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An electronics store kept track of the number of headsets sold during one week to determine how many headsets they should have in stock each day. Here is their data:
15
headsets sold on Monday,
23
on Tuesday,
15
on Wednesday,
20
on Thursday,
42
on Friday,
58
on Saturday, and
48
on Sunday.
Is there a difference between the sale of headsets on the weekdays (Monday through Friday) and the sale of headsets on the weekend days (Saturday and Sunday)? Explain your thinking.
Separate the data into weekday data and weekend data. This will help you in making the comparison.
Calculate the mean of sales on weekdays and the mean of sales on the weekend and show how the difference in means confirms your thinking in part (a).
Remember that the mean, in this case, is the number of headsets sold each day, if they sold the same
number each day. You need to find two means, one for the weekdays' sales and one for the weekend's sales.
Refer to problem 1-112 if you need additional help finding the mean.
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Step response plot of dynamic system; step response data - MATLAB step
\mathrm{sys}\left(\mathit{s}\right)=\frac{4}{{\mathit{s}}^{2}+2\mathit{s}+10}.
\begin{array}{l}\left[\begin{array}{c}{\underset{}{\overset{˙}{x}}}_{1}\\ {\underset{}{\overset{˙}{x}}}_{2}\end{array}\right]=\left[\begin{array}{cc}-0.5572& -0.7814\\ 0.7814& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]+\left[\begin{array}{cc}1& -1\\ 0& 2\end{array}\right]\left[\begin{array}{c}{u}_{1}\\ {u}_{2}\end{array}\right]\\ y=\left[\begin{array}{cc}1.9691& 6.4493\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]\end{array}
\sigma
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Refer to the graphs you made for problem 3-43 (it was a homework problem in Lesson 3.1.2). Use those graphs to help you graph each of the following inequalities.
y \leq | x |
Which region is shaded, above or below?
| y | \geq x
Which region of the graph is shaded, left or to the right?
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Combinatorial $k$-systoles on a punctured torus and a pair of pants
k
ElHadji Abdou Aziz Diop1; Masseye Gaye1; Abdoul Karim Sane1
1 Departement of Mathematics, Université Cheikh Anta Diop, Dakar, Senegal
In this paper
S
denotes a surface homeomorphic to a punctured torus or a pair of pants. Our interest is the study of combinatorial
k
-systoles, that is closed curves with self-intersection numbers greater than
k
and with least combinatorial length. We show that the maximal intersection number
{I}_{k}^{c}
of combinatorial
k
-systoles of
S
k
\underset{k\to +\infty }{lim sup}\left({I}_{k}^{c}-k\right)=+\infty
This result, in case of a pair of pants and a punctured torus, is a positive response to the combinatorial version of the Erlandsson-Parlier conjecture, originally formulated for the geometric length.
Classification: 32G15, 30F40
Keywords: closed geodesics, self-intersection,
k
-systole
ElHadji Abdou Aziz Diop 1; Masseye Gaye 1; Abdoul Karim Sane 1
author = {ElHadji Abdou Aziz Diop and Masseye Gaye and Abdoul Karim Sane},
title = {Combinatorial $k$-systoles on a punctured torus and a pair of pants},
TI - Combinatorial $k$-systoles on a punctured torus and a pair of pants
%T Combinatorial $k$-systoles on a punctured torus and a pair of pants
ElHadji Abdou Aziz Diop; Masseye Gaye; Abdoul Karim Sane. Combinatorial $k$-systoles on a punctured torus and a pair of pants. Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 29-38. doi : 10.5802/cml.76. https://cml.centre-mersenne.org/articles/10.5802/cml.76/
[1] Ara Basmajian Universal length bounds for non-simple closed geodesics on hyperbolic surfaces, J. Topol., Volume 6 (2013) no. 2, pp. 513-524 | Article | MR: 3065183
[2] Alan F. Beardon The geometry of discrete groups, Graduate Texts in Mathematics, 91, Springer-Verlag, New York, 1995, xii+337 pages (Corrected reprint of the 1983 original) | MR: 1393195
[3] Joan S. Birman; Caroline Series An algorithm for simple curves on surfaces, J. London Math. Soc. (2), Volume 29 (1984) no. 2, pp. 331-342 | Article | MR: 744104
[4] Rufus Bowen; Caroline Series Markov maps associated with Fuchsian groups, Inst. Hautes Études Sci. Publ. Math. (1979) no. 50, pp. 153-170 | MR: 556585
[5] Peter Buser Geometry and spectra of compact Riemann surfaces, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2010, xvi+454 pages (Reprint of the 1992 edition) | Article | MR: 2742784
[6] Moira Chas; Curtis T. McMullen; Anthony Phillips Almost simple geodesics on the triply-punctured sphere, Math. Z., Volume 291 (2019) no. 3-4, pp. 1175-1196 | Article | MR: 3936103
[7] Moira Chas; Anthony Phillips Self-intersection numbers of curves on the punctured torus, Experimental Mathematics, Volume 19 (2010) no. 2, pp. 129-148
[8] Moira Chas; Anthony Phillips Self-intersection numbers of curves in the doubly punctured plane, Exp. Math., Volume 21 (2012) no. 1, pp. 26-37 | Article | MR: 2904905
[9] Françoise Dal’Bo Geodesic and horocyclic trajectories, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011, xii+176 pages (Translated from the 2007 French original) | Article | MR: 2766419
[10] Pierre de la Harpe Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000, vi+310 pages | MR: 1786869
[11] ElHadji Abdou Aziz Diop; Masseye Gaye Self intersections on a pair of pants, arXiv: 2108.06796
[12] Viveka Erlandsson; Hugo Parlier Short closed geodesics with self-intersections, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 169 (2020) no. 3, pp. 623-638
[13] Thi Hanh Vo Short closed geodesics on cusped hyperbolic surfaces, arXiv:1912.09967
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Simulate approximate solution of diagonal-drift Merton jump diffusion process - MATLAB simBySolution - MathWorks España
Use simBySolution with merton Object
Quasi-Monte Carlo Simulation Using Merton Model
NNTrials
Simulate approximate solution of diagonal-drift Merton jump diffusion process
[Paths,Times,Z,N] = simBySolution(MDL,NPeriods)
[Paths,Times,Z,N] = simBySolution(___,Name,Value)
[Paths,Times,Z,N] = simBySolution(MDL,NPeriods) simulates NNTrials sample paths of NVars correlated state variables driven by NBrowns Brownian motion sources of risk and NJumps compound Poisson processes representing the arrivals of important events over NPeriods consecutive observation periods. The simulation approximates continuous-time Merton jump diffusion process by an approximation of the closed-form solution.
[Paths,Times,Z,N] = simBySolution(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.
Simulate the approximate solution of diagonal-drift Merton process.
Create a merton object.
mertonObj = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol,...
'startstat',AssetPrice)
mertonObj =
Class MERTON: Merton Jump Diffusion
StartState: 80
JumpFreq: 2
Use simBySolution to simulate NTrials sample paths of NVARS correlated state variables driven by NBrowns Brownian motion sources of risk and NJumps compound Poisson processes representing the arrivals of important events over NPeriods consecutive observation periods. The function approximates continuous-time Merton jump diffusion process by an approximation of the closed-form solution.
[Paths,Times,Z,N] = simBySolution(mertonObj, nPeriods,'nTrials', 3)
Paths(:,:,1) =
N(:,:,1) =
This example shows how to use simBySolution with a Merton model to perform a quasi-Monte Carlo simulation. Quasi-Monte Carlo simulation is a Monte Carlo simulation that uses quasi-random sequences instead pseudo random numbers.
Merton = merton(Return,Sigma,JumpFreq,JumpMean,JumpVol,'startstat',AssetPrice)
Merton =
[paths,time,z,n] = simBySolution(Merton, 10,'ntrials',4096,'montecarlomethod','quasi','QuasiSequence','sobol');
MDL — Merton model
merton object
Merton model, specified as a merton object. You can create a merton object using merton.
Example: [Paths,Times,Z,N] = simBySolution(merton,NPeriods,'DeltaTimes',dt,'NNTrials',10)
NNTrials — Simulated NTrials (sample paths)
Simulated NTrials (sample paths) of NPeriods observations each, specified as the comma-separated pair consisting of 'NNTrials' and a positive scalar integer.
Positive time increments between observations, specified as the comma-separated pair consisting of 'DeltaTimes' and a scalar or an NPeriods-by-1 column vector.
NSteps — Number of intermediate time steps within each time increment
Odd NTrials (1,3,5,...) correspond to the primary Gaussian paths.
Even NTrials (2,4,6,...) are the matching antithetic paths of each pair derived by negating the Gaussian draws of the corresponding primary (odd) trial.
If you specify an input noise process (see Z and N), simBySolution ignores the value of MonteCarloMethod.
Z — Direct specification of the dependent random noise process for generating Brownian motion vector
Direct specification of the dependent random noise process for generating the Brownian motion vector (Wiener process) that drives the simulation, specified as the comma-separated pair consisting of 'Z' and a function or an (NPeriods * NSteps)-by-NBrowns-by-NNTrials three-dimensional array of dependent random variates.
The input argument Z allows you to directly specify the noise generation process. This process takes precedence over the Correlation parameter of the input merton object and the value of the Antithetic input flag.
Specifically, when Z is specified, Correlation is not explicitly used to generate the Gaussian variates that drive the Brownian motion. However, Correlation is still used in the expression that appears in the exponential term of the log[Xt] Euler scheme. Thus, you must specify Z as a correlated Gaussian noise process whose correlation structure is consistently captured by Correlation.
random numbers from Poisson distribution with merton object parameter JumpFreq (default) | three-dimensional array | function
Dependent random counting process for generating the number of jumps, specified as the comma-separated pair consisting of 'N' and a function or an (NPeriods ⨉ NSteps) -by-NJumps-by-NNTrials three-dimensional array of dependent random variates. If you specify a function, N must return an NJumps-by-1 column vector, and you must call it with two inputs: a real-valued scalar observation time t followed by an NVars-by-1 state vector Xt.
If StorePaths is false (logical 0), simBySolution returns Paths as an empty matrix.
{X}_{t}=P\left(t,{X}_{t}\right)
simBySolution applies processing functions at the end of each observation period. These functions must accept the current observation time t and the current state vector Xt, and return a state vector that can be an adjustment to the input state.
Simulated paths of correlated state variables, returned as an (NPeriods + 1)-by-NVars-by-NNTrials three-dimensional time-series array.
For a given trial, each row of Paths is the transpose of the state vector Xt at time t. When StorePaths is set to false, simBySolution returns Paths as an empty matrix.
Z — Dependent random variates for generating the Brownian motion vector
Dependent random variates for generating the Brownian motion vector (Wiener processes) that drive the simulation, returned as a (NPeriods * NSteps)-by-NBrowns-by-NNTrials three-dimensional time-series array.
N — Dependent random variates for generating the jump counting process vector
Dependent random variates for generating the jump counting process vector, returned as an (NPeriods ⨉ NSteps)-by-NJumps-by-NNTrials three-dimensional time-series array.
This technique attempts to replace one sequence of random observations with another that has the same expected value but a smaller variance. In a typical Monte Carlo simulation, each sample path is independent and represents an independent trial. However, antithetic sampling generates sample paths in pairs. The first path of the pair is referred to as the primary path, and the second as the antithetic path. Any given pair is independent other pairs, but the two paths within each pair are highly correlated. Antithetic sampling literature often recommends averaging the discounted payoffs of each pair, effectively halving the number of Monte Carlo NTrials.
The simBySolution function simulates the state vector Xt by an approximation of the closed-form solution of diagonal drift Merton jump diffusion models. Specifically, it applies a Euler approach to the transformed log[Xt] process (using Ito's formula). In general, this is not the exact solution to the Merton jump diffusion model because the probability distributions of the simulated and true state vectors are identical only for piecewise constant parameters.
This function simulates any vector-valued merton process of the form
d{X}_{t}=B\left(t,{X}_{t}\right){X}_{t}dt+D\left(t,{X}_{t}\right)V\left(t,{x}_{t}\right)d{W}_{t}+Y\left(t,{X}_{t},{N}_{t}\right){X}_{t}d{N}_{t}
B(t,Xt) is an NVars-by-NVars matrix of generalized expected instantaneous rates of return.
D(t,Xt) is an NVars-by-NVars diagonal matrix in which each element along the main diagonal is the corresponding element of the state vector.
V(t,Xt) is an NVars-by-NVars matrix of instantaneous volatility rates.
Y(t,Xt,Nt) is an NVars-by-NJumps matrix-valued jump size function.
dNt is an NJumps-by-1 counting process vector.
simByEuler | merton
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Lua Programming/Print version - Wikibooks, open books for an open world
Lua Programming/Print version
{\displaystyle (5)_{10}=(0101)_{2}}
{\displaystyle (3)_{10}=(0011)_{2}}
{\displaystyle (1)_{10}=(0001)_{2}}
{\displaystyle (5)_{10}=(0101)_{2}}
{\displaystyle (3)_{10}=(0011)_{2}}
{\displaystyle (7)_{10}=(0111)_{2}}
{\displaystyle (5)_{10}=(0101)_{2}}
{\displaystyle (3)_{10}=(0011)_{2}}
{\displaystyle (6)_{10}=(0110)_{2}}
{\displaystyle (7)_{10}=(0111)_{2}}
{\displaystyle (8)_{10}=(1000)_{2}}
Appendix:Software testing[edit | edit source]
The term software testing refers to a number of methods and processes that are used to discover bugs and programming mistakes in computer software. Software testing can be done statically, in which case in is called static testing and is done without executing the computer software, or dynamically, in which case it is called dynamic testing and is done while the computer program that is being tested is running.
Type checking[edit | edit source]
In programming languages, a type system is a collection of rules that assign a property called a type to the various constructs—such as variables, expressions, functions or modules—a computer program is composed of. The main purpose of a type system is to reduce bugs in computer programs by defining interfaces between different parts of a computer program, and then checking that the parts have been connected in a consistent way. This checking can happen statically (at compile time), dynamically (at run time), or it can happen as a combination of static and dynamic checking. Type systems have other purposes as well, such as enabling certain compiler optimizations, allowing for multiple dispatch, providing a form of documentation, etc.
—Wikipedia, Type system
Type-checking can be done, as the extract from Wikipedia brilliantly said, at run time or at compile time. If it is done at compile time, the compiler, when compiling source code, will verify the type safety of the program and guarantee that the program satisfies certain type safety properties—generally, static type-checkers will simply verify that variables always have values of the same type and that arguments passed to functions will have the right type.
The static approach allows bugs to be discovered early in the development cycle. The dynamic approach, in contrast, consists in verifying that the program follows the type constraints when it is running. While this means that dynamic type-checkers should be able to verify more constraints, most dynamically typed languages do not have many type constraints. Lua is a dynamically typed language: in Lua, values have types, but variables do not. This means that the value of a variable can be a number at some point of the program’s execution and be a string at another point.
Lua’s type system is very simple in comparison with most other languages. It performs type checking when operators are used (attempting to add two values of which at least one is not a number and cannot be coerced to one, for example, will raise a type error) and when functions of the standard libraries are called (functions of the standard library reject arguments that do not have the right type and raise an appropriate error).
Since Lua does not have functionality for specifying a type for function parameters, the type function can be useful to verify that arguments passed to functions are of the appropriate type. This is most useful for functions that will be passed arguments provided by users while a program is running (for example, in an interactive environment for calling predefined Lua functions), since adding code for type checking to functions makes them more verbose and adds maintenance overhead.
White-box testing[edit | edit source]
The term white-box testing refers to the practice of using knowledge of the internal workings of software to create test cases to verify its functionality. It is relevant at three levels of software testing, but the one most interesting for Lua programs is the unit level, since Lua programs are usually part of a bigger application where the integration and system testing would take place.
There are multiple frameworks available for unit testing in Lua. Testing at the unit level is most appropriate for libraries, since it generally consists in writing test cases that pass specific arguments to functions and provide a warning when a function returns an unexpected value. This requires writing test cases for new functionality, but has the benefit of making errors introduced in code easier to notice when they modify the behavior of functions in a way that makes the tests not pass anymore.
There are multiple unit testing frameworks for Lua. One of them, busted, supports the standard Lua virtual machine as well as LuaJIT, and can also be used with MoonScript and Terra, the former a language that compiles to Lua and the latter a low-level language that is interoperable with Lua. Another unit testing framework for Lua, Luaunit, is written entirely in Lua and has no dependencies. Shake is a simpler test framework, initially part of the Kepler Project, that uses the assert and print functions but is no longer actively developed.
The lua-users wiki, an excellent resource to find information about Lua, provides the following material that is related to software testing. Some of these pages consist in links to other pages or to projects that can be useful for various tasks.
Lua Type Checking
Debugging Lua Code
{\displaystyle \scriptstyle a\veebar b}
{\displaystyle \scriptstyle a\veebar b}
{\displaystyle \scriptstyle a\veebar b}
{\displaystyle \scriptstyle a\land b}
{\displaystyle \scriptstyle a\land b}
{\displaystyle \scriptstyle a\lor b}
{\displaystyle \scriptstyle a\lor b}
Retrieved from "https://en.wikibooks.org/w/index.php?title=Lua_Programming/Print_version&oldid=3102386"
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Home : Support : Online Help : Connectivity : Web Features : XMLTools : IsTree
determine if an expression is an XML tree data structure
IsTree(expr)
The IsTree(expr) command tests whether a Maple expression expr is an XML tree data structure. If expr is an XML tree data structure, the value true is returned. Otherwise, false is returned.
Note: A full recursive traversal of the input is performed, so this is an extremely expensive test.
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{IsTree}\left(\mathrm{XMLElement}\left("a",["b"="c"],"d",\mathrm{XMLElement}\left("foo",["colour"="red"],"bar"\right)\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Profit maximization - Wikipedia
2 Total revenue–total cost perspective
3 Marginal revenue–marginal cost perspective
4 Case in which maximizing revenue is equivalent
5 Maximizing profits in the real world
6 Changes in total costs and profit maximization
7 Markup pricing
8 Marginal product of labor, marginal revenue product of labor, and profit maximization
9 Sub-optimal Profit maximization
{\displaystyle {\frac {\operatorname {d} ^{2}R}{{\operatorname {d} Q}^{2}}}<{\frac {\operatorname {d} ^{2}C}{{\operatorname {d} Q}^{2}}}.}
{\displaystyle {\begin{aligned}{\text{MR = }}&{\frac {\Delta {\text{TR}}}{\Delta {\text{Q}}}}\\{\text{=}}&{\frac {{\text{(P}}\Delta {\text{Q+Q}}\Delta {\text{P)}}}{\Delta {\text{Q}}}}\\=&{\text{P+}}{\frac {{\text{Q}}\Delta {\text{P}}}{\Delta {\text{Q}}}}\\\end{aligned}}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Profit_maximization&oldid=1082931076"
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Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties
2016 Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties
We define a class of noncompact Fano toric manifolds which we call admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed point sets.
We prove closed-string mirror symmetry for this class of manifolds: the Jacobian ring of the superpotential is the symplectic cohomology (not the quantum cohomology). Moreover,
S\phantom{\rule{0.3em}{0ex}}{H}^{\ast }\left(M\right)
Q\phantom{\rule{0.3em}{0ex}}{H}^{\ast }\left(M\right)
by localizing at the toric divisors. We give explicit presentations of
S\phantom{\rule{0.3em}{0ex}}{H}^{\ast }\left(M\right)
Q\phantom{\rule{0.3em}{0ex}}{H}^{\ast }\left(M\right)
, using ideas of Batyrev, McDuff and Tolman.
Assuming that the superpotential is Morse (or a milder semisimplicity assumption), we prove that the wrapped Fukaya category for this class of manifolds satisfies the toric generation criterion, ie is split-generated by the natural Lagrangian torus fibers of the moment map taken with suitable holonomies. In particular, the wrapped category is compactly generated and cohomologically finite.
We prove a generic generation theorem: a generic deformation of the monotone toric symplectic form defines a local system for which the twisted wrapped Fukaya category satisfies the toric generation criterion. This theorem, together with a limiting argument about continuity of eigenspaces, are used to prove the untwisted generation results.
We prove that for any closed Fano toric manifold, and a generic local system, the twisted Fukaya category satisfies the toric generation criterion. If the superpotential is Morse (or assuming semisimplicity), also the untwisted Fukaya category satisfies the criterion.
The key ingredients are nonvanishing results for the open-closed string map, using tools from the paper by Ritter and Smith; we also prove a conjecture from that paper that any monotone toric negative line bundle contains a nondisplaceable monotone Lagrangian torus. The above presentation results require foundational work: we extend the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in
S\phantom{\rule{0.3em}{0ex}}{H}^{\ast }\left(M\right)
S\phantom{\rule{0.3em}{0ex}}{H}^{\ast }\left(M\right)
is notoriously hard and there are very few known examples beyond the cases of cotangent bundles and subcritical Stein manifolds. So this computation is significant in itself, as well as being the key ingredient in proving the above results in homological mirror symmetry.
Alexander Ritter. "Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties." Geom. Topol. 20 (4) 1941 - 2052, 2016. https://doi.org/10.2140/gt.2016.20.1941
Primary: 53D05 , 53D20 , 53D35 , 53D37 , 57R17
Secondary: 14J33 , 14N35
Keywords: Fano , Floer cohomology , Floer homology , Fukaya category , generating , generation , generator , Jacobian ring , Lagrangian submanifold , quantum cohomology , symplectic cohomology , symplectic cohomology , symplectic geometry , symplectic topology , toric variety
Alexander Ritter "Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties," Geometry & Topology, Geom. Topol. 20(4), 1941-2052, (2016)
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Effects of Moisture Content and Plasticity Index on Duncan-Chang Model Parameters of Hydraulic Fill Soft Soil
Institute of Civil Engineering, University of Hebei, Baoding, China.
In order to explore the effects of moisture content and plasticity index on Duncan-Chang model parameters K,n, C and Rf, we selected 8 groups of soft soil with water content of 69.1% - 94.3% and plasticity index of 32.2 - 54.1 for triaxial unconsolidated undrained shear test. The results show that Cuu, K and n values all showed a downward trend, and Rf variation was not obvious with the increase of moisture content. The variation rule of each parameter is not obvious with the increase of plasticity index. When moisture content is constant, Cuu and n values do not change much, K increases with the increase of plasticity index within the range of 70% - 80% moisture content, and does not change much with the increase of plasticity index when moisture content is greater than 80%, Rf has no obvious rule. When the plasticity index is constant, Cuu, Kand n decrease with the increase of moisture content, Rf has no obvious rule. The maximum value of Cuu is 20.18 kPa, the minimum is 3.72 kPa, and the maximum to minimum ratio is 5.42. The maximum value of K is 0.517, the minimum is 0.022, and the maximum to minimum ratio is 23.5. The maximum value of n is 1.198, the minimum is 0.150, and the maximum to minimum ratio is 7.99. The maximum value of Rf is 0.872, the minimum is 0.679, and the maximum to minimum ratio is 1.28.
Moisture Content, Plasticity Index, Duncan-Chang Model, Unconsolidated Undrained Test
Chen, E. , Yan, M. , Ding, J. , Gao, C. and Gan, Y. (2019) Effects of Moisture Content and Plasticity Index on Duncan-Chang Model Parameters of Hydraulic Fill Soft Soil. World Journal of Engineering and Technology, 7, 408-417. doi: 10.4236/wjet.2019.73030.
The constitutive relation of soil is the rule to reveal the internal mechanical properties of soil. In recent years, soft soil foundation has been paid more and more attention. Especially in the coastal areas, the hydraulic filling of soft foundation is more and more widely distributed. Compared with other general soft soil and clay, hydraulic fill soft soil is a kind of artificial soft soil, characterized by high moisture content and low strength. After it is used as foundation, it will often lead to the destruction of buildings due to the excessive settlement and insufficient bearing capacity in the later stage. Therefore, it is very important to study the stress-strain relation, or constitutive relation, of soft soil.
Duncan-Chang model is a nonlinear elastic model of soil, with simple parameters and clear concept, which can be obtained by conventional triaxial shear test, and can well reflect the stress-strain law of soil. Many scholars have studied the effects of basic physical indexes on the parameters of Duncan-Chang model. Liu Xiaowen et al. [1] studied the stress-strain curves and other mechanical properties of laterite with different compactness, and obtained the Duncan-Chang model parameters. Chen Wei et al. [2] studied the stress-strain curves and other mechanical properties of compacted loess under different basic indexes through triaxial test, and obtained the Duncan-Chang model parameters. Liao Hongjian et al. [3] obtained the stress-strain curves of remolded saturated loess through triaxial test and analyzed the influencing factors of Duncan-Chang model parameters. Wu Nengsen et al. [4] studied the influence of moisture content on Duncan-Chang model parameters of granite residual soil. Yang Xuehui [5] studied the Duncan-Chang model of unsaturated remolded loess under different conditions, and gave the intensity index of unsaturated remolded loess and the change rule of Duncan-Chang model parameters with the initial conditions. Zdravkovic and Jardin [6] discussed the anisotropy of the rotation of the middle principal stress direction caused by consolidation, and pointed out that it had obvious influence on the strength and deformation of soil, and then affected the parameters of Duncan-Chang model. Cokca et al. [7] studied the effects of moisture content on the cohesion and internal friction angle of Duncan-Chang model of unsaturated clay. Cheng Ying et al. [8] analyzed the triaxial test results of marine and fluvial soft soils by using Duncan-Chang model, and gave the change rules of model parameters and dry density. Based on the triaxial test results, Wang Shuai [9] established Duncan-Chang model of coastal soft soil with different initial degree of consolidation.
It can be seen from the above results that there are few studies on the influence of moisture content and plasticity index on Duncan-Chang model parameters of soft soil. Therefore, this experiment mainly studies the influence of moisture content and plasticity index on the parameters of Duncan-Chang model.
Firstly, the basic physical indexes of selected soil samples were determined. Then, Duncan-Chang model parameters were obtained by triaxial unconsolidated and undrained test. Finally, the relationship between water content and plastic limit on Duncan-Chang model parameters was analyzed.
The test soil samples were taken from a blow fill site located in tidal flats and shallow sea areas. The landform is mainly alluvial plain, and the landform unit is delta, which belongs to the tidal landform type of estuary, sand mouth and sand island. The main strata from top to bottom include: backfill sand, silt (blowing), coarse sand (blowing), fine sand (blowing), silt—silt soil (blowing), clay—silty clay, silt soil—clay. The sampling area is mainly located in the silt distribution area, and the sampling depth is 5 - 15 m below the surface. Related physical indicators are shown in Table 1.
Plasticity is the characteristic of clay, which reflects the degree of interaction between clay and water. Figure 1 shows the plasticity diagram of the test soil sample. As can be seen from Figure 1, the soil samples are near line A, and all
Table 1. Physical and mechanical properties of soil samples.
Figure 1. Distribution diagram of soil moisture content and plasticity index.
are high-plastic soils.
3. Introduction to Duncan-Chang Model
Kondner pointed out in 1963 that the stress-strain curve of soil triaxial test can be fitted by hyperbola:
{\sigma }_{1}-{\sigma }_{3}=\frac{{\epsilon }_{1}}{a+b{\epsilon }_{1}}
\left({\sigma }_{1}-{\sigma }_{3}\right)
is deviator stress.
{\epsilon }_{1}
is the axial strain.
a,b
are the test constant, and a is the inverse of the initial tangent deformation modulus
{E}_{i}
. b is the inverse of the limiting deviant stress
{\left({\sigma }_{1}-{\sigma }_{3}\right)}_{u}
. Among them:
{E}_{i}=K{P}_{a}{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{n}
{\left({\sigma }_{1}-{\sigma }_{3}\right)}_{u}=\frac{2c\text{cos}\phi +2{\sigma }_{3}\text{sin}\phi }{{R}_{f}\left(1-\text{sin}\phi \right)}
{R}_{f}=\frac{{\left({\sigma }_{1}-{\sigma }_{3}\right)}_{f}}{{\left({\sigma }_{1}-{\sigma }_{3}\right)}_{u}}
{P}_{a}
is atmospheric pressure, with
{P}_{a}
value of 101.4 kPa, and the dimension is the same as
{\sigma }_{3}
{R}_{f}
is the damage ratio;
c,\phi
is the cohesion and internal friction Angle of the soil sample;
K,n
are the test constant;
{\left({\sigma }_{1}-{\sigma }_{3}\right)}_{f}
is the strength of the soil.
According to Equations (1)-(3), the tangent deformation modulus of Duncan-Chang model can be expressed as:
{E}_{t}=K{P}_{a}{\left(\frac{{\sigma }_{3}}{{P}_{a}}\right)}^{n}{\left[1-{R}_{f}\frac{\left(1-\text{sin}\phi \right)\left({\sigma }_{1}-{\sigma }_{3}\right)}{2c\text{cos}\phi +2{\sigma }_{3}\text{sin}\phi }\right]}^{2}
This experiment mainly studies five parameters of Duncan-Chang model:
K,n,c,\phi
{R}_{f}
The effects of moisture content and plasticity index on the parameters of Duncan-Chang model was studied by triaxial unconsolidated undrained shear test. The model parameters of the whole group are obtained by taking the experimental data of the third group of soil samples as an example. The relationship between deviator stress difference and axial strain of soil samples in group 3 is shown in Figure 2.
{C}_{uu},{\phi }_{uu}
{C}_{uu},{\phi }_{uu}
are generally obtained by drawing Mohr circle of stress under different confining pressures. As the soil sample is nearly saturated and does not drain during the shear process, the force between shear planes is assumed by the excess pore water pressure, and the friction between the soil particles does not exist, so
{\phi }_{uu}=0
. See Table 2 for details.
{R}_{f}
{R}_{f}
is obtained by drawing
{\epsilon }_{1}/\left({\sigma }_{1}-{\sigma }_{3}\right)~{\epsilon }_{1}
relation curves under different confining pressures, as shown in Figure 3.
{\epsilon }_{1}/\left({\sigma }_{1}-{\sigma }_{3}\right)~{\epsilon }_{1}
curves are fitted to obtain intercept a and slope b. Combining Equations (3) and (4),
{E}_{i}
{R}_{f}
are further obtained.
{R}_{f}
is averaged, as shown in Table 2.
\left({\sigma }_{1}-{\sigma }_{3}\right)~{\epsilon }_{1}
relation curve.
{\epsilon }_{1}/\left({\sigma }_{1}-{\sigma }_{3}\right)~{\epsilon }_{1}
4.3. K,n
through mapping the
\mathrm{lg}\left({\sigma }_{3}/{P}_{a}\right)~\mathrm{lg}\left({E}_{i}/{P}_{a}\right)
curve to obtain, as shown in Figure 4.
It can be seen from Table 2 that parameters Cuu, K and n of Duncan-Chang model of soft soil fill have a large variation range, while Rf has a small variation range. The average value of Rf is 0.773, and the coefficient of variation is 0.08. From Figures 5-7, it can be seen that Cuu, K and n basically show a downward trend with the increase of moisture content, and there is no obvious rule with the change of plasticity index.
Figures 8-10 are the cloud diagram of Duncan-Chang model parameters Cuu, K and n changing with water content and plasticity index. Figure 8 shows that when the moisture content is constant, the Cuu does not change much. When the plasticity index is constant, Cuu decreases with the increase of moisture content.
\mathrm{lg}\left({\sigma }_{3}/{P}_{a}\right)~\mathrm{lg}\left({E}_{i}/{P}_{a}\right)
Table 2. Duncan-Chang model parameter summary.
{C}_{uu}~w
relation cloud diagram.
K~w
n~w
{C}_{uu}~\left(w,{I}_{p}\right)
K~\left(w,{I}_{p}\right)
n~\left(w,{I}_{p}\right)
The maximum value of Cuu is 20.18 kPa when the moisture content is 69.1% and the plasticity index is 38.0. The minimum value of Cuu is 3.72 kPa when the moisture content is 84.9% and the plasticity index is 53.6. The maximum to minimum ratio is 5.42.
Figure 9 shows that when moisture content is constant, K increases with the increase of plasticity index within the range of 70% - 80% moisture content, and changes little with the increase of plasticity index when moisture content is greater than 80%. When the plasticity index is constant, K decreases with the increase of moisture content. The maximum value of K is 0.517 when the moisture content is 76.8% and the plasticity index is 45.1. The minimum value of K was 0.022 when the moisture content was 84.9% and the plasticity index was 53.6. The maximum to minimum ratio is 23.5.
Figure 10 shows that when the moisture content is constant, n changes little. When the plasticity index is constant, n decreases with the increase of moisture content. The maximum value of n is 1.198 when the moisture content is 69.1% and the plasticity index is 38.0. The minimum value of n was 0.150 when the moisture content was 94.3% and the plasticity index was 51.7. The maximum to minimum ratio is 7.99.
Through the triaxial unconsolidated undrained shear test of 8 groups of soft soil with moisture content of 69.1% - 94.3% and plasticity index of 32.2 - 54.1, the relationship between the four material parameters K, n, C and Rf of tangential deformation modulus of Duncan-Chang model and the moisture content and plasticity index of soil samples are analyzed, and the conclusions are as follows:
1) With the increase of moisture content, Cuu, K and n values all showed a downward trend, and Rf variation was not obvious. With the increase of plasticity index, the variation rule of each parameter is not obvious.
2) When water content is constant, Cuu and n values do not change much. K increases with the increase of plasticity index within the range of 70% - 80% moisture content, and changes little with the increase of plasticity index when the moisture content is greater than 80%, Rf has no obvious rule. When the plasticity index is constant, Cuu, K and n decrease with the increase of moisture content, Rf has no obvious rule.
3) The test constant K, whose maximum value is 23.5 times of the minimum value, is most significantly affected by moisture content and plasticity index. The influence is relatively small for Cuu and test constant n, the difference between the maximum and minimum values is 5.42 and 7.99 times respectively. The smallest influence is Rf, whose maximum value is only 1.28 times different from the minimum value.
[1] Liu, X. and Chen, X. (2012) A Triaxial Test Study on the Strength of Remolded Laterite. Journal of Nanchang University (Engineering and Technology Edition), 34, 239-242.
[2] Chen, W., Zhang, W. and Ma, Y. (2014) Triaxial Test of Compacted Loess Strength. China Earthquake Engineering Journal, 36, 239-242.
[3] Liao, H., Li, T. and Peng, J. (2011) Study on the Strength Characteristics of Loess of High and Steep Slope. Rock and Soil Mechanics, 32, 1939-1944.
[4] We, N., Lai, R. and Zou, W. (2013) Effect of Water Content on Duncan-Chang Model Parameters of Granite Residual Soil. Journal of Wuhan Institute of Technology, 35, 18-23.
[5] Yang, X. (2008) Experimental Study on Strength Characteristics of Unsaturated Remolded Loess. Northwest A&F University, Yanglin.
[6] Zdravkovic, L. and Jardine, R.J. (2001) The Effect on Anisotropy of Rocating the Principal Stress Axes during Consolidation. Geotechnique, 51, 69-83.
https://doi.org/10.1680/geot.51.1.69.39359
[7] Cokca, E., Erol, O. and Armangi, F. (2004) Effects of Compaction Moisture Content on the Shear Strength of an Unsaturated Clay. Geotechnical and Geological Engineering, 22, 285-297.
https://doi.org/10.1023/B:GEGE.0000018349.40866.3e
[8] Cheng, Y., Li, L. and Chen, H. (2011) Effects of Dry Density on Duncan-Chang Model Parameters of Marine and Fluvial Soft Soils. Highway Engineering, 36, 61-76.
[9] Wang, S. (2016) Experimental Study on Strength Variation and Constitutive Model during Soft Soil Reinforcement. Tianjin University, Tianjin.
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The school librarian is looking at her records for the number of math books that are lost each year. She knows that the average cost to replace a book is
\$75
. You need to help her decide how much money she should plan to replace lost books next year. She made the scatter plot at right using her information.
About how many books were lost in the year after the library opened?
The number of books lost is on the vertical axis.
The number of years since the library opened is on the horizontal axis.
What number of books does the first dot refer to?
In what year were the most books lost? Approximately how many books were lost that year?
This question is asking about the highest dot.
About how many books should she expect to be lost next year (
11
years after the library opened)? Based on your estimate, about how much money should she expect to spend to replace the lost books?
Overall, the dots are increasing as the years increase.
It costs about
\$75
to replace a book, so the estimated money needed for next year is between
\$2625
\$3000
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Double Atwood Machine - Maple Help
Home : Support : Online Help : Math Apps : Engineering and Applications : Double Atwood Machine
\textcolor[rgb]{0,0.329411764705882,0.501960784313725}{}
The double Atwood machine consists of an Atwood machine with one of the original masses replaced by a second Atwood machine. In total there are three masses,
{m}_{1}, {m}_{2}, {m}_{3},
suspended via massless, frictionless, inextensible cords connected to massless, frictionless pulleys. For convenience, it is assumed that the pulley system and cords have no mass or friction. A straightforward application of Newton's laws can be tedious as it requires the solution of three equations. Instead, the Lagrangian approach to deriving the equations of motion using generalized coordinates is used, which leads to only two equations to be solved.
Suppose the double Atwood machine is composed of three masses,
{m}_{1}, {m}_{2}, {m}_{3}
connected by two chords of length
L
L'
respectively through two ideal pulleys. The system has two degrees of freedom, since the height of
{m}_{3}
can be found from the height of
{m}_{2}
and the second pulley. The most convenient set of generalized coordinates to describe the motion are
{y}_{1}
: the distance from
{m}_{1 }
to the the top pulley, and
{y}_{2}
{m}_{2}
to the second pulley. Given these coordinates, the distance from
{m}_{2}
to the top pulley is
L-{y}_{1} +{y}_{2}
, and the distance from
{m}_{3}
L+L' - {y}_{1}- {y}_{2}
The kinetic energy of the system is then:
T=\frac{1}{2}{m}_{1}{\stackrel{.}{y}}_{1}^{2}+\frac{1}{2}{m}_{2}{\left(-\stackrel{.}{{y}_{1}}+\stackrel{.}{{y}_{2}}\right)}^{2}+\frac{1}{2}{m}_{3}{\left(-\stackrel{.}{{y}_{1}}-\stackrel{.}{{y}_{2}}\right)}^{2}
where the dot denotes a time derivative. By setting the potential energy to zero at the height of the top pulley, the total potential energy of the system is:
U = -{m}_{1}g {y}_{1}-{m}_{2}g\left(L-{y}_{1}+{y}_{2}\right)-{m}_{3}g\left(L-{y}_{1}+L' - {y}_{2}\right)
L = T - U
L = \frac{1}{2}{m}_{1}{\stackrel{.}{y}}_{1}^{2}+\frac{1}{2}{m}_{2}{\left(-\stackrel{.}{{y}_{1}}+\stackrel{.}{{y}_{2}}\right)}^{2}+\frac{1}{2}{m}_{3}{\left(-\stackrel{.}{{y}_{1}}-\stackrel{.}{{y}_{2}}\right)}^{2}+{m}_{1}g {y}_{1}+{m}_{2}g\left(L-{y}_{1}+{y}_{2}\right)+{m}_{3}g\left(L-{y}_{1}+L'-{y}_{2}\right)
The equations of motion then follow from evaluating the Euler-Lagrange equations:
\frac{∂}{∂t}\left(\frac{∂L}{∂{\stackrel{.}{y}}_{1}}\right)-
\frac{∂L}{∂{y}_{1}} = 0
\frac{∂}{∂t}\left(\frac{∂L}{∂{\stackrel{.}{y}}_{2}}\right)-
\frac{∂L}{∂{y}_{2}} = 0
Differentiating and simplifying these equations give the following equations of motion for the double Atwood machine:
{m}_{2}\stackrel{..}{{y}_{1}}+{m}_{2}\left(\stackrel{..}{{y}_{1}}- \stackrel{..}{{y}_{2}}\right)+{m}_{3}-g\left({m}_{1}-{m}_{2}-{m}_{3}\right)=0
{m}_{2}\left(-\stackrel{..}{{y}_{1}}+\stackrel{..}{{y}_{2}}\right)+{m}_{3}\left(\stackrel{..}{{y}_{1}}+\stackrel{..}{{y}_{2}}\right)-g\left({m}_{2}-{m}_{3}\right)=0
Rearranging terms gives the accelerations of each block as:
\stackrel{..}{{y}_{1}}=-g\frac{{m}_{1}\left({m}_{2} + {m}_{3}\right)- 4 {m}_{2}{m}_{3}}{{m}_{1}\left({m}_{2}+{m}_{3}\right) + 4 {m}_{2}{m}_{3}}
\stackrel{..}{{y}_{2}}=-g\frac{{m}_{1}\left({m}_{2} - 3 {m}_{3}\right)+ 4 {m}_{2}{m}_{3}}{{m}_{1}\left({m}_{2}+{m}_{3}\right) + 4 {m}_{2}{m}_{3}}
\stackrel{..}{{y}_{3}}=-g\frac{{m}_{1}\left({m}_{3} - 3 {m}_{2}\right)+ 4 {m}_{2}{m}_{3}}{{m}_{1}\left({m}_{2}+{m}_{3}\right) + 4 {m}_{2}{m}_{3}}
\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}
\stackrel{..}{{y}_{1}}=-\frac{\left(\stackrel{..}{{y}_{2}}+\stackrel{..}{{y}_{3}}\right)}{2}
and solved for
\stackrel{..}{{y}_{3}}
, the position of the third mass.
Left Mass,
{m}_{1}
Centre Mass ,
{m}_{2}
Right mass,
{m}_{3}
|
1Laboratoire d’Energies Thermiques Renouvelables (L.E.T.RE), Université Joseph Ki Zerbo, Ouaga, Burkina Faso.
2Laboratoire de Physique de l’Atmosphère et de l’Océan Siméon Fongang, Université Cheikh Anta Diop, Dakar-Fann, Sénégal.
3Centre Universitaire Polytechnique de Kaya (CUP-K), Kaya, Burkina Faso.
DOI: 10.4236/ojap.2019.84007 PDF HTML XML 400 Downloads 827 Views Citations
This paper deals with the climatology of aerosols in West Africa based on satellite and in situ measurements between 2001 and 2016 and covers four sites in the Sahelian zone. There are indeed Banizoumbou (13.541°N, 02.665°E), Cinzana (13.278°N, 05.934°W), Dakar (14.394°N, 16.959°W) and Ouagadougou (12.20°N, 1.40°W) located respectively in Niger, Mali, Senegal and Burkina Faso. Thus, an intercomparison between the satellite observations and the in situ measurements shows a good correlation between MODIS and AERONET with a correlation coefficient R = 0.86 at Cinzana, R = 0.85 at Banizounbou, R = 0.84 at Ouagadougou and a low correlation coefficient R = 0.66 calculated on the Dakar site. Like MODIS, SeaWiFS shows a very good correspondence with measurements of the ground photometer especially for Banizoumbou (R = 0.89), Cinzana (R = 0.88) and Dakar (R = 0.75) followed by a low correlation coefficient calculated on the Ouagadougou site (R = 0.64). The performance of these airborne sensors is also corroborated by the calculation of root mean square error (RMSE) and the mean absolute error (MAE). Following this validation, a climatological analysis based on aerosol optical depth (AOD) shows the seasonality of aerosols in West Africa strongly influenced by the climate dynamics illustrated by the MERRA model reanalysis. This seasonal spatial distribution of aerosols justifies the temporal variability of the particles observed at the different sites in the Sahel. In addition, a combined analysis of AOD and Angstrom coefficient indicates the aerosol period in the Sahel in spring (March-April-May) and summer (June-July-August). However, these aerosols are strongly dominated by desert dust whose main sources are located north in the Sahara and Sahel.
West-Africa, Aerosols, Airborne Sensors, Aeronet, MERRA Model
Bado, N. , Ouédraogo, A. , Guengané, H. , Maurice Ky, T. , Bazyomo, S. , Korgo, B. , Dramé, M. , Sall, S. , Kieno, F. and Bathiebo, D. (2019) Climatological Analysis of Aerosols Optical Properties by Airborne Sensors and in Situ Measurements in West Africa: Case of the Sahelian Zone. Open Journal of Air Pollution, 8, 118-135. doi: 10.4236/ojap.2019.84007.
{\text{AOD}}_{a}={\text{AOD}}_{b}{\left(\frac{a}{b}\right)}^{-\alpha }
In this equation, a and b denote wavelengths,
\alpha
the Angström coefficient calculated between 440 nm and 870 nm,
{\text{AOD}}_{a}
{\text{AOD}}_{b}
aerosols optical depth respectively at wavelengths a and b. The Aerosol Optical Depth (AOD) which depends on the wavelength
\lambda
and the attenuation coefficient
{\alpha }_{ext}\left(\lambda ,z\right)
\text{AOD}=\underset{\Delta z}{\int }{\alpha }_{ext}\left(\lambda ,z\right)\text{d}z
{\text{AOD}}_{\text{Sat}}=m\cdot {\text{AOD}}_{\text{AERONET}}+C
\text{RMSE}={\left[\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}{\left({\text{AOD}}_{{\text{Satellite}}_{i}}-{\text{AOD}}_{{\text{AERONET}}_{i}}\right)}^{2}\right]}^{1/2}
\text{MAE}=\frac{1}{n}\underset{i=1}{\overset{n}{\sum }}|{\text{AOD}}_{{\text{satellite}}_{i}}-{\text{AOD}}_{{\text{AERONET}}_{i}}|
on all sites especially in spring and early summer in June. These desert aerosols are indeed confirmed by the low values of the Angström coefficient
\left({\alpha }_{440-870}<0.3\right)
calculated in these periods with minima recorded in March and May around 0.18 at the sites of Ouagadougou and Banizoumbou. These same minimum values are noted in June at the sites of Cinzana and Dakar respectively around 0.17 and 0.16. This signature of the desert dust between March and June is corroborated by Dramé et al. [9] which in agreement with Mcconnell et al. [49] associates at this same period single scattering albedo (SSA) values higher than 0.9 [1]. This shows the scattering character of particle in the Sahel thus confirming their nature and primary origin (mineral dust) with a small contribution of more absorbing combustion particles. However, the values of the Angström coefficient observed between the months of July, August and September are probably due to the fine desert particles from long-range transport at medium and high troposphere as well as local combustion particles [41]. Moreover, during autumn and winter, sources of combustion are active in the Gulf of Guinea with a high intensity in winter. This generates combustion particles that are then transported by winds from the southwest to the rest of the continent. Following this, the maxima of the Angström coefficient observed in winter, particularly at the Banizoumbou, Cinzana, Dakar and Ouagadougou sites, show the state of mixing of the aerosol layer at this time in the Sahel. This layer of aerosols is in fact made up of fine desert particles linked to northeasterly winds and combustion in agreement with the intense activation of bush fires in the Gulf of Guinea [50].
|
Sudoku | Toph
The Prothom Alo newspaper sometimes publishes Sudoku for interested readers to solve. Later the solutions are also published. One day, Prothom Alo Editor Matiur Rahman thought, as everything is becoming digital, there should also be a digital version of Sudoku. Then readers will be able to submit the solution on the newspaper website. He assigned Munir Hasan for that job. Now Munir Hasan is a very busy person, he cannot possibly manage to do it himself. Besides, he have been away from programming for a long while. So he appointed a web developer for the task. Within a month the web developer implemented the system. But he was unable to write the program to check whether the Sudoku solution was correct. While being charged by Munir Hasan about that matter, he said, “Sir, I don’t know any logic”. Anyway, now your task is to rescue Munir Hasan from this situation.
A Sudoku consists of a
9 \times 9
9×9 board (with a total of
81 cells). Initially some cells are filled with a number between
1 and
9. The remaining empty cells are to be filled with numbers ranging from
1 to
9. This should be done in such a way that, no row have the same number more than once and no column have the same number more than once. Each of the nine squares of size
3 \times 3
3×3 in that
9 \times 9
9×9 board can also have any number from
1 to
9 only once.
9 \times 9
9×9 Sudoku board where each cell is filled with a number between
1 to
9.
Print Congratulations! If the Sudoku solution is correct. Otherwise print Oh No!
sbduke73Earliest, Nov '18
sbduke73Fastest, 0.0s
suvomsomLightest, 0 B
SCB-PA Inter School and College Programming Contest 2018 (College Division)
SCB-PA Inter School and College Programming Contest 2018 (School Division)
SCB-PA Inter School and College Programming Contest 2018 (Replay)
|
Cubic Hermite spline - Wikipedia
(Redirected from Hermite curve)
Cubic function used for interpolation
Cubic Hermite splines are typically used for interpolation of numeric data specified at given argument values
{\displaystyle x_{1},x_{2},\ldots ,x_{n}}
, to obtain a continuous function. The data should consist of the desired function value and derivative at each
{\displaystyle x_{k}}
. (If only the values are provided, the derivatives must be estimated from them.) The Hermite formula is applied to each interval
{\displaystyle (x_{k},x_{k+1})}
separately. The resulting spline will be continuous and will have continuous first derivative.
1 Interpolation on a single interval
1.1 Unit interval (0, 1)
1.2 Interpolation on an arbitrary interval
2 Interpolating a data set
2.3 Catmull–Rom spline
2.4 Kochanek–Bartels spline
2.5 Monotone cubic interpolation
3 Interpolation on the unit interval with matched derivatives at endpoints
Interpolation on a single interval[edit]
Unit interval (0, 1)[edit]
On the unit interval
{\displaystyle (0,1)}
, given a starting point
{\displaystyle {\boldsymbol {p}}_{0}}
{\displaystyle t=0}
and an ending point
{\displaystyle {\boldsymbol {p}}_{1}}
{\displaystyle t=1}
with starting tangent
{\displaystyle {\boldsymbol {m}}_{0}}
{\displaystyle t=0}
and ending tangent
{\displaystyle {\boldsymbol {m}}_{1}}
{\displaystyle t=1}
, the polynomial can be defined by
{\displaystyle {\boldsymbol {p}}(t)=(2t^{3}-3t^{2}+1){\boldsymbol {p}}_{0}+(t^{3}-2t^{2}+t){\boldsymbol {m}}_{0}+(-2t^{3}+3t^{2}){\boldsymbol {p}}_{1}+(t^{3}-t^{2}){\boldsymbol {m}}_{1},}
Interpolation on an arbitrary interval[edit]
Interpolating
{\displaystyle x}
in an arbitrary interval
{\displaystyle (x_{k},x_{k+1})}
is done by mapping the latter to
{\displaystyle [0,1]}
through an affine (degree-1) change of variable. The formula is
{\displaystyle {\boldsymbol {p}}(x)=h_{00}(t){\boldsymbol {p}}_{k}+h_{10}(t)(x_{k+1}-x_{k}){\boldsymbol {m}}_{k}+h_{01}(t){\boldsymbol {p}}_{k+1}+h_{11}(t)(x_{k+1}-x_{k}){\boldsymbol {m}}_{k+1},}
{\displaystyle t=(x-x_{k})/(x_{k+1}-x_{k})}
{\displaystyle h}
refers to the basis functions, defined below. Note that the tangent values have been scaled by
{\displaystyle x_{k+1}-x_{k}}
compared to the equation on the unit interval.
{\displaystyle P,Q}
be two third-degree polynomials satisfying the given boundary conditions. Define
{\displaystyle R=Q-P,}
{\displaystyle R(0)=Q(0)-P(0)=0,}
{\displaystyle R(1)=Q(1)-P(1)=0.}
{\displaystyle Q}
{\displaystyle P}
are third-degree polynomials,
{\displaystyle R}
is at most a third-degree polynomial. So
{\displaystyle R}
must be of the form
{\displaystyle R(x)=ax(x-1)(x-r).}
{\displaystyle R'(x)=ax(x-1)+ax(x-r)+a(x-1)(x-r).}
{\displaystyle R'(0)=Q'(0)-P'(0)=0,}
{\displaystyle R'(0)=0=ar,}
{\displaystyle R'(1)=Q'(1)-P'(1)=0,}
{\displaystyle R'(1)=0=a(1-r).}
Putting (1) and (2) together, we deduce that
{\displaystyle a=0}
{\displaystyle R=0,}
{\displaystyle P=Q.}
{\displaystyle {\boldsymbol {p}}(t)=h_{00}(t){\boldsymbol {p}}_{0}+h_{10}(t){\boldsymbol {m}}_{0}+h_{01}(t){\boldsymbol {p}}_{1}+h_{11}(t){\boldsymbol {m}}_{1}}
{\displaystyle h_{00}}
{\displaystyle h_{10}}
{\displaystyle h_{01}}
{\displaystyle h_{11}}
are Hermite basis functions. These can be written in different ways, each way revealing different properties:
{\displaystyle h_{00}(t)}
{\displaystyle 2t^{3}-3t^{2}+1}
{\displaystyle (1+2t)(1-t)^{2}}
{\displaystyle B_{0}(t)+B_{1}(t)}
{\displaystyle h_{10}(t)}
{\displaystyle t^{3}-2t^{2}+t}
{\displaystyle t(1-t)^{2}}
{\displaystyle {\frac {1}{3}}\cdot B_{1}(t)}
{\displaystyle h_{01}(t)}
{\displaystyle -2t^{3}+3t^{2}}
{\displaystyle t^{2}(3-2t)}
{\displaystyle B_{3}(t)+B_{2}(t)}
{\displaystyle h_{11}(t)}
{\displaystyle t^{3}-t^{2}}
{\displaystyle t^{2}(t-1)}
{\displaystyle -{\frac {1}{3}}\cdot B_{2}(t)}
The "expanded" column shows the representation used in the definition above. The "factorized" column shows immediately that
{\displaystyle h_{10}}
{\displaystyle h_{11}}
are zero at the boundaries. You can further conclude that
{\displaystyle h_{01}}
{\displaystyle h_{11}}
have a zero of multiplicity 2 at 0, and
{\displaystyle h_{00}}
{\displaystyle h_{10}}
have such a zero at 1, thus they have slope 0 at those boundaries. The "Bernstein" column shows the decomposition of the Hermite basis functions into Bernstein polynomials of order 3:
{\displaystyle B_{k}(t)={\binom {3}{k}}\cdot t^{k}\cdot (1-t)^{3-k}.}
Using this connection you can express cubic Hermite interpolation in terms of cubic Bézier curves with respect to the four values
{\displaystyle {\boldsymbol {p}}_{0},{\boldsymbol {p}}_{0}+{\frac {{\boldsymbol {m}}_{0}}{3}},{\boldsymbol {p}}_{1}-{\frac {{\boldsymbol {m}}_{1}}{3}},{\boldsymbol {p}}_{1}}
and do Hermite interpolation using the de Casteljau algorithm. It shows that in a cubic Bézier patch the two control points in the middle determine the tangents of the interpolation curve at the respective outer points.
{\displaystyle {\boldsymbol {p}}(t)=(2{\boldsymbol {p}}_{0}+{\boldsymbol {m}}_{0}-2{\boldsymbol {p}}_{1}+{\boldsymbol {m}}_{1})t^{3}+(-3{\boldsymbol {p}}_{0}+3{\boldsymbol {p}}_{1}-2{\boldsymbol {m}}_{0}-{\boldsymbol {m}}_{1})t^{2}+({\boldsymbol {m}}_{0})t+{\boldsymbol {p}}_{0}}
Interpolating a data set[edit]
A data set,
{\displaystyle (x_{k},{\boldsymbol {p}}_{k})}
{\displaystyle k=1,\ldots ,n}
, can be interpolated by applying the above procedure on each interval, where the tangents are chosen in a sensible manner, meaning that the tangents for intervals sharing endpoints are equal. The interpolated curve then consists of piecewise cubic Hermite splines and is globally continuously differentiable in
{\displaystyle (x_{1},x_{n})}
Finite difference[edit]
{\displaystyle {\boldsymbol {m}}_{k}={\frac {1}{2}}\left({\frac {{\boldsymbol {p}}_{k+1}-{\boldsymbol {p}}_{k}}{x_{k+1}-x_{k}}}+{\frac {{\boldsymbol {p}}_{k}-{\boldsymbol {p}}_{k-1}}{x_{k}-x_{k-1}}}\right)}
for internal points
{\displaystyle k=2,\dots ,n-1}
, and one-sided difference at the endpoints of the data set.
Cardinal spline[edit]
Cardinal spline example in 2D. The line represents the curve, and the squares represent the control points
{\displaystyle {\boldsymbol {p}}_{k}}
. Notice that the curve does not reach the first and last points; these points do, however, affect the shape of the curve. The tension parameter used is 0.1
{\displaystyle {\boldsymbol {m}}_{k}=(1-c){\frac {{\boldsymbol {p}}_{k+1}-{\boldsymbol {p}}_{k-1}}{x_{k+1}-x_{k-1}}}}
Catmull–Rom spline[edit]
{\displaystyle {\boldsymbol {m}}_{k}={\frac {1}{2}}{\frac {{\boldsymbol {p}}_{k+1}-{\boldsymbol {p}}_{k-1}}{x_{k+1}-x_{k-1}}}}
Kochanek–Bartels spline[edit]
A Kochanek–Bartels spline is a further generalization on how to choose the tangents given the data points
{\displaystyle {\boldsymbol {p}}_{k-1}}
{\displaystyle {\boldsymbol {p}}_{k}}
{\displaystyle {\boldsymbol {p}}_{k+1}}
, with three parameters possible: tension, bias and a continuity parameter.
Monotone cubic interpolation[edit]
Interpolation on the unit interval with matched derivatives at endpoints[edit]
Consider a single coordinate of the points
{\displaystyle {\boldsymbol {p}}_{n-1},{\boldsymbol {p}}_{n},{\boldsymbol {p}}_{n+1}}
{\displaystyle {\boldsymbol {p}}_{n+2}}
as the values that a function f(x) takes at integer ordinates x = n − 1, n, n + 1 and n + 2,
{\displaystyle p_{n}=f(n)\quad \forall n\in \mathbb {Z} .}
{\displaystyle m_{n}={\frac {f(n+1)-f(n-1)}{2}}={\frac {p_{n+1}-p_{n-1}}{2}}\quad \forall n\in \mathbb {Z} .}
{\displaystyle x=n+u,}
{\displaystyle n=\lfloor x\rfloor =\operatorname {floor} (x),}
{\displaystyle u=x-n=x-\lfloor x\rfloor ,}
{\displaystyle 0\leq u<1,}
{\displaystyle \lfloor x\rfloor }
denotes the floor function, which returns the largest integer no larger than x.
{\displaystyle {\begin{aligned}f(x)=f(n+u)&={\text{CINT}}_{u}(p_{n-1},p_{n},p_{n+1},p_{n+2})\\&={\begin{bmatrix}1&u&u^{2}&u^{3}\end{bmatrix}}\cdot {\begin{bmatrix}0&1&0&0\\-{\tfrac {1}{2}}&0&{\tfrac {1}{2}}&0\\1&-{\tfrac {5}{2}}&2&-{\tfrac {1}{2}}\\-{\tfrac {1}{2}}&{\tfrac {3}{2}}&-{\tfrac {3}{2}}&{\tfrac {1}{2}}\end{bmatrix}}\cdot {\begin{bmatrix}p_{n-1}\\p_{n}\\p_{n+1}\\p_{n+2}\end{bmatrix}}\\&={\frac {1}{2}}{\begin{bmatrix}-u^{3}+2u^{2}-u\\3u^{3}-5u^{2}+2\\-3u^{3}+4u^{2}+u\\u^{3}-u^{2}\end{bmatrix}}^{\mathrm {T} }\cdot {\begin{bmatrix}p_{n-1}\\p_{n}\\p_{n+1}\\p_{n+2}\end{bmatrix}}\\&={\frac {1}{2}}{\begin{bmatrix}u{\big (}(2-u)u-1{\big )}\\u^{2}(3u-5)+2\\u{\big (}(4-3u)u+1{\big )}\\u^{2}(u-1)\end{bmatrix}}^{\mathrm {T} }\cdot {\begin{bmatrix}p_{n-1}\\p_{n}\\p_{n+1}\\p_{n+2}\end{bmatrix}}\\&={\tfrac {1}{2}}{\Big (}{\big (}u^{2}(2-u)-u{\big )}p_{n-1}+{\big (}u^{2}(3u-5)+2{\big )}p_{n}+{\big (}u^{2}(4-3u)+u{\big )}p_{n+1}+u^{2}(u-1)p_{n+2}{\Big )}\\&={\tfrac {1}{2}}{\big (}(-u^{3}+2u^{2}-u)p_{n-1}+(3u^{3}-5u^{2}+2)p_{n}+(-3u^{3}+4u^{2}+u)p_{n+1}+(u^{3}-u^{2})p_{n+2}{\big )}\\&={\tfrac {1}{2}}{\big (}(-p_{n-1}+3p_{n}-3p_{n+1}+p_{n+2})u^{3}+(2p_{n-1}-5p_{n}+4p_{n+1}-p_{n+2})u^{2}+(-p_{n-1}+p_{n+1})u+2p_{n}{\big )}\\&={\tfrac {1}{2}}{\Big (}{\big (}(-p_{n-1}+3p_{n}-3p_{n+1}+p_{n+2})u+(2p_{n-1}-5p_{n}+4p_{n+1}-p_{n+2}){\big )}u+(-p_{n-1}+p_{n+1}){\Big )}u+p_{n},\end{aligned}}}
{\displaystyle \mathrm {T} }
denotes the matrix transpose. The bottom equality is depicting the application of Horner's method.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cubic_Hermite_spline&oldid=1086095753"
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Fundamental operation on complex numbers
{\displaystyle z}
{\displaystyle {\overline {z}}}
{\displaystyle z}
{\displaystyle a}nd
{\displaystyle b}
{\displaystyle a+bi}
{\displaystyle a-bi.}
{\displaystyle z}
{\displaystyle {\overline {z}}.}
{\displaystyle re^{i\varphi }}
{\displaystyle re^{-i\varphi }.}
{\displaystyle a^{2}+b^{2}}
{\displaystyle r^{2}}
{\displaystyle z}
{\displaystyle {\overline {z}}}
{\displaystyle z^{*}.}
{\displaystyle 2\times 2}
{\displaystyle z}
{\displaystyle w,}
{\displaystyle z}
{\displaystyle w}
{\displaystyle a+bi.}
{\displaystyle {\begin{aligned}{\overline {z+w}}&={\overline {z}}+{\overline {w}},\\{\overline {z-w}}&={\overline {z}}-{\overline {w}},\\{\overline {zw}}&={\overline {z}}\;{\overline {w}},\quad {\text{and}}\\{\overline {\left({\frac {z}{w}}\right)}}&={\frac {\overline {z}}{\overline {w}}},\quad {\text{if }}w\neq 0.\end{aligned}}}
{\displaystyle \left|{\overline {z}}\right|=|z|.}
{\displaystyle z}
{\displaystyle z.}
{\displaystyle {\overline {\overline {z}}}=z.}
{\displaystyle z{\overline {z}}={\left|z\right|}^{2}.}
{\displaystyle z^{-1}={\frac {\overline {z}}{{\left|z\right|}^{2}}},\quad {\text{ for all }}z\neq 0.}
{\displaystyle {\overline {z^{n}}}=\left({\overline {z}}\right)^{n},\quad {\text{ for all }}n\in \mathbb {Z} }
{\displaystyle \exp \left({\overline {z}}\right)={\overline {\exp(z)}}}
{\displaystyle \ln \left({\overline {z}}\right)={\overline {\ln(z)}}{\text{ if }}z{\text{ is non-zero }}}
{\displaystyle p}
{\displaystyle p(z)=0,}
{\displaystyle p\left({\overline {z}}\right)=0}
{\displaystyle \varphi }
{\displaystyle \varphi (z)}
{\displaystyle \varphi ({\overline {z}})}
{\displaystyle \varphi \left({\overline {z}}\right)={\overline {\varphi (z)}}.\,\!}
{\displaystyle \sigma (z)={\overline {z}}}
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {C} /\mathbb {R} .}
{\displaystyle \sigma }
{\displaystyle \mathbb {C} .}
{\displaystyle \mathbb {C} }
Use as a variable[edit]
{\displaystyle z=x+yi}
{\displaystyle z=re^{i\theta }}
{\displaystyle z}
{\displaystyle x=\operatorname {Re} (z)={\dfrac {z+{\overline {z}}}{2}}}
{\displaystyle y=\operatorname {Im} (z)={\dfrac {z-{\overline {z}}}{2i}}}
{\displaystyle r=\left|z\right|={\sqrt {z{\overline {z}}}}}
{\displaystyle e^{i\theta }=e^{i\arg z}={\sqrt {\dfrac {z}{\overline {z}}}},}
{\displaystyle \theta =\arg z={\dfrac {1}{i}}\ln {\sqrt {\frac {z}{\overline {z}}}}={\dfrac {\ln z-\ln {\overline {z}}}{2i}}}
{\displaystyle {\overline {z}}}
{\displaystyle \left\{z:z{\overline {r}}+{\overline {z}}r=0\right\}}
{\displaystyle {r},}
{\displaystyle z\cdot {\overline {r}}}
{\displaystyle z}
{\displaystyle {r}}
{\displaystyle u=e^{ib},}
{\displaystyle {\frac {z-z_{0}}{{\overline {z}}-{\overline {z_{0}}}}}=u^{2}}
{\displaystyle z_{0}}
{\displaystyle u.}
{\displaystyle z}
{\textstyle {\overline {\mathbf {AB} }}=\left({\overline {\mathbf {A} }}\right)\left({\overline {\mathbf {B} }}\right),}
{\textstyle {\overline {\mathbf {A} }}}
{\displaystyle \mathbf {A} .}
{\textstyle \left(\mathbf {AB} \right)^{*}=\mathbf {B} ^{*}\mathbf {A} ^{*},}
{\textstyle \mathbf {A} ^{*}}
{\textstyle \mathbf {A} .}
{\textstyle a+bi+cj+dk}
{\textstyle a-bi-cj-dk.}
{\displaystyle {\left(zw\right)}^{*}=w^{*}z^{*}.}
{\textstyle V}
{\textstyle \varphi :V\to V}
{\displaystyle \varphi ^{2}=\operatorname {id} _{V}\,,}
{\displaystyle \varphi ^{2}=\varphi \circ \varphi }
{\displaystyle \operatorname {id} _{V}}
{\displaystyle V,}
{\displaystyle \varphi (zv)={\overline {z}}\varphi (v)}
{\displaystyle v\in V,z\in \mathbb {C} ,}
{\displaystyle \varphi \left(v_{1}+v_{2}\right)=\varphi \left(v_{1}\right)+\varphi \left(v_{2}\right)\,}
{\displaystyle v_{1}v_{2},\in V,}
{\displaystyle \varphi }
{\displaystyle V.}
{\textstyle \varphi }
{\textstyle \mathbb {R} }
{\textstyle V,}
{\displaystyle V}
{\displaystyle V.}
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Early Solar System Solar Wind Implantation of 7Be into Calcium-Alumimum Rich Inclusions in Primitive Meteortites
Department of Physics, Purdue University Northwest, Westville, USA
The one time presence of short-lived radionuclides (SLRs) in Calcium-Aluminum Rich inclusions (CAIs) in primitive meteorites has been detected. The solar wind implantation model (SWIM) is one possible model that attempts to explain the catalogue of SLRs found in primitive meteorites. In the SWIM, solar energetic particle (SEP) nuclear interactions with gas in the proto-solar atmosphere of young stellar objects (YSOs) give rise to daughter nuclei, including SLRs. These daughter nuclei then may become entrained in the solar wind via magnetic field lines. Subsequently, the nuclei, including SLRs, may be implanted into CAI precursors that have fallen from the main accretion flow which had been destined for the proto-star. This mode of implanting SLRs in the solar system is viable, and is exemplified by the impregnation of the lunar surface with solar wind particles, including SLRs. X-ray luminosities have been measured to be 100,000 times more energetic in YSOs, including T-Tauri stars, than present-day solar luminosities. The SWIM scales the production rate of SLRs to nascent SEP activity in T-Tauri stars. Here, we model the implantation of 7Be into CAIs in the SWIM, utilizing the enhanced SEP fluxes and the rate of refractory mass inflowing at the X-region, 0.06 AU from the proto-Sun. Taking into account the radioactive decay of 7Be and spectral flare variations, the 7Be/9Be initial isotopic ratio is found to range from 1 × 10−5 to 5 × 10−5.
Radio-Nuclide, 7Be, Early Solar System, Solar Wind, CAI, Solar Wind Implantation Model, X-Wind
S={\stackrel{˙}{M}}_{D}\cdot {X}_{r}\cdot F
{\stackrel{˙}{M}}_{D}
{\stackrel{˙}{M}}_{D}
{\stackrel{˙}{M}}_{D}
P=p\cdot f
\frac{dF}{dE}=k{E}^{-r}
p=\underset{i}{\sum }{N}_{i}\int {\sigma }_{ij}\frac{dF\left(E\right)}{d{E}_{j}}dE
{\sigma }_{ij}\left(E\right)
\frac{dF\left(E\right)}{d{E}_{j}}dE
{N}^{7\text{Be}}=\frac{P}{S}=\frac{p\cdot f}{{\stackrel{˙}{M}}_{D}\cdot {X}_{r}\cdot F}
Bricker, G.E. (2019) Early Solar System Solar Wind Implantation of 7Be into Calcium-Alumimum Rich Inclusions in Primitive Meteortites. International Journal of Astronomy and Astrophysics, 9, 12-20. https://doi.org/10.4236/ijaa.2019.91002
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2. Bricker, G.E. and Caffee, M.W. (2010) Solar Wind Implantation Model for 10Be in CAIs. Astrophysical Journal, 725, 443-449. https://doi.org/10.1088/0004-637X/725/1/443
3. Bricker, G.E. and Caffee, M.W. (2013) Incorporation of 36Cl in Calcium-Aluminum-Rich Inclusions in the Solar Wind Implantation Model. Advances in Astronomy, 2013, Article ID: 487606. https://doi.org/10.1155/2013/487606
4. Feigelson, E.D., Garmire, G.P. and Pravdo, S.H. (2002) Magnetic Flaring in the Pre-Main-Sequence Sun and Implications for the Early Solar System. Astrophysical Journal, 572, 335-349. https://doi.org/10.1086/340340
5. Nishiizumi, K. and Caffee, M.W. (2001) Beryllium-10 from the Sun. Science, 294, 352-354. https://doi.org/10.1126/science.1062545
6. Bricker, G.E. (2013) Calculation of 10Be and 14C in the Solar Atmosphere: Implication for Solar Flare Spectral Index. International Journal of Astronomy and Astrophysics, 3, 17-20. https://doi.org/10.4236/ijaa.2013.32A003
7. Jull, A.J.T., Lal, D., McHargue, L.R., Burr, G.S. and Donahue, D.J. (2000) Cosmogenic and Implanted Radionuclides Studied by Selective Etching of Lunar Soils. Nuclear Instruments & Methods in Physics Research Section B, 172, 867-872. https://doi.org/10.1016/S0168-583X(00)00232-9
8. Jaeger, M., Wilmes, S., Kölle, V., Staudt, G. and Mohr, P. (1996) Precision Measurement of the Half-Life of 7Be. Physical Review, C54, 423-424. https://doi.org/10.1103/PhysRevC.54.423
9. Mandzhavidz, N., Ramaty, R. and Kozlovsky, B. (1997) Solar Atmospheric and Solar Flare Accelerated Helium Abundances from Gamma-Ray Spectroscopy. Astrophysical Journal, 489, L99-L102. https://doi.org/10.1086/310965
10. Chaussidon, M., Robert, F. and McKeegan, K. (2006) Li and B Isotopic Variations in an Allende CAI: Evidence for the In Situ Decay of Short-Lived 10Be and for the Possible Presence of the Short-Lived Nuclide 7Be in the Early Solar System. Geochimica et Cosmochimiva Acta, 70, 224-225. https://doi.org/10.1016/j.gca.2005.08.016
11. Mishra, R. and Marhas, K. (2018) Fossil Record of 7Be and 10Be in a Cai: Implications for the Origin and Early Evolution of Our Solar System. 81st Annual Meeting of the Meteoritical Society, Moscow, 22-27 July 2018, Article ID: 6125.
12. Nishiizumi, K., Imamura, M., Caffee, M., Southon, J., Finkel, R. and McAnich, J. (2007) Absolute Calibration of 10Be AMS Standards. Nuclear Instruments and Methods in Physics Research B: Beam Interactions with Materials and Atoms, 258, 403-413. https://doi.org/10.1016/j.nimb.2007.01.297
13. McKeegan, K., Chaussidon, M. and Robert, F. (2000) Incorporation of Short-Lived 10Be in a Calcium-Aluminum-Rich Inclusion from the Allende Meteorite. Science, 289, 1334-1337. https://doi.org/10.1126/science.289.5483.1334
14. Leya, I., Wieler, R. and Halliday, A.N. (2003) The Predictable Collateral Consequences of Nucleosynthesis by Spallation Reactions in the Early Solar System. Astrophysical Journal, 594, 605-616. https://doi.org/10.1086/376795
15. Duprat, J. and Tatischeff, V. (2008) On Non-Thermal Nucleosynthesis of Short-Lived Radionuclei in the Early Solar System. New Astronomy Reviews, 52, 463-466. https://doi.org/10.1016/j.newar.2008.06.016
16. Shu, F.H., Shang, H. and Lee, T. (1996) Toward an Astrophysical Theory of Chondrites. Science, 271, 1545-1552. https://doi.org/10.1126/science.271.5255.1545
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20. Calvet, N., Briceno, B., Hernandez, J., Hoyer, S., Hartmann, L., Sicila, Megeath, S.T. and D’Alessio, P. (2005) Disk Evolution in the Orion OB1 Association. Astronomical Journal, 129, 935-946. https://doi.org/10.1086/426910
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25. Gounelle, M., Shu, F.H., Shang, H., Glassgold, A.E., Rehm, K.E. and Lee, T. (2006) The Irradiation Origin of Beryllium Radioisotopes and Other Short-Lived Radionuclides. The Astrophysical Journal, 640, 1163-1189. https://doi.org/10.1086/500309
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Limited differential as a planetary bevel gear - Simulink - MathWorks India
\eta {T}_{d}{\omega }_{d}
\eta {T}_{1}{\omega }_{1}
\eta {T}_{2}{\omega }_{2}
\begin{array}{l}{\stackrel{˙}{W}}_{loss}= -\left({P}_{t}+{P}_{d}+{P}_{c}\right)+{P}_{s}\\ \\ {P}_{t}=\eta \left({T}_{d}{\omega }_{d}+{T}_{1}{\omega }_{1}+{T}_{2}{\omega }_{2}\right)\end{array}
{P}_{d}=-\left({b}_{1}|{\omega }_{1}|+{b}_{2}|{\omega }_{2}|+{b}_{d}|{\omega }_{d}|\right)
{P}_{c}={T}_{c}|\overline{\omega }|
{P}_{s}=-\left({\omega }_{1}{\stackrel{˙}{\omega }}_{1}{J}_{1}+{\omega }_{2}{\stackrel{˙}{\omega }}_{2}{J}_{2}+{\omega }_{d}{\stackrel{˙}{\omega }}_{d}{J}_{d}\right)
{\stackrel{˙}{\omega }}_{d}{J}_{d}=\eta {T}_{d}-{\omega }_{d}{b}_{d}-{T}_{i}
{\stackrel{˙}{\omega }}_{1}{J}_{1}=\eta {T}_{1}-{\omega }_{1}{b}_{1}-{T}_{i1}
{\stackrel{˙}{\omega }}_{2}{J}_{2}=\eta {T}_{2}-{\omega }_{2}{b}_{2}-{T}_{i2}
\begin{array}{l}\eta {T}_{1}=\frac{N}{2}{T}_{i}-\frac{1}{2}{T}_{c}\\ \eta {T}_{2}=\frac{N}{2}{T}_{i}+\frac{1}{2}{T}_{c}\end{array}
{\omega }_{d}=\frac{N}{2}\left({\omega }_{1}+{\omega }_{2}\right)
\varpi
{R}_{eff}
{R}_{o}
{T}_{c}={F}_{c}N\mu \left(|\varpi |\right){R}_{eff}\mathrm{tanh}\left(4|\varpi |\right)
{R}_{eff}=\frac{2\left({R}_{o}{}^{3}-{R}_{i}{}^{3}\right)}{3\left({R}_{o}{}^{2}-{R}_{i}{}^{2}\right)}
\varpi ={\omega }_{1}-{\omega }_{2}
\varpi ={\omega }_{1}-{\omega }_{2}
\eta {T}_{i1}= \eta {T}_{i2}=\frac{N}{2}{T}_{i}
{\omega }_{d}=\frac{N}{2}\left({\omega }_{1}+{\omega }_{2}\right)
{R}_{eff}
{R}_{eff}=\frac{2\left({R}_{o}{}^{3}-{R}_{i}{}^{3}\right)}{3\left({R}_{o}{}^{2}-{R}_{i}{}^{2}\right)}
{R}_{o}
{R}_{i}
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Schoof–Elkies–Atkin algorithm - Wikipedia
The Schoof–Elkies–Atkin algorithm (SEA) is an algorithm used for finding the order of or calculating the number of points on an elliptic curve over a finite field. Its primary application is in elliptic curve cryptography. The algorithm is an extension of Schoof's algorithm by Noam Elkies and A. O. L. Atkin to significantly improve its efficiency (under heuristic assumptions).
The Elkies-Atkin extension to Schoof's algorithm works by restricting the set of primes
{\displaystyle S=\{l_{1},\ldots ,l_{s}\}}
considered to primes of a certain kind. These came to be called Elkies primes and Atkin primes respectively. A prime
{\displaystyle l}
is called an Elkies prime if the characteristic equation:
{\displaystyle \phi ^{2}-t\phi +q=0}
{\displaystyle \mathbb {F} _{l}}
, while an Atkin prime is a prime that is not an Elkies prime. Atkin showed how to combine information obtained from the Atkin primes with the information obtained from Elkies primes to produce an efficient algorithm, which came to be known as the Schoof–Elkies–Atkin algorithm. The first problem to address is to determine whether a given prime is Elkies or Atkin. In order to do so, we make use of modular polynomials
{\displaystyle \Phi _{l}(X,Y)}
that parametrize pairs of
{\displaystyle l}
-isogenous elliptic curves in terms of their j-invariants (in practice alternative modular polynomials may also be used but for the same purpose).
If the instantiated polynomial
{\displaystyle \Phi _{l}(X,j(E))}
{\displaystyle j(E')}
{\displaystyle \mathbb {F} _{q}}
{\displaystyle l}
is an Elkies prime, and we may compute a polynomial
{\displaystyle f_{l}(X)}
whose roots correspond to points in the kernel of the
{\displaystyle l}
-isogeny from
{\displaystyle E}
{\displaystyle E'}
. The polynomial
{\displaystyle f_{l}}
is a divisor of the corresponding division polynomial used in Schoof's algorithm, and it has significantly lower degree,
{\displaystyle O(l)}
{\displaystyle O(l^{2})}
. For Elkies primes, this allows one to compute the number of points on
{\displaystyle E}
{\displaystyle l}
more efficiently than in Schoof's algorithm. In the case of an Atkin prime, we can gain some information from the factorization pattern of
{\displaystyle \Phi _{l}(X,j(E))}
{\displaystyle \mathbb {F} _{l}[X]}
, which constrains the possibilities for the number of points modulo
{\displaystyle l}
, but the asymptotic complexity of the algorithm depends entirely on the Elkies primes. Provided there are sufficiently many small Elkies primes (on average, we expect half the primes
{\displaystyle l}
to be Elkies primes), this results in a reduction in the running time. The resulting algorithm is probabilistic (of Las Vegas type), and its expected running time is, heuristically,
{\displaystyle {\tilde {O}}(\log ^{4}q)}
, making it more efficient in practice than Schoof's algorithm. Here the
{\displaystyle {\tilde {O}}}
notation is a variant of big O notation that suppresses terms that are logarithmic in the main term of an expression.
Schoof–Elkies–Atkin algorithm is implemented in the PARI/GP computer algebra system in the GP function ellap.
"Schoof: Counting points on elliptic curves over finite fields"
article on Mathworld
"Remarks on the Schoof-Elkies-Atkin algorithm"
"The Schoof-Elkies-Atkin Algorithm in Characteristic 2"
Retrieved from "https://en.wikipedia.org/w/index.php?title=Schoof–Elkies–Atkin_algorithm&oldid=999342473"
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The following table represents the Frequency Distribution and Cumulative Distributions for
The following table represents the Frequency Distribution and Cumulative Distributions for this data set: 12, 13, 17, 18, 18, 24, 26, 27, 27, 30, 30,
The following table represents the Frequency Distribution and Cumulative Distributions for this data set: 12, 13, 17, 18, 18, 24, 26, 27, 27, 30, 30, 35, 37, 41, 42, 43, 44, 46, 53, 58
\begin{array}{|cccc|}\hline \text{Class}& \text{Frequency}& \text{Relative Frequency}& \text{Cumulative Frequency}\\ \text{10 but les than 20}& 5\\ \text{20 but les than 30}& 4\\ \text{30 but les than 40}& 4\\ \text{40 but les than 50}& 5\\ \text{50 but les than 60}& 2\\ \text{TOTAL}\\ \hline\end{array}
What is the Relative Frequency for the class: 20 but less than 30? State you answer as a value with exactly two digits after the decimal. for example 0.30 or 0.35
Step 1 Given The following table represents the Frequency Distribution and Cumulative Distributions for this data set: 12, 13, 17, 18, 18, 24, 26, 27, 27, 30, 30, 35, 37, 41, 42, 43, 44, 46, 53, 58
Relative Frequency
{R}_{f}={F}_{i}\text{ }\sum \left(F\right)
Cumulative Frequency = It is the cumulative sum of frequency.
\begin{array}{|ccccc|}\hline \text{Class}& \text{Frequency}& \text{Relative Frequency}& \text{Cumulative Frequency}& \text{Percentage}\\ \text{10-19}& 5& 0.25& 0.25& 25\\ \text{20-29}& 4& 0.2& 0.4545\\ \text{30-39}& 4& 0.2& 0.65& 65\\ \text{40-49}& 5& 0.25& 0.9& 90\\ \text{50-59}& 2& 0.10& 1& 100\\ \text{TOTAL}& 20\\ \hline\end{array}
What is the Relative Frequency for the class: 20 but less than 30? Answer : 0.20
A weather forecaster predicts that the temperature in Antarctica will decrease
{8}^{\circ }F
each hour for the next 6 hours. Write and solve an inequality to determine how many hours it will take for the temperature to drop at least
{36}^{\circ }F
Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task.
\left(1,3\right),\left(2,6\right),\left(3,12\right),\left(4,24\right)
A parks and recreation department is constructing a new bike path. The path will be parallel to the railroad tracks shown and pass through the parking area al the point
\left(4,\text{ }5\right).
Write an equation that represents the path.
Create equations to represent a relationship between quantities.
Define appropriate quantities for . . . descriptive modeling
Marcie can assemble 1 puzzle in 2 hours, and Janice can assemble 2 puzzles in 5 hours. How many hours will it take Marcie and Janice to assemble a puzzle if they work together?
Discuss the importance of data modeling.
Range=max value- min value
h=r/m m=no of classes
The weights shown in data given to the nearest tenth of a pound, were obtained from a sample of 18- to 24-year-old males. Organize these data into frequency and relative-frequency distributions. Use a class width of 20 and a first cutpoint of 120.
Also calculate cumulative frequency and midpoint.
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Automorphism - formulasearchengine
5 Inner and outer automorphisms
Closure: composition of two endomorphisms is another endomorphism.
Associativity: composition of morphisms is always associative.
Identity: the identity is the identity morphism from an object to itself, which exists by definition.
Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.
{\displaystyle \mu }
is a new fifth root of unity, connected with the former fifth root
{\displaystyle \lambda }
by relations of perfect reciprocity.
Weisstein, Eric W., "Automorphism", MathWorld.
Retrieved from "https://en.formulasearchengine.com/index.php?title=Automorphism&oldid=218522"
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The positive integer just greater than (1 + .0001)10000 is
The inverse point of (1, 2) with respect to the circle x2 + y2 - 4x - 6y + 9 = 0 is
If the line lx+my=1 is a tangent line to the circle x2+y2=a2, the locus of (l,m) is
an ellipse circle
The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, then the value of n will be
Let coefficients of three consecutive terms i.e., (r + 1)th, (r + 2)th and (r + 3)th in expansion of (1 + x)n are 165.330 and 462 respectively then
coefficient of (r + 1)th term = nCr = 165
coefficient of (r + 2)th term = nCr + 1 = 330
∴ n C r + 1 n C r = n - r r + 1 = 2
or, n - r = 2(r + 1)
or, r = 1 3 (n - 2)
and n C r + 2 n C r + 1 = n - r - 1 r + 2 = 231 165
or, 165(n - r - 1) = 231 (r + 2)
or, 165n - 627 = 396r
or, 165n - 627 = 396 x 1 3 (n - 2)
or, 165n - 627 = 132 (n - 2)
[(-1+√-3)/2]3n + [(-1-√-3)/2]3n =
If x is positive, then the first negative terms in the expansion of 1 + x 27 5 is
If w is a cube root of unity, then the value of (1 + ω - ω)2 (1 - ω + ω2) is
4 log e
The area bounded by the parabola y2=4ax and the straight line y=2ax is
On solving y 2 = 4ax and y = 2ax, we get
x = 0 or 1 a
and y = 0 or 2
If m and n are integers, then what is the value of
1 m + n
1 m − n
Since sin mx ,sin nx is an odd function if m ≠ n, then ∫ 0 π sin mx . sin nx dx = 0
The particular integral of (D2 + 1)y = xe2x is equal to
The order and degree of the equation are
If y=log logx, ey(dy/dx)=
(1/xlogx)
1/logx
(d/dx)(log tan x)=
cosec (1/x)
cosec (-2x)
If the normal at the point P(θ) to the ellipse ((x2/14)+(y2/5) = 1) intersects it again at the point Q(2θ), then cosθ is equal to
If the latus rectum of an ellipse is one half of its minor axis, then its eccentricity is
The parametric equations of the hyperbola x2/a2 - y2/b2 = 1 are
x = a tan θ, y = b secθ
x = √2a, y = b
Let f(x) = tan-1 {φ(x)}, where φ(x)is monotonically increasing for 0 < x < π/2. Then f(x) is
increasing in (0, π/2)
decreasing in (0, π/2)
increasing in (0, π/4) and decreasing in (π/4, π/2)
f\prime \left(x\right)=\frac{\phi \prime \left(x\right)}{1+\left\{\phi \left(x\right){\right\}}^{2}}>\phantom{\rule{0.5em}{0ex}}\text{0 for 0}\phantom{\rule{0.5em}{0ex}}<x<\frac{\pi }{2}\phantom{\rule{0.5em}{0ex}}\text{because}\phantom{\rule{0.5em}{0ex}}\phi \prime \left(x\right)>0,\phi \left(x\right)\phantom{\rule{0.5em}{0ex}}\text{being monotonically increasing}
Sin (sin⁻11/2 + cos⁻11/2) equals
If is finite, then the values of a,b are respectively
If the value of a third order determinant is11, then the value of the square of the determinant formed by the co-factors will be
Largest value of min (2 + x2, 6 - 3x) when x > 0 is :
If a complex number lies in the IIIrd quadrant then its conjugate lies in quadrant number
The vertex of the parabola x 2 + 8 x + 12 y + 4 = 0 is
− 4 , 1
4, − 1
Axis of the parabola x 2 − 3 y − 6 x + 6 = 0 is
Out of 15 points in a plane, no three are in a straight line except 8 points which are collinear. How many triangles can be formed by joining them?
How many words can be formed from the letters of the word 'SIGNATURE' be arranged so that the vowels always come together?
A box contains n enumerated articles. All the articles are taken out one by one at random. The probability that the numbers of the selected articles are in the sequence 1, 2, .........n is
1 /n !
n /n !
1 /(n n !)
In Δ A B C , i f b = 20 , c = 21 and sin A = 3 5 , then a =
Two dice are thrown simultaneously. The probability of getting a pair of 1 is
In a Δ A B C , 2s = perimeter and R = circumradius. Then s/R is equal to
sin A + sin B + sinC
the greatest value of a non-negative real number λ for which both the equations 2x2 + (λ - 1)x + 8 = 0 and x2 - 8x + λ + 4 = 0 have real roots is :
The second drivative f"(x) of the function f(x) exists for all x in [0,1] and satisfies | f ″ x | ≤ 1. If f 0 = f 1 , then for all x in [0,1]
|f'(x)|< 1
|f'(x)|>1
|f'(x)|=1
f(x) is constant
The first derivative f'(x) exists for all x in [0,1] which implies that f(x) is continuous for all x in [0,1]. Also, it is given that f(0) = f(1)
Thus, applying Rolle's theorem on f(x) in the interval [0,1], we have
f'(c) = 0 for some c in [0,1]
The second derivative f''(x) exists for all x in [0,1] which implies that f'(x) is continuous for all x in [0,1]
Thus, applying Lagrange's theorem on f'(x) in the interval [ c , x ] c < x ≥ 1 , we have
Similarly, applying Lagrange's theorem on f’(x) in the interval [ x , c ] 0 ≤ x < c , we have
The set of values of p for which the roots of the equation 3x2 + 2x + (p - 1)p = 0 are of opposite sign, is
(-∝, 0)
(1, ∝)
The fourth, seventh and tenth terms of a G.P. are p, q and r respectively, then
If A.M. between two numbers is 5 and their G.M. is 4, then their H.M. is
If x, y and z respectively represent AM, GM and HM between two numbers a and b, then
Here x = 5, y = 4
then 16 = 5x z
Every term of a G.P. is positive and also every term is the sum of two preceding terms. Then the common ratio of the G.P. is
√5 - 1/2
√5 + 1/2
1 - √5/2
If sin (x + 3α) = 3 sin (α - x), then
tan x = tan3α
tan x = tan α
tan x = tan2 α
tan x = 3 tan α
cot⁻1(-1/2) + cot⁻1(-1/3) is equal to
The coordinates of mid-point of portion of line cut by coordinate axis are (3,2), the equation of the line is
The angle between lines 3x+y-7=0 and x+2y+9=0 is
The condition that the cubic equation x3 - px2 + qx - r = 0 has all of its three roots equal is given by___
p2 = 3qr
q2 = 3pr
r2 = 3pq
\alpha
{x}^{3}-p{x}^{2}+qx-r=0
Since, all the roots of the given equation are equal
\therefore \alpha +\alpha +\alpha =p
⇒\alpha =p∕3\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\dots \left(1\right)
\alpha .\alpha +\alpha .\alpha +\alpha .\alpha =q
⇒{\alpha }^{2}=q∕3\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\dots \left(2\right)
\alpha .\alpha .\alpha =r
⇒{\alpha }^{3}=r\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\dots \left(3\right)
\because {\left({\alpha }^{2}\right)}^{2}={\alpha }^{3}.\alpha
⇒{\left(q∕3\right)}^{2}=\left(r\right).\left(p∕3\right)
⇒{q}^{2}=3pr
The mean of the numbers a,b,8,5,10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?
A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. The maximum possible volume of the box is given by (in cm2)
The solution for the equation
On the interval [0, 1] the function
f x = x 1005 (1 − x )1002
assumes maximum value equal to
Let a,b,c be any real numbers. Suppose that there are real numbers x,y,z not all zero such that x=cy+bz, y=az+cx and z=bx+ay. Then a2+b2+c2+2abc is equal to
is a binary operation defined on Q. Find which of the following operation is Associative.
a ∗ b = a-b ∀ a, b ∈ Q
a ∗ b = a b 2 ∀ a, b ∈ Q
a ∗ b = ab2 ∀ a, b ∈ Q
In a triangle X Y Z , ∠ Z = π 2 . If tan X 2 and tan Y 2 are the roots of the equation ax2 + bx + c = 0, a ≠ 0 then
then ∑ r = 1 n Δ r is equal to
dependent n
dependent of θ
Independent of x, y, z
Statement-1 :
Statement-1 is true, Statement-2 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1
Let P(3, 2, 6) be a point in space and Q be a point on the line
Then the value of μ for which the vector P Q → is parallel to the plane x - 4y+3z = 1 is
Given the family of lines, a (3x + 4y +6) + b (x + y +2) = 0. The line of the family situated at the greatest distance from the point P(2, 3) has equation
The differential equation (x4 − 2 x y2 + y4 ) d x − 2x2 y − 4xy3 + sin y) dy = 0 has its solution as
If the function f(x) and g(x) are defined on R → R such that
The area bounded by the curves f (x) = sin − 1 (sin x) and (gx) = [ sin − 1 sin x ] in the interval [ 0 , π ] , where [ . ] is a greatest integer function, is
(π 2 − 1) 2
(π 4 − 1)2
If are any four vectors then is a vector
Along the line of intersection of two planes, one containing a → , b →
The cubes of natural number are grouped as 1 3 , 2 3 , 3 3 , 4 3 , 5 3 , 6 3 ,...Let Sn denotes the sum of cubes in the nth group, then 8Sn is divisible by
The solution of the equation cos 103 x − sin 103 x = 1 are
− π /2
In Δ A B C , c , then
B = π /3
A,B,C, are in A.P.
The point (0,0), (a, 11) and (b, 37) are the vertices of an equilateral triangle. Then
For three vectors u → , v → , w → which of the following expression is not equal to any of the remaining three?
f(x) increases in (0, 2)
f ′ x is continuous for all x ∈ R
f(x) decreases in − ∞ , 0 ∪ 2 , ∞
The interval into which the function f(x) transforms the entire real line is [ − 1 3 , 1 ]
If a → = i‸ + j ‸ + k ‸ , b → = 4 i ‸ + 3 j ‸ + 4 k ‸ and c → = i ‸ + α j ‸ + β k ‸ are linearly dependent vectors and | c | = 3 , then
α = 1 , β = − 1
α = 1 , β = +- 1
α = − 1 , β = 1
α = 1 , β = 1
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interate[e^X^4(x+x^3+2x ^5)e^x^2] - Maths - Integrals - 10495375 | Meritnation.com
interate[e^X^4(x+x^3+2x ^5)e^x^2]
In question it is easier to differentiate the options and checking with qn
Differentiating option D we get
\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{2}{\mathrm{x}}^{2}{\mathrm{e}}^{{\mathrm{x}}^{2}}{\mathrm{e}}^{{\mathrm{x}}^{4}}+\mathrm{c}\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\frac{1}{2}{\mathrm{x}}^{2}{\mathrm{e}}^{{\mathrm{x}}^{2}}{\mathrm{e}}^{{\mathrm{x}}^{4}}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}×\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^{2} . {\mathrm{e}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}×\left[{\mathrm{x}}^{2} × \left(2\mathrm{x}+4{\mathrm{x}}^{3}\right) .{\mathrm{e}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}} + {\mathrm{e}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}}. 2\mathrm{x} \right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}=\left[{\mathrm{x}}^{2}.{\mathrm{e}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}}\left(\mathrm{x}+2{\mathrm{x}}^{3}\right) + \mathrm{x} . {\mathrm{e}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}={\mathrm{x}}^{3}{\mathrm{e}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}} + 2{\mathrm{x}}^{5}{\mathrm{e}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}} + {\mathrm{xe}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}={\mathrm{e}}^{{\mathrm{x}}^{2}+{\mathrm{x}}^{4}}\left[{\mathrm{x}}^{3}+2{\mathrm{x}}^{5}+\mathrm{x}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}={\mathrm{e}}^{{\mathrm{x}}^{2} }. {\mathrm{e}}^{{\mathrm{x}}^{4}} \left[{\mathrm{x}}^{3}+2{\mathrm{x}}^{5}+\mathrm{x}\right]\phantom{\rule{0ex}{0ex}}
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Confluentes Mathematici is a mathematical research journal. Since its creation in 2009 by the Institut Camille Jordan UMR 5208 and the Unité de Mathématiques Pures et Appliquées UMR 5669 of the Université de Lyon, it reflects the wish of the mathematical community of Lyon—Saint-Étienne to participate in the new forms of scientific edittion. The journal is electronic only, fully open acces and without author charges.
The journal aims to publish high quality mathematical research articles in English, French or German. All domains of Mathematics (pure and applied) and Mathematical Physics will be considered, as well as the History of Mathematics. Confluentes Mathematici also publishes survey articles.
Authors are asked to pay particular attention to the expository style of their article, in order to be understood by all the communities concerned.
Papers published in Confluentes Mathematici are reviewed in Mathematical Reviews and Zentralblatt Math, and since 2017 are indexed in Scopus.
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Abstract: In this note, we present a result from an earlier work which shows that the so-called Lithium problem is nothing more than the consequence of several reactions being absent from the commonly used BBN software package.
Keywords: Big Bang Nucleosynthesis, Lithium Problem
The so-called Lithium problem refers to the discrepancy between the calculated density of Lithium at the end of Big Bang nucleosynthesis and the density deduced from observations with the calculated density being about 3 times greater than the observed value. The general viewpoint has been that the calculated value is correct and that some unknown process has removed the excess Lithium subsequent to nucleosynthesis. What we incidentally discovered during the development of a comprehensive new model of cosmology [1] is that it is the calculated value that is wrong and the reason for this is that several known Lithium reactions are not included in the standard BBN software package [2]. Because this discovery is buried within a discussion concerning cosmology, we thought that it would be useful to present this result on its own for the benefit of researchers concerned with the problem. The model, the method used to calculate the nucleosynthesis reaction rates, and the references for the reaction data are presented in [1] so please refer to that paper for the details. Note that in the new model, the temperature at the time that nucleosynthesis proper began is lower than the corresponding temperature assumed in the standard model. As a result, the curves shown here are similar to but not exactly the same as those of the standard model. The most notable difference is that the new model nucleosynthesis is compressed in time relative to that of the standard model.
In Table 1, we list all the reactions we were able to find by searching the internet. For each reaction entry, we include a letter “y” to indicate that the reaction is included in the standard BBN code and a blank otherwise.
During the development of our new model of cosmology, we ran a number of simulations with different initial particle densities and these are shown in [1]. A single example will be sufficient for our purpose and for this, we chose to show
Table 1. List of reactions.
Figure 1. Nucleosynthesis reaction with only BBN reactions included.
Figure 2. Nucleosynthesis reaction with all reactions included.
the results corresponding to a present-day particle density of
{n}_{part}\left({t}_{0}\right)=2\text{ }{m}^{-3}
. In Figure 1, we show the results obtained using just the reactions included in the BBN simulation model and in Figure 2, the results obtained with all the reactions included. The only difference between these two simulations is the list of reactions included. Comparing, we see that, with the exception of Lithium, the results are the same. For lithium, however, we find that the BBN calculation predicts a significantly larger density of 7Li than does the calculation including all the reactions.
The ratio is 2.8 which is exactly the value needed to explain the Lithium problem.
Cite this paper: Botke, J. (2021) The Lithium Problem—The Excess Isn’t Missing; It Was Never There. Journal of High Energy Physics, Gravitation and Cosmology, 7, 320-323. doi: 10.4236/jhepgc.2021.71015.
[1] Botke, J.C. (2020) A Different Cosmology—Thoughts from Outside the Box. Journal of High Energy Physics, Gravitation and Cosmology, 6, 573-566.
[2] Arbey, A., Auffinger, J., Hickerson, K.P. and Jenssen, E.S. (2018) AlterBBN v2: A Public Code for Calculating Big-Bang Nucleosynthesis Constraints in Alternative Cosmologies.
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Semi-Thue system - Wikipedia
(Redirected from Semi-Thue grammar)
In theoretical computer science and mathematical logic a string rewriting system (SRS), historically called a semi-Thue system, is a rewriting system over strings from a (usually finite) alphabet. Given a binary relation
{\displaystyle R}
between fixed strings over the alphabet, called rewrite rules, denoted by
{\displaystyle s\rightarrow t}
, an SRS extends the rewriting relation to all strings in which the left- and right-hand side of the rules appear as substrings, that is
{\displaystyle usv\rightarrow utv}
{\displaystyle s}
{\displaystyle t}
{\displaystyle u}
{\displaystyle v}
are strings.
The notion of a semi-Thue system essentially coincides with the presentation of a monoid. Thus they constitute a natural framework for solving the word problem for monoids and groups.
An SRS can be defined directly as an abstract rewriting system. It can also be seen as a restricted kind of a term rewriting system. As a formalism, string rewriting systems are Turing complete.[citation needed] The semi-Thue name comes from the Norwegian mathematician Axel Thue, who introduced systematic treatment of string rewriting systems in a 1914 paper.[1] Thue introduced this notion hoping to solve the word problem for finitely presented semigroups. Only in 1947 was the problem shown to be undecidable— this result was obtained independently by Emil Post and A. A. Markov Jr.[2][3]
2 Thue congruence
3 Factor monoid and monoid presentations
4 Connections with other notions
5 History and importance
A string rewriting system or semi-Thue system is a tuple
{\displaystyle (\Sigma ,R)}
Σ is an alphabet, usually assumed finite.[4] The elements of the set
{\displaystyle \Sigma ^{*}}
(* is the Kleene star here) are finite (possibly empty) strings on Σ, sometimes called words in formal languages; we will simply call them strings here.
R is a binary relation on strings from Σ, i.e.,
{\displaystyle R\subseteq \Sigma ^{*}\times \Sigma ^{*}.}
{\displaystyle (u,v)\in R}
is called a (rewriting) rule and is usually written
{\displaystyle u\rightarrow v}
If the relation R is symmetric, then the system is called a Thue system.
The rewriting rules in R can be naturally extended to other strings in
{\displaystyle \Sigma ^{*}}
by allowing substrings to be rewritten according to R. More formally, the one-step rewriting relation relation
{\displaystyle {\xrightarrow[{R}]{}}}
induced by R on
{\displaystyle \Sigma ^{*}}
for any strings
{\displaystyle s,t\in \Sigma ^{*}}
{\displaystyle s{\xrightarrow[{R}]{}}t}
{\displaystyle x,y,u,v\in \Sigma ^{*}}
{\displaystyle s=xuy}
{\displaystyle t=xvy}
{\displaystyle u\rightarrow v}
{\displaystyle {\xrightarrow[{R}]{}}}
{\displaystyle \Sigma ^{*}}
{\displaystyle (\Sigma ^{*},{\xrightarrow[{R}]{}})}
fits the definition of an abstract rewriting system. Obviously R is a subset of
{\displaystyle {\xrightarrow[{R}]{}}}
. Some authors use a different notation for the arrow in
{\displaystyle {\xrightarrow[{R}]{}}}
{\displaystyle {\xrightarrow[{R}]{}}}
) in order to distinguish it from R itself (
{\displaystyle \rightarrow }
) because they later want to be able to drop the subscript and still avoid confusion between R and the one-step rewrite induced by R.
Clearly in a semi-Thue system we can form a (finite or infinite) sequence of strings produced by starting with an initial string
{\displaystyle s_{0}\in \Sigma ^{*}}
and repeatedly rewriting it by making one substring-replacement at a time:
{\displaystyle s_{0}\ {\xrightarrow[{R}]{}}\ s_{1}\ {\xrightarrow[{R}]{}}\ s_{2}\ {\xrightarrow[{R}]{}}\ \ldots }
A zero-or-more-steps rewriting like this is captured by the reflexive transitive closure of
{\displaystyle {\xrightarrow[{R}]{}}}
{\displaystyle {\xrightarrow[{R}]{*}}}
(see abstract rewriting system#Basic notions). This is called the rewriting relation or reduction relation on
{\displaystyle \Sigma ^{*}}
induced by R.
Thue congruence[edit]
In general, the set
{\displaystyle \Sigma ^{*}}
of strings on an alphabet forms a free monoid together with the binary operation of string concatenation (denoted as
{\displaystyle \cdot }
and written multiplicatively by dropping the symbol). In a SRS, the reduction relation
{\displaystyle {\xrightarrow[{R}]{*}}}
{\displaystyle x{\xrightarrow[{R}]{*}}y}
{\displaystyle uxv{\xrightarrow[{R}]{*}}uyv}
{\displaystyle x,y,u,v\in \Sigma ^{*}}
{\displaystyle {\xrightarrow[{R}]{*}}}
is by definition a preorder,
{\displaystyle \left(\Sigma ^{*},\cdot ,{\xrightarrow[{R}]{*}}\right)}
forms a monoidal preorder.
Similarly, the reflexive transitive symmetric closure of
{\displaystyle {\xrightarrow[{R}]{}}}
{\displaystyle {\overset {*}{\underset {R}{\leftrightarrow }}}}
(see abstract rewriting system#Basic notions), is a congruence, meaning it is an equivalence relation (by definition) and it is also compatible with string concatenation. The relation
{\displaystyle {\overset {*}{\underset {R}{\leftrightarrow }}}}
is called the Thue congruence generated by R. In a Thue system, i.e. if R is symmetric, the rewrite relation
{\displaystyle {\xrightarrow[{R}]{*}}}
{\displaystyle {\overset {*}{\underset {R}{\leftrightarrow }}}}
Factor monoid and monoid presentations[edit]
{\displaystyle {\overset {*}{\underset {R}{\leftrightarrow }}}}
{\displaystyle {\mathcal {M}}_{R}=\Sigma ^{*}/{\overset {*}{\underset {R}{\leftrightarrow }}}}
{\displaystyle \Sigma ^{*}}
by the Thue congruence in the usual manner. If a monoid
{\displaystyle {\mathcal {M}}}
{\displaystyle {\mathcal {M}}_{R}}
{\displaystyle (\Sigma ,R)}
{\displaystyle {\mathcal {M}}}
We immediately get some very useful connections with other areas of algebra. For example, the alphabet {a, b} with the rules { ab → ε, ba → ε }, where ε is the empty string, is a presentation of the free group on one generator. If instead the rules are just { ab → ε }, then we obtain a presentation of the bicyclic monoid.
The importance of semi-Thue systems as presentation of monoids is made stronger by the following:
Theorem: Every monoid has a presentation of the form
{\displaystyle (\Sigma ,R)}
, thus it may be always be presented by a semi-Thue system, possibly over an infinite alphabet.[5]
In this context, the set
{\displaystyle \Sigma }
is called the set of generators of
{\displaystyle {\mathcal {M}}}
{\displaystyle R}
is called the set of defining relations
{\displaystyle {\mathcal {M}}}
. We can immediately classify monoids based on their presentation.
{\displaystyle {\mathcal {M}}}
finitely generated if
{\displaystyle \Sigma }
finitely presented if both
{\displaystyle \Sigma }
{\displaystyle R}
Connections with other notions[edit]
A semi-Thue system is also a term-rewriting system—one that has monadic words (functions) ending in the same variable as the left- and right-hand side terms,[6] e.g. a term rule
{\displaystyle f_{2}(f_{1}(x))\rightarrow g(x)}
is equivalent to the string rule
{\displaystyle f_{1}f_{2}\rightarrow g}
A semi-Thue system is also a special type of Post canonical system, but every Post canonical system can also be reduced to an SRS. Both formalisms are Turing complete, and thus equivalent to Noam Chomsky's unrestricted grammars, which are sometimes called semi-Thue grammars.[7] A formal grammar only differs from a semi-Thue system by the separation of the alphabet into terminals and non-terminals, and the fixation of a starting symbol amongst non-terminals. A minority of authors actually define a semi-Thue system as a triple
{\displaystyle (\Sigma ,A,R)}
{\displaystyle A\subseteq \Sigma ^{*}}
is called the set of axioms. Under this "generative" definition of semi-Thue system, an unrestricted grammar is just a semi-Thue system with a single axiom in which one partitions the alphabet into terminals and non-terminals, and makes the axiom a nonterminal.[8] The simple artifice of partitioning the alphabet into terminals and non-terminals is a powerful one; it allows the definition of the Chomsky hierarchy based on what combination of terminals and non-terminals the rules contain. This was a crucial development in the theory of formal languages.
In quantum computing, the notion of a quantum Thue system can be developed.[9] Since quantum computation is intrinsically reversible, the rewriting rules over the alphabet
{\displaystyle \Sigma }
are required to be bidirectional (i.e. the underlying system is a Thue system, not a semi-Thue system). On a subset of alphabet characters
{\displaystyle Q\subseteq \Sigma }
one can attach a Hilbert space
{\displaystyle \mathbb {C} ^{d}}
, and a rewriting rule taking a substring to another one can carry out a unitary operation on the tensor product of the Hilbert space attached to the strings; this implies that they preserve the number of characters from the set
{\displaystyle Q}
. Similar to the classical case one can show that a quantum Thue system is a universal computational model for quantum computation, in the sense that the executed quantum operations correspond to uniform circuit classes (such as those in BQP when e.g. guaranteeing termination of the string rewriting rules within polynomially many steps in the input size), or equivalently a Quantum Turing machine.
History and importance[edit]
Semi-Thue systems were developed as part of a program to add additional constructs to logic, so as to create systems such as propositional logic, that would allow general mathematical theorems to be expressed in a formal language, and then proven and verified in an automatic, mechanical fashion. The hope was that the act of theorem proving could then be reduced to a set of defined manipulations on a set of strings. It was subsequently realized that semi-Thue systems are isomorphic to unrestricted grammars, which in turn are known to be isomorphic to Turing machines. This method of research succeeded and now computers can be used to verify the proofs of mathematic and logical theorems.
At the suggestion of Alonzo Church, Emil Post in a paper published in 1947, first proved "a certain Problem of Thue" to be unsolvable, what Martin Davis states as "...the first unsolvability proof for a problem from classical mathematics -- in this case the word problem for semigroups."[10]
Davis also asserts that the proof was offered independently by A. A. Markov.[11]
^ Book and Otto, p. 36
^ Abramsky et al. p. 416
^ Salomaa et al., p.444
^ In Book and Otto a semi-Thue system is defined over a finite alphabet through most of the book, except chapter 7 when monoid presentation are introduced, when this assumption is quietly dropped.
^ Book and Otto, Theorem 7.1.7, p. 149
^ Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite Systems (1990) p. 6
^ D.I.A. Cohen, Introduction to Computer Theory, 2nd ed., Wiley-India, 2007, ISBN 81-265-1334-9, p.572
^ Dan A. Simovici, Richard L. Tenney, Theory of formal languages with applications, World Scientific, 1999 ISBN 981-02-3729-4, chapter 4
^ J. Bausch, T. Cubitt, M. Ozols, The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension, Ann. Henri Poincare 18(11), 2017 doi:10.1007/s00023-017-0609-7 pp. 3449-3513
^ Martin Davis (editor) (1965), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, after page 292, Raven Press, New York
^ A. A. Markov (1947) Doklady Akademii Nauk SSSR (N.S.) 55: 583–586
Ronald V. Book and Friedrich Otto, String-rewriting Systems, Springer, 1993, ISBN 0-387-97965-4.
Matthias Jantzen, Confluent string rewriting, Birkhäuser, 1988, ISBN 0-387-13715-7.
Martin Davis, Ron Sigal, Elaine J. Weyuker, Computability, complexity, and languages: fundamentals of theoretical computer science, 2nd ed., Academic Press, 1994, ISBN 0-12-206382-1, chapter 7
Elaine Rich, Automata, computability and complexity: theory and applications, Prentice Hall, 2007, ISBN 0-13-228806-0, chapter 23.5.
Samson Abramsky, Dov M. Gabbay, Thomas S. E. Maibaum (ed.), Handbook of Logic in Computer Science: Semantic modelling, Oxford University Press, 1995, ISBN 0-19-853780-8.
Grzegorz Rozenberg, Arto Salomaa (ed.), Handbook of Formal Languages: Word, language, grammar, Springer, 1997, ISBN 3-540-60420-0.
Landmark papers[edit]
Emil Post (1947) Recursive Unsolvability of a Problem of Thue, The Journal of Symbolic Logic 12: 1–11 via Project Euclid.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Semi-Thue_system&oldid=1071873546"
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YYiki: School buses are expensive
School buses are expensive
One thing that may not be visible to those who live in the US is that school buses simply don’t exist in many schools of the world because schools are so close to everyone and easily accessible by walking/biking.
US is notoriously built for cars. That means places are farther apart and there aren’t good pedestrian/bike paths. As a result, schools should cover a huge area and often it is impossible for kids to walk or bike to school. Thus, schools should run extensive school bus service.
For instance, I’m living right next to an elementary school. A large part of school district is essentially cut off by a dangerous, blind-drive hill without any pedestrian sidewalk nor safe bike path (near 10th & smith Rd.). This means everyone west of that hill should either take the bus, drive, or risk their lives everyday walking on the shoulder of a dangerous road.
Not having a safe path for walking or biking is expensive. Each school bus costs
100,000-
300,000. If it runs for 10-15 years, this is about
10,000-
30,000. Maintenance & fuel may cost
30,000-
40,000. And then we need a driver. We are not even considering the cost of acquiring and managing parking lots, support staffs, etc. All and all, this can reach up to
100,000 per bus. In other words, for every 50-70 students who cannot safely access school by walking or biking, we must pay up to
100,000 every year.
The lack of accessibility to school is expensive. Because schools are inaccessible without cars, we are paying for buses, cars, and parking lots instead of great teachers.
« articles/20220427 School bus »
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Use the 5-D Process to solve the following problem. Write an expression to represent each column of your table.
Yosemite Falls, the highest waterfall in the United States, is actually made up of three smaller falls. The Lower Yosemite Falls is
355
feet shorter than the Middle Cascades Falls. The Upper Yosemite Falls is
80
feet more than twice the Middle Cascades Falls. If the entire set of waterfalls is
2425
feet long, how tall is each of the smaller waterfalls?
Write an expression for the height of each falls.
x-355
x
80+2x
Now note that the sum of these expressions is
2425
(x-355)+(x)+(80+2x)=2425
x
x=\text{the height of the middle falls}
Use the value you found for
x
to calculate the height of the lower and upper falls.
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Index Controller - Indexed Finance
The index pool controller is a contract which tracks token values and sets portfolio targets using an adjusted capitalization-weighted formula. The NDX governance dao has the ability to create and manage token categories on the controller, which are baskets of assets with some arbitrary commonality.
Category Token Selection
The current rules for inclusion in each category can be found on their respective pages on the app.
Decentralized Finance Category
Cryptocurrency Category
Token categories are regularly sorted in descending order of the tokens' market caps using a weekly moving average of the tokens' prices. Market caps are extrapolated by taking the weekly moving average price of a token returned by the Uniswap oracle and multiplying by its total supply. In the future we plan on using more advanced metrics like float-adjusted capitalization, as is used in S&P indices, to get a more accurate representation of the value of tokens' active liquidity.
Index Token Selection
When an index is first deployed, and each month thereafter, the controller selects the top
n
tokens in its category as the target portfolio assets, where
n
is the index size set at deployment. The tokens must be sorted within the 24 hour period prior to the selection process. Further details can be found in the documentation regarding pool re-indexing.
Token Weighting Algorithm
Because tokens in the DeFi ecosystem have such a wide range of market caps, we decided to use an adjusted algorithm for computing token weights. Rather than weighing assets by market cap, we weigh them by market cap square root. This algorithm still favors tokens with larger market caps but does not result in some assets having such a massively higher weight than the others that the smaller cap tokens are effectively irrelevant, which would be the case in many indices if standard market cap weighting was used.
The algorithm to compute the weight of token
t
in an index with
l
m(T_{n})
is the extrapolated market cap of the
n^{th}
token, is:
w(t) = \frac{\sqrt{m(t)}}{\sum_{n=0}^{n < l} {\sqrt{m(T_{n})}}}
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numtheory(deprecated)/factorset - Maple Help
Home : Support : Online Help : numtheory(deprecated)/factorset
factorset(n)
Important: The numtheory package has been deprecated. Use the superseding command NumberTheory[PrimeFactors] instead.
The function factorset will compute the set of prime factors of its integer argument n.
The command with(numtheory,factorset) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{numtheory}\right):
\mathrm{factorset}\left(10\right)
{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}}
\mathrm{ifactor}\left(96\right)
{\left(\textcolor[rgb]{0,0,1}{2}\right)}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)
\mathrm{ifactors}\left(96\right)
[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}]]]
\mathrm{factorset}\left(96\right)
{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}}
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Jana’s mom gave her
\$100
to shop for some new school clothes. She is at the store and has picked out a pair of pants that cost
\$49.50
. She wants to spend the rest of her money buying various colors of a shirt that is on sale for
\$12.99
. Write an inequality that can be used to calculate the number of shirts she can buy. Solve your inequality. How many shirts can Jana buy?
Jana can spend less than or equal to
\$100
, but she has already decided on a pair of pants. This means that the money she has left over must be less than or equal to the
\$100
minus the money she is spending on the pair of pants.
\$12.99(x\ \text{shirts})\le\$100-\$49.50
Subtract and remove units to make the inequality easier to solve.
12.99x\le50.50
Now solve for the number of shirts,
x
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The new mineral species hydroxylhedyphane, ideally Ca2Pb3(AsO4)3(OH), has been discovered in the Långban Fe–Mn–(Ba–As–Pb–Sb) deposit, Filipstad district, Värmland, Sweden. It occurs as colourless prismatic crystals, up to 2.5 cm in length, forming an oriented intergrowth with a serpentine-subgroup mineral, as fracture-fillings cutting braunite and hausmannite ore. Electron-microprobe analysis yielded (mean of 16 spot analyses): P2O5 0.96(9), V2O5 0.07(4), As2O5 25.36(19), SiO2 0.91(2), CaO 7.74(11), MnO 0.03(2), BaO 2.95(10), PbO 59.81(50), Na2O 0.09(2), F 0.06(7), Cl 1.03(6), H2Ocalc 0.46, O = −(F + Cl) = –0.26, total 99.21. On the basis of 13 anions per formula unit, taking into account the crystal-structure data and the general formula of apatite-group minerals, the empirical formula of hydroxylhedyphane is M(1)(Ca1.56Pb0.41Mn0.01Na0.03)Σ2.01M(2)(Pb2.80Ba0.24Ca0.09)Σ3.13T(As2.64P0.16V0.01Si0.18)Σ2.99O12X[(OH)0.61Cl0.35F0.04]. Main diffraction lines are [d(Å) (relative intensity) hkl]: 4.354 (21) 200; 4.138 (24) 111; 3.643 (33) 002; 3.291(31) 210; 2.999 (100) 211; 2.949 (41) 112; 2.903 (86) 300; and 2.177 (23) 400. Hydroxylhedyphane is trigonal, space group P
3¯
, with a = 10.0414(3), c = 7.2752(2) Å, V = 635.28(4) Å3, Z = 2. The crystal structure has been refined to R1 = 0.034 on the basis of 1356 unique reflections with Fo > 4σ (Fo) and 67 refined parameters. It agrees with the topology of the other apatite-supergroup minerals, with a symmetry reduction from P63/m to P
3¯
and the splitting of the 4f M(1) site in the space group P63/m into two distinct 2d sites M(1) and M(1)’. This lowering of symmetry is likely related to the preferential partitioning of Pb at the M(1) site. Finally, infrared spectroscopy suggests the possible occurrence of minor CO2 in hydroxylhedyphane, but the crystal-structure refinement did not permit locating CO3 groups.
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hyperterm - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : sumtools : hyperterm
input a hypergeometric term
hyperterm(U, L, z, k)
This function is a shorthand for a hypergeometric term of variable k where U and L denote the lists of upper and lower parameters, and z is the evaluation point. The procedure Hyperterm is the corresponding inert form which remains unevaluated.
The command with(sumtools,hyperterm) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{sumtools}\right):
\mathrm{hyperterm}\left([a,b],[c],z,k\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{pochhammer}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{k}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{pochhammer}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{k}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{k}}}{\textcolor[rgb]{0,0,1}{\mathrm{pochhammer}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{k}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{!}}
\mathrm{Hyperterm}\left([a,b],[c],z,k\right)
\textcolor[rgb]{0,0,1}{\mathrm{Hyperterm}}\textcolor[rgb]{0,0,1}{}\left([\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{c}]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{k}\right)
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Lemma 10.63.4 (00LB)—The Stacks project
Lemma 10.63.4. Let $R$ be a ring, and $M$ an $R$-module. Suppose there exists a filtration by $R$-submodules
such that each quotient $M_ i/M_{i-1}$ is isomorphic to $R/\mathfrak p_ i$ for some prime ideal $\mathfrak p_ i$ of $R$. Then $\text{Ass}(M) \subset \{ \mathfrak p_1, \ldots , \mathfrak p_ n\} $.
Proof. By induction on the length $n$ of the filtration $\{ M_ i \} $. Pick $m \in M$ whose annihilator is a prime $\mathfrak p$. If $m \in M_{n-1}$ we are done by induction. If not, then $m$ maps to a nonzero element of $M/M_{n-1} \cong R/\mathfrak p_ n$. Hence we have $\mathfrak p \subset \mathfrak p_ n$. If equality does not hold, then we can find $f \in \mathfrak p_ n$, $f \not\in \mathfrak p$. In this case the annihilator of $fm$ is still $\mathfrak p$ and $fm \in M_{n-1}$. Thus we win by induction. $\square$
Comment #4945 by yogesh on February 22, 2020 at 20:34
In the proof, I think the inclusion
\mathfrak{p} \subset \mathfrak{p}_n
is going the wrong way and the rest of the proof is unnecessary. I think any nonzero element of
R/\mathfrak{p}_n
is annihilated by exactly by
\mathfrak{p}_n
Oops, sorry disregard my previous comment. I got forgot we're talkng about the annihilator as an element of
M
M/M_{n-1}
M
M/M_{n-1}
. I was thinking of using the previous result of associated primes of short exact sequences applied to
M_{n-1} \to M \to R/\mathfrak{p}_n
Ass(M) \subseteq Ass(M_{n-1} \cup \{ \mathfrak{p}_n\}
But the proof you give is basically the proof of that result, whose proof is omitted.
Comment #5967 by Maxim Mornev on March 10, 2021 at 19:57
Here is an alternative argument. By Lemma 02M3 it is enough to show that
\operatorname{Ass}(R/\mathfrak p_i) = \\{\mathfrak p_i\\}
. By Lemma 05BY the set
\operatorname{Ass}_R(R/\mathfrak p_i)
\operatorname{Ass}_{R/\mathfrak p_i}(R/\mathfrak p_i)
\operatorname{Spec} R
R/\mathfrak p_i
is a domain, so its only associated prime is
(0)
We can't use this argument as Lemma 10.63.14 comes later. The proof as we have it now is totally fine however.
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00LB. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 00LB, in case you are confused.
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Pole-zero plot of linear system approximated from nonlinear Simulink model - Simulink - MathWorks Benelux
Settling time (sec) <=
Percent overshoot <=
Damping ratio >=
Pole-zero plot of linear system approximated from nonlinear Simulink model
This block is the same as the Check Pole-Zero Characteristics block except for different default parameter settings in the Bounds tab.
Compute a linear system from a Simulink model and plot the poles and zeros on a pole-zero map.
During simulation, the software linearizes the portion of the model between specified linearization inputs and outputs, and plots the poles and zeros of the linear system.
You can specify multiple bounds that approximate second-order characteristics on the pole locations and view them on the plot. You can also check that the bounds are satisfied during simulation:
You can add multiple Pole-Zero Plot blocks to compute and plot the poles and zeros of various portions of the model.
The following table summarizes the Pole-Zero Plot block parameters, accessible via the block parameter dialog box.
(Optional) Specify bounds on pole-zero for assertion.
Include settling time bound in assertion.
Include percent overshoot bound in assertion.
Include damping ratio bound in assertion.
Include natural frequency bound in assertion.
Check that the pole locations satisfy approximate second-order bounds on the settling time, specified in Settling time (sec) <=. The software displays a warning if the poles lie outside the region defined by the settling time bound.
You can specify multiple settling time bounds on the linear system. The bounds also appear on the pole-zero plot. If you clear Enable assertion, the bounds are not used for assertion but continue to appear on the plot.
Off for Pole-Zero Plot block.
On for Check Pole-Zero Characteristics block.
Check that each pole lies in the region defined by the settling time bound, during simulation.
Do not check that each pole lies in the region defined by the settling time bound, during simulation.
Clearing this parameter disables the settling time bounds and the software stops checking that the bounds are satisfied during simulation. The bounds are also greyed out on the plot.
If you also specify other bounds, such as percent overshoot, damping ratio or natural frequency, but want to exclude the settling time bound from assertion, clear this parameter.
To only view the bounds on the plot, clear Enable assertion.
Parameter: EnableSettlingTime
Default: 'off' for Pole-Zero Plot block, 'on' for Check Pole-Zero Characteristics block.
Settling time, in seconds, of the second-order system.
[] for Pole-Zero Plot block
1 for Check Pole-Zero Characteristics block
Finite positive real scalar for one bound.
Cell array of finite positive real scalars for multiple bounds.
To assert that the settling time bounds are satisfied, select both Include settling time bound in assertion and Enable assertion.
You can add or modify settling time bounds from the plot window:
To add a new settling time bound, right-click the plot, and select Bounds > New Bound. Specify the new value in Settling time.
To modify a settling time bound, drag the corresponding bound segment. Alternatively, right-click the bound and select Bounds > Edit. Specify the new value in Settling time (sec) <.
Value: [] | 1 | finite positive real scalar| cell array of finite positive real scalars. Must be specified inside single quotes ('').
Default: '[]' for Pole-Zero Plot block, '1' for Check Pole-Zero Characteristics block.
Check that the pole locations satisfy approximate second-order bounds on the percent overshoot, specified in Percent overshoot <=. The software displays a warning if the poles lie outside the region defined by the percent overshoot bound.
You can specify multiple percent overshoot bounds on the linear system. The bounds also appear on the pole-zero plot. If you clear Enable assertion, the bounds are not used for assertion but continues to appear on the plot.
Check that each pole lies in the region defined by the percent overshoot bound, during simulation.
Do not check that each pole lies in the region defined by the percent overshoot bound, during simulation.
Clearing this parameter disables the percent overshoot bounds and the software stops checking that the bounds are satisfied during simulation. The bounds are also greyed out on the plot.
If you specify other bounds, such as settling time, damping ratio or natural frequency, but want to exclude the percent overshoot bound from assertion, clear this parameter.
Parameter: EnablePercentOvershoot
Percent overshoot of the second-order system.
10 for Check Pole-Zero Characteristics block
Real scalar for single percent overshoot bound.
Cell array of real scalars for multiple percent overshoot bounds.
The percent overshoot p.o can be expressed in terms of the damping ratio ζ, as:
p.o.=100{e}^{-\pi \zeta /\sqrt{1-{\zeta }^{2}}}.
To assert that the percent overshoot bounds are satisfied, select both Include percent overshoot bound in assertion and Enable assertion.
You can add or modify percent overshoot bounds from the plot window:
To add a new percent overshoot bound, right-click the plot, and select Bounds > New Bound. Select Percent overshoot in Design requirement type and specify the value in Percent overshoot <.
To modify a percent overshoot bound, drag the corresponding bound segment. Alternatively, right-click the bound, and select Bounds > Edit. Specify the new damping ratio for the corresponding percent overshoot value in Damping ratio >.
Value: [] | 10 | real scalar between 0 and 100 | cell array of real scalars between 0 and 100. Must be specified inside single quotes ('').
Default: '[]' for Pole-Zero Plot block, '10' for Check Pole-Zero Characteristics block.
Check that the pole locations satisfy approximate second-order bounds on the damping ratio, specified in Damping ratio >=. The software displays a warning if the poles lie outside the region defined by the damping ratio bound.
You can specify multiple damping ratio bounds on the linear system. The bounds also appear on the pole-zero plot. If you clear Enable assertion, the bounds are not used for assertion but continues to appear on the plot.
Check that each pole lies in the region defined by the damping ratio bound, during simulation.
Do not check that each pole lies in the region defined by the damping ratio bound, during simulation.
Clearing this parameter disables the damping ratio bounds and the software stops checking that the bounds are satisfied during simulation. The bounds are also greyed out on the plot.
If you specify other bounds, such as settling time, percent overshoot or natural frequency, but want to exclude the damping ratio bound from assertion, clear this parameter.
Parameter: EnableDampingRatio
Damping ratio of the second-order system.
Finite positive real scalar for single damping ratio bound.
Cell array of finite positive real scalars for multiple damping ratio bounds.
The damping ratio ζ, and percent overshoot p.o are related as:
p.o.=100{e}^{-\pi \zeta /\sqrt{1-{\zeta }^{2}}}.
To assert that the damping ratio bounds are satisfied, select both Include damping ratio bound in assertion and Enable assertion.
You can add or modify damping ratio bounds from the plot window:
To add a new damping ratio bound, right-click the plot and select Bounds > New Bound. Select Damping ratio in Design requirement type and specify the value in Damping ratio >.
To modify a damping ratio bound, drag the corresponding bound segment or right-click it and select Bounds > Edit. Specify the new value in Damping ratio >.
Parameter: DampingRatio
Value: [] | finite positive real scalar between 0 and 1 | cell array of finite positive real scalars between 0 and 1 . Must be specified inside single quotes ('').
Check that the pole locations satisfy approximate second-order bounds on the natural frequency, specified in Natural frequency (rad/sec). The natural frequency bound can be greater than, less than or equal one or more specific values. The software displays a warning if the pole locations do not satisfy the region defined by the natural frequency bound.
You can specify multiple natural frequency bounds on the linear system. The bounds also appear on the pole-zero plot. If Enable assertion is cleared, the bounds are not used for assertion but continue to appear on the plot.
Check that each pole lies in the region defined by the natural frequency bound, during simulation.
Do not check that each pole lies in the region defined by the natural frequency bound, during simulation.
Clearing this parameter disables the natural frequency bounds and the software stops checking that the bounds are satisfied during simulation. The bounds are also greyed out on the plot.
If you also specify settling time, percent overshoot or damping ratio bounds and want to exclude the natural frequency bound from assertion, clear this parameter.
Parameter: NaturalFrequencyBound
Natural frequency of the second-order system.
Finite positive real scalar for single natural frequency bound.
Cell array of finite positive real scalars for multiple natural frequency bounds.
To assert that the natural frequency bounds are satisfied, select both Include natural frequency bound in assertion and Enable assertion.
You can add or modify natural frequency bounds from the plot window:
To add a new natural frequency bound, right-click the plot and select Bounds > New Bound. Select Natural frequency in Design requirement type and specify the natural frequency in Natural frequency.
To modify a natural frequency bound, drag the corresponding bound segment or right-click it and select Bounds > Edit. Specify the new value in Natural frequency.
Parameter: NaturalFrequency
Value: [] | positive finite real scalar | cell array of positive finite real scalars. Must be specified inside single quotes ('').
|
Atomic_orbital Knowpia
Each orbital in an atom is characterized by a set of values of the three quantum numbers n, ℓ, and ml, which respectively correspond to the electron's energy, angular momentum, and an angular momentum vector component (the magnetic quantum number). Alternative to the magnetic quantum number, the orbitals are often labeled by the associated harmonic polynomials (e.g. xy, x2 − y2). Each such orbital can be occupied by a maximum of two electrons, each with its own projection of spin
{\displaystyle m_{s}}
. The simple names s orbital, p orbital, d orbital, and f orbital refer to orbitals with angular momentum quantum number ℓ = 0, 1, 2, and 3 respectively. These names, together with the value of n, are used to describe the electron configurations of atoms. They are derived from the description by early spectroscopists of certain series of alkali metal spectroscopic lines as sharp, principal, diffuse, and fundamental. Orbitals for ℓ > 3 continue alphabetically (g, h, i, k, ...),[3] omitting j[4][5] because some languages do not distinguish between the letters "i" and "j".[6]
Electron propertiesEdit
Formal quantum mechanical definitionEdit
Types of orbitalsEdit
The form of the Gaussian type orbital (Gaussians) has no radial nodes and decays as
{\displaystyle e^{-\alpha r^{2}}}
Bohr atomEdit
Modern conceptions and connections to the Heisenberg uncertainty principleEdit
Orbital namesEdit
Orbital notation and subshellsEdit
{\displaystyle X\,\mathrm {type} \ }
{\displaystyle X\,\mathrm {type} ^{y}\ }
X-ray notationEdit
Hydrogen-like orbitalsEdit
Complex orbitalsEdit
The azimuthal quantum number ℓ describes the orbital angular momentum of each electron and is a non-negative integer. Within a shell where n is some integer n0, ℓ ranges across all (integer) values satisfying the relation
{\displaystyle 0\leq \ell \leq n_{0}-1}
. For instance, the n = 1 shell has only orbitals with
{\displaystyle \ell =0}
, and the n = 2 shell has only orbitals with
{\displaystyle \ell =0}
{\displaystyle \ell =1}
The magnetic quantum number,
{\displaystyle m_{\ell }}
, describes the magnetic moment of an electron in an arbitrary direction, and is also always an integer. Within a subshell where
{\displaystyle \ell }
is some integer
{\displaystyle \ell _{0}}
{\displaystyle m_{\ell }}
ranges thus:
{\displaystyle -\ell _{0}\leq m_{\ell }\leq \ell _{0}}
The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of
{\displaystyle m_{\ell }}
{\displaystyle m_{\ell }=0}
Subshells are usually identified by their
{\displaystyle n}
{\displaystyle \ell }
{\displaystyle n}
is represented by its numerical value, but
{\displaystyle \ell }
is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with
{\displaystyle n=2}
{\displaystyle \ell =0}
Real orbitalsEdit
Animation of continuously varying superpositions between the
{\displaystyle p_{1}}
{\displaystyle p_{x}}
orbitals. Note that this animation does not utilize the Condon–Shortley phase convention.
In addition to the complex orbitals described above, it is common, especially in the chemistry literature, to utilize real atomic orbitals. These real orbitals arise from simple linear combinations of the complex orbitals. Using the Condon–Shortley phase convention, the real atomic orbitals are related to the complex atomic orbitals in the same way that the real spherical harmonics are related to the complex spherical harmonics. Letting
{\displaystyle \psi _{n,\ell ,m}}
denote a complex atomic orbital with quantum numbers
{\displaystyle n}
{\displaystyle l}
{\displaystyle m}
, we define the real atomic orbitals
{\displaystyle \psi _{n,\ell ,m}^{\text{real}}}
{\displaystyle \psi _{n,\ell ,m}^{\text{real}}={\begin{cases}{\sqrt {2}}(-1)^{m}{\text{Im}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m<0\\\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\{\sqrt {2}}(-1)^{m}{\text{Re}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m>0\end{cases}}={\begin{cases}{\frac {i}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}-(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m<0\\\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\{\frac {1}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}+(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m<0\\\end{cases}}}
{\displaystyle \psi _{n,\ell ,m}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell }^{m}(\theta ,\phi )}
{\displaystyle R_{nl}(r)}
the radial part of the orbital, this definition is equivalent to
{\displaystyle \psi _{n,\ell ,m}^{\text{real}}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell m}(\theta ,\phi )}
{\displaystyle Y_{\ell m}}
is the real spherical harmonic related to either the real or imaginary part of the complex spherical harmonic
{\displaystyle Y_{\ell }^{m}}
Real spherical harmonics are physically relevant when an atom is embedded in a crystalline solid, in which case there are multiple preferred symmetry axes but no single preferred direction[citation needed]. Real atomic orbitals are also more frequently encountered in introductory chemistry textbooks and shown in common orbital visualizations.[22] In the real hydrogen-like orbitals, the quantum numbers
{\displaystyle n}
{\displaystyle \ell }
have the same interpretation and significance as their complex counterparts, but
{\displaystyle m}
is no longer a good quantum number (though its absolute value is).
Some real atomic orbitals are given specific names beyond the simple
{\displaystyle \psi _{n,\ell ,m}}
designation. Orbitals with quantum number
{\displaystyle \ell }
{\displaystyle 0,1,2,3,4,5,6\ldots }
are referred to as
{\displaystyle s,p,d,f,g,h,\ldots }
orbitals. With this it already possible to assigns names to complex orbitals such as
{\displaystyle 2p_{\pm 1}=\psi _{2,1,\pm 1}}
where the first symbol is the
{\displaystyle n}
quantum number, the second number is the symbol for that particular
{\displaystyle \ell }
quantum number and the subscript is the
{\displaystyle m}
As an example of how the full orbital names are generated for real orbitals, we may calculate
{\displaystyle \psi _{n,1,\pm 1}^{\text{real}}}
. From the table of spherical harmonics, we have that
{\displaystyle \psi _{n,1,\pm 1}=R_{n,1}Y_{1}^{\pm 1}=\mp R_{n,1}{\sqrt {3/8\pi }}\cdot (x\pm iy)/r}
{\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}
{\displaystyle {\begin{aligned}\psi _{n,1,+1}^{\text{real}}=&R_{n,1}{\sqrt {\frac {3}{4\pi }}}\cdot {\frac {x}{r}}\\\psi _{n,1,-1}^{\text{real}}=&R_{n,1}{\sqrt {\frac {3}{4\pi }}}\cdot {\frac {y}{r}}\end{aligned}}}
Likewise we have
{\displaystyle \psi _{n,1,0}=R_{n,1}{\sqrt {3/4\pi }}\cdot z/r}
. As a more complicated example, we also have
{\displaystyle \psi _{n,3,+1}^{\text{real}}=R_{n,3}{\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}\cdot {\frac {x\cdot (5z^{2}-r^{2})}{r^{3}}}}
In all of these cases we generate a Cartesian label for the orbital by examining, and abbreviating, the polynomial in
{\displaystyle x}
{\displaystyle y}
{\displaystyle z}
appearing in the numerator. We ignore any terms in the
{\displaystyle z,r}
polynomial except for the term with the highest exponent in
{\displaystyle z}
. We then use the abbreviated polynomial as a subscript label for the atomic state, using the same nomenclature as above to indicate the
{\displaystyle n}
{\displaystyle \ell }
{\displaystyle {\begin{aligned}\psi _{n,1,-1}^{\text{real}}=&np_{y}={\frac {i}{\sqrt {2}}}\left(np_{-1}+np_{+1}\right)\\\psi _{n,1,0}^{\text{real}}=&np_{z}=2p_{0}\\\psi _{n,1,+1}^{\text{real}}=&np_{x}={\frac {1}{\sqrt {2}}}\left(np_{-1}-np_{+1}\right)\\\psi _{n,3,+1}^{\text{real}}=&nf_{xz^{2}}={\frac {1}{\sqrt {2}}}\left(nf_{-1}-nf_{+1}\right)\end{aligned}}}
Note that the expression above all use the Condon–Shortley phase convention which is favored by quantum physicists.[23][24] Other conventions for the phase of the spherical harmonics exists.[25][26] Under these different conventions the
{\displaystyle p_{x}}
{\displaystyle p_{y}}
orbitals may appear, for example, as the sum and difference of
{\displaystyle p_{+1}}
{\displaystyle p_{-1}}
, contrary to what is shown above.
Below is a tabulation of these Cartesian polynomial names for the atomic orbitals.[27][28] Note that there does not seem to be reference in the literature as to how to abbreviate the lengthy Cartesian spherical harmonic polynomials for
{\displaystyle \ell >3}
so there does not seem be consensus as to the naming of
{\displaystyle g}
orbitals or higher according to this nomenclature.
{\displaystyle \ell =0}
{\displaystyle s}
{\displaystyle \ell =1}
{\displaystyle p_{y}}
{\displaystyle p_{z}}
{\displaystyle p_{x}}
{\displaystyle \ell =2}
{\displaystyle d_{xy}}
{\displaystyle d_{yz}}
{\displaystyle d_{z^{2}}}
{\displaystyle d_{xz}}
{\displaystyle d_{x^{2}-y^{2}}}
{\displaystyle \ell =3}
{\displaystyle f_{y(3x^{2}-y^{2})}}
{\displaystyle f_{xyz}}
{\displaystyle f_{yz^{2}}}
{\displaystyle f_{z^{3}}}
{\displaystyle f_{xz^{2}}}
{\displaystyle f_{z(x^{2}-y^{2})}}
{\displaystyle f_{x(x^{2}-3y^{2})}}
Shapes of orbitalsEdit
The single s-orbitals (
{\displaystyle \ell =0}
) are shaped like spheres. For n = 1 it is roughly a solid ball (it is most dense at the center and fades exponentially outwardly), but for n = 2 or more, each single s-orbital is composed of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well). See illustration of a cross-section of these nested shells, at right. The s-orbitals for all n numbers are the only orbitals with an anti-node (a region of high wave function density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have angular momentum, and thus avoid the nucleus (having a wave node at the nucleus). Recently, there has been an effort to experimentally image the 1s and 2p orbitals in a SrTiO3 crystal using scanning transmission electron microscopy with energy dispersive x-ray spectroscopy.[29] Because the imaging was conducted using an electron beam, Coulombic beam-orbital interaction that is often termed as the impact parameter effect is included in the final outcome (see the figure at right).
Orbitals tableEdit
. . . ‡ . . . . . . ‡ . . . . . . ‡ . . . * . . . * . . . * . . . * . . . * . . . * . . . *
. . . . . . † . . . † . . . * . . . * . . . * . . . * . . . * . . . * . . . * . . . * . . . * . . . * . . . * . . . *
† The elements with this magnetic quantum number have been discovered, but their electronic configuration is only a prediction.
‡ The electronic configuration of the elements with this magnetic quantum number has only been confirmed for a spin quantum number of +1/2 (Ds, Rg and Cn are still missing).
These are the real-valued orbitals commonly used in chemistry. Only the
{\displaystyle m=0}
orbitals where are eigenstates of the orbital angular momentum operator,
{\displaystyle {\hat {L}}_{z}}
. The columns with
{\displaystyle m=\pm 1,\pm 2,\cdots }
are contain combinations of two eigenstates. See comparison in the following picture:
Qualitative understanding of shapesEdit
{\displaystyle u_{01}}
{\displaystyle u_{02}}
{\displaystyle u_{03}}
{\displaystyle r_{max}=2a_{0}}
{\displaystyle r_{max}=10a_{0}}
{\displaystyle r_{max}=20a_{0}}
{\displaystyle u_{11}}
{\displaystyle u_{12}}
{\displaystyle u_{13}}
{\displaystyle r_{max}=10a_{0}}
{\displaystyle r_{max}=20a_{0}}
{\displaystyle r_{max}=25a_{0}}
{\displaystyle u_{21}}
{\displaystyle u_{22}}
{\displaystyle u_{23}}
Orbital energyEdit
In atoms with a single electron (hydrogen-like atoms), the energy of an orbital (and, consequently, of any electrons in the orbital) is determined mainly by
{\displaystyle n}
{\displaystyle n=1}
orbital has the lowest possible energy in the atom. Each successively higher value of
{\displaystyle n}
has a higher level of energy, but the difference decreases as
{\displaystyle n}
increases. For high
{\displaystyle n}
, the level of energy becomes so high that the electron can easily escape from the atom. In single electron atoms, all levels with different
{\displaystyle \ell }
within a give{\displaystyle n}
are degenerate in the Schrödinger approximation, and have the same energy. This approximation is broken to a slight extent in the solution to the Dirac equation (where the energy depends on n and another quantum number j), and by the effect of the magnetic field of the nucleus and quantum electrodynamics effects. The latter induce tiny binding energy differences especially for s electrons that go nearer the nucleus, since these feel a very slightly different nuclear charge, even in one-electron atoms; see Lamb shift.
In atoms with multiple electrons, the energy of an electron depends not only on the intrinsic properties of its orbital, but also on its interactions with the other electrons. These interactions depend on the detail of its spatial probability distribution, and so the energy levels of orbitals depend not only o{\displaystyle n}
but also on
{\displaystyle \ell }
. Higher values of
{\displaystyle \ell }
are associated with higher values of energy; for instance, the 2p state is higher than the 2s state. When
{\displaystyle \ell =2}
, the increase in energy of the orbital becomes so large as to push the energy of orbital above the energy of the s-orbital in the next higher shell; when
{\displaystyle \ell =3}
The increase in energy for subshells of increasing angular momentum in larger atoms is due to electron–electron interaction effects, and it is specifically related to the ability of low angular momentum electrons to penetrate more effectively toward the nucleus, where they are subject to less screening from the charge of intervening electrons. Thus, in atoms of higher atomic number, the
{\displaystyle \ell }
of electrons becomes more and more of a determining factor in their energy, and the principal quantum numbers
{\displaystyle n}
The energy sequence of the first 35 subshells (e.g., 1s, 2p, 3d, etc.) is given in the following table. Each cell represents a subshell with
{\displaystyle n}
{\displaystyle \ell }
Electron placement and the periodic tableEdit
Relativistic effectsEdit
In the Bohr Model, an n = 1 electron has a velocity given by
{\displaystyle v=Z\alpha c}
, where Z is the atomic number,
{\displaystyle \alpha }
is the fine-structure constant, and c is the speed of light. In non-relativistic quantum mechanics, therefore, any atom with an atomic number greater than 137 would require its 1s electrons to be traveling faster than the speed of light. Even in the Dirac equation, which accounts for relativistic effects, the wave function of the electron for atoms with
{\displaystyle Z>137}
is oscillatory and unbounded. The significance of element 137, also known as untriseptium, was first pointed out by the physicist Richard Feynman. Element 137 is sometimes informally called feynmanium (symbol Fy).[35] However, Feynman's approximation fails to predict the exact critical value of Z due to the non-point-charge nature of the nucleus and very small orbital radius of inner electrons, resulting in a potential seen by inner electrons which is effectively less than Z. The critical Z value, which makes the atom unstable with regard to high-field breakdown of the vacuum and production of electron-positron pairs, does not occur until Z is about 173. These conditions are not seen except transiently in collisions of very heavy nuclei such as lead or uranium in accelerators, where such electron-positron production from these effects has been claimed to be observed.
pp hybridisation (conjectured)Edit
Transitions between orbitalsEdit
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Now Cecil can slide on his tightrope!
The first time that he crossed his new tightrope, he did three slides and walked five feet before going down the ladder. The second time, he did only two slides and then walked six feet to the end of the rope, as shown in the diagram at right. Write an equation to represent Cecil's two trips across the tightrope.
In Cecil's first attempt he crossed the tightrope like this:
x+x+x+5
=
length of the tightrope
In Cecil's second attempt he crossed the tightrope like this:
x+x+6=
x+x+x+5=x+x+6
x
's to make the equation
3x+5=2x+6
How far does Cecil travel in each slide? Show how you know.
Cecil travels
1
foot in each slide. Remember to show how you know.
How long is the tightrope? How can you tell?
Three slides of one foot each and
5
more feet would be
3(1)+5
. How long is this?
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Trifocal tensor - Wikipedia
In computer vision, the trifocal tensor (also tritensor) is a 3×3×3 array of numbers (i.e., a tensor) that incorporates all projective geometric relationships among three views. It relates the coordinates of corresponding points or lines in three views, being independent of the scene structure and depending only on the relative motion (i.e., pose) among the three views and their intrinsic calibration parameters. Hence, the trifocal tensor can be considered as the generalization of the fundamental matrix in three views. It is noted that despite the tensor being made up of 27 elements, only 18 of them are actually independent.
There is also a so-called calibrated trifocal tensor, which relates the coordinates of points and lines in three views given their intrinsic parameters and encodes the relative pose of the cameras up to global scale, totalling 11 independent elements or degrees of freedom. The reduced degrees of freedom allow for fewer correspondences to fit the model, at the cost of increased nonlinearity.[1]
1 Correlation slices
2 Trilinear constraints
4.1 Uncalibrated
4.2 Calibrated
Correlation slices[edit]
The tensor can also be seen as a collection of three rank-two 3 x 3 matrices
{\displaystyle {\mathbf {T} }_{1},\;{\mathbf {T} }_{2},\;{\mathbf {T} }_{3}}
known as its correlation slices. Assuming that the projection matrices of three views are
{\displaystyle {\mathbf {P} }=[{\mathbf {I} }\;|\;{\mathbf {0} }]}
{\displaystyle {\mathbf {P} }'=[{\mathbf {A} }\;|\;{\mathbf {a} }_{4}]}
{\displaystyle {\mathbf {P} ''}=[{\mathbf {B} }\;|\;{\mathbf {b} }_{4}]}
, the correlation slices of the corresponding tensor can be expressed in closed form as
{\displaystyle {\mathbf {T} }_{i}={\mathbf {a} }_{i}{\mathbf {b} }_{4}^{t}-{\mathbf {a} }_{4}{\mathbf {b} }_{i}^{t},\;i=1\ldots 3}
{\displaystyle {\mathbf {a} }_{i},\;{\mathbf {b} }_{i}}
are respectively the ith columns of the camera matrices. In practice, however, the tensor is estimated from point and line matches across the three views.
Trilinear constraints[edit]
One of the most important properties of the trifocal tensor is that it gives rise to linear relationships between lines and points in three images. More specifically, for triplets of corresponding points
{\displaystyle {\mathbf {x} }\;\leftrightarrow \;{\mathbf {x} }'\;\leftrightarrow \;{\mathbf {x} }''}
and any corresponding lines
{\displaystyle {\mathbf {l} }\;\leftrightarrow \;{\mathbf {l} }'\;\leftrightarrow \;{\mathbf {l} }''}
through them, the following trilinear constraints hold:
{\displaystyle ({\mathbf {l} }^{\prime t}\left[{\mathbf {T} }_{1},\;{\mathbf {T} }_{2},\;{\mathbf {T} }_{3}\right]{\mathbf {l} }'')[{\mathbf {l} }]_{\times }={\mathbf {0} }^{t}}
{\displaystyle {\mathbf {l} }^{\prime t}\left(\sum _{i}x_{i}{\mathbf {T} }_{i}\right){\mathbf {l} }''=0}
{\displaystyle {\mathbf {l} }^{\prime t}\left(\sum _{i}x_{i}{\mathbf {T} }_{i}\right)[{\mathbf {x} }'']_{\times }={\mathbf {0} }^{t}}
{\displaystyle [{\mathbf {x} }']_{\times }\left(\sum _{i}x_{i}{\mathbf {T} }_{i}\right){\mathbf {l} }''={\mathbf {0} }}
{\displaystyle [{\mathbf {x} }']_{\times }\left(\sum _{i}x_{i}{\mathbf {T} }_{i}\right)[{\mathbf {x} }'']_{\times }={\mathbf {0} }_{3\times 3}}
{\displaystyle [\cdot ]_{\times }}
denotes the skew-symmetric cross product matrix.
Given the trifocal tensor of three views and a pair of matched points in two views, it is possible to determine the location of the point in the third view without any further information. This is known as point transfer and a similar result holds for lines and conics. For general curves, the transfer can be realized through a local differential curve model of osculating circles (i.e., curvature), which can then be transferred as conics.[2] The transfer of third-order models reflecting space torsion using calibrated trifocal tensors have been studied,[3] but remains an open problem for uncalibrated trifocal tensors.
Uncalibrated[edit]
The classical case is 6 point correspondences[4][5] giving 3 solutions.
The case estimating the trifocal tensor from 9 line correspondences has only recently been solved.[6]
Calibrated[edit]
Estimating the calibrated trifocal tensor has been cited as notoriously difficult, and requires 4 point correspondences.[7]
The case of using only three point correspondences has recently been solved, where the points are attributed with tangent directions or incident lines; with only two of the points having incident lines, this is a minimal problem of degree 312 (so there can be at most 312 solutions) and is relevant for the case of general curves (whose points have tangents), or feature points with attributed directions (such as SIFT directions).[8] The same technique solved the mixed case of three point correspondences and one line correspondence, which has also been shown to be minimal with degree 216.
^ Martyushev, E. V. (2017). "On Some Properties of Calibrated Trifocal Tensors". Journal of Mathematical Imaging and Vision. 58 (2): 321–332. arXiv:1601.01467. doi:10.1007/s10851-017-0712-x. S2CID 1634602.
^ Schmid, Cordelia (2000). "The Geometry and Matching of Lines and Curves Over Multiple Views" (PDF). International Journal of Computer Vision. 40 (3): 199–233. doi:10.1023/A:1008135310502. S2CID 11844321.
^ Fabbri, Ricardo; Kimia, Benjamin (2016). "Multiview Differential Geometry of Curves". International Journal of Computer Vision. 120 (3): 324–346. arXiv:1604.08256. Bibcode:2016arXiv160408256F. doi:10.1007/s11263-016-0912-7. S2CID 11908870.
^ Richard Hartley and Andrew Zisserman (2003). "Online Chapter: Trifocal Tensor" (PDF). Multiple View Geometry in computer vision. Cambridge University Press. ISBN 978-0-521-54051-3.
^ Heyden, A. (1995). "Reconstruction from Image Sequences by means of Relative Depths". Proceedings of IEEE International Conference on Computer Vision. pp. 1058–1063. doi:10.1109/ICCV.1995.466817. ISBN 0-8186-7042-8. S2CID 7789642.
^ Larsson, Viktor; Astrom, Kalle; Oskarsson, Magnus (2017). "Efficient Solvers for Minimal Problems by Syzygy-Based Reduction". 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). pp. 2383–2392. doi:10.1109/CVPR.2017.256. ISBN 978-1-5386-0457-1. S2CID 13069612.
^ Nister, David; Schaffalitzky, Frederik (2006). "Four Points in Two or Three Calibrated Views: Theory and Practice". International Journal of Computer Vision. 67 (2): 211–231. doi:10.1007/s11263-005-4265-x. S2CID 10231211.
^ Fabbri, Ricardo; Duff, Timothy; Fan, Hongyi; Regan, Margaret; de Pinho, David; Tsigaridas, Elias; Wampler, Charles; Hauenstein, Jonathan; Kimia, Benjamin; Leykin, Anton; Pajdla, Tomas (23 Mar 2019). "Trifocal Relative Pose from Lines at Points and its Efficient Solution". arXiv:1903.09755 [cs.CV].
Hartley, Richard I. (1997). "Lines and Points in Three Views and the Trifocal Tensor". International Journal of Computer Vision. 22 (2): 125–140. doi:10.1023/A:1007936012022. S2CID 8979544.
Torr, P. H. S.; Zisserman, A. (1997). "Robust Parameterization and Computation of the Trifocal Tensor". Image and Vision Computing. 15 (8): 591–607. CiteSeerX 10.1.1.41.3172. doi:10.1016/S0262-8856(97)00010-3.
Visualization of trifocal geometry (originally by Sylvain Bougnoux of INRIA Robotvis, requires Java)
Matlab implementation of the uncalibrated trifocal tensor estimation and comparison to pairwise fundamental matrices
C++ implementation of the calibrated trifocal tensor estimation using optimized Homotopy Continuation code. Presently includes cases of three corresponding points with lines at these points (as in feature positions and orientations, or curve points with tangents), and also for three corresponding points and one line correspondence.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Trifocal_tensor&oldid=1071108810"
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Assignment 4, Shaders Part 5
In projects 3-1 and 3-2, we were doing all of our raytracing computation on CPU. You've likely felt the effects of this already, waiting minutes to render a single frame with any realistic lighting, even with threading. For real-time and interactive applications, which often have framerates of 60 fps (that's 60 frames per second), this is just impossibly slow. In this part, you will get a glimpse of how things may be accelerated by writing a few basic GLSL shader programs. Shaders are isolated programs that run in parallel on GPU, executing sections of the graphics pipeline, taking in an input, and outputing a single 4 dimensional vector. Recall the brief overview of shader programs given in lecture.
First let's get acquainted with GLSL, a C like language in which we will write our shaders. This is a great highlevel overview of the basic constructs of the language. Take a minute to look through the definitions. TLDR:
A GLSL shader can have functions just like C
An attribute is an input to a vertex shader (position, normal, uv coordinates)
A uniform is shared by all instances of the running program (light position, textures, transform matrices)
A varying is typically written into by the vertex shader for use in the fragment shader (transformed positions, normals, uv coordinates)
GLSL has built-in types and operations that make vector math simpler like vec3, vec4, mat4, dot, length
We will be dealing with two basic OpenGL shader types:
vertex shaders: These shaders generally apply transforms to vertices, modifying their geometric properties like position and normal vectors, writing the final position of the vertex to gl_Position in addition to writing varyings for use in the fragment shader.
fragment shaders: After rasterization, we end up with fragments, which these shaders process. These shaders generally take in geometric attributes of the fragment calculated by the vertex shader to compute and write a color into out_color.
To create a shader program, we compile and link a vertex and fragment shader, the output of the vertex shader becomes the input of the fragment shader.
NB: Because we didn't want to make a high-end GPU part of CS 184's required hardware, our shaders only use up to OpenGL 3.3 features. Shaders post OpenGL 3.3 have some much nicer features and syntax, but operate on entirely the same principles. If you're interseted, learnopengl.com is an excellent guide for modern openGL programming.
The skeleton will automatically search for and load shader programs contained in the shaders directory. A simple shader program is made of two parts:
A .vert file, which specifies a vertex shader. The vertex shader is responsible for reading and writing all per-vertex values. These per-vertex values are then interpolated via barycentric coordinates across the polygon's face.
A .frag file, which specifies a fragment shader. The fragment shader is responsible for writing all per-pixel* values. It takes as input the interpolated per-vertex values from the vertex shader and outputs the final color of that pixel.
The skeleton will scan the shaders directory for <NAME>.frag shaders, and link it to the corresponding <NAME>.vert, defaulting to Default.vert if none exists. (Shader-based final projects might want to take advantage of this behavior!)
When writing your shaders, be sure to pay extra attention to the types you are using. GLSL 1.0 will not automatically promote ints to floats or demote floats to ints. Function calls must match their declared types exactly as well, so something like max(2, 3.0) will cause the shader to fail compilation. In addition, the built ins gl_Position and gl_FragColor are both of type vec4, the first expecting homogenous coordinates and the second expecting an rgba vector. Many of our calculations will be done using vec3's, so don't forget to convert back to a vec4 (an easy way is simply use vec4(my_vec_3, w_coordinate)).
*Note: technically, a fragment shader writes "per-fragment" not "per-pixel" values. An OpenGL fragment is much closer to our notion of a sample than a pixel. Recall that in super-sampling anti-aliasing, a single pixel might represent the averaged color of multiple individual sample points. Additionally, we might take samples that are overwritten or occluded by those of other polygons. Nevertheless, for most purposes, we can think of a fragment as equivalent to a pixel.
Task 1: Diffuse Shading
shaders/Default.vert
shaders/Diffuse.frag
In project 3-1, you saw how diffuse objects appear under light in the world. Let's try to recreate this in a shader program. In Default.vert, we can see an example of a simple vertex shader. It takes in as input the model-space attributes of in_position and in_normal, both of type vec4, in addition to the uniforms u_model, u_view_projection, which are the matrices used to transform a point from model space into world space, and from world space to view space to screen space respectively. We output two values for use in the fragment shader: v_position and v_normal.
Taking a look at the main function, we see that the world space position and normal vector are written into the corresponding varyings v_position and v_normal and the screen space position is written into gl_Position.
Recall the formula for diffuse lighting from the lecture:
\mathbf{L}_d = \mathbf{k}_d\ (\mathbf{I} / r^2)\ \max(0, \mathbf{n} \cdot \mathbf{l})
Now in Diffuse.frag, output into out_color the color of the fragment. The light intensity and position are provided as uniforms to the fragment shader. You may choose your own diffuse coefficient vector (you probably just want 1).
After completing this part, you should be able to render the cloth like below:
Task 2: Blinn-Phong Shading
shaders/Phong.frag
Now let's create a shader capable of performing Blinn-Phong shading. Recall the equation for Blinn-Phong shading from lecture:
\mathbf{L} = \mathbf{k}_a\ \mathbf{I}_a\ + \mathbf{k}_d\ (\mathbf{I} / r^2)\ \max(0, \mathbf{n} \cdot \mathbf{l})\ + \mathbf{k}_s\ (\mathbf{I} / r^2)\ \max(0, \mathbf{n} \cdot \mathbf{h})^p
Notice we add both an ambient light component and specular reflection component to the diffuse lighting from the previous part to calculate the output light. As before, the light intensity and position are passed as uniforms to the shader. Complete the main function in Phong.frag to implement the Blinn-Phong shading model. You may decide on
k_a, k_d, k_s, I_a, p
to suite your tastes. After completion, you should be able to see some nice specular lighting effects:
Task 3: Texture Mapping
shaders/Texture.frag
Looking at Default.vert, we can notice that this shader also takes in a in_uv coordinate associated with the instance's vertex and writes it into v_uv for use in the fragment shader.
We can sample from the u_texture_1 uniform using the built-in function texture(sampler2D tex, vec2 uv), which samples from a texture tex at the texture space coordinate uv. In Texture.frag, complete the shader so that the sampled spectrum is output as the fragment's color.
Task 4: Displacement and Bump Mapping
shaders/Bump.frag
shaders/Displacement.vert
shaders/Displacement.frag
We can use textures for more than just determining the color on a mesh. With displacement and bump mapping, we can encode a height map in a texture to be processed and applied by a shader program. NOTE: You don't have to generate exactly the same results as we have in our reference images, just make sure the results are plausible.
4.1: Bump Mapping
In bump mapping, we modify the normal vectors of an object so that the fragment shader gives the illusion of detail (such as bumps) on an object. How can we calculate the new normals given the height map?
To make our calculations easier, we can work in object space, where all normal vectors initially point directly out of the local vertex and have a z-coordinate of 1. Given a vector in object space, we can transform it into back into model space by multiplying by the tangent-bitangent-normal (TBN) matrix.
We already know the original model-space normal vector
\mathbf{n}
as this is an input to our vertex shader. We can pre-compute the tangent vector
\mathbf{t}
from the mesh geometry, and this will also be passed as an attribute to the vertex shader. The bitangent should be orthogonal to both the tangent and normal and can be found using the cross product
\mathbf{b} = \mathbf{n} \times \mathbf{t}
. We then have:
TBN = [ \mathbf{t}\ \ \mathbf{b}\ \ \mathbf{n}]
Because we have access to the entire height map, we can compute the local space normals by looking at how the height changes as we make small changes in
u
v
h(u, v)
be a function that returns the height encoded by a height map at texture coordinates
u
v
w
h
be the width and height of our texture.
dU = (h(u + 1 / w, v) - h(u, v)) * k_h * k_n
dV = (h(u, v + 1 / h) - h(u, v)) * k_h * k_n
k_h
is a height scaling factor and
k_n
is a normal scaling factor represented in our shader by the u_height_scaling and u_normal_scaling variables. The local space normal is then just
\mathbf{n}_o = (-dU, -dV, 1)
. Our displaced model space normal is then
\mathbf{n}_d = TBN\ \mathbf{n}_o
Complete the main function in Bump.frag to calculate the displaced world space normal. The height map is stored in the u_texture_2 texture and the resolution of the texture is stored in vec2 u_texture_2_size. One such
h(u, v)
you could use would be the r component of the color vector stored in the texture at coordinates
(u, v)
. On completion, you should be able to see some realistic lighting effects on the mapped bumps:
4.2: Displacement Mapping
In displacement mapping, we modify the position of vertices to reflect the height map in addition to modifying the normals to be consistent with the new geometry. First, copy your fragment shader from Bump.frag into Displacement.frag. Modify Displacement.vert so that it also displaces the vertex positions in the direction of the original model space vertex normal scaled by the u_height_scaling variable:
\mathbf{p}' = \mathbf{p} + \mathbf{n} * h(u, v) * k_h
On completion, you should be able to notice the change in geometry:
Task 5: Environment-mapped Reflections
shaders/Mirror.frag
In the pathtracer project, we saw a simple model for a mirror material. We took the incoming eye-ray, reflected it across the surface normal to get the outgoing direction, and then sampled the environment for that direction's incoming radiance.
Here, we will approximate the incoming radiance sample using an environment map, which is a pre-computed store of the direction-to-radiance mapping we previously calculated explicitly via monte carlo integration. We do this by enclosing our scene inside of an infinitely-large room with the environment's appearance painted on the inside. If this sounds like the environment lights from the previous project, you're right! However, as a part of the approximation, we will sample the environment map without shadow rays (assumes no intersections with other parts of the scene).
Using the camera's position u_cam_pos and the fragment's position v_position, compute the outgoing eye-ray,
w_o
. Then reflect
w_o
across the surface normal given in v_normal to get
w_i
. Finally, sample the environment map u_texture_cubemap for the incoming direction
w_i
We can sample from the u_texture_cubemap uniform again using the built-in function texture(samplerCube tex, vec3 dir) overload, which samples from a texture tex at looking down the direction dir.
(For the cubemap used above and other great cubemap textures, check out Emil Persson's site.)
shaders/Custom.vert
shaders/Custom.frag
Make your own shader or scene! If you need more space to work, you can edit loadObjectsFromFile() in main and add a new object type. It can be as complex as you desire.
Add color controls to a previous shader, add a time uniform to generate procedural scenes, add a new object type or texture and see how your shaders work with them! Try combining together all 5 shader types and see what you can make!
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Labelling problems for graphs consist in building distributed data structures, making it possible to check a given graph property or to compute a given function, the arguments of which are vertices. For an inductively computable function
D
G
n
vertices and of clique-width at most
k
k
is fixed, we can associate with each vertex
G
a piece of information (bit sequence)
\mathrm{lab}\left(x\right)
O\left({log}^{2}\left(n\right)\right)
such that we can compute
D
in constant time, using only the labels of its arguments. The preprocessing can be done in time
O\left(h.n\right)
h
is the height of the syntactic tree of
G
. We perform an inductive computation, without using the tools of monadic second order logic. This enables us to give an explicit labelling scheme and to avoid constants of exponential size.
Classification : 68R10, 90C35
Mots clés : terms, graphs, clique-width, labeling schemes, inductive computation
author = {Carr\`ere, Fr\'ed\'erique},
title = {Inductive computations on graphs defined by clique-width expressions},
AU - Carrère, Frédérique
TI - Inductive computations on graphs defined by clique-width expressions
Carrère, Frédérique. Inductive computations on graphs defined by clique-width expressions. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 3, pp. 625-651. doi : 10.1051/ita/2009010. http://www.numdam.org/articles/10.1051/ita/2009010/
[1] S. Arnborg, J. Lagergren, D. Seese, Easy problems for tree-decomposable graphs. J. Algor. 12 (1991) 308-340. | MR 1105479 | Zbl 0734.68073
[2] H. Bodlaender, Treewidth: Algorithmic techniques and results, in Proceedings 22nd International Symposium on Mathematical Foundations of Computer Science. Lect. Notes Comput. Sci. 1295 (1997) 19-36. | MR 1640205 | Zbl 0941.05057
[3] R.B. Borie, R.G. Parker, C.A. Tovey, Algorithms on Recursively Constructed Graphs. CRC Handbook of Graph Theory (2003) 1046-1066.
[4] S. Chaudhuri, C.D. Zaroliagis, Optimal parallel shortest paths in small treewidth digraphs, in: Proceedings 3rd Annual European Symposium on Algorithms. Lect. Notes Comput. Sci. 979 (1995) 31-45. | MR 1460735
[5] D.G. Corneil, M. Habib, J.M. Lanlignel, B.A. Reed, U. Rotics, Polynomial time recognition algorithm of clique-width
\le 3
graphs, LATIN'00. Lect. Notes Comput. Sci. 1776 (2000) 126-134. | Zbl 0961.05062
[6] B. Courcelle, Clique-width of countable graphs: a compactness property. Discrete Math. 276 (2003) 127-148. | MR 2046629 | Zbl 1073.68062
[7] B. Courcelle, J.A. Makowsky, U. Rotics, On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Appl. Math. 108 (2001) 23-52. | MR 1804711 | Zbl 0972.05023
[8] B. Courcelle, M. Mosbah, Monadic second-order evaluations of tree-decomposable graphs. Theoret. Comput. Sci. 109 (1993) 49-82. | MR 1205621 | Zbl 0789.68083
[9] B. Courcelle, S. Olariu, Upper bounds to clique-width of graphs. Discrete Appl. Math. 101 (2000) 77-114. | MR 1743732 | Zbl 0958.05105
[10] B. Courcelle, A. Twigg, Compact forbidden-set routing, in: STACS'07. Lect. Notes Comput. Sci. 4393 (2007) 37-48. | MR 2361056
[11] B. Courcelle, R. Vanicat, Query efficient implementations of graphs of bounded clique-width. Discrete Appl. Math. 131 (2003) 129-150. | MR 2016489 | Zbl 1029.68113
[12] C. Demetrescu, G.F. Italiano, a new approach to dynamic all pairs shortest paths, in Proceedings of. the 35. th. Annual ACM Symposium on the Theory of Computing (2003) 159-166. | MR 2121050
[13] R.G. Downey, M.R. Fellows, Parametrized Complexity. Springer Verlag (1999).
[14] J. Engelfriet, G. Rozenberg, Node replacement graph grammars, in Handbook of Graph Grammars and Computing by Graph Transformation, Foundations, Vol. 1, edited by G. Rozenberg. World Scientific (1997) 1-94. | MR 1480953
[15] M.R. Fellows, F.A. Rosamond, U. Rotics, S. Szeider, Clique-width minimization is NP-hard. Proceedings of. the 38. th. Annual ACM Symposium on the Theory of Computing (2006) 354-362. | MR 2277161
[16] J. Flum, M. Grohe, Theory of parametrized complexity. Springer Verlag (2006). | MR 2238686
[17] M. Frick, M. Grohe, The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic 130 (2004) 3-31. | MR 2092847 | Zbl 1062.03032
[18] C. Gavoille, M. Katz, Nir A. Katz, C. Paul, D. Peleg, Approximate distance labeling schemes, ESA'01. Lect. Notes Comput. Sci. 2161 (2001) 476-488. | MR 1969938 | Zbl 1006.68542
[19] C. Gavoille, C. Paul, Distance labeling scheme and split decomposition. Discrete Math. 273 (2003) 115-130. | MR 2025945 | Zbl 1029.05136
[20] C. Gavoille, D. Peleg, Compact and localized distributed data structures. J. Distrib. Comput. 16 (2003) 111-120.
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-NLC graphs and polynomial algorithms. Disc. Appl. Math. 54 (1994) 251-266. | MR 1300250 | Zbl 0812.68106
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[26] D. Seese, Interpretability and tree automata: A simple way to solve algorithmic problems on graphs closely related to trees, in Tree Automata and Languages, edited by M. Nivat, A. Podelski. North-Holland (1992) 83-114. | MR 1196733 | Zbl 0798.68059
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|
General Chemistry/Gas Laws/Answers - Wikibooks, open books for an open world
< General Chemistry | Gas Laws
Answers to Gas Laws Questions[edit | edit source]
1. The Ideal Gas Law accounts for chemical change. The Combined Gas Law accounts for changes in pressure, volume, and temperature. These are physical properties. The Ideal Gas Law accounts for these properties along with molar mass. Although molar mass is a physical property as well, the amount of molecules in an isolated gas would only change in the event of a chemical reaction. Thus, the Ideal Gas Law can account for chemical reactions.
2. Density is mass divided by volume, and the number of moles equals the mass divided by molecular weight. So:
{\displaystyle D={\frac {m}{V}}}
{\displaystyle PV=nRT}
{\displaystyle {\frac {n}{V}}={\frac {P}{RT}}}
{\displaystyle D={\frac {P\times (MW)}{RT}}}
Substituting and solving gives us a density of 80.7 g/m3. Remember that hydrogen is diatomic, so its molecular mass is 2.0 g/mol.
3. Simply substitute into the Ideal Gas Law and solve. 0.144 m3.
4. Stoichometrically, one mole of H2 is needed for each mole of H2S. We must determine the number of moles in 7.4L of hydrogen sulfide, then convert that number of moles into liters of hydrogen. This is a two-step problem, both using the Ideal Gas Law.
Solving for moles of hydrogen sulfide: 0.330 mol
Solving for liters of hydrogen: 7.39 L H2
Retrieved from "https://en.wikibooks.org/w/index.php?title=General_Chemistry/Gas_Laws/Answers&oldid=1415834"
Book:General Chemistry
|
Latisha wants to get at least a
\text{B}+
in her history class. To do so, she needs to have an overall average of at least
86\%
. So far, she has taken three tests and has gotten scores of
90\%
82\%
81\%
Use the 5-D Process to help Latisha determine what percent score she needs on the fourth test to get the overall grade that she wants. The fourth test is the last test of the grading period.
Remember, you are trying to find the average. Set up an equation that will help you find the score Latisha needs to get an average of
86\%
\frac{90+82+81+?}{4}=86
91\%
The teacher decided to make the last test worth twice as much a regular test. How does this change the score that Latisha needs on the last test to get an overall average of
86\%
? Support your answer with mathematical work. You may choose to use the 5-D Process again.
Since the last test is worth twice as much, it is essentially two scores for the same test.
\frac{90+82+81+2(?)}{5} = 86
|
Normal Curve Distribution - SAGE Research Methods
Normal Curve Distribution | The SAGE Encyclopedia of Communication Research Methods
Normal curve distribution is a symmetrical distribution, which has a bell shape and identical scores for the mean (i.e., the average score), median (i.e., the middle score splitting the bottom 50% from the top 50% in the distribution), and mode (i.e., most frequent value). A bell-shaped curve (see an example in Figure 1) characterizing the normal distribution can be represented by the equation below:
\mathrm{Y}=\frac{N}{\sqrt{2\pi \sigma }}e-\frac{{\left(\mathrm{X}-\mu \right)}^{2}}{2{\sigma }^{2}}
where Y = frequency of a given value of X; X = any score in ...
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V-Lab: S&P/ASX 200 Zero Slope Spline-GARCH Volatility Analysis
S&P/ASX 200 Zero Slope Spline-GARCH Volatility Analysis
Volatility Prediction for Wednesday, May 25th, 2022:16.43% (-0.78%)
Analysis last updated: Tuesday, May 24, 2022, 07:31 AM UTC
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\beta
{\gamma }_{1}
Estimation Period:
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Home : Support : Online Help : Education : Student Packages : Multivariate Calculus : Lines and Planes : Angle
Determine the angle between lines, vectors, and planes.
Angle(x, y)
a vector, a line, or a plane
The Angle command determines the angle between two vectors, a vector and a line, a vector and a plane, two lines, a line and a plane, or two planes.
The angle between two intersecting lines can be measured at the intersection point; the angle returned is in the interval
[0,\frac{\mathrm{\pi }}{2}]
. When two lines do not intersect, we define the angle determined by them as the angle between two lines through the origin parallel to the given lines.
The angle between two planes is equal to the angle between their normals.
The angle between a line and a plane is equal to the complement of the angle between the line and the normal of the plane.
An angle involving one vector, v, is the same as if instead of the vector, you had supplied a line having v as its direction. An angle between two vectors is slightly different, in that it can attain all values in
[0,\mathrm{\pi }]
\mathrm{with}\left(\mathrm{Student}[\mathrm{MultivariateCalculus}]\right):
\mathrm{v1}≔〈1,2,3〉:
\mathrm{v2}≔〈0,0,1〉:
\mathrm{v3}≔〈a,b,c〉:
\mathrm{l1}≔\mathrm{Line}\left([0,0,0],〈1,2,4〉\right):
\mathrm{l2}≔\mathrm{Line}\left([1,1,2],〈2,3,0〉\right):
\mathrm{p1}≔\mathrm{Plane}\left([1,2,0],〈-1,1,0〉\right):
\mathrm{p2}≔\mathrm{Plane}\left([1,1,2],〈1,2,1〉\right):
\mathrm{Angle}\left(\mathrm{v1},\mathrm{v2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{14}}}{\textcolor[rgb]{0,0,1}{14}}\right)
Angle between a vector and a line
\mathrm{Angle}\left(\mathrm{v3},\mathrm{l1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{min}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{\pi }}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{21}}}{\textcolor[rgb]{0,0,1}{21}\textcolor[rgb]{0,0,1}{}\sqrt{{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{c}}^{\textcolor[rgb]{0,0,1}{2}}}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{c}\right)\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{21}}}{\textcolor[rgb]{0,0,1}{21}\textcolor[rgb]{0,0,1}{}\sqrt{{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{c}}^{\textcolor[rgb]{0,0,1}{2}}}}\right)\right)
Angle between a vector and a plane
\mathrm{Angle}\left(\mathrm{v2},\mathrm{p1}\right)
\textcolor[rgb]{0,0,1}{0}
\mathrm{Angle}\left(\mathrm{l1},\mathrm{l2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{8}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{21}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{13}}}{\textcolor[rgb]{0,0,1}{273}}\right)
\mathrm{Angle}\left(\mathrm{l2},\mathrm{p1}\right)
\frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{13}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{26}}\right)
\mathrm{Angle}\left(\mathrm{p1},\mathrm{p2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{arccos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{6}}}{\textcolor[rgb]{0,0,1}{12}}\right)
The Student[MultivariateCalculus][Angle] command was introduced in Maple 18.
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Robust Statistics - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : Statistics Package : Example Worksheets : Robust Statistics
Robust measures of central tendency and dispersion
Robust statistics seek to describe data sets that suffer from noisy measurements. In particular, they should remain meaningful when a fraction of the data is changed dramatically.
A measure of dispersion, also known as a measure of scale, is a statistic of a data set that describes the variability or spread of that data set. Two well-known examples are the standard deviation and the interquartile range. Two more measures of dispersion are called
\mathrm{Sn}
\mathrm{Qn}
X
n
Y
rn
r
X := Sample(Normal(0, 1), 1000);
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627816165327924}}
StandardDeviation(X);
\textcolor[rgb]{0,0,1}{0.990176940818334}
Y := copy(X);
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627816062026444}}
Y[1] := 10^100:
StandardDeviation(Y);
\textcolor[rgb]{0,0,1}{3.16227766016838}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{98}}
r
Y
X
InterquartileRange(X);
\textcolor[rgb]{0,0,1}{1.36849073417322}
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627816062015596}}
Y[1 .. 249] := 10^100:
InterquartileRange(Y);
\textcolor[rgb]{0,0,1}{3.30395809676134}
Y[250] := 10^100:
\textcolor[rgb]{0,0,1}{5.83333333333371}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{99}}
\frac{1}{4}
\frac{1}{4}
\frac{1}{2}
MedianDeviation(X);
\textcolor[rgb]{0,0,1}{0.686396253277771}
Y := copy(X):
MedianDeviation(Y);
\textcolor[rgb]{0,0,1}{5.37556591483202}
\textcolor[rgb]{0,0,1}{5.00000000000000}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{99}}
\frac{1}{2}
\mathrm{Sn}
\mathrm{Qn}
. Maple has an implementation of both of these, called RousseeuwCrouxSn and RousseeuwCrouxQn.
RousseeuwCrouxSn(X);
\textcolor[rgb]{0,0,1}{0.836306393200841}
RousseeuwCrouxQn(X);
\textcolor[rgb]{0,0,1}{0.453501081152612}
RousseeuwCrouxSn(Y);
\textcolor[rgb]{0,0,1}{5.64926532313226}
RousseeuwCrouxQn(Y);
\textcolor[rgb]{0,0,1}{0.0137805612041167}
\textcolor[rgb]{0,0,1}{1.00000000000000}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{100}}
\textcolor[rgb]{0,0,1}{0.00711405767220685}
\mathrm{Qn}
estimator requires a different pattern to break:
Y[1..499] := Vector(499, i -> i * 10^97):
\textcolor[rgb]{0,0,1}{5.64926532313226}
Y[500] := 500 * 10^97:
\textcolor[rgb]{0,0,1}{1.00000000000000}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{97}}
0
\frac{1}{2}
5
r
100r
5
1
100
1
. This is the number shown in the plot below for each of the five measures of dispersion discussed above.
functions := [StandardDeviation, InterquartileRange, MedianDeviation, RousseeuwCrouxSn, RousseeuwCrouxQn]:
nf := numelems(functions):
X := Sample(BetaDistribution(0.9, 1.7), 10^6):
true_values := map(f -> f(X), functions);
\textcolor[rgb]{0,0,1}{\mathrm{true_values}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{0.250714910329766}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.398535818906638}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.191972959574996}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.229873030382496}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.107739262732785}]
sample_sizes := [10, 30, 100, 300, 1000, 3000, 10000]:
nss := numelems(sample_sizes):
results := Array(1 .. nf, 1 .. nss, 0 .. 10, 1 .. 100);
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627816061991140}}
for k to 100 do
X := Sample(BetaDistribution(0.9, 1.7), max(sample_sizes));
for i to nss do
Y := X[1 .. sample_sizes[i]];
sort[inplace](Y, `>`):
Y[1 .. ceil(j * sample_sizes[i] / 20)] := 5;
for f to nf do
results[f, i, j, k] := functions[f](Y) / true_values[f];
rr := Array(1 .. nf, 1 .. nss, 0 .. 10):
rr[f,i,j] := sqrt(Moment(results[f, i, j], 2, origin = 1));
plots:-display(plots:-surfdata~([seq(convert(rr[i], Matrix), i=1 .. nf)], 1 .. nss, 0 .. 0.5,
color =~ [red, green, blue, yellow, purple], transparency = 0.2),
axis[1]=[tickmarks=[seq(i = sample_sizes[i], i = 1 .. nss)]], axis[3]=[mode=log],
view=[DEFAULT,DEFAULT, min(rr) .. 10], orientation=[116, -68, 177],
labels=[`Sample sizes`, r, `Standard deviation`],
labeldirections=[horizontal, horizontal, vertical]);
The colors are red for the standard deviation, green for the interquartile range, blue for the median absolute deviation from the median, yellow for Rousseeuw and Croux'
\mathrm{Sn}
, and purple for
\mathrm{Qn}
. Lower numbers are shown higher in the graph, and are better. We see that in the case where there is no noise (
r=0
r<0.25
r
make it, too, unusable. For larger values, the median absolute deviation from the median (blue),
\mathrm{Sn}
(yellow), and
\mathrm{Qn}
(purple) all do reasonably well.
Another interesting experiment is to see how these measures of dispersion distinguish two Cauchy distributions with different scale parameters. We can see that the values in
\mathrm{X2}
(plotted in green, below) are just a little further spread out than those in
\mathrm{X1}
(plotted in red). Indeed, one could obtain a sample of the distribution underlying
\mathrm{X2}
by multiplying a sample from the distribution underlying
\mathrm{X1}
1.1
. It would be nice if measures of dispersion reflect this fact. However, the Cauchy distribution naturally has many outliers, and indeed the standard deviation of the distribution is undefined.
X1 := Sample(Cauchy(0, 1.0), 10^5):
plots:-display(KernelDensityPlot~([X1, X2], left=-12, right=12, color =~ [red, green]));
for i to nf do
f1 := functions[i](X1);
print(convert(functions[i], 'string'), f1, f2, f2/f1);
\textcolor[rgb]{0,0,1}{"StandardDeviation"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{450.792317322618}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{369.344598687811}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.819323188295339}
\textcolor[rgb]{0,0,1}{"InterquartileRange"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.97963706221884}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2.20170072370362}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.11217392608112}
\textcolor[rgb]{0,0,1}{"MedianDeviation"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.989686310280030}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.10085271595101}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.11232488973150}
\textcolor[rgb]{0,0,1}{"RousseeuwCrouxSn"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.40300629472880}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.55680729448674}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.10962245881275}
\textcolor[rgb]{0,0,1}{"RousseeuwCrouxQn"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.822432294182147}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.912138503220176}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.10907427842098}
We see that all measures of dispersion with a breakpoint greater than
0
, that is, all of them except for the standard deviation, reproduce this ratio of
1.1
fairly closely.
Robust measures of central tendency
A measure of central tendency is a statistic that identifies a central value in a sample or distribution. Well-known examples are the Mean, the Median, and the Mode. Another measure of central tendency was invented by Hodges and Lehmann (see [2]) and independently by Sen (see [3]); it is often called the Hodges-Lehmann estimator.
We can study the breakdown point of these quantities as we did with the measures of dispersion. For the mean, the breakdown point is
0
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627816044665900}}
\textcolor[rgb]{0,0,1}{-0.0139943685387067}
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627816062015116}}
\textcolor[rgb]{0,0,1}{1.00000000000000}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{97}}
The mode is a little tricky to handle for a continuous probability distribution given by a sample. The median is clearer; its breakdown point is
\frac{1}{2}
\textcolor[rgb]{0,0,1}{-0.0168102021680020}
Y[1..499] := 10^100:
Median(Y);
\textcolor[rgb]{0,0,1}{2.99849087645744}
\textcolor[rgb]{0,0,1}{5.00000000000000}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{99}}
The Hodges-Lehmann estimator has a breakdown point of
1-\frac{\sqrt{2}}{2}
0.29
HodgesLehmann(X);
\textcolor[rgb]{0,0,1}{-0.0174989845223721}
HodgesLehmann(Y);
\textcolor[rgb]{0,0,1}{2.01242526431790}
\textcolor[rgb]{0,0,1}{5.00000000000000}\textcolor[rgb]{0,0,1}{×}{\textcolor[rgb]{0,0,1}{10}}^{\textcolor[rgb]{0,0,1}{99}}
The advantage of the Hodges-Lehmann estimator is that it converges to its limit value more quickly than the median does (at least for distributions that are symmetric about the median); that is, for relatively small sample sizes, the Hodges-Lehmann estimator has greater accuracy. We proceed as in the previous section.
functions := [Mean, Median, HodgesLehmann];
\textcolor[rgb]{0,0,1}{\mathrm{functions}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{Mean}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Median}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{HodgesLehmann}}]
\textcolor[rgb]{0,0,1}{\mathrm{true_values}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{0.346193421872074}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.302664238430270}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.334408091829171}]
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893627816057486260}}
color =~ [red, green, blue], transparency = 0.2),
We see that the mean (in red) performs best when
r=0
, but miserably otherwise. The Hodges-Lehmann estimator behaves very well for
r<0.29
. Beyond that only the median does well.
We can also reproduce the experiment with the Cauchy distribution. We now vary the location parameter between the two samples; the values in
\mathrm{X2}
(plotted in green, below) are just a little further to the right, that is, greater, than those in
\mathrm{X1}
(plotted in red). In this case, one could obtain a sample of the distribution underlying
\mathrm{X2}
0.1
to a sample from the distribution underlying
\mathrm{X1}
. It would be nice if measures of central tendency reflect this fact. However, the Cauchy distribution does not have a mean.
X1 := Sample(Cauchy(0.0, 1), 10^5):
print(convert(functions[i], 'string'), f1, f2, f2-f1);
\textcolor[rgb]{0,0,1}{"Mean"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3.77709097557069}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-0.714889535849446}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-4.49198051142014}
\textcolor[rgb]{0,0,1}{"Median"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.00112967419851748}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.107246525011298}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.106116850812780}
\textcolor[rgb]{0,0,1}{"HodgesLehmann"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.000686323260547918}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.101012603763531}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0.100326280502983}
Again, we see that the two measures of central tendency with breakpoint greater than
0
(that is, the median and the Hodges-Lehmann estimator) reproduce this difference of
0.1
correctly, whereas the mean (with breakpoint
0
) does not.
HodgesLehmann, InterquartileRange, Mean, Median, MedianDeviation, Mode, RousseeuwCrouxQn, RousseeuwCrouxSn, StandardDeviation, Statistics
[2] Hodges, Joseph L., and Lehmann, Erich L. Estimation of location based on ranks. Annals of Mathematical Statistics 34(2), 1963, pp.598–611.
[3] Sen, Pranab K. On the estimation of relative potency in dilution(-direct) assays by distribution-free methods. Biometrics 19(4), 1963, pp.532–552.
|
Thomas Dreyfus1; Charlotte Hardouin2
1 Institut de Recherche Mathématique Avancée UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg, France
2 Université Paul Sabatier - Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France
In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series
Q\left(x,y,t\right)
that count the number of walks confined in the first quadrant of the plane with a fixed set of admissible steps, called the model of the walk. While the variables
x
y
are associated to the ending point of the path, the variable
t
encodes its length. In this paper, we prove that in the unweighted case,
Q\left(x,y,t\right)
satisfies an algebraic differential relation with respect to
t
if and only if it satisfies an algebraic differential relation with respect
x
y
). Combined with [2, 3, 4, 9, 11], we are able to characterize the
t
-differential transcendence of the
79
models of walks listed by Bousquet-Mélou and Mishna.
Classification: 05A15, 30D05, 39A06
Keywords: Random walks, Difference Galois theory, Transcendence, Valued differential fields.
Thomas Dreyfus 1; Charlotte Hardouin 2
author = {Thomas Dreyfus and Charlotte Hardouin},
title = {Length derivative of the generating function of walks confined in the quarter plane},
TI - Length derivative of the generating function of walks confined in the quarter plane
%T Length derivative of the generating function of walks confined in the quarter plane
Thomas Dreyfus; Charlotte Hardouin. Length derivative of the generating function of walks confined in the quarter plane. Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 39-92. doi : 10.5802/cml.77. https://cml.centre-mersenne.org/articles/10.5802/cml.77/
[1] Matthias Aschenbrenner; Lou van den Dries; Joris van der Hoeven Asymptotic differential algebra and model theory of transseries, Annals of Mathematics Studies, 195, Princeton University Press, Princeton, NJ, 2017, xxi+849 pages
[2] Olivier Bernardi; Mireille Bousquet-Mélou; Kilian Raschel Counting quadrant walks via Tutte’s invariant method, Discrete Mathematics & Theoretical Computer Science (2020)
[3] Alin Bostan; Mark van Hoeij; Manuel Kauers The complete generating function for Gessel walks is algebraic, Proc. Amer. Math. Soc., Volume 138 (2010) no. 9, pp. 3063-3078
[4] Mireille Bousquet-Mélou; Marni Mishna Walks with small steps in the quarter plane, Algorithmic probability and combinatorics (Contemp. Math.), Volume 520, Amer. Math. Soc., Providence, RI, 2010, pp. 1-39
[5] Richard M. Cohn Difference algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965
[6] Lucia Di Vizio; Charlotte Hardouin Descent for differential Galois theory of difference equations: confluence and
q
-dependence, Pacific J. Math., Volume 256 (2012) no. 1, pp. 79-104 | Article | MR: 2928542
[7] Lucia Di Vizio; Changgui Zhang On
q
-summation and confluence, Ann. Inst. Fourier, Volume 59 (2009) no. 1, pp. 347-392
[8] Thomas Dreyfus Differential algebraic generating series of weighted walks in the quarter plane, arXiv preprint arXiv:2104.05505 (2021)
[9] Thomas Dreyfus; Charlotte Hardouin; Julien Roques; Michael F Singer On the nature of the generating series of walks in the quarter plane, Inventiones mathematicae, Volume 213 (2018) no. 1, pp. 139-203
[10] Thomas Dreyfus; Charlotte Hardouin; Julien Roques; Michael F Singer On the kernel curves associated with walks in the quarter plane, Transient Transcendence in Transylvania International Conference (2019), pp. 61-89
[11] Thomas Dreyfus; Charlotte Hardouin; Julien Roques; Michael F Singer Walks in the quarter plane: Genus zero case, Journal of Combinatorial Theory, Series A, Volume 174 (2020), p. 105251
[12] Thomas Dreyfus; Kilian Raschel Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks, Publications Mathématiques de Besançon (2019) no. 1, pp. 41-80
[13] J. Duistermaat Discrete Integrable Systems: Qrt Maps and Elliptic Surfaces, Springer Monographs in Mathematics, 304, Springer-Verlag, New York, 2010
[14] Guy Fayolle; Roudolf Iasnogorodski; Vadim Malyshev Random walks in the quarter-plane, Applications of Mathematics (New York), 40, Springer-Verlag, Berlin, 1999, xvi+156 pages (Algebraic methods, boundary value problems and applications) | Article | MR: 1691900
[15] Guy Fayolle; Kilian Raschel On the holonomy or algebraicity of generating functions counting lattice walks in the quarter-plane, Markov Process. Related Fields, Volume 16 (2010) no. 3, pp. 485-496 | MR: 2759770
[16] Jean Fresnel; Marius van der Put Rigid analytic geometry and its applications, Progress in Mathematics, 218, Birkhäuser Boston, Inc., Boston, MA, 2004, xii+296 pages
[17] Charlotte Hardouin; Michael F. Singer Differential Galois theory of linear difference equations, Math. Ann., Volume 342 (2008) no. 2, pp. 333-377
[18] Charlotte Hardouin; Michael F Singer On differentially algebraic generating series for walks in the quarter plane, Selecta Mathematica, Volume 27 (2021) no. 5, pp. 1-49
[19] Dale Husemöller Elliptic curves. With appendices by Otto Forster, Ruth Lawrence, and Stefan Theisen, 111, New York, NY: Springer, 2004, xxi + 487 pages
[20] Manuel Kauers; Rika Yatchak Walks in the quarter plane with multiple steps, Proceedings of FPSAC 2015 (Discrete Math. Theor. Comput. Sci. Proc.) (2015), pp. 25-36
[21] E. R. Kolchin Algebraic groups and algebraic dependence, Amer. J. Math., Volume 90 (1968), pp. 1151-1164 https://doi-org.prox.lib.ncsu.edu/10.2307/2373294 | Article | MR: 0240106
[22] Ellis Robert Kolchin Differential algebra & algebraic groups, 54, Academic press, 1973
[23] Irina Kurkova; Kilian Raschel On the functions counting walks with small steps in the quarter plane, Publ. Math. Inst. Hautes Études Sci., Volume 116 (2012), pp. 69-114 | Article | MR: 3090255
[24] Serge Lang Complex analysis, 103, Springer Science & Business Media, 2013
[25] Saunders MacLane The universality of formal power series fields, Bull. Am. Math. Soc., Volume 45 (1939), pp. 888-890
[26] Stephen Melczer; Marni Mishna Singularity analysis via the iterated kernel method, Combin. Probab. Comput., Volume 23 (2014) no. 5, pp. 861-888 | Article | MR: 3249228
[27] Marni Mishna; Andrew Rechnitzer Two non-holonomic lattice walks in the quarter plane, Theoret. Comput. Sci., Volume 410 (2009) no. 38-40, pp. 3616-3630 | Article | MR: 2553316
[28] Alexandre Ostrowski Sur les relations algébriques entre les intégrales indéfinies, Acta Math., Volume 78 (1946), pp. 315-318 https://doi-org.prox.lib.ncsu.edu/10.1007/BF02421605 | Article | MR: 0016764
[29] Alexey Ovchinnikov; Michael Wibmer
\sigma
-Galois theory of linear difference equations, Int. Math. Res. Not. IMRN (2015) no. 12, pp. 3962-4018
[30] Peter Roquette Analytic theory of elliptic functions over local fields, Vandenhoeck u. Ruprecht, 1970 no. 1
[31] Jacques Sauloy Systèmes aux
q
-différences singuliers réguliers: classification, matrice de connexion et monodromie, Ann. Inst. Fourier (Grenoble), Volume 50 (2000) no. 4, pp. 1021-1071 | MR: 1799737 (2001m:39043)
[32] Joseph H. Silverman Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer-Verlag, New York, 1994
[33] Joseph H Silverman The arithmetic of elliptic curves, 106, Springer Science & Business Media, 2009
[34] Snowbird lectures in algebraic geometry. Proceedings of an AMS-IMS-SIAM joint summer research conference on algebraic geometry: Presentations by young researchers, Snowbird, UT, USA, July 4–8, 2004 (Ravi Vakil, ed.), 388, Providence, RI: American Mathematical Society (AMS), 2005
[35] Marius van der Put; Michael F. Singer Galois theory of difference equations, Lecture Notes in Mathematics, 1666, Springer-Verlag, Berlin, 1997, viii+180 pages
[36] E. T. Whittaker; G. N. Watson A course of modern analysis, Mineola, NY: Dover Publications, 2020, 613 pages
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Model an Excavator Dipper Arm as a Flexible Body - MATLAB & Simulink - MathWorks India
Step 1: Define the Geometry and Material Properties of the Dipper Arm
Step 2: Specify the Locations of Interface Frames
Step 3: Create the Finite-Element Mesh
Step 4: Set up the Multipoint Constraints for the Interface Frames
Step 5: Generate the Reduced-Order Model
Step 6: Import Reduced-Order Data
Compute the Modal Damping Matrix
The Reduced Order Flexible Solid block models a deformable body based on a reduced-order model that characterizes the geometric and mechanical properties of the body. The basic data imported from the reduced-order model includes:
A list of coordinate triples that specify the position of all interface frame origins relative to a common reference frame.
A symmetric stiffness matrix that describes the elastic properties of the flexible body.
A symmetric mass matrix that describes the inertial properties of the flexible body.
There are several ways to generate the reduced-order data required by this block. Typically, you generate a substructure (or superelement) by using finite-element analysis (FEA) tools.
This example uses the Partial Differential Equation Toolbox™ to create a reduced-order model for a flexible dipper arm, such as the arm for an excavator or a backhoe. You start with the CAD geometry of the dipper arm, generate a finite-element mesh, apply the Craig-Bampton FEA substructuring method, and generate a reduced-order model. The model sm_flexible_dipper_arm uses the reduced-order data from this example. In the model, the dipper arm is mounted on top of a rotating tower as part of a test rig. For more information, see Flexible Dipper Arm.
The file sm_flexible_dipper_arm.STL contains a triangulation that defines the CAD geometry of the dipper arm. To view the geometry stored in this file, use the MATLAB® functions stlread and trisurf:
The dipper arm is constructed from steel. To represent its material properties, set these values for Young's modulus, Poisson's ratio, and mass density:
The dipper arm has three interface frames where you can connect other Simscape™ Multibody™ elements, such as joints, constraints, forces, and sensors:
The cylinder connection point, where the arm connects to a hydraulic cylinder that actuates the arm vertically.
The bucket connection point, where the arm connects to the excavator bucket.
The fulcrum point, where the arm connects to the excavator boom.
The positions of all interface frame origins are specified in meters relative to same common reference frame used by the CAD geometry.
To generate the mesh for the dipper arm, first call the createpde (Partial Differential Equation Toolbox) function, which creates a structural model for modal analysis of a solid (3-D) problem. After importing the geometry and material properties of the arm, the generateMesh (Partial Differential Equation Toolbox) function creates the mesh.
Each interface frame on the block corresponds to a boundary node that contributes six degrees of freedom to the reduced-order model. There are several ways to ensure that the FEA substructuring method preserves the required degrees of freedom. For example, you can create a rigid constraint to connect the boundary node to a subset of finite-element nodes on the body. You can also use structural elements, such as beam or shell elements, to introduce nodes with six degrees of freedom.
This example uses a multipoint constraint (MPC) to preserve the six degrees of freedom at each boundary node. To identify the geometric regions (such as faces, edges, or vertices) to associate with each MPC, first plot the arm geometry by using the function pdegplot (Partial Differential Equation Toolbox):
You can zoom, rotate, and pan this image to determine the labels for the faces corresponding to the boundary nodes. These faces define the MPCs associated with the boundary nodes in the dipper arm:
Cylinder connection point: face 1
Bucket connection point: face 27
Fulcrum point: face 23
To verify these values, plot the mesh and highlight the selected faces:
Call the function structuralBC (Partial Differential Equation Toolbox) to define the MPCs for the boundary nodes in these faces:
The function reduce (Partial Differential Equation Toolbox) applies the Craig-Bampton order reduction method and retains all fixed-interface modes up to a frequency of
1{0}^{4}
radians per second.
Store the results of the reduction in a data structure arm. Transpose the ReferenceLocations matrix to account for the different layout conventions used by Partial Differential Equation Toolbox and Simscape Multibody.
The function computeModalDampingMatrix, which is defined at the bottom of this page, computes a reduced modal damping matrix with a damping ratio of 0.05:
The boundary nodes in the reduced-order model must be specified in the same order as the corresponding interface frames on the block. This order is given by the rows of the array origins. If the order of the MPCs is different than the order specified by origins, permute the rows and columns of the various matrices so that they match the original order.
The model sm_flexible_dipper_arm uses the data structure arm to set up the parameters of the Reduced Order Flexible Solid block. In the block, these parameters import the reduced-order data:
For more information, see Flexible Dipper Arm.
This function computes a modal damping matrix associated with the stiffness matrix K and mass matrix M. This function applies a single scalar damping ratio to all of the flexible (non-rigid-body) normal modes associated with K and M.
|
Which of the equations below represent proportional relationships? If the relationship is proportional, identify the constant of proportionality. If the relationship is not proportional, explain why.
y = \frac { 3 } { 4 } x + 2
A proportional relationship should involve only multiplication. Does this equation fit that description?
Not proportional, since there is addition.
y = ( 4 \frac { 2 } { 3 } ) x
There is only multiplication involved in this equation.
Proportional; constant of proportionality is
4\frac{2}{3}
y=3(x-1)
This equation is equivalent to
y=3x-3
|
Daniela, Kieu, and Duyen decide to go to the movies one hot summer afternoon. The theater is having a summer special called Three Go Free. They will get free movie tickets if they each buy a large popcorn and a large soft drink. They take the deal and spend a total of
\$22.50
on large popcorns and soft drinks. The next week, they go back again, only this time, they each pay
\$8.00
for a ticket, they each get a large soft drink, but they share one large bucket of popcorn. This return trip costs them a total of
\$37.50
What is the price of a large soft drink and the price of a large bucket of popcorn?
Set up a system of two equations to find the prices of a large popcorn and a drink.
=\$4.50
and drink
=\$3.00
Did you write two equations or did you use another method? If you used another method, write two equations now and solve them. If you already used a system of equations, skip this part.
|
Write and solve an inequality for the following situation.
Robert is painting a house. He has
35
cans of paint. He has used
30
cans of paint on the walls. Now he needs to paint the trim. If each section of trim takes
\frac { 1 } { 2 }
can of paint, how many sections of trim can he paint? Show your answer as an inequality with symbols, in words, and with a number line. Make sure that your solution makes sense for this situation.
The total number of cans used must be less than or equal to
35
cans of paint, and only
5
cans can be used for the trim.
\frac{1}{2}\textit{x}+30\le35
x
is the number of sections of trim he can paint.
|
When a forest environment has changed because of logging or a forest fire, soil erosion from wind and water run-off is a big concern. A botanist is studying ceanothus plants, which are commonly used to help stabilize the soil on bare hillsides. The roots from the plants act like a wire mesh to keep the soil in place. One factor to consider for this type of plant is germination times. Germination time is measured from when a seed is hydrated (watered) until it produces its very first root.
Ceanothus plants will not germinate at room temperature. Cooler temperatures are required. Two batches of
30
ceanothus seeds were watered and kept at the following cool temperatures (
40^\circ
F) and their time until germination was measured (days). The boxplots are shown below (the
40^\circ
F batch is the top boxplot).
Describe the center, shape, spread, and outliers for the germination time of each batch of seeds.
Center refers to the typical germination times, so use the mean or median. Descriptions of shape include words like symmetric, uniform, or skewed. To describe the spread, use the standard deviation or IQR.
Which batch was more consistent?
How might this information be useful to firefighters who plant ceanothus seeds to prevent erosion after a fire?
Assuming that the firefighters can spread thousands of seeds, what would be better weather conditions,
40^\circ
F or
60^\circ
F? Why?
|
Lemma 10.104.8 (00NE)—The Stacks project
Lemma 10.104.8. Let $R$ be a Noetherian local Cohen-Macaulay ring of dimension $d$. Let $0 \to K \to R^{\oplus n} \to M \to 0$ be an exact sequence of $R$-modules. Then either $M = 0$, or $\text{depth}(K) > \text{depth}(M)$, or $\text{depth}(K) = \text{depth}(M) = d$.
So what if the depth of
R
is zero?
M
is the zero module then the conventions in section 10.71 say it has infinite depth.
Then every (nonzero) module has depth zero, right? So maybe the first line of the proof should be "If
\text{depth}(M)=0
\text{depth}(R)=0
then the lemma is clear."
This is an annoying but important lemma. I tried to fix it so it is actually true! Hope I succeeded this time. The fix is here.
Comment #5003 by songong on March 28, 2020 at 12:34
Isn't this just a special case of 10.72.6?
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00NE. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 00NE, in case you are confused.
|
Buybacks & Recollateralization - Chemix Ecosystem Documents
During the operation of Chemix, there may be a mismatch between the value of the collateral and the collateral ratio: when the total value of the collateral is less than the current collateral ratio of the system, the collateral needs to be increased; when the value of the collateral exceeds the ratio of the collateral ratio, the excess collateral value can be allocated to CEC holders. To quickly redistribute value back to CEC holders or increase system collateral, two functions are built into the protocol: buyback and recollateralization.
When the actual collateral ratio of the system is lower than the nominal collateral ratio, the system requires users to replenish the collateral. This process is called recollateralization. Anyone can call the recollateralization function which then checks if the total collateral value in USD across the system is below the current collateral ratio. If it is, the system allows the caller to add up to the amount needed to reach the target collateral ratio in exchange for newly minted CBT at a bonus rate. The bonus rate is set to 0.5% to quickly incentivize arbitragers to close the gap and recollateralize the protocol to the target ratio. The incentive amount can be adjusted through community governance in the subsequent operation of the agreement.
CBT_{received} = \frac{(Y_i*P_i)(1+B_r)}{P_E}
Y_i
is the amount of collateral
i
required to reach the collateral rate;
P_i
is the price for collateral
i
provided by the Chainlink Oracle;
B_r
is the percentage of the reward CEC minted during recollateralization;
P_E
is the market price of the CEC token, provided by the weighted average prices of DEX pool.
Assuming that the system is in a state of insufficient collateral at this time, it needs $500,000 of collateral to reach the target mortgage rate. The user can call the recollateralization function and provide $500,000 worth of collateral to the protocol. At this time, the user can receive CBT tokens equivalent to 500,000×1.05, and the reward amount is 5%.
Placing 500,000 BUSD($1.00/BUSD), the CEC price is $6.00/CEC, the calculation process is as follows:
CBT_{received} = \frac{(500,000×1.00)(1+0.05)}{6.00} = 87,500
The opposite scenario occurs when there is excess collateral in the system than required to hold the target collateral ratio. This can happen in several ways:
The protocol has been lowering the collateral ratio successfully keeping the price of QSD stable;
Interest bearing collateral is accepted into the protocol and its value accrues;
Minting and redemption fees are creating revenue;
In such a scenario, any CEC holder can call the buyback function to exchange the amount of excess collateral value in the system for CEC which is then burned by the protocol.
To effectively redistributes excess value back to the CEC distribution and holders, we expect the users to actively participate in buybacks to gain value since there is 1% bonus rate for the buyback function, with the consumption in buyback, CEC also has expectations of value growth. After the buybacks occurs, since some CEC will be burned, it can also bring added value to all CEC holders.
Collateral_{i,\ received} = \frac{E×P_E}{P_i}
E
is units of CEC to be burned in buyback;
P_E
is the market price of the CEC token, provided by the weighted average prices of DEX pool;
P_i
is the price in USD of collateral
i
, provided by Chainlink Oracle.
Chemix supports a variety of collateral, so when buyback happens, the system will allow users to choose the collateral by themselves. Suppose the amount of certain collateral is completely repurchased and the collateral ratio still cannot return to the target value. In that case, the user can continue to choose other collateral to repurchase until the target collateral ratio is achieved.
There is 60,000,000 QSD in circulation at a 60% collateral ratio. The total value of collateral across the BUSD, BNB, BTCB pools is 40,000,000 USD. There is 40,000,000 USD worth of excess collateral available for CEC buybacks. In the three kinds of collateral, the total value of the lowest is BTCB(3,700,000 USD). Assuming that a user chooses to exchange BTCB with CEC, then after exchanging all BTCB, the user can still exchange BUSD or BNB with CEC equal to 300,000 USD.
If the user chooses to exchange a BNB equivalent to 4,000,000 USD at the beginning, and the price of BNB is $42.5/BNB and CEC is $10/CEC, then:
CEC_{burned} = \frac{4,000,000}{10} = 400,000\ CEC
BNB_{received} = \frac{400,000×10}{42.5} = 94,117.64705882\ BNB
|
Rebalancing - Indexed Finance
Bootstrapping New Tokens
Pools can be re-weighed to adjust the composition of the current desired tokens in the pool. The current desired tokens are the underlying tokens in the pool with a target weight greater than zero. Re-weighing a pool adjusts the target weights of each asset but does not remove or add tokens.
Re-indexing assets
Pools can be re-indexed to adjust both the underlying tokens and their target weights.
Any current underlying tokens which are not assigned a new target weight in the re-index call will be assigned a target weight of 0 so that they can be gradually removed from the pool. If this occurs for a token which is bound but not initialized, the token will be unbound (see unbound token handling).
New tokens added to the pool must be assigned a minimum balance which is roughly equal to 1% of the total pool value (see bootstrapping new tokens) .
In order to rebalance through internal swaps, Indexed uses a desired weight (
D_t
) parameter which defines the target weight for an asset. If a desired weight is higher than the actual weight, the pool should increase its balance in that token. If the desired weight is lower than the actual weight, the pool should decrease its balance in that token.
Each pool has a minimum update delay, which by default is 30 minutes, and a weight change factor, which by default is 1%.
If a token with a positive weight difference (
D_t > W_t
) is swapped into the pool and it has been more than the minimum delay period since its last weight change, the pool will increase that token's weight by either the weight change factor or the token's desired weight, whichever is less.
const weightChangeFactor = 0.01;
updateWeightIn(tokenIn) {
if (tokenIn.desiredWeight > tokenIn.weight) {
tokenIn.weight = Math.min(
tokenIn.desiredWeight,
tokenIn.weight * (1 + weightChangeFactor)
If a token with a negative weight difference (
D_t < W_t
) is swapped out of the pool and it has been more than the minimum delay period since its last weight change, the pool will decrease that token's weight by either the weight change factor or the token's desired weight, whichever is less.
updateWeightOut(tokenOut) {
if (tokenOut.desiredWeight < tokenOut.weight) {
tokenOut.weight = Math.max(
tokenOut.desiredWeight,
tokenOut.weight * (1 - weightChangeFactor)
As swaps are executed and LP tokens are minted and burned, inbound tokens with desired weight increases (desired > real) and outbound tokens with desired weight decreases (real > desired) automatically adjust their weights, occasionally creating small arbitrage opportunities which move tokens toward their target balances when traders execute on them, thus rebalancing the pool.
Note: There are two exceptions to the stated weight adjustment rule:
The exitPool function, which sends out some amount of every initialized token. In order to minimize gas expenditure, this function does not adjust the weights of outbound tokens.
When an uninitialized token becomes ready, the weight can increase beyond the fee factor1.
|
Coherent Thermal Emission From Modified Periodic Multilayer Structures | J. Heat Transfer | ASME Digital Collection
Lee, B. J., and Zhang, Z. M. (April 5, 2006). "Coherent Thermal Emission From Modified Periodic Multilayer Structures." ASME. J. Heat Transfer. January 2007; 129(1): 17–26. https://doi.org/10.1115/1.2401194
Enhancement of thermal emission and control of its direction are important for applications in optoelectronics and energy conversion. A number of structures have been proposed as coherent emission sources, which exhibit a large emissivity peak within a narrow wavelength band and at a well-defined direction. A commonly used structure is the grating, in which the excited surface polaritons or surface waves are coupled with propagating waves in air, resulting in coherent emission for
polarization only. One-dimensional photonic crystals can also support surface waves and may be modified to construct coherent emission sources. The present study investigates coherent emission from a multilayer structure consisting of a
SiC
film coated atop a dielectric photonic crystal (PC). By exciting surface waves at the interface between
SiC
and the PC, coherent emission is predicted for both
p
s
polarizations. In addition to the excitation of surface waves, the emission from the proposed multilayer structure can be greatly enhanced by the cavity resonance mode and the Brewster mode.
silicon compounds, multilayers, photonic crystals, emissivity, polaritons, surface electromagnetic waves, electromagnetic, emitting, microstructures, radiation, thin films
Cavities, Crystals, Emissions, Emissivity, Excitation, Polarization (Electricity), Polarization (Light), Polarization (Waves), Resonance, Surface waves (Fluid), Wavelength, Waves, Diffraction gratings, Reflectance
A Strongly Directional Emitting and Absorbing Surface
Bidirectional Reflection Measurements of Periodically Microstructured Silicon Surfaces
Organ Pipe Radiant Modes of Periodic Micromachined Silicon Surfaces
Spectrally Selective Thermal Radiators and Absorbers with Periodic Microstructured Surface for High-Temperature Applications
Latest Results on Semitransparent Power Silicon Solar Cells
A Review of Progress in Thermophotovoltaic Generation of Electricity
Radiation Filters and Emitters for the NIR Based on Periodically Structured Metal Surfaces
Mittler-Neher
Thermally Induced Emission of Light from a Metallic Diffraction Grating, Mediated by Surface Plasmons
A Metamaterial for Directive Emission
Planar Heterogeneous Structures for Coherent Emission of Radiation
N. E. J.
Enhanced Photoluminescence by Resonant Absorption in Er-Doped SiO2∕Si Microcavities
Photonic Crystal Enhanced Narrow-Band Infrared Emitters
Three-Dimensional Photonic-Crystal Emitter for Thermal Photovoltaic Power Generation
Single-Defect Bragg Stacks for High-Power Narrow-Band Thermal Emission
, Princeton, N.J.
Kassakian
Electromagnetic Propagation in Periodic Stratified Media. I. General Theory
Surface Electromagnetic Wave Excitation on One-Dimensional Photonic Band-Gap Arrays
Gaspar-Armenta
Photonic Surface-Wave Excitation: Photonic Crystal-Metal Interface
, Fort Worth, TX.
Effects of Periodic Structures on the Coherence Properties of Blackbody Radiation
Photon Localization and Electromagnetic Field Enhancement in Laser-Irradiated, Random Porous Media
Handbook of Optical Constants of Solids II
Ramos-Mendieta
Electromagnetic Surface Modes of a Dielectric Superlattice: The Supercell Method
Surface Polaritons — Propagating Electromagnetic Modes at Interfaces
Experimental Measurement of the Effect of Termination on Surface Electromagnetic Waves in One-Dimensional Photonic Bandgap Arrays
Kollyukh
Morozhenko
Thermal Radiation of Plane-Parallel Semitransparent Layers
Manipulation of Thermal Emission by Use of Micro and Nanoscale Structures
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Introduction to Chemical Engineering Processes/Unusual Units - Wikibooks, open books for an open world
1 Standard vs. Actual Volume
1.1 Conversion of volume to volume
2 "Gauge" Pressure vs. "Absolute" Pressure
4 "Pound-mass" vs. "Pound-force"
Standard vs. Actual VolumeEdit
When specifying the volume of a gas, the pressure and temperature must be specified because the volume of a gas depends strongly on both temperature and pressure (assuming that it is in an expandable container). In order to avoid specifying a different set of conditions for each measurement, an engineer can convert to standard temperature and pressure (typically 1 atm and 0oC) which is common to all measurements. This allows direct comparisons of volume measurements, but also requires that one convert back to the actual conditions present in the system before the value can be used.
The conversion that is used assumes that the gas is ideal, so that:
{\displaystyle PV=nRT}
Conversion of volume to volumeEdit
We wish to compare a standard state to the actual state in the system. Let us consider the standard state (state "s") first. We have the ideal gas law for the standard state:
{\displaystyle P_{s}*V_{s}=n_{s}RT_{s}}
Now let us compare the standard state to the actual conditions in the system. The standard state conditions have been completely specified. The ideal gas law is assumed to hold in the actual system conditions as well:
{\displaystyle P_{a}V_{a}=n_{a}RT_{a}}
Divide this equation by the standard-state ideal gas law gives:
{\displaystyle {\frac {P_{a}V_{a}}{P_{s}{V}_{s}}}={\frac {n_{a}RT_{a}}{n_{s}RT_{s}}}}
where a is actual and s is standard. If we assume that we want to compare the same number of moles of the substance between the standard and actual states, the following conversion between the states is obtained:
Conversion from standard to actual volume
{\displaystyle V_{a}={\frac {P_{s}V_{s}}{T_{s}}}*{\frac {T_{a}}{P_{a}}}}
"Gauge" Pressure vs. "Absolute" PressureEdit
Different types of molesEdit
"Pound-mass" vs. "Pound-force"Edit
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/Unusual_Units&oldid=3325806"
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We study the relation between the standard two-way automata and more powerful devices, namely, two-way finite automata equipped with some
\ell
additional “pebbles” that are movable along the input tape, but their use is restricted (nested) in a stack-like fashion. Similarly as in the case of the classical two-way machines, it is not known whether there exists a polynomial trade-off, in the number of states, between the nondeterministic and deterministic two-way automata with
\ell
nested pebbles. However, we show that these two machine models are not independent: if there exists a polynomial trade-off for the classical two-way automata, then, for each
\ell
≥ 0, there must also exist a polynomial trade-off for the two-way automata with
\ell
nested pebbles. Thus, we have an upward collapse (or a downward separation) from the classical two-way automata to more powerful pebble automata, still staying within the class of regular languages. The same upward collapse holds for complementation of nondeterministic two-way machines. These results are obtained by showing that each pebble machine can be, by using suitable inputs, simulated by a classical two-way automaton (and vice versa), with only a linear number of states, despite the existing exponential blow-up between the classical and pebble two-way machines.
Mots clés : finite automata, regular languages, descriptional complexity
author = {Geffert, Viliam and I\v{s}to\v{n}ov\'a, L'ubom{\'\i}ra},
title = {Translation from classical two-way automata to pebble two-way automata},
AU - Geffert, Viliam
AU - Ištoňová, L'ubomíra
TI - Translation from classical two-way automata to pebble two-way automata
Geffert, Viliam; Ištoňová, L'ubomíra. Translation from classical two-way automata to pebble two-way automata. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 4, pp. 507-523. doi : 10.1051/ita/2011001. http://www.numdam.org/articles/10.1051/ita/2011001/
[1] J. Berman and A. Lingas, On the complexity of regular languages in terms of finite automata. Tech. Rep., Vol. 304, Polish Academy of Sciences (1977). | Zbl 0364.68057
[2] M. Blum and C. Hewitt, Automata on a 2-dimensional tape, in Proc. IEEE Symp. Switching Automata Theory (1967), 155-160.
[3] C. Boyer, A History of Mathematics. John Wiley & Sons (1968). | Zbl 0698.01001
[4] J.H. Chang, O.H. Ibarra, M.A. Palis and B. Ravikumar, On pebble automata. Theoret. Comput. Sci. 44 (1986) 111-121. | Zbl 0612.68045
[5] R. Chang, J. Hartmanis and D. Ranjan, Space bounded computations: Review and new separation results. Theoret. Comput. Sci. 80 (1991) 289-302. | Zbl 0745.68051
[6] M. Chrobak, Finite automata and unary languages. Theoret. Comput. Sci. 47 (1986) 149-158. (Corrigendum: Theoret. Comput. Sci. 302 (2003) 497-498). | Zbl 0638.68096
[7] W. Ellison and F. Ellison, Prime Numbers. John Wiley & Sons (1985). | Zbl 0624.10001
[8] J. Engelfriet and H.J. Hoogeboom, Tree-walking pebble automata, in Jewels Are Forever, Contributions to Theoretical Computer Science in Honor of Arto Salomaa, J. Karhumäki, H. Maurer, G. Păun and G. Rozenberg, Eds. Springer-Verlag (1999), 72-83. | Zbl 0944.68108
[9] V. Geffert, Nondeterministic computations in sublogarithmic space and space constructibility. SIAM J. Comput. 20 (1991) 484-498. | Zbl 0762.68022
[10] V. Geffert, Bridging across the
log\left(n\right)
space frontier. Inform. Comput. 142 (1998) 127-158. | Zbl 0917.68077
[11] V. Geffert, C. Mereghetti and G. Pighizzini, Converting two-way nondeterministic unary automata into simpler automata. Theoret. Comput. Sci. 295 (2003) 189-203. | Zbl 1045.68080
[12] V. Geffert, C. Mereghetti and G. Pighizzini, Complementing two-way finite automata. Inform. Comput. 205 (2007) 1173-1187. | Zbl 1121.68063
[13] N. Globerman and D. Harel, Complexity results for two-way and multi-pebble automata and their logics. Theoret. Comput. Sci. 169 (1996) 161-184. | Zbl 0874.68213
[14] J. Hartmanis, P. M. Lewis Ii and R. E. Stearns, Hierarchies of memory limited computations, in IEEE Conf. Record on Switching Circuit Theory and Logical Design (1965), 179-190. | Zbl 0229.02033
[15] J. Hopcroft, R. Motwani and J. Ullman, Introduction to Automata Theory, Languages, and Computation. Addison-Wesley (2001). | Zbl 0980.68066
[16] J. Hromkovič and G. Schnitger, Nondeterminism versus determinism for two-way nondeterministic automata: Generalizations of Sipser's separation, in Proc. Internat. Colloq. Automata, Languages and Programming. Lect. Notes Comput. Sci. 2719 (2003) 439-451. | Zbl 1039.68068
[17] Ch.A. Kapoutsis, Deterministic moles cannot solve liveness. J. Automat. Lang. Combin. 12 (2007) 215-235. | Zbl 1145.68461
[18] O.B. Lupanov, Über den Vergleich zweier Typen endlicher Quellen. Probleme der Kybernetik Akademie-Verlag, Berlin, in German, Vol. 6, 329-335 (1966). | Zbl 0168.25902
[19] C. Mereghetti and G. Pighizzini, Optimal simulations between unary automata. SIAM J. Comput. 30 (2001) 1976-1992. | Zbl 0980.68048
[20] F. Moore, On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Trans. Comput. C-20 (1971) 1211-1214. | Zbl 0229.94033
[21] M. Rabin and D. Scott, Finite automata and their decision problems. IBM J. Res. Develop. 3 (1959) 114-125. | Zbl 0158.25404
[22] W. Sakoda and M. Sipser, Nondeterminism and the size of two-way finite automata, in Proc. ACM Symp. Theory Comput. (1978), 275-286.
[23] A. Salomaa, D. Wood and S. Yu, On the state complexity of reversals of regular languages. Theoret. Comput. Sci. 320 (2004) 315-329. | Zbl 1068.68078
[24] M. Shepherdson, The reduction of two-way automata to one-way automata. IBM J. Res. Develop. 3 (1959) 198-200. | Zbl 0158.25601
[25] M. Sipser, Lower bounds on the size of sweeping automata, in Proc. ACM Symp. Theory Comput. (1979) 360-364. | Zbl 0445.68064
[26] M. Sipser, Halting space bounded computations. Theoret. Comput. Sci. 10 (1980) 335-338. | Zbl 0423.68011
[27] A. Szepietowski, Turing Machines with Sublogarithmic Space. Lect. Notes Comput. Sci. 843 (1994). | Zbl 0998.68062
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Shallow Faulting and Folding in the Epicentral Area of the 1886 Charleston, South Carolina, Earthquake | Bulletin of the Seismological Society of America | GeoScienceWorld
Thomas L. Pratt;
Thomas L. Pratt *
U.S. Geological Survey, Reston, Virginia, U.S.A.
Corresponding author: tpratt@usgs.gov
Ronald C. Counts;
Ronald C. Counts
University of Mississippi, Oxford, Mississippi, U.S.A.
Thomas L. Pratt, Anjana K. Shah, Ronald C. Counts, J. Wright Horton, Martin C. Chapman; Shallow Faulting and Folding in the Epicentral Area of the 1886 Charleston, South Carolina, Earthquake. Bulletin of the Seismological Society of America 2022; doi: https://doi.org/10.1785/0120210329
The moment magnitude (
Mw
) ∼7 earthquake that struck Charleston, South Carolina, on 31 August 1886 is the largest historical earthquake in the United States east of the Appalachian Mountains. The fault(s) that ruptured during this earthquake has never been conclusively identified, and conflicting fault models have been proposed. Here we interpret reprocessed seismic reflection profiles, reprocessed legacy aeromagnetic data, and newly collected ground penetrating radar (GPR) profiles to delineate faults deforming the Cretaceous and younger Atlantic Coastal Plain (ACP) strata in the epicentral area of the 1886 earthquake. The data show evidence for faults folding or vertically displacing ACP strata, including apparent displacements of near‐surface strata (upper ∼20 m). Aeromagnetic data show several northeast (NE)‐trending lineaments, two of which correlate with faults and folds with vertical displacements as great as 55 m on the seismic reflection and radar profiles. ACP strata show only minor thickness changes across these structures, indicating that much of the displacement postdates the shallowest well‐imaged ACP strata of Eocene age. Faults imaged on the seismic reflection profiles appear on GPR profiles to displace the erosional surface at the top of the upper Eocene to Oligocene Cooper Group, including where railroad tracks were bent during the 1886 earthquake. Some faults coincide with changes in river trends, bifurcations of river channels, and unusual river meanders that could be related to recent fault motion. In contrast to our interpreted NE fault trends, earthquake locations and some focal mechanisms in the modern seismic zone have been interpreted as defining a nearly north‐striking, west‐dipping zone of aftershocks from the 1886 earthquake. The relationship between the modern seismicity and the faults we image is therefore enigmatic. However, multiple faults in the area clearly have been active since the Eocene and deform strata in the upper 20 m, providing potential targets for field‐based geologic investigations.
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Lognormal probability density function - MATLAB lognpdf - MathWorks Australia
y = lognpdf(x)
y = lognpdf(x,mu)
y = lognpdf(x,mu,sigma)
y = lognpdf(x) returns the probability density function (pdf) of the standard lognormal distribution, evaluated at the values in x. In the standard lognormal distribution, the mean and standard deviation of logarithmic values are 0 and 1, respectively.
y = lognpdf(x,mu) returns the pdf of the lognormal distribution with the distribution parameters mu (mean of logarithmic values) and 1 (standard deviation of logarithmic values), evaluated at the values in x.
y = lognpdf(x,mu,sigma) returns the pdf of the lognormal distribution with the distribution parameters mu (mean of logarithmic values) and sigma (standard deviation of logarithmic values), evaluated at the values in x.
Compute the pdf values evaluated at the values in x for the lognormal distribution with mean mu and standard deviation sigma.
y = lognpdf(x,mu,sigma);
Values at which to evaluate the pdf, specified as a positive scalar value or an array of positive scalar values.
To evaluate the pdf at multiple values, specify x using an array. To evaluate the pdfs of multiple distributions, specify mu and sigma using arrays. If one or more of the input arguments x, mu, and sigma are arrays, then the array sizes must be the same. In this case, lognpdf expands each scalar input into a constant array of the same size as the array inputs. Each element in y is the pdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in x.
y=f\left(x|\mu ,\sigma \right)=\frac{1}{x\sigma \sqrt{2\pi }}\mathrm{exp}\left\{\frac{-{\left(\mathrm{log}x-\mu \right)}^{2}}{2{\sigma }^{2}}\right\},\text{ }\text{for}\text{\hspace{0.17em}}x>0.
lognpdf is a function specific to lognormal distribution. Statistics and Machine Learning Toolbox™ also offers the generic function pdf, which supports various probability distributions. To use pdf, create a LognormalDistribution probability distribution object and pass the object as an input argument or specify the probability distribution name and its parameters. Note that the distribution-specific function lognpdf is faster than the generic function pdf.
pdf | logncdf | logninv | lognstat | lognfit | lognlike | lognrnd | LognormalDistribution
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PLEASE SOLVE AND GIVE CORRECT ANSWER Question 75 of 90 The value of ∫-π/2π/2p sin3x+qsin4x+r sin5x dx depends on : - Maths - Integrals - 11774295 | Meritnation.com
{\int }_{-\mathrm{\pi }/2}^{\mathrm{\pi }/2}\left(p {\mathrm{sin}}^{3}x+q{\mathrm{sin}}^{4}x+r {\mathrm{sin}}^{5}x\right) dx
depends on :
4. p and q
\mathrm{We} \mathrm{know} \mathrm{that}\phantom{\rule{0ex}{0ex}}{\int }_{-\mathrm{a}}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right).\mathrm{dx}=\left\{\begin{array}{ll}0& \mathrm{if} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{odd}\\ 2{\int }_{0}^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right).\mathrm{dx}& \mathrm{if} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{even}\end{array}\right\...\left(\mathrm{i}\right)\phantom{\rule{0ex}{0ex}}\mathrm{Now}\phantom{\rule{0ex}{0ex}}{\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}} \left({\mathrm{psin}}^{3}\mathrm{x}+{\mathrm{qsin}}^{4}\mathrm{x}+{\mathrm{rsin}}^{5}\mathrm{x}\right)\mathrm{dx}\phantom{\rule{0ex}{0ex}}={\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}}\left({\mathrm{psin}}^{3}\mathrm{x}+{\mathrm{rsin}}^{5}\mathrm{x}\right).\mathrm{dx}+{\int }_{-\frac{\mathrm{\pi }}{2}}^{\frac{\mathrm{\pi }}{2}} {\mathrm{qsin}}^{4}\mathrm{x}.\mathrm{dx}\phantom{\rule{0ex}{0ex}}\mathrm{Hence} \mathrm{using} \left(\mathrm{i}\right) \mathrm{we} \mathrm{get}\phantom{\rule{0ex}{0ex}}\mathrm{I}=0+2{\int }_{0}^{\frac{\mathrm{\pi }}{2}} {\mathrm{qsin}}^{4}\mathrm{x}.\mathrm{dx}=2{\int }_{0}^{\frac{\mathrm{\pi }}{2}} {\mathrm{qsin}}^{4}\mathrm{x}.\mathrm{dx}\phantom{\rule{0ex}{0ex}}\mathrm{Hence} \mathrm{option}\left(2\right) \mathrm{is} \mathrm{correct}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
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G
e
G
M
\mathrm{μ} :G→.M
\mathrm{μ}\left(e, x\right) = x
\mathrm{μ}\left(a*b,x\right) = \mathrm{μ}\left(a, \mathrm{μ}\left(b, x\right)\right)
a, b ∈G
x ∈ M
\mathrm{μ},
{\mathrm{μ}}_{1,a}:M→M
{\mathrm{\mu }}_{1,a}\left(x\right) = \mathrm{μ}\left(a, x\right)
{\mathrm{μ}}_{2,x}:G→M
{\mathrm{\mu }}_{2,x}\left(a\right) = \mathrm{μ}\left(a, x\right)
\mathrm{μ}
{\mathrm{Γ}}_{\mathrm{μ}}
M
{\mathrm{\mu }}_{2,x}
G
{\mathrm{Γ}}_{\mathrm{μ} }
{\mathrm{\mu }}_{2,x}
and evaluating the results at the identity.
\mathrm{μ}
{\mathrm{Γ}}_{\mathrm{μ}} = \mathrm{Γ}
{\mathrm{\mu }}_{1,a}
{\mathrm{\mu }}_{2,x}
\mathrm{μ}
\mathrm{𝔤}
\mathrm{Γ}
\mathrm{𝔤}\mathit{ }
\mathrm{𝔤}
\mathrm{Γ}
\mathrm{𝔤}
\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):
\mathrm{with}\left(\mathrm{GroupActions}\right):
\mathrm{with}\left(\mathrm{LieAlgebras}\right):
\mathrm{with}\left(\mathrm{Library}\right):
\left[x,y\right].
\mathrm{DGsetup}\left([x,y],M\right):
\mathrm{Γ}
\mathrm{Gamma}≔\mathrm{evalDG}\left([\mathrm{D_x},\mathrm{D_y},y\mathrm{D_x}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{Γ}
\mathrm{LieAlgebraData}\left(\mathrm{Gamma}\right)
\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{e1}}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3}],G\right):
\mathrm{μ1}≔\mathrm{Action}\left(\mathrm{Gamma},G\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ1}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]
\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{μ1},[\mathrm{z1},\mathrm{z2},\mathrm{z3}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{newGamma}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{Γ2}≔\mathrm{evalDG}\left([y\mathrm{D_x},\mathrm{D_x},\mathrm{D_y}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ2}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{Γ2},\mathrm{Alg2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{L2}}\textcolor[rgb]{0,0,1}{:=}\left[\left[\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e3}}\right]\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\right]
\mathrm{DGsetup}\left(\mathrm{L2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Lie algebra: Alg2}}
\mathrm{Adjoint}\left(\right)
\left[\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\textcolor[rgb]{0,0,1}{,}\left[\begin{array}{rrr}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}\right]\right]
\mathrm{μ1},B≔\mathrm{Action}\left(\mathrm{Γ2},G,\mathrm{output}=["ManifoldToManifold","Basis"]\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]\textcolor[rgb]{0,0,1}{,}\left[\left[\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}\right]\textcolor[rgb]{0,0,1}{,}\left[\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}{0}\right]\textcolor[rgb]{0,0,1}{,}\left[\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}\right]\right]
\mathrm{newGamma}≔\mathrm{InfinitesimalTransformation}\left(\mathrm{μ1},[\mathrm{z1},\mathrm{z2},\mathrm{z3}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{newGamma}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{map}\left(\mathrm{DGzip},B,\mathrm{Γ2},"plus"\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{DGsetup}\left([x,y],M\right):
\mathrm{Γ3}≔\mathrm{Retrieve}\left("Gonzalez-Lopez",1,[22,17],\mathrm{manifold}=M\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ3}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\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}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}],\mathrm{G3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: G3}}
\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{Γ3},\mathrm{G3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{z5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\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}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{z4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\right]
\mathrm{InfinitesimalTransformation}\left(\mathrm{\mu },[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}]\right)
\left[{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\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}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\right]
\mathrm{DGsetup}\left([x,y,u,v],\mathrm{M4}\right):
\mathrm{Γ4}≔\mathrm{Retrieve}\left("Petrov",1,[32,6],\mathrm{manifold}=\mathrm{M4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Γ4}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right]
\mathrm{DGsetup}\left([\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}],\mathrm{G4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{frame name: G4}}
\mathrm{\mu }≔\mathrm{Action}\left(\mathrm{Γ4},\mathrm{G4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{μ}}\textcolor[rgb]{0,0,1}{:=}\left[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{z3}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{z4}}\right)\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{\mathrm{z3}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{z2}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{v}\right]
\mathrm{InfinitesimalTransformation}\left(\mathrm{\mu },[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}]\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_x}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D_u}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{D_y}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{u}\right]
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Define component equations - MATLAB - MathWorks España
Define component equations
equations begins the equation section in a component file; this section is terminated by an end keyword. The purpose of the equation section is to establish the mathematical relationships among a component’s variables, parameters, inputs, outputs, time and the time derivatives of each of these entities. All members declared in the component are available by their name in the equation section.
The equation section of a Simscape™ file is executed throughout the simulation. You can also specify equations that are executed during model initialization only, by using the (Initial=true) attribute. For more information, see Initial Equations.
The following syntax defines a simple equation.
The statement Expression1 == Expression2 is an equation statement. It specifies continuous mathematical equality between two objects of class Expression. An Expression is a valid MATLAB® expression. Expression may be constructed from any of the identifiers defined in the model declaration.
The equation section may contain multiple equation statements. You can also specify conditional equations by using if statements as follows:
{ elseif Expression
EquationList }
The total number of equation expressions, their dimensionality, and their order must be the same for every branch of the if-elseif-else statement.
You can declare intermediate terms in the intermediates section of a component or domain file and then use these terms in any equations section in the same component file, in an enclosing composite component, or in a component that has nodes of that domain type.
You can also define intermediate terms directly in equations by using let statements as follows:
The following rules apply to the equation section:
EquationList is one or more objects of class EquationExpression, separated by a comma, semicolon, or newline.
EquationExpression can be one of:
Conditional expression (if-elseif-else statement)
Let expression (let-in-end statement)
Expression is any valid MATLAB expression. It may be formed with the following operators:
Relational (with restrictions, see Use of Relational Operators in Equations)
In the equation section, Expression may not be formed with the following operators:
MATLAB functions not listed in Supported Functions
The colon operator may take only constants or end as its operands.
All members of the component are accessible in the equation section, but none are writable.
The following MATLAB functions can be used in the equation section. The table contains additional restrictions that pertain only to the equation section. It also indicates whether a function is discontinuous. If the function is discontinuous, it introduces a zero-crossing when used with one or more continuous operands.
All arguments that specify size or dimension must be unitless constants or unitless compile-time parameters.
isequal Possibly, if arguments are real and have the same size and commensurate units
isinf Yes
isfinite Yes
isnan Yes
mldivide First argument must be a scalar
mrdivide Second argument must be a scalar
floor Yes
ceil Yes
fix Yes
eq Do not use with continuous variables
ne Do not use with continuous variables
atan2 Yes
int32 Yes
uint32 Yes
reshape Expanded empty dimension is not supported
diff In the two argument overload, the upper bound on the second argument is 4, due to a Simscape limitation
The (Initial=true) attribute lets you specify equations that are executed during model initialization only:
equations (Initial=true)
The default value of the Initial attribute for equations is false, therefore you can omit this attribute when declaring regular equations.
For more information on when and how to specify initial equations, see Initial Equations.
For a component where x and y are declared as 1x1 variables, specify an equation of the form y = x2:
For the same component, specify the following piecewise equation:
y=\left\{\begin{array}{ll}x\hfill & \text{for }-1<=\text{ }x<=1\hfill \\ {x}^{2}\hfill & \text{otherwise }\hfill \end{array}
This equation, written in the Simscape language, would look like:
If a function has multiple return values, use it in a let statement to access its values. For example:
[m, i] = min(a);
x == m;
y == i;
assert | delay | der | function | integ | intermediates | tablelookup | time
|
One way of thinking about solving equations is to work to get the variable terms on one side of the equation and the constants on the other side. Consider the equation
71=9x-37
As a first step, you could subtract
71
from both sides, or divide both sides by
9
, or add
37
to both sides of the equation. Does one of these steps get all of the variable terms on one side of the equation and the constants on the other?
Try each of the operations above. Which operation isolates the variable and a constant on separate sides of the equation?
37
71=9x-37
x
Use the skills you have learned from past problems to solve this equation. Remember to show your work.
x=12
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DMCMN: Experimental/Analytical Evaluation of the Effect of Tip Mass on Atomic Force Microscope Cantilever Calibration | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
, 535 ERB, 1500 Engineering Drive, Madison, WI 53706
Hartono Sumali,
, P.O. Box 5800, Albuquerque, NM 87185
e-mail: hsumali@sandia.gov
, 250 Joanne Dr., Brookfield, WI 53005
e-mail: penegor@gmail.com
Matthew S. Allen Assistant Professor
Hartono Sumali Principal Member of Technical Staff
Peter C. Penegor Undergraduate Student
Allen, M. S., Sumali, H., and Penegor, P. C. (October 30, 2009). "DMCMN: Experimental/Analytical Evaluation of the Effect of Tip Mass on Atomic Force Microscope Cantilever Calibration." ASME. J. Dyn. Sys., Meas., Control. November 2009; 131(6): 064501. https://doi.org/10.1115/1.4000160
Quantitative studies of material properties and interfaces using the atomic force microscope (AFM) have important applications in engineering, biotechnology, and chemistry. Contrary to what the name suggests, the AFM actually measures the displacement of a microscale probe, so one must determine the stiffness of the probe to find the force exerted on a sample. Numerous methods have been proposed for determining the spring constant of AFM cantilever probes, yet most neglect the mass of the probe tip. This work explores the effect of the tip mass on AFM calibration using the method of Sader (1995, “Method for the Calibration of Atomic Force Microscope Cantilevers,” Rev. Sci. Instrum., 66, pp. 3789) and extends that method to account for a massive, rigid tip. One can use this modified method to estimate the spring constant of a cantilever from the measured natural frequency and
Q
-factor for any mode of the probe. This may be helpful when the fundamental mode is difficult to measure or to check for inaccuracies in the calibration obtained with the fundamental mode. The error analysis presented here shows that if the tip is not considered, then the error in the static stiffness is roughly of the same order as the ratio of the tip’s mass to the cantilever beam’s. The area density of the AFM probe is also misestimated if the tip mass is not accounted for, although the trends are different. The model presented here can be used to identify the mass of a probe tip from measurements of the natural frequencies of the probe. These concepts are applied to six low spring-constant, contact-mode AFM cantilevers, and the results suggest that some of the probes are well modeled by an Euler–Bernoulli beam with a constant cross section and a rigid tip, while others are not. One probe is examined in detail, using scanning electron microscopy to quantify the size of the tip and the thickness uniformity of the probe, and laser Doppler vibrometry is used to measure the first four mode shapes. The results suggest that this probe’s thickness is significantly nonuniform, so the models upon which dynamic calibration is based may not be appropriate for this probe.
atomic force microscopy, beams (structures), calibration, cantilevers, displacement measurement, elastic constants, error analysis, Q-factor, scanning electron microscopy
Atomic force microscopy, Calibration, Cantilevers, Probes, Errors, Elastic constants, Mode shapes, Stiffness, Density
Scratching the surface: Fundamental Investigations of Tribology With Atomic Force Microscopy
E. -L.
Intermolecular Forces and Energies Between Ligands and Receptors
Method for the Calibration of Atomic Force Microscope Cantilevers
Experimental Determination of Spring Constants in Atomic Force Microscopy
Characterization of Application Specific Probes for SPMs
Calculation of Thermal Noise in Atomic Force Microscopy
Calculation of Thermal Noise in an Atomic Force Microscope With a Finite Optical Spot Size
Practical Implementation of Dynamic Methods for Measuring Atomic Force Microscope Cantilever Spring Constants
Cantilever Spring Constant Calibration Using Laser Doppler Vibrometry
Application Note #94: Practical Advice on the Determination of Cantilever Spring Constants
,” Veeco Application Notes, http://www.veeco.com/libraryhttp://www.veeco.com/library.
Fast Imaging and Fast Force Spectroscopy of Single Biopolymers With a New Atomic Force Microscope Designed for Small Cantilevers
Ogisoa
Ultrasonic Force Microscopy for Nanometer Resolution Subsurface Imaging
Probing Attractive Forces at the Nanoscale Using Higher-Harmonic Dynamic Force Microscopy
Resonant Harmonic Response in Tapping-Mode Atomic Force Microscopy
Yaralioglu
High-Resolution Imaging of Elastic Properties Using Harmonic Cantilevers
Theory of Multifrequency Atomic Force Microscopy
Equivalent Point-Mass Models of Continuous Atomic Force Microscope Probes
General Scaling Law for Stiffness Measurement of Small Bodies With Applications to the Atomic Force Microscope
Vibration of a Cantilever Beam With a Base Excitation and Tip Mass
Free Vibration of Beams With Finite Mass Rigid Tip Load and Flexural-Torsional Coupling
Piecewise-Linear Restoring Force Surfaces for Semi-Nonparametric Identification of Nonlinear Systems
Parameter Identification of a Cantilever Beam Immersed in Viscous Fluids With Potential Applications to the Probe Calibration of Atomic Force Microscopes
Young’s Modulus Measurement of Electroplated Nickel Using AFM
|
Introduction to Chemical Engineering Processes/Problem considerations with molecular balances - Wikibooks, open books for an open world
1 Degree of Freedom Analysis on Reacting Systems
2.1 Independent and Dependent Reactions
2.1.1 Linearly Dependent Reactions
2.2 Extent of Reaction for Multiple Independent Reactions
2.3.1 Liquid-phase Analysis
2.3.2 Gas-phase Analysis
2.4 Special Notes about Gas Reactions
2.5 Inert Species
3 Example Reactor Solution using Extent of Reaction and the DOF
4 Example Reactor with Equilibrium
Degree of Freedom Analysis on Reacting SystemsEdit
If we have N different molecules in a system, we can write N mass balances or N mole balances, whether a reaction occurs in the system or not. The only difference is that in a reacting system, we have one additional unknown, the molar extent of reaction, for each reaction taking place in the system. Therefore each reaction taking place in a process will add one degree of freedom to the process.
This will be different from the atom balance which is discussed later.
Unfortunately, life is not ideal, and even if we want a single reaction to occur to give us only the desired product, this is either impossible or uneconomical compared to dealing with byproducts, side reactions, equilibrium limitations, and other non-idealities.
Independent and Dependent ReactionsEdit
When you have more than one reaction in a system, you need to make sure that they are independent. The idea of independent reactions is similar to the idea of linear independence in mathematics.
Lets consider the following two general parallel competing reactions:
{\displaystyle aA+bB\rightarrow cC+dD}
{\displaystyle a_{2}A+b_{2}B\rightarrow e_{2}E}
We can represent each of the reactions by a vector of the coefficients:
{\displaystyle V=[{\mbox{ A coeff, B coeff, C coeff, D coeff, E coeff}}]}
{\displaystyle v_{1}=[-a,-b,c,d,0]}
{\displaystyle v_{2}=[-a_{2},-b_{2},0,0,e_{2}]}
The site above gives a nice tool to tell whether any number of vectors are linearly dependent or not. Lacking such a tool, it is necessary to assess by hand whether the equations are independent. Only independent equations should be used in your analysis of multiple reactions, so if you have dependent equations, you can eliminate reactions from consideration until you've obtained an independent set.
By definition a set of vectors is only linearly independent if the equation:
{\displaystyle K_{1}*v_{1}+K_{2}*v_{2}=0}
where K1 and K2 are constants only has one solution:
{\displaystyle K_{1}=K_{2}=0}
Lets plug in our vectors:
{\displaystyle K_{1}*[-a,-b,c,d,0]+K_{2}*[-a_{2},-b_{2},0,0,e_{2}]=0}
Since all components must add up to 0, the following system follows:
{\displaystyle -K_{1}*a-K_{2}*a_{2}=0}
{\displaystyle -K_{1}*b-K_{2}*b_{2}=0}
{\displaystyle K_{1}*c+0=0}
{\displaystyle K_{1}*d+0=0}
{\displaystyle 0+K_{2}*e_{2}=0}
Obviously, the last three equations imply that unless c = d = 0 and e2 = 0,
{\displaystyle K_{1}=K_{2}=0}
and thus the reactions are independent.
Linearly Dependent ReactionsEdit
There is one rule to keep in mind whenever you are checking for reaction dependence or independence, which is summarized in the following box.
If any non-zero multiple of one reaction can be added to a multiple of a second reaction to yield a third reaction, then the three reactions are not independent.
Therefore, if the following reaction could occur in the same system as the two above:
{\displaystyle (a+a_{2})A+(b+b_{2})B\rightarrow cC+dD+e_{2}E}
then it would not be possible to analyze all three reactions at once, since this reaction is the sum of the first two. Only two can legitimately be analyzed at the same time.
All degree of freedom analyses in this book assume that the reactions are independent. You should check this by inspection or, for a large number of reactions, with numerical methods.
Extent of Reaction for Multiple Independent ReactionsEdit
When you are setting up extent of reaction in a molecular species balance, you must sure that you set up one for each reaction, and include both in your mole balance. So really, your mole balance will look like this:
{\displaystyle \Sigma n_{A,in}-\Sigma {n_{A},out}+\Sigma a_{k}X_{k}=0}
for all k reactions. In such cases it is generally easier, if possible, to use an atom balance instead due to the difficulty of solving such equations.
Equilibrium ReactionsEdit
In many cases (actually, the majority of them), a given reaction will be reversible, meaning that instead of reacting to completion, it will stop at a certain point and not go any farther. How far the reaction goes is dictated by the value of the equilibrium coefficient. Recall from general chemistry that the equilibrium coefficient for the reaction
{\displaystyle aA+bB\rightarrow cC+dD}
{\displaystyle K={\frac {C_{C,eq}^{c}*C_{D,eq}^{d}}{C_{A,eq}^{a}*C_{B,eq}^{b}}}}
with concentration
{\displaystyle C_{i}}
expressed as molarity for liquid solutes or partial pressure for gasses
Here [A] is the equilibrium concentration of A, usually expressed in molarity for an aqueous solution or partial pressure for a gas. This equation can be remembered as "products over reactants" .
Usually solids and solvents are omitted by convention, since their concentrations stay approximately constant throughout a reaction. For example, in an aqueous solution, if water reacts, it is left out of the equilibrium expression.
Often, we are interested in obtaining the extent of reaction of an equilibrium reaction when it is in equilibrium. In order to do this, first recall that:
{\displaystyle X={\frac {-\Delta n_{A}}{a}}}
and similar for the other species.
Liquid-phase AnalysisEdit
Rewriting this in terms of molarity (moles per volume) by dividing by volume, we have:
{\displaystyle {\frac {X}{V}}={\frac {[A]_{0}-[A]_{f}}{a}}}
Or, since the final state we're interested in is the equilibrium state,
{\displaystyle {\frac {X}{V}}={\frac {[A]_{0}-[A]_{eq}}{a}}}
Solving for the desired equilibrium concentration, we obtain the equation for equilibrium concentration of A in terms of conversion:
{\displaystyle [A]_{eq}=[A]_{0}-{\frac {aX}{V}}}
Similar equations can be written for B, C, and D using the definition of extent of reaction. Plugging in all the equations into the expression for K, we obtain:
{\displaystyle K={\frac {([C]_{0}+{\frac {cX}{V}})^{c}([D]_{0}+{\frac {dX}{V}})^{d}}{([A]_{0}-{\frac {aX}{V}})^{a}([B]_{0}-{\frac {bX}{V}})^{b}}}}
At equilibrium for liquid-phase reactions only
Using this equation, knowing the value of K, the reaction stoichiometry, the initial concentrations, and the volume of the system, the equilibrium extent of reaction can be determined.
If you know the reaction reaches equilibrium in the reactor, this counts as an additional piece of information in the DOF analysis because it allows you to find X. This is the same idea as the idea that, if you have an irreversible reaction and know it goes to completion, you can calculate the extent of reaction from that.
Gas-phase AnalysisEdit
By convention, gas-phase equilibrium constants are given in terms of partial pressures which, for ideal gasses, are related to the mole fraction by the equation:
{\displaystyle P_{A}=y_{A}P}
for ideal gasses only
If A, B, C, and D were all gases, then, the equilibrium constant would look like this:
{\displaystyle {\frac {P_{C}^{c}P_{D}^{d}}{P_{A}^{a}P_{B}^{b}}}}
Gas-Phase Equilibrium Constant
In order to write the gas equilibrium constant in terms of extent of reaction, let us assume for the moment that we are dealing with ideal gases. You may recall from general chemistry that for an ideal gas, we can write the ideal gas law for each species just as validly as we can on the whole gas (for a non-ideal gas, this is in general not true). Since this is true, we can say that:
{\displaystyle {\frac {n_{A}}{V}}=[A]={\frac {P_{A}}{RT}}}
Plugging this into the equation for
{\displaystyle {\frac {X}{V}}}
above, we obtain:
{\displaystyle {\frac {aX}{V}}=[A]-[A]_{eq}={\frac {P_{A0}}{RT}}-{\frac {P_{A,eq}}{RT}}}
{\displaystyle P_{a,eq}=P_{A0}-{\frac {aXRT}{V}}}
Similar equations can be written for the other components. Plugging these into the equilibrium constant expression:
{\displaystyle K={\frac {(P_{C0}+{\frac {cXRT}{V}})^{c}(P_{D0}+{\frac {dXRT}{V}})^{d}}{(P_{A0}-{\frac {aXRT}{V}})^{a}(P_{B0}-{\frac {bXRT}{V}})^{b}}}}
Gas Phase Ideal-Gas Equilibrium Reaction at Equilibrium
Again, if we know we are at equilibrium and we know the equilibrium coefficient (which can often be found in standard tables) we can calculate the extent of reaction.
Special Notes about Gas ReactionsEdit
You need to remember that In a constant-volume, isothermal gas reaction, the total pressure will change as the reaction goes on, unless the same number of moles are created as produced. In order to show that this is true, you only need to write the ideal gas law for the total amount of gas, and realize that the total number of moles in the system changes.
This is why we don't want to use total pressure in the above equations for K, we want to use partial pressures, which we can conveniently write in terms of extent of reaction.
Inert SpeciesEdit
Notice that all of the above equilibrium equations depend on concentration of the substance, in one form or another. Therefore, if there are species present that don't react, they may still have an effect on the equilibrium because they will decrease the concentrations of the reactants and products. Just make sure you take them into account when you're calculating the concentrations or partial pressures of each species in preparation for plugging into the equilibrium constant.
Example Reactor Solution using Extent of Reaction and the DOFEdit
Consider the reaction of Phosphene with oxygen:
{\displaystyle 4PH_{3}+8O_{2}\rightarrow P_{4}O_{10}+6H_{2}O}
Suppose a 100-kg mixture of 50%
{\displaystyle PH_{3}}
{\displaystyle O_{2}}
by mass enters a reactor in a single stream, and the single exit stream contains 25%
{\displaystyle O_{2}}
by mass. Assume that all the reduction in oxygen occurs due to the reaction. How many degrees of freedom does this problem have? If possible, determine mass composition of all the products.
It always helps to draw a flowchart:
There are four independent unknowns: the total mass (mole) flowrate out of the reactor, the concentrations of two of the exiting species (once they are known, the forth can be calculated), and the extent of reaction.
Additionally, we can write four independent equations, one on each reacting substance. Hence, there are 0 DOF and this problem can be solved.
Let's illustrate how to do it for this relatively simple system, which illustrates some very important things to keep in mind.
First, recall that total mass is conserved even in a reacting system. Therefore, we can write that:
{\displaystyle {\dot {m}}_{out}={\dot {m}}_{in}=100{\mbox{ kg}}}
Now, since component masses aren't conserved, we need to convert as much as we can into moles so we can apply the extent of reaction.
{\displaystyle {\dot {n}}_{PH_{3},in}=0.5*(100{\mbox{ kg}})*{\frac {1{\mbox{ mol}}}{0.034{\mbox{ kg}}}}=1470.6{\mbox{ moles PH}}_{3}{\mbox{ in}}}
{\displaystyle {\dot {n}}_{O_{2},in}=0.5*(100{\mbox{ kg}})*{\frac {1{\mbox{ mol}}}{0.032{\mbox{ kg}}}}=1562.5{\mbox{ moles O}}_{2}{\mbox{ in}}}
{\displaystyle {\dot {n}}_{O_{2},out}=0.25*(100{\mbox{ kg}})*{\frac {1{\mbox{ mol}}}{0.032{\mbox{ kg}}}}=781.25{\mbox{ moles O}}_{2}{\mbox{ out}}}
Let's use the mole balance on oxygen to find the extent of reaction, since we know how much enters and how much leaves. Recall that:
{\displaystyle \Sigma {\dot {n}}_{A,in}-\Sigma {\dot {n}}_{A,out}-a*X=0}
where a is the stoichiometric coefficient for A. Plugging in known values, including a = 8 (from the reaction written above), we have:
{\displaystyle 1562.5-781.25-8X=0}
{\displaystyle X=97.66{\mbox{ moles}}}
Now let's apply the mole balances to the other species to find how much of them is present:
{\displaystyle PH_{3}:1470.6-{\dot {n}}_{PH_{3},out}-4(97.66)=0\rightarrow {\dot {n}}_{PH_{3},out}=1080.0{\mbox{ moles PH}}_{3}}
{\displaystyle P_{4}H_{1}0:0-{\dot {n}}_{P_{4}O_{10},out}+1(97.66)=0\rightarrow {\dot {n}}_{P_{4}H_{1}0,out}=97.66{\mbox{ moles P}}_{4}O_{10}}
(note it's + instead of - because it's being generated rather than consumed by the reaction)
{\displaystyle H_{2}O:0-{\dot {n}}_{H_{2}O,out}+6(97.66)=0\rightarrow {\dot {n}}_{H_{2}O,out}=586.0{\mbox{ moles H}}_{2}O}
Finally, the last step we need to do is find the mass of all of these, and divide by the total mass to obtain the mass percents. As a sanity check, all of these plus 25 kg of oxygen should yield 100 kg total.
{\displaystyle PH_{3}}
out = 1080 moles * 0.034 kg\mole = 36.72 kg
{\displaystyle P_{4}O_{10}}
out = 97.66 moles * .284 kg\mole = 27.74 kg
{\displaystyle H_{2}O}
out = 586 moles * 0.018 kg\mole = 10.55 kg
Sanity check: 36,72 + 27.74 + 10.55 + 25 (oxygen) = 100 kg (total), so we're still sane.
{\displaystyle 36.72\%PH_{3},27.74\%P_{4}H_{10},10.55\%H_{2}O,25\%O_{2}}
Example Reactor with EquilibriumEdit
Suppose that you are working in an organic chemistry lab in which 10 kg of compound A is added to 100 kg of a 16% aqueous solution of B (which has a density of 57 lb/ft^3) The following reaction occurs:
{\displaystyle A+2B\leftarrow \rightarrow 3C+D}
A has a molar mass of 25 g/mol and B has a molar mass of 47 g/mol. If the equilibrium constant for this reaction is 187 at 298K, how much of compound C could you obtain from this reaction? Assume that all products and reactants are soluble in water at the design conditions. Adding 10 kg of A to the solution causes the volume to increase by 5 L. Assume that the volume does not change over the course of the reaction.
Solution: First, draw a flowchart of what we're given.
Since all of the species are dissolved in water, we should write the equilibrium constant in terms of molarity:
{\displaystyle K=187={\frac {[C]^{3}[D]}{[A][B]^{2}}}}
We use initial molarities of A and B, while we are given mass percents, so we need to convert.
Let's first find the number of moles of A and B we have initially:
{\displaystyle n_{A0}=10{\mbox{ kg A}}*{\frac {1{\mbox{ mol A}}}{0.025{\mbox{ kg A}}}}=400{\mbox{ mol A}}}
{\displaystyle n_{B0}=100{\mbox{ kg solution}}*{\frac {0.16{\mbox{ kg B}}}{\mbox{kg sln}}}=16{\mbox{ kg B}}*{\frac {1{\mbox{mol B}}}{\mbox{ 0.047 kg B}}}=340.43{\mbox{ mol B}}}
Now, the volume contributed by the 100kg of 16% B solution is:
{\displaystyle V={\frac {m}{\rho }}={\frac {100{\mbox{ kg}}}{57{\frac {lb}{ft^{3}}}*{\frac {1{\mbox{ kg}}}{2.2{\mbox{ lb}}}}*{\frac {1{\mbox{ ft}}^{3}}{28.317{\mbox{ L}}}}}}=109.3{\mbox{ L}}}
Since adding the A contributes 5L to the volume, the volume after the two are mixed is
{\displaystyle 109.3{\mbox{ L}}+5{\mbox{ L}}=114.3{\mbox{ L}}}
By definition then, the molarities of A and B before the reaction occurs are:
{\displaystyle [A]_{0}={\frac {400{\mbox{ moles A}}}{114.3{\mbox{ L}}}}=3.500M}
{\displaystyle [B]_{0}={\frac {340.42{\mbox{ moles B}}}{114.3{\mbox{ L}}}}=2.978M}
In addition, there is no C or D in the solution initially:
{\displaystyle [C]_{0}=[D]_{0}=0}
According to the stoichiometry of the reaction,
{\displaystyle a=1,b=2,c=3,d=1}
. Therefore we now have enough information to solve for the conversion. Plugging all the known values into the equilibrium equation for liquids, the following equation is obtained:
{\displaystyle 187={\frac {({\frac {3X}{114.3}})^{3}({\frac {X}{114.3}})}{(3.5-{\frac {X}{114.3}})(2.978-{\frac {2X}{114.3}})^{2}}}}
This equation can be solved using Goalseek or one of the numerical methods in appendix 1 to give:
{\displaystyle X=146.31{\mbox{ moles}}}
Since we seek the amount of compound C that is produced, we have:
{\displaystyle X={\frac {\Delta n_{C}}{c}}}
{\displaystyle c=3,n_{C0}=0,{\mbox{ and }}X=146.31}
, this yields
{\displaystyle n_{C}=3*146.31=438.93{\mbox{ moles C}}}
438.93 moles of C can be produced by this reaction.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processes/Problem_considerations_with_molecular_balances&oldid=3547422"
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|where  <math style="vertical-align: 0%">0^{-}</math> can be thought of as "really small negative numbers approaching zero." Since the handed limits do not agree, the limit as x approaches 5 does not exist.
|where  <math style="vertical-align: 0%">0^{-}</math> can be thought of as "really small negative numbers approaching zero." Since the handed limits do not agree, the limit as <math style="vertical-align: 0%">x</math> approaches 5 does not exist.
{\displaystyle g(x)={\frac {x+5}{x^{2}-25}}}
{\displaystyle g(x)}
{\displaystyle \lim _{x\rightarrow 5}g(x)}
{\displaystyle f}
{\displaystyle x_{0}}
{\displaystyle \lim _{x\rightarrow x_{_{0}}}f(x)=f\left(x_{0}\right).}
This can be viewed as saying the left and right hand limits exist, and are equal to the value o{\displaystyle f}
{\displaystyle x_{0}}
{\displaystyle g(x)\,\,=\,\,{\frac {x+5}{x^{2}-5}}\,\,=\,\,{\frac {x+5}{(x-5)(x+5)}}.}
{\displaystyle x_{0}}
{\displaystyle f(x_{0})}
{\displaystyle x=\pm 5}
{\displaystyle f}
{\displaystyle (-\infty ,-5)\cup (-5,5)\cup (5,\infty ).}
{\displaystyle {\begin{array}{rcl}{\displaystyle \lim _{x\rightarrow 5^{+}}g(x)}&=&{\displaystyle \lim _{x\rightarrow 5^{+}}{\frac {x+5}{(x-5)(x+5)}}}\\\\&=&{\displaystyle \lim _{x\rightarrow 5^{+}}{\frac {1}{x-5}}}\\\\&=&{\displaystyle {\frac {1}{\,\,0^{+}}}\rightarrow +\infty ,}\end{array}}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \lim _{x\rightarrow 5^{-}}g(x)}&=&{\displaystyle \lim _{x\rightarrow 5^{-}}{\frac {x+5}{(x-5)(x+5)}}}\\\\&=&{\displaystyle \lim _{x\rightarrow 5^{-}}{\frac {1}{x-5}}}\\\\&=&{\displaystyle {\frac {1}{\,\,0^{-}}}\rightarrow -\infty ,}\end{array}}}
{\displaystyle 0^{-}}
can be thought of as "really small negative numbers approaching zero." Since the handed limits do not agree, the limit as
{\displaystyle x}
approaches 5 does not exist.
{\displaystyle f}
{\displaystyle (-\infty ,-5)\cup (-5,5)\cup (5,\infty ).}
{\displaystyle \lim _{x\rightarrow 5}g(x)}
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Robust Loop Shaping of Nanopositioning Control System - MATLAB & Simulink
Glover-McFarlane Loop Shaping
Compensator Simplification
This example shows how to use the Glover-McFarlane technique to obtain loop-shaping compensators with good stability margins. The example applies the technique to a nanopositioning stage. These devices can achieve very high precision positioning which is important in applications such as atomic force microscopes (AFMs). For more details on this application, see [1].
The following illustration shows a feedback diagram of a nanopositioning device. The system consists of piezo-electric actuation, a flexure stage, and a detection system. The flexure stage interacts with the head of the AFM.
Load the plant model for the nanopositioning stage. This model is a seventh-order state-space model fitted to frequency response data obtained from the device.
load npfit A B C D
Typical design requirements for the control law include high bandwidth, high resolution, and good robustness. For this example, use:
Bandwidth of approximately 50 Hz
Roll-off of -40 dB/decade past 250 Hz
Gain margin in excess of 1.5 (3.5 dB) and phase margin in excess of 60 degrees
Additionally, when the nanopositioning stage is used for scanning, the reference signal is triangular, and it is important that the stage tracks this signal with minimal error in the midsection of the triangular wave. One way of enforcing this is to add the following design requirement:
A double integrator in the control loop
First try a PI design. To accommodate the double integrator requirement, multiply the plant by
1/s
. Set the desired bandwidth to 50 Hz. Use pidtune to automatically tune the PI controller.
Integ = tf(1,[1 0]);
bw = 50*2*pi; % 50 Hz in rad/s
PI = pidtune(G*Integ,'pi',50*2*pi);
C = PI*Integ;
bopt = bodeoptions;
bopt.FreqUnits = 'Hz'; bopt.XLim = [1e0 1e4];
bodeplot(G*C,bopt), grid
This compensator meets the bandwidth requirement and almost meets the roll-off requirement. Use allmargin to calculate the stability margins.
allmargin(G*C)
GainMargin: [0 1.1531 13.7832 7.4195 Inf]
GMFrequency: [0 2.4405e+03 3.3423e+03 3.7099e+03 Inf]
PMFrequency: 314.1959
DMFrequency: 314.1959
The phase margin is satisfactory, but the smallest gain margin is only 1.15, far below the target of 1.5. You could try adding a lowpass filter to roll off faster beyond the gain crossover frequency, but this would most likely reduce the phase margin.
The Glover-McFarlane technique provides an easy way to tweak the candidate compensator C to improve its stability margins. This technique seeks to maximize robustness (as measured by ncfmargin) while roughly preserving the loop shape of G*C. Use ncfsyn to apply this technique to this application. Note that ncfsyn assumes positive feedback so you need to flip the sign of the plant G.
[K,~,gam] = ncfsyn(-G,C);
Check the stability margins with the refined compensator K.
[Gm,Pm] = margin(G*K)
The ncfsyn compensator increases the gain margin to 3.7 and the phase margin to 70 degrees. Compare the loop shape for this compensator with the loop shape for the PI design.
bodeplot(G*C,G*K,bopt), grid
legend('PI design','Glover-McFarlane')
The Glover-McFarlane compensator attenuates the first resonance responsible for the weak gain margin while boosting the lead effect to preserve and even improve the phase margin. This refined design meets all requirements. Compare the two compensators.
bodeplot(C,K,bopt), grid
The refined compensator has roughly the same gain profile. ncfsyn automatically added zeros in the right places to accommodate the plant resonances.
The ncfsyn algorithm produces a compensator of relatively high order compared to the original second-order design.
You can use ncfmr to reduce this down to something close to the original order. For example, try order 4.
Kr = ncfmr(K,ord);
[Gm,Pm] = margin(G*Kr)
bodeplot(G*K,G*Kr,bopt), grid
legend('11th order','4th order')
The reduced-order compensator Kr has very similar loop shape and stability margins and is a reasonable candidate for implementation.
Salapaka, S., A. Sebastian, J. P. Cleveland, and M. V. Salapaka. “High Bandwidth Nano-Positioner: A Robust Control Approach.” Review of Scientific Instruments 73, no. 9 (September 2002): 3232–41.
ncfsyn | ncfmargin | ncfmr
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transform(deprecated)/statvalue - Maple Help
Home : Support : Online Help : transform(deprecated)/statvalue
stats[transform, statvalue]
value of each datum
stats[transform, statvalue](data)
transform[statvalue](data)
The statvalue function of the subpackage stats[transform, ...] replaces each data point of data by its value. In other words, it resets the weight of each data point to 1.
Note that repeated items will still be repeated, though each instance will have a weight of 1. If it is required that only a single instance of the items remain, use transform[tally] prior to using transform[statvalue]. Of course, the call to tally will, in general, change the order of the data.
The name of this function is chosen to avoid the clash with the value() function.
\mathrm{with}\left(\mathrm{stats}\right):
\mathrm{data1}≔[\mathrm{Weight}\left(3,2\right),\mathrm{Weight}\left(4,5\right)]
\textcolor[rgb]{0,0,1}{\mathrm{data1}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\right)]
\mathrm{transform}[\mathrm{statvalue}]\left(\mathrm{data1}\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]
For the data list (which has the item 3 appearing twice)
\mathrm{data2}≔[\mathrm{Weight}\left(3,2\right),\mathrm{Weight}\left(3,6\right),\mathrm{Weight}\left(4,5\right)]
\textcolor[rgb]{0,0,1}{\mathrm{data2}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{6}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\right)]
\mathrm{transform}[\mathrm{statvalue}]\left(\mathrm{data2}\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]
\mathrm{transform}[\mathrm{tally}]\left(\mathrm{data2}\right)
[\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{8}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{5}\right)]
\mathrm{transform}[\mathrm{statvalue}]\left(\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]
A more general example is:
\mathrm{data3}≔[\mathrm{Weight}\left(3,10\right),\mathrm{missing},4,\mathrm{Weight}\left(11..12,3\right),15..17]
\textcolor[rgb]{0,0,1}{\mathrm{data3}}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{10}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{missing}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Weight}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{17}]
\mathrm{transform}[\mathrm{statvalue}]\left(\mathrm{data3}\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{missing}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{15}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{17}]
transform(deprecated)[frequency]
transform(deprecated)[tally]
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Frequency modulation - formulasearchengine
Template:Modulation techniques
In telecommunications and signal processing, frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave. (Compare with amplitude modulation, in which the amplitude of the carrier wave varies, while the frequency remains constant.)
In analog signal applications, the difference between the instantaneous and the base frequency of the carrier is directly proportional to the instantaneous value of the input-signal amplitude.
Digital data can be encoded and transmitted via a carrier wave by shifting the carrier's frequency among a predefined set of frequencies—a technique known as frequency-shift keying (FSK). FSK is widely used in modems and fax modems, and can also be used to send Morse code.[1] Radioteletype also uses FSK.[2]
Frequency modulation is used in radio, telemetry, radar, seismic prospecting, and monitoring newborns for seizures via EEG.[3] FM is widely used for broadcasting music and speech, two-way radio systems, magnetic tape-recording systems and some video-transmission systems. In radio systems, frequency modulation with sufficient bandwidth provides an advantage in cancelling naturally-occurring noise.
Frequency modulation is known as phase modulation when the carrier phase modulation is the time integral of the FM signal.Template:Clarify
3 {{safesubst:#invoke:anchor|main}}Implementation
{\displaystyle x_{m}(t)}
{\displaystyle x_{c}(t)=A_{c}\cos(2\pi f_{c}t)\,}
, where fc is the carrier's base frequency, and Ac is the carrier's amplitude, the modulator combines the carrier with the baseband data signal to get the transmitted signal:
{\displaystyle y(t)=A_{c}\cos \left(2\pi \int _{0}^{t}f(\tau )d\tau \right)}
{\displaystyle =A_{c}\cos \left(2\pi \int _{0}^{t}\left[f_{c}+f_{\Delta }x_{m}(\tau )\right]d\tau \right)}
{\displaystyle =A_{c}\cos \left(2\pi f_{c}t+2\pi f_{\Delta }\int _{0}^{t}x_{m}(\tau )d\tau \right)}
{\displaystyle f(\tau )\,}
{\displaystyle f_{\Delta }\,}
Mathematically, a baseband modulated signal may be approximated by a sinusoidal continuous wave signal with a frequency fm. The integral of such a signal is:
{\displaystyle \int _{0}^{t}x_{m}(\tau )d\tau ={\frac {A_{m}\cos(2\pi f_{m}t)}{2\pi f_{m}}}\,}
{\displaystyle y(t)=A_{c}\cos \left(2\pi f_{c}t+{\frac {f_{\Delta }}{f_{m}}}\cos \left(2\pi f_{m}t\right)\right)\,}
{\displaystyle A_{m}\,}
of the modulating sinusoid is represented by the peak deviation
{\displaystyle f_{\Delta }\,}
As in other modulation systems, the value of the modulation index indicates by how much the modulated variable varies around its unmodulated level. It relates to variations in the carrier frequency:
{\displaystyle h={\frac {\Delta {}f}{f_{m}}}={\frac {f_{\Delta }|x_{m}(t)|}{f_{m}}}\ }
{\displaystyle f_{m}\,}
{\displaystyle \Delta {}f\,}
is the peak frequency-deviation—i.e. the maximum deviation of the instantaneous frequency from the carrier frequency. For a sine wave modulation, the modulation index is seen to be the ratio of the amplitude of the modulating sine wave to the amplitude of the carrier wave (here unity).
{\displaystyle h\ll 1}
, the modulation is called narrowband FM, and its bandwidth is approximately
{\displaystyle 2f_{m}\,}
{\displaystyle h={\frac {\Delta {}f}{f_{m}}}={\frac {\Delta {}f}{\frac {1}{2T_{s}}}}=2\Delta {}fT_{s}\ }
{\displaystyle T_{s}\,}
{\displaystyle f_{m}={\frac {1}{2T_{s}}}\,}
{\displaystyle f_{c}\,}
{\displaystyle f_{c}+\Delta {}f}
{\displaystyle f_{c}-\Delta {}f}
{\displaystyle h\gg 1}
{\displaystyle 2f_{\Delta }\,}
{\displaystyle \Delta {}f\,}
{\displaystyle f_{m}}
Frequency modulation can be classified as narrowband if the change in the carrier frequency is about the same as the signal frequency, or as wideband if the change in the carrier frequency is much higher (modulation index >1) than the signal frequency. [6] For example, narrowband FM is used for two way radio systems such as Family Radio Service, in which the carrier is allowed to deviate only 2.5 kHz above and below the center frequency with speech signals of no more than 3.5 kHz bandwidth. Wideband FM is used for FM broadcasting, in which music and speech are transmitted with up to 75 kHz deviation from the center frequency and carry audio with up to a 20-kHz bandwidth.
Since the sidebands are on both sides of the carrier, their count is doubled, and then multiplied by the modulating frequency to find the bandwidth. For example, 3 kHz deviation modulated by a 2.2 kHz audio tone produces a modulation index of 1.36. Suppose that we limit ourselves to only those sidebands that have a relative amplitude of at least 0.01. Then, examining the chart shows this modulation index will produce three sidebands. These three sidebands, when doubled, gives us (6 * 2.2 kHz) or a 13.2 kHz required bandwidth.
0 0.52 0.43 0.20 0.06 0.02
0 −0.34 −0.13 0.25 0.40 0.32 0.19 0.09 0.03 0.01
0 0.27 0.06 −0.24 −0.23 0.03 0.26 0.34 0.28 0.18 0.10 0.05 0.02
A rule of thumb, Carson's rule states that nearly all (~98 percent) of the power of a frequency-modulated signal lies within a bandwidth
{\displaystyle B_{T}\,}
{\displaystyle \ B_{T}=2(\Delta f+f_{m})\,}
{\displaystyle \Delta f\,}
{\displaystyle f(t)\,}
{\displaystyle f_{c}\,}
A major advantage of FM in a communications circuit, compared for example with AM, is the possibility of improved Signal-to-noise ratio (SNR). Compared with an optimum AM scheme, FM typically has poorer SNR below a certain signal level called the noise threshold, but above a higher level – the full improvement or full quieting threshold – the SNR is much improved over AM. The improvement depends on modulation level and deviation. For typical voice communications channels, improvements are typically 5-15 dB. FM broadcasting using wider deviation can achieve even greater improvements. Additional techniques, such as pre-emphasis of higher audio frequencies with corresponding de-emphasis in the receiver, are generally used to improve overall SNR in FM circuits. Since FM signals have constant amplitude, FM receivers normally have limiters that remove AM noise, further improving SNR.[7][8]
{{safesubst:#invoke:anchor|main}}Implementation
Direct FM modulation can be achieved by directly feeding the message into the input of a VCO.
For indirect FM modulation, the message signal is integrated to generate a phase-modulated signal. This is used to modulate a crystal-controlled oscillator, and the result is passed through a frequency multiplier to give an FM signal.[9]
{{#invoke:see also|seealso}} Many FM detector circuits exist. A common method for recovering the information signal is through a Foster-Seeley discriminator. A phase-locked loop can be used as an FM demodulator. Slope detection demodulates an FM signal by using a tuned circuit which has its resonant frequency slightly offset from the carrier. As the frequency rises and falls the tuned circuit provides a changing amplitude of response, converting FM to AM. AM receivers may detect some FM transmissions by this means, although it does not provide an efficient means of detection for FM broadcasts.
FM is also used at intermediate frequencies by analog VCR systems (including VHS) to record the luminance (black and white) portions of the video signal. Commonly, the chrominance component is recorded as a conventional AM signal, using the higher-frequency FM signal as bias. FM is the only feasible method of recording the luminance ("black and white") component of video to (and retrieving video from) magnetic tape without distortion; video signals have a large range of frequency components – from a few hertz to several megahertz, too wide for equalizers to work with due to electronic noise below −60 dB. FM also keeps the tape at saturation level, acting as a form of noise reduction; a limiter can mask variations in playback output, and the FM capture effect removes print-through and pre-echo. A continuous pilot-tone, if added to the signal – as was done on V2000 and many Hi-band formats – can keep mechanical jitter under control and assist timebase correction.
These FM systems are unusual, in that they have a ratio of carrier to maximum modulation frequency of less than two; contrast this with FM audio broadcasting, where the ratio is around 10,000. Consider, for example, a 6-MHz carrier modulated at a 3.5-MHz rate; by Bessel analysis, the first sidebands are on 9.5 and 2.5 MHz and the second sidebands are on 13 MHz and −1 MHz. The result is a reversed-phase sideband on +1 MHz; on demodulation, this results in unwanted output at 6−1 = 5 MHz. The system must be designed so that this unwanted output is reduced to an acceptable level.[10]
{{#invoke:main|main}} Edwin Howard Armstrong (1890–1954) was an American electrical engineer who invented wideband frequency modulation (FM) radio.[11] He patented the regenerative circuit in 1914, the superheterodyne receiver in 1918 and the super-regenerative circuit in 1922.[12] Armstrong presented his paper, "A Method of Reducing Disturbances in Radio Signaling by a System of Frequency Modulation", (which first described FM radio) before the New York section of the Institute of Radio Engineers on November 6, 1935. The paper was published in 1936.[13]
↑ 4.0 4.1 T.G. Thomas, S. C. Sekhar Communication Theory, Tata-McGraw Hill 2005, ISBN 0-07-059091-5 page 136
↑ "Communication Systems" 4th Ed, Simon Haykin, 2001
Template:Analogue TV transmitter topics Template:Telecommunications Template:Audio broadcasting
Retrieved from "https://en.formulasearchengine.com/index.php?title=Frequency_modulation&oldid=219496"
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QSD Redeeming - Chemix Ecosystem Documents
When redeeming, the user provides QSD tokens to the system, selects the corresponding collateral, and executes the redemption action. Through redemption, users can use QSD to exchange certain collateral and CEC tokens from the collateral pool. In the initial stage of the system startup, the CEC tokens obtained from redemption will be encapsulated by the CBT token, which can be converted into CEC tokens under certain conditions.
Since Chemix supports multiple collaterals, the system will allow users to choose specific collateral to redeem. If a certain amount of collateral is completely redeemed and the user’s redemption goal cannot be fulfilled, the user can continue to select other collateral for redemption until all redemption operations are completed.
Redeeming QSD is done by rearranging the previous system of equations, and solving for the units of collateral,
Y_i
, and the units of CBT.
Y_i = {\frac{Q*C_r}{P_{i}}}
CBT = {\frac{Q*(1-C_r)}{P_E}}
R_i
is the ratio of the total value of collateral
i
to the total value of all collateral assets;
Y_i
is the units of collateral
in the system;
P_i
is the price of collateral
i
Q
is the units of QSD redeemed;
C_r
is the ratio of the collateral in the protocol;
CBT
is the number of CBT tokens minted when redeeming;
P_E
On the Binance Smart Chain, if 10,000 QSD are needed to redeem, there are total collateral of 20,000,000 BUSD( $1/BUSD), 5,000,000 BNB($40/BNB), and 1,500 BTCB($37,000/BTCB) in the system, and the proportion of collateral ratio is 50%. The price of CEC is $0.5/CEC.
If the redemption collateral selected by the user at the beginning is BTCB, there are:
Y_{BTCB} = \frac{10,000×0.5}{37,000} = 0.135135135
CBT = \frac{10,000×(1-0.5)}{0.5} = 10,000
Under the above conditions, redeeming 10,000 QSD will get 0.135135135 BTCB, and newly-minted 10,000 CBT to the redeemer.
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Helix Angle Effect on the Helical Gear Load Carrying Capacity ()
The aim of this study is to investigate the helix angle effect on the helical gear load carrying capacity, including the bending and contact load carrying capacity. During the simulation, the transverse contact ratio is calculated with respect to the constant pressure angle. By changing the helix angle, both the overlap contact ratio and total contact ratio are calculated and simulated. The bending stress and contact stress of a helical gear are calculated and simulated with respect to the helix angle. Solid (CAD) modelling of a pinion gear was obtained using SOLIDWORKS software. The analytically obtained results and finite elements method results are compared. It is observed that increasing the helix angle causes an increase of the contact ratio of the helical gear. Furthermore, increasing the contact ratio reduces the bending stress and contact stress of the helical gear. However, with a constant transverse contact ratio, it is possible to improve the total contact ratio depending on the helix angle. It is concluded that a higher helix angle increases the helical gear bending and contact load carrying capacity.
Helical Gear, Helix Angle, Contact Ratio, Bending Stress, Contact Stress
Bozca, M. (2018) Helix Angle Effect on the Helical Gear Load Carrying Capacity. World Journal of Engineering and Technology, 6, 825-838. doi: 10.4236/wjet.2018.64055.
Gears are widely used to mechanically transmit power in automotive transmissions. The aim of the gears is to couple two shafts together; the rotation of the drive-shaft is a function of the rotation of the drive-shaft in the gear mechanism. Therefore, determining the geometric design parameters of gears is crucial.
The contact ratio is an important parameter for successful gear design. The helix angle is considered to be an effective parameter to increase the contact ratio of a helical gear. Thus, it is possible to increase the helical gear load carrying capacity, including the tooth bending stress and tooth contact stress.
One of the disadvantages of increasing the helix angle is the axial forces caused on the helical gear mechanism.
Optimisation of effective design parameters to reduce the tooth bending stress in an automotive transmission gearbox is presented. Therefore, the contact ratio effect on the tooth bending stress by changing the contact ratio with respect to the pressure angle is analysed [1] . It is concluded that a higher contact ratio results in reduced tooth bending stress, while a higher pressure angle and decreased contact ratio caused an increase in tooth bending stress and contact stress [1] .
When the helix angle is increased from 15 [˚] to 35 [˚], the corresponding bending stress and compression stress decrease [2] .
It was concluded that a helix angle increase had significant effects on the tooth-root bending stress and tooth compressive stress. Moreover, it was observed that when the helix angle increased from 0 [˚] to 22.5 [˚], both the bending stress and compression stress were reduced approximately 10% [3] .
For a given number of teeth, a smaller pressure angle may produce an undercut. However, the contact ratio increases, so the load carrying capacity may improve as the load is distributed along a longer line of contact [4] .
The contact ratio for a helical gear pair increases with the helix angle, which generates the screwed surface of the tooth face [5] .
The aim of this study is to investigate the helix angle effect on the helical gear load carrying capacity, including the bending and contact load carrying capacity. For this aim the analytically obtained results and finite elements method results are compared. It is concluded that a higher helix angle increases the helical gear bending and contact load carrying capacity.
2.1. Pinion and Wheel Gears Mechanism
In the proposed pinion and wheel gear mechanism, all pinion and wheel gears are helical and are made of 16MnCr5.
2.1.1. Helix Angle β
The helix angle, β, is the angle between the helix line and horizontal axis, as shown in Figure 1.
2.1.2. Contact Ratio
The dimensions of the helical gear are shown in Figure 2, and the contact line of the helical gear is shown in Figure 3.
A second pair of mating teeth should come into contact before the first pair is out of contact during pinion and wheel gear running [6] .
If the gear contact ratio is equal to 1, one tooth is leaving contact just as the next tooth is beginning contact. If the gear contact ratio is larger than 1, load sharing among the teeth is possible during pinion and wheel gear running [7] .
Figure 1. Pinon and wheel gear mechanism.
Figure 2. Dimensions of a helical gear.
Figure 3. Contact line of a helical gear.
When the contact ratio is equal to 2 or more, at least two pairs of teeth are theoretically in contact [7] .
If the gear profile contact ratio is less than 2.0, it is called the Low Contact Ratio (LCR). If he gear profile contact ratio equals 2.0 or greater, it is called the High Contact Ratio (HCR).
The contact ratio consists of two parts, such as the transverse contact ratio, εα, and the overlap or face contact ratio, εβ.
1) Transverse contact ratio εα
It is well-known that the average number of teeth that are in contact as the gears rotate is the contact ratio (CR). The contact ratio is calculated from the following equation.
The transverse contact ratio, εα, is calculated as follows [8] [9] [10] [11] .
{\epsilon }_{\alpha }=\frac{{g}_{\alpha }}{{p}_{et}}
{\epsilon }_{\alpha }=\frac{0.5\cdot \left(\sqrt{{d}_{a1}^{2}-{d}_{b1}^{2}}+\sqrt{{d}_{a2}^{2}-{d}_{b2}^{2}}\right)-{a}_{d}\cdot \mathrm{sin}{\alpha }_{t}}{\text{π}\cdot {m}_{t}\cdot \mathrm{cos}{\alpha }_{t}}
where gα is the path length of the contact line [mm], pet is the base pitch [mm], da1 is the addendum circle diameter of the pinion gear [mm], db1 is the base circle diameter of the pinion gear [mm], da2 is the addendum circle diameter of the wheel gear [mm], db2 is the base circle diameter of the wheel gear [mm], ad is the centre distance [mm], αt is the transverse pressure angle [˚], and mt is the transverse module [mm].
2) Overlap ratio εβ
The overlap ratio, εβ, is calculated as follows [8] [9] [10] [11] .
{\epsilon }_{\beta }=\frac{U}{{p}_{t}}
{\epsilon }_{\beta }=\frac{b\cdot \mathrm{tan}\beta }{{p}_{t}}
{\epsilon }_{\beta }=\frac{b\cdot \mathrm{sin}\beta }{\text{π}\cdot {m}_{n}}
3) Total contact ratio εγ
{\epsilon }_{\gamma }={\epsilon }_{\alpha }+{\epsilon }_{\beta }
where εα is the transverse contact ratio and εβ is the overlap ratio. Helical gears have higher load carrying capacities than spur gears because their contact ratios are larger than those of spur gears.
2.2. Calculating the Load Carrying Capacity of Helical Gears
2.2.1. Nominal Tangential Load
The nominal tangential load Ft is calculated as follows.
{F}_{t}=\frac{2\cdot {T}_{L}}{{d}_{1}}
where TL is the applied torque [N・mm] and d1 is the base diameter of the tooth diameter [mm].
2.2.2. Axial Load
Axial load Fa is calculated as follows.
{F}_{a}=\frac{{F}_{t}}{\mathrm{tan}\beta }
where β is helix angle [˚].
2.2.3. Tooth Bending Stress
The real tooth-root stress, σF is calculated as follows [8] [9] [10] [11] [12] . The bending stress of the tooth-root is shown in Figure 4.
{\sigma }_{F}=\frac{{F}_{t}}{b{m}_{n}}{Y}_{F}{Y}_{S}{Y}_{\epsilon }{Y}_{\beta }{K}_{A}{K}_{V}{K}_{F\beta }{K}_{F\alpha }
where Ft is the nominal tangential load [N], b is the face width [mm], mn is the normal module [mm], YF is the form factor [-], Ys is the stress correction factor [-], Yε is the contact ratio factor [-], KA is the application factor [-], KV is the internal dynamic factor [-], KFβ is the face load factor for tooth-root stress [-] and KFα is the transverse load factor for tooth-root stress [-]. The safety factor for bending stress SF is calculated as follows [8] [9] [10] [11] [12] .
{S}_{F}=\frac{{\sigma }_{Fp}}{{\sigma }_{F}}
where σFp is the permissible bending stress.
2.2.4. Tooth Contact Stress
The real contact stress, σH is calculated as follows [8] [9] [10] [11] [12] . The contact stress at the tooth flank is shown in Figure 5.
{\sigma }_{H}=\sqrt{\frac{{F}_{t}}{b{m}_{n}}\frac{u+1}{u}}{Z}_{H}{Z}_{E}{Z}_{\epsilon }{Z}_{\beta }\sqrt{{K}_{A}{K}_{V}{K}_{H\beta }{K}_{H\alpha }}
where u is the gear ratio [-], ZH is the zone factor [-], ZE is the elasticity factor [ ], Zε is the contact ratio factor [-], Zβ is the helix angle factor [-], KHβ is the face load factor for contact stress [-] and KHα is the transverse load factor for contact stress [-].
The safety factor for contact stress, SH is calculated as follows [8] [9] [10] [11] [12] .
{S}_{H}=\frac{{\sigma }_{Hp}}{{\sigma }_{H}}
where σHp is the permissible contact stress.
2.3. Finite Elements Model
Solid (CAD) modelling of a pinion gear was obtained using SOLIDWORKS software. A solid model is essential for finite element method (FEM) analysis [13] [14] . Solid (CAD) modelling of a pinion gear is shown in Figure 6.
The obtained Solid (CAD) model is used to obtain the finite element method (FEM) model using the SOLIDWORKS finite element tool.
Figure 6. Solid (CAD) modelling of a pinion gear.
To simulate the actual conditions of the pinion gear for analysis, the boundary conditions below were used.
a) The pinion gear was constrained in the centre of the pinion gear.
b) The applied load for bending the pinion gear tooth was considered at the tooth top surface.
During simulation, the tooth bending stress and tooth contact stress were calculated according to ISO 6336. The effects of the helix angle on the tooth bending stress and tooth contact stress are analysed by varying the helix angle. The tooth bending stress and tooth contact stress parameters are shown in Table 1.
The tooth bending stress and tooth contact stress simulation results are shown in Table 2.
3.1. Helix Angle and Overlap Contact Ratio Relation
The helix angle and overlap contact ratio relation are shown in Figure 7. As the helix angle increases from 22 [˚] to 32 [˚], the overlap contact ratio increases from 1.01 [-] to 1.43 [-].
3.2. Helix Angle and Total Contact Ratio Relation
The helix angle and total contact ratio relation are shown in Figure 8. As the helix angle increases from 22 [˚] to 32 [˚], the total contact ratio increases from 2.52 [-] to 2.88 [-].
3.3. Helix Angle and Bending Stress Relation
The helix angle and tooth bending stress relation are shown in Figure 9. As the helix angle decreases from 22 [˚] to 12 [˚], the bending stress is reduced from 365 [N/mm2] to 233 [N/mm2].
Table 1. Tooth bending stress and tooth contact stress parameters.
Table 2. Tooth bending stress and tooth contact stress simulation results.
3.4. Helix Angle and Tooth Contact Stress Relation
The helix angle and contact stress relation are shown in Figure 10. As the helix angle decreases from 22 [˚] to 12 [˚], the bending stress is reduced from 1265 [N/mm2] to 1128 [N/mm2].
Figure 7. Helix angle and overlap contact ratio relation.
Figure 8. Helix angle and total contact ratio relation.
Figure 9. Helix angle and tooth bending stress relation.
Figure 10. Helix angle and tooth contact stress relation.
3.5. Helix Angle and Axial Force Relation
The helix angle and axial force relation is shown in Figure 11. As the helix angle decreases from 22 [˚] to 12 [˚], the axial force is increased from 2319 [N] to 3280 [N].
Figure 11. Helix angle and axial force relation.
3.6. Static Structural Analysis with FEM
Static structural analysis of the pinon gear was completed for the applied load considering the Von Mises stress. The Von Mises stress is written as follows.
{\sigma }_{vM}=\sqrt{{\sigma }^{2}+3{\tau }^{2}}
Theoretical bending stress is written as follows.
{\sigma }_{FT}=\frac{{F}_{t}}{b\cdot m}{Y}_{F}
The applied load was considered in 6 different pinion gears that had 6 different helix angles. The Von-Mises stresses are shown in Figures 12-17.
The Von Mises stress obtained by finite elements analyses are shown in Table 3. A comparison between Von Mises Stress with FEM and analytical static stress depending on the helix angle is shown in Figure 12. By observing obtained finite elements method results, it is concluded that 45% increased helix angle result in 6.5% decreased Von Mises stress.
The maximum Von Mises stress of the pinion gear reaches 112, 900 [N/mm2] on the pinion tooth root for a helix angle β = 22 [˚], as shown in Figure 13.
During simulation, tooth bending stress and tooth contact stress were calculated according to ISO 6336. Solid (CAD) modelling of a pinion gear was completed
Figure 12. Comparison of Von Mises stress and Theoretical stress.
Figure 13. Von Mises stress for a helix angle β = 22 [˚].
using SOLIDWORKS software. The analytically obtained results and finite element method results were compared.
The effect of the helix angle on the tooth bending stress and tooth contact stress was analysed by varying helix angle, and the following conclusions are drawn.
Increasing the helix angle β, results in increasing the overlap contact ratio εβ. Thus, increasing the helix angle β results in increasing the total contact ratio εγ.
Increasing the helix angle β results in a reduction of the tooth bending stress σF and tooth contact stress σH.
The analytically obtained results are verified by the finite element method
Table 3. Von Mises stresses as determined by finite element analyses.
results. By considering the tooth bending stress, the analytically obtained results and finite element method results only differ by 5%.
Helical gears have higher load carrying capacities than spur gears because their contact ratios are larger than those of spur gears.
Increasing the helix angle β results in an increase in the axial force Fa. Thus, one of the disadvantages of increasing the helix angle is the increase of axial forces on the helical gear mechanism.
[1] Bozca, M. (2017) Optimisation of Effective Design Parameters for an Automotive Transmission Gearbox to Reduce Tooth Bending Stress. Modern Mechanical Engineering, 7, 35-56.
[2] Ventkatesh, B., Prabhakar, Vattikuti, S.V. and Deva Prasad, S. (2014) Investigate the Combined Effect on Gear Ratio, Helix Angle, Facewidth and Module on Bending and Compressive Stress of Steel Alloy Heical Gear. Procedia Material Science, 6, 1865-1870.
[3] Zhan, J.X. and Fard, M. (2018) Effects of Helix Angle, Mechanical Errors, and Coefficient of Friction on the Time-Varying Tooth-Root Stress of Helical Gears. Measurement, 118, 135-146.
[5] Kang, J.S. and Choi, Y.-S. (2008) Optimisation of Helix Angle for Helical Gear System. Journal of Mechanical Science and Technology, 22, 2393-2402.
[7] Norton, R.L. (2011) Machine Design. Prentice Hall, Upper Saddle River, NJ.
[8] ISO 6336-5: Calculation of Load Capacity of Spur and Helical Gears-Part 5: Strength and Quality of Materials.
[9] ISO 6336-3: Calculation of Load Capacity of Spur and Helical Gears-Part 3: Calculation of Tooth Bending Strength.
[10] Matek, R. (2005) Maschinenelemente. Vieweg & Sohn Verlag/Fachverlage, GmbH, Wiesbaden.
[11] Decker (2009) Maschinenelemente. Carl Hanser Verlag, München.
[12] Naunheimer, H., Bertsche, B., Ryborz, J. and Novak W. (2011) Automotive Transmissions. Springer-Verlag, Berlin, Heidelberg.
[13] Moaveni, S. (2003) Finite Element Analysis, Theory and Application with ANSYS. Prentice Hall, Upper Saddle River, NJ.
[14] Chandrupatla, T.R. and Belegundu, A.D. (2002) Introduction to Finite Elements in Engineering. Prentice Hall, Upper Saddle River, NJ.
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Revision as of 17:06, 6 January 2022 by Eragon4 (talk | contribs) (→Determination of stats: Home)
{\displaystyle HP={\Biggl \lfloor }{{\Biggl (}(Base+DV)\times 2+{\biggl \lfloor }{\tfrac {{\bigl \lceil }{\sqrt {STATEXP}}{\bigr \rceil }}{4}}{\biggr \rfloor }{\Biggr )}\times Level \over 100}{\Biggr \rfloor }+Level+10}
{\displaystyle OtherStat={\Biggl \lfloor }{{\Biggl (}(Base+DV)\times 2+{\biggl \lfloor }{\tfrac {{\bigl \lceil }{\sqrt {STATEXP}}{\bigr \rceil }}{4}}{\biggr \rfloor }{\Biggr )}\times Level \over 100}{\Biggr \rfloor }+5}
{\displaystyle HP={\Bigl \lfloor }{(2\times Base+IV+\lfloor {\tfrac {EV}{4}}\rfloor )\times Level \over 100}{\Bigr \rfloor }+Level+10}
{\displaystyle OtherStat={\Biggl \lfloor }{\biggl (}{\Bigl \lfloor }{(2\times Base+IV+\lfloor {\tfrac {EV}{4}}\rfloor )\times Level \over 100}{\Bigr \rfloor }+5{\biggr )}\times Nature{\Biggr \rfloor }}
Attack: Hitmonlee
Defense: Hitmonchan
HP: Hitmontop
Trade (Good Friend) 1
Trade (Great Friend) 2
Trade (Ultra Friend) 3
Weather-boosted Shadow Pokémon 4
Trade (Best Friend) 5
Shadow Pokémon from Giovanni 6
Field/Special/Timed Research 10
Lucky Trade 12
{\displaystyle Stat=(base+IV)\times cpMult}
{\displaystyle 2\times IV_{HP}+1}
{\displaystyle 2\times IV_{Attack}+1}
{\displaystyle 2\times IV_{Defense}+1}
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Review: Dockcase's 8-in-1 USB-C Hub - The Tape Drive
Review: Dockcase's 8-in-1 USB-C Hub
DockCase just launched their latest 8-in-1 USB-C hub on Kickstarter (ends Nov 26th). I was interested in reviewing DockCase’s 8-in-1 USB-C hub as I had a good experience with their previous 7-in-1 USB-C hub review unit they sent me last time they were on Kickstarter.
Besides the USB-C main device I/O port the 8-in-1 USB-C hub features
(1) 100W USB-C PD input port (power)
(1) USB-C 3.2 port
Overall I think this dock provides a good selection of ports as DockCase added a Gigabit Ethernet port and USB-C port this time around. I do miss the micro SD card slot from DockCase’s older 7-in-1 USB-C though. The main port I wish both had was a USB-C port that supported DisplayPort as I use DP for my external monitor since it supports a higher refresh rate than HDMI on my MacBook.
What makes the DockCase hub unique is its integrated display. My favorite feature of the display is its ability to show if a device is connected to a USB-A 2.0 or USB-A 3.2 port and which USB-C PD profile (volts, amps, & watts) it is using to charge the main attached device. This is especially helpful as different combinations of devices and chargers will result in different PD profile handshakes. One downside of using a hub with a 16 inch MacBook Pro is that the hub charges the MBP near its fastest speed which is to be expected since the hub takes a few watts to power itself and any accessories attached to it.
Another feature of the display new to the 8-in-1 hub is the ability to change how much power the hub should use and which mode it should optimize for. The hub has 4 main modes: video, data, charging, and ‘my’ mode. The video mode changes all the presets to enable the hub to convert a 4K@30Hz HDMI signal to 4K@60Hz at the cost of reducing the maximizing charging and data speeds. I believe this mode also supports using a Nintendo Switch in dock mode but I wasn’t able to test this. Similarly, the data mode enables the data ports to reach their fastest speeds and the charging mode reduces the power used by the dock to enable the fastest charging possible on the main port. And the ‘my’ mode can be used to provide a balance of the 3 other modes. I was concerned at first when I hooked up the new 8-in-1 hub and discovered that it was charging my MacBook Pro more slowly than the previous 7-in-1 hub. But when I switched the 8-in-1 hub to charging mode it started to charge my MacBook Pro faster than the 7-in-1 hub.
The hubs that DockCase makes are really interesting given the integrated display and the latest 8-in-1 hub builds on that with its ability to change modes. If you don’t want to spend a lot of money on Thunderbolt port and don’t hook up an external display via Display Port then this could be a good hub for you if you understand that given the limitations of this USB3.2 Gen 2 hub means that you can’t run the data ports at full speed while driving an HDMI display at 4K@60Hz. For instance, I think the ethernet port speed drops from 1000Mbps to 100Mbps if the hub is driving a 4K@60Hz display (I wasn’t able to test this scenario since I used DisplayPort). If you don’t want the hassle of switching modes (and it is a hassle since it is a button tap on the hub to change the orientation of the integrated display but a 6 second long press if you want to enter the settings of the hub and another 6 second long press to confirm a selection) or if you need to use full speed data ports and a 4K@60Hz display at the same time then you might be better off getting a Thunderbolt hub. That is if you are willing to spend
50 to
100 more than what DockCase’s 8-in-1 hub costs. I would really like to see DockCase make a Thunderbolt hub with an integrated display since I don’t think anything like that exists or even a hub with a port that supports DisplayPort.
The review unit I tested shipped with beta firmware. It will be interesting to see what features DockCase can add with new firmware though judging by the current support documentation on their site it could be a difficult process. DockCase will begin delivers of their 8-in-1 hub in January 2022.
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Gustav Fechner - formulasearchengine
Gustav Theodor Fechner (Template:IPAc-en; Template:IPA-de; April 19, 1801 – November 18, 1887), was a German philosopher, physicist and experimental psychologist. An early pioneer in experimental psychology and founder of psychophysics, he inspired many 20th century scientists and philosophers. He is also credited with demonstrating the non-linear relationship between psychological sensation and the physical intensity of a stimulus via the formula:
{\displaystyle S=K\ln I}
, which became known as the Weber–Fechner law.[1][2]
1 Early life and scientific career
2.1 The Fechner color effect
2.4 Corpus Callosum Split
2.5 Golden Section Hypothesis
2.6 The two-piece normal distribution
3.1 Fechner Day
Fechner was born at Groß Särchen, near Muskau, in Lower Lusatia, where his father was a pastor. Despite being raised by his religious father, Fechner became an atheist in later life.[3] He was educated first at Sorau (now Żary in Western Poland). In 1817 he studied of medicine at the Medizinisch-Chirurgische Akademie in Dresden and from 1818 at the University of Leipzig, the city in which he spent the rest of his life.[4] In 1834 he was appointed professor of physics. But in 1839, he contracted an eye disorder while studying the phenomena of color and vision, and, after much suffering, resigned. Subsequently recovering, he turned to the study of the mind and its relations with the body, giving public lectures on the subjects dealt with in his books.
Gustav Fechner published chemical and physical papers, and translated chemical works by J. B. Biot and Louis Jacques Thénard from the French language. A different but essential side of his character is seen in his poems and humorous pieces, such as the Vergleichende Anatomie der Engel (1825), written under the pseudonym of "Dr. Mises."
Fechner's epoch-making work was his Elemente der Psychophysik (1860). He starts from the monistic thought that bodily facts and conscious facts, though not reducible one to the other, are different sides of one reality. His originality lies in trying to discover an exact mathematical relation between them. The most famous outcome of his inquiries is the law known as the Weber–Fechner law which may be expressed as follows:
Though holding good within certain limits only, the law has been found to be immensely useful. Fechner's law implies that sensation is a logarithmic function of physical intensity, which is impossible due to the logarithm's singularity at zero; therefore, S. S. Stevens proposed the more mathematically plausible power-law relation of sensation to intensity in his famous 1961 paper entitled "To Honor Fechner and Repeal His Law."
In 1838, he also studied the still-mysterious perceptual illusion of what is still called the Fechner color effect, whereby colors are seen in a moving pattern of black and white. The English journalist and amateur scientist Charles Benham, in 1894, enabled English-speakers to learn of the effect through the invention of the spinning top that bears his name. Whether Fechner and Benham ever actually met face to face for any reason is not known.
In 1878 Fechner published a paper in which he developed the notion of the median. He later delved into experimental aesthetics and thought to determine the shapes and dimensions of aesthetically pleasing objects. He mainly used the sizes of paintings as his data base. In his 1876 Vorschule der Aesthetik he used the method of extreme ranks for subjective judgements.[5]
Fechner is generally credited with introducing the median into the formal analysis of data.[6]
In 1871 Fechner reported the first empirical survey of coloured letter photisms among 73 synesthetes.[7][8] His work was followed in the 1880s by that of Francis Galton.[9][10][11]
One of Fechner's speculations about consciousness dealt with brain. During his time, it was known that the brain is bilaterally symmetrical and that there is a deep division between the two halves that are linked by a connecting band of fibers called the corpus callosum. Ergo, Fechner speculated that if the corpus callosum were split, two separate streams of consciousness would result - the mind would become two. Yet, Fechner believed that his theory would never be tested; he was incorrect. During the mid-twentieth century, Roger Sperry and Michael Gazzaniga worked on epileptic patients with sectioned corpus callosum and observed that Fechner's idea was correct.[12]
Fechner constructed ten rectangles with different ratios of width to length and asked numerous observers to choose the "best" and "worst" rectangle shape. He was concerned with the visual appeal of rectangles with different proportions. The rectangles chosen as "best" by the largest number of participants had a ratio of 0.62 (between 3:5 and 5:8). This became known as the "golden section" and referred to the ratio of a rectangle's width to length that is most appealing to the eye. Carl Stumpf partook in this study as a participant. However, there has been some ongoing dispute on the experiment itself, as the fact that Fechner deliberately discarded results of the study ill-fitting to his needs became known, with many mathematicians including Mario Livio refuting the result of the experiment.
In his posthumously published Kollektivmasslehre (1897), Fechner introduced the Zweiseitige Gauss'sche Gesetz or two-piece normal distribution, to accommodate the asymmetries he had observed in empirical frequency distributions in many fields. The distribution has been independently rediscovered by several authors working in different fields.[13]
Though he had a vast influence on psychophysics, the actual disciples of his general philosophy were few. Ernst Mach was inspired by his work on psychophysics.[14] William James also admired his work: in 1904, he wrote an admiring introduction to the English translation of Fechner's Büchlein vom Leben nach dem Tode (Little Book of Life After Death).
Fechner's world concept was highly animistic. He felt the thrill of life everywhere, in plants, earth, stars, the total universe. Man stands midway between the souls of plants and the souls of stars, who are angels.[15] God, the soul of the universe, must be conceived as having an existence analogous to men. Natural laws are just the modes of the unfolding of God's perfection. In his last work Fechner, aged but full of hope, contrasts this joyous "daylight view" of the world with the dead, dreary "night view" of materialism. Fechner's work in aesthetics is also important. He conducted experiments to show that certain abstract forms and proportions are naturally pleasing to our senses, and gave some new illustrations of the working of aesthetic association. Charles Hartshorne saw him as a predecessor on his and Alfred North Whitehead's philosophy and regretted that Fechner's philosophical work had been neglected for so long.[16]
Fechner's position in reference to predecessors and contemporaries is not very sharply defined. He was remotely a disciple of Schelling, learnt much from Benedict de Spinoza, Gottfried Wilhelm Leibniz, Johann Friedrich Herbart, Arthur Schopenhauer, and Christian Hermann Weisse, and decidedly rejected Georg Hegel and the monadism of Rudolf Hermann Lotze.
It is claimed that, on the morning of 22 October 1850, Fechner awoke with a sudden new insight into how to study the mind. Moving away from Wundtarian introspection and basing his work on that of Weber, he developed his psychophysical Fechner scale. Each year, psychophysicists celebrate 22 October as the anniversary of Fechner's new insight as Fechner Day.[17] Celebrations to mark Fechner Day were organized by the International Society for Psychophysics and held in Fechner's home city of Leipzig in 2001.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}
Praemissae ad theoriam organismi generalem ("Advances in the general theory of organisms") (1823).
(Dr. Mises) Stapelia mixta (1824). Google (Harvard)
Resultate der bis jetzt unternommenen Pflanzenanalysen ("Results of plant analyses undertaken to date") (1829). Google (Stanford)
Maassbestimmungen über die galvanische Kette (1831).
(Dr. Mises) Schutzmittel für die Cholera ("Protective equipment for cholera") (1832). Google (Harvard) — Google (UWisc)
Repertorium der Experimentalphysik (1832). 3 volumes.
Volume 1. Google (NYPL) — Google (Oxford)
(ed.) Das Hauslexicon. Vollständiges Handbuch praktischer Lebenskenntnisse für alle Stände (1834–38). 8 volumes.
Das Büchlein vom Leben nach dem Tode (1836). 6th ed., 1906. Google (Harvard) — Google (NYPL)
REDIRECT Template:Link language On Life After Death (1882). Google (Oxford) — IA (UToronto) 2nd ed., 1906. Google (UMich) 3rd ed., 1914. IA (UIllinois)
REDIRECT Template:Link language The Little Book of Life After Death (1904). IA (UToronto) 1905, Google (UCal) — IA (Ucal) — IA (UToronto)
(Dr. Mises) Gedichte (1841). Google (Oxford)
Ueber das höchste Gut ("Concerning the Highest Good") (1846). Google (Stanford)
(Dr. Mises) Nanna oder über das Seelenleben der Pflanzen (1848). 2nd ed., 1899. 3rd ed., 1903. Google (UMich) 4th ed., 1908. Google (Harvard)
Zend-Avesta oder über die Dinge des Himmels und des Jenseits (1851). 3 volumes. 3rd ed., 1906. Google (Harvard)
Ueber die physikalische und philosophische Atomenlehre (1855). 2nd ed., 1864. Google (Stanford)
Professor Schleiden und der Mond (1856). Google (UMich)
Elemente der Psychophysik (1860). 2 volumes. Volume 1. Google (ULausanne) Volume 2. Google (NYPL)
Ueber die Seelenfrage ("Concerning the Soul") (1861). Google (NYPL) — Google (UCal) — Google (UMich) 2nd ed., 1907. Google (Harvard)
Die drei Motive und Gründe des Glaubens ("The three motives and reasons of faith") (1863). Google (Harvard) — Google (NYPL)
Einige Ideen zur Schöpfungs- und Entwickelungsgeschichte der Organismen (1873). Google (UMich)
(Dr. Mises) Kleine Schriften (1875). Google (UMich)
Erinnerungen an die letzen Tage der Odlehre und ihres Urhebers (1876). Google (Harvard)
Vorschule der Aesthetik (1876). 2 Volumes. Google (Harvard)
In Sachen der Psychophysik (1877). Google (Stanford)
Die Tagesansicht gegenüber der Nachtansicht (1879). Google (Oxford) 2nd ed., 1904. Google (Stanford)
Revision der Hauptpuncte der Psychophysik (1882). Google (Harvard)
Kollektivmasslehre (1897). Google (NYPL)
↑ Fechner, Gustav Theodor at vlp.mpiwg-berlin.mpg.de
↑ Pojman, Paul, "Ernst Mach", The Stanford Encyclopedia of Philosophy (Winter 2011 Edition), Edward N. Zalta (ed.) [2]
↑ For Hartshorne's appreciation of Fechner see his Aquinas to Whitehead – Seven Centuries of Metafysics of Religion. Hartshorne also comments that William James failed to do justice to the theological aspects of Fechner's work. Hartshorne saw also resemblances with the work of Fechner's contemporary Jules Lequier. See also: Hartshorne – Reese (ed.) Philosophers speak of God.
↑ Kreuger, L. E. (1993) Personal Communication. ref History of Psychology 4th edition David Hothersal 2004 ISBN 9780072849653
Heidelberger, M. (2001) Gustav Theodor Fechner, Statisticians of the Centuries (ed. C. C. Heyde and E. Seneta) pp. 142–147. New York: Springer Verlag, 2001.
Michael Heidelberger. Nature From Within: Gustav Theodor Fechner and his Psychophysical Worldview Trans. Cynthia Klohr. Pittsburgh, PA: University of Pittsburgh Press, 2004. ISBN 0-822-9421-00
Stephen M Stigler. The History of Statistics: The Measurement of Uncertainty before 1900, Cambridge, MA: Harvard University Press, 1986. pp. 242–254.
Works of Gustav Theodor Fechner at Projekt Gutenberg-DE. (German)
Excerpt from Elements of Psychophysics from the Classics in the History of Psychology website.
Robert H. Wozniak’s Introduction to Elemente der Psychophysik.
Biography, bibliography and digitized sources in the Virtual Laboratory of the Max Planck Institute for the History of Science
Granville Stanley 1912 'Founders of modern psychology p. 125ff archive.org
Gustav Theodor Fechner 1904 The Little Book of Life after Death Forward by William James
Gustav Theodor Fechner 1908 The Living Word
Gustav Theodor Fechner at statprob.com
Retrieved from "https://en.formulasearchengine.com/index.php?title=Gustav_Fechner&oldid=286860"
German statisticians
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Standardized Kt/V - wikidoc
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Standardized Kt/V, also std Kt/V, is a way of measuring (renal) dialysis adequacy. It was developed by Frank Gotch and is used in the USA to measure dialysis. Despite the name, it is quite different from Kt/V. In theory, both peritoneal dialysis and hemodialysis can be quantified with std Kt/V.
2 Interpretation of std Kt/V
3 Comparison to Kt/V
4 Advantages of std Kt/V
5 Criticism/disadvantages of std Kt/V
6 Calculating stdKt/V from treatment Kt/V and number of sessions per week
7 Nomogram to get stdKt/V from treatment Kt/V with different treatment schedules
Standardized Kt/V is motivated by the steady state solution of the mass transfer equation often used to approximate kidney function (equation 1), which is also used to define clearance.
{\displaystyle V{\frac {dC}{dt}}=-K\cdot C+{\dot {m}}\qquad (1)}
{\displaystyle {\dot {m}}}
is the mass generation rate of the substance - assumed to be a constant, i.e. not a function of time (equal to zero for foreign substances/drugs) [mmol/min] or [mol/s]
t is dialysis time [min] or [s]
V is the volume of distribution (total body water) [L] or [m3]
K is the clearance [mL/min] or [m3/s]
C is the concentration [mmol/L] or [mol/m3] (in the USA often [mg/mL])
{\displaystyle {\frac {dC}{dt}}}
Derivation equation 1 is described in the article clearance (medicine).
The solution of the above differential equation (equation 1) is
{\displaystyle C={\frac {\dot {m}}{K}}+\left(C_{o}-{\frac {\dot {m}}{K}}\right)e^{-{\frac {K\cdot t}{V}}}\qquad (2)}
Co is the concentration at the beginning of dialysis [mmol/L] or [mol/m3]
The steady state solution is
{\displaystyle C_{\infty }={\frac {\dot {m}}{K}}\qquad (3a)}
{\displaystyle K={\frac {\dot {m}}{C_{\infty }}}\qquad (3b)}
Equation 3b is the equation that defines clearance. It is the motivation for K' (the equivalent clearance):
{\displaystyle {K'}={\frac {\dot {m}}{C_{o}}}\qquad (4)}
K' is the equivalent clearance [mL/min] or [m3/s]
{\displaystyle {\dot {m}}}
is the mass generation rate of the substance - assumed to be a constant, i.e. not a function of time [mmol/min] or [mol/s]
Equation 4 is normalized by the volume of distribution to form equation 5:
{\displaystyle {\frac {K'}{V}}={\frac {\dot {m}}{C_{o}\cdot V}}\qquad (5)}
Equation 5 is multiplied by an arbitrary constant to form equation 6:
{\displaystyle {\mbox{const}}\cdot {\frac {K'}{V}}={\mbox{const}}\cdot {\frac {\dot {m}}{C_{o}\cdot V}}\qquad (6)}
Equation 6 is then defined as standardized Kt/V (std Kt/V):
{\displaystyle {\mbox{std}}{\frac {K\cdot t}{V}}\ {\stackrel {\mathrm {def} }{=}}\ {\mbox{const}}\cdot {\frac {\dot {m}}{C_{o}\cdot V}}\qquad (7)}
const is 7×24×60×60 seconds, the number of seconds in a week.
Interpretation of std Kt/V
Standardized Kt/V can be interpreted as a concentration normalized by the mass generation per unit volume of body water.
Equation 7 can be written in the following way:
{\displaystyle {\mbox{std}}{\frac {K\cdot t}{V}}\ {\stackrel {\mathrm {def} }{=}}{\mbox{ const}}\cdot {\frac {\dot {m}}{V}}{\frac {1}{C_{o}}}\qquad (8)}
If one takes the inverse of Equation 8 it can be observed that the inverse of std Kt/V is proportional to the concentration of urea (in the body) divided by the production of urea per time per unit volume of body water.
{\displaystyle \left[std{\frac {K\cdot t}{V}}\right]^{-1}\propto {\frac {C_{o}}{{\dot {m}}/V}}\qquad (9)}
Comparison to Kt/V
Kt/V and standardized Kt/V are not the same. Kt/V is a ratio of the pre- and post-dialysis urea concentrations. Standardized Kt/V is an equivalent clearance defined by the initial urea concentration (compare equation 8 and equation 10).
Kt/V is defined as (see article on Kt/V for derivation):
{\displaystyle {\frac {K\cdot t}{V}}=\ln {\frac {C_{o}}{C}}\qquad (10)}
Since Kt/V and std Kt/V are defined differently, Kt/V and std Kt/V values cannot be compared.
Advantages of std Kt/V
Can be used to compare any dialysis schedule (i.e. nocturnal home hemodialysis vs. daily hemodialysis vs. conventional hemodialysis)
Applicable to peritoneal dialysis.
Can be applied to patients with residual renal function; it is possible to demonstrate that Co is a function of the residual kidney function and the "cleaning" provided by dialysis.
The model can be applied to substances other than urea, if the clearance, K, and generation rate of the substance,
{\displaystyle {\dot {m}}}
, are known.[2]
Criticism/disadvantages of std Kt/V
It is complex and tedious to calculate, although web-based calculators are available to do this fairly easily.
Many nephrologists have difficulty understanding it.
Standardized Kt/V only models the clearance of urea and thus implicitly assumes the clearance of urea is comparable to other toxins. It ignores molecules that (relative to urea) have diffusion-limited transport - so called middle molecules.
It ignores the mass transfer between body compartments and across the plasma membrane (i.e. intracellular to extracellular transport), which has been shown to be important for the clearance of molecules such as phosphate.
The Standardized Kt/V is based on body water volume (V). The Glomerular filtration rate, an estimate of normal kidney function, is usually normalized to body surface area (S). S and V differ markedly between small vs. large people and between men and women. A man and a woman of the same S will have similar levels of GFR, but their values for V may differ by 15-20%. Because standardized Kt/V incorporates residual renal function into the calculations, it makes the assumption that kidney function should scale by V. This may disadvantage women and smaller patients of either sex, in whom V is decreased to a greater extent than S.
Calculating stdKt/V from treatment Kt/V and number of sessions per week
The various ways of computing standardized Kt/V by Gotch [5], Leypoldt [6], and the FHN trial network [7] are all a bit different, as assumptions differ on equal spacing of treatments, use of a fixed or variable volume model, and whether or not urea rebound is taken into effect [8]. One equation, proposed by Leypoldt and modified by Depner that is cited in the KDOQI 2006 Hemodialysis Adequacy Guidelines and which is the basis for a web calculator for stdKt/V is as follows:
{\displaystyle stdKt/V={\frac {\frac {10080\cdot (1-e^{-eKt/V})}{t}}{{\frac {1-e^{-eKtV}}{spKt/V}}+{\frac {10080}{N\cdot t}}-1}}}
where stdKt/V is the standardized Kt/V
spKt/V is the single-pool Kt/V, computed as described in Kt/V section using a simplified equation or ideally, using urea modeling, and
eKt/V is the equilibrated Kt/V, computed from the single-pool Kt/V (spKt/V) and session length (t) using, for example, the Tattersall equation [9]:
{\displaystyle ekt/V=spKt/V\cdot {\frac {t}{t+C}}}
where t is session duration in minutes, and C is a time constant, which is specific for type of access and type solute being removed. For urea, C should be 35 minutes for arterial access and 22 min for a venous access.
The regular "rate equation" [10] also can be used to determine equilibrated Kt/V from the spKt/V, as long as session length is 120 min or longer.
Nomogram to get stdKt/V from treatment Kt/V with different treatment schedules
One can use a nomogram derived from the above equation to estimate standardized Kt/V for any level of single-pool Kt/V. Because the equations are quite dependent on session length, the numbers will change substantially between two sessions given at the same schedule, but with different session lengths. For the present nomogram, a session length of 0.4 Kt/V units per hour was assumed, with a minimum dialysis session length of 2.0 hours.
[[1] - File:Std ktv.svg]
Standardized Kt/V calculator - HDCN
↑ Gotch FA. The current place of urea kinetic modelling with respect to different dialysis modalities. Nephrol Dial Transplant. 1998;13 Suppl 6:10-4. PMID 9719197. Full Text.
↑ 2.0 2.1 Gotch FA, Sargent JA, Keen ML. Whither goest Kt/V? Kidney Int Suppl. 2000 Aug;76:S3-18. PMID 10936795.
↑ Gotch FA, Sargent JA. A mechanistic analysis of the National Cooperative Dialysis Study (NCDS). Kidney Int. 1985 Sep;28(3):526-34. PMID 3934452.
↑ Gotch FA. The current place of urea kinetic modelling with respect to different dialysis modalities. Nephrol Dial Transplant. 1998;13 Suppl 6:10-4. Review. PMID 9719197
↑ Leypoldt JK, Jaber BL, Zimmerman DL. Predicting treatment dose for novel therapies using urea standard Kt/V. Semin Dial. 2004 Mar-Apr;17(2):142-5. PMID 15043617
↑ Suri RS, Garg AX, Chertow GM, Levin NW, Rocco MV, Greene T, Beck GJ, Gassman JJ, Eggers PW, Star RA, Ornt DB, Kliger AS. Frequent Hemodialysis Network (FHN) randomized trials: Study design. Kidney Int. 2007 Feb;71(4):349-59. Epub 2006 Dec 13. PMID 17164834
↑ Diaz-Buxo JA, Loredo JP. Standard Kt/V: comparison of calculation methods. Artif Organs. 2006 Mar;30(3):178-85. Erratum in: Artif Organs. 2006 Jun;30(6):490. PMID: 16480392
↑ Tattersall JE, DeTakats D, Chamney P, Greenwood RN, Farrington K. The post-hemodialysis rebound: predicting and quantifying its effect on Kt/V. Kidney Int. 1996 Dec;50(6):2094-102. PMID 8943495
↑ Daugirdas JT, Greene T, Depner TA, Leypoldt J, Gotch F, Schulman G, Star R; Hemodialysis Study Group. Factors that affect postdialysis rebound in serum urea concentration, including the rate of dialysis: results from the HEMO Study. J Am Soc Nephrol. 2004 Jan;15(1):194-203. PMID 14694173
Retrieved from "https://www.wikidoc.org/index.php?title=Standardized_Kt/V&oldid=689088"
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(→Horizontal and vertical profiles of the velocity components and Reynolds stresses)
Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient <math>\frac{\partial \langle u \rangle}{\partial z}</math> at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.
[[File:UFR3-35_ReStresses_z.png|centre|frame|Fig. 9: Vertical profiles of the Reynolds stresses <math> \langle u_i'u_j'(z)\rangle/u_{\mathrm{b}}^2</math> and the turbulent kinetic energy <math> \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2</math>]]
[[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses <math> \langle u'u'(z)\rangle/u_{\mathrm{b}}^2</math> ]]
[[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses <math> \langle u'w'(z)\rangle/u_{\mathrm{b}}^2</math> ]]
[[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy <math> \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2</math>]]
The vertical profiles of the Reynolds stresses and the resulting turbulent kinetic energy comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) is presented in Fig. 9 at <math>x_{\mathrm{adj}} = -1.5 </math>, <math>x_{\mathrm{adj}} = -1.0 </math>, and <math>x_{\mathrm{adj}} = -0.5 </math>. The accelerating jet is again indicated by a near wall peak of the <math>\langle u'u'\rangle</math> stress, while the horseshoe vortex leaves its footprint in the stresses <math>\langle u'u'\rangle</math> and <math>\langle w'w'\rangle</math> as (local) peak at <math>z_{\mathrm{V1}}/D</math>.
The vertical profiles of the Reynolds normal stresses <math> \langle u'u'\rangle </math>, the Renyolds shear stresses <math> \langle u'w'\rangle </math> and the resulting turbulent kinetic energy <math> \langle k\rangle </math> comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. The accelerating jet is again indicated by a near wall peak of the <math>\langle u'u'\rangle</math> stress, while the horseshoe vortex leaves its footprint in the stresses <math>\langle u'u'\rangle</math> and <math>\langle w'w'\rangle</math> as (local) peak at <math>z_{\mathrm{V1}}/D</math>.
The shear stress distribution <math>\langle u'w'\rangle</math> inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle \langle u\rangle }
{\displaystyle \langle w\rangle }
{\displaystyle \langle u'_{i}u'_{j}\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle ||{\vec {U}}||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {PIV} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {LES} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle -0.788}
{\displaystyle 0.03}
{\displaystyle -0.843}
{\displaystyle 0.037}
{\displaystyle -0.918}
{\displaystyle 0}
{\displaystyle -1.1}
{\displaystyle 0}
{\displaystyle -0.533}
{\displaystyle 0}
{\displaystyle -0.534}
{\displaystyle 0}
{\displaystyle -0.507}
{\displaystyle 0.036}
{\displaystyle -0.50}
{\displaystyle 0.04}
{\displaystyle -0.697}
{\displaystyle 0.051}
{\displaystyle -0.735}
{\displaystyle 0.06}
{\displaystyle -0.513}
{\displaystyle 0.017}
{\displaystyle -0.513}
{\displaystyle 0.02}
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }={\frac {x-x_{\mathrm {Cyl} }}{x_{\mathrm {Cyl} }-x_{\mathrm {V1} }}}}
{\displaystyle x_{\mathrm {Cyl} }=-0.5D}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {V1} }}
{\displaystyle \langle u(z)\rangle /u_{\mathrm {b} }}
{\displaystyle u(z)}
{\displaystyle x_{\mathrm {adj} }=-0.25}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
Fig. 9 a) Vertical profiles of the Reynolds normal stresses
{\displaystyle \langle u'u'(z)\rangle /u_{\mathrm {b} }^{2}}
Fig. 9 b) Vertical profiles of the Reynolds shear stresses
{\displaystyle \langle u'w'(z)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(z)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
The vertical profiles of the Reynolds normal stresses
{\displaystyle \langle u'u'\rangle }
, the Renyolds shear stresses
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle k\rangle }
comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. The accelerating jet is again indicated by a near wall peak of the
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle w'w'\rangle }
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle w(x)\rangle /u_{\mathrm {b} }}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle w(x)\rangle }
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }\approx -0.1}
{\displaystyle x_{\mathrm {adj} }=-0.65}
{\displaystyle \langle u_{i}'u_{j}'(x)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(x)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u_{i}'u_{j}'\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.074u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.079u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.09u_{\mathrm {b} }^{2}}
{\displaystyle 0=P+\nabla T-\epsilon +C}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle C}
{\displaystyle v}
{\displaystyle P=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}}
{\displaystyle T=\underbrace {-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle } _{\text{turbulent fluctuations}}\underbrace {-{\frac {1}{\rho }}\langle u_{i}'p'\rangle } _{\text{pressure transport}}\underbrace {+2\nu \langle u_{j}'s_{ij}\rangle } _{\text{viscous diffusion}}}
{\displaystyle \epsilon =2\nu \langle s_{ij}s_{ij}\rangle }
{\displaystyle s_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}'}{\partial x_{j}}}+{\frac {\partial u_{j}'}{\partial x_{i}}}\right)}
{\displaystyle \epsilon _{\mathrm {total} }=\epsilon _{\mathrm {res} }+\epsilon _{\mathrm {SGS} }=2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle }
{\displaystyle C=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}}
{\displaystyle D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {PIV} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {LES} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle 0.3u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {LES} }\approx 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {PIV} }\approx 0.2u_{\mathrm {b} }^{3}/D}
{\displaystyle x=-0.7D}
{\displaystyle P}
{\displaystyle \nabla T_{\mathrm {turb,PIV} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb,LES} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x=-0.75D}
{\displaystyle 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb,LES} }\approx 0.35u_{\mathrm {b} }^{3}/D}
{\displaystyle \nabla T_{\mathrm {press,LES} }=-{\frac {1}{\rho }}{\frac {\partial \langle u_{i}'p'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {visc,LES} }=2\nu {\frac {\partial \langle u_{j}'s_{ij}\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb} }}
{\displaystyle \nabla T_{\mathrm {press} }}
{\displaystyle \langle w\rangle <0}
{\displaystyle w-}
{\displaystyle w'}
{\displaystyle p'<0}
{\displaystyle \nabla T_{\mathrm {visc} }}
{\displaystyle |0.05|u_{\mathrm {b} }^{3}/D}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{\mathrm {PIV} }=2\nu \langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \epsilon _{\mathrm {LES,total} }=(2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle )\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P}
{\displaystyle \epsilon _{\mathrm {LES} }=0.066u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {max} }}
{\displaystyle \epsilon _{\mathrm {max} }}
{\displaystyle C_{\mathrm {PIV} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle C_{\mathrm {LES} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x\approx -0.63D}
{\displaystyle C}
{\displaystyle R_{\mathrm {PIV} }=P+\nabla T_{\mathrm {turb} }-\epsilon +C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle -\nabla T_{\mathrm {press,LES} }}
{\displaystyle R_{\mathrm {LES} }=P+\nabla T-\epsilon _{\mathrm {total} }+C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle <|0.01|u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb} }=-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }={\frac {\langle p\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }={\frac {\langle \tau _{\mathrm {w} }\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle z_{1}\approx 0.0036D\approx 10\mathrm {px} }
{\displaystyle z_{1}\approx 0.0005D}
{\displaystyle z-}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle |c_{\mathrm {f} }|=0.01}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle 50\times 171(n\times m)}
{\displaystyle 143\times 131(n\times m)}
{\displaystyle n\cdot m}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle -}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle v\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {press} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {visc} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \epsilon _{\mathrm {total} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle c_{\mathrm {p} }}
|
(→Streamlines)
== Streamlines ==
The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus <math> ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}</math>. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component,respectively.
The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus <math> ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}</math>. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle \langle u\rangle }
{\displaystyle \langle w\rangle }
{\displaystyle \langle u'_{i}u'_{j}\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle ||{\vec {U}}||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {PIV} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle ||{\vec {U}}_{\mathrm {LES} }||={\sqrt {\langle u^{2}\rangle +\langle w^{2}\rangle }}/u_{\mathrm {b} }}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle x/D}
{\displaystyle z/D}
{\displaystyle -0.788}
{\displaystyle 0.03}
{\displaystyle -0.843}
{\displaystyle 0.037}
{\displaystyle -0.918}
{\displaystyle 0}
{\displaystyle -1.1}
{\displaystyle 0}
{\displaystyle -0.533}
{\displaystyle 0}
{\displaystyle -0.534}
{\displaystyle 0}
{\displaystyle -0.507}
{\displaystyle 0.036}
{\displaystyle -0.50}
{\displaystyle 0.04}
{\displaystyle -0.697}
{\displaystyle 0.051}
{\displaystyle -0.735}
{\displaystyle 0.06}
{\displaystyle -0.513}
{\displaystyle 0.017}
{\displaystyle -0.513}
{\displaystyle 0.02}
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }={\frac {x-x_{\mathrm {Cyl} }}{x_{\mathrm {Cyl} }-x_{\mathrm {V1} }}}}
{\displaystyle x_{\mathrm {Cyl} }=-0.5D}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {V1} }}
{\displaystyle \langle u(z)\rangle /u_{\mathrm {b} }}
{\displaystyle u(z)}
{\displaystyle x_{\mathrm {adj} }=-0.25}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle {\frac {\partial \langle u\rangle }{\partial z}}}
{\displaystyle \langle u_{i}'u_{j}'(z)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(z)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle x_{\mathrm {adj} }=-1.5}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle x_{\mathrm {adj} }=-0.5}
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle u'u'\rangle }
{\displaystyle \langle w'w'\rangle }
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u'w'\rangle }
{\displaystyle \langle w(x)\rangle /u_{\mathrm {b} }}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle w(x)\rangle }
{\displaystyle x-}
{\displaystyle x_{\mathrm {adj} }\approx -0.1}
{\displaystyle x_{\mathrm {adj} }=-0.65}
{\displaystyle \langle u_{i}'u_{j}'(x)\rangle /u_{\mathrm {b} }^{2}}
{\displaystyle \langle k(x)\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle z_{\mathrm {V1} }/D}
{\displaystyle \langle u_{i}'u_{j}'\rangle }
{\displaystyle \langle k\rangle }
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k\rangle =0.5(\langle u'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {PIV,inplane} }\rangle =0.074u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,inplane} }\rangle =0.079u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.5(\langle u'^{2}\rangle +\langle v'^{2}\rangle +\langle w'^{2}\rangle )/u_{\mathrm {b} }^{2}}
{\displaystyle \langle k_{\mathrm {LES,total} }\rangle =0.09u_{\mathrm {b} }^{2}}
{\displaystyle 0=P+\nabla T-\epsilon +C}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle C}
{\displaystyle v}
{\displaystyle P=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}}
{\displaystyle T=\underbrace {-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle } _{\text{turbulent fluctuations}}\underbrace {-{\frac {1}{\rho }}\langle u_{i}'p'\rangle } _{\text{pressure transport}}\underbrace {+2\nu \langle u_{j}'s_{ij}\rangle } _{\text{viscous diffusion}}}
{\displaystyle \epsilon =2\nu \langle s_{ij}s_{ij}\rangle }
{\displaystyle s_{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}'}{\partial x_{j}}}+{\frac {\partial u_{j}'}{\partial x_{i}}}\right)}
{\displaystyle \epsilon _{\mathrm {total} }=\epsilon _{\mathrm {res} }+\epsilon _{\mathrm {SGS} }=2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle }
{\displaystyle C=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}}
{\displaystyle D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {PIV} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P_{\mathrm {LES} }=-\langle u_{i}'u_{j}'\rangle {\frac {\partial \langle u_{i}\rangle }{\partial x_{j}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle 0.3u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {LES} }\approx 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {PIV} }\approx 0.2u_{\mathrm {b} }^{3}/D}
{\displaystyle x=-0.7D}
{\displaystyle P}
{\displaystyle \nabla T_{\mathrm {turb,PIV} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb,LES} }=-{\frac {1}{2}}{\frac {\partial \langle u_{i}'u_{j}'u_{j}'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x=-0.75D}
{\displaystyle 0.4u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb,LES} }\approx 0.35u_{\mathrm {b} }^{3}/D}
{\displaystyle \nabla T_{\mathrm {press,LES} }=-{\frac {1}{\rho }}{\frac {\partial \langle u_{i}'p'\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {visc,LES} }=2\nu {\frac {\partial \langle u_{j}'s_{ij}\rangle }{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \nabla T_{\mathrm {turb} }}
{\displaystyle \nabla T_{\mathrm {press} }}
{\displaystyle \langle w\rangle <0}
{\displaystyle w-}
{\displaystyle w'}
{\displaystyle p'<0}
{\displaystyle \nabla T_{\mathrm {visc} }}
{\displaystyle |0.05|u_{\mathrm {b} }^{3}/D}
{\displaystyle P}
{\displaystyle \nabla T}
{\displaystyle \epsilon }
{\displaystyle \epsilon _{\mathrm {PIV} }=2\nu \langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle \epsilon _{\mathrm {LES,total} }=(2\nu \langle s_{ij}s_{ij}\rangle +2\langle \nu _{\mathrm {t} }s_{ij}s_{ij}\rangle )\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle P}
{\displaystyle \epsilon _{\mathrm {LES} }=0.066u_{\mathrm {b} }^{3}/D}
{\displaystyle P_{\mathrm {max} }}
{\displaystyle \epsilon _{\mathrm {max} }}
{\displaystyle C_{\mathrm {PIV} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle C_{\mathrm {LES} }=-\langle u_{i}\rangle {\frac {\partial k}{\partial x_{i}}}\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle x\approx -0.63D}
{\displaystyle C}
{\displaystyle R_{\mathrm {PIV} }=P+\nabla T_{\mathrm {turb} }-\epsilon +C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle -\nabla T_{\mathrm {press,LES} }}
{\displaystyle R_{\mathrm {LES} }=P+\nabla T-\epsilon _{\mathrm {total} }+C\cdot D/u_{\mathrm {b} }^{3}}
{\displaystyle <|0.01|u_{\mathrm {b} }^{3}/D}
{\displaystyle T_{\mathrm {turb} }=-{\frac {1}{2}}\langle u_{i}'u_{j}'u_{j}'\rangle }
{\displaystyle c_{\mathrm {p} }(x)}
{\displaystyle c_{\mathrm {f} }(x)}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }={\frac {\langle p\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }={\frac {\langle \tau _{\mathrm {w} }\rangle }{{\frac {\rho }{2}}u_{\mathrm {b} }^{2}}}}
{\displaystyle z_{1}\approx 0.0036D\approx 10\mathrm {px} }
{\displaystyle z_{1}\approx 0.0005D}
{\displaystyle z-}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {p} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle x_{\mathrm {adj} }=-1.0}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle |c_{\mathrm {f} }|=0.01}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle c_{\mathrm {f} }}
{\displaystyle 50\times 171(n\times m)}
{\displaystyle 143\times 131(n\times m)}
{\displaystyle n\cdot m}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle -}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle -}
{\displaystyle \epsilon {\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle -}
{\displaystyle x_{\mathrm {adj} }}
{\displaystyle {\frac {x}{D}}}
{\displaystyle {\frac {z}{D}}}
{\displaystyle {\frac {\langle u\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle v\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle w\rangle }{u_{\mathrm {b} }}}}
{\displaystyle {\frac {\langle u'u'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle w'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'v'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle u'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle {\frac {\langle v'w'\rangle }{u_{\mathrm {b} }^{2}}}}
{\displaystyle P{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle C{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {turb} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {press} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \nabla T_{\mathrm {visc} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle \epsilon _{\mathrm {total} }{\frac {D}{u_{\mathrm {b} }^{3}}}}
{\displaystyle c_{\mathrm {p} }}
|
Control of Charge Dilution in Turbocharged Diesel Engines via Exhaust Valve Timing | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
Yilmaz, H., and Stefanopoulou, A. (August 24, 2004). "Control of Charge Dilution in Turbocharged Diesel Engines via Exhaust Valve Timing." ASME. J. Dyn. Sys., Meas., Control. September 2005; 127(3): 363–373. https://doi.org/10.1115/1.1985440
In this paper we extend an existing crank angle resolved dynamic nonlinear model of a six-cylinder 12 l turbocharged (TC) Diesel engine with exhaust valve closing (EVC) variability. Early EVC achieves a high level of internal exhaust gas recirculation (iEGR) or charge dilution in Diesel engines, and thus reduces generated oxides of nitrogen
(NOx)
. This model is validated in steady-state conventional (fixed EVC) engine operating points. It is expected to capture the transient interactions between EVC actuation, the turbocharger dynamics, and the cylinder-to-cylinder breathing characteristics, although this has not been explicitly validated due to lack of hardware implementation. A nominal low order linear multi-input multi-output model is then identified using cycle-sampled or cycle-averaged data from the higher order nonlinear simulation model. Various low-order controllers that vary EVC to maximize the steady-state iEGR under air-to-fuel ratio (AFR) constraints during transient fueling demands are suggested based on different sensor sets. The difficulty in the control tuning arises from the fact that the EVC affects both the AFR and engine torque requiring coordination of fueling and EVC. Simulation results are shown on the full order model.
diesel engines, valves, compressors, MIMO systems, air pollution control
Control equipment, Cycles, Cylinders, Diesel engines, Engines, Exhaust systems, Feedback, Fuels, Intake manifolds, Pressure, Steady state, Torque, Turbochargers, Valves, Dynamics (Mechanics), Design, Flow (Dynamics), Exhaust manifolds, Feedforward control
The Potential of a Combined Miller Cycle and Internal EGR Engine for Future Heavy Duty Truck Applications
Local Linear Model Trees (LILIMOT) Toolbox for Nonlinear System Identification
Proceedings 12th IFAC Symposium on System Identification
Turbocharged Diesel Engine Modeling for Nonlinear Engine Control and Estimation
Torque Management of Engines with Variable Cam Timing
Compression Braking for Longitudinal Control of Commercial Heavy Vehicles
,” PATH Report UCB-ITS-PRR-2001-11.
Transition from Combustion to Compression Braking
,” SAE 2000-01-1228, SAE World Congress.
Brake Valve Timing and Fuel Injection: a Unified Engine Torque Actuator for Heavy-Duty Vehicles
Multivariable Feedback Control: Analysis Design
|
Large_Underground_Xenon_experiment Knowpia
The Large Underground Xenon experiment (LUX) aimed to directly detect weakly interacting massive particle (WIMP) dark matter interactions with ordinary matter on Earth. Despite the wealth of (gravitational) evidence supporting the existence of non-baryonic dark matter in the Universe,[1] dark matter particles in our galaxy have never been directly detected in an experiment. LUX utilized a 370 kg liquid xenon detection mass in a time-projection chamber (TPC) to identify individual particle interactions, searching for faint dark matter interactions with unprecedented sensitivity.[2]
The LUX experiment, which cost approximately $10 million to build,[3] was located 1,510 m (4,950 ft) underground at the Sanford Underground Laboratory (SURF, formerly the Deep Underground Science and Engineering Laboratory, or DUSEL) in the Homestake Mine (South Dakota) in Lead, South Dakota. The detector was located in the Davis campus, former site of the Nobel Prize-winning Homestake neutrino experiment led by Raymond Davis. It was operated underground to reduce the background noise signal caused by high-energy cosmic rays at the Earth's surface.
The detector was decommissioned in 2016 and is now on display at the Sanford Lab Homestake Visitor Center.[4]
The Large Underground Xenon experiment installed 1,480 m (4,850 ft) underground inside a 260 m3 (70,000 US gal) water tank shield. The experiment was a 370 kg liquid xenon time projection chamber that aimed to detect the faint interactions between WIMP dark matter and ordinary matter.
The detector was isolated from background particles by a surrounding water tank and the earth above. This shielding reduced cosmic rays and radiation interacting with the xenon.
Interactions in liquid xenon generate 175 nm ultraviolet photons and electrons. These photons were immediately detected by two arrays of 61 photomultiplier tubes at the top and bottom of the detector. These prompt photons were the S1 signal. Electrons generated by the particle interactions drifted upwards towards the xenon gas by an electric field. The electrons were pulled in the gas at the surface by a stronger electric field, and produced electroluminescence photons detected as the S2 signal. The S1 and subsequent S2 signal constituted a particle interaction in the liquid xenon.
The detector was a time-projection chamber (TPC), using the time between S1 and S2 signals to find the interaction depth since electrons move at constant velocity in liquid xenon (around 1–2 km/s, depending on the electric field). The x-y coordinate of the event was inferred from electroluminescence photons at the top array by statistical methods (Monte Carlo and maximum likelihood estimation) to a resolution under 1 cm.[5]
Particle interactions inside the LUX detector produced photons and electrons. The photons (
{\displaystyle \gamma }
), moving at the speed of light, were quickly detected by the photomultiplier tubes. This photon signal was called S1. An electric field in the liquid xenon drifted the electrons towards the liquid surface. A much higher electric field above the liquid surface pulled the electrons out of the liquid and into the gas, where they procued electroluminescence photons (in the same way that neon sign produces light). The electroluminescence photons were detected by the photomultiplier tubes as the S2 signal. A single particle interaction in the liquid xenon could be identified by the pair of an S1 and an S2 signal.
Schematic of the Large Underground Xenon (LUX) detector. The detector consisted of an inner cryostat filled with 370 kg of liquid xenon (300 kg in the inner region, called the "active volume") cooled to −100 °C. 122 photomultiplier tubes detected light generated inside the detector. The LUX detector had an outer cryostat that provided vacuum insulation. An 8-meter-diameter by 6-meter-high water tank shielded the detector from external radiation, such as gamma rays and neutrons.
Finding dark matterEdit
The LUX collaboration was composed of over 100 scientists and engineers across 27 institutions in the US and Europe. LUX was composed of the majority of the US groups that collaborated in the XENON10 experiment, most of the groups in the ZEPLIN III experiment, the majority of the US component of the ZEPLIN II experiment, and groups involved in low-background rare event searches such as Super Kamiokande, SNO, IceCube, Kamland, EXO and Double Chooz.
The LUX experiment's co-spokesmen were Richard Gaitskell from Brown University (who acted as co-spokesman from 2007 on) and Daniel McKinsey from University of California, Berkeley (who acted as co-spokesman from 2012 on). Tom Shutt from Case Western Reserve University was LUX co-spokesman between 2007 and 2012.
Detector assembly began in late 2009. The LUX detector was commissioned overground at SURF for a six-month run. The assembled detector was transported underground from the surface laboratory in a two-day operation in the summer of 2012 and began data taking April 2013, presenting initial results Fall 2013. It was decommissioned in 2016.[4]
The next-generation follow-up experiment, the 7-ton LUX-ZEPLIN has been approved,[6] expected to begin in 2020.[7]
Initial unblinded data taken April to August 2013 were announced on October 30, 2013. In an 85 live-day run with 118 kg fiducial volume, LUX obtained 160 events passing the data analysis selection criteria, all consistent with electron recoil backgrounds. A profile likelihood statistical approach shows this result is consistent with the background-only hypothesis (no WIMP interactions) with a p-value of 0.35. This was the most sensitive dark matter direct detection result in the world, and ruled out low-mass WIMP signal hints such as from CoGeNT and CDMS-II.[8][9] These results struck out some of the theories about WIMPs, allowing researchers to focus on fewer leads.[10]
In the final run from October 2014 to May 2016, at four times its original design sensitivity with 368 kg of liquid xenon, LUX saw no signs of dark matter candidate—WIMPs.[7] According to Ethan Siegel, the results from LUX and XENON1T have provided evidence against the supersymmetric "WIMP Miracle" strong enough to motivate theorists towards alternate models of dark matter.[11]
^ Beringer, J.; et al. (2012). "2012 Review of Particle Physics" (PDF). Phys. Rev. D. 86 (10001). Bibcode:2012PhRvD..86a0001B. doi:10.1103/PhysRevD.86.010001.
^ Akerib, D.; et al. (March 2013). "The Large Underground Xenon (LUX) experiment". Nuclear Instruments and Methods in Physics Research A. 704: 111–126. arXiv:1211.3788. Bibcode:2013NIMPA.704..111A. doi:10.1016/j.nima.2012.11.135. S2CID 67768071.
^ Reich, E. Dark-matter hunt gets deep Nature 21 Feb 2013
^ a b Van Zee, Al (July 20, 2017). "LUX dark matter detector now part of new exhibit at Sanford Lab". Black Hills Pioneer. Lead, South Dakota. Retrieved June 21, 2019.
^ Akerib; et al. (May 2013). "Technical results from the surface run of the LUX dark matter experiment". Astroparticle Physics. 45: 34–43. arXiv:1210.4569. Bibcode:2013APh....45...34A. doi:10.1016/j.astropartphys.2013.02.001. S2CID 118422051.
^ "Dark-matter searches get US government approval". Physics World. July 15, 2014. Retrieved February 13, 2020.
^ a b "World's most sensitive dark-matter search comes up empty handed". Hamish Johnston. physicsworld.com (IOP). 22 July 2016. Retrieved February 13, 2020.
^ Akerib, D. (2014). "First results from the LUX dark matter experiment at the Sanford Underground Research Facility" (PDF). Physical Review Letters. 112 (9): 091303. arXiv:1310.8214. Bibcode:2014PhRvL.112i1303A. doi:10.1103/PhysRevLett.112.091303. hdl:1969.1/185324. PMID 24655239. S2CID 2161650. Retrieved 30 October 2013.
^ Dark Matter Search Comes Up Empty Fox News, 2013 October 30
^ Dark matter experiment finds nothing, makes news The Conversation, 01 November 2013
^ Siegel, Ethan (February 22, 2019). "The 'WIMP Miracle' Hope For Dark Matter Is Dead". Starts With A Bang. Forbes. Archived from the original on February 22, 2019. Retrieved June 21, 2019.
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CreateEPUB - Maple Help
Home : Support : Online Help : Programming : eBookTools Package : CreateEPUB
convert Maple worksheets to eBook
CreateEPUB(book, settings)
The CreateEPUB command performs transformation of Maple worksheets into eBook (.epub) format suitable for Apple iBooks, and with further conversion, for Amazon Kindle.
\mathrm{with}\left(\mathrm{eBookTools}\right):
\mathrm{book}≔\mathrm{NewBook}\left("eBookSample","eBook Sample Book","Maplesoft, a division of Waterloo Maple Inc.","2012"\right):
\mathrm{AddChapter}\left(\mathrm{book},"legal",\mathrm{cat}\left(\mathrm{kernelopts}\left('\mathrm{datadir}'\right),"/eBookTools/Legal.mw"\right)\right):
\mathrm{AddChapter}\left(\mathrm{book},"preface",\mathrm{cat}\left(\mathrm{kernelopts}\left('\mathrm{datadir}'\right),"/eBookTools/Preface.mw"\right)\right):
\mathrm{AddChapter}\left(\mathrm{book},1,\mathrm{cat}\left(\mathrm{kernelopts}\left('\mathrm{datadir}'\right),"/eBookTools/GettingStartedWithMaple.mw"\right)\right):
\mathrm{CreateEPUB}\left(\mathrm{book}\right)
The eBookTools[CreateEPUB] command was introduced in Maple 16.
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numtheory(deprecated)/iscyclotomic - Maple Help
Home : Support : Online Help : numtheory(deprecated)/iscyclotomic
test if a polynomial is cyclotomic
iscyclotomic(m, x)
iscyclotomic(m, x, 'n')
polynomial in x over the rationals
(optional) name for the output of the order of the polynomial, if cyclotomic
Important: The numtheory package has been deprecated. Use the superseding command NumberTheory[IsCyclotomicPolynomial] instead.
The iscyclotomic(m, x) calling sequence returns true if m(x) is a cyclotomic polynomial, and false otherwise.
The iscyclotomic(m, x, 'n') calling sequence also assigns the order of the cyclotomic polynomial to n when the function returns true.
This function is part of the numtheory package, and so can be used in the form iscyclotomic(..) only after executing the command with(numtheory) or with(numtheory,iscyclotomic). The function can always be accessed in the long form numtheory[iscyclotomic](..).
\mathrm{with}\left(\mathrm{numtheory}\right):
m≔\mathrm{cyclotomic}\left(10,x\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{iscyclotomic}\left(m,x,'n'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
n
\textcolor[rgb]{0,0,1}{10}
f≔{x}^{5}+{x}^{4}+{x}^{3}+{x}^{2}+x+1
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}
\mathrm{iscyclotomic}\left(f,x\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
numtheory(deprecated)[phi]
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Simple heat transfer model between two general fluids - MATLAB - MathWorks Italia
Specific Dissipation Heat Transfer
Thermal Liquid mass flow rate vector, mdot1
Controlled fluid mass flow rate vector, mdot2
Specific dissipation table, SD(mdot1, mdot2)
Check if violating maximum specific dissipation
Simple heat transfer model between two general fluids
Simscape / Fluids / Fluid Network Interfaces / Heat Exchangers / Fundamental Components
The Specific Dissipation Heat Transfer block models the heat transfer between two fluids given only minimal knowledge of component parameters. The fluids are controlled by physical signals, with these providing the entrance mass flow rate and isobaric specific heat for each. Thermal ports set the entrance temperatures of the fluids.
The rate of heat transfer is calculated from the specific dissipation, a parameter specified in tabulated form as a function of the entrance mass flow rates. The specific dissipation quantifies the amount of heat exchanged between the fluids per unit of time when the entrance temperatures differ by one degree.
Pressure losses and other aspects of flow mechanics are ignored. To capture such effects, use the heat exchanger interface blocks provided in the same library. Combine heat transfer and heat exchanger blocks to model a custom heat exchanger. See the composite block diagrams of the heat exchanger blocks for examples.
Heat flows from the warmer fluid to the cooler fluid, at a rate proportional to the difference between the fluid entrance temperatures. The heat flow rate is positive if fluid 1 enters at a higher temperature than fluid 2—and therefore if heat flows from fluid 1 to fluid 2:
Q=SD\left({T}_{1,\text{in}}-{T}_{2,\text{in}}\right),
where T*,in are the fluid entrance temperatures, determined by the conditions at thermal port H1 for fluid 1 and H2 for fluid 2. SD is the specific dissipation obtained from the specified tabulated data at the given mass flow rates:
SD=SD\left({\stackrel{˙}{m}}_{1},{\stackrel{˙}{m}}_{2}\right),
\stackrel{˙}{m}
are the entrance mass flow rates, specified through physical signal port M1 for fluid 1 and M2 for fluid 2. The specific dissipation can be calculated for a set of entrance mass flow rates (
{\stackrel{˙}{m}}_{1}
{\stackrel{˙}{m}}_{2}
) given the experimental values of the heat transfer rate and the corresponding entrance temperature difference:
SD=\frac{Q}{{T}_{1,in}-{T}_{2,in}}.
Maximum Heat Transfer Rate
The heat transfer rate is constrained so that the specific dissipation used in the calculations can never exceed the maximum value:
S{D}_{\text{max}}=\text{min}\left({C}_{1},{C}_{2}\right),
where C* are the thermal capacity rates of the controlled fluids, each defined as:
{C}_{*}={\stackrel{˙}{m}}_{*}{c}_{p,*},
with cp,* denoting the isobaric specific heat of the fluid, specified through physical signal port CP1 for fluid 1 and CP2 for fluid 2. The constraint on the maximum heat transfer rate is implemented in the form of a piecewise function:
Q=\left\{\begin{array}{c}\begin{array}{ll}S{D}_{max}\left({T}_{1,in}-{T}_{2,in}\right),\hfill & \text{if }SD>S{D}_{max}\hfill \\ SD\left({T}_{1,in}-{T}_{2,in}\right),\hfill & \text{otherwise}\hfill \end{array}\end{array},
A warning is issued whenever the heat flow rate exceeds the maximum value,
S{D}_{max}\left({T}_{1,in}-{T}_{2,in}\right)
, if the Check if violating maximum specific dissipation block parameter is set to Warning.
CP1 — Isobaric specific heat of controlled fluid 1
Isobaric specific heat of controlled fluid 1.
M1 — Entrance mass flow rate of controlled fluid 1
Entrance mass flow rate of controlled fluid 1. Positive values indicate flow into the heat exchanger. Negative values indicate flow out of the heat exchanger.
H1 — Entrance temperature of controlled fluid 1
Entrance temperature of controlled fluid 1.
Thermal Liquid mass flow rate vector, mdot1 — Entrance mass flow rate at which to specify specific-dissipation data
[.3, .5, .6, .7, 1, 1.4, 1.9, 2.3] kg/s (default) | M-element array with units of mass/time
Array of mass flow rates at the inlet for controlled fluid 1. Each value corresponds to a row in the specific dissipation lookup table. Positive values indicate flow into the heat exchanger and negative values indicate flow out of the heat exchanger.
Controlled fluid mass flow rate vector, mdot2 — Entrance mass flow rate at which to specify specific-dissipation data
[.3, .5, 1, 1.3, 1.7, 2, 2.6, 3.3] kg/s (default) | N-element array with units of mass/time
Specific dissipation table, SD(mdot1, mdot2) — Specific dissipation values corresponding to specified mass flow rates
8-by-8 matrix with units of kW/K (default) | M-by-N matrix with units of power/temperature
Matrix of specific dissipation values corresponding to the specified mass flow rate arrays for controlled fluids 1 and 2. The block uses the tabulated data to calculate the heat transfer at the simulated operating conditions.
Mass flow rate threshold for flow reversal — Mass flow rate below which to smooth numerical data
Mass flow rate below which to initiate a smooth flow reversal to prevent discontinuities in the simulation data.
Check if violating maximum specific dissipation — Option to warn if the specific dissipation exceeds the maximum allowed value
Option to warn if the specific dissipation exceeds the maximum value described in the block description.
Simple Heat Exchanger Interface (G) | Simple Heat Exchanger Interface (TL)
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Explanation of Pressure Effect for High Temperature Superconductors Using Pressure Dependent Schrodinger Equation and String Theory
Einas Mohamed Ahmed Mohamed1,2, Nagwa Idriss Ali Ahmed2,3, Musa Ibrahim Babiker Hussein4,5, Rasha Abd Elhai Mohammad Taha6,7, Mohammed Idriss Ahmed7, Mubarak Dirar Abd-Alla7
1Department of Physics, University College (Turaba), Taif University, Taif, KSA.
2Department of Physics and Mathematics, Hantoub Faculty of Education, Algezira University, Wad Madaniin-Hantoub, Sudan.
3Department of Physics, Faculty of Science & Art (Dariyah), Qassim University, Al-Mulida, KSA.
4Department of Physics and Mathematics, Faculty of Education, Albutana University, Rufaa, Sudan.
5Department of Physics, Faculty of Science & Art, Dariyah University, Buljurashi, KSA.
6Department of Physics, College of Science, Majmaah University, Majmaah, KSA.
7Department of Physics, Faculty of Science, Sudan University of Science and Technology, Khartoum, Sudan.
A pressure dependent Schrodinger equation is used to find the conditions that lead to superconductivity. When no pressure is exerted, the superconductor resistance vanishes beyond a critical temperature related to the repulsive force potential of the electron gass, where one assuming the electron total energy to be thermal, where applying mechanical pressure destroys Sc when it exceeds a certain critical value. However when the electron total energy is an assumed to be that of the free electron model and that the pressure is thermal and mechanical, the situation is different. The quantum expression for resistance shows that the increase of mechanical pressure increases the critical temperature. Such phenomenon is observed in high temperature cupper group.
Pressure Dependent Schrodinger Equation, Superconductivity, Critical Temperature Pressure, High Temperature Superconductor
Ahmed Mohamed, E. , Ali Ahmed, N. , Babiker Hussein, M. , Mohammad Taha, R. , Ahmed, M. and Abd-Alla, M. (2020) Explanation of Pressure Effect for High Temperature Superconductors Using Pressure Dependent Schrodinger Equation and String Theory. Natural Science, 12, 28-34. doi: 10.4236/ns.2020.121004.
Superconductivity (Sc) phenomenon is one of the most interesting properties of bulk matter. In this phenomenon, the resistance to electric current vanishes beyond a certain critical temperature [1].
This leads to generation of powerful magnetic field that used in a wide variety of applications.
For instance, the magnetic property is used in magnetic resonance imaginary (MRI), super magnetic trains and in generating powerful electric energy [2].
The wide spread of Sc applications face a disaster problem. This problem is related to the fact that the operating temperature of Sc materials are far beyond 100 k, which is much lower than the ambient temperature 300 k [3,4]. Fortunately, recently new Sc materials, he so called high temperature Sc (HTSc), were discovered.
They can operate at temperatures above 130 k as [5,6]. To reach the ambient temperature operation, one needs a well-defined model that gives a clear path way that shows how to select compounds to increase the critical temperature to be above the ambient temperature.
Unfortunately there are many problems associated with (HTSc). They suffer from some long standing problem like pressure problem and isotope problem. In the former one, the application of pressure on some compound can increase or decrease the critical temperature.
In the latter one, the replacement of some compound constituent by their isotope changes the critical temperature [7,8].
Fortunately, some models were proposed to solve some of these problem. Some models solve pressure problem, while other models try to explain magnetic destruction of Sc [9,10].
This needs new alternatives that leads to a well define model that can solve all Sc problems. This paper is concerned with construction and development of a new that can be promoted to solve the problems associated with the Sc. Section 2 is concerned with constructing a new model based on string theory to explain the pressure effect. The discussion and conclusion are exhibited in Sections 3 and 4.
In a work done by many authors [11], an energy dependent expression on the potential V and pressure P was obtained from the plasma equation:
vm\frac{\text{d}v}{\text{d}x}=m\frac{\text{d}x}{\text{d}t}=-\nabla P-\nabla V=-\text{d}\left(\frac{P+V}{\text{d}x}\right)
\frac{\text{d}}{\text{d}x}\left(\frac{1}{2}m{v}^{2}+V+P\right)=0
KE+V+P=\text{Constant}
This constant of motion, which is shown to be standing for the energy of the system, is given by:
E=KE+V+P
3. THE PRESSURE DEPENDENT SCHRODINGER MODEL
The ordinary Schrodinger equation is given, according to (3), by:
i\hslash \frac{\text{d}\Psi }{\text{d}t}=-\frac{{\hslash }^{2}}{2m}{\nabla }^{2}\Psi +V\Psi +P\Psi
The time independent equation can be found by suggesting the wave function (wf) to be:
\Psi =u{\text{e}}^{-i{\omega }_{0}t}=u\left(r\right){\text{e}}^{-i{\omega }_{0}t}
A direct substitution of (5) in (4) yields:
{E}_{0}u=\hslash {\omega }_{0}u=-\frac{{\hslash }^{2}}{2m}{\nabla }^{2}u+Vu+P\Psi
Consider an electron acting a harmonic oscillator subjected to a constant potential
{V}_{1}
. For electron Schrodinger Equation (6) is written as:
{E}_{0}u=-\frac{{\hslash }^{2}}{2m}{\nabla }^{2}u+\frac{1}{2}k{x}^{2}u+Vu+P\Psi
Try solution of the form:
u=A{\text{e}}^{-\alpha {x}^{2}}
\nabla u=-2\alpha xu
{\nabla }^{2}=-2\alpha u-2\alpha x\nabla u
{\nabla }^{2}u=-2\alpha u+4{\alpha }^{2}{x}^{2}u
In view of Equation (3) c consider the potential V to result from repulsive crystal potential of the electron gass
{V}_{0}
beside the mechanical pressure
{P}_{0}
V=\frac{1}{2}k{x}^{2}+{V}_{0}
P=-{P}_{0}
Here the pressure is exerted on the system by surrounding media, inserting (8) in (7) gives:
{E}_{0}u=-\frac{{\hslash }^{2}}{2m}\left(-2\alpha +4{\alpha }^{2}{x}^{2}\right)u+\left(\frac{1}{2}k{x}^{2}+{V}_{0}-{V}_{p}\right)u
Equating the coefficients of u and x2u yields:
{E}_{0}=\frac{{\hslash }^{2}\alpha }{2m}+{V}_{0}-{V}_{p}=\frac{{\hslash }^{2}\alpha }{m}+{V}_{0}-{P}_{0}
\frac{2{\hslash }^{2}{\alpha }^{2}}{m}=\frac{1}{2}k=\frac{1}{2}m{\omega }^{2}
From (12), one gets:
{\alpha }^{2}=\frac{{m}^{2}{\omega }^{2}}{4{\hslash }^{2}}=\frac{{m}^{2}{c}^{4}}{4{\hslash }^{4}{c}^{4}}
where one assumes that the energy satisfies Einstein and Max plank relations, i.e.:
E=m{c}^{2}=\hslash \omega
{\alpha }^{2}=\frac{{\left(\hslash \omega \right)}^{4}}{4{\hslash }^{4}{c}^{4}}=\frac{1}{4}{\left(\frac{\omega }{c}\right)}^{4}=\frac{1}{4}{\left(\frac{ck}{c}\right)}^{4}=\frac{1}{4}{k}^{4}
\alpha =\frac{1}{2}{k}^{2}
There for Equation (11) gives:
{E}_{0}=\frac{{\hslash }^{2}{k}^{2}}{2m}+{V}_{0}-{V}_{P}=\frac{{\hslash }^{2}{K}^{2}}{2m}+{V}_{0}-{P}_{0}
Thus the momentum is given by:
mv=\hslash k=\sqrt{2m\left({E}_{0}-{V}_{0}+{V}_{p}\right)}
This relation can be used to define a quantum resistance by using the relation:
R=\frac{U}{I}=\frac{V}{eI}=\frac{V}{enevA}=\frac{m{v}^{2}}{2n{e}^{2}vA}=\frac{mv}{2n{e}^{2}A}
With U, I, V and n standing for the electric potential, current potential energy and concentration respectively:
R={R}_{r}+i{R}_{i}={R}_{s}+i{R}_{i}
{R}_{r},{R}_{s},{R}_{i}
stands for real, superconductor and imaginary resistance respectively.
When no pressure was applied:
{P}_{0}={V}_{0}=0
In view of Equations (17) and (18), one gets
R=\frac{\hslash k}{2n{e}^{2}A}
Using Equations (17), (20) in (21) yields:
R=\frac{\sqrt{2m\left({E}_{0}-{V}_{0}\right)}}{2n{e}^{2}A}
Assume a gain that the energy
{E}_{0}
is in the form of thermal energy:
{E}_{0}=KT
Using (23) in (22) gives:
R=\frac{\sqrt{2m\left(KT-{V}_{0}\right)}}{2n{e}^{2}A}
The resistence R become pure imaginary when:
KT<{V}_{0}
Define the critical temperature to be satisfying the relation:
{V}_{0}=K{T}_{c}
Thus Equation (25) requires:
KT<K{T}_{c}
T<{T}_{c}
Thus according to Equations (19) and (24) one gets:
R={R}_{i},{R}_{s}=0
However when the pressure was applied. Such that:
{P}_{0}>{V}_{0}
{P}_{0}\ge K{T}_{c}
Using also Equations (17), (16), (21), (23), (26) one gets:
R=\frac{\sqrt{2m\left(KT-K{T}_{c}+{P}_{0}\right)}}{2n{e}^{2}A}
In this case in view of Equations (29) and (30), one gets:
{R}_{i}=0
For all values of T, i.e. for:
T\ge 0
Therefore according to Equations (19) & (31) the superconducting resistance is no longer equal to zero, i.e.:
R={R}_{s}\ne 0
For all values of T greater than zero.
Thu applying pressure such that it exceeds a certain critical value gives by (29) and (26) destroys Sc. The new critical decreases upon increasing pressure, when:
{T}_{C}={T}_{C}-{P}_{0}
Another approach can be suggested by assuming that the pressure result from a thermal one KT exerted by the surrounding. In this case Equation (3) takes the form:
E=KE+V+KT-{P}_{0}
Assuming also the existence of very large attractive force on electrons due to the ionic cores:
V=-{V}_{0}
KE+V=KE-{V}_{0}=-{V}_{1}
where the potential energy is assumed to be much large than the kinetic energy. Using nearly electron model for solids:
E=\frac{{\hslash }^{2}{k}^{2}}{2{m}^{*}}
Thus Equations (35), (36), (37) and (38) gives:
\hslash k=\sqrt{2{m}^{*}\left(KT-{V}_{1}-{P}_{0}\right)}
Define now the critical temperature to satisfy:
K{T}_{c}={V}_{1}+{P}_{0}
\hslash k=\sqrt{2{m}^{*}k\left(T-{T}_{c}\right)}
According to Equations (17), (18), (19) and (41):
R={R}_{i}+i{R}_{s}=\frac{\sqrt{2{m}^{*}k\left(T-{T}_{c}\right)}}{2n{e}^{2}A}
Thus for all:
T\le {T}_{c}
R={R}_{I}
{R}_{s}=0
It is very interesting to note that Equation (40), shows that the critical temperature can be increased by increasing the applied mechanical pressure as observed for Cu.
Pressure dependent Schrodinger equation [see Equation (14)] is used to find a useful expression for energy, wave number and momentum.
These expressions were obtained by treating electrons as vibrating strings subjected to repulsive electron force and mechanical pressure.
Using the ordinary expression for resistance a useful expression of quantum resistance is obtained [see Equations (18), (21)] through the momentum and wave number. Assuming first electron energy to be purely thermal [Equations (23)], the Sc resistance [Equations (24)-(28)] vanishes beyond a critical temperature in the absence of the external pressure. This a grees with experiment. Applying external pressure destroys Sc when the pressure exceeds a certain critical value [see Equations (29)-(34)]. The pressure here decreases the critical temperature.
When one assume the pressure to be resulting from electron gas thermal pressure beside external mechanical pressure [see Equation (35)], the critical temperature in (40) increases upon increasing pressure. This can thus explains the behavior of high temperature Sc copper group.
The pressure dependent Schrodinger equation can be used to obtain a useful expression of quantum resistance. This expression can be used to explain the effect of external mechanical pressure on high temperature Sc. It can explain why pressure decreases or increases sometimes the critical temperature.
[1] Annett, J.F. (2006) Superconductivity, Superfluid’s and Condense States, OVP, Campridge.
[2] Poole Jr., C.P., Afrach, H. and Creswick, R.J. (2007) Superconductivity. Academic Press, London.
[3] Jay Kumarsaxena, A. (2009) High Temperature Superconductors, Rewa, India.
[4] Hammed, I.A.I., Dirar, M., Atta, N.O., Taha, R.A., Haroun, Kh.M. and Elgani, R.A. (2017) Pressure Effect on Superconducting Critical Temperature According to String Model. International Journal of Fluid Mechanics & Thermal Sciences, 3, 70-74.
https://doi.org/10.11648/j.ijfmts.20170306.12
[5] Abdalrahman, R.I., Taha, R.A.E.M., Attia, I.A. and Allah, M.D.A. (2016) New Derivation of Simple Josephson Effect Relation Using New Quantum Mecjanical Equation. Natural Science, 8, 85-88.
[6] Mubark Dirar, A., As, A., Ahmed, Z., Asma, M.E., Rawia, A. and Amel, A.A. (2006) Effect of Magnetic Field on Superconducting Complex Resistance According to Quantum Mechanics. Global Journal of Engineering Science and Researches, 3.
[7] Dirar, M., Einas, M. and Elhouri, S.A. (2015) Relation between Critical Temperature and Superconductivity Zero Resistance According to Quantum Laws. International Journal of Management Science and Engineering Management, 5.
[8] Zakaria, A., Asma, A., Dirar, M., Asma, M.E. and Amel, A. (2016) Quantum Effect of Magnetic Field in Destroying Superconductivity. International Journal of Management Science and Engineering Management, 2.
[9] Scott, A.C. (2003) Nonlinear Science: Emergence & Dynamics of Coherent Structures. 2nd Edition, Oxford University Press, New York.
[10] Sulem, C. and Sulem, P.-L. (1999) The Nonlinear Schrodinger Equation: Self-Focusing and Wave Collapse. Springer-Verlag, New York.
[11] Karpman, V.I. (1975) Nonlinear Waves in Dispersive Media. Pergamon, Oxford.
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HypergeometricRandomVariable - Maple Help
Home : Support : Online Help : Education : Student Packages : Statistics : Random Variable Distributions : HypergeometricRandomVariable
HypergeometricRandomVariable(M, X, m)
The hypergeometric random variable is a discrete probability random variable with probability function given by:
f\left(t\right)={\begin{array}{cc}0& t<0\\ 0& X<t\\ \frac{\left(\genfrac{}{}{0}{}{X}{t}\right)\left(\genfrac{}{}{0}{}{M-X}{m-t}\right)}{\left(\genfrac{}{}{0}{}{M}{m}\right)}& \mathrm{otherwise}\end{array}
0\le M,m\le M,X\le M,0\le m,0\le X,M::\mathrm{Typesetting}:-\mathrm{_Hold}\left(\left['\mathrm{integer}'\right]\right),X::\mathrm{Typesetting}:-\mathrm{_Hold}\left(\left['\mathrm{integer}'\right]\right),m::\mathrm{Typesetting}:-\mathrm{_Hold}\left(\left['\mathrm{integer}'\right]\right)
The hypergeometric random variable is a consequence of a sequence of repeated trials (such as drawing balls from an urn) whereby items drawn are not replaced after each trial. In each trial, there is assumed to be a certain number of successes remaining that could be obtained. This random variable measures the probability of achieving a certain number of successes after all trials are complete.
The Quantile and CDF functions applied to a hypergeometric distribution use a sequence of iterations in order to converge upon the desired output point. The maximum number of iterations to perform is equal to 100 by default, but this value can be changed by setting the environment variable _EnvStatisticsIterations to the desired number of iterations.
\mathrm{with}\left(\mathrm{Student}[\mathrm{Statistics}]\right):
X≔\mathrm{HypergeometricRandomVariable}\left(5,z,m\right):
\mathrm{ProbabilityFunction}\left(X,u\right)
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{u}\\ \frac{\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{z}}{\textcolor[rgb]{0,0,1}{u}}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}}{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{u}}\right)}{\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{5}}{\textcolor[rgb]{0,0,1}{m}}\right)}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{ProbabilityFunction}\left(X,2\right)
{\begin{array}{cc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{2}\\ \frac{\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{z}}{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{z}}{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}}\right)}{\left(\genfrac{}{}{0}{}{\textcolor[rgb]{0,0,1}{5}}{\textcolor[rgb]{0,0,1}{m}}\right)}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{Mean}\left(X\right)
\frac{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}}{\textcolor[rgb]{0,0,1}{5}}
\mathrm{Variance}\left(X\right)
\frac{\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{z}}{\textcolor[rgb]{0,0,1}{5}}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{m}\right)}{\textcolor[rgb]{0,0,1}{20}}
Y≔\mathrm{HypergeometricRandomVariable}\left(10,3,7\right):
\mathrm{ProbabilityFunction}\left(Y,x,\mathrm{output}=\mathrm{plot}\right)
\mathrm{CDF}\left(Y,x\right)
\textcolor[rgb]{0,0,1}{1}
\mathrm{CDF}\left(Y,3,\mathrm{output}=\mathrm{plot}\right)
The Student[Statistics][HypergeometricRandomVariable] command was introduced in Maple 18.
Statistics[Distributions][Hypergeometric]
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IsOnLine - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 2-D Euclidean : Point Functions : IsOnLine
test if a point, a list, or a set of points is on a line
IsOnLine(f, l, cond)
The routine returns true if the point f or the list/set of points f is on line l; false if it is not; and FAIL if it is unable to reach a conclusion.
In case of FAIL, if the third optional argument is given, the condition that makes f on line l is assigned to this argument. It will be either of the form
\mathrm{expr}=0
&\mathrm{and}\left(\mathrm{expr_1}=0,...,\mathrm{expr_n}=0\right)
The command with(geometry,IsOnLine) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{line}\left(\mathrm{l1},y=0,[x,y]\right),\mathrm{line}\left(\mathrm{l2},x+y=1,[x,y]\right),\mathrm{point}\left(A,\frac{1}{2},\frac{1}{2}\right):
\mathrm{IsOnLine}\left(A,\mathrm{l1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{IsOnLine}\left(A,\mathrm{l2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{point}\left(A,a,\frac{1}{2}\right),\mathrm{point}\left(B,\frac{3}{5},b\right):
\mathrm{IsOnLine}\left({A,B},\mathrm{l2},'\mathrm{cond}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
\mathrm{cond}
\left(\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0,0,1}{&and}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left(\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}{a}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\right)
make the necessary assumptions
\mathrm{assume}\left(\mathrm{op}\left(\mathrm{cond}\right)\right)
\mathrm{IsOnLine}\left({A,B},\mathrm{l2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Numerical Solution of ODE with Delay - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 2015 : Numerical Solution of ODE with Delay
Numerical Solutions of ODE with Delay
Numeric solutions for initial value problems with ODE/DAE via dsolve[numeric] has been enhanced to accommodate delay terms for the three main variable step integrators, rkf45, ck45, and rosenbrock.
Example: Harmonic oscillator with delay
\mathrm{dsys} ≔ \left\{\frac{{ⅆ}^{2}}{ⅆ {t}^{2}} y\left(t\right) + y\left(t-\frac{1}{10}\right)=0, y\left(0\right)=1, y'\left(0\right)=0\right\}
\textcolor[rgb]{0,0,1}{\mathrm{dsys}}\textcolor[rgb]{0,0,1}{:=}\left\{\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{t}}^{\textcolor[rgb]{0,0,1}{2}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{10}}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{D}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{y}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\right\}
\mathrm{dsn} ≔ \mathrm{dsolve}\left(\mathrm{dsys}, \mathrm{numeric}\right):
\mathrm{plots}:-\mathrm{odeplot}\left(\mathrm{dsn},0..10,\mathrm{size}=\left[600,"golden"\right]\right);
\mathrm{dsys_var} ≔ \left\{\frac{ⅆ}{ⅆ t} x\left(t\right) = -x\left(t-\frac{1}{2}-\frac{\mathrm{exp}\left(-t\right)}{2}\right), x\left(0\right)=1\right\}
\textcolor[rgb]{0,0,1}{\mathrm{dsys_var}}\textcolor[rgb]{0,0,1}{:=}\left\{\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{t}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{t}\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}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{t}}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{0}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\right\}
\mathrm{max_delay} ≔ \mathrm{fsolve}\left(t = \frac{1}{2}+\frac{\mathrm{exp}\left(-t\right)}{2}, t\right);
\textcolor[rgb]{0,0,1}{\mathrm{max_delay}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{0.7388350311}
\mathrm{dsn_var} ≔ \mathrm{dsolve}\left(\mathrm{dsys_var}, \mathrm{numeric}, \mathrm{delaymax}=0.74\right):
\mathrm{plots}:-\mathrm{odeplot}\left(\mathrm{dsn_var}, 0..5,\mathrm{size}=\left[600,"golden"\right]\right)
Detailed information on this feature, such as setting of initial values, controlling the storage used to retain the delay data, and use with events can be found on the dsolve[numeric][delay] help page.
dsolve/numeric, dsolve/rkf45, dsolve/ck45, dsolve/rosenbrock, dsolve/numeric/delay
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Orientation - Maple Help
Home : Support : Online Help : Getting Started : Menus : Document Menu Bar : Format Menu : Plot Menu : Orientation
The Orientation menu contains commands that change the orientation (view) of your 3-D plot.
To change the orientation of your plot:
Click the plot, and then select Plot > Orientation.
Default: This undoes all pans and scales you performed on the plot. Also, the rotation angles theta, phi, and psi are set to their default values (55, 75, and 0, respectively).
Default rotation: This sets the rotation angles theta, phi, and psi to their default values (55, 75, and 0, respectively).
Default translation: This undoes any pans you performed on the plot.
Default size: This undoes any zoom actions you performed on the plot.
X Axis: Rotates the plot so that the positive
x
-axis is facing you.
Y Axis: Rotates the plot so that the positive
y
Z Axis: Rotates the plot so that the positive
z
Projection: Change the perspective of the plot using the Set projection dialog. For more information, see Change Projection.
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To solve the following problem, use the 5-D Process. Define a variable and write an expression for each column of your table.
In the first three football games of the season, Carlos gained three times as many yards as Alston. Travis gained ten yards more than Carlos. Altogether, the three players gained a total of
430
yards. How many yards did Carlos gain?
Based on the information from the problem, two of the players, Carlos and Travis, can be directly related to Alston in yardage. If Alston's yards are represented by the variable,
x
, how can Carlos’s and Travis's yards be represented? A 5 column table, first row labeled as follows: First 3 columns, define, fourth, do, fifth, decide.
Row 2 labels added as follows: Define 1: Alston's yards, x. Define 2: Carlos's yards, 3 x. Define 3: Travis's yards, 3 x + 10. Do: Sum of the yards, x + 3, x + 3, x + 10. Decide: 430 yards.
Row 3 labels, left to right: 50, 150, 160, 50 + 150 + 160, 360 yards too low.
Now run some trials of your own to determine how many yards Carlos gained. Row 4 labels, left to right: 70, 210, 220, 70 + 210 + 220, 500 yards too high.
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Behavioral model of a timer integrated circuit - MATLAB - MathWorks Benelux
Discharge switch on-resistance
Discharge switch off-resistance
Potential divider component resistance
Behavioral model of a timer integrated circuit
The Timer block is a behavioral model of a timer integrated circuit such as the NE555.
The following figure shows the implementation structure.
The Potential divider component resistance parameter sets the values of the three resistors creating the potential divider. The two comparator inputs have infinite input resistance and zero input capacitance. The S-R Latch block provides the functionality of the set-reset latch. It includes an output capacitor and a resistor with values set to match the Propagation delay parameter value. The block models the output stage inverter using a CMOS NOT block. You define the output resistance, low-level output voltage, and high-level output voltage for the CMOS gate in the Timer block dialog box. The discharge switch approximates the NPN bipolar transistor on a real timer as a switch with defined switch on-resistance and off-resistance values.
The behavior is abstracted. Results are not as accurate as a transistor-level model.
Delay in response to changing inputs depends solely upon the RC time constant of the resistor-capacitor network at the output of the latch. In practice, the delay has a more complex dependency on the device structure. Set this value based on the output-pulse rise and fall times.
The drop in output voltage is a linear function of output current. In practice, the relationship is that of a bipolar transistor push-pull pair.
The controlled switch arrangement used by the block is an approximation of an open-collector arrangement.
The power supply connects internally within the component, and the block assumes that the GND pin is grounded.
THRES — Threshold pin
Electrical conserving port associated to the timer threshold pin
TRIG — Trigger pin
Electrical conserving port associated to the timer trigger pin
CONT — Control pin
Electrical conserving port associated to the timer control pin
RESET — Reset pin
Electrical conserving port associated to the timer reset pin
OUT — Output pin
Electrical conserving port associated to the timer output pin
DISCH — Discharge pin
Electrical conserving port associated to the timer discharge pin
Power supply voltage — Power supply voltage
The voltage value
{V}_{cc}
that the block applies internally to the timer component.
The output voltage when the timer output is low and no output current is drawn.
14.1 V (default)
The output voltage
{V}_{OH}
when the timer output is high and no current is drawn.
The ratio of output voltage drop to output current. Set this parameter to
\left({V}_{OH}-{V}_{OH}{}_{1}\right)/{I}_{OH}{}_{1}
{V}_{OH}{}_{1}
is the reduced output high voltage when the output current is
{I}_{OH}{}_{1}
100e-9 s (default)
Set this value to the input-pulse or output-pulse rise time.
Discharge switch on-resistance — Discharge switch on-resistance
A representative value is the discharge pin saturation voltage divided by the corresponding current.
Discharge switch off-resistance — Discharge switch off-resistance
500e6 Ohm (default)
A representative value is the discharge pin leakage current divided by the corresponding pin voltage.
Potential divider component resistance — Potential divider component resistance
5 kOhm (default)
A typical value for a 555-type timer is 5 kΩ. You can measure it directly across the positive supply and control pins when the chip does not connect to a circuit.
S-R Latch | Comparator
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Difference between revisions of "UFR 3-35 Test Case" - KBwiki
Difference between revisions of "UFR 3-35 Test Case"
(→Test Case Experiments)
(→Experimental set-up)
The experimental set-up is shown in Fig. 2. A high-level water tank fed the flume with a constant energy head. After the inlet, a flow straightener, a surface waves damper and vortex generators as recommended by (Counihan 1969) were installed such that the turbulent open-channel flow developed along the entry length of <math>200D</math>. A sluice gate at the end of the flume controlled the water depth before the water recirculated driven by a pump. The experimental parameters are listed in Table 1:
{| class="wikitable" style="text-align: center;" border="1"
[[File:UFR3-35_Flume.png|thumb|centre|800px|Fig. 2: Experimental set-up]]
{| class="wikitable" style="text-align: center;" border="1" "margin: auto;"
|+ style="caption-side:bottom;"|Tab. 1: Experimental parameters
|+Experimental parameters
| <math>[-]</math>
= Measurement technique =
2 General Remark
3 Test Case Experiments
6 CFD Code and Methods
The experimental and numerical setups applied in this study were described in detail by Schanderl et al. (2017b) (PIV and LES). The experiment is further described by Jenssen (2019), the numerics in Schanderl & Manhart (2016), Schanderl et al. (2017a) and Schanderl & Manhart (2018). Thus, the following shall provide a brief overview only.
In order to provide both numerical and experimental data acquired for the same flow configuration under identical (as good as possible) boundary conditions, we performed a large eddy simulation and a particle image velocimetry experiment. We studied the flow around a wall-mounted slender (
{\displaystyle D/z_{0}<0.7}
) circular cylinder with a flow depth of
{\displaystyle z_{0}=1.5D}
. The width of the rectangular channel was
{\displaystyle 11.7D}
(see Fig. 1). The investigated Reynolds number was approximately
{\displaystyle Re_{D}={\frac {u_{\mathrm {b} }D}{\nu }}=39{,}000}
, the Froude number was in the subcritical region. As inflow condition we applied a fully-developed open-channel flow.
Fig. 1: Sketch of flow configuration
The experimental set-up is shown in Fig. 2. A high-level water tank fed the flume with a constant energy head. After the inlet, a flow straightener, a surface waves damper and vortex generators as recommended by (Counihan 1969) were installed such that the turbulent open-channel flow developed along the entry length of
{\displaystyle 200D}
. A sluice gate at the end of the flume controlled the water depth before the water recirculated driven by a pump. The experimental parameters are listed in Table 1:
Fig. 2: Experimental set-up
Tab. 1: Experimental parameters
{\displaystyle D}
{\displaystyle 0.1}
{\displaystyle [\mathrm {m} ]}
Flow depth
{\displaystyle z_{0}}
{\displaystyle 0.15}
{\displaystyle [\mathrm {m} ]}
{\displaystyle b}
{\displaystyle 1.17}
{\displaystyle [\mathrm {m} ]}
{\displaystyle Q}
{\displaystyle 0.069}
{\displaystyle [\mathrm {m} ^{3}\mathrm {s} ^{-1}]}
Depth-averaged velocity of approach flow
{\displaystyle u_{\mathrm {b} }}
{\displaystyle 0.3986}
{\displaystyle [\mathrm {m} \,\mathrm {s} ^{-1}]}
{\displaystyle \nu }
{\displaystyle 1.0502\cdot 10^{-6}}
{\displaystyle [\mathrm {m} ^{2}\mathrm {s} ^{-1}]}
{\displaystyle Re_{D}}
{\displaystyle 37{,}954}
{\displaystyle [-]}
{\displaystyle Re_{z_{0}}={\frac {u_{\mathrm {b} }\cdot 4R_{\mathrm {hyd} }}{\nu }}={\frac {u_{\mathrm {b} }\cdot 4(b\cdot z_{0})/(2z_{0}+b)}{\nu }}}
{\displaystyle 181{,}162}
{\displaystyle [-]}
{\displaystyle Re_{\tau }={\frac {u_{\tau }\cdot z_{0}}{\nu }}}
{\displaystyle 2571}
{\displaystyle [-]}
The experimental data were acquired by conducting planar monoscopic 2D-2C PIV in the vertical symmetry plane upstream of the cylinder. The PIV snapshots were evaluated by the standard interrogation window based cross-correlation of
{\displaystyle 16\times 16\mathrm {px} }
. Doing so, we achieved instantaneous velocity fields of the streamwise (
{\displaystyle u}
) and the wall-normal (
{\displaystyle w}
) velocity component. From these data the time-averaged turbulent statistics were calculated in the post-processing. We used a CCD-camera with a
{\displaystyle 2048\times 2048\mathrm {px} }
square sensor. The size of a pixel was
{\displaystyle 36.86\mu \mathrm {m} }
, therefore the spatial resolution of the images was
{\displaystyle 2712\mathrm {px} /D}
. The size of the interrogation windows was
{\displaystyle 5.8976\cdot 10^{-3}D}
. The temporal resolution was
{\displaystyle 7.25\mathrm {Hz} }
, which is approximately twice the macro time scale
{\displaystyle u_{\mathrm {b} }/D=3.9\mathrm {Hz} }
. The light sheet was approximately 2mm thick provided by a
{\displaystyle 532\mathrm {nm} }
Nd:YAG laser. The f-number and the focal length of the lens were
{\displaystyle 2.8}
{\displaystyle 105\mathrm {mm} }
At the measurement section, the flume had transparent walls. Therefore, the laser light, which entered the flow from above could pass with a minimum amount of surface reflections through the bottom wall. However, an acrylic glass plate had to be mounted at the water-air interface to suppress the bow waves of the cylinder and let the light sheet enter the water body perpendicularly (see Fig. 3). The influence of this device at the water surface was tested and considered to be insignificant at the cylinder-wall junction.
Hollow glass spheres were used as seeding and had a diameter of
{\displaystyle 10\mu \mathrm {m} }
. The corresponding Stokes number was
{\displaystyle 4.7\cdot 10^{-3}}
, and therefore, the particles were considered to follow the flow precisely.
The total number of time-steps was
{\displaystyle 27{,}000}
, the time-delay between two image frames of a time-step was
{\displaystyle 700\mu \mathrm {s} }
. Therefore, the total sampling time was
{\displaystyle 27{,}000/7.25=3724\mathrm {s} }
{\displaystyle 1484D}
. During the experiment seeding and other particles accumulated along the bottom plate, which undermined the image quality by increasing the surface reflection. Therefore, the data acquisition was stopped after
{\displaystyle 1500}
images to allow surface cleaning and to empty the limited capacity of the laboratory PC's RAM. The sampling time of such a batch was
{\displaystyle 1500/7.25=207\mathrm {s} }
{\displaystyle 82D/u_{b}}
The data acquisition time and number of valid vectors was validated by the convergence of statistical moments. In the centre of the HV the number valid samples had its minimum. Therefore, the time-series at the centre of the HV was analysed as a reference for the entire flow field. The standard error of the mean was
{\displaystyle 0.0065}
times the standard deviation, the corresponding error in the fourth central moment is
{\displaystyle 0.0545}
The PIV set-up is given in detail here, including the qualitative size of the field-of-views (FOV) for investigating the approaching boundary layer as well as the flow in front of the wall-mounted cylinder.
CFD Code and Methods
We applied our in-house finite volume code MGLET with a staggered Cartesian grid. The grid was equidistant in the horizontal directions and stretched away from the wall in the vertical direction by a factor smaller than
{\displaystyle 1.01}
. The horizontal grid spacing was four times as large as the vertical one. The time integration was done by applying a third order Runge-Kutta sheme, the spatial approximation by second order central differences and the maximum of the CFL number was in the range of 0.55 to 0.82. To model the cylindrical body, a second order immersed boundary method was applied (Peller et al.2006; Peller 2010). The sub-grid scales were modelled using the Wall-Adapting Local Eddy-Viscosity (WALE) model (Nicoud & Ducros 1999). Around the cylinder, the grid was refined by three locally embedded grids (Manhart 2004), each reducing the grid spacing by a factor of two. A grid study shows the results to be converged over grid spacing (Schanderl & Manhart 2016). The resulting grid spacing in the vertical direction at the bottom plate around the cylinder was smaller than approximately 1.6 wall units (based on the local wall-shear stress) (Schanderl & Manhart 2016). The fraction of the modelled dissipation is about 30% of the total dissipation rate (Schanderl & Manhart 2018).
The setup simulated was intended to be identical to the experimental one. To model the bottom and side walls, we applied no-slip boundary conditions, whereas the free surface was modelled by a slip boundary condition. Therefore, the Froude number in the LES was infinitesimal and no surface waves occurred. By conducting a precursor simulation a fully-developed turbulent open-channel flow was achieved as inflow condition. The streamwise boundary conditions were periodic, and the precursor domain had a length of 30D. The wall resolution of the precursor grid was 7.5 wall units.
Grid arrangement of the LES (Schanderl 2018)
Applied grids in the LES
Cells per diameter
{\displaystyle \Delta {x}^{+}/\Delta {y}^{+}/\Delta {z}_{\mathrm {wall} }^{+}}
Precursor 0
{\displaystyle 60/60/15}
{\displaystyle 44\cdot 10^{6}}
{\displaystyle 31.25/125}
{\displaystyle 60/60/15}
{\displaystyle 35\cdot 10^{6}}
Grid 1 1
{\displaystyle 62.5/250}
{\displaystyle 30/30/7.5}
{\displaystyle 80\cdot 10^{6}}
{\displaystyle 125/500}
{\displaystyle 15/15/3.7}
{\displaystyle 64\cdot 10^{6}}
{\displaystyle 250/1000}
{\displaystyle 7.5/7.5/1.9}
{\displaystyle 177\cdot 10^{6}}
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Stack Overflow | Toph
Stack is a basic data structure. Where 3 operations can be done:
Push: You can push object to the stack
Pop: You can pop the object to the stack
Top: You can check the value of the top object.
For further details you can get idea here (if you really don’t know): https://en.wikibooks.org/wiki/Data_Structures/Stacks_and_Queues
Now we have a problem here, there are
N stacks in front of you. they are numbered from 1 to
N. Each of them are initially empty. Now you will have
Q operations. Each operation can be one the below 4 types:
push i x, Push a value of
x to stack numbered
i
pop i, Pop value from the stack numbered
i, if stack is empty discard the operation
put i j, Put the
j-th stack on top of the
i-th stack. So there will be no element left on the
j-th stack.
top i, Print the value of the top element of
i-th stack. If stack is empty print “Empty!”
T (
1 ≤ T ≤ 5
1≤T≤5), denoting the number of test cases.
N (
1 ≤ N ≤ 10^4
1≤N≤104),
Q (
1 ≤ Q ≤ 5 \times 10^4
1≤Q≤5×104). The next
Q lines will contain an operation like above mentioned.
1 ≤ i, j ≤ N
1≤i,j≤N
1 ≤ x ≤ 10^5
1≤x≤105
For each test case, print the case number in a single line. Then for each 4th type operation you should print the value or “Empty!” if the stack is empty.
exsurrealEarliest, Aug '15
defaltFastest, 0.0s
|
A Story of Basis and Kernel - Part II: Reproducing Kernel Hilbert Space
In the previous blog, the function basis was briefly discussed. We began with viewing a function as an infinite vector, and then defined the inner product of functions. Similar to
\mathcal{R}^n
space, we can also find orthogonal function basis for a function space.
This blog will move a step further discussing about kernel functions and reproducing kernel Hilbert space (RKHS). Kernel methods have been widely used in a variety of data analysis techniques. The motivation of kernel method arises in mapping a vector in
\mathcal{R}^n
space as another vector in a feature space. For example, imagine there are some red points and some blue points as the next figure shows, which are not easily separable in
\mathcal{R}^n
space. However, if we map them into a high-dimension feature space, we may be able to seperate them easily. This article will not provide strict theoretical definition, but rather intuitive description on the basic ideas.
2. Eigen Decomposition
For a real symmetric matrix
\mathbf{A}
, there exists real number
\lambda
\mathbf{x}
\mathbf{A} \mathbf{x} = \lambda \mathbf{x}
\lambda
\mathbf{A}
\mathbf{x}
is the corresponding eigenvector. If
\mathbf{A}
has two different eigenvalues
\lambda_1
\lambda_2
\lambda_1 \neq \lambda_2
\mathbf{x}_1
\mathbf{x}_2
\lambda_1 \mathbf{x}_1^T \mathbf{x}_2 = \mathbf{x}_1^T \mathbf{A}^T \mathbf{x}_2 = \mathbf{x}_1^T \mathbf{A} \mathbf{x}_2 = \lambda_2 \mathbf{x}_1^T \mathbf{x}_2
\lambda_1 \neq \lambda_2
\mathbf{x}_1^T \mathbf{x}_2 = 0
\mathbf{x}_1
\mathbf{x}_2
\mathbf{A} \in \mathcal{R}^{n \times n}
n
eigenvalues a long with
n
orthogonal eigenvectors. As a result,
\mathbf{A}
can be decomposited as
\mathbf{A} = \mathbf{Q} \mathbf{D} \mathbf{Q}^T
\mathbf{Q}
is an orthogonal matrix (i.e.,
\mathbf{Q} \mathbf{Q}^T = \mathbf{I}
\mathbf{D} = \text{diag} (\lambda_1, \lambda_2, \cdots, \lambda_n)
\mathbf{Q}
column by column
\mathbf{Q}=\left( \mathbf{q}_1, \mathbf{q}_2, \cdots, \mathbf{q}_n \right)
\begin{array}{rl} \mathbf{A}=\mathbf{Q} \mathbf{D} \mathbf{Q}^T &= \left( \mathbf{q}_1, \mathbf{q}_2, \cdots, \mathbf{q}_n \right) \begin{pmatrix} \lambda_1\ &&& \\ & \lambda_2\ && \\ && \ddots\ & \\ &&&\lambda_n \end{pmatrix} \begin{pmatrix} \mathbf{q}_1^T \\ \mathbf{q}_2^T \\ \vdots \\ \mathbf{q}_n^T \end{pmatrix} \\ &= \left( \lambda_1 \mathbf{q}_1, \lambda_2 \mathbf{q}_2, \cdots, \lambda_n \mathbf{q}_n \right) \begin{pmatrix} \mathbf{q}_1^T \\ \mathbf{q}_2^T \\ \vdots \\ \mathbf{q}_n^T \end{pmatrix} \\ &=\sum_{i=1}^n \lambda_i \mathbf{q}_i \mathbf{q}_i^T \end{array}
{ \{\mathbf{q}_i \} }_{i=1}^n
is a set of orthogonal basis of
\mathcal{R}^n
3. Kernel Function
f(\mathbf{x})
can be viewed as an infinite vector, then for a function with two independent variables
K(\mathbf{x},\mathbf{y})
, we can view it as an infinite matrix. Among them, if
K(\mathbf{x},\mathbf{y}) = K(\mathbf{y},\mathbf{x})
\int \int f(\mathbf{x}) K(\mathbf{x},\mathbf{y}) f(\mathbf{y}) d\mathbf{x} d\mathbf{y} \geq 0
f
K(\mathbf{x},\mathbf{y})
is symmetric and positive definite, in which case
K(\mathbf{x},\mathbf{y})
is a kernel function.
Similar to matrix eigenvalue and eigenvector, there exists eigenvalue
\lambda
and eigenfunction
\psi(\mathbf{x})
\int K(\mathbf{x},\mathbf{y}) \psi(\mathbf{x}) d\mathbf{x} = \lambda \psi(\mathbf{y})
For different eigenvalues
\lambda_1
\lambda_2
with corresponding eigenfunctions
\psi_1(\mathbf{x})
\psi_2(\mathbf{x})
, it is easy to show that
\begin{array}{rl} \int \lambda_1 \psi_1(\mathbf{x}) \psi_2(\mathbf{x}) d\mathbf{x} & = \int \int K(\mathbf{y},\mathbf{x}) \psi_1(\mathbf{y}) d\mathbf{y} \psi_2(\mathbf{x}) d\mathbf{x} \\ & = \int \int K(\mathbf{x},\mathbf{y}) \psi_2(\mathbf{x}) d\mathbf{x} \psi_1(\mathbf{y}) d\mathbf{y} \\ & = \int \lambda_2 \psi_2(\mathbf{y}) \psi_1(\mathbf{y}) d\mathbf{y} \\ & = \int \lambda_2 \psi_2(\mathbf{x}) \psi_1(\mathbf{x}) d\mathbf{x} \end{array}
< \psi_1, \psi_2 > = \int \psi_1(\mathbf{x}) \psi_2(\mathbf{x}) d\mathbf{x} = 0
Again, the eigenfunctions are orthogonal. Here
\psi
denotes the function (the infinite vector) itself.
For a kernel function, infinite eigenvalues
{ \{\lambda_i\} }_{i=1}^{\infty}
along with infinite eigenfunctions
{ \{\psi_i\} }_{i=1}^{\infty}
may be found. Similar to matrix case,
K(\mathbf{x},\mathbf{y}) = \sum_{i=0}^{\infty} \lambda_i \psi_i (\mathbf{x}) \psi_i (\mathbf{y})
which is the Mercer's theorem. Here
< \psi_i, \psi_j > = 0
i \neq j
{ \{\psi_i\} }_{i=1}^{\infty}
construct a set of orthogonal basis for a function space.
Here are some commonly used kernels:
Polynomial kernel
K(\mathbf{x},\mathbf{y}) = ( \gamma \mathbf{x}^T \mathbf{y} + C)^d
Gaussian radial basis kernel
K(\mathbf{x},\mathbf{y}) = \exp (-\gamma \Vert \mathbf{x} - \mathbf{y} \Vert^2 )
Sigmoid kernel
K(\mathbf{x},\mathbf{y}) = \tanh (\gamma \mathbf{x}^T \mathbf{y} + C )
4. Reproducing Kernel Hilbert Space
{ \{\sqrt{\lambda_i} \psi_i\} }_{i=1}^{\infty}
as a set of orthogonal basis and construct a Hilbert space
\mathcal{H}
. Any function or vector in the space can be represented as the linear combination of the basis. Suppose
f = \sum_{i=1}^{\infty} f_i \sqrt{\lambda_i} \psi_i
f
as an infinite vector in
\mathcal{H}
f = (f_1, f_2, ...)_\mathcal{H}^T
For another function
g = (g_1, g_2, ...)_\mathcal{H}^T
< f,g >_\mathcal{H} = \sum_{i=1}^{\infty} f_i g_i
For the kernel function
K
, here I use
K(\mathbf{x},\mathbf{y})
to denote the evaluation of
K
\mathbf{x},\mathbf{y}
which is a scalar, use
K(\cdot,\cdot)
to denote the function (the infinite matrix) itself, and use
K(\mathbf{x},\cdot)
\mathbf{x}
th "row" of the matrix, i.e., we fix one parameter of the kernel function to be
\mathbf{x}
then we can regard it as a function with one parameter or as an infinite vector. Then
K(\mathbf{x},\cdot) = \sum_{i=0}^{\infty} \lambda_i \psi_i (\mathbf{x}) \psi_i
\mathcal{H}
, we can denote
K(\mathbf{x},\cdot) = (\sqrt{\lambda_1} \psi_1 (\mathbf{x}), \sqrt{\lambda_2} \psi_2 (\mathbf{x}), \cdots )_\mathcal{H}^T
< K(\mathbf{x},\cdot), K(\mathbf{y},\cdot) >_\mathcal{H} = \sum_{i=0}^{\infty} \lambda_i \psi_i (\mathbf{x}) \psi_i(\mathbf{y}) = K(\mathbf{x},\mathbf{y})
This is the reproducing property, thus
\mathcal{H}
is called reproducing kernel Hilbert space (RKHS).
Now it is time to return to the problem from the beginning of this article: how to map a point into a feature space? If we define a mapping
\bold{\Phi} (\mathbf{x}) = K(\mathbf{x},\cdot) = (\sqrt{\lambda_1} \psi_1 (\mathbf{x}), \sqrt{\lambda_2} \psi_2 (\mathbf{x}), \cdots )^T
then we can map the point
\mathbf{x}
\mathcal{H}
\bold{\Phi}
is not a function, since it points to a vector or a funtion in the feature space
\mathcal{H}
< \bold{\Phi} (\mathbf{x}), \bold{\Phi} (\mathbf{y}) >_\mathcal{H} = < K(\mathbf{x},\cdot), K(\mathbf{y},\cdot) >_\mathcal{H} = K(\mathbf{x},\mathbf{y})
As a result, we do not need to actually know what is the mapping, where is the feature space, or what is the basis of the feature space. For a symmetric positive-definite function
K
, there must exist at least one mapping
\bold{\Phi}
and one feature space
\mathcal{H}
< \bold{\Phi} (\mathbf{x}), \bold{\Phi} (\mathbf{y}) > = K(\mathbf{x},\mathbf{y})
which is the so-called kernel trick.
Consider kernel function
K(\mathbf{x},\mathbf{y}) = \left( x_1, x_2, x_1 x_2 \right) \begin{pmatrix} y_1 \\ y_2 \\ y_1 y_2 \end{pmatrix} = x_1 y_1 + x_2 y_2 + x_1 x_2 y_1 y_2
\mathbf{x}=(x_1,x_2)^T, \mathbf{y}=(y_1,y_2)^T
\lambda_1=\lambda_2=\lambda_3=1
\psi_1(\mathbf{x})=x_1
\psi_2(\mathbf{x})=x_2
\psi_3(\mathbf{x})=x_1 x_2
. We can define the mapping as
\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \overset{\bold{\Phi}}{\longrightarrow} \begin{pmatrix} x_1 \\ x_2 \\ x_1 x_2 \end{pmatrix}
< \bold{\Phi} (\mathbf{x}), \bold{\Phi}(\mathbf{y}) > = \left( x_1, x_2, x_1 x_2 \right) \begin{pmatrix} y_1 \\ y_2 \\ y_1 y_2 \end{pmatrix} = K(\mathbf{x},\mathbf{y})
Support vector machine (SVM) is one of the most widely known application of RKHS. Suppose we have data pairs
{ (\mathbf{x}_i, y_i) }_{i=1}^n
y_i
is either 1 or -1 denoting the class of the point
\mathbf{x}_i
. SVM assumes a hyperplane to best seperate the two classes.
\min_{\boldsymbol{\beta}, \beta_0} \frac{1}{2} \Vert \boldsymbol{\beta} \Vert^2 + C \sum_{i=1}^n \xi_i
\text{subject to } \xi_i \geq 0, y_i (\mathbf{x}_i^T \boldsymbol{\beta} + \beta_0 ) \geq 1 - \xi_i, \forall i
Sometimes the two classes cannot be easily seperated in
\mathcal{R}^n
space, thus we can map
\mathbf{x}_i
into a high-dimension feature space where the two classes may be easily seperated. The original problem can be reformulated as
\min_{\boldsymbol{\beta}, \beta_0} \frac{1}{2} \Vert \boldsymbol{\beta} \Vert^2 + C \sum_{i=1}^n \xi_i
\text{subject to } \xi_i \geq 0, y_i (\bold{\Phi}(\mathbf{x}_i)^T \boldsymbol{\beta} + \beta_0 ) \geq 1 - \xi_i, \forall i
The Lagrange function is
L_p = \frac{1}{2} \Vert \boldsymbol{\beta} \Vert^2 + C \sum_{i=1}^n \xi_i - \sum_{i=1}^n \alpha_i [y_i (\bold{\Phi}(\mathbf{x}_i)^T \boldsymbol{\beta} + \beta_0) - (1-\xi_i)] -\sum_{i=1}^n \mu_i \xi_i
\frac{\partial L_p}{\partial \boldsymbol{\beta}} = \mathbf{0}
\boldsymbol{\beta} = \sum_{i=1}^n \alpha_i y_i \bold{\Phi}(\mathbf{x}_i)
\boldsymbol{\beta}
can be writen as the linear combination of
\mathbf{x}_i
s! We can substitute
\boldsymbol{\beta}
and get the new optimization problem. The objective function changes to:
\begin{array}{rl} &\frac{1}{2} \Vert \sum_{i=1}^n \alpha_i y_i \bold{\Phi} (\mathbf{x}_i) \Vert^2 + C \sum_{i=1}^n \xi_i \\ =& \frac{1}{2} < \sum_{i=1}^n \alpha_i y_i \bold{\Phi} (\mathbf{x}_i), \sum_{j=1}^n \alpha_j y_j \bold{\Phi} (\mathbf{x}_j) > + C \sum_{i=1}^n \xi_i \\ =& \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \alpha_i \alpha_j y_i y_j < \bold{\Phi} (\mathbf{x}_i), \bold{\Phi} (\mathbf{x}_j) > + C \sum_{i=1}^n \xi_i \\ = & \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \alpha_i \alpha_j y_i y_j K(\mathbf{x}_i, \mathbf{x}_j) + C \sum_{i=1}^n \xi_i \end{array}
The constraints changes to:
\begin{array}{rl} & y_i \left[\bold{\Phi}(\mathbf{x}_i)^T \left( \sum_{j=1}^n \alpha_j y_j \bold{\Phi}(\mathbf{x}_j) \right) + \beta_0 \right] \\ =& y_i \left[ \left( \sum_{j=1}^n \alpha_j y_j < \bold{\Phi}(\mathbf{x}_i), \bold{\Phi}(\mathbf{x}_j) > \right) + \beta_0 \right] \\ =& y_i \left[ \left( \sum_{j=1}^n \alpha_j y_j K(\mathbf{x}_i, \mathbf{x}_j) \right) + \beta_0 \right] \geq 1 - \xi_i, \forall i \end{array}
What we need to do is determining a kernel function and solve for
\boldsymbol{\alpha}, \beta_0, \xi_i
. We do not need to actually construct the feature space. For a new data
\mathbf{x}
with unknown class, we can predict its class by
\begin{array}{ccl} \hat{y} &=& \text{sign} \left[ \bold{\Phi} (\mathbf{x})^T \boldsymbol{\beta} + \beta_0 \right] \\ &=& \text{sign} \left[ \bold{\Phi} (\mathbf{x})^T \left( \sum_{i=1}^n \alpha_i y_i \bold{\Phi}(\mathbf{x}_i) \right) + \beta_0 \right] \\ &=& \text{sign} \left( \sum_{i=1}^n \alpha_i y_i < \bold{\Phi} (\mathbf{x}), \bold{\Phi}(\mathbf{x}_i) > + \beta_0 \right) \\ &=& \text{sign} \left( \sum_{i=1}^n \alpha_i y_i K(\mathbf{x},\mathbf{x}_i) + \beta_0 \right) \end{array}
Kernel methods greatly strengthen the discriminative power of SVM.
7. Summary and Reference
Kernel method has been widely utilized in data analytics. Here, the fundamental property of RKHS is introduced. With kernel trick, we can easily map the data to a feature space and do analysis. Here is a video with nice demonstration on why we can easily do classification with kernel SVM in a high-dimension feature space.
The example in Section 5 is from
Gretton A. (2015): Introduction to RKHS, and some simple kernel algorithms, Advanced Topics in Machine Learning, Lecture conducted from University College London.
Other reference includes
Paulsen, V. I. (2009). An introduction to the theory of reproducing kernel Hilbert spaces. Lecture Notes.
Daumé III, H. (2004). From zero to reproducing kernel hilbert spaces in twelve pages or less.
Friedman, J., Hastie, T., and Tibshirani, R. (2001). The elements of statistical learning. Springer, Berlin: Springer series in statistics.)
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Choosing Identified Plant Structure - MATLAB & Simulink - MathWorks
Existing Plant Models
Switching Between Model Structures
Estimating Parameter Values
Handling Initial Conditions
PID Tuner provides two types of model structures for representing the plant dynamics: process models and state-space models.
Use your knowledge of system characteristics and the level of accuracy required by your application to pick a model structure. In absence of any prior information, you can gain some insight into the order of dynamics and delays by analyzing the experimentally obtained step response and frequency response of the system. For more information see the following in the System Identification Toolbox™ documentation:
Correlation Models (System Identification Toolbox)
Frequency-Response Models (System Identification Toolbox)
Each model structure you choose has associated dynamic elements, or model parameters. You adjust the values of these parameters manually or automatically to find an identified model that yields a satisfactory match to your measured or simulated response data. In many cases, when you are unsure of the best structure to use, it helps to start with the simplest model structure, transfer function with one pole. You can progressively try identification with higher-order structures until a satisfactory match between the plant response and measured output is achieved. The state-space model structure allows an automatic search for optimal model order based on an analysis of the input-output data.
When you begin the plant identification task, a transfer function model structure with one real pole is selected by default. This default set up is not sensitive to the nature of the data and may not be a good fit for your application. It is therefore recommended that you choose a suitable model structure before performing parameter identification.
Process models are transfer functions with 3 or fewer poles, and can be augmented by addition of zero, delay and integrator elements. Process models are parameterized by model parameters representing time constants, gain, and time delay. In PID Tuner, choose a process model in the Plant Identification tab using the Structure menu.
For any chosen structure you can optionally add a delay, a zero and/or an integrator element using the corresponding checkboxes. Click Edit Parameters to view the model transfer function configured by these choices.
The simplest available process model is a transfer function with one real pole and no zero or delay elements:
H\left(s\right)=\frac{K}{{T}_{1}s+1}.
This model is defined by the parameters K, the gain, and T1, the first time constant. The most complex process-model structure choose has three poles, an additional integrator, a zero, and a time delay, such as the following model, which has one real pole and one complex conjugate pair of poles:
H\left(s\right)=K\frac{{T}_{z}s+1}{s\left({T}_{1}s+1\right)\left({T}_{\omega }^{2}{s}^{2}+2\zeta {T}_{\omega }s+1\right)}{e}^{-\tau s}.
In this model, the configurable parameters include the time constants associated with the poles and the zero, T1, Tω, and Tz. The other parameters are the damping coefficient ζ, the gain K, and the time delay τ.
When you select a process model type, PID Tuner automatically computes initial values for the plant parameters and displays a plot showing both the estimated model response and your measured or simulated data. You can edit the parameter values graphically using indicators on the plot, or numerically using the Plant Parameters editor. For an example illustrating this process, see Interactively Estimate Plant Parameters from Response Data.
The following table summarizes the various parameters that define the available types of process models.
K — Gain All transfer functions
Can take any real value.
In the plot, drag the plant response curve (blue) up or down to adjust K.
T1 — First time constant
Transfer function with one or more real poles
Can take any value between 0 and T, the time span of measured or simulated data.
In the plot, drag the red x left (towards zero) or right (towards T) to adjust T1.
T2— Second time constant Transfer function with two real poles
In the plot, drag the magenta x left (towards zero) or right (towards T) to adjust T2.
Tω — Time constant associated with the natural frequency ωn, where Tω = 1/ωn Transfer function with underdamped pair (complex conjugate pair) of poles
In the plot, drag one of the orange response envelope curves left (towards zero) or right (towards T) to adjust Tω.
ζ — Damping coefficient Transfer function with underdamped pair (complex conjugate pair) of poles
Can take any value between 0 and 1.
In the plot, drag one of the orange response envelope curves left (towards zero) or right (towards T) to adjust ζ.
τ — Transport delay Any transfer function
In the plot, drag the orange vertical bar left (towards zero) or right (towards T) to adjust τ.
Tz — Model zero Any transfer function
Can take any value between –T and T, the time span of measured or simulated data.
In the plot, drag the red circle left (towards –T) or right (towards T) to adjust Tz.
Integrator Any transfer function Adds a factor of 1/s to the transfer function. There is no associated parameter to adjust.
The state-space model structure for identification is primarily defined by the choice of number of states, the model order. Use the state-space model structure when higher order models than those supported by process model structures are required to achieve a satisfactory match to your measured or simulated I/O data. In the state-space model structure, the system dynamics are represented by the state and output equations:
\begin{array}{l}\stackrel{˙}{x}=Ax+Bu,\\ y=Cx+Du.\end{array}
x is a vector of state variables, automatically chosen by the software based on the selected model order. u represents the input signal, and y the output signals.
To use a state-space model structure, in the Plant Identification tab, in the Structure menu, select State-Space Model. Then click Configure Structure to open the State-Space Model Structure dialog box.
Use the dialog box to specify model order, delay and feedthrough characteristics. If you are unsure about the order, select Pick best value in the range, and enter a range of orders. In this case, when you click Estimate in the Plant Estimation tab, the software displays a bar chart of Hankel singular values. Choose a model order equal to the number of Hankel singular values that make significant contributions to the system dynamics.
When you choose a state-space model structure, the identification plot shows a plant response (blue) curve only if a valid estimated model exists. For example, if you change structure after estimating a process model, the state-space equivalent of the estimated model is displayed. If you change the model order, the plant response curve disappears until a new estimation is performed.
When using the state-space model structure, you cannot directly interact with the model parameters. The identified model should thus be considered unstructured with no physical meaning attached to the state variables of the model.
However, you can graphically adjust the input delay and the overall gain of the model. When you select a state-space model with a time delay, the delay is represented on the plot by a vertical orange bar is shown on the plot. Drag this bar horizontally to change the delay value. Drag the plant response (blue) curve up and down to adjust the model gain.
Any previously imported or identified plant models are listed the Plant List area.
You can define the model structure and initialize the model parameter values using one of these plants. To do so, in the Plant Identification tab, in the Structure menu, select the linear plant model you want to use for structure an initialization.
If the plant you select is a process model (idproc (System Identification Toolbox) object), PID Tuner uses its structure. If the plant is any other model type, PID Tuner uses the state-space model structure.
When you switch from one model structure to another, the software preserves the model characteristics (pole/zero locations, gain, delay) as much as possible. For example, when you switch from a one-pole model to a two-pole model, the existing values of T1, Tz, τ and K are retained, T2 is initialized to a default (or previously assigned, if any) value.
Once you have selected a model structure, you have several options for manually or automatically adjusting parameter values to achieve a good match between the estimated model response and your measured or simulated input/output data. For an example that illustrates all these options, see:
Interactively Estimate Plant Parameters from Response Data (Control System Toolbox™)
Interactively Estimate Plant from Measured or Simulated Response Data Simulink® Control Design™)
PID Tuner does not perform a smart initialization of model parameters when a model structure is selected. Rather, the initial values of the model parameters, reflected in the plot, are arbitrarily-chosen middle of the range values. If you need a good starting point before manually adjusting the parameter values, use the Initialize and Estimate option from the Plant Identification tab.
In some cases, the system response is strongly influenced by the initial conditions. Thus a description of the input to output relationship in the form of a transfer function is insufficient to fit the observed data. This is especially true of systems containing weakly damped modes. PID Tuner allows you to estimate initial conditions in addition to the model parameters such that the sum of the initial condition response and the input response matches the observed output well. Use the Estimation Options dialog box to specify how the initial conditions should be handled during automatic estimation. By default, the initial condition handling (whether to fix to zero values or to estimate) is automatically performed by the estimation algorithm. However, you can enforce a certain choice by using the Initial Conditions menu.
Initial conditions can only be estimated with automatic estimation. Unlike the model parameters, they cannot be modified manually. However, once estimated they remain fixed to their estimated values, unless the model structure is changed or new identification data is imported.
If you modify the model parameters after having performed an automatic estimation, the model response will show a fixed contribution (i.e., independent of model parameters) from initial conditions. In the following plot, the effects of initial conditions were identified to be particularly significant. When the delay is adjusted afterwards, the portion of the response to the left of the input delay marker (the τ Adjustor) comes purely from initial conditions. The portion to the right of the τ Adjustor contains the effects of both the input signal as well as the initial conditions.
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Extend the pattern by drawing Figures
0
4
5
. Then describe Figure
100
. Give as much information as you can about Figure
100
. What will it look like? How will the tiles be arranged? How many tiles will it have?
What connections do you see between the different representations (figures,
x→y
table, and graph)? How can you show these connections?
As a team, organize your work into a large poster that clearly shows each representation of your pattern, as well as a description of Figure
100
. When your team presents your poster to the class, you will need to support each statement with a reason from your observations. Each team member must explain something mathematical as part of your presentation.
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A set of equations is returned indicating the number of each element type name that occurs in the XML element represented by the xmlTree. Each equation has the form
\mathrm{elementName}=\mathrm{elementFrequency}
, where elementName is the name of the element type, and elementFrequency is a non-negative integer that indicates the number of elements of the corresponding type that occur in the element represented by xmlTree.
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{xmlTree}≔\mathrm{XMLElement}\left("a",[],\mathrm{XMLElement}\left("b",[],"b text"\right),"some text",\mathrm{XMLElement}\left("b",[],"more b text"\right)\right):
\mathrm{ElementStatistics}\left(\mathrm{xmlTree}\right)
[\textcolor[rgb]{0,0,1}{"a"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"b"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2}]
\mathrm{xmlTree}≔\mathrm{CleanXML}\left(\mathrm{ParseFile}\left("myfile.xml"\right)\right):
\mathrm{ElementStatistics}\left(\mathrm{xmlTree}\right)
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Solve system of differential equations - MATLAB dsolve - MathWorks Switzerland
\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ay}
{C}_{1} {\mathrm{e}}^{a t}
\frac{{\mathit{d}}^{2}\mathit{y}}{{\mathit{dt}}^{2}}=\mathit{ay}
{C}_{1} {\mathrm{e}}^{-\sqrt{a} t}+{C}_{2} {\mathrm{e}}^{\sqrt{a} t}
\frac{\mathit{dy}}{\mathit{dt}}=\mathit{ay}
y\left(0\right)=5
5 {\mathrm{e}}^{a t}
\frac{{\mathit{d}}^{2}\mathit{y}}{{\mathit{dt}}^{2}}={\mathit{a}}^{2}\mathit{y}
y\left(0\right)=b
{y}^{\prime }\left(0\right)=1
\frac{{\mathrm{e}}^{a t} \left(a b+1\right)}{2 a}+\frac{{\mathrm{e}}^{-a t} \left(a b-1\right)}{2 a}
\begin{array}{l}\frac{\mathit{dy}}{\mathit{dt}}=\mathit{z}\\ \frac{\mathit{dz}}{\mathit{dt}}=-\mathit{y}.\end{array}
{C}_{1} \mathrm{cos}\left(t\right)+{C}_{2} \mathrm{sin}\left(t\right)
{C}_{2} \mathrm{cos}\left(t\right)-{C}_{1} \mathrm{sin}\left(t\right)
{C}_{1} \mathrm{cos}\left(t\right)+{C}_{2} \mathrm{sin}\left(t\right)
{C}_{2} \mathrm{cos}\left(t\right)-{C}_{1} \mathrm{sin}\left(t\right)
\frac{\partial }{\partial t}y\left(t\right)={e}^{-y\left(t\right)}+y\left(t\right)
\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)={\mathrm{e}}^{-y\left(t\right)}+y\left(t\right)
{\mathrm{W}\text{lambertw}}_{0}\left(-1\right)
F\left(y\left(t\right)\right)=g\left(t\right)
\left(\begin{array}{c}\left({\int \frac{{\mathrm{e}}^{y}}{y {\mathrm{e}}^{y}+1}\mathrm{d}y|}_{y=y\left(t\right)}\right)={C}_{1}+t\\ {\mathrm{e}}^{-y\left(t\right)} \left({\mathrm{e}}^{y\left(t\right)} y\left(t\right)+1\right)=0\end{array}\right)
F\left(y\left(x\right)\right)=g\left(x\right)
{\mathrm{e}}^{y\left(x\right)}+\frac{{y\left(x\right)}^{2}}{2}={C}_{1}+{\mathrm{e}}^{-x}+\frac{{x}^{2}}{2}
\frac{\mathit{dy}}{\mathit{dt}}=\frac{\mathit{a}}{\sqrt{\mathit{y}}}+\mathit{y}
y\left(a\right)=1
{\left({\mathrm{e}}^{\frac{3 t}{2}-\frac{3 a}{2}+\mathrm{log}\left(a+1\right)}-a\right)}^{2/3}
a
\begin{array}{l}\left\{\begin{array}{cl}\left\{\begin{array}{cl}\left\{{\sigma }_{1}\right\}& \text{ if }-\frac{\pi }{2}<{\sigma }_{2}\\ \left\{{\sigma }_{1},-{\left(-a+{\mathrm{e}}^{\frac{3 t}{2}-\frac{3 a}{2}+\mathrm{log}\left(a+{\left(-\frac{1}{2}+{\sigma }_{3}\right)}^{3/2}\right)+2 \pi {C}_{2} \mathrm{i}}\right)}^{2/3} \left(\frac{1}{2}+{\sigma }_{3}\right)\right\}& \text{ if }{\sigma }_{2}\le -\frac{\pi }{2}\end{array}& \text{ if }{C}_{2}\in \mathbb{Z}\\ \varnothing & \text{ if }{C}_{2}\notin \mathbb{Z}\end{array}\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_{1}={\left(-a+{\mathrm{e}}^{\frac{3 t}{2}-\frac{3 a}{2}+\mathrm{log}\left(a+1\right)+2 \pi {C}_{2} \mathrm{i}}\right)}^{2/3}\\ \\ \mathrm{ }{\sigma }_{2}=\text{angle}\left({\mathrm{e}}^{\frac{3 {C}_{1}}{2}+\frac{3 t}{2}}-a\right)\\ \\ \mathrm{ }{\sigma }_{3}=\frac{\sqrt{3} \mathrm{i}}{2}\end{array}
\left({x}^{2}-1{\right)}^{2}\frac{{\partial }^{2}}{\partial {x}^{2}}y\left(x\right)+\left(x+1\right)\frac{\partial }{\partial x}y\left(x\right)-y\left(x\right)=0
{C}_{2} \left(x+1\right)+{C}_{1} \left(x+1\right) \int \frac{{\mathrm{e}}^{\frac{1}{2 \left(x-1\right)}} {\left(1-x\right)}^{1/4}}{{\left(x+1\right)}^{9/4}}\mathrm{d}x
x=-1
\left(\begin{array}{c}x+1\\ \frac{1}{{\left(x+1\right)}^{1/4}}-\frac{5 {\left(x+1\right)}^{3/4}}{4}+\frac{5 {\left(x+1\right)}^{7/4}}{48}+\frac{5 {\left(x+1\right)}^{11/4}}{336}+\frac{115 {\left(x+1\right)}^{15/4}}{33792}+\frac{169 {\left(x+1\right)}^{19/4}}{184320}\end{array}\right)
\infty
\left(\begin{array}{c}x-\frac{1}{6 {x}^{2}}-\frac{1}{8 {x}^{4}}\\ \frac{1}{6 {x}^{2}}+\frac{1}{8 {x}^{4}}+\frac{1}{90 {x}^{5}}+1\end{array}\right)
\left(\begin{array}{c}x-\frac{1}{6 {x}^{2}}-\frac{1}{8 {x}^{4}}-\frac{1}{90 {x}^{5}}-\frac{37}{336 {x}^{6}}\\ \frac{1}{6 {x}^{2}}+\frac{1}{8 {x}^{4}}+\frac{1}{90 {x}^{5}}+\frac{37}{336 {x}^{6}}+\frac{37}{1680 {x}^{7}}+1\end{array}\right)
\frac{\mathit{dy}}{\mathit{dx}}=\frac{1}{{\mathit{x}}^{2}}{\mathit{e}}^{-\frac{1}{\mathit{x}}}
{C}_{1}+{\mathrm{e}}^{-\frac{1}{x}}
\mathit{y}\left(0\right)=1
{\mathrm{e}}^{-\frac{1}{x}}+1
{\mathit{e}}^{-\frac{1}{\mathit{x}}}
x=0
\underset{\mathit{x}\to {0}^{+}}{\mathrm{lim}}\text{\hspace{0.17em}}{\mathit{e}}^{-\frac{1}{\mathit{x}}}=0
\underset{\mathit{x}\to {0}^{-}}{\mathrm{lim}}\text{\hspace{0.17em}}{\mathit{e}}^{-\frac{1}{\mathit{x}}}=\infty
\mathrm{lim}\text{\hspace{0.17em}}\mathit{x}\to {\mathit{x}}_{0}^{+}
O\left({\mathrm{var}}^{n}\right)
O\left({\mathrm{var}}^{-n}\right)
\mathrm{lim}x\to {x}_{0}^{+}\text{ }
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Erlang B: Details & Formula - Miscellaneous - 2022
Erlang B: details & formula
The basic Erlang calculations enabled an good idea of traffic loading in a telecommunications circuit to be analysed.
The main drawback was that it did not take account of real life in terms of variations in loading. The Erlang B seeks address this issue by looking at peak loading.
Accordingly, the Erlang B is used to calculate how many lines are required from a knowledge of the traffic figure during the busiest hour.
The Erlang B figure assumes that any blocked calls are cleared immediately. This is the most commonly used figure to be used in any telecommunications capacity calculations.
It is particularly important to understand the traffic volumes at peak times of the day. Telecommunications traffic, like many other commodities, varies over the course of the day, and also the week. It is therefore necessary to understand the telecommunications traffic at the peak times of the day and to be able to determine the acceptable level of service required. The Erlang B figure is designed to handle the peak or busy periods and to determine the level of service required in these periods.
Essentially, the Erlang B traffic model is used by telephone system designers to estimate the number of lines required for PSTN connections or private wire connections. The three variables involved are Busy Hour Traffic (BHT), Blocking and Lines.
Busy Hour Traffic (in Erlangs) is the number of hours of call traffic there are during the busiest hour of operation of a telephone system.
Blocking is the failure of calls due to an insufficient number of lines being available. E.g. 0.03 mean 3 calls blocked per 100 calls attempted.
Lines is the number of lines in a trunk group.
The Extended Erlang B is similar to Erlang B, but it can be used to factor in the number of calls that are blocked and immediately tried again.
The formula for the Erlang B calculation can be seen below:
B=\frac{\frac{{A}^{N}}{N!}}{\sum \left(\frac{{A}^{i}}{i!}\right)}
B=Erlang B loss probability
N=Number of trunks in full availability group
A=Traffic offered to group in Erlangs
The summation is undertaken from i = 0 to N
Watch the video: 이동통신 17강 interference in TDD, Erlang B formula (May 2022).
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Copyright 2022 \ Erlang B: details & formula...
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numtheory(deprecated)/sum2sqr - Maple Help
Home : Support : Online Help : numtheory(deprecated)/sum2sqr
the sum of two squares problem
sum2sqr(n)
Important: The numtheory package has been deprecated. Use the superseding command NumberTheory[SumOfSquares] instead.
The sum2sqr returns the solutions of the sum of two squares problem.
[[{a}_{1},{b}_{1}],[{a}_{2},{b}_{2}],\mathrm{...},[{a}_{n},{b}_{n}]
{a}_{i}
{b}_{i}
are non-negative integers,
{a}_{i}^{2}+{b}_{i}^{2}=n
{a}_{i}<={b}_{i},1<=i<=n
The command with(numtheory,sum2sqr) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{numtheory}\right):
\mathrm{sum2sqr}\left(17\right)
[[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4}]]
\mathrm{sum2sqr}\left(938491\right)
[]
\mathrm{sum2sqr}\left(10281960\right)
[[\textcolor[rgb]{0,0,1}{234}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3198}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1014}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3042}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1422}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2874}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{1446}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2862}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2106}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2418}]]
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VolgaCTF 2021 Writeup | y011d4.log
I participated in VolgaCTF 2021 as a member of WreckTheLine. The results was 14th/231 (within teams with positive points). I solved 2 crypto problems, "QR Codebook" and "Carry". This is a writeup for those.
QR Codebook
We were given 3 images.
img: QR which says "Helpful_Information"
img_enc: Processed QR from above img
img_flag_enc: Processed QR from FLAG QR
We didn't have any other information, so let's observe. My observation revealed the following:
To see images in a row, continuous numbers occur
ex. the first raw of img_enc is [164, 164, ..., 164, 164, 201, 201, ..., 201, 201, 164, ..., 164, 201, ..., 201, ...]
To see images in a column, the same patterns appear many times (16 cycles)
ex. the first column of img_enc is [164, 25, 22, 17, 172, 23, 87, 71, 132, 139, 59, 221, 39, 32, 173, 89, 164, 25, 22, 17, 172, 23, 87, 71, 132, 139, 59, 221, 39, 32, 173, 89, ...]
Column patterns are the same between img_enc and img_flag_enc
At each dot the RGB values are the same in img_enc
I guessed that there is one-to-one mapping from img[i: i+16, j] to img_enc[i: i+16, j, k] for all
i (i \equiv 0 \mod 16), j, k
. Let's try to find that mapping and apply it to img_flag_enc.
# 0 is black, 1 (255) is white.
img_enc = np.array(Image.open("./qr.encrypted.png"))
img_flag_enc = np.array(Image.open("./flag.encrypted.png"))
img = np.array(Image.open("./qr.png"))
img_size = img.shape[0]
assert img_size == img_enc.shape[0] == img_enc.shape[1]
enc_to_dec = {}
for i in range(0, img_size, 16):
for j in range(img_size):
tmp = tuple(img_enc[i : i + 16, j, 0])
if tmp not in enc_to_dec:
enc_to_dec[tmp] = img[i : i + 16, j]
img_flag = np.ones(img_flag_enc.shape[:2])
for i in range(0, img_flag_enc.shape[0], 16):
for j in range(img_flag_enc.shape[1]):
tmp = tuple(img_flag_enc[i : i + 16, j, 0])
if tmp in enc_to_dec:
img_flag[i : i + 16, j] = enc_to_dec[tmp]
plt.imshow(img_flag, cmap="gray")
plt.savefig("./flag.png")
There seem some incorrect dots in QR... Since each dots in QR code are bigger than 16 pixels, we can reconstruct from this image to the correct QR code in theory. But my smart phone can read this broken QR code correctly (smart!).
We were given a shift register (FCSR) and an image xor-ed by bits generated from that register. The state of the register wasn't disclosed. The implementation of FCSR was as follow:
fcsr.py
class FCSR():
def __init__(self, q: int, m: int, a: int):
self.q = q + 1
self.k = int(math.log(q, 2))
def get_i(n: int, i: int) -> int:
return (n & (0b1 << i)) >> i
def clock(self) -> int:
s = self.m
s += self.get_i(self.q, i) * self.get_i(self.a, self.k - i)
a_k = s % 2
a_0 = self.a & 0b1
self.m = s // 2
self.a = (self.a >> 1) | (a_k << (self.k - 1))
def encrypt(self, data: bytes) -> bytes:
encrypted = b''
key_byte = 0
bit = self.clock()
key_byte = (key_byte << 1) | bit
encrypted += int.to_bytes(key_byte ^ byte, 1, 'big')
Since the first 16 bytes of a PNG file are fixed, the 128 generated bits can be calculated.
[0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1]
From some experiments, I found that FCSR generated cyclic bits at short period when chosen parameters were bad. Looking at this problem carefully, I noticed that there was a cycle like [0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1].
from fcsr import FCSR
with open("./encrypted_png", "rb") as f:
png_header = b"\x89\x50\x4e\x47\x0d\x0a\x1a\x0a" + b"\x00\x00\x00\x0d\x49\x48\x44\x52"
first_bytes = xor(png_header, buf)
first_bytes_long = bytes_to_long(first_bytes)
first_bits = list(map(int, f"{first_bytes_long:0128b}")) # known bits: 16 * 8 = 128
cycle = [0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1]
bits = first_bits + ((8 * len(buf) - len(first_bits)) // len(cycle) + 1) * cycle
all_bytes = b""
all_bytes += long_to_bytes(sum([b * 2 ** (7 - j) for j, b in enumerate(bits[i: i+8])]))
ans = xor(all_bytes, buf)
with open("dec.png", "wb") as f:
VolgaCTF{0bfc16cc12effc1bae4d3766c4f2257d}
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