piping hot.
Food type categories
Name Berries Bread Cheese Creepies & Crawlies Dairy Fish Food Forage Fruit Game Meat Mushrooms Nuts Offal Poultry Sausage Sweets & Desserts Vegetables
Hungry Hat (2021-09-23) >"Satiations now increase and decrease 3x slower, as suggested here. The main functional change of this is that you should have to eat less often."
Still Brandy (2021-06-19) >"Added/Re-added drunkenness. When consuming alchoholic beverages, you now get a tiered buff, much like you did in Legacy Haven."
Retrieved from "https://ringofbrodgar.com/w/index.php?title=Food_Satiations&oldid=92871"
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The two triangles at right are similar.
x
What would the scale factor be between these two shapes?
1
cm is the corresponding side, multiply it by the scale factor to get
x
x=4
Find the area of the smaller triangle.
Use the formula for the area of a triangle:
\frac{1}{2}\left(6\ \text{cm}\right)\left(1\ \text{cm}\right)
Based on ratios of similarity, find the area of the large triangle.
When using ratios of similarity to scale a shape, the area of the old shape is multiplied by the scale factor squared to get the area of the new shape.
3(4)^2=48
Find the area of the larger triangle by using the formula for the area of a triangle.
\frac{1}{2}\left(\text{base}\right)\left(\text{height}\right)
Verify that your answers to (c) and (d) are the same.
Help (e):
Did you get the same answer? Make sure to show your work.
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(Redirected from Miles per gallon)
Distance travelled by a vehicle compared to volume of fuel consumed
{\displaystyle {\frac {235}{\rm {mpg_{US}}}}={\rm {1\;L/100\;km}}}
{\displaystyle {\frac {235}{\rm {L/100\;km}}}={\rm {1\;mpg_{US}}}}
{\displaystyle {\frac {282}{\rm {mpg_{Imp}}}}={\rm {1\;L/100\;km}}}
{\displaystyle {\frac {282}{\rm {L/100\;km}}}={\rm {1\;mpg_{Imp}}}}
{\displaystyle {\rm {8.5\;mpg_{US}=1\;km/20\;L}}}
{\displaystyle {\frac {2000}{\rm {L/100\;km}}}={\rm {1\;km/20\;L}}}
{\displaystyle 1\;{\rm {mpg_{US}={\rm {0.8327\;{\rm {mpg_{Imp}}}}}}}}
{\displaystyle 1\;{\rm {mpg_{Imp}=1.2001\;{\rm {mpg_{US}}}}}}
Fuel economy statistics[edit]
Speed and fuel economy studies[edit]
In 1998, the U.S. Transportation Research Board footnoted an estimate that the 1974 National Maximum Speed Limit (NMSL) reduced fuel consumption by 0.2 to 1.0 percent.[17] Rural interstates, the roads most visibly affected by the NMSL, accounted for 9.5% of the U.S' vehicle-miles-traveled in 1973,[18] but such free-flowing roads typically provide more fuel-efficient travel than conventional roads.[19] [20] [21]
Differences in testing standards[edit]
Energy considerations[edit]
{\displaystyle F={\frac {dW}{ds}}\propto {\text{Fuel economy}}}
Fuel economy-boosting technologies[edit]
Engine-specific technology[edit]
Other vehicle technologies[edit]
Future technologies[edit]
Fuel economy data reliability[edit]
Concerns over EPA estimates[edit]
Fuel economy maximizing behaviors[edit]
Fuel economy as part of quality management regimes[edit]
Fuel economy standards and testing procedures[edit]
10–15 mode[edit]
JC08[edit]
US Energy Tax Act[edit]
EPA testing procedure through 2007[edit]
EPA testing procedure: 2008 and beyond[edit]
Electric vehicles and hybrids[edit]
CAFE standards[edit]
Federal and state regulations[edit]
Unit conversions[edit]
Conversion from mpg[edit]
Conversion from km/L and L/100 km[edit]
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Perform transformation from three-phase (abc) signal to dq0 rotating reference frame or the inverse - Simulink - MathWorks Nordic
abc to dq0, dq0 to abc
Rotating frame alignment at wt=0
Perform transformation from three-phase (abc) signal to dq0 rotating reference frame or the inverse
The abc to dq0 block uses a Park transformation to transform a three-phase (abc) signal to a dq0 rotating reference frame. The angular position of the rotating frame is given by the input wt, in rad.
The dq0 to abc block uses an inverse Park transformation to transform a dq0 rotating reference frame to a three-phase (abc) signal. The angular position of the rotating frame is given by the input wt, in rad.
When the rotating frame alignment at wt=0 is 90 degrees behind the phase A axis, a positive-sequence signal with Mag=1 and Phase=0 degrees yields the following dq values: d=1, q=0.
\begin{array}{c}{V}_{d}=\frac{2}{3}\left({V}_{a}\mathrm{sin}\left(\omega t\right)+{V}_{b}\mathrm{sin}\left(\omega t-2\pi /3\right)+{V}_{c}\mathrm{sin}\left(\omega t+2\pi /3\right)\right)\\ {V}_{q}=\frac{2}{3}\left({V}_{a}\mathrm{cos}\left(\omega t\right)+{V}_{b}\mathrm{cos}\left(\omega t-2\pi /3\right)+{V}_{c}\mathrm{cos}\left(\omega t+2\pi /3\right)\right)\\ {V}_{0}=\frac{1}{3}\left({V}_{a}+{V}_{b}+{V}_{c}\right)\end{array}
\begin{array}{c}{V}_{a}={V}_{d}\mathrm{sin}\left(\omega t\right)+{V}_{q}\mathrm{cos}\left(\omega t\right)+{V}_{0}\\ {V}_{b}={V}_{d}\mathrm{sin}\left(\omega t-2\pi /3\right)+{V}_{q}\mathrm{cos}\left(\omega t-2\pi /3\right)+{V}_{0}\\ {V}_{c}={V}_{d}\mathrm{sin}\left(\omega t+2\pi /3\right)+{V}_{q}\mathrm{cos}\left(\omega t+2\pi /3\right)+{V}_{0}\end{array}
The block supports the two conventions used for the Park transformation:
When the rotating frame is aligned with the phase A axis at t = 0, that is, at t = 0, the d-axis is aligned with the a-axis. This type of Park transformation is also known as the cosine-based Park transformation.
When the rotating frame is aligned 90 degrees behind the phase A axis, that is, at t = 0, the q-axis is aligned with the a-axis. This type of Park transformation is also known as the sine-based Park transformation. Use this transformation in Simscape™ Electrical™ Specialized Power Systems models with three-phase synchronous and asynchronous machines.
Deduce the dq0 components from the abc signals by performing an abc to αβ0 Clarke transformation in a fixed reference frame. Then perform an αβ0 to dq0 transformation in a rotating reference frame, that is, by performing a −(ω.t) rotation on the space vector Us = uα + j· uβ.
The abc-to-dq0 transformation depends on the dq frame alignment at t = 0. The position of the rotating frame is given by ω.t, where ω represents the dq frame rotation speed.
When the rotating frame is aligned with the phase A axis, the following relations are obtained:
\begin{array}{l}{U}_{s}={u}_{d}+j\cdot {u}_{q}=\left({u}_{a}+j\cdot {u}_{\beta }\right)\cdot {e}^{-j\omega t}=\frac{2}{3}\cdot \left({u}_{a}+{u}_{b}\cdot {e}^{\frac{-j2\pi }{3}}+{u}_{c}\cdot {e}^{\frac{j2\pi }{3}}\right)\cdot {e}^{-j\omega t}\\ {u}_{0}=\frac{1}{3}\left({u}_{a}+{u}_{b}+{u}_{c}\right)\\ \left[\begin{array}{c}{u}_{d}\\ {u}_{q}\\ {u}_{0}\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{cos}\left(\omega t\right)& \mathrm{cos}\left(\omega t-\frac{2\pi }{3}\right)& \mathrm{cos}\left(\omega t+\frac{2\pi }{3}\right)\\ -\mathrm{sin}\left(\omega t\right)& -\mathrm{sin}\left(\omega t-\frac{2\pi }{3}\right)& -\mathrm{sin}\left(\omega t+\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]\end{array}
The inverse transformation is given by:
\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}\left(\omega t\right)& -\mathrm{sin}\left(\omega t\right)& 1\\ \mathrm{cos}\left(\omega t-\frac{2\pi }{3}\right)& -\mathrm{sin}\left(\omega t-\frac{2\pi }{3}\right)& 1\\ \mathrm{cos}\left(\omega t+\frac{2\pi }{3}\right)& -\mathrm{sin}\left(\omega t+\frac{2\pi }{3}\right)& 1\end{array}\right]\left[\begin{array}{c}{u}_{d}\\ {u}_{q}\\ {u}_{0}\end{array}\right]
When the rotating frame is aligned 90 degrees behind the phase A axis, the following relations are obtained:
\begin{array}{l}{U}_{s}={u}_{d}+j\cdot {u}_{q}=\left({u}_{\alpha }+j\cdot {u}_{\beta }\right)\cdot {e}^{-j\left(\omega t-\frac{\pi }{2}\right)}\\ \left[\begin{array}{c}{u}_{d}\\ {u}_{q}\\ {u}_{0}\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{sin}\left(\omega t\right)& \mathrm{sin}\left(\omega t-\frac{2\pi }{3}\right)& \mathrm{sin}\left(\omega t+\frac{2\pi }{3}\right)\\ \mathrm{cos}\left(\omega t\right)& \mathrm{cos}\left(\omega t-\frac{2\pi }{3}\right)& \mathrm{cos}\left(\omega t+\frac{2\pi }{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]\end{array}
\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{sin}\left(\omega t\right)& \mathrm{cos}\left(\omega t\right)& 1\\ \mathrm{sin}\left(\omega t-\frac{2\pi }{3}\right)& \mathrm{cos}\left(\omega t-\frac{2\pi }{3}\right)& 1\\ \mathrm{sin}\left(\omega t+\frac{2\pi }{3}\right)& \mathrm{cos}\left(\omega t+\frac{2\pi }{3}\right)& 1\end{array}\right]\left[\begin{array}{c}{u}_{d}\\ {u}_{q}\\ {u}_{0}\end{array}\right]
abc — abc signal
abc signal, specified as a vector.
dq0 — dq0 signal
dq0 signal, specified as a vector.
wt — Angular position of dq rotating frame
Angular position of the dq rotating frame, in radians, specified as a positive scalar.
dq0 signal, returned as a vector.
abc signal, returned as a vector.
Rotating frame alignment at wt=0 — Alignment of rotating frame
90 degrees behind phase A axis (default) | Aligned with phase A axis
Alignment of the rotating frame at t = 0 of the dq0 components of a three-phase balanced signal:
{u}_{a}=\mathrm{sin}\left(\omega t\right);\text{ }{u}_{b}=\mathrm{sin}\left(\omega t-\frac{2\pi }{3}\right);\text{ }{u}_{c}=\mathrm{sin}\left(\omega t+\frac{2\pi }{3}\right)
The positive-sequence magnitude equals 1.0 pu, and the phase angle equals 0 degrees.
When you select 90 degrees behind phase A axis, the dq0 components are d = 1, q = 0, and zero = 0.
The power_Transformations example shows various ways to use blocks to perform Clarke and Park transformations.
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Category:Many-body perturbation theory - Vaspwiki
Category:Many-body perturbation theory
Many-body perturbation theory includes screening and renormalization effects beyond the density-functional theory (DFT). It is based on the Green's-function formalism and can be derived and visualized in terms of a diagrammatic expansion of, e.g., the electron interacting with other electrons. Instead of describing the electrons by means of Kohn-Sham (KS) orbitals, the renormalized (or dressed) propagators yield quasiparticle orbitals.
1.1 Random-phase approximation (RPA)
1.2 Constrained random-phase approximation
1.3 GW method
1.4 Bethe-Salpeter equations (BSE)
1.5 Second-order Møller-Plesset perturbation theory (MP2)
Random-phase approximation (RPA)
GW and RPA are post-DFT methods used to solve the many-body problem approximatively.
RPA stands for the random-phase approximation and is often used as a synonym for the adiabatic connection fluctuation-dissipation theorem (ACFDT). RPA/ACFDT provides access to the correlation energy of a system and can be understood in terms of Feynman diagrams as an infinite sum of all bubble diagrams, where excitonic effects (interactions between electrons and holes) are neglected. The RPA/ACFDT is used as a post-processing tool to determine a more accurate ground-state energy.
RPA/ACFDT: Correlation energy in the Random Phase Approximation .
Constrained random-phase approximation
The constrained random-phase approximation (CRPA) is a method that allows calculating the effective interaction parameter
{\displaystyle U}
{\displaystyle J}
{\displaystyle J'}
for model Hamiltonians. The main idea is to neglect the screening effects of specific target states in the screened Coulomb interaction
{\displaystyle W}
{\displaystyle GW}
method. Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher-level theory, such as dynamical-mean-field theory (DMFT).
Formalism used for the CRPA method
The GW approximation goes hand in hand with the RPA since the very same diagrammatic contributions are taken into account in the screened Coulomb interaction of a system often denoted as W. However, in contrast to the RPA/ACFDT, the GW method provides access to the spectral properties of the system by means of determining the energies of the quasi-particles of a system using a screened exchange-like contribution to the self-energy. The GW approximation is currently one of the most accurate many-body methods to calculate band-gaps.
The GW approximation of Hedin's equations.
Bethe-Salpeter equations (BSE)
VASP offers a powerful module for solving time-dependent DFT (TD-DFT) and time-dependent Hartree-Fock equations (TDHF) (the Casida equation) or the Bethe-Salpeter (BSE) equation[1][2]. These approaches are used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional, or GW approximations. VASP also offers the TDHF and BSE calculations beyond the Tamm-Dancoff approximation (TDA)[3].
Tags and articles related to BSE calculations.
Second-order Møller-Plesset perturbation theory (MP2)
There are three implementations available:
MP2[4]: this implementation is recommended for very small unit cells, very few k-points and very low plane-wave cuttofs. The system size scaling of this algorithm is N⁵.
LTMP2[5]: for all larger systems this Laplace transformed MP2 (LTMP) implementation is recommended. Larger cutoffs and denser k-point meshes can be used. It possesses a lower system size scaling (N⁴) and a more efficient k-point sampling.
stochastic LTMP2[6]: even faster calculations at the price of statistical noise can be achieved with the stochastic MP2 algorithm. It is an optimal choice for very large systems where only relative errors per valence electron are relevant. Keeping the absolute error fixed, the algorithm exhibits a cubic scaling with the system size, N³, whereas for a fixed relative error, a linear scaling, N¹, can be achieved. Note that there is no k-point sampling and no spin polarization implemented for this algorithm.
Practical guides to different diagrammatic approximations are found on following pages:
ACFDT: ACFDT/RPA calculations.
GW: Practical guide to GW calculations.
BSE: BSE calculations.
Using the GW routines for the determination of frequency dependent dielectric matrix: GW and dielectric matrix.
MP2 method: MP2 ground state calculation - Tutorial.
↑ S. Albrecht, L. Reining, R. Del Sole, and G. Onida, Phys. Rev. Lett. 80, 4510-4513 (1998).
↑ M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 81, 2312-2315 (1998).
↑ T. Sander, E. Maggio, and G. Kresse, Phys. Rev. B 92, 045209 (2015).
↑ M. Marsman, A. Grüneis, J. Paier, and G. Kresse, J. Chem. Phys. 130, 184103 (2009).
↑ T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 146, 104101 (2017).
Pages in category "Many-body perturbation theory"
NBANDSO
NBANDSV
NBSEEIG
Retrieved from "https://www.vasp.at/wiki/index.php?title=Category:Many-body_perturbation_theory&oldid=16683"
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Data Presentation Problem Solving Practice Problems Online | Brilliant
N
The above bar chart shows the number of vehicles that were sold at a dealership in the month of October. If each square represents 13 vehicles, how many more trucks were sold than convertibles?
\frac{1}{4}
of the attendees were children. If there are 180 men who attended, how many more men than women attended?
\frac{1}{4}
of the attendees were children. If there are 114 men who attended, how many women attended?
2-3 hours 1-2 hours <1 hour
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River - Electowiki
River is a cloneproof monotonic Condorcet ambiguity resolution method with similarities to both Ranked Pairs and Schulze, but when cycles exist, can in rare cases find a different winner than either of the other two methods.
It was first proposed in 2004 by Jobst Heitzig on the Election-methods mailing list.[1][2] Jobst later refined the definition to be more similar to Ranked Pairs.[3]
Quick summary of method, which is identical to Ranked Pairs except where emphasized:
Rank defeats in descending order of winning vote strength.
Starting with the strongest defeat, affirm defeats unless a cycle is created or a candidate is defeated twice.
The result is that only sufficient defeat information to determine the winner is included.
Because not all defeats are processed, the social ordering is not linear—in general, it is a tree (or river) diagram, with the victor at the base of the river.
Example using 2004 baseball scores. This shows how a 14-candidate election winner can be determined much more quickly using River than with RP or Schulze.
Early criticism of the River method. This shows that the River method violates mono-add-top and mono-remove-bottom.
River can be interpreted as a Minmax method, Minmax(non-cyclic pairwise loss) or MMNCPL. It is similar to Minmax(winning votes) except that River elects the candidate whose greatest non-cyclic pairwise loss to another candidate is least. As in Ranked Pairs, the greatest pairwise loss (GPL) of each candidate is considered in order from largest (among all candidates) to smallest and locked. If a candidate's GPL is cyclic, it is discarded, and the next-greatest pairwise loss of that candidate is added to the list. When the non-cyclic greatest pairwise losses of (N-1) candidates have been locked, the remaining candidate is the winner.
The River method constructs a maximum spanning tree from the undirected multigraph whose adjacency matrix is based on the Condorcet matrix.[4] The wv and pairwise opposition variants use the Condorcet matrix
{\displaystyle C}
as the adjacency matrix, whereas the margins variant uses a matrix
{\displaystyle D}
{\displaystyle D_{A>B} = C_{A>B} - C_{B>A}}
Since it's possible to determine the minimum spanning tree of a dense graph in
{\displaystyle O(|E|)}
time, it is possible to determine the winner of the River method in
{\displaystyle O(c^2)}
{\displaystyle c}
is the number of candidates. Warren provides such an algorithm in his paper.[4]
Criterion compliancesEdit
River passes Condorcet, Smith, the monotonicity criterion[4], independence of clones, and independence of Pareto-dominated alternatives.[5] It fails mono-add-top, later-no-harm, and the participation criterion.
In addition, River passes Heitzig's independence of strongly dominated alternatives criterion, which is weaker than independence of uncovered alternatives and stronger than independence of Pareto-dominated alternatives.
SmithEdit
Suppose A is in the Smith set and B is not. By definition, A beats B pairwise. In the absence of any constraints, A>B will be locked before B>A. The only possible constraints preventing A>B from being locked are that someone else over B is already locked; or that B beats someone who beats A, so that locking A>B would create a cycle. Since B is not in the Smith set, the latter can't happen. If some other X>B is already locked, then either X is in the Smith set or not. If X is in the Smith set, then Smith is already satisfied (even if the winner happens to not be A). On the other hand, if X is not in the Smith set, then the proof can be repeated with X in the place of B.
Independence of clonesEdit
Suppose A is cloned into two candidates A1 and A2. Every voter who ranked some candidate X above A now ranks X over both clones, and every voter who ranked A above X now ranks both clones above X. For any other candidate, Y>A1 is thus locked iff Y>A was locked in the original election, and A1>Y is locked iff A>Y was locked in the original election. As a consequence, pairwise contests not involving the clones are locked iff they were locked in the original election. So A can't lose by being cloned, can't win by being cloned, and can't affect what other candidate X wins, by being cloned.
↑ Heitzig, Jobst (2004-04-10). "Hello again -- and a new method for you!". Election-Methods mailing list. Retrieved 2020-02-17.
↑ Heitzig, Jobst (2004-10-06). "River method -- updated summary". Election-Methods mailing list. Retrieved 2020-02-17.
↑ a b c Smith, Warren D. (2007-06-12). "Descriptions of single-winner voting systems" (PDF). pp. 12–13. Retrieved 2020-02-17.
↑ Heitzig, Jobst (2004-04-24). "River method - a refinement, minor computational evidence, and a generalized IPDA criterion ISDA". Election-Methods mailing list. Retrieved 2020-02-17.
Retrieved from "https://electowiki.org/w/index.php?title=River&oldid=8191"
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Binary Search | Brilliant Math & Science Wiki
Karleigh Moore, Satyabrata Dash, Jimin Khim, and
Binary search is an efficient algorithm that searches a sorted list for a desired, or target, element. For example, given a sorted list of test scores, if a teacher wants to determine if anyone in the class scored
80
, she could perform a binary search on the list to find an answer quickly. Binary search works by halving the number of elements to look through and hones in on the desired value. Binary search can determine if and where an element exists in a list, or determine if it is not in the list at all.
A gif showing a binary search for the number 47 in the given list.
Binary search works by comparing the target value to the middle element of the array. If the target value is greater than the middle element, the left half of the list is eliminated from the search space, and the search continues in the right half. If the target value is less than the middle value, the right half is eliminated from the search space, and the search continues in the left half. This process is repeated until the middle element is equal to the target value, or if the algorithm returns that the element is not in the list at all.
Binary search looks through a sorted list to see if a desired element is in the list. It does this efficiently by halving the search space during each iteration of the program. Basically, binary search finds the middle of the list, asks “is the element I’m looking for larger or smaller than this?” Then it cuts the search space in the list in half and searches only in the left list if the element is smaller, and searches only in the right list if the element is bigger. It repeats this process until it finds the element it is looking for (or reports back that the element isn’t in the list at all). The algorithm uses a divide and conquer (or divide and reduce) approach to search.
Visualization of the binary search algorithm where 4 is the target value.[1]
In simple terms, the algorithm works as follows:
The following assumes zero indexing, meaning that the left-most element of a list is the
0^\text{th}
Determine the middle element of a sorted list by taking the value of the floor of
\frac{\text{low + high}}{2}
, where low is the lowest index of the list, and high is the highest index in the list. So in the list
[1,2,3,4]
2
(since 2 occurs at index 1) would be the middle. In the list
[1,2,3,4,5]
3
(since 3 occurs at index 2) is the middle.
Compare the value of that middle element with the target value.
If the target value is equal to the middle element, return that it is true the element is in the list (if the position of the element in the list is desired, return the index as well).
If the target value is less than the middle element, eliminate all elements to the right of (and including) the middle element from the search, and return to step one with this smaller search space.
If the target value is greater than the middle element, eliminate all the elements to the left of (and including) the middle element from the search, and return to step one with this smaller search space.
A limitation of binary search is that it can only search in a pre-sorted list. If the list is not pre-sorted, binary search will not work. Linear search may be a better choice of search algorithm for an unsorted list.
Show the steps that binary search would perform to determine if
14
is in the following list:
A =[0,2,5,5,9,10,11,12,14,15]
. Round down when determining the middle element.
List Middle element
[0,2,5,5,9,10,11,12,14,15]
[10,11,12,14,15]
[14,15]
Since, in the third step, the middle element is equal to the target value, we can return that the element is in the list.
_\square
Binary search can be implemented either iteratively or recursively.
Here is an iterative implementation in Python.
#note: Use of // indicates floor division. Ex. 5//2 = 2
Here is a recursive implementation of binary search in Python.
def binary_search_recursive(listOfInts, elem, start=0, end=None):
end = len(listOfInts) - 1
return 'Value not found in list'
if elem == listOfInts[mid]:
if elem < listOfInts[mid]:
return binary_search_recursive(listOfInts, elem, start, mid-1)
# elem > listOfInts[mid]
return binary_search_recursive(listOfInts, elem, mid+1, end)
listOfInts = [0,1,2,3,4,5,22,33,45]
print (binary_search_recursive(listOfInts, 22))
Say you want to search the list
[0,2,5,5,9,10,11,12,14,15]
to see if the value 13 is in the list using the binary search algorithm described here.
Which of the following values in the list (not the indices) will never be a value of "first" in the search for 13?
Hint: If you get stuck, try running the code from the binary search page and add a few useful print statements to keep track of the value of the array at index "first."
Binary search has a worst-case and average-case running time of
O(\log n)
O(\log n)
comparisons, where
n
is the number of elements in the array. The worst case occurs when a given target element is not in the list at all. This means the algorithm has halved the list again and again until it reaches a list of size one. To get a list of size
n
down to a list of
1
element,
\log n
divisions must be made. Each step of the binary search halves the search space.
Binary search has a
O(1)
best case running time. This occurs when the first midpoint selected is the target value.
Binary search is a key idea in binary search trees which are very important data structures in computer science.
Binary search can be used to help estimate the square roots of numbers. With a few modifications to the basic algorithm shown in the implementation section, Newton's method can be implemented.
How could you use binary search to help you estimate the square root of a number?
x
, we know that the square root of
x
x
. Our initial search space is
[0,x]
. In other words, the "list" is the set of numbers between
0
x
The basic idea of binary search is to halve the search space with each iteration. What might this look like in the case of square roots? We can select our midpoint the same way we did in the algorithm described in the above sections.
If the midpoint (proposed root) is not equal to
x
, to determine if we should search to the left of the midpoint or to the right, compare the value of the midpoint squared to the value of
x
(\text{midpoint})^2
x
, search in the right half of the list; if
(\text{midpoint})^2
x
, search in the left half of the list.
To calculate the "first" and "last" (or start index of the search space and end index of the search space), if
(\text{midpoint})^2
x
, "first" becomes the current midpoint value, the new midpoint becomes
\frac{\text{first + last}}{2}
, and the current value of "last" remains the value of "last". If
(\text{midpoint})^2
x
, the current midpoint value becomes the new "last", the current value of "first" remains the value of "first", and the new midpoint becomes
\frac{\text{first + last}}{2}
Repeat this process until the root is found.
_\square
Below is a Python implementation of the process described above. [4]
def sqrt(n, precision=10e-8):
first, last = 0.0, max(n, 1.0)
prev, mid = 0, (first + last) / 2.0
while abs(mid - prev) > precision:
if mid ** 2 > n:
prev, mid = mid, (first + last) / 2.0
, T. Binary search into array. Retrieved May 2, 2016, from https://en.wikipedia.org/wiki/File:Binary_search_into_array.png
, . The Binary Search. Retrieved April 30, 2016, from http://interactivepython.org/runestone/static/pythonds/SortSearch/TheBinarySearch.html
, . The Binary Search. Retrieved June 5, 2016, from http://interactivepython.org/runestone/static/pythonds/SortSearch/TheBinarySearch.html
Krenzel , S. Deriving Newton's Method from Binary Search. Retrieved May 2, 2016, from http://stevekrenzel.com/articles/newtons-law
Cite as: Binary Search. Brilliant.org. Retrieved from https://brilliant.org/wiki/binary-search/
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Facts and Factors of Development I
The extraordinary development of universities in the United States is paralleled by the growth of its museums. In Washington, New York, Chicago and Pittsburgh, four museums of natural history have in a comparatively brief period taken their places among the leading institutions of the world, and in many other cities there are important and growing museums. Many of these museums, like the universities, are interesting demonstrations of the possible achievements of a democracy. On the one hand, they are supported in almost equal measure by taxation and by private gifts, on the other hand, they are devoted not primarily to the preservation of stuffed animals, but to education, research and public service.
The forty-fifth annual report of the American Museum of Natural History in New York City illustrates these remarks. The city has provided land and buildings worth many million dollars and approved a plan of development of unexampled magnitude. The city also provided last year $200,000 for maintenance. Then for exploration, research and the increase of the collections about $250,000 accrued from private endowment and gifts. The annual gifts are about equal in amount from the trustees and from members and friends, who number some 3,500.
The illustrations here reproduced show the museum and its approaches, though it should not be assumed that such crowds enter the museum every day in the year. The total attendance in 1913 was 866,633, of whom 138,375 were primarily present for lectures and scientific meetings. Part of the attendance in the galleries was due to the flower exhibition of the Horticultural Society and other temporary exhibits, which Dr. Lucas, the director, holds do not result in any real profit to a museum. We should suppose, however, that while such exhibits and the large number of lectures and scientific meetings may not greatly increase interest in the natural history collections, they enlarge the functions of a museum in a desirable manner.
This holds still more for the expeditions and research work. As in the university the professor earns his salary by teaching but is expected to advance knowledge, so in the museum the curator must care for the display of the exhibits but he should also be engaged in scientific research. So long as society provides no way of paying directly for the results of investigations having no immediate commercial value, these must be undertaken by universities and scientific institutions. This should be regarded as part of their function, but in any case it is justified by the fact that the professor or curator will do the work of teaching or caring for collections better if ha is encouraged to engage in research and publication, and under these circumstances better men can be secured for the positions.
Though the museum has been unfortunate in losing two of its most distinguished investigators, Professor Boas of Columbia University in anthropology and Professor Wheeler of Harvard University in invertebrate zoology, it produces each year an important series of contributions to scientific knowledge. Last year the sum of $25,000 was spent on publications, partly technical researches, of which Dr. P. G. Elliot's "A Review of the Primates" is the most noteworthy, and partly on popular publications, including the excellent "Museum Journal."
Visitors Entering the American Museum of Natural History.
The report of the president. Dr. Osborn, reviews the general progress of the work of the museum, noting the establishment of a contributory pension system, according to which the employee contributes to the fund three per cent, of his salary and the trustees provide an equal amount. Among installations, the collection of bronzes made in China by Dr. Laufer is especially noted. Gifts include the Mason archeological collection from Tennessee by the late Mr. J. P. Morgan, the Angelo Heilprin Exploring Fund, established by Mr. and Mrs. Paul J. Sachs, and numerous specimens from individuals and institutions.
The museum, however, must depend for its most valuable accessions on its own expeditions. The number and range of these expeditions in 1913 are shown on the chart. The expedition to Crocker Land, under Mr. McMillan, suffered from the stranding of the Diana, but has proceeded to the Arctic regions. Expeditions to the north in search of bowhead whales and to the south to secure the nearly extinct sea elephant were not successful, but other material was obtained including motion pictures of the life on the seal islands. The paleontological and ethnoogical expeditions in the west from which important collections and researches have resulted were continued. In South America Mr. Chapman and others have made ornithological surveys and collections, and the present expedition of Mr. Roosevelt is under the auspices of the museum. Africa has been explored by Messrs. Lang, Chapin, Rainsford and Rainey. Dr. Osborn, the president, has visited the French prehistoric caverns. Such expeditions not only increase in the most desirable wav the collections of a museum, but also contribute in large measure to the advancement of science.
It would perhaps be worth while to issue a number of The Popular Science Monthly consisting entirely of articles sent in by those who in Bishop Berkeley's phrase are "undebauched by learning." At first sight it might seem disquieting that there are so many people in the United States without the slightest training or appreciation of scientific methods who would like to publish their views on electricity, gravity, the ice age and similar topics, or have them endowed by the Carnegie Institution. But we may in fact regard it as a not altogether unsatisfactory symptom of universal education in a democracy, and of growing interest in science. The pseudo-science often exhibited in our daily papers and legislative halls will surely be eliminated by a comparatively small increase in education and the control of public sentiment by those who know, and we may then look to a notable advance in scientific research through the rewards and opportunities which a discriminating public would be able to bestow.
While it might be unfair to print some of the contributions sent in, it may not be amiss to quote two paragraphs which have just now been brought to our attention. The first is from a speech in the House of Representatives by Mr. Hobson of Alabama, which is being widely circulated under the congressional franking privilege. He said:
The last word of science, after exact research in all the domains, is that alcohol is a poison. It has been found to be a hydrocarbon of the formula
{\displaystyle {\ce {C2H6O}}}
, that is produced by the process of fermentation, and is the toxin or liquid excretion or waste product of the yeast or ferment germ. According to the universal law cf biology that the toxin of one form of life is a poison to all forms of life of a higher alcohol, the toxin of the low yeast germ, is a protoplasmic poison to all life, whether plant, animal or man, and to all the living tissues and organs.
After long continued drinking, even though temperate, the microscope shows that the white blood corpuscles, with the serum which contains their vegetable food continually sucked up by the dehydrating toxin, become carnivorous, and begin to feed upon the tissues and organs, like disease germs. The favorite tissue food of the degenerate corpuscles are the tender cells of latest development. In the human being the latest development is the brain. The microscope shows the degenerate corpuscles, with the goods upon them, down in their bodies the gray matter of the brain. This accounts for the tremendous mortality among heavy drinkers and for the degeneracy that will be referred to later.
The second quotation, the head lines and editorial from the April publications of a Sunday newspaper syndicate, is as follows:
WHEN THE WORLD'S BACK BROKE
EDITOR'S VOTE.—Mr. Curwood is the first writer to tell in fiction the dramatic story of that day thousands of years ago, when in the space of what was probably tin more than a few minutes the earth tilted twenty-three and a half degrees on its axis, transforming what was then a tropical world into the blackness of a night which lasted for unnumbered centuries, and out of which came what are known as the North Polar regions of today. In that one "first night" of a life that had known only perpetual day all living creatures perished; but entombed in their caskets of ice and frozen earth many of tin m have conn down tn us fifty or a hundred thousand years later, so completely preserved that the flesh of mastodons recently discovered was eaten by dogs and men. In fact, Mr. Curwood helped uncover a mastodon at Fort Migley and ate of the flesh.
QUARANTINE OF HAWAIIAN FRUIT
The office of information of the Department of Agriculture has sent out a notice in regard to the stringent regulations which have been adopted to guard against danger from the melon fly and the Mediterranean fruit fly. Any one who attempts after May 1, to bring into the United States certain Hawaiian fruits, nuts and vegetables will face a penalty of $500 fine or imprisonment for a year or both. A new order issued by the Department of Agriculture provides this punishment for attempts to violate the quarantine declared in 1912, under the plants quarantine act, against Hawaiian products which might introduce into the United States two dangerous pests, the melon fly and the Mediterranean fruit fly. ruder the new regulations importations of bananas and pineapples are per mined under stringent conditions of inspection and certification. Practically all other fruits and such vegetables as tomatoes, squashes, green peppers and string leans are absolutely excluded. Circulars are to be distributed on all incoming steamships warning passengers of the quarantine and the reason for it.
Hitherto the United states has fortunately been free from both the melon fly and the Mediterranean fruit fly. The latter in particular has proved a source of great loss, practically putting i an end to the fruit industry wherever it has obtained a good foothold. The Bermuda peach crop, for instance, is now a thing of the past. It is believed to have originated on the west coast of Africa, its name being due to the great damage it did after it had been carried to the Mediterranean. It also spread to Bermuda, South Africa, Australia and New Zealand, whence t was carried in ships' cargoes to Hawaii. In all probability the fly would lie in California to-day if it were not for the fact that no fruit is grown in the immediate vicinity of San Francisco. The great danger is that some traveler may unknowingly bring with him as a curiosity pest-infected fruit, nuts or vegetables and introduce them into a region favorable for the fly's spread.
Commercially the quarantine will not seriously interfere with Hawaiian industries. Bananas and pineapples, the only fruits which are grown in the island in commercial quantities, do not, as a rule, carry the infection. When property inspected and packed in accordance with the department's regulations, they will, therefore, be allowed admission. Other fruits, such as alligator pears, Chinese ink berries,
The Late Sir John Murray,
the distinguished Scottish oceanographer.
figs, guavas, papayas, etc., are far more dangerous. They have, however, little commercial importance. If they are taken on board at all, they must either be consumed or thrown overboard before the ship reaches the United States.
We record with regret the deaths of Dr. Edward Singleton Holden, astronomer and librarian of the United States Naval Academy, formerly director of the Lick Observatory; of Mr. George Westinghouse, the distinguished inventor and engineer; of Dr. Alexander F. Chamberlain, professor of anthropology at Clark University; of Adolph Francis Alphonse Bandelier, an authority on South American archeology, lecturer in Columbia University, and of Dr. John Henry Poynting, professor of physics at Birmingham University.
A portrait of Sir William Ramsay, painted by Mr. Mark Milbanke, has been presented to University College, London, by former colleagues and past students. Professor J. Norman Collie made the address. A replica of the portrait has been presented to Lady Ramsay.
The former students of Dr. J. McKeen Cattell, professor of psychology in Columbia University, at a dinner held in New York on April 8, presented him, in celebration of his completion of twenty-five years as professor of psychology, with a "Festschrift" in the form of reviews of his researches and of the work in psychology to which they have led. On April 6, 7 and 8, there was held at Columbia University a Conference on Individual Psychology by former students of the department of psychology, at which thirty papers were presented.
The Rockefeller Institute for Medical Research, New York, announces that it has received from Mr. John D. Rockefeller an additional endowment of $1,000,000 for the purpose of organizing a department for the study of animal diseases. A gift of $50,000 has also been received from Mr. James J. Hill, for the study of hog cholera.
Following the disastrous fire at Wellesley College the General Education Board has promised to give $750,000 to the college on condition that the balance of the $2,000,000 restoration and endowment fund is completed by January 1, 1915.
Mr. Andrew Carnegie has given $100,000 to the New York Zoological Society to provide a pension fund for the Now York Zoological Park and the Aquarium. The scientific staff and the employees will contribute annually 2 per cent, of their salaries, and any sum that may be lacking will 1 e made up by the Zoological Society.
As has already been noted in Science, the American Chemical Society held its spring meeting at Cincinnati, Ohio, during the week of April 6. Each of the sections had a full and important program. At the general session on the first day, after addresses of welcome by the mayor of the city and the president of the University of Cincinnati, and a reply by the president of the society. Professor Theodore W. Richards, the following papers were announced: Arthur L. Day, "The Chemical Problems of an Active Volcano"; L. J. Henderson, "The Chemical Fitness of the World for Life"; W. D. Bancroft, "Flame Reactions"; Irving Langmuir, "Chemical Reactions at Low Pressures."
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Unique eight-digit number used to identify a holy periodical publication
and encoded in an EAN-13 barcode with an EAN-2 add-on designatin' issue number 13
An International Standard Serial Number (ISSN) is an eight-digit serial number used to uniquely identify a holy serial publication, such as a magazine.[1] The ISSN is especially helpful in distinguishin' between serials with the bleedin' same title. ISSNs are used in orderin', catalogin', interlibrary loans, and other practices in connection with serial literature.[2]
The ISSN system was first drafted as an International Organization for Standardization (ISO) international standard in 1971 and published as ISO 3297 in 1975.[3] ISO subcommittee TC 46/SC 9 is responsible for maintainin' the bleedin' standard.
When a serial with the oul' same content is published in more than one media type, an oul' different ISSN is assigned to each media type. Whisht now and eist liom. For example, many serials are published both in print and electronic media. The ISSN system refers to these types as print ISSN (p-ISSN) and electronic ISSN (e-ISSN), respectively.[4] Consequently, as defined in ISO 3297:2007, every serial in the feckin' ISSN system is also assigned a holy linkin' ISSN (ISSN-L), typically the oul' same as the oul' ISSN assigned to the feckin' serial in its first published medium, which links together all ISSNs assigned to the oul' serial in every medium.[5]
2.1 Linkin' ISSN
The format of the ISSN is an eight-digit code, divided by a bleedin' hyphen into two four-digit numbers.[1] As an integer number, it can be represented by the bleedin' first seven digits.[6] The last code digit, which may be 0-9 or an X, is a bleedin' check digit. Formally, the feckin' general form of the oul' ISSN code (also named "ISSN structure" or "ISSN syntax") can be expressed as follows:[7]
where N is in the oul' set {0,1,2,...,9}, a digit character, and C is in {0,1,2,...,9,X}; or by a holy Perl Compatible Regular Expressions (PCRE) regular expression:[8]
For example, the oul' ISSN of the journal Hearin' Research, is 0378-5955, where the bleedin' final 5 is the oul' check digit, that is C=5. To calculate the oul' check digit, the feckin' followin' algorithm may be used:
The sum of the bleedin' first seven digits of the bleedin' ISSN is calculated and multiplied by its position in the bleedin' number, countin' from the feckin' right, that is, 8, 7, 6, 5, 4, 3, and 2, respectively:
{\displaystyle {\begin{aligned}&0\cdot 8+3\cdot 7+7\cdot 6+8\cdot 5+5\cdot 4+9\cdot 3+5\cdot 2\\&=0+21+42+40+20+27+10\\&=160\end{aligned}}}
The modulus 11 of this sum is then calculated; the bleedin' remainder is determined after dividin' the sum by 11:
{\displaystyle {\frac {160}{11}}=14{\mbox{ remainder }}6=14+{\frac {6}{11}}}
If there is no remainder the check digit is 0, otherwise the feckin' remainder value is subtracted from 11 to give the feckin' check digit:
{\displaystyle 11-6=5}
5 is the bleedin' check digit, C. For calculations, an upper case X in the oul' check digit position indicates a check digit of 10 (like an oul' Roman ten).
To confirm the oul' check digit, calculate the feckin' sum of all eight digits of the ISSN multiplied by its position in the oul' number, countin' from the oul' right (if the oul' check digit is X, then add 10 to the bleedin' sum). G'wan now. The modulus 11 of the sum must be 0, bejaysus. There is an online ISSN checker that can validate an ISSN, based on the bleedin' above algorithm.[9]
ISSNs can be encoded in EAN-13 bar codes with a bleedin' 977 "country code" (compare the bleedin' 978 country code ("bookland") for ISBNs), followed by the 7 main digits of the oul' ISSN (the check digit is not included), followed by 2 publisher-defined digits, followed by the feckin' EAN check digit (which need not match the ISSN check digit).[10]
ISSN codes are assigned by a holy network of ISSN National Centres, usually located at national libraries and coordinated by the ISSN International Centre based in Paris. C'mere til I tell ya now. The International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the feckin' French government.
Linkin' ISSN[edit]
ISSN-L is an oul' unique identifier for all versions of the feckin' serial containin' the bleedin' same content across different media. As defined by ISO 3297:2007, the feckin' "linkin' ISSN (ISSN-L)" provides a feckin' mechanism for collocation or linkin' among the different media versions of the same continuin' resource. Right so. The ISSN-L is one of a feckin' serial's existin' ISSNs, so does not change the oul' use or assignment of "ordinary" ISSNs;[11] it is based on the bleedin' ISSN of the feckin' first published medium version of the feckin' publication, begorrah. If the feckin' print and online versions of the publication are published at the oul' same time, the oul' ISSN of the print version is chosen as the feckin' basis of the oul' ISSN-L.
With ISSN-L is possible to designate one single ISSN for all those media versions of the oul' title, the shitehawk. The use of ISSN-L facilitates search, retrieval and delivery across all media versions for services like OpenURL, library catalogues, search engines or knowledge bases.
The International Centre maintains a holy database of all ISSNs assigned worldwide, the oul' ISDS Register (International Serials Data System), otherwise known as the feckin' ISSN Register, the hoor. At the bleedin' end of 2016,[update] the oul' ISSN Register contained records for 1,943,572 items.[12] The Register is not freely available for interrogation on the feckin' web, but is available by subscription. Jesus, Mary and holy Saint Joseph.
WorldCat permits searchin' its catalog by ISSN, by enterin' "issn:" before the code in the oul' query field. Jesus Mother of Chrisht almighty. One can also go directly to an ISSN's record by appendin' it to "https://www.worldcat.org/ISSN/", e.g. https://www.worldcat.org/ISSN/1021-9749, what? This does not query the ISSN Register itself, but rather shows whether any WorldCat library holds an item with the feckin' given ISSN.
ISSN and ISBN codes are similar in concept, where ISBNs are assigned to individual books. Me head is hurtin' with all this raidin'. An ISBN might be assigned for particular issues of a serial, in addition to the oul' ISSN code for the serial as a whole. An ISSN, unlike the bleedin' ISBN code, is an anonymous identifier associated with a serial title, containin' no information as to the publisher or its location. For this reason a bleedin' new ISSN is assigned to a bleedin' serial each time it undergoes an oul' major title change.
Since the feckin' ISSN applies to an entire serial a bleedin' new identifier, other identifiers have been built on top of it to allow references to specific volumes, articles, or other identifiable components (like the oul' table of contents): the bleedin' Publisher Item Identifier (PII) and the feckin' Serial Item and Contribution Identifier (SICI).
Separate ISSNs are needed for serials in different media (except reproduction microforms). Thus, the bleedin' print and electronic media versions of a serial need separate ISSNs,[13] and CD-ROM versions and web versions require different ISSNs. Holy blatherin' Joseph, listen to this. However, the oul' same ISSN can be used for different file formats (e.g, fair play. PDF and HTML) of the same online serial.
This "media-oriented identification" of serials made sense in the feckin' 1970s, fair play. In the bleedin' 1990s and onward, with personal computers, better screens, and the feckin' Web, it makes sense to consider only content, independent of media, game ball! This "content-oriented identification" of serials was a repressed demand durin' a bleedin' decade, but no ISSN update or initiative occurred. Listen up now to this fierce wan. A natural extension for ISSN, the oul' unique-identification of the oul' articles in the serials, was the feckin' main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier (DOI), an ISSN-independent initiative, consolidated in the oul' 2000s.
Only later, in 2007, ISSN-L was defined in the oul' new ISSN standard (ISO 3297:2007) as an "ISSN designated by the bleedin' ISSN Network to enable collocation or versions of a holy continuin' resource linkin' among the feckin' different media".[14]
An ISSN can be encoded as a bleedin' uniform resource name (URN) by prefixin' it with "urn:ISSN:".[15] For example, Rail could be referred to as "urn:ISSN:0953-4563". URN namespaces are case-sensitive, and the bleedin' ISSN namespace is all caps.[16] If the checksum digit is "X" then it is always encoded in uppercase in a feckin' URN.
ISSN is not unique when the bleedin' concept is "a journal is a holy set of contents, generally copyrighted content": the bleedin' same journal (same contents and same copyrights) may have two or more ISSN codes. Stop the lights! A URN needs to point to "unique content" (a "unique journal" as a holy "set of contents" reference).
Example: Nature has an ISSN for print, 0028-0836, and another for the oul' same content on the oul' Web, 1476-4687; only the feckin' oldest (0028-0836) is used as a unique identifier. Sufferin' Jaysus listen to this. As the ISSN is not unique, the bleedin' U.S, like. National Library of Medicine needed to create, prior to 2007, the NLM Unique ID (JID).[17]
ISSN does not offer resolution mechanisms like a digital object identifier (DOI) or a bleedin' URN does, so the DOI is used as a holy URN for articles, with (for historical reasons) no need for an ISSN's existence.
Example: the bleedin' DOI name "10.1038/nature13777" can be represented as an HTTP strin' by https://doi.org/10.1038/nature13777, and is redirected (resolved) to the feckin' current article's page; but there is no ISSN online service, like http://dx.issn.org/, to resolve the bleedin' ISSN of the bleedin' journal (in this sample 1476-4687).
A unique URN for serials simplifies the bleedin' search, recovery and delivery of data for various services includin', in particular, search systems and knowledge databases.[14] ISSN-L (see Linkin' ISSN above) was created to fill this gap.
The two standard categories of media in which serials are most available are print and electronic, so it is. In metadata contexts (e.g., JATS), these may have standard labels.
p-ISSN is an oul' standard label for "Print ISSN", the ISSN for the print media (paper) version of a feckin' serial. Sure this is it. Usually it is the bleedin' "default media" and so the "default ISSN".
e-ISSN (or eISSN) is a bleedin' standard label for "Electronic ISSN", the bleedin' ISSN for the feckin' electronic media (online) version of a feckin' serial.[18]
ROAD: Directory of Open Access Scholarly Resources [it] (est. Sure this is it. 2013), produced by the ISSN International Centre and UNESCO[19]
^ a b . ISSN. Retrieved 3 April 2020.
^ "3". Listen up now to this fierce wan. ISSN Manual (PDF), what? Paris: ISSN International Centre, would ye swally that? January 2015. Whisht now and listen to this wan. pp. 14, 16, 55–58. HTML version available at www.issn.org
^ Thren, Slawek Rozenfeld (January 2001). Whisht now and listen to this wan. "Usin' The ISSN (International Serial Standard Number) as URN (Uniform Resource Names) within an ISSN-URN Namespace", what? tools.ietf.org. {{cite journal}}: CS1 maint: url-status (link)
^ github.com/amsl-project/issn-resolver See p. Would ye believe this shite?ex. $pattern at source code (issn-resolver.php) of GitHub.
^ "Online ISSN Validator". Jesus, Mary and Joseph. Journal Seeker. Bejaysus. Retrieved 9 August 2014.
^ Identification with the bleedin' GTIN 13 barcode, fair play. ISSN International Centre, would ye swally that? Archived from the oul' original on 29 June 2020.
^ "Total number of records in the feckin' ISSN Register" (PDF), would ye swally that? ISSN International Centre. February 2017. Bejaysus this is a quare tale altogether. Retrieved 23 February 2017.
^ "ISSN for Electronic Serials". U.S, like. ISSN Center, Library of Congress. 19 February 2010, would ye believe it? Retrieved 12 July 2014.
^ a b "The ISSN-L for publications on multiple media". Sufferin' Jaysus listen to this. ISSN International Centre. Here's another quare one. Retrieved 12 July 2014.
^ Rozenfeld, Slawek (January 2001). "Usin' The ISSN (International Serial Standard Number) as URN (Uniform Resource Names) within an ISSN-URN Namespace". Whisht now and listen to this wan. IETF Tools. RFC 3044. Retrieved 15 July 2014.
^ Powell, Andy; Johnston, Pete; Campbell, Lorna; Barker, Phil (21 June 2006). "Guidelines for usin' resource identifiers in Dublin Core metadata §4.5 ISSN". Dublin Core Architecture Wiki. Jesus Mother of Chrisht almighty. Archived from the original on 13 May 2012.
^ "MEDLINE/PubMed Data Element (Field) Descriptions". Jaysis. U.S. National Library of Medicine, bedad. 7 May 2014. Would ye swally this in a minute now?Retrieved 19 July 2014.
^ "La nueva Norma ISSN facilita la vida de la comunidad de las publicaciones en serie", A. Would ye believe this shite?Roucolle. Jaykers! "Archived copy". Jaysis. Archived from the original on 10 December 2014. Jaysis. Retrieved 29 October 2014. {{cite web}}: CS1 maint: archived copy as title (link)
^ "Road in a nutshell". Bejaysus this is a quare tale altogether. Road.issn.org. Would ye believe this shite?Archived from the original on 5 September 2017. Jesus, Mary and Joseph. Retrieved 12 September 2017.
Gettin' an ISSN in the bleedin' UK, British Library .
Gettin' an ISSN in France (in French), Bibliothèque nationale de France
Gettin' an ISSN in Germany (in German), Deutsche Nationalbibliothek
Gettin' an ISSN in South Africa, National Library of South Africa, archived from the original on 24 December 2017, retrieved 7 January 2015
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The Summation Process - MATLAB & Simulink - MathWorks í•œêµ
Addition is the most common arithmetic operation a processor performs. When two n-bit numbers are added together, it is always possible to produce a result with n + 1 nonzero digits due to a carry from the leftmost digit.
Suppose you want to sum three numbers. Each of these numbers is represented by an 8-bit word, and each has a different binary-point-only scaling. Additionally, the output is restricted to an 8-bit word with binary-point-only scaling of 2-3.
The summation is shown in the following model for the input values 19.875, 5.4375, and 4.84375.
The sum follows these steps:
Because the biases are matched, the initial value of Qa is trivial:
{Q}_{a}=00000.000.
The first number to be summed (19.875) has a fractional slope that matches the output fractional slope. Furthermore, the binary points and storage types are identical, so the conversion is trivial:
\begin{array}{l}{Q}_{b}=10011.111,\\ {Q}_{Temp}={Q}_{b}.\end{array}
{Q}_{a}={Q}_{a}+{Q}_{Temp}=10011.111.
The second number to be summed (5.4375) has a fractional slope that matches the output fractional slope, so a slope adjustment is not needed. The storage data types also match, but the difference in binary points requires that both the bits and the binary point be shifted one place to the right:
\begin{array}{l}{Q}_{c}=0101.0111,\\ {Q}_{Temp}=convert\left({Q}_{c}\right)\\ {Q}_{Temp}=00101.011.\end{array}
Note that a loss in precision of one bit occurs, with the resulting value of QTemp determined by the rounding mode. For this example, round-to-floor is used. Overflow cannot occur in this case because the bits and binary point are both shifted to the right.
\begin{array}{c}{Q}_{a}={Q}_{a}+{Q}_{Temp}\\ \text{ }10011.111\\ =\frac{+\text{â}00101.011}{\text{â}\text{â}11001.010}\begin{array}{c}\\ =25.250.\end{array}\end{array}
Note that overflow did not occur, but it is possible for this operation.
The third number to be summed (4.84375) has a fractional slope that matches the output fractional slope, so a slope adjustment is not needed. The storage data types also match, but the difference in binary points requires that both the bits and the binary point be shifted two places to the right:
\begin{array}{l}{Q}_{d}=100.11011,\\ {Q}_{Temp}=convert\left({Q}_{d}\right)\\ {Q}_{Temp}=00100.110.\end{array}
Note that a loss in precision of two bit occurs, with the resulting value of QTemp determined by the rounding mode. For this example, round-to-floor is used. Overflow cannot occur in this case because the bits and binary point are both shifted to the right.
\begin{array}{c}{Q}_{a}={Q}_{a}+{Q}_{Temp}\\ \text{ }11001.010\\ =\frac{+\text{â}00100.110}{\text{â}\text{â}11110.000}\begin{array}{c}\\ =30.000.\end{array}\end{array}
As shown here, the result of step 7 differs from the ideal sum:
\begin{array}{c}\text{ }10011.111\\ \text{ 0}101.0111\\ =\frac{+\text{â}100.11011}{\text{â}\text{â}11110.001}\begin{array}{c}\\ =30.125.\end{array}\end{array}
Blocks that perform addition and subtraction include the Add, Gain, and Discrete FIR Filter blocks.
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Perform Cyclic Redundancy Check - MATLAB & Simulink - MathWorks Deutschland
Calculate Check Value by Hand
Calculate Check Value Programmatically
Check Message Integrity
This example shows how to perform a cyclic redundancy check (CRC) on the bits of a number. CRCs are used to detect errors in the transmission of data in digital systems. When a piece of data is sent, a short check value is attached to it. The check value is obtained by polynomial division with the bits in the data. When the data is received, the polynomial division is repeated, and the result is compared with the check value. If the results differ, then the data was corrupted during transmission.
Start with a 16-bit binary number, which is the message to be transmitted:
To obtain the check value, divide this number by the polynomial
{\mathit{x}}^{3}+{\mathit{x}}^{2}+\mathit{x}+1
. You can represent this polynomial with its coefficients: 1111.
The division is performed in steps, and after each step the polynomial divisor is aligned with the left-most 1 in the number. Because the result of dividing by the four term polynomial has three bits (in general dividing by a polynomial of length
\mathit{n}+1
produces a check value of length
\mathit{n}
), append the number with 000 to calculate the remainder. At each step, the result uses the bit-wise XOR of the four bits being operated on, and all other bits are unchanged.
Each successive division operates on the result of the previous step, so the second division is
The division is completed once the dividend is all zeros. The complete division, including the above two steps, is
The remainder bits, 110, are the check value for this message.
In MATLAB®, you can perform this same operation to obtain the check value using bit-wise operations. First, define variables for the message and polynomial divisor. Use unsigned 32-bit integers so that extra bits are available for the remainder.
Next, initialize the polynomial divisor. Use dec2bin to display the bits of the result.
Now, shift the divisor and message so that they have the correct number of bits (16 bits for the message and 3 bits for the remainder).
Perform the division steps of the CRC using a for loop. The for loop always advances a single bit each step, so include a check to see if the current digit is a 1. If the current digit is a 1, then the division step is performed; otherwise, the loop advances a bit and continues.
Shift the bits of the remainder to the right to get the check value for the operation.
You can use the check value to verify the integrity of a message by repeating the same division operation. However, instead of using a remainder of 000 to start, use the check value 110. If the message is error free, then the result of the division will be zero.
Reset the remainder variable, and add the CRC check value to the remainder bits using a bit-wise OR. Introduce an error into the message by flipping one of the bit values with bitset.
Perform the CRC division operation and then check if the result is zero.
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Counting Integers in a Range Practice Problems Online | Brilliant
n
22 \leq n \leq 49
How many integers are greater than or equal to
986
but strictly less than
1031?
n
are there satisfying the inequality
479 < n < 559?
How many positive 2 digit numbers are there, such that the units digit is strictly smaller than the tens digit?
12=012
Han Dynasty is the second imperial dynasty of China. It was established in 206 BCE and survived until 220 CE. How many years are there between its establishment and collapse inclusive of both ends?
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Calculus/Product Rule - Wikibooks, open books for an open world
Calculus/Product Rule
← Differentiation/Differentiation Defined Calculus Quotient Rule →
When we wish to differentiate a more complicated expression such as:
{\displaystyle h(x)=(x^{2}+5x+7)\cdot (x^{3}+2x-4)}
our only way (up to this point) to differentiate the expression is to expand it and get a polynomial, and then differentiate that polynomial. This method becomes very complicated and is particularly error prone when doing calculations by hand. A beginner might guess that the derivative of a product is the product of the derivatives, similar to the sum and difference rules, but this is not true. To take the derivative of a product, we use the product rule.
Derivatives of products (Product rule)
{\displaystyle {\frac {d}{dx}}\left[f(x)\cdot g(x)\right]=f(x)\cdot g'(x)+f'(x)\cdot g(x)\,\!}
It may also be stated as
{\displaystyle (f\cdot g)'=f'\cdot g+f\cdot g'\,\!}
or in the Leibniz notation as
{\displaystyle {\dfrac {d}{dx}}(u\cdot v)=u\cdot {\dfrac {dv}{dx}}+v\cdot {\dfrac {du}{dx}}}
The derivative of the product of three functions is:
{\displaystyle {\dfrac {d}{dx}}(u\cdot v\cdot w)={\dfrac {du}{dx}}\cdot v\cdot w+u\cdot {\dfrac {dv}{dx}}\cdot w+u\cdot v\cdot {\dfrac {dw}{dx}}}
Since the product of two or more functions occurs in many mathematical models of physical phenomena, the product rule has broad application in Physics, Chemistry, and Engineering.
1.1 Physics Example I: Electromagnetic induction
3 Application, proof of the power rule
Suppose one wants to differentiate ƒ(x) = x2 sin(x). By using the product rule, one gets the derivative ƒ '(x) = 2x sin(x) + x2cos(x) (since the derivative of x2 is 2x and the derivative of sin(x) is cos(x)).
One special case of the product rule is the constant multiple rule which states: if c is a real number and ƒ(x) is a differentiable function, then cƒ(x) is also differentiable, and its derivative is (c × ƒ)'(x) = c × ƒ '(x). This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear.
The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable, but only says what its derivative is if it is differentiable.)
Physics Example I: Electromagnetic induction[edit | edit source]
Faraday's law of electromagnetic induction states that the induced electromotive force is the negative time rate of change of magnetic flux through a conducting loop.
{\displaystyle {\mathcal {E}}=-{{d\Phi _{B}} \over dt},}
{\displaystyle {\mathcal {E}}}
is the electromotive force (emf) in volts and ΦB is the magnetic flux in webers. For a loop of area, A, in a magnetic field, B, the magnetic flux is given by
{\displaystyle \Phi _{B}=B\cdot A\cdot \cos(\theta ),}
where θ is the angle between the normal to the current loop and the magnetic field direction.
Taking the negative derivative of the flux with respect to time yields the electromotive force gives
{\displaystyle {\begin{aligned}{\mathcal {E}}&=-{\frac {d}{dt}}\left(B\cdot A\cdot \cos(\theta )\right)\\&=-{\frac {dB}{dt}}\cdot A\cos(\theta )-B\cdot {\frac {dA}{dt}}\cos(\theta )-B\cdot A{\frac {d}{dt}}\cos(\theta )\\\end{aligned}}}
In many cases of practical interest, only one variable (A, B, or θ) is changing so two of the three above terms are often zero.
Proving this rule is relatively straightforward, first let us state the equation for the derivative:
{\displaystyle {\frac {d}{dx}}\left[f(x)\cdot g(x)\right]=\lim _{h\to 0}{\frac {f(x+h)\cdot g(x+h)-f(x)\cdot g(x)}{h}}}
We will then apply one of the oldest tricks in the book—adding a term that cancels itself out to the middle:
{\displaystyle {\frac {d}{dx}}\left[f(x)\cdot g(x)\right]=\lim _{h\to 0}{\frac {f(x+h)\cdot g(x+h)\mathbf {-f(x+h)\cdot g(x)+f(x+h)\cdot g(x)} -f(x)\cdot g(x)}{h}}}
Notice that those terms sum to zero, and so all we have done is add 0 to the equation. Now we can split the equation up into forms that we already know how to solve:
{\displaystyle {\frac {d}{dx}}\left[f(x)\cdot g(x)\right]=\lim _{h\to 0}\left[{\frac {f(x+h)\cdot g(x+h)-f(x+h)\cdot g(x)}{h}}+{\frac {f(x+h)\cdot g(x)-f(x)\cdot g(x)}{h}}\right]}
Looking at this, we see that we can separate the common terms out of the numerators to get:
{\displaystyle {\frac {d}{dx}}\left[f(x)\cdot g(x)\right]=\lim _{h\to 0}\left[f(x+h){\frac {g(x+h)-g(x)}{h}}+g(x){\frac {f(x+h)-f(x)}{h}}\right]}
Which, when we take the limit, becomes:
{\displaystyle {\frac {d}{dx}}\left[f(x)\cdot g(x)\right]=f(x)\cdot g'(x)+g(x)\cdot f'(x)}
, or the mnemonic "one D-two plus two D-one"
This can be extended to 3 functions:
{\displaystyle {\frac {d}{dx}}[fgh]=f(x)g(x)h'(x)+f(x)g'(x)h(x)+f'(x)g(x)h(x)\,}
For any number of functions, the derivative of their product is the sum, for each function, of its derivative times each other function.
Back to our original example of a product,
{\displaystyle h(x)=(x^{2}+5x+7)\cdot (x^{3}+2x-4)}
, we find the derivative by the product rule is
{\displaystyle h'(x)=(x^{2}+5x+7)(3x^{2}+2)+(2x+5)(x^{3}+2x-4)=5x^{4}+20x^{3}+27x^{2}+12x-6\,}
Note, its derivative would not be
{\displaystyle {\color {red}(2x+5)\cdot (3x^{2}+2)=3x^{3}+15x^{2}+4x+10}}
which is what you would get if you assumed the derivative of a product is the product of the derivatives.
To apply the product rule we multiply the first function by the derivative of the second and add to that the derivative of first function multiply by the second function. Sometimes it helps to remember the memorize the phrase "First times the derivative of the second plus the second times the derivative of the first."
Application, proof of the power rule[edit | edit source]
The product rule can be used to give a proof of the power rule for whole numbers. The proof proceeds by mathematical induction. We begin with the base case
{\displaystyle n=1}
{\displaystyle f_{1}(x)=x}
then from the definition is easy to see that
{\displaystyle f_{1}'(x)=\lim _{h\rightarrow 0}{\frac {x+h-x}{h}}=1}
Next we suppose that for fixed value of
{\displaystyle N}
, we know that for
{\displaystyle f_{N}(x)=x^{N}}
{\displaystyle f_{N}'(x)=Nx^{N-1}}
. Consider the derivative of
{\displaystyle f_{N+1}(x)=x^{N+1}}
{\displaystyle f_{N+1}'(x)=(x\cdot x^{N})'=(x)'x^{N}+x\cdot (x^{N})'=x^{N}+x\cdot N\cdot x^{N-1}=(N+1)x^{N}.}
We have shown that the statement
{\displaystyle f_{n}'(x)=n\cdot x^{n-1}}
{\displaystyle n=1}
and that if this statement holds for
{\displaystyle n=N}
, then it also holds for
{\displaystyle n=N+1}
. Thus by the principle of mathematical induction, the statement must hold for
{\displaystyle n=1,2,\dots }
http://www.calculusapplets.com/prodquot.html
Retrieved from "https://en.wikibooks.org/w/index.php?title=Calculus/Product_Rule&oldid=3602205"
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Compression-ignition controller that includes air mass flow, torque, and EGR estimation - Simulink - MathWorks France
P{w}_{inj}
R{P}_{cmd}={f}_{RPcmd}\left(Tr{q}_{cmd},N\right)
EG{R}_{cmd}={f}_{EGRcmd}\left(Tr{q}_{cmd},N\right)
P{w}_{inj}=\frac{{F}_{cmd, tot}}{{S}_{inj}}
P{w}_{inj}
{F}_{cmd,tot}={f}_{Fcmd,tot}\left(Tr{q}_{cmd},N\right)
MAINSOI=f\left({F}_{cmd,tot},N\right)
{C}_{idle}\left(z\right)={K}_{p,idle}+{K}_{i,idle}\frac{{t}_{s}}{z-1}
{\stackrel{˙}{m}}_{egr,est}={\stackrel{˙}{m}}_{egr,std}\frac{{P}_{exh,est}}{{P}_{std}}\sqrt{\frac{{T}_{std}}{{T}_{exh,est}}}
{\stackrel{˙}{m}}_{egr,std}=f\left(\frac{MAP}{{P}_{exh,est}},EGRap\right)
{\stackrel{˙}{m}}_{egr,std}
{\stackrel{˙}{m}}_{egr,est}
{\stackrel{˙}{m}}_{egr,std}
{P}_{std}
{T}_{std}
{P}_{Amb}
{P}_{exh,est}={P}_{Amb}P{r}_{turbo}
\begin{array}{l}P{r}_{turbo}=f\left({\stackrel{˙}{m}}_{airstd},{N}_{vgtcorr}\right)f\left(VG{T}_{pos}\right)\\ \text{where:}\\ {N}_{vgtcorr}=\frac{{N}_{vgt}}{\sqrt{{T}_{exh,est}}}\end{array}
{\stackrel{˙}{m}}_{egr,est}
{\stackrel{˙}{m}}_{egr,std}
{\stackrel{˙}{m}}_{port,est}
{\stackrel{˙}{m}}_{airstd}
{P}_{std}
{T}_{std}
{P}_{Amb}
P{r}_{turbo}=f\left({\stackrel{˙}{m}}_{airstd},{N}_{vgtcorr}\right)
{\stackrel{˙}{m}}_{airstd}
{\stackrel{˙}{m}}_{airstd}=\left({\stackrel{˙}{m}}_{port,est}-{\stackrel{˙}{m}}_{egr,est}\right)\frac{{P}_{std}}{MAP}\sqrt{\frac{MAT}{{T}_{std}}}
{T}_{exh}={f}_{Texh}\left(F,N\right)
\begin{array}{l}{T}_{exhnom}=SO{I}_{exhteff}MA{P}_{exhteff}MA{T}_{exhteff}O2{p}_{exhteff}FUEL{P}_{exhteff}Tex{h}_{opt}\\ {T}_{exh}={T}_{exhnom}+\Delta {T}_{post}\\ \\ SO{I}_{exhteff}={f}_{SO{I}_{exhteff}}\left(\Delta SOI,N\right)\\ MA{P}_{exhteff}={f}_{MA{P}_{exhteff}}\left(MA{P}_{ratio},\lambda \right)\\ MA{T}_{exhteff}={f}_{MA{T}_{exhteff}}\left(\Delta MAT,N\right)\\ O2{p}_{exhteff}={f}_{O2{p}_{exhteff}}\left(\Delta O2p,N\right)\\ Tex{h}_{opt}={f}_{Texh}\left(F,N\right)\end{array}
{\stackrel{˙}{m}}_{fuel,cmd}=\frac{N{S}_{inj}P{w}_{inj}{N}_{cyl}}{Cps\left(\frac{60s}{min}\right)\left(\frac{1000mg}{g}\right)}
AF{R}_{est}=\frac{{\stackrel{˙}{m}}_{port,est}}{{\stackrel{˙}{m}}_{fuel,cmd}}
P{w}_{inj}
{\stackrel{˙}{m}}_{fuel,cmd}
{S}_{inj}
{\stackrel{˙}{m}}_{port,est}
{P}_{Amb}
P{w}_{inj}
{\stackrel{˙}{m}}_{fuel,cmd}
{\stackrel{˙}{m}}_{port,est}
P{w}_{inj}
EG{R}_{cmd}={f}_{EGRcmd}\left(Tr{q}_{cmd},N\right)
R{P}_{cmd}={f}_{RPcmd}\left(Tr{q}_{cmd},N\right)
{S}_{inj}
{F}_{cmd,tot}={f}_{Fcmd,tot}\left(Tr{q}_{cmd},N\right)
MAINSOI=f\left({F}_{cmd,tot},N\right)
{N}_{cyl}
Cps
{V}_{d}
{R}_{air}
{P}_{std}
{T}_{std}
{\eta }_{v}={f}_{{\eta }_{v}}\left(MAP,N\right)
{\eta }_{v}
{\stackrel{˙}{m}}_{egr,std}=f\left(\frac{MAP}{{P}_{exh,est}},EGRap\right)
{\stackrel{˙}{m}}_{egr,std}
P{r}_{turbo}=f\left({\stackrel{˙}{m}}_{airstd},{N}_{vgtcorr}\right)
{\stackrel{˙}{m}}_{airstd}
{T}_{brake}={f}_{Tnf}\left(F,N\right)
C{p}_{exh}
{T}_{exh}={f}_{Texh}\left(F,N\right)
{T}_{exh}
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Truncated_square_tiling Knowpia
tr{4,4} or
{\displaystyle t{\begin{Bmatrix}4\\4\end{Bmatrix}}}
Bowers acronym Tosquat
Dual Tetrakis square tiling
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}.
Conway calls it a truncated quadrille, constructed as a truncation operation applied to a square tiling (quadrille).
Other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges.
There are two distinct uniform colorings of a truncated square tiling. (Naming the colors by indices around a vertex (4.8.8): 122, 123.)
2 colors: 122
The truncated square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing number).[1]
The squares from the truncation can be alternately sized. In the limit, half of the vertices can remain untruncated, leading to a chamfered square tiling.
A skew equilateral form with squares into rhombi, and flattened octagons.
One variations on this pattern, often called a Mediterranean pattern, is shown in stone tiles with smaller squares and diagonally aligned with the borders. Other variations stretch the squares or octagons.
The Pythagorean tiling alternates large and small squares, and may be seen as topologically identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices.
A weaving pattern also has the same topology, with octagons flattened rectangles.
Rectangular/rhombic
The truncated square tiling is used in an optical illusion with truncated vertices divides and colored alternately, seeming to twist the grid.
The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:
The 3-dimensional bitruncated cubic honeycomb projected into the plane shows two copies of a truncated tiling. In the plane it can be represented by a compound tiling, or combined can be seen as a chamfered square tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms: square tiling, truncated square tiling, snub square tiling.
Related tilings in other symmetriesEdit
Tetrakis square tilingEdit
The tetrakis square tiling
The tetrakis square tiling is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2.
Conway calls it a kisquadrille,[2] represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille). It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.[3]
Wikimedia Commons has media related to Uniform tiling 4-8-8 (truncated square tiling).
^ Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern H
^ Stephenson, John (1970), "Ising Model with Antiferromagnetic Next-Nearest-Neighbor Coupling: Spin Correlations and Disorder Points", Phys. Rev. B, 1 (11): 4405–4409, doi:10.1103/PhysRevB.1.4405 .
Grünbaum, Branko & Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65)
Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56
Klitzing, Richard. "2D Euclidean tilings o4x4x - tosquat - O6".
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\mathbf{F}=u\left(x,y,z\right) \mathbf{i}+u\left(x,y,z\right) \mathbf{j}+w\left(x,y,z\right) \mathbf{k}
u,v,w
∇×\mathbf{F}
, the curl of F:
∇×\mathbf{F}
\left[\begin{array}{c}\frac{∂}{∂y}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}w\left(x,y,z\right)-\frac{∂}{∂z}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}v\left(x,y,z\right)\\ \frac{∂}{∂z}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}u\left(x,y,z\right)-\frac{∂}{∂x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}w\left(x,y,z\right)\\ \frac{∂}{∂x}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}v\left(x,y,z\right)-\frac{∂}{∂y}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}u\left(x,y,z\right)\end{array}\right]
Compute the divergence of the curl:
{\left({w}_{y}-{v}_{z}\right)}_{x}+{\left({u}_{z}-{w}_{x}\right)}_{y}+{\left({v}_{x}-{u}_{y}\right)}_{z}={w}_{\mathrm{yx}}-{v}_{\mathrm{zx}}+{u}_{\mathrm{zy}}-{w}_{\mathrm{xy}}+{v}_{\mathrm{xz}}-{u}_{\mathrm{yz}}
The expression on the right vanishes because of the equality of the mixed partial derivatives, guaranteed, for example, by continuity of the second partial derivatives.
Define the Cartesian vector field F
〈u\left(x,y,z\right),v\left(x,y,z\right),w\left(x,y,z\right)〉
\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{u}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\\ \textcolor[rgb]{0,0,1}{v}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\\ \textcolor[rgb]{0,0,1}{w}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\end{array}\right]
\stackrel{\text{to Vector Field}}{\to }
\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{u}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\\ \textcolor[rgb]{0,0,1}{v}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\\ \textcolor[rgb]{0,0,1}{w}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\end{array}\right]
\stackrel{\text{assign to a name}}{\to }
\textcolor[rgb]{0,0,1}{F}
Del, dot product, and cross product operators
∇·\left(∇×\mathbf{F}\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{BasisFormat}\left(\mathrm{false}\right):
\mathrm{PDEtools}:-\mathrm{declare}\left(u\left(x,y,z\right),v\left(x,y,z\right),w\left(x,y,z\right),\mathrm{quiet}\right)
\mathbf{F}≔\mathrm{VectorField}\left(〈u\left(x,y,z\right),v\left(x,y,z\right),w\left(x,y,z\right)〉\right)
\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{u}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\\ \textcolor[rgb]{0,0,1}{v}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\\ \textcolor[rgb]{0,0,1}{w}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\right)\end{array}\right]
∇·\left(∇×\mathbf{F}\right)=0
\mathrm{Divergence}\left(\mathrm{Curl}\left(\mathbf{F}\right)\right)
\textcolor[rgb]{0,0,1}{0}
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Perform transformation from αβ0 stationary reference frame to dq0 rotating reference frame or the inverse - Simulink - MathWorks Nordic
Alpha-Beta-Zero to dq0, dq0 to Alpha-Beta-Zero
Perform transformation from αβ0 stationary reference frame to dq0 rotating reference frame or the inverse
The Alpha-Beta-Zero to dq0 block performs a transformation of αβ0 Clarke components in a fixed reference frame to dq0 Park components in a rotating reference frame.
The dq0 to Alpha-Beta-Zero block performs a transformation of dq0 Park components in a rotating reference frame to αβ0 Clarke components in a fixed reference frame.
The block supports the two conventions used in the literature for Park transformation:
Rotating frame aligned with A axis at t = 0. This type of Park transformation is also known as the cosine-based Park transformation.
Rotating frame aligned 90 degrees behind A axis. This type of Park transformation is also known as the sine-based Park transformation. Use it in Simscape™ Electrical™ Specialized Power Systems models of three-phase synchronous and asynchronous machines.
Knowing that the position of the rotating frame is given by ω.t (where ω represents the frame rotation speed), the αβ0 to dq0 transformation performs a −(ω.t) rotation on the space vector Us = uα + j· uβ. The homopolar or zero-sequence component remains unchanged.
Depending on the frame alignment at t = 0, the dq0 components are deduced from αβ0 components as follows:
When the rotating frame is aligned with A axis, the following relations are obtained:
\begin{array}{l}{U}_{s}={u}_{d}+jâ
{u}_{q}=\left({u}_{a}+jâ
{u}_{\mathrm{β}}\right)â
{e}^{âj\mathrm{Ï}t}\\ \left[\begin{array}{c}{u}_{d}\\ {u}_{q}\\ {u}_{0}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}\left(\mathrm{Ï}t\right)& \mathrm{sin}\left(\mathrm{Ï}t\right)& 0\\ â\mathrm{sin}\left(\mathrm{Ï}t\right)& \mathrm{cos}\left(\mathrm{Ï}t\right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{\mathrm{β}}\\ {u}_{0}\end{array}\right]\end{array}
\begin{array}{l}{u}_{\mathrm{α}}+jâ
{u}_{\mathrm{β}}=\left({u}_{d}+jâ
{u}_{q}\right)â
{e}^{j\mathrm{Ï}t}\\ \left[\begin{array}{c}{u}_{\mathrm{α}}\\ {u}_{\mathrm{β}}\\ {u}_{0}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{cos}\left(\mathrm{Ï}t\right)& â\mathrm{sin}\left(\mathrm{Ï}t\right)& 0\\ \mathrm{sin}\left(\mathrm{Ï}t\right)& \mathrm{cos}\left(\mathrm{Ï}t\right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_{d}\\ uq\\ {u}_{0}\end{array}\right]\end{array}
When the rotating frame is aligned 90 degrees behind A axis, the following relations are obtained:
\begin{array}{l}{U}_{s}={u}_{d}+jâ
{u}_{q}=\left({u}_{\mathrm{α}}+jâ
{u}_{\mathrm{β}}\right)â
{e}^{âj\left(\mathrm{Ï}tâ\frac{\mathrm{Ï}}{2}\right)}\\ \left[\begin{array}{c}{u}_{d}\\ {u}_{q}\\ {u}_{0}\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\mathrm{sin}\left(\mathrm{Ï}t\right)& \mathrm{sin}\left(\mathrm{Ï}tâ\frac{2\mathrm{Ï}}{3}\right)& \mathrm{sin}\left(\mathrm{Ï}t+\frac{2\mathrm{Ï}}{3}\right)\\ \mathrm{cos}\left(\mathrm{Ï}t\right)& \mathrm{cos}\left(\mathrm{Ï}tâ\frac{2\mathrm{Ï}}{3}\right)& \mathrm{cos}\left(\mathrm{Ï}t+\frac{2\mathrm{Ï}}{3}\right)\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}{u}_{a}\\ {u}_{b}\\ {u}_{c}\end{array}\right]\end{array}
{u}_{\mathrm{α}}+jâ
{u}_{\mathrm{β}}=\left({u}_{d}+jâ
{u}_{q}\right)â
{e}^{j\left(\mathrm{Ï}tâ\frac{\mathrm{Ï}}{2}\right)}
The abc-to-Alpha-Beta-Zero transformation applied to a set of balanced three-phase sinusoidal quantities ua, ub, uc produces a space vector Us whose uα and uβ coordinates in a fixed reference frame vary sinusoidally with time. In contrast, the abc-to-dq0 transformation (Park transformation) applied to a set of balanced three-phase sinusoidal quantities ua, ub, uc produces a space vector Us whose ud and uq coordinates in a dq rotating reference frame stay constant.
Rotating frame alignment (at wt=0)
Select the alignment of rotating frame, when wt = 0, of the dq0 components of a three-phase balanced signal:
{u}_{a}=\mathrm{sin}\left(\mathrm{Ï}t\right);\text{ }{u}_{b}=\mathrm{sin}\left(\mathrm{Ï}tâ\frac{2\mathrm{Ï}}{3}\right);\text{ }{u}_{c}=\mathrm{sin}\left(\mathrm{Ï}t+\frac{2\mathrm{Ï}}{3}\right)
(positive-sequence magnitude = 1.0 pu; phase angle = 0 degree)
When you select Aligned with phase A axis, the dq0 components are d = 0, q = −1, and zero = 0.
When you select 90 degrees behind phase A axis, the default option, the dq0 components are d = 1, q = 0, and zero = 0.
The vectorized αβ0 signal.
The vectorized dq0 signal.
The angular position, in radians, of the dq rotating frame relative to the stationary frame.
The power_Transformations example shows various uses of blocks performing Clarke and Park transformations.
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Multiply and add using fused multiply add approach - MATLAB fma - MathWorks América Latina
Multiply and Add Three Inputs Using Fused Multiply Add
Multiply and add using fused multiply add approach
X = fma(A, B, C) computes A.*B+C using a fused multiply add approach. Fused multiply add operations round only once, often making the result more accurate than performing a multiplication operation followed by an addition.
This example shows how to use the fma function to calculate
A×B+C
using a fused multiply add approach.
Define the inputs and use the fma function to compute the multiply add operation.
Compare the result of the fma function with the two-step approach of computing the product and then the sum.
Input array, specified as a floating-point scalar, vector, matrix, or multidimensional array. When A and B are matrices, fma performs element-wise multiplication followed by addition.
Data Types: single | double | half
Input array, specified as a floating-point scalar, vector, matrix, or multidimensional array.
X — Result of multiply and add operation
Result of multiply and add operation, A.*B+C, returned as a scalar, vector, matrix, or multidimensional array.
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Programmer Indexing of Arrays, Matrices and Vectors
Operations on pairs of Sparse or Dense arrays
Round brackets can be used to index into Array, Matrix, and Vector data structures. For full details refer to ?rtable_indexing. Highlights include:
the ability to grow arrays past their initially declared bounds.
the ability to reference multi-dimensional arrays with a single integer, making looping over all elements easier and more efficient.
compatibility with the way MATLAB® indexes arrays.
The zip command now accepts a prefix option: zip[sparse], and zip[dense] indicating whether the zeros in the sparse objects should be ignored or included.
zip[sparse]((x,y)->x+y+1,Array(1..3,{1=1},storage=sparse),
Array(1..4,{2=2},storage=sparse));
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{0}\end{array}]
zip[dense]((x,y)->x+y+1,Array(1..3,{1=1},storage=sparse),
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}\end{array}]
The max and min commands now accept lists and rtable-based arrays as arguments. The list or array is scanned to find the maximum or minimum object contained inside.
max( [-5,1,17], <12,-32,4> );
\textcolor[rgb]{0,0,1}{17}
min( <12,-32,4> );
\textcolor[rgb]{0,0,1}{-32}
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Given the piecewise-defined function below, what value of
a
will make this function continuous? .
f ( x ) = \left\{ \begin{array} { l l } { \sqrt { 1 - x } + 7 } & { \text { for } x > - 3 } \\ { - a x ^ { 3 } } & { \text { for } x \leq - 3 } \end{array} \right.
The Three Conditions of Continuity
1. \lim \limits_{x \rightarrow a} f(x) \text{ exists} \\ 2. f(a) \text{ exists}\\ 3. \lim \limits_{x \rightarrow a} f(x) =f(a)
\sqrt{1-(-3)}+7=-a(-3)^{3}
a
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IX-6315 "Dawn" Electric Propulsion System - Kerbal Space Program Wiki
(Redirected from Ion engine)
Ion engine by
Research Ion Propulsion
Part configuration ionEngine.cfg
(vacuum) 4200 s
Fuel consumption 0.486 /s
Electricity required 367.098 ⚡/s
8.740 ⚡/s
On the launchpad No
In the atmosphere No
The IX-6315 "Dawn" Electric Propulsion System is a tiny-sized engine which runs on electricity and xenon gas propellant. It's modeled after the real-world Hall effect thruster.
1.1 Electrical vs Xenon consumption
This engine has a phenomenal fuel efficiency (4200 s Isp), but very low thrust and requires a substantial amount of electricity to operate. Xenon gas is provided by xenon containers like the PB-X50R Xenon Container, PB-X150 Xenon Container or PB-X750 Xenon Container. Electricity can be obtained using solar panels, radioisotope batteries (RTGs), and fuel cells. Solar panels are recommended in this case, however RTGs are very efficient when traveling to far distances such as Jool and Eeloo. The amount of electricity needed to keep one ion engine running at full thrust is roughly equivalent to half the output of one Gigantor XL Solar Array (however one array will power two engines only near peak sun exposure around Kerbin), 6 Fuel Cells, 12 PB-NUK Radioisotope Thermoelectric Generators, or 25 OX-STAT Photovoltaic Panels at peak output. When pairing with solar panels, it is highly recommended to bring more than it needs (slightly less than 9 Ec/s). Not all panels are at peak output during operations, and maximum available power falls of with the square of the distance from the star.
Batteries can be used to store the electricity since there may be times the solar panels will be blocked from the Sun by objects or the dark side of celestial bodies. For extended burns in the darkness the fuel cells happen to be a good choice. When the power is provided by the fuel cells the majority of the mass flow (about 69.2%) is liquid fuel and oxidizer used by the fuel cell. Thus the ion engine powered by the fuel cell may be seen as having much more modest but still impressive effective Isp of 1293 sec. However if the burn doesn't take more than a couple of hours the stack of RTGs providing the same amount of power (and thus thrust) tend to be heavier than the fuel cell array and its fuel tank. The RTGs should be reserved for very long low-thrust burns in the deep space.
The ion engine is good for fine tuning of orbits. It was also a popular propulsion method for planes on planets on which jet engines don't work, though its current thrust falloff reduces the value there. Due to its great fuel efficiency it is also well-suited for interplanetary travel, but maneuvers tend to take a long time to complete due to its very low thrust-to-weight ratio -- it is advised to use it for very small craft, and to use physics-warp while propelling with it. Usually the engine is used on long range craft due to its high efficiency. But when less delta-v is required, overall, it can be easily surpassed by smaller liquid fueled engines such as the 48-7S "Spark" engine with a lot better TWR.
It is incredibly difficult to build an ion-rocket which can defeat gravity on Kerbin, because the engine isn't even strong enough to lift itself against gravity, let alone itself and its fuel, a battery and a probe core. But when on a low-gravity moon like Minmus or Gilly it is possible to land, start, enter orbit and reach escape velocity with ion-propulsion alone. Since 0.23.5, it is technically possible to create an ion-powered probe, albeit with a minimum of parts, which will be able to defy Mun gravity. With that in mind, it is also possible to resist gravity with lightweight Ion craft on Duna, Moho, Dres, Eeloo, and every in-game moon with two exceptions: Laythe and Tylo have too high gravity for a single ion thruster, xenon tank, probe core, and battery.
While it is possible to build an airplane powered solely by this engine, the ion engine's efficiency is awful in the atmosphere. Unless you are going very far from the KSC, jet planes are much cheaper and more efficient.
Ion-powered "ferries" may also be useful for moving fuel, oxidizer and/or kerbonauts between two larger vessels, by keeping the large craft at such range that only one of them is within draw distance from the "ferry" at any moment, performance loss can be avoided. It is generally more fuel-efficient to move fuel and oxidizer between two ships using a ferry than it would be to dock the larger craft together using their own engines and RCS.
Because it uses only about 0.485 units of xenon per second, one PB-X50R Xenon Container with 400 units of xenon can supply the engine for almost 14 minutes. The other larger tank PB-X150 Xenon Container with 700 units of xenon has enough to supply the engine more than 24 minutes.
Electrical vs Xenon consumption
While the consumption ratios listed in the part.cfg files are normally relative mass flows (1.8 electricity and 0.1 xenon here), this breaks down somewhat with massless resources like electricity. Rather, the entire mass flow goes to the xenon, with the relative ratio (1.8 to 0.1, or 18 in total) creating a seemingly disproportionate drain.
Calculating the mass and unit flow from specific impulse and thrust:
{\displaystyle I_{sp}={\frac {F}{{\dot {m}}\cdot g_{0}}}\Rightarrow {\frac {F}{I_{sp}\cdot g_{0}}}={\dot {m}}\Rightarrow {\frac {2000N}{4200s\cdot 9.80665{\frac {m}{s}}}}=0.04856{\frac {kg}{s}}}
{\displaystyle Xeflow={\frac {\dot {m}}{\rho }}={\frac {0.04856{\frac {kg}{s}}}{0.1{\frac {kg}{l}}}}=0.4856{\frac {l}{s}}}
{\displaystyle Electricityflow=Xe*ratio=0.4856*18=8.740zaps/s}
“ By emitting ionized xenon gas through a small thruster port, Dawn can produce incredibly efficient propulsion, but with a downside of very low thrust and high energy usage. According to ISP Electronics sales reps, the rumours of this engine being powered by "dark magic" are largely exaggerated.
“ By emitting ionized xenon gas through a small thruster port, the PB-ION can produce incredibly efficient propulsion, but with a downside of very low thrust and high expense. The general perception of this engine as being powered by "witchcraft" has unfortunately given it a sour reputation.
— Ionic Protonic Electronics (0.18-0.23)
Although the 2 kN IX-6315 "Dawn" Electric Propulsion System is considered to have very low thrust in KSP, real-life Hall effect thrusters typically have orders of magnitude less thrust usually below 1 N (0.001 kN). They make up for this by having a service life of thousands of hours of continuous operation and consuming fuel extremely slowly, which would have been impractical in the game.
The name "Dawn" is a reference to NASA's Dawn spacecraft, which is NASA's first interplanetary space probe to use an ion engine for propulsion.
A Solar Glider Ready for take-off. This ship uses ion engines.
The Solar glider in action.
The Ion-Powered Space Probe stock craft, powered by an ion engine.
Moved from Utility to Engines
Moved to Propulsion from Utility.
Ionic Protonic Electronics renamed Ionic Symphonic Protonic Electronics, description changed, thrust increased from 0.5 to 2, relative electricity reduced from 12 to 1.8, description changed.
Retrieved from "https://wiki.kerbalspaceprogram.com/index.php?title=IX-6315_%22Dawn%22_Electric_Propulsion_System&oldid=100091"
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softwaremill/scala-clippy - Gitter
adamw commented #64
agilesteel commented #64
lrytz opened #65
bjchambers opened #64
kciesielski on 0.6.1
kciesielski on master
Update sbt to 0.13.7 Add a predefined constant for n… (compare)
nonsleepr opened #63
nonsleepr commented #49
Fix fatal warnings example (compare)
nafg commented #30
Adjust tests (compare)
kciesielski on v0.6.0
Support fatal warnings Update version to 0.6.0 Allow defining fatal warnings w… and 1 more (compare)
acruise opened #62
quasi-coherent commented #57
michelemauro commented #57
@kciesielski
@ShaneDelmore I merged your PR, thanks a lot for this contribution.
Could you check whether 0.4.2-SNAPSHOT fixes the issue with requiring re-creating of toolbox?
I checked yesterday and it didn't build due to the SSL errors. Have you published a new snapshot on top of my changes?
@ShaneDelmore published just now
@kciesielski Success!
I don't think the colors work as intended here:
Highlighting error messages correctly is tricky. I worked on error messages in Dotty a bit and Felix Mulder put together a really nice framework for highlighting messages there. Depending on interest we might try to backport that to clippy.
I found this issue related to Path, unless there is objection I will just make a pr and file it away under this issue. http4s/http4s#559
@ShaneDelmore Sounds very interesting with the dotty highlighter. Are you sure the http4s path issue relates to Clippy?
@ochrons Are you using windows? Because this looks like softwaremill/scala-clippy#40
We are going to add support for custom Ansi RESET color, so windows users can set it to light-grey or whatever. For some reason it doesn't work well on Windows (when used with Li Haoyi's fansi lib)
Win10 and mingw bash under ConEmu terminal
@ochrons you can use clippy 0.5.1-SNAPSHOT as a workaround for now (described in issue). We'll soon add the mentioned option as a proper fix.
ok, will try at some point :)
@kciesielski User error, that was meant for the https room
@kciesielski not at all related to clippy.
So, is there a way I can conditionally disable clippy for certain projects via my ~/…/global.sbt? Like if (sbt.project == "scala") scalacOptions += "-P:clippy:enable=false"?
I'm getting the following error again. Anyone else? Information:scalac: Unable to load/store local Clippy advice due to: sun.security.validator.ValidatorException: PKIX path building failed: sun.security.provider.certpath.SunCertPathBuilderException: unable to find valid certification path to requested target
The error cleared up. I have no idea why. Weird.
prateekmane99
@prateekmane99
I have added addCompilerPlugin("com.softwaremill.clippy" %% "plugin" % "0.5.2" classifier "bundle") in my sbt
I cant see Clippy errors in my console
What error are you using to test?
Does clippy replace scalac errors or just adds information?
@soronpo only adds information. The original error is always there
What do you think of adding an optional flag that sets if we want the original error message to replaced?
@soronpo well, what would be the use-case? :)
I'm creating a DSL and
some of its users will be less confused that way
well there are mechanisms like @implicitNotFound for customizing error messages. I'd be cautious in replacing the original error message entirely, I suppose we would need to have a good motivating example to include that
@adamw On one hand, @implicitNotFound is very limited, and on the other hand, the missing implicit messages for my library are too verbose. This is not a must feature, but if a library owner has an optional flag for that, I do not see the harm.
Crucial feature that I am currently missing is the ability to grab text from the original message and place it in the new/hint message.
I’ve started seeing this
21:06:25.829 DEBUG akka://ENSIME/user/ensime-main/tcp-server/con1/3 o.e.s.RequestHandler - received handled message SymbolDesignations(RawFile(.),List()) from Actor[akka://ENSIME/user/ensime-main/project/scalac#1727592554] in state receive
java.lang.ClassCastException: com.softwaremill.clippy.InjectReporter
anon
1 cannot be cast to scala.tools.nsc.Global$GlobalPhase
at scala.tools.nsc.Global$Run.$anonfun$compileLate$3(Global.scala:1519)
at scala.tools.nsc.Global$Run.$anonfun$compileLate$2$adapted(Global.scala:1518)
at scala.tools.nsc.Global$Run$$Lambda$1471/168254663.apply(Unknown Source)
at scala.tools.nsc.Global$Run.compileLate(Global.scala:1518)
at scala.tools.nsc.interactive.Global.parseAndEnter(Global.scala:645)
at scala.tools.nsc.interactive.Global.$anonfun$backgroundCompile$4(Global.scala:554)
at scala.tools.nsc.interactive.Global.backgroundCompile(Global.scala:551)
at scala.tools.nsc.interactive.PresentationCompilerThread.run(PresentationCompilerThread.scala:25)
interaction between clippy (or so it appear) and ensime. Commenting out clippy (which I’ve been loading via my global sbt plugins) resolves the issue but at quite a loss.
Does the issue happen with 0.5.2?
I don't use ensime but added a feature in 0.5.3 and am worried there might be a conflict.
Although I don't know anything about how the reporter is injected, that was another contributor who worked on that but I do remember it changing a few releases back when the nice coloring was added.
so that answer to that is yes, it fails with 0.5.2 in the same manner
Hmm...no idea then. Just for troubleshooting purposes you could try loading an old pre-new reporter version, say 0.3.0 to see if that still has the issue. But it sounds related to the new reporter and I have no familiarity with that area but I may be able to help in a few days.
Either way, if it hurts I say it's worth opening an issue.
so I had intended to start this conversation in emacs-ensime rather than the cats gitter. I’m guessing it’s something in ensime that changes since it didn’t properly corelate with an clippy update
I’m going to drill into ensime first and if don’t make any progress I’ll come back to clippy
@crispywalrus Sounds good, but if we need to work together with ensime to get it sorted let us know. I have a knee surgery tomorrow so I will be offline for a few days but if you don't hear back for a few know that I will get back to you.
well good luck with that. I’d like to get this working again as I find value in both tools
@crispywalrus we had a bit similar issue with SBT, as SBT was assuming a specific implementation of the reporter (which we have a custom implementation of). So maybe here the presentation compiler does the same - although I'm not using Ensime, so hard to say. If you won't find anything, please file a bug report :)
I just stumbled upon the ensime issue yesterday
I'm using ensime SNAPSHOT so maybe it's something new :)
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Category:Van der Waals functionals - Vaspwiki
Category:Van der Waals functionals
The semilocal and hybrid functionals do not include the London dispersion forces, therefore they can not be applied reliably on systems where the London dispersion forces play an important role. To account more properly of the London dispersion forces in DFT, a correlation dispersion term can be added to the semilocal or hybrid functional. This leads to the so-called van der Waals functionals:
{\displaystyle E_{\text{xc}}=E_{\text{xc}}^{\text{SL/hybrid}}+E_{\text{c,disp}}.}
There are essentially two types of dispersion terms
{\displaystyle E_{\text{c,disp}}}
that have been proposed in the literature. The first type consists of a sum over the atom pairs
{\displaystyle A}
{\displaystyle B}
{\displaystyle E_{\text{c,disp}}=-\sum _{A<B}\sum _{n=6,8,10,\ldots }f_{n}^{\text{damp}}(R_{AB}){\frac {C_{n}^{AB}}{R_{AB}^{n}}},}
{\displaystyle C_{n}^{AB}}
are the dispersion coefficients,
{\displaystyle R_{AB}}
is the distance between atoms
{\displaystyle A}
{\displaystyle B}
{\displaystyle f_{n}^{\text{damp}}}
is a damping function. Many variants of such atom-pair corrections exist and the most popular of them are available in VASP (see list below).
The other type of dispersion correction is of the following type:
{\displaystyle E_{\text{c,disp}}={\frac {1}{2}}\int \int n({\textbf {r}})\Phi \left({\textbf {r}},{\textbf {r}}'\right)n({\textbf {r}}')d^{3}rd^{3}r',}
which requires a double spatial integration and is therefore of the nonlocal type. The kernel
{\displaystyle \Phi }
depends on the electron density
{\displaystyle n}
{\displaystyle \nabla n}
as well as on
{\displaystyle \left\vert {\bf {{r}-{\bf {{r}'}}}}\right\vert }
. The nonlocal functionals are more expensive to calculate than semilocal functionals, however they are efficiently implemented by using FFTs [1].
More details on the various van der Waals types methods available in VASP and how to use them can be found at the pages listed below.
Atom-pairwise methods for van der Waals interactions: (selected with the IVDW tag):
↑ G. Román-Pérez and J. M. Soler, Phys. Rev. Lett. 103, 096102 (2009).
Pages in category "Van der Waals functionals"
Retrieved from "https://www.vasp.at/wiki/index.php?title=Category:Van_der_Waals_functionals&oldid=17467"
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f\left(x,y,z\right)
, the gradient of
f\left(x,y,z\right)
∇f=\left[\begin{array}{c}{f}_{x}\\ {f}_{y}\\ {f}_{z}\end{array}\right]
∇×\left(∇f\right)=|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ {∂}_{x}& {∂}_{y}& {∂}_{z}\\ {f}_{x}& {f}_{y}& {f}_{z}\end{array}|
\left[\begin{array}{c}{∂}_{y}({f}_{z})-{∂}_{z}({f}_{y})\\ {∂}_{z}({f}_{x})-{∂}_{x}({f}_{z})\\ {∂}_{x}({f}_{y})-{∂}_{y}({f}_{x})\end{array}\right]
\left[\begin{array}{c}{f}_{\mathrm{zy}}-{f}_{\mathrm{yz}}\\ {f}_{\mathrm{xz}}-{f}_{\mathrm{zx}}\\ {f}_{\mathrm{yx}}-{f}_{\mathrm{xy}}\end{array}\right]
\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]
The curl vanishes because of the equality of the mixed partial derivatives, guaranteed, for example, by continuity of the second partial derivatives.
Obtain the curl of the gradient of
∇×\left(∇f\left(x,y,z\right)\right)
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\end{array}\right]
\mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{BasisFormat}\left(\mathrm{false}\right):
Obtain the curl of the gradient of
\mathrm{Curl}\left(\mathrm{Gradient}\left(f\left(x,y,z\right)\right)\right)
\left[\begin{array}{r}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\end{array}\right]
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Josephus Problem | Brilliant Math & Science Wiki
Akshat Sharda, Akshay Yadav, Arjun Grandhe, and
Flavius Josephus was a famous historian of the first century. During the Jewish-Roman war, he was among a band of 41 Jewish rebels trapped in a cave by the Romans. Preferring suicide to capture, the rebels decided to form a circle and to kill every third remaining person until no one was left. But Josephus, along with an unindicted conspirator, wanted none of this suicide nonsense and therefore quickly calculated where he and his friend should stand in the circle so that they can survive.
We will start with
n
people numbered
1
n
around a circle, and we'll eliminate every second remaining person until only one survives (the main problem is to find the survivor's number).
J(n)
be the survivor's number, and as an example take
n=10
, then the order of elimination would be 2,4,6,8,10,3,7,9,1. So, number 5 survives. Therefore,
J(10)=5
It is easy to notice that
J(n)
is always an odd number because the first trip around the circle would eliminate all the even numbers. Also, if
n
is even number then we will come at the same situation except this time there are only half as many people and their numbers have changed.
Now let us take another case in which
n=2^k
for some positive integral value of
k
. In this case the person with number
1
survives in the last always or
J(n)=1
Now if we take a value
n
which may not be equal to a perfect power of
2
n=m+2^k,
2^k \leq n
m
Now we just need to count till
m
number of people are executed and then next person will be the one who will survive the deadly procedure.
Also, notice that if you want to find the number of person who was executed
2^{nd}
in the process, the number is
4
; if you want to find the number of person who was executed
3^{rd}
6
, and so on. In general the person executed
m^{th}
in the process has number is
2m
. (Note that this generalization is only valid when
m < n
, in our case its true as
m=n-2^k
The person standing next with number
2m
will have number
2m+1
\begin{aligned} J(n) &= 2m+1\\ &= 2(n-2^k)+1\\ &= 2n-2^{k+1}+1, \end{aligned}
2^k \leq n.
Cite as: Josephus Problem. Brilliant.org. Retrieved from https://brilliant.org/wiki/josephus-problem/
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(Redirected from Set (game))
For other uses, see Set.
Visualization, logical reasoning, ability to focus
Three cards from a Set deck. These cards each have a unique number, symbol, shading, and color, and are thus a "set".
Set (stylized as SET) is a real-time card game designed by Marsha Falco in 1974 and published by Set Enterprises in 1991. The deck consists of 81 unique cards that vary in four features across three possibilities for each kind of feature: number of shapes (one, two, or three), shape (diamond, squiggle, oval), shading (solid, striped, or open), and color (red, green, or purple).[1] Each possible combination of features (e.g. a card with three striped green diamonds) appears as a card precisely once in the deck.
In the game, certain combinations of three cards are said to make up a set. For each one of the four categories of features — color, number, shape, and shading — the three cards must display that feature as either a) all the same, or b) all different. Put another way: For each feature the three cards must avoid having two cards showing one version of the feature and the remaining card showing a different version.
For example, 3 solid red diamonds, 2 solid green squiggles, and 1 solid purple oval form a set, because the shadings of the three cards are all the same, while the numbers, the colors, and the shapes among the three cards are all different.
For any "set", the number of features that are all the same and the number of features that are all different may break down as 0 the same + 4 different; or 1 the same + 3 different; or 2 the same + 2 different; or 3 the same + 1 different. (It cannot break down as 4 features the same + 0 different as the cards would be identical, and there are no identical cards in the Set deck.)
3 Basic combinatorics of Set
The game evolved out of a coding system that the designer used in her job as a geneticist.[2] Set won American Mensa's Mensa Select award in 1991 and placed 9th in the 1995 Deutscher Spiele Preis.
Several games can be played with these cards, all involving the concept of a set. A set consists of three cards satisfying all of these conditions:
The rules of Set are summarized by: If you can sort a group of three cards into "two of ____ and one of ____", then it is not a set.
For example, these three cards form a set:
Given any two cards from the deck, there is one and only one other card that forms a set with them.
In the standard Set game, the dealer lays out cards on the table until either twelve are laid down or someone sees a set and calls "Set!". The player who called "Set" takes the cards in the set, and the dealer continues to deal out cards until twelve are on the table. A player who sees a set among the twelve cards calls "Set" and takes the three cards, and the dealer lays three more cards on the table. (To call out "set" and not pick one up quickly enough results in a penalty.) There may be no set among the twelve cards; in this case, the dealer deals out three more cards to make fifteen dealt cards, or eighteen or more, as necessary. This process of dealing by threes and finding sets continues until the deck is exhausted and there are no more sets on the table. At this point, whoever has collected the most sets wins.
Variants were included with the Set game that involve different mechanics to find sets, as well as different player interaction. Additional variants continue to be created by avid players of the game.[3][4]
Basic combinatorics of Set[edit]
A complete set of 81 cards isomorphic with those of the game Set showing all possible combinations of the four features. Considering each 3×3 group as a plane aligned in 4-dimensional space, a set comprises 3 cards in a (4-dimensional) row, with wrap-around. An example 20-card cap set is shaded yellow.
Given any two cards, there is exactly one card that forms a set with those two cards. Therefore, the probability of producing a Set from 3 randomly drawn cards from a complete deck is 1/79.
A Cap set is a mathematical structure describing a Set layout in which no set may be taken. The largest group of cards that can be put together without creating a set is 20.[5][6] Such a group is called a maximal cap set (sequence A090245 in the OEIS). Donald Knuth found in 2001 that there are 682344 such cap sets of size 20 for the 81-card version of Set; under affine transformations on 4-dimensional finite space, they all reduce to essentially one cap set.
{\displaystyle \textstyle {\frac {81 \choose 2}{3}}={\frac {81\times 80}{2\times 3}}=1080}
unique sets.
The probability that a set will have
{\displaystyle d}
features different and
{\displaystyle 4-d}
features the same is
{\displaystyle \textstyle {\frac {{4 \choose d}2^{d}}{80}}}
. (Note: The case where d = 0 is impossible, since no two cards are identical.) Thus, 10% of possible sets differ in one feature, 30% in two features, 40% in three features, and 20% in all four features.
The number of different 12-card deals is
{\displaystyle \textstyle {81 \choose 12}={\frac {81!}{12!69!}}=70\,724\,320\,184\,700\approx 7.07\times 10^{13}}
The odds against there being no Set in 12 cards when playing a game of Set start off at 30:1 for the first round. Then they quickly fall, and after about the 4th round they are 14:1 and for the next 20 rounds, they slowly fall towards 13:1. So for most of the rounds played, the odds are between 14:1 and 13:1.[7]
The odds against there being no Set in 15 cards when playing a game are 88:1.[7] (This is different from the odds against there being no Set in any 15 cards (which is 2700:1) since during play, 15 cards are only shown when a group of 12 cards has no Set.)
Around 30% of all games always have a Set among the 12 cards, and thus never need to go to 15 cards.[8]
The average number of available Sets among 12 cards is
{\displaystyle \textstyle {12 \choose 3}\cdot {\frac {1}{79}}\approx 2.78}
and among 15 cards
{\displaystyle \textstyle {15 \choose 3}\cdot {\frac {1}{79}}\approx 5.76}
. However, in play the numbers are smaller.
If there were 26 sets picked from the deck, the last three cards would necessarily form another 27th set.
Using a natural generalization of Set, where the number of properties and values vary, it was shown that determining whether a set exists from a collection of dealt cards is NP-complete.[9]
^ "How to Play the Daily SET Puzzle". America's Favorite Card Games®. 2015-08-11. Retrieved 2022-02-07.
^ "Set - The history of". 2006-10-21. Archived from the original on 21 October 2006. Retrieved 2022-02-07.
^ "Set Variants". magliery.com. Retrieved 2022-02-07.
^ "Get Set - A Set Variant". www.thegamesjournal.com. Retrieved 2022-02-07.
^ Edel, Yves (2004), "Extensions of generalized product caps", Designs, Codes and Cryptography, 31 (1): 5–14, doi:10.1023/A:1027365901231, MR 2031694, S2CID 10138398 .
^ Benjamin Lent Davis and Diane Maclagan. "The Card Game Set" (PDF). Archived from the original (PDF) on June 5, 2013.
^ a b "SET Probabilities Revisited".
^ "SET® Probabilities Revisited". Henrik Warne's blog. 2011-09-30. Retrieved 2022-02-07.
^ Chaudhuri, Kamalika; Godfrey, Brighten; Ratajczak, David; Wee, Hoeteck (2003). "On the Complexity of the Game of Set" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
Set Enterprises website
A (2002?) mathematic exploration of the game Set . Including 'How many cards may be laid without creating a set', as well as investigations of different types of set games (some in the Fano plane).
The Mathematics of the Card Game Set - Paola Y. Reyes - 2014 - Rhode Island College Honors Projects
Set at BoardGameGeek
There is a graphic computer solitaire version of Set written in tcl/Tk. The script can be found in a "tclapps" bundle at ActiveState Ftp://tcl.activestate.com/pub/tcl/nightly-cvs/.
Sets, Planets, and Comets. An alternate, extended version of Set
Retrieved from "https://en.wikipedia.org/w/index.php?title=Set_(card_game)&oldid=1080814142"
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Geometric Accuracy Design Method of Roller Cavity Surfaces for Net-Shape Rolling Compressor Blades
1Key Laboratory of Road Construction Technology and Equipment of MOE, Chang’an University, Xi’an, China
2The Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education, Northwestern Polytechnical University, Xi’an, China
The accurate shape of roller cavity surfaces is vital part for net-shape rolling. This paper presents a new design method of roller cavity surfaces with high accuracy for rolling compressor blades based on the geometrical inheritance and evolution of the net-shape profiles. Firstly, a process model of the blade is modeled by adding process allowance and locating basis at the CAD (Computer Aided Design) model of the blade to represent the roll formed blade, the process model inherits the net-shape profiles of the blade at the pressure and suction surfaces. Secondly, an algorithm is proposed to discretize a curve to a set of ranked points with the restriction of the maximum chord height, and a new section curve which represents the geometrical feature of the pressure and suction surfaces are distributed on the process model based on the algorithm. Finally, a mapping algorithm is proposed to transform the section curves to the cavity section curves around the roller axis based on the conjugate movement between rollers and blade, and the cavity surfaces are reconstructed based on the transformed section curves. The design method is implemented for the roller cavities of a variable cross-section compressor blade, and the accuracy of the designed cavities is checked based on the precision of the roll formed blade by the finite element method. The results reveal that the designed cavities achieved the net-shape precision at pressure and suction surfaces of the blade. The paper provides an effective method for designing rolling cavity surfaces with excellent design quality.
Rollingblade, Cavity Design, Process Model, Section Curves, Discretizing Algorithm, Transformation Algorithm
{d}_{k}\le \delta
{d}_{k}>\delta
{P}_{k}
{L}_{k}
{d}_{k}
\left\{\begin{array}{l}{x}^{\prime }=x\hfill \\ {y}^{\prime }=\frac{D}{2}-\left(\frac{D}{2}-y\right)\mathrm{cos}\left({\theta }_{k}\right)\hfill \\ {z}^{\prime }=\left(\frac{D}{2}-y\right)\mathrm{sin}\left({\theta }_{k}\right)\hfill \end{array}
\left\{\begin{array}{l}{x}^{\prime }=x\hfill \\ {y}^{\prime }=-\frac{D}{2}-\left(y-\frac{D}{2}\right)\mathrm{cos}\left(-{\theta }_{k}\right)\hfill \\ {z}^{\prime }=\left(y-\frac{D}{2}\right)\mathrm{sin}\left(-{\theta }_{k}\right)\hfill \end{array}
\left(\begin{array}{ccc}x& y& z\end{array}\right)
\left(\begin{array}{ccc}{x}^{\prime }& {y}^{\prime }& {z}^{\prime }\end{array}\right)
{\theta }_{k}
{S}_{k}
{{S}^{\prime }}_{k}
{\theta }_{0}=0
{l}_{t}
{l}_{r}
Jin, Q.C., Wang, W.H., Jiang, R.S., Cai, Z.Y. and Li, D.T. (2019) Geometric Accuracy Design Method of Roller Cavity Surfaces for Net-Shape Rolling Compressor Blades. Open Access Library Journal, 6: e5279. https://doi.org/10.4236/oalib.1105279
1. Yan, Y.X., Sun, Q., Chen, J.J. and Pan, H.L. (2015) Effect of Processing Parameters on Edge Cracking in Cold Rolling. Materials and Manufacturing Processes, 30, 1174-1178. https://doi.org/10.1080/10426914.2013.811746
2. Byon, S.M. and Lee, Y. (2014) An Investigation on the Effect of Design Parameters of Side Guide on Camber in Hot Slab Rolling. Materials and Manufacturing Processes, 29, 107-114. https://doi.org/10.1080/10426914.2013.811743
3. Dong, Y.G. and Song, J.F. (2016) Research on the Characteristics of Forward Slip and Backward Slip in Alloyed Bar Rolling by the Round-Oval-Round Pass Sequence. The Interna-tional Journal of Advanced Manufacturing Technology, 87, 3605-3617. https://doi.org/10.1007/s00170-016-8731-0
4. Kong, N., Cao, J.G., Wang, Y.P., et al. (2014) Development of Smart Contact Backup Rolls in Ultra-Wide Stainless Strip Rolling Process. Materials and Manufacturing Processes, 29, 129-133. https://doi.org/10.1080/10426914.2013.822979
5. Zhang, J., Guan, R.G., Tie, D., Wang, X., Guan, X.H., et al. (2015) Effects of Technical Parameters of Semi-Solid Stirring and Rheo-Rolling on Micro-structure of A356-5wt.% B4C Composite Strip. Materials and Manufacturing Processes, 30, 340-345. https://doi.org/10.1080/10426914.2014.941477
6. Su, X. and Xu, G.M. (2015) Microstructure Homogenization of 7075 Alloy by a Novel Electric Pulse Rheo-Rolling Process. Materials and Manufacturing Processes, 30, 1-5. https://doi.org/10.1080/10426914.2014.984204
7. Sedighi, M. and Mahmoodi, M. (2012) Pressure Distribution in Cold Rolling of Turbo-Engine Thin Compressor Blades. Materials and Manufacturing Processes, 27, 401-405. https://doi.org/10.1080/10426914.2011.560229
8. Sedighi, M. and Mahmoodi, M. (2009) An Approach to Simulate Cold Roll-Forging of Turbo-Engine Thin Compressor Blade. Aircraft Engineering and Aerospace Technology, 81, 191-198. https://doi.org/10.1108/00022660910954682
9. Lu, B., Ou, H., Armstrong, C.G. and Rennie, A. (2009) 3D Die Shape Optimisation for Net-Shape Forging of Aerofoil Blades. Materials & Design, 30, 2490-2500.
10. Sung, J.U., Na, D.H. and Lee, Y. (2014) A Study on Design Equation of Separating and Oval Roll Grooves in Rebar Manufacturing Process. Materials and Manufacturing Processes, 29, 100-106. https://doi.org/10.1080/10426914.2013.811742
11. Ou, H., Lan, J., Armstrong, C.G., et al. (2004) An FE Simulation and Optimisa-tion Approach for the Forging of Aeroengine Components. Journal of Materials Processing Technology, 151, 208-216.
12. Ma, J. and Zhang, W. (2008) A Technical Note on Profile Curves Design of Tooth-Shaped Rolls. Journal of Materials Processing Technology, 204, 508-512.
13. Kiuchi, M., Naeini, H.M. and Shintani, K. (2001) Computer Aided Design of Rolls for Reshaping Processes from Round Pipes to “Channel-Type” Pipes. Journal of Materials Processing Technology, 111, 193-197.
14. Li, R.J., Li, M.Z., Qiu, N.J., et al. (2014) Surface Flexible Rolling for Three-Dimensional Sheet Metal Parts. Journal of Materials Processing Technology, 214, 380-389.
15. Cai, Z.Y. (2005) Precision Design of Roll-Forging Die and Its Applica-tion in the Forming of Automobile Front Axles. Journal of Materials Processing Technology, 168, 95-101. https://doi.org/10.1016/j.jmatprotec.2004.11.005
16. Zhang, D.W., Zhao, S.D. and Ou, H. (2016) Motion Characteristic between Die and Workpiece in Spline Rolling Process with Round Dies. Advances in Mechanical Engineering, 8, 1-12. https://doi.org/10.1177/1687814016655961
17. Zhao, X., Zhang, Z.M. and Xue, Y. (2016) An Optimum Design on Rollers Con-taining the Groove with Changeable Inner Diameter Based on Response Surface Methodology. Advances in Mechanical Engineering, 8, 1-9.
18. Chan, I.W., Pinfold, M., Kwong, C.K., et al. (2014) Automation and Optimisation of Family Mould Cavity and Runner Layout Design (FMCRLD) Using Genetic Algorithms and Mould Layout Design Grammars. Computer-Aided Design, 47, 118-133.
19. Ali, A.T., Aziz, H.A. and Sorour, A.H. (2015) On Curvatures and Points of the Translation Surfaces in Euclidean 3-Space. Journal of the Egyptian Mathematical Society, 23, 167-172. https://doi.org/10.1016/j.joems.2014.02.007
20. Jin, Q.C., Wang, W.H., Yan, W.Y., et al. (2017) Springback and Forward Slip Compensation in Designing Roller Cavity Surfaces for Net-Shape Rolling Compressor Blades. https://doi.org/10.1080/10426914.2017.1317796
21. Ji, L.C., Yu, J., Li, W.W., et al. (2017) Study on Aerodynamic Optimal Su-per/Transonic Turbine Cascade and Its Geometry Characteristics. Proceedings of the Institution of Mechanical Engineers Part G Journal of Aerospace Engineering, 23, 435-443.
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Machine Learning in your marketing mix strategy: taking the guesswork out | Tryolabs
Designing a pricing solution involves shuffling a fundamental piece in the company's marketing mix puzzle. While the price may look like the most fundamental piece to maximize profit it can rarely be defined independently of the others.
This is why, in this post, we will dive into how Machine Learning systems can be beneficial to automate and optimize other marketing tasks, and how globally they can help improve your company's pricing strategy.
At the end of the day, it is not enough to define at what price you should sell your items but it is also necessary to define:
and when to do so
Machine Learning Models vs. Machine Learning Systems
Let's begin with an important distinction: ML models vs. ML Systems since both terms are often confused. Machine Learning models allow us to generate predictions and also allow us to anticipate or determine how something will look like given certain conditions or variables. These models learn from experience, gathered from past data.
We could build a Machine Learning model to predict:
Customer Expected LifeTime Value
SKU Expected Demand
Items categorization
Customer propensity to different actions, as: trying a new product, category expansion, propensity to buy more, to churn, to engage, to change shopping habits, and more.
The machine learning model per se does not tell you what are the best actions or decisions to take. This is where the machine learning system as a whole comes into play.
The machine learning system is fed with one or several different machine learning models and includes a certain engine or optimizer that defines the best action to take given the predictions obtained by the machine learning models.
In order to create the optimizer, one needs to mathematically formulate which is the objective function to minimize or maximize and take into consideration the existing constraints (could be operational or business ones). Based on the objective function, one will be able to define which machine learning model/s and variables are best suited for the business case.
Finally, the machine learning system must be designed in such a way that one will be able to evaluate its performance not from the technical point of view but from the business one. The machine learning system must include an experimental design capable of answering the following kind of questions:
Has this ML System improved my performance?
Did we acquire new profitable customers?
Have we increased the gross margin?
Have we reduced churn?
In how much?
So which machine learning system could you incorporate at your marketing department?
This question is a tricky one since there is no such thing as a general out-of-the-box plug-and-play machine learning system suitable for all businesses. These solutions to be successful must be built and adapted for each company's specific needs and characteristics. This is why it is fundamental to have on the same team marketing experts, who understand the specificities of your own business, and machine learning experts who will understand the technical and data requirements needed to make the solution possible.
No matter whether you develop the machine learning capabilities in-house or you hire third-party experts, the key for a successful solution is to have these people working side-by-side.
3 Machine Learning systems which can revolutionize your marketing department:
Here we exemplify 3 Machine Learning systems that take in several different Machine Learning models and go inline with the consumer's life cycle, i.e., acquisition, optimization, and retention. With these examples, you will be able to better understand what we mean by integrating the Machine Learning models, with an optimizer and an experimental design.
1. Machine Learning Acquisition System
Imagine we are standing at the initial point of your customer's journey and there are consumers who initially do not interact with your brand or product, and prefer other brands or totally different product categories. The main goal for you at this phase is to acquire new customers.
Your acquisition system will very much depend on your business model. For instance, it won't be the same acquisition system built for a grocery shop than the one built for a hotel, airline, university course or conference which have limited seats or stocks. Moreover, the acquisition system will also differ if we are dealing with a subscription-based product.
Since it is important to target customers with high value for your business and with high probability of responding to your marketing campaign, you will have to:
Identify quality potential customers
Predict their customer marketing campaign response
In other words, the objective is to maximize the expected profit by finding the subset of customers that are likely to respond in the most profitable way.
Nevertheless, it is important to also consider the probability of your customer to respond without the treatment, as we do not want to target customers who would buy your product even without the promotion or discount. This is formally known as maximizing the uplift metric or the incremental response.
The overall system overview would look something like this:
The customer segmentation model could be a look-alike model in which we learn from historical data how our target customer looks like or could also be a clustering model in which segments are identified in an unsupervised way. Or even more, we could segment our customers based on their expected LifeTime Value thinking of it as a regression problem —with a continuous assigned value— or as a classification problem in which we discretize the segment e.g., high, medium, low value.
Once we have the segments defined we need to estimate how these different segments will respond to the different campaigns. This could be a regression problem if thought as the probability of buying a new product or if thought as the amount of money spent given a certain discount.
Finally, the Customer Acquisition Optimizer will be fed with the Machine Learning models and will maximize the expected profit by finding the subset of customers that are likely to respond in the most profitable way, taking into consideration the operational and business costs and constraints for each business campaign. Once the optimal acquisition strategy is defined we need to randomly or quasi-experimentally divide the customers into two groups, i.e., treatment and control, in order to be able to measure the system's causal gain in a statistically significant way. This last step is the one that enables us to determine the what-if scenario.
How much did my gross margin increasewith the implementation of the price optimization system?
3. Machine Learning Retention System
Finally, our focus may not be on acquiring new customers nor optimizing our existing ones but could be saving customers who are likely to leave. This problem is many times modeled as identifying customers with a high probability of churn.
These churn prevention systems —or retention systems— are widely used in:
and other subscription-based domains where the continuity of a relationship is critical
Nevertheless, the problem of customer churn is relevant for most non-subscription businesses as well, including retail. Since acquiring new customers can be much more challenging and expensive than the retention of existing ones.
In fact, research done by Frederick Reichheld of Bain & Company (the inventor of the net promoter score) has shown that by increasing retention by as little as 5%, profits can be boosted by as much as 95%.
In any Machine Learning Retention System one needs to identify those customers with a high probability of churning but also identify those who are worth investing in retaining, since retaining low-value customers would be meaningless. Therefore, we could think of two important machine learning models to include:
Churn Estimation Model
LTV Estimation Model
These two models combined could be understood as the expected loss:
L = \frac{1}{2} \rho v^2 S C_L
The Churn Estimation Model could be thought of as a classical propensity model in which the output to be estimated is the probability of each customer to churn in X periods of time from now. However, we could also frame the problem as a Survival Analysis problem in which we would like to learn what is the customer churn probability in each period of time in the future. This Survival Analysis approach is useful from an explainability point of view where we could learn which are the main drivers of churn in each moment in time.
The LTV Estimation Model, as we mentioned before, could be thought of a regression problem in which we would estimate the exact value of each customer along with the whole relationship with the brand, or could be thought of as a classification problem if we would like to categorize customer in different proxy buckets (let's say high, medium and low-value customers).
Finally, we would use both Machine Learning models to feed our campaign optimizer which would target customers with the highest expected loss. The campaign optimizer would use the campaign templates and would aim to maximize the Campaign's ROI. In other words, the optimizer needs to define the optimal number of customers to include in the targeting list, or equivalently, finding the threshold score that separates these top customers (with high expected loss) from the low ones.
All in all, we would implement this solution (execute the campaigns) in a targeted and control group in order to measure the gain, and we would be able to answer this question:
How much did your churn rate decreasedue to the implementationof the Machine Learning Retention System?
Machine learning models are extraordinary tools to integrate into any marketer's toolkit. They allow you to make accurate predictions using structured and unstructured data. However, in order for these models to deliver meaningful results, they must be integrated into a Machine Learning System which is fed with the Machine Learning model's predictions and performs some sort of optimization.
The key value of Machine Learning for decision-makers does not lay on the predictions itself but lay on the ability of learning which actions are best to take. Therefore, these systems must be designed with an experimental mindset in order to be able to clearly estimate the impact of the actions taken.
To sum up, Machine Learning systems allow you to:
Automatically compare several thousand scenarios
Adapt and learn from the latest trends and data
Design easy to evaluate campaigns and actions
And finally: learn what to sell, when to do so, to which customers and at which price.
At Tryolabs we are Machine Learning experts eager to partner up with your marketing department in order to build high-value ML Systems. By working together we can crunch huge amounts of marketing data and boost your marketer's capabilities and return on investment (ROI) faster.
If you have any doubts or ideas just contact us. We are here to help you.
Learn how companies leverage the power of data to drive outstanding results. 👇
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Exponents | Brilliant Math & Science Wiki
Ashley Toh, Seth-Riley Adams, Munem Shahriar, and
To find rules for working with exponents, see: Rules of Exponents
When dealing with positive integers, exponents can be thought of as a shorthand notation for "repeated multiplication" of a number by itself several times. For example, the shorthand for multiplying 5 copies of the number 2 is
2 \times 2 \times 2 \times 2 \times 2 = 2 ^ 5.
In words, we say that "
2^ 5
2
to the fifth power." In this example, 2 is the base and 5 is the exponent.
Note that neither the base nor the exponent need to be positive integers, meaning the "repeated multiplication" idea breaks down;
2^{-5} ,
\frac{2}{3}^\frac{1}{3} ,
0.99^{0.99}
are all valid exponent expressions. Their nature can be worked out via the Rules of Exponents.
3^4
3 ^ 4 = 3 \times 3 \times 3 \times 3 = 9 \times 3 \times 3 = 27 \times 3 = 81 .
3^4 = 81
_\square
2^5
1 \times 2 = 2
2 \times 2 = 4
4 \times 2 = 8
8 \times 2 = 16
16 \times 2 = 32
2^ { 5 } = 32
_ \square
2^ {10}
For larger powers, we have to be careful as we multiply out these terms. If you are uncertain, it is best to list out the powers in order. For example, we have
\begin{array} { l | l } n & 2^n \\ \hline 1 & 2^1 = 2 \\ 2 & 2 \times 2 = 4 \\ 3 &2 \times 4 = 8 \\ 4 &2 \times 8 = 16 \\ 5 &2 \times 16 = 32 \\ 6 &2 \times 32 = 64 \\ 7 &2 \times 64 = 128 \\ 8 &2 \times 128 = 256 \\ 9 &2 \times 256 = 512 \\ 10 &2 \times 512 = 1024 \\ \end{array}
2^ { 10 } = 1024
_ \square
Just like the multiplication tables, after a while you will start to be familiar with some of these numbers, and can remember what they are without having to work them out every single time. It just takes some practice.
Click here to learn more about the rules of exponents
The inverse operation of an exponential function is a logarithm. Learn more about them here.
Cite as: Exponents. Brilliant.org. Retrieved from https://brilliant.org/wiki/calculation-exponents/
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find the inversion of a point, plane, or sphere with respect to a given sphere.
inversion(Q, P, s)
point, line, or sphere
If P is a point that is not the same as the center O of sphere
s\left(r\right)
, the inverse of P in, or with respect to, sphere
s\left(r\right)
\mathrm{OP}\mathrm{OQ}={r}^{2}
s\left(r\right)
is called the sphere of inversion, point O the center of inversion, r the radius of inversion, and
{r}^{2}
the power of inversion.
For a detailed description of Q the object created, use the routine detail (i.e., detail(Q))
The command with(geom3d,inversion) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geom3d}\right):
Define the sphere s with center (0,0,0), radius 1
\mathrm{sphere}\left(s,[\mathrm{point}\left(o,0,0,0\right),1]\right)
\textcolor[rgb]{0,0,1}{s}
Define a plane passing through A, B, C
\mathrm{plane}\left(p,[\mathrm{point}\left(A,1,0,-1\right),\mathrm{point}\left(B,0,0,-1\right),\mathrm{point}\left(C,0,1,-1\right)]\right)
\textcolor[rgb]{0,0,1}{p}
Find the inversion of the plane with respect to the sphere s
\mathrm{inversion}\left(\mathrm{s1},p,s\right)
\textcolor[rgb]{0,0,1}{\mathrm{s1}}
Sine the plane p does not pass through the center of inversion, its inversion is a sphere through the center of inversion.
\mathrm{detail}\left(\mathrm{s1}\right)
\begin{array}{ll}\textcolor[rgb]{0,0,1}{\text{name of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{s1}}\\ \textcolor[rgb]{0,0,1}{\text{form of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{sphere3d}}\\ \textcolor[rgb]{0,0,1}{\text{name of the center}}& \textcolor[rgb]{0,0,1}{\mathrm{center_s1_1}}\\ \textcolor[rgb]{0,0,1}{\text{coordinates of the center}}& [\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}]\\ \textcolor[rgb]{0,0,1}{\text{radius of the sphere}}& \frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\\ \textcolor[rgb]{0,0,1}{\text{surface area of the sphere}}& \textcolor[rgb]{0,0,1}{\mathrm{\pi }}\\ \textcolor[rgb]{0,0,1}{\text{volume of the sphere}}& \frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{6}}\\ \textcolor[rgb]{0,0,1}{\text{equation of the sphere}}& {\textcolor[rgb]{0,0,1}{\mathrm{_x}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{_y}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathrm{_z}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{_z}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\end{array}
\mathrm{IsOnObject}\left(o,\mathrm{s1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{draw}\left([s,\mathrm{s1},p\left(\mathrm{style}=\mathrm{patchnogrid},\mathrm{color}=\mathrm{maroon}\right)],\mathrm{style}=\mathrm{wireframe},\mathrm{view}=[-1..1,-1..1,-2..1],\mathrm{title}=\mathrm{`inversion of a plane with respect to a sphere`}\right)
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Double Factorials and Multifactorials | Brilliant Math & Science Wiki
Sravanth C., Ashish Menon, Pi Han Goh, and
Before reviewing this article, the readers are expected to know what factorials are first.
The double factorial of a positive integer
n
is the generalization of the factorial
n!;
this type of factorial is denoted by
n!!
. It is a type of multifactorial which will be discussed later. As far as double factorial is concerned, it ends with
2
for an even number, and
1
for an odd number:
n>0
n!!=n\times (n-2)\times \cdots\times 4\times 2;
for an odd number
n>0
n!!=n\times (n-2)\times \cdots\times 3\times 1;
n=0
0!!=1.
To clarify,
n!!
(n!)!
4!! = 4\times2=8,
(4!)! = 24! = 620448401733239439360000
. Go ahead and try out the following warm-up problem:
\big((2^2)!!\big)!
n,
\dfrac{n!}{n!!}=(n-1)!! ~~\text{ or }~~ n!=(n-1)!!×n!!.
We have the following 2 cases:
n
\dfrac{n!}{n!!}=\dfrac{n\times (n-1)\times (n-2)\times \cdots \times 3\times 2\times 1}{n\times (n-2)\times (n-4)\times \cdots 5\times 3\times 1}.
n, n-2, n-4, \ldots , 5, 3
get canceled, we are left with the equation
\dfrac{n!}{n!!}=(n-1)!!.
n
\dfrac{n!}{n!!}=\dfrac{n\times (n-1)\times (n-2)\times \cdots \times 3\times 2\times 1}{n\times (n-2)\times (n-4)\times \cdots \times 4\times 2}.
n, n-2, n-4, \ldots , 4, 2
\dfrac{n!}{n!!}=(n-1)!!.
Combining both cases, we find that for any non-negative integer
n
\dfrac{n!}{n!!}=(n-1)!!. \ _\square
n!!
n!! = \begin{cases} n \times (n-2) \times \cdots \times 5 \times 3 \times 1 &\text{if } n \text{ is odd}; \\ n \times (n-2) \times \cdots \times 6 \times 4 \times 2 &\text{if } n \text{ is even}; \\ 1 &\text{if } n = 0, - 1. \\ \end{cases}
\color{#D61F06}{\dfrac{9!}{6!!}} \div \color{#20A900}{\dfrac{9!!}{6!}}?
n,
\dfrac{(2n+1)!}{(2n)!!}=(2n+1)!!.
Here, there is no need to consider two separate cases because it makes no difference whether
n
is odd or even.
We can expand the LHS as
\dfrac{(2n+1)\times (2n)\times (2n-1)\times \cdots \times 3\times 2\times 1}{(2n)\times (2n-2)\times (2n-4)\times \cdots \times 4\times 2}.
2n, 2n-2, 2n-4, \ldots, 4, 2
\dfrac{(2n+1)!}{(2n)!!}=(2n+1)!!. \ _\square
\frac {9!}{9!!}
\frac {n!}{n!!}=(n-1)!!
, substituting the values, we get
\begin{aligned} \dfrac{9!}{9!!}&=(9-1)!!\\ &=8!!\\ &=8×6×4×2\\ &=384. \ _\square \end{aligned}
\frac {(3!)!}{3!!}
\begin{aligned} \dfrac {(3!)!}{3!!} &=\dfrac {(3×2×1)!}{3×1}\\ &=\dfrac {6!}{3}\\ &=\dfrac {6×5×4×3×2×1}{3}\\ &=\dfrac {720}{3}\\ &=240. \ _\square \end{aligned}
\Large{\color{#20A900}{\dfrac{9!}{8!!}}} \div {\color{#EC7300}{\dfrac{7!}{6!!}}} = \, ?
n!! = \begin{cases} n \times (n-2) \times \cdots \times 5 \times 3 \times 1 && \text{if } n \text{ is odd;} \\ n \times (n-2) \times \cdots \times 6 \times 4 \times 2 && \text{if } n \text{ is even;} \\ 1 && \text{if } n = 0, - 1. \\ \end{cases}
n
\dfrac{(2n-1)!}{(2n-2)!!}=(2n-1)!!.
Again here, there is no need to consider two separate cases. We can expand the LHS as
\dfrac{(2n-1)\times (2n-2)\times (2n-3)\times \cdots \times 3\times 2\times 1}{(2n-2)\times (2n-4)\times \cdots \times 4\times 2}.
2n-2, 2n-4, \ldots , 4, 2
\dfrac{(2n-1)!}{(2n-2)!!}=(2n-1)!!. \ _\square
\frac{9!}{8!!}
\begin{aligned} \dfrac { 9! } { 8!!} &= \dfrac{ (2 \times 5 - 1 ) ! } { (2 \times 5 - 2 ) !! } \\ &= (2 \times 5 - 1 )!! \\ &= 9 !! \\ &= 9 \times 7 \times 5 \times 3 \times 1 \\ &= 945. \ _\square \end{aligned}
Cite as: Double Factorials and Multifactorials. Brilliant.org. Retrieved from https://brilliant.org/wiki/double-factorials-and-multifactorials/
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Electrostatic Potential And Capacitance, Popular Questions: CBSE Class 12-science SCIENCE, Science - Meritnation
+ 15 \mu C and + 9 \mu C
\lambda
\overline{)E}={\overline{)E}}_{0}\stackrel{^}{j}
△\mathrm{\theta }
\left(\mathrm{\lambda }=1 \mathrm{kg}/\mathrm{m}, \mathrm{a}=2\mathrm{m}, \mathrm{p}=1/3 \mathrm{C}/\mathrm{m}, {\mathrm{E}}_{0}={\mathrm{\pi }}^{2} \mathrm{N}/\mathrm{C}\right)
4×{10}^{15} Hz to 8×{10}^{15 }Hz?\phantom{\rule{0ex}{0ex}}Given h= 6.4×{10}^{-34} J -s,e=1.6×{10}^{-19} C and c=3×{10}^{8} m{s}^{-1}
{R}_{1}
\frac{8}{5} A
64. \mathrm{Figure} \mathrm{shows} \mathrm{two} \mathrm{shells} \mathrm{of} \mathrm{radii} \mathrm{R} \mathrm{and} 2\mathrm{R}. \mathrm{The} \mathrm{inner} \mathrm{shell} \left(\mathrm{centre} \mathrm{at} \mathrm{A}\right) \mathrm{is} \mathrm{nonconducting} \mathrm{and} \mathrm{uniformly} \mathrm{charged} \mathrm{with} \mathrm{charge} \mathrm{Q} \mathrm{while} \mathrm{the} \mathrm{outer} \mathrm{shell} \left(\mathrm{centre} \mathrm{at} \mathrm{B}\right) \mathrm{is} \mathrm{conducting} \mathrm{and} \mathrm{uncharged}. \mathrm{The} \mathrm{potential} \mathrm{at} \mathrm{the} \mathrm{point} \mathrm{B} \mathrm{is} :\phantom{\rule{0ex}{0ex}}\left(\mathrm{a}\right) \mathrm{zero} \left(\mathrm{b}\right) \frac{\mathrm{KQ}}{\mathrm{R}} \left(\mathrm{c}\right) \frac{\mathrm{KQ}}{\mathrm{x}} \left(\mathrm{d}\right) \mathrm{None} \mathrm{of} \mathrm{these}
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An equilateral pentagon, i.e. a pentagon whose five sides all have the same length
108° (if equiangular, including regular)
In geometry, a pentagon (from the Greek πέντε pente meaning five and γωνία gonia meaning angle[1]) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
1.1 Derivation of the area formula
1.3 Chords from the circumscribed circle to the vertices
1.4 Point in plane
1.5 Geometrical constructions
1.5.1 Richmond's method
1.5.2 Carlyle circles
1.5.3 Euclid's method
1.6 Physical construction methods
1.8 Regular pentagram
2 Equilateral pentagons
3 Cyclic pentagons
4 General convex pentagons
5 Pentagons in tiling
6 Pentagons in polyhedra
10 In-line notes and references
Regular pentagons[edit]
Side (
{\displaystyle t}
), circumradius (
{\displaystyle R}
), inscribed circle radius (
{\displaystyle r}
), height (
{\displaystyle R+r}
), width/diagonal (
{\displaystyle \varphi t}
A regular pentagon has Schläfli symbol {5} and interior angles of 108°.
A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length
{\displaystyle t,}
its height
{\displaystyle H}
(distance from one side to the opposite vertex), width
{\displaystyle W}
(distance between two farthest separated points, which equals the diagonal length
{\displaystyle D}
) and circumradius
{\displaystyle R}
{\displaystyle {\begin{aligned}H&={\frac {\sqrt {5+2{\sqrt {5}}}}{2}}~t\approx 1.539~t,\\W=D&={\frac {1+{\sqrt {5}}}{2}}~t\approx 1.618~t,\\W&={\sqrt {2-{\frac {2}{\sqrt {5}}}}}\cdot H\approx 1.051~H,\\R&={\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.8507~t,\\D&=R\ {\sqrt {\frac {5+{\sqrt {5}}}{2}}}=2R\cos 18^{\circ }=2R\cos {\frac {\pi }{10}}\approx 1.902~R.\end{aligned}}}
The area of a convex regular pentagon with side length
{\displaystyle t}
{\displaystyle {\begin{aligned}A&={\frac {t^{2}{\sqrt {25+10{\sqrt {5}}}}}{4}}={\frac {5t^{2}\tan 54^{\circ }}{4}}\\&={\frac {{\sqrt {5(5+2{\sqrt {5}})}}\;t^{2}}{4}}\approx 1.720~t^{2}.\end{aligned}}}
If the circumradius
{\displaystyle R}
of a regular pentagon is given, its edge length
{\displaystyle t}
is found by the expression
{\displaystyle t=R\ {\sqrt {\frac {5-{\sqrt {5}}}{2}}}=2R\sin 36^{\circ }=2R\sin {\frac {\pi }{5}}\approx 1.176~R,}
{\displaystyle A={\frac {5R^{2}}{4}}{\sqrt {\frac {5+{\sqrt {5}}}{2}}};}
since the area of the circumscribed circle is
{\displaystyle \pi R^{2},}
the regular pentagon fills approximately 0.7568 of its circumscribed circle.
Derivation of the area formula[edit]
{\displaystyle A={\frac {1}{2}}Pr}
{\displaystyle A={\frac {1}{2}}\cdot 5t\cdot {\frac {t\tan {\mathord {\left({\frac {3\pi }{10}}\right)}}}{2}}={\frac {5t^{2}\tan {\mathord {\left({\frac {3\pi }{10}}\right)}}}{4}}}
Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by
{\displaystyle r={\frac {t}{2\tan {\mathord {\left({\frac {\pi }{5}}\right)}}}}={\frac {t}{2{\sqrt {5-{\sqrt {20}}}}}}\approx 0.6882\cdot t.}
Chords from the circumscribed circle to the vertices[edit]
Point in plane[edit]
For an arbitrary point in the plane of a regular pentagon with circumradius
{\displaystyle R}
, whose distances to the centroid of the regular pentagon and its five vertices are
{\displaystyle L}
{\displaystyle d_{i}}
respectively, we have [2]
{\displaystyle {\begin{aligned}\textstyle \sum _{i=1}^{5}d_{i}^{2}&=5\left(R^{2}+L^{2}\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{4}&=5\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{6}&=5\left(\left(R^{2}+L^{2}\right)^{3}+6R^{2}L^{2}\left(R^{2}+L^{2}\right)\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{8}&=5\left(\left(R^{2}+L^{2}\right)^{4}+12R^{2}L^{2}\left(R^{2}+L^{2}\right)^{2}+6R^{4}L^{4}\right).\end{aligned}}}
{\displaystyle d_{i}}
are the distances from the vertices of a regular pentagon to any point on its circumcircle, then [2]
{\displaystyle 3\left(\textstyle \sum _{i=1}^{5}d_{i}^{2}\right)^{2}=10\textstyle \sum _{i=1}^{5}d_{i}^{4}.}
Geometrical constructions[edit]
Richmond's method[edit]
One method to construct a regular pentagon in a given circle is described by Richmond[3] and further discussed in Cromwell's Polyhedra.[4]
To determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle is found as
{\displaystyle \scriptstyle {\sqrt {5}}/2}
. Side h of the smaller triangle then is found using the half-angle formula:
{\displaystyle \tan(\phi /2)={\frac {1-\cos(\phi )}{\sin(\phi )}}\ ,}
{\displaystyle h={\frac {{\sqrt {5}}-1}{4}}\ .}
If DP is truly the side of a regular pentagon,
{\displaystyle m\angle \mathrm {CDP} =54^{\circ }}
, so DP = 2 cos(54°), QD = DP cos(54°) = 2cos2(54°), and CQ = 1 − 2cos2(54°), which equals −cos(108°) by the cosine double angle formula. This is the cosine of 72°, which equals
{\displaystyle \left({\sqrt {5}}-1\right)/4}
Carlyle circles[edit]
Main article: Carlyle circle
Euclid's method[edit]
Euclid's method for pentagon at a given circle, using of the golden triangle, animation 1 min 39 s
Physical construction methods[edit]
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g5 subgroup has no degrees of freedom but can be seen as directed edges.
Regular pentagram[edit]
Main article: Pentagram
Equilateral pentagons[edit]
Main article: Equilateral pentagon
Cyclic pentagons[edit]
There exist cyclic pentagons with rational sides and rational area; these are called Robbins pentagons. It has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and it is conjectured that all the diagonals must be rational.[13]
General convex pentagons[edit]
For all convex pentagons, the sum of the squares of the diagonals is less than 3 times the sum of the squares of the sides.[14]: p.75, #1854
Pentagons in tiling[edit]
The best-known packing of equal-sized regular pentagons on a plane is a double lattice structure which covers 92.131% of the plane.
A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form a regular tiling (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that 360° / 108° = 31⁄3 (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons:
The maximum known packing density of a regular pentagon is approximately 0.921, achieved by the double lattice packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that the double lattice packing of the regular pentagon (which they call the "pentagonal ice-ray" packing, and which they trace to the work of Chinese artisans in 1900) has the optimal density among all packings of regular pentagons in the plane.[15] As of 2020[update], their proof has not yet been refereed and published.
There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this is that the polygons that touch the edges of the pentagon must alternate around the pentagon, which is impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126°. To find the number of sides this polygon has, the result is 360 / (180 − 126) = 62⁄3, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons.
Pentagons in polyhedra[edit]
Pentagons in nature[edit]
Another example of echinoderm, a sea urchin endoskeleton.
A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron. The faces are true regular pentagons.
A pyritohedral crystal of pyrite. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.
In-line notes and references[edit]
^ "pentagon, adj. and n." OED Online. Oxford University Press, June 2014. Web. 17 August 2014.
^ a b Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340.
^ Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (2nd ed.). CRC Press. p. 329. ISBN 1-58488-347-2.
^ DeTemple, Duane W. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon constructions" (PDF). The American Mathematical Monthly. 98 (2): 97–108. doi:10.2307/2323939. JSTOR 2323939. Archived from the original (PDF) on 2015-12-21.
^ George Edward Martin (1998). Geometric constructions. Springer. p. 6. ISBN 0-387-98276-0.
^ Fitzpatrick, Richard (2008). Euklid's Elements of Geometry, Book 4, Proposition 11 (PDF). Translated by Richard Fitzpatrick. p. 119. ISBN 978-0-6151-7984-1.
^ Weisstein, Eric W. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource. [1]
^ Robbins, D. P. (1994). "Areas of Polygons Inscribed in a Circle". Discrete and Computational Geometry. 12 (2): 223–236. doi:10.1007/bf02574377.
^ Robbins, D. P. (1995). "Areas of Polygons Inscribed in a Circle". The American Mathematical Monthly. 102 (6): 523–530. doi:10.2307/2974766. JSTOR 2974766.
^ * Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and area", Journal of Number Theory, 128 (1): 17–48, doi:10.1016/j.jnt.2007.05.005, MR 2382768 .
^ Hales, Thomas; Kusner, Wöden (September 2016), Packings of regular pentagons in the plane, arXiv:1602.07220
Wikimedia Commons has media related to Pentagons.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Pentagon&oldid=1087837558"
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Algebraic Expressions, Equations, and Inequalities - Vocabulary - Course Hero
College Algebra/Algebraic Expressions, Equations, and Inequalities/Vocabulary
number zero, which has the property that
a+0=a
additive inverse
a
-a
. The sum of a number and its additive inverse is the additive identity, zero.
Addition Property of EqualitySubtraction Property of Equality
property stating that when adding three or more numbers, the sum does not change based on the way the numbers are grouped:
a+(b+c)=(a+b)+c
property stating that when multiplying three or more real numbers, the product does not change based on the way the numbers are grouped:
a(bc)=(ab)c
property stating that when adding real numbers, the sum does not change based on the order of the numbers:
a+b=b+a
property stating that when multiplying real numbers, the product does not change based on the order of the numbers:
ab=ba
a(b+c)=ab+ac
Property stating that the sum of zero and any number is the number itself:
a+0=a
property stating that the product of
1
and any number is the given number:
a\cdot1=a
number 1, which has the property
a\cdot1=a
a
a
\frac{1}{a}
. The product of a number and its multiplicative inverse is the multiplicative identity, 1.
Multiplication Property of EqualityDivision Property of Equality
<Overview>Properties of Real Numbers
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3 Ways to Convert an Improper Fraction to Percent - wikiHow
How to Convert an Improper Fraction to Percent
2 Multiplying by 100
3 Finding an Equivalent Fraction
An improper fraction is a fraction with a larger numerator than denominator. You can convert an improper fraction to a percent the same way that you would convert any fraction to a percent. The simplest way is to divide the numerator by the denominator, then multiply by 100. It is important to keep in mind, however, that since an improper fraction denotes a number greater than 1 whole, it also denotes a percentage greater than 100%.
Converting to a Decimal Download Article
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Use a calculator to divide the numerator by the denominator. The numerator is the top number, and the denominator is the bottom number. It is best to use a calculator, because depending on the fraction, you might end up with a number that has many decimal places. Completing this calculation converts the fraction to a decimal.
Note that, because you have an improper fraction, the denominator will divide into the numerator at least 1 whole time. That means that your decimal will be greater than 1. It also means that your percent will be greater than 100%.
{\displaystyle {\frac {13}{8}}}
{\displaystyle 13\div 8=1.625}
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Multiply the decimal by 100. You can do this with a calculator, but an easy way to do this in your head is to simply move the decimal point two places to the right.[1] X Research source
{\displaystyle 1.625\times 100=162.5}
Add a percent sign. Until you add a percent sign, your number still reads like a decimal, even though it no longer is. So, add your percent sign to avoid confusion. The percent sign comes after the number.
{\displaystyle 162.5\%}
Multiplying by 100 Download Article
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Multiply the fraction by 100. To multiply a fraction by 100, multiply the numerator by 100. Keep the same denominator.[2] X Research source You can also think of multiplying the fraction by
{\displaystyle {\frac {100}{1}}}
{\displaystyle 100\div 1=100}
{\displaystyle {\frac {100}{1}}=100}
. It doesn’t matter if you multiply by the whole number or the fraction, since they mean the same thing.
{\displaystyle {\frac {13}{10}}\times 100}
{\displaystyle {\frac {13\times 100}{10}}}
{\displaystyle {\frac {1300}{10}}}
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Divide the numerator by the denominator. The numerator is the top number, the denominator is the bottom number. You will likely need a calculator to do this, or you can do it manually.
{\displaystyle 1300\div 10=130}
Add a percent sign. The percent sign goes after the number, and helps avoid confusion. Remember that a percent is not the same thing as a decimal or whole number, so you need to denote a percent with the appropriate sign.
{\displaystyle 130\%}
Finding an Equivalent Fraction Download Article
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Understand what a percent looks like as a fraction. Since a percent is equal to a certain number of hundredths, you are going to convert the fraction to an equivalent one with 100 in the denominator.
{\displaystyle {\frac {8}{5}}}
, you want to find
{\displaystyle x}
{\displaystyle {\frac {8}{5}}={\frac {x}{100}}}
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Divide 100 by the denominator of the fraction. Remember that the denominator is the bottom number. This calculation will give you a factor of change telling you how much bigger 100 is than the denominator. Note that this method only works if the denominator divides evenly into 100.
{\displaystyle {\frac {8}{5}}}
{\displaystyle 100\div 5=20}
. So, a denominator of 100 is 20 times bigger than a denominator of 5.
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Multiply the numerator and denominator by the factor of change. To maintain an equivalent fraction, whatever you do to the original denominator, you must also do to the numerator. [3] X Research source
{\displaystyle {\frac {8}{5}}={\frac {8\times 20}{5\times 20}}={\frac {160}{100}}}
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State the fraction as a percent. Remember that a percent is simply a number out of 100, or hundredths. Since your fraction is now shown as
{\displaystyle {\frac {x}{100}}}
, the numerator of your fraction is now the percent. Note that, since you are working with an improper fraction, your numerator is larger than your denominator. This means that your fraction represents more than 100 percent.
Don’t forget to include the percent sign after the number.
{\displaystyle {\frac {160}{100}}=160\%}
How do I convert a percentage to a decimal?
Drop the percent sign and divide the percentage by 100. Do this by moving the decimal point two places to the left.
How do I adjust for a percentage greater than 100%, e.g. 113%?
The conversion is the same as with percentages less than 100%: move the decimal point two places to the left. 113% equals 1.13, or 1 13/100.
↑ http://www.bbc.co.uk/schools/gcsebitesize/maths/number/fracsdecpersrev3.shtml
↑ http://www.virtualnerd.com/middle-math/ratios-proportions-percent/percents/fraction-to-percent-conversion
To convert an improper fraction to percent, start by dividing the numerator by the denominator. Then, multiply the decimal you get by 100. Finally, add a percent sign to the end of your answer. You can also multiply the improper fraction by 100 and then divide the numerator by the denominator. Then, add a percent sign and you're done! If you want to find the percentage by using an equivalent fraction, keep reading!
|
Examine the following series. Use one of the tests you have learned so far to determine if the series converges or diverges. Name the test that you used.
\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \operatorname { ln } n }
\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n }
This is a form of the harmonic series.
\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { \pi } }
π
is just a number, so this is a
p
\displaystyle\sum _ { n = 1 } ^ { \infty } 2 ( \frac { 5 } { 4 } ) ^ { n }
This is a geometric series. What is the value of
r
|
Convert linear prediction coefficients to cepstral coefficients or cepstral coefficients to linear prediction coefficients - Simulink - MathWorks Benelux
LPC to/from Cepstral Coefficients
LPC to CC
CC to LPC
Convert linear prediction coefficients to cepstral coefficients or cepstral coefficients to linear prediction coefficients
Estimation / Linear Prediction
dsplp
The LPC to/from Cepstral Coefficients block either converts linear prediction coefficients (LPCs) to cepstral coefficients (CCs) or cepstral coefficients to linear prediction coefficients. Set the Type of conversion parameter to LPCs to cepstral coefficients or Cepstral coefficients to LPCs to select the domain into which you want to convert your coefficients. The LPC port corresponds to LPCs, and the CC port corresponds to the CCs. For more information, see Algorithm.
The block input can be an N-by-M matrix or an unoriented vector. Each column of the matrix is treated as a channel. When the input is an unoriented vector, the input is treated as one channel.
Consider a signal x(n) as the input to an FIR analysis filter represented by LPCs. The output of this analysis filter, e(n), is known as the prediction error signal. The power of this error signal is denoted by P, the prediction error power.
When you select LPCs to cepstral coefficients from the Type of conversion list, you can specify the prediction error power in two ways. From the Specify P list, choose via input port to input the prediction error power using input port P. The input to the port must be a vector with length equal to the number of input channels. Select assume P equals 1 to set the prediction error power equal to 1 for all channels.
When you select LPCs to cepstral coefficients from the Type of conversion list, the Output size same as input size check box appears. When you select this check box, the length of the input vector of LPCs is equal to the output vector of CCs. When you do not select this check box, enter a positive scalar for the Length of output cepstral coefficients parameter.
When you select LPCs to cepstral coefficients from the Type of conversion list, you can use the If first input value is not 1 parameter to specify the behavior of the block when the first coefficient of the LPC vector is not 1. The following options are available:
Replace it with 1 —- Changes the first value of the coefficient vector to 1. The other coefficient values are unchanged.
Normalize — Divides the entire vector of coefficients by the first coefficient so that the first coefficient of the LPC vector is 1.
Normalize and Warn — Divides the entire vector of coefficients by the first coefficient so that the first coefficient of the LPC vector is 1. The block displays a warning message telling you that your vector of coefficients has been normalized.
Error — Displays an error telling you that the first coefficient of the LPC vector is not 1.
When you select Cepstral coefficients to LPCs from the Type of conversion list, the Output P check box appears on the block. Select this check box when you want to output the prediction error power from output port P.
The cepstral coefficients are the coefficients of the Fourier transform representation of the logarithm magnitude spectrum. Consider a sequence, x(n), having a Fourier transform X(ω). The cepstrum, cx(n), is defined by the inverse Fourier transform of Cx(ω), where Cx(ω) = logeX (ω). See the Real Cepstrum block reference page for information on computing cepstrum coefficients from time-domain signals.
When in this mode, this block uses a recursion technique to convert LPCs to CCs. The LPC vector is defined by
\left[\begin{array}{cccc}{a}_{0}& {a}_{1}& {a}_{2}& \begin{array}{cc}...& {a}_{p}\end{array}\end{array}\right]
and the CC vector is defined by
\left[\begin{array}{ccccccc}{c}_{0}& {c}_{1}& {c}_{2}& ...& {c}_{p}& ...& {c}_{n-1}\end{array}\right]
. The recursion is defined by the following equations:
{c}_{0}={\mathrm{log}}_{e}P
{c}_{m}=-{a}_{m}+\frac{1}{m}\sum _{k=1}^{m-1}\left[-\left(m-k\right)\cdot {a}_{k}\cdot {c}_{\left(m-k\right)}\right],1\le m\le p
{c}_{m}=\sum _{k=1}^{p}\left[\frac{-\left(m-k\right)}{m}\cdot {a}_{k}\cdot {c}_{\left(m-k\right)}\right],p<m<n
When in this mode, this block uses a recursion technique to convert CCs to LPCs. The CC vector is defined by
\left[\begin{array}{cccc}\begin{array}{cccc}{c}_{0}& {c}_{1}& {c}_{2}& ...\end{array}& {c}_{p}& ...& {c}_{n}\end{array}\right]
and the LPC vector is defined by
\left[\begin{array}{cccc}{a}_{0}& {a}_{1}& {a}_{2}& \begin{array}{cc}...& {a}_{p}\end{array}\end{array}\right]
. The recursion is defined by the following equations
{a}_{m}=-{c}_{m}-\frac{1}{m}\sum _{k=1}^{m-1}\left[\left(m-k\right)\cdot {c}_{\left(m-k\right)}\cdot {a}_{k}\right]
P=\mathrm{exp}\left({C}_{0}\right)
m=1,2,...,p
Choose LPCs to cepstral coefficients or Cepstral coefficients to LPCs to specify the domain into which you want to convert your coefficients.
Specify P
Choose via input port to input the values of prediction error power using input port P. Select assume P equals 1 to set the prediction error power equal to 1.
Output size same as input size
When you select this check box, the length of the input vector of LPCs is equal to the output vector of CCs.
Length of output cepstral coefficients
Enter a positive scalar that is the length of each output channel of CCs.
If first input value is not 1
Select what you would like the block to do when the first coefficient of the LPC vector is not 1. You can choose Replace it with 1, Normalize, Normalize and Warn, and Error.
Select this check box to output the prediction error power for each channel from output port P.
Papamichalis, Panos E. Practical Approaches to Speech Coding. Englewood Cliffs, NJ: Prentice Hall, 1987.
LPC to LSF/LSP Conversion DSP System Toolbox
LSF/LSP to LPC Conversion DSP System Toolbox
LPC to/from RC DSP System Toolbox
LPC/RC to Autocorrelation DSP System Toolbox
Real Cepstrum DSP System Toolbox
Complex Cepstrum DSP System Toolbox
|
Difference between revisions of "Randomness, Structure and Causality - Abstract" - Santa Fe Institute Events Wiki
Difference between revisions of "Randomness, Structure and Causality - Abstract"
(Created page with '{{Randomness, Structure and Causality}}')
'''Effective Complexity of Stationary Process Realizations'''
'''Learning Out of Equilibrium'''
'''The Transmission of Sense Information'''
Bergstrom, Carl (cbergst@u.washington.edu)
SFI & University of Washington
Biologists rely heavily on the language of information, coding, and transmission that is commonplace in the field of information theory developed by Claude Shannon, but there is open debate about whether such language is anything more than facile metaphor. Philosophers of biology have argued that when biologists talk about information in genes and in evolution, they are not talking about the sort of information that Shannon’s theory addresses. First, philosophers have suggested that Shannon’s theory is only useful for developing a shallow notion of correlation, the so-called ‘‘causal sense’’ of information. Second, they typically argue that in genetics and evolutionary biology, infor- mation language is used in a ‘‘semantic sense,’’ whereas semantics are deliber- ately omitted from Shannon’s theory. Neither critique is well-founded. Here we propose an alternative to the causal and semantic senses of information: a transmission sense of information, in which an object X conveys information if the function of X is to reduce, by virtue of its sequence properties, uncertainty on the part of an agent who observes X. The transmission sense not only captures much of what biologists intend when they talk about information in genes, but also brings Shannon’s theory back to the fore. By taking the view- point of a communications engineer and focusing on the decision problem of how information is to be packaged for transport, this approach resolves several problems that have plagued the information concept in biology, and highlights a number of important features of the way that information is encoded, stored, and transmitted as genetic sequence.
'''Optimizing Information Flow in Small Genetic Networks'''
Bialek, William (wbialek@Princeton.EDU)
'''To a Mathematical Theory of Evolution and Biological Creativity'''
Links: [[File:Darwin.pdf]]
'''Framing Complexity'''
Crutchfield, James (chaos@cse.ucdavis.edu)<br>
Links: [[http://users.cse.ucdavis.edu/~cmg/compmech/pubs.htm]]
'''The Vocabulary of Grammar-Based Codes and the Logical Consistency of Texts'''
Debowski, Lukasz (ldebowsk@ipipan.waw.pl)<br>
Polish Academy of Sciences<br>
We will present a new explanation for the distribution of words in
grammar-based encoding of the text. Secondly, the text is assumed to
be emitted by a finite-energy strongly nonergodic source whereas the
facts are binary IID variables predictable in a shift-invariant
'''Prediction, Retrodiction, and the Amount of Information Stored in the Present'''
Ellison, Christopher (cellison@cse.ucdavis.edu)<br>
<br>We introduce an ambidextrous view of stochastic dynamical systems, comparing their forward-time and reverse-time representations and then integrating them into a single time-symmetric representation. The perspective is useful theoretically, computationally, and conceptually. Mathematically, we prove that the excess entropy--a familiar measure of organization in complex systems--is the mutual information not only between the past and future, but also between the predictive and retrodictive causal states. Practically, we exploit the connection between prediction and retrodiction to directly calculate the excess entropy. Conceptually, these lead one to discover new system invariants for stochastic dynamical systems: crypticity (information accessibility) and causal irreversibility. Ultimately, we introduce a time-symmetric representation that unifies all these quantities, compressing the two directional representations into one. The resulting compression offers a new conception of the amount of information stored in the present.
'''Complexity Measures and Frustration'''
Feldman, David (dave@hornacek.coa.edu)<br>
In this talk I will present some new results applying complexity
measures to frustrated systems, and I will also comment on some
frustrations I have about past and current work in complexity
measures. I will conclude with a number of open questions and ideas
I will begin with a quick review of the excess entropy/predictive
information and argue that it is a well understood and broadly
applicable measure of complexity that allows for a comparison of
information processing abilities among very different systems. The
vehicle for this comparison is the complexity-entropy diagram, a
scatter-plot of the entropy and excess entropy as model parameters are
varied. This allows for a direct comparison in terms of the
configurations' intrinsic information processing properties. To
illustrate this point, I will show complexity-entropy diagrams for: 1D
and 2D Ising models, 1D Cellular Automata, the logistic map, an
ensemble of Markov chains, and an ensemble of epsilon-machines.
I will then present some new work in which a local form of the 2D
excess entropy is calculated for a frustrated spin system. This
allows one to see how information and memory are shared unevenly
across the lattice as the system enters a glassy state. These results
show that localised information theoretic complexity measures can be
usefully applied to heterogeneous lattice systems. I will argue that
local complexity measures for higher-dimensional and heterogeneous
systems is a particularly fruitful area for future research.
Finally, I will conclude by remarking upon some of the areas of
complexity-measure research that have been sources of frustration.
These include the persistent notions of a universal "complexity at
the edge of chaos," and the relative lack of applications of
complexity measures to empirical data and/or multidimensional systems.
These remarks are designed to provoke dialog and discussion about
interesting and fun areas for future research.
Links: [[File:afm.tri.5.pdf]] and [[File:CHAOEH184043106_1.pdf]]
'''Complexity, Parallel Computation and Statistical Physics'''
Links: [[http://arxiv.org/abs/cond-mat/0510809]]
'''Crypticity and Information Accessibility'''
Mahoney, John (jmahoney3@ucmerced.edu)<br>
'''Automatic Identification of Information-Processing Structures in Cellular Automata'''
'''Phase Transitions and Computational Complexity'''
We study EC3, a variant of Exact Cover which is equivalent to Positive 1-in-3 SAT. Random instances of EC3 were recently used as benchmarks for simulations of an adiabatic quantum algorithm. Empirical results suggest that EC3 has a phase transition from satisfiability to unsatisfiability when the number of clauses per variable r exceeds some threshold r* ~= 0.62 +- 0.01. Using the method of differential equations, we show that if r <= 0.546 w.h.p. a random instance of EC3 is satisfiable. Combined with previous results this limits the location of the threshold, if it exists, to the range 0.546 < r* < 0.644.
Links: [[http://arxiv.org/abs/cs/0508037]]
'''Statistical Mechanics of Interactive Learning'''
Still, Suzanne (sstill@hawaii.edu)<br>
Using a cocycle formulation, old and new ergodic parameters beyond the
Lyapunov exponent are rigorously characterized. Dynamical Renyi entropies
and fluctuations of the local expansion rate are related by a generalization
of the Pesin formula.
How the ergodic parameters may be used to characterize the complexity of
dynamical systems is illustrated by some examples: Clustering and
synchronization, self-organized criticality and the topological structure of
'''Hidden Quantum Markov Models and Non-adaptive Read-out of Many-body States'''
Stochastic finite-state generators are compressed descriptions of infinite time series. Alternatively, compressed descriptions are given by quantum finite- state generators [K. Wiesner and J. P. Crutchfield, Physica D 237, 1173 (2008)]. These are based on repeated von Neumann measurements on a quantum dynamical system. Here we generalise the quantum finite-state generators by replacing the von Neumann pro jections by stochastic quantum operations. In this way we assure that any time series with a stochastic compressed description has a compressed quantum description. Moreover, we establish a link between our stochastic generators and the sequential readout of many-body states with translationally-invariant matrix product state representations. As an example, we consider the non-adaptive read-out of 1D cluster states. This is shown to be equivalent to a Hidden Quantum Model with two internal states, providing insight on the inherent complexity of the process. Finally, it is proven by example that the quantum description can have a higher degree of compression than the classical stochastic one.
The Transmission of Sense Information
Links: File:Darwin.pdf
{\displaystyle n}
{\displaystyle n^{\beta }}
{\displaystyle n^{\beta }/\log n}
Links: File:Afm.tri.5.pdf and File:CHAOEH184043106 1.pdf
Measuring the Complexity of Psychological States
Tononi, Guilio (gtononi@wisc.edu)
Retrieved from "https://wiki.santafe.edu/index.php?title=Randomness,_Structure_and_Causality_-_Abstract&oldid=38821"
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Elasticity (physics) — Wikipedia Republished // WIKI 2
"Elasticity theory" redirects here. For the economics measurement, see Elasticity (economics). For the cloud computing term, see Elasticity (cloud computing).
{\displaystyle J=-D{\frac {d\varphi }{dx}}}
In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state.
The physical reasons for elastic behavior can be quite different for different materials. In metals, the atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. For rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied.
Hooke's law states that the force required to deform elastic objects should be directly proportional to the distance of deformation, regardless of how large that distance becomes. This is known as perfect elasticity, in which a given object will return to its original shape no matter how strongly it is deformed. This is an ideal concept only; most materials which possess elasticity in practice remain purely elastic only up to very small deformations, after which plastic (permanent) deformation occurs.
In engineering, the elasticity of a material is quantified by the elastic modulus such as the Young's modulus, bulk modulus or shear modulus which measure the amount of stress needed to achieve a unit of strain; a higher modulus indicates that the material is harder to deform. The SI unit of this modulus is the pascal (Pa). The material's elastic limit or yield strength is the maximum stress that can arise before the onset of plastic deformation. Its SI unit is also the pascal (Pa).
Class 11 chapter 9 || MECHANICAL PROPERTIES OF SOLIDS 01|| Elasticity : Introduction IIT JEE /NEET
Matric Part 1, Elasticity Physics - Physics Ch 7 Properties & Matter - 9th Class
Elasticity - Stress Strain
Class 11 Physics Applications of Elastic Behaviour of Materials
2 Linear elasticity
3 Finite elasticity
3.1 Cauchy elastic materials
3.2 Hypoelastic materials
3.3 Hyperelastic materials
5 Factors affecting elasticity
When an elastic material is deformed due to an external force, it experiences internal resistance to the deformation and restores it to its original state if the external force is no longer applied. There are various elastic moduli, such as Young's modulus, the shear modulus, and the bulk modulus, all of which are measures of the inherent elastic properties of a material as a resistance to deformation under an applied load. The various moduli apply to different kinds of deformation. For instance, Young's modulus applies to extension/compression of a body, whereas the shear modulus applies to its shear.[1] Young's modulus and shear modulus are only for solids, whereas the bulk modulus is for solids, liquids, and gases.
The elasticity of materials is described by a stress–strain curve, which shows the relation between stress (the average restorative internal force per unit area) and strain (the relative deformation).[2] The curve is generally nonlinear, but it can (by use of a Taylor series) be approximated as linear for sufficiently small deformations (in which higher-order terms are negligible). If the material is isotropic, the linearized stress–strain relationship is called Hooke's law, which is often presumed to apply up to the elastic limit for most metals or crystalline materials whereas nonlinear elasticity is generally required to model large deformations of rubbery materials even in the elastic range. For even higher stresses, materials exhibit plastic behavior, that is, they deform irreversibly and do not return to their original shape after stress is no longer applied.[3] For rubber-like materials such as elastomers, the slope of the stress–strain curve increases with stress, meaning that rubbers progressively become more difficult to stretch, while for most metals, the gradient decreases at very high stresses, meaning that they progressively become easier to stretch.[4] Elasticity is not exhibited only by solids; non-Newtonian fluids, such as viscoelastic fluids, will also exhibit elasticity in certain conditions quantified by the Deborah number. In response to a small, rapidly applied and removed strain, these fluids may deform and then return to their original shape. Under larger strains, or strains applied for longer periods of time, these fluids may start to flow like a viscous liquid.
For small strains, the measure of stress that is used is the Cauchy stress while the measure of strain that is used is the infinitesimal strain tensor; the resulting (predicted) material behavior is termed linear elasticity, which (for isotropic media) is called the generalized Hooke's law. Cauchy elastic materials and hypoelastic materials are models that extend Hooke's law to allow for the possibility of large rotations, large distortions, and intrinsic or induced anisotropy.
For more general situations, any of a number of stress measures can be used, and it is generally desired (but not required) that the elastic stress–strain relation be phrased in terms of a finite strain measure that is work conjugate to the selected stress measure, i.e., the time integral of the inner product of the stress measure with the rate of the strain measure should be equal to the change in internal energy for any adiabatic process that remains below the elastic limit.
Main article: Linear elasticity
As noted above, for small deformations, most elastic materials such as springs exhibit linear elasticity and can be described by a linear relation between the stress and strain. This relationship is known as Hooke's law. A geometry-dependent version of the idea[5] was first formulated by Robert Hooke in 1675 as a Latin anagram, "ceiiinosssttuv". He published the answer in 1678: "Ut tensio, sic vis" meaning "As the extension, so the force",[6][7][8] a linear relationship commonly referred to as Hooke's law. This law can be stated as a relationship between tensile force F and corresponding extension displacement x,
{\displaystyle F=kx,}
{\displaystyle \varepsilon }
{\displaystyle \sigma =E\varepsilon ,}
where E is known as the elastic modulus or Young's modulus.
The elastic behavior of objects that undergo finite deformations has been described using a number of models, such as Cauchy elastic material models, Hypoelastic material models, and Hyperelastic material models. The deformation gradient (F) is the primary deformation measure used in finite strain theory.
Main article: Cauchy elastic material
A material is said to be Cauchy-elastic if the Cauchy stress tensor σ is a function of the deformation gradient F alone:
{\displaystyle \ {\boldsymbol {\sigma }}={\mathcal {G}}({\boldsymbol {F}})}
It is generally incorrect to state that Cauchy stress is a function of merely a strain tensor, as such a model lacks crucial information about material rotation needed to produce correct results for an anisotropic medium subjected to vertical extension in comparison to the same extension applied horizontally and then subjected to a 90-degree rotation; both these deformations have the same spatial strain tensors yet must produce different values of the Cauchy stress tensor.
Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses might depend on the path of deformation. Therefore, Cauchy elasticity includes non-conservative "non-hyperelastic" models (in which work of deformation is path dependent) as well as conservative "hyperelastic material" models (for which stress can be derived from a scalar "elastic potential" function).
Main article: Hypoelastic material
A hypoelastic material can be rigorously defined as one that is modeled using a constitutive equation satisfying the following two criteria:[9]
{\displaystyle {\boldsymbol {\sigma }}}
{\displaystyle t}
depends only on the order in which the body has occupied its past configurations, but not on the time rate at which these past configurations were traversed. As a special case, this criterion includes a Cauchy elastic material, for which the current stress depends only on the current configuration rather than the history of past configurations.
{\displaystyle G}
{\displaystyle {\dot {\boldsymbol {\sigma }}}=G({\boldsymbol {\sigma }},{\boldsymbol {L}})\,,}
{\displaystyle {\dot {\boldsymbol {\sigma }}}}
{\displaystyle {\boldsymbol {L}}}
is the spatial velocity gradient tensor.
If only these two original criteria are used to define hypoelasticity, then hyperelasticity would be included as a special case, which prompts some constitutive modelers to append a third criterion that specifically requires a hypoelastic model to not be hyperelastic (i.e., hypoelasticity implies that stress is not derivable from an energy potential). If this third criterion is adopted, it follows that a hypoelastic material might admit nonconservative adiabatic loading paths that start and end with the same deformation gradient but do not start and end at the same internal energy.
{\displaystyle G}
exists. As detailed in the main hypoelastic material article, specific formulations of hypoelastic models typically employ so-called objective rates so that the
{\displaystyle G}
Main article: Hyperelastic material
Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from a strain energy density function (W). A model is hyperelastic if and only if it is possible to express the Cauchy stress tensor as a function of the deformation gradient via a relationship of the form
{\displaystyle {\boldsymbol {\sigma }}={\cfrac {1}{J}}~{\cfrac {\partial W}{\partial {\boldsymbol {F}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}
This formulation takes the energy potential (W) as a function of the deformation gradient (
{\displaystyle {\boldsymbol {F}}}
). By also requiring satisfaction of material objectivity, the energy potential may be alternatively regarded as a function of the Cauchy-Green deformation tensor (
{\displaystyle {\boldsymbol {C}}:={\boldsymbol {F}}^{\textsf {T}}{\boldsymbol {F}}}
{\displaystyle {\boldsymbol {\sigma }}={\cfrac {2}{J}}~{\boldsymbol {F}}{\cfrac {\partial W}{\partial {\boldsymbol {C}}}}{\boldsymbol {F}}^{\textsf {T}}\quad {\text{where}}\quad J:=\det {\boldsymbol {F}}\,.}
Linear elasticity is used widely in the design and analysis of structures such as beams, plates and shells, and sandwich composites. This theory is also the basis of much of fracture mechanics.
Hyperelasticity is primarily used to determine the response of elastomer-based objects such as gaskets and of biological materials such as soft tissues and cell membranes.
For isotropic materials, the presence of fractures affects the Young and the shear moduli perpendicular to the planes of the cracks, which decrease (Young's modulus faster than the shear modulus) as the fracture density increases,[10] indicating that the presence of cracks makes bodies brittler. Microscopically, the stress–strain relationship of materials is in general governed by the Helmholtz free energy, a thermodynamic quantity. Molecules settle in the configuration which minimizes the free energy, subject to constraints derived from their structure, and, depending on whether the energy or the entropy term dominates the free energy, materials can broadly be classified as energy-elastic and entropy-elastic. As such, microscopic factors affecting the free energy, such as the equilibrium distance between molecules, can affect the elasticity of materials: for instance, in inorganic materials, as the equilibrium distance between molecules at 0 K increases, the bulk modulus decreases.[11] The effect of temperature on elasticity is difficult to isolate, because there are numerous factors affecting it. For instance, the bulk modulus of a material is dependent on the form of its lattice, its behavior under expansion, as well as the vibrations of the molecules, all of which are dependent on temperature.[12]
Elasticity tensor
Tactile imaging
Elastic modulus
Linear elasticity
Rubber elasticity
^ Treloar, L. R. G. (1975). The Physics of Rubber Elasticity. Oxford: Clarendon Press. p. 2. ISBN 978-0-1985-1355-1.
^ Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 70. ISBN 978-0-1237-4446-3.
^ Atanackovic, Teodor M.; Guran, Ardéshir (2000). "Hooke's law". Theory of elasticity for scientists and engineers. Boston, Mass.: Birkhäuser. p. 85. ISBN 978-0-8176-4072-9.
^ "Strength and Design". Centuries of Civil Engineering: A Rare Book Exhibition Celebrating the Heritage of Civil Engineering. Linda Hall Library of Science, Engineering & Technology. Archived from the original on 13 November 2010. [page needed]
^ Sadd, Martin H. (2005). Elasticity: Theory, Applications, and Numerics. Oxford: Elsevier. p. 387. ISBN 978-0-1237-4446-3.
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Bitmap - WikiMili, The Best Wikipedia Reader
Data structure for mapping from some domain (for example, a range of integers) to bits
For other uses, see Bitmap (disambiguation).
Other bitmap file formats
As a noun, the term "bitmap" is very often used to refer to a particular bitmapping application: the pix-map, which refers to a map of pixels, where each one may store more than two colors, thus using more than one bit per pixel. In such a case, the domain in question is the array of pixels which constitute a digital graphic output device (a screen or monitor). In some contexts, the term bitmap implies one bit per pixel, while pixmap is used for images with multiple bits per pixel. [1] [2]
Many graphical user interfaces use bitmaps in their built-in graphics subsystems; [3] for example, the Microsoft Windows and OS/2 platforms' GDI subsystem, where the specific format used is the Windows and OS/2 bitmap file format, usually named with the file extension of .BMP (or .DIB for device-independent bitmap). Besides BMP, other file formats that store literal bitmaps include InterLeaved Bitmap (ILBM), Portable Bitmap (PBM), X Bitmap (XBM), and Wireless Application Protocol Bitmap (WBMP). Similarly, most other image file formats, such as JPEG, TIFF, PNG, and GIF, also store bitmap images (as opposed to vector graphics), but they are not usually referred to as bitmaps, since they use compressed formats internally.
In typical uncompressed bitmaps, image pixels are generally stored with a variable number of bits per pixel which identify its color, the color depth. Pixels of 8 bits and fewer can represent either grayscale or indexed color. An alpha channel (for transparency) may be stored in a separate bitmap, where it is similar to a grayscale bitmap, or in a fourth channel that, for example, converts 24-bit images to 32 bits per pixel.
For an uncompressed, packed within rows, bitmap, such as is stored in Microsoft DIB or BMP file format, or in uncompressed TIFF format, a lower bound on storage size for a n-bit-per-pixel (2n colors) bitmap, in bytes, can be calculated as:
{\displaystyle {\text{size}}={\text{width}}\cdot {\text{height}}\cdot n/8}
where width and height are given in pixels.
In the formula above, header size and color palette size, if any, are not included. Due to effects of row padding to align each row start to a storage unit boundary such as a word, additional bytes may be needed.
Main article: BMP file format
Microsoft has defined a particular representation of color bitmaps of different color depths, as an aid to exchanging bitmaps between devices and applications with a variety of internal representations. They called these device-independent bitmaps as DIBs, and the file format for them is called DIB file format or BMP file format. According to Microsoft support: [4]
Here, "device independent" refers to the format, or storage arrangement, and should not be confused with device-independent color.
The X Window System uses a similar XBM format for black-and-white images, and XPM (pixelmap) for color images. Numerous other uncompressed bitmap file formats are in use, though most not widely. [5] For most purposes standardized compressed bitmap files such as GIF, PNG, TIFF, and JPEG are used; lossless compression in particular provides the same information as a bitmap in a smaller file size. [6] TIFF and JPEG have various options. JPEG is usually lossy compression. TIFF is usually either uncompressed, or lossless Lempel-Ziv-Welch compressed like GIF. PNG uses deflate lossless compression, another Lempel-Ziv variant.
There are also a variety of "raw" image files, which store raw bitmaps with no other information; such raw files are just bitmaps in files, often with no header or size information (they are distinct from photographic raw image formats, which store raw unprocessed sensor data in a structured container such as TIFF format along with extensive image metadata).
Free space bitmap, an array of bits that tracks which disk storage blocks are in-use
JPEG is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degree of compression can be adjusted, allowing a selectable tradeoff between storage size and image quality. JPEG typically achieves 10:1 compression with little perceptible loss in image quality. Since its introduction in 1992, JPEG has been the most widely used image compression standard in the world, and the most widely used digital image format, with several billion JPEG images produced every day as of 2015.
Portable Network Graphics is a raster-graphics file format that supports lossless data compression. PNG was developed as an improved, non-patented replacement for Graphics Interchange Format (GIF) — unofficially, the initials PNG stood for the recursive acronym "PNG's not GIF".
PCX, standing for PiCture eXchange, is an image file format developed by the now-defunct ZSoft Corporation of Marietta, Georgia, United States. It was the native file format for PC Paintbrush and became one of the first widely accepted DOS imaging standards, although it has since been succeeded by more sophisticated image formats, such as BMP, JPEG, and PNG. PCX files commonly stored palette-indexed images ranging from 2 or 4 colors to 16 and 256 colors, although the format has been extended to record true-color (24-bit) images as well.
Tag Image File Format, abbreviated TIFF or TIF, is an image file format for storing raster graphics images, popular among graphic artists, the publishing industry, and photographers. TIFF is widely supported by scanning, faxing, word processing, optical character recognition, image manipulation, desktop publishing, and page-layout applications. The format was created by the Aldus Corporation for use in desktop publishing. It published the latest version 6.0 in 1992, subsequently updated with an Adobe Systems copyright after the latter acquired Aldus in 1994. Several Aldus or Adobe technical notes have been published with minor extensions to the format, and several specifications have been based on TIFF 6.0, including TIFF/EP, TIFF/IT, TIFF-F and TIFF-FX.
The BMP file format, also known as bitmap image file, device independent bitmap (DIB) file format and bitmap, is a raster graphics image file format used to store bitmap digital images, independently of the display device, especially on Microsoft Windows and OS/2 operating systems.
Truevision TGA, often referred to as TARGA, is a raster graphics file format created by Truevision Inc.. It was the native format of TARGA and VISTA boards, which were the first graphic cards for IBM-compatible PCs to support Highcolor/truecolor display. This family of graphic cards was intended for professional computer image synthesis and video editing with PCs; for this reason, usual resolutions of TGA image files match those of the NTSC and PAL video formats.
Pixel art is a form of digital art, drawn with software, whereby images are built with the exclusive and intentional placement of pixels.
Netpbm is an open-source package of graphics programs and a programming library. It is used mainly in the Unix world, where one can find it included in all major open-source operating system distributions, but also works on Microsoft Windows, macOS, and other operating systems.
Raster graphics editors can be compared by many variables, including availability.
Image file formats are standardized means of organizing and storing digital images. An image file format may store data in an uncompressed format, a compressed format, or a vector format. Image files are composed of digital data in one of these formats so that the data can be rasterized for use on a computer display or printer. Rasterization converts the image data into a grid of pixels. Each pixel has a number of bits to designate its color. Rasterizing an image file for a specific device takes into account the number of bits per pixel that the device is designed to handle.
JPEG XR is an image compression standard for continuous tone photographic images, based on the HD Photo specifications that Microsoft originally developed and patented. It supports both lossy and lossless compression, and is the preferred image format for Ecma-388 Open XML Paper Specification documents.
Bitmap is a type of memory organization or image file format used to store digital images.
A large number of image file formats are available for storing graphical data, and, consequently, there are a number of issues associated with converting from one image format to another, most notably loss of image detail.
↑ James D. Foley (1995). map+%22short+for+pixel+map%22&pg=PA13 Computer Graphics: Principles and Practice. Addison-Wesley Professional. p. 13. ISBN 0-201-84840-6. The term bitmap, strictly speaking, applies only to 1-bit-per-pixel bilevel systems; for multiple-bit-per-pixel systems, we use the more general term pix-map (short for pixel map). {{cite book}}: Check |url= value (help)
↑ V.K. Pachghare (2005). Comprehensive Computer Graphics: Including C++. Laxmi Publications. p. 93. ISBN 81-7008-185-8.
↑ Julian Smart; Stefan Csomor & Kevin Hock (2006). Cross-Platform GUI Programming with Wxwidgets. Prentice Hall. ISBN 0-13-147381-6.
↑ "DIBs and Their Uses". Microsoft Help and Support. 2005-02-11.
↑ "List of bitmap file types". Search File-Extensions.org.
↑ J. Thomas; A. Jones (2006). Communicating Science Effectively: a practical handbook for integrating visual elements. IWA Publishing. ISBN 1-84339-125-2.
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f\left(x,y,z\right)=x y z
, on the region
R
0≤x,y,z≤1
f\left(x,y,z\right)
as the integrand
f\left(x,y,z\right)=x y z
\stackrel{\text{assign as function}}{\to }
\textcolor[rgb]{0,0,1}{f}
Form a Riemann sum
q=\underset{i=1}{\overset{u}{∑}}\underset{j=1}{\overset{v}{∑}}\underset{k=1}{\overset{w}{∑}}f\left(\frac{i}{u},\frac{j}{v},\frac{k}{w}\right)\cdot \left(\frac{1}{u v w}\right)
\stackrel{\text{assign}}{\to }
Obtain an iterated limit
Apply the relevant form of the limit command.
\mathrm{limit}\left(q,\left\{u=\infty ,v=\infty ,w=\infty \right\}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{8}}
In three dimensions, Maple's limit command computes iterated limits, not a true multidimensional limit. This is in distinction to the two-dimensional case where, for certain functions, Maple can obtain a true bivariate limit. Note also that there is no convenient "syntax-free" way to implement the multidimensional limit.
The expression for the Riemann sum in three dimensions can be cumbersome. Its simplified form is displayed below.
\mathrm{simplify}\left(q\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{8}}\frac{\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{w}}
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Bertrand's Paradox | Brilliant Math & Science Wiki
Daniel Liu, Eli Ross, and Jimin Khim contributed
A chord is selected at random inside a circle. What is the probability that the length of this chord is longer than the side length of an inscribed equilateral triangle in the circle?
We will attack this problem in three different ways.
First, we set a point to be stationary, and randomly select the other point. Clearly, when the other point is contained in the far
120^{\circ}
arc, the length of it is longer than the length of the side length of the triangle (shown in the picture as green), and elsewhere, it is shorter (shown as red). Thus, the probability is
\dfrac{120^{\circ}}{360^{\circ}}=\boxed{\dfrac{1}{3}}.
We randomly choose a point, then draw a horizontal line through it to form a chord in the circle. The probability that the chord is longer than the side length of the triangle is a little harder to figure out, but still easily done with some elementary geometry:
The side length of the equilateral triangle divides the radius of the circle into halves, as shown in the above diagram. Thus, the probability of the random chord being longer than the length of the side of the equilateral triangle is
\boxed{\dfrac{1}{2}}.
We pick a random point inside the circle, and draw a chord through it such that the point is the midpoint of the chord. Note that whenever the point picked is inside the circle in the middle, then the chord has a side length larger than the side of the triangle; otherwise, smaller.
Recall that the centroid of a triangle divides the medians into
2:1
pieces. Thus,
R=2r
\dfrac{r}{R}=\dfrac{1}{2}
. Thus, the ratio of the two circles' areas is
\dfrac{\pi r^2}{\pi R^2}=\left(\dfrac{1}{2}\right)^2=\boxed{\dfrac{1}{4}}.
How can three different methods yield three different answers? Which one is the correct, and which ones are bogus?
Cite as: Bertrand's Paradox. Brilliant.org. Retrieved from https://brilliant.org/wiki/bertrands-paradox/
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(iii) Section A contains 6 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each. Section D contains 8 questions of 4 marks each.
(iv) There is no overall choice. However, an internal choice has been provided in four questions of 3 marks each and 3 questions of 4 marks each. You have to attempt only one of the alternatives in all such questions.
(v) Use of calculated is not permitted.
If x = 3 is one root of the quadratic equation x2 – 2kx – 6 = 0, then find the value of k. VIEW SOLUTION
What is the HCF of smallest prime number and the smallest composite number? VIEW SOLUTION
Find the distance of a point P(x, y) from the origin. VIEW SOLUTION
In an AP, if the common difference (d) = –4, and the seventh term (a7) is 4, then find the first term. VIEW SOLUTION
What is the value of (cos2 67° – sin2 23°)? VIEW SOLUTION
Given ∆ABC ~ ∆PQR, if
\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{1}{3},
\frac{\mathrm{ar} ∆\mathrm{ABC}}{\mathrm{ar} ∆\mathrm{PQR}}.
\sqrt{2}
\left(5+3\sqrt{2}\right)
is an irrational number. VIEW SOLUTION
Find the sum of first 8 multiples of 3. VIEW SOLUTION
Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, –3). Hence find m. VIEW SOLUTION
Two different dice are tossed together. Find the probability :
(ii) of getting a sum 10, of the numbers on the two dice. VIEW SOLUTION
(ii) not divisible by 8. VIEW SOLUTION
Find HCF and LCM of 404 and 96 and verify that HCF × LCM = Product of the two given numbers. VIEW SOLUTION
Find all zeroes of the polynomial
\left(2{x}^{4}-9{x}^{3}+5{x}^{2}+3x-1\right)
if two of its zeroes are
\left(2+\sqrt{3}\right) \mathrm{and} \left(2-\sqrt{3}\right)
If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides.
If A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD. VIEW SOLUTION
A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed. VIEW SOLUTION
If the area of two similar triangles are equal, prove that they are congruent. VIEW SOLUTION
If 4 tan θ = 3, evaluate
\left(\frac{4 \mathrm{sin} \mathrm{\theta }-\mathrm{cos} \mathrm{\theta }+1}{4 \mathrm{sin} \mathrm{\theta }+\mathrm{cos} \mathrm{\theta }-1}\right)
If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A. VIEW SOLUTION
Find the area of the shaded region in Fig. 3, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm. [Use π = 3.14]
A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 2. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm. Find the total surface area of the article.
A heap of rice is in the form of a cone of base diameter 24 m and height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover the heap? VIEW SOLUTION
The table below shows the salaries of 280 persons :
Salary (In thousand Rs) No. of Persons
Calculate the median salary of the data. VIEW SOLUTION
A motor boat whose speed is 18 km/hr in still water takes 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed. If it takes 3 hours to complete total journey, what is the original average speed? VIEW SOLUTION
The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7 : 15. Find the numbers. VIEW SOLUTION
In an equilateral ∆ ABC, D is a point on side BC such that BD =
\frac{1}{3}
BC. Prove that 9(AD)2 = 7(AB)2
Prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. VIEW SOLUTION
Draw a triangle ABC with BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct a triangle whose sides are
\frac{3}{4}
of the corresponding sides of the ∆ABC. VIEW SOLUTION
\frac{\mathrm{sin} \mathrm{A}-2 {\mathrm{sin}}^{3} \mathrm{A}}{2 {\mathrm{cos}}^{3} \mathrm{A}-\mathrm{cos} \mathrm{A}}=\mathrm{tan} \mathrm{A}
(ii) Why we should avoid the bucket made by ordinary plastic? [Use π = 3.14] VIEW SOLUTION
As observed from the top of a 100 m high light house from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the light house, find the distance between the two ships.
\left[\mathrm{Use} \sqrt{3}=1.732\right]
The mean of the following distribution is 18. Find the frequency f of the class 19 – 21.
The following distribution gives the daily income of 50 workers of a factory :
Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive. VIEW SOLUTION
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(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.
Cards marked with number 3, 4, 5, ...., 50 are placed in a box and mixed thoroughly. A card is drawn at random form the box. Find the probability that the selected card bears a perfect square number. VIEW SOLUTION
In Fig. 1, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m, find the length of the ladder.
\left(\mathrm{use}\sqrt{3}=1.73\right)
Find the 9th term from the end (towards the first term) of the A.P. 5, 9, 13, ...., 185. VIEW SOLUTION
From an external point P, tangents PA and PB are drawn to a circle with centre O. If ∠PAB = 50°, then find ∠AOB. VIEW SOLUTION
The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2, –5) and R(–3, 6), find the coordinates of P. VIEW SOLUTION
In Fig. 2, a circle is inscribed in a ΔABC, such that it touches the sides AB, BC and CA at points D, E and F respectively. If the lengths of sides AB, BC and CA and 12 cm, 8 cm and 10 cm respectively, find the lengths of AD, BE and CF.
In Fig. 3, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and ∠APB = 60°. Find the length of chord AB.
x=\frac{2}{3} \mathrm{and} x =-3
are roots of the quadratic equation ax2 + 7x + b = 0, find the values of a and b. VIEW SOLUTION
Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, –4). Also find the coordinates of the point of division. VIEW SOLUTION
How many terms of the A.P. 27, 24, 21, .... should be taken so that their sum is zero? VIEW SOLUTION
If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P. VIEW SOLUTION
A well of diameter 4 m is dug 21 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 3 m to form an embankment. Find the height of the embankment. VIEW SOLUTION
In Fig. 4, ABCD is a square of side 14 cm. Semi-circles are drawn with each side of square as diameter. Find the area of the shaded region.
\left(\mathrm{use} \mathrm{\pi }=\frac{22}{7}\right)
In Fig. 5, is a decorative block, made up two solids – a cube and a hemisphere. The base of the block is a cube of side 6 cm and the hemisphere fixed on the top has diameter of 3.5 cm. Find the total surface area of the bock.
\left(\mathrm{use\pi }=\frac{22}{7}\right)
In Fig. 6, ABC is a triangle coordinates of whose vertex A are (0, −1). D and E respectively are the mid-points of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the mid-point of BC, find the areas of ∆ABC and ∆DEF.
In Fig. 7, are shown two arcs PAQ and PBQ. Arc PAQ is a part of circle with centre O and radius OP while arc PBQ is a semi-circle drawn on PQ ad diameter with centre M. If OP = PQ = 10 cm show that area of shaded region is
25\left(\sqrt{3}-\frac{\mathrm{\pi }}{6}\right){\mathrm{cm}}^{2}.
The angles of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectively. Find the height of the tower and the horizontal distance between the tower and the building. (use
\sqrt{3}=1.73
) VIEW SOLUTION
\frac{x+1}{x-1}+\frac{x-2}{x+2}=4-\frac{2x+3}{x-2}; x\ne 1,-2, 2\phantom{\rule{0ex}{0ex}}
(ii) getting a total of 6 or 7 of the numbers on two dice VIEW SOLUTION
A right circular cone of radius 3 cm, has a curved surface area of 47.1 cm2. Find the volume of the cone. (use π 3.14). VIEW SOLUTION
A passenger, while boarding the plane, slipped form the stairs and got hurt. The pilot took the passenger in the emergency clinic at the airport for treatment. Due to this, the plane got delayed by half an hour. To reach the destination 1500 km away in time, so that the passengers could catch the connecting flight, the speed of the plane was increased by 250 km/hour than the usual speed. Find the usual speed of the plane.
What value is depicted in this question? VIEW SOLUTION
In Fig. 8, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.
Prove that the lengths of tangents drawn from an external point to a circle are equal. VIEW SOLUTION
Prove that the area of a triangle with vertices (t, t −2), (t + 2, t + 2) and (t + 3, t) is independent of t. VIEW SOLUTION
A game of chance consists of spinning an arrow on a circular board, divided into 8 equal parts, which comes to rest pointing at one of the numbers 1, 2, 3, ..., 8 (Fig. 9), which are equally likely outcomes. What is the probability that the arrow will point at (i) an odd number (ii) a number greater than 3 (iii) a number less than 9.
An elastic belt is placed around the rim of a pulley of radius 5 cm. (Fig. 10) From one point C on the belt, the elastic belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm from the point O. Find the length of the belt that is still in contact with the pulley. Also find the shaded area. (use π = 3.14 and
\sqrt{3}
A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm3. The radii of the top and bottom circular ends are 20 cm and 12 cm, respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (use π = 3.14) VIEW SOLUTION
The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are 60° and 30° respectively. Find the height of the tower. VIEW SOLUTION
Construct a triangle ABC in which BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct another triangle whose sides are
\frac{3}{4}
times the corresponding sides of ΔABC. VIEW SOLUTION
The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle. VIEW SOLUTION
A thief, after committing a theft, runs at a uniform speed of 50 m/minute. After 2 minutes, a policeman runs to catch him. He goes 60 m in first minute and increases his speed by 5 m/minute every succeeding minute. After how many minutes, the policeman will catch the thief? VIEW SOLUTION
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The slope of a line is a measure of its steepness and indicates whether it goes up or down from left to right. For example, the slope of the line segment
A
at right is
\frac{1}{2}
, while the slope of the line segment
B
-\frac{3}{4}
For each line segment below, find the slope. You may want to copy each line segment on graph paper in order to draw slope triangles.
Create a slope triangle for each segment.
y
x
for each segment.
See the example below. Note:
Δ =
\frac{\Delta y}{\Delta x}=-\frac{3}{5}
\frac{\Delta y}{\Delta x}=-\frac{0}{7}=0
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Donchian Channels Definition and Example
Career futures trader Richard Donchian developed the indicator in the mid-20th century to help him identify trends. He would later be nicknamed "The Father of Trend Following."
Donchian Channels are a technical indicator seeks to identify bullish and bearish extremes that favor reversals as well as higher and lower breakouts, breakdowns, and emerging trends.
The middle band simply computes the average between the highest high over N periods and the lowest low over N periods.
These points identify the median or mean reversion price.
\begin{aligned}&\text{UC = Highest High in Last }N\text{ Periods}\\&\text{Middle Channel} = ((UC+LC)/2)\\&\text{LC = Lowest Low in Last }N\text{ periods}\\&\textbf{where:}\\&UC=\text{Upper channel}\\&\begin{aligned}N=&\text{ Number of minutes, hours, days,}\\&\text{ weeks, months}\end{aligned}\\&\begin{aligned}\text{Period}=&\text{Minutes, hours, days, weeks,}\\&\text{months}\end{aligned}\\&LC=\text{Lower channel}\end{aligned}
UC = Highest High in Last N PeriodsMiddle Channel=((UC+LC)/2)LC = Lowest Low in Last N periodswhere:UC=Upper channelN= Number of minutes, hours, days, weeks, monthsPeriod=Minutes, hours, days, weeks,monthsLC=Lower channel
Add the lowest low print to the highest high print and divide by 2.
Donchian Channels identify comparative relationships between the current price and trading ranges over predetermined periods. Three values build a visual map of price over time, similarly to Bollinger Bands, indicating the extent of bullishness and bearishness for the chosen period. The top line identifies the extent of bullish energy, highlighting the highest price achieved for the period through the bull-bear conflict.
The center line identifies the median or mean reversion price for the period, highlighting the middle ground achieved for the period through the bull-bear conflict. The bottom line identifies the extent of bearish energy, highlighting the lowest price achieved for the period through the bull-bear conflict.
In this example, the Donchian Channel is the shaded area bounded by the upper green line and the lower red line, both of which use 20 days as the band construction (N) periods. As price moves up to its highest point in the last 20 days or more, the price bars “push” the green line higher, and as price goes down to its lowest point in 20 days or more, the price bars “push” the red line lower.
When the price decreases for 20 days from a high, the green line will be horizontal and then start dropping. Conversely, when the price rises from a low for 20 days, the red line will be horizontal for 20 days and then start rising.
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The Number System - Wikiversity
The Number System[edit | edit source]
The number system is the way we categorize numbers. There may be an infinite amount of them, but they all fall nicely into several ranges.
Except for the last one, all of these groups, or sets, are cascading inclusive: the first group is part of the second, which is part of the third, which is part of the fourth, and the pattern continues.
Natural Numbers[edit | edit source]
Natural numbers (also called counting numbers) can be formed by repeated addition of the number 1.
1, 2, 3, 4, 5, 6, 7... and so on
By most definitions, 0 is a natural number representing no value.
Whole Numbers[edit | edit source]
The group of whole numbers is another name for the natural numbers but always includes 0:
Integers include all whole numbers but also extend infinitely into the negative numbers. Except for zero (which is neither positive nor negative), all integers are assumed to be positive if they do not have a negative sign (-) marking them negative.
...-4, -3, -2, -1, 0, 1, 2, 3, 4, 5....
Rational Numbers[edit | edit source]
Rational numbers are any number that can be represented by
{\displaystyle {\frac {a}{b}}}
, where a and b are any integer and b does not equal zero.
This includes fractions such as
{\displaystyle {\frac {2}{3}}}
and whole numbers (The whole number 32 can be represented as
{\displaystyle {\frac {32}{1}}}
). Many decimals are rational numbers, too, even non-terminating repeating ones such as 0.333... and 0.412412412.... 0.333... can be expressed as
{\displaystyle {\frac {3}{9}}}
{\displaystyle {\frac {1}{3}}}
(try dividing 3 into 9, and you'll see why). 0.412412412... can be expressed as
{\displaystyle {\frac {412}{999}}}
Irrational Numbers[edit | edit source]
This group is completely exclusive from all the aforementioned groups. It is its own group. Irrational numbers are any number which can not be written as the quotient (division) of two integers, which means it can't be represented by
{\displaystyle {\frac {a}{b}}}
forms, numbers such as pi, equal to 3.14159... (with no terminating digit). The square root of any whole number other than the square of an integer (0, 1, 4, 9, 16 ...) is irrational. Irrational numbers have non-repeating decimal expansions. Any number which is a repeating decimal is rational.
Retrieved from "https://en.wikiversity.org/w/index.php?title=The_Number_System&oldid=2241210"
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(iii) Sections A contains 8 questions of one mark each, which are multiple choice type questions, section B contains 6 questions of two marks each, section C contains 10 questions of three marks each, and section D contains 10 questions of four marks each.
A ladder makes an angle of 60° with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is:
\frac{4}{\sqrt{3}}
4\sqrt{3}
2\sqrt{2}
If two different dice are rolled together, the probability of getting an even number on both dice, is:
\frac{1}{36}
\frac{1}{2}
\frac{1}{6}
\frac{1}{4}
A number is selected at random from the numbers 1 to 30. The probability that it is a prime number is:
\frac{2}{3}
\frac{1}{6}
\frac{1}{3}
\frac{11}{30}
If the points A(x, 2), B(−3, −4) and C(7, − 5) are collinear, then the value of x is:
(D) −60 VIEW SOLUTION
In Fig. 1, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is :
In Fig. 2, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals:
The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:
The first three terms of an AP respectively are 3y – 1, 3y + 5 and 5y + 1. Then y equals:
If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO. VIEW SOLUTION
Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail. VIEW SOLUTION
In fig. 3, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)
Solve the quadratic equation 2x2 + ax − a2 = 0 for x. VIEW SOLUTION
Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre. VIEW SOLUTION
The first and the last terms of an AP are 7 and 49 respectively. If sum of all its terms is 420, find its common difference. VIEW SOLUTION
In Fig 4, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of inscribed circle and the area of the shaded region. [Use π = 3.14 and
\sqrt{3}=1.73
In Fig.5, PSR, RTQ and PAQ are three semicircles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region. [Use π = 3.14]
A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely? VIEW SOLUTION
A solid metallic right circular cone 20 cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter
\frac{1}{12}\phantom{\rule{0ex}{0ex}}
cm, find the length of the wire. VIEW SOLUTION
If the seventh term of an AP is
\frac{1}{9}
\frac{1}{7}
, find its 63rd term. VIEW SOLUTION
Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle. VIEW SOLUTION
Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 m, find the distance between the two ships. [Use
\sqrt{3}=1.73
] VIEW SOLUTION
\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3x-1}; x\ne -1, x\ne \frac{1}{3}
, for x. VIEW SOLUTION
Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). Find the values of y. Hence find the radius of the circle. VIEW SOLUTION
If the points P(–3, 9), Q(a, b) and R(4, – 5) are collinear and a + b = 1, find the values of a and b. VIEW SOLUTION
The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them. VIEW SOLUTION
Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, – 3). Also find the value of x. VIEW SOLUTION
In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P. VIEW SOLUTION
Prove that a parallelogram circumscribing a circle is a rhombus. VIEW SOLUTION
Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which
\frac{2}{5}
th of the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant? VIEW SOLUTION
From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid.
\left[\mathrm{Take} \pi =\frac{22}{7}\right]
\frac{3}{28}
. Find the numbers. VIEW SOLUTION
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. VIEW SOLUTION
(iv) king VIEW SOLUTION
Find the values of k for which the quadratic equation (3k + 1) x2 + 2(k + 1) x + 1 = 0 has equal roots. Also, find the roots. VIEW SOLUTION
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