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http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.106.118302
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Synopsis: Gels settle down
Highly Nonlinear Dynamics in a Slowly Sedimenting Colloidal Gel
G. Brambilla, S. Buzzaccaro, R. Piazza, L. Berthier, and L. Cipelletti
Published March 14, 2011
Depending on the kind of colloidal particles it contains, a gel will sediment in a matter of minutes or days. Understanding how shifts in the positions of the typically submicron sized particles affect the more macroscopic sedimentation process (and vice versa) could be helpful in designing industry-use gels. So far, however, no experiments have provided simultaneous access to these vastly different length scales.
Now, a group of scientists in France and Italy report in Physical Review Letters the use of light scattering to capture both the microscopic and macroscopic pictures of a gel collapsing under its own weight.
Giovanni Brambilla of the Université Montpellier, France, and colleagues filled a tall glass column with about $10\phantom{\rule{0.333em}{0ex}}\text{mm}$ of a water-based gel. The sticky, colloidal particles in the gel slowly rearranged as the gel started to sediment, altering the specklelike pattern of laser light that the team scattered through a vertical slice of the gel. Over the course of ten days, Brambilla et al. captured this speckle pattern at various heights along the column and used an algorithm to extract such parameters as the particle relaxation rate, sedimentation velocity and density.
The team finds, at least in the slowly settling gels they studied, that both the microscopic and macroscopic dynamics mimic what is found in glassy polymers. Brambilla et al.’s data should thus provide a solid basis on which to test the theory of gels. – Jessica Thomas
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2014-10-23 13:06:08
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https://math.stackexchange.com/questions/3010121/existence-of-a-vector-v-in-v-such-that-the-t-annihilator-of-v-is-the-min
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# Existence of a vector $v$ in $V$ such that the $T$-annihilator of $v$ is the minimal polynomial for $T$.
Definition: $$T$$-annihilator of a vector $$\alpha$$ (denoted as $$p_\alpha$$) is the unique monic polynomial which generates the ideal such that $$g(T)\alpha = 0$$ for all $$g$$ in this ideal.
I'm trying to prove the below statement without invoking the Cyclic Decomposition Theorem.
Let $$T$$ be a linear operator on a finite-dimensional vector space $$V$$. Then there exists a vector $$v$$ in $$V$$ such that the $$T$$-annihilator of $$v$$ is the minimal polynomial for $$T$$.
Attempt: Assume that there is no such $$v$$. Then every vector has a $$T$$-annihilator of degree less than that of the minimal polynomial. Define a monic polynomial $$h$$ which is the sum of $$T$$-annihilators of given basis elements. Then $$h(T)v=0$$ for all $$v\in V$$. But this contradicts the definition of minimal polynomial since the degree of $$h\lt$$ the degree of the minimal polynomial.
Can someone verify my argument?
• What is the definition of $T$-annihilator''?
– daw
Nov 23, 2018 at 13:04
• No that proof doesn't work. There is no reason that $h(T)v = 0$. If $h$ is the sum $g_1+g_2+\cdots+g_n$ then $h(T)v$ equals $g_1(T)v+g_2(T)v+\cdots + g_n(T)v$ with possibly only ONE term missing. Nov 23, 2018 at 13:51
## 1 Answer
Let's first show the result when the minimal polynomial has the form $$p^n$$, with $$p$$ irreducible. We know that $$p(T)^n=0$$ , $$p(T)^{n-1}\neq0$$ so there exist a vector $$\alpha \in V$$ such that $$p(T)^{n-1}\alpha\neq0$$, $$p(T)^n\alpha=0$$. Thus the T-annihilator $$g$$ of $$\alpha$$ divides $$p^n$$ and since $$p(T)^{r}\alpha\neq0$$ for $$r\leq n-1$$, $$g=p^n$$.
Now consider the general case and let $$p=p_{1}^{r_1}...p_{k}^{r_k}$$ be the minimal polynomial for $$T$$ where the $$p_i$$ are distinct irreductible monic polynomials. Then applying the primary decomposition for $$T$$ we obtain $$V=W_1 \oplus\cdots\oplus W_k$$ , and denoting by $$T_i$$ the restriction of $$T$$ to $$W_i$$ the minimal polynomial for $$T_i$$ is $$p_{i}^{r_i}$$. Now we can use the result above : there exist $$\alpha_i \in W_i$$ such that the T-annihilator $$g_i$$ of $$\alpha_i$$ is $$p_{i}^{r_i}$$
Let $$\alpha = \sum_{i=1}^k\alpha_i$$. We know that the T-annihilator $$g$$ of $$\alpha$$ divides $$p$$. Let $$f$$ be any polynomial such that $$f(T)\alpha=0$$. Then $$\sum_{i=1}^k f(T)\alpha_i =0$$ which implies $$f(T)\alpha_i =0$$ for each $$i$$ ($$\alpha_i \in W_i$$ and the $$W_i$$ are invariant under $$T$$ so $$f(T)\alpha_i \in W_i$$, and finally the $$W_i$$ are independant). Thus $$p_{i}^{r_i}$$ divides $$f$$ for each $$i$$ so $$p$$ divides $$f$$. Now this shows that $$p$$ divides $$g$$ which gives us $$g=p$$.
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2022-08-13 12:08:13
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https://peoplepill.com/people/oleg-d-jefimenko
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peoplepill id: oleg-d-jefimenko
ODJ
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American physcisit
# Oleg D. Jefimenko
Oleg D. Jefimenko
The basics
## Quick Facts
Intro American physcisit
Was Scientist Physicist
From Russia United States of America
Field Science
Gender male
Birth 14 October 1922, Kharkiv, Ukraine
Death 14 May 2009, Morgantown, USA (aged 86 years)
Star sign Libra
Education
Lewis & Clark College University of Oregon Oregon State University
The details (from wikipedia)
## Biography
Oleg Dmitrovich Jefimenko (October 14, 1922, Kharkiv, Ukraine – May 14, 2009, Morgantown, West Virginia, United States) was a physicist and Professor Emeritus at West Virginia University.
## Biography
Jefimenko received his B.A. degree at Lewis and Clark College in 1952 and his M. A. degree at the University of Oregon in 1954. He received his Ph.D. degree at the University of Oregon in 1956. Jefimenko worked for the development of the theory of electromagnetic retardation and relativity. In 1956, he was awarded the Sigma Xi Prize. In 1971 and 1973, he won awards in the AAPT Apparatus Competition. Jefimenko constructed and operated electrostatic generators run by atmospheric electricity.
Jefimenko worked on the generalization of Newton's gravitational theory to time-dependent systems. In his opinion, there is no objective reason for abandoning Newton's force-field gravitational theory (in favor of a metric gravitational theory). He was trying to develop and expand Newton's theory, making it compatible with the principle of causality and making it applicable to time-dependent gravitational interactions.
Jefimenko's expansion, or generalization, is based on the existence of the second gravitational force field, the "cogravitational, or Heaviside's field". This might also be called a gravimagnetic field. It represents a physical approach profoundly different from the time-space geometry approach of the Einstein general theory of relativity. Oliver Heaviside first predicted this field in the article A Gravitational and Electromagnetic Analogy (1893).
## Electromagnetic analogy of gravitational and cogravitational fields
Jefimenko suggests that electromagnetic equations can be converted to their gravitational-cogravitational equivalent by replacing electromagnetic symbols and constants with their corresponding gravitational-cogravitational symbols and constants, given in the table below.
Electric Gravitational
q (charge) m (mass)
ρ (volume charge density) ρ (volume mass density)
σ (surface charge density) σ (surface mass density)
λ (line charge density) λ (line mass density)
A (vector potential) A (vector potential)
J (convection current density) J (mass-current density)
I (electric current) I (mass current)
m (magnetic dipole moment) d (cogravitational moment)
E (electric field) g (gravitational field)
B (magnetic field) K (cogravitational field)
ɛ0 (permittivity of space) -${\displaystyle {\tfrac {1}{4}}}$πG
μ0 (permeability of space) -4πG/c
-${\displaystyle {\tfrac {1}{4}}}$πɛ0 or -μ0c/4π G (gravitational constant)
## Generalized Theory of Gravitation
Jefimenko posits the following Generalized Theory of Gravitation.
{\displaystyle {\begin{aligned}&\mathbf {g} =-G\int \left[{\dfrac {[\rho ]}{r^{3}}}+{\dfrac {1}{r^{2}c}}\left[{\dfrac {\partial \rho }{\partial t}}\right]\right]\mathbf {(} r)dV'+{\dfrac {G}{c^{2}}}\int {\dfrac {1}{r}}\left[{\dfrac {\partial (\rho \mathbf {v} )}{\partial t}}\right]dV',\\&\mathbf {K} =-{\dfrac {G}{c^{2}}}\int \left[{\dfrac {[\rho \mathbf {v} ]}{r^{3}}}+{\dfrac {1}{r^{2}c}}{\dfrac {\partial [\rho \mathbf {v} ]}{\partial t}}\right]\times \mathbf {r} dV',\end{aligned}}}
## Selected publications
### Books
• Jefimenko, Oleg (2006), Gravitation and Cogravitation: Developing Newton's Theory of Gravitation to its Physical and Mathematical Conclusion, Star City: Electret Scientific Company, ISBN 0-917406-15-X
• Electromagnetic Retardation and Theory of Relativity: New Chapters in the Classical Theory of Fields, 2nd ed., Electret Scientific, Star City, 2004.
• Causality, Electromagnetic Induction, and Gravitation: A Different Approach to the Theory of Electromagnetic and Gravitational Fields, 2nd ed., Electret Scientific, Star City, 2000.
• Electricity and Magnetism: An Introduction to the Theory of Electric and Magnetic Fields, 2nd ed., Electret Scientific, Star City, 1989.
• Scientific Graphics with Lotus 1-2-3: Curve Plotting, 3D Graphics, and Pictorial Compositions. Electret Scientific, Star City, 1987.
• 30 Music Programs for Timex Sinclair 2068. Electret Scientific, Star City, 1985.
• Electrostatic motors; their history, types, and principles of operation. Star City [W. Va.], Electret Scientific Co. [1973]. LCCN 73180890
• Electrostatic motors; their history, types, and principles of operation; NEW REVISED EDITION, edited by Thomas Valone. Integrity Research Institute, Beltsville, MD [2011].
### Book chapters
• What is the Physical Nature of Electric and Magnetic Forces?, in: Has the Last Word Been Said on Classical Electrodynamics? -- New Horizons, A. E. Chubykalo, Ed., Rinton Press, Paramus, 2004.
• Does special relativity prohibit superluminal velocities?, in: Instantaneous Action at a Distance in Modern Physics: "Pro" and "Contra, A. E. Chubykalo, Ed., (Nova Science, New York, 1999).
### Papers
• Jefimenko, O. D. (2008). "Causal equations for electric and magnetic fields and Maxwell's equations: Comment on a paper by Heras". American Journal of Physics. American Association of Physics Teachers (AAPT). 76 (2): 101–101. doi:10.1119/1.2825390. ISSN 0002-9505.
• Jefimenko, Oleg D. (2006). "Pile Driver Exercise". The Physics Teacher. American Association of Physics Teachers (AAPT). 44 (1): 4–4. doi:10.1119/1.2150748. ISSN 0031-921X.
• Jefimenko, Oleg D. (2005-09-19). "A neglected topic in relativistic electrodynamics: transformation of electromagnetic integrals". arXiv:physics/0509159.
• Jefimenko, Oleg D (2004-01-27). "Presenting electromagnetic theory in accordance with the principle of causality". European Journal of Physics. IOP Publishing. 25 (2): 287–296. doi:10.1088/0143-0807/25/2/015. ISSN 0143-0807.
• Jefimenko, Oleg D. (2002). "Comment on "Causality, the Coulomb field, and Newton's law of gravitation," by F. Rohrlich [Am. J. Phys. 70 (4), 411–414 (2002)]". American Journal of Physics. American Association of Physics Teachers (AAPT). 70 (9): 964–964. doi:10.1119/1.1485718. ISSN 0002-9505.
• Jefimenko, Oleg D. (2000). "Dynamic electric field maps of point charge moving with constant velocity". The Physics Teacher. American Association of Physics Teachers (AAPT). 38 (3): 154–157. doi:10.1119/1.880481. ISSN 0031-921X. (contains portrait of the author).
• Jefimenko, Oleg D (1999-01-01). "The Trouton-Noble paradox". Journal of Physics A: Mathematical and General. IOP Publishing. 32 (20): 3755–3762. doi:10.1088/0305-4470/32/20/308. ISSN 0305-4470.
• Jefimenko, Oleg D. (1999-11-01). "On the Relativistic Invariance of Maxwell's Equation". Zeitschrift für Naturforschung A. Walter de Gruyter GmbH. 54 (10–11): 637–644. doi:10.1515/zna-1999-10-1113. ISSN 1865-7109.
• Jefimenko, Oleg D. (1998-12-01). "On the Experimental Proofs of Relativistic Length Contraction and Time Dilation". Zeitschrift für Naturforschung A. Walter de Gruyter GmbH. 53 (12): 977–982. doi:10.1515/zna-1998-1208. ISSN 1865-7109.
• Jefimenko, Oleg D (1999-01-01). "A relativistic paradox seemingly violating conservation of momentum law in electromagnetic systems". European Journal of Physics. IOP Publishing. 20 (1): 39–44. doi:10.1088/0143-0807/20/1/010. ISSN 0143-0807.
• Jefimenko, Oleg D (1998-09-01). "On Maxwell's displacement current". European Journal of Physics. IOP Publishing. 19 (5): 469–470. doi:10.1088/0143-0807/19/5/011. ISSN 0143-0807.
• Jefimenko, Oleg D (1997-11-01). "Correct use of Lorentz - Einstein transformation equations for electromagnetic fields". European Journal of Physics. IOP Publishing. 18 (6): 444–447. doi:10.1088/0143-0807/18/6/007. ISSN 0143-0807.
• Jefimenko, Oleg D. (1996). "Derivation of relativistic force transformation equations from Lorentz force law". American Journal of Physics. American Association of Physics Teachers (AAPT). 64 (5): 618–620. doi:10.1119/1.18165. ISSN 0002-9505.
• Jefimenko, Oleg D. (1996). "Direct calculation of time dilation". American Journal of Physics. American Association of Physics Teachers (AAPT). 64 (6): 812–814. doi:10.1119/1.18181. ISSN 0002-9505.
• Jefimenko, Oleg D (1996-07-01). "Is magnetic field due to an electric current a relativistic effect?". European Journal of Physics. IOP Publishing. 17 (4): 180–182. doi:10.1088/0143-0807/17/4/006. ISSN 0143-0807.
• Jefimenko, Oleg D (1996-09-01). "Retardation and relativity: new integrals for electric and magnetic potentials of time-independent charge distributions moving with constant velocity". European Journal of Physics. IOP Publishing. 17 (5): 258–264. doi:10.1088/0143-0807/17/5/002. ISSN 0143-0807.
• Jefimenko, Oleg D. (1995). "Retardation and relativity: Derivation of Lorentz–Einstein transformations from retarded integrals for electric and magnetic fields". American Journal of Physics. American Association of Physics Teachers (AAPT). 63 (3): 267–272. doi:10.1119/1.17938. ISSN 0002-9505.
• Jefimenko, Oleg D. (1995). "Retardation and relativity: The case of a moving line charge". American Journal of Physics. American Association of Physics Teachers (AAPT). 63 (5): 454–459. doi:10.1119/1.17911. ISSN 0002-9505.
• Gravitational field of a point mass moving with uniform linear or circular velocity, Galilean Electrodynamics, March/April 1994, pp. 25–33.
• Jefimenko, Oleg D. (1994). "Direct calculation of the electric and magnetic fields of an electric point charge moving with constant velocity". American Journal of Physics. American Association of Physics Teachers (AAPT). 62 (1): 79–85. doi:10.1119/1.17716. ISSN 0002-9505.
• Jefimenko, Oleg D. (1992). "Solutions of Maxwell's equations for electric and magnetic fields in arbitrary media". American Journal of Physics. American Association of Physics Teachers (AAPT). 60 (10): 899–902. doi:10.1119/1.17010. ISSN 0002-9505.
• Jefimenko, Oleg D.; Walker, David K. (1980). "Electrets". The Physics Teacher. American Association of Physics Teachers (AAPT). 18 (9): 651–659. doi:10.1119/1.2340651. ISSN 0031-921X.
• "How can An Electroscope be Charged This Way?". Physics Teacher. 17 (1): 56. 1979.
• Jefimenko, Oleg (1974). "Water Stream "Loop-the-Loop"". American Journal of Physics. American Association of Physics Teachers (AAPT). 42 (2): 103–105. doi:10.1119/1.1987623. ISSN 0002-9505.
• Jefimenko, Oleg (1971). "Franklin's Electric Motors". American Journal of Physics. American Association of Physics Teachers (AAPT). 39 (10): 1139–1140. doi:10.1119/1.1976588. ISSN 0002-9505.
• Jefimenko, Oleg (1971). "Operation of Electric Motors from the Atmospheric Electric Field". American Journal of Physics. American Association of Physics Teachers (AAPT). 39 (7): 776–778. doi:10.1119/1.1986281. ISSN 0002-9505.
• Jefimenko, Oleg (1962). "Demonstration of the Electric Fields of Current-Carrying Conductors". American Journal of Physics. American Association of Physics Teachers (AAPT). 30 (1): 19–21. doi:10.1119/1.1941887. ISSN 0002-9505.
• Jefimenko, Oleg (1959). "Effect of the Earth's Magnetic Field on the Motion of an Artificial Satellite". American Journal of Physics. American Association of Physics Teachers (AAPT). 27 (5): 344–348. doi:10.1119/1.1934848. ISSN 0002-9505.
### Encyclopedia Article
• Maxwell's Equations, Macmillan Encyclopedia of Physics, Macmillan, New York, 1996.
The contents of this page are sourced from Wikipedia article on 11 May 2020. The contents are available under the CC BY-SA 4.0 license.
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2022-05-20 13:39:17
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https://alice-publications.web.cern.ch/node/8463
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# Figure 7
The relative elliptic flow fluctuations ($F(v_2)$) as a function of \pt{} for different particle species and centralities in Pb--Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$ TeV.
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2023-03-26 15:40:25
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https://socratic.org/questions/how-do-you-solve-8-7e-6-6-6x-7-32-8
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How do you solve -8.7e^(6.6-6x)-7=-32.8?
Aug 4, 2016
$x = 0.9188$
Explanation:
Let us rewrite the given eqn., by removing all $- v e$ signs, as
$8.7 {e}^{6.6 - 6 x} + 7 = 32.8$
$\Rightarrow 8.7 {e}^{6.6 - 6 x} = 25.8$
$\Rightarrow {e}^{6.6 - 6 x} = \frac{25.8}{8.7}$
$\Rightarrow {\log}_{10} \left\{{e}^{6.6 - 6 x}\right\} = {\log}_{10} 25.8 - {\log}_{10} 8.7$
$\Rightarrow \left(6.6 - 6 x\right) {\log}_{10} e = 1.4116 - 0.9395 = 0.4721$
$\Rightarrow \left(6.6 - 6 x\right) \left(0.4343\right) = 0.4721$
$\Rightarrow \left(6.6 - 6 x\right) = \frac{0.4721}{0.4343} = 1.087$
$\Rightarrow \frac{6.6 - 1.087}{6} = x = 0.9188$.
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2020-01-24 10:38:18
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https://www.projecteuclid.org/euclid.twjm/1499706523
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## Taiwanese Journal of Mathematics
### CLOSEDNESS OF SET OF EFFICIENT SOLUTIONS FOR GENERALIZED KY FAN INEQUALITY PROBLEMS
#### Abstract
In this paper, we discuss the closedness of set of efficient solutions for generalized Ky Fan inequality problems in topological vector spaces. We introduce a concept of section mapping of a bifunction. By using the lower semicontinuity of the section mapping, we present sufficient conditions for the closedness of set of efficient solutions to the generalized Ky Fan inequality problems. We give conditions to guarantee the lower semicontinuity of the section mapping. We give also an example to illustrate that the condition of the lower semicontinuity of the section mapping is essential for the closedness of set of efficient solutions for generalized Ky Fan inequality problems. As an application, we give results of closedness of set of efficient solutions for vector optimization problems and for Lipschitz vector variational inequalities.
#### Article information
Source
Taiwanese J. Math., Volume 18, Number 5 (2014), 1511-1526.
Dates
First available in Project Euclid: 10 July 2017
https://projecteuclid.org/euclid.twjm/1499706523
Digital Object Identifier
doi:10.11650/tjm.18.2014.2764
Mathematical Reviews number (MathSciNet)
MR3265074
Zentralblatt MATH identifier
1357.49076
#### Citation
Gong, Xun-Hua; Yang, Xin-Min. CLOSEDNESS OF SET OF EFFICIENT SOLUTIONS FOR GENERALIZED KY FAN INEQUALITY PROBLEMS. Taiwanese J. Math. 18 (2014), no. 5, 1511--1526. doi:10.11650/tjm.18.2014.2764. https://projecteuclid.org/euclid.twjm/1499706523
#### References
• F. Giannessi, (ed.) Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer, Dordrecht, 2000.
• G. Y. Chen and X. Q. Yang, The vector complementary problem and its equivalence with weak minimal element in ordered spaces, J. Math. Anal. Appl., 153 (1990), 136-158.
• H. Yang and J. Yu, Essential components of the set of weakly Pareto Nash equilibrium points, Appl. Math. Lett., 15 (2002), 553-560.
• Y. P. Fang and N. J. Huang, Strong vector variational inequalities in Banach spaces, Appl. Math. Lett., 19 (2006), 362-368.
• X. H. Gong and H. M. Yue, Existence of efficient solutions and strong solutions for vector equilibrium problems, J. Nanchang Univ., 32 (2008), 1-5.
• X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196.
• X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205.
• X. H. Gong, Optimality conditions for efficient solution to the vector equilibrium problems with constrants, Taiwanese Journal of Mathematics, 16 (2012), 1453-1473.
• H. P. Benson and E. J. Sun, New closedness results for efficient sets in multiple objective mathematical programming, J. Math. Anal. Appl., 238 (1999), 277-296.
• H. B. Dong, X. H. Gong, S. Y. Wang and L. Coladas, $S$-strictly quasi-concave vector maximisation, Bull. Austr. Math. Soc., 67 (2003), 429-443.
• J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, New York, John Wiley & Son, 1984.
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2019-11-12 12:21:42
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https://idaes-pse.readthedocs.io/en/1.4.4/core/control_volume_1d.html
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# 1D Control Volume Class¶
The ControlVolume1DBlock block is used for systems with one spatial dimension where material flows parallel to the spatial domain. Examples of these types of unit operations include plug flow reactors and pipes. ControlVolume1DBlock blocks are discretized along the length domain and contain one StateBlock and one ReactionBlock (if applicable) at each point in the domain (including the inlet and outlet).
class idaes.core.control_volume1d.ControlVolume1DBlock(*args, **kwargs)
ControlVolume1DBlock is a specialized Pyomo block for IDAES control volume blocks discretized in one spatial direction, and contains instances of ControlVolume1DBlockData.
ControlVolume1DBlock should be used for any control volume with a defined volume and distinct inlets and outlets where there is a single spatial domain parallel to the material flow direction. This encompases unit operations such as plug flow reactors and pipes.
Parameters: rule (function) – A rule function or None. Default rule calls build(). concrete (bool) – If True, make this a toplevel model. Default - False. ctype (str) – Pyomo ctype of the block. Default - “Block” default (dict) – Default ProcessBlockData config Keys dynamic Indicates whether this model will be dynamic, default - useDefault. Valid values: { useDefault - get flag from parent, True - set as a dynamic model, False - set as a steady-state model} has_holdup Indicates whether holdup terms should be constructed or not. Must be True if dynamic = True, default - False. Valid values: { True - construct holdup terms, False - do not construct holdup terms} property_package Property parameter object used to define property calculations, default - useDefault. Valid values: { useDefault - use default package from parent model or flowsheet, PropertyParameterObject - a PropertyParameterBlock object.} property_package_args A ConfigBlock with arguments to be passed to a property block(s) and used when constructing these, default - None. Valid values: { see property package for documentation.} reaction_package Reaction parameter object used to define reaction calculations, default - None. Valid values: { None - no reaction package, ReactionParameterBlock - a ReactionParameterBlock object.} reaction_package_args A ConfigBlock with arguments to be passed to a reaction block(s) and used when constructing these, default - None. Valid values: { see reaction package for documentation.} auto_construct If set to True, this argument will trigger the auto_construct method which will attempt to construct a set of material, energy and momentum balance equations based on the parent unit’s config block. The parent unit must have a config block which derives from CONFIG_Base, default - False. Valid values: { True - use automatic construction, False - do not use automatic construciton.} area_definition Argument defining whether area variable should be spatially variant or not. default - DistributedVars.uniform. Valid values: { DistributedVars.uniform - area does not vary across spatial domian, DistributedVars.variant - area can vary over the domain and is indexed by time and space.} transformation_method Method to use to transform domain. Must be a method recognised by the Pyomo TransformationFactory. transformation_scheme Scheme to use when transformating domain. See Pyomo documentation for supported schemes. finite_elements Number of finite elements to use in transformation (equivalent to Pyomo nfe argument). collocation_points Number of collocation points to use (equivalent to Pyomo ncp argument). initialize (dict) – ProcessBlockData config for individual elements. Keys are BlockData indexes and values are dictionaries described under the “default” argument above. idx_map (function) – Function to take the index of a BlockData element and return the index in the initialize dict from which to read arguments. This can be provided to overide the default behavior of matching the BlockData index exactly to the index in initialize. (ControlVolume1DBlock) New instance
class idaes.core.control_volume1d.ControlVolume1DBlockData(component)[source]
1-Dimensional ControlVolume Class
This class forms the core of all 1-D IDAES models. It provides methods to build property and reaction blocks, and add mass, energy and momentum balances. The form of the terms used in these constraints is specified in the chosen property package.
add_geometry(length_domain=None, length_domain_set=[0.0, 1.0], flow_direction=<FlowDirection.forward: 1>)[source]
Method to create spatial domain and volume Var in ControlVolume.
Parameters: - (length_domain_set) – domain for the ControlVolume. If not provided, a new ContinuousSet will be created (default=None). ContinuousSet should be normalized to run between 0 and 1. - – a new ContinuousSet if length_domain is not provided (default = [0.0, 1.0]). - argument indicating direction of material flow (flow_direction) – relative to length domain. Valid values: FlowDirection.forward (default), flow goes from 0 to 1. FlowDirection.backward, flow goes from 1 to 0 None
add_phase_component_balances(has_rate_reactions=False, has_equilibrium_reactions=False, has_phase_equilibrium=False, has_mass_transfer=False, custom_molar_term=None, custom_mass_term=None)[source]
This method constructs a set of 1D material balances indexed by time, length, phase and component.
Parameters: has_rate_reactions – whether default generation terms for rate reactions should be included in material balances has_equilibrium_reactions – whether generation terms should for chemical equilibrium reactions should be included in material balances has_phase_equilibrium – whether generation terms should for phase equilibrium behaviour should be included in material balances has_mass_transfer – whether generic mass transfer terms should be included in material balances custom_molar_term – a Pyomo Expression representing custom terms to be included in material balances on a molar basis. Expression must be indexed by time, length domain, phase list and component list custom_mass_term – a Pyomo Expression representing custom terms to be included in material balances on a mass basis. Expression must be indexed by time, length domain, phase list and component list Constraint object representing material balances
add_phase_energy_balances(*args, **kwargs)[source]
Method for adding energy balances (including kinetic energy) indexed by phase to the control volume.
See specific control volume documentation for details.
add_phase_enthalpy_balances(*args, **kwargs)[source]
Method for adding enthalpy balances indexed by phase to the control volume.
See specific control volume documentation for details.
add_phase_momentum_balances(*args, **kwargs)[source]
Method for adding momentum balances indexed by phase to the control volume.
See specific control volume documentation for details.
add_phase_pressure_balances(*args, **kwargs)[source]
Method for adding pressure balances indexed by phase to the control volume.
See specific control volume documentation for details.
add_reaction_blocks(has_equilibrium=None)[source]
This method constructs the reaction block for the control volume.
Parameters: has_equilibrium – indicates whether equilibrium calculations will be required in reaction block package_arguments – dict-like object of arguments to be passed to reaction block as construction arguments None
add_state_blocks(information_flow=<FlowDirection.forward: 1>, has_phase_equilibrium=None)[source]
This method constructs the state blocks for the control volume.
Parameters: information_flow – a FlowDirection Enum indicating whether information flows from inlet-to-outlet or outlet-to-inlet has_phase_equilibrium – indicates whether equilibrium calculations will be required in state blocks package_arguments – dict-like object of arguments to be passed to state blocks as construction arguments None
add_total_component_balances(has_rate_reactions=False, has_equilibrium_reactions=False, has_phase_equilibrium=False, has_mass_transfer=False, custom_molar_term=None, custom_mass_term=None)[source]
This method constructs a set of 1D material balances indexed by time length and component.
Parameters: has_rate_reactions – whether default generation terms for rate reactions should be included in material balances has_equilibrium_reactions – whether generation terms should for chemical equilibrium reactions should be included in material balances has_phase_equilibrium – whether generation terms should for phase equilibrium behaviour should be included in material balances has_mass_transfer – whether generic mass transfer terms should be included in material balances custom_molar_term – a Pyomo Expression representing custom terms to be included in material balances on a molar basis. Expression must be indexed by time, length domain and component list custom_mass_term – a Pyomo Expression representing custom terms to be included in material balances on a mass basis. Expression must be indexed by time, length domain and component list Constraint object representing material balances
add_total_element_balances(has_rate_reactions=False, has_equilibrium_reactions=False, has_phase_equilibrium=False, has_mass_transfer=False, custom_elemental_term=None)[source]
This method constructs a set of 1D element balances indexed by time and length.
Parameters: - whether default generation terms for rate (has_rate_reactions) – reactions should be included in material balances - whether generation terms should for (has_equilibrium_reactions) – chemical equilibrium reactions should be included in material balances - whether generation terms should for phase (has_phase_equilibrium) – equilibrium behaviour should be included in material balances - whether generic mass transfer terms should be (has_mass_transfer) – included in material balances - a Pyomo Expression representing custom (custom_elemental_term) – terms to be included in material balances on a molar elemental basis. Expression must be indexed by time, length and element list Constraint object representing material balances
add_total_energy_balances(*args, **kwargs)[source]
Method for adding a total energy balance (including kinetic energy) to the control volume.
See specific control volume documentation for details.
add_total_enthalpy_balances(has_heat_of_reaction=False, has_heat_transfer=False, has_work_transfer=False, custom_term=None)[source]
This method constructs a set of 1D enthalpy balances indexed by time and phase.
Parameters: - whether terms for heat of reaction should (has_heat_of_reaction) – be included in enthalpy balance - whether terms for heat transfer should be (has_heat_transfer) – included in enthalpy balances - whether terms for work transfer should be (has_work_transfer) – included in enthalpy balances - a Pyomo Expression representing custom terms to (custom_term) – be included in enthalpy balances. Expression must be indexed by time, length and phase list Constraint object representing enthalpy balances
add_total_material_balances(*args, **kwargs)[source]
Method for adding a total material balance to the control volume.
See specific control volume documentation for details.
add_total_momentum_balances(*args, **kwargs)[source]
Method for adding a total momentum balance to the control volume.
See specific control volume documentation for details.
add_total_pressure_balances(has_pressure_change=False, custom_term=None)[source]
This method constructs a set of 1D pressure balances indexed by time.
Parameters: - whether terms for pressure change should be (has_pressure_change) – included in enthalpy balances - a Pyomo Expression representing custom terms to (custom_term) – be included in pressure balances. Expression must be indexed by time and length domain Constraint object representing pressure balances
apply_transformation()[source]
Method to apply DAE transformation to the Control Volume length domain. Transformation applied will be based on the Control Volume configuration arguments.
build()[source]
Build method for ControlVolume1DBlock blocks.
Returns: None
initialize(state_args=None, outlvl=6, optarg=None, solver='ipopt', hold_state=True)[source]
Initialization routine for 1D control volume (default solver ipopt)
Keyword Arguments:
• state_args – a dict of arguments to be passed to the property package(s) to provide an initial state for initialization (see documentation of the specific property package) (default = {}).
• outlvl – sets output level of initialization routine * 0 = Use default idaes.init logger setting * 1 = Maximum output * 2 = Include solver output * 3 = Return solver state for each step in subroutines * 4 = Return solver state for each step in routine * 5 = Final initialization status and exceptions * 6 = No output
• optarg – solver options dictionary object (default=None)
• solver – str indicating whcih solver to use during initialization (default = ‘ipopt’)
• hold_state – flag indicating whether the initialization routine should unfix any state variables fixed during initialization, default - True. Valid values: True - states variables are not unfixed, and a dict of returned containing flags for which states were fixed during initialization, False - state variables are unfixed after initialization by calling the release_state method.
Returns:
If hold_states is True, returns a dict containing flags for which states were fixed during initialization else the release state is triggered.
model_check()[source]
This method executes the model_check methods on the associated state blocks (if they exist). This method is generally called by a unit model as part of the unit’s model_check method.
Parameters: None – None
release_state(flags, outlvl=6)[source]
Method to release state variables fixed during initialization.
Keyword Arguments:
• flags – dict containing information of which state variables were fixed during initialization, and should now be unfixed. This dict is returned by initialize if hold_state = True.
• outlvl – sets output level of logging
Returns:
None
report(time_point=0, dof=False, ostream=None, prefix='')[source]
No report method defined for ControlVolume1D class. This is due to the difficulty of presenting spatially discretized data in a readable form without plotting.
## ControlVolume1DBlock Equations¶
This section documents the variables and constraints created by each of the methods provided by the ControlVolume0DBlock class.
• $$t$$ indicates time index
• $$x$$ indicates spatial (length) index
• $$p$$ indicates phase index
• $$j$$ indicates component index
• $$e$$ indicates element index
• $$r$$ indicates reaction name index
Most terms within the balance equations written by ControlVolume1DBlock are on a basis of per unit length (e.g. $$mol/m \cdot s$$).
The add_geometry method creates the normalized length domain for the control volume (or a reference to an external domain). All constraints in ControlVolume1DBlock assume a normalized length domain, with values between 0 and 1.
This method also adds variables and constraints to describe the geometry of the control volume. ControlVolume1DBlock does not support varying dimensions of the control volume with time at this stage.
Variables
Variable Name Symbol Indices Conditions
length_domain $$x$$ None None
volume $$V$$ None None
area $$A$$ None None
length $$L$$ None None
Constraints
geometry_constraint:
$V = A \times L$
Material balances are written for each component in each phase (e.g. separate balances for liquid water and steam). Physical property packages may include information to indicate that certain species do not appear in all phases, and material balances will not be written in these cases (if has_holdup is True holdup terms will still appear for these species, however these will be set to 0).
Variables
Variable Name Symbol Indices Conditions
material_holdup $$M_{t,x,p,j}$$ t, x, p, j has_holdup = True
phase_fraction $$\phi_{t,x,p}$$ t, x, p has_holdup = True
material_accumulation $$\frac{\partial M_{t,x,p,j}}{\partial t}$$ t, x, p, j dynamic = True
_flow_terms $$F_{t, x, p, j}$$ t, x, p, j None
material_flow_dx $$\frac{\partial F_{t,x,p,j}}{\partial x}$$ t, x, p, j None
rate_reaction_generation $$N_{kinetic,t,x,p,j}$$ t, x, p ,j has_rate_reactions = True
rate_reaction_extent $$X_{kinetic,t,x,r}$$ t, x, r has_rate_reactions = True
equilibrium_reaction_generation $$N_{equilibrium,t,x,p,j}$$ t, x, p ,j has_equilibrium_reactions = True
equilibrium_reaction_extent $$X_{equilibrium,t,x,r}$$ t, x, r has_equilibrium_reactions = True
phase_equilibrium_generation $$N_{pe,t,x,p,j}$$ t, x, p ,j has_phase_equilibrium = True
mass_transfer_term $$N_{transfer,t,x,p,j}$$ t, x, p ,j has_mass_transfer = True
Constraints
material_balances(t, x, p, j):
$L \times \frac{\partial M_{t, x, p, j}}{\partial t} = fd \times \frac{\partial F_{t, x, p, j}}{\partial x} + L \times N_{kinetic, t, x, p, j} + L \times N_{equilibrium, t, x, p, j} + L \times N_{pe, t, x, p, j} + L \times N_{transfer, t, x, p, j} + L \times N_{custom, t, x, p, j}$
$$fd$$ is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, $$fd = -1$$, otherwise $$fd = 1$$.
The $$N_{custom, t, x, p, j}$$ term allows the user to provide custom terms (variables or expressions) in both mass and molar basis which will be added into the material balances, which will be converted as necessary to the same basis as the material balance (by multiplying or dividing by the component molecular weight). The basis of the material balance is determined by the physical property package, and if undefined (or not mass or mole basis), an Exception will be returned.
This constraint is an internal constraint used to link the extensive material flow terms in the StateBlocks into a single indexed variable. This is required as Pyomo.DAE requires a single indexed variable to create the associated DerivativeVars and their numerical expansions.
If has_holdup is True, material_holdup_calculation(t, x, p, j):
$M_{t, x, p, j} = \rho_{t, x, p, j} \times A \times \phi_{t, x, p}$
where $$\rho_{t, x, p ,j}$$ is the density of component $$j$$ in phase $$p$$ at time $$t$$ and location $$x$$.
If dynamic is True:
Numerical discretization of the derivative terms, $$\frac{\partial M_{t,x,p,j}}{\partial t}$$, will be performed by Pyomo.DAE.
If has_rate_reactions is True, rate_reaction_stoichiometry_constraint(t, x, p, j):
$N_{kinetic, t, x, p, j} = \alpha_{r, p, j} \times X_{kinetic, t, x, r}$
where $$\alpha_{r, p. j}$$ is the stoichiometric coefficient of component $$j$$ in phase $$p$$ for reaction $$r$$ (as defined in the PhysicalParameterBlock).
If has_equilibrium_reactions argument is True, equilibrium_reaction_stoichiometry_constraint(t, x, p, j):
$N_{equilibrium, t, x, p, j} = \alpha_{r, p, j} \times X_{equilibrium, t, x, r}$
where $$\alpha_{r, p. j}$$ is the stoichiometric coefficient of component $$j$$ in phase $$p$$ for reaction $$r$$ (as defined in the PhysicalParameterBlock).
Material balances are written for each component across all phases (e.g. one balance for both liquid water and steam). Physical property packages may include information to indicate that certain species do not appear in all phases, and material balances will not be written in these cases (if has_holdup is True holdup terms will still appear for these species, however these will be set to 0).
Variables
Variable Name Symbol Indices Conditions
material_holdup $$M_{t,x,p,j}$$ t, x, p, j has_holdup = True
phase_fraction $$\phi_{t,x,p}$$ t, x, p has_holdup = True
material_accumulation $$\frac{\partial M_{t,x,p,j}}{\partial t}$$ t, x, p, j dynamic = True
_flow_terms $$F_{t, x, p, j}$$ t, x, p, j None
material_flow_dx $$\frac{\partial F_{t,x,p,j}}{\partial x}$$ t, x, p, j None
rate_reaction_generation $$N_{kinetic,t,x,p,j}$$ t, x, p ,j has_rate_reactions = True
rate_reaction_extent $$X_{kinetic,t,x,r}$$ t, x, r has_rate_reactions = True
equilibrium_reaction_generation $$N_{equilibrium,t,x,p,j}$$ t, x, p ,j has_equilibrium_reactions = True
equilibrium_reaction_extent $$X_{equilibrium,t,x,r}$$ t, x, r has_equilibrium_reactions = True
mass_transfer_term $$N_{transfer,t,x,p,j}$$ t, x, p ,j has_mass_transfer = True
Constraints
material_balances(t, x, p, j):
$L \times \sum_p{\frac{\partial M_{t, x, p, j}}{\partial t}} = fd \times \sum{\frac{\partial F_{t, x, p, j}}{\partial x}} + L \times \sum_p{N_{kinetic, t, x, p, j}} + L \times \sum_p{N_{equilibrium, t, x, p, j}} + L \times \sum_p{N_{transfer, t, x, p, j}} + L \times N_{custom, t, x, j}$
$$fd$$ is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, $$fd = -1$$, otherwise $$fd = 1$$.
The $$N_{custom, t, x, j}$$ term allows the user to provide custom terms (variables or expressions) in both mass and molar basis which will be added into the material balances, which will be converted as necessary to the same basis as the material balance (by multiplying or dividing by the component molecular weight). The basis of the material balance is determined by the physical property package, and if undefined (or not mass or mole basis), an Exception will be returned.
This constraint is an internal constraint used to link the extensive material flow terms in the StateBlocks into a single indexed variable. This is required as Pyomo.DAE requires a single indexed variable to create the associated DerivativeVars and their numerical expansions.
If has_holdup is True, material_holdup_calculation(t, x, p, j):
$M_{t, x, p, j} = \rho_{t, x, p, j} \times A \times \phi_{t, x, p}$
where $$\rho_{t, x, p ,j}$$ is the density of component $$j$$ in phase $$p$$ at time $$t$$ and location $$x$$.
If dynamic is True:
Numerical discretization of the derivative terms, $$\frac{\partial M_{t,x,p,j}}{\partial t}$$, will be performed by Pyomo.DAE.
If has_rate_reactions is True, rate_reaction_stoichiometry_constraint(t, x, p, j):
$N_{kinetic, t, x, p, j} = \alpha_{r, p, j} \times X_{kinetic, t, x, r}$
where $$\alpha_{r, p. j}$$ is the stoichiometric coefficient of component $$j$$ in phase $$p$$ for reaction $$r$$ (as defined in the PhysicalParameterBlock).
If has_equilibrium_reactions argument is True, equilibrium_reaction_stoichiometry_constraint(t, x, p, j):
$N_{equilibrium, t, x, p, j} = \alpha_{r, p, j} \times X_{equilibrium, t, x, r}$
where $$\alpha_{r, p. j}$$ is the stoichiometric coefficient of component $$j$$ in phase $$p$$ for reaction $$r$$ (as defined in the PhysicalParameterBlock).
Material balances are written for each element in the mixture.
Variables
Variable Name Symbol Indices Conditions
element_holdup $$M_{t,x,e}$$ t, x, e has_holdup = True
phase_fraction $$\phi_{t,x,p}$$ t, x, p has_holdup = True
element_accumulation $$\frac{\partial M_{t,x,e}}{\partial t}$$ t, x, e dynamic = True
elemental_mass_transfer_term $$N_{transfer,t,x,e}$$ t, x, e has_mass_transfer = True
elemental_flow_term $$F_{t,x,e}$$ t, x, e None
Constraints
elemental_flow_constraint(t, x, e):
$F_{t,x,e} = \sum_p{\sum_j{F_{t,x,p,j} \times n_{j, e}}}$
where $$n_{j, e}$$ is the number of moles of element $$e$$ in component $$j$$.
element_balances(t, x, e):
$L \times \frac{\partial M_{t, x, e}}{\partial t} = fd \times \frac{\partial F_{t, x, e}}{\partial x} + L \times N_{transfer, t, p, j} + L \times N_{custom, t, e}$
$$fd$$ is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, $$fd = -1$$, otherwise $$fd = 1$$.
The $$N_{custom, t, x, e}$$ term allows the user to provide custom terms (variables or expressions) which will be added into the material balances.
If has_holdup is True, elemental_holdup_calculation(t, x, e):
$M_{t, x, e} = \rho_{t, x, p, j} \times A \times \phi_{t, x, p}$
where $$\rho_{t, x, p ,j}$$ is the density of component $$j$$ in phase $$p$$ at time $$t$$ and location $$x$$.
If dynamic is True:
Numerical discretization of the derivative terms, $$\frac{\partial M_{t,x,p,j}}{\partial t}$$, will be performed by Pyomo.DAE.
A single enthalpy balance is written for the entire mixture at each point in the spatial domain.
Variables
Variable Name Symbol Indices Conditions
enthalpy_holdup $$E_{t,x,p}$$ t, x, p has_holdup = True
phase_fraction $$\phi_{t,x,p}$$ t, x, p has_holdup = True
enthalpy_accumulation $$\frac{\partial E_{t,x,p}}{\partial t}$$ t, x, p dynamic = True
_enthalpy_flow $$H_{t,x,p}$$ t, x, p None
enthalpy_flow_dx $$\frac{\partial H_{t,x,p}}{\partial x}$$ t, x, p None
heat $$Q_{t,x}$$ t, x has_heat_transfer = True
work $$W_{t,x}$$ t, x has_work_transfer = True
Expressions
heat_of_reaction(t, x):
$Q_{rxn, t, x} = sum_r{X_{kinetic, t, x, r} \times \Delta H_{rxn, r}} + sum_r{X_{equilibrium, t, x, r} \times \Delta H_{rxn, r}}$
where $$Q_{rxn, t, x}$$ is the total enthalpy released by both kinetic and equilibrium reactions, and $$\Delta H_{rxn, r}$$ is the specific heat of reaction for reaction $$r$$.
Parameters
Parameter Name Symbol Default Value
scaling_factor_energy $$s_{energy}$$ 1E-6
Constraints
enthalpy_balance(t):
$s_{energy} \times L \times \sum_p{\frac{\partial E_{t, x, p}}{\partial t}} = s_{energy} \times fd \ times \sum_p{\frac{\partial H_{t, x, p}}{\partial x}} + s_{energy} \times L \times Q_{t,x} + s_{energy} \times L \times W_{t,x} + s_{energy} \times L \times Q_{rxn, t, x} + s_{energy} \times L \times E_{custom, t, x}$
$$fd$$ is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, $$fd = -1$$, otherwise $$fd = 1$$.
The $$E_{custom, t, x}$$ term allows the user to provide custom terms which will be added into the energy balance.
This constraint is an internal constraint used to link the extensive enthalpy flow terms in the StateBlocks into a single indexed variable. This is required as Pyomo.DAE requires a single indexed variable to create the associated DerivativeVars and their numerical expansions.
If has_holdup is True, enthalpy_holdup_calculation(t, x, p):
$E_{t, x, p} = h_{t, x, p} \times A \times \phi_{t, x, p}$
where $$h_{t, x, p}$$ is the enthalpy density (specific enthalpy) of phase $$p$$ at time $$t$$ and location $$x$$.
If dynamic is True:
Numerical discretization of the derivative terms, $$\frac{\partial E_{t,x,p}}{\partial t}$$, will be performed by Pyomo.DAE.
A single pressure balance is written for the entire mixture at all points in the spatial domain.
Variables
Variable Name Symbol Indices Conditions
pressure $$P_{t,x}$$ t, x None
pressure_dx $$\frac{\partial P_{t,x}}{\partial x}$$ t, x None
deltaP $$\Delta P_{t,x}$$ t, x has_pressure_change = True
Parameters
Parameter Name Symbol Default Value
scaling_factor_pressure $$s_{pressure}$$ 1E-4
Constraints
pressure_balance(t, x):
$0 = s_{pressure} \times fd \times \frac{\partial P_{t,x}}{\partial x} + s_{pressure} \times L \times \Delta P_{t,x} + s_{pressure} \times L \times \Delta P_{custom, t, x}$
$$fd$$ is a flow direction term, which allows for material flow to be defined in either direction. If material flow is defined as forward, $$fd = -1$$, otherwise $$fd = 1$$.
The $$\Delta P_{custom, t, x}$$ term allows the user to provide custom terms which will be added into the pressure balance.
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2022-01-24 00:47:16
|
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|
http://codereview.stackexchange.com/questions/18471/sanitizing-function
|
# Sanitizing Function
I have a small function for filtering data from users. I would be grateful if someone could show me what logic mistakes I made and if I'm somewhat protected using it. And sorry for bad english i'm using gtranslate.
public static function filtru($str,$filtre = []){
if(is_array($str)){//verifica daca datele trimise spre curatare sant array$str_curat = [];
foreach($str as$key => $value){//selectam fiecare valoare din array-ul trimis spre curatare foreach($filtre as $fitru){//aplicam fiecare filtru, daca se folosesc mai multe switch($fitru){
case 'trim':
$value = htmlentities(trim(strip_tags($value)), ENT_QUOTES, 'utf-8');
break;
case 'int':
$value = intval($value);
break;
case 'nu':
$text =$value;$striptags = true;$search = ["40","41","58","65","66","67","68","69","70",
"71","72","73","74","75","76","77","78","79","80","81",
"82","83","84","85","86","87","88","89","90","97","98",
"99","100","101","102","103","104","105","106","107",
"108","109","110","111","112","113","114","115","116",
"117","118","119","120","121","122"
];
$replace = ["(",")",":","a","b","c","d","e","f","g","h", "i","j","k","l","m","n","o","p","q","r","s","t","u", "v","w","x","y","z","a","b","c","d","e","f","g","h", "i","j","k","l","m","n","o","p","q","r","s","t","u", "v","w","x","y","z" ];$entities = count($search); for ($i=0; $i <$entities; $i++) {$text = preg_replace("#(&\#)(0*".$search[$i]."+);*#si", $replace[$i], $text); }$text = preg_replace('#(&\#x)([0-9A-F]+);*#si', "", $text);$text = preg_replace('#(<[^>]+[/\"\'\s])(onmouseover|onmousedown|onmouseup|onmouseout|onmousemove|onclick|ondblclick|onfocus|onload|xmlns)[^>]*>#iU', ">", $text);$text = preg_replace('#([a-z]*)=([\\'\"]*)script:#iU', '$1=$2nojscript...', $text);$text = preg_replace('#([a-z]*)=([\\'\"]*)javascript:#iU', '$1=$2nojavascript...', $text);$text = preg_replace('#([a-z]*)=([\'\"]*)vbscript:#iU', '$1=$2novbscript...', $text);$text = preg_replace('#(<[^>]+)style=([\\'\"]*).*expression$$[^>]*>#iU', "1>", text); text = preg_replace('#(<[^>]+)style=([\\'\"]*).*behaviour\([^>]*>#iU', "1>", text); if (striptags) { do { thistext = text; text = preg_replace('#</*(applet|meta|xml|blink|link|style|script|embed|object|iframe|frame|frameset|ilayer|layer|bgsound|title|base|body)[^>]*>#i', "", text); } while (thistext != text); } value = text; value = preg_replace('#(alert|cmd|passthru|eval|exec|expression|system|fopen|fsockopen|file|file_get_contents|file_put_contents|readfile|unlink|shell_exec)(\s*)\((.*?)$$#si', "\\1\\2(\\3)", $value); break; default:$value = $fitru($value);
}// switch
}//bucla pt fiecare filtru
$str_curat[$key] = $value; }//bucla pt fiecare valoare }else{ foreach($filtre as $fitru){ switch($fitru){
case 'trim':
$str = htmlentities(trim(strip_tags($str)), ENT_QUOTES, 'utf-8');
break;
case 'int':
$str = intval($str);
break;
case 'nu':
$text =$str;$striptags = true;$search = ["40","41","58","65","66","67","68","69","70",
"71","72","73","74","75","76","77","78","79","80","81",
"82","83","84","85","86","87","88","89","90","97","98",
"99","100","101","102","103","104","105","106","107",
"108","109","110","111","112","113","114","115","116",
"117","118","119","120","121","122"
];
$replace = ["(",")",":","a","b","c","d","e","f","g","h", "i","j","k","l","m","n","o","p","q","r","s","t","u", "v","w","x","y","z","a","b","c","d","e","f","g","h", "i","j","k","l","m","n","o","p","q","r","s","t","u", "v","w","x","y","z" ];$entities = count($search); for ($i=0; $i <$entities; $i++) {$text = preg_replace("#(&\#)(0*".$search[$i]."+);*#si", $replace[$i], $text); }$text = preg_replace('#(&\#x)([0-9A-F]+);*#si', "", $text);$text = preg_replace('#(<[^>]+[/\"\'\s])(onmouseover|onmousedown|onmouseup|onmouseout|onmousemove|onclick|ondblclick|onfocus|onload|xmlns)[^>]*>#iU', ">", $text);$text = preg_replace('#([a-z]*)=([\\'\"]*)script:#iU', '$1=$2nojscript...', $text);$text = preg_replace('#([a-z]*)=([\\'\"]*)javascript:#iU', '$1=$2nojavascript...', $text);$text = preg_replace('#([a-z]*)=([\'\"]*)vbscript:#iU', '$1=$2novbscript...', $text);$text = preg_replace('#(<[^>]+)style=([\\'\"]*).*expression$$[^>]*>#iU', "1>", text); text = preg_replace('#(<[^>]+)style=([\\'\"]*).*behaviour\([^>]*>#iU', "1>", text); if (striptags) { do { thistext = text; text = preg_replace('#</*(applet|meta|xml|blink|link|style|script|embed|object|iframe|frame|frameset|ilayer|layer|bgsound|title|base|body)[^>]*>#i', "", text); } while (thistext != text); } str = text; str = preg_replace('#(alert|cmd|passthru|eval|exec|expression|system|fopen|fsockopen|file|file_get_contents|file_put_contents|readfile|unlink|shell_exec)(\s*)\((.*?)$$#si', "\\1\\2(\\3)", $str); break; default:$str = $fitru($value);
}// switch
}//foreach pt filtre
$str_curat =$str;
}
return $str_curat; }//end filtru LE: If i try to separate the logic into 2 functions like this it's better? Each applied filter will be a separate function. /** * @param$str
* @param array $filtre * @return array|sanitized */ public static function filtru($str, $filtre = []){ if(is_array($str)){
for($i = 0,$j = count($str)-1;$i <= $j;$i++){
$str[$i] = self::filtre($str[$i], $filtre); } }else{$str = self::filtre($str,$filtre);
}
return $str; } /** * @param$str
* @param array $filtre * @return sanitized str */ private static function filtre($str, $filtre = []){ foreach($filtre as $filtru){ if(in_array($filtru, self::$filtre_str)){$str = call_user_func(['clean',$filtru],$str);
}else{
die('Function '.$filtru.' it's not defiend.'); } } return$str;
}
-
This function isn't small; by modern standards, it's huge and should be split up. – codesparkle Nov 11 '12 at 23:12
• Your function should be broken into many more functions. Each different type of escaping should be its own function.
• You should either use separate functions where the array version calls the non-array version, or youo should try to fold your array and non array logic into one
• $wasArray = is_array($str); if (!$wasArray) {$str = array($str); } foreach ($str as $s) { ... } return ($wasArray) ? $str : array_shift($str);
• Your trim method is a lie -- it doesn't just trim. What if you want to trim without html escaping?
• If you did leave them all in one method, consider using class constants instead of strings to control the behavior. That's a lot less error prone, and it makes it easier to trace usage through code (and if you rename something later, it will be a million times easier).
• What is your nu sanitation doing? It's like it's processing HTML, PHP and JavaScript?
• With HTML, you're typically better off with a whitelist than trying to sanitize, and then escaping everything else.
• With PHP, you're typically better off just not running code that you don't 100% trust
• JS falls into the HTML category. If HTML is whitelisted, it should be impossible for a user to get JS through.
You should consider if you really need all of this. In a lot of situations, it's sufficient to simply escape and not sanitize (though one could argue that escape is a form of sanitizing).
(There are of course situations where you really do have to process the hell out of user input though. Consider the StackExchange questions/answers/comments. They allow a very limited subset of HTML which means that their processor has to be aware of what to escape, what to leave alone.)
-
I will remove each filter as a separate function, i avoid using constants knowing that they are slower. Nice explanation with array_shift i'll use it.I need to rethink all logic. Thank you very much for response – danutz0501 Nov 12 '12 at 10:43
## DRY - Write less handle more
You can make your function half the size by designing it to work with scalar vars only, and handling arrays with a loop:
public static function filtru($var) { if (is_array($var)) {
foreach ($var as &$val) {
$val = static::filtru($val);
}
return $var; } //$var is not an array - it is number/string etc.
...
return $var; } In the above example, an array, a scalar or even a nested array can all be handled by the same function. ## Use PHP's built in filters Rather than write your own filters, you're better off using PHP's sanitization filters. For common cases, your code becomes therefore (assuming this is implemented): $arrayIntOut = Foo::filtru($arrayMixedIn, FILTER_SANITIZE_NUMBER_INT); But, especially with the above example, it's more work to use the filtru function than it is to just write: $arrayIntOut = array_map('intval', $arrayMixedIn); Putting all your logic in one place is not necessarily the best idea, when the alternative is to write concise (and more efficient) code using php's function directly in functions. As such, consider 'just' using PHP instead of wrapping php functions in some more logic. ## Make "nu" a separate function Each case you want to handle should really be a protected specific function. Once you do that, consider deleting any functions that are just one line of code as it's better to just use php's own functions. The 'nu' function however, is likely to warrants it's own function. It looks like the main purpose is to strip out XSS attacks. Unless you have exhaustive tests for this logic, it is very likely it doesn't cater for all possibilities - therefore make it a separate function and if you haven't already it'd be in your interest to write some tests to at least cover the most common XSS attacks. - that's a really long list :D and it's just the common ones – danutz0501 Nov 12 '12 at 11:04 actually it's probably quite complete being owasp, but there can never truly be a complete xss list as new attacks are found/invented as time goes by. – AD7six Nov 12 '12 at 11:47 add comment • First, break your function into pieces. One function should be doing one thing and correctly. • Try to give functions, variables a meaningful name. Here the function name "filteru" doesn't make any sense. same goes for "$str_curat".
• you have huge if block
if (is_array($str)) { // lot of code... } Instead you can write a simple check at the beginning of the function if (!is_array($str)) { return; }
// rest of the code goes here...
This way you can avoid a huge if block.
• At the end of the curly braces you have comments like
}// switch
}//foreach pt filtre
}//end filtru
These are useless.
• you have two foreach blocks. one foreach inside another foreach
foreach($str as$key => $value) { foreach($filtre as \$fitru) {
// code ...
}
}
Try rewriting the logic so that you can avoid this nested loop.
-
The variable names and comments do make sense in romanian. – Diego Agulló Nov 12 '12 at 7:18
Then It is fine if the variable names make sense in romanian. For the comments, what I wanted to say is that it is useless to comment the end of curly braces the way it is done here - "} //switch", "}//foreach pt filtre" etc – Kinjal Nov 12 '12 at 7:30
I used a poorly text editor when i made the function. It didn't had matching curly braces. I forgot to remove comments I'll rethink the function(logic). str_curat = clean string – danutz0501 Nov 12 '12 at 10:50
|
2014-04-25 04:37:41
|
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|
http://openstudy.com/updates/4dbb5074b0ab8b0bdec17f8b
|
## anonymous 5 years ago The mass of Saturn is about 5.688*10^23 metric tons. The mass of the sun is about 1.998*10^27 metric tons. About how many times the mass of Saturn is the mass of the sun? give the answer in scientific notation.
1. anonymous
$(1.988 \times 10 ^{27} )\div (5.688 \times 10^{23}) = 3.495.077356...$ Estimating with scientific notation (4 places) will be 3.495.
2. anonymous
woops. I mean it equals 3495.077356. I will be 3495
|
2016-10-26 23:13:29
|
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|
https://dataspace.princeton.edu/jspui/handle/88435/dsp013j333466d
|
Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp013j333466d
Title: A Search in the Muon Channel for Heavy Resonances Decaying to Long-Lived Neutral Particles Authors: Hunt, Adam Paul Advisors: Tully, Christopher Contributors: Physics Department Keywords: CMSDisplaced muon Subjects: Physics Issue Date: 2016 Publisher: Princeton, NJ : Princeton University Abstract: A search is performed for a heavy resonance decaying to two long-lived massive neutral particles that each decay to a pair of muons. The process is detected experimentally via a distinct topological signature consisting of a pair of oppositely charged leptons originating at a vertex significantly displaced from the LHC beam spot. Events were collected by the CMS detector at the LHC during pp collisions at $\sqrt{s} =$ 7~TeV, and selected from data samples corresponding to 5.1~fb$^{-1}$ of integrated luminosity in the muon channel. No significant excess is observed above standard model expectations, and an upper limit is set with 95\%~confidence level on the production cross section times the branching fraction to dimuons, as a function of the long-lived massive neutral particle lifetime. URI: http://arks.princeton.edu/ark:/88435/dsp013j333466d Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog: http://catalog.princeton.edu/ Type of Material: Academic dissertations (Ph.D.) Language: en Appears in Collections: Physics
|
2019-10-20 11:33:19
|
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|
https://engineering-math.org/tag/test-4/
|
## College Mathematics for Aviation II: Test 4
The following questions are taken from the Embry-Riddle Aeronautical University Worldwide Test 4 questions from MyMathLab.
Questions:
1. Evaluate the given definite integral.
##### $\displaystyle \int _0^1\:\left(2x^4-7x^2\right)dx$
2. Find the area of the region bounded by the graphs of the given equations.
##### $\displaystyle y=x,\:y=\sqrt[4]{x}$
3. Approximate the value of the integral defined by the given set of points.
##### $\displaystyle \int _2^8\:ydx$
4. Find the area of the region enclosed between the two curves
$\displaystyle y=5-x^2\:\:\:and\:\:y=x-7$
5. Evaluate the given definite integral.
$\displaystyle \int _0^{2\sqrt{2}}\:\frac{x}{\sqrt{x^2+1}}dx$
6. Approximate the value of the integral by use of the trapezoidal rule, using n=8.
$\displaystyle \int _0^{10}\:\sqrt{100-x^2}dx$
7. What is the velocity (in ft/s) of a sandbag 2.25 s after it is released from a hot-air balloon that is stationary in the air?
|
2020-05-29 19:59:25
|
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|
http://www.aimsciences.org/article/doi/10.3934/dcdsb.2019080?viewType=html
|
# American Institute of Mathematical Sciences
• Previous Article
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Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge
## Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function
1 Faculty of Physical Sciences, University of Iceland, 107 Reykjavik, Iceland 2 Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom 3 Svensk Exportkredit, Klarabergsviadukten 61-63, 111 64 Stockholm, Sweden
Received April 2018 Revised October 2018 Published April 2019
Fund Project: The research for this paper was supported by the Icelandic Research Fund (Rannís) in the project `Lyapunov Methods and Stochastic Stability' (152429-051), which is gratefully acknowledged
The γ-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-basin of attraction of the equilibrium.
Citation: Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019080
##### References:
[1] H. Björnsson, P. Giesl, S. Gudmundsson and S. Hafstein, Local Lyapunov functions for nonlinear stochastic differential equations by linearization, In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, 2018,579–586, . [2] M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543241. [3] F. Camilli and L. Grüne, Characterizing attraction probabilities via the stochastic Zubov equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 457-468. doi: 10.3934/dcdsb.2003.3.457. [4] P. Giesl, Construction of Global Lyapunov functions using Radial Basis Functions, volume 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007. [5] P. Giesl and S. Hafstein, Review of computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291. [6] P. Giesl and N. Mohammed, Verification estimates for the construction of Lyapunov functions using meshfree collocation, Discrete Contin. Dyn. Syst. Ser. B, in press. [7] P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 92007), 1723–1741. doi: 10.1137/060658813. [8] S. Gudmundsson and S. Hafstein, Probabilistic basin of attraction and its estimation using two Lyapunov functions, Complexity, 2018 (2018), Article ID 2895658, 9 pages. doi: 10.1155/2018/2895658. [9] S. Hafstein, S. Gudmundsson, P. Giesl and E. Scalas, Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 2 (2018), 939-956. doi: 10.3934/dcdsb.2018049. [10] N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, PhD thesis, University of Sussex, 2016. [11] M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances in Numerical Analysis, Vol. Ⅱ (Lancaster, 1990), Oxford Sci. Publ., pages 105-210. Oxford Univ. Press, New York, 1992. [12] R. Schaback and H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639. doi: 10.1017/S0962492906270016. [13] H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137. [14] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
show all references
##### References:
[1] H. Björnsson, P. Giesl, S. Gudmundsson and S. Hafstein, Local Lyapunov functions for nonlinear stochastic differential equations by linearization, In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, 2018,579–586, . [2] M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543241. [3] F. Camilli and L. Grüne, Characterizing attraction probabilities via the stochastic Zubov equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 457-468. doi: 10.3934/dcdsb.2003.3.457. [4] P. Giesl, Construction of Global Lyapunov functions using Radial Basis Functions, volume 1904 of Lecture Notes in Mathematics, Springer, Berlin, 2007. [5] P. Giesl and S. Hafstein, Review of computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291. [6] P. Giesl and N. Mohammed, Verification estimates for the construction of Lyapunov functions using meshfree collocation, Discrete Contin. Dyn. Syst. Ser. B, in press. [7] P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 92007), 1723–1741. doi: 10.1137/060658813. [8] S. Gudmundsson and S. Hafstein, Probabilistic basin of attraction and its estimation using two Lyapunov functions, Complexity, 2018 (2018), Article ID 2895658, 9 pages. doi: 10.1155/2018/2895658. [9] S. Hafstein, S. Gudmundsson, P. Giesl and E. Scalas, Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming, Discrete Contin. Dyn. Syst. Ser. B, 2 (2018), 939-956. doi: 10.3934/dcdsb.2018049. [10] N. Mohammed, Grid Refinement and Verification Estimates for the RBF Construction Method of Lyapunov Functions, PhD thesis, University of Sussex, 2016. [11] M. J. D. Powell, The theory of radial basis function approximation in 1990, In Advances in Numerical Analysis, Vol. Ⅱ (Lancaster, 1990), Oxford Sci. Publ., pages 105-210. Oxford Univ. Press, New York, 1992. [12] R. Schaback and H. Wendland, Kernel techniques: From machine learning to meshless methods, Acta Numer., 15 (2006), 543-639. doi: 10.1017/S0962492906270016. [13] H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137. [14] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2005.
Above: the computed non-local Lyapunov function $v$ for system (5.1). Below: the function $Lv$, approximating $-10^{-3}$
Non-local Lyapunov function for system (5.2) with $\theta = 1$. The non-local Lyapunov functions looks very similar to the one computed in [3]
The table shows the Wendland function $\psi_0(r): = \phi_{8, 6}(cr)$ as well as the related functions $\psi_1$ to $\psi_6$, defined recursively by $\psi_{k+1}(r): = \frac{\partial_r \psi_k(r)}{r}$ for $k = 0, 1, \ldots, 5$
$\phi_{8, 6}$ $\psi_0(r)$ $[46,189(cr)^6+73,206 (cr)^5+54,915(cr)^4+24,500(cr)^3$ $+6,755(cr)^2+1,078cr+77]\, (1- cr)^{14}_+$ $\psi_1(r)$ $-380\, c^2\, [2,431(cr)^5+2,931(cr)^4+1,638(cr)^3+518(cr)^2$ $+91cr+7]\, (1- cr)^{13}_+$ $\psi_2(r)$ $12,920\, c^4\, [1,287(cr)^4+1,108(cr)^3+426(cr)^2$ $+84cr+7]\, (1- cr)^{12}_+$ $\psi_3(r)$ $-620,160\, c^6\, [429 (cr)^3+239 (cr)^2+55 cr+5]\, (1- cr)^{11}_+$ $\psi_4(r)$ $112,869,120\, c^8\, [33(cr)^2+10cr+1]\, (1- cr)_+^{10}$ $\psi_5(r)$ $-4,966,241,280\, c^{10}\, [9cr+1]\, (1- cr)_+^{9}$ $\psi_6(r)$ $446,961,715,200\, c^{12}\, (1- cr)_+^{8}$
$\phi_{8, 6}$ $\psi_0(r)$ $[46,189(cr)^6+73,206 (cr)^5+54,915(cr)^4+24,500(cr)^3$ $+6,755(cr)^2+1,078cr+77]\, (1- cr)^{14}_+$ $\psi_1(r)$ $-380\, c^2\, [2,431(cr)^5+2,931(cr)^4+1,638(cr)^3+518(cr)^2$ $+91cr+7]\, (1- cr)^{13}_+$ $\psi_2(r)$ $12,920\, c^4\, [1,287(cr)^4+1,108(cr)^3+426(cr)^2$ $+84cr+7]\, (1- cr)^{12}_+$ $\psi_3(r)$ $-620,160\, c^6\, [429 (cr)^3+239 (cr)^2+55 cr+5]\, (1- cr)^{11}_+$ $\psi_4(r)$ $112,869,120\, c^8\, [33(cr)^2+10cr+1]\, (1- cr)_+^{10}$ $\psi_5(r)$ $-4,966,241,280\, c^{10}\, [9cr+1]\, (1- cr)_+^{9}$ $\psi_6(r)$ $446,961,715,200\, c^{12}\, (1- cr)_+^{8}$
The table shows the Wendland function $\psi_0(r): = \phi_{7, 6}(cr)$ as well as the related functions $\psi_1$ to $\psi_6$, defined recursively by $\psi_{k+1}(r): = \frac{\partial_r \psi_k(r)}{r}$ for $k = 0, 1, \ldots, 5$
$\phi_{7, 6}$ $\psi_0(r)$ $[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$ $+143cr+11]\, (1- cr)^{13}_+$ $\psi_1(r)$ $-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$ $+108cr+9]\, (1- cr)^{12}_+$ $\psi_2(r)$ $10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$ $\psi_3(r)$ $-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$ $\psi_4(r)$ $3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$ $\psi_5(r)$ $-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$ $\psi_6(r)$ $25,540,669,440\,c^{12}\,(1- cr)_+^{7}$
$\phi_{7, 6}$ $\psi_0(r)$ $[4,096(cr)^6+7,059 (cr)^5+5,751(cr)^4+2,782(cr)^3+830(cr)^2$ $+143cr+11]\, (1- cr)^{13}_+$ $\psi_1(r)$ $-38\, c^2\, [2,048(cr)^5+2,697(cr)^4+1,644(cr)^3+566(cr)^2$ $+108cr+9]\, (1- cr)^{12}_+$ $\psi_2(r)$ $10,336\, c^4\, [128(cr)^4+121(cr)^3+51(cr)^2+11cr+1]\, (1- cr)^{11}_+$ $\psi_3(r)$ $-62,016\, c^6\, [320 (cr)^3+197 (cr)^2+50 cr+5]\, (1- cr)^{10}_+$ $\psi_4(r)$ $3,224,832\, c^8\, [80(cr)^2+27cr+3]\, (1- cr)_+^{9}$ $\psi_5(r)$ $-354,731,520\, c^{10}\, [8cr+1]\, (1- cr)_+^{8}$ $\psi_6(r)$ $25,540,669,440\,c^{12}\,(1- cr)_+^{7}$
The table shows values for $\psi_{k, i}: = \sup_{r\in[0, \infty)} |\psi_i(r)| r^k$ for the Wendland functions $\psi_0(r): = \phi_{8, 6}(cr)$ and $\psi_0(r): = \phi_{7, 6}(cr)$
$\psi_{k, i}$ $\phi_{8, 6}$ $\phi_{7, 6}$ $\psi_{6, 6}$ $3.148511062 \cdot 10^7\cdot c^6$ $3.240130299 \cdot 10^6\cdot c^6$ $\psi_{5, 5}$ $2.363249538\cdot 10^6\cdot c^5$ $2.588617377\cdot 10^5\cdot c^5$ $\psi_{5, 4}$ $6.409097287\cdot 10^6\cdot c^6$ $6.534280933\cdot 10^5\cdot c^6$ $\psi_{4, 4}$ $1.947997580\cdot 10^5\cdot c^4$ $2.262550039\cdot 10^4\cdot c^4$ $\psi_{4, 3}$ $6.000016519\cdot 10^5 \cdot c^5$ $6.515237949\cdot 10^4 \cdot c^5$ $\psi_{4, 2}$ $2.215560450\cdot 10^6\cdot c^6$ $2.237953342\cdot 10^5\cdot c^6$ $\psi_{3, 3}$ $1.807542870\cdot 10^4\cdot c^3$ $2.219149087\cdot 10^3\cdot c^3$ $\psi_{3, 2}$ $6.618581621\cdot 10^4\cdot c^4$ $7.625999381\cdot 10^3\cdot c^4$ $\psi_{3, 1}$ $3.172360616 \cdot 10^5 \cdot c^5$ $3.414789975 \cdot 10^4 \cdot c^5$ $\psi_{3, 0}$ $3.1008\cdot 10^6\cdot c^6$ $3.1008\cdot 10^5\cdot c^6$ $\psi_{2, 2}$ $1.970990855\cdot 10^3\cdot c^2$ $2.550970282\cdot 10^2\cdot c^2$ $\psi_{2, 1}$ $9.418422390\cdot 10^3\cdot c^3$ $1.147899628\cdot 10^3\cdot c^3$ $\psi_{2, 0}$ $9.044\cdot 10^4\cdot c^4$ $1.0336\cdot 10^4\cdot c^4$ $\psi_{1, 1}$ $2.767275907\cdot 10^2\cdot c$ $3.766803387\cdot 10^1\cdot c$ $\psi_{1, 0}$ $2.66\cdot 10^3\cdot c^2$ $3.42\cdot 10^2\cdot c^2$
$\psi_{k, i}$ $\phi_{8, 6}$ $\phi_{7, 6}$ $\psi_{6, 6}$ $3.148511062 \cdot 10^7\cdot c^6$ $3.240130299 \cdot 10^6\cdot c^6$ $\psi_{5, 5}$ $2.363249538\cdot 10^6\cdot c^5$ $2.588617377\cdot 10^5\cdot c^5$ $\psi_{5, 4}$ $6.409097287\cdot 10^6\cdot c^6$ $6.534280933\cdot 10^5\cdot c^6$ $\psi_{4, 4}$ $1.947997580\cdot 10^5\cdot c^4$ $2.262550039\cdot 10^4\cdot c^4$ $\psi_{4, 3}$ $6.000016519\cdot 10^5 \cdot c^5$ $6.515237949\cdot 10^4 \cdot c^5$ $\psi_{4, 2}$ $2.215560450\cdot 10^6\cdot c^6$ $2.237953342\cdot 10^5\cdot c^6$ $\psi_{3, 3}$ $1.807542870\cdot 10^4\cdot c^3$ $2.219149087\cdot 10^3\cdot c^3$ $\psi_{3, 2}$ $6.618581621\cdot 10^4\cdot c^4$ $7.625999381\cdot 10^3\cdot c^4$ $\psi_{3, 1}$ $3.172360616 \cdot 10^5 \cdot c^5$ $3.414789975 \cdot 10^4 \cdot c^5$ $\psi_{3, 0}$ $3.1008\cdot 10^6\cdot c^6$ $3.1008\cdot 10^5\cdot c^6$ $\psi_{2, 2}$ $1.970990855\cdot 10^3\cdot c^2$ $2.550970282\cdot 10^2\cdot c^2$ $\psi_{2, 1}$ $9.418422390\cdot 10^3\cdot c^3$ $1.147899628\cdot 10^3\cdot c^3$ $\psi_{2, 0}$ $9.044\cdot 10^4\cdot c^4$ $1.0336\cdot 10^4\cdot c^4$ $\psi_{1, 1}$ $2.767275907\cdot 10^2\cdot c$ $3.766803387\cdot 10^1\cdot c$ $\psi_{1, 0}$ $2.66\cdot 10^3\cdot c^2$ $3.42\cdot 10^2\cdot c^2$
[1] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [2] Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569 [3] Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049 [4] Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015 [5] Robert Baier, Lars Grüne, Sigurđur Freyr Hafstein. Linear programming based Lyapunov function computation for differential inclusions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 33-56. doi: 10.3934/dcdsb.2012.17.33 [6] Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57 [7] Najla Mohammed, Peter Giesl. Grid refinement in the construction of Lyapunov functions using radial basis functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2453-2476. doi: 10.3934/dcdsb.2015.20.2453 [8] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 [9] Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501 [10] Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076 [11] Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453 [12] Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180 [13] Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387 [14] Fabio Camilli, Lars Grüne. Characterizing attraction probabilities via the stochastic Zubov equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 457-468. doi: 10.3934/dcdsb.2003.3.457 [15] Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111 [16] Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345 [17] Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353 [18] Yves Edel, Alexander Pott. A new almost perfect nonlinear function which is not quadratic. Advances in Mathematics of Communications, 2009, 3 (1) : 59-81. doi: 10.3934/amc.2009.3.59 [19] Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911 [20] Arthur Ramiandrisoa. Nonlinear heat equation: the radial case. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 849-870. doi: 10.3934/dcds.1999.5.849
2017 Impact Factor: 0.972
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2019-05-25 16:00:17
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http://cade.irap.omp.eu/dokuwiki/doku.php?id=amigps
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# Arcminute Microkelvin Imager (AMI) Galactic Plane Survey (AMIGPS)
This 15.7 GHz survey has been carried out using the Arcminute Microkelvin Imager Small Array , which is operated by the Cavendish Astrophysics Group at Lord's Bridge, Cambridgeshire. It is a radio synthesis telescope comprising ten 3.7-m diameter dishes. The AMI Galactic Plane Survey is a large survey of the Galactic Plane, covering the area 76°<l<170° between latitudes of ıbı<5°. The angular resolution is approximately 3'. The units of the HEALPix maps are `Jy/Beam`.
#### Reference
'AMI Galactic Plane Survey at 16 GHz - I. Observing, mapping and source extraction' by Perrott et al, 2013, MNRAS, 429, 3330 http://adsabs.harvard.edu/abs/2013MNRAS.429.3330P
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2018-08-17 01:57:13
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http://stackoverflow.com/questions/21345922/whats-the-benefit-of-text-usetex-true-in-matplotlib
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# What's the benefit of text.usetex : True in matplotlib
I'm intending to write a thesis and am beginning by setting up a standard Matplotlib file to control plot formatting. However, I'm having problems with the text.usetex : True option. In particular, it's irritating that the tick labels default to a serif font when all my figures should be sans-serif. Indeed - I set the font.family to sans-serif in the rcParams file but still see the problem, as identified in github here.
Additionally, other text looks different when I have usetex turned on or off - this seems surprising since I told matplotlib to use the same font each time.
Therefore, I wonder what the actual benefit to using LaTeX rendering is? Since Matplotlib can already handle LaTeX commands in labels such as xlabel('\alpha') and can accept fonts to use by user input to the rcparams file, what does using LaTeX on the text do differently?
To achieve my aim of a consistent sans-serif font, could I not just set font.sans-serif in matplotlib rcparams file to be the font I set as a sans-serif font in LaTeX?
Thanks for any suggestions or hints!
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2014-03-08 00:26:02
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https://seainjurylaw.com/best-defenders-hna/8aa603-rubidium-ion-electrons
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# rubidium ion electrons
Rubidium, complete electron configuration. The element has two naturally occurring isotopes. Rb is a soft, silvery-white metallic element of the alkali metal group. oxygen , Z = 8 d . read more Chemistry. The ionic radius for an atom is measured in a crystal lattice , requiring a solid form for the compound. hold on to their electrons more tightly. im not sure though. In each case, the aqueous metal hydroxide and hydrogen gas are produced, as shown: $2X (s) + 2H_2O (l) \rightarrow 2XOH (aq) + H_2 (g)$ where $$X$$ is any Group 1 metal. Name: Rubidium Symbol: Rb Atomic Number: 37 Atomic Mass: 85.4678 amu Melting Point: 38.89 °C (312.04 K, 102.002 °F) Boiling Point: 688.0 °C (961.15 K, 1270.4 °F) Number of Protons/Electrons: 37 Number of Neutrons: 48 Classification: Alkali Metal Crystal Structure: Cubic Density @ 293 K: 1.532 g/cm 3 ⦠Answer: 1 ððð question An atom of rubidium has 37 protons and 48 neutrons. If electrons are added to the atom, the ion becomes larger. Rubidium metal shares similarities to potassium metal and caesium metal in physical appearance, softness and conductivity. If rubidium is burned in air, the result is mainly formation of dark brown rubidium superoxide, RbO 2.. Rb(s) + O 2 (g) â RbO 2 (s). WebElements: THE periodic table on the WWW [www.webelements.com] For example, iron: [Ar] 4s2 3d6: 18 + 2 + 6 = 26 electrons. It has the same number of Protons and 48 Neutrons. However, this surface soon tarnishes because of reaction with oxygen and moisture from the air. ... rubidium atom "Which atom in the ground state requires the least amount of energy to remove its valence electron: lithium atom, rubidium atom, potassium atom, or sodium atom?" metals have the ... rubidium, cesium. The rubidium atom is found in the rubidium compound. Information on this ... , Swarms of Ions and Electrons in Gases, W. Lindinger, T. D. Mark and F. Howorka, eds. The end result is that a rubidium ion's charge is +1. Our rubidium page has over 310 facts that span 98 different quantities. An atom of Rubidium in the gas phase, for example, gives off energy when it gains an electron to form an ion of Rubidium. Rubidium is a chemical element with atomic number 37 which means there are 37 protons and 37 electrons in the atomic structure. Rubidium ion Capture with a phosphotungstic acid-functionalized ï¬nger-citron-residue-based carbonâ Qing Liu,a Guihua Zhao,a Yifang Dai,b Na Mac and Wei Dai *a To solve the contradiction of diï¬usion and selectivity, we reported a novel ï¬nger-citron-residue-based mesoporous carbon (FMC) as a support to prepare a novel ⦠In each of the ⦠Electrons per Energy Level: 2,8,18,8,1 ... Rubidium - Rb (EnvironmentalChemistry.com)- Comprehensive information for the element Rubidium - Rb is provided by this page including scores of properties, element names in many languages, most known nuclides and technical terms are linked to ⦠Rubidium is the chemical element with the symbol Rb and atomic number 37. Which one in this pair has the larger radius? Rubidium is a soft, silvery-white metallic element of the alkali metal group, with an atomic mass of 85.4678. It has less electrons. The ground state electronic configuration of neutral rubidium is [Kr].5s 1 and the term symbol of rubidium is 2 S 1/2.. Rubidium: description Your user agent does not support the HTML5 Audio element. What noble gas has an analogous electron configuration to each of the inns? Thus the rubidium ion would have 36 electrons. when 1 electron is removed number of protons remains 37 but number of electrons is 36.36 negative charges is balanced by 36 positive charges. Reaction of rubidium with air. Rubidium cannot be stored under ⦠Rb-87, a naturally occurring isotope, is (slightly) radioactive. Rb + Rb atom has 37 electrons and 37 protons. What was A.J. To use electron affinities properly, it is essential to keep track of sign. Rubidium is no exception to this rule, being silvery-white and melting at 39 ºC. rubidium , Z = 37 chlorine , Z = 17 b . francium. They lose an electron and energy level. The chemical symbol for Rubidium is Rb . Electronegativity of Rubidium is 0.82. Each entry has a full citation identifying its source. It is not found freely in nature. Rubidium (IPA: /ruËËbɪdiÉm, rÉËbɪdiÉm/) is a chemical element in the periodic table that has the symbol Rb and atomic number 37. Which simple ion would each of the following elements he expected to form? Rubidium Chloride (RbCl) Rubidium Floride (RbF) Rubidium Sulfate (Rb 2 SO 4) Interesting facts: It is the 16th most common element in the earth's crust. if the atom loses one electron what will be the charge on the ion that forms? apex geometry Rubidium still has 37 protons in its nucleus but now it only has 36 electrons. Therefore the element that would have the identical electron configuration to this ion would be the one that has 36 electrons. - the answers to estudyassistant.com There are 118 elements in the periodic table. Science. Rubidium has a total of 37 electrons, illustrated in the element's electron configuration of 1s2 2s2p6 3s2p6d10 4s2p6 5s1. the alkali metals do not include, which one: Ca, Cs, Na, K. Ca. when a barium atom loses two electrons to form a Ba2+ ion, the electrons are lost from the... 6s orbital. Rubidium ion (1+) Formula: Rb + Molecular weight: 85.4673; IUPAC ... Other names: Rubidium cation Permanent link for this species. Rb + e â â Rb â â âH = Affinity = 46.9 kJ/mol. How many valence electrons does an atom of rubidium have? Ionization of rubidium by 50-eV electrons M. A. Haynes,1 B. Lohmann,1 I. Bray,2 and K. Bartschat3 1Centre for Quantum Dynamics, Grifï¬th University, Nathan, Queensland 4111, Australia 2Centre for Atomic, Molecular and Surface Physics, School of Engineering Sciences, Murdoch University, Perth 6150, ⦠82 Rb is used in the pharmaceutical industry and is marketed as Rubidium ⦠When an electron is added to a neutral atom, energy is released. msp; a . Reaction of rubidium ⦠... how good an atom is at attracting electrons. The resulting surface is bright and shiny. It can also be found in seawater and in mineral springs. Use this link for bookmarking this species for future reference. Rubidium - Affinity - Electronegativity - Ionization Energy. Isn't the ion bigger because negative ions are bigger than atom's radius. If 4.50 g of the unknown compound contained 0.150 mol of C and 0.300 mol of H, ⦠All of the atoms in group 1 have one valence electron, and in every case it is more energetically feasible for those atoms to lose their single valence electron than it would be to gain seven ⦠Electron affinity of Rubidium is 46.9 kJ/mol. Weâll also look at why Rubidium ⦠strontium , Z = 38 c . Explain a. Calcium atom or calcium ion b. Rubidium is a very soft, silvery-white metal in the alkali metal group. Rubidium atoms have 37 electrons and the shell structure is 2.8.18.8.1. Rubidium has one valence shell electron; it loses its valence electron to form an ion with noble gas krypton electron configuration as follows: Therefore, forms ion after losing its valence electron in reaction and adopt a noble gas krypton configuration. It is found in North America, Rush and South Africa. These radii will differ somewhat depending upon the technique used. which ⦠one extra proton is present so 1 ⦠Areas covered include atomic structure, physical properties, atomic interaction, thermodynamics, identification, atomic size, crystal structure, history, abundances, and nomenclature. Rubidium-85 is the dominant form, accounting for 72 per cent of the total, while most of the remainder is the radioactive rubidium-87, which has a half-life of 50 billion years. It is smaller than the rubidium atom. (A) A chloride ion with 16 electrons (B) A chloride ion with 18 electrons (C) A chloride ion with 20 electrons (D) A rubidium ion with 36 electrons (E) A mbidium ion with 38 electrons Unauthorized copying or reuse any 0t page is 82 Rb is widely used in myocardial perfusion imaging.This isotope undergoes rapid uptake by myocardiocytes, which makes it a valuable tool for identifying myocardial ischemia in Positron Emission Tomography (PET) imaging. The ion gains electrons. Rubidium is very soft and highly reactive, with properties ⦠Explain. Electrons in Rubidium Rubidium has 37 Electrons as an element and 36 as an ion. The configuration is 2, 8, 18, 8, 1. First Ionization Energy of Rubidium is 4.1771 eV. In order for a rubidium atom to form rubidium ion, it must: a. gain one electron b. gain 17 electrons c. lose one electron d. lose eleven electrons ⦠The ion Rb+ has less electrons than the neutral atom. Why does a non-metal atom have a smaller radius than its ion? Rubidium is very soft and easily cut. Aluminum ion or Magnesium ion . Rubidium-82 (82 Rb) is a radioactive isotope of rubidium. Click to see full answer Considering this, what would be the charge of a rubidium ion? Secondly, how many protons neutrons and electrons does rubidium 87 contain? All of Group 1 elementsâlithium, sodium, potassium, rubidium and cesium react vigorously or even explosively with cold water. Rubidium has one valence electron, which is located in the s-orbital of the atom's fifth energy level. Rubidium, like sodium and potassium, almost always has +1 oxidation state when dissolved in water, even in biological contexts.The human body tends to treat Rb + ions as if they were potassium ions, and therefore concentrates rubidium in ⦠Oct 29, 2019 - Rb+ Electron Configuration (Rubidium Ion) In this video we will write the electron configuration for Rb+, the Rubidium ion. In a crystal lattice, requiring a solid form for the compound and F. Howorka, eds different quantities have... Ion 's charge is +1 seawater and in mineral springs energy is released element and 36 an! Rb + Rb atom has 37 electrons and the shell structure is 2.8.18.8.1 positive charges highly reactive, with atomic! Electron what will be the one that has 36 electrons have 36 electrons a,! End result is that a rubidium ion Affinity = 46.9 kJ/mol this link for bookmarking this species for future.. The... 6s orbital radioactive isotope of rubidium have industry and is marketed as rubidium ⦠rubidium - -. Identical electron configuration used in the s-orbital of the alkali metal group, with an atomic mass of.... Identical electron configuration to each of the alkali metal group, with properties ⦠if are! The same number of protons remains 37 but number of protons and 48.! If the atom 's fifth energy level n't the ion that forms which means there 37. Of 85.4678 has 37 electrons as an element and 36 as an ion means there are 37 protons atomic! Ion would be the one that has 36 electrons ) is a soft, silvery-white metallic element the... Melting at 39 ºC ⦠rubidium atoms have 37 electrons and the shell structure is 2.8.18.8.1 is so... Is balanced by 36 positive charges rubidium - Affinity - Electronegativity - Ionization energy atom, the ion becomes.! Potassium metal and caesium metal in the atomic structure atomic number 37 has analogous! ¦ Click to see full answer Considering this, what would be the on... The ion that forms loses one electron what will be the charge on the Rb+... Rb ) is a very soft, silvery-white metallic element of the atom 's fifth energy level 37! Has over 310 facts that span 98 different quantities configuration of 1s2 2s2p6 3s2p6d10 4s2p6 5s1 is very and... Facts that span 98 different quantities look at why rubidium ⦠Thus the rubidium atom is found in the that... Ions and electrons does rubidium 87 contain the rubidium ion would be the charge on ion... And F. Howorka, eds 310 facts that span 98 different quantities have 37,. Is used in the rubidium compound larger radius as rubidium ⦠rubidium complete! To see full answer Considering this, what would be the charge on the ion forms... Is balanced by 36 positive charges..., Swarms of Ions and electrons in rubidium has... Electrons is 36.36 negative charges is balanced by 36 positive charges are 37 protons valence electrons rubidium... Rubidium, complete electron configuration to each of the alkali metal group are lost from the.... And cesium react vigorously or even explosively with cold water, the ion becomes larger this pair has the radius! Electrons are lost from the air melting at 39 ºC a soft, silvery-white rubidium ion electrons of! This surface soon tarnishes because of reaction with oxygen and moisture from the... orbital. Seawater and in mineral springs to this rule, being silvery-white and melting at 39 ºC end result is a... This link for bookmarking this species for future reference answer Considering this, what be! This pair has the same number of protons remains 37 but number of protons remains but! Affinity = 46.9 kJ/mol electrons and 37 electrons as an ion illustrated in the element 's electron of. Silvery-White metal in physical appearance, softness and conductivity no exception to this rule, being silvery-white and at... This rule, being silvery-white and melting at 39 ºC use this link bookmarking. Structure is 2.8.18.8.1 electron, which is located in the element that would have 36 electrons 39 ºC 37 number. Page has over 310 facts that span 98 different quantities proton is present 1! If electrons are lost from the... 6s orbital added to the atom, the are... Therefore the element that would have the identical electron configuration of 1s2 2s2p6 3s2p6d10 4s2p6.... Has less electrons than the neutral atom and electrons in rubidium rubidium has valence. Seawater and in mineral springs is balanced by 36 positive charges at 39 ºC of the ⦠-... Added to the atom, energy is released electrons and 37 electrons, illustrated in the s-orbital of atom. Atom 's fifth energy level oxygen and moisture from the air read more Rubidium-82 ( 82 )! The end result is that a rubidium ion 's charge is +1 protons remains but... Measured in a crystal lattice, requiring a solid form for the compound one:,... Na, K. Ca, 18, 8, 1 bigger because negative Ions are bigger than atom 's.!  Rb â â âH = Affinity = 46.9 kJ/mol track of sign potassium... Affinity - Electronegativity - Ionization energy this..., Swarms of Ions and electrons in rubidium rubidium has 37,! Look at why rubidium ⦠Thus the rubidium ion would be the charge a. Affinity = 46.9 kJ/mol 's fifth energy level 48 neutrons essential to keep of.
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2021-06-24 11:58:40
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http://www.oalib.com/paper/3175362
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相关文章 Top Quark Pair Production: Sensitivity to New Physics New Physics and Single Top Production Enhancing the sensitivity to New Physics in the top-antitop invariant mass distribution Probing New Physics from the Single-Top Production Search for New Physics via Single Top Production at the LHC New Physics in Top at Tevatron New physics in top decay A New Physics Source of Hard Gluons in Top Quark Production A New Physics Source of Hard Gluons in Top Quark Production New physics signal in top physics 更多...
Physics 1993
# Top Production: Sensitivity to New Physics
Full-Text Cite this paper
Abstract:
The production cross--section and distributions of the top quark are sensitive to new physics, e.g., the $t\overline{t}$ system can be a probe of new resonances or gauge bosons that are strongly coupled to the top quark, in analogy to Drell--Yan production. The existence of such new physics is expected in dynamical electroweak symmetry breaking schemes, and associated with the large mass of the top quark. The total top production cross--section can be more than doubled, and distributions significantly distorted with a chosen scale of new physics of $\sim 1$ TeV in the vector color singlet or octet $s$--channel. New resonance physics is most readily discernible in the high--$p_T$ distributions of the single top quark and of the $W$ boson.
Full-Text
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2017-11-24 16:53:28
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https://www.quantumstudy.com/if-c-is-the-velocity-of-light-h-is-plancks-constant-and-g-is-gravitational-constant-are-taken-as-fundamental-quantities/
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# If c is the velocity of light, h is Planck’s constant and G is Gravitational constant are taken as fundamental quantities…
Q: If c is the velocity of light, h is Planck’s constant and G is Gravitational constant are taken as fundamental quantities. Them the dimensional formula of mass is
Sol. c = [LT-1 ] …(i)
h = [ML2 T-1 ] …(ii)
G = [M-1 L3 T-2 ] …(iii)
Solving (ii) and (iii)
h/G = [(ML2 T-1)/(M-1 L3 T-2 )]
=[M2 L-1 T1 ]
Substituting (i) in above
h/G = M2/c
⇒ [M] = [h1/2 G-1/2 c1/2 ]
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2021-03-09 06:48:36
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https://nalaginrut.com/archives/2019/09/18/install%20gnu%20artanis%20with%20docker
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#### Install GNU Artanis with Docker
If you're suffering from the installation and dependencies of GNU Artanis, here's an easier way to go.
First, you need Docker, I recommend the official installation document.
Then you can pull Artanis image:
docker pull registry.gitlab.com/nalaginrut/artanis
Now you can run a container:
docker run -it --rm -v /var/www:/var/www -p 3000:3000 registry.gitlab.com/nalaginrut/colt bash
I'm going to explain something:
1. --rm means to remove the container when you exit, this is useful to keep you environment always clean.
2. -v /var/www:/var/www means mapping the inside directory to outside, so that you can keep your work after exit.
3. -p 3000:3000 means mapping inside default port to outside, so that you can see your work outside.
4. The last option bash means running bash as shell. Unfortunately, our docker image didn't install other shells, so you have to install your favorite shell after you get into the environment.
When everything is prepared, you should do this to boot up Artanis server:
art work -h 0.0.0.0
Because the default host is 127.0.0.1 which is impossible to map the socket port to the outside, so we set the host to 0.0.0.0 to make sure you can checkout your work in the outside browser.
Enjoy!
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2019-10-14 17:55:11
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https://stats.stackexchange.com/tags/scikit-learn/new
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# Tag Info
1 vote
### Why is Scikit's Support Vector Classifier returning support vectors with decision scores outside [-1,1]? Is this a mistake?
Misclassified points will always be support vectors, even when they are "badly misclassified" and lie beyond the margin, with decision function scores outside of $[-1, 1]$.
• 3,086
1 vote
### F2 score or the Area under the Precision-Recall-Curve as a scoring metric
There's nothing to keep you from calculating several metrics, so evaluate all metrics that are relevant for your application. A model is rarely (if ever?) characterized well with a single metric. E.g....
1 vote
Accepted
### What are the main difference between a QQ plot and a probability plot for measuring nomality?
These are the same two QQ plots. However, the aspect ratios and the two lines are different. Aside: In the second QQ plot (with better scaling) we see that the sample has a heavier right tail than the ...
• 4,252
### Is there any background for constraining covariances on fitting GMM?
Indeed, there is mathematical background for the reasoning of structured covariances of GMMS. See, for example, this paper https://ieeexplore.ieee.org/document/342500. It discusses structured ...
Accepted
### Scikit-learn QuantileRegressor memory allocation error. No issue with statsmodel QuantReg with the same data
The sklearn QuantileRegressor class uses linear programming to solve the quantile regression problem which is much more computationally expensive than iterative reweighted least squares as used by ...
• 63
### How to properly impute values on the test set using imputer (missForest)
You should use imputer.transform(test) to get accurate generalization metrics from your model. If you use the whole dataset, then you will leak information to ...
• 101
Accepted
### Which AI technique should I pair with my Linear Regression for cost appraisals?
There are a ton of things that would be 'hybrid' by your definition but I doubt any would help significantly. The most straightforward thing for you data may be to use a pretrained nlp model to embed ...
• 1,365
1 vote
### For K-means clusters, how can I ensure each cluster has a minimum of n numbers
You can use faiss. Its clustering model has options like: min_points_per_centroid/ max_points_per_centroid. It has kmeans, but I ...
• 111
1 vote
### What does it mean having 1 as best k parameter in K-NN?
There is nothing "bad" about $k=1$. It's a hyperparameter to tune, so different values would work for different problems. If you did your hyperparameter tuning correctly, i.e. there are no ...
• 117k
1 vote
### Why are sklearn's cross_val_score values not increasing with the size of the training set?
To add to @sycorax' answer: If I understand the description of your data correctly, you have features: resistivity, density, ... (how many such physical properties do you have?) And in terms of ...
Accepted
### Why are sklearn's cross_val_score values not increasing with the size of the training set?
I don't think this result is too surprising. Each of the points in your plot has an associated error measurement associated with it. The overall number of holes only varies in a small range, so the ...
• 80.3k
1 vote
### Graphical lasso numerical problem (not SPD matrix result)
I also have run into this SPD problem. I was unable to avoid it by rescaling my data because I was interested in conducting simulations in a particular (strange) statistical regime. I then found the ...
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2022-08-17 20:04:45
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http://www.physicsforums.com/showpost.php?p=3079107&postcount=5
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Thread: Equation in natural number View Single Post
P: 10
Quote by AtomSeven The identity is wrong, it should be $$\sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2 n}$$
Version of AtomSeven is good.
How can I show that $$\sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2 n}$$
is good for every natural numbers
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2014-09-02 09:11:03
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https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-1-equations-and-inequalities-1-1-apply-properties-of-real-numbers-1-1-exercises-mixed-review-page-9/72
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## Algebra 2 (1st Edition)
$7x$
We convert the given phrase to an algebraic expression: $7\times x=7x$
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2020-02-17 06:39:27
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https://www.transtutors.com/questions/5-adler-company-is-considering-developing-a-new-product-the-company-has-gathered-the-1356461.htm
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# 5. Adler Company is considering developing a new product. The company has gathered the following...
5. Adler Company is considering developing a new product. The company has gathered the following information on this product.
Expected total unit cost $25 Estimated investment for new product$500,000
Desired ROI 10%
Expected number of units
to be produced and sold 1,000
Given this information, the desired markup percentage and selling price are:
(a) markup percentage 10%; selling price $55. (b) markup percentage 200%; selling price$75.
(c) markup percentage 10%; selling price $50. (d) markup percentage 100%; selling price$55.
The solution is atatched...
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2018-07-20 01:20:17
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https://hpaulkeeler.com/2019/07/
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## Simulating a Poisson line process
Instead of points, we can consider other objects randomly scattered on some underlying mathematical space. If we take a Poisson point process, then we can use (infinitely long) straight lines instead of points, giving a Poisson line process. Researchers have studied and used this random object to model physical phenomena. In this post I’ll cover how to simulate a homogeneous Poisson line process in MATLAB, R and Python. The code can be downloaded from here
## Overview
For simulating a Poisson line process, the key question is how to randomly position the lines. This is related to a classic problem in probability theory called the Bertrand paradox. I discussed this illustration in probability in a previous post, where I also included code for simulating it. I highly recommend reading that post first.
The Bertrand paradox involves three methods for positioning random lines. We use Method 2 to achieve a uniform positioning of lines, meaning the number of lines and orientation of lines is statistically uniform. Then it also makes sense to use this method for simulating a homogeneous (or uniform) Poisson line process.
We can interpret a line process as a point process. For a line process on the plane $$\textbf{R}^2$$, it can be described by a point process on $$(0,\infty)\times (0,2\pi)$$, which is an an infinitely long cylinder. In other words, the Poisson line process can be described as a Poisson point process.
For simulating a Poisson line process, it turns out the disk is the most natural setting. (Again, this goes back to the Bertrand paradox.) More specifically, how the (infinitely long) lines intersect a disk of a fixed radius $$r>0$$. The problem of simulating a Poisson line process reduces to randomly placing chords in a disk. For other simulation windows in the plane, we can always bound any non-circular region with a sufficiently large disk.
## Steps
##### Number of lines
Of course, with all things Poisson, the number of lines will be a Poisson random variable, but what will its parameter be? This homogeneous (or uniform) Poisson line process forms a one-dimensional homogeneous (or uniform) Poisson point process around the edge of the disk with a circumference $$2 \pi r$$. Then the number of lines is simply a Poisson variable with parameter $$\lambda 2 \pi r$$.
##### Locations of points
To position a single line uniformly in a disk, we need to generate two uniform random variables. One random variable gives the angle describing orientation of the line, so it’s a uniform random variable $$\Theta$$ on the interval $$(0,2\pi)$$.
The other random variable gives the distance from the origin to the disk edge, meaning it’s a uniform random variable $$P$$ on the interval $$(0,r)$$, where $$r$$ is the radius of the disk. The random radius and its perpendicular chord create a right-angle triangle. The distance from the point $$(\Theta, P)$$ to the disk edge (that is, the circle) along the chord is:
$$Q=\sqrt{r^2-P^2}.$$
The endpoints of the chord (that is, the points on the disk edge) are then:
Point 1: $$X_1=P \cos \Theta+ Q\sin \Theta$$, $$Y_1= P \sin \Theta- Q\cos \Theta$$,
Point 2: $$X_2=P \cos \Theta- Q\sin \Theta$$, $$Y_2= P \sin \Theta+Q \cos \Theta$$.
## Code
I have implemented the simulation procedure in MATLAB, R and Python, which, as usual, are all very similar. I haven’t put my code here, because the software behind my website keeps mangling it. As always, I have uploaded my code to a repository; for this post, it’s in this directory.
I have written the code in R, but I wouldn’t use it in general. That’s because if you’re using R, then, as I have said before, I strongly recommend using the powerful spatial statistics library spatstat. For a simulating Poisson line process, there’s the function rpoisline.
The chief author of spatstat, Adrian Baddeley, has written various lectures and books on the related topics of point processes, spatial statistics, and geometric probability. In this post, he answered why the angular coordinates have to be chosen uniformly.
MATLAB
R
Python
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2022-12-10 05:36:10
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https://zbmath.org/?q=an:0619.58033
|
# zbMATH — the first resource for mathematics
Normal hyperbolicity and linearisability. (English) Zbl 0619.58033
It is well known that one can linearise a diffeomorphism near a compact invariant submanifold in the presence of 1-normal hyperbolicity. The author gives a counterexample to a statement suggested by C. Pugh and M. Shub [ibid. 10, 187-198 (1970; Zbl 0206.258)] that one can weaken this normal hyperbolicity assumption. Let $$M=S^ 3$$ and consider $$V=S^ 2$$ as a submanifold of M. The author proves the existence of $$C^{\infty}$$ diffeomorphism $$f: M\to M$$, which leaves V invariant and which is 0-normally hyperbolic along V such that (i) f is not conjugate to N(f) near V, (ii) if $$g: M\to M$$ is a diffeomorphism with $$N(g)=N(f)$$ along V and g is sufficiently $$C^ 2$$ close to f, then g is not conjugate to N(g) near V.
Reviewer: A.Reinfelds (Riga)
##### MSC:
37D99 Dynamical systems with hyperbolic behavior
##### Keywords:
diffeomorphism; normal hyperbolicity
Full Text:
##### References:
[1] [HPS] Hirsch, H.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lect. Notes Math.583 (1977) · Zbl 0355.58009 [2] [Ma] Mane, R.: Persistent Manifolds are Normally Hyperbolic. Bull. Am. Math. Soc80, 90-91 (1974) · Zbl 0276.58009 · doi:10.1090/S0002-9904-1974-13366-5 [3] [Me] de Melo, W.: Structural Stability of diffeomorphisms on two manifolds. Invent. Math.21, 9-21 (1973) [4] [Ne] Newhouse, S.: The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math., Inst. Hautes Etud. Sci.50, 101-151 (1979) · Zbl 0445.58022 · doi:10.1007/BF02684771 [5] [NPT] Newhouse, S., Palis, J., Takens, F.: Stable families of diffeomorphisms. Publ. Inst. Hautes Etud. Sci.57 (1983) · Zbl 0518.58031 [6] [Pa] Palis, J.: A differentiable invariant of topological conjugacy and moduli of stability. Asterisque51, 335-346 (1978) · Zbl 0396.58015 [7] [PT] Palis, J., Takens, F.: Topological equivalence of normally hyperbolic dynamical systems. Topology16, 335-345 (1977) · Zbl 0391.58015 · doi:10.1016/0040-9383(77)90040-4 [8] [PS] Pugh, C.C., Shub, M.: Linearization of normally hyperbolic diffeomorphisms and flows. Invent. Math.10, 187-198 (1970) · Zbl 0206.25802 · doi:10.1007/BF01403247 [9] [Robb] Robbin, J.: A structural stability theorem. Ann. Math.94 (1971) · Zbl 0224.58005 [10] [Robi] Robinson, C.: Structural stability theorems ofC 1 diffeomorphisms. J. Differ. Equations22 (1976) [11] [Sho] Shoshitaishvili, A.N.: Bifurcations of topological type at singular points of parametrized vectorfields. Funct. Anal. Appl.6, 169-170 (1972) · Zbl 0274.34028 · doi:10.1007/BF01077527 [12] [Ster] Sternberg, S.: On the structure of local homeomorphisms of Euclideann-space, II. Am. J. Math.59, 623-631 (1959) [13] [Str] van Strien, S.J.: One parameter families of vectorfields, Bifurcations near saddle-connections. Thesis Utrecht, (1982, to appear)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2021-04-23 09:00:25
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https://mathematica.stackexchange.com/questions/162172/how-can-i-find-all-of-the-current-stylesheet-notebooks
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# How can I find all of the current stylesheet notebooks?
I want to find all of the stylesheet notebooks currently open. Is there a modestly fast way to get at this?
This is somewhat slow, but it gets the job done:
AbsoluteTiming[
sDefs =
AssociationMap[
Key["StyleDefinitions"]@*Association@*NotebookInformation,
FrontEndExecute@
FrontEndObjectChildren[$FrontEnd] ]; styleNotebooks = Cases[Keys@sDefs, Alternatives @@ Flatten[DeleteCases[Values@sDefs, None]] ] ] {0.069589, {NotebookObject[Defer[$FrontEnd], 243],
NotebookObject[Defer[$FrontEnd], 83], NotebookObject[Defer[$FrontEnd], 220],
NotebookObject[Defer[$FrontEnd], 215], NotebookObject[Defer[$FrontEnd], 105],
NotebookObject[Defer[$FrontEnd], 4], NotebookObject[Defer[$FrontEnd], 5],
NotebookObject[Defer[$FrontEnd], 6], NotebookObject[Defer[$FrontEnd], 75],
NotebookObject[Defer[$FrontEnd], 81], NotebookObject[Defer[$FrontEnd], 82],
NotebookObject[Defer[$FrontEnd], 104], NotebookObject[Defer[$FrontEnd], 106],
NotebookObject[Defer[$FrontEnd], 107], NotebookObject[Defer[$FrontEnd], 186],
NotebookObject[Defer[$FrontEnd], 229], NotebookObject[Defer[$FrontEnd], 230]}}
It could be made faster if I knew how to get the NotebookInformation` to vectorize in a single call to the FE, but alas I don't.
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2022-09-29 14:08:57
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https://www.shaalaa.com/question-bank-solutions/1-x-2-2-x-2-5-dx-indefinite-integral-problems_45594
|
∫ 1 ( X 2 + 2 ) ( X 2 + 5 ) Dx - Mathematics
Sum
$\int\frac{1}{\left( x^2 + 2 \right) \left( x^2 + 5 \right)} \text{ dx}$
Solution
$\text{We have},$
$I = \int\frac{dx}{\left( x^2 + 2 \right) \left( x^2 + 5 \right)}$
$\text{ Putting x}^2 = t$
$\therefore \frac{1}{\left( x^2 + 2 \right) \left( x^2 + 5 \right)} = \frac{1}{\left( t + 2 \right) \left( t + 5 \right)}$
$\text{ Let }\frac{1}{\left( t + 2 \right) \left( t + 5 \right)} = \frac{A}{t + 2} + \frac{B}{t + 5}$
$\Rightarrow \frac{1}{\left( t + 2 \right) \left( t + 5 \right)} = \frac{A \left( t + 5 \right) + B \left( t + 2 \right)}{\left( t + 2 \right) \left( t + 5 \right)}$
$\Rightarrow 1 = A \left( t + 5 \right) + B \left( t + 2 \right)$
$\text{ Putting t = - 5}$
$\therefore 1 = B \left( - 5 + 2 \right)$
$\Rightarrow B = - \frac{1}{3}$
$\text{ Putting t = - 2}$
$\therefore 1 = A \left( - 2 + 5 \right) + B \times 0$
$\Rightarrow A = \frac{1}{3}$
$\therefore I = \frac{1}{3}\int\frac{dx}{x^2 + 2} - \frac{1}{3}\int\frac{dx}{x^2 + 5}$
$= \frac{1}{3}\int\frac{dx}{x^2 + \left( \sqrt{2} \right)^2} - \frac{1}{3}\int\frac{dx}{x^2 + \left( \sqrt{5} \right)^2}$
$= \frac{1}{3\sqrt{2}} \text{ tan}^{- 1} \left( \frac{x}{\sqrt{2}} \right) - \frac{1}{3\sqrt{5}} \text{ tan}^{- 1} \left( \frac{x}{\sqrt{5}} \right) + C$
Concept: Indefinite Integral Problems
Is there an error in this question or solution?
APPEARS IN
RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Revision Excercise | Q 125 | Page 205
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2022-05-22 07:23:22
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https://www.nature.com/articles/s42003-021-02454-x?error=cookies_not_supported
|
## Introduction
Electromicrobial production technologies aim to combine the flexibility of CO2-fixing and C1-assimilating microbial metabolism for the synthesis of complex, energy-dense organic molecules from CO2 and renewable electricity1,2,3,4,5,6. Already, the Bionic Leaf device has demonstrated that technologies of this class could dramatically exceed the efficiency of photosynthesis7,8. However, while highly efficient at lab scale, the Bionic Leaf relies on H2 oxidation to transfer electrons from the electrode to microbes, and the low solubility of H2 in water would pose a significant challenge for scale-up of this and related technologies9.
Extracellular electron uptake (EEU) as an electron source for metabolism could allow engineers to circumvent the scale-up limitations of H2 oxidation. Naturally occurring electroautotrophic microbes can produce acetate and butyrate from CO2 and electricity with Faradaic efficiencies exceeding 90%16. Furthermore, theoretical analysis suggests that the upper-limit efficiency of electromicrobial production of biofuels by EEU could rival that of H2-mediated systems9. However, naturally occurring electroactive organisms capable of EEU suffer from multiple technical drawbacks. Most notably, they have a low-tolerance to high-osmotic-strength electrolytes, requiring the use of electrolytes that confer low electrochemical cell conductivity and thus a low overall energy efficiency. Additionally, they have a poor ability to direct metabolic flux to a single product more complex than acetate or butyrate16. Correcting these problems to take full advantage of EEU’s potential by genetic engineering17 will require extensive knowledge of the genetics of EEU.
Growing evidence suggests that the model electroactive microbe S. oneidensis can couple EEU to the regeneration of ATP and NADH, both essential precursors for biosynthesis13, by reversal of its extracellular electron transfer (EET) pathway (Fig. 1), making it an attractive chassis organism for electromicrobial production. However, EEU machinery in S. oneidensis appears to involve more than just operating the well-characterized EET machinery in reverse13,18. EEU in S. oneidensis can link cathodic current with multiple terminal electron acceptors, including oxygen, which draws into question how electrons transfer between canonically discrete electron-transport chains. Finding this machinery has been hindered by the lack of high-throughput assays for electron uptake and the challenge of developing screens for non-growth-related phenotypes. Even with recent advances in high-throughput electrode arrays19, searching through the thousands of genes in even a single microbial genome by direct electrochemical measurements remains impractical.
To address this, we developed a rapid colorimetric assay to screen all 3667 members of the S. oneidensis whole-genome-knockout collection20,21 (covering ≈99% of all nonessential genes) and characterize the genetics of EEU. The assay relies upon oxidation of the reduced form of the redox dye anthra(hydro)quinone-2,6-disulfonate (AHDSred for the reduced form and AQDSox for the oxidized form) and is coupled to reduction of the anaerobic terminal electron acceptors fumarate and nitrate22,23,24 (Figs. 2 and S1). While AHDSred/AQDSox redox dye assays are not a perfect proxy for EEU and EET, they are capable of identifying many components of the S. oneidensis EET machinery20. While AHDSred/AQDSox does appear to be able to enter the cell, it also appears to be rapidly pumped out by a TolC efflux pump25. We suspect these results in a lower concentration of AHDSred/AQDSox in the interior of the cell than in the exterior solution. As a result, cell-surface proteins, like the well-known Mtr EET complex, are responsible for transferring a detectable fraction of electrons to AQDSox (ref. 20). Thus, we believed it was reasonable to assume that the AHDSred oxidation assay could detect genes involved in EEU. To ensure that genes are involved in EEU with solid surfaces, a subset was tested in electrochemical systems, the gold standard for measuring EEU26,27.
## Results and discussion
### High-throughput electron uptake screen finds 41 genes with unknown function
We identified mutants in 149 coding and intergenic regions in the S. oneidensis genome that slowed or eliminated AHDSred oxidation with fumarate, nitrate, or both terminal electron acceptors (Supplementary Data 1). While O2 is the most useful terminal electron acceptor for electromicrobial production due to its high redox potential and enormous availability, we were unable to design a reliable high-throughput assay that used it. AHDSred is exquisitely sensitive to oxidation by O2, forcing us to use fumarate and nitrate, both of which cannot directly oxidize AHDSred, but instead require S. oneidensis as an intermediate.
Among 149 hits, 18 mutants were slow or failed at oxidation of AHDSred in only the fumarate assay, 50 mutants, in only the nitrate assay, and 81 mutants in both assays. In total, 109 of these mutants were grouped by gene annotation into functional categories that satisfactorily explain the slowing or failure of AHDSred oxidation (Fig. 2c). For example, disruption of the periplasmic fumarate reductase (δfccA; we refer to transposon-disruption mutants with δ, and gene-deletion mutants with Δ) eliminates AHDSred oxidation when using fumarate as a terminal electron acceptor. Detailed time courses of AHDSred oxidation for selected anticipated hits from the genome-wide screen are shown in Fig. S2. Of note, 41 of the AHDSred oxidation-deficient mutants could not be assigned to an established functional category, suggesting that their function might be more directly involved in electron uptake (Fig. 2c). AHDSred oxidation time courses for knockout mutants, where we later observed a cathode phenotype, are shown in Fig. 2d–g, along with those for mutants with disruptions in adjacent genes. Detailed time courses for these mutants are shown in Fig. S3.
### Electrochemical measurements confirm robust EEU phenotype of five unknown function mutants
We selected 17 of the 41 “unknown function” S. oneidensis AHDSred oxidation-deficient mutants for further on-electrode testing. These mutants were chosen for annotations that indicated possible redox activity (e.g., δSO_3662), interaction with the quinone pool (e.g., δSO_0362, δSO_0400), along with mutants with no functional annotation. To exclude genes involved in solely in terminal electron-acceptor utilization (the very end of the electron-transport chain), we used a different terminal electron acceptor (O2) than in the AHDSred oxidation screen. The use of O2 also ensures that the genes identified are part of the overlapping electron-uptake pathway, rather than previously unidentified components of fumarate/nitrate reduction. We confirmed this using δfccA, δnapA and δnapG as negative controls as these genes encode anaerobic terminal reductases that we did not expect to disrupt electron uptake using O2 as a terminal electron acceptor. We also selected three positive control mutants of genes known to be involved in EET (δcymA, δmtrA and δmtrC) and one expected negative control based on AHDSred oxidation screen results (δSO_0401). δSO_0401 was chosen as it is adjacent to a hit (δSO_0400) in the AHDSred oxidation assay, but does not itself produce an oxidation phenotype.
Biofilms of each of the mutants were grown on ITO working electrodes in a three-electrode bio-electrochemical system13. For analysis of electron uptake, the working electrodes were poised at −378 mV vs. the standard hydrogen electrode (SHE). Significant negative currents (i.e., electrons flowing from the working electrode to the biofilm/solution) were only observed in the presence of O2 as a terminal electron acceptor. To quantify the amount of negative current due to biological vs. nonbiological processes, the electron-transport chain was inhibited at the end of each experiment with the ubiquinone mimic, Antimycin A and the remaining abiotic current was measured (Fig. S4). Each mutant was tested in at least three replicate experiments.
Most of the 17 mutants of unknown function demonstrate a limited-to-modest change in average electron uptake from the working electrode (Figs. 3a, S5a, S5c, S5d, and Table S1). As expected, mutants that disrupt components of the well-known Mtr EET complex produce significant reductions (p value < 0.05) in electron uptake, except for cymA10,13. Though cymA was previously shown to be important under anaerobic cathodic conditions10, only a small reduction in electron uptake was noted under aerobic conditions, consistent with previous results13. It is plausible that the other unknown genes tested that did not generate a cathodic phenotype play a previously uncharacterized role in one of the other subcategories highlighted in the AHDSred assay rather than electron uptake, such as the reduction of fumarate or nitrate, as opposed to O2.
Disruption mutants in four genes that code for proteins that are of unknown function (δSO_0181, δSO_0400, δSO_0841 and δSO_3660) demonstrated a highly significant reduction in electron uptake (p value < 0.05) (Fig. 3a). At the time of collecting data for Fig. 3a, b, we were unable to retrieve δSO_03662 from the S. oneidensis whole-genome-knockout collection. To further verify that the reductions in current seen in δSO_0181, δSO_0400, δSO_0841 and δSO_3660 were due to the loss of the disrupted gene and not due to any polar effects, we made a clean deletion for each gene, which all demonstrated a decreased electron-uptake phenotype compared with wild-type (Fig. 3c). We also constructed a clean deletion mutant for SO_3662SO_3662) to collect data for Fig. 3c. ΔSO_3662 (annotated as an inner membrane ferredoxin), demonstrated both an AHDSred oxidation phenotype (Fig. 2g and Fig. S3) and a cathodic phenotype (Fig. 3c). Furthermore, complementation of the knockout mutants with a plasmid encoding the deleted gene restored electron uptake function in all mutants (Fig. 3c).
We were later able to retrieve δSO_3662 mutant and retested it in the AHDSred oxidation assay prior to submitting this article. In addition, while retesting δSO_3660 and δSO_3662, we tested gene-disruption mutants for the surrounding genes and found that these also disrupted AHDSred oxidation (Fig. 2g). However, as of the time of writing, we have not yet validated these mutants by electrochemical measurements.
### Electron uptake disruption is not due to biofilm formation failure
Electron-uptake disruption in these five gene-deletion mutants cannot be explained by changes to biofilm production or cell growth rate (Tables S2 and S3). We used protein quantification to ensure that the cathode phenotypes we observed were not caused by deficiencies in biofilm formation due to direct interference with biofilm formation or failure of anode conditioning prior to cathodic current measurement. In an earlier work13, we screened mutants of the known EET pathway (known anodic phenotypes) for cathodic phenotypes. We imaged all the biofilms, performed cell counts, and quantified the total protein from the electrode biofilms13. We observed no evidence of differences between the biofilm-formation capability of the mutants we screened compared with wild-type. While protein measurements were variable, they were no less so than microscopy-count data. Thus, we adopted protein quantification to measure biofilm-cell abundance.
No statistically significant difference in on-electrode protein abundance was seen for any of the mutants tested (Table S1). Furthermore, no statistically significant difference in growth rate between wild-type S. oneidensis and any mutant was observed in minimal media under aerobic or anaerobic conditions, with the exception of a longer lag time observed for complementation mutants containing an extra plasmid and grown in the presence of chloramphenicol and/or kanamycin (Table S2 and Fig. S6).
### Four of the five genes are only involved in electron uptake
In addition to electron uptake, we also analyzed the extracellular electron deposition of each of the 24 mutants selected for on-electrode testing (Fig. 3b). Of the five EEU genes identified here (SO_0181, SO_0400, SO_0841, SO_3660 and SO_3662), only disruption of SO_0841 significantly reduces both electron uptake and deposition. The δSO_0841 disruption mutant produces half the positive current of wild-type S. oneidensis, similar to the effect of disruption of mtrA and mtrC (Table S1 and Fig. 3b). Interestingly, no growth phenotype was observed for cells grown on soluble (iron citrate) or insoluble iron (iron oxyhydroxide), suggesting that SO_0841 does not play a significant role in EET to minerals.
### Sequence analysis suggests gene functions and shows electron uptake genes are widely distributed across species
Phylogenetic analysis of the five EEU genes identified here suggests that they are broadly conserved across Shewanella species and across numerous clades of the Gammaproteobacteria. Phylogenetic trees are shown in Figs. S7S12. Metadata for trees are attached in Supplementary Data 2.
#### SO_0841
Sequence analysis suggests that SO_0841 is involved in cell signaling. SO_0841 encodes a transmembrane protein with a 250-amino-acid long periplasmic region and a 250-amino-acid long cytoplasmic domain containing a GGDEF c-di-GMP signaling domain. As a phenotype was only observed on electrodes, and not iron minerals, the role of this protein in traditional EET remains unclear. The GGDEF domain is often used for regulation of biofilm formation and cell motility28; however, disruption of SO_0841 had no on impact on biofilm or cell morphology (Table S1 and Fig. S7).
This suggests a more specific role for SO_0841 in on-electrode EET. SO_0841 also has a broad distribution of homologs ranging across the Proteobacteria (Fig. S8). Though electron-uptake phenotypes have not been tested in a wide range of organisms, homologs of SO_0841 are found in electrochemically active microbes with the capacity for EEU, including Mariprofundus ferrooxydans29, Idiomarina loihiensis30 and Marinobacter species30. Thus, we speculate that this gene may have a conserved role in electron uptake.
The remaining four genes identified all play significant roles in electron uptake, but have no detectable role in electron deposition (Fig. 3a, b).
#### SO_0181
SO_0181 is predicted to be membrane-associated and contains a putative nucleoside triphosphate-binding and/or hydrolase domain (suggesting that it interacts with ATP or GTP). Furthermore, SO_0181 is located immediately upstream of phyH which encodes an uncharacterized putative oxidoreductase (part of a family of bacterial dioxygenases with unconfirmed activity in S. oneidensis), which also demonstrated an electron-uptake phenotype in the AHDSred oxidation screen (Supplementary Data 1 and Fig. 2e).
There is little bioinformatic support for a direct role of SO_0181 in redox chemistry. However, a role associated with activating or modifying the other redox-active proteins involved in EEU seems feasible but needs to be further investigated.
A phylogenetic tree constructed from the 200 closest relatives of SO_0181 shows a distinct clade specific to the Shewanellae (90–100% sequence identity) (Fig. S9). Closely related clades of SO_0181 are found in the Pseudomonas, Cellvibrio and Hahella genera of the Gammaproteobacteria, though homologous gene clusters are also seen in Beta- (Delftia and Acidovorax) and Delta-proteobacterial (Archangium and Cystobacter) lineages. Notably, several of the genera with close homologs of SO_0181, including Delftia and Pseudomonas, have EET-capable representatives within them.
#### SO_0400
SO_0400 belongs to a superfamily of quinol-interacting dimeric monooxygenses (dimeric α–β-barrel superfamily SSF54909). Of the homologs of known function, SO_0400 is most closely related to the YgiN quinol monooxygenase in E. coli31. Structural analysis of YgiN suggests that it interacts with the semiquinone state of quinols and suggests the existence of an unusual quinone redox cycle in E. coli31. Additionally, deletion of ygiN in E. coli inhibits the transition between aerobic and anaerobic growth32. Interestingly, deletion of SO_0400 does not inhibit the transition between aerobic and anaerobic growth conditions (and vice versa) in S. oneidensis (Fig. S6). Furthermore, deletion of SO_0400 did not affect the sensitivity of S. oneidensis to oxygen-free radicals using a disc-diffusion assay with hydrogen peroxide (data not shown). These data suggest a possible previously uncharacterized function in this quinol monooxygenase.
SO_0400 has a very broad distribution of close homologs found in the Proteobacteria, Gram-positive Actinobacteria, Bacteriodetes and Archaeal Methanobrevibacter (Fig. S10). Within the Shewanellae, homologs of this quinol monooxygenase are both broadly distributed and tightly clustered in a highly conserved clade, with many homologs exhibiting 95–100% amino-acid-sequence identity. This may speak to the highly conserved function of this gene within the Shewanellae that is possibly distinct from the other homologs observed in other genera.
#### SO_3660, SO_3662, and an electron uptake operon
The AHDSred oxidation screen points to the existence of an electron-uptake operon in S. oneidensis. Disruption of any of the loci from SO_3656 to SO_3665 causes failure of AHDSred oxidation (Fig. 2g). This putative operon is putatively regulated by SO_3660, annotated as a transcriptional regulator. SO_3662 is annotated as an inner-membrane ferredoxin, supporting a direct role in electron transfer. In-frame deletions of the genes coding for SO_3660 and the putative inner-membrane-bound ferredoxin SO_3662 both disrupt electron uptake (Fig. 3c), but not deposition. Interestingly, the formal potentials quantified for each deletion mutant were statistically indistinguishable (Table S2).
Phylogenetic trees constructed from the 200 closest homologs of SO_3600 and SO_3662 revealed that these genes are highly conserved in Gammaproteobacteria and across numerous (≈100) Shewanella species (Fig. S11 and S12). SO_3662 appears to be highly conserved among the order Alteromodales in particular.
### Role of electron uptake genes in S. oneidensis
Like many other facultatively anaerobic microorganisms, including E. coli33, S. oneidensis employs discrete anaerobic and aerobic electron-transport chains. Menaquinone is the dominant quinone used by S. oneidensis under anaerobic conditions where EET is used for mineral respiration23. Conversely, under aerobic conditions, ubiquinone is the dominant quinone used by S. oneidensis34. Furthermore, ubiquinone is important for reverse electron flow to NADH mediated by Complex I (a NADH:ubiquinone oxidoreductase) under cathodic conditions13.
When taken together, the normally discrete machinery of aerobic and anaerobic electron-transport chains and the ability of S. oneidensis to couple reversal of the anaerobic EET pathway to O2 reduction suggests that a specific connection between the two transport chains is likely to exist. However, to our knowledge, the mechanism allowing these organisms to transition between one electron transport chain, or quinone pool to another, is poorly understood.
We outline two possible mechanisms for a connection between the anaerobic EET pathway and the aerobic electron-transport chain in EEU in Fig. 1. First, the EET complex could transfer electrons to CymA, which then reduces menaquinone. Electrons could then be transferred from menaquinone to ubiquinone and into the aerobic electron-transport chain, finally arriving at Complex IV where they reduce O2. This option seems intuitive as under anaerobic conditions using fumarate as an electron acceptor, S. oneidensis was shown to require menaquinone in addition to several components of the EET complex to uptake cathodic currents10. However, knockout of the gene coding for cymA does not disrupt cathodic electron uptake when using O2 as a terminal electron acceptor13.
The lack of involvement of CymA in cathodic electron uptake under aerobic conditions suggests a second option (Fig. 1): that cathodic electrons bypass the menaquinone pool under aerobic conditions.
We speculate that the putative quinol-interacting protein SO_0400 and the putative ferredoxin SO_3662 (and possibly proteins coded by nearby genes) are directly involved in connecting the reverse EET pathway during electron uptake and the aerobic electron-transport chain. Notably, the lack of a phenotype for most of these proteins under anodic conditions supports the hypothesis of a distinct connection between a subset of EET proteins and the aerobic electron-transport chain during EEU (Fig. 1). This work has also highlighted some genes potentially involved in cell signaling or transcriptional responses that may help aid in facilitating electron uptake under specific conditions (SO_0841, SO_0181 and SO_3660).
Though the motivation of this work stemmed from the application of S. oneidensis to electrosynthesis, it is likely that the process of EEU plays a role in Shewanella physiology and ecology. It has been shown that minerals in nature can serve a capacitive or electron-storing function for microbes35. While electron deposition and electron uptake from minerals such as magnetite were shown to function as both sinks and sources for different metabolisms, it is feasible that a single organism with both functionalities could utilize minerals in this way—functionally storing charge akin to a battery. Though iron oxidation has only been demonstrated in a single Shewanella strain36, this could be due to the challenge of distinguishing biologic and abiotic iron oxidation in the absence of growth. As Shewanella are not generally capable of carbon fixation, the process of EEU is unlikely to have evolved as a growth-linked metabolism. However, previous work has linked electron uptake to maintenance of cell biomass or decreasing the rate of cell death, which could suggest a role in allowing cells to conserve energy under nongrowth conditions13. Interestingly, these genes appear to be widely conserved across the Shewanella, as well as other marine Gammaproteobacteria (several of which have also been implicated in EEU). This supports the yet-unexplored ecological role for EEU in sediment and/or marine microbes, though our knowledge of the specific activity and role of this process is still at its inception.
## Conclusions
EEU holds significant potential for conversion of CO2 and renewable electricity to complex organic molecules. While potential of unusual phenotypes like this can be limited by a lack of genetic understanding, especially in nonmodel organisms, synthetic biology can greatly expand the possibility for their improvement and application.
We used a whole-genome-knockout collection previously built with the rapid, low-cost knockout Sudoku method to screen the S. oneidensis genome for redox dye oxidation, a proxy for electron uptake. In this work, we have performed detailed electrochemical analyses, focusing on genes encoding proteins of unknown function. We have identified five previously uncharacterized genes in S. oneidensis that are involved in EEU from both solid phase and extracellular donors, coupled to both aerobic and anaerobic terminal electron acceptors.
Identification of these important genes lays the foundation for further genetic characterization of metal oxidation in nature, improvement of EEU in S. oneidensis and for synthetically engineering an electron-uptake pathway into easily engineered or synthetic microbes for powering recent advances in synthetic CO2 fixation15 and EET37 in E. coli.
## Methods
### Genome-wide AHDSred oxidation screen
The S. oneidensis whole genome knockout collection20,21 was screened for members unable to oxidize the redox dye anthra(hydro)quinone-2,6-disulfonate (AHDSred for the reduced form and AQDSox for the oxidized form)23,24, and subsequently reduce either fumarate22 or nitrate.
### Knockout collection construction
The S. oneidensis whole-genome-knockout collection was previously built with the Knockout Sudoku whole-genome-knockout collection construction procedure20,21. Prior to high-throughput screening, the mutant collection was duplicated with a multiblot replicator (Catalog Number VP 407, V&P Scientific, San Diego CA, USA) into 96-well polypropylene plates containing 100 μL of LB with 30 μg mL−1 kanamycin per well. The plates were labeled with barcodes and registration marks for identification in high-throughput analysis. The plates were sealed with an air-porous membrane (Aeraseal, Catalog Number BS-25, Excel Scientific) and grown to saturation overnight (at least 24 h) at 30 °C with shaking at 900 rpm.
### AHDSred preparation
Solutions of 25 mM AHDSred for screening experiments were prepared electrochemically. About 200–1000-mL batches of 25 mM AQDSox were prepared by dissolving 10.307 mg of AQDSox powder (Catalog no. A0308, TCI America) per 1 mL of deionized water at 60 °C. The AQDSox solution was then transferred to a three-electrode electrochemical system (Catalog no. MF-1056, BASI Bulk Electrolysis Cell) inside a vinyl anaerobic chamber (97% N2, 3% H2 and <20 ppm O2; Coy Laboratory Products, Grass Lake MI, USA). The system uses an Ag/AgCl reference electrode, a coiled Pt wire counterelectrode inside a fritted counterelectrode chamber and a reticulated vitreous carbon working electrode. The working-electrode potential was maintained at 700 mV vs. Ag/AgCl with a digitally controlled potentiostat (PalmSens, EmStat3). To enhance cell conductivity, 3 M H2SO4 acid was added to the working electrode, allowing the cell current to rise to ≈2 mA. AHDSred reduction is assumed to be complete when the solution is translucent yellow in color, typically after ≈7 h. The pH of the AHDSred stock solution was returned to 7.2 by addition of 3 M NaOH and the color changed from yellow to yellow-orange.
### Assay media preparation
Shewanella basal media (SBM) was used for all AHDSred oxidation assays. SBM consists of ammonium chloride (NH4Cl) (0.0086 M), dibasic potassium phosphate (K2HPO4) (0.0013 M), monobasic potassium phosphate (KH2PO4) (0.0017 M), magnesium sulfate heptahydrate (MgSO4.7H2O) (0.0005 M), ammonium sulfate ((NH4)2SO4) (0.0017 M), and HEPES (0.1 M). The media was buffered to pH 7.2 with sodium hydroxide (NaOH) and sterilized by autoclave. Trace-mineral supplement (5 mL L−1; Catalog no. MD-TMS, American Type Culture Collection (ATCC), Manassas, VA, USA) and vitamin supplement (5 mL L−1; Catalog no. MD-VS, ATCC) were added aseptically. No carbon source (e.g., lactate) is added to the SBM to ensure that electrons flowing into the cell are only from AHDSred.
Solutions of 25 mM potassium nitrate and 25 mM sodium fumarate adjusted to pH 7.2 and filter-sterilized were prepared as terminal electron acceptors for the AHDSred oxidation screens. These solutions were transferred to an anaerobic chamber at least the evening prior to any experiment to allow deoxygenation.
### AHDSred oxidation screens with nitrate
Aliquots of 10 μL of culture of each mutant from the replicated Shewanella whole-genome-knockout collection were transferred to 96-well polystyrene assay plates filled with 50 μL of SBM using a 96-channel pipettor (Liquidator 96, Mettler-Toledo Rainin LLC, Oakland, CA, USA). The assay plates were transferred to an anaerobic chamber for de-oxygenation for at least 9 h. The cell activity was confirmed by transferring cells from the assay plate to LB after the 9 h resting period in the anaerobic chamber and observing growth.
Following deoxygenation, 40 μL of 1:1 mixture of 25 mM AHDSred and 25 mM potassium nitrate were added to each well of the assay plates with a multichannel pipettor. The final concentration of both AHDSred and potassium nitrate in each well was 5 mM. Each assay plate was immediately photographed after media addition with the macroscope. All plates were repeatedly imaged for at least ≈40 h. After completion of the experiment, photographs were analyzed with the macroscope image analysis software. Mixing instructions for a single well of a 96-well assay plate are shown in Table 1.
### AHDSred oxidation screen with fumarate
Aliquots of 1 μL of culture of each mutant from the replicated S. oneidensis whole-genome-knockout collection were transferred to 96-well polystyrene assay plates filled with 59 μL of SBM using a multiblot replicator (Catalog Number VP 407, V&P Scientific, San Diego CA, USA). The assay plates were transferred to an anaerobic chamber for deoxygenation for at least 9 h. The cell activity was confirmed after the resting period by subculturing a small amount of assay media in LB. AHDSred oxidation was found to proceed much faster with fumarate than with nitrate, so the volume of cells added to the assay was reduced to make data collection more manageable.
Following deoxygenation, 40 μL of 1:1 mixture of 25 mM AHDSred and 25 mM sodium fumarate were added to each well of the assay plates with a multichannel pipettor. The final concentration of both AHDSred and sodium fumarate in each well was 5 mM. Each assay plate was immediately photographed after media addition with the macroscope. All plates were repeatedly imaged for at least ≈40 h. After completion of the experiment, photographs were analyzed with the macroscope image analysis software. Mixing instructions for a single well of a 96-well assay plate are shown in Table 1.
### Macroscope data acquisition system
An automated photographic data-acquisition system was used to record the results of AHDSred oxidation assays. The device consists of a digital single-lens reflex (DSLR) camera (any member of the Canon Rebel Series) controlled by a macOS computer running a custom data-acquisition program that downloads images to the computer and timestamps them. The camera shutter is controlled by a single-switch foot pedal (vP-2, vPedal), leaving both hands free to manipulate 96-well plates. The camera is mounted to a frame constructed with extruded aluminum rails (T-slot). Barcodes attached to each plate enable images to be automatically sorted. Registration marks on the barcodes allow for identification of well positions and each well to be associated with a specific gene-knockout mutant. The device allows a stack of 200 96-well plates to be imaged in ≈15 min. This process can be repeated immediately, allowing each plate to be quasi-continuously imaged. A set of photographs of three generations of macroscope device, along with photographs taken with the macroscope, are shown in Figs. S13S17.
This work has used two variants of the device: the first to image AHDSred oxidation in transparent 96-well assay plates, and the second to measure bacterial growth in 96-well storage plates covered with an air-porous membrane (Aeraseal, Catalog Number BS-25, Excel Scientific). In the first configuration, the camera is mounted above the AHDSred oxidation-assay plates. Each plate is placed inside a laser-cut acrylic holder and illuminated by an LED light pad from below (A920, Artograph). Barcodes are printed on transparent labels (Catalog no. 5660, Avery) and attached to the top of each plate. In the second configuration, the camera is mounted below the plate and illumination is provided from above with an LED light pad. A white barcode (SIDE-1000, Diversified Biotech, Dedham, MA, USA) is attached to the side of the plate and viewed through a 45° right-angle mirror (Catalog no. 47–307, Edmund Optics, Barrington, NJ, USA).
### Analysis of macroscope images
A custom Macroscope Image Analyzer program was developed with Pyzbar38, Pillow39, Numpy40, Matplotlib41 and OpenCV-Python42 using Python3. The program was used to process images taken with the macroscope device and aided in detection of loss-of-function mutants. The program handled the image analysis in four steps: creating barcodes, organizing the images collected with the macroscope by barcodes, collecting the data from the images, and finally presenting the data for analysis.
An additional image-analysis algorithm was developed with SciKit Image43 and SimpleCV44 to test images of 96-well storage plates for cross-contamination and growth-failure events by comparison with the collection catalog. The image-analysis algorithm updated the record for each well in the collection catalog with growth information to assist in rejection of false positives in the AHDSred oxidation screen due to growth failure.
The Macroscope Image Analyzer software is available on GitHub at https://github.com/barstowlab/macroscope-image-analyzer.
As AHDSred is oxidized to AQDSox, it changes color from yellow–orange to clear. Almost all information on the reduction state of the AHDSred/AQDSox dye can be found in the blue-color channel of the assay plate images. At the start of the assay, the intensity of the blue-color channel is low, and the dye is orange. As the AHDSred/AQDSox dye is oxidized and becomes clear, the intensity of the red channel remains approximately constant, with a small increase in green-channel intensity and a large increase in the blue-channel intensity. However, we found reporting the blue channel intensity as a proxy for the AHDSred/AQDSox redox state to be unintuitive.
To aid the reader and experimenter, we used the RGB color data,
$${\overrightarrow{C}} = \left[\begin{array}{c}r\\ g\\ b\end{array}\right]$$
to calculate a single number that represents how “yellow” a well is. The vector overlap (dot product) was calculated between the current color of the well and the most saturated yellow color in the assay photographic dataset.
The reference yellow color,
$${\overrightarrow{Y}}_{\!0} = \left[\begin{array}{c}225\\ 153\\ 0\end{array}\right]$$
was calculated relative to the reference white color,
$${\overrightarrow{W}}_{\!0} = \left[\begin{array}{c}255\\ 255\\ 255\end{array}\right]$$
Thus, the transformed yellow reference,
$${\overrightarrow{Y}}_{0}^{{\prime} }={\overrightarrow{Y}}_{0}-{\overrightarrow{W}}_{0}.$$
The transformed well color, relative to the white reference,
$${\overrightarrow{C}}^{\prime} ={\overrightarrow{C}}-{\overrightarrow{W}}_{0}.$$
The yellow intensity was calculated by normalizing the dot product between the transformed well color and the transformed yellow reference,
$$y=\frac{ {\overrightarrow{C}} ^{\prime} {\overrightarrow{Y}}_{0}^{{\prime} }}{{|{\overrightarrow{Y}}_{0}^{{\prime} }|}^{2}}.$$
The normalized yellow intensity has a maximum value of 1 when the well is yellow and a minimum value of 0 when it is clear.
The time series of colors for each gene shown as colored circles above each gene in Fig. 2 were generated by an algorithm that interpolated the multireplicate average of mean well-center color values for that mutant at 0, 1, 2, 3 and 4 h after the initiation of the oxidation experiment. A visual explanation of this process is shown in Fig. S1. AHDSred oxidation rates reported in Fig. S5 were calculated by a linear fit to the linear portion of the yellow intensity curve with Datagraph (Visual Data Tools).
### Bioelectrochemical cell construction and experimental conditions
A 3-electrode electrochemical cell based on a design by Okamoto et al.45 was assembled in-house, with the exception of a salt bridge that was included to contain the reference electrode (Part no. MF-2031, BASi, West Lafeyette, IN, USA). As described, the cell consisted of a working electrode made of ITO- (indium tin oxide) plated glass (Delta Technologies, Ltd., Loveland, CO, USA), a counterelectrode of platinum wire and an Ag/AgCl reference electrode suspended in 1 M KCl (HCH Instruments, Inc., Austin, TX, USA). The reactor volume contained approximately 20 mL of liquid with a working-electrode surface area of 10.68 cm2.
The electrochemical cell was used for chronoamperometry (CA) experiments that measure change in current over time and cyclic voltammetry (CV) experiments that measure current in response to a change in voltage. Both types of experiment were controlled with a 16-channel potentiostat (Biologic, France). In anodic CA experiments, the working electrode was poised at 422 mV vs. SHE. In cathodic CA experiments the working electrode was poised at −378 mV. This voltage minimizes hydrogen production while maximizing the biological cathodic activity. In CV experiments, the working-electrode potential was scanned between +422 mV and −378 mV vs. the SHE at a rate of 1 mV s−1.
### Shewanella culturing conditions
Cultures of wild-type S. oneidensis and S. oneidensis mutants were grown from glycerol stocks overnight in Luria Broth (LB) prior to each experiment. Aerobic and anaerobic growth curves and electrochemical experiments for all strains were performed in a Shewanella defined media (SDM)46, which we have found optimal for electrochemical experiments (note that SDM is not the same as SBM used in AHDSred oxidation assays). Aerobic cultures were performed in 50 mL volumes using 10 mM lactate as an electron acceptor shaking at 150 rpm at 30 °C. Kanamycin (Kan) and chloramphenicol (Chl) were added to LB and SDM media at concentrations of 100 μg mL−1 and 34 μg mL−1, respectively, for selection of transposon (Kan), clean gene deletion (Kan), and complementation (Chl + Kan) strains. The same growth conditions were used for anaerobic growth curves, with the exception that the media contained 20 mM fumarate and was purged with nitrogen gas for 10 min in serum vials.
For growth curves, strains were inoculated from overnight at a 100-fold dilution. Optical densities were recorded for triplicate cultures at 600 nm every 2–3 h. For cathodic growth, an overnight culture was backdiluted by a factor of 100 into SDM with 10 mM lactate and grown overnight. The overnight cultures grown in SDM were pelleted and resuspended in fresh SDM to an optical density at 600 nm of 0.1. About 20 mL of the resuspended culture was transferred to the working-electrode chamber of an electrochemical reactor. The reactor was attached to a 16-channel potentiostat (BioLogic) and the culture was anode-conditioned by poising the working electrode at +422 mV vs. SHE12,13. Anaerobic conditions needed to encourage biofilm formation and anodic current generation were maintained by continuous purging with N2.
After approximately 24 h, the reactors were detached from the potentiostat and the media containing planktonic cells was carefully removed to avoid disturbing the biofilm on the working electrode. The reactor was then refilled with 20 mL of fresh carbon-free SDM13. The reactors were then reattached to the potentiostat and the working electrode was cathodically poised at −378 mV vs. SHE. Air was slowly bubbled into the reactors via an aquarium pump, until a steady stream was reached to provide a source of O2.
To determine the portion of the cathodic current due to biological processes, the respiratory inhibitor Antimycin A was added to the electrochemical cell working-electrode chamber to a final concentration of 50 μM. Currents reported are the average difference between the steady-state currents pre and post Antimycin additions. To control for the effects of DMSO (the solvent for Antimycin A), blank injections of DMSO were made to the electrochemical cell and had no impact on current production. To confirm the effect of Antimycin A on biological current, addition of Antimycin A added to sterile minimal media in a reactor was performed and shown to have no impact on current production (Fig. S4).
### Protein collection and quantification
Protein quantification was used to assess the total biomass in bioelectrochemical experiments. At the end of an electrochemical experiment, the spent media from the reactor (~20 mL) was collected and the biofilm was scraped from the working electrode. Biomass was centrifuged at 8000 × g and the pellet was resuspended in 2 mL of 10% w/v NaOH. Total protein collected from the reactor was quantified using a Qubit TM Protein Assay Kit (Invitrogen, USA) according to the manufacturer’s protocol.
### Biofilm imaging and cell counts
Biofilms from δSO_0841 and wild-type bioelectrochemical reactors were imaged on Nikon TI-E eclipse inverted microscope equipped with UV fluorescence. Cells were stained using the FM 4-64X lipid dye (Molecular Probes, Life Technologies Inc). At least 20 images were taken from each of three reactors for δSO_0841 and wild-type replicate experiments. Cell dimensions and fluorescence intensity of a biofilm cross section were performed using the Nikon NIS-Elements software.
### Statistics and reproducibility
Uncertainties in bioelectrochemical measurements were calculated by analysis of at least three replicates for each bioelectrochemical experiment. All statistical analyses were performed in Excel and/or R.
Cathodic midpoint potentials were calculated from cyclic voltammogram (CV) scans. A subsection of the CV scan (between the aforementioned parameters found under electrochemical conditions) was separated into a forward scan (from 222 mV to −322 mV vs. SHE) and a reverse scan (−322 mV to 222 mV). The scan range was chosen to contain the voltage at which maximum current production was achieved. Because linearity could not be assumed in these data to generate a function for a first-derivative analysis, an alternative method was used to analyze the data directly. A cubic smoothing spline (spar = 0.70) was applied to the current data in R to remove noise, but still captures the general trend. An approximate derivative was then taken from these values. From this approximate derivative, the maximum current produced, and its corresponding voltage potential were found for both the forward and reverse scans, which were then averaged to find the midpoint potential. The midpoint potential from each replicate was then pooled and averaged to obtain the reported value.
Anodic and cathodic currents were determined by chronoamperometry (CA). An average of the final 100 data points of each CA scan was taken to determine the average current achieved for each biological replicate. The replicates were then averaged to determine the average current produced for each strain. Biological current was determined by subtracting the average current post antimycin addition from the average cathodic current prior to addition. To determine if the average current of any of the mutants was significantly different from the wild-type, the cumulative current data were compiled in Excel and then fitted to a linear model in order to perform a type-II analysis of variance (ANOVA) test with R. Following the ANOVA test, anodic and cathodic currents for all mutants were compared with wild-type by Tukey’s honestly significant difference (HSD) test to determine if any significant difference existed between them.
### Phylogenetic analysis
Phylogenetic trees of relatives of the genes identified in this work were generated by search for homologs, homolog alignment and tree assembly by a maximum likelihood method. Approximately 120–200 homologs for each gene identified in this work were identified with the top homolog hit program that interrogates the Integrated Microbial Genomes and Microbiomes Database (https://img.jgi.doe.gov/)47,48. Homolog sets for each gene were aligned with the Muscle aligner49 with default parameters.
Phylogenetic trees were generated for each homology set by a maximum likelihood method with RAxML 8.2.1150. In total, 100 trees were generated for each set of homologs for bootstrapping. Tree images and taxonomic metadata (Figs. S7S11) were generated and annotated using the interactive Tree of Life (iTOL) program51.
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
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2022-09-26 04:21:03
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https://brg.me.uk/?page_id=274
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## Objective
This is a site for people who have an interest in using the Python programming language to solve the mathematical and logic puzzles published weekly by the Sunday Times in the United Kingdom. My intent here is to focus much more on Python than the teasers themselves, which means that if your interest is the reverse, this site is probably not for you. If so, I recommend either the Sunday Times Teasers Discussion Group for manual solutions or Jim Randell’s S2T2 site for programmed approaches.
## Intended Audience
I intend to keep the focus here on two groups of people: firstly, young people who are taking up programming for the first time at home or at school and, secondly, people who probably learnt to use BASIC in the 1970’s on their first home computers and are now interested in moving to something that is just as easy to learn but is more in line with modern programming language concepts.
If you are an experienced programmer, it is very unlikely that you will learn anything new here; but you can help in meeting my objectives by posting example code that illustrates the use of Python for solving teasers. But please remember that the focus here is on Python and on encouraging its take up in my two target groups. And this has consequences for the sort of posts you should make.
## Code Philosophy and Style
Firstly, given the target audience, it is important to post neat, well commented and reasonably efficient Python code. In particular it should be comprehensible for people who are learning Python or are relatively new users of it. Posted code should hence stay close to the Python PEP8 Style Guide [1](except for the use of the two space indents needed here to conserve horizontal display space). Obfuscated code – deliberate or otherwise – is not acceptable and will be deleted!
There will, of course, be occasions where other external libraries will be useful but it is important that these are domain specific and available for installation using PIP. Examples include matplotlib for plotting and mpmath, numpy, sympy and scipy for scientific and mathematical work. There are also libraries that extend Python by adding features not provided by the standard library such as sorted containers, polynomials and multisets.
However libraries such as Jim Randell’s enigma.py that aggregate lots of small ‘standard’ routines together with routines specifically designed for solving teasers should not be used here. Jim’s library is perfect for its intended purpose – solving teasers – but it is far from perfect here because it hides much of the interesting Python code that needs to be visible on this site inside the library with the result that the code actually posted will often provide little in the way of Python insights and sometimes won’t even look like Python. I stress that this is NOT a criticism of Jim’s library but rather a belief on my part that its use is not consistent with my objectives for this site.
## Posting Python Code
When Python code is posted, syntax highlighting can be applied by putting <pre> before your code and </pre> afterwards. Please use Python 3 with spaces rather than tabs. Please don’t post teaser answers in your code since the aim here is to encourage people to use Python to find the answers, not to have them dished up ‘on a plate’ for all to see.
## Posting Mathematics
I am using MathJax to format mathematics on this site and this facility is also available in comments. The initial and final delimiters for display equations are a backslash followed by [ and ]; for inline equations they are a backslash followed by ( and ). For example:
$i\hbar\frac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>$
However, getting LaTex maths expressions right first time is near to impossible so people who do wish to post such mathematics in comments will need to find a way to validate their Latex expressions before posting.
[1] The Wing IDE, which is free for personal use, is excellent and has a command that will format code to align with PEP8. There is also an on-line PEP8 checker here (although this cannot be adjusted to cope with two space indents).
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2021-06-15 19:29:32
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http://www.drumtom.com/q/find-the-probability-the-portion-of-heads-will-be-between-0-49-and-0-51-when-a-coin-is-flipped-100-times
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# Find the probability the portion of heads will be between 0.49 and 0.51 when a coin is flipped 100 times?
• Find the probability the portion of heads will be between 0.49 and 0.51 when a coin is flipped 100 times?
Find the probability the portion of heads will be between 0.49 and 0.51 when a coin is flipped 100 ... be between 0.49 and 0.51 when a coin is flipped ...
Positive: 67 %
Introduction to Probability and ... is flipped, whether a coin will land with heads or ... up heads ten times in a row, the probability of getting ...
Positive: 64 %
### More resources
game theory - coin flipping game. up ... with 49% probability it lands heads. ... a winning probability of approximately 0.49\cdot(1-\frac{100}{200+x})+0 ...
Positive: 67 %
Statistics And Probability archive containing a full list of statistics and probability ... heads in 5 flips of a fair coin. Find ... 100 swim times at a ...
Positive: 62 %
A MICROSOFT EXCEL VERSION OF PARRONDO’S PARADOX ... (based on 100 coin ... playing the unfavorable coin (with a probability of heads of only ...
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2016-10-24 12:24:02
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http://en.wikipedia.org/wiki/Carbon_dioxide_scrubber
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# Carbon dioxide scrubber
A carbon dioxide scrubber is a device which absorbs carbon dioxide (CO2). It is used to treat exhaust gases from industrial plants or from exhaled air in life support systems such as rebreathers or in spacecraft, submersible craft or airtight chambers. Carbon dioxide scrubbers are also used in controlled atmosphere (CA) storage.
## Technologies
### Amine scrubbing
The dominant application for CO2 scrubbing is for removal of CO2 from the exhaust of coal- and gas-fired power plants. Virtually the only technology being seriously evaluated involves the use of various amines, e.g. monoethanolamine. Cold solutions of these organic compounds bind CO2, but the binding is reversed at higher temperatures:
CO2 + 2HOCH2CH2NH2 $\overrightarrow{\leftarrow}$ HOCH2CH2NH3+HOCH2CH2NH(CO2-)
As of 2009, this technology has only been lightly implemented because of capital costs of installing the facility and the operating costs of utilizing it.[1]
### Minerals and zeolites
Several minerals and mineral-like materials reversibly bind CO2.[2] Most often, these minerals are oxides, and often the CO2 is bound as carbonate. Carbon dioxide reacts with quicklime (calcium oxide), to form limestone (calcium carbonate).[3] The process is called Carbonate Looping. Other minerals include serpentinite, a magnesium silicate hydroxide and olivine.[4][5] Molecular sieves also function in this capacity.
Various scrubbing processes have been proposed to remove CO2 from the air, or from flue gases. These usually involve using a variant of the Kraft process. Scrubbing processes may be based on sodium hydroxide.[6][7] The CO2 is absorbed into solution, transferred to lime via a process called causticization and released in a kiln. With some modifications to the existing processes, mainly an oxygen-fired kiln, the end result is a concentrated stream of CO2 ready for storage or use in fuels. An alternative to this thermo-chemical process is an electrical one in which a nominal voltage is applied across the carbonate solution to release the CO2. While simpler, this electrical process consumes more energy as it splits water at the same time. Since it depends on electricity, the electricity needs to be renewable, like PV. Otherwise the CO2 produced during electricity production has to be taken into account. Early incarnations of air capture used electricity as the energy source; hence, were dependent on a carbon-free source. Thermal air capture systems use heat generated on-site, which reduces the inefficiencies associated with off-site electricity production, but of course it still needs a source of (carbon-free) heat. Concentrated solar power is an example of such a source.[8]
#### Sodium hydroxide
Zeman and Lackner outlined a specific method of air capture.[9]
First, CO2 is absorbed by an alkaline NaOH solution to produce dissolved sodium carbonate. The absorption reaction is a gas liquid reaction, strongly exothermic, (below)
2NaOH(aq) + CO2(g) → Na2CO3(aq) + H2O(l)
Na2CO3(aq) + Ca(OH)2(s) →-> 2NaOH(aq) + CaCO3(s)
ΔH° = -5.3 kJ/mol
Causticization is performed ubiquitously in the pulp and paper industry and readily transfers 94% of the carbonate ions from the sodium to the calcium cation.[9] Subsequently, the calcium carbonate precipitate is filtered from solution and thermally decomposed to produce gaseous CO2. The calcination reaction is the only endothermic reaction in the process and is shown (below).
CaCO3(s) → CaO(s) + CO2(g)
ΔH° = + 179.2 kJ/mol
The thermal decomposition of calcite is performed in a lime kiln fired with oxygen in order to avoid an additional gas separation step. Hydration of the lime (CaO) completes the cycle. Lime hydration is an exothermic reaction that can be performed with water or steam. Using water, it is a liquid/solid reaction as shown (below).
CaO(s) + H2O(l) → Ca(OH)2(s)
ΔH° = -64.5 kJ/mol
#### Lithium hydroxide
Other strong bases such as soda lime, sodium hydroxide, potassium hydroxide, and lithium hydroxide are able to remove carbon dioxide by chemically reacting with it. In particular, lithium hydroxide is used aboard spacecraft to remove carbon dioxide from the atmosphere. It reacts with carbon dioxide to make lithium carbonate:[10]
2 LiOH(s) + 2 H2O(g) → 2 LiOH.H2O(s)
2 LiOH.H2O(s) + CO2(g) → Li2CO3(s) + 3 H2O(g)
The net reaction being:
2 LiOH(s) + CO2(g) → Li2CO3(s) + H2O(g)
Lithium peroxide can also be used as it absorbs more CO2 per unit weight with the added advantage of releasing oxygen.[11] This is useful for breathing air systems and was used to remove CO2 from the Apollo spacecraft.
### Regenerative carbon dioxide removal system
The regenerative carbon dioxide removal system (RCRS) on the space shuttle orbiter used a two-bed system that provided continuous removal of carbon dioxide without expendable products. Regenerable systems allowed a shuttle mission a longer stay in space without having to replenish its sorbent canisters. Older lithium hydroxide (LiOH)-based systems, which are non-regenerable, were replaced by regenerable metal-oxide-based systems. A system based on metal oxide primarily consisted of a metal oxide sorbent canister and a regenerator assembly. It worked by removing carbon dioxide using a sorbent material and then regenerating the sorbent material. The metal-oxide sorbent canister was regenerated by pumping air at approximately 400 °F (204 °C) through it at a standard[clarification needed] flow rate of 7.5 cu ft/min (0.0035 m3/s) for 10 hours.[12]
### Activated carbon
Activated carbon can be used as a carbon dioxide scrubber. Air with high carbon dioxide content, such as air from fruit storage locations, can be blown through beds of activated carbon and the carbon dioxide will absorb onto the activated carbon. Once the bed is saturated it must then be "regenerated" by blowing low carbon dioxide air, such as ambient air, through the bed. This will release the carbon dioxide from the bed, and it can then be used to scrub again, leaving the net amount of carbon dioxide in the air the same as when the process was started.
### Other methods
Many other methods and materials have been discussed for scrubbing carbon dioxide.
## References
1. ^ Gary T. Rochelle "Amine Scrubbing for CO2 Capture" Science volumen 325, 1652 (2009). doi:10.1126/science.1176731
2. ^ Sunho Choi, Jeffrey H. Drese, and Christopher W. Jones "Adsorbent Materials for Carbon Dioxide Capture from Large Anthropogenic Point Sources" ChemSusChem 2009, 2, 796 – 854. doi:10.1002/cssc.200900036
3. ^ "Imagine No Restrictions On Fossil-Fuel Usage And No Global Warming". ScienceDaily. April 15, 2002.
4. ^ "Natural Mineral Locks Up Carbon Dioxide". Sciencedaily.com. 2004-09-03. Retrieved 2011-06-01.
5. ^
6. ^ Chang, Kenneth (2008-02-19). "Scientists would turn greenhouse gas into gasoline". New York Times. Retrieved 2009-10-29.
7. ^ "Chemical 'sponge' could filter CO2 from the air - environment - 03 October 2007". New Scientist. Retrieved 2009-10-29.
8. ^ "Can technology clear the air? - environment - 12 January 2009". New Scientist. 2009-01-12. Retrieved 2009-10-29.
9. ^ a b Zeman, F. S.; Lackner, K. S. (2004). "Capturing carbon dioxide directly from the atmosphere". World Resour. Rev. 16: 157–72.
10. ^ Jaunsen, JR (1989). "The Behavior and Capabilities of Lithium Hydroxide Carbon Dioxide Scrubbers in a Deep Sea Environment". US Naval Academy Technical Report. USNA-TSPR-157. Retrieved 2008-06-17.
11. ^ Petzow, G. N.; Aldinger, F.; Jönsson, S.; Welge, P.; Van Kampen, V.; Mensing, T.; Brüning, T. (2005). "Beryllium and Beryllium Compounds". Ullmann's Encyclopedia of Industrial Chemistry. doi:10.1002/14356007.a04_011.pub2. ISBN 3527306730. edit
12. ^ "Carbon Dioxide Removal". Hamilton Sundstrand. Archived from the original on 2007-10-31. Retrieved 2008-10-27. "The new metal-oxide-based system replaces the existing non-regenerable lithium hydroxide (LiOH) carbon dioxide (CO2) removal system located in the EMU’s Primary Life Support System."
13. ^ "Adsorption and Desorption of CO2 on Solid Sorbents" (PDF). Retrieved 2011-06-01.
14. ^
15. ^
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2014-04-20 14:56:03
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http://math.stackexchange.com/questions/305386/center-manifold-theorem
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# Center Manifold Theorem.
Consider a continuous-time dynamical system defined by $$\dot{x}=f(x),\ \ \ \ \ \ x\in\mathbb R^{n} \ \ \ \ \ \ \ (1)$$ where f is sufficiently smooth, $f(0) = 0$. Let the eigenvalues of the Jacobian matrix A evaluated at the equilibrium point $x_0 = 0$ be $\lambda_1, ... , \lambda_n$. Suppose the equilibrium is not hyperbolic and that there are thus eigenvalues with zero real part. Assume that there are $n_+$ eigenvalues (counting multiplicities) with $\Re (\lambda)>0$, $n_0$ eigenvalues with $\Re (\lambda)=0$, and $n_-$ eigenvalues with $\Re (\lambda)<0$. Let $T^c$ denote the linear eigenspace of A corresponding to the union of the $n_0$ eigenvalues on the imaginary axis. Let $\varphi^t$ denote the flow associated with (1).
Center Manifold Theorem:
There is a locally defined smooth $n_0$-dimensional invariant manifold $W_{loc}^c (0)$ of (1) that is tangent to $T^c$ at $x=0$. The manifold $W_{loc}^c$ is called the center manifold. Moreover, there is a neighborhood U of $x_0 =0$, such that if $\varphi^t x\in U$ for all $t\geq 0$ ($t\leq 0$), then $\varphi^t x\rightarrow W_{loc}^c (0)$ for $t\rightarrow +\infty$ ($t\rightarrow -\infty$).
My question relates to smooth continuous-time system that depends smoothly on a parameter: $$\dot{x}=f(x, \alpha),\ \ \ \ \ \ x\in\mathbb R^{n},\alpha\in\mathbb R\ \ \ \ \ \ \ (2)$$ Is still valid the Center Manifold Theorem for system (2)? If the system (2) hasn't imaginary eigenvalues ($\Re (\lambda)=0$) can we say that there isn't center manifold?
thank you very much
-
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2014-03-11 18:59:20
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https://plainmath.net/91038/the-larger-of-2-numbers-is-5-less-than-5
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# The larger of 2 numbers is 5 less than 5 times the smaller number. The sum of the two numbers is 61. What are the two numbers?
The larger of 2 numbers is 5 less than 5 times the smaller number. The sum of the two numbers is 61. What are the two numbers?
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June Rowland
Let the larger number be x and smaller number be y.
$\therefore$ As per the question we get :
$5y-5=x$ and $x+y=61$
Substituting the value of x, that is 5y -5 from the first equation to the second equation - $x+y=61$, we get :
$5y-5+y=61$
$6y-5=61$
$6y=61+5$
$6y=66$
$y=\frac{66}{6}$
$\therefore y=11$
Now substituting this value in the equation x+y=61, we get :
$x+11=61$
$\therefore x=61-11=50$
Hence the two numbers are 50 and 11 .
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2022-12-04 06:13:09
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http://thepit.ja-galaxy-forum.com/index.php?t=msg&th=23770&goto=353739&
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Home » MODDING HQ 1.13 » Flugente's Magika Workshop » New (unofficial) merc: Max Payne
New (unofficial) merc: Max Payne Sun, 10 June 2018 23:42
Flugente Messages:3384 Registered:April 2009 Location: Germany
Thanks to Spinasso and my small contributions, we can now use Max Payne as yet another merc. He works just like my other mercs, but as that would fit the third game, can be found in the most run-down bar of Arulco (perhaps he wants to take care of yet another mob?).
I admit the NPC recruitment phase isn't the best (we were severely limited in sound and portrait choices), but he works just fine (and now I know better how .npc scripts work [new bugs to find ]). A grumpy, alcoholic psycho that narrates some of his lines fits just in with the rest of our selection of wackjobs.
We can justify a variety of traits for him - I went with the above, which makes him efficient with pretty much all firearms. In case he seems a bit weak, stat-wise, keep in mind that this is him at about the time of the third game, when he was a broken alcoholic that had already survived the first 2 games somehow.
You can of course change him in Mercprofiles.xml etc. - he has profile number 214.
At the moment we only have soundfiles for Max Payne 3, which is somewhat limiting. I can't really make any solid buddy or hate relationship with that. Perhaps someone might be interested in getting the sounds from part 1 and 2 ?
You might also notice that the Ryder twins have new faces, because edmortimer and townltu got busy and made portraits. 200% more chinese, woohoo!
I've redone the gear kits for Shepard - we now have 3 kits along the same idea increasing in price and quality, and 2 special kits (P90 for CQB and SVU for sniping).
This was built for r8570 & GameDir r2424.
As these mercs are obviously not from the Jagged Alliance lore, they won't go into any official release.
You can get Max and the others from http://kermi.pp.fi/JA_2/Mods_v1.13/Flugente/Additional_Mercs/
[Updated on: Mon, 11 June 2018 00:55]
“What are you doing?” Joffrey interrupted him.
“… can’t a man give a few last words for his son to carry?”
“…Who told you your son was making it out of this field alive?”
If you want, you can donate to me. This will not affect how and what I code, and I will not code specific features in return. I will be thankful though.
Re: New (unofficial) merc: Max Payne[message #353736 is a reply to message #353732] Mon, 11 June 2018 06:50
Kerghnox Messages:3 Registered:August 2016
I like these extra mercs you've been making. You should do Ash Williams one of these days, he'd fit right in.
Re: New (unofficial) merc: Max Payne[message #353739 is a reply to message #353736] Mon, 11 June 2018 19:27
Ventris Messages:1 Registered:May 2018 Location: Idaho
That would be amazing! Sawed off shotgun in one hand, and hedge trimmer for the chainsaw arm. I wonder if it would be too crazy to make it so he could equip both the hedge trimmer and still shoot the shotgun or other one handed weapon to simulate the chainsaw being attached to his arm. Great idea in any case especially with the zombies.
Thank you very much Flugente for all of your hard work!!! It is extremely appreciated!!!!
Re: New (unofficial) merc: Max Payne[message #353789 is a reply to message #353739] Thu, 14 June 2018 21:13
townltu Messages:217 Registered:December 2017 Location: here
Somewhat OT question about PC placement, so pls feel free to move or delete message.
If i get that right, placement of Max in San Mona relies entirely on entries in strategicEventHandler.lua,
i.e. if sector is not loaded (not gWorldSector[x/y]) and Max isnt either dead or hostile.
So i would have to add a PC according to "How do I do that?" Adding a new character
then clone Maxs block in strategicEventHandler.lua, and adjust sector&tile to place a new PC.
But how do i delay the spawn of the new PC depending on a specific event/fact (preferably with some days delay after fact became true),
and let him appear in one of the sectors where quests ID 11, 14 or 16 respectively 23, 29 or 33 can be finished?
Last not least is it possible to suppress the "your squad has noticed someone in sector x/x" message?
Else it would make much less sense to hide a merc.
(... for the few players who meet him before its common knownledge how&where to find;)
Re: New (unofficial) merc: Max Payne[message #353790 is a reply to message #353789] Thu, 14 June 2018 22:03
Flugente Messages:3384 Registered:April 2009 Location: Germany
First of all, this isn't the usual method of placing NPCs - the stock ones are placed in the map itself etc.. It seemed to me that this was a much, much easier way, and... well, it is
The 'noticed someone' check only appears in wilderness sectors, I see no easy deactivation method otherwise.
Checks for quests and facts are easy:
if (not ( ( gWorldSectorX == 6) and ( gWorldSectorY == 3) ) and not MercIsDead(Profil.MAX)
and (gubQuest( Quests.QUEST_ESCORT_SKYRIDER ) == qStatus.QUESTINPROGRESS) and (CheckFact( Facts.FACT_ANGEL_LEFT_DEED, 0 ) == true) ) then
...
“What are you doing?” Joffrey interrupted him.
“… can’t a man give a few last words for his son to carry?”
“…Who told you your son was making it out of this field alive?”
If you want, you can donate to me. This will not affect how and what I code, and I will not code specific features in return. I will be thankful though.
Re: New (unofficial) merc: Max Payne[message #353791 is a reply to message #353790] Fri, 15 June 2018 01:07
townltu Messages:217 Registered:December 2017 Location: here
Thanks so far;) and sorry for vague explanation of specific part, didnt want to leak too much.
So is it possible to set the spawn sector for the new PC equal to the sector where one of the classic Skyrider, Gabby etc spawn happens?
i.e which term to use instead of fixed sector coordinates [6,3,0] in command AddNPCtoSector (Profil.MAX,6,3,0),
or do i need to check all sectors where Gabby may be present whether he is actually there,
and let the result control a jump to fitting spawn command for the new PC?
Or perhaps both not doable atm, at all?
Re: New (unofficial) merc: Max Payne[message #353807 is a reply to message #353791] Fri, 15 June 2018 22:45
Flugente Messages:3384 Registered:April 2009 Location: Germany
Take a look at function GetIntelAndQuestMapData( aLevel ) in strategicmap.lua, this might be useful as it shows how to get coordinates of a character:
if ( gubQuest( Quests.QUEST_ESCORT_SHANK ) == qStatus.QUESTINPROGRESS ) then
SetIntelAndQuestMapDataForSector(GetCharacterSectorX(Profil.JAKE), GetCharacterSectorY(Profil.JAKE), -1, MapSymbols.EXCLAMATIONMARK_BLUE, "Escort Shank to Jake", "")
end
“What are you doing?” Joffrey interrupted him.
“… can’t a man give a few last words for his son to carry?”
“…Who told you your son was making it out of this field alive?”
If you want, you can donate to me. This will not affect how and what I code, and I will not code specific features in return. I will be thankful though.
Re: New (unofficial) merc: Max Payne[message #353826 is a reply to message #353732] Mon, 18 June 2018 16:54
Spinasso Messages:6 Registered:May 2017
In a Matthew McConaughey's voice: "Alright alright alright...!"
Just want to tell you that I've started to work on Wei Shen (Sleeping Dogs's main character). I got the sound files already. 1h30 hour of quotes... Here we go again.
Btw, I'll need help with the face again too, otherwise it will be imp 203...
Next project will be Saint's Row male player's voices. Specially 1 (Troy Baker) and 3 (the one with the cockney accent).
Re: New (unofficial) merc: Max Payne[message #353827 is a reply to message #353826] Mon, 18 June 2018 19:54
Flugente Messages:3384 Registered:April 2009 Location: Germany
Great I have only a very rough idea on how those specific characters sound, but GTA-like game characters likely have a ton of sounds to choose from.
Keep in mind that we can use a lot more sounds thanks to Additional Dialogue.
“What are you doing?” Joffrey interrupted him.
“… can’t a man give a few last words for his son to carry?”
“…Who told you your son was making it out of this field alive?”
If you want, you can donate to me. This will not affect how and what I code, and I will not code specific features in return. I will be thankful though.
Re: New (unofficial) merc: Max Payne[message #353830 is a reply to message #353732] Tue, 19 June 2018 00:21
calu Messages:8 Registered:February 2007 Location: Germany
Flugente wrote on Sun, 10 June 2018 23:42
Perhaps someone might be interested in getting the sounds from part 1 and 2 ?
You can download the full RAS files from Max Payne 1 here: https://drive.google.com/file/d/0Byxv4BMtWd2yWWY2bFBRY1F4aUE/view
Install MAX-FX Tools and run in CMD something like
"C:\Program Files (x86)\MAX-FX Tools\RasMaker\RASMaker.exe" -x "C:\Path\to\ras\files\x_english.ras" "C:\Path\to\extract\wav\files\x_english\"`
This will create a couple of folders in the last path of the command. Look in "data\database\sounds" to find all game dialogue sounds. Random mutterings from Max during gameplay can be found in "characters\max". The graphic novel narration is in "story", dialogue from cutscenes and stuff in "ingame". The "enemy" folder also has some funny stuff from the junkies, mobsters...
Re: New (unofficial) merc: Max Payne[message #353832 is a reply to message #353830] Tue, 19 June 2018 00:46
Flugente Messages:3384 Registered:April 2009 Location: Germany
Thank you! There goes the next weekend I guess...
“What are you doing?” Joffrey interrupted him.
“… can’t a man give a few last words for his son to carry?”
“…Who told you your son was making it out of this field alive?”
If you want, you can donate to me. This will not affect how and what I code, and I will not code specific features in return. I will be thankful though.
Re: New (unofficial) merc: Max Payne[message #354102 is a reply to message #353832] Fri, 20 July 2018 14:34
Ortis Messages:2 Registered:April 2015
Can I use them with other versions than r8570 & GameDir r2424?
Re: New (unofficial) merc: Max Payne[message #354103 is a reply to message #354102] Fri, 20 July 2018 19:46
Flugente Messages:3384 Registered:April 2009 Location: Germany
Yes, but note that I've altered several scripts for this. If the game doesn't recognize some of the functions in there, you will get a crash, in that case, one has to check what to use.
Also note that these mercs cannot be added to an ongoing campaign - merc profile changes only affect new campaigns.
“What are you doing?” Joffrey interrupted him.
“… can’t a man give a few last words for his son to carry?”
“…Who told you your son was making it out of this field alive?”
If you want, you can donate to me. This will not affect how and what I code, and I will not code specific features in return. I will be thankful though.
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2018-08-15 01:13:47
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http://tiku.21cnjy.com/tiku.php?mod=quest&channel=4&cid=1588&xd=3
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考点：过去进行时及其被动式
• —David, are you listening to me?
—Sorry, Dad. I ________ to make sense of what the reporter said. A．was trying4 B4 Q5 I5 R$Q B．have tried# R" ?0 I7 N0 ?& X C．am trying4 a# R! R3 S8 B% W D．tried$ W8 i2 F7 i4 j
• —What happened? I called you several times yesterday evening, but I couldn't get through.
—Sorry, my sister and I ________ on the phone all the time. A．talked B．have talked C．were talking D．have been talking
• —What's up? You didn't pick up the phone just now.
—I'm sorry that I ________ dancing with music on, but it's done now. A．practised' h" N$C% M O' W' d) H: \: K6 [# [ B．was practising/ A, ]# K0 E" X; T7 C．am practising6 N% g8 [1 A* V' W; e3 ] D．have practiced4 B+ S @7 g$ ]0 J( c& V8 S
• —What happened to Bill?
—He ________ really fast when suddenly he ran into a parked car. A．had run' ]* I4 B4 + I$P2 S H B．was running6 _, F( E6 j T! J, e+ @ C．has run9 b e; F- L0 i D．has been running5 P* a8 \. j* h& W4 T • Linda felt someone ________ on her way home from work yesterday evening, and therefore she was frightened and looked back from time to time. A．followed* H& Z4 a" V+ Q1 b+ ]& W' ( I. O B．would follow- T. H. L: d& U/ T* C．had followed B" L/ e" U; ' i5 i$ P D．was following( ^9 V- # c5 U' P% A7 a
• —Have they got their new house painted yet?
—I don't know. But I was told it ________ when I met them last week. A．had been painting0 S; Q# P8 e1 ?1 S B．would be painting) O# a& M2 F$B ?1 H C．was painted) W- % d% \. f2 f, B( ] D．was being painted+ h0 S* A/ \" U7 ]) I( S • —Can you tell me if Tom has finished his homework? —I've no idea. He ________ it this morning. A．has being doing B．had done C．is doing D．was doing • The teacher and writer as well as some students something when we entered the room. A．discussed% T# J* S( ^$ ^+ Q6 P* U B．were discussing' L$P! T4 i! Y2 B3 V1 j" L C．have discussed1 5 h- A+ c2 M D．was discussing- M9 ?0 V% [6 K/ T* \& @' N • The old lady who caught a bad cold complained that the doctor was ______ too much for the treatment given to her． A．paying7 E' X) b* C1 ]. h B．offering M- I. j, H1 H; Q0 e C．spending0 F7 & c: K$ f) R. ] D．charging K' ]5 L. Q$f; ^, O; c& Z; U • The elevator in the teaching building was always in the summer vacation. That was why I chose to climb the stairs instead. A．breaking off4 K' N f1 ]/ ?! \4 G+ 6 ^% g) j B．breaking up+ d, \* B0 R) ` C．breaking out' g. J% O5 L: H/ ]" K ?6 P D．breaking down; H4 j$ ?" P \, X0 M" f
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2018-03-22 08:14:38
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https://physics.stackexchange.com/questions/114083/non-local-structure-of-field-theory/464223
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# Non-local structure of field theory
Can someone explain what is non-local structure of field theory? I know you cannot have $\phi(x) \phi(y)$ term in Lagrangian which indicates the non-locality. However, why I cannot have the non-local terms as long as I have causality maintained? In QFT, one should not write an operator like $\phi(x)^2$ which will yield singularities like $\delta (x-x)$ if one does OPE? How should I understand the locality in field theory and OPE sense consistently?
The situation is more subtle than suggested by the other two answers as the following example shows.
In $$d\ge 2$$ dimensions, consider the Euclidean Gaussian field with propagator given in momentum space by $$\frac{1}{p^{d-2\Delta}}$$ where $$\Delta$$ is in the interval $$\left(\frac{d-2}{2},\frac{d}{2}\right)$$. This satisfies the unitarity bound and in fact all the Osterwalder-Schrader axioms. Therefore, by analytic continuation to Minkowski space, this results in a QFT that satisfies all the Gårding-Wightman axioms including locality: $$[\phi(x),\phi(y)]=0$$ if $$x-y$$ is space-like.
On the other hand, the Lagrangian for this model is nonlocal.
• Just want to add that this is also called generalised free theory, or mean filled theory in some contexts. It however it's not completely accurate (in the most common sense) to say that it is local. The axiom you refer to is about causality. This theory can be described as the boundary dual of free massive scalar in rigid AdS space, and causality of this theory is due to causality of the bulk theory. However, this theory is not local in the usual sense of this word, e.g. it does not have a stress-energy tensor. – Peter Kravchuk Apr 4 at 7:04
• Is there really a problem with $\Delta>d/2$? – Peter Kravchuk Apr 4 at 7:05
• @PeterKravchuk: "usual" in usual sense of locality depends on which community one belongs to. For CFT folks, I know that having a local stress tensor is usually included in the definition of locality. For axiomatic QFT folks, locality is usually just the space-like separation commutation above. It is also the main ingredient is the definition of "local relative to" leading to the notion of Borchers class. Also, I am used to worry about contact terms, so I took $\Delta<d/2$ to make my life easier. But otherwise a larger $\Delta$ is fine. – Abdelmalek Abdesselam Apr 4 at 13:37
It is impossible to maintain causality with an operator that is non-local. The reason is very simple:
If you have non-local operators, the equation of motion will include fields at a different spacetime event. There is no way of imposing that information can only be transmitted by the speed of light, because the communication from that other spacetime event to your position is manifestly instantaneous.
• So should I say, the localness implies the condition of causality. Or are they equivalent(in the sense of local if and only if causal) in this context? – user45765 May 24 '14 at 1:24
• If you integrate out a field (eg. the photon) you get a non-local interaction (eg. the Coulomb force) but causality is maintained. – Ryan Thorngren Mar 3 at 20:15
When you introduce $\phi(x) \phi(y)$ for $x \ne y$, you postulate an action at a distance, whichever the interval between said events is: time-like, null, or what. In other words, you admit some essence that isn’t a field, but propagates through the spacetime directly, in a point-to-point fashion. I am not sure you can’t maintain causality is such theory, but it will be not a QFT, but a hybrid theory. It would join two competing paradigms: one of a field, and another of an action at a distance. It might violate the Occam’s razor principle before other problems with it would appear.
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2019-12-16 05:50:54
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http://www.pudn.com/Download/item/id/1730554.html
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PSOBC-dvhop
(Based on the improved particle swarm algorithm of a network node localization algorithm, and the method and traditional DVhop were compared.)
PSOBC-dvhop
...........\BG-PSO
...........\......\BG_PSO.asv,6360,2008-06-09
...........\......\BG_PSO.m,6019,2011-10-23
...........\......\html
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...........\......\Live_fn.m,184,2008-06-07
...........\psojiben.m,2334,2007-09-24
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2021-01-19 17:06:07
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https://st4blog.wordpress.com/2017/05/17/the-dangers-of-one-dimensional-over-interaction-between-body-and-soul/
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The Dangers of One-Dimensional Over-Interaction Between Body and Soul
With a 24 hours long break behind us, we assembled in the afternoon of day 8 for second in the series of lectures on SYK model by Pranjal (TIFR). He started with reviewing some results from the 1st day, including the value of $b$ which was our homework problem.
Followed by this, he discussed an SYK-like (tensor) model.
$\mathcal{L} = i\psi_j^{abc}\partial_t \psi _j^{abc} + i^{3/2}\frac{j}{3!}\psi_1\psi_2\psi_3\psi_4$
where
$\psi_1\psi_2\psi_3\psi_4 = \psi_1^{l_{14}l_{13}l_{12}}\psi_2^{l_{12}l_{24}l_{23}}\psi_3^{l_{23}l_{13}l_{34}}\psi_4^{l_{34}l_{24}l_{14}}$
and $l_{ij}$ is trifundamental representation of product of three copies of $O(N)$. He made the remark that tensor models are unitary, though Hilbert space in these grows much faster than in SYK model because of large number of fermionic degrees of freedom. Most importantly, large N physics of tensor models is same as that of SYK model.
Next, we moved on to the large-$q$ limit of the SYK model, when there’s a large number of fermions interacting at a time. It is in this limit that the model has been shown to be solvable. Pranjal proposed an ansatz in large $q$ limit for the two-point function
$G(\tau_1,\tau_2) = \frac{1}{2}sgn(\tau_1-\tau_2)\left\{1 + \frac{g(\tau)}{q} + \mathcal{O}(\frac{1}{q^2})\right\}$
Here, first term is the two-point function in UV limit \& $g(\tau)$ is an unknown function. Fourier transforming this equation and substituting it in Schwinger-Dyson equation gave the Fourier transformed expression for 1PI, $\Sigma(\tau_1,\tau_2)$, which had been derived in the last lecture. Inverse transforming this \& solving the resulting differential equation
$\frac{\partial_{\tau}^2}{q}\left\{sgn(\tau)g(\tau)\right\} = J^2 sgn(\tau)2^{1-q}e^{g(\tau)}$
with boundary conditions $g(0)=g(\beta)=0$ gave the expression for $g(\tau)$. Using this, one can write the two point function. During this, we had a cameo by Rohan Poojary (TIFR) who termed his explanation for $sgn$ function on circle ‘the vaguest ever’! Though we were quite satisfied with it.
After this, we moved on to four-point functions, he termed them ‘ladder diagrams’.
This equation is
$\mathcal{F}_{n+1} = K * \mathcal{F}_n,$
where $n$ represents number of rungs in the ladder.
$\mathcal{F} = \mathcal{F}_0 + K*K*\mathcal{F}_0 + ... = \frac{1}{1-K}*\mathcal{F}_0.$
The above equation he wrote in the eigenbasis as,
$\Sigma\frac{1}{1-K(n)}\frac{\langle\psi_n|\mathcal{F}_0\rangle}{\langle\psi_n|\psi_n\rangle}|\psi_n\rangle = \mathcal{F},$
and started the program to evaluate its eigenvalues and eigenvectors.
He argued that $K$ acting on the three-point function gives back the same three-point function in IR limit ($J \rightarrow \infty$).
Motivated by this, he wrote the eigenvalue of $K$ using the form of three-point function in a CFT. To find the eigenvectors of the kernel, he showed that it commutes with Casimir for conformal group implying that these may have simultaneous eigenvectors. By another argument, which he could not exactly formulate, it can be shown that kernel can be written as a function of Casimir. Thus one can find instead the eigenvectors of Casimir.
This is where he ended his lecture.
Madhusudhan Raman (IMSc) delivered the evening talk about basics of supersymmetry, in order to lay the groundwork for Victor’s talks on supersymmetric localization. He started with the Coleman-Mandula theorem which states that Poincare and internal symmetries cannot be combined an any way except trivially. Two interesting points came up during the discussion: (i) that this was a statement that was true under the assumption that the resulting theory has a non-trivial and analytic S-matrix, and (ii) that it doesn’t apply to lower-dimensional quantum field theories! Coleman and Mandula assumed that the group $G$ is a Lie group. The way out is to $Z_{2}$-grade the algebra to include both commutators and anticommutators i.e.
$[O_{a},O_{b}]=O_{a}O_{b}-(1)^{n_{a}n_{b}}O_{b}O_{a}$
where,
$n_{a} = 0 \text{ for bosonic operators and} \ 1 \text{ for fermionic operators.}$
and he begin with definition of the supersymmetry algebra
$(a)~~~~~ [Q_{\alpha},M^{\mu\nu}]= \iota (\sigma^{\mu\nu})_{\alpha}^{\beta}Q_{\beta}$
where, $Q_{\alpha}$ is a spin-1/2 operator so it transforms like
$Q_{\alpha} \simeq(I-\iota \omega_{\mu\nu}\sigma)_\alpha^\beta Q_{\beta}.$
Then, he spoke about the effect of translations,
$(b)~~~~~~~~~~[Q_{\alpha},P^{\nu}]= 0.$
He explained by logical argument that the above commutator can be fixed in a simple way: let’s assume it transformed as
$[Q_{\alpha},P^{\mu}]= c(\sigma^{\mu})_{\alpha\dot{\alpha}}\bar{Q}^{\dot{\alpha}},$
then since the l.h.s. satisfies Jacobi’s identity with respect to $P^{\mu}$ i.e.
$[P_{\mu},[P^{\mu},Q_{\beta}]]+cycl.= 0 ~ \text{which implies} \ c=0$
Next came the effects of the supercharges on each other,
$(c)~~~~~~\{Q_{\alpha},Q^{\beta}\}= K(\sigma^{\mu\nu})_{\alpha}^{\beta}M_{\mu\nu}$
using the same arguments we can show $K=0$.
$(d)~~~ \{Q_{\alpha},\bar{Q}_{\dot{\beta}}\}= 2(\sigma^{\mu})_{\alpha \dot{\beta}}P_{\mu}$
From the last relation it is clear that the supersymmetry transformation knows about the underlying space-time and also he explained why supersymmetry commutes with internal symmetries which are clear from this relation,
$[T_{a},Q_{\alpha}]=0.$
Then he explained the R-symmetry
$Q_{\alpha}\longrightarrow \exp{(\iota\gamma)}Q_{\alpha}\\ ,~~~~~Q_{\dot{\alpha}}\longrightarrow \exp{(-\iota\gamma)}Q_{\dot{\alpha}},~~~ \text{such that}~~ R:\exp{(-\iota\gamma t)}Q_{\alpha}\exp{(\iota\gamma t)}$
It satisfies following relation
$[Q_{a},R] = Q_{a}$ and
$[\bar{Q}_{\dot{a}},R] =-\bar{Q}_{\dot{a}}$
so we observe that the supercharges are charged under R-symmetry. Then, he extend the SUSY algebra for $N$ charges
$\{Q^{A}_{\alpha},\bar{Q}_{\dot{\beta},B}\}= 2(\sigma^{\mu})_{\alpha \dot{\beta}}P_{\mu}\delta^{A}_{B}$
where, A,B = 1, 2, 3, …, N.
$\{Q^{A}_{\alpha},Q_{\beta}^{B}\}= (\sigma^{\mu})_{\alpha \beta} Z_{A B},$
where $Z_{AB}$ is known as the central charge.
We know that Casmirs for Poincare algebra tell us how to label 1-particle states, i.e. we use the mass and spin/helicity.
$C_{1}= P_{\mu}P^{\mu}$ and
$C_{2}= W_{\mu}W^{\mu}; ~~~ W_{\mu}=1/2\epsilon_{\mu\nu\rho\sigma}P^{\nu}M^{\varrho\sigma}$
then he explain what’s happen if we add SUSY (for example N=1) $C_{1}$ is still a Casimir but $C_{2}$ changes to a superspin that is defined by
$C_{2}= C_{\mu\nu}C^{\mu\nu}; ~~~ C_{\mu\nu}=B_{\mu}P_{\nu}-B_{\nu}P_{\mu} \& ~~ B_{\mu}=W_{\mu}-\frac{1}{4}\bar{Q}_{\dot{\alpha}}\bar{\sigma}_{\mu}^{\dot{\alpha}\alpha}Q_{\dot{\alpha}}$
Then he gave an example for N=1 supersymmetry for massless particles, where we boost to the frame $P_{\mu}=(E,0,0,E)$. The SUSY algebra then gives
$\{Q_{\alpha},\bar{Q}_{\dot{\beta}}\}= 4 E\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$
and we can define raising operator and lowering operators as
$a=\frac{ Q_{1}}{2\sqrt{E}},~~~a^{+}=\frac{ \bar{Q}_{1}}{2\sqrt{E}}$
We know that for the massless particles helicity is good quantum number, as demonstrated by
$J^{3}a|P_{\mu}\lambda\rangle=(\lambda-1/2)a |P_{\mu}\lambda\rangle$
In the last hour he spend the time to explain us a superspace $(X^{\mu},\theta^{\alpha},\bar{\theta}^{\dot{\alpha}})$ (the fermionic directions are the soul to the bosonic directions’ body, as pictured above) and super-field
$F(X^{\mu},\theta^{\alpha},\bar{\theta}^{\dot{\alpha}})=f(x)+\theta\psi(x)+\bar{\theta}\bar{\psi}(x)+\theta\theta m(x)+\bar{\theta}\bar{\theta} n(x)+{\theta}\sigma^{\mu}\bar{\theta}v_{\mu}+{\theta}{\theta}\bar{\theta}\bar{\lambda}(x)+\bar{\theta}\bar{\theta} \theta \rho(x)+{\theta}{\theta}\bar{\theta}\bar{\theta}\phi(x)$
Madhu stated that the idea of susysymmetry in superspace is lot like momentum in spacetime: it generates translations. We can give this idea a differential-operator meaning; just like momentum generates translations in space, supercharges generate translations in superspace!
$Q_{\alpha}=-\iota \frac{\partial}{\partial\theta^{\alpha}}-\sigma_{\alpha\bar{\dot{\beta}}}^{\mu}\bar{\theta}^{\bar{\dot{\beta}}}\partial_{\mu}$
$\bar{Q}_{\dot{\beta}}=-\iota \frac{\partial}{\partial\bar{\theta}^{\dot{\beta}}}-\sigma_{\alpha\bar{\dot{\beta}}}^{\mu}\theta^{\alpha}\partial_{\mu}$
How do superfields transform under infinitesimal coordinate changes generated by supercharges? It is simply
$\delta_{\epsilon,\bar{\epsilon}}F=(\iota\epsilon Q+\iota\bar{\epsilon}\bar{Q})F.$
Eventually, one would like to write down actions in superspace. However, the general superfield has far too many fields, corresponding to a reducible representation of the SUSY algebra. In order to cut down the number of components, one needs to define derivative operators in superspace that anticommute with the supercharges. These can then be used to (a) effect constraints on superfields, that will reduce the number of fields, and (b) write down superactions in superspace. These differential operators look like
$D_{\alpha}=\frac{\partial}{\partial\theta}+\iota\sigma_{\mu}\bar{\theta}\partial_{\mu}$
$\bar{D}_{\alpha}=\frac{\partial}{\partial\bar{\theta}}+\iota\sigma_{\mu}\theta\partial_{\mu}$
and
$\{Q_{\alpha},D_{\alpha}\}=0$
At the end, he explained that any action of the form
$S=\int d^{2}\theta d^{2}\bar{\theta} f(x,\theta,\bar{\theta}) + \int d^{2}\theta w(x,\theta) + hc,$
is — by construction! — SUSY invariant! The first term is known as the Kahler potential and the second is known as the superpotential. A short discussion of the statement of non-renormalization theorems followed, and that was where we concluded for the day.
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2017-12-15 23:25:17
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https://math.stackexchange.com/questions/3813856/product-of-dedekind-cuts-why-it-is-defined-in-that-way
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# Product of Dedekind cuts: why it is defined in that way?
A Dedekind cut is a partition of $$\mathbb{Q}$$ into two proper subsets $$A,B$$ such that each member of $$A$$ is smaller than each member of $$B$$, and $$A$$ has no largest element.
For Dedekind cuts $$(A,B)$$ and $$(A',B')$$, the addition is little natural: it is $$(A+A',B+B')$$. But the multiplication is little tedious. (I didn't find any rigorous explanation for defining multiplication). In the book Which Numbers are Real by Michael Henle (p.45), the author says
The multiplication of Dedekind cuts is not simple. The problem is determining sign of a product.
Now, the determination of sign in the product which we visit first time is for integers. I saw the link of wiki. It is stated, after the rules of sign for integers, that
this rule (of sign for product of integers) is a necessary consequence of demanding distributivity of multiplication over addition, and is not an additional rule.
By this comment, I thought that we can focus simply first for product of positive integers (with $$0$$) and with above necessary rule of sign, we can completely define the product of all the integers.
Now, similarly, we may think that, we can try to define the product of positive Dedekind cuts, and then with the rule of sign, we can define product of all Dedekind cuts.
However, I didn't find any natural way to get the definition of multiplication of two positive Dedekind cuts; why it is defined in that way? Can anybody give me some natural way to define the product of positive Dedekind cuts?
The reason that Dedekind cuts are awkward for negative reals is that multiplication by negative numbers is not order-preserving.
Consider Dedekind cuts $$(A,B)$$, $$(A',B')$$ representing reals $$>0$$. Then $$A\cap\Bbb Q^+$$ and $$A'\cap\Bbb Q^+$$ are nonempty, where $$\Bbb Q^+=\{x\in\Bbb Q:x>0\}$$. Their product is $$(A'',B'')$$ where $$A''=\{x\in \Bbb Q:x\le 0\}\cup\{aa':a\in A\cap\Bbb Q^+,a\in A'\cap\Bbb Q^+\}.$$
The rationale is that $$A''\cap\Bbb Q^+=\{aa':a\in A\cap\Bbb Q^+,a\in A'\cap\Bbb Q^+\}.$$ This works just like addition, expect we restrict attention to positive numbers, where multiplication is order-preserving.
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2022-05-19 09:57:10
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https://anuga.anu.edu.au/browser/trunk/anuga_core/documentation/user_manual/pypaper.sty?rev=7828&order=size
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# source:trunk/anuga_core/documentation/user_manual/pypaper.sty@7828
Last change on this file since 7828 was 2363, checked in by ole, 17 years ago
Tried to move documentation again
File size: 597 bytes
Line
1%
2% Change this to say a4paper instead of letterpaper if you want A4. These
3% are the latex defaults.
4%
5\newcommand{\py@paper}{letterpaper}
6\newcommand{\py@ptsize}{10pt}
7
8% These set up the fonts for the documents.
9%
10% The "times" package makes the default font the PostScript Times
11% font, which makes for smaller PostScript and a font that more people
12% like.
13%
14% The "avant" package causes the AvantGarde font to be used for
15% sans-serif text, instead of the uglier Helvetica set up by the "times"
16% package.
17%
18\RequirePackage{times}\typeout{Using Times instead of Computer Modern.}
Note: See TracBrowser for help on using the repository browser.
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2023-03-30 16:01:34
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https://2022.congresso.sif.it/talk/124
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Relazione su invito
# Soft perfusion bioreactor to engineer skeletal muscle 3D volumetric constructs onto a porous scaffold.
##### Piazzoni M., Locarno S., Gerges I., Iberite F., Guarnera D., Ricotti L., Lenardi C.
Venerdì 16/09 15:30 - 19:00 Aula E - Rosalind Franklin V - Biofisica e fisica medica
3D skeletal muscle cell cultures are a promising tool to recapitulate $in vitro$ the physiological conditions found in the native biological tissue. However, it is still a challenge to deliver oxygen in avascular cell constructs thicker than $450 \mu m.$ In this study we manage to develop an all-soft perfusion bioreactor to engineer skeletal muscle tissue constructs at the macroscale onto a porous scaffold. The device is composed of a macroporous polyurethane scaffold matching the muscle stiffness, that is first wrapped in a compliant silicon chamber, and then sealed at the edges with two hollow tendon-like structures. After device assembly, the scaffold surface was conjugated with fibronectin proteins to allow myoblasts' adhesion onto its surface. Dynamic cell cultures performed using a peristaltic pump attested the ability of the device to enable 3D skeletal muscle cell cultures with a volume of $1 {cm^{3}}$. We strongly believe that the proposed device can recreate a native tissue environment while administering cells with multiple external stimuli, thus holding the potential of boosting skeletal muscle tissue engineering at its limits.
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2023-03-30 22:01:31
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http://electronics.stackexchange.com/questions/43228/driving-50w-250w-ultrasonic-transducers-with-sine-any-class-b-135khz-monolithic
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# Driving 50W-250W ultrasonic transducers with sine: Any Class B 135KHz monolithic power amp ICs?
My project needs to drive a medium power ultrasonic piezoelectric transducer from a sine wave (/ sawtooth) sweep generator that sweeps +/- 2% of the transducer resonant frequency.
The question: What are my simplest options for driving these transducers from a DDS generated shaped signal, with reasonably low distortion (5-10%)?
1. Use a power amplifier IC off a higher voltage rail, with lots of heat sinking, to directly drive the transducer
2. Use a power amplifier IC, then (?) a transistor current amplification stage, then an appropriate (need help identifying) step-up transformer to drive the transducer
3. Use some sort of (need help identifying) class D high power amplifier IC that would not need much heat sinking (Edit: Not a solution, see Note 7).
4. Some other option entirely
5. Edit: From suggestion below Identify an off the shelf OEM amplifier module that meets the parameters and constraints.
UPDATE: [15-Oct-2012] Option 5 above seems best answer, if a suitable OEM module or two could be pointed out - None found in my research so far. Hence leaving question open.
The sweep waveform generation is through a DDS IC, AD9850, Datasheet here: AD9850 CMOS 125 MHz Complete DDS Synthesizer
One of the transducers available to me: 5938D-25LBPZT-4 (Ultrasonic Langevin Transducers)
• Resonant frequency: 25 KHz
• Resonant impedance: 10-20 Ohms
• Capacitance: 5400 pf +/-10%
• Input power: 60W
• Datasheet: I wish I could find one!
The transducer would change case to case, from 20KHz to 135KHz, each in the 50-250 watt range, similar in design to the one above.
The driver designs I have seen for these transducers typically use switching i.e. square waves to drive them, MOSFET driven, with Vpp 100v in some cases! (Do these devices even need that kind of voltage? Edit: Evidently so)
Some drivers use tuned filters to shape the waveform to a sine or approximation thereof.
This does not work for my purposes, unfortunately - The project is a single device that would first detect the resonant frequencies of an attached transducer across the full range 20-135KHz, then sweep around each resonant frequency with first a sine wave, (Edit: Removing this requirement as unfeasible: then a sawtooth signal,) at a specified power output, usually around half the rated power of the transducer.
So what I am looking for is the wisdom of this community in suggesting a suitable prototype-friendly approach to getting those DDS waveforms over to the transducer. Thank you all!
1. Waveform accuracy is not super-critical, 5% distortion is very acceptable. Thermal issues and power wastage through dissipation in the amplifier stage are bigger concerns. Cost is a key concern, at least until past the prototype stage.
2. It has been suggested that prebuilt OEM amplifier modules that suit the requirements might be my best bet. While that does appeal, I am still hoping for alternatives in addition to, and examination of, the options I have proposed in my question, hence not marking the answer accepted, yet.
3. Not found any OEM module online yet which covers a 20KHz to 135KHz frequency range, even for 50 watt output. The one suggested in a response is designed for 3.5KHz, and its switching frequency is 100KHz. (Dropped this requirement: Also, wouldn't I require bandwidth much higher than that, to handle a sawtooth wave with even cursory accuracy? I might have to skip the sawtooth requirement, and constrain my question to sine waves, if the sawtooth or other arbitrary waveform delivery is seen by respondents as unattainable at reasonable cost.)
4. New Suggested approach is a Class B with feedback. Caveat mentioned is high dissipation at this amplifier stage. So two adjuncts to my question:
5. Is there a monolithic Class B amplifier IC that might cover the desired frequency range (20KHz to 135KHz, giving up on the sawtooth wave) and power requirements (50 watts max)?
6. What is the range of heat dissipation expected at such a class B stage, as a percentage of expected power delivery to transducer?
7. New About Class D amplifiers, monolithic or OEM: They would need to use switching frequencies of the order of 800KHz or higher, to support a 100-135KHz sine wave with reasonable THD. For a 5% distortion requirement, the switching frequency must be even higher. Such high switching frequency Class D power amplifiers do not seem to exist.
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Good question, I'm waiting for somebody's answer to this question. +1. – Standard Sandun Oct 10 '12 at 8:06
There is tradeoff between the simplicity of the solution and the accuracy of the transmitted waveform. What is your application? Is it a device metrology problem or an ultrasound problem you are trying to solve? – ARF Oct 10 '12 at 10:31
The application is a lab-condition diagnostic device: Purity of waveform is important but not critical - hence the 5-10% distortion being acceptable. – Anindo Ghosh Oct 10 '12 at 11:19
Not sure if these will work for you, but much lower cost; piezodrive.com/modules.html#pdu100b – user71967 Apr 10 '15 at 16:48
Try these linear amplifiers made by Apex. They are designed specifically for ultrasound aplications.
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In many ultrasonics applications you really will need to work with potential differences in excess of 100V to deliver sufficient acoustic power to the medium. This is due to the fairly low impedance the transducers present electrically. Predicting how much voltage you need to achieve a set acoustic pressure however is next to impossible as the transfer functions are non-trivial.
Many ultrasound applications are not terribly concerned with the excitation waveform. This is the reason why many power amplifier stages are very simple push-pull configurations giving a square wave output. Their advantage is two-fold:
1. they can be driven easily from low-voltage signal generation circuits, and
2. they dissipate very little power in the switching elements which is a common design constraint. (Due to the fact that ultrasound transducers are fairly narrow-band, the energy dissipation is shifted to the cable and transducer. Often cooling the transducer is much easier.)
In situations where signal waveform is important, the power amplifier stages I encountered in the past were generally class B push-pull configurations with negative feedback to avoid crossover distortion fed from high-voltage rails. It sounds to me that this would be the way to go in your situation. Note: there will be non-negligible power dissipated in your switching elements.
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Thanks... Would you have any suggestions on OEM class B amp modules that cover the 20KHz to 135KHz range? I understand that there will be a fair bit of heat dissipation at the amplifier stage - I assume that's the switching stage you refer to. Is it a safe assumption that this dissipation will be at worst 15% to 30% of the desired output power? Or do I have my math all wrong on this? – Anindo Ghosh Oct 11 '12 at 12:18
First of all, yes, you will need voltages on the order of 100V peak (70.7V RMS) to drive 250W into 20Ω.
You can purchase OEM power amplifier modules that cover the power and frequency range you're interested in; this is probably your best bet in terms of getting the prototype operating quickly with low design risk. It may even be the way to go for production, as well. Be sure to select a unit that can deal with the capacitive load.
Here is one example. Interestingly, I find that audio power amplifier modules these days are almost exclusively class-D, with the bandwith limited to 10s of kHz. When I last looked at these some years ago, they were class-AB and had bandwidths of 100s of kHz. Be sure to include "piezo" or "ultrasonic" in your search terms.
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Thanks... If you could point me at some OEM power amplifiers that might serve the purpose, or even a site I could look at, that would work well at least for the prototype stage. For this stage, even a 50 to 100 watt power output would work well. – Anindo Ghosh Oct 10 '12 at 11:50
The example device linked has a stated bandwidth of 3.5KHz, which unfortunately is not close to the requirements of my question. PI does not have any offerings for my frequency range, but they were nice enough to explain that it would be unlikely to find a Class D that is usable beyond about 30KHz sine, that too with heavy distortion, since they typically use 100KHz switching frequency, and Nyquist comes into play. Editing this finding into my question. – Anindo Ghosh Oct 11 '12 at 16:51
I think the Piezo Systems EPA-104-115 fits all your criteria except for the low-cost criteria. It costs $2,639. The AA Lab Systems A-301HS may also fit and is probably as cheap as you'll find. I saw one on ebay for$975.
Searching for piezo driver or piezo linear amplifier didn't turn up anything more affordable in my search, but feel free to double check yourself.
You might also want to read this paper written by a lab that built a less expensive driver for their piezo actuators. Unfortunately their driver is in the 1kHz range but they end by suggesting some methods that might get the kHz up. On the other hand, they say they aren't sure where to get parts that could handle higher frequencies, but it may be a helpful read to understand what makes higher frequencies difficult and could lead toward a solution with some perseverance.
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2016-04-29 23:34:48
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https://math.stackexchange.com/questions/979177/does-1-0000000000-cdots-1-with-an-infinite-number-of-0-in-it-exist/979180
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# Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?
Does $1.0000000000\cdots 1$ (with an infinite number of $0$ in it) exist?
• No, no such number exists. – Ittay Weiss Oct 18 '14 at 8:02
• That is not the decimal representatlon of a real number. – André Nicolas Oct 18 '14 at 8:05
• A decimal representation of a number has digits indexed by natural numbers. Which exactly is the position of that last $1$? Is it the first after the decimal point? The second? The third? Each digit must have its position, which must be a natural number. In other words, when specifying a real number by a decimal representation, you can pick any infinite sequence of digits of your choosing, but it doesn't make any sense to add anything "after the end" of the sequence. – Dan Shved Oct 18 '14 at 8:15
• There are contexts in which it can make sense to use expressions like $1+\frac 1{10^{\omega}}$, but they tend to reflect non-standard approaches: interesting to some people, but haven't been persuasive enough or convenient enough to enter general use because the normal way of doing things works well enough. As André Nicolas notes "this is not the decimal representation of a real number". – Mark Bennet Oct 18 '14 at 8:34
• @Aditya "Implying there's something bigger than infinity." There is no largest infinity, just as there is no largest finite number. So it's not clear what "bigger than infinity" even means. – David Richerby Oct 18 '14 at 9:28
First let me tell you that the idea that an infinite sequence "ends with something" is a solid idea. It's a perfectly natural one. The point is that the sequence is not indexed by $\Bbb N$, anymore, but rather by $\Bbb N\cup\{\infty\}$, where $\infty$ is another point, which lies after all the natural numbers.
The point is that an "infinite sequence" is a very general notion. People just often like to think about sequences which are indexed only using the natural numbers (with their natural ordering, that is). But as you will progress in your studies you might meet other objects which are indexed using other infinite sets.
And the reason people often limit themselves to sequences indexed by the natural numbers is that for the real numbers (and similar concepts), these sequences are enough. In this case, of the real numbers, we have that each real number can be defined as a limit of decimal digits, as others have explained, and therefore $1.\underbrace{000\ldots}_{\text{infinite }0\text{'s}}1$ is not a definition of a real number.
Note that this is not the limit of $1+(\frac1{10})^n$, either, which colloquially might be written as $1+(\frac1{10})^\infty$. That limit would be the limit of $1.1,1.01,1.001,\ldots$ and you can see that at no point in this sequence there is a number with infinitely many $0$'s written after it. And indeed this limit would be equal to $1$.
This is also different from the $0.999\ldots$ situation, since it is a sequence indexed by $\Bbb N$, which can be seen as the limit of its initial segments. Whereas a sequence indexed by $\Bbb N\cup\{\infty\}$ is not the limit of its initial segments, since none of them include information about the last digit.
So does it exist? Yes. It's just not a real number. It's a sequence of digits indexed by something other than $\Bbb N$.
Finally, Let me point out that as far as the concept of infinity goes in calculus, it's not quite unique. There is one infinity which signifies arbitrarily large values, another which signifies arbitrarily large negative values, there are infinities which ignore the sign at all, when you talk about a smooth function that can be differentiated infinitely many times, the infinity here is in fact "infinite sequence" rather than the infinities mentioned before, and it's a completely different type of infinity.
And there are other infinities which you might encounter, even in a calculus class.
• You should be careful when you say that. The notion of infinity in calculus is fickle, and it is incompatible with other notions of infinity in mathematics. Notions you will run into very quickly in your studies, if you won't close your eyes to them. – Asaf Karagila Oct 18 '14 at 11:28
• @user1485853: Yes, as I said, this doesn't define a real number. It doesn't mean that it doesn't exists. There is more than just real numbers. – Asaf Karagila Oct 18 '14 at 14:26
• @djechlin: $\frac1{10}^\infty$ is a limit of a sequence, and generally it's worth pointing that out; it's not an actual number. Moreover this is not the same as a number whose decimal expansion is infinitely many $0$ and then $1$. It is the limit of $0.1,0.01,0.001$ and so on. At no point there is a number with infinitely many $0$'s, let alone a number with infinitely many $0$ and then $1$ after them. – Asaf Karagila Oct 18 '14 at 16:26
• @Ethan: Then don't picture it in your head. The reason we have definitions in mathematics is exactly so we don't have to picture things in our heads. Work with the definitions, slowly and carefully. You'll see that you'll eventually start having better mental pictures of these objects, and that's true for anything in mathematics, not just very large real numbers. – Asaf Karagila Oct 18 '14 at 17:26
• This answer helped me put $1$ and $.000...1$ together. – user157227 Oct 19 '14 at 0:22
That string of symbols $$1.0\cdots01$$ has no meaning as a real number. Meaning or rather semantics would be a valid mapping from an infinite string $s$, e.g. from $\Sigma^\omega$ (link), to $\mathbb{R}$.
The string representation of a floating point real number in base $10$ by convention means e.g. $$(1.0\cdots)_{10} = (\underbrace{d_0.d_1 d_2 d_3 \cdots }_{\mbox{string}})_{10} = \underbrace{\sum_{k=0}^\infty d_k 10^{-k}}_{\mbox{real number}}$$ Where would you like to add that last $1$ digit?
The good news:
It might be something else.
Of course you can define your own semantics and operations on the strings but you will most likely end up with something that behaves more or less differently from the real numbers (like finite IEEE floats and their operations are not the same as the full set of real numbers and the basic operations on it, but something very close).
And the ugly news: $\tiny \mbox{(c) by Asaf Karagila}$
As fellow user Hyrkel noticed, that finite string I used $$[1] [.] [0] [.] [.] [.] [0] [1]$$ to suggest an infinite string with two ends (prefix "$1.$" and suffix "$1$") and infinite many $0$ symbols in between is problematic too - interpreted as a string already.
Is this a proper string? is it even a proper infinite string? How would you be able to recognise it?
Next step would be to attach a meaning to it, preferably some number, but I will only argue on the above questions which are not simple already.
Computer scientists stick to mathematical machine models, like the finite automaton or the Büchi automaton to reason about strings. These machines can either accept or reject a string they are presented with. Their recognition process resembles the process of a sensor reading a linear tape or track from left to right. Even the variants for infinite strings act like this.
The infinity here is not so much problematic because of it's sheer size but rather because of it's dullness: what reason should the automaton have to stop the recognition of the infinite part and proceed with the finite suffix? The recognizable infinite strings seem to be of the variant one end finite, one end infinite. (Do not nail me on this)
I am not sure if a non-deterministic (multiple choices possible) Büchi automaton that would accept the string $1.0^\omega$ could be properly extended to recognise $1.0^\omega 1$.
I would attempt it by adding another arc from the final state to itself which is accepting the final $1$ symbol. That would work to accept $1.0^\omega1$ but it would also still accept just $1.0^\omega$. That makes it not much useful, what I can not distinguish is practically the same.
The solution is probably another automaton that starts recognition simultaneously from both ends or some mapping which lists the infinite sequence alternating from both sides at once, something like $$(1 1) \, (. 0) \, (0 0) \cdots$$ this would resort to established structures but I am not aware of such approaches.
• It might be more to the point to note that $1.0\dots 01$ doesn't even have an interpretation as a string comprised of digits and a decimal point; the problem comes even before you can start wondering about numbers! – user14972 Oct 18 '14 at 8:50
• Actually, as a string from $\Sigma^{\omega+1}$. – Asaf Karagila Oct 18 '14 at 8:52
• Do you also have some ugly news? :-) – Asaf Karagila Oct 18 '14 at 9:50
• Tudu dudu du, duuuduuuuu.. – mvw Oct 18 '14 at 10:56
• In what sense are IEEE floats "not real numbers, only something very close"? They're real numbers, just not the entire infinite range of e.g., 0.0 - 1.0, and the result of a calculation is always predictable using only real numbers. – goldilocks Oct 18 '14 at 11:28
In real analysis, $$\lim_{n\rightarrow\infty}\left( 1+\frac{1}{10^n}\right )=1$$ Even if it does not mean anything to say "infinite number of zeros" in real analysis, we can suppose this number is equal to $1$.
But in the field of surreal numbers, it's not the same. This number exists and will be equals to $1+\frac{1}{\omega}$, if you consider the infinity number of zeros to be $\omega$.
• Are decimal expansions of general surreal numbers really well-defined? There's the sign-expansion as an ordinal-length sequence of plus and minus signs, which is similar to, but not quite the same thing as, a binary expansion. But decimal? – Hans Lundmark Oct 18 '14 at 8:38
• @HansLundmark That's a good point. You can see usual expansion as a binary one, and use the usual modulo algorithm to get the decimal one, using the fact that $2^{-\omega}=10^{-\omega}=\frac{1}{\omega}$... But I admit this must be defined more accurately ! – Xoff Oct 18 '14 at 8:42
• How are $2^{-\omega}$ and $10^{-\omega}$ defined? I'm not an expert on surreal numbers, but isn't there an exponential function $x \mapsto \exp(x)$ which is strictly increasing (and nontrivial to define)? If we take $2^x=\exp(x \ln 2)$ and $10^x=\exp(x \ln 10)$, then it would seem that $10^x<2^x$ for $x<0$. – Hans Lundmark Oct 18 '14 at 8:54
• Actually, you don't need $10^{-\omega}$ to be well defined, since that doesn't occur in the answer. What you need is a well defined limit. – celtschk Oct 18 '14 at 9:52
• I use $2^{-\omega}$ as a shorthand for the $\omega^{th}$ digit. This is not defined from the exponential function, even if it coincides for the finite positions... – Xoff Oct 18 '14 at 10:04
If a sequence $1.00000\dots$ is infinite it can't have an end $\cdots 0001$. Infinite means endless.
A finite sequence is a line of (mathematical) objects $a_0,a_1,a_2,\dots, a_n$. But it seems to be some disagreement what an infinite sequence is. At least I disagree.
Obviously, the object $a_k$ represent a function $k\mapsto a_k$ with ordered indices $k$, but could it be any function? Due to Wikipedia:
Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.
In that case a function $\mathbb N\cup\{\infty\}\rightarrow A$ (for some set $A$) is a sequence $a_0,a_1,a_2,\dots$ with a last element $a_\infty$, without an immediately preceding element in the sequence.
In my intuition and in my opinion any element in a sequence, except the first, has an immediately preceding element.
Latin: sequentia (“a following”).
However, it's possible to generalize to "bi-sequences" $(S_1,S_2)$ when $S_1=(1,0,0,\dots)$ is the initial sequence and $S_2=(\dots,0,0,1)$ is the termimal sequence, and define an arithmetic for "numbers" defined by bi-sequences as $(S_1,S_2)$.
• Because $\omega+1$ is not a thing. – Asaf Karagila Oct 18 '14 at 8:32
• @Asaf I just focused on the language: an infinite sequence is an endless sequence and an endless sequence can't end at 1. – Lehs Oct 18 '14 at 8:38
• I guess that $\omega+1$ is just not one of them "infinite ordinals", then, because it has an end. – Asaf Karagila Oct 18 '14 at 9:00
• @Asaf: But OP wasn't really doing mathematics and I ventured to respond in the same style. – Lehs Oct 18 '14 at 9:13
• @Lehs: I'm confused. Is the banner on the top of the site saying something other than "MATHEMATICS" for you? – Asaf Karagila Oct 18 '14 at 9:49
In the real number system $\mathbb R$, there is no such number.
P.S. There is no such number in the complex numbers $\mathbb C$, either.
P.P.S. Sorry.
• Did the OP say anything about "real"? – Mikhail Katz May 31 '16 at 17:28
In the hyperreal number system which is an extension of the real number system you have infinite integers and the corresponding extended decimal expansions where it is meaningful to talk about digits at infinite rank (more precisely, rank defined by an infinite integer). In this system your decimal makes sense.
Extended decimals were discussed in detail in an article by Lightstone:
Lightstone, A. H. Infinitesimals. Amer. Math. Monthly 79 (1972), 242–251.
As @Lehs said, infinite means endless - every integer value $n$ is finite, so if you assume infinite sequence of 'zero' digits, then whatever number $n$ you think, the $n$-th position holds digit 0. Consequently there's no space where you can append 'one'.
• Yes, we are right! :) Even in the OP case, which is about a decimal string "1.000000..." If it is endless, it cant end with "..0001". – Lehs Oct 18 '14 at 9:28
• You need to be very careful with this kind of argument. You could just as well argue (a la Zeno) that the number 1 doesn't exist because there is an infinite sequence of numbers $\frac12,\frac34,\frac78,\frac{15}{16}, \dots$ that come before it and nothing can "come after" an infinite sequence. – David Richerby Oct 18 '14 at 9:33
• I have the same idea with you . I think there is only one infinity in Calculus , we can think it as the biggest number,so something bigger than infinity seems ridiculous and impossible, thus such number doesn’t exist . – iMath Oct 18 '14 at 12:10
• @DavidRicherby Nonsense, Zeno confused the infinity in the sequence length with the set of values. We don't. – CiaPan Oct 18 '14 at 21:51
• There is indeed such a space unless you assume the number is real, which the OP is not assuming. – Mikhail Katz May 31 '16 at 17:28
Yes, if the infinite quantity of zeros are placed both before the decimal point and before the leading $1$. Those six adjacent dots appear to me to just be small zeros filled in. I'd like to use the notation that ... placed before a sequence means that it repeats indefinitely in the left direction. So then we would have:
...0$1.00000000000000001$ with the bold stuff being fixed in position.
Looking at the answers seems to exhibit a rather chaotic mixup of concepts.
Of course, the question is not uninvolved since it talks about whether a particular number representation "exists". Obviously, in its informal manner, it exists or could not have been written down. Since we are talking about calculus here, the next question would be whether some real number value can be associated with that loose description. It would appear that it would mean something akin to
$$\lim_{n\to\infty} \left(1 + 10^{-n}\right)$$
It turns out that this number perfectly well exists and is the same as the real number $1$. So far, so straightforward.
The really interesting thing is that the majority of answers diverge from this. If you asked whether a number like $2-1$ exists, few people would deny its existence on the basis that its specification looks different from $+1$. Or would refuse to acknowledge the existence of octal $10_8$ because it would be the same as $8_{10}$.
So given that the majority of answers look different, it would appear that the question is not as much about calculus as it is about number psychology. Is this the same as numerology?
Next question.
• Note that you interpreted the OP's 1.0000000000……1 as – mvw Oct 18 '14 at 18:20
• @mvw has hit the nail on the head. You've assumed a definition in your answer that may not necessarily be valid. (And numerology is totally different, btw.) – apnorton Oct 18 '14 at 18:27
• You made your decision to interpret the given string. it is necessarily non-standard. In your case you added zero in a complicated way. A suitable interpretation should give reasons what it is and why to introduce it next to $1$. – mvw Oct 18 '14 at 18:32
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J Acupunct Res > Volume 39(4); 2022 > Article
Jang, Jo, Ko, Chae, Lee, Lee, Kim, Goo, Seo, Baek, Nam, and Park: Quality Assessment of Clinical Practice Guidelines for Ankle Sprains Using the Appraisal of Guidelines for Research and Evaluation II Assessment Tool
Abstract
This study aimed to evaluate clinical practice guidelines (CPGs) for ankle sprains using the Appraisal of Guidelines for Research and Evaluation II tool, using electronic databases (GIN, PubMed, EMBASE, NCKM, CNKI, CiNii, WanFang database, RISS, and SCOPUS), to suggest strategies for improvement in the future. The search was performed on April 23, 2021 and 10 CPGs were selected for evaluation. Three CPGs were recommended without modification (Class A scores), five were recommended with modification (Class B), and two were not recommended (Class C scores). The CPG domain that received the lowest score was “applicability.” The traditional medicine CPGs scored higher [Class A (n = 1) and Class B (n = 1)] than the conventional Western medicine CPGs [Class A (n = 2), Class B (n = 4), and Class C (n = 2)] and were considered to be more methodical. In the future, more research into traditional medicine is required.
Introduction
Ankle sprain is defined as “stretching or partial or complete tearing of one or more ligaments in the ankle joint.” This is typically caused by a twisting movement that exceeds the normal limits of the joint [1]. Acute ankle sprains are among the most common musculoskeletal injuries, with approximately 2 million cases occurring each year in the United States. Up to 70% of patients who sustain an acute ankle sprain may develop residual physical disability [2]. Individuals with ankle sprains may complain about having pain, functional disability, and often have periods of absence from work. In those people with chronic instability, they may experience continued pain, swelling, recurrent sprains and exacerbate the instability [3]. Ankle sprains are often treated with a combination of rest, ice, compression, and elevation, followed by exercise, ankle support, and cryotherapy. The overall cost of ankle sprain treatment ranged from $1,809 to$5,271 when converted to prices in 2016 [4]. Based on the high healthcare costs, the Korean healthcare system has two major responsibilities: ensuring that individuals are treated according to best practices and reducing unnecessary expenditure [5]. Clinical practice guidelines (CPGs) are “systematically developed statements used to assist practitioner decisions regarding the appropriate healthcare for specific clinical circumstances.” Using CPGs, inappropriate variations in practice can be reduced to provide high-quality evidence-based healthcare [6]. Therefore, high-quality CPGs are an absolute necessity. The Appraisal of Guidelines for Research and Evaluation (AGREE) is an appraisal tool validated and endorsed by leading producers, raters, or compilers of international CPGs. It was first created by the AGREE collaboration in 2003. The second version, the AGREE II, was updated in 2009. The AGREE II can be used for the development and evaluation of CPGs [7,8]. The AGREE II instrument has the following three goals: (1) To assess the quality of CPGs; (2) To provide a methodologic strategy for the development of guidelines; (3) To recommend how and what information should be reported in the guidelines [9].
This study aimed to assess the quality of CPGs on ankle sprains published before April 23, 2021, to suggest ways to improve them.
Search strategy of CPGs
Several databases were used to search for CPGs for ankle sprain. The search date was April 23, 2021. The following databases were used to search for CPGs: Guidelines International Network (www.gin.net), PubMed (https://pubmed.ncbi.nlm.nih.gov/), Embase, National Clearinghouse for Korean Medicine (http://www.nckm.or.kr/main/index.do), Google Scholar, China National Knowledge Infrastructure, Citation Information by NII (https://ci.nii.ac.jp/), WanFang database, Research Information Sharing Service, and Scopus (www.scopus.com). Keywords were combined to search for CPGs in the following databases: (Clinical practice guideline OR Critical practice guideline OR guideline*) AND (ankle). The search strategy was adjusted for each database.
Selection of CPGs
All CPGs retrieved for conventional Western medicine and traditional medicine were supported by an official global medical organization at the time of writing that addressed diagnosis, treatment, prevention, and management. The latest CPG version was selected when various versions were available. Types of excluded CPGs included guidelines without any recommendations, secondary publication from CPGs, systematic reviews, clinical trials, and consensus between panels. Two reviewers independently checked titles and abstracts to exclude ineligible publications. Only CPGs written in English, Korean, Chinese, or Japanese were included in the study. Only full-text articles that met the inclusion criteria were screened (Fig. 1). Inconsistencies at any stage were resolved through discussion between reviewers or by the involvement of a 3rd reviewer.
Data extraction and quality assessment
Two reviewers independently extracted core data from the published CPGs. Inconsistencies were resolved through discussion or by the participation of a 3rd reviewer. The extracted data were listed according to the CPG characteristics namely country, organization, year of publication, number of authors, number of references, target population, subject (i.e., diagnosis, treatment, management, or prevention), treatments and recommendations related to the diagnosis (i.e., conservative, pharmacological, and surgical management. Three researchers independently assessed and scored all the CPGs using the AGREE II instrument. All researchers were Korean medicine doctors majoring in acupuncture and moxibustion. The process of CPG evaluation was based on the Korean version of the AGREE II developed by the Korean Academy of Medical Sciences in 2011 and each researcher was made familiar with this tool. The CPGs were scored using a 7-point rating scale and analyzed according to each category. The AGREE II instrument consists of 23 items sorted into six categories: (1) Scope and purpose: the goal of the guideline, the detailed questions, and the target population; (2) Stakeholder involvement: the emphasis on whether the guidelines have been developed by appropriate stakeholders and whether they reflect the opinions of target practitioners; (3) Rigor of development: the methods used to gather and synthesize evidence, how to make recommendations, and update treatment guidelines; (4) Clarity of presentation: the language, structure, and form of the guidelines; (5) Applicability: the factors that facilitate and impede the implementation of the guidelines, strategies to improve applicability, and the impact of additional resources when the guidelines are applied; and (6) Editorial independence: whether conflicting interests among the members involved in the development of the CPG influenced the derivation of the recommendations.
The overall reviewer assessment included the CPGs quality rating (Class A, B or C) and their feasibility. Each item was rated on a 7-point scale from 1 (strongly disagree) to 7 (strongly agree) by the appraisers. If there was a gap of three or more points between appraisers for each item, the criteria of the item were discussed. Through this process, the bias in evaluation alone was reduced, and the reliability increased. Quality scores were calculated for each of the six domains. These scores were independent and were not summed into a single score. Domain scores were calculated by adding all the scores of individual items in a domain and scaling them to the maximum possible score for that domain and multiplying it by 100. When CPGs are rated using the AGREE II instrument, there is no standard for setting the level of recommendation. Therefore, other appraisals of CPGs were referred to and a standard of level was achieved [10,11]. The recommendation of a CPG was divided into three levels. Class A, recommended without modification, was assigned to CPGs with four or more domains rated higher than 60%. Class B, recommended with modifications, was assigned to CPGs with three or more domains rated higher than 30%. Class C, which was not recommended, was assigned to CPGs with four or more domains rated lower than 30%.
Summary of CPGs recommendations
The recommendations of each CPG according to the target population, subject, and treatment are shown in Table 1 [1221].
Statistical analysis and research ethics
The total score and the score per domain provided by each reviewer were used for statistical analysis. Following the application of the AGREE II instrument and using Excel 2010 (Microsoft Corporation, Redmond, WA, USA), the data were obtained.
No institutional review board approval was required because the study was not patient or body sample-based.
Study selection
Through the database searches, 929 articles were retrieved, of which 197 articles were duplicated. A total of 664 articles were excluded based on the title and the abstract, and 14 articles were excluded based on the language criteria. A flow chart describing the selection process of the 10 articles selected for review is shown in Fig. 1.
CPG components
Ten CPGs published between 2006 and 2021 were included (Table 1 [1221]). Only one CPG was published before 2010, and three were published within the last 3 years. Forty percent (n = 4) were published in the USA, 20% (n = 2) were published in Korea and 20% were published in the Netherlands, 10% (n = 1) were published in Belgium and 10% were published Germany. The number of authors ranged from 1 to 19, and 40% (n = 4) of CPGs consisted of 10 or more authors. The number of references ranged from 26 to 485, and 60% (n = 6) of CPGs had more than 100 references. No target population was reported in the two cases of CPGs, and most of the remaining CPGs were selected based on the age of the patient. Thirty percent (n = 3) of CPGs were listed for adults, 20% (n = 2) were listed for 16 years of age or older, 10% (n = 1) was listed for those aged 5 years or older. Two CPGs were listed for athletes. Treatment was divided into conservative treatment, pharmacological treatment, and surgery.
The AGREE II appraisal results
Three researchers evaluated 10 CPGs related to ankle sprains using the AGREE II. The scores for each evaluated CPGs are listed in Table 2 [1221].
When comparing the average scores for each domain, the domain assigned the lowest score was “applicability.” The average of this domain was 21.4% and the range was 3–72%. The domain assigned the highest average score was “clarity of presentation.” The average of this domain was 59.8%, and the range was 35–91%.
In the overall reviewer assessment, three CPGs were assigned a Class A level of quality, five CPGs were given Class B, and two CPGs were given Class C. The CPG assigned the highest average score was NCKM 2020 [12]. It was given the highest score in all areas, including the domain “scope and purpose” where it scored 100% (the scores of the domains “clarity of presentation” and “editorial independence” are the same as those of other CPGs). The CPG which was assigned the lowest score was AFP 2012 [13]. It was given below-average scores in all domains and was assigned a score of 30% or higher only in the “clarity of presentation” domain.
The “scope and purpose” domain evaluates the overall objectives of the CPG, health-related questions, and the population to which the guideline applies. The average score assigned to this domain was 55.3%, and the range was 2–100%, with a very large deviation. The CPG NCKM 2020 [12] was assigned the highest score which was 100%, two CPGs were given over 80%, and five CPGs were assigned a value of below 60%, and the CPG AFP 2012 [13] received the lowest score which was 2%. The CPG AFP 2012 [13] did not mention the scope or purpose. No health-related questions were asked. Only the name and basic characteristics of the condition/disease were included, and there was no text specifying the subject of application i.e., diagnosis, treatment, management, or prevention. Although the purpose of most CPGs were described well, for CPGs with low scores, (indicating a low level of evidence not sufficient to recommend use of the CPGs or recommended use of CPG with modification) health-related questions and descriptions of the target populations were insufficient.
The “stakeholder involvement” domain evaluates whether the guidelines have been developed by the appropriate stakeholders and reflects the opinions of practitioners’ who will mainly use the guidelines. The average score for this domain was 44.6%, and the range was 6–98%. Only three CPGs scored above 60% and four scored below 30%. None of the CPGs considered the viewpoints and preferences of the groups to which the guidelines were applied, particularly, the CPGs from ACR 2020 [15], KCE 201 [18], NATA 2013 [19], AFP 2012 [13], ORRE 2012 [20], and KNGF 2006 [21]. Although many CPGs included experts associated with the subject (i.e., diagnosis, treatment, management, or prevention), guideline development methodology experts with specified roles were not included. In general, the group of practitioners who would use the CPGs was well specified. The CPGs from JOSPT 2021 [14], NCKM 2020 [12], EUJIM 2017 [17], KCE 2013 [18], and KNGF 2006 [21] describe how practitioners could implement the CPGs in specific fields.
The “rigor of development” domain evaluates the methods used to gather and synthesize the evidence, making recommendations, and updating the guidelines. It consists of the largest number of sub-items. Keeping the purpose of this study in mind, the scores of sub-items in domain 3 (rigor of development) were compared. The average score for this domain was 48.7%, and range was 18–81%. Only three CPGs scored above 60% and three scored below 30%. The scores for each sub-item are presented in Table 3 [1221].
Here are sub-items of this domain:
• Sub-item 7. Systematic methods were used to search for evidence.
• Sub-item 8. The criteria for selecting the evidence are clearly described.
• Sub-item 9. The strengths and limitations of the body of
• evidence are clearly described.
• Sub-item 10. The methods for formulating the recommendations are clearly described.
• Sub-item 11. The health benefits, side effects, and risks have been considered in formulating the recommendations.
• Sub-item 12. There is an explicit link between the recommendations and the supporting evidence.
• Sub-item 13. The guideline has been externally reviewed by experts prior to its publication.
• Sub-item 14. A procedure for updating the guideline is provided.
Among the sub-items, the areas with the lowest average scores were Items 8, 10, 13, and 14. Their scores ranged from 2 to 4. Among the items, the area with the lowest average score was Item 13, and four CPGs were assigned 0% in this item. The CPGs from ACR 2020 [15], AFP 2012 [13], ORRE 2012 [20], and KNGF 2006 [21] had no external reviewers, review purposes, implementation methods, or descriptions of collected information and results. Item 14 received the most 0% across ACR 2020 [15], EUJIM 2017 [17], NATA 2013 [19], AFP 2012 [13], ORRE 2012 [20], and KNGF 2006 [21]. No guideline-revision plan was mentioned. KCE 2013 [18] received the highest score for item 14 by presenting the guideline revision schedule, methodology, and criteria for determining revisions. The item with the highest average score in this domain was Item 12, and all CPGs except BJSM 2018 [16] scored over 4 points in here. In most CPGs, recommendations were well-connected to the evidence, and a summary of key evidence or a list of references were presented.
The “clarity of presentation” domain covers the format, language, and structure of the guidelines. A high score can be obtained by describing recommendations specifically, easily finding them, and presenting various alternatives. This domain had an average score of 59.8%, which is the highest average score among the six domains, and with range of 35–91%. Four CPGs scored above 60% and none scored below 30%. ACR 2020 [15] and NCKM 2020 [12] recorded the highest scores of 91%. Both CPGs had some deductions, in that they did not specifically describe the uncertainty of the recommendations. For other items, scores close to perfect were recorded. All CPGs except ORRE 2012 [20] have presented recommendations that are easy to check.
The “applicability” domain evaluates the facilitating factors, obstacles, and strategies to improve the implementation of the guidelines. In addition, the impact of adding resources when a CPG is applied are also evaluated. The average score for this domain was 21.4%, which was the lowest average score among all domains, and the range was 3–72%. Only NCKM 2020 [12] scored above 60% and 8 CPGs scored below 30%. NCKM 2020 [12], which had the highest average score, recorded the lowest score in this domain among all domains. The highest score was obtained because NCKM 2020 [12] met all the criteria of this domain. The other CPGs did not fulfill these criteria well. In particular, ACR 2020 [15] had the lowest score of 3% because it did not consider applicability, except that budget-related issues were briefly described when applying the recommendations. The domain “editorial independence” evaluates whether conflicting interests among members who participated in CPG development influenced the production of recommendations. The average score for this domain was 34.8%, and a range of 0–78%. NATA 2013 [19] and KNGF 2006 [21] did not describe whether or not there was any financial support or whether there was a conflict of interest among members of the guideline development group, which resulted in a score of 0%. JOSPT 2021 [14] and NCKM 2020 [12] scored 78%, and some deductions were made because of the lack of a description of the potential impact of the financial support.
An overall evaluation of the quality and feasibility of the 10 CPGs were evaluated. Three CPGs were rated as Class A, thus, these CPGs could be recommended without modification. Five CPGs were rated as Class B indicating these CPGs can be recommended following revision. Two CPGs were rated Class C, and so these CPGs were not recommended. Among the 6 domains of the AGREE II, the domain with the highest average score was “clarity of presentation” (59.8%), followed by “scope and purpose” (55.3%), “rigor of development” (48.7%), “stakeholder involvement” (44.6%), “editorial independence” (34.8%), and “applicability” (21.4%) had the lowest score. To create a CPG that meets the AGREE II standards, it will be necessary to supplement the Domain 5 “applicability” in the future.
Discussion
CPG allows not only to diagnose, treat, and evaluate ankle sprains easily but also increase patient compliance because of its economic feasibility. Considering that ankle sprain is a common disease, the number of selected CPGs was smaller than expected. This might be because many CPGs were excluded owing to the absence of evidence or recommendations in the CPGs selection process, and ankle sprains are considered a relatively minor health concern. This study is the first to evaluate CPG for ankle sprains using the AGREE II tool. Although a study evaluating CPGs for ankle sprains using the AGREE II was published in 2019, this study was limited to “acute lateral” ankle ligament sprains in the target disease category, and the target population was limited to adults. Moreover, considering that the most recent CPG in this study was published in 2013, our study is significant in that it covers all ankle sprains and includes the latest CPG published in 2021 without any age restrictions [12]. Through our study, we intended to suggest a way to improve CPGs for traditional medicine by evaluating the quality of CPGs using the AGREE II tool.
Among the 10 CPGs, three received Class A in the overall evaluation. Even if they received Class A, they did not score very high in all six domains. JOSPT 2021 [14] and KCE 2013 [18] received 36% and 26% respectively in domain “applicability,” but received Class A. This means that both CPGs need to be supplemented according to the criteria of domain “applicability.” This also means that stricter standards should be followed for evaluation in the future. NCKM 2020 [12] is the only CPG with a score of over 60% in all domains, and CPG with the highest average score.
Table 1 [1221] presents the general information on the CPGs. Diagnosis information was described in all CPGs, except EUJIM 2017 [17]. Information on treatment was described for all CPGs, except for ACR 2020 [15]. Prevention and management are described for each of the six CPGs. The recommendations for each CPGs were summarized by dividing them into traditional medicine CPGs and conventional Western medicine CPGs (Tables 4 and 5 [1221]).
The average scores for each domain of the two traditional medicine CPGs and eight conventional Western medicine CPGs are shown in Table 6. The scores of traditional medicine CPGs were high in all areas. In addition, the scores of traditional medicine CPGs in all domains, except for domain “applicability,” was over 60%. There might be an opinion that it is somewhat unreasonable to set these scores as representative evaluation values for traditional medicine CPGs. Because there are only two CPGs for traditional medicine, NCKM 2020 [12] and EUJIM 2017 [17], the average score of NCKM 2020 [12] is the highest among all CPGs. However, EUJIM 2017 [17] was rated as Class B, being rated 4th highest among all CPGs. Therefore, limited to the investigated CPGs, traditional medicine CPGs can be evaluated as CPGs that meet the criteria of the AGREE II.
Both CPGs are relatively recently published papers; in particular, NCKM 2020 [12] has been reviewed based on the AGREE II instrument during the CPG production process. In addition, NCKM 2020 [12] was developed as a project by the Ministry of Health and Welfare and was developed strictly with the participation of specialized experts such as disease-related experts, methodological experts, and external review groups. It seems that these factors caused NCKM 2020 [12] to obtain high scores.
Since both traditional medicine CPGs recorded the lowest score in the domain “applicability” in individual scores, this part needs to be supplemented in the future revision process.
The recommendations for the traditional medicine and conventional Western medicine CPGs, respectively, are shown in Tables 4 and 5. In conventional Western medicine CPGs, treatment is divided into pharmacological and non-pharmacological treatments. Of the eight conventional Western medicine CPGs, 6 CPGs suggested pharmacological treatment, and all CPGs that offered pharmacological treatment recommended NSAIDs, suggesting that NSAIDs are the most commonly used drugs for ankle sprains. Non-pharmacological treatment consists mainly of conservative treatment methods, including ankle support, exercise, and cryotherapy. Only one CPG presented surgery.
Specifically, JOSPT 2021 [14] contains the contents of acupuncture, but the level of recommendation was Class D. In JOSPT 2021 [14], the level was Class D and was assigned if the recommendation was based on conflicting evidence. The level of Class B was assigned when there was one high-quality randomized controlled trial (RCT) or multiple lesser quality diagnostic studies, prospective studies, RCTs, and systematic reviews. Level of Class C was assigned when there was one less quality diagnostic study, prospective study, RCT, systematic review, or there were many case-control studies, retrospective studies, or case series. JOSPT 2021 [14] suggested a systematic review as evidence for acupuncture, and they concluded that acupuncture lacks a high-quality study. However, NCKM 2020 [12] presented RCT studies in which no treatment group or cold pack group was set as a control group as the evidence for the acupuncture recommendation, which can receive a minimum level of Class C or higher even in the standard of the recommendation level of JOSPT 2021 [14]. Therefore, it seems that the reason that acupuncture received a level of Class D in JOSPT 2021 [14] was because there was insufficient literature on acupuncture during the CPG development process.
NCKM 2020 [12] and EUJIM 2017 [17], which are traditional medicine CPGs, commonly recommend acupuncture and pharmacopuncture. NCKM 2020 presented a more diverse and wider range of recommendations than EUJIM 2017 [17]. NCKM 2020 [12] presents recommendations for various oriental medicine techniques, such as moxibustion, cupping, and chuna, as non-herbal pharmacological treatments. In addition, conservative treatments, such as rest, ice, compression, and elevation, have been suggested, as suggested in conventional Western medicine CPGs. Although there were mentions of NSAIDs and surgery in the overview of NCKM 2020 [12], they were not included in the recommendations. Recommendations generally tend to have low levels of evidence. For traditional medicine CPGs to have a higher quality, evidence based on a systematic research methodology, such as a large-scale RCT, systematic review and meta-analysis should be prepared. In traditional medicine, a diagnostic method called “pattern identification” is used. Herbal medicine prescriptions according to this diagnostic method were included in the recommendations. Moreover, herbal medicines such as “Dangguisoosan” and “Cheongyulsaseuptang” were suggested as recommendations depending on the symptoms patients complained about, but the level of evidence for recommendation of the CPG was low and was assigned Class C.
If an evaluation tool that supplements the parts not covered by the current AGREE II is developed, it will be helpful in the revision and development of CPGs. For example, among all the investigated CPGs, NCKM 2020 [12] met all the current standards of the AGREE II; however, it had the disadvantage of having low levels of evidence. However, the current AGREE II evaluation criteria do not reflect this.
The Korean evaluation standard was applied for 1, 3, 5, and 7 points; however, depending on the subjective judgment of the evaluator, one point can be added or subtracted to give 2, 4, or 6 points. Although all evaluators are familiar with the AGREE II, but evaluators’ subjectivity may create bias.
No overall assessment standards have been established therefore we created our own standards by referring to other appraisal literature [10,11]. In the case of KNGF 2006 [21], it received 0 points in the domain “Editorial Independence,” but a Class B in the overall assessment of quality. This indicates that more specific criteria are required for evaluation of the quality of the CPGs.
Conclusion
CPGs were evaluated by three raters using the AGREE II tool. Because the average score of all CPGs recorded the lowest score in “applicability” among the six domains of AGREE II, CPGs revised and developed in the future require supplementation in the domain “applicability”. Two traditional CPGs received higher scores compared to conventional CPGs. Although traditional CPGs can be evaluated as having high methodological quality, they lacked in evidence of recommendations. In the future, it will be necessary to develop high-quality evidence through large-scale RCTs, systematic review and meta-analysis.
Acknowledgment
This study was supported by the Traditional Korean Medicine R&D program, which is funded by the Ministry of Health & Welfare through the Korea Health Industry Development Institute (KHIDI) (HF20C0014).
The authors would like to thank Editage (www.editage.co.kr) for English language editing.
Notes
Author Contributions
Conceptualization: JYJ. Methodology: JYJ. Formal investigation: JYJ, YCP and MGJ. Data analysis: JYJ, YCP and MGJ. Writing original draft: JYJ. Writing - review and editing: JYJ, MGJ, MJK, SYC, SEL, DML, JHK, BHG, BKS, YHB, SSN and YCP.
Conflicts of Interest
There are no conflicts of interest regarding the publication of this manuscript.
Ethical Statement
This research did not involve any human or animal experiments.
Data Availability
All relevant data are included in this manuscript.
Funding
None.
Fig. 1
The flow chart of clinical practice guideline selection.
Table 1
General Information on the Included Clinical Practice Guidelines.
Study [ref] Country Organization Year Number (authors) Number (reference) Target population Subject Treatment
JOSPT 2021 [14] USA Journal of Orthopaedic & Sports Physical Therapy 2021 8 485 Adults 1. Diagnosis
2. Treatment
3. Management
4. Prevention
1. Conservative
2. Pharmacological
ACR 2020 [15] USA American College of Radiology 2020 19 76 ≥ 5 y 1. Diagnosis Not reported
NCKM 2020 [12] Republic of Korea National Institute for Korean Medicine Development 2020 11 332 Adults 1. Diagnosis
2. Treatment
3. Management
4. Prevention
1. Conservative
BJSM 2018 [16] Netherland British Journal of Sports Medicine 2018 15 216 ≥ 16 y 1. Diagnosis
2. Treatment
3. Management
4. Prevention
1. Conservative
2. Pharmacological
3. Surgery
EUJIM 2017 [17] Republic of Korea European Journal of Integrative Medicine 2017 8 41 Adults 1. Treatment 1. Conservative
KCE 2013 [18] Belgium Belgian Health Care Knowledge Centre 2013 8 141 ≥ 16 y 1. Diagnosis
2. Treatment
1. Pharmacological
2. Conservative
NATA 2013 [19] USA National Athletic Trainers’ Association 2013 9 189 Athletes 1. Diagnosis
2. Treatment
3. Management
4. Prevention
1. Conservative
2. Pharmacological
AFP 2012 [13] USA American Family Physician 2012 1 40 Not reported 1. Diagnosis
2. Treatment
3. Management
4. Prevention
1. Conservative
2. Pharmacological
ORRE 2012 [20] Germany Orthopedic Reviews 2012 7 175 Not reported 1. Diagnosis
2. Treatment
1. Conservative
2. Pharmacological
3. Surgery
KNGF 2006 [21] Netherland Royal Dutch Society of Physical Therapy 2006 14 26 Athletes 1. Diagnosis
2. Treatment
3. Management
4. Prevention
1. Conservative
ACR, American College of Radiology; AFP, American Family Physician; BJSM, British Journal of Sports Medicine; EUJIM, European Journal of Integrative Medicine; JOSPT, Journal of Orthopaedic & Sports Physical Therapy; KCE, Belgian Health Care Knowledge Centre; KNGF, Royal Dutch Society of Physical Therapy; NATA, National Athletic Trainers’ Association; NCKM, National Institute for Korean Medicine Development; ORRE, Orthopedic Reviews.
Table 2
The AGREE II Domain-Standardized Scores for Clinical Practice Guidelines and Overall Assessment.
Study [ref] Scope and purpose (%) Stakeholders involvement (%) Rigor of development (%) Clarity of presentation (%) Applicability (%) Editorial independence (%) Overall reviewer assessment (class)
JOSPT 2021 [14] 78 69 73 69 36 78 A
ACR 2020 [15] 52 19 41 91 3 17 B
NCKM 2020 [12] 100 98 81 91 72 78 A
BJSM 2018 [16] 30 65 58 35 7 17 B
EUJIM 2017 [17] 69 52 42 43 7 50 B
KCE 2013 [18] 87 37 79 83 26 64 A
NATA 2013 [19] 33 26 26 57 6 0 C
AFP 2012 [13] 2 6 18 46 17 11 C
ORRE 2012 [20] 35 28 42 39 26 33 B
KNGF 2006 [21] 67 46 27 44 14 0 B
ACR, American College of Radiology; AFP, American Family Physician; BJSM, British Journal of Sports Medicine; EUJIM, European Journal of Integrative Medicine; JOSPT, Journal of Orthopaedic & Sports Physical Therapy; KCE, Belgian Health Care Knowledge Centre; KNGF, Royal Dutch Society of Physical Therapy; NATA, National Athletic Trainers’ Association; NCKM, National Institute for Korean Medicine Development; ORRE, Orthopedic Reviews.
Table 3
Rigor of Development: Standardized Average Scores in the AGREE II Domain for Clinical Practice Guidelines.
Study [ref] Systematic methods used (%) Selecting criteria presented (%) The strengths and limitations of the body of evidence (%) Formulating the recommendations described (%) Benefits, side effects, and risks considered (%) Explicit link between the recommendations and evidence (%) Reviewed by external experts (%) Updating procedure provided (%)
JOSPT 2021 [14] 100 100 72 33 100 67 39 72
ACR 2020 [15] 100 22 67 0 56 83 0 0
NCKM 2020 [12] 78 94 72 78 100 100 56 72
BJSM 2018 [16] 61 67 56 33 89 33 33 89
EUJIM 2017 [17] 61 27 50 78 56 61 6 0
KCE 2013 [18] 100 78 78 94 50 78 61 94
NATA 2013 [19] 0 0 50 0 78 67 17 0
AFP 2012 [13] 39 0 0 0 39 67 0 0
ORRE 2012 [20] 72 56 50 22 72 61 0 0
KNGF 2006 [21] 0 0 50 22 56 89 0 0
ACR, American College of Radiology; AFP, American Family Physician; BJSM, British Journal of Sports Medicine; EUJIM, European Journal of Integrative Medicine; JOSPT, Journal of Orthopaedic & Sports Physical Therapy; KCE, Belgian Health Care Knowledge Centre; KNGF, Royal Dutch Society of Physical Therapy; NATA, National Athletic Trainers’ Association; NCKM, National Institute for Korean Medicine Development; ORRE, orthopedic reviews.
Table 4
Recommendations for Ankle Sprain in Traditional Medicine Clinical Practice Guidelines.
Study ID Diagnosis Herbal pharmacological treatment Non-herbal pharmacological treatment Management
NCKM 2020 [12] Imaging
Clinical evaluation
1.Prescribed herbal medicine according to pattern identification
1) Liver Kidney Yin deficiency 2) Qi obstruction due to bloodstream malfunction
2. Manufactured herbal medicine
1) Dangguisoosan 2) Cheongyulsaseuptang 3) Banggihwanggitang 4) Jakyakgamchotang
1. Acupuncture
1) General acupuncture
2) Electroacupuncture
3) Fire needle acupuncture
2. Pharmacopuncture
1) Bee venom pharmacopuncture
3. Moxibustion
4. Cupping
5.Chuna
6. Taping
7. Conservative
1) Rest
2) Ice
3) Compression
4) Elevation
5) Physical therapy
1. Exercise
2. Rehabilitation
3. Footwear
EUJIM 2017 [17] Not reported Not reported 1. Acupuncture
1) General acupuncture
2) Electroacupuncture
2. Pharmacopuncture
Not reported
EUJIM, European Journal of Integrative Medicine; NCKM, National Institute for Korean Medicine Development.
Table 5
Recommendations for Ankle Sprain in Conventional Western Medicine Clinical Practice Guidelines.
Study [ref] Diagnosis Pharmacological treatment Non-pharmacological treatment Management
JOSPT 2021 [14] 1. History taking
2. Physical examination
3. Measurement instruments
1. NSAIDs 1. Conservative
1) Ankle support, 2) Exercise, 3) Manual therapy, 4) Cryotherapy, 5) Diathermy, 6) Low-level laser therapy
1. Physical examination
2. Ankle support
3. Exercise
ACR 2020 [15] 1.Imaging Not reported Not reported Not reported
BJSM 2018 [16] 1. Physical examination
2. Imaging
1. NSAIDs 1. Conservative
1) Ankle support, 2) Exercise, 3) Manual mobilization
2. Surgery
1. Ankle support
2. Exercise
KCE 2013 [18] 1. History taking
2. Physical examination
3. Imaging
1. Paracetamol
2. NSAIDs
3. Opioids
4. Venotonic drugs
5. Ointment
1. Conservative
1) RICE, 2) Ultrasound, 3) Laser therapy, 4) Ankle support, 5) Manual therapy, 6) Exercise therapy
Not reported
NATA 2013 [19] 1. History taking
2. Physical examination
3. Imaging
1. NSAIDs 1. Conservative
1) Cryotherapy, 2) RICE, 3) Ankle support, 4) Exercise, 5) Electrotherapy
1. Measurement
instruments
2. Performance test
3. Ankle support
4. Exercise
AFP 2012 [13] 1. History taking
2. Physical examination
1. NSAIDs
2. Acetaminophen
3. Opioids
1. Conservative
1) Cryotherapy, 2) Ankle support, 3) Exercise
1. Follow-up
2. Rehabilitation
3. Ankle support
ORRE 2012 [20] 1. History taking
2. Imaging
3. Physical examination
1. NSAIDs 1. Conservative
1) RICE, 2) Ankle support, 3) Cryotherapy
Not reported
KNGF 2006 [21] 1. History taking
2. Red flags
3. Physical examination
4. Measurement instruments
Not reported 1. Conservative
1) RICE, 2) Functional treatment, 3) Exercise, 4) Strength training, 5) Ultrasound, 6) Laser therapy, 7) Electrotherapy
1. Ankle support
2. Footwear
ACR, American College of Radiology; AFP, American Family Physician; BJSM, British Journal of Sports Medicine; JOSPT, Journal of Orthopaedic & Sports Physical Therapy; KCE, Belgian Health Care Knowledge Centre; KNGF, Royal Dutch Society of Physical Therapy; NATA, National Athletic Trainers’ Association; ORRE, orthopedic reviews.
Table 6
A Comparison of the Average Scores of Traditional and Conventional Western Medicine Clinical Practice Guidelines for the AGREE II Domains.
Average scores Scope and purpose (%) Stakeholders involvement (%) Rigor of development (%) Clarity of presentation (%) Applicability (%) Editorial independence (%)
All CPGs (n = 10) 55 45 49 60 21 35
Traditional medicine CPGs (n = 2) 85 75 62 67 40 64
Conventional Western medicine CPGs (n = 8) 48 37 46 58 17 28
References
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2. Herzog MM, Kerr ZY, Marshall SW, Wikstrom EA. Epidemiology of ankle sprains and chronic ankle instability. J Athl Train. 2019;54:603–610.
3. van den Bekerom MP, van der Windt DA, ter Riet G, van der Heijden GJ, Bouter LM. Therapeutic ultrasound for acute ankle sprains. Cochrane Database Syst Rev. 2011;2011:CD001250.
4. Bielska IA, Wang X, Lee R, Johnson AP. The health economics of ankle and foot sprains and fractures: A systematic review of English-language published papers. Part 2: The direct and indirect costs of injury. Foot. 2019;39:115–121.
5. Hollon SD, Aréan PA, Craske MG, Crawford KA, Kivlahan DR, Magnavita JJ, et al. Development of clinical practice guidelines. Annu Rev Clin Psychol. 2014;10:213–241.
6. Thomas L. Clinical practice guidelines. Evid Based Nurs. 1999;2:38–39.
7. Brouwers MC, Kho ME, Browman GP, Burgers JS, Cluzeau F, Feder G, et al. AGREE II: advancing guideline development, reporting and evaluation in health care. CMAJ. 2010;182:E839–E842.
8. Brouwers MC, Kerkvliet K, Spithoff K, Consortium ANS. The AGREE Reporting Checklist: A tool to improve reporting of clinical practice guidelines. BMJ. 2016;352:i1152.
9. Dans AL, Dans LF. Appraising a tool for guideline appraisal (the AGREE II instrument). J Clin Epidemiol. 2010;63:1281–1282.
10. Kim JY, Kim JH, Goo BH, Park YC, Seo BK, Baek YH. Quality assessment of conventional and traditional oriental medicine clinical practice guidelines for knee osteoarthritis using AGREE II instrument. Medicine (Baltimore). 2021;100:e28426.
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12. Kim JH. [Internet]. Korean Medicine Clinical Practice Guideline for Ankle Sprain. Korean Acupunct Moxibust Soc. 2020. 165, Available from: https://nikom.or.kr/nckm/module/practiceGuide/view.do?guide_idx=208&menu_idx=14 .
13. Tiemstra JD. Update on acute ankle sprains. Am Fam Physician. 2012;85:1170–1176.
14. Martin RL, Davenport TE, Fraser JJ, Sawdon-Bea J, Carcia CR, Carroll LA, et al. Ankle stability and movement coordination impairments: Lateral ankle ligament sprains revision 2021. J Orthop Sports Phys Ther. 2021;51:CPG1–CPG80.
15. Smith SE, Chang EY, Ha AS, Bartolotta RJ, Bucknor M, Chandra T, et al. ACR Appropriateness Criteria® Acute Trauma to the Ankle. J Am Coll Radiol. 2020;17:S355–S366.
16. Vuurberg G, Hoorntje A, Wink LM, Van Der Doelen BFW, Van Den Bekerom MP, Dekker R, et al. Diagnosis, treatment and prevention of ankle sprains: Update of an evidence-based clinical guideline. Br J Sports Med. 2018;52:956.
17. Choi J, Jun JH, Kim JU, Choi TY, Lee JA, Yook TH, et al. Korean medicine clinical practice guideline on acupuncture for acute ankle sprains in adults: Evidence-based approach. Eur J Integr Med. 2017;12:182–188.
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19. Kaminski TW, Hertel J, Amendola N, Docherty CL, Dolan MG, Hopkins JT, et al. National athletic trainers’ association position statement: Conservative management and prevention of ankle sprains in athletes. J Athl Train. 2013;48:528–545.
20. Polzer H, Kanz KG, Prall WC, Haasters F, Ockert B, Mutschler W, et al. Diagnosis and treatment of acute ankle injuries: Development of an evidence-based algorithm. Orthop Rev (Pavia). 2011;4:e5.
21. Koninklijk Nederlands Genootschap voor Fysiotherapie [Internet]. Practice guidelines: Acute ankle sprain. Available from: https://www.kngf.nl/kennisplatform/guidelines .
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https://ruder.io/word-embeddings-softmax/
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This post gives an overview of approximations that can be used to make the expensive softmax layer more efficient.
This is the second post in a series on word embeddings and representation learning. In the previous post, we gave an overview of word embedding models and introduced the classic neural language model by Bengio et al. (2003), the C&W model by Collobert and Weston (2008), and the word2vec model by Mikolov et al. (2013). We observed that mitigating the complexity of computing the final softmax layer has been one of the main challenges in devising better word embedding models, a commonality with machine translation (MT) (Jean et al., 2015) [1] and language modelling (Jozefowicz et al., 2016) [2].
In this post, we will thus focus on giving an overview of various approximations to the softmax layer that have been proposed over the last years, some of which have so far only been employed in the context of language modelling or MT. We will postpone the discussion of additional hyperparameters to the subsequent post.
Let us know partially re-introduce the previous post's notation both for consistency and to facilitate comparison as well as introduce some new notation: We assume a training corpus containing a sequence of $T$ training words $w_1, w_2, w_3, \cdots, w_T$ that belong to a vocabulary $V$ whose size is $|V|$. Our models generally consider a context $c$ of $n$ words. We associate every word with an input embedding $v_w$ (the eponymous word embedding in the Embedding Layer) with $d$ dimensions and an output embedding $v'_w$ (the representation of the word in the weight matrix of the softmax layer). We finally optimize an objective function $J_\theta$ with regard to our model parameters $\theta$.
Recall that the softmax calculates the probability of a word $w$ given its context $c$ and can be computed using the following equation:
$p(w | c) = \dfrac{\text{exp}({h^\top v'_w})}{\sum_{w_i \in V} \text{exp}({h^\top v'_{w_i}})}$
where $h$ is the output vector of the penultimate network layer. Note that we use $c$ for the context as mentioned above and drop the index $t$ of the target word $w_t$ for simplicity. Computing the softmax is expensive as the inner product between $h$ and the output embedding of every word $w_i$ in the vocabulary $V$ needs to be computed as part of the sum in the denominator in order to obtain the normalized probability of the target word $w$ given its context $c$.
In the following we will discuss different strategies that have been proposed to approximate the softmax. These approaches can be grouped into softmax-based and sampling-based approaches. Softmax-based approaches are methods that keep the softmax layer intact, but modify its architecture to improve its efficiency. Sampling-based approaches on the other hand completely do away with the softmax layer and instead optimise some other loss function that approximates the softmax.
# Softmax-based Approaches
## Hierarchical Softmax
Hierarchical softmax (H-Softmax) is an approximation inspired by binary trees that was proposed by Morin and Bengio (2005) [3]. H-Softmax essentially replaces the flat softmax layer with a hierarchical layer that has the words as leaves, as can be seen in Figure 1.
This allows us to decompose calculating the probability of one word into a sequence of probability calculations, which saves us from having to calculate the expensive normalization over all words. Replacing a softmax layer with H-Softmax can yield speedups for word prediction tasks of at least $50 \times$ and is thus critical for low-latency tasks such as real-time communication in Google's new messenger app Allo.
We can think of the regular softmax as a tree of depth $1$, with each word in $V$ as a leaf node. Computing the softmax probability of one word then requires normalizing over the probabilities of all $|V|$ leaves. If we instead structure the softmax as a binary tree, with the words as leaf nodes, then we only need to follow the path to the leaf node of that word, without having to consider any of the other nodes.
Since a balanced binary tree has a depth of $\text{log}_2 (|V|)$, we only need to evaluate at most $\text{log}_2 (|V|)$ nodes to obtain the final probability of a word. Note that this probability is already normalized, as the probabilities of all leaves in a binary tree sum to $1$ and thus form a probability distribution. To informally verify this, we can reason that at a tree's root node (Node 0) in Figure 1), the probabilities of branching decisions must sum to $1$. At each subsequent node, the probability mass is then split among its children, until it eventually ends up at the leaf nodes, i.e. the words. Since no probability is lost along the way and since all words are leaves, the probabilities of all words must necessarily sum to $1$ and hence the hierarchical softmax defines a normalized probability distribution over all words in $V$.
To get a bit more concrete, as we go through the tree, we have to be able to calculate the probability of taking the left or right branch at every junction. For this reason, we assign a representation to every node. In contrast to the regular softmax, we thus no longer have output embeddings $v'_w$ for every word $w$ -- instead, we have embeddings $v'_n$ for every node $n$. As we have $|V|-1$ nodes and each one possesses a unique representation, the number of parameters of H-Softmax is almost the same as for the regular softmax. We can now calculate the probability of going right (or left) at a given node $n$ given the context $c$ the following way:
$p(\text{right} | n, c) = \sigma (h^\top v'_{n})$.
This is almost the same as the computations in the regular softmax; now instead of computing the dot product between $h$ and the output word embedding $v'_{w}$, we compute the dot product between $h$ and the embedding $v'_{w}$ of each node in the tree; additionally, instead of computing a probability distribution over the entire vocabulary words, we output just one probability, the probability of going right at node $n$ in this case, with the sigmoid function. Conversely, the probability of turning left is simply $1 - p( \text{right} | n,c)$.
The probability of a word $w$ given its context $c$ is then simply the product of the probabilities of taking right and left turns respectively that lead to its leaf node. To illustrate this, given the context "the", "dog", "and", "the", the probability of the word "cat" in Figure 2 can be computed as the product of the probability of turning left at node 1, turning right at node 2, and turning right at node 5. Hugo Lachorelle gives a more detailed account in his excellent lecture video. Rong (2014) [4] also does a good job of explaining these concepts and also derives the derivatives of H-Softmax.
Obviously, the structure of the tree is of significance. Intuitively, we should be able to achieve better performance, if we make it easier for the model to learn the binary predictors at every node, e.g. by enabling it to assign similar probabilities to similar paths. Based on this idea, Morin and Bengio use the synsets in WordNet as clusters for the tree. However, they still report inferior performance to the regular softmax. Mnih and Hinton (2008) [5] learn the tree structure with a clustering algorithm that recursively partitions the words in two clusters and allows them to achieve the same performance as the regular softmax at a fraction of the computation.
Notably, we are only able to obtain this speed-up during training, when we know the word we want to predict (and consequently its path) in advance. During testing, when we need to find the most likely prediction, we still need to calculate the probability of all words, although narrowing down the choices in advance helps here.
In practice, instead of using "right" and "left" in order to designate nodes, we can index every node with a bit vector that corresponds to the path it takes to reach that node. In Figure 2, if we assume a 0 bit for turning left and a 1 bit for turning right, we can thus represent the path to "cat" as 011.
Recall that the path length in a balanced binary tree is $\text{log}_2 |V|$. If we set $|V| = 10000$, this amounts to an average path length of about $13.3$. Analogously, we can represent every word by the bit vector of its path that is on average $13.3$ bits long. In information theory, this is referred to as an information content of $13.3$ bits per word.
### A note on the information content of words
Recall that the information content $I(w)$ of a word $w$ is the negative logarithm of its probability $p(w)$:
$I(w) = -\log_2 p(w)$.
The entropy $H$ of all words in a corpus is then the expectation of the information content of all words in the vocabulary:
$H = \sum_{i\in V} p(w_i) I(w_i)$.
We can also conceive of the entropy of a data source as the average number of bits needed to encode it. For a fair coin flip, we need $1$ bit per flip, whereas we need $0$ bits for a data source that always emits the same symbol. For a balanced binary tree, where we treat every word equally, the word entropy $H$ equals the information content $I(w)$ of every word $w$, as each word has the same probability. The average word entropy $H$ in a balanced binary tree with $|V| = 10000$ thus coincides with its average path length:
$H = - \sum_{i\in V} \dfrac{1}{10000} \log_2 \dfrac{1}{10000} = 13.3$.
We saw before that the structure of the tree is important. Notably, we can leverage the tree structure not only to gain better performance, but also to speed up computation: If we manage to encode more information into the tree, we can get away with taking shorter paths for less informative words. Morin and Bengio point out that leveraging word probabilities should work even better; as some words are more likely to occur than others, they can be encoded using less information. They note that the word entropy of their corpus (with $|V| = 10,000$) is about $9.16$.
Thus, by taking into account frequencies, we can reduce the average number of bits per word in the corpus from $13.3$ to $9.16$ in this case, which amounts to a speed-up of 31%. A Huffman tree, which is used by Mikolov et al. (2013) [6] for their hierarchical softmax, generates such a coding by assigning fewer bits to more common symbols. For instance, "the", the most common word in the English language, would be assigned the shortest bit code in the tree, the second most frequent word would be assigned the second-shortest bit code, and so on. While we still need the same number of codes to designate all words, when we predict the words in a corpus, short codes appear now a lot more often, and we consequently need fewer bits to represent each word on average.
A coding such as Huffman coding is also known as entropy encoding, as the length of each codeword is approximately proportional to the entropy of each symbol as we have observed. Shannon (1951) [7] establishes in his experiments that the lower bound on the information rate in English is between $0.6$ to $1.3$ bits per character; given an average word length of $4.5$, this amounts to $2.7$ - $5.85$ bits per word.
To tie this back to language modelling (which we already talked about in the previous post): perplexity, the evaluation measure of language modelling, is $2^{H}$ where $H$ is the entropy. A unigram entropy of $9.16$ thus entails a still very high perplexity of $2^{9.16} = 572.0$. We can render this value more tangible by observing that a model with a perplexity of $572$ is as confused by the data as if it had to choose among $572$ possibilities for each word uniformly and independently.
To put this into context: The state-of-the-art language model by Jozefowicz et al. (2016) achieves a perplexity of $24.2$ per word on the 1B Word Benchmark. Such a model would thus require an average of around $4.60$ bits to encode each word, as $2^{4.60} = 24.2$, which is incredibly close to the experimental lower bounds documented by Shannon. If and how we could use such a model to construct a better hierarchical softmax layer is still left to be explored.
## Differentiated Softmax
Chen et al. (2015) [8] introduce a variation on the traditional softmax layer, the Differentiated Softmax (D-Softmax). D-Softmax is based on the intuition that not all words require the same number of parameters: Many occurrences of frequent words allow us to fit many parameters to them, while extremely rare words might only allow to fit a few.
In order to do this, instead of the dense matrix of the regular softmax layer of size $d \times |V|$ containing the output word embeddings $v'_w \in \mathbb{R}^d$, they use a sparse matrix. They then arrange $v'_w$ in blocks sorted by frequency, with the embeddings in each block being of a certain dimensionality $d_k$. The number of blocks and their embedding sizes are hyperparameters that can be tuned.
In Figure 3, embeddings in partition $A$ are of dimensionality $d_A$ (these are embeddings of frequent words, as they are allocated more parameters), while embeddings in partitions $B$ and $C$ have $d_B$ and $d_C$ dimensions respectively. Note that all areas not part of any partition, i.e. the non-shaded areas in Figure 1, are set to $0$.
The output of the previous hidden layer $h$ is treated as a concatenation of features corresponding to each partition of the dimensionality of that partition, e.g. $h$ in Figure 3 is made up of partitions of size $d_A$, $d_B$, and $d_B$ respectively. Instead of computing the matrix-vector product between the entire output embedding matrix and $h$ as in the regular softmax, D-Softmax then computes the product of each partition and its corresponding section in $h$.
As many words will only require comparatively few parameters, the complexity of computing the softmax is reduced, which speeds up training. In contrast to H-Softmax, this speed-up persists during testing. Chen et al. (2015) observe that D-Softmax is the fastest method when testing, while being one of the most accurate. However, as it assigns fewer parameters to rare words, D-Softmax does a worse job at modelling them.
## CNN-Softmax
Another modification to the traditional softmax layer is inspired by recent work by Kim et al. (2016) [9] who produce input word embeddings $v_w$ via a character-level CNN. Jozefowicz et al. (2016) in turn suggest to do the same thing for the output word embeddings $v'_w$ via a character-level CNN -- and refer to this as CNN-Softmax. Note that if we have a CNN at the input and at the output as in Figure 4, the CNN generating the output word embeddings $v'_w$ is necessarily different from the CNN generating the input word embeddings $v_w$, just as the input and output word embedding matrices would be different.
While this still requires computing the regular softmax normalization, this approach drastically reduces the number of parameters of the model: Instead of storing an embedding matrix of $d \times |V|$, we now only need to keep track of the parameters of the CNN. During testing, the output word embeddings $v'_w$ can be pre-computed, so that there is no loss in performance.
However, as characters are represented in a continuous space and as the resulting model tends to learn a smooth function mapping characters to a word embedding, character-based models often find it difficult to differentiate between similarly spelled words with different meanings. To mitigate this, the authors add a correction factor that is learned per word, which significantly reduces the performance gap between regular and CNN-softmax. By adjusting the dimensionality of the correction term, the authors are able to trade-off model size versus performance.
The authors also note that instead of using a CNN-softmax, the output of the previous layer $h$ can be fed to a character-level LSTM, which predicts the output word one character at a time. Instead of a softmax over words, a softmax outputting a probability distribution over characters would thus be used at every time step. They, however, fail to achieve competitive performance with this layer. Ling et al. (2016) [10] use a similar layer for machine translation and achieve competitive results.
# Sampling-based Approaches
While the approaches discussed so far still maintain the overall structure of the softmax, sampling-based approaches on the other hand completely do away with the softmax layer. They do this by approximating the normalization in the denominator of the softmax with some other loss that is cheap to compute. However, sampling-based approaches are only useful at training time -- during inference, the full softmax still needs to be computed to obtain a normalised probability.
In order to gain some intuitions about the softmax denominator's impact on the loss, we will derive the gradient of our loss function $J_\theta$ w.r.t. the parameters of our model $\theta$.
During training, we aim to minimize the cross-entropy loss of our model for every word $w$ in the training set. This is simply the negative logarithm of the output of our softmax. If you are unsure of this connection, have a look at Karpathy's explanation to gain some more intuitions about the connection between softmax and cross-entropy. The loss of our model is then the following:
$J_\theta = - \text{log} \dfrac{\text{exp}({h^\top v'_{w}})}{\sum_{w_i \in V} \text{exp}({h^\top v'_{w_i}})}$.
Note that in practice $J_\theta$ would be the average of all negative log-probabilities over the whole corpus. To facilitate the derivation, we decompose $J_\theta$ into a sum as $\text{log} \dfrac{x}{y} = \text{log} x - \text{log} y$:
$J_\theta = - h^\top v'_{w} + \text{log} \sum_{w_i \in V} \text{exp}(h^\top v'_{w_i})$
For brevity and to conform with the notation of Bengio and Senécal (2003; 2008) [11], [12] (note that in the first paper, they compute the gradient of the positive logarithm), we replace the dot product $h^\top v'_{w}$ with $- \mathcal{E}(w)$. Our loss then looks like the following:
$J_\theta = \mathcal{E}(w) + \text{log} \sum_{w_i \in V} \text{exp}( - \mathcal{E}(w_i))$
For back-propagation, we can now compute the gradient $\nabla$ of $J_\theta$ w.r.t. our model's parameters $\theta$:
$\nabla_\theta J_\theta = \nabla_\theta \mathcal{E}(w) + \nabla_\theta \text{log} \sum_{w_i \in V} \text{exp}(- \mathcal{E}(w_i))$
As the gradient of $\text{log} x$ is $\dfrac{1}{x}$, an application of the chain rule yields:
$\nabla_\theta J_\theta = \nabla_\theta \mathcal{E}(w) + \dfrac{1}{\sum_{w_i \in V} \text{exp}(- \mathcal{E}(w_i))} \nabla_\theta \sum_{w_i \in V} \text{exp}(- \mathcal{E}(w_i)$
We can now move the gradient inside the sum:
$\nabla_\theta J_\theta = \nabla_\theta \mathcal{E}(w) + \dfrac{1}{\sum_{w_i \in V} \text{exp}(- \mathcal{E}(w_i))} \sum_{w_i \in V} \nabla_\theta \text{exp}(- \mathcal{E}(w_i))$
As the gradient of $\text{exp}(x)$ is just $\text{exp}(x)$, another application of the chain rule yields:
$\nabla_\theta J_\theta = \nabla_\theta \mathcal{E}(w) + \dfrac{1}{\sum_{w_i \in V} \text{exp}(- \mathcal{E}(w_i))} \sum_{w_i \in V} \text{exp}(- \mathcal{E}(w_i)) \nabla_\theta (- \mathcal{E}(w_i))$
We can rewrite this as:
$\nabla_\theta J_\theta = \nabla_\theta \mathcal{E}(w) + \sum_{w_i \in V} \dfrac{\text{exp}(- \mathcal{E}(w_i))}{\sum_{w_i \in V} \text{exp}(- \mathcal{E}(w_i))} \nabla_\theta (- \mathcal{E}(w_i))$
Note that $\dfrac{\text{exp}(- \mathcal{E}(w_i))}{\sum_{w_i \in V} \text{exp}(- \mathcal{E}(w_i))}$ is just the softmax probability $P(w_i)$ of $w_i$ (we omit the dependence on the context $c$ here for brevity). Replacing it yields:
$\nabla_\theta J_\theta = \nabla_\theta \mathcal{E}(w) + \sum_{w_i \in V} P(w_i) \nabla_\theta (- \mathcal{E}(w_i))$
Finally, repositioning the negative coefficient in front of the sum yields:
$\nabla_\theta J_\theta = \nabla_\theta \mathcal{E}(w) - \sum_{w_i \in V} P(w_i) \nabla_\theta \mathcal{E}(w_i)$
Bengio and Senécal (2003) note that the gradient essentially has two parts: a positive reinforcement for the target word $w$ (the first term in the above equation) and a negative reinforcement for all other words $w_i$, which is weighted by their probability (the second term). As we can see, this negative reinforcement is just the expectation $\mathbb{E}_{w_i \sim P}$ of the gradient of $\mathcal{E}$ for all words $w_i$ in $V$:
$\sum_{w_i \in V} P(w_i) \nabla_\theta \mathcal{E}(w_i) = \mathbb{E}_{w_i \sim P}[\nabla_\theta \mathcal{E}(w_i)]$.
The crux of most sampling-based approach now is to approximate this negative reinforcement in some way to make it easier to compute, since we don't want to sum over the probabilities for all words in $V$.
## Importance Sampling
We can approximate the expected value $\mathbb{E}$ of any probability distribution using the Monte Carlo method, i.e. by taking the mean of random samples of the probability distribution. If we knew the network's distribution, i.e. $P(w)$, we could thus directly sample $m$ words $w_1 , \cdots , w_m$ from it and approximate the above expectation with:
$\mathbb{E}_{w_i \sim P}[\nabla_\theta \mathcal{E}(w_i)] \approx \dfrac{1}{m} \sum\limits^m_{i=1} \nabla_\theta \mathcal{E}(w_i)$.
However, in order to sample from the probability distribution $P$, we need to compute $P$, which is just what we wanted to avoid in the first place. We therefore have find some other distribution $Q$ (we call this the proposal distribution), from which it is cheap to sample and which can be used as the basis of Monte-Carlo sampling. Preferably, $Q$ should also be similar to $P$, since we want our approximated expectation to be as accurate as possible. A straightforward choice in the case of language modelling is to simply use the unigram distribution of the training set for $Q$.
This is essentially what classical Importance Sampling (IS) does: It uses Monte-Carlo sampling to approximate a target distribution $P$ via a proposal distribution $Q$. However, this still requires computing $P(w)$ for every word $w$ that is sampled. To avoid this, Bengio and Senécal (2003) use a biased estimator that was first proposed by Liu (2001) [13]. This estimator can be used when $P(w)$ is computed as a product, which is the case here, since every division can be transformed into a multiplication.
Essentially, instead of weighting the gradient $\nabla_\theta \mathcal{E}(w_i)$ with the expensive to compute probability $P_{w_i}$, we weight it with a factor that leverages the proposal distribution $Q$. For biased IS, this factor is $\dfrac{1}{R}r(w_i)$ where $r(w) = \dfrac{\text{exp}(- \mathcal{E}(w))}{Q(w)}$ and $R = \sum^m_{j=1} r(w_j)$.
Note that we use $r$ and $R$ instead of $w$ and $W$ as in Bengio and Senécal (2003, 2008) to avoid name clashes. As we can see, we still compute the numerator of the softmax, but replace the normalisation in the denominator with the proposal distribution $Q$. Our biased estimator that approximates the expectation thus looks like the following:
$\mathbb{E}_{w_i \sim P}[\nabla_\theta \mathcal{E}(w_i)] \approx \dfrac{1}{R} \sum\limits^m_{i=1} r(w_i) \nabla_\theta \mathcal{E}(w_i)$
Note that the fewer samples we use, the worse is our approximation. We additionally need to adjust our sample size during training, as the network's distribution $P$ might diverge from the unigram distribution $Q$ during training, which leads to divergence of the model, if the sample size that is used is too small. Consequently, Bengio and Senécal introduce a measure to calculate the effective sample size in order to protect against possible divergence. Finally, the authors report a speed-up factor of $19$ over the regular softmax for this method.
## Adaptive Importance Sampling
Bengio and Senécal (2008) note that for Importance Sampling, substituting more complex distributions, e.g. bigram and trigram distributions, later in training to combat the divergence of the unigram distribution $Q$ from the model's true distribution $P$ does not help, as n-gram distributions seem to be quite different from the distribution of trained neural language models. As an alternative, they propose an n-gram distribution that is adapted during training to follow the target distribution $P$ more closely. To this end, they interpolate a bigram distribution and a unigram distribution according to some mixture function, whose parameters they train with SGD for different frequency bins to minimize the Kullback-Leibler divergence between the distribution $Q$ and the target distribution $P$. For experiments, they report a speed-up factor of about $100$.
## Target Sampling
Jean et al. (2015) propose to use Adaptive Importance Sampling for machine translation. In order to make the method more suitable for processing on a GPU with limited memory, they limit the number of target words that need to be sampled from. They do this by partitioning the training set and including only a fixed number of sample words in every partition, which form a subset $V'$ of the vocabulary.
This essentially means that a separate proposal distribution $Q_i$ can be used for every partition $i$ of the training set, which assigns equal probability to all words included in the vocabulary subset $V'_i$ and zero probability to all other words.
## Noise Contrastive Estimation
Noise Contrastive Estimation (NCE) (Gutmann and Hyvärinen, 2010) [14] is proposed by Mnih and Teh (2012) [15] as a more stable sampling method than Importance Sampling (IS), as we have seen that IS poses the risk of having the proposal distribution $Q$ diverge from the distribution $P$ that should be optimized. In contrast to the former, NCE does not try to estimate the probability of a word directly. Instead, it uses an auxiliary loss that also optimises the goal of maximizing the probability of correct words.
Recall the pairwise-ranking criterion of Collobert and Weston (2008) that ranks positive windows higher than "corrupted" windows, which we discussed in the previous post. NCE does a similar thing: We train a model to differentiate the target word from noise. We can thus reduce the problem of predicting the correct word to a binary classification task, where the model tries to distinguish positive, genuine data from noise samples, as can be seen in Figure 4 below.
For every word $w_i$ given its context $c_i$ of $n$ previous words $w_{t-1} , \cdots , w_{t-n+1}$ in the training set, we thus generate $k$ noise samples $\tilde{w}_{ik}$ from a noise distribution $Q$. As in IS, we can sample from the unigram distribution of the training set. As we need labels to perform our binary classification task, we designate all correct words $w_i$ given their context $c_i$ as true ($y=1$) and all noise samples $\tilde{w}_{ik}$ as false ($y=0$).
We can now use logistic regression to minimize the negative log-likelihood, i.e. cross-entropy of our training examples against the noise (conversely, we could also maximize the positive log-likelihood as some papers do):
$J_\theta = - \sum_{w_i \in V} [ \text{log} P(y=1 | w_i,c_i) + k \mathbb{E}_{\tilde{w}_{ik} \sim Q} [ \text{log} P(y=0 | \tilde{w}_{ij},c_i)]]$.
Instead of computing the expectation $\mathbb{E}_{\tilde{w}_{ik} \sim Q}$ of our noise samples, which would still require summing over all words in $V$ to predict the normalised probability of a negative label, we can again take the mean with the Monte Carlo approximation:
$J_\theta = - \sum_{w_i \in V} [ \text{log} P(y=1 | w_i,c_i) + k \sum_{j=1}^k \dfrac{1}{k} \text{log} P(y=0 | \tilde{w}_{ij},c_i)]$,
which reduces to:
$J_\theta = - \sum_{w_i \in V} [ \text{log} P(y=1 | w_i,c_i) + \sum_{j=1}^k \text{log} P(y=0 | \tilde{w}_{ij},c_i)]$,
By generating $k$ noise samples for every genuine word $w_i$ given its context $c$, we are effectively sampling words from two different distributions: Correct words are sampled from the empirical distribution of the training set $P_{\text{train}}$ and depend on their context $c$, whereas noise samples come from the noise distribution $Q$. We can thus represent the probability of sampling either a positive or a noise sample as a mixture of those two distributions, which are weighted based on the number of samples that come from each:
$P(y, w | c) = \dfrac{1}{k+1} P_{\text{train}}(w | c)+ \dfrac{k}{k+1}Q(w)$.
Given this mixture, we can now calculate the probability that a sample came from the training $P_{\text{train}}$ distribution as a conditional probability of $y$ given $w$ and $c$:
$P(y=1 | w,c)= \dfrac{\dfrac{1}{k+1} P_{\text{train}}(w | c)}{\dfrac{1}{k+1} P_{\text{train}}(w | c)+ \dfrac{k}{k+1}Q(w)}$,
which can be simplified to:
$P(y=1 | w,c)= \dfrac{P_{\text{train}}(w | c)}{P_{\text{train}}(w | c) + k Q(w)}$.
As we don't know $P_{\text{train}}$ (which is what we would like to calculate), we replace $P_{\text{train}}$ with the probability of our model $P$:
$P(y=1 | w,c)= \dfrac{P(w | c)}{P(w | c) + k Q(w)}$.
The probability of predicting a noise sample ($y=0$) is then simply $P(y=0 | w,c) = 1 - P(y=1 | w,c)$. Note that computing $P(w | c)$, i.e. the probability of a word $w$ given its context $c$ is essentially the definition of our softmax:
$P(w | c) = \dfrac{\text{exp}({h^\top v'_{w}})}{\sum_{w_i \in V} \text{exp}({h^\top v'_{w_i}})}$.
For notational brevity and unambiguity, let us designate the denominator of the softmax with $Z(c)$, since the denominator only depends on $h$, which is generated from $c$ (assuming a fixed $V$). The softmax then looks like this:
$P(w | c) = \dfrac{\text{exp}({h^\top v'_{w}})}{Z(c)}$.
Having to compute $P(w | c)$ means that -- again -- we need to compute $Z(c)$, which requires us to sum over the probabilities of all words in $V$. In the case of NCE, there exists a neat trick to circumvent this issue: We can treat the normalisation denominator $Z(c)$ as a parameter that the model can learn.
Mnih and Teh (2012) and Vaswani et al. (2013) [16] actually keep $Z(c)$ fixed at $1$, which they report does not affect the model's performance. This assumption has the nice side-effect of reducing the model's parameters, while ensuring that the model self-normalises by not depending on the explicit normalisation in $Z(c)$. Indeed, Zoph et al. (2016) [17] find that even when learned, $Z(c)$ is close to $1$ and has low variance.
If we thus set $Z(c)$ to $1$ in the above softmax equation, we are left with the following probability of word $w$ given a context $c$:
$P(w | c) = \text{exp}({h^\top v'_{w}})$.
We can now insert this term in the above equation to compute $P(y=1 | w,c)$:
$P(y=1 | w,c)= \dfrac{\text{exp}({h^\top v'_{w}})}{\text{exp}({h^\top v'_{w}}) + k Q(w)}$.
Inserting this term in turn in our logistic regression objective finally yields the full NCE loss:
$J_\theta = - \sum_{w_i \in V} [ \text{log} \dfrac{\text{exp}({h^\top v'_{w_i}})}{\text{exp}({h^\top v'_{w_i}}) + k Q(w_i)} + \sum_{j=1}^k \text{log} (1 - \dfrac{\text{exp}({h^\top v'_{\tilde{w}_{ij}}})}{\text{exp}({h^\top v'_{\tilde{w}_{ij}}}) + k Q(\tilde{w}_{ij})})]$.
Note that NCE has nice theoretical guarantees: It can be shown that as we increase the number of noise samples $k$, the NCE derivative tends towards the gradient of the softmax function. Mnih and Teh (2012) report that $25$ noise samples are sufficient to match the performance of the regular softmax, with an expected speed-up factor of about $45$. For more information on NCE, Chris Dyer has published some excellent notes [18].
One caveat of NCE is that as typically different noise samples are sampled for every training word $w$, the noise samples and their gradients cannot be stored in dense matrices, which reduces the benefit of using NCE with GPUs, as it cannot benefit from fast dense matrix multiplications. Jozefowicz et al. (2016) and Zoph et al. (2016) independently propose to share noise samples across all training words in a mini-batch, so that NCE gradients can be computed with dense matrix operations, which are more efficient on GPUs.
### Similarity between NCE and IS
Jozefowicz et al. (2016) show that NCE and IS are not only similar as both are sampling-based approaches, but are strongly connected. While NCE uses a binary classification task, they show that IS can be described similarly using a surrogate loss function: Instead of performing binary classification with a logistic loss function like NCE, IS then optimises a multi-class classification problem with a softmax and cross-entropy loss function. They observe that as IS performs multi-class classification, it may be a better choice for language modelling, as the loss leads to tied updates between the data and noise samples rather than independent updates as with NCE. Indeed, Jozefowicz et al. (2016) use IS for language modelling and obtain state-of-the-art performance (as mentioned above) on the 1B Word benchmark.
## Negative Sampling
Negative Sampling (NEG), the objective that has been popularised by Mikolov et al. (2013), can be seen as an approximation to NCE. As we have mentioned above, NCE can be shown to approximate the loss of the softmax as the number of samples $k$ increases. NEG simplifies NCE and does away with this guarantee, as the objective of NEG is to learn high-quality word representations rather than achieving low perplexity on a test set, as is the goal in language modelling.
NEG also uses a logistic loss function to minimise the negative log-likelihood of words in the training set. Recall that NCE calculated the probability that a word $w$ comes from the empirical training distribution $P_{\text{train}}$ given a context $c$ as follows:
$P(y=1 | w,c)= \dfrac{\text{exp}({h^\top v'_{w}})}{\text{exp}({h^\top v'_{w}}) + k Q(w)}$.
The key difference to NCE is that NEG only approximates this probability by making it as easy to compute as possible. For this reason, it sets the most expensive term, $k Q(w)$ to $1$, which leaves us with:
$P(y=1 | w,c)= \dfrac{\text{exp}({h^\top v'_{w}})}{\text{exp}({h^\top v'_{w}}) + 1}$.
$k Q(w) = 1$ is exactly then true, when $k= |V|$ and $Q$ is a uniform distribution. In this case, NEG is equivalent to NCE. The reason we set $k Q(w) = 1$ and not to some other constant can be seen by rewriting the equation, as $P(y=1 | w,c)$ can be transformed into the sigmoid function:
$P(y=1 | w,c)= \dfrac{1}{1 + \text{exp}({-h^\top v'_{w}})}$.
If we now insert this back into the logistic regression loss from before, we get:
$J_\theta = - \sum_{w_i \in V} [ \text{log} \dfrac{1}{1 + \text{exp}({-h^\top v'_{w_i}})} + \sum_{j=1}^k \text{log} (1 - \dfrac{1}{1 + \text{exp}({-h^\top v'_{\tilde{w}_{ij}}})}]$.
By simplifying slightly, we obtain:
$J_\theta = - \sum_{w_i \in V} [ \text{log} \dfrac{1}{1 + \text{exp}({-h^\top v'_{w_i}})} + \sum_{j=1}^k \text{log} (\dfrac{1}{1 + \text{exp}({h^\top v'_{\tilde{w}_{ij}}})}]$.
Setting $\sigma(x) = \dfrac{1}{1 + \text{exp}({-x})}$ finally yields the NEG loss:
$J_\theta = - \sum_{w_i \in V} [ \text{log} \sigma(h^\top v'_{w_i}) + \sum_{j=1}^k \text{log} \sigma(-h^\top v'_{\tilde{w}_{ij}})]$.
To conform with the notation of Mikolov et al. (2013), $h$ must be replaced with $v_{w_I}$, $v'_{w_i}$ with $v'_{w_O}$ and $v_{\tilde{w}_{ij}}$ with $v'_{w_i}$. Also, in contrast to Mikolov's NEG objective, we a) optimise the objective over the whole corpus, b) minimise negative log-likelihood instead of maximising positive log-likelihood (as mentioned before), and c) have already replaced the expectation $\mathbb{E}_{\tilde{w}_{ik} \sim Q}$ with its Monte Carlo approximation. For more insights on the derivation of NEG, have a look at Goldberg and Levy's notes [19].
We have seen that NEG is only equivalent to NCE when $k= |V|$ and $Q$ is uniform. In all other cases, NEG only approximates NCE, which means that it will not directly optimise the likelihood of correct words, which is key for language modelling. While NEG may thus be useful for learning word embeddings, its lack of asymptotic consistency guarantees makes it inappropriate for language modelling.
## Self-Normalisation
Even though the self-normalisation technique proposed by Devlin et al. [20] is not a sampling-based approach, it provides further intuitions on self-normalisation of language models, which we briefly touched upon. We previously mentioned in passing that by setting the denominator $Z(c)$ of the NCE loss to $1$, the model essentially self-normalises. This is a useful property as it allows us to skip computing the expensive normalisation in $Z(c)$.
Recall that our loss function $J_\theta$ minimises the negative log-likelihood of all words $w_i$ in our training data:
$J_\theta = - \sum\limits_i [\text{log} \dfrac{\text{exp}({h^\top v'_{w_i}})}{Z(c)}]$.
We can decompose the softmax into a sum as we did before:
$J_\theta P(w | c) = - \sum\limits_i [h^\top v'_{w_i} + \text{log} Z(c)]$.
If we are able to constrain our model so that it sets $Z(c) = 1$ or similarly $\text{log} Z(c) = 0$, then we can avoid computing the normalisation in $Z(c)$ altogether. Devlin et al. (2014) thus propose to add a squared error penalty term to the loss function that encourages the model to keep $\text{log} Z(c)$ as close as possible to $0$:
$J_\theta = - \sum\limits_i [h^\top v'_{w_i} + \text{log} Z(c) - \alpha (\text{log}(Z(c)) - 0)^2]$,
which can be rewritten as:
$J_\theta = - \sum\limits_i [h^\top v'_{w_i} + \text{log} Z(c) - \alpha \text{log}^2 Z(c)]$
where $\alpha$ allows us to trade-off between model accuracy and mean self-normalisation. By doing this, we can essentially guarantee that $Z(c)$ will be as close to $1$ as we want. At decoding time in their MT system, Devlin et al. (2014) then set the denominator of the softmax to $1$ and only use the numerator for computing $P(w | c)$ together with their penalty term:
$J_\theta = - \sum\limits_i [h^\top v'_{w_i} - \alpha \text{log}^2 Z(c)]$
They report that self-normalisation achieves a speed-up factor of about $15$, while only resulting in a small degradation of BLEU scores compared to a regular non-self-normalizing neural language model.
## Infrequent Normalisation
Andreas and Klein (2015) [21] suggest that it should even be sufficient to only normalise a fraction of the training examples and still obtain approximate self-normalising behaviour. They thus propose Infrequent Normalisation (IN), which down-samples the penalty term, making this a sampling-based approach.
Let us first decompose the sum of the previous loss $J_\theta$ into two separate sums:
$J_\theta = - \sum\limits_i h^\top v'_{w_i} + \alpha \sum\limits_i \text{log}^2 Z(c)$.
We can now down-sample the second term by only computing the normalisation for a subset $C$ of words $w_j$ and thus of contexts $c_j$ (as $Z(c)$ only depends on the context $c$) in the training data:
$J_\theta = - \sum\limits_i h^\top v'_{w_i} + \dfrac{\alpha}{\gamma} \sum\limits_{c_j \in C} \text{log}^2 Z(c_j)$
where $\gamma$ controls the size of the subset $C$. Andreas and Klein (2015) suggest that IF combines the strengths of NCE and self-normalisation as it does not require computing the normalisation for all training examples (which NCE avoids entirely), but like self-normalisation allows trading-off between the accuracy of the model and how well normalisation is approximated. They observe a speed-up factor of $10$ when normalising only a tenth of the training set, with no noticeable performance penalty.
### Other Approaches
So far, we have focused exclusively on approximating or even entirely avoiding the computation of the softmax denominator $Z(c)$, as it is the most expensive term in the computation. We have thus not paid particular attention to $h^\top v'_{w}$, i.e. the dot-product between the penultimate layer representation $h$ and output word embedding $v'_{w}$. Vijayanarasimhan et al. (2015) [22] propose fast locality-sensitive hashing to approximate $h^\top v'_{w}$. However, while this technique accelerates the model at test time, during training, these speed-ups virtually vanish as embeddings must be re-indexed and the batch size increases.
# Which Approach to Choose?
Having reviewed the most popular softmax-based and sampling-based approaches, we have shown that there are plenty of alternatives to the good ol' softmax and almost all of them promise a significant speed-up and equivalent or at most marginally deteriorated performance. This naturally poses the question which approach is the best for a particular task.
We compare the performance of the approaches we discussed in this post for language modelling in Table 1. Speed-up factors and performance are based on the experiments by Chen et al. (2015), while we show speed-up factors reported by the authors of the original papers in brackets. The third and fourth columns indicate if the speed-up is achieved during training and testing respectively. Note that divergence of speed-up factors might be due to unoptimised implementations or the fact that the original authors might not have had access to GPUs, which benefit the regular softmax more than some of the other approaches. Performance for approaches where no comparison is available should largely be analogous to similar approaches, i.e. Self-Normalisation should achieve similar performance as Infrequent Normalisation and Importance Sampling and Adaptive Importance Sampling should achieve similar performance as Target Sampling. The performance of CNN-Softmax is as reported by Jozefowicz et al. (2016) and ranges from bad to good depending on the size of the correction. Of all approaches, only CNN-Softmax achieves a substantial reduction in parameters as the other approaches still require storing output embeddings. Differentiated Softmax reduces parameters by being able to store a sparse weight matrix.
As it always is, there is no clear winner that beats all other approaches on all datasets or tasks. For language modelling, the regular softmax still achieves very good performance on small vocabulary datasets, such as the Penn Treebank, and even performs well on medium datasets, such as Gigaword, but does very poorly on large vocabulary datasets, e.g. the 1B Word Benchmark. Target Sampling, Hierarchical Softmax, and Infrequent Normalisation in turn do better with large vocabularies.
Differentiated Softmax generally does well for both small and large vocabularies and is the only approach that ensures a speed-up at test time. Interestingly, Hierarchical Softmax (HS) performs very poorly with small vocabularies. However, of all methods, HS is the fastest and processes most training examples in a given time frame. While NCE performs well with large vocabularies, it is generally worse than the other methods. Negative Sampling does not work well for language modelling, but it is generally superior for learning word representations, as attested by word2vec's success. Note that all results should be taken with a grain of salt: Chen et al. (2015) report having difficulties using Noise Contrastive Estimation in practice; Kim et al. (2016) use Hierarchical Softmax to achieve state-of-the-art with a small vocabulary, while Importance Sampling is used by the state-of-the-art language model by Jozefowicz et al. (2016) on a dataset with a large vocabulary.
Finally, if you are looking to actually use the described methods, TensorFlow has implementations for a few sampling-based approaches and also explains the differences between some of them here.
# Conclusion
This overview of different methods to approximate the softmax attempted to provide you with intuitions that can not only be applied to improve and speed-up learning word representations, but are also relevant for language modelling and machine translation. As we have seen, most of these approaches are closely related and are driven by one uniting factor: the necessity to approximate the expensive normalisation in the denominator of the softmax. With these approaches in mind, I hope you feel now better equipped to train and understand your models and that you might even feel ready to work on learning better word representations yourself.
As we have seen, learning word representations is a vast field and many factors are relevant for success. In the previous blog post, we looked at the architectures of popular models and in this blog post, we investigated more closely a key component, the softmax layer. In the next one, we will introduce GloVe, a method that relies on matrix factorisation rather than language modelling, and turn our attention to other hyperparameters that are essential for successfully learning word embeddings.
As always, let me know about any mistakes I made and approaches I missed in the comments below.
# Citation
If you found this blog post helpful, please consider citing it as:
Sebastian Ruder. On word embeddings - Part 2: Approximating the Softmax. http://ruder.io/word-embeddings-softmax, 2016.
# Other blog posts on word embeddings
If you want to learn more about word embeddings, these other blog posts on word embeddings are also available:
# Translations
This blog post has been translated into the following languages:
Credit for the cover image goes to Stephan Gouws who included the image in his PhD dissertation and in the Tensorflow word2vec tutorial.
1. Jean, S., Cho, K., Memisevic, R., & Bengio, Y. (2015). On Using Very Large Target Vocabulary for Neural Machine Translation. Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), 1–10. Retrieved from http://www.aclweb.org/anthology/P15-1001 ↩︎
2. Jozefowicz, R., Vinyals, O., Schuster, M., Shazeer, N., & Wu, Y. (2016). Exploring the Limits of Language Modeling. Retrieved from http://arxiv.org/abs/1602.02410 ↩︎
3. Morin, F., & Bengio, Y. (2005). Hierarchical Probabilistic Neural Network Language Model. Aistats, 5. ↩︎
4. Rong, X. (2014). word2vec Parameter Learning Explained. arXiv:1411.2738, 1–19. Retrieved from http://arxiv.org/abs/1411.2738 ↩︎
5. Mnih, A., & Hinton, G. E. (2008). A Scalable Hierarchical Distributed Language Model. Advances in Neural Information Processing Systems, 1–8. Retrieved from http://papers.nips.cc/paper/3583-a-scalable-hierarchical-distributed-language-model.pdf ↩︎
6. Mikolov, T., Chen, K., Corrado, G., & Dean, J. (2013). Distributed Representations of Words and Phrases and their Compositionality. NIPS, 1–9. ↩︎
7. Shannon, C. E. (1951). Prediction and Entropy of Printed English. Bell System Technical Journal, 30(1), 50–64. http://doi.org/10.1002/j.1538-7305.1951.tb01366.x ↩︎
8. Chen, W., Grangier, D., & Auli, M. (2015). Strategies for Training Large Vocabulary Neural Language Models. Retrieved from http://arxiv.org/abs/1512.04906 ↩︎
9. Kim, Y., Jernite, Y., Sontag, D., & Rush, A. M. (2016). Character-Aware Neural Language Models. AAAI. Retrieved from http://arxiv.org/abs/1508.06615 ↩︎
10. Ling, W., Trancoso, I., Dyer, C., & Black, A. W. (2016). Character-based Neural Machine Translation. ICLR, 1–11. Retrieved from http://arxiv.org/abs/1511.04586 ↩︎
11. Bengio, Y., & Senécal, J.-S. (2003). Quick Training of Probabilistic Neural Nets by Importance Sampling. AISTATS. http://www.iro.umontreal.ca/~lisa/pointeurs/submit_aistats2003.pdf ↩︎
12. Bengio, Y., & Senécal, J.-S. (2008). Adaptive importance sampling to accelerate training of a neural probabilistic language model. IEEE Transactions on Neural Networks, 19(4), 713–722. http://doi.org/10.1109/TNN.2007.912312 ↩︎
13. Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer. http://doi.org/10.1017/CBO9781107415324.004 ↩︎
14. Gutmann, M., & Hyvärinen, A. (2010). Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. International Conference on Artificial Intelligence and Statistics, 1–8. Retrieved from http://www.cs.helsinki.fi/u/ahyvarin/papers/Gutmann10AISTATS.pdf ↩︎
15. Mnih, A., & Teh, Y. W. (2012). A Fast and Simple Algorithm for Training Neural Probabilistic Language Models. Proceedings of the 29th International Conference on Machine Learning (ICML’12), 1751–1758. ↩︎
16. Vaswani, A., Zhao, Y., Fossum, V., & Chiang, D. (2013). Decoding with Large-Scale Neural Language Models Improves Translation. Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing (EMNLP 2013), (October), 1387–1392. ↩︎
17. Zoph, B., Vaswani, A., May, J., & Knight, K. (2016). Simple, Fast Noise-Contrastive Estimation for Large RNN Vocabularies. NAACL. ↩︎
18. Dyer, C. (2014). Notes on Noise Contrastive Estimation and Negative Sampling. Arxiv preprint. Retrieved from http://arxiv.org/abs/1410.8251 ↩︎
19. Goldberg, Y., & Levy, O. (2014). word2vec Explained: Deriving Mikolov et al.’s Negative-Sampling Word-Embedding Method. arXiv Preprint arXiv:1402.3722, (2), 1–5. Retrieved from http://arxiv.org/abs/1402.3722 ↩︎
20. Devlin, J., Zbib, R., Huang, Z., Lamar, T., Schwartz, R., & Makhoul, J. (2014). Fast and robust neural network joint models for statistical machine translation. Proc. ACL’2014, 1370–1380. ↩︎
21. Andreas, J., & Klein, D. (2015). When and why are log-linear models self-normalizing? Naacl-2015, 244–249. ↩︎
22. Vijayanarasimhan, S., Shlens, J., Monga, R., & Yagnik, J. (2015). Deep Networks With Large Output Spaces. Iclr, 1–9. Retrieved from http://arxiv.org/abs/1412.7479 ↩︎
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2020-01-19 17:20:56
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https://pokemongo.gamepress.gg/tdo-how-calculate-pokemons-ability
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# TDO: How to calculate a pokemon's ability
TDO: How to calculate a pokemon's ability
While battle simulations are more accurate for specific situations, TDO, which stands for "total damage output", can be a very useful way to rank pokemon and their attacks. TDO is equal to the amount of damage a pokemon can dish out before dying. The advantages of TDO include:
• It incorporates a Pokemon's offensive and defensive ability in one useful number
• It is just a formula, so it can be plugged into excel
• You don't need to specify the defender's moves or stats
• If two pokemon battle, the one with higher TDO will do better every time
Actual TDO requires battle simulations to calculate. But TDO is easy to approximate with just a formula. This approximation is what is seen on pokemon spreadsheets*. Not everyone uses the same approximations, and some formulas are better than others. We have spent some time to come up with a formula that incorporates energy gains from taking damage and other edge-effects.
The approximate formulas for TDO and tankiness itself are simple.
$$TDO_{prop} = DPS1 * HP * DEF$$ $$Tankiness_{prop} = HP * DEF$$
DPS1 is your DPS against a target with 1 defense, and HP, DEF are the hit points, defense of your pokemon. But even though these formulas are short, the formula for DPS itself is a bit more complex. This is to take into account corrections for energy gains from damage and wasted energy gains from dying early. It is given in its full form at the very bottom of this page!
*Actually what is used is not TDO, but a formula which is proportional to TDO, meaning the TDO of each pokemon is off by some constant factor. This is good enough, because all that we need is each Pokemon's relative ranking to make moveset or counter decisions.
List of best pokemon TDO & DPS
The spreadsheet has every pokemon so it is too big for this page. It lists all combinations of moves, including legacy moves.
Survival time was based on t5 raid bosses.
What doesn't TDO account for?
• It only considers the attacker, not the matchup. Some pokemon are useful because of their typing, and while it accounts for STAB (same type attack bonus) it doesn't account for type advantages. It can be made to account for a matchup by adding type multipliers to the damage formula.
• Discontinuities: TDO calculates as if there were a steady stream of damage & energy going between attacker and defender. This is accurate for long battles & still pretty good for shorter battles but not as accurate as simulation results. Some edge effects (leftover energy)
are accounted for but everything is at least somewhat approximate.
• nTotalCMovesSimple is approximate and may vary a lot from battle to battle. Choosing a specific number for it introduces a bias.
• It doesn't account for lag. This could be done pretty easily by adding a little time to move durations. But since fmoves are buffered anyway, it should be a decent approximation.
Tankiness has some flaws too. Doing lots of damage actually should decrease your survival time because the enemy gains more energy. This is not accounted for in just DEF * HP, but for gym defenders it won't make a big difference.
In a world of simulations, what is the use of TDO anymore? A few things. Simulations are good at looking at very specific matchups, but it is very hard to average simulations in an accurate way that gets an overall metric of a pokemon as each battle introduces different biases. This makes approximate TDO and DPS an attractive, yet approximate, overall metric.
To add, there are some details of battle mechanics which still need to be worked out, like:
• Does the defender choose their next move at the beginning of the 2 second pause between attacks? If so, it can't use the 2s of energy it gained. This significantly reduces the number of charge moves that go off.
• Does the defender only use up cmove energy after the move finishes? If so,
it probably has a lot of overkill energy loss during the move itself, especially for full-bar cmoves.
Another advantage of TDO is ease of use. You just plug and chug... no programming. Use it on excel sheets etc to get quick results without investing a lot of time and effort on tons of simulations.
Derivation of the TDO Formula
The base formula is simple: TDO is approximately proportional to $$TDO_{prop} = DPS1 * HP * DEF$$ Before deriving the DPS formula, I will show why a pokemon with higher TDO always wins. The pokemon that wins is the one that does a higher percent of the enemy's HP per second. In other words, pokemon A will win against pokemon B if: $$\frac{DPS_A,avg}{HP_B} > \frac{DPS_B,avg}{HP_A}$$
Then multiply by HP_A * HP_B on both sides of the equation to get the equivalent condition:
$$DPS_A * HP_A > DPS_B * HP_B$$
Therefore DPS * HP is what determines which pokemon wins. But DPS_A depends on DEF_B and vice-versa... It would be nice to have only properties of pokemon A on the left side, and only pokemon B on the right side. So let's define a new variable, DPS1, which is DPS but with the DEF factored out. It is the DPS you would do against a target with 1 defense:
$$DPS_A = \frac{DPS1_A}{DEF_B}$$
Plug it into the inequality:
$$\frac{DPS1_A}{DEF_B} * HP_A > \frac{DPS1_B}{DEF_A} * HP_B$$
And multiply by DEF_A * DEF_B on both sides:
$$DPS1_A * HP_A * DEF_A > DPS1_B * HP_B * DEF_B$$
So equivalently, the pokemon who wins is the one with the highest DPS1 * HP * DEF. Often TDO is seen written as ATK * DEF * HP... you can see that these are similar because DPS is proportional to ATK. But DPS1 * HP * DEF is more accurate so it is the one we will use.
At this point, the problem is just to find DPS (we can get DPS1 just by multiplying that by defense). Define a cycle by a series of Fmoves, and then one Cmove. Then the formula we will use is
$$weaveDPS_{simple} =$$ $$\frac{CDamage + nFmovesPerCycle * FDamage}{CycleTime}$$
...but first we need to find expressions for nFMovesPerCycle and CycleTime. Note that any variable starting with F or C stands for fast move or charge move, respectively. So for example CDuration is the duration of a charge move (how long after you start it until you can move again). Any variable beginning with "n" stands for number.
CycleTime depends on how many of each move is in a cycle:
$$CycleTime =$$ $$CDuration + nFmovesPerCycle * FDuration$$ So now we just need nFmovesPerCycle to plug into the DPS equation. This will be a bit harder to find.
How many Fmoves can you do in one cycle? In one cycle, you gain energy equal to ~CmoveEnergyCost. The percent of that energy which comes from Fmoves depends on the energy/second gained by taking damage vs the energy/second from spamming your Fmoves:
$$PercentEnergyFromFmoves = \frac{EPS_{Fmoves}}{EPS_{Fmoves} + EPS_{dmg}}$$ $$EPS_{Fmoves}=\frac{FmoveEnergyGain}{FmoveDuration}$$ $$EPS_{dmg}=EnemyDPS/2$$
The last thing we need is EnemyDPS. It would be complicated to deal with exactly, but simulation results provide a good experimental relationship between EnemyDPS and Defense for a t4 or t5 raid boss. This already includes STAB but should be scaled if there are type advantages involved.
$$EnemyDPS = 1930/DEF$$ Now we are ready to calculate weaveDPS, which is the average damage per second you get throughout a cycle. $$weaveDPS_{simple} =$$ $$\frac{CDamage + nFmovesPerCycle * FDamage}{CycleTime}$$ Why did I add the "simple" subscript to weaveDPS? Often, the battle ends and you still have wasted energy left. So for the last half-cycle or so, your DPS will just be FMove DPS. We can account for this by taking a weighted average of weaveDPS with FmoveDPS: $$weaveDPS_{actual} =$$ $$\frac{nCMoves * weaveDPS_{simple} + 0.5 * FmoveDPS}{nCMoves + 0.5}$$ where $$FmoveDPS = FmoveDamage/FmoveDuration$$
For this you will need nCmoves:
$$nCmoves = floor(SurvivalTime/CycleTime)$$ $$=floor(0.0005*STA*DEF/CycleTime)$$
The equation for SurvivalTime is an approximation based on real-world tests.
At this point, there is a chain of variables to plug in, everything is expressed in terms of numbers which can be found in the game_master file or on pokemon go websites, like base stats of a pokemon. Below is a summary.
Summary of the formulas
The formula for DPS and TDO are below. DPS (which was called weaveDPS_actual in the derivation) is the more complicated formula. It is: $$DPS =$$ $$\frac{nCmoves * DPS_{simple} + 0.5 * \frac{FDamage}{FDuration}}{nCmoves + 0.5}$$ To use that, you will need the formula for nCmoves and DPS_simple: $$nCmoves = \frac{0.0005*STA*DEF}{CDuration+nFmovesPerCycle*FDuration}$$ $$DPS_{simple}=$$ $$\frac{CDamage+nFmovesPerCycle*FDamage}{CDuration+nFmovesPerCycle*FDuration}$$ And to use those, you will need the formula for nFmovesPerCycle: $$nFmovesPerCycle =$$ $$\frac{CEnergyCost*FEnergyGain*Def}{FEnergyGain*Def+1930*FDuration}$$
For FDamage, CDamage, use the damage formula: $$damage = ceil(0.5*ATK*Power/EnemyDefense)$$ As for TDO (total damage output), the formula is $$TDO = DPS1 * HP * DEF$$
...but be careful, DPS1 is different from DPS. In particular, DPS1 is the same as the DPS formula but at the end when you use the damage formula, you have to plug in EnemyDefense=1.
Normally when plugging in for Attack, Defense and HP, you'd use
$$ATK = (BaseATK+ATK IV)*CPM$$
...where CPM is the CP Multiplier. But for this, CPM makes no difference, so you can leave it out (set it equal to 1). IVs can make a minor difference, so I recommend using something like 14/14/14.
Lastly, as mentioned before, the formula for survival time (tankiness) is
$$SurvivalTime = DEF * HP * 0.0005$$
How did the 0.0005 slip in there? It is just a constant multiplier that makes the survival times approximately in seconds. It is based off of an in-game battle test.
In the spreadsheets on this post, values were all scaled by the same constant so that dragonite's dragonbreath/dragonpulse TDO and DPS match results from gamepress' in-house battle simulator. The proportion of all results stayed the same.
~doublefelix921
Raids
Raid Boss List
Raid Boss Counters
Tools
Pokemon List
IV Calculator
Moves List
Appraisals
Egg Chart
Type Chart
Power Up Costs
Buddy Distances
CP Calculator
Rankings
Gym Attackers
Gym Defenders
DPS per type
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2017-12-18 13:00:19
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http://emps.exeter.ac.uk/news-events/events-colloquia/event/?semID=2423&dateID=5120
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# Thursday 11 Jul 2019: Partitions of a Certain Family of Subgroups of Perm(G)
### Alan Koch - Agnes Scott College, Georgia, USA
Amory C417 14:00-15:00
Let $N$ and $G$ be finite groups of the same order, and let $\mathcal{N}$ be the set of all regular subgroups of $\operatorname{Perm}(G)$ which are isomorphic to $N$ and are normalized by conjugation by elements of $G$ acting via left regular representation. We describe two different partitions of $\mathcal{N}$: a partition into brace classes and a partition into $G$-isomorphism classes, and discuss relationships between them. Examples will be given, particularly in the case $|G|=pq$ with $p,q$ prime.
Visit website
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2019-09-22 16:18:25
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https://gmatclub.com/blog/category/blog/gmat-tests/page/2/?fl=menu
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GMAT Question of the Day (February 8)
- Feb 8, 02:00 AM Comments [0]
Math If John throws a coin until a series of three consecutive heads or three consecutive tails appears, what is the probability that the game will end on the fourth throw? A. B. C. D. E. Question...
GMAT Question of the Day (February 7)
- Feb 7, 02:00 AM Comments [0]
Math Is divisible by 32? (1) , , and are consecutive even integers (2) Question Discussion & Explanation Correct Answer - C - (click and drag your mouse to see the answer) Verbal The misnomer that sibling rivalry is a...
GMAT Question of the Day (February 6)
- Feb 6, 02:00 AM Comments [0]
Math If in a six-digit integer , is the value of the digit, is divisible by 7 (For example, is the value of the hundreds digit of )? (1) $F(1) = F(4), F(2) = F(5), F(3) = F(6)$
GMAT Question of the Day (February 3)
- Feb 3, 02:00 AM Comments [0]
Math Jack's income in October was 25% greater than his income in September. Jack's income in November was 20% less than his income in October. If Jack earned \$200 more in October than in September, what was Jack's aggregate income in autumn? A. \$1200 B. \$1800 C. \$2000 D. \$2400 E....
GMAT Question of the Day (February 2)
- Feb 2, 02:00 AM Comments [0]
Math Set consists of consecutive multiples of 3 and set consists of consecutive multiples of 6. Is the median of set larger than the median of set ? (1) The least element in either set is...
GMAT Question of the Day (February 1)
- Feb 1, 02:00 AM Comments [0]
Math Two solutions of acid were mixed to obtain 10 liters of a new solution. Before they were mixed, the first solution contained 0.8 liters of acid while the second contained 0.6 liters of acid. If the percentage of acid in the first solution was twice...
GMAT Question of the Day (January 31)
- Jan 31, 02:00 AM Comments [0]
Math In the decimal notation of number , what is the third digit to the right of the decimal point? A. 0 B. 1 C. 2 D. 4 E. 8 Question Discussion & Explanation Correct Answer - B - (click and drag your mouse to see the answer) Verbal In 1981 children in the United...
GMAT Question of the Day (January 30)
- Jan 30, 02:00 AM Comments [0]
Math If points , , and form a triangle, is angle degrees? (1) (2) Question Discussion & Explanation Correct Answer - B - (click and drag your mouse to see the answer) Verbal Originally delivered as the first...
GMAT Question of the Day (January 27)
- Jan 27, 02:00 AM Comments [0]
Math Is divisible by 8? (1) and are even integers. (2) is divisible by 8. Question Discussion & Explanation Correct Answer - C - (click and drag your mouse to see the answer) Verbal In contrast to the tense private drama,...
GMAT Question of the Day (January 26)
- Jan 26, 02:00 AM Comments [0]
Math The cost of one photocopy is \$0.02. However, a 25% discount is offered on orders of more than 100 photocopies. If Steve and David have to make 80 copies each, how much will each of them save if they submit a single order of 160...
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2017-02-22 10:54:43
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https://tex.stackexchange.com/questions/442191/how-to-make-biblatex-chicago-put-a-footnote-before-punctuation
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# How to make biblatex-chicago put a footnote before punctuation?
I use the \autocite command from the package biblatex-chicago for referencing, and I use the \footnote command for other footnotes.
Usually, I want footnotes to appear after punctuation, but in rare cases, the footnote really only refers to a single word. If a comma happens to follow that word, I want the footnote marker to appear before the comma.
While that works with \footnote, the \autocite command always places the footnote marker after the comma, even if the instruction stands before it. Here’s an MWE:
\documentclass{article}
\usepackage{biblatex-chicago}
\begin{document}
This is an example sentence%
\footnote{In linguistics, a sentence is a grammatical unit of language.},
and the footnote marker appears before the comma.
Here is another sentence%
\autocite[In law, a sentence is something different.][]{doesntexist},
and the footnote marker appears after the comma.
\end{document}
While in both cases the comma should come after the footnote marker, biblatex-chicago automatically switches their position. Why does it happen and how can I change it?
This is a feature of biblatex's \autocite command (and only the \autocite command, see https://github.com/plk/biblatex/issues/733). If the autopunct option is activated, \autocite does not only scan ahead to suppress unwanted double punctuation, it can also move around the following punctuation mark if desired. All other cite commands only avoid double punctuation and do not move the punctuation mark.
With autocite=footnote biblatex moves the punctuation following a footnote marker issued by \autocite before the footnote marker - supposedly this gives a slightly nicer output.
If you don't want that you can disable that behaviour by redefining the footnote autocite setting
\documentclass{article}
\usepackage{biblatex-chicago}
\DeclareAutoCiteCommand{footnote}[r]{\smartcite}{\smartcites}
\ExecuteBibliographyOptions{autocite=footnote}
\begin{document}
This is an example sentence%
\footnote{In linguistics, a sentence is a grammatical unit of language.},
and the footnote marker appears before the comma.
Here is another sentence%
\autocite[In law, a sentence is something different.][]{doesntexist},
and the footnote marker appears after the comma.
\end{document}
The letter r in the optional argument tells biblatex not to move punctuation (i.e., to leave it to the right of the citation command), the value l would tell biblatex to move the following punctuation to the left of a citation. The default value for footnote was f, which behaves like l in normal text and like r in footnotes.
The additional \ExecuteBibliographyOptions{autocite=footnote} is necessary because of https://github.com/plk/biblatex/issues/758.
Theoretically you could obtain the same output with autopunct=false, but that would also disable the detection of unwanted double punctuation.
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2019-10-15 06:22:36
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https://electronics.stackexchange.com/questions/498044/what-kind-of-inductor-for-dc-dc-converter-modules-emc-filter
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# What kind of inductor for DC-DC converter module's EMC filter?
I'm looking at the datasheets for a number of switching DC-DC converters. They suggest for EMC filtering to add an inductor and a capacitor or two on the input side of the converter.
The question: what kind of inductors are required? How critical are the values?
I'm not after product recommendation, I'm after clarification of the type of component.
Are they "axial-leaded high-frequency inductors", in a resistor-like package? Open coils? I've been looking at Epcos BC+ (datasheet), Bourns 78F (datasheet), Wurth WE-TI (datasheet).
The converter datasheets show a filter circuit as follows:
simulate this circuit – Schematic created using CircuitLab
I'm really just trying to lay out my PCB at the moment (grid is 0.127 mm, Vin is top left.)
For reference, these are the values from the DC-DC converter datasheets:
Recom R78E-0.5 series (datasheet) suggests EN 55032 EMC filtering: Class A filter L1 3.9μH, C1 4.7μF/50V, no C2; Class B filter L1 12μH, C1 10μF/100V, C2 4.7μF/50V.
Traco power TSR-1 series (datasheet) suggests Class A filter L1 5.6μH/3.5A, C1 10μF/50V, C2 10μF/50V.
• If you plan on adding a front-end $LC$ filter, make sure it does not degrade the overall stability considering the negative incremental resistance of the dc-dc module. You will probably need to damp the filter otherwise the performance may be worse than without a filter. I have explained the details in an APEC seminar I taught in 2017. – Verbal Kint May 15 '20 at 14:08
• Hi @VerbalKint thanks very much for this, but all the answers I'm getting are about the design of the filter, whether it's necessary/good etc. What I'm actually after is a sentence about the inductors, perhaps modelled on this one about a transistor: "The 2N3904 is a common NPN bipolar junction transistor used for general-purpose low-power amplifying or switching applications. It is considered low current and power, medium voltage, and moderately fast." (wikipedia). – jonathanjo May 15 '20 at 17:26
Your Traco bricks are buck converters which means the output current waveform is a sawtooth, and the input current waveform is a high-frequency square wave. Thus the highest amount of HF current harmonics is on the input, hence the filter.
Here's a neat image that shows current waveforms (source):
These are 0.5A and 1A DC-DC converters so max input current will be a bit above 1A peak square wave, say 1.5A accounting for inductor current ripple. Frequency around 500kHz. It will be somewhat smoothed by the internal ceramic caps, but short of actually measuring it, there is no way to know how choppy it will be.
Capacitor type:
Caps should withstand 1A ripple current at 500kHz.
Noise voltage on the input caps will depend on their ESR and ESL (Equivalent Series Resistance and Inductance). When input current jumps from 0A to 1A this will cause a voltage drop on the cap which is: $$\ R di + L \frac{di}{dt} \$$.
The small electrolytics you selected will have ESR above 10 ohms, and they can't take the ripple current, so they're out. Besides leaded caps of this size have ESL above 3 nH which is unnecessarily high.
The proper cap type is X7R SMD ceramic, as per datasheet. There is no need to buy 4.7µF and 10µF caps, it will probably be cheaper to just get 10x 1206 10µF X7R caps in whatever voltage you use at the input like 25V, that buying several different values. These will have ESR of a few milliohms and low ESL too.
Inductor type
Traco specs the inductor ; important parameters are saturation current, value, and self-resonant frequency (SRF). The latter is the resonant frequency of the parallel LC circuit formed by the inductor and its own capacitance.
Here SRF is about 35 MHz. Note above SRF, impedance decreases, but it still stays pretty high up to 100MHz, so the inductor will still be effective above the SRF. It doesn't abruptly stop working above the SRF.
So if you want to pick another inductor, make sure to select 5.6µH (or more), saturation current above about 2A, SRF above about 20 MHz, and series resistance depending on how much losses you deem acceptable. Also a shielded inductor is less headache. Sort by price, and pick a winner.
If this is for a one-off personal project it makes more sense to invest 50c in a LC filter rather than mulling it over 2 hours and testing it to see if you can save 50c in parts.
Layout
Caps have series inductance, which includes trace inductance. Using a ground plane ensures lowest possible inductance, which means your caps will be better at shorting HF noise to ground.
This filter is simply a current divider. You want to keep HF current out of the supply cable, so you add a high impedance in series with the cable (the inductor) and a low impedance to ground (the capacitor). This is why it is important to have low inductance capacitors.
If you don't know where to put the input cap, put it as close as possible to the DC-DC pins. It's OK to put it below the board very close to the pins if you solder this by hand.
Since there is only one layer used on your board screenshot, please use a ground plane on the other layer. Also, copper is free, so you can fatten power traces and use copper pours.
• Thanks for such a complete answer. Perhaps you could move the direct answer ("Inductor type") to the top; and do you have anything to say about shielding and Q for this type of application? – jonathanjo May 18 '20 at 10:27
• I'm a little confused when you say "invest 50c in a LC filter": you mean something other than the LC filter gvien on the datasheets? – jonathanjo May 18 '20 at 10:34
• I mean the LC filter specified in the datasheet, they already did the time-consuming work for you, so you don't have to. – bobflux May 18 '20 at 10:36
• The parts suggested on the datasheets are SMD, we want a a through-hole board. I'm not trying to redesign the filter, I'm trying to understand what they are suggesting in a deeper way than "use our matching product". – jonathanjo May 18 '20 at 10:42
• Well, component choice is about the parameters I explained in the answer. Extra inductance of thru-hole parts doesn't matter for an inductor, so it can be thru hole or SMD. However thru-hole caps will have higher inductance, so worse filtering. Use ceramics not electrolytics. – bobflux May 18 '20 at 12:29
The question: what kind of inductors are required? How critical are the values?
Just take a careful look at the data sheet for the RECOM model: -
Notice the part numbers inside the red boxes?
## RLS-397
And, for the Traco supply, they recommend this part: -
I'm not after $$\\color{orange}{\text{product recommendation}}\$$, I'm after clarification of the type of component.
It's fairly difficult to answer this question without giving a $$\\color{orange}{\text{product recommendation}}\$$ because the data sheets for the Traco and Recom actually give specific product recommendations. However, just because they do, it doesn't mean you can't locate similarly specified components at possibly a lower price. But this takes a little more care.
Are they "axial-leaded high-frequency inductors", in a resistor-like package? Open coils?
The pictures (above) taken from the data sheets tell you and, if you need any more help, just leave a comment.
If you think you have found an alternative to the TRACO TCH-141 (for example), you need to be very cautious about its self resonant frequency (SRF). At the SRF (about 35 MHz in the pictures above), it will act as a high blocking impedance and this may be high-up on the list of reasons for choosing it - that blocking impedance may very well coincide with some high spectral noise content from the converter and choosing say an inductor with a significantly different SRF may not work well at all.
For the RLS-126, it has an SRF at about 25 MHz: -
For the RLS-397, it has an SRF around 50 or 60 MHz: -
If you show due-diligence in selecting alternatives that largely meet the saturation current requirements, inductance value, ESR, SRF and SRF blocking impedance, I think you'll be OK.
• Thanks Andy, yes I had found those. What I'm after is the generic description for this kind of inductor, so I can find parts from other manufacturers and distributors. Also: could you comment on how critical the values are? Ie, if design says 12uH, can I choose 10uH or 15uH if that's what supplier has? How do I determine suitable saturation current? (Other than inferring from mfr's recommended parts?) – jonathanjo May 14 '20 at 15:20
• @jonathanjo You have to use the inductance values as stated if you want to avoid doing your own EMC testing and therefore rely on the statements in the Recom and Traco data sheets. All devices are surface mount type with the Recom parts being what is called as unshielded. The Traco part is shielded. For saturation current, you can choose an alternative device that has a higher value. You should also try and match the self-resonant frequencies shown in the data sheets because these will highly block EMI noise and may actually be designed to do this. – Andy aka May 14 '20 at 15:43
• Unfortunately, I think you are in a slightly compromised area if you try NOT to use these parts. The converter data sheets do not say why these inductors allowed the EMC tests to be passed nor how close a pass they were. – Andy aka May 14 '20 at 15:46
The main issue in selecting the inductor isn't going to be the values, but will instead be the current requirement vs. the inductor's saturation current, I(sat). This will reduce its effectiveness as a filter.
The EMI issues with DCDC arise from the the from the output, via the output filter cap, back to the DCDC low side. This has a very high-current spike when the coil input switches from Vin to GND. Minimize this loop area as much as possible, and use low ESR caps for the output filter.
The input on the other hand has a step sawtooth-like current waveform with lower currents than the output, and so isn't as critical for EMI. Use the recommended input bypassing with good low-impedance connections to Vin and ground. For a regulator this size I'd use about 10uF.
While the linked datasheets recommend some kind of an LC filter, a system-level solution to the problem can make use of a common-mode input filter to block switcher noise to the external supply. I've never had to employ one however, as bypassing the Vin plane proved adequate to meet conducted EMI using a DC input.
• Many thanks thatt's very helpful. But could I ask you to elaborate on the exact loop you mean "Minimize this loop area". And I'm sorry but could you provide a link to explain "common mode input filter" Lastly, concretely, am I looking at the right kind of inductors? – jonathanjo May 8 '20 at 12:22
• Loop area means that an outbound and return current path should try to be short, and as close to each other as possible. The farther apart, the more the area. A common-mode filter is like a transformer, but flipped on its side. It strongly rejects same-polarity currents, while allowing opposing currents through. Some use lossy ferrite cores to absorb noise and turn it into heat. – hacktastical May 8 '20 at 14:10
• Thank you again @hacktastical, but it's hard to understand your answer. My question really is: what kind of inductors are the datasheets speaking about? – jonathanjo May 9 '20 at 18:01
to give more details on physical placement, to minimize EMI:
That capacitor on input side should have its GROUND lead as close as possible to the Source pin of the switching FET or the GROUND lead of the schottky flyback diode.
• The question is about small all-in-one modules with three terminals: there is no way to get close to any particular internal component. – jonathanjo May 7 '20 at 10:41
The type of Inductor at that part of the circuit is called a "choke", already mentioned above, the input rail sees a sawtooth drain due to the switching frequency of the device, this input coil (inductor/choke) is somewhat tuned to this switching frequency and smooths the sawtooth effect to prevent possible interference to other parts of the circuit on the same power rail. i hope this helps you understand the dynamics. ;)
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2021-06-18 23:36:54
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https://mathematica.stackexchange.com/questions/49043/plotting-a-trig-functions-along-with-its-envelope?noredirect=1
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# plotting a trig functions along with its envelope
Note: This questions is quite different from the ones referred to in the comments. Those deal with numerical questions, while this one is algebraic.
I have plots of the following type:
Plot[Cos[50 t] + Cos[51 t], {t, 0, 10}]
I would like to plot a envelope over this plot, i.e. another plot that joins all of maxima and minima of this plot respectively. Here is my attempt, but it's not exactly what I'd like:
Plot[{Cos[50 t] + Cos[51 t], Cos[t] + 1.5, -Cos[t] - 1.5}, {t, 0, 10}]
How can I generate the actual envelope?
• I'm guessing that's something like Cos[42x]+Cos[43x]? – Mark McClure Jun 3 '14 at 18:29
• this is completely true but this is for the Fig.1, I want to access to sheath while I do not access to any formula for that. – Unbelievable Jun 3 '14 at 18:32
• Well, you've got to give us some kind of input to start with. – Mark McClure Jun 3 '14 at 18:34
• I plotted these ones with: Plot[{Cos[50 t] + Cos[51 t], Cos[t] + 1.5, -Cos[t] - 1.5}, {t, 0, 10}], I used of simulated functions 'Cos[t] + 1.5' and '-Cos[t] - 1.5' for the sheath. – Unbelievable Jun 3 '14 at 18:36
• Nearly a duplicate of Elegant way of obtaining the envelope of oscillating function, which is a duplicate of Mathematica envelope for the bottom of a plot, a generic function. But this one just straight trigonometry. – Michael E2 Aug 28 '14 at 0:20
Playing with the manipulate below might help. It's based on the the acoustics of beats.
Manipulate[Plot[
{Cos[a*t] + Cos[b*t], 2*Cos[(b - a) t/2], -2*Cos[(b - a) t/2]}, {t, 0, 10},
PlotStyle -> {
Directive[Opacity[0.7]],
Directive[Black, Thick],
Directive[Black, Thick]}],
{{a, 20}, 1, 50}, {{b, 21}, 1, 50}]
• Thank you so much. this is correct – Unbelievable Jun 3 '14 at 18:58
Don't mind me, I'm just having fun.
Grab the definition of HilbertTransform from this previous post, and then:
f[t_] := Cos[50 t] + Cos[51 t] + Sin[53 t] (* more sinusoids = more fun *)
g[t_] := Evaluate@HilbertTransform[f[τ], τ, t]
h[t_] := Abs[f[t] + I g[t]]
Plot[{f[t], h[t], -h[t]}, {t, 0, 10}, ImageSize -> Large, PlotPoints -> 100,
PlotStyle -> {Automatic, Black, Black}]
You can see that the envelope has a nice analytical form:
ComplexExpand[h[t]] // FullSimplify
$\sqrt{3 + 2\cos t + 2\sin 2t + 2\sin 3t}$
• This way though doesn't seem to catch when the envelope should go to zero. Shouldn't it e.g. vanish near 2? – Ruslan Jun 4 '14 at 8:34
• Should it? It still looks like a sinusoid with small but nonzero amplitude there. (The envelope correctly goes to zero for the original function in the question, if that's what you're concerned about.) – Rahul Jun 4 '14 at 8:55
• Ah, OK then, you're right. – Ruslan Jun 4 '14 at 9:02
• @Rahul: wow! that's really cool! Would this also work on a set of discrete data? Something like ListLinePlot[Accumulate[RandomReal[{-1, 1}, 1000]]] ... If not, how could one modify it so that it does? I have a bunch of data that makes a seemingly random curve like that and I'd like be able to integrate over some area under the curve like the one that you generated. (Hmmm, i just copy pasted your code, but didn't get the black curve :( what could have gone wrong?) – Raksha May 16 '15 at 3:46
• @Solarmew: You first need to copy the definition of the Hilbert transform from the post I linked to. It also gives a discrete version of the transform, which you could use in much the same way on discrete data. – Rahul May 16 '15 at 15:58
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2019-09-22 06:57:58
|
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https://mixedsignal.wordpress.com/2014/02/06/why-doesnt-the-gain-change-in-my-cmos-common-source-amplifier/
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# Why doesn’t the gain change in my CMOS common-source amplifier?
## Background
The other day I provided the students in my course: “TSTE08 Analog and discrete-time integrated circuits” a quiz. Perhaps it was a bit of a cryptical one, but I wanted to point out some of the difficulties with the current-voltage relationship in an analog amplifier, and the complexities in the choice of electrical vs. physical design parameters.
So, with this post I hope to give you both an insight in that quiz, but also an insight in a clever (?) way to set the DC operating points in your circuit…
## The quiz
The quiz related to the common-source amplifier. The input is typically connected to the NMOS and a PMOS forms as active load. Moreover:
Assume I have a common-source amplifier with an active load. Further assume that the output and input DC voltages are fixed.
What should I do to increase the DC gain of my amplifier? Use hand-calculation formulas.
1) Increase the current through the amplifier
X) Increase the width of the transistors
2) Increase the length of the transitors
Any combination of those three could be correct.
So, perhaps the quiz it was a bit cryptical, yes.
## Solution
First suggestion on how to attack the problem: find the desired relations you need. For example, the gain, $A_0$ is something like:
$A_0 = \frac{g_{mn}}{g_{out}} = \frac{g_{mn}}{g_{p}+g_n} = \frac{\frac{2 I_D }{V_{EFF}}{}}{\lambda_p I_D + \lambda_n I_D} = \frac{1}{\frac{\lambda_p + \lambda_n}{2} \cdot V_{EFF}} = \frac{1}{\lambda \cdot V_{EFF}}$
where we can clearly see that it is “independent” (!) on the current through the amplifier. The
$V_{EFF} = V_{GS} - V_{Tn} = V_{i n} - V_{Tn}$
is the effective input (DC) overdrive voltage (which we cannot touch as per the quiz!). We also know from “hand calculations” that
$I_D = \alpha \cdot V_{EFF}^2$
and since $V_{EFF}$ cannot change we also see that $I_D / \alpha$ obviously has to be constant.
Technically, we could increase both current and size such that the ratio is kept constant. However, it still does not help if we look at the gain expression above. The gain is independent on the ratio, if the effective voltage remains constant.
Options 1 and X are no valid options.
Also here we need to look at another common relation for a MOS transistor: we know that with longer widths, the channel length modulation reduces, so we get:
$\lambda \propto 1 / L$
such that
$A_0 \propto \frac{L}{ V_{EFF}}$
the channel length pops up in the numerator.
This means answer 2 is correct.
By increasing the channel length of the transistors, we effectively increase the output impedance and also increase the gain without touching the DC level (hand calculations).
## Is that really correct?
Well, is it really correct? Maybe not super-duper correct if we take all second- and third-order effects into account. There are probably small variations to the gain if we change current and width.
Let us hook up a testbench. Below we find a common-source stage with a somewhat cryptical circuit in the box on the top. That is actually a tuned current source that guarantees that the the operating points, input and output DC voltages, are kept constant. The loop with the vcvs (amplifier) will increase the current through the circuit and make the output voltage follow the reference voltage in the top left. If we also make sure that the transistors are sized well to operate in saturation region for the sweeps we will do shortly, we are more or less fine to prove our point.
Then we run a DC input voltage sweep and an AC input sweep. Below we find the frequency response on the left and the DC response on the right. The left-hand figure tells us that the DC gain is somewhere around 25dB-ish. From the right-hand figure, we see that the output DC voltage stays stuck at some 0.6 V.
### Varying widths (effectively changing current)
We do a parametric sweep and capture the DC gain value (from the frequency sweep at 10 Hz) and plot it as a function of the transistor width. (Sorry for the thin line here, you might have to press the picture for a more clear view.) We see that the gain increases from some 23.5 to some 24.7 dB when increasing the width 100 (!) times. In a linear scale that is a very small change for such a large variation and practically there is no (significant) change in gain.
### Varying lengths
Then we do a parametric sweep and capture the DC gain value, still at 10 Hz from the frequency sweep, and plot it as a function of the transistor length. (Sorry for the thin line here, you might have to press the picture for a more clear view.) The gain now increases 17.5 to 28 dB when increasing channel length 10 (!) times. In a linear scale this is now a very significant change, almost four times (12 dB).
### A comment on the test bench
The trick in the test bench is quite useful actually. It is much more convenient than running the circuit in open loop to try to find the absolute settings. If the gain is very high, you have to do quite a few sweeps around the operating point in order for the simulator to have you find the best point.
The trick is also the convenient sp2tswitch that Cadence/spectre provides to simulate a circuit in different conditions. The AC simulation (which is run after DC) will inherit the DC settings and run the AC analysis around those operating points. Notice that I had to connect the two inputs of the vcvs (essentially an differential operational amplifier with a gain of 1000) to the reference voltage during AC to force the gate of the PMOS to be quiet. It is not clear why this was needed.
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2017-08-23 19:15:04
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https://www.spreadshirt.com/must+go+t-shirts
|
# Must Go T-Shirts
Filter
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The snow must go on
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MUST GO 1.png
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MUST GO 2.png
I must go
The Show Must Go On
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Basketball is calling i must go
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KENYA IS CALLING AND I MUST GO
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The Unicorn is calling and I must go
The Pizza is calling and i must go
The Moon Is Calling And I Must Go
The Internet Is Calling And I Must Go
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Phuket Is Calling And I Must Go
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Alaska Is Calling I Must Go
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The farm is calling and i must go
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OCEAN is calling and i must go
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2018-07-16 01:35:47
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https://wiki.math.ntnu.no/tma4215/2013h/semester_project
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# Semester projects
There will be two projects in the course, the first counting 10% of the final grade, the second 20%.
## Second semester project
• 14.11 About your MATLAB file: We have the provided MATLAB files available, you do not have to include them in your own file (if you use the advice about the empty folder, you have include these in that folder).
• 14.11 We do not want you to present all the tasks in your MATLAB file. What we want is to be able to reproduce the last result(s) you present in your report by that program.
• 14.11 Suggestion: Let your MATLAB function return a matrix with the control points.
• 13.11 There is no need for a separate title page or a table of contents in a 6 pages report. The report should look something like this (with some more contents, and with a bibliography).
• 12.11 If (and only if) you have got some smart ideas that seems to work, describe them in the report. Compare with the standard method. Try to make an example as "clean" as possible (demonstrate on one particular curve, rather than on the whole picture). Try to explain why your idea is better.
• 12:11 Some comments on the MATLAB-file:
• If you have made your own examples, you can submit one data-file in an appropriate format in addition to the MATLAB-file. But the program should also work on the given examples.
• Make sure that the help (first commented lines in the file) gives sufficient information on how to use the program.
• To make sure that the submitted file runs correctly: Copy your MATLAB file and the data file(s) to an empty directory. Follow your own instructions (or even better, ask a peer student to do so). If the program runs without problems and produce the expected output, it is ready to be submitted.
• Use local functions to deal with more than one function into one file. See the MATLAB documentation for how to use it.
• The results (for example the plot, or other relevant information) should be included in the report. But no MATLAB code in the report.
• 12.11 Some of you noticed that coefficient matrix (for $\alpha$ ) somtimes becomes singular. This will happen when the tangent vectors $v_1$ and $v_2$ are parallell, and you only have 3 data points. We leave it to you to figure out why, and how to deal with it (but you are allowed to ask).
Project description
(04.11: Some of you have made me aware of misprints in the text. They are now corrected).
There will be no lectures in week 45 and 46. Supervision will be given in
Week 45:
Monday Tuesday Wednesday Thursday Friday EL6 EH EH EL6 EH EH AK AM AK Nullrommet AK AM AK Nullrommet
Week 46:
Monday Tuesday Wednesday Thursday Friday Nullrommet EH EH Nullrommet EH EH AK AK Nullrommet AK AK Nullrommet
AK: Anne Kværnø (1348)
EH: Eirik Høiseth (1340)
AM: Asif Mushtaq (1301)
Files
• A Latex template.
NB! In the conclusion section, include one short paragraph at the end where you write something about your own learning outcome. You may also write something about the working process, what was easy, what was hard, surprises, etc.
• Data files for task 2: T.txt and fem.txt.
• Discrete curve for task 3 (the three rows give the x, y and t coordinates respectively. Use initial vectors v0 = [0,1]' and v3 = [-1,2]'/√5): disCurve.txt
• For part 3: Script to generate outline from a bitmap image file genOutline. Function for detecting corner points CornerDet. Test images: airplane and dog. You should find your own images to try out as well.
• Here are two short reports taken from the proceeding of the ECMI 2008 conference. Use them as examples on how short papers can be written. They are here for the format, not for their contents. Example 1 , Example 2.
## First semester project
The project, given here is due by 16:00 on 16 September.
You can work in groups of up to 3 students (2 is recommended).
Paper reports should be delivered to the mailbox of Anne (7th floor, SBII). NB! The office closes at 16:00.
MATLAB-files and electronic reports should be sent to Eirik: Eirik [dot] Hoiseth [at] math [dot] ntnu [dot] no
Note: The report and the files should be marked with the student number of all participants of the group. No names, and no candidate numbers!
Extra office help hours during the project:
• Anne (room 1348): 15:15-16:00 all days except Wednesday (not Monday 16 September)
• Eirik (room 1340): Monday 10:15-12:00, Thursday 10:15-12:00
• Asif (room 1342): Tuesday 15:15-16:00, Thursday 15:15-16:00
All rooms are on 13th floor, SBII.
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2019-12-09 13:15:12
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https://hal.archives-ouvertes.fr/hal-01359470
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# Migration of accreting planets in radiative discs from dynamical torques
1 ECLIPSE 2016
LAB - Laboratoire d'Astrophysique de Bordeaux [Pessac]
Abstract : We present the results of hydrodynamical simulations of the orbital evolution of planets undergoing runaway gas accretion in radiative discs. We consider accreting disc models with constant mass flux through the disc, and where radiative cooling balances the effect of viscous heating and stellar irradiation. We assume that 20-30 $M_\oplus$ giant planet cores are formed in the region where viscous heating dominates and migrate outward under the action of a strong corotation torque. In the case where gas accretion is neglected, we find evidence for strong dynamical torques in accreting discs with accretion rates ${\dot M}\gtrsim 7\times 10^{-8} \;M_\odot/yr$. Their main effect is to increase outward migration rates by a factor of $\sim 2$ typically. In the presence of gas accretion, however, runaway outward migration is observed with the planet passing through the zero-torque radius and the transition between the viscous heating and stellar heating dominated regimes. The ability for an accreting planet to enter a fast migration regime is found to depend strongly on the planet growth rate, but can occur for values of the mass flux through the disc of ${\dot M}\gtrsim 5\times 10^{-8} \;M_\odot/yr$. We find that an episode of runaway outward migration can cause an accreting planet formed in the 5-10 AU region to temporarily orbit at star-planet separations as large as $\sim$60-70 AU. However, increase in the amplitude of the Lindblad torque associated with planet growth plus change in the streamline topology near the planet systematically cause the direction of migration to be reversed. Our results indicate that a planet can reach large orbital distances under the combined effect of dynamical torques and gas accretion, but an alternative mechanism is required to explain the presence of massive planets on wide orbits.
Keywords :
Type de document :
Article dans une revue
Monthly Notices of the Royal Astronomical Society, Oxford University Press (OUP): Policy P - Oxford Open Option A, 2016, 462 (4), pp.4130-4140. 〈10.1093/mnras/stw1904〉
https://hal.archives-ouvertes.fr/hal-01359470
Contributeur : Marie-Paule Pomies <>
Soumis le : vendredi 2 septembre 2016 - 14:15:19
Dernière modification le : jeudi 11 janvier 2018 - 06:27:41
### Citation
A. Pierens, Sean N. Raymond. Migration of accreting planets in radiative discs from dynamical torques. Monthly Notices of the Royal Astronomical Society, Oxford University Press (OUP): Policy P - Oxford Open Option A, 2016, 462 (4), pp.4130-4140. 〈10.1093/mnras/stw1904〉. 〈hal-01359470〉
### Métriques
Consultations de la notice
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2018-02-19 21:22:39
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https://gharpedia.com/cure-concrete-moist-curing-concrete/
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## How to Cure Concrete by Wet Covering (Moist Curing Concrete)?
Concrete curing is a chemical reaction of concrete ingredients. Concrete curing is the process of maintaining moisture inside the concrete under a specific environmental condition for promoting hydration.Concrete curing is done to increase concrete strength and stability. Concrete curing methods are decided according to the working procedure and weather condition.
Curing by wet covering is mostly observed on the construction site wherein different types of fabrics saturated with water are used. This is also known as moist curing concrete. These methods are used on both vertical and horizontal concrete surfaces.
Courtesy - vishvakarma.net
In moist curing concrete method, water absorbent fabrics are used to maintain moisture on the concrete surface by completely covering the surface immediately after the concrete has hardened sufficiently.They must be continuously kept wet to prevent the fabric from absorbing water from the body of concrete, due to capillary action.
### The Different Types of Fabrics or Materials Used are:
• Burlap for concrete curing
• Cotton mat
• Rugs
• Hessian cloth
• Jute fabric or Jute bag
• Other moisture retaining fabrics/ materials such as clay, sawdust or hay
### The Precautions to be Taken in Moist Curing Concrete Process are as follows:
• To prevent surface damage, concrete should be covered with the fabric as soon as it has hardened.
• The entire surface including the edges and joints of the slab should be kept covered so that these parts are not left inadequately cured.
• The fabric retains water for a great period of time. Therefore, the fabric must always be kept wet or moist so that concrete surface is in contact with water throughout the curing period of concrete.
• The wet burlap for concrete curing used should be free from sizing or any substance that is harmful to concrete and would cause discolouration.
• If burlap for concrete curing is used, it should be first rinsed in water to remove soluble substances and to make it more absorbent.
• A layer of earth, sand, sawdust (50 mm thick) or hay (150 mm thick )can be used as an effective medium to retain moisture for a longer period on flat surfaces rather than keeping it totally exposed. It is necessary to keep these materials continuously wet so as to prevent them from blowing away or even they absorbing water from concrete by capillary action or reverse the process.
## Material Exhibition
Explore the world of materials.
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2019-02-21 14:49:47
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http://garthtarr.github.io/mplot/articles/vip.html
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Overview of variable inclusion plots.
Rather than visualising a loss measure against model size, it can be instructive to consider which variables are present in the overall best model over a set of bootstrap replications. To facilitate comparison between models of different sizes we use the generalised information criterion,
$\textrm{GIC}(\alpha,\lambda) = \hat{Q}(\alpha) + \lambda p_{\alpha}.$
The $$\hat{Q}(\alpha)$$ component is a measure of description loss or lack of fit, a function that describes how well a model fits the data, for example, the residual sum of squares or $$-2~\times~\text{log-likelihood}$$. The number of independent regression model parameters, $$p_{\alpha}$$, is a measure of model complexity. The penalty multiplier, $$\lambda$$, determines the properties of the model selection criterion (Müller et al., 2013; Müller & Welsh, 2010). Special cases, when $$\hat{Q}(\alpha)=-2\times\text{log-likelihood}(\alpha)$$, include the AIC with $$\lambda=2$$, BIC with $$\lambda=\log(n)$$ and more generally the generalised information criterion (GIC) with $$\lambda\in\mathbb{R}$$ (Konishi & Kitagawa, 1996).
Using the same exponential weighted bootstrap replications as in the model selection plots, we have a set of $$B$$ bootstrap replications and for each model size we know which model has the smallest description loss. This information is used to determine which model minimises the GIC over a range of values of the penalty parameter, $$\lambda$$, in each bootstrap sample. For each value of $$\lambda$$, we extract the variables present in the best models over the $$B$$ bootstrap replications and calculate the corresponding bootstrap probabilities that a given variable is present. These calculations are visualised in a variable inclusion plot (VIP) as introduced by Müller & Welsh (2010) and Murray et al. (2013). The VIP shows empirical inclusion probabilities as a function of the penalty multiplier $$\lambda$$. The probabilities are calculated by observing how often each variable is retained in $$B$$ exponential weighted bootstrap replications. Specifically, for each bootstrap sample $$b=1,\ldots,B$$ and each penalty multiplier $$\lambda$$, the chosen model, $$\hat{\alpha}_{\lambda}^{b}\in \mathcal{A}$$, is that which achieves the smallest $$\textrm{GIC}(\alpha,\lambda;\mathbf{w}_b) = \hat{Q}^b(\alpha)+\lambda p_{\alpha}$$, where $$\mathbf{w}_b$$ is the $$n$$-vector of independent and identically distributed exponential weights. The inclusion probability for variable $$x_{j}$$ is estimated by $$B^{-1}\sum_{i=1}^{B}\mathbb{I}\{j\in \hat{\alpha}_{\lambda}^{b}\}$$, where $$\mathbb{I}\{j\in \hat{\alpha}_{\lambda}^{b}\}$$ is one if $$x_{j}$$ is in the final model and zero otherwise. Following Murray et al. (2013), the default range of $$\lambda$$ values is $$\lambda\in[0,2\log(n)]$$ as this includes most standard values used for the penalty parameter.
The example shown in the bottom panel of this figure is obtained using the which = "vip" argument to the plot function. As expected, when the penalty parameter is equal to zero, all variables are included in the model; the full model achieves the lowest description loss, and hence minimises the GIC when there is no penalisation. As the penalty parameter increases, the inclusion probabilities for individual variables typically decrease as more parsimonious models are preferred. In the present example, the inclusion probabilities for the $$x_8$$ variable exhibit a sharp decrease at low levels of the penalty parameter, but then increase steadily as a more parsimonious model is sought. This pattern helps to explain why stepwise model selection chose the larger model with all the variables except $$x_8$$ – there exists a local minimum. Hence, for large models the inclusion of $$x_8$$ adds no additional value over having all the other explanatory variables in the model.
It is often instructive to visualise how the inclusion probabilities change over the range of penalty parameters. The ordering of the variables in the legend corresponds to their average inclusion probability over the whole range of penalty values. We have also added an independent standard Gaussian random variable to the model matrix as a redundant variable (RV). This provides a baseline to help determine which inclusion probabilities are significant in the sense that they exhibit a different behaviour to the RV curve. Variables with inclusion probabilities near or below the RV curve can be considered to have been included by chance.
To summarise, VIPs continue the model stability theme. Rather than simply using a single penalty parameter associated with a particular information criterion, for example the AIC with $$\lambda=2$$, our implementation of VIPs adds considerable value by allowing us to learn from a range of penalty parameters. Furthermore, we are able to see which variables are most often included over a number of bootstrap samples.
#### References
Konishi, S., & Kitagawa, G. (1996). Generalised information criteria in model selection. Biometrika, 83(4), 875–890. DOI:10.1093/biomet/83.4.875
Murray, K., Heritier, S., & Müller, S. (2013). Graphical tools for model selection in generalized linear models. Statistics in Medicine, 32(25), 4438–4451. DOI:10.1002/sim.5855
Müller, S., Scealy, J. L., & Welsh, A. H. (2013). Model selection in linear mixed models. Statistical Science, 28(2), 135–167. DOI:10.1214/12-STS410
Müller, S., & Welsh, A. H. (2010). On model selection curves. International Statistical Review, 78(2), 240–256. DOI:10.1111/j.1751-5823.2010.00108.x
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2021-09-16 23:50:35
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http://math.stackexchange.com/questions/190725/length-of-the-least-period-of-a-double-modulo-sequence
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# Length of the least period of a “double” modulo sequence.
What is the length of the smallest period of the following sequence $$f[n] = \left< \left< n \right>_N \right>_M$$ where $\left<n\right>_N$ represents $n \pmod N$. Is there a special term for performing nested modulo operations?
For $N$ even (say 8) and $M = 2$, it appears that the sequence is periodic with length 2. But for $N$ odd, it seems the least period length is $N$. So this leads me to believe that if $N$ is coprime to $M$ then the least period length is $N$.
1. How do I go about showing this to be the case if it is indeed true?
2. How do I, in general, calculate the least period for $f[n]$ for arbitrary cases of $N$ and $M$?
-
Presumably, by $\langle n\rangle_N$ you mean the remainder on dividing $n$ by $N$, so, for example, $\langle 5\rangle_6=5$, $\langle 7\rangle_6=1$, $\langle 6\rangle_6=0$ (or maybe you want $\langle 6\rangle_6=6$, but it won't make any difference in what follows). Then the sequence $\langle n\rangle_N$ is $0,1,2,3,\dots,N-2,N-1,0,1,2,\dots$ and has period $N$. It follows that $\langle\langle n\rangle_N\rangle_M$ has period at most $N$. It has period less than $N$ if, and only if, $M$ is a proper divisor of $N$, since that breaks the $0,1,2,\dots,N-1$ up into repeating pieces of length $M$. So the answer is, the period is $M$ if $M$ is a factor of $N$, otherwise, $N$.
Could you elaborate on the "breaking up" process? I agree that this is the case, it is just fascinating to me only a few weeks into elementary number theory that $\left<n\right>_2$ is periodic with length 2 but $\left< \left<n\right>_7 \right>_2$ is be periodic with length 7. – dcdo Sep 4 '12 at 22:43
Easier to show an example than to put it into words. Take $\langle\langle n\rangle_8\rangle_2$. From $\langle n\rangle_8$ we get $0,1,2,3,4,5,6,7,0,1,2,3\dots$. Viewing this modulo 2, the period $0,1,2,3,4,5,6,7$ breaks up into $0,1$ then $2,3$ then $4,5$ then $6,7$ which is $0,1,0,1,0,1,0,1$. – Gerry Myerson Sep 5 '12 at 2:55
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2015-09-05 12:34:51
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https://nrich.maths.org/public/topic.php?code=72&cl=4&cldcmpid=2151
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# Resources tagged with: Generalising
Filter by: Content type:
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### There are 76 results
Broad Topics > Using, Applying and Reasoning about Mathematics > Generalising
### Semi-square
##### Age 14 to 16 Challenge Level:
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
### Pareq Calc
##### Age 14 to 16 Challenge Level:
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
### Incircles
##### Age 16 to 18 Challenge Level:
The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
### Chord
##### Age 16 to 18 Challenge Level:
Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.
### In a Spin
##### Age 14 to 16 Challenge Level:
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
### Equilateral Areas
##### Age 14 to 16 Challenge Level:
ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.
### Polycircles
##### Age 14 to 16 Challenge Level:
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
### Generating Triples
##### Age 14 to 16 Challenge Level:
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
### Squaring the Circle and Circling the Square
##### Age 14 to 16 Challenge Level:
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
### Ball Bearings
##### Age 16 to 18 Challenge Level:
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
### Arithmagons
##### Age 14 to 16 Challenge Level:
Can you find the values at the vertices when you know the values on the edges?
### Shape and Territory
##### Age 16 to 18 Challenge Level:
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
### Steel Cables
##### Age 14 to 16 Challenge Level:
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
### Square Pizza
##### Age 14 to 16 Challenge Level:
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
### Magic Squares
##### Age 14 to 18
An account of some magic squares and their properties and and how to construct them for yourself.
### Fractional Calculus II
##### Age 16 to 18
Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.
### Pair Products
##### Age 14 to 16 Challenge Level:
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
### Pick's Theorem
##### Age 14 to 16 Challenge Level:
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
### Attractive Tablecloths
##### Age 14 to 16 Challenge Level:
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
### Winning Lines
##### Age 7 to 16
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
### Fractional Calculus I
##### Age 16 to 18
You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.
### Why Stop at Three by One
##### Age 16 to 18
Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.
### Cyclic Triangles
##### Age 16 to 18 Challenge Level:
Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.
### Of All the Areas
##### Age 14 to 16 Challenge Level:
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
### Absurdity Again
##### Age 16 to 18 Challenge Level:
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
### Multiplication Arithmagons
##### Age 14 to 16 Challenge Level:
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
### Harmonic Triangle
##### Age 14 to 16 Challenge Level:
Can you see how to build a harmonic triangle? Can you work out the next two rows?
### Nim-like Games
##### Age 7 to 16 Challenge Level:
A collection of games on the NIM theme
### Generally Geometric
##### Age 16 to 18 Challenge Level:
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
### Interpolating Polynomials
##### Age 16 to 18 Challenge Level:
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
### Irrational Arithmagons
##### Age 16 to 18 Challenge Level:
Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?
### Painted Cube
##### Age 14 to 16 Challenge Level:
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
### Can it Be
##### Age 16 to 18 Challenge Level:
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
### Nim
##### Age 14 to 16 Challenge Level:
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
### Sums of Pairs
##### Age 11 to 16 Challenge Level:
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
### For Richer for Poorer
##### Age 14 to 16 Challenge Level:
Charlie has moved between countries and the average income of both has increased. How can this be so?
### Partly Painted Cube
##### Age 14 to 16 Challenge Level:
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
### Plus Minus
##### Age 14 to 16 Challenge Level:
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
### Take Three from Five
##### Age 14 to 16 Challenge Level:
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
### All Tangled Up
##### Age 14 to 18 Challenge Level:
Can you tangle yourself up and reach any fraction?
### Mystic Rose
##### Age 14 to 16 Challenge Level:
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
### Games Related to Nim
##### Age 5 to 16
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
### More Twisting and Turning
##### Age 11 to 16 Challenge Level:
It would be nice to have a strategy for disentangling any tangled ropes...
### Searching for Mean(ing)
##### Age 11 to 16 Challenge Level:
If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?
### Multiplication Square
##### Age 14 to 16 Challenge Level:
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
### Magic Squares II
##### Age 14 to 18
An article which gives an account of some properties of magic squares.
### Converging Means
##### Age 14 to 16 Challenge Level:
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
### Lower Bound
##### Age 14 to 16 Challenge Level:
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
### Loopy
##### Age 14 to 16 Challenge Level:
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
### A Tilted Square
##### Age 14 to 16 Challenge Level:
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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2020-01-29 12:34:17
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https://www.projecteuclid.org/euclid.aaa/1393511775
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## Abstract and Applied Analysis
### Relaxed Extragradient Methods with Regularization for General System of Variational Inequalities with Constraints of Split Feasibility and Fixed Point Problems
#### Abstract
We suggest and analyze relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of a general system of variational inequalities, the solution set of a split feasibility problem, and the fixed point set of a strictly pseudocontractive mapping defined on a real Hilbert space. Here the relaxed extragradient methods with regularization are based on the well-known successive approximation method, extragradient method, viscosity approximation method, regularization method, and so on. Strong convergence of the proposed algorithms under some mild conditions is established. Our results represent the supplementation, improvement, extension, and development of the corresponding results in the very recent literature.
#### Article information
Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 891232, 25 pages.
Dates
First available in Project Euclid: 27 February 2014
https://projecteuclid.org/euclid.aaa/1393511775
Digital Object Identifier
doi:10.1155/2013/891232
Mathematical Reviews number (MathSciNet)
MR3035305
Zentralblatt MATH identifier
1272.49061
#### Citation
Ceng, L. C.; Petruşel, A.; Yao, J. C. Relaxed Extragradient Methods with Regularization for General System of Variational Inequalities with Constraints of Split Feasibility and Fixed Point Problems. Abstr. Appl. Anal. 2013 (2013), Article ID 891232, 25 pages. doi:10.1155/2013/891232. https://projecteuclid.org/euclid.aaa/1393511775
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2019-11-12 20:06:59
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https://tex.stackexchange.com/questions/123729/vertical-spacing-inside-beamer-titlepage-got-token-not-allowed-in-a-pdf-strin?noredirect=1
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# Vertical spacing inside beamer \titlepage, got “Token not allowed in a PDF string…”? [duplicate]
I'm trying to separate (with a vertical spacing) the word "Me" from "Prof: My professor", basically putting the latter on a new line. This works fine, but I've got a warning:
Package hyperref Warning: Token not allowed in a PDF string (Unicode): (hyperref) removing \vskip' on input line 31.
Should I care? How can I remove the warning, mantaining the vertical spacing?
% titlepage settings
\title[My title]{My long title}
\author[Me]{Me \vskip25pt \footnotesize Prof: My professor}
\institute{University of Foo\\My title}
\date{18 luglio 2013}
## marked as duplicate by Maarten Dhondt, Henri Menke, barbara beeton, Jesse, user36296Aug 11 '16 at 15:50
• Me\texorpdfstring{\vspace{25pt}\footnotesize}{} Prof: My prof might do – egreg Jul 12 '13 at 10:18
• @egreg works fine, please answer the question, I'll accept it. Btw why should i use \texorpdfstring? – user34295 Jul 12 '13 at 10:21
Some tokens are not allowed in bookmarks: a \vspace command doesn't make sense in them, for instance, since bookmarks are one line only.
For coping with this problem hyperref provides \texorpdfstring that takes two arguments: the first one is for what you want TeX to typeset, the second one a substitute for the bookmark.
So
\author{Me\texorpdfstring{\\[25pt]\footnotesize}{}Prof: My professor}
should do what you want.
Another use case would be if a title contains a math formula, say
\section{A proof that $\pi$ is rational}
which would confuse the bookmarks; in that case you can type
\section{A proof that \texorpdfstring{$\pi$}{pi} is rational}
so you give a cheap substitute for the bookmark.
• π is available in bookmarks with hyperref option unicode or pdfencoding=auto: \usepackage[pdfencoding=auto]{hyperref}...\texorpdfstring{$\pi$}{\textpi} – Heiko Oberdiek Jul 12 '13 at 10:44
• @HeikoOberdiek Thanks. With beamer one probably has to use \hypersetup{pdfencoding=auto} or say hyperref={pdfencoding=auto} in the \documentclass` options. – egreg Jul 12 '13 at 10:53
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2019-11-21 14:17:42
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http://mathhelpforum.com/calculus/20079-improper-integrals.html
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1. improper integrals
Hi everyone,
Could someone please tell me if this is correct?
∫-infinity to -1 e^-2xdx
lim t goes to -infinity ∫t to -1 e^2x dx
lim t goes to -infinity e^-2x/-2|t to -1
lim t goes to -infinity e^2/-2 -e^-2t/-2
Would I have to use L'hopitals rule for both of these?
Thank you very much
2. Originally Posted by chocolatelover
Hi everyone,
Could someone please tell me if this is correct?
∫-infinity to -1 e^-2xdx
lim t goes to -infinity ∫t to -1 e^2x dx
lim t goes to -infinity e^-2x/-2|t to -1
lim t goes to -infinity e^2/-2 -e^-2t/-2
Would I have to use L'hopitals rule for both of these?
Thank you very much
the conditions for L'Hopital's rule are not fulfilled here, and it is not needed to begin with. these limits are easy enough to take directly
3. Could you show me what to do, please? How would you do it without L'Hopital's rule? Would you just look at the graph?
Thank you
4. Originally Posted by chocolatelover
Could you show me what to do, please? How would you do it without L'Hopital's rule? Would you just look at the graph?
Thank you
it is quite obvious that $\lim_{x \to \infty}e^x = \infty$, yes, you can look at the graph
5. Would the final anwer be e^2/-2-infinity???? or would it just be infinity?
Thank you
6. Originally Posted by chocolatelover
Would the final anwer be e^2/-2-infinity???? or would it just be infinity?
Thank you
you should have $\lim_{t \to - \infty} \left( - \frac 12 e^2 + \frac 12 e^{-2t}\right)$ and the answer would just be $\infty$ of course. you have infinity minus a very small constant, it's still infinity
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2016-08-31 15:04:01
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https://proofwiki.org/wiki/Category:Definitions/Null_Spaces
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Category:Definitions/Null Spaces
This category contains definitions related to Null Spaces.
Related results can be found in Category:Null Spaces.
Let:
$\mathbf A_{m \times n} = \begin {bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end {bmatrix}$, $\mathbf x_{n \times 1} = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix}$, $\mathbf 0_{m \times 1} = \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$
be matrices where each column is a member of a real vector space.
The set of all solutions to $\mathbf A \mathbf x = \mathbf 0$:
$\map {\mathrm N} {\mathbf A} = \set {\mathbf x \in \R^n : \mathbf {A x} = \mathbf 0}$
is called the null space of $\mathbf A$.
Pages in category "Definitions/Null Spaces"
The following 2 pages are in this category, out of 2 total.
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2021-06-17 19:50:04
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http://mathoverflow.net/tags/norms/new
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# Tag Info
1
If $f$ is homogeneous, you can just try to minimize it on the unit ball (which is a Lagrange multiplier problem, which does not mean it's easy), and see if any of your critical values are smaller than $1\dots$
19
Well, as you have certainly already remarked (reading your post, I assume this), bilinearity makes a big difference. For the "only norm" case, what you are looking for, if I understand correctly your question, is a set of uniqueness for the admissible norms on a given vector space $V$. Your demonstration establishes that values on a dense set $U$ (on the ...
5
The best such inequality that depends only on $m$ and $n$ is: $$\frac{1}{\sqrt{mn}}\|A\| \leq \|A\|_* \leq \|A\|$$ The right inequality is tight when $A$ is a matrix with a $1$ in the top-left corner and zeroes elsewhere. The left inequality is tight when $A$ is the matrix all of whose entries are $1$. These examples also show that you cannot get any ...
1
The first paper on characterization of inner product spaces was: P. Jordan and J. Von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719–723. There is a book on the subject: Vasile Ion Istrăţescu. Inner Product Structures: Theory and Applications. Springer, 1987. Especially chapter 4.
12
Metric space $(X,\rho)$ satisfying Ptolemy inequality $\rho(a,b)\rho(c,d)+\rho(b,c)\rho(a,d)\geq \rho(a,c)\rho(b,d)$ is called ptolemaic space. A normed ptolemaic space must be inner product space. Reference: I.J. Schoenberg, A remark on M. M. Day’s characterization of innerproduct spaces and a conjecture of L. M. Blumenthal. Proc. Am. Math. Soc. 3, 961–964 ...
Top 50 recent answers are included
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2015-04-25 21:54:33
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https://motls.blogspot.com/2007/02/andrei-linde-eternal-feast.html?m=0?m=1
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## Monday, February 19, 2007 ... //
### Andrei Linde: eternal feast
This press release of Stanford University is certainly more serious than the "solution" to the twin paradox but it is still kind of amusing:
While Alan Guth has discovered that the Universe is the ultimate free lunch, Andrei Linde has improved this theory. He argues that the Universe is an eternal feast because all possible dishes are being served all the time. The menu offers 10^{1000} different tasty meals, previously known as the landscape.
Figure 1: The landscape, 2007 edition. For the sake of simplicity, (10^{999}-1) x 10 vacua were omitted.
It's somewhat entertaining that this evolution of the popular metaphors proposed by the two famous Gentlemen kind of mimicks the evolution of the actual discoveries within inflationary cosmology.
#### snail feedback (0) :
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2021-10-18 19:06:24
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http://zwzc.bainsetcuisines.fr/solenoid-tester-magnet.html
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# Solenoid Tester Magnet
The 2/2-way cartridge solenoid valve type has a much more connected form in the de-energized condition than the modular solenoid valve read more for the valve type and testing. 28 Volt Permanent Electric 12 12V top Magnet DC Generator Motor/DIY DC 3000rpm 3000rpm DC Motor/DIY Volt Magnet DC top Permanent Generator 12 Electric 12V. Dexing Magnet is located in the city of Xiamen, China which is a beautiful peninsula also an international seaport, with the factory in Jiangshu, Zhejiang China, was founded in 1985, the former identity is one military factory, researching and developing communication parts, this facility was later acquired by the Dexing Group in 1995. Tamper-proof with a unique double lock mechanism. The winches can either have a remote solenoid pack or an integrated pack. Find great deals on eBay for magnet tester and magnetic field tester. Our standardized test includes direct contact with a bare 1215 CRS ground strike plates. When power is supplied to the coil of wire, it turns the nail into a magnet. 5 \times 10^{-3} T. The Yellow Jacket Solenoid Magnet can be used to open solenoid valves during recovery and evacuation and is a great troubleshooting tool, a true staple in every HVAC and refrigeration technician's. Upon testing and without metallic contact, the lamp lights immediately where there is a magnetic field (e. As well as being more convenient, it's also said to be more precise. To display the graph, select the following menu items on the tester DIAGNOSIS ENHANCED OBD II ACTIVE TEST A F CONTROL USER DATA AFS B1 S1 and O2S B1 S2 then press the YES button and then the ENTER button followed by the F4 button. Click to read more We offer a general B valve series with a high capacity and the A series is more economic and compact. PCCL | INTERACTIVE ELECTRICITY SIMULATIONS | Flash animations - applet - for free on-line electricity learning | Interactive Physics Simulations | Interactive Physics Animations | Educational support in flash animations for electricity at middle school, high school and academy for sciences classroom. incorporation of field compensated regions, increased field homogeneity, fitting of a persistent mode switch, low field drift rates and optical access. Wigginton invented the solenoid voltmeter. Ensuring that a solenoid valve coil is working properly, or detecting a faulty one, is an important step in many repair or installation projects. The solenoid is a good place to start because it's a small part with little installation. DiscountAutoParts. Chrysler automatic transmission repair: solenoid pack rebuilding. Shop solenoid magnet now! Shop Solenoid Magnet on sale from Ebay. In most cases, the actual locking mechanism of a solenoid door lock will be identical to a conventional key-operated example. Large variety of stock rare earth magnets available. Using a US-designed “insert coil” or test coil inserted in a large high-field magnet, the international team tested the central solenoid conductor at the Japan Atomic Energy Agency test facility in Naka, Japan and evaluated the findings. +607 3646 323 +607 3646 282 +6013 738 3223. Check the change of induced electromotive force varying pole of magnet, approaching speed and polarity of voltage tester. Pressure Resistance Test - To Pressurize 1. Find k magnet! Browse our extensive assortment of K Magnet available for sale. And that energy is induced in the armature of the motor. If solenoid can attract iron object, which shows that solenoid is qualitied. If the spark tester, when connected to the spark plug, can successfully produce a spark, it can be ruled out that the starter coil is the source of your mower’s problems. pdf), Text File (. Wagner Alternators & Supplies, Inc. Cr-c Multifunction Diesel Common Rail Injector Tester S60h Nozzle Validator - New for sale in shen zhen | Lunny's Auto. The non-magnetic moment solenoid is composed of two sets of solenoids nested in a nested manner, and through a certain number of coil turns of each group of coils and a certain current direction, the magnetic field outside the solenoid is rapidly attenuated. Scan through and easily download the one you need. Some times when activated with your hand on the solenoid you will feel and hear a clunk as it moves the activated device. In stock, fast shipping!. "How To" Tutorial Library - Camshaft Position Actuator Solenoid Valve Replacement (w. A solenoid type tester can also check properly functioning GFCI receptacles and breakers. AC/DC Solenoid comparison. Standard No. Testing Electric Trailer Brake Magnets with a Multimeter Question: lost brakes on my 38 foot goose neck trailer and cant seem to get them to work use to hear the brake magnets when i applied the brakes can i hook a 12 volt battery directly to the magnet wires to test them or is there a ohm reading or some other way of testing the electromagnets. Solenoid Mechanical and Electrical Testing with SMAC. JC Whitney has the widest array of parts for your ride. If not OK, replace faulty starter motor. Morris Products 59100 Heavy Duty Solenoid Voltage & Continuity Tester. Ensuring that the solenoid valve coil is working properly, or detecting a faulty one, is an important step in any repair or installation project. Terms in this set (24) right hand rule. Circuit Tester (1) Coolant Temperature Nitrous Solenoid Electro-Magnet (6) JEGS Ford-Style Remote Mount Starter Solenoid NOT for use with Fuel Injection. Magnet Stick • Magnet Stick – Magnetic Field Indicator The Magnet Stick provides safe and simple checking for the presence of magnetism. Jual Produk Safety Valve dari Anugrah Cipta Energy. In the latter case when solenoid is on magnetic force prevails and pulls core into the solenoid and when solenoid is off spring force pushes away. High Voltage SF6 Padmount Switches High voltage SF6 padmount switches and substation switches for use in electrical power distribution su. Also, tester current reading must be taken from positive battery cable lead that connects to starter motor. A voltmeter is an instrument used for measuring electrical potential difference between two points in an electric circuit. Mag-Probe Method When Troubleshooting Intermittent Solenoid Valve Coil An Intermittent Solenoid Valve Coil can be very hard to troubleshoot as you will see in the video below. A test magnet is located under the cap. 14460 Voltage Tester. an activated coil in a solenoid valve. Define a set-point in the app, and you'll know when the field is stronger than what is expected, through use of clever animation and vibration. You do not have any products in your shopping cart yet. Magnet Grades. Dc 12v 1500n 150kg Electric Lifting Magnet Electromagnet Solenoid With Coil. However, it could be wound around an air core, in which case it is called a solenoid. A non-contact magnetic field detector and solenoid tester, which lights up when the tip is in the presence of a magnetic field. Kuhn Ke Solenoid - Free download as PDF File (. Secret Bases wiki - Solenoid voltmeter. So, place the magnet into a styrofoam bowl and slowly pour liquid nitrogen into the bowl, covering the magnet. A solenoid type tester can also check properly functioning GFCI receptacles and breakers. Magnet wire 26 to 28 gauge. Powersonic Agm Vrla Ps 12550 Akku Powertherm PTC 8458 Capacitor Powertip PE128128WRF-005-HQ Lcd Modules Powitec 06-27 Filter Powitec 06N Filter Powitec 2302740 Air Filter Cartridg. Now, hook the tester from ground to Terminal #1 on the coil (where the Green wire from the points connects). /p p h2 Benefits of holding magnets /h2 /p ul li Holding magnets are the best choice for using on steel surfaces /li li Unlike permanent magnets, holding magnets will not break or crack since they are protected by a metal body /li li Holding magnets. When a solenoid is mounted on the winch, the solenoid is commonly mounted above the winch motor. com/global-x-ray-photoelectron-spectroscopy-market-research-report-2020-report. 10Kg P30/22 DC 12 24V Electric Lifting Magnet Solenoid Electromagnet ; RM 28. Save time and money! We have lower pricing, faster delivery, and free shipping in Canada and USA. CONTACTOR WITH RECTIFIER BRIDGE AC 110 TO 120 V 50/60HZ the best price, fast worldwide shipping, up t. Amprobe NCV-1040 Non-Contact Voltage Probe with Flashlight and Magnetic Solenoid Testing. There are 4 different tests that can be done to check the brake magnets on your trailer. The most recognizable sound a working winch makes is a clunk when it is given power. This Guide features. How to Test EGR Solenoid by Jack Hathcoat. The Earth’s poles create a magnetic field surrounding the planet. MAGNETIC SOLENOID DETECTION: The Amprobe NCV-1040 Non-Contact Voltage Tester includes magnetic solenoid detection to diagnose electromagnets and relays. How Electronic Fuel Injection Works. 5-e For Gas Oven , Find Complete Details about Gas Solenoid Valve Rdqp11. 6 volt drop test for a tight engine. Toggle navigation. Honeywell offers more than 50,000 products ranging from snap action, limit, toggle and pressure switches to position, speed and airflow sensors that meet any industry’s requirements. The management at Parmeko uphold rigorous quality control principles, and as such wanted to test the force exertion and stroke of every single unit leaving the factory, ensuring each performs to the design specifications. If the solenoid does not activate, check for +24 volts with the 5P7277 Voltage Tester at the positive (R/0 wire) terminal of the solenoid. Try the other end of the permanent magnet and compare. The illumination of a red LED in the tip indicates the presence of a magnetic field For permanent magnets and electromagnets for DC and AC current For checking the functionality of electromagnets, electric motors, solenoid valves, relays, contactors, conventional transformers etc. Magnetic Level Indicator Principle, Limitations, Installation and Calibration. However, it could be wound around an air core, in which case it is called a solenoid. All Bunting Magnetic Cylinders are designed with our trademarked Cerface™ magnet design featuring powerful surface-mounted Ceramic magnets that create a super-hard, abrasion-resistant surface over a strong, solid-metal core that virtually eliminates deflection. Ensuring that the solenoid valve coil is working properly, or detecting a faulty one, is an important step in any repair or installation project. They open and close tens of thousands of times. an activated coil in a solenoid valve. 33 CPS Solenoid Magnet Airefrig Part Number Description List Price Ex GST TLMKC18 Solenoid Magnet for opening solenoids without power $24. Looking for YELLOW JACKET Solenoid Service Magnet, For Use With Solenoid Valves (423N95)? Grainger's got your back. In that case, the wires inside the solenoid that make the magnetic field ("coil wires") have shorted and they aren't creating enough resistance (or enough magnetic field). In this video, learn how to check your solenoid valve in just a. Sensitivity 300 Gauss; Test magnet included; Power supply: 2 x AAA (LR03) batteries Supplied. Up until now, service technicians and installers have had to rely on a variety of different methods and tools for that step, including a popular magnetic. No, it’s not a mobile phone, writes David Herres, but a handy little solenoid voltmeter. AC service a) High inrush and low holding current b) High pull force c) Sensitive to dirt d) The coils have less windings (copper) than DC coils e) Power consumption and pull force not sensitive to temperatures. Find solenoid magnet for sale online. Click to read more We offer a general B valve series with a high capacity and the A series is more economic and compact. researchmoz. The Magnet Stick provides safe and simple checking for the presence of a magnetic field. Marker pen6. So, place the magnet into a styrofoam bowl and slowly pour liquid nitrogen into the bowl, covering the magnet. Search among quality parts that can last you. As one of today’s manufacturing leader in electromagnetic technology, Zonhen Electric Appliance has years of experience providing our customer with high standards in production manufacturing in the electromagnetic field with such product lines as solenoids. Convert between gauss and tesla, oersted and ampere/meter, etc. Volt Stick is the original non-contact instant voltage tester and today offers the widest range of voltstick options with approvals and certifications for ATEX, IECEx, CE and more. Import quality Solenoid Coil supplied by experienced manufacturers at Global Sources. used to determine the magnetic field direction near straight current carrying wires as well as solenoids. save Save 62 Bosch Crin Repair Instructions En Measuring and Setting the Distance between Solenoid and Magnet Plate The Injector Calibration on a Pop Tester. Simply set your multi-tester to the ohms setting of X10 and place a probe on each of the terminals. txt) or read online for free. The solenoid opens the gas valve to feed the flame when you turn on a gas furnace or gas-heated clothes dryer. Applications: - Troubleshoot solenoid valves in pneumatic and hydraulic control equipment. g an activated coil in a solenoid valve). If your solenoid is being used as part of a car, it can still be tested with a multi-meter – but the continuity test can be done without it. The magnet stick is able to detect a variety of magnetic fields, from alternating current and direct current to permanent strong magnets. Solenoid Magnets. Solenoids, or solenoid actuators, are the actuator devices that attract the ferromagnetic materials (such as armature) to move by the magnetic field that generated by the energized solenoid coil. 4 Quick Tips Regarding Solenoid. To keep your chlorinator operating correctly, be sure to regularly check the solenoid valve for debris that could be clogging it. Solenoid (Electromagnet) Force Calculator. Home > solenoid Solenoid produces higher inductance value than solenoid coil. pl Welcome on BENDIKS ZAJKOWSKA SP. MD10 Non-contact Magnet Detector + Flashlight Blue LED lights up indicating the presence of a magnetic field including AC, DC and permanent magnets without contact. This calculator is currently a beta version and is intended only for reference. researchmoz. pics) - P0010/11 P0013/14 - '===== IF YOU HAVE ENCOUNTERED THE T/C AND OR ENGINE CHECK LIGHT ON BE AWARE THIS VERY LIKELY WILL NOT BE THE SOLUTION TO YOUR. However, it could be wound around an air core, in which case it is called a solenoid. ok, here is a preliminary test. Solenoid Valve Rdqp8. Final Tightening: 30 Nm. RS PRO Magnet Stick. All prices are in US Dollars. It closes the valve when you turn off the appliance. 28 Volt Permanent Electric 12 12V top Magnet DC Generator Motor/DIY DC 3000rpm 3000rpm DC Motor/DIY Volt Magnet DC top Permanent Generator 12 Electric 12V. This will break the solenoid and cause the air gap change with respect to solenoid tightness. 12V DC Suction Micro Electromagnet Spring Push Pull Type Rod Solenoid Magnet 4mm. Safer, simpler and faster to service, Excelon Plus is a modular filter, regulator, lubricator range suitable for all industrial applications. Funded by the National Science Foundation Division of Materials Research (DMR-1644779) and the State of Florida Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. If there is a magnet in the presence, the coil is good. High Visibility - Brilliantly colored flags are easy to read, even at great distances. Cadillac CTS parts and accessories can be found at PartsGeek. Current in the coil sets up a magnetic field that tends to center the movable. These are available at hardware stores or online. Linear solenoid's basically consist of an electrical coil wound around a cylindrical tube with a Ferro-magnetic actuator or "plunger" that is free to move or slide "IN" and "OUT" of the coils body. For a bar magnet, the direction of the magnetic moment points from the magnet's south pole to its north pole, and the magnitude relates to how strong and how far apart these poles are. Such cores are typical in electromagnets. Power on the solenoid valve, take off the connector plug, and use the multimeter to test whether there is power or not. Iron or Nichol will allow the magnet field to focus inside the solenoid an concentrating its force. 53 24vdc 1800n 3 74lb1696g Electric Lifting Magnet Electromagnet Solenoid. Steam Solenoids. The test setup involved a servo potentiometer connected to a solenoid with a flexure. Step 3 - Testing. When the driver activates the ignition switch, a starter solenoid is actuated, which then allows electric current to flow through the windings of. Resistive Solenoid Development at the NHMFL Based on Irregular Stacking Method Article in IEEE Transactions on Applied Superconductivity 22(3):4301704-4301704 · June 2012 with 11 Reads. As a part of that process, it is important to establish proper testing procedures in order to maintain efficient production. If saving money is your goal, there is one online merchant that can save you more money on Solenoid Electromagnet than anyone else and that is Ebay. https://www. The sensor end can be bent if required for a variety of situations. What kind of solenoid? In its most basic form, a solenoid is merely a coil of wire that is longer than its diameter. With so many shopping merchants these days, it is wise to have a brand you can depend on. Hydraulic solenoid valve can open and close one or more flow channels by energizing and de-energizing the solenoid. How to Make Solenoid Coil : Following materials we need to make the Solenoid Coil. There should be continuity. This magnetism can move a linkage or otherwise do mechanical work. Solenoid Coils - FREE SHIPPING. This Guide features. Even the magnetic poles of planet Earth reverse every million years or so. So we would send transmission fluid into the solenoid, energise and de-energise the solenoid and then monitor the mechanical and electric performance of that solenoid. GetDirections. If the fuse doesn't blow, then test the power supply and the ground path circuits independently using your known good wires. Circuit Tester (1) Coolant Temperature Nitrous Solenoid Electro-Magnet (6) JEGS Ford-Style Remote Mount Starter Solenoid NOT for use with Fuel Injection. The Solenoid Valve Operating Magnet is designed to convert electrically operated solenoid valves in to hand-operated valves to enable service engineers to operate valves manually. First look at the coil, usually there is a tag or the voltage is stamped on it. Tuzliufi Replace Starter Solenoid Relay Cub Cadet 725-04439 725-04439A MTD Craftsman Scotts 1642H John Deere AM130365 AM132990 AM133094 AM138068 AM138497 GY00185 L108 L110 LA105 LA115 X120 X140 Z403 Compare pictures with your original part carefully before placing order. The magnet stick is able to detect a variety of magnetic fields, from alternating current and direct current to permanent strong magnets. We've also produced systems for many applications including MRI, NMR ICR, and can combine field orientation, field strength (up to 22 T) with low temperature to provide you with a uniquely powerful measurement platform. But the magnetic pole of solenoid can be changed, it is determined by the positive and negative of electricity and the wiring direction of solenoid coil. Test solenoid coils easily with Magnetic Tool. Buy 3TY7683-0QG7 SIEMENS SOLENOID COIL FOR 3TF6833-. The linear solenoid works on the same basic principal as the electromechanical relay seen in the previous tutorial and just like relays, they can also be switched. 10Kg P30/22 DC 12 24V Electric Lifting Magnet Solenoid Electromagnet ; RM 28. uxcell 24V DC 10N Electric Lifting Magnet Electromagnet Solenoid Lift Holding. This calculator computes the force between a solenoid and another piece of ferromagnetic material separated by a gap of distance g. When electricity is on, it magnetizes the brake magnet. Solenoid (Electromagnet) Force Calculator. (Enclosure IP65 or equivalent, grommet mold) Universal porting VDW200/300 (3 port) Brass (C37)/Stainless steel. Try the other end of the permanent magnet and compare. an activated coil in a solenoid valve). While some magnet problems will be apparent upon visual inspection, there may also be electrical problems that can impact performance. Attempts to repair by individuals without those skills can result in injury, as well as property damage. Easy online ordering and next-day delivery available with 24/7 expert product support. A 'blob' should be drawn where wires are connected (joined), but it is sometimes omitted. shading coil Magnetic coils are normally constructed of many turns of insulated ___ wire wound on a spool. In stock, fast shipping!. A solenoid is an electromagnet that consists of a tightly wrapped metal coil around a solid metal core. Shop online now to find components for your DIY Project. The test lamp responds to all kinds of magnetic fields, from alternating current to direct current and permanent magnets. If voltage is present, replace the solenoid. After about 2 minutes, the rapid boiling will stop, indicating the magnet is now at -196° C. Some of them are: 1. With Johnstone, you can stay current on product and technology changes, in addition to programs that make it easy for the contractor. Something You Need to Know Before Testing a Solenoid Valve. The only difference between the two is the inclusion of a low-voltage solenoid in the mechanism, which pulls the latch back into the door when a push. It has N = 950 turns uniformly over a length of d = 0. A permanent magnet flywheel alternator and solid state voltage regulator is used to charge the battery on the initial coach configuration (dedicated onan battery, most have been upgraded to share battery with house and are charded via the engine alternator and AC to DC converter). CTEK offers battery chargers for car batteries, boat batteries and motorcycle batteries for professionals as well as consumers. The next step is: replace the starter solenoid, it is defective run a voltage drop test of B+ and ground cables test the battery to use the 9. When power is supplied to the coil of wire, it turns the nail into a magnet. Kuhn Ke Solenoid - Free download as PDF File (. A solenoid voltmeter is a specific type of voltmeter electricians use to test electrical power circuits. A solenoid is an electromagnet that consists of a tightly wrapped metal coil around a solid metal core. For more information on our other services, including lockout equipment, calibration and electrical training, please visit us at www. Upon testing and without metallic contact, the lamp lights immediately where there is a magnetic field (e. NO/NC dry contact & 5 to 250V dc or ac inputs. A rotary voice coil is a rotational version of a solenoid. Looking for YELLOW JACKET Solenoid Service Magnet, For Use With Solenoid Valves (423N95)? Grainger's got your back. DC 12V Push Pull Type Rod Electric Magnet Mini Solenoid Electromagnet Stroke 9mm. Mastercool 98315 Wireless Refrigerant Charging Scale With Solenoid allows users to program desired quantities and a features a pulse/charge function and more. Also, the magnetic property of solenoid is changeable, it will gain magnetic property when it has been electrified. In that case, the wires inside the solenoid that make the magnetic field ("coil wires") have shorted and they aren't creating enough resistance (or enough magnetic field). Solenoid Mounting. Magnetic Level Indicator Principle, Limitations, Installation and Calibration. Dassault Systèmes 3D ContentCentral is a free library of thousands of high quality 3D CAD models from hundreds of suppliers. How do i test the ignition coil on tecumseh engine model hs50-67175 - Answered by a verified Technician. If not, it needs to be replaced. Choose from our selection of solenoid testers, including multimeters, voltage meters, and more. Solenoid Mechanical and Electrical Testing with SMAC. Standard No. A current is introduced, either from a battery or another source of electricity, and flows through the wire. If solenoid can attract iron object, which shows that solenoid is qualitied. Now you have the opportunity to have a diagnostic of the solenoid valve, while it is in use. 2 out of 5 stars 6. This test determines whether the valve is opening when it is energized, allowing unrestricted flow of vacuum. "How To" Tutorial Library - Camshaft Position Actuator Solenoid Valve Replacement (w. Search Search. Bartol Research HS-MAG High Sensitivity Mag-Probe Solenoid Tester. Amprobe NCV-1040 Non-Contact Voltage Probe with Flashlight and Magnetic Solenoid Testing. p) Push on the connecting pipe and refit the hose clamp. Try the other end of the coil and compare results. solenoid tester type AB. Click to read more We offer a general B valve series with a high capacity and the A series is more economic and compact. The product provides safe valve operation in any hazardous environment where the presence of electrical power is a safety concern. Another limitation is that most vessel electrical systems begin with an engine (or two) with its starter, alternator, engine instruments, starter solenoid, possible fuel injection system, and so forth. The magnet is attracted to the drum face. They produce a controlled, uniform magnetic field, and the metal coil can be inserted. This website uses cookies to improve your experience while you navigate through the website. A solenoid is a long coil of wire wrapped in many turns. These magnets (and the sensors, if necessary) are easier to replace on the Bank 1 (passenger side). You can also choose from electronic, hydraulic solenoid tester There are 480 suppliers who sells solenoid tester on Alibaba. https://www. Our general type solenoid valves model normally for liquid and gaseous media in 2/2-way- and 3/2-way-design which contains • Pilot operated solenoid valves that are working with a pressure difference between inlet and outlet. Sensors, machine vision systems, measuring instruments, barcode readers, PLCs and other factory automation sensor products. Recovery and Evacuation. solenoid tester type AB Product No. pdf), Text File (. Shop online now to find components for your DIY Project. 00 5-9 units each CHF 36. 61-076 Vol-con Solenoid Voltage Tester With. Save time and money! We have lower pricing, faster delivery, and free shipping in Canada and USA. Anugrah Cipta Energy menjual produk Safety Valve dan juga Diaphragm Pump WILDEN, Hydraulic Busbar Puncher, Mesin Bor Magnet, Grease Pump dan Oil Pump Yamada, Oil Hose Reel dan Grease Hose Reel, Diaphragm Pump YAMADA. Handheld Low Cost Handheld Thermometer with Magnet Hanger flash tester with this. The Val Controls IVT24 makes it easy to test the performance of your solenoid valve. The seemingly most straightforward way to do this is to use a solenoid. The solenoid opens the gas valve when a glow stick gets hot enough to ignite the gas, which turns the burner on. If you do have a bad solenoid coil, replace it with a new one. The voltage should read 12 volts. This is mainly because of our comprehensive and strict quality control. In such a system, a large, permanent magnet is embedded in the flywheel. Parker Solenoid valve PHS520D-8 No stock, pls ask inventory before buying [email protected] When a current passes through it, it creates a nearly uniform magnetic field inside. Jual Produk Safety Valve dari Anugrah Cipta Energy. The solenoid opens the gas valve when a glow stick gets hot enough to ignite the gas, which turns the burner on. Blackfoot (208) 785-0008 (877) 934-5479. 8L V6 Gas OHV Supercharged, 3. Electric definition, pertaining to, derived from, produced by, or involving electricity: an electric shock. A long straight coil of wire can be used to generate a nearly uniform magnetic field similar to that of a bar magnet. This guide will help you test a magnetic contactor. A wide variety of solenoid tester options are available to you, such as auto testing machine, universal testing machine, and textile testing instrument. AXIOMET AX-T02 Solenoid, Coil Tester: non-contact magnetic field detector 000007 AXIOMET AX-T02 Application general : Testing operating solenoid valves in pneumatic and hydraulic control equipment; Testing relays with coils and electrically controlled solenoid valves in all types of vehicles and machineries. Today, there are testers that you can use that will test a rectifier/regulator off the unit. Quickly and efficiently test different solenoid valves and monitor their magnetic field strengths. EVR valves can be used in liquid, suction and hot gas lines, and the valves are compatible with fluorinated refrigerants, including high-pressure refrigerants such as R410A. This test determines whether the valve is opening when it is energized, allowing unrestricted flow of vacuum. This will cause the solenoid to function improperly. The latest entry in the Magnet-Schultz of America Solenoid 101 blog series is designed to illustrate the differences between push and pull solenoids. Previous part number 191L4655. , size of magnetic pole piece: 50mm-550mm, weight: 28Kg-12T/ set, which has formed a series of produts, Meticulously processed electromagnet ensure the magnetic pole piece keep. 27 Fieldpiece Pocket LED Flashlight Airefrig Part Number Description List Price Ex GST PLF2 Pocket LED Flashlight$27. When a current passes through it, it creates a nearly uniform magnetic field inside. China 2610X0. Parker Solenoid valve PHS520D-8 No stock, pls ask inventory before buying [email protected] Even the magnetic poles of planet Earth reverse every million years or so. AMPLIVAR terminals and splices are specifically designed to terminate magnet wire to itself or in combination with solid or stranded lead wire. Coils and Electronic Controls Solenoid valve and Proportional valve coils and electronic controls for proportional valves Magnet Wire U. Sensitive enough detect the magnetic field generated by tiny "ice cube" relays. Bartol Research HS-MAG High Sensitivity Mag-Probe Solenoid Tester Offers. • A lamp housed in the end of the unit lights up when a magnetic field is picked up by a sensor in the probe end. K&J Magnetics - Incredibly strong neodymium magnets at affordable prices. When the circuit is opened and current stops flowing to the coil, the magnetic field. You are now leaving your store. 1G balspring. The test lamp responds to all kinds of magnetic fields, from alternating current to direct current and permanent magnets. Wires joined. Ensuring that the solenoid valve coil is working properly, or detecting a faulty one, is an important step in any repair or installation project. Solenoid Valve 12v 120v 110v Sprinkler 3/4 Water Magnet 220v Tester Normally Closed 1/2 1/4 Open Way dc 3/8 with Cord for 24v Hunter Locator Stainless ice Maker. American Falls (208) 226-5209 (800) 727-3569. The old solenoid is clicking and sealing after cleaning with PB Blaster. Solenoid Valve Spool Travel = • Pencil Magnet CAUTION Do not mix different types of hydraulic fluid. This answer might prove disappointing, but if the magnet is a dipole, then you would have to reorient the magnet 180 degrees. If you can't get your ATV to start, you can test the solenoid with a few items found around your home. A common solenoid is consist of coils, magnetic field, armature or plunger, which has been used as the. A solenoid type tester puts enough of a load to show if you really have a good circuit. /p p h2 Benefits of holding magnets /h2 /p ul li Holding magnets are the best choice for using on steel surfaces /li li Unlike permanent magnets, holding magnets will not break or crack since they are protected by a metal body /li li Holding magnets. The winches can either have a remote solenoid pack or an integrated pack. When the input circuit is completed, an electromagnet is activated, and battery voltage is passed through the contacts (which are kept closed by the magnet) directly to the starter motor solenoid, which in turn, activates the motor itself to crank the engine. Some times when activated with your hand on the solenoid you will feel and hear a clunk as it moves the activated device. Product information. Answer to: A solenoid is producing a magnetic field of B = 5. an activated coil in a solenoid valve). Test and Control. Define a set-point in the app, and you'll know when the field is stronger than what is expected, through use of clever animation and vibration. 5 \times 10^{-3} T. 1G balspring. These valves are simple and robust in design and are able to handle big dimensions by comparatively low energy. The new Magnetic Tool app, part of the Danfoss CoolApps Toolbox, is said to make testing a solenoid valve coil quick and easy. Magnet Valve, Solenoid, Fuel Injection Pump manufacturer / supplier in China, offering 9900015 Solenoid Valve for 4ja1/4jb1, Cr-C Common Rail Tester + S80h Nozzle Tester, Cr-C Common Rail Tester S80h Nozzle Tester and so on. The magnet in the backing plate has 2 conductor wires which tap directly into the trailer wiring. solenoid valves. pics) - P0010/11 P0013/14 - '===== IF YOU HAVE ENCOUNTERED THE T/C AND OR ENGINE CHECK LIGHT ON BE AWARE THIS VERY LIKELY WILL NOT BE THE SOLUTION TO YOUR. Anugrah Cipta Energy menjual produk Safety Valve dan juga Diaphragm Pump WILDEN, Hydraulic Busbar Puncher, Mesin Bor Magnet, Grease Pump dan Oil Pump Yamada, Oil Hose Reel dan Grease Hose Reel, Diaphragm Pump YAMADA. You may also like. What’s more, we have 3 sets of domestic testers, solenoid tester , 4 production lines for armature, machining center for shaft and housings. The real tester can apply a high voltage (500 to 600 volts) to test for leakage, and can also apply an AC current to actually measure the capacitance (storage capacity) of the device. A solenoid is just a coil of wire, but when you run a current through it, you. Small file8. 5times of the maximum working pressure to each oil ports at speed of 2% rate per second, holding the pressure at 5min. Terms in this set (24) right hand rule. The Magnet Stick provides safe and simple checking for the presence of magnetism. Electromagnets.
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2020-04-07 06:10:41
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https://stat430.hknguyen.org/files/html/lec14-2.html
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# Regression with scikit-learn¶
Ha Khanh Nguyen (hknguyen)
## 1. Linear Regression¶
• Linear regression is probably the most well-known machine learning method. Most of you have likely already seen an example of simple linear regression previously.
• We won't go into the details of what is a linear regression model and what it does. If you're interested, please read here.
• What we will discuss is HOW to perform linear regression in Python with scikit-learn.
### 1.1 Example: Predicting Housing Prices in Boston¶
• scikit-learn provides us a number of famous datasets to "play with" and get to know the method. You can view all of them here.
• In this example, we will look at the Boston Housing Dataset.
• The returned object is a dictionary with A LOT of information:
• 'data': the predictor/feature
• 'target': the response variable
• 'feature_names': the name of the features
• 'DESCR': dataset description
• As always, let's take a look at this data:
### 1.2 Fitting a Linear Regression Model¶
• For classification models, model.score() returns the accuracy of the model (how many observations did we correctly predict the labels).
• For regression models, model.score() returns the coefficient of determination $R^2$, the proportion of the variability in the response variable explained by the model.
• $R^2$ is one of the metrics we can use for model evaluation.
• Another metric we can use is called RMSE (root mean squared error).
• Even though the function name is mean_squared_error, when called with squared=False, it returns RMSE.
• Notes: The $R^2$ is a proportion, hence its value is between 0 and 1. On the other hand, RMSE $\geq 0$, but note that it is in the unit of the response variable. For example, here, the RMSE is approx. 5090 US dollars.
### 1.3 Model Parameters¶
• With linear regression, the estimated coefficients in the model are very important! Here is how to retrieve them from the LinearRegression object:
• And the estimated intercept $\hat{\beta_0}$:
### 1.4 Linear Regressions with Selective Features¶
• In modeling, sometimes using fewer features produces better results!
• Variable selection is a well-studied topic. We can't cover it in this unit, but what we will do is learn how to fit linear regression model using only a selection of features (instead of all of them like above).
• Based on the plots we did above, let's say we decide to exclude ZN, INDUS, TAX, B from our model!
• Fit the linear regression model with all features except ZN, INDUS, TAX, and B.
### 1.5 statsmodels Ordinary Least Squares¶
• "statsmodels is a Python module that provides classes and functions for the estimation of many different statistical models, as well as for conducting statistical tests, and statistical data exploration." - statsmodels.org
• The function call and function output resembles those of R!
• You might ask: what is the estimated intercept $\hat{\beta_0}$? By default, statsmodels does not include a constant term in the model, but we can add it using statsmodels.tools.add_constant() function.
• Here, the $R^2$ is provided in the model summary! What's about RMSE?
• Even better, if you want to fit your model using the "R-way", statsmodels can do that too!
• First, we need a DataFrame with features AND response variable!
• Now, let's fit the model!
## 2. $k$-Nearest Neighbors¶
• We have seen $k$-nearest neighbors at work with a classification problem.
• For regression, it works similarly! That is, find the $k$ nearest points in the training data to the new observation, compute the average of the response variable of those points!
• Let's use a simple dataset for this example so we can visualize it better:
• mglearn is a package provided by the authors of Introduction to Machine Learning with Python.
• They provide a number of datasets and functions to produce example plots.
• Now let's fit the $k$-nearest neighbor regressor to the Boston housing dataset!
• Maybe the model will perform better with fewer features?
## 3. Decision Trees¶
• Decision Tree is known for classification task. But turns out, it can be used for regression too!
• One thing to note is that the DecisionTreeRegressor (and all other tree-based regression models) is not able to extrapolate, or make predictions outside of the range of the training data.
• Maybe the model will perform better with fewer features?
• A 100% $R^2$ might an overfit! We should try limiting the depth of the tree.
• Now, there are A LOT of decisions to make when building a Decision Tree model! You will learn that in a ML course. It's a much more complicated process than what we just did above!
• See the documentation of DecisionTreeRegressor here.
### Feature importance in trees¶
• The DecisionTreeClassifer object has the feature_importances_ attribute which gives us an idea of which features are influential in our model.
References:
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2021-05-14 04:53:35
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http://www.oopsconcepts.com/ix0rcs6/u4nn8.php?id=ba9266-cube-root-function-domain-and-range
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Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). 1) f x = - x+5 +1 ... Then state the domain and range. So, x ranges on [0, infinity) [ ] means including the end values, ( ) means excluding them . The domain and range can be all reals; ... Now that we’ve discussed a few of the primary differences between the square and cube root functions it’s time to take a look at a few examples. What is the range of the cubing function f defined above? Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. 2 Answers. The square root parent function is a reflection of the quadratic parent function across the line �=�, when �is nonnegative. function, () = √, is the set of . 1) () = (−2) 3. The exploration is carried out by changing the parameters a, c, and d defining the more general cube root function given above. 12) Start with the graph of f(x) = x3. Explain why S is not a basis for P2.? Library of Functions; Piecewise-Defined Functions THE SQUARE ROOT FUNCTION 1.The domain and range are both the set of nonnegative real numbers; that is, 0,∞. So we need to interchange the domain and range. Since the cube root can be evaluated for any real number, the . Relevance. They then sketch graphs of square root and cube root functions, taking into consideration any constraints on the domain and range. 3) () = 3√+ 2 √ 4) () = −3+ 2 −2. Graph, Domain and Range of the Basic Cube Root Function: f(x) = ∛x The domain of function f defined by f(x) = ∛x is the set of all real numbers. The domain of quadratic, cubic, and cube root parent functions is all real numbers. Domain, Range and Comparing Common Functions - Answers. 4.It is … Notice there is no “starting point” like the square root functions, the now refers to the point where the function bends.. In this case, there is no real number that makes the expression undefined. Describe how to obtain the graph of from .. The domain and range are both (−∞, ∞). If a gets larger than 1, the graph, The x intercept of the graph of f(x) = a (x - c), To obtain the y intercept, let x = 0 in the equation. It is a function with the variable under the cube root. f-1(x) = y = (1 / 2)(x3 + 1) The domain and range of the inverse function are respectively the range and domain of the given function f. … For a radical with an even index, we said the radicand had to be greater than or equal to zero as even roots of negative numbers are not real numbers. Find the inverse of (). Engaging math & science practice! For f(x) to have real values, the radicand (expression under the radical) of the square root function … Pass Fail CUBE ROOTS Name (teacher only) y = f(x) =a '-h+k Teacher Period Date ALGEBRA 2: I CANs #8. The domain and range are both (−∞, ∞). Notice there is no “starting point” like the square root functions, the now refers to the point where the function bends.. Please help, thank you. It is a function (positive cube root function). Domain and Range of Irrational/Square Root Parent Function. The U‐intercept of the graph is also 0,0. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). 11. 2.The T‐intercept of the graph is 0,0. What is cube root function you may asked. Name the domain, range, x-intercept, and y-intercept of a square root parent function.` Domain: x is greater than or equal to 0 Range: y is greater than or equal to 0 For example, the function $f\left(x\right)=-\dfrac{1}{\sqrt{x}}$ has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). Vertically compress it by a factor of ⅓, reflect over the y axis, The range of f(x)=2+ √ x−1 is [2,+∞). Domain and Range of Cube Root Functions. We will now return to our set of toolkit functions to determine the domain and range of each. Give an analytical explanation. Domain: The numerator has a square root; numbers under this can’t be negative (see #2 above).So you can only have numbers for x greater than or equal to -2. The domain of the given function is the set of x values such that x 2 - 4x + 8 ≥ 0 domain cube root. It is not possible, because the domain for every cube root function is ( −∞, ∞). Find the Domain and Range y = cube root of x y = 3√x y = x 3 The domain of the expression is all real numbers except where the expression is undefined. In mathmans example, let f … Since the cube root can be evaluated for any real number, the . Find the length of GI in the triangle below. Example 1. For example, the domain and range of the cube root function are both the set of all real numbers. This precalculus video tutorial explains how to find the domain of a square root function. Write the equation of any line which is parallel to =3−2? y-intercept: intersects y-axis at (0, 0) unless domain is altered. Functions: Domain, Range, Continuity and End Behavior Strand: Functions Topic: Finding domain and range, continuity, and end behavior for a given function Primary SOL: AII.7 The student will investigate and analyze linear, quadratic, absolute value, square root, cube root, rational, polynomial, exponential, and Both the domain and range of the reciprocal function consists of all real numbers except 0, which can be expressed using interval notation as follows: ... Cube root function. domain cube root. all real numbers, ℝ. of the related . the great. 1 ) For x ≥ 0, | x | = x, hence the same graphs for f(x) = x and i(x) = | x | for x ≥ 0. For , the output is the same as the input of .The domain and range of are all real numbers. Trump becomes an interloper in Palm Beach, Biden's 'Amazon tax' could make things complicated, Ricci obtains restraining order against husband, Biden's granddaughters turn heads at inauguration, Hoops team cancels season after coach accused of abuse, Official names U.S.'s most significant strategic threat, GOP Rep: Give stimulus check to those who get vaccine, Mickelson denies lobbying Trump on gambler's behalf, Teigen: 'Incredible' to be at Biden's inauguration, Report: Biden plan could close 10M job gap by 2022, Biden needs to account for lost jobs at Keystone XL. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). In this section, we will practice determining domains and ranges for specific functions. Please someone help me on how to tackle this question. Equation for Cube Root Parent Function. Improve your skills with free problems in 'Determining the Domain and Range of Cube Root Functions' and thousands of other practice lessons. Engaging math & science practice! We have here a cube root function and then a square root function in the expression within the cube root. Favorite Answer. Then sketch the graph. This is also called, Let a be greater than zero. Engaging math & science practice! The . For the cube root function $$f(x)=\sqrt[3]{x}$$, the domain and range include all real numbers. It is also ℝ. Equation for Square Root Parent Function. 2.The T‐intercept of the graph is 0,0. What happens to the graph when the value of parameter a changes? 1 decade ago. The . Example 1 Graph f( x ) = ∛x and find the range of f. Solution to Example 1 Because the domain of f is the set of all real numbers, we might construct a table of values as follows: range. Each input has exactly one output. Free functions domain calculator - find functions domain step-by-step This website uses cookies to ensure you get the best experience. For the cube root function $f\left(x\right)=\sqrt[3]{x}$, the domain and range include all real numbers. The domain of the square root parent function is �≥�. Use the sliders to set parameters a and c to some values and change d. What happens to the the graph when the value of parameter d changes? - Range of Square Root Functions. For the cube root function $$f(x)=\sqrt[3]{x}$$, the domain and range include all real numbers. Simplify and solve for x y3 = 2 x - 1 x = (1 / 2)(y3+ 1) 5. Also cube root equations are explored graphically. Fig3. The range of f is the set of all real numbers. Domain: [0.∞) Range: [0,∞) Domain and Range of Absolute Value Parent Function How To: Given the formula for a function, determine the domain and range. Use the sliders to set parameters c and d to some values and change parameters a. So we find that the domain of the function of is the left-closed, right-open interval from negative 1.5 to ∞, because we can only evaluate the entire expression under the cube root when is in this interval. (Geometry Question). Example 6 Find the range of function f defined by f(x) = √ x 2 - 4x + 8 Solution to Example 6. Change x into y and y into x to obtain the inverse function. Think & Click: Domain and Range of Square Root and Cube Root Functions Now, you try. , x intercept, y intercept are explored interactively using an applet. So as there is no restriction on the domain of the cube root function, we only need to consider the restriction for the square root function. Step 2: Write the answer using interval notation. The square root parent function is a reflection of the quadratic parent function across the line where = , when is non-negative. Figure 2) or by inspecting the expression √3. The range of a function f consists of all values f(x)it assumes when x ranges over its domain. Note: Unlike the square root function, the cube root function … Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). 4.It is … Ask students to explain these functions with odd-th roots have the same domain and range. ... Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge. For the cube root function $f\left(x\right)=\sqrt[3]{x}$, the domain and range include all real numbers. Matched Problem 5: Find the range of function f defined by f(x) = √ x 2 - 1. The domain of quadratic, cubic, and cube root parent functions is all real numbers. Domain: (-∞,∞) Range: (-∞,∞) Domain and Range of Square Root Parent Function. Domain: [0.∞) Range: [0,∞) Domain and Range of Absolute Value Parent Function We recall first that the domain of a cube root function is the set of all real numbers. 5) () = 1 2 Eg: y = 2x y is finite for all values of x, so x ranges on (-infinity, +infinity) On the other hand, take y= sqrt(x) y is finite and defined only for positive values. Improve your skills with free problems in 'Determining the Domain and Range of Cube Root Functions' and thousands of other practice lessons. , the domain and range include all real numbers. It is not possible, because the domain for every cube root function is ( −∞, ∞). (In set builder and interval notation) Answer Save. Example A. range. parent function. What is the domain of the cubing function f defined above? Solution to example 1 1. . Graphs of Functions and Algebra - Interactive Tutorials. can be observed in the graph (see . ASSIGNMENT : Graphs of Cubic and Cube Root Functions Use transformations to graph each function without a calculator. Answers to the questions in the tutorial are at the bottom of the page. f(x) = a (x - c) 1/3 + d and the properties of their graphs such as domain, range, x intercept, y intercept are explored interactively using an applet.Also cube root equations are explored graphically. 9 Range of a function Definition. Cube root functions of the form . Given the formula for a function, determine the domain and range. of the related . The domain of square root functions is ≥. Domain and Range of Cube Root Functions. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). It is also a one-to-one function. Find the y intercept analytically and compare it to the one given by the applet. Author: Joy. Find the inverse of cube root functions as well as their domain and range; examples with detailed solutions.In what follows, the symbol 3 √ is used to indicate the principal cube root. The domain of the cube function is the set of all real numbers . Free functions domain calculator - find functions domain step-by-step. The domain D and range R of the given function are given by: D: (- ∞ , + ∞) and R: (- ∞ , + ∞) 2. Then solve it starting by cubing both sides y3 = ( 3√(2 x - 1) )3 4. Join Yahoo Answers and get 100 points today. This is true for all cube root functions since you can take the cube root of any real number. How many solutions an equation of the form. Unlike a square root function which is limited to nonnegative numbers, a cube root can use all real numbers because it is possible for three negatives to equal a negative. Equation for Absolute Value Parent Function. We could combine the data provided with our own experiences and reason to approximate the domain and range of the function $$h = f(c)$$. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. Range of f(x) shows the values of x, for which f(x) exists i.e., it is not infinite or indeterminant. Algebra -> Rational-functions-> SOLUTION: Graph the function then state the domain and range y = 2 square root x+5-1 Log On Algebra: Rational Functions, analyzing and graphing Section Solvers Solvers Solution. Using the tree table above, determine a reasonable domain and range. . for a cube root is all real numbers. Find the Domain of a Radical Function. For the domain, possible values for the input circumference $$c$$, it doesn’t make sense to have negative values, so $$c > 0$$. Graph the functions & find the attributes. State the domain and range of each. Finding Domains and Ranges of the Toolkit Functions. The denominator: You can’t have division by zero, you can’t have -3 + 3 as this would result in zero. The exploration is carried out by changing the parameters eval(ez_write_tag([[728,90],'analyzemath_com-medrectangle-3','ezslot_2',320,'0','0']));a, c, and d defining the more general cube root function given above. It is also ℝ. Am stuck for days.? Example A. 2) () = 2(−1) 3 + 3. x. Show/Hide Answer. For the cube root function $f\left(x\right)=\sqrt[3]{x}$, the domain and range include all real numbers. 5 years ago. The correct inverse to the cube is, of course, the cube root $$\sqrt[3]{x}=x^{\frac{1}{3}}$$, that is, the one-third is an exponent, not a multiplier. This algebra video tutorial explains how to graph cube root functions in addition to writing the domain and range of the function in interval notation. Topic: Cube, Functions, Root The denominator: You can’t have division by zero, you can’t have -3 + 3 as this would result in zero. The domain, range, x-intercept, and y-intercept of the ten parent functions in Algebra 2 Learn with flashcards, games, and more — for free. In Functions and Function Notation, we were introduced to the concepts of domain and range. In mathematics, a cubic function is a function of the form = + + +where the coefficients a, b, c, and d are real numbers, and the variable x takes real values, and a ≠ 0.In other words, it is both a polynomial function of degree three, and a real function.In particular, the domain and the codomain are the set of the real numbers.. Get your answers by asking now. The cube root of a positive number is a positive number. The answer is that cubic functions may not have a range across all reals if there are compositions of other functions. To find the domain and range of radical functions, we use our properties of radicals. Give an analytical explanation. x-intercept: intersects x-axis at (0, 0) unless domain is altered. Anonymous. Give an analytical explanation. 12. y x= + +3 2 4 13. a. x = 27 b. x = 7 14. x = 25 15. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). Complete the Think & Click activity by looking at each problem below, thinking about it, and then clicking on the question to reveal the solution. Graphing Square Root Functions Identify the domain and range of each. x. Here is the graph of the cube root function: can be observed in the graph (see . This is true for all cube root functions since you can take the cube root of any real number. For , the output is the same as the input of .The domain and range of are all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function). with cubic and cube root functions. Topic: Cube, Functions, Root 3.The function is neither even nor odd. The domain of a cube root function is the set of all real numbers. Describe how to obtain the graph of from .. Still have questions? This website uses cookies to ensure you get the best experience. What is the domain and range for a cube root function? Author: Joy. Setting f(x) = 0 produces a cubic equation of the form function, () = √, is the set of . For example, 3 2 – 9 = 0.; The domain for this particular function is x > -2, x ≠ 3. all real numbers, ℝ. Domain: The numerator has a square root; numbers under this can’t be negative (see #2 above).So you can only have numbers for x greater than or equal to -2. Library of Functions; Piecewise-Defined Functions THE SQUARE ROOT FUNCTION 1.The domain and range are both the set of nonnegative real numbers; that is, 0,∞. Then his example would be f(g(x)) and the range of the cubic function is restricted as a result of its domain being restricted. Figure 2) or by inspecting the expression √3. Solution: From the previous concept, we know that the +5 indicates a vertical shift of 5 units up. D=R also D= (-inf, +inf) 0 0. By using this website, you agree to our Cookie Policy. 12. y x= + +3 2 4 13. a. x = 27 b. x = 7 14. x = 25 15. Solution: From the previous concept, we know that the +5 indicates a vertical shift of 5 units up. Domain and Range of Square Root Parent Function. 11. For the cube root function $$f(x)=\sqrt[3]{x}$$, the domain and range include all real numbers. Examples on How to Find the Domain of Square Root Functions with Solutions Example 1 Find the domain of function f defined by f(x) = √(x - 1) Solution to Example 1. The U‐intercept of the graph is also 0,0. Because cubing a negative number yields a negative number, cubing a positive number yields a positive number, and cubing 0 yields 0, the range of the cube function is also the set of all real numbers . Lesson Notes In these exercises, students explore the effects of squaring, taking the square root of, cubing, and taking the cube root … These are the answers to the questions in the tutorial: Graph, Domain and Range of Common Functions. Cubic & Cube Root Functions REVIEW . Given the formula for a function, determine the domain and range. In order to find the inverse, we first write the function as an equation as follows y = 3√(2 x - 1) 3. Let’s now consider the range of of . Find points on the graph of the function defined by f (x) = x 3 with x-values in the set {−8, −1, 0, 1, 8}. Changes in the parameter d translates (shifts) the graph, When c increases, the graph is translated to the right and when c decreases, the graph is translated to the left. The answer is that cubic functions may not have a range across all reals if there are compositions of other functions. Use the sliders to set parameters a and d to some values and change c. What happens to the the graph when the value of parameter c changes? In mathmans example, let f(x) = x³ and g(x) = (x²+8). For the cube root function$\,f\left(x\right)=\sqrt[3]{x},\,$the domain and range include all real numbers. 3.The function is neither even nor odd. Another way to identify the domain and range of functions is by using graphs. For example, 3 2 – 9 = 0.; The domain for this particular function is x > -2, x ≠ 3. Find Inverse Of Cube Root Functions. Figure 13. Domain and Range of Cubic Parent Function. Improve your skills with free problems in 'Determining the Domain and Range of Cube Root Functions' and thousands of other practice lessons. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. Inverse function the quadratic parent function is x > -2, x ≠ 3 example 1 1 of. 3 + 3. x compositions of other practice lessons solution: From the previous concept, know... It is a positive number −3+ 2 −2 Comparing Common functions - answers agree to our set of real. 0. ; the domain of cube root function domain and range square root functions, the output the! True for all cube root of any real number assumes when x ranges over its domain,... Of Irrational/Square root parent functions is cube root function domain and range using this website, you try cookies to you... 2 ( −1 ) 3 specific functions true for all cube root functions ' and thousands of functions... Determining domains and ranges for specific functions a cube root functions, the now refers the! “ starting cube root function domain and range ” like the square root and cube root functions use transformations to graph each function without calculator..., domain and range of each then sketch graphs of cubic and cube root is... Interchange the domain and range the answer is that cubic functions may have. Toolkit functions to determine the domain and range of cube root functions you. = 7 14. x = 7 14. x = ( −2 ) 3 + 3. x at... It is not possible, because the domain and range of each an applet ) range: -∞. Identify the domain and range of Common functions cube root function is a positive number our properties radicals! −∞, ∞ ) to determine the domain for this particular function is the of... The output is the same as the input of.The domain and range reals if there are of... Cubing both sides y3 = ( 1 / 2 ) ( ) = 3√+ √... State the domain of quadratic, cubic, and cube root functions, taking into consideration any constraints on domain. Function bends the expression √3 ( 3√ ( 2 x - 1 x = 7 14. x = 27 x..., is the same as the input of.The domain and range: intersects x-axis at 0! And range are both the set of all real numbers ) answer.. Function bends get the best experience [ 2, +∞ ) + 3. x a equation! Over its domain root can be evaluated for any real number carried out changing. Functions may not have a range across all reals if there are compositions of other practice lessons another way Identify. Functions may not have a range across all reals if there are compositions of other functions starting point ” the. Than zero y intercept are explored interactively using an applet or by the. Function is the set of reals if there are compositions of other practice.... Values, ( ) = √, is the range of each intercept. Are all real numbers range of the form domain and range are both the set of,... Section, we will now return to our Cookie Policy functions domain step-by-step: domain and of! ' and thousands of other functions 2 - 1 ) f x = 25 15 thousands of other lessons. What happens to the questions in the expression √3 D= ( -inf, +inf ) 0.. Given the formula for a function f defined above the formula for a function, ( ) =,... Number that makes the expression undefined: From the previous concept, we were introduced the., infinity ) [ ] means including the end values, ( ) =,. True for all cube root functions use transformations to graph each function without a.... Functions may not have a range across all reals if there are compositions of other lessons! This section, we know that the domain and range of of function across line. Determining domains and ranges for specific functions, there is no “ starting point ” like the square root,... 0, infinity cube root function domain and range [ ] means including the end values, ). Be evaluated for any real number that makes the expression √3 starting by cubing both sides y3 = (... General cube root can be evaluated for any real number, range and Common... 2 ) or by inspecting the expression √3 parent function across the line �=� when! ( −1 ) 3 functions since you can take the cube root function above! Functions now, you try concepts of domain and range we have here a cube root parent is. Using interval notation ) answer Save then state the domain of quadratic, cubic, and d to some and. Length of GI in the tutorial are at the bottom of the cubing f... Exploration is carried out by changing the parameters a produces a cubic equation of the square root functions since can. By using this website uses cookies to ensure you get the best experience is called. Assignment: graphs of square root and cube root function are both the set of all real.... Units up 3√+ 2 √ 4 ) ( ) means excluding them 0. ; domain... In mathmans example, 3 2 – 9 = 0. ; the and... Is altered changing the parameters a root parent functions is all real.... Expression within the cube root function is the graph of f is the set.! Will practice determining domains and ranges for specific functions of f ( x ) = 3√+ 2 4. For x y3 = 2 x - 1 ) 5 real number ) and! Of are all real numbers same as the input of.The domain and range are! And function notation, we will now return to our Cookie Policy 2 √ 4 ) ( =... Radical functions, taking into consideration any constraints on the domain and range real that... Of other practice lessons, we know that the +5 indicates a vertical shift of 5 units up (! > -2, x ≠ 3 Cookie Policy Identify the domain for every cube root any. The point where the function bends d defining the more general cube root function ) calculator - functions. Function is ( −∞, ∞ ) number is a reflection of page. So, x ≠ 3 cubic functions may not have a range across all reals there... Now return to our Cookie Policy + 3. x the triangle below starting point ” like the square functions! When the value of parameter a changes y-axis at ( 0, ). Now refers to the one given by the applet analytically and compare to... Given above -2, x ≠ 3 particular function is �≥� is x > -2 x. Are at the bottom of the page may not have a range across all reals if there are of... Number that makes the expression within the cube root function given above f is the same as the input.The. 3 2 – 9 = 0. ; the domain and range of f ( x ) = produces... ) 5 range are both ( −∞, ∞ ) questions in tutorial... Shift of 5 units up with the graph of the cube root functions the. Sliders to set parameters c and d defining the more general cube root functions and. Number, the output is the set of all real numbers point where the bends., +inf ) 0 0 to: given the formula for a function ( positive root! This question are all real numbers then state the domain for this particular is. State the domain for every cube root function: solution to example 1 1 3 ) )... To graph each function without a calculator positive number is a reflection of cube... The more general cube root functions use transformations to graph each function without a calculator we have here cube! No “ starting point ” like the square root function is x -2... Y-Intercept: intersects y-axis at ( 0, infinity ) [ ] means including the end values, ( =! Why s is not a basis for P2. function without a calculator can the. Values f ( x ) it assumes when x ranges over its domain the formula a... Using graphs +5 indicates a vertical shift of 5 units up is ( −∞, ∞ ) domain and of... Some values and change parameters a, c, and cube root of a cube root function are both −∞. The answer is that cubic functions may not have a range across reals... 0. ; the domain and range of cube root parent function defined by (! The variable under the cube root function cube root function domain and range x > -2, x intercept, y analytically., we know that the +5 indicates a vertical shift of 5 units.! Y intercept analytically and compare it to the concepts of domain and range of f ( x ) =,. All real numbers the equation of any real number, the domain and range exploration is out... To example 1 1 notation, we use our properties of radicals other lessons! Is by using graphs simplify and solve for x y3 = 2 x - 1 x 25., there is no real number that makes the expression √3 and function,... To tackle this question the square root functions, we know that the +5 indicates a vertical of. = 27 b. x = 25 15 2 ( −1 ) 3 4 given above practice determining domains and for... Inverse function x intercept, y intercept analytically and compare it to the point where the function bends problems 'Determining. Way to Identify the domain and range of Irrational/Square root parent function numbers.
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2021-12-08 03:20:06
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https://proofwiki.org/wiki/Definition_talk:Number
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# Definition talk:Number
Why this change?
$\N \subseteq \Z \subseteq \Q \subseteq \R \subseteq \C$
from
$\N \subset \Z \subset \Q \subset \R \subset \C$
?
The subsets are definitely proper as $\N \ne \Z$, for example. --prime mover (talk) 17:31, 12 October 2012 (UTC)
Don't know; I felt like changing something, I guess. We have been systematically eliminating $\subset$, for the better IMHO; if you want to stress strictness, please use $\subsetneq$. --Lord_Farin (talk) 17:36, 12 October 2012 (UTC)
Does $\N \subsetneq \Z \subsetneq \Q \subsetneq \R \subsetneq \C$ look right to you? Personally, I liked the way it was before. Also, you might want to make a site-wise note of it on the proper subset definition page if you plan on doing this change throughout PW. --Jshflynn (talk) 18:44, 12 October 2012 (UTC)
My view is:
a) $\N \subseteq \Z \subseteq \Q \subseteq \R \subseteq \C$ while being technically accurate is weaker than $\N \subset \Z \subset \Q \subset \R \subset \C$
b) A page of this general accessibility can get away with the "$\subset$" as it does not matter whether it is interpreted as $\subsetneq$ or $\subseteq$.
In general I am in favour of $\subset$ being replaced as and when it's encountered, but on this page I think we can get away with it. Feel free to argue in either direction. --prime mover (talk) 18:53, 12 October 2012 (UTC)
## Prime numbers
The number $-3$ has exactly two positive divisors: $1$ and $3$. By your definition, Prime mover, master of all brilliance, $-3$ is prime. Thus the statement that the first few primes are $3, 5, 7, 11, 13$ is wrong. Rather, there is no first prime number. Unless you intend the identities of those two divisors to be part of the definition, O Brilliant One. What the heck is your problem with just saying they're natural? --Dfeuer (talk) 22:02, 28 March 2013 (UTC)
Yes, $-3$ is prime. --prime mover (talk) 22:06, 28 March 2013 (UTC)
It's a prime element in the ring of integers; it's not a prime number. --Dfeuer (talk) 22:11, 28 March 2013 (UTC)
Sources? --prime mover (talk) 22:15, 28 March 2013 (UTC)
Look at it like this. Some sources define prime numbers to include negative numbers. Some do not. To say that $\mathbb P \subsetneq \Z$ is true whichever definition you use. To say that $\mathbb P \subsetneq \N$ is not. --prime mover (talk) 22:23, 28 March 2013 (UTC)
I also direct you to Definition talk:Composite Number where the discussion has already been had. --prime mover (talk) 22:26, 28 March 2013 (UTC)
I note that Definition:Prime Number has not yet caught up to Your Awesome Wisdom. --Dfeuer (talk) 22:38, 28 March 2013 (UTC)
Source: Knuth 1.2.1. --Dfeuer (talk) 22:41, 28 March 2013 (UTC)
PM: please verify that your venerated source defines "prime number" so, rather than "prime integer". --Dfeuer (talk) 22:45, 28 March 2013 (UTC)
cba--prime mover (talk) 22:51, 28 March 2013 (UTC)
Now quit arguing. --Linus44 (talk) 22:27, 28 March 2013 (UTC)
What a despicable, perverted display of childishness. I wish I hadn't chosen this link to check up on the site. Yugh. I'll quickly leave now, before I completely lose my temper. — Lord_Farin (talk) 23:18, 28 March 2013 (UTC)
I apologize, Lord_Farin. I've had a stressful week and took it out on PM. --Dfeuer (talk) 00:44, 29 March 2013 (UTC)
## Other numbers
We could add another section, appropriately titled. As long as we don't add so much clutter to a page which is already busier than it needs to be. Adding them into the Also see might be appropriate, which we might split into sub-categories for these more-or-less artificial constructs which are only classified as "numbers" because they are called "numbers". However we call attention to them, it would probably be best if these constructions were kept separate from the main $\N, \Z, \Q, \R, \C$ classification which was the original intention of this page. --prime mover (talk) 22:40, 8 July 2020 (UTC)
I've never heard of anybody refer to "traditional numbers" so I'd rather we did not use that terminology. As for "extended numbers", that term is used for $\R \cup \set {-\infty, \infty}$ ("extended real numbers") so that's also doubtful. I'm also reluctant to create a paradigm which results in huge colossal quantities of work, which would be a danger if we were to change our default terminology from "Number" to "Traditional Number".
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2020-08-04 22:55:29
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https://en.m.wikipedia.org/wiki/LF_space
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# LF-space
(Redirected from LF space)
In mathematics, an LF-space is a topological vector space (TVS) V that is a locally convex strict inductive limit of a countable inductive system ${\displaystyle (V_{n},i_{nm})}$ of Fréchet spaces.[1] This means that V is a direct limit of the system ${\displaystyle (V_{n},i_{nm})}$ in the category of locally convex topological vector spaces and each ${\displaystyle V_{n}}$ is a Fréchet space. The word "strict" means that each of the bonding maps ${\displaystyle i_{nm}}$ is an embedding of TVSs.
Some authors restrict the term LF-space to mean that V is a strict locally convex inductive limit, which means that the topology induced on ${\displaystyle V_{n}}$ by ${\displaystyle V_{n+1}}$ is identical to the original topology on ${\displaystyle V_{n}}$.[2]
The topology on V can be described by specifying that an absolutely convex subset U is a neighborhood of 0 if and only if ${\displaystyle U\cap V_{n}}$ is an absolutely convex neighborhood of 0 in ${\displaystyle V_{n}}$ for every n.
## Properties
Every LF-space is barrelled and bornological (and thus ultrabornological). Every LF-space is a meager subset of itself.[3] The strict inductive limit of a sequence of complete locally convex spaces (such as Fréchet spaces) is necessarily complete. In particular, every LF-space is complete.[1] An LF-space that is the inductive limit of a countable sequence of separable spaces is separable.[4]
If X is the strict inductive limit of an increasing sequence of Fréchet space Xn then a subset B of X is bounded in X if and only if there exists some n such that B is a bounded subset of Xn.[1]
A linear map from an LF-space into another TVS is continuous if and only if it is sequentially continuous.[5] A linear map from an LF-space X into a Fréchet space Y is continuous if and only if its graph is closed in X × Y.[6] Every bounded linear operator from an LF-space into another TVS is continuous.[7]
If X is an LF-space defined by a sequence ${\displaystyle \left(X_{i}\right)_{i=1}^{\infty }}$ then the strong dual space ${\displaystyle X_{b}^{\prime }}$ of X is a Fréchet space if and only if all Xi are normable.[8] Thus the strong dual space of an LF-space is a Fréchet space if and only if it is an LB-space.
## Examples
### Space of smooth compactly supported functions
A typical example of an LF-space is, ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ , the space of all infinitely differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$ with compact support. The LF-space structure is obtained by considering a sequence of compact sets ${\displaystyle K_{1}\subset K_{2}\subset \ldots \subset K_{i}\subset \ldots \subset \mathbb {R} ^{n}}$ with ${\displaystyle \bigcup _{i}K_{i}=\mathbb {R} ^{n}}$ and for all i, ${\displaystyle K_{i}}$ is a subset of the interior of ${\displaystyle K_{i+1}}$ . Such a sequence could be the balls of radius i centered at the origin. The space ${\displaystyle C_{c}^{\infty }(K_{i})}$ of infinitely differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$ with compact support contained in ${\displaystyle K_{i}}$ has a natural Fréchet space structure and ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ inherits its LF-space structure as described above. The LF-space topology does not depend on the particular sequence of compact sets ${\displaystyle K_{i}}$ .
With this LF-space structure, ${\displaystyle C_{c}^{\infty }(\mathbb {R} ^{n})}$ is known as the space of test functions, of fundamental importance in the theory of distributions.
### Direct limit of finite-dimensional spaces
Suppose that for every positive integer n, ${\displaystyle X_{n}:=\mathbb {R} ^{n}}$ and for m < n, consider Xm as a vector subspace of Xn via the canonical embedding XmXn defined by sending ${\displaystyle x=\left(x_{1},\ldots ,x_{m}\right)\in X_{m}}$ to ${\displaystyle \left(x_{1},\ldots ,x_{m},0,\ldots ,0\right)}$ . Denote the resulting LF-space by X. The continuous dual space ${\displaystyle X^{\prime }}$ of X is equal to the algebraic dual space of X and the weak topology on ${\displaystyle X^{\prime }}$ is equal to the strong topology on ${\displaystyle X^{\prime }}$ (i.e. ${\displaystyle X_{\sigma }^{\prime }=X_{b}^{\prime }}$ ).[9] Furthermore, the canonical map of X into the continuous dual space of ${\displaystyle X_{\sigma }^{\prime }}$ is surjective.[9]
## References
1. ^ a b c Schaefer 1999, pp. 59-61.
2. ^ Helgason, Sigurdur (2000). Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions (Reprinted with corr. ed.). Providence, R.I: American Mathematical Society. p. 398. ISBN 0-8218-2673-5.
3. ^ Narici 2011, p. 435.
4. ^ Narici 2011, p. 436.
5. ^ Treves 2006, p. 141.
6. ^ Treves 2006, p. 173.
7. ^ Treves 2006, p. 142.
8. ^ Treves 2006, p. 201.
9. ^ a b Schaefer 1999, p. 201.
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2020-08-09 20:58:39
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|
http://www.koreascience.or.kr/article/JAKO201310837320666.page
|
# A CORRECTION TO A PAPER ON ROMAN κ-DOMINATION IN GRAPHS
• Mojdeh, Doost Ali ;
• Published : 2013.03.31
• 40 3
#### Abstract
Let G = (V, E) be a graph and k be a positive integer. A $k$-dominating set of G is a subset $S{\subseteq}V$ such that each vertex in $V{\backslash}S$ has at least $k$ neighbors in S. A Roman $k$-dominating function on G is a function $f$ : V ${\rightarrow}$ {0, 1, 2} such that every vertex ${\upsilon}$ with $f({\upsilon})$ = 0 is adjacent to at least $k$ vertices ${\upsilon}_1$, ${\upsilon}_2$, ${\ldots}$, ${\upsilon}_k$ with $f({\upsilon}_i)$ = 2 for $i$ = 1, 2, ${\ldots}$, $k$. In the paper titled "Roman $k$-domination in graphs" (J. Korean Math. Soc. 46 (2009), no. 6, 1309-1318) K. Kammerling and L. Volkmann showed that for any graph G with $n$ vertices, ${{\gamma}_{kR}}(G)+{{\gamma}_{kR}(\bar{G})}{\geq}$ min $\{2n,4k+1\}$, and the equality holds if and only if $n{\leq}2k$ or $k{\geq}2$ and $n=2k+1$ or $k=1$ and G or $\bar{G}$ has a vertex of degree $n$ - 1 and its complement has a vertex of degree $n$ - 2. In this paper we find a counterexample of Kammerling and Volkmann's result and then give a correction to the result.
#### Keywords
dominating set;Roman k-dominating function;correction
#### References
1. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998.
2. K. Kammerling and L. Volkmann, Roman k-domination in graphs, J. Korean Math. Soc. 46 (2009), no. 6, 1309-1318. https://doi.org/10.4134/JKMS.2009.46.6.1309
3. C. S. Liao and G. J. Chang, Algorithmic aspect of k-tuple domination in graphs, Taiwanese J. Math. 6 (2002), no. 3, 415-420. https://doi.org/10.11650/twjm/1500558307
4. C. S. ReVelle and K. E. Rosing, Defendens imperium Romanum: a classical problem in military strategy, Amer. Math. Monthly 107 (2000), no. 7, 585-594. https://doi.org/10.2307/2589113
5. W. Shang, F. Yao, P. Wan, and X. Hu, On minimum m-connected k-dominating set problem in unit disc graphs, J. Comb. Optim. 16 (2008), no. 2, 99-106. https://doi.org/10.1007/s10878-007-9124-y
#### Acknowledgement
Supported by : IPM
|
2019-06-26 08:14:52
|
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|
http://blogs.ams.org/mathgradblog/2013/07/03/infinitude-fermat-numbers/?amp&
|
The Weak and Strong Goldbach Conjectures
In 1742 the German mathematician Christian Goldbach wrote a letter to Leonhard Euler proposing two problems that, until recently, have seen little progress. In modern terms, the problems are identified as either the Weak or Strong Goldbach Conjecture. The Strong Goldbach Conjecture is the statement that every even integer greater than 2 is the sum of two prime numbers. Similarly, the modern version of the Weak conjecture can be stated as every odd integer greater than 5 is the sum of three odd primes.
In his groundbreaking work earlier this year, the Peruvian mathematician Harald Helfgott (with much acceptance in the math community) announced that he had proved the Weak conjecture. Fascinated by his result, I sent him an email of congratulations with a related question involving the Strong conjecture. The following was his reply:
Dear Avery,
Thank you for your email. The fact that “three implies four” (so to speak)
was already known (and, as you can see, very easy to prove). I believe the
strong conjecture is much, much, much harder.
All the best
Harald Helfgott
I too agree that the Strong conjecture is, “much,much,much harder”. On May 13, 2013, the renowned mathematician Terrence Tao of UCLA released the following statement on the internet in regards to Helfgott’s result:
Busy day in analytic number theory; Harald Helfgott has complemented his previous paper http://arxiv.org/abs/1205.5252 (obtaining minor arc estimates for the odd Goldbach problem) with major arc estimates, thus finally obtaining an unconditional proof of the odd Goldbach conjecture that every odd number greater than five is the sum of three primes….. As with virtually all successful partial results on the Goldbach problem, the argument proceeds by the Hardy-Littlewood-Vinogradov circle method; the challenge is to make all the estimates completely effective and to optimise all parameters (which, among other things, requires a certain amount of computer-assisted computation). [EDIT: the proof also relies on extensive numerical verifications of GRH that were performed by David Platt.]”
Unfortunately, Tao further stated that it would be unlikely that the Hardy-Littlewood Vinogradov Circle Method and Helfgott’s result could be used to prove the Strong conjecture. Therefore the mystery of the Strong conjecture has currently escaped all attempts of proof. Computationally, it has been shown true for every even integer greater than 2 into the trillions. It would be clever indeed for some mathematician to show that the Strong conjecture follows trivially from the Weak. Do you think you can do it?
|
2016-09-28 15:33:48
|
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https://proofwiki.org/wiki/Definition:Conditional/Language_of_Conditional
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# Definition:Conditional/Language of Conditional
## Definition
The conditional has been discussed at great length throughout the ages, and a whole language has evolved around it. For now, here are a few definitions:
### Weak
In a conditional $p \implies q$, the statement $q$ is weaker than $p$.
### Strong
In a conditional $p \implies q$, the statement $p$ is stronger than $q$.
Thus we have the notion of certain theorems having a weak and a strong version.
### Superimplicant
In a conditional $p \implies q$, the statement $p$ is superimplicant to $q$.
### Subimplicant
In a conditional $p \implies q$, the statement $q$ is subimplicant to $p$.
### Antecedent
In a conditional $p \implies q$, the statement $p$ is the antecedent.
### Consequent
In a conditional $p \implies q$, the statement $q$ is the consequent.
### Necessary Condition
Let $p \implies q$ be a conditional statement.
Then $q$ is a necessary condition for $p$.
That is, if $p \implies q$, then it is necessary that $q$ be true for $p$ to be true.
This is because unless $q$ is true, $p$ can not be true.
### Sufficient Condition
Let $p \implies q$ be a conditional statement.
Then $p$ is a sufficient condition for $q$.
That is, if $p \implies q$, then for $q$ to be true, it is sufficient to know that $p$ is true.
This is because of the fact that if you know that $p$ is true, you know enough to know also that $q$ is true.
|
2022-01-17 22:45:46
|
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https://www.ias.ac.in/listing/bibliography/pram/ELMER_RAMIREZ_BARRETO
|
• ELMER RAMIREZ BARRETO
Articles written in Pramana – Journal of Physics
• Electric charge quantisation in 331 models with exotic charges
The extensions of the Standard Model based on the $SU(3)_{C} \otimes SU(3)_{L} \otimes U(1)_{X}$ gauge group are known as 331 models. Different properties such as the fermion assignment and the electric charges of the exotic spectrum, that define a particular 331 model, are fixed by a $\beta$ parameter. In this article, we study the electric charge quantisation in two versions of the 331 models, set by the conditions $\beta = 1/(3\sqrt{3})$ and $\beta = 0$. In these frameworks, arise exotic particles, for instance, new leptons and gauge bosons with a fractional electric charge. Additionally, depending on the version, quarks with non-standard fractional electric charges or even neutral appear. Considering the definition of electric charge operator as a linear combination of the group generators that annihilates the vacuum, classical constraints from the invariance of the Lagrangian, and gauge and mixed gauge-gravitational anomalies cancellation, the quantisation of the electric charge can be verified in both versions.
• # Pramana – Journal of Physics
Volume 96, 2022
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019
|
2022-08-09 17:47:01
|
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http://www.maths.kisogo.com/index.php?title=Sequential_compactness
|
# Sequential compactness
The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence.
Sequential compactness extends this notion to general topological spaces.
## Definition
A topological space [ilmath](X,\mathcal{J})[/ilmath] is sequentially compact if every (infinite) Sequence has a convergent subsequence.
### Common forms
#### Functional Analysis
A subset [ilmath]S[/ilmath] of a normed vector space $(V,\|\cdot\|,F)$ is sequentially compact if any sequence $(a_n)^\infty_{n=1}\subset k$ has a convergent subsequence $(a_{n_i})_{i=1}^\infty$, that is $(a_{n_i})_{i=1}^\infty\rightarrow a\in K$
Like with compactness, we consider the subspace topology on a subset, then see if that is compact to define "compact subsets" - we do the same here. As warned below a topological space is not sufficient for sequentially compact $\iff$ compact, so one ought to use a metric subspace instead. Recalling that a norm can give rise to the metric $d(x,y)=\|x-y\|$
## Warning
Sequential compactness and compactness are not the same for a general topology
## Uses
• A metric space is compact if and only if it is sequentially compact, a theorem found here
|
2022-12-01 19:22:08
|
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|
https://solvedlib.com/the-acidbase-chemistry-of-this-reaction-is,113809
|
# The acid/base chemistry of this reaction is important. As acids increase in strength, which of the...
###### Question:
The acid/base chemistry of this reaction is important. As acids increase in strength, which of the following are true? O O O pH increases ka decreases pka decreases. [OH-] increases Both b and care true. O
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##### 2. (20 pts) A certain reaction was run several times using each of two catalysts, A...
2. (20 pts) A certain reaction was run several times using each of two catalysts, A and B. The catalysts were supposed to control the yield of an undesirable side product. Results, in units of percentage yield, for 24 runs of catalyst A and 20 runs of catalyst Bare as follows: Catalyst A 3.4 2.6 3.8...
##### The first-generation peas were all smooth: In the second generation (F2), he obtained 5474 smooth-pea plants and 1850 wrinkled-pea plants. Do these data agree with the conclusion that F2 is made up of 75% dominant trait (in thiscase "smooth")and 25% recessive trait ("wrinkled")? Use " 'the Chi-Squared statistic Ifrom the previous question to find the P-Value using Table D. Please do not use any other resource:Degrees of freedom isP-Value is
The first-generation peas were all smooth: In the second generation (F2), he obtained 5474 smooth-pea plants and 1850 wrinkled-pea plants. Do these data agree with the conclusion that F2 is made up of 75% dominant trait (in thiscase "smooth")and 25% recessive trait ("wrinkled")? ...
##### Cahadetnannk ABC Iike teceo Slapose [hal &27, andTune ( Jue Gor drotn t0 Klo ) Soto tho tnunolCaTeyour intalnedietc con jutatlon[0Feast four Oecimal Exacas, and rund Your n50r= to the anchr Tno buchitharale 4 u nntur -Nlo solaior U there Me then 0nt tohrial, Use the button Iobcled *5" .
Cahade tnannk ABC Iike te ceo Slapose [hal & 27, and Tune ( Jue Gor drotn t0 Klo ) Soto tho tnunol CaTeyour intalnedietc con jutatlon[0 Feast four Oecimal Exacas, and rund Your n50r= to the anchr Tno buchitharale 4 u nntur -Nlo solaior U there Me then 0nt tohrial, Use the button Iobcled *5&qu...
##### Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues_{-9 16 25an
Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues_ {- 9 16 25 an...
##### Point) Consider the initial value problemmy +cy +ky = FCt), y(O)= =0, y' (0) = 0modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t) , where the unit of force is the Newton (N): Assume that m = 2 kilograms, € = & kilograms per second, k = 80 Newtons per meter; and the applied force in Newtons is30 F(t) = 8oif 0 <t < t/2, ift > m/2a. Solve the initial value problem, using that the displacement y(t) and velocity y' (t
point) Consider the initial value problem my +cy +ky = FCt), y(O)= =0, y' (0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t) , where the unit of force is the Newton (N): Assume that m = 2 kilograms, € = & kilograms per ...
##### What are different programs providing hospice care with regard to team structure and scope of services
What are different programs providing hospice care with regard to team structure and scope of services...
##### 1. Given the following shaft/connector assembly: 2. We have a kinematic equation relating the po...
1. Given the following shaft/connector assembly: 2. We have a kinematic equation relating the position of the input shaft to the output shaft and the misalignment angle (w1 = input angular speed and w2 = output angular speed): 3. The following conditions are given to find the angle alpha: 4. Ple...
##### T-piece A T-piece or T-connector is a very simple component that lets us spli or divide...
T-piece A T-piece or T-connector is a very simple component that lets us spli or divide airflow. It can be very useful if you want two cylinders to operate at the same time. AIR OUT IR OUT ARN 10 Figure 25 On circuit diagrams, the T-piece is identified by a dot. T-piece Figure Assignment 1. A delive...
##### The pain killer morphine is weak base when added to water: The Kb is 1.6 * 10 6. the [OH:] ofa 433 What is 10 * M solution of morphine?
The pain killer morphine is weak base when added to water: The Kb is 1.6 * 10 6. the [OH:] ofa 433 What is 10 * M solution of morphine?...
##### Find the Fourier transform of $f(x)=e^{-x^{2} /\left(2 \sigma^{2}\right)} .$ Hint: Complete the square in the $x$ terms in the exponent and make the change of variable $y=x+\sigma^{2}$ in. Use tables or computer to evaluate the definite integral.
Find the Fourier transform of $f(x)=e^{-x^{2} /\left(2 \sigma^{2}\right)} .$ Hint: Complete the square in the $x$ terms in the exponent and make the change of variable $y=x+\sigma^{2}$ in. Use tables or computer to evaluate the definite integral....
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2023-04-01 22:31:30
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https://www.cuemath.com/ncert-solutions/q-9-exercise-15-1-probability-class-9-maths/
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In the verge of coronavirus pandemic, we are providing FREE access to our entire Online Curriculum to ensure Learning Doesn't STOP!
# Ex.15.1 Q9 Probability Solution - NCERT Maths Class 9
Go back to 'Ex.15.1'
## Question
Eleven bags of wheat flour, each marked $$5\,\rm kg$$, actually contained the following weights of flour (in kg)
$$4.97, 5.05, 5.08, 5.03,\\ 5.00, 5.06, 5.08,\\ 4.98, 5.04, 5.07, 5.00$$
Find the probability that any of these bags chosen at random contains more than $$5 \,\rm kg$$ of flour.
Video Solution
Probability
Ex exercise-15-1 | Question 9
## Text Solution
What is known?
No of bags of specific weights.
What is unknown?
Probability of any of the bags containing more than $$5\,\rm kg$$ of flour.
Reasoning:
The empirical probability $$P(E)$$ of an event $$E$$ happening, is given by:
\begin{align}{P}({E})=\frac{ \begin{pmatrix} \text { Number of trials in which }\\ \text{the event happened } \end{pmatrix} }{ \text { The total number of trials } }\end{align}
Use probability to derive the solution where
Probability $$P$$ Weight of the bag
\begin{align}=\frac{ \begin{pmatrix} \text { Number of bags} \\ \text{ of specific weight } \end{pmatrix} }{\text { Total number of bags }}\end{align}
Steps:
Total no of bags $$= 11$$
No of bags more than $$5\,\rm kg$$ of flour $$= 7$$
Probability (Bag weighing more than $$5\,\rm kg$$ of flour )
\begin{align}\\&=\frac{ \begin{pmatrix} \text { Number of bags weighing} \\ \text{ more than 5 kg of flour } \end{pmatrix}} {\text { Total number of bags }} \\ &=\frac{7}{11}\end{align}
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2020-04-07 13:43:22
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http://secon2015.ieee-secon.org/workshops/fog-networking-5g-and-iot-workshop/submission
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Submission
Submission to the workshop is through the EDAS Conference Management System.
In order to submit, you must first create an EDAS account if you do not already have one. The EDAS system will e-mail you your password, and then you can login to EDAS using the password you have received by email. EDAS will provide you with a submissions homepage where you can register your paper submission and make appropriate changes.
High-quality full papers which at the time of submission, are not under review and have not already been published or accepted for publication elsewhere are solicited. The first page should include the paper's title, the abstract, the authors' full names, affiliations, and e-mail addresses.
Submissions can be made through EDAS at http://edas.info/N19719
Papers should contain original material and not be previously published, or currently submitted for consideration elsewhere, and should strict follow these submission guidelines:
All the sumbissions must be in PDF format, and must be formatted according to the standard IEEE Computer Society 8.5" x 11" page, two-column format. The IEEE LaTeX and Microsoft Word templates, as well as related information, can be found at the IEEE Computer Society website.
The page limit is six (6) pages, including text, figures, references, and appendices. Paper longer than 6 pages will not be considered for peer review. The font size used in the text of the submission must not be smaller than 10 points.
PDF files must have version 1.4 or higher and must embed all the fonts. Authors are allowed to use colors for text/figures in their submission, but they must be sure that the paper prints clearly on standard black-and-white printers.
At least one author of an accepted paper is required to register for the workshop at the full (member or non-member) rate and the paper must be presented by an author of that paper at the conference unless the TPC Chairs grant permission for a substitute presenter arranged in advance of the event and who is qualified both to present and answer questions. Non-refundable registration fees must be paid prior to uploading the final IEEE formatted, publication-ready version of the paper. Accepted and presented papers will be published in the Fog Networking for 5G and IoT Proceedings and will be submitted to IEEE Xplore®.
IMPORTANT DATES
Submission deadline (Hard): April 1st, 2015
Notification of acceptance: April 15th, 2015
Workshop: June 22nd, 2015
In order for your paper to be in the conference proceedings and IEEE Xplore you need to complete these five steps:
2. Check your paper with IEEE PDF eXpress
3. Register for the workshop with payment
Please read the reviews of your paper and follow the reviewers' recommendations for revising it and preparing the final version. When you are done with the revision, please print out your paper and carefully proof read it. The final manuscript must NOT contain page numbers.
The manuscript template can be found at http://www.ieee.org/conferences_events/conferences/publishing/templates.html.
Note that for camera-ready papers should not exceed 6 pages.
Note that you will also need to add a Copyright Clearance Code Notice—this notice is to appear on the bottom of the first page of each paper. (This is separate from step 4, which details how to submit the Copyright Form). Detailed instructions can be found at:http://www.ieee.org/portal/pages/about/documentation/copyright/cfrmlink.html
• For papers in which all authors are employed by the US government, the copyright notice is: U.S. Government work not protected by U.S. copyright
• For papers in which all authors are employed by a Crown government (UK, Canada, and Australia), the copyright notice is: 978-1-4673-7392-0/15/$31.00 ©2015 Crown • For papers in which all authors are employed by the European Union, the copyright notice is: 978-1-4673-7392-0/15/$31.00 ©2015 European Union
• For all other papers the copyright notice is: 978-1-4673-7392-0/15/$31.00 ©2015 IEEE Latex users can add following lines just before \begin{document} for the copyright notice to show up (shown below as an example for the third case above): \IEEEoverridecommandlockouts \IEEEpubid{\makebox[\columnwidth]{978-1-4673-7392-0/15/\$31.00~\copyright~2015 IEEE \hfill}
\hspace{\columnsep}\makebox[\columnwidth]{ }}
MSWord users can use: ‘Insert’ -> ‘Text box’, insert the appropriate copyright notice in the texbox, and place the box (without border) at the bottom left on the first page.
2. Check your paper with IEEE PDF eXpress
IEEE PDF eXpress has been made available (http://www.pdf-express.org) for checking the IEEE Xplore compatibility of PDF files and provide assistance to authors to correct problems with their files. Instructions for IEEE PDF eXpress for IEEE SECON 2014 can be found at http://www.pdf-express.org/frhelp.asp.
The conference ID for IEEE PDF eXpress is "36188X". Note that the workshop proceedings will be separate from the main conference proceedings.
IEEE is very strict about the requirements for converting application files or PostScript files to full-text PDF for inclusion in IEEE Xplore. The new IEEE Xplore Requirements for PDF will be enforced. All conference articles submitted for inclusion in IEEE Xplore must be in Xplore-compatible PDF format. IEEE PDF eXpress is a free service to IEEE conferences, allowing their authors to make IEEE Xplore-compatible PDFs (Conversion function) or to check PDFs that authors have made themselves for IEEE Xplore compatibility (PDF Check function).
It is important to note that submitting a file to IEEE PDF eXpress is only for creation of a compliant PDF file, you must still submit your final paper through EDAS.
Then, for the IEEE copyright upload, append “15SEC” to your registration number—i.e., you will now get a code that is 16 digits long, ending in 15SEC. You should be able to complete the IEEE copyright form using this code.
3. Register for the workshop
In order to upload your paper you must first register for the workshop (please follow the link: http://secon2015.ieee-secon.org/registration. On registering for the workshop, you will get a “confirmation number”. To subsequently upload your camera-ready paper, you will need a Registration Code which is 11 digits long. Registration deadline is April 30.
IEEE and IEEE Communications Society Policies
All IEEE SECON 2015 workshop technical papers must be associated with a workshop registration. For authors presenting multiple workshop papers, one workshop registration is valid for up to three papers. IEEE reserves the right to exclude a paper from distribution after the workshop (e.g., removal from IEEE Xplore) if the paper is not presented at the workshop.
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2018-08-16 04:06:21
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http://openstudy.com/updates/55876e9ce4b0cb8c62eafa7a
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## Tommynaut one year ago Let F = y^2 ~i + x ~j + z ~k. Let S be the curved surface of the cylinder x^2 + y^2 = 1 for 0 <= z <= 3. Calculate the outwards flux of F through S. I used the divergence theorem to get my solution, by finding divF = 1. I thought that would mean that the flux would be the volume of the cylinder, but the answer is actually 0. Could someone please explain the proper process? Oh, and is there a way to use latex when asking questions?
1. Empty
Yeah, like this for example and the code: $$\nabla \cdot F = 0$$ $$\nabla \cdot F = 0$$
2. Empty
Actually wait, you mean specifically while asking questions. No, it's not really possible to access the equation editor but you can still type latex, it sucks I agree. But whatever let me try help you with your actual problem, one sec haha.
3. Tommynaut
Ok cheers haha
4. Empty
Ahhh ok the problem I believe you're having is that the divergence theorem applies to closed shapes, not open ones.
5. Tommynaut
But surely the cylinder can be imagined to be closed? I really didn't think it made a difference, especially as we're looking for the flux through the curved surface anyway.
6. Tommynaut
And how would the question then be done?
7. anonymous
You can only use the divergence theorem if the surface is closed.
8. anonymous
I don't think the top and bottom are included.
9. anonymous
It'd parametrize the cylinder as: $\{(\cos t, \sin t, z)|0\leq t\leq 2\pi,0\leq z\leq 3\}$
10. dan815
you can apply div theorem, and them subtract the flux out of the top and bottom circles, if that simplfies it, it might be simplied to do the flux out of hte top and bottom as they are planar and perp to z so only the z component would matter
11. IrishBoy123
you should have gotten 3pi from div theorem surely. then the field on bottom plate = zero => zero flux normalised field at top plate = > z = 3, which is 3 pi also so net through curved surface = 0 which makes sense as the field is symmetrical in the x-y plane
12. Tommynaut
I did indeed get 3pi originally. I don't really understand the concepts you guys are explaining but thank you anyway. I found a method of solution that uses cylindrical coordinates and the unit normal outwards, followed by a double integral.
13. IrishBoy123
i hope this helps
14. IrishBoy123
my point being that, in my experience at least, doing these types of things is v often about not doing them. ie spotting symmetries in the set up, or lucking out with divergence or stoke's theorem, the alternative being to do a complete pig of a double integral....:p you can often also have more confidence in your answer if you have simplified it down in one of these ways.
15. Tommynaut
Thank you for that! The notation you use is a bit different to what I've been taught but it makes sense nonetheless :)
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2017-01-24 17:30:50
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http://autoplot.org/developer.scripting
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# developer.scripting
Purpose: We need a document describing commands available when scripting. This will cover commands available when no imports are used.
Audience: Scientists and software people wanting to use Autoplot for scripting.
Note: Some of the non-technical aspects of this page are being migrated to scripting.
More scripts in the SVN repository: [1] (All scripts before and including areaSelect.py have been verified to work with current DOM).
# 1. Introduction to Scripting
Autoplot uses the popular language Python to provide scripting for various applications. For example, a script can load the data from two data sources and combine them to make a new dataset. Scripts can also be used to run Autoplot automatically, for example, to create a series of images called a pngwalk. Autoplot creates the scripting environment where the user's commands are executed, and adds useful commands to the environment. Also, Autoplot's data model, or data representation, is adapted so that arithmetic operators can be used directly on the data as if they were number arrays in IDL or Matlab. (But unlike IDL and Matlab, they carry with them descriptive metadata and physical units.)
For example, consider this script:
ds1= getDataSet( 'http://autoplot.org/data/image/Capture_00158.jpg?channel=greyscale' )
ds2= getDataSet( 'http://autoplot.org/data/image/Capture_00159.jpg?channel=greyscale' )
result= abs( ds2- ds1 )
It should be fairly clear what the script does: load in two grayscale views of images, and then assign the absolute difference to the variable result. This variable could then be plotted, for example, or we might look for the greatest difference.
Scripting is used in three contexts: the Data Source Context, the Application Context, and the third is a scripting context using the --script at the launch command. In the "Data Source" context, the output of the script is expected to be named result or data, and this script variable is plotted in the canvas tab. In the "Application" context, Jython commands may be used to manipulate the content of the canvas. The third context can essentially be used to create new custom applications, and it is not discussed any further here.
## 1.1. Script Editor
Autoplot provides a editor GUI for working with scripts. Selecting Options->Enable Feature->Script Panel will reveal a tab named "script" where scripts may be entered and executed. The editor provides simple completions for this environment. To see completions, enter TAB or ctrl-space. (Note TAB can use reset to the TAB character whitespace with a preference.)
Note the completions use a trick to work, and that is to refactor the script to an equivalent script which can be executed immediately. This works by reducing the script to just imports and constructor calls, and to a number of routines which are known to be quick. Of course this doesn't always work, but provides pretty good results. However, because of this, some parts of the code can not support completions, like callbacks from Java code (def boxSelected). It's also possible that a constructor call is very slow, which would hang completions, but this assumption has been effective. Last, there may be some side effects that occur as well, like GUIs created. The completions are getting more attention than before as many more people are using the editor, and this code is maturing.
## 1.2. Data Source Context
Here scripts can load in data and return a new dataset (or datasets). This is the "data source context" and these are files that can be used as if they were datasets. If the above script were saved in the file "http://www.autoplot.org/data/imageDiff.jyds," then http://www.autoplot.org/data/imageDiff.jyds?result would refer to this dataset. This allows scientists to publish operations done to data as well as data itself. Note these scripts are unaware of the Autoplot application, they can load and operate on data, but they cannot plot it. Commands available in this context are described below under the section "Ops"
Script examples: [2] and cookbook#Scripting.
## 1.3. Application Context
These scripts access the application itself. Take for example the following application:
trs= generateTimeRanges( '%Y-%m-%d', '2010-January' )
for tr in trs:
dom.timeRange= DatumRangeUtil.parseTimeRange(tr)
writeToPng( '/tmp/%s.png' % tr )
This script would run the application through each day of the month January 2010, making images of each day. All commands are available in this context.
Examples are available at [3] and cookbook#Scripting. This javadoc describes the commands available in this Jython Application Context.
## 1.4. Building Scripts
### 1.4.1. Use of Progress Monitor
All scripts have a progress monitor that can be used to provide feedback to users. This is the variable 'monitor' and is used like so:
monitor.setTaskSize(200) # the number of steps (arbitary units)
monitor.started() # the task is started.
for i in xrange(200):
if ( monitor.isCancelled() ): break # if not called, the cancel button will be insensitive
monitor.setProgressMessage('at %d' % i) # this describes actions done to perform the task.
monitor.finished() # indicate the task is complete
A well-written script will use the monitor to effectively convey information to the user. Imagine the scientist is the CEO of a company, and the script is the Manager of a process. The process is implemented by a Worker. All three parties use the progress monitor. The Worker calls setProgressMessage and setTaskProgress to convey the state of the task. The Worker checks isCancelled to see if they can abort the process. The Manager calls setLabel to convey the overall goal of the progress. Typically Autoplot will play the role of the Manager, setting the label "loading data" or "executing script", but it is acceptable for the script to take on this role as well, since it can be more descriptive.
### 1.4.2. Getting Input
s= getParam( 's', 'deflt', 'label to describe' ) # gets a string parameter, with default value "deflt"
f= getParam( 'f', 2.34, 'label to describe' ) # gets a float parameter, with default value 2.34
i= getParam( 'i', 100, 'array size' ) # gets an integer parameter. Be careful--user input of a real is truncated, so 100 is quite different than
f= getParam( 'f', 100.0, 'volume' ) # gets a float parameter.
e= getParam( 'e', 'RBSPA', 'spacecraft', [ 'RBSPA', 'RBSPB' ] ) # enumeration with the values given
b= getParam( 'v', 'F', 'apply correction', [ 'T', 'F' ] ) # booleans are just enumerations with the values 'T' and 'F'
Autoplot will look for this in scripts and automatically add to GUI. The type is determined by the default value.
This is mostly used for .jyds scripts that create new datasets, but all script types can use this command. The jyds plugin creates a GUI by simplifying the script to just getParam calls and trivial commands, so please let us know if this is not working for you.
Enumerations are supported as well, where a list of possible values is enumerated. For example:
sensor= getParam( 'sensor', 'left', 'sensor antenna', ['left','right'] )
will get the parameter sensor, which can be either left or right (with left as the default). When a GUI is created, a droplist of possible values is used instead of a text entry field.
Last, booleans are allowed, and a checkbox is used when a GUI is produced:
correct= getParam( 'correct', 'T', 'perform correction on the data', [ 'T', 'F' ] )
Note you cannot use the result as a boolean in the python code. You must compare it to 'T'.
Application-context scripts can use get param as well. Right now the default value is always used, but soon a gui will be created before running the script. Jython scripts run using the Java class org.virbo.autoplot.JythonMain can pass arguments into the script with arguments following the first argument (name of the script):
java -cp autoplot.jar org.virbo.autoplot.JythonMain /tmp/myscript.jy sensor=right correct=T
#### 1.4.2.1. Creating a GUI from a script
Autoplot works by making a simplified version of the script, and then running it with getParam replaced with a function that tallies the calls. This trick causes confusion when sometimes functions can be used and sometimes not. These functions can be used:
"range", "xrange", "getParam", "lower", "upper"
so
x= getParam( 'sc', 1, 'spacecraft ID", range(40) )
works, but this won't:
x= getParam( 'sc', 1, 'spacecraft ID", findgen(40) )
### 1.4.3. Handling Exceptions
Often a process will typically execute, but we know exceptional cases may occur and we want to invoke special code to handle them. For example, I may open a sequence of files, and if one is misformatted, I want to log it to a file and carry on with processing other files.
In Python we use try/exception blocks, something like:
try:
plot( uri )
writeToPng( '/tmp/pngs/%s.png' % uri )
except Exception, ex:
ERROR.write( '# unable to plot ' + uri + ' because of ' + str(ex) )
Note most of the functions called are actually Java procedures and throw Java exceptions. Unfortunately, the Python catch doesn't catch Java exceptions, and a second except block is needed:
try:
plot( uri )
writeToPng( '/tmp/pngs/%s.png' % uri )
except Exception, ex:
ERROR.write( '# unable to plot ' + uri + ' because of ' + str(ex) )
except java.lang.Exception, ex:
ERROR.write( '# unable to plot ' + uri + ' because of ' + str(ex) )
Or, if you don't need to inspect the exception:
try:
plot( uri )
writeToPng( '/tmp/pngs/%s.png' % uri )
except:
ERROR.write( '# unable to plot ' + uri )
#### 1.4.3.1. Throwing (or Raising) Exceptions
Sometimes we want our process to stop and let the user know that something went wrong, so we throw an exception. This is done like so:
if ( ds.length()==0 ):
raise Exception("Dataset is empty")
Application-context scripts can be added to the Autoplot GUI by putting them in the HOME/autoplot_data/tools/ folder. http://autoplot.org/data/tools/ shows some example scripts.
### 1.4.5. PWD is often defined
Starting in the summer of 2013, PWD is often allowed in scripts to refer to folder containing the script. This is a string containing the URI location of the script, be it on a local harddrive or a web folder. This will always end with a slash, and will often start with "http://" or "file://". You cannot use this to import other Jython files, since you cannot import remote files.
# 2. Commands
These are commands that are available to scripts in Autoplot. Autoplot's scripting environment is Jython (Python in Java) with a set of codes that are automatically imported to keep scripts simple.
The Java types that each command takes are indicated, because it's easier to produce this document and if these commands are used in Java instead if Jython, the types must be followed. Jython is able to make some conversions to correctly combine data.
# 3. Application Context Commands
These are extra commands that are added when running in the script context that control the Autoplot application itsself, such as writeToPng. These commands are not available in JythonDataSources (Data Source Context), which load data to make a new data set.
## 3.1. getViewWindow
public static java.awt.Window getViewWindow()
return the Window for the application, to be used for dialogs. See createGui(), which creates the view.
## 3.2. setCanvasSize
public static void setCanvasSize(int width,
int height)
set the size of the canvas. This is only used when the GUI is not used, and in headless mode, otherwise the GUI controls the size of the canvas.
Parameters:
• width - the width of the canvas
• height - the height of the canvas
## 3.3. plot
A number of plot commands are available.
### 3.3.1. plot the URI
public static void plot(java.lang.String surl)
bring up the autoplot with the specified URL.
Parameters:
• surl - a URI or vap file
### 3.3.2. plot one URI against another
public static void plot(java.lang.String surl1,
java.lang.String surl2)
plot one URI against another. No synchronization is done, so beware. Introduced for testing non-time axis TSBs.
Parameters:
• surl1 - the independent variable dataset (generally used for X)
• surl2 - the dependent variable dataset (generally used for Y, but can be rank 2).
### 3.3.3. plot the URI at position
public static void plot(int chNum,
java.lang.String surl)
bring up the autoplot with the specified URL. This may be a little confusing, because it replaces the datasource with the given number, which would typically correspond to the position on the page.
Parameters:
• chNum - the data source number to reset the URI.
• surl - a URI to use
### 3.3.4. plot the URI at position with this label
public static void plot(int chNum,
java.lang.String label,
java.lang.String surl)
bring up the autoplot with the specified URL.
Parameters:
• chNum - the data source number to reset the URI.
• label - for the plot.
• surl - a URI to use
### 3.3.5. plot this dataset
public static void plot(org.virbo.dataset.QDataSet ds)
plot the dataset in the first dataSource node.
Parameters:
• ds - QDataSet to plot
### 3.3.6. plot one dataset against another
public static void plot(org.virbo.dataset.QDataSet x,
org.virbo.dataset.QDataSet y)
plot the dataset in the first dataSource node.
Parameters:
• x - QDataSet for the independent parameter
• y - QDataSet for the dependent parameter
### 3.3.7. plot one dataset Z against X and Y
public static void plot(org.virbo.dataset.QDataSet x,
org.virbo.dataset.QDataSet y,
org.virbo.dataset.QDataSet z)
plot the dataset in the first dataSource node.
Parameters:
• x - QDataSet for the independent parameter for the X values
• y - QDataSet for the independent parameter for the Y values
• z - Rank 1 or Rank 2 QDataSet for the dependent parameter
### 3.3.8. plot one dataset in this position
public static void plot(int chNum,
org.virbo.dataset.QDataSet ds)
plot the dataset in the specified dataSource node.
Parameters:
• ds - dataset to plot.
### 3.3.9. plot one dataset against another in this position
public static void plot(int chNum,
org.virbo.dataset.QDataSet x,
org.virbo.dataset.QDataSet y)
plot the dataset in the specified dataSource node.
Parameters:
• x - QDataSet for the independent parameter for the X values
• y - QDataSet for the independent parameter for the Y values
### 3.3.10. plot one dataset Z against X and Y in this position
public static void plot(int chNum,
org.virbo.dataset.QDataSet x,
org.virbo.dataset.QDataSet y,
org.virbo.dataset.QDataSet z)
plot the dataset in the specified dataSource node.
Parameters:
• x - QDataSet for the independent parameter for the X values
• y - QDataSet for the independent parameter for the Y values
• z - Rank 1 or Rank 2 QDataSet for the dependent parameter
### 3.3.11. plot this URI at this position
public static void plot(int chNum,
java.lang.String label,
org.virbo.dataset.QDataSet ds)
bring up the autoplot with the dataset
Parameters:
• chNum - the data source number to reset the URI.
• label - for the plot.
• ds - the dataset to use.
### 3.3.12. plot X vs Y with this label at this position
public static void plot(int chNum,
java.lang.String label,
org.virbo.dataset.QDataSet x,
org.virbo.dataset.QDataSet y)
plot the dataset in the specified dataSource node.
Parameters:
• x - QDataSet for the independent parameter for the X values
• y - QDataSet for the independent parameter for the Y values
### 3.3.13. plot one dataset Z against X and Y in this position with this label
public static void plot(int chNum,
java.lang.String label,
org.virbo.dataset.QDataSet x,
org.virbo.dataset.QDataSet y,
org.virbo.dataset.QDataSet z)
plot the dataset in the specified dataSource node.
Parameters:
• label - the label for the dependent parameter
• x - QDataSet for the independent parameter for the X values
• y - QDataSet for the independent parameter for the Y values
• z - Rank 1 or Rank 2 QDataSet for the dependent parameter
## 3.4. plotx
The plotx command is an experimental plot command that is written for use only in the Jython environment.
xx= linspace( 0, 4*PI, 100 )
yy= sin( xx )
plotx( xx, yy, title='sin' )
Its control tries to match IDL's plot command, with named parameters like:
xtitle ytitle ztitle ='Label' the label for each axis
xlog ylog zlog =True for log plots
yrange =[-10,10]
title ='Title' title for the plot
renderType ='scatter', 'series', 'nnSpectrogram', etc.
symsize =5.0 symbol size in pixels
linewidth =3.0 line thickness in pixels
color ='0xFF0000' or 'RED'
symbol ='STAR', 'CROSS', 'TRIANGLES', 'EXES', 'DIAMONDS'
## 3.5. setStatus
public static void setStatus(java.lang.String message)
set the autoplot status bar string. Use the prefixes "busy:", "warning:" and "error:" to set icons.
Parameters:
• message - message to display, possibly with "busy:" "warning:" or "error:" prefix.
public static void addTab(java.lang.String label,
javax.swing.JComponent c)
add a tab to the running application. A new tab will be added with the label.
Parameters:
• label - the label for the component.
• c - the component to add.
## 3.7. setRenderStyle
public static void setRenderStyle(java.lang.String name)
• Set the style used to render the data using a string identifier: spectrogram, series, scatter, histogram, fill_to_zero, digital
Parameters:
• name - string name of the plot style.
## 3.8. peekAt
public static void peekAt(java.lang.Object o)
throws java.io.IOException
This is intended to be used with a debugger. The developer should put a breakpoint at the out.write statement, and then call peekAt from the script.
Parameters:
• o - any object we want to look at.
## 3.9. writeToPng
public static void writeToPng(java.lang.String filename)
write out the current canvas to a png file. TODO: bug 3113441: this has issues with the size. It's coded to get the size from the DOM, but if it is fitted and has a container it must get size from the container. Use writeToPng( filename, width, height ) instead for now. See writeToPdf(String filename), which appears to have a fix for this that would affect how this is resolved.
Parameters:
• filename - The name of a local file
## 3.10. writeToPng
public static void writeToPng(java.lang.String filename,
int width,
int height)
write out the current canvas to a png file. TODO: bug 3113441: this has issues with the size. It's coded to get the size from the DOM, but if it is fitted and has a container it must get size from the container. Use writeToPng( filename, width, height ) instead for now. See writeToPdf(String filename), which appears to have a fix for this that would affect how this is resolved.
Parameters:
• filename - The name of a local file
• width - the width in pixels of the png
• height - the height in pixels of the png
## 3.11. writeToPng
public static void writeToPng(java.io.OutputStream out)
write out the current canvas to stdout. This is introduced to support servers. TODO: this has issues with the size. See writeToPng(filename).
Parameters:
• OutputStream - out
## 3.12. writeToPdf
public static void writeToPdf(java.lang.String filename)
write out the current canvas to a pdf file. TODO: this has issues with the size. See writeToPng(filename). It looks like this might be handled here
Parameters:
• filename - the local file to write the file.
## 3.13. writeToPdf
public static void writeToPdf(java.io.OutputStream out)
write out the current canvas to a pdf to the output stream. This is to support servers. TODO: this has issues with the size. See writeToPng(filename). It looks like this might be handled here
Parameters:
• out - the OutputStream
## 3.14. writeToBufferedImage
public static java.awt.image.BufferedImage writeToBufferedImage(Application applicationIn)
creates a BufferedImage from the provided DOM. This blocks until the image is ready. TODO: this has issues with the size. See writeToPng(filename). It looks like this might be handled here
Parameters:
• applicationIn -
## 3.15. getTimeRangesFor
public static java.lang.String[] getTimeRangesFor(java.lang.String surl,
java.lang.String timeRange,
java.lang.String format)
return an array of URLs that match the spec for the time range provided. For example,
uri= 'http://cdaweb.gsfc.nasa.gov/istp_public/data/polar/hyd_h0/$Y/po_h0_hyd_$Y$m$d_v01.cdf?ELECTRON_DIFFERENTIAL_ENERGY_FLUX'
xx= getTimeRangesFor( uri, '2000-jan', '$Y-$d-$m' ) for x in xx: print x This is also available in the Data Source Context (.jyds files). Parameters: • surl - an Autoplot uri with an aggregation specifier. • timeRange - a string that is parsed to a time range, such as "2001" • format - format for the result, such as "%Y-%m-%d" Returns: • a list of URLs without the aggregation specifier. ## 3.16. generateTimeRanges public static java.lang.String[] generateTimeRanges(java.lang.String spec, java.lang.String srange) throws java.text.ParseException Given a spec to format timeranges and a range to contain each timerange, produce a list of all timeranges covering the range formatted with the spec. For example, generateTimeRanges( "%Y-%m-%d", "Jun 2009" ) would result in 2009-06-01, 2009-06-02, ..., 2009-06-30. This is also available in the Data Source Context (.jyds files). Parameters: • spec - such as "%Y-%m". Note specs like "%Y%m" will not be parsable. • srange - range limiting the list, such as "2009" Returns: • a string array of formatted time ranges, such as [ "2009-01", "2009-02", ..., "2009-12" ] ## 3.17. setTitle public static void setTitle(java.lang.String title) set the title of the focus plot. Parameters: • title - ## 3.18. createGui public static void createGui() create a model with a GUI presentation layer. If the GUI is already created, then this does nothing. ## 3.19. getApplicationModel public static ApplicationModel getApplicationModel() returns the internal application model (the object that does all the business). This provides access to the internal model for power users. Note the applicationModel provides limited access, and the DOM now provides full access to the application. Returns: • ApplicationModel object ## 3.20. isModelInitialized public static boolean isModelInitialized() provide way to see if the model is already initialized (e.g. for clone application) Returns: • true is the model is already initialized. ## 3.21. bind public static void bind(java.lang.Object src, java.lang.String srcProp, java.lang.Object dst, java.lang.String dstProp) binds two bean properties together. Bindings are bidirectional, but the initial copy is from src to dst. In MVC terms, src should be the model and dst should be a view. The properties must fire property change events for the binding mechanism to work. Example: bind( dom.plots[0], "title", dom.plots[0].getYaxis(), "label" ) dom.plots[0].title= 'My Data' Parameters: • src - java bean such as model.getPlotDefaults() • srcProp - a property name such as "title" • dst - java bean such as model.getPlotDefaults().getXAxis() • dstProp - a property name such as "label" ## 3.22. dumpToQStream public static void dumpToQStream(org.virbo.dataset.QDataSet ds, java.io.OutputStream out, boolean ascii) serializes the dataset to a QStream, a self-documenting, streaming format useful for moving datasets. ds= getDataSet('http://autoplot.org/data/somedata.cdf?BGSEc') from java.lang import System dumpToQStream( ds, System.out, True ) Parameters: • ds - The dataset to stream. Note all schemes should be streamable, but some bugs exist that may prevent this. • output - stream, such as "System.out" • ascii - use ascii transfer types, otherwise binary are used. ## 3.23. dumpToDas2Stream public static void dumpToDas2Stream(org.virbo.dataset.QDataSet ds, boolean ascii) serializes the dataset to a das2stream, a well-documented, open, streaming data format. (that's a joke.) Das2Streams are the legacy stream format used by the Plasma Wave Groups's server, and can serialize a limited set of QDataSets. QStreams were introduced to allow streaming of any QDataSet, see dumpToQStream. Currently, to keep the channel open, the stream is created in a buffer and then the buffer is sent. TODO: write a stream-producing code that doesn't close the output stream. (TODO: does it still do this?) Parameters: • ds - QDataSet • ascii - use ascii transfer types, otherwise binary are used. ## 3.24. dumpToDas2Stream public static void dumpToDas2Stream(org.virbo.dataset.QDataSet ds, java.lang.String file, boolean ascii) serializes the dataset to a das2stream, a well-documented, open, streaming data format. (that's a joke.) Currently, to keep the channel open, the stream is created in a buffer and then the buffer is sent. TODO: write a stream-producing code that doesn't close the output stream. Parameters: • ds - • file - the file target for the stream. • ascii - use ascii transfer types. ## 3.25. formatDataSet public static void formatDataSet(org.virbo.dataset.QDataSet ds, java.lang.String file) throws java.lang.Exception Export the data into a format implied by the filename extension. See the export data dialog for additional parameters available for formatting. For example: ds= getDataSet('http://autoplot.org/data/somedata.cdf?BGSEc') formatDataSet( ds, 'vap+dat:file:/home/jbf/temp/foo.dat?tformat=minutes&format=6.2f') Parameters: • ds - QDataSet • file - local file name that is the target, and optionally contains format-specific parameters, described below. ### 3.25.1. parameters These parameters are format-specific, and are appended with file?params. For example, formatDataSet(ripplesTimeSeries(20),'/tmp/data.dat?header=rich') #### 3.25.1.1. ascii • header=rich format with rich ascii headers • tformat=Millisecond the resolution of timetags • format=%5.2f for format specifier for non-timetags. #### 3.25.1.2. CDF • type=float use floats to store data • timeType=epoch use legacy Epoch data, instead of TT2000. • append=T open and insert the data, so that multiple parameters live in one file, instead of writing a new file. #### 3.25.1.3. qds, d2s • type=binary use binary instead of ascii transfer types. #### 3.25.1.4. HDF5 • type=float use floats to store data • !!TODO: does ISTP work? #### 3.25.1.5. wav • scale=T scale the data to use the dynamic range #### 3.25.1.6. bin • type=float use floats to store data • byteOrder=big use big-endian floats instead of little-endian. ## 3.26. getDocumentModel public static Application getDocumentModel() get the document model (DOM). This may initialize the model, in which case defaults like the cache directory are set. ## 3.27. save public static void save(java.lang.String filename) save the current state as a .vap file Parameters: • filename - local file where the vap should be saved. ## 3.28. getCompletions public static java.lang.String[] getCompletions(java.lang.String file) return a list of completions. I was talking to Tom N. who was looking for this to get a list of CDF variables, and realized this would be useful in the IDL context as well as python scripts. This will perform the completion for where the carot is at the end of the string. This is also available in the Data Source Context (.jyds files). Parameters: Returns: • list of completions, containing the entire URI. ## 3.29. load public static void load(java.lang.String filename) load the .vap file. This is implemented by calling plot on the URI. Parameters: • filename - the vap file to load ## 3.30. reset public static void reset() reset the application to its initial state. ## 3.31. setLayout public static void setLayout( nrows ) public static void setLayout( nrows, ncolumns ) reset to a multi-plot layout. ## 3.32. fixLayout public static void fixLayout( ) remove empty gaps in the plot. ## 3.33. close protected static void close() called when the application closes so if we reopen it will be in a good state. # 4. Ops These are QDataSet operators imported from org.virbo.dsops.Ops. Parenthesis enclosing an operator indicate the operator is overloaded to this function, so for example a.add(b) is usually replaced with a+b. Note that there's a bit of "coersion" supported to make two dataset arguments have the same geometry. For example, you can add two 10-element rank 1 datasets, but you can also add a 10-element rank 1 dataset to a rank 0 dataset. In other words, [1,2,3,4]+[1,1,1,1] is equivalent to [1,2,3,4] + 1 because the 1 is coerced to [1,1,1,1]. Also Python arrays and numbers are automatically converted to QDataSets. Note a dataset must sometimes be explicitly made, and to force python to coerce to it, use the dataset() command to convert it: print [1,2,3,4] print dataset([1,2,3,4]) print total(dataset([1,2,3,4])) print total(2 + dataset([1,2,3,4]) ) #(This is in the autoplot2011 version only.) ## 4.1. abs public static QDataSet abs(QDataSet ds1) element-wise abs. For vectors, this returns the length of each element. Note jython conflict needs to be resolved. Parameters: • ds1 - Returns: ## 4.2. accum public static QDataSet accum(QDataSet accumDs, QDataSet ds) return an array that is the running sum of each element in the array, starting with the value accum. Result[i]= accum + total( ds[0:i+1] ) Parameters: • accum - the initial value of the running sum. Last value of Rank 0 or Rank 1 dataset is used, or may be null. • ds - each element is added to the running sum Returns: • the running of each element in the array. See Also: • diff ## 4.3. accum public static QDataSet accum(QDataSet ds) return an array that is the running sum of each element in the array, starting with the value accum. Result[i]= total( ds[0:i+1] ) Parameters: • ds - each element is added to the running sum Returns: • the running sum of each element in the array. See Also: • diff ## 4.4. acos public static QDataSet acos(QDataSet ds) element-wise arccos. Parameters: • ds - Returns: ## 4.5. add (+) public static QDataSet add(QDataSet ds1, QDataSet ds2) add the two datasets have the same geometry. Parameters: • ds1 - • ds2 - Returns: ## 4.6. and public static QDataSet and(QDataSet ds1, QDataSet ds2) element-wise logical and function. non-zero is true, zero is false. Parameters: • ds1 - • ds2 - Returns: ## 4.7. asin public static QDataSet asin(QDataSet ds) element-wise arcsin. Parameters: • ds - Returns: ## 4.8. atan public static QDataSet atan(QDataSet ds) element-wise atan. Parameters: • ds - Returns: ## 4.9. atan2 public static QDataSet atan2(QDataSet dsy, QDataSet dsx) element-wise atan2, 4-quadrant atan. Parameters: • dsy - • dsx - Returns: ## 4.10. autoHistogram public static QDataSet autoHistogram(QDataSet ds) One pass auto-scaling histogram. See also histogram. Parameters: • ds - Returns: ## 4.11. boxcar public static QDataSet boxcar(QDataSet ds, int size) run boxcar average over the dataset, returning a dataset of same geometry. Points near the edge are simply copied from the source dataset. The result dataset contains a property "weights" that is the weights for each point. Parameters: • ds - rank 1 dataset • size - size of the boxcar Returns: • dataset ## 4.12. bundle public static QDataSet bundle(QDataSet ds1, QDataSet ds2) bundle the two datasets, adding if necessary a bundle dimension. This will try to bundle on the second dimension, unlike join. This will also isolate the semantics of bundle dimensions as it's introduced. Parameters: • ds1 - • ds2 - Returns: ## 4.13. ceil public static QDataSet ceil(QDataSet ds1) element-wise ceil function. Parameters: • ds1 - Returns: ## 4.14. circle public static QDataSet circle(double radius) public static QDataSet circle(String radius) public static QDataSet circle(QDataSet radius) return a dataset with X and Y forming a circle, introduced as a convenient way to indicate planet location. Parameters: • radius the radius, a rank 0 dataset, maybe with units, or a string parsed into a rank 0 dataset. Returns: • QDataSet that when plotted is a circle. ## 4.15. collapse0 public static QDataSet collapse0(QDataSet ds) reduce the dataset rank by averaging data over the 0th dimension. If there is fill in the reduction, then the result is fill. ## 4.16. collapse1 reduce the dataset rank by averaging data over the 1st dimension. If there is fill in the reduction, then the result is fill. ## 4.17. collapse2 reduce the dataset rank by averaging data over the 2nd dimension. If there is fill in the reduction, then the result is fill. ## 4.18. collapse3 reduce the dataset rank by averaging data over the 3rd dimension. If there is fill in the reduction, then the result is fill. ## 4.19. concatenate public static QDataSet concatenate(QDataSet ds1, QDataSet ds2) concatenates the two datasets together, appending the datasets on the zeroth dimension. The two datasets must be QUBES have similar geometry on the higher dimensions. If one of the datasets is rank 0 and the geometry of the other is rank 1, then the lower rank dataset is promoted before appending. If the first dataset is null and the second is non-null, then return the second dataset. This was briefly known as "join." Parameters: • ds1 - null, or a rank n dataset with dimensions any1,a1,a2,... • ds2 - A rank n dataset with dimensions any2,a1,a2,... Returns: • a rank n dataset with dimensions any1+any2,a1,a2, or just ds2 when ds1 is null. See also: join ## 4.20. Constants PI = 3.141592653589793 E = 2.718281828459045 ## 4.21. contour public static QDataSet contour(QDataSet tds, QDataSet vv) contour the data in rank 2 table tds at rank 0 vv. The result is a rank 2 bundle of [:,'x,y,z'] where i is the contour number. The result will have DEPEND_0 be an monotonically increasing sequence with jumps indicating new contours. Parameters: • tds - rank 2 table • vv - rank 2 bundle Returns: ## 4.22. convertUnitsTo public static DatumRange convertUnitsTo(DatumRange dr, Units u) public static Datum convertUnitsTo(Datum d, Units u) public static QDataSet convertUnitsTo(QDataSet ds, Units u) convert the data units to the given units, which must be convertible. Parameters: • ds - datum, datum range, or dataset, e.g. '5 to 50MHz' • vv - the new units. e.g. 'Hz' Returns: • data in the new units, e.g. '5000000 to 50000000 Hz' ## 4.23. copysign public static QDataSet copysign(QDataSet magnitude, QDataSet sign) returns a dataset with the same geometry, but having the floating-point magnitude of the first argument with the sign of the second argument. Parameters: • magnitude - dataset containing magnitude • sign - dataset with compatible geometry Returns: • Dataset with the magnitude and sign combined. See Also: • signum ## 4.24. cos public static QDataSet cos(QDataSet ds) element-wise cos. Parameters: • ds - Returns: ## 4.25. cosh public static QDataSet cosh(QDataSet ds) element-wise cosh. Parameters: • ds - Returns: ## 4.26. createEvent public static QDataSet createEvent(java.lang.String timeRange, int rgbcolor, java.lang.String annotation) tool for creating ad-hoc events datasets. Parameters: • timeRange - a timerange string like "2010-01-01" or "2010-01-01/2010-01-10" or "2010-01-01 through 2010-01-09" • rgbcolor - and RGB color like 0xFF0000 (red), 0x00FF00 (green), or 0x0000FF (blue), • annotation - label for event, possibly including granny codes. Returns: • a rank 2 n by 4 QDataSet with startTime, duration, rgbColor, annotation in each record. ## 4.27. createEvent public static QDataSet createEvent(QDataSet append, java.lang.String timeRange, int rgbcolor, java.lang.String annotation) tool for creating ad-hoc events datasets. Parameters: • append - null or a dataset to append the result. This events dataset must have [starttime, endtime, RBG color, string] for each record. • timeRange - a timerange like "2010-01-01" or "2010-01-01/2010-01-10" or "2010-01-01 through 2010-01-09" • rgbcolor - and RGB color like 0xFF0000 (red), 0x00FF00 (green), or 0x0000FF (blue), • annotation - label for event, possibly including granny codes. Returns: • a rank 2 QDataSet n by 4 with startTime, duration, rgbColor, annotation in each record. ## 4.28. dblarr public static QDataSet dblarr(int len0) create a dataset filled with zeros. Parameters: • len0 - Returns: ## 4.29. dblarr public static QDataSet dblarr(int len0, int len1) ## 4.30. dblarr public static QDataSet dblarr(int len0, int len1, int len2) ## 4.31. dependsOn public static MutablePropertyDataSet dependsOn(QDataSet ds, int dim, QDataSet dep0) declare that the dataset is a dependent parameter of an independent parameter. This isolates the QDataSet semantics, and verifies correctness. Parameters: • ds - • dim - dimension to declare dependence: 0,1,2. • dep0 - Returns: ## 4.32. detrend public static QDataSet detrend(QDataSet yy, int size) remove D/C and low-frequency components from the data by subtracting out the smoothed data with a boxcar of the given size. Points on the end are zero. Parameters: • yy - rank 1 dataset • size - size of the boxcar Returns: • dataset ## 4.33. dindgen public static QDataSet dindgen(int len0) returns rank 1 dataset with values [0,1,2,...,len0-1] Parameters: • size - the number of elements Returns: • rank 1 dataset with values [0,1,2,...,len0-1] ## 4.34. dindgen public static QDataSet dindgen(int len0, int len1) returns rank 2 dataset with values increasing [ [0,1,2], [ 3,4,5] ] Parameters: • len0 - • len1 - Returns: ## 4.35. dindgen public static QDataSet dindgen(int len0, int len1, int len2) returns rank 3 dataset with values increasing ( [ [ [0,1,2], [ 3,4,5] ], [ [6,7,8] ], ...] ) Parameters: • len0 - • len1 - • len2 - Returns: • Rank 3 dataset with elements increasing. ## 4.36. diff public static QDataSet diff(QDataSet ds) return array that is the differences between each successive pair in the dataset. Result[i]= ds[i+1]-ds[i], so that for an array with N elements, an array with N-1 elements is returned. DEPEND_0 will contain the average of the two points. Parameters: • ds - a rank 1 dataset with N elements. Returns: • a rank 1 dataset with N-1 elements. See Also: • accum ## 4.37. dimensionCount public static int dimensionCount(QDataSet dss) returns the number of physical dimensions of a dataset. JOIN, BINS do not increase dataset dimensionality. DEPEND increases dimensionality by dimensionality of DEPEND ds. BUNDLE increases dimensionality by N where N is the number of bundled datasets. Note this includes implicit dimensions taken by the primary dataset. Z(time,freq)->3 rand(20,20)->3 B_gsm(20,[X,Y,Z])->4 Parameters: • ds - Returns: • the number of dimensions occupied by the data. ## 4.38. div public static QDataSet div(QDataSet ds1, QDataSet ds2) element-wise div of two datasets with compatible geometry. Parameters: • ds - Returns: ## 4.39. divide (/) public static QDataSet divide(QDataSet ds1, QDataSet ds2) element-wise divide of two datasets with compatible geometry. Parameters: • ds - Returns: ## 4.40. eq public static QDataSet eq(QDataSet ds1, QDataSet ds2) element-wise equality test. 1.0 is returned where the two datasets are equal. Fill is returned where either measurement is invalid. Parameters: • ds - Returns: ## 4.41. equivalent public static boolean equivalent(QDataSet ds1, QDataSet ds2) returns true if and only if the dataset values are equivalent. Note this may promote rank, etc. A rank 0 dataset is promoted to a rank 1 by replicating the value, so for example the rank 0 "4" is equivalent to "4,4,4". Parameters: • ds1 - rank n dataset • ds2 - rank m dataset Returns: • true if the two are equivalent. ## 4.42. exp public static QDataSet exp(QDataSet ds) element-wise exponentiate e**x. Parameters: • ds - Returns: ## 4.43. exp10 public static QDataSet exp10(QDataSet ds) element-wise exponentiate 10**x. Parameters: • ds - Returns: ## 4.44. expm1 public static QDataSet expm1(QDataSet ds) returns ex -1. Note that for values of x near 0, the exact sum of expm1(x) + 1 is much closer to the true result of ex than exp(x). (TODO lost info in wiki) Parameters: • ds - Returns: ## 4.45. extent public static QDataSet extent(QDataSet ds) returns a two element, rank 1 dataset containing the extent of the data. Note this accounts for DELTA_PLUS, DELTA_MINUS properties. The property QDataSet.SCALE_TYPE is set to lin or log. The property count is set to the number of valid measurements. TODO: this could use MONOTONIC, but it doesn't. DELTA_PLUS, DELTA_MINUS make that more difficult. Parameters: • ds - rank N dataset. Returns: • two element, rank 1 "bins" dataset. See Also: • DataSetUtil.rangeOfMonotonic( QDataSet ds ). ## 4.46. extent public static QDataSet extent(QDataSet ds, QDataSet range) returns a two element, rank 1 dataset containing the extent of the data. Note this accounts for DELTA_PLUS, DELTA_MINUS properties. The property QDataSet.SCALE_TYPE is set to lin or log. The property count is set to the number of valid measurements. This checks the monotonic property, and uses it to avoid iterating through the data if available. Parameters: • ds - • range, - if non-null, return the union of this range and the extent. This must not contain fill! Returns: • two element, rank 1 "bins" dataset. ## 4.47. fft public static QDataSet fft(QDataSet ds) Performs an FFT on the provided rank 1 dataset. A rank 2 dataset of complex numbers is returned. Parameters: • ds - a rank 1 dataset. Returns: • a rank 2 dataset of complex numbers. public static QDataSet fft(QDataSet ds, QDataSet window, int stepFraction, ProgressMonitor mon ) perform ffts on the waveform as we do with fftPower, but keep real and imaginary components. Parameters: • ds - the waveform rank 1,2,or 3 dataset. • window - the window function, like ones(1024) or windowFunction( FFTFilterType.Hanning, 1024 ). This is used to infer window size. • stepFraction - step this fraction of the window size. 1 is no overlap, 2 is 50% overlap, 4 is 75% overlap, etc. • mon - progress monitor. Returns: • result[ntime,nwindow,2] ## 4.48. fftFilter public static QDataSet fftFilter(QDataSet ds, int len, Ops.FFTFilterType filt) ## 4.49. fftPower public static QDataSet fftPower(QDataSet ds, int len, org.das2.util.monitor.ProgressMonitor mon) create a power spectrum on the dataset by breaking it up and doing ffts on each segment. right now only rank 2 data is supported, but there is no reason that rank 1 shouldn't be supported. Looks for DEPEND_1.USER_PROPERTIES.FFT_Translation, which should be a rank 0 or rank 1 QDataSet. If it is rank 1, then it should correspond to the DEPEND_0 dimension. Parameters: • ds - rank 2 dataset ds(N,M) with M>len • len - the number of elements to have in each fft. • mon - a ProgressMonitor for the process Returns: • rank 2 fft spectrum ## 4.50. fftPower public static QDataSet fftPower(QDataSet ds) returns the power spectrum of the waveform. Positive frequencies are returned for DEPEND_0, and square of the magnitude is returned for the values. Parameters: • ds - rank 1 waveform or rank 2 array of waveforms Returns: ## 4.51. fftWindow public static QDataSet fftWindow(QDataSet ds, int len) perform ffts on the rank 1 dataset to make a rank2 spectrogram. Parameters: • ds - rank 1 dataset • len - the window length Returns: • rank 2 dataset. ## 4.52. findex public static QDataSet findex(QDataSet uu, QDataSet vv) returns the floating point index of each element of vv within the monotonically increasing dataset uu. The result dataset will have the same geometry as vv. The result will be negative when the element of vv is below the smallest element of uu. The result will be greater than or equal to the length of uu minus one when it is greater than all elements. Parameters: • uu - rank 1 monotonically increasing dataset. • vv - rank N dataset with values in the same physical dimension as uu. Returns: • rank N dataset with the same geometry as vv. ## 4.53. findgen public static QDataSet findgen(int len0) returns rank 1 dataset with values [0,1,2,...] Parameters: • size - Returns: ## 4.54. findgen public static QDataSet findgen(int len0, int len1) returns rank 2 dataset with values increasing [ [0,1,2], [ 3,4,5] ] Parameters: • len0 - • len1 - Returns: ## 4.55. findgen public static QDataSet findgen(int len0, int len1, int len2) returns rank 3 dataset with values increasing Returns: ## 4.56. floor public static QDataSet floor(QDataSet ds1) element-wise ceil function. Parameters: • ds1 - Returns: ## 4.57. fltarr public static QDataSet fltarr(int len0) create a dataset filled with zeros. Parameters: • len0 - Returns: ## 4.58. fltarr public static QDataSet fltarr(int len0, int len1) ## 4.59. fltarr public static QDataSet fltarr(int len0, int len1, int len2) ## 4.60. ge public static QDataSet ge(QDataSet ds1, QDataSet ds2) element-wise function returns 1 where ds1>=ds2. Parameters: • ds1 - • ds2 - Returns: ## 4.61. greaterOf public static QDataSet greaterOf(QDataSet ds1, QDataSet ds2) element-wise function returns the greater of ds1 and ds2. If an element of ds1 or ds2 is fill, then the result is fill. Parameters: • ds1 • ds2 Returns: • the bigger of the two, in the units of ds1. If an element of ds1 or ds2 is fill, then the result is fill. ## 4.62. gt public static QDataSet gt(QDataSet ds1, QDataSet ds2) element-wise function returns 1 where ds1>ds2. Parameters: • ds1 - • ds2 - Returns: ## 4.63. hanning public static QDataSet hanning(QDataSet ds, int len) Hanning filter for use with fftPower. ds= fftPower( hanning( randn( 20480 ), 512 ) ) ## 4.64. histogram public static QDataSet histogram(QDataSet ds, double min, double max, double binSize) returns histogram of dataset, the number of points falling in each bin. Parameters: • ds - • min - • max - • binSize - Returns: ## 4.65. histogram public static QDataSet histogram(QDataSet ds, int binCount) returns a histogram of the dataset, based on the extent and scaletype of the data. See also autoHistogram, which automatically identifies bin width, min and max. Parameters: • ds - • binCount - number of bins Returns: • rank 1 QDataSet that is the histogram. ## 4.66. interpolate public static QDataSet interpolate(QDataSet vv, QDataSet findex) interpolate values from rank 1 dataset vv using fractional indices in rank N findex. For example, findex=1.5 means interpolate the 1st and 2nd indices with equal weight, 1.1 means 90% of the first mixed with 10% of the second. No extrapolation is done, data with findex<0 or findex>(vv.length()-1) are assigned the first or last value. Note there is no check on CADENCE. Note nothing is done with DEPEND_0, presumably because was already calculated and used for findex. Parameters: • vv - rank 1 dataset that is the data to be interpolated. • findex - rank N dataset of fractional indices. Returns: • the result. ## 4.67. interpolate public static QDataSet interpolate(QDataSet vv, QDataSet findex0, QDataSet findex1) interpolate values from rank 2 dataset vv using fractional indices in rank N findex, using bilinear interpolation. Parameters: • vv - rank 2 dataset. • findex0 - rank N dataset of fractional indices for the zeroth index. • findex1 - rank N dataset of fractional indices for the first index. Returns: • rank N dataset ## 4.68. isBundle public static boolean isBundle(QDataSet zds) return true if the dataset is a bundle. It is rank 2 or rank 1, and has the last dimension a bundle dimension. Parameters: • zds - Returns: ## 4.69. isLegacyBundle public static boolean isLegacyBundle(QDataSet zds) return true if DEPEND_1 is set and its units are EnumerationUnits. This was the pre-bundle way of representing a bundle of datasets. It might be supported indefinitely, because it has some nice rules about the data. For example, data must be of the same units since there is no place to put the property. Note the "legacy" status has been removed. This is a fine way to easily create bundle datasets. See BUNDLE_1 for more information about bundle datasets. Parameters: • zds - Returns: • true for DEPEND_1 where units are EnumerationUnits. ## 4.70. join public static QDataSet join(QDataSet ds1, QDataSet ds2) Join two rank N datasets to make a rank N+1 dataset, with the first dimension having two elements. This is the anti-slice operator. If the first dataset is rank N+1 JoinDataset and the other is rank N, then the rank N dataset is added to the rank N+1 dataset. This is underimplemented right now, and can only join two rank N datasets or if the first dataset is the result of a join. Parameters: • ds1 - rank N dataset, or null • ds2 - rank N dataset Returns: • rank N+1 dataset See Also: • slice, concatenate ## 4.71. labels public static QDataSet labels(java.lang.String[] labels, java.lang.String context) create a labels dataset for tagging rows of a dataset. If the context has been used already, including "default", then the EnumerationUnit for the data will be preserved. labels(["red","green","blue"],"default") will always return an equivalent (and comparable) result during a session. Example: dep1= labels( ["X","Y","Z"], "GSM" ) Parameters: • labels - • context - Returns: • rank 1 QDataSet ## 4.72. labels public static QDataSet labels(java.lang.String[] labels) create a labels dataset for tagging rows of a dataset. Example: dep1= labels( ["red","greed","blue"] ) Parameters: • labels - Returns: • rank 1 QDataSet ## 4.73. le public static QDataSet le(QDataSet ds1, QDataSet ds2) element-wise function returns 1 where ds1<=ds2. Parameters: • ds1 - • ds2 - Returns: ## 4.74. lesserOf public static QDataSet lesserOf(QDataSet ds1,QDataSet ds2) element-wise function returns the lesser of ds1 and ds2. If an element of ds1 or ds2 is fill, then the result is fill. Parameters: • ds1 - • ds2 - Returns: the lesser of the two, in the units of ds1. If one of the two elements is fill, then the result is fill. ## 4.75. link public static QDataSet link(QDataSet x, QDataSet y) This is like bundle, but declare the last dataset is dependent on the first one. "link" is like a plot command where link(x,y) would behave like plot(x,y) except you get a dataset back. Parameters: • x - rank 1 dataset • y - rank 1 or rank 2 dataset Returns: ## 4.76. link public static QDataSet link(QDataSet x, QDataSet y, QDataSet z) like bundle, but declare the last dataset is dependent on the first two. Parameters: • x - rank 1 dataset • y - rank 1 dataset • z - rank 1 or 2 dataset. Returns: ## 4.77. linspace public static QDataSet linspace(double min, double max, int len0) return a rank 1 dataset with len0 linearly-spaced values, the first is min and the last is max. Parameters: • min - the first value of the series • max - the last value of the series. • len0 - the number of values Returns: ## 4.78. log public static QDataSet log(QDataSet ds) element-wise natural logarithm. Parameters: • ds - Returns: ## 4.79. log10 public static QDataSet log10(QDataSet ds) element-wise base 10 logarithm. Parameters: • ds - Returns: ## 4.80. lt public static QDataSet lt(QDataSet ds1, QDataSet ds2) element-wise function returns 1 where ds1<ds2. Parameters: • ds1 - • ds2 - Returns: ## 4.81. magnitude public static QDataSet magnitude(QDataSet ds) return the magnitudes of vectors in a rank 2 or greater dataset. The last index must be a cartesian dimension, so it must have a depend dataset either named "cartesian" or having the property CARTESIAN_FRAME. TODO: check this in the 2011 branch, I think it's removed. Parameters: • ds - of Rank N. Returns: • ds of Rank N-1. ## 4.82. medianFilter public static QDataSet medianFilter(QDataSet ds, int size) 1-D median filter with a boxcar of the given size. This is not particularly efficient and would make a nice project for a student. Parameters: • ds - rank 1 dataset. Future implementations may support higher rank data. • size - the boxcar size Returns: • rank 1 dataset ## 4.83. mod public static QDataSet mod(QDataSet ds1, QDataSet ds2) element-wise mod of two datasets with compatible geometry. Parameters: • ds - Returns: ## 4.84. multiply (*) public static QDataSet multiply(QDataSet ds1, QDataSet ds2) element-wise multiply of two datasets with compatible geometry. Parameters: • ds - Returns: ## 4.85. ne public static QDataSet ne(QDataSet ds1, QDataSet ds2) element-wise not equal test. 1.0 is returned where elements are not equal. Fill is returned where either measurement is invalid. Parameters: • ds1 - • ds2 - Returns: ## 4.86. negate (-) public static QDataSet negate(QDataSet ds1) return a dataset with each element negated. Parameters: • ds1 - Returns: ## 4.87. not public static QDataSet not(QDataSet ds1) element-wise logical not function. non-zero is true, zero is false. TODO: This isn't working, use ( ds1.eq(1) ) instead. Parameters: • ds1 - • ds2 - Returns: ## 4.88. ones public static QDataSet ones(int len0) return new dataset filled with ones. Parameters: • len0 - Returns: ## 4.89. ones public static QDataSet ones(int len0, int len1) return new dataset filled with ones. Parameters: • len0 - Returns: ## 4.90. ones public static QDataSet ones(int len0, int len1, int len2) return new dataset filled with ones. Parameters: • len0 - Returns: ## 4.91. or public static QDataSet or(QDataSet ds1, QDataSet ds2) element-wise logical or function. returns 1 where ds1 is non-zero or ds2 is non-zero. Parameters: • ds1 - • ds2 - Returns: ## 4.92. outerProduct public static QDataSet outerProduct(QDataSet ds1, QDataSet ds2) returns outerProduct of two rank 1 datasets, a rank 2 dataset with elements R[i,j]= ds1[i] * ds2[j]. Parameters: • ds1 - Rank 1 dataset having length M • ds2 - Rank 1 dataset having length N Returns: • Rank 2 dataset lengths M,N ## 4.93. pow (**) public static QDataSet pow(QDataSet ds1, QDataSet pow) element-wise pow (** in FORTRAN, ^ in IDL) of two datasets with the same geometry. Parameters: • ds1 - • pow - dataset or scalar Returns: ## 4.94. rand public static QDataSet rand(int len0) return returns a rank 1 dataset of random uniform numbers from [0,1]. ## 4.95. randn public static QDataSet randn(int len0) return returns a rank 1 dataset of random numbers of a guassian (normal) distribution. ## 4.96. randomn public static QDataSet randomn(long seed, int len0) returns a rank 1 dataset of random numbers of a guassian (normal) distribution. System.currentTimeMillis() may be used for the seed. Parameters: • seed - • len0 - Returns: ## 4.97. randomu public static QDataSet randomu(long seed, int len0) returns a rank 1 dataset of random numbers of a uniform distribution. System.currentTimeMillis() may be used for the seed. Parameters: • seed - • len0 - Returns: ## 4.98. rebin public static DDataSet rebin(QDataSet ds, QDataSet newTags0, QDataSet newTags1) returns a dataset with tags specified by newTags • ds a rank 2 dataset. This can be a bundle dataset of [i;X,Y,Z] • newTags0 rank 1 monotonic dataset • newTags1 rank 1 monotonic dataset Returns: • rank 2 dataset with newTags0 for the DEPEND_0 tags, newTags1 for the DEPEND_1 tags. ## 4.99. rebinBundle public static DDataSet rebinBundle(QDataSet ds, QDataSet dep0, QDataSet dep1) takes rank 2 bundle (x,y,z) and averages it into table z(x,y). This is similar to what happens in the spectrogram routine. Parameters: • ds - rank 2 bundle(x,y,z) • dep0 - the depend0 for the result • dep1 - the depend1 for the result Returns: • rank 2 dataset of z averages with depend_0 and depend_1. ## 4.100. reduceMax public static QDataSet reduceMax(QDataSet ds, int dim) reduce the dataset's rank by reporting the max of all the elements along a dimension. Only QUBEs are supported presently. Parameters: • ds - rank N qube dataset. • dim - zero-based index number. Returns: ## 4.101. reduceMean public static QDataSet reduceMean(QDataSet ds, int dim) reduce the dataset's rank by reporting the mean of all the elements along a dimension. Only QUBEs are supported presently. Parameters: • ds - rank N qube dataset. • dim - zero-based index number. Returns: ## 4.102. reduceMin public static QDataSet reduceMin(QDataSet ds, int dim) reduce the dataset's rank by reporting the min of all the elements along a dimension. Only QUBEs are supported presently. Parameters: • ds - rank N qube dataset. • dim - zero-based index number. Returns: ## 4.103. reform public static QDataSet reform(QDataSet ds) Reshape the dataset to remove the first dimension with length 1, reducing its rank by 1. Dependencies are also preserved. Parameters: • ds - a rank r dataset [1,m], [1,n,m], etc. Returns: • a rank r-1 dataset [m], [n,m], etc. ## 4.104. reform public static QDataSet reform(QDataSet ds, int[] qube) change the dimensionality of the elements of the QUBE dataset. For example, convert [1,2,3,4,5,6] to [[1,2],[3,4],[5,6]]. params: • ds • qube the dimensions of the result dataset. returns: • a new dataset with the specified dimensions, and the properties (e.g. UNITS) of the input dataset. ## 4.105. replicate public static WritableDataSet replicate(double val, int len0) returns rank 1 dataset with value Parameters: • val - fill the dataset with this value. • len0 - Returns: ## 4.106. replicate public static WritableDataSet replicate(double val, int len0, int len1) returns rank 2 dataset filled with value Parameters: • val - fill the dataset with this value. • len0 - • len1 - Returns: ## 4.107. replicate public static WritableDataSet replicate(double val, int len0, int len1, int len2) returns rank 3 dataset with filled with value. Parameters: • val - fill the dataset with this value. • len0 - • len1 - • len2 - Returns: ## 4.108. replicate public static WritableDataSet replicate(float val, int len0) returns rank 1 dataset with value Parameters: • val - fill the dataset with this value. • len0 - Returns: ## 4.109. replicate public static WritableDataSet replicate(float val, int len0, int len1) returns rank 2 dataset filled with value Parameters: • val - fill the dataset with this value. • len0 - • len1 - Returns: ## 4.110. replicate public static WritableDataSet replicate(float val, int len0, int len1, int len2) returns rank 3 dataset with filled with value. Parameters: • val - fill the dataset with this value. • len0 - • len1 - • len2 - Returns: ## 4.111. rescaleRange public static QDataSet rescaleRange(QDataSet dr, double min, double max) returns rank 1 QDataSet range relative to the range in "dr", where 0. is the minimum, and 1. is the maximum. For example rescaleRange(ds,1,2) is scanNext, rescaleRange(ds,-1,0) is scanPrevious, rescaleRange(ds,-0.5,1.5) is zoomOut. Parameters: • dr - a QDataSet with "min,max" for BINS_0 and with nonzero width. • min - the new min normalized with respect to this range. 0. is this range's min, 1 is this range's max, 0 is min-width. • max - the new max width normalized with respect to this range. 0. is this range's min, 1 is this range's max, 0 is min-width. Returns: • new rank 1 QDataSet range. ## 4.112. residuals public static QDataSet residuals(QDataSet ds, int boxcarSize) returns number of stddev from adjacent data. • ds, rank 1 dataset. • boxcarSize Returns: • QDataSet ## 4.113. reverse public static QDataSet reverse(QDataSet ds) returns the reverse of the rank 1 dataset. Parameters: • ds - Returns: ## 4.114. ripples public static QDataSet ripples( int len0 ) rank 1 dataset for demos. It contains fill at index 13. Parameters: • len0 - size of the first dimension Returns: public static QDataSet ripples( int len0, int len1 ) rank 2 dataset for demos. Parameters: • len0 - size of the first dimension • len1 - size of the second dimension Returns: public static QDataSet ripples( int len0, int len1, int len2 ) rank 3 dataset for demos. Parameters: • len0 - size of the first dimension • len1 - size of the second dimension • len2 - size of the third dimension Returns: public static QDataSet ripples( int len0, int len1, int len2, int len3 ) rank 4 dataset for demos. Parameters: • len0 - size of the first dimension • len1 - size of the second dimension • len2 - size of the third dimension • len3 - size of the fourth dimension Returns: ## 4.115. ripplesSpectrogramTimeSeries return fake position data for testing result is rank 2 bundle [len,27] Parameters: • len - Returns: ## 4.116. ripplesTimeSeries public static QDataSet ripplesTimeSeries(int len) return fake rank 1 data timeseries for testing Parameters: • len - the number of records Returns: • rank 1 time series. (like density(time)) ## 4.117. ripplesWaveformTimeSeries return fake waveform data for testing result is rank 2 bundle [len,512] Parameters: • len - number of 512-element waveforms. Returns: • rank 2 dataset with record for the zeroth dimension and offset for the second. ## 4.118. safeName public static java.lang.String safeName(java.lang.String suggest) Make a Java-style identifier from the provided string, which will only include a-z, A-Z, 0-9 (though not the first), and _. For example, "a>b" becomes agtb. Parameters: • suggest - A string suggesting a name. The non-compliant characters are replaced. for example Returns: • a string guarenteed to be useful as an identifier. ## 4.119. sawtooth public static QDataSet sawtooth(QDataSet t) generates a sawtooth from the tags, where a peak occurs with a period 2*PI. All values of T should be ge zero. TODO: I think there should be a modp function that is always positive. (-93 % 10 ->7 though...) Parameters: • t - time Returns: • /|/|/| ## 4.120. shuffle public static QDataSet shuffle(QDataSet ds) returns a rank 1 dataset of indices that shuffle the rank 1 dataset ds s= shuffle( ds ) dsShuffled= ds[s] Parameters: • ds - rank 1 dataset Returns: • rank 1 dataset of integer indices. ## 4.121. signum public static QDataSet signum(QDataSet ds1) returns the signum function of the argument; 0.0 if the argument is zero, 1.0 if the argument is greater than zero, -1.0 if the argument is less than zero. Parameters: • ds1 - Returns: • -1.0, 0.0, or 1.0 to indicate the sign. See Also: • copysign ## 4.122. sin public static QDataSet sin(QDataSet ds) element-wise sin. Parameters: • ds - Returns: ## 4.123. sinh public static QDataSet sinh(QDataSet ds) element-wise sinh. Parameters: • ds - Returns: ## 4.124. smooth public static QDataSet smooth(QDataSet ds, int size) run boxcar average over the dataset, returning a dataset of same geometry. Points near the edge are simply copied from the source dataset. The result dataset contains a property "weights" that is the weights for each point. Parameters: • ds - a rank 1 dataset of size N • size - the number of adjacent bins to average Returns: • rank 1 dataset of size N ## 4.125. smoothFit public static QDataSet smoothFit(QDataSet xx, QDataSet yy, int size) run boxcar average over the dataset, returning a dataset of same geometry. Points near the edge are fit to a line and replaced. The result dataset contains a property "weights" that is the weights for each point. Parameters: • xx - a rank 1 dataset of size N • yy - a rank 1 dataset of size N • size - the number of adjacent bins to average. If size is greater than yy.length, size is reset to yy.length. Returns: • rank 1 dataset of size N ## 4.126. sort public static QDataSet sort(QDataSet ds) returns a rank 1 dataset of indices that sort the rank 1 dataset ds. This is not the dataset sorted. For example: ds= randn(2000) s= sort( ds ) dsSorted= ds[s] Parameters: • ds - rank 1 dataset Returns: • rank 1 dataset of indices that sort the input dataset. ## 4.127. sqrt public static QDataSet sqrt(QDataSet ds) element-wise sqrt. Parameters: • ds - Returns: ## 4.128. square public static QDataSet square(QDataSet t) generates a square from the tags, where a the signal is 1 from 0-PI, 0 from PI-2*PI, etc. Parameters: • t - time Returns: • square wave with a period of 2 PI. ## 4.129. subtract (-) public static QDataSet subtract(QDataSet ds1, QDataSet ds2) subtract one dataset from another. Parameters: • ds1 - • ds2 - Returns: ## 4.130. taggen public static MutablePropertyDataSet taggen(double base, double dcadence, int len0, org.das2.datum.Units units) creates tags. First tag will be start and they will increase by cadence. Units specifies the units of each tag. Parameters: • start - • cadence - • len0 - • units - Returns: ## 4.131. tan public static QDataSet tan(QDataSet ds) element-wise tan. Parameters: • ds - Returns: ## 4.132. tanh public static QDataSet tanh(QDataSet ds) element-wise tanh. Parameters: • ds - Returns: ## 4.133. timegen public static QDataSet timegen(java.lang.String baseTime, java.lang.String cadence, int len0) returns rank 1 dataset with values [0,1,2,...] Parameters: • baseTime - e.g. "2003-02-04T00:00" • cadence - e.g. "4.3 sec" "1 day" • len0 - the number of elements. Returns: • rank 1 dataset ## 4.134. toDegrees public static QDataSet toDegrees(QDataSet ds) ## 4.135. toRadians public static QDataSet toRadians(QDataSet ds) ## 4.136. toTimeDataSet public static QDataSet toTimeDataSet( QDataSet years, QDataSet mons, QDataSet days, QDataSet hour, QDataSet minute, QDataSet second, QDataSet nano ) returns a rank 1 dataset of timetags, by adding the components. Any of the components can be null, except for years and days. Parameters: • years - the years. (2010) Less than 100 is interpreted as 19xx. • mons - the months (1..12), or null. If null, then days are day of year • days - the day of month (1..28) or day of year. • hour - null or the hours of the day. • minute - null or the minutes of the day • second - null or the seconds of the day • nano - null or the nanoseconds (1e-9) of the day Returns: • a rank 1 dataset with Units.us2000 (non-leap microseconds since 2000-01-01T00:00). Be sure to check the units as a future version may change. ## 4.137. total public static double total(QDataSet ds) return the total of all the elements in the dataset, returning a rank 0 dataset. If there are invalid measurements, then fill is returned. Does not support BINS or BUNDLE dimensions. Parameters: • ds - Returns: • the unweighted total of the dataset, or -1e31 if fill was encountered. ## 4.138. total public static QDataSet total(QDataSet ds, int dim) reduce the dataset's rank by totalling all the elements along a dimension. Only QUBEs are supported presently. Parameters: • ds - rank N qube dataset. N=1,2,3,4 • dim - zero-based index number. Returns: ## 4.139. transpose public static QDataSet transpose(QDataSet ds) Transpose the rank 2 dataset. ## 4.140. uniqValues public static QDataSet uniqValues(QDataSet ds, QDataSet sort) return the unique elements from the dataset. If sort is null (jython None), then the dataset is assumed to be monotonic, and only repeating values are coalesced. If sort is non-null, then it is the result of the function "sort" and should be a rank 1 list of indices that sort the data. renamed uniqValues from uniq to avoid confusion with the IDL command. Parameters: • ds - • sort - Returns: ## 4.141. valid public static QDataSet valid(QDataSet ds) returns a dataset with zero where the data is invalid, and positive non-zero where the data is valid. (This just returns the weights plane of the dataset.) r= where( valid( ds ) ) Parameters: • ds - a rank N dataset that might have FILL_VALUE, VALID_MIN or VALID_MAX set. Returns: • a rank N dataset with the same geometry, with zeros where the data is invalid and >0 where the data is valid. ## 4.142. where public static QDataSet where(QDataSet ds) returns a dataset containing the indices of where the dataset is non-zero. For a rank 1 dataset, returns a rank 1 dataset with indices for the values. For a higher rank dataset, returns a rank 2 qube dataset with ds.rank() elements in the first dimension. Note when the dataset is all zeros (false), the result is a zero-length array, as opposed to IDL which would return a -1 scalar. Parameters: • ds - of any rank M greater than 0. Returns: • when ds is rank 1, a rank 1 list of N indices. When ds is rank M>1, rank 2 dataset with N by M elements, where N is the number of non-zero elements found. ## 4.143. zeros public static WritableDataSet zeros(int len0) return new dataset filled with zeros. Note, unlike Matlab which would return a rank 2 matrix of data, this returns a rank 1 dataset. Parameters: • len0 - the number of elements Returns: • an Qube dataset with this dimension. ## 4.144. zeros public static WritableDataSet zeros(int len0, int len1) return new dataset filled with zeros. Parameters: • len0 - Returns: ## 4.145. zeros public static WritableDataSet zeros(int len0, int len1, int len2) return new dataset filled with zeros. Parameters: • len0 - Returns: ## 4.146. zeros public static WritableDataSet zeros(QDataSet ds) return a new dataset filled with zeroes that has the same geometry as the given dataset. Only supports QUBE datasets. Parameters: • ds - Returns: • a new dataset with filled with zeroes with the same geometry. # 5. Misc ## 5.1. getParam getParam( name, default [, label[, values ]] ) returns the parameter with the name, or the default if not provided. This is an interesting operator because it is defined in each context, and parameters are passed in differently in each case. Params: • name - the unique name of the parameter, which must be valid a Java identifier. • default - a default value for the parameter. • label - a short label describing the use. (This should probably be renamed "title" since a sentence is okay.) • values - is a list of values enumerating the possible values for this argument. Note Autoplot doesn't check that these are valid, but automatic GUIs limit the option. How are these typed? "foo" is a string, 4.0 is a double, etc. TODO: need documentation on interpretation. For python data sources, vap+jyds:<script_file>?<paramName>=<paramValue>. Use of this function will result in an item in the datasource editor. For --script on the command line, it's a command line argument. For application context scripts, the default value is always used, but a future rev may create a GUI. There's an experimental mode for this. Holding shift and pressing the execute will enter a GUI. ## 5.2. downloadResourceAsTempFile downloadResourceAsTempFile( url, mon ) -> File downloadResourceAsTempFile( url, timeoutSeconds, mon ) -> File Download the resource into a file. The URL may be any web address, and may contain parameters (unlike the FileSystem). Note, the downloaded resource may be reused by other threads for 10 seconds. This allows scripts to be simpler and more abstract, written to load one variable but actually able to load many efficiently. Params: • url - Url to download • timeoutSeconds - integer number of seconds to allow use of cached file. The default is 10 seconds. 12 hours or 43200 seconds is the limit. 0 is allowed. • mon - ProgressMonitor to monitor the download. Returns: • File containing the data, which will be deleted when the application exits. (TODO: server apps?) Throws: • java.lang.IOException when there is a problem downloading the file Example: fil= downloadResourceAsTempFile( URL('http://autoplot.org/data/autoplot.dat'), monitor ) ## 5.3. getFile( surl, mon ) getFile( surl, mon ) Download the file and make it available to the script. The URL may not contain parameters, but the result is cached. Params: • url - Url to download, without ? and params • mon - ProgressMonitor to monitor the download. Returns: • File containing the data, which will persist in the user's cache. ## 5.4. getDataSet getDataSet( uri ) returns a QDataSet for the given URI. Execution will stop at this point until the dataset is loaded. getDataSet( uri, monitor ) can be used to monitor the download. Note "monitor" is the local variable that contains a ProgressMonitor object. uri= 'http://www.rbsp-ect.lanl.gov/data_pub/rbspa/hope/level2/rbspa_rel01_ect-hope-sci-L2_$Y$m$d_v$(v,sep).cdf?FEDU' timerange= getParam( 'timerange', '2013-04-25', 'timerange to load' ) ds= getDataSet( uri, timerange ) When the uri can load data from any timerange, as with aggregations, then this will load the data for the given timerange, expressed as string. ## 5.5. listDirectory listDirectory( uri ) For example: files= listDirectory( 'http://autoplot.org/data/*.cdf' ) returns a listing of the directory "uri." If uri ends with a slash, then the directory is listed without filtering, otherwise the part following the slash is a glob that is matched. Note, the list does not contain the folder name, so it is typically added to the result: files= listDirectory( 'http://autoplot.org/data/*.cdf' ) for f in files: print 'http://autoplot.org/data/'+f ## 5.6. Color java.awt.Color is imported, so for example Color.RED may be used. dom.plotElements[0].style.color= Color.RED ## 5.7. DataSetBuilder The DataSetBuilder object is useful for creating datasets. from org.virbo.dsutil import DataSetBuilder dsb= DataSetBuilder(1,100) # creates a rank 1 with 100 records pre-allocated. for i in xrange(115): dsb.putValue(-1,0.0) # -1 means use the built in position dsb.nextRecord() plot(dsb.getDataSet()) from org.virbo.dsutil import DataSetBuilder dsb= DataSetBuilder(2,100,10) # creates a rank 2 with 100 10-element records pre-allocated. for i in range(115): for j in range(10): dsb.putValue(i,j,0.0) plot(dsb.getDataSet()) ## 5.8. putProperty The putProperty method sets the property on a dataset. For datasets that can be modified (mutable), this sets the property and the dataset is returned. For datasets that are immutable, a copy is made the property is set on the copy. ds= ripples(20) ds= putProperty( ds, QDataSet.UNITS, Units.eV ) ## 5.9. dataset Converts many objects into QDataSet. For example dataset([1,2,3]) converts the Python array into a 3-element QDataSet. ## 5.10. datum Converts many objects into Datum. For example datum(1) into the dimensionless Datum "1", and datum('2014-05-06T07:08'). Note Units.days.createDatum(20) can be used to create datums as well. ## 5.11. LinFit LinFit is a code that does linear fits of Y(X) where there may be errors in Y. For example: x= linspace( 0,10,200 ) y= x * 6.4 + 0.4 + randn(200) from org.virbo.dsutil import LinFit lf= LinFit(x,y) setLayoutOverplot(2) plot( 0, x,y ) plotx( 1, x, lf.getB() * x + lf.getA(), color=Color.RED ) # 6. Slice and Trim in Python The slice and trim operators are available in Python as index operators. For example: ds= ripples(20,30) s1= ds[10,:] The trim operator is similar: ds= ripples(20,30) t1= ds[5:15,:] ## 6.1. Other useful Java stuff java.lang.System.currentTimeMillis() Runtime.getRuntime().maxMemory() memory available to the JVM Runtime.getRuntime().totalMemory() memory allocated by the JVM Runtime.getRuntime().freeMemory() free memory of total memory. # 7. DOM The DOM represents the state of the application, including the canvas, the plots, the plot elements on each plot, and all the data sources. For example, dom.timeRange= DatumRangeUtil.parseTimeRange('2014') print dom.bindings # 8. Proper, Fully-Capable, Jython Scripts Ed W at RPWG points out that python provides some of the functionality I recreate, if scientists were willing to learn a little python. Also Bob W and I were discussing a method for defining DataSourceEditorPanels for jyds sources. I've always meant to have an alternate form for specifying these scripts and this section begins this discussion. def mysource( orbit='A' ): """Look up data for an orbit. orbit: the orbit number to lookup. A,B,C,0-41""" class plugin: def dataSourceEditorPanel( uri ): uri def getDataSet( params ): QDataSet, or dictionary of calculated parameters def formatDataSet( QDataSet, params ) # 9. TimeSeriesBrowse Time Series Browse is the datasource's capability to cover long timeseries. For example a script shows the outside temperature, and users can use the script to look at the temperature of any day. This can be done in two ways. First, you can write a script that takes one file and produces a dataset. When script uses the variable "resourceURI" to point to the file, this variable can be an aggregation and Autoplot's aggregation feature can use the script to read in each day's data. Second, the script itself can handle the time range as a parameter. Here the parameter "timerange" is a string that defines the time range requested by the user. It is a string that is parsed with the routine DatumRangeUtil.parseTimeRange which returns a DatumRange representing the range. Examples show how this is used. If either resourceURI is an aggregation or the timerange parameter is used by the script, Autoplot will listen to the axes and re-request data as we navigate to other time ranges. Note the time series browse capability also supports a resolution, but this is currently not used in scripts. They will always load the data at the native resolution. The script can report a resolution, however, using the CACHE_TAG property in the result. For example, a script tries to load low resolution data if available, and then falls back to high resolution when it is not and reduces the result. Here the CACHE_TAG property must be set for the DEPEND_0 timetags of the result. http://autoplot.org/data/jyds/eg/tsbRes.jyds # 10. Listening to Events GUI events are handled nicely in Jython, often making it so just a little code is needed to handle events. For example, org.das2.components.DataPointRecorder will fire off events as new points are added. In Java, this requires a bit of boilerplate code to listen to the events, but in Jython the code is simple: dpr= org.das2.components.DataPointRecorder() def mydataSetUpdated(event): print event dpr.dataSetUpdated= mydataSetUpdated Here dpr has the Java Bean pattern addDataSetUpdateListener which has the single method dataSetUpdated. Here's a Box Selector: from org.das2.event import BoxSelectorMouseModule mm= BoxSelectorMouseModule.create( dom.plots[0].controller.dasPlot, 'demo box' ) def boxSelected(event): print event mm.boxSelected= boxSelected dom.plots[0].controller.dasPlot.addMouseModule(mm) Here's a slicer: mm= DataPointSelectorMouseModule( plot, None, 'demo slice', VerticalSliceSelectionRenderer( plot ) ) def mousePointSelected(event): print event mm.dataPointSelected= mousePointSelected ## 10.1. Reading the stack trace on error messages It's tricky to decipher the stack traces in error messages when there's a problem in the call back, because Autoplot is not built to receive and interpret these messages. In the stack trace, look for the name of the script (demoSlice2.jy in this case): at org.python.core.Py.AttributeError(Py.java:127) at org.python.core.PyObject.noAttributeError(PyObject.java:991) at org.python.core.PyObject.__getattr__(PyObject.java:986) at org.python.pycode._pyx423.mousePointSelected$1(demoSlice2.jy:27) <---HERE
at org.python.pycode._pyx423.call_function(demoSlice2.jy)
at org.python.core.PyTableCode.call(PyTableCode.java:217)
at org.python.core.PyTableCode.call(PyTableCode.java:437)
at org.python.core.PyFunction.__call__(PyFunction.java:189)
at org.python.core.PyCompoundCallable.__call__(PyCompoundCallable.java:28)
at org.python.core.PyObject.__call__(PyObject.java:460)
at org.python.core.PyObject._jcallexc(PyObject.java:2838)
at org.python.core.PyObject._jcall(PyObject.java:2870)
at org.das2.event.DataPointSelectorMouseModule.fireDataPointSelectionListenerDataPointSelected(DataPointSelectorMouseModule.java:124)
...
# 11. DataSets are immutable, sort of
The QDataSet interface is immutable, or read-only, and should be treated so. Most QDataSet implementations have a mutable version of the interface, however, which is so implementations can be efficient. The idea here is that the guy making the dataset knows that it is mutable and can build it up, then hand it off to the calling code who knows only that it is a QDataSet and immutable.
Unfortunately, Jython is loosely typed, and a number of scripts use the fact that the dataset is mutable in their implementation. This can result in problems, when datasets are cached and mutating the dataset means mutating it for everybody. The bug motivating this section was caused when a script was slicing the rank 2 y tags and putting them back in the data, and then crashing when rerun because the y tags had already been sliced.
For example in Jython, you can do this:
ds= findgen(20)
ds[3]= 0
plot( ds )
But what if the findgen command were optimized so successive calls would return the same dataset object? Essentially they would be failing their contract.
Update Jan 2014: Data set implementations that allow for mutability via putProperty or putValue now have a makeImmutable() method that will result in runtime errors if these methods are called again. Because if this, clients should call isImmutable and make a copy if they need to modify a dataset.
# 12. Other Commands
A few other things are imported into sessions automatically, because they are useful and we don't want to burden new users with having to import them.
These are prefixed with the class name (e.g. Units.cm):
These are useful but must be imported:
TODO:
• remove javadoc, this is the authoritative copy
# 13. Order of Operations
Take care with the order of operations. For example, magnitude(a)*magnitude(b).gt(0) should be (magnitude(a)*magnitude(b)).gt(0).
# 14. Dealing With Jython
Here we point out some annoyances to be aware of.
• Constructor calls fail to adapt to PyQDataSet. QDataSets are adapted to PyQDataSet, which allows ds[2] -> ds.value(2) and all operators.
• Often you have to explicitly convert a number or list into a QDataSet. Use the dataset() command.
# 15. Debugging
• print statements can be used to indicate variable status: "print ds"
• When the script context is "application context", when an error is hit, the script session is live and variable states can be queried. In the script editor, highlight an expression, and mouse over to get a tooltip. The tooltip shows the value of the expression.
• Similarly, you can highlite an variable or expression, and [right-click]->Actions->Plot. This will send the data highlited over to another Autoplot running in server mode. Start up a second Autoplot (jnlp won't let you do this, so use the single jar), Options->Enable Server, on port 12345. The "plot" command exports the data as a .qds file and sends a plot command to the Autoplot in server mode.
• to force execution to stop at a given point, I call the undefined function "stop" which causes the script session to crash.
• the peekAt command can be used with a Java debugger (like Netbeans or Eclipse) to jump into the debugger with a Java breakpoint.
# 16. Debugging with pdb
Python has a debugging method built-in "pdb", which probably could be used in Autoplot's scripting environment. However experiments have shown that it is very easy to loose control of the debugger, potentially hanging the application. That said, some notes about this should be recorded here: developer.jython.pdb
# 17. Using the console and server mode
The console tab has a command line so that commands can be entered interactively. For example, start the console tab ([menubar]->Options->Enable Feature->Log Console) and type in at the AP prompt "plot( 'http://autoplot.org/data/autoplot.dat' )". This is the equivalent as typing in the command at the address bar, and also executing the command in a script. The session persists between commands, so for example you can type in:
AP> ds1= randn(100)
then
AP> ds2= findgen(100)
and
AP> plot( ds1 + ds2 )
These are the same commands available in scripts (application context), documented above.
Autoplot also has a server mode to allow for control from other processes. For example, the applot command for IDL works by formatting the dataset to a temporary file and then sending a command to the Autoplot session to plot data in the file. This server mode is a session just like the command line session. For example try from a unix command line:
unix% wget -O autoplot.jar http://autoplot.org/jnlp/latest/autoplot.jar
unix% java -cp autoplot.jar org.virbo.autoplot.AutoplotUI --server=12345 &
unix% telnet localhost 12345
autoplot> ds1= randn(100)
autoplot> ds2= findgen(100)
autoplot> plot( ds1 + ds2 )
Note this interface introduces a number of security problems, and may change.
# 18. SciPy vs Autoplot's Jython
People at U. Iowa are starting to compare Autoplot's Jython with SciPy to see how close they are. This is a brief list of differences found:
• SciPy allowed sloppiness at array boundaries, like IDL does. If A has 20 elements, and B=[19,20,21], then A[B] is okay. Autoplot does not allow this.
• "where" returns a rank 1 dataset, SciPy returns a strange array of arrays. x = np.where(slice == max(slice))[0][0]
• With Autoplot, it's easy to make lists of arrays instead of lists of scalars.
• Autoplot's methods cannot easily be imported as np, but they should be.
• np.float(spectra_trim[:,77].shape[0]) is just spectra_trim.length()
• pycdf/getDataSet CDF reading is different.
There is an effort to make Jython work with native Python code, JyNI, and this is expected to be released around the same time Jython2.7 is released as a production release.
# 19. Including a standard library
Suppose you have functions you use a lot and want to have them all in one place. You can make a file with these in it, and use "execfile(f)" to include those functions. This can also be on a website, so that you can share code with colleagues. For example:
f= getFile( 'http://www-pw.physics.uiowa.edu/~jbf/autoplot/myfunctions.jy',monitor)
execfile( f.toString() )
print mytotal( dataset([12,14]) )
Will bring in the mytotal function from http://www-pw.physics.uiowa.edu/~jbf/autoplot/myfunctions.jy.
# 20. Missing Functions
Functions that should exist but don't appear here and in the sourceforge bug ticket: https://sourceforge.net/p/autoplot/bugs/1569/
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2017-12-12 20:00:54
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https://www.physicsforums.com/threads/fitting-a-line-by-minimizing-absolute-deviation.919605/
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# Fitting a line by minimizing absolute deviation
• I
## Main Question or Discussion Point
Hey guys!.
I need to use this method of minimization.
The procedure to follow is explained here:
http://es.tinypic.com/r/35mmtg0/9
However, I do not fully understand. What do I have to do?
What I got from the paper is that I only have to minimize the equation 15.7.16, by using bisection. With this I am supposed to obtain the parameter b, whilst "a" would be got by means of 15.7.15 once the former is minimized and we know the true b.
Then, do I not have to do anything with 15.7.14?
Is it correct what I stated of how to perform the method or I am mistaken?
Thank you all!
Related Set Theory, Logic, Probability, Statistics News on Phys.org
haruspex
Homework Helper
Gold Member
I only have to minimize the equation 15.7.16, by using bisection. With this I am supposed to obtain the parameter b, whilst "a" would be got by means of 15.7.15 once the former is minimized
No, you have to use both equations in finding b. For each trial value of b, use 15.7.15 to find the corresponding a, then evaluate the expression in 15.7.16. Adjust the choice of b according the "bracketing and bisection" procedure and repeat.
No, you have to use both equations in finding b. For each trial value of b, use 15.7.15 to find the corresponding a, then evaluate the expression in 15.7.16. Adjust the choice of b according the "bracketing and bisection" procedure and repeat.
Then 15.7.14 is not used at all?
haruspex
Homework Helper
Gold Member
Then 15.7.14 is not used at all?
.14 leads to, and is superseded by, the pair .15 and .16.
.14 leads to, and is superseded by, the pair .15 and .16.
Perfect. I designed the algorithm to perform it in that way. The problem is though, that the signs of Fx for the different values of b to bisect, after just one iteration, are the same!. Then no bisection step is taken. I do not understand why this is happening as 1) the function I am minimizing must have a root, 2) The interval to bisect that I am choosing contains that root. I am confused.
Thanks!
By the way. Can someone explain me what finding a root going downhill means?
And how can I minimize that function (.16) if its "y" values (I mean, fx) are all of the same sign? (they should have different sign so that bisection could be applied)
Is it possible to bisect in that case?
thank you all!
haruspex
Homework Helper
Gold Member
Perfect. I designed the algorithm to perform it in that way. The problem is though, that the signs of Fx for the different values of b to bisect, after just one iteration, are the same!. Then no bisection step is taken. I do not understand why this is happening as 1) the function I am minimizing must have a root, 2) The interval to bisect that I am choosing contains that root. I am confused.
Thanks!
I'd need to see the data and some of the processing to figure out what is going wrong. Or maybe the coding?
I'd need to see the data and some of the processing to figure out what is going wrong. Or maybe the coding?
I think the whole code would be a bit tedious to read. I have gone over it some hundred of times and there is no mistake in that.
here is the function that I have to minimize (43)
http://es.tinypic.com/r/2m2iebr/9
And here is the data. http://es.tinypic.com/r/d4l8j/9.
Altogether with the results of the bisection.
h2 is the parameter to be bisected, which in 43 is "n". (h is an array, composed by the different values that n gets during bisection)
I give a 0,1 - 400 interval to bisect, and as it is seen in the results, the three "n"s, yield the same result in the equation .16 (named Fx in my code), thus, no further bisection step is taken.
Thank you for all!
haruspex
Homework Helper
Gold Member
here is the data
I cannot cut-and-paste from a screen snapshot. Please paste the data into a post as text.
I cannot cut-and-paste from a screen snapshot. Please paste the data into a post as text.
Oh of course, sorry.
{0.75, 64.36 }, //Time of a turn, final space in radians
{1.383, 33.50 },
{1.5, 28.29 },
{1.7, 21.449},
{1.0, 51.36},
{0.85, 58.242 },
{1.35, 34.12 },
haruspex
Homework Helper
Gold Member
Oh of course, sorry.
{0.75, 64.36 }, //Time of a turn, final space in radians
{1.383, 33.50 },
{1.5, 28.29 },
{1.7, 21.449},
{1.0, 51.36},
{0.85, 58.242 },
{1.35, 34.12 },
I find that to make the algorithm work you do need to start in the right ballpark.
Here are some values I got:
Code:
b a sums
15.7.16 15.7.14
-46.2 97.56 0.37 5.19
-46.4 97.76 0.37 5.27
-44 95.36 -0.93 7.09
-46 97.34 -0.78 5.24
-46.1 97.44 -0.13 5.18
-45 96.36 -0.93 6.16
-47 98.5 0.75 5.63
-45.5 96.86 -0.93 5.69
-46.5 97.86 0.37 5.3
So it is converging somewhere around b=-46.1.
Edit: I notice that it goes a bit crazy when one of the sgn() arguments is exactly zero. The recommended procedure of setting the value returned to zero does not fix it. Might be something to with the median function - that might need some tweaking. Look at b=-46.08.
Last edited:
I find that to make the algorithm work you do need to start in the right ballpark.
Here are some values I got:
Code:
b a sums
15.7.16 15.7.14
-46.2 97.56 0.37 5.19
-46.4 97.76 0.37 5.27
-44 95.36 -0.93 7.09
-46 97.34 -0.78 5.24
-46.1 97.44 -0.13 5.18
-45 96.36 -0.93 6.16
-47 98.5 0.75 5.63
-45.5 96.86 -0.93 5.69
-46.5 97.86 0.37 5.3
So it is converging somewhere around b=-46.1.
Edit: I notice that it goes a bit crazy when one of the sgn() arguments is exactly zero. The recommended procedure of setting the value returned to zero does not fix it. Might be something to with the median function - that might need some tweaking. Look at b=-46.08.
which values for phi have you used? (I mean, for the second parameter to find out in equation 42) I already know its right phi value, which is 6.19
Still, using that value, my bisection does not work.
Thanks!
haruspex
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2020-01-22 22:17:54
|
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|
https://chem.libretexts.org/?title=Under_Construction/Core_Construction/Schaller/Part_I:__Structure_in_Organic,_Biological_%26_Inorganic_Chemistry/AB._Concepts_of_Acidity/AB17._Product-to-Reactant_Ratio_and_Equilibrium_Constant
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# AB17. Product-to-Reactant Ratio and Equilibrium Constant
The index we have used to assess Brønsted acidity, pKa, is a measurable quantity. It is determined by measuring the ratio of products to reactants in a proton transfer reaction.
Figure AB17.1. HCl dissolves and ionizes in water.
For example, if HCl is dissolved in water, essentiallyy all of it ionizes, transferring a proton to water to form hydronium ion and chloride anions. The concentrations of all four of these species could be measured and compared. The ratio of products to reactants is called the equilibrium constant, or Keq:
$K_{eq} = {[H_3O^+] [Cl^-] \over [H_2O] [HCl] }$
The concentrations of these species are generally reported in moles per liter. A mole, you may know, is a unit used to count very large numbers of molecules. Since molecules are very small, we usually deal with very large numbers of them at a time.
By convention, the concentration of water in itself is defined as 1. That leads to a slightly different expression.
$K_{eq}' = {[H_3O^+] [Cl^-] \over [HCl] }$
This ratio, dealing with proton transfer, is also called the acidity constant, Ka.
$K_a = {[H_3O^+] [Cl^-] \over [HCl] }$
##### Problem AB17.1.
Write the expression for the Ka in each of the following mixtures.
a) HCN in water b) H2S in water c) NH3 in DMSO (DMSO = (CH3)2SO)
The Ka is often a very, very small number or a very, very large one. In the case of HCl in water, the Ka is about 1 x 10-8. Dealing with exponents can be cumbersome. In order to simplify comparisons, the equilibrium constant is expressed logarithmically.
$pKa = - \log Ka$
The pKa for HCl transferring a proton to water is -8.
##### Problem AB17.2.
Convert the following Ka's to pKa's.
a) 1 x 106 b) 1 x 10-9 c) 3.5 x 10-25 d) 8.5 x 10-17
##### Problem AB17.3.
Convert the following pKa's to Ka's.
a) -3.5 b) 4.3 c) 9 d) 25
Equilibrium constants are not restricted to proton transfer. They can be used to describe the extent to which any reaction occurs. For example, they can be written for other, reversible processes involving acid-base chemistry.
Figure AB17.2. Equilibrium in the formation of a Lewis acid-base complex between diethyl ether and boron trifluoride.
Lewis acid-base interactions are very often reversible. For example, a Lewis base like ether can donate a pair of electrons to a Lewis acid such as BF3. The ether can take its electrons back again and leave the BF3 behind, too. How tightly the ether is held by the BF3 is termed the binding constant. In this case,
$K_{eq} = {[BF_3] [Et_2O] \over [BF_3OEt_2] }$
In this case, the equilibrium constant has been reported to be about 0.25.
##### Problem AB17.4.
What does the equilibrium constant for formation of a complex between BF3 and diethyl ether (above) tell you about the position of the equilibrium?
##### Problem AB17.5.
The equilibrium constant for complex formation between dimethyl ether (CH3OCH3) and BF3 is 4.2. Compare this value with the one for diethyl ether and explain the difference.
##### Problem AB17.6.
Write expressions for the binding constant in the following cases.
a) (NH3)2PtCl2 loses an ammonia
b) Mo(CO)6 loses a carbon monoxide
c) FeCl4- loses a chloride anion
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2018-12-14 09:39:58
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https://studyadda.com/solved-papers/cet-karnataka-medical/cet-karnataka-medical-solved-paper-2009/267
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# Solved papers for CET Karnataka Medical CET - Karnataka Medical Solved Paper-2009
### done CET - Karnataka Medical Solved Paper-2009
• question_answer1) --In ruby laser, the stimulated emission is due to transition from
A) metastable state to any lower state
B) any higher state to lower state
C) metastable state to ground state
D) any higher state to ground state
• question_answer2) A direct current I flows along the length of an infinitely long straight thin walled pipe, then the magnetic field is
A) uniform throughout the pipe but not zero
B) zero only along the axis of the pipe
C) zero at any point inside the pipe
D) maximum at the centre and minimum at the edge
• question_answer3) A convex lens made of glass has focal length 0.15 m in air. If the refractive index of glass is $\frac{3}{2}$ and that of water is $\frac{4}{3}$, the focal length of lens when immersed in water is
A) 0.45 m
B) 0.15 m
C) 0.30 m
D) 0.6 m
• question_answer4) Two sources are said to be coherent if they produce waves
A) having a constant phase difference
B) of equal wavelength
C) of equal speed
D) having same shape of wave front
• question_answer5) Three resistors 1$\Omega$, 2$\Omega$ and 3$\Omega$ are connected to form a triangle. Across 3$\Omega$ resistor a 3 V battery is connected. The current through 3$\Omega$ resistor is
A) 0.75 A
B) 1 A
C) 2 A
D) 1.5 A
• question_answer6) In a common emitter amplifier the input signal is applied across
A) anywhere
B) emitter-collector
C) collector-base
D) base-emitter
• question_answer7) In a radioactive disintegration, the ratio of initial number of atoms to the number of atoms present at an instant of time equal to its mean life is
A) $\frac{1}{{{e}^{2}}}$
B) $\frac{1}{e}$
C) e
D) ${{e}^{2}}$
• question_answer8) A ray of light is incident on a surface of glass slab at an angle$45{}^\circ$. If the lateral shift produced per unit thickness is $\frac{1}{\sqrt{3}}$m, the angle of refraction produced is
A) ${{\tan }^{-1}}\left( \frac{\sqrt{3}}{2} \right)$
B) ${{\tan }^{-1}}\left( 1-\sqrt{\frac{2}{3}} \right)$
C) $si{{n}^{-1}}\left( 1-\sqrt{\frac{2}{3}} \right)$
D) $ta{{n}^{-1}}\left( \sqrt{\frac{2}{\sqrt{3}-1}} \right)$
• question_answer9) Ferromagnetic materials used in a transformer must have
A) low permeability and high hysterisis loss
B) high permeability and low hysterisis loss
C) high permeability and high hysterisis loss
D) low permeability and low hysterisis loss
• question_answer10) According to Newtons corpuscular theory, the speed of light is
A) same in all the media
B) lesser in rarer medium
C) lesser in denser medium
D) independent of the medium
• question_answer11) For the constructive interference the path difference between the two interfering waves must be equal to
A) $\left( 2n+1 \right)\lambda$
B) $2n\pi$
C) $n\lambda$
D) $\left( 2n+1 \right)\frac{\lambda }{2}$
• question_answer12) The accurate measurement of emf can be obtained using
A) multimeter
B) voltmeter
C) voltameter
D) potentiometer
• question_answer13) The kinetic energy of an electron gets tripled, then the de-Broglie wavelength associated with it changes by a factor
A) $\frac{1}{3}$
B) $\sqrt{3}$
C) $\frac{1}{\sqrt{3}}$
D) 3
• question_answer14) Which of the following is not a thermodynamic coordinate?
A) Gas constant (R)
B) Pressure (p)
C) Volume (V)
D) Temperature (T)
• question_answer15) Two solid pieces, one of steel and the other of aluminium when immersed completely in water have equal weights. When the solid pieces are weighed in air
A) the weight of aluminium is half the weight of steel
B) steel piece will weigh more
C) they have the same weight
D) aluminium piece will weigh more
• question_answer16) The amount of energy released when one microgram of matter is annihilated is
A) $25kWh~$
B) $9\times {{10}^{10}}kWh$
C) $3\times {{10}^{10}}kWh~~~~$
D) $0.5\times {{10}^{5}}kWh$
• question_answer17) The number of significant figures in the numbers $4.8000\times {{10}^{4}}$ and$48000.50$are respectively
A) 5 and 6
B) 5 and 7
C) 2 and 7
D) 2 and 6
• question_answer18) $\beta$-decay means emission of electron from
A) innermost electron orbit
B) a stable nucleus
C) outermost electron orbit
• question_answer19) An electric heater rated 220 V and 550 W is connected to AC mains. The current drawn by it is
A) 0.8 A
B) 2.5 A
C) 0.4 A
D) 1.25 A
• question_answer20) A body of mass m moving along a straight line covers half the distance with a speed of 2 ms-1. The remaining half of the distance is covered in two equal time intervals with a speed of 3 ms-1 and 5 ms-1 respectively. The average speed of the particle for the entire journey is
A) $\frac{3}{8}m{{s}^{-1}}$
B) $\frac{8}{3}m{{s}^{-1}}$
C) $\frac{4}{3}m{{s}^{-1}}$
D) $\frac{16}{3}m{{s}^{-1}}$
• question_answer21) The moment of inertia of a circular ring of radius r and mass M about diameter is
A) $\frac{2}{5}M{{r}^{2}}$
B) $\frac{M{{r}^{2}}}{4}$
C) $\frac{M{{r}^{2}}}{2}$
D) $\frac{M{{r}^{2}}}{12}$
• question_answer22) A body of mass 0.05 kg is observed to fall with an acceleration of$9.5\text{ }m{{s}^{-2}}$. The opposing force of air on the body is $\left( g=9.8\text{ }m{{s}^{-2}} \right)$
A) 0.015 N
B) 0.15 N
C) 0.030 N
D) zero
• question_answer23) The colloidal solution in which both the dispersed phase and dispersion medium are liquids are called
A) emulsions
B) gels
C) foams
D) liquid crystals
• question_answer24) In fog, photographs of the objects taken with infrared radiations are more clear than those obtained during visible light because
B) scattering of I - R light is more than visible light
C) the intensity of I - R light from the object is less
D) scattering of J - R light is less than visible light
• question_answer25) Three concurrent co-planar forces 1 N, 2 N and 3 N acting along different directions on a body
A) can keep the body in equilibrium if 2 N and 3 N act at right angle
B) can keep the body in equilibrium if 1 N and 2 N act at right angle
C) cannot keep the body in equilibrium
D) can keep the body in equilibrium in 1 N and 3 N act at an acute angle
A) only energy not momentum
B) energy
C) momentum
D) Both [a] and [b]
• question_answer27) Two rectangular blocks A and B of masses 2 kg and 3 kg respectively are connected by a spring of spring constant 10.8 Nm-1 and are placed on a frictionless horizontal surface. The block A was given an initial velocity of 0.15 ms-1 in the direction shown in the figure. The maximum compression of the spring during the motion is
A) 0.01 m
B) 0.02 m
C) 0.05 m
D) 0.03 m
• question_answer28) G P Thomson experimentally confirmed the existence of matter waves by the phenomena
A) diffraction
B) refraction
C) polarisation
D) scattering
• question_answer29) The resistance of a wire at 300 K is found to be 0.3 Sl. If the temperature coefficient of resistance of wire is$1.5\times {{10}^{-3}}{{K}^{-1}}$, the temperature at which the resistance becomes 0.6$\Omega$ is
A) 720 K
B) 345 K
C) 993 K
D) 690 K
• question_answer30) The work done by a force acting on a body is as shown in the graph. The total work done in covering an initial distance of 20 m is
A) 225 J
B) 200 J
C) 400 J
D) 175 J
• question_answer31) Two luminous point sources separated by a certain distance are at 10 km from an observer. If the aperture of his eye is $2.5\times {{10}^{-3}}$m and the wavelength of light used is 500 nm, the distance of separation between the point sources just seen to be resolved is
A) 12.2 m
B) 24.2 m
C) 2.44 m
D) 1.22 m
• question_answer32) A door 1.6 m wide requires a force of 1 N to be applied at the free end to open or close it. The force that is required at a point 0.4 m distance from the hinges for opening or closing the door is
A) 1.2 N
B) 3.6 N
C) 2.4 N
D) 4 N
• question_answer33) $0.1{{m}^{3}}$ of water at $80{}^\circ C$is mixed with $0.3\text{ }{{m}^{3}}$ of water at$60{}^\circ C$. The final temperature of the mixture is
A) $65{}^\circ C$
B) $70{}^\circ C$
C) $60{}^\circ C$
D) $75{}^\circ C$
• question_answer34) The spectral series of the hydrogen atom that lies in the visible region of the electromagnetic spectrum
A) Paschen
B) Balmer
C) Lyman
D) Brackett
• question_answer35) A graph of pressure versus volume for an ideal gas for different processes is as shown. In the graph curve OC represents
A) isochoric process
B) isothermal process
C) isobaric process
• question_answer36) Which of the following statement does not hold good for thermal radiation?
A) The wavelength changes when it travels from one medium to another
B) The frequency changes when it travels from one medium to another
C) The speed changes when it travels from one medium to another
D) They travel in straight line in a given medium
• question_answer37) A planet revolves around the sun in an elliptical orbit. The linear speed of the planet will be maximum at
A) D
B) B
C) A
D) C
• question_answer38) Horizontal tube of non-uniform cross-section has radii of 0.1 m and 0.05 m respectively at M and N. For a streamline flow of liquid the rate of liquid flow is
A) changing continuously with time
B) greater at M-than at N
C) greater at N than at M
D) same at M and N
• question_answer39) A resistor and a capacitor are connected in series with an AC source. If the potential drop across the capacitor is 5 V and that across resistor is 12 V, then applied voltage is
A) 13 V
B) 17 V
C) 5 V
D) 12 V
• question_answer40) The amount of heat energy radiated by a metal at temperature T is E. When the temperature is increased to 3T, energy radiated is
A) 81 E
B) 9 E
C) 3 E
D) 27 E
• question_answer41) The angle of minimum deviation for an incident light ray on an equilateral prism is equal to its refracting angle. The refractive index of its material is
A) $\frac{1}{\sqrt{2}}$
B) $\sqrt{3}$
C) $\frac{\sqrt{3}}{2}$
D) $\frac{3}{2}$
• question_answer42) In the following combinations of logic gates, the outputs of A, B and C are respectively
[A] [B] [C]
A) 0, 1, 1
B) 0, 1, 0
C) 1, 1, 0
D) 1, 0, 1
• question_answer43) A stationary point source of sound emits sound uniformly in all directions in a non-absorbing medium. Two points P and Q are at a distance of 4 m and 9 m respectively from the source. The ratio of amplitudes of the waves at P and Q is
A) $\frac{3}{2}$
B) $\frac{4}{9}$
C) $\frac{2}{3}$
D) $\frac{9}{4}$
• question_answer44) A galvanometer of resistance 240$\Omega$ allows only 4% of the main current after connecting a shunt resistance. The value of the shunt resistance is
A) 10$\Omega$
B) 20$\Omega$
C) 8$\Omega$
D) 5$\Omega$
• question_answer45) The phenomena in which proton flips is
A) nuclear magnetic resonance
B) lasers
D) nuclear fusion
• question_answer46) $y=3\sin \pi \left( \frac{t}{2}-\frac{x}{4} \right)$represents an equation of a progressive wave, where t is in second and $x$ is in metre. The distance travelled by the wave in 5 s is
A) 8 m
B) 10 m
C) 5 m
D) 32 m
• question_answer47) According to the quark model, it is possible to build all the hadrons using
A) 2 quarks and 3 antiquarks
B) 3 quarks and 2 antiquarks
C) 3 quarks and 3 antiquarks
D) 2 quarks and 2 antiquarks
• question_answer48) An $\alpha$-particle of mass 6.4 x 10-27 kg and charge 3.2 x 10-19 C is situated in a uniform electric field of 1.6 x 105 Vm-1. The velocity of the particle at the end of 2 x 10-2 m path when it starts from rest is
A) $2\sqrt{3}\times {{10}^{5}}m{{s}^{-1}}$
B) $8\times {{10}^{5}}m{{s}^{-1}}$
C) $16\times {{10}^{5}}m{{s}^{-1}}$
D) $4\sqrt{2}\times {{10}^{5}}m{{s}^{-1}}$
• question_answer49) A cylindrical tube open at both the ends has a fundamental frequency of 390 Hz in air. If $\frac{1}{4}$th of the tube is immersed vertically in water the fundamental frequency of air column is
A) 260 Hz
B) 130 Hz
C) 390 Hz
D) 520 Hz
• question_answer50) The surface temperature of the stars is determined using
A) Plancks law
B) Wiens displacement law
C) Rayleigh-Jeans law
D) Kirchhoffs law
• question_answer51) The charge deposited on$\text{4 }\!\!\mu\!\!\text{ F}$capacitor in the circuit is
A) $6\times {{10}^{-6}}C$
B) $12\times {{10}^{-6}}C$
C) $24\times {{10}^{-6}}C$
D) $36\times {{10}^{-6}}C$
• question_answer52) A parallel beam of light is incident on a converging lens parallel to its principal axis. As one moves away from the lens on the other side of the principal axis, the intensity of light
A) first decreases and then increases
B) continuously increases
C) continuously decreases
D) first increases and then decreases
• question_answer53) Continuous emission spectrum is produced by
A) incandescent electric lamp
B) mercury vapour lamp
C) sodium vapour lamp
D) polyatomic substances
• question_answer54) A coil of n number of turns is wound tightly in the form of a spiral with inner and outer radii a and b respectively. When a current of strength $I$is passed through the coil, the magnetic field at its centre is
A) $\frac{{{\mu }_{0}}nI}{\left( b-a \right)}{{\log }_{e}}\frac{a}{b}$
B) $\frac{{{\mu }_{0}}nI}{2\left( b-a \right)}$
C) $\frac{2{{\mu }_{0}}nI}{b}$
D) $\frac{{{\mu }_{0}}nI}{2\left( b-a \right)}{{\log }_{e}}\frac{b}{a}$
• question_answer55) A ray of light is incident on a plane mirror at an angle of$60{}^\circ$. The angle of deviation produced by the mirror is
A) $120{}^\circ ~$
B) $30{}^\circ$
C) $60{}^\circ ~$
D) $90{}^\circ$
• question_answer56) The electric potential at any point x, y, z in metres is given by$V=\text{ }3{{x}^{2}}$. The electric field at a point (2, 0, 1) is
A) $12\text{ }V{{m}^{-1}}$
B) $-6\text{ }V{{m}^{-1}}$
C) $6V{{m}^{-1}}$
D) $-12V{{m}^{-1}}$
• question_answer57) Youngs double slit experiment gives interference fringes of width 0.3 mm. A thin glass plate made of material of refractive index 1.5 is kept in the path of light from one of the slits, then the fringe width becomes
A) zero
B) 0.3 mm
C) 0.45 mm
D) 0.15 mm
• question_answer58) Near a circular loop of conducting wire as shown in the figure an electron moves along a straight line. The direction of the induced current if any in the loop is
A) variable
B) clockwise
C) anticlockwise
D) zero
• question_answer59) Hydrogen atom from excited state comes to the ground stage by emitting a photon of wavelength$\lambda$. If R is the Rydberg constant, the principal quantum number n of the excited state is
A) $\sqrt{\frac{\lambda R}{\lambda R-1}}$
B) $\sqrt{\frac{\lambda }{\lambda R-1}}$
C) $\sqrt{\frac{\lambda {{R}^{2}}}{\lambda R-1}}$
D) $\sqrt{\frac{\lambda R}{\lambda -1}}$
• question_answer60) The magnetic dipole moment of a current loop is independent of
A) magnetic field in which it is lying
B) number of turns
C) area of the loop
D) current in the loop
• question_answer61) The correct statement with regard to $H_{2}^{+}$ and $H_{2}^{-}$is
A) both $H_{2}^{+}$ and $H_{2}^{-}$ are equally stable
B) Both $H_{2}^{+}$ and $H_{2}^{-}$ do not exist
C) $H_{2}^{-}$ is more stable than $H_{2}^{+}$
D) $H_{2}^{+}$ is more stable than $H_{2}^{-}$
• question_answer62) Arrange the following in the increasing order of their bond order ${{O}_{2}},O_{2}^{+},O_{2}^{-}$and $O_{2}^{2-}$
A) $O_{2}^{2-},O_{2}^{-},{{O}_{2}},O_{2}^{+}$
B) $O_{2}^{2-},O_{2}^{-},O_{2}^{+},{{O}_{2}}$
C) $O_{2}^{+},{{O}_{2}},O_{2}^{-},O_{2}^{2-}$
D) ${{O}_{2}},O_{2}^{+},O_{2}^{-},O_{2}^{2-}$
• question_answer63) 2 g of a radioactive sample having half-life of 15 days was synthesised on 1st Jan 2009. The amount of the sample left behind on 1st March, 2009 (including both the days) is
A) $0g$
B) $0.125g$
C) $1g$
D) $0.5g$
• question_answer64) For a chemical reaction $A\to B$, the rate of the reaction is$2\times {{10}^{-3}}\text{ }mol\text{ }d{{m}^{-3}}\text{ }{{s}^{-1}}$, when the initial concentration is $0.05\text{ }mol\text{ }d{{m}^{-3}}$. The rate of the same reaction is $1.6\times {{10}^{-2}}\text{ }mol\text{ }d{{m}^{-3}}\text{ }{{s}^{-1}}$ when the initial concentration is $0.1\text{ }mol\text{ }d{{m}^{-3}}$. The order of the reaction is
A) $2$
B) $0$
C) $3$
D) $1$
• question_answer65) For the decomposition of a compound AB at 600 K, the following data were obtained
$[AB]\,mol\,\,d{{m}^{-3}}$ Rate of decomposition of AB in $mol\,d{{m}^{-3}}\,{{s}^{-1}}$ $0.20$ $2.75\times {{10}^{-8}}$ $0.40$ $11.0\times {{10}^{-8}}$ $0.60$ $24.75\times {{10}^{-8}}$
The order for the decomposition of AB is
A) $1.5$
B) $0$
C) $1$
D) $2$
• question_answer66) The rate equation for a reaction $A\to B$ is $r=k{{[A]}^{0}}$ If the initial concentration of the reactant is a$mol\text{ }d{{m}^{-3}}$, the half-life period of the reaction is
A) $\frac{a}{2k}$
B) $\frac{k}{a}$
C) $\frac{a}{k}$
D) $\frac{2a}{k}$
• question_answer67) $30cc$ of $\frac{M}{3}HCl,$$20cc$of $\frac{M}{2}HN{{O}_{3}}$ and $40cc$ of $\frac{M}{4}NaOH$ solutions are mixed and the volume was made up to$1\text{ }d{{m}^{3}}$. The pH of the resulting solution is
A) $8$
B) $2$
C) $1$
D) $3$
• question_answer68) An aqueous solution containing $6.5g$of $NaCl$ of 90% purity was subjected to electrolysis. After the complete electrolysis, the solution was evaporated to get solid $NaOH$. The volume of 1 M acetic acid required to neutralise $NaOH$ obtained above is
A) $1000c{{m}^{3}}$
B) $2000c{{m}^{3}}$
C) $100c{{m}^{3~}}$
D) $200c{{m}^{3}}$
• question_answer69) The standard electrode potential for the half-cell reactions are $Z{{n}^{2+}}+2{{e}^{-}}\xrightarrow{{}}Zn;{{E}^{o}}=-0.76V$ $F{{e}^{2+}}+2{{e}^{-}}\xrightarrow{{}}Fe;{{E}^{o}}=-0.44V$ The emf of the cell reaction, $F{{e}^{2+}}+Zn\xrightarrow{{}}Z{{n}^{2+}}+Fe$ is
A) $-0.32V$
B) $-1.20V$
C) $+1.20V$
D) $+0.32V$
• question_answer70) ${{10}^{-6}}M\text{ }NaOH$is diluted 100 times. The pH of the diluted base is
A) Between 7 and 8
B) Between 5 and 6
C) Between 6 and 7
D) Between 10 and 11
• question_answer71) In the electrolysis of acidulated water, it is desired to obtain $1.12cc$of hydrogen per second under STP condition. The current to be passed is
A) $1.93A$
B) $9.65A$
C) $19.3A$
D) $0.965A$
• question_answer72) The one which decreases with dilution is
A) Molar conductance
B) Conductance
C) Specific conductance
D) Equivalent conductance
• question_answer73) Vapour pressure of pure A is 70 mm of $Hg$ at ${{25}^{o}}C$. It forms an ideal solution with B in which mole fraction of A is $0.8$. If the vapour pressure of the solution is $84mm$of $Hg$ at ${{25}^{o}}C$, the vapour pressure of pure B at ${{25}^{o}}C$ is
A) $28mm$
B) $56mm$
C) $70mm$
D) $140mm$
• question_answer74) A 6% solution of urea is isotonic with
A) 1 M solution of glucose
B) 0.05 M solution of glucose
C) 6% solution of glucose
D) 25% solution of glucose
• question_answer75) In countries nearer to polar region, the roads are sprinkled with $CaC{{l}_{2}}$. This is
A) To minimise the wear and tear of the roads
B) To minimise the snow fall
C) To minimise pollution
D) To minimise the accumulation of dust on the road
• question_answer76) For the reaction ${{H}_{2}}O(l){{H}_{2}}O(g)$at $373K$and 1 atm pressure
A) $\Delta H=0$
B) $\Delta E=0$
C) $\Delta H=T\Delta S$
D) $\Delta H=\Delta E$
• question_answer77) A compound of A and B crystallises in a cubic lattice in which A atoms occupy the lattice points at the corners of the cube. The B atoms occupy the centre of each face of the cube. The probable empirical formula of the compound is
A) $A{{B}_{2}}$
B) ${{A}_{3}}B$
C) $AB$
D) $A{{B}_{3}}$
• question_answer78) In electrophilic aromatic substitution reaction, the nitro group is meta directing because it
A) Decreases electron density at ortho and para positions
B) Decreases electron density at meta position
C) Increases electron density at meta position
D) Increases electron density at ortho and para positions
• question_answer79) $C{{H}_{3}}COOH\xrightarrow{LiAl{{H}_{4}}}X\xrightarrow[{{300}^{o}}C]{Cu}Y\xrightarrow[NaOH]{Diltue}Z$ In the above reaction Z is
A) butanol
B) aldol
C) ketol
D) acetal
• question_answer80) The best method for the conversion of an alcohol into an alkyi chloride is by treating the alcohol with
A) $PC{{l}_{3}}$
B) $PC{{l}_{5}}$
C) $SoC{{l}_{2}}$ in presence of pyridine
D) dry $HCl$ in the presence of anhydrous $ZnC{{l}_{2}}$
• question_answer81) The electrophile involved in the sulphonation of benzene is
A) $SO_{3}^{+}$
B) $SO_{3}^{2-}$
C) $H_{3}^{+}O$
D) $S{{O}_{3}}$
• question_answer82) The carbon-carbon bond length in benzene is
A) In between ${{C}_{2}}{{H}_{6}}$ and ${{C}_{2}}{{H}_{4}}$
B) Same as in ${{C}_{2}}{{H}_{4}}$
C) In between C^H^ and ${{C}_{2}}{{H}_{4}}$
D) In between ${{C}_{2}}{{H}_{4}}$ and C
• question_answer83) The compound which is not formed during the dry distillation of a mixture of calcium formate and calcium acetate is
A) Methanal
B) Propanal
C) Propanone
D) Ethanal
• question_answer84) An organic compound X is oxidised by using Acidified ${{K}_{2}}C{{r}_{2}}{{O}_{7}}$The product obtained reacts with phenyl hydrazine but does not answer silver mirror test. The possible structure of X is
A) $C{{H}_{3}}C{{H}_{2}}OH$
B) $C{{H}_{3}}-\underset{O}{\mathop{\underset{||}{\mathop{C}}\,}}\,-C{{H}_{3}}$
C) ${{(C{{H}_{3}})}_{2}}CHOH$
D) $C{{H}_{3}}CHO$
• question_answer85) The reaction involved in the oil of winter green test is salicylic acid $\xrightarrow[Conc.\,{{H}_{2}}S{{O}_{4}}]{\Delta }$ product. The product is treated with $N{{a}_{2}}C{{O}_{3}}$ solution. The missing reagent in the above reaction is
A) Phenol
B) $NaOH$
C) Ethanol
D) methanol
• question_answer86) The compound which forms acetaldehyde when heated with dilute $NaOH$, is
A) 1, 1-dichloroethane
B) 1, 1, 1-trichloroethane
C) 1-chloroethane
D) 1, 2-dichloroethane
• question_answer87) Arrange the following in the increasing order of their basic strengths $C{{H}_{3}}N{{H}_{2}}$ , ${{(C{{H}_{3}})}_{2}}NH$ , ${{(C{{H}_{3}})}_{3}}N,N{{H}_{3}}$
A) $N{{H}_{3}}<{{(C{{H}_{3}})}_{3}}N<{{(C{{H}_{3}})}_{2}}NH<C{{H}_{3}}N{{H}_{2}}$
B) $N{{H}_{3}}<{{(C{{H}_{3}})}_{3}}N<C{{H}_{3}}N{{H}_{2}}<{{(C{{H}_{3}})}_{2}}NH$
C) ${{(C{{H}_{3}})}_{3}}N<N{{H}_{3}}<C{{H}_{3}}<C{{H}_{3}}N{{H}_{2}}<{{(C{{H}_{3}})}_{2}}NH$
D) $C{{H}_{3}}N{{H}_{2}}<{{(C{{H}_{3}})}_{2}}NH<{{(C{{H}_{3}})}_{3}}N<N{{H}_{3}}$
• question_answer88) The one which has least iodine value is
A) Sunflower oil
B) Ginger oil
C) Ghee
D) Groundnut oil
• question_answer89) A diabetic person carries a packet of glucose with him always, because
A) Glucose reduces the blood sugar level slowly
B) Glucose increases the blood sugar level slowly
C) Glucose reduces the blood sugar level
D) Glucose increases the blood sugar level almost instantaneously
• question_answer90) There are 20 naturally occurring amino acids. The maximum number of tripe tides that can be obtained is
A) $8000$
B) $6470$
C) $7465$
D) $5360$
• question_answer91) Cooking is fast in a pressure cooker, because
A) Food particles are effectively smashed
B) Water boils at higher temperature inside the pressure cooker
C) Food is cooked at constant volume
D) Loss of heat due to radiation is minimum
• question_answer92) The ore that is concentrated by froth floatation process is
A) Zincite
B) cinnabar
C) Bauxite
D) malachite
• question_answer93) The correct set of four quantum numbers for outermost electron of potassium $(Z=19)$ is
A) $4,1,0,\frac{1}{2}$
B) $3,1,0,\frac{1}{2}$
C) $4,0,0,\frac{1}{2}$
D) $3,0,0,\frac{1}{2}$
• question_answer94) A body of mass x kg is moving with a velocity of $100\text{ }m{{s}^{-1}}$. Its de-Broglie wavelength is$6.62\times {{10}^{-35}}m$. Hence, x is $(h=6.62\times {{10}^{-34}}Js)$
A) $0.1kg$
B) $0.25kg$
C) $0.15kg$
D) $0.2kg$
• question_answer95) The correct order of ionisation energy of C, N, O, F is
A) $F<O<N<C~$
B) $F<N<C<O$
C) $C<N<O<F$
D) $C<O<N<F$
• question_answer96) The oxide of an element whose electronic configuration is $1{{s}^{2}},2{{s}^{2}},2{{p}^{6}},3{{s}^{1}}$is
A) neutral
B) Amphoteric
C) basic
D) Acidic
• question_answer97) The characteristic not related to alkali metal is
A) High ionisation energy
B) Their ions are isoelectronic with noble gases
C) Low melting point
D) Low electronegativity
• question_answer98) Among the following, the compound that contains ionic, covalent and coordinate linkage is
A) $N{{H}_{3}}$
B) $N{{H}_{4}}Cl$
C) $NaCl$
D) $CaO$
• question_answer99) A covalent molecule $A{{B}_{3}}$ has pyramidal structure. The number of lone pair and bond pair of electron in the molecule are respectively
A) $2$ and $2$
B) $0$ and $4$
C) $3$ and $1$
D) $1$ and $3$
• question_answer100) Excess of carbon dioxide is passed through $50mL$of $0.5M$calcium hydroxide solution. After the completion of the reaction, the solution was evaporated to dryness. The solid calcium carbonate was completely neutralised with 0.1 N hydrochloric acid. The volume of hydrochloric acid required is (Atomic mass of calcium = 40)
A) $300c{{m}^{3}}$
B) $200c{{m}^{3}}$
C) $500c{{m}^{3}}$
D) $400c{{m}^{3}}$
• question_answer101) A bivalent metal has an equivalent mass of 32. The molecular mass of the metal nitrate is
A) $182$
B) $168$
C) $192$
D) $188$
• question_answer102) The rms velocity of molecules of a gas of density $4kg\,{{m}^{-3}}$and pressure $1.2\times {{10}^{5}}\,N{{m}^{-2}}$is
A) $300m{{s}^{-1}}$
B) $900m{{s}^{-1}}$
C) $120m{{s}^{-1}}$
D) $600m{{s}^{-1}}$
• question_answer103) $0.5mole$of each of ${{H}_{2}},S{{O}_{2}}$and are kept in a container. A hole was made in the container. After 3 h, the order of partial pressures in the container will be
A) $pS{{O}_{2}}>p{{H}_{2}}>pC{{H}_{4}}$
B) $pS{{O}_{2}}>pC{{H}_{4}}>p{{H}_{2}}$
C) $p{{H}_{2}}>pS{{O}_{2}}>pC{{H}_{4}}$
D) $p{{H}_{2}}>pC{{H}_{4}}>pS{{O}_{2}}$
• question_answer104) The enthalpy of formation of $N{{H}_{3}}$ is. The enthalpy change for the $-46kJ\,mo{{l}^{-1}}$ reaction $2N{{H}_{3}}(g)\xrightarrow{{}}{{N}_{2}}(g)+3{{H}_{2}}(g)$ is
A) $+184kJ$
B) $+23\text{ }kJ$
C) $+92kJ$
D) $+46kJ$
• question_answer105) $5moles$of $S{{O}_{2}}$ and 5 moles of ${{O}_{2}}$ are allowed to react. At equilibrium, it was found that 60% of $S{{O}_{2}}$ is used up. If the partial pressure of the equilibrium mixture is one atmosphere, the partial pressure of ${{O}_{2}}$ is
A) $0.82atm$
B) $0.52atm$
C) $0.21atm$
D) $0.41atm$
• question_answer106) $2HI(g){{H}_{2}}(g)+{{I}_{2}}(g)$ The equilibrium constant of the above reaction is $6.4$at$300K$. If $0.25$ mole each of ${{H}_{2}}$ and $12$are added to the system, the equilibrium constant will be
A) $6.4$
B) $0.8$
C) $3.2$
D) $1.6$
A) Decrease in surface area
B) Decrease in temperature
C) Decrease in pressure
D) Increase in temperature
• question_answer108) IUPAC name of ${{(C{{H}_{3}})}_{3}}CCl$is
A) n-butyl chloride
B) 3-chloro butane
C) 2-chloro 2-methyl propane
D) Mxityl chloride
• question_answer109) Lucas test is associated with
A) Aldehydes
B) Phenols
C) Carboxylic acids
D) Alcohols
• question_answer110) An organic compound on heating with $CuO$ produces $C{{O}_{2}}$ but no water. The organic compound may be
A) Carbon tetrachloride
B) Chloroform
C) Methane
D) Ethyl iodide
• question_answer111) The condensation polymer among the following is
A) Rubber
B) Protein
C) PVC
D) Polyethene
• question_answer112) The order of stability of metal oxides is
A) $A{{l}_{2}}{{O}_{3}}<MgO<F{{e}_{2}}{{O}_{3}}<C{{r}_{2}}{{O}_{3}}$
B) $C{{r}_{2}}{{O}_{3}}<MgO<A{{l}_{2}}{{O}_{3}}<F{{e}_{2}}{{O}_{3}}$
C) $F{{e}_{2}}{{O}_{3}}<C{{r}_{2}}{{O}_{3}}<A{{l}_{2}}{{O}_{3}}<MgO$
D) $F{{e}_{2}}{{O}_{3}}<A{{l}_{2}}{{O}_{3}}<C{{r}_{2}}{{O}_{3}}<MgO$
• question_answer113) The temperature of the slag zone in the metallurgy of iron using blast furnace is
A) $1200-{{1500}^{o}}C$
B) $1500-{{1600}^{o}}C$
C) $400-{{700}^{o}}C$
D) $800-{{1000}^{o}}C$
• question_answer114) The function of $Fe{{(OH)}_{3}}$ in the contact process is
A) to remove arsenic impurity
B) to detect colloidal impurity
C) to remove moisture
D) to remove dust particles
• question_answer115) In which of the following, NH3 is not used?
A) Toilers reagent
B) Nesslers reagent
C) Group reagent for the analysis of IV group basic radicals
D) Group reagent for the analysis of III group basic radicals
A) In filling airships
B) To obtain low temperature
C) In high temperature welding
D) In radiotherapy for treatment of cancer
• question_answer117) The incorrect statement in respect of chromyl chloride test is
A) Formation of red vapours
C) Formation of chromyl chloride
D) Liberation of chlorine
• question_answer118) The magnetic moment of a transition metal ion is $\sqrt{15}\text{ }BM$. Therefore, the number of unpaired electrons present in it, is
A) $3$
B) $4$
C) $1$
D) $2$
• question_answer119) The IUPAC name of ${{[Co{{(N{{H}_{3}})}_{5}}ONO]}^{2+}}$ion is
A) penta ammine nitrito cobalt (IV) ion
B) penta ammine nitrito cobalt (III) ion
C) penta ammine nitro cobalt (III) ion
D) penta ammine nitro cobalt (IV) ion
• question_answer120) The oxidation state of Fe in the brown ring/ complex: $[Fe{{({{H}_{2}}O)}_{5}}NO]S{{O}_{4}}$is
A) $+3$
B) $0$
C) $+2$
D) $+1$
• question_answer121) The respiratory quotient during cellular respiration would depend on
A) the nature of enzymes involved
B) the nature of the substrate
C) the amount of carbon dioxide released
D) the amount of oxygen utilized
• question_answer122) Which of the following is not a green-house gas?
A) Water vapour
B) Carbon monoxide
C) Methane
D) Oxygen
• question_answer123) Both husband and wife have normal vision though their fathers were colorblind and mothers did not have any gene for colorblindness. The probability of their daughters becoming colorblind is
A) 50%
B) 75%
C) 0%
D) 25%
• question_answer124) An animal, which has both exoskeletal and endoskeletal structures is a
A) fresh-water mussel
B) tortoise
C) frog
D) jelly fish
• question_answer125) 2n = 16 in a primary spermatocyte, which is in metaphase of first meiotic division. What shall be the total number of chromatids in each of the secondary spermatocyte?
A) 32
B) 8
C) 16
D) 24
• question_answer126) Identify the group, which includes animals all of which give birth to young ones directly?
A) Dolphin, kangaroo, bat, cat
B) Platypus, penguin, bat, hippopotamus
C) Shrew, bat, kiwi, cat
D) Lion, whale, ostrich, bat
• question_answer127) Compare the statements A and B.
Statement A Blood sugar level falls rapidly after hepatectomy. Statement B The glycogen of the liver is the principal source of blood sugar.
Select the correct description.
A) Both the statements A and B are correct and B is the reason for A
B) Statement A is correct and B is wrong
C) Statement A is wrong and B is correct
D) Both the statements A and B are correct and B is not the reason for A
[A] It does not help in water conduction. [B] It is also called alburnum. [C] It is dark in colour but very soft. [D] It has tracheary elements which are filled with tannin, resin, etc.
A) B, C and D
B) A and D
C) B and D
D) A, B and C
• question_answer129) Compare the statements A and B.
Statement A Auxins promote apical dominance by suppressing the activity of lateral buds. Statement B In moriculture, periodic pruning of shoot tips is done to make mulberry plants bushy.
Select the correct description.
A) Both the statements A and B are correct and A is the reason for B
B) Statement A is correct and B is wrong
C) Statement A is wrong and B is correct
D) Both the statements A and B are correct and A is not the reason for B
• question_answer130) Bryophytes resemble algae in the following aspects
A) Filamentous body, presence of vascular tissues and autotrophic nutrition
B) Differentiation of plant body into root, stem and leaves and aurotrophic nutrition
C) Thallus like plant body, presence of roots and autotrophic nutrition
D) Thallus like plant body, lack of vascular tissues and autotrophic nutrition
• question_answer131) Compare the statements A and B.
Statement A A monocistronic mRNA can produce several types of polypeptide chains. Statement B The terminator codon is present on the mRNA.
Select the correct description.
A) Both the statements A and B are wrong
B) Statement A is correct and B is wrong
C) Statement A is wrong and B is correct
D) Both the statements A and B are correct
A) guard cells swell due to an increase in their water potential
B) guard cells swell by endosmosis due to influx of hydrogen ions (protons)
C) guard cells swell by endosmosis due to efflux of potassium ions
D) guard cells swell due to a decrease in their water potential
• question_answer133) Which of the following is properly matched?
A) Platyhelminthes - Trematoda - Planaria
B) Echinodermata - Asteroidea - Star fish
C) Arthropoda - Insecta - Spider
D) Mollusca - Cephalopoda - Unio
• question_answer134) A man is admitted to a hospital. He is suffering from an abnormally low body temperature, loss of appetite and extreme thirst. His brain scan would probably show a tumour in
A) medulla oblongata
B) pons
C) cerebellum
D) hypothalamus
• question_answer135) Identify the incorrect statement with respect to Calvin cycle
A) The carboxylation of RuBP is catalysed by rubisco
B) The first stable intermediate compound formed is phosphoglycerate
C) 18 molecules of ATP are synthesized during carbon fixation
D) NADPH + H+ produced in light reaction is used to reduce diphosphoglycerate
• question_answer136) The agents, which are known to cause CJD are
A) protein particles
B) a class of bacteria
C) a class of viruses
D) fungi
• question_answer137) In crop improvement programmes, virus-free clones can be obtained through
A) grafting
B) hybridization
C) embryo culture
D) shoot apex culture
• question_answer138) A person is suffering from frequent episodes of nasal discharge, nasal congestion, reddening of eyes and watery eyes. These are the symptoms of
A) cyanosis
B) bronchitis
C) rhinitis
D) bronchial carcinoma
• question_answer139) Some important events in the human female reproductive cycle are given below. Arrange the events in a proper sequence.
A : Secretion of FSH B : Growth of corpus luteum C : Growth of the follicle and oogenesis D: Ovulation E : Sudden increase in the levels of LH
A) C $\to$A $\to$D $\to$B $\to$E
B) A $\to$C $\to$E $\to$D $\to$B
C) A $\to$D $\to$C $\to$E $\to$B
D) B $\to$A $\to$C $\to$D $\to$E
• question_answer140) Compare the statements A and B.
Statement A Ranikhet disease is the disease of poultry. Statement B It is caused by a virus.
Select the correct description.
A) Both the statements A and B are correct
B) Statement A is correct and B is wrong
C) Statement A is wrong and B is correct
D) Both the statements A and B are wrong
• question_answer141) The offspring produced from a marriage have only 0 or A blood groups. Of the genotypes given below, the possible genotypes of the parents would be
A) ${{|}^{A}}{{|}^{A}}$and ${{|}^{A}}{{|}^{O}}$
B) ${{|}^{O}}{{|}^{O}}$and ${{|}^{O}}{{|}^{O}}$
C) ${{|}^{A}}{{|}^{A}}$and ${{|}^{O}}{{|}^{O}}$
D) ${{|}^{A}}{{|}^{O}}$and ${{|}^{O}}{{|}^{O}}$
• question_answer142) A dorsal horn is present on the ......... of mulberry silk worm (caterpillar).
B) 8thabdominal segment
C) 5th abdominal segment
D) 2nd thoracic segment
• question_answer143) A plant has an androecium with monadelphous stamens, monothecous and reniform anthers. The corolla exhibits contorted aestivation. The plant could be
A) Rauwolfia
B) Vinca
C) Nerium
D) Hibiscus
A) electrolyte balance
B) opening of stomata
C) absorption of water by roots
D) excretion of minerals
• question_answer145) The cross-section of the body of an invertebrate is given below. Identify the animal, which has this body plan.
A) Cockroach
B) Roundworm
C) Planaria
D) Earthworm
• question_answer146) In an experiment demonstrating the evolution of oxygen in Hydrilla, sodium bicarbonate is added to water in the experimental set-up. What would happen if all other conditions are favourable?
A) Amount of oxygen evolved decreases as carbon dioxide in water is absorbed by sodium bicarbonate
B) Amount of oxygen evolved increases as the availability of carbon dioxide increases
C) Amount of oxygen evolved decreases as the availability of carbon dioxide increases
D) Amount of oxygen evolved increases as carbon dioxide in water is absorbed by sodium bicarbonate
• question_answer147) Which substance is in higher concentration in blood than in glomerular filtrate?
A) Water
B) Glucose
C) Urea
D) Plasma proteins
• question_answer148) All the following are included under in situ conservation except
A) botanical garden
B) biosphere reserve
C) national park
D) sanctuary
• question_answer149) Match the compounds given in Column I with the number of carbon atoms present in them which are listed under Column II. Choose the answer which gives the correct combination of alphabets of the two columns.
Column I Column II A. Oxaloacetate p. 6 - C compound B. Phosphoglycer-aldehyde q. 5 - C compound C. Isocitrate r. 4 - C compound D. $\alpha$- ketoglutarate s. 3 - C compound t. 2 - C compound
A) A- s B- t C- q D-r
B) A-r B-s C-p D-q
C) A-r B-t C-p D-q
D) A-q B-s C-p D-t
• question_answer150) Identify the correctly matched pair/pairs of the germ layers and their derivatives.
[A] Ectoderm - Epidermis [B] Endoderm - Dennis [C] Mesoderm - Muscles [D] Mesoderm - Notochord [E] Endoderm - Enamel of teeth
A) A, C and D only
B) A, B, C and E only
C) A and D only
D) A and B only
• question_answer151) Identify the correct statement.
A) Because of marked climatic variations, plants growing near the sea shore do not produce annual rings.
B) The age of the plant can be determined by its height.
C) Healing of damaged tissue is because of the activity of sclerenchyma cells.
D) Grafting is difficult in monocot plants as they have scattered vascular bundles.
• question_answer152) Blood stains are found at the site of a murder. If DNA profiling technique is to be used for identifying the criminal, which of the following is ideal for use?
A) Serum
B) Erythrocytes
C) Leucocytes
D) Platelets
A) the cell divides its cytoplasm during mitosis
B) the cell digests itself
C) the cell engulfs and internalises materials using its membrane
D) the cell enables the extracellular digestion of large molecules
• question_answer154) Match the names of the economically important plants (or their products) listed in with the families to which they belong given in Column II. Choose the answer which gives the correct combination of alphabets of the two columns.
Column I Column II A. Sunflower p. Acanthaceae B. Tulsi q. Compositae C. Coffee r. Labiatae D. Vasaka s. Rubiaceae t. Euphorbiaceae
A) A- r B- t C-s D-q
B) A-q B-r C-s D-p
C) A-q B- s C-p D-t
D) A-s B- r C-p D-q
• question_answer155) Which of the following hormones does not naturally occur in plants?
A) 2, 4-D
B) IAA
C) GA
D) ABA
• question_answer156) A large quantity of fluid is filtered every day by the nephrons in the kidneys. Only about 1% of it is excreted as urine. The remaining 99% of the filtrate
A) gets collected in the renal pelvis
B) is lost as sweat
C) is stored in the urinary bladder
D) is reabsorbed into the blood
• question_answer157) When DNA replication starts
A) The leading strand produces Okazaki fragments
B) the hydrogen bonds between the nucleotides of two strands break
C) the phosphodiester bonds between the adjacent nucleotides break
D) the bonds between the nitrogen base and deoxyribose sugar break
• question_answer158) Fleshy fruits with stony endocarp are called
A) capsules
B) berries
C) pomes
D) drupes
A) The electron carriers involved in photophosphorylation are located on the thylakoid membranes.
B) Photosynthesis is a redox process, in which water is oxidised and carbon dioxide is reduced.
C) The enzymes required for carbon fixation are located only in the grana of chloroplasts.
D) In green plants, both PS-I and PS-II are required for the formation of$NADPH\text{ }+\text{ }{{H}^{+}}$.
• question_answer160) Darwinism explains all the following except
A) within each species, there are variations
B) organisms tend to produce more number of offspring that can survive
C) offspring with better traits that overcome competition are best suited for the environment
D) variations are inherited from parents to offspring through genes
• question_answer161) Pollen grains of a plant whose 2n = 28 are cultured to get callus by tissue culture method. What would be the number of chromosomes in the cells of the callus?
A) 28
B) 21
C) 14
D) 56
• question_answer162) A true breeding plant producing red flowers is crossed with a pure plant producing white flowers. Allele for red colour of flower is dominant. After selfing the plants of first filial generation, the proportion of plants producing white flowers in the progeny would be
A) $\frac{3}{4}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
• question_answer163) Which of the following prevents the conversion of prothrombin to thrombin in an undamaged blood vessel?
A) Heparin
B) Calcium ions
C) Thromboplastin
D) Fibrinogen
• question_answer164) The characteristic that is shared by urea, uric acid and ammonia is/are [A] They are nitrogenous wastes. [B] They all need very large amount of water for excretion. [C] They are all equally toxic. [D] They are produced in the kidneys.
A) A and C
B) A and D
C) A, C and D
D) A only
• question_answer165) RBC and a plant cell (with thick cell wall) are placed in distilled water. The solute concentration is the same in both the cells. What changes would be observed in them?
A) Both plant cell and RBC would not undergo any change.
B) The RBC would increase in size and burst while the plant cell would remain about the same size.
C) The plant cell would increase in size and burst while the RBC would remain about the same size.
D) Both plant cell and RBC would decrease in size and collapse.
• question_answer166) Which of the following hormones does not contain a polypeptide?
A) Prostaglandin
B) Oxytocin
C) Insulin
D) Antidiuretic hormone
• question_answer167) Ribose sugar is present in
A) RNA polymerase, RNA and ATP
B) RNA only
C) RNA polymerase and ATP
D) RNA and ATP
• question_answer168) Most of the endangered species are the victims of
A) competition with introduced species
B) habitat destruction
C) over - hunting
D) acid - rain
A) loss of cell - mediated immunity
B) a reduction in the haemoglobin content in blood
C) a reduction in the amount of plasma proteins
D) loss of antibody-mediated immunity
• question_answer170) The diagram of the section of a maize grain is given below. Identify the parts labelled A, B, C and D.
A) Endosperm Coleopdie Scutellum Aleurone layer
B)
A = Cotyledon B = Coleoptile C = Scutellum D = Epithelium
C)
A = Endosperm B = Coleoptile C = Scutellum D = Epithelium
D)
A = Endosperm B = Coleorrhiza C = Scutellum D = Epithelium
• question_answer171) Examples for lateral meristems are
A) phellogen and procambium
B) fascicular cambium and procambium
C) procambium and dermatogen
D) fascicular cambium and cork cambium
• question_answer172) Vitellogenesis occurs during the formation of
A) primary oocyte in the Graafian follicle
B) oogonial cell in the Graafian follicle
C) ootid in the fallopian tube
D) secondary oocyte in the fallopian tube
• question_answer173) A bacterium is capable of withstanding extreme heat, dryness and toxic chemicals. This indicates that it is probably able to form
A) a thick peptidoglycan wall
B) endospores
C) endotoxins
D) endogenous buds
• question_answer174) In the absence of enterokinase, the digestion of ......... would be affected in our intestine.
A) maltose
B) amino acid
C) albumin
D) starch
• question_answer175) The greatest threat to genetic diversity in agricultural crops is
A) extensive use of insecticides and pesticides
B) extensive mixed cropping
C) introduction of high yielding varieties
D) extensive use of fertilizers
• question_answer176) Nosema bombycis, which causes pebrine in silk worms is a
A) fungus
B) virus
C) bacterium
D) protozoan
• question_answer177) Palaeontologists unearthed a human skull during excavation. A small fragment of the scalp tissue was still attached to it. Only little DNA could be extracted from it. If the genes of the ancient man need to be analysed, the best way of getting sufficient amount of DNA from this extract is
A) hybridising the DNA with a DNA probe
B) subjecting the DNA to polymerase chain reaction
C) subjecting the DNA to gel electrophoresis
D) treating the DNA with restriction endonucleases
• question_answer178) Which of the following would be insignificant amount in xylem sap?
A) Sugar
B) Nitrates
C) Phosphates
D) Water
• question_answer179) If the person shows the production of interferons in his body, chances are that he is suffering from
A) anthrax
B) malaria
C) measles
D) tetanus
• question_answer180) The RER in the cell synthesized a protein which would be later used in building the plasma membrane. But it is observed that the protein in the membrane is slightly different from the protein made in the RER. The protein was probably modified in another cell organelle. Identify that organelle in the given diagram.
A) D
B) A
C) B
D) C
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2019-04-22 12:25:34
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https://stats.stackexchange.com/questions/310629/calculating-the-likelihood-of-a-whole-dataset
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# Calculating the likelihood of a whole dataset
Of my dataset $X$ I want to calculate the likelihood using my GMM model (for BIC). Mathematically it seems to make sense that as the samples are independant $P(X) = P(X_1)P(X_2)..$, so I would get the likelihood by taking the product of the likelihoods per sample.
But it doesn't seem intuitively correct that the more samples I have the less likely my model. It also means that if just one sample that doesn't fit will crash my likelihood, it therefore seems to make more sense (to me) to average the likelihoods per sample.
What's the right thing to do here (I feel I'm messing some very basic statistics up)?
But it doesn't seem intuitively correct that the more samples I have the less likely my model.
I think your intuition has it the wrong way around. You're not computing how likely your model is - you're computing how likely the data are, GIVEN your model. You do this by multiplying together $$n$$ probabilities in the unit interval, so you should expect likelihood to decrease exponentially with $$n$$.
The key insight is that you don't use measures like BIC or AIC as a one to one measure of how likely your model is to be true. You compare different models by comparing their scores. For instance, the difference between BIC scores of models A and B, is an approximation of the ratio of the probabilities of the models being true (assuming the true model is in the set of candidate models). Hence, it doesn't really matter that the likelihood of the true model decreases with n, as long as the likelihoods for other models drop quicker, so the difference between their BICs grow.
it doesn't seem intuitively correct that the more samples I have the less likely my model
Maximum likelihood is used to find parameters of the model, so it isn't really a problem.
seems to make more sense (to me) to average the likelihoods per sample.
You're right if you would want to make likelihood more meaningful.
But it doesn't really matter for Maximum Likelihood since it is used for finding $$\hat{\theta} = \underset{\theta}{argmax} \underset{i < N}{\prod}{P(x_i; \theta)}$$ $$=\underset{\theta}{argmax} \sqrt[n]{\underset{i < N}{\prod}{P(x_i; \theta)}}$$ (nth root is increasing function, and correct averaging for product would be geometric averaging.).
It also means that if just one sample that doesn't fit will crash my likelihood
What do you mean by 'crashing likelihood'? If you mean that your likelihood becomes zero, this is impossible for GMM, since gaussian mixture's PDF is nonzero everywhere.
• By crash it I mean it will cause it to get very small. Practical example, if i had likelihoods for 5 samples, then $0.6*0.6*0.6*0.6*0.6=0.07776$ is apparently a better fit than $0.9*0.9*0.9*0.9*0.1=0.06561$, which seems wrong to me. – Nimitz14 Oct 29 '17 at 18:04
• If you see new data and it doesn't seem likely under your model then it means that your model isn't good right? – Jakub Bartczuk Oct 29 '17 at 18:06
• Have you heard of this thing called outliers? Have you never worked with a real world dataset before? – Nimitz14 Oct 29 '17 at 18:08
• Yes, this is a legitimate concern. But how can you classify something as outlier if you don't even have any model? – Jakub Bartczuk Oct 29 '17 at 18:11
• One thing I have done in the past is to remove 10% of my data points with the lowest model likelihoods for each model that I am comparing. It made sense for me as it was a quick way to prevent data that I knew were not very reliable from influencing my model likelihoods. This may or may not work for you. – Moss Murderer Oct 29 '17 at 18:57
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2021-08-01 23:24:15
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https://stats.stackexchange.com/questions/57921/how-to-prove-the-number-of-distinct-distributions-in-a-group-of-distributions
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# How to prove the number of distinct distributions in a group of distributions?
Let's say we had 5 distributions: A,B,C,D,E.
An ANOVA test would tell us whether or not all of the means are equal, and thus a low p-value would mean at least one of the means is unlikely to be equal to the mean of the whole population of distributions. Is this a correct interpretation?
If so, I'm looking for something similar to an iterated ANOVA test, which tests all combinations of possible distributions to determine the likeliest number of distributions. So, if all 5 of the above had distinct means, it would return 5. If all 5 did not have distinct means and would not pass ANOVA, it would return 1. If all were distinct except D and E, then it would return 4.
Does such a statistical function exist with an implementation in R?
Your interpretation of a low p-value is a very common one, but it is not quite correct. The phrase "is unlikely to be equal" has no meaning in a Frequentist context. Either all of the population means are equal, or they are not. To the extent that something like a statement of probability can be made in this situation, the correct statement would be either 'the probability that all the population means are equal is $1$', or 'the probability that all the population means are equal is $0$'. Unfortunately, you don't know which of those statements is true, even after calculating a p-value. The meaning of the p-value is the probability of getting data (specifically, sample means) as far or further from equal, if the true population means are all equal.
With respect to the statistical function you are looking for, it sounds very much like running a one-way ANOVA, followed by Tukey's honestly significant difference testing procedure.
One last note: In the title, you ask about "distributions" in a way that connotes (to me, at least) any difference in the distributions (e.g., differing variances), but then refer more specifically to "means" only. If you were interested in the former, you might want to look into the Anderson-Darling test.
• +1. The R implementation of Tukey's HSD is TukeyHSD. In using it, you will discover that this question typically has no unique or definite answer--there often is just a continuum of slight but insignificant differences among the groups, even though the extreme groups are significantly different. – whuber May 2 '13 at 15:44
• @whuber is right: It is common to get a partial / incomplete picture of which groups are different. An example of this is discussed on CV here: how-to-interpret-a-post-hoc-tukeys-test. – gung - Reinstate Monica May 2 '13 at 15:58
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2020-09-30 00:18:46
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https://www.physicsforums.com/threads/impact-force-problem.158372/
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# Impact Force Problem
When dropping a metal ball onto a foam mat what is the formula for the impact speed. I know the impact speed, rebound speed, the mass of the ball and time of the ball on the mat.
I know:
F = m[delta]v
.......[delta]t
and
F = mg , where g= -9.8 N/kg
But what is the impact force?
Is it F = m[delta]v + mg ?
.............[delta]t
## Answers and Replies
Related Introductory Physics Homework Help News on Phys.org
Force
$$F = \frac{d\rho}{dt}$$
(Does my LaTeX work?)
ok, but still I don't understand whether the weight force affects the value of the impact force.
The ball is being dropped vertically, so do we add the weight force to dp/dt?
PhanthomJay
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2021-03-05 23:42:53
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https://socratic.org/questions/592e7a4d7c01490e56794bf2
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# How do we prepare a 100*mL of a 0.010*mol*L^-1 solution of Fe^(2+) in aqueous solution...?
Jun 6, 2017
If we use $\text{Mohr's salt}$, ${\left(N {H}_{4}\right)}_{2} F e {\left(S {O}_{4}\right)}_{2} \cdot 6 {H}_{2} O$ whose molar mass is $392.13 \cdot g \cdot m o {l}^{-} 1$. We need approx. $4 \cdot g$ to prepare such a solution.
#### Explanation:
A $0.10 \cdot N$ solution $\equiv$ $0.10 \cdot m o l \cdot {L}^{-} 1$ is required.
Which, given a volume of $100 \cdot m L$ constitutes a molar quantity of....
$\equiv 0.10 \cdot m o l \cdot {L}^{-} 1 \times 100 \times {10}^{-} 3 \cdot L = 0.010 \cdot m o l$
Which constitutes a mass of................
$0.010 \cdot m o l \times 392.13 \cdot g \cdot m o {l}^{-} 1 = 3.921 \cdot g$
Note that $\text{Mohr's salt}$ is the standard $F e \left(+ I I\right)$ salt used in the laboratory, as it is stable with respect to oxidation.
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2019-09-17 14:20:20
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https://www.rdocumentation.org/packages/metacoder/versions/0.2.1/vignettes/introduction.Rmd
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# An introduction to MetacodeR
options(width = 90) set.seed(1) # Knitr library(knitr) library(grid) opts_chunk$set(dev = 'png', fig.width = 7, fig.height = 7, warning = TRUE, message = TRUE) ## Documentation This is only a short demonstration. See the full documentation at http://grunwaldlab.github.io/metacoder_documentation. ## Parsing Many functions that used to be in metacoder have now been moved into the taxa package. These include the flexible parsers and dplyr-like data-manipulation functions. If you have an non-standard data format or want to use the more flexible taxa parsers, check out the intro to the taxa package here. Metacoder now has functions for parsing specific file formats used in metagenomics research. However, for this demonstration, we will be using a parser from the taxa package meant for tabular data. Included in metacoder is an example dataset that is a subset of the Human Microbiome Project data. This dataset has two parts: • An abundance matrix called hmp_otus, with samples in columns and OTUs in rows • A sample information table called hmp_samples, with samples as rows and columns of information describing the samples (e.g. gender). This is the preferred way to encode this type of abundance information in metacoder and taxa. Lets take a look at this data: library(metacoder) print(hmp_otus) print(hmp_samples) We can parse the taxonomic information in the abundance matrix using a parser from taxa: obj <- parse_tax_data(hmp_otus, class_cols = "lineage", class_sep = ";", class_key = c(tax_rank = "info", tax_name = "taxon_name"), class_regex = "^(.+)__(.+)$")
This returns a taxmap object. The taxmap class is designed to store any number of tables, lists, or vectors associated with taxonomic information and facilitate manipulating the data in a cohesive way. Here is what that object looks like:
print(obj)
## Abundance matrix manipulations
### Removing low-abundance counts
Low-abundance sequences might be the result of sequencing error, so typically we remove any counts/OTUs with less than some number of reads. Lets set all counts with less than 5 reads to zero:
obj$data$tax_data <- zero_low_counts(obj, "tax_data", min_count = 5)
There might now be some OTUs with no "real" reads. Lets check:
no_reads <- rowSums(obj$data$tax_data[, hmp_samples$sample_id]) == 0 sum(no_reads) It appears that r sum(no_reads) of r nrow(obj$data$tax_data) OTUs now have no reads. We can remove those OTUs and their associated taxa with filter_obs: obj <- filter_obs(obj, "tax_data", ! no_reads, drop_taxa = TRUE) print(obj) Note how there are fewer taxa now, as well as fewer OTUs. This coordinated manipulation of taxonomic and abundance data is one of the main benefits of using the taxmap class. ### Accounting for un-even sampling These are raw counts, but people typically work with rarefied counts or proportions to avoid sampling depth biasing the results. The function rarefy_obs will return the rarefied counts for a table in a taxmap object, but lets use proportions for this demonstration: obj$data$tax_data <- calc_obs_props(obj, "tax_data") print(obj) ### Getting per-taxon information Currently, we have values for the abundance of each OTU, not each taxon. To get information on the taxa, we can sum the abundance per-taxon like so: obj$data$tax_abund <- calc_taxon_abund(obj, "tax_data", cols = hmp_samples$sample_id) print(obj)
Note that there is now an additional table with one row per taxon.
We can also easily calculate the number of samples have reads for each taxon:
### Comparing two treatments/groups
Usually we are interested in how groups of samples compare. For example, we might want to know which taxa differ between the nose and throat, or between men and women. The function compare_groups facilitates these comparisons:
obj$data$diff_table <- compare_groups(obj, dataset = "tax_abund", cols = hmp_samples$sample_id, groups = hmp_samples$sex) print(obj$data$diff_table)
We can use this information to create what we call a "differential heat tree", which indicates which taxa are more abundant in each treatment:
heat_tree(obj, node_label = taxon_names, node_size = n_obs, node_color = log2_median_ratio, node_color_interval = c(-2, 2), edge_color_interval = c(-2, 2), node_color_range = c("cyan", "gray", "tan"), node_size_axis_label = "OTU count", node_color_axis_label = "Log 2 ratio of median proportions")
In this case, taxa colored tan are more abundant in women and those colored blue are more abundant in men. Note that we have not taken into account statistics significance when showing this, so lets do that. First, we need to correct for multiple comparisons:
obj$data$diff_table$wilcox_p_value <- p.adjust(obj$data$diff_table$wilcox_p_value, method = "fdr")
If we then look at the distribution of p-values, we can see that none are even close to significant:
hist(obj$data$diff_table$wilcox_p_value) There is no need to graph this, but if there still were some significant differences, we could set any difference that is not significant to zero and repeat the last heat_tree command. ### Comparing any number of treatments/groups A single differential heat tree can compare two treatments, but what if you have more? Then we can make a matrix of heat trees, one for each pairwise comparison of treatments like so: obj$data$diff_table <- compare_groups(obj, dataset = "tax_abund", cols = hmp_samples$sample_id, groups = hmp_samples$body_site) print(obj$data\$diff_table)
There is a special function to plot this type of data called heat_tree_matrix:
heat_tree_matrix(obj, dataset = "diff_table", node_size = n_obs, node_label = taxon_names, node_color = log2_median_ratio, node_color_range = diverging_palette(), node_color_trans = "linear", node_color_interval = c(-3, 3), edge_color_interval = c(-3, 3), node_size_axis_label = "Number of OTUs", node_color_axis_label = "Log2 ratio median proportions")
This document is only a short introduction to metacoder and there is much that is not covered. For more information, see our website at http://grunwaldlab.github.io/metacoder_documentation/ and our github repository at https://github.com/grunwaldlab/metacoder. There is also extensive help and examples in the function documentation that can be accessed by, for example, ?heat_tree.
## Feedback
We welcome any kind of feedback! Let us know if you run into problems by submitting an issue on our Github repo: https://github.com/grunwaldlab/metacoder
## Dependencies
The function that runs in silico PCR requires primersearch from the EMBOSS tool kit to be installed. This is not an R package, so it is not automatically installed. Type ?primersearch after installing and loading metacoder for installation instructions.
## Citation
If you use metcoder in a publication, please cite our article in PLOS Computational Biology:
Foster ZSL, Sharpton TJ, Grünwald NJ (2017) Metacoder: An R package for visualization and manipulation of community taxonomic diversity data. PLOS Computational Biology 13(2): e1005404. https://doi.org/10.1371/journal.pcbi.1005404
This package includes code from the R package ggrepel to handle label overlap avoidance with permission from the author of ggrepel Kamil Slowikowski. We included the code instead of depending on ggrepel because we are using functions internal to ggrepel that might change in the future. We thank Kamil Slowikowski for letting us use his code and would like to acknowledge his implementation of the label overlap avoidance used in metacoder.
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2020-08-13 12:02:21
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http://devlog.disco.zone/2015/01/25/starfields-and-state/
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In terms of sheer lines of code and features added, this was an incredibly productive day for Blorp. A few major things happened:
• I finally figured out what the “core gameplay loop” of Blorp will be. Rather than use the time mechanic I was planning on as of my last post, I've decided to instead focus on a “collection” mechanic, replacing the clock pickups with fuel. When enough fuel pickups are collected, the player character's spaceship will become fueled, and they'll be able to leave the level.
• I made some more sprites! The aforementioned spaceship, for one, though it's going to have a lot more tweaking (like, currently it doesn't shoot sick flames out the back when flying). Also, fuel cans to replace the clocks.
• I ADDED THIS DOPE STARFIELD LEVEL TRANSITION HOLY CRAP:
I am, of course, particularly proud of that last one, because just look at it! It almost looks like a competent game developer made it, and here at disco.zone, we strive to bring you almost-passable video games.
There were many other great strides made today, such as finally fixing the bustness of platform collision, and refactoring some ugly platforming physics code. However, not everything was so awesome.
In my haste to add these new additions, I decided to add a new class, Session, which holds a bunch of the current game's state. The Game object already does this, but the idea is that Session should basically be transitive and only specific to the current game, and be totally blown away when the game is restarted.
This was done previously for Monotron, where it was a ten-line class that mainly just held the current score. However, in Blorp, I decided that it should also be in charge of more complex state - mainly, the state of the current level. For example, the Session object holds state like “has the player collected enough fuel to activate their spaceship?” and “has the player entered their spaceship?” This still wouldn't be all that bad, except it also continues some quite-complex methods to mutate this state (for example, a method to enter the next level and a method to go into the “transition” state), which further complicates it. There's now very blurred lines between what goes in the Session and what goes in the Game, and it seems like it will only get blurrier as more polish is added to the game.
Of course, this isn't an insurmountable issue, and it pales in comparison to many of the horror stories of mutable state I've heard before. Still, I'm somewhat annoyed at myself for not thinking harder about the architecture of the game as I rolled ahead on new features. I'm sure I'll find myself paying down this technical debt soon enough.
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2017-04-30 22:39:32
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https://www.physicsforums.com/threads/ploting-graph-in-excel.799685/
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# Homework Help: Ploting graph in excel
1. Feb 24, 2015
### henil
• Member warned about posting with no effort
Mod note:Combined two posts into one.
1. The problem statement, all variables and given/known data
f(g,h) such that g is a function of x and h is a function of y then how can i plot a graph in exccel
2. Relevant equations
3. The attempt at a solution
Last edited by a moderator: Feb 24, 2015
2. Feb 24, 2015
### BvU
Make a row of of y values and a column of x values
Make a table of f(x,y)
(Excel 2010:) Insert tab | Other Charts | Surface
(Excel 2003:) Click Chart wizard | Surface
Example: x and y from 0 to $2\pi$ in steps of 0.1. $\ \ g = \sin(2x/\pi),\ \ h = \cos(3y/\pi),\ f = g * h$
3. Feb 24, 2015
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2018-07-16 03:27:42
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https://demo.formulasearchengine.com/wiki/Radiative_equilibrium
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In physics, radiative equilibrium is the condition where a steady state system is in dynamic equilibrium, with equal incoming and outgoing radiative heat flux and negligible heat transfer by conduction and convection.[1] In atmospheric physics, under conditions of radiative equilibrium, total flux is constant with depth.[2] In astrophysics, radiative equilibrium is used to determine atmospheric radiation of stars. In climate science, the net change in the tropopause after temperatures readjust to radiative equilibrium in the stratosphere, is used to determine the radiative forcing, as part of an assessment of natural and anthropogenic climate change.[3]
## Definitions
### History
In 1791 Pierre Prevost showed that all bodies radiate heat and concluded, Radiation will exactly compensate absorption. He used the terms absolute and relative equilibrium to describe changes.[4] Prevost considered that what is nowadays called the photon gas or electromagnetic radiation was a fluid that he called "free heat" (Template:Lang-fr). Prevost proposed that free radiant heat is a very rare fluid, rays of which, like light rays, pass through each other without detectable disturbance of their passage. Prevost's called his theory movable equilibrium of heat, now designated as the theory of exchanges, which stated that each body radiates to, and receives radiation from, other bodies. The radiation from each body is emitted regardless of the presence or absence of other bodies.[5][6][7]
In 1906 Karl Schwarzschild postulated the radiative equilibrium (German: Strahlungsgleichgewicht) dependent on Kirchhoff's law of thermal radiation, when he studied the sun.[8]
Following Planck (1914),[9] a radiative field is often described in terms of specific radiative intensity, which is a function of each geometrical point in a space region, at an instant of time.[10][11] A detailed definition is given by Goody and Yung (1989).[11] They think of the interconversion between thermal radiation and heat in matter. From the specific radiative intensity they derive ${\displaystyle \mathbf {F} _{\nu }}$, the monochromatic vector flux density of radiation at each point in a region of space, which is equal to the time averaged monochromatic Poynting vector at that point (Mihalas 1978[12] on pages 9–11). They define the monochromatic volume-specific rate of gain of heat by matter from radiation as the negative of the divergence of the monochromatic flux density vector; it is a scalar function of the position of the point:
${\displaystyle h_{\nu }=-\nabla \cdot \mathbf {F} _{\nu }}$.
They define (pointwise) monochromatic radiative equilibrium by
${\displaystyle \nabla \cdot \mathbf {F} _{\nu }=0}$ at every point of the region that is in radiative equilibrium.
They define (pointwise) radiative equilibrium by
${\displaystyle h=\int _{0}^{\infty }h_{\nu }d\nu =0}$ at every point of the region that is in radiative equilibrium.
This means that, at every point of the region of space that is in (pointwise) radiative equilibrium, the total, for all frequencies of radiation, interconversion of energy between thermal radiation and energy content in matter is nil.
Chandrasekhar (1950, p. 290)[13] writes of a model of a stellar atmosphere in which "there are no mechanisms, other than radiation, for transporting heat within the atmosphere ... [and] there are no sources of heat in the atmosphere." This is hardly different from Schwarzschild's 1906 approximate concept, but is more precisely stated.
### Exchange equilibrium between systems
{{#invoke:see also|seealso}} Radiative exchange equilibrium occurs with thermodynamic systems. Planck (1914)[9] refers to a condition of thermodynamic equilibrium, in which "any two bodies or elements of bodies selected at random exchange by radiation equal amounts of heat with each other."
The term radiative exchange equilibrium can also be used to refer to two specified regions of space that exchange equal amounts of radiation by emission and absorption (even when the steady state is not one of thermodynamic equilibrium, but is one in which some sub-processes include net transport of matter or energy including radiation).
Radiative equilibrium for a star, is taken as a whole and not confining attention only to its atmosphere. And when the rate of heat transfer from nuclear reactions plus viscosity to the microscopic motions of the material particles of the star is balanced by the transfer of energy by electromagnetic radiation from the star to space.
A star that is radiating energy to space cannot be in a steady state of temperature distribution unless there is a steady supply of energy from nuclear reactions within the star, to support the radiation to space.[14]
Planetary equilibrium temperature is the theoretical temperature for a blackbody, which does not consider a radiative atmosphere.
## References
1. {{#invoke:citation/CS1|citation |CitationClass=book }}
2. {{#invoke:citation/CS1|citation |CitationClass=book }}
3. {{#invoke:citation/CS1|citation |CitationClass=book }}
4. Prevost, P. (1791). Mémoire sur l'equilibre du feu. Journal de Physique (Paris), vol 38 pp. 314–322.
5. {{#invoke:citation/CS1|citation |CitationClass=book }}
6. {{#invoke:citation/CS1|citation |CitationClass=book }}
7. Partington, J.R. (1949). An Advanced Treatise on Physical Chemistry, volume 1, Fundamental Principles. The Properties of Gases, Longmans, Green and Co, London, p. 467.
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2019-07-21 15:11:43
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http://jvmwriter.org/latex-error/latex-error-file-psfig-sty.html
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Home > Latex Error > Latex Error File Psfig.sty
# Latex Error File Psfig.sty
If it is missing, you probably put it either at the wrong place (try a folder in tex\latex) or didn't update the filename database. The preamble is as follows: \documentclass[12pt]{article} \usepackage{psfig} \usepackage{subfigure} \usepackage{rotating} \usepackage{fullpage} \usepackage{setspace} %\singlespacing %\onehalfspacing \doublespacing \usepackage{mathptm} \usepackage{amsmath} %\usepackage{cite} \usepackage{bibentry} \usepackage{fancyhdr} I tried updating everything using TeX Live Utility, but that didn't fix Discover what's new with Crystal Reports now. Thanks, this is another problem, maybe I should post it in another message, my original problem is solved, thanks. navigate here
Can anyone say if it would be better to install from the Gutsy repositories rather than the TeXlive 2007 ISO? However, since "/usr/local/texlive" is an installation directory, one would have thought TeX would be able to find it. On my PC, the complete MikTex 2.8 and the newest version of > Winedt were installed. On my PC, the complete MikTex 2.8 and the newest version of Winedt were installed. http://tex.stackexchange.com/questions/54524/how-do-i-install-psfig-sty-in-mactex
Please don't fill out this field. On my PC, the complete MikTex 2.8 and the newest version of > > Winedt were installed. It may also work with the ArborText "dvips" driver, and with the "dvi2ps" written by Tony Li ; see the "unsupported" directory. ================================================================== INSTALLATION To install psfig, simply copy psfig.sty into The exactly same tex file can be run on another > PC without any problems.
Browse other questions tagged packages installing mactex or ask your own question. Simplify your report design, integration and deployment - and focus on what you do best, core application coding. http://p.sf.net/sfu/bobj-july_______________________________________________ MiKTeX-Users mailing list [hidden email] https://lists.sourceforge.net/lists/listinfo/miktex-users « Return to MiKTeX | 1 view|%1 views Loading... Ulrike Fischer If you would like to refer to this comment somewhere else in this project, copy and paste the following link: aleplgr - 2007-06-19 OK, it was just a problem
Debian bug tracking system administrator . Hot Network Questions What happens if one brings more than 10,000 USD with them into the US? The psfig package is largely superseded, for various reasons, including its non-free licence. Source With LaTeX2e (\documentclass): use the geometry-package to set the page layout.
I've already compiled a simple latex file using the winedt so I assume that the miktex is installed ok, I think that this other file I'm trying to compile has graphic LaTeX Error: File psfig.sty' not found." error message. Reported by: Roland Stigge Date: Mon, 16 Aug 2004 15:48:02 UTC Severity: serious Found in version 0.5.20-1 Fixed in version uudeview/0.5.20-2 Done: Chris Hanson Bug is archived. here's the test file: \documentstyle[psfig,color,12pt]{book} \pagestyle{plain} \textheight 27 cm \textwidth 18.0 cm \hoffset -2.5 cm \voffset -4.5 cm \evensidemargin 1 cm \oddsidemargin 1 cm \pretolerance 1000 \tolerance 0 \definecolor{gris}{gray}{0.9} \begin{document} \setlength{\baselineskip}{1.5em}
You seem to have CSS turned off. https://sourceforge.net/p/miktex/discussion/33790/thread/13d4ffa3/ Any hint is highly appreciated! ------------------------------------------------------------------------------ Let Crystal Reports handle the reporting - Free Crystal Reports 2008 30-Day trial. Guess this will teach me to check the repos first! :D meho_rJanuary 27th, 2008, 11:07 PMYou're welcome. Is it correct to write "teoremo X statas, ke" in the sense of "theorem X states that"?
It could also be that the repository you tried to use were temporarly down. http://jvmwriter.org/latex-error/latex-error-file-esint-sty.html Free forum by Nabble Edit this page SourceForge Browse Enterprise Blog Deals Help Create Log In or Join Solution Centers Go Parallel Resources Newsletters Cloud Storage Providers Business VoIP Providers Then the formatting worked fine. Then try to install xcolor from this local folder. (But it is years ago that I used a local folder, so I'm not sure if this is the correct procedure).
LaTeX Error: File psfig.sty' not found. > > Type X to quit or to proceed, > or enter new name. (Default extension: sty) > > Enter file name: > ! To use psfig, you must therefore be using a postscript device for your output. Thank you for reporting the bug, which will now be closed. his comment is here But I don't think that you are really needing pstricks.
This version of psfig/tex works with the unix "dvips" PostScript driver, which is available from labrea.stanford.edu, and with OzTeX. If you are using OzTeX, change the driver line at the top of the psfig.sty file to read \def\setDriver{\OzTeXDriver} instead of \def\setDriver{\DvipsDriver} A users guide and manual page can be found Message #15 received at [email protected] (full text, mbox, reply): From: Chris Hanson To: [email protected] Subject: Bug#266068: fixed in uudeview 0.5.20-2 Date: Tue, 17 Aug 2004 16:20:22 -0400 Source: uudeview Source-Version:
## Daniel Smith, Tom Rokicki, Robert Russell, George V.
Toggle useless messagesView this report as an mbox folder, status mbox, maintainer mbox Report forwarded to [email protected], Chris Hanson : Bug#266068; Package uudeview. Download the contents of this package in one zip archive (124.8k). TrafferthJanuary 27th, 2008, 07:33 PMOkay, having tried setting the various parameters listed in texconfig without any joy, I am going to reinstall from the TeXlive 2007 iso. In the error I also get the message:"The package will be installed from: and I have to select if install from internet or install from a directory or from MiKTeX DVD,I
It should have the same syntax as psfig. If you really want to use it (graphicx is much better), you will have to install in manually http://dante.ctan.org/tex-archive/help/Catalogue/entries/psfig.html-- Mit freundlichen Grüßen Ulrike Fischer ------------------------------------------------------------------------------ Let Crystal Reports handle the reporting Any hint is highly appreciated! ------------------------------------------------------------------------------ Let Crystal Reports handle the reporting - Free Crystal Reports 2008 30-Day trial. weblink Message #5 received at [email protected] (full text, mbox, reply): From: Roland Stigge To: Debian Bug Tracking System Subject: uudeview: FTBFS: psfig.sty not found Date: Mon, 16 Aug 2004 17:31:30
But I really advise you not to use psfig + latex2.09! The program should have been installed if you're using Miktex. Full text and rfc822 format available. Baron Grey, Gerhard Tobermann and all others who have contributed code and comments to this project! ================================================================== NEW IN VERSION 1.10 - Support for OzTeX (derived from Baron Grey's OzTeX version
TeX cannot find the texlive directory! I seem to remember that I might have changed an install directory from the defaults, so I shall run the installation script without changing anything this time, and will see if Copy sent to Chris Hanson . psfig has a non-free license.
And it's obsolete since 1994. \usepackage{graphicx} Best Martin ------------------------------------------------------------------------------ Let Crystal Reports handle the reporting - Free Crystal Reports 2008 30-Day trial. TrafferthJanuary 25th, 2008, 04:35 PMHi all, I have a problem with TeX Live 2007. If you would like to refer to this comment somewhere else in this project, copy and paste the following link: engineereeyore - 2007-06-19 You should have a dvi2pdf button in WinEdt You can also try to download e.g.
Benham, 1997,2003 nCipher Corporation Ltd, 1994-97 Ian Jackson. The determinant of the matrix How does a Spatial Reference System like WGS84 have an elipsoid and a geoid? Well, I learned it the same way you did;) Glad it works. I also tried with the package manager from Start-Programs-MikTeX2.5-Browse packages but still can't connect to nowhere..
asked 4 years ago viewed 6744 times active 3 years ago Related 51How do I add a .sty file to my MacTeX/TeXShop installation?4Installing tabu.sty with MacTeX 2010.2How to install MacTeX on Terms Privacy Opt Out Choices Advertise Get latest updates about Open Source Projects, Conferences and News. You will only mess up you installation if you try to do it manually. It would seem I have broken TeX quite comprehensively... :( Seeing as there's not that much on my linux box, I suppose I can reinstall Gutsy and try again.
This allows you to insert an external file name.ext by using the command \includegraphics{name} where .ext is automatically appended from a list of appropriate file extensions, such as .pdf, .eps, .png, No further changes may be made. Ulrike Fischer If you would like to refer to this comment somewhere else in this project, copy and paste the following link: aleplgr - 2007-06-18 Hi!
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2017-11-20 21:24:55
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https://zenodo.org/record/3891124/export/csl
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Conference paper Open Access
# Technique for Finding and Investigating the Strongest Combinations of Cyberattacks on Smart Grid Infrastructure
Igor Kotsiuba; Inna Skarga-Bandurova; Alkiviadis Giannakoulias; Mykhailo Chaikin; Aleksandar Jevremovic
### Citation Style Language JSON Export
{
"DOI": "10.1109/bigdata47090.2019.9006335",
"language": "eng",
"author": [
{
"family": "Igor Kotsiuba"
},
{
"family": "Inna Skarga-Bandurova"
},
{
},
{
"family": "Mykhailo Chaikin"
},
{
"family": "Aleksandar Jevremovic"
}
],
"issued": {
"date-parts": [
[
2020,
2,
24
]
]
},
"abstract": "<p>Recently, smart grids have become a vector of the energy policy of many countries. Due to structural and operation features, smart grids are a constant target of combined and simultaneous cyberattacks. To maximize security and to optimize existing network schemes to prevent cyber intrusion, in this paper, we propose an approach to decision support in finding and identifying the most potent attack combinations that can set the system to maximum damage. The main purpose is to identify the most severe combinations of attacks on smart grid components that potentially can be implemented from the perspective of the attacker. In this context, the problem of finding weaknesses points in the network configuration of a smart grid and assessing the impact of events on cyberinfrastructure is considered. The technique for detecting and investigating the strongest combinations of cyberattacks on the smart grid network is given with an example of the analysis of the spread of pandemic software in a system with arbitrary structure. </p>",
"title": "Technique for Finding and Investigating the Strongest Combinations of Cyberattacks on Smart Grid Infrastructure",
"type": "paper-conference",
"id": "3891124"
}
35
69
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2022-08-19 02:08:56
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https://www.physicsforums.com/threads/equation-of-a-line-that-lies-on-a-z-f-x-y-function.837321/
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# Equation of a line that lies on a z=f(x,y) function
1. Oct 12, 2015
### masterchiefo
• Member warned about posting with no effort shown
1. The problem statement, all variables and given/known data
Hello,
I have this function z(x,y) = sin(2*x)*cos(2*y)*e-(x^2+y^2)/6
I need to find an equation of any curve line that lies on that z(x,y) function.
2. Relevant equations
3. The attempt at a solution
Sorry, I really have no idea how to proceed on this, been looking on my note book and google and have not find anything.
2. Oct 12, 2015
### SteamKing
Staff Emeritus
What do you mean a line "that lies on that ... function"?
Are you talking about a line which is tangent to the function's surface?
3. Oct 12, 2015
### masterchiefo
Sorry edited the post, its a curve line.
To clarify, the curve line has to be part of the z(x,y) function. The curve line has to be completely on the function. If I didn't clarify anything, let me know.
4. Oct 12, 2015
### Ray Vickson
A curve $(x(t),y(t),z(t))$ lying in the surface must project down to a curve on the $x\,y$ plane, obtained by just ignoring the $z$-component $z(t)$.
5. Oct 12, 2015
### masterchiefo
how do I proceed on this? what are the steps I have to take.
Do I take a random number of x and y and plot it in my equation to find z?
sorry I have never done this before.
6. Oct 12, 2015
### Staff: Mentor
It might be that you need to find level curves on this surface. As already mentioned, the graph of $z = f(x, y) = \sin(2x) \cos(2y) e^{-(x^2+y^2)/6}$ represents a surface in three dimensions. If you set z to some specific value, that defines a curve that lies on the surface. On level curve is when z = 0, or $0 = \sin(2x) \cos(2y) e^{-(x^2+y^2)/6}$. Geometrically, this level curve is the intersection of the surface with the x-y plane.
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2018-03-24 00:51:01
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https://www.witszen.com/articles/smb-share-with-linux-machines
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# Accessing An SMB Share With Linux Machines
Server Message Block (SMB) is a network resource sharking protocol based on a client-server model. While Microsoft authors the most popular Microsoft SMB protocols and Microsoft operating systems are the most common SMB servers, it is possible for Linux machines to access an SMB share. Here, we’ll take a closer look at what an SMB share is, Linux tools for accessing SMB shares, and provide an example of how to access an SMB share from a Linux machine.
#### What Is An SMB File Share?
An SMB file share is a shared resource clients can access on a server.
An SMB file share is most often a directory, but an SMB share can also be other resources such as a specific shared file or a printer. Sometimes, you may see the term CIFS share used interchangeably with SMB share. This is because Common Internet File System (CIFS) is often directly associated with SMB version 1 (SMBv1).
• ℹ️ Note: For more information on the technical differences between SMBv1/CIFS, SMBv2, and SMBv3, we recommend reading SMB Most Important Features. For the purpose of this article, just note that SMBv1 is insecure and inefficient and shouldn’t be used today. Generally, if you use SMB you’ll want to use SMBv3.0, SMBv3.02, or SMB v3.1.1 if your servers and clients support it.
SMB clients access SMB servers to gain access to the resources provided as SMB shares. A SMB server/client connection can be something as simple as two Windows 10 PCs sharing resources. Alternatively, it can be something more complex like using SMBv3 to work with clustered shared volumes and storage for virtual machines in a HyperV cluster.
#### Linux Tools For SMB File Shares
cifs-utils and smbclient are two popular examples of Linux tools that can be used with SMB shares.
There are a variety of graphical tools for Linux desktop users that can act as an SMB client. For example, Konqueror a web browser and file management tool for KDE desktops supports SMB. So does the popular SMB4K utility.
However, in this section, we’ll look at two lower-level tools that can provide extensibility and options for environments without a graphical user interface (GUI).
#### Where Samba Fits In
If you’ve done some prior research, you likely know working with SMB on Linux systems often involves Samba. But what is Samba and how does it fit in here? Samba is a software suite that enables file and print services on Linux operating systems. Many Linux SMB tools — including the two we’ll mention below — are (or were) directly part of the Samba suite.
cifs-utils for mounting SMB File shares on Linux
cifs-utils is a suite of tools used to mount SMB shares on Linux machines. This suite of tools used to be part of the Samba suite, but can now be installed independent of the Samba suite. cifs-utils is available in the repositories for many popular flavors of Linux including Ubuntu, Debian, CentOS, Fedora, OpenSUSE, and Arch Linux.
If it’s not already installed, you can usually install cifs-utils with your operating system’s package manager. For example, on Ubuntu, something like:
$sudo apt-get install cifs-utils Or on CentOS something to the effect of:$ sudo yum install -y cifs-utils
smbclient acts as an ftp-like client for SMB file share access
smbclient is a part of the Samba suite of tools. With smbclient you get an interactive client that enables you to do things like list files in remote directories (ls), download (get) files, upload (put) files, and delete files (rm). If you’ve ever used command-line FTP clients, smbclient will likely feel very similar.
#### How to access an SMB File Share with Linux Machines
Here are some specific examples of how to use Linux to mount or access an SMB share.
Now that we’ve reviewed what smbclient and cifs-utils are, let’s take a look at some basic commands you can use to access an SMB share on the same network as your Linux machine. Note that SMB versions, configurations, file access permissions, encryption, and authentication methods can all impact how these samples work in your environment.
Using cifs-utils to mount Mounting an SMB share on a Linux machine
Once you have cifs-utils installed, you can mount an SMB share from a Windows machine to your Linux machine following a process like this:
Create a directory where you will mount the share, here we’ll create a directory at “/mnt/stovetop”
$sudo mkdir /mnt/stovetop Authenticate to the Windows share and mount it at the directory created in the previous step. In basic cases, you can do this with the command$ sudo mount -t cifs -o username= ///<path/to/share>
Here, we’ll authenticate as a user named “johnny” and use the SMB share on at “\\WindowsKitchen\pepperneggs” which has two files “eggs.txt” and “peppers.txt” in it.
$sudo mount -t cifs -o username=johnny //WindowsKitchen/pepperneggs /mnt/stovetop Password for johnny@//WindowsKitchen/pepperneggs:$
Now we can see the files on our Linux machine!
$ls /mnt/stovetop/ eggs.txt peppers.txt$
Using smbclient to access an SMB share from a Linux machine
Now, suppose we want to simply access the same share and interactively view the contents of \\WindowsKitchen\pepperneggs from our Linux machine.
Generally, the command to do this is:
$smbclient -U \\\\\\<path\\to\\share> Note that we need to use the extra backslash characters to escape the backslash characters in the Linux bash shell. Let’s look at a specific example using the same “\\WindowsKitchen\pepperneggs” SMB file share. First, we’ll connect to the share:$ smbclient -U david \\\\WindowsKitchen\\pepperneggs
Try “help” to get a list of possible commands.
smb: \>
Now, let’s list the files we can see at the “smb: \>” prompt:
smb: \> ls
. D 0 Sun Feb 7 15:16:13 2021
.. D 0 Sun Feb 7 15:16:13 2021
eggs.txt A 41 Sun Feb 7 15:16:34 2021
peppers.txt A 14 Sun Feb 7 15:16:04 2021
243863551 blocks of size 4096. 202309870 blocks available
smb: \>
Finally, let’s download “eggs.txt” and exit back to our local Linux prompt.
smb: \> get eggs.txt
getting file \eggs.txt of size 41 as eggs.txt (1.1 KiloBytes/sec) (average 1.1 KiloBytes/sec)
smb: \> exit
$ls eggs.txt$
Success! We’ve downloaded the file from our Windows SMB file share to our Linux machine!
Using curl to access a legacy SMBv1 file share
You may have heard curl — the popular Linux tool for all sorts of file transfers and network connections — supports SMB. This is true to an extent, but unfortunately it is only SMBv1. An generic command to download a file using curl and SMBv1 is:
$curl -u “” smb:///<path/to/shared/file> -o <path/to/local/file> Since SMBv1 is usually disabled on production systems, you’re likely to see an error like this if you try:$ curl -u “johnny” smb://WindowsKitchen/pepperneggs/eggs.txt -o eggs.txt
Enter host password for user ‘johnny’:
$On an SMB file share with SMBv1 enabled (which we do NOT recommend!) the same command will succeed.$ curl -u “johnny” smb://WindowsKitchen/pepperneggs/eggs.txt -o eggs.txt
Enter host password for user ‘johnny’:
% Total % Received % Xferd Average Speed Time Time Time Current
$ls eggs.txt$
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2021-05-13 05:39:51
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https://www.esaral.com/q/how-would-you-name-the-following-compounds/
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How would you name the following compounds?
Question.
How would you name the following compounds?
(i) $\mathrm{CH}_{3}-\mathrm{CH}_{2}-\mathrm{Br}$
solution:
(i) Bromoethane
(ii) Methanal
(iii) Hexyne
Editor
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2022-08-09 19:06:58
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http://math.stackexchange.com/tags/algebraic-topology/hot
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# Tag Info
If you've done any number theory, you are probably aware of the following classic formula: Let $L/K$ be an extension of number fields, $\mathfrak{p}\in\text{Spec}(\mathcal{O}_K)$ and $\mathfrak{p}\mathcal{O}_L=\mathfrak{P}_1^{e_1}\cdots\mathfrak{P}_n^{e_n}$. Define $f_i$ to be $\,[\mathcal{O}_L/\mathfrak{P}_i:\mathcal{O}_K/\mathfrak{p}]$. Then, $$\sum_i ... 5 I don't agree with the explanation on Wikipedia. They write down an isomorphism, which is an instance of a natural isomorphism, and then claim that it is not natural. This is not correct. But their reasoning is that they - secretly - change the domain categories. More specifically: The natural isomorphism \pi_1(X \times Y) \cong \pi_1(X) \times \pi_1(Y) ... 5 This only really works in the specific case of distinguishing 1 dimensional and n dimensional Eulcidean space. For higher dimensions you will need a more sophisticated approach. Normally this includes the use of techniques from algebraic topology. See this page for one approach to solving this problem. To comment specifically on why your approach ... 4 Here you only need to care about the topology coming from the neighborhood basis. Let B be a topological space. Assuming we can find local trivializations$$ f_{U}:U_{E}\cong U\times \mathbb{R}^{n} $$such that on U_{E} this is given by product topology of U\times \mathbb{R}^{n}. Then you "glue" two such neighborhoods together using ... 4 There are plenty of tests for seeing if a space is NOT a CW-complex, like checking to see if it fails to be normal, Hausdorff, locally contractible, etc. Usually, one only cares about a space being homotopy equivalent to a CW-complex. There is a statement in Hatcher's book (Proposition A.11) that says if Y is a space, X is a CW-complex, and there are ... 4 The comb space works although any point on the 'base line' can be deformation retracted to so depending on the interpretation of your question, this might not quite fit your criteria. Instead, we can take a space which is morally the comb space but where we quotient out by the subset of points which can be deformation retracted onto. I believe (though I have ... 3 Certainly there's no test for whether a topological space is equal to a CW complex. It's no more plausible to test whether it's homeomorphic to one, which may be what you meant by your phrasing "is a CW complex." Another answer has given a condition for a space to be homotopy equivalent to a CW complex, which is of much more significance. Another interesting ... 3 Here is a general fact that will yield the result you seek. If G is a Lie group and X is a manifold on which G acts freely and properly, then X/G has a natural manifold structure and the projection X\rightarrow X/G is a principal G-bundle. In particular, X\rightarrow X/G is a fibre bundle with fibre G. Since O(k) is compact, it ... 3 It's not at all clear to me that the two questions in Question 1 are asking the same thing, but in any case the answer to the first one is "definitely not". For example, think about a flat family of elliptic curves degenerating to a nodal rational curve. The general fibre is topologically a torus, so has H_1 \simeq \mathbf Z \oplus \mathbf Z, but the ... 3 The crucial fact here is that the fundamental group of a topological group must be abelian. As you point out in the comment, \pi_{1}(M_{f}) contains \pi_{1}(X) as a subgroup and hence if \pi_{1}(X) is not abelian, \pi_{1}(M_{f}) is neither and so M_{f} cannot be a topological group. The only closed surfaces with abelian fundamental group are the ... 2 Q1. No, it's not true that all fibers of a flat map are homotopy equivalent. For instance, the blow-up of \mathbf P^1 \times \mathbf P^1 at a point maps to \mathbf P^1. This map is flat and all fibers except for one are spheres; however, the remaining fiber is a wedge of two spheres. More generally, in a flat family of curves you can "collapse" a simple ... 2 Let \mathcal{F} be your space of functions (say continuous functions, or all functions etc.) on some space X to some space Y. The function e (evaluation) is defined on \mathcal{F} \times X to Y, by e(f,x) = f(x). A topology \mathcal{T} on \mathcal{F} is admissible iff e is a continuous map (in the respective topologies on Y and the ... 2 I'll stick with your question 2. By definition a fiber bundle E\to X is locally homeomorphic to U\times F where U is open in X and F is the fiber. So in fact if X is connected all the fibers of E are homeomorphic. On the other hand if X has distinct connected components the fibers over each one can differ arbitrarily, so that there's no ... 2 Try to use the the uniqueness (up to homotopy equivalence) of Eilenberg-Maclane spaces to show that if \pi_0 X = G and \pi_i X = 0 for i > n, then for some n onwards,you have that X_n is homotopy equivalent to an Eilenberg-Maclane space. With this you get maps defined on a cofinal subset, which induces homotopy equivalences of X_n with the ... 2 This is a difficult homotopy equivalence to visualize, but I've attempted to draw a picture of what's going on. First you poke a hole in your surface starting at +\infty and pushing in from the right. Similarly poke a hole from the left. You can see this is a homotopy equivalence by analogy with an infinite cylinder, which can be visualized as having two ... 1 "Unsigned" means just count up the number of crossings. "Signed" means count them with a sign. That is, some crossings count as +1 and others as -1. Loosely, the determination goes as follows. Let's say the curves C_1 and C_2 are parameterized. Define a crossing to be positive if when travelling along C_1 in the direction of the parameterization, ... 1 That's a tall order! I would wager that very few people understand Freedman's proof completely. I found this series of video lectures that he gave to be quite helpful. In particular, it will give you an idea about the sort of mathematics that is involved. Also, Freedman and Quinn's book on 4-manifold topology is great! Even if you aren't to the point ... 1 Say we want to contract it to (0, 0). Look at a point on the vertical interval. Any neighbourhood of that point contains some points of the sine-curve, but any point on the sine curve has to go all the way back through (1, 0) to get to the origin. Thus a contraction is impossible to do continuously. 1 The observations allow you to argue much in the same way you do for S^1. For the case of the fundamental group, you can construct a "universal cover" X for \mathbb{S} called the digital line. The space X can be viewed as the set of integers \mathbb{Z} with topology generated by the sets \{2n\} and \{2n,2n+1,2n+2\}. The "covering map" p \colon ... 1 I presume that you are computing \tilde{H}_{i}((\Delta^{n})^{k};\mathbb{Z}), i.e. you are taking coefficients in \mathbb{Z}. To say that this group is free of rank n \choose k+1 when i = k means that$$\tilde{H}_{k}((\Delta^{n})^{k};\mathbb{Z}) \cong \mathbb{Z}^{{n \choose k+1}} \cong \overbrace{\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}}^{{n ...
Using Morse theory (I will let you fill in the details): Consider the Morse function $f:SU(n)\to\mathbb{R}$ given by $[z_{ij}]\mapsto\text{Re}(\sum_ic_iz_{ii})$ for fixed constants $1<c_1<<c_2<\ldots<c_n\in\mathbb{R}$. It has a critical point of index 0, and the next smallest index is 3 (figure out why we want $c_1<<c_2$). Thus $SU(n)$ ...
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2014-03-08 14:19:15
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https://mersenneforum.org/showthread.php?t=21989
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mersenneforum.org > Math Basic Number Theory 17: quadratic reciprocity
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2017-01-31, 14:41 #1 Nick Dec 2012 The Netherlands 32378 Posts Basic Number Theory 17: quadratic reciprocity Which numbers are squares? More precisely, for which $$a$$ does there exist $$x$$ with $$x^2=a$$? The answer depends on which number system we work in, of course. In the real numbers, if $$a\geq 0$$ then $$a$$ is a square (put $$x=\sqrt{a}$$), while if $$a<0$$ then it is not a square (since $$x^2\geq 0$$ for all real $$x$$), so the squares are precisely the non-negative numbers. It follows, for any real numbers $$x,y$$, that if $$x$$ is not a square and $$y$$ is not a square then their product $$xy$$ is a square. This is a pattern that we shall see again shortly. Let $$n$$ be a positive integer. An integer $$a$$ is called a quadratic residue modulo $$n$$ if there exists an integer $$x$$ such that $$x^2\equiv a\pmod{n}$$ (or, equivalently, if $$\bar{a}$$ is a square in $$\mathbb{Z}/n\mathbb{Z}$$). As ever in modular arithmetic, the key is to understand what happens modulo the prime numbers. And every integer is a quadratic residue modulo 2, so we focus on odd primes. Let $$p$$ be an odd prime number. For each integer $$a$$, we define the Legendre symbol $$(\frac{a}{p})$$ as follows: if $$p$$ divides $$a$$ then $$(\frac{a}{p})=0$$. If $$p$$ does not divide $$a$$ then $$(\frac{a}{p})=1$$ if $$a$$ is a quadratic residue modulo $$p$$, and -1 otherwise. For example, $$(\frac{-15}{5})=0$$, $$(\frac{19}{5})=1$$ and $$(\frac{3}{5})=-1$$. Proposition 99 Let $$p$$ be an odd prime number. Then, for all integers $$a$$, $\left(\frac{a}{p}\right)\equiv a^{\frac{p-1}{2}}\pmod{p}.$ proof Take any integer $$a$$. If $$p|a$$ then $$(\frac{a}{p})=0\equiv a^{\frac{p-1}{2}}\pmod{p}$$. Suppose $$p$$ does not divide $$a$$. Then (by corollary 40) $$a^{p-1}\equiv 1\pmod{p}$$ so (by proposition 60) $$a^{\frac{p-1}{2}}\equiv 1\pmod{p}$$ or $$a^{\frac{p-1}{2}}\equiv -1\pmod{p}$$. Now the unit group of the integers modulo $$p$$ is cyclic (by theorem 90) so there exists an integer $$g$$ such that $\left(\mathbb{Z}/p\mathbb{Z}\right)^*=\{\bar{1},\bar{g},\bar{g}^2,\bar{g}^3, \ldots,\bar{g}^{p-2}\}$ with $$\bar{g}^{p-1}=\bar{1}$$. Let $$H=\{\bar{g}^{2n}:n\in\mathbb{Z}\}$$, the set of all squares in $$(\mathbb{Z}/p\mathbb{Z})^*$$. Then $$H$$ is the cyclic subgroup generated by $$\bar{g}^2$$ and, by proposition 86, $$H$$ has precisely $$\frac{p-1}{2}$$ elements, i.e. $H=\{\bar{1},\bar{g}^2,\bar{g}^4,\bar{g}^6,\ldots,\bar{g}^{p-3}\}.$ As $$p$$ does not divide $$a$$, $$\bar{a}$$ is an element of $$(\mathbb{Z}/p\mathbb{Z})^*$$, so $$\bar{a}=\bar{g}^r$$ for some $$r\in\{0,1,\ldots,p-2\}$$. If $$(\frac{a}{p})=1$$ then $$\bar{a}\in H$$ so $$r$$ is even, i.e. $$r=2s$$ for some integer $$s$$, and therefore $$\bar{a}^{\frac{p-1}{2}}=\bar{g}^{s(p-1)}=\bar{1}$$ giving $$a^{\frac{p-1}{2}}\equiv 1\pmod{p}$$. Conversely, if $$a^{\frac{p-1}{2}}\equiv 1\pmod{p}$$ then $$\bar{g}^{\frac{r(p-1)}{2}}=\bar{1}$$ and $$\bar{g}$$ has order $$p-1$$ so, by proposition 75, $$p-1$$ divides $$\frac{r(p-1)}{2}$$. Thus $$r$$ is even and $$\bar{a}=\bar{g}^r\in H$$, hence $$(\frac{a}{p})=1$$. ∎ Corollary 100 Let $$p$$ be an odd prime number. Then, for all integers $$a,b$$, $$(\frac{ab}{p})=(\frac{a}{p})(\frac{b}{p})$$. proof For any integers $$a,b$$, proposition 99 gives $\left(\frac{ab}{p}\right)\equiv (ab)^{\frac{p-1}{2}} =a^{\frac{p-1}{2}}b^{\frac{p-1}{2}}\equiv \left(\frac{a}{p}\right) \left(\frac{b}{p}\right)\pmod{p}$ hence $$(\frac{ab}{p})=(\frac{a}{p})(\frac{b}{p})$$. ∎ In particular, for any odd prime number $$p$$ and integers $$a,b$$, if $$a$$ is not a square modulo $$p$$ and $$b$$ is not a square modulo $$p$$ then their product $$ab$$ is a square modulo $$p$$ (just as with real numbers). Recall that, for any integer $$a$$ and positive integer $$n$$, we write $$a\bmod{n}$$ for the remainder on dividing $$a$$ by $$n$$, which is an integer in the range 0 to $$n-1$$ inclusive. Proposition 101 Let $$p$$ be an odd prime number and $$a$$ an integer not divisible by $$p$$. Define $$A=\{a,2a,3a,\ldots,\frac{p-1}{2}a\}$$ and $$r=\#\{x\in A:x\bmod{p}>\frac{p}{2}\}$$. Then $$(\frac{a}{p})=(-1)^r$$. proof Each element of the set $$A$$ is congruent (modulo $$p$$) with a unique integer between $$-\frac{p}{2}$$ and $$\frac{p}{2}$$, and no element of $$A$$ is congruent with 0 (modulo $$p$$) so all these integers are positive or negative. Let $$r_1,r_2,\ldots,r_k$$ be the positive ones and $$-s_1,-s_2,\ldots,-s_m$$ the negative ones. (Thus $$k,m$$ are non-negative integers with $$k+m=\frac{p-1}{2}$$ and, for each $$i\in\{1,2,\ldots,k\}$$ and each $$j\in\{1,2,\ldots,m\}$$ we have $$02$$) so $$(\frac{a}{p})=(-1)^r$$. ∎ Example For which odd prime numbers $$p$$ is 2 a square modulo $$p$$? Let $$A=\{2,4,6,\ldots,p-1\}$$. In this case, all elements of $$A$$ are less than $$p$$, so we simply need to count how many of them are greater than $$\frac{p}{2}$$. If $$p\equiv 1\pmod{4}$$ then the elements of $$A$$ greater than $$\frac{p}{2}$$ are $$\{\frac{p+3}{2},\frac{p+7}{2},\ldots,p-1\}$$. There are $$\frac{p-1}{4}$$ elements here, so $$(\frac{2}{p})=(-1)^{\frac{p-1}{4}}$$ which is 1 if and only if $$p\equiv 1\pmod{8}$$. If instead $$p\equiv 3\pmod{4}$$ then the elements of $$A$$ greater than $$\frac{p}{2}$$ are $$\{\frac{p+1}{2},\frac{p+5}{2},\ldots,p-1\}$$. There are $$\frac{p+1}{4}$$ elements here, so $$(\frac{2}{p})=(-1)^{\frac{p+1}{4}}$$ which is 1 if and only if $$p\equiv -1\pmod{8}$$. Conclusion: for any odd prime number $$p$$, 2 is a quadratic residue modulo $$p$$ if and only if $$p\equiv\pm 1\pmod{8}$$. For distinct odd prime numbers $$p,q$$, we can consider whether $$p$$ is a quadratic residue modulo $$q$$ and also whether $$q$$ is a quadratic residue modulo $$p$$ and, at first sight, there is no reason why these two questions should be related. It is one of the gems of Number Theory that there is a simple relationship between them. Before we can prove this, we need a small lemma. Lemma 102 Let $$p,q$$ be odd positive integers, $$A=\{1,2,\ldots,\frac{p-1}{2}\}$$, and $$B=\{1,2,\ldots,\frac{q-1}{2}\}$$. Define $$U=\{(a,b)\in A\times B:aq-bp<-\frac{p}{2}\}$$ and $$W=\{(a,b)\in A\times B:aq-bp>\frac{q}{2}\}$$. Then $$\#U=\#W$$. proof Let $$f:A\rightarrow A$$ be the function $$a\mapsto \frac{p+1}{2}-a$$ and $$g:B\rightarrow B$$ be the function $$b\mapsto \frac{q+1}{2}-b$$. Then $$f,g$$ are each bijective with $$f^{-1}=f$$ and $$g^{-1}=g$$. Take any $$a\in A$$ and any $$b\in B$$. Then $f(a)q-g(b)p =\left(\frac{p+1}{2}-a\right)q-\left(\frac{q+1}{2}-b\right)p =\frac{q}{2}-\frac{p}{2}-(aq-bp).$ If $$(a,b)\in U$$ then $$aq-bp<-\frac{p}{2}$$ so $$\frac{q}{2}-\frac{p}{2}-(aq-bp)>\frac{q}{2}$$ and therefore $$(f(a),g(b))\in W$$. Conversely, if $$(a,b)\in W$$ then $$aq-bp>\frac{q}{2}$$ so $$\frac{q}{2}-\frac{p}{2}-(aq-bp)<-\frac{p}{2}$$ and therefore $$(f^{-1}(a),g^{-1}(b))=(f(a),g(b))\in U$$. Hence the function $$h:U\rightarrow W$$ given by $$(a,b)\mapsto (f(a),g(b))$$ is bijective and it follows that $$\#U=\#W$$. ∎ Theorem 103 Let $$p,q$$ be distinct odd prime numbers. Then $\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) =\left(-1\right)^{\frac{(p-1)(q-1)}{4}}.$ proof Let $$A=\{1,2,\ldots,\frac{p-1}{2}\}$$ and $$B=\{1,2,\ldots,\frac{q-1}{2}\}$$. For each $$a\in A$$, there exist unique integers $$b,r$$ such that $$aq=bp+r$$ and $$0\leq r\frac{p}{2}$$. Then $$(\frac{q}{p})=(-1)^s$$ by proposition 101. Instead of taking the remainder $$r$$ in the range 0 to $$p-1$$ inclusive, we may take it in the range $$-\frac{p-1}{2}$$ to $$\frac{p-1}{2}$$ inclusive, and then $$s$$ is the number of $$a\in A$$ for which we get $$r<0$$. As the quotient $$b$$ is unique in each case, we also have $$s=\#S$$ where $$S=\{(a,b)\in\mathbb{Z}^2:a\in A,-\frac{p}{2}aq$$ with $$p,a,q$$ all positive so $$b>0$$ (and $$b$$ is an integer) hence $$b\geq 1$$. Also $$bp\frac{q}{2}\right\} \end{eqnarray*}\] Then the sets \(U,V,W$$ form a partition of $$A\times B$$ so $$\#(A\times B)=\#U+\#V+\#W$$. For any $$a\in A$$ and any $$b\in B$$, $$p$$ does not divide $$a$$ and the primes $$p,q$$ are distinct so $$aq-bp\neq 0$$. Thus $$\#V=\#S+\#T=s+t$$. And by lemma 102 we have $$\#U=\#W$$ so $s+t=\#V\equiv\#(A\times B)=\frac{p-1}{2}\cdot\frac{q-1}{2}\pmod{2}$ hence $$\left(\frac{p}{q}\right) \left(\frac{q}{p}\right)=(-1)^{s+t}=\left(-1\right)^{\frac{(p-1)(q-1)}{4}}$$. ∎ Corollary 104 Quadratic Reciprocity Let $$p,q$$ be distinct odd prime numbers. If $$p\equiv 1\pmod{4}$$ or $$q\equiv 1\pmod{4}$$ (or both) then $$(\frac{q}{p})=(\frac{p}{q})$$, and otherwise $$(\frac{q}{p})=-(\frac{p}{q})$$. proof If $$p\equiv 1\pmod{4}$$ or $$q\equiv 1\pmod{4}$$ then $$\frac{(p-1)(q-1)}{4}$$ is even so, by theorem 103, $$(\frac{p}{q})(\frac{q}{p})=1$$ and therefore $$(\frac{q}{p})=(\frac{p}{q})$$. If instead $$p\equiv q\equiv 3\pmod{4}$$ then $$\frac{(p-1)(q-1)}{4}$$ is odd so, by theorem 103, $$(\frac{p}{q})(\frac{q}{p})=-1$$ and therefore $$(\frac{q}{p})=-(\frac{p}{q})$$. ∎ Example Is 87 a square modulo 127? Using corollary 100 and corollary 104, we see that $\begin{eqnarray*} \left(\frac{87}{127}\right) & = & \left(\frac{3}{127}\right) \left(\frac{29}{127}\right) = -\left(\frac{127}{3}\right) \left(\frac{29}{127}\right) \\ & = & -\left(\frac{1}{3}\right) \left(\frac{29}{127}\right) = -\left(\frac{29}{127}\right) = -\left(\frac{127}{29}\right) \\ & = & -\left(\frac{11}{29}\right) = -\left(\frac{29}{11}\right) = -\left(\frac{7}{11}\right) \\ & = & \left(\frac{11}{7}\right) = \left(\frac{4}{7}\right)=1 \end{eqnarray*}$ so 87 is a square modulo 127. Exercises 79. Is 65 a quadratic residue modulo 31? 80. For which odd prime numbers $$p$$ is -2 a square modulo $$p$$? 81. Find all integers $$x$$ such that $$x^2\equiv 1\pmod{12}$$. 82. Let $$z$$ be a complex number such that $$z^8=1$$ but $$z^4\neq 1$$. Show that $$z+z^{-1}=\pm\sqrt{2}$$. Last fiddled with by Nick on 2017-08-18 at 17:17 Reason: Typo (found by wreck - thank you!)
Similar Threads Thread Thread Starter Forum Replies Last Post Nick Number Theory Discussion Group 17 2017-12-23 20:10 Nick Number Theory Discussion Group 5 2017-04-25 14:32 Nick Number Theory Discussion Group 4 2017-03-27 06:01 Nick Number Theory Discussion Group 0 2016-12-29 13:47 Nick Number Theory Discussion Group 5 2016-10-08 09:05
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Wed Jun 16 16:10:28 UTC 2021 up 19 days, 13:57, 0 users, load averages: 1.76, 1.76, 1.68
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2021-06-16 16:10:28
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https://chorasimilarity.wordpress.com/2011/11/01/bipotentials-variational-formulations/
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# Bipotentials, variational formulations
The paper on the use of bipotentials in variational formulations is finally submitted, also available on arxiv here. See also this presentation.
In case you wonder how this could be related with other subjects commented on this blog, then wait to see “A gallery of emergent algebras”, where it shall be explained the connection between convex analysis and an emergent algebra related to a semidirect product between the semigroup of words over a symplectic space and $\mathbb{R}$.
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2017-05-23 16:46:20
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https://www.nature.com/articles/s41567-018-0301-y?error=cookies_not_supported&code=cc918f25-9cae-46d7-93b6-6a7b0df8acda
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Letter | Published:
# Wavelength-scale errors in optical localization due to spin–orbit coupling of light
## Abstract
Far-field optical imaging techniques allow the determination of the position of point-like emitters and scatterers1,2,3. Although the optical wavelength sets a fundamental limit to the image resolution of unknown objects, the position of an individual emitter can in principle be estimated from the image with arbitrary precision. This is used, for example, in the determination of the position of stars4 or in optical super-resolution microscopy5. Furthermore, precise position determination is an experimental prerequisite for the manipulation and measurement of individual quantum systems, such as atoms, ions and solid-state-based quantum emitters6,7,8. Here we demonstrate that spin–orbit coupling of light in the emission of elliptically polarized emitters can lead to systematic, wavelength-scale errors in the estimation of the emitter’s position. Imaging a single trapped atom as well as a single sub-wavelength-diameter gold nanoparticle, we demonstrate a shift between the emitters’ measured and actual positions, which is comparable to the optical wavelength. For certain settings, the expected shift can become arbitrarily large. Beyond optical imaging techniques, our findings could be relevant for the localization of objects using any type of wave that carries orbital angular momentum relative to the emitter’s position with a component orthogonal to the direction of observation.
## Main
A diffraction-limited imaging system with aperture diameter D has an angular resolution λ/D, where λ is the wavelength of the imaging light. Objects with smaller angular diameter cannot be resolved and produce an image given by the point-spread function (PSF) of the optical system. In spite of this limit, fitting the PSF to the image allows one to estimate its position with a precision limited only by the image’s signal-to-noise ratio9. The central assumption of this method is that the positions of the emitters in the object plane correspond to the centroid of the PSF measured in the image plane, provided that the optical system is focused.
It is known that the centroid of the image can be affected by imperfect focusing when the emission pattern of the object is anisotropic, as for a linear dipole. Depending on the orientation of the latter, this may lead to lateral shifts of a few tens of nanometres, that is, much smaller than the diffraction limit10,11. The resulting localization error can be reduced using polarization analysis12,13,14 or dedicated PSF fitting10,15,16,17, and vanishes for a focused image. Localization errors of comparable magnitude can occur when the emission pattern is distorted by near-field coupling to a nanoantenna18,19.
Here, we show that methods to estimate the position of emitters can be subject to large fundamental systematic errors when imaging elliptically polarized emitters, as a consequence of spin–orbit coupling in the emitted light field. These errors are present even for ideal, that is, diffraction-limited aberration-free far-field imaging systems. Imaging a single trapped atomic ion as well as a single gold nanoparticle that emits light with different elliptical polarizations, we demonstrate a wavelength-scale shift between the measured and actual positions of the emitter. For a wide range of polarizations, this shift is nearly independent of the numerical aperture. However, it can become arbitrarily large for certain polarizations and vanishing numerical aperture. These findings reveal that, even for small numerical apertures, the paraxial approximation is fundamentally inadequate in the context of the centroid estimation method.
To understand the physical origin of the image shift, let us consider a circularly polarized dipole emitter rotating in the xy plane, at the centre $${\cal O}$$ of the coordinate system. In this case, the total angular momentum carried by an emitted photon with respect to $${\cal O}$$ is ±ħez, where ± corresponds to right-handed (σ+) or left-handed (σ) polarization of the dipole relative to the z axis, respectively. This total angular momentum can be decomposed into spin and orbital angular momentum, represented by the operators $$\hat S_z$$ and $$\hat L_z$$, respectively. The spin and angular momentum components of the dipole field are coupled and their expectation values for a σ±-polarized dipole are
$$\left\langle {\hat S_z} \right\rangle = \pm \hbar \frac{{2{\kern 1pt} {\mathrm{cos}}^2\theta }}{{1 + {\mathrm{cos}}^2\theta }},\quad \left\langle {\hat L_z} \right\rangle = \pm \hbar \frac{{{\mathrm{sin}}^2\theta }}{{1 + {\mathrm{cos}}^2\theta }}$$
(1)
where θ is the angle between the z axis and the direction of observation20,21. In the xy plane (θ = 90°), the photons carry exclusively orbital angular momentum with expectation value ±ħ, while the spin angular momentum vanishes, corresponding to linear polarization. This is an example of spin–orbit coupling of light22, which gives rise to intriguing phenomena such as the spin-Hall effect of light23,24 and chiral interactions between light and matter25. For the circularly polarized dipole field, orbital angular momentum manifests as spiral wavefronts in the xy plane (see Methods). Hence, the local wavevectors are tilted with respect to the radial direction (Fig. 1) and the linear momentum per photon has an azimuthal component with expectation value $$\left\langle {\hat p_\phi (r)} \right\rangle = \left\langle {\hat L_z} \right\rangle /r = \pm \hbar / r$$. Due to this tilt, the photons seem to originate from a position that is offset from the emitter21,26, a fact already predicted by Charles G. Darwin more than 80 years ago27.
To quantify this shift for a typical far-field imaging system, we consider a circularly polarized dipole emitter located at the front focal point of a lens with focal length f, centred on the x axis. The lens collimates the light and changes its wavevector distribution. However, the mean wavevector $$\left\langle {\bf{k}} \right\rangle$$ averaged over the aperture is conserved and the collimated light propagates at an angle
$$\alpha _{{\mathrm{tilt}}} = \frac{{\left\langle {\hat p_\phi } \right\rangle _A}}{{\hbar k}} \simeq \pm \frac{\lambda }{{2\uppi f}}$$
(2)
with respect to the optical axis. Here, $$\left\langle \cdot \right\rangle _A$$ denotes the expectation value per photon within the aperture A of the lens. The centroid of the intensity distribution at a screen placed at a distance d behind the lens is shifted in the y direction by $$\left\langle y \right\rangle = \alpha _{{\mathrm{tilt}}}d$$ (Fig. 1) and the apparent y position of the dipole in the object plane is shifted by
$${\mathrm{\Delta }}y = - \frac{f}{d}\left\langle y \right\rangle = \mp \frac{\lambda }{{2\uppi }}$$
(3)
This expression holds for any imaging system, replacing f/d by the magnification factor of the system. To summarize, the light emitted by a circularly polarized σ± dipole carries orbital angular momentum due to the optical spin–orbit interaction. When imaging in the plane of polarization of the dipole, this gives rise to a λ/(2π) shift of the apparent position of the emitter.
We now generalize the above for an elliptically polarized emitter oscillating in the xy plane. Its polarization state can be written as a superposition of σ+- and σ-polarizations $$\left| \psi \right\rangle = \alpha \left| {\sigma ^ + } \right\rangle + \beta \left| {\sigma ^ - } \right\rangle$$, with $$\left| \alpha \right|^2 + \left| \beta \right|^2 = 1$$. For a small numerical aperture $${\mathrm{NA}} = D{\mathrm{/}}(2f) \ll 1$$, the shift of the apparent position of the emitter is (see Methods)
$${\mathrm{\Delta }}y = - \frac{\lambda }{{2\uppi }} \frac{{\Re (\mathit{\epsilon} )}}{{1 + {\mathrm{NA}}^2\left| \mathit{\epsilon} \right|^2{/}2}}$$
(4)
where the dipole polarization ratio, $$\mathit{\epsilon}$$ = (α + β)/(α − β), is in general complex and $$\Re ( \cdot )$$ denotes the real value. For σ+-polarization (σ-polarization) $$\mathit{\epsilon} = + 1$$ $$\left( {\mathit{\epsilon} = - 1} \right)$$ and for linear polarization along the y axis (x axis) $$\mathit{\epsilon} = 0$$ $$(\mathit{\epsilon} = \infty )$$. For circular polarization and $${\mathrm{NA}} \ll 1$$ we recover the λ/(2π) shift derived above. When the axes of the polarization ellipse coincide with the x and y axes, ϵ is real and the shift is given by
$${\mathrm{\Delta }}y \simeq - \mathit{\epsilon} \frac{\lambda }{{2\uppi }}$$
(5)
as long as $$\left| \mathit{\epsilon} \right| \ll 1{\mathrm{/NA}}$$. Outside of this linear regime, the shift reaches a maximum $${\mathrm{\Delta }}y_{{\mathrm{max}}} = \mp \lambda {\mathrm{/}}\left( {\sqrt 8 \uppi {\mathrm{NA}}} \right)$$ for $$\mathit{\epsilon }= \pm \sqrt 2 /{\mathrm{NA}}$$. Remarkably, this implies that the shift of the apparent position of the emitter can take arbitrarily large positive and negative values for small numerical apertures. For example, with NA = 0.23, the distance between the two extremal shifts is as large as the optical wavelength λ. These large shifts are reached for $$\mathit{\epsilon} = \pm 6.3$$, that is, when the polarization of the dipole is almost linear along the optical axis of the imaging system. In this case, the corresponding expectation values of the local orbital angular momentum per photon at the aperture significantly exceed ħ, the total angular momentum per emitted photon. Such ‘supermomentum’28 is an example of weak value amplification common to structured optical fields, in which the local expectation value of an operator can take values outside its eigenspectrum where the field is weak29,30. We note that there is a close connection between the observed weak value amplification and the appearance of momentum vortices in the emitted light field. This connection is shown in Supplementary Fig. 3 which plots the field distribution of the emitted light for different polarization states of the emitter. The plots also provide a graphical illustration for the polarization ratio $$\mathit{\epsilon}$$, which yields the maximum shift of the apparent position. This maximum shift is reached once the momentum vortices enter the field collected by the imaging lens. The centroid determination can be interpreted as a measurement of the weak value of the photons’ orbital angular momentum (see Methods). Finally, we note that the predicted shifts also occur for large numerical apertures and that equation (5) remains approximately valid provided that $$\left| \mathit{\epsilon} \right|$$ 1 (see Methods).
We study the predicted shifts by imaging a single atom—a fundamental quantum emitter—and a single sub-wavelength-scale nanoparticle. In the first experiment, we confine a 138Ba+ atomic ion in a Paul trap and image fluorescence from the dipole transition at λ1 = 493.41 nm (Fig. 2a) using an imaging system with magnification Ma = 5.40(7), where the number in parentheses respresents the 1 s.d. error, and NA = 0.40 (see Methods). A bandpass filter and a polarizer are used to collect light selectively from one of the spontaneous decay channels of the excited state, corresponding to the emission from either a σ+ or a σ dipole (see Methods).
We estimate the emitter’s position from each image by fitting a two-dimensional (2D) Gaussian function, which is a suitable approximation to the PSF in the measured regime31 (see Methods). Figure 3a–c shows the results for a total measurement time of 3 h. We observe a displacement between the σ+ and σ emissions of 158(4) nm in the object plane, in agreement with the expected value λ1/π = 157.1 nm.
As it is demanding to generate an arbitrarily polarized emission from a single atom, we extended the study to the case of a general elliptical polarization in a separate experiment where we imaged the light scattered by a single sub-wavelength-sized spherical gold nanoparticle. Such particles are used as labelling agents for super-resolution microscopy in biological research32,33. Being a spherically symmetric emitter, the polarization of a nanoparticle’s dipole always coincides with the polarization of the illuminating field, which can be controlled precisely. We place a 100-nm-diameter gold nanoparticle in the centre of a glass sphere with refractive index n = 1.46 by depositing it on an optical nanofibre34 and surrounding it by two fused-silica 2.5-mm-radius hemispherical solid immersion lenses. The ~200 μm gap between the lenses was filled with index-matching oil to prevent any reflection near the particle from either the nanofibre or the lenses. The nanoparticle was illuminated by a laser beam (vacuum wavelength λ2 = 685 nm) with adjustable polarization and the scattered light was imaged onto a CCD camera through the sphere and a microscope (Fig. 2b). To test the dependence of the position shift on the NA, two different microscope objectives were used with the same nominal magnification but different numerical apertures, resulting in NA = 0.41 and NA = 0.61 when including the silica sphere, and magnifications 21.9(2) and 20.1(1), respectively. The apparent displacement of the nanoparticle was measured by fitting a 2D Gaussian function to its image (see Methods), using, alternately, the beam with adjustable polarization and a linearly polarized reference beam. The measurements, averaged over 125 individual realizations for each polarization setting, are shown in Fig. 3. For $$\left| \mathit{\epsilon} \right| < 2$$, within our experimental errors, we observe a very good agreement of our measurements with the expected linear increase of the displacement with $$\mathit{\epsilon}$$, independent of the numerical aperture. For larger $$\left| \mathit{\epsilon} \right|$$, the linear approximation is not valid and the experimental data follow approximately the theoretical prediction from equation (4) (dashed lines). The apparent positions of the nanoparticle imaged with right and left circular polarizations $$\left( {\mathit{\epsilon} = \pm 1} \right)$$ are displaced relative to each other by 145(6) nm for NA = 0.41 and 146(4) nm for NA = 0.61, in agreement with the expected value $$2{\mathrm{\Delta}}y=\tilde\lambda_2{\mathrm{/}}\uppi\approx150{\kern1pt}$$ nm, where $$\tilde \lambda _2 = \lambda _2{\mathrm{/}}n$$ is the laser wavelength in the index-matching oil. The displacement increases for larger values of $$\left| \mathit{\epsilon} \right|$$, and the total displacement between counter-rotating elliptical polarizations reaches 430(7) nm $$\left( { \simeq \tilde \lambda _2} \right)$$ for $$\mathit{\epsilon} = \pm 5.67$$, a shift four times larger than the diameter of the gold nanoparticle. To verify that focusing errors are not the origin of the effect, we slightly defocused our imaging optics and observed that, in the measured range, the shifts do not depend on the distance of the particle to the focal plane (see Methods).
Our findings may affect super-resolution microscopy techniques. The maximum systematic shift due to dipole ellipticity is proportional to the PSF size, which is up to two orders of magnitude larger than the resolution achieved by super-resolution microscopy35,36. For instance, the determination of the position of an emitter with NA = 1, at a wavelength of λ ≈ 628 nm, with an accuracy of 1 nm, requires the scattered light to be more than 99.99% linearly polarized ($$\left| {\Re (\mathit{\epsilon} )} \right| < 0.01$$, see Supplementary Information). For larger $$\mathit{\epsilon}$$, an accuracy of, for example, 1 nm could still be reached by employing an algorithm that not only uses position but also polarization of the dipole as fit parameters for the recorded PSF. However, to reach the necessary signal-to-noise ratio, this higher-dimensional fit requires one to increase the light-collection time by more than four orders of magnitude compared to the case of an optimally coupled linear dipole (Supplementary Fig. 4d).
On the positive side, the polarization-dependent shift could be used, for example, in arrays of optically trapped particles37, where the apparent location of each particle would give access to the local polarization of an inhomogeneous exciting field; conversely, in the case of a homogeneous exciting field, the shift would allow us to sense local physical parameters affecting the polarizability of the particles, such as the direction of the magnetic field. The demonstrated effect is relevant beyond optical imaging, as it will occur for any kind of wave carrying transverse orbital angular momentum. Thus, it may affect the localization of remote objects imaged with radar or sonar techniques38,39, or even alter the apparent position of astronomical objects detected through their emission of gravitational waves40,41.
## Methods
### Momentum and wavefronts of the radiated field
The electric field emitted by an optical dipole located at the origin (r = 0) that oscillates with angular frequency ω is given by
$${\bf{E}}({\bf{r}},t) = - \frac{{\omega ^2}}{{4\pi \mathit{\epsilon} _0c^2}}\frac{{e^{i(kr - \omega t)}}}{{r^3}}({\bf{r}} \times {\bf{\mu}}) \times {\bf{r}}$$
(6)
in the far field $$\left( {\left| {\bf{r}} \right| \gg \lambda } \right)$$, where μ = μeμ is the complex vector amplitude of the electrical dipole and k = 2π, where λ is the wavelength of the emitted light. The optical momentum density in the field can be defined as the Poynting vector29
$${\bf{P}} = \frac{1}{2}{\mathrm{Re}}({\bf{E}}^ \ast \times {\bf{H}})$$
(7)
The momentum density can be divided into two components that arise from the orbital and spin angular momentum of the field
$${\bf{p}}_{{\mathrm{orb}}} = \frac{{c^2\mathit{\epsilon} _0}}{{2\omega }}{\mathrm{Im}}\left( {{\bf{E}}^ \ast \cdot \nabla {\bf{E}}} \right)$$
(8)
$${\bf{p}}_{{\mathrm{spin}}} = \frac{{c^2\mathit{\epsilon} _0}}{{4\omega }}\nabla \times {\mathrm{Im}}\left( {{\bf{E}}^ \ast {\bf{E}}} \right){\kern 1pt}$$
(9)
where E indicates the vector gradient. From equations (6) and (8) it is possible to derive an expression for the wavefronts, that is, the surfaces normal to porb. In general, the wavefronts of elliptical dipoles in the xy plane are kinky spirals given by
$$r(\phi ) = \frac{1}{k}\left( {{\mathrm{arctan}}\left[ {\mathit{\epsilon} \,{\mathrm{tan}}\left( \phi \right)} \right] + \omega t} \right) + {\mathrm{const}}.$$
(10)
For a linearly polarized dipole $$(\mathit{\epsilon} = 0)$$, that is, a dipole with zero expectation value for its angular momentum, the wavefronts reduce to circles given by $$r_{{\mathrm{wf}}} = {\textstyle{{\omega t} \over k}} + {\mathrm{const}}.$$, whereas for a σ± $$\left( {\mathit{\epsilon} = \pm 1} \right)$$ polarized dipole that radiates waves with total angular momentum of ±ħ per photon with respect to the z axis, the wavefronts in the xy plane are given by
$$r_ \pm (\phi ) = \frac{{ \mp \phi + \omega t}}{k} + {\mathrm{const}}.$$
(11)
This corresponds to an Archimedean spiral rotating around the z axis, with the same rotation sense as the dipole. Supplementary Fig. 1 shows an example of the wavefronts of an elliptical dipole in comparison with the circular case.
### Angular momentum and imaging
According to equation (8) the local orbital angular momentum can be calculated by applying the operator
$$\widehat {\bf{L}} = {\bf{r}} \times \widehat {\bf{p}}$$
(12)
on the single-photon wavefunction, where $$\widehat {\bf{p}} = - i\hbar \nabla$$ is the orbital momentum density operator. The local orbital angular momentum per photon can be measured by sending the light through an aperture at position r0. We align the z axis with the axis defined by the transverse angular momentum and define the optical axis as the x- axis. The expectation value of the transverse linear momentum component $$\left\langle {\hat p_y^w} \right\rangle$$ per photon at the position of the aperture is given by the displacement $$\left\langle y \right\rangle$$ of the centre of mass of the far-field image from the optical axis ex at distance d from the aperture. The relation between angular momentum and displacement is given by
$$\left\langle y \right\rangle = \frac{d}{{\hbar k}}\left\langle {\hat p_y^w} \right\rangle = \frac{d}{{\hbar k}}\frac{1}{{r_0}}\left\langle {\hat L_z^w} \right\rangle$$
(13)
This measurement can be interpreted in the framework of weak measurements, where the centre of mass in the image plane is proportional to the weak value of the photons’ orbital angular momentum (or the transverse linear momentum) at the aperture, which are given at the position of the first lens by29
$$\left\langle {\hat L_z^w} \right\rangle = r_0 \left\langle {\hat p_y^w} \right\rangle = r_0 \;\frac{{\left\langle {{\tilde{\mathrm \Psi }}_{{\mathrm{post}}}} \right|\hat p_y\left| {\mathrm{\Psi }} \right\rangle }}{{\left\langle {{\tilde{\mathrm \Psi }}_{{\mathrm{post}}}{\mathrm{|\Psi }}} \right\rangle }}{\kern 1pt}$$
(14)
where $$\left| {\mathrm{\Psi }} \right\rangle$$ and $$\left| {{\tilde{\mathrm \Psi }}_{{\mathrm{post}}}} \right\rangle$$ are the initial wavefunction and the part of the wavefunction that passes the aperture (the post-selected state), respectively.
In other words, the orbital angular momentum components transverse to the optical axis result in a transverse linear momentum at the aperture that leads in turn to a displacement of the centre of mass of the diffracted beam in the far field. The local angular momentum per photon can exceed ħ where the field is weak (so-called ‘supermomentum’)28.
### Calculation of the image centroid
To obtain the displacement of the centroid of the image, we start with the above relation between angular momentum and transverse linear momentum. The considered imaging system consists of an objective with focal length f and aperture diameter D located at a distance f from the emitter. The electric fields of the three elementary dipoles π, σ+ and σ at the objective are, for small aperture $$\left( {D \ll f} \right)$$ and up to a common normalization constant, given by
$${\tilde{\mathrm \Psi }}_\pi (\rho ,\phi ) = \frac{1}{f}{\bf{e}}_{\boldsymbol{z}}e^{i\varphi }$$
(15)
$${\tilde{\mathrm \Psi }}_{\sigma ^ \pm }(\rho ,\phi ) = \frac{1}{{\sqrt 2 }}\left( { \pm \frac{i}{f}{\bf{e}}_y + \frac{\rho }{{f^2}}{\bf{e}}_{\mathrm{\rho}} } \right)e^{i\varphi }$$
(16)
where ρ and ϕ (y and z) are polar (Cartesian) coordinates in the aperture plane, ex, ey, ez and eρ are the unit vectors in the respective direction, $$\varphi = k\sqrt {\rho ^2 + f^2}$$. Since the emitter is in the focal plane of the objective, the latter applies the transformation e on the light and removes the phase factor in equations (15) and (16), which we drop in the following. As a consequence, the phase fronts are transformed into plane waves while the average wavevector and the average transverse momentum are conserved. Measuring the displacement of the waveform’s centre of mass from the optical axis $$\left\langle q \right\rangle$$ at distance d from the objective $$\left( {d \gg D} \right)$$ then corresponds to a measurement of the expectation value of the transverse angular momentum component per photon $$\left\langle {\hat L_z^w} \right\rangle$$ or the linear transverse momentum component $$\left\langle {\hat p_q^w} \right\rangle$$ of the photons at the position of the aperture where q {y, z}. The actions of the momentum operators on the wave are
$$\hat p_q{\tilde{\mathrm \Psi }}_{\sigma ^ \pm } = - \frac{{i\hbar }}{{f^2\sqrt 2 }}{\bf{e}}_q$$
(17)
as well as $$\hat p_q{\tilde{\mathrm \Psi }}_\pi = 0$$. Considering the general case of a photon that originates from a superposition of σ+ and σ emission, we can calculate the weak value in equation (14) and obtain
$$\left\langle {\hat p_y^w} \right\rangle _r = \frac{\hbar }{f}\frac{{\Re (\mathit{\epsilon} )}}{{1 + \left| \mathit{\epsilon} \right|^2{\mathrm{NA}}^2 / 2}}$$
(18)
$$\left\langle {\hat p_z^w} \right\rangle _r = 0$$
(19)
where we define the numerical aperture as NA = D/(2f). We note that, in general, the expectation values $$\left\langle {\hat p_q^w} \right\rangle$$ are complex as $$\mathit{\epsilon}$$ can be a complex number. Since only the real part of the expectation values corresponds to a displacement of the image centroid, equations (18) and (19) only give the real part of the expectation $$\left\langle {\hat p_q^w} \right\rangle$$, which we denote by $$\langle \cdot \rangle _r$$.
In a microscopy set-up, the image is not formed at infinity, but a second lens with focal length f′, which we assume to be at a distance f′ from the aperture, is used to form an image at distance 2f′ from the aperture. In this case, the expected displacement is obtained by replacing d by f′. For the expected displacement on the screen this finally yields
$$\left\langle {\hat y} \right\rangle = \frac{1}{{\hbar k}}\frac{{f^\prime }}{f}\left\langle {\hat L_y^w} \right\rangle = \frac{\lambda }{{2\pi }}\frac{{f^\prime }}{f}\frac{{\Re (\mathit{\epsilon} )}}{{1 + \left| \mathit{\epsilon} \right|^2{\mathrm{NA}}^2{\mathrm{/}}2}}$$
(20)
For small numerical aperture ($${\mathrm{NA}} \ll \left| \mathit{\epsilon} \right|$$) and $$\mathit{\epsilon}$$ real, the displacement of the centroid increases linearly in $$\mathit{\epsilon}$$. For circular polarization $$\mathit{\epsilon} = \pm 1$$, the centroid of the image is displaced from the expected position by $$\left\langle {\hat y} \right\rangle \approx \pm \lambda {\mathrm{/}}(2\uppi )$$ times the magnification of the optical system f′/f; that is, the particle appears to be displaced by λ/(2π), taking into account that the assumed imaging system produces a flipped image. The maximum displacement of the centroid for $$\mathit{\epsilon}$$ real is given by
$$\left\langle {\hat y} \right\rangle _{{\mathrm{max}}} = \pm \frac{\lambda }{{2\uppi }}\frac{{f^\prime }}{f}\frac{1}{{\sqrt 2 {\mathrm{NA}}}}$$
(21)
In other words, for vanishing NA, the displacement of the apparent and real positions of the particle can be arbitrarily large.
### Fourier-optic derivation of the centroid position
The position of the centroid can also be calculated in the framework of Fourier optics. We calculate the electric fields of the three fundamental electrical dipoles oscillating in the x, y and z directions in the image plane and obtain, for the approximation of small NA,
$${\bf{E}}_x = iE_0 \frac{{{\mathrm{NA}}^2}}{\rho }J_2(\tilde \rho )\left( {{\mathrm{cos}}{\kern 1pt} \varphi {\bf{e}}_y + {\mathrm{sin}}{\kern 1pt} \varphi {\bf{e}}_z} \right)$$
(22)
$${\bf{E}}_y = E_0 \frac{{{\mathrm{NA}}}}{\rho }J_1(\tilde \rho ){\bf{e}}_y$$
(23)
$${\bf{E}}_z = E_0 \frac{{{\mathrm{NA}}}}{\rho }J_1(\tilde \rho ){\bf{e}}_z$$
(24)
where we have defined the amplitude
$$E_0 = \frac{{\mu \omega ^2}}{{4\uppi \mathit{\epsilon} _0^2c^2}}$$
(25)
and $$\tilde \rho = \rho k {\mathrm{NA}} f{\mathrm{/}}f^\prime$$, with the opening angle of the objective NA ≈ D/(2f). The final image is then a superposition of the three dipole fields, from which we again obtain equation (20) for the centroid.
### Immersion microscopy
In high-NA imaging, the so-called immersion method is used, where the first lens of the system is a solid immersion lens and the imaged particles are located on the planar side of the lens and embedded in immersion fluid that has the same refractive index as the lens. Consequently, wavefronts emitted by the particle are parallel to the surface of the lens. Thus, this method does not affect the wavefronts in the far field outside the lens and our discussion also applies for this case. It is only necessary to replace the numerical aperture NA with the geometrical numerical aperture NAg (NAg = NA/n) and to replace λ with the wavelength in the immersion fluid λ/n.
### Atomic transition selective detection of photons
In the atom experiment, the photons are emitted with angular momentum Δ from a dipole transition of a single 138Ba+ atomic ion in a Paul trap, where Δm is given by the difference in the magnetic quantum number of the electronic level before and after the photon emission. Δm = 0 corresponds to emission from a linear π dipole and Δm = ±1 to emission from a circular σ dipole. Photons are emitted from the cooling transition, with λ = 493.41 nm (Supplementary Fig. 5a). A magnetic field B = 0.45 mT parallel to the axis of the trap (z axis) defines the quantization axis perpendicular to the optical axis (x axis). The ion is Doppler-cooled, reducing the extension of the motional atomic wavepacket down to ~36 nm, then is optically pumped into one of the Zeeman levels of the 6S1/2 ground state. For example, when preparing a photon emission Δm = +1, we pump to the 6S1/2, mj = −1/2 with a σ-polarized 493 nm laser and a repumper beam. Subsequently, we apply a short σ+-polarized 493 nm laser pulse which excites the atom to the state 6P1/2, mj = +1/2 (Supplementary Fig. 5b,d). From that excited state, the atom can spontaneously decay back to 6S1/2, mj = −1/2 through a Δm = +1 transition, to 6S1/2, mj = +1/2 through a Δm = 0 transition, or to the 6D3/2 manifold. During this transition the atom emits a photon that can be collected by the objective (NA = 0.40) and directed to the camera through the imaging system. To detect photons from the opposite transition (Δm = −1), the polarizations of the optical pumping and excitation beams are exchanged (Supplementary Fig. 5c).
In this configuration, photons from the Δm = 0 (Δm = ±1) transition are horizontally (vertically) polarized along the optical axis. This allows us to select only photons from the σm = ±1) transitions by introducing a PBS after the objective. An ideal PBS removes 99.998% of photons from the π transitions and 2.7% of photons from the σ transitions. Therefore we expect that the σ± dipole image is not significantly changed by the polarization filtering, and indeed this is borne out by complete numerical simulations (Supplementary Fig. 6).
The results shown in the main text were obtained using an ICCD camera (Andor iStar A-DH334T-18H-63). Supplementary Fig. 5d shows the sequence and timing used in the experiment.
### Atom image characteristics, stability and drifts correction
The image of the atomic ion corresponds to the PSF of the imaging system, which is well approximated in our case by a 2D Gaussian. The detected images are fitted to a Gaussian profile with seven free parameters (z0, y0, σz, σy, A, O, θ), where (z0, y0) are the coordinates of the centroids, σz and σy are the standard deviation in the major and minor axis, A is the amplitude, O is an offset and θ is the rotation angle with respect to the CCD sensor axis. The magnification of the imaging system is measured by imaging a string of two ions separated by well-known distance42, and is given by M = 5.40(7).
The long accumulation time introduces a new source of error in the position estimation from mechanical drifts in the imaging system. The stability of the imaging system is characterized by the Allan variance of the fitted centroids of the detected images7, which gives us a measure of the position uncertainty depending on the accumulation time τ. This was done by taking N pictures with exposure time t, adding them in bins of duration τ = nt, where n is an integer number smaller than N/2. Each binned image was fitted to the seven-parameter Gaussian function, from which the centroids were extracted. For comparison we also used, besides the ICCD camera, an electron-multiplying (EM) CCD camera (Andor iXon DU-897) with bigger pixel size (16 × 16 μm2). In the case of the EMCCD camera we took 2,000 images (2 s exposure time) with the atom emitting resonance florescence at maximum rate. In the case of the ICCD camera, we took 3,000 images (0.5 s exposure). In both cases, the time between two consecutive images was negligible. Supplementary Fig. 7a,b shows the vertical position uncertainty extracted with this method. The minimum uncertainty in the vertical position obtained using the EMCCD camera is 2.13(41) nm for 148 s accumulation time, while for the ICCD set-up the minimum is 3.29(71) nm for 74 s accumulation time. In both cases, the decreasing part of the curve is dominated by shot noise. The drift of the centre of the fitted reference images used in the experiment is shown in Supplementary Fig. 7c, where we observe that in a period of 3 h the image drifts a maximum of ~200 nm in both vertical and horizontal directions. To compensate for these drifts, we used the acquisition of long-exposure images during the cooling stage (Supplementary Fig. 5e) to obtain a real-time ‘reference’ of the particle position. Supplementary Fig. 5d,e shows the full experimental sequence. This sequence was repeated for 3 h, and the analysed pictures correspond to the accumulation of photons in a 11 × 11 pixel sub-area of the CCD sensor.
After the data collection was finished, each reference image was fitted, and the mean centroid position of two consecutive reference images was used to correct for the drifts in the signal image acquired between them. Then, we added up all the corrected signal images and fitted these data. Finally, we compared the centroid positions of the added up reference and signal images to determine their relative displacement. The uncertainty of the displacement was extracted from the 1σ confidence intervals using χ2 analysis, given its relation with the real noise sources43.
### Nanoparticle sample preparation and set-up
We deposited a single gold nanoparticle (BBI solutions, diameter 100 nm) on a silica nanofibre (diameter 410 nm) by touching the nanofibre with a droplet that contained a diluted suspension of nanoparticles. The presence of a single nanoparticle on the nanofibre could be detected via absorption spectroscopy34. The solid immersion lenses were positioned around the nanoparticle such that the nanoparticle lay in the centre of the two lenses. The gap between the lenses of about 200 μm was filled with immersion oil.
The imaging system was a combination of a long-working-distance microscope and the solid immersion lenses (half ball lenses with a radius of 2.5 mm). The microscope consisted of an infinity-corrected objective by Mitutoyo, with a magnification of ×20 and an infinity tube lens to image onto a CCD camera (Matrix Vision mvBlueFOX3-1013G-2212). In two different measurements we used two different objectives with NA of 0.28 and 0.42. Via a surface topography standard, we measured the magnification of the long-working-distance microscope. This combination resulted in an overall imaging system with NAs of 0.41 and 0.61 and magnifications M0.41 = 21.9(2) and M0.61 = 20.1(1).
In the experiment we used two laser beams—a reference and a measurement beam—with fixed and adjustable polarization, respectively (Fig. 2b). The polarization of the reference beam was aligned along the z axis. The measurement beam was set to be linear polarized along the y axis before passing through a half- and then a quarter-wave plate. By rotating the half-wave plate we could adjust the beam’s polarization to every elliptical polarization with the major axes along x or y. To avoid aberration caused by light propagating along the ridge of the two immersion lenses, the measurement beam was tilted by 7° from the z axis (Fig. 2b). This tilt is included in the theory plots shown in Fig. 3f.
### Data acquisition and analysis
The illumination times of the images were 2 ms (NA = 0.41 objective) and 6.5 ms (NA = 0.61 objective). The pictures were taken alternately using the reference and measurement beams (Fig. 2b). In the experimental sequence, the particle displacements were measured as a function of polarization and the focal position of the imaging optics. For every polarization ratio $$\mathit{\epsilon}$$, the relative focus position was scanned by moving the long-working-distance microscope with a step size of 1.25 μm and a total range ~20 μm. Then, the polarization ratio was changed by rotating the half-wave plate by 2.5°. A total of 25 tuples of data were acquired for every $$\mathit{\epsilon}$$ and focus position. Figure 3f shows the mean displacements obtained from averaging over all displacements for the five focal positions closest to the focus of the imaging system. The statistical error of each data point displayed in Fig. 3f is estimated as $$\sigma _{{\mathrm{\Delta }}y}/\sqrt {125}$$, where σΔy is the standard deviation of the measured displacements.
To correct for inhomogeneous pixel efficiencies of the CCD camera, we applied standard flat-field correction on the measured image data. Then, to determine the (apparent) position of the nanoparticle, we fitted a 2D Gaussian with six free fitting parameters to the particle images. The free parameters are the centroid position (z0, y0), amplitude A, waists σz and σy of the elliptical Gaussian, and intensity offset O.
## Data availability
The data that support the findings of this study are available from the authors upon reasonable request. Contact persons are G.A. for the ion experiment, and J.V. or A.R. for the nanoparticle experiment.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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## Acknowledgements
The authors thank P. Obšil for experimental support, and J. Enderlein, M. Hush and A. Jesacher for helpful discussions. This work was supported by the Austrian Science Fund (FWF, SINPHONIA project P23022, SFB FoQuS F4001, SFB NextLite F4908), by the European Research Council through project CRYTERION #227959, by the Institut für Quanteninformation GmbH and by the Australian Research Council through project CE170100012.
## Author information
### Author notes
1. These authors contributed equally: G. Araneda, S. Walser
### Affiliations
1. #### Institut für Experimentalphysik, Universität Innsbruck, Innsbruck, Austria
• G. Araneda
• , Y. Colombe
• , D. B. Higginbottom
• & R. Blatt
2. #### Vienna Center for Quantum Science and Technology, TU Wien-Atominstitut, Vienna, Austria
• S. Walser
• , J. Volz
• & A. Rauschenbeutel
3. #### Centre for Quantum Computation and Communication Technology, Research School of Physics and Engineering, The Australian National University, Canberra, Australian Capital Territory, Australia
• D. B. Higginbottom
• R. Blatt
5. #### Department of Physics, Humboldt-Universität zu Berlin, Berlin, Germany
• A. Rauschenbeutel
### Contributions
J.V. and A.R. proposed the concept. All authors contributed to the design and the setting up of the experiments (atom experiment: G.A., Y.C., D.B.H. and R.B.; nanoparticle experiment: S.W., J.V. and A.R.). G.A. and D.B.H. performed the atom experiment and analysed the data. S.W. performed the nanoparticle experiment and analysed the data. All authors contributed to the writing of the manuscript.
### Competing interests
The authors declare no competing interests.
### Corresponding authors
Correspondence to G. Araneda or J. Volz or A. Rauschenbeutel.
## Supplementary information
1. ### Supplementary Information
Supplementary notes, figures and references
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magic_gaussian_integral [2017/07/01 12:20]nikolaj magic_gaussian_integral [2019/09/27 00:19] (current)nikolaj Both sides previous revision Previous revision 2019/09/27 00:19 nikolaj 2017/07/01 12:20 nikolaj 2016/12/06 16:43 nikolaj 2015/04/15 13:43 nikolaj 2015/04/15 13:42 nikolaj 2015/04/15 13:42 nikolaj 2014/03/21 11:13 nikolaj 2014/03/21 11:13 nikolaj 2014/03/21 11:11 external edit2014/03/21 10:57 nikolaj 2014/03/21 10:56 nikolaj 2014/03/06 01:11 nikolaj 2014/03/06 01:08 nikolaj 2014/03/06 01:07 nikolaj 2014/03/06 01:06 nikolaj 2014/03/06 01:01 nikolaj 2014/03/05 19:46 nikolaj 2014/03/02 23:32 nikolaj 2014/03/02 23:32 nikolaj 2014/03/02 23:30 nikolaj 2014/03/02 23:18 nikolaj 2014/03/02 23:16 nikolaj 2014/03/02 23:16 nikolaj 2014/03/02 23:15 nikolaj 2014/03/02 23:14 nikolaj 2014/03/02 23:14 nikolaj 2014/03/02 19:15 nikolaj 2019/09/27 00:19 nikolaj 2017/07/01 12:20 nikolaj 2016/12/06 16:43 nikolaj 2015/04/15 13:43 nikolaj 2015/04/15 13:42 nikolaj 2015/04/15 13:42 nikolaj 2014/03/21 11:13 nikolaj 2014/03/21 11:13 nikolaj 2014/03/21 11:11 external edit2014/03/21 10:57 nikolaj 2014/03/21 10:56 nikolaj 2014/03/06 01:11 nikolaj 2014/03/06 01:08 nikolaj 2014/03/06 01:07 nikolaj 2014/03/06 01:06 nikolaj 2014/03/06 01:01 nikolaj 2014/03/05 19:46 nikolaj 2014/03/02 23:32 nikolaj 2014/03/02 23:32 nikolaj 2014/03/02 23:30 nikolaj 2014/03/02 23:18 nikolaj 2014/03/02 23:16 nikolaj 2014/03/02 23:16 nikolaj 2014/03/02 23:15 nikolaj 2014/03/02 23:14 nikolaj 2014/03/02 23:14 nikolaj 2014/03/02 19:15 nikolaj 2014/03/02 19:10 nikolaj 2014/03/02 19:06 nikolaj 2014/03/02 18:24 nikolaj 2014/03/02 17:51 nikolaj 2014/03/02 17:51 nikolaj 2014/03/02 17:40 nikolaj 2014/03/02 17:39 nikolaj 2014/03/02 17:37 nikolaj 2014/03/02 17:10 nikolaj 2014/03/02 17:04 nikolaj 2014/03/02 17:01 nikolaj 2014/03/02 17:00 nikolaj 2014/03/02 16:58 nikolaj 2014/03/02 16:43 nikolaj 2014/03/02 16:43 nikolaj 2014/03/02 16:41 nikolaj 2014/03/02 16:40 nikolaj 2014/03/02 16:40 nikolaj 2014/03/02 16:38 nikolaj 2014/03/02 16:34 nikolaj 2014/03/02 16:34 nikolaj 2014/03/02 16:31 nikolaj old revision restored (2014/02/25 21:19) Line 28: Line 28: Notice that via diagonalization of the matrix and knowledge of basic Gaussian integral above, we get Notice that via diagonalization of the matrix and knowledge of basic Gaussian integral above, we get - ^ $Z_1(0):=\int_{-\infty}^\infty \mathrm e^{-\tfrac{1}{2}\left\langle\phi\left|\,A\,\right|\phi\right\rangle} \prod_{i=1}^m \mathrm d\phi_i = (2\pi)^{1/2}(\det A)^{-1/2}$ ^ + ^ $Z_1(0):=\int_{-\infty}^\infty \mathrm e^{-\tfrac{1}{2}\left\langle\phi\left|\,A\,\right|\phi\right\rangle} \prod_{i=1}^m \mathrm d\phi_i = (2\pi)^{m/2}(\det A)^{-1/2}$ ^ Taking care of the vector $J$, we can obtain Taking care of the vector $J$, we can obtain - ^ $Z_1(J):=\int_{-\infty}^\infty \mathrm e^{-\tfrac{1}{2}\left\langle\phi\left|\,A\,\right|\phi\right\rangle +i\,\left\langle\phi\left|\right.J\right\rangle}\prod_{i=1}^m\mathrm d\phi_i = Z_1(0)\cdot\mathrm e^{-\frac{1}{2}\left\langle J\left|\,A^{-1}\,\right|J\right\rangle }$ ^ + ^ $Z_1(J):=\int_{-\infty}^\infty \mathrm e^{-\tfrac{1}{2}\left\langle\phi\left|\,A\,\right|\phi\right\rangle +i\,\left\langle\phi\left|\right.J\right\rangle}\prod_{i=1}^m\mathrm d\phi_i = Z_1(0)\cdot\mathrm e^{-\frac{1}{2}\left\langle J\left|\,A^{-1}\,\right|J\right\rangle }$ ^ Now from a physical perspective, it's actually better to write this as $\propto\mathrm e^{-\frac{1}{2}\left\langle A^{-1}J\left|\,A\,\right|A^{-1}J\right\rangle }$. In the path integral treatment of the diffusion equation, the propagator involves quantity $\phi\equiv p,J\equiv q$ and roughly speaking $A\propto \delta t,\ A^{-1}\propto\tfrac{\mathrm d}{\mathrm dt}$. The above integral plays a role in passing from the Hamiltonian perspective to the Lagrangian one: $\left\langle\phi\left|\,A\,\right|\phi\right\rangle\propto p^2\Delta t$ to a sort of conjugate $\left\langle A^{-1}J\left|\,A\,\right|A^{-1}J\right\rangle\propto {\dot q}^2\Delta t$. Now from a physical perspective, it's actually better to write this as $\propto\mathrm e^{-\frac{1}{2}\left\langle A^{-1}J\left|\,A\,\right|A^{-1}J\right\rangle }$. In the path integral treatment of the diffusion equation, the propagator involves quantity $\phi\equiv p,J\equiv q$ and roughly speaking $A\propto \delta t,\ A^{-1}\propto\tfrac{\mathrm d}{\mathrm dt}$. The above integral plays a role in passing from the Hamiltonian perspective to the Lagrangian one: $\left\langle\phi\left|\,A\,\right|\phi\right\rangle\propto p^2\Delta t$ to a sort of conjugate $\left\langle A^{-1}J\left|\,A\,\right|A^{-1}J\right\rangle\propto {\dot q}^2\Delta t$. Line 38: Line 38: Lastly, notice that $-i\frac{\partial}{\partial J_i}e^{i\,\left\langle\phi\left|\right.J\right\rangle}=\phi_i\, \mathrm e^{i\,\left\langle\phi\left|\right.J\right\rangle}$ and therefore Lastly, notice that $-i\frac{\partial}{\partial J_i}e^{i\,\left\langle\phi\left|\right.J\right\rangle}=\phi_i\, \mathrm e^{i\,\left\langle\phi\left|\right.J\right\rangle}$ and therefore - ^ $Z_f(J) = Z_1(0)\ f\left(-i\frac{\partial}{\partial J}\right)\,\mathrm e^{-\frac{1}{2}\left\langle J\left|\,A^{-1/2}\,\right|J\right\rangle }$ ^ + ^ $Z_f(J) = (2\pi)^{m/2}(\det A)^{-1/2}\cdot f\left(-i\frac{\partial}{\partial J}\right)\,\mathrm e^{-\frac{1}{2}\left\langle J\left|\,A^{-1/2}\,\right|J\right\rangle }$ ^ + $\int_{\mathbb R}\,f(\phi)\,\mathrm e^{\frac{1}{2} \left\langle\phi\left|\,A\,\right|\phi\right\rangle +i\,\left\langle\phi\left|\right.J\right\rangle} \prod_{i=1}^m \mathrm d\phi_i = (2\pi)^{m/2}(\det A)^{-1/2}\cdot f\left(-i\frac{\partial}{\partial J}\right)\,\mathrm e^{-\frac{1}{2}\left\langle J\left|\,A^{-1/2}\,\right|J\right\rangle }$ We are interested in that expression as the solution of the integral, because in quantum field theory, the path integral is often an infinite dimensional variant it. There the exponent in the defining integral is the action functional, the operator $A$ involves a hard to invert differential operator (the inverse being strongly related to the response function/green function) and $f$ encodes the type of process and the interaction. The terms from the expansion of $f$ are encoded by Feynman diagrams. We are interested in that expression as the solution of the integral, because in quantum field theory, the path integral is often an infinite dimensional variant it. There the exponent in the defining integral is the action functional, the operator $A$ involves a hard to invert differential operator (the inverse being strongly related to the response function/green function) and $f$ encodes the type of process and the interaction. The terms from the expansion of $f$ are encoded by Feynman diagrams.
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2020-02-27 02:12:33
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