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Sector bound for control system tuning - MATLAB - MathWorks América Latina
TuningGoal.ConicSector class
SectorMatrix
Constrain Input and Output Trajectories to Conic Sector
Sector bound for control system tuning
A conic sector bound is a restriction on the output trajectories of a system. If for all nonzero input trajectories u(t), the output trajectory z(t) = (Hu)(t) of a linear system H satisfies:
{\int }_{0}^{T}z{\left(t\right)}^{\text{T}}Q\text{\hspace{0.17em}}z\left(t\right)dt<0,
for all T ≥ 0, then the output trajectories of H lie in the conic sector described by the symmetric indefinite matrix Q. Selecting different Q matrices imposes different conditions on the system response.
When tuning a control system with systune, use TuningGoal.ConicSector to restrict the output trajectories of the response between specified inputs and outputs to a specified sector. For more information about sector bounds, see About Sector Bounds and Sector Indices.
Req = TuningGoal.ConicSector(inputname,outputname,Q) creates a tuning goal for restricting the response H(s) from inputs inputname to outputs outputname to the conic sector specified by the symmetric matrix Q. The tuning goal constrains H such that its trajectories z(t) = (Hu)(t) satisfy:
{\int }_{0}^{T}z{\left(t\right)}^{\text{T}}Q\text{\hspace{0.17em}}z\left(t\right)dt<0,
for all T ≥ 0. (See About Sector Bounds and Sector Indices.) The matrix Q must have as many negative eigenvalues as there are inputs in H.
To specify frequency-dependent sector bounds, set Q to an LTI model that satisfies Q(s)T = Q(–s).
A matrix, for constant sector geometry. Q is a symmetric square matrix that is ny on a side, where ny is the number of signals in outputname. The matrix Q must be indefinite to describe a well-defined conic sector. An indefinite matrix has both positive and negative eigenvalues. In particular, Q must have as many negative eigenvalues as there are input channels specified in inputname (the size of the vector input signal u(t)).
An LTI model, for frequency-dependent sector geometry. Q satisfies Q(s)T = Q(–s). In other words, Q(s) evaluates to a Hermitian matrix at each frequency.
Sector geometry, specified as a matrix or an LTI model. The Q input argument sets initial value of SectorMatrix when you create the tuning goal, and the same restrictions and characteristics apply to SectorMatrix as apply to Q.
Regularization parameter, specified as a real nonnegative scalar value.
Given the indefinite factorization of the sector matrix,
Q={W}_{1}{W}_{1}^{\text{T}}-{W}_{2}{W}_{2}^{\text{T}},\text{ }{W}_{1}^{\text{T}}{W}_{2}=0
the sector bound
H{\left(-j\omega \right)}^{\text{T}}Q\text{\hspace{0.17em}}H\left(j\omega \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}\text{\hspace{0.17em}}0
{H}_{1}{\left(j\omega \right)}^{\text{H}}{H}_{1}\left(j\omega \right)<{H}_{2}{\left(j\omega \right)}^{\text{H}}{H}_{2}\left(j\omega \right),
{H}_{1}={W}_{1}^{\text{T}}H
{H}_{2}={W}_{2}^{\text{T}}H
, and (•)H denotes the Hermitian transpose. Enforcing this condition might become numerically challenging when other tuning goals drive both H1(jω) and H2(jω) to zero at some frequencies. This condition is equivalent to controlling the sign of a 0/0 expression, which is intractable in the presence of rounding errors. To avoid this condition, you can regularize the sector bound to
H{\left(-j\omega \right)}^{\text{T}}Q\text{\hspace{0.17em}}H\left(j\omega \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\epsilon }^{2}I,
{H}_{1}{\left(j\omega \right)}^{\text{H}}{H}_{1}\left(j\omega \right)+{\epsilon }^{2}I<{H}_{2}{\left(j\omega \right)}^{\text{H}}{H}_{2}\left(j\omega \right).
This regularization prevents H2(jω) from becoming singular, and helps keep evaluation of the tuning goal numerically tractable. Use the Regularization property to set the value of ε to a small (but not negligible) fraction of the typical norm of the feedthrough term in H. For example, if you anticipate the norm of the feedthrough term of H to be of order 1 during tuning, try:
Req.Regularization = 1e-3;
Input signal names, specified as a cell array of character vectors. The input signal names specify the inputs of the constrained response, initially populated by the inputname argument.
Output signal names, specified as a cell array of character vectors. The output signal names specify the outputs of the constrained response, initially populated by the outputname argument.
Create a tuning goal that restricts the response from an input or analysis point 'u' to an output or analysis point 'y' in a control system to the following sector:
S=\left\{\left(y,u\right):0.1{u}^{2}<uy<10{u}^{2}\right\}.
Use this Q matrix to create the tuning goal.
TG = TuningGoal.ConicSector('u','y',Q)
ConicSector with properties:
SectorMatrix: [2x2 double]
Regularization: 0
Input: {'u'}
Set properties to further configure the tuning goal. For example, suppose the control system model has an analysis point called 'OuterLoop', and you want to enforce the tuning goal with the loop open at that point.
TG.Openings = 'OuterLoop';
Before or after tuning, use viewGoal to visualize the tuning goal.
viewGoal(TG)
The goal is met when the relative sector index R < 1 at all frequencies. The shaded area represents the region where the goal is not met. When you use this requirement to tune a control system CL, viewGoal(TG,CL) shows R for the specified inputs and outputs on this plot, enabling you to identify frequency ranges in which the goal is not met, and by how much.
Suppose that the signal u is marked as an analysis point in a Simulink model or genss model of the control system. Suppose also that G is the closed-loop transfer function from u to y. Create a tuning goal that constrains all I/O trajectories {u(t),y(t)} of G to satisfy:
{\int }_{0}^{T}{\left(\begin{array}{c}y\left(t\right)\\ u\left(t\right)\end{array}\right)}^{T}Q\phantom{\rule{0.2777777777777778em}{0ex}}\left(\begin{array}{c}y\left(t\right)\\ u\left(t\right)\end{array}\right)dt<0,
T\ge 0
. For this example, use sector matrix that imposes input passivity with index 0.5.
Q = [0 -1;-1 2*nu];
Constraining the I/O trajectories of G is equivalent to restricting the output trajectories
z\left(t\right)
H=\left[G;I\right]
to the sector defined by:
{\int }_{0}^{T}z{\left(t\right)}^{T}Q\phantom{\rule{0.2777777777777778em}{0ex}}z\left(t\right)dt<0.
(See About Sector Bounds and Sector Indices for more details about this equivalence.) To specify this constraint, create a tuning goal that constrains the transfer function
H=\left[G;I\right]
, which the transfer function from input
u
to outputs
\left\{y;u\right\}
TG = TuningGoal.ConicSector('u',{'y';'u'},Q);
When you specify the same signal 'u' as both input and output, the conic sector tuning goal sets the corresponding transfer function to the identity. Therefore, the transfer function constrained by TG is
H=\left[G;I\right]
as intended. This treatment is specific to the conic sector tuning goal. For other tuning goals, when the same signal appears in both inputs and outputs, the resulting transfer function is zero in the absence of feedback loops, or the complementary sensitivity at that location otherwise. This result occurs because when the software processes analysis points, it assumes the input is injected after the output. See Mark Signals of Interest for Control System Analysis and Design for more information about how analysis points work.
The conic sector tuning goal requires that
{W}_{2}^{\text{T}}H\left(s\right)
be square and minimum phase, where H(s) is the transfer function between the specified inputs and outputs, and W2 spans the negative invariant subspace of the sector matrix, Q:
Q={W}_{1}{W}_{1}^{\text{T}}-{W}_{2}{W}_{2}^{\text{T}},\text{ }{W}_{1}^{\text{T}}{W}_{2}=0
(See Algorithms.) This means that the stabilized dynamics for this goal are not the poles of H, but rather the transmission zeros of
{W}_{2}^{\text{T}}H\left(s\right)
. The MinDecay and MaxRadius options of systuneOptions control the bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, use systuneOptions to change these defaults.
Q={W}_{1}{W}_{1}^{\text{T}}-{W}_{2}{W}_{2}^{\text{T}},\text{ }{W}_{1}^{\text{T}}{W}_{2}=0
be an indefinite factorization of Q. When
{W}_{2}^{\text{T}}H\left(s\right)
is square and minimum phase, then the time-domain sector bound on trajectories z(t) = Hu(t),
{\int }_{0}^{T}z{\left(t\right)}^{\text{T}}Q\text{\hspace{0.17em}}z\left(t\right)dt<0,
is equivalent to the frequency-domain sector condition,
H{\left(-j\omega \right)}^{\text{T}}Q\text{\hspace{0.17em}}H\left(j\omega \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}\text{\hspace{0.17em}}0
for all frequencies. The TuningGoal.ConicSector goal uses this equivalence to convert the time-domain characterization into a frequency-domain condition that systune can handle in the same way it handles gain constraints. To secure this equivalence, TuningGoal.ConicSector also makes
{W}_{2}^{\text{T}}H\left(s\right)
minimum phase by making all its zeros stable.
For sector bounds, the R-index plays the same role as the peak gain does for gain constraints (see About Sector Bounds and Sector Indices). The condition
H{\left(-j\omega \right)}^{\text{T}}Q\text{\hspace{0.17em}}H\left(j\omega \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}\text{\hspace{0.17em}}0
is satisfied at all frequencies if and only if the R-index is less than one. The viewGoal plot for TuningGoal.ConicSector shows the R-index value as a function of frequency (see sectorplot).
When you tune a control system using a TuningGoal object to specify a tuning goal, the software converts the tuning goal into a normalized scalar value f(x), where x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint.
For the sector bound
H{\left(-j\omega \right)}^{\text{T}}Q\text{\hspace{0.17em}}H\left(j\omega \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}\text{\hspace{0.17em}}0
TuningGoal.ConicSector uses the objective function given by:
f\left(x\right)=\frac{R}{1+R/{R}_{\mathrm{max}}},\text{ }{R}_{\mathrm{max}}={10}^{6}.
R is the sector-bound R-index (see getSectorIndex for details).
The dynamics of H affected by the minimum-phase condition are the stabilized dynamics for this tuning goal. The MinDecay and MaxRadius options of systuneOptions control the bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, use systuneOptions to change these defaults.
systune | systune (for slTuner) (Simulink Control Design) | getSectorIndex | viewGoal | evalGoal | slTuner (Simulink Control Design)
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To Charles Lyell 1 April [1862]1
I am not quite sure that I understand your difficulty, so I must give what seems to me the explanation of the glacial-lake-theory at some little length.2 You know that there is a rocky outlet at the level of all the shelves.— Please look at my map:3 I suppose whole valley of Glen Spean filled with ice; then water would escape from outlet at Loch Spey & the highest shelf would be first formed. Secondly ice began to retreat, & water would flow for short time over its surface; but as soon as it retreated from behind hill marked Craig Dhu, where the outlet on level of 2d shelf was discovered by Milne, the water would flow from it, & the second shelf would be formed.4 This supposes that a vast barrier of ice still remains, under Ben Nevis, along all the lower part of the Spean. Lastly I suppose the ice disappeared, everywhere along L. Laggan, L Treig & Glen Spean, except close under Ben Nevis, where it still formed a Barrier, the water flowing out at level of lowest Shelf, by the pass of Muckul at head of L. Laggan.— This seems to me to account for everything. It presupposes that the shelves were formed towards close of Glacial period.—
I come up to London to read on Thursday short paper at Linn. Soc.y 5 Shall I call on Friday morning at 9
\frac{1}{2}
& sit half an hour with you? Pray have no scruple to send a line to Q. Anne St to say “no”—if it will take anything out of you.—6 If I do not hear, I will come.—
The year is established by the reference to CD’s reading a paper at the Linnean Society (see n. 5, below).
See letter from Charles Lyell, [26–31 March 1862].
CD refers to the map accompanying his paper, ‘Parallel roads of Glen Roy’.
Milne 1849, p. 398.
CD read his paper, ‘Three sexual forms of Catasetum tridentatum’, before the Linnean Society of London on 3 April 1862 (Journal of the Proceedings of the Linnean Society of London (Botany) 6: lxiv). According to Emma Darwin’s diary (DAR 242), CD left for London on Wednesday 2 April and returned home on Friday 4 April 1862.
CD’s brother Erasmus Alvey Darwin lived at 6 Queen Anne Street, London.
Explains how melting of ice in Glen Spean could have successively freed two lower cols, thus establishing the water-levels that determined the two lower shelves in Glen Roy.
Plans to read a paper to the Linnean Society ["Sexual forms of Catasetum", Collected papers 2: 63–70].
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Tensor_product_of_algebras Knowpia
{\displaystyle A\otimes _{R}B}
{\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=a_{1}a_{2}\otimes b_{1}b_{2}}
There are natural homomorphisms from A and B to A ⊗R B given by[4]
{\displaystyle a\mapsto a\otimes 1_{B}}
{\displaystyle b\mapsto 1_{A}\otimes b}
{\displaystyle {\text{Hom}}(A\otimes B,X)\cong \lbrace (f,g)\in {\text{Hom}}(A,X)\times {\text{Hom}}(B,X)\mid \forall a\in A,b\in B:[f(a),g(b)]=0\rbrace ,}
where [-, -] denotes the commutator. The natural isomorphism is given by identifying a morphism
{\displaystyle \phi :A\otimes B\to X}
on the left hand side with the pair of morphisms
{\displaystyle (f,g)}
on the right hand side where
{\displaystyle f(a):=\phi (a\otimes 1)}
{\displaystyle g(b):=\phi (1\otimes b)}
{\displaystyle X\times _{Y}Z=\operatorname {Spec} (A\otimes _{R}B).}
The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the
{\displaystyle \mathbb {C} [x,y]}
{\displaystyle \mathbb {C} [x,y]/f}
{\displaystyle \mathbb {C} [x,y]/g}
, then their tensor product is
{\displaystyle \mathbb {C} [x,y]/(f)\otimes _{\mathbb {C} [x,y]}\mathbb {C} [x,y]/(g)\cong \mathbb {C} [x,y]/(f,g)}
, which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C.
{\displaystyle A}
is a commutative ring and
{\displaystyle I,J\subseteq A}
are ideals, then
{\displaystyle {\frac {A}{I}}\otimes _{A}{\frac {A}{J}}\cong {\frac {A}{I+J}}}
, with a unique isomorphism sending
{\displaystyle (a+I)\otimes (b+J)}
{\displaystyle (ab+I+J)}
Tensor products can be used as a means of changing coefficients. For example,
{\displaystyle \mathbb {Z} [x,y]/(x^{3}+5x^{2}+x-1)\otimes _{\mathbb {Z} }\mathbb {Z} /5\cong \mathbb {Z} /5[x,y]/(x^{3}+x-1)}
{\displaystyle \mathbb {Z} [x,y]/(f)\otimes _{\mathbb {Z} }\mathbb {C} \cong \mathbb {C} [x,y]/(f)}
Tensor products also can be used for taking products of affine schemes over a field. For example,
{\displaystyle \mathbb {C} [x_{1},x_{2}]/(f(x))\otimes _{\mathbb {C} }\mathbb {C} [y_{1},y_{2}]/(g(y))}
{\displaystyle \mathbb {C} [x_{1},x_{2},y_{1},y_{2}]/(f(x),g(y))}
which corresponds to an affine surface in
{\displaystyle \mathbb {A} _{\mathbb {C} }^{4}}
if f and g are not zero.
^ Kassel (1995), p. 32.
Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics, vol. 155, Springer, ISBN 978-0-387-94370-1 .
Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. Vol. 21. Springer. ISBN 0-387-95385-X.
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FAQ - Hedging Agents - Angle
Common questions about HAs and perpetual futures
How come perpetual futures insure the Core module?
Let's take a simple example where users bring an amount y of collateral to get y worth of stablecoins, and Hedging Agents back up exactly this amount y of collateral (after bringing a margin of x in collateral).
If the initial price was p and the current price is q, now to be able to sustain the convertibility, an amount y.p/q of collateral is needed to reimburse users.
According to the formula of the position of the HA written, besides the x of collateral initially brought, the amount of collateral due to (or taken from) the HA is now y(1-p/q).
Hence while there was initially x+y in the Core module, what is now due to HAs and users is:
x+ y\cdot \frac{p}{q} + y\cdot (1-\frac{p}{q}) = x+y
Since the Core module is fully hedged by Hedging Agents with perpetual futures, it can sustain convertibility and keep the stablecoins stable.
Is there a cost for holding perpetual futures as a HA?
No, there is no funding rate like in centralized exchanges proposing perpetual futures. The only fees paid with Angle perpetuals are entry and exit fees.
There are no fees for removing or adding collateral as margin to a position.
Can HAs always open new positions with the Core module?
Within Angle Core module, Hedging Agents are here to cover the volatility of the collateral that was brought by users. If the amount of collateral from users is worth x of stablecoins and HAs already cover this amount, then new ones will not be able to enter.
The ratio between what HA currently hedge and what they hedge in total can be seen in the analytics. If we take the USDC/EUR pair as example, HA can't open positions anymore when the hedge ratio reaches the target hedge ratio.
To put it in other words, Angle Core module can be seen as a marketplace between stability and volatility seekers. If the supply of volatility is fully taken by HAs, then the Core module cannot offer more leveraged positions (and thus more volatility) than what it has already offered.
How is the amount to hedge by HAs computed?
The amount to hedge for a given collateral corresponds to the quantity of stablecoins issued against this collateral, multiplied by a target ratio fixed by the Core module. It refers to the total position HAs can open.
For instance, if a user minted 20000 agEUR with wETH, then the amount to hedge is 20000 EUR of wETH. Then, if a user burns 15000 agEUR against wETH, the amount to hedge becomes 5000 EUR of wETH.
If the target proportion that HAs are allowed to hedge is 90%, then after the burn the amount to hedge by HAs is 5000 * 0.9 = 4500 EUR worth of stablecoins. This means that the sum of the position size (in collateral) multiplied by the entry oracle rates of HAs should less or equal to 4500 EUR.
Why is that quantity expressed in stablecoins? Let's say that there is one HA in the Core module that entered at a time where the price of collateral with respect to the stablecoin was p_e and commits to an amount of collateral of c. HAs are not allowed to hedge more collateral that is in the Core module. To this extent, c must be collateral in the Core module's reserves, brought by users minting stablecoins.
If the HA cashes out at a moment in which the collateral is worth p, if we imagine that there was only c of collateral in the module before the HA came in, then it ends up with:
c - \texttt{PnL} = c \cdot \frac{p_e}{p}
This means that the Core module has enough in reserves to burn:
c \cdot \frac{p_e}{p} \cdot p = c \cdot p_e \texttt{ stablecoins}
How is the amount hedged by HAs computed?
For a given stablecoin-collateral pair, it is the sum of the product between the amount hedged by each HA perpetual and the entry rate when this perpetual was opened. It is expressed in stablecoin value.
This amount is updated whenever a HA position is opened, closed, or liquidated.
This amount should always remain inferior to the target amount HAs should hedge (as specified above). Above this amount hedged, HAs are no longer allowed to open perpetual positions. Additionally, if HAs hedge significantly more than the target hedge amount, then keepers can force-close some positions.
What is exactly implied by hedging ratio?
The hedging ratio refers to the portion of the amount to hedge by Hedging Agents that is actually hedged.
If one user mints 2000 agEUR for 1 wETH, and the target hedging ratio is set to 90%, then the amount HAs should hedge for agEUR is 1800 agEUR of wETH. If one HA enters when wETH is worth 1000 EUR, brings 2 wETH and decides to commit to the variation of 0.8 wETH (this HA has a leverage of x1.4), then 800 out of the 1800 wETH in stablecoin value are hedged: the hedging ratio is 44.44%.
What happens if there are too many HAs with respect to the amount to hedge from the Core module?
At any given point in time, HAs could hedge more than the target hedge amount.
In case this happens, Angle Core module has two parameters: the target hedging ratio, and the limit hedging ratio.
The target heding ratio gives the amount to be hedged by HAs. Above this ratio, HAs can't open positions anymore. The limit hedging ratio is the maximum hedging ratio possible in the Core module. If HAs start hedging more than this ratio because of users burning stablecoins, their positions can be automatically cashed out. This is called force-closing.
Imagine the limit hedging ratio is defined at 95%, then if 0.9x/(x-y) becomes superior to 0.95, then some HAs positions could be force-closed until the amount covered by HAs is back in the bounds again (below the target hedging ratio).
When a HA position is force-closed, the owner gets back its margin plus any unrealized PnL. Keepers are responsible for force-closing HAs positions in such cases. They are incentivized to force-close positions in a way that the hedge ratio comes back as close as possible to the target. There is no other conditions that could depend on the position size. More details in the Keepers page.
What happens if the Core module does not have enough reserves to close a HA position?
It is possible, because of users bringing one collateral, getting stablecoins and directly burning it against another collateral or because some of the reserves are lent and thus not immediately available, that when a HA tries to close their positions there is not enough collateral in the pool to which they contributed.
In this case, the HA gets everything that can be given from the collateral pool, and the rest is returned in Standard Liquidity Providers' tokens (called sanTokens).
Can I open multiple perpetuals?
A HA can own multiple perpetuals across a similar pool. For example, a HA can choose to have two positions on the wETH/EUR pair, one with a x5 leverage, and the other with a x2 leverage. Each position has its dedicated margin (isolated margin perpetuals).
Once the amount hedged by a HA has been set, it can no longer be modified. For instance if a HA opens a position of 5 wETH with 1 wETH of collateral as margin, the HA can add or remove collateral from this perpetual to change the leverage, but it can never modify this amount committed (also defined as the position size). The solution would be to close this position and to open a new one.
A HA can also own perpetuals across multiple pairs for different stablecoins. For instance, a HA could have a perpetual on the pair USDC/EUR, and another perpetual on the pair wETH/EUR.
In the Core module, positions are represented by NFTs: they can be transferred from one address to another and are non fungible.
What's the minimum amount of time I can stay as a HA?
To reduce the risk of front-running with the oracle, after perpetual creation, a HA cannot exit or remove collateral from the position within the hour that follows.
This is a parameter that can be modified by Angle governance.
Is there a maintenance margin like in centralized exchanges?
Yes. If the margin ratio goes below a certain threshold, your position should get liquidiated by keepers. The maintenance margin depends on the stablecoin/collateral pair concerned. For instance for wETH/EUR pair, the maintenance margin should be set at 6.25%.
Are there minimum or maximum leverage as a HA?
There is no minimal leverage, meaning you can come as a HA in the Core module, bring 2 wETH and open a position of only 0.001 wETH (thus hedging only that amount).
There is a maximum leverage though. This parameter will depend on the volatility of the pairs.
For instance, for a perpetual on the pair wETH/EUR, the maximum leverage allowed will be x10. For perpetuals on forex pairs like USDC/EUR, the maximum leverage is set at x100.
Are there slippage protections for HAs?
Yes. When opening a perpetual in Angle Core module, HAs can specify the maximum oracle value they are willing to accept as entry price.
The same goes for HAs cashing out their perpetual: they can specify the smallest oracle value below which they do not want their transaction executed.
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Conserved quantity - Wikipedia
Find sources: "Conserved quantity" – news · newspapers · books · scholar · JSTOR (March 2010) (Learn how and when to remove this template message)
In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant along each trajectory of the system.[1]
Not all systems have conserved quantities, and conserved quantities are not unique, since one can always apply a function to a conserved quantity, such as adding a number.
Since many laws of physics express some kind of conservation, conserved quantities commonly exist in mathematical models of physical systems. For example, any classical mechanics model will have mechanical energy as a conserved quantity as long as the forces involved are conservative.
For a first order system of differential equations
{\displaystyle {\frac {d\mathbf {r} }{dt}}=\mathbf {f} (\mathbf {r} ,t)}
where bold indicates vector quantities, a scalar-valued function H(r) is a conserved quantity of the system if, for all time and initial conditions in some specific domain,
{\displaystyle {\frac {dH}{dt}}=0}
Note that by using the multivariate chain rule,
{\displaystyle {\frac {dH}{dt}}=\nabla H\cdot {\frac {d\mathbf {r} }{dt}}=\nabla H\cdot \mathbf {f} (\mathbf {r} ,t)}
so that the definition may be written as
{\displaystyle \nabla H\cdot \mathbf {f} (\mathbf {r} ,t)=0}
which contains information specific to the system and can be helpful in finding conserved quantities, or establishing whether or not a conserved quantity exists.
Hamiltonian mechanics[edit]
For a system defined by the Hamiltonian
{\displaystyle {\mathcal {H}}}
, a function f of the generalized coordinates q and generalized momenta p has time evolution
{\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\{f,{\mathcal {H}}\}+{\frac {\partial f}{\partial t}}}
and hence is conserved if and only if
{\textstyle \{f,{\mathcal {H}}\}+{\frac {\partial f}{\partial t}}=0}
{\displaystyle \{f,{\mathcal {H}}\}}
denotes the Poisson bracket.
Suppose a system is defined by the Lagrangian L with generalized coordinates q. If L has no explicit time dependence (so
{\textstyle {\frac {\partial L}{\partial t}}=0}
), then the energy E defined by
{\displaystyle E=\sum _{i}\left[{\dot {q}}_{i}{\frac {\partial L}{\partial {\dot {q}}_{i}}}\right]-L}
{\textstyle {\frac {\partial L}{\partial q}}=0}
, then q is said to be a cyclic coordinate and the generalized momentum p defined by
{\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}}}
is conserved. This may be derived by using the Euler–Lagrange equations.
^ Blanchard, Devaney, Hall (2005). Differential Equations. Brooks/Cole Publishing Co. p. 486. ISBN 0-495-01265-3. {{cite book}}: CS1 maint: multiple names: authors list (link)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Conserved_quantity&oldid=1058124904"
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EUDML | Ring epimorphisms and . EuDML | Ring epimorphisms and .
Ring epimorphisms and
C\left(X\right)
Barr, Michael, Burgess, W. D., and Raphael, R.. "Ring epimorphisms and .." Theory and Applications of Categories [electronic only] 11 (2003): 283-308. <http://eudml.org/doc/123695>.
author = {Barr, Michael, Burgess, W. D., Raphael, R.},
keywords = {epimorphism; ring of continuous functions; category of rings},
title = {Ring epimorphisms and .},
AU - Burgess, W. D.
AU - Raphael, R.
TI - Ring epimorphisms and .
KW - epimorphism; ring of continuous functions; category of rings
Ronnie Levy, M. Matveev, Functional separability
A. R. Olfati, On a question of
{C}_{c}\left(X\right)
epimorphism, ring of continuous functions, category of rings
C
{C}^{*}
-embedding
Articles by Burgess
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EUDML | Algebraically closed and existentially closed substructures in categorical context. EuDML | Algebraically closed and existentially closed substructures in categorical context.
Algebraically closed and existentially closed substructures in categorical context.
Hebert, Michel. "Algebraically closed and existentially closed substructures in categorical context.." Theory and Applications of Categories [electronic only] 12 (2004): 269-298. <http://eudml.org/doc/125972>.
@article{Hebert2004,
author = {Hebert, Michel},
keywords = {pure embeddings; algebraically closed embeddings; elementary embeddings; existentially closed embeddings; -presentable morphisms; locally presentable categories; locally presentable factorization systems; -presentable morphisms},
title = {Algebraically closed and existentially closed substructures in categorical context.},
AU - Hebert, Michel
TI - Algebraically closed and existentially closed substructures in categorical context.
KW - pure embeddings; algebraically closed embeddings; elementary embeddings; existentially closed embeddings; -presentable morphisms; locally presentable categories; locally presentable factorization systems; -presentable morphisms
pure embeddings, algebraically closed embeddings, elementary embeddings, existentially closed embeddings,
\lambda
-presentable morphisms, locally presentable categories, locally presentable factorization systems,
\lambda
-presentable morphisms
Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
Accessible and locally presentable categories
Articles by Hebert
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Virtual Networks All the Way Down
For almost as long as there have been packet-switched networks, there have been ideas about how to virtualize them, starting with virtual circuits. But what exactly does it mean to virtualize a network?
Virtual memory is a helpful example. Virtual memory creates an abstraction of a large and private pool of memory, even though the underlying physical memory may be shared by many applications and considerably smaller that the apparent pool of virtual memory. This abstraction enables programmers to operate under the illusion that there is plenty of memory and that no-one else is using it, while under the covers the memory management system takes care of things like mapping the virtual memory to physical resources and avoiding conflict between users.
A key point is the virtualization of computing resources preserves the abstractions and interfaces that existed before they were virtualized. This is important because it means that users of those abstractions don't need to change—they see a faithful reproduction of the resource being virtualized. Virtualization also means that the different users (sometimes called tenants) cannot interfere with each other. So what happens when we try to virtualize a network?
VPNs, as described in Section 3.2, were one early success for virtual networking. They allowed carriers to present corporate customers with the illusion that they had their own private network, even though in reality they were sharing underlying links and switches with many other users. VPNs, however, only virtualize a few resources, notably addressing and routing tables. Network virtualization as commonly understood today goes further, virtualizing every aspect of networking. That means that a virtual network should support all the basic abstractions of a physical network. In this sense, they are analogous to the virtual machine, with its support of all the resources of a server: CPU, storage, I/O, and so on.
To this end, Virtual LANs (VLANs) are how we typically virtualize an L2 network. Supporting VLANs required a fairly simple extension to the original 802.1 header specification, inserting a 12-bit VLAN ID (VID) field between the SrcAddr and Type fields, as shown in Figure 1. (This VID is typically referred to as a VLAN Tag.) There are actually 32-bits inserted in the middle of the header, but the first 16-bits are used to preserve backwards compatibility with the original specification (they use Type = 0x8100 to indicate that this frame includes the VLAN extension); the other four bits hold control information used to prioritizing frames. This means it is possible to map 2
^{12}
= 4096 virtual networks onto a single physical LAN.
802.1Q VLAN tag embedded within an Ethernet (802.1) header.
VLANs proved to be quite useful to enterprises that wanted to isolate different internal groups (e.g., departments, labs), giving each of them the appearance of having their own private LAN. VLANs were also seen as a promising way to virtualize L2 networks in cloud datacenters, making it possible to give each tenant their own L2 network so as to isolate their traffic from the traffic of all other tenants. But there was a problem: the 4096 possible VLANs was not sufficient to account for all the tenants that a cloud might host, and to complicate matters, in a cloud the network needs to connect virtual machines rather than the physical machines that those VMs run on.
To address this problem, another standard called Virtual Extensible LAN (VXLAN) was introduced. Unlike the original approach, which effectively encapsulated a virtualized ethernet frame inside another ethernet frame, VXLAN encapsulates a virtual ethernet frame inside a UDP packet. This means a VXLAN-based virtual network (which is often referred to as an overlay network) runs on top of an IP-based network, which in turn runs on an underlying ethernet (or perhaps in just one VLAN of the underlying ethernet). VXLAN also makes it possible for one cloud tenant to have multiple VLANs of their own, which allows them to segregate their own internal traffic. This means it is ultimately possible to have a VLAN encapsulated in a VXLAN overlay encapsulated in a VLAN.
The powerful thing about virtualization is that when done right, it should be possible to nest one virtualized resource inside another virtualized resource, since after all, a virtual resource should behave just like a physical resources and we know how to virtualize physical resources! Said another way, being able to virtualize a virtual resource is the best proof that you have done a good job of virtualizing the original physical resource. To re-purpose the mythology of the World Turtle: It's virtual networks all the way down.
The actual VXLAN header is simple. It includes a 24-bit Virtual Network Id (VNI), plus some flag bits. It also implies a particular setting of the UDP source and destination port fields (see Section 5.1), with the destination port 4789 officially reserved for VXLANs. Figuring out how to uniquely identify virtual LANs (VLAN tags) and virtual networks (VXLAN VIDs) is the easy part. This is because encapsulation is the fundamental cornerstone of virtualization; all you need to add is an identifier that tells you which of many possible users this encapsulated packet belongs to.
The hard part is grappling with the idea of virtual networks being nested (encapsulated) inside virtual networks, which is networking’s version of recursion. The other challenge is understanding how to automate the creation, management, migration, and deletion of virtual networks, and on this front there is still a lot of room for improvement. Mastering this challenge will be at the heart of networking in the next decade, and while some of this work will undoubtedly happen in proprietary settings, there are open source network virtualization platforms (e.g., the Linux Foundation's Tungsten Fabric project) leading the way.
To continue reading about the cloudification of the Internet, see The Cloud is Eating the Internet.
To learn more about the maturation of virtual networks, we recommend:
Network Heresy, 2012.
Tungsten Fabric, 2018.
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Scales, fields, and a problem of Hurewicz | EMS Press
Lyubomyr Zdomskyy
Menger's basis property is a generalization of
\sigma
-comp\-actness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non
\sigma
-compact sets of reals with Menger's property. Special instances of these constructions give known counterexamples to conjectures of Menger and Hurewicz. We obtain the first explicit solution to the Hurewicz 1927 problem, that was previously solved by Chaber and Pol on a dichotomic basis. The constructed sets generate nontrivial subfields of the real line with strong combinatorial properties, and most of our results can be stated in a Ramsey-theoretic manner. Since we believe that this paper is of interest to a diverse mathematical audience, we have made a special effort to make it self-contained and accessible.
Boaz Tsaban, Lyubomyr Zdomskyy, Scales, fields, and a problem of Hurewicz. J. Eur. Math. Soc. 10 (2008), no. 3, pp. 837–866
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EUDML | Notes on algebraic functions. EuDML | Notes on algebraic functions.
Notes on algebraic functions.
Guan, Ke-Ying; Lei, Jinzhi
Guan, Ke-Ying, and Lei, Jinzhi. "Notes on algebraic functions.." International Journal of Mathematics and Mathematical Sciences 2003.13 (2003): 835-844. <http://eudml.org/doc/52140>.
author = {Guan, Ke-Ying, Lei, Jinzhi},
title = {Notes on algebraic functions.},
AU - Guan, Ke-Ying
TI - Notes on algebraic functions.
Singularities, monodromy, local behavior of solutions, normal forms
{}_{2}{F}_{1}
Articles by Guan
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Inventory Systems and Inventory Costing Methods - Course Hero
Principles of Accounting/Inventory/Inventory Systems and Inventory Costing Methods
Learn all about inventory systems and inventory costing methods in just a few minutes! Fabio Ambrosio, CPA, instructor of accounting at the Central Washington University, explains perpetual inventory systems, periodic inventory systems, and the first in, first out (FIFO), last in, first out (LIFO) and weight average methods of valuing inventory.
In a perpetual inventory system, each sale and purchase transaction related to merchandise is recorded directly to the inventory account as well as the corresponding subsidiary ledger as records are constantly updated in real time. Inventory can be tracked using various means, such as real-time databases, bar-code scanners, and point-of-sale terminals. Thus, the amount of inventory on hand and available is always current. In order to properly establish an inventory costing system, it is important to note this fact because it will be a consideration in determining how costs flow.
Under the perpetual cost system, depending on when the goods are sold, the company may have to use different cost points. For example, if the company sells 5 units on 3/30/18, then the range of costs is between $1.25 and $1.40. However, if the company sells 5 units on 4/30/18, then the range of costs is now between $1.25 and $1.55. It is important for the accountant to note the date of the sales and the date of the purchases. Therefore, with each sale or purchase, the accountant has a new basis on which to determine the cost of each unit. As such, the cost of sales is calculated on a rolling basis to provide updated information as needed.
Perpetual Sale of Goods Illustration
Under the perpetual inventory system, the inventory account is constantly up to date. Therefore, with each sale or purchase, the accountant has a new basis from which to determine the cost of the unit.
When using the periodic inventory system, a business's records are updated when a physical count of the inventory on hand is done, after everything has been counted. The inventory account is not systematically updated throughout a period, as is done when using the perpetual inventory system. The process of physical counting takes time, even with the aid of scanning devices and computers. So, with the periodic inventory system, the count may be done when merchandise inventory is low, perhaps during the off-season when there is less to count. Doing a physical count can serve as an accuracy check for a merchandising business using the perpetual inventory system. However, because inventory numbers are not updated until period-end, current sales and purchases data throughout the period would not be available in real time for determining the cost of units. Costs would be determined at the end of the designated period—often yearly—after the physical inventory is conducted. Just the ending inventory balance is updated in the general ledger.
For example, Company ABC starts with a beginning inventory of 10 units. Goods are purchased and sold throughout the period of one month—January. Relevant information to determine the cost of the goods sold includes the amount of sales, the gross profit, and the amount that will remain in inventory at the end of the month.
Information to Determine COGS with the Periodic FIFO Method
Jan. 1 Beginning inventory 10 $1.25 $12.50
Jan. 10 Purchase 5 $1.35 $6.75
Jan. 31 Total goods available 21 $27.95
Jan. 31 Less physical inventory 9 $11.98
Jan. 31 Cost of goods sold 12 $15.97
FIFO Method of Valuing Inventory
With the FIFO inventory valuation system, the units that are purchased first will be the first to be sold. The physical flow of the units is not followed, but the flow of costs is followed. The first set of costs acquired will be the first costs used in determining the cost of the goods sold.
For example, assume that Company ABC is using the perpetual inventory system. Its beginning inventory includes 10 units. Goods are purchased and sold throughout the period of one month—January. Relevant information to determine the cost of the goods sold includes the amount of sales, the gross profit, and the amount that will remain in inventory at the end of the month.
Information to Determine COGS with the Perpetual FIFO Method
Jan. 4 Sale at $3.00/unit 7
Jan. 22 Sale at $3.00/unit 3
Cost Flow Example
Using the first-in, first-out (FIFO) method, the flow of costs is followed to determine cost of goods sold, not the physical flow of units.
This illustration provides the same information found in the previous information table. In this illustration of cost flow, the beginning inventory and purchases are separated from the sales. This is important because the goal is to determine the proper costs of the inventory sold. The sale dates and number of units of goods sold determine which cost level will be used to arrive at the cost of goods sold. Referring to the table, at each sale the accountant must determine which goods are still currently available for sale. For the first sale of 7 units on 1/4/18 shown in the previous table, only 10 items are available in the beginning inventory. The 5 units purchased on 1/10/2018 and 6 units purchased on 1/30/2018 would not be relevant for the sale on 1/4/18, because those purchases have not occurred yet. Therefore, all 7 units purchased would have to come from the beginning inventory, which was bought at $1.25 per unit. As you can see in the next illustration, 3 units sold on 1/22/18 come from the remaining beginning inventory units. The 2 units sold on 1/28/18 come from the purchased units on 1/10/2018. The FIFO method starts from the top with the oldest costs, the first ones in.
The first-in, first-out (FIFO) cost flow starts from the top down. The first units (7 units) that are assumed to be sold are the oldest units with the oldest costs: in this case, $1.25 per unit.
Therefore, the cost of goods sold under the FIFO method would comprise all of the beginning inventory units and 2 units from the 1/10/2018 purchase. The remaining 3 units purchased on 1/10/2018 as well as the goods purchased on 1/30/2018 would be left over in ending inventory.
Valuing Inventory Using FIFO Method
Date Description Units Cost Total Cost
Remaining Goods (Ending Inventory)
Remember: the ending inventory now consists of 3 units at $1.35 each and 6 units at $1.45 each that will become the beginning inventory in the next period of a month. Thus, February begins with two different cost levels of inventory—$1.35 and $1.45.
LIFO Method of Valuing Inventory
The LIFO method of inventory valuation in some respects will be the opposite of FIFO. However, the challenge with using LIFO in a perpetual inventory system is that when using the most recent costs acquired, as new purchases are made, the cost levels used will always change.
The last-in, first-out (LIFO) cost flow starts from the bottom and works its way up. The first units that are assumed to be sold are the newest units with the most recent costs.
Based on the same information given in the FIFO example, the sale of the 7 units on 1/4/2018 must come from the beginning inventory level because it was the only level that existed at that point in time. However, when choosing which costs to use for the sales on 1/22/2018 and 1/28/2018, the most recent costs acquired are now from the purchase on 1/10/2018. Because this is LIFO, which starts from the bottom, the business must begin with the cost at the bottom at the time of the sale. In this example, the purchase that took place on 1/30/2018 is still not relevant because it was made after the sale dates. Therefore, the units in blue will be left in ending inventory. Those units will comprise all 6 units purchased on 1/30/2018 as well as 3 of the units from beginning inventory.
Valuing Inventory Using LIFO Method
Jan. 1 Beginning inventory 7 $1.25 $8.75
Similar to the FIFO method, at the end of the period two cost levels remain in ending inventory. Therefore, at the beginning of the next month, inventory contains two levels of costs: 3 units at $1.25 each and 6 units at $1.45 each.
Weighted Average Method of Valuing Inventory
In times of rising inventory costs, the weighted average method serves as a middle ground between the FIFO and LIFO methods so that inventory values and cost amounts will fall between FIFO and LIFO values. The illustration for weighted average will be different than FIFO and LIFO methods even though the same information is used. Using the same information, the new cost structure is recalculated after each purchase.
Inventory Information to Calculate Cost
Remaining Units 3 $1.25 $3.75
A new average does not need to be calculated for the first sale. For example, on January 10, Company ABC purchased 5 additional units at $1.35, each creating the need to calculate the average cost. Three units still remain at the $1.25 level.
Weighted Average Cost Calculation
Jan. 4 Remaining inventory 3 $1.25 $3.75
Jan. 10 Purchased 5 $1.35 $6.75
Total 8 $10.50
\text{Weighted Average Cost}=\text{Total Cost}/\text{Total Units}
\text{Weighted Average Cost}=\$10.50/8\;\text{units}
Weighted Average Cost = $1.31
Ideally, the cost will calculate to a round number; however, if it does not, the number should be rounded to four or five digits after the decimal point to ensure the averages are as accurate as possible. Rounding to fewer digits could cause a variation in the amounts because of rounding error. For example, $1.31 per unit, rounded from $1.3125, is used as the weighted average cost per unit. Thus, units sold on 1/22/2018 and 1/28/2018 will cost $1.31 per unit.
Valuing Inventory with Weighted Average Method
Weighted Average Summary
Jan. 4 Sale from inventory 7 $1.25 $8.75
Jan. 22 Sale at Weighted average 3 $1.31 $3.93
Total Cost of Goods $15.30
The next step is to calculate the cost of the remaining items. Remember that after every purchase, the average cost must be recalculated. Therefore, after the 1/30/2018 purchase of 6 units at $1.45 each, the new weighted average cost per unit is $1.40.
Ending Inventory Weighted Average Cost
Jan. 10 Remaining inventory 3 $1.31 $3.93
\text{Weighted Average Cost}=\text{Total Cost}/\text{Total Units}
\text{Weighted Average Cost}=\$12.63/9\;\text{units}
By recalculating the new weighted average cost per unit after the January 30 purchase, the ending inventory is calculated as $12.60, beginning the next month with one level of costs.
Recalculated Weighted Average Cost
Jan. 30 Weighted average after purchase 9 $1.40 $12.60
Remaining Goods (Ending Inventory) $12.60
<Inventory Cost Flow Assumptions>Comparing Inventory Costing Methods
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Circom Implementation - Electron Labs
Zk-proof of a single ED25519 signature
We will now discuss how we can define the ED25519 maths in the R1CS model, by using Circom. We will then use snarkJS to generate zk-proofs of ED25519 signatures. This section assumes a working knowledge of zk-snarks and Circom/snarkJS.
Step1: Defining ED25519 prime field inside the altbn prime field
Both Circom/zk-snarks and ED25519 are defined under a finite field. This means all math operations are performed under modulo a prime number.
Prime Number for ED25519: p = 2^255 - 19
Prime Number for Circom/zk-snarks: altbn_P = 21888242871839275222246405745257275088548364400416034343698204186575808495617
Note that the prime p used by ED25519 is different and greater in value than altbn_P. Hence, we must define an approach that allows us to perform ED25519 operations while keeping all numbers within the snark prime field.
One way to solve this problem is to represent all numbers as an array of base2 numbers (binary). This would make sure that no single element exceeds altbn_P but the entire array could represent larger numbers. In fact, we can choose any base b such that b < altbn_P
Max value of any number in ED25519 is pand it can be represented in a max of 255 bits. In base2^51, we need a max of 5 elements to represent all ed25519 numbers. The number of constraints is the same when multiplying 1*1 or 2^51*2^51. Hence, it makes sense to use the largest base possible.
We have chosen to use base 2^51. (although base2^85 is even better). Note that 2^51 < altbn_P which is necessary.
Next, we must define addition, multiplication, and modulus operator in base2^51 inside circom, so that we can start doing ED25519 maths.
Step2: Defining adder, multiplier, and modulus in base 2^51
Inputs and outputs to these circuits are arrays where each element of the array is less than 2^51
Find the implementation of base2^51 Adder here
Find the implementation of base2^51 Multiplier here. The algorithm is basically standard long multiplication, where the carry is handled in the end.
Find the implementation of base2^51 Modulus here. This template takes a base2^51 array as input and applied modulo p = 2^255 - 19. In the code, p is also represented as base2^51.
We perform modulo in such a way that long division is avoided. The algorithm is as follows -
define c = 2^255
Given input x s.t max(x) < p^2, calculate x%p
x%c = r (remainder) (note that max(r) < c)
x//c = q (quotient) (note that max(q) < p^2/c)
x = q*c + r
x = q*(p+19) + r
x%p = 19q%p + r%p
Now, calculate r%p -
if (r>=p): r%p = r-p
if (r<p): r%p = r
The same logic is extended to calculate 19q
In the circuit, we can't use if/else conditionals but a multiplexer. In the circuit, we apply the above algorithm recursively, and hence x can be of arbitrary size.
Step3: Defining ED25519 Point Addition in Circom
First, we must represent all the ED25519 operations as polynomials so that they can be represented in the R1CS constraint model.
Let's start with point addition. Re-arranging the point addition equation discussed in the previous section, we get:-
x_3 = x_1y_2 + x_2y_1 + (\small 121665/121666\normalsize )x_1x_2x_3y_1y_2 \newline y_3 + (\small 121665/121666\normalsize )x_1x_2y_1y_2y_3 = y_1y_2 + x_1x_2
Hence, if three points (x1, y1), (x2, y2) and (x3,y3) satisfy these two polynomials, we can say that the third point is the sum of the first two points.
In our circom implementation, rather than using cartesian coordinates, we use the radix format as defined in the reference implementation (given here). This changes the polynomials a bit, but the core logic is the same. Please see the code here.
Step4: Defining ED25519 Scalar Multiplication in Circom
Step5: Defining Full Signature Verification in Circom
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Rubber cis-polyprenylcistransferase - Wikipedia
Rubber cis-polyprenylcistransferase
In enzymology, a rubber cis-polyprenylcistransferase (EC 2.5.1.20) is an enzyme that catalyzes the chemical reaction
poly-cis-polyprenyl diphosphate + isopentenyl diphosphate
{\displaystyle \rightleftharpoons }
diphosphate + a poly-cis-polyprenyl diphosphate longer by one C5 unit
Thus, the two substrates of this enzyme are poly-cis-polyprenyl diphosphate and isopentenyl diphosphate, whereas its two products are diphosphate and poly-cis-polyprenyl diphosphate longer by one C5 unit.
This enzyme belongs to the family of transferases, specifically those transferring aryl or alkyl groups other than methyl groups. The systematic name of this enzyme class is poly-cis-polyprenyl-diphosphate:isopentenyl-diphosphate polyprenylcistransferase. Other names in common use include rubber allyltransferase, rubber transferase, isopentenyl pyrophosphate cis-1,4-polyisoprenyl transferase, cis-prenyl transferase, rubber polymerase, and rubber prenyltransferase. This enzyme participates in biosynthesis of steroids.
Archer BL, Cockbain EG (1969). "Rubber transferase from Hevea brasiliensis latex". Methods Enzymol. Methods in Enzymology. 15: 476–480. doi:10.1016/S0076-6879(69)15018-1. ISBN 978-0-12-181872-2.
McMullen AI, McSweeney GP (October 1966). "The biosynthesis of rubber: Incorporation of isopentenyl pyrophosphate into purified rubber particles by a soluble latex serum enzyme". The Biochemical Journal. 101 (1): 42–7. PMC 1270063. PMID 16742418.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Rubber_cis-polyprenylcistransferase&oldid=918616180"
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Exponentiation - Simple English Wikipedia, the free encyclopedia
It has been suggested that the text on Exponent be merged into (added to) this article. (Discuss) Proposed since February 2021.
In mathematics, exponentiation (power) is an arithmetic operation on numbers. It can be thought of as repeated multiplication, just as multiplication can be thought of as repeated addition.
{\displaystyle x}
{\displaystyle y}
, the exponentiation of
{\displaystyle x}
{\displaystyle y}
{\displaystyle x^{y}}
, and read as "
{\displaystyle x}
{\displaystyle y}
", or "
{\displaystyle x}
{\displaystyle y}
th power".[1][2] Other methods of mathematical notation have been used in the past. When the upper index cannot be written, people can write powers using the ^ or ** signs, so that 2^4 or 2**4 means
{\displaystyle 2^{4}}
Here, the number
{\displaystyle x}
is called base, and the number
{\displaystyle y}
is called exponent. For example, in
{\displaystyle 2^{4}}
, 2 is the base and 4 is the exponent.
{\displaystyle 2^{4}}
, one simply multiply 4 copies of 2. So
{\displaystyle 2^{4}=2\cdot 2\cdot 2\cdot 2}
, and the result is
{\displaystyle 2\cdot 2\cdot 2\cdot 2=16}
. The equation could be read out loud as "2 raised to the power of 4 equals 16."
More examples of exponentiation are:
{\displaystyle 5^{3}=5\cdot {}5\cdot {}5=125}
{\displaystyle x^{2}=x\cdot {}x}
{\displaystyle 1^{x}=1}
for every number x
If the exponent is equal to 2, then the power is called square, because the area of a square is calculated using
{\displaystyle a^{2}}
{\displaystyle x^{2}}
{\displaystyle x}
Similarly, if the exponent is equal to 3, then the power is called cube, because the volume of a cube is calculated using
{\displaystyle a^{3}}
{\displaystyle x^{3}}
is the cube of
{\displaystyle x}
If the exponent is equal to -1, then the power is simply the reciprocal of the base. So
{\displaystyle x^{-1}={\frac {1}{x}}}
If the exponent is an integer less than 0, then the power is the reciprocal raised to the opposite exponent. For example:
{\displaystyle 2^{-3}=\left({\frac {1}{2}}\right)^{3}={\frac {1}{8}}}
If the exponent is equal to
{\displaystyle {\tfrac {1}{2}}}
, then the result of exponentiation is the square root of the base, with
{\displaystyle x^{\frac {1}{2}}={\sqrt {x}}.}
{\displaystyle 4^{\frac {1}{2}}={\sqrt {4}}=2}
Similarly, if the exponent is
{\displaystyle {\tfrac {1}{n}}}
, then the result is the nth root, where:
{\displaystyle a^{\frac {1}{n}}={\sqrt[{n}]{a}}}
If the exponent is a rational number
{\displaystyle {\tfrac {p}{q}}}
, then the result is the qth root of the base raised to the power of p:
{\displaystyle a^{\frac {p}{q}}={\sqrt[{q}]{a^{p}}}}
In some cases, the exponent may not even be rational. To raise a base a to an irrational xth power, we use an infinite sequence of rational numbers (xn), whose limit is x:
{\displaystyle x=\lim _{n\to \infty }x_{n}}
{\displaystyle a^{x}=\lim _{n\to \infty }a^{x_{n}}}
There are some rules which make the calculation of exponents easier:[3]
{\displaystyle \left(a\cdot b\right)^{n}=a^{n}\cdot {}b^{n}}
{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}},\quad b\neq 0}
{\displaystyle a^{r}\cdot {}a^{s}=a^{r+s}}
{\displaystyle {\frac {a^{r}}{a^{s}}}=a^{r-s},\quad a\neq 0}
{\displaystyle a^{-n}={\frac {1}{a^{n}}},\quad a\neq 0}
{\displaystyle \left(a^{r}\right)^{s}=a^{r\cdot s}}
{\displaystyle a^{0}=1}
It is possible to calculate exponentiation of matrices. In this case, the matrix must be square. For example,
{\displaystyle I^{2}=I\cdot I=I}
Commutativity[change | change source]
Both addition and multiplication are commutative. For example, 2+3 is the same as 3+2, and 2 · 3 is the same as 3 · 2. Although exponentiation is repeated multiplication, it is not commutative. For example, 2³=8, but 3²=9.
Inverse Operations[change | change source]
Addition has one inverse operation: subtraction. Also, multiplication has one inverse operation: division.
But exponentiation has two inverse operations: The root and the logarithm. This is the case because the exponentiation is not commutative. You can see this in this example:
If you have x+2=3, then you can use subtraction to find out that x=3−2. This is the same if you have 2+x=3: You also get x=3−2. This is because x+2 is the same as 2+x.
If you have x · 2=3, then you can use division to find out that x=
{\textstyle {\frac {3}{2}}}
. This is the same if you have 2 · x=3: You also get x=
{\textstyle {\frac {3}{2}}}
. This is because x · 2 is the same as 2 · x
If you have x²=3, then you use the (square) root to find out x: you get the result that x =
{\textstyle {\sqrt[{2}]{3}}}
. However, if you have 2x=3, then you can not use the root to find out x. Rather, you have to use the (binary) logarithm to find out x: you get the result that x=log2(3).
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Exponentiation&oldid=7390927"
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Inner_automorphism Knowpia
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. They can be realized via simple operations from within the group itself, hence the adjective "inner". These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup is the definition of the outer automorphism group.
If G is a group and g is an element of G (alternatively, if G is a ring, and g is a unit), then the function
{\displaystyle {\begin{aligned}\varphi _{g}\colon G&\to G\\\varphi _{g}(x)&:=g^{-1}xg\end{aligned}}}
is called (right) conjugation by g (see also conjugacy class). This function is an endomorphism of G: for all
{\displaystyle x_{1},x_{2}\in G,}
{\displaystyle \varphi _{g}(x_{1}x_{2})=g^{-1}x_{1}x_{2}g=\left(g^{-1}x_{1}g\right)\left(g^{-1}x_{2}g\right)=\varphi _{g}(x_{1})\varphi _{g}(x_{2}),}
where the second equality is given by the insertion of the identity between
{\displaystyle x_{1}}
{\displaystyle x_{2}.}
Furthermore, it has a left and right inverse, namely
{\displaystyle \varphi _{g^{-1}}.}
{\displaystyle \varphi _{g}}
is bijective, and so an isomorphism of G with itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation.[1]
When discussing right conjugation, the expression
{\displaystyle g^{-1}xg}
is often denoted exponentially by
{\displaystyle x^{g}.}
This notation is used because composition of conjugations satisfies the identity:
{\displaystyle \left(x^{g_{1}}\right)^{g_{2}}=x^{g_{1}g_{2}}}
{\displaystyle g_{1},g_{2}\in G.}
This shows that conjugation gives a right action of G on itself.
Inner and outer automorphism groupsEdit
The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of G is a group, the inner automorphism group of G denoted Inn(G).
Inn(G) is a normal subgroup of the full automorphism group Aut(G) of G. The outer automorphism group, Out(G) is the quotient group
{\displaystyle \operatorname {Out} (G)=\operatorname {Aut} (G)/\operatorname {Inn} (G).}
The outer automorphism group measures, in a sense, how many automorphisms of G are not inner. Every non-inner automorphism yields a non-trivial element of Out(G), but different non-inner automorphisms may yield the same element of Out(G).
Saying that conjugation of x by a leaves x unchanged is equivalent to saying that a and x commute:
{\displaystyle a^{-1}xa=x\iff ax=xa.}
Therefore the existence and number of inner automorphisms that are not the identity mapping is a kind of measure of the failure of the commutative law in the group (or ring).
An automorphism of a group G is inner if and only if it extends to every group containing G.[2]
By associating the element a ∈ G with the inner automorphism f(x) = xa in Inn(G) as above, one obtains an isomorphism between the quotient group G/Z(G) (where Z(G) is the center of G) and the inner automorphism group:
{\displaystyle G/Z(G)\cong \operatorname {Inn} (G).}
This is a consequence of the first isomorphism theorem, because Z(G) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism (conjugation changes nothing).
Non-inner automorphisms of finite p-groupsEdit
A result of Wolfgang Gaschütz says that if G is a finite non-abelian p-group, then G has an automorphism of p-power order which is not inner.
It is an open problem whether every non-abelian p-group G has an automorphism of order p. The latter question has positive answer whenever G has one of the following conditions:
G is nilpotent of class 2
G is a regular p-group
G/Z(G) is a powerful p-group
The centralizer in G, CG, of the center, Z, of the Frattini subgroup, Φ, of G, CG ∘ Z ∘ Φ(G), is not equal to Φ(G)
The inner automorphism group of a group G, Inn(G), is trivial (i.e., consists only of the identity element) if and only if G is abelian.
The group Inn(G) is cyclic only when it is trivial.
At the opposite end of the spectrum, the inner automorphisms may exhaust the entire automorphism group; a group whose automorphisms are all inner and whose center is trivial is called complete. This is the case for all of the symmetric groups on n elements when n is not 2 or 6. When n = 6, the symmetric group has a unique non-trivial class of outer automorphisms, and when n = 2, the symmetric group, despite having no outer automorphisms, is abelian, giving a non-trivial center, disqualifying it from being complete.
If the inner automorphism group of a perfect group G is simple, then G is called quasisimple.
Lie algebra caseEdit
An automorphism of a Lie algebra 𝔊 is called an inner automorphism if it is of the form Adg, where Ad is the adjoint map and g is an element of a Lie group whose Lie algebra is 𝔊. The notion of inner automorphism for Lie algebras is compatible with the notion for groups in the sense that an inner automorphism of a Lie group induces a unique inner automorphism of the corresponding Lie algebra.
If G is the group of units of a ring, A, then an inner automorphism on G can be extended to a mapping on the projective line over A by the group of units of the matrix ring, M2(A). In particular, the inner automorphisms of the classical groups can be extended in that way.
^ S., Dummit, David (2004). Abstract algebra. Foote, Richard M., 1950- (3. ed.). Hoboken, NJ: Wiley. p. 45. ISBN 9780471452348. OCLC 248917264.
^ Schupp, Paul E. (1987), "A characterization of inner automorphisms" (PDF), Proceedings of the American Mathematical Society, American Mathematical Society, 101 (2): 226–228, doi:10.2307/2045986, JSTOR 2045986, MR 0902532
Abdollahi, A. (2010), "Powerful p-groups have non-inner automorphisms of order p and some cohomology", J. Algebra, 323 (3): 779–789, arXiv:0901.3182, doi:10.1016/j.jalgebra.2009.10.013, MR 2574864
Abdollahi, A. (2007), "Finite p-groups of class 2 have noninner automorphisms of order p", J. Algebra, 312 (2): 876–879, arXiv:math/0608581, doi:10.1016/j.jalgebra.2006.08.036, MR 2333188
Deaconescu, M.; Silberberg, G. (2002), "Noninner automorphisms of order p of finite p-groups", J. Algebra, 250: 283–287, doi:10.1006/jabr.2001.9093, MR 1898386
Gaschütz, W. (1966), "Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen", J. Algebra, 4: 1–2, doi:10.1016/0021-8693(66)90045-7, MR 0193144
Liebeck, H. (1965), "Outer automorphisms in nilpotent p-groups of class 2", J. London Math. Soc., 40: 268–275, doi:10.1112/jlms/s1-40.1.268, MR 0173708
Remeslennikov, V.N. (2001) [1994], "Inner automorphism", Encyclopedia of Mathematics, EMS Press
Weisstein, Eric W. "Inner Automorphism". MathWorld.
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Draw a diagram that could be represented by each of the following number expressions. Then circle the terms and calculate the value of each expression.
4+2(5)+(-3)
Remember the Order of Operations when solving this kind of problem.
4+2(5)+(-3)
4+2(5)+(-3)
4+10+(-3)
14+(-3)
11
There are many possible diagrams for this problem.
\begin{array}{l} + \; +\; +\; + \qquad +\; +\; +\; +\; + \\ -\; -\; - \quad \qquad \; +\; +\; + \; + \; + \end{array}
4+2(5+(−3))
Don't forget to do everything inside the parentheses first before you simplify the rest of the expression.
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Voltage-controlled switch with hysteresis - MATLAB - MathWorks América Latina
Hysteresis voltage, VH
Control voltage for on state, VON
Control voltage for off state, IOFF
Voltage-controlled switch with hysteresis
The Voltage-Controlled Switch block represents the electrical characteristics of a switch whose state is controlled by the voltage across the input ports (the controlling voltage).
This block models either a variable-resistance or a short-transition switch. For a variable-resistance switch, set the Switch model parameter to Smooth transition between Von and Voff. For a short-transition switch, set Switch model to Abrupt transition after delay.
When the controlling voltage is greater than or equal to the sum of the Threshold voltage, VT and Hysteresis voltage, VH parameter values, the switch is closed and has a resistance equal to the On resistance, RON parameter value.
When the controlling voltage is less than the Threshold voltage, VT parameter value minus the Hysteresis voltage, VH parameter value, the switch is open and has a resistance equal to the Off resistance, ROFF parameter value.
When the controlling voltage is greater than or less than the Threshold voltage, VT parameter value by an amount less than or equal to the Hysteresis voltage, VH parameter value, the voltage is in the crossover region and the state of the switch remains unchanged.
If the Hysteresis voltage, VH parameter value is less than 0, the block models a variable-resistance switch independently of the value you set for the Switch model parameter.
When the Control voltage for on state, VON is greater than the Control voltage for off state, VOFF:
If the controlling voltage is greater than the Control voltage for on state, VON parameter value, the switch is closed and has a resistance equal to the On resistance, RON parameter value.
If the controlling voltage is less than the Control voltage for off state, VOFF parameter value, the switch is open and has a resistance equal to the Off resistance, ROFF parameter value.
If the controlling voltage is greater than the Control voltage for off state, VOFF parameter value or less than the Control voltage for on state, VON parameter value, the resistance is defined by:
{R}_{s}=\mathrm{exp}\left[{L}_{m}+3{L}_{r}\left(\frac{{V}_{c}-{V}_{m}}{2{V}_{d}}\right)-2{L}_{r}\left(\frac{{\left({V}_{c}-{V}_{m}\right)}^{3}}{{V}_{d}^{3}}\right)\right]
{L}_{m}=ln\left[{\left({R}_{ON}*{R}_{OFF}\right)}^{\frac{1}{2}}\right]
{L}_{r}=ln\left(\frac{{R}_{ON}}{{R}_{OFF}}\right)
Vc is the controlling voltage.
{V}_{m}=\frac{{V}_{ON}+{V}_{OFF}}{2}
is the mean of the control voltages.
{V}_{d}={V}_{ON}-{V}_{OFF}
is the difference between the control voltages.
When the Control voltage for on state, VON is less than the Control voltage for off state, VOFF:
If the controlling voltage is less than the Control voltage for on state, VON parameter value, the switch is closed and has a resistance equal to the On resistance, RON parameter value.
If the controlling voltage is greater than the Control voltage for off state, VOFF parameter value, the switch is open and has a resistance equal to the Off resistance, ROFF parameter value.
If the controlling voltage is less than the Control voltage for off state, VOFF parameter value or greater than the Control voltage for on state, VON parameter value, the resistance is defined by:
{R}_{s}=\mathrm{exp}\left[{L}_{m}+3{L}_{r}\left(\frac{{I}_{c}-{I}_{m}}{2{I}_{d}}\right)-2{L}_{r}\left(\frac{{\left({I}_{c}-{I}_{m}\right)}^{3}}{{I}_{d}^{3}}\right)\right]
Increase the Hysteresis voltage, VH parameter value to reduce switch chatter.
Electrical conserving port associated with the voltage controlled switch positive input.
Electrical conserving port associated with the voltage controlled switch negative input.
Electrical conserving port associated with the voltage controlled switch positive output.
Electrical conserving port associated with the voltage controlled switch negative output.
Voltage above which the block interprets the controlling voltage as HIGH.
The controlling voltage must differ from the threshold voltage by at least the Hysteresis voltage, VH parameter value to change the state of the switch.
Hysteresis voltage, VH — Hysteresis voltage
Amount by which the controlling voltage must exceed or fall below the Threshold voltage, VT parameter value to change the state of the switch.
Control voltage for on state, VON — Control voltage for on state
Switch control voltage for the on state.
To enable this parameter, set Switch model to Smooth transition between Von and Voff.
Control voltage for off state, IOFF — Control voltage for off state
Switch control voltage for the off state.
Set Switch model to Smooth transition between Von and Voff. The default value is 1e6.
Environment Parameters | Current-Controlled Switch
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Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone
Lei Qiao, "Nontangential Limits for Modified Poisson Integrals of Boundary Functions in a Cone", Journal of Function Spaces, vol. 2012, Article ID 825240, 9 pages, 2012. https://doi.org/10.1155/2012/825240
Lei Qiao1
1Department of Mathematics and Information Science, Henan University of Economics and Law, Zhengzhou 450002, China
Academic Editor: Dachun Yang
Our aim in this paper is to deal with non-tangential limits for modified Poisson integrals of boundary functions in a cone, which generalized results obtained by Brundin and Mizuta-Shimomura.
Let 𝐑 and 𝐑+ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by 𝐑𝑛(𝑛≥2) the 𝑛-dimensional Euclidean space. A point in 𝐑𝑛 is denoted by 𝑃=(𝑋,𝑥𝑛), where 𝑋=(𝑥1,𝑥2,…,𝑥𝑛−1). The Euclidean distance of two points 𝑃 and 𝑄 in 𝐑𝑛 is denoted by |𝑃−𝑄|. Also |𝑃−𝑂| with the origin 𝑂 of 𝐑𝑛 is simply denoted by |𝑃|. The boundary, the closure, and the complement of a set 𝐒 in 𝐑𝑛 are denoted by 𝜕𝐒, 𝐒, and 𝐒𝑐, respectively.
We introduce a system of spherical coordinates (𝑟,Θ), Θ=(𝜃1,𝜃2,…,𝜃𝑛−1), in 𝐑𝑛 which are related to cartesian coordinates (𝑥1,𝑥2,…,𝑥𝑛−1,𝑥𝑛) by 𝑥𝑛=𝑟cos𝜃1.
For positive functions ℎ1 and ℎ2, we say that ℎ1≲ℎ2 if ℎ1≤𝑀ℎ2 for some constant 𝑀>0. If ℎ1≲ℎ2 and ℎ2≲ℎ1, we say that ℎ1≈ℎ2.
For 𝑃∈𝐑𝑛 and 𝑅>0, let 𝐵(𝑃,𝑅) denote the open ball with center at 𝑃 and radius 𝑅 in 𝐑𝑛. The unit sphere and the upper half unit sphere are denoted by 𝐒𝑛−1 and 𝐒+𝑛−1, respectively. For simplicity, a point (1,Θ) on 𝐒𝑛−1 and the set {Θ;(1,Θ)∈Ω} for a set Ω, Ω⊂𝐒𝑛−1 are often identified with Θ and Ω, respectively. For two sets Ξ⊂𝐑+ and Ω⊂𝐒𝑛−1, the set {(𝑟,Θ)∈𝐑𝑛;𝑟∈Ξ,(1,Θ)∈Ω} in 𝐑𝑛 is simply denoted by Ξ×Ω. In particular, the half space 𝐑+×𝐒+𝑛−1={(𝑋,𝑥𝑛)∈𝐑𝑛;𝑥𝑛>0} will be denoted by 𝐓𝑛.
By 𝐶𝑛(Ω), we denote the set 𝐑+×Ω in 𝐑𝑛 with the domain Ω on 𝐒𝑛−1. We call it a cone. Then 𝑇𝑛 is a special cone obtained by putting Ω=𝐒+𝑛−1. We denote the sets 𝐼×Ω and 𝐼×𝜕Ω with an interval on 𝐑 by 𝐶𝑛(Ω;𝐼) and 𝑆𝑛(Ω;𝐼). By 𝑆𝑛(Ω) we denote 𝑆𝑛(Ω;(0,+∞)) which is 𝜕𝐶𝑛(Ω)−{𝑂}.
Let Ω be a domain on 𝐒𝑛−1 with smooth boundary. Consider the Dirichlet problem: Λ𝑛+𝜆𝜑=0onΩ,𝜑=0on𝜕Ω,(1.1) where Λ𝑛 is the spherical part of the Laplace operator Δ𝑛∶Δ𝑛=𝑛−1𝑟𝜕+𝜕𝜕𝑟2𝜕𝑟2+Λ𝑛𝑟2.(1.2) We denote the least positive eigenvalue of this boundary value problem by 𝜆Ω and the normalized positive eigenfunction corresponding to 𝜆Ω by 𝜑Ω(Θ), Ω𝜑2Ω(Θ)𝑑𝜎Θ=1,(1.3) where 𝑑𝜎Θ is the surface area on 𝑆𝑛−1. We denote the solutions of the equation 𝑡2+(𝑛−2)𝑡−𝜆Ω=0 by 𝛼Ω,−𝛽Ω (𝛼Ω,𝛽Ω>0). If Ω=𝐒+𝑛−1, then 𝛼Ω=1,𝛽Ω=𝑛−1, and 𝜑1(Θ)=(2𝑛𝑠𝑛−1)1/2cos𝜃1, where 𝑠𝑛 is the surface area 2𝜋𝑛/2(Γ(𝑛/2))−1 of 𝐒1.
To simplify our consideration in the following, we will assume that if 𝑛≥3, then Ω is a 𝐶2,𝛼-domain (0<𝛼<1) on 𝐒𝑛−1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see ([1], pages 88-89) for the definition of 𝐶2,𝛼-domain). Then by modifying Miranda’s method ([2], pages 7-8), we can prove the following inequality: 𝜑Ω(Θ)≈dist(Θ,𝜕Ω)(Θ∈Ω).(1.4)
For any (1,Θ)∈Ω, we have (see [3]) 𝜑Ω(Θ)≈dist(1,Θ),𝜕𝐶𝑛(Ω),(1.5) which yields that 𝛿(𝑃)≈𝑟𝜑Ω(Θ),(1.6) where 𝛿(𝑃)=dist(𝑃,𝜕𝐶𝑛(Ω)) and 𝑃=(𝑟,Θ)∈𝐶𝑛(Ω).
Let 𝐺Ω(𝑃,𝑄)(𝑃=(𝑟,Θ),𝑄=(𝑡,Φ)∈𝐶𝑛(Ω)) be the Green function of 𝐶𝑛(Ω). We define the Poisson kernel 𝐾Ω(𝑃,𝑄) by 𝐾Ω1(𝑃,𝑄)=𝑐𝑛𝜕𝜕𝑛𝑄𝐺Ω(𝑃,𝑄),(1.7) where 𝑐𝑛=(2𝜋𝑛=2,𝑛−2)𝑠𝑛𝑛≥3,(1.8)𝑄∈𝑆𝑛(Ω) and 𝜕/𝜕𝑛𝑄 denotes the differentiation at 𝑄 along the inward normal into 𝐶𝑛(Ω).
In this paper, we consider functions 𝑓∈𝐿𝑝(𝜕𝐶𝑛(Ω)), where 1≤𝑝<∞. Then the Poisson integral 𝑊Ω𝑓(𝑃)(𝑃∈𝐶𝑛(Ω)) is defined by 𝑊Ω𝑓(𝑃)=𝑆𝑛(Ω)𝐾Ω(𝑃,𝑄)𝑓(𝑄)𝑑𝜎𝑄,(1.9) where 𝑑𝜎𝑄 is the surface area element on 𝑆𝑛(Ω).
Remark 1.1. Let Ω=𝐒+𝑛−1. Then 𝐺𝐒+𝑛−1||(𝑃,𝑄)=log𝑃−𝑄∗||||||||||−log𝑃−𝑄𝑛=2,𝑃−𝑄2−𝑛−||𝑃−𝑄∗||2−𝑛𝑛≥3,(1.10) where 𝑄∗=(𝑌,−𝑦𝑛), that is, 𝑄∗ is the mirror image of 𝑄=(𝑌,𝑦𝑛) with respect to 𝜕𝑇𝑛. Hence, for the two points 𝑃=(𝑋,𝑥𝑛)∈𝑇𝑛 and 𝑄=(𝑌,𝑦𝑛)∈𝜕𝑇𝑛, we have 𝑐𝑛𝐾𝐒+𝑛−1𝜕(𝑃,𝑄)=𝜕𝑛𝑄𝐺𝐒+𝑛−12||||(𝑃,𝑄)=𝑃−𝑄−2𝑥𝑛||||𝑛=2,2(𝑛−2)𝑃−𝑄−𝑛𝑥𝑛𝑛≥3.(1.11)
We fix an open, nonempty, and bounded set 𝐺(Ω)⊂𝜕𝐶𝑛(Ω). In 𝐶𝑛(Ω), we normalise the extension, with respect to 𝐺(Ω), by 𝒫ℐΩ𝑊𝑓(𝑃)=Ω𝑓(𝑃)𝑊Ω𝜒𝐺(Ω)(,𝑃)(1.12) where 𝜒𝐺(Ω) denotes the characteristic function of 𝐺(Ω).
Let Γ(Ω,𝜁)=𝑃=(𝑟,Θ)∈𝐶𝑛||||(Ω)∶(𝑟,Θ)−𝜁≲𝛿(𝑃)(1.13) be a nontangential cone in 𝐶𝑛(Ω) with vertex 𝜁∈𝜕𝐶𝑛(Ω).
We define ℵ𝑝1(𝑓,𝑙,𝑃)=𝑙𝑛−1𝐵(𝑃,𝑙)||||𝑓(𝑄)𝑝𝑑𝜎𝑄1/𝑝,𝔼𝑝𝑓(𝐺(Ω))=𝑃∈𝐺(Ω)∶ℵ𝑝.(𝑓−𝑓(𝑃),𝑙,𝑃)⟶0as𝑙⟶0(1.14)
Note that, if 𝑓∈𝐿𝑝(𝜕𝐶𝑛(Ω)), then |𝐺(Ω)⧵𝔼𝑝𝑓(𝐺(Ω))|=0 (a.e. point is a Lebesgue point).
In 𝑇𝑛, the following conclusion was proved by Brundin (see ([4], pages 11–16)) and Mizuta and Shimomura (see ([5], Theorem 3)), respectively. In the unit disc, about related results, we refer the readers to the papers by Sjögren (see [6, 7]), Rönning (see [8]), and Brundin (see [9]).
Theorem A. For a.e. 𝜁∈𝐺(𝐒+𝑛−1), 𝒫ℐ𝐒+𝑛−1𝑓(𝑃)→𝑓(𝜁) (see Remark 1.1 for the definition of 𝒫ℐ𝐒+𝑛−1𝑓(𝑃)) as 𝑃→𝜁 along Γ(𝐒+𝑛−1,𝜁).
Our aim is to generalize Theorem A to the conical case.
Theorem 1.2. For any 𝜁∈𝔼𝑝𝑓(𝐺(Ω)) (in particular, for a.e. 𝜁∈𝐺(Ω)) one has that 𝒫ℐΩ𝑓(𝑃)→𝑓(𝜁) as 𝑃→𝜁 along Γ(Ω,𝜁).
Lemma 2.1. One has 𝐾Ω(𝑃,𝑄)≈𝑟−𝛽Ω𝑡𝛼Ω−1𝜑Ω(Θ),resp.𝐾Ω(𝑃,𝑄)≈𝑟𝛼Ω𝑡−𝛽Ω−1𝜑Ω,(Θ)(2.1) for any 𝑃=(𝑟,Θ)∈𝐶𝑛(Ω) and any 𝑄=(𝑡,Φ)∈𝑆𝑛(Ω) satisfying 0<𝑡/𝑟≤4/5(resp.0<𝑟/𝑡≤4/5); 𝐾Ω(𝑃,𝑄)≈𝑟𝜑Ω(Θ)||||𝑃−𝑄𝑛,(2.2) for any 𝑃=(𝑟,Θ)∈𝐶𝑛(Ω) and any 𝑄=(𝑡,Φ)∈𝑆𝑛(Ω;(4𝑟/5,5𝑟/4)).
Proof. These immediately follow from ([10], Lemma 2), ([11], Lemma 4 and Remark), and (1.4).
Lemma 2.2. One has 𝑊Ω1(𝑃)=𝑂(1)as𝑃⟶𝜁∈𝐺.(2.3)
Proof. Write 𝑊Ω1(𝑃)=𝐸1+𝐸2+𝐸3=𝑈1(𝑃)+𝑈2(𝑃)+𝑈3(𝑃),(2.4) where 𝐸1=𝑆𝑛4Ω;0,5𝑟,𝐸2=𝑆𝑛5Ω;4𝑟,∞,𝐸3=𝑆𝑛4Ω;55𝑟,4𝑟.(2.5)
By (2.1), we have the following estimates 𝑈1(𝑃)≈𝑟−𝛽Ω𝜑Ω(Θ)𝐸1𝑡𝛼Ω−1𝑑𝜎𝑄≈𝑠𝑛𝛽Ω45𝛽Ω𝜑Ω𝑈(Θ),(2.6)2𝑠(𝑃)≈𝑛𝛼Ω45𝛼Ω𝜑Ω(Θ).(2.7)
Next, we will estimate 𝑈3(𝑃). Take a sufficiently small positive number 𝑘 such that 𝐸3⊂𝑃=(𝑟,Θ)∈Λ(𝑘)𝐵1𝑃,2𝑟,(2.8) where Λ(𝑘)=𝑃=(𝑟,Θ)∈𝐶𝑛(Ω);inf𝑧∈𝜕Ω||||(1,Θ)−(1,𝑧)<𝑘,0<𝑟<∞.(2.9)
Since 𝑃→𝜁∈𝐺, we only consider the case 𝑃∈Λ(𝑘). Now put 𝐻𝑖(𝑃)=𝑄∈𝐸3;2𝑖−1𝛿||||(𝑃)≤𝑃−𝑄<2𝑖𝛿.(𝑃)(2.10)
Since 𝑆𝑛(Ω)∩{𝑄∈𝐑𝑛∶|𝑃−𝑄|<𝛿(𝑃)}=Ø, we have 𝑈3(𝑃)≈𝑖(𝑃)𝑖=0𝐻𝑖(𝑃)𝑟𝜑Ω(Θ)||||𝑃−𝑄𝑛𝑑𝜎𝑄,(2.11) where 𝑖(𝑃) is a positive integer satisfying 2𝑖(𝑃)−1𝛿(𝑃)≤𝑟/2<2𝑖(𝑃)𝛿(𝑃).
By (1.6) we have 𝐻𝑖(𝑃)𝑟𝜑Ω(Θ)||||𝑃−𝑄𝑛𝑑𝜎𝑄≈𝑟𝜑Ω(Θ)𝐻𝑖(𝑃)1𝛿(𝑃)𝑑𝜎𝑄=𝑟𝜑Ω(Θ)𝑠𝛿(𝑃)𝑛2𝑖(𝑃)≈𝑠𝑛2𝑖(𝑃)(2.12) for 𝑖=0,1,2,…,𝑖(𝑃).
So 𝑈3(𝑃)≈𝑂(1).(2.13)
Combining (2.6)–(2.13), Lemma 2.2 is proved.
Lemma 2.3. One has 𝑊Ω𝜒𝐺(Ω)(𝑃)=𝑊Ω1(𝑃)+𝑂(1)as𝑃⟶𝜁∈𝐺(Ω).(2.14)
Proof. In fact, we only need to prove 𝑈4(𝑃)=𝑆𝑛(Ω)−𝐺(Ω)𝐾Ω(𝑃,𝑄)𝑑𝜎𝑄≲𝑂(1).(2.15)
Write 𝑈4(𝑃)=(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸1+(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸2+(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸3=𝑈5(𝑃)+𝑈6(𝑃)+𝑈7(𝑃),(2.16) where 𝐸1, 𝐸2, and 𝐸3 are sets on 𝑆𝑛(Ω) used in Lemma 2.2.
Obviously, 𝑈5(𝑃)≲𝑈1𝑈(𝑃)≈𝑂(1),(2.17)6(𝑃)≲𝑈2(𝑃)≈𝑂(1).(2.18)
Further, we have by (2.2) 𝑈7(𝑃)≈𝑟𝜑Ω(Θ)(𝑆𝑛(Ω)−𝐺(Ω))∩𝐸31||||𝑃−𝑄𝑛𝑑𝜎𝑄≲𝑠𝑛𝑑||𝜁||𝜑Ω(Θ)(𝑃⟶𝜁∈𝐺(Ω)),(2.19) where 𝑑=inf𝑄∈𝜕𝐶𝑛(Ω)−𝐺(Ω)||||.𝑄−𝜁(2.20)
Combining (2.17)–(2.19), (2.15) holds which gives the conclusion.
3. Proof of the Theorem 1.2
As 𝑃→𝜁∈𝐺(Ω), 𝑊Ω𝜒𝐺(Ω)(𝑃)=𝑂(1) from Lemmas 2.2 and 2.3.
Now, let 𝑓∈𝐿𝑝(𝜕𝐶𝑛(Ω)) and 𝜁∈𝔼𝑝𝑓(𝐺(Ω)) be given. We may, without loss of generality, assume that 𝑓(𝜁)=0. Furthermore, we assume that 𝑃=(𝑟,Θ)∈Γ(Ω,𝜁). For short, let 𝑠=|(𝑟,Θ)−𝜁|. We write 𝑊Ω𝑓(𝑃)=𝐸1+𝐸2+𝐸3∩𝐵(𝜁,2𝑠)+𝐸3∩𝐵𝑐(𝜁,2𝑠)=𝑉1𝑓(𝑃)+𝑉2𝑓(𝑃)+𝑉3𝑓(𝑃)+𝑉4𝑓(𝑃),(3.1) where 𝐸1, 𝐸2, and 𝐸3 are sets on 𝑆𝑛(Ω) used in Lemma 2.2.
By using Hölder’s inequality, (2.1), we have the following estimates ||𝑉1||𝑓(𝑃)≲𝑟−𝛽Ω𝜑Ω(Θ)𝐸1𝑡𝛼Ω−1𝑓(𝑄)𝑑𝜎𝑄≲𝑟(1−𝑛)/𝑝‖𝑓‖𝑝,||𝑉2||𝑓(𝑃)≲𝑟(1−𝑛)/𝑝‖𝑓‖𝑝.(3.2)
Similar to the estimate of 𝑈3(𝑃) in Lemma 2.2, we only consider the following inequality by (1.6) 𝐻𝑖(𝑃)𝑟𝜑Ω(Θ)||||𝑃−𝑄𝑛𝑑𝜎𝑄≈𝑟𝜑Ω(Θ)𝐻𝑖(𝑃)12𝑖−1𝛿(𝑃)𝑛𝑑𝜎𝑄≲𝑟𝛼Ω𝜑Ω(Θ)𝐸2𝑡−𝛽Ω−1||𝑓||(𝑄)𝑑𝜎𝑄≲𝑟(1−𝑛)/𝑝‖𝑓‖𝑝(3.3) for 𝑖=0,1,2,…,𝑖(𝑃), which is similar to the estimate of 𝑉2𝑓(𝑃).
So ||𝑉3||𝑓(𝑃)≲𝑟(1−𝑛)/𝑝‖𝑓‖𝑝.(3.4)
Notice that |𝑃−𝑄|>(1/2)|𝜁−𝑄| in the case 𝑄∈𝐸3∩𝐵𝑐(𝜁,2𝑠). By (1.6) and (2.2), we have ||𝑉4||𝑓(𝑃)≲𝛿(𝑃)𝐸3∩𝐵𝑐(𝜁,2𝑠)||||𝑓(𝑄)||||𝑃−𝑄𝑛𝑑𝜎𝑄≲𝛿(𝑃)∞𝑖=1𝐸3∩(𝐵(𝜁,2𝑖+1𝑠)⧵𝐵(𝜁,2𝑖𝑠))||𝑓||(𝑄)||||𝜁−𝑄𝑛𝑑𝜎𝑄≲𝛿(𝑃)∞𝑖=112𝑖𝑠𝑛𝐸3∩𝐵(𝜁,2𝑖+1𝑠)||||𝑓(𝑄)𝑑𝜎𝑄≲𝛿(𝑃)∞𝑖=1ℵ1𝑓,2𝑖+1𝑠,𝜁≲𝛿(𝑃)∞𝑖=12𝑖+2𝑠2𝑖+1𝑠ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙≲𝛿(𝑃)∞𝑠ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙≲𝛿(𝑃)∞𝛿(𝑃)ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.5)
Thus, it follows that ||𝒫ℐΩ||≲1𝑓(𝑃)𝑂||𝑉(1)1||+||𝑉𝑓(𝑃)2||+||𝑉𝑓(𝑃)3||+||𝑉𝑓(𝑃)4||𝑓(𝑃)≲𝑟(1−𝑛)/𝑝‖𝑓‖𝑝+𝛿(𝑃)∞𝛿(𝑃)ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.6)
Using the fact that 𝑠≲𝛿(𝑃)≲𝑟𝜑Ω(Θ), we get ||𝒫ℐΩ||𝑓(𝑃)≲ℵ1(𝑓,2𝑠,𝜁)+𝛿(𝑃)∞𝛿(𝑃)ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙.(3.7)
It is clear that ∞𝛿(𝑃)ℵ1(𝑓,𝑙,𝜁)𝑙𝑑𝑙(3.8) is a convergent integral, since ℵ1(𝑓,l,𝜁)𝑙≲𝑠−1−𝑛𝑠𝑛/𝑞‖𝑓‖𝑝≲𝑠−1−(𝑛/𝑝)‖𝑓‖𝑝(3.9) from the Hölder’s inequality.
Now, as 𝛿(𝑃)→0, we also have 𝑠→0. Since 𝑓(𝜁)=0 and since we have assumed that 𝜁∈𝔼𝑝𝑓(𝐺(Ω)) (and thus that 𝜁∈𝔼1𝑓(𝐺(Ω))), it follows that 𝒫ℐΩ𝑓(𝑃)→0=𝑓(𝜁) as 𝑃=(𝑟,Θ)→𝜁 along Γ(Ω,𝜁). This concludes the proof.
This paper is supported by SRFDP (no. 20100003110004) and NSF of China (no. 11071020).
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Schwinger limit - Wikipedia
A Feynman diagram (box diagram) for photon–photon scattering; one photon scatters from the transient vacuum charge fluctuations of the other.
In quantum electrodynamics (QED), the Schwinger limit is a scale above which the electromagnetic field is expected to become nonlinear. The limit was first derived in one of QED's earliest theoretical successes by Fritz Sauter in 1931[1] and discussed further by Werner Heisenberg and his student Hans Heinrich Euler.[2] The limit, however, is commonly named in the literature[3] for Julian Schwinger, who derived the leading nonlinear corrections to the fields and calculated the rate of electron–positron pair production in a strong electric field.[4] The limit is typically reported as a maximum electric field or magnetic field before nonlinearity for the vacuum of
{\displaystyle E_{\text{c}}={\frac {m_{\text{e}}^{2}c^{3}}{q_{\text{e}}\hbar }}\simeq 1.32\times 10^{18}\,\mathrm {V} /\mathrm {m} }
{\displaystyle B_{\text{c}}={\frac {m_{\text{e}}^{2}c^{2}}{q_{\text{e}}\hbar }}\simeq 4.41\times 10^{9}\,\mathrm {T} ,}
where me is the mass of the electron, c is the speed of light in vacuum, qe is the elementary charge, and ħ is the reduced Planck constant. These are enormous field strengths. Such an electric field is capable of accelerating a proton from rest to the maximum energy attained by protons at the Large Hadron Collider in only approximately 5 micrometers. The magnetic field is associated with birefringence of the vacuum and is exceeded on magnetars.
In a vacuum, the classical Maxwell's equations are perfectly linear differential equations. This implies – by the superposition principle – that the sum of any two solutions to Maxwell's equations is another solution to Maxwell's equations. For example, two intersecting beams of light should simply add together their electric fields and pass right through each other. Thus Maxwell's equations predict the impossibility of any but trivial elastic photon–photon scattering. In QED, however, non-elastic photon–photon scattering becomes possible when the combined energy is large enough to create virtual electron–positron pairs spontaneously, illustrated by the Feynman diagram in the adjacent figure. This creates nonlinear effects that are approximately described by Euler and Heisenberg's nonlinear variant of Maxwell's equations.
A single plane wave is insufficient to cause nonlinear effects, even in QED.[4] The basic reason for this is that a single plane wave of a given energy may always be viewed in a different reference frame, where it has less energy (the same is the case for a single photon). A single wave or photon does not have a center-of-momentum frame where its energy must be at minimal value. However, two waves or two photons not traveling in the same direction always have a minimum combined energy in their center-of-momentum frame, and it is this energy and the electric field strengths associated with it, which determine particle–antiparticle creation, and associated scattering phenomena.
Photon–photon scattering and other effects of nonlinear optics in vacuum is an active area of experimental research, with current or planned technology beginning to approach the Schwinger limit.[5] It has already been observed through inelastic channels in SLAC Experiment 144.[6][7] However, the direct effects in elastic scattering have not been observed. As of 2012, the best constraint on the elastic photon–photon scattering cross section belonged to PVLAS, which reported an upper limit far above the level predicted by the Standard Model.[8]
Proposals were made to measure elastic light-by-light scattering using the strong electromagnetic fields of the hadrons collided at the LHC.[9] In 2019, the ATLAS experiment at the LHC announced the first definitive observation of photon–photon scattering, observed in lead ion collisions that produced fields as large as 1025 V/m, well in excess of the Schwinger limit.[10] Observation of a cross section larger or smaller than that predicted by the Standard Model could signify new physics such as axions, the search of which is the primary goal of PVLAS and several similar experiments. ATLAS observed more events than expected, potentially evidence that the cross section is larger than predicted by the Standard Model, but the excess is not yet statistically significant.[11]
The planned, funded ELI–Ultra High Field Facility, which will study light at the intensity frontier, is likely to remain well below the Schwinger limit[12] although it may still be possible to observe some nonlinear optical effects.[13] The Station of Extreme Light (SEL) is another laser facility under construction which should be powerful enough to observe the effect.[14].Such an experiment, in which ultra-intense light causes pair production, has been described in the popular media as creating a "hernia" in spacetime.[15]
^ F. Sauter (1931), "Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs", Zeitschrift für Physik (82 ed.), 69 (11–12): 742–764, Bibcode:1931ZPhy...69..742S, doi:10.1007/BF01339461, S2CID 122120733
^ W. Heisenberg and H. Euler (1936), "Folgerungen aus der Diracschen Theorie des Positrons", Zeitschrift für Physik (98 ed.), 98 (11–12): 714–732, Bibcode:1936ZPhy...98..714H, doi:10.1007/BF01343663, S2CID 120354480 {{citation}}: CS1 maint: uses authors parameter (link) English translation
^ M. Buchanan (2006), "Thesis: Past the Schwinger limit", Nature Physics (2 ed.), 2 (11): 721, Bibcode:2006NatPh...2..721B, doi:10.1038/nphys448, S2CID 119831515
^ a b J. Schwinger (1951), "On Gauge Invariance and Vacuum Polarization", Phys. Rev. (82 ed.), 82 (5): 664–679, Bibcode:1951PhRv...82..664S, doi:10.1103/PhysRev.82.664
^ Stepan S. Bulanov, Timur Zh. Esirkepov, Alexander G. R. Thomas, James K. Koga, and Sergei V. Bulanov (2010), "On the Schwinger limit attainability with extreme power lasers", Phys. Rev. Lett. (105 ed.), 105 (22): 220407, arXiv:1007.4306, doi:10.1103/PhysRevLett.105.220407, PMID 21231373, S2CID 36857911 {{citation}}: CS1 maint: uses authors parameter (link)
^ C. C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. Boege, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, W. Ragg, D. L. Burke, R. C. Field, G. Horton-Smith, A. C. Odian, J. E. Spencer, D. Walz, S. C. Berridge, W. M. Bugg, K. Shmakov, and A. W. Weidemann (1996), "Observation of Nonlinear Effects in Compton Scattering", Phys. Rev. Lett. (76 ed.), 76 (17): 3116–3119, Bibcode:1996PhRvL..76.3116B, doi:10.1103/PhysRevLett.76.3116, PMID 10060879 {{citation}}: CS1 maint: uses authors parameter (link)
^ C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, D. A. Reis, W. Ragg, C. Bula, K. T. McDonald, E. J. Prebys, D. L. Burke, R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berridge, W. M. Bugg, K. Shmakov, and A. W. Weidemann (1999), "Studies of nonlinear QED in collisions of 46.6 GeV electrons with intense laser pulses", Phys. Rev. D (60 ed.), 60 (9): 092004, Bibcode:1999PhRvD..60i2004B, doi:10.1103/PhysRevD.60.092004 {{citation}}: CS1 maint: uses authors parameter (link)
^ G. Zavattini et al., "Measuring the magnetic birefringence of vacuum: the PVLAS experiment", Accepted for publication in the Proceedings of the QFEXT11 Benasque Conference, [1]
^ D. d'Enterria, G. G. da Silveira (2013), "Observing Light-by-Light Scattering at the Large Hadron Collider", Phys. Rev. Lett. (111 ed.), 111 (8): 080405, arXiv:1305.7142, Bibcode:2013PhRvL.111h0405D, doi:10.1103/PhysRevLett.111.080405, PMID 24010419, S2CID 43797550 {{citation}}: CS1 maint: uses authors parameter (link)
^ ATLAS Collaboration, "ATLAS observes light scattering off light", [2], (2019)
^ The ATLAS Collaboration, "Observation of light-by-light scattering in ultraperipheral Pb+Pb collisions with the ATLAS detector", [3], (2019)
^ T. Heinzl, "Strong-Field QED and High Power Lasers", Plenary talk QFEXT11 Benasque Conference, [4] [5]
^ G. Yu. Kryuchkyan and K. Z. Hatsagortsyan (2011), "Bragg Scattering of Light in Vacuum Structured by Strong Periodic Fields", Phys. Rev. Lett. (107 ed.), 107 (5): 053604, arXiv:1102.4013, Bibcode:2011PhRvL.107e3604K, doi:10.1103/PhysRevLett.107.053604, PMID 21867070, S2CID 25991919 {{citation}}: CS1 maint: uses authors parameter (link)
^ Berboucha, Meriame. "This Laser Could Rip Apart Empty Space". Forbes. Retrieved 2021-02-18.
^ I. O'Neill (2011). "A Laser to Give the Universe a Hernia?". Discovery News. Archived from the original on November 3, 2011.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Schwinger_limit&oldid=1088239341"
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Reduced-Order Model-Based Feedback Control of Flow Over an Obstacle Using Center Manifold Methods | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
Coşku Kasnakoğlu,
, Ankara 06560, Turkey
e-mail: kasnakoglu@etu.edu.tr
R. Chris Camphouse,
Carlsbad Programs Group, Performance Assessment and Decision Analysis Department,
, Carlsbad, NM 88220
Kasnakoğlu, C., Camphouse, R. C., and Serrani, A. (December 8, 2008). "Reduced-Order Model-Based Feedback Control of Flow Over an Obstacle Using Center Manifold Methods." ASME. J. Dyn. Sys., Meas., Control. January 2009; 131(1): 011011. https://doi.org/10.1115/1.3023122
In this paper, we consider a boundary control problem governed by the two-dimensional Burgers’ equation for a configuration describing convective flow over an obstacle. Flows over obstacles are important as they arise in many practical applications. Burgers’ equations are also significant as they represent a simpler form of the more general Navier–Stokes momentum equation describing fluid flow. The aim of the work is to develop a reduced-order boundary control-oriented model for the system with subsequent nonlinear control law design. The control objective is to drive the full order system to a desired 2D profile. Reduced-order modeling involves the application of an
L2
optimization based actuation mode expansion technique for input separation, demonstrating how one can obtain a reduced-order Galerkin model in which the control inputs appear as explicit terms. Controller design is based on averaging and center manifold techniques and is validated with full order numerical simulation. Closed-loop results are compared to a standard linear quadratic regulator design based on a linearization of the reduced-order model. The averaging∕center manifold based controller design provides smoother response with less control effort and smaller tracking error.
boundary layers, control system synthesis, feedback, flow, flow control, Navier-Stokes equations, nonlinear control systems, open loop systems
Design, Feedback, Flow (Dynamics), Manifolds, Modeling, Optimization, Control equipment, Dynamics (Mechanics), Separation (Technology)
Boundary and Distributed Control of the Viscous Burgers Equation
Second Order Methods for Boundary Control of the Instationary Navier–Stokes System
Nonlinear Boundary Control of Coupled Burgers’ Equations
Contr. Cybernet.
Boundary Control of the Navier–Stokes Equation by Empirical Reduction of Modes
On Global Stabilization of Burgers’ Equation by Boundary Control
Control of Mixing by Boundary Feedback in 2D Channel Flow
Computational Fluid Dynamics: The Basics With Applications
Active Control of Pressure Fluctuations Due to Flow Over Helmholtz Resonators
On Low-Dimensional Galerkin Models for Fluid Flow
Turbulence, Coherent Structures, Dynamical System, and Symmetry
A Global Stability Analysis of the Steady and Periodic Cylinder Wake
Monketwitz
The Need for a Pressure-Term Representation in Empirical Galerkin Models of Incompressible Shear Flows
Reconstruction Equations and the Karhunen–Loeve Expansion for Systems With Symmetry
Flow Control in a Driven Cavity Incorporating Excitation Phase Differential
Feedback Control of Subsonic Cavity Flows Using Reduced-Order Models
Boundary Feedback Control Using Proper Orthogonal Decomposition Models
A Snapshot Decomposition Method for Reduced Order Modeling and Boundary Feedback Control
Low Dimensional Modelling and Dirichlet Boundary Controller Design for Burgers Equation
Kasnakoglu
Control Input Separation by Actuation Mode Expansion for Flow Control Problems
Numerical Simulation (DNS and LES) of Manipulated Turbulent Boundary Layer Flow Over a Surface-Mounted Fence
Oscillation Supression in Galerkin Systems Using Center-Manifold and Averaging Techniques
Analysis and Nonlinear Control of Galerkin Models Using Averaging and Center Manifold Theory
Feedback Control for a Two-Dimensional Burgers Equation System Model
Second AIAA Flow Control Conference
Feedback Control of Particle Dispersion in Bluff Body Wakes
Low-Dimensional Models for Feedback Flow Control. Part I: Empirical Galerkin Models
Proceedings of the Second AIAA Flow Control Conference
Low-Dimensional Models for Feedback Flow Control. Part II: Controller Design And Dynamic Estimation
Integral Averaging and Bifurcation
, NY, 2nd edition.
A Control Problem for Burgers Equation With Bounded Input∕Output
Output Regulation Design for a Class of Nonlinear Systems With an Unknown Control Direction
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Pincement de polynômes | EMS Press
Pincement de polynômes
f_{0} : \mathbb{C} \to \mathbb{C}
be a semi-hyperbolic polynomial in the sense of Carleson-Jones-Yoccoz with an attracting point. The goal of this paper is to show that one can define a semi-hyperbolic deformation
(f_{t})_{t\ge 0}
such that the attracting cycle becomes parabolic for the limit polynomial
f_{\infty}
f_{0}
f_{\infty}
are semi-conjugate. This deformation is defined by pinching curves in appropriate quotient spaces.
Peter Haïssinsky, Pincement de polynômes. Comment. Math. Helv. 77 (2002), no. 1, pp. 1–23
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Table 4 E. coli correctness on GAGE metrics
Number of structural differences
Number of indels > 5 bp
E. coli K12 MiSeq 100× 139 113,852 1 59 49.00 4
454 50× 93 117,490 2 26 52.45 0
MiSeq 100× + 200× CLR 2 2,367,319 2 11 56.26 2
454 50× + 200× CLR 1 4,649,004 2 11 56.26 2
CCS 25X + 200× CLR 1 4,653,267 0 14 55.22 2
200× CLR 1 4,653,486 0 14 55.22 2
Organism: the genome being assembled. Assembled by: the short-read data used for correction and/or assembly. Number of contigs: the number of contigs comprising the assembly. N50: N such that 50% of the genome is contained in contigs of length ≥N. Number of structural differences: the sum of inversions, relocations, and translocations versus the reference using GAGE metrics [37]. Number of discordant bases: total number of different bases when compared to the reference. QV: estimated from the number of indels and SNPs as
{log}_{10}\phantom{\rule{0.5em}{0ex}}\left(\frac{\mathrm{assembly}\phantom{\rule{0.12em}{0ex}}\mathrm{length}}{#\mathrm{incorrect}\phantom{\rule{0.12em}{0ex}}\mathrm{bases}}\right)*10
. Indels >5 bp: the number of indels >5 bp reported by GAGE metrics [37]. Assemblies were generated as in Tables 2 and 3.
|
On Topological Singular Set of Maps with Finite 3-Energy into
S^3
M.R. Pakzad
Representation Formulas for the General Derivatives of the Fundamental Solution to the Cauchy-Riemann Operator in Clifford Analysis and Applications
Weighted Hölder Continuity of Hyperbolic Harmonic Bloch functions
Guangbin RenUwe Kähler
Tensor Algebras and Displacement Structure II: Non-Commutative Szegö Polynomials
Tiberiu ConstantinescuJ. L. Johnson
Index Transforms Associated with Bessel and Lommel Functions
On Resonant Differential Equations with Unbounded Non-Linearities
A. M. Krasnosel'skiiN. A. KuznetsovD. Rachinskii
Exact Solution of a System of Generalized Hopf Equations
K.T. Joseph
Potential-Type Operators in
L^{p(x)}
David E. EdmundsAlexander Meskhi
Level Sets of Hölder Functions and Hausdorff Measures
Comparison of Non-Commutative 2- and
p
-Summing Operators from
B(l_2)
OH
L. Mezrag
Problem of Functional Extension and Space-Like Surfaces in Minkowski Space
E.G. GrigoryevaA.A. KlyachinV.M. Miklyukov
Marlène FrigonDonal O'Regan
A Unified Approach to Linear Differential Algebraic Equations and their Adjoints
K. BallaRoswitha März
A Certain Series Associated with Catalan’s Constant
V.S. Adamchik
|
I thank you for sending me your Cheque & I enclose stamped receipt for it.2 Also I enclose Statement of the Sales of your “Orchids” made up to the present time showing a balance of £36.19.6 coming to you as your share of half profits for wch amount you will find my Cheque within—
Annexed to this account is a statement of the Stocks of your different works—3
I have not settled with you for the actual Edition of Origin of Species because not 300 copies have yet been sold out of 2000 printed—4
As for Expression I rashly printed 2000 copies—in the full tide of success when the book was selling like wild fire. At that very time came a sudden stagnation—& not only have I not yet touched the Edn but I have 404 copies left out of the 7000 previously printed & paid for. We must have patience— My Sale in Nov may alter the state of affairs
I shall be prepared for your new works notwithstanding
Dr Darwin on Orchids Cr
June To Printing 1500 No 89 1 9 June By 1500 Copies
\frac{1}{4}
Reams paper 51 17 4 5 Stats Hall5
" Binding 1000 Copies 30 4 2 12 Allowed Author
" Sowerby.6 Illustrations 21 – – 30 Reviews
" Cooper7 do 53 2 – 732 685 on Hand June 1862
" Advertising 27 4 – 768 Sold viz
" Comn to agents 2 8 - 358 Trade 25 as 24 6/ 103 4 –
410 do " " 6/5 126 8 2
June By Balance deficiency 45 5 1
June To Balance deficiency 45 5 1 June By 685 on hand
Sep " Electros 3 6 – 4 Reviews
June " Binding 100 Copies 3 – 5 520 516 On Hand June 1867
" Advertising 20 10 – 165 Sold viz
52 Trade 25 as 24 6/ 15 – –
113 do " " 6/5 34 19 5
June To balance defy 22 2 1 June June By 516 on hand
June " Electros 2 10 – 1 Author in Sheets
June " Advertising 6 years 22 – – 82 81 On Hand June 1873
" Binding 400 Copies 12 1 8 434 Sold viz
" Comn allowd agents – 12 2 252 Trade 25 as 24 6/ 72 12 –
" Balance profit 73 19 – 182 do " " 6/5 56 2 11
May By Reinwald for 4 10 –
a Set of Electros8 – – –
June By 81 on hand
" Author’s
\frac{1}{3}
Profit 36 19 6
Stocks on Hand. June 30. 1873
* 1716 “Origin of species” 7/6 Edition 2000 No printed March 1873 for which Author has not yet been paid 26 “Variations Animals” &c 1250 No printed in 1868 75 “Man” 1000 No " 1871 2404 “Emotions” 7000 No printed Decr 1872 for which Author was paid £1050— 2000 No printed 1873 for which Author has not yet been paid
* Memm 57 Copies 14/– edition still on Hand
Verso 1st sheet: ‘[‘State’ del] Sale of my Books Sept 25 1873.—’ red crayon
Verso 2d sheet: ‘Number of all my publications Sept. 1873’ red crayon
Stocks on Hand: ‘26’, ‘75’, ‘2404’ underl red crayon
The year is established by the relationship between this letter and the letter to John Murray, 24 September [1873].
See letter to John Murray, 24 September [1873] and n. 2. CD was paying for the presentation copies of his works.
Murray’s statement for Orchids and the stocks of CD’s other works is in the Darwin Archive–CUL (DAR 210.11: 1).
Origin 6th ed. had been published in February 1872 (Freeman 1977).
Under the 1842 Copyright Act, four copies of any new work published in the United Kingdom and the British Dominions were to be delivered to an officer of the Stationers’ Company for distribution on demand to the Bodleian Library in Oxford, the ‘Public Library’ (now the University Library) in Cambridge, the Faculty of Advocates at Edinburgh, and the library of Trinity College, Dublin. A copy was also to be delivered to the British Museum in London. See Seville 1999, pp. 233, 262.
George Brettingham Sowerby Jr.
Cooper has not been identified.
Charles-Ferdinand Reinwald was the publisher of the French translation of Orchids (Rérolle trans. 1870). Electros: electrotypes of the illustrations.
Acknowledges CD’s cheque.
Sends CD cheque for profits on Orchids and a statement of stock on hand of CD’s works [missing].
Origin and Expression sales are stagnant.
DAR 171: 437, DAR 210.11: 1
|
Model Comparison Tests - MATLAB & Simulink - MathWorks Benelux
The primary goal of model selection is choosing the most parsimonious model that adequately fits your data. Three asymptotically equivalent tests compare a restricted model (the null model) against an unrestricted model (the alternative model), fit to the same data:
Likelihood ratio (LR) test
Lagrange multiplier (LM) test
Wald (W) test
For a model with parameters θ, consider the restriction
r\left(\theta \right)=0,
which is satisfied by the null model. For example, consider testing the null hypothesis
\theta ={\theta }_{0}.
The restriction function for this test is
r\left(\theta \right)=\theta -{\theta }_{0}.
The LR, LM, and Wald tests approach the problem of comparing the fit of a restricted model against an unrestricted model differently. For a given data set, let
l\left({\theta }_{0}^{MLE}\right)
denote the loglikelihood function evaluated at the maximum likelihood estimate (MLE) of the restricted (null) model. Let
l\left({\theta }_{A}^{MLE}\right)
denote the loglikelihood function evaluated at the MLE of the unrestricted (alternative) model. The following figure illustrates the rationale behind each test.
Likelihood ratio test. If the restricted model is adequate, then the difference between the maximized objective functions,
l\left({\theta }_{A}^{MLE}\right)-l\left({\theta }_{0}^{MLE}\right),
should not significantly differ from zero.
Lagrange multiplier test. If the restricted model is adequate, then the slope of the tangent of the loglikelihood function at the restricted MLE (indicated by T0 in the figure) should not significantly differ from zero (which is the slope of the tangent of the loglikelihood function at the unrestricted MLE, indicated by T).
Wald test. If the restricted model is adequate, then the restriction function evaluated at the unrestricted MLE should not significantly differ from zero (which is the value of the restriction function at the restricted MLE).
The three tests are asymptotically equivalent. Under the null, the LR, LM, and Wald test statistics are all distributed as
{\chi }^{2}
with degrees of freedom equal to the number of restrictions. If the test statistic exceeds the test critical value (equivalently, the p-value is less than or equal to the significance level), the null hypothesis is rejected. That is, the restricted model is rejected in favor of the unrestricted model.
Choosing among the LR, LM, and Wald test is largely determined by computational cost:
To conduct a likelihood ratio test, you need to estimate both the restricted and unrestricted models.
To conduct a Lagrange multiplier test, you only need to estimate the restricted model (but the test requires an estimate of the variance-covariance matrix).
To conduct a Wald test, you only need to estimate the unrestricted model (but the test requires an estimate of the variance-covariance matrix).
All things being equal, the LR test is often the preferred choice for comparing nested models. Econometrics Toolbox™ has functionality for all three tests.
You can conduct a likelihood ratio test using lratiotest. The required inputs are:
Value of the maximized unrestricted loglikelihood,
l\left({\theta }_{A}^{MLE}\right)
Value of the maximized restricted loglikelihood,
l\left({\theta }_{0}^{MLE}\right)
Number of restrictions (degrees of freedom)
Given these inputs, the likelihood ratio test statistic is
{G}^{2}=2×\left[l\left({\theta }_{A}^{MLE}\right)-l\left({\theta }_{0}^{MLE}\right)\right].
When estimating conditional mean and variance models (using arima, garch, egarch, or gjr), you can return the value of the loglikelihood objective function as an optional output argument of estimate or infer. For multivariate time series models, you can get the value of the loglikelihood objective function using estimate.
The required inputs for conducting a Lagrange multiplier test are:
Gradient of the unrestricted likelihood evaluated at the restricted MLEs (the score), S
Variance-covariance matrix for the unrestricted parameters evaluated at the restricted MLEs, V
Given these inputs, the LM test statistic is
LM={S}^{\prime }VS.
You can conduct an LM test using lmtest. A specific example of an LM test is Engle’s ARCH test, which you can conduct using archtest.
The required inputs for conducting a Wald test are:
Restriction function evaluated at the unrestricted MLE, r
Jacobian of the restriction function evaluated at the unrestricted MLEs, R
Variance-covariance matrix for the unrestricted parameters evaluated at the unrestricted MLEs, V
Given these inputs, the test statistic for the Wald test is
W={r}^{\prime }{\left(RV{R}^{\prime }\right)}^{-1}r.
You can conduct a Wald test using waldtest.
You can often compute the Jacobian of the restriction function analytically. Or, if you have Symbolic Math Toolbox™, you can use the function jacobian.
For estimating a variance-covariance matrix, there are several common methods, including:
Outer product of gradients (OPG). Let G be the matrix of gradients of the loglikelihood function. If your data set has N observations, and there are m parameters in the unrestricted likelihood, then G is an N × m matrix.
{\left({G}^{\prime }G\right)}^{-1}
is the OPG estimate of the variance-covariance matrix.
For arima, garch, egarch, and gjr models, the estimate method returns the OPG estimate of the variance-covariance matrix.
Inverse negative Hessian (INH). Given the loglikelihood function
l\left(\theta \right),
the INH covariance estimate has elements
\mathrm{cov}\left(i,j\right)={\left(-\frac{{\partial }^{2}l\left(\theta \right)}{\partial {\theta }_{i}\partial {\theta }_{j}}\right)}^{-1}.
The estimation function for multivariate models, estimate, returns the expected Hessian variance-covariance matrix.
If you have Symbolic Math Toolbox, you can use jacobian twice to calculate the Hessian matrix for your loglikelihood function.
lmtest | waldtest | lratiotest
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In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.
Formally: if
{\displaystyle f:S^{n}\to \mathbb {R} ^{n}}
is continuous then there exists an
{\displaystyle x\in S^{n}}
{\displaystyle f(-x)=f(x)}
{\displaystyle n=1}
can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space.
{\displaystyle n=2}
is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space.
The Borsuk–Ulam theorem has several equivalent statements in terms of odd functions. Recall that
{\displaystyle S^{n}}
is the n-sphere and
{\displaystyle B^{n}}
is the n-ball:
{\displaystyle g:S^{n}\to \mathbb {R} ^{n}}
is a continuous odd function, then there exists an
{\displaystyle x\in S^{n}}
{\displaystyle g(x)=0}
{\displaystyle g:B^{n}\to \mathbb {R} ^{n}}
is a continuous function which is odd on
{\displaystyle S^{n-1}}
(the boundary of
{\displaystyle B^{n}}
), then there exists an
{\displaystyle x\in B^{n}}
{\displaystyle g(x)=0}
2.1 With odd functions
2.2 With retractions
3.1 1-dimensional case
3.2.1 Algebraic topological proof
3.2.2 Combinatorial proof
According to Jiří Matoušek (2003, p. 25) harvtxt error: no target: CITEREFJiří_Matoušek2003 (help), the first historical mention of the statement of the Borsuk–Ulam theorem appears in Lyusternik & Shnirel'man (1930). The first proof was given by Karol Borsuk (1933), where the formulation of the problem was attributed to Stanislaw Ulam. Since then, many alternative proofs have been found by various authors, as collected by Steinlein (1985).
Equivalent statementsEdit
The following statements are equivalent to the Borsuk–Ulam theorem.[1]
With odd functionsEdit
{\displaystyle g}
is called odd (aka antipodal or antipode-preserving) if for every
{\displaystyle x}
{\displaystyle g(-x)=-g(x)}
The Borsuk–Ulam theorem is equivalent to the following statement: A continuous odd function from an n-sphere into Euclidean n-space has a zero. PROOF:
If the theorem is correct, then it is specifically correct for odd functions, and for an odd function,
{\displaystyle g(-x)=g(x)}
{\displaystyle g(x)=0}
. Hence every odd continuous function has a zero.
For every continuous function
{\displaystyle f}
, the following function is continuous and odd:
{\displaystyle g(x)=f(x)-f(-x)}
. If every odd continuous function has a zero, then
{\displaystyle g}
has a zero, and therefore,
{\displaystyle f(x)=f(-x)}
. Hence the theorem is correct.
With retractionsEdit
Define a retraction as a function
{\displaystyle h:S^{n}\to S^{n-1}.}
The Borsuk–Ulam theorem is equivalent to the following claim: there is no continuous odd retraction.
Proof: If the theorem is correct, then every continuous odd function from
{\displaystyle S^{n}}
must include 0 in its range. However,
{\displaystyle 0\notin S^{n-1}}
so there cannot be a continuous odd function whose range is
{\displaystyle S^{n-1}}
Conversely, if it is incorrect, then there is a continuous odd function
{\displaystyle g:S^{n}\to {\mathbb {R}}^{n}}
with no zeroes. Then we can construct another odd function
{\displaystyle h:S^{n}\to S^{n-1}}
{\displaystyle h(x)={\frac {g(x)}{|g(x)|}}}
{\displaystyle g}
has no zeroes,
{\displaystyle h}
is well-defined and continuous. Thus we have a continuous odd retraction.
1-dimensional caseEdit
The 1-dimensional case can easily be proved using the intermediate value theorem (IVT).
{\displaystyle g}
be an odd real-valued continuous function on a circle. Pick an arbitrary
{\displaystyle x}
{\displaystyle g(x)=0}
then we are done. Otherwise, without loss of generality,
{\displaystyle g(x)>0.}
{\displaystyle g(-x)<0.}
Hence, by the IVT, there is a point
{\displaystyle y}
{\displaystyle x}
{\displaystyle -x}
{\displaystyle g(y)=0}
Algebraic topological proofEdit
{\displaystyle h:S^{n}\to S^{n-1}}
is an odd continuous function with
{\displaystyle n>2}
{\displaystyle n=1}
is treated above, the case
{\displaystyle n=2}
can be handled using basic covering theory). By passing to orbits under the antipodal action, we then get an induced continuous function
{\displaystyle h':\mathbb {RP} ^{n}\to \mathbb {RP} ^{n-1}}
between real projective spaces, which induces an isomorphism on fundamental groups. By the Hurewicz theorem, the induced ring homomorphism on cohomology with
{\displaystyle \mathbb {F} _{2}}
coefficients [where
{\displaystyle \mathbb {F} _{2}}
denotes the field with two elements],
{\displaystyle \mathbb {F} _{2}[a]/a^{n+1}=H^{*}\left(\mathbb {RP} ^{n};\mathbb {F} _{2}\right)\leftarrow H^{*}\left(\mathbb {RP} ^{n-1};\mathbb {F} _{2}\right)=\mathbb {F} _{2}[b]/b^{n},}
{\displaystyle b}
{\displaystyle a}
. But then we get that
{\displaystyle b^{n}=0}
is sent to
{\displaystyle a^{n}\neq 0}
, a contradiction.[2]
One can also show the stronger statement that any odd map
{\displaystyle S^{n-1}\to S^{n-1}}
has odd degree and then deduce the theorem from this result.
The Borsuk–Ulam theorem can be proved from Tucker's lemma.[1][3][4]
{\displaystyle g:S^{n}\to \mathbb {R} ^{n}}
be a continuous odd function. Because g is continuous on a compact domain, it is uniformly continuous. Therefore, for every
{\displaystyle \epsilon >0}
{\displaystyle \delta >0}
such that, for every two points of
{\displaystyle S_{n}}
which are within
{\displaystyle \delta }
of each other, their images under g are within
{\displaystyle \epsilon }
Define a triangulation of
{\displaystyle S_{n}}
with edges of length at most
{\displaystyle \delta }
. Label each vertex
{\displaystyle v}
of the triangulation with a label
{\displaystyle l(v)\in {\pm 1,\pm 2,\ldots ,\pm n}}
The absolute value of the label is the index of the coordinate with the highest absolute value of g:
{\displaystyle |l(v)|=\arg \max _{k}(|g(v)_{k}|)}
The sign of the label is the sign of g, so that:
{\displaystyle l(v)=\operatorname {sgn}(g(v))|l(v)|}
Because g is odd, the labeling is also odd:
{\displaystyle l(-v)=-l(v)}
. Hence, by Tucker's lemma, there are two adjacent vertices
{\displaystyle u,v}
with opposite labels. Assume w.l.o.g. that the labels are
{\displaystyle l(u)=1,l(v)=-1}
. By the definition of l, this means that in both
{\displaystyle g(u)}
{\displaystyle g(v)}
, coordinate #1 is the largest coordinate: in
{\displaystyle g(u)}
this coordinate is positive while in
{\displaystyle g(v)}
it is negative. By the construction of the triangulation, the distance between
{\displaystyle g(u)}
{\displaystyle g(v)}
{\displaystyle \epsilon }
, so in particular
{\displaystyle |g(u)_{1}-g(v)_{1}|=|g(u)_{1}|+|g(v)_{1}|\leq \epsilon }
{\displaystyle g(u)_{1}}
{\displaystyle g(v)_{1}}
have opposite signs) and so
{\displaystyle |g(u)_{1}|\leq \epsilon }
. But since the largest coordinate of
{\displaystyle g(u)}
is coordinate #1, this means that
{\displaystyle |g(u)_{k}|\leq \epsilon }
{\displaystyle 1\leq k\leq n}
{\displaystyle |g(u)|\leq c_{n}\epsilon }
{\displaystyle c_{n}}
is some constant depending o{\displaystyle n}
{\displaystyle |\cdot |}
The above is true for every
{\displaystyle \epsilon >0}
{\displaystyle S_{n}}
is compact there must hence be a point u in which
{\displaystyle |g(u)|=0}
No subset of
{\displaystyle \mathbb {R} ^{n}}
{\displaystyle S^{n}}
The ham sandwich theorem: For any compact sets A1, ..., An in
{\displaystyle \mathbb {R} ^{n}}
we can always find a hyperplane dividing each of them into two subsets of equal measure.
Equivalent resultsEdit
Above we showed how to prove the Borsuk–Ulam theorem from Tucker's lemma. The converse is also true: it is possible to prove Tucker's lemma from the Borsuk–Ulam theorem. Therefore, these two theorems are equivalent. There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column.[5]
In the original theorem, the domain of the function f is the unit n-sphere (the boundary of the unit n-ball). In general, it is true also when the domain of f is the boundary of any open bounded symmetric subset of
{\displaystyle \mathbb {R} ^{n}}
containing the origin (Here, symmetric means that if x is in the subset then -x is also in the subset).[6]
Consider the function A which maps a point to its antipodal point:
{\displaystyle A(x)=-x.}
{\displaystyle A(A(x))=x.}
The original theorem claims that there is a point x in which
{\displaystyle f(A(x))=f(x).}
In general, this is true also for every function A for which
{\displaystyle A(A(x))=x.}
[7] However, in general this is not true for other functions A.[8]
Kakutani's theorem (geometry)
^ a b Prescott, Timothy (2002). "Extensions of the Borsuk–Ulam Theorem (Thesis)". Harvey Mudd College. CiteSeerX 10.1.1.124.4120. {{cite journal}}: Cite journal requires |journal= (help)
^ Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 12 for a full exposition.)
^ Freund, Robert M; Todd, Michael J (1982). "A constructive proof of Tucker's combinatorial lemma". Journal of Combinatorial Theory, Series A. 30 (3): 321–325. doi:10.1016/0097-3165(81)90027-3.
^ Simmons, Forest W.; Su, Francis Edward (2003). "Consensus-halving via theorems of Borsuk–Ulam and Tucker". Mathematical Social Sciences. 45: 15–25. doi:10.1016/s0165-4896(02)00087-2. hdl:10419/94656.
^ "Borsuk fixed-point theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
^ Yang, Chung-Tao (1954). "On Theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobo and Dyson, I". Annals of Mathematics. 60 (2): 262–282. doi:10.2307/1969632. JSTOR 1969632.
^ Jens Reinhold, Faisal; Sergei Ivanov. "Generalization of Borsuk-Ulam". Math Overflow. Retrieved 18 May 2015.
Borsuk, Karol (1933). "Drei Sätze über die n-dimensionale euklidische Sphäre" (PDF). Fundamenta Mathematicae (in German). 20: 177–190. doi:10.4064/fm-20-1-177-190.
Lyusternik, Lazar; Shnirel'man, Lev (1930). "Topological Methods in Variational Problems". Issledowatelskii Institut Matematiki I Mechaniki Pri O. M. G. U. Moscow.
Matoušek, Jiří (2003). Using the Borsuk–Ulam theorem. Berlin: Springer Verlag. doi:10.1007/978-3-540-76649-0. ISBN 978-3-540-00362-5.
Steinlein, H. (1985). "Borsuk's antipodal theorem and its generalizations and applications: a survey. Méthodes topologiques en analyse non linéaire". Sém. Math. Supér. Montréal, Sém. Sci. OTAN (NATO Adv. Study Inst.). 95: 166–235.
Su, Francis Edward (Nov 1997). "Borsuk-Ulam Implies Brouwer: A Direct Construction" (PDF). The American Mathematical Monthly. 104 (9): 855–859. CiteSeerX 10.1.1.142.4935. doi:10.2307/2975293. JSTOR 2975293. Archived from the original (PDF) on 2008-10-13. Retrieved 2006-04-21.
Who (else) cares about topology? Stolen necklaces and Borsuk-Ulam on YouTube
Retrieved from "https://en.wikipedia.org/w/index.php?title=Borsuk–Ulam_theorem&oldid=1048853626"
|
Chromatic aberration — Wikipedia Republished // WIKI 2
Failure of a lens to focus all colors on the same point
Not to be confused with Chromosome aberration.
Optical aberration
Spherical aberration
Petzval field curvature
Photographic example showing high quality lens (top) compared to lower quality model exhibiting transverse chromatic aberration (seen as a blur and a rainbow edge in areas of contrast.)
In optics, chromatic aberration (CA), also called chromatic distortion and spherochromatism, is a failure of a lens to focus all colors to the same point.[1] It is caused by dispersion: the refractive index of the lens elements varies with the wavelength of light. The refractive index of most transparent materials decreases with increasing wavelength.[2] Since the focal length of a lens depends on the refractive index, this variation in refractive index affects focusing.[3] Chromatic aberration manifests itself as "fringes" of color along boundaries that separate dark and bright parts of the image.
What is Chromatic Aberration? (And why?)
When Good Lenses Go Bad - Chromatic Aberration - How to Test & Fix
Chromatic Aberration (Geometrical optics lecture :32)
chromatic aberration in hindi
2 Minimization
2.1 Mathematics of chromatic aberration minimization
3 Image processing to reduce the appearance of lateral chromatic aberration
4 Photography
4.1 Black-and-white photography
5 Electron microscopy
Comparison of an ideal image of a ring (1) and ones with only axial (2) and only transverse (3) chromatic aberration
There are two types of chromatic aberration: axial (longitudinal), and transverse (lateral). Axial aberration occurs when different wavelengths of light are focused at different distances from the lens (focus shift). Longitudinal aberration is typical at long focal lengths. Transverse aberration occurs when different wavelengths are focused at different positions in the focal plane, because the magnification and/or distortion of the lens also varies with wavelength. Transverse aberration is typical at short focal lengths. The ambiguous acronym LCA is sometimes used for either longitudinal or lateral chromatic aberration.[2]
The two types of chromatic aberration have different characteristics, and may occur together. Axial CA occurs throughout the image and is specified by optical engineers, optometrists, and vision scientists in diopters.[4] It can be reduced by stopping down, which increases depth of field so that though the different wavelengths focus at different distances, they are still in acceptable focus. Transverse CA does not occur in the center of the image and increases towards the edge. It is not affected by stopping down.
In digital sensors, axial CA results in the red and blue planes being defocused (assuming that the green plane is in focus), which is relatively difficult to remedy in post-processing, while transverse CA results in the red, green, and blue planes being at different magnifications (magnification changing along radii, as in geometric distortion), and can be corrected by radially scaling the planes appropriately so they line up.
Chromatic correction of visible and near infrared wavelengths. Horizontal axis shows degree of aberration, 0 is no aberration. Lenses: 1: simple, 2: achromatic doublet, 3: apochromatic and 4: superachromat.
In the earliest uses of lenses, chromatic aberration was reduced by increasing the focal length of the lens where possible. For example, this could result in extremely long telescopes such as the very long aerial telescopes of the 17th century. Isaac Newton's theories about white light being composed of a spectrum of colors led him to the conclusion that uneven refraction of light caused chromatic aberration (leading him to build the first reflecting telescope, his Newtonian telescope, in 1668.[5])
Modern telescopes, as well as other catoptric and catadioptric systems, continue to use mirrors, which have no chromatic aberration.
There exists a point called the circle of least confusion, where chromatic aberration can be minimized.[6] It can be further minimized by using an achromatic lens or achromat, in which materials with differing dispersion are assembled together to form a compound lens. The most common type is an achromatic doublet, with elements made of crown and flint glass. This reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction. By combining more than two lenses of different composition, the degree of correction can be further increased, as seen in an apochromatic lens or apochromat. Note that "achromat" and "apochromat" refer to the type of correction (2 or 3 wavelengths correctly focused), not the degree (how defocused the other wavelengths are), and an achromat made with sufficiently low dispersion glass can yield significantly better correction than an achromat made with more conventional glass. Similarly, the benefit of apochromats is not simply that they focus three wavelengths sharply, but that their error on other wavelengths is also quite small.[7]
Many types of glass have been developed to reduce chromatic aberration. These are low dispersion glass, most notably, glasses containing fluorite. These hybridized glasses have a very low level of optical dispersion; only two compiled lenses made of these substances can yield a high level of correction.[8]
The use of achromats was an important step in the development of optical microscopes and telescopes.
An alternative to achromatic doublets is the use of diffractive optical elements. Diffractive optical elements are able to generate arbitrary complex wave fronts from a sample of optical material which is essentially flat.[9] Diffractive optical elements have negative dispersion characteristics, complementary to the positive Abbe numbers of optical glasses and plastics. Specifically, in the visible part of the spectrum diffractives have a negative Abbe number of −3.5. Diffractive optical elements can be fabricated using diamond turning techniques.[10]
Chromatic aberration of a single lens causes different wavelengths of light to have differing focal lengths
Diffractive optical element with complementary dispersion properties to that of glass can be used to correct for color aberration
For an achromatic doublet, visible wavelengths have approximately the same focal length
Mathematics of chromatic aberration minimization
For a doublet consisting of two thin lenses in contact, the Abbe number of the lens materials is used to calculate the correct focal length of the lenses to ensure correction of chromatic aberration.[11] If the focal lengths of the two lenses for light at the yellow Fraunhofer D-line (589.2 nm) are f1 and f2, then best correction occurs for the condition:
{\displaystyle f_{1}\cdot V_{1}+f_{2}\cdot V_{2}=0}
where V1 and V2 are the Abbe numbers of the materials of the first and second lenses, respectively. Since Abbe numbers are positive, one of the focal lengths must be negative, i.e., a diverging lens, for the condition to be met.
The overall focal length of the doublet f is given by the standard formula for thin lenses in contact:
{\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}}
and the above condition ensures this will be the focal length of the doublet for light at the blue and red Fraunhofer F and C lines (486.1 nm and 656.3 nm respectively). The focal length for light at other visible wavelengths will be similar but not exactly equal to this.
Chromatic aberration is used during a duochrome eye test to ensure that a correct lens power has been selected. The patient is confronted with red and green images and asked which is sharper. If the prescription is right, then the cornea, lens and prescribed lens will focus the red and green wavelengths just in front, and behind the retina, appearing of equal sharpness. If the lens is too powerful or weak, then one will focus on the retina, and the other will be much more blurred in comparison.[12]
Image processing to reduce the appearance of lateral chromatic aberration
In some circumstances, it is possible to correct some of the effects of chromatic aberration in digital post-processing. However, in real-world circumstances, chromatic aberration results in permanent loss of some image detail. Detailed knowledge of the optical system used to produce the image can allow for some useful correction.[13] In an ideal situation, post-processing to remove or correct lateral chromatic aberration would involve scaling the fringed color channels, or subtracting some of a scaled versions of the fringed channels, so that all channels spatially overlap each other correctly in the final image.[14]
As chromatic aberration is complex (due to its relationship to focal length, etc.) some camera manufacturers employ lens-specific chromatic aberration appearance minimization techniques. Almost every major camera manufacturer enables some form of chromatic aberration correction, both in-camera and via their proprietary software. Third party software tools such as PTLens are also capable of performing complex chromatic aberration appearance minimization with their large database of cameras and lens.
In reality, even a theoretically perfect post-processing based chromatic aberration reduction-removal-correction systems do not increase image detail as a lens that is optically well corrected for chromatic aberration would for the following reasons:
Rescaling is only applicable to lateral chromatic aberration but there is also longitudinal chromatic aberration
Rescaling individual color channels result in a loss of resolution from the original image
Most camera sensors only capture a few and discrete (e.g., RGB) color channels but chromatic aberration is not discrete and occurs across the light spectrum
The dyes used in the digital camera sensors for capturing color are not very efficient so cross-channel color contamination is unavoidable and causes, for example, the chromatic aberration in the red channel to also be blended into the green channel along with any green chromatic aberration.
The above are closely related to the specific scene that is captured so no amount of programming and knowledge of the capturing equipment (e.g., camera and lens data) can overcome these limitations.
The term "purple fringing" is commonly used in photography, although not all purple fringing can be attributed to chromatic aberration. Similar colored fringing around highlights may also be caused by lens flare. Colored fringing around highlights or dark regions may be due to the receptors for different colors having differing dynamic range or sensitivity – therefore preserving detail in one or two color channels, while "blowing out" or failing to register, in the other channel or channels. On digital cameras, the particular demosaicing algorithm is likely to affect the apparent degree of this problem. Another cause of this fringing is chromatic aberration in the very small microlenses used to collect more light for each CCD pixel; since these lenses are tuned to correctly focus green light, the incorrect focusing of red and blue results in purple fringing around highlights. This is a uniform problem across the frame, and is more of a problem in CCDs with a very small pixel pitch such as those used in compact cameras. Some cameras, such as the Panasonic Lumix series and newer Nikon and Sony DSLRs, feature a processing step specifically designed to remove it.
On photographs taken using a digital camera, very small highlights may frequently appear to have chromatic aberration where in fact the effect is because the highlight image is too small to stimulate all three color pixels, and so is recorded with an incorrect color. This may not occur with all types of digital camera sensor. Again, the de-mosaicing algorithm may affect the apparent degree of the problem.
Color shifting through corner of eyeglasses.
Severe purple fringing can be seen at the edges of the horse's forelock, mane, and ear.
This photo taken with the lens aperture wide open resulting in a narrow depth-of-field and strong axial CA. The pendant has purple fringing in the near out-of-focus area and green fringing in the distance. Taken with a Nikon D7000 camera and an AF-S Nikkor 50mm f/1.8G lens.
Chromatic aberration also affects black-and-white photography. Although there are no colors in the photograph, chromatic aberration will blur the image. It can be reduced by using a narrow-band color filter, or by converting a single color channel to black and white. This will, however, require longer exposure (and change the resulting image). (This is only true with panchromatic black-and-white film, since orthochromatic film is already sensitive to only a limited spectrum.)
Chromatic aberration also affects electron microscopy, although instead of different colors having different focal points, different electron energies may have different focal points.[15]
Aberration in optical systems
Achromatic lens – A fix for chromatic aberration
Achromatic telescope
Apochromatic lens
Cooke triplet
Chromostereopsis – Stereo visual effects due to chromatic aberration
Theory of Colours
^ Marimont, D. H.; Wandell, B. A. (1994). "Matching color images: The effects of axial chromatic aberration" (PDF). Journal of the Optical Society of America A. 11 (12): 3113. Bibcode:1994JOSAA..11.3113M. doi:10.1364/JOSAA.11.003113.
^ a b Thibos, L. N.; Bradley, A; Still, D. L.; Zhang, X; Howarth, P. A. (1990). "Theory and measurement of ocular chromatic aberration". Vision Research. 30 (1): 33–49. doi:10.1016/0042-6989(90)90126-6. PMID 2321365. S2CID 11345463.
^ Kruger, P. B.; Mathews, S; Aggarwala, K. R.; Sanchez, N (1993). "Chromatic aberration and ocular focus: Fincham revisited". Vision Research. 33 (10): 1397–411. doi:10.1016/0042-6989(93)90046-Y. PMID 8333161. S2CID 32381745.
^ Aggarwala, K. R.; Kruger, E. S.; Mathews, S; Kruger, P. B. (1995). "Spectral bandwidth and ocular accommodation". Journal of the Optical Society of America A. 12 (3): 450–5. Bibcode:1995JOSAA..12..450A. CiteSeerX 10.1.1.134.6573. doi:10.1364/JOSAA.12.000450. PMID 7891213.
^ Hall, A. Rupert (1996). Isaac Newton: Adventurer in Thought. Cambridge University Press. p. 67. ISBN 978-0-521-56669-8.
^ Hosken, R. W. (2007). "Circle of least confusion of a spherical reflector". Applied Optics. 46 (16): 3107–17. Bibcode:2007ApOpt..46.3107H. doi:10.1364/AO.46.003107. PMID 17514263.
^ "Chromatic Aberration". hyperphysics.phy-astr.gsu.edu
^ Elert, Glenn. "Aberration." – The Physics Hypertextbook.
^ Zoric N.Dj.; Livshits I.L.; Sokolova E.A. (2015). "Advantages of diffractive optical elements application in simple optical imaging systems". Scientific and Technical Journal of Information Technologies, Mechanics and Optics. 15 (1): 6–13. doi:10.17586/2226-1494-2015-15-1-6-13.
^ Amako, J; Nagasaka, K; Kazuhiro, N (2002). "Chromatic-distortion compensation in splitting and focusing of femtosecond pulses by use of a pair of diffractive optical elements". Optics Letters. 27 (11): 969–71. Bibcode:2002OptL...27..969A. doi:10.1364/OL.27.000969. PMID 18026340.
^ Sacek, Vladmir. "9.3. DESIGNING DOUBLET ACHROMAT." telescope-optics.net
^ Colligon-Bradley, P (1992). "Red-green duochrome test". Journal of Ophthalmic Nursing & Technology. 11 (5): 220–2. PMID 1469739.
^ Hecht, Eugene (2002). Optics. 4. ed. Reading, Mass. Addison-Wesley
^ Kühn, J; Colomb, T; Montfort, F; Charrière, F; Emery, Y; Cuche, E; Marquet, P; Depeursinge, C (2007). "Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition". Optics Express. 15 (12): 7231–42. Bibcode:2007OExpr..15.7231K. doi:10.1364/OE.15.007231. PMID 19547044.
^ Misell, D. L.; Crick, R. A. (1971). "An estimate of the effect of chromatic aberration in electron microscopy". Journal of Physics D: Applied Physics. 4 (11): 1668–1674. Bibcode:1971JPhD....4.1668M. doi:10.1088/0022-3727/4/11/308.
Wikimedia Commons has media related to Chromatic aberration.
Methods to correct chromatic aberrations in lens design
Explanation of chromatic aberration by Paul van Walree
PanoTools Wiki article about chromatic aberration
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EUDML | C*-Algebras of Dynamical Systems on the Non-Commutative Torus. EuDML | C*-Algebras of Dynamical Systems on the Non-Commutative Torus.
C*-Algebras of Dynamical Systems on the Non-Commutative Torus.
Carla Farsi; Neil Watling
Farsi, Carla, and Watling, Neil. "C*-Algebras of Dynamical Systems on the Non-Commutative Torus.." Mathematica Scandinavica 75.1 (1994): 101-110. <http://eudml.org/doc/167307>.
@article{Farsi1994,
author = {Farsi, Carla, Watling, Neil},
keywords = {-algebras of dynamical systems; trace; -theoretical invariants; crossed products; affine automorphisms; fixed point subalgebra},
title = {C*-Algebras of Dynamical Systems on the Non-Commutative Torus.},
AU - Farsi, Carla
AU - Watling, Neil
TI - C*-Algebras of Dynamical Systems on the Non-Commutative Torus.
KW - -algebras of dynamical systems; trace; -theoretical invariants; crossed products; affine automorphisms; fixed point subalgebra
{C}^{*}
-algebras of dynamical systems, trace,
K
-theoretical invariants, crossed products, affine automorphisms, fixed point subalgebra
{C}^{*}
{W}^{*}
K
Articles by Carla Farsi
Articles by Neil Watling
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Vaults - Angle
Whitelisting and volatile assets
Token Reactors
Angle Borrowing Module agTokens Vaults
The core mechanism of this module relies on a vault system. Users can deposit collateral in VaultManager contracts, and borrow a certain amount of agTokens from this vault as a debt that will have to be repaid later. By doing so, they can keep their exposure to the tokens deposited as collateral, while being able to spend the borrowed funds. They can also use this mechanism to increase their exposure to the collateral they own, on-chain and in one transaction.
Angle Borrowing module vault-based system lets you:
Borrow agTokens from tokens deposited as collateral in the protocol
Leverage collateral exposure in one transaction
Transfer your debt between vaults to avoid liquidation
Perform different actions on your vault in a single transaction and in a capital-efficient manner
Take self-repaying loans
Borrowing agTokens
The main feature of vaults is the ability to borrow Angle stablecoins. A vault is opened when users deposit tokens as collateral into a VaultManager contract. When doing so, they can choose to borrow a certain amount of agTokens against their collateral. The agTokens borrowed are minted and deposited into their wallets, for them to use however they want. For instance, one may want to borrow stablecoins to profit from stablecoins yield, while keeping exposure to their collateral.
Users with borrowed stablecoins should monitor their vaults' health factor. This metric keeps track of the "health" of the vault: it compares the collateral ratio of the vault with the minimum accepted. If the value of the collateral with respect to the agTokens borrowed decreases so much that the health factor goes below 1, then the vault can get liquidated.
Angle Vaults
With an opened vault, there are many different actions that can be performed beyond borrowing stablecoins: adding collateral, removing collateral from it, repaying the agToken debt. All these actions can be combined with one another in a single transaction.
For more details on how to combine transactions in a modular way with the protocol, you can check this page of our developers documentation.
The advantage of this design is that it enables capital-efficient interactions with Angle vaults.
Capital-efficient debt repayment
An application of this is that users can repay their debt without any capital requirement.
At some point, users may want to close their debt towards the protocol. Instead of having to get back the agTokens they initially borrowed, they can just use the collateral that is in the vault to have it swapped against their debt tokens. In this case, users can just get back the remaining collateral after the debt has been fully paid back.
Capital Efficient Debt Repayment
On a similar note, it is this capital-efficient design that allows liquidators to participate in liquidations without bringing any stablecoin from their pocket. More is explained here.
Leveraging collateral exposure
Users can also take advantage of Angle Borrowing module capital-efficiency features to borrow agTokens to leverage their collateral exposure in one transaction.
When users deposit collateral to open a vault, they can choose the Leverage feature and input the amount of additional exposure they want to the collateral token, up to a certain threshold. What the protocol does is that it mints the necessary quantity of agTokens, swaps it against the desired collateral, and deposits it back into the vault.
Angle Leverage exposure
Leveraging collateral exposure is a very useful feature for users wanting to safely increase exposure to their collateral token on chain. It is however reserved to more advanced users as it increases liquidation risks.
Optimized Liquidations
Liquidations in this borrowing system are designed to protect borrowers, making sure they're less affected than what they would be in other protocols by such events. This is possible thanks to the special liquidation features introduced by Angle including variable liquidation amounts, dynamic discount for liquidators based on Dutch auctions.
For more insight on liquidations, check out this page.
Governance could vote to accept any collateral that can easily be liquidated on any chain for this module. As such, any yield-bearing asset could be used, meaning users could take loans with an interest rate smaller than what they are earning thanks to their yield-bearing asset in collateral: Angle Borrowing module hence opens the way to self-repaying loans.
Vaults are defined by a specific set of information:
A collateral token that is deposited
A debt token that is borrowed (the stablecoin)
And a set of parameters:
Collateral factor
Mint fee
Stability fee
Repay fee
Liquidation surcharge
Dust amount
The most defining parameter of a vault is its Collateral Ratio. It defines the ratio between the collateral's value in stablecoin, and the value of the stablecoins borrowed.
\texttt{Collateral Ratio} = \frac{\texttt{collateral value}}{\texttt{debt value}}
The higher the collateral ratio, the "safer" the vault, as the price decline of the collateral needed to get liquidated and lose the collateral is much bigger.
Additionnally, each vault type has its own collateral factor (CF) parameter. It dictates the minimum collateral ratio allowed for vaults of this type. If this ratio drops below this quantity defined by the CF (1/CF more precisely), the vault risks being liquidated.
For example, if the CF of a vault type is at 2/3, or 150% of min collateral ratio, users need to deposit at least x1.5 more than what they want to borrow. In practice, this means that users wanting to borrow 1,000 agEUR need to deposit at least 1,500€ of ETH for example. If the value of their ETH deposit drops, pushing their CR below 150%, they are in risk of getting liquidated.
Isolated positions & debt transfer
Various Collateral Types
There is one different smart contract per collateral type in Angle Borrowing module: these contracts are called VaultManager contracts. There could be for the same collateral asset different VaultManager contracts, corresponding to different parameters.
For instance, we could imagine for the same agToken ETH as a collateral with a minimum collateral ratio of 150% and a 1% interest rate on borrows and ETH with a minimum collateral ratio of 120% and an interest rate of 3%.
Isolated positions & NFTs
Each VaultManager contract follows the standard of NFT contracts. As such, each vault position is represented by a NFT. This makes them isolated and transferrable. Additionally, this means that a position of an address in a vault is totally separated from a position of the same address in another vault, within the same VaultManager contract or within another contract.
One vault liquidation has no impact on the others.
Thanks to the isolation of positions, one could become under a risk of liquidation, while the other has plenty collateral and a much higher collateral ratio.
In this case, users are able to perform a debt transfer, meaning that they will transfer part of their debt from one of their vault to another, without actually transferring stablecoins. Debt transfer is only possible between vaults related to the same stablecoin, but can be done with vaults linked to different collateral types.
A debt transfer operation increases the health ratio of the first vault, as it has less debt, and decreases that of the second. It allows users to rebalance their debt and collateral ratio between the different vaults they own in an efficient manner, as the transaction just involves an accounting operation.
Interacting with multiple vaults at the same time
Angle is one of the first protocols to implement a permit function on a NFT/ERC721 contract to grant or revoke approval to an address for all the vaults of a contract with a gasless permit signature.
This allows users to interact with multiple vaults from different collaterals in just one transaction using a router contract. In this situation, by just granting approval with a signature to a router contract, they can for instance repay the debt from all their vaults at once, or perform multiple swap/wrapping transactions (e.g. from ETH to wstETH) before adding collateral to a vault.
To make sure the protocol doesn't accumulate bad debt, the protocol needs to enforce a minimum amount of agTokens to be borrowed. This is required to make sure that each position has enough debt so that it's always worth to liquidate it.
It's important to keep in mind that these debt-based models similar to Maker rely heavily on liquidations to remain robust and collateralized. If the amounts to liquidate are too small, the profit made by liquidators could be too low so that it doesn't even cover gas cost, and is not profitable anymore. In that case, the protocol would be left holding under-collateralized positions.
Keepers & Oracle
Angle Borrowing module only relies on two types of keepers: liquidators, which erase risky positions and are key to the health of the system, and keepers to push the surplus stored in the Treasury contract to the protocol.
Angle Borrowing module uses Chainlink price feeds, and may sometimes also rely on on-chain data. For the case of WStETH for example, the protocol needs to call some functions of the StETH contract besides the Chainlink feeds to get the EUR price of WStETH.
The next sections will dive in more details into some aspects of the vaults, such as fees, liquidations, the whitelisting of specific addresses, and token reactors.
Angle Borrowing Module - Previous
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Identifying Sustainable Practices for Tapping and Sap Collection from Birch Trees: Optimum Timing of Tapping Initiation and the Volume of Nonconductive Wood Associated with Taphole Wounds ()
1Proctor Maple Research Center, The University of Vermont, Underhill, VT, USA.
2The University of Vermont Extension, Underhill, VT, USA.
Experiments were conducted to determine two pieces of information essential to identify practices necessary to ensure tapping trees for birch sap collection is both sustainable and profitable—the selection of the time to initiate tapping birch trees to obtain maximum yields, and the volume of nonconductive wood (NCW) associated with taphole wounds in birch trees. The yields obtained from various timing treatments varied between sapflow seasons, but indicate that using test tapholes to choose the appropriate time to initiate tapping is likely to result in optimum yields from birch trees. The volume of NCW associated with taphole wounds in birch trees was highly variable and generally quite large, averaging 220 times the volume of the taphole drilled, and requiring relatively high radial growth rates to maintain NCW at sustainable levels over the long-term. However, more conservative tapping practices, including reduced taphole depth and increased dropline length, as well as thinning and other stand management practices, can be used to reduce the minimum growth rates required. Producers can use this information to ensure that they use tapping practices that will result in sustainable outcomes and obtain the maximum possible sap yields from their trees.
van den Berg, A. , Isselhardt, M. and Perkins, T. (2018) Identifying Sustainable Practices for Tapping and Sap Collection from Birch Trees: Optimum Timing of Tapping Initiation and the Volume of Nonconductive Wood Associated with Taphole Wounds. Agricultural Sciences, 9, 237-246. doi: 10.4236/as.2018.93018.
\stackrel{¯}{\mathrm{Y}}{\text{}}_{\text{Early}}
\stackrel{¯}{\mathrm{Y}}{\text{}}_{\text{Testtaps}}
\stackrel{¯}{\mathrm{Y}}{\text{}}_{\text{Endofmaple}}
[1] Farrell, M. (2013) The Sugarmaker’s Companion. Chelsea Green, White River Junction, USA.
[2] Svanberg, I., Soukand, R., Luczaj, L., Kalle, R., Zyryanova, O., Denes, A., Papp, N., Nedelcheva, A., Seskauskaite, D., Kolodziejska-Degorska, I. and Kolosova, V. (2012) Uses of Tree Saps in Northern and Eastern Parts of Europe. Acta Societatis Botanicorum Poloniae, 81, 343-357.
[3] van den Berg, A.K., Perkins, T.D., Isselhardt, M.L. and Wilmot, T.R. (2016) Growth Rates of Sugar Maple Trees Tapped for Maple Syrup Production Using High-Yield Sap Collection Practices. Forest Science, 62, 107-114.
[4] van den Berg, A.K. and Perkins, T.D. (2014) A Model of the Tapping Zone. Maple Syrup Digest, 26, 18-27.
[5] Chapeskie, D., Wilmot, T.R., Chabot, B. and Perkins, T.D. (2006) Maple Sap Production—Tapping, Collection, and Storage. In: Heiligmann, R.B., Koelling, M.R. and Perkins, T.D., Eds., North American Maple Syrup Producers Manual, The Ohio State University, Columbus, 81-116.
[6] Marov, G.J. and Dowling, J.F. (1990) Sugar in Beverages. In: Pennington, N.L. and Baker, C.W., Eds., Sugar, a User’s Guide to Sucrose, The Sugar Association, Inc., New York, 198-211.
[7] Rasband, W.S., (1997-2016) ImageJ. U. S. National Institutes of Health, Bethesda, Maryland, USA.
[8] Houston, D.R. (1971) Discoloration and Decay in Red Maple and Yellow Birch: Reduction through Wound Treatment. Forest Science, 17, 402-406.
[9] Shigo, A.L. and Sharon, E.M. (1968) Discoloration and Decay in Hardwoods Following Inoculation with Hymenomycetes. Phytopathology, 58, 1493-1498.
[10] Bauch, J., Shigo, A.L. and Starck, M. (1980) Wound Effects in the Xylem of Acer and Betula Species. Holzforschung, 24, 153-160.
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(Redirected from Inverse square)
{\displaystyle {\text{intensity}}\ \propto \ {\frac {1}{{\text{distance}}^{2}}}\,}
{\displaystyle {\frac {{\text{intensity}}_{1}}{{\text{intensity}}_{2}}}={\frac {{\text{distance}}_{2}^{2}}{{\text{distance}}_{1}^{2}}}}
{\displaystyle {\text{intensity}}_{1}\times {\text{distance}}_{1}^{2}={\text{intensity}}_{2}\times {\text{distance}}_{2}^{2}}
{\displaystyle F=k_{\text{e}}{\frac {q_{1}q_{2}}{r^{2}}}}
{\displaystyle I={\frac {P}{A}}={\frac {P}{4\pi r^{2}}}.\,}
{\displaystyle p\ \propto \ {\frac {1}{r}}\,}
{\displaystyle v\,}
{\displaystyle p\,}
{\displaystyle v\ \propto {\frac {1}{r}}\ \,}
{\displaystyle I\ =\ pv\ \propto \ {\frac {1}{r^{2}}}.\,}
{\displaystyle I\propto {\frac {1}{r^{n-1}}},}
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Filter outliers using Hampel identifier - MATLAB - MathWorks Deutschland
The dsp.HampelFilter System object™ detects and removes the outliers of the input signal by using the Hampel identifier. The Hampel identifier is a variation of the three-sigma rule of statistics that is robust against outliers. For each sample of the input signal, the object computes the median of a window composed of the current sample and
\frac{Len-1}{2}
adjacent samples on each side of current sample. Len is the window length you specify through the WindowLength property. The object also estimates the standard deviation of each sample about its window median by using the median absolute deviation. If a sample differs from the median by more than the threshold multiplied by the standard deviation, the filter replaces the sample with the median. For more information, see Algorithms.
Local median —
{m}_{i}=\mathrm{median}\left({x}_{i-k},{x}_{i-k+1},{x}_{i-k+2},\dots ,{x}_{i},\dots ,{x}_{i+k-2},{x}_{i+k-1},{x}_{i+k}\right)
Standard deviation —
{\sigma }_{i}=\kappa \mathrm{median}\left(|{x}_{i-k}-{m}_{i}|,\dots ,|{x}_{i+k}-{m}_{i}|\right),
\kappa =\frac{1}{\sqrt{2}{\mathrm{erfc}}^{-1}\left(1/2\right)}\approx 1.4826
|{x}_{i}-{m}_{i}|>{n}_{\sigma }{\sigma }_{i}
Compares the current sample with nσ × σi, where nσ is the threshold value. If
|{x}_{s}-{m}_{i}|>{n}_{\sigma }×{\sigma }_{i}
, the filter identifies the current sample, xs, as an outlier and replaces it with the median value, mi.
In this example, the Hampel filter slides a window of length 5 (Len) over the data. The filter has a threshold value of 2 (nσ). To have a complete window at the beginning of the frame, the filter algorithm prepends the frame with Len – 1 zeros. To compute the first sample of the output, the window centers on the
{\left[\frac{Len-1}{2}+1\right]}^{\text{th}}
sample in the appended frame, the third zero in this case. The filter computes the median, median absolute deviation, and the standard deviation over the data in the local window.
Median absolute deviation:
ma{d}_{i}=\mathrm{median}\left(|{x}_{i-k}-{m}_{i}|,\dots ,|{x}_{i+k}-{m}_{i}|\right)
. For this window of data,
mad=\mathrm{median}\left(|0-0|,\dots ,|1-0|\right)=0
Standard deviation: σi = κ × madi = 0, where
\kappa =\frac{1}{\sqrt{2}{\mathrm{erfc}}^{-1}\left(1/2\right)}\approx 1.4826
\left[|{x}_{s}-{m}_{i}|=0\right]>\left[\left({n}_{\sigma }×{\sigma }_{i}\right)=0\right]
Repeat this procedure for every succeeding sample until the algorithm centers the window on the
{\left[End-\frac{Len-1}{2}\right]}^{\text{th}}
sample, marked as End. Because the window centered on the last
\frac{Len-1}{2}
samples cannot be full, these samples are processed with the next frame of input data.
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A New Bi-Frequency Soil Smart Sensing Moisture and Salinity for Connected Sustainable Agriculture ()
LAAS-CNRS, Université de Toulouse, CNRS, INSA, UT2J, Toulouse, France.
DOI: 10.4236/jst.2019.93004 PDF HTML XML 457 Downloads 994 Views Citations
Optimizing water consumption is a major challenge for more sustainable agriculture with respect for the environment. By combining micro and nanotechnologies with the offered solutions of IoT connection (Sigfox and LoRa), new sensors allow the farmer to be connected to his agricultural production by mastering in real time the right contribution needed in water and fertilizer. The sensor designed in this research allows a double measurement of soil moisture and salinity. In order to minimize the destructuring of the ground to insert the sensor, we have designed a cylindrical sensor, easy to insert, with its electronics inside its body to propose a low power electronic architecture capable of measuring and communicating wireless with a LoRa or Sigfox network or even the farmer’s cell phone. This new smart sensor is then compared to the current leaders in agriculture to validate its performance. Finally, the sensor has better performance than commercials, a better response time, a better precision and it will be cheaper. For the salinity measure, it can detect the level of fertilizer in the soil according to the need of farmers.
Smart Sensing, Connected Agriculture, Bi-Frequency Sensor
Roux, J. , Escriba, C. , Fourniols, J. and Soto-Romero, G. (2019) A New Bi-Frequency Soil Smart Sensing Moisture and Salinity for Connected Sustainable Agriculture. Journal of Sensor Technology, 9, 35-43. doi: 10.4236/jst.2019.93004.
In the context of agriculture modernization, the farmers need tools to develop the smart farming [1] [2] . For that, it is necessary to develop new sensors which can be deployed closest to the plants. To control the irrigation, the sensors measure the soil moisture near the crops [3] [4] . This new intelligent sensor combines non-contact moisture and salinity measurement [5] , exploiting a capacitive reading between two spiral form factor electrodes on a cylindrical preform to optimize contact surfaces for minimal electrode volume. Thus, by reducing the cost of the measuring point, it is possible to deploy more sensors in the ground and thus obtain an observation in the field more in agreement with the variations of the behavior of the grounds. The two optimized parameters are, on the one hand, soil moisture, which measured at different depths allows to know the mechanism of absorption of the crop plant; and on the other hand, the salinity of the soil which gives information on the amount of soil nutrients necessary for the development of the plant [6] [7] .
In order to reduce the cost of manufacture and the constraints of placement in the ground, we did not retain the resistive measurement by contact [8] [9] [10] but privileged a capacitive measurement. The main advantage of this type of sensor is that it possesses a response time of less than a minute, which gives it the ability to monitor the hydric condition of the soil in close to real time [11] . Existing solutions [12] [13] [14] include sensors based on a small-sized (<1 mm) detection cell), which limits the volume of soil that can be tested. On the other hand, their complex structures [15] [16] [17] do not make them easy to use and require some time, several months, to restructure the soil, which can be damaging to farmers. Moreover, some commercials (Enviroscan, Meteor) is available but their costs limit their deployment and do not permit a reliable cover of the fields or others (Divine) can’t be implemented for a season of culture and provide just a portable measurement.
The article presented will be oriented in three parts. The form factor of the electrodes will not be detailed in order to privilege on the one hand the electronic architecture of reading bi-frequency retained, then we will demonstrate by experimentation the usable frequency bands to observe respectively salinity and humidity [18] . Finally, we will present the results of the assembled sensor and will position its measurement sensitivity with respect to Decagon™ and Sentek™ sensors used by the farmers.
2. Sensor’s Model
We chose the capacitive method to measure so the sensor is a capacity. But because of the integration, parasite capacities are created as shown in Figure 1.
Figure 1. Electric model of the sensor.
In this model, we can see the variable capacity C1 dependent to soil properties (humidity or salinity) and also fixed capacities created by the electrodes’ interferences C4 and the plastic protection C2 and C3. This model can be simplified by a single variable capacity in parallel with a single fixed capacity.
3. Measurement Architecture
The capacitive variation induced by the lining of soil properties in response to a variation of humidity and/or salinity is exploited by a Colpitts oscillator which generates a sinusoidal signal of adjustable frequency to sweep the spectrum of measurements to search for the spectral band containing the information. In Figure 2, we can see the schematic of this Colpitts oscillator and its output frequency that is inversely proportional to the soil moisture.
The oscillation frequency can be read by an analog acquisition chain (Figure 3) associating an integrated frequency/voltage converter and a voltage reading using a CAN built into a microcontroller. Therefore, thinking microcontroller forces us to compare this architecture with an all-digital solution (Figure 4) with a dedicated embedded algorithm.
Three levels of salinity have been defined; the frequency sweep performed by the Colpitts oscillator shows the variation of capacity on the interval 100 kHz to
Figure 2. Frequency response of the Colpitts oscillator by the soil moisture.
Figure 3. Analog sensor’s architecture.
Figure 4. Digital sensor’s architecture.
10 MHz (Figure 5).
Two areas of interest appear on this curve:
· An area where the capacitance variation of the electrodes is insensitive to the salinity: above 4 MHz, the curves merge regardless of the salinity.
· An area where the capacity varies proportionally with salinity. Thus, the measurement of the salinity is defined at the frequency of 500 kHz.
By observing (Figure 6) the variation of capacity as a function of humidity, the reading frequency ranges are not superimposed with the exploitable area of the 500 kHz dedicated to the salinity measurement. We can, therefore, define 8 MHz frequency for humidity measurement and thus obtain two operating ranges of our electrodes that will de-worm two totally uncorrelated salinity and soil moisture observations.
To make the good choice between an analogue or all-digital architecture, it is necessary to compare the stabilization of the measurement with the variations of environment of which one of the principal parameters is the variation of temperature in the soil which varies between 5˚C to 50˚C (Figure 7). While the time
Figure 5. Sensor’s capacity by the soil salinity.
Figure 6. Sensor’s capacity by soil moisture.
Figure 7. Thermal response of both architectures.
frequency converter has at a sensitivity of 0.3%∙˚C−1 for the analog architecture; the TCXO choice allows the digital architecture to guarantee robustness to temperature variations. With this quartz cost constraint, all digital solution is chosen and can offer versatile corrections linked to soil characteristics variations (type of soil size of aggregates…).
To allow the insertion in the ground and to ensure the solidity, we decide to place the electronic card in the sensor tube even if the presence of components close to the electrodes will modify the electromagnetic behavior. The contact between the electrodes of the tube and the sensor is made using contact zones positioned on the top of the card. Connections with the outside (data + energy) are made on the top of the card and are solidified by cable holes. Complete integration is shown in Figure 8 with its characteristics in Table 1.
To be available by the farmers, the sensor’s data have to be online. For that, the sensor has to be connected. Several links exist but two of them stand out, Sigfox and LoRa. These protocols are low rate but long range and low energy. With these links, the sensor can transmit at more than 10 km while remaining autonomous.
The sensor is buried on the soil so it cannot emit itself, he needs a part aboveground. This part concentrates the data of four sensors to measure at four depths. It will transmit the data to a server and a web application posts them to the farmers.
Figure 8. Sensor integration
Table 1. Our sensor’s performance compared to market leader.
To reproduce field conditions in laboratory, we sample soil in culture. This soil is heated to eliminate all water contained. To evacuate all the water, the soil is weighed before and after the heated phase. If the weight does not change, the soil is dry. If not, a new phase is begun. Then the soil is separated in calibrated samples.
To obtain a precise moisture range, a calibrated quantity of water relative to the weight of the sample is added to the samples to create different levels of moisture.
For the salinity, a calibrated quantity of fertilizer is added to the water but the sample receives the same quantity of water.
The samples are stocked on airtight jar to avoid evaporation and then modifications of the samples.
To realize the measurement, we buried the sensor in the jar. As the pot is made of glass, the electromagnetic field does not distribute and the measure is correct (Figure 9). We use an impedance-meter which can sweep the frequency of measurement to obtain the results presented in this paper.
Then, Figure 10 shows the sensor measuring salinity variations. To obtain
Figure 9. Measurement in a jar.
Figure 10. Sensor’s output by soil salinity.
these variations, soil samples are moistened to the same humidity but in addition, nitrogen, a common fertilizer, is dissolved to modify soil salinity.
We can observe that output can be approached by the following linear relation:
{V}_{s}\left(V\right)=0.0657*\text{Nitrogenadded}\left(%\right)-0.0023
In fact, the sensor has not to be very precise. The farmers set a threshold below which it is needed to add fertilizer. The threshold changes with the type of culture. The salinity measurement is validated.
6. Sensor Measurements
Observe and compare the behavior of our sensor over long periods with industrialized sensors on an orchard and cornfield culture (Figure 11 & Figure 12). Look at the moisture response following the water supplies confirmed by the rain gauge. We can notice that we are more precise on the observation of the soil drying dynamics.
We can observe on this graph the similarity of response between the Decagon™ and our sensor: response time to a water intake is immediate; drying dynamics are identical. In addition, note in the purple box area: our sensor detects
Figure 11. Cornfield culture sensing.
Figure 12. Apple orchard sensing.
a water intake not seen by the Decagon™ sensor, which, in this example, reflects a better sensitivity.
Using capacitive technology for our sensor, we develop a new smart sensor able to measure soil moisture and also soil salinity. For this purpose, double helix electrodes are formed to optimize the relationship between the sensor and the ground. Bases on Colpitts oscillator we develop dual-frequency electronics and a full digital signal processing to reduce cost. Regarding the humidity measurement, we obtain a sensor that has the same performance as the market leaders but for lower cost and a new functionality.
Now for a large deployment, new tests have to be made to test the resistance of the sensor. For the tests, even if they were realized in actual exploitations, they were not manipulated by farmers. With these tests, a new mechanical shape may be designed. Moreover, these tests will permit to monitor the plastic protection’s usury. Our longer test lasts 6 months so we don’t know if the sensor can be reused several seasons of culture.
This work was realized within the framework of the IRRIS project which is funded by the French Midi Pyrenees council and the French Ministry of Agriculture. Great thanks are due to TCSD (Telecommunication Service & Distribution) who lead the project and helped us to deploy our system into actual agricultural exploitation. We would thank Gilbert SA for their industrial support and their advice for the mechanical integration.
[1] Nabayi, A., The, C.B.S., Mohd Hanif, A.H. and Sulaiman, Z. (2018) Plant Growth, Nutrient Content and Water Use of Rubber (Hevea brasiliensis) Seedlings Grown Using Root Trainers and Different Irrigation Systems. Pertanika Journal of Tropical Agricultural Science, 41, 251-270.
[2] Pigford, A.-A.E., Hickey, G.M. and Klerkx, L. (2018) Beyond Agricultural Innovation Systems? Exploring an Agricultural Innovation Ecosystems Approach for Niche Design and Development in Sustainability Transitions. Agricultural Systems, 164, 116-121.
[3] Gao, Z., Zhu, Y., Liu, C., Qian, H., Cao, W. and Ni, J. (2018) Design and Test of a Soil Profile Moisture Sensor Based on Sensitive Soil Layers. Sensors, 18, 1648.
[4] Faybishenko, B. (2000) Tensiometer for Shallow and Deep Measurements of Water Pressure in Vadose Zone and Groundwater. Soil Science, 165, 473-482.
[5] Visacro, S., Alipio, R., Murta Vale, M.H. and Pereira, C. (2011) The Response of Grounding Electrodes to Lightning Currents: The Effect of Frequency-Dependent Soil Resistivity and Permittivity. IEEE Transactions on Electromagnetic Compatibility, 53, 401-406.
[6] Oriol, P., Rapanoelina, M. and Gaudin, R. (1995) Le pilotage de l’irrigation de la canne à sucre par tensiomètres. Agricultural Development, 6, 39-48.
[7] Moutonnet, P., Brandy-Cherrier, M. and Chambon, J. (1981) Possibilites d’utilisation des tensiometres pour l’automation de l’irrigation. Plant Soil, 59, 335-345.
[8] Zhou, Q.Y., Shimada, J. and Sato, A. (2001) Three-Dimensional Spatial and Temporal Monitoring of Soil Water Content Using Electrical Resistivity Tomography. Water Resources Research, 37, 273-285.
[9] Dean, T.J., Bell, J.P. and Baty, A.J.B. (1987) Soil Moisture Measurement by an Improved Capacitance Technique, Part I. Sensor Design and Performance. Journal of Hydrology, 93, 67-78.
[10] da Costa, E., de Oliveira, N., Morais, F., Carvalhaes-Dias, P., Duarte, L., Cabot, A. and Siqueira Dias, J. (2017) A Self-Powered and Autonomous Fringing Field Capacitive Sensor Integrated into a Micro Sprinkler Spinner to Measure Soil Water Content. Sensors, 17, 575.
[11] Chakraborty, M., Kalita, A. and Biswas, K. (2018) PMMA-Coated Capacitive Type Soil Moisture Sensor: Design, Fabrication, and Testing. IEEE Transactions on Instrumentation and Measurement, 68, 189-196.
[12] Kalita, H., Palaparthy, V.S., Baghini, M.S. and Aslam, M. (2016) Graphene Quantum Dot Soil Moisture Sensor. Sensors and Actuators B: Chemical, 233, 582-590.
[13] Palaparthy, V.S., Kalita, H., Surya, S.G., Baghini, M.S. and Aslam, M. (2018) Graphene Oxide-Based Soil Moisture Microsensor for in Situ Agriculture Applications. Sensors and Actuators B: Chemical, 273, 1660-1669.
[14] Boudaden, J., Steinmal, M., Endres, H.E., Drost, A., Eisele, I., Kutter, C. and Müller- Buschbaum, P. (2018) Polyimide-Based Capacitive Humidity Sensor. Sensors, 18, 1516.
[15] Xiao, D., Feng, J., Wang, N., Luo, X. and Hu, Y. (2013) Integrated Soil Moisture and Water Depth Sensor for Paddy Fields. Computers and Electronics in Agriculture, 98, 214-221.
[16] Dias, P.C., Cadavid, D., Ortega, S., Ruiz, A., França, M.B., Morais, F.J.O., Ferreira, E.C. and Cabot, A. (2016) Autonomous Soil Moisture Sensor Based on Nanostructured Thermosensitive Resistors Powered by an Integrated Thermoelectric Generator. Sensors and Actuators A: Physical, 239, 1-7.
[17] Pichorim, S., Gomes, N. and Batchelor, J. (2018) Two Solutions of Soil Moisture Sensing with RFID for Landslide Monitoring. Sensors, 18, 452.
[18] Weil, N.C., Brady, R.R. and Weil, R.R. (2016) The Nature and Properties of Soils. Fifteenth Edition, Pearson, Columbus.
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Zeal - The RuneScape Wiki
The Old School RuneScape Wiki also has an article on: osrsw:Zeal points
No (intangible)
Soul Wars Zeal is obtained as a reward after a Soul Wars game. The amount of points given depends on the game result (see table below). It's advised to use Zeal on high skills to get more experience per Zeal.
Points earned after a game
1.2 Charm rewards
1.3.1 Slayer pets
1.3.2 Random loot
1.3.3 Imbuing rings
Main article: Soul Wars/Rewards
Nomad, or Zimberfizz after completing Nomad's Requiem, or Zimberfizz ashes or Zanik after completing Nomad's Elegy exchanges Zeal into combat or Slayer experience, Summoning charms or slayer pets. Players may also gamble their points for a random item.
Main article: Soul Wars/Rewards § Charts
{\displaystyle Z}
{\displaystyle L}
{\displaystyle \left\lfloor {\frac {L^{2}}{600}}\right\rfloor \times 525\times Z}
{\displaystyle \left\lfloor {\frac {L^{2}}{600}}\right\rfloor \times 480\times Z}
{\displaystyle \left\lfloor {\frac {L^{2}}{600}}\right\rfloor \times 270\times Z}
{\displaystyle \left\lfloor 7.5\times 1.1048^{L-1}\right\rfloor \times Z}
{\displaystyle (\left\lfloor {\frac {L^{2}}{349}}\right\rfloor +1)\times 45\times Z}
For calculators for the experience gained for Zeal, see Soul Wars/Rewards#Calculators.
Charm rewards[edit | edit source]
Main article: Soul Wars/Rewards § Charms
Gold, green, crimson plus blue charms can be bought. A set of gold charms costs 4 Zeal, green charms 5 points, crimson charms 12 points and blue charms 30 points. The number of charms received depends on the player's combat levels.
Other rewards[edit | edit source]
Main article: Soul Wars/Rewards § Other
Other rewards include Slayer pets, the option to gamble for random items, and the ability to imbue rings.
Slayer pets[edit | edit source]
Main article: Soul Wars/Rewards § Slayer pets
You will need a trophy dropped by a Slayer monster (stuffed or unstuffed) and a number of Zeal respective to the monster's Slayer level.
Fire cape TzRek Jad 100
You will not reclaim a trophy when removing it from your player-owned house's skill hall.
Main article: Soul Wars/Rewards § Random loot
The Random loot option costs two Zeal and rewards the player with a random stack of items related to Prayer or Summoning. A full list of possible rewards can be found here.
Main article: Soul Wars/Rewards § Imbuing rings
The Imbue option allows players to have a selection of rings imbued with power. Imbuing rings increases their damage bonuses by a small amount. A full list of rings imbuable with Zeal can be found here.
Zeal rewards were tripled on 28 October 2011 to celebrate the bot nuke.
Retrieved from ‘https://runescape.wiki/w/Zeal?oldid=35684500’
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Field (mathematics) - Simple English Wikipedia, the free encyclopedia
In mathematics, a field is a certain kind of algebraic structure. In a field, one can add (
{\displaystyle x+y}
), subtract (
{\displaystyle x-y}
), multiply (
{\displaystyle x\cdot y}
) and divide (
{\displaystyle x/y}
) two numbers (with division only possible if
{\displaystyle y}
is non-zero). A field is a special ring in which division is possible. Both the set of rational numbers and the set of real numbers are examples of fields. One can also use the symbol
{\displaystyle \mathbb {F} }
as a variable for a field.[1]
Since a field is always a ring, it consists of a set (represented here with the letter R) with two operations: addition (+) and multiplication (•). The properties of a ring are:
Closure: If an operation is used on any elements in the ring, the element that is formed will also be part of the ring.
For all a, b in R, the result of the operation a + b is also in R.
For all a, b in R, the result of the operation a • b is also in R.
Additive Identity element: One element of the ring is special. It is called the additive identity element named zero. If we add zero to any element, that element will not change.
There exists an element 0 in R, such that for all elements a in R, the equation 0 + a = a + 0 = a holds.
Associativity of Addition: When addition is done many times, it does not matter how it is grouped, the result will be the same.
For all a, b and c in R, the equation (a + b) + c = a + (b + c) holds.
Commutativity of addition: When addition is done, it does not matter which element is on the right and left, the result will be the same.
For all a and b in R, the equation a + b = b + a holds.
Additive Inverse element: Every element in the ring has another element in the ring, such that when addition is performed between them, the result is zero. This is known as its additive inverse.
For each a in R, there exists an element -a in R such that a + (-a) = 0, where 0 is the additive identity element.
Associativity of multiplication: When multiplication is done many times, it does not matter how it is grouped, the result will be the same.
For all a, b and c in R, the equation (a • b) • c = a • (b • c) holds.
Distribution: Addition and multiplying mix together nicely:
For all a, b and c in R, the equation (a + b) • c = (a • c) + (b • c) holds.
For all a, b and c in R, the equation a • (b + c) = (a • b) + (a • c) holds.
A field is a ring (so it follows the rules above), with these extra rules:
Commutativity of multiplication: When multiplication is done, it does not matter which element is on the right and left, the result will be the same.
For all a and b in R, the equation a • b = b • a holds.
Multiplicative Identity element: One element of the ring is special. It is called the multiplicative identity element named one, and is different to zero. If we multiply one with any element, that element will not change.
There exists an element 1 (different to 0) in R, such that for all elements a in R, the equation 1 • a = a • 1 = a holds.
Multiplicative Inverse element: Every element in the ring apart from the zero element has an inverse. Multiplying an element by its inverse will give one.
For each a in R, apart from 0, there exists an element a-1 such that a • a-1 = a-1 • a = 1 holds.
Another way of describing a field is this:
The set R using the operation + is an abelian (commutative) group with the zero element as its identity element.
The set R, without the zero element, using the operation • is also an abelian group, with the unit element as its identity element.
The field also satisfies the distribution rules, listed above.
Examples of fields[change | change source]
Examples of fields include:[2]
The set of rational numbers
{\displaystyle \mathbb {Q} }
The set of real numbers
{\displaystyle \mathbb {R} }
The set of complex numbers
{\displaystyle \mathbb {C} }
In fact, among the subsets of complex numbers, the largest field is
{\displaystyle \mathbb {C} }
and the smallest field is
{\displaystyle \mathbb {Q} }
.[3] The set of integers,
{\displaystyle \mathbb {Z} }
, is not a field, because the result of a division is not always an integer.
↑ Weisstein, Eric W. "Field". mathworld.wolfram.com. Retrieved 2020-10-07.
↑ "Algebra - Fundamental concepts of modern algebra". Encyclopedia Britannica. Retrieved 2020-10-07.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Field_(mathematics)&oldid=8211719"
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Real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in C3
April, 2000 Real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in
{C}^{3}
We show that the real Seifert form determines the weights for nondegenerate quasihomogeneous polynomials in
{C}^{3}
. Consequently the real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in
{C}^{3}
. As a corollary, we obtain the topological invariance of weights for nondegenerate quasihomogeneous polynomials in
{C}^{3}
, which has already been proved by the author [Sael] and independently by Xu and Yau [Yal], [Ya2], [XY1], [XY2]. The method in this paper is totally different from their approaches and gives some new results, as corollaries, about holomorphic function germs in
{C}^{3}
which are connected by
\mu
-constant deformations to nondegenerate quasihomogeneous polynomials. For example, we show that two semiquasihomogeneous functions of three complex variables have the same topological type if and only if they are connected by a
\mu
-constant deformation.
Osamu SAEKI. "Real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in
{C}^{3}
Secondary: 14B05 , 32S25 , 32S55
Keywords: $\mu$-constant deformation , quasihomogeneous polynomiaj weights , Real Seifert form , spectrum
Osamu SAEKI "Real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in
{C}^{3}
|
EUDML | Sheaves associated to holomorphic first integrals EuDML | Sheaves associated to holomorphic first integrals
ℱ:L\to TS
be a foliation on a complex, smooth and irreducible projective surface
S
ℱ
admits a holomorphic first integral
f:S\to {ℙ}^{1}
{h}^{0}\left(S,{𝒪}_{S}\left(-n{𝒦}_{S}\right)\right)>0
n\ge 1
we prove the inequality:
\left(2n-1\right)\left(g-1\right)\le {h}^{1}\left(S,{ℒ}^{\text{'}-1}\left(-\left(n-1\right){K}_{S}\right)\right)+{h}^{0}\left(S,{ℒ}^{\text{'}}\right)+1
S
is rational we prove that the direct image sheaves of the co-normal sheaf of
ℱ
f
are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.
Zamora, Alexis García. "Sheaves associated to holomorphic first integrals." Annales de l'institut Fourier 50.3 (2000): 909-919. <http://eudml.org/doc/75443>.
abstract = {Let $\{\cal F\}: L \rightarrow TS$ be a foliation on a complex, smooth and irreducible projective surface $S$, assume $\{\cal F\}$ admits a holomorphic first integral $f:S \rightarrow \{\Bbb P\}^1$. If $h^0(S,\{\cal O\}_S(-n\{\cal K\}_S))>0$ for some $n\ge 1$ we prove the inequality: $(2n-1)(g-1) \le h^1(S, \{\cal L\}^\{\prime -1\}(-(n-1)K_S)) +h^0 (S, \{\cal L\}^\prime ) +1$. If $S$ is rational we prove that the direct image sheaves of the co-normal sheaf of $\{\cal F\}$ under $f$ are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.},
author = {Zamora, Alexis García},
keywords = {holomorphic foliations; first integrals},
title = {Sheaves associated to holomorphic first integrals},
AU - Zamora, Alexis García
TI - Sheaves associated to holomorphic first integrals
AB - Let ${\cal F}: L \rightarrow TS$ be a foliation on a complex, smooth and irreducible projective surface $S$, assume ${\cal F}$ admits a holomorphic first integral $f:S \rightarrow {\Bbb P}^1$. If $h^0(S,{\cal O}_S(-n{\cal K}_S))>0$ for some $n\ge 1$ we prove the inequality: $(2n-1)(g-1) \le h^1(S, {\cal L}^{\prime -1}(-(n-1)K_S)) +h^0 (S, {\cal L}^\prime ) +1$. If $S$ is rational we prove that the direct image sheaves of the co-normal sheaf of ${\cal F}$ under $f$ are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.
KW - holomorphic foliations; first integrals
[1] W. BARTH, C. PETERS, and A. VAN DE VEN, Compact Complex Surfaces, Springer Verlag, 1984. Zbl0718.14023MR86c:32026
[2] M. BRUNELLA, Feuilletages holomorphes sur les surfaces complexes compactes, Ann. scient. Ec. Norm. Sup., 30 (1997), 569-594. Zbl0893.32019MR98i:32051
[3] X. GOMEZ-MONT, and R. VILA, On Meromorphic Integrals of Holomorphic Foliations in Surfaces, Unpublished.
[4] P. GRIFFITHS, and J. HARRIS, Principles of Algebraic Geometry, John Wiley & Sons, 1978. Zbl0408.14001MR80b:14001
[5] G. KEMPF, Algebraic Varieties, Cambridge University Press, 1993. Zbl0780.14001MR94k:14001
[6] D. MUMFORD, Abelian Varieties, Oxford University Press, 1970. Zbl0223.14022MR44 #219
[7] H. POINCARÉ, Sur l'intégration algébrique des équations différentielles du primer ordre, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191. JFM23.0319.01
[8] A. SEIDENNBERG, Reduction of Singularities of the Differential Equation Ady - Bdx, Am. Journal of Math., (1968), 248-269. Zbl0159.33303
[9] A.G. ZAMORA, Foliations on Algebraic Surfaces having a Rational First Integral, Public. Mat de la Universitá Aut. de Barcelona, 41 (1997), 357-373. Zbl0910.32039MR98m:32048
holomorphic foliations, first integrals
Holomorphic foliations and vector fields
Articles by Alexis García Zamora
|
Basic state - zxc.wiki
The basic state of a quantum mechanical or quantum field theoretical system is its state with the lowest possible energy (see also energy level ).
When the only electron of the hydrogen atom can no longer give off any energy, it is in the ground state (bottom line)
The basic state of a system is always stable, since there is no state of lower energy into which it could pass ( decay ). A system in a state of higher energy (an excited state ) can, in accordance with the law of conservation of energy , pass into its basic state or a less highly excited state, if this is not prevented by certain laws, such as other laws of conservation ( selection rules ).
However, the term system is somewhat arbitrary. You can z. For example, consider the particles into which a radioactive atomic nucleus decays (e.g. nitrogen-14 nucleus + electron + antineutrino) as just another state of the original system (here carbon-14 nucleus); in this sense, the basic state of a radionuclide is not stable either.
The basic state of a quantum mechanical system does not have to be unique. If there are several states with the same lowest energy, this is called a degenerate ground state. One example is the spontaneous breaking of symmetry , where the symmetry of the system is reduced by the degeneracy of the ground state .
Since the temperature is a monotonically increasing function of the energy of the individual particles, systems in a "cold" environment are usually in their ground state. For most systems, e.g. B. Atoms , room temperature is already a cold environment.
A system in its ground state can still contain a surprising amount of energy. This can be seen in the example of the Fermi distribution of conduction electrons in a metal: the Fermi temperature of the most energetic electrons near the Fermi edge is a few 10,000 K - even if the metal has cooled down well below room temperature. However, this energy cannot be extracted from the metal and used because the electron gas can not adopt an even lower energy state.
{\ displaystyle T_ {f} \! \,}
In quantum field theory , the ground state is often referred to as vacuum state , vacuum or quantum vacuum . The ground state on the flat Minkowski space-time is defined by its invariance under Poincaré transformations, in particular under time translation. Since the Poincaré group is not a symmetry group for curved spacetime , quantum fields in curved spacetime have i. General no clear ground state. To be more precise, there is only a unique ground state if there is a one-parameter isometric group of time translations of space-time.
Peter W. Milonni: The quantum vacuum - an introduction to quantum electrodynamics. Acad. Press, San Diego 1994, ISBN 0-12-498080-5
This page was last edited on 9 September 2018, at 10:38 PM.
This page is based on the copyrighted Wikipedia article "Grundzustand" (Authors); it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License. You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA.
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Finite State Machines | Brilliant Math & Science Wiki
Karleigh Moore and Dishant Gupta contributed
A finite state machine (sometimes called a finite state automaton) is a computation model that can be implemented with hardware or software and can be used to simulate sequential logic and some computer programs. Finite state automata generate regular languages. Finite state machines can be used to model problems in many fields including mathematics, artificial intelligence, games, and linguistics.
A system where particular inputs cause particular changes in state can be represented using finite state machines. This example describes the various states of a turnstile. Inserting a coin into a turnstile will unlock it, and after the turnstile has been pushed, it locks again. Inserting a coin into an unlocked turnstile, or pushing against a locked turnstile will not change its state.
There are two types of finite state machines (FSMs): deterministic finite state machines, often called deterministic finite automata, and non-deterministic finite state machines, often called non-deterministic finite automata. There are slight variations in ways that state machines are represented visually, but the ideas behind them stem from the same computational ideas. By definition, deterministic finite automata recognize, or accept, regular languages, and a language is regular if a deterministic finite automaton accepts it. FSMs are usually taught using languages made up of binary strings that follow a particular pattern. Both regular and non-regular languages can be made out of binary strings. An example of a binary string language is: the language of all strings that have a 0 as the first character. In this language, 001, 010, 0, and 01111 are valid strings (along with many others), but strings like 111, 10000, 1, and 11001100 (along with many others) are not in this language.
You can walk through the finite state machine diagram to see what kinds of strings the machine will produce, or you can feed it a given input string and verify whether or not there exists a set of transitions you can take to make the string (ending in an accepting state).
A deterministic finite automaton (DFA) is described by a five-element tuple:
(Q, \Sigma, \delta, q_0, F)
Q
= a finite set of states
\Sigma
= a finite, nonempty input alphabet
\delta
= a series of transition functions
q_0
= the starting state
F
= the set of accepting states
There must be exactly one transition function for every input symbol in
\Sigma
from each state.
DFAs can be represented by diagrams of this form:
Write a description of the DFA shown above. Describe in words what it does.
Q = \{s_1, s_2\}
\Sigma\ = \{0,1\}
\delta
current state input symbol new state
s_1
s_1
s_1
s_2
s_2
s_2
s_2
s_1
q_0 = s_1
F= {s_1}
This DFA recognizes all strings that have an even number of 0’s (and any number of 1’s). This means that if you run any input string that has an even number of 0’s, the string will finish in the accepting state. If you run a string with an odd number of 0’s, the string will finish in
s_2
, which is not an accepting state.
abacdaac abac aaaaac aaaacd
What string cannot be generated by the finite state machine below?
Here is a DFA diagram that describes a few simple moves that a character in a video game can do: stand, run, and jump. The buttons that a player can use to control this particular character are "Up," "A," or the player can press no button.
Using the state diagram for the video game character above, describe how a player can control their character to go from standing to running to jumping.
In the standing state, the player can press nothing and remain in the standing state, then, to transition to the running state, the user must press the "Up" button. In the running state, the user can continue to make their character run by pressing the "Up" button, and then to transition to the jump state, the user must press "A."
Draw a diagram for a DFA that recognizes the following language: The language of all strings that end with a 1.
Similar to a DFA, a nondeterministic finite automaton (NDFA or NFA) is described by a five-element tuple:
(Q, \Sigma, \delta, q_0, F)
Q
\Sigma
\delta
q_0
F
Unlike DFAs, NDFAs are not required to have transition functions for every symbol in
\Sigma
, and there can be multiple transition functions in the same state for the same symbol. Additionally, NDFAs can use null transitions, which are indicated by
\epsilon
. Null transitions allow the machine to jump from one state to another without having to read a symbol.
An NDFA accepts a string
x
if there exists a path that is compatible with that string that ends in an accept state.
NDFAs can be represented by diagrams of this form:
Describe the language that the NDFA above recognizes.
The NDFA above recognizes strings that end in “10” and strings that end in “01.”
a
is the start state, and from there, we can create a string with however many 1’s and 0’s in any order, and then transfer to state
b
or state
, or we can immediately transfer to state
b
or state
. In any case, the NDFA will only accept a string that reaches state
d
g
. In order to reach state
d
g
, the string must end with a “01” (for state
d
) or a “10” (for state
g
For example, the following strings are all recognized by this NDFA.
Draw a diagram for the NDFA that describes the following language: The language of all strings that end with a 1.
One might think that NDFAs can solve problems that DFAs cannot, but DFAs are just as powerful as NDFAs. However, a DFA will require many more states and transitions than an NDFA would take to solve the same problem. To see this, examine the example in the proof below.
NDFAs are equivalent to DFAs
To convert a DFA into an NDFA, just define an NDFA that has all the same states, accept states, transitions, and alphabet symbols as the DFA. Essentially, the NDFA “ignores” its nondeterminism because it does not use null transitions and has exactly one transition per symbol in each state.
DFAs are equivalent to NDFAs
If an NDFA uses
n
states, a DFA would require up to
2^n
states in order to solve the same problem. To translate an NDFA into a DFA, use powerset construction. For example, if an NDFA has 3 states, the corresponding DFA would have the following state set:
\{\emptyset, \{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}
. Each element in the state set represents a state in the DFA. We can determine the transition functions between these states for each element of the set.
For example, in the DFA, state
\{a\}
\{a,b\}
on input 0 since in the NDFA above, state
\{a\}
goes into both state
and state
b
. This process is repeated for the remaining states in the set of the DFA.
Since DFAs are equivalent to NDFAs, it follows that a language is regular if and only if it is recognized by an NDFA.
With a few additional proofs, one can show that NDFAs and DFAs are equivalent to regular expressions.
By definition, a language is regular if and only if there is a DFA that recognizes it. Since DFAs are equivalent to NDFAs, it follows that a language is regular if and only if there is an NDFA that recognizes it. Therefore, DFAs and NDFAs recognize all regular languages.
One limitation of finite state machines is that they can only recognize regular languages. Regular languages make up only a small, finite portion of possible languages. See context free grammars and Turing machines.
Aaronson, S. 6.045J Automata, Computability, and Complexity: Lecture 3, Spring 2011. Retrieved April,1, 2016, from http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-045j-automata-computability-and-complexity-spring-2011
Cite as: Finite State Machines. Brilliant.org. Retrieved from https://brilliant.org/wiki/finite-state-machines/
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EUDML | On approximation of compact multivalued maps in topological vector spaces. EuDML | On approximation of compact multivalued maps in topological vector spaces.
On approximation of compact multivalued maps in topological vector spaces.
Gajić, Ljiljana
Gajić, Ljiljana. "On approximation of compact multivalued maps in topological vector spaces.." Novi Sad Journal of Mathematics 26.1 (1996): 19-24. <http://eudml.org/doc/228959>.
@article{Gajić1996,
author = {Gajić, Ljiljana},
keywords = { paranormed spaces; paranormed spaces},
title = {On approximation of compact multivalued maps in topological vector spaces.},
AU - Gajić, Ljiljana
TI - On approximation of compact multivalued maps in topological vector spaces.
KW - paranormed spaces; paranormed spaces
{\phi }^{\text{'}}
paranormed spaces,
{\varphi }^{\text{'}}
paranormed spaces
Articles by Gajić
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Home : Support : Online Help : Science and Engineering : Units : Environments : Natural : Equalities and Inequalities
Equalities and Inequalities in the Natural Units Environment
In the Natural Units environment, the equality and inequality operators (=, <, <=, >, >=, and <>) are modified so that they perform the necessary operations on expressions with units.
The properties of the five arithmetic operations are:
Aa=Bb->A=rB
Aa<Bb->A<rB
Aa<=Bb->A<=rB
Aa>Bb->A>rB
Aa>=Bb->A>=rB
Aa<>Bb->A<>rB
where a and b are units; A and B are coefficients; and r is the conversion factor from the unit
b
to the unit a.
3*ft=yd;
\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3}
evalb((1));
\textcolor[rgb]{0,0,1}{\mathrm{true}}
1.60*m < mi;
\textcolor[rgb]{0,0,1}{1.60}\textcolor[rgb]{0,0,1}{<}\frac{\textcolor[rgb]{0,0,1}{201168}}{\textcolor[rgb]{0,0,1}{125}}
\textcolor[rgb]{0,0,1}{\mathrm{true}}
30.48*cm = ft;
\textcolor[rgb]{0,0,1}{30.48}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{762}}{\textcolor[rgb]{0,0,1}{25}}
\textcolor[rgb]{0,0,1}{\mathrm{true}}
3 * ft + 3*`in` = m;
\frac{\textcolor[rgb]{0,0,1}{4953}}{\textcolor[rgb]{0,0,1}{5000}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}
\textcolor[rgb]{0,0,1}{\mathrm{false}}
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Logic - Wackypedia - The nonsensical encyclopedia anyone can mess up
Logic is the mythical belief that things make sense. It is completely untrue.
1 Proof one
2 Proof two
4 Use as a trash can
Proof one[edit]
For every number n and every formula F(y), where y is a free variable, we define q(n,G(F)), a relation between two numbers n and G(F), such that it corresponds to the statement "n is not the Gödel number of a proof of F(G(F))". Here, F(G(F)) can be understood as F with its own Gödel number as its argument.
Note that q takes as an argument G(F). the Gödel number of F. In order to prove either q(n,G(F)), or
{\displaystyle \lnot }
q(n,G(F)), it is necessary to perform number-theoretic operations on G(F) that mirror the following steps: decode the number G(F) into the formula F, replace all occurrences of y in F with the number G(F), and then compute the Gödel number of the resulting formula F(G(F)).
Note that for every specific number n and formula F(y), q(n,G(F)) is a straightforward (though complicated) arithmetical relation between two numbers n and G(F), building on the relation PF defined earlier. Further, q(n,G(F)) is provable if the finite list of formulas encoded by n is not a proof of F(G(F)), and
{\displaystyle \lnot }
q(n,G(F)) is provable if the finite list of formulas encoded by n is a proof of F(G(F)). Given any numbers n and G(F), either q(n,G(F)) or
{\displaystyle \lnot }
q(n,G(F)) (but not both) is provable.
Any proof of F(G(F)) can be encoded by a Gödel number n, such that q(n,G(F)) does not hold. If q(n,G(F)) holds for all natural numbers n, then there is no proof of F(G(F)). In other words,
{\displaystyle \forall y\,q(y,G(F))}
, a formula about natural numbers, corresponds to "there is no proof of F(G(F))".
We now define the formula P(x)=
{\displaystyle \forall y\,q(y,x)}
, where x is a free variable. The formula P itself has a Gödel number G(P) as does every formula.
This formula has a free variable x. Suppose we replace it with G(F), the Gödel number of a formula F(z), where z is a free variable. Then, P(G(F))=
{\displaystyle \forall y\,q(y,G(F))}
corresponds to "there is no proof of F(G(F))", as we have seen.
Consider the formula P(G(P))=
{\displaystyle \forall y\,q(y,G(P))}
. This formula concerning the number G(P) corresponds to "there is no proof of P(G(P))". We have here the self-referential feature that is crucial to the proof: A formula of the formal theory that somehow relates to its own provability within that formal theory. Very informally, P(G(P)) says: "I am not provable".
We will now show that neither the formula P(G(P)), nor its negation
{\displaystyle \lnot }
P(G(P), is provable.
Suppose P(G(P))=
{\displaystyle \forall y\,q(y,G(P))}
is provable. Let n be the Gödel number of a proof of P(G(P)). Then, as seen earlier, the formula
{\displaystyle \lnot }
q(n,G(P)) is provable. Proving both
{\displaystyle \lnot }
q(n,G(P)) and
{\displaystyle \forall y\,q(y,G(P))}
violates the consistency of the formal theory. We therefore conclude that P(G(P) is not provable.
Consider any number n. Suppose
{\displaystyle \lnot }
q(n,G(P)) is provable. Then, n must be the Gödel number of a proof of P(G(P)). But we have just proved that P(G(P) is not provable. Since either q(n,G(P)) or
{\displaystyle \lnot }
q(n,G(P)) must be provable, we conclude that, for all natural numbers n, q(n,G(P)) is provable.
Suppose the negation of P(G(P)),
{\displaystyle \lnot }
P(G(P)) =
{\displaystyle \exists x\,\lnot q(x,G(P))}
, is provable. Proving both
{\displaystyle \exists x\,\lnot q(x,G(P))}
, and q(n,G(P)), for all natural numbers n, violates ω-consistency of the formal theory. Thus if the theory is ω-consistent,
{\displaystyle \lnot }
P(G(P)) is not provable.
We have sketched a proof showing that:
For any formal, recursively enumerable (i.e. effectively generated) theory of Peano Arithmetic,
if it is consistent, then there exists an unprovable formula (in the language of that theory).
if it is ω-consistent, then there exists a formula such that both it and its negation are unprovable.
Proof two[edit]
A = A (Rand's theorem)
Anything = Anything (substituting Anything for A)
Pineapple = Orange (This step relies on the fact that Pineapple, Orange, etc. are subtypes of Anything.)
But pineapples are clearly not oranges. Therefore logic is wrong.
Use as a trash can[edit]
Yes, logic can be used to dispose of trash.
Retrieved from "https://wackypedia.risteq.net/w/index.php?title=Logic&oldid=124520"
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What Is the Day Rate (Oil Drilling)?
Day rate refers to all in daily costs of renting a drilling rig. The operator of a drilling project pays a day rate to the drilling contractor who provides the rig, the drilling personnel and other incidentals. The oil companies and the drilling contractors usually agree on a flat fee per contract, so the day rate is determined by dividing the total value of the contract by the number of days in the contract.
The Formula for Day Rate (Oil Drilling) Is
\text{Day Rate} = \frac{\text{Total Contract Value}}{\text{Number of Days in Contract}}
Day Rate=Number of Days in ContractTotal Contract Value
How to Calculate Day Rate (Oil Drilling)
To calculate the day rate (oil drilling), divide the total value of the contract by the number of days in the contract.
What Does Day Rate (Oil Drilling) Tell You?
Day rate (oil drilling) is a metric that investors in the oil and gas industry watch to evaluate the overall health of the industry. The day rate makes up roughly half the cost of an oil well. Of course, the price of oil is the most important metric by far in the oil and gas industry.
That said, investors can gain insights into the oil supply and demand picture by watching metrics like day rate and rig utilization in addition to global inventories. Day rate fluctuations, which can be wide, are used by investors as an indicator of the health of the drilling market. For example, if day rates fall, investors may take it as a sign to exit oil and gas positions.
The day rate includes the cost of drilling an oil well, including the cost to run the rig, supplies, and employees.
These costs generally make up half the total cost of an oil well.
Tends to have a positive correlation with oil prices and rig utilization rates.
Example of How to Use Day Rate (Oil Drilling)
Day rates can be used to assess the current demand for oil, ultimately gleaming insight into where oil prices are headed. An increase in the price of oil increases the number of projects that can recover their extraction costs, making difficult formations and unconventional oil reserves feasible to extract. The more projects greenlit on an economic basis, the more competition there is for the finite number of oil rigs available for rent – so the day rate rises. When oil prices waver and fall, the day rate that rigs can command drops.
As an example of actual day rates – Transocean signed a contract in December 2018 with Chevron to provide drilling services. The contract is for one rig, will span five years and is worth $830 million. The effective day rate for the rig is $455,000:
\$830 \text{ mill.} \div (5 \text{ years} \times 365 \text{ days}) = \$455,000
$830 mill.÷(5 years×365 days)=$455,000
The Difference Between Day Rate (Oil Drilling) and Utilization Rate
Like the day rate, the rig utilization rate is a key metric for determining the overall health of the oil and gas sector. The day rate lays out a large part of the costs of drilling a well, while the utilization rate is how many wells are being used.
Investors use both of these metrics and a fall in each could signal a slowdown in oil demand. High utilization rates mean a company is using a large part of its fleet, suggesting oil demand, and ultimately, oil prices are on the rise. There is a positive correlation between oil prices and both day rates and rig utilization.
Limitations of Using Day Rate (Oil Drilling)
The strength of the correlation between oil prices and day rates is not consistent. The correlation is strong when oil prices and rig utilization are both high. In this situation, day rates increase almost in lockstep with prices. In an environment of rising oil prices and high utilization, the day rates in a long-term contract will shoot up even faster than short term contracts as rig operators demand a premium for being locked in on a project.
In a low price environment with falling utilization, however, the day rate may plunge much faster than the oil prices as rigs enter low bids on long contracts just to keep busy in a potential slowdown. Due to the volatility and the varying strength of the correlation, investors and traders can flip between seeing day rates as a leading or a lagging indicator for oil prices and the health of the oil and gas industry as a whole.
Learn More About Day Rate (Oil Drilling)
To learn more about day rates and the oil drilling industry, check out Investopedia’s guide to the oil services industry.
Rig Utilization Rate Definition
The rig utilization rate describes the number of oil drilling rigs being used by a company as a percentage of a company's total fleet.
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Home : Support : Online Help : Education : Student Packages : Statistics : Probability
Probability(X, numeric_option, inert_option)
The Probability function computes the probability of the event X.
By default, the probability of an event is computed according to the rules mentioned above. To always compute the probability numerically, specify the numeric or numeric = true option.
\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):
Compute the probability of the normal random variable with parameters
and
b
N≔\mathrm{NormalRandomVariable}\left(a,b\right):
\mathrm{Probability}\left(x<N\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{erf}}\textcolor[rgb]{0,0,1}{}\left(\frac{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}}\right)}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Probability}\left(\mathrm{NormalRandomVariable}\left(2,3\right)<x\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{\mathrm{erf}}\textcolor[rgb]{0,0,1}{}\left(\frac{\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{6}}\right)}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Probability}\left(\mathrm{NormalRandomVariable}\left(2,3\right)<1\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{erf}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{6}}\right)}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{Probability}\left(\mathrm{NormalRandomVariable}\left(2,3\right)<1,'\mathrm{numeric}'\right)
\textcolor[rgb]{0,0,1}{0.369441340181764}
Compute the probability that a exponential random variable lies in the range of (4,7). Instead of calling the function as Probability(4 < E < 7), the right way of using the function is Probability({4<E,E<7}).
E≔\mathrm{ExponentialRandomVariable}\left(5\right):
\mathrm{Probability}\left({4<E,E<7}\right)
\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{7}}{\textcolor[rgb]{0,0,1}{5}}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{4}}{\textcolor[rgb]{0,0,1}{5}}}
Consider a random variable that is made up by two other random variables.
X≔2\mathrm{BernoulliRandomVariable}\left(\frac{1}{2}\right)+\mathrm{ExponentialRandomVariable}\left(3\right):
\mathrm{Probability}\left(X<5.0\right)
\textcolor[rgb]{0,0,1}{0.7216224780}
\mathrm{Probability}\left(X<5,\mathrm{inert}\right)
\underset{\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{\to }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{-}}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{lim}}\textcolor[rgb]{0,0,1}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{\mathrm{_k}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\left({\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{_k}}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{\mathrm{_k}}\\ \frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}\right)\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}^{\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{_k}}}\left({\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{\mathrm{_t}}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{0}\\ \frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{\mathrm{_t}}}{\textcolor[rgb]{0,0,1}{3}}}}{\textcolor[rgb]{0,0,1}{3}}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{\mathrm{_t}}\right)
The Student[Statistics][Probability] command was introduced in Maple 18.
Statistics[Probability]
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Sulfinoalanine decarboxylase - Wikipedia
Cysteine sulfinic acid decarboxylase homodimer, Human
In enzymology, a sulfinoalanine decarboxylase (EC 4.1.1.29) is an enzyme that catalyzes the chemical reaction
3-sulfino-L-alanine
{\displaystyle \rightleftharpoons }
hypotaurine + CO2
Hence, this enzyme has one substrate, 3-sulfino-L-alanine (also known as Cysteine sulfinic acid), and two products, hypotaurine and CO2.
This enzyme belongs to the family of lyases, specifically the carboxy-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is 3-sulfino-L-alanine carboxy-lyase (hypotaurine-forming). Other names in common use include cysteine-sulfinate decarboxylase, L-cysteinesulfinic acid decarboxylase, cysteine-sulfinate decarboxylase, CADCase/CSADCase, CSAD, cysteic decarboxylase, cysteinesulfinic acid decarboxylase, cysteinesulfinate decarboxylase, sulfoalanine decarboxylase, and 3-sulfino-L-alanine carboxy-lyase. This enzyme participates in taurine metabolism. It employs one cofactor, pyridoxal phosphate.
As of late 2007, only one structure has been solved for this class of enzymes, with the PDB accession code 2JIS.
Guion-Rain MC, Portemer C, Chatagner F (1975). "Rat liver cysteine sulfinate decarboxylase: purification, new appraisal of the molecular weight and determination of catalytic properties". Biochim. Biophys. Acta. 384 (1): 265–76. doi:10.1016/0005-2744(75)90115-1. PMID 236774.
JACOBSEN JG, THOMAS LL, SMITH LH (1964). "Properties and Distribution of Mammalian L-Cysteine Sulfinate Carboxy-Lyases". Biochim. Biophys. Acta. 85: 103–16. doi:10.1016/0926-6569(64)90171-3. PMID 14159288.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sulfinoalanine_decarboxylase&oldid=1042904933"
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Calculating the Producer Surplus as an Area - Course Hero
Microeconomics/Consumer and Producer Surplus/Calculating the Producer Surplus as an Area
When the supply curve is drawn as a step diagram, each seller's producer surplus (the area above their minimum price and below the market price) is also the area of a rectangle. To calculate this area, the base of the rectangle is multiplied by its height. Here, Rise 'N' Shine's surplus is $2.00 (market price) minus $0.50 (price at which Rise 'N' Shine sells), which is $1.50. Similarly, Super Cup's surplus is
\$2.00-\$1.00=\$1.00
, and Hot Cuppa's surplus is
\$2.00-\$1.50=\$0.50
. Thus, total producer surplus for this market is $3 (
\$1.50+\$1.00+\$0.50=\$3.00
To calculate the producer surplus for sellers in this market, multiply the base of their step (the quantity) by the height of their step (market price minus seller's price). The base of each step in this case is 1 cup of coffee. Total producer surplus in this market is the sum of the seller surpluses.
A supply curve drawn as a step diagram is useful when there are only a few potential producers in the market. With a big market that has a large number of producers in it, those individual steps will smooth out resulting in a supply curve that is a upward sloping line. For example, there are around 24,000 coffee shops in the United States. The average price of a cup of coffee in a coffee shop in this market is $2.38, with about 400 million cups of coffee sold per day. The producer surplus is calculated based on quantity of cups sold and price per cup.
The U.S. coffee market serves around 400 million cups per day at an average price (market price) of $2.38 per cup. The producer surplus for this market is the area above the supply curve and below the market price.
Because of the size of the market, the producer surplus (the area below the curve and above the market price) is now the area of a triangle:
\frac{1}{2}\times \text{Base}\times \text{Height}
In the case of the U.S. coffee market, the producer surplus is:
\$276\;\text{million}=\frac12\times400\;\text{million}\times\$1.38
<Producer Surplus>How Changing Prices Affect Producer Surplus
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Upgraded Markdown cheatsheet · Life is a marathon
Upgraded Markdown cheatsheet
Let’s start with a informative paragraph. This text is bolded. But not this one! How about italic text? Cool right? Ok, let’s combine them together. Yeah, that’s right! I have code to highlight, so ThisIsMyCode(). What a nice! Good people will hyperlink away, so here we go or https://www.example.com.
The theme comes ready with mathjax support built in, allowing for both simple inline equations like
ax^2 + bx + c = 0
and much more complex mathematical expressions such as equation
\eqref{eq:sample}
Footnote number one yeah baby! Long sentence test of footnote to see how the words are wrapping between each other. Might overflowww! ↩︎
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Trans esterification is the process of :
conversion of an aliphatic acid to ester
conversion of an aromatic acid to ester
conversion of one ester to another ester
conversion of an ester into its components namely acid and alcohol
Transesterification is the process of conversion of one ester to another ester.
The epoxide ring consists of which of the following
three membered ring with two carbon and one oxygen
four membered ring with three carbon and one oxygen
five membered ring with four carbon and one oxygen
six membered ring with five carbon and one oxygen
The structural formula of epoxide is . It consists three membered ring with two carbon and one oxygen.
In the Grignard reaction, which metal forms an organometallic bond?
In the Grignard reaction magnesium metal forms an organometallic bond.
\mathrm{RX}\quad +\quad \mathrm{Mg}\quad \stackrel{\mathrm{Dry}\hspace{0.17em}\mathrm{ether}}{\to }\quad \underset{\mathrm{Grignard}\quad \mathrm{reagents}}{\mathrm{R}\quad -\quad \mathrm{Mg}\quad -\quad \mathrm{X}}
Due to o- and p- directing nature of CH3 group.
The electrophile involved in the above reaction is
dichloromethyl cation (C+HCl )
dichlorocarbene ( :CCl2)
trichloromethyl anion (C-Cl3 )
formyl cation (C+HO)
Phenols are more acidic than alcohols because
phenoxide ion is stabilised by resonance
phenols are more soluble in polar solvents
phenoxides ions do not exhibit resonance
alcohols do not lose H atoms at all
Phenol is more acidic than alcohol because phenoxide ion is stabilised by resonance.
Resonance structures of phenoxide ion of phenol
The structure of the compound that gives a tribromo derivative on treatment with bromine water is
compound ‘D’ is
therefore the compound D is n-propyl alcohol
Phenol is more acidic than :
Phenol is more acidic than ethanol due to resonance stabilisation of phenoxide ion.
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Synthetic Routes: Definition & Examples | StudySmarter
We will discuss many synthetic routes elaborating on how to make a chemical compound from another chemical compound. We will learn as well how to go about the chemical reactions, and what reagents and catalysts to use.
You will have to recollect what you learnt in Organic Chemistry and Organic Synthesis.
A synthetic route is a series of steps to be followed in order to make a chemical compound from smaller and less complex chemicals.
In this article, you will learn what a synthetic route is.
You will learn how to map a synthetic route when given a starting material and a target compound.
You will look at the overview of synthetic routes for aliphatic compounds and aromatic compounds.
The synthetic route overview will help map the steps to take when converting one organic compound into another - the reagents, catalysts, and conditions required for each reaction.
We will discuss and map some examples of synthetic routes to help you understand how to use the information given in this article.
Synthetic routes - aliphatic compounds
Let us start by listing out all the functional groups we know. We are familiar with:
Halogenalkanes/haloalkanes
That's quite a few! We can draw a flow chart of which functional groups can be converted into other groups. This will make it easier to understand and remember.
Organic synthesis routes, Olive [Odagbu] StudySmarter
The arrows represent which groups can be synthesised from other groups using the right reagents, catalysts, and conditions.
Synthetic route: 1-bromopropane to propanoic acid
Let us consider the synthesis of propanoic acid from 1-bromopropane. Propanoic acid has a carboxylic acid functional group, while 1-bromopropane is a haloalkane.
Propanoic acid, Kanishk Singh, StudySmarter Originals
1-bromopropane, Kanishk Singh, StudySmarter Originals
Looking at the flowchart, the easiest route to take from haloalkane to carboxylic acid is through an alcohol. Thus, we can convert 1-bromopropane into an alcohol and then into a carboxylic acid. Let's go through the steps of this reaction.
Haloalkanes can be converted to alcohols through hydrolysis. Hydrolysis of haloalkanes is a nucleophilic substitution reaction. In this reaction, the nucleophile is H2O. Since H2O alone is a weak nucleophile, this reaction is slow. Therefore, NaOH is added and the solution is heated under reflux. The halogen is replaced by the -OH group.
To convert alcohol to carboxylic acid, it needs to be oxidised. For this reaction, acidified Potassium Dichromate (K2Cr2O7/H+) is added. K2Cr2O7 is an orange-coloured substance which is a strong oxidising agent. The reaction needs to take place with heat under reflux.
Since we're starting with 1-bromopropane, this molecule would be called the starting material in this particular route, while propanoic acid is the target compound. All compounds formed in between the starting material and the target compound are called intermediate compounds. Propanol is called the intermediate compound.
The synthetic route of Propanoic acid from 1-bromopropane can be described like this -
Synthetic Route of 1-bromopropane to Propanoic acid, Kanishk Singh, StudySmarter Originals
Synthetic route: ethene to propylamine
Let us consider another synthetic route. We can synthesise propylamine from ethene. Ethene is an alkene with a double bond between 2 carbon atoms. Propylamine has an amine group attached at the end of a 3-Carbon chain.
Ethene, Kanishk Singh StudySmarter Originals
Propylamine, Kanishk Singh, StudySmarter Originals
Look at the flowchart again. To get to an Amine from an Alkene, you can take the route of alkene → haloalkane → amine. But if we take this route, the end product will be ethylamine, which is a 2- carbon compound. Where do we get an extra carbon from? We will have to take a longer route through the nitrile. So, the final route would be alkene → haloalkane → nitrile → amine. The nitrile group (CN) will give us the extra carbon we need. Let us go through the steps of this synthesis route:
Alkenes can be converted to haloalkanes by adding a hydrogen halide at 20oC. Thus, adding a Hydrogen halide (HBr for example) to ethene and holding the reaction at 20oC will give us 1-bromoethane. The double bond of ethene is replaced by hydrogen on one side and bromine on the other side.
We will now replace the halogen with a nitrile group. Haloalkanes can be converted to nitrile by reacting them with a solution of sodium cyanide (NaCN) or potassium cyanide (KCN) in ethanol, and heating under reflux. With this reaction, 1-bromoethane will be converted to propanenitrile.
Next, we need to convert the nitrile group to an amine group. The nitrile group consists of a triple bond between carbon and nitrogen (
\mathrm{C}\equiv \mathrm{N}
). This can be reduced by catalytic hydrogenation (reaction with hydrogen in the presence of a catalyst). The catalyst used is usually a metal catalyst such as palladium, platinum, or nickel. Through this reaction,
\mathrm{C}\equiv \mathrm{N}
will give us
\mathrm{C}-{\mathrm{NH}}_{2}
Propanenitrile is also called ethyl cyanide, and propionitrile.
This diagram summarises the reactions discussed above, and represents the synthetic route of ethene to propylamine.
Synthetic route of Ethene to Propylamine, Kanishk Singh, StudySmarter Originals
Synthetic route: ethene to ethanol (hydration of alkene)
Let us now consider a simple synthetic route. We shall synthesise ethanol from ethene. Ethene is an alkene with a double bond between 2 carbon atoms. Ethanol is an alcohol with an -OH group attached to a 2-carbon chain.
Ethene, Kanishk Singh, StudySmarter Originals
Ethanol | Kanishk Singh, StudySmarter Originals
Looking at the flowchart, we can convert an alkene to an alcohol in a single step i.e. single reaction. This reaction is called hydration and is the direct, simpler method. In hydration, the alkene is treated with steam at high temperature and pressure, in the presence of a catalyst. The catalyst used in this process is phosphoric acid (H3PO4).
But there is also a two-step hydration process of alkenes. Let us see the reactions of the two-step hydration of ethene.
In the first step, ethene is treated with sulphuric acid (H2SO4). This reaction yields Ethyl Hydrogen Sulphate. It is also called the 'Ethyl Ester of Sulphuric Acid'.
Then, ethyl hydrogen sulphate is hydrolysed i.e. reacted with water. This reaction produces ethanol, and also regenerates the sulphuric acid.
Synthetic Route of Ethene to Ethanol, Kanishk Singh, StudySmarter Originals
This two-step hydration of alkenes is traditionally used in the industrial production of ethanol.
Synthetic routes for all aliphatic compounds
The synthetic routes overview diagram given below shows all possible synthetic routes between functional groups found in aliphatic organic compounds. The diagram shows the reagents and conditions required for each conversion. You can map any synthetic route between any starting organic material to any target organic compound. To map a synthetic route between a starting material and a target compound, check for common intermediate functional groups from the flowchart below. Then, list the reactions required to be done, as well as the reagents, catalysts, and conditions required for each intermediate compound.
Synthetic Route Overview for Aliphatic Compounds | StuDocu
Looking at the synthetic routes overview diagram, there are always multiple routes to get from a starting material to a target compound. You should always try to minimise the steps it takes to get from the starting material to the target compound, to maximise the product yield.
Synthetic routes - aromatic compounds
Similar to the synthetic routes overview diagram for aliphatic compounds, we can draw a synthetic routes overview diagram for aromatic compounds. Thankfully, it is much smaller than for aliphatic compounds, and much easier to remember.
Synthetic Routes Overview for Aromatic Compounds | StuDocu
You know that the parent member of all aromatic compounds is benzene. All aromatic compounds can be synthesised from benzene, and that already makes it easier to remember. As an example, let us try to synthesise one aromatic compound from benzene.
Synthetic route: 4-bromo-3-nitroacetophenone from benzene
Let us draw the starting material and the target compound first.
Benzene as the starting material, 4-bromo-3-nitroacetophenone as the target compound, Kanishk Singh, StudySmarter Originals
In the target compound, there are 3 functional groups attached to the parent compound (benzene). Looking at the overview of the synthetic routes, we can see that there is no direct route to get from our starting material to the target compound. So, this will be a 3-step route, in which each step will be a reaction to add 1 functional group to the parent compound. To map this route, we will employ the technique of retrosynthesis.
Retrosynthesis is the process of figuring out the synthetic route in the reverse direction - from target compound to the starting material.
We look at the target compound, thinking about which reaction was done last to obtain this compound. To determine this, we have to look at the functional groups.
We know that Br is an ortho-, para-director because of the lone pairs of electrons on it.
We also know that the nitro group is a meta-director because of the +1 formal charge.
We also know that the acyl group is a meta-director because of the partial positive charge on the carbon double-bonded to oxygen.
Considering these points, it makes sense that the nitro group was added last because:
It is in the ortho position to the ortho-director bromine.
It is meta to the meta-director acyl group.
So, the precursor to the target compound was 4-bromoacetophenone, on which the nitration reaction was done. We can write the nitration reaction as below:
Nitration of 4-bromoacetophenone, Kanishk Singh, StudySmarter Originals
Following the same procedure, we know that bromine is an ortho-, para-director and the acyl group is a meta-director. Therefore, it makes sense that the precursor to 4-bromoacetophenone was bromobenzene and the acyl group was added to the ortho position by the ortho-director bromine. The reaction to add an acyl group is called Friedel–Crafts acylation reaction.
Friedel-Crafts Acylationof Bromobenzene, Kanishk Singh, StudySmarter Originals
You might be wondering; since bromine is a deactivation group on the benzene ring, how can we add an acyl group? The answer to this is that bromine is only a weakly deactivating group. We can't do a Friedel-Crafts reaction when there is a moderately or highly deactivating group present on the ring.
Finally, the only group left is bromine. The reaction is a bromination reaction:
Bromination of Benzene, Kanishk Singh, StudySmarter Originals
We have discussed the individual steps of the synthetic route of 4-bromo-3-nitroacetophenone from benzene. This diagram shows the complete synthetic route.
Synthetic Route of 4-bromo-3-nitroacetophenone from Benzene, Kanishk Singh, StudySmarter Originals
For your exam, you are expected to know the synthetic route for any starting material to any target compound, and the reagents, catalysts, and conditions required. There are numerous possibilities! The easiest way is to remember the two synthetic route overview diagrams given in this article (one for aliphatic compounds and the other for aromatic compounds) and construct the route for any given starting and target compound. You'll need a lot of practice, but you can handle this!
Synthetic Routes - Key takeaways
In a synthetic route, each step is a chemical reaction.
The final chemical in the route i.e., the chemical compound to be made is called the target compound. The chemical from which you start the route is called the starting material.
All chemical compounds formed between the starting material and the target compound are called intermediate compounds.
While mapping synthetic routes, always try to minimise the number of steps to maximise the product yield.
For your exam, you are expected to know the synthetic routes - reagents, catalysts, and the conditions required for each reaction - for any given starting and target compound.
How do you make a synthetic route?
To make a synthetic route between a starting material and the target compound -
List all the molecules that can be made from the starting material.
List all the molecules from which the target molecule can be made.
Check for common intermediate compounds between the starting material and the target compound.
What is a synthetic route?
How do you remember synthetic routes?
The easiest way to map synthetic routes on your own is to remember the two synthetic route overview diagrams given in this article (one for aliphatic compounds and the other from aromatic compounds), and construct the route for any given starting and target compound from there.
Final Synthetic Routes Quiz
How do you carry out hydration of alkenes?
Alkenes are treated with steam at 300oC and 60-70 atm in the presence of phosphoric acid.
What is the product of hydration of alkenes?
What do you get when you oxidize an alcohol?
To convert 1-bromopropane to propaneamine, which reaction do you require?
Reaction with excess ethanolic ammonia and heat.
What do you get when you hydrolyse haloalkanes?
Which of the following chemicals is not the same as the others?
Propanenitril
How many steps would it take to synthesize propylamine from ethene?
How many intermediate compounds will be formed during the synthesis of propylamine from ethene?
2 intermediate compounds.
How many steps would it take to synthesize propanoic acid from propanenitrile?
How do you convert propanenitrile to propanoic acid?
Reaction with dil. HCl and water, and heat under reflux
How do you convert an alcohol to an alkene?
Reaction of alcohol with concentrated phosphoric acid and heat at 170oC.
How do you convert carboxylic acid to a primary amide?
First, convert carboxylic acid to acyl chloride by reaction with SOCl2.
Then, add ammonia at 20oC to get the primary amide.
What do get after reduction of an alkene?
How can you reduce an alkene to get an alkane?
Reaction of alkene with hydrogen in the presence of a metal catalyst (Pd, Pt, or Ni) at 1500
How do you convert an aldehyde group to a hydroxynitrile group?
Reaction with hydrogen cyanide (HCN)
of the users don't pass the Synthetic Routes quiz! Will you pass the quiz?
Acylation Learn
Alcohol Elimination Reaction Learn
Alcohols Learn
Aldehydes and Ketones Learn
Alkanes Learn
Alkenes Learn
Amide Learn
Amines Learn
Amines Basicity Learn
Amino Acids Learn
Anti-Cancer Drugs Learn
Aromatic Chemistry Learn
Benzene Structure Learn
Biodegradability Learn
Carbon -13 NMR Learn
Carbonyl Group Learn
Carboxylic Acids Learn
Chlorination Learn
Chromatography Learn
Column Chromatography Learn
Combustion Learn
Condensation Polymers Learn
Cracking (Chemistry) Learn
Elimination Reactions Learn
Esters Learn
Fractional Distillation Learn
Gas Chromatography Learn
Halogenoalkanes Learn
Hydrogen -1 NMR Learn
Infrared Spectroscopy Learn
Isomerism Learn
NMR Spectroscopy Learn
Nucleophilic Substitution Reactions Learn
Optical Isomerism Learn
Organic Compounds Learn
Organic Synthesis Learn
Oxidation of Alcohols Learn
Ozone Depletion Learn
Paper Chromatography Learn
Polymerisation Reactions Learn
Preparation of Amines Learn
Production of Ethanol Learn
Properties of Polymers Learn
Reaction Mechanism Learn
Reactions of Aldehydes and Ketones Learn
Reactions of Alkenes Learn
Reactions of Benzene Learn
Reactions of Carboxylic Acids Learn
Reactions of Esters Learn
Thin-Layer Chromatography Learn
Understanding NMR Learn
Uses of Amines Learn
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Puiseux series - MATLAB series - MathWorks España
Find Puiseux Series Expansion
Plot Puiseux Series Approximation
Specify Direction of Expansion
series(f,var)
series(f,var,a)
series(___,Name,Value)
series(f,var) approximates f with the Puiseux series expansion of f up to the fifth order at the point var = 0. If you do not specify var, then series uses the default variable determined by symvar(f,1).
series(f,var,a) approximates f with the Puiseux series expansion of f at the point var = a.
series(___,Name,Value) uses additional options specified by one or more Name,Value pair arguments. You can specify Name,Value after the input arguments in any of the previous syntaxes.
Find the Puiseux series expansions of univariate and multivariate expressions.
Find the Puiseux series expansion of this expression at the point x = 0.
series(1/sin(x), x)
x/6 + 1/x + (7*x^3)/360
Find the Puiseux series expansion of this multivariate expression. If you do not specify the expansion variable, series uses the default variable determined by symvar(f,1).
f = sin(s)/sin(t);
series(f)
sin(s)/t + (7*t^3*sin(s))/360 + (t*sin(s))/6
To use another expansion variable, specify it explicitly.
series(f, s)
s^5/(120*sin(t)) - s^3/(6*sin(t)) + s/sin(t)
Find the Puiseux series expansion of psi(x) around x = Inf. The default expansion point is 0. To specify a different expansion point, use the ExpansionPoint name-value pair.
series(psi(x), x, 'ExpansionPoint', Inf)
log(x) - 1/(2*x) - 1/(12*x^2) + 1/(120*x^4)
Alternatively, specify the expansion point as the third argument of series.
series(psi(x), x, Inf)
Find the Puiseux series expansion of exp(x)/x using different truncation orders.
Find the series expansion up to the default truncation order 6.
f = exp(x)/x;
s6 = series(f, x)
\frac{x}{2}+\frac{1}{x}+\frac{{x}^{2}}{6}+\frac{{x}^{3}}{24}+\frac{{x}^{4}}{120}+1
s7 = series(f, x, 'Order', 7)
\frac{x}{2}+\frac{1}{x}+\frac{{x}^{2}}{6}+\frac{{x}^{3}}{24}+\frac{{x}^{4}}{120}+\frac{{x}^{5}}{720}+1
\frac{x}{2}+\frac{1}{x}+\frac{{x}^{2}}{6}+\frac{{x}^{3}}{24}+\frac{{x}^{4}}{120}+\frac{{x}^{5}}{720}+\frac{{x}^{6}}{5040}+1
Plot the original expression f and its approximations s6, s7, and s8. Note how the accuracy of the approximation depends on the truncation order.
fplot([s6 s7 s8 f])
legend('approximation up to O(x^6)','approximation up to O(x^7)',...
'approximation up to O(x^8)','exp(x)/x','Location', 'Best')
title('Puiseux Series Expansion')
Find the Puiseux series approximations using the Direction argument. This argument lets you change the convergence area, which is the area where series tries to find converging Puiseux series expansion approximating the original expression.
Find the Puiseux series approximation of this expression. By default, series finds the approximation that is valid in a small open circle in the complex plane around the expansion point.
series(sin(sqrt(-x)), x)
(-x)^(1/2) - (-x)^(3/2)/6 + (-x)^(5/2)/120
Find the Puiseux series approximation of the same expression that is valid in a small interval to the left of the expansion point. Then, find an approximation that is valid in a small interval to the right of the expansion point.
series(sin(sqrt(-x)), x, 'Direction', 'left')
series(sin(sqrt(-x)), x, 'Direction', 'right')
- x^(1/2)*1i - (x^(3/2)*1i)/6 - (x^(5/2)*1i)/120
x^(1/2)*1i + (x^(3/2)*1i)/6 + (x^(5/2)*1i)/120
Try computing the Puiseux series approximation of this expression. By default, series tries to find an approximation that is valid in the complex plane around the expansion point. For this expression, such approximation does not exist.
series(real(sin(x)), x)
Error using sym/series>scalarSeries (line 90)
Unable to compute series expansion.
However, the approximation exists along the real axis, to both sides of x = 0.
series(real(sin(x)), x, 'Direction', 'realAxis')
Expansion variable, specified as a symbolic variable. If you do not specify var, then series uses the default variable determined by symvar(f,1).
Expansion point, specified as a number, or a symbolic number, variable, function, or expression. The expansion point cannot depend on the expansion variable.
You also can specify the expansion point as a Name,Value pair argument. If you specify the expansion point both ways, then the Name,Value pair argument takes precedence.
Example: series(psi(x),x,'ExpansionPoint',Inf,'Order',9)
You can also specify the expansion point using the input argument a. If you specify the expansion point both ways, then the Name,Value pair argument takes precedence.
Order — Truncation order of Puiseux series expansion
Truncation order of Puiseux series expansion, specified as a positive integer or a symbolic positive integer.
series computes the Puiseux series approximation with the order n - 1. The truncation order n is the exponent in the O-term: O(varn).
Direction — Direction for area of convergence of Puiseux series expansion
'complexPlane' (default) | 'left' | 'right' | 'realAxis'
Direction for area of convergence of Puiseux series expansion, specified as:
'left' Find a Puiseux series approximation that is valid in a small interval to the left of the expansion point.
'right' Find a Puiseux series approximation that is valid in a small interval to the right of the expansion point.
'realAxis' Find a Puiseux series approximation that is valid in a small interval on the both sides of the expansion point.
'complexPlane' Find a Puiseux series approximation that is valid in a small open circle in the complex plane around the expansion point. This is the default value.
If you use both the third argument a and the ExpansionPoint name-value pair to specify the expansion point, the value specified via ExpansionPoint prevails.
pade | taylor
|
While Mrs. Poppington was visiting the historic battleground at Gettysburg, she talked with the landscapers who were replacing the sod in a park near the visitor center. The part of the lawn being replaced is shown in the diagram at right. Measurements are in yards. The diagram is not drawn to scale, but all corners are right angles.
Use your ruler to make a scale drawing of the lawn that is being replaced. Let
\frac { 1 } { 4 }
inch equal
1
Draw the diagram on your paper. Be sure to scale it correctly.
How many square yards of sod will be needed?
Find the area of the diagram in square yards.
Split the diagram into two rectangles. Find the area of each rectangle respectively and add them together to find the total area.
\text{Total area} =3(10)+4(6)
\text{Total area}=54\text{ square yards}
\$2.43
a square yard, how much will the sod cost?
Multiply the cost per square yard by the total square yards of sod needed.
The area will be surrounded by a temporary fence which costs
\$0.25
per linear foot. How many feet will be needed, and how much will it cost?
Find the perimeter of the diagram to find how many feet will be needed and multiply that by $0.25 to find how much it will cost.
What will be the total cost of materials for the sod and fencing?
Add your answers from parts (c) and (d).
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multiple - Maple Help
Home : Support : Online Help : Graphics : 2-D : multiple
Plot multiple functions
multiple(plotcommand, plotargs1, ..., plotargsn, options)
plot command to apply
plotargs1, ..., plotargsn
The multiple command builds a composite plot.
The parameter plotcommand is a Maple plotting command, applied to the plot arguments plotargsi.
Each parameter plotargsi is a list of arguments for plotcommand, the plotting command. To plot the sine function for two different frequencies, you can use:
plots[multiple]( plot, [sin(2*x), x=-Pi..Pi], [sin(3*x), x=-Pi..Pi] );
The remaining arguments to the multiple command are interpreted as options. They are equations of the form option = value, and are passed on to the plots[display] command.
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{multiple}\left(\mathrm{plot3d},[x+y,x=0..1,y=0..1],[1,t=0..1,u=0..1]\right)
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Wilkinson Notation - MATLAB & Simulink - MathWorks Nordic
Formula Specification
Random-Effects and Mixed-Effects Models
Linear Model Examples
Intercept and Two Predictors
No Intercept and Two Predictors
Intercept, Two Predictors, and an Interaction Term
Intercept, Three Predictors, and All Interaction Effects
Intercept, Three Predictors, and Selected Interaction Effects
Intercept, Three Predictors, and Lower-Order Interaction Effects Only
Linear Mixed-Effects Model Examples
Random Effect Intercept, No Predictors
Random Intercept and Fixed Slope for One Predictor
Random Intercept and Random Slope for One Predictor
Generalized Linear Model Examples
Generalized Linear Mixed-Effects Model Examples
Repeated Measures Model Examples
Three Predictors and an Interaction Term
Wilkinson notation provides a way to describe regression and repeated measures models without specifying coefficient values. This specialized notation identifies the response variable and which predictor variables to include or exclude from the model. You can also include squared and higher-order terms, interaction terms, and grouping variables in the model formula.
Specifying a model using Wilkinson notation provides several advantages:
You can include or exclude individual predictors and interaction terms from the model. For example, using the 'Interactions' name-value pair available in each model fitting functions includes interaction terms for all pairs of variables. Using Wilkinson notation instead allows you to include only the interaction terms of interest.
You can change the model formula without changing the design matrix, if your input data uses the table data type. For example, if you fit an initial model using all the available predictor variables, but decide to remove a variable that is not statistically significant, then you can re-write the model formula to include only the variables of interest. You do not need to make any changes to the input data itself.
Statistics and Machine Learning Toolbox™ offers several model fitting functions that use Wilkinson notation, including:
Linear models (using fitlm and stepwiselm)
Generalized linear models (using fitglm)
Linear mixed-effects models (using fitlme and fitlmematrix)
Generalized linear mixed-effects models (using fitglme)
Repeated measures models (using fitrm)
Cox proportional hazards model (using fitcox)
A formula for model specification is a character vector or string scalar of the form y ~ terms, where y is the name of the response variable, and terms defines the model using the predictor variable names and the following operators.
Predictor Terms in Model
no intercept –1
x1, x2 x1 + x2
x1, x2, x1x2 x1*x2 or x1 + x2 + x1:x2
x1x2 x1:x2
x1, x12 x1^2
x12 x1^2 – x1
Wilkinson notation includes an intercept term in the model by default, even if you do not add 1 to the model formula. To exclude the intercept from the model, use -1 in the formula.
The * operator (for interactions) and the ^ operator (for power and exponents) automatically include all lower-order terms. For example, if you specify x^3, the model will automatically include x3, x2, and x. If you want to exclude certain variables from the model, use the – operator to remove the unwanted terms.
For random-effects and mixed-effects models, the formula specification includes the names of the predictor variables and the grouping variables. For example, if the predictor variable x1 is a random effect grouped by the variable g, then represent this in Wilkinson notation as follows:
(x1 | g)
For repeated measures models, the formula specification includes all of the repeated measures as responses, and the factors as predictor variables. Specify the response variables for repeated measures models as described in the following table.
Response Terms in Model
y1, y2, y3 y1,y2,y3
y1, y2, y3, y4, y5 y1–y5
For example, if you have three repeated measures as responses and the factors x1, x2, and x3 as the predictor variables, then you can define the repeated measures model using Wilkinson notation as follows:
y1,y2,y3 ~ x1 + x2 + x3
y1-y3 ~ x1 + x2 + x3
If the input data (response and predictor variables) is stored in a table or dataset array, you can specify the formula using the variable names. For example, load the carsmall sample data. Create a table containing Weight, Acceleration, and MPG. Name each variable using the 'VariableNames' name-value pair argument of the fitting function fitlm. Then fit the following model to the data:
MPG={\beta }_{0}+{\beta }_{1}Weight+{\beta }_{2}Acceleration
tbl = table(Weight,Acceleration,MPG, ...
'VariableNames',{'Weight','Acceleration','MPG'});
mdl = fitlm(tbl,'MPG ~ Weight + Acceleration')
The model object display uses the variable names provided in the input table.
If the input data is stored as a matrix, you can specify the formula using default variable names such as y, x1, and x2. For example, load the carsmall sample data. Create a matrix containing the predictor variables Weight and Acceleration. Then fit the following model to the data:
MPG={\beta }_{0}+{\beta }_{1}Weight+{\beta }_{2}Acceleration
X = [Weight,Acceleration];
mdl = fitlm(X,y,'y ~ x1 + x2')
x1 -0.0082475 0.00059836 -13.783 5.3165e-24
x2 0.19694 0.14743 1.3359 0.18493
The term x1 in the model specification formula corresponds to the first column of the predictor variable matrix X. The term x2 corresponds to the second column of the input matrix. The term y corresponds to the response variable.
Use fitlm and stepwiselm to fit linear models.
For a linear regression model with an intercept and two fixed-effects predictors, such as
{y}_{i}={\beta }_{0}+{\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2}\text{\hspace{0.17em}}+{\epsilon }_{i},
specify the model formula using Wilkinson notation as follows:
'y ~ x1 + x2'
For a linear regression model with no intercept and two fixed-effects predictors, such as
{y}_{i}={\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2}+{\epsilon }_{i}\text{\hspace{0.17em}},
'y ~ -1 + x1 + x2'
For a linear regression model with an intercept, two fixed-effects predictors, and an interaction term, such as
{y}_{i}={\beta }_{0}+{\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2}+{\beta }_{3}{x}_{i1}{x}_{i2}+{\epsilon }_{i}\text{\hspace{0.17em}},
'y ~ x1*x2'
'y ~ x1 + x2 + x1:x2'
For a linear regression model with an intercept, three fixed-effects predictors, and interaction effects between all three predictors plus all lower-order terms, such as
{y}_{i}={\beta }_{0}+{\beta }_{1}x{i}_{1}+{\beta }_{2}{x}_{i2}+{\beta }_{3}{x}_{i3}+{\beta }_{4}{x}_{1}{x}_{i2}+{\beta }_{5}{x}_{1}{x}_{i3}+{\beta }_{6}{x}_{2}{x}_{i3}+{\beta }_{7}{x}_{i1}{x}_{i2}{x}_{i3}+{\epsilon }_{i}\text{\hspace{0.17em}},
'y ~ x1*x2*x3'
For a linear regression model with an intercept, three fixed-effects predictors, and interaction effects between two of the predictors, such as
{y}_{i}={\beta }_{0}+{\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2}+{\beta }_{3}{x}_{i3}+{\beta }_{4}{x}_{1}{x}_{i2}+{\epsilon }_{i}\text{\hspace{0.17em}},
'y ~ x1*x2 + x3'
'y ~ x1 + x2 + x3 + x1:x2'
For a linear regression model with an intercept, three fixed-effects predictors, and pairwise interaction effects between all three predictors, but excluding an interaction effect between all three predictors simultaneously, such as
{y}_{i}={\beta }_{0}+{\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2}+{\beta }_{3}{x}_{i3}+{\beta }_{4}{x}_{1}{x}_{i2}+{\beta }_{5}{x}_{i1}{x}_{i3}+{\beta }_{6}{x}_{i2}{x}_{i3}+{\epsilon }_{i}\text{\hspace{0.17em}},
'y ~ x1*x2*x3 - x1:x2:x3'
Use fitlme and fitlmematrix to fit linear mixed-effects models.
For a linear mixed-effects model that contains a random intercept but no predictor terms, such as
{y}_{im}={\beta }_{0m}\text{\hspace{0.17em}},
{\beta }_{0m}={\beta }_{00}+{b}_{0m}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{b}_{0m}\sim N\left(0,{\sigma }_{0}^{2}\right)
and g is the grouping variable with m levels, specify the model formula using Wilkinson notation as follows:
'y ~ (1 | g)'
For a linear mixed-effects model that contains a fixed intercept, random intercept, and fixed slope for the continuous predictor variable, such as
{y}_{im}={\beta }_{0m}+{\beta }_{1}{x}_{im}\text{\hspace{0.17em}},
{\beta }_{0m}={\beta }_{00}+{b}_{0m}\text{\hspace{0.17em}},\text{\hspace{0.17em}}{b}_{0m}\sim N\left(0,{\sigma }_{0}^{2}\right)
'y ~ x1 + (1 | g)'
For a linear mixed-effects model that contains a fixed intercept, plus a random intercept and a random slope that have a possible correlation between them, such as
{y}_{im}={\beta }_{0m}+{\beta }_{1m}{x}_{im}\text{\hspace{0.17em}},
{\beta }_{0m}={\beta }_{00}+{b}_{0m}
{\beta }_{1m}={\beta }_{10}+{b}_{1m}
\left[\begin{array}{c}{b}_{0m}\\ {b}_{1m}\end{array}\right]\sim N\left\{0,{\sigma }^{2}D\left(\theta \right)\right\}
and D is a 2-by-2 symmetric and positive semidefinite covariance matrix, parameterized by a variance component vector θ, specify the model formula using Wilkinson notation as follows:
'y ~ x1 + (x1 | g)'
The pattern of the random effects covariance matrix is determined by the model fitting function. To specify the covariance matrix pattern, use the name-value pairs available through fitlme when fitting the model. For example, you can specify the assumption that the random intercept and random slope are independent of one another using the 'CovariancePattern' name-value pair argument in fitlme.
Use fitglm and stepwiseglm to fit generalized linear models.
In a generalized linear model, the y response variable has a distribution other than normal, but you can represent the model as an equation that is linear in the regression coefficients. Specifying a generalized linear model requires three parts:
Distribution of the response variable
The distribution of the response variable and the link function are specified using name-value pair arguments in the fit function fitglm or stepwiseglm.
The linear predictor portion of the equation, which appears on the right side of the ~ symbol in the model specification formula, uses Wilkinson notation in the same way as for the linear model examples.
A generalized linear model models the link function, rather than the actual response, as y. This is reflected in the output display for the model object.
For a generalized linear regression model with an intercept and two predictors, such as
\mathrm{log}\left({y}_{i}\right)={\beta }_{0}+{\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2},
Use fitglme to fit generalized linear mixed-effects models.
In a generalized linear mixed-effects model, the y response variable has a distribution other than normal, but you can represent the model as an equation that is linear in the regression coefficients. Specifying a generalized linear model requires three parts:
The distribution of the response variable and the link function are specified using name-value pair arguments in the fit function fitglme.
The linear predictor portion of the equation, which appears on the right side of the ~ symbol in the model specification formula, uses Wilkinson notation in the same way as for the linear mixed-effects model examples.
A generalized linear model models the link function as y, not the response itself. This is reflected in the output display for the model object.
The pattern of the random effects covariance matrix is determined by the model fitting function. To specify the covariance matrix pattern, use the name-value pairs available through fitglme when fitting the model. For example, you can specify the assumption that the random intercept and random slope are independent of one another using the 'CovariancePattern' name-value pair argument in fitglme.
For a generalized linear mixed-effects model that contains a fixed intercept, random intercept, and fixed slope for the continuous predictor variable, where the response can be modeled using a Poisson distribution, such as
\mathrm{log}\left({y}_{im}\right)={\beta }_{0}+{\beta }_{1}{x}_{im}+{b}_{i}\text{\hspace{0.17em}},
{b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)
Use fitrm to fit repeated measures models.
For a repeated measures model with five response measurements and one predictor variable, specify the model formula using Wilkinson notation as follows:
'y1-y5 ~ x1'
For a repeated measures model with five response measurements and three predictor variables, plus an interaction between two of the predictor variables, specify the model formula using Wilkinson notation as follows:
'y1-y5 ~ x1*x2 + x3'
[1] Wilkinson, G. N., and C. E. Rogers. "Symbolic description of factorial models for analysis of variance." J. Royal Statistics Society 22, pp. 392–399, 1973.
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C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, R. P. Stutz
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, midcircuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a 10-qubit quantum charge-coupled device trapped-ion quantum computer to encode a single logical qubit using the
\left[\left[7,1,3\right]\right]
color code, first proposed by Steane [Phys. Rev. Lett. 77, 793 (1996)]. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical state preparation and measurement (SPAM) error of
1.7\left(2\right)×{10}^{-3}
, compared to the average physical SPAM error
2.4\left(4\right)×{10}^{-3}
of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a
\overline{T}|+{⟩}_{L}
magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudothreshold.
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Structural Engineering - Wikibooks, open books for an open world
This book requires that you first read Statics.
2 Structural Engineering Activities
3.1 Analysis of Statically Determinate Structures
3.1.1 Idealized Structures
3.1.1.1 Supports for Coplanar Structures
3.2 Analysis of Statically Determinate Trusses
3.3 Internal Loadings Developed in Structural Members
3.4 Cables and Arches
3.6.1 Elastic-Beam Theory
3.6.2 The Double Integration Method
3.6.3 Castigliano's theorems
3.7 First theorem
3.8 Second theorem
What is Structural Engineering?Edit
Structural engineering is a branch of engineering which deals with the analysis and design of various structural systems. Although this branch of engineering has influence on various other disciplines like mechanical or aeronautical engineering, etc., it is more commonly identified with civil engineering. Structural engineering deals with conception, design, and construction of the structural systems that are needed in support of human civil engineering
"Structural engineering is the art of molding materials we do not really understand into shapes we cannot really analyze, so as to withstand forces we cannot really assess, in sucha a way that the public does not really suspect." British structural engineer Dr. E. H. Brown, from his book Structural Analysis [1]
Engineers are the invisible presence that brings the architect or designer's concept into reality. In a broad sense, architects create the skin defining the usable flow of space that we live and work - occasionally appearing to defy gravity. The Engineer relies on the physical sciences to create a "skeleton" of interconnected elements that support the shell in the physical world. He or she uses the laws of physics (equilibrium) to design the structure (in part and in whole) to safely support the weight of the permanent building materials, changing movement of people, and short term applications of snow and water. These are defined as the vertical "gravity" loads (NOTE: Refer to topics related to; live, dead, and short-term loads) while resisting the "horizontal" forces of nature due to the worst case comparing; wind and seismic events that may act simualtaneously. Equilibrium is achieved by anchoring the structure to an adequate foundation based on the recommendations of a geotechnical (soils) engineer. Starting at the top and working down to the foundation, the engineer designs each element and the connections supporting the combination of elements to create the "structure" - a sum of its elements (whether a building, soil retaining wall or a bridge). The engineer follows building codes, enacted in state congress to create law that provides a minimum requirement to protect life safety. Working with different building materials (wood, steel, concrete, masonry and/or proprietary connectors and structural systems). LQEngineer (talk) 05:13, 1 September 2009 (UTC)
Structural Engineering ActivitiesEdit
Structural engineering is mainly involved with two activities
Structural AnalysisEdit
It deals with analyzing a particular structural system. A structural system may vary from simple systems (like beams, columns, slabs, etc.) to more complex systems (like frames, bridges, piers, foundations, retaining walls, etc.). The objective behind analysis is to estimate or find resultant stresses (or forces) so that these elements can be designed to withstand the load that comes over it.
Analysis of Statically Determinate StructuresEdit
Idealized StructuresEdit
In reality the exact analysis of a structure can never be carried out. Idealizing a structure is a method of conservatively simplifying the components of the structural system, while keeping its behavior under loading the same. This is done in order to simplify calculations. Without an idealized structure, design could take a massively longer amount of time... and sometimes becomes impossible. It is important for a structural engineer to develop methods to idealize a structure in order to carry out analysis.
Supports for Coplanar StructuresEdit
Structural Members are joined together by supports depending on the intent of the engineer. Three types of joints most often used are pin connections, roller supports, and fixed joints. Pin connections provide vertical and lateral support, but cannot provide a moment reaction. A roller support can only supply a reaction of a force in one direction. A fixed joint provides vertical, lateral as well as a moment reaction.
Analysis of Statically Determinate TrussesEdit
Typically done using method of joints, method of sections or graphical methods.
Internal Loadings Developed in Structural MembersEdit
Internal loadings are found by "cutting" the member and applying the equations of equilibrium to an isolated section.
Cables and ArchesEdit
What are Cables and Arches?
Cables are structural elements that can only withstand loads in tension.
Arches are structural elements that withstand loads in compression.
Influence Lines for Statically Determinate StructuresEdit
Influence lines are a diagram that represents variation of a loading function at one location on a structure. For example, a truck drives across a bridge:
When the truck is at the left side on top of the bridge, very little loading is felt at the right support. The function represented by influence lines changes as the truck moves towards the right end of the bridge. The loading at the right support grows as the truck nears it, until the truck reaches the far end, where the end realizes the maximum loading. Influence lines for statically determinate structures consist of a straight line.
DeflectionsEdit
Deflections occur naturally in structures from various sources. Loads and temperature changes are sources of deflections that structural engineers have to design for because they are unavoidable. Designs must be made in order to avoid cracking of the materials used. Fabrication or Design Errors can lead to failure of structures and should be avoided through careful planning and analysis.
Structures in most cases are built with materials that can withstand the designed loading and only have a linear elastic response. Under these conditions loads may cause deflections, but when the load is removed the structure will return to it original shape and strength. Overloading beyond the linear elastic response may cause damage and failure of the structure. This is referred to as "plastic deformation".
Elastic-Beam TheoryEdit
If a material is homogeneous and behaves in a linear elastic manner we can derive a nonlinear second order differential equation that when solved through the double integration method can give a solution deflection as a function of x. We must assume
{\displaystyle dv/dx=0}
relative to the length of the beam in the x-axial direction.
{\displaystyle {\frac {d^{2}v}{dx^{2}}}={\frac {M}{EI}}}
M = the internal moment in the beam E = the material's modulus of elasticity I = the beam's moment of inertia computed about the neutral axis v = deflection of beam
The Double Integration MethodEdit
When Moment can be expressed as a function of position x, then performing single integration will yield the beam's slope as a function of x, and the equation of the elastic curve. Performing double integration will yield the beam's deflection for any position of x
The two integrations will yield two constants of integration. Using the boundary conditions the constants can be solved for.
Castigliano's theoremsEdit
First theoremEdit
If the strain energy of an elastic structure can be expressed as a function of generalized displacement qi; then the partial derivative of the strain energy with respect to generalized displacement gives the generalized force Qi.
Using this theorem the generalized forces acting on the structure can be calculated.
Second theoremEdit
If the strain energy of a linearly elastic structure can be expressed as a function of generalized force Qi; then the partial derivative of the strain energy with respect to generalized force gives the generalized displacement qi in the direction of Qi.
Using this theorem the generalized displacement of the structure can be calculated. Castigliano's method usually refers to the application of his second theorem.
Structural DesignEdit
Structural design is the process of selecting members of required dimensions such that they provide adequate stability under service loads. There are two conditions that a structural designer must keep in mind. One is "stability" and the other is "serviceability". Stability of a structure means that it can resist the loads acting on it satisfactorily and that the structure will not collapse immediately (that is, it provides enough time to escape to safety). Serviceability refers to certain conditions that are required so that the structure remains serviceable. For example, consider a bridge that can resist service loads without collapse. This is a "stable" structure. Now assume that this bridge shows abnormal deflections. The deflections could be such that the bridge feels bouncy and could lead to steering problems for vehicles crossing it at high speeds. As such this may not cause the structure to collapse. So we can say that the structure is stable, but according to serviceability criterion it is not serviceable because people could feel afraid of using the bridge.
In the design and construction of the foundation and framing for buildings and bridges, the main materials used are concrete, steel, timber, and masonry. Steel can further be subdivided into two subsections: hot-rolled steel and cold-formed steel. Cold-formed steel applies to material of approximately 1/8" or less in thickness that is either folded or roll-formed from flat sheets into structural shapes while at room temperature.
Safety FactorsEdit
All structures must be designed to carry all foreseeable loads with a suitable factor of safety. Clearly it would be unsafe to walk on a structure that was adequate but had no margin of safety built in. With this in mind, most countries have standards that prescribe the required minimum safety factors for structures. The minimum is usually a factor of about 1.7. This is not a factor to allow for overloading or poor workmanship. If there is a possibility of overloading or poor workmanship, the design loads must be increased to account for the overloading and the strengths of the materials upon which the design is based must be reduced to account for the poor workmanship. The safety factor must remain complete, as it is there to account for the unexpected events and unforeseen circumstances. If a structure becomes worn, loose, cracked or corroded, it should always be repaired so the safety factor is preserved.
↑ https://books.google.com/books?id=YJoSY5QT6LYC&pg=PA133&lpg=PA133&dq=%22e.h.+brown%22+molding+materials+-facebook&source=bl&ots=XFZqHCjI9W&sig=ACfU3U0z7L1NpqrSK5-fy-2wpSuy949_xA&hl=en&sa=X&ved=2ahUKEwiys8eDxLfkAhULbawKHSYzDdMQ6AEwCnoECAkQAQ#v=onepage&q=%22e.h.%20brown%22%20molding%20materials%20-facebook&f=false
Retrieved from "https://en.wikibooks.org/w/index.php?title=Structural_Engineering&oldid=3568660"
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EUDML | A bijection between classes of fully packed loops and plane partitions. EuDML | A bijection between classes of fully packed loops and plane partitions.
A bijection between classes of fully packed loops and plane partitions.
Di Francesco, P., Zinn-Justin, P., and Zuber, J.-B.. "A bijection between classes of fully packed loops and plane partitions.." The Electronic Journal of Combinatorics [electronic only] 11.1 (2004): Research paper R64, 11 p.-Research paper R64, 11 p.. <http://eudml.org/doc/124090>.
@article{DiFrancesco2004,
author = {Di Francesco, P., Zinn-Justin, P., Zuber, J.-B.},
keywords = {full pached loop configurations; plane partitions},
title = {A bijection between classes of fully packed loops and plane partitions.},
AU - Zinn-Justin, P.
AU - Zuber, J.-B.
TI - A bijection between classes of fully packed loops and plane partitions.
KW - full pached loop configurations; plane partitions
Philippe Di Francesco, Paul Zinn-Justin, Jean-Bernard Zuber, Determinant formulae for some tiling problems and application to fully packed loops
full pached loop configurations, plane partitions
2
Articles by Di Francesco
Articles by Zinn-Justin
Articles by Zuber
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EUDML | A Composition Algebra for Multiplace Functions. EuDML | A Composition Algebra for Multiplace Functions.
A Composition Algebra for Multiplace Functions.
H.I. WHITLOCK
WHITLOCK, H.I.. "A Composition Algebra for Multiplace Functions.." Mathematische Annalen 157 (1964/65): 167-178. <http://eudml.org/doc/161231>.
@article{WHITLOCK1964/65,
author = {WHITLOCK, H.I.},
title = {A Composition Algebra for Multiplace Functions.},
AU - WHITLOCK, H.I.
TI - A Composition Algebra for Multiplace Functions.
Isidore Fleischer, Functional representation of preiterative/combinatory formalism
Kazimierz Głazek, A certain Galois connection and weak automorphisms
R. Gorton,
A
Articles by H.I. WHITLOCK
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Maja has an unfair coin which is weighted so that, when flipped, it has a
\frac{2}{3}
chance of landing on heads and
\frac{1}{3}
chance of landing on tails.
The first scenario was an example of probability. We knew the coin was unfair, with a
\frac{2}{3}
chance of heads and a
\frac{1}{3}
chance of tails on every flip. We used that rule to figure out the chance that the same side would come up for two flips in a row.
If our data leads us to make an estimate that the true number of fish in every lake in the United States is 9 billion
\pm
0.7 billion, what does that mean?
|
M. Krameret al.
Continued timing observations of the double pulsar PSR J0737–3039A/B, which consists of two active radio pulsars (A and B) that orbit each other with a period of 2.45 h in a mildly eccentric (
e=0.088
) binary system, have led to large improvements in the measurement of relativistic effects in this system. With a 16-yr data span, the results enable precision tests of theories of gravity for strongly self-gravitating bodies and also reveal new relativistic effects that have been expected but are now observed for the first time. These include effects of light propagation in strong gravitational fields which are currently not testable by any other method. In particular, we observe the effects of retardation and aberrational light bending that allow determination of the spin direction of the pulsar. In total, we detect seven post-Keplerian parameters in this system, more than for any other known binary pulsar. For some of these effects, the measurement precision is now so high that for the first time we have to take higher-order contributions into account. These include the contribution of the A pulsar’s effective mass loss (due to spin-down) to the observed orbital period decay, a relativistic deformation of the orbit, and the effects of the equation of state of superdense matter on the observed post-Keplerian parameters via relativistic spin-orbit coupling. We discuss the implications of our findings, including those for the moment of inertia of neutron stars, and present the currently most precise test of general relativity’s quadrupolar description of gravitational waves, validating the prediction of general relativity at a level of
1.3×{10}^{-4}
with 95% confidence. We demonstrate the utility of the double pulsar for tests of alternative theories of gravity by focusing on two specific examples and also discuss some implications of the observations for studies of the interstellar medium and models for the formation of the double pulsar system. Finally, we provide context to other types of related experiments and prospects for the future.
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\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\mathrm{f3}{!}^{k}\dots
i,j,k
\mathrm{f1}
\mathrm{f2}
\mathrm{f3}
\sqrt{\mathrm{\pi }}=\mathrm{\Gamma }\left(\frac{1}{2}\right)
0<i
j,k<0
\mathrm{f1}-\mathrm{f2}-\mathrm{f3}=n
\frac{\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\mathrm{f2}}\right)c\mathrm{f2}!\mathrm{f3}!}{\mathrm{f1}!}
c
is a correction factor depending on
\mathrm{f3}
i,0<j
k<0
\mathrm{f3}-\mathrm{f1}-\mathrm{f2}=n
\frac{c\mathrm{f3}!}{\mathrm{f1}!\mathrm{f2}!\left(\genfrac{}{}{0}{}{\mathrm{f2}}{\mathrm{f1}}\right)}
\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\dots
i,j
\frac{\mathrm{f1}}{\mathrm{f2}}
r
1<|r|
\frac{\mathrm{f2}!\left(\genfrac{}{}{0}{}{\mathrm{f1}}{\mathrm{f2}}\right)\left(\mathrm{f1}-\mathrm{f2}\right)!}{\mathrm{f1}!}
|r|<1
\frac{\mathrm{f2}!}{\mathrm{f1}!\left(\genfrac{}{}{0}{}{\mathrm{f2}}{\mathrm{f1}}\right)\left(\mathrm{f2}-\mathrm{f1}\right)!}
a≔\frac{n!}{k!\left(n-k\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{!}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{n}}{\textcolor[rgb]{0,0,1}{k}}\right)
a≔\frac{n\left({n}^{2}+m-k+2\right)\left({n}^{2}+m\right)!}{k!\left({n}^{2}+m-k+2\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\right)\textcolor[rgb]{0,0,1}{!}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\frac{\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{k}}\right)}{{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}
a≔\frac{{m!}^{3}}{\left(3m\right)!}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{!}}^{\textcolor[rgb]{0,0,1}{3}}}{\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}\right)\textcolor[rgb]{0,0,1}{!}}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{m}}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{m}}\right)}
a≔\frac{\mathrm{\Gamma }\left(m+\frac{3}{2}\right)}{\mathrm{sqrt}\left(\mathrm{\pi }\right)\mathrm{\Gamma }\left(m\right)}
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{\Gamma }}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\right)}
\mathrm{convert}\left(a,\mathrm{binomial}\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}}\right)
|
Per-unit discrete-time single-phase induction machine field-oriented control - Simulink - MathWorks í•œêµ
T*=\left({K}_{p_\mathrm{Ï}}+{K}_{i_\mathrm{Ï}}\frac{{T}_{s}z}{zâ1}\right)\left({\mathrm{Ï}}_{ref}â{\mathrm{Ï}}_{r}\right)
where ωr is the rotor angular mechanical speed in rad/s.
\mathrm{Ï}*=\frac{2\mathrm{Ï}{f}_{n}{\mathrm{Ï}}_{n}}{p\mathrm{min}\left(|{\mathrm{Ï}}_{r}|,\frac{2\mathrm{Ï}{f}_{n}}{p}\right)}
ψn is the rated flux.
{i}_{q}*=\frac{\left({L}_{ms}+{L}_{lar}\right)T*}{pa{L}_{ms}\mathrm{Ï}*}
{i}_{d}*=\frac{\mathrm{Ï}*}{{a}^{2}{L}_{ms}}
\frac{d\mathrm{θ}}{dt}=p{\mathrm{Ï}}_{r}+\frac{{a}^{3}{L}_{ms}{R}_{ar}}{{\mathrm{Ï}}^{*}\left({L}_{ms}+{L}_{lar}\right)}
\left(\begin{array}{c}{i}_{a}\\ {i}_{b}\end{array}\right)=\left(\begin{array}{cc}{a}^{2}\mathrm{sin}\left(\mathrm{θ}\right)& {a}^{2}\mathrm{cos}\left(\mathrm{θ}\right)\\ \mathrm{cos}\left(\mathrm{θ}\right)& â\mathrm{sin}\left(\mathrm{θ}\right)\end{array}\right)\left(\begin{array}{c}{i}_{d}\\ {i}_{q}\end{array}\right)
Reference | Rotor velocity | θ | iab | iabRef | TqRef | FluxRef
θ, in radians
|
CECIL’S LATEST TRICK
To amaze the audience, Cecil set up four tightropes in a rectangle, each connected at a pole as shown in the diagram at right. A ladder down from the ropes is located at point
A
If Cecil starts at point
A
and must travel completely around the rectangle, how far must Cecil travel?
The distance Cecil must travel is the perimeter of the rectangle made by the four tightropes.
54
How can Cecil’s distance be represented by an expression?
Perimeter can be written as
2(\text{width})+2(\text{length})
. Use this expression, substituting the appropriate values for ''width'' and ''length.''
12 \text{ feet}
15 \text{ feet}
|
Key Combinations for 2-D Math Notation - MapleSim Help
Home : Support : Online Help : MapleSim : Using MapleSim : Building a Model : Annotating a Model : Key Combinations for 2-D Math Notation
In parameter values, parameter names, and annotations, you can enter text in 2-D math notation, which is a formatting option that allows you to add mathematical elements such as subscripts, superscripts, and Greek characters. This option also provides a symbol completion feature, which automatically suggests mathematical symbols to insert as you enter a phrase in 2-D math notation.
To enter 2-D math notation in a text annotation, select Math ( ) in the text formatting toolbar.
To move the cursor out of a denominator, superscript, or subscript position:
The following table lists the key combinations for 2-D math notation:
Symbol completion (parameter values and annotations only)
1. Enter the first few characters of a symbol name.
2. Use one of the following platform-specific key combinations:
3. From the pop-up menu, select the symbol or command that you want to insert.
Enter a subscript for a variable
Ctrl (or Command for Mac) + Shift + underscore ( _ )
{x}_{a}
Enter a superscript (parameter values and annotations only)
{x}^{2}
Enter a fraction
\frac{1}{4}
Enter an underscript
Ctrl (or Command) + '
\underset{x}{y}
Enter an overscript
Ctrl (or Command) + Shift + "
\stackrel{x}{y}
Enter a pre-superscript
Ctrl (or Command) + Shift + ^
{}^{a}x
Enter a pre-subscript
Ctrl (or Command) + Shift + _
\mathrm{x__a__x__}
Enter a Greek character
Enter the first few characters of a Greek character name (for example, "phi"), and use your platform-specific key combination for symbol completion.
Press Ctrl (or Command) + Shift + G and then enter a Greek character according to key mappings. For more information, see Key Mappings for Greek Characters.
\mathrm{α}, \mathrm{φ}
Display backslash, underscore, or caret characters in 2-D math notation
backslash (\) + backslash (\) or underscore ( _ ) or caret (^)
Move the cursor one level out of a nested structure
Ctrl (or Command) + [
Move the cursor one level in a nested structure
Ctrl (or Command) + ]
Return the cursor to the baseline
Ctrl (or Command) + /
Adding Text Annotations to a Model
Key Mappings for Greek Characters
|
\mathrm{restart}
\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):
c
f
f\left(x\right)=\underset{n=0}{\overset{\infty }{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∑}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{a}_{n}{\left(x-c\right)}^{n}
{a}_{n}
{a}_{n}=\frac{{f}^{\left(n\right)}\left(c\right)}{n!}
{f}^{\left(n\right)}\left(c\right)
is the nth derivative of
c
f
c
{ⅇ}^{x}
{ⅇ}^{x}=\underset{n=0}{\overset{\infty }{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∑}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{x}^{n}}{n!}
ⅇ=\underset{n=0}{\overset{\infty }{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∑}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{1}{n!}
\mathrm{TaylorApproximation}\left({x}^{7}-5{x}^{5}+4{x}^{4}-7{x}^{2}+3,x=1,\mathrm{order}=3\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}
x=1
{x}^{7}-5{x}^{5}+4{x}^{4}-7{x}^{2}+3
{x}^{3}-15{x}^{2}+11x-1
\mathrm{TaylorApproximation}\left({x}^{7}-5{x}^{5}+4{x}^{4}-7{x}^{2}+3,x=1,\mathrm{order}=3,\mathrm{output}=\mathrm{plot}\right)
\mathrm{TaylorApproximation}\left(\mathrm{sin}\left(x\right),x=1,\mathrm{output}=\mathrm{animation},\mathrm{order}=1..16\right)
I
-I
1
\mathrm{TaylorApproximation}\left(\mathrm{arctan}\left(x\right),x=0,\mathrm{output}=\mathrm{animation},\mathrm{order}=1..20\right)
\mathrm{TaylorApproximationTutor}\left(\right)
\mathrm{FunctionChart}\left({x}^{4}+2{x}^{3}-9{x}^{2}-3x+6,x=-5..4\right)
\mathrm{FunctionChart}\left(\mathrm{sin}\left(x\right),x=0..2\mathrm{π}\right)
\mathrm{FunctionChart}\left(\frac{{x}^{3}-2{x}^{2}-4x+2}{x-4},x=-3..3\right)
\mathrm{CurveAnalysisTutor}\left(\right)
and an expression
f\left(x\right)
x
a
f\left(a\right)
f\left(x\right)
\mathrm{Tangent}\left(f\left(x\right),x=a,\mathrm{output}=\mathrm{line}\right)
\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{a}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{-}\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{a}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}
\mathrm{collect}\left(\mathrm{convert}\left(,\mathrm{D}\right),\mathrm{D}\left(f\right)\left(a\right)\right)
\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D}}\left(\textcolor[rgb]{0,0,1}{f}\right)\left(\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)
\mathrm{solve}\left(=0,x\right)
\frac{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{D}}\left(\textcolor[rgb]{0,0,1}{f}\right)\left(\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{\mathrm{D}}\left(\textcolor[rgb]{0,0,1}{f}\right)\left(\textcolor[rgb]{0,0,1}{a}\right)}
\mathrm{expand}\left(\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)}{\textcolor[rgb]{0,0,1}{\mathrm{D}}\left(\textcolor[rgb]{0,0,1}{f}\right)\left(\textcolor[rgb]{0,0,1}{a}\right)}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}
F\left(x\right)={x}^{2}-1
x=2.0
F:=x→{x}^{2}-1
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{→}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}
\mathrm{aroot}:=2.0-\frac{F\left(2.0\right)}{\mathrm{D}\left(F\right)\left(2.0\right)}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.250000000}
9
\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}5\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{aroot}:=\mathrm{aroot}-\frac{F\left(\mathrm{aroot}\right)}{\mathrm{D}\left(F\right)\left(\mathrm{aroot}\right)}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.025000000}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.000304878}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.000000046}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.000000000}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.000000000}
\mathrm{NewtonsMethod}\left(F\left(x\right),x=2,\mathrm{output}=\mathrm{sequence}\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.250000000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.025000000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.000304878}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.000000046}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.000000000}
\mathrm{NewtonsMethod}\left(F\left(x\right),x=2,\mathrm{output}=\mathrm{plot}\right)
\mathrm{NewtonsMethod}\left(\frac{\mathrm{sin}\left(x\right)}{x},x=1,\mathrm{output}=\mathrm{plot}\right)
\mathrm{NewtonsMethod}\left(\frac{\mathrm{sin}\left(x\right)}{x},x=2,\mathrm{output}=\mathrm{plot}\right)
30
7
\mathrm{Digits}:=30
\mathrm{NewtonsMethod}\left({x}^{4}-4{x}^{3}+4{x}^{2}-3x+3,x=1,\mathrm{output}=\mathrm{sequence},\mathrm{iterations}=10\right)
\mathrm{Digits}:=10
\textcolor[rgb]{0,0,1}{\mathrm{Digits}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{30}
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.33333333333333333333333333333}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28318584070796460176991150443}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231623816647766759714731909}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363411166690275078928}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754743}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754709}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754707}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754707}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754707}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754707}
\textcolor[rgb]{0,0,1}{\mathrm{Digits}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{10}
\mathrm{NewtonsMethodTutor}\left(\right)
|
Probability Problem on Central Limit Theorem: Magic Box Game - Worranat Pakornrat | Brilliant
Magic Box Game
You're in a game show called "Magic Box," where you're given 4 identical balls to throw into
4
boxes randomly. Each box will earn you the point(s) written on it multiplied by the number of balls thrown into it. For example, if you put one ball into each box, you'll get
1+2+3+4 =10
What is the expected value of points earned from this game?
|
Hermite interpolation - Wikipedia
Polynomial interpolation using derivative values
In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than n that takes the same value at n given points as a given function. Instead, Hermite interpolation computes a polynomial of degree less than mn such that the polynomial and its m − 1 first derivatives have the same values at n given points as a given function and its m − 1 first derivatives.
Hermite's method of interpolation is closely related to the Newton's interpolation method, in that both are derived from the calculation of divided differences. However, there other methods for computing a Hermite interpolating polynomial. One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see Chinese remainder theorem § Hermite interpolation.
2.3.1 Quintic Hermite interpolation
Statement of the problem[edit]
Hermite interpolation consists of computing a polynomial of degree as low as possible that matches an unknown function both in observed value, and the observed value of its first m derivatives. This means that n(m + 1) values
{\displaystyle {\begin{matrix}(x_{0},y_{0}),&(x_{1},y_{1}),&\ldots ,&(x_{n-1},y_{n-1}),\\(x_{0},y_{0}'),&(x_{1},y_{1}'),&\ldots ,&(x_{n-1},y_{n-1}'),\\\vdots &\vdots &&\vdots \\(x_{0},y_{0}^{(m)}),&(x_{1},y_{1}^{(m)}),&\ldots ,&(x_{n-1},y_{n-1}^{(m)})\end{matrix}}}
must be known. The resulting polynomial has a degree less than n(m + 1). (In a more general case, there is no need for m to be a fixed value; that is, some points may have more known derivatives than others. In this case the resulting polynomial has a degree less than the number of data points.)
Let us consider a polynomial P(x) of degree less than n(m + 1) with indeterminate coefficients; that is, the coefficients of P(x) are n(m + 1) new variables. Then, by writing the constraints that the interpolating polynomial must satisfy, one gets a system of n(m + 1) linear equations in n(m + 1) unknowns.
In general, such a system has exactly one solution. Charles Hermite proved that this is effectively the case here, as soon as the xi are pairwise different,[citation needed] and provided a method for computing it, which is described below.
Simple case[edit]
When using divided differences to calculate the Hermite polynomial of a function f, the first step is to copy each point m times. (Here we will consider the simplest case
{\displaystyle m=1}
for all points.) Therefore, given
{\displaystyle n+1}
{\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n}}
, and values
{\displaystyle f(x_{0}),f(x_{1}),\ldots ,f(x_{n})}
{\displaystyle f'(x_{0}),f'(x_{1}),\ldots ,f'(x_{n})}
{\displaystyle f}
that we want to interpolate, we create a new dataset
{\displaystyle z_{0},z_{1},\ldots ,z_{2n+1}}
{\displaystyle z_{2i}=z_{2i+1}=x_{i}.}
Now, we create a divided differences table for the points
{\displaystyle z_{0},z_{1},\ldots ,z_{2n+1}}
. However, for some divided differences,
{\displaystyle z_{i}=z_{i+1}\implies f[z_{i},z_{i+1}]={\frac {f(z_{i+1})-f(z_{i})}{z_{i+1}-z_{i}}}={\frac {0}{0}}}
which is undefined. In this case, the divided difference is replaced by
{\displaystyle f'(z_{i})}
. All others are calculated normally.
In the general case, suppose a given point
{\displaystyle x_{i}}
has k derivatives. Then the dataset
{\displaystyle z_{0},z_{1},\ldots ,z_{N}}
contains k identical copies of
{\displaystyle x_{i}}
. When creating the table, divided differences of
{\displaystyle j=2,3,\ldots ,k}
identical values will be calculated as
{\displaystyle {\frac {f^{(j)}(x_{i})}{j!}}.}
{\displaystyle f[x_{i},x_{i},x_{i}]={\frac {f''(x_{i})}{2}}}
{\displaystyle f[x_{i},x_{i},x_{i},x_{i}]={\frac {f^{(3)}(x_{i})}{6}}}
{\displaystyle f(x)=x^{8}+1}
. Evaluating the function and its first two derivatives at
{\displaystyle x\in \{-1,0,1\}}
, we obtain the following data:
ƒ'(x)
ƒ''(x)
Since we have two derivatives to work with, we construct the set
{\displaystyle \{z_{i}\}=\{-1,-1,-1,0,0,0,1,1,1\}}
. Our divided difference table is then:
{\displaystyle {\begin{array}{llcclrrrrr}z_{0}=-1&f[z_{0}]=2&&&&&&&&\\&&{\frac {f'(z_{0})}{1}}=-8&&&&&&&\\z_{1}=-1&f[z_{1}]=2&&{\frac {f''(z_{1})}{2}}=28&&&&&&\\&&{\frac {f'(z_{1})}{1}}=-8&&f[z_{3},z_{2},z_{1},z_{0}]=-21&&&&&\\z_{2}=-1&f[z_{2}]=2&&f[z_{3},z_{2},z_{1}]=7&&15&&&&\\&&f[z_{3},z_{2}]=-1&&f[z_{4},z_{3},z_{2},z_{1}]=-6&&-10&&&\\z_{3}=0&f[z_{3}]=1&&f[z_{4},z_{3},z_{2}]=1&&5&&4&&\\&&{\frac {f'(z_{3})}{1}}=0&&f[z_{5},z_{4},z_{3},z_{2}]=-1&&-2&&-1&\\z_{4}=0&f[z_{4}]=1&&{\frac {f''(z_{4})}{2}}=0&&1&&2&&1\\&&{\frac {f'(z_{4})}{1}}=0&&f[z_{6},z_{5},z_{4},z_{3}]=1&&2&&1&\\z_{5}=0&f[z_{5}]=1&&f[z_{6},z_{5},z_{4}]=1&&5&&4&&\\&&f[z_{6},z_{5}]=1&&f[z_{7},z_{6},z_{5},z_{4}]=6&&10&&&\\z_{6}=1&f[z_{6}]=2&&f[z_{7},z_{6},z_{5}]=7&&15&&&&\\&&{\frac {f'(z_{6})}{1}}=8&&f[z_{8},z_{7},z_{6},z_{5}]=21&&&&&\\z_{7}=1&f[z_{7}]=2&&{\frac {f''(z_{7})}{2}}=28&&&&&&\\&&{\frac {f'(z_{7})}{1}}=8&&&&&&&\\z_{8}=1&f[z_{8}]=2&&&&&&&&\\\end{array}}}
and the generated polynomial is
{\displaystyle {\begin{aligned}P(x)&=2-8(x+1)+28(x+1)^{2}-21(x+1)^{3}+15x(x+1)^{3}-10x^{2}(x+1)^{3}\\&\quad {}+4x^{3}(x+1)^{3}-1x^{3}(x+1)^{3}(x-1)+x^{3}(x+1)^{3}(x-1)^{2}\\&=2-8+28-21-8x+56x-63x+15x+28x^{2}-63x^{2}+45x^{2}-10x^{2}-21x^{3}\\&\quad {}+45x^{3}-30x^{3}+4x^{3}+x^{3}+x^{3}+15x^{4}-30x^{4}+12x^{4}+2x^{4}+x^{4}\\&\quad {}-10x^{5}+12x^{5}-2x^{5}+4x^{5}-2x^{5}-2x^{5}-x^{6}+x^{6}-x^{7}+x^{7}+x^{8}\\&=x^{8}+1.\end{aligned}}}
by taking the coefficients from the diagonal of the divided difference table, and multiplying the kth coefficient by
{\displaystyle \prod _{i=0}^{k-1}(x-z_{i})}
, as we would when generating a Newton polynomial.
Quintic Hermite interpolation[edit]
The quintic Hermite interpolation based on the function (
{\displaystyle f}
), its first (
{\displaystyle f'}
) and second derivatives (
{\displaystyle f''}
) at two different points (
{\displaystyle x_{0}}
{\displaystyle x_{1}}
) can be used for example to interpolate the position of an object based on its position, velocity and acceleration. The general form is given by
{\displaystyle {\begin{aligned}p(x)&=f(x_{0})+f'(x_{0})(x-x_{0})+{\frac {1}{2}}f''(x_{0})(x-x_{0})^{2}+{\frac {f(x_{1})-f(x_{0})-f'(x_{0})(x_{1}-x_{0})-{\frac {1}{2}}f''(x_{0})(x_{1}-x_{0})^{2}}{(x_{1}-x_{0})^{3}}}(x-x_{0})^{3}\\&+{\frac {3f(x_{0})-3f(x_{1})+\left(2f'(x_{0})+f'(x_{1})\right)(x_{1}-x_{0})+{\frac {1}{2}}f''(x_{0})(x_{1}-x_{0})^{2}}{(x_{1}-x_{0})^{4}}}(x-x_{0})^{3}(x-x_{1})\\&+{\frac {6f(x_{1})-6f(x_{0})-3\left(f'(x_{0})+f'(x_{1})\right)(x_{1}-x_{0})+{\frac {1}{2}}\left(f''(x_{1})-f''(x_{0})\right)(x_{1}-x_{0})^{2}}{(x_{1}-x_{0})^{5}}}(x-x_{0})^{3}(x-x_{1})^{2}.\end{aligned}}}
Call the calculated polynomial H and original function f. Evaluating a point
{\displaystyle x\in [x_{0},x_{n}]}
, the error function is
{\displaystyle f(x)-H(x)={\frac {f^{(K)}(c)}{K!}}\prod _{i}(x-x_{i})^{k_{i}},}
where c is an unknown within the range
{\displaystyle [x_{0},x_{N}]}
, K is the total number of data-points, and
{\displaystyle k_{i}}
is the number of derivatives known at each
{\displaystyle x_{i}}
plus one.
Newton series, also known as finite differences
Neville's schema
Bernstein form of the interpolation polynomial
Burden, Richard L.; Faires, J. Douglas (2004). Numerical Analysis. Belmont: Brooks/Cole.
Spitzbart, A. (January 1960), "A Generalization of Hermite's Interpolation Formula", American Mathematical Monthly, 67 (1): 42–46, doi:10.2307/2308924, JSTOR 2308924
Hermites Interpolating Polynomial at Mathworld
Retrieved from "https://en.wikipedia.org/w/index.php?title=Hermite_interpolation&oldid=1078078491"
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Scouting - MetaSoccer
As in real soccer, as an Owner, you can send a Youth Scout in search of promising young players to join your team. Right after performing the action, the new players will appear in your squad, but their information will not be revealed until after 6 days – unless the scout has the special ability “Master of Relations”, in which case the process takes 3 days. Once the search is complete, you will be able to reveal the identity of your new players.
After each Scouting, the scout will gain more experience and their attributes will improve. Each Scouting performed by the scout will improve its Knowledge values.
Players found and Cost
The number of players you find in each scouting will also vary according to their knowledge, as finding better players is harder and more expensive.
Scout Overall Knowledge
Players Found
Cost in $MSU
Cost in $MSC
Players' Role and Scout Specific Role
Scouts will only find players that play within the same role of their expertise. This means that depending on their role being Goalkeeper, Defender, Midfielder or Forward, they will only be able to find players for that general positions.
The first player found in the Scouting, where the scout specific role (if any) is applied, will always have the scout specific role. For the rest of the players (if the scout is to find more than one), probabilities will depend on the scout’s role, according to the tables below.
Scout with Goalkeeper Role
Player Specific Role
Scout with Defender Role
Scout with Midfielder Role
Scout with Forward Role
Players' Age
The age of the players found by each scout will depend on his characteristics.
The usual age range of players found in scouting will be from 16 to 19 years old, with a 25% probability of each age. However, there is one scout special ability, which is ‘Youth specialist’, that allows the scout to find players from 14 to 19 years old. In that case, the probabilities are:
18: years old: 12%
Additionally, it is important to understand that age also impacts the current abilities of the player. Older players are more developed than younger players, and based on that, some ratios are applied to the potential abilities to get the initial abilities when the scouting is done.
Current Ability = f(Potential Ability, Specific Role, Age)
According to that, the age ratios, which also depend on the overall knowledge of the scout, are the following ones:
Scout Overall Knowledge / Age
There is however a Scout Special Ability which will have an impact on the previous table. Any scout with ‘Developed players’ special ability will increase, for any player at any age, 0,10 more points to the previous table. For example, a scout with ‘Developed players’ with an overall of 55 that finds a player that is 16 years old will have a ratio of 0.71 (0.61 from the previous table + 0.10 additional percentage points).
Players will achieve their peak in between 2–10 years depending on the initial age and the scout’s overall knowledge, giving the user many time to enjoy the player at his maximum potential.
Players' Abilities
The characteristics of the scout also influence the type of abilities the players found will have.
Based on the scout overall knowledge, a potential value for each players’ ability is calculated accordingly. Each specific role has its calculations, meaning that there are some abilities more linked to some specific roles than others. For the same overall potential, for example, RWB and RB will have better potential in the crossing and dribbling abilities than CB.
However, CB will have better strength. This means that if a Midfielder Scout with an overall of 75 has the best knowledge Defense and second best in Attacking, he will find players with a Potential similar to 75 but with better Attacking-related abilities than Physical & Mental. If the Scout has the best knowledge of Defense and second Physical & Mental, the players will be better in physical-related abilities than Defense (and still an overall of 75).
A balanced process converts scouts’ knowledge overall to players’ abilities. So, given a scout with Overall 80 that has mental knowledge of 45, probably there will be a balance based on the specific role when finding the players. Finally, mental-related abilities can be slightly increased, depending on the importance in the specific role.
All processes to find the potentials of each ability have some randomness added, but always take into account that the overall potential of the player is similar to the overall of the scout.
Players' Special Abilities
Every scout has a chance to find one of these special players, but some of them have a better eye for it.
Scouts with players' special abilities will be more likely to discover players with special abilities. The overall knowledge also has some influence in the probability of finding players with special abilities.
The first player found is the one with the higher chance to have a special ability. Also, it may happen that the players discovered have different abilities than those of the scout.
Scout without Players' Special Abilities
When a youth scout has no Player Special Abilities, the probability that the first player found has a special ability is low.
The rest of the players found in the Scouting of a scout without players' special abilities will have no special abilities.
Scout with 1 Player Special Ability
If a youth scout has one player special ability, the chances of the first player being discovered to have a special ability as well increases. There is a small chance that the ability is not the same as the scout.
The rest of the players found will have each one a 5% chance of having one special ability.
Scout with 2 Player Special Abilities
Youth scouts with two player special abilities have the same probability of finding a first player with special abilities as the Scouts that only have one special ability. The main difference is that they also have the chance to discover a player with two abilities. There is a small chance that one of the abilities is not the same as the scout.
The rest of the players found will have, each one, a 5% chance of having one special ability and a 1% of having two special abilities (94% chance of having 0 special abilities).
Scouts that have as a special ability being a ‘Scouting Expert’ are the ones that have higher probabilities of finding players with special abilities; in fact, it adds 20 percentage points more (up to a maximum of 100) to the chance that the first player they discover has a special ability.
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(→P-LOCAL: Polylogarithmically many rounds in the LOCAL model)
===== <span id="plocal" style="color:red">P-LOCAL</span>: Polylogarithmically many rounds in the LOCAL model =====
===== <span id="plocal" style="color:red">P-LOCAL</span>: Polylogarithmic rounds in the LOCAL model =====
In the local model of computation, there is a graph with n vertices, each with its own, unique Θ(log(n)) bit ID. Each vertex starts with the state of knowing only its ID and has unbounded computational ability. At each round, every vertex synchronously sends and recieves one unbounded message from each neighbor. At the end of algorithm, each vertex should give an output satisfying the problem. For instance, the vertices may output a valid coloring.
In the local model of computation, there is a graph with n vertices, each with its own, unique Θ(log(n)) bit ID. Each vertex starts with the state of knowing only its ID and has unbounded, deterministic, computational ability. At each round, every vertex synchronously sends and receives one unbounded message from each neighbor. At the end of the algorithm, each vertex gives an output satisfying the problem. For instance, the vertices may output a valid [[Complexity_Garden#coloring|coloring]].
Proven in [[zooref#RG20|[RG20]]] that P-LOCAL = P-RLOCAL.
Proven in [[zooref#RG20|[RG20]]] that P-LOCAL = [[#prlocal|P-RLOCAL]].
===== <span id="prlocal" style="color:red">P-RLOCAL</span>: Polylogarithmic rounds in the Randomized LOCAL model =====
The same as [[#plocal|P-LOCAL]], except now the vertices may use randomness in the messages they send and their final output.
Proven in [[zooref#RG20|[RG20]]] that [[#plocal|P-LOCAL]] = P-RLOCAL.
===== <span id="pls" style="color:red">PLS</span>: Polynomial Local Search =====
The subclass of [[Complexity Zoo:T#tfnp|TFNP]] function problems that are guaranteed to have a solution because of the lemma that "every finite directed acyclic graph has a sink."
{\displaystyle k}
{\displaystyle n^{O(1)}}
{\displaystyle O(n)}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle F:X_{1}\times X_{2}\times \cdots \times X_{k}\to \{0,1\}}
{\displaystyle i\in [1..k]}
{\displaystyle X_{k}\in \{0,1\}^{n}}
{\displaystyle F}
{\displaystyle k}
{\displaystyle X_{i}}
{\displaystyle O\left(\mathrm {poly} (\log n)\right)}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle {\mathsf {P}}^{\mathrm {SAT} [1]}={\mathsf {P}}^{\mathrm {SAT} [2]}\Rightarrow {\mathsf {PH}}\subseteq {\mathsf {NP}}}
The same as P-LOCAL, except now the vertices may use randomness in the messages they send and their final output.
{\displaystyle c}
{\displaystyle 2^{c}}
{\displaystyle 2^{n^{O(1)}}}
{\displaystyle n^{O(1)}}
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Rewrite symbolic expression in terms of common subexpressions - MATLAB subexpr - MathWorks España
Rewrite Expression Using Abbreviations
Customize Abbreviation Variables
Rewrite symbolic expression in terms of common subexpressions
[r,sigma] = subexpr(expr)
[r,var] = subexpr(expr,'var')
[r,var] = subexpr(expr,var)
[r,sigma] = subexpr(expr) rewrites the symbolic expression expr in terms of a common subexpression, substituting this common subexpression with the symbolic variable sigma. The input expression expr cannot contain the variable sigma.
[r,var] = subexpr(expr,'var') substitutes the common subexpression by var. The input expression expr cannot contain the symbolic variable var.
[r,var] = subexpr(expr,var) is equivalent to [r,var] = subexpr(expr,'var'), except that the symbolic variable var must already exist in the MATLAB® workspace.
This syntax overwrites the value of the variable var with the common subexpression found in expr. To avoid overwriting the value of var, use another variable name as the second output argument. For example, use [r,var1] = subexpr(expr,var).
Solve the following equation. The solutions are very long expressions. To display the solutions, remove the semicolon at the end of the solve command.
solutions = solve(a*x^3 + b*x^2 + c*x + d == 0, x, 'MaxDegree', 3);
These long expressions have common subexpressions. To shorten the expressions, abbreviate the common subexpression by using subexpr. If you do not specify the variable to use for abbreviations as the second input argument of subexpr, then subexpr uses the variable sigma.
[r, sigma] = subexpr(solutions)
\begin{array}{l}\left(\begin{array}{c}\sigma -\frac{b}{3 a}-\frac{{\sigma }_{2}}{\sigma }\\ \frac{{\sigma }_{2}}{2 \sigma }-\frac{b}{3 a}-\frac{\sigma }{2}-{\sigma }_{1}\\ \frac{{\sigma }_{2}}{2 \sigma }-\frac{b}{3 a}-\frac{\sigma }{2}+{\sigma }_{1}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}=\frac{\sqrt{3} \left(\sigma +\frac{{\sigma }_{2}}{\sigma }\right) \mathrm{i}}{2}\\ \\ \mathrm{ }{\sigma }_{2}=\frac{c}{3 a}-\frac{{b}^{2}}{9 {a}^{2}}\end{array}
{\left(\sqrt{{\left(\frac{d}{2 a}+\frac{{b}^{3}}{27 {a}^{3}}-\frac{b c}{6 {a}^{2}}\right)}^{2}+{\left(\frac{c}{3 a}-\frac{{b}^{2}}{9 {a}^{2}}\right)}^{3}}-\frac{{b}^{3}}{27 {a}^{3}}-\frac{d}{2 a}+\frac{b c}{6 {a}^{2}}\right)}^{1/3}
Solve a quadratic equation.
solutions = solve(a*x^2 + b*x + c == 0, x)
\left(\begin{array}{c}-\frac{b+\sqrt{{b}^{2}-4 a c}}{2 a}\\ -\frac{b-\sqrt{{b}^{2}-4 a c}}{2 a}\end{array}\right)
Use syms to create the symbolic variable s, and then replace common subexpressions in the result with this variable.
[abbrSolutions,s] = subexpr(solutions,s)
abbrSolutions =
\left(\begin{array}{c}-\frac{b+s}{2 a}\\ -\frac{b-s}{2 a}\end{array}\right)
\sqrt{{b}^{2}-4 a c}
Alternatively, use 's' to specify the abbreviation variable.
[abbrSolutions,s] = subexpr(solutions,'s')
\left(\begin{array}{c}-\frac{b+s}{2 a}\\ -\frac{b-s}{2 a}\end{array}\right)
\sqrt{{b}^{2}-4 a c}
Both syntaxes overwrite the value of the variable s with the common subexpression. Therefore, you cannot, for example, substitute s with some value.
subs(abbrSolutions,s,0)
\left(\begin{array}{c}-\frac{b+s}{2 a}\\ -\frac{b-s}{2 a}\end{array}\right)
To avoid overwriting the value of the variable s, use another variable name for the second output argument.
[abbrSolutions,t] = subexpr(solutions,'s')
\left(\begin{array}{c}-\frac{b+s}{2 a}\\ -\frac{b-s}{2 a}\end{array}\right)
\sqrt{{b}^{2}-4 a c}
\left(\begin{array}{c}-\frac{b}{2 a}\\ -\frac{b}{2 a}\end{array}\right)
expr — Long expression containing common subexpressions
Long expression containing common subexpressions, specified as a symbolic expression or function.
var — Variable to use for substituting common subexpressions
character vector | symbolic variable
Variable to use for substituting common subexpressions, specified as a character vector or symbolic variable.
subexpr throws an error if the input expression expr already contains var.
r — Expression with common subexpressions replaced by abbreviations
Expression with common subexpressions replaced by abbreviations, returned as a symbolic expression or function.
var — Variable used for abbreviations
Variable used for abbreviations, returned as a symbolic variable.
children | simplify | subs
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The maximum number of electrons in a sub-energy level is equal to
4\left(2l+1\right)
2\left(2l+1\right)
2{n}^{2}
{n}^{3}
Multimedia and Graphics MCQ Calendar MCQ Biological Classification MCQ Alkyl and Aryl Halides MCQ Simplification MCQ Building Materials MCQ Photo-Chemistry MCQ Phylum - Aves (Birds) MCQ Odd Man Out MCQ Chemical Bonding and Molecular Structure MCQ Transition Element MCQ Design of Masonry Structures MCQ
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Home : Support : Online Help : Education : Student Packages : Linear Algebra : Computation : Queries : IsSimilar
IsSimilar(A, B, options)
(optional) parameters; for a complete list, see LinearAlgebra[IsSimilar]
The IsSimilar(A, B) command returns false if A and B are not similar, and otherwise returns the expression sequence true, P, where P is the similarity transformation matrix, that is,
A={P}^{\left(-1\right)}·B·P
If either of the Matrices has a floating-point data type, then each must evaluate to a purely floating-point Matrix. In this case, the result of IsSimilar is achieved by making a numeric comparison of the eigenvalues of each Matrix.
\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):
\mathrm{IsSimilar}\left(〈〈1,2〉|〈3,4〉〉,〈〈9,17〉|〈-2,-4〉〉\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\end{array}]
A≔〈〈3,1〉|〈2,2〉〉
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
B≔\mathrm{DiagonalMatrix}\left([1,4]\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}\end{array}]
S≔\mathrm{IsSimilar}\left(A,B\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{3}}& \frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{3}}\\ \frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{3}}& \frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{3}}\end{array}]
{S[2]}^{-1}·B·S[2]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
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Convert model from discrete to continuous time - MATLAB d2c - MathWorks Italia
H\left(z\right)=\frac{z-1}{{z}^{2}+z+0.3}
{T}_{s}=0.1s
Discrete-time model, specified as a dynamic system model such as tf, ss, or zpk.
'matched' — Zero-pole matching method (for SISO systems only). See [1].
Does not include the estimated parameter covariance of sysd. If you want to translate the covariance while converting the model, use translatecov (System Identification Toolbox).
{x}_{c}\left(k{T}_{s}\right)=G\left[\begin{array}{c}{x}_{d}\left[k\right]\\ u\left[k\right]\end{array}\right].
{x}_{c}\left(0\right)=G\left[\begin{array}{c}{x}_{0}\\ {u}_{0}\end{array}\right].
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Vizing-like conjecture for the upper domination of Cartesian products of graphs -- the proof.
Brešar, Boštjan — 2005
The cube polynomial and its derivatives: The case of median graphs.
Brešar, Boštjan; Klavžar, Sandi; Škrekovski, Riste — 2003
On Vizing's conjecture
Bostjan Bresar — 2001
A dominating set D for a graph G is a subset of V(G) such that any vertex in V(G)-D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G. For the Cartesian product G ⃞ H Vizing's conjecture [10] states that γ(G ⃞ H) ≥ γ(G)γ(H) for every pair of graphs G,H. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prove that the conjecture holds when γ(G) = γ(H) = 3.
Arboreal structure and regular graphs of median-like classes
Bostjan Brešar — 2003
We consider classes of graphs that enjoy the following properties: they are closed for gated subgraphs, gated amalgamation and Cartesian products, and for any gated subgraph the inverse of the gate function maps vertices to gated subsets. We prove that any graph of such a class contains a peripheral subgraph which is a Cartesian product of two graphs: a gated subgraph of the graph and a prime graph minus a vertex. Therefore, these graphs admit a peripheral elimination procedure which is a generalization...
Edge-connectivity of strong products of graphs
Bostjan Bresar; Simon Spacapan — 2007
The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either
{x}_{i}={y}_{i}
{x}_{i}{y}_{i}\in E\left({G}_{i}\right)
. In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.
On partial cubes and graphs with convex intervals
Boštjan Brešar; Sandi Klavžar — 2002
A graph is called a partial cube if it admits an isometric embedding into a hypercube. Subdivisions of wheels are considered with respect to such embeddings and with respect to the convexity of their intervals. This allows us to answer in negative a question of Chepoi and Tardif from 1994 whether all bipartite graphs with convex intervals are partial cubes. On a positive side we prove that a graph which is bipartite, has convex intervals, and is not a partial cube, always contains a subdivision...
Tree-like isometric subgraphs of hypercubes
Bostjan Brešar; Wilfried Imrich; Sandi Klavžar — 2003
Tree-like isometric subgraphs of hypercubes, or tree-like partial cubes as we shall call them, are a generalization of median graphs. Just as median graphs they capture numerous properties of trees, but may contain larger classes of graphs that may be easier to recognize than the class of median graphs. We investigate the structure of tree-like partial cubes, characterize them, and provide examples of similarities with trees and median graphs. For instance, we show that the cube graph of a tree-like...
The periphery graph of a median graph
Boštjan Brešar; Manoj Changat; Ajitha R. Subhamathi; Aleksandra Tepeh — 2010
The periphery graph of a median graph is the intersection graph of its peripheral subgraphs. We show that every graph without a universal vertex can be realized as the periphery graph of a median graph. We characterize those median graphs whose periphery graph is the join of two graphs and show that they are precisely Cartesian products of median graphs. Path-like median graphs are introduced as the graphs whose periphery graph has independence number 2, and it is proved that there are path-like...
Median and quasi-median direct products of graphs
Boštjan Brešar; Pranava K. Jha; Sandi Klavžar; Blaž Zmazek — 2005
Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G×P₃ is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G×K₂ are proved, and it is shown that the only nonbipartite quasi-median direct product is K₃×K₃.
7 Brešar, B
4 Klavžar, S
2 Bresar, B
1 Changat, M
1 Imrich, W
1 Jha, PK
1 Spacapan, S
1 Subhamathi, AR
1 Tepeh, A
1 Zmazek, B
1 Škrekovski, R
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Difference between revisions of "Complexity Zoo:B" - Complexity Zoo
Difference between revisions of "Complexity Zoo:B"
(→BQP: Bounded-Error Quantum Polynomial-Time: Added another old result due FR98)
m (→BP�NP: Probabilistic NP: more html escapes)
===== <span id="bpnp" style="color:red">BP•NP</span>: Probabilistic [[Complexity Zoo:N#np|NP]] =====
===== <span id="bpnp" style="color:red">BP•NP</span>: Probabilistic [[Complexity Zoo:N#np|NP]] =====
Equals [[Complexity Zoo:A#am|AM]].
{\displaystyle k}
βP: Limited-Nondeterminism NP
βkP is the class of decision problems solvable by a polynomial-time Turing machine that makes O(logkn) nondeterministic transitions, with the same acceptance mechanism as NP. Equivalently, the machine receives a purported proof of size O(logkn) that the answer is 'yes.'
Then βP is the union of βkP over all constant k.
Defined in [KF84]. See also the survey [GLM96].
There exist oracles relative to which basically any consistent inclusion structure among the βkP's can be realized [BG98].
β2P contains LOGNP and LOGSNP.
BC=P: Bounded-Error C=P
If the answer is 'yes' then exactly 1/2 of the computation paths accept.
If the answer is 'no' then either at most 1/4 or at least 3/4 of the computation paths accept.
(Here all computation paths have the same length.)
Defined in [Wat15], where it was shown that BC=P admits efficient amplification and is closed under union, intersection, disjunction, and conjunction, and that coRP ⊆ BC=P ⊆ BPP.
BH: Boolean Hierarchy Over NP
The smallest class that contains NP and is closed under union, intersection, and complement.
BH1 = NP.
BH2i is the class of languages that are the intersection of a BH2i-1 language with a coNP language.
BH2i+1 is the class of languages that are the union of a BH2i language with an NP language.
Then BH is the union over all i of BHi.
Defined in [WW85].
For more detail see [CGH+88].
Contained in Δ2P and indeed in PNP[log].
If BH collapses at any level, then PH collapses to Σ3P [Kad88].
See also: DP, QH.
BPd(P): Polynomial Size d-Times-Only Branching Program
Defined in [Weg88].
The class of decision problems solvable by a family of polynomial size branching programs, with the additional condition that each bit of the input is tested at most d times.
BPd(P) strictly contains BPd-1(P), for every d > 1 [Tha98].
Contained in PBP.
See also: P-OBDD, k-PBP.
BPE: Bounded-Error Probabilistic E
Has the same relation to E as BPP does to P.
BPEE: Bounded-Error Probabilistic EE
Has the same relation to EE as BPP does to P.
BPHSPACE(f(n)): Bounded-Error Halting Probabilistic f(n)-Space
The class of decision problems solvable in O(f(n))-space with error probability at most 1/3, by a Turing machine that halts on every input and every random tape setting.
Contains RHSPACE(f(n)).
Is contained in DSPACE(f(n)3/2) [SZ95].
BPL: Bounded-Error Probabilistic L
Has the same relation to L as BPP does to P. The Turing machine has to halt for every input and every randomness.
Contained in SC [Nis92] and in PL.
BP•NP: Probabilistic NP
Equals AM.
BPP: Bounded-Error Probabilistic Polynomial-Time
If the answer is 'yes' then at least 2/3 of the computation paths accept.
If the answer is 'no' then at most 1/3 of the computation paths accept.
Often identified as the class of feasible problems for a computer with access to a genuine random-number source.
Contained in Σ2P ∩ Π2P [Lau83], and indeed in ZPPNP [GZ97].
If BPP contains NP, then RP = NP [Ko82,Gil77] and PH is contained in BPP [Zac88].
If any problem in E requires circuits of size 2Ω(n), then BPP = P [IW97] (in other words, BPP can be derandomized).
Indeed, any proof that BPP = P requires showing either that NEXP is not in P/poly, or else that #P requires superpolynomial-size arithmetic circuits [KI02].
BPP is not known to contain complete languages. [Sip82], [HH86] give oracles relative to which BPP has no complete languages.
There exist oracles relative to which P = RP but still P is not equal to BPP [BF99].
In contrast to the case of P, it is unknown whether BPP collapses to BPTIME(nc) for some fixed constant c. However, [Bar02] and [FS04] have shown hierarchy theorems for BPP with a small amount of advice.
A zero-one law exists stating that BPP has p-measure zero unless BPP = EXP [Mel00].
Equals Almost-P.
See also: BPPpath.
BPPcc: Communication Complexity BPP
The analogue of Pcc for bounded-error probabilistic communication complexity.
Does not equal Pcc, and is not contained in NPcc, because of the EQUALITY problem.
If the classes are defined in terms of partial functions, then BPPcc
is not contained in PNPcc [PPS14];
does not contain NPcc ∩ coNPcc [Kla03].
{\displaystyle k}
cc: BPPcc in NOF model,
{\displaystyle k}
Has the same relation to BPPcc and BPP as P
{\displaystyle k}
{\displaystyle k}
{\displaystyle k}
{\displaystyle k\leq (1-\delta )\cdot \log n}
{\displaystyle \delta >0}
[DP08].
BPPKT: BPP With Time-Bounded Kolmogorov Complexity Oracle
BPP with an oracle that, given a string x, returns the minimum over all programs P that output xi on input i, of the length of P plus the maximum time taken by P on any input.
A similar class was defined in [ABK+02], where it was also shown that in BPPKT one can factor, compute discrete logarithms, and more generally invert any one-way function on a non-negligible fraction of inputs.
See also: PK.
BPP/log: BPP With Logarithmic Karp-Lipton Advice
The class of problems solvable by a semantic BPP machine with O(log n) advice bits that depend only on the input length n. If the advice is good, the output must be correct with probability at least 2/3. If it is bad, the machine must provide some answer with probability at least 2/3. See the discussion for BQP/poly.
Contained in BPP/mlog.
BPP/mlog: BPP With Logarithmic Deterministic Merlin-Like Advice
The class of problems solvable by a syntactic BPP machine with O(log n) advice bits that depend only on the input length n. If the advice is good, the output must be correct with probability at least 2/3. If it is bad, it need not be.
Contained in BPP/rlog.
BPP//log: BPP With Logarithmic Randomness-Dependent Advice
The class of problems solvable by a BPP machine that is given O(log n) advice bits, which can depend on both the machine's random coin flips and the input length n, but not on the input itself.
Defined in [TV02], where it was also shown that if EXP is in BPP//log then EXP = BPP, and if PSPACE is in BPP//log then PSPACE = BPP.
BPP/rlog: BPP With Logarithmic Deterministic Merlin-Like Advice
The class of problems solvable by a syntactic BPP machine with O(log n) random advice bits whose probability distribution depends only on the input length n. For each n, there exists good advice such that the output is correct with probability at least 2/3.
Contains BPP/mlog. The inclusion is strict, because BPP/rlog contains any finitely sparse language by fingerprinting; see the discussion for ALL.
Contained in BPP//log.
BPP-OBDD: Polynomial-Size Bounded-Error Ordered Binary Decision Diagram
Same as P-OBDD, except that probabilistic transitions are allowed and the OBDD need only accept with probability at least 2/3.
Does not contain the integer multiplication problem [AK96].
Strictly contained in BQP-OBDD [NHK00].
BPPpath: Threshold BPP
Same as BPP, except that now the computation paths need not all have the same length.
Defined in [HHT97], where the following was also shown:
BPPpath contains MA and PNP[log], and is contained in PP and BPPNP.
BPPpath is closed under complementation, intersection, and union.
If BPPpath = BPPpathBPPpath, then PH collapses to BPPpath.
If BPPpath contains Σ2P, then PH collapses to BPPNP.
There exists an oracle relative to which BPPpath is not contained in Σ2P [BGM02].
An alternate characterization of BPPpath uses the idea of post-selection. That is, BPPpath is the class of languages
{\displaystyle L}
for which there exists a pair of polynomial-time Turing machines
{\displaystyle A}
{\displaystyle B}
such that the following conditions hold for all
{\displaystyle x}
{\displaystyle x\in L}
{\displaystyle \Pr _{r\in \{0,1\}^{\mathrm {poly} (\left\vert x\right\vert )}}\left[A(x,r)\mid B(x,r)\right]\geq {\frac {2}{3}}}
{\displaystyle x\notin L}
{\displaystyle \Pr _{r\in \{0,1\}^{\mathrm {poly} (\left\vert x\right\vert )}}\left[A(x,r)\mid B(x,r)\right]<{\frac {1}{3}}}
{\displaystyle \Pr _{r\in \{0,1\}^{\mathrm {poly} (n)}}\left[B(x,r)\right]>0}
{\displaystyle B}
is the post-selector. Intuitively, this characterization allows a BPP machine to require that its random bits have some special but easily verifiable property. This characterization makes the inclusion NP ⊆ BPPpath nearly trivial.
See Also: PostBQP (quantum analogue).
BPQP: Bounded-Error Probabilistic QP
Equals BPTIME(2O((log n)^k)); that is, the class of problems solvable in quasipolynomial-time on a bounded-error machine.
Defined in [CNS99], where the following was also shown:
If either (1) #P does not have a subexponential-time bounded-error algorithm, or (2) EXP does not have subexponential-size circuits, then the BPQP hierarchy is strict -- that is, for all a < b at least 1, BPTIME(2(log n)^a) is strictly contained in BPTIME(2(log n)^b).
BPSPACE(f(n)): Bounded-Error Probabilistic f(n)-Space
The class of decision problems solvable in O(f(n))-space with error probability at most 1/3, by a Turing machine that halts with probability 1 on every input.
Contains RSPACE(f(n)) and BPHSPACE(f(n)).
BPTIME(f(n)): Bounded-Error Probabilistic f(n)-Time
Same as BPP, but with f(n)-time (for some constructible function f) rather than polynomial-time machines.
BPTIME(nlog n) does not equal BPTIME(2n^ε) for any ε>0 [KV88]. Proving a stronger time hierarchy theorem for BPTIME is a longstanding open problem; see [BH97] for details.
[Bar02] has shown the following:
If we allow a small number of advice bits (say log n), then there is a strict hierarchy: for every d at least 1, BPTIME(nd)/(log n) does not equal BPTIME(nd+1)/(log n).
In the uniform setting, if BPP has complete problems then BPTIME(nd) does not equal BPTIME(nd+1).
BPTIME(n) does not equal NP.
Subsequently, [FS04] managed to reduce the number of advice bits to only 1: BPTIME(nd)/1 does not equal BPTIME(nd+1)/1. They also proved a hierarchy theorem for HeurBPTIME.
For another bounded-error hierarchy result, see BPQP.
BQNC: Alternate Name for QNC
BQNP: Alternate Name for QMA
BQP: Bounded-Error Quantum Polynomial-Time
The class of decision problems solvable in polynomial time by a quantum Turing machine, with at most 1/3 probability of error.
One can equivalently define BQP as the class of decision problems solvable by a uniform family of polynomial-size quantum circuits, with at most 1/3 probability of error [Yao93]. Any universal gate set can be used as a basis; however, a technicality is that the transition amplitudes must be efficiently computable, since otherwise one could use them to encode the solutions to hard problems (see [ADH97]).
BQP is often identified as the class of feasible problems for quantum computers.
Contains the factoring and discrete logarithm problems [Sho97], the hidden Legendre symbol problem [DHI02], the Pell's equation and principal ideal problems [Hal02], and some other problems not thought to be in BPP.
Defined in [BV97], where it is also shown that BQP contains BPP and is contained in P with a #P oracle.
BQPBQP = BQP [BV97].
[ADH97] showed that BQP is contained in PP, and [FR98] showed that BQP is contained in AWPP.
There exist oracles relative to which:
BQP does not equal to BPP [BV97] (and by similar arguments, is not in P/poly).
BQP is not contained in MA [Wat00].
BQP is not contained in PH [RT18] (see also [Wu18]).
BQP is not contained in ModpP for prime p [GV02].
NP, and indeed NP ∩ coNP, are not contained in BQP with probability 1 relative to a random oracle and a random permutation oracle, respectively [BBB+97].
SZK is not contained in BQP [Aar02].
BQP is not contained in SZK (follows easily using the quantum walk problem in [CCD+03]).
BQP is not contained in BPPpath [Aar10] (see also [Che16]).
PPAD is not contained in BQP [Li11].
P=BQP and PH is infinite [FR98].
If P=BQP relative to a random oracle then BQP=BPP [FR98].
BQP/log: BQP With Logarithmic-Size Karp-Lipton Advice
Same as BQP/poly except that the advice is O(log n) bits instead of a polynomial number.
Contained in BQP/mlog.
BQP/poly: BQP With Polynomial-Size Karp-Lipton Advice
Is to BQP/mpoly as ∃BPP is to MA. Namely, the BQP machine is required to give some answer with probability at least 2/3 even if the advice is bad. Even though BQP/mpoly is a more natural class, BQP/poly follows the standard definition of advice as a class operator [KL82].
Contained in BQP/mpoly and contains BQP/log.
BQP/mlog: BQP With Logarithmic-Size Deterministic Merlin-Like Advice
Same as BQP/mpoly except that the advice is O(log n) bits instead of a polynomial number.
Strictly contained in BQP/qlog [NY03].
BQP/mpoly: BQP With Polynomial-Size Deterministic Merlin-Like Advice
The class of languages recognized by a syntactic BQP machine with deterministic polynomial advice that depends only on the input length, such that the output is correct with probability 2/3 when the advice is good.
Can also be defined as the class of problems solvable by a nonuniform family of polynomial-size quantum circuits, just as P/poly is the class solvable by a nonuniform family of polynomial-size classical circuits.
Referred to with a variety of other ad hoc names, including BQP/poly on occassion.
Contains BQP/qlog, and is contained in BQP/qpoly.
Does not contain ESPACE [NY03].
BQP/qlog: BQP With Logarithmic-Size Quantum Advice
Same as BQP/mlog except that the advice is quantum instead of classical.
Strictly contains BQP/mlog [NY03].
Contained in BQP/mpoly.
BQP/qpoly: BQP With Polynomial-Size Quantum Advice
The class of problems solvable by a BQP machine that receives a quantum state ψn as advice, which depends only on the input length n.
As with BQP/mpoly, the acceptance probability does not need to be bounded away from 1/2 if the machine is given bad advice. (Thus, we are discussing the class that [NY03] call BQP/*Qpoly.) Indeed, such a condition would make quantum advice unusable, by a continuity argument.
Does not contain EESPACE [NY03].
[AD14] showed that BQP/qpoly = YQP/poly.
There exists an oracle relative to which BQP/qpoly does not contain NP [Aar04b].
A classical oracle separation between BQP/qpoly and BQP/mpoly is presently unknown, but there is a quantum oracle separation [AK06]. An unrelativized separation is too much to hope for, since it would imply that PP is not contained in P/poly.
Contains BQP/mpoly.
BQP-OBDD: Polynomial-Size Bounded-Error Quantum Ordered Binary Decision Diagram
Same as P-OBDD, except that unitary (quantum) transitions are allowed and the OBDD need only accept with probability at least 2/3.
Strictly contains BPP-OBDD [NHK00].
BQPSPACE: Bounded-Error Quantum PSPACE
Equals PSPACE and PPSPACE.
BQTIME(f(n)): Bounded-Error Quantum f(n)-Time
Same as BQP, but with f(n)-time (for some constructible function f) rather than polynomial-time machines.
BQPCTC: BQP With Closed Time Curves
Same as BQP with access to two sets of qubits: causality-respecting qubits and CTC qubits.
Defined in [Aar05c], where it was shown that PSPACE is contained in BQPCTC, which in turn is contained in SQG = PSPACE. Therefore, BQPCTC = PSPACE; this was also shown in [AW09].
See also PCTC.
BQPtt/poly: BQP/mpoly With Truth-Table Queries
Same as BQP/mpoly, except that the machine only gets to make nonadaptive queries to whatever oracle it might have.
Defined in [NY03b], where it was also shown that P is not contained in BQPtt/poly relative to an oracle.
k-BWBP: Bounded-Width Branching Program
Alternate name for k-PBP.
Retrieved from "https://complexityzoo.net/index.php?title=Complexity_Zoo:B&oldid=6702"
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(1-hydroxycyclohexan-1-yl)acetyl-CoA cyclohexanone-lyase (acetyl-CoA-forming) Wikipedia
(1-hydroxycyclohexan-1-yl)acetyl-CoA cyclohexanone-lyase (acetyl-CoA-forming)
(1-hydroxycyclohexan-1-yl)acetyl-CoA lyase
In enzymology, a (1-hydroxycyclohexan-1-yl)acetyl-CoA lyase (EC 4.1.3.35) is an enzyme that catalyzes the chemical reaction
(1-hydroxycyclohexan-1-yl)acetyl-CoA
{\displaystyle \rightleftharpoons }
acetyl-CoA + cyclohexanone
Hence, this enzyme has one substrate, (1-hydroxycyclohexan-1-yl)acetyl-CoA, and two products, acetyl-CoA and cyclohexanone.
This enzyme belongs to the family of lyases, specifically the oxo-acid-lyases, which cleave carbon-carbon bonds. The systematic name of this enzyme class is (1-hydroxycyclohexan-1-yl)acetyl-CoA cyclohexanone-lyase (acetyl-CoA-forming). This enzyme is also called (1-hydroxycyclohexan-1-yl)acetyl-CoA cyclohexanone-lyase.
Ougham HJ, Trudgill PW (June 1982). "Metabolism of cyclohexaneacetic acid and cyclohexanebutyric acid by Arthrobacter sp. strain CA1". Journal of Bacteriology. 150 (3): 1172–82. PMC 216338. PMID 7076617.
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Overview of General Relativity Computations - Maple Help
Home : Support : Online Help : Mathematics : DifferentialGeometry : Overview of General Relativity Computations
An Overview of General Relativity computations with DifferentialGeometry
2. Algebraic Manipulations of Tensors
5. Curvature Tensors
6. Working with Orthonormal Tetrads
7. Working with Null Tetrads and the Newman-Penrose Formalism
8. Two Component Spinor Formalism
10. Electro-Vac Spacetimes
11. Exact Solution Database
The GRTensor package, developed by K. Lake, P. Musgrave and D. Polleny, has long provided the general relativity community with an excellent collection of command for symbolic computations in general relativity. This functionality is also now available within the DifferentialGeometry package and its subpackage Tensor, as well as within the Physics package.
This worksheet provides a quick introduction to General Relativity computations using DifferentialGeometry.
In this section we give a quick illustration of the use of the DifferentialGeometry software to perform a standard calculation in General Relativity. We shall solve the Einstein vacuum equations for spherically symmetric metrics in 4-dimensions. We check that the solution is of Petrov type D and we show that there are 4 Killing vectors. First load the DifferentialGeometry package, the Tensor subpackage and the Tools subpackage.
with(DifferentialGeometry): with(Tensor): with(Tools):
All calculations in DifferentialGeometry begin by using the DGsetup command to initialize the computational environment. For our illustration we shall use standard spherical coordinates We use the name
\mathrm{M1}
to designate the computational environment (in this case a 4-dimensional manifold) which we are creating.
DGsetup([t, r, theta, phi], M1, verbose);
\textcolor[rgb]{0,0,1}{\mathrm{The following coordinates have been protected:}}
{\textstyle \left[\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{r}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{\varphi }}\right]}
\textcolor[rgb]{0,0,1}{\mathrm{The following vector fields have been defined and protected:}}
{\textstyle \left[\textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''vector''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\left[\textcolor[rgb]{0,0,1}{}\right]\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{1}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\right]\right]\right)\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''vector''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\left[\textcolor[rgb]{0,0,1}{}\right]\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{2}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\right]\right]\right)\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''vector''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\left[\textcolor[rgb]{0,0,1}{}\right]\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{3}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\right]\right]\right)\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''vector''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\left[\textcolor[rgb]{0,0,1}{}\right]\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{4}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\right]\right]\right)\right]}
\textcolor[rgb]{0,0,1}{\mathrm{The following differential 1-forms have been defined and protected:}}
{\textstyle \left[\textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''form''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{1}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\right]\right]\right)\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''form''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{2}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\right]\right]\right)\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''form''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{3}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\right]\right]\right)\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''form''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{4}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\right]\right]\right)\right]}
\textcolor[rgb]{0,0,1}{\mathrm{frame name: M1}}
The coordinate vectors and the 1-forms are now defined and these can be used to construct general vector fields, differential forms and tensors. Use +/- for addition and subtraction, * for scalar multiplication, &t for tensor product and &w for wedge product. The command evalDG is used to insure that these operations are properly performed in DifferentialGeometry.
Here is the usual form for the static (time-independent), rotational invariant metric that is used as the starting ansatz for solving the Einstein equations. The two functions
A\left(r\right)
B\left(r\right)
will be determined by the Einstein equations
g := evalDG( A(r) *dt &t dt + B(r)* dr &t dr + r^2*(dtheta &t dtheta + sin(theta)^2* dphi &t dphi));
{\textstyle \textcolor[rgb]{0,0,1}{\mathrm{\_DG}}\textcolor[rgb]{0,0,1}{}\left(\left[\left[\textcolor[rgb]{0,0,1}{''tensor''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{\mathrm{M1}}\textcolor[rgb]{0,0,1}{\,}\left[\left[\textcolor[rgb]{0,0,1}{''cov\_bas''}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{''cov\_bas''}\right]\textcolor[rgb]{0,0,1}{\,}\left[\textcolor[rgb]{0,0,1}{}\right]\right]\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\left[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{1}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\right)\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{2}\right]\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{r}\right)\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{3}\right]\textcolor[rgb]{0,0,1}{\,}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\right]\textcolor[rgb]{0,0,1}{\,}\left[\left[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{\,}\textcolor[rgb]{0,0,1}{4}\right]\textcolor[rgb]{0,0,1}{\,}{\textcolor[rgb]{0,0,1}{r}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\θ}}\right)}^{\textcolor[rgb]{0,0,1}{2}}\right]\right]\right]\right)}
To solve the Einstein equations for this metric, we first calculate the Einstein tensor.
Ein := EinsteinTensor(g);
xmlns="http://www.w3.org/1998/Math/MathML">_DGtensor,M1,con_bas,con_bas,,1,1,−Br2+diffBr,rr−BrArr2Br2,2,2,−ArBr−diffAr,rr−ArA
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Erdős–Szekeres-type statements: Ramsey function and decidability in dimension 1
15 September 2014 Erdős–Szekeres-type statements: Ramsey function and decidability in dimension
1
Boris Bukh, Jiří Matoušek
A classical and widely used lemma of Erdős and Szekeres asserts that for every
n
N
N
-term sequence
\underline{a}
of real numbers contains an
-term increasing subsequence or an
-term nonincreasing subsequence; quantitatively, the smallest
N
with this property equals
\left(n-1{\right)}^{2}+1
. In the setting of the present paper, we express this lemma by saying that the set of predicates
\mathbit{\Phi }=\left\{{x}_{1}<{x}_{2},{x}_{1}\ge {x}_{2}\right\}
is Erdős–Szekeres with Ramsey function
{ES}_{\mathbit{\Phi }}\left(n\right)=\left(n-1{\right)}^{2}+1
In general, we consider an arbitrary finite set
\mathbit{\Phi }=\left\{{\Phi }_{1},\dots ,{\Phi }_{m}\right\}
of semialgebraic predicates, meaning that each
{\Phi }_{j}={\Phi }_{j}\left({x}_{1},\dots ,{x}_{k}\right)
is a Boolean combination of polynomial equations and inequalities in some number
k
of real variables. We define
\mathbit{\Phi }
to be Erdős–Szekeres if for every
n
N
N
\underline{a}
of real numbers has an
-term subsequence
\underline{b}
such that at least one of the
{\Phi }_{j}
holds everywhere on
\underline{b}
{\Phi }_{j}\left({b}_{{i}_{1}},\dots ,{b}_{{i}_{k}}\right)
holds for every choice of indices
{i}_{1},{i}_{2},\dots ,{i}_{k}
1\le {i}_{1}<{i}_{2}<\cdots <{i}_{k}\le n
{ES}_{\mathbit{\Phi }}\left(n\right)
for the smallest
N
with the above property.
We prove two main results. First, the Ramsey functions in this setting are at most doubly exponential (and sometimes they are indeed doubly exponential): for every
\mathbit{\Phi }
that is Erdős–Szekeres, there is a constant
C
{ES}_{\mathbit{\Phi }}\left(n\right)\le {2}^{{2}^{Cn}}
. Second, there is an algorithm that, given
\mathbit{\Phi }
, decides whether it is Erdős–Szekeres; thus,
1
-dimensional Erdős–Szekeres-style theorems can in principle be proved automatically.
We regard these results as a starting point in investigating analogous questions for
d
-dimensional predicates, where instead of sequences of real numbers, we consider sequences of points in
{\mathbb{R}}^{d}
(and semialgebraic predicates in their coordinates). This setting includes many results and problems in geometric Ramsey theory, and it appears considerably more involved. Here we prove a decidability result for algebraic predicates in
{\mathbb{R}}^{d}
(i.e., conjunctions of polynomial equations), as well as for a multipartite version of the problem with arbitrary semialgebraic predicates in
{\mathbb{R}}^{d}
Boris Bukh. Jiří Matoušek. "Erdős–Szekeres-type statements: Ramsey function and decidability in dimension
1
." Duke Math. J. 163 (12) 2243 - 2270, 15 September 2014. https://doi.org/10.1215/00127094-2785915
Boris Bukh, Jiří Matoušek "Erdős–Szekeres-type statements: Ramsey function and decidability in dimension
1
," Duke Mathematical Journal, Duke Math. J. 163(12), 2243-2270, (15 September 2014)
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Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : IDBasis : constructor
construct an IDBasis object
IDBasis(S, Vs)
a LHPDE object that is of finite type (see IsFiniteType)
a list of Vectors, a Matrix, a list of linear combination of parametric derivatives of S, or the string "standardBasis", representing a change-of-basis for the initial data of S
Let S be a LHPDE object of finite type whose solution dimension is r. The IDBasis command constructs an IDBasis object representing a basis for the initial data of S.
The IDBasis object is managed with respect to a fixed standard basis, namely the parametric derivatives of the LHPDE object S.
The second input argument, Vs, represents a change-of-basis matrix to give a new basis.
If Vs is a list of Vectors, then there must be r Vectors of length r in the list.
If Vs is a Matrix, then it must be r x r invertible matrix.
If Vs is a list of linear combination of parametric derivatives of S, then there must be r items in the list.
The call IDBasis(S, "standardBasis") returns a IDBasis object containing the standard initial data basis (i.e. the change-of-basis matrix is the identity matrix).
Some methods become available once a valid IDBasis object is constructed. See Overview of the IDBasis object for more detail.
This command can be used in the form IDBasis(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used by executing LieAlgebrasOfVectorFields:-IDBasis(...).
\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):
\mathrm{Typesetting}:-\mathrm{Suppress}\left([\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)]\right):
\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):
\mathrm{E2}≔\mathrm{LHPDE}\left([\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0],\mathrm{dep}=[\mathrm{\xi },\mathrm{\eta }],\mathrm{indep}=[x,y]\right)
\textcolor[rgb]{0,0,1}{\mathrm{E2}}\textcolor[rgb]{0,0,1}{≔}[{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{indep}}\textcolor[rgb]{0,0,1}{=}[\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}{\mathrm{dep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
\mathrm{ParametricDerivatives}\left(\mathrm{E2}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
The solution dimension of E2 is 3, so we must have three vectors of length 3.
B≔\mathrm{IDBasis}\left(\mathrm{E2},[〈1,0,0〉,〈-y,-x,-1〉,〈0,1,0〉]\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}]
B looks like a list but it is really an IDBasis object.
\mathrm{type}\left(B,'\mathrm{list}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left(B,'\mathrm{IDBasis}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The IDBasis can be constructed from a change-of-basis 3 by 3 matrix
\mathrm{IDBasis}\left(\mathrm{E2},\mathrm{Matrix}\left([[1,-y,0],[0,-x,1],[0,-1,0]]\right)\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}]
... or can be constructed from three linear combinations of the parametric derivatives of E2
\mathrm{IDBasis}\left(\mathrm{E2},[\mathrm{\xi }\left(x,y\right)-y\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{\eta }\left(x,y\right)-x\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)]\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}]
Or we can construct a standard IDBasis object whose change-of-basis is the identity matrix.
\mathrm{IDBasis}\left(\mathrm{E2},"standardBasis"\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
The LieAlgebrasOfVectorFields[IDBasis] command was introduced in Maple 2020.
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The Zeno - Lightbook
A medium of exchange for digital assets
For a thing of value – an asset – to be traded, it needs a medium of exchange. In the blockchain space, the nature of ‘asset’ and ‘coin’ is often confused, since most cryptocurrencies treat coins as the assets themselves. With a real economy in the digital space, with distinct assets in the form of digital files, the coins become a medium that enables asset trade.
Implementing this approach avoids the paradox whereby the ‘token economics’ has no basis in the value of assets (and uses as a medium of exchange the ‘thing of value’ itself – the token (or coin) – creating something of a circular economic motivation). With proper blockchain governance of files and the cryptocurrency used as a medium of exchange, tokenomics can refer to the actual economics of a cryptocurrency; one that is used to pay for goods & services.
Zenotta’s tokenomic policy emphasises fairness and maximization of distribution, with long mining timelines (approx. 200 yrs), a large number of coins (10 billion) and a more inclusive mining algorithm (see Chapter 4). The desire or demand for assets rather than purely for coins creates a highly liquid market, and since the Zeno is the only currency traded for Smart Data, there is a strong incentive to use the currency rather than merely ‘hodl’ it. This increases distribution drastically over standard cryptocurrencies.
Zeno issuance follows a fixed supply cap mechanic according to the properties outlined in the following table:
The total number of Zeno coins reached will be 10 billion, with 2.5 billion reserved for a treasury, to be used for stakeholders, a development fund, and an economic activity fund. The remaining 7.5 billion is mined out via a smoothed issuance curve based on the emission approach employed by the CryptoNote protocol. The smoothed issuance is designed to minimize the volatility seen in standard halving mechanics whereby the reward drops suddenly by half on a particular date. The Bitcoin stock-to-flow model lends support to the idea that this sudden reduction in supply is at least partly responsible for the extreme bull market/bear market cycles.
The sub-unit of the Zeno is the zent. A number with a large number (90) of divisors was chosen as the conversion factor between the zent and the Zeno:
This particular conversion factor is the 24th antiprime, or highly composite number, in the sequence of positive integers with a greater number of divisors than any smaller positive integer. This antiprime number has 90 divisors, and 9 prime factors. The use of a number with a large number of divisors facilitates the use of fractional payments without the need for rounding. At the same time we stay within the bounds of the u64 integer type employed in our codebase for the total cap in zenoshis of 10 billion x 25200 = 2.52e14.
The block reward in zenoshis is calculated using the total Zeno supply and the current Zeno supply in the market via the following formula:
The equation determining Zeno issuance, based on the CryptoNote protocol
In a practical network protocol, issuance must be implemented using bitwise operators, in order to ensure consistent performance across different hardware types. This allows operations to be performed on the bit level and therefore proceed at the maximum possible speed. Additionally, this approach ensures the consistency of floating point operations across different architecture types. In Bitcoin and bitcoin-like protocols this manifests requiring the block reward to drop by half (the ‘halving’) every n blocktimes. This is due to the bitshift operator being applied to the block reward (the left-hand side of the equation). We apply it to the recursive right-hand side of the issuance equation, which allows for a smoothed curve without the sudden halving jumps that likely have undesirable economic properties due to sudden supply shocks. Therefore, the block reward in zenoshis (for a 60 second blocktime) is given in practical terms by
\text{reward} = (\text{total supply − current supply}) >> 25
The issuance curve is shown in graphical form in the figure below. The emission is divided into two parts, with the miners taking the majority and the compute and storage nodes being allocated a small percentage that varies over time.
The issuance schedule for the Zeno in terms of the block reward, divided into miners (blue line) and the compute & storage nodes (magenta line) with the total for the network (black line). The issuance is smooth, rather than proceeding via the standard halving mechanic, in order to minimize the chance of extreme bull/bear market cycles driven by supply/demand shocks.
The fraction of the issuance allocated to the compute and storage nodes follows a reverse logistic function, starting at 15% and varying smoothly to 5% over a period of approx. 100 years. This is to reward the compute and storage nodes for the larger role that they play in the initial years of the network, and to recoup large capital costs, and to incentivize miners to play a larger role in the network later on.
Fraction of issuance allocated to the compute and storage nodes (magenta line in Figure 13) with time. The fraction allocation follows a reverse logistic function from an initial value of 15% and levelling out at 5%.
The form of this function is shown in the figure above. The total number of Zeno coins mined over time is shown in the figure below, along with the specific number mined by the mining, compute and storage nodes.
The total number of Zeno coins mined with time, showing the mining hard cap limit of 7.5 billion (the hard cap limit minus the treasury allocation; 10 billion - 2.5 billion). The black line shows the total earned by the network through mining, divided into two groups: the miners (blue line) and the compute & storage nodes (magenta line).
The economic policy of Zenotta is focused around the goals of (i) coin value as a function of real economic activity and (ii) maximization of distribution. The Zenotta data protocol that allows for the creation of Smart Data assets is intrinsically linked to the Zenotta network protocol and the miners via a Socratic approach to democracy – if you put the work in to help secure the ledger, you earn the right to create tradeable assets. This keeps incentives aligned and ensures a healthy level of informed participation.
An economy based on (distributed) goods & services
Boom-bust cycles in the mainstream global economy are largely due to a form of money creation that is funnelled into financial speculation rather than the creation of goods & services or productive innovation. Financial crashes – the ‘popping of the bubble’ – harm everyone, even those who were the architects of the bubble, unless they are among the lucky few to see it coming and get out at the right time. The trend of the ‘too big to fail’ banks creating money for risky, speculative investments that do not increase GDP has eroded the function of banks from helping the economy to harming it.
“More than 70% of all lending – actually way more than that – is money creation for financial transactions; for asset transactions; for purchasing ownership rights. Now, then you have a problem. Why? Because you are creating new money, but you are not creating new goods & services. You are simply giving somebody new purchasing power over existing assets, and therefore you must push up asset prices [...] and that also creates the inequality – when the banking sector has focused too much on unproductive lending."
– Prof. Richard Werner, episode of RT UK.
The crypto economy is still new and largely untested; however, it is thriving. The problem, though, is that money creation in the crypto economy has similar problems to those described above – when new tokens are created, usually as part of the block reward, there is no corresponding increase in goods services, beyond the function of the relevant blockchain network as an effective, fair, and efficient money transmitter. Ethereum’s world computer (and subsequent variants that have emerged) can arguably be said to funnel money creation into innovation (although most of that innovation to date has been to develop more ways to create money or increase speculative activity).
In the Smart Data economy provided by Zenotta, the creation of new money through mining can quickly and easily feed into the creation of Smart Data goods and services. Smart Data is itself created through the process of either mining or verifying the ledger, through a system of create credits whereby miners or those running a full verifying node are awarded the right to create Smart Data assets of their choosing (in other words, they are awarded with the ability to convert files into Smart Data files that can be traded on the Zenotta blockchain). In order to facilitate trade of Smart Data assets (goods & services) the Zenotta blockchain ledger is a new design of ledger that incorporates a dual double entry accounting of both the payment and the asset, such that the trade proceeds two-way, with both halves notarized by the blockchain. This makes the Zenotta blockchain a blockchain for trade rather than merely for payments.
The nature of data assets; namely, files, as the digital good being traded for in the blockchain ledger ensures a real and considerable level of a priori distribution and de- centralization. Everyone with a computer has a substantial number of files, and through the create credits system of validating or mining a small part of the network, coupled with the balanced mining approach outlined in the next section that reduces the vast inequality in mining power to acceptable limits, access to the ‘thing of value’ – the file – is far more distributed than any other type of good or service in any other economy (crypto or otherwise). Therefore, the economy provided by the Zenotta Digital System contains within it the ideal of equality of opportunity from the very start.
The value of an asset in any economy (for example a coin, or a token) can be expressed by the well-known formula below:
\text{token value} = \frac{\text{economic activity}}{\substack{{\text{circulating}}\ {\text{token supply}}} \times \substack{{\text{consumer}}\ {\text{token velocity}}}}
In the crypto space, the numerator of this equation is something of a problem. Since there is no (or very little) economic activity by the usual definition, assigning value to a crypto coin is difficult. Ultimately it is set by the market, but this is a price-based definition of value rather than a fundamental one. In the Zenotta digital economy, Zeno coins are used to purchase Smart Data and to provide the ‘gas’ for Smart Data contracts. With the Zeno coin, the economic activity is real, and therefore the value of the Zeno has a direct link to the value of the Smart Data being traded. The total value in the Zeno currency is therefore more akin to that of GDP, namely the GDP of the Smart Data economy.
A distributed mint
For a blockchain-based economy the ‘mint’ is not a central body. Coins are created through the Proof-of-Work (PoW) consensus protocol and awarded to the miners upon winning the block. In this way we have a distributed mint, and so coin distribution begins in an already distributed state (relative to a central bank driven economy). We further optimise this distribution at the mint level by employing a thermodynamic-like protocol that brings the mining power of the entities in the network closer to balance, via a peer-to-peer algorithm. This approach increases the decentralization and distribution of mining power, which increases the security and the efficiency of the network, as well as the decentralization and distribution of the mining reward.
The full technical description of the balancing protocol is the subject of a separate paper, but briefly, the block processing power of the network (the ‘node temperature’) is moved towards a homogeneous distribution by decreasing the effective hashrate of the ‘hot’ nodes (e.g., the ASICs) and increasing it for the ‘cold’ nodes (e.g., the CPUs). This is done smoothly, allowing our node temperature quantity η to change between nodes pairwise using a simple differential equation, which for miner A takes the form:
\frac{d\eta_A}{dt} = -\alpha \Delta \eta
where ∆η ≡ηA−ηB, and α is a normalisation constant. As a result, the algorithm avoids a centralised controller and operates autonomously.
The desired end-state of the network is not one where the effective hashrate is completely balanced (which would bring its own undesirable economic properties and increase the attack surface for Sybil attacks on the network) but rather in-between the two extremes of the proportional model (where the probability of finding a block is proportional to hashrate) and the homogeneous model (where the probability of finding a block is equal for all participants) at a point where distribution is maximised and excess energy usage is minimised without increasing the net attack surface.
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EUDML | Signatures of singular branched covers. EuDML | Signatures of singular branched covers.
Signatures of singular branched covers.
Gilmer, Patrick. "Signatures of singular branched covers.." Mathematische Annalen 295.4 (1993): 643-660. <http://eudml.org/doc/165063>.
@article{Gilmer1993,
author = {Gilmer, Patrick},
keywords = {branched cyclic -fold covers; Hirzebruch -class; branched cyclic cover of P along a collection of hyperplanes; closed smooth oriented manifold; branching index; rational homology manifolds; eigenspace signatures; singular algebraic varieties; stable tangent bundle},
title = {Signatures of singular branched covers.},
AU - Gilmer, Patrick
TI - Signatures of singular branched covers.
KW - branched cyclic -fold covers; Hirzebruch -class; branched cyclic cover of P along a collection of hyperplanes; closed smooth oriented manifold; branching index; rational homology manifolds; eigenspace signatures; singular algebraic varieties; stable tangent bundle
branched cyclic
d
-fold covers, Hirzebruch
L
-class, branched cyclic cover of
ℂ
\left(2k\right)
along a collection of hyperplanes, closed smooth oriented manifold, branching index, rational homology manifolds, eigenspace signatures, singular algebraic varieties, stable tangent bundle
Articles by Patrick Gilmer
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Deconvolution and polynomial division - MATLAB deconv - MathWorks Nordic
[q,r] = deconv(u,v)
[q,r] = deconv(u,v) deconvolves a vector v out of a vector u using long division, and returns the quotient q and remainder r such that u = conv(v,q) + r. If u and v are vectors of polynomial coefficients, then deconvolving them is equivalent to dividing the polynomial represented by u by the polynomial represented by v.
Create two vectors u and v containing the coefficients of the polynomials
2{x}^{3}+7{x}^{2}+4x+9
{x}^{2}+1
, respectively. Divide the first polynomial by the second by deconvolving v out of u, which results in quotient coefficients corresponding to the polynomial
2x+7
and remainder coefficients corresponding to
2x+2
row or column vectors
Input vectors, specified as either row or column vectors. u and v can be different lengths or data types.
If one or both of u and v are of type single, then the output is also of type single. Otherwise, deconv returns type double.
The lengths of the inputs should generally satisfy length(v) <= length(u). However, if length(v) > length(u), then deconv returns the outputs as q = 0 and r = u.
q — Quotient
Quotient, returned as a row or column vector such that u = conv(v,q)+r.
r — Remainder
Remainder, returned as a row or column vector such that u = conv(v,q)+r.
conv | residue
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Hydraulic variable orifice shaped as rectangular slot - MATLAB - MathWorks í•œêµ
Orifice with Variable Area Slot
Hydraulic variable orifice shaped as rectangular slot
The block models a variable orifice created by a cylindrical sharp-edged spool and a rectangular slot in a sleeve. The flow rate through the orifice is proportional to the orifice opening and to the pressure differential across the orifice. The flow rate is determined according to the following equations:
q={C}_{D}â
A\left(h\right)\sqrt{\frac{2}{\mathrm{Ï}}}\frac{\mathrm{Î}p}{{\left(\mathrm{Î}{p}^{2}+{p}_{\text{Cr}}^{2}\right)}^{1/4}},
\mathrm{Î}p={p}_{\text{A}}â{p}_{\text{B}},
h={x}_{0}+x·or
A\left(h\right)=\left\{\begin{array}{ll}{A}_{leak}\hfill & \text{if }bâ
h<{A}_{leak}\hfill \\ bâ
h\hfill & \text{otherwise }\hfill \end{array}
A(h) Instantaneous orifice passage area
b Width of the orifice slot
x0 Initial opening
or Orifice orientation indicator. The variable assumes +1 value if a spool displacement in the globally assigned positive direction opens the orifice, and –1 if positive motion decreases the opening.
{p}_{cr}=\frac{\mathrm{Ï}}{2}{\left(\frac{{\mathrm{Re}}_{cr}â
\mathrm{ν}}{{C}_{D}â
{D}_{H}}\right)}^{2}
{D}_{H}=\sqrt{\frac{4A}{\mathrm{Ï}}}
DH Instantaneous orifice hydraulic diameter
The block positive direction is from port A to port B. This means that the flow rate is positive if it flows from A to B and the pressure differential is determined as
\mathrm{Î}p={p}_{\text{A}}â{p}_{\text{B}},
. Positive signal at the physical signal port S opens or closes the orifice depending on the value of the parameter Orifice orientation.
The width of the rectangular slot. The default value is 1e-2 m.
Annular Orifice | Constant Area Hydraulic Orifice | Fixed Orifice | Orifice with Variable Area Round Holes | Variable Area Hydraulic Orifice | Variable Orifice
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O
X
O
Andy and Bob are playing a game on a
9 \times 9
checkerboard. They take turns to move a move: on his turn, Andy places an X in an empty square while Bob places an O. When the entire checkerboard is filled, Andy scores a point for each row or column that contains more X's than O's, while Bob scores a point for each row or column that contains more O's than X's. The winner of the game is the person with (strictly) more points.
Given that Andy makes the first move, who has the winning strategy?
Note: If they both scored 9 points, then it is considered a draw.
Andy Neither Bob Insufficient information
Daniel and Cody are playing a game on an infinite unit square grid. Daniel places unit square sticky notes with a letter
D
, while Cody places unit square sticky notes with a letter
C
. They take turns placing sticky notes on squares in the grid, with Daniel going first. Cody wants to have four of his sticky notes form the corners of a perfect square with sides parallel to the grid lines, while Daniel wants to prevent him from doing so.
Can Daniel succeed?
Assume that both players make optimal moves.
Yes It depends No
by Daniel Liu
Alice and Carla are playing a game often learned in elementary school known as Say 16. The rules for the game are as follows:
Each player takes turns saying between 1 and 3 consecutive numbers, with the first player starting with the number 1. For example, Player 1 could say the numbers 1 and 2, then Player 2 can say "3, 4, 5", then Player 1 can say "6" and so on.
The goal of the game is to be the one to say "16".
Carla decides that she'll go first and that Alice will go second. Is there a way to tell which player is going to win before the game even starts?
Yes, Carla will win No, there is no way to tell who will win Yes, Alice will win
You're playing tic tac toe with an opponent who is
X
while you are
O
The initial gameplay is as shown below:
Assuming that your opponent will play optimally from now on, where must you place your next mark in order to obtain a winning position?
Upper-right corner Either upper-left corner or lower-right corner Either left side or lower side Middle tile
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dual number - Wiktionary
dual number (plural dual numbers)
(grammar) Grammatical number denoting a quantity of exactly two.
1840, George Grey, A Vocabulary of the Dialects of South Western Australia, T. & W. Boone, 2nd Edition, page xi,
This is particularly the case with regard to the pronouns of the dual number, which form one of the peculiarities of the language.
1998, William Blake Tyrrell, Larry J. Bennett, Recapturing Sophocles' Antigone, Rowman & Littlefield, page 43,
Eteocles and Polyneices are enemy brothers, that is, they are alike in spite of the difference erected by being on opposite sides of the Theban battlements.1 By uniting them with the dual number in the description of their deaths, Sophocles has the elders speak of them as indistinguishable in their hatred, spear work, and deaths.
2009, Giuliano Lancioni, Formulaic models and formulaicity in Classical and Modern Standard Arabic, Roberta Corrigan, Edith A. Moravcsik, Hamid Ouali, Kathleen M. Wheatley (editors), Formulaic Language, Volume 1: Distribution and historical change, John Benjamins Publishing Company, page 225,
Classical Arabic has a fully functional dual number, which requires dual agreement in verbs, adjectives and pronouns.
(algebra) An element of an algebra (the algebra of dual numbers) which includes the real numbers and an element ε which satisfies ε ≠ 0 and ε² = 0.
Hypernym: hypercomplex number
If f(x) is a polynomial or a power series then its derivative can be obtained “automatically” using dual numbers as follows:
{\displaystyle f'(x)={f(x+\epsilon )-f(x) \over \epsilon }}
{\displaystyle f(x+\epsilon )}
should be expanded using the algebra of dual numbers. The role of the ε is therein similar to that of an infinitesimal.
1993, Ivan Kolář, Peter W. Michor, Jan Slovák, Natural Operations in Differential Geometry, Springer, page 336,
In the special case of the algebra
{\displaystyle \mathbf {D} }
of dual numbers we get the vertical tangent bundle
{\displaystyle V}
1996, Roger Cooke (translator), Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevich, Mathematics of the 19th Century: Geometry, Analytic Function Theory, Birkhäuser, page 87,
These numbers are now called dual numbers (a term due to Study) and can be defined as expressions of the form
{\displaystyle a+b\epsilon }
{\displaystyle \epsilon ^{2}=0}
. When this is done as Kotel'nikov and Study showed, the dual distance between two points of this sphere representing two lines, forming an angle
{\displaystyle \varphi }
and having shortest distance
{\displaystyle d}
, is equal to the dual number
{\displaystyle \varphi +\epsilon d}
, while the motions of Euclidean space are represented by rotations of the dual sphere, and consequently depend on three dual parameters.
2002, Jiří Tomáš, Some Classification Problems on Natural Bundles Related to Weil Bundles, Olga Gil-Medrano, Vicente Miquel (editors), Differential Geometry, Valencia 2001: Proceedings of the International Conference, World Scientific, page 297,
Thus we investigate natural
{\displaystyle T}
-functions defined on
{\displaystyle T^{*}T^{\mathbf {D} \otimes A}M}
to determine all natural operators
{\displaystyle T\to TT^{*}T^{A}}
{\displaystyle \mathbf {D} }
denoting the algebra of dual numbers.
grammatical number denoting a quantity of exactly two — see dual
element of the algebra of dual numbers
Polish: liczba dualna f
Welsh: (please verify) rhif deuol m
Dual (grammatical number) on Wikipedia.Wikipedia
Dual number on Wikipedia.Wikipedia
Retrieved from "https://en.wiktionary.org/w/index.php?title=dual_number&oldid=54187731"
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Per Capita GDP and Living Standards - Course Hero
Macroeconomics/Economic Growth/Per Capita GDP and Living Standards
A nation's real GDP, the total output of the economy adjusted for inflation, is often considered a measure of its income because the production of goods and services generates income for workers and owners of firms. When a customer spends two dollars to purchase a bottle of soda, each penny of that two dollars goes to someone in the form of income: the money paid to the producers of the soda and the bottle, the employees of the store, the driver who delivered it from the factory to the point of purchase, the stockholders of the soda company. Therefore, the growth rate of real GDP can be used to see what is happening to output in an economy as well as to measure income.
To look at how economic growth impacts the people living in a country, it may be more useful to consider how living standards in a country have changed. This refers to the level of material comfort that people living in a country have, such as the quality and availability of food and housing. The real GDP per capita reflects living standards because it is a measure of the output per person in a country and is calculated by dividing real GDP (RGDP) by the population. This gives a better idea of the average level of income in the economy. For example, the 2017 GDP of the United States was $18.5 trillion, whereas the GDP of Luxembourg was only $63.5 billion. However, the United States has a much larger population than Luxembourg, so a smaller GDP doesn't mean that people are worse off. In fact, the per capita GDP in Luxembourg was $108,000, while the per capita GDP in the United States was just over half that at $57,000. This shows the importance of having different measures to discuss an economy and to understand what that means for the population.
\text{Real GDP per Capita}=\frac{\text{RGDP}}{\text{Population}}
When considering the impact on individuals in an economy, economic growth is measured by the percentage change in real GDP per capita from one year (
t
t + 1
\text{Rate of Growth from Year}\;t\;\text{to Year }t+1=\frac{\text{RGDP per Capita}_{t+1}-\text{RGDP per Capita}_{t}}{\text{RGDP per Capita}_{t}}\times100\%
However, even real GDP per capita does not provide a complete picture of the well-being of all residents within a country. Not every person in an economy has the same income, so some people will make more than the real GDP per capita and others will make less. Depending on the distribution of incomes in a country, some people may be much better or worse off than what is reflected in the real GDP per capita. Other measurement standards that account for other aspects of the economy, such as equal wealth distribution throughout the population, provide a more complete description of the standard of living for a country's population.
Economic growth is important in determining future living standards of a population. It indicates developments such as technological improvements, more use of natural resources, and increased worker productivity, all of which can increase the real GDP. If a country has a growing population, then the real GDP must increase in order for the real GDP per capita to even stay the same. If the real GDP does not grow, or if it grows at a slower rate than the population, the real GDP per capita will go down and living standards will decline accordingly because people will, on average, have less purchasing power.
Even without a growing population, economic growth is important for maintaining and improving living standards. Increasing the real GDP enables people on the low end of the income distribution to be lifted out of poverty and allows the government to increase spending on improving infrastructure. It can also allow more people access to technological developments, such as electricity or cars during the 20th century in the United States, which then become normalized and raise the expected standard of living in a country.
<The Business Cycle>The Rule of 70
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The act of "unlevering" beta involves extracting the impact of debt obligations of a company before evaluating an investment's risk in comparison to the market. Unlevered beta is considered useful because it helps to show a firm's asset risk compared with the overall market. For this reason, the unlevered beta is also sometimes referred to as the asset beta.
Beta is unlevered through the following calculation:
\frac{ \text{Levered Beta}}{1\ +\ \big((1\ -\ \text{Tax Rate})\ \times\ \big(\frac{\text{Total Debt}}{\text{Total Equity}}\big)\big)}
1 + ((1 − Tax Rate) × (Total EquityTotal Debt))Levered Beta
Beta vs. Unlevered Beta
In technical terms, beta is the slope coefficient of a publicly traded stock’s returns that have been regressed against market returns (usually the S&P 500). By regressing a stock's return relative to a broad index, investors are able to gauge how sensitive any given security might be to macroeconomic risks.
As companies add more and more debt, they are simultaneously increasing the uncertainty of future earnings. In essence, they are assuming financial risk that is not necessarily representative of the type of market risk that beta aims to capture. Investors can then accommodate for this phenomenon by extracting the potentially misleading debt impact. Since a company with debt is said to be leveraged, the term used to describe this extraction is called "unlevering."
All of the information necessary to unlever beta can be found in a company's financial statements where tax rates and the debt/equity ratio can be calculated. In general, the unlevered beta will always be lower than the standard, levered beta if the company has debt. Extracting the debt from the equation through the use of the debt/equity ratio will always result in a lower risk beta.
Investors can easily calculate the asset beta using the equation above. The unlevered beta is typically used in analysis alongside the levered beta to compare the risk of a stock to the market. Just like with the levered beta, the baseline is 1. When the levered beta is unlevered it may move closer to 1 or even be less than 1.
Professional analysts in investment management, investment banking, and equity research often use the asset beta to build out modeling reports that provide more than just a basic scenario. Thus, anytime beta is used in a modeling project’s calculations, an additional scenario can be added which shows effects with the asset beta.
Overall, the use of asset beta provides a deeper analysis of a company’s risks with a focus on the company’s debt. It is important when making any type of beta comparison that the betas aren’t confused and identical metrics are used. Typically, the use of asset beta is also further accompanied by a look at the company’s debt. If a company has a relatively higher debt to equity ratio but the debt is AAA-rated it may be of less concern than a high debt to equity ratio with a junk debt issuance.
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Uplink sounding reference signal - MATLAB lteSRS - MathWorks América Latina
lteSRS
Generate Uplink SRS Values
Generate SRS Symbols for Two Antennas
Uplink sounding reference signal
seq = lteSRS(ue,chs)
[seq,info] = lteSRS(ue,chs)
seq = lteSRS(ue,chs) returns a complex matrix, seq, containing uplink sounding reference signal (SRS) values and information structure array given structures containing UE-specific settings, and signal transmission configuration settings. For more information, see SRS Processing and TS 36.213 [1], Section 8.2.
[seq,info] = lteSRS(ue,chs) also returns an SRS information structure array, info.
This example generates SRS values for 1.4 MHz bandwidth using the default SRS configuration.
chs.CyclicShift = 0;
chs.SeqGroup = 0;
chs.SeqIdx = 0;
Generate Uplink SRS resource element values.
srs = lteSRS(ue,chs);
srs(1:4)
Generate the SRS symbols for two transmit antenna paths. Display the information structure.
Initialize UE-specific and channel configuration structures (ue and chs) for 3 MHz bandwidth and two antennas using the default SRS configuration. Generate SRS symbols and the information structure (ind and info).
[ind,info] = lteSRS(ue,chs);
Since there are two antennas, the SRS symbols are output as a two column vector and the info output structure contains two elements.
SeqIdx: 0
RootSeq: -1
NZC: -1
UE-specific settings, specified as a structure containing these following fields.
CyclicPrefixUL — Cyclic prefix length for uplink
Cyclic prefix length for uplink, specified as 'Normal' or 'Extended'.
Initial frame number, specified as a nonnegative integer.
Example: 'TDD'
Uplink or downlink configuration, specified as an integer from 0 to 6. Only required for TDD duplex mode.
Special subframe configuration, specified as an integer from 0 to 9. Only required for TDD duplex mode.
CyclicPrefix — Cyclic prefix length in the downlink
Cyclic prefix length in the downlink, specified as 'Normal' or 'Extended'.
BWConfig — SRS bandwidth configuration
SRS bandwidth configuration, specified as an integer from 0 to 7. (CSRS)
UE-specific SRS bandwidth, specified as an integer from 0 to 3. (BSRS)
7 (default) | integer from 0 to 644 | optional
UE-specific cyclic shift, specified as an integer from 0 to 7. (
{n}_{SRS}^{cs}
SeqGroup — SRS sequence group number
0 (default) | integer from 0 to 29 | optional
SRS sequence group number, specified as an integer from 0 to 29. (u)
Base sequence number, specified as either 0 or 1. (v)
seq — Uplink SRS values
Uplink SRS values, returned as a complex matrix. The symbols for each antenna are in the columns of the matrix, seq. The symbols for each antenna are in the columns of seq, with the number of columns determined by the number of transmission antennas configured. For more information, see SRS Processing.
Reference signal cyclic shift, returned as a numeric scalar. (α)
SRS sequence group number, returned as an integer from 0 to 29. (u)
Base sequence number, returned as 0 or 1. (v)
Root Zadoff-Chu sequence index, returned as an integer. (q)
{N}_{ZC}^{RS}
lteSRSIndices | lteSRSInfo | lteCellRS | lteCSIRS | lteDMRS | ltePRS
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They place three heaps of coins on a table. The first heap has
10
coins, the second heap has
7
coins, and the third heap has
9
The second player adds another pile of coins on the table, having at most
10
n
\left(P_{n}, P_{n-1}, \cdots P_1 \right)
with the strict order of superiority
\left(P_{n} > P_{n-1} > \cdots > P_1 \right)
They find 100 gold coins. They must distribute the coins among themselves according to the following rules:
The superior-most surviving pirate proposes a distribution.
The pirates, including the proposer, then vote on whether to accept this distribution.
In case of a tie vote the proposer has the casting vote.
If the distribution is accepted, the coins are disbursed and the game ends. If not, the proposer is thrown overboard from the pirate ship and dies (leaving behind
(n-1)
pirates and their original ordering), and the next most superior pirate makes a new proposal to begin the game again.
Every pirate knows that every other pirates is perfectly rational, which means:
They will always be able to deduce something logically deducible.
Their first preference is that they want to survive.
Their second preference is that they want to maximize the number of coins they receive.
Other things remaining same, they prefer to kill a pirate rather than having him alive.
n
such that the superior-most pirate can propose no distribution of coins that will let him survive?
Ivan and Jake are playing a game with the following rules:
The game starts with a parameter N which is a positive integer.
Ivan always plays first.
In each turn, the player can either choose to add or subtract the largest prime smaller than N, to N.
The loser is the person who cannot continue. The other person wins by default.
Ivan: N -> N + 7 = 17
Jake: N -> N - 13 = 4
Ivan: N -> 4 - 3 = 1
Ivan wins because Jake cannot continue the game.
Now, suppose that Ivan and Jake know how to play optimally. For how many starting values of
N < 10^4
does Jake win?
Mursalin and Trevor decide to play a game where Trevor gets to pick any positive integer
n<1000
After that the game begins.
First Mursalin names an integer
x
1
n
[both inclusive]. Then Trevor has to name an integer from
1
n
[both inclusive] that does not divide
x
. Then it's Mursalin's turn again and so on. On each turn the player has to select an integer from
1
n
[both inclusive] that doesn't divide all the numbers selected so far. The first person unable to name an integer under these constraints loses. Assuming that both players play optimally, for how many
n
's does Mursalin have a winning strategy?
5 \times 5
grid. A token starts in the bottom left corner of the grid. On each turn, a player can move the token one or two units to the right, or to the leftmost square of the row immediately above it. The last player who is able to move wins.
Lino and Calvin decide that they want to make the game more interesting and instead of playing with a single token, they will play with two tokens, one red, and one blue, and on a turn a player moves either of the tokens. They also decided that the tokens will start in random positions on the board. Of the
25 \times 25 =625
possible starting positions for the 2 tokens, how many of these are winning positions for the first player if he plays optimally?
Clarification: Both tokens are allowed to be on the same square at any point during the game.
For the edge case where both tokens start in the top right corner, we declare that the second player wins.
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Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : IDBasis : Overview
Overview of the IDBasis Object
List of IDBasis object Methods
The IDBasis object is designed and created to represent an initial data basis for linear homogeneous partial differential equations (LHPDEs).
The IDBasis object provides possibility of changing the initial data basis on the LHPDE object.
A LHPDE object must be provided in order to construct an IDBasis object. Here the LHPDE object must be of finite type (i.e. there are a finite number of parametric derivatives). See LieAlgebrasOfVectorFields[IDBasis] for more detail.
The IDBasis object is exported by the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.
The default initial data basis is based on the parametric derivatives of a LHPDE object. Then the IDBasis object stores the change-of-basis matrix (and its inverse) with respect to this fixed basis.
The data attributes of the IDBasis object are "change-of-basis matrix" and the "parametric derivatives". These can be accessed via the GetChangeBasis and GetParametricDerivatives methods.
After an IDBasis object B is successfully constructed, each method in the IDBasis object can be accessed by either the short form method(B, arguments) or the long form B:-method(B, arguments).
Three methods are available after an IDBasis object is constructed:
\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):
\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):
Inserting option `static` gives a list of exports of the IDBasis object.
\mathrm{exports}\left(\mathrm{IDBasis},'\mathrm{static}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{GetChangeBasis}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{GetParametricDerivatives}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Copy}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ModulePrint}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ModuleCopy}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{ModuleApply}}
\mathrm{Typesetting}:-\mathrm{Suppress}\left([\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right),\mathrm{\alpha }\left(x,y\right),\mathrm{\beta }\left(x,y\right)]\right)
\mathrm{E2}≔\mathrm{LHPDE}\left([\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0],\mathrm{dep}=[\mathrm{\xi },\mathrm{\eta }],\mathrm{indep}=[x,y]\right)
\textcolor[rgb]{0,0,1}{\mathrm{E2}}\textcolor[rgb]{0,0,1}{≔}[{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\eta }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{indep}}\textcolor[rgb]{0,0,1}{=}[\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}{\mathrm{dep}}\textcolor[rgb]{0,0,1}{=}[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
Constructing an IDBasis B relevant to the LHPDEs system E2,
B≔\mathrm{IDBasis}\left(\mathrm{E2},[〈1,0,0〉,〈-y,-x,-1〉,〈0,1,0〉]\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}]
\mathrm{GetChangeBasis}\left(B\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}]
\mathrm{GetChangeBasis}\left(B,'\mathrm{output}'="matrix"\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}& \textcolor[rgb]{0,0,1}{0}\end{array}]
Returns the inverse of the change-of-basis matrix
\mathrm{GetChangeBasis}\left(B,"newToOld"\right)
[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\end{array}]
\mathrm{Copy}\left(B,[\mathrm{\alpha },\mathrm{\beta }]\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\beta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\alpha }}}_{\textcolor[rgb]{0,0,1}{y}}]
The IDBasis B is constructed based on the parametric derivatives of the LHPDE object E2,
\mathrm{GetParametricDerivatives}\left(B\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
\mathrm{ParametricDerivatives}\left(\mathrm{E2}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}]
The initial data basis of a LHPDEs system E2 can be recorded as being the IDBasis object B.
\mathrm{SetIDBasis}\left(\mathrm{E2},B\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}]
\mathrm{GetIDBasis}\left(\mathrm{E2}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{\xi }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{\eta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{\xi }}}_{\textcolor[rgb]{0,0,1}{y}}]
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Construction of K-stable foliations for two-dimensional dispersing billiards without eclipse
July, 2004 Construction of
K
-stable foliations for two-dimensional dispersing billiards without eclipse
T
be the billiard map for a two-dimensional dispersing billiard without eclipse. We show that the nonwandering set
{\Omega }^{+}
T
has a hyperbolic structure quite similar to that of the horseshoe. We construct a sort of stable foliation for
\left({\Omega }^{+},T\right)
each leaf of which is a
K
-decreasing curve. We call the foliation a
K
-stable foliation for
\left({\Omega }^{+},T\right)
. Moreover, we prove that the foliation is Lipschitz continuous with respect to the Euclidean distance in the so called
\left(r,\varphi \right)
-coordinates. It is well-known that we can not always expect the existence of such a Lipschitz continuous invariant foliation for a dynamical system even if the dynamical system itself is smooth. Therefore, we keep our construction as elementary and self-contained as possible so that one can see the concrete structure of the set
{\Omega }^{+}
and why the
K
-stable foliation turns out to be Lipschitz continuous.
Takehiko MORITA. "Construction of
K
-stable foliations for two-dimensional dispersing billiards without eclipse." J. Math. Soc. Japan 56 (3) 803 - 831, July, 2004. https://doi.org/10.2969/jmsj/1191334087
Keywords: billiard map , Lipschitz continuous invariant foliation
Takehiko MORITA "Construction of
K
-stable foliations for two-dimensional dispersing billiards without eclipse," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 56(3), 803-831, (July, 2004)
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Results object containing estimation results from least-squares regression - MATLAB LeastSquaresResults object - MathWorks Benelux
LeastSquaresResults object
Package: SimBiology.fit
Results object containing estimation results from least-squares regression
The LeastSquaresResults object is a superclass of two results objects: NLINResults object and OptimResults object. These objects contain estimation results from fitting a SimBiology® model to data using sbiofit with any supported algorithm.
If sbiofit uses the nlinfit estimation algorithm, the results object is the NLINResults object. If sbiofit uses any other supporting algorithm, then the results object is an OptimResults object. See the sbiofit function for the list of supported algorithms.
boxplot Create box plot showing the variation of estimated SimBiology model parameters
fitted Return simulation results of SimBiology model fitted using least-squares regression
plot Compare simulation results to the training data, creating a time-course subplot for each group
plotActualVersusPredicted Compare predictions to actual data, creating a subplot for each response
plotResidualDistribution Plot the distribution of the residuals
plotResiduals Plot residuals for each response, using time, group, or prediction as x-axis
predict Simulate and evaluate fitted SimBiology model
random Simulate SimBiology model, adding variations by sampling error model
summary Return structure array that contains estimated values and fit quality statistics
GroupName Categorical variable representing the name of the group associated with the results, or [] if the 'Pooled' name-value pair argument was set to true when you ran sbiofit.
Beta Table of estimated parameters where the jth row represents the jth estimated parameter βj. It contains transformed values of parameter estimates if any parameter transform is specified.
Standard errors of these parameter estimates (StandardError) are calculated as: sqrt(diag(COVB)).
It can also contain the following variables:
Bounds — the values of transformed parameter bounds that you specified during fitting
CategoryVariableName — the names of categories or groups that you specified during fitting
CategoryValue — the values of category variables specified by CategoryVariableName
This table contains one row per distinct parameter value.
ParameterEstimates Table of estimated parameters where the jth row represents the jth estimated parameter βj. This table contains untransformed values of parameter estimates.
Standard errors of these parameter estimates (StandardError) are calculated as: sqrt(diag(CovarianceMatrix)).
Bounds — the values of parameter bounds that you specified during fitting
This table contains sets of parameter values that are identified for each individual or group.
J Jacobian matrix of the model, with respect to an estimated parameter, that is,
J\left(i,j,k\right)={\frac{\partial {y}_{k}}{\partial {\beta }_{j}}|}_{{t}_{i}}
where ti is the ith time point, βj is the jth estimated parameter in the transformed space, and yk is the kth response in the group of data.
COVB Estimated covariance matrix for Beta, which is calculated as: COVB = inv(J'*J)*MSE.
CovarianceMatrix Estimated covariance matrix for ParameterEstimates, which is calculated as: CovarianceMatrix = T'*COVB*T, where T = diag(JInvT(Beta)).
JInvT(Beta) returns a Jacobian matrix of Beta which is inverse transformed accordingly if you specified any transform to estimated parameters.
For instance, suppose you specified the log-transform for an estimated parameter x when you ran sbiofit. The inverse transform is: InvT = exp(x), and its Jacobian is: JInvT = exp(x) since the derivative of exp is also exp.
R Residuals matrix where Rij is the residual for the ith time point and the jth response in the group of data.
AIC Akaike Information Criterion (AIC), calculated as AIC = 2*(-LogLikelihood + P), where P is the number of parameters.
BIC Bayes Information Criterion (BIC), calculated as BIC = -2*LogLikelihood + P*log(N), where N is the number of observations, and P is the number of parameters.
MSE Mean squared error.
SSE Sum of squared (weighted) errors or residuals.
Weights Matrix of weights with one column per response and one row per observation.
It has one row per error model.
The ErrorModelInfo.Properties.RowsNames property identifies which responses the row applies to.
The table contains three variables: ErrorModel, a, and b. The ErrorModel variable is categorical. The variables a and b can be NaN when they do not apply to a particular error model.
y=f+ae
y=f+b|f|e
y=f+\left(a+b|f|\right)e
y=f\ast \mathrm{exp}\left(ae\right)
EstimationFunction Name of the estimation function.
DependentFiles File names to include for deployment.
Loglikelihood, AIC, and BIC properties are empty for LeastSquaresResults objects that were obtained before R2016a.
NLINResults object | OptimResults object | sbiofit | sbiofitmixed
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K-semistability is equivariant volume minimization
1 November 2017 K-semistability is equivariant volume minimization
This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over
\mathbb{Q}
-Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler–Einstein metrics on Fano varieties. In particular, we prove that for a
\mathbb{Q}
-Fano variety
V
, the K-semistability of
\left(V,-{K}_{V}\right)
is equivalent to the condition that the normalized volume is minimized at the canonical valuation
{\mathrm{ord}}_{V}
{\mathbb{C}}^{\ast }
-invariant valuations on the cone associated to any positive Cartier multiple of
-{K}_{V}
. In this case, we show that
{\mathrm{ord}}_{V}
is the unique minimizer among all
{\mathbb{C}}^{\ast }
-invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over
V
Chi Li. "K-semistability is equivariant volume minimization." Duke Math. J. 166 (16) 3147 - 3218, 1 November 2017. https://doi.org/10.1215/00127094-2017-0026
Received: 5 May 2016; Revised: 31 March 2017; Published: 1 November 2017
Secondary: 13A18 , 52A27 , 53C25
Keywords: Ding semistability , Kähler–Einstein metrics , K-semistability , normalized volume , real valuations , volume minimization
Chi Li "K-semistability is equivariant volume minimization," Duke Mathematical Journal, Duke Math. J. 166(16), 3147-3218, (1 November 2017)
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Each validator maintains an effective balance in addition to its actual balance.
The validator's influence in the protocol is proportional to its effective balance, as are its rewards and penalties.
The effective balance tracks the validator's actual balance, but is designed to change much more rarely. This is an optimisation.
A validator's effective balance is capped at 32 ETH.
The beacon chain maintains two separate records of each validator's balance: its actual balance and its effective balance.
A validator's actual balance is straightforward. It is the sum of any deposits made for it via the deposit contract, plus accrued beacon chain rewards, minus accrued penalties. Withdrawals are not yet possible, but will be subtracted from this balance when available. The actual balance is rapidly changing, being updated at least once per epoch for all active validators, and every slot for sync committee participants. It is also fine-grained: units of the actual balance are Gwei, that is,
10^{-9}
A validator's effective balance is derived from its actual balance in such a way that it changes much more slowly. To achieve this, the units of effective balance are whole Ether (see EFFECTIVE_BALANCE_INCREMENT), and changes to the effective balance are subject to hysteresis.
Using the effective balance achieves two goals, one to do with economics, the other purely engineering.
Economic aspects of effective balance
The effective balance was first introduced to represent the "maximum balance at risk" for a validator, capped at 32 ETH. A validator's actual balance could be much higher, for example if a double deposit had been accidentally made a validator would have an actual balance of 64 ETH but an effective balance of only 32 ETH. We could envisage a protocol in which each validator has influence proportional to its uncapped actual balance, but that would complicate committee membership among other things. Instead we cap the effective balance and require stakers to deposit for more validators if they wish to stake more.
The scope of effective balance quickly grew, and now it completely represents the weight of a validator in the consensus protocol.
All of the following consensus-related matters are proportional to the effective balance of a validator:
the probability of being selected as the beacon block proposer;
the validator's weight in the LMD-GHOST fork choice rule;
the validator's weight in the justification and finalisation calculations calculations; and
the probability of being included in a sync committee.
Correspondingly, the following rewards, penalties, and punishments are also weighted by effective balance:
the base reward for a validator, in terms of which the attestation rewards and penalties are calculated;
the inactivity penalties applied to a validator as a consequence of an inactivity leak; and
both the initial slashing penalty and the correlated slashing penalty.
However, the block proposer reward is not scaled in proportion to the proposer's effective balance. Since a validator's probability of being selected to propose is proportional to its effective balance, the reward scaling with effective balance is already taken care of. For the same reason sync committee rewards are not proportional to the participants' effective balances either.
Engineering aspects of effective balance
We could achieve all of the above simply by using validators' actual balances as their weights, capped at 32 ETH. However, we can gain significant performance benefits by basing everything on effective balances instead.
For one thing, effective balances are updated only once per epoch, which means that we need only calculate things like the base reward per increment once and we can cache the result for the whole epoch, irrespective of any changes in actual balances.
But the main feature of effective balances is that they are designed to change much more rarely than that. This is achieved by making them very granular, and by applying hysteresis to any updates.
One of the big performance challenges in calculating the beacon chain state transition is generating the hash tree root of the entire state. The Merkleization process allows parts of the state that have not been changed to be cached, providing a significant performance boost.
The list of validator records in the state is a large data structure. Were we to store the validators' actual balances within those records they would be frequently changing and the whole data structure would need to be re-hashed at least once per epoch.
The first approach to addressing this simply moved the validators' balances out of the validator records into a dedicated list in the state. This reduces the amount of re-hashing required as the whole validator list does not need to be re-hashed when only the validators' balances change.
However, that leads to a performance issue elsewhere. Light clients needing information on validators' balances would now need to acquire data from two different parts of the state – both the validator record and the validator balance list. This requires two Merkle proofs rather than one, significantly increasing their bandwidth costs.
A way round this is to store a slowly changing version of the balances in the validators' records – meaning that they need to be re-hashed infrequently – and to store the fast-changing actual balances in a separate list, a much smaller structure to re-hash.
From the notes for an early attempt at a kind of effective balance implementation:
[Effective balances are an] "approximate balance" that can be used by light clients in the validator_registry, reducing the number of Merkle branches per validator they need to download from 3 to 2 (actually often from ~2.01 to ~1.01, because when fetching a committee the Merkle branches in active_index_roots are mostly shared), achieving a very significant decrease in light client bandwidth costs
The point is that light clients will not need to access the list of actual balances that is stored separately in state, only the validator records they were downloading anyway.
In summary, adding effective balances to validators' records allows us to achieve two performance goals simultaneously: avoiding the workload of frequently re-hashing the validator list in the state while not increasing the workload of light clients.
Although effective balances are denominated in Gwei they can only be whole multiples of EFFECTIVE_BALANCE_INCREMENT, which is 1 ETH (
10^9
Gwei). Actual balances can be any number of Gwei.
This multiple is known in the spec as an "increment" and shows up in places like calculating the base reward, and other rewards and penalties calculations. Being a handy 1 ETH, it's easy to mentally substitute "Ether" for "increment" to gain some intuition.
It would probably be cleaner to store effective balance in terms of increments instead of Gwei. It would certainly reduce the amount of dividing and multiplying by EFFECTIVE_BALANCE_INCREMENT that goes on, and the associated danger of arithmetic overflows. But the current version evolved over time, and it would be intrusive and risky to go back and change things now.
Effective balances are guaranteed to vary much more slowly than actual balances by adding hysteresis to their calculation.
In our context, hysteresis means that if the effective balance is 31 ETH, the actual balance must rise to 32.25 ETH to trigger an effective balance update to 32 ETH. Similarly, if the effective balance is 31 ETH, then the actual balance must fall to 30.75 ETH to trigger an effective balance update to 30 ETH.
The following chart illustrates the behaviour.
The actual balance and the effective balance both start at 32 ETH.
Initially the actual balance rises. Effective balance is capped at 32 ETH, so it does not get updated.
Only when the actual balance falls below 31.75 ETH does the effective balance get reduced to 31 ETH.
Although the actual balance rises and oscillates around 32 ETH, no effective balance update is triggered and it remains at 31 ETH.
Eventually the actual balance rises above 32.25 ETH, and the effective balance is updated to 32 ETH.
Despite the actual balance falling again, it does not fall below 31.75 ETH, so the effective balance remains at 32 ETH.
Illustration of the relationship between the actual balance (solid line) and the effective balance (dashed line) of a validator. The dotted lines are the thresholds at which the effective balance gets updated - the hysteresis.
The hysteresis levels are controlled by the hysteresis parameters in the spec:
These are applied at the end of each epoch during effective balance updates. Every validator in the state (whether active or not) has its effective balance updated as follows:
If actual balance is less than effective balance minus 0.25 ( = HYSTERESIS_DOWNWARD_MULTIPLIER / HYSTERESIS_QUOTIENT) increments (ETH), then reduce the effective balance by an increment.
If actual balance is more than effective balance plus 1.25 ( = HYSTERESIS_UPWARD_MULTIPLIER / HYSTERESIS_QUOTIENT) increments (ETH), then increase the effective balance by an increment.
The effect of the hysteresis is that the effective balance cannot change more often than it takes for a validator's actual balance to change by 0.5 ETH, which would normally take several weeks or months.
Historical note: the initial implementation of hysteresis effectively had QUOTIENT = 2, DOWNWARD_MULTIPLIER = 0, and UPWARD_MULTIPLIER = 3. This meant that a validator starting with 32 ETH actual balance but suffering a minor initial outage would immediately drop to 31 ETH effective balance. To get back to 32 ETH effective balance it would need to achieve a 32.5 ETH actual balance, and meanwhile the validator's rewards would be 3.1% lower due to the reduced effective balance. This seemed unfair, and incentivised stakers to "over-deposit" Ether to avoid the risk of an initial effective balance drop. Hence the change to the current parameters.
The presets that constrain the effective balance, MAX_EFFECTIVE_BALANCE and EFFECTIVE_BALANCE_INCREMENT.
The parameters that control the hysteresis.
The function process_effective_balance_updates() for the actual calculation and application of hysteresis.
Validator objects store the effective balances. The registry in the beacon state contains the list of validators alongside a separate list of the actual balances.
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A Fixed Point Result for Boyd-Wong Cyclic Contractions in Partial Metric Spaces
Hassen Aydi, Erdal Karapinar, "A Fixed Point Result for Boyd-Wong Cyclic Contractions in Partial Metric Spaces", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 597074, 11 pages, 2012. https://doi.org/10.1155/2012/597074
Hassen Aydi1 and Erdal Karapinar 2
1Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Université de Sousse, Route GP1-4011, Hammam Sousse, 4002 Sousse, Tunisia
2Department of Mathematics, Atilim University, 06836 İncek, Turkey
Academic Editor: Billy Rhoades
A fixed point theorem involving Boyd-Wong-type cyclic contractions in partial metric spaces is proved. We also provide examples to support the concepts and results presented herein.
Partial metric spaces were introduced by Matthews [1] to the study of denotational semantics of data networks. In particular, he proved a partial metric version of the Banach contraction principle [2]. Subsequently, many fixed points results in partial metric spaces appeared (see, e.g., [1, 3–19] for more details).
Throughout this paper, the letters and will denote the sets of all real numbers and positive integers, respectively. We recall some basic definitions and fixed point results of partial metric spaces.
Definition 1.1. A partial metric on a nonempty set is a function such that for all (p1),(p2),(p3), (p4). A partial metric space is a pair such that is a nonempty set and is a partial metric on .
If is a partial metric on , then the function given by is a metric on .
Example 1.2 (see, e.g., [1, 3, 11, 12]). Consider with . Then, is a partial metric space. It is clear that is not a (usual) metric. Note that in this case .
Example 1.3 (see, e.g., [1]). Let , and define . Then, is a partial metric space.
Example 1.4 (see, e.g., [1, 20]). Let , and define by Then, is a complete partial metric space.
Each partial metric on generates a topology on , which has as a base the family of open -balls , where for all and .
Definition 1.5. Let be a partial metric space and a sequence in . Then, (i) converges to a point if and only if ,(ii) is called a Cauchy sequence if exists and is finite.
Definition 1.6. A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point , such that .
Lemma 1.7 (see, e.g., [3, 11, 12]). Let be a partial metric space. Then,(a) is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space , (b) is complete if and only if the metric space is complete. Furthermore, if and only if
Lemma 1.8 (see, e.g., [3, 11, 12]). Let be a partial metric space. Then, (a)if , then , (b)if , then .
Remark 1.9. If , may not be .
Lemma 1.10 (see, e.g., [3, 11, 12]). Let as in a partial metric space where . Then, for every .
Let be the set of functions such that(i) is upper semicontinuous (i.e., for any sequence in such that as , we have ),(ii) for each .
Recently, Romaguera [21] obtained the following fixed point theorem of Boyd-Wong type [22].
Theorem 1.11. Let be a complete partial metric space, and let be a map such that for all where and Then, has a unique fixed point.
In 2003, Kirk et al. [23] introduced the following definition.
Definition 1.12 (see [23]). Let be a nonempty set, a positive integer, and a mapping. is said to be a cyclic representation of with respect to if (i) are nonempty closed sets, (ii).
Recently, fixed point theorems involving a cyclic representation of with respect to a self-mapping have appeared in many papers (see, e.g., [24–28]).
Very recently, Abbas et al. [24] extended Theorem 1.11 to a class of cyclic mappings and proved the following result, but with being a continuous map.
Theorem 1.13. Let be a complete partial metric space. Let be a positive integer, nonempty closed subsets of , and . Let be a mapping such that(i) is a cyclic representation of with respect to , (ii)there exists such that is continuous and for each , satisfying for any , , , where and is defined by (1.5). Then, has a unique fixed point .
In the following example, , but it is not continuous.
Example 1.14. Define by for all and for , . Then, is upper semicontinuous on with for all . However, it is not continuous at for all .
Following Example 1.14, the main aim of this paper is to present the analog of Theorem 1.13 for a weaker hypothesis on , that is, with . Our proof is simpler than that in [24]. Also, some examples are given.
Theorem 2.1. Let be a complete partial metric space. Let be a positive integer, nonempty closed subsets of , and . Let be a mapping such that(1) is a cyclic representation of with respect to ,(2)there exists such that for any , , , where and is defined by (1.5).Then, has a unique fixed point .
Proof. Let . Consider the Picard iteration given by for . If there exists such that , then and the existence of the fixed point is proved.
Assume that , for each . Having in mind that , so for each , there exists such that and . Then, by (2.1) where Therefore, If for some , we have , so by (2.2) which is a contradiction. It follows that Thus, from (2.2), we get that Hence, is a decreasing sequence of positive real numbers. Consequently, there exists such that . Assume that . Letting in the above inequality, we get using the upper semicontinuity of which is a contradiction, so that , that is, By (1.1), we have for all , and then from (2.9) Also, by , In the sequel, we will prove that is a Cauchy sequence in the partial metric space . By Lemma 1.7, it suffices to prove that is Cauchy sequence in the metric space . We argue by contradiction. Assume that is not a Cauchy sequence in . Then, there exists for which we can find subsequences and of with such that Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (2.12). Then, We use (2.13) and the triangular inequality Letting in (2.14) and using (2.10), we find On the other hand Letting in the two above inequalities and using (2.10) and (2.15), Similarly, we have Also, by (1.1), (2.11), and (2.15)–(2.18), we may find On the other hand, for all , there exists , , such that . Then, (for large enough, ) and lie in different adjacently labeled sets and for certain . Using the contractive condition (2.1), we get where As (2.19), using (2.9), we may get By (2.22) and (2.23), we get that Letting in (2.20), we get using (2.22), (2.24), and the upper semicontinuity of which is a contradiction.
This shows that is a Cauchy sequence in the complete subspace equipped with the metric . Thus, there exists . Notice that the sequence has an infinite number of terms in each , , so since is complete, from each , one can extract a subsequence of that converges to . Because , are closed in , it follows that Thus, .
For simplicity, set . Clearly, is also closed in , so it is a complete subspace of and then is a complete partial metric space. Consider the restriction of on , that is, . Then, satisfies the assumptions of Theorem 1.11, and thus has a unique fixed point in .
We give some examples illustrating our results.
Example 3.1. Let and . It is obvious that is a complete partial metric space.
Set , , and . Define by
Notice that and , and hence . Analogously, and , and hence .
Take Clearly, satisfies condition (2.1). Indeed, we have the following cases.
Case 1. and . Inequality (2.1) turns into which is necessarily true.
Case 2. and . Inequality (2.1) becomes It is clear that for all . Hence, (3.4) holds.
Case 3. and . Inequality (2.1) turns into which is true again by the fact that for all .
Case 4. and . Inequality (2.1) becomes Let use examine all possibilities: Thus, (2.1) holds for .
The rest of the assumptions of Theorem 2.1 are also satisfied. The function has as a unique fixed point.
However, since is not a continuous function, we could not apply Theorem 1.13.
Example 3.2. Let and for all . Then, is a complete partial metric space. Take . Define by . Consider given by Example 1.14.
For all , we have Then all the assumptions of Theorem 2.1 are satisfied. The function has as a unique fixed point.
Similarly, Theorem 1.13 is not applicable.
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User Trading Rewards - Entropy
Overview of the User Trading Rewards program.
30.00% of the initial token supply (300,000,000 ENT) will be distributed to all users participating in the Entropy contract transaction in each period (14 days) within five years, and 2,301,369 ENT will be distributed in each period.
Accelerate market liquidity and overall product usage.
ENT will be distributed to traders according to a formula that rewards the combination of fees paid on perpetual contracts and open positions.
The Cobb-Douglas function is used to compute how much ENT is awarded to each trader during each epoch:
Reward for a specific trader.
Total reward to be split between all traders in the pool for the epoch.
Total fees paid by a trader in this epoch.
Individual trader score.
{\sum\limits _{n} w_{n}}
Sum of all trader scores.
A trader’s average open interest (measured every minute) across all markets in this epoch.
Total number of traders in this epoch.
A constant in the range that determines the weight of fees vs open interest. The initial value is α=0.7.
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Fekete-Szegő Inequality for a Subclass of p-Valent Analytic Functions
2013 Fekete-Szegő Inequality for a Subclass of
p
-Valent Analytic Functions
Mohsan Raza, Muhammad Arif, Maslina Darus
The main object of this paper is to study Fekete-Szegő problem for the class of
p
-valent functions. Fekete-Szegő inequality of several classes is obtained as special cases from our results. Applications of the results are also obtained on the class defined by convolution.
Mohsan Raza. Muhammad Arif. Maslina Darus. "Fekete-Szegő Inequality for a Subclass of
p
-Valent Analytic Functions." J. Appl. Math. 2013 1 - 7, 2013. https://doi.org/10.1155/2013/127615
Mohsan Raza, Muhammad Arif, Maslina Darus "Fekete-Szegő Inequality for a Subclass of
p
-Valent Analytic Functions," Journal of Applied Mathematics, J. Appl. Math. 2013(none), 1-7, (2013)
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Weak key - Wikipedia
In cryptography, a weak key is a key, which, used with a specific cipher, makes the cipher behave in some undesirable way. Weak keys usually represent a very small fraction of the overall keyspace, which usually means that, if one generates a random key to encrypt a message, weak keys are very unlikely to give rise to a security problem. Nevertheless, it is considered desirable for a cipher to have no weak keys. A cipher with no weak keys is said to have a flat, or linear, key space.
2 Weak keys in DES
3 List of algorithms with weak keys
4 No weak keys as a design goal
Virtually all rotor-based cipher machines (from 1925 onwards) have implementation flaws that lead to a substantial number of weak keys being created. Some rotor machines have more problems with weak keys than others, as modern block and stream ciphers do.
The first stream cipher machines were also rotor machines and had some of the same problems of weak keys as the more traditional rotor machines. The T52 was one such stream cipher machine that had weak key problems.
The British first detected T52 traffic in Summer and Autumn of 1942. One link was between Sicily and Libya, codenamed "Sturgeon", and another from the Aegean to Sicily, codenamed "Mackerel". Operators of both links were in the habit of enciphering several messages with the same machine settings, producing large numbers of depths.
There were several (mostly incompatible) versions of the T52: the T52a and T52b (which differed only in their electrical noise suppression), T52c, T52d and T52e. While the T52a/b and T52c were cryptologically weak, the last two were more advanced devices; the movement of the wheels was intermittent, the decision on whether or not to advance them being controlled by logic circuits which took as input data from the wheels themselves.
In addition, a number of conceptual flaws (including very subtle ones) had been eliminated. One such flaw was the ability to reset the keystream to a fixed point, which led to key reuse by undisciplined machine operators.
Weak keys in DES[edit]
The block cipher DES has a few specific keys termed "weak keys" and "semi-weak keys". These are keys that cause the encryption mode of DES to act identically to the decryption mode of DES (albeit potentially that of a different key).
In operation, the secret 56-bit key is broken up into 16 subkeys according to the DES key schedule; one subkey is used in each of the sixteen DES rounds. DES weak keys produce sixteen identical subkeys. This occurs when the key (expressed in hexadecimal) is:[1]
Alternating ones + zeros (0x0101010101010101)
Alternating 'F' + 'E' (0xFEFEFEFEFEFEFEFE)
'0xE0E0E0E0F1F1F1F1'
'0x1F1F1F1F0E0E0E0E'
If an implementation does not consider the parity bits, the corresponding keys with the inverted parity bits may also work as weak keys:
all zeros (0x0000000000000000)
all ones (0xFFFFFFFFFFFFFFFF)
'0x1E1E1E1E0F0F0F0F'
Using weak keys, the outcome of the Permuted Choice 1 (PC-1) in the DES key schedule leads to round keys being either all zeros, all ones or alternating zero-one patterns.
Since all the subkeys are identical, and DES is a Feistel network, the encryption function is self-inverting; that is, despite encrypting once giving a secure-looking cipher text, encrypting twice produces the original plaintext.
DES also has semi-weak keys, which only produce two different subkeys, each used eight times in the algorithm: This means they come in pairs K1 and K2, and they have the property that:
{\displaystyle E_{K_{1}}(E_{K_{2}}(M))=M}
where EK(M) is the encryption algorithm encrypting message M with key K. There are six semi-weak key pairs:
0x011F011F010E010E and 0x1F011F010E010E01
0x01E001E001F101F1 and 0xE001E001F101F101
0x01FE01FE01FE01FE and 0xFE01FE01FE01FE01
0x1FE01FE00EF10EF1 and 0xE01FE01FF10EF10E
0x1FFE1FFE0EFE0EFE and 0xFE1FFE1FFE0EFE0E
0xE0FEE0FEF1FEF1FE and 0xFEE0FEE0FEF1FEF1
There are also 48 possibly weak keys that produce only four distinct subkeys (instead of 16). They can be found in a NIST publication.[2]
These weak and semi-weak keys are not considered "fatal flaws" of DES. There are 256 (7.21 × 1016, about 72 quadrillion) possible keys for DES, of which four are weak and twelve are semi-weak. This is such a tiny fraction of the possible keyspace that users do not need to worry. If they so desire, they can check for weak or semi-weak keys when the keys are generated. They are very few, and easy to recognize. Note, however, that currently DES is no longer recommended for general use since all DES keys can be brute-forced it's been decades since the Deep Crack machine was cracking them on the order of days, and as computers tend to do, more recent solutions are vastly cheaper on that time scale. Examples of progress are in Deep Crack's article.
List of algorithms with weak keys[edit]
DES, as detailed above.
RC4. RC4's weak initialization vectors allow an attacker to mount a known-plaintext attack and have been widely used to compromise the security of WEP.[3]
GMAC. Frequently used in the AES-GCM construction. Weak keys can be identified by the group order of the authentication key H (for AES-GCM, H is derived from the encryption key by encrypting the zero block).
RSA and DSA. August 2012 Nadia Heninger, Zakir Durumeric, Eric Wustrow, J. Alex Halderman found that TLS certificates they assessed share keys due to insufficient entropy during key generation, and were able to obtain DSA and RSA private keys of TLS and SSH hosts knowing only the public key.[4]
No weak keys as a design goal[edit]
The goal of having a 'flat' keyspace (i.e., all keys equally strong) is always a cipher design goal. As in the case of DES, sometimes a small number of weak keys is acceptable, provided that they are all identified or identifiable. An algorithm that has unknown weak keys does not inspire much trust.[citation needed]
However, weak keys are much more often a problem where the adversary has some control over what keys are used, such as when a block cipher is used in a mode of operation intended to construct a secure cryptographic hash function (e.g. Davies–Meyer).
^ FIPS, Guidelines for Implementing and Using the NBS Data Encryption Standard, FIPS-PUB 74, http://www.itl.nist.gov/fipspubs/fip74.htm
^ NIST, Recommendation for the Triple Data Encryption Algorithm (TDEA) Block Cipher, Special Publication 800-67, page 14
^ Fluhrer, S., Mantin, I., Shamir, A. Weaknesses in the key scheduling algorithm of RC4. Eighth Annual Workshop on Selected Areas in Cryptography (August 2001), http://citeseer.ist.psu.edu/fluhrer01weaknesses.html
^ "Research Paper - factorable.net". factorable.net. Retrieved 2020-06-26.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Weak_key&oldid=1053599381"
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Units in Physics Calculations - MATLAB & Simulink Example - MathWorks India
Define and Solve Equation of Motion
Find Unit of Drag Constant
Estimate Terminal Velocity
Convert Velocity to Imperial Units
Plot Velocity over Time
This example shows how to work with units in physics calculations. Calculate the terminal velocity of a falling paratrooper in both SI and imperial units. Solve the motion of the paratrooper, taking into account the gravitational force and the drag force.
Imagine a paratrooper jumping out of an airplane. Assume there are only two forces acting on the paratrooper: the gravitational force and an opposing drag force from the parachute. The drag force is proportional to the velocity squared of the paratrooper.
The net force acting on the paratrooper can be expressed as
\mathrm{mass}\cdot \mathrm{acceleration}=\mathrm{drag}\text{\hspace{0.17em}}\mathrm{force}-\mathrm{gravitational}\text{\hspace{0.17em}}\mathrm{force}
\mathit{m}\frac{\partial }{\partial \mathit{t}}\mathit{v}\left(\mathit{t}\right)={\mathit{c}}_{\mathit{d}}{\mathit{v}\left(\mathit{t}\right)}^{2}-\mathit{m}\text{\hspace{0.17em}}\mathit{g}
m
is the mass of the paratrooper
v\left(t\right)
is the velocity of the paratrooper
{c}_{d}
is the drag constant
Define the differential equation describing the equation of motion.
m \frac{\partial }{\partial t}\mathrm{ }v\left(t\right)+g m={c}_{d} {v\left(t\right)}^{2}
Assume that the parachute opens immediately at
\mathit{t}=0
so that the equation eq is valid for all values of
\mathit{t}\ge 0
. Solve the differential equation analytically using dsolve with the initial condition
\mathit{v}\left(0\right)=0
. The solution represents the velocity of the paratrooper as a function of time.
-\frac{\sqrt{g} \sqrt{m} \mathrm{tanh}\left(\frac{\sqrt{{c}_{d}} \sqrt{g} t}{\sqrt{m}}\right)}{\sqrt{{c}_{d}}}
Find the SI unit of the drag constant
{c}_{d}
The SI unit of force is the Newton
\left(\mathit{N}\right)
. In terms of the base units, the Newton is
\left(\frac{\mathrm{kg}\cdot \mathit{m}}{{\mathit{s}}^{2}}\right)
. Since these are equivalent, they have a unit conversion factor of 1.
1
The drag force
{\mathit{c}}_{\mathit{d}}{\mathit{v}\left(\mathit{t}\right)}^{2}
must have the same unit in Newton
\left(\mathit{N}\right)
as the gravitational force
m\phantom{\rule{0.16666666666666666em}{0ex}}g
. Using dimensional analysis, solve for the unit of
{c}_{d}
1 \frac{\mathrm{kg}\mathrm{"kilogram - a physical unit of mass."}}{\mathrm{m}\mathrm{"meter - a physical unit of length."}}
Describe the motion of the paratrooper by defining the following values.
Mass of the paratrooper
\mathit{m}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}70\text{\hspace{0.17em}}\mathrm{kg}
\mathit{g}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}9.81\text{\hspace{0.17em}}\mathit{m}/{\mathit{s}}^{2}
{\mathit{c}}_{\mathit{d}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}40\text{\hspace{0.17em}}\mathrm{kg}/\mathit{m}
Substitute these values into the velocity equation and simplify the result.
-\frac{\mathrm{tanh}\left(\frac{t \sqrt{40 \frac{\mathrm{kg}\mathrm{"kilogram - a physical unit of mass."}}{\mathrm{m}\mathrm{"meter - a physical unit of length."}}} \sqrt{\frac{981}{100} \frac{\mathrm{m}\mathrm{"meter - a physical unit of length."}}{{\mathrm{s}\mathrm{"second - a physical unit of time."}}^{2}}}}{\sqrt{70 \mathrm{kg}\mathrm{"kilogram - a physical unit of mass."}}}\right) \sqrt{70 \mathrm{kg}\mathrm{"kilogram - a physical unit of mass."}} \sqrt{\frac{981}{100} \frac{\mathrm{m}\mathrm{"meter - a physical unit of length."}}{{\mathrm{s}\mathrm{"second - a physical unit of time."}}^{2}}}}{\sqrt{40 \frac{\mathrm{kg}\mathrm{"kilogram - a physical unit of mass."}}{\mathrm{m}\mathrm{"meter - a physical unit of length."}}}}
-\frac{3 \sqrt{763} \mathrm{tanh}\left(\frac{3 \sqrt{763} t}{35} \frac{1}{\mathrm{s}\mathrm{"second - a physical unit of time."}}\right)}{20} \frac{\mathrm{m}\mathrm{"meter - a physical unit of length."}}{\mathrm{s}\mathrm{"second - a physical unit of time."}}
Compute a numerical approximation of the velocity to 3 significant digits.
-4.14 \mathrm{tanh}\left(2.37 t \frac{1}{\mathrm{s}\mathrm{"second - a physical unit of time."}}\right) \frac{\mathrm{m}\mathrm{"meter - a physical unit of length."}}{\mathrm{s}\mathrm{"second - a physical unit of time."}}
The paratrooper approaches a constant velocity when the gravitational force is balanced by the drag force. This is called the terminal velocity and it occurs when the drag force from the parachute cancels out the gravitational force (there is no further acceleration). Find the terminal velocity by taking the limit of
\mathit{t}⟶\infty
-4.14 \frac{\mathrm{m}\mathrm{"meter - a physical unit of length."}}{\mathrm{s}\mathrm{"second - a physical unit of time."}}
Finally, convert the velocity function from SI units to imperial units.
-13.6 \mathrm{tanh}\left(2.37 t \frac{1}{\mathrm{s}\mathrm{"second - a physical unit of time."}}\right) \frac{\mathrm{ft}\mathrm{"foot - a physical unit of length."}}{\mathrm{s}\mathrm{"second - a physical unit of time."}}
Convert the terminal velocity.
-13.6 \frac{\mathrm{ft}\mathrm{"foot - a physical unit of length."}}{\mathrm{s}\mathrm{"second - a physical unit of time."}}
To plot the velocity as a function of time, express the time t in seconds and replace t by T s, where T is a dimensionless symbolic variable.
-4.14 \mathrm{tanh}\left(2.37 T\right) \frac{\mathrm{m}\mathrm{"meter - a physical unit of length."}}{\mathrm{s}\mathrm{"second - a physical unit of time."}}
-13.6 \mathrm{tanh}\left(2.37 T\right) \frac{\mathrm{ft}\mathrm{"foot - a physical unit of length."}}{\mathrm{s}\mathrm{"second - a physical unit of time."}}
Separate the expression from the units by using separateUnits. Plot the expression using fplot. Convert the units to strings for use as plot labels by using symunit2str.
The velocity of the paratrooper approaches steady state when
\mathit{t}>1
. Show how the velocity approaches terminal velocity by plotting the velocity over the range
0\text{\hspace{0.17em}}\le \mathit{T}\text{\hspace{0.17em}}\le 2
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Algebra Problem on Newton's Identities: Newton's Sums - Sharky Kesa | Brilliant
p(x) = x^3 + 3x^2 + 4x - 8
\color{#D61F06}{a}
\color{#3D99F6}{b}
\color{#69047E}{c}
\color{#D61F06}{a}^2 \big(1 + \color{#D61F06}{a}^2\big) + \color{#3D99F6}{b}^2 \big(1 + \color{#3D99F6}{b}^2\big) + \color{#69047E}{c}^2 \big(1 + \color{#69047E}{c}^2\big)?
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How to export to PDF & LaTeX using Curvenote - Curvenote
Writing up research for submission to a particular conference, journal, or preprint service is a major task. Researchers need to prepare their hypothesis, methods, data, results, and conclusions to best represent their research, its outcome, and impacts. On top of this, they need to deal with the formatting and layout requirements of the organization where they are publishing that work. The task of formatting and layout is not a core part of research work yet it can be a major effort and time sink because of factors like strict requirements, uncooperative tools, or additional learning curves — like having to learn
\LaTeX
while writing in the context of a specific template.
Formatting and finalizing a publication should be easy and Curvenote aims to be a writing environment where researchers can move from their day-to-day notes, notebooks, and reports into their manuscripts with a minimum of rework. Curvenote lets you write a manuscript, share it and gather feedback and then export it to multiple different formats. From conference abstract to a preprint through to a journal article, Curvenote aims to let you export your work to the formats you need as your research evolves.
#Open source LaTeX templates
\LaTeX
, PDF, and Microsoft Word documents represent the predominant formats for the submission of academic work. Curvenote allows you to export to
\LaTeX
and PDF based on a specific
\LaTeX
template (we are working on Microsoft Word currently!). We started with
\LaTeX
and PDF, as typically journal articles, preprint services, and universities provide templates to ensure that submissions are correctly laid out and meet their requirements.
Still, working with a template, especially larger templates with many features can be onerous. At Curvenote we’ve done two things:
Our export engine is based on a templating system that allows us to load various templates and provide a user interface to configure them. We’ll cover this a more detail in the rest of this post.
To be able to accept and integrate a large number of templates, we have created an open template repository where template specifics and our modifications can be easily read and where we can accept both requests and contributions from others.
If you are writing or planning to start a paper, report, or thesis and want to do your writing in Curvenote — you can open an issue or pull request on this repo and we will make sure your template gets integrated.
#Export to PDF or
\LaTeX
Once templates are tested and accepted in the repository, they are available for export and appear directly in the Curvenote when you click “Export As” on an article.
Figure 2:Export from Curvenote - the first step is to select the target LaTeX or PDF and choose the template to use for the layout
Both PDF and
\LaTeX
options use the same template, but the latter will create a .zip file containing a
\LaTeX
“project” — a folder containing a main.tex and main.bib file as well as definition files and image assets. This enables you to customize and build locally, should you want to.
But that is not our goal!
We want to be able to export submission-ready PDF documents directly from Curvenote — to avoid building
\LaTeX
locally at all. We’ve made this possible through Template Options and Tagged Blocks. Setup on a per template basis, these allow you to provide additional information and any specific content required via the user interface during export.
#Template Options
Once you’ve selected a template, the next screen you are shown is a dynamic form showing the options for that template. When integrating the template we look at the author guidelines, nominating required and optional items appropriately — so you’ll need to meet the required options before you can export.
Here’s an example of options for the AGU 2019 template with two required and one optional field.
We are able to proceed once we’ve entered the required information. This template has relatively few options — other templates have more and/or conditional options that only appear based on the value of certain fields, a typical case being additional options for different journals based on the Journal Name you select.
#Tagged Blocks
Beyond template options, that are appropriate for setting metadata, additional author information, or switching on/off certain template features and styles. Templates also often require specific content from the user that is handled differently in the typesetting process — a well-known example would be the Abstract for a paper.
Since Curvenote also gives you an easy to publish an interactive online version of your paper, that you can share and link back to as well as generating the PDF for submission, it makes total sense to keep content like the abstract within the document.
In an abstract, we are dealing with more than a line of text or a link. We usually have a paragraph or two of content that may contain maths and symbols but that is also fundamentally part of the document rather than metadata.
We’ve achieved this through the use of the tags that are available on any block in Curvenote — let’s see how this works. After entering the template options, you move onto the following screen.
This is essentially a summary of the special content requirements of this template along with flags on whether they have been met or not. When integrating templates, we include as much information from the original author guidelines as possible, as show that information here.
In the AGU 2019 template example, we can set content for an Abstract, Plain Language Summary, and Acknowledgments. Furthermore, an Abstract is required and in this case, we’ve not provided one.
To fix this we head back to our manuscript, select the block containing our abstract, and set the tag appropriately.
Tags can be set to any string we like, in the UI we suggest some of the common tags you’ll find in templates, but any tag you need can be added. (Note that the word Abstract appears as a decoration on the block, that’s a special case as we are still deciding how to best show the block tags while keeping the page uncluttered).
Now when we head back to export, we can see we’ve provided our abstract and can proceed.
We can of course provide information in any of the optional fields too, and this will be added appropriately, by the template.
#Write once, export to any format
We’ve built our export process to take away the pain from the submission process. We want Curvenote to help its users focus on science and writing while taking care of the formatting and packaging concerns of your paper, report, or thesis.
This is something we’ll be perfecting over time, but by making our templates configurable like this, we are now able to get people to submission-ready PDFs without having to dive into a
\LaTeX
build — which we think is a solid step forward.
The philosophy behind Curvenote is to support your day-to-day work in one place and make it easy to reuse your writing, notebooks, figures, and plots across documents, presentations, and papers. This saves you time and effort and keeps you focussed on your research.
When it comes to writing your paper, you can easily move one manuscript forward and export that one piece of content to multiple formats for your internal group review, preprint service submission, and ultimately conference or journal submission. At the same time, you can easily share supporting work online securely with collaborators or publicly and attach it to your Curvenote profile.
#Want to change your writing experience?
Try out Curvenote and tell us about which conference or journal paper that you are aiming for — we’ll add the template for you. If you are writing your thesis we can also help, we can get you started now and we will commit to adding your university thesis template in time for your submission. Book a demo with us to find out more.
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