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Combinatorics of the Teichmüller TQFT
Rinat Kashaev1
1 Section de mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211 Genève 4, Suisse
Based on the lectures given by the author at the School on braids and low dimensional topology “Winter Braids VI”, University of Lille I, 22-25 February 2016, we review the combinatorics underlying the Teichmüller TQFT, a new type of three-dimensional TQFT with corners where the vector spaces associated with surfaces are infinite dimensional. The geometrical ingredients and the semi-classical behaviour suggest that this theory is related with hyperbolic geometry in dimension three.
Rinat Kashaev 1
author = {Rinat Kashaev},
title = {Combinatorics of the {Teichm\"uller} {TQFT}},
TI - Combinatorics of the Teichmüller TQFT
%T Combinatorics of the Teichmüller TQFT
Rinat Kashaev. Combinatorics of the Teichmüller TQFT. Winter Braids Lecture Notes, Volume 3 (2016), Talk no. 2, 16 p. doi : 10.5802/wbln.13. https://wbln.centre-mersenne.org/articles/10.5802/wbln.13/
[1] Jørgen Ellegaard Andersen and Rinat Kashaev. A TQFT from Quantum Teichmüller Theory. Comm. Math. Phys., 330(3):887–934, 2014. | Article | Zbl: 1305.57024
[2] Stéphane Baseilhac and Riccardo Benedetti. Quantum hyperbolic invariants of 3-manifolds with
\mathrm{PSL}\left(2,ℂ\right)
-characters. Topology, 43(6):1373–1423, 2004. | Article | MR: 2081430 | Zbl: 1065.57008
[3] Rinat Kashaev, Feng Luo, and Grigory Vartanov. A TQFT of Turaev-Viro type on shaped triangulations. Ann. Henri Poincaré, 17(5):1109–1143, 2016. | Article | MR: 3486430 | Zbl: 1337.81105
[4] Rinat M. Kashaev. On realizations of Pachner moves in 4d. J. Knot Theory Ramifications, 24(13):1541002, 13, 2015. | Article | MR: 3434541 | Zbl: 1337.57064
[5] W. B. R. Lickorish. Simplicial moves on complexes and manifolds. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), volume 2 of Geom. Topol. Monogr., pages 299–320 (electronic). Geom. Topol. Publ., Coventry, 1999. | Article | Zbl: 0963.57013
[6] S. V. Matveev. Transformations of special spines, and the Zeeman conjecture. Izv. Akad. Nauk SSSR Ser. Mat., 51(5):1104–1116, 1119, 1987. | Article | MR: 925096 | Zbl: 0642.57003
[7] Sergei Matveev. Algorithmic topology and classification of 3-manifolds, volume 9 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2003. | Article | Zbl: 1048.57001
[8] John Milnor. Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (N.S.), 6(1):9–24, 1982. | Article | MR: 634431 | Zbl: 0486.01006
[9] Udo Pachner. P.L. homeomorphic manifolds are equivalent by elementary shellings. European J. Combin., 12(2):129–145, 1991. | Article | Zbl: 0729.52003
[10] Riccardo Piergallini. Standard moves for standard polyhedra and spines. Rend. Circ. Mat. Palermo (2) Suppl., (18):391–414, 1988. Third National Conference on Topology (Italian) (Trieste, 1986). | Zbl: 0672.57004
[11] G. Ponzano and T. Regge. Semiclassical limit of Racah coefficients. In Spectroscopic and group theoretical methods in physics, pages 1–58. North-Holland Publ. Co., Amsterdam, 1968.
[12] V. G. Turaev. Quantum invariants of knots and 3-manifolds, volume 18 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994. | Article | Zbl: 0812.57003
[13] V. G. Turaev and O. Ya. Viro. State sum invariants of
3
-manifolds and quantum
6j
-symbols. Topology, 31(4):865–902, 1992. | Article | MR: 1191386 | Zbl: 0779.57009
|
Lateral Fluid Flow in a Compacting Sand-Shale Sequence: South Caspian Basin: ERRATUM | AAPG Bulletin | GeoScienceWorld
Rashid D. Djevanshir;
Rashid D. Djevanshir
Kenneth R. Belitz
https://doi.org/10.1306/703C9A1E-1707-11D7-8645000102C1865D
John D. Bredehoeft, Rashid D. Djevanshir, Kenneth R. Belitz; Lateral Fluid Flow in a Compacting Sand-Shale Sequence: South Caspian Basin: ERRATUM. AAPG Bulletin 1988;; 72 (12): 1525. doi: https://doi.org/10.1306/703C9A1E-1707-11D7-8645000102C1865D
The article “Lateral Fluid Flow in a Compacting Sand-Shale Sequence: South Caspian Basin” by John D. Bredehoeft, Rashid D. Djevanshir, and Kenneth R. Belitz (AAPG Bulletin, v. 72, no. 4, April 1988, p. 416-424) needs the following corrections.
There are several typographical errors that were the authors’ error.
Equation 2 should be
DΦDt=∂Φ∂t+∂Φ∂z∂z∂t.
∂Φ∂t=−∂Φ∂z∂z∂t.
Although the equations were incorrect in the article, the associated discussion and equations 4 and 5 are based upon the correct expressions, as shown above.
Table 3, the calculated regional sand permeabilities, is also incorrect. The hydraulic conductivity, which in our case has the units of meters per second, was incorrectly converted to millidarcys; values in meters per second were multiplied by the standard conversion 1 m/sec equals 1.04 × 108 md. In retrospect, this conversion was a conceptual error. Table 3 should read
Calculated Regional Sand Permeabilities
Depth Interval (m)
Hydraulic Conductivity (m/sec)
1,000-2,000 3.4 × 10-7 26.0
3,000-4,000 7.6 × 10-8 3.4
|
\mathrm{with}\left(\mathrm{ThermophysicalData}\right):
\mathrm{h_f_C4H10}≔\mathrm{Chemicals}:-\mathrm{Property}\left("HeatOfFormation","C4H10\left(l\right),n-buta",\mathrm{useunits}\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{150.6640000}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{\mathrm{kJ}}}{\textcolor[rgb]{0,0,1}{\mathrm{mol}}}⟧
\mathrm{h_N2}≔\mathrm{Chemicals}:-\mathrm{Property}\left("Hmolar","N2\left(g\right)","temperature"=\mathrm{T}\right):
\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{h_O2}≔\mathrm{Chemicals}:-\mathrm{Property}\left("Hmolar","O2\left(g\right)","temperature"=\mathrm{T}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\mathrm{h_H2O}≔\mathrm{Chemicals}:-\mathrm{Property}\left("Hmolar","H2O\left(g\right)","temperature"=\mathrm{T}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\mathrm{h_CO2}≔\mathrm{Chemicals}:-\mathrm{Property}\left("Hmolar","CO2\left(g\right)","temperature"=\mathrm{T}\right):
\mathrm{H_reactants}≔1⟦\mathrm{mol}⟧\cdot \mathrm{h_f_C4H10}
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{150.6640000}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{kJ}}⟧
\mathrm{H_products}≔\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}4 ⟦\mathrm{mol}⟧\cdot \mathrm{h_CO2}+ 5 ⟦\mathrm{mol}⟧ \cdot \mathrm{h_H2O}+ 24.44 ⟦\mathrm{mol}⟧ \cdot \mathrm{h_N2}
\textcolor[rgb]{0,0,1}{\mathrm{H_products}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Chemicals}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{Property}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"Hmolar"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"CO2\left(g\right)"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"temperature"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{T}\right)\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{mol}}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Chemicals}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{Property}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"Hmolar"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"H2O\left(g\right)"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"temperature"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{T}\right)\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{mol}}⟧\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{24.44}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{Chemicals}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{Property}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"Hmolar"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"N2\left(g\right)"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"temperature"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{T}\right)\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{mol}}⟧
\mathrm{fsolve}\left(\mathrm{H_reactants}=\mathrm{H_products},\mathrm{T}=2000⟦\mathrm{K}⟧\right)
\textcolor[rgb]{0,0,1}{2379.853026}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{K}⟧
\mathrm{restart}:
\mathrm{obj}≔\frac{1}{2}\cdot 4\cdot \mathrm{\pi }\cdot {\mathrm{R}}^{2}+2\cdot \mathrm{\pi }\cdot \mathrm{R}\cdot \mathrm{L}+\mathrm{\pi }\cdot \mathrm{R}\cdot \sqrt{{\mathrm{H}}^{2}+{\mathrm{R}}^{2}}:
\mathrm{cons1}≔\frac{1}{2}\frac{4}{3}\mathrm{\pi }\cdot {\mathrm{R}}^{3}+\mathrm{\pi } {\mathrm{R}}^{2}\cdot \mathrm{L}+\frac{1}{3}\cdot \mathrm{\pi }\cdot {\mathrm{R}}^{2}\cdot \mathrm{H}=3⟦{\mathrm{m}}^{3}⟧:
\mathrm{cons2}≔0≤\mathrm{R},0≤\mathrm{L},0≤\mathrm{H}:
\mathrm{dimensions}≔\mathrm{Optimization}:-\mathrm{Minimize}\left(\mathrm{obj},\left\{\mathrm{cons1},\mathrm{cons2}\right\},\mathrm{initialpoint}=\left\{\mathrm{H}=1⟦\mathrm{m}⟧,\mathrm{L}=1⟦\mathrm{m}⟧,\mathrm{R}=1⟦\mathrm{m}⟧\right\}\right)
[\textcolor[rgb]{0,0,1}{10.2533536615869920}\textcolor[rgb]{0,0,1}{}⟦{\textcolor[rgb]{0,0,1}{m}}^{\textcolor[rgb]{0,0,1}{2}}⟧\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.785093823049978}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{L}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.392546902492684}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.877761593519080}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧]]
P≔\mathrm{ThermophysicalData}:-\mathrm{Property}\left("pressure","methane","temperature"=350⟦K⟧,"density"=\frac{1}{V}\right):
We then perform the numeric integration
{∫}_{1.0 {m}^{3} {\mathrm{kg}}^{-1}}^{0.5 {m}^{3} {\mathrm{kg}}^{-1}}P ⅆV
to calculate the work done.
-\mathrm{int}\left(P,V=1.0 ⟦{m}^{3} {\mathrm{kg}}^{-1}⟧.. 0.5 ⟦{m}^{3} {\mathrm{kg}}^{-1}⟧,\mathrm{numeric}\right)
\textcolor[rgb]{0,0,1}{125.4345869}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{\mathrm{kJ}}}{\textcolor[rgb]{0,0,1}{\mathrm{kg}}}⟧
\mathrm{with}\left(\mathrm{ThermophysicalData}:-\mathrm{Chemicals}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{Property}\left("Hmolar","C2H5OH\left(L\right)","temperature"=320⟦\mathrm{K}⟧\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{274957.7153}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{J}}{\textcolor[rgb]{0,0,1}{\mathrm{mol}}}⟧
\mathrm{Property}\left("Hmolar","C2H5OH\left(L\right)","temperature"=320⟦\mathrm{K}⟧\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{274.9577153}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{\mathrm{kJ}}}{\textcolor[rgb]{0,0,1}{\mathrm{mol}}}⟧
The list of suggested unit conversions now gives you
typeset math, e.g.
\frac{\mathrm{J}}{\mathrm{mol} \mathrm{K}}
instead of J/mol/K
sensible suggestions for unit conversions, e.g. selecting a result with units of
\frac{\mathrm{kg} {\mathrm{m}}^{2}}{{\mathrm{s}}^{2} \mathrm{mol}}
\frac{\mathrm{J}}{\mathrm{mol}}
\frac{\mathrm{kJ}}{\mathrm{mol}}
as potential conversions
Ultimate Strength of a Bolt Group
Spring Design Optimization
|
Whirlpool_(hash_function) Knowpia
Whirlpool (hash function)
Version changesEdit
The block cipher W consists of an 8×8 state matrix
{\displaystyle S}
of bytes, for a total of 512 bits.
The encryption process consists of updating the state with four round functions over 10 rounds. The four round functions are SubBytes (SB), ShiftColumns (SC), MixRows (MR) and AddRoundKey (AK). During each round the new state is computed as
{\displaystyle S=AK\circ MR\circ SC\circ SB(S)}
SubBytesEdit
ShiftColumnsEdit
MixRowsEdit
The MixRows operation is a right-multiplication of each row by an 8×8 matrix over
{\displaystyle GF({2^{8}})}
. The matrix is chosen such that the branch number (an important property when looking at resistance to differential cryptanalysis) is 9, which is maximal.
AddRoundKeyEdit
Whirlpool hashesEdit
Even a small change in the message will (with an extremely high probability of
{\displaystyle 1-10^{-154}}
) result in a different hash, which will usually look completely different just like two unrelated random numbers do. The following demonstrates the result of changing the previous input by a single letter (a single bit, even, in ASCII-compatible encodings), replacing d with e:
^ Florian Mendel1, Christian Rechberger, Martin Schläffer, Søren S. Thomsen (2009-02-24). The Rebound Attack: Cryptanalysis of Reduced Whirlpool and Grøstl (PDF). Fast Software Encryption: 16th International Workshop. {{cite conference}}: CS1 maint: multiple names: authors list (link)
^ Barreto, Paulo S. L. M. & Rijmen, Vincent (2003-05-24). "The WHIRLPOOL Hashing Function". Archived from the original (ZIP) on 2017-10-26. Retrieved 2018-08-09. {{cite journal}}: Cite journal requires |journal= (help)
^ Kyoji, Shibutani & Shirai, Taizo (2003-03-11). "On the diffusion matrix employed in the Whirlpool hashing function" (PDF). Retrieved 2018-08-09. {{cite journal}}: Cite journal requires |journal= (help)
|
On complete monotonicity of the Riemann zeta function | Journal of Inequalities and Applications | Full Text
On complete monotonicity of the Riemann zeta function
Ruiming Zhang1
Under the assumption of the Riemann hypothesis for the Riemann zeta function and some Dirichlet L-series we demonstrate that certain products of the corresponding zeta functions are completely monotonic. This may provide a method to disprove a certain Riemann hypothesis numerically.
MSC:30E15, 33D45.
\zeta \left(s\right)
\zeta \left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{{n}^{s}},\phantom{\rule{1em}{0ex}}\mathrm{\Re }\left(s\right)>1,
and on the rest of the complex plane by analytic continuation. It is known that the extended
\zeta \left(s\right)
is meromorphic with infinitely many zeros at
-2n
n\in \mathbb{N}
(a.k.a trivial zeros) and with infinitely many zeros within the vertical strip
0<\mathrm{\Re }\left(s\right)<1
(nontrivial zeros). The Riemann hypothesis for
\zeta \left(s\right)
says that all nontrivial zeros are actually on the critical line
\mathrm{\Re }\left(s\right)=\frac{1}{2}
z\in \mathbb{C}
\mathrm{\Gamma }\left(z\right)
be Euler’s Gamma function defined by [1–8]
\frac{1}{\mathrm{\Gamma }\left(z\right)}=z\prod _{j=1}^{\mathrm{\infty }}\left(1+\frac{z}{j}\right){\left(1+\frac{1}{j}\right)}^{-z}.
Then, the Riemann
\mathrm{\Xi }\left(z\right)
\mathrm{\Xi }\left(z\right)=-\frac{1+4{z}^{2}}{8}{\pi }^{-\frac{1+2iz}{4}}\mathrm{\Gamma }\left(\frac{1+2iz}{4}\right)\zeta \left(\frac{1+2iz}{2}\right)
is an even entire function of order 1. The celebrated Riemann hypothesis is equivalent to the statement that
\mathrm{\Xi }\left(z\right)
has only real zeros.
\chi \left(n\right)
be a real primitive character with modulus m; the function
L\left(s,\chi \right)
is defined by [3, 8]
L\left(s,\chi \right)=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi \left(n\right)}{{n}^{s}},\phantom{\rule{1em}{0ex}}\mathrm{\Re }\left(s\right)>1.
\alpha =\left\{\begin{array}{cc}0,\hfill & \chi \left(-1\right)=1,\hfill \\ 1,\hfill & \chi \left(-1\right)=-1,\hfill \end{array}
\mathrm{\Xi }\left(z,\chi \right)={\left(\frac{\pi }{m}\right)}^{-\left(1+2\alpha +2iz\right)/4}\mathrm{\Gamma }\left(\frac{1+2\alpha +2iz}{4}\right)L\left(\frac{1+2iz}{2},\chi \right)
is an even entire function of order 1. The Riemann hypothesis for
L\left(s,\chi \right)
\mathrm{\Xi }\left(z,\chi \right)
having only real zeros.
Given real numbers a, b with
a<b
and an indefinite differentiable real valued function
f\left(x\right)
\left(a,b\right)
f\left(x\right)
is called completely monotonic on
\left(a,b\right)
{\left(-1\right)}^{m}{f}^{\left(m\right)}\left(x\right)\ge 0
x\in \left(a,b\right)
m=0,1,\dots
. In this work, under the assumptions of the Riemann hypothesis for the Riemann zeta function and certain L-series, we apply the ideas from [8, 9] to prove that some products of these zeta functions are completely monotonic. This complete monotonicity may provide a method to disprove a certain Riemann hypothesis via numerical methods.
Lemma 1 Given a non-increasing sequence of positive numbers such that
\sum _{n=1}^{\mathrm{\infty }}|{\lambda }_{n}|<\mathrm{\infty },
then, the entire function
f\left(x\right)=\prod _{n=1}^{\mathrm{\infty }}\left(1-x{\lambda }_{n}\right)
is completely monotonic on
\left(-\mathrm{\infty },{\lambda }_{1}^{-1}\right)
Proof It is a direct consequence of Theorem 1 of [8]. □
Assuming the Riemann hypothesis is true, we list all positive zeros of
\mathrm{\Xi }\left(z\right)
{z}_{1}\le {z}_{2}\le \cdots \le {z}_{n}\le \cdots ,
{z}_{1}
is approximately 14.1347. Then,
\mathrm{\Xi }\left(z\right)=\mathrm{\Xi }\left(0\right)\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{{z}^{2}}{{z}_{n}^{2}}\right).
\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{{z}_{n}^{2}}\right)=\frac{\mathrm{\Xi }\left(\sqrt{z}\right)}{\mathrm{\Xi }\left(0\right)},
\frac{\mathrm{\Xi }\left({z}^{\frac{1}{4}}\right)\mathrm{\Xi }\left(i{z}^{\frac{1}{4}}\right)}{{\mathrm{\Xi }}^{2}\left(0\right)}=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{{z}_{n}^{4}}\right),
\frac{\mathrm{\Xi }\left({z}^{\frac{1}{6}}\right)\mathrm{\Xi }\left(\rho {z}^{\frac{1}{6}}\right)\mathrm{\Xi }\left({\rho }^{2}{z}^{\frac{1}{6}}\right)}{{\mathrm{\Xi }}^{3}\left(0\right)}=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{{z}_{n}^{6}}\right)
0\le arg\left(z\right)<2\pi
\rho ={e}^{\frac{2\pi i}{3}}
. In fact, for any positive integer
\ell >1
{\rho }_{\ell }
is a primitive ℓ th root of unity; then we have
\frac{{\prod }_{j=1}^{\ell }\mathrm{\Xi }\left({\rho }_{\ell }^{j}{z}^{\frac{1}{2\ell }}\right)}{{\mathrm{\Xi }}^{\ell }\left(0\right)}=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{{z}_{n}^{2\ell }}\right).
Corollary 2 Under the Riemann hypothesis, let
{z}_{1}
be the least positive zeros of
\mathrm{\Xi }\left(z\right)
; then the function
\mathrm{\Xi }\left(\sqrt{z}\right)
is completely monotonic for
z\in \left(-\mathrm{\infty },{z}_{1}^{2}\right)
\mathrm{\Xi }\left({z}^{\frac{1}{4}}\right)\mathrm{\Xi }\left(i{z}^{\frac{1}{4}}\right)
z\in \left(-\mathrm{\infty },{z}_{1}^{4}\right)
\mathrm{\Xi }\left({z}^{\frac{1}{6}}\right)\mathrm{\Xi }\left(\rho {z}^{\frac{1}{6}}\right)\mathrm{\Xi }\left({\rho }^{2}{z}^{\frac{1}{6}}\right)
z\in \left(-\mathrm{\infty },{z}_{1}^{6}\right)
{\rho }_{\ell }
be a primitive ℓth root of unity for some positive integer ℓ; then
{\prod }_{j=1}^{\ell }\mathrm{\Xi }\left({\rho }_{\ell }^{j}{z}^{\frac{1}{2\ell }}\right)
z\in \left(-\mathrm{\infty },{z}_{1}^{2\ell }\right)
\mathrm{\Xi }\left(0\right)
is a positive constant, and the claims are obtained by applying Corollary 1 to equations (2.5)-(2.8). □
Assuming the Riemann hypothesis for
L\left(s,\chi \right)
, we list all the positive zeros for
\mathrm{\Xi }\left(z,\chi \right)
{z}_{1}\left(\chi \right)\le {z}_{2}\left(\chi \right)\le \cdots \le {z}_{n}\left(\chi \right)\le \cdots .
\mathrm{\Xi }\left(z,\chi \right)=\mathrm{\Xi }\left(0,\chi \right)\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{{z}^{2}}{{z}_{n}{\left(\chi \right)}^{2}}\right).
\mathrm{\Xi }\left(0,\chi \right)\ne 0,
\mathrm{\Xi }\left(z,\chi \right)\equiv 0
, which is clearly false. Thus,
\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{{z}_{n}{\left(\chi \right)}^{2}}\right)=\frac{\mathrm{\Xi }\left(\sqrt{z},\chi \right)}{\mathrm{\Xi }\left(0,\chi \right)}
0\le arg\left(z\right)<2\pi
\frac{\mathrm{\Xi }\left({z}^{\frac{1}{4}},\chi \right)\mathrm{\Xi }\left(i{z}^{\frac{1}{4}},\chi \right)}{{\mathrm{\Xi }}^{2}\left(0\right)}=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{{z}_{n}^{4}\left(\chi \right)}\right)
\frac{\mathrm{\Xi }\left({z}^{\frac{1}{6}},\chi \right)\mathrm{\Xi }\left(\rho {z}^{\frac{1}{6}},\chi \right)\mathrm{\Xi }\left({\rho }^{2}{z}^{\frac{1}{6}},\chi \right)}{{\mathrm{\Xi }}^{3}\left(0\right)}=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{{z}_{n}^{6}\left(\chi \right)}\right)
0\le arg\left(z\right)<2\pi
\rho ={e}^{\frac{2\pi i}{3}}
{\rho }_{\ell }
be a primitive ℓ th root of unity for some positive integer ℓ; then we have
\frac{{\prod }_{j=1}^{\ell }\mathrm{\Xi }\left({\rho }_{\ell }^{j}{z}^{\frac{1}{2\ell }},\chi \right)}{{\mathrm{\Xi }}^{\ell }\left(0,\chi \right)}=\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{{z}_{n}^{2\ell }\left(\chi \right)}\right).
Corollary 3 Assume that the Riemann hypothesis is true for
L\left(s,\chi \right)
{z}_{1}\left(\chi \right)
is the least positive zero of
\mathrm{\Xi }\left(z,\chi \right)
\frac{\mathrm{\Xi }\left(\sqrt{z},\chi \right)}{\mathrm{\Xi }\left(0,\chi \right)}
z\in \left(-\mathrm{\infty },{z}_{1}^{2}\left(\chi \right)\right)
\frac{\mathrm{\Xi }\left({z}^{\frac{1}{4}},\chi \right)\mathrm{\Xi }\left(i{z}^{\frac{1}{4}},\chi \right)}{{\mathrm{\Xi }}^{2}\left(0\right)}
z\in \left(-\mathrm{\infty },{z}_{1}^{4}\left(\chi \right)\right)
\frac{\mathrm{\Xi }\left({z}^{\frac{1}{6}},\chi \right)\mathrm{\Xi }\left(\rho {z}^{\frac{1}{6}},\chi \right)\mathrm{\Xi }\left({\rho }^{2}{z}^{\frac{1}{6}},\chi \right)}{{\mathrm{\Xi }}^{3}\left(0\right)}
z\in \left(-\mathrm{\infty },{z}_{1}^{6}\left(\chi \right)\right)
{\rho }_{\ell }
be a primitive ℓth root of unity for some positive integer ℓ, then
{\prod }_{j=1}^{\ell }\mathrm{\Xi }\left({\rho }_{\ell }^{j}{z}^{\frac{1}{2\ell }},\chi \right)
z\in \left(-\mathrm{\infty },{z}_{1}^{2\ell }\left(\chi \right)\right)
Proof These are consequences of Lemma 1 and equations (2.12)-(2.15). □
Abramowitz M, Stegun IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Natl. Bur. of Standards, New York; 1972. Tenth printing with corrections
Davenport H: Multiplicative Number Theory. Springer, New York; 2000.
Erdélyi A: Higher Transcendental Functions, Vol. I. Krieger, Malabar; 1985.
Erdélyi A: Higher Transcendental Functions, Vol. II. Krieger, Malabar; 1985.
Erdélyi A: Higher Transcendental Functions, Vol. III. Krieger, Malabar; 1985.
Titchmarsh EC: The Theory of Riemann Zeta Function. 2nd edition. Clarendon Press, New York; 1987.
Zhang R: Sums of zeros for certain special functions. Integral Transforms Spec. Funct. 2010,21(5):351-365. 10.1080/10652460903286231
Ismail, MEH, Zhang, R: Completely monotonic Fredholm determinants. J. Approx. Theory (submitted)
This research is partially supported by National Natural Science Foundation of China, grant No. 11371294.
Correspondence to Ruiming Zhang.
Zhang, R. On complete monotonicity of the Riemann zeta function. J Inequal Appl 2014, 15 (2014). https://doi.org/10.1186/1029-242X-2014-15
|
2021 Limit theorems for trawl processes
Mikko S. Pakkanen,1 Riccardo Passeggeri,2 Orimar Sauri,3 Almut E. D. Veraart1
1Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK
2Department of Statistical Sciences, University of Toronto, 700 University Ave., Toronto, ON, M5G 1X6, Canada
3Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220, Aalborg, Denmark
{\left({X}_{i{\mathrm{\Delta }}_{n}}\right)}_{i=0}^{⌊nt⌋-1}
n↑\mathrm{\infty }
{\mathrm{\Delta }}_{n}↓0
n{\mathrm{\Delta }}_{n}\to \mathrm{\mu }\in \left[0,+\mathrm{\infty }\right]
R. P. acknowledges the support provided by the Fondation Sciences Mathématiques de Paris (FSMP) fellowship, held at LPSM (Sorbonne University). O. S. would like to thank the Villum Fonden for providing partial funding for this research as part of the project number 11745 titled “Ambit Fields: Probabilistic Properties and Statistical Inference”.
Mikko S. Pakkanen. Riccardo Passeggeri. Orimar Sauri. Almut E. D. Veraart. "Limit theorems for trawl processes." Electron. J. Probab. 26 1 - 36, 2021. https://doi.org/10.1214/21-EJP652
Received: 28 October 2020; Accepted: 22 May 2021; Published: 2021
Keywords: Functional limit theorem , moving average , partial sum , stable convergence , trawl process
Mikko S. Pakkanen, Riccardo Passeggeri, Orimar Sauri, Almut E. D. Veraart "Limit theorems for trawl processes," Electronic Journal of Probability, Electron. J. Probab. 26(none), 1-36, (2021)
|
Exponential function - Simple English Wikipedia, the free encyclopedia
Three different functions: Linear (red), Cubic (blue) and Exponential (green).
In mathematics, the exponential function is a function that grows quicker and quicker. More precisely, it is the function
{\displaystyle \exp(x)=e^{x}}
, where e is Euler's constant, an irrational number that is approximately 2.71828.[1][2][3]
3 Relation to the mathematical constant e
Because exponential functions use exponentiation, they follow the same exponent rules. Thus,
{\displaystyle e^{x+y}=\exp(x+y)=\exp(x)\exp(y)=e^{x}e^{y}}
This follows the rule that
{\displaystyle x^{a}\cdot x^{b}=x^{a+b}}
The natural logarithm is the inverse operation of an exponential function, where:
{\displaystyle \ln(x)=\log _{e}(x)={\frac {\log(x)}{\log(e)}}}
The exponential function satisfies an interesting and important property in differential calculus:
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}e^{x}=e^{x}}
This means that the slope of the exponential function is the exponential function itself, and as a result has a slope of 1 at
{\displaystyle x=0}
. These properties are the reason it is an important function in mathematics.
The general exponential function, where the base is not necessarily
{\displaystyle e}
, is among the most useful of mathematical functions. It is used to represent exponential growth, which has uses in virtually all scientific disciplines and is also prominent in finance. Another application of the exponential function is exponential decay, which occurs in radioactive decay and the absorption of light.
One example of an exponential function in real life would be interest in a bank. If a person deposits £100 into an account which gets 3% interest a month, then the balance each month (assuming the money is untouched) would be as follows:
January £100.00 July £119.41
February £103.00 August £122.99
March £106.09 September £126.68
April £109.27 October £130.48
May £112.55 November £134.39
June £115.93 December £138.42
Here, notice how the extra money from interest increases each month, in that the greater the original balance, the more interest the person will get.
Two mathematical examples of exponential functions (with base a) are shown below.
{\displaystyle x}
{\displaystyle x}
Relation to the mathematical constant e[change | change source]
Even though the base (
{\displaystyle a}
) can be any number bigger than zero, for example, 10 or 1/2, often it is a special number called e. The number e cannot be written exactly, but it is almost equal to 2.71828.
The number e is important to every exponential function. For example, a bank pays interest of 0.01 percent every day. One person takes his interest money and puts it in a box. After 10,000 days (about 30 years), he has 2 times as much money as he started with. Another person takes his interest money and puts it back into the bank. Because the bank now pays him interest on his interest, the amount of money is an exponential function.
In fact, after 10,000 days, he does not have 2 times as much money as he started with, but he has 2.718145 times as much money as he started with. This number is very close to the number e. If the bank pays interest more often, so the amount paid each time is less, then the number will be closer to the number e.
A person can also look at the picture to see why the number e is important for exponential functions. The picture has three different curves. The curve with the black points is an exponential function with a base slightly smaller than e. The curve with the short black lines is an exponential function with a base slightly bigger than e. The blue curve is an exponential function with a base exactly equal to e. The red line is a tangent to the blue curve. It touches the blue curve at one point without crossing it. A person can see that the red curve crosses the x-axis, the line that goes from left to right at -1. This is true only for the blue curve. This is the reason that the exponential function with the base e is special.
e is the unique number a, such that the value of the derivative of the exponential function f (x) = ax (blue curve) at the point x = 0 is exactly 1. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 (red).
↑ Weisstein, Eric W. "Exponential Function". mathworld.wolfram.com. Retrieved 2020-08-28.
↑ "Exponential Function Reference". www.mathsisfun.com. Retrieved 2020-08-28.
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Exponential_function&oldid=8044690"
|
Correlating Microscale Thermal Conductivity of Heavily-Doped Silicon With Simultaneous Measurements of Stress | J. Eng. Mater. Technol. | ASME Digital Collection
Ming Gan,
e-mail: tomar@purdue.edu
Gan, M., and Tomar, V. (October 20, 2011). "Correlating Microscale Thermal Conductivity of Heavily-Doped Silicon With Simultaneous Measurements of Stress." ASME. J. Eng. Mater. Technol. October 2011; 133(4): 041013. https://doi.org/10.1115/1.4004699
The functioning and performance of today’s integrated circuits and sensors are highly affected by the thermal properties of microscale silicon structures. Due to the well known size effect, the thermal properties of microscale silicon structures are not the same as those of the bulk silicon. Furthermore, stress/strain inside microscale silicon structures can significantly affect their thermal properties. This article presents the first thermal conductivity measurements of a microscale silicon structure under applied compressive stress at 350 K. Atomic force microscope (AFM) cantilevers made of doped single-crystal silicon were used as samples. A resistance temperature detector (RTD) heater attached to another RTD sensor was used as the heating module, which was mounted onto a nanoindentation test platform. This integrated system applied compressive load to the cantilever in the longitudinal direction while supplying heat. The thermal conductivity of the cantilevers was calculated using steady state heat conduction equation. The result shows that the measured thermal conductivity varies from
110 W/(m·K)
140 W/(m·K)
as compressive strain varies from 0.1% to 0.3%. Thermal conductivity is shown to increase with increase in compressive strain. These results match with the published simulation values. The measured thermal conductivity and stress values vary in the similar manner as a function of applied strain.
atomic force microscopy, cantilevers, elemental semiconductors, nanoindentation, semiconductor doping, silicon, stress-strain relations, thermal conductivity, thermal resistance, thermal conductivity measurement, stress/strain, micro/nano-scale, silicon structures, RTD sensor, Fourier’s law
Cantilevers, Microscale devices, Silicon, Stress, Thermal conductivity, Atomic force microscopy, Heating, Sensors, Temperature, Nanoindentation
Temperature measurements of semiconductor devices—A Review
, 20th Annual IEEE.
International Technology Roadmap for Semiconductors, 2010.
Prediction and Measurement of Temperature-Fields in Silicon-on-Insulator Electronic-Circuits
Thermal Conductivity of Silicon + Germanium From 3 Degrees K to Melting Point
Phys. Rev. A—Gen. Phys.
Thermal Conductivity-Metallic Elements and Alloys
The TPRC Data Series
Strain Effect Analysis on Phonon Thermal Conductivity of Two-Dimensional Nanocomposites
Size Effect on the Thermal Conductivity of Nanowires
Phonon Heat Transport in Silicon Nanowires
Thermal-Conductivity of Crystals—A Molecular-Dynamics Study of Heat-Flow in a Two-Dimensional Crystal
Size Effect of Lattice Thermal Conductivity Across Nanoscale Thin Films by Molecular Dynamics Simulations
Molecular Dynamics Study on Thermal Conductivity of Nanoscale Thin Films
Heat Transfer From Freely Suspended Biomaterial Microcantilevers
Thermal Conductivity of Heavily Doped LPCVD Polysilicon
Electron Devices Meeting
The Effects of Processing Conditions on the Thermal Conductivity of Polycrystalline Silicon Films
Micro-Raman Spectroscopy to Study Local Mechanical Stress in Silicon Integrated Circuits
Role of Mechanical Strain on Thermal Conductivity of Nanoscale Aluminum Films
Influence of Applied In-Plane Strain on Transverse Thermal Conductivity of 0°/90° and Plain Weave Ceramic Matrix Composites
The Effect of Residual Strain on the Thermal Conductivity of Nuclear Graphite
Er. -W.
Effect of Doping on the Raman Scattering of 6H-SiC Crystals
Temperature Characterization of Silicon Substrates for Gas Sensors by Raman Spectroscopy
|
'''A Geometric Approach to Complexity'''
I discuss several complexity measures of random fields from a geometric perspective. Central to this approach is the notion of multi-information, a generalization of mutual information. As
demonstrated by Amari, information geometry allows to decompose this measure in a natural way. In my talk I will show how this decomposition leads to a unifying scheme of various approaches to complexity. In particular, connections to the complexity measure of Tononi, Sporns, and Edelman and also to excess entropy (predictive information) can be established. In the second part of my talk, the interplay between complexity and causality (causality in Pearl's sense) will be discussed. A generalization of Reichenbach's common cause principle will play a central role in this regard.
'''Information Aggregation in Correlated Complex Systems and Optimal Estimation'''
Information is a peculiar quantity. Unlike matter and energy, which are conserved by the laws of physics, the aggregation of knowledge from many sources can in fact produce more information (synergy) or less (redundancy) than the sum of its parts, provided these sources are correlated. I discuss how the formal properties of information aggregation - expressed in information theoretic terms - provide a general window for explaining features of organization in several complex systems. I show under what circumstances collective coordination may pay off in stochastic search problems, how this can be used to estimate functional relations between neurons in living neural tissue and more generally how it may have implications for other network structures in social and biological systems.
Links: [[Media:Darwin.pdf| Paper]]
'''Framing Complexity''' [[Media:CrutchfieldTalkSlides.pdf|PDF]]
Links: [[Media:afm.tri.5.pdf| Paper 1]] and [[Media:CHAOEH184043106_1.pdf| Paper 2]]
'''Introduction to the Workshop''' [[Media:MachtaWorkshopIntro.pdf|PDF]]
In this talk I argue that a fundamental measure of physical complexity is obtained from the parallel computational complexity of sampling states of the system. After motivating this idea, I will briefly review relevant aspects of computational complexity theory, discuss the properties of the proposed measure of physical complexity and illustrate the ideas with some examples from statistical physics.
'''Thermodynamics and Local Complexity of Domino Gases'''
'''Dominos, Ergodic Flows'''
'''Ergodic Parameters and Dynamical Complexity''' [[Media:VilelaMendezTalksSlides.pdf|PDF]]
Wiesner, Karoline (k.wiesner@bristol.ac.uk) [[Media:WiesnerTalkSlides.pdf|PDF]]
I discuss several complexity measures of random fields from a geometric perspective. Central to this approach is the notion of multi-information, a generalization of mutual information. As demonstrated by Amari, information geometry allows to decompose this measure in a natural way. In my talk I will show how this decomposition leads to a unifying scheme of various approaches to complexity. In particular, connections to the complexity measure of Tononi, Sporns, and Edelman and also to excess entropy (predictive information) can be established. In the second part of my talk, the interplay between complexity and causality (causality in Pearl's sense) will be discussed. A generalization of Reichenbach's common cause principle will play a central role in this regard.
Information Aggregation in Correlated Complex Systems and Optimal Estimation
Framing Complexity PDF
{\displaystyle n}
{\displaystyle n^{\beta }}
{\displaystyle n^{\beta }/\log n}
Introduction to the Workshop PDF
Ergodic Parameters and Dynamical Complexity PDF
Wiesner, Karoline (k.wiesner@bristol.ac.uk) PDF
|
scoring: computing various performance metrics - mlxtend
Example 1 - Classification Error
A function for computing various different performance metrics.
from mlxtend.evaluate import scoring
P
N
Both the prediction error (ERR) and accuracy (ACC) provide general information about how many samples are misclassified. The error can be understood as the sum of all false predictions divided by the number of total predications, and the the accuracy is calculated as the sum of correct predictions divided by the total number of predictions, respectively.
True and False Positive Rates
The True Positive Rate (TPR) and False Positive Rate (FPR) are performance metrics that are especially useful for imbalanced class problems. In spam classification, for example, we are of course primarily interested in the detection and filtering out of spam. However, it is also important to decrease the number of messages that were incorrectly classified as spam (False Positives): A situation where a person misses an important message is considered as "worse" than a situation where a person ends up with a few spam messages in his e-mail inbox. In contrast to the FPR, the True Positive Rate provides useful information about the fraction of positive (or relevant) samples that were correctly identified out of the total pool of Positives.
Precision, Recall, and the F1-Score
Precision (PRE) and Recall (REC) are metrics that are more commonly used in Information Technology and related to the False and True Prositive Rates. In fact, Recall is synonymous to the True Positive Rate and also sometimes called Sensitivity. The F
_1
-Score can be understood as a combination of both Precision and Recall.
Sensitivity (SEN) is synonymous to Recall and the True Positive Rate whereas Specificity (SPC) is synonymous to the True Negative Rate -- Sensitivity measures the recovery rate of the Positives and complimentary, the Specificity measures the recovery rate of the Negatives.
Matthews correlation coefficient (MCC) was first formulated by Brian W. Matthews [3] in 1975 to assess the performance of protein secondary structure predictions. The MCC can be understood as a specific case of a linear correlation coefficient (Pearson's R) for a binary classification setting and is considered as especially useful in unbalanced class settings. The previous metrics take values in the range between 0 (worst) and 1 (best), whereas the MCC is bounded between the range 1 (perfect correlation between ground truth and predicted outcome) and -1 (inverse or negative correlation) -- a value of 0 denotes a random prediction.
Average Per-Class Accuracy
y_targ = [1, 1, 1, 0, 0, 2, 0, 3]
res = scoring(y_target=y_targ, y_predicted=y_pred, metric='error')
print('Error: %s%%' % (res * 100))
|
How to Find Derivatives in 3 Steps | Outlier
What is a derivative, and how can we calculate it? In this article, we’ll discuss the definition of a derivative and 3 steps to differentiate functions. Then, you can test your knowledge with some examples.
3 Steps to Find Derivatives
Derivatives measure the instantaneous rate of change of a function. When we talk about rates of change, we’re talking about slopes.
The instantaneous rate of change of a function at a point is equal to the slope of the function at that point. When we find the slope of a curve at a single point, we find the slope of the tangent line. The tangent line to a function at a point is a line that just barely touches the function at that point.
So, the derivative of a function at a single point is equal to the slope of the tangent line at that point.
To visualize the tangent line, let’s look at an example where the equation for the tangent line has already been calculated. Consider the function
f(x) = \ln{(x)}
, indicated by the blue line.
Suppose we want to find the derivative of
f(x)
(1, 0)
. The tangent line to the curve
f(x)
(1, 0)
is represented by the red line,
f(x) = x - 1
. Notice how this line touches
f(x) = \ln{(x)}
at just one point,
(1, 0)
f(x) = x - 1
is given in the slope-intercept form
f(x) = mx +b
m
is the slope. So, we can easily see that the slope of
f(x) = x - 1
This means that the instantaneous rate of change, or derivative, of the function
f(x) = \ln{(x)}
(0, 1)
The instantaneous rate of change of
a
, or the derivative of
a
, is represented by the notation
f’(a)
. We read aloud the symbol
f’(a)
as either “the derivative of
a
f
a
The general derivative function of
y = f(x)
is usually represented by either
f’(x)
\frac{dy}{dx}
. (You can read more about the meaning of dy/dx if needed.) This function tells us the instantaneous rate of change of
x
at any point on the curve.
You can watch Dr. Hannah Fry explain more about what a derivative is below.
Next, we’ll learn exactly how to find the derivative of a function. So far, we’ve learned that the slope at a point on a curve is called the slope of the tangent line or the instantaneous rate of change.
By contrast, the slope between two separate points on a curve is called the slope of the secant line. This slope value is also referred to as the average rate of change. The average rate of change will help us calculate the derivative of a function.
To find the average rate of change, we divide the change in the output values (y-values) by the change in the input values (x-values). The delta symbol
\Delta{x}
represents the "change in
x
," which is the value that
x
is changing by. The average rate of change of the function
f
[a, b]
\text{Average Rate of Change} = \frac{\Delta{y}}{\Delta{x}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(b)-f(a)}{b-a}
\Delta{x}
approach 0, we can find the instantaneous rate of change. Take a look at the limit definition of a derivative below.
The derivative of
x
is equal to the limit of the average rate of change of
[x, x +\Delta{x}]
\Delta{x}
approaches 0. This limit is given by:
f’(x) = \mathop {\lim }\limits_{\Delta{x} \to 0}\frac{\Delta{y}}{\Delta{x}}=\mathop{\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x\right)}}{\Delta{x} }=L
f(x)
is differentiable, and the derivative of the function
f
x
L
. For a brief review of limits, read What Are Limits? and How to Find Limits.
Here are 3 simple steps to calculating a derivative:
Substitute your function into the limit definition formula.
Let’s walk through these steps using an example. Suppose we want to find the derivative of
f(x) = 2x^2
First, we need to substitute our function
f(x) = 2x^2
into the limit definition of a derivative. Substituting the first term of the limit definition’s numerator correctly can be tricky at first. The key is to simply substitute
x
(x + \Delta{x})
x
appears in the function.
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} }
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{2(x + \Delta{x})^2 - 2x^2}{\Delta{x}}
Next, we simplify our function as much as we can. First, we’ll expand the term
2(x + \Delta{x})^2
, and then combine like terms. Then, since
\Delta{x}
is present in all terms of the numerator and denominator, we can divide by
\Delta{x}
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{2(x^2 + 2x\Delta{x} + \Delta{x}^2) - 2x^2}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{2x^2 + 4x\Delta{x} + 2\Delta{x}^2 - 2x^2}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{4x\Delta{x} + 2\Delta{x}^2}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0}4x + 2\Delta{x}
Finally, we can evaluate the limit as
\Delta{x}
approaches 0. Since we’re left with a polynomial function and polynomials are always continuous, we can simply substitute
\Delta{x} = 0
into the function we’re left with.
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0}4x + 2\Delta{x}
= 4x + 2(0)
= 4x
Thus, the general derivative formula of
f(x) = 2x^2
4x
. If you want to find the derivative at a single point, you can simply plug
x = a
f’(x) = 4x
f’(1) = 4(1) = 4
. This value represents the slope of the tangent line at
x = 1
f’(2) = 4(2) = 8
x = 2
f’(100) = 4(100) = 400
x = 100
Standard Derivative Rules
Now that you’re familiar with the limit definition of a derivative, you can begin to memorize the standard derivative rules below. These derivative rules are derived from the limit definition of a derivative. They allow us to evaluate derivatives much faster.
Here are some of the most common derivative rules to know:
\frac{d}{dx}c = 0
\frac{d}{dx}(x^n) = nx^{n-1}
\frac d{dx}(x)=1
\frac d{dx}(c\cdot f(x))=c\cdot f'(x)
\frac{d}{dx}f(g(x)) = f’(g(x))g’(x)
\frac{d}{dx}[f(x) \cdot g(x)] = f’(x) \cdot g(x) + f(x)\cdot g’(x)
\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}
\frac{d}{dx}[f(x) \pm g(x)] = f’(x) \pm g’(x)
\frac{d}{dx}(\sin{(x)}) = \cos{(x)}
\frac{d}{dx}(\cos{(x)}) = -\sin{(x)}
\frac{d}{dx}(\tan{(x)}) = \sec ^2 (x)
\frac{d}{dx} (\ln{x}) = \frac{1}{x}
\frac{d}{dx}(e^x) = e^x
Dr. Tim Chartier explains more about the Product and Quotient derivative rules.
f(x) = 7x - 1
. Using the limit definition of a derivative, find
f’(x)
Substituting your function into the limit definition can be the hardest step for functions with multiple terms. Remember to double-check your answer, use parentheses where necessary, and distribute negative signs appropriately.
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{7(x + \Delta{x}) - 1 - (7x - 1)}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{7x + 7\Delta{x} - 1 - 7x + 1}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{7\Delta{x}}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} 7
=7
f’(x) = 7
f(x) = \frac{1}{x}
f’(x)
After substituting our function into the limit definition, we’ll need to combine the two fractions in the numerator by finding a common denominator and then multiplying appropriately.
f’(x)= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\frac{1}{x + \Delta{x}} - \frac{1}{x}}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\frac{x}{x(x + \Delta{x})} - \frac{(x+ \Delta{x})}{x(x+ \Delta{x})}}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\frac{x-x-\Delta{x}}{x(x+\Delta{x})}}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{-\Delta{x}}{x(x+ \Delta{x})} \cdot \frac{1}{\Delta{x}}
= \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{-1}{x(x+\Delta{x})}
= \frac{-1}{x(x+0)}
= \frac{-1}{x \cdot x}
= -\frac{1}{x^2}
f’(x) = -\frac{1}{x^2}
f(x) = 3x^2 + e^{cos{(x)}}
. Using the derivative rules, find
f’(x)
For this problem, we’ll need to use the Sum Rule. The Sum Rule states that the derivative of a sum of functions is equal to the sum of their derivatives. To find the derivative of each separate function, we can use the Power Rule and the Constant Multiple Rule for the first term, and the Chain Rule, trigonometry rules, and the exponential rule for the second term.
For the first term, we use the Power Rule with
n=2
, combining that with the Constant Multiple Rule where
c=3
. Using these, we see that the derivative of the first term is
f'(x)=3\cdot2x^{2-1}=3\cdot2x=6x
For the second term, we also have a composition of functions with
e^{\cos{(x)}}
. The Chain Rule says that the derivative of a composition of functions is found by first taking the derivative of the "outside" function and leaving the “inside” unchanged, and then multiplying by the derivative of the "inside" function.
So, to find the derivative of
e^{\cos{(x)}}
, we can simply use the exponential rule and then multiply by
-\sin{(x)}
, which is the derivative of the inside function
\cos{(x)}
f’(x) = 6x + e^{\cos{(x)}}\cdot (-\sin{(x)})
f’(x) = 6x -\sin{(x)}e^{\cos{(x)}}
f’(x) = 6x -\sin{(x)}e^{\cos{(x)}}
f(x) = 2x\sin{(5x)}
f’(x)
For this problem, we’ll need to use the Product Rule. The product rule states that the derivative of a product of functions is the sum of the first function times the derivative of the second and the second function times the derivative of the first.
The first function in our product is
2x
. To find the derivative of this we use the special case of the Product Rule with
n=1
as well as the Constant Multiple Rule with
c=2
. Applying these rules, we find that the derivative of
2x
\frac d{dx}\left(2x\right)=2\cdot1=2
For the second function in our product, we have a composition of functions with
\sin{(5x)}
\sin{(5x)}
, we can simply use the trigonometry rules for sin and then multiply by
5
5x
. The derivative of the inside function is found by using the special case of the Power Rule for
n=1
and the Constant Multiple Rule.
Plugging the derivatives we found into the Product Rule, we get:
f’(x) = 2x \cdot \cos{(5x)} \cdot 5 + \sin{(5x)} \cdot 2
f’(x) = 10x\cos{(5x)} + 2\sin{(5x)}
f’(x) = 10x\cos{(5x)} + 2\sin{(5x)}
|
Hemodilution of Prostate-Specific Antigen Levels Among Obese Men | Cancer Epidemiology, Biomarkers & Prevention | American Association for Cancer Research
Hemodilution of Prostate-Specific Antigen Levels Among Obese Men
Catherine Richards;
Department of Epidemiology, Mailman School of Public Health and Department of Medicine and the Herbert Irving Comprehensive Cancer Center, Columbia University, New York City, NY
Andrew Rundle, Catherine Richards, Alfred I. Neugut; Hemodilution of Prostate-Specific Antigen Levels Among Obese Men. Cancer Epidemiol Biomarkers Prev 1 August 2009; 18 (8): 2343. https://doi.org/10.1158/1055-9965.EPI-09-0441
To the Editor: A recent article by Grubb and colleagues (1) shows that the inverse relationship between prostate-specific antigen (PSA) test results and body mass index (BMI) among men from the Prostate, Lung, Colorectal, and Ovarian Cancer Screening Trial can be explained by the higher blood volume in the obese, an effect called hemodilution (2). Their finding is consistent with our analyses in the EHE International, Inc. clinical population (3). We suggested new PSA cut points for clinical follow-up for obese and morbidly obese men, but agree with Grubb et al. (1) that using body surface area in conjunction with PSA test results to set clinical cut points has merit (3). We suggest an equation based on hemodilution models to standardize PSA test results for differences in body surface. The equation is an algebraic rearrangement of the formulas used by Grubb et al. (1), and ourselves, standardizing the PSA test results to a man who has the mean height for the U.S. (175.77 cm) and whose weight is equivalent to a BMI of 25 for that height (3). The adjusted PSA value can then be compared with standard thresholds for clinical decision making regarding follow-up.
AdjustedPSA=((0.007184*(heightincm0.725)*(weightinkg0.425))*1578*PSAtestscore)(0.007184*(175.77cm0.725)*(77.25kg0.425)*1578)
Using this approach, a man who is 183 cm tall (6 foot), 100 kg (220 pounds), with a PSA of 3 ng/mL, would have an adjusted PSA of 3.45; if he were 143 kg (313 pounds), the adjusted PSA would be 4.0.
Grubb et al. (1) also discuss the possibility that the inverse association between PSA and BMI could be due to hormonal disturbances caused by obesity, rather than a hemodilution effect. However, our recently published analyses show that both lean and fat mass are inversely associated with PSA test scores (4). The finding of an association between PSA test results and lean mass is inconsistent with a hormone disturbance theory, which focuses on the effects of fat mass on PSA. However, because both lean and fat mass require a blood supply the finding is consistent with the hemodilution theory.
We are gratified to see that others have observed results regarding hemodilution and PSA similar to those observed by Banez and colleagues and ourselves (2, 3). We suggest the formula for calculating a hemodilution adjusted PSA score will be useful for physicians working with obese men.
Body composition, abdominal fat distribution, and prostate-specific antigen test results
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Systematic Study on Thermo-Mechanical Durability of Pb-Free Assemblies: Experiments and FE Analysis | J. Electron. Packag. | ASME Digital Collection
CALCE Electronic Products and Systems Center, Mechanical Engineering Department,
, McKinney, TX 75069
Hector Pallavicini
Zhang, Q., Dasgupta, A., Nelson, D., and Pallavicini, H. (January 6, 2005). "Systematic Study on Thermo-Mechanical Durability of Pb-Free Assemblies: Experiments and FE Analysis." ASME. J. Electron. Packag. December 2005; 127(4): 415–429. https://doi.org/10.1115/1.2098812
As the ban of the Pb use in electronics products is approaching due to the waste electrical and electronic equipment (WEEE) and restriction of hazardous substances (ROHS) directives, electronics companies start to deliver the products using the Pb-free solders. There are extensive databases of mechanical properties, durability properties (for both mechanical and thermal cycling), and micromechanical characteristics for Sn-Pb solders. But similar databases are not readily yet available for Pb-free solders to predict its mechanical behavior under environmental stresses. In this study, the thermo-mechanical durability of the Pb-free Sn3.8Ag0.7Cu solder is investigated by a systematic approach combining comprehensive thermal cycling tests and finite element modeling. A circuit card assembly (CCA) test vehicle was designed to analyze several design and assembly process variables when subjected to environmental extremes. The effects of mixed solder systems, device types, and underfill are addressed in the thermal cycling tests. The thermal cycle profile consisted of temperature extremes from
−55to+125°
Celsius with a
15min
dwell at hot, a
10min
dwell at cold, and a 5–10° Celsius per minute ramp. Thermal cycling results show that Sn3.8Ag0.7Cu marginally outperforms SnPb for four different components under the studied test condition. In addition, the extensive detailed three-dimensional viscoplastic finite element stress and damage analysis is conducted for five different thermal cycling tests of both Sn3.8Ag0.7Cu and Sn37Pb solders. Power law thermo-mechanical durability models of both Sn3.8Ag0.7Cu and Sn37Pb are obtained from thermal cycling test data and stress and damage analysis. The results of this study provide an important basis of understanding the thermo-mechanical durability behavior of Pb-free electronics under thermal cycling loading and environmental stresses.
tin alloys, silver alloys, copper alloys, durability, thermomechanical treatment, solders, assembling, viscoplasticity, printed circuit testing, surface mount technology, finite element analysis
Durability, Finite element analysis, Solder joints, Solders, Thermomechanics, Stress, Damage, Cycles, Vehicles, Lead-free solders, Failure, Manufacturing, Modeling
Overview of Lead-Free Solders for Slectronics and Microelectronics
Surface Mount International Conference and Exposition
, Surface Mount Int., Edina, MN, USA, pp.
A Manufacturable Lead-Free Surface-Mount Process?
Circuits Assem.
Lead-Free Reflow Soldering for Electronics Assembly
Surf. Mount. Technol.
52nd Electronic Components and Technology Conference
, CA, May 28–31 2002, pp.
Sunappan
Lead-Free Solder Process Implementation for PCB Assembly
, 2002, “Six Cases of Reliability Study of Pb-Free Solder Joints in Electron Packaging Technology,”
, R38, pp.
Manufacturing and Reliability of Chip Scale Area Array Packaging in Avionics Environments
Proceedings of the 2002 SMTA International Conference
, Rosemont, IL, September 22–26, pp.
Durability of Lead-Free Solder Connections for Area-Array Packages
IPC SMEMA Council APEX 2001
, LF2–7.
, Personal Communications, 2003.
Solder Joint Reliability of BGA, Flip Chip, CSP, and Fine Pitch SMT Assemblies
Chip Scale Package (CSP) Solder Joint Reliability and Modeling
Isothermal Mechanical and Thermo-Mechanical Durability Characterization of Selected Pb-Free Solders
, Ph.D. Dissertation, University of Maryland, College Park, MD.
Test Method Considerations for SMT Solder Joint Durability
, Autumn, p.
Mechanical Behaviors of 60∕40 Tin-Lead Solder Lap Joints
Fatigue of Solder Joints in Surface Mount Devices
, eds., ASTM STP, Philadelphia, p.
Energy-Based Methodology for the Fatigue Life Prediction of Solder Materials
Low Cycle Fatigue of Surface Mounted Chip Carrier/Printed Wiring Board Joints
Proceedings IEEE 39th ECC
Design and Reliability of Solders and Solder Interconnections
Calibration of Virtual Qualification Model for Leaded Packages With Pb-Free Solder
,” CALCE Internal Report.
Isothermal Mechanical Durability of Three Selected PB-Free Solders: Sn3.9Ag0.6Cu, Sn3.5Ag, and Sn0.7Cu
Effect of Dwell Times and Ramp Rates on the Thermal Cycling Reliability of Pb-Free Wafer-Level Chip Scale Packages: Experiments and Modeling
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8.2: What Is a Model? - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F08%253A_Fitting_Models_to_Data%2F8.02%253A_What_Is_a_Model
In statistics, a model is meant to provide a similarly condensed description, but for data rather than for a physical structure. Like physical models, a statistical model is generally much simpler than the data being described; it is meant to capture the structure of the data as simply as possible. In both cases, we realize that the model is a convenient fiction that necessarily glosses over some of the details of the actual thing being modeled. As the statistician George Box famously said: “All models are wrong but some are useful.”
data = model + error
This expresses the idea that the data can be described by a statistical model, which describes what we expect to occur in the data, along with the difference between the model and the data, which we refer to as the error.
8.2: What Is a Model? is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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Mt. Rose Middle School collected canned food to donate to a local charity. Each classroom kept track of how many cans it collected. The number of cans in each room were:
107, 55, 39, 79, 86, 62, 65, 70, 80
77
What is the range of the data? Are there any outliers?
The range is the length of the interval from the lowest to the highest data point. An outlier is data that is much higher or lower than the others. In other words, it does not fit in with the others.
68
Outliers:
39
107
could be outliers
55
74
67
On graph paper, make a new box plot that includes this data. Clearly label the median and the upper and lower quartiles.
55
74
67
58.5
79.5
Remember, the median is the middle value that divides the upper and lower halves. The lower quartile is the middle value of the lower half of the data, while the upper quartile is the middle value of the upper half of the data.
Use the eTool below to create the new plot.
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Relatively Elastic Demand: A Complete Overview | Outlier
5 Different Types of Price Elasticity of Demand
What Is Relative Elasticity?
What Causes Relative Elasticity?
4 Different Factors That Influence Elasticity
Elasticity is the percent change in one variable relative to a change in another variable. In economics, we use elasticity to measure the responsiveness of buyers and sellers relative to the change in the price of goods and services. Before explaining what is relatively elastic, let's review how elasticity works and the five different types of elasticity.
Although elasticity of demand can sound a bit technical or intimidating at first, we will see throughout this article that elasticity is very intuitive. That is because elasticity is an outcome of the well-known law of demand.
From the law of demand, we know there is an inverse relationship between the price of a product and the quantity demanded. If the price of goods increases, the demand will decrease. Similarly, if the price decreases, more people will want to buy the goods and demand will increase. The price elasticity of demand measures the percentage change of the quantity demanded relative to its price change.
As we can imagine, certain goods are more sensitive to a change in price than others. For example, when the price of medicine changes, it has a minor effect on the demand for medicine than the price change of an airline ticket to a vacation destination. This is because medicine is usually a necessary good, so consumers are not very responsive to price fluctuations. However, a slight price change of an airline ticket for vacation will have a much more significant effect on the quantity of demand since consumers are much more price-conscious when buying an airplane ticket to go on vacation.
To understand these dynamics more, let’s go over what the following five types of price elasticity are:
Unit elasticity means that the percentage change in price causes an equal percentage change in the quantity demanded.
Economists consider something is unit elastic to always equal to one.
\frac{\text{Percentage Change in Quantity}}{\text{Percentage Change in Price}} = \text{ED 1}
A value of one indicates that the percentage change in price has the same exact percentage change in the quantity demanded.
2. Elastic Demand
Elastic demand means consumers are responsive to price changes. Goods that have a percentage change in price which results in a greater percentage change in the quantity demanded are elastic.
Economists consider elasticity with a numerical value greater than 1 to be elastic.
\frac{\text{Percentage Change in Quantity}}{\text{Percentage Change in Price}} = \text{ED > 1}
A value of 1 or higher indicates that the percentage change in price results in a more significant percentage change in quantity demanded.
We can think about elasticity like a rubber band. The more you can stretch a rubber band the more elastic it is. A tighter rubber band that does not stretch is inelastic. The same idea applies to the demand curve for goods and services in the economy. The more sensitive consumers are to a small change in price, the more elastic they will be.
Inelastic demand means consumers are not very responsive to price changes. If the percentage change in price does not have a significant impact on the quantity demand then those goods are inelastic.
Economists consider elasticity with a numerical value less than 1 to be inelastic.
\frac{\text{Percentage Change in Quantity}}{\text{Percentage Change in Price}} = \text{ED < 1}
A value less than 1 indicates that the percentage change in price does not result in a significant percentage change in quantity demanded.
Generally, goods that are more of a necessity like gasoline or medicine tend to be inelastic. This makes intuitive sense, consumers do not have many choices if the price of gas or medicine goes up. Therefore, the demand curve is not significantly impacted by a price change
Perfectly inelastic demand means that no matter what the price change will be there will be absolutely no change in the quantity of demand.
Economists consider elasticity with a numerical value of zero to be perfectly inelastic.
\frac{\text{Percentage Change in Quantity}}{\text{Percentage Change in Price}} = \text{ED 0}
A value of zero indicates that the percentage change in price has no effect on the quantity demanded.
Perfectly inelastic goods are rare but can happen if the good is an absolute necessity. For example, no matter how much the price will change for a life-saving drug, the demand will stay the same. The graph below shows how a perfectly inelastic demand curve will be a vertical line. The straight vertical demand curve indicates that the quantity demanded remains unchanged no matter the different prices.
Perfectly elastic demand means that even the slightest price change will deter consumers from buying the product.
Perfectly elastic demand is infinity because the slightest price increase will make demand go to zero.
\frac{\text{Percentage Change in Quantity}}{\text{Percentage Change in Price}} = \text{ED ∞ }
Perfectly elastic demand happens when there are plenty of close substitutes in a competitive market. For example, if two ice cream trucks in front of the park sell the same exact product. If one truck increases the price, people will buy ice cream from the other (cheaper) ice cream truck, so the demand for ice cream from the more expensive truck will be zero. A perfectly elastic demand curve is horizontal. This indicates that even a slight price increase will cause the demand to zero.
Most luxury products are elastic goods. Since they are not essential to living life, consumers are picky about the price. Another example that often has elastic demand is goods purchased only a few times throughout one's lifetime, like a washing machine or car; these tend to be elastic goods because consumers can shop around for a better price.
What Is the Price Elasticity for an Airline Ticket to Disneyland?
The graph below shows the demand curve for an airline ticket to Disneyland in Orlando, FL, from New York City.
At a price of $200 (P1) the quantity demanded is 300 (Q1). If the price rises to $240 (P2), the quantity demanded falls to 200 (Q2). This is elastic demand because a 20% increase in the price causes a 33% decrease in the quantity demanded.
We know the formula for Price Elasticity of Demand (ED):
\text{Price of ED} = \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}}
So we get ED:
\frac{\text{33 Percent Change in Quantity Demanded}}{\text{22 Percent Change in Price}} = 1.65
Since the price elasticity for an airline ticket to Orlando is 1.65, the demand curve is elastic (ED > 1 = elastic demand).
Relative elasticity means that if comparing the demand curve of two different goods and one has a greater consumer response to a price change, then the other is relatively elastic. In the real world of business, the two extremes of perfectly elastic and perfectly inelastic are not very common. However, economists and business managers often use the relative elasticity of two different demand curves to measure how consumers will react to price changes in different goods.
Relative elasticity means that consumers have a certain amount of responsiveness to price changes for every product type. Therefore, we’ll compare two different demand curves to get the relative elasticity. By comparing the demand curve of one good relative to another product, we can determine what is more elastic—i.e., what goods are more sensitive to price changes. The graph on the left is the demand curve for gasoline, and the graph on the right is the demand curve for Boba Tea.
As we can see, the demand curve for Boba tea is more horizontal than the demand curve for gasoline. This indicates a significant change in demand with each price change. On the other hand, the demand curve for gasoline is relatively a vertical line because the demand is not too sensitive to gasoline prices.
Without calculating how much the elasticity is at each point on the graph, the difference in elasticity here makes intuitive sense. Since as we can imagine, even if the price of gas goes up, people who drive cars don't have many options (at least in the short-run), so they will still have to buy gas at a higher price. But, on the other hand, the price for Boba tea is relatively elastic. So if one tea shop decides to raise the price, it will affect the size of the demand.
Now that we understand the definition of elasticity, we will discuss several determinants that cause the market of certain goods to be more elastic than others. The causes which influence relative elasticity are external factors that influence people’s response to price changes:
1. Necessity of a Good
How essential goods are will be a significant factor in determining the relative elasticity. For example, the demand for basic foods like bread, milk, and eggs does not change much depending on price because these are things that are needed to live. Conversely, luxury goods or even something that we can skip buying once in a while, like candy or soda, are more sensitive to price since consumers are more price-conscious with unessential goods.
The availability of substitutes will determine the elasticity of something. If there is nothing that can easily replace a particular product or service, then higher prices will not automatically deter consumers from buying it. However, if there are easy ways to substitute that good, consumers will move away from products with high prices.
Time has a major role in determining demand elasticity. An increase in time will always increase the elasticity of a good, and the decrease of time – i.e., a time constraint – will always decrease the elasticity. Let's use an example of transportation to understand this better. Let's assume that the train that takes you to work every day increases its price one day. The first few days after the price increase, one might not have any other commuting options, so they are ready to pay the higher train price. After some time, however, people will start to find alternative possibilities; maybe they will buy a bike or arrange a carpool. The point here is that the more time consumers have while purchasing something, the more elastic the price will be. The same will apply the other way around; the less time one has when buying something, the less sensitive they will be to the price.
The general income level of people has a major effect on the elasticity of goods. The more money one earns, the less price-sensitive they are. So we can assume someone with a low income has a more significant response to price changes than someone with a higher income.
Another way economists like to think about the income level effect on elasticity is that goods which take up a large percentage of your income (like rent or a car lease) are more likely to be elastic. Meaning to say that when it comes to larger purchases, people are more sensitive to price changes because it's consuming a large percentage of their income.
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bootstrap: The ordinary nonparametric boostrap for arbitrary parameters - mlxtend
Example 1 -- Bootstrapping the Mean
Example 2 - Bootstrapping a Regression Fit
An implementation of the ordinary nonparametric bootstrap to bootstrap a single statistic (for example, the mean. median, R^2 of a regression fit, and so forth).
The bootstrap offers an easy and effective way to estimate the distribution of a statistic via simulation, by drawing (or generating) new samples from an existing sample with replacement. Note that the bootstrap does not require making any assumptions about the sample statistic or dataset being normally distributed.
Using the bootstrap, we can estimate sample statistics and compute the standard error of the mean and confidence intervals as if we have drawn a number of samples from an infinite population. In a nutshell, the bootstrap procedure can be described as follows:
Draw a sample with replacement
Compute the sample statistic
Repeat step 1-2 n times
Compute the standard deviation (standard error of the mean of the statistic)
Or, in simple terms, we can interpret the bootstrap a means of drawing a potentially endless number of (new) samples from a population by resampling the original dataset.
Note that the term "bootstrap replicate" is being used quite loosely in current literature; many researchers and practitioners use it to define the number of bootstrap samples we draw from the original dataset. However, in the context of this documentation and the code annotation, we use the original definition of bootstrap repliactes and use it to refer to the statistic computed from a bootstrap sample.
[1] Efron, Bradley, and Robert J. Tibshirani. An introduction to the bootstrap. CRC press, 1994. Management of Data (ACM SIGMOD '97), pages 265-276, 1997.
This simple example illustrates how you could bootstrap the mean of a sample.
x = rng.normal(loc=5., size=100)
original, std_err, ci_bounds = bootstrap(x, num_rounds=1000, func=np.mean, ci=0.95, seed=123)
print('Mean: %.2f, SE: +/- %.2f, CI95: [%.2f, %.2f]' % (original,
ci_bounds[0],
ci_bounds[1]))
This example illustrates how you can bootstrap the
R^2
of a regression fit on the training data.
from mlxtend.data import autompg_data
X, y = autompg_data()
def r2_fit(X, model=lr):
x, y = X[:, 0].reshape(-1, 1), X[:, 1]
pred = lr.fit(x, y).predict(x)
return r2_score(y, pred)
original, std_err, ci_bounds = bootstrap(X, num_rounds=1000,
func=r2_fit,
The number of bootstrap samnples to draw where each bootstrap sample has the same number of records as the original dataset.
For more usage examples, please see http://rasbt.github.io/mlxtend/user_guide/evaluate/bootstrap/
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Piecewise linear diode in electrical systems - MATLAB - MathWorks United Kingdom
Piecewise linear diode in electrical systems
The Diode block models a piecewise linear diode. If the voltage across the diode is bigger than the Forward voltage parameter value, then the diode behaves like a linear resistor with low resistance, given by the On resistance parameter value, plus a series voltage source. If the voltage across the diode is less than the forward voltage, then the diode behaves like a linear resistor with low conductance given by the Off conductance parameter value.
When forward biased, the series voltage source is described with the following equation:
V=Vf\left(1-{R}_{on}{G}_{off}\right),
Ron On resistance
Goff Off conductance
The Ron.Goff term ensures that the diode current is exactly zero when the voltage across it is zero. The reverse behavior is given byi/Goff, which is also zero at zero current.
Minimum voltage that needs to be applied for the diode to become forward-biased. The default value is 0.6 V.
The resistance of a forward-biased diode. The default value is 0.3 Ω.
The conductance of a reverse-biased diode. The default value is 1e-8 1/Ω.
Electrical conserving port associated with the diode positive terminal.
Electrical conserving port associated with the diode negative terminal.
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Notes on Topological Quantum Field Theories
Francesco Costantino1
1 Institut de Mathématiques de Toulouse (IMT) 118, Route de Narbonne 31062 Toulouse, France
These notes are the outcome of a mini-course on TQFTs held at the edition of Winter Braids in Pau in February 2015. We define the notion of TQFT and provide the first basic examples obtained via the universal construction and via Frobenius algebras. After recalling some basic notions on the mapping class groups of surfaces, we concentrate on the Reshetikhin-Turaev construction via the skein theoretical approach: we first define the skein module of a
3
-manifold and the RT invariants; then we apply the universal construction to get the RT
\mathrm{SU}\left(2\right)
-TQFTs. We conclude with an overview of the main results on these TQFTs and on some recent developments. An appendix summarizes the basic notions and facts in category theory used here.
Francesco Costantino 1
author = {Francesco Costantino},
title = {Notes on {Topological} {Quantum} {Field} {Theories}},
TI - Notes on Topological Quantum Field Theories
%T Notes on Topological Quantum Field Theories
Francesco Costantino. Notes on Topological Quantum Field Theories. Winter Braids Lecture Notes, Volume 2 (2015), Talk no. 1, 45 p. doi : 10.5802/wbln.7. https://wbln.centre-mersenne.org/articles/10.5802/wbln.7/
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[2] J.E. Andersen, Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups, Ann. of Math. (2), 163(1):347–368, 2006. | Article | MR: 2195137 | Zbl: 1157.53049
[3] J.E. Andersen and J. Fjelstad, Reducibility of quantum representations of mapping class groups, Lett. Math. Phys., 91(3):215–239, 2010. | Article | MR: 2595924 | Zbl: 1197.57030
[4] J.E. Andersen, G. Masbaum and K. Ueno, Topological quantum field theory and the Nielsen-Thurston classification of
M\left(0,4\right)
Math. Proc. Cambridge Philos. Soc., 141(3) :477–488, 2006. | Article | MR: 2281410 | Zbl: 1110.57009
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[8] C. Blanchet, F. Costantino, N. Geer, B. Patureau, Non semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants, arXiv:1404.7289, to appear in Advances in Mathematics. | Article | MR: 3539369 | Zbl: 1412.57025
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[10] K. Brown, Cohomology of groups, Graduate Texts in Mathematics, No. 87, Springer Verlag. | Article
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SL2\left(ℂ\right)
-characters and the Kauffman bracket skein module, Comment. Math. Helv., 72(4):521–542, 1997. | Article | MR: 1600138 | Zbl: 0907.57010
[12] L. Charles, J. Marché, Multicurves and regular functions on the representation variety of a surface in SU(2), Commentarii Mathematici Helvetici, Vol 87 (2012), p. 409–431 arXiv:0901.3064. | Article | MR: 2914854 | Zbl: 1246.57022
[13] F. Costantino, N. Geer, B. Patureau, Quantum invariants of
3
-manifolds via link surgery presentations and non semi-simple categories, Journal of Topology, vol 7. Issue 1, pp. 1-49 (2014). | Article | MR: 3286896 | Zbl: 1320.57016
[14] F. Costantino, B. Martelli, An analytic family of representations for the mapping class group of punctured surfaces, Geometry & Topology, vol. 3, no 18 (2014), 1485-1538. | Article | MR: 3228457 | Zbl: 1311.57041
[15] R. Dijkgraaf, A geometric approach to two dimensional conformal field theory, PhD thesis, University of Utrecht, 1989.
[16] D. B. A. Epstein, Curves on 2-manifolds and isotopies, Acta Math., 115:83–107, 1966. | Article | MR: 214087 | Zbl: 0136.44605
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[18] B. Farb, D. Margalit, A primer on mapping class groups, Princeton Mathematical Series, 2011. | Article | Zbl: 1245.57002
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4
-manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999) | Article | Zbl: 0933.57020
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-manifolds, Topology 35 no 2, pp 363-376, 1996. | Article | MR: 1380504 | Zbl: 0857.57021
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[33] W. B. R. Lickorish, Skeins and handlebodies, Pacif. Journal of Math. 159 no. 2 (1993), 337–349. | Article | MR: 1214075 | Zbl: 0805.57010
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Elasticity of Demand: Meaning, Formula & Examples | Outlier
Elasticity of Demand: Meaning, Formula & Examples
The article explains what elasticity of demand is and what it means in economics. It also explains the different types and the main differences between elastic and inelastic demand.
The Demand Curve and the Price Elasticity of Demand
Elasticity of demand measures the responsiveness of demand to a change in some other factor in the market. For example, if the price of a product changes, the price elasticity of demand tells you how much demand will change in response to that price change.
Demand can either be elastic or inelastic. When demand is elastic, it is more sensitive to the changes it is being measured against. Inelastic goods are less sensitive to the changes they are being measured against.
In economics, there are different types of elasticities of demand. The ones you are most likely to encounter in undergraduate microeconomics and macroeconomics courses are:
Price elasticity of demand measures the percentage change in quantity demanded of a good relative to a percentage change in its price. It is also called own-price elasticity of demand, E
_{D}
or PED. Price elasticity of demand is measured as the absolute value of the ratio of these two changes.
E_{D} = | \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}}|
2. Cross Price Elasticity of Demand
Cross price elasticity of demand measures the percentage change in the quantity demanded of one good relative to a percentage change in the price of another good. It is also called XED.
\text{XED} = \frac{\text{Percentage Change in Quantity Demanded of Good X}}{\text{Percentage Change in Price of Good Y}}
Income elasticity of demand measures the percentage change in demand for a good relative to a percentage change in consumer incomes. It is also called E
_{I}
E_{I} = \frac{\text{Percentage Change in Quantity Demanded }}{\text{Percentage Change in Consumer Income}}
Elasticity can also be applied to the supply side of the market. For supply, you can measure:
The price elasticity of supply measures the percentage change in quantity supplied of a good relative to a percentage change in its price. It is also referred to as E
_{S}
E_{S} = \frac{\text{Percentage Change in Quantity Supplied}}{\text{Percentage Change in Price}}
The calculations for each type of elasticity are slightly different, but the intuition behind all elasticities is the same. In every case, elasticity measures the responsiveness of one factor—typically the quantity demanded or supplied of a good—relative to a percentage change in some other factor such as price or income.
Price elasticity of demand is closely related to the slope of the demand curve. In your very first economics course, you probably learned the law of demand, which states that consumers will demand a higher quantity of goods at cheaper prices, and a lower quantity of goods at higher prices. The law of demand explains why demand curves are downward sloping.
Price elasticity of demand is related to the steepness of the demand curve. It explains the extent to which demand changes when price increases or price decreases. The steeper the demand curve, the more inelastic demand is — meaning a small percentage change in price will not have a very big impact on the quantity demanded. The flatter the demand curve is, the more elastic demand is. When a demand curve is flat, even a small percentage change in price will have a large effect on quantity demanded
Because the price elasticity of demand is related to the slope of the demand curve, it’s easy to confuse the two, but the slope of the demand curve and the price elasticity of demand are not the same things. The slope of the demand curve is approximated by the change in price divided by the change in quantity. The price elasticity of demand is calculated as the percentage change in quantity divided by the percentage change in price!
Demand is elastic or inelastic, but economists further separate elasticity into five zones.
The measured value of elasticity is sometimes called the elasticity coefficient. When measured, the price elasticity of demand will have an elasticity coefficient greater than or equal to 0 and can be divided into five zones depending on the value of the coefficient.
PRICE ELASTICITY OF DEMAND DESCRIPTION ELASTICITY COEFFICIENT
Perfectly Elastic Demand Even a small change in price results in demand dropping to zero Elasticity = ∞
Elastic Demand A change in price results in a relatively large change in demand Elasticity > 1
Unit Elastic Demand (also called unitary elasticity) A change in price results in a proportional change in demand Elasticity = 1
Inelastic Demand A change in price results in a relatively small change in demand Elasticity < 1
Perfectly Inelastic Demand A change in price has no effect on demand. Price is not a determinant of demand Elasticity = 0
The cross price elasticity of demand ranges from negative infinity to infinity and can also be divided into five zones of elasticity. The zones of elasticity can help you determine whether the two goods being compared are complements or substitutes.
CROSS PRICE ELASTICITY OF DEMAND DESCRIPTION ELASTICITY COEFFICIENT
Perfectly Elastic Demand The two goods being compared are perfect complements when XED = -∞ and perfect substitutes when XED = ∞ Elasticity = -∞ or ∞
Elastic Demand The two goods being compared are close complementary goods when XED < -1 and close substitute goods when XED > 1 Elasticity < -1 or > 1
Unit Elastic Demand The percent change in quantity demanded of Good X will be equal to the percent change in the price of Good Y Elasticity = -1 or 1
Inelastic Demand The two goods being compared are weak complements when -1< XED < 0 and weak substitutes when 0 < XED < 1 -1 < Elasticity < 1 but not equal to 0
Perfectly Inelastic Demand The two goods being compared are unrelated. Good Y’s price is not a determinant of Good X’s demand Elasticity = 0
Like the cross price elasticity of demand, income elasticity can be positive or negative. The income effect tells us that demand for normal goods will increase as income increases and decrease when income decreases. The income effect also tells us that demand for inferior goods will decrease as income increases and increases as income decreases. Using the income effect and the income elasticity of demand, you can determine whether a good is a normal or inferior good.
INCOME ELASTICITY OF DEMAND DESCRIPTION ELASTICITY
Negative Elasticity An increase in income leads to a decrease in the quantity demanded, indicating that the good is an inferior good Elasticity < 0
Inelastic Demand for Normal Goods A percentage change in income will lead to a relatively small percentage change in quantity demanded. The good is a normal good and is likely to be a good that is a necessity 0 < Elasticity < 1
Unit Elastic Demand for Normal Goods A percentage change in income will lead to an equivalent percentage change in quantity demanded. The good is a normal good Elasticity = 1
Elastic Demand for Normal Goods A percentage change in income will lead to a relatively large change in the quantity demanded. The good is a normal good and is likely to be a luxury good Elasticity > 1
Perfectly Inelastic Demand Changes in income have no effect on quantity demanded. Income is not a determinant of demand Elasticity = 0
Why small choices have big impact.
How money moves our world.
How data describes our world.
Understanding the Supply Curve & How It Works
Learn about what a supply curve is, how a supply curve works, examples, and a quick overview of the law of demand and supply.
Perfect Competition: The Theory and Why It Matters
This article gives a quick overview of perfect competition in microeconomics with examples.
5 Things That Can Shift a Demand Curve
This article is a comprehensive guide on the causes for a demand curve to change. Included are five common demand shifter examples.
Mendy Wolff
A Guide to Price Elasticity of Demand
Price Floors, Explained: A Microeconomics Tool With Macro Impact
A Thorough Guide to the Production Possibilities Frontier
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Subset of Regressors Approximation for GPR Models - MATLAB & Simulink - MathWorks Italia
Predictive Variance Problem
The subset of regressors (SR) approximation method consists of replacing the kernel function
k\left(x,{x}_{r}|\theta \right)
in the exact GPR method by its approximation
{\stackrel{^}{k}}_{SR}\left(x,{x}_{r}|\theta ,\mathcal{A}\right)
, given the active set
\mathcal{A}\subset \mathcal{N}=\left\{1,2,...,n\right\}
. You can specify the SR method for parameter estimation by using the 'FitMethod','sr' name-value pair argument in the call to fitrgp. For prediction using SR, you can use the 'PredictMethod','sr' name-value pair argument in the call to fitrgp.
For the exact GPR model, the expected prediction in GPR depends on the set of
\mathcal{N}
{\mathcal{S}}_{\mathcal{N}}=\left\{k\left(x,{x}_{i}|\theta \right),i=1,2,\dots ,n\right\}
\mathcal{N}=\left\{1,2,...,n\right\}
is the set of indices of all observations, and n is the total number of observations. The idea is to approximate the span of these functions by a smaller set of functions,
{\mathcal{S}}_{\mathcal{A}}
\mathcal{A}\subset \mathcal{N}=\left\{1,2,...,n\right\}
is the subset of indices of points selected to be in the active set. Consider
{\mathcal{S}}_{\mathcal{A}}=\left\{k\left(x,{x}_{j}|\theta \right),j\in \mathcal{A}\right\}
. The aim is to approximate the elements of
{\mathcal{S}}_{\mathcal{N}}
as linear combinations of the elements of
{\mathcal{S}}_{\mathcal{A}}
Suppose the approximation to
k\left(x,{x}_{r}|\theta \right)
using the functions in
{\mathcal{S}}_{\mathcal{A}}
\stackrel{^}{k}\left(x,{x}_{r}|\theta \right)=\sum _{j\in \mathcal{A}}{c}_{jr}k\left(x,{x}_{j}|\theta \right),
{c}_{jr}\in ℝ
are the coefficients of the linear combination for approximating
k\left(x,{x}_{r}|\theta \right)
C
is the matrix that contains all the coefficients
{c}_{jr}
C
|\mathcal{A}|×n
C\left(j,r\right)={c}_{jr}
. The software finds the best approximation to the elements of
{\mathcal{S}}_{\mathcal{N}}
using the active set
\mathcal{A}\subset \mathcal{N}=\left\{1,2,...,n\right\}
by minimizing the error function
E\left(\mathcal{A},C\right)=\sum _{r=1}^{n}{‖k\left(x,{x}_{r}|\theta \right)-\stackrel{^}{k}\left(x,{x}_{r}|\theta \right)‖}_{ℋ}^{2},
ℋ
is the Reproducing Kernel Hilbert Spaces (RKHS) associated with the kernel function k [1], [2].
The coefficient matrix that minimizes
E\left(\mathcal{A},C\right)
{\stackrel{^}{C}}_{\mathcal{A}}=\text{ }K{\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)}^{-1}K\left({X}_{\mathcal{A}},X|\theta \right),
and an approximation to the kernel function using the elements in the active set
\mathcal{A}\subset \mathcal{N}=\left\{1,2,...,n\right\}
\stackrel{^}{k}\left(x,{x}_{r}|\theta \right)=\sum _{j\in \mathcal{A}}{c}_{jr}k\left(x,{x}_{j}|\theta \right)=\text{ }K\left({x}^{T},{X}_{\mathcal{A}}|\theta \right)C\left(:,r\right).
The SR approximation to the kernel function using the active set
\mathcal{A}\subset \mathcal{N}=\left\{1,2,...,n\right\}
{\stackrel{^}{k}}_{SR}\left(x,{x}_{r}|\theta ,\mathcal{A}\right)=\text{ }K\left({x}^{T},{X}_{\mathcal{A}}|\theta \right){\stackrel{^}{C}}_{\mathcal{A}}\left(:,r\right)=K\left({x}^{T},{X}_{\mathcal{A}}|\theta \right)K{\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)}^{-1}K\left({X}_{\mathcal{A}},{x}_{r}^{T}|\theta \right)
and the SR approximation to
K\left(X,X|\theta \right)
{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)=\text{ }K\left(X,{X}_{\mathcal{A}}|\theta \right)\text{ }K{\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)}^{-1}\text{ }K\left({X}_{\mathcal{A}},X|\theta \right).
K\left(X,X|\theta \right)
{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)
in the marginal log likelihood function produces its SR approximation:
\begin{array}{ll}\mathrm{log}{P}_{SR}\left(y|X,\beta ,\theta ,{\sigma }^{2},\mathcal{A}\right)=\hfill & -\frac{1}{2}{\left(y-H\beta \right)}^{T}{\left[{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}{I}_{n}\right]}^{-1}\left(y-H\beta \right)\hfill \\ \hfill & -\frac{N}{2}\mathrm{log}2\pi -\frac{1}{2}\mathrm{log}|{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}{I}_{n}|\hfill \end{array}
\stackrel{^}{\beta }\left(\theta ,{\sigma }^{2}\right)
\beta
\theta
{\sigma }^{2}
\theta
{\sigma }^{2}
\beta
-profiled marginal log likelihood. The SR estimate to
\beta
\theta
{\sigma }^{2}
{\stackrel{^}{\beta }}_{SR}\left(\theta ,{\sigma }^{2},\mathcal{A}\right)={\left[\underset{*}{\underbrace{{H}^{T}{\left[{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}{I}_{n}\right]}^{-1}H}}\right]}^{-1}\underset{**}{\underbrace{{H}^{T}{\left[{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}{I}_{n}\right]}^{-1}y}},
\begin{array}{l}{\left[{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}{I}_{n}\right]}^{-1}=\frac{{I}_{N}}{{\sigma }^{2}}-\frac{K\left(X,{X}_{\mathcal{A}}|\theta \right)}{{\sigma }^{2}}{A}_{\mathcal{A}}^{-1}\frac{K\left({X}_{\mathcal{A}},X|\theta \right)}{{\sigma }^{2}},\\ {A}_{\mathcal{A}}=K\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)+\frac{K\left({X}_{\mathcal{A}},X|\theta \right)K\left(X,{X}_{\mathcal{A}}|\theta \right)}{{\sigma }^{2}},\\ *=\frac{{H}^{T}H}{{\sigma }^{2}}-\frac{{H}^{T}K\left(X,{X}_{\mathcal{A}}|\theta \right)}{{\sigma }^{2}}{A}_{\mathcal{A}}^{-1}\frac{K\left({X}_{\mathcal{A}},X|\theta \right)H}{{\sigma }^{2}},\\ **=\frac{{H}^{T}y}{{\sigma }^{2}}-\frac{{H}^{T}K\left(X,{X}_{\mathcal{A}}|\theta \right)}{{\sigma }^{2}}{A}_{\mathcal{A}}^{-1}\frac{K\left({X}_{\mathcal{A}},X|\theta \right)y}{{\sigma }^{2}}.\end{array}
And the SR approximation to the
\beta
-profiled marginal log likelihood is:
\begin{array}{l}\mathrm{log}{P}_{SR}\left(y|X,{\stackrel{^}{\beta }}_{SR}\left(\theta ,{\sigma }^{2},\mathcal{A}\right),\theta ,{\sigma }^{2},\mathcal{A}\right)=\\ \begin{array}{ll}\hfill & -\frac{1}{2}{\left(y-H{\stackrel{^}{\beta }}_{SR}\left(\theta ,{\sigma }^{2},\mathcal{A}\right)\right)}^{T}{\left[{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}{I}_{n}\right]}^{-1}\left(y-H{\stackrel{^}{\beta }}_{SR}\left(\theta ,{\sigma }^{2},\mathcal{A}\right)\right)\hfill \\ \hfill & -\frac{N}{2}\mathrm{log}2\pi -\frac{1}{2}\mathrm{log}|{\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}{I}_{n}|.\hfill \end{array}\end{array}
The SR approximation to the distribution of
{y}_{new}
y
X
{x}_{new}
P\left({y}_{new}|y,X,{x}_{new}\right)=\mathcal{N}\left({y}_{new}|h{\left({x}_{new}\right)}^{T}\beta +{\mu }_{SR},{\sigma }_{new}^{2}+{\Sigma }_{SR}\right),
{\mu }_{SR}
{\Sigma }_{SR}
are the SR approximations to
\mu
\Sigma
shown in prediction using the exact GPR method.
{\mu }_{SR}
{\Sigma }_{SR}
are obtained by replacing
k\left(x,{x}_{r}|\theta \right)
by its SR approximation
{\stackrel{^}{k}}_{SR}\left(x,{x}_{r}|\theta ,\mathcal{A}\right)
\mu
\Sigma
{\mu }_{SR}=\underset{\left(1\right)}{\underbrace{{\stackrel{^}{K}}_{SR}\left({x}_{new}^{T},X|\theta ,\mathcal{A}\right)}}\underset{\left(2\right)}{\underbrace{{\left({\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}\text{ }{I}_{N}\right)}^{-1}}}\left(y-H\beta \right).
\begin{array}{l}\left(1\right)=K\left({x}_{new}^{T},{X}_{\mathcal{A}}|\theta \right)\text{ }K{\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)}^{-1}K\left({X}_{\mathcal{A}},X|\theta \right)\hfill \end{array},
\left(2\right)=\frac{{I}_{N}}{{\sigma }^{2}}-\frac{K\left(X,{X}_{\mathcal{A}}|\theta \right)}{{\sigma }^{2}}{\left[\text{ }K\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)+\frac{K\left({X}_{\mathcal{A}},X|\theta \right)\text{ }K\left(X,{X}_{\mathcal{A}}|\theta \right)}{{\sigma }^{2}}\right]}^{-1}\frac{K\left({X}_{\mathcal{A}},X|\theta \right)}{{\sigma }^{2}},\text{ }
and from the fact that
{I}_{N}-\text{ }B{\left(\text{ }A+\text{ }B\right)}^{-1}=\text{ }A{\left(\text{ }A+\text{ }B\right)}^{-1}
{\mu }_{SR}
\begin{array}{ll}{\mu }_{SR}\hfill & =\text{ }K\left({x}_{new}^{T},{X}_{\mathcal{A}}|\theta \right){\left[K\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)+\frac{K\left({X}_{\mathcal{A}},X|\theta \right)K\left(X,{X}_{\mathcal{A}}|\theta \right)}{{\sigma }^{2}}\right]}^{-1}\frac{K\left({X}_{\mathcal{A}},X|\theta \right)}{{\sigma }^{2}}\left(y-H\beta \right)\hfill \end{array}.
{\Sigma }_{SR}
is derived as follows:
{\Sigma }_{SR}=\underset{*}{\underbrace{{\stackrel{^}{k}}_{SR}\left({x}_{new},{x}_{new}|\theta ,\mathcal{A}\right)}}-\underset{**}{\underbrace{{\stackrel{^}{K}}_{SR}\left({x}_{new}^{T},X|\theta ,\mathcal{A}\right)}}\underset{***}{\underbrace{{\left({\stackrel{^}{K}}_{SR}\left(X,X|\theta ,\mathcal{A}\right)+{\sigma }^{2}{I}_{N}\right)}^{-1}}}\underset{****}{\underbrace{{\stackrel{^}{K}}_{SR}\left(X,{x}_{new}^{T}|\theta ,\mathcal{A}\right)}}.
*\text{ = }K\left({x}_{new}^{T},{X}_{\mathcal{A}}|\theta \right)K{\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)}^{-1}K\left({X}_{\mathcal{A}},\text{ }{x}_{new}^{T}|\theta \right),
\begin{array}{l}**=K\left({x}_{new}^{T},{X}_{\mathcal{A}}|\theta \right)K{\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)}^{-1}K\left({X}_{\mathcal{A}},X|\theta \right),\\ ***=\left(2\right)\text{ in the equation of }{\mu }_{SR},\end{array}
****\text{ }=\text{ }K\left(X,{X}_{\mathcal{A}}|\theta \right)K{\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)}^{-1}K\left({X}_{\mathcal{A}},\text{ }{x}_{new}^{T}|\theta \right),
{\Sigma }_{SR}
is found as follows:
{\sum }_{SR}=K\left({x}_{new}^{T},{X}_{\mathcal{A}}|\theta \right){\left[\text{ }K\left({X}_{\mathcal{A}},{X}_{\mathcal{A}}|\theta \right)+\frac{K\left({X}_{\mathcal{A}},X|\theta \right)\text{ }K\left(X,{X}_{\mathcal{A}}|\theta \right)\right)}{{\sigma }^{2}}\right]}^{-1}K\left({X}_{\mathcal{A}},\text{ }{x}_{new}^{T}|\theta \right).
One of the disadvantages of the SR method is that it can give unreasonably small predictive variances when making predictions in a region far away from the chosen active set
\mathcal{A}\subset \mathcal{N}=\left\{1,2,...,n\right\}
. Consider making a prediction at a new point
{x}_{new}
that is far away from the training set
X
. In other words, assume that
K\left({x}_{new}^{T},X|\theta \right)\approx 0
For exact GPR, the posterior distribution of
{f}_{new}
y
X
{x}_{new}
would be Normal with mean
\mu =0
\Sigma =k\left({x}_{new},{x}_{new}|\theta \right)
. This value is correct in the sense that, if
{x}_{new}
X
, then the data
\left(X,y\right)
does not supply any new information about
{f}_{new}
and so the posterior distribution of
{f}_{new}
y
X
{x}_{new}
should reduce to the prior distribution
{f}_{new}
{x}_{new}
, which is a Normal distribution with mean
0
k\left({x}_{new},{x}_{new}|\theta \right)
For the SR approximation, if
{x}_{new}
is far away from
X
(and hence also far away from
{X}_{\mathcal{A}}
{\mu }_{SR}=0
{\Sigma }_{SR}=0
. Thus in this extreme case,
{\mu }_{SR}
\mu
from exact GPR, but
{\Sigma }_{SR}
is unreasonably small compared to
\Sigma
from exact GPR.
The fully independent conditional approximation method can help avoid this problem.
[2] Smola, A. J. and B. Schökopf. "Sparse greedy matrix approximation for machine learning." In Proceedings of the Seventeenth International Conference on Machine Learning, 2000.
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Model mixer and local oscillator described by rfdata object - Simulink - MathWorks France
Source of input power data
Model mixer and local oscillator described by rfdata object
The General Mixer block models the mixer described by an RF Toolbox™ data (rfdata.data) object.
Data source — Data source that describes the mixer behavior
Data file (default) | RFDATA object
Data source that describes the mixer behavior, specified as a Data file or an RFDATA object.
Data file — Name of file that contains mixer data
default.s2d (default) | string | character vector
Name of file that contains the mixer data, specified as a string or a character vector. The file name must include the extension. If the file is not in your MATLAB® path, specify the full path to the file or click the Browse button to find the file.
If the data file contains an intermodulation table, the General Mixer block ignores the table. Use RF Toolbox software to ensure the cascade has no significant spurs in the frequency band of interest before running a simulation.
To enable this parameter, choose Data file in Data source.
RFDATA object — RF data object that contains mixer data
read(rfdata.data, 'default.s2p') (default) | rfdata.data object
RF data object that contains the mixer data, specified as an RF Toolbox rfdata.data object, an RF Toolbox command that creates the rfdata.data object, or a MATLAB expression that generates such an object.
{P}_{1dB,out}
{P}_{sat,out}
G{C}_{sat}
Extracted from data source (default) | User-specified
Frequency data source, specified as Extracted from data source or User-specified.
Source of input power data — Input power data source
Extracted from data source (default)
Input power data source, specified as Extracted from data source.
Input power data (dBm) — Input power data
[0:19] (default) | vector
Input power data, specified as a vector with each element unit in dBm.
X-Y plane S11, S12, S21, S22, GroupDelay, OIP3, NF, NFactor, NTemp, Fmin, GaammaOPT, RN, and PhaseNoise.
S11 | S12 | S21 | S22 | GroupDelay | OIP3 | NF | ...
Magnitude (decibels) (default) | Magnitude (linear) | Angle(degrees) | Angle(radians) | Real | Imaginary | ...
S11, S12, S21, S22, and GaammaOPT. Magnitude (decibels), Magnitude (linear), Angle(degrees), Angle(radians), Real, and Imaginary.
NFactor and RN. None
Fmin None and Magnitude (decibels)
Agilent® P2D and S2D files define block parameters for several operating conditions. Operating conditions are the independent parameter settings that are used when creating the file data. By default, the blockset defines the block behavior using the parameter values that correspond to the operating conditions that appear first in the file. To use other property values, you must select a different operating condition in the General Mixer block dialog box.
The network parameter values all refer to the mixer input frequency. If network parameter data and corresponding frequencies exist as S-parameters in the rfdata.data object, the General Mixer block interpolates the S-parameters to determine their values at the modeling frequencies. If the block contains network Y- or Z-parameters, the block first converts them to S-parameters. See Map Network Parameters to Modeling Frequencies for more details.
{b}_{1}
{b}_{2}
\left[\begin{array}{c}{b}_{1}\left({f}_{in}\right)\\ {b}_{2}\left({f}_{out}\right)\end{array}\right]=\left[\begin{array}{cc}{S}_{11}& {S}_{12}\\ {S}_{21}& {S}_{22}\end{array}\right]\left[\begin{array}{c}{a}_{1}\left({f}_{in}\right)\\ {a}_{2}\left({f}_{out}\right)\end{array}\right]
{f}_{in}
{f}_{out}
{a}_{1}
{a}_{2}
Spot noise data in the data source.
Spot noise data in the block dialog box.
Spot noise data (rfdata.noise) object in the block dialog box.
Noise figure, noise factor, or noise temperature value in the block dialog box.
Frequency-dependent noise figure data (rfdata.nf) object in the block dialog box.
The latter four options are only available if noise data does not exist in the data source.
If you specify block noise as spot noise data, the block uses the data to calculate noise figure. The block first interpolates the noise data for the modeling frequencies, using the specified Interpolation method. It then calculates the noise figure using the resulting values.
The General Mixer block applies phase noise to a complex baseband signal. The block first generates additive white Gaussian noise (AWGN) and filters the noise with a digital FIR filter. It then adds the resulting noise to the angle component of the input signal.
If power data exists in the data source, the block extracts the AMAM/AMPM nonlinearities from it.
If the data source contains no power data, then you can introduce nonlinearities into your model by specifying parameters in the Nonlinearity Data tab of the General Mixer block dialog box. Depending on which of these parameters you specify, the block computes up to four of the coefficients
{c}_{1}
{c}_{3}
{c}_{5}
{c}_{7}
{F}_{AM/AM}\left(s\right)={c}_{1}s+{c}_{3}{|s|}^{2}s+{c}_{5}{|s|}^{4}s+{c}_{7}{|s|}^{6}s
s
{c}_{1}
OIP3
IIP3
{P}_{1dB,out}
{P}_{sat,out}
{P}_{sat,out}
G{C}_{sat}
G{C}_{sat}
\begin{array}{c}{P}_{sat,out}+G{C}_{sat}={P}_{sat,in}+{G}_{lin}\\ {P}_{1dB,out}+1={P}_{1dB,in}+{G}_{lin}\\ OIP3=IIP3+{G}_{lin}\end{array}
{G}_{lin}
{c}_{1}
{c}_{3}
{c}_{5}
{c}_{7}
\begin{array}{c}\sqrt{{P}_{sat,out}}={c}_{1}\sqrt{{P}_{sat,in}}+{c}_{3}{\left(\sqrt{{P}_{sat,in}}\right)}^{3}+{c}_{5}{\left(\sqrt{{P}_{sat,in}}\right)}^{5}+{c}_{7}{\left(\sqrt{{P}_{sat,in}}\right)}^{7}\\ \sqrt{{P}_{1dB,out}}={c}_{1}\sqrt{{P}_{1dB,in}}+{c}_{3}{\left(\sqrt{{P}_{1dB,in}}\right)}^{3}+{c}_{5}{\left(\sqrt{{P}_{1dB,in}}\right)}^{5}+{c}_{7}{\left(\sqrt{{P}_{1dB,in}}\right)}^{7}\\ 0=\frac{{c}_{1}}{IIP3}+{c}_{3}\end{array}
{F}_{AM/AM}\left(s\right)
\left(\sqrt{{P}_{sat,in}},\sqrt{{P}_{sat,out}}\right)
\left(\sqrt{{P}_{1dB,in}},\sqrt{{P}_{1dB,out}}\right)
{c}_{7}
{c}_{5}
{c}_{7}
rfdata.data | S-Parameters Mixer | Y-Parameters Mixer | Z-Parameters Mixer
|
Electrical Performance of PEM Fuel Cells With Different Gas Diffusion Layers | J. Electrochem. En. Conv. Stor | ASME Digital Collection
P. Gallo Stampino,
P. Gallo Stampino
Dipartimento di Chimica, Materiali e Ingegneria Chimica “G. Natta,”
, piazza L. da Vinci 32, 20133 Milano, Italy
L. Omati,
L. Omati
G. Dotelli
Stampino, P. G., Omati, L., and Dotelli, G. (March 28, 2011). "Electrical Performance of PEM Fuel Cells With Different Gas Diffusion Layers." ASME. J. Fuel Cell Sci. Technol. August 2011; 8(4): 041005. https://doi.org/10.1115/1.4003630
The microporous layer (MPL) is a key component of polymer electrolyte membrane fuel cells (PEM-FCs), and it is in charge of the gas and water management at the electrode-gas diffusion layer (GDL) interfaces. A MPL was prepared and coated onto two different commercial GDLs: a carbon paper (woven-non-woven (WNW)) and a carbon cloth (CC). Electrical performances of the so-obtained gas diffusion media (GDM), i.e., GDL coated with the MPL, were investigated in single cell testing (steady-state polarization curves) using a Nafion® catalyst coated membrane with a platinum loading of
0.5 mg/cm2
both for the anode and the cathode. Moreover, in order to better understand the polarization phenomena during the running of the FC, impedance spectroscopy was carried out in galvanostatic mode at different current densities. In particular, the effect of the air relative humidity (RH 100%, 80%, and 60%) was investigated, while the hydrogen was fed always fully humidified (100%). The WNW substrate has demonstrated to be superior to CC in a vast range of current densities (from open circuit voltage to
0.8 A/cm2
). However, at high current density, the WNW GDM has some problems in water management.
electrochemical electrodes, proton exchange membrane fuel cells
Flow (Dynamics), Gas diffusion layers, Polarization (Electricity), Polarization (Light), Polarization (Waves), Proton exchange membrane fuel cells, Current density, Electrodes, Carbon, Electrochemical impedance spectroscopy, Catalysts, Diffusion (Physics), Membranes, Pressure, Anodes, Hydrogen, Water resource management, Circuits, Textiles, Spectra (Spectroscopy), Water, Platinum, Steady state
Economic & Commercial Viability of Hydrogen Fuel Cell Vehicles From an Automotive Manufacturer Perspective
Applications of Proton Exchange Membrane Fuel Cell Systems
Gallo Stampino
Dotelli
AC Impedance Technique in PEM Fuel Cell Diagnosis—A Review
|
GMS 6803 Data Science for Clinical Research
Implementations and Comparisons
Ensembles (combine models) can give you a boost in prediction accuracy
Three most popular ensemble methods:
- Bagging: build multiple models (usually the same type) from different subsamples of the training dataset
- Boosting: build multiple models (usually the same type) each of which learns to fix the prediction errors of a prior model in the sequence of models
- Voting: build multiple models (usually different types) and simple statistics (e.g. mean) are used to combine predictions
Take multiple samples from your training dataset (with replacement) and train a model for each sample
The final output prediction is averaged across the predictions of all of the sub-models
Performs best with algorithms that have high variance (e.g. decision trees)
Run in parallel because each bootstrap sample does not depend on others
- bagged decision trees
with reduced correlation between individual classifiers
a random subset of features are considered for each split
- extra trees
further reduce correlation between individual classifiers
cut-point is selected fully at random, independently of the outcome
Creates a sequence of models that attempt to correct the mistakes of the models before them in the sequence
Build a model from the training data, then create a second model that attempts to correct the errors from the first model
Once created, the models make predictions which may be weighted by their demonstrated accuracy and the results are combined to create a final output prediction
Models are added until the training set is predicted perfectly or a maximum number of models are added
Works in sequential manner
Weight instances in the dataset by how easy or difficult they are to predict
Allow the algorithm to pay more or less attention to them in the construction of subsequent models
Gradient Boosting (Stochastic Gradient Boosting)
Boosting algorithms as iterative functional gradient descent algorithms
At each iteration of the algorithm, a base learner is fit on a subsample of the training set drawn at random without replacement
Initialize observation weights:
For m=1 to M
- fit a classifier Gm(x) to training data
w_i=1/N
err_m= {{\sum^N_{i=1}w_iI(y_i \ne G_m(x_i))}\over {\sum^N_{i=1}w_i}}
err_m= {{\sum^N_{i=1}w_iI(y_i \ne G_m(x_i))}\over {\sum^N_{i=1}w_i}}
\alpha_m=log({1-err_m\over err_m})
\alpha_m=log({1-err_m\over err_m})
w_i<-w_i\times exp[\alpha_m \times I(y_i \ne G_M(x_i))], i=1,2,...,N
w_i<-w_i\times exp[\alpha_m \times I(y_i \ne G_M(x_i))], i=1,2,...,N
Intuitive sense: weights will be increased for incorrectly classified observation
- give more focus to next iteration
- weights will be reduced for correctly classified observation
Instead of reweighting observations in adaptive boosting, gradient boosting make some corrections to prediction errors directly
Learn a model -> compute the error residual -> learn to predict the residual
Compute residuals
Learn sequence of models
Combination of models is increasingly accurate and increasingly complex
XGBoost uses presorted algorithm and histogram-based algorithm to compute the best split, while LightGBM uses gradient-based one-side sampling to filter out observations for finding a split value
How they handle categorical variables:
- XGBoost cannot handle categorical features by itself, therefore one has to perform various encodings such as label encoding, mean encoding or one-hot encoding before supplying categorical data to XGBoost
- LightGBM can handle categorical features by taking the input of feature names. It does not convert to one-hot coding, and is much faster than one-hot coding. LGBM uses a special algorithm to find the split value of categorical features
- CatBoost has the flexibility of giving indices of categorical columns so that it can be one-hot encoded or encoded using an efficient method that is similar to mean encoding
import pandas as pd, numpy as np, time
data = pd.read_csv("flights.csv")
data = data.sample(frac = 0.1, random_state=10)
A Kaggle dataset of flight delays for the year 2015.
Approximately 5 million rows. A 10% subset of this data ~ 500k rows.
- MONTH, DAY, DAY_OF_WEEK: int
- AIRLINE and FLIGHT_NUMBER: int
- ORIGIN_AIRPORT and DESTINATION_AIRPORT: string
- DEPARTURE_TIME: float
- ARRIVAL_DELAY: binary outcome indicating delay of more than 10 minutes
- DISTANCE and AIR_TIME: float
# Parameter Tuning
grid_search = GridSearchCV(model, param_grid=param_dist, cv = 3,
verbose=10, n_jobs=-1)
grid_search.fit(train, y_train)
model = xgb.XGBClassifier(max_depth=50, min_child_weight=1, n_estimators=200,\
n_jobs=-1 , verbose=1,learning_rate=0.16)
model.fit(train,y_train)
auc(model, train, test)
grid_search = GridSearchCV(lg, n_jobs=-1, param_grid=param_dist, cv = 3, scoring="roc_auc", verbose=5)
grid_search.fit(train,y_train)
d_train = lgb.Dataset(train, label=y_train)
params = {"max_depth": 50, "learning_rate" : 0.1, "num_leaves": 900, "n_estimators": 300}
# Without Categorical Features
model2 = lgb.train(params, d_train)
auc2(model2, train, test)
#With Catgeorical Features
model2 = lgb.train(params, d_train, categorical_feature = cate_features_name)
params = {'depth': [4, 7, 10],
cb = cb.CatBoostClassifier()
cb_model = GridSearchCV(cb, params, scoring="roc_auc", cv = 3)
cb_model.fit(train, y_train)
#Without Categorical features
clf = cb.CatBoostClassifier(eval_metric="AUC", depth=10, iterations= 500, l2_leaf_reg= 9, learning_rate= 0.15)
auc(clf, train, test)
#With Categorical features
clf = cb.CatBoostClassifier(eval_metric="AUC",one_hot_max_size=31, \
depth=10, iterations= 500, l2_leaf_reg= 9, learning_rate= 0.15)
|
N-bit successive approximation register (SAR) based ADC - Simulink - MathWorks América Latina
SAR Frequency (Hz)
N-bit successive approximation register (SAR) based ADC
Successive Approximation Register (SAR) based ADC consists of a sample and hold circuit (SHA), a comparator, an internal digital to analog converter (DAC), and a successive approximation register.
When the ADC receives the start command, SHA is placed in hold mode. The most significant bit (MSB) of the SAR is set to logic 1, and all other bits are set to logic 0.
The output of the SAR is fed back to a DAC, whose output is compared with the incoming input signal. If the DAC output is greater than the analog input, MSB is reset, otherwise it is left set. The next MSB is now set to 1, and the process is repeated until every bit the SAR is compared. The final value of the SAR at the end of this process corresponds to the analog input value. The end of the conversion process is indicated by the ready signal.
Converted digital output signal, returned as a scalar.
8 (default) | positive real integer in the range [1, 26]
Select to connect to an external start conversion clock. By default, this option is selected. If you deselect this option, a Sampling Clock Source block inside the SAR ADC is used to generate the start conversion clock.
Frequency of internal start conversion clock, specified as a positive real scalar in Hz. Conversion start frequency determines the rate of the ADC.
RMS aperture jitter added as an impairment to the start conversion clock, specified as a real nonnegative scalar in s. Set RMS aperture jitter value to zero if you want a clean clock signal.
SAR Frequency (Hz) — Frequency of SAR clock
Frequency of the SAR clock, specified as a real scalar in Hz. SAR Frequency (Hz) must be high enough to allow the ADC to perform Nbits comparison, where Nbits is the Number of bits of the ADC. The block has one cycle overhead due to algebraic loop removal. So, the clock must run for one additional cycle before the output is ready. So, the SAR Frequency (Hz) (fSAR) is given by the equation
{f}_{\text{SAR}}\ge \left(Nbits+1\right){f}_{\text{start}}
, where fstart is the Conversion start frequency.
Use get_param(gcb,'SARFreq') to view the current value of SAR Frequency (Hz).
Use set_param(gcb,'SARFreq',value) to set SAR Frequency (Hz) to a specific value.
Shifts quantization steps by specific value, specified as a scalar in least significant bit (LSB) or %.
Use get_param(gcb,'OffsetError') to view the current value of Offset error (LSB).
Use set_param(gcb,'OffsetError',value) to set Offset error (LSB) to a specific value.
Flash ADC | Aperture Jitter Measurement | ADC DC Measurement | ADC AC Measurement | ADC Testbench
|
Microwaving Lunch Boxes
After suffering from the deficit in summer camp, Ainu7 decided to supply lunch boxes instead of eating outside for Algospot.com winter camp.
He contacted the famous packed lunch company "Doosot" to prepare N
lunch boxes for N
participants. Due to the massive amount of order, Doosot was not able to prepare the same menu. Instead, they provided different N
lunch boxes. Ainu7 put all the lunch boxes to a refrigerator.
The lunch time has come, and suddenly Ainu7 noticed that there is only one microwave available. As all lunch boxes are not the same, they need a different amount of time to microwave and eat. Specifically, i
-th lunch box needs M_i
seconds to microwave and E_i
seconds to eat.
Ainu7 needs to schedule microwave usage order to minimize lunch time. Lunch time is defined as the duration from the beginning of microwaving of any lunch box to the end of eating for all participants. Write a computer program that finds minimum lunch time to help Ainu7. Note that substituting lunch while microwave is turned on is totally unnecessary, because the lunch will be cooled down.
The first line of the input contains one integer T
Each test case consists of three lines. The first line of each test case contains N\,(1 \le N \le 10000)
N\,(1 \le N \le 10000)
, the number of the participants.
N
integers will follow on the second line. They represent M_1
, M_2
\cdots
, M_N
Similarly, N
integers will follow on the third line, representing E_1
, E_2
\cdots
, E_N
For each test case, print the minimized lunch time in one line. It is guaranteed that the answer is always strictly less than 2^{31}
2^{31}
헐 이문젠 왜 영어예여? 영어 되게 잘하넹
icpc '모의고사' 니까요...
나중에 다시풀어보려고 붙입니다.
전자레인지가 열약해서 1번에 1개씩만 데울수있다.
도시락은 빨리 식기 때문에 한번에 다 데우고 바로 먹기 시작한다.
모든 사람이 점심을 다먹는데 걸리는 시간은 얼마인가?
테스트케이스 갯수제한이 없는데, 정상인건가요?
왜 오답이야 ㅡㅡ
이거 문제가 좀.... 뭐라 그래야 하나 이상하다고 해야 하나 애매하다고 해야 하나요. 입이 하나가 아닌데 전자레인지는 하나이기 때문에 계속 헤매는 문제 같네요
안되신 분들 이 반례 해보세요
|
j=1..\mathrm{num}
\mathrm{with}\left(\mathrm{combstruct}\right):
u
\mathrm{eqns0}≔\mathrm{agfeqns}\left({T=\mathrm{Prod}\left(Z,\mathrm{Set}\left(T\right)\right)},{h\left(T\right)=\mathrm{Prod}\left(0,\mathrm{Set}\left(h\left(T\right)\right)+1\right)},\mathrm{labeled},z,[[u,h]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{eqns0}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\right)}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{Z}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}]
\mathrm{eqns1}≔\mathrm{gfeqns}\left({E=\mathrm{Ε},T=\mathrm{Prod}\left(Z,\mathrm{Prod}\left(E,\mathrm{Set}\left(T\right)\right)\right)},\mathrm{labeled},z,[[u,E]]\right)
\textcolor[rgb]{0,0,1}{\mathrm{eqns1}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{E}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\right)}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{Z}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}]
u
u
\mathrm{agfmomentsolve}\left(\mathrm{eqns0},0\right)
{\textcolor[rgb]{0,0,1}{T}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{Z}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}}
\mathrm{agfmomentsolve}\left(\mathrm{eqns0},1,\mathrm{new}\right)
{{\textcolor[rgb]{0,0,1}{T}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}}{\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{T}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}}{\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{Z}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{Z}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}}
\mathrm{new}
[{\textcolor[rgb]{0,0,1}{T}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{T}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{T}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{T}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{Z}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{Z}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{z}]
{z}^{n}
T\left(z\right)
\mathrm{coeff}\left(T\left(z\right),z,n\right)
nodes, and
\mathrm{coeff}\left({T}_{2}\left(z\right),z,n\right)
nodes for the number of internal nodes. The average number of internal nodes for a tree on
\frac{\mathrm{coeff}\left({T}_{2}\left(z\right),z,n\right)}{\mathrm{coeff}\left(T\left(z\right),z,n\right)}
\mathrm{sol}≔\mathrm{agfmomentsolve}\left(\mathrm{eqns0},2\right):
\mathrm{subs}\left(\mathrm{sol},T[2,2]\left(z\right)\right)
\frac{{\left({\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\right)}{{\left({\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\right)}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\left({\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{LambertW}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}\right)}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}}
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Train support vector machine (SVM) classifier for one-class and binary classification - MATLAB fitcsvm - MathWorks América Latina
G\left({x}_{j},{x}_{k}\right)=\mathrm{exp}\left(-{‖{x}_{j}-{x}_{k}‖}^{2}\right)
G\left({x}_{j},{x}_{k}\right)={x}_{j}\prime {x}_{k}
G\left({x}_{j},{x}_{k}\right)={\left(1+{x}_{j}\prime {x}_{k}\right)}^{q}
\left\{\begin{array}{l}{\alpha }_{j}\left[{y}_{j}f\left({x}_{j}\right)-1+{\xi }_{j}\right]=0\\ {\xi }_{j}\left(C-{\alpha }_{j}\right)=0\end{array}
f\left({x}_{j}\right)=\varphi \left({x}_{j}\right)\prime \beta +b,
0.5\sum _{jk}{\alpha }_{j}{\alpha }_{k}G\left({x}_{j},{x}_{k}\right)
{\alpha }_{1},...,{\alpha }_{n}
\sum {\alpha }_{j}=n\nu
0\le {\alpha }_{j}\le 1
f\left(x\right)=x\prime \beta +b,
2/‖\beta ‖.
‖\beta ‖
0.5{‖\beta ‖}^{2}+C\sum {\xi }_{j}
{y}_{j}f\left({x}_{j}\right)\ge 1-{\xi }_{j}
{\xi }_{j}\ge 0
0.5\sum _{j=1}^{n}\sum _{k=1}^{n}{\alpha }_{j}{\alpha }_{k}{y}_{j}{y}_{k}{x}_{j}\prime {x}_{k}-\sum _{j=1}^{n}{\alpha }_{j}
\sum {\alpha }_{j}{y}_{j}=0
0\le {\alpha }_{j}\le C
\stackrel{^}{f}\left(x\right)=\sum _{j=1}^{n}{\stackrel{^}{\alpha }}_{j}{y}_{j}x\prime {x}_{j}+\stackrel{^}{b}.
\stackrel{^}{b}
{\stackrel{^}{\alpha }}_{j}
\stackrel{^}{\alpha }
\text{sign}\left(\stackrel{^}{f}\left(z\right)\right).
0.5\sum _{j=1}^{n}\sum _{k=1}^{n}{\alpha }_{j}{\alpha }_{k}{y}_{j}{y}_{k}G\left({x}_{j},{x}_{k}\right)-\sum _{j=1}^{n}{\alpha }_{j}
\sum {\alpha }_{j}{y}_{j}=0
0\le {\alpha }_{j}\le C
\stackrel{^}{f}\left(x\right)=\sum _{j=1}^{n}{\stackrel{^}{\alpha }}_{j}{y}_{j}G\left(x,{x}_{j}\right)+\stackrel{^}{b}.
{C}_{j}=n{C}_{0}{w}_{j}^{\ast },
{x}_{j}^{\ast }=\frac{{x}_{j}-{\mu }_{j}^{\ast }}{{\sigma }_{j}^{\ast }},
\begin{array}{c}{\mu }_{j}^{\ast }=\frac{1}{\sum _{k}{w}_{k}^{*}}\sum _{k}{w}_{k}^{*}{x}_{jk},\\ {\left({\sigma }_{j}^{\ast }\right)}^{2}=\frac{{v}_{1}}{{v}_{1}^{2}-{v}_{2}}\sum _{k}{w}_{k}^{*}{\left({x}_{jk}-{\mu }_{j}^{\ast }\right)}^{2},\\ {v}_{1}=\sum _{j}{w}_{j}^{*},\\ {v}_{2}=\sum _{j}{\left({w}_{j}^{*}\right)}^{2}.\end{array}
\sum _{j=1}^{n}{\alpha }_{j}=n\nu .
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Initial Inflation Rate:
8\%
Dis-inflation Rate:
-15\%
Long-term Inflation Rate:
1.5\%
In the above graph we see the annual inflation rate [
\%
] over time, given the inflation parameters proposed above.
\%~\text{SOL Staked} = \frac{\text{Total SOL Staked}}{\text{Total Current Supply}}
This parameter must be estimated because it is a dynamic property of the token holders and staking incentives. The values of % of Staked SOL presented here range from
60\% - 90\%
, which we feel covers the likely range we expect to observe, based on feedback from the investor and validator communities as well as what is observed on comparable Proof-of-Stake protocols.
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Alpha vs. Beta: What’s the Difference? – Investing News Hubb
Beta indicates how volatile a stock’s price has been in comparison to the market as a whole.
Professional portfolio managers calculate alpha as the rate of return that exceeds the model’s prediction or comes short of it. They use a capital asset pricing model (CAPM) to project the potential returns of an investment portfolio.
<br /> begin{aligned} &text{Alpha} = frac{ text{End Price} + text{DPS} – text{Start Price} }{ text{Start Price} } \ &textbf{where:}\ &text{DPS} = text{Distribution per share} \ end{aligned}<br />
Because alpha represents the performance of a portfolio relative to a benchmark, it represents the value that a portfolio manager adds or subtracts from a fund’s return. The baseline number for alpha is zero, which indicates that the portfolio or fund is tracking perfectly with the benchmark index. In this case, the investment manager has neither added nor lost any value.
Often referred to as the beta coefficient, beta is an indication of the volatility of a stock, a fund, or a stock portfolio in comparison with the market as a whole. A benchmark index (most commonly the S&P 500) is used as the proxy measurement for the market. Knowing how volatile a stock’s price is can help an investor decide whether it is worth the risk.
The baseline number for beta is one, which indicates that the security’s price moves exactly as the market moves. A beta of less than 1 means that the security is less volatile than the market, while a beta greater than 1 indicates that its price is more volatile than the market.
If a stock’s beta is 1.5, it is considered to be 50% more volatile than the overall market.
<br /> begin{aligned} &text{Beta} = frac{ text{CR} }{ text{Variance of Market’s Return} } \ &textbf{where:}\ &text{CR} = text{Covariance of asset’s return with market’s return} \ end{aligned}<br />
Variance refers to how far a stock moves relative to its mean. It is frequently used to measure the volatility of a stock’s price over time.
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Category:1001 General Requirements for Material - Engineering_Policy_Guide
1007CAAB, Initial Evaluation Suitable for only Agg Base.
{\displaystyle \mu \,}
{\displaystyle \mu \,}
{\displaystyle \mu \,}
1The last letter of this template is subject to change if the template is revised. The most current template should have the designation "NEWEST" before its description.
{\displaystyle \mu \,}
{\displaystyle \mu \,}
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Category:801 Lime and Fertilizer - Engineering_Policy_Guide
Category:801 Lime and Fertilizer
Refer to Sec 801 for MoDOT’s specifications.
For Laboratory testing and sample reporting procedures, refer to EPG 801.2 Laboratory Testing Guidelines for Sec 801.
1 801.1 Inspection Guidance for Sec 801
1.1 Description (Sec 801.1)
1.2 Material (Sec 801.2)
1.3 Equipment (Sec 801.3)
1.4 Construction Requirements (Sec 801.4)
1.5 Method of Measurement (Sec 801.5)
1.6 Basis of Payment (Sec 801.6)
801.1 Inspection Guidance for Sec 801
Roadside Development, where related to living plants, can only be defined in general terms. Successful inspection of seeding, fertilizing, sodding, mulch and planting largely depends on two conditions:
Strict adherence to specifications, and
Quality inspection based on experience, practical knowledge, and good judgment applied within the intent of specifications.
Fertilizing furnishes basic nutrients essential for plant growth and increases availability of other nutrients. Soil neutralization neutralizes soil acidity. The area to be limed and fertilized will be the area specified within the limits of construction. The contractor shall provide certification as per specification for lime and fertilizer.
Material (Sec 801.2)
Bagged fertilizer may be accepted on the basis of bag label analysis. The guaranteed analysis on the bag label is to be shown on the acceptance report. Samples of commercial bagged fertilizer are not necessary as the Missouri laws governing the labeling of fertilizer and penalties for breaching those laws are adequate to ensure compliance to the bag label analysis. When acceptance is by bag label analysis, the label should be removed from a sack for each type furnished, (i.e. 12-12-12 or 10-20-10). These figures represent the percent of each component in the total. This label should be retained in permanent records with proper notation in the inspector's diary.
Bulk fertilizer may be accepted on the basis of the supplier's or manufacturer's certification. The certification shall include the project number, route, county, supplier's name, a certifying statement, a guaranteed analysis of each component, and shall be signed by responsible personnel of the supplier. The bill of lading or truck ticket accompanied by a certification statement may be used if all the information required for a certification is shown. A copy of the certification is to be retained in the district office.
Liquid fertilizer shall be accepted on the basis of the supplier's certification and sampling and testing of the mixture in the tanks. Random sampling of the shipments shall be performed at the discretion of the District Construction and Materials Engineer with a minimum of 10 percent of the shipments to be sampled. A sample of liquid fertilizer is to consist of approximately 1/2 gallon and is to be submitted to the Laboratory in a 1/2 gallon plastic jug (jug and packaging to be as described in EPG 1070 Water) accompanied by a record in AASHTOWARE Project (AWP). Certifications and the distribution of the certifications are to be as described in Paragraph 801.2.1 of this Section.
Agricultural lime will be accepted on the basis of the certification of analysis furnished to the lime producer or supplier by the Director of the Missouri Agriculture Experiment Station at Columbia, Missouri, in accordance with the Missouri Agriculture Liming Materials Act. The certification of analysis will include the calcium carbonate equivalent, the fineness factor, and the effective neutralizing material per ton of lime. A copy of the certification of analysis shall be made available by the producer or supplier at the source where agricultural lime is presented for inspection. The amount of agricultural lime to be applied is determined from the effective neutralizing material per ton. The certification of analysis can be verified by checking the Missouri University Agricultural Experiment Station reports at the following Internet link:
http://aes.missouri.edu/pfcs/aglime/index.stm
The only Missouri Department of Transportation sample and test anticipated at a source engaged in the production and sale of agriculture limestone would be a field gradation check for the amount of plus No. 8 (2.36 mm) material and that only needs to be done when visual examination indicates a probable deviation.
Equipment (Sec 801.3)
Depending on what type of a grading job is contracted (light grading-shallow cuts and light fills or major grading - deep cuts and high fills), the contract rates are based on soil samples that may or may not be representative of the actual material that will be seeded; therefore the fertilization rates provided in the contract are the minimum quantities to be applied.
It is the contractor’s responsibility to determine whether or not additional fertilizer and/or lime may be required. No direct payment will be made for any additional fertilizer required to meet this specification. The performance standard shall be met before acceptance of the work. (See Sec 805.4 of the Missouri Standard Specifications.)
The following examples illustrate methods that may be used to determine rate of application to provide specified quantities of nitrogen, phosphoric acid and potash.
Assume Contract requirements:
nitrogen (N), 100 lbs. per acre,
phosphoric acid (P2O5,), 200 lbs. per acre,
potash (K2O), 100 lbs. per acre.
The first case to be considered is for material furnished as a mixture with certified or acceptable test analysis for 10-20-10, (10% nitrogen, 20% phosphoric acid, and 10% potash). To compute weight of mixture to be placed per acre the following method is suggested.
{\displaystyle Nitrogen:\,(Total\,weight,\,in\,lbs.,\,of\,mixture\,to\,be\,applied\,per\,acre)}
{\displaystyle ={\frac {Required\,wt.\,(in\,lbs.)\,Nitrogen\,per\,acre}{\%\,Nitrogen\,in\,mix\,(expressed\,as\,a\,decimal)}}}
{\displaystyle Total\,weight\,mixture\,per\,acre={\frac {100}{0.10}}=1000\,lbs.}
{\displaystyle Phosphoric\,Acid:\,(Total\,weight,\,in\,lbs.,\,of\,mixture\,to\,be\,applied\,per\,acre)}
{\displaystyle ={\frac {Required\,wt.\,(in\,lbs.)\,Phosphoric\,Acid\,per\,acre}{\%\,Phosphoric\,Acid\,in\,mix\,(expressed\,as\,a\,decimal)}}}
{\displaystyle Total\,weight\,mixture\,per\,acre={\frac {200}{0.20}}=1000\,lbs.}
{\displaystyle Potash:\,(Totalvweight,\,in\,lbs.,\,of\,mixture\,to\,be\,applied\,per\,acre)}
{\displaystyle ={\frac {Required\,wt.\,(in\,lbs.)\,Potash\,per\,acre}{\%\,Potash\,in\,mix\,(expressed\,as\,a\,decimal)}}}
{\displaystyle Total\,weight\,mixture\,per\,acre={\frac {100}{0.10}}=1000\,lbs.}
Application of 1,000 lbs. of mixed commercial fertilizer will, therefore, furnish the specified amount of each required component.
The contractor may sometimes ask to use a mixed commercial fertilizer that will supply more of at least one required component than specified. For example, assume contract requirements are, as in the previous example, nitrogen 100 lbs. per acre, phosphoric acid 200 lbs. per acre, potash 100 lbs. per acre, and the contractor asks to use a 12-12-12 fertilizer. Following the methods in the above example:
(N) 100 / 0.12 = 833 lbs. of mix to be applied
P2O5, 200 / 0.12 = 1667 lbs. of mix to be applied
K2O 100 / 0.12 = 833 lbs. of mix to be applied
This shows that 1667 lbs. of the mixture must be applied to obtain the required phosphoric acid; however, this amount will furnish twice the required nitrogen and potash. This could be an undesirable application and must be reviewed by the district office before permitting its use.
The next example assures the supplier furnishes three separate fertilizers to the project. The material will thus supply the required nitrogen, phosphoric acid, and potash as individual components. Assume specified amounts of 100 lbs. per acre of nitrogen, 200 lbs. per acre of phosphoric acid, and 100 lbs. per acre of potash as in the previous examples. Material furnished to supply nitrogen is certified and/or tested to be 33.5-0-0 fertilizer. Material furnished to supply phosphoric acid is certified and/or inspected as 0-46-0. Material furnished to supply potash is tested and/or certified as 0-0-62.
{\displaystyle Nitrogen:\,Weight\,in\,lbs.\,of\,Nitrogen\,bearing\,material\,required\,per\,acre)}
{\displaystyle ={\frac {Required\,wt.\,(in\,lbs.)\,Nitrogen\,per\,acre}{\%\,Nitrogen\,in\,mix\,(expressed\,as\,a\,decimal)}}}
{\displaystyle Total\,weight\,mixture\,per\,acre={\frac {100}{0.335}}=229\,lbs.\,per\,acre}
{\displaystyle Phosphoric\,Acid:\,Weight\,in\,lbs.\,of\,Phosphoric\,Acid\,bearing\,material\,required\,per\,acre)}
{\displaystyle ={\frac {Required\,wt.\,(in\,lbs.)\,Phosphoric\,Acid\,per\,acre}{\%\,Phosphoric\,Acid\,in\,mix\,(expressed\,as\,a\,decimal)}}}
{\displaystyle Total\,weight\,mixture\,per\,acre={\frac {200}{0.46}}=435\,lbs.\,per\,acre}
{\displaystyle Potash:\,Weight\,in\,lbs.\,of\,Potash\,bearing\,material\,required\,per\,acre)}
{\displaystyle ={\frac {Required\,wt.\,(in\,lbs.)\,Potash\,per\,acre}{\%\,Potash\,in\,mix\,(expressed\,as\,a\,decimal)}}}
{\displaystyle Total\,weight\,mixture\,per\,acre={\frac {100}{0.62}}=161\,lbs.\,per\,acre}
Using weights determined above, the contractor could apply components individually or could mix them in proper proportions by weight to be placed in one application.
Mixture to provide required nutrients:
nitrogen (N)bearing material= 299 lbs.
phosphoric acid (P2O5,) bearing material= 435 lbs.
potash (K2O) bearing material= 161 lbs.
Total mixture per acre= 896 lbs.
If the contractor elects to mix materials the inspector should observe the operation since "hot spots" could result if mixing is not carefully done to assure a homogeneous mixture. Improper proportions or improper mixing could lead to an excess of one component and deficiencies of others, detrimental to establishment of vegetation.
As previously stated, the inspector must have a certification stating mix components. State law regulates tolerances of any commercial fertilizer. The supplier's certification is a statement that these regulatory tolerances have been met. The supplier's certification for bulk fertilizer should contain the following information with each lot or shipment:
(1) Name, brand, and trademark under which fertilizer is sold.
(2) Name, and address of person guaranteeing the fertilizer.
(3) Guaranteed chemical composition of the fertilizer expressed in the following terms:
(a) Percent total nitrogen
(b) Percent available phosphoric acid
(c) Percent of soluble potash
(4) Project, Route, County, Contractor's name.
Bag label analysis should have items (1) and (3). The resident engineer should prepare a field acceptance report showing the quantity, the analysis and the basis of acceptance.
Earth shoulders, medians, and the entire roadway outside the roadbed limits, excluding rock or surfaced areas, are fertilized and limed.
Separate payment is not made for fertilizing and liming areas to be sodded or seeded by contract.
Articles in "801 Lime and Fertilizer"
Retrieved from "https://epg.modot.org/index.php?title=Category:801_Lime_and_Fertilizer&oldid=50216"
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Summarize threshold-switching dynamic regression model estimation results - MATLAB summarize - MathWorks India
Display Estimation Summary for One State
Return Estimation Summary Table
Summarize threshold-switching dynamic regression model estimation results
summarize(Mdl,state)
results = summarize(___)
summarize(Mdl) displays a summary of the threshold-switching dynamic regression model Mdl.
If Mdl is an estimated model returned by estimate, then summarize displays estimation results to the MATLAB® Command Window. The display includes:
Estimated threshold transitions
Fit statistics, which include the effective sample size, number of estimated submodel parameters and constraints, loglikelihood, and information criteria (AIC and BIC)
A table of submodel estimates and inferences, which includes coefficient estimates with standard errors, t-statistics, and p-values
If Mdl is an unestimated threshold-switching model returned by tsVAR, summarize prints the standard object display (the same display that tsVAR prints during model creation).
summarize(Mdl,state) displays only summary information for the submodel with name state.
results = summarize(___) returns one of the following variables using any of the input argument combinations in the previous syntaxes.
If Mdl is an estimated threshold-switching model, results is a table containing the submodel estimates and inferences.
If Mdl is an unestimated model, results is a tsVAR object that is equal to Mdl.
summarize does not print to the Command Window
{\mathit{y}}_{\mathit{t}}={\epsilon }_{\mathit{t}}
{\mathit{y}}_{\mathit{t}}=2+{\epsilon }_{\mathit{t}}
{\epsilon }_{\mathit{t}}\sim Ν\left(0,1\right)
\mathit{d}=1
{\mathit{y}}_{\mathit{t}-1}
\mathit{d}=1
\left[\begin{array}{c}{y}_{1,t}\\ {y}_{2,t}\end{array}\right]=\left[\begin{array}{c}-1\\ -4\end{array}\right]+\left[\begin{array}{cc}-0.5& 0.1\\ 0.2& -0.75\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ {y}_{2,t-1}\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\end{array}\right]
\left[\begin{array}{c}{y}_{1,t}\\ {y}_{2,t}\end{array}\right]=\left[\begin{array}{c}1\\ 4\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\end{array}\right]
\left[\begin{array}{c}{y}_{1,t}\\ {y}_{2,t}\end{array}\right]=\left[\begin{array}{c}1\\ 4\end{array}\right]+\left[\begin{array}{cc}0.5& 0.1\\ 0.2& 0.75\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ {y}_{2,t-1}\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\end{array}\right]
\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\end{array}\right]\sim {N}_{2}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}2& -1\\ -1& 1\end{array}\right]\right)
{\mathit{y}}_{2,\mathit{t}-4}<-3
-3\le {\mathit{y}}_{2,\mathit{t}-4}<3
DIsplay a summary of the unestimated DGP.
summarize(MdlDGP)
summarize prints an object display.
{\mathit{y}}_{2,\mathit{t}-4}
Display an estimation summary for state 3 only.
State 3 Constant(1) 1.0621 0.095701 11.098 1.2802e-28
State 3 Constant(2) 3.8707 0.068772 56.284 0
State 3 AR{1}(2,2) 0.7568 0.013102 57.761 0
{\mathit{y}}_{\mathit{t}}={\epsilon }_{\mathit{t}}
{\mathit{y}}_{\mathit{t}}=2+{\epsilon }_{\mathit{t}}.
{\epsilon }_{\mathit{t}}\sim Ν\left(0,1\right)
\mathit{d}
= 1. In other words, the threshold variable is
{\mathit{y}}_{\mathit{t}-1}
\mathit{d}=1
EstMdl = estimate(Mdl,tt0,y);
Return an estimation summary table.
results is a table containing estimates and inferences for all submodel coefficients.
Identify significant coefficient estimates.
results.Properties.RowNames(results.PValue < 0.05)
{'State 2 Constant(1)'}
Mdl — Threshold-switching dynamic regression model
Threshold-switching dynamic regression model, specified as a tsVAR object returned by estimate or tsVAR.
state — State to summarize
integer in 1:Mdl.NumStates (default) | state name in Mdl.StateNames
State to summarize, specified as an integer in 1:Mdl.NumStates or a state name in Mdl.StateNames.
The default summarizes all states.
Example: summarize(Mdl,3) summarizes the third state in Mdl.
Example: summarize(Mdl,"Recession") summarizes the state labeled "Recession" in Mdl.
table | tsVAR object
Model summary, returned as a table or tsVAR object.
If Mdl is an estimated threshold-switching model returned by estimate, results is a table of summary information for the submodel parameter estimates. Each row corresponds to a submodel coefficient. Columns correspond to the estimate (Estimate), standard error (StandardError), t-statistic (TStatistic), and the p-value (PValue).
When the summary includes all states (the default), results.Properties stores the following fit statistics:
Description Model summary description (character vector)
EffectiveSampleSize Effective sample size (numeric scalar)
NumConstraints Number of equality constraints (numeric scalar)
When results is a table, it contains only submodel parameter estimates:
Mdl.Switch contains estimates of threshold transitions.
Threshold-switching models can have one or more residual covariance matrices. When Mdl has a model-wide covariance, Mdl.Covariance contains the estimated residual covariance. Otherwise, Mdl.Submodels(j).Covariance contains the estimated residual covariance of state j. For details, see tsVAR.
estimate searches over levels and rates for estimated threshold transitions while solving a conditional least-squares problem for submodel parameters, as described in [2]. The standard errors, loglikelihood, and information criteria are conditional on optimal parameter values in the estimated threshold transitions Mdl.Switch. In particular, standard errors do not account for variation in estimated levels and rates.
tsVAR | threshold
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PairwiseSummation - Maple Help
Home : Support : Online Help : Mathematics : Mathematical Functions : MathematicalFunctions Package : Evalf : PairwiseSummation
PairwiseSummation
perform a summation using a pairwise algorithm, known to substantially reduce the accumulated round-off error when adding floating-point numbers if compared with adding them one at a time
PairwiseSummation(U)
PairwiseSummation(U, m, n)
one-dimensional Array or procedure of one argument
(optional) indicates the lower summation limit
(optional) indicates the upper summation limit
The PairwiseSummation command performs a summation using a pairwise algorithm, known to substantially reduce the accumulated round-off error when adding floating-point numbers if compared with adding them one at a time. Although there exist other techniques that can have even smaller round-off errors, pairwise summation provides sufficient advantage while having a much lower computational cost than slightly better algorithms.
The first argument, U, is expected to be either a one-dimensional Array or a procedure of one single argument - say
j
- such that
U\left(j\right)
gives the
{j}^{\mathrm{th}}
term to be added. When U is an Array, if the (optional) summation limits m and n where not indicated, the Array dimensions (as returned by the ArrayDims command) are taken for summation limits. If however U is a procedure, the values of m and n should be indicated.
When the value of Digits is smaller or equal to the value of evalhf(Digits) and provided that the procedure or Array U satisfy required conditions (see evalhf), the pairwise summation is performed under evalhf, that is using the floating-point hardware of the underlying system and done in double precision. This can significantly speed up the summation process. Otherwise, the summation is performed anyway using option hfloat.
\mathrm{with}\left(\mathrm{MathematicalFunctions}:-\mathrm{Evalf}\right);\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right):
{\textcolor[rgb]{0,0,1}{\mathrm{Add}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Evalb}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Zoom}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{QuadrantNumbers}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Singularities}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{GenerateRecurrence}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{PairwiseSummation}}}
Set the value of Digits to 15 and create an Array with 40,000 random complex numbers, organized into four Arrays, with 1/4 of these numbers in each of the four quadrants of the complex plane, and with all the numbers having absolute value in between 1/10000 and 1/2 - see QuadrantNumbers
\mathrm{Digits}≔15
\textcolor[rgb]{0,0,1}{\mathrm{Digits}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{15}
A ≔ \mathrm{QuadrantNumbers}\left(\mathrm{around} = \mathrm{map}\left(u → \left(u/10000\right), \left[$\left(1 .. 5000\right)\right]\right)\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{1..4 x 1..10001}}\textcolor[rgb]{0,0,1}{\mathrm{Array}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type:}}\textcolor[rgb]{0,0,1}{\mathrm{anything}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Storage:}}\textcolor[rgb]{0,0,1}{\mathrm{rectangular}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Order:}}\textcolor[rgb]{0,0,1}{\mathrm{Fortran_order}}\end{array}\right]
Concatenate the four Arrays into one containing the 40,000 numbers
B≔\mathrm{ArrayTools}:-\mathrm{Concatenate}\left(2,{A}_{1},{A}_{2},{A}_{3},{A}_{4}\right)
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{1 .. 40004}}\textcolor[rgb]{0,0,1}{\mathrm{Array}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type:}}\textcolor[rgb]{0,0,1}{\mathrm{anything}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Storage:}}\textcolor[rgb]{0,0,1}{\mathrm{rectangular}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Order:}}\textcolor[rgb]{0,0,1}{\mathrm{Fortran_order}}\end{array}\right]
Add now these numbers, using both the standard Maple add command and the PairwiseSummation command: compare the precision (higher with PairwiseSummation) and also the time consumed to perform the summations
\mathrm{_t0}≔\mathrm{time}\left(\right):
\mathrm{add}\left(B\right);
\mathrm{`time consumed`}=\mathrm{time}\left(\right)-\mathrm{_t0}
\textcolor[rgb]{0,0,1}{8.63950358284711}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{38.2988008918779}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}
\textcolor[rgb]{0,0,1}{\mathrm{time consumed}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.093}
\mathrm{_t0}≔\mathrm{time}\left(\right):
\mathrm{PairwiseSummation}\left(B\right);
\mathrm{`time consumed`}=\mathrm{time}\left(\right)-\mathrm{_t0}
\textcolor[rgb]{0,0,1}{8.63950358331158}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{38.2988008918301}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}
\textcolor[rgb]{0,0,1}{\mathrm{time consumed}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.047}
Note that, depending on the random sample of numbers, the difference in time consumed for adding 40,000 complex numbers is either advantageous for PairwiseSummation or negligible, due to performing the pairwise summation under evalhf. Now, we know from the theory [1] that the pairwise summation result has less round-off error but how could we verify that? By performing the same addition with a higher value of Digits in order to diminish the accumulation of round-off error. For instance, perform the addition with Digits = 20, then round the result to Digits = 15 and compare with the results (5) and (6):
\mathrm{evalf}\left[15\right]\left(\mathrm{evalf}\left[20\right]\left(\mathrm{add}\left(B\right)\right)\right)
\textcolor[rgb]{0,0,1}{8.63950358331122}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{38.2988008918305}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}
\mathrm{evalf}\left[15\right]\left(\mathrm{evalf}\left[20\right]\left(\mathrm{PairwiseSummation}\left(B\right)\right)\right)
\textcolor[rgb]{0,0,1}{8.63950358331123}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{38.2988008918305}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}
We see that the result (8) by PairwiseSummation only changed the last couple of digits of (6) while the result (7) by add changed various trailing digits of (5) and actually came closer to (6) by PairwiseSummation. Since these last two results (7) and (8) are assured to have less round-off error that (5) and (6), we see that (6) using PairwiseSummation has less less round-off error (2 to 4 more correct digits) than (5).
In conclusion from this experiment, depending on the random sample of complex numbers, for example for Digits = 15 we can expect say from two to four more correct digits when adding using pairwise summation than when adding the same numbers one at a time. In practice, even adding a small amount of numbers with Digits = 10, we can expect one to two more correct digits when performing the addition using the pairwise technique.
[1] Wikipedia, "Pairwise summation". http://en.wikipedia.org/wiki/Pairwise_summation
The MathematicalFunctions[Evalf][PairwiseSummation] command was introduced in Maple 2017.
ArrayTools:-Concatenate
Evalf:-Add
Evalf:-GenerateRecurrence
Evalf:-QuadrantNumbers
|
Resolution - Global Math Week
Here’s the division problem from the last station:
\dfrac {x^{3}-3x+2} {x+2}
And here is the picture for it in a
1 \leftarrow x
We need to find all the copies of
x + 2
(one dot next to two dots) anywhere in the picture of
x^{3}-3x+2
, but we can't find any.
And we can’t unexplode dots to help us out as we don’t know the value of
x
. (We don’t know how many dots to draw when we unexplode.)
The situation seems hopeless at present.
But I have a piece of advice for you, a general life lesson in fact. It’s this.
If there is something in life you want, make it happen!
(and deal with the consequences.)
Right now, is there anything in life we want?
Look at that single dot way at the left. Wouldn’t it be nice to have two dots in the box next to it to make a copy of
x+2
So let’s just put two dots into that empty box! That’s what I want, so let’s make it happen!
But there are consequences: that box is meant to be empty. And in order to keep it empty, we can put in two antidots as well!
Finish up computing
x^{3}-3x+2 \div x+2
on the on the
1 \leftarrow x
machine to make sure you can see how the app works.
(Hint: Can you make antidots have the pattern we want?)
If you are looking for some practice problems, feel free to try these. Try them with pencil and paper, and then with the app perhaps.
\dfrac {x<sup>{3}-3x</sup>{2}+3x-1} {x-1}
\dfrac {4x<sup>{3}-14x</sup>{2}+14x-3} {2x-3}
Aside: Is there a way to conduct the dots and boxes approach with ease on paper? Rather than draw boxes and dots, can one work with tables of numbers that keep track of coefficients? (The word synthetic is often used for algorithms one creates that are a step or two removed from that actual process at hand.)
Once you’ve read this lesson, check out Levi’s video here for more explanation and practice!
|
Some Inevitable Remarks on “Tripled Fixed Point Theorems for Mixed Monotone Kannan Type Contractive Mappings”
Hamed H. Alsulami, Antonio-Francisco Roldán-López-de-Hierro, Erdal Karapınar, Stojan Radenović, "Some Inevitable Remarks on “Tripled Fixed Point Theorems for Mixed Monotone Kannan Type Contractive Mappings”", Journal of Applied Mathematics, vol. 2014, Article ID 392301, 7 pages, 2014. https://doi.org/10.1155/2014/392301
Hamed H. Alsulami,1 Antonio-Francisco Roldán-López-de-Hierro,2 Erdal Karapınar,1,3 and Stojan Radenović4
We advise that the proof of Theorem 12 given by Borcut et al. (2014) is not correct, and it cannot be corrected using the same technique. Furthermore, we present some similar results as an approximation to the opened question if that statement is valid.
The definition of coupled fixed point was firstly given by Guo and Lakshmikantham in [1]. This concept, in the context of metric space, was reconsidered by Gnana Bhaskar and Lakshmikantham [2] in 2006 and by Lakshmikantham and Ciric [3] in the coincidence case. Later, Karapınar investigated this notion in the context of cone metric space. After that, Berinde and Borcut [4] presented the notion of tripled fixed point obtaining similar results, and the same authors extended their work to the coincidence case in [5] (see also, e.g., [6–9]).
A coupled fixed point of is a point such that and . In order to ensure existence and uniqueness of coupled fixed points, Bhaskar and Lakshmikantham introduced the concept of mapping having the mixed monotone property. Henceforth, let be a partial order on . The mapping is said to have the mixed monotone property (with respect to ) if is monotone nondecreasing in and monotone nonincreasing in ; that is, for any ,
Inspired by the previous notions, Berinde and Borcut defined the concepts of tripled fixed point and mixed monotone property as follows. A tripled fixed point of is a point such that , , and . The mapping is said to have the mixed monotone property (with respect to ) if is monotone nondecreasing in and , and it is monotone nonincreasing in ; that is, for any , The second equation that defines a tripled fixed point, that is, , uses the point twice in the arguments of . This fact is necessary to ensure the existence of tripled fixed points of a nonlinear contration because, in such a case, the mixed monotone property is applicable.
Very recently, as a continuation of their pioneering works in the tripled case, Borcut et al. announced in [10] the following result.
Theorem 1 (Borcut et al. [10], Theorem 12). Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Assume that there exists a such that for all , , . Also suppose that either(a) is continuous, or(b) has the following properties:(i)if a nondecreasing sequence , then for all ;(ii)if a nonincreasing sequence , then for all .
If there exist such that then has a tripled fixed point in ; that is, there exist such that
This note is to advise that the proof given by the authors of the previous result is not correct, and it cannot be corrected using the same technique. Furthermore, we present some similar results as an approximation to the opened question if the previous theorem is valid.
2. A Review of the Incorrect Proof
Let us review the lines of their proof. Based on , the authors defined, recursively, for all , and they proved that and were monotone nondecreasing sequences and was a monotone nonincreasing sequence in . Then, they defined, for all , Using the contractivity condition (3), they proved that, taking into account that , , and , Based on this inequality, the authors immediately announced that (see [10, page 4, inequalities and ]). However, these last two inequalities are false. In fact, we can only prove that However, comparing (9) with (11), we notice that does not necessarily coincide with and, similarly, is not necessarily equal to . Therefore, inequality (9) cannot be ensured.
Exactly in the same way, it can be possible to see that (10) cannot be proved using the contractivity condition (3). In such a case, the proof given by the authors, which decisively used inequalities (9) and (10), is false.
3. Some Berinde and Borcut’s Type Tripled Fixed Point Theorems
For the moment, the question about whether Theorem 1 is valid is opened. The following results are some approximations to this problem, using contractivity conditions that are inspired in (3). The main aim of this section is to show some results in this line of research using a well-known result by Ćirić [11]. Our technique is based on some very recent works which showed that most of coupled/tripled/quadrupled fixed point results can be reduced to their corresponding unidimensional theorems in different frameworks (see, for instance, [12–18]). Before that, let us introduce some notation and basic results.
Given a binary relation on , let us define If is a partial order on , then is also a partial order on .
Given a metric on , let us define , for all , by Then and are metrics on . In addition to this, if is complete, then and are also complete.
Given a mapping , let us denote by the mapping Notice that a tripled fixed point of is nothing but a fixed point of . If is -continuous, then is -continuous. Furthermore, if has the mixed monotone property with respect to , then is nondecreasing with respect to (see [12]). We also recall the following result.
Definition 2. Let be a metric on and let be a partial order on . We will say that is regular if it verifies the following two properties:(i)if a nondecreasing sequence , then for all ;(ii)if a nonincreasing sequence , then for all .
Lemma 3. If is regular, then is also regular.
The first version of the following theorem was given by Ćirić in 1972 (see [11]) in the case of metric spaces that were not necessarily partially ordered. A partially ordered version can be found, for example, in [19]. Our main results will be consequences of the next result.
Theorem 4 (see e.g., [19]). Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a nondecreasing mapping and let be such that for all such that . Also assume that is continuous or is regular. If there exist such that , then has a fixed point.
In the following result, we found some terms that play an important role in the contractivity condition (3).
Theorem 5. Let be a partially ordered set and suppose there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that there exists such that for all such that , , and . Also assume that is continuous or is regular. If there exist such that then has a tripled fixed point in ; that is, there exist such that
Proof. As has the mixed monotone property on with respect to , it follows that is -nondecreasing. Let us define, for all , Assume that , , and ; that is, . In this case, the contractivity condition (16) guarantees that Taking into account that , , and , we also find the same upper bound: Furthermore, as , , and , Joining the last three inequalities, we deduce that, for all such that , As , Theorem 4 guarantees that has a fixed point; that is, has a tripled fixed point.
Theorem 6. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that there exists such that for all such that , , and . Also assume that is continuous or is regular. If there exist such that then has a tripled fixed point in ; that is, there exist such that
Proof. Following the lines of the previous proof, consider provided with the metric . Assume that , , and ; that is, . In this case, the contractivity condition (24) guarantees that and the same upper bound is valid for . Moreover, as , , and ,
Therefore Theorem 4 guarantees that has a fixed point; that is, has a tripled fixed point.
The following particularization is also inspired by some Berinde and Borcut’s results.
Corollary 7. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that there exists such that and for all such that , , and . Also assume that is continuous or is regular. If there exist such that then has a tripled fixed point in ; that is, there exist such that
Proof. Let us define . If are such that , then so the previous theorem is applicable.
The following result presents a contractivity condition more similar to (3) than (16). It follows from the previous result using .
Corollary 8. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Let be a mapping having the mixed monotone property on . Suppose that there exists such that for all such that , , and . Also assume that is continuous or is regular. If there exist such that then has a tripled fixed point in ; that is, there exist such that
This research was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors offer thanks to anonymous referees for their remarkable comments, suggestion, and ideas which helped to improve this paper. Antonio-F. Roldán-López-de-Hierro has been partially supported by Junta de Andalucía by Project FQM-268 of the Andalusian CICYE.
V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 4889–4897, 2011. View at: Publisher Site | Google Scholar | MathSciNet
M. Borcut and V. Berinde, “Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces,” Applied Mathematics and Computation, vol. 218, no. 10, pp. 5929–5936, 2012. View at: Publisher Site | Google Scholar | MathSciNet
H. Aydi, E. Karapinar, and S. Radenovic, “Tripled coincidence fixed point results for Boyd-Wong and Matkowski type contractions,” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales A, vol. 107, no. 2, pp. 339–353, 2013. View at: Publisher Site | Google Scholar | MathSciNet
H. Aydi, E. Karapinar, and W. Shatanawi, “Tripled coincidence point results for generalized contractions in ordered generalized metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 101, 2012. View at: Publisher Site | Google Scholar
H. Aydi, E. Karapınar, and C. Vetro, “Meir-Keeler type contractions for tripled fixed points,” Acta Mathematica Scientia, vol. 32, no. 6, pp. 2119–2130, 2012. View at: Publisher Site | Google Scholar | MathSciNet
H. Aydi and E. Karapnar, “New Meir-Keeler type tripled fixed-point theorems on ordered partial metric spaces,” Mathematical Problems in Engineering, vol. 2012, Article ID 409872, 17 pages, 2012. View at: Publisher Site | Google Scholar
M. Borcut, M. Păcurar, and V. Berinde, “Tripled fixed point theorems for mixed monotone Kannan type contractive mappings,” Journal of Applied Mathematics, vol. 2014, Article ID 120203, 8 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
L. B. Ćirić, “Fixed point theorems for mappings with a generalized contractive iterate at a point,” Publications de l'Institut Mathématique, vol. 13(27), pp. 11–16, 1972. View at: Google Scholar | MathSciNet
A. Roldán, J. Martínez-Moreno, C. Roldán, and E. Karapnar, “Some remarks on multidimensional fixed point theorems,” To appear in Fixed Point Theory. View at: Google Scholar
M. A. Khamsi, “Remarks on cone metric spaces and fixed point theorems of contractive mappings,” Fixed Point Theory and Applications, vol. 2010, Article ID 315398, 7 pages, 2010. View at: Publisher Site | Google Scholar | MathSciNet
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G
-metric spaces and fixed point theorems,” Fixed Point Theory and Applications, vol. 2012, article 210, 7 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet
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This problem is a checkpoint for factoring quadratic expressions. It will be referred to as Checkpoint 5.
4x^{2} −1
4x^{2} + 4x + 1
2y^{2} + 5y + 2
3m^{2} − 5m −2
If you needed help factoring these expressions correctly, then you need more practice. Review the Checkpoint 5 materials and try the practice problems. Also, consider getting help outside of class time. From this point on, you will be expected to factor expressions like these quickly and easily.
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Differentiable Function: Meaning, Formulas and Examples | Outlier
In this article, we’ll discuss the definition of differentiable. Using graphed examples, we’ll learn how to spot where a function is non-differentiable. Then, we’ll review the difference between differentiable and continuous.
What Makes a Function Non-Differentiable?
What is the Difference Between Differentiable and Continuous?
Common Derivative Formulas
What does differentiable mean? If a function is differentiable, its derivative exists at every point in its domain. If a function is differentiable at a point
x
, the limit of the average rate of change of
[x, x +\Delta{x}]
\Delta{x}
approaches 0 exists.
f’(x) = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\Delta{y}}{\Delta{x}} = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} } = L
Let’s unpack what the limit means. The function inside this limit probably looks familiar. The average rate of change over an interval, otherwise known as the difference quotient, measures a function’s slope between two points.
This slope value represents how fast a function’s output values (y-values) change with respect to its input (x-values). The delta symbol
\Delta{x}
is used to represent the value that a variable changes by.
The formula for the average rate of change of the function
f
[a, b]
is below.
\text{Average Rate of Change} = \frac{\Delta{y}}{\Delta{x}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(b)-f(a)}{b-a}
The difference quotient is also commonly represented like this:
\frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} }
\frac{{f\left( {x + h } \right) - f\left( x \right)}}{h }
h = \Delta{x}
Here, the delta symbol
\Delta{x}
represents the value that
x
changes by. When we make
\Delta{x}
approach 0 in the limit below, we can find the derivative of a function or instantaneous rate of change. This value also represents the slope of the tangent line.
f’(x) = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{\Delta{y}}{\Delta{x}} = \mathop {\lim }\limits_{\Delta{x} \to 0} \frac{{f\left( {x + \Delta{x} } \right) - f\left( x \right)}}{\Delta{x} } = L
If this limit exists, then
L
f(x)
. This is denoted by
f’(x)
\frac{dy}{dx}
Dr. Hannah Fry dives more into what a derivative is:
Let’s look at some differentiable examples of functions.
f(x) = 4x^3 - 7x
f(x) = 12
f(x) = \sin{(x)}
f(x) = \cos{(x)}
f(x) = e^x
All polynomial functions are differentiable everywhere, as are constant functions. A rational function is differentiable except at the x-value that makes its denominator 0.
Now, let’s learn how to find where a function is not differentiable. If a function has any discontinuities, it is not differentiable at those points. In order to be differentiable, a function must be continuous. The output value must be defined for each input value.
Second, the limit as
\Delta{x}
approaches the difference quotient must exist in order for a function to be differentiable at a point
x
. The limit does not exist if the limit as
\Delta{x}
approaches 0 from the left does not equal the limit as
\Delta{x}
approaches 0 from the right.
This might happen if a function is not continuous at
x
, or if the function’s graph has a corner point, cusp, or vertical tangent.
Knowing what corner points, cusps, vertical tangents, and discontinuities look like on a graph can help you pinpoint where a function is not differentiable. Let’s examine some non-differentiable graph examples below.
f(x)=\cos^{-1}\left(\cos\left(x\right)\right)
is an example of a function with corner points, such as at
x = \pi
. A corner point looks like two linear sections of a function that meet at a sharp point.
The slope of the tangent line to the left of a corner point is different from the slope to the right of the corner point. Because of this, a function’s slope is not defined at a corner point, so its derivative cannot be calculated there.
f(x) = 2\left(x-1\right)^{\left(\frac{2}{3}\right)}
is an example of a function with a cusp at
x = 1
. A cusp looks like two curves that meet at a sharp point.
The slopes of the tangent lines to the left of the cusp approach
-\infty
, while the slopes of the tangent lines to the right of the cusp approach
+\infty
. Because of this, a function’s slope is not defined at a cusp, so its derivative cannot be calculated there.
f(x)=\sqrt[3]{x}
is an example of a function with a vertical tangent. At
x = 0
, the slope of the tangent line approaches infinity.
f
has a vertical tangent at
x
if
x
and if the slope of the tangent line at
x
approaches either negative infinity or positive infinity.
f(x)=\frac{1}{(x^{2})}+u\left(x-2\right)+u\left(x-9\right)
u
represents the unit-step function, is an example of a function with a discontinuity. For example, there are jump continuities at
x = 2
x = 9
Since the limit as
x
approaches these points from the left does not equal the limit as
x
approaches these points from the right,
f(x)
is not differentiable at these points.
A differentiable function must be continuous. However, the reverse is not necessarily true. It’s possible for a function to be continuous but not differentiable. (If needed, you can review our full guide on continuous functions.)
Let’s examine what it means to be a differentiable versus continuous function. For example, consider the absolute value function
f(x) = \vert x \vert
This function is continuous everywhere because we can draw its curve without ever lifting a hand. Its curve has no holes, breaks, jumps, or vertical asymptotes. However, at
x = 0
, the function is not differentiable.
How can we tell this function is not differentiable?
We know it’s non-differentiable because there’s a corner point at
x = 0
. This makes it impossible to draw the tangent line to
f(x) = \vert x \vert
x = 0
. More precisely, the absolute value function fails the limit definition of differentiability at
x = 0
Let’s verify that the absolute value function fails the limit definition of differentiability at
x = 0
f(x) = \vert x \vert
x = 0
into the limit definition of a derivative formula. So that we don’t confuse
x
\Delta{x}
, we’ll substitute the variable
h
\Delta{x}
We are calculating
\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h } \right) - f\left( x \right)}}{h }
f(x) = \vert x \vert
x = 0
\mathop {\lim }\limits_{h \to 0} \frac{\vert x + h \vert - \vert x \vert}{h}
x = 0
\mathop {\lim }\limits_{h \to 0} \frac{\vert 0 + h \vert - \vert 0 \vert}{h} = \mathop {\lim }\limits_{h \to 0} \frac{\vert h \vert}{h}
Let’s stop and take a closer look at the function
\frac{\vert h \vert}{h}
, which can be written as a piecewise function. This piecewise function represents
f’(x)
, the derivative of our function
f(x) = \vert x \vert
We can use this piecewise function to finish evaluating our limit and to understand why
f’(x)
is non-differentiable at
x = 0
First, let’s take the limit as
h
approaches 0 from the right. Imagine
h
as a slightly positive value, so that
h >0
Looking at our piecewise function, we can plug in
\frac{\vert h \vert}{h} = \frac{h}{h} = 1
h > 0
. Remember that the limit as
h
approaches a constant value is simply the constant value itself.
\mathop {\lim }\limits_{h \to 0^+} \frac{\vert h \vert}{h} = \mathop {\lim }\limits_{h \to 0^+} 1 = 1
Now, let’s take the limit as h approaches 0 from the left. Imagine h as a slightly negative value, so that h < 0. Looking at our piecewise function, we can plug in for h < 0 the following:
\frac{\vert h \vert}{h}
\frac{-h}{h} = -1
\mathop {\lim }\limits_{h \to 0^-} \frac{\vert h \vert}{h} = \mathop {\lim }\limits_{h \to 0^+} = -1
\mathop {\lim }\limits_{h \to 0^+} \not = \mathop {\lim }\limits_{h \to 0^-} \frac{\vert h \vert}{h}
1 \not = -1
. In order for a limit to exist, the right-handed limit must equal the left-handed limit.
On our graph of
f’(x) = \frac{\vert x \vert}{x}
below, this looks like a jump discontinuity.
\mathop {\lim }\limits_{h \to 0} \frac{{f\left( {x + h } \right) - f\left( x \right)}}{h }
f(x) = \vert x \vert
x = 0
does not exist. Thus,
f(x) = \vert x \vert
x = 0
So, although
f(x) = \vert x \vert
is continuous everywhere, it is not differentiable at
x = 0
. The same is true for many other functions, so make sure you understand the difference.
Once you’ve learned how to determine if a function is differentiable, you can start to become familiar with the most common derivative formulas and their rules.
Here is a list of the most useful derivative rules to memorize:
\frac{d}{dx}c = 0
\frac{d}{dx}(x^n) = nx^{n-1}
\frac{d}{dx}f(g(x)) = f’(g(x))g’(x)
\frac{d}{dx}[f(x) \cdot g(x)] = f’(x) \cdot g(x) + f(x)\cdot g’(x)
\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{g(x)f’(x)-f(x)g’(x)}{(g(x))^2}
\frac{d}{dx}[f(x) \pm g(x)] = f’(x) \pm g’(x)
\frac{d}{dx}(\sin{(x)}) = \cos{(x)}
\frac{d}{dx}(\cos{(x)}) = -\sin{(x)}
\frac{d}{dx}(\tan{(x)}) = \sec ^2 (x)
\frac{d}{dx} (\ln{x}) = \frac{1}{x}
\frac{d}{dx}(e^x) = e^x
Dr. Tim Chartier discusses the Product and Quotient derivative rules more in depth:
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Create capacitively loaded monopole antenna over rectangular ground plane - MATLAB - MathWorks í•œêµ
TopHatLength
TopHatWidth
Create and View Top-Hat Monopole
Calculate Impedance of Top-Hat Monopole Antenna
Compare Impedance of Top-Hat Monopole Antenna and Monopole Antenna
Top-Hat Monopole with Multiple Dielectric Layers
Create capacitively loaded monopole antenna over rectangular ground plane
The monopoleTopHat object is a top-hat monopole antenna mounted over a rectangular ground plane. The monopole always connects with the center of top hat. The top hat builds up additional capacitance to ground within the structure. This capacitance reduces the resonant frequency of the antenna without increasing the size of the element.
The width of the monopole is related to the diameter of an equivalent cylindrical monopole by the expression
w=2d=4r
For a given cylinder radius, use the cylinder2strip utility function to calculate the equivalent width. The default top-hat monopole is center-fed. The feed point coincides with the origin. The origin is located on the xy- plane.
mth = monopoleTopHat
mth = monopoleTopHat(Name,Value)
mth = monopoleTopHat creates a capacitively loaded monopole antenna over a rectangular ground plane.
mth = monopoleTopHat(Name,Value) creates a capacitively loaded monopole antenna with additional properties specified by one or more name-value pair arguments. Name is the property name and Value is the corresponding value. You can specify several name-value pair arguments in any order as Name1, Value1,..., NameN, ValueN. Properties not specified retain their default values.
Monopole height, specified as a scalar in meters. By default, the height is chosen for an operating frequency of 75 MHz.
TopHatLength — Top hat length along x-axis
Top hat length along x-axis, specified as a scalar in meters.
Example: 'TopHatLength',4
TopHatWidth — Top hat width along y-axis
Top hat width along y-axis, specified as a scalar in meters.
Example: 'TopHatWidth',4
Type of dielectric material used as the substrate, specified as a dielectric object. You can also specify multiple dielectric layers. When creating multiple dielectric layers, in the dielectric function, specify the name, thickness, loss tangent, and relative permittivity of each layer. For more information, see dielectric. For more information on dielectric substrate meshing, see Meshing.
Example: d = dielectric('FR4'); mth = monopoleTopHat('Substrate',d)
Example: d = dielectric('FR4'); mth = MonopoleTopHat; mth.Substrate = d
Example: d = dielectric('Name',{'FR4','Teflon'},'Thickness',[0.5 0.5],'LossTangent',[0.002 0.002],'EpsilonR',[4.8 2.1]); mth = monopoleTopHat('Substrate',d)
Example: mth.Load = lumpedElement('Impedance',75)
Create and view a top-hat monopole with 1 m length, 0.01 m width, groundplane dimensions 2mx2m and top hat dimensions 0.25mx0.25m.
th = monopoleTopHat
monopoleTopHat with properties:
GroundPlaneLength: 2
GroundPlaneWidth: 2
TopHatLength: 0.2500
TopHatWidth: 0.2500
Calculate and plot the impedance of a top-hat monopole over a frequency range of 40 MHz-80 MHz.
th = monopoleTopHat;
impedance(th,linspace(40e6,80e6,41));
Impedance comparison between a monopole of similar dimensions and the top-hat monopole in example 2.
impedance(m,linspace(40e6,80e6,41));
Create a top-hat monopole with default dimensions and a substrate with two dielectric layers.
mth = monopoleTopHat;
d = dielectric('Name',{'FR4','Teflon'},'Thickness',[0.5 0.5],'LossTangent',[0.002 0.002],'EpsilonR',[4.8 2.1]);
mth.Substrate = d
mth =
View the top-hat monopole antenna.
show(mth)
monopole | dipole | patchMicrostrip
|
Zou, J. (2019) How Inflation Affects the Management Earnings Forecasts. American Journal of Industrial and Business Management, 9, 21-48. doi: 10.4236/ajibm.2019.91003.
\text{Accuracy}=\left|\frac{\left(\text{the lower forecast EPS}+\text{the upper forecast EPS}\right)\times 0.5-\text{actual EPS}}{\text{stock price of 2}2\text{days before forecast date}}\right|
\text{The Lower}\left(\text{Upper}\right)\text{Forecast EPS}=\frac{\text{the lower}\left(\text{upper}\right)\text{forecast EPS}}{\text{total shares outstanding}}
\text{Actual EPS}=\frac{\text{net profit attibuted to parent company}}{\text{total shares outstanding}}
\text{Bias}=\left|\frac{\left(\text{the lower forecast EPS}+\text{the upper forecast EPS}\right)\times 0.5-\text{actual EPS}}{\text{actualEPS}}\right|
\begin{array}{l}\text{logit}/\text{reg}\left({\text{Dependent Variable}}_{q}\right)={\alpha }_{q}+\beta \times {\text{Inf}}_{q-1}+{{\displaystyle \sum }}^{\text{}}{\gamma }_{i}\times {\text{ControlVariable}}_{q-1}\\ \text{}+{{\displaystyle \sum }}^{\text{}}\text{Year}+{{\displaystyle \sum }}^{\text{}}\text{Quarter}+{{\displaystyle \sum }}^{\text{}}\text{Industry}+{\epsilon }_{q}\end{array}
\text{Accuracy}1=\left|\frac{\left(\text{thelowerforecastEPS}+\text{thehigherforecastEPS}\right)\times 0.5-\text{actualEPS}}{\text{actualEPS}}\right|
\text{Bias}=1\text{ifAccuracy}>\text{MeanAccuracy}+\text{Std}\left(\text{Accuracy}\right),\text{otherwiseis}0
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Investigation of Fatigue Crack Propagation in Line Pipes Containing an Angled Surface Flaw | J. Pressure Vessel Technol. | ASME Digital Collection
Lichun Bian,
Lichun Bian
Department of Civil and Resource Engineering,
, Halifax, NS, B3J 1Z1, Canada
e-mail: farid.taheri@dal.ca
Bian, L., and Taheri, F. (January 24, 2008). "Investigation of Fatigue Crack Propagation in Line Pipes Containing an Angled Surface Flaw." ASME. J. Pressure Vessel Technol. February 2008; 130(1): 011405. https://doi.org/10.1115/1.2826416
The angled crack problem has been given special attention in the recent years by fracture mechanics investigators due to its close proximity to realistic conditions in engineering structures. In the present paper, an investigation of fatigue crack initiation and propagation in line pipes containing an inclined surface crack is presented. The inclined angle of the surface crack with respect to the axis of loading varies between
0deg
90deg
. Based on the concept of the effective stress intensity factor range
Δkeff
, the rate of fatigue crack propagation
db∕dN
is postulated to be a function of the effective strain energy density factor range
ΔSeff
. This concept is applied to predict the crack growth due to fatigue loading. Furthermore, the threshold condition for nongrowth of the initial crack was established and assessed based on the experimental data.
fatigue cracks, fracture mechanics, pipes, structural engineering, fatigue, mixed mode, crack growth rate, stress intensity factor, strain energy density
Density, Fatigue cracks, Fracture (Materials), Stress, Pipes, Tension
Some Aspects of Crack Propagation Under Monotonic and Cyclic Load
Effective Stress Intensity Factor and Contact Stress for a Partially Closed Griffith Crack in Bending
Some Basic Problems in Fracture Mechanics and New Concepts
New Mixed Mode Fracture Criterion. Maximum Tangential Strain Energy Density Criterion
An Introduction to AC Potential Drop (ACPD)
Mixed Mode Fatigue Crack Growth Predictions
Fatigue Crack Growth Under Mixed Mode I and II Loading
Three Dimensional Crack Problems
|
LogisticRegression: A binary classifier - mlxtend
Example 3 - Stochastic Gradient Descent w. Minibatches
A logistic regression class for binary classification tasks.
from mlxtend.classifier import LogisticRegression
Related to the Perceptron and 'Adaline', a Logistic Regression model is a linear model for binary classification. However, instead of minimizing a linear cost function such as the sum of squared errors (SSE) in Adaline, we minimize a sigmoid function, i.e., the logistic function:
z
is defined as the net input
The net input is in turn based on the logit function
p(y=1 \mid \mathbf{x})
is the conditional probability that a particular sample belongs to class 1 given its features
\mathbf{x}
. The logit function takes inputs in the range [0, 1] and transform them to values over the entire real number range. In contrast, the logistic function takes input values over the entire real number range and transforms them to values in the range [0, 1]. In other words, the logistic function is the inverse of the logit function, and it lets us predict the conditional probability that a certain sample belongs to class 1 (or class 0).
After model fitting, the conditional probability
p(y=1 \mid \mathbf{x})
is converted to a binary class label via a threshold function
g(\cdot)
Objective Function -- Log-Likelihood
In order to parameterize a logistic regression model, we maximize the likelihood
L(\cdot)
(or minimize the logistic cost function).
We write the likelihood as
under the assumption that the training samples are independent of each other.
In practice, it is easier to maximize the (natural) log of this equation, which is called the log-likelihood function:
One advantage of taking the log is to avoid numeric underflow (and challenges with floating point math) for very small likelihoods. Another advantage is that we can obtain the derivative more easily, using the addition trick to rewrite the product of factors as a summation term, which we can then maximize using optimization algorithms such as gradient ascent.
Objective Function -- Logistic Cost Function
An alternative to maximizing the log-likelihood, we can define a cost function
J(\cdot)
to be minimized; we rewrite the log-likelihood as:
$$ J\big(\phi(z), y; \mathbf{w}\big) = $$
As we can see in the figure above, we penalize wrong predictions with an increasingly larger cost.
Gradient Descent (GD) and Stochastic Gradient Descent (SGD) Optimization
Gradient Ascent and the log-likelihood
To learn the weight coefficient of a logistic regression model via gradient-based optimization, we compute the partial derivative of the log-likelihood function -- w.r.t. the jth weight -- as follows:
As an intermediate step, we compute the partial derivative of the sigmoid function, which will come in handy later:
Now, we re-substitute back into in the log-likelihood partial derivative equation and obtain the equation shown below:
Now, in order to find the weights of the model, we take a step proportional to the positive direction of the gradient to maximize the log-likelihood. Futhermore, we add a coefficient, the learning rate
\eta
to the weight update:
Note that the gradient (and weight update) is computed from all samples in the training set in gradient ascent/descent in contrast to stochastic gradient ascent/descent. For more information about the differences between gradient descent and stochastic gradient descent, please see the related article Gradient Descent and Stochastic Gradient Descent.
The previous equation shows the weight update for a single weight
j
. In gradient-based optimization, all weight coefficients are updated simultaneously; the weight update can be written more compactly as
Gradient Descent and the logistic cost function
In the previous section, we derived the gradient of the log-likelihood function, which can be optimized via gradient ascent. Similarly, we can obtain the cost gradient of the logistic cost function
J(\cdot)
and minimize it via gradient descent in order to learn the logistic regression model.
The update rule for a single weight:
The simultaneous weight update:
As a way to tackle overfitting, we can add additional bias to the logistic regression model via a regularization terms. Via the L2 regularization term, we reduce the complexity of the model by penalizing large weight coefficients:
In order to apply regularization, we just need to add the regularization term to the cost function that we defined for logistic regression to shrink the weights:
For more information on regularization, please see Regularization of Generalized Linear Models.
Bishop, Christopher M. Pattern recognition and machine learning. Springer, 2006. pp. 203-213
lr = LogisticRegression(eta=0.1,
minibatches=1, # for Gradient Descent
plot_decision_regions(X, y, clf=lr)
plt.title('Logistic Regression - Gradient Descent')
plt.plot(range(len(lr.cost_)), lr.cost_)
Predicting Class Labels
print('Last 3 Class Labels: %s' % y_pred[-3:])
Last 3 Class Labels: [1 1 1]
y_pred = lr.predict_proba(X)
Last 3 Class Labels: [ 0.99997968 0.99339873 0.99992707]
plt.title('Logistic Regression - Stochastic Gradient Descent')
Here, we set minibatches to 5, which will result in Minibatch Learning with a batch size of 20 samples (since 100 Iris samples divided by 5 minibatches equals 20).
minibatches=5, # 100/5 = 20 -> minibatch-s
LogisticRegression(eta=0.01, epochs=50, l2_lambda=0.0, minibatches=1, random_seed=None, print_progress=0)
Note that this implementation of Logistic Regression expects binary class labels in {0, 1}.
l2_lambda : float
Regularization parameter for L2 regularization. No regularization if l2_lambda=0.0.
The number of minibatches for gradient-based optimization. If 1: Gradient Descent learning If len(y): Stochastic Gradient Descent (SGD) online learning If 1 < minibatches < len(y): SGD Minibatch learning
List of floats with cross_entropy cost (sgd or gd) for every epoch.
For usage examples, please see http://rasbt.github.io/mlxtend/user_guide/classifier/LogisticRegression/
Class 1 probability : float
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Laplace transform - MATLAB laplace - MathWorks Switzerland
Laplace Transform of Symbolic Expression
Laplace Transforms of Dirac and Heaviside Functions
Relation Between Laplace Transform of Function and Its Derivative
Laplace Transform of Array Inputs
Laplace Transform of Symbolic Function
If Laplace Transform Cannot Be Found
laplace(f,transVar)
laplace(f,var,transVar)
laplace(f) returns the Laplace Transform of f. By default, the independent variable is t and the transformation variable is s.
laplace(f,transVar) uses the transformation variable transVar instead of s.
laplace(f,var,transVar) uses the independent variable var and the transformation variable transVar instead of t and s, respectively.
Compute the Laplace transform of 1/sqrt(x). By default, the transform is in terms of s.
f = 1/sqrt(x);
pi^(1/2)/s^(1/2)
Compute the Laplace transform of exp(-a*t). By default, the independent variable is t, and the transformation variable is s.
Specify the transformation variable as y. If you specify only one variable, that variable is the transformation variable. The independent variable is still t.
laplace(f,y)
1/(a + y)
Specify both the independent and transformation variables as a and y in the second and third arguments, respectively.
laplace(f,a,y)
1/(t + y)
Compute the Laplace transforms of the Dirac and Heaviside functions.
syms a positive
laplace(dirac(t-a),t,s)
exp(-a*s)
laplace(heaviside(t-a),t,s)
exp(-a*s)/s
Show that the Laplace transform of the derivative of a function is expressed in terms of the Laplace transform of the function itself.
syms f(t) s
Df = diff(f(t),t);
laplace(Df,t,s)
s*laplace(f(t), t, s) - f(0)
Find the Laplace transform of the matrix M. Specify the independent and transformation variables for each matrix entry by using matrices of the same size. When the arguments are nonscalars, laplace acts on them element-wise.
laplace(M,vars,transVars)
[ exp(x)/a, 1/b]
[ 1/(c^2 + 1), 1i/d^2]
If laplace is called with both scalar and nonscalar arguments, then it expands the scalars to match the nonscalars by using scalar expansion. Nonscalar arguments must be the same size.
laplace(x,vars,transVars)
[ x/a, 1/b^2]
[ x/c, x/d]
Compute the Laplace transform of symbolic functions. When the first argument contains symbolic functions, then the second argument must be a scalar.
laplace([f1 f2],x,[a b])
[ 1/(a - 1), 1/b^2]
If laplace cannot transform the input then it returns an unevaluated call.
f(t) = 1/t;
laplace(1/t, t, s)
Return the original expression by using ilaplace.
t (default) | symbolic variable
Independent variable, specified as a symbolic variable. This variable is often called the "time variable" or the "space variable." If you do not specify the variable then, by default, laplace uses t. If f does not contain t, then laplace uses the function symvar to determine the independent variable.
s (default) | z | symbolic variable | symbolic expression | symbolic vector | symbolic matrix
Transformation variable, specified as a symbolic variable, expression, vector, or matrix. This variable is often called the "complex frequency variable." If you do not specify the variable then, by default, laplace uses s. If s is the independent variable of f, then laplace uses z.
The Laplace transform F = F(s) of the expression f = f(t) with respect to the variable t at the point s is
F\left(s\right)=\underset{{0}^{–}}{\overset{\infty }{\int }}f\left(t\right)\text{\hspace{0.17em}}{e}^{-st}dt.
If any argument is an array, then laplace acts element-wise on all elements of the array.
To compute the inverse Laplace transform, use ilaplace.
The Laplace transform is defined as a unilateral or one-sided transform. This definition assumes that the signal f(t) is only defined for all real numbers t ≥ 0, or f(t) = 0 for t < 0. Therefore, for a generalized signal with f(t) ≠ 0 for t < 0, the Laplace transform of f(t) gives the same result as if f(t) is multiplied by a Heaviside step function.
For example, both of these code blocks:
return 1/(s^2 + 1).
fourier | ifourier | ilaplace | iztrans | ztrans
|
Reversed version of a generalized Aczél’s inequality and its application | Journal of Inequalities and Applications | Full Text
Reversed version of a generalized Aczél’s inequality and its application
Jing-Feng Tian1
In this paper, we give a reversed version of a generalized Aczél’s inequality which is due to Wu and Debnath. As an application, an integral type of the reversed version of the Aczél-Vasić-Pečarić inequality is obtained.
In 1956, Aczél [1] established the following inequality which is of wide application.
{a}_{i}
{b}_{i}
i=1,2,\dots ,n
) are positive numbers such that
{a}_{1}^{2}-{\sum }_{i=2}^{n}{a}_{i}^{2}>0
{b}_{1}^{2}-{\sum }_{i=2}^{n}{b}_{i}^{2}>0
\left({a}_{1}^{2}-\sum _{i=2}^{n}{a}_{i}^{2}\right)\left({b}_{1}^{2}-\sum _{i=2}^{n}{b}_{i}^{2}\right)\le {\left({a}_{1}{b}_{1}-\sum _{i=2}^{n}{a}_{i}{b}_{i}\right)}^{2}.
It is well known that Aczél’s inequality (1) plays an important role in the theory of functional equations in non-Euclidean geometry. Various refinements, generalizations and applications of inequality (1) have appeared in literature (see, e.g., [2–12], [13] and the references therein).
One of the most important results in the works mentioned above is the exponential generalization of (1) asserted by Theorem B.
Theorem B Let p and q be real numbers such that
p,q\ne 0
\frac{1}{p}+\frac{1}{q}=1
{a}_{i}
{b}_{i}
i=1,2,\dots ,n
) be positive numbers such that
{a}_{1}^{p}-{\sum }_{i=2}^{n}{a}_{i}^{p}>0
{b}_{1}^{q}-{\sum }_{i=2}^{n}{b}_{i}^{q}>0
p>1
{\left({a}_{1}^{p}-\sum _{i=2}^{n}{a}_{i}^{p}\right)}^{\frac{1}{p}}{\left({b}_{1}^{q}-\sum _{i=2}^{n}{b}_{i}^{q}\right)}^{\frac{1}{q}}\le {a}_{1}{b}_{1}-\sum _{i=2}^{n}{a}_{i}{b}_{i}.
p<1
p\ne 0
), we have the reverse inequality.
p>1
of Theorem B was proved by Popoviciu [8]. The case
p<1
was given in [10] by Vasić and Pečarić.
In another paper [11], Vasić and Pečarić presented the following extension of inequality (1).
{a}_{rj}>0
{\lambda }_{j}>0
{a}_{1j}^{{\lambda }_{j}}-{\sum }_{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}>0
r=1,2,\dots ,n
j=1,2,\dots ,m
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}\ge 1
\prod _{j=1}^{m}{\left({a}_{1j}^{{\lambda }_{j}}-\sum _{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}\le \prod _{j=1}^{m}{a}_{1j}-\sum _{r=2}^{n}\prod _{j=1}^{m}{a}_{rj}.
Recently, it comes to our attention that an interesting generalization of Aczél’s inequality, which was established by Wu and Debnath in [14], is as follows.
{a}_{rj}>0
{\lambda }_{j}>0
{a}_{1j}^{{\lambda }_{j}}-{\sum }_{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}>0
r=1,2,\dots ,n
j=1,2,\dots ,m
\rho =min\left\{{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}},1\right\}
\prod _{j=1}^{m}{\left({a}_{1j}^{{\lambda }_{j}}-\sum _{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}\le {n}^{1-\rho }\prod _{j=1}^{m}{a}_{1j}-\sum _{r=2}^{n}\prod _{j=1}^{m}{a}_{rj},
{a}_{1j}={n}^{\frac{1}{{p}_{j}}}{a}_{2j}=\cdots ={n}^{\frac{1}{{p}_{j}}}{a}_{nj}
j=1,2,\dots ,m
\rho <1
\frac{{a}_{11}^{{\lambda }_{1}}}{{a}_{1j}^{{\lambda }_{j}}}=\frac{{a}_{21}^{{\lambda }_{1}}}{{a}_{2j}^{{\lambda }_{j}}}=\cdots =\frac{{a}_{n1}^{{\lambda }_{1}}}{{a}_{nj}^{{\lambda }_{j}}},\phantom{\rule{1em}{0ex}}j=2,3,\dots ,m\mathit{\text{for }}\rho =1.
The purpose of this work is to give a reversed version of inequality (4). As application, an integral type of the reversed version of the Aczél-Vasić-Pečarić inequality is obtained.
2 Reversed version of a generalized Aczél’s inequality
We need the following lemmas in our deduction.
{x}_{i}\ge 0
{\lambda }_{i}>0
i=1,2,\dots ,n
0<p\le 1
\sum _{i=1}^{n}{\lambda }_{i}{x}_{i}^{p}\le {\left(\sum _{i=1}^{n}{\lambda }_{i}\right)}^{1-p}{\left(\sum _{i=1}^{n}{\lambda }_{i}{x}_{i}\right)}^{p}.
p\ge 1
p<0
. In each case, the sign of the equality holds if and only if
{x}_{i}={x}_{j}
i,j=1,2,\dots ,n
Lemma 2.2 [11] (Generalized Hölder’s inequality)
{a}_{rj}>0
j=1,2,\dots ,m
r=1,2,\dots ,n
{\lambda }_{1}\ne 0
{\lambda }_{j}<0
j=2,3,\dots ,m
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}\le 1
\sum _{r=1}^{n}\prod _{j=1}^{m}{a}_{rj}\ge \prod _{j=1}^{m}{\left(\sum _{r=1}^{n}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}.
The sign of the equality holds if and only if the m sets
\left({a}_{r1}\right),\left({a}_{r2}\right),\dots ,\left({a}_{rm}\right)
{a}_{rj}>0
r=1,2,\dots ,n
j=1,2,\dots ,m
{\lambda }_{1}\ne 0
{\lambda }_{j}<0
j=2,3,\dots ,m
\tau =max\left\{{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}},1\right\}
\sum _{r=1}^{n}\prod _{j=1}^{m}{a}_{rj}\ge {n}^{1-\tau }\prod _{j=1}^{m}{\left(\sum _{r=1}^{n}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}.
\left({a}_{r1}\right),\left({a}_{r2}\right),\dots ,\left({a}_{rm}\right)
are proportional for
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}\le 1
{a}_{1j}={a}_{2j}=\cdots ={a}_{nj}
j=1,2,\dots ,m
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}>1
Proof Case (I). When
{\lambda }_{1}<0
\tau =1
. Obviously, inequality (7) is equivalent to inequality (6).
Case (II). When
{\lambda }_{1}>0
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}\ge 1
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}=t
t\ge 1
{\sum }_{j=1}^{m}\frac{1}{t{\lambda }_{j}}=1
. By inequality (6), we have
\begin{array}{rcl}{\left(\sum _{r=1}^{n}\prod _{j=1}^{m}{a}_{rj}\right)}^{2}& =& \sum _{s=1}^{n}\left(\prod _{i=1}^{m}{a}_{si}\right)\sum _{r=1}^{n}\prod _{j=1}^{m}{a}_{rj}\\ \ge & \sum _{s=1}^{n}\left(\prod _{i=1}^{m}{a}_{si}\right)\left[\prod _{j=1}^{m}{\left(\sum _{r=1}^{n}{a}_{rj}^{t{\lambda }_{j}}\right)}^{\frac{1}{t{\lambda }_{j}}}\right]\\ =& \sum _{s=1}^{n}\left\{{\left({a}_{s1}^{t{\lambda }_{1}}\sum _{r=1}^{n}{a}_{r1}^{t{\lambda }_{1}}\right)}^{\frac{1}{t{\lambda }_{1}}-{\sum }_{j=2}^{m}\frac{1}{t{\lambda }_{j}}}×\left[\prod _{j=2}^{m}{\left({a}_{s1}^{t{\lambda }_{1}}\sum _{r=1}^{n}{a}_{rj}^{t{\lambda }_{j}}\right)}^{\frac{1}{t{\lambda }_{j}}}\right]\\ ×\left[\prod _{j=2}^{m}{\left({a}_{sj}^{t{\lambda }_{j}}\sum _{r=1}^{n}{a}_{r1}^{t{\lambda }_{1}}\right)}^{\frac{1}{t{\lambda }_{j}}}\right]\right\}.\end{array}
Consequently, according to
\left(\frac{1}{t{\lambda }_{1}}-{\sum }_{j=2}^{m}\frac{1}{t{\lambda }_{j}}\right)+\frac{1}{t{\lambda }_{2}}+\frac{1}{t{\lambda }_{3}}+\cdots +\frac{1}{t{\lambda }_{m}}+\frac{1}{t{\lambda }_{2}}+\frac{1}{t{\lambda }_{3}}+\cdots +\frac{1}{t{\lambda }_{m}}=1
, by using inequality (6) on the right side of (8), we observe that
\begin{array}{rcl}{\left(\sum _{r=1}^{n}\prod _{j=1}^{m}{a}_{rj}\right)}^{2}& \ge & {\left(\sum _{s=1}^{n}\sum _{r=1}^{n}{a}_{s1}^{t{\lambda }_{1}}{a}_{r1}^{t{\lambda }_{1}}\right)}^{\frac{1}{t{\lambda }_{1}}-{\sum }_{j=2}^{m}\frac{1}{t{\lambda }_{j}}}\\ ×\left[\prod _{j=2}^{m}{\left(\sum _{s=1}^{n}\sum _{r=1}^{n}{a}_{s1}^{t{\lambda }_{1}}{a}_{rj}^{t{\lambda }_{j}}\right)}^{\frac{1}{t{\lambda }_{j}}}\right]\left[\prod _{j=2}^{m}{\left(\sum _{s=1}^{n}\sum _{r=1}^{n}{a}_{sj}^{t{\lambda }_{j}}{a}_{r1}^{t{\lambda }_{1}}\right)}^{\frac{1}{t{\lambda }_{j}}}\right].\end{array}
Additionally, using Lemma 2.1 together with
t\ge 1
Combining inequalities (9) and (10) leads to inequality (7) immediately.
Case (III). When
{\lambda }_{1}>0
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}\le 1
The condition of the equality for inequality can easily be obtained by Lemma 2.1 and Lemma 2.2. This completes the proof of Lemma 2.3. □
Remark 2.4 It is clear that the generalized Hölder inequality (6) is a simple consequence of Lemma 2.3 presented in this article.
{a}_{rj}>0
{\lambda }_{1}\ne 0
{\lambda }_{j}<0
j=2,3,\dots ,m
{a}_{1j}^{{\lambda }_{j}}-{\sum }_{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}>0
r=1,2,\dots ,n
j=1,2,\dots ,m
\tau =max\left\{{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}},1\right\}
\prod _{j=1}^{m}{\left({a}_{1j}^{{\lambda }_{j}}-\sum _{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}\ge {n}^{1-\tau }\prod _{j=1}^{m}{a}_{1j}-\sum _{r=2}^{n}\prod _{j=1}^{m}{a}_{rj},
and the equality holds if and only if
{a}_{1j}={n}^{\frac{1}{{\lambda }_{j}}}{a}_{2j}=\cdots ={n}^{\frac{1}{{\lambda }_{j}}}{a}_{nj}
j=1,2,\dots ,m
\tau >1
\frac{{a}_{11}^{{\lambda }_{1}}}{{a}_{1j}^{{\lambda }_{j}}}=\frac{{a}_{21}^{{\lambda }_{1}}}{{a}_{2j}^{{\lambda }_{j}}}=\cdots =\frac{{a}_{n1}^{{\lambda }_{1}}}{{a}_{nj}^{{\lambda }_{j}}},\phantom{\rule{1em}{0ex}}j=2,3,\dots ,m\mathit{\text{for }}\tau =1.
{a}_{1j}^{{\lambda }_{j}}-\sum _{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}={x}_{j}^{{\lambda }_{j}},
\prod _{j=1}^{m}{a}_{1j}-{n}^{\tau -1}\sum _{r=2}^{n}\prod _{j=1}^{m}{a}_{rj}={n}^{\tau -1}\prod _{j=1}^{m}{x}_{j}.
By using inequality (7), we have
\prod _{j=1}^{m}{a}_{1j}=\prod _{j=1}^{m}{\left({x}_{j}^{{\lambda }_{j}}+\sum _{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}\le {n}^{\tau -1}\left(\prod _{j=1}^{m}{x}_{j}+\sum _{r=2}^{n}\prod _{j=1}^{m}{a}_{rj}\right),
{\left({x}_{m}^{{\lambda }_{m}}+\sum _{r=2}^{n}{a}_{rm}^{{\lambda }_{m}}\right)}^{\frac{1}{{\lambda }_{m}}}\prod _{j=1}^{m-1}{\left({x}_{j}^{{\lambda }_{j}}+\sum _{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}\le {n}^{\tau -1}\left(\prod _{j=1}^{m}{x}_{j}+\sum _{r=2}^{n}\prod _{j=1}^{m}{a}_{rj}\right).
Therefore, from (12), (13) and (15), we obtain
{\left({x}_{m}^{{\lambda }_{m}}+\sum _{r=2}^{n}{a}_{rm}^{{\lambda }_{m}}\right)}^{\frac{1}{{\lambda }_{m}}}\prod _{j=1}^{m-1}{a}_{1j}\le \prod _{j=1}^{m}{a}_{1j}.
{x}_{m}^{{\lambda }_{m}}\ge {a}_{1m}^{{\lambda }_{m}}-\sum _{r=2}^{m}{a}_{rm}^{{\lambda }_{m}},
{x}_{m}\le {\left({a}_{1m}^{{\lambda }_{m}}-\sum _{r=2}^{m}{a}_{rm}^{{\lambda }_{m}}\right)}^{\frac{1}{{\lambda }_{m}}}.
\begin{array}{rcl}\prod _{j=1}^{m}{x}_{j}& \le & {\left({a}_{1m}^{{\lambda }_{m}}-\sum _{r=2}^{m}{a}_{rm}^{{\lambda }_{m}}\right)}^{\frac{1}{{\lambda }_{m}}}\prod _{j=1}^{m-1}{x}_{j}\\ =& {\left({a}_{1m}^{{\lambda }_{m}}-\sum _{r=2}^{m}{a}_{rm}^{{\lambda }_{m}}\right)}^{\frac{1}{{\lambda }_{m}}}\prod _{j=1}^{m-1}{\left({a}_{1j}^{{\lambda }_{j}}-\sum _{r=2}^{m}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}\\ =& \prod _{j=1}^{m}{\left({a}_{1j}^{{\lambda }_{j}}-\sum _{r=2}^{m}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}.\end{array}
By using (13), we immediately obtain the desired inequality (11). The condition of the equality for inequality (11) can easily be obtained by Lemma 2.3. The proof of Theorem 2.5 is completed. □
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}\le 1
, then from Theorem 2.5, we obtain the following reversed version of inequality (3).
{a}_{rj}>0
{\lambda }_{1}\ne 0
{\lambda }_{j}<0
j=2,3,\dots ,m
{\sum }_{j=1}^{m}\frac{1}{{\lambda }_{j}}\le 1
{a}_{1j}^{{\lambda }_{j}}-{\sum }_{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}>0
r=1,2,\dots ,n
j=1,2,\dots ,m
\prod _{j=1}^{m}{\left({a}_{1j}^{{\lambda }_{j}}-\sum _{r=2}^{n}{a}_{rj}^{{\lambda }_{j}}\right)}^{\frac{1}{{\lambda }_{j}}}\ge \prod _{j=1}^{m}{a}_{1j}-\sum _{r=2}^{n}\prod _{j=1}^{m}{a}_{rj}.
m=2
{\lambda }_{1}=p\ne 0
{\lambda }_{2}=q<0
{a}_{r1}={a}_{r}
{a}_{r2}={b}_{r}
r=1,2,\dots ,n
), then from Theorem 2.5, we obtain
{a}_{r}>0
{b}_{r}>0
r=1,2,\dots ,n
{a}_{1}^{p}-{\sum }_{r=2}^{n}{a}_{r}^{p}>0
{b}_{1}^{q}-{\sum }_{r=2}^{n}{b}_{r}^{q}>0
p\ne 0
q<0
\rho =max\left\{\frac{1}{p}+\frac{1}{q},1\right\}
. Then the following inequality holds:
{\left({a}_{1}^{p}-\sum _{r=2}^{n}{a}_{r}^{p}\right)}^{\frac{1}{p}}{\left({b}_{1}^{q}-\sum _{r=2}^{n}{b}_{r}^{q}\right)}^{\frac{1}{q}}\ge {n}^{1-\rho }{a}_{1}{b}_{1}-\sum _{r=2}^{n}{a}_{r}{b}_{r}.
\frac{1}{p}+\frac{1}{q}=1
, inequality (21) reduces to the famous Aczél-Vasić-Pečarić inequality (2).
As application of the above results, we establish here an integral type of the reversed version of the Aczél-Vasić-Pečarić inequality.
{\lambda }_{1}>0
{\lambda }_{j}<0
j=2,3,\dots ,m
{\sum }_{j=1}^{m}{\lambda }_{j}=1
{A}_{j}>0
j=1,2,\dots ,m
{f}_{j}\left(x\right)
j=1,2,\dots ,m
\left[a,b\right]
{A}_{j}^{{\lambda }_{j}}-{\int }_{a}^{b}{f}_{j}^{{\lambda }_{j}}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x>0
\prod _{j=1}^{m}{\left({A}_{j}^{{\lambda }_{j}}-{\int }_{a}^{b}{f}_{j}^{{\lambda }_{j}}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right)}^{\frac{1}{{\lambda }_{j}}}\ge \prod _{j=1}^{m}{A}_{j}-{\int }_{a}^{b}\prod _{j=1}^{m}{f}_{j}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x.
Proof For any positive integer n, we choose an equidistant partition of
\left[a,b\right]
\begin{array}{c}a<a+\frac{b-a}{n}<\cdots <a+\frac{b-a}{n}k<\cdots <a+\frac{b-a}{n}\left(n-1\right)<b,\hfill \\ {x}_{k}=a+\frac{b-a}{n}k,\phantom{\rule{2em}{0ex}}\mathrm{\Delta }{x}_{k}=\frac{b-a}{n},\phantom{\rule{1em}{0ex}}k=1,2,\dots ,n.\hfill \end{array}
Since the hypothesis
{A}_{j}^{{\lambda }_{j}}-{\int }_{a}^{b}{f}_{j}^{{\lambda }_{j}}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x>0
j=1,2,\dots ,m
{A}_{j}^{{\lambda }_{j}}-\underset{n\to \mathrm{\infty }}{lim}\sum _{k=1}^{n}{f}_{j}^{{\lambda }_{j}}\left(a+\frac{k\left(b-a\right)}{n}\right)\frac{b-a}{n}>0\phantom{\rule{1em}{0ex}}\left(j=1,2,\dots ,m\right),
{A}_{j}^{{\lambda }_{j}}-\sum _{k=1}^{n}{f}_{j}^{{\lambda }_{j}}\left(a+\frac{k\left(b-a\right)}{n}\right)\frac{b-a}{n}>0\phantom{\rule{1em}{0ex}}\text{for all }n>N\text{ and }j=1,2,\dots ,m.
By using Theorem 2.5, we obtain that for any
n>N
\sum _{j=1}^{m}\frac{1}{{\lambda }_{j}}=1,
In view of the hypotheses that
{f}_{j}\left(x\right)
j=1,2,\dots ,m
\left[a,b\right]
{\prod }_{j=1}^{m}{f}_{j}\left(x\right)
{f}_{j}^{{\lambda }_{j}}\left(x\right)
\left[a,b\right]
. Passing the limit as
n\to \mathrm{\infty }
on both sides of inequality (24), we obtain inequality (22). The proof of Theorem 3.1 is completed. □
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The author would like to express his sincere thanks to the anonymous referees for their great efforts to improve this paper. This work was supported by the NNSF of China (Grant No. 61073121), and the Fundamental Research Funds for the Central Universities (No. 11ML65).
College of Science and Technology, North China Electric Power University, Baoding, Hebei Province, 071051, P.R. China
Correspondence to Jing-Feng Tian.
Tian, JF. Reversed version of a generalized Aczél’s inequality and its application. J Inequal Appl 2012, 202 (2012). https://doi.org/10.1186/1029-242X-2012-202
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A note on a class of Hardy-Rellich type inequalities | Journal of Inequalities and Applications | Full Text
A note on a class of Hardy-Rellich type inequalities
Yanmei Di1,
Liya Jiang1,
Shoufeng Shen1 &
Yongyang Jin1
In this note we provide simple and short proofs for a class of Hardy-Rellich type inequalities with the best constant, which extends some recent results.
It is well known that Hardy’s inequality and its generalizations play important roles in many areas of mathematics. The classical Hardy inequality is given by, for
N\ge 3
{\int }_{{R}^{N}}{|\mathrm{\nabla }u\left(x\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dx\ge {\left(\frac{N-2}{2}\right)}^{2}{\int }_{{R}^{N}}\frac{{|u\left(x\right)|}^{2}}{{|x|}^{2}}\phantom{\rule{0.2em}{0ex}}dx,
u\in {C}_{0}^{\mathrm{\infty }}\left({R}^{N}\right)
{\left(\frac{N-2}{2}\right)}^{2}
is optimal and not attained.
Recently there has been a considerable interest in studying the Hardy-type and Rellich-type inequalities. See, for example, [1–7]. In [8] Caffarelli, Kohn and Nirenberg proved a rather general interpolation inequality with weights. That is the following so-called Caffarelli-Kohn-Nirenberg inequality. For any
u\in {C}_{0}^{\mathrm{\infty }}\left({R}^{N}\right)
C>0
{\parallel {|x|}^{\gamma }u\parallel }_{{L}^{r}}\le C{\parallel {|x|}^{\alpha }|\mathrm{\nabla }u|\parallel }_{{L}^{p}}^{a}\cdot {\parallel {|x|}^{\beta }u\parallel }_{{L}^{q}}^{1-a},
\frac{1}{r}+\frac{\gamma }{N}=a\left(\frac{1}{p}+\frac{\alpha -1}{N}\right)+\left(1-a\right)\left(\frac{1}{q}+\frac{\beta }{N}\right)
In [9] Costa proved the following
{L}^{2}
-case version for a class of Caffarelli-Kohn-Nirenberg inequalities with a sharp constant by an elementary method. For all
a,b\in R
u\in {C}_{0}^{\mathrm{\infty }}\left({R}^{N}\mathrm{\setminus }\left\{0\right\}\right)
\stackrel{ˆ}{C}{\int }_{{R}^{N}}\frac{{|u|}^{2}}{{|x|}^{a+b+1}}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|u|}^{2}}{{|x|}^{2a}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{2}}{\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{2}}{{|x|}^{2b}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{2}},
\stackrel{ˆ}{C}=\stackrel{ˆ}{C}\left(a,b\right):=\frac{|N-\left(a+b+1\right)|}{2}
On the other hand, the Rellich inequality is a generalization of the Hardy inequality to second-order derivatives, and the classical Rellich inequality in
{R}^{N}
states that for
N\ge 5
u\in {C}_{0}^{\mathrm{\infty }}\left({R}^{N}\setminus \left\{0\right\}\right)
{\int }_{{R}^{N}}{|\mathrm{\Delta }u\left(x\right)|}^{2}\phantom{\rule{0.2em}{0ex}}dx\ge {\left(\frac{N\left(N-4\right)}{4}\right)}^{2}{\int }_{{R}^{N}}\frac{{|u\left(x\right)|}^{2}}{{|x|}^{4}}\phantom{\rule{0.2em}{0ex}}dx.
\frac{{N}^{2}{\left(N-4\right)}^{2}}{16}
is sharp and never achieved. In [10] Tetikas and Zographopoulos obtained a corresponding stronger versions of the Rellich inequality which reads
{\left(\frac{N}{2}\right)}^{2}{\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{2}}{{|x|}^{2}}\phantom{\rule{0.2em}{0ex}}dx\le {\int }_{{R}^{N}}{|\mathrm{△}u|}^{2}\phantom{\rule{0.2em}{0ex}}dx
u\in {C}_{0}^{\mathrm{\infty }}
N\ge 3
. In [11] Costa obtained a new class of Hardy-Rellich type inequalities which contain (1.5) as a special case. If
a+b+3\le N
\stackrel{ˆ}{C}{\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{2}}{{|x|}^{a+b+1}}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|\mathrm{△}u|}^{2}}{{|x|}^{2b}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{2}}{\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{2}}{{|x|}^{2a}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{2}},
\stackrel{ˆ}{C}=\stackrel{ˆ}{C}\left(a,b\right):=|\frac{N+a+b-1}{2}|
The goal of this paper is to extend the above (1.3) and (1.6) to the general
{L}^{p}
case for
1<p<\mathrm{\infty }
by a different and direct approach.
In this section, we will give the proof of the main theorems.
a,b\in R
u\in {C}_{0}^{\mathrm{\infty }}\left({R}^{N}\setminus \left\{0\right\}\right)
C{\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{a+b+1}}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{ap}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{b\frac{p}{p-1}}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{p-1}{p}},
1<p<\mathrm{\infty }
C=|\frac{N-\left(a+b+1\right)}{p}|
u\in {C}_{0}^{\mathrm{\infty }}\left({R}^{N}\setminus \left\{0\right\}\right)
a,b\in R
\lambda =a+b+1
. By integration by parts and the Hölder inequality, one has
\begin{array}{rcl}{\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx& =& \frac{1}{N-\lambda }{\int }_{{R}^{N}}{|u|}^{p}div\left(\frac{x}{{|x|}^{\lambda }}\right)\phantom{\rule{0.2em}{0ex}}dx\\ =& -\frac{1}{N-\lambda }{\int }_{{R}^{N}}pu{|u|}^{p-2}\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx\\ \le & |\frac{-p}{N-\lambda }|{\int }_{{R}^{N}}\frac{|x\cdot \mathrm{\nabla }u|}{{|x|}^{\lambda }}{|u|}^{p-1}\phantom{\rule{0.2em}{0ex}}dx\\ \le & |\frac{p}{N-\lambda }|{\int }_{{R}^{N}}\frac{|\mathrm{\nabla }u|{|u|}^{p-1}}{{|x|}^{a+b}}\phantom{\rule{0.2em}{0ex}}dx\\ \le & |\frac{p}{N-\lambda }|{\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{ap}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{b\frac{p}{p-1}}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{p-1}{p}}.\end{array}
|\frac{N-\lambda }{p}|{\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{ap}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{b\frac{p}{p-1}}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{p-1}{p}}.
It remains to show the sharpness of the constant. By the condition with equality in the Hölder inequality, we consider the following family of functions:
{u}_{\epsilon }\left(x\right)={e}^{-\frac{{C}_{\epsilon }}{\beta }{|x|}^{\beta }},\phantom{\rule{1em}{0ex}}\text{when }\beta =a-\frac{b}{p-1}+1\ne 0
{u}_{\epsilon }\left(x\right)=\frac{1}{{|x|}^{{C}_{\epsilon }}},\phantom{\rule{1em}{0ex}}\text{when }\beta =a-\frac{b}{p-1}+1=0,
{C}_{\epsilon }
is a positive number sequence converging to
|\frac{N-\left(a+b+1\right)}{p}|
\epsilon \to 0
. By direct computation and the limit process, we know the constant
\frac{|N-\left(a+b+1\right)|}{p}
is sharp. □
p=2
, the inequality (2.1) covers the inequality (2.4) in [9].
a=0
b=p-1
, the inequality (2.1) is the classical
{L}^{p}
Hardy inequality:
{\left(\frac{N-p}{p}\right)}^{p}{\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{p}}\phantom{\rule{0.2em}{0ex}}dx\le {\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx.
When we take special values for a, b, the following corollary holds.
Corollary 1 (i) When
b=\left(a+1\right)\left(p-1\right)
, the inequality (2.1) is just the weighted Hardy inequality:
|\frac{N-p\left(a+1\right)}{p}{|}^{p}{\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{\left(a+1\right)p}}\phantom{\rule{0.2em}{0ex}}dx\le {\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{ap}}\phantom{\rule{0.2em}{0ex}}dx.
a+b+1=ap
, according to the inequality (2.1), we have
|\frac{N-ap}{p}|{\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{ap}}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{ap}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{ap-\frac{p}{p-1}}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{p-1}{p}}.
a=-p
a+b+1=0
\frac{N}{p}{\int }_{{R}^{N}}{|u|}^{p}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p}{|x|}^{{p}^{2}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}\frac{{|u|}^{p}}{{|x|}^{p}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{p-1}{p}}.
By a similar method, we can prove the following
{L}^{p}
case Hardy-Rellich type inequality.
1<p<N
\frac{p-N}{p-1}\le a+b+1\le 0
u\in {C}_{0}^{\mathrm{\infty }}\left({R}^{N}\setminus \left\{0\right\}\right)
\stackrel{ˆ}{C}{\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{a+b+1}}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|{\mathrm{△}}_{p}u|}^{p}}{{|x|}^{ap}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{q}}{{|x|}^{bq}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{q}},
\frac{1}{p}+\frac{1}{q}=1
\stackrel{ˆ}{C}=\left(\frac{N-p+\left(p-1\right)\left(a+b+1\right)}{p}\right)
{\mathrm{△}}_{p}u=div\left({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\right)
is the p-Laplacian operator.
\lambda =a+b+1
\begin{array}{rl}{\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx& =\frac{1}{N-\lambda }{\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p}div\left(\frac{x}{{|x|}^{\lambda }}\right)\phantom{\rule{0.2em}{0ex}}dx\\ =-\frac{1}{N-\lambda }{\int }_{{R}^{N}}\frac{p}{2}{|\mathrm{\nabla }u|}^{p-2}\frac{x}{{|x|}^{\lambda }}\cdot \mathrm{\nabla }\left({|\mathrm{\nabla }u|}^{2}\right)\phantom{\rule{0.2em}{0ex}}dx\\ =\frac{p}{2\left(\lambda -N\right)}{\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p-2}\frac{x\cdot \mathrm{\nabla }\left({|\mathrm{\nabla }u|}^{2}\right)}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx.\end{array}
\begin{array}{rcl}{\int }_{{R}^{N}}{\mathrm{△}}_{p}u\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx& =& {\int }_{{R}^{N}}div\left({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\right)\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx\\ =& -{\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\cdot \mathrm{\nabla }\left(\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}\right)\phantom{\rule{0.2em}{0ex}}dx\\ =& -{\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p-2}\left(\frac{{|\mathrm{\nabla }u|}^{2}}{{|x|}^{\lambda }}+\frac{\frac{1}{2}x\cdot \mathrm{\nabla }\left({|\mathrm{\nabla }u|}^{2}\right)}{{|x|}^{\lambda }}-\lambda \frac{{\left(x\cdot \mathrm{\nabla }u\right)}^{2}}{{|x|}^{\lambda +2}}\right)\phantom{\rule{0.2em}{0ex}}dx,\end{array}
Then, we can deduce from (2.8) and (2.9)
\frac{N-p-\lambda }{p}{\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx+\lambda {\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p-2}\frac{{\left(x\cdot \mathrm{\nabla }u\right)}^{2}}{{|x|}^{\lambda +2}}\phantom{\rule{0.2em}{0ex}}dx={\int }_{{R}^{N}}{\mathrm{△}}_{p}u\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx.
{\int }_{{R}^{N}}{\mathrm{△}}_{p}u\frac{x\cdot \mathrm{\nabla }u}{{|x|}^{\lambda +2}}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|{\mathrm{△}}_{p}u|}^{q}}{{|x|}^{aq}}\right)}^{\frac{1}{q}}{\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{bp}}\right)}^{\frac{1}{p}},
\frac{p-N}{p-1}\le \lambda \le 0
\frac{N-p+\left(p-1\right)\lambda }{p}{\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{p}}{{|x|}^{\lambda }}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|{\mathrm{△}}_{p}u|}^{p}}{{|x|}^{ap}}\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}\frac{{|\mathrm{\nabla }u|}^{q}}{{|x|}^{bq}}\right)}^{\frac{1}{q}}.
We mention that we do not know whether the constant
\left(\frac{N-p+\left(p-1\right)\left(a+b+1\right)}{p}\right)
in (2.7) is optimal or not. □
a+b+1=0
, we have the following inequalities:
a=-1
b=0
, the inequality (2.7) is equivalent to the inequality
{\left(\frac{N-p}{p}\right)}^{p}{\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx\le {\int }_{{R}^{N}}{|{\mathrm{△}}_{p}u|}^{p}{|x|}^{p}\phantom{\rule{0.2em}{0ex}}dx.
a=1
b=-2
\left(\frac{N-p}{p}\right){\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}\frac{{|{\mathrm{△}}_{p}u|}^{p}}{{|x|}^{p}}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{q}{|x|}^{2q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{q}}.
a=0
b=-1
\left(\frac{N-p}{p}\right){\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx\le {\left({\int }_{{R}^{N}}{|{\mathrm{△}}_{p}u|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}}{\left({\int }_{{R}^{N}}{|\mathrm{\nabla }u|}^{q}{|x|}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{q}}.
Adimurthi AS: Role of the fundamental solution in Hardy-Sobolev type inequalities. Proc. R. Soc. Edinb., Sect. A 2006, 136: 1111–1130. 10.1017/S030821050000490X
Garofalo N, Lanconelli E: Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann. Inst. Fourier (Grenoble) 1990, 40: 313–356. 10.5802/aif.1215
Goldstein JA, Kombe I: Nonlinear degenerate parabolic equations on the Heisenberg group. Int. J. Evol. Equ. 2005, 1: 1–22.
Goldstein JA, Zhang QS: On a degenerate heat equation with a singular potential. J. Funct. Anal. 2001, 186: 342–359. 10.1006/jfan.2001.3792
Jin Y, Han Y: Weighted Rellich inequality on H -type groups and nonisotropic Heisenberg groups. J. Inequal. Appl. 2010., 2010: Article ID 158281
Jin Y, Zhang G: Degenerate p -Laplacian operators and Hardy type inequalities on h -type groups. Can. J. Math. 2010, 62: 1116–1130. 10.4153/CJM-2010-033-9
García Azorero JP, Peral Alonso I: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 1998, 144: 441–476. 10.1006/jdeq.1997.3375
Caffarelli L, Kohn R, Nirenberg L: First order interpolation inequalities with weights. Compos. Math. 1984, 53: 259–275.
Costa DG: Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities. J. Math. Anal. Appl. 2008, 337: 311–317. 10.1016/j.jmaa.2007.03.062
Tertikas A, Zographopoulos NB: Best constants in the Hardy-Rellich inequalities and related improvements. Adv. Math. 2007, 209: 407–459. 10.1016/j.aim.2006.05.011
Costa DG:On Hardy-Rellich type inequalities in
{R}^{N}
. Appl. Math. Lett. 2009, 22: 902–905. 10.1016/j.aml.2008.02.018
This work is supported by NNSF of China (11001240), ZJNSF (LQ12A01023) and the foundation of the Zhejiang University of the Technology (20100229).
Department of Mathematics, Zhejiang University of Technology, Hangzhou, P.R. China
Yanmei Di, Liya Jiang, Shoufeng Shen & Yongyang Jin
Yanmei Di
Liya Jiang
Di, Y., Jiang, L., Shen, S. et al. A note on a class of Hardy-Rellich type inequalities. J Inequal Appl 2013, 84 (2013). https://doi.org/10.1186/1029-242X-2013-84
Caffarelli-Kohn-Nirenberg inequality
|
The mysterious geometry of Artin groups
Jon McCammond1
1 Dept. of Math., University of California, Santa Barbara, CA 93106
Artin groups are easily defined but most of them are poorly understood. In this survey I try to highlight precisely where the problems begin. The first part reviews the close connection between Coxeter groups and Artin groups as well as the associated topological spaces used to investigate them. The second part describes the location of the border between the Artin groups we understand at a very basic level and those that remain fundamentally mysterious. The third part highlights those collections of Artin groups (and their relatives) that are not currently understood but which we are likely to understand sometime soon.
Keywords: Artin groups
Jon McCammond 1
author = {Jon McCammond},
title = {The mysterious geometry of {Artin} groups},
TI - The mysterious geometry of Artin groups
%T The mysterious geometry of Artin groups
Jon McCammond. The mysterious geometry of Artin groups. Winter Braids Lecture Notes, Volume 4 (2017), Talk no. 1, 30 p. doi : 10.5802/wbln.17. https://wbln.centre-mersenne.org/articles/10.5802/wbln.17/
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|
Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One | J. Comput. Nonlinear Dynam. | ASME Digital Collection
Mechanical Engineering and Energy Processes,
, Carbondale, Illinois 62901
e-mail: om@engr.siu.edu
Kumar, P., and Agrawal, O. P. (December 16, 2005). "Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One." ASME. J. Comput. Nonlinear Dynam. April 2006; 1(2): 178–185. https://doi.org/10.1115/1.2166147
This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order
α
1<α<2
which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra-type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra-type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve two problems, linear and nonlinear, using two different types of polynomials, cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results for problem 1 and the numerical results obtained using another scheme for problem 2. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.
differential equations, polynomials, integral equations
Algorithms, Differential equations, Errors, Polynomials, Integral equations
Algorithms for that Fractional Calculus: A Selection of Numerical Methods
A Cubic Scheme for Numerical Solution of Fractional Differential Equations
Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference, ENOC
The Numerical Solution of Linear Multiterm Fractional Differential Equations: Systems of Equations
The Approximate Solution of Fractional Differential Equations of order Greater than 1
,” On the WWW, May. URL http://www.chester.ac.uk/maths/nevillepub.htmlhttp://www.chester.ac.uk/maths/nevillepub.html.
Multiorder Fractional Differential Equations and their Numerical Solution
Simulation of Fractional Systems: A Benchmark
Proceedings of the first IFAC Workshop on Fractional Differentiation and its Applications
, FDA04.
Collection Methods for Volterra Integral and Related Functional Differential Equations
Fedstein
Smoothness of Solutions of Volterra Equations with Weakly Singular Kernels
Rungetta-Kutta theory for Volterra and Abel Integral Equations of the Second Kind
Initialization Issue of the Caputo Fractional Derivative
Proceedings of IDETC/CIE 2005
, New Jersey, Vol.
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Democracy provides the perfect conditions for pandemics. Covid-19 can be used as an index/indicator of how much freedom people (if democracy means freedom) have in their respective countries (re how China tackled Covid "so well" & compare that to Europe's & America's "failure" to do so).
Also, don't forget the Covid conspiracy theories - they're big in democracies, further undermining attempts to check the pandemic. Anyhow, Covid conspiracy theorists have managed to connect the dots: (more) pandemics
\rightarrow
(more) freedom. Diseases (especially those that can cause pandemics) are an enemy of freedom as measures against them tend to be authoritarian in character.
Of course, one might say Covid originated in communist China. Firstly, is China really communist and secondly, I'm talking about dissemination/spread and not source.
Who here has had a 'booster' dose of the vaccine? How was it?
Go drinking with your buddies. — Agent Smith
Daughter visited this weekend and mentioned that the alcoholics in the hospital where she works routinely steal and drink the hand sanitiser. Now that's dedication.
Daughter visited this weekend and mentioned that the alcoholics in the hospital where she works routinely steal and drink the hand sanitiser. Now that's dedication. — unenlightened
:smile: I'm a chain-smoker. I know the feeling.
This is anecdotal but a colleague of mine told me (today) that his friend tested negative for Covid for the whole month he (the friend) was drinking and when he underwent another test, at the end of this Dionysian month, his result came out positive i.e. he had Covid. Go figure!
Zolenskify
↪The Opposite
I got the Moderna booster, and I had the J&J single dose as my initial vaccine. I can remember that night after the J&J being hell, and was expecting something like that for the booster. To my surprise though, I was completely fine. I think the strategy to avoiding side effects is to drink a lot of water the day before your vaccine, and continue to do so even after you've been given the shot.
Zolenskify OptionsShare
↪Zolenskify
I'm not sure if I even need to get a booster. I had both doses of AstraZeneca last year, and omicron doesn't seem to be severe...
I had covid back when there was only one version, then 2 astra jabs and a pfizer boost and a few weeks earlier, the flu jab. And when I get offered another booster, I'll be there for it, because my mild covid was no fun at all.
And at the moment I have an ordinary cold - sore throat runny nose temperature, and that's not very much fun... but that's sod's law.
Well I am not too familiar with that particular vaccine, and it's effectiveness, but I will say that there are more daily Covid cases than ever. I think ultimately though, if you get vaccinated (and maybe a booster here or there, depending on your stance, state mandates, and its workability with your main vaccine), eat healthy, and maintain a... stable mental wellbeing, then you'll be fine.
↪unenlightened
@Zolenskify OK I will get a booster
CO2 emission reduction due to Covid (1 year)=
2.3 \times 10^9
tons.
1 person emits 2 tons of CO2 per year.
Death toll from Covid (humans as individuals): 5,000,000 and counting.
Actual death toll from Covid (humans in terms of CO2 emission):
\frac{2.3 \times 10^9}{2} = 1.15 \times 10^9
= 1 billion!
StyleGAN (This Person does not Exist)
Here they are disbanding the corona restrictions and even the local health officials don't see worth in continuing to have corona passports. Cases are multitude far higher than ever before, hospitalizations have stayed low and deaths as sporadic as they been all the time. The obvious fact that even the officials have admitted is that omicron isn't at all so lethal as variants before.
I would assume this pandemic will end as T.S. Elliot wrote "not with a bang, but with a wimper":
First it was about the leaders of countries holding press conferences and issuing dramatic restrictions. Then the presidents and prime ministers have other more important things to do and the press conferences are held by a health minister or the sort. Then it's just some official. Then even the media doesn't participate. Then it becomes an issue that you can read at a government web page just like about the new flu variants and new seasonal flu shots. They do recommend people to get seasonal flu shots, you know.
That's where the covid-19 will be buried and will stay for perhaps decades if not longer.
And of course, this long thread will be buried somewhere in the backpages.
↪jorndoe
Snow shovels are needed here. Actually have to go and do some shoveling after this...
as usual politicians are late to the party. https://thephilosophyforum.com/discussion/comment/627691
Benkei OptionsShare
↪Benkei
Better to be late than never to come to the party.
But yes, again good points from you earlier.
Here's a non-technical overview of the possibilities for the further evolution of covid. That it becomes rapidly insignificant to humans does not seem the most likely scenario.
I'm not sure the measure should be how it continues to affect people but how it affects society and particularly the health care system. It might be we permanently need to increase health care capacity to manage both Covid and influenza seasonality.
Not that our newly formed government is thinking about that. They're actually planning on saving 6 billion in 4 years. How's that for a stab in the back after all the work doctors and nurses did in the past 2 years?
I wonder if it's possible to manufacture a genetically modified variant that is almost non-lethal but has a transmission level hundreds of times that of Omicron?
In order to bypass human stupidity, lower slow and costly distribution and build up herd immunity fast, wouldn't a genetically modified virus be a better way towards that since it will distribute itself? It would bypass anyone who's stupid and doesn't understand how vaccines work, it would bypass slow and bureaucratic distribution chains, bypass corporate profits and be equal between poor and rich nations.
If there was a way to remove lethality and increase transmission rates, that would be a much more effective distribution towards herd immunity than any kind of vaccine. So modifying the virus towards that and intentionally setting it loose could be a very controversial but more efficient way of ending a pandemic.
how it affects society and particularly the health care system. — Benkei
I think we need to adapt our society as fast as the virus adapts its DNA. In particular, there are intersections with climate change measures that seem like obviously sensible precautions. Big reduction in international travel, a big move to level up access to basic hygiene, food, and medicine worldwide, routine hand washing and mask wearing when in close contact. A lot more care over domestic animal hygiene, and more protection for wilderness. There's probably more...
But at the moment, the priorities are saving travel and tourism, levelling down, profiting from vaccine sales, and 'getting back to normal'. :death:
But at the moment, the priorities are saving travel and tourism, levelling down, profiting from vaccine sales, and 'getting back to normal' — unenlightened
Perhaps there are reasons also for that.
The economic recession due to the pandemic was just papered over by the central banks, which made the statistics simply not make sense. And now thanks to that we have inflation. (Which I estimate will not be as transitory as they say).
Back when it all began we talked about why people of color seemed to be at a higher risk. Now we know at least one factor is the prevalence of vitamin D deficiency among POC.
Vitamin D deficiency is associated with more severe disease. Here.
Thanks for that, useful info.
Perhaps there are reasons also for that. — ssu
Reasons for being unreasonable:
https://foreignpolicy.com/2010/10/15/five-zombie-economic-ideas-that-refuse-to-die/
There are people profiteering from crisis, and from increasing disasters, and unfortunately they are dominating the world. Here in the UK, the government trumpets economic growth while presiding on a huge decline in living standards.
The alternative is central planning. That would also be full of woe and dastardly deeds. It's just how we are. Pick your poison.
It's just how we are. — frank
Let's change! — unenlightened
Gotta let them play out their story.
Sweden just announced that it's lifting all pandemic restrictions.
People might remember that Sweden chose a different path from other EU or Western countries. Now it's signalling basically the end of restrictions due to the pandemic.
The Czech Republic is on a similar path:
I'm hoping my country follows the similar path. They have already basically abandoned the corona passport requirement.
May as well. Sooner or later everyone is going to get the Omicron vaccine.
The pandemic is still going. Groan.
I remember the bad old days when there were "no spitting" notices on public transport and other public spaces. What an outrageous curtailment of freedom that was in the name of public health! Almost as bad as the obligation to list the ingredients on food packaging, that still prevents us from selling ground glass enriched bread and so on. Cry 'God for Harry, England, and Saint George!
“Haha! No spitting in public! That’s all we did to you, honest!”
AJJ OptionsShare
Canada’s panty-waisted despot just invoked the Emergencies Act to quell the so-called Freedom Convoy protests. The act gives the federal government sweeping powers, such as to regulate and freeze an individual’s bank account or call in the military. Whatever piddling rights Canada offers its citizens are effectively gone, for now.
The protesters have blocked some important border-crossings, impeding the government’s bottom dollar, and stewing more fear in the ruling class than any Molotov-throwing rioter ever could. All they demand was an end to public health mandates, but Canada’s state-run television likens them to insurrectionists, as is fashionable these days. The worrying part is that none of this is surprising.
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29 | Digest: Magic: the Gathering is Turing Complete | Peter Murphy
This post aims to summarize the 900 IQ construction of a Turing Machine using Magic: the Gathering (MTG) mechanics described in the paper Magic: the Gathering is Turing Complete authored by Churchill, Biderman, and Herrick.
In this paper we show that optimal play in real-world Magic is at least as hard as the Halting Problem, solving a problem that has been open for a decade. To do this, we present a methodology for embedding an arbitrary Turing machine into a game of Magic such that the first player is guaranteed to win the game if and only if the Turing machine halts. Our result applies to how real Magic is played, can be achieved using standard size tournament-legal decks, and does not rely on stochasticity or hidden information. Our result is also highly unusual in that all moves of both players are forced in the construction. This shows that even recognizing who will win a game in which neither player has a non-trivial decision to make for the rest of the game is undecidable.
Embeddings, Universal Turing Machines
Before we jump into the nitty gritty of the construction, we need to define some language to describe Turing Machines and their applications.
For our (and the authors') purposes, an embedding is an arrangement of a subset of rules of a system such that they simulate the workings of a Turing Machine.
Additionally, we'll define that Universal Turing Machine (UTM) is a special kind of Turing Machine which can simulate other Turing Machines. It can perform any computation that any other computer program can perform given the right inputs.
If you can embed a UTM into a system, it means that your system is capable of simulating any other computer. Furthermore, it means your system can perform any computable task.
A UTM is a Finite State Machine with a read and write head, an infinitely long tape broken up into cells, and a controller. By moving along the tape, reading and writing symbols, a UTM can emulate (or rather, it is by definition) a Finite State Machine. These two capacities are sufficient to describe any (non-quantum) computer that we would typically use.
While there are several different configurations of UTMs, this paper employs a specific variety described by Yurii Rogozhin called a
UTM(2,18)
. The arguments here mean that the UTM has 2 states, and 18 symbols.
The Church-Turing Thesis states:
a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine.
running time not withstanding.
The Halting Problem, proved to be unsolvable in general by Turing states that:
From a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. A general algorithm to solve the halting problem for all possible program-input pairs cannot exist.
One such example of the Halting Problem is Goldbach's Conjecture which states that every even whole number greater than 2 is the sum of two prime numbers. Alternatively, every integer can be expressed as the sum of primes. It remains a conjecture since we cannot prove that it is true.
\begin{aligned} 47 &= \underbrace{40}_{ \tt{prime?} ❌} + \underbrace{7}_{ \tt{prime?} ✅} \\ 48 &= \underbrace{41}_{ \tt{prime?} ✅} + \underbrace{7}_{ \tt{prime?} ✅} \end{aligned}
But can we determine if the program to determine if this conjecture is true ever halts?
Consider some program
H
which determines if another program halts e.g.
\begin{aligned} H: (A, \bullet) \rightarrow \{True, False\} \end{aligned}
A
is some other program, and
\bullet
are arbitrary arguments for
A
A
could be another Turing Machine with arguments which could be symbols on a tape.
Turing proved that such a program
H
does not exist. Suppose you have some other program:
\begin{aligned} \overline{H}: (A, \bullet) \rightarrow \{True, False\} \end{aligned}
which tells you the opposite of
H
H
is true, and its input program
A
\overline H
runs forever. But if
H
does not halt, then
\overline H
terminates immediately. What would happen if we gave
\overline{H}
itself as the input:
\begin{aligned} \overline{H}(\overline{H}, \bullet) = ??? \end{aligned}
\overline{H}
runs forever on
\overline{H}
H(\overline{H})
must have halted
\overline{H}
halted, then
H
must have run forever
But how can we tell that
H
ran forever? That means that
\overline{H}(\overline{H})
both halted and ran forever ⚔️.
Okay, so how does this all come together and what the heck does it have to do with Magic?
Finally, Magic
I'd preface this by saying I have not played Magic myself since I was at Scout camp nearly a decade ago and I got stomped, here's a 5 minute breakdown of how the game is played.
MTG is a popular, famously complicated tabletop card game. A simple premise of Magic is that each card that can be played changes or breaks the initial rules in some interesting way.
There are two basic types of cards:
Creatures which have a subtype, power/toughness stats (ATK/DEF), as well as some flavor text describing the modification of the rules to be enacted once the creature enters play
Land mana to cast your spells. There are 5 types, or colors, of land
Most cards are spells, playable via the mana available to a player on their turn. Once a card has been used, it becomes tap'd (making it unavailable for the rest of their turn, pending other rules). After a combat interaction, dead creatures go to a graveyard after a stat comparison (power/toughness).
For the sake of the paper, combat encounters are irrelevant to the UTM, but creatures' stats, which are modifiable by other cards, are the gateway to the central proof.
The authors are trying to embed a UTM into Magic to gain access to the halting problem and everything that accompanies it.
They use the aforementioned Rogozhin
UTM(2,18)
which is sort of a minimum viable project to simulate any computer in the world (non-quantum). Rogozhin's
UTM(2,18)
has two states
\{ q_1, q_2 \}
q_i
is a transition between states involving all or none of the options available to a Turing Machine: read, write, move. Additionally the 18 states are:
\begin{aligned} \big\{\overleftharpoon{1}, 1, \overrightharpoon{1}, \overleftharpoon{1}_1, \overrightharpoon{1}_1, b, \overleftharpoon{b}, \overrightharpoon{b}, \overleftharpoon{b}_1, \overrightharpoon{b}_1, b_2, b_3, c, \overleftharpoon{c}, \overrightharpoon{c}, \overleftharpoon{c}_1, \overrightharpoon{c}_1, c_2 \big\} \end{aligned}
All these don't really matter, the point is that we can represent them using Magic cards.
It's important to note that some games are obviously Turing Complete, like Minecraft. However, MTG is notably not intentionally Turing Complete.
Now, the authors assert that it is possible to compute the next board state given the current board state and a legal move:
\begin{aligned} f: (\text{board state, legal move}) \rightarrow \text{next board state} \end{aligned}
However, given the nature of MTG, there is no trivial
\tt is\_legal(move, board\_state)
function or check. The authors acknowledge that previous work has shown that, working cooperatively (and making known, legal moves), players can construct a Turing Machine, but it is far more interesting to show that in a limited, competitive game, it is not possible to predict how a game will end.
The authors introduce our two arch-nemeses: Alice and Bob along with the 3 elements of a Turing Machine that need to be mapped to the game of Magic: the tape, controller, and read/write head.
The tape is intuitively challenging to embed since there are no geometric or physical metrics present in the game. All we have in a game of Magic are stats, counters, modifiers, etc.
Nonetheless, lots of creatures, organized by color (with
\color{#0C0} \text{green}
to the left of the head, and
\overbrace{\color{#FFF}\text{white}}^{lol}
to the right) yields a directional notation.
The origin of the tape starts is a Rotlung reanimator (2/2) and the read/write head of the Turing Machine is lethal as, in order to traverse the tape, it must slay a creature.
Expanding outwards in either direction from the 2/2 origin are 3/3, 4/4, ..., n/n creatures. The board might look something like this:
\begin{aligned} \color{#0C0} (n/n) , ... ,(4/4), (3/3), \color{#000} \underbrace{(2/2)}_{head}, \color{#BBB} (3/3), (4/4), ..., (n/n) \end{aligned}
The authors select 18 creature types to correspond to Rogozhin's
UTM(2,18)
symbols. Notably, each of these 2/2 creatures spawns a another one of the 18 symbolic creatures upond death. (NATO hates them, check out how these three computational theory researchers destroyed the standard phonetic alphabet):
The authors map the controller using black cards, like the Rotlung Reanimator whose flavor text reads:
Whenever Rotlung Reanimator or another Cleric is put into a graveyard from play, put a 2/2 black Zombie creature token into play.
this card represents the origin of the board.
Some cards, like Artificial Evolution modify the text of other cards, for example: duplicate the Rotlung Reanimator, and modify its flavor text:
Using a slew of other cards, the authors demonstrate how it is possible to map the 3 key properties of UTM: a read/write head, a tape, and the ability to change state via the controller.
Computation begins as follows:
At the beginning of a computational step, it is Alice’s turn and she has the card Infest in hand. Her library consists of the other cards she will cast during the computation (Cleansing Beam, Coalition Victory, and Soul Snuffers, in that order). Bob’s hand and library are both empty. The Turing machine is in its starting state and the tape has already been initialized.
When Alice casts Infest (-2/-2) it kills all 2/2 creatures which, as we noted above, is just the origin of our tape (which happens to belong to Bob, RIP).
This kills one creature: the tape token at the position of the current read head, controlled by Bob. This will cause precisely one creature of Bob’s to trigger – either a Rotlung Reanimator or a Xathrid Necromancer... This Reanimator or Necromancer will create a new 2/2 token to replace the one that died. The new token’s creature type represents the symbol to be written to the current cell, and the new token’s colour indicates the direction for the machine to move: white for left or green for right.
Upon casting Infest, the center card at the head of the tape dies, but in order to traverse the tape, we must add a +1/+1 and -1/-1 buff and debuff respectively to the current and adjacent creature in order to move, as well as all
2n
other creatures on either side as well...
This is cleverly accomplished by modifying creatures of a specific color.
On Alice’s second turn, she casts Cleansing Beam, which reads “Cleansing Beam deals 2 damage to target creature and each other creature that shares a color with it.”
On the last turn of the cycle, Alice casts Soul Snuffers, a 3/3 black creature which reads “When Soul Snuffers enters the battlefield, put a −1/−1 counter on each creature.”
To ensure that the creatures providing the infrastructure (such as Rotlung Reanimator) aren’t killed by the succession of −1/−1 counters each computational step, we arrange that they also have game colours green, white, red and black, using Prismatic Lace, “Target permanent becomes the color or colors of your choice. (This effect lasts indefinitely.)” Accordingly, each cycle Cleansing Beam will put two +1/+1 counters on them, growing them faster than the −1/−1 counters shrink them.
In this way, Cleansing Beam and Soul Snuffer effectively facilitate the movement of the head across the tape.
The full construction includes details about several other cards which Alice and Bob possess in order to maintain the infrastructure prevented, but the embedding is hopefully clear at this point.
Using almost entirely arbitrary information present with MTG, along with Rogozhin's UTM(2,18) in order to "compute" a program , the authors showed that the outcome of a game of Magic is non-computable. this is accomplished by importing everything we described about the Halting problem earlier.
It's important to distinguish between the outcome being simply non-deterministic versus non-computable. Non-deterministic outcomes are computable (albeit hindered by combinatorically exploding possibilities), whereas the outcome here is not even in principle a question that we can ask in a way that makes it computable.
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A Simplified Descriptor System Approach to Delay-Dependent Stability and Robust Performance Analysis for Discrete-Time Systems with Time Delays
Fengying Xu, Daxin Li, "A Simplified Descriptor System Approach to Delay-Dependent Stability and Robust Performance Analysis for Discrete-Time Systems with Time Delays", Mathematical Problems in Engineering, vol. 2013, Article ID 810271, 6 pages, 2013. https://doi.org/10.1155/2013/810271
Fengying Xu 1 and Daxin Li1
1Physical Education Department, Qufu Normal University, Qufu, Shandong 273165, China
Academic Editor: Yang Yi
A simplified descriptor system approach is proposed for discrete-time systems with delays in terms of linear matrix inequalities. In comparison with the results obtained by combining the descriptor system approach with recently developed bounding technique, our approach can remove the redundant matrix variables while not reducing the conservatism. It is shown that the bounding technique is unnecessary in the derivation of our results. Via the proposed method, delay-dependent results on quadratic cost and performance analysis are also presented.
In the past decades, considerable attention has been paid to the problems of stability analysis and control synthesis of time-delay systems. Many methodologies have been proposed, and a large number of results have been established (see, e.g., [1, 2] and the references therein). All these results can be generally divided into two categories: delay-independent stability conditions [3, 4] and delay-dependent stability conditions [5–11]. The delay-independent stability condition does not take the delay size into consideration and thus is often conservative especially for systems with small delays, while the delay-dependent stability condition makes fully use of the delay information and thus is less conservative than the delay-independent one. Very recently, in order to provide less conservative delay-dependent stability criteria, a descriptor system approach was proposed in [12, 13], while a new bounding technique has been presented in [14] (also called Moon's inequality). By combining the descriptor system approach with the bounding technique, novel delay-dependent sufficient conditions for the existence of a memoryless feedback guaranteed cost controller are derived for a class of discrete-time systems with delays in [6, 7].
Although the descriptor system approach proposed in [12, 13] is powerful to deal with the stability analysis of time-delay systems, there are too many matrix variables introduced. In [15], a simplified but equivalent descriptor system approach to delay-dependent stability analysis was established for the continuous-time systems with delays. It is shown in [15] that the bounding technique in [14] is not necessary when deriving the delay-dependent stability results. It should be pointed out that the result in [15] is only applicable to continuous-time systems with delays. In this paper, we focus our attention upon deriving a simplified descriptor system approach to delay-dependent stability analysis in the context of discrete-time systems with delays. It is shown that the results derived by our approach are also equivalent to those obtained in [6, 7] but with fewer variables to be determined. It is also proved that, for discrete-time systems, the bounding technique in [14] will introduce some redundant variables and thus is unnecessary. Via the proposed method, delay-dependent results on quadratic cost and performance analysis are also presented. It is worth mentioning that through the approach proposed in this paper, the delay-dependent guaranteed cost control conditions in [6, 7] obtained by the descriptor system approach and the bounding technique can also be simplified.
Notations. Throughout this paper, for real symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite). The superscript “” represents the transpose. is an identity matrix with appropriate dimension. denotes a diagonal matrix. refers to the space of square summable infinite vector sequences. In symmetric block matrices, we use an asterisk “” to represent a term that is induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.
In order to introduce the simplified descriptor system approach, we consider the following discrete time-delay system where is the state, is the initial condition, the scalar is an upper bound on the time delays , and , are known real constant matrices.
Throughout this paper, we make the following assumption.
Assumption 1. are unknown but satisfy for all
Now, we are in a position to present the main result of this paper.
Theorem 2. Under Assumption 1, the time-delay system is asymptotically stable for all , satisfying (2) if there exist matrices , , , , and , such that the following LMI holds: where
Proof. For all , , satisfying (2), it can be verified that (3) implies that Let It is easy to see that Then, the system can be transformed into an equivalent descriptor form Now, choose a Lyapunov functional candidate as where Then, where .
Furthermore, from (11), we obtain After some manipulations, we get Combining (12) with (13) yields where is given in (5) and Therefore, the time-delay system is asymptotically stable for all , satisfying (2) by the Lyapunov stability theory. This completes the proof.
Remark 3. It is noted that only two time delays are considered for the sake of simplicity. However, the results in Theorem 2 can be extended to the case of multiple delays. The simplified approach in Theorem 2 can also be used to tackle with the discrete time-delay systems with uncertainties, such as norm-bounded parameter uncertainties and linear fractional uncertainties.
Remark 4. Note that the delays considered here satisfy (2). From the proof of Theorem 2, the delay-dependent results in this paper can be extended to the case of interval delays (see [16] for more details), where the delays vary between a lower bound (may be not zero) and an upper bound.
By the method proposed in Theorem 2, the quadratic cost analysis result derived by using the descriptor system approach, together with the inequality in [14] as shown in [6, 7], can also be simplified. To make it clear, introduce the following quadratic cost function Then, by Theorem 2, we have the following result.
Theorem 5. There exist matrices , , , , and , such that the following LMI holds: where , with and being defined in (4), then the system is asymptotically stable, and the cost function in (16) satisfies where .
In the next, via the method proposed in Theorem 2, we will present the performance analysis result.
Consider the following time-delay system: where is the output and is the disturbance signal which is assumed to be in .
Then, the following delay-dependent result on performance analysis can be obtained by Theorem 2.
Theorem 6. Given a scalar . Then, under Assumption 1, the time-delay system :(i)is asymptotically stable with ,(ii)satisfies under zero-initial condition for all nonzero if there exist matrices , such that the following LMI holds:
where , with and being defined in (4).
In this section, we present a numerical example to the effectiveness of the proposed algorithm. In order to show the comparison,we choose and .
Example 7. Consider the system with Based on Theorem 2, we seek the maximum value of . Compared with three methods, which are in [6, 17, 18], respectively; we can illustrate the advantage of the proposed algorithm in this paper. Table 1 presents the result of comparison.
References [18] [17] [6] Theorem 2
the maximum delay bound of .
In this paper, we have proposed a simplified delay-dependent stability condition for discrete-time systems with delays. The given condition has fewer variables compared with those established using the descriptor system approach with Moon's bounding technique. It has been shown that Moon's bounding technique is unnecessary when deriving the delay-dependent stability conditions. By the proposed method in this paper, the delay-dependent results on quadratic cost and performance analysis have also been provided.
This work was supported by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20113705120003.
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AR Conference - Wikiversity
This learning resource is about the requirements and constraints of Augmented Reality (AR) Conferences.
AR.js Example with Kanji Marker with Water Molecule as 3D object
Kanji Marker[1] for AR.js - places the animated 3D model of an AR Conference participant on the marker in camera image. Users can place the participants according to the markers in the camera image.
Joseph Gatt wearing the Mo-Cap suit for Kratos during production of God of War II and III in April 2010
Two repetitions of a walking sequence recorded using a motion-capture system[2]. Watch the reference points and regard the reference point for the moving body as input stream of an AR conference to place the member of the conference in the room and assign the real movements of the AR conference participant on the AR model in AR conference.
2 History of Learning Resource
3 Requirements and Constraints
The learning resouce elaborates on basic concepts of AR conferences, requirements and constraints within the use of web-based technologies. The AR conference should be usable in low-bandwidth environments is available.
History of Learning ResourceEdit
The concept of the learning resource was driven by use of standard video conferencing systems in times of COVID-19 and missing physical presence in lecture room, where you can look around and watching the different participants speak or other students in the seminar room or classroom. Learning environment using video conference are screen focused, while classroom or seminar room is an open space where learners interact with material in room look at eachother while interacting with the learning environment. This comparison of the real classroom situation and learning environment with a video conference leads to this learning environment about Augmented Reality Conferences
Requirements and ConstraintsEdit
the learning resource follows the Open Community Approach and share the content of the learning resource on Wikiversity and
uses OpenSource to implement prototypes and small test scenarios to explore the basic constitutent of AR conferences
(3D Modelling) Explore the learning resource about 3D Modelling and learn about basic concepts of generating 3D models of the lecture room or classroom
(3D Design of Classrooms) Explore the OpenSource software Sweet Home 3D for generating the classroom in 3D.
(Augmented Reality with Markers) Assume that all participants are in larger room, which is sufficient for 5 participants. Explore marker based Augmented Reality and assume that you place a marker for all of the 5 people and everyone places the marker at a chair for the camera position or the position of the single participant in the room. Assume that the face animation of the remote participant is displayed on the marker in room and the other participants wear a head mounted display for the web-based application (e.g. with AR.js) so the other people are projected into the real camera image.
(Audio-Video-Compression) Analyze the learning resource about Audio-Video-Compression and identify possibilty to reduce the bandwidth, e.g. using a GIF animation for silence and talking projected on a plane in the Aframe model.
Use silence detection in the audio stream to select which GIF animation is displayed in the Aframe plane of the AR.js environment.
3D modelling needs a lot of client performance of rendering photo realistic faces. Therefore we use in the learning resource GIF animation of short videostream.
Assume you have multiple GIF animations for different emotions. Is is possible to detect emotions in an audio stream and display the appropriate GIF animation in AFrame model for the AR conference.
Assume you use phonem recognition and transmit just the phonem sequence in low-bandwidth enviroments, how do you use the animation features in Aframe model the mouth appropriate for the speaking the words e.g. the "o" in "onion" or the "sh" in "shark".
(Motion Capture) Motion capturing is a standard method of transfering movements of an actor to an digital model (e.g. a robot, an fictional character in a movie, a dinosaur, ...). How can motion capturing be used in Open Source software. Assume you transmit the position of markers on the face instead of the real camera image. What is the compression rate roughly for an HD video stream?
(Read Words from Face Expression) Some people that have difficulties to hear someone speaking have the skill to read the spoken word from the face. How can this expertise be used for improving speech recognition and sending the recognized word instead of the audio stream. How this feature be used for supporting handicapped people in AR video conference?
(AR Conference on Mars) We use the remote location on planet Mars to explain the concept of Augmented Reality video conference in the context of a learning environments. Assume we use a Mars rover with a stereoscopic 360-Degree image (for visual information for left and right eye).
5 learners and a teacher can meet in physical room (Greenbox) with Motion Capture option of the 6 people in the green room. All people in the room wear a head mounted display and the green screen method will replace the background with a real 360 degree image from the mars (keep in mind the 8min latency from Mars to Earth).
Now we replace the real stereoscopic camera image from the Mars rover with 3D model of the Mars surface (see Digital Elevation Model (DEM)). Now teachers and learners can jump to different location on the Mars explore surfaces and the Modelling allows to view the situation on Mars when water was available on Mars.
Transfer the remote AR conference on Mars to an island that is affected from climate change and learner can explore the situation on that island in an AR conference. Compare AR conference with a real visit on that island and talking to people, that live on that island and where exposed to rising sea level. Discuss also the carbon footprint of learning environments.
(Mathematical Modelling) This learning task was created for learners with a mathematical background. Assume you have a body position encoded with 25 points and 25 frames per seconds. Every point
{\displaystyle P_{(k,t)}:=(x_{(k,t)},y_{(k,t)},z_{(k,t)})\in \mathbb {R} ^{3}}
{\displaystyle k\in \{1,...,25\}}
{\displaystyle t\in \{1,...,100\}}
consists of a
{\displaystyle x,\,y}
{\displaystyle z}
coordinate. One single coordinate is represented by a real value (stored as float variable in a programming language). The movement is recorded for 4 sec (i.e. 100 frames). Calculate the number of real values that you need to encode the body movement 4 sec. Compare the required storage to the storage of the whole body surface encoded as 3D points. What are the benefits and drawback of such an encoding? How would you apply that on facial motion capture of an AR Conference?
Audio-Video-Compression
AR with Markers
↑ Kanji Marker provided by ARToolkit on Github (accessed 2017/12/12) - https://github.com/artoolkit/artoolkit5/blob/master/doc/patterns/Kanji%20pattern.pdf
↑ Olsen, NL; Markussen, B; Raket, LL (2018), "Simultaneous inference for misaligned multivariate functional data", Journal of the Royal Statistical Society Series C, 67 (5): 1147–76, arXiv:1606.03295, doi:10.1111/rssc.12276
Retrieved from "https://en.wikiversity.org/w/index.php?title=AR_Conference&oldid=2273296"
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Growth/Personalized first day/Structured tasks/Add a link - MediaWiki
This page describes the Growth team's work on the "add a link" structured task, which is a type of structured task that the Growth team offers through the newcomer homepage. This page contains major assets, designs, open questions, and decisions. Most incremental updates on progress will be posted on the general Growth team updates page, with some large or detailed updates posted here.
As of August 2021, the first iteration of this task is deployed to half of all new accounts being created in Arabic, Czech, Vietnamese, Bengali, Polish, French, Russian, Romanian, Hungarian, and Persian Wikipedias. We have analyzed data from the first two weeks of the feature's deployment, and we find that newcomers are making many of these edits, and that they have low revert rates. Learnings from this analysis have led us to make improvements to the feature, and the results are encouraging us to broaden the deployment of the feature to more wikis.
Screen from a design concept for the "add a link" structured task
You can see what we are building in these interactive prototypes. Note that because they are prototypes, not all buttons work:
Members of the team presented on the background, algorithm, implementation, and results of this work at Wikimania 2021. See the video here. and the slides here.
To try this feature, see Help:Growth/Tools/Add a link
3 Why wikilinks?
4.1 Comparative review
4.3 Mobile mockups: August 2020
2020-01-07: first evaluation of feasibility of link recommendation algorithm
2020-02-24: evaluation of improved link recommendation algorithm
2020-05-11: community discussion on structured tasks and link recommendations
2020-08-27: backend engineering begins
2020-09-07: first round of user testing of mobile designs
2020-10-19: second round of user testing of mobile designs
2020-10-21: first round of user testing of desktop designs
2020-10-29: frontend engineering begins
2020-11-02: second round of user testing desktop designs
2020-11-10: call for feedback on designs from Arabic, Vietnamese, and Czech communities
2021-04-19: added sections on Terminology and Measurement
2021-05-10: feature is being tested in production on our four pilot wikis
2021-05-27: deployed to half of newcomers on Arabic, Vietnamese, Czech, and Bengali Wikipedias
2021-07-21: deployed to half of newcomers on Polish, Russian, French, Romanian, Hungarian, and Persian Wikipedias.
2021-07-23: posted analysis from first two weeks of feature's deployment.
2021-08-15: presentation at Wikimania about the background, implementation, algorithm, and results.
Next: minor improvements, continued deployments to more wikis, and deeper analysis.
Structured tasks are meant to break down editing tasks into step-by-step workflows that make sense for newcomers and make sense on mobile devices. The Growth team believes that introducing these new kinds of editing workflows will allow more new people to begin participating on Wikipedia, some of whom will learn to do more substantial edits and get involved with their communities. After discussing the idea of structured tasks with communities, we decided to build the first structured task: "add a link". This task will use an algorithm to point out words or phrases that may be good wikilinks, and newcomers can accept or reject the suggestions. With this project, we want to gain learnings on these questions:
Are structured tasks engaging to newcomers?
Do newcomers succeed with structured tasks on mobile?
Do they generate valuable edits?
Do they lead some newcomers to increase their involvement?
Why wikilinks?
The below is excerpted from the structured tasks page, explaining why we chose to build "add a link" as the first structured task.
This section contains our current design thinking. To look into the full set of thinking around designs for the "add a link" structured task, see this slideshow, which contains background, user stories, and initial design concepts.
Our designs evolved through several rounds of user tests and iterations. As of December 2020, we have settled on the designs that we'll engineer for the first version of this feature. You can see them in these interactive prototypes. Note that because they are prototypes, not all buttons work:
When we design a feature, we look into similar features in other software platforms outside of the Wikimedia world. These are some highlights from comparative reviews done in preparation for Android’s suggested edits feature, which remain relevant for our project.
Mobile mockups: August 2020
Slides showing the full set of Concept A and B mockups (in English)
In discussing these designs, our team is hoping for input on a set of essential questions:
Post-deployment Add a Link 290 28 9.7%
Unstructured 63 22 34.9%
Pre-deployment Add a Link 958 49 5.1%
{\displaystyle \chi ^{2}=16.5,df=1,p\ll 0.001}
Post-deployment 597 72.4 125 15.2 103 12.5 825
Pre-deployment 1,464 65.1 595 26.5 189 8.4 2,248
Post-deployment ≥1 edit 96 31 32.3
Pre-deployment ≥1 edit 64 10 15.6
≥5 edits 19 1 5.3
Post-deployment 178 96 53.9
Pre-deployment 101 64 63.4
Desktop Almost everyone knows what it is 2,732 53.0%
Linking to wrong article 1,377 26.7%
Text should include more or fewer words 378 7.3%
Mobile Almost everyone knows what it is 1,835 53.3%
Linking to wrong article 791 23.0%
Undefined 62 1.8%
Retrieved from "https://www.mediawiki.org/w/index.php?title=Growth/Personalized_first_day/Structured_tasks/Add_a_link&oldid=5103626"
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Topographic Response to Simulated Mw 6.5–7.0 Earthquakes on the Seattle Fault | Bulletin of the Seismological Society of America | GeoScienceWorld
Topographic Response to Simulated Mw 6.5–7.0 Earthquakes on the Seattle Fault
Ian Stone;
Ian Stone *
U.S. Geological Survey, Earthquake Science Center, Seattle, Washington, U.S.A.
Corresponding author: istone@usgs.gov
Erin A. Wirth;
Erin A. Wirth
Ian Stone, Erin A. Wirth, Arthur D. Frankel; Topographic Response to Simulated
Mw
6.5–7.0 Earthquakes on the Seattle Fault. Bulletin of the Seismological Society of America 2022;; 112 (3): 1436–1462. doi: https://doi.org/10.1785/0120210269
We explore the response of ground motions to topography during large crustal fault earthquakes by simulating several magnitude 6.5–7.0 rupture scenarios on the Seattle fault, Washington State. Kinematic simulations are run using a 3D spectral element code and a detailed seismic velocity model for the Puget Sound region. This model includes realistic surface topography and a near‐surface low‐velocity layer; a mesh spacing of ∼30 m at the surface allows modeling of ground motions up to 3 Hz. We simulate 20 earthquake scenarios using different slip distributions and hypocenter locations on a planar fault surface. Results indicate that average ground motions in simulations with and without topography are similar. However, shaking amplification is common at topographic highs, and more than a quarter of all sites experience short‐period (≤2 s) ground‐motion amplification greater than 25%–35%, compared with models without topography. Comparisons of peak ground velocity at the top and bottom of topographic features demonstrate that amplification is sensitive to period, with the greatest amplifications typically manifesting near a topographic feature’s estimated resonance frequency and along azimuths perpendicular to its primary axis of elongation. However, interevent variability in topographic response can be significant, particularly at shorter periods (<1 s). We do not observe a clear relationship between source centroid‐to‐site azimuths and the strength of topographic amplification. Overall, our results suggest that although topographic resonance does influence the average ground motions, other processes (e.g., localized focusing and scattering) also play a significant role in determining topographic response. However, the amount of consistent, significant amplification due to topography suggests that topographic effects should likely be considered in some capacity during seismic hazard studies.
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Reachability relation, transitive closure, and transitive reductionEdit
Topological orderingEdit
Combinatorial enumerationEdit
{\displaystyle a_{n}=\sum _{k=1}^{n}(-1)^{k-1}{n \choose k}2^{k(n-k)}a_{n-k}.}
Related families of graphsEdit
Topological sorting and recognitionEdit
Construction from cyclic graphsEdit
Transitive closure and transitive reductionEdit
Closure problemEdit
Path algorithmsEdit
Data processing networksEdit
Causal structuresEdit
Genealogy and version historyEdit
Citation graphsEdit
The Price model is too simple to be a realistic model of a citation network but it is simple enough to allow for analytic solutions for some of its properties. Many of these can be found by using results derived from the undirected version of the Price model, the Barabási–Albert model. However, since Price's model gives a directed acyclic graph, it is a useful model when looking for analytic calculations of properties unique to directed acyclic graphs. For instance, the length of the longest path, from the n-th node added to the network to the first node in the network, scales as[52]
{\displaystyle \ln(n)}
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Ella and her study team are arguing about the slope of the line in the graph below. They have come up with four different answers:
\frac { 3 } { 4 }
- \frac { 4 } { 3 }
- \frac { 3 } { 4 }
\frac { 4 } { 3 }
. Which slope is correct? Justify your answer.
Remember that the slope can be written as
\frac{\text{change in}\ y}{\text{change in}\ x}
On this line as the
x
-value increases, does the
y
-value decrease or increase?
Decreasing makes the slope negative.
Increasing makes the slope positive.
Try answering this problem by going from the point
\left(−3, 3\right)
\left(1, 0\right)
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PHC6937-MCH-Fall2017
Epidemiological evidence on the impact of environment on pregnancy, birth and child health outcomes
Impact of Environment on Maternal and Child Health
Methodological Challenges in Studying Environmental Impacts on Maternal and Child Health
The Total Environment and Hypertensive Disorders of Pregnancy
Environment is an important health determinant
Phenotypes are a function of inherited and environmental factors
T2DM, Cancer, LBW, PTD
There are exquisite tools that have been developed to sequence the human genome and to interrogate individual susceptibility through genome-wide association studies (GWAS)
Thousands of GWAS
However, there was a lack of comparable tools in relation to exposure assessment
- almost uniquely focused on single exposure-health effect relationships and no global view of how various types of exposures co-exist and jointly affect health
A similar platform for discovery should exist for E
Heritability: the range of phenotypic variability attributed to genetic variability in a population
- indicator of the proportion of phenotypic differences attributed to G
\sigma_P^2=\sigma_G^2+\sigma_E^2
\sigma_P^2=\sigma_G^2+\sigma_E^2
H^2={{\sigma^2_G}\over {\sigma^2_P}}
H^2={{\sigma^2_G}\over {\sigma^2_P}}
To draw attention to the critical need for more complete environmental exposure assessment
- environment is defined in this context as 'non-genetic' factors
- exposome complements the genome
The exposome is composed of every exposure to which an individual is subjected from conception to death
- the nature of those exposures
- their changes over time
Three domains of exposome
Three broad domains of non-genetic exposures:
- specific external
- general external
Source: Wild CP. The exposome: from concept to utility. International journal of epidemiology. 2012 Feb 1;41(1):24-32.
Three domains of exposome (continued)
There is overlap in the three domains
- physical activity can either be internal or specific external
The domains can also be considered as intertwined
- the internal may at least partially be a response to the external
Measures in one domain or another may reflect to differing degrees one component of the exposome:
- the urban environment (general external)
- air pollution (specific external)
- inflammation (internal)
Individual's Health
Behavior and Intrinsic Biological Factors
May cause skin cancer and melanoma
Recent studies also showed associations between UVR and increased mortality due to CVD, cancer, and respiratory diseases
Natural radioactive decay of uranium
6 criteria air pollutants - PM, SO2, CO, NOx, O3, Pb
Indoor air pollution: tobacco smoke, combustion products, radon gas
Extreme temperature and precipitation
Associated with increased mortality and morbidity
Food access, Walkability, and Greenness
Associated with physical activity, obesity, and cardiometabolic outcomes
No safe level of lead exposure has been identified
Even low levels of lead can cause neuropsychiatric function and potentially lead to behavior problems
Education, Poverty, and Safety
Important health determinants
Environmental Impacts on Maternal and Child Health
Pregnant women, developing fetus, and children are often especially susceptible to environmental exposures
Multiple environmental factors found to be associated with adverse pregnancy and birth outcomes
- persistent organic pollutants (POPs)
- residential greenness
Similar associations found for ASD, ADHD, learning capacity, and brain growth and development among children.
Environmental Contributions to Disparities in Maternal and Child Health
Persisting large disparities
- Largely driven by individuals' SES?
Large disparities observed among WIC participants
Mediation analyses to determine how neighborhood environmental factors contribute to racial disparities in HDP:
The Inverse Odds Ratio Weighting (IORW):
Environmental Contributions to Disparities in Maternal and Child Health (continued)
Hu, H., Ha, S., & Xu, X. (2017). Ozone and hypertensive disorders of pregnancy in Florida: Identifying critical windows of exposure. Environmental research, 153, 120-125.
Methodological advantages to study environmental impacts on maternal and child health
High spatio-temporal variability of environmental exposures
It is relatively easier to study environmental impacts on maternal and child health outcomes because of the relatively short and clearly defined exposure windows
E.g. pregnancy period for pregnancy and birth outcomes
Life course approach is increasingly popular to study chronic disease epidemiology, but it poses great challenges to study environmental expsoures
Lack of data on residential history and activity patterns
Most environmental epidemiological studies on pregnancy and birth outcomes are based on residential addresses at delivery
Autocorrelated exposure
Traditional methods are not able to address the autocorrelations
Unique selection bias
Fixed cohort bias
Residential history and activity patterns
For birth cohorts, the population at risk is constantly changing as new pregnancies start and existing pregnancies end.
In retrospective birth cohorts, when using a study period based on date of birth (e.g. all births from Jan 1, 2010 to Dec 31, 2010), the population at risk is different at the start and end of the cohort.
Fixed cohort bias:
- only the longer pregnancies at the start of the study and only the shorter pregnancies at the end of the study are included
- e.g. if there was an unusually hot month in the first trimester of the included women, then we will observe that high temperature is wrongly associated with longer gestations
Fixed Cohort Bias (continued)
Fixed cohort bias not only bias the estimated effects of season, but also bias the estimated effects of seasonal exposures (e.g. many environmental factors such as temperature, air pollution, etc.)
Method 1: Assuming the shortest gestation is 16 weeks, and the longest gestation is 44 weeks. We can then remove pregnancies that were conceived earlier than 16 weeks prior to the start of the study period and later than 44 weeks after before the end of the study period
Method 2: Use a study period based on date of conception (instead of date of birth)
It is important to identify critical windows of exposure to environmental factors that impact maternal and child health
E.g. when assessing the impacts of temperature on low birth weight, we want to know what pregnancy periods are more susceptible
Many environmental factors have high spatio-temporal variability
- place to place
Why traditional models cannot be used?
Distributed Lag Models (continued)
A class of models that is developed to describe associations in which the dependency between an exposure and an outcome is lagged in time
"Exposure-response" + "lag-response" -> "Exposure-lag-response"
Firstly applied in time series analysis, and recently generalized beyond time series design
The general idea is to weight past exposures through specific functions whose parameters are estimated by the data
- two sets of basis functions that can independently model the exposure and lag-response relationships
git clone https://github.com/benhhu/DLM.git
https://github.com/benhhu/DLM
Hypertensive Disorders of Pregnancy (HDP)
Most common pregnancy complication (up to 10%)
Major cause of morbidity and mortality in both mothers and babies
Image: dunyanews.tv
Large racial disparities
Individual's HDP Risk
Only a few factors in the environment have been assessed, and usually separately without considering the totality of environment
Florida Vital Statistics Birth Records
Who and Where future interventions should be focused
How environmental inequity contributes to racial disparities
What risk factors require interventions
Individual-level Risk Factors Identification and Recommendation
Neighborhood-level Risk Prediction and Needs Assessment
Slides for Guest Lecture, Fall 2017, PHC6937 Maternal and Child Health Epidemiology
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Multidimensional inverse fast Fourier transform - MATLAB ifftn - MathWorks España
{X}_{{p}_{1},{p}_{2},...,{p}_{N}}=\sum _{{j}_{1}=1}^{{m}_{1}}\frac{1}{{m}_{1}}{\omega }_{{m}_{1}}^{{p}_{1}{j}_{1}}\sum _{{j}_{2}=1}^{{m}_{2}}\frac{1}{{m}_{2}}{\omega }_{{m}_{2}}^{{p}_{2}{j}_{2}}...\sum _{{j}_{N}=1}^{{m}_{N}}\frac{1}{{m}_{N}}{\omega }_{{m}_{N}}^{{p}_{N}{j}_{N}}{Y}_{{j}_{1},{j}_{2},...,{j}_{N}}.
{\omega }_{{m}_{k}}={e}^{2\pi i/{m}_{k}}
g\left(a,b,c,...\right)
g\left(a,b,c,...\right)={g}^{*}\left(-a,-b,-c,...\right)
. However, the fast Fourier transform of a multidimensional time-domain signal has one half of its spectrum in positive frequencies and the other half in negative frequencies, with the first row, column, page, and so on, reserved for the zero frequencies. For this reason, for example, a 3-D array Y is conjugate symmetric when all of these conditions are true:
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What Is Tax Incidence and How Does It Works? | Outlier
What Is Tax Incidence and How it Works
Here’s an overview about what tax incidence is, how it works, and how it relates to price elasticity.
How Does Elasticity Determine the Tax Incidence?
Example of Elastic Supply and Inelastic Demand
Example of Inelastic Supply and Elastic Demand
Tax Incidence Calculation
For every economic transaction, the government imposes a tax. But who should pay that tax? Does the tax burden fall onto the buyer or the seller? Or maybe both? This article will discuss tax incidence, the economic analysis that determines how the overall cost of taxes is distributed between buyers and sellers.
Tax incidence is how the tax burden is divided between buyers and sellers. This division of the tax expense is primarily determined by the relative elasticity of the supply and demand for the goods or services we are discussing.
Usually, the tax incidence falls on both the consumers and producers. However, to calculate which side will pay the majority of the tax, we analyze the demand and supply elasticity. The tax cost will be more significant for the more inelastic side of demand and supply. If the demand for a good is more inelastic than the supply, the buyers will then burden more of the tax cost. Inversely, if the supply side is more inelastic than the demand, the producers or sellers will pay most of the tax.
Tax incidence is how the tax burden is divided between buyers and sellers.
Let us quickly review how elasticity works, and then we will see how it determines which party in a transaction will pay the tax. Elasticity is the percentage change of the quantity demanded or quantity supplied relative to its price change.
Simply put, the more sensitive buyers are to a price change of any good or service, the more elastic the demand will be. The demand side is inelastic when the percentage change in the quantity demanded is not significantly affected by a change in price. Meaning a big move in the price will not cause a big difference in the quantity demanded. The same would be on the supply side of the market.
Elasticity is determined by how sensitive the willingness or ability of producers is to supply the goods or services. The more susceptible they are to different factors, the more elastic the supply will be. Supply is considered inelastic when the percentage change in the quantity being supplied is less than the percent change in price. With this in mind, we can now see that the burden of taxation is passed to the more inelastic side of the market because that is the side that is less sensitive to a price change.
To see how tax incidence is determined by the relative elasticity of demand and supply, let's use the example of medicine or healthcare. Consumers of medical treatment or drugs are not too sensitive to a price change, so it's inelastic. Even if the price of a particular drug rises sharply, the people dependent on it will still have to buy it.
So the quantity of demand for that drug will not change even with a significant price change. Therefore, if the government imposes a tax on that drug, producers of the drug will pass the tax cost onto buyers because the demand for it is inelastic. Even if the price of the medicine rises, it will not significantly affect the quantity of demand.
Another example where most of the tax expense falls on the buyer is cigarettes. Although there is a pretty high tax on cigarettes, the buyers—not the sellers—pay most of it. Since many smokers are somewhat addicted, they are not that sensitive to a price change. If sellers increase the price of cigarettes to capture the entire tax cost, it will not significantly change the quantity demanded. So the tax incidence here again shows us that the cost is distributed to the more inelastic side of the market.
Pe
is the equilibrium price before taxes are introduced. The distance between the price paid by consumers
Pc
and the price producers receive
Pp
is the tax revenue per unit sold. In this diagram, the supply is more elastic than demand, so the tax incidence falls more on consumers than on producers as we see
Pc–Pe
Pp–Pe
Now let's look at an example where the supply is inelastic. Where producers are willing to supply relatively the same amount no matter the price, they can sell it. On the other hand, the demand is elastic; consumers are sensitive to the price, so a slight price increase will cause a significant decrease in the demand.
An example of this is luxury goods like jewelry, art, or expensive cars or furniture. Since these things are not a necessary good—as opposed to medicine or healthcare—buyers are sensitive to any price changes. A slight increase in price causes a more significant percentage decrease in demand.
Thus, if a tax is imposed on such a good, the producers will have to bear most of the cost. Since the consumption of luxury goods is elastic, the producers or sellers cannot pass the cost of taxes onto the consumers without affecting the quantity of demand. So the tax incidence in such a case falls more on the sellers than on the buyers.
Pe
Pc
Pp
is the tax revenue per unit sold. In this diagram, the demand is more elastic than supply, so the tax incidence falls more on producers than on consumers as we see
Pp–Pe
Pc–Pe
With most goods and services, demand and supply are not entirely inelastic or elastic; the entire tax burden does not fall on only one side of the market. Typically, this means that the producer or seller can pass at least some of the tax expense onto the buyer as higher prices. Whatever portion of the tax is not captured in the sale price, falls onto the cost of producers or sellers.
The Tax Incidence of a New Tax on Soda
To see how the cost of tax gets divided between buyers and sellers, let's imagine the government imposes a new tax on soda. Let's assume the new tax will be $0.25 on every bottle of soda sold. The first step for us to see who will end up paying the new 25 cent expense is to calculate the price elasticity of demand and supply.
The Formula for Calculating Elasticity
\text{Price Elasticity of Demand} = \frac{\text{Percent Change in Quantity}}{\text{Percent Change in Price}}
So let's assume that for a 10 percent increase in the price of soda, the rate of demand drops about 7%.
So our demand elasticity is:
\text{Price Elasticity of Demand} = \frac{\text{Percent Change in Quantity}}{\text{Percent Change in Price}}= \frac{-7}{10} = -0.7
\text{Price Elasticity of Supply} = \frac{\text{Percent Change in Quantity}}{\text{Percent Change in Price}}
Let us assume the supply of soda is somewhat inelastic. For every 2% change in the price, there is a 1% change in the quantity demanded.
\text{Price Elasticity of Supply} = \frac{\text{Percent Change in Quantity}}{\text{Percent Change in Price}}= \frac{1}{2} = 0.5
Now that we have the relative elasticity of both the demand and supply, we can go ahead and calculate the tax incidence for soda:
\frac{\text{Price Elasticity of Supply}}{\text{Price Elasticity of Supply}-\text{Price Elasticity of Demand}}
\frac{0.5}{0.5-(-0.7)}=\frac{0.5}{1.2}
100% – 41% = 59% is the amount of tax incidence paid by the seller.
What we see here is that the party with the greater elasticity—the producers of soda—ends up paying a bigger portion of the tax.
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Time-varying flow resistance - MATLAB - MathWorks Switzerland
Variable Local Restriction (2P)
Time-varying flow resistance
The Variable Local Restriction (2P) block models the pressure drop due to a time-varying flow resistance such as a valve. Ports A and B represent the restriction inlet and outlet. Port AR sets the time-varying restriction area, specified as a physical signal.
The restriction consists of a contraction followed by a sudden expansion in flow area. The contraction causes the fluid to accelerate and its pressure to drop. The expansion recovers the lost pressure though only in part, as the flow separates from the wall, losing momentum in the process.
The mass balance equation is
{\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,
{\stackrel{˙}{m}}_{A}
{\stackrel{˙}{m}}_{B}
are the mass flow rates into the restriction through port A and port B.
The energy balance equation is
{\varphi }_{A}+{\varphi }_{B}=0,
ϕA and ϕB are the energy flow rates into the restriction through port A and port B.
The local restriction is assumed to be adiabatic and the change in specific total enthalpy is therefore zero. At port A,
{u}_{A}+{p}_{A}{\nu }_{A}+\frac{{w}_{A}^{2}}{2}={u}_{R}+{p}_{R}{\nu }_{R}+\frac{{w}_{R}^{2}}{2},
while at port B,
{u}_{B}+{p}_{B}{\nu }_{B}+\frac{{w}_{B}^{2}}{2}={u}_{R}+{p}_{R}{\nu }_{R}+\frac{{w}_{R}^{2}}{2},
uA, uB, and uR are the specific internal energies at port A, at port B, and the restriction aperture.
pA, pB, and pR are the pressures at port A, port B, and the restriction aperture.
νA, νB, and νR are the specific volumes at port A, port B, and the restriction aperture.
wA, wB, and wR are the ideal flow velocities at port A, port B, and the restriction aperture.
The ideal flow velocity is computed as
{w}_{A}=\frac{{\stackrel{˙}{m}}_{ideal}{\nu }_{A}}{S}
at port A, as
{w}_{B}=\frac{{\stackrel{˙}{m}}_{ideal}{\nu }_{B}}{S}
at port B, and as
{w}_{R}=\frac{{\stackrel{˙}{m}}_{ideal}{\nu }_{R}}{{S}_{R}},
inside the restriction, where:
{\stackrel{˙}{m}}_{ideal}
is the ideal mass flow rate through the restriction.
S is the flow area at port A and port B.
SR is the flow area of the restriction aperture.
The ideal mass flow rate through the restriction is computed as:
{\stackrel{˙}{m}}_{ideal}=\frac{{\stackrel{˙}{m}}_{A}}{{C}_{D}},
CD is the flow discharge coefficient for the local restriction.
Local Restriction Variables
The change in momentum between the ports reflects in the pressure loss across the restriction. That loss depends on the mass flow rate through the restriction, though the exact dependence varies with flow regime. When the flow is turbulent:
\stackrel{˙}{m}={S}_{\text{R}}\left({p}_{\text{A}}-{p}_{\text{B}}\right)\sqrt{\frac{2}{|{p}_{\text{A}}-{p}_{\text{B}}|{\nu }_{\text{R}}{K}_{\text{T}}}},
where KT is defined as:
{K}_{\text{T}}=\left(1+\frac{{S}_{\text{R}}}{S}\right)\left(1-\frac{{\nu }_{\text{in}}}{{\nu }_{\text{out}}}\frac{{S}_{\text{R}}}{S}\right)-2\frac{{S}_{\text{R}}}{S}\left(1-\frac{{\nu }_{\text{out}}}{{\nu }_{\text{R}}}\frac{{S}_{\text{R}}}{S}\right),
in which the subscript in denotes the inlet port and the subscript out the outlet port. Which port serves as the inlet and which serves as the outlet depends on the pressure differential across the restriction. If pressure is greater at port A than at port B, then port A is the inlet; if pressure is greater at port B, then port B is the inlet.
When the flow is laminar:
\stackrel{˙}{m}={S}_{\text{R}}\left({p}_{\text{A}}-{p}_{\text{B}}\right)\sqrt{\frac{2}{\Delta {p}_{\text{Th}}{\nu }_{\text{R}}{\left(1-\frac{{S}_{\text{R}}}{S}\right)}^{2}},}
where ΔpTh denotes the threshold pressure drop at which the flow begins to smoothly transition between laminar and turbulent:
\Delta {p}_{\text{Th}}=\left(\frac{{p}_{\text{A}}+{p}_{\text{B}}}{2}\right)\left(1-{B}_{\text{L}}\right),
in which BLam is the Laminar flow pressure ratio block parameter. The flow is laminar if the pressure drop from port A to port B is below the threshold value; otherwise, the flow is turbulent.
The pressure at the restriction area, pR likewise depends on the flow regime. When the flow is turbulent:
{p}_{\text{R,L}}={p}_{\text{in}}-\frac{{\nu }_{\text{R}}}{2}{\left(\frac{\stackrel{˙}{m}}{{S}_{\text{R}}}\right)}^{2}\left(1+\frac{{S}_{\text{R}}}{S}\right)\left(1-\frac{{\nu }_{\text{in}}}{{\nu }_{\text{R}}}\frac{{S}_{\text{R}}}{S}\right).
{p}_{\text{R,L}}=\frac{{p}_{\text{A}}+{p}_{\text{B}}}{2}.
Area normal to the flow path at the restriction aperture when the restriction is in the fully closed state. The area obtained from physical signal AR saturates at this value. Input values smaller than the minimum restriction area are ignored and replaced by the value specified here. The default value of 1e-10 m^2.
Area normal to the flow path at the restriction aperture when the restriction is in the fully open state. The area obtained from physical signal AR saturates at this value. Input values greater than the maximum restriction area are ignored and replaced by the value specified here. The default value is 0.005 m^2.
Area normal to the flow path at the restriction ports. The ports are assumed to be identical in cross-section. The default value, 0.01 m^2, is the same as the restriction aperture area.
Ratio of the actual to the theoretical mass flow rate through the restriction. The discharge coefficient is an empirical parameter used to account for non-ideal effects such as those due to restriction geometry. The default value is 0.64.
Ratio of the outlet to the inlet port pressure at which the flow regime is assumed to switch from laminar to turbulent. The prevailing flow regime determines the equations used in simulation. The pressure drop across the restriction is linear with respect to the mass flow rate if the flow is laminar and quadratic (with respect to the mass flow rate) if the flow is turbulent. The default value is 0.999.
A pair of two-phase fluid conserving ports labeled A and B represent the restriction inlet and outlet. A physical signal input port labeled AR controls the cross-sectional area of the restriction aperture, located between the restriction inlet and outlet.
Local Restriction (2P)
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Compute H2 optimal controller - MATLAB h2syn - MathWorks España
Stabilizing Controller for MIMO Plant
Mixed-Sensitivity H2 Loop Shaping
Compute H2 optimal controller
[K,CL,gamma] = h2syn(P,nmeas,ncont)
[K,CL,gamma] = h2syn(P,nmeas,ncont,opts)
[K,CL,gamma,info] = h2syn(___)
[K,CL,gamma] = h2syn(P,nmeas,ncont) computes a stabilizing H2-optimal controller K for the plant P. The plant has a partitioned form
\left[\begin{array}{c}z\\ y\end{array}\right]=\left[\begin{array}{cc}{P}_{11}& {P}_{12}\\ {P}_{21}& {P}_{22}\end{array}\right]\left[\begin{array}{c}w\\ u\end{array}\right],
nmeas and ncont are the number of signals in y and u, respectively. y and u are the last outputs and inputs of P, respectively. h2syn returns a controller K that stabilizes P and has the same number of states. The closed-loop system CL = lft(P,K) achieves the performance level gamma, which is the H2 norm of CL (see norm).
[K,CL,gamma] = h2syn(P,nmeas,ncont,opts) specifies additional computation options. To create opts, use h2synOptions.
[K,CL,gamma,info] = h2syn(___) returns a structure containing additional information about the H2 synthesis computation. You can use this argument with any of the previous syntaxes.
Stabilize a 5-by-4 unstable plant with three states, two measurement signals, and one control signal.
In practice, P is an augmented plant that you have constructed by combining a model of the system to control with appropriate
{\mathit{H}}_{2}
weighting functions. For this example, use the following model.
A = [5 6 -6
-6 5 4];
B = [0 4 0 0
4 0 0 -3];
C = [-6 0 8
0 -15 7];
D = [0 0 0 0
8 0 -7 0];
Confirm that P is unstable by examining its poles, some of which lie in the right half-plane.
Design the stabilizing controller. h2syn assumes that the nmeas measurement signals and the ncont control signals are the last outputs and last inputs of P, respectively.
[K,CL,gamma] = h2syn(P,nmeas,ncont);
Examine the closed-loop system to confirm that the controller K stabilizes the plant.
Shape the singular value plots of the sensitivity
S=\left(I+GK{\right)}^{-1}
and complementary sensitivity
T=GK\left(I+GK{\right)}^{-1}
To do so, find a stabilizing controller K that minimizes the
{H}_{2}
norm of:
Assume the following plant and weights:
G\left(s\right)=\frac{s-1}{s-2},{W}_{1}=\frac{0.1}{100s+1},{W}_{2}=0.1,{W}_{3}=0.
Using those values, construct the augmented plant P, as illustrated in the mixsyn reference page.
G = 10*(s-1)/(s+1)^2;
G.u = 'u2';
W1 = 0.1/(100*s+1);
W1.u = 'y2';
W1.y = 'y11';
W2 = tf(0.1);
W2.u = 'u2';
S = sumblk('y2 = u1 - y');
P = connect(G,S,W1,W2,{'u1','u2'},{'y11','y12','y2'});
Use h2syn to generate the controller. This system has one measurement signal and one control signal, which are the last output and input of P, respectively.
[K,CL,gamma] = h2syn(P,1,1);
Examine the resulting loop shapes.
S = inv(1+L);
sigmaplot(L,'k-.',S,'r',T,'g')
legend('open-loop','sensitivity','closed-loop')
\begin{array}{c}dx=Ax+{B}_{1}w+{B}_{2}u\\ z={C}_{1}x+{D}_{11}w+{D}_{12}u\\ y={C}_{2}x+{D}_{21}w+{D}_{22}u.\end{array}
If P is a generalized state-space model with uncertain or tunable control design blocks, then the function uses the nominal or current value of those elements.
For the H2 synthesis problem to be solvable, (A,B2) must be stabilizable, and (A,C2) must be detectable. The plant is further restricted in that P12 and P21 must have no zeros on the imaginary axis (continuous-time plants) or the unit circle (discrete-time plants). In continuous time, this restriction means that
\left[\begin{array}{cc}A-j\omega & {B}_{2}\\ {C}_{1}& {D}_{12}\end{array}\right]
has full column rank for all frequencies ω. By default, h2syn automatically adds extra disturbances and errors to the plant to ensure that the restriction on P12 and P21 is met. This process is called regularization. If you are certain your plant meets the conditions, you can turn off regularization using the Regularize option of h2synOptions.
h2synOptions object
Additional options for the computation, specified as an options set you create using h2synOptions. Available options include turning off automatic scaling and regularization. For more information, see h2synOptions.
Controller, returned as a state-space (ss) model object. The controller stabilizes P and has the same number of states as P. The controller has nmeas inputs and ncont outputs.
Closed-loop transfer function, returned as a state-space (ss) model object or []. The closed-loop transfer function is CL = lft(P,K) as in the following diagram.
Controller performance, returned as a nonnegative scalar value. This value is the performance achieved using the returned controller K, and is the H2 norm of CL (see norm).
Additional synthesis data, returned as a structure. info has the following fields.
Solution of state-feedback Riccati equation, returned as a matrix.
Solution of observer Riccati equation, returned as a matrix.
State feedback gain of in the observer form of controller K returned as a matrix. For more information about the observer-form controller, see Tips.
Observer gains of the observer form of controller K, returned as matrices. For more information about the observer-form controller, see Tips.
Regularized plant used for h2syn computation, returned as a state-space (ss) model object. By default, h2syn automatically adds extra disturbances and errors to the plant to ensure that it meets certain conditions (see the input argument P). The field info.Preg contains the resulting plant model.
Costs for the synthesized controller, returned in a vector of the form [FI OE DF FC], where:
FI is the full-information control cost.
OE is the output-estimation cost.
DF is the disturbance-feedforward cost.
FC is full control cost.
These quantities are related by FI^2 + OE^2 = DF^2 + FC^2 = gamma^2. For more details on these norms, see sections 14.8 and 14.9 of [1].
Full-information state-feedback gain, returned as a matrix. The full-information problem assumes full knowledge of the state x and disturbance w, and looks for an optimal state-feedback control of the form:
u(t) = KFI*[x(t);w(t)] in continuous time. In continuous time, u depends only on x. The entries in KFI corresponding to w are zero.
u[k] = KFI*[x[k];w[k]] in discrete time.
For more information, see section 14.8 of [1].
GFI Full-information closed-loop transfer from w to z with the controller KFI, returned as a state-space (ss) model. The H2 norm of GFI is FI.
HAMX,HAMY X Hamiltonian matrix (state feedback) and Y Hamiltonian matrix (Kalman filter). These values are provided for reference, but h2syn does not use them to compute the Riccati solutions. Instead, h2syn uses the implicit solvers icare and idare.
h2syn gives you state-feedback gain and observer gains that you can use to express the controller in observer form. The observer form of the controller K is:
\begin{array}{c}d{x}_{e}=A{x}_{e}+{B}_{2}u+{L}_{x}e\\ u={K}_{u}{x}_{e}+{L}_{u}e.\end{array}
Here, the innovation term e is:
e=y-{C}_{2}{x}_{e}-{D}_{22}u.
h2syn returns the state-feedback gain Ku and the observer gains Lx and Lu as fields in the info output argument.
h2syn uses the methods described in Chapter 14 of [1].
[1] Zhou, K., Doyle, J., Glover, K, Robust and Optimal Control. Upper Saddle River, NJ: Prentice Hall, 1996.
hinfsyn | h2synOptions | mixsyn | loopsyn | ncfsyn | norm
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How to Find the Median | Outlier
In this article, we’ll look at medians, one of the fundamental concepts for ordering statistical data. You’ll learn how to calculate them both by hand and with the assistance of statistical software, as well as how to apply them within different statistical contexts.
Finding the Median Using Statistical Software
Why Are Medians Important? And How Are They Different from the Mean?
Finding the Median with Ordinal Data
The median is the value that falls in the middle of a data set when the values are arranged in increasing order. The median marks the data’s 50th percentile, dividing the bottom half of the data from the top half.
Median, mean, and mode are all measures of center, but they do not represent the same thing:
The median is the middle of your data, and it marks the 50th percentile.
The mean is the arithmetic average of your data.
The mode is the value that occurs most frequently in your data.
To find the median, arrange your data from smallest to largest and look for the middle value. If the total number of data points is odd, there will be one number that sits directly in the middle of your data set, and this will be your median. If the total number of data points is even, look for the two middle values and average them. This average will be your median.
Example 1. Finding the median in an odd-numbered data set
Say you have the following odd-numbered data set where n=9:
You can calculate the median in 3 easy steps:
1. Arrange the data from smallest to largest
2. Identify the position of the median.
If your data set is small, you can identify the middle position just by looking at the data. If your data set is large, you can identify the middle position by dividing the total number of data points by two and rounding up to the nearest whole number.
\tfrac{n}{2}
\tfrac{9}{2}
Rounding up we get 5.
The median will be the 5th value in the data set.
2. Find the median based on the position identified in the previous step.
From the previous step, we know that the median is the 5th value in the data set. The 5th value, in this case, is 120. We can confirm that this is the median by checking to see if there is an equal number of observations above and below it.
Example 2. Finding the median with an even-numbered data set
Say you have the following even-numbered data set where n=8:
The first step is the same, but the second and third steps are slightly different.
1. Arrange the data in numerical order from smallest to largest
2. Identify the position of the two middle values.
When n is even, there will be two values in the middle of your data. You can identify the position of these two values using the following formulas:
The position of the first value is equal to
\tfrac{n}{2}
The position of the second value is equal to
\tfrac{n}{2}
+ 1.
In this case, the middle values are the 4th and 5th values:
\tfrac{8}{2}
\tfrac{8}{2}
3. Find the median by averaging the two values positioned in the middle of your data.
The fourth and fifth values of this data are 8 and 10. We find the median by averaging them.
While it’s helpful to know how to calculate the median by hand, it is often easier to find the median using statistical software.
In Excel or Google sheets, use the formula =MEDIAN(). The list of your data should be included inside the parentheses. For example, if your data has ten values in cells A1 through A10, the formula would be =MEDIAN(A1:A10).
In Desmos, use the function median() to find the median. The values in your data set should be included inside the parentheses. To find the median from Example 1 above, you would type: median(115, 138, 133, 120, 117, 125, 130, 100, 112).
In R, you can also use the command median(). You should include a list of your data or the name of your random variable inside the parentheses.
For practice, try calculating the median from the examples above using one or all of these options.
The median is a measure of center that is not affected by outliers or the skewness of data.
If you have a roughly symmetric data set, the mean and the median will be similar values, and both will be good indicators of the center of the data. When the distribution of data is not symmetric—when data is skewed heavily to the right or to the left—the median is the preferred measure of centrality. This is because the median is resistant to extreme values and long tails in the data, while the mean is not.
To demonstrate this, imagine ten people grabbing a drink at a local tavern. All ten have similar annual incomes represented in the data below.
$55K, $58K, $62K, $65K, $67K, $70K, $73K, $74K, $77K, $83K
The mean income of this group is $68,400. The median income is $68,500. Notice that the mean and the median are not so different and that both are good indicators of where the data is centered.
Out of the blue, in walks Oprah Winfrey. Let’s say Oprah makes $300 million a year. What happens to the mean and the median? The median income is now $70,000. Not too different from what it was before. The mean income has shot up to $27,335,000. Oprah’s income is an extreme outlier, and it skews the data far to the right.
If someone told you that the mean income of patrons in a bar was just above $27 million, you might imagine a bar filled with millionaires and billionaires. The mean no longer gives us a good sense of where the data is centered, but the median still does.
Ordinal data is data that is not quite qualitative and not quite quantitative, as it consists of categories with a natural order or prescribed rank. A good example is survey data that asks respondents to rank their satisfaction on a scale of 1 to 5. Even though ordinal data has a clear ranking (5 is better than 4 is better than 3, etc.), the scale or distance between each ranking can be uneven or unknown. 5 may be a lot better than 4, but 4 may only be a bit better than 3. Alternatively, different respondents might have subjective interpretations of what constitutes each ranking.
Medians cannot be calculated for qualitative data, but you can calculate a median for an odd-numbered ordinal data set. Let’s say you have a survey asking respondents to rank a movie on a scale from 1 to 10. In this case, you could calculate the median following the exact same method we showed above.
If you have an even-numbered ordinal data set, technically speaking, you cannot calculate a median. This is because you cannot average two ranks that have an uneven or unknown scale.
Consider the following health survey question. Each response to this question is coded with a ranked number:
2 to 4 times a month (2)
4 or more times a week (4)
Imagine you survey eight people with this question, and the resulting ordinal data is: {0, 1, 2, 3, 4, 4, 4, 4}. In this case, the two middle values of the data are 3 and 4, but there is no way to take the average of “2 to 3 times a week” and “4 or more times a week.”
Despite the technical ambiguities, some people believe that calculating a median for ordinal data is meaningful and necessary, even for even-numbered data. In spite of the equivocal results, you can choose to calculate the median of ordinal data, even when the data set is even-numbered. To do this, you can assign the median to the lower of your two middle values, or you can go ahead and take the average of the two middle rankings. If you want to calculate a median for an ordinal data set, think carefully about what it means to do so and always keep in mind that some people might challenge your calculations.
The ability to clearly visualize data is critical to getting the most out of your statistical analysis. In this article, you’ll learn step-by-step how to construct a box plot, one of the simplest and most efficient tools for conveying information regarding a data set.
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Find sources: "Revolutions per minute" – news · newspapers · books · scholar · JSTOR (July 2016) (Learn how and when to remove this template message)
International System of Units[edit]
{\displaystyle {\begin{aligned}1~&{\text{rad/s}}&&=&{\frac {1}{2\pi }}~&{\text{Hz}}&&=&{\frac {60}{2\pi }}~&{\text{rpm}}\\[9pt]2\pi ~&{\text{rad/s}}&&=&1~&{\text{Hz}}&&=&60~&{\text{rpm}}\\[9pt]{\frac {2\pi }{60}}~&{\text{rad/s}}&&=&{\frac {1}{60}}~&{\text{Hz}}&&=&1~&{\text{rpm}}\end{aligned}}}
{\displaystyle \omega =2\pi f\,,\qquad f={\frac {\omega }{2\pi }}\,.}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Revolutions_per_minute&oldid=1040583813"
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The slug is a derived unit of mass in a weight-based system of measures, most notably within the British Imperial measurement system and the United States customary measures system. Systems of measure either define mass and derive a force unit or define a base force and derive a mass unit[1] (cf. poundal, a derived unit of force in a force-based system). A slug is defined as the mass that is accelerated by 1 ft/s2 when a net force of one pound (lbf) is exerted on it.[2]
1 slug in ... ... is equal to ...
BGS base units
1 ft−1⋅lbf⋅s2
{\displaystyle 1~{\text{slug}}=1~{\text{lbf}}{\cdot }{\frac {{\text{s}}^{2}}{\text{ft}}}\quad \Longleftrightarrow \quad 1~{\text{lbf}}=1~{\text{slug}}{\cdot }{\frac {\text{ft}}{{\text{s}}^{2}}}}
One slug is a mass equal to 32.1740 lb (14.59390 kg) based on standard gravity, the international foot, and the avoirdupois pound.[3] At the Earth's surface, an object with a mass of 1 slug exerts a force downward of approximately 32.2 lbf or 143 N.[4][5]
The slug is part of a subset of units known as the gravitational FPS system, one of several such specialized systems of mechanical units developed in the late 19th and the 20th century. Geepound was another name for this unit in early literature.[6]
The name "slug" was coined before 1900 by British physicist Arthur Mason Worthington,[7] but it did not see any significant use until decades later.[8] It is derived from the meaning "solid block of metal", not from the slug mollusc.[9] A 1928 textbook says:
No name has yet been given to the unit of mass and, in fact, as we have developed the theory of dynamics no name is necessary. Whenever the mass, m, appears in our formulae, we substitute the ratio of the convenient force-acceleration pair (w/g), and measure the mass in lbs. per ft./sec.2 or in grams per cm./sec.2.
— Noel Charlton Little, College Physics, Charles Scribner's Sons, 1928, p. 165.
The slug is listed in the Regulations under the Weights and Measures (National Standards) Act, 1960. This regulation defines the units of weights and measures, both regular and metric, in Australia.
The blob is the inch version of the slug (1 blob is equal to 1 lbf⋅s2/in, or 12 slugs)[3][12] or equivalent to 386.0886 pounds (175.1268 kg). This unit is also called slinch (a portmanteau of the words slug and inch).[13][14] Similar terms include slugette[15] and snail.[16]
Similar metric units include the glug in the centimetre–gram–second system, and the mug, par, or MTE in the metre–kilogram–second system.[17]
British Engineering Units
^ See Elementary High School physics and chemistry text books/fundamentals.
^ Collins, Danielle (May 2019). "How to convert between mass and force — in metric and English units". Linear Motion Tips. Retrieved 18 January 2021.
^ a b Shigley, Joseph E. and Mischke, Charles R. Mechanical Engineering Design, Sixth ed, pp. 31–33. McGraw Hill, 2001. ISBN 0-07-365939-8.
^ Beckwith, Thomas G., Roy D. Marangoni, et al. Mechanical Measurements, Fifth ed, pp. 34-36. Addison-Wesley Publishing, 1993. ISBN 0-201-56947-7.
^ Shevell, R.S. Fundamentals of Flight, Second ed, p. xix. Prentice-Hall, 1989.
^ gee Archived 2018-01-27 at the Wayback Machine. unit2unit.eu
^ Worthington, Arthur Mason (1900). Dynamics of Rotation: An Elementary Introduction to Rigid Dynamics (3rd ed.). Longmans, Green, and Co. p. 9.
^ Gyllenbok, Jan (April 11, 2018). Encyclopaedia of Historical Metrology, Weights, and Measures: Volume 1. Birkhäuser. ISBN 9783319575988 – via Google Books.
^ Society, Digital Equipment Computer Users (September 4, 1965). "Papers and Presentations" – via Google Books.
^ Norton, Robert L. Cam Design and Manufacturing Handbook, p. 13. Industrial Press Inc., 2009. ISBN 0831133678.
^ Slug Archived 2016-11-30 at the Wayback Machine. DiracDelta Science & Engineering Encyclopedia
^ "1 blob". Wolfram Alpha Computational Knowledge Engine. Retrieved 27 October 2011.
^ Celmer, Robert. Notes to Accompany Vibrations II. Version 2.2. 2009.
^ Rowlett, Russ. "How Many? A Dictionary of Units of Measurement". unc.edu, September 1, 2004. Retrieved January 26, 2018.
^ Cardarelli, François (1999). Scientific Units, Weights and Measures. Springer. pp. 358, 377. ISBN 1-85233-682-X.
"What is a Slug?" on phy-astr.gsu.edu
Retrieved from "https://en.wikipedia.org/w/index.php?title=Slug_(unit)&oldid=1074764809"
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Capital efficiency - Jarvis Network - Synthereum
As explained previously, rather than using an AMM, buying and selling jFIATs are performed using a mint and burn mechanism. This design allows for greater capital efficiency for liquidity providers.
In the current version of Synthereum, there is only one liquidity provider; a second version of the protocol will allow anyone to become a liquidity provider.
Since all the actions (minting and selling, buying and burning) happen within the same transaction, they can be compensated. Following the previous example:
For buying, instead of the pool having to spend 110 USD to mint jEUR and then receiving 100 USD when selling them, the pool utilizes the 100 USD of the buyer and adds 10 USD to form the collateral. Therefore, 1 USD in the liquidity pool allows for buying $10 worth of jEUR. The minting CR can be changed, allowing for a bigger or smaller buying capacity, to adapt to a specific situation.
BuyingCapacity = \frac{Liquidity}{CR-1}
For selling, instead of spending 100 USD to buy back the jEUR, and then retrieve 110 USD, the pool utilizes the jEUR to redeem the 110 USD, and then uses them to pay 100 USD to the seller, and keeps the rest. Therefore, there is no liquidity needed for selling jEUR; the maximum amount of jFIATs that can be sold is the maximum amount of jFIATs that have been minted with the pool.
Single asset pool
Thanks to this burn and mint mechanism, the pool only needs to hold collateral (only USD at the moment) to facilitate the buying and selling of jFIATs.
The entire liquidity is available at the reference price, and therefore cannot guarantee the execution of any size of trades, unlike bonding curve-based systems.
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Home : Support : Online Help : Programming : Document Tools : Layout : Image
generate XML for a Image element
Image( img, opts )
content for the Image element, as either a file name or a numeric Array image as used with the ImageTools package
captionalignment : identical(left,center,right):=left Indicates how the caption is horizontally aligned.
captionposition : identical(below,above):=below Indicates whether the caption is positioned below or above the image.
drawcaption : truefalse:=false Indicates whether the caption will be shown. The form of the caption is automatically generated and numbered by the interface upon display.
height : posint:=200 ; The height in pixels to be used. If img specifies a filename then by default the height used is 200. If img is an Array then by default the first dimension's size is used as the height.
width : posint:=200 ; The width in pixels to be used. If img specifies a filename then by default the width used is 200. If img is an Array then by default the second dimension's size is used as the width.
The Image command in the Layout Constructors package returns an XML function call which represents a Image element of a worksheet.
\mathrm{with}\left(\mathrm{DocumentTools}\right):
\mathrm{with}\left(\mathrm{DocumentTools}:-\mathrm{Layout}\right):
This example uses an Array with a single 2-dimensional layer for a grayscale image.
A≔\mathrm{Array}\left(1..20,1..20,\left(i,j\right)↦\mathrm{evalhf}\left(\mathrm{argument}\left(\mathrm{sin}\left(\frac{i}{2}-5+\frac{I\cdot \left(j-10\right)}{2}\right)\right)\right),\mathrm{datatype}=\mathrm{float}[8]\right):
\mathrm{img}≔\mathrm{ImageTools}:-\mathrm{FitIntensity}\left(A\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627811500728556}}
Executing the Image command produces a function call, in which the supplied image has been encoded.
g≔\mathrm{Image}\left(A,\mathrm{height}=150,\mathrm{width}=150\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Image}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"captionalignment"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"1"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"captionposition"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"1"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"drawcaption"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"false"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"height"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"150"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"width"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"150"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"zoomable"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"false"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"TUZOV3RLVWI8b2I8Uj1NRExDZE5CP0o+SjpcXFRrRW5iazphTlxcQE5kXFxrZUhQVEhQajxQdnl5eXl4YVl5YWFLaj1ycmhrW3lRPXl5eXhRWHVtV3hrSlxcVnJ5Sz90a0tgclJ0eV08bWptTE1cXG0/YEpqUEtcXFxceFpcXHM7cW9MVG9aTFJ1TGx4bVY+QG5DYE51XFxsP2BuTFlTRT1rV0htS0hqO0hMUz1sPWhrQ0hMPGlWRmlWWj1Wcnl5aXl0YVh5YUhMTGFraUVOO2B4ZEFUa0xsWlxcdUhdTj1ATGhhbkp5UkJNdTt2cj5GX2ZpXFxLcVtRXFw6RmM/b2M+b288P2Y8Mzw="}\right)
\mathrm{xml}≔\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(g\right)\right)\right)\right):
\mathrm{InsertContent}\left(\mathrm{xml}\right):
A color image can also be supplied. The next example uses an Array with three 2-dimensional layers for a color image. The first layer initially represents the Hue layer in HSV colorspace, and the Array is recomputed in the RGB colorspace before being passed to the Image element constructor.
By default the dimensions of the Array are used to determine the height and width of the Image.
A≔\mathrm{Array}\left(1..200,1..100,\left(i,j\right)↦\mathrm{evalhf}\left(\mathrm{argument}\left(\mathrm{sin}\left(\frac{i}{20}-5+\frac{I\cdot \left(j-50\right)}{10}\right)\right)\right),\mathrm{datatype}=\mathrm{float}[8]\right):
A≔\mathrm{ImageTools}:-\mathrm{FitIntensity}\left(A\right)\cdot 360:
\mathrm{img}≔\mathrm{Array}\left(1..200,1..100,1..3,1.0,\mathrm{datatype}=\mathrm{float}[8]\right):
\mathrm{img}[..,..,1]≔A:
\mathrm{img}≔\mathrm{ImageTools}:-\mathrm{HSVtoRGB}\left(\mathrm{img}\right):
g≔\mathrm{Image}\left(\mathrm{img}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(g\right)\right)\right)\right)\right):
The image may also be specified with a file name, as a string.
\mathrm{img}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/images/antennas.jpg"\right):
g≔\mathrm{Image}\left(\mathrm{img}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(g\right)\right)\right)\right)\right):
The height and width options can be used to control the displayed dimensions of the Image.
g≔\mathrm{Image}\left(\mathrm{img},\mathrm{height}=150,\mathrm{width}=300\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(g\right)\right)\right)\right)\right):
\mathrm{img2}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/images/undercarriage.jpg"\right):
\mathrm{A2}≔\mathrm{ImageTools}:-\mathrm{Read}\left(\mathrm{img2}\right):
h,w≔\mathrm{ceil}\left(\frac{\mathrm{ImageTools}:-\mathrm{Height}\left(\mathrm{A2}\right)}{2}\right),\mathrm{ceil}\left(\frac{\mathrm{ImageTools}:-\mathrm{Width}\left(\mathrm{A2}\right)}{2}\right)
\textcolor[rgb]{0,0,1}{h}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{w}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{200}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{300}
g≔\mathrm{Image}\left(\mathrm{img2},\mathrm{height}=h,\mathrm{width}=w\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(g\right)\right)\right)\right)\right):
An Image element can also be used as the content of a Table cell.
g≔\mathrm{Image}\left(\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/images/antennas.jpg"\right)\right):
T≔\mathrm{Table}\left(\mathrm{Row}\left(g,g\right),\mathrm{exterior}=\mathrm{none},\mathrm{interior}=\mathrm{none},\mathrm{alignment}=\mathrm{center}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
Note that Table cells' contents are automatically rescaled, on display, to fit the cell width. In this next example the Table is specified to have a width of 25 percent of the visible worksheet.
\mathrm{g1}≔\mathrm{Image}\left(\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/images/antennas.jpg"\right)\right):
\mathrm{g2}≔\mathrm{Image}\left(\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/images/lichtenstein.jpg"\right)\right):
T≔\mathrm{Table}\left(\mathrm{Row}\left(\mathrm{g1},\mathrm{g2}\right),\mathrm{Row}\left("Some text.",\mathrm{InlinePlot}\left(\mathrm{plot}\left(\mathrm{sin}\right)\right),\mathrm{alignment}=\mathrm{center}\right),\mathrm{Row}\left(\mathrm{g2},\mathrm{g1}\right),\mathrm{width}=25,\mathrm{alignment}=\mathrm{center}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
A numbered caption can be used.
g≔\mathrm{Image}\left(\mathrm{img},\mathrm{drawcaption}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(g\right)\right)\right)\right)\right):
The caption position is optional.
g≔\mathrm{Image}\left(\mathrm{img},\mathrm{drawcaption},\mathrm{captionposition}=\mathrm{above},\mathrm{captionalignment}=\mathrm{left}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(g\right)\right)\right)\right)\right):
The DocumentTools:-Layout:-Image command was introduced in Maple 2015.
|
SMS:Create Datasets - XMS Wiki
SMS:Create Datasets
The Create Datasets dialog is obsolete as of SMS 10.1. It has been replaced by the Dataset Toolbox.
Create Datasets dialog
Automatically generate commonly used datasets
The Create Datasets dialog is used to create functions for the entire mesh or active scatter set. The option is in Data | Create Datasets in both the Scatter and Mesh modules.
Each function that is toggled on will be created. All of the available functions can be turned on by pushing All On. All of the functions can be turned off by pushing All Off. The Gravity can be set and is used in several of the function calculations. The functions that can be created include:
Grid Spacing – Creates a function that gives the average distance between a node and its neighbors.
Gradient – Creates a function that gives the gradient at each node. The gradient is calculated at each point by averaging the normals of the faces connected to that point. For vertices in a TIN, this includes all the triangles connected to the vertex. For nodes in a mesh, it includes all the elements connected to the node. Currently no adjustment is made to account for the varied area of these faces. The gradient is defined as the run divided by the rise.
Gradient Angle – Creates a function that gives the direction in degrees of the maximum gradient at each point. (See above for the method used to compute the gradient.)
Directional Derivative – Creates a vector function that gives the gradient (run/rise) in the x and y directions. (See above for the method used to compute the gradient.)
Shallow Wavelength/Celerity – Creates two functions that calculate the celerity and wavelength at each node in shallow water. The celerity is calculated as:
{\displaystyle Celerity=(Gravity*NodalElevation)^{0.5}}
. The Wavelength is:
{\displaystyle Wavelength=Period*Celerity}
Local Wavelength/Celerity – Creates two functions that calculate the celerity and wavelength at each node for any depths.
Gravity Wave Courant Number – Creates a function that gives the courant number for each node given the Time Step. The equation is:
{\displaystyle CourantNumber=TimeStep*(Gravity*NodalElevation)^{0.5}/NodalSpacing}
Gravity Wave Time Steps – Creates a function that calculates the gravity wave time step given the Courant Number. The equation is the same as for the Gravity Wave Courant Number, solved for the Courant Number.
The final three functions can be created when the current numeric model is set to ADCIRC.
Advective Courant Number – Creates a function that calculates the courant number given the Time Step and a velocity function. The velocity function can be selected by clicking on the Advective Functions&ldots; button. This brings up a Select Dataset dialog that lists the vector functions currently in memory. The courant cumber is calculated as:
{\displaystyle CourantNumber=NodalVelocityMagnitude*TimeStep/NodalSpacing}
This option is disabled if no vector functions exist.
Advective Time Steps – Creates a function that calculates the time step given the Courant Number and a velocity function. The velocity function can be selected as described above in the description of the Advective Courant Number. The equation is the same as for the Advective Courant Number, solved for the time step. This option is disabled if no vector functions exist.
Harmonic – Creates a scalar harmonic function and/or a vector harmonic function. Pushing the Options button brings up the Harmonic Options dialog. The name of the function(s) to be created can be set in the Name fields. The frequencies to be used in creating the function can be chosen be double-clicking on a frequency name shown in the Scalar Frequencies window or be clicking on the name and by pushing Select. A frequency can be unselected by double-clicking on the name again or by selecting it and pushing Unselect. The time values that will be used in calculating the time steps can be set in the fields at the bottom of the dialog.
Mesh Module Data Menu
Scatter Module Data Menu
Particle Module Data Menu
Retrieved from "https://www.xmswiki.com/index.php?title=SMS:Create_Datasets&oldid=140267"
SMS Dataset Dialogs
|
Lifting a Filter Bank - MATLAB & Simulink - MathWorks India
This example shows how to use lifting to progressively change the properties of a perfect reconstruction filter bank. The following figure shows the three canonical steps in lifting: split, predict, and update.
The first step in lifting is simply to split the signal into its even- and odd-indexed samples. These are called polyphase components and that step in the lifting process is often referred to as the "lazy" lifting step because you really are not doing that much work. You can do this in MATLAB™ by creating a "lazy" lifting scheme using liftingScheme with default settings.
Use the lifting scheme to obtain the level 1 wavelet decomposition of a random signal.
[ALazy,DLazy] = lwt(x,'LiftingScheme',LS,'Level',1);
MATLAB indexes from 1 so ALazy contains the odd-indexed samples of x and DLazy contains the even-indexed samples. Most explanations of lifting assume that the signal starts with sample 0, so ALazy would be the even-indexed samples and DLazy the odd-indexed samples. This example follows that latter convention. The "lazy" wavelet transform treats one half of the signal as wavelet coefficients, DLazy, and the other half as scaling coefficients, ALazy. This is perfectly consistent within the context of lifting, but a simple split of the data does really sparsify or capture any relevant detail.
The next step in the lifting scheme is to predict the odd samples based on the even samples. The theoretical basis for this is that most natural signals and images exhibit correlation among neighboring samples. Accordingly, you can "predict" the odd-indexed samples using the even-indexed samples. The difference between your prediction and the actual value is the "detail" in the data missed by the predictor. That missing detail comprises the wavelet coefficients.
In equation form, you can write the prediction step as
{d}_{j}\left(n\right)={d}_{j-1}\left(n\right)-P\left({a}_{j-1}\right)
{d}_{j-1}\left(n\right)
are the wavelet coefficients at the finer scale and
{a}_{j-1}
is some number of finer-scale scaling coefficients.
P\left(\cdot \right)
is the prediction operator.
Add a simple (Haar) prediction step that subtracts the even (approximation) coefficient from the odd (detail) coefficient. In this case the prediction operator is simply
\left(-1\right){a}_{j-1}\left(n\right)
. In other words, it predicts the odd samples based on the immediately preceding even sample.
The above code says "create an elementary prediction lifting step using a polynomial in
z
with the highest power
{z}^{0}
. The coefficient is -1. Update the lazy lifting scheme.
LSN = addlift(LS,ElemLiftStep);
Apply the new lifting scheme to the signal.
[A,D] = lwt(x,'LiftingScheme',LSN,'Level',1);
Note that the elements of A are identical to those in ALazy. This is expected because you did not modify the approximation coefficients.
[A ALazy]
If you look at the elements of D{1}, you see that they are equal to DLazy{1}-ALazy.
Dnew = DLazy{1}-ALazy;
[Dnew D{1}]
Compare Dnew to D. Imagine an example where the signal was piecewise constant over every two samples.
v = [1 -1 1 -1 1 -1];
u = repelem(v,2)
u = 1×12
1 1 -1 -1 1 1 -1 -1 1 1 -1 -1
Apply the new lifting scheme to u.
[Au,Du] = lwt(u,'LiftingScheme',LSN,'Level',1);
Du{1}
You see that all the Du are zero. This signal has been compressed because all the information is now contained in 6 samples instead of 12 samples. You can easily reconstruct the original signal
urecon = ilwt(Au,Du,'LiftingScheme',LSN);
max(abs(u(:)-urecon(:)))
In your prediction step, you predicted that the adjacent odd sample in your signal had the same value as the immediately preceding even sample. Obviously, this is true only for trivial signals. The wavelet coefficients capture the difference between the prediction and the actual values (at the odd samples). Finally, use the update step to update the even samples based on differences obtained in the prediction step. In this case, update using the following
{a}_{j}\left(n\right)={a}_{j-1}\left(n\right)+{d}_{j-1}\left(n\right)/2
. This replaces each even-indexed coefficient by the arithmetic average of the even and odd coefficients.
elsUpdate = liftingStep('Type','update','Coefficients',1/2,'MaxOrder',0);
LSupdated = addlift(LSN,elsUpdate);
Obtain the wavelet transform of the signal with the updated lifting scheme.
[A,D] = lwt(x,'LiftingScheme',LSupdated,'Level',1);
If you compare A to the original signal, x, you see that the signal mean is captured in the approximation coefficients.
[mean(A) mean(x)]
In fact, the elements of A are easily obtainable from x by the following.
for ii = 1:2:numel(x)
meanz(n) = mean([x(ii) x(ii+1)]);
Compare meanz and A. As always, you can invert the lifting scheme to obtain a perfect reconstruction of the data.
xrec = ilwt(A,D,'LiftingScheme',LSupdated);
It is common to add a normalization step at the end so that the energy in the signal (
{\ell }^{2}
norm) is preserved as the sum of the energies in the scaling and wavelet coefficients. Without this normalization step, the energy is not preserved.
norm(x,2)^2
norm(A,2)^2+norm(D{1},2)^2
Add the necessary normalization step.
LSsteps = LSupdated.LiftingSteps;
LSscaled = liftingScheme('LiftingSteps',LSsteps,'NormalizationFactors',[sqrt(2)]);
[A,D] = lwt(x,'LiftingScheme',LSscaled,'Level',1);
{\ell }^{2}
norm of the signal is equal to the sum of the energies in the scaling and wavelet coefficients. The lifting scheme you developed in this example is the Haar lifting scheme.
Wavelet Toolbox™ supports many commonly used lifting schemes through liftingScheme with predefined predict and update steps, and normalization factors. For example, you can obtain the Haar lifting scheme with the following.
lshaar = liftingScheme('Wavelet','haar');
To see that not all lifting schemes consist of single predict and update lifting steps, examine the lifting scheme that corresponds to the bior3.1 wavelet.
lsbior3_1 = liftingScheme('Wavelet','bior3.1')
lsbior3_1 =
Coefficients: -0.3333
MaxOrder: -1
|
bootstrap_point632_score: The .632 and .632+ boostrap for classifier evaluation - mlxtend
Example 1 -- Evaluating the predictive performance of a model via the classic out-of-bag Bootstrap
Example 2 -- Evaluating the predictive performance of a model via the .632 Bootstrap
Example 3 -- Evaluating the predictive performance of a model via the .632+ Bootstrap
An implementation of the .632 bootstrap to evaluate supervised learning algorithms.
from mlxtend.evaluate import bootstrap_point632_score
Originally, the bootstrap method aims to determine the statistical properties of an estimator when the underlying distribution was unknown and additional samples are not available. Now, in order to exploit this method for the evaluation of predictive models, such as hypotheses for classification and regression, we may prefer a slightly different approach to bootstrapping using the so-called Out-Of-Bag (OOB) or Leave-One-Out Bootstrap (LOOB) technique. Here, we use out-of-bag samples as test sets for evaluation instead of evaluating the model on the training data. Out-of-bag samples are the unique sets of instances that are not used for model fitting as shown in the figure below [1].
The figure above illustrates how three random bootstrap samples drawn from an exemplary ten-sample dataset (
X_1,X_2, ..., X_{10}
) and their out-of-bag sample for testing may look like. In practice, Bradley Efron and Robert Tibshirani recommend drawing 50 to 200 bootstrap samples as being sufficient for reliable estimates [2].
In 1983, Bradley Efron described the .632 Estimate, a further improvement to address the pessimistic bias of the bootstrap cross-validation approach described above [3]. The pessimistic bias in the "classic" bootstrap method can be attributed to the fact that the bootstrap samples only contain approximately 63.2% of the unique samples from the original dataset. For instance, we can compute the probability that a given sample from a dataset of size n is not drawn as a bootstrap sample as
which is asymptotically equivalent to
\frac{1}{e} \approx 0.368
n \rightarrow \infty.
Vice versa, we can then compute the probability that a sample is chosen as
P (\text{chosen}) = 1 - \bigg(1 - \frac{1}{n}\bigg)^n \approx 0.632
for reasonably large datasets, so that we'd select approximately
0.632 \times n
uniques samples as bootstrap training sets and reserve
0.368 \times n
out-of-bag samples for testing in each iteration.
Now, to address the bias that is due to this the sampling with replacement, Bradley Efron proposed the .632 Estimate that we mentioned earlier, which is computed via the following equation:
\text{ACC}_{train}
is the accuracy computed on the whole training set, and
\text{ACC}_{h, i}
is the accuracy on the out-of-bag sample.
.632+ Bootstrap
Now, while the .632 Boostrap attempts to address the pessimistic bias of the estimate, an optimistic bias may occur with models that tend to overfit so that Bradley Efron and Robert Tibshirani proposed the The .632+ Bootstrap Method (Efron and Tibshirani, 1997). Instead of using a fixed "weight"
\omega = 0.632
we compute the weight
\gamma
where R is the relative overfitting rate
(Since we are plugging
\omega
into the equation for computing that we defined above, and
\text{ACC}_{train}
still refer to the out-of-bag accuracy in the ith bootstrap round and the whole training set accuracy, respectively.)
Further, we need to determine the no-information rate
\gamma
in order to compute R. For instance, we can compute
\gamma
by fitting a model to a dataset that contains all possible combinations between samples
x_{i'}
and target class labels
y_{i}
— we pretend that the observations and class labels are independent:
Alternatively, we can estimate the no-information rate
\gamma
p_k
is the proportion of class
k
samples observed in the dataset, and
q_k
k
samples that the classifier predicts in the dataset.
[1] https://sebastianraschka.com/blog/2016/model-evaluation-selection-part2.html
[2] Efron, Bradley, and Robert J. Tibshirani. An introduction to the bootstrap. CRC press, 1994. Management of Data (ACM SIGMOD '97), pages 265-276, 1997. [3] Efron, Bradley. 1983. “Estimating the Error Rate of a Prediction Rule: Improvement on Cross-Validation.” Journal of the American Statistical Association 78 (382): 316. doi:10.2307/2288636.
[4] Efron, Bradley, and Robert Tibshirani. 1997. “Improvements on Cross-Validation: The .632+ Bootstrap Method.” Journal of the American Statistical Association 92 (438): 548. doi:10.2307/2965703.
The bootstrap_point632_score function mimics the behavior of scikit-learn's `cross_val_score, and a typically usage example is shown below:
scores = bootstrap_point632_score(tree, X, y, method='oob')
acc = np.mean(scores)
print('Accuracy: %.2f%%' % (100*acc))
lower = np.percentile(scores, 2.5)
upper = np.percentile(scores, 97.5)
print('95%% Confidence interval: [%.2f, %.2f]' % (100*lower, 100*upper))
95% Confidence interval: [87.71, 100.00]
scores = bootstrap_point632_score(tree, X, y)
scores = bootstrap_point632_score(tree, X, y, method='.632+')
95% Confidence interval: [91.86, 98.92]
bootstrap_point632_score(estimator, X, y, n_splits=200, method='.632', scoring_func=None, predict_proba=False, random_seed=None, clone_estimator=True)
Implementation of the .632 [1] and .632+ [2] bootstrap for supervised learning
An estimator for classification or regression that follows the scikit-learn API and implements "fit" and "predict" methods.
method : str (default='.632')
The bootstrap method, which can be either - 1) '.632' bootstrap (default) - 2) '.632+' bootstrap - 3) 'oob' (regular out-of-bag, no weighting) for comparison studies.
scoring_func : callable,
Score function (or loss function) with signature scoring_func(y, y_pred, **kwargs). If none, uses classification accuracy if the
estimator is a classifier and mean squared error if the estimator is a regressor.
predict_proba : bool
Whether to use the predict_proba function for the estimator argument. This is to be used in conjunction with scoring_func which takes in probability values instead of actual predictions. For example, if the scoring_func is :meth:sklearn.metrics.roc_auc_score, then use predict_proba=True. Note that this requires estimator to have predict_proba method implemented.
clone_estimator : bool (default=True)
Clones the estimator if true, otherwise fits the original.
scores : array of float, shape=(len(list(n_splits)),)
Array of scores of the estimator for each bootstrap replicate.
|
paired_ttest_resample: Resampled paired *t* test - mlxtend
paired_ttest_resample: Resampled paired t test
Example 1 - Resampled paired t test
Resampled paired t test procedure to compare the performance of two models
from mlxtend.evaluate import paired_ttest_resample
Resampled paired t test procedure (also called k-hold-out paired t test) is a popular method for comparing the performance of two models (classifiers or regressors); however, this method has many drawbacks and is not recommended to be used in practice [1], and techniques such as the paired_ttest_5x2cv should be used instead.
To explain how this method works, let's consider to estimator (e.g., classifiers) A and B. Further, we have a labeled dataset D. In the common hold-out method, we typically split the dataset into 2 parts: a training and a test set. In the resampled paired t test procedure, we repeat this splitting procedure (with typically 2/3 training data and 1/3 test data) k times (usually 30). In each iteration, we train A and B on the training set and evaluate it on the test set. Then, we compute the difference in performance between A and B in each iteration so that we obtain k difference measures. Now, by making the assumption that these k differences were independently drawn and follow an approximately normal distribution, we can compute the following t statistic with k-1 degrees of freedom according to Student's t test, under the null hypothesis that the models A and B have equal performance:
p^{(i)}
i
p^{(i)} = p^{(i)}_A - p^{(i)}_B
\overline{p}
\overline{p} = \frac{1}{k} \sum^k_{i=1} p^{(i)}
\alpha=0.05
\alpha
split dataset into training and test subsets
fit models A and B to the training set
compute the performances of A and B on the test set
record the performance difference between A and B
compute p value from t-statistic with k-1 degrees of freedom
compare p value to chosen significance threshold
The problem with this method, and the reason why it is not recommended to be used in practice, is that it violates the assumptions of Student's t test [1]:
p^{(i)} = p^{(i)}_A - p^{(i)}_B
p^{(i)}_A
p^{(i)}_B
p^{(i)}
's themselves are not independent because of the overlapping test sets; also, test and training sets overlap as well
\alpha=0.05
for rejecting the null hypothesis that both algorithms perform equally well on the dataset and conduct the paired sample t test:
from mlxtend.evaluate import paired_ttest_resampled
t, p = paired_ttest_resampled(estimator1=clf1,
p > t
\alpha=0.05
p < 0.001
\alpha
|
Out of electric field vector E and magnetic field vector B in an e m wave , which is more - Physics - Electromagnetic Waves - 9970145 | Meritnation.com
Out of electric field vector E and magnetic field vector B in an e.m wave , which is more effective and why?
Force on the charge particle due to magnetic field will be less than the force due to electric field due to presence of the
{c}^{2}
term in the denominator. But otherwise both are equally important.
|
A Problem! - Global Math Week
Okay. Now that we are feeling really good about doing advanced algebra, I have a confession to make.
I’ve been fooling you!
Do you see that I have been avoiding negative numbers all this time?
Can we solve a problem like this one:
\dfrac {x^{3}-3x+2} {x+2}
Do you see any groups of
x+2
in the picture?
We are looking for one dot next to two dots in the picture of
x^{3}-3x+2
. I don’t see any.
So what can we do, besides weep a little?
Do you have any ideas? Is there anything on the app that will help?
It is tempting to say that we should just unexplode some dots. That’s a brilliant idea! Except ... we don’t know the value for x and so don’t know how many dots to draw when we unexplode. Bother!
We need some amazing flash of insight for something clever to do. Or maybe polynomial problems with negative numbers just can’t be solved with this dots and boxes method.
|
Kmeans: k-means clustering - mlxtend
Example 1 - Three Blobs
A implementation of k-means clustering.
from mlxtend.cluster import Kmeans
Clustering falls into the category of unsupervised learning, a subfield of machine learning where the ground truth labels are not available to us in real-world applications. In clustering, our goal is to group samples by similarity (in k-means: Euclidean distance).
The k-means algorithms can be summarized as follows:
Randomly pick k centroids from the sample points as initial cluster centers.
Assign each sample to the nearest centroid
\mu(j), \; j \in {1,...,k}
Move the centroids to the center of the samples that were assigned to it.
Repeat steps 2 and 3 until the cluster assignments do not change or a user-defined tolerance or a maximum number of iterations is reached.
MacQueen, J. B. (1967). Some Methods for classification and Analysis of Multivariate Observations. Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press. pp. 281–297. MR 0214227. Zbl 0214.46201. Retrieved 2009-04-07.
Load some sample data:
plt.scatter(X[:, 0], X[:, 1], c='white')
Compute the cluster centroids:
km = Kmeans(k=3,
print('Iterations until convergence:', km.iterations_)
print('Final centroids:\n', km.centroids_)
Iteration: 2/50 | Elapsed: 00:00:00 | ETA: 00:00:00
Iterations until convergence: 2
Visualize the cluster memberships:
y_clust = km.predict(X)
plt.scatter(X[y_clust == 0, 0],
X[y_clust == 0, 1],
plt.scatter(X[y_clust == 1,0],
X[y_clust == 1,1],
plt.scatter(km.centroids_[:,0],
km.centroids_[:,1],
plt.legend(loc='lower left',
Kmeans(k, max_iter=10, convergence_tolerance=1e-05, random_seed=None, print_progress=0)
K-means clustering class.
Added in 0.4.1dev
max_iter : int (default: 10)
Number of iterations during cluster assignment. Cluster re-assignment stops automatically when the algorithm converged.
convergence_tolerance : float (default: 1e-05)
Compares current centroids with centroids of the previous iteration using the given tolerance (a small positive float)to determine if the algorithm converged early.
Set random state for the initial centroid assignment.
Prints progress in fitting to stderr. 0: No output 1: Iterations elapsed 2: 1 plus time elapsed 3: 2 plus estimated time until completion
centroids_ : 2d-array, shape={k, n_features}
Feature values of the k cluster centroids.
custers_ : dictionary
The cluster assignments stored as a Python dictionary; the dictionary keys denote the cluster indeces and the items are Python lists of the sample indices that were assigned to each cluster.
iterations_ : int
For usage examples, please see http://rasbt.github.io/mlxtend/user_guide/classifier/Kmeans/
fit(X, init_params=True)
|
A Comparison of Solar Photocatalytic Inactivation of Waterborne E. coli Using Tris (
2,2′
-bipyridine)ruthenium(II), Rose Bengal, and
TiO2
Rengifo-Herrera, J. A., Sanabria, J., Machuca, F., Dierolf, C. F., Pulgarin, C., and Orellana, G. (December 12, 2005). "A Comparison of Solar Photocatalytic Inactivation of Waterborne E. coli Using Tris (
2,2′
TiO2
." ASME. J. Sol. Energy Eng. February 2007; 129(1): 135–140. https://doi.org/10.1115/1.2391319
Background. The development of alternative processes to eliminate pathogenic agents in water is a matter of growing interest. Current drinking water disinfection procedures, such as chlorination and ozonation, can generate disinfection by-products with carcinogenic and mutagenic potential and are not readily applicable in isolated rural communities of less-favored countries. Solar disinfection processes are of particular interest to water treatment in sunny regions of the Earth. Solar light may be used to activate a photocatalyst or photosensitizer that generates, in the presence of molecular oxygen dissolved in water, reactive oxygen species (ROS), such as the
HO•
radical, singlet oxygen
(O21)
, or superoxide
(O2•)
, which are toxic to waterborne microorganisms. Method of Approach. Wild and collection-type Escherichia coli have been selected as model bacteria. Inactivation of such bacteria by either
TiO2
nanoparticles, water-soluble tris(
2,2′
-bipyridine)ruthenium(II) dichloride or Rose bengal (RB) subject to simulated sunlight have been compared. Although
TiO2
is the prototypical material for heterogeneous photocatalysis, the other two dyes are known to generate significant amounts of
O21
by photosensitization but have different chemical structures. The concentration of dye, illumination time, photostability, presence of scavengers, and post-treatment regrowth of bacteria have been investigated. Results. After
1hr
of solar illumination the Ru(II) complex produced a strong loss of E. coli culturability monitored with solid selective agars. Both the collection- and wild-type bacteria are sensitive to the treatment with
2-10mgL−1
of dye. This photosensitizer showed a better inactivation effect than
TiO2
and the anionic organic dye RB due to a combination of visible light absorption, photostability, and production of
O21
and other ROS when bound to the bacterial membrane. A complete loss of culturability was observed when the initial concentration was
103CFUmL−1
, with no bacteria regrowth detected after
24hr
of the water treatment. At higher initial microorganism levels, culturability still remains and regrowth is observed. Scavengers show that the
HO•
radical is not involved in bacteria inactivation by photosensitization. Conclusions. A higher quantum yield of ROS generation by the sensitizing dyes compared to the semiconductor photocatalyst determines the faster sunlight-activated water disinfection of photodynamic processes. The homogeneous nature of the latter determines a more efficient interaction of the toxic intermediates with the target microorganisms. Solid supporting of the Ru(II) dye is expected to eliminate the potentials problems associated to the water-soluble dye.
|
a+0=a
0+b=b
\mathrm{degree}\left(a\right)≺\mathrm{degree}\left(b\right)
≺
denotes the strict ordering of ordinals, then
a+b=b
a=l+{\mathbf{\omega }}^{e}\cdot m
l=0
\mathrm{tdegree}\left(l\right)≻e
b={\mathbf{\omega }}^{e}\cdot n+r
e≻\mathrm{degree}\left(r\right)
a+b=l+{\mathbf{\omega }}^{e}\left(m+n\right)+r
\mathrm{tdegree}\left(a\right)≻\mathrm{degree}\left(b\right)
a+b
is just the concatenation (formal sum) of all terms of
and
b
Mathematically, addition of two ordinals
a+b
corresponds to the disjoint union
a⊔b
and
b
, respectively, such that every element of
b
is strictly larger than every element of
a
a,b,\mathrm{...}
contain at least one ordinal data structure, that is, an ordinal number greater or equal to
\mathbf{\omega }
\mathrm{with}\left(\mathrm{Ordinals}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{`+`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`.`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`<`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{<=}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Add}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Base}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Dec}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Decompose}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Div}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Eval}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Factor}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Gcd}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Lcm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LessThan}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Max}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Min}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Mult}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Ordinal}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Power}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Split}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Sub}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{`^`}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{degree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{log}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{lterm}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{\omega }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{quo}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{rem}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tcoeff}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tdegree}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{tterm}}]
a≔\mathrm{Ordinal}\left([[\mathrm{\omega },1],[3,2],[1,4],[0,5]]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}
b≔\mathrm{Ordinal}\left([[2,3],[0,2]]\right)
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
c≔\mathrm{Ordinal}\left([[3,3],[2,1],[1,7]]\right)
\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}
\mathrm{Add}\left(a,b,c\right)=a+b+c
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}
a+c+b
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
c+b+a=a
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}
\mathrm{result}≔\left(a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&+\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}b\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&+\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}c:
\mathrm{result}=\mathrm{value}\left(\mathrm{result}\right)
\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{+}}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\mathbf{+}}\left({\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}\right)\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}
Any of the arguments can be a nonnegative integer. It will be absorbed if the term to the right is an ordinal greater or equal to
\mathbf{\omega }
b+2
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}
b+2+c=c
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}
d≔\mathrm{Ordinal}\left([[3,x],[2,1],[1,7]]\right)
\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}
d+b
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}
b+d
b+\mathrm{Eval}\left(d,x=x+1\right)
{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{\cdot }\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{\mathbf{\omega }}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathbf{\omega }}\textcolor[rgb]{0,0,1}{\cdot }\textcolor[rgb]{0,0,1}{7}
|
Welders wear special goggles or face masks with glass windows to protect their eyes from electromagnetic radiation Name the - Physics - Electromagnetic Waves - 8174125 | Meritnation.com
Welders wear special goggles or face masks with glass windows to protect their eyes from Ultra Violet Rays (UV-rays).
Frequency range of UV- rays is
7.5×{10}^{14} \mathrm{to} 5×{10}^{17} \mathrm{Hz} \left(800 \mathrm{THz} \mathrm{to} 30 \mathrm{PHz}\right)
|
From Heegaard splittings to trisections; porting $3$-dimensional ideas to dimension $4$
From Heegaard splittings to trisections; porting
3
-dimensional ideas to dimension
4
David T Gay1
1 Euclid Lab 160 Milledge Terrace Athens, GA 30606 Department of Mathematics University of Georgia Athens, GA 30602
These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan “Trisections are to
4
–manifolds as Heegaard splittings are to
3
–manifolds”, focusing on and clarifying the distinction between three ways of thinking of things: the basic definitions as decompositions of manifolds, the Morse theoretic perspective and descriptions in terms of diagrams. We also lay out these themes in two important relative settings:
4
–manifolds with boundary and
4
–manifolds with embedded
2
–dimensional submanifolds.
David T Gay 1
author = {David T Gay},
title = {From {Heegaard} splittings to trisections; porting $3$-dimensional ideas to dimension $4$},
TI - From Heegaard splittings to trisections; porting $3$-dimensional ideas to dimension $4$
%T From Heegaard splittings to trisections; porting $3$-dimensional ideas to dimension $4$
David T Gay. From Heegaard splittings to trisections; porting $3$-dimensional ideas to dimension $4$. Winter Braids Lecture Notes, Volume 5 (2018), Talk no. 4, 19 p. doi : 10.5802/wbln.24. https://wbln.centre-mersenne.org/articles/10.5802/wbln.24/
[1] R. İnanç Baykur; Osamu Saeki Simplified broken Lefschetz fibrations and trisections of 4-manifolds, Proceedings of the National Academy of Sciences, Volume 115 (2018) no. 43, pp. 10894-10900 http://www.pnas.org/content/115/43/10894 | http://www.pnas.org/content/115/43/10894.full.pdf | Article | MR: 3871793 | Zbl: 1421.57026
[2] Nickolas A. Castro Relative trisections of smooth 4-manifolds with boundary (2016) (Ph. D. Thesis)
[3] Nickolas A. Castro; David T. Gay; Juanita Pinzón-Caicedo Diagrams for relative trisections, Pacific J. Math., Volume 294 (2018) no. 2, pp. 275-305 | Article | MR: 3770114 | Zbl: 1394.57015
[4] David Gay; Robion Kirby Trisecting 4-manifolds, Geom. Topol., Volume 20 (2016) no. 6, pp. 3097-3132 | Article | MR: 3590351 | Zbl: 1372.57033
[5] David Gay; Jeffrey Meier Doubly pointed trisection diagrams and surgery on 2-knots, 2018 | arXiv:1806.05351
[6] András Juhász Holomorphic discs and sutured manifolds, Algebr. Geom. Topol., Volume 6 (2006), pp. 1429-1457 | Article | MR: 2253454 | Zbl: 1129.57039
[7] Robion Kirby; Abigail Thompson A new invariant of 4-manifolds, Proceedings of the National Academy of Sciences, Volume 115 (2018) no. 43, pp. 10857-10860 http://www.pnas.org/content/115/43/10857 | http://www.pnas.org/content/115/43/10857.full.pdf | Article | MR: 3871787 | Zbl: 1421.57031
[8] Peter Lambert-Cole Bridge trisections in
{\mathrm{ℂℙ}}^{2}
and the Thom conjecture, 2018 | arXiv:1807.10131
[9] Peter Lambert-Cole; Jeffrey Meier Bridge trisections in rational surfaces, 2018 | arXiv:1810.10450
[10] François Laudenbach A proof of Reidemeister-Singer’s theorem by Cerf’s methods, Ann. Fac. Sci. Toulouse Math. (6), Volume 23 (2014) no. 1, pp. 197-221 | Article | Numdam | MR: 3204738 | Zbl: 1322.57020
[11] François Laudenbach; Valentin Poénaru A note on
4
-dimensional handlebodies, Bull. Soc. Math. France, Volume 100 (1972), pp. 337-344 | Article | Numdam | MR: 0317343 (47 #5890) | Zbl: 0242.57015
[12] Jeffrey Meier Trisecting surfaces filling transverse links (in preparation)
[13] Jeffrey Meier; Alexander Zupan Bridge trisections of knotted surfaces in
{S}^{4}
, Trans. Amer. Math. Soc., Volume 369 (2017) no. 10, pp. 7343-7386 | Article | MR: 3683111 | Zbl: 1376.57025
[14] Jeffrey Meier; Alexander Zupan Bridge trisections of knotted surfaces in 4-manifolds, Proceedings of the National Academy of Sciences, Volume 115 (2018) no. 43, pp. 10880-10886 http://www.pnas.org/content/115/43/10880 | http://www.pnas.org/content/115/43/10880.full.pdf | Article | MR: 3871791 | Zbl: 1418.57017
[15] Kurt Reidemeister Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg, Volume 9 (1933) no. 1, pp. 189-194 | Article | MR: 3069596 | Zbl: 0007.08005
[16] Adam Saltz Invariants of knotted surfaces from link homology and bridge trisections, 2018 | arXiv:1809.06327
[17] James Singer Three-dimensional manifolds and their Heegaard diagrams, Trans. Amer. Math. Soc., Volume 35 (1933) no. 1, pp. 88-111 | Article | MR: 1501673 | Zbl: 0006.18501
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Organic content and maturation effects on elastic properties of source rock shales in the Central North Sea | Interpretation | GeoScienceWorld
Jørgen André Hansen;
. E-mail: j.a.hansen@geo.uio.no; manzar.fawad@geo.uio.no.
Nazmul Haque Mondol;
. E-mail: nazmulh@geo.uio.no.
Jørgen André Hansen, Nazmul Haque Mondol, Manzar Fawad; Organic content and maturation effects on elastic properties of source rock shales in the Central North Sea. Interpretation 2019;; 7 (2): T477–T497. doi: https://doi.org/10.1190/INT-2018-0105.1
We have investigated the effects of organic content and maturation on the elastic properties of source rock shales, mainly through integration of a well-log database from the Central North Sea and associated geochemical data. Our aim is to improve the understanding of how seismic properties change in source rock shales due to geologic variations and how these might manifest on seismic data in deeper, undrilled parts of basins in the area. The Tau and Draupne Formations (Kimmeridge shale equivalents) in immature to early mature stages exhibit variation mainly related to compaction and total organic carbon (TOC) content. We assess the link between depth, acoustic impedance (AI), and TOC in this setting, and we express it as an empirical relation for TOC prediction. In addition, where S-wave information is available, we combine two seismic properties and infer rock-physics trends for semiquantitative prediction of TOC from
VP/VS
and AI. Furthermore, data from one reference well penetrating mature source rock in the southern Viking Graben indicate that a notable hydrocarbon effect can be observed as an addition to the inherently low kerogen-related velocity and density. Published Kimmeridge shale ultrasonic measurements from 3.85 to 4.02 km depth closely coincide with well-log measurements in the mature shale, indicating that upscaled log data are reasonably capturing variations in the actual rock properties. Amplitude variation with offset inversion attributes should in theory be interpreted successively in terms of compaction, TOC, and maturation with associated generation of hydrocarbons. Our compaction-consistent decomposition of these effects can be of aid in such interpretations.
Draupne Formation
Egersund Basin
Asta Graben
Ling Depression
|
Analyzing Indifference Curves: Purpose, Types, and Shape | Outlier
How do you compare one indifference curve to another?
Are indifference curves realistic?
Let’s say you have a little brother who’s an avid consumer of two things: chocolate bars and gummy bears. It might not be apparent at first glance that there’s any relationship between how much he likes one compared to the other. However, the field of microeconomics has given us a number of tools designed specifically to illuminate the complicated and often hidden relationships which shape the ways in which people not only consume goods, but also how they make choices between more abstract things, such as their preference between leisure (hours of free time) and the income gained from working. One of the foundational tools of the discipline is the indifference curve, which is defined as a curve connecting different combinations of two goods that each produce the same level of utility (or satisfaction) for a particular individual.
Take for example, the indifference curve below, which connects different combinations of chocolate bars and gummy bears that provide him with the exact same level of utility. According to this indifference curve, 3 chocolate bars and 2 packs of gummy bears is just as good as 2 chocolate bars and 4 packs of gummy bears. If you asked your brother to choose between any of the combinations along this curve, he’d simply be indifferent.
Likewise, an analysis of the figure below shows us that Individual A is indifferent between having 9 hours of leisure in a day with daily earnings of $225 and having 6 hours of leisure per day with daily earnings of $300.
Indifference curves are subjective
Indifference curves represent individual tastes and preferences. They’re subjective in the sense that they will look different from person to person. You might look at the indifference curve in the example above and feel differently about the amount of chocolate you want relative to packs of gummy bears. If that’s the case, the indifference curves representing your preferences would look different from those of your hypothetical brother.
What does the shape of an indifference curve tell us?
One thing to notice when you’re analyzing an indifference curve is its slope. The slope of an indifference curve at any given point along the curve is called the marginal rate of substitution (MRS). Remember, a slope can be approximated as the change in Y over the change in X, so we can interpret the marginal rate of substitution as the amount of Good Y that a person is willing to give up in exchange for one additional unit of Good X.
In the figure below, the marginal rate of substitution at Point A is 10. As you move from Point A to Point B, you are willing to give up 10 units of Good Y (40 units to 30 units) for one additional unit of Good X (5 units to 6 units). Keep in mind that this way of calculating the MRS is just an approximation. To find the exact MRS you would use calculus to calculate the derivative at Point A.
The Marginal Rate of Substitution (MRS) is the slope of an indifference curve at a given point. The MRS can be approximated and interpreted as the amount of a good that an individual is willing to part with in exchange for one additional unit of another while remaining on the same indifference curve.
Knowing what a marginal rate of substitution is, we can learn more about the overall shape of indifference curves. A relatively flat indifference curve (like the one shown in the figure on the left below) demonstrates a lower overall willingness to give up the good on the y-axis in exchange for more of the good on the x-axis. The individual whose preferences are represented on the left would have to be compensated with a lot of Good X just to part with a single unit of Good Y. Conversely, a relatively steep indifference curve demonstrates a stronger overall willingness to give up some of the good on the y-axis in exchange for more of the good on the x-axis. The individual whose preferences are represented on the right is willing to give up a relatively large amount of Good Y just to get one additional unit of Good X.
Interpreting the MRS
A high marginal rate of substitution (a steeper slope on the indifference curve) indicates that an individual is willing to give up a large amount of the good plotted on the y-axis in exchange for just one additional unit of the good plotted along the x-axis.
A low marginal rate of substitution (a flatter slope on the indifference curve) indicates that an individual is only willing to give up a small amount of the good plotted on the y-axis in exchange for just one additional unit of the good plotted along the x-axis. In other words, the lower MRS indicates more of a reluctance to part with the good plotted on the y-axis.
All of the indifference curves so far, whether steep or flat, have been convex to the origin. This means the curve is bowed outwards away from the origin, and that the curve gets steeper as it approaches higher and higher values of Good Y and flatter as it approaches higher and higher values of Good X. Almost all of the indifference curves you see will have this general shape.
It makes sense that an indifference curve would be convex, rather than concave, to the origin if you make the following assumption:
When you have a lot of Good Y but very little Good X, you are more willing to part with Good Y for an additional unit of Good X (the MRS will be high and the indifference curve will be steep). On the other hand, if you have plenty of Good X but very little of Good Y, you will be less willing to give up a unit of Good Y for one additional unit of Good X (the MRS will be lower and the indifference curve flatter).
So far we have only looked at one indifference curve at a time, but below you can see what a typical graph of indifference curves looks like. You’ll notice that there are multiple indifference curves on this graph. Each indifference curve (
IC_1
IC_2
,...) represents combinations of goods that provide the same amount of utility, but the level of utility varies from curve to curve. As you move towards the right corner of the graph, each indifference curve represents higher and higher levels of utility. An indifference curve above and to the right of another indifference curve will always be preferred. To understand why this is the case, compare combinations of goods between any two indifference curves in the graph. Combinations along the indifference curve above and to the right will always offer a better deal than those on the lower curve.
Indifference curves are a useful tool for modeling individual preferences, but like all economic models, indifference curves are not always realistic and make strong assumptions about the way people behave which do not always hold up against empirical study of the manner in which individuals actually behave. Some applications of indifference curves have been criticized for deviating too far from how people act in real life to be useful.
When using indifference curves, economists often make the following strong assumptions regarding preferences, regardless of their distance from lived behavior:
Transitive preferences. If a person prefers A to B and B to C, then they must prefer A to C.
Complete preferences. People always have a clear sense of what their preferences are and are never indecisive.
Behavior in line with preferences. People always act in accordance with their preferences. Once preferences are established they don’t change from moment to moment.
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Logarithmic Equations - Course Hero
College Algebra/Solving Exponential and Logarithmic Equations/Logarithmic Equations
A logarithmic equation is an equation with a logarithmic expression that contains a variable. A logarithmic equation may have logarithms on one side, such as
\log_{3}x=7
, or on both sides, such as
\log_{2}(x+1)=\log_{2} 6
Solving a logarithmic equation with logarithms on both sides may involve the equality property: For
x>0
y>0
\log_{b} x=\log_{b} y
x=y
The argument of a logarithm is the expression the logarithm applies to. For example, the argument of
\log_4(x-3)
x-3
. When solving equations involving logarithms, check the answers in the original equation to make sure the argument of the logarithm is positive.
Solve a Simple Equation with Logarithms on Both Sides
\log_{2}(x+1)=\log_{2}6
The logarithms have the same base, so the arguments are equal.
\begin{aligned}\log_{2}(x+1)&=\log_{2} 6\\x+1&=6\end{aligned}
x=5
x
in the original equation to make sure all arguments are positive.
\begin{aligned}\log_{2}(5+1)&=\log_{2}6\\\log_{2}6&=\log_{2}6\end{aligned}
The arguments are positive. So,
x=5
To solve a logarithmic equation with logarithms on one side, use the relationship between exponents and logarithms: If
\log_{b} x=\log_{b} y
b^y=x
Solve a Simple Equation with a Logarithm on One Side
\log_{3}x=7
\begin{aligned}\log_{b}y&=x &\rightarrow &&b^{x}&=y\\ \log_{3}x&=7&\rightarrow &&3^{7}&=x\end{aligned}
Simplify to determine the value of
x
\begin{aligned}3^{7}&=x\\2\text{,}187&=x\end{aligned}
Make sure the argument is positive by substituting the value of
x
in the original equation.
\begin{aligned}\log_{3}x&=7\\\log_{3}2\text{,}187&=7\end{aligned}
The argument is positive. So,
x=2\text{,}187
Solving Multistep Logarithmic Equations
To solve multistep logarithmic equations, first use properties of logarithms to simplify the equation.
\log_{b}{xy}=\log_{b}{x}+\log_{b}{y}
\log_{b}{\frac{x}{y}}=\log_{b}{x}-\log_{b}{y}
\log_{b}{x^p}=p\cdot\log_{b}{x}
\log_{b}{x}=\frac{\log_{a}{x}}{\log_{a}{b}}
After simplifying the equation, if there is a single logarithm on both sides of the equation, use the equality property: If
\log_{b} x=\log_{b} y
x=y
. If there is a single logarithm on only one side of the equation, write the equation in exponential form: If
\log_{b} x= y
b^{y}=x
. Make sure to check all answers to ensure that the argument of the logarithm is positive.
Solve a Multistep Equation with Logarithms on Both Sides
\log_{4}8x-\log_{4}(x-2)=\log_{4}x
Apply a property of logarithms.
The equation involves a difference of logarithms, and all logarithms have the same base. Use the quotient rule.
\begin{aligned}\log_{4}8x-\log_{4}(x-2)&=\log_{4}x\\\log_{4}{\left ( \frac{8x}{x-2} \right )}&=\log_{4}{x}\end{aligned}
\frac{8x}{x-2}=x
x-2
\begin{aligned}\frac{8x}{x-2}&=x\\\left(\frac{8x}{x-2}\right)(x-2)&=x(x-2)\\8x&=x(x-2) \end{aligned}
x
\begin{aligned}8x&=x(x-2) \\8x&=x^{2}-2x\end{aligned}
Set the equation equal to zero by subtracting
8x
\begin{aligned}8x=x^{2}-\phantom{1}2x\\\underline{-8x=\phantom{x^2}-\phantom{1}8x}\\\phantom{-8}0=x^{2}-10x\end{aligned}
\begin{aligned}0&=x^{2}-10x\\0&=x(x-10)\end{aligned}
\begin{aligned}x=0\quad \text{ or }\quad x-10&=0\\x&=10\end{aligned}
x
x=0
\log_{4}(8\cdot 0)-\log_{4}(0-2)=\log_{4}0
x=0
is not valid because
\log_{4} 0
is undefined. Check
x=10
\begin{aligned}\log_{4}(8\cdot 10)-\log_{4}(10-2)&=\log_{4}10&&\text{Substitute }10 \text{ for }x\text{.}\\\log_{4}80-\log_{4}8&=\log_{4}10&&\text{Simplify.}\\\log_{4}\frac{80}{8}&=\log_{4}10&&\text{Use the quotient rule of logarithms.}\end{aligned}
x=10
is a valid solution. It is the only valid solution for the equation.
Solve a Multistep Equation with Logarithms on One Side
Solve the logarithm:
\log(x) + \log(x-15)=2
The equation involves a sum of logarithms. No base is shown, so all the logarithms in the equation have a base of 10. Use the product rule.
\begin{aligned}\log(x) + \log(x-15)&=2\\\log[(x)(x-15)]&=2\end{aligned}
\log(x^2-15x)=2
10^2=x^2-15x
\begin{aligned}&\begin{aligned}100&=x^2-15x &&\text{Simplify.}\\0&=x^2-15x-100 &&\text{Set the equation equal to zero.}\\0&=(x+5)(x-20)&&\text{Factor.}\end{aligned}\end{aligned}
\begin{aligned} x+5&=0\\x&=-5\end{aligned} \hspace{10pt} \text{or} \hspace{10pt} \begin{aligned}x-20&=0\\x&=20\end{aligned}
x
x=-5
\log(-5) + \log(-5-15)=2
x=-5
is not valid because the arguments of
\log(-5)
\log(-5-15)
are negative. Check
x=20
\begin{aligned}\log(20) + \log(20-15)&=2\\\log(20)+\log(5)&=2\\\log(20 \cdot 5)&=2\\\log100&=2\end{aligned}
x=20
is valid. It is the only valid solution for the equation.
Some logarithmic equations can be solved by replacing a logarithmic expression with a temporary variable. Solve for the temporary variable, and then replace the temporary variable with the logarithmic expression and solve for
x
Solve a Logarithmic Equation Using Substitution
\ln^2(x)+3\ln(x)=-2
Use a temporary variable to represent the logarithmic expression.
u
\ln{x}
u^2 + 3u =-2
\begin{aligned}u^2+&3u+2=0\;\;\;\;\;&&\text{Set the equation equal to zero.}\\\;(u + 2)(&u+1)= 0\;\;\;\;&&\text{Factor.}\end{aligned}
Use the zero product property to set each factor equal to zero and then solve each equation.
\begin{aligned}u+2&=0\\u&=-2\end{aligned}\hspace{10pt}\text{or}\hspace{10pt}\begin{aligned} u+1&=0\\u&=-1\end{aligned}
u
with the expression it represents.
\ln{x}
u
in the solutions.
\ln(x) = -2 \hspace{10pt}\text{or}\hspace{10pt}\ln(x) = -1
Write the equations in exponential form.
x = e^{-2}\hspace{10pt} \text{or} \hspace{10pt} x = e^{-1}
Write the exponential expressions with positive exponents.
x = \frac{1}{e^2} \hspace{10pt}\text{or} \hspace{10pt}x = \frac{1}{e^1}
x
Since
\frac{1}{e^2}
\frac{1}{e}
x = \frac{1}{e^2}
\begin{aligned}\ln^2\left(\frac{1}{e^2}\right)+3\ln\left(\frac{1}{e^2}\right)&=-2\\4+(-6)&=-2\end{aligned}
x = \frac{1}{e^2}
is valid. Check
x = \frac{1}{e}
\begin{aligned}\ln^2\left(\frac{1}{e}\right)+3\ln\left(\frac{1}{e}\right)&=-2\\1+(-3)&=-2\end{aligned}
x = \frac{1}{e}
is also valid.
x = \frac{1}{e^2}
x = \frac{1}{e}
are valid. The approximate solutions are
x \approx0.14
x \approx0.37
Solving Logarithmic Equations by Graphing
To solve a logarithmic equation by graphing, graph the related functions for both sides of the equation and look for points of intersection. This process is similar to solving a system of equations in two variables. If the logarithmic equation is set equal to zero, graph the related function for the logarithmic expression and look for the zeros. A zero of a function is any input value of a function that makes the output of the function equal zero.
Solve a Logarithmic Equation by Graphing
x
\ln{(x+5)}=1
Use a graphing calculator or other graphing utility to graph the functions:
f(x)=\ln{(x+5)} \hspace{20pt} f(x)=1
x
The graphs intersect where
x \approx -2.28
<Vocabulary>Exponential Equations
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Separation of Variables: What Is It & How to Do It | Outlier
Separation of Variables: What Is It & How to Do It
In this article, we discuss the meaning of separation of variables. Then, we learn how to apply it to ordinary differential equations using a step-by-step guide. Finally, we introduce how this method can be used to solve differential equations with two variables.
Separation of Variables for Ordinary Differential Equations
A differential equation is an equation that relates an unknown function
y = f(x)
with its derivatives.
Solving differential equations is an essential prediction tool in:
Here are some simple examples of differential equations to help you understand what a differential equation might look like:
\frac{dy}{dx} = 2xy
y’’ - 2y’ + y = 3x^2 - 8
2x^2 - 3\frac{dy}{dx} = 0
You might be used to solving equations that require a single number as its solution. It’s important to note that a solution to a differential equation is a function instead of a number. You can use this function to forecast future values. For example, the solution to a differential equation could be a function that is used to predict animal populations.
A solution to a differential equation is a function that satisfies the differential equation. So, to check if a function
f(x)
is a solution to a differential equation, we can simply substitute
f(x)
and its derivatives into the differential equation and verify that both sides of the equation are the same.
For example, consider the differential equation
4x^3 - 2 \frac{dy}{dx} = 0
. Suppose we want to check if the function
y = \frac{x^4}{2} + 5
is a solution to the given differential equation.
Well, taking the derivative of
y
\frac{dy}{dx} = 2x^3
\frac{dy}{dx}
into our differential equation, we get:
4x^3 - 2(2x^3) = 0
4x^3 - 4x^3 = 0
0 = 0
Since both sides of the equation agree,
y
is a solution to the differential equation.
So, how exactly can we find all the functions that satisfy a given differential equation? Separation of variables is one method for solving differential equations. Differential equations that can be solved using separation of variables are called separable differential equations.
\frac{dy}{dx} = \frac{f(x)}{g(y)}
. If we multiply both sides by the denominators of each side, we get the equation
g(y)dy = f(x)dx
You can write separable differential equations in the form
g(y)dy = f(x)dx
given above. This means that all terms involving
y
are on one side of the equation, while all terms involving
x
are on the other side of the equation. Since
x
y
are separated, this is why we call such an equation a separable differential equation.
After separating the variables, separable differential equations can be solved by integrating both sides of the equation with respect to the variable on that side. Don’t forget to add the constant of integration to your equation.
\int g(y) \, dy = \int f(x) \, dx
G(y) = F(x) + C
F
G
are the antiderivative functions of
g
respectively. This is our general solution to
\frac{dy}{dx} = \frac{f(x)}{g(y)}
To check our work and verify that
G(x) = F(x) + C
is a solution to our original differential equation
\frac{dy}{dx} = \frac{f(x)}{g(y)}
, we can simply differentiate each side implicitly with respect to
x
and hope to get our original equation back.
Remember, to differentiate implicit functions with respect to
x
, we must treat
y
x
. This means that we’ll have to use the Chain Rule when we come across terms involving
y
\frac{dy}{dx}
. Then, we can solve for
\frac{dy}{dx}
A differential equation is an equation that relates an unknown function y = f(x) with its derivatives.
Here’s what this process looks like. Since we’re differentiating
G(y)
x
, we have to use the Chain Rule and multiply
G'(y)
\frac{dy}{dx}
. By the definition of an antiderivative function, we know that
G’(y) = g(y)
F’(x) = f(x)
G’(y) \frac{dy}{dx} = F’(x)
g(y) \frac{dy}{dx} = f(x)
\frac{dy}{dx} = \frac{f(x)}{g(y)}
Since we were able to obtain our original equation
\frac{dy}{dx} = \frac{f(x)}{g(y)}
, this verifies that
G(x) = F(x) + C
is the general solution to the differential equation.
To recap, here are three simple steps to solve differential equation using separation of variables:
Separate the variables of the equation so that all the
y
-terms are on one side of the equation and all the
x
-terms are on the other side of the equation.
Integrate each side of the equation with respect to the variable present on that side. Don’t forget to add the constant of integration to one side of the equation.
Simplify where necessary.
Separable differential equations are differential equations that can be solved using separation of variables.
The above process details the separation of variables method for ordinary differential equations. An ordinary differential equation involves functions of only one variable and the derivatives of those functions. It contains no partial derivatives.
Let’s go through a few separation of variables examples together.
4x^3 - 2 \frac{dy}{dx} = 0
. We previously verified that
y = \frac{x^4}{2} + 5
is one solution to the differential equation, but how can we determine the general solution? We’ll follow the three steps given above.
Separating the variables, we get:
4x^3 - 2\frac{dy}{dx} = 0
4x^3 = 2\frac{dy}{dx}
4x^3 \, dx = 2 \, dy
2. Using the power rule to integrate both sides of the equation with respect to each side’s respective variable, we get:
\int 4x^3 \, dx = \int 2 \, dy
\frac{4x^4}{4} = 2y + C
x^4 = 2y + C
3. Simplifying, we get
y = \frac{x^4}{2} + C
. This is our general solution for the differential equation
4x^3 - 2 \frac{dy}{dx} = 0
One of the most significant ordinary differential equations that can be solved using separation by variables is the exponential differential equation
\frac{dP}{dt} = kP
P
represents the size of some population,
\frac{dP}{dt}
represents the rate at which the population is changing with respect to time, and
k
represents the growth and decay constant. Solving this differential equation allows us to derive the formula for exponential growth and decay.
We’ll follow the three steps given earlier.
\frac{dP}{dt} = kP
dP = kP \, dt
\frac{dP}{P} = k \, dt
Now we can integrate both sides of the equation with respect to each side’s respective variable. Using the power and logarithm rules for integration, we get:
\int \frac{1}{P} \, dP = \int k \, dt
\ln{P} = kt + C
In order to simplify, we need to get
P
by itself. To do this, we can raise
\ln{P}
power. Also, notice
e^C
is just a constant, so we can say
C = e^C
e^{\ln{P}} = e^{kt+C}
P(t) = e^{kt} \cdot e^C
P(t) = Ce^{kt}
Let’s stop for a minute and examine
P(0)
, which is the initial population size at
t = 0
P(0) = Ce^0 = C \cdot 1 = C
C = P(0)
C
is always equal to the population at
t = 0
. We’ll denote this by
P_0
. Now we have:
P(t) = P_0 e^{kt}
This is the formula for exponential growth and decay, which you can use to determine the population at any time
t
k
value indicates growth, while a negative
k
value indicates decay.
Retired NFL lineman and matematician John Urschel discusses exponential decay and how exponential equations fit into the larger story of Calculus:
Separation of variables can also be used to solve some partial differential equations. A partial differential equation is an equation that involves an unknown function and its derivatives, which depend on two or more independent variables.
Explaining how to solve partial differential equations using separation of variables will require a separate tutorial. In this guide, we will walk through one example that solves for the product solution of a partial differential equation. This will help you understand how to solve partial differential equations using separation of variables.
We will find the product solution of Laplace’s Equation, given below.
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
u
is a function of two variables,
x
y
. The first step to solving a partial differential equation using separation of variables is to assume that it is separable. We must assume that it can be separated into separate functions, each with only one independent variable. If this assumption is false, the procedure to solve the differential equation will fail part way through.
So, we will make the assumption that:
u(x, y) = X(x)Y(y)
Next, we substitute the above product of functions into the differential equation and differentiate accordingly to obtain the terms required in the original partial differential equation. So, substituting
u(x, y) = X(x)Y(y)
\frac{\partial^2}{\partial x^2}X(x)Y(y) + \frac{\partial^2}{\partial y^2}X(x)Y(y) = 0
\frac{\partial^2}{\partial x^2}X(x)Y(y)
is taking a second partial derivative with respect to
x
, so we can treat
y
as a constant. So, we can say
\frac{\partial^2}{\partial x^2}X(x)Y(y) = Y(y)X’’(x)
Similarly, the term
\frac{\partial^2}{\partial y^2}X(x)Y(y)
y
x
is kept fixed. So, we can say
\frac{\partial^2}{\partial y^2}X(x)Y(y) = X(x)Y’’(y)
Now, we’ve gotten rid of the partial derivatives. Our current equation is:
Y(y)X’’(x) + X(x)Y’’(y) = 0
The next step is to move the
x
-terms to one side of the equation and
y
-terms to the other side of the equation, if possible. Luckily, this is possible for our example. If it is not possible, the partial differential equation is non-separable and separation by variables cannot be used.
All we have to do to separate our function is divide each term by
X(x)Y(y)
. Then, we can cancel similar terms and simplify:
Y(y)X’’(x) + X(x)Y’’(y) = 0
\frac{Y(y)X’’(x)}{X(x)Y(y)} + \frac{X(x)Y’’(y)}{X(x)Y(y)} = \frac{0}{X(x)Y(y)}
\frac{X’’(x)}{X(x)} + \frac{Y’’(y)}{Y(y)} = 0
\frac{X’’(x)}{X(x)} = - \frac{Y’’(y)}{Y(y)}
Next, since
X(x)
y
Y(y)
x
, both sides must yield the same constant value. We’ll call this constant
-\lambda^2
. Setting the equation equal to
-\lambda^2
seems random, but it will make the resulting ordinary differential equations easier to solve.
\frac{Y(y)X’’(x)}{X(x)Y(y)} + \frac{X(x)Y’’(y)}{X(x)Y(y)} = \frac{0}{X(x)Y(y)}
Setting our equation equal to a constant allows us to form two ordinary differential equations.
\frac{X’’(x)}{X(x)} + \lambda^2 = 0
X’’(x) + \lambda^2 X(x) = 0
\frac{Y’’(y)}{Y(y)} - \lambda^2 = 0
Y’’(y) - \lambda^2 Y(y) = 0
The general solution for an ordinary differential equation of the form
y’’ + w^2y = 0
y(x) = A \cos{(wx)} + B \sin{(wx)}
. So, the general solution of
X’’(x) + \lambda^2 X(x) = 0
X(x) = A \sin{(kx)} + B \cos{(kx)}
Note that the auxiliary equation of
Y’’(y) - \lambda^2 Y(y) = 0
r^2 - \lambda^2 = 0 = (r+\lambda)(r-\lambda)
r_1 = \lambda
r_2 = -\lambda
. The general solution for a homogeneous differential equation with two real distinct roots is
y(x) = c_1e^{r_1x} + c_2e^{r_2x}
Y’’(y) - \lambda^2 Y(y) = 0
Y(y) = Ce^{ky} + De^{-ky}
Thus the equations
X(x) = A \sin{(kx)} + B \cos{(kx)}
Y(y) = Ce^{ky} + De^{-ky}
give us the product solution of the partial differential equation, so that
u = (A \sin{(kx)} + B \cos{(kx)}) \cdot (Ce^{ky} + De^{-ky})
Note, the product solution might not be the complete solution. Finding the complete solution requires further mathematical context surrounding boundary conditions, initial conditions, and superposition.
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Interface between gas and mechanical rotational networks - MATLAB - MathWorks Switzerland
Rotational Mechanical Converter (G)
Partial Derivatives for Perfect and Semiperfect Gas Models
Partial Derivatives for Real Gas Model
Interface between gas and mechanical rotational networks
The Rotational Mechanical Converter (G) block models an interface between a gas network and a mechanical rotational network. The block converts gas pressure into mechanical torque and vice versa. It can be used as a building block for rotary actuators.
The converter contains a variable volume of gas. The pressure and temperature evolve based on the compressibility and thermal capacity of this gas volume. The Mechanical orientation parameter lets you specify whether an increase in the gas volume results in a positive or negative rotation of port R relative to port C.
Port A is the gas conserving port associated with the converter inlet. Port H is the thermal conserving port associated with the temperature of the gas inside the converter. Ports R and C are the mechanical rotational conserving ports associated with the moving interface and converter casing, respectively.
Mass conservation equation is similar to that for the Constant Volume Chamber (G) block, with an additional term related to the change in gas volume:
\frac{\partial M}{\partial p}\cdot \frac{d{p}_{I}}{dt}+\frac{\partial M}{\partial T}\cdot \frac{d{T}_{I}}{dt}+{\rho }_{I}\frac{dV}{dt}={\stackrel{˙}{m}}_{A}
\frac{\partial M}{\partial p}
is the partial derivative of the mass of the gas volume with respect to pressure at constant temperature and volume.
\frac{\partial M}{\partial T}
is the partial derivative of the mass of the gas volume with respect to temperature at constant pressure and volume.
pI is the pressure of the gas volume. Pressure at port A is assumed equal to this pressure, pA = pI.
TI is the temperature of the gas volume. Temperature at port H is assumed equal to this temperature, TH = TI.
ρI is the density of the gas volume.
V is the volume of gas.
\stackrel{˙}{m}
A is the mass flow rate at port A. Flow rate associated with a port is positive when it flows into the block.
Energy conservation equation is also similar to that for the Constant Volume Chamber (G) block. The additional term accounts for the change in gas volume, as well as the pressure-volume work done by the gas on the moving interface:
\frac{\partial U}{\partial p}\cdot \frac{d{p}_{I}}{dt}+\frac{\partial U}{\partial T}\cdot \frac{d{T}_{I}}{dt}+{\rho }_{I}{h}_{I}\frac{dV}{dt}={\Phi }_{A}+{Q}_{H}
\frac{\partial U}{\partial p}
\frac{\partial U}{\partial T}
ФA is the energy flow rate at port A.
QH is the heat flow rate at port H.
hI is the specific enthalpy of the gas volume.
The partial derivatives of the mass M and the internal energy U of the gas volume, with respect to pressure and temperature at constant volume, depend on the gas property model. For perfect and semiperfect gas models, the equations are:
\begin{array}{l}\frac{\partial M}{\partial p}=V\frac{{\rho }_{I}}{{p}_{I}}\\ \frac{\partial M}{\partial T}=-V\frac{{\rho }_{I}}{{T}_{I}}\\ \frac{\partial U}{\partial p}=V\left(\frac{{h}_{I}}{ZR{T}_{I}}-1\right)\\ \frac{\partial U}{\partial T}=V{\rho }_{I}\left({c}_{pI}-\frac{{h}_{I}}{{T}_{I}}\right)\end{array}
cpI is the specific heat at constant pressure of the gas volume.
For real gas model, the partial derivatives of the mass M and the internal energy U of the gas volume, with respect to pressure and temperature at constant volume, are:
\begin{array}{l}\frac{\partial M}{\partial p}=V\frac{{\rho }_{I}}{{\beta }_{I}}\\ \frac{\partial M}{\partial T}=-V{\rho }_{I}{\alpha }_{I}\\ \frac{\partial U}{\partial p}=V\left(\frac{{\rho }_{I}{h}_{I}}{{\beta }_{I}}-{T}_{I}{\alpha }_{I}\right)\\ \frac{\partial U}{\partial T}=V{\rho }_{I}\left({c}_{pI}-{h}_{I}{\alpha }_{I}\right)\end{array}
β is the isothermal bulk modulus of the gas volume.
α is the isobaric thermal expansion coefficient of the gas volume.
The gas volume depends on the rotation of the moving interface:
V={V}_{dead}+{D}_{\mathrm{int}}{\theta }_{\mathrm{int}}{\epsilon }_{\mathrm{int}}
Vdead is the dead volume.
Dint is the interface volume displacement.
θint is the interface rotation.
εint is the mechanical orientation coefficient. If Mechanical orientation is Pressure at A causes positive rotation of R relative to C, εint = 1. If Mechanical orientation is Pressure at A causes negative rotation of R relative to C, εint = –1.
If you connect the converter to a Multibody joint, use the physical signal input port q to specify the rotation of port R relative to port C. Otherwise, the block calculates the interface rotation from relative port angular velocities. The interface rotation is zero when the gas volume is equal to the dead volume. Then, depending on the Mechanical orientation parameter value:
If Pressure at A causes positive rotation of R relative to C, the interface rotation increases when the gas volume increases from dead volume.
If Pressure at A causes negative rotation of R relative to C, the interface rotation decreases when the gas volume increases from dead volume.
Torque balance across the moving interface on the gas volume is
{\tau }_{\mathrm{int}}=\left({p}_{env}-{p}_{I}\right){D}_{\mathrm{int}}{\epsilon }_{\mathrm{int}}
τint is the torque from port R to port C.
penv is the environment pressure.
The converter casing is perfectly rigid.
There is no flow resistance between port A and the converter interior.
There is no thermal resistance between port H and the converter interior.
The moving interface is perfectly sealed.
The block does not model mechanical effects of the moving interface, such as hard stop, friction, and inertia.
Gas conserving port associated with the converter inlet.
Thermal conserving port associated with the temperature of the gas inside the converter.
Select the alignment of moving interface with respect to the converter gas volume:
Pressure at A causes positive rotation of R relative to C — Increase in the gas volume results in a positive rotation of port R relative to port C.
Pressure at A causes negative rotation of R relative to C — Increase in the gas volume results in a negative rotation of port R relative to port C.
Rotational offset of port R relative to port C at the start of simulation. A value of 0 corresponds to an initial gas volume equal to Dead volume.
Interface volume displacement — Displaced gas volume per unit rotation
0.01 m^3/rad (default)
Displaced gas volume per unit rotation of the moving interface.
Dead volume — Volume of gas when the interface rotation is 0
Volume of gas when the interface rotation is 0.
The cross-sectional area of the converter inlet, in the direction normal to gas flow path.
Select a specification method for the environment pressure:
Atmospheric pressure — Use the atmospheric pressure, specified by the Gas Properties (G) block connected to the circuit.
Pressure outside the converter acting against the pressure of the converter gas volume. A value of 0 indicates that the converter expands into vacuum.
Constant Volume Chamber (G) | Translational Mechanical Converter (G) | Rotational Multibody Interface
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ftest: F-test for classifier comparisons - mlxtend
Example 1 - F-test
F-test for comparing the performance of multiple classifiers.
from mlxtend.evaluate import ftest
In the context of evaluating machine learning models, the F-test by George W. Snedecor [1] can be regarded as analogous to Cochran's Q test that can be applied to evaluate multiple classifiers (i.e., whether their accuracies estimated on a test set differ) as described by Looney [2][3].
More formally, assume the task to test the null hypothesis that there is no difference between the classification accuracies [1]:
\{C_1, \dots , C_M\}
be a set of classifiers which have all been tested on the same dataset. If the
M
classifiers do not perform differently, then the F statistic is distributed according to an F distribution with
(M-1)
(M-1)\times n
degrees of freedom, where
n
is the number of examples in the test set. The calculation of the F statistic consists of several components, which are listed below (adopted from [2]).
We start by defining
ACC_{avg}
as the average of the accuracies of the different models
The sum of squares of the classifiers is then computed as
G_j
n
examples classified correctly by classifier
j
The sum of squares for the objects is calculated as follows:
M_j
is the number of classifiers out of
M
that correctly classified object
\mathbf{x}_j \in \mathbf{X}_{n}
, where is the test dataset on which the classifiers are tested on.
Finally, we compute the total sum of squares,
so that we then can compute the sum of squares for the classification--object interaction:
To compute the F statistic, we next compute the mean SSA and mean SSAB values:
From the MSA and MSAB, we can then calculate the F-value as
After computing the F-value, we can then look up the p-value from a F-distribution table for the corresponding degrees of freedom or obtain it computationally from a cumulative F-distribution function. In practice, if we successfully rejected the null hypothesis at a previously chosen significance threshold, we could perform multiple post hoc pair-wise tests -- for example, McNemar tests with a Bonferroni correction -- to determine which pairs have different population proportions.
[1] Snedecor, George W. and Cochran, William G. (1989), Statistical Methods, Eighth Edition, Iowa State University Press.
[2] Looney, Stephen W. "A statistical technique for comparing the accuracies of several classifiers." Pattern Recognition Letters 8, no. 1 (1988): 5-9.
[3] Kuncheva, Ludmila I. Combining pattern classifiers: methods and algorithms. John Wiley & Sons, 2004.
## Dataset:
# ground truth labels of the test dataset:
y_true = np.array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
# predictions by 3 classifiers (`y_model_1`, `y_model_2`, and `y_model_3`):
y_model_1 = np.array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0,
Assuming a significance level
\alpha=0.05
, we can conduct Cochran's Q test as follows, to test the null hypothesis there is no difference between the classification accuracies,
p_i: H_0 = p_1 = p_2 = \cdots = p_L
f, p_value = ftest(y_true,
y_model_1,
y_model_3)
print('F: %.3f' % f)
print('p-value: %.3f' % p_value)
Since the p-value is smaller than
\alpha
, we can reject the null hypothesis and conclude that there is a difference between the classification accuracies. As mentioned in the introduction earlier, we could now perform multiple post hoc pair-wise tests -- for example, McNemar tests with a Bonferroni correction -- to determine which pairs have different population proportions.
ftest(y_target, y_model_predictions)*
F-Test test to compare 2 or more models.
*y_model_predictions : array-likes, shape=[n_samples]
Variable number of 2 or more arrays that contain the predicted class labels from models as 1D NumPy array.
f, p : float or None, float
Returns the F-value and the p-value
For usage examples, please see http://rasbt.github.io/mlxtend/user_guide/evaluate/ftest/
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Determine Cointegration Rank of VEC Model - MATLAB & Simulink - MathWorks Italia
This example shows how to convert an n-dimensional VAR model to a VEC model, and then compute and interpret the cointegration rank of the resulting VEC model.
The rank of the error-correction coefficient matrix, C, determines the cointegration rank. If rank(C) is:
Zero, then the converted VEC(p) model is a stationary VAR(p - 1) model in terms of
\Delta {y}_{t}
, without any cointegration relations.
n, then the VAR(p) model is stable in terms of
{y}_{t}
The integer r such that
0<r<n
r
cointegrating relations. That is, there are
r
linear combinations that comprise stationary series. You can factor the error-correction term into the two n-by- r matrices
C=\alpha {\beta }^{\prime }
\alpha
contains the adjustment speeds, and
\beta
the cointegration matrix. This factorization is not unique.
For more details, see Cointegration and Error Correction and [139], Chapter 6.3.
Consider the following VAR(2) model.
{y}_{t}=\left[\begin{array}{ccc}1& 0.26& 0\\ -0.1& 1& 0.35\\ 0.12& -0.05& 1.15\end{array}\right]{y}_{t-1}+\left[\begin{array}{ccc}-0.2& -0.1& -0.1\\ 0.6& -0.4& -0.1\\ -0.02& -0.03& -0.1\end{array}\right]{y}_{t-2}+{\epsilon }_{t}.
Create the variables A1 and A2 for the autoregressive coefficients. Pack the matrices into a cell vector.
A1 = [1 0.26 0; -0.1 1 0.35; 0.12 -0.5 1.15];
A2 = [-0.2 -0.1 -0.1; 0.6 -0.4 -0.1; -0.02 -0.03 -0.1];
Var = {A1 A2};
Compute the autoregressive and error-correction coefficient matrices of the equivalent VEC model.
Because the degree of the VAR model is 2, the resulting VEC model has degree
q=2-1
. Hence, Vec is a one-dimensional cell array containing the autoregressive coefficient matrix.
Determine the cointegration rank by computing the rank of the error-correction coefficient matrix C.
r = rank(C)
The cointegrating rank is 2. This result suggests that there are two independent linear combinations of the three variables that are stationary.
vec2var | var2vec
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{\displaystyle n}
{\displaystyle n^{\beta }}
{\displaystyle n^{\beta }/\log n}
Links: [[10]] and [[11]]
Thermodynamics and Local Complexity of Domino Gases
We determine the local Shannon-Boltzmann entropy for a domino gas model by exactly counting configurations. This allows us to compute the entropy from a local template, comparing it with known classical values for monomer-dimer tilings. We then study the diffusion of heat through the system via a source of dominos at one end and a sink at the other. In this setting we estimate the density gradient of the nonequilibrium steady state, using various statistics to measure a macroscopic "conductivity".
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Hydraulic pipeline with resistive, fluid inertia, and fluid compressibility properties - MATLAB
Segmented Pipeline
Hydraulic pipeline with resistive, fluid inertia, and fluid compressibility properties
The Segmented Pipeline block models hydraulic pipelines with circular cross sections. Hydraulic pipelines, which are inherently distributed parameter elements, are represented with sets of identical, connected in series, lumped parameter segments. It is assumed that the larger the number of segments, the closer the lumped parameter model becomes to its distributed parameter counterpart. The equivalent circuit of a pipeline adopted in the block is shown below, along with the segment configuration.
V=\frac{\pi ·{d}^{2}}{4}\frac{L}{N}
The Constant Volume Hydraulic Chamber block is placed between two branches, each consisting of a Hydraulic Resistive Tube block and a Fluid Inertia block. Every Hydraulic Resistive Tube block lumps (L+L_ad)/(N+1)-th portion of the pipe length, while Fluid Inertia block has L/(N+1) length (L_ad denotes additional pipe length equal to aggregate equivalent length of pipe local resistances, such as fitting, elbows, bends, and so on).
The nodes to which Constant Volume Hydraulic Chamber blocks are connected are assigned names N_1, N_2, …, N_n (n is the number of segments). Pressures at these nodes are assumed to be equal to average pressure of the segment. Intermediate nodes between Hydraulic Resistive Tube and Fluid Inertia blocks are assigned names nn_0, nn_1, nn_2, …, nn_n. The Constant Volume Hydraulic Chamber blocks are named ch_1, ch_2, …, ch_n, Hydraulic Resistive Tube blocks are named tb_0, tb_1, tb_2, …, tb_n, and Fluid Inertia blocks are named fl_in_0, fl_in_1, fl_in_2, …, fl_in_n.
N>\frac{4L}{\pi ·c}\omega
The table contains an example of simulation of a pipeline where the first four true eigenfrequencies are 89.1 Hz, 267 Hz, 446 Hz, and 624 Hz.
The error between simulated and actual eigenfrequencies is less than 5% if an eight-segment model is used.
The flow rate through the pipeline is positive if it is directed from port A to port B. The pressure differential is positive if pressure is higher at port A than at port B.
Coefficient that establishes relationship between the pressure and the internal diameter at steady-state conditions. This coefficient can be determined analytically for cylindrical metal pipes or experimentally for hoses. The parameter is used if the Pipe wall type parameter is set to Flexible Wall, and the default value is 2e-10 m/Pa.
Time constant in the transfer function that relates pipe internal diameter to pressure variations. By using this parameter, the simulated elastic or viscoelastic process is approximated with the first-order lag. The value is determined experimentally or provided by the manufacturer. The default value is 0.008 s.
Initial pressures at model nodes
Lets you specify the initial condition for pressure inside the pipe segments. The parameter can have one of two values:
The same initial pressure for all nodes — The initial pressure in all pipe segments is the same, and is specified by the Initial pressure parameter value. This is the default.
Custom — Lets you specify initial pressure individually for each pipe segment, by using the Initial pressure vector parameter. The vector size must be equal to the number of pipe segments, defined by the Number of segments parameter value.
Specifies the initial pressure in all pipe segments. The parameter is used if the Initial pressures at model nodes parameter is set to The same initial pressure for all nodes, and the default value is 0.
Initial pressure vector
Lets you specify initial pressure individually for each pipe segment. The parameter is used if the Initial pressures at model nodes parameter is set to Custom. The vector size must be equal to the number of pipe segments, defined by the Number of segments parameter value.
Specifies the initial volumetric flow rate through the pipe. The default value is 0 m^3/s.
Hydraulic Pipeline | Linear Hydraulic Resistance | Hydraulic Resistive Tube
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(-)-limonene synthase Wikipedia
Not to be confused with (R)-limonene synthase.
In enzymology, a (4S)-limonene synthase (EC 4.2.3.16) is an enzyme that catalyzes the chemical reaction
{\displaystyle \rightleftharpoons }
(−)-(4S)-limonene + diphosphate
Hence, this enzyme has one substrate, geranyl diphosphate, and two products, (−)-(4S)-limonene and diphosphate.
This enzyme belongs to the family of lyases, specifically those carbon-oxygen lyases acting on phosphates. The systematic name of this enzyme class is geranyl-diphosphate diphosphate-lyase [cyclizing, (−)-(4S)-limonene-forming]. Other names in common use include (−)-(4S)-limonene synthase, 4S-(−)-limonene synthase, geranyldiphosphate diphosphate lyase (limonene forming), geranyldiphosphate diphosphate lyase [cyclizing,, and (4S)-limonene-forming]. This enzyme participates in monoterpenoid biosynthesis.
Bohlmann J, Steele CL, Croteau R (1997). "Monoterpene synthases from grand fir (Abies grandis). cDNA isolation, characterization, and functional expression of myrcene synthase, (−)-(4S)-limonene synthase, and (−)-(1S,5S)-pinene synthase". J. Biol. Chem. 272 (35): 21784–92. doi:10.1074/jbc.272.35.21784. PMID 9268308.
Colby SM, Alonso WR, Katahira EJ, McGarvey DJ, Croteau R (1993). "4S-limonene synthase from the oil glands of spearmint (Mentha spicata). cDNA isolation, characterization, and bacterial expression of the catalytically active monoterpene cyclase". J. Biol. Chem. 268 (31): 23016–24. PMID 8226816.
Yuba A, Yazaki K, Tabata M, Honda G, Croteau R (1996). "cDNA cloning, characterization, and functional expression of 4S-(−)-limonene synthase from Perilla frutescens". Arch. Biochem. Biophys. 332 (2): 280–7. doi:10.1006/abbi.1996.0343. PMID 8806736.
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num_permutations: number of permutations for creating subsequences of *k* elements - mlxtend
num_permutations: number of permutations for creating subsequences of k elements
Example 1 - Compute the number of permutations
A function to calculate the number of permutations for creating subsequences of k elements out of a sequence with n elements.
from mlxtend.math import num_permutations
Permutations are selections of items from a collection with regard to the order in which they appear (in contrast to combinations). For example, let's consider a permutation of 3 elements (k=3) from a collection of 5 elements (n=5):
In the example above the permutations 1a, 1b, and 1c, are the "same combination" but distinct permutations -- in combinations, the order does not matter, but in permutation it does matter.
To compute the number of permutations with replacement, we simply need to compute
n^k
c = num_permutations(n=20, k=8, with_replacement=False)
print('Number of ways to permute 20 elements'
Number of ways to permute 20 elements into 8 subelements: 5079110400
c = num_permutations(n=20, k=8, with_replacement=True)
Number of ways to combine 20 elements into 8 subelements (with replacement): 25600000000
It is often quite useful to track the progress of a computational expensive tasks to estimate its runtime. Here, the num_combination function can be used to compute the maximum number of loops of a permutations iterable from itertools:
max_iter = num_permutations(n=len(items), k=3,
for idx, i in enumerate(itertools.permutations(items, r=3)):
num_permutations(n, k, with_replacement=False)
Function to calculate the number of possible permutations.
permut : int
Number of possible permutations.
For usage examples, please see http://rasbt.github.io/mlxtend/user_guide/math/num_permutations/
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Invested Capital Turnover | Intrinio
Invested capital is the total amount of money raised by a company by issuing securities to shareholders and bondholders, and invested capital is calculated by adding the total debt and capital lease obligations to the amount of equity issued to investors. Invested capital is not a line item in company's financial statement because debt, capital leases and stockholders equity are each listed separately in the balance sheet.
\text{investedcapitalturnover =}\frac{\text{totalrevenue}}{\text{investedcapital}}
investedcapitalturnover
Type Efficiency
Screenable? No
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KASUMI - Wikipedia
(Redirected from KASUMI (block cipher))
KASUMI is a block cipher used in UMTS, GSM, and GPRS mobile communications systems. In UMTS, KASUMI is used in the confidentiality (f8) and integrity algorithms (f9) with names UEA1 and UIA1, respectively.[1] In GSM, KASUMI is used in the A5/3 key stream generator and in GPRS in the GEA3 key stream generator.
KASUMI was designed for 3GPP to be used in UMTS security system by the Security Algorithms Group of Experts (SAGE), a part of the European standards body ETSI.[2] Because of schedule pressures in 3GPP standardization, instead of developing a new cipher, SAGE agreed with 3GPP technical specification group (TSG) for system aspects of 3G security (SA3) to base the development on an existing algorithm that had already undergone some evaluation.[2] They chose the cipher algorithm MISTY1 developed[3] and patented[4] by Mitsubishi Electric Corporation. The original algorithm was slightly modified for easier hardware implementation and to meet other requirements set for 3G mobile communications security.
KASUMI is named after the original algorithm MISTY1 — 霞み (hiragana かすみ, romaji kasumi) is the Japanese word for "mist".
In January 2010, Orr Dunkelman, Nathan Keller and Adi Shamir released a paper showing that they could break Kasumi with a related-key attack and very modest computational resources; this attack is ineffective against MISTY1.[5]
1.2.1 Function FL
1.2.2 Function FO
1.2.3 Function FI
1.2.4 Substitution boxes
KASUMI algorithm is specified in a 3GPP technical specification.[6] KASUMI is a block cipher with 128-bit key and 64-bit input and output. The core of KASUMI is an eight-round Feistel network. The round functions in the main Feistel network are irreversible Feistel-like network transformations. In each round the round function uses a round key which consists of eight 16-bit sub keys derived from the original 128-bit key using a fixed key schedule.
The 128-bit key K is divided into eight 16-bit sub keys Ki:
{\displaystyle K=K_{1}\|K_{2}\|K_{3}\|K_{4}\|K_{5}\|K_{6}\|K_{7}\|K_{8}\,}
Additionally a modified key K', similarly divided into 16-bit sub keys K'i, is used. The modified key is derived from the original key by XORing with 0x123456789ABCDEFFEDCBA9876543210 (chosen as a "nothing up my sleeve" number).
Round keys are either derived from the sub keys by bitwise rotation to left by a given amount and from the modified sub keys (unchanged).
The round keys are as follows:
{\displaystyle {\begin{array}{lcl}KL_{i,1}&=&{\rm {ROL}}(K_{i},1)\\KL_{i,2}&=&K'_{i+2}\\KO_{i,1}&=&{\rm {ROL}}(K_{i+1},5)\\KO_{i,2}&=&{\rm {ROL}}(K_{i+5},8)\\KO_{i,3}&=&{\rm {ROL}}(K_{i+6},13)\\KI_{i,1}&=&K'_{i+4}\\KI_{i,2}&=&K'_{i+3}\\KI_{i,3}&=&K'_{i+7}\end{array}}}
Sub key index additions are cyclic so that if i+j is greater than 8 one has to subtract 8 from the result to get the actual sub key index.
KASUMI algorithm processes the 64-bit word in two 32-bit halves, left (
{\displaystyle L_{i}}
{\displaystyle R_{i}}
). The input word is concatenation of the left and right halves of the first round:
{\displaystyle {\rm {input}}=R_{0}\|L_{0}\,}
In each round the right half is XOR'ed with the output of the round function after which the halves are swapped:
{\displaystyle {\begin{array}{rcl}L_{i}&=&F_{i}(KL_{i},KO_{i},KI_{i},L_{i-1})\oplus R_{i-1}\\R_{i}&=&L_{i-1}\end{array}}}
where KLi, KOi, KIi are round keys for the ith round.
The round functions for even and odd rounds are slightly different. In each case the round function is a composition of two functions FLi and FOi. For an odd round
{\displaystyle F_{i}(K_{i},L_{i-1})=FO(KO_{i},KI_{i},FL(KL_{i},L_{i-1}))\,}
and for an even round
{\displaystyle F_{i}(K_{i},L_{i-1})=FL(KL_{i},FO(KO_{i},KI_{i},L_{i-1}))\,}
The output is the concatenation of the outputs of the last round.
{\displaystyle {\rm {output}}=R_{8}\|L_{8}\,}
Both FL and FO functions divide the 32-bit input data to two 16-bit halves. The FL function is an irreversible bit manipulation while the FO function is an irreversible three round Feistel-like network.
Function FL[edit]
The 32-bit input x of
{\displaystyle FL(KL_{i},x)}
is divided to two 16-bit halves
{\displaystyle x=l\|r}
. First the left half of the input
{\displaystyle l}
is ANDed bitwise with round key
{\displaystyle KL_{i,1}}
and rotated left by one bit. The result of that is XOR'ed to the right half of the input
{\displaystyle r}
to get the right half of the output
{\displaystyle r'}
{\displaystyle r'={\rm {ROL}}(l\wedge KL_{i,1},1)\oplus r}
Then the right half of the output
{\displaystyle r'}
is ORed bitwise with the round key
{\displaystyle KL_{i,2}}
and rotated left by one bit. The result of that is XOR'ed to the left half of the input
{\displaystyle l}
to get the left half of the output
{\displaystyle l'}
{\displaystyle l'={\rm {ROL}}(r'\vee KL_{i,2},1)\oplus l}
Output of the function is concatenation of the left and right halves
{\displaystyle x'=l'\|r'}
Function FO[edit]
{\displaystyle FO(KO_{i},KI_{i},x)}
is divided into two 16-bit halves
{\displaystyle x=l_{0}\|r_{0}}
, and passed through three rounds of a Feistel network.
In each of the three rounds (indexed by j that takes values 1, 2, and 3) the left half is modified to get the new right half and the right half is made the left half of the next round.
{\displaystyle {\begin{array}{lcl}r_{j}&=&FI(KI_{i,j},l_{j-1}\oplus KO_{i,j})\oplus r_{j-1}\\l_{j}&=&r_{j-1}\end{array}}}
The output of the function is
{\displaystyle x'=l_{3}\|r_{3}}
Function FI[edit]
The function FI is an irregular Feistel-like network.
The 16-bit input
{\displaystyle x}
{\displaystyle FI(Ki,x)}
is divided to two halves
{\displaystyle x=l_{0}\|r_{0}}
{\displaystyle l_{0}}
is 9 bits wide and
{\displaystyle r_{0}}
is 7 bits wide.
Bits in the left half
{\displaystyle l_{0}}
are first shuffled by 9-bit substitution box (S-box) S9 and the result is XOR'ed with the zero-extended right half
{\displaystyle r_{0}}
to get the new 9-bit right half
{\displaystyle r_{1}}
{\displaystyle r_{1}=S9(l_{0})\oplus (00\|r_{0})\,}
Bits of the right half
{\displaystyle r_{0}}
are shuffled by 7-bit S-box S7 and the result is XOR'ed with the seven least significant bits (LS7) of the new right half
{\displaystyle r_{1}}
to get the new 7-bit left half
{\displaystyle l_{1}}
{\displaystyle l_{1}=S7(r_{0})\oplus LS7(r_{1})\,}
The intermediate word
{\displaystyle x_{1}=l_{1}\|r_{1}}
is XORed with the round key KI to get
{\displaystyle x_{2}=l_{2}\|r_{2}}
{\displaystyle l_{2}}
{\displaystyle r_{2}}
{\displaystyle x_{2}=KI\oplus x_{1}}
Bits in the right half
{\displaystyle r_{2}}
are then shuffled by 9-bit S-box S9 and the result is XOR'ed with the zero-extended left half
{\displaystyle l_{2}}
to get the new 9-bit right half of the output
{\displaystyle r_{3}}
{\displaystyle r_{3}=S9(r_{2})\oplus (00\|l_{2})\,}
Finally the bits of the left half
{\displaystyle l_{2}}
are shuffled by 7-bit S-box S7 and the result is XOR'ed with the seven least significant bits (LS7) of the right half of the output
{\displaystyle r_{3}}
to get the 7-bit left half
{\displaystyle l_{3}}
of the output.
{\displaystyle l_{3}=S7(l_{2})\oplus LS7(r_{3})\,}
The output is the concatenation of the final left and right halves
{\displaystyle x'=l_{3}\|r_{3}}
Substitution boxes[edit]
The substitution boxes (S-boxes) S7 and S9 are defined by both bit-wise AND-XOR expressions and look-up tables in the specification. The bit-wise expressions are intended to hardware implementation but nowadays it is customary to use the look-up tables even in the HW design.
S7 is defined by the following array:
int S7[128] = {
54, 50, 62, 56, 22, 34, 94, 96, 38, 6, 63, 93, 2, 18,123, 33,
55,113, 39,114, 21, 67, 65, 12, 47, 73, 46, 27, 25,111,124, 81,
53, 9,121, 79, 52, 60, 58, 48,101,127, 40,120,104, 70, 71, 43,
20,122, 72, 61, 23,109, 13,100, 77, 1, 16, 7, 82, 10,105, 98,
117,116, 76, 11, 89,106, 0,125,118, 99, 86, 69, 30, 57,126, 87,
112, 51, 17, 5, 95, 14, 90, 84, 91, 8, 35,103, 32, 97, 28, 66,
102, 31, 26, 45, 75, 4, 85, 92, 37, 74, 80, 49, 68, 29,115, 44,
64,107,108, 24,110, 83, 36, 78, 42, 19, 15, 41, 88,119, 59, 3
167,239,161,379,391,334, 9,338, 38,226, 48,358,452,385, 90,397,
183,253,147,331,415,340, 51,362,306,500,262, 82,216,159,356,177,
175,241,489, 37,206, 17, 0,333, 44,254,378, 58,143,220, 81,400,
95, 3,315,245, 54,235,218,405,472,264,172,494,371,290,399, 76,
501,407,249,265, 89,186,221,428,164, 74,440,196,458,421,350,163,
344,300,276,242,437,320,113,278, 11,243, 87,317, 36, 93,496, 27,
487,446,482, 41, 68,156,457,131,326,403,339, 20, 39,115,442,124,
475,384,508, 53,112,170,479,151,126,169, 73,268,279,321,168,364,
363,292, 46,499,393,327,324, 24,456,267,157,460,488,426,309,229,
439,506,208,271,349,401,434,236, 16,209,359, 52, 56,120,199,277,
465,416,252,287,246, 6, 83,305,420,345,153,502, 65, 61,244,282,
173,222,418, 67,386,368,261,101,476,291,195,430, 49, 79,166,330,
132,225,203,316,234, 14,301, 91,503,286,424,211,347,307,140,374,
35,103,125,427, 19,214,453,146,498,314,444,230,256,329,198,285,
50,116, 78,410, 10,205,510,171,231, 45,139,467, 29, 86,505, 32,
72, 26,342,150,313,490,431,238,411,325,149,473, 40,119,174,355,
185,233,389, 71,448,273,372, 55,110,178,322, 12,469,392,369,190,
1,109,375,137,181, 88, 75,308,260,484, 98,272,370,275,412,111,
336,318, 4,504,492,259,304, 77,337,435, 21,357,303,332,483, 18,
414,486,394, 96, 99,154,511,148,413,361,409,255,162,215,302,201,
485,422,248,297, 23,213,130,466, 22,217,283, 70,294,360,419,127,
312,377, 7,468,194, 2,117,295,463,258,224,447,247,187, 80,398,
284,353,105,390,299,471,470,184, 57,200,348, 63,204,188, 33,451,
97, 30,310,219, 94,160,129,493, 64,179,263,102,189,207,114,402,
438,477,387,122,192, 42,381, 5,145,118,180,449,293,323,136,380,
43, 66, 60,455,341,445,202,432, 8,237, 15,376,436,464, 59,461
In 2001, an impossible differential attack on six rounds of KASUMI was presented by Kühn (2001).[7]
In 2003 Elad Barkan, Eli Biham and Nathan Keller demonstrated man-in-the-middle attacks against the GSM protocol which avoided the A5/3 cipher and thus breaking the protocol. This approach does not attack the A5/3 cipher, however.[8] The full version of their paper was published later in 2006.[9]
In 2005, Israeli researchers Eli Biham, Orr Dunkelman and Nathan Keller published a related-key rectangle (boomerang) attack on KASUMI that can break all 8 rounds faster than exhaustive search.[10] The attack requires 254.6 chosen plaintexts, each of which has been encrypted under one of four related keys, and has a time complexity equivalent to 276.1 KASUMI encryptions. While this is obviously not a practical attack, it invalidates some proofs about the security of the 3GPP protocols that had relied on the presumed strength of KASUMI.
In 2010, Dunkelman, Keller and Shamir published a new attack that allows an adversary to recover a full A5/3 key by related-key attack.[5] The time and space complexities of the attack are low enough that the authors carried out the attack in two hours on an Intel Core 2 Duo desktop computer even using the unoptimized reference KASUMI implementation. The authors note that this attack may not be applicable to the way A5/3 is used in 3G systems; their main purpose was to discredit 3GPP's assurances that their changes to MISTY wouldn't significantly impact the security of the algorithm.
A5/1 and A5/2
^ "Draft Report of SA3 #38" (PDF). 3GPP. 2005.
^ a b "General Report on the Design, Speification and Evaluation of 3GPP Standard Confidentiality and Integrity Algorithms" (PDF). 3GPP. 2009.
^ Matsui, Mitsuru; Tokita, Toshio (Dec 2000). "MISTY, KASUMI and Camellia Cipher Algorithm Development" (PDF). Mitsubishi Electric Advance. Mitsubishi Electric corp. 100: 2–8. ISSN 1345-3041. Archived from the original (PDF) on 2008-07-24. Retrieved 2010-01-06.
^ US 7096369, Matsui, Mitsuru & Tokita, Toshio, "Data Transformation Apparatus and Data Transformation Method", published Sep. 19, 2002, issued Aug. 22, 2006
^ a b Orr Dunkelman; Nathan Keller; Adi Shamir (2010-01-10). "A Practical-Time Attack on the A5/3 Cryptosystem Used in Third Generation GSM Telephony". {{cite journal}}: Cite journal requires |journal= (help)
^ "3GPP TS 35.202: Specification of the 3GPP confidentiality and integrity algorithms; Document 2: Kasumi specification". 3GPP. 2009.
^ Kühn, Ulrich. Cryptanalysis of Reduced Round MISTY. EUROCRYPT 2001. CiteSeerX 10.1.1.59.7609.
^ Elad Barkan, Eli Biham, Nathan Keller. Instant Ciphertext-Only Cryptanalysis of GSM Encrypted Communication (PDF). CRYPTO 2003. pp. 600–616. {{cite conference}}: CS1 maint: multiple names: authors list (link)
^ Elad Barkan, Eli Biham, Nathan Keller. "Instant Ciphertext-Only Cryptanalysis of GSM Encrypted Communication by Barkan and Biham of Technion (Full Version)" (PDF). {{cite web}}: CS1 maint: multiple names: authors list (link)
^ Eli Biham, Orr Dunkelman, Nathan Keller. A Related-Key Rectangle Attack on the Full KASUMI. ASIACRYPT 2005. pp. 443–461. Archived from the original (ps) on 2013-10-11. {{cite conference}}: CS1 maint: multiple names: authors list (link)
Nathan Keller's homepage
Retrieved from "https://en.wikipedia.org/w/index.php?title=KASUMI&oldid=1084899455"
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I was first introduced to similar types of computing through a YouTube video about Conway's Game of Life. Conway's Game of Life is actually turing-complete, which means in simple terms that it can simulate what a computer can do. In this case, gliders can represent bits, and the interaction between many gliders can different logic gates such as AND/OR/XOR/etc. For this post, I will be performing some binary additions with the help of many dominoes! I also recommend reading about some bitwise operators if you are not familiar with those.
As noted above, in order be turing-complete, we need to make sure we can construct proper logic gates with dominoes to execute certain bit operations. Let's go over the basic logic of addition below first.
Let's start with 11 + 01:
1 + 1 = 10
0 + 0 = 0
1 + 0 = 1
0 + 1 = 1
in binary. Perhaps you might have thought about using the XOR with these bits, and you would be correct! Since XOR only returns 1 when there's exactly one 1 bit, it is very fit for this task. However, we still need carry if there are two 1 bits. Thus, we can use the AND gate to represent whether we carry or not. Shown below is our current logic for single digit addition.
After creating the logic for the first digits, we can essentially chain these on for additional digits with minor adjustments. One thing we do need to watch out for when chaining is to take care of the carry sum from previous chains. To accommodate for this, we can use XOR again with the already XOR-ed sum, and OR (we can guarantee there aren't two 1 bits in the carry) the two carry digits together for the third digit. The diagram below will make way more sense.
Reading the 3 boxes at the bottom from right to left will get you the binary sum of the two binary numbers.
Logic with Dominoes
We have figured out the basic logic needed to add two 2-digit binary numbers, and we just have to build our domino contraption now!
One of the easiest gates we need is the OR gate. We need to make it such that if one stream of dominoes fall, then the output also falls. In this case, we have inputs/outputs, if it falls, then it is represented as 1, otherwise, 0. We can use this sort of arrangement shown below.
As you can see, if either side gets knocked down (1), the output will also be knocked over (0).
This gate is slightly more complicated than the OR gate. The basic idea is that we need to block off the dominoes that can knock over the output stream if there is one or less 1 bit. Otherwise, that path should stay open. Below is an arrangement that can be used for an AND gate.
It is important to note that the right side has more dominoes! This is especially important because we want to make sure there is time to block off the path if there is less than 2 1 bits.
This is the last gate we will use for our domino computer! It is slightly more complex to understand, but easier to build. We just need to terminate the stream if we have 2 1 bits; otherwise, we should let either one of the streams knock over the output. Below is an implementation for a XOR gate.
You can observe that if both inputs are knocked over, it will terminate. However, if only 1 input is knocked over, the dominoes will continue to knock over the output.
The three logic gates that are utilized in this computer have been covered! Now you can follow the full logic diagram in the above section to build your own! Make sure you keep track of the timing of the dominoes as many of the gates need a somewhat precise timing to function. More specifically, you might need to tinker with the length of connection dominoes between different gates. This will be somewhat tedious to fine tune and will require some time and many dominoes. If it does become hard, try to build a single digit computer and extend off of that! I hope you enjoyed this post, and happy dominoes!
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What Is Multivariable Calculus? | Outlier
This article is an introductory guide to multivariable calculus. We’ll discuss the meaning of multivariable calculus and how to solve basic multivariable differentiation problems. We’ll also provide a brief overview of different concepts in multivariable calculus and some real-world applications.
What Does Multivariable Mean?
How to Solve Multivariable Calculus
Basic vs. Advanced Multivariable Calculus
So far, our study of calculus has been limited to functions of a single variable. Multivariable calculus studies functions with two or more variables. Functions that take two or more input variables are called “multivariate.” These functions depend on two or more input variables to produce an output.
f(x, y) = x^2 + y
is a multivariate function.
How can we calculate derivatives in multivariable calculus? The derivative or rate of change in multivariable calculus is called the gradient. The gradient of a function
f
is computed by collecting the function’s partial derivatives into a vector. The gradient is one of the most fundamental differential operators in vector calculus. Vector calculus is an important component of multivariable calculus that is concerned with the study of vector fields.
The notation for the gradient vector is
\nabla f
. For a function with two variables, the gradient looks like this:
\nabla f(x, y) = \langle \frac{\partial{f(x, y)}}{\partial{x}}, \frac{\partial{f(x, y)}}{\partial{y}} \rangle
\frac{\partial{f(x, y)}}{\partial{x}}
is the partial derivative of
x
\frac{\partial{f(x, y)}}{\partial{x}}
x
as a variable and
y
as a constant, and then differentiating.
\frac{\partial{f(x, y)}}{\partial{y}}
is the partial derivative of
y
\frac{\partial{f(x, y)}}{\partial{y}}
y
x
Partial derivatives help us to understand the behavior of a multivariate function when one variable changes while the rest are held constant.
Let’s do one example together. We’ll calculate the gradient of
f(x, y) = x^2 + y
(1, 2)
. The plot of
f(x, y) = x^2 + y
can be seen below.
First, we’ll calculate
\frac{\partial{f(x, y)}}{\partial{x}}
, the partial derivative of
x
. Treating
y
as a constant and differentiating normally, we find that
\frac{\partial{f(x, y)}}{\partial{x}} = 2x
. Remember that the derivative of any constant is zero.
Next, we’ll calculate
\frac{\partial{f(x, y)}}{\partial{y}}
, the partial derivative of
y
x
\frac{\partial{f(x, y)}}{\partial{y}} = 1
Collecting these components into a vector, we find that
\nabla f(x, y) = \langle 2x, 1 \rangle
Plugging in our point (1, 2), we find that
\nabla f(1, 2) = \langle 2, 1 \rangle
. Thus, the gradient of
at (1, 2) is
\langle 2, 1 \rangle
What insight does the gradient give us? Recall that the gradient is a vector. Vectors have both a magnitude and a direction. The gradient of a function
f
points in the direction of greatest increase of
. The direction of greatest increase can also be thought of as the direction of steepest ascent, the direction of the greatest rate of change, and the direction where
f
increases the fastest.
Both basic and advanced multivariable calculus concepts will involve the study of multivariable integration and multivariable differentiation. What are some of the most basic concepts in basic multivariable calculus?
Below is a list of 4 fundamental multivariable calculus concepts.
Partial derivatives show us how a multivariate function behaves when we change just one variable while the rest are held constant. In multivariable calculus, partial derivatives are used to calculate the gradient.
2. Gradient Vectors
\nabla f(x, y)
is the collection of partial derivatives of
. The gradient points in the direction of the greatest increase of
. In the same vein, the opposite direction
-\nabla f(x, y)
is the direction of the greatest decrease of
Recall that the definite integral
\int_a ^b f(x)
is used to calculate the area under a curve. Similarly, the double integral
\int \int\limits_R f(x, y)dy dx
may be used to calculate the volume under a surface, where
R
is the region of integration.
Any study of double integrals will cover Fubini’s Theorem, which states that we can evaluate double integrals using iterated integrals and that the order of integration does not matter in its computation. We can visualize iterated integrals below in Fubini’s Theorem.
\int \int\limits_R f(x, y)dy dx = \int_a ^b \int_c ^d f(x, y) dy dx = \int_c ^d \int_a ^b f(x, y) dx dy
(You can also review our beginner’s guide to integrals if needed).
4. Relative Maxima and Minima
Relative maxima and minima are “peaks” on a multivariate function’s graph. At a relative maximum, moving in any direction will decrease the value of the function. At a relative minimum, moving in any direction will increase the value of the function. You can visualize relative maxima and minima in multivariable calculus as stalagmites and stalactites in a cave. These points are called extrema, and their gradients are 0.
Critical points are points on a function’s graph where the derivative is either zero or undefined. Not all critical points of a multivariate function are relative extrema, but the relative extrema of a multivariate function are always critical points. Much of calculus is concerned with finding and classifying critical points.
What kind of advanced concepts can you expect to cover in a multivariable calculus course?
Below is a list of 6 advanced multivariable calculus concepts.
Second-Order Partial Derivatives and Higher-Order Partial Derivatives
To compute second-order partial derivatives, we calculate the partial derivatives of the partial derivatives of a function. These higher-order derivatives can be used to check the concavity and extrema of a multivariate function.
While double integrals integrate over a two-dimensional region, triple integrals integrate over a three-dimensional region.
The multivariable chain rule is derived from the chain rule in single-variable calculus. Suppose that we have two multivariable functions,
x = g(u, v)
y = h(u, v)
z = f(x, y) = f(g(u, v)), h(u, v))
Then the multivariable chain rule states that the partial derivatives of
f(g(u, v)), h(u, v))
(u,v)
\frac{\partial{z}}{\partial{u}} = \frac{\partial{z}}{\partial{x}} \frac{\partial{x}}{\partial{u}} + \frac{\partial{z}}{\partial{y}}\frac{\partial{y}}{\partial{u}}
\frac{\partial{z}}{\partial{v}} = \frac{\partial{z}}{\partial{x}} \frac{\partial{x}}{\partial{v}} + \frac{\partial{z}}{\partial{y}}\frac{\partial{y}}{\partial{v}}
Four Critical Theorems
The following four theorems are some of the most important theorems in multivariable calculus:
All four theorems are concerned with multivariable integration. While they require more context than is appropriate for this brief overview, they are exciting theorems to look forward to in your study of multivariable calculus.
The Jacobian matrix is the matrix of all the first-order partial derivatives of a function. When the Jacobian matrix is square, meaning that it has the same number of rows and columns, then its determinant is called the Jacobian determinant. The Jacobian determinant at a given point provides very valuable information about the function’s behavior and invertibility near that point.
More Vector Calculus and Matrices
Many multivariable calculus or Calculus 3 courses include a vector calculus component.
Vector calculus is a subdivision of calculus underneath the broader umbrella category of multivariable calculus and involves:
Parametric functions and surfaces
Many phenomena require more than one input variable to construct a sufficient mathematical model. Because of this, multivariable calculus is useful in many disciplines. Here are four examples of real-world applications of multivariable calculus.
Multivariable calculus is useful in business and finance. For example, a company’s profit depends on multiple independent variables, such as pricing, sales, the cost of materials, and overhead costs. Multivariable calculus is necessary to calculate profit maximization.
Multivariable calculus is essential to electrical engineering. Maxwell’s equations, a set of four partial differential equations, make up the backbone of electromagnetism. These equations, which explain the behavior of the different frequency ranges in the electromagnetic spectrum, encouraged the development of vector calculus in physics. Multivariable calculus is fundamental to understanding the behavior of electric fields and optimizing new electric technologies.
We use multivariable calculus in graphics, programming, animation, and game development to model 3D worlds. For example, multivariable calculus is handy for modeling the movement of different curves and surfaces and developing lighting models.
Most algorithms in machine learning depend on multiple variables. Many algorithms themselves rely on multivariable calculus concepts. For example, an algorithm called gradient descent is an optimization algorithm used to minimize a cost function.
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(Redirected from Working fluids)
For fluid power, a working fluid is a gas or liquid that primarily transfers force, motion, or mechanical energy. In hydraulics, water or hydraulic fluid transfers force between hydraulic components such as hydraulic pumps, hydraulic cylinders, and hydraulic motors that are assembled into hydraulic machinery, hydraulic drive systems, etc. In pneumatics, the working fluid is air or another gas which transfers force between pneumatic components such as compressors, vacuum pumps, pneumatic cylinders, and pneumatic motors. In pneumatic systems, the working gas also stores energy because it is compressible. (Gases also heat up as they are compressed and cool as they expand; this incidental heat pump is rarely exploited.) (Some gases also condense into liquids as they are compressed and boil as pressure is reduced.)
For passive heat transfer, a working fluid is a gas or liquid, usually called a coolant or heat transfer fluid, that primarily transfers heat into or out of a region of interest by conduction, convection, and/or forced convection (pumped liquid cooling, air cooling, etc.).
The working fluid of a heat engine or heat pump is a gas or liquid, usually called a refrigerant, coolant, or working gas, that primarily converts thermal energy (temperature change) into mechanical energy (or vice versa) by phase change and/or heat of compression and expansion. Examples using phase change include water↔steam in steam engines, and chlorofluorocarbons in most vapor-compression refrigeration and air conditioning systems. Examples without phase change include air or hydrogen in hot air engines such as the Stirling engine, air or gases in gas-cycle heat pumps, etc. (Some heat pumps and heat engines use "working solids", such as rubber bands, for elastocaloric refrigeration or thermoelastic cooling and nickel titanium in a prototype heat engine.)
Working fluids other than air or water are necessarily recirculated in a loop. Some hydraulic and passive heat-transfer systems are open to the water supply and/or atmosphere, sometimes through breather filters. Heat engines, heat pumps, and systems using volatile liquids or special gases are usually sealed behind relief valves.
Properties and statesEdit
This section reads like a textbook. Please improve this article to make it neutral in tone and meet Wikipedia's quality standards. (May 2010)
The working fluid's properties are essential for the full description of thermodynamic systems. Although working fluids have many physical properties which can be defined, the thermodynamic properties which are often required in engineering design and analysis are few. Pressure, temperature, enthalpy, entropy, specific volume, and internal energy are the most common.
Pressure–volume diagram showing state (p,V)
The working fluid can be used to output useful work if used in a turbine. Also, in thermodynamic cycles energy may be input to the working fluid by means of a compressor. The mathematical formulation for this may be quite simple if we consider a cylinder in which a working fluid resides. A piston is used to input useful work to the fluid. From mechanics, the work done from state 1 to state 2 of the process is given by:
{\displaystyle W=-\int _{1}^{2}\mathbf {F} \cdot \mathrm {d} \mathbf {s} }
where ds is the incremental distance from one state to the next and F is the force applied. The negative sign is introduced since in this case a decrease in volume is being considered. The situation is shown in the following figure:
Work input on a working fluid by means of a cylinder–piston arrangement
{\displaystyle {\begin{aligned}W&=-\int _{1}^{2}PA\cdot \mathrm {d} \mathbf {s} \\&=-\int _{1}^{2}P\cdot \mathrm {d} V\end{aligned}}}
Where A⋅ds = dV is the elemental change of cylinder volume. If from state 1 to 2 the volume increases then the working fluid actually does work on its surroundings and this is commonly denoted by a negative work. If the volume decreases the work is positive. By the definition given with the above integral the work done is represented by the area under a pressure–volume diagram. If we consider the case where we have a constant pressure process then the work is simply given by
{\displaystyle {\begin{aligned}W&=-P\int _{1}^{2}\mathrm {d} V\\&=-P\cdot \left(V_{2}-V_{1}\right)\end{aligned}}}
Constant pressure process on a p–V diagram
Main article: Working fluid selection
Rankine cycles Water↔steam, pentane, toluene
Eastop & McConkey (1993). Applied Thermodynamics for Engineering Technologists (5th ed.). Singapore: Prentice Hall. pp. 9–12. ISBN 0-582-09193-4.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Working_fluid&oldid=1087578081"
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Salami and More Deli sells a
6
-foot sandwich for parties. It weighs
8
pounds. Assume the weight per foot is constant.
How much does a sandwich
0
feet long weigh?
0
Draw a graph showing the weight of the sandwich (vertical axis) compared to the length of the sandwich (horizontal axis). Label the axes with appropriate units.
Graph starting with the point where
x=0
Your graph should show a line with positive slope.
Units on the vertical axis should be in pounds.
Units on horizontal axis should be in feet.
Use your graph to estimate the weight of a
1
-foot sandwich.
y
x=1
Write a proportion to find the length of a
12
-pound sandwich.
Compare sandwich length and height.
\frac{6}{8}=\frac{\it x}{12}
9
Use the eTool below to help you solve the problem.
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Cheers. The fact that the immoral literal interpretation is held to be true by a great number of your fellow-travellers, despite your sophistic brilliance, remains.
There is no easy answer at the ground level. — Ennui Elucidator
Yet this is the level that matters.
People typically know little more about their religion than they do their government or political party - they are just engaged in tribalistic behavior
I refuse to just let religious people get away with this.
But no matter how you feel about Christians, stop dictating what religion is, was, or can be. Especially stop questioning the legitimacy of someone's religion because it doesn't comport with your understanding of bad religions.
Oh, I don't think they are "bad" religions. It seems more likely they are accurate, evolutionarily advantageous religions.
If Tom can't handle the religious bullying in his otherwise nominally secular workplace, and quits his job and accepts a worse, less payed one, then this is a victory for the religious. Who laugh at him.
Religion will long outlive us both, maybe we should be fostering better religion (however you understand that) and not just kicking it.
You're mistaking me for someone else. I believe religions are generally evolutionarily advantageous, in that they provide justifications for the Darwinian struggle for survival.
Your pointing out plain (and I mean screamingly obvious) absurdities in the Bible, as if believers could not have seen them as absurdities had it not been for your helpful guidance, must be missing something, unless you truly are baffled as to why such a large segment of the population could be so very blind to the obvious.
The best source I can cite to you for the position I'm arguing is The Case for God, by Karen Armstrong, which I've begun reading recently, whose position seems very much aligned with what I've been arguing.
From a review of her book at: https://religiondispatches.org/religion-is-not-about-belief-karen-armstrongs-ithe-case-for-godi/
“Until well into the modern period,” Armstrong contends, “Jews and Christians both insisted that it was neither possible nor desirable to read the Bible literally, that it gives us no single, orthodox message and demands constant reinterpretation.” Myths were symbolic, often therapeutic, teaching stories and were never understood literally or historically. But that all changed with the advent of modernity. Precipitated by the rediscovery of Aristotle and the rise of scholasticism in the late middle ages, rational systematization took center stage, preparing the way for a modern period that would welcome both humanistic individualism and the eventual triumph of reason and science." — Hanover
This discussion has been hopelessly hampered by political correctness.
The fact that the immoral literal interpretation — Banno
It's still not clear why you consider it immoral.
Is the Darwinian struggle for survival immoral? If you don't think it is, then based on what can you consider the literal interpretation of the Bible immoral??
It's still not clear why you consider it immoral. — baker
Why not? It's in the original article, in the one mentioned above and I've presented it several times myself, as well as even Hanover's repeating it in agreement just above.
Are you having trouble with your comprehension?
https://www.academia.edu/35166134/David_Lewis_on_Divine_Evil?email_work_card=title
A good God (like a good human) would ensure that a
person’s punishment should be
to their crime. 2.
Since humans only live for a finite number of years (and can commit only a finite number of evils during this time), they can commit only a finite amount of evil. 3.
Hell involves an
amount of punishment, and would not be appropriate for
crime. 4.
So, an all-good God would never sentence people to Hell. If
the “God” of Christianity and Islam does this
, then it is a highly immoral being that does not deserve to be worshipped.
(Or, God simply doesn’t exist).
I'm not going to reformat it for you.
The fact that the immoral literal interpretation is held to be true by a great number of your fellow-travellers, despite your sophistic brilliance, remains. — Banno
I don't travel with the literalists. They're your kin.
But sure, to the extent there are those advocating throwing stones at little girls' heads, I stand opposed. Such a radical position for a theist, I know.
Since humans only live for a finite number of years (and can commit only a finite number of evils during this time), they can commit only a finite amount of evil.
Is that how that works? We count how many evil acts you’ve committed? More is worser?
God could have done a better job letting people know about it (for example, God could have given Hitler, Stalin, etc. a few more hints on what would happen if they continued on their evil path.)
1. It would be appallingly unfair of God to allow Hitler and Stalin to experience eternal damnation (in any of the several forms contemplated, including annihilation).
2. At most they should get a lot of damnation, but not an infinite amount.
3. Honestly, they probably shouldn’t even get that, because how could they possibly know — really know — there would be a price to pay in the afterlife.
4. The whole system was rigged against Hitler (and Stalin!) from the beginning.
5. Guy that would set up a system like this, basically to entrap Hitler (and Stalin!), that’s not a good guy.
6. Anyone who thinks it’s okay to treat Hitler (and Stalin!) so shabbily, is also morally suspect.
If there is a god then David Lewis should qualify for eternal damnation for writing such an asinine article.
You think the bible's bad? you should try Aesop's fables.
Childish pranks punished by being eaten by wolves, Grasshopper attention deficit punished by being cast out of society and left to freeze and starve. You won't believe how immoral this collection is.
↪Joshs
David Lewis said "Let there be shite" and then there was this thread.
There are enough people out there that assent to neverending damnation, not just in the US.
Anyone that doesn't - kudos - good for you. :up:
On occasion someone asks you
Why not accept the free gift of salvation?
only to threaten with the above if someone puts them on the spot.
Call them fringe or cult if you like, or distance yourself from them.
Has little bearing on Lewis' point.
If anyone wants to witch-hunt the assenters, then they're no better.
At some point someone ought be/come better, and yaay some have. :up:
+\infty
(God) and you get
-\infty
. No?
Is God a mathematician? — Mario Livio
That someone's moral character can be judged on this basis is questionable. It is evident that not all Christians, or more generally, all who worship a monotheistic God do not all have the same moral character.
But the assumption that the Bible represents a single, unchanging, universal God is simply wrong. We cannot begin in the beginning. The stories in the Bible are not ex nihilo. They were told and retold in various ways, by various authors from various cultures with various beliefs and values. They are more representative of those authors than of some single entity that informs their stories.
In addition there is a history of interpretation, often quite contentious.
Actually, the majority of modern translations of the Bible are flawed, at least in English and French. A literal translation of the Greek word 'aion', with respect to the passages concerning damnation, would be more appropriately rendered as 'age' and not 'eternity'. It is quite a severe error that drastically alters the meaning of the passages where it appears.
Therefore, for those who are aware of the original Greek meaning the concept of punishment is finite. Although admittedly, most Christians do indeed believe that hell is eternal.
If you have some free time and the inclination watch this video for further detail.
emancipate OptionsShare
6. Anyone who thinks it’s okay to treat Hitler (and Stalin!) so shabbily, is also morally suspect. — Srap Tasmaner
What the OP @Banno and D. Lewis are forgetting is that in no major monotheistic religion is killing, raping, and pillaging an automatic disqualifier from getting into heaven eternal (!!!). It just isn't.
You can kill, rape, and pillage and still get to heaven just fine.
Now how's that for "divine evil"!
In actual monotheistic religions, what is said to be the cause for eternal damnation is the act of rejecting god. Different monotheistic religions specify different criteria for what exactly counts as a rejection of god, but they do agree on this one point.
This way, for example, a person who has lived a pious, harmless life (one without killing, raping, and pillaging) but has a change of heart on their deathbed and rejects god, goes to hell, forever doomed to suffering, while even serial killers can go to heaven and enjoy eternal happiness in heaven as long as they repent and accept god.
Rejecting god is an infinite offense, an infinite evil.
O, the infinitely evil Buddhists; all destined for eternal damnation! :rofl:
in no major monotheistic religion is killing, raping, and pillaging an automatic disqualifier from getting into heaven eternal (!!!). It just isn't.
You can kill, rape, and pillage and still get to heaven just fine. — baker
That's an accurate assessment and probably well covered by the likes of Hitchens and Dawkins.
That is correct. Contrary to popular belief, we'll all make it to paradise, even me!
That is correct. Contrary to popular belief, we'll all make it to paradise, even me! — Olivier5
Yep but not without a period of punishment. This is the universalist position and there is even Biblical evidence for this. It's pretty uncommon among Christians but I have met a few people who believe this.
That's an accurate assessment and probably well covered by the likes of Hitchens and Dawkins. — Tom Storm
Hitchens and Dawkins covered this but they presented it as though a murderer could simply repent on their deathbed and get instantly into heaven. That's a strawman of Christianity that conveniently leaves out the idea of purgatory.
we'll all make it to paradise, even me!
— Olivier5
Yep but not without a period of punishment. — emancipate
Not according to St Polnareff.
↪Olivier5
Il n'est pas un saint, lui.
His song saved me when I was a kid, though.
Hitchens and Dawkins covered this but they presented it as though a murderer could simply repent on their deathbed and get instantly into heaven. That's a strawman of Christianity that conveniently leaves out the idea of purgatory. — emancipate
Further, it's a strawman that leaves out that someone who has lived their life killing, raping, and pillaging isn't likely to repent on their deathbed.
Anyway, the point is that in monotheistic religions, killing, raping, and pillaging isn't the kind of automatic disqualifier from living a good life (forever) they way it is in a humanistic outlook.
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Home : Support : Online Help : Mathematics : Optimization : Simplex Linear Optimization : convexhull
find convex hull enclosing the given points
convexhull(P)
convexhull(P, output=area)
convexhull(P, output=hull)
convexhull(P, output=plot)
list or set of points in 2 dimensions
The command convexhull(P) returns a list of points defining the convex hull of the points in P in counter-clockwise order.
Every point in P must be a list of an x and y coordinate, both of which must be numbers (of type numeric).
If the option output=area is specified, the area of the convex hull of the points is returned.
If the option output=hull is specified, the convex hull is is returned as a list of points. This is the default return value.
If the option output=plot is specified, a plot of the points and the convex hull is returned.
Multiple outputs can be specified, for example, output=[hull,area] returns a sequence of two values.
n\mathrm{log}\left(n\right)
algorithm computing tangents of pairs of points is used.
For a slightly more geometrical interpretation of the same command, see geometry[convexhull]. The points input and output are geometric points, and the output is considered to be a convex polygon.
The command with(simplex,convexhull) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{simplex}\right):
P≔{[0,0],[1,0],[1,1],[1,\frac{1}{2}],[2,0],[\frac{1}{2},\frac{1}{2}]}
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{≔}{[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\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}}]}
\mathrm{convexhull}\left(P\right)
[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]
\mathrm{convexhull}\left(P,\mathrm{output}=\mathrm{area}\right)
\textcolor[rgb]{0,0,1}{1}
H,A≔\mathrm{convexhull}\left(P,\mathrm{output}=[\mathrm{hull},\mathrm{area}]\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[[\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}
Execute the following to get a plot of the points and the convex hull.
\mathrm{convexhull}\left(P,\mathrm{output}=\mathrm{plot}\right)
This creates 50 points uniformly distributed at random on [0,1).
R≔\mathrm{rand}\left({10}^{10}\right):
U01 := proc() Float( R(), -10 ) end proc:
P≔{\mathrm{seq}\left([\mathrm{U01}\left(\right),\mathrm{U01}\left(\right)],i=1..50\right)}:
\mathrm{convexhull}\left(P,\mathrm{output}=\mathrm{area}\right)
\textcolor[rgb]{0,0,1}{0.7570693924}
\mathrm{convexhull}\left(P,\mathrm{output}=\mathrm{plot}\right)
|
Troubleshooting Model Estimation - MATLAB & Simulink - MathWorks Switzerland
About Troubleshooting Models
Model Order Is Too High or Too Low
Substantial Noise in the System
Unstable Models
Unstable Linear Model
Unstable Nonlinear Models
When an Unstable Model Is OK
Missing Input Variables
Nonlinearity Estimator Produces a Poor Fit
During validation, models can exhibit undesirable characteristics or a poor fit to the validation data.
Use the tips in these sections to help improve your model performance. Some features, such as low signal-to-noise ratio, varying system properties, or nonstationary disturbances, can produce data for which a good model fit is not possible.
A poor fit in the Model Output plot can be the result of an incorrect model order. System identification is largely a trial-and-error process when selecting model structure and model order. Ideally, you want the lowest-order model that adequately captures the system dynamics. High-order models are more expensive to compute and result in greater parameter uncertainty.
Start by estimating the model order as described in Preliminary Step – Estimating Model Orders and Input Delays. Use the suggested order as a starting point to estimate the lowest possible order with different model structures. After each estimation, monitor the Model Output and Residual Analysis plots, and then adjust your settings for the next estimation.
When a low-order model fits the validation data poorly, estimate a higher-order model to see if the fit improves. For example, if the Model Output plot shows that a fourth-order model gives poor results, estimate an eighth-order model. When a higher-order model improves the fit, you can conclude that higher-order linear models are potentially sufficient for your application.
Use an independent data set to validate your models. If you use the same data set for both estimation and validation, the fit always improves as you increase the model order and you risk overfitting. However, if you use an independent data set to validate your model, the fit eventually deteriorates if the model orders are too high.
Substantial noise in your system can result in a poor model fit. The presence of such noise is indicated when:
A state-space model produces a better fit than an ARX model. While a state-space structure has sufficient flexibility to model noise, an ARX structure is unable to independently model noise and system dynamics. The following ARX model equation shows that A couples the dynamics and the noise terms by appearing in the denominator of both:
y=\frac{B}{A}u+\frac{1}{A}e
A residual analysis plot shows significant autocorrelation of residuals at nonzero lags. For more information about residual analysis, see the topics on the Residual Analysis page.
To model noise more carefully, use either an ARMAX or the Box-Jenkins model structure, both of which model the noise and dynamics terms using different polynomials.
You can test whether a linear model is unstable is by examining the pole-zero plot of the model, which is described in Pole and Zero Plots. The stability threshold for pole values differs for discrete-time and continuous-time models, as follows:
For stable continuous-time models, the real part of the pole is less than 0.
For stable discrete-time models, the magnitude of the pole is less than 1.
Linear trends in estimation data can cause the identified linear models to be unstable. However, detrending the model does not guarantee stability.
If your model is unstable, but you believe that your system is stable, you can.
Force stability during estimation — Set the Focus estimation option to a value that guarantees a stable model. This setting can result in reduced model quality.
Allow for some instability — Set the stability threshold advanced estimation option to allow for a margin of error:
For continuous-time models, set the value of Advanced.StabilityThreshold.s. The model is considered stable if the pole on the far right is to the left of s.
For discrete-time models, set the value of Advanced.StabilityThreshold.z. The model is considered stable if all of the poles are inside a circle with a radius of z that is centered at the origin.
For more information about Focus and Advanced.StabilityThreshold, see the various commands for creating estimation option sets, such as tfestOptions, ssestOptions, and procestOptions.
To test if a nonlinear model is unstable, plot the simulated model output on top of the validation data. If the simulated output diverges from measured output, the model is unstable. However, agreement between model output and measured output does not guarantee stability.
In some cases, an unstable model is still useful. For example, if your system is unstable without a controller, you can use your model for control design. In this case, you can import the unstable model into Simulink® or Control System Toolbox™ products.
If modeling noise and trying different model structures and orders still results in a poor fit, try adding more inputs that can affect the output. Inputs do not need to be control signals. Any measurable signal can be considered an input, including measurable disturbances.
Include additional measured signals in your input data, and estimate the model again.
If a linear model shows a poor fit to the validation data, consider whether nonlinear effects are present in the system.
You can model the nonlinearities by performing a simple transformation on the input signals to make the problem linear in the new variables. For example, in a heating process with electrical power as the driving stimulus, you can multiply voltage and current measurements to create a power input signal.
If your problem is sufficiently complex and you do not have physical insight into the system, try fitting nonlinear black-box models to your data, see About Identified Nonlinear Models.
For nonlinear ARX and Hammerstein-Wiener models, the Model Output plot does not show a good fit when the nonlinearity estimator has incorrect complexity.
Specify the complexity of piece-wise-linear, wavelet, sigmoid, and custom networks using the NumberOfUnits nonlinear estimator property. A higher number of units indicates a more complex nonlinearity estimator. When using neural networks, specify the complexity using the parameters of the network object. For more information, see the Deep Learning Toolbox™ documentation.
To select the appropriate nonlinearity estimator complexity, first validate the output of a low-complexity model. Next, increase the model complexity and validate the output again. The model fit degrades when the nonlinearity estimator becomes too complex. This degradation in performance is only visible if you use independent estimation and validation data sets
|
Megan went walking for
2
hours and walked at a steady pace of
3
How far did she walk?
Make a careful graph of the situation with the horizontal axis having units of hours and the vertical axis having units of miles per hour. Shade the area under the curve. You should get a rectangle for the area under the curve. (The area “under” a curve means the area between the curve and the
x
Divide the rectangle into six smaller rectangles by drawing vertical lines at the
1
hour and
2
hour marks, and horizontal lines at the
1
mph and
2
mph marks.
Explain why each of the smaller rectangles that you created in part (c) represents a distance of
1
mile that Megan walked. Hint: What are the units of the sides of the rectangles?
2\text{ hours}\cdot\frac{3\text{ miles}}{\text{hour}}=?
|
Gibbs - Monte-Carlo
Robert and Casella (2013) gives the following algorithm,
while Liu, Jun S. (2008) introduces two types of Gibbs sampling strategy.
Bivariate Gibbs sampler
It is easy to implement this sampler:
## Julia program for Bivariate Gibbs sampler
## author: weiya <[email protected]>
function bigibbs(T, rho)
x = ones(T+1)
y = ones(T+1)
x[t+1] = randn() * sqrt(1-rho^2) + rho*y[t]
y[t+1] = randn() * sqrt(1-rho^2) + rho*x[t+1]
bigibbs(100, 0.5)
Completion Gibbs Sampler
We can use the following Julia program to implement this algorithm.
## Julia program for Truncated normal distribution
# Truncated normal distribution
function rtrunormal(T, mu, sigma, mu_down)
x = ones(T)
z = ones(T+1)
# set initial value of z
z[1] = rand()
if mu < mu_down
z[1] = z[1] * exp(-0.5 * (mu - mu_down)^2 / sigma^2)
x[t] = rand() * (mu - mu_down + sqrt(-2*sigma^2*log(z[t]))) + mu_down
z[t+1] = rand() * exp(-(x[t] - mu)^2/(2*sigma^2))
rtrunormal(1000, 1.0, 1.0, 1.2)
Robert and Casella (2013) introduces the following slice sampler algorithm,
and Liu, Jun S. (2008) also presents the slice sampler with slightly different expression:
In my opinion, we can illustrate this algorithm with one dimensioanl case. Suppose we want to sample from normal distribution (or uniform distribution), we can sample uniformly from the region encolsed by the coordinate axis and the density function, that is a bell shape (or a square).
Consider the normal distribution as an instance.
It is also easy to write the following Julia program.
## Julia program for Slice sampler
function rnorm_slice(T)
w = ones(T+1)
w[t+1] = rand() * exp(-1.0 * x[t]^2/2)
x[t+1] = rand() * 2 * sqrt(-2*log(w[t+1])) - sqrt(-2*log(w[t+1]))
rnorm_slice(100)
A special case of Completion Gibbs Sampler.
Let's illustrate the scheme with grouped counting data.
And we can obtain the following algorithm,
But it seems to be not obvious to derive the above algorithm, so I wrote some more details
Liu, Jun S. (2008) also presents the DA algorithm which based on Bayesian missing data problem.
m
\mathbf y_{mis}
in each iteration is not really necessary. And briefly summary the DA algorithm:
It seems to agree with the algorithm presented by Robert and Casella (2013).
It seems that we do not need to derive the explicit form of
g(x, z)
, if we can directly obtain the conditional distribution. We can use the following Julia program to sample.
## Julia program for Grouped Multinomial Data (Ex. 7.2.3)
# call gamma function
#using SpecialFunctions
# sample from Dirichlet distributions
function gmulti(T, x, a, b, alpha1 = 0.5, alpha2 = 0.5, alpha3 = 0.5)
z = ones(T+1, size(x, 1)-1) # initial z satisfy `z <= x`
mu = ones(T+1)
eta = ones(T+1)
# sample from g_1(theta | y)
dir = Dirichlet([z[t, 1] + z[t, 2] + alpha1, z[t, 3] + z[t, 4] + alpha2, x[5] + alpha3])
sample = rand(dir, 1)
mu[t+1] = sample[1]
eta[t+1] = sample[2]
# sample from g_2(z | x, theta)
bi = Binomial(x[i], a[i]*mu[t+1]/(a[i]*mu[t+1]+b[i]))
z[t+1, i] = rand(bi, 1)[1]
bi = Binomial(x[i], a[i]*eta[t+1]/(a[i]*eta[t+1]+b[i]))
return mu, eta
a = [0.06, 0.14, 0.11, 0.09];
b = [0.17, 0.24, 0.19, 0.20];
x = [9, 15, 12, 7, 8];
gmulti(100, x, a, b)
Reversible Data Augmentation
Reversible Gibbs Sampler
Random Sweep Gibbs Sampler
Random Gibbs Sampler
Hybrid Gibbs Samplers
Metropolization of the Gibbs Sampler
Let us illuatrate this algorithm with the following example.
|
Password Protection - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 2017 : Password Protection
Distribute Password Protected Content
...While Still Allowing Execution
To open this worksheet as a workbook, click here.
In a workbook, you can password-protect worksheets, documents and application but still
pass parameters into the protected content and run its code
and get results out.
If you were to open the workbook version of this help page, as suggested in the first paragraph, you would see that the workbook contains a protected worksheet. In the Navigator panel, you can see two worksheets.
Main.mw is the worksheet you're reading right now
Ideal Brayton Cycle.mw is the protected worksheet
If you double click on Ideal Brayton Cycle.mw, Maple asks you for a password.
However, you can send parameters into this worksheet and get results out with RunWorksheet.
If you want to view Ideal Brayton Cycle.mw, you need to supply a password (the password is "maple").
\mathrm{DocumentTools}:-\mathrm{RunWorksheet}\left("this:///Ideal Brayton Cycle.mw",\left[\mathrm{Tmin}=300⟦\mathrm{degC}⟧,\mathrm{Tmax}=1300⟦\mathrm{degC}⟧,\mathrm{Pmin}=101⟦\mathrm{kPa}⟧,\mathrm{r}=8\right]\right)
\textcolor[rgb]{0,0,1}{0.4086652304}
|
Estimate default probability using time-series version of Merton model - MATLAB mertonByTimeSeries - MathWorks India
mertonByTimeSeries
Compute Probability of Default Using the Time-Series Approach to the Merton Model
Compute Probability of Default Using the Time-Series Approach to the Merton Model With Drift
Merton Model for Time Series
Estimate default probability using time-series version of Merton model
[PD,DD,A,Sa] = mertonByTimeSeries(Equity,Liability,Rate)
[PD,DD,A,Sa] = mertonByTimeSeries(___,Name,Value)
[PD,DD,A,Sa] = mertonByTimeSeries(Equity,Liability,Rate) estimates the default probability of a firm by using the Merton model.
[PD,DD,A,Sa] = mertonByTimeSeries(___,Name,Value) adds optional name-value pair arguments.
Compute the default probability by using the time-series approach of Merton's model.
[PD,DD,A,Sa] = mertonByTimeSeries(Equity,Liability,Rate);
plot(Dates,PD)
Compute the plot for the default probability values by using the time-series approach of Merton's model. You compute the PD0 (blue line) by using the default values. You compute the PD1 (red line) by specifying an optional Drift value.
PD0 = mertonByTimeSeries(Equity,Liability,Rate);
PD1 = mertonByTimeSeries(Equity,Liability,Rate,'Drift',0.10);
plot(Dates, PD0, Dates, PD1)
Equity — Market value of firm's equity
Market value of the firm's equity, specified as a positive value.
Liability — Liability threshold of firm
Liability threshold of the firm, specified as a positive value. The liability threshold is often referred to as the default point.
Rate — Annualized risk-free interest rate
Annualized risk-free interest rate, specified as a numeric value.
Example: [PD,DD,A,Sa] = mertonByTimeSeries(Equity,Liability,Rate,'Maturity',4,'Drift',0.22,'Tolerance',1e-5,'NumPeriods',12)
Maturity — Time to maturity corresponding to liability threshold
1 year (default) | positive numeric value
Time to maturity corresponding to the liability threshold, specified as the comma-separated pair consisting of 'Maturity' and a positive value.
Drift — Annualized drift rate
risk-free interest rate defined in Rate (default) | numeric value
Annualized drift rate, expected rate of return of the firm's assets, specified as the comma-separated pair consisting of 'Drift' and a numeric value.
NumPeriods — Number of periods per year
250 trading days per year (default) | positive integer
Number of periods per year, specified as the comma-separated pair consisting of 'NumPeriods' and a positive integer. Typical values are 250 (yearly), 12 (monthly), or 4 (quarterly).
Tolerance — Tolerance for convergence of the solver
Tolerance for convergence of the solver, specified as the comma-separated pair consisting of 'Tolerance' and a positive scalar value.
MaxIterations — Maximum number of iterations allowed
Maximum number of iterations allowed, specified as the comma-separated pair consisting of 'MaxIterations' and a positive integer.
PD — Probability of default of firm at maturity
Probability of default of the firm at maturity, returned as a numeric.
DD — Distance-to-default
Distance-to-default, defined as the number of standard deviations between the mean of the asset distribution at maturity and the liability threshold (default point), returned as a numeric.
A — Value of firm's assets
Value of firm's assets, returned as a numeric value.
Sa — Annualized firm's asset volatility
Annualized firm's asset volatility, returned as a numeric value.
In the Merton model, the value of a company's equity is treated as a call option on its assets, and the liability is taken as a strike price.
Given a time series of observed equity values and liability thresholds for a company, mertonByTimeSeries calibrates corresponding asset values, the volatility of the assets in the sample's time span, and computes the probability of default for each observation. Unlike mertonmodel, no equity volatility input is required for the time-series version of the Merton model. You compute the probability of default and distance-to-default by using the formulae in Algorithms.
Given the time series for equity (E), liability (L), risk-free interest rate (r), asset drift (μA), and maturity (T), mertonByTimeSeries sets up the following system of nonlinear equations and solves for a time series asset values (A), and a single asset volatility (σA). At each time period t, where t = 1...n:
\begin{array}{l}{A}_{1}=\left(\frac{{E}_{1}+{L}_{1}{e}^{-{r}_{1}{T}_{1}}N\left({d}_{2}\right)}{N\left({d}_{1}\right)}\right)\\ {A}_{t}=\left(\frac{{E}_{t}+{L}_{t}{e}^{-{r}_{t}{T}_{t}}N\left({d}_{2}\right)}{N\left({d}_{1}\right)}\right)\\ ...\\ {A}_{n}=\left(\frac{{E}_{n}+{L}_{n}{e}^{-{r}_{n}{T}_{n}}N\left({d}_{2}\right)}{N\left({d}_{1}\right)}\right)\end{array}
where N is the cumulative normal distribution. To simplify the notation, the time subscript is omitted for d1 and d2. At each time period, d1, and d2 are defined as:
{d}_{1}=\frac{\mathrm{ln}\left(\frac{A}{L}\right)+\left(r+0.5{\sigma }_{A}^{2}\right)T}{{\sigma }_{A}\sqrt{T}}
{d}_{2}={d}_{1}-{\sigma }_{A}\sqrt{T}
The formulae for the distance-to-default (DD) and default probability (PD) at each time period are:
DD=\frac{\mathrm{ln}\left(\frac{A}{L}\right)+\left({\mu }_{A}-0.5{\sigma }_{A}^{2}\right)T}{{\sigma }_{A}\sqrt{T}}
PD=1-N\left(DD\right)
[1] Zielinski, T. Merton's and KMV Models In Credit Risk Management.
[2] Loeffler, G. and Posch, P.N. Credit Risk Modeling Using Excel and VBA. Wiley Finance, 2011.
[3] Kim, I.J., Byun, S.J, Hwang, S.Y. An Iterative Method for Implementing Merton.
[4] Merton, R. C. “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance. Vol. 29. pp. 449 – 470.
mertonmodel | asrf
|
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